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[; ]();a,;nolistsep 1.3in***ATT*** 𝕀 ℙ 𝔼 𝕍 X⃗ x⃗ t⃗ Z⃗ z⃗ y⃗ ℝ thmTheorem DefDefinition CorCorollary LemmaLemma AssumptionAssumption AssumptionPrime[1][Assumption]#1 definition exExampleConverting High-Dimensional Regression to High-Dimensional Conditional Density Estimation Rafael IzbickiDepartment of Statistics, Federal University of So Carlos, Brazil.and Ann B. LeeDepartment of Statistics, Carnegie Mellon University, USA.============================================================================================================================================================== There is a growing demand for nonparametric conditional density estimators (CDEs) in fields such as astronomy and economics.In astronomy, for example, one can dramatically improve estimates of the parameters that dictate theevolution of the Universe by working with full conditional densities instead of regression (i.e., conditional mean) estimates. More generally, standard regression falls short in any prediction problem where the distribution of the response is more complex with multi-modality, asymmetry or heteroscedastic noise. Nevertheless, much of the work on high-dimensional inference concerns regression and classification only, whereas research on density estimation has lagged behind. Here we propose FlexCode, a fully nonparametric approach to conditional density estimation that reformulates CDE as a non-parametric orthogonal series problem where the expansion coefficients are estimated by regression. By taking such an approach, one can efficiently estimate conditional densities and not just expectations in high dimensions by drawing upon the success in high-dimensional regression. Depending on the choice of regression procedure, our method can adapt to a variety of challenging high-dimensional settings with different structures in the data (e.g., a large number of irrelevant components and nonlinear manifold structure) as well as different data types (e.g., functional data, mixed data types and sample sets). We study the theoretical and empirical performance of our proposed method, and we compare our approach with traditional conditional density estimators on simulated as well as real-world data, such as photometric galaxy data, Twitter data, andline-of-sight velocities in a galaxy cluster. § INTRODUCTIONA challenging problem in modern statistical inference is how to estimate a conditional density of a random variable Z ∈given a high-dimensional random vector X⃗∈^D, f(z|).This quantity plays a key role in several statistical problems in the sciences where the regression function [Z|] is not informative enough due to multi-modality and asymmetry of the conditional density. For example, several recent works incosmology <cit.> have shown that one can significantly reduce systematic errors in cosmological analyses by using the full probability distribution of photometric redshifts z (a key quantity that relates the distance of a galaxy to the observer)given galaxy colors x⃗ (i.e., differences of brightness measures at two different wavelengths). Other fields where conditional density estimation plays a key role are time series forecasting in economics <cit.> and approximate Bayesian methods <cit.>.Conditional densities can also be used toconstruct accurate predictive intervals for new observations in settings with complicated sources of errors <cit.> or multimodal distributions (see Fig. <ref> and Fig. <ref> for examples).Nevertheless,whereasa large literature has been devoted to estimatingthe regression [Z|], statisticians have paid far less attention to estimating the full conditional density f(z|), especially whenis high-dimensional.Most attempts to estimate f(z|) can effectively only handle up to about 3 covariates (see, e.g., ). In higher dimensions, such methods typically rely on a prior dimension reduction step which, as is the case with any data reduction, can result in significant loss of information. Contribution. There is currently no general procedure for converting successful regression estimators (that is, estimators of the conditional mean [Z|]) to estimators of the full conditional density f(z|) — indeed, this is a non-trivial problem. In this paper, we propose a fully nonparametric approach to conditional density estimation, which reformulates CDE as an orthogonal series problem where the expansion coefficients are estimated by regression. By taking such an approach, one can efficiently estimate conditional densities in high dimensions by drawing upon the success in high-dimensional regression. Depending on the choice of regression procedure, our method can exploit different types of sparse structure in the data, as well as handle different types of data.For example, in a setting with submanifold structure, our estimator adapts to the intrinsic dimensionality of the data with a suitably chosen regression method;such as, nearest neighbors, local linear, tree-based or spectral series regression <cit.>. Similarly, ifthe number of relevant covariates (i.e., covariates thataffect the distribution of Z) is small, one can construct a good conditional density estimator using lasso, SAM, Rodeo or other additive-based regression estimators <cit.>.Because of the flexibility of our approach, the method is able to overcome thethe curse of dimensionality in a variety of scenarios with faster convergence rates and better performance than traditional conditional density estimators; see Sections <ref>-<ref> for specific examples and analysis.By choosing appropriate regression methods, the method can also handle different types of covariates that represent discrete data, mixed data types, functional data, circular data, and so on, which generally require hand-tailoredtechniques (e.g., ).Most notably, Sec. <ref> describes an entirely new area of conditional density estimation (here referred to as “Distribution CDE”) where a predictor is an entire sample set from an underlying distribution. We call our general approach FlexCode, which stands for Flexible nonparametric conditional density estimation via regression.Existing Methodology.With regards to existing methods for estimating f(z|x⃗), several nonparametric estimators have been proposed when x⃗lies ina low-dimensional space. Many of these methods are based on first estimating f(z,x⃗) and f(x⃗) with, for example, kernel density estimators <cit.>, and then combining the estimates according to f(z|x⃗)=f(z,x⃗)/f(x⃗). Several works further improve upon such an approach by using different criteria and shortcuts to tune parameters as well as creating fast shortcuts toimplement these methods (e.g., ). Other approaches to conditional density estimation in low dimensions include using locally polynomial regression <cit.>,least squares approaches <cit.> and density estimation through quantile estimation <cit.>; see <cit.> and references therein for other methods. For moderate dimensions, <cit.> propose a method for tuning parameters in kernel densityestimators which automatically determines which components of x⃗ arerelevant to f(z|x⃗). The method produces good results but is not practical for high-dimensional data sets: Because it relies on choosing a different bandwidth for each covariate, it hasa high computational cost that increases with both the sample size n and the dimension D, with prohibitive costs even for moderate n's and D's.Similarly,<cit.> propose a conditional estimator that selects relevant components but under the restrictive assumption that f(z|) has an additive structure; moreover the method scales as O(D^3),which is also computationally prohibitive for moderate dimensions. Another framework is developed by <cit.>, who proposes an orthogonal series estimator that automatically performs dimension reduction on x⃗ when several components of this vector are conditionally independent of the response. Unfortunately, the method requires one to compute D+1 tensor products, which quickly becomes computationally intractable even for as few as 10 covariates.More recently, <cit.> propose an alternative orthogonal series estimator that uses a basis that adapts to the geometry of the data. They show that their approach, called Spectral Series CDE,as well as the k-nearest neighbor method by <cit.>,work well in high dimensions when there is submanifold structure. These methods, however, do not perform well whenhas irrelevant components.FlexCode, on the other hand, is flexible enough to overcome the difficulties of other methods under a large variety of situations because it makes use of the many existing regression methods for high-dimensional inference. As an example, Fig. <ref> shows the level sets of the estimated conditional density in achallenging problem that involves ≈500 covariates. Here we estimate f(|), whereis the content of a tweet andis the location where it was posted (latitude and longitude). While FlexCode is able to estimate the location of tweets even in ambiguous cases (there is a Long Beach both in California and in Connecticut, which is reflected by the results in thebottom right plot in Fig. <ref>), no other existing fully nonparametricmethods we are aware of are able to estimate this quantity with reasonable precision. FlexCode, based on sparse additive regression, is able to estimate the location of tweets even in ambiguous cases (there is a Long Beach both in California and in Connecticut, which is reflected by the results in thebottom right plot in Fig. <ref>); we are not aware of any other existing fully nonparametricmethod that are able to estimate this quantity with reasonable precision as well as attach meaningful measures of uncertainty. See more details about this example in Sec. <ref>.In Section <ref>, we describe our method in detail, andpresent connections with existingliterature on Varying Coefficient methods andSpectral Series CDE. Section <ref> presents several applications of FlexCode. Section <ref>discusses convergence rates of the estimator, and Section <ref> concludes the paper. § METHODSAssume we observe i.i.d. data (_1,Z_1),…,(_n,Z_n), where the covariates ∈ℝ^D with D potentially large, and the response Z ∈ [0,1]. [More generally,can represent functional data, distributions, as well as mixed continuous and discrete data; see Sec. <ref> for examples. The response z can also be multivariate (Sec. <ref>) or discrete <cit.>.] Our goal is to estimate the full density f(z|) rather than, e.g., only the conditional mean [Z|] and conditional variance [Z|]. We propose a novel “varying coefficient” series approach, where we start by specifying an orthonormal basis (ϕ_i)_i ∈ℕfor ℒ^2(). This basis will be used to model the density f(z|) as a function of z. As we shall see, each coefficient in the expansion can be directly estimated via a regression.Note that there is a wide range of (orthogonal) bases one can choose from to capture any challenging shape of the density function of interest<cit.>. For instance, a natural choice for reasonably smooth functions f(z|) is the Fourier basis:ϕ_1(z)=1;ϕ_2i+1(z)=√(2)sin(2π iz ),i∈ℕ; ϕ_2i(z)=√(2)cos(2π iz ),i∈ℕ Alternatively, one can use wavelets or related bases to capture inhomogeneities in the density (see Sec.<ref> for an example), and indicator functions to model discrete responses <cit.>. Smoothing using orthogonal functions is per se not a new concept <cit.>. The novelty in FlexCode is that we, by using an orthogonal series approach for the response variable, can convert a challenging high-dimensional conditional density estimation problem to a simpler high-dimensional regression (point estimation) problem.For fixed ∈^D,we write f(z|)=∑_i ∈ℕβ_i ()ϕ_i(z).Note that our model is fully nonparametric: Equation <ref> holdas long as, for every , f(z|) is ℒ^2() integrable as a function of z. Furthermore, because the {ϕ_i}_i ∈ℕbasis functions are orthogonal to each other, the expansion coefficients are given byβ_i () =⟨ f(.|),ϕ_i⟩ = ∫_ϕ_i(z) f(z|) dz = [ϕ_i(Z)|].That is, each “varying coefficient” β_i () in Eq. <ref> is a regression function, or conditional expectation.This suggests that we, for fixed i, estimate β_i () by regressing ϕ_i(z) onusing the sample (_1,ϕ_i(Z_1)),…,(_n,ϕ_i(Z_n)). We define our FlexCode estimator of f(z|)asf(z|)=∑_i=1^Iβ_i()ϕ_i(z),where the results from the regression,β_i () = [ϕ_i(Z)|],model how the density varies in covariate space. The cutoff I in the series expansion is a tuning parameter that controls the bias-variance tradeoff in the final density estimate. Generally speaking, the smoother the density, the smaller the value of I; see Sec. <ref> Theory for details. In practice, we use cross-validation or data splitting (Sec. <ref>) to tune parameters. With FlexCode, the problem of high-dimensional conditional density estimation boils down to choosing appropriate methods for estimating theregression functions [ϕ_i(Z)|], i=1,…,I. The key advantage of FlexCode is its flexibility: By taking advantage of new and existing regression methods, we can adapt to different structures in the data (e.g., manifolds, irrelevant covariates as well as different relationships betweenand the response Z), and we can handle different types of data (e.g. mixed data, functional data, and so on).We will further explore this topic in Secs. <ref>-<ref>.§.§ Loss Function and Tuning of Parameters For a given estimator f(z|x⃗), we measure the discrepancy betweenf(z|x⃗) and f(z|x⃗) via the loss function L(f,f)= ∬(f(z|x⃗)-f(z|x⃗))^2dP(x⃗)dz =∬f^2(z|x⃗)dP(x⃗)dz-2∬f(z|x⃗)f(z,x⃗) dx⃗dz+C,where C is a constant that does not depend on the estimator. To tune the parameter I, we split the data into a training and a validation set. We use the training set to estimate each regression function β_i(). We then use the validation set (z'_1,x⃗'_1),…, (z'_n',x⃗'_n') toestimate the loss (<ref>) (up to the constant C) according to:L(f,f)=∑_i=1^I 1/n'∑_k=1^n'β^2_i(_k') -21/n'∑_k=1^n'f(z'_k|x⃗'_k),This estimator is consistent because of the orthogonality of the basis{ϕ_i}_i. We choose the tuning parameterswith the smallest estimated loss L(f,f).Algorithm 1 summarizes our procedure.In line 3,we split the training data in two parts to tune the parameters associated with the regression using the standard ℒ^2() regression loss, i.e., [(W-β_i ())^2]. In terms of computational efficiency, FlexCode is typically faster than existing methods for conditional density estimation (see Section <ref>), especially in high dimensions and for massive data sets. If the FlexCode estimator is based on a scalable regression procedure(e.g., ), then the resulting conditional density estimator will scale as well. Furthermore, FlexCode is naturally suited for parallel computing, as one can estimate each of the I regression functionsseparately and then combine the estimates according to Eq. (<ref>). Our implementation of FlexCodeis available at <https://github.com/rizbicki/FlexCoDE>, and implements a parallel version of the estimator. For the final density estimate (Step 9 in Algorithm 1), we apply the same techniques as in <cit.> to remove potentially negative values and spurious bumps.§.§ Extension to Vector-Valued Responses By tensor products, one can directly extend FlexCode to cases where the response variableis vector-valued.For instance,if ∈^2, consider the basis {ϕ_i,j(z⃗)=ϕ_i(z_1)ϕ_j(z_2): i,j∈ℕ}, where =(z_1,z_2), and {ϕ_i(z_1)}_i and {ϕ_j(z_2)}_j are bases for functions in 𝔏^2().Then, letf(|)=∑_i,j ∈ℕβ_i,j()ϕ_i,j(), where the expansion coefficients β_i,j() =⟨ f(.|),ϕ_i,j⟩ = ∫_^2ϕ_i,j() f(|) d = [ϕ_i,j()|]. Note that each β_i () is a regression function of a scalar response. In other words,the FlexCode framework allows one to estimate multivariate conditional densities by only using regression estimators of scalar responses.Remark: To avoid tensor products, one can alternatively compute a spectral basis {ϕ_i()}_i≥ 0<cit.>. This basis is orthonormal with respect to the density f() and adapts to the density's intrinsic geometry.The expansion coefficientsare then givenbyβ_i() =⟨ f(.|),ϕ_i⟩_f() = ∫_^dϕ_i() f(|) f() d = [ϕ_i()f()|], in which case one needs to estimate f() as well. §.§ Connection to Other MethodsVarying-Coefficient Models.The model f(z|)=∑_i ∈ℕβ_i ()ϕ_i(z) can be viewed as a fully nonparametric varying-coefficient model. Varying-coefficient models <cit.> are often seen as semi-parametric models or as extensions of classical linear models, in which a function η is modeled as η=∑_i=1^dβ_i()u_i, where β_i() are smooth functions of the predictors , and u_1,…,u_d are other predictors. In our case, we have a fully nonparametric model, because d ⟶∞ and(u_i)_i≥1:={ϕ_i(z)}_i≥1 is a basis of ℒ^2().Traditional Series CDE. If each β_i() is estimated using a standard orthogonal series regression estimator, FlexCode recovers the standard orthogonal series CDE from <cit.>. Indeed, let (ψ_j)_j be an orthonormal basis for(not necessarilythe same as (ϕ_i)_i). A standard orthogonal series estimator is based on the fact that theconditional density can be expanded as f(z|)=∑_i≥1∑_j≥1β_i,jϕ_i(z) ψ_j(), where β_i,j= [ϕ_i(Z)ψ_j()/f()].One typically estimates β_i,j using n^-1∑_k=1^nϕ_i(Z_k)ψ_j(_k)(f(_k))^-1, where f is an estimate of the marginal density of the covariates.Now,the standard orthogonal series regression estimator for β_i() is based on the expansionβ_i()=∑_j≥1γ^(i)_j ψ_j(),where γ^(i)_j= ∫β_i() ψ_j() d = ∫[ϕ_i(Z)|] ψ_j() d == ∫[ϕ_i(Z)ψ_j()/f() |]f() d = [ϕ_i(Z)ψ_j()/f()]. One typically estimates γ^(i)_j using n^-1∑_k=1^nϕ_i(Z_k)ψ_j(_k)(f(_k))^-1, where f is an estimate of the marginal density of the covariates. By comparing the estimators of β_i,j and γ^(i)_j (Eqs. <ref> and <ref>), we see these methods are equivalent. Spectral Series CDE. If each β_i() is estimated using a spectral series regression estimator <cit.>, FlexCode recovers thespectral series CDE of <cit.> as a special case of FlexCode. The reason is analogous to that used for showing the connection between FlexCodeand standard orthogonal series regression. The difference is that the basis (ψ_j())_j used in spectral series regression is orthogonal with respect to P() rather than Lebesgue measure, and the expansion coefficients from Eq. <ref> become β_i,j= [ϕ_i(Z)ψ_j()], but also do γ^i_j in Eq. <ref>. See more details in <cit.>.Spectral Series CDE. FlexCode recovers the spectral series conditional density estimator of <cit.> if each β_i() is estimated via a spectral series regression <cit.>. Indeed, let{ψ_j}_j be a spectral basis for , where by construction ∫_𝒳ψ_i(x⃗)ψ_j(x⃗)dP(x⃗)=δ_i,j=(i=j) <cit.>. In spectral series CDE, one writes the conditional density asf(z|)=∑_i≥1∑_j≥1β_i,jϕ_i(z) ψ_j(), where the coefficientsβ_i,j= ∬ f(z|x⃗)ϕ_i(z) ψ_j()dP(x⃗)dz = [ϕ_i(Z)ψ_j()].Now,a spectral series regression for β_i()=[ϕ_i(Z)|] is based on the model β_i()=∑_j≥1γ^(i)_j ψ_j(),where γ^(i)_j= ∫β_i() ψ_j()dP(x⃗) = ∫[ϕ_i(Z)|] ψ_j()dP(x⃗) == ∫[ϕ_i(Z)ψ_j() |]dP() = [ϕ_i(Z)ψ_j()].By inserting β_i() into Eq. <ref>, we see that Spectral Series CDE <cit.> is a special case of FlexCode. Henceforth, we will refer to this version of FlexCode as FlexCode-Spec. Remark:Using similar arguments, one can show that FlexCode recovers the orthogonal series CDE of <cit.> if each β_i() is estimated via traditional orthogonal series regression. However, as discussed in<cit.>, traditional series approaches via tensor products quickly become intractable in high dimensions.Nevertheless, it is interesting to note that FlexCode forms a very large family of CDE approaches that includes Spectral Series CDE and traditional orthogonal series CDE as special cases.§ EXPERIMENTS In what follows, we compare the following estimators: * FlexCode is our proposed series approach. We implementsix versions of FlexCode, where we use different regression methods to compute the coefficients β_i () = [ϕ_i(Z)|] in Eq. <ref>. FlexCode-SAM is based on Sparse Additive Models <cit.>.[Sparse additive regression models can be useful even if the true coefficients β_i () are not additive, because of the curse of dimensionality and the ability of sparse additive models to identify irrelevant coefficients without too restrictive assumptions.] FlexCode-NN is based onNearest Neighbors regression <cit.>. FlexCode-Spec usesSpectral Series regression<cit.> andis, as shown in Sec. <ref>, the same as Spectral Series CDE, the conditional density estimator in <cit.>. For mixed data types, we implementFlexCode-RF, which estimates the regression functions via random forests <cit.>,and for functional data, we use FlexCode-fKR, where the coefficients in the model are estimated via functional kernel regression <cit.>. Finally, in Sec. <ref>,we illustrate how FlexCode-SDM can extend Support Distribution Machines (SDM; ) and other distribution regression methods to estimating conditional densities on sample setsor groups of vectors. * KDE is the kernel density estimator f(z|):=f(z,)/f(), wheref(z,) and f() are standard multivariate kernel density estimators. We rescale the data to have the same mean and variance in each direction, and we assume an isotropic Gaussian kernel for bothand z, i.e., f(z|) = ∑_i=1^n K_h_x(-_i)K_h_z(z-Z_i)/∑_i=1^n K_h_x(-_i) , where K_h(t)=h^-d K(t/h) denotes an isotropic Gaussian kernel with bandwidth h in d dimensions. * KDE_ is the multivariate kernel density estimator f(z|):=f(z,)/f(), where the estimators f(z,) and f() have a different bandwidthfor each component of<cit.>; i.e., f(z|) = ∑_i=1^nK_h(-_i) K_h_z(z-Z_i)/∑_i=1^n K_h(-_i) , where K_h(-_i)=(h_1 … h_d)^-1∏_j=1^d K(x_j-X_ij/h_j) fordata _i=(X_i1,…, X_id) anda bandwidth vector h=(h_1,…, h_d). We use the R package<cit.> with kd-trees and Epanechnikov kernels for computational efficiency <cit.>. * kNN is a kernel nearest neighbors approach <cit.> to conditional density estimation; it is defined as f(z|x⃗)∝∑_j ∈𝒩_k(x⃗) K_ϵ(z-Z_j), where 𝒩_k(x⃗) is the set of the k closest neighbors toin the training set, and K_ϵ is a multivariate (isotropic) Gaussian kernel with bandwidth ϵ. * fkDE is a nonparametric conditional density estimator for functional data <cit.>. It is defined as f(z|) = 1/h_z∑_i=1^nK(d(x,X_i)/h_x) K_0(z-Z_i/h_z) /∑_i=1^nK(d(x,X_i)/h_x), where d is a distance measure in the (functional) space of the data, K and K_0 are isotropic kernel functions, and h_x and h_z are tuning parameters. Note that for regression, SAM is designed to work well when there is a small number of relevant covariates, and both Spectral Series Regression and Nearest Neighbors Regression perform well when the covariates exhibit a low intrinsic dimensionality. To our knowledge, KDE_ isthe only CDE method that can handle mixed data types. §.§ Toy ExamplesBy simulation, we create toy versions of common scenarios with different structures in data and different types of data. We use 700 data points for training, 150 for validation and 150 for testing the methods. Each simulation is repeated 200 times.§.§.§ Different structures in data. * Irrelevant Covariates. In this example, we generate data according to Z|∼ N(x_1,0.5), where =(X_1,…,X_d) ∼ N(0⃗,I_d), that is, only the first covariate influences the response. * Data on Manifold. Here we let Z|∼ N(θ(),0.5), where =(x_1,…,x_d) lie on a unit circle embedded in a D-dimensional space, and θ() is the angle corresponding to the position of . For simplicity, we assume that the data are uniformly distributed on the manifold; i.e., we let θ() ∼ Unif(0,2π). * Non-Sparse Data. Finally, we consider data with no sparse (low-dimensional) structure. We assume Z|∼ N(,0.5), where =(X_1,…,X_d) ∼ N(0⃗,I_d). §.§.§ Different types of data. * Mixed Data Types. Few existing CDE methods can handle mixed data types; the only other method the authors are aware of is KDE_. For our study, we generate mixed categorical and continuous data, where the categorical covariates (X_1,…,X_D/2) are i.i.d. Unif{c_1,c_2,c_3,c_4,c_5}, and the continuous covariates (X_D/2+1,…,X_D) are i.i.d. N(0,1). The response is given by Z|∼ N(x_D/2+1,0.5) ifx_1∈{c_1,c_2} 10+ 2N(x_D/2+2,0.5)ifx_1∈{c_3,c_4,c_5} * Functional Data. We also consider spectrometric data for finely chopped pieces of meat. These high-resolution spectra are available[<http://lib.stat.cmu.edu/datasets/tecator>; the original data source is Tecator AB] as a benchmark for functional regression models (see, e.g., <cit.>), where the task is to predict the fat content of a meat sample on the basis of its near infrared absorbance spectrum. In our study, we use 215 samples to estimate conditional densities. The covariates are spectra of light absorbance as functions of the wavelength, and the response is the fat content of a piece of meat. We compare the functional kernel density estimator (fKDE) with a FlexCode approach(FlexCode-fKR), where the coefficients in the model are estimated via functional kernel regression <cit.>. We follow <cit.> and implement both methods with the kernel function K(u)=1-u^2 and the ℒ^2() norm between the second derivatives of the spectra as a distance measure. We use 70% of the data points for training, 15% for validation and 15% for testing; the experiment is repeated 100 times by randomly splitting the data. §.§.§ ResultsFigures <ref>-<ref> show the results for the toy data. Our main observations are: * Irrelevant Covariates. In terms of estimated loss(Fig. <ref>, top left),both FlexCode-SAM andKDE_outperform the other methods. However, in terms of computational time (Fig. <ref>, bottom left), FlexCode-SAMis clearly faster than KDE_ as the dimension D of the data grows. When D=17, eachfit with KDE_already takes an average of 240 seconds (4 minutes) on an Intel i7-4800MQ CPU 2.70GHz processor, compared to 22 seconds for FlexCode-SAM.Fig. <ref>, left, shows that the loss of FlexCode-SAM remains the same even for large D ∼ 1000, although fitting the estimator becomes computationally more challenging in high dimensions. Nevertheless, fitting KDE_would be unfeasible for D>50. * Data on Manifold. FlexCode-Spec has the best statistical performance, followed by FlexCode-NN and KDE_ (Fig. <ref>, top center). As before, the computational timeof KDE_ increases rapidly with the dimension (Fig. <ref>, center bottom). For these data, FlexCode-SAM is slow as well even for moderate D, perhaps because SAM cannot find sparse representations of the regression functions.On the other hand (see Fig. <ref>, right), FlexCode-Spec has a computational time that is almost constant as a function of D and the statistical performance remains the same even for large D. The latter result is consistent with our previous findings that spectral series adapt to the intrinsic dimension of the data <cit.>. * Non-Sparse Data. For this example, FlexCode-Spec and FlexCode-SAM are the best estimators. * Mixed Data Types.FlexCode-RF yields better results than KDE_ (its only competitor in this setting) both in terms of estimated loss and computational time; see Fig. <ref>. The computational advantage is especially obvious for larger values of D. When the dimension D=56, eachfitof KDE_ takes an average of 2250 seconds (≈ 37 minutes) on an Intel i7-4800MQ CPU 2.70GHz processor, compared to 304seconds (≈ 5 minutes) for FlexCoDR-RF. * Functional Data. FlexCode via Functional kernel regression improves upon the results of thetraditional Functional kernel density estimator with an estimated loss of -2.78 (0.07) instead of -2.08 (0.03). §.§ Photometric Redshift Estimation Our first application is photometric redshift estimation. Redshift (a proxy for a galaxy's distance from the Earth) is a key quantity for inferring cosmological model parameters. Redshift can be estimated with high precision via spectroscopy but the resource considerations of large-scale sky surveys call for photometry – a much faster measuring technique, where the radiation from an astronomical objects is generally coarsely recorded via ∼5-10 broad-band filters. In photometric redshift estimation, the goal is to estimate the redshift zof a galaxy based on its observed photometric covariates x⃗, using a sample of galaxies with spectroscopically confirmed redshifts. Because of degeneracies(two galaxies with different redshifts can have similar photometric signatures) and because of complicated observational noise, probability densities of the form f(z|) better describe the relationship betweenand z than the regression (z|) does. In this example, we test our CDE methods onn=752galaxies from COSMOS, with D=37 covariates derived from a variety of photometric bands (these data were obtained from T. Dahlen 2013, private communication; seefor additional details).Figure <ref> summarizes the results. All versions of FlexCode improve upon the traditional estimators. The best performance is achieved for FlexCode via Sparse Additive Models (FlexCode-SAM), which indicates that only a subset of the 37 covariates are relevant for redshift estimation; for these data, FlexCode-SAM selected ≈ 18variables in each regression, and three out of the 37 covariates were present in more than 75% of the regressions. §.§ Twitter Data Twitter is a social network where each user is able to post a small text (a tweet) containing at most 140 characters. Information about the location of the post is available upon user permission, but only a few users allow this information to be publicly shared. Here we use samples with knownlocations to estimate the location of tweets where this information has not been shared publicly. Note that most literature on the topic concerns creatingpoint estimates for locations(see, e.g., and references therein). In this work, we estimate the full conditional distribution of latitude and longitude given the content of the tweet; that is, we estimate f(|), whereare covariates extracted from the tweets and =(z_1,z_2) is the pair latitude/longitude. Our data set contains ≈ 8000 tweets in the USA from July 2015 with the word “beach". We extract 500 covariates via a bag-of-words method with the most frequent unigrams and bigrams <cit.>. As we only expect a few of the 500 covariates to be relevant to locating the tweets, we implement FlexCode via sparse additive models. Figure <ref> shows two examples of estimated densities; see Supplementary material for additional examples. To our knowledge, no other fully nonparametric conditional density estimation method can be directly applied tothese types of data where there are many irrelevant variables. Moreover, because FlexCode-SAM is based on sparse additive models, we can find out which covariates are most relevant for predicting location. For the example in Fig.<ref>, left, the expressions “beachin",“boardwalk", and “daytona" are included in at least 33% of the estimated regression functions. For the example to the right, the relevant covariates are “long beach", “island", “long", and “haven". §.§ From Distribution Regression to “Distribution CDE”: Estimating the Mass of a Galaxy Cluster from Sample Sets of Galaxy Velocities Distribution regression and classification is a recent emerging field of machine learning. Instead of treating individual data points (or feature vectors) as covariates, these methods operate on sample sets, where each set is a sample from some underlying feature distribution; see <cit.> and references within.Here we show that FlexCode extends to sample sets as well; our application is estimation of the mass of a galaxy cluster given the line-of-sight velocities of the galaxies in thecluster. Galaxy clusters, the most massive gravitationally bound systems in the Universe, can contain up to ∼1000 galaxies. These structures are a rich source of information on astrophysical processes and cosmological parameters, but to use galaxy clusters as cosmological probes one needs to accurately measure their masses. A standard approach is to employ the classical virial theorem and directly relate the mass of a cluster tothe line-of-sight (LOS) galaxy velocity dispersion, i.e., the variance of the measured galaxy velocities in the cluster<cit.>. Recently, <cit.> and <cit.> have shown that one can significantly improve such mass predictions by taking advantage of the entire LOS velocity distribution of galaxies instead of only the dispersion (i.e., a summary of the distribution). Here we show that FlexCode can further improve these results. The general set-up is that we observe data of the form (^(1)_1, …,_1^(J_1),z_1),…,(_I^(1), …,_I^(J_I),z_I), where z_i is the mass of the i-th cluster for i=1,…,I; and _i^(j) is a vector of galaxy observables (such as LOS velocity and the projected distance from the cluster center) for the j-th galaxy in the i-th cluster. Note that different clusters i contain different numbers J_i of galaxies. The key idea behind Support Distribution Machines (SDMs; proposed for this application by ) as well as other “distribution regression” methods <cit.>, is to treat each sequence _i^(1), …,_i^(J_i) as a sample from a probability distribution p_i, and to construct an appropriate kernel matrix on these sample sets. The task is then to predict a scalar (z_i) from a distribution (p_i) by estimating 𝔼[Z|p]. Here we show how FlexCode extends regression on distributions to conditional density estimation on distributions; i.e., instead of providing a point estimate (and standard error) of the mass of a galaxy cluster, we estimate the full probability density f(z|p) of the unknown mass of a galaxy cluster given galaxy observables. In our application, the response z_i is the logarithm of the cluster mass (log M) and the observables {x_i^j}_j=1^J_i are scalar quantities that represent the absolute values of galaxy velocities along one line-of-sight. The challenge in this particular problem is that the observed samples are given by (^(1)_1, …,_1^(I_1),z_1),…,(_n^(1), …,_n^(I_n),z_n), where z_i is the log mass of the i-th cluster and _i^(j) is the vector of observed absolute values of the velocities of the j-th galaxy of the i-th cluster along each line-of-sight. Notice differentclusters may have a different number of galaxies on them. It is therefore not possible to use standard conditional density estimators; in fact even the regression task is challenging here. The only regression method we are aware of that has been applied to this problem is the so-called Support Distribution Machine <cit.>. The key idea is to treat each sequence ^(i)_1, …,_i^(I_i) as a sample from a distribution p_i, and build an appropriate kernel matrix based on this. Here we show how one can easily adapt this idea for conditional density estimation via FlexCode. All the approaches we compare here are based on the similarity matrix build by <cit.>, which we describe in details in the sequence. Like <cit.>, we use the Kullback-Leibler (KL) divergence to measure similarity between pairs of velocity distributions, and we estimate the divergence from the observed galaxy velocities with the estimator from <cit.>. The details are as follows: Let p_A and p_B denote velocity distributions for clusters A and B, respectively. Define the kernel k(p_A,p_B)=exp(-(p_A, p_B)/σ^2), where (p_A, p_B) is the Kullback-Leibler divergence between p_A and p_B. We estimate the KL divergence via Wang et al's k nearest neighbors method for k=2. That is, let X_A denote the set of LOS velocities associated with the n galaxies of cluster A, and let X_B denote the set of velocities associated with the m galaxies of cluster B. The estimated KL divergence from p_A to p_B is given by _n,m(X_A,X_B)=d/n∑_i=1^n logν_k(i)/ρ_k(i)+logm/n-1, where ν_k(i) is the Euclidean distance from the covariates (in this case, the LOS velocity) of the i-th galaxy in X_A to its k-th nearest neighbor in X_B, ρ_k(i) is the Euclidean distance from the covariates (the LOS velocity) of the i-th galaxy in X_A to its k-th nearest neighbor in X_A, and d is the number of galaxy observables (in this example, d=1). As the computed kernel matrix k(X_A,X_B)=exp(-_n,m(X_A,X_B)/σ^2) may not be positive semi-definite (PSD), we project the matrix to the closest PSD matrix in Frobenius norm <cit.>. Using the PSD kernel matrix, we then estimate the conditional density f(z|p). We compare four approaches to conditional density estimation on distributions, which as in the rest of the paper use a Fourier basis in z; * Functional KDE: the functional kernel density estimator <cit.>, * FlexCode-NN: FlexCode with Nearest Neighbors regression, * FlexCode-Spec: FlexCode with Spectral Series regression, * FlexCode-SDM: FlexCode with SDM regression. In the experiments, we also include a FlexCode estimator that use a wavelet basis in z; * FlexCode_W-SDM: FlexCode with SDM regression in x, and Daubechies wavelets with 3 vanishing moments in z. Our data consist of simulations of n=5028 unique galaxy clusters with minimum mass of 1 × 10^14 M_⊙ h^-1; see <cit.> for details. All four methods above are based on the same distance computation _n,m(X_A,X_B) with k=2, and we use data splitting and the loss (<ref>) for selecting tuning parameters. For simplicity, we only consider one LOS for each cluster (the x-axis LOS in the catalog). It is clear from Table <ref> that the FlexCode-SDMand FlexCode_W-SDM estimates of conditional density are more accurate than the results from any other method. The coverage plots (see Appendix <ref> for the definition) in the bottom panel of Fig <ref> also verify that these density estimates fit the observed data well. The top left panel of Figure <ref> shows examples of density estimates from FlexCode_W-SDM for 16 randomly chosen clusters.Several of these distributions are bimodal, in which case regression estimates are not very informative. This can be further illustrated by Fig. <ref>. The left panel shows a scatter plot of the observed log massesversus the estimated conditional mean [Z|p]:=∫ z f(z|p)dz for unimodal versus multimodal cases. The right panel shows a boxplot ofthe absolute fractional mass error |ε|for the two populations; the fractional mass error ε is defined as<cit.> ε = M_ pred-M/M, whereM is the observed cluster mass and M_ pred is the predicted cluster mass. Much of the scatter can indeed be attributed to multimodal densities and non-standard prediction settings. Finally, we notice that both the mean and the mode of FlexCode-SDM as well as FlexCode_W-SDMdensities improve upon plain SDM regression. Table <ref>compares the fractional mass error distributions of the predictions.By taking the mode of the FlexCode density we reduce theε 68% scatter[Theε 68% scatter, Δε, is the 68% quantile of the distribution of |ε|] from Δε≈ 0.24 for standard SDM down to a width of ≈0.15 for FlexCode-SDM and of ≈0.17 for FlexCode_W-SDM with a mode estimator. To summarize: FlexCode extends SDM to conditional density estimation on distributions, and the estimated densities produce better point estimates of cluster masses. The real advantage with FlexCode, however, is that we can more accurately quantify the uncertainty in the predictions and potentially improve inference for outliers or cases that are not well described by one-number summaries. For example, we can use the estimated densities to construct more informative highest predictive density (HPD) regions of the cluster mass, i.e., regions of the form {z:f(z|)≥ K}, where K is chosen in such a way that the regions have the desired coverage level (e.g., 95%). The toppanels of Figure <ref> shows some examples of multimodal densities and their 95% HPD regions.In many cases, returning a predictive region for the cluster mass is a better alternative to just taking the mean or mode of the density. The coverage plot in the bottom right panel also indicates that the empirical coverage of these regions is indeed close to 95%. § THEORYIn this section, we derive bounds and rates for FlexCode; that is, the conditional density estimator in Eq. <ref>. We use the notation f_I(z|) to indicate its dependence on the cutoff I. We assume that f belongs to a set of functions which are not too “wiggly”.For every s>1/2 and 0<c<∞, let W_ϕ(s,c)={f = ∑_i≥ 1θ_i ϕ_i :∑_i≥ 1 a_i^2 θ^2_i ≤ c^2 }, where a_i ∼ (π i)^s, denote the Sobolev space. For the Fourier basis{ϕ_i}_i, this is the standard definition of Sobolev space <cit.>; it isthespace of functions that have their s-th weak derivative bounded by c^2 and integrable in ℒ^2(). We enforce smoothness in the z-direction by requiring f(z|) to be in a Sobolev space for all x⃗. This is formally stated as Assumption <ref>, where β and C are used to link the Sobolev spaces at different x. [Smoothness in z direction] ∀x⃗∈𝒳,f(z|) ∈ W_ϕ(s_x⃗,c_x⃗), where f(z|) is viewed as a function of z, and s_x⃗ and c_x⃗ are such thatinf_x⃗ s_x⃗=β>1/2 and∫_𝒳 c_x⃗^2dx⃗= C <∞.We also assume that eachfunction β_i() is estimated using a regression method with convergence rate O(n^-2α/(2α+d)), where typically α is a parameter related to the smoothness of the β_i() function, and d is either thenumber of relevant covariates or the intrinsic dimension of .In other words, we assume that each regression adapts tosparse structure in the data. This is formally stated as Assumption <ref>. [Regression convergence] For every i ∈ℕ, there exists some d ∈ℕ and α>0 such that [∫(β_i()-β_i() )^2d]=O(n^-2α/(2α+d)) Note that the smoothness parameter α must be the same for every i ∈ℕ.Typically this assumption will hold because in many applications it is reasonable to assume that (i) if _1 is close to _2, then f(z|_1) is also close to f(z|_2) for every z ∈ (in other words, f(z|) is smooth as a function of ), and (ii) there is some structure in x (e.g., low intrinsic dimensionality) or in the relationship between x and z (e.g., sparsity), which the regression method for estimating β_i takes advantage of. Here are some examples where Assumption <ref> holds: * β_i is the k-nearest neighbors estimator<cit.>, d is the intrinsic dimension of the covariate space and,for every z ∈ [0,1],f(z|) is L-Lipschitz in(in this case, α=1);* β_i is a local polynomial regression <cit.>, d is the intrinsic dimension of the covariate space and, for every z ∈ [0,1], f(z|) is α times differentiable with all partial derivatives up to order α inare bounded; * β_i is the Rodeo estimator<cit.>, d is the number of variables that affect the distribution of Z and, for everyz,all partial derivatives of f(z|) up to fourth orderinare bounded(in this case, α=2); * β_i is the regression estimator from <cit.>, d is the number of variables that affect the distribution of Z,[That is, there exists a subset R ⊆{1, …,D} with |R|=d such thatf(z|)=f(z|(x_i)_i ∈ R)] and, for every z ∈ [0,1],f(z|) is α-Hölderian in ; * β_i is the Spectral series regression<cit.>, d is the intrinsic dimension of the covariate space and, for every z ∈ [0,1],f(z|) is smooth with respect to P_X according to ∫ ||∇ f(z|)||^2dS()<∞ for a smoothed version S() of f (in which case α=1);* β_i is a local linear functional regression <cit.>, the predictor X is a function talking values in ℒ^2([0,1]), X is fractal of order τ, and, for every z ∈ [0,1], f(z|) is twicedifferentiable with a continuous second derivative (yielding rates with α=2 and d=τ.) In essence, Assumption 2holds for examples E1-E6 because smoothness in f(z|) (seen as a function of ) implies smoothness of the β_i() functions in FlexCode. We refer to Appendix A1for details and proofs. (See also, e.g., <cit.> and references therein for other adaptive regression methods.) We also note that the converge rates may vary depending on the choice of basis. Under Assumptions 1-2, we bound the bias and variance of f_I(z|) separately. [Bias Bound] Under Assumption <ref>,∑_i>I∫(β_i() )^2d=O(I^-2β)[Variance Bound] From Assumption <ref>, it follows that∑_i=1^I [∫(β_i()-β_i() )^2d]=IO(n^-2α/(2α+d))Our main result follows. Under Assumptions <ref> and <ref>, an upper bound on the risk of the CDEfrom Equation <ref>is [∬(f_I(z|)-f(z|))^2dzd] ≤IO(n^-2α/(2α+d))+O(I^-2β) See Appendix <ref> for proofs. Under Assumptions <ref> and <ref>, it is optimal to take I ≍ n^2α/(2α+d)(2β+1),which yields the rate O(n^-2β/2β+d2β+1/2α+1)for the estimator in Equation <ref>. To summarize: The convergence rate of FlexCode only depends on d, the “true” dimension of the problem. Moreover, the rate is near minimax with regards to d: In the isotropic setting whereand z have the same degree of smoothness, i.e., α=β, the rate becomesO(n^-2α/2α+d2α+1/2α+1),which is close to the minimax rate O(n^-2α/(2α+1+d)) of a conditional density estimator with d covariates <cit.>. The difference is the multiplicative factor 2α+1/2α, which gets closer to 1, the smoother f is.Although FlexCode's rate isslightly slower than the optimal rate,[The reason may be that that we optimize the tuning parameters of each regression β_i() so as to have optimal regression estimates (Assumption <ref>) rather than an optimal estimate of f(z|).] the estimator is still considerably faster than O(n^-2α/(2α+1+D)), the usual minimax rate of a nonparametric conditional density estimator in ℝ^D.In other words, even though there are D covariates, our estimator can overcome the curse-of-dimensionality and behave as if there are only d ≪ D covariates. Finally, note that although we here restrict our examples to cases where either (i) the intrinsic dimension is small or (ii) several covariates are irrelevant, the theory we develop can easily be applied to other settings for high-dimensional regression estimation. For instance,<cit.> introduce a third type of sparse structure: in their paper,r may depend on all D covariates, but admits an additive structurer = ∑_s=1^k r_s, where each component function r_s depends on a small number d_sof predictors. The authors then show that an additive Gaussian process regression achieves good rates of convergence in such a setting. It follows that FlexCode can achieve good rates under a similar additive settingf(z|) = ∑_s=1^k f_s(z|)if one estimates the expansion coefficients via additive Gaussian process regression.§ CONCLUSIONSWith FlexCode, one can use any regression methodology to estimate a conditional density. In other words, FlexCode is a powerful inference and data analysis tool that converts prediction to the problem of understanding the role of covariates in explaining the outcome, with meaningful measures of uncertainty attached to the predictions. Because of the flexibility of the method, one can construct estimators for a range of different scenarios with complex, high-dimensional data. In the paper, we emphasized examples where several redundant covariates are correlated, and examples where only a small number of covariates influence the distribution of the response. We showed that FlexCode has good theoretical properties and empirical performance comparable tostate-of-the-art approaches in a wide variety of settings, including cases with mixed data types and functional data.In the paper, we restricted most analyses to Fourier bases in the outcome space, but for distributions that are inhomogeneous with respect to the response variable, one may benefit from nonlinear approximations in a wavelet basis <cit.>. We will explore this aspect further in a separate paper, as well as extensions of FlexCode to approximate likelihood computationfor structured data and complex simulation models. Another interesting direction for future work is variable selection via FlexCode. For example, FlexCode-Forest and FlexCode-SAM currently perform a separate variable selection for each coefficient β_i() in FlexCode (Eq. <ref>), but one can unify these results to define a common support for the final FlexCode estimate.In this work we propose a promising new conditional density estimator which takes advantage of the large literature on regression function estimation, FlexCode. The estimator is able to overcome the curse-of-dimensionality in several scenarios. We emphasized two example: (i) when the covariates are very redundant, and (ii) when only a small number of covariates influence the distribution of the response. We showed that FlexCode has good theoretical properties, and empirical performance comparable tostate-of-the-art approaches in a wide variety of settings, including mixed data types and functional data examples. Moreover, it is generally faster thanmany of such approaches. Finally, we argued that it is extremely flexible. FlexCode is a very powerful tool: it extends any regression estimator to a conditional density estimator, i.e., given any regression methodology, our framework allows one to use the same methodology to estimate a conditional density.Acknowledgments. We thank Michelle Ntampaka and Hy Trac for sharing the data for the galaxy cluster mass example, and Peter E. Freeman for his help with the photometric redshift and galaxy cluster mass studies. This work was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (2014/25302-2), NSF DMS-1520786, and the National Institute of Mental Health grant R37MH057881. plainnat Appendix§ DIAGNOSTIC TEST OFCONDITIONAL DENSITY ESTIMATESTo assess how well a model actually fits the observed data, we use coverage plots that are based on Highest-Predictive Density (HPD) regions.* (Q-Q Plot) For every c in a grid of values on [0,1] and for every observation i in the test sample, compute Q_i^c = F_z|x⃗_i^-1(c). Definec=1/n∑_i=1^n(Z_i ≤ Q_i^c). We plot the values of c against the corresponding values of c. If the distributions F_z|x⃗ and F_z|x⃗ are similar, then the points in the Q-Q plot will approximately lie on the line c = c. * (P-value)For every test data point i, let U_i=F_z|x⃗_i(Z_i).If the data are really distributed according to F_z|x⃗, then U_1,…,U_n ∼ Unif(0,1). Hence, we compute the p-value for a Kolmogorov-Smirnoff test that compares the distributions of these statistics to the uniform distribution.* Coverage Plots and HPD Regions.Let f_z|x⃗_i denote the estimated conditional density function for z given x⃗_i. For every α in a grid of values in[0,1]and for every data point i in the test sample, we define a set A_i such that ∫_A_if(z|x⃗_i)dz=α. Here we choose the set A_i with the smallest area: A_i={z:f(z|_i)>t} where t is such that∫_A_if(z|x⃗_i)dz=α; i.e., A_iis a Highest Predictive Density region. Let α_i=1/n∑_i=1^n (Z_i ∈ A_i).If f_z|x⃗ and the true density f_z|x⃗ are similar, then α_i ≈α_i.Hence, as a diagnostic tool, we graph α_i versus α_i for the test set, and assess how close these points are to the line α = α. For each α_i, we also include a 95% confidence intervalbased on a normal approximation to the binomial distribution.§ ADDITIONAL TWITTER DATA Here we consider 5000 geotagged tweets posted in July 2015 that include either the keywordfrio or the keyword calor; these words mean cold and hot in Spanish as well as in Portuguese. As in Sec. <ref>, the goal is to predict the latitude and longitude of a tweet, , based on its content . Using the same methodology (FlexCode-SAM) as before, we estimate f(|). Fig. <ref> shows the results for three tweets. In the tweet corresponding to the left plot, the user mentions “beach" and “heat". Because (i) July is a summer month in the north hemisphere, (ii) the tweet is in Spanish, and (iii) it mentions “beach", FlexCode automatically assigns high probability to the coast of Spain. For the example corresponding to the middle plot, on the other hand, the word “beach" does not occur, but the tweet is in Spanish and it mentions hot weather. As a result, our density model assigns high probability to the interior of Spain. Our final example, corresponding to the right plot in the figure, is a tweet in Portuguese about cold weather. Our FlexCode model here assigns high probability to big cities in Brazil, which is consistent with July being a winter month in the south hemisphere. We also notice that it in the winter rains a lot in Recife, the northernmost city that are colored red in the density plot. This is why FlexCode assigns a high probability to this location despite the city being much smaller than Sao Paulo and Rio de Janeiro. § PROOFS AND ADDITIONAL RESULTS To prove that the estimators in examples E1-E6 in Sec. <ref> satisfy Assumption <ref>, we only need to show that smoothness in the conditional densityf(z|) (seen as a function of ) implies smoothness for each varying coefficientβ_i().Assumption <ref> then follows directly from known convergence results for regression. For E3 and E4, note that if there exists a subset R ⊆{1, …,D} with |R|=d such thatf(z|)=f(z|(x_i)_i ∈ R) (i.e., there are only d relevant covariates), then β_i()=β_i((x_i)_i ∈ R). Different estimators use different notions of smoothness. In <cit.>, the authors show that k-NN regressors converge at rates the depend only on the intrinsic dimension of data if the target function is Lipschitz. Hence, for example E1, we use the Lipschitz notion of smoothness: Let {ϕ_i}_i be the Fourier basis. If, for every fixed z ∈, f(z|) is L-Lipschitz function, then β_i() is √(2)L-Lipschitz for all i ∈ℕ. Let ,∈^D. Then |β_i()-β_i()|= |∫ϕ_i(z)f(z|)dz-∫ϕ_i(z)f(z|)dz |≤∫| ϕ_i(z)| | f(z|)-f(z|)|dz≤L ||-||∫| ϕ_i(z)|dz ≤√(2) L ||-||∫| ϕ_i(z)|^2 dz = √(2) L ||-|| Local polynomial regression <cit.> and Rodeo <cit.> use the notion of bounded partial derivatives. Hence, we use the following result: Let {ϕ_i}_i be the Fourier basis. If for every fixed z ∈, f(z|) has all partial derivatives of order p bounded by K, then β_i() has all partial derivatives of order p bounded by √(2)K Let ∈^D and a_1,…,a_p ∈{1,2,…,D}. Then | ∂/∂ x_a_1…∂ x_a_pβ_i() |= |∂/∂ x_a_1…∂ x_a_p∫ϕ_i(z)f(z|)dz |≤∫| ϕ_i(z)| | ∂/∂ x_a_1…∂ x_a_p f(z|) |dz≤√(2) K The notion of smoothness in <cit.> is based on Hölderian classes. Hence: Let {ϕ_i}_i be the Fourier basis and 𝒫_l(f)(· ,) be Taylor polynomial of order l associated with f at the point . If, for every fixed z ∈, f_z():=f(z|) belongs to Σ(α,L), the α-Hölderian class, i.e., |f_z()-𝒫_l(f_z)( ,)|≤ L ||-||_1^α where l =⌊α⌋, then β_i() belongs to Σ(α,√(2)L) for all i ∈ℕ. Because β_i()=∫ϕ_i(z)f(z|)dz, then 𝒫_l(β_i)( ,)=∫ϕ_i(z)𝒫_l(f_z)(,) dz.Hence, we have that |β_i()-𝒫_l(β_i)( ,)| ≤∫| ϕ_i(z)||f(z|)-𝒫_l(f_z)(,)| dz ≤√(2) L ||-||_1^α The spectral series estimator <cit.> assumes that the regression function is smooth with respect to P. Hence: Let {ϕ_i}_i be the Fourier basis and assumethat, for every fixed z ∈, ∫ ||∇ f(z|)||^2dS()<∞. Then, for all i ∈ℕ, ∫ ||∇β_i()||^2dS() < ∞. Because β_i()=∫ϕ_i(z)f(z|)dz, then ∫ ||∇β_i()||^2dS() = ∫∇∫ϕ_i(z)f(z|)dz^2dS() = ∫∫ϕ_i(z)∇ f(z|)dz^2dS()≤∫( ∫ϕ_i^2(z) dz ) (∫ || ∇ f(z|)||^2dz ) dS() = ∫(∫ || ∇ f(z|)||^2 dS() )dz < ∞ Finally, the local linear functional regression estimator <cit.> assumes that the regression function has continuous second derivatives. Hence: Let {ϕ_i}_i be the Fourier basis and assumethat ∈ℒ^2([0,1]) and that, for every fixed z ∈, f(z|) has continuous second derivative. Then β_i()also has continuous second derivative for every i∈ℕ. Because β_i()=∫ϕ_i(z)f(z|)dz, then d^2 β_i()/d^2 = ∫ϕ_i(z) d^2 f(z|)/d^2 dz We now present the proofs of the other results presented in the paper. §.§ Proof of Lemma 1 Because f(z|) belongs to W_ϕ(s_x⃗,c_x⃗) for all z, and f(z|)=∑_i ≥ 1β_i()ϕ_i(z), we have that ∑_i ≥ I I^2s_x⃗(β_i())^2 ≤∑_i ≥ I i^2s_x⃗(β_i())^2 ≤c_x⃗^2. Hence ∑_i ≥ I∫(β_i())^2 dx⃗≤∫ c_x⃗^2/I^2s_x⃗dx⃗ = O(I^-2 β). §.§ Proof of Theorem 1: ∬(f_I(z|)-f(z|))^2dzd=∬(∑_i=1^Iβ_i()ϕ_i(z)-∑_i≥ 1β_i()ϕ_i(z))^2dzd=∬(∑_i=1^I( β_i() -β_i())ϕ_i(z)-∑_i>Iβ_i()ϕ_i(z))^2dzd(*)=∫( ∑_i=1^I ( β_i() -β_i())^2 +∑_i>I (β_i())^2 ) d =∑_i=1^I ∫ ( β_i() -β_i())^2 d+∑_i>I∫ (β_i())^2d, where step (*) follows from expanding the square and the fact that the Fourier basis is orthonormal (i.e., the cross products in the expansion are zero). The final result follows from Lemmas <ref> and <ref>. | http://arxiv.org/abs/1704.08095v1 | {
"authors": [
"Rafael Izbicki",
"Ann B. Lee"
],
"categories": [
"stat.ME",
"stat.ML"
],
"primary_category": "stat.ME",
"published": "20170426132143",
"title": "Converting High-Dimensional Regression to High-Dimensional Conditional Density Estimation"
} |
Sumário Figuras Tabelas Resumo Apêndice Referências Bibliografia Índice remissivo Figura Tabela Página veja veja também arabic empty 5pt Calores específicos dos gases ideais degenerados(Specific heats of degenerate ideal gases)Francisco Caruso^1,2,Vitor Oguri^2 e Felipe Silveira^2^1 Coordenação de Física de Altas EnergiasCentro Brasileiro de Pesquisas FísicasRua Dr. Xavier Sigaud, 150 – Urca, Rio de Janeiro, RJ – 22290-180 ^2 Departamento de Física Nuclear e Altas Energias Instituto de Física Armando Dias TavaresUniversidade do Estado do Rio de JaneiroRua São Francisco Xavier, 524 – Maracanã, Rio de Janeiro, RJ – 20550-900 arabic myheadingsCalores específicos dos gases ideais degenerados(Specific heats of degenerate ideal gases) Resumo A partir de argumentos baseados no princípio da incerteza de Heisenberg e no princípio de exclusão de Pauli, estimam-se os calores específicos molares dos gases ideais degenerados em baixas temperaturas, com resultados compatíveis com o princípio de Nerst-Planck (a 3lei da Termodinâmica). É apresentado, ainda, o fenômeno da condensação de Bose-Einstein com base nocomportamento do calor específico de gases de bósons massivos e não relativísticos.Palavras-chave: calor específico, gases degenerados, condensação de Bose-Einstein.AbstractFrom arguments based on Heisenberg's uncertainty principle and Pauli's exclusion principle, the molar specific heats of degenerate ideal gases at low temperatures are estimated, giving rise to values consistent with the Nerst-Planck Principle (third law of Thermodynamics).The Bose-Einstein condensation phenomenon based on the behavior of specific heat of massive and non-relativistic boson gases is also presented. Keywords: specific heat, degenerate gases, Bose-Einstein condensation. § INTRODUÇÃO Após um longo tempo de ensino dos fenômenos térmicos em cursos de Física, é fácil perceber a necessidade de uma abordagem que facilite aos alunos a transição da Termodinâmica para a Física Estatística.A abordagem estatístico-probabilística, tanto em seus aspectos formais como conceituais, não é do domínio da maioria dos alunos, mesmo daqueles que já superaram o ciclo básico do ensino superior. Por exemplo, muitos estudantes têm dificuldades em compreender a explicação do calor específico dos gases e dos sólidos baseada no princípio da equipartição de energia, pois ainda não sabem utilizar a distribuição de Maxwell-Boltzmann, ou qualquer distribuição de probabilidades, para o cálculo de valores médios.[ Essa mesma dificuldade, por mais estranha que pareça, é encontrada na Mecânica Quântica, após a interpretação probabilística de Born.]O calor específico expressa a capacidade de uma substância absorver energia quando excitada por algum agente externo. Quanto maior o número de modos pelos quais é possível essa absorção maior o calor específico de uma substância. Por esse motivo, o calor específico de um gás monoatômico é menor do que a de um gás poliatômico, ou de um sólido.Desde sua descoberta <cit.>, os estudos e as medidas dos calores específicos têm contribuído de forma determinante para a compreensão da estrutura da matéria. Por exemplo, as medidas dos calores específicos dos sólidos por P.L. Dulong e A.T. Petit (1819) <cit.> permitiram queJ.J. Berzelius <cit.> corrigisse o peso atômico de vários elementos químicos ao longo do século XIX.Esse trabalho sistemático do químico sueco foi fundamental para que D. Mendeleiev <cit.> pudesse elaborar a sua Tabela Periódica, em torno de 1869. Embora para gases monoatômicos os primeiros resultados, baseados no princípio da equipartição da energia, tenham sido satisfatórios, o mesmo não ocorreu paralíquidos,sólidos e gases de moléculas mais complexas.A comprovação de que os calores específicos variavam com a temperatura, ao contrário da lei de Dulong-Petit, exigiu a revisão crítica de vários conceitos físicos, levando a modificações profundas, não dos fundamentos da Física Estatística, como se acreditava, mas da própria Mecânica Clássica <cit.>. Foram as medidas desses calores específicos dos sólidos a baixas temperaturas que permitiram testar a então nova teoria atômica da matéria.As bases da Física Estatística foram estabelecidas por L. Boltzmann (1884) <cit.> e J. W. Gibbs (1901) <cit.>, a partir do conceito clássico de estado de um sistema como pontos de um contínuo.No entanto, a descoberta de Planck (1900) <cit.>, generalizada pela Mecânica Quântica <cit.>, de que os estados de um sistema, confinado em um volume, constituem um conjunto discreto associado a um espectro discreto de energia, implica a discretização do próprio espaço de fase, independentemente de qualquer conceito estatístico. Do ponto de vista quântico, as partículas de um sistema, em baixas temperaturas, tendem a se agrupar pelos estados associados aos menores valores de energia. Nesse limite há uma maior organização e uma diminuição da capacidade de excitação do sistema, o que implica decréscimo da entropia e do calor específico de um sistema. Esse é o conteúdo da 3 lei da Termodinâmica.[ A 3 lei da Termodinâmica, ou lei de Nernst, estabelece que a entropia de um sistema se aproxima de um valor constante no limite T → 0, o qual, segundo Planck, é nulo. Desse modo, o calor específico também é nulo quando T=0.] Como é mostrado nesse artigo, a definição de quantidades e parâmetros característicos de um gás ideal, baseada em princípios quânticos, permite estabelecer a divisão dos gases em duas categorias: os gases de férmions e os gases de bósons. A partir dessa divisão, obtêm-se estimativas para os calores específicos dos gases ideaisem baixas temperaturas, compatíveis com a 3 lei da Termodinâmica. Ao final do artigo, com base no comportamento do calor específico de um gás de bósons massivos e não relativísticos, o fenômeno da condensação de Bose-Einstein é apresentado como uma transição de fase para um estado mais ordenado, o condensado de Bose-Einstein.§ LIMITES DOS GASES IDEAIS Os gases molecularesem condiçõesambientaistêmdensidades que variam da ordem de 10^-5 g/cm^3 a 10^-3 g/cm^3, e se comportam como gases ideais para os quais a energia interna depende apenas da temperatura e o calor específico molar é constante <cit.>.Sendo N o número total das partículas (átomos ou moléculas) constituintes de um gás ideal, e Go número de estados associadosàs partículas,[ O estado de uma partícula é qualquer condição possível, caracterizada porvalores de grandezas como a posição e o momentum, ou a energia, associada à partícula.] a razão (N/G) entre esses números permite a divisão dos gases ideais em duas classes <cit.>: não degeneradose degenerados. {[ N/G ≪ 1;; N/G ≥ 1; ]. Enquanto as propriedades dos gases não degenerados não dependem da natureza de suas partículas constituintes, os gases degenerados evidenciam a natureza quântica de suas partículas, a qual se reflete em seu comportamento macroscópico.Em geral, o comportamento degenerado manifesta-se em baixas temperaturas ou altas densidades de um gás, quando as partículas têm de competirpelos estados acessíveis. Em altas temperaturas ou baixas densidades, quando o número de estados acessíveis é muito maior que o número de partículas, a competição pelos estados praticamente não existe, e o gás exibe comportamento não degenerado.§ NÚMERO E DENSIDADE DE ESTADOSUtilizando-seuma abordagem semiclássica para o cálculo donúmero de estadosacessíveisàs partículasquase livres de um gás,[ O intervalo de tempo da interação de umapartícula de um gás com outras partículas é bem pequeno em relação ao intervalo de tempo no qual que a partícula se movelivremente.] contidas em um volume V e com momenta que variam de zero a um dado valorp, os limites que caracterizam o comportamento dos gases ideais podem ser apropriadamente quantificados.Do ponto de vista da Mecânica Clássica <cit.>, o estado de uma única partículaé caracterizado por sua posição e momentum. Assim,a evolução ao longo do tempo do estado da partícula pode ser representada em um espaçode dimensão seis, denominado espaço de fase da partícula, no qual cada ponto de coordenadas (x, y, z, p_x, p_y, p_z) caracteriza o estado dinâmico dessa partícula.Se a partícula move-se ao longo de uma direção x, entre 0 e L, em qualquer sentido, com momentum menor que um valor p, sua evolução é visualizada em um plano no qual cada ponto (x, p_x), dentro do retângulo de lados L e 2 p (Figura <ref>), representa um possível estado acessível à partícula. A área desse retângulo é proporcional ao número de estados acessíveis à partícula.Nesse contexto, segundo argumento originalmente proposto por O. Sackur e H. Tetrode <cit.> em 1912,[ Esse argumento foi utilizado também por S. Bose <cit.> em 1924, ao deduzir a fórmula de Planck para a radiação de corpo negro.] o número de estados (G) acessíveis a cada uma das partículasde um gás contido em um recipiente de volume V pode ser expresso pelarazão entre o volume no espaço de fase de dimensão seis, associado a uma única partícula, cujo momentum varia de zero até umvalor p, e um volume mínimo igual a h^3, G = f1/h^3∫∫∫∫∫∫ dxdydzdp_xdp_ydp_z =f V/h^3( 4π/3 p^3 ),sendo h ≃ 6,626 × 10^-34 J·s a constante de Planck.O fator f depende da natureza das partículas,[ Para um gás molecular ideal, f = 1; para qualquer partícula de spin 1/2 ou partículas não massivas de spin 1 (fóton ou o fônon) , f = 2. Esse fator indica a multiplicidade dos estados associados às partículas.] e o termo entre parênteses é o volume da esfera de raio p= √(p_x^2+p_y^2+p_z^2) no espaço dos momenta. A existência de um volume mínimo no espaço de fase pode ser justificado a partir doprincípio da incerteza de Heisenberg <cit.>, o qual estabelece uma correlação entre o momentum (p__x) e aposição (x) de uma partícula em uma dada direção, talque o limite mínimo para o produto das incertezas (Δ_x, Δ_p_x)associadas às medidas desses pares de variáveis é da ordem da constante de Planck.Para as três direções espaciais, pode-se escrever( Δ_x·Δ_p_x)_∼h, ( Δ_y·Δ_p_y)_∼ h ( Δ_z·Δ_p_z)_∼ h.De acordo com as relações entre o momentum (p) e a energia (ϵ) de uma partícula livre,{[ p = √(2mϵ) ;;/;p = ϵ / c ;;]. pode-se expressar o número de estados acessíveis eq. (<ref>) a cada partícula de um gás como função da energia por G(ϵ) = {[ (4π f/ 3) V/ h^3( 2mϵ)^3/2,;;(4π f/ 3) V/ (ch)^3 ϵ^3.; ].Para um gás molecular nas condições ambientais, o número de estados acessíveis é da ordem de10^27 e o número de moléculas, cerca de 10^23. Nessas condições, o estado do gás é não degenerado.Definindo-se a densidade de estados g(ϵ) de uma partícula livre por g(ϵ) = G/ϵ = {[ 2π f V(2m/ h^2)^3/2ϵ^1/2,;; 4π f V/ (ch)^3 ϵ^2,; ].o número de estados acessíveis em qualquer intervalo de energia pode ser determinado integrando-se g(ϵ) em relação à energia.§ GASES IDEAIS NÃO DEGENERADOSDe acordo com os experimentos sobre o comportamento os gases <cit.>, anteriores ao século XX, a energia interna (U) de um gás ideal monoatômico, em condições ambientais, à temperatura T e, portanto, não degenerado, é igual aU= 3/2 NkT= 3/2n R T, e o calor específico molar a volume constante (c__V),[ O calor específico molar a volume constante é definido porc__V = 1/n ( ∂ U/∂ T)_V. ]c__V= 3/2R sendo k ≃ 1,38 × 10^-23 J/K a constante de Boltzmann, R ≃ 8,3 J·mol^-1·K^-1 a constante dos gases, N o número de moléculas e n o número de mols.Em baixas temperaturas, esse comportamento clássico do calor específico não é compatível coma 3 lei da Termodinâmica <cit.>, segundo a qual o calor específico a 0 K deve-se anular, lim_T → 0 c__V → 0. § TEMPERATURAS CRÍTICASUma vez que a energia média por partícula ϵ̅ = U/N de um gás não degenerado é cerca deϵ̅≃ kT, o número de estados ocupados pelas partículas de um gás não degenerado é cerca de G(ϵ̅) e, portanto, o critério para a não degenerescência pode ser expresso como N/G≃{[ (3/ 4 π f) (h^2/ 2mk)^3/2(N/ V)1/ T^3/2;;(3/ 4π f) (ch/ k)^3 (N/ V)1/ T^3; ]. ≪ 1.A figura <ref> mostra o compromisso entre a densidade de partículas (N/V) e a temperaturade um gás de partículas massivas não relativísticas – a linha pontilhada[ Essa linha, de acordo com a primeira das eqs. (<ref>),é definida por (N/V)Λ^3= 1, em que Λ = ( 3/8π)^1/3h/√(2mkT).]define os valores críticos que separam as regiões nas quais o gas édegenerado ou não degenerado.Para temperaturas e densidades abaixo dessa linha, o estado do gás é degeneradoe a natureza quântica de suaspartículas deve ser considerada. Para temperaturas e densidades acima da linha crítica, o estado do gás é não degenerado e pode ser tratado como um sistema clássico. A Tabela <ref> mostra os valorescríticos de temperaturae dasdensidades de alguns gases moleculares.Em função da densidade de partículas (N/V), as temperaturas críticas (T_c), definidas pela condição N/G=1, são dadas por T_c = {[ T__F = (3/ 4π f)^2/3(h^2/ 2mk) (N/ V)^2/3,; ;T__D = (3/ 4π f)^1/3(ch/ k) (N/ V)^1/3, ]. em queT__Fe T__Dsão usualmente denominadas temperatura de Fermi e temperatura de Debye.Assim, o critério para a não degenerescência de um gás é usualmente expresso como T/T_c≫ 1SeT ≤ T_c, o gás é dito degenerado.Os gases ideais, não degenerados ou degenerados, são idealizações que não correspondem exatamente a nenhum sistema macroscópico. Apesar disso, a ênfase observada em seu estudo, bastante simples, decorre do fato de que em um grande número de casos alguns sistemas macroscópicos podem ser representados por conjuntos de constituintes quase independentes, ou seja, como gases.A Tabela <ref> mostra, de acordo com as eqs. (<ref>), as temperaturas críticas relativas a vários sistemas que se comportamcomo gasesideais. Pode-se observar, assim, que, mesmo para baixas temperaturas (T ∼ 10 K), os gases moleculares comportam-se como gases ideais não degenerados; já os elétrons de condução nos metais constituem um sistema degenerado em qualquer temperatura, uma vez que a 10^5 K não existe matéria sólida. No caso dos elétrons relativísticos provenientes dos átomos de hélio ionizados de uma estrela anã branca a 10^7 K, a temperatura da estrela ainda é muito menor queo valorcrítico e, portanto, o sistema comporta-se como um gás degenerado relativístico. Por outro lado, os elétrons em semicondutorese os osciladores atômicos nos cristais não são degenerados nas condições ambientais,sendo degenerados apenasem baixas temperaturas (T ∼ 10 K).Em termosdas temperaturas críticas, os números e as densidades de estados podem ser escritos comoG(ϵ) = {[ N (ϵ/ kT__F)^3/2,;; N (ϵ/ kT__D)^3,; ]. ⟹ g(ϵ) = {[ 3/ 2N/ (kT__F)^3/2ϵ^1/2,; ; 3 N/ (kT__D)^3ϵ^2.;].§ GASES IDEAIS DEGENERADOSO princípio de exclusão de Pauli <cit.> estabelece uma outra correlação quântica tal que partículas idênticas de spin semi-inteiro não podem compartilhar o mesmo estado. Assim, os gases ideais degenerados dividem-se em duas classes: aqueles constituídos porpartículas de spin semi-inteiro, denominadas férmions, e aqueles constituídos por partículas de spin inteiro, denominadas bósons. Nos gases não degenerados, quando efetivamente não há competição pelos estados acessíveis por suas partículas constituintes, tal distinção não é necessária. Em temperaturas próximasa 0 K, quando o calor específico e a entropia de um sistema tendem a se anular,as partículas de um gás distribuem-se entre os estados acessíveis tal que a energia total seja mínima. Diz-se que o sistema encontra-se em seu estado fundamental.Para temperaturas finitas, mas ainda muito menores que a temperatura crítica (T ≪ T_c), uma pequena fração das partículas do gás são excitadas além do estado fundamental. Essas partículas excitadas comportam-se, praticamente, como um subsistema não degenerado responsável pelas propriedades térmicas do gás.§.§ Férmionsnão relativísticos fortemente degeneradosDevido ao princípio de Pauli, para um gás ideal de férmions completamente degenerado (T=0 K) em seu estado fundamental, as partículas se acomodam nos estados acessíveis de tal forma quecada estado seja ocupado por apenas uma partícula. Nessas condições, o número de estados ocupados é igual ao número de partículas.Assim, para um gás de férmions não relativísticos completamente degenerado no estado fundamental,N/G(ϵ__F) = 1, sendo ϵ__F o maior valor de energia associado aos estados ocupados pelas partículas. De acordo com as eqs. (<ref>), esse valor, denominado energia de Fermi, é dado por ϵ__F = kT__F. Segundo a Tabela <ref>, para o gás de elétrons de condução de um metal à temperatura ambiente (T ≪ T__F), a energia de Fermi é da ordem deϵ__F≃ 10^-18 ≃ 10 ,e a energia dos elétrons excitados cerca deϵ≃ kT ≃ 25 ≪ϵ__F (T ∼ 300 K ). Para temperaturas acima de 0 K, mas ainda muito menores que a temperatura de Fermi (T ≪ T__F),[ Para os elétrons de condução em um metal, essa condição sempre se verifica.] algumas partículas são excitadas acima do nível de Fermi. Essas partículas excitadas comportam-se como um subsistema não degenerado com energia média por partícula da ordem de kT ≪ϵ__F.Nessas condições, o número de partículas excitadas (N_) é aproximadamente igual ao número de estados acessíveis no intervalo de energia (ϵ__F , ϵ__F + kT), ou seja,N_ ≃ ∫_ϵ__F^ϵ__F+kT3/2N/ϵ__F^3/2ϵ^1/2 ϵ=N/ϵ__F^3/2[ (ϵ__F+kT)^3/2 - ϵ__F^3/2] ≃ N [ (1+kT/ϵ__F)^3/2 - 1 ] ≃ 3/2 N T/T__F. Pode-se obter a energia interna(U) do gásadicionando-se a energia do estado fundamental (U_∘) para T=0 K à energia das partículas excitadas (N_ kT),U≃U_∘ + N_ kT= U_∘ +3/2 N k T^2/T__F,e o calor específico molar a volume constante,por[ Segundo Sommerfeld <cit.>, c__V =π^2/2RT/T__F.] c__V ≃3 RT/T__F (T ≪ T__F)Esse comportamento, obtido por Sommerfeld em 1928, compatível com a 3 lei da Termodinâmica, é o esperado para a variação do calor específico dos metais em temperaturas próximas a 0 K, quando as propriedades térmicas dos metais são atribuídas ao movimento dos elétrons de condução <cit.>.§.§ Bósons ideais degeneradosUma vez que os bósons de um sistema podem ser associados a estados de mesma energia, todos podem ocupar um único estado. Em geral, a energia do estado fundamental de um gás de bósons completamente degenerado (T = 0 K) pode ser considerada nula.De modo análogo aos férmions, para temperaturas um pouco acima de 0 K, mas ainda muito menores que a temperatura crítica (T≪ T__D), algumas partículas são excitadas com energia da ordem de kT.Dependendo da massa e do caráter relativístico, o número de partículas excitadas (N_ϵ > 0), bem como a energia interna (U) e o calor específico molar (c__V), são calculados de modos distintos. §.§.§ Bósons não massivos fortemente degeneradosNo caso de bósons não massivos, apesar de ser pequena afração de partículas que abandonam o estado fundamental quandoT ≪ T__D, o número de partículas excitadas com energia média da ordem de kT émaior do que o número de estados com energia até kT. Assim,N_ϵ > 0= α∫_0^kT 3 N/(kT__D)^3 ϵ^2ϵ_G(ϵ >0) = N α ( T/T__D)^3, sendo αum parâmetro da ordem de 10, que depende da razão T/T__D.[ α = π^4/5 (T ≪ T__D).]Desse modo, a energia interna corresponde aU = N_ϵ > 0 kT =N k α T^4/T__D^3, e o calor específico molar a volume constante, a c__V =4 R α( T/T__D)^3 (T ≪ T__D)A dependência da temperatura (∝ T^3), estabelecida por P. Debye <cit.>, em 1912, é compatível com a 3 lei da Termodinâmica, e descreve o comportamento do calor específico molar dos sólidosem baixas temperaturas (T ∼ 4) <cit.>.[ Para os sólidos, o calores específicos a volume e a pressão constantes são praticamente iguais, principalmente em baixas temperaturas.] §.§.§ Os sólidos cristalinosA lei empírica de Dulong e Petit, de que o valor do calor específico dos sólidos seria uma constante independente da temperatura, apesar dos muitos desvios observados, só começa a declinar, realmente, no início do século XX. A partir de misturas refrigerantes, foram alcançadas temperaturas baixas o suficiente para evidenciar a dependência do calor específico com a temperatura. Verificou-se, experimentalmente <cit.>, que, em baixas temperaturas, o calor específico de um sólido obedecia à 3 lei da Termodinâmica, variando com a temperatura segundo a chamada lei de Debye, eq. (<ref>), a seguir.A representação das vibrações atômicas nos sólidos cristalinos[ Um sólido cristalino é constituído pela repetição de uma unidade básica de padrão geométrico regular, na qual os átomosexecutam pequenas vibrações em torno de posições relativas mais ou menos fixas.] como um gás de bósons não massivos, chamados fônons, baseia-se na aproximação harmônica da energia potencial efetiva,a qual descreve as interações de cadaátomocom seus vizinhos. Desse modo, cada átomo é associado a um oscilador harmônico independente dos demais, cujo espectro de energia, ϵ_n(ν), segundo a Mecânica Quântica, é dado por ϵ_n (ν) = ( n + 1/2) h ν= ϵ_∘ + n h ν (n=0,1, … ), em que ν é a frequência natural de vibração, e ϵ_∘ = h ν/2 é a energia do estado fundamental de um particular oscilador.A energia de cada átomo em um sólido cristalino em equilíbrio térmico, portanto,é composta por uma parcela constante, a energia do estado fundamental,[ Como a energia é definida a menos de uma constante, essa parcela pode ser considera nula.] e uma parcela que depende do grau de excitação do átomo, ou seja, da temperatura do cristal.Considerando que cada estado excitado de um oscilador corresponde a n partículas independentes não massivas, cada qual com energia ϵ = h ν, o conjunto de osciladores que representam as vibrações atômicasdo cristal pode ser associado a um sistema de partículas independentes que se comportam como um gás de bósons não massivos, pois o número (n) de partículas associado a cada estado depende da temperatura e, portanto, não obedece ao princípio de exclusão de Pauli. 13cm =10ptEm baixas temperaturas, as vibrações atômicas em um sólido cristalino são equivalentes a um gás degenerado de bósons não massivos, os fônons. Como cada átomo corresponde a 3 osciladores independentes, o número total de osciladores no cristal é 3N, e o calor específico molar é dado pela lei de Debye <cit.>,[ A rigor a lei de Debye é válida em temperaturas menores que T__D/50 <cit.>.] c__V =12 R α( T/T__D)^3 (T ≪ T__D) O resultado do experimento pioneiro de P. H. Keesom e N. Pearlman <cit.>, de 1955, apresentado no gráfico c__V/TversusT^2da figura <ref>, mostra queo comportamento do calor específico dos sólidos não metálicosa baixas temperaturas(T < T__D/50 ∼ 4 K) écompatível com a lei de Debye.Em altas temperaturas(T ≫ T__D), os sólidos comportam-se como um sistema não degenerado cujo calor específico molar (c__V) obedece à lei de Dulong-Petit <cit.>,c__V = 3R ≃ 6 cal· mol^-1· K^-1≃ 25 J· mol^-1· K^-1(T ≫ T__D)Tanto do ponto de vistateórico, como experimentalmente, o calor específico dos sólidos cresce suavemente com a temperatura até o valor limite dado pela lei de Dulong-Petit (figura <ref>).§.§.§ O calor específico dos metaisOs metais constituem uma classe especial de sólidos cristalinos. Além dos íons que constituem a rede cristalina, e vibram em torno de suas posições de equilíbrio, possuem também praticamente o mesmo número de elétrons, os elétrons de condução, que não estão associados a nenhum particular íon.A hipótese de que algumas das propriedades de um metal pudessem ser obtidas a partir do modelo do gás de elétrons remontam à época de P. Drude (1900). Tal hipótese apoia-se no fato de que em um cristal os íons positivos do metal estabelecem um campo eletromagnético que tende a anular a ação dos outros elétrons sobre um determinado elétron. No entanto, ao se considerar os elétrons de condução como umgásnão degenerado, os resultados obtidos não foram compatíveis com o comportamento térmico observado. Apenas quando Sommerfeldconsiderou-os como um gás degenerado de férmions de spin 1/2, os resultados teóricos tornaram-se compatíveis com os experimentos <cit.>. Assim,o calor específicode um metal possui uma componente devida às vibrações da rede, ou aum gás degenerado de fônons, eoutra devida ao gás degenerado de elétrons de condução do metal, c_ = c_ +c_. Comoà temperatura ambienteT ≪ T__Fpara qualquermetal, acontribuição eletrônicasó é relevante em temperaturas muito menores que a temperaturade Debye do metal, quando o calor específico pode ser expresso comoc_ =α T^3+γ T ( T ≪ T__D), em que o parâmetros α e γ determinam, respectivamente,a temperatura de Debyee atemperatura de Fermi.Esse comportamento dos metais, já observado por Keesom (1935), foi estabelecido nos experimentos realizados por W. Corak, M. P. Garfunkel, C. B. Satterthwaite eA. Wexler em 1955 <cit.>(figura <ref>). §.§.§ A radiação de corpo negro De maneira análoga às vibrações, o campo eletromagnético associado à radiação de corpo negro em equilíbrio térmico pode serrepresentado por um gás de bósons não massivos – osfótons com energias hν(ν=0,…∞) <cit.>.Nasvibrações atômicas, a energia do estado fundamental a 0 Ktem valor constante e finito determinado pelo número de osciladores do cristal; no caso da radiação de corpo negro, o número total de osciladores associados ao campo eletromagnético não é finito. Assim, para se contornar o problema de um número infinito de termos contribuir para a energia do estado fundamentaldo sistema a 0 K, considera-se que a energia do sistema é a energia dos estados excitados, ou seja, a energia dos fótons. Uma vez que não existem restrições quanto ao número de fótons, poisessenúmeropode ser infinito, a radiação de corpo negro sempre constitui umsistema degenerado de bósons não massivos em qualquer temperatura. Substituindo-se a expressão para a temperatura de Debye (a segunda das eqs. (<ref>)na eq. (<ref>),a densidade de energia (u = U/V) da radiação de corpo negro pode ser escrita na forma usual da lei de Stefan-Boltzmann <cit.>, u = 8 πα/3 k^4/(ch)^3T^4 = a T^4, sendo a ≃7,6 × 10^-16 J· m^-3· K^-4, para α= π^4/5.§.§.§ Bósons massivos não relativísticos fortemente degenerados econdensação de Bose-Einstein O número de partículas excitadas em um gás de bósons é diferente quando eles são massivos ou não, devido, principalmente, ao fenômeno de criação e aniquilação de partículas. Tanto a energia daradiação eletromagnética de corpo negro, quanto das oscilações atômicas nos cristais, podem ser descritas pela soma das energias debósons não massivos, respectivamente denominados fótons e fônons.Uma vez que a quantidade dessesbósons não massivos não é fixa, dependendo fortemente da temperatura (∝ T^3), diz-se que fótons e fônons podem ser criados ou aniquilados.Diferentemente dosbósons não massivos, os bósons massivos não relativísticos não podem ser criados ou aniquilados, o que implica conservação do número de partículas.[ No caso de bósons massivos relativísticos, deve-se considerar também os processos de criação e aniquilação.] Desse modo, a energia do estado fundamental é nula e pode-se admitir que em baixas temperaturas o número de partículas excitadas com energia média da ordem de kT seja dado pelo número de estados com energia até kT,[ A temperatura crítica e a densidade de estados para bósons massivos não relativísticos são calculadas do mesmo modo que para os férmions não relativísticos, pois a relação entre a energia (ϵ) e o momentum (p) de partículas não relativísticas de massa m é dada por p = √(2mϵ).]N_ϵ > 0 = ∫_0^kT3/2N/(kT_c)^3/2ϵ^1/2 ϵ = N ( T/T_c)^3/2, e a energia interna e o calor específico molar a volume constante, compatível com a 3 lei da Termodinâmica, sejam, aproximadamente, U ≃ N_ϵ > 0 kT =N kT^5/2/T_c^3/2,c__V≃ R( T/T_c)^3/2. Onúmero de bósonsno estado fundamental(N_∘ = N - N_ϵ > 0), por sua vez, é dado porN_∘ = N [ 1 - ( T/T_c)^3/2]. De acordo com a eq. (<ref>), abaixo da temperatura crítica o gás se aproxima de um estado em que o número de bósons massivos agrupados nas vizinhanças do estado fundamental cresce rapidamente (figura <ref>). Com base nesse comportamento, Einstein supõe que na temperatura crítica ocorra um novo fenômeno, no qual o sistema de bósons massivos atinja um estado mais organizado da matéria, em uma transição do tipo desordem–ordem. Em analogia com a condensação de um gás ordinário, o fenômeno é conhecido, desde então, como condensação de Bose-Einstein <cit.>.Sendo nula a energia do estado fundamental, a grande maioria das partículas do chamado condensado de Bose-Einstein possui momentum nulo em baixas temperaturas (T/T_c < 0,4). Em maior número que as demais, taispartículas contribuiriam com maior peso para as propriedades macroscópicas do sistema, como pressão e viscosidade.Essas conclusõesforam estabelecidas por Einstein em 1924e 1925 <cit.>, a partir do métodode contagem de fótons da radiação de corpo negro utilizado por Bose.Bose considerou que os fótons não obedeciam ao princípio de Pauli, à época desconhecido, e eram indistinguiveis.[ Bose também admitiu corretamente que, devido a polarização da luz,o fatorde multiplicação de estados era igual a 2(f=2).] Ao estender o procedimento de Bose aos gasesmoleculares, Einstein obtém uma teoria quântica dos gases ideais degenerados de bósons massivos não relativísticos.A descoberta da existência de um novo estado condensado da matéria foi considerada, inicialmente, apenas um resultado matemático, sem possibilidade de verificação. Devido ao baixíssimo valor da temperatura crítica de um gás molecular (T_c < 0,1 K), qualquer gás real a tão baixa temperatura estaria no estado líquido.Apesar dos argumentos contrários à condensação de Bose-Einstein, F. London, em 1938, estabelece que o calor específico molar a volume constante de um gás de bósons massivos não relativísticos para temperaturas até o valorcrítico é dado por <cit.>c__V = 1,93 R( T/T_c)^3/2 (T ≤ T_c) Como o valor do calor específico molar à temperatura crítica(1,93R) excede o valor clássico (1,5R) para o qual deve se aproximar assintoticamente para T > T_c, nas vizinhanças da temperatura crítica o calor específico do gás apresenta um comportamento não suave (figura <ref>), o que implica descontinuidade em sua derivada.Foi exatamente a partir desse comportamento que Londondefendeu a hipótese de Einstein sobre a temperatura crítica de um gás de bósons massivos não relativísticos estar associada a uma transição de fase. Rebatendo as críticas de que o fenômeno da condensação de Bose-Einstein resultaria de um artifício matemático, London sugere a transição líquido-superfluido do hélio (He^4) como um exemplo de condensação de Bose-Einstein que ocorreria na natureza.[ Apesar de ocorrer também a concentração de partículas no estado fundamental de um gás de bósons não massivos em baixas temperaturas, a energia do estado fundamental não é nula e o calor específico varia suavemente (figura <ref>), não havendo descontinuidade em sua derivadae, portanto, não caracterizando uma transição de fase.] §.§.§ OHe^4 líquidoA particularidade mais marcante do He^4líquido (T ∼ 4 K)quando é resfriado em equilíbrio comsua pressão de vapor, após alcançar a temperatura crítica da ordem de 2 K, é a súbita capacidade de fluir em tubos capilares sem exercer pressão, como se deixasse de terviscosidade. Essa mudança brusca na viscosidade, acompanhada também de uma variação brusca no calor específico, é a propriedade quecaracteriza o fenômeno como uma transição de fase líquido-superfluido.Uma vez que os átomos de He^4possuem seis partículas (2 prótons, 2 nêutrons e 2 elétrons) de spin semi-inteiros, F. London supõe, em primeira aproximação, que o hélio líquido constituído dos isótopos He^4pode ser considerado um sistema departículas independentes de spin inteiros, ou seja, como um gás de bósons massivosnão relativísticos, e que a transiçãolíquido-superfluido tivesse relação com o fenômeno de condensação de Bose-Einstein.Essa hipótese é reforçada quando, ao considerar-se He^4 líquido como um gás de bósons, o valor da temperatura crítica é cerca de 3 K, conforme Tabela <ref>. Apesar da similaridade e do estabelecimento posterior do fenômeno como um exemplo de condensação de Bose-Einstein,[ A observação da população de átomos de He^4com momentum nulo foi realizadaem 1982 <cit.>.] por não ser exatamente um sistema de partículas que não interagem, o comportamento do calor específico doHe^4 líquido é muito diferente do gás ideal degenerado de bósons massivos.A procura de um sistema gasoso que exibisse a condensação de Bose-Einstein só teve êxito em 1995, quando vapores deátomos de Rb, a uma densidade de 10^12 cm^-3, foram resfriados a baixíssima temperatura, cerca de 100 nK, por um grupo do JILA.[ Joint Institute for Laboratoty Astrophysics, um instituto conjunto da University of Colorado e do NIST (National Institute of Standards and Technology).] Desde então, váriosexperimentos sobre esse estranho comportamento da matéria foram realizados, constituindo-se em uma ativa área de pesquisa da Física de ultra baixas temperaturas <cit.>. § CONSIDERAÇÕES FINAISDo ponto de vista dinâmico, ainda que em equilíbrio térmico, o número de partículas associado a cada nível de energia (ϵ) de um gás varia incessantemente. Devido a essas flutuações e ao grande número de partículas, a distribuição das partículas pelos estados associados aos níveis de energia do gás são caracterizadas pelos números médios de ocupação, ⟨ n _ϵ⟩, ou população média dos estados.A distribuição das partículas depende do estado de degenerescência do gás e, portanto, da natureza de suas partículas constituintes. Além das correlações decorrentes do princípio de exclusão de Pauli, deve-se considerar que as partículas, bósons e férmions, de um gás ideal são, entre si, indistinguíveis.As três distribuições usuais para os gases ideais podem ser sintetizadas como <cit.>⟨ n_ϵ⟩ =1/λ^ - 1 e^ϵ/kT +a, em que λ é a fugacidade do gás.[ Para partículas massivas a fugacidade (λ) é determinada pela restrição ao número de partículas; para bósons não massivos não há restrição associada ao número de partículas, eno caso dos fônons, o espectro de energia tem um limite determinado pela temperatura de Debye (ϵ_ = k T__D).] A distribuição de Maxwell-Boltzmann (a=0) descreve o comportamento da população média em um gás ideal não degenerado, a distribuição de Fermi-Dirac (a=1), a população média em um gás ideal de férmions, e a distribuição de Bose-Einstein (a=-1), a população média em gases ideais de bósons não massivos (λ=1) e massivos.[ Para bósons não massivos, principalmente, para fótons da radiação de corpo negro, a distribuição é chamada também distribuição de Planck.]Assim, as propriedades macroscópicas de um gás em equilíbrio térmico, como a energia média U, ou energia interna, e o número total de partículas N, satisfazem as seguintes relações{[ U =∫_0^∞ϵg(ϵ)⟨ n_ϵ⟩ ϵ,; ;N = ∫_0^∞ g(ϵ)⟨ n_ϵ⟩ ϵ, ].sendog(ϵ)adensidade de estados. A partir dessas expressões, eqs. (<ref>), são obtidas estimativas mais acuradas sobre o comportamento dos gases, correspondentes a um amplo domínio de temperaturas, desde as mais baixas (T ≪ T_c) às mais altas (T ≫ T_c). -.4em-.8ex 90MCKIE D. McKie & N.H. de V. Heathcote, The Discovery of Specific and Latent Heats, London: Edward Arnols, 1935. DULONG A.T. Petit, P.L. Dulong, “Recherches sur quelques points importants de la Théorie de la Chaleur”,Annales de Chimie et de Physique v. 10, p. 395-413 (1819). BERZELIUSJ.J. Berzelius, “Essay on the Cause of Chemical Proportions, and Some Circumstances Relating to Them: Together with a Short and easy Method of Expressing Them”,Annals of Philosophy v. 2, p. 443-454 (1813).MENDELEIEV D. Mendeleiev, “The Relation between the Properties and Atomic Weights of the Elements",Journal of theRussian Physical Chemical Society v. 1, p. 60-77 (1869).CARUSO-OGURI F. Caruso & V. Oguri, Física Moderna: Origens Clássicas e Fundamentos Quânticos, 2. ed., Rio de Janeiro: LTC, 2016.BOLTZ L. Boltzmann,Lectures on Gas Theory:1896-1898,New York: Dover Pub. Inc., 1995.GIBBS J.W. Gibbs, Elementary Principles in Statistical Mechanics: 1902, New York: Dover Pub. Inc., 2014. PLANCK M. Planck & H. Kangro (annotations), Planck's original papers in quantum physics: German and English edition, London: Taylor & Francis Ltd. 1972.DIRAC P.A.M. Dirac, The Principles of Quantum Mechanics: 1930, 4th. ed., Oxford: Oxford University Press, 1958.FERMI1 E. Fermi, Molecules, Crystals, and Quantum Statistics, New York: W.A. Benjamin Inc., 1966.SOMMER A. Sommerfeld, Thermodynamics and Statistical Mechanics, Lectures on Theoretical Physics, vol. V, Londo, New York: Academic Press, 1955.ZEMANSKY M.W. Zemansky, R.H. Diettman,Heat and Thermodynamics, 7th. ed.,New York: McGraw-Hill, 1997. MARIO M.J. de Oliveira, Termodinâmica, 2. ed., São Paulo: Livraria da Física, 2012.Epifanov G.I. Epifanov, Solid State Physics, Moscow: Mir Publisher, 1979.GRIMUS W. Grimus, On the 100th anniversary of the Sackur-Tetrode equation, arXiv:1112.3748v2 [physics.hist-ph], 23 Jan 2013.BOSE S. Bose, artigo de 1924 traduzido para o português por F. Caruso & V. Oguri, “A lei de Planck e a hipótese dos quanta de luz”, Revista Brasileira de Ensino de Física v. 27, n. 3, p. 463-465 (2005).DEBYE P. Debye,“Zur theorie der spezifischen Wärme”, Annalen der Physik v. 39, p. 789-839 (1912). KEESOM P.H. Keesom,N. Pearlman,“The Heat Capacity of KCl below4 ^∘K”,Physical Review v. 98, n. 6, p. 1699-1707 (1955).Black M. Blackman,“The Theory of the Specific Heat of Solids”,Reports Progress in Phys. v. VIII (1941).CORAK W. Corak, M.P. Garfunkel, C.B. Satterthwaite & A. Wexler,“Atomic Heats of Copper, Silver, and Gold from1 ^∘K to5 ^∘K”,Physical Review v. 98, n. 6, p. 1699-1707 (1955). STUDART N. Studart, “A Invenção do Conceito de Quantum de Energia segundo Planck”, Revista Brasileira de Ensino de Física, v. 22, n. 4, p. 523-535 (2000).LONDON F. London, “On the Bose-Einstein Condensation”. Physical Review v. 54, p. 947-954 (1938). DAHMEN2 S.R. Dahmen, “Bose e Einstein: Do nascimento da estatística quântica à condensação sem interação II”, Revista Brasileira de Ensino de Física v. 27, n. 2, p. 283-298 (2005).BAGNATO1 V.S. Bagnato, “A Condensação de Bose-Einstein”, Revista Brasileira de Ensino de Física v. 19, n. 1, p. 11-26 (1997).EINSTEIN1 A. Einstein, artigo de 1924traduzido para o inglês por L. Amendola, “Quantum Theory of a Monoatomic Ideal Gas”,<http://www.thphys.uni-heidelberg.de/ amendola/otherstuff/einstein-paper-v2.pdf>, acessado em 26/04/2017.EINSTEIN2 A. Einstein, artigo de 1925 traduzido para o português por S.R. Dahmen, “Teoria quântica de gás ideal monoatômico – segundo tratado” Revista Brasileira de Ensino de Física v. 27, n. 1, p. 113-120 (2005).BAGNATO2E.A.L. Henn, J.A. Seman, G.B. Seco, E.P. Olimpio, P. Castilho, G. Roati, D.V. Magalhães, K.M.F. Magalhãesand V.S. Bagnato, “Bose-Einstein Condensation in ^87Rb: Characterization of the Brazilian Experiment”, Brazilian Journal of Physics, v. 38, n. 2, p. 279-286 (2008). | http://arxiv.org/abs/1704.08946v1 | {
"authors": [
"Francisco Caruso",
"Vitor Oguri",
"Felipe Silveira"
],
"categories": [
"physics.gen-ph"
],
"primary_category": "physics.gen-ph",
"published": "20170426174537",
"title": "Specific heats of degenerate ideal gases"
} |
1Janusz Gil Institute of Astronomy, University of Zielona Góra, ul. Z. Szafrana 2, PL-65-516 Zielona Góra, Poland2DLR Institute of Space Systems, Robert Hooke Str. 7, 28359 Bremen, Germany [email protected] present the results of our radio interferometric observations of pulsars at325 MHz and 610 MHz using the Giant Metrewave Radio Telescope (GMRT). We usedthe imaging method to estimate the flux densities of several pulsars atthese radio frequencies. The analysis of the shapes of the pulsar spectraallowed us to identify five new gigahertz-peaked spectra (GPS) pulsars. Usingthe hypothesis that the spectral turnovers are caused by thermal free-freeabsorption in the interstellar medium, we modeled the spectra of all knownobjects of this kind. Using the model, we were able to put some observationalconstrains on the physical parameters of the absorbing matter, which allows usto distinguish between the possible sources of absorption.We also discuss the possible effectsof the existence of GPS pulsars on future search surveys, showing that the optimal frequency rangefor finding such objects would be from a few GHz (for regular GPS sources) to possibly10 GHz for pulsars and radio-magnetars exhibiting very strong absorption.§ INTRODUCTIONThe radio spectra in pulsars, typically characterized by an inverse power law,have been important in understanding the non-thermal origin of radio emission in pulsars. In a majority of pulsars the spectral nature is described by a steeppower-law function with an average spectral index of -1.8 <cit.>. In recent years a new class of pulsars have been identified with a distinct spectral nature showing a turnover around 1 GHz <cit.>. These sources, classified the gigahertz-peaked spectra (GPS) pulsars, exhibit the typical power law spectrum at higher frequencies, but their observed flux decreases with frequency and the corresponding spectral index becomespositive at frequencies < 1 GHz. A systematic study was conducted by<cit.> and <cit.> where they reportednine pulsars and two magnetars to exhibit GPS characteristics. The GPSphenomenon appears to be associated with relatively young pulsars found in oraround peculiar environments such as pulsar wind nebulae (PWNe), H II regions, etc. This motivated<cit.> to suggest that the spectral turnover (∼ 1 GHz) is a result of theinteraction of the radio emission with the pulsars' environments. The unique binary system of PSR B1259-63 and Be star LS 2883 provides awindow into the GPS phenomenon <cit.>. The spectrum of B1259-63 at various orbital phases mimics the spectrum of a GPS pulsar. <cit.> considered two mechanisms that might influence the observed radioemission: free-free absorption and cyclotron resonance.Most GPS pulsars have no companion and thus there is no direct analogy between them and the binary system. However, the appearance of the GPS in isolated pulsars may be caused, like in the case of PSR B1259-63/LS 2883, by their peculiar environments (like pulsar wind nebulae, supernova remnants filaments).The interaction of the radio emission from pulsars with their environments was investigated in detail by <cit.>. They showed that thephysical properties of certain of environments suggest that thermal absorptioncan cause the observed spectra to turnover at gigahertz frequencies. Thedifference between the binary system and a typical GPS pulsar is that for PSRB1259-63 the intensity of the effect changes due to variable amount of matter that the pulsar radiation has to pass through, <cit.>, whereas for theisolated pulsar the geometry of the absorber remains static, producing astable, GPS-type spectrum. Similar approach was also used by<cit.> for a selected sample of six pulsars and by<cit.> to explain the apparent variability of the spectra inPSR B1800-21.The statistical studies of the past pulsar search surveys (see for example) indicate that the pulsars exhibiting the GPS phenomenon can amount up to 10% of the entire pulsar population. However, the relativelysmall sample of GPS pulsars (11 known cases) is understandable given the pulsarspectra are not well studied at low radio frequencies (below 1 GHz). The fluxmeasurement becomes particularly challenging in pulsars with suspected GPScharacteristics. The specialized environments around these pulsars imply thatthey generally have relatively high (> 200 pc cm^-3) Dispersion Measure(DM) and their profiles are significantly smeared by interstellar scattering;see <cit.> and <cit.> for recent studies on scattering. As explained in <cit.>, the traditional flux measurement technique using single dish orphased-array telescopes is inadequate to measure the flux in highly scatteredpulsars leading to gross underestimation of the pulsar flux. Interferometricimaging is the only method to securely measure the pulsar flux in such cases.Additionally, <cit.> carried out a detailed comparison between thetwo flux measurement schemes and demonstrated that the interferometric imagingis a vastly superior technique to determine the pulsar flux.The primary objectives of this paper are twofold. Firstly, we have measuredthe flux in a large number of pulsars at low radio frequencies using theinterferometric methods. This is intended at extending the potential sample ofGPS pulsars and also verifying the GPS nature in a number of cases as acontinuation of the studies initiated in <cit.>. Secondly, we have carried out detailed modeling using the thermal free-free absorption following the suggestion of <cit.>. The model fits were carried out for all pulsars exhibiting GPScharacteristics and expanding the study of <cit.> and<cit.>. We used the data fitted models to estimate the physicalproperties of the absorbing electron gas - its density, temperature andpossible sizes of the absorber. § OBSERVATIONS AND DATA ANALYSISWe have conducted extensive observations using the Giant Metrewave RadioTelescope (GMRT) located near Pune, India. The GMRT consists of an array of 30distinct dishes, each with a diameter of 45 meters andspreadout over a region of ∼ 27 km in a Y-shaped array. The GMRT operates mostly in the meter wavelengths, between 150 MHz and 1.4 GHz, and is ideally suited tomeasure the pulsar flux at the low radio frequencies and check their spectral shape. We used the 325 MHz and 610 MHz frequency bands for our studies andobserved nine pulsars at 325 MHz and eight pulsars at 610 MHz, respectively. The data were recorded in the interferometric observing mode with eachfrequency band having a bandwidth of 33 MHz spread over 256 spectral channels. The 610 MHz observations were conducted between December 2013 and January,2014 while the 325 MHz observation was carried out in January 2015. Eachsource was observed for roughly one hour during each observing run at both frequencies. All observations were repeated twice and the observing sessions wereseparated by a week to account for intrinsic flux variations as well as systematics.The observations were carried out using standard interferometric schemes withstrategically spaced calibrators interspersed with the sources. We recordedflux calibrators 3C48 and 3C286 at the start and end of observations for around 8–10 minutes. In addition, a number of phase calibrators, spatially close tothe pulsars, were selected and observed for 3–4 minutes every hour to accountfor the temporal gain variations in each antenna. The phase calibrators werealso used to correct for fluctuations in the frequency band. During the 325 MHz observations two phase calibrators were used, 1714-252 and 1822-096, whilethe 610 MHz observations utilized five different phase calibrators 1822-096, 1830-360, 1924+334, 2021+233 and 2047-026. The imaging analysis was carried out using the Astronomical Image Processing System (AIPS), similarly to Dembska et al. (2015a) and <cit.>. The flux scales of the calibrators 3C286 and3C48 were set using the estimates of <cit.> and used to calculatethe flux of the different phase calibrators (see Table <ref>). Wereached noise levels of 0.2–0.5 mJy at 325 MHz ensuring detections of allsources with flux in excess of 1.0–2.5 mJy (5σ detections). All thepulsars observed at 325 MHz could be detected in our observations, the onlyexception was B1822-14 where the presence of a nearby strong source increased the noise levels and the pulsar was below detection limit on the firstobserving session. The noise levels in the maps at 610 MHz were between0.1–0.2 mJy ensuring detection limits between 0.5–1 mJy. However, we wereonly able to detect four of the seven sources at 610 MHz with the remainingpulsars below the detection limit. The detailed results and implications ofthese measurements are discussed in the subsequent sections.§.§ The thermal absorption model In this section we present the basic tenets of the thermal free-freeabsorption to model the turnover in the spectra ().To model the spectra we used the approach proposed by <cit.> and used by <cit.>. In this model, the intrinsic pulsar spectrum is assumed to be a single power-law, and to estimate the optical depth we used an approximate formula for thermal free-free absorption <cit.>. The observed spectrum can be thus written as:S(ν) = A ( ν/ν_0)^α e^-B ν^-2.1 , here A is the intrinsic pulsar flux scaling factor (i.e. the flux density at at the frequency of ν_0=10 GHz),α is the intrinsic spectral index of the pulsar, and the frequency ν is in GHz. The optical depth τ was expressed by a product of the frequencydependent term and the frequency independent parameterB =0.08235 × T_e^-1.35 EM,where T_e isthe electron temperature and EM is the Emission Measure (in pc cm^-6). The best fits to themeasured spectra were obtained using the Levenberg-Marquardt least squaresalgorithm. Compared to the earlier attempt at modeling of the GPSpulsar spectra by <cit.>, the model we use here is more adequate, since we use a full 3-parameter fit, while they performed a 2-parameter fit, using high frequencyflux measurements as an “anchor point”, instead of fitting for the intrinsic pulsar flux amplitude A.Table <ref> shows the best fit values forparameters A, B and α, along with the normalized χ^2 and the resulting peak frequency,i.e. the observing frequency at which the model reaches maximum flux. We also included the values of reduced χ^2 for a single power law fit for comparison, and as one can see for all pulsars all but one the thermal absorption model provides significantly better fit to the spectra. Theuncertainties of the fitted parameters were obtained using 3-dimensionalχ^2 mapping. The dashed lines in the spectra plots shown in the next section represent the envelopes of all the models that lie within 1σ boundary of a best fit model for thegiven spectrum[for a given spectrum all the models that agree with the best fit to1σ level will in their entirety lie within the envelope.] § RESULTS Table <ref> shows the results of our flux density measurements, both for individual observations at both frequencies, as well as the average value. As mentioned in the previous section, the pulsars J1834-0812,J1856+0245 and J1916+0748 were below our detection limit at 610 MHz. Using the newly acquired data and the previously published flux measurements (see figure captions) we constructed the spectra for all pulsars that either exhibit the GPS behavior or had been suspectedto do so before our recent observations.The spectra are shown in Figures <ref> to <ref>, and the new measurements are denoted by filled symbols. The spectra were divided into 4 groups: the new GPS pulsars (Fig. <ref>), objects for which the GPS feature was confirmed earlier (Fig. <ref>), the spectra that resemble a simple power-law (Fig. <ref>; these pulsars were usually suspected, or previously claimed to show a high frequency turnover), and finally the spectra of two pulsars that still require further investigation (Fig. <ref>), which we consider to be promising GPS candidates. In all of these figures except the last one we are also showing the results of spectral modeling fits. In our analysis we omitted some of the archival measurements for pulsars B1822-14 and B1823-13 that were published by <cit.>. These measurements were indicated by the authors of that work as suspicious and possibly affected by interstellar scattering. As we mentioned in the Introduction, and also discussed previously in<cit.> and <cit.> that may lead to severe underestimation of the received flux density, when measured by a standard pulse profile based method [it is always best to check the profile on which the flux measuremets was based, for example in the EPN Database: http://www.jb.man.ac.uk/research/pulsar/Resources/epn/.]. The theoretical aspects of the influence of interstellar scattering on the observed pulsar flux were also recently discussed by <cit.>. This calls for caution when using archival(or catalog) flux density measurements, especially at low frequencies, where the effects of scattering-induced flux underestimation will be strongest.§.§ New and confirmed GPS pulsars Figure <ref> shows the spectra of five pulsars for which the GPS effect is presented for the first time. These objects are characterized by therelatively high DM values (>190 pc cm^-3), except PSR J1841-0345 which has DM = 56 pc cm^-3. Most of the pulsars are relatively young τ <10^6 year (see Table <ref>, the basic pulsar parameters were taken from the ATNF Catalog[Available at http://www.atnf.csiro.au/people/pulsar/psrcat/,<cit.>]).For pulsars J1723-3659, J1835-1020 and J1901+0510 our new measurements were only the third in their respective spectra. Having three data points makes it impossible to model the spectrum with three parameters. In the case of these three pulsars, we have limited the number of fitted parameters by an additional assumption that the pulsar's intrinsic spectrum has a spectral index of -1.8 (the average value for the non-recycled pulsar population, see ), and estimatedonlyA and B parameters from the fits.Figure <ref> presents the spectra of four objects which were classified as GPS pulsars by <cit.> and <cit.>. The objects were included in our sample to confirm the shape of the spectra and put better constrains on the thermal absorption model by adding new measurements at low observing frequencies (where the free-free absorption manifests itself the strongest). In all four cases the new measurements confirm the earlier claims of high frequency turnovers. §.§ Objects with typical spectra and GPS candidate pulsarsThe pulsars whose spectra are shown in Figure <ref> were suspected to show the GPS phenomenon in our earlier studies and hence were included in our observing sample. However, our new measurements (together with the previously published flux density values) clearly show that pulsars B1832-06, J1834-0731 and B1904+06 exhibit a regular power-law spectra down to 300 MHz.Figure <ref> presents spectra of two pulsars we attempted to measure in our observations but only obtained upper limits for their flux density at 610 MHz. We still believe that these objects are good candidates for GPS pulsars.PSR J1834-0812 exhibits a very high dispersion measure (DM=1020 pc cm^-3). For PSR J1916+0748, the upper limit for the flux density clearly suggests a positive spectral index, however we can not explain why our limit is lower than the earlier measured flux at 400 MHz.§.§ Radiomagnetars and other GPS pulsars Figures <ref> and <ref> present the spectra of the GPS pulsars andradio magnetars that were published earlier <cit.>. We did not add any new data points to these spectra, however, for completeness, we decided to apply the thermal absorption model to these spectra as well. Theresults of our fits are included in Tables <ref>. In the case of PSR B1054-62 there are some results that allow for a power law spectrum to within a 1σ level. This is indicatedby the shapes of the 1σ envelopes, and the fact that the uncertainty inB parameter (see Table <ref>) extends down to include the value of B=0 within 1σ level, which means no absorption.Similar is the case of PSR J1809-1917, the only pulsar in Table <ref>, for which the reduced χ^2 from a power-law fit is lower than for the thermal absorption fit. Compared to the spectrum that was shown earlier in <cit.> we addeda measurement from <cit.> at 6.5 GHz. The large discrepancy between this measurement and our previous 5 GHz measurement causes the fit to be less conclusive. This is because in our modeling the high frequency part of the spectrum is primarily used to ascertain the slope of the intrinsic pulsar spectrum. Hence the discrepancy between two high frequency measurement translates to large uncertainties, as we can not exclude the possibility that the intrinsic spectrum is relatively flat. We decided, however to include this pulsar as a GPS source - the low frequency part of the spectrum clearly indicates a turnover, as the 1170 MHz flux is almost 2.5 times smaller than the 1.4 GHz flux.<cit.> used the thermal absorption model to explain the observed GPS-like evolution of the Sgr A* radio magnetar spectrum (PSR J1745-2900). Here we show the spectra of another two radio magnetars that were pointed out by <cit.> to show GPS characteristics. The results of our modeling clearly show that the high frequency turnovers in the spectra are undeniable, since - as it is indicated by the shape of 1σenvelope - no single power-law model will agree with our results to within 1σ uncertainty. §.§ Break or turnover in the spectrum of the PSR B1828-11 In Fig. <ref> we present the spectrum of PSR B1828-11. This pulsar was previously studied by <cit.>, and was classified as a flat-normal, or in other words, a broken spectrum, with the spectral index of -2.4 in the high frequency range, and 0.5 in the low frequency range, below 1.2 GHz.Despite having no new observations for the source, we decided to apply our thermal absorption model to its spectrum (see Table <ref> for the results of our fit).We believe that the results of our modeling show that the broken spectra of pulsars can be, at least in some of the cases, explained with the absorption model. The lack of a spectral turnover in such a case may be the result of the limited range of frequencies over which the pulsar flux density measurements were done. To test that hypothesis detailed observations at lower frequencies are necessary. § DISCUSSIONThe interferometric imaging technique is a much superior technique to estimatethe pulsar flux (see ), and provides the only means to estimatethe flux of pulsars which are affected by scattering (see discussion in ).Given the robustness of our flux measurements, we were able to estimate the low frequency spectra in several pulsars and found five newcases of gigahertz-peaked spectra: PSR B1641-45, PSR J1723-3659,PSR J1835-1020, PSR J1841-0345 and PSR J1901+0510. In addition, we also verified the GPS phenomenon in another four pulsars. The spectral turnover in fifteen GPS pulsarswas successfully explainedusing the thermal absorption model (see Table <ref>). In the remainder of this section we explore the physical implications of these results. The spectra of six GPS pulsars were previously modeled using the thermal absorption hypothesis by<cit.>. Their models were using our data from <cit.> and <cit.>. For five of these sources we are showing a model based on spectra in a wider frequency range, as we added new flux density measurements, mainly at low frequencies. This, in conjunctionwith using a full three parameter model, allows us to obtain much more reliable estimates of the absorbers physical parameters. §.§ Thermal absorption as the source of spectral turnovers The idea of the thermal free-free absorption as the source of the low frequency spectral turnovers in pulsars was first proposed by <cit.> and was proposed to explain the GPS spectra of pulsars and magnetars by <cit.>.<cit.> showed that indeed the peculiar environments of some pulsars may provide sufficient amounts of absorption to cause the spectra to turnover at gigahertz frequencies. Until now we were aware of the low frequency turnovers (around 100 MHz), and the gigahertz-peaked spectra in which the peak frequency was reported to be around 1 GHz (see ), however, the results we present here show rather a continuous range of peak frequencies (see Table <ref>), and not merely bi-modal. Also, owing to the use of an actual physical model, instead of purely morphological attempts (using for example , log-parabolic fits), the peak frequencies we obtained are lower than the values estimated earlier. This is to be expected, since the spectral profile of the absorption will be always asymmetric, with the peak shiftedtowards lower frequencies.The fact that the peak frequencies in the pulsar spectra affected by thermal absorption cover a wide range of values should come as no surprise, since in reality both the ISM and the immediate pulsar surroundings can show a wide, and roughly continuous range of physical parameters relevant to this phenomenon. Depending on the amount of absorption, which in itself is bound to the physical properties of the absorbing matter (electron density, temperature and its extent along the line of sight), one can expect that different configurations will cause the spectra to peak at different frequencies - below 300 MHz for pure ISM, around 1 GHz for pulsars obscured by dense supernova remnant (SNR) filaments, but also the entire range between these two possibilities should be covered by ionized clouds - regions that have electron densities higher than neutral ISM, but not as high as the compact clumps of matter in supernova remnants.§.§ The absorber physical parametersFollowing <cit.> and <cit.>, we have to point out thatthe free electrons in the ISM contribute to both the observed dispersion of the pulsar signal as well as to the absorption. In the case of the dispersion effect, the contribution from any region of space is proportional to its electron density n_e, but the same region contributes to the optical depth as its n_e^2. Using our assumption of a dense uniform region being the source of the absorption, we can estimate its emission measure along the line of sight to beEM=n_e^2 × s , where s is the width of the absorber. On the other, hand this absorber will also contribute to the pulsar's dispersion measure providing ΔDM=n_e × s. Our best model fits provideparameter B which is dependent on the temperature of the electrons and the emission measure.Using the above, it can be written as B = 0.08235 × T_e^-1.35× n_e^2 × s, which gives us the first equation binding these 3 parameters. The dispersion measure provides the second equation, butsince the DM's dependence on the electron density is not as steep as in the case of thermal absorption, one can not negate the DM contribution from the general ISM (i.e. region outside of the absorber). Obviously, without additional information about the electron density and the size of the absorbing region, we can not reliably estimate the fraction of the total observed DM that comes from the absorber. Following<cit.> and <cit.>, for the purposes of our calculations, we assumed that for the pulsars exhibiting the GPS phenomenon, hence the ones that have a well defined absorber along their line-of-sight, the contribution to the DM from the absorber is equal to half of the observed DM value. This allows us to write the second equation concerning the physical parameters of the absorber: 0.5DM = n_e× s. Since we have only two equations and three free parameters (s, n_e and T_e), it is impossible to solve for the values of the parameters, without additional information about at least one of them, or without additional assumptions. In the case of pulsar observations the additional information is usually not available, as it is extremely hard to identify the actual absorber, or - in the case of known PWNe - to extract the physical parameters of the region that will contribute to radio wave absorption. Therefore, following <cit.>, we decided to continue our analysis for three distinct categories of possible absorbers: * dense filaments in an SNR with typical s = 0.1 pc,* cometary shaped tail in a PWN with s = 1.0 pc, and* H II region with s = 10 pc. Table <ref>, in addition to basic and relevant pulsar parameters (i.e. DM, characteristic age and possible associations) shows the results of our calculations for the physical properties of the absorbing region using the three cases described above, i.e. for every pulsar the electron density and temperature was calculated for hypothetical absorbers with physical width of 0.1, 1.0 and 10 parsecs. Based on this information one can attempt to identify the type of absorbing region, by excluding the unphysical or unlikely parameter combinations as explored below.§.§ Constrains on possible astrophysical absorbers <cit.> pointed out that it is extremely difficult to estimate the electron densities and temperatures in the pulsar surroundings, using observational data which are, as a result, extremely rare. In the case of the GPS pulsars identified so far such data are not available. Additionally, the situation is even more futile in the case of pulsar spectral observations, since from such studies one can not ascertain the geometry of the absorber: its actual location and size (i.e. its extent along the line-of-sight) remains unknown. In fact, the location of the absorber is completely irrelevant to the amount of the observed absorption. We can only assume that the absorption happens in the vicinity of the pulsar, since pulsars (especially the young ones),are often located in environments with relatively high electron density.For the above reasons we are, at the moment, unable to predict the actual physical parameters of the absorbing matter that causes the turnover in pulsar spectra, we can, however, put some observational constrains on these absorbers. This allows us, at least to some degree, to distinguish between the different kinds of astrophysical sources of absorption and point out the most likely case, or at the least exclude the non-viable possibilities. In Table <ref>, we present the results of our calculations of the physical properties of the absorbers that would cause the observed amount of absorption for each of the GPS pulsars. For each of the sources, we calculated the required electron density and temperature assuming three different sizes of the hypothetical absorber. These represent an absorption in an H II region (10 pc in size), a shell of a PWN, or a small H II region (1 pc) and a dense SNR filament (0.1 pc). The corresponding derived values of density and temperature of electrons can be used to estimate the likelihood of a particular type of absorber being the reason of the GPS behavior. For example, in virtually all the cases presented in Table <ref>, the assumption of the H II absorber yields somewhat realistic values of the electron density, up to few tens of electrons per cubic centimeter, however the corresponding electron temperatures are extremely low (below 100 K) which is unphysical for this type of ionized region (typical electron temperatures in H II regions are of the order of a few thousand kelvins). Therefore we can confidently exclude them as possible absorbers.In general, we believe that our calculations clearly show that extended H II regions are the most unlikely causes for the appearance of the GPS spectra. The high electron temperature in such regions causes the ISM to be more transparent. To provide enough absorption in such a case the electron density would have to be much higher. While the high value of density on its own is not an issue, sincesome of the H II regions exhibit densities up to 10^5 particles per cm^3, one has to realize that the same region would also provide the dispersion of the pulsar signal. The value of the observed DM is easy to calculate: a 10 pc region with an electron density of 1000 particles per cm^3 would provide 10000 pc cm^-3 of dispersion, making the pulsar very unlikely, or even impossible, to be discovered in a regular pulsar survey (even provided that the survey would be conducted at high frequency, wherethe thermal absorption would not play a significant role).As for the other types of absorbers, the interpretation is usually not as straightforward, since these kinds of pulsar environments can exhibit a range of physical parameters. As<cit.> pointed out, the dense filaments in a supernova remnant can be dense enough (up to a few thousand particles per cubic centimeter) that, even when considering their high temperature (up to a few thousand kelvins), they still provide enough absorption to cause spectral turnover at gigahertz frequencies, while being only a fraction of a parsec in size. Looking at the data in Table <ref> a number of pulsars seems to fit the criteria: for example PSR B1054-62 (n_e = 1601 cm^-3, T_e=5080 K), or PSR B1822-14 (n_e = 1785 cm^-3, T_e=10100 K), and several others. Obviously, not all of the GPS pulsars listed in the table that would fit the SNR filament scenario were actually observed within an SNR, however this can be caused by the fact thatthe distant remnants are especially difficult to detect. Additionally, one has to remember that the absorber does not have to be physically or evolutionary connected to the remnant - just like in the case of PSR B1800-21, studied by <cit.>, an object whose line-of-sight apparently crosses an unrelated W30 remnant nebula.Distinguishing between absorption in a dense filament and absorption in a shell (or rather “tail”) of a pulsar wind nebula may be more difficult.<cit.> showed that the measurements of the temperatures and densities in cometary-shaped PWNe are very sparse, at least when it comes to these parts of the nebulae that could possibly provide free-free absorption in the radio regime. They also show that if one considers densities of the order of a few hundred particles per cm^3, then for roughly 1pc size of the absorber in the PWN tail, one would need temperatures as low as few hundred kelvins, to provide significant absorption at gigahertz frequencies. Based on the data we show in Table <ref>, several pulsars would qualify in this category, like for examplePSR B1641-45 (n_e=239 cm^-3, T_e =983 K). However, for these sources the SNR filament scenario also provides reasonable parameters, making it extremely difficult to decide which of these two possibilities we are dealing with for a particular pulsar. To solve this one would need some additional observations, like the detection of a PWN, or measurements of the associated/coincident SNR, however in the case of the confirmed GPS pulsars such observations are not available, and even in the cases where we have a confirmation of a pulsar wind nebula or supernova remnant - the data are not detailed enough to provide information that would allow us to estimate the parameters relevant to the thermal absorption of radio waves. Finally, since the effect of absorption in our model does not depend on the actual location of the absorber, we expect that in some cases we may see the pulsars through absorbers that are not in their immediate vicinity, which makes the interpretation even harder; the possibility of multiple absorbers (with different physical parameters) cannot also be excluded. §.§ Finding GPS pulsars in future surveysBased on the values of peak frequency we obtained (see ν_p in Table <ref>) we created a histogram of the peak frequency distribution which is shown in Figure <ref>. To create the histogram we included all the pulsars and radio-magnetars presented in this work, as well as the data from the binary pulsar B1259-63 <cit.>, the Galactic center radio-magnetar J1745-2900 <cit.>and PSR B1800-21 <cit.>. While the statistics is still small, looking at the histogram one can note a difference between the regular GPS pulsars and the peak frequencies in the spectra of radio magnetars. This may be at least partially explained by the differences in the methods of discovery. The radio-magnetars are always found by targeted searches following an X-ray outburst of the source, and the X-ray data often provides the rotational period, which significantly narrows the parameter-space that needs to be searched to findthe radio counterpart. On the other hand finding a GPS pulsar with a peak frequency higher than, say, 1.5 GHz will be difficult. An optimal frequency range for a GPS pulsar search is definitely much higher than its peak frequency. For example, if ν_0.9is the frequency at which one would see 90% of the pulsar's flux, then using simple algebra one can show that ν_0.9/ν_p ≈√(4.5 ×α) (where α is the intrinsic pulsar spectral index). For a typical pulsar (α≈ 2) it means that only in the frequency range above 3 times the peak frequency the observer receives almost all of the pulsar intrinsic flux, making this range optimal for detection. This would explain the lack of regular pulsars withν_p above 2 GHz, since the search for such sources would be most effective at frequencies higher than 6 GHz.There were only a few limited attempts to search for pulsars in this range, like for example the Parkes Methanol Multibeam Survey <cit.>. One of the main reasons why such searches are not attempted more often is that blind surveys at such high frequencies are extremely time consuming. This is due to the decreased telescope beam size, which in turn decreases the Galaxy volume that may be searched in a given time. Additionally, GPS pulsars with such a high absorption would also most likely exhibit high values of DM, and that would happen regardless of their distance, since the absorption and its associated high DM contribution is most probably local to the pulsar. A possibility of a large DM value may significantly increase the parameter space that has to be searched, however, since the dispersion smearing is inversely proportional to the observing frequency squared this may not be a big issue for such high frequency searches.<cit.> recently published a detailed study of past and proposed high frequency pulsar surveys, calculating the expected discovery probabilities while taking into account the effects of thermal free-free absorption andscattering. They show that the optimal frequency range for finding pulsars in the central regions of the Milky Way would be 9 to 13 GHz, and they claim that the main limiting factor in this case will be rather the interstellar scattering, not absorption. However, in their calculation of thermal absorption they used an emission measure (EM) of 5× 10^5 pc cm^-6 for the Galactic Center pulsars. As we show in Table <ref> in some scenarios the GPS spectrum may be explained by regions with EM exceeding 10^6 pc cm^-6 (see the radio-magnetars in the0.1 pc absorber case), which means that for very dense environments the thermal absorption may play much stronger role in the probability of detection, than <cit.> used for the proposed Galactic Center surveys. For the above reasons we believe, that there is still a high chance that we can discover GPS pulsars in some of the less studied PWNe or SNRs, by the means of targeted searches at frequencies close to and above 10 GHz. Additional targets for such searches may be the unresolved X-ray sources that show characteristics of a PWN (for example with regard to their spectra). Also, there is a chance that a number of GPS pulsars still remains undetected in the larger/more dense/cooler H II regions, and possibly in the center of the Milky Way. Blind surveys performed at higher frequencies may also yield occasional discoveries, however they will be much more time consuming. §.§ GPS radio magnetars We also included in our studies two radio magnetars, J1550-5418 and J1622-4950 that were previously reported by <cit.> to exhibit GPS type spectra, with the apparent peaks at very high frequencies. Another case was studied by<cit.>, namely the Sgr A* radio magnetar, located very close to the central black hole of our Galaxy, and this object was also exhibiting GPS characteristics, which evolved with time - since, according to the model shown by the authors, the amount of absorption was decreasing with time.One has to note a significant difference between our results and the study of theSgr A* magnetar: the spectra that were analysed for this object were obtained for individual epochs, while the spectra we show here forJ1550-5418 and J1622-4950 were obtained from all available radio flux density measurements, regardless of the date of observations. Therefore a certain degree of caution is required when interpreting the results. If the radio spectra of the magnetars indeed change over time, the spectra we used for our models do not correspond to any given moment in time and their shape may be heavily affected by the actual evolution. For that reason, the physical parameters inferred from the models should be treated only as rough/averageestimates. <cit.> noted that the peak frequencies for the two radio magnetars are much higher than the values obtained for the remaining GPS pulsars, suggesting that the magnetars have to be in even more extreme environments compared to the other GPS pulsars.This would be explained by the fact, that these magnetars are extremely young objects (see Table <ref>, the ages of 1410 and 4030 years), which would indicate that one can expect much more extreme conditions in their surroundings, especially when it comes to the electron density. The density influence will be somewhat offset by higher temperatures of these surroundings, however since the dependence on density (through the emission measure) is much stronger, that would explain higher amounts of absorption and higher peak frequencies. We also have to note that the peak frequencies obtained from our thermal absorption fits are significantly lower than the ones reported before, i.e. 3.27 and 3.7 GHz instead of 5.0 and 8.3 GHz (see also Fig. <ref>). This discrepancy comes from the fact that the previous paper used the purely morphological model of parabolic spectral shape proposed by <cit.>. The application of the real physical model suggests that the radio magnetars have relatively flat spectra: the intrinsic spectral indexes we obtained from our modeling are the lowest in the sample (-0.46 and -0.54) which combined with the thermal absorption profile causes only very small changes in the observed flux density over a relatively wide range of frequencies. The use of the actual model moves the peak frequency from the middle of that “almost flat” spectral region towardslower frequencies, yielding the values around 3 GHz.<cit.> in their study of the Sgr A* radio magnetar obtained a similar value (which can be inferred from their spectra plots),about 3.5 GHz at 40 days since the magnetar outburst. Also <cit.>observed the spectrum of that magnetar, and their plots suggest a peak frequency of about 2 to 2.5 GHz, and their observations were made approximately 100 days after the outburst. All of these studies clearly indicate that if the thermal absorption is the cause of the spectral turnovers in radio magnetars, then indeed the parameters of the absorbers must be more extreme than in the case of regular GPS pulsars. In Section <ref> we discussed some basic aspects of GPS pulsar searches. Based on that we believe, that the main reason for the fact that we know of magnetars with such high peak frequencies, but we did not discover pulsars exhibiting such high amount of absorption is purely due to the discovery bias: the magnetars are discovered in the radio regime only after their initial X-ray outburst discovery (which often provides the rotational parameters as well), while for regular pulsars we do not have that option and we are forced to perform a full search, which (as we discussed above) can succeed only if performed at sufficiently high observing frequency. It is possible, that there are pulsars located in similarly extreme environments that simply were not discovered yet, because no one searched for them at observing frequencies around 10 GHz and above. Moreover, there is a possibility that the reason that some of the magnetars do not exhibit radio emission at all may be due to thermal absorption that is even more extreme than in the cases ofJ1550-5418 and J1622-4950. However, given the nearly flat intrinsic radio-magnetar spectra, these objects would be easier to find at frequencies around and above 10 GHz, and we know that such attempts were made, at least in some selected cases. Hence we believe, that the free-free absorption as the explanation for radio-quiet magnetars is not very likely.§ SUMMARYWe used the interferometric imaging technique to estimate the low radio frequency flux in fifteen pulsars. The high sensitivity of the measurementsallowed us to construct the spectral shape of these pulsars and in the processwe identified five new pulsars with gigahertz-peaked spectra (GPS).Additionally, our measurements resulted in tighter constrains on the lowfrequency spectra of four pulsars and confirmed their GPS characteristics.Summarizing, the GPS pulsar population now consists of seventeen sources. The GPS phenomenon in pulsars is most likely a result of thermal absorption ofthe pulsar flux in specialized environments along the line of sight. A detailed study was conducted where the low frequency turnover in the spectral shape was successfully modeled using thermal absorption, using 3-parameter fit procedure similar to the one employed by <cit.>. We presented the results of our modeling for fourteen GPS pulsars (see Figs. <ref>, <ref>, <ref>, <ref>).The remaining three GPS neutron stars: PSR B1800-21, PSR B1259-63 in the binary system with Be star LS 2883 and SGR J1745-2900 have been modeled in separate works (see section 4.4). A detailed study of PSR J1740+1000, a source which was claimed to show GPS characteristics by <cit.>, is in progress.The spectral shape arising due to the thermal absorption is not sufficient to fully characterize the physical properties like temperature, electron density andsize of the absorbing region. However, using basic physical arguments, we haveexplored the nature of the absorber and ruled out some of the potential candidates. We also discussed the strategies for finding GPS pulsars in the future search surveys. The optimal frequency rangeis usually a few times larger than the peak frequency, meaning that for a normal GPS pulsar (with a peak frequency close to 1 GHz) the optimal frequency would be greater than 3 GHz, while for the sources with much stronger absorption, similar to radio-magnetars (ν_p close to 3 GHz), the optimal frequency would be much higher, possibly close to 10 GHz. Therefore we believe that a targeted search surveys of known PWNe, H II regions and supernova remnants should be the way to go. However, we have to alsopoint out thatthere isa possibility, that a lot of GPS sources still hides in the known pulsar population. 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[pages=1-last]ACM_2017_final.pdf | http://arxiv.org/abs/1704.08356v1 | {
"authors": [
"Saurav Talukdar",
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Multi-cut Solutions in Chern-Simons Matrix Models18pt Takeshi Morita[ E-mail address: [email protected] ] and Kento Sugiyama[ E-mail address: [email protected]] Department of Physics, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, JapanWe elaborate the Chern-Simons (CS) matrix models at large N. The saddle point equations of these matrix models have a curious structure which cannot be seen in the ordinary one matrix models. Thanks to this structure, an infinite number of multi-cut solutions exist in the CS matrix models. Particularly we exactly derive the two-cut solutions at finite 't Hooft coupling in the pure CS matrix model. In the ABJM matrix model, we argue that some of multi-cut solutions might be interpreted as a condensation of the D2-brane instantons. § INTRODUCTION Matrix models play crucial roles in various topics in theoretical physics.(See, for example, recent review <cit.>.) Particularly, in string theory, there are several proposals that matrixes may provide the non-perturbative formulation of non-critical strings <cit.> and critical strings <cit.>, and, many remarkable results have been obtained in this direction. Besides, since matrixes are also related to the large-N gauge theories, they play special roles in quantum gravity through the gauge/gravity correspondence <cit.>. Recently, the importance of matrix models is significantly increasing in supersymmetric gauge theories and related mathematical physics too. In supersymmetric theories, the dynamical degree of freedom drastically reduce due to the cancellation between the bosons and fermions, and, in some special cases, the theories are effectively described by zero dimensional matrix models <cit.>. Especially, through the development of the localization technique <cit.>, many correspondences between the supersymmetric gauge theories and matrix models have been found. (See reviews <cit.>.)Among these matrix models, we focus on the U(N) Chern-Simons (CS) matrix models in this article. The partition function of the pure CS matrix model is given by <cit.>Z_N^ CS(k) =1/N!∫∏_i du_i/2π e^-N/4π i λ∑_i u^2_i∏_i<j[ 2 sinhu_i-u_j/2]^2.Here λ≡ N/k is the 't Hooft coupling and k is an integer representing the CS level. The computation of this integral has been already doneand we know the exact value of this partition function as a function of k and N <cit.>. However investigating this model is still valuable. As we will see, this model shows curious properties which have not been seen in any other one matrix models. In addition, the varieties of the pure CS matrix model are being actively studied, and understanding this model may help us in developing insights into these models. For example, the ABJM theory <cit.> on S^3 is described by a similar matrix integral <cit.>. Also the three dimensional 𝒩=2 supersymmetric CS matter theory coupled to matters with arbitrary R-charge on S^3 is described by a related matrix model <cit.>. Especially the 't Hooft limit of these models are important in string theory, and we study these CS matrix models under this limit.When we take the 't Hooft limit (N →∞, λ fixed), the saddle point approximation may be applicable. The saddle point equation of the pure CS matrix model (<ref>) is given byu_i=2π i λ/N∑_j ≠ iu_i-u_j/2,(i=1,⋯, N).The solution of this equation at finite λhas been found, and it is characterized by a single cut of the eigenvalue distribution <cit.>. (See figure <ref> (left).)Then a natural question is whether the solution is unique. We find that the answer is no. The saddle point equation (<ref>) has an interesting structure which allows an infinite number of solutions characterized by multi-cut eigenvalue distributions.See figure <ref> (right) for a two-cut solution. Moreover these multi-cut solutions would ubiquitously exist in the varieties of the CS matrix models too. (Actually such multi-cut solutions were first found in the CS matrix model coupled to adjoint matters through a numerical analysis <cit.>.)In one matrix models, the multi-cut solutions are usually related to the large-N instantons <cit.>. We argue that indeed some of the multi-cut solutions in the ABJM theory might be interpreted as a condensation of the D2-brane instantons <cit.>. Besides, some of the multi-cut solutions in the CS matrix model coupled to adjoint matter case might be related to the poles in the Borel plane <cit.>. However, we have not understood the physical interpretations of other multi-cut solutions. They might be related to some non-perturbative objects in the CS matrix models and string theory. Although we find the new multi-cut solutions in the CS matrix models, we have to emphasize that we just evaluate the saddle point equation (<ref>). It means that we cannot answer whether these solutions should be summed up through the path integral (<ref>). This is a crucial question but we leave this problem for future work. The organization of this paper is as follows: In section <ref>, we explain the basic idea of the derivation of the multi-cut solutions in the pure CS matrix model at weak coupling. (We show the derivation at finite coupling in appendix <ref>.) In section <ref>, we consider the CS matrix model coupled to adjoint matters. There, we argue the relation to the poles in the Borel plane in this model. In section <ref>, we argue the multi-cut solutions in the ABJM matrix model and show the connection to the D2-brane instantons. Section <ref> list some discussions for future explorations. § MULTI-CUT SOLUTIONS IN PURE CS MATRIX MODEL §.§ One-cut solution We first briefly review how the one-cut solutionin the pure CS matrix model (<ref>) which has been investigated in the previous studies is obtained. To see it quickly, we take a weak coupling limit |λ| ≪ 1 of the saddle point equation (<ref>), although we can solve it even at finite λ<cit.>. Under this limit, the quadratic potential of the path integral (<ref>) dominates and the eigenvalues may be strongly confined around u_i ≃ 0. Thus we can approximate the right hand side of the saddle point equation (<ref>) asV'_ G (u_i)=2λ/N∑_j ≠ i1/u_i-u_j,(i=1,⋯ N),V_ G (u) ≡1/4 π iu^2.This is the well-known saddle point equation for the Hermitian matrix model with a (complex) Gaussian potential, and we can solve this equation at large Nby using the resolvent. Although the derivation of the solution is well known, since we will use this technique to obtain the multi-cut solutions too, we briefly explain it here.To solve equation (<ref>) at large N, we introduce the eigenvalue density ρ(z) and the resolvent v(z) which are defined byρ(z) = 1/N∑_i^N δ(z-u_i) ,v(z)=∫_a^b dwρ(w)/z-w.Here a and b denote the location of the end points of the eigenvalue distribution, which will be determined later. By using the resolvent, the saddle point equation (<ref>) becomes[ ϵ may be a complex, since the cut does not need to be on the real axis in the CS matrix models.]V_ G'(z) =λlim_ϵ→ 0[ v(z+iϵ)+v(z-iϵ) ] , (z ∈ [a,b] )and the eigenvalue density can be read off from the resolvent ρ(z) =-1/2π ilim_ϵ→ 0[ v(z+iϵ)-v(z-iϵ) ] , (z ∈ [a,b] ) .Also, through the definition of the resolvent (<ref>), v(z) shows the asymptotic behaviorv(z) →1/z,(z →∞).We will use this equation as the boundary condition of v(z). The one-cut solution of equation (<ref>) can be obtained by using the following ansatz <cit.>v(z)=1/λ∫_C_1dw/4π iV_ G'(w)/z-w√((z-a)(z-b)/(w-a)(w-b)),where the integral contour C_1goes around the cut:[a,b] in a counter-clockwise way. By performing this integral and imposing the boundary condition (<ref>), we obtainv(z) = z/ 4 π i λ - 1/4 π i λ√(z^2 - 8 π i λ),where a and b have been determined b=-a= √(8 π i λ). Obviously this resolvent satisfies the equation (<ref>). From this result, we obtain the eigenvalue density,ρ(z) = 1/4 π^2 i λ√( 8 π i λ- z^2), (z ∈[-√(8 πi λ),√(8 πi λ)] ).Thus the eigenvalues are distributed in the complex plane as shown in figure <ref> (left) [In addition to this cut, its images appears on z ∈[-√(8 πi λ) +2 π in,√(8 πi λ) +2 π in ], (n ∈𝐙) due to the periodicity of the coth in the saddle point equation (<ref>).In this paper, we do not count these images as the number of the cuts.]. §.§ Two-cut solutionNow we show that the saddle point equation (<ref>) even at weak coupling has a non-trivial structure and other solutions exist. As a warm up, we first consider the saddle point equation (<ref>) at N=2.u_1/2π i λ=1/2u_1-u_2/2,u_2/2π i λ=-1/2u_1-u_2/2.By summing these equations, we immediately find u_2=-u_1 and the equations reduce tou_1/2π i λ=1/2 u_1.Obviously this equation has an infinite number of solutions. If λ is small, we can perturbatively solve this equation and obtainu_1 = ±√(π i λ) + ⋯ , u_1 = π i n + λ/n+ ⋯ ,where n is a non-zero integer. If we take N large, the first equation would correspond to the one-cut solution (<ref>). On the other hand, the second solution indicates the existence of another class of the solutions[A related speculation was first done in Ref.<cit.>.].Let us try to find the solution corresponding to the second solution at large N. By regarding the result (<ref>), it is natural to take the following ansatz,u_i= π i n +x_i , (i=1, ⋯ , N/2), u_N/2+j= -π i n +y_j , (j=1, ⋯ , N/2).We can choose the integer n positive without loss of generality. By substituting this ansatz into the saddle point equation (<ref>), we obtainn/2 +1/2π i x_i =λ/N∑_ j ≠ i^N/2x_i-x_j/2 + λ/N∑_j =1^N/2x_i-y_j/2,(i=1, ⋯ , N/2) ,-n/2+1/2π i y_i =λ/N∑_ j ≠ i^N/2y_i-y_j/2 + λ/N∑_j =1^N/2y_i-x_j/2 , (i=1, ⋯ , N/2).From now, we focus on the weak coupling limit |λ | ≪ 1 and consider the leading order in the expansion. (We show the result at finite λ in appendix <ref>.) We assume that the eigenvalues locate around x_i = 0 and y_j = 0 at weak coupling similar to the one-cut solution in the previous section. (Actually we will later see that x_i, y_j ∼ O(λ) similar to the N=2 case (<ref>).) Then equation (<ref>) reduce ton/2=2λ/N∑_ j ≠ i^N/21/x_i-x_j + 2λ/N∑_j =1^N/21/x_i-y_j, (i=1, ⋯ , N/2), -n/2=2λ/N∑_ j ≠ i^N/21/y_i-y_j + 2λ/N∑_j =1^N/21/y_i-x_j,(i=1, ⋯ , N/2). Here we argue how “forces" act on the eigenvalues { x_i } and { y_j }. The expression of the right hand side of these equations are similar to the Vandermonde repulsive force between the eigenvalues in the Hermitian matrix model (<ref>). The difference is that they also work between x_i and y_j as if they locate nearby, although they are actually separated by 2 π i nin the complex plane. The left hand side of equation (<ref>) is an external force acting on the eigenvalue x_i and y_j. It drags x_i towards the left direction (x_i → - ∞) and y_j towards the right direction (y_j →∞). These directions of the forces can be read off from the potential of the CS matrix model (<ref>) u_i^2/4 π i too. At u_i =π i n+ x_i, this potential can be expanded as u_i^2/4 π i= π in^2/4+nx_i/2+ ⋯which is a linear potential for x_i at the leading order.From the sign of this potential, we confirm that x_i is indeed dragged towards x_i → - ∞.Thus x_i and y_j tend to move toward left and right, respectively, and they repulse each other.Hence, if we set x_i right and y_j left, they may balance similar to the N=2 solution (<ref>). By regarding it, we assume that x_i and y_j satisfyRe( y_j )≤ 0 ≤ Re( x_i ).This assumption allows us to combine the saddle point equations for x_i and y_j (<ref>) into a single equationV'(z_i)=2 λ/N∑_ j ≠ i^N1/z_i-z_j,V'(z)=n[ θ( Re( z)) - 1/2],(i=1, ⋯ , N),where z_i = x_i for 1 ≤ i ≤ N/2 and z_i+N/2 = y_i for 1 ≤ i ≤ N/2. We have used the step function θ(x) in order to combine the left hand sides of the equation (<ref>). Now the problem becomes much simpler. This is merely a standard potential problem (<ref>) with the singular potential V(z)= n z [θ( Re( z )) -1/2].We plot the profile of V(z) in figure <ref> (right). Although V(z) is singular, this potential has the unique minimum at z=0 and the solution would be given by an “one-cut" configuration.Here “one-cut" means “one-cut" of z_i and it indeed describes the two-cuts of the original variable u_i as shown in figure <ref>. Hence we introduce the resolvent v(z) as in the previous section and apply the ansatz for the one-cut solutionv(z)= 1/λ∫_C_1dw/4π iV'(w)/z-w√((z-a)(z-b)/(w-a)(w-b))= 1/λ∫_0^b dw/2π in/z-w√((z-a)(z-b)/(w-a)(w-b))+ 1/λ∫_C_1dw/4π i-n/2/z-w√((z-a)(z-b)/(w-a)(w-b)) .Here we have defined a and b as the end points of the cut and C_1 as the contour around them. We have taken Re(b)>0 and Re(a)<0 obeying the assumption (<ref>).By regarding the parity symmetry z ↔ -z of the potential V(z) (<ref>), we assume a=-b. Then, by performing the integrals in (<ref>), we obtainv(z)=n/πλarctan( √(z+b/z-b)) -n/4 λ.We can confirm that this resolvent satisfies the equation (<ref>) with V'(z) given by (<ref>)[To see that the resolvent (<ref>) satisfies the equation (<ref>), the relation arctan x = i/2log( i+x/i-x) is useful. ]. Thus the ansatz (<ref>) works even in such a singular potential case too.Now we take the limit z →∞ and determineb so that v(z) satisfies the boundary condition (<ref>). By using the expansion arctan (1+x)= π/4 +x/2 +O(x^2), we obtainv(z)= nb/2 πλ1/z + O(1/z^2),(z →∞).Hence we fixb= 2 πλ/n, and we finally obtainv(z)=n/πλarctan( √(n z+2πλ/n z-2πλ)) -n/4 λ.The result is plotted in figure <ref> (left). From this plot, we can read off the eigenvalue density of the solution. Interestingly the profile of the density is an “onion head" shape, and it is quite different from the one-cut solution (<ref>) obeying the semi-circle law. Note that, although the density diverges at z=0 as ρ(z) ∼log |z|,ρ(z) is correctly normalized ∫ρ(z) dz=1.§.§.§ Effective potential and free energyThrough the standard matrix model technique <cit.>, we compute the effective potential V_eff(z) which is the potential for the single eigenvalue of the matrix in the two-cut solution (<ref>). For z > 0, it becomesV_eff(z)= ∫^z dw ( V'(w)-2λ v(w) ) =nz-2nz/πarctan( √(n z+2πλ/n z-2πλ)) - 2λlog( z+√(z^2-4 π ^2 λ^2 /n^2)) .The result is plotted in figure <ref> (right). As usual, a plateau appears on the cut. Thus, although the eigenvalue density has the singularity at z=0, the effective potential is not so different from the ordinary solutions in the one matrix models <cit.>. We can also compute the free energy of this solution,F^ 2-cut(λ,N)=N^2 n^2 π i /4λ+N^2/λ[ ∫_-b^b dz V(z) ρ(z)-λ/2 PV∫_-b^b dz ∫_-b^b dw ρ(z) ρ(w) log(z-w)^2] = N^2 n^2π i/4 λ+ N^2[ -1/2- log( 2πλ/n)].Here the first term is from the original classical potential u_i^2/4π i λ. Similarly we can calculate the free energy of the one-cut solution (<ref>),F^ 1-cut(λ,N) = -N^2/2logλ +𝒪(λ).Thus the real part of the free energy of the one-cut solution is lower than the two-cut solution at weak coupling, and it will dominate the path integral (<ref>). §.§.§ Other multi-cut solutionsWe have obtained the two-cut solution inspired by the N=2 analysis (<ref>). If we change N as N=3,4,5,⋯ in the saddle point equation (<ref>), we would easily find various non-trivial solutions. These solutions indicate the existence of further multi-cut solutions in this system. For example, at weak coupling, we can approximately superpose the one-cut solution (<ref>) and the two-cut solution (<ref>), if n is odd. See figure <ref> (left). (When n is odd, the coth interaction between these configurations are negligible.)Besides, we find the asymmetric two-cut solution in which each cuts consist of N_1 and N_2(=N-N_1) eigenvalues. In this solution, the cut for the N_1 eigenvalues appears around u=2π i n N_2/N and the cut for the N_2 eigenvalues does around u= - 2π i n N_1/N, where n is a non-zero integer. See figure <ref> (center). We show the derivation of the asymmetric two-cut solution at finite λ in appendix <ref>.In this way,the saddle point equation of the pure CS matrix model allows an infinite number ofmulti-cut solutions. This is a remarkable property of the CS matrix model which is quite distinct from the ordinary one matrix models. From now we show that similar multi-cut solutions ubiquitously exist in other CS matrix models too. § 𝒩=2 ADJOINT MATTERWe consider 𝒩 =2 supersymmetric CS matter theory coupled to N_A adjoint matters on S^3. Through the localization technique, the partition function of this theory is effectively given by the following matrix model <cit.>,Z_N^ CSM(k,h,N_A)=1/N!∫∏_i du_i/2π e^-k/4π i∑_i u_i^2∏_i<j[ 2 sinhu_i-u_j/2]^2 ∏_i , j e^N_A l ( 1-h+iu_i-u_j/2π),where h is the R-charge of the adjoint matter which takes 0 < h < 1. l(z) is the function defined in Ref.<cit.> and it satisfiesl'(z)=-π z π z.From this partition function, we obtain the saddle point equation u_i = 2π i λ/N∑_j ≠ iu_i-u_j/2 + N_A λ/N∑_j ≠ i[ ( π(1-h)+i u_i-u_j/2) ( π(1-h)+i u_i-u_j/2)]- N_A λ/N∑_j ≠ i[ ( π(1-h)-i u_i-u_j/2) ( π(1-h)-i u_i-u_j/2)].As argued in Ref.<cit.>, non-trivial multi-cut solutions would exist in this model too.Here we explore for two-cut solutions. First we apply the N=2 analysis in the pure CS matrix model to the saddle point equation (<ref>) too. Then we find u_1=-u_2 again, and the saddle point equation becomesu_1 = π i λu_1 + N_A λ/2( π( 1-h) +iu_1 ) ( π(1-h)+i u_1 ) - N_A λ/2( π( 1-h) -iu_1 ) ( π(1-h)-i u_1 ) .At weak coupling, we find the solutionu_1 = ±√(π i λ) + ⋯ ,u_1= π i n + λ/n+ ⋯,u_1= ±π i ( 1-h +n )± N_A λ/2n/1-h+n + ⋯.The first and second solutions are the same solutions to the pure CS matrix model at this order, while the last one is new. Thus a novel class of two-cut solutions associated with this solution may exist in this model. To find this new two-cut solution at large N, we use the ansatzu_i= π i ( 1-h +n ) +x_i , (i=1, ⋯ , N/2), u_N/2+j= - π i ( 1-h +n )+y_j , (j=1, ⋯ , N/2).By substituting them into the saddle point equation (<ref>) at weak coupling, we obtainπ i ( 1-h +n )= 4π i λ/N∑_j ≠ i1/x_i-x_j+ n N_A /24π i λ/N∑_j =1^N/21/x_i-y_j, -π i( 1-h +n )= 4π i λ/N∑_j ≠ i1/y_i-y_j+ n N_A /24π i λ/N∑_j =1^N/21/y_i-x_j .Here we have assumed that x_i, y_j ∼ O(λ). Finding the two-cut solution from these equations is still not easy. However, in the nN_A=2 case (N_A = 2 and n = 1 or N_A = 1 and n = 2), the equations become almost identical to equation (<ref>), and the solution is given by (<ref>) with n → 1-h+n.See figure <ref> (right) for the schematic plot of this solution.In additon to this solution, we can perturbatively obtain the two-cut solution associated withu_1= π i n +O(λ)in (<ref>) too. The solution at the leading order is the same to the pure CS matrix model case (<ref>).Lastly we argue a possible interpretation of the multi-cut solutions in this model. Recently Ref.<cit.> pointed out that this model may be non-Borel summable in the 1/k expansion due to the singularities in the Borel plane. These singularities are related to the configuration u_i-u_j= 2π i ( 1-h +n ). Our two-cut solution (<ref>) indeed satisfy this relation and it indicates that our solutions might be related to the singularities in the Borel plane. Usually singularities in the Borel plane correspond to some non-perturbative objects, and thus we expect that our solutions might describe such objects in this model.§MULTI-CUT SOLUTION IN ABJM MATRIX MODELWe consider the ABJM matrix model <cit.>. The partition function of this model is defined byZ_N^ ABJM(k) = 1/(N!)^2∫∏_i=1^N d μ_i/2π e^-k/4π iμ_i^2 ∏_j=1^N d ν_i/2π e^k/4π iν_j^2 ∏_i<j^N[ 2sinhμ_i-μ_j/2]^2 ∏_i<j^N[ 2sinhν_i-ν_j/2]^2 /∏_i,j=1^N[ 2coshμ_i-ν_j/2]^2.From this partition function, we obtain the saddle point equation,μ_i = 2π i λ/N[ ∑_j ≠ i^Nμ_i-μ_j/2- ∑_j=1^Ntanhμ_i-ν_j/2], -ν_i = 2 π i λ/N[ ∑_j≠ i^Nν_i-ν_j/2-∑_j=1^Ntanhν_i-μ_j/2] ,where we have defined 't Hooft coupling λ≡ N/k. By summing over these saddle point equations, we obtain the relation∑_i=1^N μ_i = ∑_j=1^N ν_j.In the ABJM matrix model, the parameter κ which is related to the 't Hooft coupling λ byλ(κ) = κ/8 π_3F_2 ( 1/2, 1/2, 1/2; 1, 3/2; - κ^2/16)is useful <cit.>. Particularly this relation becomesλ = 1/8πκ +O(κ^2),(|λ| ≪ 1),λ = 1/2π^2(logκ)^2 + 1/24+O(κ^-2),(|λ| ≫ 1) , at weak and strong coupling, respectively. One solution of the saddle point equation (<ref>) at large N has been derived byDrukker, Marino and Putrov <cit.>.In this article, we call this solution “DMP solution". The spectral curve of theDMPsolution for {μ_i } is given by Y=e^y,X=e^x,Y+ X^2/Y-X^2 + i κ X-1=0.This curve describes two cuts along [-α,α] and [-β+π i ,β+π i][In addition to these two cuts, there are images of them: [-α+2 π in,α+2 π in] and [ -β+π i+2 π in,β+π i+2 π in], (n ∈ Z).]. (The eigenvalue {μ_i } and {ν_i } distribute along the region [-α,α] and [-β,β], respectively. See figure <ref> (left).) The locations ofα and βare determined through the relationA= e^α,B=e^β,A+ 1/A+B+1/B=4,A+ 1/A-B-1/B=2 i κ.By using the relation (<ref>), they becomeα = √(8 π i λ)+⋯, β= i√(8 π i λ)+⋯, (|λ| ≪ 1) ,α = π√(2 λ̂)+ π/2 i -2i e^-π√(2 λ̂)+⋯, β= π√(2λ̂)- π/2 i+2i e^-π√(2 λ̂)+⋯, (|λ| ≫ 1),where λ̂≡λ- 1/24.§.§ Multi-cut solution Now we consider finding other solutions. Similar to the previous CS matrix models, various multi-cut solutions would exist in the ABJM matrix model too. The simplest solution at weak coupling is the superposition of the solutions of the pure CS matrix model (<ref>). To see this solution, we use the following ansatzμ_i= π i n +x_i(i=1, ⋯ , N/2), μ_N/2+j = -π i n +y_j(j=1, ⋯ , N/2),ν_i= π i m +z_i(i=1, ⋯ , N/2), ν_N/2+j = -π i m +w_j(j=1, ⋯ , N/2),where n and m are integers.When n ± m is even, the tanh interaction between {x_i, y_j} and {w_i, z_j} are negligibleat weak coupling, and the saddle point equation (<ref>) becomesn/2=2λ/N∑_ j ≠ i^N/21/x_i-x_j + 2λ/N∑_j =1^N/21/x_i-y_j, (i=1, ⋯ , N/2), -n/2=2λ/N∑_ j ≠ i^N/21/y_i-y_j + 2λ/N∑_j =1^N/21/y_i-x_j , (i=1, ⋯ , N/2), -m/2=2λ/N∑_ j ≠ i^N/21/z_i-z_j + 2λ/N∑_j =1^N/21/z_i-w_j, (i=1, ⋯ , N/2),m/2=2λ/N∑_ j ≠ i^N/21/w_i-w_j + 2λ/N∑_j =1^N/21/w_i-z_j , (i=1, ⋯ , N/2).These are merely two set of the saddle point equations for the two-cut solutions in the pure CS matrix model (<ref>). (λ→ - λ for {w_i, z_j}.) Thus the solution is given by (<ref>). See figure <ref> for the m=n=1 case. (The plot is a numerical result at N=100 and λ=10.)In addition to this solution, various multi-cut solutions would exist in the ABJM matrix model. However obtaining analytic solution is not simple generally. Hence we numerically solve the saddle point equations (<ref>) at finite N and λ through the method developed in Refs.<cit.> in order to find other multi-cut solutions. See figure <ref> for some examples. §.§ Connection to D2-brane instantonsIn the DMP solution at strong coupling, two kinds of non-perturbative effects are important: the world sheet instanton and the D2-brane instanton. Their instanton actions are given byworld sheet instanton: 2π√(λ), D2-brane instanton:π N √(2/λ).Remarkably, it was pointed outthat the instanton action of the D2-brane instanton is obtained through a sophisticated cycle integral of the spectral curve (<ref>) <cit.>.Here we argue the relation between the multi-cut solution and the D2-brane instanton. Among the multi-cut solutions we have discussed in the previous section, we focus on the one plotted in figure <ref> (right). This solution can be regarded as a deformation of the DMP solution (figure <ref> (left)). In this solution, the original cut [-α, α] of the DMP solution is divided into three cuts, and the left cut is roughly shifted by - 2 πi n and the right one is shifted by2 πi n, where n is a positive integer[ In our numerical analysis, we could not find the multi-cut solution with negative n. This may be the same mechanism that the cuts in the two-cut solution of the pure CS matrix model (<ref>) never appear on the lower right quadrant of the complex plane. Besides, to retain the relation (<ref>), we tune the shifted two cuts symmetric under μ↔ - μ.There may be asymmetric solutions also as in the pure CS matrix model case if we tune the cuts appropriately.].(Figure <ref> is for the n=1 case.) Since the coth interactions between {μ_j } in the saddle point equation (<ref>) have a periodicity μ→μ + 2πi, the interactions between the cuts are still significant even after the 2πi n shift, and such a multi-cut configuration can be a solution.To see the connection to the D2-brane instanton, we investigate the dynamics of the single eigenvalue, say μ_N, in the DMP solution.The effective potential for the eigenvalue μ_N is given by V_ eff (μ_N)=N/4π i λμ^2_N + V_ int (μ_N) -V_ eff(α), V_ int (μ_N)≡ -∑_j =1 ^N-1log[ 2sinhμ_N-μ_j/2]^2+∑_j=1^Nlog[ 2coshμ_N-ν_j/2]^2 .Here we fix {μ_i } (i ≠ N) and{ν_j }to be the DMP solution and we ignorethe back reaction[Note that ignoring the back reaction of μ_N may be subtle, since it breaks the relation (<ref>).But we can avoid this issue if we set another eigenvalue, say μ_1, as μ_1=-μ_N.This prescription makes the value of the effective action twice.]of μ_N. -V_ eff(α) in (<ref>) is added so that V_ eff (μ_N=α)=0, since we are interested in the deviation from the DMP solution.Suppose μ_N is shifted fromμ_N=α to α+ 2πi n [ One question is whether μ_N=α+ 2πi n is a solution of the saddle point equation for μ_N derived from the effective potential (<ref>) which is given by 0=N μ_N/2 π i λ +V'_ int(μ_N). Since we have assumed that μ_N= α is the DMP solution,it should satisfy0=N α/2 π i λ +V'_ int(α). However it immediately means that μ_N=α+2π i n is not a solution due to the periodicity V'_ int(μ +2πi n )=V'_ int(μ). (The same result can be seen through the equation y=0, where y is the curve (<ref>).) On the other hand, the multi-cut solutions in the numerical analysis indicates the existence of the solution near μ_N=α+2π i n. It implies that the back reaction to other eigenvalues are important to construct the solution. In this paper, we do not consider the back reaction and just evaluate the value of the effective potential (<ref>) atμ_N=α+2π i n, since the correction from the back reaction to the value of the effective potential may be higher order in the 1/N expansion.On the other hand, the back reaction may correct the total free energy. It would be an interesting future work to find the multi-cut solution analytically by taking into account the back reaction and evaluate the free energy. ]. Then this configuration can be regarded as a single point version of the multi-cut solution discussed above. In this case, the effective potential becomes V_ eff (α+ 2πi n)=N/4π i λ(α+ 2π i n )^2 + V_ int (α+ 2πi n) -V_ eff(α) = n Nα/λ- iN π n^2 /λ= nNπ√(2 /λ)+ ⋯,(|λ| ≫ 1).Here we have used the periodicity of the interaction V_ int (α+ 2πi n)=V_ int (α), and equation (<ref>). Therefore the value of the effective potential with n=1 precisely agrees with the instanton action of the D2-brane instanton at strong coupling |λ| ≫ 1 (<ref>). This quantitative agreement indicates that the multi-cut solution may be interpreted as a condensation of the D2-brane instantons. § DISCUSSIONSWe have revealed that the CS matrix models in the 't Hooft limit have remarkable properties and an infinite number of multi-cut solutions exit. These solutions might illuminate non-perturbative aspects of the CS theories and string theory. However, as mentioned in the introduction,we just evaluated the saddle point equations of the matrix models, and we should understand whether these solutions contribute to the path integral or not. On the other hand, we have seen some possible connections between the multi-cut solutions and other known non-perturbative objects: the singularities in the Borel plane in the CS matter theory <cit.> and the D2-brane instantons in the ABJM theory <cit.>. Hence some of the multi-cut solutions might be indeed physical. To develop understanding the role of the multi-cut solutions, it would be important to compare our results and other non-perturbative properties of the CS matrix models further.In the pure CS matrix model (<ref>), the following instanton factor has been obtained through an A-cycle integral <cit.>,N/2π i λ( 4π^2 λ n - 4 π^2 n m ),where n and m are integers which are concerned with the periodicity of coth in the saddle point equation (<ref>). Thus the periodicity of coth is crucial in both this instanton and our multi-cut solutions. However this instanton is obtained through the A-cycle integral, and the instanton factor (<ref>) may not be related to the free energy of the multi-cut solution.In the ABJM matrix model, so-called Fermi gas approach is quite powerful <cit.>.By using this technique, various important results on the membrane instantons have been obtained <cit.>. (See a review <cit.>.) Thus comparing these results with our multi-cut solutions may be important. However, in order to do this quantitatively, we need to develop our computation technique and obtain the analytic expressions for the multi-cut solutions.Another important direction of investigation is finding the gravity duals of the multi-cut solutions through the AdS/CFT correspondence. Lastly we argue possible instanton factors in the CS matrix models. In section <ref>, we have seen that the instanton factor of the D2-brane instanton at strong coupling is reproduced by evaluating the difference of the effective potential through the shift of the eigenvalue α→α + 2π i,where α is the end point of the cut. Therefore it might be possible to extractsimilar instanton factors in other CS matrix models through the 2 π i shift of the eigenvalue at the end point of the cut.Particularly, if the interaction terms have the periodicity u_i → u_i + 2π i as in the pure CS and ABJM case, the instanton factor arises from the classical action V_ classical(u) only, and it becomesV_ classical(α+ 2 πi )-V_ classical(α),where α is the end point of the cut in a suitable solution of the saddle point equation[This instanton factor differs from (<ref>) in the pure CS matrix model, since this factor is related to B-cycle whereas (<ref>) is related to A-cycle.]. Hence once we know the location of the end point of the cut, we may easily obtain the instanton factor[In the adjoint matter case (<ref>),the term ∑_i,j l(1-h+i (u_i-u_j)/2 π ) induces the additional cuts at 2π i (± h+n) (n ∈𝐑). Thus there might be instantons corresponding to the shift of the eigenvalue α→α + 2π i (± h+n).]. It may be valuable to explore these instantons in the CS matrix models.Acknowledgements The authors would like to thank Yasuyuki Hatsuda, Masazumi Honda, Kazumi Okuyama, Takao Suyama and Asato Tsuchiya for valuable discussions and comments. The work of T. M. is supported in part by Grant-in-Aid for Scientific Research (No. 15K17643) from JSPS.§ EXACT TWO-CUT SOLUTION AT FINITE Λ IN PURE CS MATRIX MODELWe calculate thetwo-cut solutions in the pure CS matrix model at large N with a finite λ. In section <ref>, we assumed that each cut is composed by N/2 eigenvalues. Here we relax this condition and consider the two cuts composed by N_1 and N_2=N-N_1 eigenvalues. (See figure <ref> (center).) By summing the saddle point equation (<ref>), we obtain a relation∑_i=1^N u_i=0.By regarding this equation, we use the following ansatz for the two-cut solutionu_i=2 π i nN_2/N +x_i , (i=1, ⋯ , N_1), u_N_1+j= - 2 π i nN_1/N+y_j , (j=1, ⋯ , N_2),where n is an integer. We can choose this integer n positive without loss of generality. By substituting this ansatz into the saddle point equation (<ref>), we obtainN_2/N n+1/2π i x_i =λ/N∑_ j ≠ i^N_1x_i-x_j/2 + λ/N∑_j =1^N_2x_i-y_j/2,(i=1, ⋯ , N_1), -N_1/Nn+1/2π i y_i =λ/N∑_ j ≠ i^N_2y_i-y_j/2 + λ/N∑_j =1^N_1y_i-x_j/2 , (i=1, ⋯ , N_2).In order to solve these equations, we use the following assumptionRe( y_j )≤ Re( z_0 ) ≤ Re( x_i ),where z_0 is a complex number representing the boundary of { x_i } and { y_j }. This assumption is the same to the assumption (<ref>) if z_0=0. Since the system is asymmetric if N_1 ≠ N_2, the boundary z_0 may be non-zero in general. Thanks to this assumption, we can combine the saddle point equations (<ref>) into the single equation by using the variable z_i = x_i for 1 ≤ i ≤ N_1 and z_i+N_1 = y_i for 1 ≤ i ≤ N_2,2n θ( Re( z -z_0 )) - N_1/N 2n + 1/π i z_i=2 λ/N∑_ j ≠ i^Nz_i-z_j/2.If the first two terms on the left hand side did not exist, the problem is solving the saddle point equation in the original pure CS matrix model (<ref>). Hence we can apply the technique developed there <cit.>. We define new variable Z_i ≡ e^z_i and Z_0 ≡ e^z_0, and rewrite equation (<ref>) as U'(Z_i) =2 λ/N∑_ j ≠ i^NZ_i+Z_j/Z_i-Z_j,U'(Z) ≡ 2n θ( |Z| - |Z_0|) - N_1/N 2n + 1/π ilog Z .Now we introduce the eigenvalue density ρ(Z) and resolvent v(Z) asρ(Z) ≡1/N∑_i^N δ(Z-Z_i) ,v(Z)=∫_A^B dZ' ρ(Z') Z+Z'/Z-Z',where A and B are the locations of the end points of the eigenvalue distribution, which satisfy 0<|A| ≤ |Z_0| and |Z_0| ≤ |B| because of the assumption (<ref>). By using this resolvent, we rewrite the saddle point equation (<ref>) asU'(Z)= λlim_ϵ→ 0( v(Z+ i ϵ)+v(Z- i ϵ) ).Therefore the equations which we want to solve are similar to those of the Hermitian matrix model.The difference is only the boundary condition of the resolvent. From the definition of the resolvent (<ref>), it should satisfyv(Z) → 1(Z →∞),v(Z) → -1(Z → 0).Note that the eigenvalue density is now derived as ρ(Z) =-1/4π i Zlim_ϵ→ 0[ v(Z+iϵ)-v(Z-iϵ) ] , (Z ∈ [A,B] ) ,through (<ref>).As in the analysis at the weak coupling, we apply the “one-cut" ansatz to the resolvent,v(Z)= 1/λ∫_C_1dZ'/4π iU'(Z')/Z-Z'√((Z-A)(Z-B)/(Z'-A)(Z'-B))= 1/2π i λ∫_Z_0^B dZ' 2n/Z-Z'√((Z-A)(Z-B)/(Z'-A)(Z'-B))-1/4π i λ∫_C_1 dZ' N_1/N 2n/Z-Z'√((Z-A)(Z-B)/(Z'-A)(Z'-B))+1/4π i λ1/π i∫_C_1 dZ' logZ'/Z-Z'√((Z-A)(Z-B)/(Z'-A)(Z'-B)),where the integral contour C_1goes around the cut:[A,B] in a counter-clockwise way. Then we can calculate each integrals as∫_Z_0^B dZ' 1/Z-Z'√((Z-A)(Z-B)/(Z'-A)(Z'-B)) =2i arctan( √(Z-A/Z-B)√(B-Z_0/Z_0-A)),∫_C_1 dZ' 1/Z-Z'√((Z-A)(Z-B)/(Z'-A)(Z'-B)) =2π i ,∫_C_1 dZ' logZ'/Z-Z'√((Z-A)(Z-B)/(Z'-A)(Z'-B)) = 4π i log( Z+√(AB)-√((Z-A)(Z-B))/√(A)+√(B)).Hence the resolvent becomesv(Z) = 2n/πλarctan( √(Z-A/Z-B)√(B-Z_0/Z_0-A)) -n/λN_1/N+1/π i λlog( Z+√(AB)-√((Z-A)(Z-B))/√(A)+√(B)).We can confirm that this resolvent satisfies the equation (<ref>) as in the weak coupling case. Now we determine the constant A, B and Z_0. Firstly, the solution (<ref>) has to satisfy the boundary condition (<ref>) which is evaluated as2narctan( √(B-Z_0/Z_0-A)) - π n N_1/N -i log( √(A)+√(B)/2)=πλ , 2n arctan(√(A/B)√(B-Z_0/Z_0-A))- π n N_1/N+ ilog( √(A)+√(B)/2 √(AB))=-πλ .In addition, the numbers of the eigenvalues on the cut [Z_0,B] and[A,Z_0] should be N_1 and N_2 respectively. Through the expression of ρ(Z) (<ref>), this condition becomesN_1/N = ∫_Z_0^B dZ ρ(Z) = 1/4π i∫_C_2 dZ v(Z)/Z,where the integral contour C_2goes around the cut:[Z_0,B] in a counter-clockwise way[Through (<ref>), ρ(Z) on the cut:[Z_0,B] becomesρ(Z)= 1/2 π^2λ Z( n log[1 ±√(Z-A/B-Z)√(B-Z_0/Z_0-A)/1 ∓√(Z-A/B-Z)√(B-Z_0/Z_0-A)] -log( Z+√(AB)+i √((Z-A)(B-Z))/√(A)+√(B)) + 1/2log Z).Here we have used the relation( Z+√(AB)+i √((Z-A)(B-Z))/√(A)+√(B)) (Z+√(AB)-i √((Z-A)(B-Z))/√(A)+√(B)) =Z . ]. Note that the condition for the cut:[A,Z_0]is automatically satisfied if these conditions are satisfied. From these conditions, we can fix the constant A, B and Z_0 at least numerically. We plot the result for the N_1:N_2=2:3 and n=1 case in figure <ref>. §.§ Weak coupling analysis At weak coupling, we can solve equation (<ref>) and (<ref>) perturbatively in λ. At the leading order, the boundary condition (<ref>) can be solvedb =z_0+2πλ/ntan( π/2N_1/N) +O(λ^2),a =z_0-2πλ/ntan( π/2N_2/N)+O(λ^2),where a ≡log A and b ≡log B. Now we need to fix z_0 by solving (<ref>). Through equation (<ref>) and (<ref>), equation (<ref>) becomes the following relation∫_a^b dzz ρ(z)=0. We can perturbatively solve this equation, and obtainz_0= -λπ/2n( tan( π/2N_1/N)-tan( π/2N_2/N)) +O(λ^2).Therefore, if, for example, N_1<N_2, z_0 becomes positive as shown in figure <ref> (center). This is consistent with the numerical result shown in figure <ref>.§.§ N_1=N_2 at finite λIn the case of N_1=N_2, due to the symmetry z ↔ -z, Z_0=1 and A=1/B would be satisfied. (They imply z_0=0 and a=-b in the original z variable.) Then the condition (<ref>) and (<ref>) reduce to ( i-√(B)/i+√(B))^n √(B)+1/√(B)/2 = exp(π i (λ +n/2)).Here we haveused the relation arctan x = i/2log( i+x/i-x). We can analytically solve this equation for n=1,2,3 and 4.For example, the solution for the n=1 case is given by√(B) = -i ( e^π i λ-1 ) + √(2e^π i λ-e^2π i λ).Here we have chosen the solution which satisfies |B|>|Z_0|=1. JHEP | http://arxiv.org/abs/1704.08675v2 | {
"authors": [
"Takeshi Morita",
"Kento Sugiyama"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170427173927",
"title": "Multi-cut Solutions in Chern-Simons Matrix Models"
} |
Anomaly-free local horizontal symmetry and anomaly-full rare B-decays Tsutomu T. Yanagida December 30, 2023 =====================================================================empty empty Recent regulatory changes proposed by the Federal Communications Commission (FCC) permitting unlicensed use of television white space (TVWS) channels present new opportunities for designing wireless networks that make efficient use of this spectrum. The favorable propagation characteristics of these channels and their widespread availability, especially in rural areas, make them well-suited for providing broadband services in sparsely populated regions where economic factors hinder deployment of such services on licensed spectrum.In this context, this paper explores the deployment of an outdoor Wi-Fi-like network operating in TVWS channels, referred to commonly as a Super Wi-Fi network. Since regulations governing unlicensed use of these channels allow (a) mounting fixed devices up to a height of 30 m and operation at transmit powers of up to 4 W EIRP, and (b) operation at transmit powers of up to 100 mW EIRP for portable devices, such networks can provide extended coverage and higher rates than traditional Wi-Fi networks. However, these gains are subject to the viability of the uplink from the portable devices (clients) to the fixed devices (access points (AP)) because of tighter restrictions on transmit power of clients compared to APs. This paper leverages concepts from stochastic geometry to study the performance of such networks with specific focus on the effect of (a) transmit power asymmetry between APs and clients and its impact on uplink viability and coverage, and (b) the interplay between height and transmit power of APs in determining the network throughput. Such an analysis reveals that (a) maximum coverage of no more than 700 m is obtained even when APs are deployed at 30 m height, and (b) operating APs at transmit power of more than 1 W is beneficial only at sparse deployment densities when rate is prioritized over coverage. § INTRODUCTIONIn light of rapidly growing mobile broadband traffic, providing additional spectrum is an important policy goal for spectrum regulators worldwide. With Internet of Things (IoT) and Machine to Machine (M2M) communications rising up the horizon, there is a need for ubiquitous connectivity. Meeting these requirements, especially in rural areas, is challenging because of geographic and monetary constraints. In this context, this paper investigates the viability of deploying an outdoor Wi-Fi-like network using television white space (TVWS) channels in rural/suburban areas for providing broadband connectivity. Such a network is typically referred to as a Super Wi-Fi network <cit.>. TVWS channels are unused TV channels that can be opportunistically used on a secondary basis in the absence of primary transmissions. In the USA, these channels are 6 MHz wide and span from 54 MHz to 698 MHz. In particular, the channels in the 512-698 MHz range allow the secondary devices to be either fixed or portable, as in Table <ref> <cit.>. As seen in Fig. <ref>, TVWS channels are known to be locally under-utilized, especially in rural/suburban areas. Further, the relatively low frequency of TVWS channels comes with some significant advantages like lower path loss and better wall penetration <cit.>. The under-utilization of TVWS channels in rural areas, along with their favorable propagation characteristics, motivates investigating the feasibility of deploying outdoor Super Wi-Fi networksfor broadband connectivity in such regions where providing access solutions continues to be exorbitantly expensive. In particular, this work envisions deploying a large number of fixed wireless access points (APs) over a rural/suburban area with channel access mediated by carrier sense multiple access with collision avoidance (CSMA/CA) while adhering to TVWS regulations.This work complements recent interest in utilizing TVWS channels for providing backhaul solutions <cit.> where a wide-area network of cellular base stations over TVWS channels for backhaul is studied. To mitigate any potential impact on the primary users, secondary usage of TVWS channels is regulated by the Federal Communications Commission (FCC) in the USA. In particular, Table <ref> specifies two important regulations that significantly impact the operation of a secondary Wi-Fi-like network—the first governs the maximum height of secondary devices and the second governs the maximum transmit power. In particular, the regulations allow mounting a fixed device at a height of up to 30 m while also allowing it to operate at up to 4 W EIRP (Equivalent Isotropically Radiated Power)—thus permitting a much larger coverage area when compared to a typical Wi-Fi AP that is mounted at much lower heights and is restricted to transmit at no more than 100 mW EIRP. However, since the portable devices are restricted to a much lower transmit power of 100 mW, an increase in downlink coverage from AP to client is not reciprocated by an equivalent increase in uplink coverage. In other words, there may be scenarios where the uplink may not be viable even though the downlink is. This presents a major point of difference between the network under consideration and traditional Wi-Fi networks.Given the crucial role played by uplink association request and acknowledgment packets in determining AP-client association and successful downlink transmissions, the significant difference in operating parameters between APs and clients makes it extremely important to factor in uplink viability. Additionally, the potentially large downlink coverage can have a detrimental impact on the AP transmission probabilities when using CSMA/CA. Thus, although an increase in AP transmit power and/or height might seem beneficial, the above issues highlight the difficulty in choosing the right set of operating parameters for striking the right balance between coverage and throughput in such networks. With the broad goal of understanding the performance of large-scale outdoor Super Wi-Fi networks, this work uses concepts from stochastic geometry to obtain an analytical characterization of SINR coverage and rates. Tools from stochastic geometry are used to first characterize the probability of transmission of a Super Wi-Fi AP and subsequently study the area spectral efficiency (ASE) of such a network as a function of (a) AP deployment density, (b) AP height, and (c) AP transmission power. Due to the unequal transmission powers between APs and clients, this analysis explicitly requires the uplink to be viable when computing the downlink throughput. It is primarily in this respect that the current work significantly differs from existing literature on analyzing such networks <cit.>. The results of this work show that it is not always beneficial to operate at high AP transmit power (P_AP) and AP height (h_AP). It is consistently observed across different deployment densities that operating at high P_AP and h_AP values leads to a sharp drop in probability of transmission for APs, in turn decreasing the ASE. At deployment densities of less than 1AP/km^2, ASE is maximized when P_AP and h_AP are close to 1 W and 1.5 m, respectively. At higher deployment densities of 10 APs/km^2, ASE is maximized when P_AP and h_AP are close to 0.1 W and 1.5 m, respectively. On the contrary, for optimal coverage, maximizing h_AP proves beneficial (note that coverage is determined by uplink viability and hence is independent of P_AP). Setting P_AP greater than 1 W is observed to be useful only in sparse deployment densities, when rate is prioritized over coverage, with APs mounted at less than 6 m height. The rest of the paper is organized as follows — related work is presented in Section <ref>, the system model and parameters involved in the analysis are described in Section <ref>, characterization of the network throughput is given in Section <ref>, a discussion of the results obtained is provided in Section IV and the concluding remarks are mentioned in Section <ref>.§ RELATED WORKThe release of TVWS channels for unlicensed use promoted active research in investigating the feasibility of deploying cognitive radio networks in these channels <cit.>. The possibility of using TVWS channels for Super Wi-Fi operation is explored extensively in <cit.>. The authors of <cit.> propose to enhance the coverage of public Wi-Fi networks operating in 2.4 GHz by extending their operation to TVWS channels. The authors of <cit.> built a prototype for Super Wi-Fi (called White-Fi) networking and modified the medium access control (MAC) protocol to factor in spatial and temporal variations of TVWS channels. In <cit.>, a quantitative study of Super Wi-Fi networks is provided and it is observed that TVWS is an attractive alternative for providing connectivity in outdoor rural areas. A real world deployment of Super Wi-Fi networks is presented in <cit.> where the potential of using TVWS for bringing broadband connectivity to unconnected areas is established. The issue of transmit power asymmetry in TVWS networks has been studied in <cit.>, albeit in a vehicular connectivity set up, in which the authors propose to extend the range of uplink from clients by using existing cellular paths. However, in the current work, rural areas are the target environments and hence no form of connectivity is likely to be preexistent. While these efforts provide the motivation to better utilize this technically and economically significant frequency range, a theoretical study of the performance of a large scale Super Wi-Fi network is not available to the best of our knowledge. Additionally, such an analysis should consider the various regulatory constraints imposed on the operating parameters, specifically transmission powers and heights, as described in Table <ref>. The current work provides a theoretical framework to serve this purpose by employing concepts from stochastic geometry. Among the first efforts to theoretically analyze traditional Wi-Fi networks were the analyses presented in <cit.> and <cit.> to accurately model the 802.11 protocol. While these efforts captured finer aspects of the CSMA/CA protocol (e.g., exponential backoff), spatial aspects of the wireless medium are not modeled. Stochastic geometry provides a natural framework to analyze wireless networks while retaining the spatial characteristics of signal propagation. The use of stochastic geometry for modeling and analyzing wireless networks started with the extensive analysis of ALOHA <cit.>. In particular, <cit.> studies CSMA-based networks using a Matern hard-core point processes where each AP was assumed to have a disc of fixed radius around itself within which there are no other APs. A modification to this analysis that modeled the backoff procedure in CSMA/CA and included fading was presented in <cit.>. The basic framework of <cit.> to analyze CSMA/CA forms the foundation for the current effort. Subsequent analysis of interference due to concurrent AP transmissions is modeled using the methodology proposed in <cit.>.A comprehensive overview of using stochastic geometry to model a wide variety of wireless networks is given in <cit.>. The mathematical tools and theory of point processes used in the current analysis are presented in <cit.> and <cit.>.§ SYSTEM MODELConsider a large set of APs whose locations are fixed and drawn from a homogeneous Poisson point process (PPP) of intensity λ. The set of AP locations is given by Φ_A = {x⃗_1, x⃗_2, ..., x⃗_k, ...}. The notation ||(x⃗_i-x⃗_j)|| is used to represent distances between APs at two locations x⃗_i and x⃗_j. The APs are assumed to only serve clients located within its Voronoi cell (provided the uplink from client to AP is viable), with client locations being uniformly distributed within the Voronoi cell. In such a setting, the distribution of AP-Client distance r (without factoring uplink viability) is given by f_r(r) =2πλre^-λπr^2.It is assumed that all APs have at least one associated client to serve at any instance. The APs have access to one TVWS channel (6 MHz wide) and all APs contend to get access to this channel. APs are bound by TVWS regulations that govern fixed devices while the clients are bound by the regulations governing portable devices. Thus, while APs can transmit at any transmit power P_AP≤4W, the clients are assumed to transmit at P_C = 0.1W. The APs are also assumed to be mounted at any height h_AP≤ 30m, while clients are always assumed to be at 1 m height. Isotropic antennas are assumed at both APs and clients. The analysis in this paper assumes a persistent downlink traffic and negligible uplink traffic.§.§ Radio Propagation Model The power received at a point y⃗ from an AP located at x⃗ is given byP(x⃗,y⃗) = P_AP ρ(x⃗,y⃗) F(x⃗,y⃗), where ρ(x⃗,y⃗) is the pathloss encountered by the transmission between x⃗ and y⃗, and F(x⃗,y⃗) is the fading coefficient between x⃗ and y⃗. F(x⃗,y⃗) is modeled as an i.i.d. exponential random variable with mean μ = 1. The notation P(d), ρ(d) and F(d) are used when referring to received power, pathloss and fading coefficient between two generic locations that are at a distance `d' from each other. Two different pathloss models are used to define AP-AP and AP-Client transmission links. The two pathloss models are drawn from the dual-slope model specified in <cit.> for suburban environments. This particular pathloss model is chosen as it is sensitive to transmitter and receiver heights and is applicable to a wide range of sub-GHz frequencies. The dual-slope model is given by ρ(x⃗,y⃗)in dB =ρ_LOS + 20 + 25 log(dR_bp), if d < R_bp ρ_LOS + 20 + 40 log(dR_bp), if d ≥ R_bpwhere* d is the distance between x⃗ and y⃗,* ρ_LOS is the line-of-sight pathloss (in dB), given by ρ_LOS = | 20 log(λ^2/8π h_t h_r)|,* R_bp is the breakpoint distance (in meters), given byR_bp = 1/λ√((Σ^2 - Δ^2)^2 - 2(Σ^2 + Δ^2)(λ/2)^2 + (λ/2)^4). The different parameters involved in the above model are* λ - wavelength (m),* h_t - height of transmitting antenna (m), * h_r - height of receiving antenna (m),* Σ = h_t + h_r,* Δ = h_t - h_r.For modeling AP-AP transmissions, h_t and h_r are set to AP antenna height h_AP. For modeling AP-client transmissions, h_t = h_AP and h_r = 1 m.§.§ Channel Contention Model Channel access in the current network is governed by CSMA/CA. In CSMA/CA, an AP gets access to a channel when there are no other contending APs in its neighborhood (i.e., the channel is sensed to be idle), otherwise an exponential back-off procedure is initiated. The channel is sensed to be idle when the received signal strength from all neighboring APs is below the clear-channel-assessment (CCA) threshold. In conventional Wi-Fi networks, the CCA threshold is typically set to -82 dBm and the same threshold is used in the current work. §.§ Uplink Viability Uplink viability determines the ability of a client to associate with a neighboring AP. If the association request messages from the client do not reach an AP, the client cannot be served. This scenario may often occur in Super Wi-Fi networks as clients transmit at powers lower than APs. Note that even though uplink viability also affects the receipt of acknowledgment (ACK) packets from the client indicating successful downlink transmission, the assumption in the current work is that once association is established between an AP-client pair, the channel remains time invariant. This paper defines uplink viability as follows.The uplink transmission between client y⃗ and its AP x⃗ is said to be viable if the received signal strength from the client to the AP exceeds a certain threshold γ, i.e.,P_C ρ(x⃗,y⃗) G(x⃗,y⃗) > γ. Note that channel reciprocity is not assumed and hence G(x⃗,y⃗) and F(x⃗,y⃗) are two independent random variables. In this paper, the threshold γ is set to be equal to the CCA threshold σ. Although the criterion (<ref>) only accounts for uplink packet detection and not successful decoding, it simplifies the subsequent analysis while retaining the essential characteristics of the network under consideration.Using the above definition, uplink viability is computed as p_U(r) = ℙ[P_C ρ(x⃗,y⃗)G(x⃗,y⃗) > γ] = e^-μγ/P_Cρ(x⃗,y⃗). where r = ||x⃗ - y⃗|| is the distance between AP x⃗ and client y⃗. In particular, coverage range of an AP is defined as the largest AP-client distance d such that p_U(d) ≥ 0.1where p_U(r) is defined in (<ref>). §.§ Transmission ModelThe signal to interference-plus-noise ratio (SINR) observed at a client y⃗ and associated with an AP x⃗ is given by SINR(x⃗,y⃗) = ρ(x⃗,y⃗)/N_0 + ∑_z⃗∈Φ_T∖x⃗I(z⃗,y⃗)where* N_0 is the noise variance,* Φ_T is the set of concurrently transmitting APs,* ∑_z⃗∈Φ_T∖x⃗I(z⃗,y⃗) is the cumulative interference at client y⃗ due to all concurrently transmitting APs except AP x⃗. The transmitted data rate from AP x⃗ to client y⃗ is then given by log_2(1+SINR(x⃗,y⃗)). § THROUGHPUT MODELING AND ANALYSIS This section focuses on characterizing the performance of the network under consideration through metrics such as AP transmission probability and area spectral efficiency. Area spectral efficiency is defined as the average throughput of an AP multiplied by the density of the AP deployment. It is assumed that all APs actively contend for the channel and when channel access is granted, use the channel for a fixed period of time to transmit to one of their associated clients. For an AP to serve a client the following three transmissions must be successfully received: (a) association request packets at the AP, (b) transmission payload packets at the client, and (c) acknowledgment packets at the AP. Clearly, (a) and (c)both require uplink viability, and it is assumed that as long as the uplink received signal strength exceeds the threshold γ, both these transmissions are successful. Under these assumptions, the probability of an AP serving a user at a distance r is equal to the probability of an uplink viable client being located at a distance r and is given byf_R(r|I_u=1)=f_R(r)ℙ(I_u=1|R=r)/ℙ(I_u=1),where I_u is a binary random variable indicating uplink viability. In particular, I_u = 1(P_C ρ(r) G(r) > γ), and ℙ(I_u=1|R=r)=ℙ (G(r) > γP_C ρ(r) )=e^-μγ/P_c ρ(r). Given the random deployment of APs and clients, the average throughput is computed over all possible AP and and client locations. In particular, when the AP has channel access and serves an uplink-viable client located at a distance r, the average throughput to that client is given by 𝒯(r)=𝔼_(SINR|R=r)[log(1+SINR)]where the expectation is over the distribution of SINR conditioned on the client being at a distance of r from the AP. Note that the SINR distribution is independent of uplink viability.Thus, the average throughput of an AP after accounting for the probability of transmission can be written as𝒯 = ∫_0^∞p_T(r)𝒯(r) f_R(r|I_u=1)drwhere p_T(r) is the transmission probability of the AP conditioned on serving a client that is at a distance of r. Just as the SINR distribution, p_T(r) is also independent of uplink viability.The rest of the section focuses on computing p_T(r) and 𝒯(r). The methodology adopted is similar to the framework presented in <cit.>. §.§ Probability of AP Transmission: p_T(r) Probability of an AP transmitting is governed by CSMA/CA and the exponential backoff procedure. As proposed in <cit.>, the exponential backoff procedure used by an AP when the channel is busy can be approximately modeled by tagging each AP in the Poisson field with an independent mark. This mark decides the backoff time of that AP. In particular, each AP x⃗ in Φ_A is assigned an independent mark m_x⃗ uniformly distributed in [0,1]. Defining the neighborhood of an AP x⃗ as 𝒩(x⃗) = {y⃗∈Φ_A : P(x⃗,y⃗) > σ}, an AP transmits if no other AP in its neighborhood has a smaller mark than itself. Thus, the set of concurrently transmitting APs can now be defined asΦ_T = {x⃗∈Φ_A : m_x⃗ < m_y⃗, ∀ y⃗∈𝒩(x⃗) }.This model captures the fact that CSMA/CA grants channel access to that AP with minimal back-off time (equivalent to having lowest mark) among all APs in its neighborhood and that an AP abstains from transmitting if another AP in its neighborhood is already transmitting.Note that this approximate model ignores collisions, the exponential nature of back-off, and the history of timers. Nevertheless, as shown by the authors in <cit.>, through ns-2 simulations, this model provides fairly accurate results. Without loss of generality, we focus on an AP located at the origin and denoted as AP 0⃗. Let AP 0⃗ serve a client y⃗ located at a distancer. Computing the transmission probability is equivalent to computing the probability that among the APs with a mark less that m_0⃗, none of them are in the neighborhood, i.e.,p_T(r)=ℙ (m_x⃗≥ m_0⃗ ∀ x⃗∈𝒩(0⃗)).It can be shown using the results in<cit.> that the transmission probability, as defined above, can be computed asp_T(r) = ∫_0^1 e^-λ m_0∫_ℝ^2∖ℬ(y⃗,r)S(x⃗) dx⃗ dm_0 = 1-e^-λ∫_ℝ^2∖ℬ(y⃗,r)S(x⃗)dx⃗/λ∫_ℝ^2∖ℬ(y⃗,r)S(x⃗)dx⃗where ℬ(y⃗,r) is a ball of radius r with the client at its center (which by hypothesis cannot contain any AP other than AP 0⃗) and S(x⃗) is the probability of AP 0⃗ detecting an AP at x⃗. S(x⃗) can be computed asS(x⃗) = ℙ[P_AP ρ(0⃗,x⃗) F(0⃗,x⃗) > σ] = e^-μσ/P_AP ρ(||x||) The expression in (<ref>) can be computed in a straightforward manner using standard numerical techniques. §.§ Average Throughput to a Client: 𝒯(r)Computing the average throughput delivered by an AP to its associated client at a distance r requires determining the distribution of SINR at the client. This in turn requires the computation of the cumulative interference caused at the client due to all other APs concurrently transmitting with AP 0⃗. To compute the SINR distribution, the methodology used in <cit.> is adopted. In particular, the complementary cumulative distribution function (CCDF) of SINR can be expressed using Laplace functionals and written asℙ(SINR(r)> β) = ψ_I(s)ψ_N(s),where s = μβ/P_APρ(r), and ψ_I(·) and ψ_N(·) are the Laplace functionals of the interference from other AP transmissions and additive noise, respectively.Switching to a user-centric perspective by shifting the origin to the location of client and assuming AP 0⃗ to now be located at (r,0) (in polar coordinates), (<ref>) can be approximated asℙ(SINR> β)≈ e^-sN_0e^-λ∫_0^2π∫_r^∞q(b(v,θ))[1-ϕ_F(sρ(v))]v dv dθ where s is as before, and * q(d) is the probability that two APs separated by a distance d transmit concurrently (computation of q(d) is given in Appendix I),* b(v,θ) = v^2 + r^2 - rv cos(θ) is the distance between the serving AP at (r,0) and a generic interfering AP located at (v,θ), * ϕ_F - Laplace transform of the fading random variable ϕ_F(x) = 1/1+x as fading is exponentially distributed, * N_0 - Thermal noise variance. Evaluating (<ref>) at s = μβ/P_APρ(r) using standard numerical techniques yields the distribution of SINR at the client. Once the distribution of SINR is obtained, the expected rate delivered by the AP can be computed as 𝔼[log(1+SINR)]) where the expectation is computed over the distribution of SINR. § NUMERICAL RESULTS AND DISCUSSIONThis section presents the results obtained using the methodology outlined earlier and highlights key takeaways on the design of Super Wi-Fi networks. Some of the key parameters used in the computations are given in Table <ref>. §.§ Validation of Methodology To validate the proposed stochastic-geometry-based model of Super Wi-Fi networks, results obtained using such an approach are compared against simulation results obtained using OPNET, an industry-standard packet-based network simulation tool <cit.>.In particular, due to computational complexity of large scale simulations in OPNET and lack of in-built support for rate adaptivity, OPNET is used to simulate the performance of a network with a single AP and a single client. This result is then compared against results obtained using the stochastic-geometry-based model in a sparse deployment setup where effects of channel contention are minimized and the AP transmissions can be assumed to be independent of each other. For the OPNET simulation h_AP and P_AP are set to 30 m and 0.1 W or 1 W, respectively. Due to lack of support for rate adaptivity, modulation-and-coding-scheme (MCS) index was varied manually to identify the best index for a given setup. While the pathloss model used in OPNET is different from the one listed Table III, the simulation tool is used as a means of validating the stochastic geometry analysis developed in this paper. Specifically, OPNET uses the Suburban Hata model, where the pathloss after substitution of h_AP = 30 m is given by ρ(d)= 124.3 + 35.23log(d), where d is in km. For comparison, the stochastic-geometry-based model is also set to use the Suburban Hata model with AP density set to 0.1 AP/km^2. Fig. <ref> shows the plots of AP throughput (in Mbps) per channel as a function of AP-client distance (in m) for three cases - (i) OPNET simulation, (ii) stochastic-geometry model when uplink viability is factored in, and (iii) stochastic-geometry model under the assumption that uplink is always viable. Note that in OPNET fading is not modeled, which is evident from the abrupt fall in throughput at ∼ 550 m in Fig. <ref>. It is seen from Fig. <ref> that the current model closely follows the throughput obtained using OPNET. Further, even though uplink and downlink powers are the same in this case, factoring uplink viability produces more accurate results when compared to the existing models for Wi-Fi networks. The impact of uplink viability is even more pronounced in Fig. <ref>. Note that the proposed model and the OPNET results both indicate that clients beyond 550 m are incapable of being served. At shorter distances, while restrictions on MCS indices cap the maximum throughput in OPNET, no such restriction is placed on the proposed model. Due to lack of fading in OPNET simulations, OPNET predicts larger throughputs in the 400 m to 550 m range than the proposed model. It is clear that models that do not factor in uplink viability are particularly inaccurate at larger distances and predict much larger coverage than is realistically possible. These results along with the comparison in <cit.> against ns-2 simulations further validate this model. §.§ Results on Throughput and Coverage Analysis The section presents the projected performance of Super Wi-Fi networks from the perspective of (a) transmission probability, (b) coverage of an AP, and (c) area spectral efficiency (network throughput). A well-designed Super Wi-Fi network requires striking the right balance between all of the above three attributes.§.§.§ Transmission Probability This section examines the impact of AP transmit power (P_AP) and height (h_AP) on AP transmission probabilities at various deployment densities. Fig. <ref> plots average p_T as a function of h_AP for three different densities. The average is computed over all AP-client distances r and is given byp̅_T = 𝔼_r[p_T(r)] = ∫_0^∞p_T(r) f_R(r|I_u=1) dr.Fig. <ref> illustrates the interplay betweenP_AP and h_AP in deciding how often an AP transmits and as expected, transmission probabilities decrease with increasing density. Note that at densities ≤ 1 AP/km^2 and low antenna heights (1.5 to 3 m), transmit power does not play a significant role in determining the transmission probability. Further, it is seen that at higher transmission powers, p̅_T drops significantly as h_AP increases. For instance, at an AP deployment density of 1 AP/km^2, when P_AP = 4 W, an AP transmits with a probability of ≈ 0.85 when operated at a height of 1.5 m but this probability drops below 0.1 when operated at a height of 15 m. In fact, at high densities (≥ 10 APs/km^2), operating at any height above 3 m does not seem optimal. On the other hand, given a target p̅_T and deployment density, and multiple (P_AP, h_AP) pairs that meet the target p̅_T, choosing the pair with the highest AP height is advisable, as increasing AP height benefits both uplink and downlink, while increasing P_AP only aids downlink, leading to greater asymmetry. Thus, when designing Super Wi-Fi networks, careful consideration must be given to the choice of P_AP and h_AP, with particular attention paid to uplink-downlink asymmetry.§.§.§ Coverage Analysis Unlike existing work on characterizing coverage by computing the probability that SINR exceeds a given threshold, this work defines coverage via uplink viability and determines a client to be in coverage if the uplink packets are received above the CCA threshold. When defined in this manner, coverage becomes independent of downlink transmit power. This alternate definition is particularly appropriate for Super Wi-Fi networks where clients are restricted to transmit at 0.1 W, but APs are allowed to transmit up to 4 W. Fig. <ref> plots uplink viability p_U as a function of AP-client distance, assuming clients to transmit at 0.1 W. It can be observed that even at an AP height of 30 m, uplink is no longer viable beyond 700 m. In fact, for users who are 400 to 700 m away, uplink is viable less than 60% of times—further restricting the range of an AP if reliable transmission is desired. This coverage limitation leads to large coverage gaps in a sparse deployment, as seen in Fig. <ref>. Assuming clients to be uniformly distributed, Fig. <ref> shows that at deployment densities of 1 AP/km^2, over 50% of clients cannot associate with any AP. Thus, Super Wi-Fi networks with deployment densities less than 1 AP/km^2 are only able to provide localized coverage while densities greater than 10 APs/km^2 are required to ensure pervasive coverage over a wide area.This observation is illustrated in Figs. <ref> and <ref>. Fig. <ref> shows coverage and downlink range (defined analogous to coverage range; determines channel contention radius) in a Super Wi-Fi network with λ=0.1 AP/km^2. APs are assumed to be at a height of 30 m and transmit at 4 W. It is clear that at such densities only localized coverage is possible. Further, the figure also illustrates why operating at P_AP = 4 W is not optimal even at such low AP deployment densities. High AP transmit powers lead to unnecessary enlargement of the contention radius, causing a drop in p̅_T and a potential decrease in networkperformance (as seen in the next subsection).Fig. <ref> represents a Super Wi-Fi network with λ=10 APs/km^2, that is capable of providing pervasive coverage. Parameters h_AP and P_AP are set to 1 W and 10 m, respectively. At this height more than 80% of the area is covered. Once again it is seen that higher transmit powers lead to a large contention radius that can be detrimental to network performance. §.§.§ Area Spectral EfficiencyThroughput 𝒯 of an AP is as defined in (<ref>). Area spectral efficiency is the product 𝒯λ, and reflects the total number of bits transmitted over the wireless medium in a given area.The following discussion is split into three cases, depending on the deployment density.Fig. <ref>, plots performance of a very sparse deployment with a density of 0.1 AP/km^2. At such densities, only localized coverage is possible and for each curve in Fig. <ref>, the coverage characteristics remain the same. It is seen that for a fixed transmit power, ASE increases with decreasing height, owing to reduced coverage area, thereby serving only those clients who are at a very close proximity to the AP. On the other hand, if h_AP is held fixed, then ASE increases with transmit power at lower heights, but decreases when h_AP exceeds 9 to 10 m. This observation can be attributed to the fact that at lower AP heights, AP transmission probabilities are only a weak function of P_AP, and the gains in downlink SINR do not get negated by a drop in p̅_̅T̅, as is the case for higher AP heights. In the case when maximum coverage is sought by setting h_AP to 30 m, each AP delivers close to 40 Mbps over a 6 MHz TVWS channel when operating at a transmit power of 0.1 W. The key takeaway here is that when seeking to maximize localized coverage (by setting h_AP>10 m), increasing AP transmit power is unlikely to yield better performance due to the sharp drop in AP transmission probability. Fig. <ref> considers a medium deployment density of 1 AP/km^2.Once again two different behaviors are seen depending on whether h_AP exceeds 3 m or not. It is clear that h_AP exceeding 10 m has a detrimental impact on network performance. Even at such densities, pervasive coverage is difficult to achieve. When maximum coverage is sought, each AP delivers close to 12 Mbps per TVWS channel, with a total of 120 Mbps/km^2. Fig. <ref> considers a dense deployment scenario with a density of 10 APs/km^2. In this case, it is possible to achieve pervasive coverage when h_AP exceeds 15 m. Unlike the previous two cases, for a fixed h_AP, ASE decreases with increasing P_AP suggesting that changes in p̅_T plays a more important role than changes in SINRs. Maximizing coverage is not as important as before, and setting 9 m ≤ h_AP≤ 15 m suffices to ensure that more than 80% of the clients are under coverage. When h_AP = 9 m, an ASE of 240 Mbps/km^2 over one TVWS channel can be achieved. The striking similarity between Fig. <ref> and Fig. <ref> suggests that this network is interference limited where interference dominates over noise and an increase in transmit power leads to an equal amount of increase in signal and interference strength, leaving SINR unchanged. The key takeaways from this section are summarized in Table <ref> where λ denotes the number of APs/km^2. Note that pervasive coverage is achieved only at high deployment density and when coverage per AP is prioritized. §.§ Network Planning for a Suburban use-case This section explores a network plan for providing broadband connectivity to a suburban region using results from the previous discussion. The region of interest is Sharon Springs in Wallace County, Kansas, USA.Fig. <ref> shows the distribution of houses in Sharon Springs obtained from the records of 2010 US Census.There are around 400 households spread out over an area of 3 km^2. Suppose each household is to be supported with a data rate of 10 Mbps, the required network throughput is 400×10/3 = 1330 Mbps/km^2. In this area, 37 TVWS channels amounting to a total of 222 MHz are currently available (from Google spectrum database <cit.>).Consider deploying a Super Wi-Fi network with a deployment density of 10 APs/km^2. With the objective of attaining a coverage probability of at least 75% and with the ease of mounting antennas on street light poles in mind, AP height is chosen to be h_AP = 6 m. Using results from Fig. <ref>, AP transmit power P_AP is set to 0.1 W to obtain an ASE of 12 bps/Hz/km^2, translating to a network throughput of 72 Mbps/km^2 per TVWS channel. Thus, it is possible to meet the demands of this suburban region using at most 18 of the 37 available channels. Further network efficiency and better coverage can be achieved using a more careful AP deployment leading to a more economical use of available bandwidth. These computations suggest that such a network can be a reasonable access alternative to satellite-based internet service which tends to be the dominant means of connectivity in such rural areas. Backhaul services for the deployed APs can also be provided using TVWS channels as investigated in <cit.>.§ CONCLUSIONThis paper uses a stochastic geometry analysis to study the feasibility of utilizing TVWS channels to provide broadband connectivity in rural and under-served regions using a Wi-Fi-like network. Regulations on transmit power and antenna height for both APs and clients present situations in which the downlink may be viable but the uplink from client to AP is not. The performance of such a network operating in TVWS channels is analyzed using stochastic geometry while explicitly factoring in uplink viability. Such an analysis is used to characterize AP transmission probabilities, coverage, and area spectral efficiency. These results show that operating APs at high transmit powers and heights is not always beneficial to the performance of the network, even at low AP deployment densities. It is however seen that APs deployed at higher heights significantly improve uplink viability. This exemplifies the rate-coverage trade-off in these networks. The choice of operating parameters for such a network will depend on the desired balance between coverage and rate. § ACKNOWLEDGMENTThis work is supported in part by a grant from the U.S. Office of Naval Research (ONR) under grant number N00014-15-1-2168.§ DERIVATION OF PROBABILITY OF CONCURRENT AP TRANSMISSIONSLet AP x⃗ represent an AP that is at a distance d from AP 0⃗. Then, q(d) represents the probability that AP 0⃗ and AP x⃗ transmit at the same time, and can bewritten as q(d) = ℙ_Φ_A^0⃗,x⃗{x⃗∈Φ_T | 0⃗∈Φ_T }= ℙ_Φ_A^0⃗,x⃗{x⃗∈Φ_T , 0⃗∈Φ_T }/ℙ_Φ_A^0⃗,x⃗{0⃗∈Φ_T }. To compute the numerator of (<ref>), let m_0⃗ and m_x⃗ as the marks chosen by AP 0⃗ and AP x⃗ respectively and assume m_0⃗ < m_x⃗, without loss of generality. Denote z⃗ as a potential interferer. To compute the joint probability that both AP 0⃗ and AP x⃗ concurrently transmit, two `classes' of APs need to be considered—those with mark m < m_0⃗, distributed as a PPP of intensity λ m_0⃗, which prevent both AP 0⃗ and AP x⃗ from transmitting, and those with mark m_0⃗ < m < m_x⃗, distributed as a PPP of intensity λ (m_x⃗ - m_0⃗), which prevent only AP x⃗ from transmitting. Using these observations, the numerator of (<ref>) can be computed asℙ_Φ_A^0⃗,x⃗{x⃗∈Φ_T , 0⃗∈Φ_T}=2(1-e^-μσ/P_APPL(x))∫_0^1 [∫_m_0⃗^1 e^-λ(m_x⃗-m_0⃗)∫_ℝ^2S_x⃗(z⃗) dz⃗dm_x⃗ ] =× e^-λm_0⃗∫_ℝ^2S_0⃗orx⃗(z⃗) dz⃗ dm_0⃗ where 1-e^-μσ/P_APρ(x⃗,0⃗) is the probability that AP x⃗ is not in the neighborhood of AP 0⃗. The factor of two accounts for the case when m_x⃗ < m_0⃗. S_x⃗(z⃗) is the probability that AP x⃗ senses the transmission of AP z⃗ and S_0⃗orx⃗(z⃗) = 1 - (1-S_x⃗(z⃗))(1-S_0⃗(z⃗)) is the probability that the interfering AP z⃗ is sensed by at least one of AP 0⃗ or AP x⃗.In a similar manner, the denominator of (<ref>) can be computed as ℙ_Φ_A^0⃗,x⃗{0⃗∈Φ_T } = ∫_0^1∫_0^m_x⃗ e^-λm_0⃗∫_ℝ^2S_0⃗(z⃗) dz⃗ dm_0⃗+∫_m_x⃗^1 (1-e^-μσ/P_APPL(x))e^-λm_0⃗∫_ℝ^2S_0⃗(z⃗) dz⃗ dm_0⃗dm_x⃗.In the above expression, the first term considers the case when m_0⃗ < m_x⃗, where AP 0⃗ can transmit regardless of whether AP x⃗ is transmitting or not, while the second terms considers the case when m_0⃗ > m_x⃗, in which case, AP 0⃗ can transmit only if the AP 0⃗ cannot sense AP x⃗'s transmissions.IEEEtran | http://arxiv.org/abs/1704.08152v1 | {
"authors": [
"Neelakantan Nurani Krishnan",
"Gokul Sridharan",
"Ivan Seskar",
"Narayan Mandayam"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170426150051",
"title": "Coverage and Rate Analysis of Super Wi-Fi Networks Using Stochastic Geometry"
} |
δ̣ϵφϰ̨∂Γ⊙ AGILE Fermi LATMAGICH.E.S.S. VeritasPKS 2155-304PKS 1510-089501Mkn 501IC 3104̧543C 454.33C 27987M87^1Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin 2, Ireland ^2Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany ^3National Research Nuclear University “MEPhI”, Kashirskoje Shosse, 31, 115409 Moscow, Russia ^4Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6, D-15738 Zeuthen, Germany ^5Astrophysical Big Bang Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan ^6Department of Physics and Astronomy, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907-2036, USA ^7Department of Physics, Rikkyo University, Nishi-Ikebukuro 3-34-1, Toshima-ku, Tokyo 171-8501, JapanWe analyze three scenarios to address the challenge of ultrafast gamma-ray variability reported from active galactic nuclei. We focus on the energy requirements imposed by these scenarios: (i) external cloud in the jet, (ii) relativistic blob propagating through the jet material, and (iii) production of high-energy gamma rays in the magnetosphere gaps.We show that while the first two scenarios are not constrained by the flare luminosity, there is a robust upper limit on the luminosity of flares generated in the black hole magnetosphere. This limit depends weakly on the mass of the central black hole and is determined by the accretion disk magnetization, viewing angle, and the pair multiplicity. For the most favorable values of these parameters, the luminosity for 5-minute flares is limited by 2×10^43 erg s^-1, which excludes a black hole magnetosphere origin of the flare detected from .In the scopes of scenarios (i) and (ii), the jet power, which is required to explain theflare, exceeds the jet power estimated based on the radio data. To resolve this discrepancy in the framework of the scenario (ii), it is sufficient to assume that the relativistic blobs are not distributed isotropically in the jet reference frame. A realization of scenario (i) demands that the jet power during the flare exceeds by a factor 10^2 the power of the radio jet relevant to a timescale of 10^8 years. Scenarios for ultrafastgamma-ray variability in AGN F.A. Aharonian^1,2,3, M.V. Barkov^4,5,6,D. Khangulyan^7 December 30, 2023 ===============================================================§ INTRODUCTION The hypothesis of supermassive black holes (SMBHs) as powerhouses of active galactic nuclei (AGN) has been proposed <cit.> to explain the immense luminositiesof AGN and quasars by the release of the gravitational energy through the process of gas accretion.The radiation power of the accreting plasma is limited by the Eddington luminosity,L_ Edd = 1.3 × 10^46 M_8 erg s^-1, whereM_8=M_ bh/10^8 M_⊙ is the mass of the black hole in the unities of 10^8 solar masses.Theapparent luminosities of radiationof many AGN,L_ app=4 π D_L^2 f(f is the detected energy flux and D_L is the source luminosity distance)may exceed theEddington luminosity of SMBHsbyorders ofmagnitude. However, the “energy crisis”can be overcomeif one assumes that the observedradiation is highly anisotropic, namely,that it is produced ina collimated outflow(jet)close to the line of sight <cit.>.The concept of relativistically beamed emission offers not onlyan elegant scheme for the unification of various classes ofAGN <cit.>,but also provides a natural interpretation ofenormous fluxes of their nonthermal emission. Indeed, the assumption of production of radiation in a source relativistically moving toward the observer with a Doppler factor δ≫ 1allows physically reasonable intrinsic luminosities ofAGN dubbedblazers,reducing them by orders of magnitude compared to the apparent luminosity, L_ app=δ^4 L_ int.This assumption also allows a larger (more “comfortable”)size of the productionregion that is demanded by the observed variability of radiation: l ≤ c Δ t_ varδ.These relations applyto allelectromagnetic wavelengths, but they are crucial, first of all,for gamma-ray loud AGN,the apparent luminositiesof which during strong flares, e.g., in3C454.3 <cit.>and <cit.>, can achievethe level of L_γ∼ 10^49-50erg s^-1. Strong Doppler boosting is also needed for prevent severe internal gamma-ray absorption,especially at VHE energies<cit.>.The gamma-ray emission of blazers is strongly variable, with fluxes that match the sensitivity of the FermiLarge Area Telescope (LAT)in the MeV/GeV band well, and the current arrays of imaging atmospheric Cherenkov telescopes(IACT), , ,, in the VHE band.During the strongest flares of BL Lac objectslike Mkn 421, Mkn 501 and PKS 2155-304,the energy fluxesof VHE gamma-raysoften exceedf_ VHE=10^-10erg cm^-2 s^-1.Such fluxes can bestudied with IACT arrays with huge detection areas that are as large as10^5 m^2 in almost backgroundfree regime,allowingvariability studies on timescales ofminutes. Although thefluxesof flaringpowerful quasars at MeV/GeV energiescan be significantly larger,f_ VHE=10^-8erg cm^-2 s^-1,because of the smalldetection areaof space-borne instruments (≃ 1 m^2), the capabilityof the latter of probing the brevity of suchstrong AGN flareshas until recentlybeen limited by timescales of hours and days.However, after the release of thelatestsoftware tools by thecollaboration, whivh allow a significant increase in the gamma-ray photon statistics,the variability studies at GeV energies forexceptionallybright flarescan be extended down tominute timescales. This potential recently has recently been demonstratedby thecollaborationforthegiant 2015 June outburst of 3C 273<cit.>. It is straightforward to comparethese timescales with the minimumtime that characterizesa black hole system as an emitter, namely, thelight-crossing time of the gravitational radius of the black hole:τ_0=r_g/c ≈ 5×10^2M_8s. Note that r_g=GM_ bh/c^2=1.5× 10^13M_8cmisthe gravitational radius corresponding to the extreme Kerr black hole, i.e.twice smaller than the Schwarzschild radius. Thus, for the mass range of black holes M ≥ 10^8 M_⊙, the current gamma-ray detectorshave a potential to explore the physics of AGN that is close to the event horizon on timescalesshorterthan τ_0.Such ultrafastgamma-ray flares[For a recent summary of ultrafast gamma-ray flares of AGN see <cit.>] have previously been detected from four AGN: <cit.>, 501 <cit.>, and <cit.> at TeV energies, andat GeV energies <cit.>. In addition, a flare with a duration comparable to the BH horizonlight-crossing time, ∼2τ_0, was observed from a missaligned radio galaxy 87, in which the jet Doppler factor is expected to besmall<cit.>. For comparison, it is interesting to note that the characteristic timescales of the even the shortest GRBs<cit.>, which are most likely associated with solar mass black holes, exceed τ_0by several orders of magnitude.The detection of variable VHE gamma-ray emission from AGN on timescales significantly shorter than τ_0 is an extraordinary result and requires a careful treatment and interpretation. The masses of SMBHs in distant AGN are typically derived from the empirical Faber-Jackson law <cit.>. Although this statistical method is characterized by a small dispersion, scatter for individual objects may be significant, which consequently leads to uncertainties of τ_0. On the other hand, it follows from Eq. (<ref>) that for the minute-scale flares reported fromand , the variability time can exceed τ_0 only for masses of the BHs than are lower than 3×10^7M_. For both objects, different methods of estimating M_ bh give significantly higher values, and therefore τ < τ_0. If the emission is produced in a relativistically moving source with a velocity β_ em, the variability time-scale for the observer is shortened by the Doppler factor _̣ em=1/_ em(1-β_ emcosθ_ em); _ em=1/√((1-β_ em^2)) is the Lorentz factor and θ_ em is the angle between the source velocity and the line of sight.Thus if we wish to increase the proper size of the emitter R' (the source size in the comoving reference frame) to a physically reasonable value of R' ≥ r_g, the Doppler factor should be large, _̣ em>10. For example, in the case of , where the mass of SMBH is expected to be high,[ To overtake this constraint, some models involve a BH binary system as the central engine in <cit.>]M_8∼10, the VHE variability sets a lower limit on the value of the Doppler factor: _̣ em≥25.However, there is another issue of conceptual importance that cannot be ignored.The problem is that if the perturbations originate in the central engine and then propagate in the jet, e.g.in the form of sequences of blobs ejected with different Lorentz factors (leading to internal shocks), the size of the emitter in thelaboratory frame, R=R'/_ j, would not depend on the Doppler factor and it should exceed the gravitational radius: R≥ r_g.Let us present the proper size of the production region as R'=λ_ jr_g, where _ j is the jet bulk Lorentz factor, and λ is a dimensionless parameter, which corresponds to the ratio of the production region size in the laboratory frame to the gravitational radius.The causality condition provides a limitation on the variability timescale t_ var≥τ_0 λ_ j_ em . The variability of t_ var=0.04τ_0 inferred from the VHE flares of <cit.> requires _ em≃25λ_ j, i.e., the emitter should move relativistically in the frame of the jet, which in turn moves relativistically toward the observer.The jet-in-jet model suggested by <cit.> can be considered as a possible realization of this general scenario. Alternatively, if the source of the flare does not move relativistically relative to the jet (_ em≃_ j), the size of the source in the laboratory frame should be much smaller than the black hole gravitational radius: λ≃0.04.If the emission site is located in the jet and formed by some perturbations propagating from the BH, one should expect λ>1. Thus, the condition of λ<1 implies that the perturbations in the jet that result in a flare should have anexternal origin, i.e., are not directly linked to the central black hole. This scenario can be realized when a star or a gas cloud of radius R_*≪ r_g enters the jet from outside and initiates perturbations on scales smaller than the black hole gravitation radius r_g<cit.>.Finally, it has been suggested that the flares can be produced in the BH magnetosphere <cit.>. In this case, the production site does not move relativistically with respect to the observer, and Eq. (<ref>) is reduced to t_ var>τ_0λ, where λ=R/r_g. Thus, the flare originates in a compact region that occupies a small fraction of the black hole magnetosphere. An analogy for this possibility could be the emission of radio-loud pulsars.It is believed that in these objects the radio pulses are produced in the polar cap region, which constitutes only a small part of the pulsar surface. Note that for the typical pulsar radius R_ psr of 10 km, τ_0=R_ psr/c ∼ 30 s is too small to be probed through the variability of the radio emission. We note here that although the production site of relativistic motion does not allow to reducing the minimum variability time (see Eq. (<ref>)), the relativistic beaming effect allows significant relaxation of the energetics required to produce the flare. Thus, magnetospheric scenarios should have higher energy requirements then the jet scenarios.In this paper we discuss in rather general terms three possible scenarios for the production of ultrafast ("subhorizon" scale)variability in AGNs: (i) The source of the flare is a magnetospheric gapoccupying a small volume in the proximityof the black hole close to the event horizon <cit.>.(ii) The emittermoves relativistically in the jet reference frame. The most feasible energy source for this motion is magnetic field reconnection in ahighly magnetized jet <cit.>. (iii) Flares areinitiated bypenetrationof externalobjects (stars or clouds) into the jet <cit.>.Apparently,any model designed to explain the ultrafast variabilityon timescalest_ var<τ_0should address some other key issues. In particular, therequired overall energy budget should be feasible, thesourceshould be optically thin forgamma-rays, and of course, the proposed radiation mechanism(s)shouldbe able to explain thereported spectralfeatures of gamma-ray emission.§ ADDRESSING THE“SUBHORIZON” SCALE VARIABILITY§.§ The Magnetospheric ModelMagnetospheres of the central SMBHs in AGN can be sites of production of gamma-rays with spectra extending to VHE energies<cit.>. At low accretion rates, the injection of charges into the BH magnetosphereis not sufficient for a full screening of the electric field induced by the rotation of the compact object. The regions with unscreenedelectric field, referred to as gaps, are capable of effective acceleration of charged particles. Such a scenario may result in a variabilityof the source on “subhorizon” timescales since the size of the gap is much smaller than the gravitational radius. The attractiveness ofthis scenario is its applicability to the non-blazar-type AGN.On the other hand, because of both the low accretion rate and the lack ofDoppler boosting,the gamma-ray luminosities of such objects are expected to be quite modest when compared to blazers. Therefore the detectability of the black hole magnetospheric radiation is most likely limited by a few nearby objects. In particular, the radio galaxyM87, as well as the compact radio source Sgr A* in the center of our Galaxy, can be considered as suitable candidates for the realizationof such a scenario <cit.>.The energy release in the entire magnetosphere is limited by the BZ luminosity. Below we follow a simplified treatment that allows us to estimate the energy release in a thin vacuum gap formed in the SMBH magnetosphere.The rotation of a magnetized neutron star or BH in vacuum induces an electric field, 𝐄_0, in the surrounding space <cit.>.If a charge enters this region, the electric field should accelerate it.In an astrophysical context the unscreened electric field is usually strong enough to boost the particle energy to the domain where the particle starts to interact with the background field and thus initiates an electron-positron pair cascade. The secondary particles move in the magnetosphere in a way that tends to screen the electric field <cit.>. Eventually, an electric-field-free configuration of the magnetosphere can be formed.However, one should note that there are differences between the structures of the pulsar and BH magnetospheres, and consequently, the theoretical results obtained for pulsar magnetospheres cannot be directly applied to the BH magnetosphere. In particular, while in the case of the pulsar magnetosphere the source of the magnetic field is well defined, in the BH magnetosphere the magnetic field is generated by currents in the disk and magnetosphere. The configuration of the field is determined by the structure of the accretion flow. Thus, a change of the accretion flow can result in the formation of charge-starved regions (gaps) in the BH magnetosphere. The charge density required for the screening is known as the Goldreich-Julian density (GJ), ρ_ GJ<cit.>.However, the process of the pair creation is expected to be highly non-stationary <cit.>, thuseven if am electric-field-freestate of the magnetosphere is possible, it cannot be stable <cit.>. The gaps, i.e., regions in which the charge density is not sufficient for the electric field screening, may appear sporadically in the magnetosphere, for example, in the vicinity of the stagnation surface, i.e., at the boundary that separates acretion and ejection trends in the flow.The vacuum electric field strength 𝐄_0 determines the maximum electric field in the gap, thus the maximum acceleration rate of a particle with charge e in the gap is mc^2γ̇< e c E_0. The total power of particle acceleration can be expressed asĖ<∫_ gapdV e (n_e+n_e^+) c E_0 ,where n_ e and n_e^+ are densities of electrons and positrons. If all the energy gained by the particles in the gap is emitted in gamma rays,Eq. (<ref>) also corresponds to the upper limit of the gamma-ray luminosity. Note that electrons and positrons move in opposite directionsin the gap,and only one of these species generates emission detectable by a distant observer. For a thin spherical gap, R<r<R+h, the luminosity upper limit isL_γ<4π R^2 h e n_ e c E_0 ,where the electrons are assumed to emit outward.The particle density can be expressed as a fraction of the Goldreich-Julian density: e n_ e=κρ_ GJ, where κ is the multiplicity. The condition for the electric field screening, e|n_ e-n_ e^+|=ρ_ GJ, allows charge configurations with high multiplicity and still non-screened electric field. To obtain a more detailed estimate of the generated pairs' influence on the electric field in the gap, it is necessary to consider the electromagnetic cascade in the gap.The numerical simulations of <cit.> and <cit.> demonstrate an important tendency. When the multiplicity becomes significant, κ∼1, the charges in the gap start to generate an electric field that is comparable to E_0, and the accelerating field vanishes.Thus, for effective charge acceleration, the following condition should be fulfilled: κ≪ 1.Thus, the total energy release in the gap of thickness, h, can be estimated as L_γ<4π R^2 hκρ_ GJ c E_0 .The electrical field in the gap is estimated as E_0 ≈B_ gRΩ_ Fsinθ c , where B_ g is the magnetic field in the vacuum gap, Ω_ F is the angular velocity of the frame, R is the radius, and θ is the polar angle. In fact, the actual electric field in the drop is smaller by a factor h/R than the value given by Eq. (<ref>)<cit.>. This factor accounts for the influenceof the magnetospheric charges located outside the gap.Eq. (<ref>) does not account for this contribution. Since these charges, evenif they remain outside the gap, tend to decrease the electrical field in the gap, Eq. (<ref>) provides a strict upper limit on the gap electric field strength.The Goldreich-Julian density is also determined by the same parameters:ρ_ GJ=Ω_ FB_ gsinθ/(2π c) .For a Kerr BHwith the maximum angular momentum,the angular velocity Ω_ F isestimated asΩ_ F c≃1/4r_g .SubstitutingEq. (<ref>) - (<ref>) to Eq. (<ref>), one obtains L_γ<1/8B_ g^2 R^3 r^2_ gκ h csin^2θ . The upper limit on the luminosity from a vacuum gap depends on the factor R^3B_ g^2 that is expected to decrease with R. For sake of simplicity below it is adoptedthatR^3B_ g^2≃ r_g^3B_ bh^2, where B_ bh is the magnetic field at the BH horizon. Thus, one obtainsL_γ<1/8 B_ bh^2 r_gκ h csin^2θ . We should note that for h → r_g, the luminosity estimate provided by Eq. (<ref>) (after averaging over the polar angle θ) exceeds the Blandford-Znajek (BZ) luminosity <cit.> by a factor of 2. This is imposed by several simplifications in our treatment. The most important contribution is caused by the usage of the electric field upper limit, Eq. (<ref>), as the accelerating field. Thus, Eq. (<ref>) can be considered as a safe upper limit for the luminosity of magnetospheric flares.A similar estimate has been obtained by <cit.> and <cit.>. However, the numerical expression in <cit.>contains some uncertain geometrical factor <cit.>. Eq. (<ref>) allows us toestimate its value: this geometrical factor should be small, ∼10^-2<cit.>. Finally, <cit.>argued that for the full screening of the electric field in a thin gap, the charge density should exceedthe Goldreich-Julian value by a factor[Note that <cit.> used a different notation for the gap thickness, Δ.]R/h, which should lead to an enhancement of the gap radiation. To illustrate the physical reason for the existence of this factor,<cit.> computed the divergence of the electric field in the gap. However, as the gap electric field they adopted a field determinedby an expression similar to Eq. (<ref>), i.e., a value that overestimates the true field by the factor R/h<cit.>.Thus, the factor suggested by <cit.> seems to be strongly overestimated.Finally, the thickness of the gap, h, in Eq. (<ref>) is constrained by thevariability time scale,h ∼ t_ varc. To production the emission variable on a 5-minute time-scale, t_ var= 5 t_ var,5 min,the gap thickness, h=10^13t_ var,5 cm, should be smaller than the gravitational radius of the SMBH with a mass M_8>1. Thus, the estimated gamma-ray luminosity cannot exceed the following value: L_γ<5×10^43κ B_4^2 M_8t_ var,5sin^2θ erg s^-1 ,where B_ bh=10^4B_4G.Eq. (<ref>) contains two parameters that are determined by properties of the advection flow in the close vicinity of the BH:pair multiplicity, κ, and the magnetic field strength, B. Importantly, these parameters are essentially defined by the same property of the flow, more specifically, by the accretion rate. The magnetic field at the BH horizon needs to be supported bythe accretion flow. Therefore the field strength is directly determined by the accretion rate. The accretion rate also definesthe intensity of photon fields in the magnetosphere, and consequently, the density of electron-positron pairs produced through gamma-gammainteraction <cit.>. If the multiplicity parameter, κ, approaches unity, the gap electric field vanishes<cit.>.This sets an upper limit on the accretion rate, and consequently on the magnetic field strength.In previous studies <cit.> the maximum accretion rate compatible with the existence of a vacuum gap in the magnetospherewas estimated asṁ<3×10^-4M_8^-1/7 ,where ṁ is the accretion rate in the Eddington units:Ṁ_ edd = 4π m_p G M_ bh/η c σ_t .Here m_p, σ_t, and η are the proton mass, the Thompson cross-section, and the accretion efficiency factor, respectively. To derive the estimate provided by Eq. (<ref>), <cit.> adopted a value of η=0.1 and estimated the magnetic field strength at the BH horizon asB_ bh=1.3×10^5(ṁ/M_8)^1/2G ,where we rescaled the numerical coefficient to the normalization used throughout our paper.For this magnetic field strength, Eq. (<ref>) yields L_γ<3×10^42κ M_8^-1/7t_ var,5sin^2θ erg s^-1 . In some cases, e.g., for , the energy requirements are rather close to the obtained upper limit, therefore we consider a somewhat more accurate treatment of the case of a magnetosphere around a Kerr BH below.The strength of the magnetic field at the BH horizon can be obtained by extrapolating the field at the last marginally stable orbit.Let us define the magnetic field in the disk as B_ d=√(8πβ_ m p_ g) ,where β_ m and p_ g are the disk magnetization and gas pressure in the accretion disk that confines the magnetic field at the horizon.The gas pressure can be estimated using the solution for a radiatively inefficient accretion flow <cit.>[A more accurate treatment of the accretion flow reveals a correction by less than 30%<cit.> as compared to the height-averaged treatment in <cit.>] as p_ g=√(10)Ṁ√(G M_ bh)/12πα_ss R^5/2 ,where α_ss is the nondimensional viscosity of the disk <cit.>. Eq. (<ref>) for R→ r_g provides an estimate for the magnetic field at the BH horizon: B_ bh=1.5β_ m^1/2 (Ṁc)^1/2/(α_ss)^1/2 r_g . The magnetic field strength provided by Eq. (<ref>) together with Eq. (<ref>) yields in L_γ < √(10)/12β_ mκ (h/r_g)sin^2θṀ c^2/α_ss . The multiplicity parameter, κ, at the Kerr radius is determined as ( see Appendix <ref> for details)κ≡n_±/n_ GJ≈ 6×10^6ṁ^7/2M_8^1/2/(ηα_ss)^7/2β_ m^1/2.The condition κ<1 determines an upper limit on the accretion rate:ṁ<10^-2ηα_ssβ_ m^1/7/M_ bh^1/7.Eq. (<ref>) and (<ref>) substituted intoEqs. (<ref>) and (<ref>) give an upper limit for the magnetic field that is consistent with the existence of vacuum gaps: B_ bh< 7× 10^3 (β_ m/M_8)^4/7 ,and consequently, the maximum luminosity of particles accelerated in the gap does not depend on α_ss and η: L_γ <2×10^43β_ m^8/7κ t_ var,5 M_8^-1/7sin^2θ ^-1 .This estimate is obtained for the thick-disk accretion (in the ADAF-like regime). The limit on the accretion rate given by Eq. (<ref>) is consistent with the realization of this accretion regime. For higher accretion rates, ṁ≥ 0.1, the accretion flow is expected to converge to the thin-disk solution <cit.>. In this regime, the temperature of the disk is expected to be significantly below 1MeV, thus the pair creation by photons supplied by the accretion disk should be cease. This effectively mitigates the constraints imposed by the accretion rate. However, the change of the accretion regime also significantly weakens the strength of the magnetic field at the BH horizon <cit.>, and consequently decreases the available power for acceleration in the gap.To deriveEq.(<ref>), we assumed that the gap thickness is determined by the variability time-scale; this corresponds to the energeticallymost feasible configuration. In a more realistic treatment, one should also take into account the interaction of the particles that are acceleratedin the gap with the background radiation field. For high and ultrahigh energies of electrons, E>1 TeV, the characteristic time ofthe inverse Compton scattering appears to be shorter than the minute-scale typical for the short TeV flares (see Appendix <ref>).For the hot target photon field, as expected from a thick accretion disk, the pair-production process should also be very efficient,λ_γγ≤λ_ IC. Thus, computation of the TeV emission requires a detailed modeling of the electromagneticcascade <cit.>. Furthermore, the production and evacuation of the cascade-generated pairs may follow a cyclicpattern and the inductive electric field may become comparable to the vacuum field <cit.>. A detailed consideration of this complexdynamics is beyond the scope of this paper, but we note that the characteristic length of such a cascade-moderated gap should be small,∼√(λ_ ICλ_γγ), resulting in a reduction of the available power <cit.>.Eq. (<ref>) determines the maximum luminosity of vacuum gaps that can collapse quicker than t_ var. It has been assumed for its derivation that the magnetic field is determined by an accretion regime that its in turn determines the intensity of the photon field in the magnetosphere. In the case of a steady accretion, this seems to be a very feasible approximation. However,this may look less certain in the case of a rapidly changing accretion rate, since the processes that govern the variation of the accretion rateand escape of the magnetic field from the BH horizon may have different characteristic timescales.Therefore we provide some estimatesfor these two timescales below. The dominant contribution to the photon field comes from plasma located at distances r∼ 2r_g, and the characteristic viscous accretiontime (density decay time in the flow) is t_ρ, decay≃2r_g cα_ ss≃10^4α_ ss,-1^-1M_8 s .When the accretion fades, the decay of the magnetic field is determined by the magnetic field reconnection rate <cit.>: t_ B, decay≃π r_g 0.3 c∼10^4M_8 s .Since these two time-scales are essentially identical, it is natural to expect that the field strength and the disk density will decay simultaneously. Thus, Eq. (<ref>) should also be valid for the time-dependent accretion regime. §.§ Relativistically Moving BlobsThe properties of radiation generated in jets may be significantly affected if some jet material moves relativistically with respect to the jet local comoving frame. For example, the magnetic field reconnection may be accompanied by the formation of slow shocks <cit.> that in the magnetically dominated plasma produce relativistic flows <cit.>. If such a process is realized in AGN jets, it can lead to gamma-ray flares in blazar-type AGN with variability timescale significantly shorter than r_g/c<cit.>.Another implication of this scenario is related to short gamma-ray flares detected from missaligned radio galaxies <cit.>. Indeed, the conservation of momentum requires that for each plasmoid directed within the jet-opening cone, there should exist a counterpart that is directed outside the jet-beaming cone.While the radiation of the plasmoid directed along the jet appears as a short flare, the emission associated with its counterpart outflow can be detected as a bright flare by an off-axis observer. The latter process may have a direct implication on the interpretation of flares from nearby missaligned radio galaxies, e.g., 87 <cit.>. If a process, operating in a region of the jet with comoving volume V', results in the ejection of plasmoids,some fraction, ξ, of the energy contained in the volume is transferred to the outflow. The conservationof energy can be written asξ V' ϵ'_ j= S_ co_ co^2 v_ coΔ t' (4/3ϵ̃_ e) .Here ϵ'_ j and Δ t' are the energy density of the jet plasma and duration of the ejection, as seen in the jetcomoving reference frame. _ co=1/√(1-(v_ co/c)^2), S_ co, and ϵ̃ are the plasmoid Lorentz factor,the outflow cross-section, and the internal energy, respectively. The outflowcross-section can be estimated asS_ co≃ S/(2_ co^2), where S≃ V'/(Δ t' v_ co) is the surface of the volume V'. Thus,one obtains an estimate for the energy density of the ϵ'_ j≃2/3ξϵ̃_ e .For simplicity, in what follows we take ξϵ'_ j≃ϵ̃_ e. The efficiency of the energy transfer,ξ, depends on a specific realization of the scenario.For example, it seems that the for thereconnection of the magnetic field, the efficiency might be high ξ∼1,as follows froman analytic treatment by <cit.> andthe results of numerical simulations by <cit.>[From Figure 2 of <cit.> it follows that n_ lab≃σ n_0 and <γ>_ labn_ lab≃σ^2 n_0, thus the internal energy in the plasmoid isϵ̃≃ñ<γ̃>≃σ n_0mc^2.].We note, however, that in the presence of a guiding field,the magnetization of the ejected plasmoids should be high <cit.>.On the other hand, the energy density in the plasmoid can be estimated through the emission variability time and the luminosity level <cit.>:ϵ̃_ e=E_ em_ eml̃_ em^3 ,where the variability time-scale determines the size of the production region: l̃_ em=cΔ t _ em, and the flux level defines the energy content in the plasmoid: E_ em=L_γΔ t/(4f_ em^2) (here f<1 defines the fraction of the plasmoid energy transferred to the flare emission; this factor is dropped in what follows, and its contribution is accounted for in the value of the factor ξ). Thus, one obtainsϵ̃_ e=L_γ 4_ em^6c^3Δ t^2 .On the other hand, the energy density in the jet isϵ'_ j=L_ jΔΩ r^2 c _ j^2 ,where ΔΩ≃π/_ j^2 is the jet propagation solid angle.The comparison of these equations allows us to estimate the required true luminosity of the jet asL_ j=L_γ_ em^6π r^2 4ξ c^2Δ t^2 .The above equation is consistent with Eq. (10) from <cit.>. Note, however, a difference in the notations: throughout this paper, L_ j is the true jet luminosity, while in <cit.>L_ j corresponds to the isotropic luminosity.If the viewing angle is small, the mini-jet Lorentz factor can be expressed as_ em=2_ j_ co/(1+α^2) where α=θ_ j is the viewing angleexpressed through the jet-opening angle (see Appendix <ref>).Thus, the above equation can be simplified as L_ j=L_γ_ co^6_ j^6(1+α^2)^6256π r^2ξ c^2Δ t^2 or L_ j=1.4×10^-5L_γ(1+α^2 4)^6_ co,1^-6_ j,1^-6ξ_-1^-1r_2^2M_8^2t_ var,5^-2 .Here it was assumed that the flare originates at a distance r_2=100 r_g from the central BH with mass M_ bh=10^8M_⊙ M_8. The above estimate describes the jet luminosity requirement to generate a single short flare of duration t_ var. Observations in HE and VHE regimes show that AGNs often demonstrate a rather long period of activity (as compared to theduration of a single peak): T≫ t_ var. If the mini-jets are isotropically distributed in the jet comoving frame, the probability for an observer to be in the mini-jet beaming cone depends weakly on the observer viewingangle[If Γ_ co>Γ_ j, this statement is correct for observers located inθ_ view<π/2, otherwise for tanθ_ view<v_ co/√(1/_ co^2-1/_ j^2).],and this probability can be estimated asP≃(2_ co)^-2<cit.>. If the mini-jet formation istriggered by somespontaneous process, then the comoving size of the region responsible for the flare is l'_0=_̣ jTc, and the energy contained in this region is E'=Sl'_0e'_ j (here S is the jet cross-section). The energy of a single mini-jet in the comoving frame isE'_ mj=L_γt_ var_ co 4ξ_ em^3 .The total number of mini-jets during a flaring episode can be estimated as N≈Φ T/P t_ var,where Φ is the so-called filling factor.The total dissipated energy for the flare should be smaller than the energy that is contained in the dissipation region:E'_ mjΦ T Pt_ var <L_ jT_̣ j_ j^2 .This implies a requirement for the jet luminosityL_ j> 0.1Φζ^2 _̣ j,1^-2 L_γξ_-1^-1 ,here ζ=_ j/_̣ j, or L_ j> 0.006 Φ(1+α^2)^4_ j,1^-2L_γξ_-1^-1 ,where the small viewing angle limit was used for the ratio of Lorentz and beaming factors: ζ=_ j/_̣ j≃(1+α^2)/2 (see appendix <ref>). The requirement imposed by Eq. (<ref>) significantly exceeds the limit related to the shortest variability time, Eq. (<ref>). The derived lower limit for the jet luminosity contains the parameter ξ, which accounts for the conversion efficiency from jet material to the outflow, and from outflow to the radiation. While the letter can be high, ∼1, if a good target for nonthermal particles exists, the value of the former efficiency depends on the process behind the outflow formation. For example, it was argued that if the outflow is formed by the Petschek-type relativistic reconnection <cit.>, the energytransfer is expected[This follows from Eq.(6) in <cit.>, i.e., ϵ̃_ e= σρ_ j'c^2≃ϵ_ j' for σ≫1.] to be high, ∼1.However, the efficiency of the transfer can be significantly suppressed if the guiding field is present in the reconnection domain <cit.>.On the other hand, this requirement can be somewhat relaxed if the velocity direction of the plasmoids is not random, e.g., is controlled by the large-scale magnetic field <cit.>, or is triggered by some perturbation propagating from the base of the jet. In the former case the mini-jet detection probability, P, may be higher, and in the latter case, the comoving distance between the mini-jets may be larger than l'_0. Let us assume that the flare trigger propagates with Lorentz factor '_ tr in the jet comoving frame, then the comoving region size islarger by a factor of '_ tr.§.§ Cloud-in-Jet ModelIn the framework of the cloud-in-jet scenario, we deal with the nonthermal emission generated at the interaction of a jet with some external obstacle, e.g., a BLR cloud or a star <cit.>. Debris of the obstacle matter, produced at such an interaction, can be caught by the jet flow. This debris should form dense blobs or clouds in the jet, and the emission generated during their acceleration may be detected as a flare <cit.>.If this interpretation is correct, each peak of the light curve can be associated with emission produced at the acceleration of some individual blob.The peak profile and its duration are determined by the condition of how quickly this blob can be involved into the the jet motion, i.e., by the dimension and mass of the blob. Light blobs with amass satisfying the condition M_ c c^2 <P_ jπ R_ c^2r_04_ j^3(here P_ j and R_ c are the jet ram pressure and the cloud radius, respectively)are accelerated on length scales smaller than the distance to the SMBH, r_0<cit.>.If this condition is fulfilled, the variability time scale can be estimated as <cit.>t_ var≃4cM_ c_ j^2 P_ jπ R_ c^2 _̣ j .This estimate ignores perturbations in the jet that are generated by the obstacle-jet interaction, which is probably very complex, and their influence can be explored only by numerical simulations, which are beyond the scope of this paper <cit.>. In the case of a cloud with a mass that satisfies to Eq. (<ref>), the variability can be very short: t_ var<r_0/(c_ j_̣ j), but obviously it cannot be shorter than the light-crossing time of the blob: t_ var>R'_ c/(c_̣ j). This implies, through Eq. (<ref>), that the minimum mass of blobs below which the dynamics of the blobs is very quick and the variability is limited by the blob light-crossing limit.On the other hand, the mass of the blob determines the energy transferred by the jet to the blob during its acceleration, and consequently, the apparent energy emitted in the corresponding peak of the light curve <cit.>: E_γ≃ξ M_ cc^2 _̣ j^3,where the factor ξ accounts for the fraction of energy transferred to the gamma-ray emitting particles and some dynamical factor <cit.>. Thus, if the cloud dynamics determines the variability, then the luminosity of the emission appears to be independent of the mass of the cloud: L_γ≃c P_0π R_ c^2 ξ_̣ j^4 4_ j^2 .Since L_ j>c P_0π R_ c^2, the aboveequation allows us to obtain a lower limit on the jet luminosity required for the operation of the star-jet interaction scenario: L_ j > 0.4ζ^2_̣ j,1^-2L_γξ_-1^-1 ,orL_ j > 0.025 (1+α^2)^4_ j,1^-2L_γξ_-1^-1 ,which is a factor of 4/Φ larger than the estimate for the jet-in-jet scenario (see Eq. (<ref>)).Eqs. (<ref>) and (<ref>) contain four parameters P_0, _ j (we treat _̣ j as a related parameter), M_ c, and R_ c, and therefore formally allow a solution even if two of these parameters are fixed. For example, for given properties of the jet (i.e., for specific values of the parameters _ j and P_0) and the parameters characterizing the flare (the total energy and the variability), the characteristics of the cloud can be determined as M_ cc^2=E_γξ_̣ j^3 ,π R_ c^2=4ζ^2_̣ j^2E_γξ t_ var cP_0 .However, the determined parameters of the cloud may not necessary be physical, and their feasibility should be examined by dynamical estimates.The first dynamical limitation is related to the ability of a cloud to penetrate the jet and become involved in the jet motion.According to the estimates given by <cit.> and <cit.>, for the typical jet parameters these constraints do not impose any strong limitations. The heaviest blobs that can be accelerated by a jet with luminosity 10^43 erg s^-1can result in flares with a total energy release of 10^54 erg. If the cloud is light enough to be caught by the jet, then one should consider two main processes: the cloud expansion, and its acceleration. At the initial stage, the cloud cross-section is not sufficiently large to provide its acceleration to relativistic velocities. On the other hand, the intense jet-cloud interaction at this stage leads to a rapid heating and expansion of the cloud.The cloud size-doubling time can be estimated ast_ exp≈ A(M_ c/γ_g R_ c P_ j)^1/2 .where γ_g=4/3 is the adiabatic index and A is a constant of about a few <cit.>. When the size of the cloud becomes large enough for acceleration to relativistic energies, the intensity of the jet-cloud interaction fades away,and the cloud expansion proceeds in the linear regime. Since the time scale for acceleration to relativistic velocity ist_ ac≃M_ cc^2π R_ c^2 cP_ j ,the size of the cloud relevant for the flare generation can be obtained by balancing Eqs. (<ref>) and (<ref>): R_ c = A_ exp(M_ cc^2/P_ jγ_ gπ^2 A^2)^1/3 .Here the constant A_ exp accounts for the cloud expansion in the linear regime.The dynamical limitation given by Eq. (<ref>) together with Eq. (<ref>) allows determination of the jet ram pressure:P_ j = π A^4/ξγ_ g^2 A_ exp^6(2ζ)^6E_γ/t_ var^3 c^3 .The actual value of the coefficient in the above equation, in particular the value of A_ exp, can be revealed only through the numerical simulations given the complexity of the jet-cloud interaction. However, if one assumes that the expansion proceeds very efficiently, i.e., the cloud size achieves a value close to the light-crossing limit, R_ c≃_̣ jt_ varc, then the expression for the jet ram pressure becomes P_ j =(2ζ_̣ j^2)^2E_γ/πξ t_ var^3 c^3 . Since each flaring episode should correspond to specific jet parameters[We note, however, that across a magnetically driven jet one may expect strong gradients of the jet ram pressure <cit.>).], the above equation implies that the energy emitted in an individual peak of a flare should be proportional to the cube of its duration: E_γ∝ t_ var^3 (or L_γ∝ t_ var^2). Obviously, the study of individual peaks in a statistically meaningful way requires a detailed light curve that can be obtained with future observations, in particular with CTA <cit.>.§.§ Energetic Constraints for Detected Exceptional FlaresSo far, several super-fast gamma-ray flares have been detected in the VHE or HE regimes from different types of AGNs. The peculiarity of the signal is related both to the duration of the flare and to the released energy. Below we consider several cases that are summarized in Table <ref>.§.§.§ The July 2006 flare ofis characterized by a very short variability of 180 s and the a intrinsic VHE gamma-ray luminosity at the level of 10^47 erg s^-1<cit.>.The mass of the central BH is estimated to be M_ BH,8≃ 10<cit.>.Together with the short variability time, this constrains the luminosity of (potential) gamma-ray flares produced by magnetosphere gaps at the level of L_γ, ms<10^43 erg s^-1and thus excludes any magnetospheric origin of these flares. is a typical representative of high-energy peaked BL Lacs. It is expected that the jet is aligned along the line of sight:α≈0. For a typical value of the jet Lorentz factor, Γ_ j=20, the jet power required for the realization of thejet-in-jet scenario isL_ j,jj>10^44Φ_-0.2_ j,1.3^-2ξ_-1^-1 .It follows from the comparison of Eqs. (<ref>) and (<ref>) that the lower limit of the jet powerin the cloud-in-jet scenario is higher by a factor ∼4/Φ, i.e., L_ j, cj > 6×10^44Γ_ j,1.3^-2ξ_-1^-1erg s^-1 .We note here that constraints imposed by the radiation mechanism enhance the required jet power by a factor of ∼10 for the external inverseCompton scenario and by ∼10^2 for proton synchrotron emission <cit.>, which exceeds the Eddington luminosity.§.§.§In 2012 November, thecollaboration detected a bright flare from <cit.>. The flare consisted of two sharp peaks with a typical duration of ∼5 min. The measured spectra were hard, with a photon index ≲2, extending up to ∼ 10 TeV. The energy released during this event has been estimated to be at a level of 2×10^44 erg s^-1.The mass of the BH powering activity ofhas been estimated to be M_=(3_-2^+4)×10^8M_<cit.>, i.e., the measured variability time scale is as short as 20% of τ_0.According to the estimate provided by Eq. (<ref>), the luminosity of flares generated in the BH magnetosphere depends weakly on the mass of the BH and is determined by the disk magnetization, the viewing angle, and the pair multiplicity[Eq. (<ref>) does not account for relativistic effects that should be small unless the gap is formed close to the horizon. However, if the vacuum gap is close to the horizon, then the gravitational redshift should make more robust the constraints imposed by the variability time.]. Since all these parameters are smaller than unity, from Eq. (<ref>) we have L_γ, ms<2×10^43 erg s^-1 This upper limit is an order of magnitude below the required value <cit.>. Thus, we conclude that the ultrafast flare detectedfrom this source cannot have a magnetospheric origin.Assuming that mini-jets are distributed isotropically in the jet frame and that the detection of two pulses is not a statistical fluctuation, one can estimate the true jet luminosity using Eq. (<ref>). For the relevant flare parameters (i.e., t_ var=4.8min, L_γ=2×10^44 erg s^-1) and M_8=3L_ j,jj> 10^44Φ_-1(1+α^25)^4_ j,1^-2ξ_-1^-1 .If the mini-jets are not distributed isotropically,the requirement on the jet power can be a few orders of magnitude weaker; see Eq. (<ref>).The cloud-in-jet scenario requires a higher jet luminosity; from Eq. (<ref>) it follows that L_ j, cj > 3× 10^45Γ_ j,1^-2(1+α^25)^4ξ_-1^-1erg s^-1 .§.§.§ 87 In 2010, a bright flare has been recorded during a multiintrument campaign in the VHE energy band <cit.>. The variability time during the VHE transient was about 0.6 day and the flux level achieved 10^42erg s^-1. This source is characterized by a large jet-viewing angle of θ_ j≈ 15^o and a Lorentz factor of about Γ_ j≈ 7<cit.>,and the SMBH mass is ∼6×10^9M_<cit.>. Given the heavy central BH and the relatively long duration of the VHE flare, which allows high values of the gap size,the energy constraint in the magnetosphere scenario is quite modest: L_γ, ms<2×10^45 erg s^-1 .87 might be an interesting candidate for a detection of magnetosphere flares. For the flare parameters(i.e., t_ var=0.6d, L_γ=10^42 erg s^-1) and M_8=60, Eq. (<ref>) constrains the required jet true luminosity at the level L_ j,jj> 10^42Φ_-0.5(1+α^25)^4_ j,1^-2ξ_-1^-1 . On the other hand, the mulitwavelength properties of the gamma-ray flares detected from 87 seem to be quite diverse, with no detected robust counterparts at other wavelengths. Thus, if the VHE emission is produced by a single mini-jet,then a much weaker constraint, provided by Eq. (<ref>), is applied. In this case, the variability detected with Cherenkovtelescopes should correspond to the mini-jet variability, thus the mini-jet comoving size should be l̃_ em=Δ t c Γ_ em=2Δ t c Γ_ jΓ_ co 1+α^2∼10^17 cm ,which is about the jet cross-section at a parsec distance from the central BH. We should also note that the typical spectra emitted by plasmoidsare dominated by synchrotron radiation, which seems to be inconsistent with the multiwavelength observations of 87. Moreover,the peculiar light curve that has been detected withhas not yet been explained in the framework of the jet-in-jet scenario. Formally, for the parameters of the flare detected from 87, the minimum jet luminosity required by the cloud-in-jet scenario isL_ j, cj > 2×10^43Γ_ j,1^-2(1+α^25)^4ξ_-1^-1erg s^-1 .However, it has been argued that the light curve and the VHE spectrum is best explained if the TeV is produced through p-pinteractions induced by the jet collision with a dense cloud. In this case, the required jet power is aboutL_ j≈5×10^44 erg s^-1<cit.>. §.§.§ 4̧54 and In 2010 November, an exceptionally bright flare was detected from4̧54byand <cit.>.The minumum detected variability time and gamma-rayluminosity were 4.5 hours and 2× 10^50 erg s^-1, respectively. Several similarlybright flares were detected formin the period from2013 December to2014 April <cit.>. The GeV gamma-ray luminosity reached a level of 6× 10^48 erg s^-1, and the flux varied on a time scale of 0.7 hr. The magnetosphere gap luminosity is limited by 10^45 erg s^-1 and 2×10^44 erg s^-1 for 4̧54 and , respectively.Therefore the magnetospheric origin of these flares is excluded for both sources.and 4̧54 are distant quasars, thus it is safe to fix α=0 for both cases. By adopting a standard jet Lorentz factor,Γ_ j=20, one can obtain the jet luminosity required for the realization of the jet-in-jet scenario: L_ j,jj> 10^47Φ_-0.2_ j,1.3^-2ξ_-1^-1for 4̧54, andL_ j,jj> 3×10^45Φ_-0.5_ j,1.3^-2ξ_-1^-1for . Both these estimates appears to be below the corresponding Eddington luminosity limits of 1.3×10^47 erg s^-1 and 6×10^46 erg s^-1for M_ BH,8=10 and M_ BH,8=5 in 4̧54 and , respectively.The star-in-jet scenario requires higher jet luminosities, which seem to exceed the Eddington limit for 4̧54. Namely, one obtains L_ j, cj > 10^48Γ_ j,1.3^-2ξ_-1^-1erg s^-1 .This value agrees with the estimate L_ j, cj≈ 10^49erg s^-1 that is obtained within a more accurate model that provides also provides an interpretation for the so-called plateau phase and the spectrum <cit.>. Such a high luminosity of the jet, L_ j≈ 0.1 L_γ, has also beenobtained in the framework of the one-zone external Compton model of <cit.>.For , the lower limit on the jet luminosity isL_ j, cj > 4×10^46Γ_ j,1.3^-2ξ_-1^-1erg s^-1 ,which is close to the Eddington limit. A similar estimate was obtained by <cit.>.§ DISCUSSION AND CONCLUSIONS§.§ Gamma-ray Flare Detected from IC 310is a radio galaxy with redshift z≃0.0189<cit.>. Radio observations revealed an extended jet with a viewingangle of θ_ j< 30^∘<cit.>. Arguments based on the absence of the a contra-jet and assuming thatthe true length of the jet is smaller than 1 Mpc allowed to further constrain the viewing angle10^∘<θ_ j<20^∘<cit.>. Finally, observations of superluminal motion allowed us to constraint the Doppler factor to _̣ j∼5.The radio luminosity has been estimated to be at the level ofL_,radio≃10^41erg s^-1<cit.>. This implies a minimum energy in relativistic electrons of5.61×10^57 erg, thus the power required for the supply of emitting electrons yieldsL_,e≃2×10^42erg s^-1, and the total jet luminosity can be estimated asL_≃10^43 erg s^-1<cit.>. We note that these estimates represent values for the minimum required energeticsaveraged over 10^8 years.In 2012 November,detected a bright flare from <cit.>. The flare consisted of two sharp peaks with a typicalduration of ∼5 min. The measured spectrum was hard, with a photon index of≲2, extending up to ∼ 10TeV.The energy released during that event has been estimated to be at a level of 2×10^44 erg s^-1.The mass of the BH powering activity ofhas been determined to be M_=(3_-2^+4)×10^8M_<cit.>, i.e., the measured variability time scale is as short as 20% of τ_0; therefore one can expecta realization of some unconventional mechanism for VHE emission production. <cit.> have considered possible scenarios(see Sect. <ref>) for the flare production and found that jet-in-jet and star-in-jet interaction models face certain difficulties.Based on this, one concluded that the magnetosphere origin remains the only possible option, and no further verification of that scenariohas been provided.<cit.> have performed simulations of the gamma-ray spectrum produced in a stationary gap for different accretion rates and concluded that the emission generated in the vacuum gap could closely reproduce the spectral properties of the TeV emission detected during the flare. The strength of the magnetic field has been fixed at a level of 10^4G. <cit.> have emphasized that the generation of such a strong magnetic field requires an accretion rate exceeding the values compatible with the existenceof vacuum gaps by a factor of 100. In addition, we note that if the thickness of the gap is determined by the variability time, h≃ t_ varc, this scenario requires an even higher efficiency <cit.>.Since one does not expect any significant focusing or enhancement of the emission produced in the magnetosphere,the measured energy should correspond to the real energetics of the processes responsible for the emission generation.Thus, the feasibility of generating such a powerful flare in the vacuum gap is closely related to the general efficiencyof processes taking place in BH magnetosphere. Currently, the BZ mechanism <cit.> representsthe most prominent energy extraction mechanism that can operate in the BH magnetosphere. The efficiency of this mechanism isdetermined by the strength of the magnetic field that is accumulated at the BH horizon, which in turn is determined by the accretion rate.<cit.> argued that the efficiency of the BZ mechanism can be very high, up to a level of 900%, as compared tothe accretion rate Ṁc^2.This assumption is based on 2D simulations presented by <cit.>. However, 3D simulations for a similar setup presented inthe same paper reveal a significantly lower efficiency, ∼300%. We note that Eq. (<ref>) with α_ ss=0.1corresponds to an efficiency of 250%, which is very close to the results of the 3D simulations by <cit.>.According to the estimate provided by Eq. (<ref>), the possible luminosity of flares generated in BH magnetosphere dependsvery weakly on the mass of the BH and is determined by disk magnetization, viewing angle, and pair multiplicity[Eq. (<ref>)does not account for relativistic effects, which should be small unless the gap is formed close to the horizon. If the vacuum gap is closeto the horizon, then gravitational redshift should even strengthen the constraints imposed by the variability time.]. Since all theseparameters are smaller than one, then the numerical coefficient in Eq. (<ref>) can be taken as a strict upper limit for the flareluminosity for the given variability time. This upper limit appears to be approximately an order of magnitude below the value measured with<cit.>. Thus, we conclude that it seems very unfeasible that these processes are indeed behind the bright flaring activityrecorded from <cit.>.Our simplified analysis does not allow us to robustly rule out two other scenarios for the flare production in . If one adopts the minimum averaged jetluminosity as a reasonable constraint for the present jet luminosity <cit.>, then both scenariosformally do not allow reproducing the observed properties. However, if one assumes that there is some anisotropy in the mini-jet distribution,then jet-in-jet model provides an energetically feasible scenario. We note that the spectral energy distributions currently obtained forthe jet-in-jet models feature the dominant excess in the UV band <cit.>, which is not consistent with the observations from .This might be either a fundamental constraint or just a systematic underestimation of the inverse Compton contribution that is dueto the small scale of the simulations.Thus, it seems that more detailed large-scale simulations are required to verify the applicability of the jet-in-jet scenario for .On the other hand, if the present-day jet luminosity is significantly higher than the averaged value, the star-in-jet scenario may also meetthe energetic requirements. §.§ Comparison of Scenarios for Ultrafast VariabilityIn this paper we considered three scenarios for the production of ultrafast AGN flares with variability times shorter than the Kerrradius light-crossing time: gamma-ray emission of gaps in the SMBH magnetosphere <cit.>, the jet-in-jetrealization <cit.>, and the emission caused by penetration ofexternal dense clouds <cit.>.The production of gamma rays in the BH magnetosphere has several unique properties. In particular, this scenario can be invokedto explain emission from off-axis AGNs and orphan gamma-ray flares. On the other hand, the luminosity of the magnetospheric gap has a robust upperlimit that depends weakly on the SMBH mass. Moreover, the magnetospheric emission is not enhanced by the Doppler-boosting effect,and this seems to be crucial for explaining short flares from distant AGN. On the other hand, some nearby SMBHs<cit.>, e.g.,the Sagittarius A star or 87,might be very promising candidates to produce gamma-ray flares <cit.>.In general terms, there can be little doubt that the nonthermal radiation of powerful AGN is related, in one way or another, to relativistic jets. The ultrafast gamma-ray flares might be linked to the formation of relativistically moving features (plasmoids or mini-jets) inside the major outflow, the jet originating from the central black hole.Depending on the orientation of the mini-jets to the jet axis, the radiation of the mini-jet can be focused within the jet cone or outside.This scenario has been suggested to interpret the variable emission from AGN <cit.>. It has been shown that under certain conditions, magnetic field reconnection can result in the formation of relativistic outflows <cit.>. We note, however, that formation of a relativistic outflow is not an indispensable feature of reconnection. Thus, ejection of relativisitcally moving plasmoids may require a specific configuration of the magnetic field. Independently, to form outflows with large Lorentz factors, _ co≥10, an initial configuration with high magnetization, σ≃_ co^2≥100, is required. Such a high magnetization of the jet at the flare production site requires an even higher initial jet magnetization, σ_ init≫10^3. Jets with such a high magnetization should have an extremely low mass load, which seems to be inconsistent with the properties of AGN jets at large distances <cit.>.Finally, the SED of the emission produced by plasmoids formed at reconnection contains a dominating synchrotron component that peaks in the UV energy band <cit.>.This feature is not consistent with the SEDs obtained from AGNs during the ultrafast flares. The presence of a guiding magnetic field can significantly enhance the magnetization of plasmoids, resulting in a further enhancement of the synchrotron component and perhaps in the extension of the synchrotron component to the gamma-ray band. The examination of this scenario requires detailed modeling, since theguiding filed also impacts the Lorentz factor of plasmoids.The jet-in-jet scenario quantitatively implies a modest requirement for the jet intrinsic luminosity, however; it can be even further relaxed if one assumes that the mini-jets are not distributed isotropically in the major jet comoving frame. Such an anisotropy can be realized, for example, by focusing the outflow along the direction of the reconnecting magnetic field. An important issue to realize the jet-in-jet scenario what is the triggering mechanism for the reconnection. If the jet is launched by the BZ mechanism, it is expected to be magnetically dominated at the initial stage, thus reconnection is a thermodynamicaly favored process. However, the reasoning based on equipartition arguments, without any particular energy transfer mechanism, can hardly be valid. The time-scale of such thermodynamic processes may be enormous; thus they might be irrelevant for astrophysical jets. In recent years, significant progress has been achieved in PIC simulations for the reconnection in magnetic field configurations with alternating polarities. Such configurations should naturally appear in the pulsar outflows close to the current sheet. However, it is less obvious how such regions would form in AGN jets. Several scenarios can be considered.The first is a change in the magnetic field polarity in the jet caused by a change in magnetic field polarity in the accretion disk and, consequently, in the BH magnetosphere <cit.>. However, such a change takes a long time, of about ∼10^3r_g/c, and it is hard to expect ultrafast variability caused by such a configuration of the magnetic field. An arrangement with alternating magnetic field polarities can also be a result of a growth of MHD instabilities in the jet <cit.>. However, an intense instability growth leads to the a flow disruption on a scale of several dynamical lengths <cit.>. So it is hard to obtain an intensive reconnection event close to the base of a jet that extends a significant distance beyond the reconnection region. Finally, the reconnection can be caused by a sudden compression and mixing of a small part of the jet, which, for example, can be due to an external obstacles in the jet.In such a case, a short but intensive local reconnection episode may occur without disrupting the entire flow. This specific case represents an interesting synergy of two models: the formation of a relativistic mini-jet by reconnection of the magnetic field triggered by a star in the jet. The feasibility of this scenario needs to be tested with detailed numerical simulations.The star-in-jet scenario, the third possibility considered in the paper, requires significantly higher jet luminosity than the jet-in-jet scenario. In many cases, the jet luminosity, needed to realize the star-in-jet scenario, exceeds the Eddington limit. It was also shown that some details of the GeV light curve obtained from 4̧54 with , e.g., the plateau phase, can be readily interpreted in the framework of the star-in-jet scenario <cit.>. It is also important to note that the emission produced by the interaction of a cloud with the AGN jet should be characterized by a universal relation between the luminosity and the duration of individual peaks of the flare: L^1/2∝Δ t. To verify this relation observationally, a high photon statistics is required, which may possibly be achieved with future observations with CTA.§ CALCULATION OF THE PAIR MULTIPLICITY FROM A RADIATIVELY INEFFICIENT ACCRETION FLOW <cit.> have estimated the density of electron-positron pairs produced by photon-photon annihilation in a radiatively inefficient accretion flow <cit.>. For the sake of consistency, a similar consideration is present below, but for a Kerr BH and explicitly accounting for the nondimensional viscosity parameter α_ ss and radiation efficiency off an accretion flow η. Identically to <cit.>, we relie on the solution obtained by <cit.>, assuming that the advection parameter is small: 1-f≪1<cit.>.In particular, the radial velocity of the accretion flow is taken as v_r=3α_ ss/5(G M_ bh/r)^1/2 .The ion density n_i can be estimated as n_i(r)=Ṁ/4π rH m_p v_r = 5√(10)/6ṁ/ηα_ss(G M_ bh)^1/2/c σ_ T r^3/2 ,given the revealed height of the accretion disk, H≃ c_ s/Ω_ k<cit.>.The total cooling rate of the ion-electron plasma can be estimated <cit.> asq_ff= q_ee+q_ei≈ 10^-21 n_e^2θ_e ^-1^-3 for relativistic electron temperature θ_e = kT_e/m_e c^2≳1. Since the density of the ions exceeds the GJ density, the pair production does not provide any sensible contribution to the disk electron density in configurations allowing the existence of vacuum gaps, thus in what follows we assume the number densities of electrons and ions to be equal, n_i = n_e.For such a hot electron plasma the emission appears in the MeV energy band, and the luminosity of the inner part of the accretionflow is <cit.>L_ff≈∫_r_g^2r_g 2π r^2 q_ff dr .A lower limit on the number density of these MeV photons isn_γ≈L_ff/4π c (2r_g)^2 e_γ≈0.7 q_ff r_g^3/c r_g^2 e_γ∼ 10^9ṁ^2/α_ss^2η^2 M_8^-3,where e_γ=3θ_e m_e c^2. The production rate of e^± pairs inside the magnetosphere due to γγ-annihilation is approximately σ_γγ n_γ^2c(4π/3)(2r_g)^3, where σ_γγ≈σ_ T/5 is the cross-section of two-photon pair production.In steady state, this rate is balanced by the escape rate ∼ 4π c (2r_g)^2 n_±, allowung us to estimate the n_±≳ 10^6 ṁ^4/η^4α_ ss^4 M_ 8^-3.The GJ density (ignoring its polar angle dependence) is determined by the accretion rate, ṁ, via Eqs. (<ref>) and (<ref>):n_ GJ=Ω B/2π e c = 0.4 (β_m ṁ/ηα_ ss M_ bh,8^3)^1/2^-3.Thus, the multiplicity parameter is κ≳n_±/n_ GJ≈ 6× 10^6 ṁ^7/2 M_ bh,8^1/2/η^7/2α_ ss^7/2β_m^1/2 .The condition κ<1 place upper limit on the accretion rate as ṁ≲ 10^-2ηα_ ssβ_m^1/7/M_8^1/7.Eq. (<ref>) allows an estimation of the characteristic inverse Compton cooling time of electrons in the photon field provided by the disk. For an electron with energy E=10^6m_ec^2γ_6 λ_ IC= 3/4σ_tmc^2n_γ e_γ 10^6γ_6 ∼α_ ss^2η^2M_8θ_e4×10^8cmγ_6 ṁ^2≳0.2 r_ gγ_6M_8^2/7β_m^2/7θ_e .§ SMALL ANGLE LIMITIt is believed that the emission from AGNs is mostly detected by observers located within the jet-beaming cone θ≤_ j^-1, but there are also some examples when the flares are detected from off-axis radio galaxies, e.g., 87 and . Sometimes it is convenient to measure the viewing angle in the jet-opening units:θ=α_ j^-1 .In particular, this parameter gives a simple relation between the jet Doppler factor and the Lorentz factor:_̣ j=1_ j(1-β_ jcosθ)=2_ j1+α^2 ,which is valid for α≪_ j.If the production region moves relativistically in the jet, then its Lorentz factor with respect to the observer is <cit.> _ em=_ j_ co(1+β_ jβ_ cocosθ') ,where θ' is the angle between the jet velocity and the outflow velocity.This angle is related to the angle in the observer frame via the aberration formula,tanθ=β_ cosinθ'_ j(β_ cocosθ'+β_ j) .The latter equation allows us to express the angle in the comoving reference framecosθ'=-α^2β_ j/β_ co±√(α^4β_ j^2/β_ co^2+(1-α^2β_ j^2/β_ co^2)(1+α^2)) 1+α^2 .One should account for the kinematic constraints that naturally appear in the above equation: |cosθ'|≤1. Taking the solution withthe + sign (which corresponds to a stronger enhancement in the region where two solutions are allowed) in the limit _ j,co≫1,one obtainsβ_ cocosθ'≃1-α^2/1+α^2-1/2_ co^2 ,and consequently, the emitter Lorentz factor is_ em≃2_ j_ co 1+α^2 . § ACKNOWLEDGMENTSThe authors are gratefulto the anonymous referee forthe insightful comments and helpful suggestions.We would like to thank Andrey Timokhin and Frank Rieger for productive discussions. The authors appreciate the support by the Russian Science Foundation under grant 16-12-10443.D.K. acknowledges financial support by a grant-in-aid for Scientific Research (KAKENHI, No. 24105007-1) from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT). M.B. acknowledges partialsupportby the JSPS (Japan Society for the Promotion of Science): No.2503786, 25610056, 26287056, 26800159. M.B. also acknowledges MEXT: No.26105521 and for partial supportby NSFgrant AST-1306672 and DoE grant DE-SC0016369.apj | http://arxiv.org/abs/1704.08148v2 | {
"authors": [
"F. A. Aharonian",
"M. V. Barkov",
"D. Khangulyan"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170426145618",
"title": "Scenarios for ultrafast gamma-ray variability in AGN"
} |
III-V nanostructures have the potential to revolutionize optoelectronics and energy harvesting. For this to become a reality, critical issues such as reproducibility and sensitivity to defects should be resolved. By discussing the optical properties of MBE grown GaAs nanomembranes we highlight several features that bring them closer to large scale applications. Uncapped membranes exhibit a very high optical quality, expressed by extremely narrow neutral exciton emission, allowing the resolution of the more complex excitonic structure for the first time. Capping of the membranes with an AlGaAs shell results in a strong increase of emission intensity but also to a shift and broadening of the exciton peak. This is attributed to the existence of impurities in the shell, beyond MBE-grade quality, showing the high sensitivity of these structures to the presence of impurities. Finally, emission properties are identical at the sub-micron and sub-millimeter scale, demonstrating the potential of these structures for large scale applications.Keywords: GaAs/AlGaAs nano mebranes, photoluminescence, electronic and optical properties of ensemble vs single nano membraneNanowires (NWs) are filamentary crystals with a diameter in the sub-micrometer down to nanometer range. Their special morphology, dimensions and high surface-to-volume ratio are often translated into advantageous optical and electrical properties. As a consequence, they have been widely used in electronics <cit.>, optoelectronics<cit.>, solar cells<cit.> and sensors <cit.>. If not adequately passivated, the surface recombination can limit the optical performance of the NWs<cit.>. In addition, surface depletion can also affect the volume distribution and separation of the carriers in the NW<cit.>. Different passivation methods have been employed in the past, notably capping of the free surfaces with a higher bandgap shell around the nanowire<cit.>. Nevertheless, capping also modifies the nature of the surface. Several effects have been reported, such as band bending at the interface leading to the accumulation of the charge at the interface or piezo electric strain<cit.>. In addition, the AlGaAs alloy typically used for capping GaAs nanowires is generally inhomogeneous, with directed and random segregation of Ga and Al forming respectively Al-rich ridges and Ga-rich nanoscale islands <cit.>. Simultaneously, III-V NWs can suffer from twin defects and polytypism<cit.>, which adversely affect their electronic and optical properties<cit.>. With a judicious optimization of growth conditions, single NWs with a pure zinc-blende or wurzite structure can be obtained <cit.>. Still, the optical and electronic properties tend to fluctuate considerably from NW to NW, which precludes the proper control of the response of an ensemble of nanowires. Recently, alternative approaches to obtain defect-free nano structures have been proposed. Particularly promising is the inversion of polarity from B to A as well as template assisted and nano-membrane assisted selective epitaxy (TASE and MASE, respectively). All these techniques provide defect free III-V nano structures by blocking the formation of twinning defects<cit.>. An additional advantage of these approaches is the possibility to engineer the shape, so that membranes<cit.>, sails <cit.> or sheets<cit.> can be grown. Nanoscale membranes show relatively long minority carrier diffusion length of 180 nm at 4.2 K, which is significantly larger than the diffusion lengths found in nanowires <cit.>. Moreover, by introducing passivation and/or doped structures, the design can be further sophisticated <cit.>. The transfer of NW optoelectronic devices to industry requires achieving highly reproducible and uniform structures through a large surface area, so that the properties of ensemble and single object are indistinguishable. For instance, in photoluminescence this implies indistinguishability in terms of line width and emission energy and spectral shape. Growing the nano structures using TASE and MASE turned out to be the most promising direction to achieve large area highly uniform systems. In this work we demonstrate, by using optical techniques, that GaAs nanoscale membranes provide the settings for extremely high quality nanostructures, both from the structural and functional point of view. We elucidate how the improvement in functional properties is homogeneous across the whole wafer. This shows the potential of these nanostructures for nanotechnology and opens the path towards large scale nano-manufacturing. Furthermore, we provide very strong evidence that capping of the membranes, despite increasing the emission efficiency, unexpectedly leads to the degradation of their optical properties. Nanomembranes have been grown using selective area epitaxy (for growth details see methods and reference<cit.>). In Fig. <ref>(a) a tilted SEM image of a GaAs nanomembrane array consisting of 10μm long and 100 nm wide nanomembranes with 500 nm pitch, used in the further optical experiments, are shown. Pitch is defined as the distance between the membranes, as depicted in Fig. <ref>(b). Nanomembranes are oriented in ⟨21̅1̅⟩ direction which is perpendicular to (111)B and (1̅1̅0) directions and expose the facets shown in Figure <ref>(c). Most of the facets belong to {110} family except high index top facets of (1̅13̅) and (1̅3̅1). Adjusting the membrane orientation to ⟨21̅1̅⟩ and growth conditions, it is possible to obtain pure zinc-blende structures with high-aspect ratio with Molecular Beam Epitaxy (MBE)<cit.>. Detailed growth conditions are given in the Method section. The reported shape is the result of an hour growth with 1 Å/s growth rate. If growth is continued long enough, the morphology of the membrane evolves into a triangular shape. During the growth of AlGaAs shell the (1̅1̅0) facet transforms to (2̅2̅1).Typical normalized μPL spectra of a single uncapped GaAs and capped GaAs/Al_xGa_1-xAs nano membranes are presented in Fig. <ref>(a). For the capped membranes the data have been taken for three different compositions of the shell (x=15,33,50%). Overall, the emission spectra are composed of two bands, around 1515 and 1490 meV. The higher energy band corresponds to the band-edge luminescence of the GaAs membranes, while the lower energy emission can be attributed to the donor acceptor transitions due to carbon impurities normally present in commercial GaAs substrates<cit.>, which was further observed in detailed cathodoluminescence studies. Our spectra are comparable to previously reported optical emission in nano membranes with 33% Al composition in the shell<cit.>. The peak related to the carbon impurities can be used as a reference for the luminescence intensity. After capping, the emission from the GaAs membrane increases dominating the carbon related PL.The dramatic increase of the emission from the membrane is a direct consequence of the surface passivation that reduces the non radiative surface recombination.The detailed nature of the emission is very different for capped and uncapped samples. For uncapped membranes the spectrum is composed of three lines (see Fig<ref>(b)). The peak at the highest energy of ∼1515.5 meV corresponds to the free exciton emission, while the two peaks at lower energies are related to neutral donor bound exciton emission (D^0-X) and acceptor bound exciton emission (A^0-X) with emission energies which are typical for bulk GaAs <cit.>. This result rules out any possible quantum confinement in the nano membranes. This is not unexpected since the exciton Bohr radius of ≃ 14 nm in GaAs is much smaller than the size of the membrane<cit.>. In contrast, the typical emission spectra for the GaAs nano membranes capped with Al_xGa_1-xAs layer (Fig. <ref>(b)) are composed of a single line, which we attribute to the neutral exciton recombination. Emission lines from D^0-X and A^0-X are completely absent. The neutral exciton emission energy red shifts and broadens with increasing Al shell content. To quantify this effect we have measured the power dependence of the energy and full width at the half maximum (FWHM) of the neutral exciton emission. In Fig <ref>(c) the emission energy is plotted as a function of excitation power. For membranes with high aluminium shell content (x ≥ 0.3) a blue shift is observed with increasing excitation power which quickly saturates for powers above a few μW. For powers of 10μW and above the emission energy is independent of the excitation power. There is a clear and systematic decrease in the emission energy (red-shift) with increasing Al content. This is illustrated in the inset in the Fig. <ref>(c), where the energy difference between uncapped and capped emission Δ E is plotted as a function of the shell aluminium composition x for the same excitation power. In Fig. <ref>(d) the FWHM of the emission is plotted as a function of the excitation power. The line widths increase slightly with increasing power, but this is negligible compared to the increase in the FWHM with increasing Al content of the shell. In the inset of Fig. <ref>(d) we plot the FWHM versus the shell Al content for an excitation power of 10μW. The linewidth is multiplied by roughly a factor of 5 between the uncapped membrane and the membrane with a 50% Al content cap layer. Thus, while capping the membranes reduces non-radiative surface recombination, leading to enhanced neutral exciton emission, it also detrimentally affects the optical properties of the GaAs core, leading to a significantly broadened emission.We turn now to the effect of the red-shift of the excitonic emission upon capping the membranes with AlGaAs. In fact, a similar effect has been observed previously for a simple AlGaAs/GaAs interface <cit.>, InP nanowires, <cit.> and for GaAs nanowires capped with AlGaAs shell <cit.>. For simple AlGaAs/GaAs the band bending at the interface forms a pocket for the electrons or holes <cit.>. Such confined carriers at the interface will recombine with the free carriers (of the opposite species) in the valence or conduction band at a sufficient distance from the interface that flat-band conditions have been re-established. As the charges are spatially separated, emission has a spatially indirect character and is red shifted in comparison to the simple excitonic emission observed in uncapped GaAs. Moreover, the band bending can be screened by photo created carriers decreasing the overall effect with the increase of the excitation power. For InP nanowires a similar picture has been proposed, where the band bending was induced by a pinning of the Fermi level<cit.>. Finally, for GaAs nanowires capped with AlGaAs shell, the mechanism of the band bending can be enriched by strain, related to the shell thickness <cit.>. However, the strain plays a significant role only for rather thick shells. In the case of the nano membranes the core is much thicker than the shell. Additional confirmation of the negligible role of the strain in our structures is given by the Raman spectroscopy. If the shift we observe originated from strain, it would imply a significantly lower Al composition than the nominal composition <cit.>. Our Raman measurements, (see SI), confirm that the Al composition corresponds very well to the nominal composition in the nano membranes and the lack of strain in the membrane core.We attribute the observed red shift of the emission to the indirect nature of the exciton recombination at the capping interface. Due to residual doping in the AlGaAs shell, band bending occurs at the AlGaAs/GaAs interface. To this end, we illustrate in Fig 3(a) the position of the valence and conduction band edges as a function of the distance from the membrane surface. Our hypothesis is that the AlGaAs shell contains some oxygen impurities, associated with the addition of aluminum. Secondary ion mass spectroscopy measurements on AlGaAs layers indeed indicate a slight O-contamination associated with Aluminum (see SI). This contamination is still better than the purity specifications of MBE-grade Aluminum, 6N5, which implies that nanostructures are much more sensitive than bulk structures to impurities. Thus, the optical response of high quality nano structures provides a sensitive means to detects extremely low levels of impurities. The red-shift of the luminescence at high excitation powers is larger for higher Al content (see Figure <ref>(a)). This shows, that the band bending increases with the Al content in the shell as the exciton recombination becomes more indirect.Our observations are further supported by the simulation of the band bending at the AlGaAs/GaAs interface by solving Poisson and Schrödinger equations self-consistently with the software nextnano3. In the model we have included the presence of p-type interface states between GaAs and AlGaAs shell, which increases with increasing Al content. Our experimental data fits well with 2 × 10^9, 6 × 10^9 and 8 × 10^9 cm^-2 interface dopants for an Al concentration of 15%, 30% and 50% respectively. Fig. <ref>(b) shows the resulting band bending at the tip of the membrane as a function of the distance to the surface and for the three Al contents. Here is evident the presence of a triangular potential in the valence band at the interface GaAs/AlGaAs where holes can be trapped. We can also observe an increase of the height of the potential with Al content, which results in a red-shift of the indirect transition.It is worth noting, that the red shift observed in our samples is of comparable magnitude with that observed by Songmuang at al <cit.> but much smaller than that reported by Dhaka et al <cit.>. This discrepancy can be partly ascribed to the MetalOrganic Vapor Phase Epitaxy (MOVPE) employed by Dhaka et al <cit.> to grow their nanowires.MOVPE involves the use of metalorganic species as group III precursors, which might introduce an unintentionally high concentration of impurities.The small blue shift observed at low powers, which saturates around 10 μW has been also observed for GaAs nanowires capped with AlGaAs shell <cit.> and it was associated with the presence of some negatively charged traps at the interface, which are filled by photo created carriers in the AlGaAs shell, which migrates towards interface. Once filled, they can no longer modify the band bending at the interface, which explains the saturation of the blue-shift of the emission energy above 10 μW, indicating that the band bending is the dominant effect in our nanomembranes.The special geometry of the nanoscale membranes requires some further modeling. First, the non-flat geometry of the interface should result into a spatially dependent band bending. In addition, the vertical nature of the membranes can additionally lead to non-homogeneous light absorption <cit.>. Fig. <ref>(c) shows the 2D valence and the conduction band maps for an Al concentration of 50%. We can observe a band-bending at the interface which is significantly larger at the top corner of the nanomembrane. We have simulated the electromagnetic field distribution using the package Meep, a freely available software implementing the Finite Difference in Time Domain Method <cit.> taking into consideration the exact geometry of the core/shell nanomembrane with a shell of 30% of Al. The dielectric constant is taken from Ref. [<cit.>]. Fig. <ref>(d) shows the cross-sectional map of the computed electric field energy density for a nanomembrane under the presence of a monochromatic wave coming from the top and with parallel polarization. It is clearly seen that the field energy is not distributed evenly across the cross-section but is rather confined at the top edge of the nanomembrane. This means that our μ photoluminescence experiments mainly probe the exciton properties at the tip, where the band bending is more pronounced. The results of this simulation explains also the broadening of the emission with the increasing Al content. Although emission is probed locally, the probed region can contain non homogenies band bending leading to the broadening of the emission peak. This is in perfect agreement with the observation that the effect is the strongest for the highest Al composition.Finally, we come to perhaps the most striking and novel property of these nano membranes, namely their reproducibility and large scale uniformity.While epitaxial MBE provides highly uniform growth, this is not the case for the self organized growth of quantum dots or NWs, where nucleation events in growth follow poissonian statistics that lead to a distribution in the properties. As an example, in NWs this leads to a twinning or stacking fault density that varies from NW to NW (complete defect-free structures are rare). As a result, the optical properties vary from NW to NW and macro-photoluminescence measurements of the ensemble normally do not match micro photoluminescence of a single NW. We have recently shown that MBE growth using selective area epitaxy can produce arrays of defect free nano membranes<cit.>. However, optical investigations were limited to PL of a single nano membrane so that the uniform optical properties of an ensemble has never been demonstrated. To demonstrate large scale uniformity, we compare the emission spectra of a single membrane with the ensemble emission of around 250 membranes measured using macro PL, achieved here by defocussing the laser spot. Representative PL spectra are shown in Fig. <ref> for the capped and uncapped membranes. Defocussing increases the contribution of the substrate which is reflected in the slightly increased amplitude of the carbon related emission which can be seen in Figure <ref>. The substrate PL is dominated by the carbon related emission and free exciton emission is not observed from the substrate.We have mapped the luminescence properties of the membranes by cathodoluminescence in a previous work <cit.>. These measurements confirm that the carbon-related peak originates solely from the substrate. Surprisingly, the PL originating from single membrane is almost identical to the ensemble emission. The carbon impurity emission is slightly enhanced in the ensemble emission of the capped samples (≃ 20% for the 50% Al membrane). This is probably due to the inhomogeneous distribution of the carbon impurities across the substrate. In contrast, the neutral exciton emission is strictly identical in both the energy of the emission and the line width for all samples. In the uncapped sample, the neutral and bound exciton emission is also almost indistinguishable (see inset Figure <ref>(a)). The identical emission from a single and an ensemble of membranes unequivocally demonstrates the very high quality of the crystal structure and extremely high reproducibility of the nano membranes, which has never been observed for the classical radial nanowires. Moreover, data measured at different places on the same membrane, and on different membranes, vary very little in intensity, energy position or line width, suggesting an excellent crystal quality of the uncapped membranes. In conclusion, we have demonstrated luminescence properties of GaAs membranes which are on a par with the best two-dimensional layers obtained with MBE.Upon capping of the membranes with an AlGaAs layer the PL emission is strongly enhanced, but also unexpectedly accompanied by a degradation of the optical properties with a significant broadening of the exciton emission. Capping also leads to a red shift of the emission which has been attributed to the residual carbon doping of Al-containing layers which leads to band bending at the AlGaAs/GaAs interface. The quality of the membrane growth process is further supported by ensemble measurements, which are almost indistinguishable from the single membrane results. Additionally, our results show an extreme high sensitivity of the optical response of the nano membranes on impurities concentration that goes beyond what is possible in terms of state of the art high purity MBE. ASSOCIATED CONTENTAdditional characterization of membranes, inelastic light scattering (Raman), secondary ion mass spectroscopy (SIMS) profile (PDF). ACKNOWLEDGEMENTSThis work was partially supported by ANR JCJC project milliPICS, the Région Midi-Pyrénées under contract MESR 13053031, BLAPHENE project under IDEX program Emergence § METHODS GaAs nanomembranes used in that study are grown with a DCA solid source Molecular Beam Epitaxy (MBE) system. Substrates are PECVD deposited SiO_2 masked (111)B GaAs. The oxide thickness is 30 nm. The growth mask is patterned with a combination of e-beam lithography and fluorine based dry etching as reported earlier <cit.>. The growth temperature is 635 ^∘C, the growth rate is 1 Å/s and the V/III ratio is 10 for the GaAs core. The length of nanomembranes and the distance between them are defined by patterning the SiO_2 mask. We focused our characterization on structures having structures with 100 nm width, 500 nm pitch and 10μm length are studied. In the case of capped GaAs nanomembranes, the structures are capped with a shell of Al_xGa_1-xAs. ( x = 0.15, 0.3 and 0.5) The substrate temperature is reduced to 460 ^∘C and As flux is increased to 1 × 10^-5 torr for shell growth. Nominal thickness of AlGaAs shell is always 50 nm and it is protected with 10 nm of GaAs against oxidization. Aluminum ratios and nominal AlGaAs layer thicknesses are deduced from RHEED calibrations performed on (100) GaAs substrates. For the measurements the samples were placed in a helium flow cryostat with optical access. The cryostat was mounted on the motorized x-y translation stages, which allows high resolution spatial mapping. A microscope objective 50×with a numerical aperture NA = 0.55 was used to focus the excitation beam and collect the PL from the nano membranes. The laser spot could be focussed down to a diameter of ≃ 0.5μm (diffraction limit), which enabled us to optically address single (or a maximum of two in a worst case scenario). To investigate many membranes the laser spot was defocussed. The steady-state μPL signal was excited with a 532 nm laser and the spectra were recorded using a spectrometer equipped with a liquid nitrogen cooled CCD camera. All the measurements presented here have been performed at T=4.2 K. | http://arxiv.org/abs/1704.08477v1 | {
"authors": [
"Z. Yang",
"A. Surrente",
"G. Tutuncuoglu",
"K. Galkowski",
"M. Cazaban-Carraze",
"F. Amaduzzi",
"P. Leroux",
"D. K. Maude",
"A. Fontcuberta i Morral",
"P. Plochocka"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170427084229",
"title": "Revealing large-scale homogeneity and trace impurity sensitivity of GaAs nanoscale membranes"
} |
I. [][60]Fe transport to the solar system by turbulent mixing of ejecta from nearby supernovae into a locally homogeneous interstellar medium Zentrum für Astronomie und Astrophysik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany [email protected] Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstraße 12–14, 69120 Heidelberg, GermanyNumerical studies on the link between radioisotopic signatures on Earth and the formation of the LB. I M. M. Schulreich et al.The discovery of radionuclides like [][60]Fe with half-lives of million years in deep-sea crusts and sediments offers the unique possibility to date and locate nearby supernovae.We want to quantitatively establish that the [][60]Fe enhancement is the result of several supernovae which are also responsible for the formation of the Local Bubble, our Galactic habitat. We performed three-dimensional hydrodynamic adaptive mesh refinement simulations (with resolutions down to subparsec scale) of the Local Bubble and the neighbouring Loop I superbubble in different homogeneous, self-gravitating environments. For setting up the Local and Loop I superbubble, we took into account the time sequence and locations of the generating core-collapse supernova explosions, which were derived from the mass spectrum of the perished members of certain stellar moving groups. The release of [][60]Fe and its subsequent turbulent mixing process inside the superbubble cavities was followed via passive scalars, where the yields of the decaying radioisotope were adjusted according to recent stellar evolution calculations. The models are able to reproduce both the timing and the intensity of the [][60]Fe excess observed with rather high precision, provided that the external density does not exceed [0.3]cm^-3 on average. Thus the two best-fit models presented here were obtained with background media mimicking the classical warm ionised and warm neutral medium. We also found that [][60]Fe (which is condensed onto dust grains) can be delivered to Earth via two physical mechanisms: either through individual fast-paced supernova blast waves, which cross the Earth's orbit sometimes even twice as a result of reflection from the Local Bubble's outer shell, or, alternatively, through the supershell of the Local Bubble itself, injecting the [][60]Fe content of all previous supernovae at once, but over a longer time range.Numerical studies on the link between radioisotopic signatures on Earth and the formation of the Local Bubble M. M. Schulreich<ref> D. Breitschwerdt<ref> J. Feige<ref> C. Dettbarn<ref> Received / Accepted =============================================================================================================§ INTRODUCTION Our solar system resides in a low-density region of the interstellar medium (ISM), partially filled with hot, soft X-ray emitting plasma, known as the Local Bubble (LB). While the existence of this superbubble (SB) is widely accepted today, not least because it was found to be responsible for about 60 % of the 1/4-keV flux in the Galactic plane <cit.>, its exact origin is still a matter of ongoing debate. The LB extends roughly [200]pc in the Galactic plane, and [600]pc perpendicular to it, with an inclination of about 20^∘ relative to the axis of Galactic rotation <cit.>. If we assume that it was created by several nearby supernova (SN) explosions in the last ≃[10]Myr <cit.>, we are faced with the problem that no young stellar cluster could be found inside its boundaries.One obvious way out of this dilemma was to scan the solar neighbourhood for moving groups of stars that might have once crossed theboundary of the LB and, additionally, provided an adequate number of SNe in order to explain both the present LB extent and the detected soft X-ray emission. For that purpose, <cit.> calculated stellar trajectories of the nearby Pleiades subgroup B1 backwards in time. Applying an integer-binned initial mass function (IMF) they inferred the number of SNe in the past, determined their explosion sites from the trajectories, and their explosion times from their main-sequence life times (assuming that all stars in the cluster are born coevally). Similar conclusions were drawn from an independent study by <cit.>. Later, <cit.> performed an ab initio search scrutinising all stars with sufficient kinematical data available within a heliocentric volume of [400]pc diameter. Using the HIPPARCOS catalogue and a compilation of radial velocities, they came up with a sample of B stars concentrated in both real and velocity space, which were then identified as members of the Sco-Cen OB association. Tracing back in time this sample,found that the stars had entered the present LB volume rather off-centre 10 to [15]Myr ago, and that since then, 14 to 20 SN explosions should have occurred, which delivered enough energy to explain the current size of the LB and its ionisation state.A deeper understanding of the nonlinear evolution and structure of the ISM in general and the LB in particular can only be gained from solving the full-blown set of hydrodynamical (HD) and magnetohydrodynamical (MHD) equations through numerical simulations. While early studies of the global ISM could only consider very few physical processes on small, two-dimensional domains at low numerical resolution <cit.>, the advance in computational capabilities not only allowed for the inclusion of additional physics (like magnetic fields) and larger-scale domains <cit.>, but eventually also the leap into the third dimension <cit.>. More recent works can be roughly divided into three classes according to their box size (and the highest resolution associated with it): small-scale <cit.>, meso-scale <cit.>, and large-scale <cit.>. A physically realistic three-dimensional model that covers all dynamical ranges of an entire galaxy is still out of reach, though there are efforts to achieve this to some degree by coupling the adaptive mesh refinement (AMR) technique with zoom-ins to specific regions <cit.>.Taking their modelled ISM as a typical background medium, <cit.> further tested the aforementioned LB formation scenario under more realistic conditions. They found that the bubble's extension, as well as its ion column density ratios <cit.>, matched ultraviolet data, as obtained e.g. with FUSE <cit.>, remarkably well. Another interesting way to validate evolution models for the LB is to search for traces of the involved SNe on Earth, manifesting themselves as geological isotope anomalies. <cit.> identified several possible target isotopes for such a study, including [][60]Fe <cit.>, which is believed to be produced primarily in core-collapse and electron-capture SNe of massive stars, with mass M≳[8]M_, via successive neutron capture on preexisting Fe isotopes, both before and during the explosion <cit.>. Other suggested sources are asymptotic giant branch <cit.> and super-AGB <cit.> stars, as well as a rare class of high-density thermonuclear SNe <cit.>.Owing to its comparatively long half-life of t_1/2≃[2.6]Myr <cit.>, ejected [][60]Fe survives long enough not only to travel significant distances (a few hundred pc) through the ISM, but also to be detectable by its β^- decay via [][60]Co and gamma-ray emission at 1173 and [1333]keV. Measurements with the INTEGRAL satellite <cit.> revealed an average photon flux per gamma-ray line of about [4.4× 10^-5]cm^-2 s^-1 rad^-1 for the inner Galaxy region, which might be translated into a steady-state Galactic [][60]Fe mass of [1.53]M_ <cit.>. Very recently, <cit.> succeeded in directly detecting [][60]Fe nuclei in Galactic cosmic rays, showing that [][60]Fe sources must have been within the distance the high-energy particles can travel for the duration of t_1/2, which is typically less than about [1]kpc.However, the half-life of [][60]Fe is very short when compared to the age of the solar system (≃[4.5]Gyr), implying that the [][60]Fe, which, as we know from the analysis of primitive meteorites <cit.>, was present at about that time, has completely decayed away. This adds to the fact that the isotope's natural terrestrial background, arising mainly from the influx of pre-enriched extraterrestrial material (e.g. dust and meteorites) onto the Earth, and to a lesser extent from cosmogenic spallative production due to the steady cosmic-ray flux into the Earth's atmosphere, as well as in situ production via the penetrating muon and neutron flux, is low <cit.>. Motivated by this, <cit.> used accelerator mass spectrometry to search for an [][60]Fe excess in a ferromanganese (FeMn) crust from the South Pacific Ocean floor and indeed found one in two layers dating back about [6]Myr. In a more detailed analysis of another FeMn crust sample (termed 237KD), stemming from the equatorial Pacific, <cit.> narrowed down the [][60]Fe anomaly to the time interval 1.7–[2.6]Myr before present <cit.>. This definitive signal was corroborated in another study of 237KD conducted by <cit.>, who also looked into a different geological reservoir: marine sediments. These grow about a thousand times faster than FeMn crusts and thus have a far better time resolution, albeit the isotopic concentration of each layer is lower. Unfortunately the sediment core analysed by <cit.>, which originates from the North Atlantic Ocean and spans the same time interval as the crust 237KD, did not show a compatible [][60]Fe signal.In contrast, <cit.> found a signal consistent with the expected time from the FeMn crust study in four sediment cores from the Indian Ocean. Remarkably, the signal is broader, rather pointing towards an accumulation of SNe than a single event. Apart from the sediment cores,also analysed two FeMn crusts (237KD not included) and two FeMn nodules stemming from the Pacific and Atlantic oceans, respectively. Since each of these deep-sea archives contain elevated levels of [][60]Fe for the time range 1.5–[3.2]Myr ago, the authors further concluded that the [][60]Fe signal must be global. For one of the two FeMn crusts they also detected an additional smaller [][60]Fe peak 6.5–[8.7]Myr ago, whose exact origin is yet rather elusive. Remarkably, a collision of asteroids in the main belt [8.3]Myr ago, connected to a boosted, possibly [][60]Fe enriched, influx of interplanetary dust particles and micrometeorites <cit.> falls within this particular time range.Returning briefly to the subject of marine sediments it should be added that these also harbour so-called magnetotactic bacteria, which gather iron from their environment to create nano-sized crystals of magnetite (Fe_3O_4). The latter are used by the microbes to orient themselves within Earth's magnetic field, and thus to navigate to their preferred conditions – a behaviour known as magnetotaxis. <cit.> speculated that nearby SN acitivity should have left its mark also in the fossilised remains of such bacteria. In order to prove this hypothesis, they analysed magnetofossils of a sediment core from the eastern equatorial Pacific. The results <cit.> indeed show a comparatively weaker [][60]Fe signal in layers spanning a time range of about [1]Myr, with the maximum again located at ≃[2.2]Myr ago.In view of all these discoveries it seems little surprising that several of the lunar rocks that were recovered in the course of the Apollo missions also showed an enhanced concentration of [][60]Fe <cit.>. Unfortunately, time-resolved studies are impossible as our cosmic neighbour is regularly hit by a broad range of impactors <cit.>, which continuously churn up its surface and mix different layers. Besides this so-called gardening, there is also almost not sedimentation on the Moon. Lunar samples can hence only provide a first hint for the existence of a signature <cit.>.Theoretical attempts to link the FeMn crust measurements to the formation of the LB, but also to a single SN event, were made by <cit.> and <cit.>, respectively, both on the basis of analytical calculations.particularly used an SN model developed by <cit.>, which does not assume a homogeneous density distribution for the blast wave to expand in, but rather a medium that had already been modified by a previous SN event. More specifically it was assumed that the first SN runs into a homogeneous ISM, whereas all subsequent SNe occur within the spherical region excavated by the first SN, while being exposed to a radial density profile of the form ρ(R)∝ R^9/2. Following <cit.>, an IMF with one star per mass bin was used, and the most probable explosion sites were calculated. Including isotopic yields from stellar evolution calculations, the measured [][60]Fe profile could be reproduced with surprisingly high accuracy.Results of such analytical models should however be treated with some caution, since the similarity solutions applied have a number of serious deficiencies. First, the external pressure is required to be small compared to the pressure in the expanding bubble. This poses no strong limitations for the modelling of a second or subsequent SN event, whose blast wave takes only little time to move through the hot and thin pre-stratified medium. However, the situation is completely different for a single SN, or the very first one in a series, whose blast wave must penetrate through a totally inhomogeneous ISM and thus would soon undergo strong distortions. By the same token, the density distribution of the ambient medium has to be either homogeneous or obey a power law, implying that possible dense obstacles are not taken into account. This includes the LB's outer shell, which was neglected in the preliminarily work by <cit.>, but was later implemented as a <cit.> wind bubble for the improved analytical study contained in <cit.>. Second, the external medium is taken to be constant over time. In reality, however, the medium changes after each SN.(and also ) at least switched from a homogeneous to a power-law density distribution after the first SN went off. Nevertheless, they assumed that all subsequent explosions occur in the same environment. Last but not least, turbulent mixing and mass loading, as present in real SBs, are not taken into account in those models. A definitive answer to the question whether the observed [][60]Fe excess is explainable in the context of the LB formation can therefore only be given through three-dimensional high-resolution numerical simulations. This paper details on our first step working towards that goal <cit.>. For this purpose we built upon the numerical simulations of , presenting an additional homogeneous model (see Table <ref>, model A). As it will be shown this sets a lower limit for possible homogeneous media. We particularly used the masses of the same 16 SN progenitor stars, together with the explosion times, total ejected masses, and [][60]Fe mass fractions derived from there. The applied SN explosion sites, on the other hand, result from the most probable stellar trajectories (see also ).Like <cit.>, we studied the formation and evolution of the LB not in isolation, but in concert with the neighbouring SB Loop I. For the sake of conciseness, we currently neglected the complex multiphase nature of the (local) ISM and considered instead only homogeneous background media, including the one already used in . Deeper conclusions can be drawn now, however, as we not only traced the global[][60]Fe dynamics in our solar neighborhood but also the contributions of the individual SNe. This was achieved by tagging each of them with their own passive scalar (or tracer) – a quantity that behaves like a drop of dye when dispersed in a liquid. The influence of the inhomogeneous SN-driven environment, already teased in , will be extensively analysed in the forthcoming Paper II, which will also contain a discussion of the properties of turbulence in such a self-consistently evolved medium. The article is organised as follows. Section <ref> describes the model setup, in Sect. <ref> our results with respect to SB evolution and comparison to the FeMn crust measurements are discussed, and we close with a general discussion and conclusions in Sect. <ref>.§ MODEL DESCRIPTION Our HD simulations were carried out with the tree-based AMR code Ramses <cit.>, which allows for solving the discretised Euler equations in their conservative form by means of a second-order unsplit Godunov method for perfect gases. In particular, we employed the MUSCL-Hancock scheme<cit.> together with the Local Lax-Friedrichs (or Rusanov) approximate Riemann solver <cit.> and the MinMod slope limiter <cit.>. The computational domain is cubic with [3]kpc side length and covered by a base cartesian grid of 64^3 uniform cells. In regions of steep density and pressure gradients, the grid was recursively refined up to six additional levels, resulting into a peak spatial resolution of [0.7]pc. This high resolution was also kept at the coordinate origin, which marked the location of Earth (or Sun; for a distinction one would require several additional levels of grid refinement, and thus much more computing time). The grid was initially filled with a uniform and static medium. The adopted values for the gas temperature T, particle density n, and metallicity (in solar units) Z/Z_ in our two best-fit models are given in Table <ref>. They actually mimic two specific `phases' of the classical three-phase ISM model by <cit.>, namely the warm ionised medium (model A) and the warm neutral medium (model B). The SB-generating SNe were initialised as blast waves right at the start of their Sedov-Taylor phase, each liberating a canonical explosion energy of E_SN=[10^51]erg. The self-gravitating gas was exposed to an isotropic radiation field (that includes absorption through the Galactic plane as well as the cosmic microwave background), but was also allowed to loose energy via various interstellar cooling processes. This energy gain and loss was modelled under the assumption of collisional ionisation equilibrium (CIE) using the temperature-dependent net cooling functions generated for several gas densities with the spectral synthesis code Cloudy <cit.>. Theboundaries of our computational domain were formally periodic, but were actually never reached by expanding shells during the time frame considered. As a further `boundary condition', we set up a neighbouring SB, intended to represent Loop I. This is required since ROSAT observations have shown that soft X-rays are absorbed by a nearby neutral `wall' enclosed by an even denser ring-like structure, which is most likely the result of an interaction between the LB and the Loop I SB <cit.>.§.§ Setting up the Loop I superbubble For identifying possible SN progenitor stars of Loop I, we performed a search for B stars analogous to the one of the LB <cit.>, but for a spherical search volume of twice the size. Since less kinematical data is available for stars at larger distances, it becomes increasingly difficult to find bright stars. It turns out that Tr 10 and the association Vel OB2 have recently passed through the present volume of Loop I rather off-centre. Like for the LB this is not a problem per se, as the extension and morphology of evolved SBs are primarily determined by the ambient density and pressure distribution.Given a sample of 80 stars that might have entered this volume about 12.3 Myr ago (main sequence lifetime of a still existing [8.2]M_ star), we first needed to estimate how many have already exploded. For that purpose, we fitted an IMF that is typical for young massive stars with initial masses M,d𝒩/dM=. d𝒩/dM|_0M^Γ-1 ,where Γ=-1.1± 0.1 <cit.>. With the lower and upper limit of the relevant mass range being defined by A0 and B0 stars, respectively (i.e. M_l=[2.6]M_ and M_u=[8.2]M_), we obtained the normalisation constant via integration of Eq. (<ref>);. d𝒩/dM|_0 =𝒩[∫_2.6^8.2M^-2.1 dM]^-1 = 80[2.6^-1.1-8.2^-1.1/1.1]^-1= 351 . The lifespan of the first and most massive star that exploded in Loop I might be expressed asτ_M_t'=τ_u-t' ,where τ_u is the lifetime of a star of mass M_u and t' is the time at which the association entered the present volume of Loop I. When quantifying the main sequence lifetime of stars in the mass range 2≤ M≤[67]M_ by a fit to the isochrone data of <cit.>,τ=τ_0 M^-β ,where τ_0=[1.6× 10^8]yr and β=0.932 <cit.>, Eq. (<ref>) translates intoM_t'=(M_u^-β-t'/τ_0)^-1/β .As t'=[12.3]Myr, the mass of the most massive star that has already exploded is M_t'=[19.2]M_. One therefore finds for the number of missing stars𝒩_SN =∫_8.2^19.2. d𝒩/dM|_0M^-2.1 dM=351[8.2^-1.1-19.2^-1.1/1.1]=19 . Obviously the masses of these SN progenitors should be based on the IMF introduced above. In order to obtain the statistically most probable distribution <cit.>, we performed the mass binning by assigning exactly one star to each mass interval of the IMF, where the actual stellar masses are taken to be the average mass in each bin. Analogously, <cit.> (and also ) derived the initial masses of the LB stars, which we also employed in this work. Using Eq. (<ref>), we obtained for the mass of a particular star lying in the mass bin between M_1 and M_2 (>M_1),M_* =1/2(M_1+M_2)=1/2{M_1+[M_1^-1.1-1.1(. d𝒩/dM|_0)^-1]^-1/1.1} . These initial masses served as a basis for determining the total ejected masses <cit.> as well as the [][60]Fe mass fractions (see Sect. <ref>). The explosion times, on the other hand, are defined by t_exp=τ-τ_c, where τ is as given in Eq. (<ref>) and τ_c is the age of the cluster, which should be comparable to the lifespan of a [8.2]M_ star, i.e. τ_c=[23]Myr. For arranging the stellar explosion centres in space, we calculated the trajectories of all known B stars of Tr 10 and Vel OB2. Since not all of the member stars pass through the Loop I volume we selected for the 19 SN progenitors only the closest to the centre. A full list of the Loop I input parameters is given in Table <ref> (available electronically only). §.§ Calculating the amount of SN-released [][60]Fe that arrives on Earth According to stellar evolution models <cit.>, there is a close link between the [][60]Fe content of a star and its mass. For the considered stellar mass range of ≃10–[30]M_, the calculated [][60]Fe yields however exhibit a large scatter between ≃10^-6 and [10^-4]M_. This is due to the sensitivity of nucleosynthesis to a variety of factors, including cross-sections, mass loss, and rotation. For our model, we neglected these subtleties and instead used the exponential fit to recent stellar evolution data provided in .Owing to its low concentration, [][60]Fe should have no dynamical influence on the (turbulent) flow. We therefore treated it as a passive scalar, obeying an advection-diffusion equation of the form <cit.>∂ Z/∂ t+(u⃗·∇)Z=α∇^2 Z ,where Z is the [][60]Fe mass fraction, u⃗ the fluid velocity, and α the diffusivity of the contaminant (which is assumed here to be isotropic). Since for the case of the ISM the Peclet number, Pe=UL/α, is large (U and L denote characteristic velocity and length scales), diffusive effects are restricted to the very small, so-called microscale of turbulence. At larger scales, the right-hand side of Eq. (<ref>) can hence be neglected. The scalar Z then acts like a marker that tags the chemically enriched fluid parcels. However, there is still a need for some diffusive process ultimately operating at small scales, even though its exact value is quite insignificant for the turbulent mixing at the larger scales, as demonstrated by <cit.> in ISM simulations with different resolutions. Like them we did not implement any physical diffusion term, but rather left the mixing to numerical diffusion, which is generally faster and operates on a larger scale than its physical counterpart. As a consequence, the timescale of mixing in our simulations rather poses a lower limit to the mixing time resulting from physical diffusion.When generalising the analysis to the spatiotemporal evolution of m passive scalars with concentrations Z_i (i = 1,...,m), the time-dependent Euler equations in conservation-law form read as <cit.>∂U⃗/∂ t+∂F⃗(U⃗)/∂ x+∂G⃗(U⃗)/∂ y+∂H⃗(U⃗)/∂ z=S⃗(U⃗) ,with the state vectorU⃗= [ρ,ρ u,ρ v,ρ w, E,ρ Z_1,…,ρ Z_m]^T ,the vectors of fluxes in the corresponding coordinate directionsF⃗ =[ρ u,ρ u^2+P,ρ uv,ρ uw, u(E+P),ρ u Z_1,…,ρ u Z_m]^T , G⃗ =[ρ v,ρ uv,ρ v^2+P,ρ vw, v(E+P),ρ v Z_1,…,ρ v Z_m]^T , H⃗ =[ρ w,ρ uw,ρ vw,ρ w^2+P, w(E+P),ρ w Z_1,…,ρ w Z_m]^T ,and S⃗(U⃗) denoting an appropriate source term vector. Here, ρ is the fluid density, u, v, w the components of the velocity vector u⃗, E the total energy, and P the (thermal) pressure. Imprinted in this evolution is the radioactive decay of [][60]Fe, for which we also accounted in our simulations. The quantitative comparison between our model results and the FeMn crust measurements was then accomplished via the following steps.First, we determined the local interstellar fluence, ℱ, which is the flux of the isotope at the Earth's position integrated over time. Since this position coincides with the point of contact of eight, highest resolved grid cells, with the HD variables being always defined at the cell centres, we had to average over the [][60]Fe mass fluxes, ρ |u⃗|Z, in those cells. Hence,ℱ=(ρ |u⃗|Z)_VA/A m_uΔ t ,where (…)_VA denotes the volume average, A=60 the mass number of [][60]Fe, m_u≃[1.66× 10^-24]g the atomic mass unit, and Δ t the last simulation time step. For the remaining steps we took the fall-out on Earth to be isotropic, which was demonstrated by <cit.> to be a suitable assumption. The surface density of [][60]Fe atoms deposited at time t before present in the FeMn crust can then be expressed as[Sometimes in the literature the quantity Σ is referred to as the `observed fluence' <cit.>.] <cit.>Σ =fU/4ℱexp(-λ t) ,where the factor 1/4 stems from relating Earth's cross-section (R_⊕^2π) to its surface area (4 R_⊕^2π), the exponential term describes the decay of the radionuclide during its final disposal in the FeMn crust (λ≡ln(2)/t_1/2), and the product fU represents the [][60]Fe survival fraction. Specifically, f denotes the fraction of [][60]Fe atoms that reaches the Earth's orbit after overcoming a variety of filtering processes, while being condensed into dust grains. After estimating the dust condensation of [][60]Fe at departure from source, as well as its destruction through interstellar passage, heliosphere, and solar radiation pressure, <cit.> concluded that f≃ 0.01. The other, so-called uptake factor, U, accounts for the fact that only a certain fraction of [][60]Fe spread over Earth's surface actually finds its way into the FeMn crust. The underlying chemical selection process is only poorly understood, implying that the value of U is, as of f, very uncertain. <cit.> quantified U using the relative concentrations of iron and manganese in water and the FeMn crust, and the uptake of manganese (4 %), while implicitly assuming that f=1. They found fU≡ U≃ 0.006. More recent studies, however, suggest that U=0.5–1 <cit.>. Hence, by taking U=0.6 and f as derived by <cit.>, one retains the numerical value claimed by <cit.>, but still adds some (very simplistic) sub-grid dust physics to the problem.We finally obtained the number density of [][60]Fe for each crust layer by summing Σ in the time bins lying within the corresponding time intervals and then dividing by the thickness of the layer. Measurements demand to relate these densities to the one of stable iron, i.e. n_^60Fe/n_Fe, or, in short, [][60]Fe/Fe. The required normalisation constant follows from <cit.>n_Fe=w_Feρ N_A/A ,where w_Fe≃ 0.1527 is the weight fraction of iron in the FeMn crust sample whose mass density ρ is [1.5]g cm^-3, N_A≃[6.022× 10^23]mol^-1 is the Avogadro constant, and A≃[55.845]g mol^-1 is the molar mass for iron. We thus have n_Fe≃[2.470× 10^21]cm^-3. §.§ Overview of input dataTo conclude this section we give a full list of the input data and recapitulate how it is obtained: * Sample of cluster stars: selected according to compactness in real and velocity space from astrometric catalogue data (such as HIPPARCOS) for a heliocentric sphere with diameter D∼[400]pc (LB) and [800]pc (Loop I)* Cluster age (τ_c): derived by comparison of the cluster turn-off point with isochrones of <cit.>* Number and initial masses of SN progenitors (𝒩_SN, M_*): estimated via IMF fitting of sample stars <cit.>, assuming one star per mass bin with the bin's average mass for the perished stars [Eqs. (<ref>) and (<ref>)]* Main sequence lifetimes of SN progenitors (τ): calculated as a function of mass only via Eq. (<ref>)* SN explosion times (t_exp): calculated from subtracting τ_c from τ, thus assuming coeval star formation in the cluster* SN explosion centres (x,y,z): derived from the center-of-mass stellar trajectories, as determined via the epicyclic equations of motion <cit.>; for the LB, the most probable ones are calculated for a Gaussian error distribution in the HIPPARCOS positions and proper motions* Total ejecta mass and [][60]Fe mass fractions (M_ej, Z): taken, corresponding to M_*, from stellar evolution models <cit.>* [][60]Fe survival fraction (fU): quantified by combining estimates from a dust survival model <cit.> and measurements in terrestrial archives <cit.>; the dynamics of dust and gas is assumed to be perfectly coupled, where the latter is traced via passive scalars§ RESULTS §.§ Evolution and properties of the Local and Loop I superbubbles The SN explosions generate a coherent bubble structure whose evolution is followed over [12.6]Myr (age of the LB) until the present time. In Figs. <ref> and <ref> we present snapshots of the total gas column density for the models A and B, respectively. Note that in these, as well as in all remaining maps, the x, y, and z-axes point towards the Galactic centre, into the direction of the Galactic rotation, and towards the Galactic north pole, respectively.After their almost coeval formation, the two SBs evolve at first completely independently from each other. The shocked ambient medium cools very quickly and gets compressed into thin shells. These supershells are subject to thermal and fluid dynamical instabilities, which give rise to the formation of cold and dense clumps. In particular, during the `breaks' between two consecutive SN explosions, when the supershell is decelerating, Rayleigh-Taylor instabilities develop at the contact discontinuity that separates the shocked SN ejecta from the cooler and denser swept-up ambient medium. The filaments of dense gas produced thereby grow into the cavity where they can mix with the hot bubble gas. In addition, an overstability sets in that arises from a mismatch between thermal and ram pressure at the shell surface. This `Vishniac instability' <cit.> can cause growing, oscillating ripples on that surface. Although such structures are particularly prominent for extinct SBs, whose shell motion has stalled due to the lack of energy sources <cit.>, they may already be visible at the late stages of the simulations presented in this work; see e.g. the spike protruding from the shell of Loop I at (x,y)=[(240,-600)]pc in the lower right panel of Fig. <ref>. It is also interesting to note a criss-cross network of shocks as a result of sequential SN explosions. This may lead to stronger coalescing shock waves and thus to local pressure enhancements, which may also be responsible for local deviations from sphericity of the LB and Loop I despite the homogeneous ambient medium.Certainly an effect of the latter is that the supershells retain their spherical shape on large scales. After about [3]Myr (t≃[9.6]Myr before present; model A) and [4.6]Myr (t≃[8]Myr before present; model B) the shells collide, giving rise to an even denser and cooler interaction layer. Also this interface becomes eventually Rayleigh-Taylor-unstable, since the pressure in the two SB volumes is not equal. This leads to the formation of cold and dense cloudlets that travel into the direction of lower pressure (i.e. the LB interior) as soon as the instability becomes fully non-linear <cit.>. In model A, the interaction shell breaks up after an evolution time of about [6.5]Myr (t≃[6.1]Myr before present), allowing for immediate exchange of hot gas between the SBs. In model B, on the other hand, no break-up occurs during the whole evolution time considered. The final size of the LB is about 800 by [600]pc (model A) and 580 by [480]pc (model B) in the (x,y)-plane, and about [760]pc (model A) and [540]pc (model B) perpendicular to it. The atomic hydrogen number density and temperature of the LB interior, particularly in the solar neighbourhood, is about 10^-4.2–[10^-3.9]cm^-3 and 10^6.9–[10^7.1]K in model A, and about 10^-4.2–[10^-3]cm^-3 and 10^5.8–[10^7]K in model B. If the ambient medium is homogeneous, it is generally the first SN that determines the final appearance of the SB most. This is because within a few sound-crossing times the bubble has a homogeneous pressure acting on the shell of the first explosion. The centre of the final SB should consequently lie very close to the explosion centre of the earliest SN. That this is indeed the case, also in our models, can be seen best in the bottom panels of Figs. <ref> and <ref> (the largest black circle marks the locations of the earliest explosion). Furthermore, since we used the same SN data for both models, the relative arrangement of the two SBs does not change either.It should be noted that the calculated final x- and y-extensions only poorly match the sizes derived from observations. Also the elongation in z-direction is highly underestimated. This is because the background medium is assumed to be homogeneous, which is certainly an oversimplification of the actual conditions in the solar neighbourhood. As seen from the extended, three-dimensional differential opacity maps of the local ISM by <cit.>, the LB is bound today by a series of dense clouds located in the first and fourth quadrants, as well as in the anti-centre region of the Galactic plane. These clouds, which also possess non-negligible vertical extensions, are known to be part of Gould's belt. Moreover, there is a 500–[1000]pc wide cavity in the third quadrant, as well as a small, elongated cavity in the opposite direction. The larger cavity is centred below the Galactic plane and was identified as the counterpart of the so-called GSH 238+00+09 supershell, which is visible in radio wavelengths <cit.>. Assuming that these structures were present in a similar fashion as observed today throughout the entire evolution time, the LB's outer shell would probably have experienced early on a greater `resistance' in the second, fourth, and (partly) the first quadrant, and would thus not have grown to such a great extent in those directions. This should particularly hold for model A, in which the SBs expand more rapidly. The evolution in the third quadrant is more difficult to predict since in our models, this region is partly occupied by Loop I. The presence of the dense interstellar cloud at a distance of ≃[200]pc and longitude 240^∘ might nevertheless restrict the LB growth in a similar fashion as the developing interaction layer in our simulations. The combined effect of all these structures might also `shift' the solar system closer to the centre of the then smaller LB. Further knowledge can only be gained from additional simulations whose initial conditions are tailored to match these recent observations. These will be the subject of a forthcoming paper. Current results however suggest that the exact size and shape of the LB does not play a great role for modelling the [][60]Fe amount that arrives on Earth, except for the case in which [][60]Fe is carried to the solar system by a reflected shock (see Sec. <ref>). Generally it is necessary that the solar system lies within the LB so that it can be reached by the expanding SN remnants. For this, the gas-dynamical properties of the medium between the SN explosion centres and the locus of the Earth are crucial, as they determine the arrival time of the LB's outer shell and hence set the date for the earliest possible [][60]Fe signal in the terrestrial record. We further note that the relative locations of the Local and the Loop I SB are not the same as in the joint-evolution model by <cit.>. This is because they used different stellar trajectories for the LB progenitors, and, more importantly, a different formation scenario for Loop I, based on other moving group stars from the Sco-Cen cluster for which they assumed a single explosion centre that is somewhat arbitrarily shifted [200]pc relative to the centre of the LB in positive x-direction. Therefore, and contrary to our model, their LB is generated by 19 SNe, and their Loop I SB by 39 SNe. Apart from that there is some discussion in the literature whether the large circular feature in the radio continuum sky, known as the North Polar Spur, which is associated with Loop I, is part of the local ISM at all or rather belongs to the Galactic centre region <cit.>. The presence of nearby young stars in the direction of the Galactic centre, however, strongly argues for SN explosions that generated the Loop I SB.Using a passive scalar as discussed in Sect. <ref>, we followed the spatiotemporal evolution of the SN-released [][60]Fe. Corresponding simulation snapshots are shown in Figs. <ref> and <ref> for models A and B, respectively. From these maps it is seen that inhomogeneities in the [][60]Fe distribution, arising from recent SNe, are smoothed out over time. Responsible for this mixing process are turbulent shear flows resulting from SN blast waves that run into the surrounding supershell and thus generate asymmetric reflected shock waves. The zones of underpressure left behind by these reflected discontinuities manifest themselves as the blue `lakes' particularly visible in the lower panels of Figs. <ref> and <ref>. With a characteristic large-eddy size of ℓ≃[100]pc and an average sound speed after an SN of the order of a≃[10^2]km s^-1, we obtained for the mixing timescale within the SB cavity, τ_m∼ℓ/a, a value of about [1]Myr. Considering that the last SN occurred roughly [1.5]Myr ago, it is therefore not surprising that the [][60]Fe mass is distributed quite uniformly in the LB cavity at present time. §.§ Comparison with crust measurements At each simulation time step, the total local interstellar fluence of [][60]Fe atoms was calculated according to Eq. (<ref>). Figure <ref> shows the resulting profiles for both models. Upon closer inspection (see inlay), the signals come in three different types, which are all embedded in a `background noise' of ≃[10^5]cm^-2 amplitude. The first type are sharp and intense (ℱ≃ 0.3–[2.4× 10^6]cm^-2) sawtooth waves that occur seven times in model A, but only once in model B. Such a wave form is characteristic of the density profile of an SN remnant in the Sedov-Taylor phase. These signals must hence originate from the shells of individual remnants that cross the Earth's orbit while expanding into the LB volume. The time of direct deposition of the SN ejecta on Earth, given by the (scale) width of the signal, is a measure of the remnant's shell thickness. The time range of direct exposure in our models, Δ t≃ 70–[130]kyr, roughly lies between the values proposed by <cit.> and <cit.>, but almost perfectly matches the predictions given in <cit.>, which are based on the SN model by <cit.>.The second type, which appears to be unique to model A, are signals that occur after almost every sawtooth wave (note that the time axis is from right to left), while being always more extended and weaker than the waves they succeed, almost as if they were the SN explosions' `echoes'. Remarkably, the time lag between the two signals increases the closer the present time is approached. The third type is a joint feature of both profiles, occurring only once and right at the beginning (t≃ 6.1 and [2.2]Myr in model A and B, respectively). Here, the signals are quite broad (Δ t≃ 300 and [450]kyr in model A and B, respectively) and possess intermediate amplitudes of about 2.9× 10^5 (model A) and [9.6× 10^5]cm^-2 (model B). Applying the rebinning procedure described in Sect. <ref> allowed us to do a direct comparison with the FeMn crust measurements of <cit.> and <cit.>, as shown in Fig. <ref>. The [][60]Fe survival fraction, fU, was either set to 0.006 (light grey histograms), as originally suggested by , or 0.005 (dark grey histograms), corresponding to a combination of a dust fraction of f=0.01 with the newly derived lower limit for U (cf. Sect. <ref>). It is seen that for both models, measurements are best matched if the slightly lower uptake factor is employed. This is particularly true for the (global) maximum in the crust layer corresponding to t≃[2.2]Myr, whose presence was lately confirmed by the study of <cit.>, and which is perfectly reproduced by both models (max (^60Fe/Fe)≃ 2.3× 10^-15). It is also interesting to note that model A suggests contributions in five more layers than model B, with two of those, namely for t≃ 3.1 and [5.7]Myr, lying within the error bars of the measurements. The local maximum calculated for t≃[3.9]Myr is however absent in the FeMn crust data available. To explore the exact origin of each signal in the fluence profiles, we rerun the simulations with all parameters unchanged, except that we assigned a passive scalar to each individual SN event. Formally, this implies adding an index i with 1≤ i≤ 16 to the variables ℱ, Z, and Σ in Eqs. (<ref>) and (<ref>). Results are plotted in Fig. <ref>, where identification numbers correspond to the explosion order. Inspecting these profiles, together with the left column panels of Figs. <ref> and <ref>, reveals that the earliest signals of both models, i.e. the quite broad peaks of intermediate amplitude already known from Fig. <ref>, are produced when the LB's supershell runs over the solar system. Separated from the shocked ambient medium by a contact discontinuity, the shell harbours a large amount of [][60]Fe atoms that have by then been expelled in previous SN events (SNe #01–08 and #01–15 in model A and B, respectively) and that have not yet decayed (see also the inlays of Fig. <ref>). Since the supershell of model B arrives later at the Earth's orbit, it could already accumulate more SN ejecta (and thus enriched material), which is the reason why the profile of the total local interstellar fluence (Fig. <ref>) has a first signal that is both broader and higher than in model A. Also note that the first signals do not set in abruptly but in a smooth, ramp-like manner. This is due to the combined effects of numerical diffusion present in all Godunov-like advection schemes and the Rayleigh-Taylor instability operating at the contact discontinuity inside the supershell (see Sect. <ref>), allowing for mixing across the discontinuity. Even the pulses generated by single SN remnants actually entrain fractions of previously released [][60]Fe, since the average time between successive explosions is short when compared to the half-life of [][60]Fe. This is nicely confirmed by Fig. <ref>, showing that the high peaks associated with recent SN events indeed superpose a series of peaks with lower amplitudes. The same can be observed for the weaker follow-up signals, which are produced when the SN blast waves return after being reflected from the surrounding supershell; so in this sense they are indeed SN `echoes'. Finally, since the supershell moves away from Earth, the time lag between an explosion and its `echo' becomes greater for later times. Putting all this information together, we arrive at the following interpretation of Fig. <ref>. Starting with the most prominent feature, the (global) maximum in the layer for t≃[2.2]Myr, we can now safely state that, in model A, it is produced by SN #14 and 15 and their reflected shocks, whereas in model B, it is due to yet undecayed material from SNe #01–15 reaching Earth almost simultaneously while being gathered within the LB's supershell (see also the middle and left column panels of Figs. <ref> and <ref>, respectively). The additional local maximum at t≃[3.9]Myr, which is unique to model A, is mainly due to SN #12 and 13. The upper crust layers (corresponding to t ≲[1.7]Myr) obtain their [][60]Fe content almost exclusively from SN #16. There is at first the arrival of the SN blast wave itself (seen for model B in the middle column panel of Fig. <ref>) that generates the enhancement in the layer for t≃[1.3]Myr[Note that in model B, the corresponding peak also contains the contribution of the reflected shock, which could already return due to the supershell's proximity, making clear why the signal has such a high amplitude. In contrast, there is a time lag of about [230]kyr in model A.]. Later on, turbulent motions induced by the blast wave of the last SN lead to the enhancement in the layer for t≃[0.4]Myr. Particularly, all the `background noise' in the modelled fluence profiles is due to turbulence inside the LB cavity. We hence conclude that the measured [][60]Fe excess in the FeMn crust could be either due to the passages of dust-loaded shells from nearby SN remnants over the Earth's orbit, sweeping up also enriched material, which was present at that time in the LB volume (model A), or, alternatively, due to the arrival of the supershell of the LB that incorporates the material from several previous SN events (model B). We consider the LB formation scenario of model B to be physically more probable than that of A. Not only are the derived and measured ^60Fe/Fe ratios in better agreement (there is particularly no second maximum), but also the size and (more importantly for simulations that are based on homogeneous background media) the gas-dynamical properties of the fully developed SB are a better match to observations.We further emphasise that the additional smaller peak at t≃6.5–[8.7]Myr, partly present in the data of <cit.>, does not occur in our profiles. If this enhancement is not of meteoritic origin, as mentioned earlier, but actually connected to the formation of the LB, it would be an indication for a clustering of stars between 12 and [15]M_, thus differing from the most probable binning used for the present study. However, this hypothesis needs further testing and simulations, and is beyond the scope of this work, but will be addressed in a forthcoming paper.§ DISCUSSION AND CONCLUSIONS In this paper we presented three-dimensional HD simulations (with resolutions adaptively adjusted down to subparsec scale) aiming to reproduce an [][60]Fe excess measured in a deep-sea FeMn crust in the context of the formation of the LB. Parameters for the LB-generating SNe are based on an IMF normalised to still existing low mass stars of the parent stellar moving group identified by <cit.>. Like in the analytical model by <cit.> (see also ), ejected masses, lifetimes and thus explosion times were derived from the initial masses of the progenitor stars, whereas computations of the most probable paths for the individual perished members of the moving group provided most probable explosion sites. For estimating the particular [][60]Fe mass ejecta, we resorted to predictions from stellar evolution models. The formation of the LB, as well as of the neighbouring Loop I SB, was studied in two homogeneous, self-gravitating environments, sharing similarities with the classical warm ionised (model A) and warm neutral medium (model B). Interstellar heating and cooling processes were included via standard CIE net cooling functions. By utilising passive scalars for tagging the [][60]Fe enriched gas released into the SB cavities, we recorded the flux of [][60]Fe atoms at the Earth's orbit (usually termed local interstellar fluence) over the whole simulation time, and then calculated time averages over a range matching the crust layers in order to do comparisons with the measurements of <cit.> and <cit.>. Our models are able to reproduce both the observed timing and intensity of the [][60]Fe excess with rather high precision. However, the underlying physical processes are different. In model A the signal arises due to the fast-paced blast waves of two individual SN remnants (#14 and 15 out of 16 in total), which cross the Earth's orbit twice as a result of reflection from the LB's outer shell. In model B, on the other hand, it is the supershell of the LB itself that injects the yet undecayed [][60]Fe content of all previous SNe (#01–15 in this case) at once, but over a longer time range. For both scenarios it is important to note that [][60]Fe does not reach Earth in gaseous form, but condensed onto dust grains that have to survive a variety of filtering processes throughout their journey. These were summarised by us into the combined multiplicative factor fU, for which we found a better agreement with the crust data if fU=0.005 rather than 0.006. Both values occur in the current literature. The observed extension of the Local Bubble as well as its gas-dynamical properties were better matched by model B. The age of the LB has been estimated from the main sequence lifetime of a still existing [8.2]M_ star <cit.>.Primarily for the sake of better comparison with analytical calculations (, ) our model does currently not include the stellar winds blown by the moving group members. For a locally homogeneous ISM, as considered in this work, these winds would have carved an extended, egg-like shaped cavity around the cluster's center-of-mass trajectory, already before the first stars explode <cit.>. As soon as this happens, however, the dynamics of the wind-blown bubble is quickly overtaken by the SNe, as is clear from comparing the mechanical luminosities of the winds and SNe <cit.>. The former cannot be larger than 1/2Ṁ_totv_∞^2≃[2.5× 10^36]erg s^-1, where Ṁ_tot≃[2×10^-6]M_⊙ yr^-1 is the maximum stellar mass loss rate (only reached if the most massive star is still alive) and v_∞≃[2000]km s^-1 is the terminal wind velocity. The latter, on the other hand, is E_SN/Δτ≃[4.2× 10^37]erg s^-1, with Δτ≃[7.4× 10^5]yr being the average time interval between two consecutive SNe. Note that we only consider cumulative mechanical energy output rates here, which should be justified because the mean separation between the massive stars (and hence the SN explosion centres) of ≃[44]pc is most of the time smaller than the size of the SB. So the expansion of mature SBs is determined by SNe, rather than by stellar winds (and also ionising photons), which dominate the early phase of evolution <cit.>.Although our simulations draw a quite conclusive picture of the LB formation scenario, there are nevertheless some caveats that have to be addressed. First, the production and expulsion of [][60]Fe is a complicated process that strongly depends on the mixing processes in the interiors of massive stars, and the depth of their convection zones. Predictions of the [][60]Fe yields can thus vary by a factor of a few, even for the same stellar masses, depending on which stellar evolution model is used <cit.>.Second, the uptake factor, quantifying the sedimentation through the Earth's atmosphere onto the ocean floor, is only loosely constrained. More precise estimates, either from measuring [][60]Fe at other sites or from the analysis of other radioisotopes, are therefore highly welcome, since its value has a profound effect on our quantative results.The third concerns heliospheric filtering. [][60]Fe-enriched plasma could only reach the solar system via the poles because of strong ram pressure of the solar wind in the equatorial plane. Due to its higher mass this does not affect [][60]Fe-dust. A comprehensive description of the formation and survival of SN dust at different compositions and sizes is therefore desirable, as it is particularly questionable whether the single factor applied in this work can fully account for all processes involved. In principle, similar arguments apply for the previous two points. An obvious advantage of employing simple input parameters however lies in the fact that they can be easily adapted when the underlying physics is better understood or measurements have become more precise. Still, it should be emphasised that these uncertainties affect the absolute values of [][60]Fe, but not relative ones, giving the model a solid foundation.Last but not least, there is the dependence on the background medium, as well as on the explosion sites in the stellar moving group. For the former we can say from our current studies with a homogeneous background medium that the average external density may not exceed ≃[0.3]cm^-3. Otherwise, the LB supershell would arrive later and the [][60]Fe overabundance would no longer appear in the crust layer in which it was discovered. Yet strictly speaking, this is only true for the SN explosion centres applied in this work, which we left unchanged as these are the most probable ones for the considered stellar moving group (see also ). One can however check whether any relevant changes in the calculated [][60]Fe/Fe profiles (Fig. <ref>) would result if an SN would occur by a maximum of 1σ away from its most probable explosion site lying at distance d. In order to estimate this spread in distance, δ R, we traced back the paths of 1000 hypothetical stars that would explode on exactly the same location [10]Myr into the past, while taking into account the error in their mean velocities (≃[3]km s^-1 for each component). We obtained δ R≃[20]pc. As the blast wave velocity in the LB volume is of the order of [10^4]km s^-1, this translates to a spread in the arrival time of a thus generated [][60]Fe signal of ≃[2]kyr, which lies far below the average time resolution of the FeMn crust sample (≃[0.6]Myr, see the bin sizes in Fig. <ref>). Relevant for the height of the signal is the size of the SN shells when they are crossing our solar system. When assuming for the sake of simplicity perfect sphericity and neglecting the clumpiness of these shells as a result of instabilities, the factor by which the height of the signal would change lies between (1±δ R/d)^2. For the last three SNe in the LB volume d≃[100]pc, yielding for the factor's lower and upper limit0.64 and 1.44, respectively.Directly related to the background medium is the question about the amount of [][60]Fe in the region where the LB began to form. The steady-state Galactic [][60]Fe mass of [1.53]M_⊙ given by <cit.> was derived for the diffuse, extended ISM and thus does not allow conclusions to be drawn on local values in the solar neighborhood today, and even less on those of more than [10]Myr ago, where also radioactive decay should play a crucial role. The only serious way to obtain such information would be long-term chemical evolution simulations of the (local) Galaxy that particularly include chemical mixing. As such models are currently unavailable, we set in our simulations the initial amount of [][60]Fe equal to zero everywhere, thus always representing the lower limit case. We note in passing that a completely analogous problem arises in the context of explaining the origin of [][60]Fe in the early solar system <cit.>, demonstrating that such studies are strongly needed.Also worth mentioning is the alternative explanation for the [2.2]Myr-old [][60]Fe signal given by <cit.> and <cit.>. Their idea is based on the fact that [][60]Fe is not only produced in SNe but also by spallation reactions between Galactic cosmic rays and nickel inmeteorites and interplanetary dust particles. Via collisions of asteroids, e.g. in the main belt, these small objects could have been released in large amounts, thus exposing Earth to a boosted isotopic influx. However, the concentration of nickel measured within the FeMn crust is too low to explain the [][60]Fe signal within the framework of this scenario <cit.>. On the other hand, the cosmic-ray protons, antiprotons, and positrons that would have been copiously released by a nearby SN would certainly have led to a temporal increase of radioisotopes in meteoritic particles. As a matter of fact, there are evidences for one (or more) such events about [2]Myr ago in the cosmic ray energy spectra <cit.>, fitting nicely into the picture drawn in this work.Another strong hint for the SN origin of the [][60]Fe signal are the experimental [][60]Fe and [][53]Mn activities recently obtained from lunar samples <cit.>. Like [][60]Fe, the long-lived radionuclide [][53]Mn is generated from cosmic-ray spallation reactions on iron in meteorites. Hence, if the [][60]Fe signal was produced from an enhanced influx of micrometeorites, one would expect the lunar activity ratios of [][60]Fe/[][53]Mn to be consistent with meteoritic values. Yet the lunar soil samples carry an enhanced [][60]Fe signature that exceeds the activity created by cosmic rays.We can actually test our numerical calculations against these lunar soil measurements, stating an observed fluence in the range between 10^7 and [6× 10^7]cm^-2. As their data is not time-resolved, we have to sum the value of Σ (cf. Eq. (<ref>)) in all time bins available, spanning a total range of [12.6]Myr. When setting U=f=1, we obtain fluences of [6.18× 10^8]cm^-2 for model A and [3.36× 10^8]cm^-2 for model B. While it is almost certain that f<1 (cf. Sect. <ref>), the Moon's uptake factor should be, due to the lack of an atmosphere, equal or very close to unity. Taking our simulations as a basis, we estimate a range of 0.016 and 0.179 for the [][60]Fe survival fraction fU. The lower limit would be actually compatible to an uptake of 100 %, if the value of f were only slightly higher than estimated by <cit.>, namely 0.016 instead of 0.010. Therefore also the measurements of [][60]Fe in lunar samples are in agreement with our previously performed simulations. M.M.S. and D.B. acknowledge funding by the DFG priority program 1573 “Physics of the Interstellar Medium”. We thank the anonymous referee for her/his valuable comments which helped to improve the manuscript. aa | http://arxiv.org/abs/1704.08221v1 | {
"authors": [
"Michael Mathias Schulreich",
"Dieter Breitschwerdt",
"Jenny Feige",
"Christian Dettbarn"
],
"categories": [
"astro-ph.HE",
"astro-ph.GA"
],
"primary_category": "astro-ph.HE",
"published": "20170426171607",
"title": "Numerical studies on the link between radioisotopic signatures on Earth and the formation of the Local Bubble. I. 60Fe transport to the solar system by turbulent mixing of ejecta from nearby supernovae into a locally homogeneous ISM"
} |
IFLP-Departamento de Física-FCE, Universidad Nacional de La Plata, C.C. 67, La Plata (1900), ArgentinaWe discuss the relation between fermion entanglement and bipartite entanglement. We first show that an exact correspondence between them arises when the states are constrained to have a definite local number parity. Moreover, for arbitrary states in a four dimensional single-particle Hilbert space, the fermion entanglement is shown to measure the entanglement between two distinguishable qubits defined by a suitable partition of this space. Such entanglement can be used as a resource for tasks like quantum teleportation. On the other hand, this fermionic entanglement provides a lower bound to the entanglement of an arbitrary bipartition although in this case the local states involved will generally have different number parities. Finally the fermionic implementation of the teleportation and superdense coding protocols based on qubits with odd and even number parity is discussed, together with the role of the previous types of entanglement. pacs03.67.Mn, 03.65.Ud, 05.30.FkBipartite entanglement infermion systems N. Gigena, R. Rossignoli – Dedicated to John Butcher, on the occasion of his 84-th birthday – ========================================================================§ INTRODUCTION Entanglement is a fundamental feature of quantum mechanics, and its quantification and characterization has been one of the main goals of quantum information theory for the last decades <cit.>. It is also at the heart of quantum information processing <cit.>, being recognized as the key ingredient for quantum state teleportation <cit.> and the resource that makes some pure state based quantum algorithms exponentially faster than their classical counterparts <cit.>.Although entanglement has been extensively studied for systems of distinguishable constituents, less attention has been paid to the case of a system of indistinguishable fermions. Only in recent years the topic has gained an increasing strength <cit.>. Mainly two different approaches may be recognized in the attempts of generalizing the definition of entanglement to fermion systems: The first is entanglement between modes <cit.>, where the system and subsystems consist of some collection of single-particlemodes that can be shared. This approach requires to fix some basis of the single-particle state space and then to specify the modes that constitute each subsystem. The other approach is known as entanglement between particles <cit.>, where the indistinguishable constituents of the system are taken as subsystems and entanglement is defined beyond symmetrization. In a previous work <cit.> we defined an entropic measure of mode entanglement in fermion systems which is shown to be a measure of entanglement between particles after an optimization over bases of the single-particle (sp) state space is performed. Moreover, when the sp state space dimension is four and the particle number is fixed to two, this entanglement measure reduces to the Slater correlation measure defined in <cit.>. In the present work we first show that the entanglement between two distinguishable qubits is the same as that measured by this fermionic entanglement entropy when the fermionic states are constrained to have a fixed local number parity in the associated bipartition of the sp space. Then we use this correspondence to show that, in fact, any state of a fermion system with a 4-dimensional sp Hilbert space may be seen as a state of two distinguishable qubits for a suitable bipartition of the sp space, with its entanglement measured by the fermionic entanglement entropy. On the other hand, for an arbitrary bipartition involving no fixed local number parity the fermionic entanglement entropy is shown to provide a lower bound to the associated bipartite entanglement.As application we use these results to show that qubit-based quantum circuits may be rewritten as mode-based fermionic circuits if we impose the appropriate restriction to the occupation numbers, recovering reversible classical computation when the input states are Slater determinants (in the basis of interest). Two types of fermionic qubit representations, based on odd or even number parity qubits, are seen to naturally emerge. Finally, we show that the extra bipartite entanglementthat can be obtained by relaxing this local parity restriction can in principlebe used for protocols such as superdense coding. The formalism and theoretical results are provided in sec. <ref>, while their applications are discussed in sec. <ref>. Conclusions are finally provided in sec. <ref>.§ FORMALISM§.§ Fermionic entanglement entropy and concurrence We will consider a fermion system with a single-particle (sp) Hilbert space H. We will deal with pure states |ψ⟩ which do not necessarily have a fixed particle number, although the number parity will be fixed, in agreement with the parity superselection rule <cit.>: P|ψ⟩=±|ψ⟩, with P=exp[iπ∑_j c^†_j c_j] the number parity operator. Herec_j, c^†_j denote fermion annihilation and creationoperators satisfying the usual anticommutation relations{c_i,c_j}=0,{c_i,c_j^†}=δ_ij . In <cit.> we defined aone-body entanglement entropy for a general pure fermion state |ψ⟩,S^ sp(|ψ⟩)= Tr h(ρ^ sp),where ρ^ sp_ij=⟨ c^†_j c_i⟩≡⟨ψ|c^†_j c_i|ψ⟩ is the one body density matrix of the system and h(p)=-plog_2 p-(1-p)log_2(1-p). Eq.(<ref>) is proportional to the minimum, over all sp bases of H, of the average entanglement entropy between a sp mode and its orthogonal complement (which in turn arises from a properly defined measurement of the occupation of a sp mode), andvanishes iff |ψ⟩ is a Slater Determinant (SD), i.e. |ψ⟩=c^†_1… c^†_k|0⟩. This definitionis easily extended to quasiparticle modes, in which case <cit.>S^ qsp(|ψ⟩)=- Tr ρ^ qsplog_2 (ρ^ qsp) ,where ρ^ qsp is now the extended one-body density matrixρ^ qsp= 1-⟨([ c; c^† ])([ c^† c ])⟩= ([ρ^ spκ;-κ̅ 1-ρ̅^ sp ]) ,with κ_ij=⟨ c_jc_i⟩,-κ̅_ij=⟨ c^†_j c^†_i⟩ and (1-ρ̅^ sp)_ij=⟨ c_j c^†_i⟩. Eq. (<ref>) vanishes iff |ψ⟩ is a quasiparticle vacuum or SD and satisfies S^ qsp(|ψ⟩)≤ S^ sp(|ψ⟩),withS^ qsp(|ψ⟩)=S^ sp(|ψ⟩) iff κ=0.While Eq. (<ref>) is invariant under unitary transformations c_i→∑_k U̅_kic_k, UU^†=I, which lead to ρ^ sp→ U^†ρ^ sp U,Eq. (<ref>) remains invariant under general Bogoliubov transformationsc_i→ a_i=∑_k U̅_ki c_k+V_kic^†_k ,where matrices U and V satisfy UU^† +VV^†= 1 and UV^T+VU^T=0 in order that {a_i,a^†_i} fulfill the fermionic anticommutation relations <cit.>. In this case ρ^ qsp→ W^†ρ^ qspW, with W=(^U V_V̅ U̅) a unitary matrix. In terms of theoperatorsdiagonalizing ρ^ qsp, we then have1-⟨([ a; a^† ])([ a^† a ])⟩ = ([ f 0; 0 1-f ]) ,with f_kl=f_kδ_kl and f_k, 1-f_kthe eigenvalues of ρ^ qsp.For a sp space H of dimension 4, ρ^ qsp becomes an 8× 8 matrix, and it was shown that its eigenvalues for a pure state|ψ⟩ arefourfold degenerate and can be written as <cit.>f_±=1±√(1-C^2(|ψ⟩))/2 ,where C(|ψ⟩)=2√(f_+f_-)∈[0,1] is called fermionic concurrence, in analogy with that defined for two-qubits <cit.>. Eq. (<ref>) becomesthen an increasing function of C(|ψ⟩), vanishing iff the latter vanishes.This fermionic concurrence can also be explicitly evaluated: Writing a general even numberparity pure state in such a spaceas |ψ⟩= (α_0+1/2∑_i,jα_ijc^†_ic^†_j+ α_4 c^†_1c^†_2c^†_3c^†_4)|0⟩ ,where α_ij=-α_ji, i,j=1,…,4and |α_0^2|+|α_4^2| + 1/2 tr α^†α=1,then ρ^ sp=αα^†+|α_4|^2 1, κ=α_0^*α+α_4 α̃^*,with α̃_ij=1/2∑_k,lϵ_ijklα_kl(ϵ_ijkl denotes the fully antisymmetric tensor)and it can be shown that <cit.> C(|ψ⟩)=2|α_12α_34-α_13α_24 +α_14α_23-α_0α_4| .For two-fermion states (α_0=α_4=0) it reduces to the Slater correlation measuredefined in <cit.>, for which κ=0 and f_± become the eigenvalues(two-fold degenerate) of ρ^ sp. An expression similar to (<ref>) holds for an oddnumber parity state (see <cit.> and sec. <ref>). Moreover, in such sp space the concurrenceand the associated entanglement of formation can also be explicitly determined for arbitrary mixedstates<cit.>. A four-dimensional sp space (which generates an eight-dimensional state space for each value of theparity P) becomes thenexactly solvable, being as well the first non-trivial dimension since fordimH ≤ 3 any definite parity pure state can be written as a SD or quasiparticlevacuum <cit.>, as verified from (<ref>) (C(|ψ⟩)=0 ifone of the sp sates is left empty).It is also physically relevant, since it can accommodate thebasic situation of two spin 1/2fermions at two different sites or, more generally, states ofspin 1/2 fermions occupying just twoorbital states, as in a double well scenario. The relevant sp space in these cases isH_S⊗ H_O, with H_S thespin space andH_O the two-dimensionalsubspace spanned by the two orbital states. In particular, just foursp states are essentially used inrecent proposals for observing Bell-violation from single electron entanglement <cit.>. §.§ Bipartiteentanglement as two-fermion entanglementLet us now consider a system of two distinguishable qubits prepared in a pure state α_+|00⟩+α_-|11⟩, i.e.|ψ⟩_AB=α_+|↑⟩_A⊗|↑⟩_B+ α_-|↓⟩_A⊗|↓⟩_B ,where |α_+^2|+|α_-^2|=1 and the notation indicates a possible realization in terms of two spin 1/2 particles located at different sites A,B, with their spins aligned parallel or antiparallel to a given (z) axis. We can also consider this last stateas a two-fermion state of a spin 1/2 fermion system with spspace H= H_S⊗ H_O:|ψ⟩_f=(α_+ c^†_A↑c^†_B↑+ α_- c^†_A↓c^†_B↓)|0⟩ ,with |0⟩ the fermionic vacuum. A measurement of spin “A” or “B” along z can be described in the fermionic representation by the operators Π_Sμ=c^†_Sμc_Sμ, S=A,B, μ=↑,↓, which satisfyΠ_Sμ^2=Π_Sμ and [Π_Sμ,Π_S'μ']=0, with ∑_μΠ_Sμ|ψ⟩_f= |ψ⟩_f. Furthermore, we can describe any “local” operator onA or B in terms ofPauli operatorsif we define, for S=A,B, σ_S x =c^†_S↑c_S↓+c^†_S↓c_S↑ , σ_Sy =-i(c^†_S↑c_S↓-c^†_S↓c_S↑) , σ_Sz = c^†_S↑c_S↑-c^†_S↓c_S↓ ,which verify the usualcommutation relations [σ_Sj,σ_S'k]=2iδ_SS'ϵ_jkl,(ϵ_jkl is the antisymmetric tensor),withσ_Sj^2|ψ⟩_f=|ψ⟩_f.It is also apparent that the state (<ref>) is separable iff α_+=0 or α_-=0, whichisprecisely the conditionwhich ensures that the state (<ref>) is a SD. Moreover, the standard concurrence <cit.> of the state (<ref>) isidentical with thefermionic concurrence (<ref>) of the state (<ref>):C(|ψ⟩_AB)=2|α_+α_-|=C(|ψ⟩_f) ,with f_±=|α_±^2| in (<ref>).Entangled two-qubit states(<ref>) correspond then to two-fermion states (<ref>) which are not SD's, and vice-versa.Such correspondence remains of course valid for any bipartite two-qubit state|ψ⟩_AB=∑_μ,να_μν|μ⟩_A⊗ |ν⟩_B , which in the fermionic representation becomes|ψ⟩_f=∑_μ,να_μνc^†_Aμc^†_Bν |0⟩ .We now obtain C(|ψ⟩_AB)=2| det α|=C(|ψ⟩_f), according to thestandard and fermionic (Eq. (<ref>)) expressions. These states can in fact be taken tothe previous Schmidt forms (<ref>)-(<ref>) (with|α_±| the singular values of the matrix α) by means of local unitary transformations, which in the fermionic representation becomec_Sμ→∑_νU̅^S_νμc_Sν. Previous considerations remain also valid for general bipartite states of systems of arbitrary dimension (μ=1,…,d_A, ν=1,…,d_B in (<ref>)–(<ref>)),if the sp space of the associated fermionicsystem (of dimension d_A+d_B)is decomposed asH_A⊕ H_B. The spdensity matrix ρ^ sp derivedfrom the state (<ref>) takes in the general case the blocked formρ^ sp=[αα^† 0; 0 α^Tα̅ ] =[ ρ_A 0; 0 ρ_B ] ,i.e. ⟨ c^†_Sνc_S'μ⟩=δ_SS'(ρ_S)_μν, where ρ_A(B)are the local reduced density matrices Tr_B(A)|ψ⟩_AB⟨ψ| of the state (<ref>)in the standard basis. Hence, in the fermionic settingρ^ sp contains the information of both local states and its diagonalization implies that ofboth ρ_A and ρ_B. Its eigenvalues will then be those of these matrices, beinghence two-fold degenerate and equal tothe square of the singular valuesof the matrix α (becoming f_±=|α_±|^2 in the two-qubit case). In the general case, the entanglement entropy of thesate (<ref>) can then be written asE(A,B)=S(ρ_A)=S(ρ_B)=1/2S(ρ^ sp) ,which holds for the von Neumann entropy S(ρ)=- Tr ρlog_2ρ as well as for any trace form entropy <cit.> S(ρ)= Tr f(ρ) (f concave,f(0)=f(1)=0). Thus, the entanglement entropy of the general bipartite state(<ref>) is just proportional to the fermionic entanglement entropy (as defined in (<ref>))of the associated state (<ref>). Hence, for any dimension there is an exact correspondencebetween the bipartite states (<ref>)and the two-fermion states (<ref>), with local operators represented by linear combinations ofone-body local fermionoperators c^†_Sνc_Sμ (satisfying [c^†_Aμc_Aν,c^†_Bμ'c_Bν']=0) and|ψ⟩_AB entangled iff |ψ⟩_f is not a SD. This equivalence holds also for mixed states i.e., convex combinations of the states (<ref>)and (<ref>). The bipartite states will be separable, i.e., convex combinations of product states <cit.> iff the associated fermionic mixed state can be written as a convex combination of SD's of the form (<ref>). In particular, for two-qubit states a four-dimensional sp fermion space suffices andthe standard mixed state concurrence <cit.> will coincide exactly with the fermionic mixed state concurrence<cit.> of mixtures of states(<ref>).§.§ Bipartite entanglement as quasiparticle fermion entanglement Other fermionic representations of the state (<ref>) with similar properties are also feasible. For instance, in the two qubit case we can perform a particle hole-transformation of the fermion operators with spin down,c^†_S↑⟶ c^†_S↑, c^†_S↓⟶c_S↓, S=A,Bsuch that the aligned state |↓⟩_A⊗ |↓⟩_B corresponds now to the vacuum of the new operators (|0⟩⟶ c^†_A↓c^†_B↓|0⟩), with the new c^†_S↓ creating a hole. The remaining states of the standard basis become one and two particle-hole excitations. We can then rewrite the state (<ref>) as|ψ̃⟩_f = (α_-+ α_+c^†_A↑c^†_A↓c^†_B↑c^†_B↓) |0⟩ ,i.e., as a superposition of the vacuum plus two particle-hole excitations,with each “side” having now either 0 or two fermions, i.e., an even local number parity(e^iπ N_S=1 for S=A,B,N_S=∑_μc^†_Sμ c_Sμ). It is apparent that the state (<ref>) is a a quasiparticlevacuum or SD iffα_+=0 or α_-=0. Moreover, for the state (<ref>) Eq. (<ref>) leads again to C(|ψ̃⟩_f) =2|α_+α_-|,implying the equivalence (<ref>) between the bipartite and the presentgeneralized fermionic concurrrence, invariant under Bogoliubov (and hence particle-hole) transformations.Thelocal Pauli operators (<ref>) become nowσ̃_S x = c^†_S↑c^†_S↓+c_S↓c_S↑ , σ̃_Sy =-i(c^†_S↑c^†_S↓-c_S↓c_S↑) , σ̃_Sz = c^†_S↑c_S↑+c^†_S↓c_S↓-1 ,whichverify the same SU(2) commutation relations [σ̃_Sj,σ̃_S'k]= 2iδ_SS'ϵ_jklσ̃_Sl, withσ̃_Sj^2| ψ̃⟩_f=|ψ̃⟩_f ∀ j. Any local operationcan be written in termsof these operators, which represent now local paticle-hole creation or annihilationand counting. Similarly, we may write the general two-qubit state (<ref>) as|ψ̃⟩_f = ∑_μ,να_μν (c^†_A↑c^†_A↓)^n_μ (c^†_B↑c^†_B↓)^n_ν |0⟩ ,where μ,ν=± and n_-=0, n_+=1. This state can be brought back to the “Schmidt” form (<ref>) by means of“local” Bogoliubov transformationsc_S↑→ u_S c_S↑+v_S c^†_S↓, c_S↓→ u_S c_S↓-v_S c^†_S↑, |u_S^2|+|v_S^2|=1,which will diagonalize ρ^ qsp (see below) and change the vacuumas |0⟩→ [∏_S=A,B(u_S-v_Sc^†_S↑c^†_S↓)]|0⟩.It is again verified that for this state Eq. (<ref>) leads to C(|ψ̃⟩_f) =2| det α|=2|α_+α_-|, with |α_±| the singular values of thematrix α. The state (<ref>) is then entangled iff the state (<ref>) is not aquasiparticle vacuum or SD (C(|ψ̃⟩_f)>0). In this case the extended density matrix ρ^ qsp is to be considered, with elements⟨ c^†_Sν c_S'μ⟩=δ_SS'δ_μνp_S, ⟨ c_Sνc_S'μ⟩=δ_SS'δ_ν,-μ(-1)^n_μq_S, where p_A(B)=|α_++|^2+|α_+-(-+)|^2, q_A(B)=α_++α^*_-+(+-)+α_+-(-+)α^*_–. For the Schmidtform (<ref>), ρ^ qsp becomes diagonal ((p_A(B)=|α_+|^2, q_A(B) = 0). Reduced states ρ_A(B) are now to be recovered as particular blocks of ρ^ qsp:ρ_S = 1/2[ 1+⟨σ̃_Sz⟩⟨σ̃_Sx⟩-i⟨σ̃_Sy⟩; ⟨σ̃_Sx⟩+ i⟨σ̃_Sy⟩ 1-⟨σ̃_Sz⟩ ]= [⟨ c^†_S↑c_S↑⟩⟨ c_S↓c_S↑⟩; ⟨ c^†_Si↑c^†_S↓⟩⟨ c_S↑c^†_S↑⟩ ] .Diagonalization of ρ^ qsp will, nevertheless, still implythat of ρ_Aand ρ_B. It is verified that its eigenvalues are f_±=|α_±|^2,four-fold degenerate, with |α_±| the singular values of the matrix α. We then have E(A,B)=S(ρ_A)=S(ρ_B)=1/4 S(ρ^ qsp) ,again valid for any trace-form entropy S(ρ)= Tr f(ρ). And for convex mixtures of states of the form (<ref>) (whose rank will be at most 4), themixed state fermionic concurrence, as defined in <cit.>, will again coincideexactly with the standard two-qubit concurrence.The same considerations hold for general bipartite states (<ref>) of systems of arbitrary dimensionif a particle hole transformation (or in general, a Bogoliubov transformation) is applied to the original fermion operators in (<ref>). In such a case Eq.(<ref>) is valid for entropic functions satisfying f(p)=f(1-p) (a reasonable assumption as p represents an average occupation number of particle or hole), since ρ^ qsp will have eigenvalues f_k and 1-f_k, now two-fold degenerate, with f_kthose of the local states ρ_A(B). A final remark is thatthe representations (<ref>) and (<ref>) of Pauli operators can coexist independentlysince[σ_Sj,σ̃_S'k]=0 ,∀ j,k for both S'≠ S and S'=S (SU(2)× SU(2) structure <cit.> at each side A or B).Moreover, the even local parity states (<ref>) belong to the kernel of the operators (<ref>), while the odd local parity states (<ref>) (e^i π N_S=-1)belong to the kernel of the operators (<ref>): σ_Sj|ψ̃⟩_f=σ̃_Sj|ψ⟩_f=0 ,for S=A,B and j=x,y,z. Hence, unitary operators e^i∑_jλ_jσ_Sj (e^i∑_jλ_jσ̃_Sj) will become identities when applied to states |ψ̃⟩_f (|ψ⟩_f). A fermion system with a sp space of dimension 4 can then accommodate two distinct two-qubit systems, one for each value of the local number parity, keeping the total number parity fixed (e^iπ(N_A+N_B)=1). §.§ Bipartite entanglement with no fermion entanglementPrevious examples show an exact correspondence between bipartite and fermion entanglement. The representations considered involve not only a fixed value of the global parity, but also of the local number parity. It is apparent, however, that it is also possible to obtain bipartite entanglement from SD's by choosing appropriate partitions of the sp space, although in this case the local parity will not be fixed. For instance, the single fermionstate |ψ⟩_f=(α c^†_A↑+β c^†_B↑)|0⟩ ,where the fermion is created in a state with no definite position if αβ≠ 0,leadsobviously to S(ρ^ sp)=0 but corresponds to an entangled stateα |↑⟩_A⊗|0⟩_B+β|0⟩_A⊗ |↓⟩_B.However, the local states at each side have different number parity. The same occurs with thetwo-fermion SD (α c^†_A↑+β c^†_B↑) (α' c^†_A↓ +β 'c^†_B↓)|0⟩, which has zero fermionic concurrence butcorresponds to the entangled state αβ'|↑⟩_A⊗ |↓⟩_B -α'β|↓⟩_A⊗ |↑⟩_B +αα'|↑↓⟩_A⊗ |0⟩_B+ββ'|0⟩_A⊗ |↑↓⟩_B. Hence, although there is entanglement with respect to the (A,B) partition,it is not possibleto make arbitrary linear combinations of the eigenstates of ρ_A or ρ_B, since they maynot have a definite number parity. Whilesuch entanglement may be sufficient for observing Bellinequalities violation, as proposed in <cit.>, it can exhibitlimitations for other tasksinvolving superpositions of local eigenstates, as discussed in sec.<ref>. This effect willoccur whenever one of the fermions is created in a state which is “split” by the chosenpartition of the sp space. With the restriction of a fixed number parity at each “side”an equivalence between bipartite and fermionic entanglement can become feasible, as discussed next.Notice that such restriction directly impliesblocked sp density matricesρ^ spand ρ^ qsp, since all contractions ⟨ c^†_Aic_Bj⟩ and⟨ c^†_Aic^† _Bj⟩ linking both sides do not conserve the local parityand will therefore vanish ∀ i,j.§.§ Fermion entanglement as two-qubit entanglement Let us now return to the two-fermion state(<ref>).The reason why the two particles become distinguishable is that the “position” observable allows us to split the sp state space H as the direct sum of two copies of the spin space H_S, H= H_ S_A⊕ H_ S_B,with _A⟨μ|μ⟩_B=⟨ 0| c_Aμc^†_Bμ|0⟩=0 for μ=↑ or ↓.This last condition ensures in fact that there is just one fermion at each side(N_A(B)|ψ⟩_f=|ψ⟩_f). However, for a more general two-fermionstate, like that considered in the previous section, it is no longer possible toperform a measurement of the spin of only one particle by coupling it with positionsince both particles may be found at the same site. But now nothing prevents us from turning back the argument and state that if for an arbitrary state |ψ⟩_f, it is possible to split H as H= H_A⊕ H_B, where H_A and H_B contain just one fermion (N_A|ψ⟩_f=N_B|ψ⟩_f=1), then we recover again a system of two distinguishable qubits.This last feature leads us to the following important result: Lemma 1: Let |ψ⟩_f be an arbitrary pure state of a fermion system with a 4-dimensional sp space H, having definite number parity yet not necessarily fixed fermion number. Then the entropy (<ref>) of the corresponding density matrix ρ^ qsp is proportional to the entanglement entropy between the two distinguishable qubits that can be extracted just by measuring the appropriate observables.We start with a general state |ψ⟩_f with even number parity, whichin this space will have the form (<ref>). For generalα_ij, α_0and α_4 in (<ref>), the basis of the sp space H determined by thefermion operators {c_i,c^†_i} cannot be split in order to measure only oneparticle at each part. This fact remains true even if α_0=α_4=0,as α is a general antisymmetric matrix. However, as proved in <cit.>,it is always possible to find another basis of H, determined by fermion operators {a_i,a^†_i} related to {c_i,c^†_i}through a Bogoliubov transformation, such that the state (<ref>) can be rewritten as|ψ⟩_f=(α_+a^†_1 a^†_2+α_-a^†_3a^†_4)|0⟩ ,which is analogous to Eq. (<ref>). Here |α_±|^2=f_± are just the distinct eigenvalues(<ref>)of the extended density matrixρ^ qsp determined by the state (<ref>), whereas {a_i,a^†_i} are suitable quasiparticle operatorsdiagonalizing ρ^ qsp. The concurrence (<ref>) becomesC(|ψ⟩_f)=2|α_+α_-|. We then recognize (<ref>) as the Schmidt decomposition (<ref>) of a two-qubit state written in the fermionic representation (<ref>),since, for instance, the sets {a^†_1, a^†_3}and {a^†_2, a^†_4} (analogous to {a^†_A↑,a^†_A↓} and {a^†_B↑,a^†_B↓}) span subspaces H_A and H_B with N_A=N_B=1 (N_A(B)|ψ⟩_f=|ψ⟩_f).And because the Schmidt coefficients |α_±|^2 coincide with the eigenvalues of ρ^ qsp, we obtain again S(ρ_A)=S(ρ_B)=1/4S(ρ^ qsp) (Eq.(<ref>)), with the fermionic concurrence coinciding exactly with the standard one.The case of general odd parity states, which in this sp space are linearcombinations of states with one and three fermions,|ψ⟩_f=∑_i=1^4β_i c^†_i|0⟩+β̃_i c_i|0̅⟩ ,where |0̅⟩=c^†_1c^†_2c^†_3c^†_4|0⟩ andc_i|0̅⟩=1/3!∑_j,k,lϵ_ijklc^†_jc^†_kc^†_l|0⟩, can be treated in a similar way, as they can be converted to even parity states of the form (<ref>) by a particle-hole transformation of one of the states (i.e.,c^†_1→ c_1, |0⟩→ c^†_1|0⟩, leading to α_0=β_1, α_4=-β̃_1, α_1j=-β_j, and α_ij=∑_kϵ_ijk1β̃_k for i,j=2,3,4 in Eq.(<ref>)). They can then be also written in the form (<ref>), in terms ofsuitable quasiparticle operators diagonalizing ρ^ qsp, so that the previousconsiderations still hold. The concurrence of the states (<ref>), given by <cit.>C(|ψ⟩_f)=2|∑_i=1^4 β_iβ̃_i|, becomes again2|α_+α_-|. Some further comments are here in order. First, just the subspaces of H generated by {a^†_1, a^†_2} and {a^†_3, a^†_4} are defined by (<ref>), since any unitary transformation a^†_1(2)→∑_k=1,2U_k,1(2)a^†_k (and similarly for a^†_3(4)) will leave itunchanged (except for phases in α_±).Secondly, we may also reinterpret the state (<ref>) as a two-fermion state with even local number parity if side A is identified with operators {a^†_1,a^†_2} and B with {a^†_3,a^†_4}, such that each side has either 0 or two fermions (even number parity qubits). Still witheven local number parity we may as well rewrite it in the form (<ref>),i.e.,|ψ⟩_f=(α_- +α_+a^†_1a^†_3a^†_2a^†_4)|0⟩ ,through a transformation a^†_i→ a_i for i=3,4, with |0⟩→ a^†_3 a^†_4|0⟩. Here just the vacuum |0⟩ and the completely occupied state |0̅⟩ are defined, since (<ref>) remains invariant (up to a phase in α_+)by any unitary transformation a^†_i→∑_kU_kia^†_k of the operators a^†_i.Finally, if|ψ⟩_f is a two-fermion state 1/2∑_ijα_ijc^†_i c^†_j|0⟩, the previous considerations remain valid for a sp space H of arbitrary dimension. In this case κ=0 and it is always possible to rewrite |ψ⟩_f as <cit.>|ψ⟩_f=∑_kα_k a^†_ka^†_k̅|0⟩ ,where |α_k^2| are the eigenvalues of ρ^ sp=αα^† and {a_k,a_k̅} are suitable fermion operators diagonalizing this matrix, obtained through a unitary transformation a_k(k̅)=∑_i U̅_ik(k̅) c_i(satisfying <cit.>U^†αU̅=α' with α'a block diagonal matrix with2× 2 blocks α_k[01; -10 ]). The sp space can then be written as H_A⊕ H_B with H_A(B) the subspaces spanned by the sets {a^†_k(k̅)}, containing each one fermion.We thus obtainS(ρ_A)=S(ρ_B)=1/2S(ρ^ sp) (Eq. (<ref>)). §.§ Fermion entanglement as minimum bipartite entanglement We now demonstrate a second general result, concerning the mode entanglement associated with generaldecompositionsH= H_A⊕ H_B of a four-dimensional sp space. Any many-fermion state can be written as|ψ⟩_f=∑_μ,να_μν|μν⟩, where μ(ν)labels orthogonal SD's on H_A (H_B) and |μν⟩= [∏_i∈ H_A (c^†_i)^n_i^μ] [∏_j∈ H_B (c^†_j)^n_j^ν] |0⟩ is a SD on H,with n_i^μ=0,1the occupation of sp state i in the state μ. The ensuing reduced statesρ_A=∑_μ,μ'(αα^†)_μμ'|μ⟩⟨μ'| and ρ_B=∑_ν,ν'(α^Tα̅)_νν'|ν⟩⟨ν'|satisfy Tr ρ_A(B) O_A(B)=_f⟨ψ|O_A(B)|ψ⟩_ffor any operator depending just on the local fermions {c_i,c^†_i, i∈ H_A(B)}. The entanglement entropy associated with such bipartitionis then <cit.> E(A,B)=S(ρ_A)=S(ρ_B).In the present case we may have either2+2 bipartitions (dimH_A = dimH_B=2), or1+3 bipartitions (dimH_A=1,dimH_B=3). In the latter the entanglement is determined just bythe average occupation of the single state of H_A <cit.>and correspondsto the case where A has access to just one of the sp states possibly occupied in|ψ⟩_f. A realization of a 2+2 partition is just thatof spin 1/2 fermions which can be at two-different sites (one accessible to Aliceand the other to Bob), while a 1+3 bipartition could be one whereAlice has accessto one site and just one spin direction, i.e., to the knowledge of the occupation ofthe sp state A_↑. It could also apply to any asymmetric situation like thatwhere spins are all up (i.e., aligned along the field direction) but the fermions can bein four different locations or orbital states, with only one accessible to Alice.Lemma 2: Let |ψ⟩_f be a general definite number parity fermion state in a sp space H of dimension 4, and let H= H_A⊕ H_B be an arbitrary decomposition of H with H_A and H_B of finite dimension. The entanglement entropy associated with such bipartitionsatisfiesS(ρ_A)=S(ρ_B)≥1/4S(ρ^ qsp) .Eq. (<ref>) holds for any entropic form S(ρ)= Trf(ρ) (f concave, f(0)=f(1)=0). Hence, the fermionic entanglement represents the minimum bipartite entanglement that can be obtained in such a space, which is reached for those bipartitions arising from the normal forms (<ref>) or (<ref>). The greater entanglement in a 2+2 bipartitionis obtained at the expense of loosing a fixed number parity in the local reduced states. Note that S(ρ^ qsp) vanishes only if |ψ⟩_f is a quasiparticlevacuum or SD in some sp basis, while S(ρ_A(B)) does so only when the previouscondition holds in a basis compatible with the chosen bipartition. We will actually show the equivalent majorization <cit.> relation λ(ρ_A(B))≺ (f_+,f_-) ,where λ(ρ_A(B)) denotes the spectrum of ρ_A or ρ_B sorted in decreasing order and f_+,f_-=1-f_+≤ f_+ are the distinct eigenvalues (<ref>) (fourfold degenerate)of ρ^ qsp. Eq.(<ref>) is then equivalent to the condition λ_ max≤ f_+, withλ_ maxthe largest eigenvalue of ρ_A(B), andimplies (<ref>), while (<ref>) implies (<ref>) if valid for any entropic functionf <cit.>.Consider first a general even parity state (<ref>) and a 2+2 decompositionH= H_A⊕ H_B, with H_A≡ H_12,H_B≡ H_34 and H_ij the subspace generated by {c^†_i, c^†_j}.Changing to the notation A_1, A_2,B_1,B_2 for sp states 1,2,3,4, we can rewrite (<ref>)as a sum of states of the form (<ref>) and (<ref>) (Fig. <ref>):|ψ⟩_f =∑_μ,νβ_μνc^†_A_μc^†_B_ν|0⟩+ ∑_μ,νβ̃_μν (c^†_A_1c^†_A_2)^n_μ(c^†_B_1c^†_B_2)^n_ν |0⟩ ,whereμ,ν=1,2, β_μν=α_μ,ν+2,n_μ=μ-1, β̃_11=α_0, β̃_22=α_4, β̃_12=α_34, and β̃_21=α_12.The first (second) sum in (<ref>) is the odd (even) local number parity component.After local unitary transformations c_Sμ→∑_νU̅^S_νμc_Sν, S=A,B, which will not affect the vacuum nor the even local parity component (except for phases in β̃_μν, determined by det U^S), we can set β_μν diagonal. Similarly, after local Bogoliubov transformations c_S_1→ u_S c_S_1+v_S c^†_S_2, c_S_2→ u_S c_S_2-v_S c^†_S_1, |u_S^2|+|v_S^2|=1, with |0⟩→[∏_S=A,B(u_S-v_Sc^†_S_1c^†_S_2)]|0⟩, we can set β̃_μν diagonal as discussed in sec. <ref>. Though modifying the vacuum, they will not change the form of the odd local parity component except forphases inβ_μν. Thus, by local transformations it is possible to rewrite (<ref>) as|ψ⟩_f=(β_1 c^†_A_1c^†_B_1+ β_2c^†_A_2c^†_B_2+β̃_1+β̃_2c^†_A_1 c^†_A_2c^†_B_1c^†_B_2)|0⟩ ,where |β_μ| and |β̃_μ| are the singular values of the 2× 2 matrices β and β̃ in (<ref>). Eq.(<ref>) is the Schmidt decomposition for this partition, with (|β_1^2|,|β_2^2|, |β̃_1^2|, |β̃_2^2|) the eigenvalues of the reduced density matrices ρ_A and ρ_B ofmodes(A_1,A_2) and (B_1,B_2) respectively. Now, suppose λ_ max=|β_1^2|.We have|β_1|^2≤ |β_1|^2+|β̃_2|^2=⟨ c^†_A_1c_A_1⟩ .But ⟨ c^†_A_1c_A_1⟩=∑_k=1^8|W_A_1,k|^2f_k, where f_k are the eigenvalues of ρ^ qsp (equal to f_+ or f_-)and W the unitary matrix diagonalizing ρ^ qsp (∑_k=1^8|W_A_1,k|^2=1).Therefore,f_-≤⟨ c^†_A_1c_A_1⟩≤ f_+ .Eqs. (<ref>)–(<ref>) imply |β_1|^2≤ f_+, which demonstrates Eq. (<ref>) and hence (<ref>) for a general 2+2 bipartition H_A⊕ H_B. Forλ_ max equal to any other coefficientthe proofis similar.Moreover, Eq. (<ref>) also shows that the sorted spectrum λ(ρ_A_1(A_2,B))=(⟨ c^†_A_1c_A_1⟩,1-⟨ c^†_A_1c_A_1⟩)^↓ associated with the 1+3 bipartition H_A_1⊕ H_A_2,B satisfies λ(ρ_A_1,(A_2,B))≺ (f_+,f_-). In the latter S(ρ_A_1) is the entanglement between the sp mode A_1 and its orthogonal complement as defined in <cit.>, determined by the average occupation ⟨ c^†_A_1c_A_1⟩ of the mode. Hence,Eqs.(<ref>)–(<ref>) hold as well for any 1+3 bipartition.And equality in (<ref>) for all entropic functions is evidently reached only for those bipartitions arising fromthe normal forms (<ref>)–(<ref>): Considering the non-trivial case f_+<1, if equality in (<ref>) is to hold for all entropies, necessarily ρ_A(B) should be of rank 2 with λ(ρ_A(B))=(f_+,f_-).For a 1+3 bipartition, this identity directly implies ⟨ c^†_A_1c_A_1⟩=f_+ or f_- and hence a bipartition arising from a normal form (<ref>)–(<ref>), where A≡ A_1 is one of the sp states of the normal basis.And for a 2+2bipartition, it implies that the two eigenstates of ρ_A with non-zero eigenvalues f_±should have the same number parity, since otherwise Eq. (<ref>) would implyC(|ψ⟩_f)=0 and therefore f_+=1, in contrast with the assumption. Hence such bipartition must arise from a normal form (<ref>) or (<ref>).The demonstration of previous results for odd global number parity states is similar, as they can be rewritten as even parity states after a particle-hole transformation. Some further comments are also in order. We may rewrite the state (<ref>) as|ψ⟩_f=√(p_-)|ψ_-⟩_f+√(p_+)|ψ_+⟩_f ,where |ψ_-⟩_f=1/√(p_-)(β_1 c^†_A_1c^†_B_1+β_2c^†_A_2c^†_B_2)|0⟩, |ψ_+⟩_f=1/√(p_+)(β̃_1+β̃_2c^†_A_1 c^†_A_2c^†_B_1c^†_B_2) |0⟩ are the normalized odd and even local parity componentsand p_-=|β_1^2|+|β_2^2|, p_+=|β̃_1^2|+|β̃_2^2|=1-p_-. We then see that for the von Neumann entropy, we obtainS(ρ_A)=S(ρ_B)=p_-S(ρ_A^-)+p_+S(ρ_A^+)+S(p) ,where the first two terms represent the average of the entanglement entropies of the odd and even local parity components (S(ρ_A^-)=-∑_μ|β_μ^2|/p_-log_2|β_μ^2|/p_-, S(ρ_A^+)=-∑_μ|β̃_μ^2|/p_+log_2|β̃_μ^2|/p_+) while S(p)=-∑_ν=± p_νlog_2 p_ν is theadditional entropy arising from the mixture of both local parities. We then have 0≤ S(ρ_A)≤ 2, with the maximum S(ρ_A)=2 reached iff S(ρ_A^±)=1 and p_±=1/2.On the other hand, the fermionic concurrence (<ref>) of the state (<ref>) is justC(|ψ⟩_f)=2|β_1β_2+β̃_1β̃_2| .It then satisfies|p_-C_–p_+C_+|≤ C(|ψ⟩_f)≤ p_-C_-+p_+C_+ ,where C_±=C(|ψ_±⟩_f)=2(_|β_1β_2|/p_-^|β̃_1 β̃_2|/p_+) are the concurrences of the even and odd local parity components. We then see, for instance, that formaximum bipartite entanglement S(ρ_A)=2, C_±=1 and hence C(|ψ⟩_f) can take any value between 0 and 1, according to the relative phase between the even and odd local parity components.Finally, it is obviously possible to rewrite the Schmidt form (<ref>) as a two-fermion state by means of suitable local particle-hole transformations (i.e. c_B_μ→ c^†_B_μ, μ=1,2, with |0⟩→ c^†_B_1c^†_B_2|0⟩). After some relabelling, weobtain the equivalent form|ψ⟩_f=(β_1c^†_A_1c^†_B_1+ β_2c^†_A_2c^†_B_2+ β̃_2c^†_A_1c^†_A_2- β̃_1c^†_B_1c^†_B_2)|0⟩ ,where terms with two fermions at the same side side are added to the form (<ref>). Therefore, all previous considerations (<ref>)–(<ref>) can be realized with a fixed total number of fermions, with expression (<ref>) still valid.§ APPLICATIONThe formalism of the previous sections may now be used to rewrite a qubit-based quantum circuit as a circuit based on fermionic modes. It is easy to see by now that any pair of fermionic modes, say i,j, prepared in such a way that their total occupation is constrained to N_ij=c^†_ic_i+c^†_jc_j=1, is essentially a qubit. Therefore, a collection of nsuch pairs of modes constitutes a system of n qubits. Furthermore any single-qubit operation can be performed on each pair just by using unitaries in H linking only these two modes, and these unitaries can be always written in terms of the effective Pauli operators (<ref>), i.e., σ^ij_x=c^†_ic_j+c^†_jc_i, σ^ij_y=i(c^†_jc_i-c^†_ic_j), σ^ij_z=c^†_ic_i-c^†_jc_j. The last ingredient for universal computation is the CNOT gate, which in the tensor product space A⊗ Bcan be written as U__ CNOT=|0⟩⟨ 0|⊗ I+|1⟩⟨ 1|⊗σ_x=exp[iπ/4(1-σ_z)⊗(1-σ_x)]. In the fermionic representation, if A is spanned by modes ij and Bby the different modes kl, for states having one fermion at each pair of modes it canbe written asU^f__ CNOT=exp[ iπ/4(1-σ_z^ij)(1-σ_x^kl)] .Since just an even number of fermion operators c per pair are involved, its action is not affected by the state of intermediate pairs. It is then possible to implement any qubit-based quantum circuit using fermion states.As an example, in Fig. <ref> we show the teleportation protocol adapted to be implemented using an entangled fermion state as resource, and a two mode sate to be teleported. Alice has the modes {|A_1⟩, |A_2⟩, |A_3⟩, |A_4⟩} while Bob is in possession of {|B_1⟩, |B_2⟩}. The first two modes of Alice are entangled with those of Bob, being in the joint state |β_00⟩=1/√(2)(c^†_A_1c^†_B_1+c^†_A_2 c^†_B_2)|0⟩, and the remaining modes of Alice are in the state |ψ⟩=(αc^†_A_3+ βc^†_A_4)|0⟩, |α|^2+|β^2|=1. The input state is therefore|ψ_i⟩=1/√(2)(αc^†_A_3+ βc^†_A_4) (c^†_A_1c^†_B_1+c^†_A_2c^†_B_2)|0⟩and it is straightforward to see that the output state is|ψ_o⟩ = 1/2[ c^†_A_4c^†_A_2(αc^†_B_1+βc^†_B_2) +c^†_A_4c^†_A_1(αc^†_B_2+βc^†_B_1)+ [c^†_A_3c^†_A_2(-αc^†_B_1+βc^†_B_2) +c^†_A_3c^†_A_1(-αc^†_B_2+βc^†_B_1)] |0⟩The controlled operations on Bob's modes depicted in Fig. <ref> then ensure that his output will be the state |ψ⟩.Considering now a general circuit, if the input states are restricted to be SD's in the previous basis, with one fermion for each pair, we recover a classical circuit. The CNOT gate in (<ref>) reduces for these states to a classical controlled swap or Fredkin gate, which implies thatreversible classical computation can be done with SD's as input states.On the other hand, if the occupation number restriction N_ij=1 (i.e., odd number parity for each pair)is relaxed, so that the building blocks of the circuit are no longer single fermionsthat can be found in two possible states, but rather the fermionic modes themselves, other possibilities arise. For instance, if now the input states contain either 0 or two fermions for each pair (evennumber parity qubits), such that modes i,j are either both empty or both occupied, then we should use the σ̃_μ^ij operators as defined in (<ref>), i.e., σ̃^ij_x=c^†_ic^†_j+c_jc_i, σ̃^ij_y=i(c_jc_i-c^†_ic^†_j), σ̃^ij_z=c^†_ic_i+c^†_jc_j-1. In thiscase the operator Ũ^f__ CNOT should be constructed as in Eq. (<ref>) with the σ̃_μ operators, while the operator (<ref>), and in fact any unitary gate built with the σ^ij_μ operators,will become an identity for these states, as previously stated. Hence, by adding the appropriate gates, the same modes can in principle be used for even and odd number parity qubits independently.For example, in the even local parity setting the input state for the teleportation protocol would be |ψ̃_i⟩=1/√(2)(β+αc^†_A_3c^†_A_4) (1+ c^†_A_1c^†_A_2c^†_B_1c^†_B_2)|0⟩ .If |0⟩ stands for a reference SD (Fermi sea), then this state involves 0, one, two and three particle hole excitations, withA_4,A_2,B_2, standing for holes. The output state becomes|ψ̃_o⟩ = 1/2[ (β+αc^†_B_1c^†_B_2) +c^†_A_1c^†_A_2(α+β c^†_B_1c^†_B_2) +c^†_A_3c^†_A_4×[ (β-α c^†_B_1c^†_B_2) +c^†_A_1c^†_A_2c^†_A_3c^†_A_4(-α+β c^†_B_1c^†_B_2)] |0⟩ ,so that if Alice measures which of her modes are occupied and sendsthe result to Bob,he can reconstruct the original state by applying the pertinentX̃≡ i e^-iπ/2σ̃^12_x and Z̃≡ i e^-iπ/2σ̃^12_z operators.Finally, let us consider the case ofsuperdense coding <cit.>.It is clear from the previous discussion that it can be implemented with the fermionic |β_00⟩ state of the teleportation example and performing exactly thesame local operations of the usual case, but viewed now as two-mode operations. Now a general state with even global parity of the four modes {|A_1⟩, |A_2⟩, |B_1⟩, |B_2⟩} is a combinationof eight states as in Eq. (<ref>): six two-particle states, the vacuum |0⟩ and the completely occupied state |0̅⟩, as shown inFig. <ref>. Four of the six two-particle states (top of Fig. <ref>),haveN_A=N_B=1 and can be used toreproduce the known results of the standardprotocol. But the four remaining states, which have even local parity, may be used aswell for superdense coding if the proper local operations expressed in terms of theσ̃^AB_μ are performed. A general even parity state (<ref>) may then be thought of as a superposition of states of two different two-qubit systems, like in Eqs.(<ref>) and (<ref>). Defining the maximally entangled orthogonal definitelocal parity states |β_00 10⟩ = γ(c^†_A_1c^†_B_1± c^†_A_2c^†_B_2)|0⟩, |β̃_00 10⟩=γ(c^†_A_1c^†_A_2 c^†_B_1c^†_B_2± 1)|0⟩ |β_01 11⟩ = γ(c^†_A_1c^†_B_2± c^†_A_2c^†_B_1)|0⟩, |β̃_01 11⟩=γ(c^†_A_1c^†_A_2± c^†_B_1c^†_B_2)|0⟩with γ=1/√(2), we may consider for instance the state|Ψ_00⟩=1/√(2)(|β_00⟩ +|β̃_00⟩) .By implementing on (<ref>) the identity and the local operationsie^-iπ/2(σ^A_μ+σ̃^A_μ)=σ_μ+σ̃_μ, μ=x,y,z, and taking into account Eq. (<ref>), Alice can generate four orthogonal states: |Ψ_00⟩ and|Ψ_01⟩ = ie^-iπ/2(σ^A_x+σ̃^A_x) |Ψ_00⟩=1/√(2)(|β_01⟩+|β̃_01⟩) ,|Ψ_10⟩ = ie^-iπ/2(σ^A_z+σ̃^A_z)|Ψ_00⟩ =1/√(2)(|β_10⟩+|β̃_10⟩) ,|Ψ_11⟩ = -e^-iπ/2(σ^A_y+σ̃^A_y) |Ψ_00⟩ =1/√(2)(|β_11⟩+|β̃_11⟩) .But shecan also perform these operations with a local parity gate P^A=-exp[iπ N_A] that changes the sign of local even parity states. This allows her to locally generate another set of four orthogonal states,|Ψ̃_ij⟩ = P^A|Ψ_ij⟩=1/√(2) (|β_ij⟩-|β̃_ij⟩) ,i,j=0,1 ,which are orthogonal to each other and to the states (<ref>)–(<ref>). Hence,by relaxing the occupation number constraint on the partitionsit is possible for Aliceto send8 orthogonal states to Bob, i.e. three bits of information, using only two modesand local unitary operations that preserve the local parity, while with one type of qubits andthe same operations she can send only two bits. Of course, ifparity restrictions were absentand she could change the local (and hence the global) parity shecould send four bits (in agreementwith the maximum capacity for twod=4 qudits, which islog_2 d^2 <cit.>). A fixed globalparity constraint reduces the total number of orthogonal states she can send to Bob by half. On the other hand, since the state (<ref>) does not have a definite local number parity,the ensuing bipartite entanglement is not restricted by the fermionic entanglement as shownin sec. <ref>. In fact all previous 8 states (<ref>), (<ref>) and (<ref>) havemaximum bipartite entanglememt, leading to maximally mixed reduced states ρ_A(B):S(ρ_A)=S(ρ_B)=2,while by applying Eq. (<ref>) it is seen that the fermionic concurrence of the previous states is C(|Ψ_ij⟩)=C(|Ψ̃_ij⟩)=1. The unitary operations applied by Alice are local and hence cannot change the bipartite entanglement, while they are also one-body unitaries (i.e., exponents of quadratic fermion operators)so that they cannot change the fermionic concurrence and entanglement (i.e., the eigenvalues ofρ^ qsp) either. In fact, the fermionic entanglement is here not required. By changing theseed state (i.e., |Ψ'_00⟩=1/√(2)(|β_00⟩+|β̃_10⟩)), it is possible for Alice to generate locally 8 orthogonal states with the same bipartite entanglement yet no fermion entanglement (C(|Ψ'_00⟩)=0).Therefore the entanglement built with local states with different number parity plays the role of a resource for superdense coding. In fact even the state(<ref>) with α=β=1/√(2), which has obviously null concurrence,can in principle be used for sending two bits if Alice can perform the parity preserving operationsP^A=-exp[iπ N_A], σ_x+σ̃_x and P^A(σ_x+σ̃_x). It is worth noting, however, that the same state cannot be directly used as a resource for teleportation with the standard protocol without violating the parity superselection rule,since Bob's two local states have opposite parity and cannot be superposed. After a measurement of Alice's modes Bob's reduced state will collapse to a state of definite parity in a realizable protocol, so that it will be impossible for him to recover a general state |ψ⟩.We have so far considered just the number parity restriction. If other superselection rules(like charge or fermion number) also apply for a particular realization they will imply strongerlimitations on the capacity of states like (<ref>). Nonetheless, even local parity qubitswith no fixed fermion number remain realizable through particle-hole realizations, i.e.,excitations over a reference Fermi sea in a many-fermion system.We also mention that a basic realization of four dimensional sp space-based fermionic qubitsis that of a pair of spin 1/2 fermions in the two lowest states of a double well scenarioin a magnetic field, which would control the energy gap between both spin directions and thetransitions between them. For single occupation of each well we would have odd local parityqubits, while allowing double or zero occupancy through hopping between wells we could alsohave even local parity qubits. § CONCLUSIONS We have first shown that there is an exact correspondence between bipartite states and two-fermion states of the form (<ref>) having afixed local number parity. Entangled states are represented by fermionic states which are not Slater Determinants, and reduced local states correspond to blocks of thesp density matrix. In particular, qubits can be represented by pairs of fermionic modes with occupation number restricted to 1 (odd number parity qubits).This result allows to rewrite qubit-based quantum circuits as fermionic circuits.But in addition, a fermionic system also enables zero or double occupancy ofthese pairs, which gives rise to a second type of qubit (even number parity qubits). Dual type circuits can then be devised, as the gates for one parity becomeidentities for the other parity. And even though both types of qubits cannotbe locally superposed due to the parity superselection rule, they can contributeto the entanglement in a global fixed parity state. We have then demonstrated rigorous properties of the basic but fundamental caseof a four-dmensional sp space. First, there is always a single-particle (or quasiparticle)basis in which any pure state can be seen as a state of two distinguishable qubits,withthe fermionic concurrence determining the entanglement between these two qubits(Eq. (<ref>)). Such entanglement is “genuine”, in the sense that the local statesinvolved have a definite parity and can therefore be combined. Secondly, such fermionicentanglement was shown to providealways a lower bound to the entanglement obtainedwithany other bipartition of this sp space, although the extra entanglement arisesfrom the superposition of states with different local parity. While its capacity forprotocols involving superpositions of local states is limited, such entanglementcan nevertheless still be useful for other tasks such as superdense coding. The authors acknowledge support from CONICET (N.G.) and CIC (R.R.) of Argentina. Work supported by CIC and CONICET PIP 112201501-00732.999 A.08 L. Amico, R. Fazio, A. Osterloh,V. Vedral,Rev. Mod. 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"authors": [
"N. Gigena",
"R. Rossignoli"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170426130825",
"title": "Bipartite entanglement in fermion systems"
} |
[email protected] 1NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA 2University of Maryland, Baltimore County,1000 Hilltop Cir, Baltimore, MD 21250, USA 3Laboratoire d'astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geoffroy Saint-Hilaire, F-33615 Pessac, France 4Department of Astronomy, Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA 5Sagan FellowPlanets are thought to form via accretion from a remnant disk of gas and solids around a newly formed star. During this process material in the disk either remains bound to the star as part of either a planet, a smaller celestial body, or makes up part of the interplanetary medium; falls into the star; or is ejected from the system. Herein we use dynamical models to probe the abundance and properties of ejected material during late-stage planet formation and estimate their contribution to the free-floating planet population. We present 300 N-body simulations of terrestrial planet formation around a solar-type star, with and without giant planets present, using a model that accounts for collisional fragmentation. In simulations with Jupiter and Saturn analogs, about one-third of the initial (∼5 M_⊕) disk mass is ejected, about half in planets more massive than Mercury but less than 0.3 M_⊕, and the remainder in smaller bodies. Most ejections occur within 25 Myr, which is shorter than the timescale typically required for Earth-mass planets to grow (30–100 Myr). When giant planets are omitted from our simulations, almost no material is ejected within 200 Myr and only about 1% of the initial disk is ejected by 2 Gyr. We show that about 2.5 terrestrial-mass planets are ejected per star in the Galaxy. We predict that the space-borne microlensing search for free-floating planets from the Wide-Field Infra-Red Space Telescope (WFIRST) will discover up to 15 Mars-mass planets, but few free-floating Earth-mass planets.The Demographics of Rocky Free-Floating Planets and their Detectability by WFIRST Thomas Barclay1,2, Elisa V. Quintana1, Sean N. Raymond3, and Matthew T. Penny4,5 Received: date / Revised version: date ====================================================================================§ INTRODUCTIONIn the classical picture of planet formation, planets grow via accretion from material within a protoplanetary disk that remains around a newly formed star. During this process, material is either accreted by planets or smaller bodies, resides in the interplanetary medium, falls into the central star, or is imparted with enough angular momentum to be ejected from the system. So-called free-floating planets (or rogue planets, wandering planets, etc.) have been observed by gravitational microlensing surveys <cit.> and by optical and infrared wide-field surveys <cit.>. Estimates have been made that there are as many as 2 free-floating Jupiter-mass planets for every star in the Galaxy <cit.>. Although it is unlikely that all the giant free-floating planets were ejected from planetary systems during planet formation <cit.> because this requires a higher occurrence rate of bound giant planets than is observed, it is probable that ejected planets make up some fraction of the free-floating population. It is also possible that binary stars produce the majority of the free-floating giant planet population <cit.>.Hitherto, the detected free-floating planets have been giant worlds that could potentially represent the tail-end of the stellar mass distribution <cit.>. However, the microlensing experiment <cit.> from the K2 mission <cit.> is potentially sensitive to super-Earth-mass bodies and above <cit.>. The Wide-Field Infra-Red Survey Telescope (WFIRST) from NASA, due to launch in 2024 <cit.>, will be sensitive to bodies less massive than Earth, and the ESA mission Euclid will be sensitive to sub-Earth-mass planets <cit.>.Ejected material is a natural outcome of the planet formation process. The process begins when a molecular cloud collapses to form a star that is subsequently surrounded by a remnant disk of gas and dust. The growth of planets from within this disk occurs in several distinct phases that are typically addressed independently due to their different physical processes. In the early phase, dusty material coalesces into 1-10 km-sized planetesimals in a process that isn't yet fully understood. The planetesimals accrete material within their gravitational zone of influence and form embryos that are typically Mars-mass in the terrestrial region <cit.>. Up to this point, gas is still present in the disk and massive cores can accrete gaseous envelopes to form giant planets <cit.>. In the final stages of planet formation, the gas in the disk is dispersed and the process becomes dominated by gravitational interactions and collisions. The solid material in the disk is stirred as bodies grow or scatter off each other, and this process continues for tens of millions of years until final planets form on stable, widely separated orbits – the so-called `giant impact phase' <cit.>. This description applies to Sun-like stars, but it is expected that similar processes occur for planets forming around other star types, albeit on different timescales.If gas giants are present while the terrestrial planets are forming, as was the case in our Solar System, their presence will dominate the dynamics and excite the eccentricities and inclinations of the protoplanets in the disk <cit.>. The degree of excitation is sensitive to the architecture of the giant planets. Nearby, more massive and/or more eccentric giant planets will cause stronger perturbations that can lead to crossing orbits among the bodies and ultimately to ejections.Numerical N-body models have been widely used to study terrestrial planet formation from a range of initial conditions – star types, disk mass distributions, etc. <cit.>. While a large number of studies have been performed, details of the ejected material from these studies is typically not well documented, as the focus of these studies has been to form and characterize terrestrial planets and not necessarily to track the fate of ejected mass. In some cases the fraction of disk mass that was ejected during a simulation is reported <cit.>, but the size distribution and timescales of the ejected material has not been analyzed. Regardless, the number of realizations per system has typically not been large enough to draw statistically meaningful conclusions. Historically, these N-body accretion models have been limited in two key ways, both attributed to the computationally intensive nature of these types of simulations. First, collisions have been treated as perfect mergers, meaning two bodies that collide stick together and conserve mass and momentum. Second, a relatively small number of realizations have typically been performed for a given star/disk configuration. The recent development of an analytical prescription for collision outcomes in the gravity-dominated regime <cit.> provided a feasible way to realistically model `hit-and-run' events <cit.> and fragmentation from energetic impacts <cit.>. The model was incorporated into the popular Mercury N-body integration package <cit.>. <cit.> showed using this version of Mercury that the inclusion of fragmentation produced a dramatic improvement in studying the aspects of planet formation that were sensitive to the fate and evolution of material compared with older work that assumed perfect accretion. The second recent innovation was to utilize the NASA Pleiades Supercomputer to address the problem of small sampling of highly stochastic processes (such as these N-body accretion models) by performing hundreds of simultaneous simulations with near-identical initial conditions <cit.>. These improvements allow for probabilistic predictions based on a relatively large number of samples as well as quantifying the occurrence rates of somewhat uncommon outcomes.In this work we present the results from 300 N-body simulations of late-stage terrestrial planet formation around a Sun-like star that we performed using our fragmentation model. Half of these simulations include giant planets analogous to Jupiter and Saturn, while half lack giant planet companions. Our simulations examine growth from a disk of hundreds of protoplanets within 4 AU from the star, and we track the fate of all bodies as the systems are evolved for 2 Gyr. We quantify the ejected material to make a prediction on the mass distribution of free-floating planets that result from the planet formation process. We also analyze results from a suite of 152 simulations performed by <cit.> that began with a disk of protoplanets that extended out to 10 AU and included three giant planets to estimate the abundance of ejected terrestrial-mass planet in systems with unstable giant planets.Finally, we discuss the implications these results have for the WFIRST microlensing experiment.§ NUMERICAL MODELThe simulations described here were performed to explore the fate of planets that formed from material that originated within the terrestrial planet region (within 4 AU). We follow the accretion of solids during the final stages of the planet formation process at an epoch that corresponds to about 10 Myr after the start of the Solar System's formation <cit.>. At this epoch, tens of Mars-sized embryos are thought to have formed from solid material in the disk along with a large number of Moon-sized and smaller planetesimals. All gas in the disk has been dispersed, therefore giant planets, if included, are assumed to be fully formed. In 150 of our simulations we include giant planets analogous to Jupiter and Saturn on orbits comparable to their present locations. Another set of 150 simulations were performed without giant planets.The growth of Mars-sized embryos is supported by simulations of earlier stages of planet formation <cit.> as well as the predicted isolation mass in theoretical planet formation models of our Solar System <cit.>. We adopt the bimodal disk model from <cit.> in which half of the disk mass is comprised of 26 approximately Mars-mass bodies (0.0933 M_⊕) and the other half is in 260 approximately lunar-mass bodies (0.00933 M_⊕). The initial disk contains 4.85 M_⊕ of solid material extending from 0.3 to 4 AU from a 1 M_⊙ central star. The initial bodies are spaced to fulfill a surface-density profile proportional to the semimajor axis raised to the -3/2 power, which results in bodies spaced by 3–6 Hill radii. This disk model is based on the `minimum mass Solar nebula' <cit.>, a model derived by essentially smoothing each of the eight planets in our Solar System into concentric rings and fitting a curve to estimate the surface-density profile of solid material that ultimately formed the planets in our Solar System. Disk models that follow this model with a bimodal mass distribution have been successful in numerically reproducing the broad characteristics of the terrestrial planets in our Solar System <cit.>. The small bodies are a proxy for what should ideally be millions of objects with masses ranging from about a lunar mass down to dust grains, where the mode of the distribution is thought to be at about 150 km sized bodies <cit.>. The resolution of 260 small bodies is chosen to keep the simulations computationally tractable (since for N-body models the computation time scales with the square of the number of bodies). Numerical simulations have shown that this resolution is sufficient to provide dynamical friction (i.e., the damping of eccentricities and inclinations of larger bodies due to the swarm of smaller bodies), which is an important mechanism to include in models of late-stage planet formation <cit.>.All simulations were evolved forward in time using the modified version of the Mercury N-body integrator <cit.> that accounts for collisional fragmentation <cit.>. If a planet is eroded during a two-body collision, the lost mass is broken up into fragments that each have a mass equal to a chosen minimum fragmentation mass. The minimum mass for the fragments is necessary in order to constrain the total number of bodies in the integration. We use a value of 0.38 Moon-mass (0.005 M_⊕) that was used in previous simulations <cit.> and shown to be computationally tractable. Our simulations assume a material density of 3 g/cm^3, which gives radii for the embryos, planetesimals and fragments of 0.56 R_⊕ (3500 km), 0.26 R_⊕ (1600 km), and 0.2 R_⊕ (1300 km), respectively. All simulations used virtually the same initial mass distribution for each disk, but each realization was minutely altered by perturbing a single body at approximately 1 AU by one meter in order to account for chaos. Of the 150 simulations with giant planets, 140 are the same simulations published by <cit.> with an additional 10 simulations performed for this work.We allow these simulations to evolve for 2 Gyrs with a 7-day time-step allowing all initial bodies and fragments created during collisions to gravitationally interact. Any body that travels within 1 Solar radius of the host star is considered accreted by the central star. Any body that travels farther than 100 AU from the central star is deemed to be ejected from the system and is no longer tracked in the simulation. While a small fraction of these bodies that travel farther from their star than 100 AU may remain bound, this ejection distance was chosen because planets beyond this distance would be classified as free-floating planets to microlensing surveys such as the one WFIRST will perform <cit.>. At each collision or ejection event, the masses, orbits, and collision parameters of the bodies involved are recorded. § SIMULATION RESULTSA summary of the aggregate properties of material ejected is presented in Table <ref>. Following <cit.>, we define a `planet' as one that grew at least as massive as the planet Mercury (0.06 M_⊕). Although planets can be smaller than this <cit.>, we keep this definition for consistency with previous work. A single initial large body or seven initial small bodies would satisfy the mass constraint for a planet. This definition allows us to distinguish between bodies whose number and mass are properly tracked in the simulation and contribute to our planet count, and the planetesimal and fragmented material that remain in the system. There are very dramatic differences between simulations with and without giant planets. With giant planets a total of 16320 bodies were ejected in the 150 simulations, of which 1192 were planets. This is an average of 7.9 large bodies per system. In contrast, not a single planet was ejected from systems without giant planets, and a total of just 1697 small bodies were ejected. With giant planets, an average of 1.64±0.15 M_⊕ of material of the initial ∼5 M_⊕ disk was ejected during each 2 Gyr simulation. For simulations without giant planets, the central 90th percentile of the mass ejected ranges from 0.01–0.14 M_⊕. In addition to the material ejected from the system, 1.0 M_⊕ per simulation with giant planets, and 0.05 M_⊕ per simulation without fell into the central star. The upper panel in Figure <ref> shows the number of bodies ejected in various mass bins, normalized by the number of simulations. Far more small bodies are ejected than large bodies in simulations with giant planets (green shaded bars), but as shown in the lower panel of Figure <ref> the ejected mass is distributed roughly equally among planets and planetesimals. Although many Mars-mass planets are ejected, no planets more than three Mars-masses are ever ejected from these simulations. Without giant planets (red shaded bars), both the number of ejected bodies and their masses are significantly lower compared to systems with giant planets. The time when ejections occur also differs radically between the simulations with and without giant planets. Figure <ref> shows distributions of the ejection times for both sets in log-space. The simulations with giant planets (green histograms) can be approximated by a mixture of two log-normal distributions. The first log-normal from Figure <ref> peaks at around 10^6 yr and a second peak appears around 10^8 yr. The red histograms in this plot show the simulations without giant planets. The first ejections don't occur until after 10^6 yr, peaking at around 500 Myr before a slow decline. With giant planets, very few ejections occur after several hundred million years, in contrast to the simulations without giant planets where ejections are still occurring at the end of the 2 Gyr simulations, albeit at a rate of ∼0.5 ejections of a planetesimal/fragment per 100 Myr. The first peak in the distribution of mass ejected as a function of ejection times (lower panel in Figure <ref>) is still present and is the result of ejections of both planetesimals and planets. The second peak is much less pronounced in mass density compared with number density, with few planets being ejected at this time. Without giant planets, the low mass of the ejected bodies is evident.In Figure <ref> we look at the types of bodies ejected as a function of time for the simulations with giant planets. The upper panel shows the number density and the lower panel shows the mass density. As we suspected from the lower panel of Figure <ref>, we see that the second peak in number density is primarily due to a population of fragmented material. As fragments are created early on in the simulations, those created within the first 20 Myr are likely reaccreted by the embryos and planetesimals fairly quickly, whereas later, when the planets are mostly formed, any material that results in fragmentation can be more easily perturbed out of the system. The mass density of material is roughly evenly split between planets, planetesimals and fragments at the time of the second peak. The peak time of ejection for planetesimals is a little earlier than for planets, but the tails of the distributions are comparable. While the number of Mars-sized free-floating planet detections predicted in this work is not inconsistent with previous predictions, we predict a severely depressed occurrence in Earth-sized planets. In the simulations with giant planets most bodies are ejected in the first tens of millions of years. This timescale explains the mass distribution of the material ejected: with most of the ejections occurring early in the simulations, there simply isn't the time required to build larger planets via pair-wise accretion. By the time Earth-like planets have largely finished forming, 30–100 Myr or so <cit.>, the dynamical interactions that cause planetary ejections are scarce. The two timescales of ejections shown in Figure <ref> reflect different populations of ejected material. The first ejected population can be thought of as a primordial material - the planetesimals and embryos that were the remnants of star and giant planet formation. A second population of debris from collisions and original planetesimals that have experienced impacts themselves are ejected at this later time with on average less than one massive body being ejected from the system after about 200 Myr.The first bodies to be ejected from the simulations with giant planets are those that had initial orbital distances closest to the orbits of the giant planets. The blue dots in Figure <ref> show the initial position of all the ejected bodies and the time when they were ejected. This plot demonstrates that the time when ejections occur follows a power-law with the first ejections consisting of material close to 4.0 AU and occurring at around 100,000 yrs into the simulation. The ejection times also show two distinct clusters with a break at an initial semimajor axis of 2.0 AU. The outer material is ejected first and is primarily primordial material while the inner population is material that has been reprocessed through collisions. In simulations without giant planets, ejections typically don't start until about 200 Myr into the simulations, primarily because ejecting a planet requires a significant amount of angular momentum to be imparted to the ejected body. It is only once relatively massive terrestrial planets have begun to form that ejection of the low-mass material is possible. The ejections occur roughly uniformly with time throughout the simulations, with perhaps a drop by a factor of four between 500 Myr and 2 Gyr. The ejected bodies, as shown in the right panel of Figure <ref>, do not show any strong dependence on initial orbital distance. The planetesimal disk is well mixed by 100 Myr and the ejected material has no `memory' of where it began. § ALTERNATIVE GIANT PLANET CONFIGURATIONS We have thus far only considered two giant planet configurations: one with Jupiter and Saturn near their present orbits, and another with no giant planets. This binary choice is clearly not representative of the huge diversity of planetary system already discovered but it does provide two relatively extreme cases. As an intermediate example, <cit.> looked at the effect of a 10 Earth-mass planet in Jupiter's orbit (near 5.2 AU) on terrestrial planet formation, assuming perfect accretion during collisions, using the same initial disk as that presented here. They found that the total mass of material ejected was between that of their simulations with no giant planets and with Jupiter and Saturn, as expected. While these types of systems with outer companions (super-Earths and ice giants) could conceivably significantly increase the number of ejected planets, they are unlikely to change mass distribution of the ejected material. We next examine the effects of various outer giant planets, some of which become unstable, on a disk of material that extends beyond Jupiter's orbit. §.§ Simulations with unstable gas giantsWe complement our analysis of ejected material with a separate set of simulations from <cit.>. The most important difference between this set of simulations and those presented in Section <ref> is that these simulations included three gas giants whose orbits often became unstable. This allows us to connect our results with a large population of extra-solar planetary systems. The broad observed eccentricity distribution of giant exoplanets <cit.> can be reproduced if the vast majority (at least 75-90%) of systems containing giant exoplanets go unstable <cit.>. Instabilities are characterized by a series of scattering events between giant planets, and the surviving planets' stretched-out orbits are essentially scars left behind by this process <cit.>. The initial setup of the simulations from <cit.> was divided into three zones. The inner zone – corresponding to the terrestrial planet region – was similar in structure to our other simulations, but included more total mass. The inner disk was populated by roughly 50 planetary embryos and 500 planetesimals. A total of 9 M_⊕ was equally divided between the embryos (0.05-0.12 M_⊕ each) and planetesimals (0.009 M_⊕ each) and distributed following an r^-1 surface-density profile. In the middle zone, exterior to the terrestrial planet region, three giant planets were included in each simulation. The innermost gas giant was placed at 5.2 AU, and two additional giants were placed farther out. The spacing between adjacent gas giants was 4-5 mutual Hill radii, placing the planets in a marginally stable configuration <cit.>. The gas giant masses varied among the different sets of simulations <cit.>. In the fiducial case, planet masses were chosen following their observed distribution <cit.> between 1 Saturn mass and 3 Jupiter masses. A 10 AU wide belt containing 50 M_⊕ in 1000 planetesimals was placed exterior to the gas giants. The inner edge of the belt was 4 Hill radii beyond the outermost gas giant. This setup was inspired by models for the Solar System's primordial planetesimal disk, which may have been disrupted by a late instability in the giant planets' orbits <cit.>. The total batch from <cit.> contains roughly 500 simulations. For simplicity, we focus on the fiducial set of simulations, which has the advantage of being well characterized and reproducing the giant exoplanets' eccentricity distribution with no free parameters. In 96 out of 152 simulations (63%) the giant planets went unstable but in the other 56 simulations they remained stable. The survival or destruction of terrestrial planets in unstable systems was linked directly to the strength of the giant planet instability, which can be measured in terms of either the closest approach of a giant planet during the scattering phase or by the eccentricity of the surviving giant planets <cit.>, in particular of the innermost giant.Figure <ref> shows the mass distribution and the time when bodies were ejected from the 152 fiducial simulations. The clump in mass between 0.05 and 0.12 M_⊕ corresponds to the initial distribution of embryo masses. The relatively small number of ejected bodies with a mass above this indicates that most ejected embryos were primordial, meaning that they had not undergone any large collisions between the start of the simulation and the time of their ejection. Although systems with unstable gas giants are far more destructive in general, they ejected somewhat fewer embryos than systems with stable gas giants. In systems with unstable gas giants, the median number of ejected embryos was 14, versus a median of 18 embryos ejected in the simulations with stable gas giants. The unstable systems ejected a mean of 1.4 M_⊕ in embryos per system vs. 1.9 M_⊕ per stable system. In contrast, a median of 23 embryos collided with the central star in the unstable systems, versus a median of just 1 embryo colliding with the star in stable systems.The bodies ejected from unstable systems can be divided by the time of ejection relative to the start of the giant planet instability. Given that there was a distribution of instability timescales, this does not translate into an obvious absolute timescale. In these simulations, the median instability timescale was roughly 10^5 years, with a tail extending out to ∼100 Myr. As shown in Figure <ref>, ejections are common prior to the giant planets becoming unstable; the instability itself is not a significant driver of ejection, although by the nature of having a dynamically excited system seems to skew ejections to earlier times. Nonetheless, the most massive ejected planet in Fig. <ref> was a 1.04 M_⊕ planet ejected during an instability. The results from these simulations – in particular those with stable giant planets – are broadly consistent with those from the simulations from Section <ref> with giant planets on Solar System-like orbits. Specifically, the embryos that were ejected tended to be between one and two Mars-masses. The vast majority of ejected embryos had undergone no accretion, and only a small fraction had grown to more than twice their initial mass. The most massive embryo ejected in a simulation with stable giant planets was 0.54 M_⊕. Despite the differences in numerical resolution, embryo mass, and integration method, the concordance between this set of simulations and those presented in Section <ref> inspires confidence in our results.§ ESTIMATING THE GALACTIC TERRESTRIAL-MASS FREE-FLOATING POPULATION The contrast between properties of the ejected material in the three different simulations is intriguing. If giant planets are ubiquitous, then there exists a very significant population of Mars-sized material to make up a population of free-floating planets – here we show as many as 8 per star with giant planets in the Galaxy from inner planetary systems (c.f. Table <ref>) and about double the number from larger orbital distances. However, there are strong indications that the occurrence rate of giant planets is significantly lower than unity; <cit.> estimate an occurrence of just 6.2% of giant planets between 3 and 7 AU from their stars. This low occurrence doesn't seem particularly outlandish, particularly considering that the most common spectral type – M-dwarfs – very rarely host giant planets <cit.>. To apply our results to the Galactic population, we divide the known systems of exoplanets into three categories (in order of increasing abundance).* Systems like our own, with gas giants on wide, near-circular orbits. Estimates of the occurrence rates for giant planets range from 5–20% with perhaps a third of these giant planets on wide, low-eccentricity orbits <cit.>. We assume that this subset makes up 6% of stars in the Galaxy.* Systems with gas giants on eccentric orbits. The remainder of the giant exoplanets have orbits that are either too close-in or too eccentric to be considered Jupiter-like. This comprises around two-thirds of the entire giant exoplanet population, approximately 12% of stars.* Systems without gas giants. Most stars have no gas giants. Of course, a large fraction may host super-Earths or ice giants <cit.>, but, as we showed in Section <ref>, we do not expect these systems to contribute significantly to the population of rocky free-floating objects.For each class of systems, one of our sets of simulations allows us to crudely estimate the rate of ejection of rocky planet-sized bodies (with the various caveats discussed in Section <ref>). We can then produce a simple linear combination of these estimates, weighted by the occurrence rate of each type of system, e.g.η =F_giants, circular× FF_giants, circular+ F_giants, eccentric× FF_giants, eccentric+ F_no giants× FF_no giantswhere η is the number of terrestrial-mass free-floating planets ejected per star, F is the fraction of stars falling into the three categories (circular giants, eccentric giants, and no giant planets), and FF is the number of free-floating planets ejected from that category. We do not include systems with hot Jupiters in this estimate because they only account for approximately 1% <cit.> of stars and therefore are unlikely to significantly affect the overall occurrence rate. Our results from Section <ref> demonstrated that systems without gas giants are unlikely to contribute to the population of free-floating rocky planets. In our simulations of the inner Solar System we found that approximately 8 Mars-mass planets are ejected per simulation, while in the simulations from <cit.> about twice this number of Mars-mass bodies are ejected (likely because they use a more massive initial disk), so we will take the average of these simulations. Finally, in the simulations with unstable gas giants, about an average of 14 Mars-mass embryos were ejected. So, this would imply (0.5 * (8+18))×0.06 bodies in stable simulations, 14×0.12 in unstable simulations for a total of 2.5 Mars-mass embryos per star in the Galaxy. Owing to relatively poor constraints on in the mass in the solids in the terrestrial region of the initial disk, this estimate probably has an uncertainty of a factor of several.§ IMPLICATIONS FOR WFIRST Previous estimates of the number of detections from WFIRST have primarily been based on the population of bound planets, either the Solar System population or the observed population from microlensing <cit.>. Figure <ref> shows a comparison between the mass distribution of bound and ejected planets for simulations with (green) and without (red) giant planets from our simulations of the inner planetary systems. Both simulation types show peaks in the bound planet mass distribution at around one Earth-mass, while there is no ejected material more massive than 0.3 M_⊕. A clear takeaway from this work is that the bound population is radically different from the ejected population.We estimated 2.5 free-floating terrestrial-mass planets per star in the Galaxy (c.f. Section <ref>). Now we look at how many of these planets would be detectable by the microlensing component of the WFIRST mission. The sensitivity of WFIRST to detecting microlensing events was estimated by <cit.>, who show estimates of the free-floating population based on simulations performed by <cit.>.<cit.> predict that with a single ejected planet per star, WFIRST would find 5.7 Mars-sized free-floating planets. Taking this estimate from WFIRST and combining it with our simulation results, we find a number three times greater at 14.3 planets detected by WFIRST. While this estimate is based on relatively simplistic assumptions, it is suggestive of a significant number of microlensing events in WFIRST data.§ LIMITATIONS AND THE EFFECT OF INITIAL CONDITIONSAn important consideration for all studies of this type is the effect of the initial conditions on the final results. Parameters such as the total disk mass, size, and mass distribution of protoplanets and the presence and proximity of outer giant planet companions can weigh heavily on the final results. Because these types of N-body simulations are computationally intensive and therefore a large number of simulations are needed due to their stochastic nature, a full exploration of parameter space for all initial conditions is not feasible. We therefore limit our primary study of ejected material from the terrestrial region (Section <ref>) to a single star/disk model that is motivated by decades of research on Solar System formation <cit.> and vary the presence of giant planets.There is very likely a great diversity of disk mass distributions, however, that result from the star formation process, as evidenced by studies of Kepler planetary systems <cit.>. While ALMA observations are revolutionizing our understanding of the earlier stages of protoplanetary disks <cit.>, we have relatively poor constraints on the total mass and distribution of solids in disks that are old enough to have lost their gas component. However, there is evidence that our radial distribution is at least plausible <cit.>. Simulations using the minimum mass Solar nebula model, which our disk is based on, have been successful at reproducing systems of terrestrial planets with properties comparable to those in our Solar System. Many numerical studies have explored the effects of varying the disk with these properties on the final planets that form. While these studies have used perfect-accretion N-body models, they are still valuable for getting a sense of the parameters that have stronger effects on the abundance and timescale of ejected mass. <cit.> examined the impact of varying the disk surface-density profile on the total mass ejected in systems with Jupiter and Saturn and found a difference of about a factor of three in ejected mass between very shallow and very steep (compared to the minimum mass Solar nebula model) profile disks. We use an intermediate slope surface-density profile which is unlikely to bias the total ejected mass by more than a factor of about two and is therefore unlikely to be the dominant source of uncertainty in our estimates.<cit.> showed that for a given total disk mass, the properties of the final planets that form are not highly sensitive to the number, mass, or bulk density of the planetary embryos that compose the disk. Variations in total disk mass, however, do influence the types of planets that form. More massive disks tend to form fewer, more massive planets <cit.>, although the timescale problem for forming and ejecting Earth-mass planets still applies. With more massive disks having more material available to eject, our predictions on the number of free-floating planets that WFIRST will detect is fairly conservative. If significantly more massive disks than we have used are the norm, we would expect to see more Mars-sized planets. The bimodal mass distribution that we consider in our initial disk is the result of a runaway and oligarchic growth phase of protoplanets during the earlier stages of planet formation <cit.>. Prior to runaway growth, the sizes of protoplanets in the disk are comparable and their eccentricities and inclinations remain low due to dynamical friction. While the abundance of ejected material during these earlier stages is difficult to estimate, the bodies ejected during this epoch are likely too small to be detectable (they only reach Mars-size in the post-oligarchic growth phase).The sizes of our initial planetesimals (∼1 lunar-mass, or 0.009 M_⊕) and the minimum fragment size (∼0.4 lunar-mass, or 0.005 M_⊕) were chosen to keep our simulations computationally tractable while retaining enough bodies to mimic a size distribution that provides dynamical friction to the embryos and final planets. Therefore, we do not draw any conclusions on the specific size distribution of ejected bodies smaller than the Moon, we class all this mass as `small' and treat this set as a whole. <cit.> used the same initial disk mass and surface-density profile as our model, but with a factor of four more planetesimals. They found that a near identical fraction of the initial mass was ejected from their simulations as those presented here. We therefore conclude that the resolution of the planetesimal population does not strongly affect the abundance of the ejected mass. Our simulations run for 2 Gyr. While in the giant planet simulations formation and evolution of the terrestrial planets is essentially complete, the simulations with no giant planets show ongoing activity. We opted not to continue these simulations beyond 2 Gyr because just running this far took several months. However, we find it unlikely that the evolution of these system will vary dramatically over the next 2–10 Gyr. There may be more collisions, but it is unlikely to cause a significant change in the mass distribution of the ejected material or the ejection frequency.Finally, we do not consider external forces such as passing stars that can dynamically drive ejections. However, although passing stars in dense clusters can stir up material and cause planets to become unstable, most planetary systems do not experience stars passing close enough to have a significant impact on the inner region of planetary systems where terrestrial planets form <cit.>.§ CONCLUSIONSIn this study we focus on quantifying the mass that is ejected from a star/disk system during the planet formation process using state-of-the-art numerical N-body simulations. Our primary study involves running hundreds of simulations for two sets of initial configurations: a disk of protoplanets around a Sun-like star with and without outer giant planets analogous to Jupiter and Saturn. We use the same initial disk of protoplanets for all simulations, but with small changes in the initial conditions of one body to account for chaos. Our disk model is based on the distribution of solids that are thought to be present about 10 Myr after the birth of our Solar System, an epoch in which the gas component of the disk has dispersed, giant planets are formed, and Moon-to-Mars-sized protoplanets have accreted <cit.>. Specifically, our disk includes about 5 M_⊕ of solids in 26 Mars-sized (∼0.1 M_⊕) embryos and 260 Moon-sized planetesimals (∼0.01 M_⊕) all within 4 AU from a Sun-like star. The goal of these simulations is to estimate the abundance and mass distribution of free-floating planets that originated in the terrestrial region (i.e., `Earth-like'). Our simulations were performed using an N-body integrator that allows both accretion and fragmentation <cit.> and we set the resolution (minimum allowed mass during fragmentation) to 0.4 lunar-mass (0.005 M_⊕).We find that in simulations with giant planets, roughly half the ejected material is composed of bodies with masses greater than 0.06 M_⊕ and half in smaller bodies. While many Mars-mass bodies were ejected during the final stages of planet formation that we simulate, no planets more massive than 0.3 M_⊕ were ejected. The primary reason is that nearly all ejections occur within the first 30 Myr, which is prior to the giant-impact phase of planet formation when planets accrete enough material to grow to Earth size. Most of the ejections essentially mirrored our initial bimodal disk mass distribution, but with embryos that only had time to grow to about one-third of an Earth mass and many planetesimals and fragments that didn't accrete a substantial amount of mass. With no giant planets, all the ejected mass is in planetesimals and amounts to just 1% of the initial disk. The vast majority of solids in the disk remains bound to the star throughout our 2 Gyr simulations. The material that is ejected does not leave the system until Earth-mass planets have had time to form, contrary to the set of simulations with giant planets. However, without giant planet perturbations, the systems lack a mechanism to impart enough angular momentum to cause planet ejections. This implies that stars that lack outer giant planet companions contribute very little to the free-floating planet population.We also examined ejected mass from simulations performed previously <cit.> that explored a different parameter space than our initial study. About 150 simulations were presented that included a disk with about 9 M_⊕ of solids in the terrestrial region (so almost double the mass of our primary study), three giant planets (one in Jupiter's orbit, the others exterior), and a population of 50 M_⊕ of solids in 1000 planetesimals exterior to the giant planets and out to 10 AU. This set allowed us to examine the effects of different giant planet architectures, giant planet instabilities, and a more massive disk on the population of ejected material. While these simulations include more mass in embryos, a wider disk, and additional giant planets, their results on the mass distribution of ejected embryos are consistent with the population ejected from the terrestrial region in our primary study. Interestingly, the abundance of embryos ejected in simulations where the giant planets became unstable was lower than when the giant planets remained stable, primarily because the instabilities tended to perturb a bulk of the lost material into the star rather than into interstellar space. With the exception of two objects, all of the ejected bodies in this set of simulations had masses lower than 0.4 M_⊕. Taken together, these simulation imply that few Earth-mass free-floating planets are ejected during the planet formation process. The population of free-floating planets is likely composed of Mars-sized bodies, many of which may have originated from the outer cooler regions of a protoplanetary disk.Our results suggest that the abundance of terrestrial planets that originate from the terrestrial region (<4 AU) detected by WFIRST will depend on the frequency of giant planets. We combined the results from all simulations with the occurrence rates of different giant planet architectures and stellar types to estimate a Galactic population. We predict that roughly 2.5 terrestrial-mass planets are ejected per star. However, the vast majority of these come from the <20% of stars that host giant planets. The WFIRST microlensing program will be sensitive to planets less massive than Mars. We predict that WFIRST will find approximately 15 Mars-mass free-floating planets, and the number could be even larger if typical disk masses are significantly more massive than we used in this work.E.V.Q.'s research was partially supported by a NASA Postdoctoral Program Senior Fellowship at the NASA Ames Research Center, administered by Universities Space Research Association under contract with NASA. The simulations presented here were performed using the Pleiades Supercomputer provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. The authors would like to thank Chris Henze for his expertise and assistance in performing simulations on the Pleiades Supercomputer. apj | http://arxiv.org/abs/1704.08749v2 | {
"authors": [
"Thomas Barclay",
"Elisa V. Quintana",
"Sean N. Raymond",
"Matthew T. Penny"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170427211748",
"title": "The Demographics of Rocky Free-Floating Planets and their Detectability by WFIRST"
} |
fancy plain plain [C] < g r a p h i c s >[L][R]< g r a p h i c s > iblabel[1]#1 akefntext[1] [0pt][r]thefnmark #1 1.125 *§0pt4pt4pt * §.§0pt15pt1pt [LO,RE]< g r a p h i c s >[CO] [CE]< g r a p h i c s >[RO]1–LastPage [LE] 1–LastPage[ \begin@twocolumnfalse https://doi.org/10.1039/c6sm02719adoi:10.1039/c6sm02719a Investigating the role of boundary bricks in DNA brick self-assemblyHannah K. Wayment-Steele,^a, Daan Frenkel^a and Aleks Reinhardt^a< g r a p h i c s >In the standard DNA brick set-up, distinct 32-nucleotide strands of single-stranded DNA are each designed to bind specifically to four other such molecules. Experimentally, it has been demonstrated that the overall yield is increased if certain bricks which occur on the outer faces of target structures are merged with adjacent bricks. However, it is not well understood by what mechanism such `boundary bricks' increase the yield, as they likely influence both the nucleation process and the final stability of the target structure. Here, we use Monte Carlo simulations with a patchy particle model of DNA bricks to investigate the role of boundary bricks in the self-assembly of complex multicomponent target structures. We demonstrate that boundary bricks lower the free-energy barrier to nucleation and that boundary bricks on edges stabilize the final structure. However, boundary bricks are also more prone to aggregation, as they can stabilize partially assembled intermediates. We explore some design strategies that permit us to benefit from the stabilizing role of boundary bricks whilst minimizing their ability to hinder assembly; in particular, we show that maximizing the total number of boundary bricks is not an optimal strategy. \end@twocolumnfalse]§^a Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, United Kingdom. Present address: Department of Chemistry, Stanford University, Stanford, California 94305, USA.§ INTRODUCTION Since their discovery,<cit.> two-dimensional DNA tiles and three-dimensional DNA bricks have gathered interest as a completely modular DNA nanomaterial. In the DNA brick set-up, short, 32-nucleotide long single-stranded DNA molecules have sequences chosen such that they hybridize specifically with four other distinct single-stranded molecules. The interactions are chosen such that favourable bonding occurs when these molecules are arranged in a target structure. If a big cubic structure is designed in this way in the first instance, other structures can rapidly be designed using the same set of starting bricks by merely omitting a subset of the bricks.<cit.> While DNA origami<cit.> is currently the most popular strategy for fabricating DNA nanomaterials, unlike with DNA bricks, DNA origami entails a long `scaffold' single-stranded DNA molecule which is linked with shorter `staple' molecules to fold the scaffold strand into the target shape, and designing a new target structure therefore requires starting from scratch with an entirely new set of staple strands.DNA brick self-assembly is also perhaps the best example of a viable addressable<cit.> self-assembled system: each subunit in the target structure is unique, and knowing the identity of a particle therefore means knowing its location, and vice versa. Systems with addressable complexity and their potential for designing structures with arbitrary shape and complexity have great promise in fields ranging from medical applications to nanoelectronics.<cit.>A recent application of DNA bricks as a nano-breadboard for chromophore-based excitonic gates<cit.> exemplifies the benefits of DNA bricks over DNA origami. In excitonic devices, where the FRET radius is less than 5, it is necessary to have nanometre-scale control over the placement of chromophores. DNA bricks have an advantage over DNA origami by having twice the spatial resolution: it is difficult to functionalize the scaffold strand in DNA origami, and thus only the staple strands, one out of two strands in any helix, are available for functionalization.<cit.> By contrast, in DNA brick structures, all strands are available for functionalization. In such technologies, the excitonic transmission behaviour is challenging to predict and the modular nature of DNA bricks allows for straightforward modification of structures, permitting a number of possible layouts to be tested and screened rapidly.<cit.> Moreover, these benefits may prove useful for other applications as well, for instance in scaffolding for multi-enzyme complexes for single-molecule reactions,<cit.> molecular rulers,<cit.> inorganic nanoparticle synthesis<cit.> and nano-robots.<cit.>In DNA brick structures, the final structure is designed to be the thermodynamic product, a benefit over folding assembly structures, where it is difficult to predict if the designed target structure is the preferred equilibrium structure.<cit.> However, DNA brick assembly has a much more complex pathway to assembly.<cit.> Because there are a vast number of intermediate states that all have similar energies, DNA bricks are very prone to kinetic traps.<cit.> This is a disadvantage in comparison to folding assembly, where the constraint offered by having all interacting particles on a backbone offers more direction to the final assembled state. Experimentally, typical DNA brick yields range from a few per cent to 30 per cent,<cit.> whereas yields for some DNA origami structures are approaching 99.<cit.> However, because we have control over interactions between DNA brick subunits, it should be possible to design interactions that can direct the assembly to avoid kinetic traps.One design strategy implemented for increasing the yield in experiment was to include larger bricks at the surfaces of target structures.<cit.> Because bricks are staggered in the xz and yz planes of the structures, at the faces of a structure, in alternating rows a brick must be bisected (Fig. afig-BB-schematic), leaving behind 16-nucleotide half-bricks in every other row (Fig. bfig-BB-schematic). These half-bricks were then connected to the bricks in the row preceding them to form larger 48-nucleotide bricks, termed `boundary bricks' (BBs) (Fig. cfig-BB-schematic). The use of BBs was shown to increase the yield by a factor of 1.4 and was implemented for all subsequent structures in the experiments of Ke et al.<cit.>Despite the effectiveness of BBs in increasing the yield, the cause of this observed effect has not been well studied. There are two principal mechanisms one can envisage by which BBs could lead to an increase of the yield. Firstly, because they are larger and have more interaction domains than regular bricks, they could serve as a larger seed particle to promote nucleation. Secondly, they may stabilize the final structure by binding the edge half-bricks that have fewer interaction points to bind to the rest of the structure.To investigate the effects of BBs on the DNA brick nucleation and assembly process, in this work we extend a simulation-based model previously used with success to describe DNA bricks<cit.> to include BBs. With this model, we use Monte Carlo simulations to show that depending on the location, BBs may differ in their contribution to increasing the nucleation rate or stabilizing the final structure. We also demonstrate that BBs are more prone to aggregation than regular bricks, and we suggest a method to overcome this whilst still benefiting from the stabilizing effect of incorporating BBs.§ METHODS We perform Metropolis Monte Carlo<cit.> simulations on a cubic lattice in the canonical ensemble with `virtual moves'<cit.> accounting for the motion of clusters. To be able to probe the time and length scales needed to observe assembly behaviour in a computational context, we model DNA bricks as spheres with four `sticky' interacting patches, representing the four 8-nucleotide sequence domains of DNA bricks, placed equidistantly on the sphere's surface to form a tetrahedral shape.<cit.> The dihedral angles in the DNA brick structures are roughly 90, which means that the centres of mass of each DNA brick in the final structure form a distorted diamond lattice,<cit.> so describing each brick as a tetrahedron serves as a reasonable first-order approximation of the experimental geometry.<cit.>In an extension of the previous model,<cit.> we model boundary bricks as dimers of these particles, i.e. as two patchy particles that are connected by a rigid bond of length corresponding to their distance in the target structure (Fig. <ref>), with (at least) two patches on one of the particles given a poly-T sequence<cit.> to passivate them (cf. Fig. afig-BB-schematic).<cit.> The two particles connected in this way are also fixed in their orientation with respect to one another.<cit.>To investigate the effect of BBs on structure nucleation and final stability, we perform simulations of a DNA brick structure with 374 bricks in the canonical ensemble (i.e. at a constant number of particles, volume and temperature).In the majority of simulations reported here, the density of each type of brick was set to 1/(62a)^3, where a is the lattice grid parameter, and a single copy of each brick that appears in the target system was present in the simulation box. The simulated structures contain up to 84 boundary bricks on the faces in the xz and yz planes (Fig. <ref>), as described below. Simulations were run for structures with DNA-specific interactions between bricks. For each set of bonded patches in the final target structure, complementary sequences were randomly generated and assigned. The interaction strength of hybridization between two complementary DNA strands largely depends on the proportion of paired guanine/cytosine (GC) pairs in the sequence. The average GC content for the structure studied was 44.6, with standard deviation 14.2. Each patch can interact with every other patch provided the patches point at each other and the corresponding particles are diagonally adjacent to each other, and the energy of interaction corresponds to the hybridization free energy obtained from the SantaLucia thermodynamic model.<cit.>In simulations investigating whether BBs in different locations of the final structure had differing impacts on structure nucleation and final stability, we chose to include various sets of BBs out of the total possible 84 BBs for this structure. One subset consisted of 26 BBs on the edges of the cubic structure, henceforth referred to as the `edge-BB structure', shown in red in Fig. <ref>. The other subset included 26 BBs at the centre of the faces of the cubic structure, henceforth referred to as the `face-BB structure', with BBs shown in green in Fig. <ref>. The structure with all possible BBs<cit.> (red, green and blue) will be referred to as the `all-BB structure'. In the original experimental design, bricks with sequence domains on the external faces of the structure had those domains either removed or replaced with non-interacting poly-T sequences.<cit.> Our model similarly passivates patches of bricks that are on the external face by assigning them a poly-T sequence. We ensured that there is no significant difference in the number of passivated patches on BBs between the edge and face conditions.In addition to simulations of structures with specific DNA interactions, we have also run simulations of structures with designed interactions all having the same fixed interaction energy. This simplification still retains the specificity of interactions required for addressable self-assembly, since patches still have a unique identity and only interact with specific patches with which they were designed to bond in the target structure, but removes variation in interaction strengths that arises from DNA sequence dependence. In such simulations, all designed interactions were assigned a fixed interaction energy of ε/k_B =-4000, which corresponds roughly to an average sequence interaction strength at 320.<cit.> All other (`incidental') patch–patch interactions were set to zero.§ RESULTS AND DISCUSSION§.§ Effect of BBs on the nucleation rate We ran brute-force Monte Carlo (MC) simulations of the self-assembly of target structures from a vapour of monomers (corresponding in reality to a dilute solution of monomers) at a range of fixed temperatures to observe the dependence of assembly behaviour on temperature, since the self-assembly process of DNA brick systems has been shown to be particularly sensitive to temperature.<cit.> We have run such simulations on four sets of building blocks, namely the edge-, face- and all-BB structures as defined above and a system with no boundary bricks. We show the size of the largest correctly assembled cluster in the system as a function of Monte Carlo time in Fig. afig-nuc-rate. We find that, for the structure considered here, at temperatures below approximately 315, assembly is dominated by unintended aggregation. Because lower temperatures favour both correct bonding and incorrect bonding, and there are statistically many more ways to bond incorrectly, the structure nucleates and assembles to some degree but quickly becomes kinetically trapped in a misassembled state and is then unable to assemble correctly any further. Optimal assembly is observed between about 317 and 318. In this range, the all-BB structure grows the most rapidly, followed by the face-BB structure and then the edge-BB structure. The no-BB structure takes the longest to nucleate and grow. The same trend was observed in structures with fixed designed interactions (FDI), confirming that this was not an artefact of any possible difference in GC content of BBs in the edge and face structures. At 319 and above, the edge-BB and no-BB structures take significantly longer to nucleate, but the all-BB structure largely assembles up to about 326, since the large number of BBs makes the bonding much more favourable for this system.We can estimate the effect of BBs on increasing the nucleation rate by measuring the mean first-passage time (MFPT). This approach is commonly used when computing nucleation rates in molecular dynamics simulations;<cit.> the mean first-passage time corresponds to the time needed on average for a stochastic process to reach a certain state for the first time. Although MC simulations do not faithfully reproduce the various time scales that may be involved in a dynamic process, we can nevertheless estimate the relative effect of BBs on the nucleation rate by calculating the MFPT for the different BB structures, using MC steps as a time step approximation.In order to compute the MFPT for nucleation in our simulations, we recorded the number of MC steps required for the size of the largest correctly assembled cluster in the system to reach 8 particles and large-scale growth to begin. This cluster size was chosen since we have previously shown that the critical cluster at temperatures at which DNA bricks can nucleate usually comprises 8 bricks,<cit.> and the largest cluster in the system typically grows rather than shrinks once clusters grow beyond this size. The reason for this well-defined critical cluster size is that 9 bricks are required to form a subunit with two closed `cycles' (i.e. closed loops of particles that are bonded to one another), and, as a monomer comes in to close a cycle, two bonds are formed simultaneously, energetically stabilizing the resulting structure at roughly the same entropic cost. The critical cluster structure comprises one brick less than this stabilized bicyclic motif.<cit.>The difference in the free-energy barrier height relative to the system with no boundary bricks, G^⋆≡ G^⋆- G^⋆_no-BB, is calculated from the ratios of the average nucleation rate R, which is the reciprocal of the MFPT. We use the classical nucleation theory relation<cit.>R=N_S Z j exp( -β G^⋆),where N_S is the number of nucleation sites, Z is the Zeldovich factor,<cit.> j is the rate at which molecules attach to the nucleus, and G^⋆ is the free energy required to self-assemble the critical nucleus from a dilute solution. Although we do not know N_S, Z or j for this system, we assume that they are roughly the same for all systems, regardless of the number of BBs present, particularly as the dependence on the nucleation free-energy barrier is exponential and the remaining terms are not. If we take the ratio of nucleation rates, these terms thus (approximately) cancel out, giving a ratio ofR/R_no-BB =exp( -β G^⋆).The mean first-passage times and the corresponding values of G^⋆ are shown in Fig. bfig-nuc-rate and cfig-nuc-rate.The relative changes in the free-energy barriers are in agreement with the trends observed qualitatively from monitoring the largest cluster over time, with the edge BBs having the smallest reduction in the free-energy barrier, followed by face BBs, and finally followed by the system with all possible BBs. Although the latter system has 3.2 times as many boundary bricks as do the edge and face-BB systems, there is only a relatively small decrease in the free-energy barrier from the edge and face-BB systems to the all-BB system. This is perhaps not particularly surprising, since the reduction of the free-energy barrier is affected principally by the bricks first involved in nucleation, not their overall number.The difference in the MFPT between the face-BB structures and edge-BB structures is interesting, as in many ways besides the obvious difference in the location of the BBs, the structures are identical. Both have 26 BBs, and they have no significant difference in the number of interacting patches or GC content. The same trend is also observed in FDI simulations, in which all designed interactions have a fixed interaction energy, indicating this is not an effect of GC content in the structures. The increased nucleation rate in face-BB structures likely arises because there are more interactions between face BBs and non-face bricks than there are interactions between edge BBs and non-edge bricks, since the latter have fewer neighbouring molecules.§.§ Nucleation locationIn order to understand better how BBs are involved in initial nucleation and growth, we have identified which bricks are involved in the nucleated clusters for the structures studied. We chose to investigate clusters comprising 9 monomers, since such clusters are post-critical, but sufficiently small to reflect the nucleation event.<cit.> For each MC trajectory, the identities of the bricks in the largest cluster were recorded at the last time step at which the largest cluster comprised 9 particles, and tallied over 60 independent simulations to give the frequency of brick types in the largest cluster. These frequencies were analysed at the temperatures at which self-assembly was `optimal' for both the fixed designed interactions and the DNA-specific interactions, 323 and 318, respectively. These temperatures correspond to the lowest temperature at which structures self-assembled to large sizes without significant misassembly.We first consider the frequencies of bricks in initial nucleation clusters for simulations of structures with assigned DNA sequences, where every pair of patches can interact with an interaction energy based on the SantaLucia thermodynamic model. Intuitively, we would expect that the stronger the bonding of a particle's patches is, the more likely it is for a particle to be found in the initial nucleus. Indeed, this is what is largely observed (Fig. (i)(a)fig-BBs-nucleationSites): bricks with a higher average GC content (and hence stronger bonding) are more likely to be present in the critical nucleus. Intriguingly, it is not the bricks with the highest GC content that dominate; instead, nucleation tends to occur in regions where several neighbouring bricks have a high GC content. In other words, it appears that designing preferential nucleation pathways would require a careful analysis of not only the bonding strength of individual particles, but how they come together in the final structure, making it a more difficult task than it might first appear. We propose to investigate this interplay of factors more systematically in future work.However, boundary bricks have a dominant effect as far as nucleation is concerned, and as soon as boundary bricks are added to a system, the small random variations in GC content that seem to determine the nucleation behaviour for the system with no BBs (Fig. (i)(a)fig-BBs-nucleationSites) no longer play any significant role in determining a particle's nucleation propensity. For the structure with DNA-specific interactions and all 84 BBs present (Fig. (i)(b)fig-BBs-nucleationSites), BBs are predominant in the initial nucleation cluster: the increased number of interactions per BB when compared to a `monomer' brick favours boundary bricks as preferred sites for nucleation to occur.The same observation holds for the edge-BB and face-BB structures (Fig. (i)(c)fig-BBs-nucleationSites and Fig. (i)(d)fig-BBs-nucleationSites). These simulations demonstrate that it is possible to tune the nucleation site to different parts of the structure depending on where the BBs are located. The edge-BB structure nucleates essentially only at the edge BBs, and the face-BB structure nucleates only at the face BBs. The locations with the highest nucleation frequencies did contain the single BB among each subset with the highest GC content; however, beyond this, there was no significant correlation between nucleation frequency and GC content, perhaps indicating that the BB with the highest GC content is fastest to nucleate.The nucleation location in structures with SantaLucia interactions is driven by both the location of BBs and the location of bricks with high GC content. BBs have a stronger proclivity for nucleation than monomers, and by selecting which bricks are bonded to others as BBs, we are able to control the nucleation site on the structure. For both the case without BBs and with BBs, bricks with higher GC content are involved in nucleation, though the exact effect of GC content on nucleation cannot be well understood from only one structure. Nevertheless, this finding could be a very useful tool in the rational design of self-assembly pathways of DNA brick structures.We can investigate the underlying behaviour that is solely due to boundary bricks notwithstanding the effect of having varying interaction strengths across the system by considering the frequencies of bricks in initial nucleation clusters for the case of fixed designed interactions (Fig. iifig-BBs-nucleationSites). This system allows us to focus on only the size and geometry effects of BBs on nucleation, without the complication of non-uniform interaction strengths of DNA sequences. For the FDI structure with no BBs (Fig. (ii)(a)fig-BBs-nucleationSites), nucleation appears to be dispersed throughout the volume of the structure, with bricks on the faces somewhat less likely to be involved in nucleation clusters. Since such bricks have non-interacting patches on the outside and therefore fewer possibilities for bonding, this behaviour is entirely consistent with expectations.For the FDI structure with all 84 BBs present (Fig. (ii)(b)fig-BBs-nucleationSites), nucleation is again largely confined to the faces and edges of the cube, where the BBs are located. Notably, nucleation essentially never occurs in the body of the structure. The presence of BBs, because they are larger units with more interaction sites per unit than regular bricks, causes nucleation to shift to the outer regions of the cube. This is further demonstrated by the FDI structures with only edge BBs (Fig. (ii)(c)fig-BBs-nucleationSites) and only face BBs (Fig. (ii)(d)fig-BBs-nucleationSites), where nucleation occurs primarily on the edges and faces, respectively, and is consistent with the behaviour seen in simulations with full sequence-dependent interactions: the nucleation propensities shown in Figs (i)(b–d)fig-BBs-nucleationSites for full sequence-dependent interaction simulations largely correspond to those of Figs (ii)(b–d)fig-BBs-nucleationSites of the analogous FDI simulations, demonstrating that the influence of GC content on the nucleation location is minimal as soon as boundary bricks are included.§.§ Effect of BBs on the degree of assemblyAlthough BBs lower the nucleation barrier for assembly, this is not necessarily beneficial to achieving successful self-assembly, as a lowered nucleation barrier may also lead to unwanted aggregation. While the all-BB system gets very close to growing to completion at 318 (Fig. afig-nuc-rate), none of the structures simulated quite reach the full size of the intended target of 374 particles at a temperature at which nucleation and designed growth occur. This is expected for a fixed temperature simulation, as partial assembly is entropically favoured,<cit.> and a temperature ramp is necessary to achieve full assembly, since at lower temperatures, the additional energetic stabilization drives the structure to assemble despite the entropic cost of full assembly.Since the entropic cost of binding a boundary brick is comparable to that of binding a non-brick monomer, but the degree of bonding can be greater, we expect that boundary bricks will stabilize the target structure in the sense that it can grow to a larger size even with a fixed-temperature growth protocol. In single-target simulations, the boundary bricks behave largely in the way we would expect them to: the all-BB structure grows to the largest final size in both fixed-temperature and gradually cooled set-ups (Fig. <ref>), as it has the largest number of boundary bricks to stabilize it.However, the all-BB structure initially nucleates at very high temperatures, and it is only at approximately 318 that the target structure is nearly complete with few errors. Below this temperature, monomer nucleation is evidently too facile, which prevents successful assembly later on in the self-assembly process, as the probability of exactly aligning and forming all the right bonds to connect two larger clusters is prohibitively low. At low temperatures, self-assembly in simulations with all brick types becomes less and less favourable.Interestingly, even though the edge-BB structures nucleate less rapidly than the face-BB structures, edge-BB structures had the second highest degree of final assembly across all optimal assembly temperatures (Fig. <ref>). This suggests that edge BBs are able to stabilize the final structure and bind bricks on the edge that would be entropically favoured to be unbound,<cit.> and are able to do so more effectively than face BBs. In Fig. afig-final-assembly-sizes, we show a typical example of a large correctly assembled cluster for each of the BB structures formed in constant-temperature simulations. In particular, the face-BB structure shown is missing all four edges. Of course, this is not wholly surprising, since the edge monomers typically have fewer bonds than the face monomers, and boundary bricks therefore play a much more significant role by comparison. This observation is supported by the fact that the face-BB structures, which only have normal monomer bricks on their edges, have nearly identical assembly sizes as the no-BB structures once this size has reached a plateau in constant-temperature simulations. §.§ Simulations of multiple target structuresWe have shown that both the nucleation behaviour of boundary bricks and their structure stabilization properties largely follow our naïve expectations: the more boundary bricks there are, the higher the nucleation temperature will be, and the more stabilized the target structure. Of course, in reality, more than a single target structure is normally assembled in solution; this complicates matters somewhat. In order to provide some insight into what the effect of boundary bricks might be in solution, we have therefore performed simulations in which several copies of the target structure are present.<cit.> In particular, we have simulated systems with 6 copies of each brick in the target structure at a density of 6/(100a)^3. Results from these simulations are particularly interesting because unlike for single-target simulations, face-BB and edge-BB structures exhibit more facile self-assembly than the system with all possible BBs, as depicted in Fig. <ref>. The average cluster sizes corresponding to the conditions of Fig. <ref> are shown in Table <ref>.Of the systems studied, simulations with face-BB structures exhibit nucleation and growth over the largest range of temperatures. The temperature largely controls the number of large nuclei in the system: at 322, only a single structure grows to an appreciable size (Fig. afig-multistruct-pics), whilst at 320, up to 4 nearly complete structures self-assemble. At 319, many simulations result in the successful self-assembly of roughly the same number of clusters (Fig. bfig-multistruct-pics), but in a number of cases, these clusters merge incorrectly, and so the resulting structure can be classed as a kinetic aggregate. The behaviour of systems with edge-BB structures is similar, and, in keeping with the monomer results (see e.g. Figs cfig-nuc-rate and bfig-final-assembly-sizes), nucleation occurs at somewhat lower temperatures. However, aggregation is not shifted by the same amount in temperature, and so the range over which self-assembly occurs is narrower (roughly 319320), and the number of large structures that grow successfully at this lower temperature is also smaller (typically only 2 at 319, Fig. cfig-multistruct-pics). As with single-target simulations, the protocol used for self-assembly is important: although clusters grow to medium sizes in multiple-target simulations after successful nucleation has occurred, the largest clusters can be made to grow essentially to completion if the system is subsequently cooled.Finally, in simulations with all possible BBs, nucleation begins at very high temperatures (∼330), consistent with single-target simulations. However, the clusters do not grow significantly at such high temperatures. As the temperature is decreased, the self-assembly process becomes very error-prone; whilst a single target structure typically grows much larger than the remaining structures, it often lacks the necessary components that have been used up to form other, smaller clusters already (see Table <ref>), and so structures grow with large sections missing. For example, in Fig. dfig-multistruct-pics, showing simulation results at 327, several of the walls have nucleated separately from the rest of the target structure, making further growth very difficult. At 326, several simulations resulted in the successful growth of a single target structure (out of a possible 6 that could grow from the monomers), but in a similar number of simulations, no single target structure grew to a large size. Of course, since there are many monomers in solution, it is in some sense easier to assemble a single copy of the target structure than in single-target simulations; however, assembling multiple target structures simultaneously is difficult, since too many clusters nucleate and it is not straightforward then for them to meet in the correct geometry to form larger structures, and it appears Ostwald ripening is also not particularly fast. The choice of which boundary bricks to include when assembling a given target structure therefore appears to be a very important consideration in DNA brick self-assembly, and it appears from our simulations that opting for all possible boundary bricks is not the most favourable design choice.One possible way of reducing the propensity for nucleation when many boundary bricks are included is to reduce their concentration relative to the remaining monomer bricks. To a first approximation, it is reasonable to assume that the chemical potential of a species appears in the same place in the hamiltonian as the binding energies. Increasing or decreasing the chemical potential of a species, for example by changing the species concentration, is thus effectively equivalent to shifting the strength of all the interactions of that species. We have run simulations of the all-BB system with only half the boundary bricks present. In keeping with expectations, the point at which nucleation occurs in brute-force simulations shifts to lower temperatures, and, with fewer clusters forming, multiple clusters can grow to larger sizes. Choosing an appropriate ratio of initial concentrations is therefore a further control parameter that can be tuned to improve assembly yields.In our simulations, we observe both point defects as well as larger misbonded aggregates and missing features in the target structure. As far as we are aware, experiments on DNA brick systems have not thus far focussed on characterizing the nature of assembly errors in self-assembly, and indeed such defects may be rather difficult to probe experimentally; however, if we wish to ensure a faithful assembly of the complete target structure, this issue may be of great importance for the future of the field.§ CONCLUSIONS DNA brick structures have increasingly been studied over the last few years, since they provide a platform both for theoretical advancement in studying addressable self-assembly and for practical applications, such as creating nanostructures with nano-scale complexity for medicine, computing and nanoelectronics. In this work, we have extended a previously introduced patchy-particle model for DNA bricks to account for boundary bricks, which have been hypothesized to be an essential component of the experimental set-up for increasing the yield,<cit.> but the effects of which had not previously been modelled.It is important to bear in mind that our results correspond to a simple `toy model' of DNA bricks.In reality, many effects that we have neglected may also be important, yet the system sizes involved are such that they make simulations with a more realistic potential intractable at present. However, the simplicity of our model suggests that our findings reflect the fundamental underlying physics of addressable self-assembly.By simulating structures with varying placement of BBs in the canonical ensemble, we have shown that BBs located on the faces of the cubic target structure were primarily responsible for increasing the nucleation rate, whilst BBs located on the edges of the structure were primarily responsible for the stability of the final target structure. However, we have also found that structures that included BBs on all four possible faces were prone to misassembly, particularly in multiple-target simulations, as nucleation is too facile and multiple competing nuclei can grow and are subsequently unable to come together in the correct manner. This indicates that a strategy where all possible DNA strands that can be fused into boundary bricks actually are, as was done in previous experimental work, may not in fact be the optimal choice; a more careful consideration of the possible mechanisms of assembly and misassembly is warranted.While the self-assembly pathways behave in fairly predictable ways in simulations where all patch–patch interactions are of the same strength, further complications arise when DNA sequences are taken into account, since the dominant nucleation locations depend on the strengths of the nearby interactions. We have briefly investigated this effect by examining the GC content of the brick structures, and we found that it was regions with a higher than average GC content that were most likely to nucleate first, rather than necessarily single bricks with an especially high GC content. It would be particularly interesting to investigate this behaviour further and determine whether any simple rules that govern the nucleation location as a function of interaction strength can be identified. However, although the GC content seems to play a significant role for structures without boundary bricks, as soon as BBs are included, the nucleation location is almost completely dominated by the BBs: in structures where BBs are localized to the edges, the edges were involved in nucleation, whilst in structures where BBs are localized to the faces, nucleation occurred primarily on the faces.However, we have shown that BBs affect more than just nucleation. Since they entail the formation of more bonds, the bonding of a BB to the growing structure results in a more favourable enthalpic contribution to the free energy than a monomer brick would give, whilst the loss of entropy is only marginally more disfavourable. This means that target structures can grow larger at a given temperature than they would for a system without boundary bricks. In particular, boundary bricks can stabilize any `fine structure' on the surface of the target structure, which could be especially important for those practical applications for which the assembled structure must be as perfect as possible.Although boundary bricks do allow us to construct structures that are more `perfect' in their final assembled state, including them can be something of a double-edged sword, since they not only stabilize the final structure, but are also easier to nucleate, which means they are more prone to misassembly and aggregation. This may to a significant extent negate the benefit of brick self-assembly being a nucleation-initiated process. In practice, it might be necessary to balance the two effects. For example, it may be possible to increase the yield by keeping the concentration of BBs lower than that of the monomer bricks or keeping the temperature higher for longer in order to keep nucleation a sufficiently rare event. It could also be possible to make the average bonding strength in boundary bricks weaker than that of the remaining monomer bricks, reducing the propensity for premature nucleation, whilst still allowing a degree of stabilization of the final structure. However, we have found that in our simulations, even when multiple structures were allowed to compete with one another in the same simulation box, there was a range of temperatures at which nucleation was rate limiting, but nevertheless sufficiently common for multiple target structures to grow essentially to completion even when boundary bricks were included.Including all possible bricks was not particularly advantageous for assembly in multiple-target simulations, and including only face or only edge BBs resulted in self-assembly that was much less error prone. We envisage that a careful consideration of the types of boundary brick to include to maximize the yield and the quality of the target structures will be particularly important when looking at more complex structures than the ones we considered here. We have only looked at a cubic target structure in this work, as such a system is easiest to study systematically. When target structures include a complex array of peaks and troughs, the choice of the types of boundary brick which will stabilize the target structure whilst minimizing misassembly is considerably less straightforward. Having a clear design strategy is even more important for such systems, but intuition alone may not be enough; indeed, a simple simulation with our coarse-grained potential may well provide a convenient design tool for this purpose. Our simple coarse-grained potential may provide a useful first approximation when faced with a realistic design problem involving DNA bricks, and we hope that our work will provide useful insight to experimentalists interested in the practical applications of such systems.§ ACKNOWLEDGEMENTSWe thank Martin Sajfutdinow, David M. Smith, William M. Jacobs and Thomas E. Ouldridge for helpful discussions. This work was supported by the Engineering and Physical Sciences Research Council [Programme Grant EP/I001352/1]. 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URSS, 1943, 18, 1–22 [FootNoteNuclLocation()]FootNoteNuclLocation The same analysis was performed for clusters comprising 8, 10, 15, and 20 bricks, with similar frequencies. [FootNoteBoundaryCount()]FootNoteGrandCanonical In some sense, this is closer to experimental reality than grand-canonical simulations, since monomer depletion can be a significant problem. However, this depletion effect is likely to be somewhat overestimated in small-scale simulations such as the ones we have performed. Moreover, if we wanted to estimate bulk yields numerically from small-scale simulations, we would need to apply a correction.<cit.> We have also performed grand-canonical simulations for completeness; these also confirm that nucleation is somewhat more difficult than in single-target canonical simulations. [Ouldridge(2012)]Ouldridge2012 T. E. Ouldridge, Inferring bulk self-assembly properties from simulations of small systems with multiple constituent species and small systems in the grand canonical ensemble, J. Chem. Phys., 2012, 137, 144105, https://dx.doi.org/10.1063/1.4757267doi:10.1063/1.4757267 | http://arxiv.org/abs/1704.08293v1 | {
"authors": [
"Hannah Wayment-Steele",
"Daan Frenkel",
"Aleks Reinhardt"
],
"categories": [
"cond-mat.soft",
"physics.chem-ph"
],
"primary_category": "cond-mat.soft",
"published": "20170426184731",
"title": "Investigating the role of boundary bricks in DNA brick self-assembly"
} |
APS/123-QED [email protected]^1Extreme Materials Initiative, Geophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road NW, Washington, DC 20015, USA ^2State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China ^3Department of Earth- and Environmental Sciences, Ludwig Maximilians Universität, Munich 80333, GermanyWe investigated the stability and mechanical and electronic properties of fifteen metastable mixed sp^2-sp^3 carbon allotropes in the family of interpenetrating graphene networks (IGNs) using density functional theory (DFT) within the generalized gradient approximation (GGA). IGN allotropes exhibit non-monotonic bulk and linear compressibilities before their structures irreversibly transform into new configurations under large hydrostatic compression. The maximum bulk compressibilities vary widely between structures and range from 3.6 to 306 TPa^-1. We find all the IGN allotropes have negative linear compressibilities with maximum values varying from -0.74 to -133 TPa^-1. The maximal negative linear compressibility of Z33 (-133 TPa^-1 at 3.4 GPa) exceeds previously reported values at pressures higher than 1.0 GPa. IGN allotropes can be classified as either armchair- or zigzag-type, and these two types of IGNs exhibit different electronic properties. Zigzag-type IGNs are node-line semimetals, while armchair-type IGNs are either semiconductors or node-loop or node-line semimetals. Experimental synthesis of these IGN allotropes might be realized since their formation enthalpies relative to graphite are only 0.1 - 0.5 eV/atom (that of C_60 fullerene is about 0.4 eV/atom), and energetically feasible binary compound pathways are possible.61.50.-f, 62.20.-x, 62.50.-p, 71.20.-b, 71.55.Ak Interpenetrating Graphene Networks: Three-dimensional Node-line Semimetals with Massive Negative Linear Compressibilities R. E. Cohen^1,3 December 30, 2023 =========================================================================================================================§ INTRODUCTION Known carbon allotropes with mixed sp^2 and sp^3 hybridizations are usually amorphous <cit.>. Multiple carbon crystals with mixed sp^2 and sp^3 hybridization have been proposed over the past decades <cit.>, although none of them have been convincingly confirmed by experiments. Recent high-resolution transmission electron microscopy (TEM) images, however, suggest that interpenetrating graphene-like networks might exist locally within compressed glassy carbons <cit.>. Interpenetrating graphene networks (IGNs) are a family of pure carbon allotropes consisting of cross-linked graphene sheets in three dimensions (3D). 3D connectivity of sheets is achieved with sp^3 nodes that link graphene sheets and create open pores in the structures. The open pores are rectangular prisms with parallel sp^3 carbon chains along the edges, which join sp^2 carbon ribbons of variable widths on the sides.Similar to carbon nanotubes<cit.>, IGNs can be classified into armchair (A) and zigzag (Z) types according to the sp^3 chain and sp^2 sheet connectivity along the pore direction (Fig. <ref>). There are two pairs of parallel sp^2 carbon ribbons on the four sides of IGN pores. In Z-type IGNs, sp^3 carbon atoms form six-atom rings with sp^2 carbon atoms on both pairs of parallel sides. In A-type IGNs, sp^3 carbon atoms form six-atom rings with sp^2 carbon atoms on one pair of parallel sides, but form four- and eight-atom rings with sp^2 carbon atoms on the others. The carbon ribbons on all sides can be described as a number of armchair or zigzag chains. In this work we designate A-type IGNs as Aij (Fig. <ref>a), where i denotes the number of armchair chains in ribbons with four- and eight-atom rings, and j denotes the number of zigzag chains in ribbons with all six-atom rings. Similarly, Z-type IGNs are denoted as Zij (Fig. <ref>b), where i and j denote the number of zigzag chains in the parallel pore ribbons. Zij and Zji are identical according to crystallographic symmetry.Zhao<cit.> explored five IGN allotropes (Z11, Z12, Z13, Z23, and Z14) using density functional theory (DFT) and demonstrated that they are energetically metastable with respect to graphite, but exhibit mechanical stability. Later, Jiang et al.<cit.> studied the mechanical and electronic properties of six kinds of IGN allotropes (A11, A22, A33, Z11, Z22 and Z33) and concluded that these structures are semiconducting and that only Z-type IGNs have negative linear compressibilities. Recently, Chen et al.<cit.> demonstrated that Z11 is not a semiconductor, but actually a semimetal based on detailed numerical computations within DFT and theoretical analysis. In addition to their special mechanical and electronic properties, IGN topologies are calculated to be low-energy metastable structures in high-pressure carbides with composition MC_6 (M=metal). For example, it might be possible to obtain Z11 by removing Li or Ca from metastable LiC_6<cit.> or CaC_6<cit.>, in a similar fashion to metal removal from zeolite-type silicon structures<cit.>. In this work, we have discovered six additional energetically competitive and mechanically stable IGN allotropes (A12, A21, A13, A31, A23 and A32) and have investigated the detailed electronic and mechanical properties of the entire IGN family (including 15 structures up to A33 and Z33).§ COMPUTATIONAL METHODS The electronic band structures and the fixed-pressure properties were calculated using density functional theory with the projector augmented-wave (PAW) method<cit.> within the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)<cit.>. The phonon vibrational frequencies were computed using density-functional perturbation theory (DFPT). All the DFT and DFPT computations were performed using PWSCF and phonon codes as implemented in QUANTUM ESPRESSO<cit.>. The plane-wave kinetic-energy cutoff was 80 Ry (1088 eV). In the fixed-pressure relaxations, dense Monkhorst-Pack (MP) k-point meshes were adopted for convergence of the relative enthalpies within several meV per carbon atom.It is known that different kinds of exchange-correlation functionals in DFT give different lattice parameters and zero-pressure stabilities relative to graphite and diamond <cit.>. The local-density approximation (LDA) and GGA are the two most widely used approximations for carbon allotropes <cit.> and many other crystalline systems <cit.>. We chose GGA as the primary method in this work because it gives better pressure-dependent phase stability predictions than LDA in some crystalline systems <cit.> and it correctly predicts that graphite is more stable than diamond at ambient pressure <cit.>. Although GGA significantly over-predicts the zero-pressure volume of graphite (30 percent larger than LDA), it gives much smaller deviations for the zero-pressure volumes and lattice parameters for IGN allotropes (only 4 percent larger than LDA). Unless otherwise specified, all of the results and discussion in this paper are based on GGA-PBE calculations. For comparison, we also list results from LDA computations in the supporting information (Table SI and Figs. S1-S3).For each IGN allotrope, we computed the enthalpy and volume (V) after relaxation at approximately 50 pressures. We then calculated the bulk and linear compressibilities using the definitions β_B=-(1/V)(∂ V/∂ p)_T and β_L=-(1/l)(∂ l/∂ p)_T (V is volume and l is lattice distance), respectively, at different pressures. For a given IGN allotrope, we computed the detailed electronic properties and phonon dispersion at one volume corresponding to 0 GPa (1 atm). The k-point meshes used in the electronic properties calculations were very dense compared with those used in structure relaxation so that energy band contacts and Fermi surfaces could be examined in detail.§ RESULTS AND DISCUSSION§.§ Structure and Stability All of the IGN allotropes studied previously by Jiang et al.<cit.> were symmetric with respect to the pore edge lengths that are normal to the pore direction (i.e., A11, Z11, etc.). Here we expand the number of structures with asymmetric pore lengths originally proposed by Zhao<cit.> by adding armchair or zigzag chains to the known structures. For example, we obtained A12 or A21 by inserting one armchair chain to the primitive cell of A11, and obtained Z12 by inserting one zigzag chain to the primitive cell of Z11 (Fig. <ref>). In this way, we obtained six additional A-type allotropes (A12, A13, A21, A23, A31, and A32) and three additional Z-type allotropes (Z12, Z13, and Z23) (see in Fig. <ref>). We note that an infinite number of structures could be built within this family by increasing the graphene nanoribbon widths. The 15 allotropes examined here have valuable information as to the general trends of properties within the entire IGN family. The detailed structure information of all these 15 IGN allotropes can be found in Table SII in the supporting information. At 0 GPa, the phonon vibrational frequencies in all of the new structures are positive (see Figs. S4-S6 in the supporting information), which indicates that they are all mechanically stable. From 0 to 16 GPa, the enthalpies all the 15 IGNs relative to graphite are in the range of 0.1 - 0.5 eV/atom (Fig. <ref>). At 0 GPa, the formation enthalpy of Z33 is only 0.123 eV/atom, which is smaller than the formation enthalpy of diamond (0.139 eV/atom) at the same pressure. With the same pore length on each side, the Z-type IGNs are energetically more favorable than the A-type. Among all 15 IGNs, Z33, Z13 and Z11 are the most energetically favorable ones at pressures of <1.7 GPa, 1.7-9.7 GPa and >9.7 GPa, respectively. These low formation enthalpies are in a plausible range for experimental synthesis: C_60, an experimentally known carbon allotrope is metastable with respect to graphite by 0.4 eV/atom<cit.>.The IGN structures are mechanically stable over a broad pressure range during cold compression. We do find, however, that all IGN allotropes transform irreversibly into new structures after compression to very high pressures. At these high pressures, new bonds form between atoms on the neighboring or opposite sides of IGN pores (Figs. S7-S11 in the supporting information), and the formation of these new bonds is irreversible during cold decompression. The maximum pressures for mechanical stability are structure dependent and vary widely amongst the IGN allotropes (see p_ir in Table <ref>). Among all 15 IGNs, A32 loses stability most easily from 18 to 20 GPa, whereas the IGN structures of Z11, Z22 and Z33 remain mechanically stable to pressures higher than 150 GPa. Most of the irreversibly transformed structures (Table SIII in the supporting information) are completely sp^3 bonded carbon allotropes except A12, A21, A32 and A23, which still contain a fraction of sp^2 bonds. Upon cold-compression, A21 transforms to the mC16 structure mentioned by Hu et al. <cit.>, Z11 transforms to the 3D-(4,0) structure by Zhao et al. <cit.>, Z12 transforms to so-called “M-Carbon”<cit.>, and Z13 transforms to the P2_1/m structure mentioned by Zhang et al. <cit.> The other transformed structures from A11, A12, A13, A22, A23, A31, A32, A33, Z22, Z23 and Z33 are different from any previously reported carbon allotropes <cit.>, including those listed in the SACADA database <cit.>. §.§ Bulk and Linear Compressibilities Although the unit cell volumes of IGN allotropes decrease with increasing pressure, as required by thermodynamics, their compressibilities are unusually non-monotonic and anisotropic (Figs. <ref>-<ref>). That is, the compressibilities of all IGNs change dramatically with pressure and are extremely sensitive to the magnitude of applied pressure. For this reason, we do not describe zero-pressure bulk moduli (as typically done for carbon allotropes), but rather discuss the pressure-dependent bulk and linear compressibilities of these phases. As pressure is initially applied from 0 GPa, the bulk compressibilities all increase except Z11 and Z12. For Z11 and Z12, bulk compressibilites decrease slightly in a specific range of pressure (0-26 GPa for Z11 and 0-6 GPa for Z12) before increasing at high pressures. The bulk compressibility reaches a maximum at a structure-dependent value, and then decreases as pressure is increased further. For Z11, this local maximum compressibility happens near 32 GPa and is insignificant compared with the other structures. Similar to graphite and diamond, the highest bulk compressibility (β_B,m in Table <ref>) of Z11 occurs at 0 GPa (negative pressures were not considered here), whereas finite pressures for maximum bulk compressibility (p_m in Table <ref>) were observed for other IGN allotropes. A general tendency within the same structure type (armchair or zigzag) is that the highest bulk compressibility increases with pore size, while p_m decreases with increasing pore size (as mentioned above, Z11 is an exception). Differences in the highest bulk compressibilities between different IGN allotropes can vary by orders of magnitude. For example, among all the 15 IGN allotropes, the highest bulk compressibility of A33 is 306 TPa^-1, however that of Z11 is only 3.6 TPa^-1.Jiang et al. <cit.> found that only Z22 and Z33 have negative linear compressibilities. Here, we show that this behavior is actually general to the entire IGN family. There are thirteen different linear directions within the primitive cell of a crystal, and for monoclinic structures the principal compression axes are not necessarily coincident with the conventional lattice directions. The linear compressibilities in all directions of IGNs are diverse. With A13 and Z13 as examples (see Fig. <ref>), there are three directions ([110],[111], and [111̅]) along which expansion is observed over a certain pressure range. This increase in lattice parameter gives rise to negative linear compressibilitiy (NLC). Meanwhile, the lattice parameters corresponding to other directions decrease with pressure and the resulting linear compressibilities are positive. Each IGN has one direction with a most negative linear compressibility and another with a most positive linear compressibillity (PLC). Similar to bulk compressibility, the linear comepressibilities in the most positive and most negative directions are pressure dependent, and the most positive and negative linear compressibilities are also non-monotonic (Fig. <ref>).Z33 has the largest PLC and NLC among these 15 IGNs (Table <ref>). Their values (PLC: 407 TPa^-1, NLC: -133 TPa^-1) pass beyond the reported “giant” linear compressibilities in Ag_3[Co(CN)_6] (PLC: 115TPa^-1, NLC: -76 TPa^-1)<cit.> and Zn[Au(CN)_2]_2 (PLC: 52 TPa^-1, NLC: -42 TPa^-1)<cit.>. We noticed that the linear compressibilites of Ag_3[Co(CN)_6] were obtained in the pressure range of 0-0.19 GPa<cit.> and those of Zn[Au(CN)_2]_2 were between 0-1.8 GPa<cit.>, while the largest linear compressibilities in Z33 were calculated at 3.6 GPa for PLC and 3.4 GPa for NLC. In the pressure range of 0-2.0 GPa, A33 (among 15 IGNs) has the largest linear compressibilites (PLC: 389 TPa^-1 at 1.0 GPa, NLC: -103 TPa^-1 at 0.8 GPa). Both the positvie and negative linear compressibilities of Z33 are larger than any previously reported high-pressure (>1.0 GPa) values for crystals, despite the fact that none of them exceed the ambient-pressure values (PLC: 430 TPa^-1, NLC: -260 TPa^-1) forCsH_2PO_4<cit.> calculated by Cairns and Goodwin<cit.> derived from elastic stiffness components determined by ultrasonic velocity measurements. §.§ Electronic Properties Previous reports of the electronic properties of IGN allotropes are in stark contrast. Jiang et al.<cit.> concluded Z11 is a semiconductor with a band gap between 0.36-0.49 eV depending on the type of functional used (Heyd-Scuseria-Ernzerhof hybrid functionals (HSE06) or Perdew-Burke-Ernzerhof functionals (PBE)). Chen et al.<cit.>, on the other hand, showed that Z11 is semimetal from both first-principles DFT calculations (PBE) and tight-binding modelling. Here, we confirm that Z11 is indeed a node-line semimetal based on our own DFT-PBE computations, which are in agreement with the results of Chen et al.<cit.>. In addition, we investigated the detailed electronic properties of all 15 IGN allotropes using densities of states (DOS) analysis, one-dimensional electronic band dispersions, and the Fermi surface and band contacts for the semimetallic structures. At the Fermi energy level, we found that the density of states in Z13, for example, is very small (on the order of 10^-3 states/cell/eV) but not zero, and the highest valence band and the lowest conduction band contact in the Gamma to Y and D to E directions (Fig. <ref>a). Since the contact points are not located at high-symmetry points, they could be easily missed without using dense k-point grids. Similar situations exist for all of the Z-type and some A-type (A11,A12,A13,A21,A31) IGNs (Fig. <ref>b and Figs. S12-S15 in the supporting information). Thus, all of the Z-type and five of the A-type IGNs are semimetals (no band gap, but vanishingly small density of states at the Fermi level). In contrast to this behavior, we found that some of the large-pore, A-type IGNs are semiconducting. For A33, we found a band gap of 0.48 eV in the band dispersion relations, which was also confirmed using the density of states. This is similar to the findings of Jiang et al.<cit.>, but the magnitude of the gap is different. We attribute the difference (including the finding that Z11 is actually a semimetal) to a finer sampling of the Brillioun zone. Similar to A33, A22, A23 and A32 are also found to be semiconductors with band gaps of 0.92, 0.96, and 0.66 eV, respectively, at the DFT (PBE) level (Figs. S10 and S11). The semimetallic structures all show band dispersion features similar to graphene. We extracted the Fermi surfaces for all semimetallic allotropes to further analyze their electronic structures. This process requires an extremely dense k-point grid in reciprocal space in order to obtain a clear picture of the Fermi surface. With A13 as an example of the A-type IGNs, we found that the Fermi surface exists within non-connected local areas (Fig. <ref>a). Very fine k-point grids are required to delineate the Fermiology. We used a k-point mesh of 28×16×48, corresponding to a spacing of 0.004, 0.004 and 0.0025 Bohr^-1 in the b_1, b_2 and b_3 directions, respectively. The Fermi surface became clearer (Fig. <ref>b) when we used a spacing of 0.0004, 0.0004 and 0.0001 Bohr^-1. The Fermi surface is comprised of four thin Fermi arcs, similar to the Fermi arcs observed in the Weyl semimetal TaAs <cit.> (Fig. S16 in the supporting information). The isoenergy surface, derived from the energy difference between the highest-energy valance band and the lowest-energy conduction band, looks like a circular loop in reciprocal space, indicating contact points (nodes where the energy difference between bands is zero). Within this contact loop, there are 4 points with band energies that are exactly the same as Fermi energy. Thus, A33 can be described as a node-loop semimetal.Looking at Z13 as an example for Z-type IGNs, the Fermi surface was also very unclear using a spacing of 0.005 Bohr^-1 in all b_1, b_2 and b_3 directions (Fig. <ref>a). It became clearer using a spacing of 0.002, 0.002 and 0.001 Bohr^-1, but still displayed an intermittent pattern (Fig. <ref>b). Using an even smaller spacing of 0.001, 0.001 and 0.0005 Bohr^-1 (Fig. <ref>c and d), we conclude that the Fermi surface of Z13 is actually connected. The Fermi surface of Z13 is formed by two symmetric lines. Each line is connected by Fermi arcs, also similar to the Fermi arcs in the Weyl semimetals TaAs <cit.> (Fig. S16 in the supporting information). Different from the case of A13, the isoenergy difference surface of Z13 looks like hollow lines, which indicates that the contact points form two lines in reciprocal space. Within these contact lines, there are four points whose band energies are exactly the same as the Fermi energy. Thus, Z13 can be described as a node-line semimetal.Using the same procedure described above for A13 and Z13, we found that all Z-type IGNs, as well as A21, are node-line semimetals, while A11, A12, A13, and A31 are node-loop semimetals (Table <ref>, Fig. <ref>, and Figs. S17-S24 in the supporting information).Now we come back to Z11, the first IGN allotrope suggested to besemimetallic<cit.>. Similar to Z13, the Fermi surface for Z11 is also formed by two symmetric lines, and the band contact points also form two lines in reciprocal space (Fig. <ref>a and b). In addition to the isoenergy difference surface (an indirect way of showing band contact properties), we directly show that the bands contact on a line by examining the two-dimensional energy band dispersion (Fig. <ref>c and d). Although we can also see that the bands contact in the one-dimensional dispersion plot (Fig. S14 in the supporting information), we can only observe isolated single points. In general, the whole contact line cannot be visualized by way of a two-dimensional dispersion plot, but by taking into account the crystallographic symmetry, the whole contact line in Z11 can be observed in a plane with fixed values in the b_2-b_1 direction (Fig. <ref>c-f). The electronic bands of Z11 are linearly dispersive (typical characteristic for Dirac and Weyl semimetals around the Dirac or Weyl points) in planes with fixed values in the b_1+b_2 direction (Fig. <ref>f). Thus, we also demonstrate that Z11 is a node-line semimetal in both indirect and direct ways.§ CONCLUSION In this work, we demonstrate that interpenetrating graphene networks are metastable pure carbon allotropes with relatively low formation enthalpies (0.1-0.5 eV/atom). Among all 15 IGN allotropes with mechanical stability at 1 atm, Z33 is the most energetically favorable IGN allotrope at P<1.7 GPa and Z11 is the most energetically favorable one at pressures P>9.7 GPa. Between 1.7<P<9.7, Z13 is the most energetically favorable.Non-monotonic bulk and negative linear compressibilities are two typical characteristics of IGNs, which are unusual compared with crystals of other carbon allotropes and most materials in general. The highest bulk compressibilities and the largest negative linear compressibilities depend on the specific structures.All Z-type IGNs are node-line semimetals. For A-type IGNs, A22, A23, A32 and A33 are semiconductors with band gaps of 0.92, 0.96, 0.66, and 0.48 eV, respectively. A21 is a node-line semimetal, while A11, A12, A13, A31 are all node-loop semimetals.These novel carbon allotropes offer attractive multifunctional properties that might see experimental realization through synthetic strategies such as metal removal from high-pressure MC_6 carbides.§ ACKNOWLEDGMENTS This work is supported by DARPA under grant No. W31P4Q1310005. Our DFT computations were performed on the supercomputer Copper of DoD HPCMP Open Research Systems under project No. ACOMM35963RC1 as well as on XSEDE supercomputers and local clusters (mw and memex) of Carnegie Institution for Science. REC is supported by the Carnegie Institution for Science and by the European Research Council Advanced Grant ToMCaT. The authors wish to thank D. Vanderbilt, J. Liu and I. 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Ding,10.1103/PhysRevX.5.031013 journal journal Phys. Rev. X volume 5, pages 031013 (year 2015)NoStop | http://arxiv.org/abs/1704.08753v1 | {
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S. N. Bose National Centre for Basic Sciences, JD Block, Sector-III, Salt Lake City, Kolkata - 700 106, India Second law for an autonomous information machine connected with multiple baths Shubhashis Rana[email: [email protected]] December 30, 2023 ==============================================================================⟨ ⟩ [2]∂#1/∂#2 [3]#3 Phys. Rev. Lett. #1, #2 [3]#3 Phys. Rev. E #1, #2 [3]#3 Phys. Rev. A #1, #2 [3]#3 J. Stat. Mech. #1, #2 [3]#3 Eur. Phys. J. B #1, #2 [3]#3 Rev. Mod. Phys. #1, #2 [3]#3 Europhys. Lett. #1, #2 [3]#3 J. Stat. Phys. #1, #2 [3]#3 Prog. Theor. Phys. Suppl. #1, #2 [3]#3 Physics Today #1, #2 [3]#3 Adv. Phys. #1, #2 [3]#3 J. Phys. Condens. Matter #1, #2 [3]#3 J. Phys. A: Math theor #1, #2 In an Information machinesystem'sdynamicsgetsaffectedby the attached information reservoir.Second lawof thermodynamics can be apparently violated for this case. In this articlewe have derived second law for an information machine, when thesystem is connected to multiple heat baths along with a work source anda single information reservoir. Here a sequence of bits written on a tape is considered as an information reservoir. We find thatthe bath entropy production during a timeinterval is restricted by the change of Shannon entropy of the compositesystem (system +information reservoir) during that interval. We have also given several examples where thislaw can be applicable and shown that our bound is tighter.Keywords: information processing, exact results, stochastic processes§ INTRODUCTIONSecond lawis a fundamental law in thermodynamics and always valid on an average <cit.>. According to the law average entropy production is always positive. Recentdevelopment of fluctuation theorems dictates that it is possible to find negative entropy productionfor an individual event for anyduration although their probabilities areexponentially small. Howevervalidity of the second law was questioned by Maxwell even more than a century ago,when he proposed a thought experiment involving an intelligent being known as Maxwell demon <cit.>. In this gedanken experiment only by knowing the velocity of gas particles confined in a box, a demon can separate them into hotter (consists of faster molecules) and colder (consists of slower molecules) part without doing any work.Half a century later, Szilard proposed another thought experiment <cit.> where he showed that it is always possible to extract heat from single heat bath and perform useful work cyclicallywhen a gas molecule, confined in a box, is treated by a certain protocol which involves measurement of the state of the system. To understand these puzzles, the last century has witnessedseveral wonderful research worksestablishing the connection between information theory and thermodynamics.<cit.>. In fact one needs to take into accountthe cost of information during the process andabove all, the process would be completed only whenthe informationcontained in the memory register of the demon will be erased. Now, according toLandauer<cit.>, one need to do at leastk_B T ln 2 work to erase one bit of information (k_B represents Boltzmann constant). Hence the second law is saved when one takes into account the effect of information.There are mainly two cases in the framework of information processing when the second law is apparently violated <cit.>. In the first one, measurement isperformed and depending on the measurement outcome, the protocol is altered. In another approach,the information contained in an information reservoir ischanged when it is allowedto interact sequentially with the system <cit.>. A sequence of bits written on a tapecan be considered as an information reservoir. Recentlythe second type of approach draws many attention and second law has been derived consequently<cit.>. The performance ofautonomous information machinehas been explained just takingbath entropy productionrestricted by the configuration entropy change of the tape<cit.>. However the inequality shouldcontain the correlation between demon and tape which makes the inequality stronger<cit.>.In <cit.> it is shown that the work done is bounded by the change in Kolmogorov-Sinai dynamical entropy rate of the tape when system is connected to a heat bath, a work source and an information reservoir(tape). Here, the derivation is done by taking into account the correlations within the input string andthose in the output stringgenerated during its evolution connecting with the demon.Note that, this bound is strongerwhen input is uncorrelated or the system(ratchet or demon) is memoryless (i.e., it has no internal states), compared to the earlier version <cit.> where statistical correctionbetween the bits are neglected and work is bounded by the marginal configuration Shannon entropy of the individual bits. Although the result is generally valid even when input is correlated and the ratchet has memory there is some concern asdiscussed in <cit.>. Besidesthe second law is derivedin<cit.> exactly in astraight forwardway by simply addingthe inequalities for each individual cycles.Whena system is connected to a heat bath, a work source and an information reservoir,the correspondingsecond law for single cycle is given by β W ≤ℋ(Π_τ) - ℋ(Π_0).Here ℋ represents the Shannon entropy of the joint system consisting of interacting bit and system. If that joint distribution at any time t is given by Π_t then corresponding Shannon entropy is denoted by ℋ(Π_t)=-∑Π_tlnΠ_t where the sum is done over all possible states. The above equation dictates that the averageextracted work W during the time interval 0≤ t<τis restricted by the changeof Shannon entropy of the joint system during that interval. Motivated by this,we would like to study an information machine which is connected to multiple baths.We have obtained corresponding second lawwhich is exact and most general as well as the bound is more compact. The derivation here is done by scrutinizinghow an autonomous information machine processed a tape sequentially during its operation. Moreover the obtained inequalityreduced to the earlier results in spacial cases. The organization of the paper is as follows. First wedescribe the model and derive our result. Then we compare the result with the earlier studies. After that we give several examples where this law can be applicable. Finally taking a simple model, we numerically showed that our bound isstronger. § THE MODELThe model consists of a demon(system) which is attached with an information reservoir and a work source.A tape, where the information is written asdiscrete symbols, acts as an information reservoir.The input tape is formed by a sequence of symbols x_1,x_2,x_3,...,x_N which is taken from a finite set χ (for binary symbols χ=(0,1)). Each symbolinteracts one by one sequentially with the demon.As a result, the demon state is going through internal statess_1, s_2, s_3,...,s_N which is taken from a finite set S. On the other hand, after interaction, the outgoing tape consists of another sequence of symbols y_1,y_2,y_3,...,y_Nwhich are elements of same set χ. Note that there are no intrinsic transitions in the tape. Only during its interaction with the demon the tape state may change.The total entropy of the incoming tape is given by H(X^N)=-∑_x^N∈χP(x^N)ln P(x^N).Here, P(x^N)=P(x_1,x_2,x_3,...,x_N) is the probability distribution of that sequence of the input.Now, If the input sequence is correlated thenH(X^N)=-∑_x^N∈χP(x^N)ln P(x^N) =-∑_x^N∈χ P(x^N)ln[P(x_N|x^N-1)......P(x_3|x_2,x_1)P(x_2|x_1)P(x_1)] =-∑_x^N∈χ P(x^N)ln P(x_N|x^N-1)-.....-∑_x^N∈χ P(x^N)ln P(x_3|x_2,x_1) -∑_x^N∈χ P(x^N) ln P(x_2|x_1)-∑_x^N∈χ P(x^N)ln P(x_1) =-∑_x^N P(x^N)ln P(x_N|x^N-1)-.....-∑_x_3,x_2,x_1 P(x_3,x_2,x_1)ln P(x_3|x_2,x_1) -∑_x_2,x_1 P(x_2,x_1) ln P(x_2|x_1)-∑_x_1 P(x_1)ln P(x_1) =∑_n=1^N H(X_n|X^n-1).Where H(X_n|X^n-1)=-∑_x^n P(x^n)ln P(x_n|x^n-1). Similarly if P(y^N)=P(y_1,y_2,y_3,...,y_N)represents the probability distribution of the output sequence of the tape, then its entropy is given by H(Y^N)=-∑_y^N∈χP(y^N)ln P(y^N)=∑_n=1^N H(Y_n|Y^n-1). The demon can interact with the nearest symbol of the tape at a time. Now if the tape moves with constant speed v then each symbol gets τ time to interact, after that the nextsymbol arrives. During that time, the joint state of demon and tape evolves with time.As an example, in n^th interaction interval in between time(n-1) τ≤ t < n τ,the input joint state (x_n,s_n) evolves and finally reaches to (y_n,s_n+1).In the beginning of the next cycle, the demon state does not change but the tape is advanced by one unit. As a result the next cyclestarts fromthe joint state (x_n+1,s_n+1).After time τ, the state(x_n+1,s_n+1)evolves to(y_n+1,s_n+2) and this process continues untilthe tapepasses completely. Note that there is two types of dynamics. One is discreteand only deals with the input and output states ( (x_1,s_1) → (y_1,s_2);(x_2,s_2) → (y_2,s_3);.... ).Another one is continuous and deals howa output state is evolved from the input state duringthe time interval τ.Consider the states of the composite system of demon and interacting tape are takenform a product set (χ× S) that containsM + 1 elements which are denoted by (σ_0,σ_1,.....,σ_M).Now at the starting ofn^th cyclethe input state is related to the joint state by (x_n,s_n)≡σ^(n-1)τ.Note that the superscript only denotes the time at which the state appears. The state σ^(n-1)τ then evolves according to the dynamics and finally at nτ ^- it reaches to another element, say σ^nτ^-≡ (y_n,s_n+1).At time nτ the tape is forwarded by one unit and the bit state is changed.As a result, the next cycle starts from (x_n+1,s_n+1)≡σ^nτ which is again an element of that set and the process continues. It can be mentioned that if the output and next input symbol (bit for binary sequence) is same then theactual state does not changeat the time of this switching, if not, then it starts fromanother element of σ. Nextwe will describe the dynamics and the evolution of the joint systemin a particular interval in detail.§ DERIVATION OF SECOND LAW IN PRESENCE OF MULTIPLE BATHS The energy difference between the states σ_0 and σ_1 is E_1 (fig.<ref>). Similarlyforσ_1 and σ_2 it is E_2 and so on. Note that, we have taken σ_0 asground state and corresponding energy is taken as 0.Each of these consecutive states can exchange energy with only one bath and transition can only happen between the consecutive states. As an example, transition betweenσ_0and σ_1can only happen by exchanging heat from a bath with inverse temperature β_1. Hence corresponding transition rates satisfythe required detailed balance <cit.> R_σ_0→σ_1/ R_σ_1→σ_0=exp^-β_1 E_1. Similarly for σ_1and σ_2 the transition can happen when heat is exchanged from thebath with inverse temperature β_2 and corresponding transition rates satisfy detailed balanceR_σ_1→σ_2/ R_σ_2→σ_1=exp^-β_2 E_2,and so on. Note that according to the construction of the model there is no other transitionpossible from or to σ_1. Moreover, the network of all the states form a linear chain and only back and forth excursions (transitions) along the chain is possible.During any time interval τ, the probability distribution of joint states of demon and bits Π_t(σ)evolves according to the master equation dΠ_t(σ)/dt=ℛΠ_t(σ),where ℛ denotes the transition rate matrix whose elements are R_σ_i→σ_j. At τ→∞ the composite system eventually reaches to the steady state where the probability distributiondoes not alter with time and can be determined easily taking ℛΠ_s(σ)=0.Now according to the construction of themodel,thosesteady state probability distributionsof two successive levelsare related byΠ_s(σ_1)= exp^-β_1 E_1Π_s(σ_0), Π_s(σ_2)= exp^-β_2 E_2Π_s(σ_1),Π_s(σ_3)= exp^-β_3 E_3Π_s(σ_2), and so on. Therefore the steady state distribution for any state (σ_i) can be written asΠ_s(σ_i)= exp^-β_i E_iΠ_s(σ_i-1) = exp^-(β_i E_i+.....+β_2 E_2 + β_1E_1)Π_s(σ_0) =e^-∑_j=1^i β_j E_j/Z.Where in the last line, we have used the normalization condition andZ=1+ ∑_i=1^M e^-∑_j=1^i β_j E_j. It is well known that astime passes,Π_t (the probability density at any time t) will approachmonotonically towards Π_s and their distance will reduce(Here for notational simplicity we have taken 0 ≤ t < τ. However it will be true for any interval(n-1)τ≤ t < nτ). Hence one can write <cit.> D(Π_τ||Π_s) ≤ D(Π_0||Π_s). Here the Kullback-Leibler divergence is defined asD(Π_t_1||Π_t_2)=∑_i=0^M Π_t_1(σ_i)lnΠ_t_1(σ_i)/Π_t_2(σ_i). We can easily expand the above inequality and rewrite it to the form:∑_i=0^M [Π_τ(σ_i)-Π_0(σ_i)] ln1/Π_s(σ_i)≤ℋ(Π_τ) - ℋ(Π_0), where ℋ represents Shannon entropy whichhave been already defined earlier. Thereforethe right hand side represents change of Shannon entropy during the evolution. Again we have ln1/Π_s(σ_i)=ln Z + ∑_j=1^i β_j E_j. Then left hand side of eq.<ref> becomes ∑_i=0^M [Π_τ(σ_i)-Π_0(σ_i)] ln1/Π_s(σ_i) =∑_i=1^M [Π_τ(σ_i)-Π_0(σ_i)] ∑_j=1^i β_j E_j =∑_i=1^M [Π_τ(σ_i)-Π_0(σ_i)]β_1 E_1+∑_i=2^M [Π_τ(σ_i)-Π_0(σ_i)]β_2 E_2+.....+∑_i=j^M [Π_τ(σ_i)-Π_0(σ_i)]β_j E_j +....+ [Π_τ(σ_M)-Π_0(σ_M)]β_M E_M . Note that ∑_i=j^M [Π_τ(σ_i)-Π_0(σ_i)] represents the net change ofprobability of all the states above (σ_j) including (σ_j) during the time of operation 0≤ t<τ. As all the allowed transitions form a linear chain, these netprobability change can only happen if same amount of transitions occurs from (σ_j-1) state to (σ_j). Now for each of these transitions E_j amount of heat will be absorbed from thebath β_j. Hence the average amount of heat that is absorbed from this bath along the evolutionduring time τ can be written as q_j=∑_i=j^M [Π_τ(σ_i)-Π_0(σ_i)] E_j. Therefore one can rewrite eq.<ref> as β_1 q_1 + β_2 q_2 +.....+β_M q_M≤ℋ(Π_τ) - ℋ(Π_0). This is the second law for each individual cycle. The left side of the equation is related to the bath entropy production while the right side represents entropy change of the joint system. Now if the system is connected with a single bath and all the transitions are happening byexchanging energy with this bath, thenβ_1=β_2=...=β_M=β and the above equation will simply reduce toβ q ≤ℋ(Π_τ) - ℋ(Π_0).where q represents total heat absorbed from the bath. In <cit.> it is assumed that each energy level is associated to a work source, as a result,the amount of heat absorbed in each transition from the singlebath is equal to same amount of work extraction. Then the above result will be reduced to Eq.<ref>as obtained in <cit.>. Note that in <cit.> transition between any two states is allowed. However in our model, we have restricted it to accommodate multiple baths which act simultaneouslyon the system. To understand the applicability of our model, weconsider different examples in next section.But before going there, we will try to relate the right hand side of eq.<ref> with theentropy change of the tape for completeness of the paper. A detailed comparison had been provided in <cit.>.Taking the notation of discrete process for n^th cycle, eq.<ref>can be written as β_1 q_1(n) + β_2 q_2(n) +.....+β_M q_M(n)≤ H(Y_n,S_n+1) - H(X_n,S_n). Here q_i(n) represents heat absorbed from the i^th bath inn^th cycle. Now for N cycles, the above inequality becomes ∑_n=1^N [ β_1 q_1(n) + β_2 q_2(n) +.....+β_M q_M(n)] ≤∑_n=1^N [H(Y_n,S_n+1) - H(X_n,S_n)]= ∑_n=1^N [H(Y_n|S_n+1) - H(X_n|S_n)] + ∑_n=1^N [H(S_n+1)-H(S_n))] = ∑_n=1^N [H(Y_n|S_n+1) - H(X_n|S_n)] + H(S_N+1)-H(S_1)) ≈∑_n=1^N [H(Y_n|S_n+1)- H(X_n|S_n)].In the last line, it is assumed thatN is very large compared to the total number of joint states (N≫ M).As a result, the contribution of the system (demon) entropy (which will be order of ln M) becomesnegligible compared to the other terms. Note that,the right hand side of the above equation is not equal to the entropy change of the tape which is given by H(Y^N)-H(X^N)=∑_n=1^N [H(Y_n|Y^n-1)- H(X_n|X^n-1)]. Hence our result differs from the earlier result <cit.>, where the entropy production rate of the bath is restricted by the change of these Shannon entropy rate (which includes all the correlation present in a stream of bits)between input tape and the processed output tape.Althoughthe correlation in the output tape may implicitly contain the informationhow it is processed,the inequality in <cit.>had been derivedignoring the detailed methodology for thegeneration of the output bits.This concern has beenpointed out in<cit.> and consequently the second law has been derived.Moreover, it is shown that the obtainedinequality is tight and can be approachedarbitrary close towards equality<cit.>. Note that, in<cit.> the performance of the autonomous information machinehad been described only taking the configuration entropy change of the tape; ignoring the correlation among the bits or the possible correlation between the output tape and the demon that might be generatedduring the operation.Now, it is generally assumed that the input sequence of the tape is not correlated with the demon state i.e, P(x_n|s_n)=P(x_n) then the right hand side of eq.(<ref>) becomes∑_n=1^N [H(Y_n|S_n+1)- H(X_n|S_n)]=∑_n=1^N [H(Y_n|S_n+1)- H(X_n)].For simplicity we take uncorrelated input sequence and try to find out the differences between our result withthat of <cit.>.§.§ uncorrelated input sequence If the input sequence does not have any correlation, then P(x^N)=P(x_N).....P(x_3)P(x_2)P(x_1) and thetotal entropy of the input tape now becomesH(x^N)=∑_n=1^N H(X_n)=N H(X). where H(X_n)=-∑_x_nP(x_n)ln P(x_n). In the last step, it is assumed that the individual probability of each element in a particularposition of the sequence (say n^th)is independent of its position. Again, eq.(<ref>) can be rewritten in the form: ∑_n=1^N [ β_1 q_1(n) + β_2 q_2(n) +.....+β_M q_M(n)] ≤∑_n=1^N [H(Y_n|S_n+1) - H(X_n|S_n)] + H(S_N+1)-H(S_1)) =∑_n=1^N [H(Y_n) - H(X_n)] - ∑_n=1^N I(S_n+1,Y_n)+ H(S_N+1)-H(S_1)).In last line it is again assumed that the input sequenceis uncorrelated with the demon states. I(S_n+1,Y_n) represents the correlation between S_n+1 and Y_n and is given by I(S_n+1,Y_n)=∑_s_n+1∑_ y_n P(s_n+1,y_n) ln(P(s_n+1,y_n) /P(s_n+1)P(y_n) ). Note that, I(S_n+1,Y_n) is always positive. Neglecting the contribution of demon state (which becomes zero for large N) the above inequality shows that our bound is more compact compared to the earlier one<cit.>. Although in <cit.> only single heat bath has been taken, we are comparing the other part except the bath entropy. Note that in <cit.> the author mentioned about the mutual information between the demon and tape but the exact expression has not been given.Now if the demon performs in steady state, then there is no need to concern about each individual cycle (the average heat absorbed from i^th bath in n^th cycleq_i(n) will be independent of the cycle i.e, q_i(n)=q_i). On the other hand entropy change ofdemonwill also be zero. Then the second law for uncorrelated independent sequencebecomesβ_1 q_1 + β_2 q_2 +.....+β_M q_M ≤ H(Y) - H(X)- 1/N∑_n=1^N I(S_n+1,Y_n)). . For large τ, the joint system may reach to thesteady state where probability distributions will take theform as shown in eq.<ref>. Now if the energy of each joint state can be written as sum of demon state energy and tape state energy, then the corresponding total probability density can be expressed in terms of product of demon state probability andtape state probability. For this case, the correlation after the evolution at τ betweendemon state and tape state I(S_n+1,Y_n) vanishes.Hencethesecond law in steady state for uncorrelated independent sequence in largeτ limit takes the form: β_1 q_1 + β_2 q_2 +.....+β_M q_M ≤ H(Y) - H(X).In next section we will talk about few examples where our law can be applicable.§ EXAMPLES§.§ Example 1 First we consider the Maxwell refrigerator model <cit.>.In this model,a two level system is coupled with an information bath and two thermal baths.A simple binary tape is taken as an information reservoir. Hence depending on the system(demon) state andthe bit state there will be four joint states 0d, 1d, 0u and 1u. Each bit can interact with the demon for a time τ before the next bit arrives. The incoming bit can changeits state only when it is interacting with the demon. After the interaction, for a time τ, the bitretains its last state as an output and the tape is forwarded. The rule for the transitions during the interaction time is as follows: When transition takes place with theexchange of heat with hot bath T_h, the bit state does not change, i.e., transition between 0u and 0d, similarly between 1u and 1d energy is exchanged with bath T_h. But for 0d and 1u energy is exchanged with the bath T_c. No other transition is permitted. Hence the joint statesform a linear chain and they are connected by the allowed transition:0u ⇆ 0d ⇆ 1u ⇆ 1d. Thereforewe can apply our model for this case. Note that, in this model 0d state can exchange energy with bath T_h in one end and it can also exchange energy with bath T_c in another end. In τ→∞ corresponding steady statedensity is given by P_s(1u)=exp^-E/T_hP_s(1d),P_s(0d)=exp^E/T_cP_s(1u),P_s(0u)=exp^-E/T_hP_s(0d).Here, the second law for each cycle becomes β_1 q_1 + β_2 q_2+β_3 q_3 ≤ℋ(Π_τ) - ℋ(Π_0). Note that for this case, first and third bath is same i.e., β_1=β_3=1/k_B T_hand second bathis denoted by β_2=1/k_B T_c. Then heat absorbed from hot and coldbath is given by q_h=q_1+q_3 andq_c=q_2 respectively. When the incoming bits are uncorrelated, then, inthe steady statethe second law simply reduces toq_h/T_h+q_c/T_c≤ k_B[H(Y) - H(X) - 1/N∑_n=1^N I(S_n+1,Y_n)]. §.§ Example 2 In next autonomousinformation machine model <cit.>, there is an additional work source along withthe information reservoir and two heat baths.The demon(system) consists of three states A, B and C.Hence depending on bit state and demon(system) state there will be six joint states.For any transition between the energy levelsA and B,E_1 amount of energy is exchanged with bath T_c.This is true for any transition between B and C. Note that during these transitions bit state is not changed. However, transition between A and C is restricted and depends on the interacting bit.Whentransition occurs from (to) C0 to (from) A1, E amount of energy isabsorbed (released)from (to) bath T_h and w amount of work is done on (extracted from) the system.But Transition between C1 and A0 is restricted. Hence the allowed transitions, from one state to another, form a linearchain and is given by A0 ⇆ B0 ⇆ C0 ⇆ A1 ⇆ B1 ⇆ C1.Therefore we can apply eq.<ref> also for this case. Similar to the earlier example, all the heat exchanged with the bath T_ccan be summed up to q_c and heatabsorbed during the transition C0 and A1 is taken as q_h. Then the secondlaw takes the form: q_h/T_h+q_c/T_c≤ k_B[H(Y) - H(X)-1/N∑_n=1^N I(S_n+1,Y_n)].As average energy of the demon does not change here, hence the first law becomes q_h+q_c=W, where W representswork extraction. On the other hand, there is no work sourcein the earlier example and first law takes theform q_h+q_c=0, although second law is same. §.§ Example 3 Now If we take E_1=0, then the above problem will be reduced to the Maxwell demon model <cit.>.For any transition from A to B or B to C and vice versa, no energy is exchanged. Therefore, bath T_c does not have any significance. Then the above second law will be reduced to q_h ≤k_B T_h[H(Y) - H(X)- 1/N∑_n=1^N I(S_n+1,Y_n)]. As total heat absorption q_h will be equal to work extraction W then W ≤ k_B T_h[H(Y) - H(X)- 1/N∑_n=1^N I(S_n+1,Y_n)].Note that k_B[H(Y) - H(X)] denotes the entropy change of the tape. As I is always positive,maximum extractable work becomes less than that was previously thought<cit.> while writing sameamount of information on tape. On the other hand, erasing same amount of information we needto do more work to compensate the term I.§ NUMERICAL RESULTS In this section we will prove our results numerically by considering the second example. Lets define the weight parameter ℰ = tanh(E/2 T_h). Note that -1≤ℰ≤ 1. Consider δ represents excessof 0 in the input tapecompared to the 1, i.e.,δ=p(0)-p(1). Here p(0) and p(1) denotes the probability of 0 and 1 respectively in the incoming bit stream. Note that -1≤δ≤1. As the consecutive bits are uncorrelated with each other, the Shannon entropy of the incoming tape becomes H(X)=-p(0)ln p(0) -p(1)ln p(1). which denotes information content per bit. Similarly,if the probability to get 0 and 1 in the outgoing bit stream are represented by p'(0) and p'(1); then the corresponding Shannon entropy will be H(Y)=-p'(0)ln p'(0) -p'(1)ln p'(1). Then entropy change of the tape is given by Δ S= k_B (H(Y) - H(X)). As q_h and q_c represents average heat absorption to the hot bath T_h and cold bath T_cper unit cycle in steady state, thenbath entropy productionwill beS_B=-q_h/T_h - q_c/T_c. The hidden entropy generated due to the correlation of output bits and the demon per unit cycle insteady state is given byΔ S_cor=- k_B/N∑_n=1^N I(S_n+1,Y_n)=-k_BI(S_n+1,Y_n).Hence the second law for this case can be rewritten asS_tot=S_B+Δ S +Δ S_cor≥ 0.In numerical simulation we have set T_h=1.0, T_c=0.5 and E_1=0.5. We have also setBoltzmann constant k_B=1 and the interaction time of the each bit with the demon τ=0.4. In fig.<ref> we have plotted total entropy production S_tot for different parameterset δ and ℰ. We have found that itremain always positive. Hence, it provesthe second law. Moreover the figure clearly indicates thereversible region, whereS_tot→ 0, by simply connecting the white dots. The relative errorR_err between the the apparent entropy production S_app=S_B+Δ S and S_tot is defined asR_err=S_app-S_tot/S_tot. When the joint system behaves reversibly entropy production iszero and R_err becomes undefined. The imaginary line connecting white dots in the fig.<ref> denotes where these phenomena is occurring. Form the figure we have found that for quite large region R_err can take value greater than 10% even it can exceed 50% in certain region (δ∼1 and high ℰ). Hence we can not neglectS_cor term and S_tot represents the proper bound. § CONCLUSIONSIn summary we have studied anautonomous Information engine model connecting with multiple baths, worksource and information reservoir. Thisis the most general scenario and second law has been obtainedconsequently. First we have derived the second law for each individual cycle. Then we sum those inequalities to get the net inequality. The derivation is done taking into account the exact mechanism how an autonomous information machine evolves connecting to the information reservoir. We have found that this inequality is tighter compared to the earlier results<cit.> which only includesconfiguration entropy change of the tape between itsoutput and input sequences. Besides thereis a significant differencesbetween our result with<cit.> where second law has beenderived ignoring the detailed operation.Moreover we have shown several examples where this law can be applicable. Finally from numerical simulation, we foundthatthe correction term which depends on the correlation between output tape and final state of the demon in each cycle, is quite significant compared with the total entropy production and hence it should not be neglected. AcknowledgementsSR thanks A M Jayannavar for broad discussions throughout the work. SR also thanksDeepak Dhar for useful discussions. SR thanks Department of Science and Technology (DST), India for the financial support through SERB NPDF.10 jar97 Jarzynski C, equality for free energy differences, 7826901997. cro98 Crooks G E, Nonequilibrium measurements of free energy differences for microscopically reversible markovian systems, 1998 J. Stat. Phys. 90 1481. sai12 Saifert U, Stochastic thermodynamics, fluctuation theorems and molecular machines, 2012 Rep. Prog. Phys. 75, 126001. max71Maxwell J C, Theory of Heat,1871 Longmans, London. szi29 Szilard L, On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings, 1929 Z. Phys. 53, 840. lan61Landauer R, Irreversibility and Heat Generation in the Computing Process, 1961, IBM J. Res. Dev. 5, 183. ben82Bennett C H, The thermodynamics of computation - a review, 1982 Int. J. Theor. Phys. 21, 905.zur89Zurek W H, Thermodynamic cost of computation, algorithmic complexity and the information metric, 1989Nature 341, 119. leff03Leff H S and Rex A F,Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing, 2003 Institute of Physics Publishing, Bristol. ved09Maruyama K,Nori F, andVedral V, Colloquium: The physics of Maxwell's demon and information, 2009Rev. Mod. Phys. 81, 1. man12Mandal D andJarzynski C, Work and information processing in a solvable model of Maxwell's demon, 2012 Proceedings of the National Academy of Sciences, 109, 11641.bar13 Barato A C andSeifert U, An autonomous and reversible Maxwell's demon, 101600012013. bar14 Barato A C andSeifert U, Unifying three perspectives on information processing in stochastic thermodynamics, 1120906012014. jar13Deffner S andJarzynski C, Information processing and the second law of thermodynamics: an inclusive, Hamiltonian approach, 2013 Phys Rev X 3, 041003. sag09 Sagawa T, Ueda M, Minimal energy cost for thermodynamic information processing:measurement and information erasure,1022506022009. par15 Parrondo J M R,Horowitz J M andSagawa T, Thermodynamics of information, 2015 Nature Physics 11, 131.man13 Mandal D,Quan H T and Jarzynski C , Maxwell's refrigerator: An exactly solvable model, 1110306022013.ran16 Rana S, Jayannavar A. M., A multipurpose information engine that can go beyond the Carnot limit, 2016, J. Stat Mech: Theo. and Exp, 10, 103207. boyd16 Boyd A B, Mandal D, Crutchfield J P, Identifying functional thermodynamics in autonomous Maxwellian ratchets,2016 New Journal of Physics 18, 023049. boyd17Boyd A B, Mandal D, Crutchfield J P, Correlation-powered information engines and the thermodynamics of self-correction, 950121522017.mer15Merhav N, Sequence complexity and work extraction, 2015 J. Stat Mech: Theo. and Exp. 06, 06037. mer16 Merhav N, Relation between work and entropy production for general information driven finite state engines, 2017 Journal of Statistical Mechanics: Theory and Experiment2, 023207. vanN. G. van Kampen,Stochastic Processes in Physics and Chemistry, (Elsevier, Amsterdam, 2007), Chap V, 3^rd ed. cover T. M. Cover and J. A. Thomas, Elements of Information Theory(Wiley-Interscience, Hoboken, New Jersey, 2006). | http://arxiv.org/abs/1704.08639v3 | {
"authors": [
"Shubhashis Rana"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170427161530",
"title": "Second law for an autonomous information machine connected with multiple baths"
} |
[email protected] Centre de Physique Théorique, École polytechnique, CNRS, Université Paris-Saclay, F-91128 Palaiseau, France [email protected] Dipartimento di Fisica, Università degli Studi di Pavia, I-27100 Pavia, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Pavia,I-27100 Pavia, Italy [email protected] Dipartimento di Fisica, Università degli Studi di Pavia, I-27100 Pavia, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Pavia,I-27100 Pavia, Italy [2]We discuss in detail the spatial distribution of angular momentum inside the nucleon. We show that the discrepancies between different definitions originate from terms that integrate to zero. Even though these terms can safely be dropped at the integrated level, they have to be taken into account at the density level. Using the scalar diquark model, we illustrate our results and, for the first time, check explicitly that the equivalence between kinetic and canonical orbital angular momentum persists at the density level, as expected in a system without gauge degrees of freedom. Spatial distribution of angular momentum inside the nucleon Barbara Pasquini December 30, 2023 ===========================================================§ INTRODUCTION Understanding how the spin of the nucleon originates from the spin and orbital motion of its constituent is one of the current key questions in hadronic physics. While this problem may seem rather straightforward in the context of ordinary quantum mechanics, it becomes quite challenging in the context of hadronic physics where one has to include relativistic, gauge-symmetry and non-perturbative aspects. One of the main conceptual issues is that the decomposition of the nucleon spin is not unique <cit.>. This intrinsic ambiguity is sometimes considered as a sign indicating that the question is not physical. It actually reflects the fact that any decomposition necessarily relies on how one defines the degrees of freedom. The problem remains physical as long as the various contributions can in principle be accessed by experiments. Ji has shown that the (kinetic) total angular momentum of quarks and gluons can be expressed in terms of generalized parton distributions (GPDs) <cit.>. This triggered an intense experimental program since GPDs can be extracted from exclusive processes like deeply virtual Compton scattering and hard meson exclusive electroproduction <cit.>. Interestingly, the connection between GPDs and angular momentum has been clearly established only at the level of integrated quantities over all space. As shown by Burkardt <cit.>, GPDs contain information about the spatial distribution of quarks and gluons inside the nucleon. It is therefore conceivable that GPDs contain also the information about the spatial distribution of angular momentum. The problem now is to determine how this information is precisely encoded.Polyakov provided the first attempt to answer this question <cit.>, but he required the nucleon to be infinitely massive, so as to avoid relativistic corrections. The infinite mass assumption can actually be relaxed, provided that one works within the light-front formalism, as sketched in the review <cit.>. Recently, Adhikari and Burkardt compared different definitions of the angular momentum density and reached the conclusion that none of the definitions agree at the density level. They attributed some of the discrepancies to missing total divergence terms, as it had been pointed out earlier in Refs. <cit.>.The purpose of the present paper is to revisit the work of Polyakov, discuss in more detail the alternative approach based on the light-front formalism, and identify all the missing terms that hinder the proper comparison of the various definitions of angular momentum. The rest of the paper is organized as follows. In <ref> we recall the connection between the energy-momentum tensor and angular momentum. We stress in particular that, unlike in General Relativity, the energy-momentum tensor is generally not symmetric in Particle Physics, owing to the presence of a spin density. In <ref> we derive three-dimensional densities of angular momentum in the Breit frame. We show that by projecting these densities onto a two-dimensional plane, they can be considered in the more general class of elastic frames. In <ref> we discuss the densities in the light-front formalism and observe that they coincide (for the longitudinal component of angular momentum) with the two-dimensional densities in the elastic frame. We illustrate our results within the scalar diquark model in <ref> and, for the first time, check explicitly that kinetic and canonical orbital angular momentum coincide at the density level in absence of gauge bosons. Finally, in <ref> we summarize our findings and draw our conclusions.§ ENERGY-MOMENTUM AND GENERALIZED ANGULAR MOMENTUM TENSORS In field theory, the conserved current associated with the invariance of the theory under Lorentz transformations, known as generalized angular momentum tensor, can be written in general as the sum of two contributionsJ^μαβ(x)=L^μαβ(x)+S^μαβ(x) .Each one of these tensors is antisymmetric under α↔β. The first contribution readsL^μαβ(x)= x^αT^μβ(x)-x^βT^μα(x) ,where T^μν(x) is the Energy-Momentum Tensor (EMT) density associated with the system, which accounts for the fact that the fields are affected by Lorentz transformations owing to their dependence on space-time points. The second contribution S^μαβ(x) accounts for the fact that fields have in general many components, which can also be affected by Lorentz transformations. The three generators of rotations are obtained when α,β=i,j are spatial components. In this case, Eq. (<ref>) simply indicates that the total Angular Momentum (AM) is the sum of Orbital Angular Momentum (OAM) and spinJ=L+Swith J^i=1/2 ϵ^ijk∫^3r J^0jk, likewise for L^i and S^i. §.§ Belinfante-improved tensors The energy-momentum tensor obtained by following the procedure in Noether's theorem is referred to as the canonical EMT, and is in general neither gauge invariant nor symmetric. Belinfante and Rosenfeld <cit.> proposed to add a so-called superpotential term to the definition of both the energy-momentum and generalized angular momentum tensors, defining the Belinfante-improved tensors as T^μν_Bel(x) = T^μν(x)+∂_λG^λμν(x), J^μαβ_Bel(x) = J^μαβ(x)+∂_λ[x^αG^λμβ(x)-x^βG^λμα(x)],wherethe superpotential G^λμν is given by the combinationG^λμν(x)=1/2[S^λμν(x)+S^μνλ(x)+S^νμλ(x)]=-G^μλν(x) .The effect of such a term is to modify the definition of the local density without changing the total charge. The Belinfante-improved tensors (<ref>)-(<ref>) are conserved and usually turn out to be gauge invariant. Moreover, the particular choice (<ref>) allows us to write the total AM in a pure orbital formJ^μαβ_Bel(x)=x^αT_Bel^μβ(x)-x^βT_Bel^μα(x).Since the new tensors are conserved, it follows from this expression that the Belinfante-improved EMT is symmetric. §.§ Kinetic tensors As discussed in Refs. <cit.>, the requirement of a symmetric EMT is usually motivated by General Relativity. In that context, the notion of spin is not accounted for from the beginning, and it is natural to consider AM as purely orbital. From a Particle Physics perspective, however, one naturally includes a spin contribution to the total AM as in Eq. (<ref>). It then follows from the conservation of both T^μν(x) and J^μαβ(x) that the EMT is in general asymmetric, the antisymmetric part being given by the divergence of the density of spinT^[αβ](x)=-∂_μS^μαβ(x),where a^[μb^ν]=a^μ b^ν- a^ν b^μ. We see the Belinfante-improved tensors as effective densities, where the effects of spin are mimicked by an obscure new contribution to momentum. Interestingly, recent developments in optics also seem to demote the Belinfante-improved expressions from their status as fundamental densities <cit.>.Instead of the Belinfante-improved tensors, Ji <cit.> proposed to use in the context of QCD the kinetic EMTT^μν_kin(x)=T^μν_kin,q(x)+T^μν_kin,g(x),where the gauge-invariant quark and gluon contributions are given by <cit.>T^μν_kin,q(x) =1/2 ψ(x)γ^μiD^νψ(x),T^μν_kin,g(x) =-2 Tr[G^μλ(x)G^ν_νλ(x)]+1/2 g^μν Tr[G^ρσ(x)G_ρσ(x)] ,with D^μ=∂^μ-igA^μ and ∂^μ=∂^μ-∂^μ, and the field-strength tensor G_μν(x)=∂_μ A_ν(x)-∂_ν A_μ(x)-ig[A_μ(x),A_ν(x)]. The kinetic generalized AM tensor readsJ^μαβ_kin(x)=L^μαβ_kin,q(x)+S^μαβ_q(x)+J^μαβ_kin,g(x)with L^μαβ_kin,q(x) =x^α T^μβ_kin,q(x)-x^β T^μα_kin,q(x) ,S^μαβ_q(x) =1/2 ε^μαβλ ψ(x)γ_λγ_5ψ(x) ,J^μαβ_kin,g(x) =x^α T^μβ_kin,g(x)-x^β T^μα_kin,g(x) .and the convention ε_0123=+1. Contrary to the quark total AM, the gluon total AM cannot be split into orbital and spin contributions which are at the same time gauge-invariant and local <cit.>. The kinetic and Belinfante-improved tensors in QCD are related as followsT^μν_kin,q(x) = T^μν_Bel,q(x)-1/2 ∂_λS^λμν_q(x) ,L^μαβ_kin,q(x)+S^μαβ_q(x) = J^μαβ_Bel,q(x)-1/2 ∂_λ[x^αS^λμβ_q(x)-x^βS^λμα_q(x)] ,the gluon contributions being the same in both cases, T^μν_kin,g(x)= T^μν_Bel,g(x) and J^μαβ_kin,g(x)=J^μαβ_Bel,g(x). Using the conservation of the total AM J^μαβ_kin and the symmetry of T^μν_kin,g(x), one can relate the antisymmetric part of the quark kinetic EMT to the quark spin divergenceT^[αβ]_kin,q(x)=-∂_μ S^μαβ_q(x),or more explicitlyψ(x)γ^[αiD^β]ψ(x)=-ε^αβμλ ∂_μ[ψ(x)γ_λγ_5ψ(x)] ,as one can also derive directly from the QCD equations of motion. It then follows that the Belinfante-improved EMT just coincides with the symmetric part of the kinetic EMT1/2 T^{μν}_kin,a(x)=T^μν_Bel,a(x), a=q,gwhere a^{μb^ν}=a^μ b^ν+ a^ν b^μ. This simple relation holds only owing to the total antisymmetry of the spin contribution. Since kinetic and Belinfante-improved tensors differ by superpotential terms, they lead to the same charges. For this reason, the superpotentials are often dropped from the discussions in the literature. However, once one goes back to the density level, it is crucial to pay attention to these terms. §.§ Parametrization in terms of form factors We are interested in the matrix elements of the above-mentioned density operators. It will be sufficient to consider the operators evaluated at x=0, since the general case is recovered simply through a translation of fields. Moreover, since the average position is the Fourier conjugate variable to the momentum transfer Δ, we need to consider off-forward matrix elements. As shown by Bakker, Leader and Trueman <cit.>, the matrix elements of the general local asymmetric energy-momentum tensor for a spin-1/2 target are parametrized in terms of five form factors <cit.>:⟨ p', s'| T^μν(0) | p, s⟩= u(p', s')[P^μP^ν/M A(t)+P^μiσ^νλΔ_λ/4M (A+B+D)(t). .+Δ^μΔ^ν-g^μνΔ^2/M C(t) +Mg^μν C̅(t)+P^νiσ^μλΔ_λ/4M (A+B-D)(t)]u(p, s) , where M is the nucleon mass, the three-vectors s and s' (with s^2=s'^2=1) denote the rest-frame polarization of the initial and final states, respectively, andP=p'+p/2, Δ=p'-p , t=Δ^2 .The onshell conditions for initial and final states p^2=p'^2=M^2 are equivalent toP^2=M^2-Δ^2/4,P·Δ=0. Beside the EMT, we also need a parametrization of the matrix elements of the quark spin operator S^μαβ_q(0). Owing to Eq. (<ref>), we can write⟨ p', s'| S^μαβ_q(0)| p, s⟩ =1/2 ε^μαβλ u(p', s')[γ_λγ_5G^q_A(t)+Δ_λγ_5/2M G^q_P(t)]u(p, s),where G^q_A(t) and G^q_P(t) are the axial-vector and induced pseudoscalar form factors, respectively. It then follows from the QCD identity (<ref>) that the form factor associated with the antisymmetric part of the quark EMT is related to the axial-vector form factor <cit.>D_q(t)=-G^q_A(t).§ DENSITIES IN INSTANT FORM Inspired by Sachs' interpretation of the electromagnetic form factors in the Breit frame <cit.>, Polyakov and collaborators discussed the spatial distribution of angular momentum in instant form based on the Belinfante form of the EMT <cit.>. We revisit this discussion in more detail, using this time the more general asymmetric EMT. From now on we drop the label “kin” in all kinetic quantities, as well as the reference to quarks and gluons.§.§ 3D densities in the Breit frame Let us start with the definition of kinetic OAM distribution in four-dimensional position space⟨ L^i⟩(x)=ε^ijk x^j∫^3Δ/(2π)^3 2√(p'^0p^0) ⟨ p', s| T^0k(x)| p, s⟩= ε^ijk x^j∫^3Δ/(2π)^3 e^iΔ· x ⟨ T^0k⟩,where we introduced for convenience⟨ T^μν⟩≡⟨ p', s| T^μν(0)| p, s⟩/2√(p'^0p^0) .Notice that the energy transfer Δ^0 is not an independent variable but a function of the three-momentum transfer Δ through the onshell conditions (<ref>)Δ^0=P·Δ/P^0, P^0=1/2[√(( P+Δ/2)^2+M^2)+√(( P-Δ/2)^2+M^2)] .Using integration by parts, and disregarding as usual the surface term, we rewrite Eq. (<ref>) as⟨ L^i⟩(x)=ε^ijk∫^3Δ/(2π)^3 e^iΔ· x[-i∂⟨ T^0k⟩/∂Δ^j+x^0/2(p'^j/p'^0+p^j/p^0)⟨ T^0k⟩].The second term is in general different from zero. Its explicit time dependence comes from the non-conservation of the individual contributions to the total AM of the system. One way to get rid of this term, along with the x^0 dependence in Eq. (<ref>), is to restrict ourselves to the Breit (or “brick-wall”) frame (BF), defined by the condition P=0. This implies in particular Δ^0=0 and P^0=√(Δ^2/4+M^2). We can then define the spatial density of kinetic OAM as[We note in passing that the incorrect sign for the Fourier transform was used in <cit.>.]⟨ L^i⟩(x)=-iε^ijk∫^3Δ/(2π)^3 e^-iΔ·x.∂⟨ T^0k⟩/∂Δ^j|_BF.This is indeed consistent with a density interpretation since p'=- p implies that the initial and final wave functions undergo the same Lorentz contraction. Using the general parametrization (<ref>) and taking the same rest-frame polarization three-vector s for both the initial and final states, we find that the kinetic OAM density reads (see Appendix <ref> for more details)⟨ L^i⟩(x)=∫^3Δ/(2π)^3 e^-iΔ·x[s^iL(t)+[(Δ·s)Δ^i-Δ^2s^i] L(t)/ t]_t=-Δ^2,where we introduced for convenience the combination of energy-momentum form factorsL(t)=1/2[A(t)+B(t)+D(t)].Similarly, for the spin density we find that⟨ S^i⟩(x) =1/2 ε^ijk∫^3Δ/(2π)^3 e^-iΔ·x.⟨ S^0jk⟩|_BF=∫^3Δ/(2π)^3 e^-iΔ·x [s^i/2G_A(t)-(Δ·s)Δ^i/4G(t)/ t]_t=-Δ^2,where we introduced for convenienceG(t)/ t=1/2P^0[G_A(t)/P^0+M+G_P(t)/M]. Polyakov and collaborators <cit.> considered the Belinfante-improved form of the EMT. Recalling that T^μν_Bel=1/2T^{μν}, it is easy to see that the density of Belinfante-improved total AM assumes the same structure as in Eq. (<ref>), but now without the D(t) contribution⟨ J^i_Bel⟩(x)=∫^3Δ/(2π)^3 e^-iΔ·x[s^iJ(t)+[(Δ·s)Δ^i-Δ^2s^i] J(t)/ t]_t=-Δ^2,where we used Polyakov's form factorJ(t)=1/2[A(t)+B(t)].We can compare this expression with the kinetic total AM density ⟨ J^i⟩(x)=⟨ L^i⟩(x)+⟨ S^i⟩(x). From Eqs. (<ref>) and (<ref>) and taking into account that D(t)=-G_A(t), we find⟨ J^i⟩ (x)=∫^3Δ/(2π)^3 e^-iΔ·x [s^i J(t)+[(Δ·s)Δ^i-Δ^2s^i] L(t)/ t-(Δ·s)Δ^i/4G(t)/ t]_t=-Δ^2.Therefore we have at the density level⟨ J^i⟩(x)≠⟨ J^i_Bel⟩(x) ,while⟨ J^i⟩ =∫^3x ⟨ J^i⟩(x)=∫^3x ⟨ J^i_Bel⟩(x)=s^i J(0),which is nothing but th Ji relation <cit.> in the rest frame of the target. The reason for this mismatch is the total divergence in Eq. (<ref>). We obtain for the corresponding density⟨ M^i⟩(x) =1/2 ε^ijk∫^3Δ/(2π)^3 e^-iΔ·x Δ^l.∂⟨ S^l0k⟩/∂Δ^j|_BF=-∫^3Δ/(2π)^3 e^-iΔ·x[[(Δ·s)Δ^i-Δ^2s^i]/2G_A(t)/ t+(Δ·s)Δ^i/4 G(t)/ t]_t=-Δ^2,leading then to⟨ J^i⟩(x)=⟨ J^i_Bel⟩(x)+⟨ M^i⟩(x) ,as expected.Notice that, since integrating over x is equivalent to setting Δ=0, from Eqs. (<ref>), (<ref>) and (<ref>) we can also writeJ^i=s^i∫^3x∫^3Δ/(2π)^3 e^-iΔ·x J(-Δ^2) .It may therefore be tempting to interpret the Fourier transform of the form factor J(t) as the density of total angular momentum. We see however from Eqs. (<ref>) and (<ref>) that, in both the Belinfante's and in the kinetic case, other terms explicitly depending on Δ do also contribute at the density level. More precisely, for the kinetic total AM we can introduce the following decomposition:⟨ J^i⟩(x)=⟨ J^i⟩_naive(x)+⟨ J^i⟩_corr(x)into a “naive” contribution⟨ J^i⟩_naive(x)=∫^3Δ/(2π)^3 e^-iΔ·x s^iJ(-Δ^2)and a correction⟨ J^i⟩_corr(x)= ∫^3Δ/(2π)^3 e^-iΔ·x [[(Δ·s)Δ^i-Δ^2s^i] L(t)/ t-(Δ·s)Δ^i/4G(t)/ t]_t=-Δ^2,satisfying∫^3x⟨ J^i⟩_naive(x)=⟨ J^i⟩, ∫^3x⟨ J^i⟩_corr(x)=0.Finally, in order to establish the connection with the results of <cit.>, we decompose Eq. (<ref>)⟨ J^i_Bel⟩(x)=⟨ J^i_Bel⟩_mono(x)+⟨ J^i_Bel⟩_quad(x),into monopole and quadrupole contributions⟨ J^i_Bel⟩_mono(x) =s^i∫^3 Δ/(2π)^3 e^-iΔ·x[J(t)+2t/3J(t)/ t]_t=-Δ^2 ,⟨ J^i_Bel⟩_quad(x) =s^j∫^3 Δ/(2π)^3 e^-iΔ·x [Δ^iΔ^j-1/3 δ^ijΔ^2]. J(t)/ t|_t=-Δ^2.The monopole contribution is the expression used by Polyakov and collaborators. As explained in the Appendix H of <cit.>, they discarded the quadrupole contribution because they interpreted it as an artifact originating from the non-covariance of the light-front formalism. Here we clearly see that the quadrupole term has actually nothing to do with the light-front formalism and simply arises from the breaking of spherical symmetry down to axial symmetry due to the polarization of the state. In conclusion, although the quadrupole term does not contribute once integrated over all space, it cannot be discarded when we consider densities. §.§ 2D densities in the elastic frame The Breit frame allows one to define 3D densities for P= 0. If we want to consider the case where P≠ 0, the only densities we can define are necessarily two-dimensional. Indeed, in order to preserve the condition Δ^0=0 which ensures that both the initial and final states are affected by the same Lorentz contraction factor, we have to restrict Δ to the subspace orthogonal to P. We define the elastic frames (EF) by the condition P·Δ=0. They constitute a class of frames characterized by the fact that there is no energy transferred to the system, i.e. Δ^0=0; the energy of the system is then given by P^0=√( P^2+Δ^2/4+M^2). The Breit frame appears as a particular element of this class. Since P distinguishes a particular spatial direction, it is convenient to write three-vectors in terms of longitudinal and transverse components. Without loss of generality, we choose the spatial axes so that P lies along the z axisP=( 0_⊥,P),Δ=(Δ_⊥,0).In order to get rid of the time dependence in Eq. (<ref>), we will restrict ourselves to the longitudinal component of angular momentum only. To comply with standard notations <cit.>, we will denote the Fourier conjugate variable to Δ_⊥ by b_⊥ instead of x_⊥. We then define the impact-parameter densities of kinetic OAM and spin as ⟨ L^z⟩(b_⊥) =-iε^3jk∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥.∂⟨ T^0k⟩/∂Δ^j_⊥|_EF=s^z∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥[L(t)+tL(t)/ t]_t=-Δ^2_⊥, ⟨ S^z⟩(b_⊥) =1/2 ε^3jk∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥.⟨ S^0jk⟩|_EF=s^z/2∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥ G_A(-Δ^2_⊥) .Similarly, for the impact-parameter densities of Belinfante-improved total AM and total divergence, we find⟨ J^z_Bel⟩(b_⊥) =-iε^3jk∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥.∂⟨ T^0k_Bel⟩/∂Δ^j_⊥|_EF=s^z∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥[J(t)+tJ(t)/ t]_t=-Δ^2_⊥, ⟨ M^z⟩(b_⊥) =1/2 ε^3jk∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥ Δ^l_⊥.∂⟨ S^l0k⟩/∂Δ^j_⊥|_EF=-s^z/2∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥[tG_A(t)/ t]_t=-Δ^2_⊥.The 2D distributions (<ref>)-(<ref>) are axially symmetric and, remarkably, appear to be independent of P. The reason is that longitudinal boosts do not mix longitudinal components of angular momentum. As a consequence, 2D distributions in the elastic frame can be directly compared with 3D distributions in the Breit frame. Since Δ is Fourier conjugate to x, setting Δ^3=0 amounts to integrating over x^3. In other words, the 2D distributions in the elastic frame are just the projections onto the transverse plane of the corresponding 3D distributions in the Breit frame⟨ j^z⟩(b_⊥)=∫ x^3 ⟨ j^z⟩( x)|_ x=( b_⊥,x^3)with b_⊥=| b_⊥| and j^z=L^z,S^z,J^z_Bel,M^z, as can be readily checked. Once again, we have⟨ J^z⟩(b_⊥)=⟨ L^z⟩(b_⊥)+⟨ S^z⟩(b_⊥)=⟨ J^z_Bel⟩(b_⊥)+⟨ M^z⟩(b_⊥) .Note also that the dependence on the induced pseudoscalar form factor G_P(t) has disappeared because the latter is multiplied by Δ^3=0.Defining the 2D Fourier transform of the form factors asF̃(b_⊥)=∫^2 Δ_⊥/(2π)^2 e^-iΔ_⊥· b_⊥F(-Δ^2_⊥) ,we can write⟨ L^z⟩(b_⊥) =-s^z/2 b_⊥ L̃(b_⊥)/ b_⊥, ⟨ S^z⟩(b_⊥) =s^z/2 G̃_A(b_⊥) , ⟨ J^z_Bel⟩(b_⊥) =-s^z/2 b_⊥ J̃(b_⊥)/ b_⊥, ⟨ M^z⟩(b_⊥) =s^z/2[G̃_A(b_⊥)+1/2 b_⊥G̃_A(b_⊥)/ b_⊥] .The impact-parameter density of kinetic total AM ⟨ J^z⟩(b_⊥)=⟨ L^z⟩(b_⊥)+⟨ S^z⟩(b_⊥) will differ from the “naive” density⟨ J^z⟩_naive(b_⊥)=s^zJ̃(b_⊥) by a correction term⟨ J^z⟩_corr(b_⊥)=-s^z[L̃(b_⊥)+1/2 b_⊥ L̃(b_⊥)/ b_⊥] .We can also project the 3D monopole and quadrupole contributions to the Belinfante-improved total AM (<ref>) and (<ref>) onto the transverse plane. This gives⟨ J^z_Bel⟩_mono(b_⊥) =s^z/3[J̃(b_⊥)-b_⊥ J̃(b_⊥)/ b_⊥], ⟨ J^z_Bel⟩_quad(b_⊥) =-s^z/3[J̃(b_⊥)+1/2 b_⊥ J̃(b_⊥)/ b_⊥] .Clearly, the total divergence (<ref>), the correction (<ref>) and the quadrupole (<ref>) terms vanish once integrated over b_⊥2π∫ b_⊥ b_⊥ ⟨ M^z⟩(b_⊥)=2π∫ b_⊥ b_⊥ ⟨ J^z⟩_corr(b_⊥)=2π∫ b_⊥ b_⊥ ⟨ J^z_Bel⟩_quad(b_⊥)=0,as one can see using integration by parts. This explains why the naive J̃(b_⊥), the Polyakov-Goeke ρ^PG_J(b_⊥) and the infinite-momentum frame ρ^IMF_J(b_⊥) definitions considered by Adhikari and Burkardt <cit.> (corresponding in our notations to ⟨ J^z⟩_naive(b_⊥), ⟨ J^z_Bel⟩_mono(b_⊥) and ⟨ J^z_Bel⟩(b_⊥), respectively) are different, even though they lead to the same integrated total angular momentum.§ DENSITIES IN FRONT FORM As discussed by Burkardt <cit.>, the density interpretation in the Breit frame is valid only when relativistic effects associated with the motion of the target can be neglected. An elegant way of getting rid of these relativistic corrections is to switch to the front-form dynamics <cit.>. In this formalism, the subgroup of Lorentz transformations associated with the transverse plane is Galilean <cit.>. As a consequence, there is no need for relativistic corrections as long as we restrict ourselves to the transverse plane.We introduce light-front coordinates a^μ=[a^+,a^-,a_⊥], with a^±=1/√(2)(a^0± a^3). Once again we focus on the longitudinal component of angular momentum. Similarly to the instant-form case, we start with the definition of kinetic OAM distribution in four-dimensional position space⟨ L^z⟩(x)= ε^3jk x^j_⊥∫^2Δ_⊥ Δ^+/(2π)^3 e^iΔ· x ⟨ T^+k⟩_LF,where⟨ T^μν⟩_LF≡⟨ p', s| T^μν(0)| p, s⟩/2√(p'^+p^+) .Using the onshell conditions, we can express the light-front energy transfer Δ^- in terms of the three-momentum transfer (Δ^+,Δ_⊥) asΔ^-=P_⊥·Δ_⊥-P^-Δ^+/P^+, P^-=1/2[( P_⊥+Δ_⊥/2)^2+M^2/2(P^++Δ^+/2)+( P_⊥-Δ_⊥/2)^2+M^2/2(P^+-Δ^+/2)].Using integration by parts, and disregarding as usual the surface term, we rewrite Eq. (<ref>) as⟨ L^z⟩(x)=ε^3jk∫^2Δ_⊥ Δ^+/(2π)^3 e^iΔ· x[-i∂⟨ T^+k⟩_LF/∂Δ^j_⊥+x^+/2(p'^j_⊥/p'^++p^j_⊥/p^+)⟨ T^+k⟩_LF].Densities in the light-front formalism are defined in the Drell-Yan (DY) frame where Δ^+= 0 and P_⊥= 0_⊥. This amounts to integrating the four-dimensional distributions over the longitudinal light-front coordinate x^-. In such a frame, the dependence on the light-front time x^+ in Eq. (<ref>) drops out. The impact-parameter densities of kinetic OAM and spin in the light-front formalism are then given by (see Appendix <ref> for more details)⟨ L^z⟩(b_⊥) =-iε^3jk∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥.∂⟨ T^+k⟩_LF/∂Δ^j_⊥|_DY=s^z∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥[L(t)+tL(t)/ t]_t=-Δ^2_⊥, ⟨ S^z⟩(b_⊥) =1/2 ε^3jk∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥.⟨ S^+jk⟩_LF|_DY=s^z/2∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥G_A(-Δ^2_⊥) .Similarly, for the impact-parameter densities of Belinfante-improved total angular momentum and total divergence, we find⟨ J^z_Bel⟩(b_⊥) =-iε^3jk∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥.∂⟨ T^+k_Bel⟩_LF/∂Δ^j_⊥|_DY=s^z∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥[J(t)+tJ(t)/ t]_t=-Δ^2_⊥,⟨ M^z⟩(b_⊥) =1/2 ε^3jk∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥ Δ^l_⊥.∂⟨ S^l+k⟩_LF/∂Δ^j_⊥|_DY=-s^z/2∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥·b_⊥[tG_A(t)/ t]_t=-Δ^2_⊥.These light-front densities in the Drell-Yan frame coincide with the corresponding instant-form densities in the elastic frame. This should not be too surprising based on the following arguments. Indeed, the Drell-Yan frame is nothing but the elastic frame with P defining the light-front direction. Moreover, instant form and front form coincide in the infinite-momentum frame where P_z→∞. Since the 2D densities we considered do not depend on P_z, they should be the same in both instant form and front form.§ ILLUSTRATION WITHIN THE SCALAR DIQUARK MODEL For illustrative purposes, we calculate explicit expressions for the impact-parameter densities in the framework of the scalar-diquark model <cit.>. This simple model depicts the nucleon as formed by an active quark and a spectator system described by a scalar diquark.The quark Light-Front Wave Functions (LFWFs) Ψ^Λ_λ(x, k_⊥), where Λ=± and λ=± denote the helicity of the nucleon and of the quark, respectively, readΨ^+_+(x, k_⊥) =Ψ^-_-(x, k_⊥)=(M+m/x)ϕ(x, k^2_⊥) , Ψ^+_-(x, k_⊥) =-[Ψ^-_+(x, k_⊥)]^*=-k^x+ik^y/x ϕ(x, k^2_⊥),where ϕ(x, k^2_⊥)=-g x√(1-x)/ k^2_⊥+u(x,m_D^2)andu(x,μ^2)=xμ^2+(1-x)m^2-x(1-x)M^2.Here g is the Yukawa coupling constant, while m, M and m_D are the mass of the quark, nucleon and diquark, respectively.We define the 2-dimensional Fourier transform of LFWFs from momentum to impact-parameter space as <cit.>Ψ^Λ_λ(x, b_⊥)=1/1-x∫^2k_⊥/(2π)^2 e^i k_⊥· b_⊥/(1-x) Ψ^Λ_λ(x, k_⊥).Writing b_⊥=b_⊥(cosϕ_b,sinϕ_b), we obtainΨ^+_+(x, b_⊥) =Ψ^-_-(x, b_⊥)=-g (xM+m)/2π √(1-x) K_0(Z) ,Ψ^+_-(x, b_⊥) =[Ψ^-_+(x, b_⊥)]^*=ig √(u(x,m_D^2)) e^iϕ_b/2π √(1-x) K_1(Z),where K_n is the n-th order modified Bessel function of the second kind and Z=√(u(x,m_D^2)) b_⊥/(1-x).GPDs can be computed using the following LFWF overlap representation in impact-parameter space <cit.>ℋ(x,b_⊥) =1/2(2π)[|Ψ^+_+(x, b_⊥)|^2+|Ψ^+_-(x, b_⊥)|^2], -1/2M(i∂/∂ b^x+∂/∂ b^y)ℰ(x,b_⊥) =1/2(2π)[Ψ^+*_+(x, b_⊥)Ψ^-_+(x, b_⊥)+Ψ^+*_-(x, b_⊥)Ψ^-_-(x, b_⊥)] ,ℋ̃(x,b_⊥) =1/2(2π)[|Ψ^+_+(x, b_⊥)|^2-|Ψ^+_-(x, b_⊥)|^2] ,where the Fourier transforms of GPDs are defined as ℱ(x,b_⊥)=∫^2Δ_⊥/(2π)^2 e^-iΔ_⊥· b_⊥ F(x,0,-Δ^2_⊥). Using Eqs. (<ref>)-(<ref>), we findℋ(x,b_⊥) =g^2/2(2π)^3(1-x){(xM+m)^2[K_0(Z)]^2+ u(x,m^2_D)[K_1(Z)]^2},ℰ(x,b_⊥) =g^2/2(2π)^3 2M(xM+m)[K_0(Z)]^2 ,ℋ̃(x,b_⊥) =g^2/2(2π)^3(1-x){(xM+m)^2[K_0(Z)]^2-u(x,m^2_D)[K_1(Z)]^2}.Taking the second Mellin moment of these expression, we obtain the EMT form factors in impact-parameter space <cit.>:∫_0^1dx x[ℋ(x,b_⊥)+ℰ(x,b_⊥)]=Ã(b_⊥)+B̃(b_⊥) , ∫_0^1dx ℋ̃(x,b_⊥)=G̃_A(b_⊥)=-D̃(b_⊥)which can then be inserted in Eqs (<ref>)-(<ref>) to get the various contributions to the density of AM. In Fig. <ref> we plot the above-mentioned densities as functions of the modulus b_⊥ of the impact parameter for a longitudinally polarized target s=(0,0,1). We choose the same mass parameters as Adhikari and Burkardt <cit.>, namely M=m=m_D=1 fm^-1. In order to regulate the ultraviolet divergences b_⊥→ 0, we adopt the Pauli-Villars regularization, using the diquark mass m_D as a regulator. More precisely, for each one of the functions ⟨ j^z⟩(b_⊥;m^2_D) considered, we plotb_⊥[⟨ j^z⟩(b_⊥;m^2_D)-⟨ j^z⟩(b_⊥;M^2_D)] ,with M_D^2=10 m_D^2. The extra factor of b_⊥ comes from the Jacobian of the transformation to polar coordinates.In the first plot we present the kinetic total AM ⟨ J^z⟩(b_⊥)=⟨ L^z⟩(b_⊥)+⟨ S^z⟩(b_⊥) as the sum of kinetic OAM and spin contributions. In the scalar diquark model, both contributions appear to be positive. In the second plot, we compare the kinetic total AM ⟨ J^z⟩(b_⊥) with the Belinfante-improved total AM ⟨ J^z_Bel⟩(b_⊥), the difference being attributed to the ⟨ M^z⟩(b_⊥) term in Eq. (<ref>), which originates fromthe total-divergence termin Eq. (<ref>). In the third plot, we compare the kinetic total angular momentum ⟨ J^z⟩(b_⊥) with the naive density J̃(b_⊥). Their difference is given by the correction term ⟨ J^z⟩_corr(b_⊥) in Eq. (<ref>). In the fourth and last plot, we decomposed the Belinfante-improved total AM⟨ J^z_Bel⟩(b_⊥)=⟨ J^z_Bel⟩_mono(b_⊥)+⟨ J^z_Bel⟩_quad(b_⊥) into its monopole and quadrupole contributions. The monopole contribution is what Adhikari and Burkardt called the Polyakov-Goeke definition <cit.>. Once again, although the total divergence term ⟨ M^z⟩(b_⊥), the correction term ⟨ J^z⟩_corr(b_⊥) and the quadrupole contribution ⟨ J^z_Bel⟩_quad(b_⊥) integrate to zero, they need to be taken into account when comparing different definitions for the density of angular momentum.There is no gauge field in the scalar diquark model we considered. As a consequence, it is expected that the kinetic OAM should coincide with the canonical (or Jaffe-Manohar) OAM <cit.>. The latter can be expressed in terms of the following LFWF overlap representation in impact-parameter spaceℒ^z(b_⊥)=1/2(2π)∫_0^1 x (1-x) |Ψ^+_-(x, b_⊥)|^2 .Using Eq. (<ref>), we findℒ^z(b_⊥)=g^2/2(2π)^3∫_0^1 x u(x,m^2_D)[K_1(Z)]^2 . Note that the canonical OAM can alternatively be defined in terms of generalized transverse momentum parton distributions <cit.>, leading to the same expression as in Eq. (<ref>) in the scalar diquark model <cit.>. Thishas to be compared with the expression for the kinetic OAM ⟨ L^z⟩(b_⊥) that we obtain from Eq. (<ref>), using (<ref>)-(<ref>):⟨ L^z⟩(b_⊥) =g^2/2(2π)^3 1/2∫_0^1 x 1/1-x {[(1-x)(x^2M^2-m^2)+(1+x) u(x,m_D^2)]Z K_0(Z)K_1(Z) +(1+x) u(x,m_D^2)[K_1(Z)]^2}.Using integration by parts, one can show that ⟨ L^z⟩(b_⊥)=ℒ^z(b_⊥) for b_⊥>0, see Appendix <ref>. To the best of our knowledge, this is the first time that the equality between kinetic and canonical OAM is checked explicitly at the density level. We also understand the failure to observe the equality in Ref. <cit.> as coming from the fact that the authors incorrectly defined the density of kinetic OAM asL^z_IMF(b_⊥)≡⟨ J^z_Bel⟩(b_⊥)-⟨ S^z⟩(b_⊥) which misses the total divergence term ⟨ M^z⟩(b_⊥) as one can see from Eq. (<ref>). § CONCLUSIONS In this work, we addressed the question of the definition of angular momentum at the density level.One often makes use of the freedom offered by superpotentials to deal with a symmetric energy-momentum tensor, as motivated by General Relativity. In the context of Particle Physics, however, spin densities play a fundamental role and make the energy-momentum tensor asymmetric. In particular, we showed that for a spin-1/2 target the form factor accounting for the antisymmetric part of the energy-momentum tensor coincides (up to a sign) with the axial-vector form factor. This provides an interesting new way of calculating the latter on the lattice. While superpotential terms do not play any role at the level of integrated quantities, it is of crucial importance to keep track of them at the density level.We revisited Polyakov's work on the three-dimensional distribution of angular momentum in the Breit frame. Working with an asymmetric energy-momentum tensor allowed us to derive directly the correct density of orbital angular momentum. Densities in the Breit frame can be extended to the more general class of elastic frame, provided one projects onto a two-dimensional plane. Thanks to this generalization, we were able to establish a simple connection between instant-form densities defined in the Breit frame and light-front densities defined in the Drell-Yan frame for the longitudinal components of angular momentum. We used the scalar diquark model to illustrate our results. We showed explicitly that when all the terms integrating to zero are included in the expressions, no discrepancies are found between the different definitions of angular momentum. In particular, we checked for the first time explicitly that the canonical and kinetic angular momentum do coincide at the density level, as expected in a system without gauge bosons.§ ACKNOWLEDGMENTS This work was partially supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 647981, 3DSPIN).§ DIRAC BILINEARS In this Appendix, we collect the Dirac bilinears involved in the calculation of the various matrix elements of <ref> and <ref>. In the Breit frame, we usedu(Δ2, s)γ_5 u(-Δ2, s) =-(Δ· s), u(Δ2, s)γ^kγ_5 u(-Δ2, s) =2P^0s^k-Δ^k(Δ· s)/2(P^0+M), u(Δ2, s)iσ^kλΔ_λ u(-Δ2, s) =-2M iϵ^klmΔ^ls^m.In the elastic frame, we usedu(P_z,Δ_⊥2, s)γ^3γ_5 u(P_z,-Δ_⊥2, s) =2P^0 s^z, u(P_z,Δ_⊥2, s)iσ^kλΔ_λ u(P_z,-Δ_⊥2, s) =-2M iϵ^kl3Δ^ls^z, k=1,2.In the Drell-Yan frame with light-front spinors, we usedu_LF(P^+,Δ_⊥2, s)γ^+γ_5 u_LF(P^+,-Δ_⊥2, s) =2P^+s^z, u_LF(P^+,Δ_⊥2, s)iσ^kλΔ_λ u_LF(P^+,-Δ_⊥2, s) =-2M iϵ^kl3Δ_⊥^ls^z, k=1,2. § KINETIC AND CANONICAL ORBITAL ANGULAR MOMENTUM Proving the equality between Eqs. (<ref>) and (<ref>) amounts to establishing the following identity∫_0^1 x 1/1-x[(1-x)(x^2M^2-m^2)+(1+x) u]Z K_0(Z)K_1(Z)=∫ x 1-3x/1-x u[K_1(Z)]^2 ,Using1/Z ∂ Z/∂ x=1/2u ∂ u/∂ x+1/1-x, x ∂ u/∂ x=u+x^2M^2-m^2,we find that1/1-x[(1-x)(x^2M^2-m^2)+(1+x) u]Z=2ux ∂ Z/∂ x.Noting now that(Z^2[K_1(Z)]^2)/ Z=-2Z^2K_0(Z)K_1(Z),we can rewrite the LHS of Eq. (<ref>) as∫_0^1 x 1/1-x[(1-x)(x^2M^2-m^2)+(1+x) u]Z K_0(Z)K_1(Z)=-∫_0^1xu/Z^2 ∂(Z^2[K_1(Z)]^2)/∂ x.Integrating by parts, the boundary term vanishes identically for b_⊥>0 and we obtain the RHS of Eq. (<ref>). myrevtex | http://arxiv.org/abs/1704.08557v1 | {
"authors": [
"Cédric Lorcé",
"Luca Mantovani",
"Barbara Pasquini"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170427132735",
"title": "Spatial distribution of angular momentum inside the nucleon"
} |
Departament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, IEEC-UB, Martí i Franquès 1, E08028 Barcelona, Spain Instituto Argentino de Radioastronomía (IAR, CCT La Plata, CONICET; CICPBA), C.C.5, (1984) Villa Elisa, Buenos Aires, Argentina Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, 1900, La Plata, Argentina Joint Institute for VLBI ERIC, Postbus 2, 7990 AA Dwingeloo, The Netherlands Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany F. L. Vieyro [email protected] of a repeating fast radio burstVieyro, Romero, Bosch-Ramon, et al.Fast radio bursts, or FRBs, are transient sources of unknown origin. Recent radio and optical observations have provided strong evidence for an extragalactic origin of the phenomenon and the precise localization of the repeating FRB 121102. Observations using the Karl G. Jansky Very Large Array (VLA) and very-long-baseline interferometry (VLBI) have revealed the existence of a continuum non-thermal radio source consistent with the location of the bursts in a dwarf galaxy. All these new data rule out severalmodels that were previously proposed, and impose stringent constraints to new models. We aim to model FRB 121102 in light of the new observational results in the active galactic nucleus (AGN) scenario. We propose a model for repeating FRBs in which a non-steady relativistic e^±-beam, accelerated by an impulsive magnetohydrodynamic (MHD)-driven mechanism, interacts with a cloud at the centre of a star-forming dwarf galaxy. The interaction generates regions of high electrostatic field called cavitons in the plasma cloud. Turbulence is also produced in the beam. These processes, plus particle isotropization, the interaction scale, and light retardation effects, provide the necessary ingredients for short-lived, bright coherent radiation bursts. The mechanism studied in this work explains the general properties of FRB 121102, and may also be applied to other repetitive FRBs. Coherent emission from electrons and positrons accelerated in cavitons provides a plausible explanation of FRBs.A model for the repeating FRB 121102 in the AGN scenario F. L. Vieyro1,2, G. E. Romero2,3, V. Bosch-Ramon1, B. Marcote4, M. V. del Valle5 December 30, 2023 ====================================================================================§ INTRODUCTION Fast radio bursts (FRBs) are bright transient flashes of cosmic origin with durations of a few milliseconds detected at radio wavelengths. They were discovered by <cit.> and found to exhibit large dispersion measures (DM).These dispersions are in excess of the contribution expected from the electron distribution of our Galaxy, hence suggesting an extragalactic, cosmological origin. Most of the eighteen known bursts have been detected so far with the Parkes radio telescope <cit.>.Only a couple of them were found with the Arecibo and Green Bank telescopes <cit.>. The physical origin of FRBs remains a mystery. The putative extragalactic distances and the extremely rapid variability imply brightness temperatures largely beyond the Compton limit for incoherent synchrotron radiation <cit.>. Thus a coherent origin of the radiation seems certain. Models proposed so far can be divided into those of catastrophic nature, in which the source does not survive the production of the burst, and those that can repeat. A non-unique FRB population is possible, with different types of sources, as is the case with the gamma-ray bursts <cit.>. The unambiguous identification of the counterparts of FRBs at other wavelengths is a very difficult task because of the extremely short life span of the events at radio frequencies, their appearance from random directions in the sky, and the large uncertainties in the determination of their precise positions. A huge step towards the clarification of the origin and nature of these happenings was the recent direct localization of an FRBand its host by <cit.>. These authors achieved the sub-arc second localization of FRB 121102, the only known repeating FRB, using high-time-resolution radio interferometric observations that directly imaged the bursts. They found that FRB 121102 originates very close to a faint and persistent radio source with a continuum spectrum consistent with non-thermal emission and a faint optical counterpart. This latter optical source has been identified by <cit.> as a low-metallicity, star-forming, dwarf galaxy at a redshift of z = 0.19273(8), corresponding to a luminosity distance of 972 Mpc. Further insights on the persistent radio source were provided by <cit.> through very-long-baseline radio interferometric observations. Marcote et al. were able to simultaneously detect and localize both, the bursts and the persistent radio source, on milliarcsecond scales. The bursts are found to be consistent with the location of the persistent radio source within a projected linear separation of less than 40 pc, 12 mas angular separation, at 95% confidence, and thus both are likely related. The unambiguous association of FRB 121102 with persistent radio and optical counterparts, along with the identification of its host galaxy, impose, for the first time, very strict constraints upon theoretical models for FRBs, beyond the general limits imposed by variability timescales and energy budgets.<cit.> show that under certain conditions, a turbulent plasma hit by a relativistic jet can emit short bursts consistent with the ones observed in FRBs. In this paper we apply the latter model to FRB 121102, based on the idea that the multiple observed bursts are the result of coherent phenomena excited in turbulent plasma by the interaction of a sporadic relativistic e^±-beam or jet, which originates from a putative somewhat massive black hole in the central region of the observed dwarf galaxy and ambient material. Our model can account for the different properties known so far for FRB 121102 and can be tested through observations of other repetitive FRBs. In what follows we first detail the known features of FRB 121102 and its host that are relevant for the involved physics (Sect. <ref>), then we describe our model (Sect. <ref>) and its application to FRB 121102 (Sect. <ref>), and we finally offer some discussion and our conclusions (Sects. <ref> and <ref>). § MAIN FACTS ABOUT FRB 121102 Fast radio burst 121102 is the only known source of its class that presents repeated bursts with consistent DM and sky localization <cit.>. This implies that the source is not annihilated by the production mechanism of the bursts. Most of the individual bursts have peak flux densities in the range 0.02–0.3 Jy at 1.4 GHz. The wide range of flux densities seen at Arecibo, some near the detection threshold, suggests that weaker bursts are also produced, likely at a higher rate <cit.>. Although the bursts do not show any periodicity, they appear to cluster in time, with some observing sessions showing multiple bright bursts and others showing none.The European very-long-baseline interferometry network (EVN) observations performed by <cit.> were simultaneous with the detection of four new bursts. One of them, dubbed burst # 2, was an order of magnitude stronger than the others. The luminosity of this burst, at the estimated distance of the host galaxy observed by <cit.>, is of approximately 6×10^42 erg s^-1 at 1.7 GHz. Its associated brightness temperature is approximately 2.5 × 10^35 K, meaning more than twenty three orders of magnitude above the Compton limit, clearly indicating that the emission is coherent. The radio observations reported by <cit.> and <cit.> show a compact source with a persistent emission ofapproximately 180Jy at 1.7 GHz, implying a radio luminosity ofapproximately 3 ×10^38 erg s^-1, with a bandwidth of 128 MHz. No significant, short-term changes in the flux density occur after the arrival of the bursts. Its 1–10-GHz radio spectrum is flat, with an index of α=-0.20 ± 0.09, S_ν∝ν^α. The projected linear diameter of the persistent radio source is measured to be less than 0.7 pc at 5.0 GHz, and it is found to be spatially coincident with the FRB 121102 location within a projected distance of 40 pc. This kind of a close proximity strongly suggests that there is a direct physical link between the bursts and the persistent source. The observed properties of the persistent source cannot be explained by a stellar or intermediate black hole, either in a binary system or not, a regular supernova remnant (SNR), or a pulsar wind nebula such as the Crab <cit.>.There is a 5-σ X-ray upper limit in the 0.5-10 keV band for the radio source of L_X≤ 5.3× 10^41 erg s^-1 <cit.>. Hence, the ratio of the radio to the X-ray flux is log R_X > -2.4,consistent with those observed in low-luminosity AGNs <cit.>.The host galaxy of the bursts and the persistent radio source is a dwarf star-forming galaxy with a diameter of less than 4 kpc and a high star-forming rate of 0.4M_⊙ yr^-1. The stellar mass of the galaxy is estimated to be in the range 4–7×10^7 M_⊙. Current evidence supports the idea that the supermassive black hole-to-galaxy mass ratio lies within 0.01-0.05 <cit.>; there are few exceptions for which this ratio can be as high as 0.15 <cit.>. Therefore, the presence of a supermassive black hole with mass M_ BH >10^7M_⊙ in the host galaxy of is unlikely, because it would already have a mass larger than, or of the order of, the total stellar mass of the galaxy. This leads to an estimation of the mass of the putative black hole of M_ BH∼ 10^5–10^6M_⊙.§ MODEL AND EMISSION MECHANISM A model for FRB 121102 should be capable of accounting for the fast bursts, their plurality, and the compactness of the continuum radio source. In addition, in the context of an AGN, the model should account for the modest energy budget inferred from the moderate M_ BH-range allowed, and the stringent X-ray upper limit. A young supernova remnant powered by a strongly magnetized and rotating neutron star faces at least three major problems: the constant DM observed in the bursts from 2012 to 2016, the lack of change of the radio flux due to rapid expansion expected in all very young SNRs, and finally the absence of an X-ray counterpart that in the case of a pulsar-powered remnant is unavoidable<cit.>. We favour, instead, a model based on interactions between a relativistic (magnetized) e^±-jet launched by amoderately massive black hole with material in the centre of the host galaxy. §.§ Jet-cloudinteraction We assumed that the galaxy where FRB 121102 occurs hosts a low luminosity active galactic nucleus (AGN), as suggested by <cit.>. Accretion onto the central black hole results in the launching of a relativistic jet, which we assume takes place in an episodic fashion. This discontinuous jet may not be resolvable at radio frequencies; the outflow may become smooth on pc-scales, however, and be responsible for the observed persistent radio source of size <0.7 pc. In the jet innermost regions, on the other hand, the sporadic jet can interact with material accumulated on its way while jet activity was off. For example, at spatial scales ofapproximately 10^13 cm from a central black hole of M_ BH∼ 10^5M_⊙ (see below), clouds moving at approximatelyKeplerian velocities could fill the channel opened by a previous jet on day timescales. The interaction between an episodic ejection with a cloud can take place without a cloud penetration phase into the jet. In fact, avoiding this kind of a phase is required, because it would last much longer than the actual FRB duration for any reasonable parameter choice. Another condition for the interaction is that the cloud boundaries or edges must be sharp enough for a quick jet-cloud effective interaction. In principle, the thermal or the ram pressure of the environment confining the cloud can provide this sharpening. For the same reason, the jet-leading edge should be also sharp, whereas the magnetic field should be weak enough so as to avoid rapid e^±-beam isotropization. These two conditions can naturally occur in thescenario adopted here. A standard ejection mechanism in AGNs is the production of magnetized jets from accreting rotating black holes <cit.>, in which the innermost regions of jets consist of strongly magnetized, relativistic e^±-beams. No significant presence of protons is expected at the base of this kind of a jet, although barions are thought to be entrained farther out <cit.>.The different ejections of the intermittent jet should have a configuration akin to that of a relativistic ejection driven by an impulsive magnetohydrodynamic (MHD) mechanism, meaning a weakly magnetized thin leading shell moving with a high Lorentz factor driven by strong magnetic pressure gradients, followed by the strongly magnetized jet bulk <cit.>. This kind of impulsive MHD acceleration mechanism allows the ejecta to achieve higher bulk Lorentz factors, γ≥ 100, than steady outflows under similar conditions. Because the ejection leading shell is dominated by its bulk kinetic energy, the electrons and positrons (electrons hereafter) propagate following quasi-straight trajectories as a cold e^±-beam. Thus, the electron and beam Lorentz factors can be considered equal in the laboratory frame (LF), γ. When this ultra-relativistic jet-leading edge reaches the target cloud, electrons propagate in a straight line until electric and magnetic fields cause a significant deflection. This stage, in which electrons move in quasi-straight trajectories within the cloud, presents suitable conditions for highly beamed, strong coherent emission, as long as the particles' mean free path is not too short (see below). §.§ Caviton formation and coherent emission §.§.§ Beaming and light retardation effects The penetration of a relativistic e^±-beam into a denser target plasma results in the formation of concentrations of electrostatic plasma waves called cavitons. The electrons crossing this caviton-filled region produce the coherent emission (see Sec. <ref>). This emission is strongly beamed towards the observer if the line of sight coincides with the electron direction of motion. When an electron crosses a caviton, it emits a pulse of Bremsstrahlung-like radiation within a solid angle of approximately 1/γ^2 in its direction of motion, such as the observer direction of motion. This yields a beaming factor for the radiation of approximately γ^2. When particle deflection is included, however, the beaming factor towards the initial electron direction gets reduced. For instance, assuming an uniform magnetic field B, the average beaming factor along a distance covered by the electron equal to its gyro-radius (r_ g=γ m_ec^2/qB) yields a factor approximately γ instead of γ^2. It should be noted thatr_ g is the electron mean free path in the case of an uniform B-field. In addition to relativistic geometric beaming, light retardation effects also strongly enhance the radiation luminosity in the electron direction of motion, because the apparent time in which electrons radiate is shortened by a factor of 1/2γ^2 with respect to the LF. In our scenario, this factor does not need to be averaged along the electron trajectory. The reason for this is that most of the radiation towards the observer is actually produced before the electron is deflected by an angle ≳ 1/γ[For this very same reason, the effective beaming factor is ∼γ^2 rather than ∼γ, for source statistics purposes.].Both effects, the averaged beaming and the light retardation, lead to an enhancement of the apparent luminosity when looking at the beam on axis by a factor approximately γ^3 compared to the LF luminosity. It should be noted that the factor γ^3 is actually a lower limit. This is due to the assumption of an uniform B-field, which produces the electron strongest possible deflection. In the case of negligible electron deflection, the apparent luminosity would be enhanced by δ_ D^4 ≈ 16γ^4, where δ_ D is the Doppler factor. This is because the beaming factor would be constant along the straight electron trajectory, and of the order ofγ^2. §.§.§ The coherent emission mechanism Collective effects lead to coherent radiation, enhancing the emitted power <cit.>. For a uniform jet, the emission between any two electrons scattered by a caviton will not present any phase coherence. If, on the contrary, the jet presents density fluctuations that are correlated, then the radiation can be coherent, and therefore strongly enhanced. This correlation in the density fluctuations is the result of turbulence generated by the coupling between the background plasma and the beam: electrons from the beam perturb the plasma, producing the two-stream instability, then cavitons form, and beam-electron bunching is also generated <cit.>. Turbulence development should not significantly affect the cold nature of the beam as long as the turbulence-associated electron velocities do not themselves become relativistic in the flow frame. Caviton formation takes place on very short timescales,≪ 1 ms in the LF <cit.>, and the temperature of the target plasma is expected to rise similarly fast. Thus the mechanism could work even for the very short timescales of FRBs.§.§.§ Electron deflection and flow isotropization The residual magnetic field dragged along by the jet-leading thin shell eventually deflects the energetic electrons propagating through the target plasma. The interpenetration of the jet-leading thin shell with the target cloud leads to the formation of a contact layer between both, the region in which the coherent emission is produced, in which electrons tend to isotropize. This layer is approximately at rest in the LF, and a strong enhancement of the perpendicular B-component (B_⊥) dragged by the beam is expected there. For this reason, it seems natural to assume that the mean free path of the electrons in the interpenetration region will be approximately r_ g, with B∼ B_⊥. On the other hand, the electrostatic field E_0 inside the cavitons, expected to be randomly oriented, will induce pitch-angle diffusion to the emitting electrons. The value of E_0 cannot be too high, because the mean free path of electrons should be larger than the caviton size, meaning r_ g>D, if the mechanism is to work. More precisely, r_ g≫ D if pitch-angle diffusion is realized. For the same reason, the magnetic field in the beam cannot be too high, and a similar constraint applies to the cloud magnetic field, the geometry of which may be between a strongly ordered B-field and the chaotically oriented E_0-field. It is also worth noting that, even if the electron mean free path in the interpenetration region is >D, it should also belong enough as to ensure that the emitting volume is sufficiently large to produce the observed fluxes.The time required for particle isotropization in the beam is δ t_ iso≳ r_ g/c. After this time, the perturbation originated by the jet-cloud interaction can affect the incoming electrons even before it reaches the cloud, isotropizing them and the electromagnetic fields they carry. At this stage, the isotropized electrons may still interact with cavitons around the jet-cloud contact discontinuity through some level of jet-cloud interpenetration. However, the related coherent emissivity is strongly reduced by a factor of ≳ 1/γ^3 (see Sect. <ref>). Moreover, as the bulk of the jet, which is more magnetized than its leading edge, reaches the cloud, the expected higher magnetic field filling the region can stop the coherent radiation completely by making the electron mean free paths smaller than the cavitons. We conclude then that δ t_ iso determines the duration of the coherent emission phase for electrons from the jet-leading thin shell interacting simultaneously with the cloud, such that δ t_ obs∼ 1ms≳δ t_ iso. §.§.§ Event duration Despite δ t_ iso playing a major role in the FRB scenario proposed here, it seems likely that the cloud will present an irregular surface. Assuming that Δ r is the relevant irregularity scale of the target cloud, the ultra-relativistic thin shell must cover a distance Δ r in a time δ t_ cross≈Δ r/c in the LF for full interaction. Unless B is extremely low, it will be the case in our scenario that δ t_ iso≪ 1 ms, in which case Δ r could determine the event duration. This implies that, for the observer, δ t_ obs≈δ t_ cross/2γ^2≈Δ r/2cγ^2≲ r_ j/2cγ^2, where Δ r≈ r_ j is the largest effective irregularity scale, and light retardation effects have been taken into account, meaning: δ t_ obs≈δ t_ cross/2γ^2.Therefore, unless the magnetic field is very low, it is the case that δ t_ iso/c≪ 1 ms, and the duration of the burst will be determined by the crossing time of the irregularity scale corrected by light propagation effects. The dynamical timescale of the intermittent ejections does not affect the duration of the burst if it is ≳ 1 ms, which is expected given that the light-crossing time of the central black hole is approximately 1 (M_ BH/10^5 M_⊙) s. Even if the jet ejection lasts for ≫ 1 ms, the isotropization of the beam particles will stop, or at least strongly reduce, the coherent emission.§.§ Radiation properties The spectrum of the coherent emission presents two main components, a line at the plasma frequency ω_e,and a broad-band tail that inherits the power-law behaviour of the density fluctuation spectrum of the e^±-beam. The existence of this type of emission is well known from controlled plasma experiments <cit.>. The broad-band component of the spectrum extends from the plasma frequencyto ω_max = 2 γ^2 c/D, which is the highest frequency emitted by electrons with Lorentz factor γ being scattered by cavitons of size D. The size D of thecavitons induced in the plasma by the relativistic e^±-beam can be estimated as approximately 15 λ_ D <cit.>, whereλ_ D is the Debye length of the plasma, such that λ_ D= 6.9√(T_ c/n_ c) cm, with [T_ c] = K and [n_ c] = cm^-3.For a e^±-beam interacting with a denser target plasma, the required condition for collective emission is q = n_ b/n_ c≥ 0.01, with n_ b and n_ c being the beam and target densities, respectively. The radiated power per electron in the LF in the form of coherent emission is given by <cit.>:P_e =E_0^2 σ_T c/8 π4 n_bπ D^3/327 π/4 f [1 + ( Δ n_b/n_b)^2 0.24 ln( 2 γ^2 c/D ω_p)],where σ_T is the Thomson cross-section, Δ n_b/n_b is the fluctuation-to-mean-density ratio of the relativistic electrons, and f is the fraction of the cloud volume that is filled with cavitons, for which we adopted f=0.1 <cit.>. The second factor in Eq. (<ref>), 4 n_bπ D^3/3, is the number of electrons inside a caviton, and it is known as the coherence factor <cit.>. The first term in Eq. (<ref>) represents the power emitted at the plasma frequency, whereas the second term is the power emitted in the tail of the spectrum. Emission at the plasma frequency is likely to be absorbed <cit.>, hence the relevant term is the one associated with the broad-band emission. It is natural for adensity fluctuation spectrum with power-law behaviour to arise as the result of turbulence in the plasma, for example a Kolmogorov spectrum has an index α = -5/3. Density fluctuations show a large range of variation, and can reach values of order unity, meaningΔ n_b/n_b∼ 1 <cit.>.For these kinds of density fluctuations, the radiation spectrum behaves as ν P_ν∝ν^α+1, and the power emitted in the broad-band component is comparable to the power emitted at the line, with most of the former being radiated in the low-frequency part of the power-law spectrum. Thus, although the radiation produced at the plasma frequency may be easily absorbed by the emitting plasma, the radiation at slightly higher frequencies has a comparable luminosity. The total power P_ t is simply the power emitted per particle times the number of relativistic electrons inside the region filled with cavitons.§ APPLICATION TO FRB 121102 In the following, we consider a cloud with an irregular surface with irregularity scale Δ r≈ r_ j and total jet-interaction section π r_ j^2 (see Fig. <ref>). As explained in Sect. <ref>, Δ r and, consequently, r_ j are constrained by the duration of the event:Δ r ≈ r_ j≈ 6 × 10^11( γ/100)^2 cm . Assuming a half-opening angle of a few degrees, this kind of an r_ j-value would correspond to approximately 10^3 R_ G∼ 10^13 cm for a 10^5M_⊙ black hole, where R_ G=GM/c^2.The coherent emission is broad-band, extending from the plasma frequency ω_e up to ω_max = 2 γ^2 c/D. The observed frequency of 1.7 GHz should be within the emission range ω_e/2π < 1.7GHz <ω_max/2π. The first condition, ω_e /2π < 1.7 GHz, results in n_ c≤ 4 × 10^10 cm^-3, where the subscript c indicates the region where cavitons form. Because the emission radiated at the plasma frequency will likely be absorbed, we adopted a lower value of the plasma density, n_ c = 10^10 cm^-3. With this choice, the peak of the emission is at ν = 900 MHz. If one adopts a Kolmogorov spectrum for the turbulence, ν P_ν∝ν^-2/3, the luminosity at the observed frequency will be approximately 70% of the luminosity of the peak. It is worth mentioning that calculations in the weak turbulence regime suggest that ambient plasma might also, in principle, produce an attenuation of the coherent signal by Raman scattering <cit.>. However, experiments show that these effects are suppressed in the strong turbulence case, where there is no theory available <cit.>. For the coherent emission to escape, free-free absorption should be also minor within the cloud. Adopting 6× 10^11 cm for the cloud size and n_ c= 10^10 cm^-3, the cloud is optically thin as long as its (pre-interaction) temperature is ≳ 10^8 K; in principle this is possible because the virial temperature at 10^3 R_ G is approximately 10^10 K. The second condition, 1.7 GHz < ω_max /2π, results in:γ^2 √( T_ c /n_ c)≥ 73.7 ,with [T_ c] = K and [n_ c] = cm^-3. The impact of the jet can heat the cloud up to a temperature of T_ c = q m_e c^2 γ/k_ B. For the adopted n_ c-value, and a density ratio of q = 1, T_ c = 6 × 10^11 K; this T_ c-value together with the adopted one for n_ c fulfils Eq. (<ref>). Cavitons formed in a plasma with these characteristics have a size of D ∼ 800 cm; the number of electrons inside cavitons results in 2 × 10^19. The obtained value for D is similar to the values found in particle-in-cell (PIC) numerical simulations <cit.>. It is worth recalling that coherent emission is only possible if r_ g is larger than the size of cavitons, as discussed in Sect. <ref>.The jet power can be obtained from the jet particle density in the LF at the interaction location, n_ j, through the relation:γ n_ j m_e c^2≈L_ j/π r_ j^2 c .Using the radius jet constrained by the event duration given by Eq. <ref>, Eq. (<ref>) yields a jet power L_ j∼ 3 × 10^40 erg s^-1.When electrons enter the cloud, they strongly radiate towards the observer until they isotropize, which occurs on a time δ t_ iso≳ r_ g/c∼ 6× 10^-8 γ/B s. Assuming that the magnetic energy density is a fraction ξ of the jet kinetic energy density, the magnetic field in the LF can be obtained from:B^2_ j,eq/8 π = 1/2ξ L_ j/π r_ j^2.At the interaction location, this yields an equipartition field (i.e. ξ =1) of B_ j,eq∼ 3200 G. We adopted a magnetic field well below equipartition (ξ< 1) in the interacting shell, as explained in Sect. <ref>; in particular, we consider B = 5 × 10^-2B_ j,eq.To obtain E_0 in Eq. (<ref>), we imposed the condition that the pitch-angle diffusion timescale, given byt_ diff = λ_E_0^2/D c,with λ_E_0 = γ m_e c^2/eE_0, should be longer than δ t_ iso = r_ g/c, as discussed in Sect. <ref>). This results in E_0 ∼ 45 G, yielding W = E_0^2/8 π n_ c k_ BT_c∼ 10^-4, which is within the range values obtained in numerical simulations <cit.>.Typically, δ t_ iso<< δ t_ cross, and therefore not all electrons will radiate simultaneously. By the time the jet-leading edge has crossed all of the cloud irregular surface, the electrons that interacted first are already isotropized and their coherent emission has been suppressed. To derive the observer luminosity, the LF luminosity (Eq. <ref>) must be corrected by a factor of δ t_ iso/δ t_ cross to account for a smaller simultaneously emitting volume. In addition, to account for light retardation effects, another factor t_ cross/t_ obs must be considered. All this renders a factorofδ t_ iso/δ t_ obs that for the adopted values of Δ r and B is ≳ 3× 10^-5. In addition tot_ iso/t_ obs, Doppler boosting must be considered to obtain the observer luminosity, as explained in Sect. <ref>; this results in:L_ obs≈γ(P_e N_e ) δ t_ iso/δ t_ obs , where N_e is the number of electrons in the volume of the emitting region, meaning approximately π r_ j^2 r_ g. Using the adopted parameter values, the predicted FRB observer luminosity is approximately 3 × 10^42 erg s^-1, comparable to the luminosity of burst #2 of FRB 121102. The values of the parameters adopted above are just one of several possible choices, because different combinations of plausible values can also explain burst #2 and other bursts with different properties.We note that for extremely low B-values, Δ r<r_ g, in which case δ t_ iso and not δ t_ cross would determine δ t_ obs. This would correspond to a plane jet-cloud interface, in which case no weighting by δ t_ iso/δ t_ obs should be applied. Another possibility that cannot be ruled out is that the transition region between the unisotropized, meaning unshocked, and the isotropized, meaning shocked, beam could be the coherent emitter<cit.>. However, the magnetic field in that location might be too strong if the bulk of the jet is highly magnetized. In addition, the duration of the FRB in this scenario would have to be determined by some ad hoc mechanism. §.§ Alternative scenario: a lighthouse effect An alternative to the scenario presented here, in which the FRBs occur at the onset of a jet's ejections, is that of a jet changing its direction <cit.>. Occasionally, this wandering jetwould intercept a cloud[In the model discussed in <cit.> the jet does not necessarily intercept a cloud; in this case, the jet might sweep through the existing medium or the radiation might be produced internally in the jet.] whilepointing to the observed and then producing an FRB. In that case, the timescale of the event will be the time needed by the jet to change direction by 1/ γ rad while it is interacting with the cloud. This situation resembles that of a lighthouse, in which the event duration is not affected by causality constraints.A lighthouse effect combined with coherent emission from jet-target interactions cannot in principle be discarded, and dispenses us with the need to assume a discontinuous jet with a sharp leading edge interacting with an irregular target. On the other hand, a modest magnetic field and a high Lorentz factor are still required. In fact, most of the details of the model given in Sect. <ref> still hold in this alternative scenario, but now the crossing and the observer timescales are the same. Nevertheless, there is drawback in the lighthouse scenario. The change in direction by 1/ γ rad of the emitting electrons in just 1 ms requires that the properties of the leading thin shell substantially change along the jet axis on a spatial scaleof approximately 1ms· c∼ 3× 10^7 cm, which is ≪ r_ j. This kind of a jet configuration is in principle possible but requires the flow to be very cold, meaning a Mach number ≫ 1/γ. Otherwise, electrons will tend to homogenize their properties on these small scales, and the very short-scale jet-bending coherence will be lost. In addition, the fast changes in direction require angular velocities of approximately 10 (100/ γ) rad s^-1, which may not be feasible for a jet-launching engine that has already a size ≳ 1 (M_ BH/10^5 M_⊙) light-second. The constraints mentioned can be relaxed for smaller central engines and thus smaller black hole masses, although smaller black holes imply tighter energy limits. If the source were super-Eddington, for example similar to SS 433, the energetics might fulfil the minimum requirements <cit.>, but this kind of a scenario requires a dedicated study.§ DISCUSSION We propose that FRB 121102 and similar events are the result of coherent radio emission produced by a relativistic, turbulent e^±-beam interacting with plasma cavitons. An advantage of this mechanism is that it might operate in different scenarios involving relativistic jets. For instance,<cit.> discuss possible settings involving long gamma-ray bursts and mini-jets produced by magnetic reconnection inside a larger outflow. Even single-event FRB may be explained in the basic framework of the proposed model as long as the recurrence time of the events is very long, depending on sensible factors such as beam orientation, Lorentz factor, propagation length within the plasma, and the interaction scale. Here, we investigate the mechanism in the context of an extragalactic episodic jet interacting with the environment, to check whether the model can account for FRB 121102 in light ofnew observational evidence. In what follows we further comment on a number of important assumptions of the model. §.§ Sporadic ejections The proposed scenario requires episodic jet launching. Episodic ejections are known to take place in several astrophysical sources. The hydrogen ionization instability is responsible for switches between periods of outburst and quiescence in dwarf novae. The state transition observed in numerous X-ray binaries is also proof of a variable accretion regime <cit.>. In fact, multiple variability timescales are common for the radiation associated with galactic and extragalactic jets. Non-steady jet production may be behind this variability and therefore render it a somewhat natural phenomenon, at least at the relatively small spatial scales relevant in our scenario, meaning approximately 10^3 R_ G. At larger scales this sporadic jet activity does not affect the persistent radio source. §.§ High jet Lorentz factors As indicated, strongly magnetized sporadic ejections can be efficiently accelerated by a MHD-driven impulsive mechanism to high bulk Lorentz factors such as γ≥ 100 <cit.>. Non-stationary magnetized outflows have also been proposed to explain the apparent disagreement between the typical AGN Lorentz factors inferred from radio data, 5 ≲γ≲ 40 <cit.>, and the higher Lorentz factors invoked to explain the rapid TeV-variability observed in blazars <cit.>. In this context, the short timescale flares observed at TeV energies would be associated with variable emission from these shells, whereas the radio data would be associated with the emission from a larger scale, smoother flow with a lower Lorentz factor <cit.>. §.§ Cloud origin Clouds from the AGN broad-line region (BLR) present densities of 10^10-10^11 cm^-3 at distances of 10^3-10^4 R_ G to the central black hole <cit.>.The presence of material for jet interaction might be also related to the accretion phenomenon itself <cit.>. §.§ Other observational aspects§.§.§ The persistent radio sourceIn our model, the observed continuum flat-spectrum radio source would correspond to the synchrotron radiation ofthejet, which is the result of the averaged intermittent ejections. The synchrotron luminosity expected at radio wavelengthsfor a jet with L_ j=3 × 10^40 erg s^-1 can be roughly estimated as <cit.>: L(∼ 1.7GHz)≈η_ NTL_ jδ_ D^4/γ^2t_ esc/t_ syn, where η_ NT is the non-thermal-to-total energy density ratio in the emitter, δ_ D is the Doppler factor, and t_ esc and t_ syn are the electron escape and cooling times, respectively. As discussed in Sect. <ref>, for a highly relativistic jet pointing towards the observer, δ_ D∼ 2γ. The jet magnetic field can be estimated assuming again a certain value for the equipartition fraction ξ, not necessarily the same as in the FRB-emitting region. From all this, plus adopting a jet distance to the black hole of z ∼ 1 pc and a jet half-opening angle of 1/γ=0.1 (10/γ) rad, one obtains: L(∼ 1.7GHz)≈ 6 × 10^39erg s^-1(η_ NT/10^-1) (γ/10)^2 (ξ/10^-1)^3/4. This luminosity is well above the persistent radio luminosity mentioned in Sect. <ref>, indicating that the scenario under typical assumptions is consistent from the energetic point of view even when considering duty cycles of jet activity ≲ 10%. It is worth noting that the jet luminosity obtained in Sect. <ref> is also compatible with the X-ray upper limit.§.§.§ Black hole mass The stellar mass of the galaxy is in the range 4–7×10^7 M_⊙ <cit.>. Little is known about the existence of massive black holes in these kinds of small galaxies. If we extrapolate from the scaling relation given by <cit.>, determined in the range 10^8 ≤ M_ stellar/ M_⊙≤ 10^12, we obtain a black hole mass of approximately10^5 M_⊙, within the allowed range (see Sect. <ref>). Although most of the estimations of the masses for black holes in the centre of galaxies are above 10^6 M_⊙, there is evidence of the presence of black holes with masses ofapproximately 10^5 M_⊙ in some AGNs <cit.>. If this kind of a black hole accretes at 1% of the Eddington rate, its luminosity wouldbe approximately 10^41 erg s^-1, a value comparable to the one adopted in our model, and of the order of the X-ray upper limit.§.§.§ Polarization FRBs present an additional challenge concerning polarization. There is no evidence of polarized emission from the repeater <cit.>; however, FRB 150807 presented linear polarization <cit.>, whereas a high degree of circular polarization was measured inFRB 140514 <cit.>. The radiation mechanism proposed in this work might produce linear polarization in the presence of a magnetic field, whereas intrinsic circular polarization is not expected <cit.>.§.§.§ High-energy emission An analysis of the multi-wavelength non-thermal emission associated with the mechanism discussed for FRB 121102 is under way and will be presented elsewhere. We can put forward however a general framework in which the cloud impacted by the jet and the shock in the jet itself may lead to efficient particle acceleration, and to high-energy emission that could be detectable very briefly, seconds to minutes, if the beam is fast enough, even for modest energetic budgets<cit.>. § CONCLUSIONS The model proposed in this work, based on a mechanism of coherent emission in beam-excited plasma cavitons, is able to explain the diverse properties of FRBs: the extragalactic origin, the energy budget, the high brightness temperature, the repetitions with no apparent periodicity, and the counterparts and upper flux limits obtained in different wavelengths. The very short duration of the events is explained by the dynamical timescale of the process corrected by light retardation effects, although the isotropization timescale of the beam particles plays also an important role, and may determine the event duration for very low magnetic fields.§ ACKNOWLEDGMENTS The authors are grateful to Jordi Miralda-Escudé for his very useful comments and suggestions. This work was supported by the Argentine Agency CONICET (PIP 2014-00338) and the Spanish Ministerio de Economía y Competitividad (MINECO/FEDER, UE) under grants AYA2013-47447-C3-1-P and AYA2016-76012-C3-1-P with partial support by the European Regional Development Fund (ERDF/FEDER), MDM-2014-0369 of ICCUB (Unidad de Excelencia `María de Maeztu'), and the Catalan DEC grant 2014 SGR 86. V.B.R. also acknowledges financial support from MINECO and European Social Funds through a Ramón y Cajal fellowship. This research has been supported by the Marie Curie Career Integration Grant 321520. M.V.d.V acknowledges support from the Alexander von Humboldt Foundation. aa | http://arxiv.org/abs/1704.08097v1 | {
"authors": [
"Florencia L. Vieyro",
"Gustavo E. Romero",
"Valentí Bosch-Ramon",
"Benito Marcote",
"María V. del Valle"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170426132441",
"title": "A model for the repeating FRB 121102 in the AGN scenario"
} |
1Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599, USA 2Hubble Fellow 3Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA 4Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil 5Department of Physics, University of Warwick, Coventry CV4 7AL, UK 6Department of Astronomy, University of Texas at Austin, Austin, TX 78712, USA 7Wesleyan University Astronomy Department, Van Vleck Observatory, 96 Foss Hill Drive, Middletown, CT 06459, USA [email protected] patterns in the oscillation frequencies of a white dwarf observed by K2, we have measured the fastest rotation rate (1.13±0.02 hr) of any isolated pulsating white dwarf known to date. Balmer-line fits to follow-up spectroscopy from the SOAR telescope show that the star (SDSSJ0837+1856, EPIC 211914185) is a 13,590±340 K, 0.87±0.03white dwarf. This is the highest mass measured for any pulsating white dwarf with known rotation, suggesting a possible link between high mass and fast rotation. If it is the product of single-star evolution, its progenitor was a roughly 4.0main-sequence B star; we know very little about the angular momentum evolution of such intermediate-mass stars. We explore the possibility that this rapidly rotating white dwarf is the byproduct of a binary merger, which we conclude is unlikely given the pulsation periods observed. § INTRODUCTIONThanks to long-baseline monitoring enabled by space missions like CoRoT and Kepler, we now have deep insight into the angular momentum evolution of low-mass stars <cit.>. Asteroseismology enables measuring core and surface rotation rates for numerous 1-3stars along the main sequence (e.g.,and references therein) and along their first ascent up the red giant branch (e.g., ), as well as in core-helium-burning, secondary clump giants <cit.>.As powerful as the Kepler seismology has been, it has so far determined internal rotation rates for just a few intermediate-mass stars (3 < M < 8 ) on the main sequence (e.g.,and references therein). Therefore we have few constraints on the past or future evolution of angular momentum in Cepheids; rotation can have a significant evolutionary impact on these standard candles <cit.>.As with all low-mass stars, intermediate-mass stars below roughly 8will end their lives as white dwarfs. We can therefore constrain the final stages of angular momentum evolution of intermediate-mass stars by observing white dwarfs. The majority of field white dwarfs have an overall mass narrowly clustered around 0.62 , as determined by fits to the pressure-broadened Balmer lines of hydrogen-atmosphere (DA) white dwarfs <cit.>. Initial-final mass relations calibrated using white dwarfs in clusters suggest that 0.62white dwarfs evolved from roughly 2.2main-sequence progenitors (e.g., ).Currently known white dwarf rotation rates lead to expected rotational broadening well below currently measured upper limits, usually vsini<10<cit.>. Thus, our best insights into the rotation rates of white dwarfs evolving without binary influence come from asteroseismology. To date, rotation of roughly 20 white dwarfs has been measured from their pulsations, with rotation periods spanning 0.4-2.2 d <cit.>. All but two have masses less than 0.73 , suggesting they generally represent the endpoints of <3.0progenitors.Thanks to its tour of many new fields along the ecliptic, the second-life of the Kepler space telescope, K2, is rapidly increasing the number of white dwarfs with nearly uninterrupted, multi-month light curves suitable to measure interior rotation rates in these stellar remnants (e.g., ). We present here the discovery of pulsations in a massive white dwarf — likely the descendent of a roughly 4.0main-sequence star — which we find to be the most rapidly rotating pulsating white dwarf known to date.§ PULSATION PERIODS FROM K2We targeted the star SDSSJ083702.16+185613.4 (hereafter , EPIC 211914185) during K2 Campaign 5 as part of a search for transits around white dwarfs using the shortest-cadence observations possible (program GO5073).was not observed as part of our GO program to search for pulsations in white dwarfs, as we believed it too faint (K_p=18.9 mag). It was selected as a candidate white dwarf from the photometric catalog of <cit.>, based on its blue colors and relatively high proper motion.We produced an extracted light curve from the processed target pixel file using the PyKE software package <cit.>, with a fixed aperture of 5 pixels. We removed K2 motion-induced noise with the kepsff task <cit.>. Our final 74.84 d light curve has 108,454 points and a duty cycle of nearly 98.7%, after iteratively clipping all points falling >5σ from the mean with the VARTOOLS software package <cit.>. All phases in our light curve are relative to the mid-time of the first exposure: 2457139.6008052 BJD_ TDB.We display in Figure <ref> a Fourier transform (FT) of our K2 observations, extending past the Nyquist frequency near 8496 , based on our 58.85 s sampling rate. Our FTs have all been oversampled by a factor of 20; all instrumental harmonics of the long-cadence sampling rate (see ) have been fit and removed. We show the confidence threshold of the FT determined by simulating 10,000 light curves, wherein we kept the time sampling but randomly shuffled the fluxes, as outlined in <cit.>. 99.0% of these synthetic FTs do not have a peak exceeding 0.92 ppt (where 1 ppt =0.1%), which we adopt as our 99% confidence threshold (shown as a red dotted line in Figure <ref>). This is close to five times the average amplitude of the entire FT: 5⟨ A⟩=0.98 ppt.Notably, two significant peaks in the FT fall within 800of the Nyquist frequency; in fact, both peaks mirrored above the Nyquist frequency have more than 5% higher amplitudes than their sub-Nyquist counterparts. To resolve this Nyquist ambiguity we obtained follow-up, time-series photometry at higher cadence over four nights in 2017 January, from both the 2.1-m Otto Struve telescope at McDonald Observatory in West Texas and the 4.1-m SOAR telescope at Cerro Pachón in Chile. All observations were obtained using blue-broadband, red-cutoff filters.Our 2.1-m McDonald data were taken over three nights with the frame-transfer ProEM camera, using 10 s exposures taken through a 3mm BG40 filter: on 2017 January 23 (4.8 hr long, variable ∼1.6 seeing, thin clouds), 2017 January 24 (2.4 hr long, variable ∼2.0 seeing, clear skies), and 2017 January 26 (2.0 hr long, variable ∼2.3 seeing, clear skies). In addition, we obtained 2.7 hr of time-series photometry on 2017 January 26 (stable ∼1.8 seeing, clear skies) on the 4.1-m SOAR telescope using the Goodman spectrograph in imaging mode, using 17 s exposures through an S8612 filter. We began collecting our SOAR light curve 2.6 hr before the McDonald data that night, giving us a two-site duty cycle of more than 16% over 72 hr.Light curves of the ground-based photometry were extracted with circular aperture photometry and corrected to the Solar System barycenter using WQED <cit.>. An FT of the ground-based data is shown in cyan in Figure <ref>, including the 99% confidence threshold of 3.6 ppt, calculated in the same way as the K2 data. The ground-based data have two significant peaks: one at 5116.53±0.30(4.7±0.7 ppt) and another at 9177.56±0.38(3.7±0.7 ppt), confirming that the super-Nyquist signals from the K2 data are in fact those in the star. In both cases, daytime observing gaps have conspired to raise an alias peak to the highest peak in the ground-based dataset (the frequency uncertainties quoted are not appropriate estimates of the actual uncertainties due to the presence of aliases).We note that the amplitudes of the ground-based data are at least 20% higher than the K2 amplitudes. This is partly a result of different limb-darkening in our bluer ground-based filters than in the Kepler bandpass <cit.>, as well as flux dilution from a nearby (<4.5, ΔK_p∼2.8 mag) galaxy. More significantly, we expect phase smearing from the 58.85 s exposures to suppress the K2 amplitudes around f_1 by more than 14% and f_2 by more than 40%. lrrcr 5 0.45 Frequencies present inID Frequency Period Amplitude Phase(μHz) (s) (ppt) (rad/2π)f_1a 5112.5995(41) 195.59522 3.23 0.3312(77)f_1b 5250.6035(72) 190.45430 1.85 0.837(13)5rf_1b-f_1a=138.004(11)f_1c 5389.852(13) 185.53384 1.03 0.832(24)5rf_1c-f_1b=139.249(20)f_2a 9037.205(17) 110.65368 0.76 0.921(32)f_2b 9161.6178(94) 109.15103 1.40 0.704(18)5rf_2b-f_2a=124.413(27)f_2c 9286.287(10) 107.68566 1.37 0.136(18)5rf_2c-f_2b=124.669(19) Informed by our higher-cadence photometry from McDonald and SOAR, we display in Table <ref> all six pulsation frequencies that we detect in , marking in bold the m=0 components. We include one mode — f_2a — for which we have slightly relaxed the significance threshold, since it falls where we would expect for a component of a rotationally split multiplet (see Section <ref>); our synthetic FTs estimate a 13% confidence in f_2a. The values in Table <ref> have been computed with a simultaneous non-linear least squares fit for the frequency, amplitude, and phase to the K2 data using PERIOD04 <cit.>. The amplitudes have a formal uncertainty of 0.16 ppt. Our period determination is not in the stellar rest frame, but is at high enough precision that it should be corrected for the gravitational redshift and line-of-sight motion of the white dwarf (e.g., ).§ MASS DETERMINATION FROM SPECTROSCOPYWith just two independent pulsation modes at 109.15103 s and 190.45430 s, we have only limited asteroseismic constraint on the properties of . Therefore, we obtained low-resolution spectra of as many Balmer lines as possible using the Goodman spectrograph on the 4.1-m SOAR telescope <cit.>. Our setup covers the wavelength range 3600-5200 Å with a dispersion of roughly 0.8 Å pixel^-1. We used a 3 slit, so our spectral resolution is seeing limited, roughly 4 Å in 1.4 seeing.We obtained spectra ofover two nights with SOAR. On the night of 2016 February 14 we obtained consecutive 2×600 s exposures in 1.4 seeing at an airmass of 1.8, giving us a signal-to-noise ratio (S/N) of 26 per resolution element in the continuum around 4600 Å. Subsequently, we obtained 9×600 s exposures on the night of 2017 January 25 in 2.0 seeing at an airmass of 1.4, yielding S/N =61.We optimally extracted all spectra <cit.> with the pamela software package. We subsequently used molly <cit.> to wavelength calibrate and apply a heliocentric correction. We flux calibrated the 2016 February spectra with the spectrophotometric standard GD 71 and the 2017 January spectra with Feige 67.We fit the six Balmer lines Hβ-H9 for each epoch of spectroscopy to pure-hydrogen, 1D model atmospheres for white dwarfs that employ the ML2/α = 0.8 prescription of the mixing-length theory; the models and fitting procedures are described in <cit.> and were convolved to match the resolution set by the seeing. For both epochs we find atmospheric parameters indicating a relatively hot and massive white dwarf: For 2016 February 14 we find 1D parameters of=13,010±370 K,=8.503±0.072, and for 2017 January 25 we find=14,020±300 K,=8.412±0.044. We display the Balmer-line fits in Figure <ref>.A weighted mean of both epochs yields 1D atmospheric parameters of=13,620±340 K,=8.437±0.052. We can correct these for the three-dimensional dependence of convection <cit.>, which slightly modifies the temperature to 13,590 K and surface gravity to=8.434.The 3D-corrected parameters ofcorrespond to a white dwarf mass of 0.87±0.03using the models of <cit.>. If it evolved in isolation,would have descended from a roughly 4.0±0.5B-star progenitor <cit.>.§ ASTEROSEISMIC MEASUREMENT OF ROTATIONWhite dwarfs oscillate in non-radial g-modes. In the absence of rotation, pulsations with the same angular degree, ℓ, and radial overtone, n, have the same frequency, independent of the azimuthal order, m. Rotation can lift this m degeneracy and decompose a mode into 2ℓ+1 components <cit.>. Due to geometric cancellation of high-ℓ modes, we most commonly observe dipole ℓ=1 modes in white dwarfs, which separate into triplets of m=-1,0,1 components.Thanks to the nearly unblinking, 74-d stare of K2, we see clearly that the two main modes ofare each composed of nearly symmetrically split triplets, shown in detail in Figure <ref>. We can use the frequency splittings within these modes to estimate the rotation rate of this massive white dwarf. The weighted mean of the frequency splittings for each mode are δf_1=138.626±0.031and δf_2=124.541±0.046 .To first order, we can connect an identified frequency splitting (δf) to the overall stellar rotation (Ω) by the relation δ f=m(1-C_n,ℓ)Ω, where C_n,ℓ represents the effect of the Coriolis force on the pulsations as formulated by <cit.>. In white dwarfs, C_n,ℓ is usually close to an asymptotic value of 1/ℓ(ℓ+1) = 0.5 for ℓ=1 (e.g., ). Some modes of lower radial order (especially n<5) are strongly affected by abrupt chemical transitions in the layering of the white dwarf, which effectively trap modes to different depths of the star. This trapping causes C_n,ℓ to deviate below the asymptotic value, and is likely why the mean frequency splittings for δf_1 and δf_2 indiffer by more than 14(10%).With just two independent modes, our models are only weakly constrained by asteroseismology. However, guided by our spectroscopically determined effective temperature and overall mass, we have used the evolutionary sequences described in <cit.> to explore model-dependent values for C_n,ℓ. We identify f_1 at 109.151 s as an ℓ=1,n=1 mode and f_2 at 190.454 s as an ℓ=1,n=2 mode. The 13,590 K, 0.87model with a canonically thick hydrogen-layer mass computed in <cit.> predicts C_1,1=0.495 and C_2,1=0.438 (with mode periods of 98.05 s and 170.78 s, respectively). From these model-based C_n,ℓ values, each triplet independently yields a rotation period of exactly 1.13 hr, from both δf_1 and δf_2.Our uncertainties on the rotation rate are dominated by the model uncertainties in computing C_n,ℓ, which consistently predict the ℓ=1,n=2 mode is highly trapped. The most deviant model from <cit.> within our spectroscopic uncertainties (13,590 K, 0.85 ) predicts C_1,1=0.495 and C_2,1=0.428, yielding a rotation period of 1.15 hr using f_2. Therefore, we adopt a rotation period of 1.13±0.02 hr for .Finally, the frequency splittings between prograde (m=0 to m=+1) relative to retrograde (m=-1 to m=0) components are asymmetric in . That is, the m=0 component is not exactly centered between the m=±1 components. For f_1, the observed m=0 component is displaced to lower frequency by 0.622±0.043 ; for f_2 the value is 0.128±0.073 . The asymmetry is <0.5% and does not significantly affect our inferred rotation rate, but is noteworthy because it likely represents second-order rotation effects, which are expected to be present for such a rapid rotator <cit.>. A systematic shift of the m=0 components can also constrain the presence of a magnetic field too weak to detect from Zeeman splitting of spectroscopy <cit.>.§ DISCUSSION AND CONCLUSIONSUsing K2, we have discovered a=13,590±340 K, 0.87±0.03white dwarf with a rotation period of 1.13±0.02 hr, faster than any known isolated pulsating white dwarf.To put this rotation in context, we show in Figure <ref> asteroseismically deduced rotation rates of apparently isolated white dwarfs with cleanly identified pulsations from the literature, as compiled by <cit.>: GD 154, HL Tau 76, KUV 11370+4222, HS 0507+0434, L19-2, LP 133-144, GD 165, R548, G185-32, G226-29, EC14012-1446, KIC 4552982 <cit.>, and KIC 11911480 (DAVs); PG 0122+200, PG 2131+066, NGC 1501, PG 1159-035, and RX J2117.1+3412 (DOVs); and the DBVs KIC 8626021 and PG 0112+104 <cit.>. We have excluded the 0.60white dwarf SDSSJ1136+0409, which rotates at 2.49±0.53 hr but is currently in a 6.9-hr binary with a detached, nearby dM companion; it underwent binary interaction via a common-envelope event <cit.>.All 20 of the white dwarfs with previously published rotation rates likely evolved in isolation, and span the range of rotation periods between roughly 0.4-2.2 d. All have spectroscopically deduced atmospheric parameters; the sample has a mean mass of 0.64and a standard deviation of 0.08 , including 3D corrections <cit.>. This is remarkably similar to the 0.62mean mass of field white dwarfs <cit.>, suggesting that the majority of isolated, canonical-mass white dwarfs rotate at 0.4-2.2 d.Figure <ref> brings one interesting fact into focus: The three most massive single white dwarfs with rotation measured via asteroseismology are also among the fastest rotators. SDSSJ161218.08+083028.2 (hereafter SDSSJ1612+0830) has significant pulsation periods at 115.17 s and 117.21 s <cit.>. These pulsation modes are not cleanly identified, but are too close together to be different radial orders. The simplest explanation is that they are components of a single ℓ=1 mode split by either 75.6or 151.2 , indicating a rotation period of roughly 2.0 hr or 1.0 hr, respectively; we mark both solutions in Figure <ref>. We have refit the SDSS spectrum of SDSSJ1612+0830 using the same models and 3D corrections described in Section <ref> and find it has=11,800±170 K,=8.281±0.048, corresponding to a mass of 0.78±0.03 . Other than , the only star in Figure <ref> more massive than SDSSJ1612+0830 is G226-29 (0.83±0.03 ), which has a single ℓ=1 mode centered at 109.28 s with splittings that indicate 8.9-hr rotation <cit.>.It is also possible to measure the rotation of white dwarfs from magnetic spots. There are four spotted white dwarfs with rotation rates shorter than 1 hr: RE J0317-853 (12.1 min, ), NLTT 12758B (22.6 min, ), SDSSJ152934.98+292801.9 (38.1 min, ), and G99-47 (58.2 min, ). RE J0317-853 is especially noteworthy since it has a mass inferred from a parallax distance of at least 1.28<cit.>; its extremely high magnetic field (>200 MG) suggests it could be the outcome of a binary merger (e.g., ).From population synthesis estimates, roughly 7-23% of all apparently single white dwarfs are expected to be the byproducts of mergers <cit.>. Therefore, it is possible that the white dwarf discovered here, , is not the descendent of single-star evolution. Asteroseismology may rule out this scenario. The 0.877model from <cit.>, highlighted in their Figure 6, shows it is difficult to observe an ℓ=1, n=1, m=0 mode with a pulsation period below 110 s without the white dwarf having a canonically thick hydrogen layer (≳10^-5 M_ H/). For this reason, we prefer a single-star evolutionary model; however, more asteroseismic analysis and modeling is required to definitively rule out a binary-merger origin.Finally, we note thatnow supplants HS 1531+7436 as the hottest known isolated DAV; HS 1531+7436 has 3D-corrected atmospheric parameters of=13,270±290 K,=8.49±0.06, found using the same model atmospheres <cit.>. However, the best-fit effective temperatures forfrom spectra taken on two different nights differ by more than 1000 K, a >2σ disagreement (the surface gravities are consistent within the uncertainties). Unfortunately the photometric colors — (u-g, g-r) = (0.36±0.03, -0.21±0.02) — do not strongly prefer one solution over the other <cit.>. Given the defining rolemay play in setting the blue edge of the DAV instability strip where pulsations driven by hydrogen partial-ionization finally reach observable amplitudes, it is worth more detailed follow-up spectroscopy to obtain a reliably accurate effective temperature.We acknowledge helpful comments from the anonymous referee, as well as useful discussions with Conny Aerts and Jamie Tayar.Support for this work was provided by NASA through Hubble Fellowship grant #HST-HF2-51357.001-A, awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555; NASA K2 Cycle 4 Grant NNX17AE92G; NASA K2 Cycle 2 Grant NNX16AE54G; CNPq and PRONEX-FAPERGS/CNPq (Brazil); NSF grants AST-1413001 and AST-1312983; the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 320964 (WDTracer); and the European Union's Horizon 2020 Research and Innovation Programme / ERC Grant Agreement n. 677706 (WD3D). 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"authors": [
"J. J. Hermes",
"Steven D. Kawaler",
"A. D. Romero",
"S. O. Kepler",
"P. -E. Tremblay",
"Keaton J. Bell",
"B. H. Dunlap",
"M. H. Montgomery",
"B. T. Gaensicke",
"J. C. Clemens",
"E. Dennihy",
"S. Redfield"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170427180000",
"title": "Evidence from K2 for rapid rotation in the descendant of an intermediate-mass star"
} |
[email protected] ^1Institute of Physics, Sachivalaya Marg, Bhubaneswar - 751005, India. ^2Department of Physics, Bharathiar University, Coimbatore - 641046, India. ^3Saha Institute of Nuclear Physics, 1/AF,Bidhannagar, Kolkata - 700064, India. ^4Homi Bhabha National Institute, Anushakti Nagar, Mumbai - 400094, India. We study the binary mass distribution for the recently predicted thermallyfissile neutron-rich uranium and thorium nuclei using statistical model.The level density parameters needed for the studyare evaluated from the excitation energies of temperature dependent relativistic mean field formalism. The excitation energy and the level density parameter for a given temperature are employed in the convolution integral method toobtain the probability of the particular fragmentation. As representative cases, we present the results for the binary yield of ^250U and ^254Th. The relative yields arepresented for three different temperatures T = 1, 2 and 3 MeV.25.85.-w, 21.10.Ma, 21.10.Pc, 24.75.+i Relative mass distributions of neutron-rich thermally fissile nuclei within statistical model S. K. Patra^1,4 December 30, 2023 =============================================================================================§ INTRODUCTION Fission phenomenon is one of the most interesting subject in the field of nuclear physics. To study the fissionproperties, a large number of models have been proposed. The fissioning of a nucleus is successfully explained by the liquid drop model and the semi-empirical mass formula isthe best and simple oldest tool to get a rough estimation of the energy released in a fission process. The pioneering work of Vautherin and Brink <cit.>, who has applied the Skyrme interaction in a self-consistent method for the calculation of ground state properties of finite nuclei opened a new dimension in the quantitative estimation of nuclear properties. Subsequently, the Hartree-Fock and time dependent Hartree-Fockformalisms <cit.> are also implemented to study the properties of fission. Most recently, the microscopicrelativistic mean field approximation, which is another successful theory in nuclear physics is also used for the study of nuclear fission <cit.>.From last few decades, the availability of neutron rich nuclei in various laboratories across the globe opened up new research in the field of nuclear physics, because of their exotic decay properties. The effort for the synthesis of superheavy nuclei in the laboratories like, Dubna (Russia),GSI (Germany), RIKEN (Japan) and BNL (USA) is also quite remarkable. Due to all these, the periodic table is extended till date upto atomic number Z = 118 <cit.>. The decay modes of these superheavy nucleiare very different than the usual modes. Mostly, we understand that, a neutron rich nucleus has a large number of neutron than the light or medium mass region of the periodic table. The study of these neutron-rich superheavy nuclei is very interesting, because of their ground state structures and various mode of decays, including multi-fragment fission (more than two) <cit.>. Another interesting feature of some neutron rich uranium and thorium nuclei is that similar to ^233U, ^235U and ^239Pu, the nuclei ^246-264Uand ^244-262Th are also thermally fissile, which are extremely important for the energy productionin fission process. If the neutron rich uranium and thorium nuclei are the viable sources, then these nuclei will be more effective to achieve the critical condition in a controlled fission reaction. Now the question arises, how we can get a reasonable estimation of the mass yield in thespallation reaction of these neutron rich thermally fissile nuclei. As mentioned earlier in this section, there are many formalisms available in the literature to study these cases. Here, we adopt the statistical model developed by Fong <cit.>. The calculation is further extended by Rajasekaran and Devanathan <cit.> to study the binary mass distributions using the single particle energiesof the Nilsson model. The obtained results are well in agreement with the experimental data. In the present study, we would like to replace the single particle energies with the excitation energies of a successful microscopic approach, the relativistic mean field (RMF) formalism. For last few decades, the relativistic mean field (RMF) formalism <cit.> with various parameter sets have successfully reproduced the bulk properties, such as binding energies, root mean square radii, quadrupole deformation etc. not only for nuclei near the β-stability line but also for nuclei away from it. Further, the RMF formalism is successfully applied to the study of clusterization of known cluster emitting heavy nucleus <cit.> and the fission of hyper-hyper deformed ^56Ni <cit.>. Rutz et. al. <cit.> reproduced the double, triplehumped fission barrier of ^240Pu, ^232Th and the asymmetric groundstates of ^226Ra using RMF formalism. Moreover, the symmetric and asymmetric fission modes are also successfully reproduced. Patra et. al. <cit.> studied the neck configuration in the fission decay of neutron rich U and Th isotopes.The main goal of this present paper is to understand the binary fragmentationyield of such neutron rich thermally fissile superheavy nuclei. ^250U and ^254Th are taken forfurther calculations as the representative cases. The paper is organized as follows: In Section <ref>, the statistical model and relativistic mean field theory are presented briefly. In subsection A of this section, the level density parameter and it's relation with the relative mass yield are outlined. In subsection B of <ref>, the equation of motion of the nucleon and meson fields obtained from the relativistic mean field Lagrangian and the temperature dependent of the equations are adopted through the occupation number of protons and neutrons. The results are discussed in Section <ref> and compared with the finite range droplet model (FRDM) predictions. The summary and concluding remarksare given in Section <ref>.§ FORMALISM The possible binary fragments of the considered nucleus is obtained by equating the charge to massratio of the parent nucleus to the fission fragments as <cit.>:Z_P/A_P≈Z_i/A_i,with A_P, Z_P and A_i, Z_i (i = 1 and 2) correspond to massand charge numbers of the parent nucleus and the fission fragments <cit.>. Theconstraints, A_1 + A_2 = A, Z_1 + Z_2 = Z and A_1 ≥ A_2 are imposed to satisfy the conservation of charge and mass number in a nuclear fissionprocess and to avoid the repetition of fission fragments.Another constraint i.e., the binary charge numbers from Z_2 ≥ 26 to Z_1 ≤ 66 is also taken into consideration from the experimental yield <cit.> to generate the combinations, assuming that the fission fragments lie within these charge range.§.§ Statistical theoryThe statistical theory <cit.> assumes that the probability of the particular fragmentation is directly proportional to the folded level density ρ_12of that fragments with the total excitation energy E^*, i.e., P(A_j,Z_j) ∝ρ_12(E^*). Where,ρ_12(E^*) = ∫_0^E^*ρ_1(E^*_1) ρ_2(E^* - E_1^*)dE^*_1, and ρ_i is the level density of two fragments (i = 1, 2). The nuclear level density <cit.> is expressed as a function of fragment excitation energy E^∗_i and the single particle level density parameter a_i which is given as:ρ_i(E^∗_i) = 112(π^2a_i)^1/4 E^∗ (-5/4)_iexp(2√(a_iE^∗_i)).In Refs. <cit.>, we calculate the excitation energies of the fragments using the ground state single particle energies of finite range droplet model(FRDM) <cit.> at a given temperature T keeping the total number of proton and neutron fixed. In the present study, we apply the self consistent temperature dependent relativistic mean field theory to calculate the E^* of the fragments. The excitation energy is calculatedas,E^*_i(T) = E_i(T) - E_i(T = 0).The level density parameter a_i is given as,a_i = E^*_i/T^2.The relative yield is calculated as the ratio of the probability of a given binary fragmentation to the sum of the probabilities of all the possible binary fragmentations and it is given by,Y(A_j,Z_j)=P(A_j,Z_j)/∑_j P(A_j,Z_j),where A_j and Z_j are referred to the binary fragmentations involving two fragments with mass and charge numbers A_1, A_2 and Z_1, Z_2 obtained from Eq. (<ref>).The competing basic decay modes such as neutron/proton emission, α decay and ternary fragmentation are not considered. In addition to these approximations, we have also not included the dynamics of the fission reaction, which are really important to get a quantitative comparison with the experimental measurements. The presented results are the prompt disintegration of a parent nucleus into two fragments (democratic breakup). The resulting excitation energy would be liberated as prompt particle emission or delayed emission, but such secondary emissions are also ignored. §.§ RMF FormalismThe RMF theory assume that the nucleons interact with each other via meson fields. The nucleon - meson interaction is given by the Lagrangian density <cit.>, L = ψ_i{iγ^μ∂_μ-M}ψ_i +1/2∂^μσ∂_μσ -1/2m_σ^2σ^2-1/3g_2σ^3 -1/4g_3σ^4 -g_σψ_iψ_iσ -1/4Ω^μνΩ_μν+1/2m_w^2V^μV_μ -g_wψ_iγ^μψ_i V_μ -1/4B⃗^μν.B⃗_μν+1/2m_ρ^2R⃗^μ .R⃗_μ -g_ρψ_iγ^μτ⃗ψ_i.R⃗^⃗μ⃗ -1/4F^μνF_μν-eψ_iγ^μ(1-τ_3i)/2ψ_iA_μ.Where, ψ_i is the single particle Dirac spinor. The arrows over the letters in the above equation represent the isovector quantities. The nucleon, the σ, ω, and ρ meson masses are denoted by M, m_σ, m_ω and m_ρ respectively. The meson and the photon fields are termed asσ, V_μ, R^μ and A_μ for σ, ω, ρ- mesons and photon respectively.The g_σ, g_ω, g_ρ and e^2/4π are the coupling constants for the σ,ω, ρ-mesons and photon fields with nucleons respectively. The strength of the constantsg_2 and g_3 are responsible for the nonlinear couplings of σmeson (σ^3 and σ^4).The field tensors of the isovector mesons and the photon are given by,Ω^μν=∂^μ V^ν - ∂^ν V^μ, B⃗^μν=∂^μR⃗^ν - ∂^νR⃗^μ - g_ρ (R⃗^μ×R⃗^ν),F^μν=∂^μ A^ν - ∂^ν A^μ.The classical variational principle gives the Euler-Lagrange equation andwe get theDirac-equation with potential terms for the nucleons and Klein-Gordan equations withsource terms for the mesons. We assume the no-sea approximation, so we neglect theantiparticle states. We are dealing with the static nucleus, so the time reversalsymmetry and the conservation of parity simplifies the calculations. After simplifications,the Dirac equation for the nucleon is given by,{ - iα.▽ + V(r) + β[M + S(r)]}ψ_i = ϵ_iψ_i,where V(r) represents the vector potential and S(r) is the scalar potential,V(r)=g_ωω_0 + g_ρτ_3ρ_0(r)+ e (1-τ_3)/2 A_0(r), [2mm] S(r) =g_σσ(r),which contributes tothe effective mass,M^*(r) = M + S(r). The Klein-Gordon equations for the mesons and the electromagnetic fields with the nucleon densities as sources are, {- + m_σ^2}σ(r) = -g_σρ_s(r)-g_2σ^2(r) -g_3σ^3(r), {- + m_ω^2}ω_0(r) = g_ωρ_v(r), {- + m_ρ^2}ρ_0(r) = g_ρρ_3(r), - A_0(r) = eρ_c(r). The corresponding densities such as scalar, baryon (vector), isovector and proton (charge) are given asρ_s(r) =∑_i n_iψ_i^†(r)ψ_i(r),ρ_v(r) =∑_i n_iψ_i^†(r)γ_0ψ_i(r),ρ_3 (r) =∑_i n_iψ_i^†(r) τ_3 ψ_i(r),ρ_ c(r) =∑_i n_iψ_i^†(r) (1 -τ_3/2)ψ_i(r).To solve the Dirac and Klein-Gordan equations, we expand the Boson fields and the Dirac spinor in an axially deformed harmonic oscillator basis with β_0 as the initial deformation parameter. The nucleon equation along with different meson equations form a set of coupled equations, which can be solved by iterative method. The center of mass correction is calculated with the non-relativistic approximation. The quadrupole deformation parameter β_2 is calculated from the resulting quadrupole moments of the proton and neutron. The total energy is given by <cit.>, E(T) = ∑_i ϵ_i n_i + E_σ + E_σ NL + E_ω + E_ρ + E_C +E_pair + E_c.m. - AM,withE_σ= -1/2g_σ∫ d^3r ρ_s(r) σ(r), E_σ NL = -1/2∫ d^3r { 2/3g_2σ^3(r) + 1/2g_3 σ^4(r) }, E_ω =-1/2g_ω∫ d^3r ρ_v(r)ω^0(r),E_ρ=-1/2g_ρ∫ d^3rρ_3(r)ρ^0(r), E_C =-e^28π∫ d^3rρ_c(r)A^0(r), E_pair = - ∑_i>0u_iv_i = -^2/G, E_c.m.= -3/4× 41A^-1/3.Here, ϵ_i is the single particle energy, n_i is the occupation probability and E_pair is the pairing energy obtained from the simple BCS formalism. §.§ Pairing and temperature dependent RMF formalism The pairing correlation plays a distinct role in open-shell nuclei. The effect of pairing correlation is markedly seen with increase in mass number A. Moreover it helps in understanding the deformation of medium and heavy nuclei. It has a lean effect on both bulk and single particles properties of lighter mass nuclei because of the availability of limited pairs near the Fermi surface. We take the case of T=1 channel of pairing correlation i.e, pairing between proton- proton and neutron-neutron. In this case, a nucleon ofquantum states | jm_z⟩ pairs with another nucleons having same I_z value with quantum states | j-m_z ⟩,since it is the time reversal partner of the other. In both nuclear and atomic domain the ideology of BCS pairing is the same. The even-odd mass staggeringof isotopes was the first evidence of its kind for the pairing energy.Considering the mean-field formalism, the violation of the particle number is seen only due to the pairing correlation. We find terms like ψ^†ψ (density) in the RMF Lagrangian density but we put an embargo on terms of the form ψ^†ψ^† or ψψ since it violates the particle number conservation.We apply externally the BCS constant pairing gap approximation for our calculation to takethe pairing correlation into account. Thepairing interaction energy in terms of occupationprobabilities v_i^2 and u_i^2=1-v_i^2 is writtenas <cit.>: E_pair=-G[∑_i>0u_iv_i]^2, with G is the pairing force constant.The variational approach with respect to the occupation number v_i^2 gives the BCS equation<cit.>: 2ϵ_iu_iv_i-(u_i^2-v_i^2)=0, with the pairing gap =G∑_i>0u_iv_i. The pairing gap () of proton and neutron is taken from the empirical formula <cit.>: = 12 × A^-1/2.The temperature introduced in the partial occupancies in the BCS approximation is given by,n_i=v_i^2=1/2[1-ϵ_i-λ/ϵ̃_̃ĩ[1-2 f(ϵ̃_̃ĩ,T)]],with f(ϵ̃_̃ĩ,T) = 1/(1+exp[ϵ̃_̃ĩ/T])andϵ̃_̃ĩ = √((ϵ_i-λ)^2+^2). The function f(ϵ̃_̃ĩ,T) represents the Fermi Dirac distribution for quasi particle energy ϵ̃_̃ĩ. The chemical potential λ_p (λ_n) for protons (neutrons) is obtained from the constraints of particle number equations ∑_i n_i^Z= Z,∑_i n_ i^N =N.The sum is taken over all proton and neutron states. The entropy is obtained by,S = - ∑_i [n_iln(n_i) + (1 - n_i)ln (1- n_i)]. The total energy and the gap parameter are obtained by minimizing the free energy,F = E - TS.In constant pairing gap calculations, for a particular value of pairing gapand force constant G, the pairing energy E_pair diverges, if it is extended to an infinite configuration space. In fact, in all realistic calculations with finite range forces,is not constant, but decreases with large angular momenta states above the Fermi surface. Therefore, a pairing window in all the equations are extended up-to the level |ϵ_i-λ|≤ 2(41A^-1/3) as a function of the single particle energy. The factor 2 has been determined so as to reproduce the pairing correlation energy for neutrons in ^118Sn using Gogny force <cit.>.§ RESULTS AND DISCUSSIONSIn our very recent work <cit.>, we have calculated the ternary mass distributionsfor ^252Cf, ^242Pu and ^236U with the fixed third fragments A_3 = ^48Ca, ^20Oand ^16O respectively for the three different temperatures T = 1, 2 and 3 MeV within the TRMFformalism. The structure effects of binary fragments are also reported in Ref. <cit.>.In this article, we study the mass distribution of ^250U and ^254Th as a representative cases from the range of neutron-rich thermally fissile nuclei ^246-264U and ^244-262Th. Because of the neutron-rich nature of these nuclei, a large number of neutrons emit during the fission process. These nucleons help to achieve the critical condition much sooner than the normal fissile nuclei. To assure the predictability of the statisticalmodel, we also study the binary fragmentation of naturally occurring^236U and ^232Th nuclei. The possible binary fragments are obtained using the Eq. (<ref>). To calculate the total binding energy at a given temperature, we use the axially symmetric harmonicoscillator basis expansion N_F and N_B for the Fermion and Boson wave-functions to solve the Dirac Eq. (<ref>) and the Klein Gordon Eqs. (<ref> - <ref>) iteratively. It is reported <cit.> that the effect of basisspace on the calculated binding energy, quadrupole deformation parameter (β_2) and therms radii of nucleus arealmost equal for the basis setN_F = N_B = 12 to 20 in themassregion A ∼ 200 . Thus, we use the basis space N_F = 12 and N_B= 20 to study the binaryfragments up to mass number A ∼ 182. The binding energy is obtained by minimizing the free energy, which gives the most probable quadrupole deformation parameter β_2 and the proton (neutron) pairing gaps _p (_n) for the given temperature. At finite temperature,the continuum corrections due to the excitation of nucleons to be considered. The level density in the continuum depends on the basis space N_F and N_B <cit.>. It is shown that the continuum corrections need not be included in the calculationsof level densities up-to the temperature T ∼ 3 MeV <cit.>. §.§ Level density parameter and level density within TRMF and FRDM formalisms In TRMF, the excitation energies E^* and the level density parameters a_i of the fragments areobtained self consistently from Eqns. (<ref>) to (<ref>).The FRDM calculations arealso done for comparison. In this case, level densityof the fragments are evaluated from the ground state single particle energies of the finite range droplet model(FRDM) of Möller et. al. <cit.> which are retrieved from the Reference InputParameter Library (RIPL-3) <cit.>.The total energy at a given temperature is calculated as E(T) = ∑ n_i ϵ_i; ϵ_i are theground state single particle energies and n_i are the Fermi-Dirac distribution function. The T dependentenergies are obtained by varying the occupation numbers at a fixed particle number for a given temperature and given fragment. The level density parameter a is a crucial quantity in the statistical theory for theestimation of yields. These values of a for the binary fragments of ^236U, ^250U, ^232Thand ^254Th obtained from TRMF and FRDM are depicted in Fig. <ref>. The empirical estimation a=A/K are also given for comparison, with K, the inverse level density parameter. In general, the K value varies from 8 to 13 with theincreasing temperature. However, the level density parameter is considered to beconstant up-to T ≈ 4 MeV.Hence, we take the practical value of K = 10as mentioned in Ref. <cit.>.The a values of TRMF are close to the empirical level density parameter. The FRDM level density parameters are appreciably lower than the referenced a. Further, in both models at T = 1 MeV, there are morefluctuations in the level density parameter due to the shell effects of the fragments. At T = 2 and 3 MeV,the variations are small. This may be due to the fact that the shell become degenerate at the higher temperatures. All fragments becomes spherical at temperature T ≈ 3 MeVas shown in Ref. <cit.>.The level density parameter a is evaluated in two different ways using excitation energy and the entropy ofthe system as:a_E = E^*T^2, a_S = S2T.For instance, the inverse level density parameters K_E and K_S of ^236U, ^250U, ^232Thand ^254Th within TRMF formalism are depicted in Fig. <ref>. Both K_S and K_E have maximum fluctuation upto 30 MeV at T =1 MeV. These values reduce to 10-13 MeV at temperature T = 2 MeV or above. It is to be notedthat at T = 3 MeV, the inverse level density parameter substantially lower around the mass numberA ∼ 130 in all cases. This may be due to the neutron closed shell (N = 82) in the fission fragmentsof ^236U and ^232Th and the neutron-rich nuclei ^250U and ^254Th. The level density for the fission fragments of ^236U, ^250U, ^232Th and ^254Th are plotted asa function of mass number in Fig. <ref> within the TRMF and FRDM formalisms at three differenttemperatures T = 1, 2 and 3 MeV. The level density ρ has maximum fluctuations at T = 1 MeV for allconsidered nuclei in TRMF model similar to the level density parameter a. The ρ values aresubstantially lower at mass number A ∼ 130 for all nuclei. In Fig. <ref>, one can notice that thelevel density has small kinks in the mass region A ∼ 71-81 of ^236Uand A ∼ 77-91 of ^250U, comparing with the neighboring nuclei at temperature T = 2 MeV. Consequently, the correspondingpartner fragments have also higher ρ values.A further inspection reveals that the level density of the closed shell nucleus around A ∼ 130 hashigher value than the neighboring nuclei for both ^236,250U, but it has lower yield due to the smaller level density of the corresponding partners.At T = 3 MeV, the level density of the fragments around mass number A ∼72 and130 havelarger values compared to other fragments of ^236U. On the other hand, the level densityin the vicinity of neutron number N = 82 and proton number Z = 50 for the fragments of the neutron-rich^250U nucleus is quite high, because of the close shell of the fragments. This is evident from the small kink in the level density of ^130Cd (N = 82), ^132In(N ∼ 82) and ^135Sn (Z = 50).Again, for ^232Th, the level densities are found to be maximum at around mass numberA ∼81 and 100for T = 2 MeV. In case of ^254Th, the ρ values are found to be large for the fragments around A ∼ 78 and 97 at T = 2 MeV. Their corresponding partners have also similar behavior. For higher temperatureT = 3 MeV, the higher ρ values of ^232Th fragments are notable aroundmass number A ∼ 130. Similarly, for ^254Th, the fission fragments around A ∼ 78 has higher level density at T = 3 MeV. In general, the level density increases towards the neutron closed shell (N = 82) nucleus.§.§ Relative fragmentation distribution in binary systems In this section, the mass distributions of ^236U, ^232Th and the neutron rich nuclei ^250U and^254Th are calculated at temperatures T = 1, 2 and 3 MeV using TRMF and FRDM excitation energies andthe level density parameters a as explained in Sec. <ref>. The binary mass distributions of ^236,250Uand ^232,254Th are plotted in Figs. <ref> and <ref>. The total energy at finite temperatureand ground state energy are calculated using the TRMF formalism as discussed in the section <ref>. From the excitation energy E^∗ and the temperature T, the level density parameter a and the leveldensity ρ of the fragments are calculated using Eq. <ref>. From the fragment level densities ρ_i,the folding density ρ_12 is calculated using the convolution integral as in Eq. <ref> and the relativeyields are calculated using Eq. <ref>. The total yields are normalized to the scale 2.The mass yield of normal nuclei ^236U and ^232Th are briefly explains first, followed by the detailed description of the neutron rich nuclei. The resultsof most favorable fragments yield of ^236,250U and ^232,254Th are listed in Table <ref> at three different temperaturesT = 1, 2 and 3 MeV for both TRMF and FRDM formalisms. From Figs. <ref> and <ref>, it is shown that the mass distributions for ^236U and ^232Th are quite different than the neutron-rich ^250U and ^254Th isotopes.The symmetric binary fragmentation ^118Pd +^118Pd for ^236U is the most favorable combination.In TRMF, the fragments with close shell (N = 100 and Z = 28) combinations are more probable at thetemperature T = 2 MeV. The blend region of neutron and proton close shell (N ≈ 82 and Z ≈ 50)has the considerable yield values at T = 3 MeV. The fragmentations ^151Pr +^85As, ^142Cs +^94Rband ^144Ba +^92Kr are the favorable combinations at temperature T = 1 MeV in FRDM formalism. For highertemperatures T = 2 and 3 MeV, the closed shell or near closed shell fragments (N =82, 50 and Z = 28)have larger yields. From Fig. <ref> in TRMF formalism, the combinations ^118Pd +^114Ruand ^140Xe +^92Kr are the possible fragments at T = 1 MeV for the nucleus ^232Th.At T = 2 MeV, we find maximum yields for the fragments with the close shell or near close shellcombinations (N = 82, 50). For higher temperature T = 3 MeV, near the neutron close shell (N ∼ 82), ^132Sb +^100Y is the mostfavorable fragmentation pair compared with all other yields. Similar fragmentations are found in the FRDM formalismat T = 2 and 3 MeV. In addition, the probability of the evaluation of ^129Sn +^103Zr is also quitesubstantial in the fission process.For T = 1 MeV, the yield is more or less similar with the TRMF model.From Fig. <ref>, for ^250U the fragment combinations ^140,141Te +^110,109Zr have the maximum yields at T = 1 MeV in TRMF. This is also consistent with the evolution of the sub-close shell proton Z = 40 in Zr isotopes <cit.>.Contrary to this almost symmetric binary yield, the mass distribution of this nucleus in FRDM formalism have the asymmetricevolution of the fragment combinations like ^160,159Pr +^90,91As, ^163,162Nd +^87,88Ge and ^150Cs +^100Rb. Interestingly, at T = 2 and 3 MeV, the more favorable fragment combinations have one of the closed shellnuclei. At T = 2 MeV, ^159Pr +^91As, ^162Nd +^88Ge and ^173Gd +^77Ni are the more probable fragmentations (see Fig. <ref>(c)). It is reported by Satpathy et al <cit.> and experimentally verified by Patel et al <cit.>that N = 100 is a neutron close shell for the deformed region, where Z = 62 acts like a magic number.In FRDM, ^128Ag +^122Rh, ^132In +^118Tc, ^140Te +^110Zr and ^173Gd +^77Ni have larger yieldat temperature T = 2 MeV. In TRMF method, the most favorable fragments are confined in the single region(A ≈ 114-136)which is the blend of vicinity of neutron (N = 82) and proton (Z = 50) closed shell nuclei at T = 3 MeV.The fragment combinations ^130Cd +^120Ru, ^132In +^118Tc and ^135Sn +^115Mo are the major yields for ^250U at T=3 MeV in TRMF calculations.In FRDM method, at T = 3 MeV, more probable fragmentsare similar that at T = 2 MeV. A comparison between Fig. <ref>(c) and<ref>(d) clears that, although the predictionof FRDM and TRMF at T = 3 MeV are qualitatively similar, but it is quantitatively very different at T = 2 MeVin both the predictions. Also, from Fig. <ref>, it is inferred that the yields of the fragment combinations in blend region increases and inother regions decreases at T = 2 MeV.In the present study, the total energy of the parent nucleus A ismore than the sum of the energies of thedaughters A_1 and A_2. Here, the dynamics of entire process starting fromthe initial stage upto the scission are ignored. As a result, the energy conservation in the spallation reaction does not taken intoaccount. The fragment yield can be regarded as the relative fragmentationprobability, which is obtained from Eq. <ref>. Now we analyze the fragmentation yields for Th isotopes and the results are depicted inFig. <ref> and Table <ref>. In this case, one can see that the mass distribution broadly spreads through out the region A_i=66-166. Again, the most concentrated yields can be divided into two regions I(A_1 = 141-148 and A_2 = 106-113) andII (A_1 = 152-158 and A_2 = 102-96) for ^254Th in TRMF formalism at the temperature T = 1 MeV.The most favorable fragmentation ^142Sn +^112Zr is obtained from region I. The other combinations in thatregion have also considerable yields. In region II, the isotopes of Ba and Cs appears curiouslyalong with their corresponding partners.Categorically, in FRDM predictions, region I has larger yields at T = 1 MeV. The other possiblefragmentations are ^163Ce +^91Ge, ^168Nd +^86Zn and ^181Gd +^73Fe (See Fig. <ref> (b,d)).The mass distribution is different with different temperature and the maximum yields at T = 2 MeV in TRMF formalism are ^174,175,176Sm +^80,79,78Ni. Apart from these combinations, there are other considerable yields can be seen in Fig. <ref> for region II. The prediction of maximum probability of the fragments production in FRDM method are^144Sb +^110Y,^178Eu +^76Co and ^127Rh +^127Rh at T = 2 MeV. Besides these yields, one can find other notable evolution of masses in region I due to the vicinity of the proton close shell. Interestingly, atT = 3 MeV, symmetric binary combination ^127Rh +^127Rh has the largestyield due to the neutronclose shell (N = 82) of the fragment ^127Rh.The other yield fragments have exactly/nearly a magicnucleon combination, mostly neutron (N = 82) as one of the fragment. A considerable yield is also seen for the proton close shell (Z = 28) Ni or/and (Z=62) Sm isotopes supporting our earlier prediction <cit.>.This confirms the prediction of Sm as a deformed magic nucleus <cit.>.Another observation of the present calculations show that the yields of theneutron-rich nuclei agree with the symmetric mass distribution of Chaudhuri et. al. <cit.> at large excitation energy, which contradict the recent prediction of large asymmetric mass distribution of neutron-deficient Th isotopes <cit.>. These two results <cit.> along with our present calculations confirm that the symmetric or asymmetricmass distribution at different temperature depends on the proton and neutron combination of the parent nucleus. In general, both TRMF and FRDM predict maximum yields for both symmetric/asymmetric binary fragmentations followed byother secondary fragmentations emission depending on the temperature as well as the mass number of the parent nucleus. Thus, the binary fragments have larger level density ρ comparing with other nuclei because of neutron/protonclose shell fragment combinations at T = 2 and 3 MeV. This results ascertain the fact that most favorablefragments have larger phase space than the neighboring nuclei as reported earlier <cit.>.To this end,it may be mentioned that the differences in the mass distributions or the relative yields calculated using TRMF and FRDM approachesmainlyarise due to the differences in the level densities associated with these approaches. The mean values and the fluctuations in the level density parameter and the corresponding level density are even qualitatively different in both the approaches considered. This is possibly stemming from the fact that the single -particle energiesin the FRDM based model are temperature independent. The temperature dependence of the excitation energy as required to calculate the level density parametercomes only from the modification of the single-particle occupancy due to the Fermi distribution. In the TRMF approach, the excitation energy for each fragment at a given temperature is calculated self-consistently. Therefore, the deformation and the single-particle energies changes with temperature.For the neutron-rich nuclei, the fragments having neutron/proton close shell N = 50, 82 and 100 have maximum possibility of emission at T = 2 and 3 MeV (for both nuclei ^250U and ^254Th). This is a general trend, we could expect for allneutron-rich nuclei. It is worthy to mention some of the recent reports and predictions of multi-fragment fission for neutron-rich uranium and thorium nuclei. When such a neutron-rich nucleus breaks into nearly two fragments, the products exceed the drip-line leaving few nucleons (or light nuclei) free. As a result, these freeparticles along with the scission neutrons enhance the chain reaction in a thermonuclear device. These additionalparticles (nucleons or light nuclei) responsible to reach the critical condition much faster than the usual fissionfor normal thermally fissile nucleus. Thus, the neutron-rich thermally fissile nuclei, which are in the case of ^246-264U and ^244-262Th will be very useful for energy production. § SUMMARY AND CONCLUSIONS The fission mass distributions of β-stable nuclei ^236U and ^232Th and the neutron-richthermally fissile nuclei ^250U and ^254Th are studied within the statistical theory. The possiblecombinations are obtained by equating the charge to mass ratio of the parents to that of the fragments.The excitation energies of fragments are evaluated from the temperature dependent self-consistent binding energies at the given T and the ground state binding energies which are calculated from therelativistic mean field model. The level densities and the yields combinationsare manipulated from the convolution integral approach. The fission mass distributions of the aforementionednuclei are also evaluated from the FRDM formalism for comparison. The level density parameter a andinverse level density parameter K are also studied to see the difference in results with these two methods. Besides fission fragments, the level densities are also discussed in the present paper. For^236U and ^232Th, the symmetric and nearly symmetric fragmentations are more favorable at temperatureT = 1 MeV. Interestingly, in most of the cases we find one of the favorable fragment is a close shell or near close shell configuration (N = 82,50 and Z = 28) at temperature T = 2 and 3 MeV. This resultascertains with our earlier predictions. Further, Zr isotopes has larger yield values for ^250U and ^254Thwith their accompanied possible fragments at T = 1 MeV. The Ba and Cs isotopes with their partners arealso more possible for ^254Th. This could be due to the deformed close shell in the region Z=52-66of the periodic table <cit.>. The Ni isotopes and the neutron close shell (N ∼ 100) nuclei are some of the prominent yields for both ^250U and ^254Th at temperature T = 2 MeV. At T = 3 MeV, the neutron close shell (N = 82) is one of the largest yield fragments. 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"authors": [
"Bharat Kumar",
"M. T. Senthil kannan",
"M. Balasubramaniam",
"B. K. Agrawal",
"S. K. Patra"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170426133918",
"title": "Relative mass distributions of neutron-rich thermally fissile nuclei within statistical model"
} |
Propagating elastic vibrations dominate thermal conduction in amorphous silicon Austin J. Minnich December 30, 2023 =============================================================================== Current measures of machine intelligence are either difficult to evaluate or lack the ability to test a robot's problem-solving capacity in open worlds. We propose a novel evaluation framework based on the formal notion of MacGyver Test which provides a practical way for assessing the resilience and resourcefulness of artificial agents.§ INTRODUCTION Consider a situation when your only suit is covered in lint and you do not own a lint remover. Being resourceful, you reason that a roll of duct tape might be a good substitute. You then solve the problem of lint removal by peeling a full turn's worth of tape and re-attaching it backwards onto the roll to expose the sticky side all around the roll.By rolling it over your suit, you can now pick up all the lint. This type of everyday creativity and resourcefulness is a hallmark of human intelligence and best embodied in the 1980s television series MacGyver which featured a clever secret service agent who used common objects around him like paper clips and rubber bands in inventive ways to escape difficult life-or-death situations.[As a society, we place a high value on our human ability to solve novel problems and remain resilient while doing so. Beyond the media, our patent system and peer-reviewed publication systems are additional examples of us rewarding creative problem solving and elegance of solution.]Yet, current proposals for tests of machine intelligence do not measure abilities like resourcefulness or creativity, even though this is exactly what is needed for artificial agents such as space-exploration robots, search-and-rescue agents, or even home and elder-care helpers to be more robust, resilient, and ultimately autonomous.In this paper we thus propose an evaluation framework for machine intelligence and capability consisting of practical tests for inventiveness, resourcefulness, and resilience.Specifically, we introduce the notion of MacGyver Test (MT) as a practical alternative to the Turing Test intended to advance research. § BACKGROUND: TURING TEST AND ITS PROGENY Alan Turing asked whether machines could produce observable behavior (e.g., natural language) that we (humans) would say required thought in people <cit.>. He suggested that if an interrogator was unable to tell, after having a long free-flowing conversation with a machine whether she was dealing with a machine or a person, then we can conclude that the machine was “thinking”. Turing did not intend for this to be a test, but rather a prediction of sorts <cit.>. Nevertheless, since Turing, others have developed tests for machine intelligence that were variations of the so-called Turing Test to address a common criticism that it was easy to deceive the interrogator.Levesque et al. designed a reading comprehension test, entitled the Winograd Schema Challenge, in which the agent is presented a question having some ambiguity in the referent of a pronoun or possessive adjective. The question asks to determine the referent of this ambiguous pronoun or possessive adjective, by selecting one of two choices <cit.>. Feigenbaum proposed a variation of the Turing Test in which a machine can be tested against a team of subject matter specialists through natural language conversation <cit.>. Other tests attempted to study a machine's ability to produce creative artifacts and solve novel problems <cit.>.Extending capabilities beyond linguistic and creative, Harnad's Total Turing Test (T3) suggested that the range of capabilities must be expanded to a full set of robotic capacities found in embodied systems <cit.>. Schweizer extended the T3 to incorporate species evolution and development over time and proposed the Truly Total Turing Test (T4) to test not only individual cognitive systems but whether as a species the candidate cognitive architecture in question is capable of long-term evolutionary achievement <cit.>.Finding that the Turing Test and its above-mentioned variants were not helping guide research and development, many proposed a task-based approach. Specific task-based goals were designed couched as toy problems that were representative of a real-world task <cit.>. The research communities benefited greatly from this approach and focused their efforts towards specific machine capabilities like object recognition, automatic scheduling and planning, scene understanding, localization and mapping, and even game-playing. Many public competitions and challenges emerged that tested the machine's performance in applying these capabilities – from image recognition contests and machine learning contests. Some of these competitions even tested embodiment and robotic capacities, while combining multiple tasks. For example, the DARPA Robotics Challenge tested a robot's ability to conduct tasks relevant to remote operation including turning valves, using a tool to break through a concrete panel, opening doors, remove debris blocking entryways.Unfortunately, the Turing Test variants as well as the task-based challenges are not sufficient as true measures of autonomy in the real-world. Autonomy requires a multi-modal ability and an integrated embodied system to interact with the environment, and achieve goals while solving open-world problems with the limited resources available. None of these tests are interested in measuring this sort of intelligence and capability, the sort that is most relevant from a practical standpoint. § THE MACGYVER EVALUATION FRAMEWORK The proposed evaluation framework, based on the idea of MacGyver-esque creativity, is intended to answer the question whether embodied machines can generate, execute and learn strategies for identifying and solving seemingly-unsolvable real-world problems. The idea is to present an agent with a problem that is unsolvable with the agent's initial knowledge and observing the agent's problem solving processes to estimate the probability that the agent is being creative: if the agent can think outside of its current context, take some exploratory actions, and incorporate relevant environmental cues and learned knowledge to make the problem tractable (or at least computable) then the agent has the general ability to solve open-world problems more effectively.[Note that the proposed MT is a subset of Harnad's T3, but instead of requiring robots to do “everything real people do”, MT is focused on requiring robots to exhibit resourcefulness and resilience.MT is also a subset of Schweizer's T4 which expands T3 with the notion of species-level intelligence.] This type of problem solving framework is typically used in the area of automated planning for describing various sorts of problems and solution plans and is naturally suited for defining a MacGyver-esque problem and a creative solution strategy.We are now ready to formalize various notions of the MacGyver evaluation framework. §.§ Preliminaries - Classical Planning We define ℒ to be a first order language with predicates p(t_1, … , t_n) and their negations ¬ p(t_1, … , t_n) , where t_i represents terms that can be variables or constants. A predicate is grounded if and only if all of its terms are constants. We will use classical planning notions of a planning domain in ℒ that can be represented as Σ = (S, A, γ ), where S represents the set of states, A is the set of actions and γ are the transition functions. A classical planning problem is a triple 𝒫=(Σ , s_0, g), where s_0 is the initial state and g is the goal state. A plan π is any sequence of actions and a plan π is a solution to the planning problem if g ⊆γ (s_0,π ). We also consider the notion of state reachability and the set of all successor states Γ̂(s), which defines the set of states reachable from s.§.§ A MacGyver Problem To formalize a MacGyver Problem (MGP), we define a universe and then a world within this universe. The world describes the full set of abilities of an agent and includes those abilities that the agent knows about and those of which it is unaware. We can then define an agent subdomain as representing a proper subset of the world that is within the awareness of the agent. An MGP then becomes a planning problem defined in the world, but outside the agent's current subdomain. (Universe).We first define a Universe 𝕌 = (S,A,γ) as a classical planning domain representing all aspects of the physical world perceivable and actionable by any and all agents, regardless of capabilities. This includes all the allowable states, actions and transitions in the physical universe. (World).We define a world 𝕎^t = (S^t, A^t, γ^t) as a portion of the Universe 𝕌 corresponding to those aspects that are perceivable and actionable by a particular species t of agent. Each agent species t ∈ T has a particular set of sensors and actuators allowing agents in that species to perceive a proper subset of states, actions or transition functions. Thus, a world can be defined as follows:𝕎^t = {(S^t, A^t, γ^t) | ((S^t ⊆ S)(A^t ⊆ A)(γ^t ⊆γ))¬ ((S^t = S)(A^t = A)(γ^t = γ))} (Agent Subdomain).We next define an agent Σ_i^t = (S_i^t, A_i^t, γ_i^t) of type t, as a planning subdomain corresponding to the agent's perception and action within its world 𝕎^t. In other words, the agent is not fully aware of all of its capabilities at all times, and the agent domain Σ_i^t corresponds to the portion of the world that the agent is perceiving and acting at time i. Σ_i^t = {(S_i^t, A_i^t, γ_i^t) | ((S_i^t ⊂ S^t)(A_i^t ⊂ A^t)(γ_i^t ⊂γ^t))¬ ((S_i^t = S^t)(A_i^t = A^t)(γ_i^t = γ^t))} (MacGyver Problem). We define a MacGyver Problem (MGP) with respect to an agent t, as a planning problem in the agent's world 𝕎_t that has a goal state g that is currently unreachable by the agent. Formally, an MGP 𝒫_M = (𝕎^t, s_0, g), where: * s_0 ∈ S^t_i is the initial state of the agent * g is a set of ground predicates * S_g = {s ∈ S | g ⊆ s } Where g ⊆ s^', ∀ s^'∈Γ̂_𝕎^t(s_0) ∖Γ̂_Σ^t_i(s_0)It naturally follows that in the context of a world 𝕎_t, the MGP 𝒫_M is a classical planning problem which from the agent's current perspective is unsolvable. We can reformulate the MGP as a language recognition problem to be able to do a brief complexity analysis. (MGP-EXISTENCE). Given a set of statements D of planning problems, let MGP-EXISTENCE(D) be the set of all statements P ∈ D such that P represents a MacGyver Problem 𝒫_M, without any syntactical restrictions. MGP-EXISTENCE is decidable. The proof is simple. The number of possible states in the agent's subdomain Σ_i^t and the agent's world 𝕎^t are finite. So, it is possible to do a brute-force search to see whether a solution exists in the agent's world but not in the agent's initial domain. MGP-EXISTENCE is EXPSPACE-complete. (Membership). An MGP amounts to looking to see if the problem is a solvable problem in the agent-domain. Upon concluding it is not solvable, the problem then becomes one of determining if it is a solvable problem in the world corresponding to the agent's species. Each of these problems are PLAN-EXISTENCE problems, which are in EXPSPACE for the unrestricted case <cit.>. Thus, MGP-EXISTENCE is in EXPSPACE.(Hardness). We can reduce the classical planning problem P(Σ,s_0,g) to an MGP (PLAN-EXISTENCE ≤_m^p MGP-EXISTENCE), by defining a new world 𝕎. To define a new world, we extend the classical domain by one state, defining the new state as a goal state, and adding actions and transitions from every state to the new goal state. We also set the agent domain to be the same as the classical planning domain. Now, P(Σ,s_0,g) ∈ PLAN-EXISTENCE iff P(𝕎,s_0,g) ∈ MGP-EXISTENCE for agent with domain Σ. Thus, MGP-EXISTENCE is EXPSPACE-hard.§.§ Solving a MacGyver Problem From Theorems <ref> and <ref>, we know that, while possible, it is intractable for an agent to know whether a given problem is an MGP. From an agent's perspective, solving an MGP is like solving any planning problem with the additional requirement to sense or learn some previously unknown state, transition function or action. Specifically, solving an MGP will involve performing some actions in the environment, making observations, extending and contracting the agent's subdomain and exploring different contexts.§.§.§ Solution Strategies(Agent Domain Modification). A domain modification Σ^t*_j involves either a domain extension or a domain contraction[In the interest of brevity we will only consider domain extensions for now.]. A domain extension Σ^t+_j of an agent is an Agent-subdomain at time j that is in the agent's world 𝕎^t but not in the agent's subdomain Σ^t_i in the previous time i, such that Σ^t_i ≼Σ^t_j. The agent extends its subdomain through sensing and perceiving its environment and its own self - e.g., the agent can extend its domain by making an observation, receiving advice or an instruction or performing introspection. Formally,Σ^t+_j = {(S_j^t+, A_j^t+, γ_j^t+) | (S_j^t+⊂ S^t ∖ S_i^t) (A_j^t+⊂ A^t ∖ A_i^t)(γ_j^t+⊂γ^t ∖γ_i^t)} The agent subdomain that results from a domain extension is Σ_j^t = Σ_i^t ∪Σ^t+_j A domain modification set Δ_Σ^t_i= {Σ^t*_1, Σ^t*_2, … ,Σ^t*_n } is a set of n domain modifications on subdomain Σ^t_i. Let Σ_Δ^t be the subdomain resulting from applying Δ_Σ^t_i on Σ^t_i (Strategy and Domain-Modifying Strategy). A strategy is a tuple ω = (π, Δ) of a plan π and a set Δ of domain modifications. A domain-modifying strategy ω^C involves at least one domain modification, i.e., Δ≠∅.(Context). A context is a tuple ℂ_i = (Σ_i,s_i) representing the agent's subdomain and state at time i. We are now ready to define an insightful strategy as a set of actions and domain modifications that the agent needs to perform to allow for the goal state of the problem to be reachable by the agent. (Insightful Strategy). Let ℂ_i = (Σ^t_i, s_0) be the agent's current context. Let 𝒫_M=(𝕎^t, s_0, g) be an MGP for the agent in this context. An insightful strategy is a domain-modifying strategy ω^I= (π^I, Δ^I) which when applied in ℂ_i results in a context ℂ_j= (Σ^t_j, s_j), where Σ^t_j = Σ^t_Δ^I such that g ⊆ s^', ∀ s^'∈Γ̂_Σ_j^t(s_j). Formalizing the insightful strategy in this way is somewhat analogous to the moment of insight that is reached when a problem becomes tractable (or in our definition computable) or when solution plan becomes feasible. Specifically, solving a problem involves some amount of creative exploration and domain extensions and contractions until the point when the agent has all the information it needs within its subdomain to solve the problem as a classical planning problem, and does not need any further domain extensions. We can alternatively define an insightful strategy in terms of when the goal state is not only reachable, but a solution can be discovered in polynomial time. We will next review simple toy examples to illustrate the concepts discussed thus far.§ OPERATIONALIZING A MACGYVER PROBLEM We will consider two examples that will help operationalize the formalism presented thus far. The first is a modification of the popular Blocks World planning problem. The second is a more practical task of tightening screws, however, with the caveat that certain common tools are unavailable and the problem solver must improvise. We specifically discuss various capabilities that an agent must possess in order to overcome the challenges posed by the examples. §.§ Toy Example: Block-and-Towel World Consider an agent tasked with moving a block from one location to another which the agent will not be able to execute without first discovering some new domain information.Let the agent subdomain Σ consist of a set of locations l = {L1, L2, L3}, two objects o = {T,B} a towel and a block, and a function 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑂𝑓:o → l representing the location of object o. Suppose the agent is aware of the following predicates and their negations: -0.2em* at(o,l): an object o is at location l* near(l): the agent is near location l* touching(o): the agent is touching object o* holding(o): the agent is holding the object o We define a set of actions A in the agent domain as follows: -0.2em* reach(o,l): Move the robot arm to near the object o precond: {at(o,l)} effect: {near(l)}* grasp(o,l): Grasp object o at l precond: {near(l),at(o,l)} effect: {touching(o)}* 𝑙𝑖𝑓𝑡(o,l): Lift object o from l precond: {touching(o),at(o,l)} effect: {holding(o),¬ at(o,l),¬ near(l)}* carryTo(o,l): Carry object o to l precond: {holding(o)} effect: {¬ holding(o), at(o,l)}* release(o,l): Release object o at l precond: {touching(o),at(o,l)} effect: {¬ touching(o), at(o,l)}Given an agent domain Σ, and a start state s_0 as defined below, we can define the agent context ℂ = (Σ, s_0) as a tuple with the agent domain and the start state.s_0 = {at(T,L1), at(B,L2), ¬ holding(T), ¬ holding(B),¬ near(L1), ¬ near(L2), ¬ near(L3), ¬ touching(T),¬ touching (B), 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑂𝑓(B)=L2,𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑂𝑓(T)=L1 } Consider a simple planning problem for the Block-and-Towel World in which the agent must move the block B from location L2 to L3. The agent could execute a simple plan as follows to solve the problem:π_1 = {reach(B,𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑂𝑓(B)), grasp(B,𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑂𝑓(B)), lift(B,𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑂𝑓(B)), carryTo(B,L3), release(B,L3)} During the course of the plan, the agent touches and holds the block as it moves it from location L2 to L3. Using a similar plan, the agent could move the towel to any location, as well.Now, consider a more difficult planning problem in which the agent is asked to move the block from L2 to L3 without touching it. Given the constraints imposed by the problem, the goal state, is not reachable and the agent must discover an alternative way to move the block. To do so, the agent must uncover states in the world 𝕎^t that were previously not in its subdomain Σ. For example, the agent may learn that by moving the towel to location L2, the towel “covers" the block, so it might discover a new predicate covered(o1,o2) that would prevent it from touching the block. The agent may also uncover a new action 𝑝𝑢𝑠ℎ(o,l1,l2) which would allow it to push the object along the surface. To uncover new predicates and actions, the agent may have to execute an insightful strategy ω^I. Once the agent's domain has been extended, the problem becomes a standard planning problem for which the agent can discover a solution plan for covering the block with the towel and then pushing both the towel and the block from location L2 to L3. In order to autonomously resolve this problem, the agent must be able to recognize when it is stuck, discover new insights, and build new plans. Additionally, the agent must be able to actually execute this operation in the real-world. That is, the agent must have suitable robotic sensory and action mechanisms to locate and grasp and manipulate the objects.§.§ Practical Example: Makeshift Screwdriver Consider the practical example of attaching or fastening things together, a critical task in many domains, which, depending on the situation, can require resilience to unexpected events and resourcefulness in finding solutions.Suppose an agent must affix two blocks from a set of blocks b = {B1,B2}. In order to do so, the agent has a tool box containing a set of tools t = {screwdriver, plier, hammer} and a set of fasteners f= {screw,nail}. In addition, there are other objects in the agent's environment o ={towel,coin,mug,duct tape}. Assume the agent can sense the following relations (i.e., predicates and their negations) with respect to the objects [Given space limitations, we have not presented the entire domain represented by this example. Nevertheless, our analysis of the MacGyver-esque properties should still hold.]: -0.2em * isAvailable(t): tool t is available to use * 𝑓𝑎𝑠𝑡𝑒𝑛𝑊𝑖𝑡ℎ(t,f): tool t is designed for fastener f * grabWith(t): tool t is designed to grab a fastener f * isHolding(t): agent is holding tool t * isReachable(t,f): tool t can reach fastener f * isCoupled(t,f): tool t is coupled to fastener f * isAttachedTo(f,b1,b2): fastener f is attached to or inserted into blocks b1 and b2 * isSecured(f,b1,b2): fastener f is tightly secured into blocks b1 and b2.We can also define a set of actions in the agent subdomain as follows: -0.2em * select(t,f): select/grasp a tool t to use with fastener f precond: {isAvailable(t), fastenWith(t,f)} effect: {isHolding(t)} * grab(t,f): Grab a fastener f with tool t. precond: {isHolding(t),grabWith(t)} effect: {isCoupled(t,f)} * placeAndAlign(f,b1,b2): Place and align fastener f, and blocks b1 and b2 effect: {isAttachedTo(f,b1,b2)} * reachAndEngage(t,f): Reach and engage the tool t with fastener f precond: {isHolding(t),fastenWith(t,f), isReachable(t,f)} effect: {isCoupled(t,f)} * install(f,t,b1,b2): Install the fastener f with tool t precond: {isCoupled(t,f),isAttachedTo(f,b1,b2)} effect: {isSecured(f,b1,b2} Now suppose a screw has been loosely inserted into two blocks (isAttachedTo(screw,B1,B2)) and needs to be tightened (¬ isSecured(screw,B1,B2)). Tightening a screw would be quite straightforward by performing actions select(),reachAndEngage(), install(). But for some reason the screwdriver has gone missing (¬ isAvailable(screwdriver)).This is a MacGyver problem because there is no way for the agent, given its current subdomain of knowledge, to tighten the screw as the goal state of isSecured(screw,B1,B2) is unreachable from the agent's current context. Hence, the agent must extend its domain. One approach is to consider one of the non-tool objects, e.g., a coin could be used as a screwdriver as well, while a mug or towel might not. The agent must be able to switch around variables in its existing knowledge to expose previously unknown capabilities of tools. For example, by switching grab(t,f) to grab(t,o) the agent can now explore the possibility of grabbing a coin with a plier. Similarly, by relaxing constraints on variables in other relations, the agent can perform a reachAndEngage(o,f) action whereby it can couple a makeshift tool, namely the coin, with the screw.What if the screw was in a recessed location and therefore difficult to access without an elongate arm? While the coin might fit on the head of the screw, it does not have the necessary elongation and would not be able to reach the screw. An approach here might be to grab the coin with the plier and use that assembly to tighten the screw, maybe even with some additionally duct tape for extra support. As noted earlier, generally, the agent must be able to relax some of the pre-existing constraints and generate new actions. By exploring and hypothesizing and then testing each variation, the agent can expand its domain.This example, while still relatively simple for humans, helps us highlight the complexity of resources needed for an agent to perform the task. Successfully identifying and building a makeshift screwdriver when a standard screwdriver is not available shows a degree of resilience to events and autonomy and resourcefulness that we believe to be an important component of everyday creativity and intelligence. By formulating the notion of resourcefulness in this manner, we can better study the complexity of the cognitive processes and also computationalize these abilities and even formally measure them.§.§.§ Agent Requirements: Intelligence and Physical Embodiment When humans solve problems, particularly creative insight problems, they tend to use various heuristics to simplify the search space and to identify invariants in the environment that may or may not be relevant <cit.>. An agent solving an MGP must possess the ability to execute these types of heuristics and cognitive strategies. Moreover, MGPs are not merely problems in the classical planning sense, but require the ability to discover when a problem is unsolvable from a planning standpoint and then discover, through environmental exploration, relevant aspects of its surroundings in order to extend its domain of knowledge. Both these discoveries in turn are likely to require additional cognitive resources and heuristics that allow the agent to make these discoveries efficiently. Finally, the agent must also be able to remember this knowledge and be able to, more efficiently, solve future instances of similar problems.From a real-world capabilities standpoint, the agent must possess the sensory and action capabilities to be able to execute this exploration and discovery process, including grasping and manipulating unfamiliar objects. These practical capabilities are not trivial, but in combination with intelligent reasoning, will provide a clear demonstration of agent autonomy while solving practical real-world problems.These examples provide a sense for the types of planning problems that might qualify as an MGP. Certain MGPs are more challenging than others and we will next present a theoretical measure for the difficulty of an MGP. §.§ Optimal Solution and M-Number Generally, we can assume that a solvable MGP has a best solution that involves an agent taking the most effective actions, making the required observations as and when needed and uncovering a solution using the most elegant strategy. We formalize these notions by first defining optimal solutions and then the M-Number, which is the measure of the complexity of an insightful strategy in the optimal solution. (Optimal Solutions). Let 𝒫_M=(𝕎^t, s_0, g) be an MGP for the agent. Let π̂ be an optimal solution plan to 𝒫_M. A set of optimal domain modifications is a set of domain modifications Δ̂ is the minimum set of domain modifications needed for the inclusion of actions in the optimal solution plan π̂. An optimal solution strategy is a solution strategy ω̂ = (π̂,Δ̂), where Δ̂ is a set of optimal domain modifications. (M-Number). Let 𝒫_M=(𝕎^t, s_0, g) be an MGP for the agent. Let Ω̂ = {ω̂_1, …, ω̂_n} be the set of n optimal solution strategies. For each ω̂_i ∈Ω̂, there exists an insightful strategy ω̂_i^I ⊆ω̂_i. Let Ω̂^I = {ω̂_1^I, …, ω̂_n^I} be the set of optimal insightful strategies. The set Ω̂^I can be represented by a program p on some prefix universal Turing machine capable of listing elements of Ω̂^I and halting. We can then use Kolmogorov complexity of the set of these insightful strategies, K(Ω̂^I) := min_p ∈𝔹^*{|p| : 𝒰(p) Ω̂^I} <cit.>. We define the intrinsic difficulty of the MGP (M-Number or ℳ) as the Kolmogorov complexity of the set of optimal insightful strategies Ω̂^I, ℳ = K(Ω̂^I). As we have shown MGP-EXISTENCE is intractable and measuring the intrinsic difficulty of an MGP is not computable if we use Kolmogorov complexity. Even if we instead choose to use an alternative and computable approximation to Kolmogorov complexity (e.g., Normalized Compression Distance), determining the M-Number is difficult to do as we must consult an oracle to determine the optimal solution. In reality, an agent does not know that the problem it is facing is an MGP and even if it did know this, the agent would have a tough time determining how well it is doing. §.§ Measuring Progress and Agent SuccessWhen we challenge each other with creative problems,we often know if the problem-solver is getting closer (“warmer") to the solution. We formalize this idea using Solomonoff Induction. To do so, we will first designate a “judge" who, based on a strategy currently executed by the agent, guesses the probability that, in some finite number of steps, the agent is likely to have completed an insightful strategy.Consider an agent performing a strategy ω to attempt to solve an MGP 𝒫_M and a judge evaluating the performance of the agent. The judge must first attempt to understand what the agent is trying to do. Thus, the judge must first hypothesize an agent model that is capable of generating ω.Let the agent be defined by the probability measure μ (ω|𝒫_M, ℂ), where this measure represents the probability that an agent generates a strategy ω given an MGP 𝒫_M when in a particular context ℂ. The judge does not know μ in advance and the measure could change depending on the type of agent. For example, a random agent could have μ(ω) = 2 ^-|ω|, whereas a MacGyver agent could be represented by a different probability measure. Not knowing the type of agent, we want the judge to be able to evaluate as many different types of agents as possible. There are infinitely many different types of agents and accordingly infinitely many different hypotheses μ for an agent. Thus, we cannot simply take an expected value with respect to a uniform distribution, as some hypotheses must be weighed more heavily than others.Solomonoff devised a universal distribution over a set of computable hypotheses from the perspective of computability theory <cit.>. The universal prior of a hypothesis was defined: P(μ) ≡∑_p: 𝒰(p,ω)=μ(ω) 2^-|p|The judge applies the principle of Occam's razor - given many explanations (in our case hypotheses), the simplest is the most likely, and we can approximate P(μ) ≡ 2^-K(μ), where K(μ) is the Kolmogorov complexity of measure μ. To be able to measure the progress of an agent solving an MGP, we must be able to define a performance metric R_μ. In this paper, we do not develop any particular performance metric, but suggest that a performance metric be proportional to the level of resourcefulness and creativity of the agent. Generally, measuring progress may depend on problem scope, control variables, length and elegance of the solution and other factors. Nevertheless, a simple measure of this sort can serve as a placeholder to develop our theory.We are now ready to define the performance or progress of an agent solving an MGP. (Expected Progress). Consider an agent in context ℂ = (Σ_i^t, s_0) solving an MGP 𝒫_M = (𝕎^t, s_0, g). The agent has executed strategy ω comprising actions and domain modifications. Let U be the space of all programs that compute a measure of agent resourcefulness. Consider a judge observing the agent and fully aware of the agent's context and knowledge and the MGP itself. Let the judge be prefix universal Turing machine 𝒰 and let K be the Kolmogorov complexity function. Let the performance metric, which is an interpretation of the cumulative state of the agent resourcefulness in solving the MGP, be R_μ. The expected progress of this agent as adjudicated by the judge is:M(ω) ≡∑_μ∈ U2^-K(μ)· R_μNow, we are also interested in seeing whether the agent, given this strategy ω is likely to improve its performance over the next k actions. The judge will need to predict the continuation of this agent's strategy taking all possible hypotheses of the agent's behavior into account. Let ω^+ be a possible continuation and let ^ represent concatenation. M(ω^ω^+ |ω ) = M(ω^ω^+)/M(ω) The judge is a Solomonoff predictor such that the predicted finite continuation ω^+ is likely to be one in which ω^ω^+ is less complex in the Kolmogorov sense. The judge measures the state of the agent's attempts at solving the MGP and can also predict how the agent is likely to perform in the future.§ CONCLUSION AND FUTURE WORK In the Apollo 13 space mission, astronauts together with ground control had to overcome several challenges to bring the team safely back to Earth <cit.>. One of these challenges was controlling carbon dioxide levels onboard the space craft: “For two days straight [they] had worked on how to jury-rig the Odyssey’s canisters to the Aquarius’s life support system. Now, using materials known to be available onboard the spacecraft – a sock, a plastic bag, the cover of a flight manual, lots of duct tape, and so on – the crew assembled a strange contraption and taped it into place. Carbon dioxide levels immediately began to fall into the safe range.” <cit.>.We proposed the MacGyver Test as a practical alternative to the Turing Test and as a formal alternative to robotic and machine learning challenges.The MT does not require any specific internal mechanism for the agent, but instead focuses on observed problem-solving behavior akin to the Apollo 13 team. It is flexible and dynamic allowing for measuring a wide range of agents across various types of problems. It is based on fundamental notions of set theory, automated planning, Turing computation, and complexity theory that allow for a formal measure of task difficulty. Although Kolmogorov complexity and the Solomonoff Induction measures are not computable, they are formally rigorous and can be substituted with computable approximations for practical applications.In future work, we plan to develop more examples of MGPs and also begin to unpack any interesting aspects of the problem's structure, study its complexity and draw comparisons between problems. We believe that the MT formally captures the concept of practical intelligence and everyday creativity that is quintessentially human and practically helpful when designing autonomous agents. Most importantly, the intent of the MT and the accompanying MGP formalism is to help guide research by providing a set of mathematically formal specifications for measuring AI progress based on an agent's ability to solve increasingly difficult MGPs. We thus invite researchers to develop MGPs of varying difficulty and design agents that can solve them. named | http://arxiv.org/abs/1704.08350v1 | {
"authors": [
"Vasanth Sarathy",
"Matthias Scheutz"
],
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"cs.AI"
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"primary_category": "cs.AI",
"published": "20170426210527",
"title": "The MacGyver Test - A Framework for Evaluating Machine Resourcefulness and Creative Problem Solving"
} |
[pages=1-last]ECML_submission_002.pdf | http://arxiv.org/abs/1704.08488v1 | {
"authors": [
"Dieter Hendricks",
"Stephen J. Roberts"
],
"categories": [
"q-fin.CP",
"cs.LG",
"stat.ML"
],
"primary_category": "q-fin.CP",
"published": "20170427092850",
"title": "Optimal client recommendation for market makers in illiquid financial products"
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[email protected] Max-Born-Institut, Max Born Strasse 2A, D-12489 Berlin, GermanyRussian Quantum Center, Skolkovo 143025, RussiaMoscow Institute of Physics and Technology, 9 Institutsky lane, Dolgoprudny, Moscow region 141700, RussiaRussian Quantum Center, Skolkovo 143025, RussiaDepartment of Physics, Moscow State University, 119991 Moscow, RussiaMax-Born-Institut, Max Born Strasse 2A, D-12489 Berlin, GermanyTechnische Universitaet Berlin, Ernst-Ruska-Gebaeude, Hardenbergstr. 36A,10623, Berlin, Germany [email protected] Max-Born-Institut, Max Born Strasse 2A, D-12489 Berlin, GermanyBlackett Laboratory, Imperial College London, South Kensington Campus, SW7 2AZ London, United KingdomDepartment of Physics, Humboldt University, Newtonstrasse 15, 12489 Berlin, GermanyThis Letter brings together two topics that, until now, have been the focus of intense but non-overlapping research efforts. The first concerns high harmonic generation in solids, which occurs when intense light field excites highly non-equilibrium electronic response in a semiconductor or a dielectric. The second concerns many-body dynamics in strongly correlated systems such as the Mott insulator. Here we show that high harmonic generation can be used to time-resolve ultrafast many-body dynamics associated with optically driven phase transition, with accuracy far exceeding one cycle of the driving light field.Our work paves the way for time-resolving highly non-equilibrium many body dynamics in strongly correlated systems, with few femtosecond accuracy.78.47.J, 71.27.+a, 42.65.KyHigh harmonic imaging of ultrafast many-body dynamics in strongly correlated systems M. Ivanov December 30, 2023 ====================================================================================High harmonic emission provides the frequency domain view of charge dynamics in quantum systems <cit.>. Complete characterization of the emitted harmonic light – its spectrum, polarization, and spectral phase – allows one to decode the underlying charge dynamics in atoms and molecules with resolution <0.1 femtosecond (fs), well below a single cycle of the driving laser field, opening the new field of ultrafast high harmonic imaging <cit.>.Here we bring high harmonic imaging to many-body dynamics in strongly correlated solids, focusing on an ultrafast phase transition. We consider the breakdown of the Mott insulating state in the canonical model of a strongly correlated solid, the Fermi-Hubbard model. We show how the complex many-body charge dynamics underlying this transitionare recorded by high harmonic emission, with few fs accuracy.Combining high harmonic generation in bulk solids <cit.> with robust techniques for characterizing the emitted light <cit.> has demonstrated the capabilities of high harmonic imaging in solids. Recent results include the demonstration of the dynamical Bloch oscillations <cit.>, the reconstruction of the band structure in ZnO <cit.>, the visualization of strong field-induced effective band structure <cit.>, and direct visualization of the asymmetric charge flow driven by MIR fields <cit.>. Yet, all of these studies have been confined to systems well described by single-particle band structure, and single-particle pictures have dominated the analysis <cit.>. The role of electron-electron correlation has been relegated to empirically introduced (and unusually short, only a few fs) relaxation times <cit.>. Yet, electron-electron correlations go well beyond mere dephasing, generating rich physics of strongly correlated systems, such as pre-thermalization and the formation of extended Gibbs ensembles <cit.>, superfluid to Mott insulator transition <cit.>, to name but a few.Studies of non-equilibrium many body dynamics have lead to the concepts of the dynamical <cit.> and light-induced phase transitions <cit.>. In particular, the Mott insulator-to-metal transition was recently achieved experimentally in VO_2 <cit.>. Resolving such transitions with few-femtosecond accuracy remains, however, elusive. Our results show that high harmonic imaging is ideally suited to address this challenge, offering detailed view of the underlying dynamics.We use one-dimensional Fermi-Hubbard model with half-filling, i.e. with averaged particle density equal to one per site (see Methods for details). A particle could freely hop to an adjacent site with a rate t_0, yielding the metallic state of the system. Hopping can be obstructed by another particle already residing on the adjacent site, via the energy U of the repulsive on-site interaction. In the strong coupling limit U≫ t_0, the Mott insulating ground state has short-range antiferromagnetic order <cit.> (the electron spins at the adjacent sites tend to be anti-parallel). The elementary charge excitations, called doublon-hole pairs <cit.>, are separated by an optical gap Δ. Deep into repulsive regime (U≫ t_0), Δ scales linearly with U. We focus on this regime, and consider the driving field ω_L≪Δ(U) in the mid-IR range, with ω_L=32.9 THz identical to that in recent experiments <cit.>, and a modest peak amplitude F_0=10 MV/cm. The hopping rate is set to t_0=0.52 eV to mimic Sr_2CuO_3 <cit.>, and U is varied to demonstrate the trends and the general nature of our conclusions. Details of our simulations are described in the Methods section.To induce and resolve the Mott transition, we apply a light pulse where the field amplitude F_0(t) increases smoothly.This field can excite the doublon-hole pairs, which play the role of carrier chargers. The density of doublon-hole pairs may change during the pulse, depending on the field amplitude F_0. As F_0 crosses the threshold F_TH, the density of charge carriers exceeds critical value, leading to the breakdown of the Mott insulator – the system becomes conducting. The transition is followed by the destruction of local magnetic order and a paramagnetic liquid-like state is formed <cit.>. Smooth variation of F_0(t) during the pulse allows us to track the transition as a function of time, with high harmonic response providing sub-cycle accuracy (see below). The transition is mathematically similar to strong-field ionization in atoms <cit.>. In particular, the parameter γ=ħω_L/ξ F_0 (where ξ is the correlation length <cit.>) serves as the analogue of the Keldysh adiabaticity parameter <cit.>. In the `tunnelling' regime γ≪1 the threshold field is <cit.>F_ TH=Δ/2eξAs the insulator-to-metal transition is marked by the increased density of charge carriers and the destruction of short-range magnetic order, we will characterize the state of the system via the two parameters describing the charge and spin degrees of freedom: the next-neighbor spin-spin correlation function η=1/L⟨∑_j=1^LS⃗_j.S⃗_j+1⟩and the average number of doublon-hole pairs per site, D=1/L⟨∑_j=1^Lc_j,↑^†c_j,↑c_j,↓^†c_j,↓⟩Here j labels the site, up-down arrows the spin, L=12 is the number of cites, c^†,c are the creation and annihilation operators.The destruction of short range antiferromagnetic order during the transition is shown in Fig. 1 (a): within a cycle, the spin-spin correlation function drops to nearly zero (limited by the finite size of the system). The second signature of the transition is the rise in the number of doublon-hole pairs, Fig.1 (b), which is linked to the loss of spin-spin correlation (compare Figs. 1 (a,b)). After the transition, the system reaches a photo-induced saturated state <cit.> and the number of pairs remains constant. The abrupt nature of the transition is shown in Fig. 1(c): for fields crossing F_ TH the overlap probability with the initial state W(t)=|⟨Ψ_0|Ψ(t)⟩|^2 drops to zero within a laser cycle, stressing the need for sub-cycle resolution.Naturally, the rise of optical charge excitations has to manifest in the optical response. Indeed, we find that the transition is accompanied by very characteristic high-harmonic emission, see Fig.2.Fig. <ref> (a) shows harmonic spectra for two different values of U/t_0. In the conducting limit U/t_0=0 the emission is typical for single band tight binding model <cit.>, demonstrating clear low-order Bloch oscillation-type harmonics associated with intra-band current (the intra-band harmonics). As expected, the harmonics are narrow and well defined. In the case of U/t_0≫1, the spectrum is quite unusual.First, the intra-band harmonics are strongly suppressed, in stark contrast with systems described by single-particle band structures (e.g. <cit.>). Second, for the same F_0, the spectrum becomes much broader and shifts towards orders N∼ U/ω_L. This characteristic change is summarized in Fig. <ref> (b), where we scan U/t_0 to demonstrate the trend. For U/t_0≫1, the spectrum peaks near the characteristic energies of doublon-hole excitation. In the half-filled system, the first allowed charge excitations are states with single doublon-hole pairs with energies between Δ and Δ+8t_0 <cit.>. Fig.2(b) shows that these excitations are the ones responsible for the harmonic emission. Indeed, their range, shown with red lines in Fig. <ref> (b), defines the lower and upper frequencies for the harmonic emission. Thus, the emission corresponds to the one-photon transition that brings the excited system back to its initial ground state via the doublon-hole recombination.Third, the regular structure of the harmonic lines is lost in the strong coupling limit. This stands in stark contrast to weakly correlated systems, where electron-electron correlation is expected to introduce fast dephasing, the latter yielding regular, narrow harmonic lines <cit.>. Figs.2(c,d) clarify the physics responsible for the irregular structure of the spectrum.Figs. <ref> (c,d) show the time profile of the harmonic emission, obtained via the Gabor transform (see Methods). Note that complete time-domain reconstruction of the emitted harmonic light with ∼1-2 fs accuracy is fully within the available experimental technology <cit.>, for the same laser parameters as in our calculation. We see that (i) the onset of the harmonic emission is synchronized with the breakdown of the insulating Mott state and the rise in the number of doublon-hole pairs, and (ii) the fall of the emission follows the depletion of the insulating state (Fig.2(c,d)), following the fidelity Ξ(t)=|⟨Ψ_0|Ψ(t)⟩|. The Gabor profiles in Fig. 2(c,d) show that the emission takes about 50-70 fsec, i.e. only about 1-2 cycles of the driving field. The temporal restriction of the emission to a couple of cycles of the driving field explains the lack of clear peaks at odd harmonics. The complexity of the spectrum affirms the strongly aperiodic many-body dynamics, in contrast to the periodic intraband motion in the limit U/t_0≪1 (see Fig.2(a)). The top panels in Figs. 2(c,d) confirm the conclusions drawn from Fig. 2(b): the emission relies on the coherence created between the Mott insulator ground state and the doublon-hole states. This is why it starts when the doublon-hole pairs are created and ends when the ground, Mott insulator state, is destroyed.The lack of low-order harmonics after the phase transition leads to another important conclusion: the new many-body state created by the photo-induced transition does not support Bloch-like oscillations of doublon-hole pairs. Indeed, these would have generated strong low-order harmonics familiar from one-electron-type excitations in a conventional conduction band. Loosely speaking, this happens because all quasi-momenta states for the doublon-hole pairs are occupied upon the transition in the tunneling regime <cit.>.In contrast to high harmonic emission in systems with single-particle band structure <cit.>, the cutoff of the harmonic signal associated with a phase transition in a strongly correlated system does not scale linearly with the electric field. In our case the harmonic emission has threshold behavior and, at F_0≥ F_ TH, covers all energies between Δ and Δ+8t_0 irrespective of the field. We also find that strong electron-electron correlations do not necessarily lead to the emergence of regular harmonic spectra with well defined lines, as expected for weakly correlated systems. Highly irregular harmonic spectra imply highly aperiodic dynamics, in line with the dramatic change in the state of the system during a phase transition.High harmonic generation has been pioneered three decades ago <cit.>, evolving from an unusual table-top source of bright, coherent XUV light to the technological backbone of attosecond science <cit.> and a unique tool for imaging ultrafast dynamics with attosecond to few-femtosecond temporal resolution <cit.>. Yet, throughout these decades, the analysis of high harmonic generation has been rooted in effectively single-electron pictures. Our work is the first to bring fundamental strongly correlated many-body dynamics squarely into its view.We gratefully acknowledge fruitful discussions with Dr. Takashi Oka, Dr. Bruno Amorim, and Dr. Peter Hawkins. M. I. and R. E. F. S. acknowledge the support from the MURI programme. § METHODS We study high harmonic generation in the one-dimensional, half-filled Fermi-Hubbard model by solving the time dependent Schrdinger's equation (TDSE) numerically exactly, fully including the electron-electron correlations in the system interacting with intense light field. We use the 1D Fermi-Hubbard Hamiltonian <cit.> Ĥ(t) = -t_0∑_σ,j=1^L(e^-iΦ(t)c_j,σ^†c_j+1,σ+e^iΦ(t)c_j+1,σ^†c_j,σ) + U∑_j=1^Lc_j,↑^†c_j,↑c_j,↓^†c_j,↓.where the laser electric field F(t)=-dA(t)/dt enters through the time-dependent Peierls phase eaF(t)=-dΦ(t)/dt, a is the lattice constant and A(t) is the field vector potential. The hopping parameter t_0 is set to t_0=0.52 eV to mimic Sr_2CuO_3 <cit.>, and U>0 is the on-site Coulomb repulsion. In the calculations, we use the periodic boundary conditions c_j,σ=c_j+L,σ with L=N=12, N being the number of particles, and focus on the S_z=0 subspace. Starting at t=0 from the ground state of the Hamiltonian, we apply the pulse with A(t)=A_0f(t)sin(ω_Lt) at the carrier wavelength of 9.11μm (ω_L=32.9 THz) and the peak amplitude F_0=ω_LA_0=10 MV/cm. All the parameters of the pulse are well within the experimental reach. The pulse has total duration of 10 optical cycles and a sin^2 envelope, and is shown in Fig.1To compute the harmonic emission, we first use the electric current operator, defined as <cit.> J(t) = -ieat_0∑_σ∑_j=1^L(e^-iΦ(t)c_j,σ^†c_j+1,σ-h.c.).to compute the time-dependent current. The harmonic spectrum is calculated as the square of the Fourier transform of the dipole acceleration, a(t)=d/dtJ(t). 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"authors": [
"R. E. F. Silva",
"Igor V. Blinov",
"Alexey N. Rubtsov",
"O. Smirnova",
"M. Ivanov"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170427081744",
"title": "High harmonic imaging of ultrafast many-body dynamics in strongly correlated systems"
} |
inst1]Bogang Juninst1]Flávio L. Pinheiroinst2,inst3] Tobias Buchmanninst4]Seung-kyu Yiinst1]César A. Hidalgocor1 [cor1]Email address: [email protected] [inst1]Collective Learning Group, MIT Media Lab, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [inst2]Chair for Innovation Economics, University of Hohenheim, D-70593 Stuttgart, Germany [inst3]Center for Solar Energy and Hydrogen Research Baden-Württemberg (ZSW), D-70565 Stuttgart, Germany [inst4]Technology Management, Economics and Policy Program, Seoul National University, Seoul, South Korea In 1990, Germany began the reunification of two separate research systems. In this study, we explore the factors predicting the East-West integration of academic fields by examining the evolution of Germany's co-authorship network between 1974 and 2014. We find that the unification of the German research network accelerated rapidly during the 1990s, but then stagnated at an intermediate level of integration. We then study the integration of the 20 largest academic fields (by number of publications prior to 1990), finding an inverted U-shaped relationship between each field's East or West "dominance" (a measure of the East-West concentration of a field's scholarly output prior to 1990) and the fields' subsequent level of integration. We checked for the robustness of these results by running Monte Carlo simulations and a differences-in-differences analysis. Both methods confirmed that fields that were dominated by either West or East Germany prior to reunification integrated less than those whose output was balanced. Finally, we explored the origins of this inverted U-shaped relationship by considering the tendency of scholars from a given field to collaborate with scholars from similarly productive regions. These results shed light on the mechanisms governing the reintegration of research networks that were separated by institutions.Knowledge DiffusionCollective LearningResearch Collaboration German Reunification § INTRODUCTIONHistory contains many examples of two independent states forming a new unified entity. Take, for instance, the unification of Spain in the fifteenth century and Italy in the nineteenth century, the creation of Yugoslavia in 1918, the unification of Vietnam in 1975, the formation of the United Arab Emirates in 1971 or, more recently, the reunification of Germany in 1990. Among these examples, the reunification of Germany, which is associated with the final years of the Cold War, is somewhat unique, since it provides an opportunity to understand the consequences of the unification of two states with separate research systems, and institutions, but a common language and geography.According to <cit.>, the reunification of Germany through the so-called Two-Plus-Four Treaty of 1990 was politically and economically beneficial to the country in the sense that it recovered its sovereignty and became the strongest economy in the EU. Nevertheless, the integration of the two blocks was neither easy nor cheap.Since 1990, the German government has spent 270 billion on a reunification process that is aimed primarily at equalizing standards of living between the East and West; however, despite such immense expenditure, in 2014 the nominal per capita GDP of East Germany was two-thirds that of West Germany (24,324 versus 36,280) <cit.>. Furthermore, East Germany has experienced a significantly higher unemployment rate than West Germany.But what factors can explain the prevalence of the East-West German gap? One possibility, according to <cit.>, is that the reunification process is governed not just by formal institutions, but also by organic social and economic forces, like those governing the formation of social networks. Social network formation is ruled by mechanisms, such as triadic closure, homophily, and shared social foci <cit.>, which cannot be easily modified through government intervention. According to these theories, we should expect German reunification to be a relatively slow process constrained by pre-existing social structures and the mechanisms known to rule the formation of social and professional networks.Inprinciple, the unification of two previously independent states affects multiple social and economic outcomes. Here, we focus on just one of them: the reunification of Germany's research-collaboration network. To explore this reunification we conducted a longitudinal analysis of this network using pairs of NUTS 3 regions (small geographical areas that divide Germany into 429 regions) as nodes, and measured the volume of collaboration (co-authorships) between scholars from East and West Germany for each pair of NUTS 3 regions as links. We chose to focus on co-authorship networks because these are considered to be good proxies for social and professional relationships <cit.>. According to <cit.>, deep social interactions facilitate learning, trust, and the ability to communicate complex ideas, all of which are prerequisites for effective scholarly collaborations.In this paper, we specifically explore the conditions that led to the successful reunification of academic fields in Germany by examining the evolution of the co-authorship network connecting scholars from East and West Germany between 1974 and 2014. To begin, we measure the unification of this network by studying the evolution of its modularity (a method to estimate the compartmentalization of a network). Our results show that the reunification of Germany's research network initially occurred rapidly during the 1990s, with many new collaborations between East- and West-German institutions, but has stagnated since then, remaining at an intermediate level of integration.Next, we examined the 20 largest academic fields (which were determined by analyzing the number of publications produced in each field prior to reunification) and studied whether fields dominated by East or West Germany prior to reunification featured more successful integration of their co-authorship networks. Consequently, we found that neither West- nor East-German-dominated fields integrated most successfully. The fields that integrated the mostboasted a more balanced output between the two sides prior to reunification. This finding is supported by our differences-in-differences (DID) analysis and Monte Carlo simulations. The latter were used to test whetherthe balance of field, and not just its size or spatial distribution, was a significant predictor of subsequent integration. Finally, we explored why the balance of a field should be associated with its subsequent integration by checking the tendency of scholars to co-author papers with other scholars from similarly ranked regions. Using numerical simulations, we found that, when this tendency was absent, the U-shaped relationship disappeared, suggesting that the homophily observed in these mixing patterns contributes to the different speeds of reunification between fields with balanced and unbalanced outputs. § HISTORICAL BACKGROUND OF THE REUNIFICATION OF THE GERMAN RESEARCH SYSTEMGermany's research system was divided after the Second World War; however, in 1990 a reunification process suddenly began. After Germany lost the Second World War, the country was occupied by allied forces and divided into four military occupation zones, which were divided between the United States, the Soviet Union, France, and the United Kingdom <cit.>. Later, as the clash between communism and capitalism manifested into bitter competition between the United States and the Soviet Union, both countries realized that they needed "their" Germany to represent an example of European economic recovery and, by extension, their form of governance <cit.>.In 1949, the Soviet Union took steps to consolidate its influence in Europe and transformed its zone of occupation into the German Democratic Republic (GDR). By this time the GDR already possessed a centrally planned economy and a centralized research system. In the same year, the three occupation zones occupied by the United States, the United Kingdom, and France were unified into the Federal Republic of Germany (FRG), and a social market economy and a federal system that included research and education were introduced. During the 1950s, an increasing number of people seeking a better future migrated from East to West Germany; however, this border-crossing movement was eventually halted by the socialist regime with the construction of tough border controls and the construction of the Berlin Wall in 1961. While the socialist economic and scientific system achieved some success, East Germany never reached the strength of West Germany, and after almost 30 years of physical separation, it collapsed in 1989 <cit.>. During this momentous year, the GDR celebrated its 40th anniversary on October 7th, and on November 9th, only one month later, a peaceful revolution occurred, which resulted in East Germany unexpectedly opening its borders and allowing its citizens to enter West Berlin and West Germany. These developments led to German reunification in 1990 <cit.>. The above shows that the reunification process occurred somewhat suddenly, and was not greatly anticipated by economic or scientific actors.In the aftermath, a treaty concerning the economic, monetary, and social union was adopted. This facilitated economic reunification and served as a master plan for the introduction of the social market economy in the former GDR. However, the process of convergence did not occur in the expected manner. East Germany had inefficient industry, obsolete equipment, and most of its products were not competitive in the international market; thus, former East German regions experienced the liquidation of many businesses, high rates of unemployment, and the transfer of skilled labor to Western Germany. Although in 1991 the government implemented a support program for East German regions called "Gemeinschaftswerk Aufschwung Ost," which was designed to foster the economic recovery of the East. Article 38 of the 1990 German Unification Treaty stipulates the creation of an integrated German R&D system that will perform as well as the old West German system. This shows that elements of the GDR system that performed well should have been preserved. In July 1990, West and East German research ministers agreed on the basic elements of a unified research system: "It is our aim to create an integrated research system in a unified Germany. Moreover, the incorporation of the institutions combined under the East German Academy of Sciences into this research scene will be a central task" <cit.>. As evidenced by the rapid reduction in R&D personnel, these goals were not realized in the short term, and part of the research structure has since been dissolved.Between 1989 and 1993, the number of R&D staff in East Germany decreased to a level as low as 30% of its initial figure. Indeed, the level of employment in R&D relative to the total number of people employed in East Germany quickly fell below 50% of its West German equivalent whereas, in 1989, there were no major differences in the ratios for the two states. Most existing universities were preserved; however, research at universities was reduced by 30%, meaning East Germany accounted for, at best, 10% of the total research capacity in the German higher education system, despite the region accommodating 20% of the population. In the GDR, non-university R&D institutions dominated; in particular, the East German Academy of Sciences, the Academy of Agricultural Sciences, and the Building Academy. For example, the Academy of Sciences' mission was to conduct basic research and use the results in applied research. This organization's important role was substantiated by the fact that R&D-intensive industries, such as the chemical industry and electronics and mechanical engineering, had less R&D personnel than equivalent West German sectors; consequently, academy scientists often performed contract research for different industries. However, once the total number of R&D personnel is taken into account, the performance of the East German R&D system is found to be significantly lower when compared to that of West Germany <cit.>.Numerous programs were implemented in an attempt to accelerate the slow convergence between the two systems. According to <cit.> and the Federal Ministry of Education and Research <cit.> of Germany, the period after reunification in 1990 can be divided into three policy regimes: (i) the reconstruction of the old system between 1990 and 1997, (ii) the introduction of a new system between 1998 and 2006, and (iii) its respective stabilization since 2007. During the initial years, these programs mainly aimed at restructuring the scientific landscape of Eastern Germany. Only in the second and, in particular, the third phase was greater emphasis placed on collaboration and knowledge transfer between east and west. These observations support the design of our study as a kind of natural experiment of the integration of two formerly separated networks. Indeed, until the end of the 1990s, innovation politics focused on fostering firms' R&D activities and innovative entrepreneurship in East Germany. Since then, however, a re-adjustment of the policy approach based on knowledge transfer and network formation has been implemented <cit.>. One good example of this new policy approach is the InnoRegio program, which operated from 1999 to 2006, as this aimed to boost competition among 23 networks of firms and research facilities. This prominent policy was based on the idea that innovation is not driven by a single individual or single Schumpeterian entrepreneur, but rather by networks consisting of various participants, organizations, and institutions. Its main objective was to improve the transfer of knowledge and technology between East German regions by building networks that had a special focus on SMEs (since the main actors in East Germany were SMEs), rather than on the large companies that had been the main actors in West Germany. In another development, the Innovationsforen program was introduced, which also supported the early phases of innovation networks in East Germany in an attempt to strengthen the development of a thematic focus and collaborative relations. Another example is a program called Entrepreneurial Regions (Unternehmen Region), which was implemented in 2004. This policy also aimed to form strong interlinked regions to promote a free exchange of knowledge. Key elements of this renewed strategy are lateral thinking, cooperation, strategic planning, and entrepreneurial action. To implement this new strategy, several initiatives have been deployed to promote national, inter-, trans-, and multidisciplinary cooperation between partners and encourage openness and transparency. Under these initiatives, networks are to be formed across East Germany that feature one or more partners from West Germany but which include a project leader from the east. Our expectation is that these initiatives have already triggered many new collaborative ties between East and West German researchers.§ DATA As material for this study, we used publication data from the journals listed in the Science Citation Index (SCI) of Web of Science (WoS). We considered all article types (journal articles, conference proceedings, reviews, letters, news, and book reviews) published between 1972 and 2014. Our final data contain all papers for which at least one author had a German address. Further, we also collected article IDs, author's addresses, the fields of study, and the institutions to which the authors belong. In total, our dataset consists of 2,897,527 papers. Since we were focusing on papers that connect regions, we generally used papers that included at least two authors based in two different regions (constituting 1,371,639 of the total number), but when we calculated the level of dominance of a region in certain field (we will explain the measure of dominance later in this paper), we used all data, including papers with a single author. Moreover, we noticed a large, unexpected jump in the number of publications between 1973 and 1974 and, consequently, we decided to omit data from 1972 and 1973.In our analysis, two regions were connected when authors from those regions had published a research article together. Therefore, in our collaboration network, nodes represent regions and links correspond to the number of papers that include authors from each region, as depicted in Figure <ref>. Throughout the remainder of this paper, the regions that belonged to former GDR regions are labeled as "East Germany," while former FRG regions are labeled as "West Germany." Considering that our research interest concerns collaborations between East and West, we ignored other collaborations with researchers from foreign regions. For example, if a paper was co-authored by three authors, with addresses in West Germany, East Germany, and the US; we recorded the collaboration between West and East Germany and ignored the collaboration between West Germany and the US and between East Germany and the US. We agree that the role of such third parties is interesting, but it goes beyond the scope of our current study. During the time period analyzed, we found that the number of German regions linked with NUTS 3 regions in at least one paper increased from 169 (39%) in 1974 to 391 (91%) in 2014. Additionally, the number of pairs of German regions with at least one co-authored paper also increased during this period, from 703 (0.7% of all possible pairs) in 1974 to 12,240 (13.3% of all possible pairs) in 2014. Figure <ref> shows the network of relationships between German regions that existed in 1985 and in 2000. White lines represent the links between East and West, while red and blue lines represent the collaboration links within East and West, respectively. As can be seen from the figure, there are few white lines (East-West collaborations) in 1985, but many in 2000. To depict the number of collaborations over time, we plot Figure <ref>, showing the evolution of the collaborations within East Germany (red) and West Germany (blue), and between West and East Germany (black) from 1974 to 2014. Additionally, the gray line shows the number of collaborations between West and East Germany estimated by a null model created through Monte Carlo simulations. In this null model, real data is used to connect each region to the other regions (i.e., each node's degree remains consistent between the real network and the null model), but collaborations are assigned at random, subject to this constraint (further information concerning the null model is provided in Appendix B). By comparing the gray and black lines we can determine if the volume of collaboration is higher or lower than our estimations. The true figure obtained shows that the volume of East-West collaborations was drastically lower than that of the null model prior to 1990 and then increased between 1990 and 1997 (approaching the value of the null model); since then, it has remained below the null model, implying that the maximum potential level of collaboration between East and West Germany was not achieved. § METHODS AND RESULTS§.§ Measuring the speed of network unification using modularitySince our aim is to analyze the relationship between the dominance of a field and the speed of unification, we measure the speed of unification and the level of dominance for each field. First, to measure the unification of the German research network, we employ a network-analysis tool called modularity <cit.>. Modularity <cit.> is a tool for the detection and characterization of communities in networks (groups of densely interconnected nodes with relatively few connections to other groups). Community detection is a popular topic among network scientists since communities may correspond to social or functional units in networks <cit.>. We estimate modularity following the seminal work of <cit.>. <cit.>'s theory is that "if the number of links between groups is significantly less than we expect by chance, or equivalent if the number of within groups is significantly more," we can say that a network exhibits a community structure. Consequently, modularity Q, is "the number of edges falling within groups minus the expected number in an equivalent network with edges placed at random." Formally, modularity Q is defined as: :𝒬 = 1/4∑_ij( A_ij - k_i k_j/2 m) ( s_i s_j + 1 ) where A_ij is the adjacency matrix of the network; k_i and k_i are the degree of nodes i and j, respectively; and m is the total number of links, which is equal to 1/2∑_ik_i. Supposing two groups exist in a network, s_i is +1 if node i belongs to group 1, and s_j is -1 if node j belongs to group 2.To study the modularity of the German research network we use equation (<ref>) with s_i=+1 for nodes belonging to West Germany and s_j=-1 for nodes belonging to East Germany. To establish a benchmark value of modularity, we created maximum and random modularity by allocating s_i and s_j to either maximize modularity, or at random. In both benchmarks we keep the same number of s_i = +1 and s_j = -1 than in the original network. To smooth out fluctuations, we generate 500 observations of random modularity for each year and average values and estimate a 95% confidence interval. Figure <ref> shows the evolution of the modularity of the German research network in comparison with two benchmarks: maximum and random modularity. As shown in Figure <ref>, before reunification, the level of modularity was closer to the maximum benchmark, but it began decreasing after reunification in 1990. After 2000, however, modularity increased again and has been between the two benchmarks ever since. Even though the total modularity of the German research network, and the benchmarks, has been decreasing during the last 15 years. Figures <ref> and <ref>, tell us that the quick integration in the 1990s stagnated at the end of the twentieth century. To measure the speed of unification for each field, we define the speed as a percentage decrease in modularity, since lower value of modularity is corresponding to higher lever of integration. Speed_p = - 𝒬_t+Δ t-𝒬_t/𝒬_t· 100where Speed_p is the speed of unification of field p, 𝒬_t and 𝒬_t+Δ t are modularity at time t and t+Δ t, respectively. §.§ Measuring the balance and dominance of academic fields Next, we explore the factors that explain which fields achieved higher levels of reunification. We begin by measure the balance or dominance of each field prior to 1990. We do this by first identifying the regions which had revealed comparative advantage (RCA) in each field, following the method described in <cit.> and <cit.>. The RCA of a region for a particular field is defined as the ratio between the observed number of publications for that region in that field and the number of publications expected, given the total academic output of that region and the total number of papers produced in that field. Formally, the RCA of region r in field p is (RCA is identical to what urban planners refer to as location quotient: LQ):RCA_r,p = .x_r,p/∑_px_r,p/ ∑_rx_r,p/∑_r,px_r,p.where x_r,p is the number publications of region r in field p. Using this method, we estimated the RCA for each region and the 39 broad subject areas featured on the WoS.We then selected the 20 fields with the largest number of publications before 1990. These fields are: clinical medicine; biological science; chemical science; physical science; basic medicine; materials engineering; other engineering and technologies; mathematics; health sciences; mechanical engineering; electrical, electronic, and information engineering; earth sciences and related environmental sciences; other natural sciences; veterinary science; agriculture, forestry, and fisheries; chemical engineering; computer and information sciences; medical engineering; environmental engineering; and animal and dairy science.1.5 Figure <ref> shows the regions with a positive RCA in chemical engineering (A), material engineering (B), and veterinary science (C). Each dot in Figure <ref> represents a region with RCA greater than or equal to 1, with regions in East Germany depicted by red dots and those in West Germany with blue dots. Material engineering is an example of a field dominated by West Germany prior to reunification, while Veterinary Science/ Chemical engineering is an example of a field dominated by East/West Germany.Next, we estimated the dominance of each region for each academic subject by determining if, prior to 1990, most regions with a positive RCA in that field were located in East or West Germany. Formally, we defined the East or West dominance of a field as:Dominance_p= # regions in East with RCA ≥ 1/71 -# regions in West with RCA ≥ 1/270 In equation <ref>, we adjust for the different sizes of East and West Germany by the number of regions in West (270) and East (71) Germany that had published at least one article in those fields before 1990. §.§ Connecting dominance and modularity 1.5 Next, we connect the change in modularity of a field with its dominance. Figure <ref> shows in red the relationship between the reduction in modularity in a field (y-axis) (a measure of subsequent integration) and its dominance prior to 1990 (x-axis). As a counterfactual, we present the values produced by the null model created through the Monte Carlo simulations (Green). In this Monte Carlo simulation we rewire links between regions for each field by preserving their total connectivity. This helps us compare the changes in modularity observed with those expected for the same regions if they had randomly connected to others.Comparing the results obtained from the original data with the Monte Carlo counterfactual reveals an inverted U-shaped relationship between the reduction in modularity of a field and the balance of that field prior to reunification. West- or East-skewed fields, such as chemical engineering, health science, agriculture, and veterinary science, experienced smaller changes in modularity afterreunification. More balanced fields, such as basic medicine, bio science, and material engineering, tended to reduce their modularity more. The Monte Carlo simulations help us test the significance and robustness of these results by checking whether the inverse U relationship should be expected simply from the variation in balance between the fields. For instance, one could expect that a field that is dominant in the West would simply have no regions to collaborate with in the East, and hence, could not experience a large change in its modularity. The Monte Carlo simulations tell us that the regions that were active in these fields, while preserving the number of connections they had, would have experienced much larger decreases in modularity than if they would have connected randomly to others. The lack of an inverse U relationship in the Monte Carlo simulation help us unsure that these reductions are characteristics of the evolution of the network of research collaborations, and not a relationship that should be expected from the definition of the variables or the network's starting conditions (see Appendix for details). Moreover, we check for robustness by repeating the exercise using the share of East Germany in the total number of publications in each field as a measure of dominance, obtaining a similar result (Figure <ref>).§.§ Checking robustness using the differences-in-differences method Next, we check for the robustness of the inverted U-shaped by using a differences-in-differences (DID) regression. To perform a DID analysis we requires two groups: a treatment and a control group. In this paper, the treatment group includes all publications in the 10 most balanced fields, while the control group uses data on the 10 fields where the output was skewed to either East or West Germany.One might worry about the possibility of selection bias in establishing the two groups, since DID is most appropriately used when an event is close to random and features conditional time and group-fixed effects <cit.>. In our case, the level of dominance that determines the group of fields is calculated without considering collaboration networks and by focusing only on the number of papers produced in each region and in each field before 1990, while the increase in the number of collaborations between East and West after 1990 is the dependent variable in our DID-based empirical model. Therefore, the manner in which we calculate the level of dominance does not directly guarantee faster collaboration after reunification. Moreover, as seen in Appendix C, other variables, such as population density, per capita GDP, and distances between regions, are similar between the two groups. To check the different effects of reunification, we selected two years, 1985 and 2005: one before and one after reunification. Basically, we used the same data that we used in previous sections, but we only considered the West-East collaboration, ignoring the collaborations within the West and within the East. However, we were unable to obtain data for other control variables, such as population density and per capita GDP before the reunification, even for the NUTS 2 regions. Consequently, to control for other factors we analyzed the different effects of reunification on the number of collaborations between 1995 and 2005. Furthermore, we controlled for the geographic/geodesic distance between the two regions using the regions' coordinates. Summary statistics in this regard are provided in Table <ref> and <ref>.Then, is the inverted U-shaped relationship statistically significant? In this section, we examine the robustness of the finding that fields with balanced capabilities achieved faster unification in their collaboration networks after 1990 compared to the fields with skewed research capabilities using a differences-in-differences (DID) method. This method can verify that the effect of reunification in 1990 on the collaboration networks is statistically different between the fields with balanced dominance and those with skewed dominance. Because we look at not the change in modularity but the change in real number of collaboration, this robustness check ensures that the inverted U-shaped relationship is not driven by the usage of modularity. DID requires two groups: a treatment and a control group. In this paper, the treatment group includes all publications in 10 fields of which the level of capabilities of the West and East are balanced, as we have already presented in Figure <ref>. Those fields are located around zero of the x-axis in Figure <ref>. On the other hand, our control group covers the other 10 fields of study that represent West- or East-skewed fields in terms of their dominance in Figure <ref>. The DID method is widely used to estimate the causal effects of events or policy interventions. However, the aim of this study is not to measure the effect of reunification, but to compare the different levels of effects to the speed of unification in terms of the capability balance of fields. Therefore, our focus is not on checking the cause of the event, which is the reunification of 1990, but to verify that the effect is significantly different for the two groups: treatment and control.One might worry about the possibility of selection bias in establishing the two groups, since DID is appropriate to use when an event is close to random and conditional on time and group-fixed effects <cit.>. In our case, the level of dominance that determines the group of fields is calculated without considering collaboration networks and by focusing only on the number of papers in each region and in each field before 1990, while the increase in the number of collaborations between East and West after 1990 is our dependent variable in our empirical model using DID. Therefore, the way that we calculate the level of dominance does not directly guarantee faster collaboration after reunification. Moreover, as seen in Appendix C, the other variables, such as population density, per capita GDP, and distance between the regions, are similar between the two groups. To check the different effects of reunification, we selected two years, 1985 and 2005: one is before and the other is after reunification. Basically, we used the same data that we used in previous sections, but we only considered the West-East collaboration, ignoring the collaborations within West and within East. However, we could not find data for other control variables, such as population density and per capita GDP before the reunification, even in the NUTS 2 region. To control for other factors, therefore, we analyzed the different effects of reunification on the number of collaborations for the years 1995 and 2005. Also, we controlled for the geographic/geodesic distance between the two regions using the regions' coordinates. Table <ref> and <ref> provides summary statistics. The empirical specification for the ordinary least squares (OLS) estimation using the DID method is provided below, with y_ij denoting the number of collaborations between regions i and j, and 𝐗 including other control variables. Again, because we only consider collaborations between East and West, i and j refer to region i in West Germany and region j in East Germany, respectively.y_ij=α+β(Treat_ij*After_ij)+σ Treat_ij+λ After_ij +𝐁𝐗^'+ε_i,j In this empirical model, the positive sign of the coefficient of the interaction term, β, means that the probability of further collaboration increases for more balanced fields. The coefficients σ and λ explain the differences in the number of collaborations between the two groups and between the two time periods, with one before and one after reunification. Figure <ref> and Table <ref> show the result of the DID analysis. Figure <ref> informs us that there were few collaborations between East and West in 1985 but that this number increased after reunification. The most balanced fields increased their number of East-West collaborations between two to three more (2.14 to 2.79 times more) than the less balanced fields.This means that, in the fields whose output was more balanced betweenEast and West, scientists were more likely to establish collaborations after reunification. Table <ref> shows that this difference is statistically significant. The result holds even when controlling for demographic and economic factors (which are available starting in 1995), such as the differences in GDP and population density, as well as geographic distance, which is the shortest distance between the two regions, as seen in columns (3) through (7).§.§ Why do they meet in the middle?1.5 Finally, we explore one mechanism that could help explaining why the more balanced fields integrated more effectively than those with a more skewed output. This is the homophily principle: the idea that "contact between similar people occurs at a higher rate than among dissimilar people," <cit.>. The homophily principle has been shown to operate in various types of relationships, such as marriage <cit.>, friendship <cit.>, career support <cit.>, networking in social media <cit.>, and even mutual attraction in a public place <cit.>. Since we cannot trace individuals scholars, we test the role of homophily by studying the mixing patterns of NUTS3 regions. That is, we explore whether regions were more or less likely to establish collaborations with other regions with the same levels of productivity (as measured by the fraction of papers they produced in a field). Figure <ref> shows the mixing patterns of networks with respect to the scientific productivity of regions during the years of 1985, 1994, 2004, and 2014. Following <cit.>, the matrices show a measure of Homophily (σ) comparing the number of co-authorships between regions after controlling for the expected number of co-authorships, and binned by their number of publications. σ_i,j=P(i,j)-P(i)P(j)/std(P(i)P(j))The fact that most positive values (yellow, orange, and red), concentrate along the diagonal (Figure <ref>) tells us that regions with similar levels of productivity were more likely to collaborate than what we would expect by random chance. Next, we check this homophily characteristic by focusing only on East-West collaborations. This time we calculate the RCA of each region in a field, so we can look at the number of collaborations among regions with similar, or different, levels of RCA. For each field we separate regions into three categories, "low," "medium," and "high," depending on their RCA value using quartiles ("low" for those below the first quartile, "high" for those above the third quartile, and "medium" for those within the second and third quartile). We then count the number of collaborations among these three groups, and determine the number of collaborations among each pair of RCA categories (low-low, low-middle, etc.). Again, we normalize the total number of observed collaborations with its expected number, obtained from the average of 100 Monte Carlo simulations. Figure <ref> shows whether each region had more or less collaborations than expected by chance between "low," "medium," and "high" comparative advantage regions. In the case of balanced fields (Figure <ref> B), we observe a strong tendency for regions to collaborate with other regions with a similar level of comparative advantage. Once again, this provides evidence of homophily in the research network, suggesting that a researcher in a balanced field might be able to find a similar researcher in terms of research capacity from another part of Germany, which could lead to further collaboration between West and East Germany.§ CONCLUSION In this paper, we explored the evolution of co-authorship network among regions in East and West Germany between 1974 and 2014. We find that the integration of the German research network, after 1990, was initially fast, but then stagnated at an intermediate level of integration. Then we explored the factors affecting the level of integration of a field by studying the reduction in modularity of each field and comparing it with the balance of its output prior to reunification. We found that the reduction of modularity in a field followed an inverse U-relationship as a function of a field's dominance. In other words, fields dominated by either West or East Germany prior to reunification integrated less than fields that possessed a more balanced output. This was confirmed by using Monte Carlo simulations and a DID analysis. Finally, we explored whether homophily could have contributed to this inverse U relationship between a field's level of integration and its balance prior to 1990. We found an assortative mixing pattern, meaning that regions are more likely to collaborate with regions with a similar productivity, even after controlling for the overall productivity of each region in each field. This suggests that after 1990, it may have been easier for researchers from the balanced fields to find collaborators from the other side of Germany, while those in the skewed fields might have experienced difficulty, leading to a slower speed of integration. While encouraging, our results should be interpreted in light of their limitations. For instance, we aggregate the data into the NUTS 3 regional level, even though our data include additional author information, such as the names of institutions the authors were affiliated with. As such, this aggregation prevents us from examining various reunification channels relating to the German research network; i.e., we do not provide a micro-level pattern of research collaboration. For example, we cannot capture instances where a researcher in West Germany moved to East Germany after reunification and held an Eastern German address. We believe this to be an important limitation since East-West migration was large after, and hence, many of the collaboration links could also be a consequence of migration. Our results do not contradict the migration hypothesis, but are not able to provide evidence of it because of the resolution of our data (it is not disambiguated at the author level). Additionally, our data is limited. As the data only captures publications from the journals listed on the Science Citation Index of WoS, other forms of research collaboration, especially those in the German language, are missing. Considering that Germany has a strong research tradition in its own language, we cannot say that our study encompasses all major research collaboration.Nevertheless, our findings on the evolution of research collaboration help to expand the body of literature on research collaboration, knowledge diffusion, and the unification of a research system when two systems are merged. Moreover, unveiling the factors that affect the speed of unification could have beneficial policy implications for the countries that are still awaiting political unification, such as the two Koreas.§ ACKNOWLEDGMENTS We wish to acknowledge the support received from the MIT Media Lab Consortia and from the Masdar Institute of Technology. We would also like to thank Xiaowen Dong, Mary Kaltenberg, Aamena Alshamsi, Jian Gao, and Juan Carlos Rocha for their valuable comments. This work was supported by Korea Institute of S&T Evaluation and Planning (KISTEP), the Center for Complex Engineering Systems (CCES) at King Abdulaziz City for Science and Technology (KACST), and the Massachusetts Institute of Technology (MIT). 1.5elsarticle-harv authoryear § APPENDIX A. NETWORK REPRESENTATION OF RESEARCH COLLABORATION BETWEEN WEST AND EAST GERMANY We use the publication data to build a network where nodes are German regions (at the NUTS3 level; Eurostat 2003) and links connect pairs of regions when the scientists of these regions have published a paper together. For the network representation in Figure <ref> and the calculation of modularity without considering different fields of study in Figure <ref>, we focus only on links that are statistically significant, since the links between German regions vary widely in the number of co-authorships involved in them. When we examine the 20 fields of study, we consider all the links because of smaller number of observations. To identify significant links, we use filtering techniques from the network science literature that are based on comparing the observed strength of a link with its expected value. This allows us to separate the links that are overexpressed–and represent strong connections between regions–from those that are underexpressed, and could be explained by chance. We estimate the strength of each link by calculating theϕ-correlation coefficient associated with a pair of regions, which is Pearson's correlation for binary variables. The ϕ-correlation can be expressed mathematically as:ϕ_ij = C_ij N - N_i N_j/√(N_i N_j ( N - N_i ) ( N - N_j ))where C_ijis the number of links between region i and j; N is the total number of authors who publish in that year; and N_i and N_j is the number of authors in region i and j, respectively. We can determine the significance of ϕ≠0 using a t-test, where t is: t =ϕ_ij√(n - 2)/√(1 - ϕ_ij^2)where n is the number of observations used to calculate ϕ. In this study, n is the total number of authors in each year. As a rule of thumb, when n is greater than 1,000, any link with is significant at the 5% level, while those withare significant at the 1% level. So we focus only on links with t greater or equal to 2.58. The distribution and frequency of ϕ values representing all links where C_ij > 0 are presented in Figure <ref>. § APPENDIX B. REWIRING NETWORK BY THE MONTE CARLO SIMULATION To have a benchmark network in evolution of collaboration network, we build a null model by randomization of network with the Monte Carlo simulation. In every iteration, we select four nodes by random sampling, and rewire its links as described in Figure <ref> with keeping the degree same. This means that we randomize not the number of collaboration of each region or regions' research capabilities, but selection of partners for collaboration. For generating one rewired observation, we randomize the network with 1,000 times of edge list's length as a number of iteration. We create hundred observations, and consider its mean and standard errors.§ APPENDIX C. DIFFERENCES-IN-DIFFERENCES ANALYSIS | http://arxiv.org/abs/1704.08426v1 | {
"authors": [
"Bogang Jun",
"Flávio Pinheiro",
"Tobias Buchmann",
"Seung-kyu Yi",
"Cesar Hidalgo"
],
"categories": [
"physics.soc-ph"
],
"primary_category": "physics.soc-ph",
"published": "20170427040331",
"title": "Meet me in the middle: The reunification of Germany's research network"
} |
Particle-in-Cell Simulations of Electron Beam Production from Infrared Ultra-intense Laser Interactions W. M. Roquemore December 30, 2023 =======================================================================================================empty § INTRODUCTION Materials crystallising in the ThCr_2Si_2-type structure comprise of such prominent compounds as the first heavy-fermion superconductor CeCu_2Si_2 <cit.> and BaFe_2As_2, a parent compound to the iron-based high-temperature superconductors <cit.>, making them especially attractive for solid-state research in the past decades. The discovery of heavy-fermion superconductivity inCeCu_2Si_2 resulted in extensive studies which became crucial for the understanding of unconventional superconductivity. In Ce-based heavy-fermion systems the strength of the hybridisation between the Ce-4f electrons and the conduction electrons is particularly important for the physical behaviour at low temperatures. There, the competition between Kondo effect and Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction along with crystalline electric field (CEF) effects lead to a large variety of different ground-state properties, which might be tuned using external control parameters such as chemical substitution, magnetic field and hydrostatic pressure <cit.>. A large number of the Ce-based compounds crystallising in the ThCr_2Si_2-type structure order antiferromagnetically at low temperatures. Their magnetism is commonly determined by a large magneto-crystalline anisotropy. This leads to the presence of distinct field-induced metamagnetic transitions <cit.>. Depending on the strength of the magnetic anisotropy, the nature of the metamagnetic transition(s) may differ. Additionally, the spin structure in the antiferromagnetic (AFM) phase plays an important role in the field-induced metamagnetic transition(s). In this regard, CePd_2As_2 offers the opportunity to study magnetism in a localised moment antiferromagnet with a huge magneto-crystalline anisotropy. Recently, the physical properties of polycrystalline CePd_2As_2, which crystallises in the ThCr_2Si_2-type structure, were reported <cit.>. CePd_2As_2 undergoes an antiferromagnetic (AFM) ordering at T_N≈ 15 K and shows evidence of a metamagnetic transition. However, a detailed investigation on single crystalline samples is necessary in order to understand the magnetic properties. In this work, we report on the magnetic anisotropy of single crystalline CePd_2As_2. To this end, we carried out magnetic susceptibility, magnetisation, electrical-transport and specific-heat measurements. Our results reveal an Ising-type magnetic anisotropy which accounts for a text-book-like spin-flip metamagnetic transition. The CEF level scheme could be fully resolved based on our experimental data. Furthermore, our analysis suggests a simple, collinear A-type antiferromagnetic spin structure in the AFM state.§ RESULTS §.§ Magnetic susceptibility and heat capacityThe temperature dependence of magnetic susceptibility χ of CePd_2As_2 with magnetic fields applied parallel (χ_) and perpendicular (χ_⊥) to the crystallographic c-axis are depicted in Fig. <ref>. χ(T) shows a sharp peak at T_N=14.7 K for both field orientations indicating the AFM transition, in good agreement with the results previously reported on polycrystalline samples <cit.>. Remarkably, χ_ is two orders of magnitude larger than χ_⊥ implying the presence of a strong magnetic anisotropy. The inverse magnetic susceptibility, χ_^-1(T) and χ_⊥^-1(T) are plotted in the inset of Fig. <ref>. Above room temperature, the susceptibility data can be fit by a Curie-Weiss law, χ(T)=C/(T-θ_ W), where C and θ_ W are the Curie constant and the Weiss temperature, respectively. We find θ^_ W=86 K andμ_ eff= 2.56μ_B for H∥ c and θ^⊥_ W= -130 Kandμ_ eff= 2.65μ_Bfor H⊥ c. The obtained effective moments are slightly enhanced compared with the calculated value of 2.54 μ_B for a free Ce^3+ ion. The deviation of χ_∥ , ⊥(T) from a Curie-Weiss law below room temperature can be attributed to CEF effects.For a free Ce^3+ ion with total angular momentum J=5/2, the ground-state consists of 6-fold degenerate levels. In the presence of a CEF with a tetragonal symmetry, these degenerate levels split into three doublets which are energetically separated from each other. The physical properties of CePd_2As_2 are greatly influenced by the relative thermal population of these energy levels. In CePd_2As_2, the magnetic contribution to the entropy, estimated from the specific heat data, reaches ∼85% of Rln2 at T_N (see Fig. <ref>). This indicates that the ground-state is a doublet well-separated from the excited CEF levels and that the Kondo effect is rather weak. Further evidence for the localised character of the Ce moments comes from the magnetisation data discussed below. §.§ Magnetisation Figure <ref> presents the temperature dependence of the magnetisation measured under various magnetic fields applied parallel and perpendicular to the crystallographic c-axis. For H∥ c, T_N shifts to lower temperatures upon increasing the magnetic field, which is expected for an antiferromagnet. As the magnetic field approaches 1 T, the peak in M(T) corresponding to the AFM transition disappears and a broad step-like feature with a saturation of M(T) toward low temperatures develops. However, for H⊥ c the position of peak corresponding to T_N is independent of the magnetic field and still clearly visible at 7 T. These different behaviours reflect the large magnetic anisotropy present in CePd_2As_2. We note that, the sudden decrease in the magnetisation below T_N for H∥ c compared to that of H⊥ c suggests that the crystallographic c-direction is the magnetic easy-axis.The isothermal magnetisation M(H) at 2 K, shown in Fig. <ref>, displays a sudden jump at μ_0H_c≈ 1 T for H∥ c followed by an immediate saturation. The observed saturation moment of 2.0 μ_B/ Ce is in reasonable agreement with the theoretical value of g_J J=2.14μ_B (where g_J = 6/7) expected for a free Ce^3+ ion. The sudden jump in magnetisation to the saturation value is a typical signature of a spin-flip metamagnetic transition. In the spin-flip process, the spins in the AFM sublattice, which are antiparallel to the field direction, are flipped at H_c. Hence, the antiferromagnetism changes to a field-polarised phase in a sudden, single step. The sharp nature of the jump in magnetisation with a small hysteresis point to a first-order type transition.At higher temperatures, the metamagnetic transition in M(H) broadens and saturates at much higher fields. In the case ofH⊥ c (inset of Fig. <ref>), the magnetisation increases monotonously and reaches at 7 T only 2.5% of the saturation value for H∥ c. Furthermore, magnetisation measurements in pulsed fields up to 60 T show a linear increase without any tendency to saturation (not shown). This suggests the absence of any metamagnetic transition for H⊥ c, which stipulates the huge magnetic anisotropy in CePd_2As_2.§.§ Electrical transportThe electrical resistivity ρ(T) of CePd_2As_2 upon cooling displays a metallic behaviour with a broad curvature at intermediate temperatures, before showing a pronounced kink at about 15 K indicating the AFM transition (inset of Fig. <ref>). The broad curvature in the resistivity may be due either to interband scattering or to weak additional spin scattering originating from thermal population of excited CEF levels. At low temperatures, the AFM ordering leads to a sudden decrease in ρ(T) due to the loss of spin-disorder scattering contribution below T_N. Figure <ref> shows the ρ(T) data recorded at different magnetic fields applied along the c-axis. Upon increasing the field up to 1 T, the kink indicating T_N shifts to lower temperatures and becomes washed out, in good agreement with the results from the magnetic susceptibility. Moreover, above 1 T the residual resistivity shows a sudden reduction which coincides with the metamagnetic critical field. The field and angular dependencies of the resistivity, plotted in Fig. <ref>(a-c), give further insights into the nature of the metamagnetic transition. At low temperatures, upon increasing the magnetic field ρ(H) suddenly drops at the onset of the metamagnetic transition at the critical field μ_0H_c ≈ 1 T. This feature can be attributed to a Fermi-surface reconstruction while going from the AFM to the field-polarised phase. At higher temperatures, the metamagnetic transition in ρ(H) broadens. A small increase in ρ(H) is observed just below H_c in the AFM phase for temperatures close to T_N. This could be due to an increased scattering during the spin-flip process associated with the transition from the AFM to field-polarised state<cit.>.Above the AFM transition temperature, ρ(H) displays a gradual decrease upon increasing magnetic field, suggesting a crossover from the paramagnetic to the field-polarised phase. The variation of ρ as function of the angle (θ) between the magnetic field (μ_0H=2 T) and the crystallographic c-axis at different temperatures is shown in Fig. <ref>(b).The step-like behaviour at lower temperatures changes to a gradual decrease in resistivity above T_N, where the system undergoes a crossover from paramagnetic to the field-polarised phase. Above 30 K, the resistivity becomes independent of the field orientation. Finally, Fig. <ref>(c) presents the resistivity as a function of field for different angles θ. The metamagnetic critical field H_c increases upon increasing θ and diverges for θ→ 90^∘. No drop in ρ(H) is observed up to 14 T for field perpendicular to the c-axis. This is consistent with our magnetisation experiments. § DISCUSSIONIn CePd_2As_2, the temperature dependence of the magnetic susceptibility below room temperature strongly deviates from a Curie-Weiss behaviour. This can be attributed to CEF effects. In order to establish the CEF scheme and learn more about the magnetic anisotropy in CePd_2As_2, we performed a detailed CEF analysis based on our magnetic susceptibility data. For a Ce atom in a tetragonal site symmetry, the CEF Hamiltonian can be written as, ℋ_CEF = B^0_2O^0_2 + B^0_4O^0_4 + B^4_4O^4_4, where B^n_m and O^n_m are the CEF parameters and the Stevens operators, respectively<cit.>. The magnetic susceptibility including the Van Vleck contribution is calculated as, χ_CEF,i = N_A(g_Jμ_B)^21/Z(∑_m≠ n 2|⟨m|J_i|n⟩| ^2 1-e^-β(E_n-E_m)/E_n-E_me^-β E_n+∑_n|⟨n|J_i|n⟩| ^2 β e^-β E_n), where Z = ∑_n e^-β E_n, β = 1/k_BT and i=x,y,z. The inverse magnetic susceptibility including the molecular field contribution λ_i is calculated as χ_i^-1=χ_CEF,i^-1-λ_i. χ_i^-1(T) is fitted simultaneously to the experimental data for both field orientations (see Fig. <ref>). The data in the paramagnetic phase are well reproduced by the CEF model with a doublet ground-state |*⟩Γ_7^(1) = 0.99|±5/2⟩+0.16|∓3/2⟩ and the excited doublet states |*⟩Γ_7^(2) = 0.99|±3/2⟩-0.16|∓5/2⟩ and|Γ_6⟩ = |±1/2⟩ at 290 K and 330 K, respectively. An illustration of the CEF level scheme is shown in the inset of Fig. <ref>. The crystal field parameters extracted from the model are B^0_2=-18.66 K, B^0_4=-0.22 K and B^4_4=1.67 K, with a molecular field contribution λ_c=-8 emu^-1mol along the c-axis. It is clear from our CEF analysis that the ground-state is an almost pure |±5/2⟩ CEF doublet which is well-separated from the excited doublets. The saturation magnetisation along the c-axis for the obtained CEF ground-state is 2.06 μ_ B/ Ce, which is in good agreement with the experimental saturation magnetisation of 2.0 μ_B/ Ce. Furthermore, the CEF parameter B^0_2 is directly related to the paramagnetic Curie-Weiss temperatures θ^⊥_ W and θ^_ W, along both principal crystallographic directions, as θ^⊥_ CW-θ^_ CW=3/10B^0_2(2J-1)(2J+1) <cit.>. Using the experimental values of θ^⊥_ W and θ^_ W, we obtain B^0_2=-22.5 K in good agreement to B^0_2=-18.66 K from the CEF-model fit.Deeper insights into the magnetic structure of the ordered phases in CePd_2As_2 can be obtained from the magnetisation and electrical resistivity data measured at different orientations of the magnetic field. The magnetisation data suggest that the crystallographic c-direction is the easy-axis of the magnetisation. Moreover, the small magnetisation in the ab-plane compared with the large magnetisation along the c-axis indicates an AFM structure with the spins pointing along the c-axis. In addition, the spins are locked along the c-axis by the magneto-crystalline anisotropy, as indicated by the absence of a metamagnetic transition for magnetic field up to 60 T applied perpendicular to the c-axis. These observations confirm that the moments in CePd_2As_2 are Ising-type. The Ising-nature of the spins is also supported by the angular dependence of the metamagnetic critical field extracted from the electrical resistivity data shown in Fig. <ref>(c). The resulting angular dependence of H_c is displayed in Fig. <ref>(d). H_c(θ) increases sharply for θ→90^∘ and H_c is not detected for field oriented perpendicular to the c-axis. In order to understand the angular dependence of H_c(θ), we fit the data by the equation, H_c = H_c0/cos(θ) where H_c0 is the critical field for field parallel to the c-axis. Equation <ref> describes the experimental data very well with μ_0H_c0 = 0.95 T. In other words, the metamagnetic transition occurs only when the component of magnetic field along the c-axis reaches the value of H_c0. Based on these results, we can conclude the following scenario for the spin structure of CePd_2As_2: in the AFM phase, the Ce moments are aligned along the c-axis and are locked along this axis by the magneto-crystalline anisotropy. When the component of external magnetic field along the c-axis exceeds H_c0, the anti-parallel spins undergo a spin-flip transition to the field-polarised ferromagnetically ordered phase. The single, sharp jump in the magnetisation with a weak hysteresis at the first-order metamagnetic transition from the AFM to the ferromagnetically polarised state points at a simple spin structure of the AFM phase. The small value of the critical field H_c0, compared to the value of T_N, indicates that, in terms of a Heisenberg model with a few different inter-site exchange interactions, the AFM ones are much weaker than the ferromagnetic (FM) ones. Because of the topology of the tetragonal body centered Ce sublattice, a strong FM interaction between atoms in adjacent layers competing with a weak in-plane AFM interaction would always result in a FM ground-state. In contrast, a strong FM in-plane interaction with a weak AFM inter-plane interaction can easily account for all observations. In addition, we note that isovalent substitution of P for As results in a FM ground-state <cit.>. Thus all these properties provide strong indication that the AFM structure of CePd_2As_2 is just a simple AFM stacking of FM layers. Substituting P for As is just turning the inter-plane exchange from weakly AFM to FM. Therefore, we propose a magnetic structure with weakly antiferromagnetically coupled FM layers of Ising-spins in the AFM state of CePd_2As_2, as illustrated in the inset of Fig. <ref>(d). A mean-field approximation based on a two-sublattice model can appropriately describe such a spin system. According to this model, the spin-flip occurs when the applied magnetic field is able to overcome the inter-layer AFM coupling. Therefore, the metamagnetic critical field can be expressed as H_c = λ_ AFM M, where λ_ AFM is the inter-sublattice molecular field constant and M is the magnetisation of the ferromagnetic state<cit.>. Similarly, the intra-sublattice molecular field constant λ_ FM can be extracted from the relation T_N = 1/2 C(λ_ FM-λ_ AFM), where C is the Curie constant C = N_Aμ_ 0g_J^2J(J+1)μ_B^2/3k_ B. By using the experimentally obtained values H_c0, M_S and T_N, the inter-layer AFM exchange strength (z_ AFMJ_ AFM) and intra-layer FM exchange strength (z_ FMJ_ FM) are calculated as -0.25 K and 9.83 K, respectively. Here, z_ AFM and z_ FM are the number of nearest-neighbour spins participating in the respective interactions. The large intra-layer FM exchange strength is consistent with the experimental observations and plays a crucial role in the first-order nature of the metamagnetic transition.The T-H phase diagram of CePd_2As_2 for H∥ c, presented inFig. <ref>, summarises our results. At low temperatures, application of a magnetic field induces a metamagnetic transition at μ_0H_c0=0.95 T resulting in a field-polarised phase. Above T_N, CePd_2As_2 shows a crossover behaviour from the paramagnetic to the field-polarised phase, reflected by the broad features in magnetisation and electrical resistivity. Additional information on the nature of the transitions between the various phases can be obtained from specific heat data. The temperature dependencies of the specific heat C_p of CePd_2As_2 for three representative magnetic fields are plotted in insets of Fig. <ref>. A cusp in C_p(T) indicates the transition form the paramagnetic to AFM phase at 0.84 T. The second order nature of this transition is evidenced by the absence of any thermal hysteresis in the data. In contrast, a strong thermal hysteresis and a spike-like anomaly in C_p(T) at 1 T confirms the first-order nature of the metamagnetic transition from the AFM to the field-polarised phase. Finally, the crossover from the paramagnetic to the field-polarised phase at higher magnetic fields is reflected by a broad, hump-like feature in C_p(T).Because of its strong Ising anisotropy and simple magnetic behaviour, CePd_2As_2 is a nice example to illustrate a misinterpretation frequently encountered in the analysis and discussion of magnetic properties of Ce- and Yb-based compounds. The Weiss temperatures determined from Curie-Weiss fits to the high temperature part of the susceptibility are frequently argued to reflect the anisotropy, the sign and the magnitude of the exchange interactions. In CePd_2As_2 this would lead to the conclusion that the exchange in the basal plane is strongly antiferromagnetic while the exchange along the c-direction is weaker and ferromagnetic. Our analysis clearly demonstrates that this conclusion would be completely wrong, because the Weiss temperatures determined from Curie-Weiss fits at high temperatures are dominated by the effect of the CEF. Except for special cases, CEF generally result in a seemingly AFM, negative θ_ W for the direction of the small CEF ground-state moment and an apparently FM, positive θ_ W for the direction of the large CEF ground-state moment.§ SUMMARY We have investigated the magnetic properties and the CEF scheme of CePd_2As_2 by detailed temperature, magnetic field and angular dependent magnetic, thermodynamic and electrical transport studies on single crystalline samples. The detailed CEF analysis based on the magnetic-susceptibility data indicates an almost pure |±5/2⟩ CEF ground-state doublet with the dominantly |±3/2⟩ and the |±1/2⟩ doublets at 290 K and 330 K, respectively. CePd_2As_2 orders antiferromagnetically in a simple A-type order below T_N=14.7 K. Our results imply a uniaxial AFM structure with spins locked along crystallographic c-axis. An external magnetic field applied along the c-axis induces a metamagnetic spin-flip transition at μ_0H_c0=0.95 T leading to a ferromagnetic spin alignment. No metamagnetic transition is observed for a magnetic field perpendicular to the c-axis, proving the huge Ising-like anisotropy in CePd_2As_2. § METHODS Single crystals of CePd_2As_2 were synthesised by a self-flux method. Initially, polycrystalline CePd_2As_2 was obtained by a solid-state reaction as reported previously <cit.>. Then the polycrystalline pellet was loaded into an alumina crucible and sealed in an evacuated quartz ampule. The ampule was heated up to 1160^∘C and held at this temperature for 24 hours, followed by slow cooling to 900^∘C at the rate of 1.6^∘C/h. Shiny plate-like single crystals of CePd_2As_2 were obtained. The crystal orientation and chemical homogeneity were checked by x-ray diffraction (XRD) and energy dispersive x-ray analysis (EDX), respectively. XRD measurements were carried out on a PANalytical X'pert MRD diffractometer with Cu K_α1 radiation and a graphite monochromator. Magnetisation measurements were carried out in the temperature range 1.8 K-400 K and in magnetic field up to 7 T using a SQUID-VSM (MPMS3, Quantum Design). High-field magnetisation measurements up to 60 T in pulsed magnetic fields were performed at the Dresden High Magnetic Field Laboratory, Germany. The electrical transport experiments were carried out in the temperature range 2 K-300 K and magnetic fields up to 14 T using a Physical Property Measurement System (PPMS, Quantum Design). The electrical resistivity was measured using a standard four-terminal method, where electrical contacts to the sample were made using 25 μm gold wires and silver paint. The temperature dependence of specific heat was also measured using a PPMS. § ACKNOWLEDGEMENTS We thank Yurii Skoursky for his support in conducting the pulsed-field magnetisation measurement. This work was partly supported by Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group GRK 1621. The work at Zhejiang University was supported by the National Natural Science Foundation of China (U1632275), National Key Research and Development Program of China (No. 2016YFA0300202) and the Science Challenge Project of China. RDdR acknowledges financial support from the Brazilian agency CNPq (Brazil). | http://arxiv.org/abs/1704.08103v1 | {
"authors": [
"M. O. Ajeesh",
"T. Shang",
"W. B. Jiang",
"W. Xie",
"R. D. dos Reis",
"M. Smidman",
"C. Geibel",
"H. Q. Yuan",
"M. Nicklas"
],
"categories": [
"cond-mat.str-el",
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.str-el",
"published": "20170426133325",
"title": "Ising-type Magnetic Anisotropy in CePd$_2$As$_2$"
} |
[email protected] Department of Physics and Centre for Neural Dynamics, University of Ottawa, 598 King Edward, Ottawa K1N 6N5, CanadaCentre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UKDepartment of Physics and Centre for Neural Dynamics, University of Ottawa, 598 King Edward, Ottawa K1N 6N5, CanadaThe theoretical description of non-renewal stochastic systems is a challenge. Analytical results are often not available or can only be obtained under strong conditions, limiting their applicability. Also, numerical results have mostly been obtained by ad-hoc Monte–Carlo simulations, which are usually computationally expensive when a high degree of accuracy is needed. To gain quantitative insight into these systems under general conditions, we here introduce a numerical iterated first-passage time approach based on solving the time-dependent Fokker–Planck equation (FPE) to describe the statistics of non-renewal stochastic systems.We illustrate the approach using spike-triggered neuronal adaptation in the leaky and perfect integrate-and-fire model, respectively. The transition to stationarity of first-passage time moments and their sequential correlations occur on a non-trivial timescale that depends on all system parameters. Surprisingly this is so for both single exponential and scale-free power-law adaptation. The method works beyond the small noise and timescale separation approximations. It shows excellent agreement with direct Monte Carlo simulations, which allows for the computation of transient and stationary distributions. We compare different methods to compute the evolution of the moments and serial correlation coefficients (SCC), and discuss the challenge of reliably computing the SCC which we find to be very sensitive to numerical inaccuracies for both the leaky and perfect integrate-and-fire models. In conclusion, our methods provide a general picture of non-renewal dynamics in a wide range of stochastic systems exhibiting short and long-range correlations.Evolution of moments and correlations in non-renewal escape-time processes André Longtin December 30, 2023 ==========================================================================§ INTRODUCTION A general property of diverse systems, ranging from superconducting quantum interference devices (SQUIDs) <cit.>, to lasers <cit.> to excitable cells <cit.> is that time intervals between specific events are not statisticallyindependent. The theoretical description of such non-renewal stochastic processes <cit.> poses a significant challenge, as it implies that the present state of the system depends, in general, on the whole past evolution or parts of it, and not just on the previous state. Analytical approximations to tackle such memory effects have included the assumption of stationarity <cit.>, small stochasticity <cit.>and time-scale separation <cit.> between stochastic and deterministic parts of the dynamics.Even if these approximations allow for some insight into the parameter dependence of e.g. serial correlations and can be used to understand experimental data, as exemplified in <cit.> in the context of excitable systems, it is desirable to understand the statistics of model systems without making simplifying assumptions. Regarding stationarity, real systems rarely operate in a stationary state due to transients that arise from deterministic or random perturbations. A prominent example are cortical neurons. An average cortical neuron receives random inputs from approximately 10^4 other neurons, whose activity is modulated by non-stationary sensory and other inputs, resulting in transient neuronal dynamics <cit.> that only become stationary after a certain time. It is therefore important to understand how statistical properties of inter-event times evolve and become invariant following a transient regime due to internal dynamics and external inputs. Keeping with illustrations from neural dynamics, it is well known that physiologically relevant processes underlying neural coding rarely only have one well-defined timescale <cit.>. This has lead researchers in various theoretical fields to consider multiple time-scale dynamics <cit.>. An important example of a system with multiple time scales is neuronal adaptation, where a neuron's firing rate adjusts in response to a stimulus. Adaptation with multiple timescales, or even no time scale as in the case of power-law adaptation <cit.>, is now known to be biophysically relevant, and even optimal for some tasks <cit.>. Recently, it was also shown that a neuron model with adaptive firing thresholds exhibiting multiple timescales is the optimal choice for the prediction of spike times in cortical neurons <cit.>. Therefore, a theoretical description of adaptation without a single well-defined timescale is an important goal.In this paper, we show how to describe two-dimensional non-renewal dynamics by an iterated first-passage time (iFPT) approach. This approach allows us to determine stationary statistical properties of the system as well as providing a description of the transition to stationarity. We furthermore show how to compute serial correlations in the time series generated by the firing times of the system. While our approach is general and applicable to any system where first-passage times <cit.> play a role, we illustrate its versatility with two important examples, namely spike-triggered neuronal adaptation with a single exponential current and a power-law current without an intrinsic timescale, respectively. Using the underlying time-dependent FPE to describe the system, we only need to apply mathematically convenient standard absorbing boundary conditions to obtain stationary distributions, e.g. that of the adaptation current upon firing. Moreover, the methods developed here can easily be extended to models with correlation-generating deterministic input currents as recently considered in <cit.>.§ MODEL We consider a stochastic differential equation (SDE) driven by an external signal s(t):X̣(t) = μ(X(t))d t + ϕ(X(t))Ẉ(t) - s(t) ṭ. X is defined on the domain (-∞, ]. If X reaches , the system is said to have generated and event, and X is instantaneously reset to 0. For all examples in this study, we chose the Ornstein–Uhlenbeck process (OUP) given its prominence in the field of stochastic systems. For the OUP, we fix the correlation time τ_m = 1/γ, bias current I_0 and noise intensity σ as follows: μ(X(t)) = γ(I_0 - X(t)), ϕ(X(t)) =σγ. W(t) is a standard Brownian motion and we set = 1. Given that the OUP is the basis for integrate-and-fire (IF) neuron models, which are among the most popular neuron descriptions <cit.>, we refer to events as spikes and to s(t) as a time-dependent adaptation current in the present study. The general dynamics of s(t) obeys a single autonomous ordinary differential equation (ODE) ṡ = ω(s), and s is increased by a fixed amount κ when X=: s → s + κ, which is the mechanism for spike-triggered adaptation <cit.>. When s(t) is also reset to its starting value s(0), the model is a renewal model and its firing statistics may be studied using standard techniques, see e.g. <cit.> for a recent review. Here we focus on two forms of the adaptation current. The first one is power-law adaptation, for whichω(s) = -1/α s^2(t) . This ODE has the general solution s(t) = ( t/α + 1/s(0))^-1. Therefore, the current s in this case has a power-law time dependence with no intrinsic time scale <cit.>. The second adaptation current is given by a single exponential decay with time scale τ_a: ω(s) = -1/τ_a s(t) , which has the general solution s(t) = s(0)e^-t/τ_a.The time to the first spike event is the following first-passage time (FPT): T_1 = inf(t>0: X(t) >| X(0) =0, s(t=0) = s(0)) .In the non-renewal case we are studying here, subsequent firing times will in general not have the same distribution as T_1. We define the kth interspike interval (ISI) asT_k = inf(t- ∑_i=1^k-1T_i : X(t) ≥, t > ∑_i=1^k-1T_i) . The first moment of the kth ISI is given by τ^1_k= 𝔼(T_k). The second moment of the kth ISI will be denoted by τ^2_k= 𝔼((T_k)^2) and the kth firing rate is given by the inverse of the corresponding mean ISI r_k = 1/τ^1_k. The kth standard deviation m_2(k) is then given by m_2(k) = √(τ^2_k-(τ^1_k)^2). The values of the peak adaptation current after the kth firing are defined for k≥ 1 as s_0^(k) = ( s(t^-) + κ : t = ∑_i=1^kT_i) , where t^- indicates that we take the left-sided limit. For simplicity, we choose s to be started from a point (s_0^(0) = κ), instead of from a biophysically more realistic initial distribution. However, the methods we are going to describe in the following are also valid when s is initially started from a distribution.The central challenge is to obtain the distributions 𝒢_k and ℱ_k for k ≥ 1, which are the distributions of s_0^(k) and T_k defined by <ref> and <ref>, respectively. An example realization for the case of power-law adaptation is shown in <ref>. The knowledge of these distributions is key to understanding the non-renewal dynamics, as they form a hidden Markov model of the underlying non-Markovian dynamics<cit.>. Therefore, once these distributions are known, the non-renewal dynamical system breaks up into coupled renewal dynamical systems, which are much more tractable mathematically. This gives rise to the iFPT approach which we now explain.§ THE IFPT APPROACHBeing a diffusion process, the system given by <ref> and <ref> is governed by a two-dimensional time-dependent FPE <cit.>. The FPE for the probability density function p(t;x,s)x̣ṣ = ℙ(X(t) ∈ (x, x+ x̣), s(t) ∈ (s, s + ṣ) | X(0) = x_0, s(0) = s_0) reads (we use 𝐱^⊤ = (x,s) for brevity) ∂_t p(t; 𝐱) = ∇· (𝐀(𝐱)∇ p(t; 𝐱)) -∇·(𝐅(𝐱) p(t; 𝐱)) , where 𝐀 is the diffusion matrix and 𝐅 the drift vector, which can be obtained in a straightforward way from the SDE for X, <ref>,and the ODE for s, <ref>. Explicitly, we have𝐀(𝐱)= [ ϕ(x)^2/20;00 ],and𝐅(𝐱) = (μ(x) -s,ω(s) )^⊤. The IF property of X entails that we have an absorbing boundary at X = for all times t: p(t; x = , s) =0. With this boundary condition, we can compute the cumulative distribution function (CDF) of the first-passage time T_1 , CDF_1(t) = ∫_0^tℱ_1(λ) λ̣, by time evolution of the FPE on a computational domain Ω⊂ℝ^2, which we choose to be a rectangle extending to sufficiently negative values in the x-direction <cit.>: CDF_1(t) = 1 - ∫_Ω p_1(t;𝐳) z, where p_1 is the solution to <ref> with the initial condition p_1(0;x,s) = δ(𝐱 - 𝐱_0), where 𝐱_0 = (0, κ)^⊤. For the computation of ℱ_1, we then only need to differentiate <ref>.To describe adaptation, one needs to compute the statistics of the peak adaptation currents, as defined by <ref>. Hence, we need to characterize the hidden Markov model generated by the ISIs T_k and the peak adaptation currents s^(k)_0. Whereas the dynamics of s and X as a whole is non-Markovian, the distribution of the ISI T_k is completely determined by the distribution 𝒢_k-1, and the sequence {T_k, s_0^(k-1)}_k=1^∞ is therefore Markovian. Knowing the values of T_k and s_0^(k-1), the value of s_0^(k) is fixed, and the distribution of the ISI T_k+1 can be obtained by solving an FPT problem with s_0^(k) as initial condition for s.The central observation now is that for the second ISI, X again starts from 0, whereas s starts from a distribution 𝒢_1. This is because X evolves stochastically, and therefore reaches the thresholdat different times T_1, corresponding to different values of s(t= T_1) + κ immediately after the firing event. To compute the second ISI, we therefore need to know 𝒢_1. This can be iterated: to compute the distribution ℱ_k of the kth ISI, we need the distribution 𝒢_k-1. This is the central idea of the iFPT approach. To set up the iFPT approach, we first observe that between threshold crossings of X, s evolves deterministically. Therefore, when we know the PDF of the first FPT, by conservation of probability, we also know the distribution of s after the first firing event: 𝒢_1(s-κ) = |ṭ(s)/ṣ|ℱ_1(t(s)) , where t(s) is the inverse function of s. The support of 𝒢_1 is shifted, because of the jump of size κ that s undergoes when X reaches its threshold .For the second ISI T_2, s is started from the distribution 𝒢_1(s) instead of a point, whereas X is started from a point again (<ref>). This means that to obtain ℱ_2, the FPE is started from a distribution: p_2(0;𝐱) ∝𝒢_1(s)δ(x). This generalizes to values of k larger than 1. For the kth ISI distribution ℱ_k(t), we therefore must choosep_k(0;𝐱) ∝𝒢_k-1(s)δ(x) . We show how to obtain the distributions 𝒢_k for k>1 in Section <ref>. Linear splines are used to create a mesh function approximating <ref> on the computational domain Ω. Due to this approximation, <ref> then has to be normalized appropriately, so that∫_Ω p_k(0;𝐳) z = 1. The FPE is then solved again, and the distributions ℱ_k(t) are obtained analogouslyto <ref>: CDF_k(t) = 1 - ∫_Ω p_k(t;𝐳) z, i.e. by timestepping the FPE to obtain the CDF of the kth ISI followed by a numerical differentiation. This constitutes the iFPT approach. To quantify the accuracy of our numerical methods, we also compute the relative disagreement Δ between results obtained by the iFPT approach and direct MC simulations. It is defined for a quantity Z by Δ(Z) = |Z_iFPT- Z_MC|/Z_iFPT. We performed MC simulations for two different simulation setups, the first one without any boundary correction (plain MC), and the second one with a boundary correction according to Giraudo and Sacerdote (MC-GS) <cit.>. This boundary correction is applied to reduce the systematic overestimation of FPTs when using the Euler–Maruyama scheme. We compute the relative disagreement given by <ref> using either MC simulations with or without boundary correction; we observed that the order of the relative disagreement is unchanged, but in general, the plain MC algorithm gives rise to larger disagreements than MC-GS. A decrease in the relative disagreement is expected, because the GS correction method should yield an improved weak error of 𝒪(h) <cit.>, in contrast to the plain MC simulation, which has a weak error of 𝒪(h^1/2) <cit.>, where h is the time step for the discretization of the SDE, <ref>. In the following, the timestep for MC simulations is chosen to be h = 10^-3 and we choose M = 10^6 independent realizations. The plain Euler-Maruyama scheme then gives rise to a weak error of 𝒪(h^1/2) ≈ 3 · 10^-2 when estimating moments of first passage times, which is one order of magnitude larger than the MC error proportional to 1/√(M) = 10^-3. Therefore, in our simulations, the plain MC error is negligible in comparison to the error introduced by the finite-time discretization of the SDE (<ref>). For the numerical solution of the FPE (<ref>), we choose a finite element discretization method <cit.> and evolve the system using either a stabilized Crank–Nicolson (CN) scheme <cit.> in <ref> or an Euler timestepping scheme <cit.> in <ref>.The relative disagreement between MC simulations and finite-element solutions stays largely constant across different lags k when we use the CN scheme instead of the Euler scheme as can be seen by comparing the lower panels of <ref> and <ref>; the sizes of the disagreement are comparable in magnitude. This suggests that the remaining small discrepancy between MC simulations and PDE results can be largely explained with the errors associated with the MC simulation method. In particular, note that for the examples we show, the MC-GS weak error of size 𝒪(h) is comparable in magnitude to the numerator of Δ(Z) (<ref>), i.e. the absolute disagreement. We will see what effects this has on the computation of correlations in Section <ref>. § TRANSITION TO STATIONARITY We show the evolution of the rate and standard deviation of ISIs in <ref> and <ref>. The rates decreases, whereas the standard deviation increases until both quantities reach a stationary value. Given that these quantities are derived from moments of the ISI distributions, the distributions also converge towards a stationary form. The convergence towards stationarity of ISI and peak adaptation current distributions (ℱ_k and 𝒢_k, respectively) is shown in <ref> and <ref>. As expected for adapting models, the ISIs (whose distributions are shown in <ref>) increase with higher k, which means that the rate r_k decreases. This is well captured by the iFPT approach, with a maximal relative disagreement smaller than 3 % in <ref> and smaller than 2 % in <ref>. Also, the width of both the ISI distributions and the peak adaptation current distributions (<ref>) increases, which is reflected by the increase of the variances of ℱ_k and 𝒢_k shown in <ref> and <ref>. The mean of the peak adaptation currents shifts to the right as stationarity is reached. Moreover, stationarity is reached with varying speed, i.e. for different values of the lag k (compare <ref> with <ref>). Generally, the speed of adaptation can be controlled by adjusting the bias current I_0 and the noise level σ as well as the adaptation strength (size of the kick κ and, in the case of single exponential adaptation, the timescale τ_a). We have carried out additional MC simulations (not shown) to obtain insight into how these parameters influence the speed of the transition to stationarity. A higher bias current and a higher noise level will in general lead to a less rapid transition to stationarity.This can be understood as follows: X is driven to threshold more rapidly, causing the inhibition to build up quickly, reaching values that are higher than those typically found around the peak of the stationary distribution. This slows down the transition to stationarity, because s needs to decay first. Moreover, a large kick size κ and a rapidly decaying adaptation current will cause a quick transition. For the latter case, this is easily understood as we are then nearly dealing with a renewal system: after a short initial period, the effect of the adaptation current on the firing time statistics is negligible. For the former case, we note that the larger the kick size κ, the more pronounced the inhibitory effect of adaptation within one ISI, which means that X takes longer to reach threshold before a large out-of-equilibrium average value of the current s (a value that is larger than those typically found around the mode of the stationary distribution) can build up. The system reaches stationarity rapidly because it is quasi-deterministic as the dynamics of s dominates the system, and the stochastic fluctuations of X will only cause small perturbations. For both power-law and single exponential adaptation currents, it is possible to reach the stationary regime already after one or two firing events as in <ref>, or to have a long transient regime as in <ref>. The initial condition s^(0)_0 for the adaptation current can also be chosen to control the speed of the transition. If it is placed far away from the mean of the stationary distribution, the transition will take a longer time; also, it is possible to obtain a non-monotonic behaviour of the rate as a function of the interval number when s^(0)_0 is placed far above the aforementioned mean. The first mean ISI will then be the longest statistically, in contrast to the examples we show in <ref> and <ref>.§ STATISTICS OF THE KTH ISIThe iFPT approach can be iterated beyond the first two firing events to obtain the distribution for the third ISI, ℱ_3(t). However, for the computation of the third ISI, no equation similar to <ref> can be used to obtain 𝒢_2 because s was started from a distribution to obtain ℱ_2. Indeed, for one fixed time T_2, there are many different starting values s^(1)_0 due to the stochastic dynamics of X. Importantly, the ISI T_2 and the initial condition s^(1)_0 are not independent random variables (for a large value of s^(1)_0, a large ISI T_2 is more probable and vice versa) so that we can obtain the value that s reaches after the second firing by the following observation (focusing on power-law adaptation): given thatT_2 = λ and s^(1)_0 = ν, we have s^(2)_0= 1/λ/α+ 1/ν+ κ. We have included the jump of size κ due to the definition of s^(2)_0 (see <ref>). This emphasizes that once two values in the triplet (T_2, s^(1)_0, s^(2)_0) are fixed, the third one is determined. In the following, we again use λ to denote a fixed FPT and ν to denote a fixed initial value of the adaptation current s. Analogously, we generally have for s^(k)_0: given T_k=λ ands_0^(k-1)=ν, s^(k)_0=f(λ,ν) is determined. The function f determining the subsequent value of the peak adaptation current given the previous ISI λ and the previous peak value of the adaptation current ν reads for power-law adaptation (<ref>): f(λ, ν) = κ + 1/λ/α + 1/ν,whereas for exponential adaptation (<ref>), we havef(λ, ν) = κ+ νexp(-λ/τ_a) . An alternative way is to fix the value of s^(k)_0 and then put a constraint on the time T_k when s_0^(k-1) is fixed: given s_0^(k-1)=ν and s^(k)_0 = θ, T_k = h(ν, θ) is determined, where, for power-law adaptation, we have the ISI as a function of the previous and subsequent adaptation values:h(ν, θ) = α(1/θ-κ - 1/ν). The function h is defined by solving the equation f(λ, ν) = θ for λ. To actually compute the density 𝒢_k, we need to relate the above observations to densities that we can compute with the iFPT approach. To that end, we now define the conditional density ℋ:ℋ(λ, ν) λ̣≡ℙ(T_1∈ (λ, λ + λ̣)|s^(0)_0 = ν) .We have used T_1 and s^(0)_0 in the definition <ref> to stress that, for the purpose of the computation of ℋ, we only need to solve the FPT problem for T_1 using different values of the initial condition s^(0)_0. It will become apparent below that we only need to compute ℋ once, because it does not depend on the firing index k. For a fixed value of ν, ℋ(λ, ν) is an FPT probability density. Our notation emphasizes that ℋ is a function of two variables. ν sets the level of initial inhibition, i.e. the starting value of s. We show the function ℋ for both power-law and exponential adaptation in <ref>. We see that with increasing starting value ν, the mode of the FPT distribution shifts to larger times. For power-law adaptation, the shape of the FPT distributions does not change much, whereas for exponential adaptation, the distributions become broader with increasing ν. With this at hand, we now show how to practically compute the distributions 𝒢_k for k>1. We can obtain the CDF of s_0^(k) by observing that: ℙ(s^(k)_0≤θ) = ∫_𝒟^(k-1)(θ)ℋ(λ, ν) 𝒢_k-1(ν) λ̣ν̣, with 𝒟^(k-1)(θ) =( λ, ν > 0 ν∈supp(𝒢_k-1), h(ν,θ) ≤λ≤ T_max) . The function h defined in <ref> ensures that for a fixed value of ν, we collect all times λ so that s_0^(k)≤θ, which ensures that f( λ, ν ) ≤θ for fixed values of θ and ν. T_max is chosen so that ℋ(T_max, ν) ≈ 0 ∀ν∈supp(𝒢_k-1). This means that T_max should be chosen in the tail of the FPT distribution. Note that for the iFPT approach, one only has to compute ℋ(λ,ν) over the support of 𝒢_k for k ≥ 1 once [The support of 𝒢_k is the open interval (0, s_0^(0) + k ·κ).], and thenmultiply it with the adaptation current distribution of the previous iteration 𝒢_k-1. This function then needs to be integrated according to <ref>, and the PDF 𝒢_k can be obtained by numerical differentiation. We show results for 𝒢_2 and 𝒢_3 using <ref> in <ref>. The agreement between MC simulations and <ref> is excellent.Iterating these ideas into the stationary regime, the ideas of the previous paragraph can be used to directly compute the stationary density of the peak adaptation current. We assume that stationarity is reached after k^∗-1 firing events. Thens_0^(k^∗) and s_0^(k^∗ +1) have the same distribution.Consequently, the stationary density of the peak adaptation current after firing satisfies the two-dimensional integral equation (cf. <ref>) 𝒬(θ) = ∫_𝒟(θ)ℋ(λ, ν) 𝒬^'(ν) λ̣ν̣, where 𝒬(θ) ≡ℙ(s^(k^∗)_0≤θ) denotes the CDF of the stationary peak value for the adaptation current s (<ref>). We have checked that this equation is indeed satisfied by the stationary distributions for the peak adaptation current obtained from MC simulations (data not shown). Therefore, <ref> can serve as a tool to check whether a given distribution for the peak adaptation current is stationary, or alternatively as a way to compute 𝒬(θ) directly if the function ℋ is known. § CORRELATIONS BETWEEN INTERSPIKE INTERVALS We now show how to compute serial correlations with the iFPT approach. We define the SCC <cit.> between the nth ISI T_n and the (n+k)th ISI T_n+k according toSCC(n, k) = 𝔼( T_n T_n+k) -𝒬_1(n,k)/𝒬_2(n,k), where 𝒬_1(n,k) = 𝔼 (T_n) 𝔼(T_n+k) , and𝒬_2(n,k) = m_2(k)m_2(n+k) =√(Var(T_n)Var(T_n+k)). Here, Var(T_n) denotes the variance of the nth ISI distribution, and m_2 is the standard deviation given by <ref>. Note that the definition <ref> does not make use of the notion of stationarity, so that the SCC depends on both the position n of the ISI in the spike train as well as on the lag k between ISIs. Since we have already computed the distributions of the kth ISI, we can readily compute the variances and means in <ref>, i.e. the terms given by <ref> and <ref>. It is slightly more complicated to compute the first term in the numerator, 𝔼( T_n T_n+k), because we need the joint densityp_2(T_n, T_n+k) of T_n and T_n+k. In the present study, we focus on k=1. By definition, we have 𝔼(T_nT_n+1) =∫d T_ndT_n+1 T_nT_n+1p_2(T_n,T_n+1),=∫dT_n dT_n+1 T_n T_n+1p_1(T_n+1|T_n)ℱ_n(T_n), where as previously ℱ_n(T_n) is the density of the nth ISI and p_1(T_n+1|T_n) is the conditional density of T_n+1 given T_n. Because T_n+1 is statistically determined only by s_0^(n), we can define this conditional density as p_1(T_n+1|T_n) =∫dy p(T_n+1,y|T_n), where p(T_n+1,y|T_n) denotes the joint density of T_n+1 and s_0^(n)=y conditioned on the previous ISI T_n. We can rewrite this as follows: p(T_n+1,y|T_n) =p_3(T_n+1,y,T_n)/p(T_n),=p(T_n+1|y,T_n)p(y,T_n)/p(T_n),=p(T_n+1|y,T_n)p(y|T_n)p(T_n)/p(T_n),=p(T_n+1|y,T_n)p(y|T_n).Now, as we have previously shown, the statistics of T_n+1 is completely determined when s_0^(n)=y is fixed, hence p(T_n+1|y,T_n)=p(T_n+1|y)≡ℋ(T_n+1,y). Therefore, we have 𝔼(T_nT_n+1)= ∫dT_ndT_n+1 dy T_n T_n+1ℋ(T_n+1,y)p(y|T_n)ℱ_n(T_n) . This can be further simplified by noting that p(y|T_n) = p(y, T_n)/ℱ_n(T_n) and therefore 𝔼(T_nT_n+1)=∫d T_n dT_n+1 dy T_n T_n+1 ℋ(T_n+1,y)p(y,T_n) . For n=1, <ref> can be simplified because s_0^(1) is a deterministic function of T_1 (see <ref>), so that p(s_0^(1) = y, T_1 = x) = δ(y-f(x, s_0^(0))) ℱ_1(x), where f is defined by <ref> and <ref>. Hence, we have for n=1 𝔼(T_1T_2)=∫d T_1dT_2T_1T_2 ℋ(T_2,f(T_1, s_0^(0)))ℱ_1(T_1). We have shown in Section <ref> how to obtain the conditional FPT density ℋ. To evaluate <ref> for general n, we still need to compute the joint density p(s_0^(n) = y,T_n). This can be achieved by means of an MC simulation, where we fix a value of n and then record the frequency with which pairs of s_0^(n) and T_n are generated by the system. We show an example of these densities in <ref>. The most notable feature is an inverse proportionality between s_0^(n) and T_n. The longer e.g. T_2, the less likely it is for the value of s after the second firing, s_0^(2), to attain a high value. <ref> is formally correct, but not very practical for actual computations. This is because to apply the iFPT approach, it is desirable to obtain all quantities needed for the SCC using solutions of the FPE only, and no MC simulations. These, however, are required to obtain an approximation for the joint density p(s_0^(n), T_n) in <ref>. We therefore propose an approximation to compute 𝔼( T_n T_n+1) using the available densities ℱ, 𝒢 and ℋ only. To that end, we note that p_3(T_n+1, T_n, y) = p(T_n+1,T_n |y) p(y) . If we now assume that T_n and s_0^(n) are independent, we can approximate this as follows: p_3(T_n+1, T_n, y)≈ p(T_n+1 |y) p(y) p(T_n) = ℋ(T_n+1,y) 𝒢_n(y) ℱ_n(T_n) . This results in an alternative, approximative expression for the expectation 𝔼(T_nT_n+1): 𝔼( T_n T_n+1) = ∫Ṭ_nṬ_n+1ỵT_nT_n+1 ℋ(T_n+1, y) 𝒢_n(y) ℱ_n(T_n). <ref> is therefore equivalent to <ref> if p(y, T_n) = 𝒢_n(y) ℱ_n(T_n). Although <ref> demonstates that p does not factorise (the joint density is negatively sloped), we show below that <ref> approximates <ref> very well. Given that <ref> does not require additional MC simulations, the small error introduced by <ref> is well offset by the large reduction in computational cost.There exists a third alternative expression for the expectation of the product of ISIs suggested for a different, but related, model, in <cit.>. It reads in our notation [Note that 3.17 in <cit.> is in the stationary state: 𝔼( T_n T_n+1) = ∫Ṭ_nṬ_n+1ỵ T_n T_n+1ℋ(T_n+1, f(T_n,y))ℋ(T_n,y ) 𝒢^∗(y).] 𝔼( T_n T_n+1) = ∫Ṭ_n Ṭ_n+1 ỵT_n T_n+1 ℋ(T_n+1, f(T_n, y)) ℋ(T_n, y) 𝒢_n-1(y) , where f is given by <ref> or <ref>. The term ℋ(T_n, y) 𝒢_n-1(y) that appears in <ref> is the same as the one on the right-hand side in <ref>, which we used to obtain the CDF of s_0^(n).For n=1, <ref> reads 𝔼( T_1 T_2)=∫Ṭ_1 Ṭ_2 T_1 T_2 ℋ(T_2, f( T_1, s_0^(0))) ℋ(T_1, s_0^(0))_=ℱ_1(T_1), because s is started from a point s_0^(0), so that formally 𝒢_0(y) = δ(y-s_0^(0)), which collapses the integration over y in <ref>. Thus, for n=1, <ref> and <ref> coincide. This is also true for higher values of n. A proof for this equivalence is presented in Appendix <ref>. <ref> only makes use of the quantities ℋ and 𝒢, which can be computed using the iFPT approach as explained in the previous section.We show comparisons between MC simulations and the three expressions for 𝔼(T_nT_n+1), <ref>, <ref> and <ref>, in <ref>. The results presented in <ref> are in agreement with the observation that the two expressions <ref> and <ref> are equivalent. We find that the agreement of <ref> with MC simulations is comparable to <ref>, particularly for exponential adaptation. Interestingly, the formally correct <ref> and the approximate <ref> give comparable results; <ref> slightly deviates from MC simulations and <ref> when n gets larger. The maximal relative disagreement between MC and iFPT results is less than 2 %(<ref>, bottom panels). We will see below that the SCC is best approximated by using the exact result <ref> (or equivalently <ref>), as we expect. We attribute the discrepancy between MC simulations and the exact result <ref> to the error caused by the numerical integration over the MC approximation of the joint density p(y, T_n). We checked that applying a kernel density estimation <cit.> to the MC results for p(y, T_n) did not alter these results. Similar results for 𝒬_1 and 𝒬_2 (<ref> and <ref>) are shown in <ref> and <ref>. The agreement is good, with the maximal relative disagreement always less than 5 %. The relative disagreement for the statistics of the product of two adjacent ISIs, 𝒬_1(n,1), is in general larger than the error for the moments, as can be seen by comparing <ref> and <ref> with <ref>. Indeed, for the case of power-law adaptation, we observe an increase of roughly one order of magnitude in the relative error even when the more accurate CN scheme is used (see e.g. left panels of <ref>). An exception is the computation of the joint expectation, shown in <ref>, where, depending on which methods are compared, the relative disagreement is comparable in size to the one for the computation of the moments shown in <ref> and <ref>. This increase of the relative disagreement makes the computation of the SCC using the iFPT approach hard, because the two expressions in the numerator of <ref> are quite close to one another for the parameter values we have chosen here, meaning that the numerator is small and indeed of the same magnitude or even smaller as the relative disagreement, e.g. -2.6· 10^-2 in the left panel and -8.6 · 10^-4 in the right panel of <ref> for n=1 for the MC-GS simulation method. The increase in the relative discrepancy is caused by error propagation, because for the second-order statistics, one has to multiply two quantities that both come with an individual error. Therefore, whereas the iFPT approach can in principle also be used to compute serial correlations present in the spike train, obtaining reliable results can in general be a computational challenge. When the negative serial correlations are stronger, so that the difference in the numerator of <ref> is larger, the iFPT approach should give more accurate results. We stress that the dominant source of error is not the computation of the joint expectation 𝔼(T_nT_n+1) of ISIs, but the product of the expectation of ISIs and the variances, which can be seen by comparing the lower panels of <ref> with those of <ref> and <ref>. Finally, we show the SCC at lag 1 obtained by MC simulations and PDE numerics in <ref>. The agreement is worse than for all previously considered quantities, but still reasonable. To verify the MC simulations, we checked that our MC simulation setup was able to reproduce known analytical results for the SCC obtained in <cit.> for certain limiting cases.The anti-correlations between adjacent ISIs (SCC(n,1) <0) strengthen until they reach a stationary value. Thus, we see that MC and PDE results for the SCC in general do not agree as well as one would expect from the good agreement of the expecations 𝔼(T_nT_n+1) in <ref>.The deviation is likely morepronounced for parameters that lead to small negative SCCs, which we have for both models considered in this section. In the next Section, we will compare this with results for the perfect integrate-and-fire model, where parameter values are chosen so that the SCCs are more negative and hence the agreement is better. This is because the two terms in the numerator of <ref> are close to each other for small SCCs, and hence a small error in them impacts the accuracy of the SCC computation quite dramatically.We show in the next section that our methods reproduce known stationary analytical results for the SCC when we consider the perfect integrate-and-fire model with single exponential adaptation in a parameter regime where we have large negative correlations, thus demonstrating that our methods are sound, but SCC calculations are very sensitive to numerical inaccuracies.§.§ The perfect integrate-and-fire model The adapting perfect integrate-and-fire (PIF) model driven by white Gaussian noise and a single exponential adaptation current is one of the simplest models for spike-triggered adaptation. For small noise intensity, analytical expressions for the stationary SCC exist. We here study this stationary limit case and compare analytical formulas to results obtained with the iFPT approach. The model reads (we follow the notation of <cit.> and <cit.>) X̣ = (I_0 -s) ṭ + √(2D)Ẉ(t) , ṣ/ṭ = -s/τ_a. The adaptation mechanism works in analogy to the previous model (<ref>): whenever X reaches the threshold X = 1, s receives a kick of size Δ≡Δ/τ_a and X is instantaneously reset to 0. The stationary SCC at lag 1 for this model under the assumption of small noise (i.e. D ≪ 1) reads <cit.> SCC(k=1) = -α(1-θ)(1-α^2θ)/1 + α^2 - 2α^2θ, where α = s^∗ - Δ/s^∗, θ = I_0 - s^∗/I_0 - s^∗ + Δ,T^∗ = 1 + Δ/I_0,s^∗ = Δ/1 - exp(-T^∗/τ_a). Thus, we can compute the SCC in closed analytical form as a function of the system parameters. This formula serves as an important benchmark for our numerical results. In particular, we expect that after the described transition to stationarity, the SCC given by <ref> will approach the stationary SCC given by <ref>. This is confirmed in <ref>. In particular, the agreement between MC simulations and the exact formula <ref> is very good (the relative disagreement between PDE numerics and the analytical result is less than 6 % for the stationary value); the agreement of MC simulations with the approximation <ref> is a bit worse, but still reasonable. Thus, we conclude that our methodology can be used more generally to compute the evolution of the moments and SCCs. However, as seen in the previous section, to obtain a good agreement between MC simulations and PDE numerics, the computational effort might be rather large. In particular, we note that the PIF example shown in <ref> gives rise to stronger negative SCCs, which means that the error propagation has less of an effect, but is still present, even when moments of firing times between MC and PDE numerics disagree by less than 1 % (data not shown). We finally note that it is also possible to analytically compute the stationary SCC at higher lags and for different models (e.g. the leaky integrate-and-fire model in the presence of weak noise or for small adaptation currents) using the approach described in <cit.>, or, using a different approach, in <cit.>.§ SUMMARY AND CONCLUSIONS In this paper, we have developed a numerical method for the computation of moments and correlations in general two-dimensional non-renewal escape time processes. Our approach relies on the numerical solution of a two-dimensional time-dependent FPE with initial conditions obtained from marginal distributions of previous states of the system. Crucially, the computation scheme presented in this study is general insofar as it can be applied to any stochastic process with a known reset condition (<ref>) and any deterministic signal (<ref>). As an important application, we have described the transition to stationarity in a stochastic IF neuron model with spike-triggered adaptation, which causes non-trivial ISI correlations. A different mechanism for introducing positive correlations between ISIs has recently been reported in <cit.> and can equally well be analyzed with the presented methodology. Moreover, our approach enables us to determine the non-trivial timescale of transition to a stationary adapted state by counting the number of intervals needed for this transition. Experimentally, the transition to stationarity is often characterized by the behaviour of the instantaneous firing rate <cit.> [Note that in <cit.>, the timescale for single exponential adaptation was inferred from the time course of the numerically obtained instantaneous firing rate, showing that the time course of the rate is not described by the same single exponential time course of the adaptation current. A similar observation is made in <cit.>.]. The instantaneous firing rate is usually obtained by averaging the neuronal activity bin-wise for a fixed time. This differs from the firing rate used here as given by the inverse of the mean ISI (<ref>). In other words, while the instantaneous firing rate is measured in real time, our firing rate relates to interval numbers. This entails that for a given time t, the firing rate contains contributions from, in general, past firing events that may have occurred at any point k in the spike train. Knowing the joint distributions of all ISIs T_k, it is at least in principle possible to reconstruct the instantaneous firing rate, whereas given the instantaneous firing rate, we cannot reconstruct the joint distributions of the individual ISIs T_k. Despite the difference in the definition of the firing rate, it might be an interesting topic for further study to classify the time scales of the transition to stationarity both experimentally and based on the theory presented here.The computation of ISI moments using the iFPT approach is computationally inexpensive, giving rise to small relative disagreements between solutions of the FPE and direct MC simulations. In contrast, the computation of correlations is harder. We observed that we lost one order of magnitude in accuracy compared to the simulation of the moments for the quantities 𝒬_1 (<ref>) and 𝒬_2 (<ref>), which makes the reliable computation of SCCs a computationally challenging task. We conclude that even a relative disagreement of ISI moments between Monte Carlo estimations and PDE solutions of the order 10^-3 is not enough to reliably estimate the SCC using PDE numerics only (but this might be specific for the examples we have considered), indicating that more refined numerical methods or larger computational ressources, or indeed both, are needed. When the difference between the two terms in the numerator of <ref> is large, the small error made by the numerical solution of the PDE should have a less detrimental influence on the final result. The need for more refined numerical methods is further substantiated by the fact that the more accurate asymptotically stable CN timestepping scheme did not result in a significant decrease in the relative disagreement between PDE results and both plain MC and MC-GS simulations, for both moments of firing intervals and the SCC. In this paper, we have only discussed the error associated with MC simulations, because it is readily available. The numerical solution of the FPE is of course also subject to numerical errors and future work will likely benefit from a discussion about how to systematically reduce these errors. In this context, it might be beneficial to compare the finite-element methods used here to other methods for solving PDEs, such as finite difference and finite volume methods <cit.>. A systematic error estimation study might be made more difficult by the fact that the diffusion matrix (<ref>) is not positive definite <cit.>.There is an alternative method to compute the ISI distributions given the distributions of the peak adaptation currents using the formula ℱ_k(t) = ∫_supp(𝒢_k-1)ℋ(t,y) 𝒢_k-1(y) ỵ. This is an integral equation frequently used in the context of randomized FPT problems <cit.>, where usually ℱ_k and the kernel ℋ are given, and one tries to find a matching distribution 𝒢_k-1 of starting points. Using <ref>, we do not have to solve a time-dependent PDE for each ISI, but must compute ℋ once as the solution of a time-dependent FPE with varying initial conditions for s, similar to the computation of ℱ_1. The averaging that the iFPT approach amounts to is particularly clear in this formulation. The densities 𝒢_k are obtained as discussed above (see <ref>). The approach relying on <ref> might be computationally less expensive, but we found that it is not as exact as solving a time-dependent FPE for each ISI, especially at larger times. This is likely caused by errors when computing ℋ, as the numerical integration in <ref> can be performed accurately and efficiently. However, <ref> could be useful for analytical explorations when ℋ is known.We finally emphasize that our approach did not use the complicated boundary conditions for stationary IF models, where the probability flux at threshold gives rise to a discontinuity of the probability flux at reset <cit.>. In contrast, our approach allows for the computation of transient and stationary distributions of the adaptation dynamics in an iterative fashion, requiring the solution of a two-dimensional time-dependent PDE. The only boundary condition that has to be taken into account is an absorbing boundary condition for the probability density at the threshold . This makes the problem tractable using finite-element approximation methods for time-dependent PDEs, resulting in a general description of two-dimensional IF models with spike-triggered adaptation. The approach we have described in this paper can in principle also be used to gain analytical insight into these system, however, quantities such as ℋ and the solution of a two-dimensional time-dependent PDE seem to be unavailable in closed analytical form except in the most simple cases.We would like to thank Alexandre Payeur for insightful discussions and helpful comments (especially about the equivalence of <ref> and <ref>) and NSERC Canada for funding this work. Furthermore, we would like to thank an anonymous referee for comments that helped us to improve the manuscript. § EQUIVALENCE OF <REF> AND <REF> We here show that <ref> and <ref> are equivalent.We recall <ref>:𝔼(T_nT_n+1)=∫Ṭ_n Ṭ_n+1 ṣ_0^(n) T_n T_n+1 ℋ(T_n+1,s_0^(n))p(s_0^(n),T_n) . We re-write<ref> as follows:𝔼( T_n T_n+1) = ∫Ṭ_n Ṭ_n+1 ỵT_n T_n+1 ℋ(T_n+1, f(T_n, y)) p(y, T_n), where y=s_0^(n-1) and we have replaced ℋ(T_n, y) 𝒢_n-1(y) = p(y, T_n). Note that in <ref>, p is the joint density of s_0^(n) and T_n, whereas p is the joint density of s_0^(n-1) and T_n in <ref>.By inspection, the two expressions are identical if we can show that ∫ṣ_0^(n)ℋ(T_n+1, s_0^(n)) p(s_0^(n), T_n) = ∫ṣ_0^(n-1)ℋ(T_n+1, f(T_n, s_0^(n-1)))p(s_0^(n-1), T_n) , for T_n and T_n+1 fixed.Starting from the second line in <ref>, we change the integration variable from s_0^(n-1) to s_0^(n) by observing that from s_0^(n) = f(T_n, s_0^(n-1)), we have ṣ_0^(n)/ṣ_0^(n-1) = ∂ f/∂ s_0^(n-1) and therefore ṣ_0^(n-1) = ṣ_0^(n)(∂ f/∂ s_0^(n-1))^-1. We need to assume that f is invertible with respect to the second argument, which is the case for both power-law (<ref>) and exponential adaptation (<ref>) considered in this paper. The integral then becomes∫Ṭ_n Ṭ_n+1 ṣ_0^(n)T_n T_n+1 ℋ(T_n+1, s_0^(n))) p[f^-1(T_n, s_0^(n)), T_n] (∂ f/∂ s_0^(n-1))^-1.But p[f^-1(T_n, s_0^(n)), T_n)] (∂ f/∂ s_0^(n-1))^-1 is nothing but the transformation from p(s_0^(n-1), T_n) to p(s_0^(n), T_n). Indeed, we have (fixing T_n) p(s_0^(n), T_n) ∂ s_0^(n)/∂ s_0^(n-1)=p(s_0^(n-1), T_n) , so that p(s_0^(n), T_n)= p[f^-1(T_n, s_0^(n)), T_n](∂ f/∂ s_0^(n-1))^-1. Therefore, <ref> and <ref> are equivalent. | http://arxiv.org/abs/1704.08669v1 | {
"authors": [
"Wilhelm Braun",
"Rüdiger Thul",
"André Longtin"
],
"categories": [
"q-bio.NC",
"math.PR",
"physics.comp-ph",
"physics.data-an"
],
"primary_category": "q-bio.NC",
"published": "20170427173332",
"title": "Evolution of moments and correlations in non-renewal escape-time processes"
} |
=1mn2e -0.8in Section Section Pre-Supernova Outbursts] Pre-Supernova Outbursts via Wave Heating in Massive Stars I: Red SupergiantsFuller et al.] Jim Fuller^1,2Email: [email protected]^1TAPIR, Walter Burke Institute for Theoretical Physics, Mailcode 350-17, Caltech, Pasadena, CA 91125, USA^2Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, CA 93106, USA[ [ December 30, 2023 =====================Early observations of supernovae (SNe) indicate that enhanced mass loss and pre-SN outbursts may occur in progenitors of many types of SNe. We investigate the role of energy transport via waves driven by vigorous convection during late-stage nuclear burning of otherwise typical 15 M_⊙ red supergiant SNe progenitors. Using MESA stellar evolution models including 1D hydrodynamics, we find that waves carry ∼ 10^7 L_⊙ of power from the core to the envelope during core neon/oxygen burning in the final years before core collapse. The waves damp via shocks and radiative diffusion at the base of the hydrogen envelope, which heats up fast enough to launch a pressure wave into the overlying envelope that steepens into a weak shock near the stellar surface, causing a mild stellar outburst and ejecting a small (≲ 1 M_⊙) amount of mass at low speed (≲ 50 km/ s) roughly one year before the SN. The wave heating inflates the stellar envelope but does not completely unbind it, producing a non-hydrostatic pre-SN envelope density structure different from prior expectations. In our models, wave heating is unlikely to lead to luminous type IIn SNe, but it may contribute to flash-ionized SNe and some of the diversity seen in II-P/II-L SNe.§ INTRODUCTIONThe connection between the diverse population of core-collapse supernovae (SNe) and their massive star progenitors is of paramount importance for the fields of both SNe and stellar evolution. Over the past decade, substantial evidence has emerged for enhanced pre-SN mass loss and outbursts in the progenitors of several types of SNe. The inferred mass loss rates are typically orders of magnitude larger than those measured in local group massive stars, and the mass loss appears to systematically occur in the last centuries, years, or weeks of the stars' lives. This deepening mystery cannot be explained by standard stellar evolution/wind theories, and its solution lies at the heart of the SNe-massive star connection. Type IIn SNe provide the most obvious evidence for pre-supernova mass loss, and it is well known that these SNe are powered by interaction between the supernova ejecta and dense circumstellar material (CSM). However, type IIn SNe are very heterogeneous ( classifies them into ten subtypes), as some appear to require interaction with ∼ 10 M_⊙ of CSM ejected in the final years of their progenitor's life, while others require mass loss rates of only ∼ 10^-4M_⊙/ yr but lasting for centuries before the explosion <cit.>. These mass loss rates are much larger than predicted by standardmass-loss prescriptions. In some cases, pre-SN outbursts resulting in mass ejection have been observed directly, famous examples being SN 2009ip (which did not explode until 2012, ), 2010mc <cit.>, LSQ13zm <cit.>, and SN 2015bh <cit.>, which show resemblance with luminous blue variable (LBV) star outburts. Pre-SN outbursts now appear to be common for type IIn SNe <cit.>.Enhanced pre-SN mass loss has also been inferred from observations of other types of SNe. Type Ibn SNe (e.g., SN 2006jc which had a pre-SN outburst, ; and SN 2015U, ) show interaction with He-rich material ejected soon before core-collapse.SN 2014C was a type Ib SNe that transitioned into a type IIn SNe after the ejecta collided with a dense shell of H-rich CSM ejected by its progenitor in its final ∼decades of life <cit.>. Early spectra of type IIb SN 2013cu reveal emission lines from a flash-ionized wind <cit.> with inferred mass loss rates over 10^-3M_⊙/ yr<cit.>. Many bright type II-P/II-L SNe also show flash-ionized emission lines in early time spectra indicative of a thick stellar wind <cit.>, while even relatively normal II-P SNe sometimes exhibit peaks in their early light curves that may be produced by shock cooling of an extremely dense stellar wind <cit.>. Recently, <cit.> found that the otherwise normal type II-P SN2013fs showed emission lines only within the first several hours after explosion, indicating that modest mass ejection of ∼ 10^-3M_⊙ in the final ∼year of the progenitor's life is common for type II-P SNe.One of the most promising explanations for pre-SN outbursts and mass loss was proposed by <cit.>, who investigated the impacts of convectively driven hydrodynamic waves during late-phase nuclear burning. Convectively driven waves are a generic consequence of convection that are routinely observed in hydrodynamic simulations. <cit.> showed that the vigorous convection of late burning stages (especially Ne/O burning) can generate waves carrying in excess of 10^7 L_⊙ of power to the outer layers of the stars, potentially depositing more than 10^47erg in the envelope of the star over its last months/years of life. <ref> provides a cartoon picture of the wave heating process. <cit.> then showed that the wave heating is generally more intense but shorter-lived in more massive stars, and could occur in a variety of SN progenitor types. More recently, <cit.> have examined the effect of super-Eddington heat deposition (e.g., due to wave energy) near the surface of a star, showing that the heat can drive a dense wind with a very large mass-loss rate.In this paper, we examine wave heating effects in otherwise “typical" M_ ZAMS =15 M_⊙ red supergiants (RSGs) that may give rise to type II-P, II-L, or IIn supernovae depending on the impact of wave heating. We quantify how wave heating alters the stellar structure, luminosity, and mass-loss rate using MESA simulations <cit.> including the effects of wave heating due to convectively driven waves. After carbon shell burning, we use the 1D hydrodynamic capabilities of MESA to account for the pressure waves, shocks, and hydrodynamic/super-Eddington mass loss that can result from intense wave heating.§ WAVE ENERGY TRANSPORT§.§ Wave Generation Gravity waves are low frequency waves that can propagate in radiative regions of stars where their angular frequency ω is smaller than the Brunt-Väisälä frequency N (see <ref>). They are excited at the interface between convective and radiative zones, carrying energy and angular momentum into the radiative zone which is sourced from the kinetic energy of turbulent convection. The energy carried by gravity waves is a small fraction of the convective luminosity, scaling roughly as <cit.> L_ wave∼ℳ_ con L_ con,where L_ con is the luminosity carried by convection and ℳ_ con is a typical turbulent convective Mach number. In most phases of stellar evolution, ℳ_ con≲ 10^-3 within interior convection zones, and the energy carried by gravity waves is negligible. Equation <ref> has been approximately verified by multidimensional simulations ().<ref> shows the quantity L_ wave within the interior of a M_ ZAMS = 15 M_⊙ stellar model from core carbon burning onward. Details and parameters of our MESA models can be found in <ref>. Before carbon shell burning, L_ wave is much less than the surface luminosity of L ≃ 10^5 L_⊙, and wave energy transport is negligible. However, after carbon burning, neutrino cooling becomes very efficient within the core, which falls out of thermal equilibrium with the envelope. To maintain thermal pressure support, burning luminosities increase and become orders of magnitude larger than the surface luminosity. Convective mach numbers also increase, and consequently L_ wave during late burning phases can greatly exceed the surface luminosity, allowing wave energy redistribution to produce dramatic effects.To estimate wave luminosities in our 1D models, we proceed as follows. First, we calculate L_ wave at each radial coordinate as shown in <ref>. Next, we calculate a characteristic convective turnover frequency at each radial coordinate viaω_ con = 2 πv_ con/2 α_ MLT H,wherev_ con=[L_ con/(4 πρ r^2)]^1/3,is the RMS convective luminosity according to mixing length theory (MLT), α_ MLT is the mixing length parameter, and H is a pressure scale height. The turbulent mach number is ℳ_ con = v_ con/c_s, where c_s is the adiabatic sound speed. Remarkably, these estimates of convective velocities and turnover frequencies typically match those seen in 3D simulations of a variety of burning phases (e.g., ) to within a factor of two.In reality, a spectrum of waves with different angular frequencies ω and angular wavenumbers k_⊥ = √(l(l+1))/r are excited by each convective zone, where l is the spherical harmonic index of the wave. Rather than model the wave spectrum, we find the maximum value of ω_ con (usually located a fraction of a scale height below the zone's outer radius), and assume that all the wave power is put into waves at this frequency ω_ wave = ω_ con,max,and angular wave numbers l=1. Simulations show that realistic wave spectra are peaked around ω = ω_ wave and l=1, even for fairly thin shell convection like that in the Sun (see ), at least for waves not immediately damped, so these approximations are reasonable. Waves at lower frequencies are typically much more strongly damped, while waves at higher frequencies contain much less power. Waves at higher values of l contain comparable or less power and are more strongly damped, so we ignore their contribution. At each time step in our simulations, we find the radial location of ω_ max within the core (usually located within the innermost convective burning zone), and then compute v_ con, ω_ wave, and L_ wave at that point using equations <ref>, <ref>, and <ref>.§.§ Wave Propagation and Dissipation The next step is to calculate how waves of frequency ω_ wave and l=1 will propagate and dissipate within the star. Typical waves at ω=ω_ wave during late burning phases are gravity waves in the core of the star, but in the envelope they are acoustic waves (see <ref>). In order to propagate into the envelope, the waves must tunnel through one or more intervening evanescent zones, the largest of which is often created by the convective helium burning shell. Apart from wave evanescence, we ignore wave interactions with convection in these regions because their convective energy fluxes and turnover frequencies are generally much smaller than the core convection that launches the waves, although some interaction may take place. Before tunneling out of the core, the waves may reflect multiple times and can be damped by neutrino emission or by breaking near the center of the star, dissipating some of their energy within the core. In <ref>, we provide details of how to calculate these effects in order to determine the fraction of wave energy f_ esc which is able to escape from the core and propagate into the envelope as acoustic waves.The wave energy that heats the envelope is thenL_ heat = η f_ esc L_ wave.Here, η is an efficiency parameter (with nominal value η =1 unless stated otherwise) we will adjust to explore the dependence of our results on the somewhat uncertain wave flux. We find typical values of f_ esc∼ 0.5 during core neon/oxygen burning, and f_ esc∼ 0.1 during shell burning phases because more wave energy is lost by tunneling into the core. We do not compute the effect of wave heating within the core because its binding energy is much larger than integrated wave heating rates, and because neutrinos can efficiently remove much of this thermal energy.<ref> shows the nuclear energy generation rate L_ nuc (not including energy carried away by neutrinos) of our stellar model as a function of time, along with the envelope wave heating rate L_ heat and the surface luminosity L_ surf.Important burning phases are labeled. Although the fraction of nuclear energy converted into waves that escape the core is generally very small (<10^-3), the value of L_ heat can greatly exceed L_ surf. In our models, L_ surf remains smaller than L_ heat during later burning phases because most of the wave heat remains trapped under the H envelope and is not radiated by the photosphere, which we discuss more in <ref>.After determining L_ heat, we must determine where within the envelope the wave energy will damp into thermal energy. This calculation is detailed in <ref>, where we calculate wave damping via thermal diffusion and describe how we add wave heat into our stellar model. The most important feature of diffusive wave damping is that it is strongly dependent on density and sound speed, with a characteristic damping mass M_ damp∝ρ^3 (equation <ref>). In RSGs, the density falls by a factor of ∼ 10^6 from the helium core to the base of the hydrogen envelope (see <ref>). Hence, acoustic waves at frequencies of interest are essentially undamped in the helium core but quickly damp as they propagate into the hydrogen envelope, and they always thermalize their energy in a narrow shell of mass at the base of the hydrogen envelope.In the late stages of preparing this manuscript, <cit.> demonstrated that acoustic waves will generally steepen into shocks before damping diffusively, causing them to thermalize their energy deeper in the star. Using their equation 6 and calculating wave amplitudes from the value of L_ heat, we find shock formation in our models occurs at somewhat larger (by a factor of a few) density than radiative diffusion, but at very similar mass coordinates and overlying binding energies. The reason is that the density cliff at the edge of the He core promotes both shock formation and diffusion. We therefore suspect that wave energy thermalization via shock formation will only marginally affect our results, but we plan to account for it in future work.Our wave heating calculations during shell Ne/O burning and core Si burning are less reliable due to an inadequate nuclear network in our models, and increasing wave non-linearity. These burning phases occur less than an envelope dynamical time before core collapse, giving waves little time to alter envelope structure. For these reasons, we do not closely examine these phases in this work, but large wave luminosities during these phases may affect some progenitors.§ EFFECTS ON PRE-SUPERNOVA EVOLUTION In our models, wave heating is most important during late C-shell burning, core Ne burning, and core O burning. To quantify the effects of wave heating on the pre-SN state of the stellar progenitor, we construct MESA models and evolve them from the main sequence to core-collapse. At each time step, we add wave heat L_ heat as described in <ref> and <ref>. Just before C burning, we utilize the 1D hydrodynamic capabilities of MESA (see <ref>) which is essential for capturing the non-hydrostatic dynamics that result from wave heating.Relative timescales are important for understanding wave heating effects. We define a local wave heating timescale t_ heat = c_s^2/ϵ_ heatwhere ϵ_ heat is the wave heat deposited per unit mass and time. This can be compared with a thermal cooling timescalet_ therm = 4 πρ r^2 H c_s^2/Lwhere H is the pressure scale height and L is the local luminosity. We also define a local dynamical time scale t_ dyn = H/c_s.Finally, all of these should be considered in relation to the time until core-collapse, t_ col. The first key insight is that wave energy is deposited at the base of the hydrogen envelope, above which t_ therm is comparable to (but generally larger than) t_ col (see <ref>). Consequently, wave heat cannot be thermally transported to the stellar surface before core-collapse, and the surface luminosity L_ surf is only modestly affected (<ref>). We therefore do not expect very luminous (L ≳ 10^6 L_⊙) pre-SN outbursts to be driven by wave heating in RSGs. The second key insight is that wave heating timescales can be very short. In the slow heating regime with t_ heat≳ t_ therm≳ t_ dyn, wave heat can be thermally transported outward without affecting the local pressure. In the moderate heating regime with t_ therm≳ t_ heat≳ t_ dyn, wave heat cannot be thermally transported outward, but the star can expand nearly hydrostatically to accomodate the increase in pressure (see discussion in ). However, we find wave heating can be so intense that it lies in the dynamical regime t_ heat≲ t_ therm,t_ dyn. In this case, wave heat and pressure build within the wave damping region, exciting a pressure wave which propagates outward at the sound speed (<ref>). This pressure wave crosses the stellar envelope on a global dynamical timescale t_ dyn,glob∼√(R^3/GM)≃ 0.5 yr for our stellar model.In our models, the most important envelope pressure wave arises from wave heating during core Ne burning and a third C-shell burning phase (later waves do not reach the surface before core-collapse). As these pressure waves approach the surface where the density and the sound speed drop, they steepen into a weak shock (ℳ≲ 3). When the shock wave breaks out of the surface, it produces a sudden spike in surface temperature and luminosity (see Figures <ref> and <ref>), akin to SN shock breakout <cit.> but with much smaller energy, E ∼ 10^47erg. This shock breakout is similar to that expected from failed SNe in RSGs <cit.>, but even less energetic and luminous, and preceding core-collapse by months or years. Unlike SNe or failed SNe, the shock in our models is not strong enough to unbind the entire RSG envelope, but it can still drive a small outflow (M_ out≲ 1 M_⊙, see <ref>) with speeds comparable tothe escape speed v_ esc. After shock breakout, the envelope expands and cools, but is not able to settle back to its quiescent state before core-collapse, or before a subsequent pressure wave is launched by a later burning phase. <ref> shows the evolution of our model in the HR diagram during its last century. The pressure wave breakout creates a jump in surface temperature and luminosity followed by envelope expansion and cooling. The rebrightening just before core-collapse occurs as a second pressure wave (driven by wave heating during C-shell burning) approaches the photosphere. <ref> shows the corresponding evolution in surface temperature and photospheric radius. Core O-burning produces a markedly different result from Ne-burning because the wave heating is both stronger and lasts longer, depositing nearly an order of magnitude more energy into the envelope (<ref>). In our models, the pressure increase in the wave heating region is large enough to accelerate material upward and out of the heating region at supersonic velocities (exceeding 10^3 km/ s, see <ref>) such that a cooling timescale by advection becomes shorter than a local dynamical timescale, limiting the buildup of pressure. This material decelerates when it runs into the massive overlying envelope.As mass is accelerated out of the heating region, a peculiar structure develops: a dense helium core surrounded by an evacuated cavity filled by the low density wind, contained by a higher density but nearly stationary overlying envelope (<ref>). In essence, the wave heating blows a nearly empty bubble at the base of the hydrogen envelope. As material is blown out of the heating region, it is replaced by upwelling material from beneath. The heating region digs down toward the helium core, and the mass coordinate of the base of the heating region decreases with time. Consequently, wave heat is distributed over a larger amount of mass (∼ 10^-2M_⊙ in our models) than it would be otherwise. The effective heating time (integrated over all mass that has absorbed wave energy) increases, becoming smaller than a dynamical time. For this reason, no strong pressure wave is driven into the envelope. Instead, the bubble inflates slowly, lifting the overlying envelope nearly hydrostaticly. We caution that multi-dimensional effects are likely to drastically alter this scenario and the resulting density profile of the star, which we discuss further in <ref>. Nonetheless, the density structure of the RSG may be substantially altered by wave heating, with likelyimplications for the lightcurve of its subsequent SN.§ DISCUSSION§.§ Implications for Subsequent Supernovae Our results have significant implications for SNe resulting from RSGs affected by wave heating. We have shown that waves can deposit ∼ 10^48erg of energy into the stellar envelope (an amount comparable to its binding energy) in the last months to years of the star's life. Because this energy is negligible compared to the core binding energy, wave heating is unlikely to greatly alter the core structure or SN explosion mechanics (also, neutrinos can cool wave heated regions in the core).The effect on the envelope structure, however, may be dramatic. The first crucial event in our models is the pressure wave breakout that results from wave heating during core Ne burning. For our nominal wave heating efficiency, a small amount of mass (∼ 10^-1M_⊙) is ejected at roughly one half the escape speed (see <ref>). Much of this mass falls back toward the star before core-collapse, and the resulting surface structure is neither hydrostatic nor does it have a steady wind density profile. However, we also note that several physical effects in the outflowing envelope material (e.g., treatment of convection, radiative transfer, non-spherical shock fronts, line-driven winds, molecule/dust formation) have not been properly treated in our models, and it is possible the outflow could have a component with somewhat higher velocity that extends to larger radii. For our optimistic wave heating efficiency (η=3), the outburst is strong enough to eject ∼ 1 M_⊙ at v ∼ v_ esc, producing a dense outflow up to the moment of core-collapse. Nominal outflow velocities of ∼ 30 km/ s and timescales of ∼ 1 yr imply the CSM is confined within ∼ 10^14cm of the progenitor photosphere at the time of core-collapse.The second crucial event occurs during core O-burning. In our models, O-burning inflates an evacuated bubble at the base of the H-envelope that lifts the overlying envelope to larger radii. The density structure of the envelope is substantially altered. The main effects (when plotting density vs. mass coordinate, see <ref>) are to increase the envelope volume and decrease its density, and to flatten the density profile of the envelope. The wave-induced mass ejection events could substantially alter early SN spectra, and are a very compelling mechanism to produce the growing class of flash-ionized Type II-P/L SNe <cit.> which show recombination lines from CSM at early times. The wave model predicts large (but not extreme) mass loss rates of 10^-3M_⊙/ yr≲Ṁ≲ 10^0 M_⊙/ yr, and slow velocities of v ≲ 100 km/ s similar to those that have been measured or inferred. Crucially, the wave model explains why outbursts occur in the last months or years of the progenitor's life, which also accounts for the confinement of the CSM to small distances from the progenitor.The altered density structure will also affect the SNe lightcurve. Shock cooling from a dense wind could create a faster rise time <cit.> that may alleviate the tension between measured galactic RSG radii and the suprisingly small radii inferred from shock cooling models without a wind <cit.>. The dense wind can also create early peaks in type IIP <cit.>, and can cause the SNe to appear more IIL-like <cit.>. Our optimistic wave efficiency produces CSM masses and density profiles similar to those inferred by <cit.>, although our nominal wave efficiency does not appear to eject mass in a wind-like density profile due to mass fallback. Additionally, the flatter density profile of our models relative to non-heated models (see <ref>) will result in a more steeply declining lightcurve <cit.>, again making the SN more II-L-like. We speculate that the altered density profile contributes substantially to the observed diversity of type II-P/II-L lightcurves, but more sophisticated SN light curve modeling will be needed for detailed predictions.Supernova shock breakout could appear different from prior expectations in the presence of wave-induced mass ejection. In contrast to the steep density profiles near the photospheres of stellar models, detected shock breakouts <cit.> appear to emerge from a more extended photosphere or wind with a shallower density profile. Wave-induced mass loss can produce this sort of density structure (<ref>). However, even in the absence of wave heating, significant “coronal" material may exist at the base of the wind-launching region <cit.> and may also contribute to extended UV shock breakout and the optical SNe features discussed above.Finally, it is unlikely that wave heating in “normal" RSGs will lead to luminous type IIn SNe. The main reason is that there is not enough time to eject material to large radii of ∼10^15-10^16cm needed for a luminous IIn event. Even optimistic ejection speeds of 10^7cm/ s and durations of 10^8s cannot quite propel material to large enough distances (but see <ref>). §.§ Comparison with Existing Observations It is well established from pre-SN imaging that most type II-P SNe arise from RSG progenitors with inferred masses M ≲ 20 M_⊙<cit.>. In many cases, progenitor characteristics have been measured from archival ground-based orHubble Space Telescope data that predates the SN by more than ∼ 10 years. In suchcases, we do not expect wave heating to significantly impact the appearance of the progenitor or its inferred mass. However, we encourage caution when inferring progenitor masses from pre-SN imaging. Our models predict that progenitors could be more luminous than expected, causing masses to be overestimated, at least when pre-SN imaging occurs after the onset of Ne/O burning. In a few cases (e.g., SN2003gd, ; SN2004A, , SN2008bk, ; ASASSN-16fq, ;), pre-explosion imaging was obtained within a few years of explosion. In most of these cases the SN progenitor was faint (L<10^5 L_⊙), and the inferred mass was low (M ≲ 11 M_⊙), significantly smaller than the 15 M_⊙ model explored here (the inference of M ∼ 17 M_⊙ for the progenitor of SN2012aw byhas since been revised downward to ≈ 12 M_⊙, see ). Note also the convective overshoot in our model made it behave like a slightly more massive star of ≈ 17 M_⊙, compared with other stellar evolution codes with less internal mixing. Future modeling of low-mass RSG SN progenitors will be needed to determine whether wave heating can strongly affect their pre-SN properties. SN2004A was imaged roughly 3 years before the SN, and was significantly brighter (and slightly cooler) than some of the other progenitors, possibly arising from a higher mass star <cit.>. We suggest the pre-SN properties of this star may have been affected by wave heating.Multi-epoch photometry of the progenitor of ASASSN-16fq disfavors significant variability like that predicted in<ref><cit.>. The progenitor was estimated to be low-mass (8 M_⊙≲ M ≲ 12 M_⊙), again significantly less massive than our model. These observations indicate that wave heating effects in that star were smaller than we have predicted for our higher mass model, or that pre-SN variability/outbursts only occur in a subset of type II-P progenitors. Preliminary wave heating calculations indicate that pre-SN variability may be smaller in progenitors with M ∼ 10 M_⊙ due to longer evolution timescales and lower wave heating rates. Future work examining wave heating in lower mass RSG progenitors will be necessary for detailed observational comparisons. §.§ Predictions The strongest prediction of our work is that mild pre-SN outbursts will be common in RSG progenitors of type II SNe. Although we have not explored the entire parameter space of RSG masses and properties, our otherwise “normal" model suggests similar effects to those explored here will operate in many RSGs. In lower mass RSGs, there may be multiple smaller amplitude outbursts spread over the last ∼decade of the star's life due to multiple core burning phases. Higher mass RSGs are expected to exhibit fewer but larger amplitude outbursts, occurring in the final ∼months of life.We also predict that most RSG outbursts will exhibit modest luminosity excursions of less than ∼2 magnitudes. We expect peak bolometric luminosities to remain under ∼ 10^6L_⊙. Ejecta masses will likely be small, M_ ej≲ 1 M_⊙, and with low velocities v ≲ 50 km/ s. These mild outbursts will be missed by most current transient surveys, but upcoming surveys with greater sensitivity and higher cadence (e.g., ZTF, BlackGem,LSST) may verify or rule out our predictions.We predict wave heating to increase the luminosity of the resulting SN due to the inflated progenitor radius. Analytic scalings predict plateau luminosities of <cit.>L_p ∝ E_ SN^5/6 M_ env^-1/2 R^2/3and plateau durations t_p ∝ E_ SN^-1/6 M_ env^1/2 R^1/6,where E_ SN is the SN explosion energy, M_ env is the envelope mass, and R is the pre-SN stellar radius. Hence, we expect the plateau duration to be insensitive to wave heating, but the SN luminosity may be significantly larger (L ∝ R^2/3) for the same explosion energy. Alternatively, the larger progenitor radii (by a factor of ∼ 2) of our models would require smaller explosion energies, all else being equal. §.§ Rayleigh-Taylor Instabilities The density profiles shown in <ref> are unrealistic because of multi-dimensional effects, in particular because of the Rayleigh-Taylor instabilities (RTI) that will exist real stars. RTI can operate when pressure and density gradients have the opposite sign <cit.>, for instance, in massive star atmospheres where density inversions predicted by 1D models are altered by RTI <cit.>. In our case, RTI will occur at the surface of the wind-blown bubble during O-burning. The interface between the inflated cavity (high pressure, low density) and overlying envelope (low pressure, high density) will give rise to RTI which will likely act to smooth the density profiles shown in <ref>. The mixing produced by RTI may allow more envelope material to mix downward into the heating region, and allow more heated material to mix upward into the envelope. The net effect on the RSG envelope structure is unclear, but the very large and low density cavities in <ref> will likely shrink and increase in density. Nonetheless, the envelope density profile may be strongly altered by wave heating during O-burning. §.§ Caveats Because this is one of the first investigations of the hydrodynamic/observational details of wave-driven heating, there are a number of uncertainties and caveats that must be considered.§.§.§ Wave Excitation Probably the largest uncertainty in our calculations is the amplitude and spectrum of gravity waves excited by convection in nuclear burning zones. We have approximated the gravity waves as monochromatic in both temporal and horizontal wavenumber (one frequency and spherical harmonic index ℓ) which is clearly a gross simplification. If the waves are excited to lower amplitudes (e.g., because we have calculated L_ wave at an inappropriate location) or higher amplitudes (e.g., because wave luminosity scales as ℳ_ con^5/8 as suggested by ), the wave heating effects will be significantly altered, as demonstrated by the reduced and enhanced wave efficiency factors η in Figures <ref>, <ref>, and <ref>. The wave frequency spectrum excited by convection is not well understood, as <cit.> argue for excitation at ω_ wave∼ω_ con due to bulk Reynolds stresses, while <cit.> argues for excitation via plume incursion that adds a substantial high frequency (ω_ wave > ω_ con) tail to the spectrum. If our estimates of wave frequencies are too high/low, then we have likely over/underestimated the fraction of wave energy that heats the envelope (f_ esc, equation <ref>) because high/low frequency waves are less/more subject to neutrino damping and usually have a higher/lower transmission coefficient into the envelope (see <ref>). Finally, if waves are mostly excited at higher angular wavenumbers than ℓ=1, heating rates will be substantially reduced because higher angular wavenumbers are more strongly damped and have smaller transmission coefficients. §.§.§ Nonlinear Effects All calculations in this work assume wave amplitudes are small enough for linear wave physics to apply, which may be reasonable where k_r ξ_r ≳ 1, with k_r the radial wavenumber and ξ_r the radial displacement. Preliminary waveform solutions indicate this criterion is satisfied for high frequency waves with ω≳ 2 ω_ con, but not for lower frequency waves. These waves may be attenuated by non-linear wave breaking in the core, so if the wave power spectrum contains most of its power at frequencies less than ∼ 2 ω_ con, our wave heating rates will be significantly overestimated. We intend to investigate this more thoroughly in a future publication. Additionally, non-linear coupling and instabilities are known to operate at smaller amplitudes (see e.g., ) in various contexts. If non-linear coupling in the g mode cavity is able to prevent waves from being transmitted into the envelope, this further could suppress wave heating.§.§.§ Convection and Radiative Transfer Our one dimensional simulations implement MLT for convective energy transport, and the diffusion approximation for radiative energy transport. The former approximation is calibrated for stars in hydrostatic and thermal equilibrium, which is not the case in the outflowing near-Eddington envelope of our models. In the wave heating region, we have utilized acceleration-limited convective velocities (see <ref>), with a maximum acceleration of the mixing length velocity equal to the local gravitational acceleration, g. However, if it can accelerate faster, convection at the base of the hydrogen envelope could carry more wave heat outward because the maximum convective luminosity L_ max = 2 π r^2 ρ c_s^3 ≫ L_ heat in this region. We have performed experiments without limiting convective acceleration, finding the pressure wave launched during Ne burning and the final stellar radius are only are weakly affected. However, during O burning, convection carries most of the wave energy outward from the heating region, causing the star to reach much higher surface luminosities of ≳ 10^6 L_⊙. The wave-inflated cavity still exists but is smaller and less evacuated. A better understanding of convection's ability to respond to sudden heating is needed for robust predictions of the stellar luminosity and density evolution.In addition to affecting the background envelope structure, the use of MLT will affect the luminosity during the pressure wave breakout. It is not immediately clear how to treat convective energy transport in the regime where bulk velocities are a significant fraction of the sound speed.We have experimented with different treatments of convection (e.g., limiting maximum convective velocities), finding they produce modest quantitative alterations of our results but do not change the basic picture. The use of the diffusion approximation may also produce errors in our predicted pressure wave breakout luminosity evolution, which we hope to re-examine in future work.§.§.§ Rotation and Flows We have ignored effects of rotation in this preliminary analysis, which is justified in the slowly rotating stellar envelope. Rotation could significantly affect wave excitation and propagation in the core if its rotation rate is comparable to wave angular frequencies, but late stage core rotation rates are poorly constrained. Rapid core rotation will probably not eliminate wave heating because it is difficult to suppress both prograde and retrograde waves with reasonable rotation profiles, although the wave heating efficiency could be reduced.In this work we did not include background flows in equations governing wave propagation, even though we showed that waves can generate supersonic flows within the stellar envelope. Our approximation is valid during core Ne burning when induced velocities are small compared to wave group velocities. During core O burning, however, some wave energy damps in regions where flow velocities are comparable to the sound speed (e.g., near 10 R_⊙ in <ref>). Such flows will alter wave propagation/dissipation, but we leave this for future work in light of the additional effects of shock formation and Rayleigh-Taylor instabilities that will also alter flow velocities (see below). §.§ Magnetic Fields Background magnetic fields may be important in some stars. We do not expect them to greatly alter the envelope dynamics where the waves are acoustic in nature and the flow velocities are mostly radial. However, sufficiently strong magnetic fields can prevent gravity wave propagation in the core <cit.>. Such fields would likely confine wave energy to the core of the star and prevent wave heating outbursts. We discuss this possibility in <ref>. §.§ BinariesBinary interactions may contribute to pre-SN mass loss <cit.> but need to be finely tuned to occur in the final years of evolution. It might be possible, however, for the combination of wave heating and binary interactions to produce IIn SNe in a small fraction of RSGs. If the RSG has been partially stripped of its H-envelope, wave heat will be concentrated in a smaller amount of mass and larger ejection speeds may be possible. Furthermore, outburst luminosities in stripped stars will be much larger due the smaller thermal time of the envelope (Fuller 2017, in prep). Finally, envelope inflation via waves could induce a common envelope event for an appropriately placed binary companion, potentially ejecting more mass at larger speeds and creating a IIn event <cit.>. §.§ Relation to other Theories of Pre-SN Outbursts The notable feature of wave-driven outbursts is its generality: it can occur in low-mass (M< 20 M_⊙) stars that are the most common SNe progenitors. Below, we discuss other mass-loss mechanisms that have been proposed, but note that many are restricted to small regions of SN progenitor parameter space or do not yet yield quantitative predictions.One possible mechanism for pre-SN outbursts is instabilities during late stage (C/Ne/O) convective shell burning. In a series of papers <cit.>, Meakin, Arnett, and collaborators have investigated the properties of convection during late phase (carbon shell burning and beyond) nuclear burning. They find that the convective burning shells exhibit some interesting properties not predicted by mixing length theory (therefore not typically implemented in 1D stellar evolution codes), such as entrainment and energy generation rate fluctuations. However, it remains unknown whether convective fluctuations can grow large enough to produce any detectable effect at the stellar surface, nor is it clear what the observational signature would be and how often this process should occur.<cit.> examined linear instabilities during late burning phases, finding no instabilities growing fast enough to produce large effects. <cit.> showed that degenerate Si-burning flashes in ≃ 10 M_⊙ stars could produce shock waves that eject part of the stellar envelope, which may account for some fraction of IIn SNe. Additionally, pair instabilities in very massive stars (M≳ 60 M_⊙) may produce some outbursts and interacting SNe <cit.>, but again the rarity of these events and distinct light curve features makes them unlikely to be responsible for most type IIn SNe. <cit.> and <cit.> show that envelope pulsational growth rates increase after core helium depletion, potentially driving a superwind during the last tens of thousands of years of a star's life, although this theory cannot explain very high (>10^-3M_⊙/ yr) mass loss rates in the last years of a star's life (except perhaps in very massive stars, ). <cit.> suggest intense core dynamo activity can generate outbursts through the buoyant rise of magnetic flux tubes, but they neglect to account for stable stratification in radiative shells which can strongly hinder the radial motion of flux tubes and prevent them from rising into the envelope. Mass loss can be triggered by the loss of gravitational binding energy due to neutrino emission <cit.>, but this can only occur for stars extremely close to the Eddington limit and can only yield Ṁ>10^-3M_⊙/ yr during the last ∼month of the star's life. § CONCLUSIONSWe have modeled the evolution of a 15 M_⊙ red supergiant (RSG) model in the final decades before core-collapse, accounting for energy transport by convectively excited waves. Our goal was to determine whether wave energy transport can affect the pre-supernova (SN) structure of the star or produce pre-SN outbursts as suggested by <cit.>. We used the MESA stellar evolution code <cit.> to model the effect of wave heating on the stellar structure, implementing its 1D hydrodynamical capabilities to capture shocks and outflows resulting from wave heating.During late nuclear burning phases (core Ne and O burning in particular), convective luminosities of L_ con∼ 10^10L_⊙ will excite gravity waves which carry energy fluxes of L_ wave∼ 2 × 10^7L_⊙. We calculate that much of this energy will be transmitted into acoustic waves that propagate out of the core and into the envelope, carrying a flux of L_ heat∼ 10^7 L_⊙. The acoustic waves damp into thermal energy near the base of the hydrogen envelope due to the large drop in density at that location. In our models, wave heating during core Ne burning launches a pressure wave that propagates toward the stellar surface, steepening into a weak shock that creates a mild outburst ∼ 1 yr before core-collapse. The outburst is dim by SN standards (L ∼ 3 × 10^5 L_⊙, <ref>), and ejects a small amount of mass (M_ ej≲ 1 M_⊙) at low velocities (v ≲ 50 km/ s, <ref>).In our models, wave heating during core O burning drives a wind off the surface of the He core, inflating a low density bubble that gradually lifts off the overlying H envelope. However, we expect Rayleigh-Taylor instabilities to strongly modify these dynamics, potentially leading to another outburst during O burning. Regardless, the H envelope can be significantly inflated, with a non-hydrostatic density profile differing from prior expectations. We do not expect wave heating to lead to very luminous type IIn SNe in “normal" M ≲ 20 M_⊙ RSG progenitors because the modest amount of ejected mass is confined at small distances (≲ 10^15cm) from the RSG. However, we find wave heating is a compelling mechanism to produce flash ionized type II-P/II-L SNe (e.g., ) showing emission lines in early spectra. The altered density structure will affect the resulting SN luminosity, potentially producing an early peak or a more II-L-like light curve, contributing to the diversity of type II SNe.The physics of wave-driven outbursts is rich, involving complex hydrodynamic processes spanning nearly 20 orders of magnitude in density. Our results are thus subject to numerous caveats discussed in <ref> that can be improved with future work. It will also be necessary to examine wave heating in other SN progenitors (e.g., different stellar masses, metallicities, rotation rates, binarity, degree of envelope stripping, etc.) to understand how wave-driven outbursts contribute to the enormous diversity of core-collapse SNe. § ACKNOWLEDGMENTS We thank Matteo Cantiello, Bill Paxton, Stephen Ro, Maria Drout, Nathan Smith, Schuyler Van Dyk, Jeremiah Murphy, Eliot Quataert, and Lars Bildsten for useful discussions. JF acknowledges partial support from NSF under grant no. AST-1205732 and through a Lee DuBridge Fellowship at Caltech. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1125915, and by the Gordon and Betty Moore Foundation through Grant GBMF5076.§ MASSIVE STAR MODELS WITH MESA§.§ Evolving to Carbon Burning We created stellar models using the MESA stellar evolution code <cit.>, version 9393. Our model evolution proceeded in three steps. First, we evolved a 15 M_⊙ model from the main sequence to just before the onset of core carbon burning. Most model settings are default values, and the models are non-rotating with Z = 0.02.One notable change is to add a significant amount of overshooting to our models via the inlist settingand using the same overshoot/undershoot values for H,He, and Z core/shell burning. This corresponds to an exponential overshoot parameter of f_ ov≃ 0.015. We use the following mass-loss prescription settings:This model has He core mass M_ He = 5.38 M_⊙ and total mass M = 12.31 M_⊙ at the onset of carbon burning. The helium core mass is somewhat larger than models not including overshoot, and make our model behave like a slightly more massive star compared to some other stellar evolution codes.We add a small amount of element diffusion (comparable to what has been asteroseismicly inferrred, ) to our models to slightly smooth sudden composition/density jumps, which produce large (possibly unphysical) spikes in the Brunt-Väisälä frequency N, usingAdditionally, we restrict changes in composition at each timestep due to nuclear burning withwhich helps ensure more accurate composition profiles as nuclear burning processes begin and end within the core. This helps prevent the occurrence of, e.g., unphysical violent burning flashes when Ne ignites due to residual unburnt carbon.We add wave heating (described below) throughout the entire evolution, however the wave energy is totally negligible (orders of magnitude below the surface luminosity) at all points proceeding carbon burning.§.§ Preparing for Hydrodynamics Before the onset of carbon burning, we save a model as our basepoint for the evolutions presented in this paper. We then load and run this model, with the followingcommand: which allows the model to evolve material above the photosphere out to an optical depth τ = 10^-4. After relaxation, we evolve the model with a maximum timestep of one year for twenty-five models, the small timestep assuring the model is very close to hydrostatic equilibrium. §.§ Running with Hydrodynamics After relaxing our model, we turn on the hydrodynamics capabilities of MESA withThis introduces a very small transient in surface temperature and luminosity, but we caution that a non-relaxed model may exhibit much larger transients and struggle converge when hydrodynamics are first turned on. At the outer boundary of our model, we let mass flow outward by removing it below a density of ρ_ min = 2 × 10^-14g/ cm^3 to avoid equation of state problems for matter at lower densitiesalthough none of our models actually reach outer boundary densities this small. We use the following settings to limit the convective energy transport via MLT in MESA:The first three commands limit the changes in convective velocities/fluxes due to sudden developments of temperature gradients, e.g., in the wave heating region or near shocks. Failure to limit convective velocities will allow convection to transport energy toward the surface and across shocks at unphysically large rates. This prescription may not be optimal, but is more realistic than allowing instantaneous increases in convective fluxes.The following commands control the hydro equations and boundary conditions solved at each timestepWe find these outer boundary conditions to be fairly stable. Experiments with other boundary conditions appear to produce similar results, but are much more likely to cause the code to crash or to produce unphysical jumps in surface temperature, especially when a shock is propagating near the photosphere.Spatial gridding and error tolerances are adjusted with the following controlsIt is necessary to adjust the grid weights, otherwise very low density regions above the photosphere and within the empty cavity during O-burning are not well-resolved.During core O-burning, an instability develops within the supersonic wind at the base of the H-envelope. The instability appears to stem from the sonic point of the flow, such that the flow below the sonic point is smooth, but large velocity/density inhomogeneities develop above. Although radial and nonradial instabilities may exist <cit.>, we believe the instability in MESA is a numerical artifact, because it is largely suppressed in the absence of convection. In our runs, we prevent convection at this sonic point by adding the following command to MESA's MLT module:which prevents convection in regions with velocities larger than 50 km/ s. Convection can still operate near the surface where velocities are typically smaller than this limit. We have performed simulations with and without this fix, and it does not appear to strongly affect the development of the wind, except that using the fix prevents the formation of internal shocks within the wind and allows the code to run much faster. We defer a more detailed analysis because the entire wind configuration will likely be altered by RTI as discussed in <ref>.Finally, we add a small amount of numerical viscosity beginning during O-burning (after the Ne pressure wave breakout):This helps the code run faster in the presence of strong shocks that can develop at interfaces between the wave-driven wind and overlying envelope.§ WAVE PROPAGATION Here we derive the fraction of wave energy which is able to tunnel into the envelope and dissipate into thermal energy.§.§ Wave Damping via Neutrinos The wave entropy perturbation per unit mass due to neutrinos is <cit.>i ω T δ S_ν = ϵ_ν[ ( ∂lnϵ_ν/∂ln T)_ρδ T/T + ( ∂lnϵ_ν/∂lnρ)_Tδρ/ρ] .Here, ϵ_ν is the neutrino cooling rate per unit mass, the terms in parentheses are its partial derivatives with respect to temperature and density, and δ T and δρ are the Lagrangian perturbations in temperature and density produced by the wave. The energy loss rate (when integrating over a wave cycle) per unit mass is thenδϵ_ν = δ T d δ S/d t= ϵ_νδ T/T[ ( ∂lnϵ_ν/∂ln T)_ρδ T/T + ( ∂lnϵ_ν/∂lnρ)_Tδρ/ρ] . Now, in the nearly adiabatic limit of interest, the temperature perturbation isδ T/T = Γ_1 ∇_ ad/c_s^2( r ω^2 ξ_⊥ - g ξ_r ) .where the thermodynamic quantities have their usual meaning, ξ_r is the radial wave displacement, and ξ_⊥ is the horizontal displacement. Essentially all of the wave neutrino losses occur in the radiative core where the waves are well approximated as WKB gravity waves. For gravity waves, ξ_r ∼ωξ_⊥/N ∼ω c_s ξ_⊥/g, and ω≪ c_s/r. Therefore, the second term in equation <ref> dominates, andδ T/T≃Γ_1 ∇_ ad g/c_s^2ξ_r .Additionally, neutrino loss rates are usually much more sensitive to temperature than density, so the first term in brackets in equation <ref> dominates. The energy loss rate via neutrinos is then δϵ_ν≃Γ_1^2 ∇_ ad^2 g^2/N^2 c_s^4ω^2 ξ_⊥^2 (∂lnϵ_ν/∂ln T)_ρϵ_ν .For gravity waves, the wave energy per unit mass is ε≃ω^2 ξ_⊥^2. So the wave energy damping rate per unit time isγ_ν = δϵ_ν/ε≃Γ_1^2 ∇_ ad^2 g^2/N^2 c_s^4(∂lnϵ_ν/∂ln T)_ρϵ_ν .§.§ Wave Tunneling into the Envelope Calculating the wave energy flux tunneling into the envelope as acoustic waves is not straightforward because there may be multiple evanescent zones separating the generated waves from the envelope. Additionally, wave energy may damp out via neutrinos along the way. Thus, it is important to keep track of where wave energy builds up and how fast it damps out.To calculate the amount of energy tunneling into the envelope, we can treat the star as a series of wave cavities separated by intervening evanescent regions. Within each wave cavity, the wave energy flux is conserved unless damping processes operate. At each evanescent region, only a fraction T^2 of the incident wave energy is able to tunnel through, where T^2 is the squared transmission coefficient of the evanescent region, which is approximately equal to <cit.> T_1,2^2 =exp( -2 ∫^r_2_r_1 |k_r| dr ) where r_1 and r_2 are the radial boundaries of the evanescent region, and the radial wavenumber isk_r^2 = (N^2 - ω_2 )(L_l^2 - ω^2)/ω^2 c_s^2.Note that k_r is imaginary in evanescent zones.In the limit of a thin evanescent region, equation <ref> needs to be slightly modified <cit.>, although we shall see below that thick evanescent regions dominate the wave trapping.In a steady state, the amount of energy entering and exiting each wave cavity is equal. The energy transfer rate from cavity 1 to cavity 2 through an evanescent region from r_1 to r_2 isĖ_1,2 = T_1,2^2/2 t_1 E_1 where E_1 is the wave energy within cavity 1 and t_1 = ∫ dr/v_g is the wave crossing time across cavity 1. Similarly, the energy transfer rate from cavity 2 to cavity 1 from r_2 to r_1 is Ė_2,1 = T_1,2^2/2 t_2 E_2 where we have used the fact that T_1,2^2 = T_2,1^2. The steady-state approximation is justified by the fact that the wave crossing timescales in the core of the star are typically much smaller than the nuclear burning timescales.Consider the cavity (labeled as cavity 1) overlying the wave generation region, which has a wave energy input L_ wave. We will also consider damping processes within cavity 1 such that the energy loss to wave damping is Ė_1, damp = E_1 γ_1. Then balancing energy input and energy losses for cavity 1 yieldsL_ wave + Ė_2,1 = Ė_1,2 + E_1 γ_1 .In our problem, neutrino damping is always largest closest to the wave generation site (cavity 1) where temperature and density are highest, so we ignore damping in overlying cavities. The net energy flux through overlying cavities is then L_ heat = L_ wave-Ė_1, damp, and our goal is to calculate L_ heat. The energy balance for cavity 2 isL_ wave - E_1 γ_1 + Ė_3,2 = Ė_2,3,and a similar equation holds for overlying cavities. Rearranging equation <ref>,E_2/2t_2 = 1/T_2,3^2[ L_ heat + Ė_3,2] ,and substituting into equation <ref>, we haveL_ heat + T_1,2^2/T_2,3^2[ L_ heat + Ė_3,2] = Ė_1,2.We can perform a similar procedure to substitute in for Ė_3,2 and all overlying cavities up to cavity n, with the boundary condition of no wave flux entering from above, Ė_n+1,n = 0. Then we have L_ heat +L_ heat T_1,2^2 ∑_2^n 1/T_n,n+1^2 = Ė_1,2.Now, using E_1 γ_1 = L_ wave - L_ heat, we haveL_ heat +L_ heat T_1,2^2 ∑_2^n 1/T_n,n+1^2 = L_ wave-L_ heat/2 γ_1 t_1 T_1,2^2.which can be rewritten asL_ heat= L_ wave[ 1 + 2 γ_1 t_1 ∑_1^n T_n,n+1^-2]^-1. Equation <ref> is the desired result, it allows us to compute the wave energy escaping into the envelope, L_ heat relative to the wave energy input rate L_ wave. All quantities on the right hand side can be computed from the stellar structure. Terms with large transmission coefficients (T^2 ≃ 1) should be replaced with the value T^2 → -ln(1-T^2)<cit.>. However, terms with small values of T^2 dominate the sum in the right hand side of equation <ref>. In practice, the thickest evanescent zone usually dominates the sum, which can be well approximated byL_ heat= L_ wave[ 1 + 2 γ_1 t_1/T_ min^2]^-1,where T_ min^2 is the minimum transmission coefficient between the side of wave generation in the core and wave dissipation in the envelope. In our models, this evanescent zone is usually created by the convective He burning shell.The value of γ_1 accounts for damping throughout cavity 1. For neutrinos, the local damping rate is given by γ_ν in equation <ref>. Upon traversing cavity 1, the wave energy is attenuated by a factor f_ν = e^x_ν = exp[ 2 ∫^r_1+_r_1-γ_ν dr/v_g] = exp[2 ∫^r_1+_r_1-γ_ν√(l(l+1))N dr/ω^2 r] .where v_g ≃ω^2 r/(√(l(l+1)) N) is the radial group velocity of gravity waves, and r_1+ and r_1- are the upper and lower boundaries of cavity 1. Then the time-averaged damping rate of the wave due to neutrino damping in cavity 1 is γ_1,ν = 1-f_ν^-1/2 t_1≃x_ν/2 t_1.The second equality arises from the fact that in our models x_ν in equation <ref> is small, and f_ν≃ 1 + x_ν.Additional damping can occur during shell burning phases, when convectively excited waves tunnel into the radiative core. In this case, the wave amplitudes near the center of the star are large enough to induce non-linear wave breaking (seeand references therein). Thus, waves entering the central radiative region will be lost, which could occur if the waves excited from shell convection reflect from an overlying evanescent zone and then tunnel back through the burning shell and into the core. This effect can be modeled as an additional source of damping in cavity 1, γ_1, core = T_ shell^2/2 t_1,where T_ shell^2 is the transmission coefficient through the burning shell that excites the wave.Accounting for both neutrino damping in cavity 1 and wave tunneling into the core, the effective damping rate in cavity 1 is γ_1 = γ_1,ν + γ_1, core. Using equation <ref>, we arrive at our final expression determining the wave flux entering the envelopeL_ heat = f_ esc L_ wave = [ 1 + T_ shell^2 + x_ν/T_ min^2]^-1 L_ wave,In our stellar models, L_ wave is calculated as described in <ref>, T_ shell is calculated from equation <ref> (with the r locations corresponding to the edge of the burning shell, and T_ shell=0 for core burning phases), and x_ν is the integral in the exponent of equation <ref>. Our code calculates the transmission coefficients of all evanescent zones overlying the wave generation zone, and T_ min is the minimum transmission coefficient found in each model. Note that in the limit of no damping in the core (x_ν = T_ shell=0), all of the wave energyescapes into the envelope.§.§ Wave Damping via Radiative Diffusion Away from evanescent regions, waves are well approximated by the WKB limit, in which the wave damping rate isL̇_ wave/L_ wave = γ = k_r^2 K where K is the thermal diffusivityK = 16 σ_ SB T^3/3 ρ^2 c_p κand σ_ SB is the Stephan-Boltzmann constant, T is temperature, c_p is specific heat at constant pressure, and κ is the Rosseland mean opacity. We find radiative diffusion is only important in the envelope of the stars where waves are well approximated as WKB acoustic waves with k_r = ω/c_s. In this limit, the waves travel at group speed v_g = c_s and we can define a damping length l_ damp = v_g/γ = c_s^3/(ω^2 K). Then the waves damp after traversing a mass M_ damp = 4 πρ r^2 l_ damp, which evaluates to M_ damp = 4 πρ r^2 c_s^3/ω^2 K = 3 πρ^3 r^2 c_s^3 c_p κ/4 σ_ SBω^2 T^3.Equating M_ damp with a the mass in one scale height roughly reproduces the damping criterion of equation 7 of <cit.>. As waves propagate upward, they damp out at a rated L_ wave/d M = -L_ wave/M_ damp.In our numerical implementation, after calculating the fraction of energy escaping into the envelope as acoustic waves, we damp out wave energy such that the decrease in wave luminosity L_ wave across a cell of mass Δ m is Δ L_ wave = -L_ waveΔ m/M_ damp. The corresponding amount of heat added to the cell per unit mass per unit time is thusϵ_ heat = L_ wave/M_ damp.The most important feature of equation <ref> is its strong dependence on density (other factors tend to somewhat cancel each other out). As waves propagate out of the core and into the envelope, the density drops by several orders of magnitude just outside the helium core (see <ref>). At this location, the damping mass drops from a value that is orders of magnitude larger than the interior mass to a value orders of magnitude smaller than the exterior mass. This means that waves are essentially undamped below this region, but totally damped when they propagate into this region. The waves tunneling out of the core will thus deposit all their energy as heat near the base of the hydrogen envelope. A simplification of our method is to ignore the wave propagation time between excitation and damping. This approximation is reasonable because propagation time scales to the base of the hydrogen envelope are hours to days, whereas stellar evolution timescales are months to years for waves excited during Ne/O burning. However, the propagation delay will need to be included to model wave heating during late O-shell burning and Si burning, when wave propagation times are comparable to evolution time scales. § EFFECTS OF MAGNETIC FIELDS Magnetic fields larger than a critical value <cit.> B_c ∼√(πρ/2)ω^2 r/Nwill prevent gravity wave propagation in stably stratified regions, converting gravity waves into Alfvén-like waves, with a slight dependence on magnetic field geometry. In <ref>, we plot the value of B_c in our model during core O-burning for the wave frequency ω_ wave = 5 × 10^-3. At this stage, a magnetic field of B ≳ 2 × 10^7G in the radiative C/O/Ne shell above the convective core would be sufficient to suppress gravity wave propagation and alter wave heating. This magnetic flux is comparable to that found in young pulsars, magnetic white dwarfs, and magnetic Ap/Bp stars, and may plausibly exist in massive stellar cores.Unfortunately, it is very difficult to estimate core magnetic field strengths of massive stars. If magnetic fields generated during previous convective core burning phases survive beyond C-burning, they can account for the required magnetic flux. There is evidence in lower mass stars that core fields frequently survive after being generated by a main sequence core dynamo (see discussion in ), although it is not clear whether they would survive subsequent convective phases like those in massive stars.If strong core fields do exist, gravity wave energy will be converted in Alfvén wave energy within the core. The fate of this energy is uncertain and depends on the global magnetic field topology. However, we speculate field strengths will be much smaller at larger mass coordinates with lower densities. This may cause Alfvén waves to damp in the outer core before reaching the hydrogen envelope. In this case, wave heating energy will probably have a negligible affect on the stellar structure due to the large binding energy of the core relative to the wave energy, and a pre-SN outburst would be suppressed. | http://arxiv.org/abs/1704.08696v2 | {
"authors": [
"Jim Fuller"
],
"categories": [
"astro-ph.SR",
"astro-ph.HE"
],
"primary_category": "astro-ph.SR",
"published": "20170427180004",
"title": "Pre-Supernova Outbursts via Wave Heating in Massive Stars I: Red Supergiants"
} |
Department of Physics, North Carolina State University, Raleigh, NC 27695 USA;Department of Physics, Haverford College, Haverford, PA 19041 USA In granular physics experiments, it is a persistent challenge to obtain the boundary stress measurements necessary to provide full a rheological characterization of the dynamics. Here, we describe a new technique by which the outer boundary of a 2D Couette cell both confines the granular material and provides spatially- and temporally- resolved stress measurements.This key advance is enabled by desktop laser-cutting technology, which allows us to design and cut linearly-deformable walls with a specified spring constant. By tracking the position of each segment of the wall, we measure both the normal and tangential stress throughout the experiment. This permits us to calculate the amount of shear stress provided by basal friction, and thereby determine accurate values ofμ(I).Granular rheology: measuring boundary forces with laser-cut leaf springs Zhu [email protected] Theodore A. Brzinski1,[email protected] E. [email protected] December 30, 2023 ========================================================================================================================§ INTRODUCTIONIt is an open question what constitutive equations best describe flows of dense cohesionless granular materials <cit.>. There has come to be a consensus that two dimensionless parameters play a key role: interial number I and the friction μ. Each of these can be defined at the particle scale. The inertial number is given byI ≡|γ̇|d/√(P/ρ)where ρ is density of the solid granular material, d is their diameter, γ̇ is the local shear rate, and P is the local pressure.Higher values of I correspond to rapid flows, and lower values to slower flows.The ratioμ≡τ/Pis the local ratio of tangential stress τ to the normal stress (pressure P), and μ(I) is observed to be a good empirical descriptor of the rheology of the system <cit.> for well-developed flows. However, bothγ̇ and μ can exhibit strong spatial and temporal gradients. Recently, non-local extensions to the μ(I) rheology <cit.> aim to provide a theoretical framework for capturingsuch features as thetransition from inertial to creeping flow <cit.>, boundary-driven shear-banding <cit.>, and fluidization due to non-local perturbations <cit.>.However, direct comparisons between experiments and theory have been hampered by the difficulty measuring the tensorial stress within a granular material. Here, we report a new design for a 2D annular Couette cell which can measure the shear and normal stresses at the boundaries (see Fig. <ref>). This design has several advantages. First, the spring walls can be cut from standard acrylic sheets, making them cheaper and more convenient than using photoelastic particles. Second, the shape of the wall can be easilycustomized to have a particular spring constant by changing the thickness or length of the springs. Third, because photoelastic materials are no longer required, experiments on ordinary granular materials are made possible.Below, we provide a description of the method for making quantitative boundary stress measurements using walls of this type. This involves (1) calibrating a single leaf spring, (2) using image cross-correlation to measure the displacement of each spring tip, and (3)calculating the stress as a function of time and azimuthal position by combining these two measurementsat multiple positions around the outer wall. We close by presenting sample measurements of the μ(I) rheology made using this method. § METHOD §.§ Apparatus We develop our technique using a standard annular Couettegeometry, which has the advantage of allowing continuous shear of a granular material to arbitrary total strain.The apparatus consists of a rotating inner disk (R_i=15 cm) and a fixed outer wall (R_o=28 cm) made up of52 leaf springs which can dilate slightly (a few mm) and thereby provide a stress measurements at the outer wall. The granular material is about 5000 circular and elliptical disks, of diameter d≈5 mm and thickness 3 mm. A photograph of the apparatus from above is shown in Fig. <ref>.§.§ Calibrating the spring wall To provide a calibration for our experiments, we cut a single spring of the same acrylic, but with a short “handle” attached, and performed force-displacement measurements using an Instron materials tester. This process is shown in Fig. <ref>. One set of measurements was taken with the single spring oriented for normal compression, and a second set with tangential shear, while simultaneously recording a video of the dynamics. In both cases, we measured the x- and y-displacements of the spring tip as a function of applied force and observed a linear response.A least-squares fit to the data provides values for the calibration constants. These areC_n,x= 1.43 ± 0.02 mm/N (x-deformation under normal force),C_n,y= 5.06 ± 0.05 mm/N (y-deformation under normal force),C_t,x= 3.49 ± 0.05 mm/N (x-deformation under tangential force), andC_t,y= 6.86 ± 0.08 mm/N (y-deformation under tangential force). All errors are reported from the fit.§.§ Measuring wall deformationTo determine the normal pressure and the shear stress on the rim, we can make measurements of tip displacements (dx,dy, rotated into the appropriate coordinate system) for each of theleaf springs around the rim. Here, we illustrate the principle using a single leaf spring. First, we extract a subregion of the overhead image (see Fig. <ref>) in the vicinity of the tip. To determine its displacement, we also extract the same subregion from an image of the experiment taken without particles. Sample images are shown in Fig. <ref>ab. However, these raw tip-images also contain a piece of the neighboring spring. Using image-segmentation and masking, we remove the neighboring spring (cd). Using these two images, a simple image cross-correlation (e) determines thetip displacements (dx,dy) by fitting with subpixel resolution.§.§ Measuring wall stressesFor each measured pair of tip deformations(dx,dy), we can use the calibration from <ref> to calculate the vector force on the tip: [C_t,x F_t + C_n,x F_n =dx; C_t,y F_t + C_n,y F_n =dy. ]For the example show in Fig. <ref>, this provides F_n=0.899 Nand tangential force F_t=0.251 N. Since the thickness of the spring wall is w=3 mm, and the segment length of each leaf spring is L=33 mm, we can convert this to the shear stress τ(R_o) measured at the outer wall, and pressure P:[ τ(R_o)=F_t/wL; P=F_n/wL. ]This provides a segment-averaged estimate for the normal and shear stresses along that particular leaf spring.§ RESULTS To illustrate how this method can be used to provide rheological measurements, we perform 3 sample runs for two packing fractions (ϕ) and two rotation rates (specified by the speed v of the inner wall at its rim). For Case 1 and 2, ϕ =0.816 ± 0.003, and for Case 3, ϕ = 0.840 ± 0.003. The error is propaged from errors in the particle and apparatus size measurements.Case 1 and 3 are taken at v=0.2 d/s, and Case 2 at v=0.02 d/s, to provide a set of controlled comparisons among the three experiments. §.§ Stress measurements Using the methods of <ref>, we measure the pressure and the shear stress for a single spring as a function of time (see Fig. <ref>a). We observe that both values fluctuate around a well-defined mean value, punctuated by brief spikes in both. These can be seen near 1000 s and 1200 s, as indicated by the arrows. These are likely due to the transient loading of force chains, which have strong spatial variations on length scales similar to that of a single spring. Measurements at other springs have different average values that this sample, further indicating the presence of spatial heterogeneities. Future work combining boundary measurements with photoelastic measurements are planned. For the three sample cases, we can determine how the time-averaged normal and tangential stresses vary according to ϕ and v. This data is shown in Fig. <ref>b.We observe that Case 1 and 2 (same ϕ) have similar values for both normal and tangential stress. We observe that Case 1 and 3 (same v), illustrate that both stress values increase with packing fraction, as would be expected.§.§ Rheological measurementsBy combining the wall stress measurements with particle-tracking, we can determine the μ(I) rheology throughout the granular material. We demonstrate this using values for the two wall stresses measured atfour equally-spaced leaf springs around the outer wall. Improved statistics would be obtained for using all 52 available springs (for which code is under development). In addition, we record the shear stress S measured at the inner wall via a Cooper Instruments torque sensor placed in line with the drive shaft.If there were no friction with the supporting plate, the tangential component of the stress would from a maximum at the inner wall according to τ(r) =S(R_i/r)^2, where R_i is the radius of the inner wall. Empirically, wemodel the effect of this friction with [ τ(r) =S(R_i/r)^2+τ_f ]whereτ_f is a constant chosen so that τ(R_o) matches the measured value for the tangential stress at the outer wall. From force-balance, we approximate that the average pressure P is independent of radial position r. Thus, the dimensionless stress ratio is μ(r)=τ(r)/P.For the three cases, we find the following values for the frictional stress τ_f: Case 1 has1870 N/m^2; Case 2 has 1630 N/m^2, and Case 3 has 3580 N/m^2. The errors on all are± 10 N/m^2, calculated from the standard error in Fig. <ref>a, averaged over the 4 sensors. To measure the inertial number I (Eq. <ref>), we track the particles using video taken at 1 Hz over the whole system<cit.>. We azimuthally-average the velocity profile v(r) and then use Fourier deriviates to calculateγ̇(r) = 1/2( ∂ v/∂ r- v/r).We use the value of P determined from the wall stress measurements, and plot μ(r)=τ(r)/P parametrically for all values of r. As shown in Fig. <ref>, the μ(I) rheology depends on both the packing fraction and the rotation rate of the inner disk. We observe that for the same inner wall speed (Case 1 and 3), the μ(I) curves agree for large I (close to the driving wall). When the inertial number is low, all curves (but particularly Case 1 and 2) approach a constant value μ≈ 0.3 which is dominated by the basal friction. § CONCLUSION We find that laser-cut leaf springs provide a convenient method to both confine a granular material, and measure the boundary wall stresses. In future work, we are expanding the image-processing to provide measurements over the full outer wall, and performing a comparison with photoelastic force measurements. Finally, the μ(I) measurements provided by Fig. <ref> will allow for quantitative investigation of the utility of nonlocal rheology models <cit.> to describe granular rheology of real materials. § ACKNOWLEDGEMENTS We thank Michael Shearer, Dave Henann, and Ken Kamrin for useful discussions about the project, and Austin Reid for help creating the boundary wall designs.We are grateful to the National Science Foundation (NFS DMR-1206808) and International Fine Particle Research Institute (IFPRI) for financial support. woc | http://arxiv.org/abs/1704.08295v1 | {
"authors": [
"Zhu Tang",
"Theodore A. Brzinski",
"Karen E. Daniels"
],
"categories": [
"physics.flu-dyn"
],
"primary_category": "physics.flu-dyn",
"published": "20170426185501",
"title": "Granular rheology: measuring boundary forces with laser-cut leaf springs"
} |
Perron–based algorithms for the multilinear PageRank Beatrice Meini[Dipartimento di Matematica, Università di Pisa, Italy.] and Federico Poloni[Dipartimento di Informatica, Università di Pisa, Italy. .] ========================================================================================================================================================= We consider the multilinear PageRank problem studied in [Gleich, Lim and Yu, Multilinear PageRank, 2015], which is a system of quadratic equations with stochasticity and nonnegativity constraints. We use the theory of quadratic vector equations to prove several properties of its solutions and suggest new numerical algorithms. In particular, we prove the existence of a certain minimal solution, which does not always coincide with the stochastic one that is required by the problem. We use an interpretation of the solution as a Perron eigenvector to devise new fixed-point algorithms for its computation, and pair them with a continuation strategy based on a perturbative approach. The resulting numerical method is more reliable than the existing alternatives, being able to solve a larger number of problems. Perron–based algorithms for the multilinear PageRank Beatrice Meini[Dipartimento di Matematica, Università di Pisa, Italy.] and Federico Poloni[Dipartimento di Informatica, Università di Pisa, Italy. .] =========================================================================================================================================================§ INTRODUCTION Gleich, Lim and Yu <cit.> consider the following problem, arising as an approximate computation of the stationary measure of an order-2 Markov chain: given ∈̌ℝ^n, R ∈ℝ^n× n^2, α∈ℝ with ≥̌0, R≥ 0, α∈ (0,1) and _n^⊤=̌ 1, _n^⊤ R = _n^2^⊤,solve forthe equation= α R(⊗) + (1-α) .̌The solution of interestis stochastic, i.e., ≥ 0and _n^⊤ = 1. Heredenotes a column vector of all ones (with an optional subscript to specify its length), and inequalities between vectors and matrices are intended in the componentwise sense. In the paper <cit.>, they prove some theoretical properties, consider several solution algorithms, and evaluate their performance. In the more recent paper <cit.>, the authors improve some results concerningthe uniqueness of the solution.This problem originally appeared in <cit.>, and is a variation of problems related to tensor eigenvalue problems and Perron–Frobenius theory for tensors; see, e.g., <cit.>. However, it also fits in the framework of quadratic vector equations derived from Markovian binary tree models introduced in <cit.> and later considered in <cit.>. Indeed, the paper <cit.> considers a more general problem, which is essentially (<ref>) without the hypotheses (<ref>). Hence, all of its results apply here, and they can be used in the context of multilinear PageRank. In particular, <cit.> considers the minimal nonnegative solution of (<ref>) (in the componentwise sense), which is not necessarily stochastic as the one sought in <cit.>. In this paper, we use the theory of quadratic vector equations in <cit.> to better understand the behavior of the solutions of (<ref>) and suggest new algorithms for computing the stochastic solution. More specifically, we show that if one considers the minimal nonnegative solution of (<ref>) as well, the theoretical properties of (<ref>) become clearer, even if one is only interested in stochastic solutions. Indeed we prove that there always exists a minimal nonnegative solution, which is the unique stochastic solutionwhen α≤ 1/2. When α > 1/2,the minimal nonnegative solutionis not stochasticand there is at least one stochastic solution ≥.Note that <cit.> already proves that when α≤1/2 thestochastic solution is unique and <cit.> slightly improves this bound; our results give a broader characterization.All this is in Section <ref>.When α≤ 1/2, as already pointed out in <cit.>, computing the stochastic solution of (<ref>) is easy. Indeed, this is also due to the fact that the stochastic solution is the minimal solution, and for instance the numerical methods proposed in <cit.> perform very well. The most difficult case is when α>1/2, in particular when α≈ 1. Since the minimal solutionof (<ref>)can be easily computed, the idea is to compute and deflate it, with a similar strategy to the one developed in <cit.>, hence allowing us to compute stochastic solutions even when they are not minimal.The main tool in this approach is rearranging (<ref>) to show that (after a change of variables) a solutioncorresponds to the Perron eigenvector of a certain matrix that depends onitself. This interpretation in terms of Perron vector allows to devise new algorithms based on fixed point iteration and on Newton's method. Sections <ref> and <ref> describe this deflation technique and the algorithms based on the Perron vector computation.Finally, we propose in Section <ref> a continuation strategy based on a perturbative approach that allows one to solve the problem for values α̂ < α in order to obtain better starting values for the more challenging cases when α≈ 1.We report several numerical experiments in Section <ref>, to show the effectiveness of these new techniques for the set of small-scale benchmark problems introduced in <cit.>, and draw some conclusions in Section <ref>.§ PROPERTIES OF THE NONNEGATIVE SOLUTIONSIn this section, we show properties of the nonnegative solutions of the equation (<ref>). In particular, we prove that there always exists a minimal nonnegative solution, which is stochastic when α≤ 1/2. These properties can be derived by specializing the results of <cit.>, which apply to more general vector equations defined by bilinear forms. We introduce the map G() : = α R(⊗) + (1-α) ,̌and its Jacobian G'_ := α R(⊗ I_n) + α R(I_n ⊗). We have the following result. Consider the fixed-point iteration _k+1 = G(_k),k=0,1,…, started from _0=0. Then the sequence of vectors {_k} is such that 0≤_k≤_k+1≤, there exists lim_k→∞_k= and is the minimal nonnegative solution of (<ref>), i.e., equation (<ref>) has a (unique) solution ≥ 0 such that ≤ for any other possible solution ≥ 0. The map G() is weakly positive, i.e., G()≥ 0, G() 0 whenever ≥0, 0. Moreover, if 0≤≤ then 0≤ G()≤. Therefore Condition A1 of <cit.> is satisfied which, according to Theorem 4 of <cit.>,implies that the sequence of vectors {_k} is bounded and converges monotonically to a vector , which is the minimal nonnegative solution of (<ref>). §.§ Sum of entries and criticalityIn this specific problem, the hypotheses (<ref>) enforce a stronger structure on the iterates of (<ref>): the sum of the entries of G() is a function of the sum of the entries ofonly. Let g(u):=α u^2 + (1-α). Then,^⊤ G() = g(^⊤) for any ∈ℝ^n.^⊤ G()= ^⊤ (α R(⊗) + (1-α) )̌ = α^⊤ R(⊗) + (1-α) ^⊤ = α^⊤(⊗) + (1-α) = α (^⊤)^2 + (1-α). This fact has important consequences for the sum of the entries of the solutions of (<ref>).Foreach solutionof (<ref>), ^⊤ is one of the two solutions of the quadratic u = g(u), i.e., u=1 or u=1-α/α. Let u be one of the solutions of u = g(u) and define the level setℓ_u = {: ^⊤=u, ≥ 0}. Since ℓ_u is convex and compact, and since G() maps ℓ_u to itself by Lemma <ref>, thenthe Brouwer fixed-point theorem implies the following result. There exists at least asolution ≥ 0 to (<ref>) with ^⊤ = 1 and a solution ≥0 with ^⊤ = 1-α/α.Hence we can have two different settings, for which we borrow the terminology from <cit.>. Subcritical case α≤1/2, hence the minimal nonnegative solution = is the unique stochastic solution. Supercritical case α > 1/2, hence the minimal nonnegative solutionsatisfies ^⊤ = 1-α/α < 1 and there is at least one stochastic solution ≥. Note that <cit.> already proves that when α≤1/2 thestochastic solution is unique; these results give a broader characterization.The tools that we have introduced can already be used to determine the behavior of simple iterations such as (<ref>). Consider the fixed-point iteration (<ref>), with a certain initial value _0≥ 0, for the problem (<ref>) with α > 1/2. Define z_k := ^⊤_k. Then, * If z_0 ∈ (0, 1-α/α], then lim_k→∞ z_k = 1-α/α, and the iteration (<ref>) can converge only to the minimal solution(if it converges). * If z_0 ∈ (1-α/α, 1], then lim_k→∞ z_k = 1, hence the iteration (<ref>) can converge only to a stochastic solution (if it converges). * If z_0 ∈ (1, +∞), then lim_k→∞ z_k = +∞, hence the iteration (<ref>) diverges.Thanks to Lemma <ref>, the quantity z_k := ^⊤_k evolves according to z_k+1 = g(z_k). So the result follows by the theory of scalar fixed-point iterations, since this iteration converges to 1-α/α for z_0 ∈ (0, 1-α/α], to 1 for z_0 ∈ (1-α/α, 1], and diverges for z_0 ∈ (1, +∞).An analogous result holds for the subcritical case.The papers <cit.> describe several methods to compute the minimal solution . In particular, all the ones described in <cit.> exhibit monotonic convergence, that is, 0 = _0 ≤_1 ≤_2 ≤⋯≤_k ≤⋯≤. Due to the uniqueness and the monotonic convergence properties, computing the minimal solutionis typically simple, fast, and free of numerical issues. Hence in the subcritical case the multilinear PageRank problem is easy to solve. The supercritical case is more problematic.Among all available algorithms to compute the minimal solution , we recall Newton's method, which is one of the most efficient ones. The Newton–Raphson method applied to the function F() =- G() generates the sequence (I-G'__k)_k+1 = (1-α) -̌α R(_k ⊗_k),k=0,1,….The following result holds <cit.>. Suppose that > 0, and that G'_ is irreducible. Then, Newton's method (<ref>) starting from _0=0 is well defined and converges monotonically to(i.e., 0 = _0 ≤_1 ≤_2 ≤⋯≤_k ≤⋯≤).Algorithm (<ref>) shows a straightforward implementation of Newton's method as described above. | http://arxiv.org/abs/1704.08072v2 | {
"authors": [
"Beatrice Meini",
"Federico Poloni"
],
"categories": [
"math.NA",
"65F99, 15A69, 05C82, 90B15"
],
"primary_category": "math.NA",
"published": "20170426121611",
"title": "Perron-based algorithms for the multilinear pagerank"
} |
empty USTC-ICTS-15-12 Flatness of Minima in Random Inflationary Landscapes Yang-Hui He^a,b,c,[], Vishnu Jejjala^d,[], Luca Pontiggia^d,[], Yan Xiao^a,[], Da Zhou^a,e,[] ^a Department of Mathematics, City, University of London,Northampton Square, London EC1V 0HB, UK 0.25cm^b Merton College, University of Oxford, OX1 4JD, UK 0.25cm^c School of Physics, NanKai University, Tianjin, 300071, P.R. China, 0.25cm^d Mandelstam Institute for Theoretical Physics, NITheP, CoE-MaSS, and School of Physics,University of the Witwatersrand, Johannesburg, WITS 2050, South Africa 0.25cm^e The Interdisciplinary Center for Theoretical Study,University of Science and Technology of China, Hefei, Anhui, 230026, ChinaWe study the likelihood for relative minima of random polynomial potentials to support the slow-roll conditions for inflation. Consistent with renormalizability and boundedness, the coefficients that appear in the potential are chosen to be order one with respect to the energy scale at which inflation transpires. Investigation of the single field case illustrates a window in which the potentials satisfy the slow-roll conditions. When there are two scalar fields, we find that the probability depends mildly on the choice of distribution for the coefficients. A uniform distribution yields a 0.05% probability of finding a suitable minimum in the random potential whereas a maximum entropy distribution yields a 0.1% probability.§ INTRODUCTIONIn order to solve the well known horizon and flatness problems, cosmological inflation <cit.> posits that the Universe underwent a period of exponential expansion early in its history. To date, there is no uniquely compelling realization of how inflation transpired. The literature abounds with numerous and varied proposed mechanisms <cit.>. Paradigmatic models involve scalar fields which dynamically roll until arriving at the (relative) minimum of some potential.While a model of physics that purports to approximate our world must correctly trace out the cosmological history of the Universe, these are not the only considerations in selecting a theory. The Standard Model of particle physics establishes that at low energies the particles in Nature organize themselves into three generations of chiral fields that transform in representations of the SU(3)× SU(2)_L× U(1)_Y gauge group. Top down realizations of low energy gauge theories from a fundamental theory such as string theory typically augment the symmetries of the S-matrix with supersymmetry. The simplest scenario for preserving N=1 supersymmetry in four dimensions involves the compactification of the heterotic string in ten dimensions on a Calabi–Yau threefold <cit.>. This effort has led to a number of constructions that reproduce the matter spectrum and Yukawa interactions observed in the Standard Model <cit.>. Again, we have an abundance of models that are a priori indistinguishable on the basis of experiments.As we do not have a sui generis path to the real world, we propose to study a large class of models at once and incorporate inputs of both cosmology and particle physics. The most important characterization of a Calabi–Yau threefold is a pair of topological invariants h^1,1 and h^2,1. There are h^1,1 Kähler and h^2,1 complex structure parameters that describe the size and the shape of the geometry. In the most naïve setup, these deformation parameters supply candidates for the scalar fields in inflation. The largest available catalog of Calabi–Yau threefolds is derived from the Kreuzer–Skarke database of reflexive polytopes <cit.>. Using the methods of Batyrev and Borisov <cit.>, each consistent triangulation of a reflexive polytope yields a toric Calabi–Yau manifold. In <cit.>, topological and geometric data are tabulated for the Calabi–Yau threefolds thus obtained for low values of h^1,1. Heterotic Standard Model constructions in string theory typically employ Calabi–Yau geometries with small values of the Hodge numbers. For example, <cit.> uses a manifold with (h^1,1,h^2,1)=(3,3). Where there are explicit candidates for particle physics from string theory, we expect only a small number of moduli to appear in the low energy effective action. Motivated by this fact, these are the models that we investigate in this article.We aim to provide statistics for how many (possibly metastable) vacua support slow-roll constraints on inflation. Working in effective field theory, we examine random polynomial potentials for inflation with a small number of scalar fields. The justification for examining these models derives from string constructions of de Sitter like metastable vacua, e.g., the KKLT <cit.> and Large Volume Scenarios <cit.>. In the latter class of models, the number of flat directions in the low energy effective potential is given by the number of parametrically large four cycles. (In fact, the number of flat directions is one less than the number of large cycles <cit.>.) There are 69 explicit Calabi–Yau geometries with two large cycles and one known Calabi–Yau geometry with three large cycles <cit.>. These are candidate manifolds for bona fide cosmological model building in string theory that correspond to one field and two field inflation. In light of this, we study random potentials relevant to these cases in particular.The potentials we study are sums of monomials in the scalar fields. We truncate the expansion to focus on the interactions that are relevant or marginal from the perspective of a four dimensional low energy effective action. As higher order monomials in the fields are irrelevant operators, we expect these to be mass suppressed and neglect them for the purposes of our investigations. We will study models where the time scale for inflation if t_GUT. Correspondingly, the energy scale in the problem is M_GUT.Invoking naturalness <cit.>, we choose coefficients in the potential to be order one with respect to this scale. (It is not always required that we choose t_GUT as the natural time scale in the problem; in fact, certain models <cit.> employ energy scales which are lower than M_GUT.) In discussing single field models with order one coefficients, our work is analytic. In multi-field models, we construct random potentials whose coefficients are selected from distributions.At the outset, we should note that the cases we analyze where there are few moduli may well represent an atypical class of string compactifications. While there are 473 800 776 reflexive polyhedra in four dimensions, there are only 30 108 pairs of Hodge numbers that appear in the threefold dataset. The number of reflexive polytopes in the Kreuzer–Skarke list peaks at the Hodge numbers (h^1,1,h^2,1) = (27,27). There are 910 113 such polytopes. Indeed, there are significant and surprising patterns in the distribution of Calabi–Yau geometries close to this maximum <cit.>. Reflexive polytopes with low Hodge numbers are sparse in the Kreuzer–Skarke database. Flux compactifications on Calabi–Yau threefolds yield, in principle, an enormously large number of potential vacua for string theory <cit.>. There are, however, to date no explicit constructions of the Standard Model on a geometry with Hodge numbers that correspond to those of a typical Calabi–Yau manifold. As there are good reasons to be skeptical about anthropic resolutions to the cosmological constant problem and there are potential issues regarding the stability of the flux vacua <cit.>, we adopt an agnostic attitude. We simply note that if a construction is stable in this context, a generic compactification on a typical Calabi–Yau manifold will most likely involve a large number of moduli fields. As we review below, the large-N limit of inflaton fields is studied in complementary work.The organization of the paper is as follows. In Section <ref>, we discuss the setup for random inflation. In Section <ref>, we investigate the case of single field inflation with O(1) coefficients. This analysis is a completely analytic study of polynomial equations. In Section <ref>, we examine the case of two scalar fields with couplings up to quartic order. Again, the coefficients are O(1). We choose coefficients using a uniform distribution and a Gaussian distributions for coefficients of indefinite sign and a gamma distribution for coefficients that are positive. In Section <ref>, we remark on future investigations in the context of semi-realistic string models.§ RANDOM POTENTIALS FOR INFLATION The action we consider takes the formI = - ∫ d^4x √(-g) ( R/16π G + 1/2 g^μν∂_μ·∂_ν - V() ) ,where the Einstein–Hilbert term is supplemented by a matter sector that consists of k scalar fields,= (ϕ^1, ϕ^2, ⋯, ϕ^k) .For simplicity, we assume that the metric in field space is the identity matrix, i.e.,∂_μ·∂_ν = δ_ij∂_μϕ^i ∂_νϕ^j .The scalar potential V() determines the model and can be expanded as a polynomial in the fields ϕ^i.This setup lets us examine cosmological inflation. Deducing the form of the potential is a long standing problem; many scenarios present attractive phenomenological features, and to date observation has provided only limited guidance in selecting V(). In models with a single inflation field, famously the WMAP <cit.> and Planck <cit.> observations disfavor the simplest quadratic potential. Other scenarios are variously consistent with the data. See, for example, <cit.> for a recent review.One of the simplest multi-field models, hybrid inflation <cit.>, involves coupling two fields according to the potentialV(ϕ,ψ) = 1/4λ(λψ^2 - M^2)^2 +1/2m^2ϕ^2 + λ'/2ϕ^2ψ^2 .Here, λ and λ' are couplings and M and m are the masses of ψ and ϕ, respectively. We require that V(ϕ)=1/2m^2ϕ^2 ≪M^4/4λ. This guarantees that the inflationary energy density of the false vacuum associated to the symmetry breaking potential V(ψ)=λ/4(ψ^2 - M^2)^2 dominates. The effective mass for the ψ field is M_eff^2 = -M^2 + λ' ϕ^2, which vanishes at ϕ_*^2 = M^2/λ'. Starting from ϕ^2≫ M^2, the minimum is at ψ = 0. This is morally a single field model with an effective potential of the formV_eff = λ/4M^4 +1/2m^2ϕ^2 .The field rolls until it reaches ϕ_*. The ψ = 0 locus is then unstable, and the field rolls again into the true minima at ϕ = 0, ψ = ± M. Interest in the model stems from its versatility and success in predicting certain features of inflation, such as the power law behavior of the perturbation spectrum. By tweaking the model in various ways, one can deal with inflation with or without first order phase transitions. While this is a prototype multi-field model, there is a built-in hierarchy to the coefficients. (See also <cit.>.)By constraining the inflationary scenario at a level matching the accuracy of current experimental data, <cit.> presents an encyclopædia of 74 satisfactory models. In our work, we adopt a slightly different approach and address the question of how generic or specific the models should be in order to satisfy experimental constraints. For this purpose, we consider randomly generated multi-field models (with the inflationary potential being given by polynomials with random coefficients) and verify whether the models can satisfy a certain set of conditions. In particular, we demand that the scalar potential has a parameter window such that slow-roll conditions are satisfied.Suppose there are some minima that satisfy the slow-roll conditions. What are the global features of the potentials that accommodate this? Turning the question around, given a large set of potentials (which may have some distribution in the function space), how likely is it that the potential has regions that satisfy slow-roll conditions? How often can slow-roll inflation be accommodated with O(1) coefficients? These are the issues we aim to address below.In an analysis of multi-field inflation, the need to establish the behavior of random potentials is almost compulsory. Generic compacticiations, can have hundreds of scalar fields <cit.>. Since these theories describe physics at energy scales close to the inflationary scale, there is considerable interest in analyzing their dynamics. Considering random potentials with large-N fields has a considerable history <cit.>. As multi-field models have an almost infinite number of ways to inflate, the task of understanding how the potential energy driving inflation is distributed among all these fields becomes an incredibly difficult one. This problem is often referred to as the measure problem, and it deals with attempting to handle the possible initial conditions <cit.>. Multidimensional landscapes may also be afflicted by instabilities <cit.>. In general, the approach that random multi-field inflation adopts, is to study the dynamics of inflation by creating an ensemble of random potentials. Then, through a statistical analysis, one can comment on the inflationary landscape produced by the respective models. Related studies have recently appeared in the context of Gaussian models <cit.> and non-minimal kinetic terms <cit.>.[ As we were completing this work, a similarly themed investigation appeared in <cit.>. This work examines inflationary landscapes corresponding to one dimensional potentials.] The study of random potentials is of course not limited to inflation. It is useful to borrow techniques for the generation of random potentials from other fields in physics, in particular string theory and quantum field theory <cit.>, and adapt these ideas to the cosmological context.In Section <ref>, we analyze the single field case analytically. In Section <ref>, we investigate the statistics of random inflation by examining a large set of sample potentials for two field inflation. In each potential, the coefficients are random numbers that fall within a particular range. For each sample potential, we examine whether it has slow-roll regions. We calculate the fraction of potentials that do have slow-roll regions and examine what features they have in common. For succinctness, in the following we will use the term “slow-roll potentials” to refer to those potentials that satisfy the slow-roll conditions in some region of the field space. We assume the potential term V() is a polynomial in ϕ^i up to degree four and is bounded below. In this paper, we only consider single field and two field inflation models. In the former case, we shall denote =φ̃, and in the latter, =(φ̃,ψ̃). Both cases are developed in the following sections.§ SINGLE FIELD MODELS The polynomial potential up to degree four for single field inflationary models has the formV_a,b(φ̃) = φ̃^2/2 (M^2 - a M φ̃+ b φ̃^2) ,where M≲ 10^16 GeV is the mass of the inflaton φ̃, and a and b are two dimensionless random numbers. Note that in order for the potential to be bounded from below, the quartic term must be positive, which means we presume b is positive. Another feature about this potential is the symmetryV_-a,b(φ̃) = V_a,b(-φ̃) ,which indicates if V_a,b is a slow-roll potential, V_-a,b must also be slow-roll. So again we only need to assume that a is positive.Now, to factor out the parameter M, we perform a rescaling, φ̃=Mφ, which also makes φ dimensionless. Then the potential can be recast asV(φ) = M^4/2φ^2 (1 - aφ + bφ^2) ,where we have omitted the two subscripts a and b on V.[ Adding a zeroth order term to the potential will shift the energy of the relative minimum. Though the potential appears in the denominator of the slow-roll conditions, we assume constant terms in the potential do not greatly affect flatness and therefore neglect such a term in writing the potential.] Consequently, the two slow-roll parameters areϵ = M_Pl^2/2( V'(φ̃)/V(φ̃))^2 = 1/2μ( V'(φ)/V(φ))^2 , η =M_Pl^2 V”(φ̃)/V(φ̃) = 1/μV”(φ)/V(φ) ,where M_Pl is the Planck mass and μ=M^2/M_Pl^2 is the square of the ratio between the mass of the inflaton and the Planck mass. The Planck 2015 <cit.> data tells us that the scalar spectral index is measured to be n_s = 0.9655± 0.0062 and the slow-roll parameters are deduced to satisfyϵ < 0.012 , η = -0.0080^+0.0088_-0.0146 .Noting that n_s - 1 = 2η - 6ϵ, in our analysis we demand that the slow-roll parameters are O(10^-2):ϵ < 0.01 , |η| < 0.01 .Given the definition of μ, the two slow-roll parameters are actually independent of the specific value of inflaton mass M because the M^4 term in (<ref>) appears in both the numerator and denominator of (<ref>) and therefore cancels.If we define a new variable y=aφ, the whole analysis will only depend on the ratio of b to a^2, which shall be dubbed β, instead of the explicit values of a and b. So we can define an auxiliary potential,v(y) = 2a^2/M^4 V(φ) = y^2 (1 - y + β y^2) , β = b/a^2 > 0 ,and two new slow-roll parameters which only depend on one parameter β,ϵ̅= 1/2( v'/v)^2 = ν/0.01ϵ ,η̅= v”/v = ν/0.01η ,wherev' ≡dv/dy , v”≡d^2v/dy^2 ,ν≡0.01μ/a^2 = 0.01/a^2M^2/M_Pl^2 .The slow-roll conditions becomeϵ̅< ν , |η̅| < ν .Of course, we have assumed a≠ 0 in (<ref>) and (<ref>), and the special a=0 case can be approximated by setting a to be an extremely small nonzero number, then a→ 0 corresponds to the β→∞ case, which is a special situation that will be discussed in Section <ref>.From (<ref>) one can see that as y→±∞, v∼ y^4 while v'∼ y^3 and v”∼ y^2, so (<ref>) is always satisfied. That means, there exists a y_0>0 and a y'_0<0 such that (<ref>) holds true for all y>y_0 or all y<y'_0. In other words, in any cases there are always at least two trivial slow-roll regions, (-∞,y'_0) and (y_0,∞). Our search for an inflationary scenario excludes these regions where the (<ref>) is satisfied simply due to the largeness of the potential v.[ This is not to say that regions where inflation transpires by virtue of a large denominator v must always be disregarded.Models such as chaotic inflation can use these trivial regions of the potential.]We aim to isolate other, perhaps more realistic scenarios which satisfy the slow-roll conditions with a flat v (i.e., small v' and v”) region of finite length. From (<ref>) we havedϵ̅/dy = v'/v^3 (v” v - v'^2) = - y^2/v^3 v' [ 2 - 4y + (3+2β) y^2 - 6by^3 + 4β^2 y^4 ] .For y<0, we have v>0, v'<0, and the expression within the brackets is positive, so dϵ̅/dy>0. This means, if we find a y'_0<0 such that ϵ̅(y'_0)=0.01, then (-∞,y'_0) is a trivial slow-roll region and (y'_0,0) does not contain a non-trivial slow-roll region. Hereafter, we are only interested in the region with y>0.In deducing the regions that satisfy the slow-roll conditions in single field inflation, we do not need to perform a Monte Carlo analysis or scan over potentials with random coefficients. It suffices to analytically examine a system of polynomial equations. We look for the true minimum of the potential. Complementary investigations (see, for example, <cit.>) examines relative minima in random landscapes. In the following subsection, we will see that different intervals for β exhibit characteristic behavior. §.§ Behavior of slow-roll parameters Graphically, we can draw the slow-roll parameters, which are functions of y given a specific value of β, on the plane and use the horizontal lines ϵ̅=ν and η̅=±ν to intercept curves of the slow-roll parameters ϵ̅ and η̅ respectively, then from the interception one can easily read off whether there are slow-roll regions for the corresponding potential. The classification of different behaviors of ϵ̅ and η̅ will be represented below.To determine the behavior of slow-roll parameters ϵ̅ and η̅, we compute their partial derivatives with respect to y,∂ϵ̅/∂ y = y^2 v'/v^3 f(y, β),∂η̅/∂ y = 2y/v^2 g(y, β) ,wherev':=2y - 3y^2 +4β y^3 ,f(y,β):=-2 + 4y - (3+2β) y^2 + 6β y^3 - 4β^2 y^4 ,g(y,β):=-2 + 6y - (6+4β) y^2 + 15β y^3 - 12β^2 y^4 . For β>9/32 we have v>0 and v'>0, so the signature of ∂ϵ̅/∂ y or ∂η̅/∂ y depends on the signature of f(y,β) or g(y,β). In order to determine the signature of f and g, we differentiate them with respect to y,∂ f/∂ y =4 - 2(3+2β)y + 18β y^2 -16β^2 y^3 , ∂^2 f/∂ y^2 =-48(β y - 3/8)^2 - (4β - 3/4) < 0 , ∂ g/∂ y =6 - 2(6+4β)y + 45β y^2 - 48β^2 y^3 , ∂^2 g/∂ y^2 =-144 (β y -5/6)^2 - (8β - 33/16) < 0 .Since ∂^2 f/∂ y^2<0 and ∂^2 g/∂ y^2<0, ∂ f/∂ y and ∂ g/∂ y are monotonically decreasing functions of y. In addition, ∂ f/∂ y(y=0)>0 and ∂ f/∂ y(y=∞)<0, so ∂ f/∂ y has one root in (0,∞). By the same token, ∂ g/∂ y also has one root in (0,∞). We illustrate the previous analysis in Figure <ref>.From Figure <ref> we can see that both functions f and g have one and only one maximum which is y̅ (respectively y̅') in (0,∞). As f(0,β)=g(0,β)=-2 and f(∞,β) or g(∞,β)<0, we conclude, (1) if f(y̅,β)<0 (respectively, g(y̅',β)<0), ϵ̅ (respectively, η̅) is monotonically decreasing in (0,∞); (2) if f(y̅,β)>0 (respectively, g(y̅',β)>0), ϵ̅ (respectively, η̅) has one local minimum and one local maximum in (0,∞). Between these two cases, there is an intermediate stage which is f(y̅,β)=0 or g(y̅',β)=0. As a result, we need to solve following two sets of equations,{[ f(y,β)=0; ∂/∂ y f(y,β)=0 ]. and{[ g(y,β)=0; ∂/∂ y g(y,β)=0 ]. .The two sets of equations are reduced, by a Groebner basis elimination, to(784β^3 - 846β^2 + 270β - 27) (4β - 1) = 0and(8192β^3 - 10368β^2 + 3591β - 378) (25β - 6) = 0respectively. Eq. (<ref>) gives β=0.778890, which supplies the bounds for the interval in Section <ref>, and (<ref>) gives β=0.602103, which then supplies the bounds for the interval in Section <ref>. §.§.§ β≥ 0.7789For β≥ 0.7789, both ϵ̅ and η̅ are monotonically decreasing for y∈(0,∞), which is shown in Figure <ref>.From this graph it can be readily seen that given any ν there is only one trivial slow-roll region for y>0 which in this graph is (y_0,∞).§.§.§ 0.6021≤β<0.7789For β in this region, ϵ̅ is still a monotonically decreasing function for y∈(0,∞) while η̅ is not any longer. The shape of these two slow-roll parameters are shown in Figure <ref>.From this graph one can easily read off, given that ν_min<ν<ν_max, the two slow-roll regions, one of which is (y_0,∞) which is trivial, and the other is (y_1,y_2) which has finite length and is thus the kind of slow-roll region we are searching for. From this graph we can also see that there are an upper bound and a lower bound for ν beyond which there is still only one trivial slow-roll region. In fact, this is a common feature, which will be justified in Sections <ref>, <ref>, and <ref>. Therefore, all these bounds of ν corresponding to different β render a window opening to non-trivial slow-roll regions, which shall be plotted in Section <ref>.§.§.§ 9/32≤β<0.6021For β<0.6021, both ϵ̅ and η̅ are not monotone functions of y in (0,∞), thus we should expect, on the whole, a wider range of ν that opens to non-trivial slow-roll regions. In particular, for β∈[9/32,0.6021), the potential v is still a monotonically increasing function for y>0, which means there is no local minimum of v in y∈(0,∞) (the cases that v has a local minimum in y∈(0,∞) shall be dealt with in the following two subsections). The typical shapes of ϵ̅ and η̅ are presented in Figure <ref>. From this figure, we can see that given ν∈(ν_min,ν_max), there is a non-trivial slow-roll region, (y_1,y_2), apart from the trivial one, (y_0,∞).§.§.§ 1/4<β<9/32In this interval, v>0 still holds, but v' is not positive definite any longer. So ∂ϵ̅/∂ y has two more roots which are the roots of v'. When β<9/32, the potential v(y) has a minimum on the right half y-v plane. In particular, for β>1/4 this minimum is a local minimum (see the first graph of Figure <ref>).Physically, we prefer the Universe to not being inflating at the minimum of the potential; the Universe should be reheating and the field should be oscillating. To ensure this, we look for |η̅(y_min)|>ν, where y_min is the local minimum point of v.Because of this extra filter, the upper bound of ν (viz., ν_max) is not necessarily equal to the local maximum of η̅ (viz., η̅_max), which is illustrated in the second graph of Figure <ref>. In that graph, the lower bound of ν, namely ν_min, is also not at the minimum of ϵ̅ which is 0. This is the consequence of the restriction η̅>-ν. §.§.§ 0<β≤ 1/4Finally, when β<1/4, v in the denominator of (<ref>). This contributes two extra singularities. The potential v has a true vacuum in the right half y-v plane, which is shown in the first graph of Figure <ref>.At this true vacuum, the potential is negative (or zero for β=1/4) and thus potential v(y) has two (or one when β=1/4) roots, dubbed y' and y” respectively.As a result, the slow-roll parameters ϵ̅ and η̅ are singular at these two roots of v, which can be seen from the second graph of Figure <ref>. The bounds of ν, namely ν_min and ν_max are also denoted in that graph. §.§ The windowNow that we have worked out all possible combinations of β and ν that opens a window to non-trivial slow-roll potentials whose procedure can be algorithmized by computer programs, we plot the numerical results in Figure <ref>.There are two salient features in this figure. First, at β=1/4, the upper bound of ν blows up, which means for any large enough ν the potential always has a non-trivial slow-roll region. Second, at β=9/32, there is a discontinuity (jump) in the upper bound of ν. This is because, from β>9/32 to β<9/32, a local minimum y_min appears suddenly in the potential, and the subsequent introduction of the extra constraint |η̅(y_min)|>ν which ensures that the Universe does not inflate at the minimum of the potential makes the bound for ν discontinuous.§.§.§ The likelihood of potentials being slow-rollSince we have made the assumption that the energy scale of inflation is at M_GUT, from (<ref>) we see that ν is around 10^-8 a^-2. If a is of order 𝒪(1), ν should be an extremely small number, then Figure <ref> indicates the likelihood of (β, ν) falling into the window must be quite small. To provide some evidence of our rough estimation, we made a statistical experiment using Monte Carlo method. We first assume that a and b are normally distributed and exponentially distributed variables respectively,a ∼𝒩(0, 1), and b ∼Exp(1).We than randomize (a, b) over 400 000 samples and found 312 slow-roll instances, which evaluates to a likelihood of approximately 0.078%.A second method of estimating the success rate is to compute the ratio of the area within the window to the area of the parameter space being surveyed. Because the window goes off to infinity, we introduce a cutoff, corresponding to excluding the region around a=0. For a cutoff at ν = 10000, the ratio is ∼0.15%; for a cutoff at 20000, the ratio is ∼0.14%; for a cutoff at 40000, it is ∼0.13%. This ratio is similar to the result obtained using the Monte Carlo method.§ MULTI-FIELD MODELS In this section, we investigate which potentials accommodate the slow-roll conditions for inflation with two fields. The form of the potentials we have isV(φ̃,χ̃)= μ^2/2 M^2 φ̃^2 +ρ^2/2 M^2 χ̃^2 + a_1 M φ̃^3 + a_2 M φ̃^2 χ̃ + a_3 M φ̃χ̃^2 + a_4 M χ̃^3 + b_1 φ̃^4 + b_2 φ̃^2 χ̃^2 + b_3 χ̃^4 ,where the masses μ M and ρ M for the fields ϕ̃ and χ̃ are defined in terms of M, the GUT mass, the a_i M are cubic couplings, and the b_i are quartic couplings. We will assume that the masses are around the GUT scale (∼ 10^16 GeV). We motivate the quartic potential from a Wilsonian perspective wherein higher order terms are suppressed by the energy scale at which new physics enters. We assume this is the string scale or Planck scale (∼ 10^19 GeV). Terms higher than quartic order, as they are suppressed by this higher energy scale, are neglected in the analysis. The coefficients a_i and b_i are order one numbers. The terms that appear in (<ref>) are dictated by the fact that we demand all slow-roll potentials to be bounded from below. With this in mind, the a_i can be positive or negative and the b_i are positive. When both fields tend to -∞, the quartic terms should have no odd powers in any of the two variables. We can rescale (<ref>) similarly to what we did in the single field case. With φ̃ = M φ and χ̃ = M χ, we havev(φ,χ)= μ^2/2φ^2 +ρ^2/2χ^2 + a_1 φ^3 + a_2 φ^2 χ + a_3 φχ^2 + a_4 χ^3 +b_1 φ^4 + b_2 φ^2 χ^2 + b_3 χ^4.Now all parameters and variables in potential v are dimensionless and it is sensible to talk about the magnitude of parameters. Note that V = M^4 v. These methods can readily be generalized to having more scalar fields. We simply require that the superpotential is renormalizable and bounded from below. As adding more scalars and studying the potentials explicitly in the finite field case is computationally more intensive, we do not extend the analysis beyond the two field level in this work. When searching for minima in the potential one will encounter both false and true vacua. We allow for slow-roll regions around false vacua (local minima) and not only the true vacua (global minima). The reason that we allow for both false and true vacua for slow-roll in two field case is based on computational and physical grounds. If we only look for global minima to test our slow-roll constraints, we firstly need to find global minima using methods such as steepest gradient descent subject to some arbitrary initial conditions. This will greatly increase the computational time and render the numerical test impossible in a reasonable time frame. Moreover, on physical grounds as long as the false vacuum is sufficiently long lived, the Universe may be in a metastable state. We do not analyze the lifetime to exclude short lived false vacua as this analysis depends on details of, say, particle physics and the presence of other nearby minima.§.§ Slow-roll conditions for multi-field inflation It is important to discuss the slow-roll conditions for multi-field inflation models as they are fundamentally different from those of single field case. The conditions are discussed in detail in <cit.>. We shall demand the following:ϵ ≡ -Ḣ/H^2 = 3(ϕ̇_̇i̇^2/V) =M_Pl^2 (∂_i V)^2/2V^2≪ 1 , ξ ≡√(V̂_̂1̂·V_2·V_2·V̂_̂1̂)≪ 1 ,withV̂_̂1̂≡∂_i V/|∂_i V| , V_2≡M_Pl^2(∂_i∂_jV)/V ,for fields = (φ,χ). Here the conditions are derived from the approximation 3Hϕ̇_i ≈ -∂_i V, which is essentially the consistent second slow-roll condition. This comes down to neglecting ϕ̈_i compared to ∂_iV. But when comparing two vectors, it is sensible only to compare their norms. Therefore we have the strong second slow-roll condition |ϕ̈_i| ≪ |∂_iV|. The reason it is called the strong second slow-roll condition is because its smallness impliesη≡V̂_̂1̂·V_2·V̂_̂1̂≪ 1 ,where η is defined to be1/ϵ Hdϵ/dt = 4ϵ - 2 V̂_̂1̂·V_2·V̂_̂1̂ = 4ϵ - 2η≪ 1 .Therefore the slow gradient flow by the fields defined in (<ref>) is not the only way to get a slowly-varying quasi-de Sitter expanding phase. §.§ Numerical tests In this section, we numerically determine whether a potential of the form (<ref>) satisfies the slow gradient flow condition in (<ref>). Because we now have the free parameters a⃗ and b⃗, we will adopt the Monte Carlo paradigm to characterize the shapes of potentials and quantify the rate of success.§.§.§ Setup for numericsThe experiment is set up as follows. *The coefficients a⃗ and b⃗ in cubic and quartic terms in (<ref>) are first sampled from a uniform distribution within range [-3,3] and [0,5] respectively. In addition, we also sampled the a⃗ coefficients from a Gaussian distribution with mean 0 and variance 1, and b⃗ coefficients from a exponential distribution with λ = 1.Let us briefly justify these choices of parameters. The experiments with the uniform distribution are performed in the spirit of Monte Carlo simulations, where parameters are chosen essentially at random.The choice of the uniform distribution is further justified by the fact that we do not know the region where slow-roll solutions reside in the seven dimensional parameter space of the potential coefficients. On the other hand, the choice of normal distribution with particular mean and variance will center our data around that mean and therefore may miss possible slow-roll regions. The choice of uniform distribution reflects the fact that we have no knowledge on the region of slow-roll samples within the parameter space a priori. In addition, the parameters are chosen to be of O(1) with respect to GUT scale. This comes from the fact that the higher order terms of the potential do not get corrections from quantum gravity effects, thereby, the potential is written in this particular quartic form. Note that this polynomial potential allows vertices that mix the two inflatons. This rules out the models such as assisted inflation where potential takes steep exponential <cit.> due to the fact that our potentials are polynomials.In the second set of experiments, the Gaussian distribution for a⃗ is motivated by the Central Limit Theorem. If we suppose our coefficients can be observed, the averages of n measurements of each coefficient then approach Gaussian distribution when n →∞. Meanwhile,the mean and variance are chosen on the grounds of naturalness. A priori, the coefficients should be order one numbers at the scale determined by the masses of the inflatons, which we set to GUT scale. Therefore, our choice of Gaussian distribution for a⃗ can be seen from previous arguments. On the other hand, we have b⃗ follow a Gamma distribution. We demand each of the elements of b⃗ to be positivein order to ensure that the potential is bounded from below.Just as the Gaussian distribution is a maximum entropy probability distribution for positive and negative real numbers, the Gamma distribution isthe maximum entropy probability distribution for positive real numbers. The Gamma distribution therefore becomes the natural candidate for selecting coefficients. In particular, we use an exponential distribution with λ = 1, which is a Gamma distribution with shape parameter k = 1 and scale parameter θ = λ. Again, the choice of 1 is motivated on the grounds of naturalness. The Central Limit Theorem requires large n. It is not clear that this applies when only a small number of scalar fields participate in inflation, so the comparison between the two possibilities is useful. * For each set of random coeffcients a⃗ and b⃗, we search for a point that satisfies conditions (<ref>) within a particular range for fields ϕ_i by using the Mathematica <cit.> function[Of course, we wrote more code than just this one line function.]FindInstance[< slow-roll conditions>, {φ, χ}].The search region within field space is rectangular with origin in the middle. The size of both sides of this region is twice the maximum of distances between origin and any stationary points of the potential. [To be precise, we usedto solve for zeros of the gradients of the potentials to find the extrema. However, this method actually turns the cubic polynomials into higher power polynomials (sometimes as high as 9-th order) thus making numerical solution highly sensitive to small change of gradient. For future work, we suggest to usedirectly instead and this might change the results slightly.] This is justified because we want a slow-roll region that is near a stationary point and the potential becomes steep far out from the origin in our potentials that are bounded from below. We do not want to falsely classify solutions as slow-roll simply by virtue of the fact the denominator, which is determined by the value of the potential, is large near infinity in field space. * For particular conditions in (<ref>), we have observational constraints ofϵ < 0.01 andξ < 0.01from measurements of scalar spectral index n_s that directly restricts slow-roll parameters. We also note that since the ξ condition implies η, the results from imposing ξ should be smaller than those from η. * The inflaton mass parameters are defined as μ = m_φ/M_GUT and ρ = m_ψ/M_GUT, where m_φ and m_ψ are the inflaton masses. Here, we set them both to be of GUT scale, so μ and ρ are of order O(1).With the mass parameters fixed, we take N=314000 uniformly distributed random coefficient samples, (a⃗_1, b⃗_1), …, (a⃗_N, b⃗_N). To be precise, we first of all generate 157000 samples using the Mathematica function RandomReal[{-3, 3}, 4] for a⃗ and RandomReal[{0, 5}, 3] for b⃗. Then we apply the slow-roll conditions in (<ref>) to these coefficients using the approach and constraint described in (<ref>) and (<ref>) to obtain instances[The FindInstance function in Mathematica is not capable of finding all desired slow-roll instances due to the mathematical complications of the slow-roll conditions and the internal algorithms designed for this task in Mathematica. Our experimental comparison of FindInstance with the more comprehensive but slower NSolve function indicate that results from using the two options are similar.]of relative minima that satisfy the necessary conditions. By noticing that the potential (<ref>) is symmetric under the following transformation for μ=ρ,φ↔χ , a_1 ↔ a_4 , a_2 ↔ a_3 , b_1 ↔ b_3,we apply this symmetry to the slow-roll coefficients found previously in the 157 000 samples. That is equivalently getting all slow-roll coefficients in 314 000 random samples. In addition, this is also done for 145 000, or equivalently, 290 000 samples drawn from Gaussian/Gamma distributions with means and shape parameters chosen as unity.§.§.§ Numerical results Of all the slow-roll potentials we obtained from choosing coefficients using both the uniform and Gaussian distributions, all of the slow-roll points found with function FindInstance have their distances from the origin greater than 30 000. The absolute values of potentials corresponding to these point instances have order of magnitude above 𝒪(10^16). Figure <ref> depicts two typical slow-roll potentials we have found for the uniform distribution. v(φ,χ)= φ^2/2 + χ^2/2 + 1.85634 φ^3 - 2.75233 φ^2χ + 0.59967 φχ^2 + 0.655031 χ^3 + 4.61147 φ^4 + 3.281 φ^2χ^2 + 0.00123865 χ^4,has three real stationary points,(-0.091192, -396.138),(-0.028419, -0.497133) and (0, 0),the furthest of which is around 400 units of distance away from the origin. However, the slow-roll point we found is at (-46745.5, -426), which is far beyond the region we expected. The potential at the slow-roll point is 2.2× 10^19, and the gradient is -1.9× 10^15 and -6.1× 10^12 in φ and χ directions respectively. One can check that, the two slow-roll parameters,ϵ = 0.0037 andξ = 0.0055,satisfy the constraints in (<ref>). However, from the careful inspection of this specific example, we can see that the slow-roll (and its adjacent region) satisfy the ϵ≪ 1 and η≪ 1 conditions because the potential at this distance from the origin is large and the two slow-roll parameters are both roughly inversely proportional to some power of the potential. All the other slow-roll instances we found in our random samples are similar to this specific example. We stated in Section <ref> that we wanted to exclude those regions where (<ref>) was satisfied simply due to the largeness of the potential. However, now that these are the only slow-roll regions we have found for the two-field case, the best statistics we can get is from this kind of slow-roll potential.To get a better idea about the slow-roll potential, we stack all the successful potentials in a single plot in Figure <ref>. As the figure shows, all the slow-roll potentials have a steep and a flat direction.Although multi-field models typically allow for the presence of isocurvature modes, Figure <ref> indicates that such modes are heavily suppressed as the two-field inflation appears to be dominated by a single flat direction. This qualitative shape of the potentials recalls those found in models of hybrid inflation type <cit.>. Generic initial conditions in the neighborhood of a minimum would first fix the field in the direction perpendicular to the valley (the steep direction), and then the field would roll along the valley to the minimum at the bottom. Unlike the potential for hybrid inflation discussed in the introduction, informed by our choices of random coefficients, the parameters that appear in the potentials here are all typically of order one with respect to the GUT scale. The statistics of slow-roll potentials we found in potentials from the uniform and Gaussian/Gamma distribution are listed in Table <ref>. From Table <ref>, we can see that in all randomly generated coefficients, around 0.05% of them correspond to slow-roll potentials and 0.1% for Gaussian/Gamma distribution. This supplies a lower bound for the percentage of relative minima that accommodate the slow-roll conditions for inflation with two scalar fields.We also draw distributions of values of coefficients (a⃗, b⃗) that correspond to slow-roll potentials in Figure <ref>. These figures show that in spite of the uniform distribution that we presume as priors for a_i and b_i, the slow-roll conditions pick the coefficients according to the mass distributions in these histograms. The distributions of all components of a⃗ and b⃗ deviate noticeably from the uniform distribution. This indicates that the slow-roll conditions set some constraints on the values of coefficients in a probabilistic sense. Deviations may also be a product of having a low number of slow-roll regions, only around 70, despite starting with a minimum of 150 000 samples. A greater computational analysis would be required to obtain more concrete statistics for the distribution of slow-roll regions. It is notable that the results are roughly the same irrespective of what distribution we employ to select the coefficients.In addition to the results from uniform distribution, we also present the histogram plots for coefficients of slow-roll potentials for Gaussian/Gamma distribution in Figure <ref>. We can see from the plot that the Gaussian nature of the plot still is still present. This shows that the slow-roll conditions we choose respects the Gaussianity of initial samples.Preliminary experiments with varying the mass parameters for the scalar fields over several orders of magnitude do not significantly change the percentages of slow-roll solutions.Finally, in Figure <ref> we present two histograms for e-foldings from different initial conditions for the two potentials in Figure <ref>. The calculation for e-foldings is set up as follows. * The number of e-foldings is a function of the initial conditions. In a single field model, we may compute N(ϕ) = 1/M_Pl^2∫_ϕ_*^ϕ_i dϕ V(ϕ)/V'(ϕ)≃∫_ϕ_*^ϕ_idϕ/ϕ 1/η , where ϕ_i is the initial value and ϕ_* is the critical value corresponding to the minimum of the potential. * Since all the potentials in the two field case that we consider have a steep direction and a flat direction, we set component of the field vector in the steep direction to a fixed value andperform the integration along the flat direction to simplify this into an effective single field problem. * Let us denote the steep direction ϕ and the flat direction by ψ. For each potential, we find its extrema and identify (ϕ_*,ψ_*) with the field configuration of the global minimum. Then we want to see how the perturbation in the path of integration affect the values of e-foldings. This is done by sampling through the final values ϕ_f, for the integration in the range of ϕ_f ∈ [ϕ_*-|100ϕ_*|,ϕ_*+|100ϕ_*|] in steps of |ϕ_*|. The field that is being integrated over is thenthe component in the flat direction ψ. The starting value ψ_i is half the value of the distance between the nearest extremum to the global minimum. For example, the bottom potential in Figure <ref> has its global minimum at (-186.663,-0.311574), and the nearest extremum in the flat direction is at (-0.152773,-0.0523367). We then choose the initial and final value for the integration in the flat direction to be (-186.663,-90).With the previous setup, we obtain the histograms in Figure <ref>, and we see that the values of e-foldings are a few magnitudes smaller compared to the phenomenological required value of ∼ 60. Therefore, even if we find certain potentials supports slow-roll at particular points, the potential can fail to sustain period of inflation long enough to have the experimental values of e-folding. It is important to note that these are preliminary results. The focus of our paper is the preliminary issue of whether slow-roll is even possible given a random potential. § DISCUSSION AND OUTLOOKWith the advent of Big Data in theoretical physics and ever-increasing computational power, we are gaining further glimpses into the various landscapes of theories ranging from string vacua to cosmological scenarios. In this paper, we have been motivated by the question of the probability of having slow-roll inflation within the landscape of effective potentials for inflatons.We started with the case of a single field with the most generic form of the potential up to degree four, subject to the constraints of slow-roll. Here, there are only twoparameters, which we have dubbed β and ν, and which can be expressed in terms of the couplings. We can solve the problem numerically to arrive at Figure <ref>. The figure depicts a non-trivial region of parameter values which satisfy the slow-roll conditions.With two inflatons, the situation is understandably more intricate. Here, up to degree four, there are seven parameters. The slow-roll conditions then translate to a polynomial system in the fields and in the parameters. This is then a problem in a potential landscape sculpted by these parameters. We find that the slow-roll conditions for multi-field are insensitive to the distribution we have used, i.e., we find that they give the same percentage of slow-roll instances. Moreover, initial experiments that change the mass scale yield similar results. The inflatons for slow-roll inflation traverse far into field space so that the slow-roll conditions are satisfied in part due to the largeness of V. The potential has a characteristic shape in which the fields would roll first down a steep direction and then follow a valley to the minimum. Based on this shape, the generation of isocurvature modes is not expected.In the single field case, the condition V'V”' ≪ V^2M_P^-4 accounts for constraints on the running of the spectral index. It would be reasonable to develop the equivalent third derivative condition for multi-field case and include this within the framework of random potentials. Preliminary analysis of example potentials that admit slow-roll minima suggest that an insufficient number of e-foldings transpires. While we can exclude certain minima as being unable to support the necessary number of e-foldings, the suitability of other minima for this purpose depends critically on the choice of initial conditions for the inflaton fields. In this work we do not perform a systematic study of the number of e-foldings for each of the potentials that support slow-roll inflation. There is no obvious a priori selection criteria for this informed by realistic string constructions of the Standard Model.We defer a more detailed investigation for future work. In general, with an arbitrary number of inflatons and a potential up to a specified degree, the slow-roll constraints will produce a large polynomial system with still larger number of parameters.For example, in the heterotic string Standard Models to which we alluded in the introduction, the contribution to the number of moduli fields come from the geometry — roughly the sum of the Hodge numbers — and from the bundle — roughly the number of endomorphisms <cit.>. For the (3,3) Calabi–Yau threefold studied in <cit.> the total number of moduli is 6+19 = 25.Because it is doubly exponential in the number of parameters, the usual Groebner basis <cit.> approach to analyzing such systems will soon become rather prohibitive, and even numerical algebraic geometry <cit.> will find this challenge daunting. Our approach of randomization over parameters is thus the standard technique, and the statistics over the landscape is an enlightening overview of how special or generic our universe is. Aided by work done in <cit.>, one could find a relationship between the general statistics of successful inflation in random polynomial potentials with the general statistics of stationary points in the same potentials. They note that the variance of positive real roots both increases with the degree of the polynomial and converges with sample size. Knowing this may give, for example, a convenient saturation limit of test samples, shifting a greater proportion of computational resource to just the search for relationships between minima and slow roll regions. Finding this would allow us to derive properties of the inflationary landscape from the statistics of the minima themselves. More work is needed hover in developing actual “search and test" techniques for inflation within these minima.Large-N field models are difficult to study when looking for successful inflation. Although the upper bound probability of critical points of a random polynomial grows as ∼N/2 <cit.>, the probability of inflation in the large N field cases appears to decrease as seen in work done in <cit.>, which provides a heuristic argument that in a multifield landscape probability distribution to realize slow-roll inflation looks like P(N_e) = AN^-α_e ,where A is exponentially small and α∼ 3. (See <cit.>.) Since out work was limited to two and three fields, it is difficult to make conclusive statements or extrapolate the probabilities we have obtained to the large N limit. We expect that the more fields we have, the harder it becomes to realize slow-roll conditions using a random potential.It has recently been conjectured that constraints on cosmological inflation are two-fold <cit.>: (1) the distance traversed in field space during inflation should be O(1); and (2) the gradient of the potential during inflation satisfies∇ V/V = √(G^ij∂_i V ∂_j V)/V≳ O(1) , ∂_i = ∂/∂ϕ^i .In the single field case, whereas we can choose parameters to satisfy (1), generically, we have not found examples that satisfy (2) within our allowed window from Figure <ref>. In the two field case, in the examples we have examined, the shape of the potential fails to satisfy the second criterion by roughly two orders of magnitude.One should mention also that there is a branch of mathematics known as random algebraic geometry where the usual quantities such topological invariants and cohomology, which govern the physics, have their analogues in the stochastic sense. It will certainly be interesting to study polynomial systems arising from the potential landscape under this light. Furthermore, with the advancement of machine learning techniques, it may be instructive to see how a neural network could be applied to finding slow-roll points in a potential. Rather than teaching a neural network to look for successful points of slow-roll inflation, one could look at the methodology these neural networks use to minimize the cost function. For a single feature in a neural net, one would use a simple gradient descent method to recursively find a local minima. This is equivalent to looking for the local minima of a single field polynomial potential. Typically, however, a neural network has an enormous amount of features, or input variables, which all determine the cost functions, and thus minimizing this cost function, is equivalent to finding a critical point of a multivariate polynomial, or in our language, a multifield potential. Neural networks do this rather efficiently, and thus one possible avenue is to tweak how these networks find critical points and ask them to find critical points that allow for slow-roll inflation. This is work in progress. § ACKNOWLEDGEMENTSWe thank Cyril Matti for collaboration during an early stage of this work. YHH would like to thank the Science and Technology Facilities Council, UK, for grant ST/J00037X/1, the Chinese Ministry of Education, for a Chang-Jiang Chair Professorship at NanKai University as well as the City of Tian-Jin for a Qian-Ren Scholarship, and Merton College, Oxford, for her enduring support. VJ and LP are supported by the South African Research Chairs Initiative, which is funded by the Department of Science and Technology and the National Research Foundation of South Africa. YX is support by the Doctoral Studentship of City University of London. DZ is supported by the China National Natural Science Foundation under contract No. 11105138 and 11575177. 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B 784, 271 (2018) [arXiv:1806.09718 [hep-th]]. | http://arxiv.org/abs/1704.08351v3 | {
"authors": [
"Yang-Hui He",
"Vishnu Jejjala",
"Luca Pontiggia",
"Yan Xiao",
"Da Zhou"
],
"categories": [
"hep-th",
"astro-ph.CO",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170426210632",
"title": "Flatness of Minima in Random Inflationary Landscapes"
} |
= 15 truecm= 23 truecm = - 0.5 truecm=-2 truecm 0.3ex<-0.75em-1.1ex∼ 0.3ex>-0.75em-1.1ex∼plainLAPTH-005/17LPT-Orsay 16-88 to 1 truecm Photon-jet correlations in Deep-Inelastic Scattering 1 truecmP. Aurenche^1,a, M. Fontannaz^2,b 3 truemm^1 LAPTh, Université Savoie Mont Blanc, CNRSBP 110, 9 Chemin de Bellevue, 74941 Annecy-le-Vieux Cedex, France ^a e-mail : [email protected] truemm^2 Laboratoire de Physique Théorique, UMR 8627 CNRS,Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay Cedex, France ^b e-mail : [email protected] truecmThe reaction e + p →γ + jet + X is studied in QCD at the next-to-leading order. Previous studies on inclusive distributions showed a good agreement with ZEUS data. To obtain a finer understanding of the dynamics of the reaction, several correlation functions are evaluated for ZEUS kinematics.= 24 pt This note is the continuation of the study of the Deep Inelastic Scattering (DIS) reaction e + p →γ + jet + X with a photon and a jet in the final state. In a preceeding paper <cit.> we calculated the next-to-leading order (NLO ) QCD cross sections describing the photoproduction of a large-p_ photon (p_^γ , η_γ) accompanied by a jet. In particular, we calculated the transverse momentum and rapidity distributions of the photon with the jet constrained in a given large range as well as the jet transverse momentum spectrum in events with a detected photon. In the present note we concentrate on the distributions in the following correlation variables: x_γ = (p_^γ e^- η_γ + p_ ^jet e^-η_jet)/(2 y E_e), x̃_p = (p_^γ e^η_γ + p_ ^jet e^η_jet)/(2 E_p) (in the laboratory frame E_e, E_p are respectively the incident energies of the electronand the proton and y is the DIS inelasticity), Δη = η _γ - η_jet and Δϕ = ϕ_electron - ϕ_γ. Note that x̃_p, as defined above, does not correspond to x_p, the fraction of momentum carried by a parton in the proton, since we are not in a collinear frame. The correlations should provide a finer understanding of the underlying production mechanism than the inclusive cross sections.Originally the NLO calculation <cit.> was performed in the γ^*-p center of mass (γ^* is the virtual photon) in which a large scale is provided by a large value of p_^*γ, the final γ transverse momentum in that frame. Production cross sections of hadrons and jets were studied in this frame and successfully described by the NLO calculation <cit.>. On the other hand, in the ZEUS experiment <cit.> the reaction e + p →γ + jet + X isstudied in the laboratory frame for which the original NLO calculation must be adapted. Indeed a large p_^γ in the laboratory does not necessarily correspond to a large p_^*γ in the γ^* - p frame. Therefore a cut-off E_ cut^* must be introduced in the calculation with p_^*γ > E_ cut^* in order to remain in a perturbative domain. In the preceding paperwe considered the valuesE_ cut^*= 2.5 GeV and E_ cut^* = .5 GeV. A good agreement was found between theory and the ZEUS data with the cut-off E_ cut^* = 2.5 GeV. We also found that there are kinematical domains in which the NLO cross sections were little dependent on E_ cut^*, for instance for small ratios (Q/p_^γ)^2 or (Q/p_^jet)^2. This result demonstrates that the theoretical calculations are almost model independent in the corresponding domains. A detailed discussion of this problem can be found in ref. <cit.>. In this note we pursue the study of the reaction e+p →γ +jet + X by calculating the cross sections dσ /dx_γ, dσ /dx_x̃_p, dσ /dΔη and dσ /dΔϕ with the cut-offs E_ cut^* = 2.5 GeV and E_ cut^* = .5 GeV, and, following ZEUS <cit.>, we consider two Q^2-ranges, namely 10 < Q^2 < 350 GeV^2 and 10 < Q^2 < 30 GeV^2. As in ref. <cit.> we adopt the ZEUS kinematics <cit.> with √(s) = 319 GeV; the photon momentum has to lie in the ranges 4 < p_^γ < 15 GeV and -.7 < η_γ < .9. For the jet momentum we have 2.5 < p_^jet < 35 GeV and -1.5 < η_jet < 1.8. Constraints on the final electron are : E_e' > 10 GeV, θ_e > 140^∘ (the z axis is pointing toward the proton direction). As in the ZEUS experiment, the photon is isolated using the democratic k_⊥-algorithm <cit.> where the photon is treated on the same footing as partons. We refer to ref. <cit.> for details on the isolation criteria and other parameters and conventions used in the calculation.We do not consider here the emission of photons by the electron. Therefore, this cross section should be added to those calculated in this paper to reconstruct the experimental cross section measured by the ZEUS collaboration. We recall that the cross section is the sum of four building blocks. The "direct" component where the initial virtual photon is coupled to the hard partonic process and the "resolved" component where it interacts via its structure fonction. In each of these the final real photon couples directly to the hard process or is a fragment of a parton produced at large transverse momentum (fragmentation component). Due to the photon isolation criteria the fragmentation components are small, typically 10% to 20% of the total depending on the kinematics. We use the virtual photon structure function presented in ref. <cit.> and the CTEQ6M parton distributions in the proton <cit.>. The fragmentation function is that of the BFG collaboration (set II) <cit.>. All the components are calculated at NLO and details can be found in ref. <cit.>. We work in the MS scheme for factorisation and renormalisation with Λ_MS(4) = 236 MeV and N_f = 4. All scales are taken equal to √(p_^*γ^2 + Q^2). The numerical calculations are carried out using the adaptive Monte Carlo code BASIS <cit.>.The results of the calculations are displayed in Fig. 1 to Fig. 4. Each figure contains four curves corresponding to two different values of E_ cut^* and to two Q^2-integration domains. The shapes of the curves corresponding to the small domain 10 < Q^2< 30 GeV^2 (red curves) and to 10 < Q^2< 350 GeV^2 are similar, but the curves of the small Q^2-domain are a factor 2 to 3 smaller than those of the full Q^2 range. For each Q^2 domain two values of E_ cut^* have been used, E_ cut^*=.5 GeV and E_ cut^*=2.5 GeV. It clearly appears that the low Q^2 data are globally less sensitive to the transverse momentum cut-off E_ cut^* : indeed, in the laboratory, the transverse momentum of the virtual photon is √(Q^2 (1-y)) and, therefore, at small Q^2 the values of p_^γ and p_^*γ are closer and the laboratory constraint p_^γ > 4. GeV is more efficient in cutting off the small p_^*γ configurations.For the large Q^2 case the effect of the E_ cut^*-variation is irregular and strongly depends on the values of the kinematical variables. Therefore the shapes of the distributions are sensitive to E_ cut^*. But there are kinematical domains in which the distributions are little affected by E_ cut^*. For instance, dσ /dx_γ is rather independent on E_ cut^* for x_γ.7. This can be understood in the following way : a small value of x_γ corresponds to a large value of y and consequently a small transverse momentum of the initial virtual photon. Also the parton with a small x_γ[The Born direct contribution corresponds to x_γ = 1.] carries away a small transverse momentum. These two effects explain the weak sensitivity of dσ /dx_γ to E_ cut^* for x_γ.7. Likewise, the positive Δη = η _γ - η_jet domain is rather stable :in this domain the cross section is dominated by the direct processes, less sensitive to E_ cut^* than the resolved contributions.As for dσ /dx̃_p a similar reason leads to a stability region for large x̃_p. A stable behavior is observable for dσ /dΔϕ in the domain Δϕ < 140^∘, which can be explained by kinematical reasons. At Δϕ = 0, for instance, p_⊥^*γ is always larger than p_^γ > 4 GeV and an E_ cut^* is not necessary. When there is a large sensitivity, we found in ref. <cit.>that inclusive data were well described with the choice E_ cut^* = 2.5 GeV.Concerning the sensitivity of the cross-sections to the factorisation and renormalisation scales, we have studied it for one representative point of the distributions in the case of 10 < Q^2 < 30 GeV^2. For the renormalisation and the proton factorisation we useμ = M = c √(Q^2 + p_^*γ^2) while in the virtual photon structure function we take M_F = √(Q^2 + (c p_^*γ)^2). This last choice is natural since M_F cannot be smaller than √(Q^2) in the photon structure function <cit.>. The sample points are .6 < x_γ < .7, .6 < Δη < 1.3 and 80^∘ < Δϕ < 110^∘, domains in which the cross sections are almost insensitive to E_ cut^*. We observe that in the range .5 < c < 2. the cross section varies by at most 15%. For example, for the distribution in x_γ this variation corresponds to the thickness of the line in Fig. 1, while for the distribution in η_γ - η_jet it is displayed on Fig. 2, at the value η_γ - η_jet = .95. We notice that this relative stability is the result of a huge compensation between the direct and the resolved terms whose size varies by a factor 2 to 4 under the scales changes. Finally, concerning the accuracy of the theoretical calculations, let us discuss another point besides the sensitivity to the cut E_ cut^* and to the scale variations. When calculating the higher order corrections to the resolved component, we assume that the photon structure function, proportional to lnp_^*2 + Q^2Q^2, is large, so that we can neglect terms not proportional to this logarithm. We have no way to estimate the importance of such terms in the ZEUS experiment in which the condition Q^2/p_^*2≪ 1 is not always verified. To test the validity of the approximation it would be interesting to compare theory and data in the two ranges 10<Q^2<30 GeV^2 and 30<Q^2<350 GeV^2 <cit.>.In conclusion, in the ZEUS experimental configuration for observables constructed with the laboratory kinematics we have a good theoretical stability under changes of cut-offs in a relatively large domainof the variables x_γ< 0.7, Δη > 0 and Δϕ < 130^∘when 10 < Q^2< 350 GeV^2. Outside these ranges the physics becomes sensitive to non perturbative effects. In the small Q^2 domain the QCD predictions are independent on the cut-offs in the above quoted ranges and rather insensitive elsewhere. As for the scale uncertainties they are less than ± 8 % when probing the standard scale ranges. Comparison of these NLO predictions with preliminary ZEUS data are in good agreement after the photon emission from the electron is taken into account <cit.>.AcknowledgmentsWe thank Peter Bussey for useful discussions. 991r P. Aurenche and M. Fontannaz, EPJC75 (2015) 64. 2r P. Aurenche, R. Basu, M. Fontannaz, R. M. Godbole, EPJC42 (2005) 43. 3r P. Aurenche, R. Basu, M. Fontannaz, EPJC71 (2011) 1616. 4r H1 Collab., A. Aktas et al., EPJC 36, 441 (2004). 5r H1 Collab., F. D. Aaron et al., EPJC 65, 363 (2010). 6r ZEUS Collab., S. Chekanov et al., Nucl. Phys. B 786, 152 (2007). 7r ZEUS Collab., H. Abramowicz et al., Phys. Lett.715, 88 (2012). 7rPB Peter Bussey, private communication. GloverMorgan E.W.N. Glover, A.G.Morgan, Z. PhysC62 , 311 (1994). 9r M. Fontannaz, EPJC38, 297 (2004). 8r J. Pumplin, D.R. Stump, J. Huston, H.L. Lai, P.M. Nadolsky, W.K. Tung, JHEP0207, 012 (2002). 10r L. Bourhis, M. Fontannaz, J. Ph. Guillet, EPJC2, 529 (1998). 11r P. Aurenche, Rahul Basu, M. Fontannaz, R. M. Godbole EPJC34, 277 (2004). 12r S. Kawabata, Comput. Phys. Comm.88, 309 (1995). 13r O. Hlushchenko, for the ZEUS collaboration, talk at DIS-2017, Birmingham (2017). p^jet_ min < p_^jet < p_ max^jet and η_ min^jet < η^jet < η_max^jet These domains correspond to configurations close to the photoproduction configuration. The transverse momenta p_^*γ and p_^γ are not too different and the domain in which the perturbation calculation is valid can be controlled by the value of p_^γ. Therefore the cut-offE_ cut^* is not necessary. In general we notice that the E_ cut^*-sensitivity is much smaller when the values of Q^2 are small. In this case we are close to the photoproduction kinematics with a large value of p_^γ 2 Q^x. Therefore in this Q^2-domain the large scale can be controlled by p_^γ. | http://arxiv.org/abs/1704.08074v1 | {
"authors": [
"P. Aurenche",
"M. Fontannaz"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170426121812",
"title": "Photon-jet correlations in Deep-Inelastic Scattering"
} |
1]MichałCzakon 2]David Heymes 2]Alexander Mitov[1]Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, D-52056 Aachen, Germany [2]Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UKfastNLO tables for NNLO top-quark pair differential distributions [=================================================================Preprint numbers: Cavendish-HEP-17/08, TTK-17-1010mmWe release fastNLO tables with NNLO QCD top-quark pair differential distributions corresponding to 8 TeV ATLAS <cit.> and CMS <cit.> measurements. This is the first time fastNLO tables with NNLO QCD accuracy have been made publicly available. The tables are indispensable in pdf fits and allow, for the first time, very fast calculation of differential distributions with any pdf set and for different values of . The numerical accuracy of the resulting differential distributions is high and comparable to the accuracy of all publicly available NNLO top-quark differential calculations. We intend to keep producing tables corresponding to existing and future LHC measurements at various collider energies.§ INTRODUCTIONCalculations of LHC processes at next-to-next-to-leading order (NNLO) in perturbative QCD are numerically challenging and at present require computing times of 𝒪(10^4-10^5) CPU hours for a single calculation. Typically, calculations are performed for a pre-decided specific set of bin(s), values of relevant parameters (like masses and coupling constants) and parton distribution functions (pdf). In the following we will refer to such calculations as direct. Once direct results have been produced, typically in the form of binned histograms, they cannot be changed any more. In other words, if one wishes to perform the same calculation but for different values of , pdf, mass or set of bins – even for the same distribution – one has to repeat the calculation from scratch. Alternative flexible formats for storing results that allow one to reuse existing calculations in future applications are, therefore, highly desirable. For the extraction of pdfs from LHC data where up to 𝒪(10^5) recalculations of the hadronic cross section for different pdfs are required, the speed of the calculation becomes a major additional requirement.A number of formats with different levels of flexibility have been proposed and are commonly used in NLO calculations. The n-tuples event-file format for (weighted) partonic NLO calculations <cit.> stores all the information for every partonic event/counterevent. This format allows one to reasonably fast recompute arbitrary observables and modify parameters like the renormalization and factorization scales, pdf's and . While the n-tuple format provides high level of flexibility, it has one disadvantage: event-files tend to be very large with sizes approaching 𝒪(1) TB (depending on the process). This disadvantage becomes more significant at NNLO due to the larger number of events/counterevents that need to be stored for numerically stable results. A first study of the size of the n-tuples event-file format for NNLO computations in e^+e^- collisions has been performed in ref. <cit.>.An alternative approach to storing and recomputing NLO calculations is offered by the fastNLO <cit.> and APPLgrid <cit.> formats. This approach produces and stores for later use an accurate interpolation of the partonic cross-section in a pdf and α_s independent way. This approach enables the extremely fast recalculation of fixed observables with different pdf sets, values ofand, possibly, factorization and renormalization scales. These interpolation formats are particularly useful for extracting pdfs from data. Importantly, the interpolation approach can be simply extended to NNLO computations and a fastNLO interface to event generators at NNLO is publicly available. In contrast to storing event-files, the required disk space for fastNLO tables, even at NNLO, does not exceed 𝒪(100) MB and is a convenient way of storing and distributing differential results at NNLO. A drawback of this approach is that one cannot change the distributions or bins.As a first application of fastNLO at NNLO in this work we produce fastNLO tables for four one-dimensional top-quark pair differential distributions for LHC at 8 TeV. These samedistributions were computed in refs. <cit.> as binned histograms and for specific pdf sets. To produce the tables we have straightforwardly interfaced the fastNLO library to an in-house Monte Carlo (MC) generator which implements the Stripper approach for NNLO calculations <cit.>. The table files are publicly available <cit.> and can be used, among others, for pdf extractions <cit.>, strong coupling variation and determination and new physics searches. § FASTNLO TABLES FOR TOP-QUARK PAIR PRODUCTION AT 8 TEV The binning of the produced tables corresponds to the common binning used by ATLAS <cit.> and CMS <cit.> in the lepton plus jets LHC 8 TeV measurements. Tables are produced for the following four distributions: invariant mass of the top-quark pair , transverse momentum of the averaged top/antitop quark , rapidity of the average top/antitop quarkand rapidity of the top-quark pair . Following ref. <cit.>, the renormalization scale μ_R and factorization scale μ_F have been set to m_T/2 for the transverse momentum distribution of the average top/antitop quark and to H_T/4 for all other distributions. We would like to emphasize that our tables contain only the central values for μ_F,R and cannot be used for performing scale variation or for changing the functional form of the two scales. The provided information should be more than adequate for most applications, like pdf fitting, where fastNLO tables are indispensable. If an estimate of the scale error for calculations performed with our fastNLO tables is desired, one could construct such an estimate from the direct calculation in ref. <cit.> which is available in electronic form.[Since results for several pdf sets are readily available in electronic form <cit.>, one could easily verify the extend to which the scale variation for each distribution and each bin is pdf-independent.] A summary of the produced fastNLO tables is given in table <ref>. Their calculation was performed at the University of Cambridge's Darwin cluster <cit.>. The computing time needed to produce the tables did not significantly exceed the time it took to perform our previous direct calculation <cit.>. For reasons of computational efficiency[We hope that for future fastNLO table releases we will be able to provide the results for each distribution within a single table.]two tables instead of one are provided for the average top/antitop rapidity distribution (each one of the other three distributions is contained within a single table). The full result foris obtained after summing the output of the two tables for each perturbative order. The fastNLO software and instructions for using fastNLO tables can be found on the fastNLO website <cit.>. An example for their usage is given in appendix <ref>.§ QUALITY OF THE FASTNLO TABLESPresently, the fastNLO table format does not have a facility for estimating Monte Carlo errors. To control the quality of our tables we perform two tests: we estimate the interpolation error of the tables as well as the accuracy of the distributions by comparing predictions derived from the tables with three independent calculations. To check the interpolation error inside our NNLO fastNLO tables in fig. <ref> we compare the four differential distributions derived from the tables with direct NNLO calculations that were performed simultaneously with the tables, using exactly the same set of partonic events. This way, the same information from each generated partonic event is passed to the table and to the histogram, ensuring the calculations of the table and the direct histogram are fully correlated. The final table is then convoluted with the same pdf set used in the direct histogram and the two are compared in fig. <ref>. To ensure pdf-independence of the test, we simultaneously compute two histograms that are based on different pdf sets. We have used NNPDF30 <cit.> and CT14 <cit.> which were chosen to have different values of the strong coupling: α_s(m_Z) = 0.118 and 0.111, respectively. We interpret the relative difference between the table-based and direct calculation for a given pdf set as due to table interpolation. The comparison for all four distributions can be found in the upper panels of fig. <ref>. The interpolation error is about 1 per mille for all bins for all four distributions and is almost independent of the pdf set. This error is negligible since it is much smaller than the Monte Carlo error of the direct NNLO calculation, which can be found in the lower panels of fig. <ref>. To check the accuracy of the distributions obtained from the NNLO fastNLO tables we compare these distributions with statistically independent NNLO calculations of the same differential distributions which were published in ref. <cit.> for three different pdf sets: NNPDF30, CT14 and MMHT14 <cit.> (all three pdf sets have α_s(m_Z) = 0.118). The comparisons are shown in the upper panels of fig. <ref>. The relative difference between the two calculations is small; for all four distributions and all three pdf sets it does not exceed 0.6% in any bin (and for almost all bins it is about half that value). As can be seen in the lower panels of fig. <ref>, this difference is comparable to the relative MC error of either the corresponding prior direct calculation or the correlated with the table NNPDF30-based direct calculation, discussed above in the context of the interpolation error estimate (recall that the dashed line in fig. <ref> exactly corresponds to the blue line in fig. <ref>). As figs. <ref>,<ref> demonstrate, distributions derived from our fastNLO tables are as accurate as the ones obtained from a direct calculation and can be readily used to obtain NNLO predictions with any pdf set without additional loss of numerical accuracy.§ SUMMARY AND OUTLOOKIn this work we produce fastNLO tables for four top-quark pair differential distributions at NNLO corresponding to the ATLAS and CMS 8 TeV measurements <cit.>. The tables are publicly available and can be downloaded here <cit.>. These are the first publicly released fastNLO tables at NNLO. The tables allow very fast calculation of these distributions with any pdf set and for different values ofthrough the LHAPDF interface. The tables will be indispensable in pdf fits as well as in any calculation of top-quark differential distributions with future pdf sets. We have verified the numerical accuracy of the NNLO differential distributions. It is high and comparable to all publicly available top-quark differential calculations. We intend to keep producing tables corresponding to other existing and future LHC measurements at 8 and 13 TeV. The most up-to-date set of released fastNLO tables can be found at the website <cit.>. § ACKNOWLEDGEMENTThe authors would like to thank Daniel Britzger and the fastNLO collaboration for help with fastNLO and useful discussion. The work of M.C. is supported in part by grants of the DFG and BMBF. The work of D.H. and A.M. is supported by the UK STFC grants ST/L002760/1 and ST/K004883/1. A.M. is also supported by the European Research Council Consolidator Grant “NNLOforLHC2".§ USING THE TABLES To obtain the full NNLO differential cross section the tables need to be convoluted with a pdf set. For this purpose, a version of the fastNLO toolkit is required. The tables have been tested for the latest public version (Version 2.3 pre-2212) which can be found on the fastNLO website <cit.>. Here is a command line example for convoluting thetable with the NNPDF30 pdf set through the LHAPDF <cit.> interface: fnlo-tk-cppread LHC8-Mtt-HT4-173_3-bin1.tab NNPDF30_nnlo_as_0118 1 LHAPDF no The outputted cross-section for each bin, in pb/GeV, reads ——————————————————– LO cross section NLO cross sectionNNLO cross section——————————————————– 7.38589896926E-011.00686397386E+001.08054272971E+00 7.76050226541E-011.01684892793E+001.06831476452E+00 4.72181816638E-016.14208165337E-016.51937131158E-01 2.37714769748E-013.09500005873E-013.32361059337E-01 9.50531653713E-021.24354657190E-011.34756258089E-012.30309358260E-023.02533902041E-023.31987429211E-02 2.66871047208E-033.51131256484E-033.94448314912E-03 00 Aad:2015mbvG. Aad et al. [ATLAS Collaboration],Eur. Phys. J. C 76, no. 10, 538 (2016) [arXiv:1511.04716 [hep-ex]].Khachatryan:2015oqaV. Khachatryan et al. [CMS Collaboration],Eur. Phys. J. C 75, no. 11, 542 (2015) [arXiv:1505.04480 [hep-ex]].Bern:2013zjaZ. Bern, L. J. Dixon, F. Febres Cordero, S. Hoeche, H. Ita, D. A. Kosower and D. Maitre,Comput. Phys. Commun.185, 1443 (2014) [arXiv:1310.7439 [hep-ph]].Heinrich:2016jadD. Maitre, G. Heinrich and M. Johnson,PoS LL 2016, 016 (2016) [arXiv:1607.06259 [hep-ph]].Kluge:2006xsT. Kluge, K. Rabbertz and M. Wobisch,hep-ph/0609285.Wobisch:2011ijM. Wobisch et al. [fastNLO Collaboration],arXiv:1109.1310 [hep-ph].Britzger:2012bsD. Britzger et al. [fastNLO Collaboration],arXiv:1208.3641 [hep-ph].Carli:2010rwT. Carli, D. Clements, A. Cooper-Sarkar, C. Gwenlan, G. P. Salam, F. Siegert, P. Starovoitov and M. Sutton,Eur. Phys. J. C 66, 503 (2010) [arXiv:0911.2985 [hep-ph]].Czakon:2016dgfM. Czakon, D. Heymes and A. Mitov,JHEP 1704, 071 (2017) [arXiv:1606.03350 [hep-ph]].Czakon:2015owfM. Czakon, D. Heymes and A. Mitov,Phys. Rev. Lett.116, no. 8, 082003 (2016) [arXiv:1511.00549 [hep-ph]].Czakon:2010tdM. Czakon,Phys. Lett. B 693, 259 (2010) [arXiv:1005.0274 [hep-ph]].Czakon:2011veM. Czakon,Nucl. Phys. B 849, 250 (2011) [arXiv:1101.0642 [hep-ph]].Czakon:2014omaM. Czakon and D. Heymes,Nucl. Phys. B 890, 152 (2014) [arXiv:1408.2500 [hep-ph]].web-tablesRepository with t fastNLO tables: <http://www.precision.hep.phy.cam.ac.uk/results/ttbar-fastnlo/> Czakon:2016oljM. Czakon, N. P. Hartland, A. Mitov, E. R. Nocera and J. Rojo,JHEP 1704, 044 (2017) [arXiv:1611.08609 [hep-ph]].DarwinThe Cambridge HPC Cluster Darwin: <http://www.hpc.cam.ac.uk/services/darwin> fastNLOHomepage of the fastNLO project: <http://fastnlo.hepforge.org> Ball:2014uwaR. D. Ball et al. [NNPDF Collaboration],JHEP 1504, 040 (2015) [arXiv:1410.8849 [hep-ph]].Dulat:2015mcaS. Dulat et al.,Phys. Rev. D 93, no. 3, 033006 (2016) [arXiv:1506.07443 [hep-ph]].Harland-Lang:2014zoaL. A. Harland-Lang, A. D. Martin, P. Motylinski and R. S. Thorne,Eur. Phys. J. C 75, no. 5, 204 (2015) [arXiv:1412.3989 [hep-ph]].Buckley:2014anaA. Buckley, J. Ferrando, S. Lloyd, K. Nordstrom, B. Page, M. Ruefenacht, M. Schoenherr and G. Watt,Eur. Phys. J. C 75, 132 (2015) [arXiv:1412.7420 [hep-ph]]. | http://arxiv.org/abs/1704.08551v1 | {
"authors": [
"Michal Czakon",
"David Heymes",
"Alexander Mitov"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170427131831",
"title": "fastNLO tables for NNLO top-quark pair differential distributions"
} |
-1.0cm -0.8cm -0.8cmemptyA symmetry for ϵ_K Leandro Da Rold, Iván A. DavidovichCentro Atómico Bariloche, Instituto Balseiro and CONICET Av. Bustillo 9500, 8400, S. C. de Bariloche, Argentina Abstract We show a symmetry that, in the context of a composite Higgs with anarchic flavor, can suppress the dominant CP violating contributions to K-K̅ mixing. Based on previous extensions of SU(3)_c, we consider the case in which the composite sector has a global SU(6) symmetry, spontaneously broken to a subgroup containing SU(3)×SU(3). We show that the interactions with the Standard Model can spontaneously break the remaining symmetry to the diagonal subgroup, identified with the group of color interactions, and naturally suppress ϵ_K. We consider this scenario in the context of the Minimal Composite Higgs Model based on SO(5)/SO(4) for the electroweak sector. By working in the framework of 2-site models, we compute the scalar potential, determine the conditions for a successful breaking of the symmetries and calculate the spectrum of lightest states. We find that ϵ_K can be suppressed and the top mass reproduced for a large region of the parameter space where the symmetries are dynamically broken. Besides other new resonances, the model predicts the presence of a new singlet scalar state, generally lighter than the Higgs, that could have evaded detection at colliders. E-mails: [email protected], [email protected]§ INTRODUCTIONThe presence of a new strongly coupled field theory (SCFT) containing a Higgs-like resonance can provide a solution to the electroweak-Planck hierarchy problem as well as a rationale for the flavor hierarchies. One of the most attractive possibilities corresponds to the Higgs being a pseudo Nambu Goldstone Boson (pNGB) arising from the strong dynamics. Besides the Higgs, the SCFT provides new resonances at a scale of few TeV that cut off the momentum integrals in the Higgs potential. Generically, the presence of these states induce corrections to the electroweak (EW) observables, that are in conflict with EW precision tests. Many of the constraints can be alleviated by the use of suitable symmetries <cit.>.A key ingredient to obtain flavor hierarchies in composite Higgs models is partial compositeness <cit.>. In partial compositeness hierarchies in the fermion masses can be easily obtained by assuming that the degree of compositeness of the fermions are hierarchical, without hierarchies or relations in the flavor structure of the SCFT. This assumption leads to non-universal couplings between the SM fermions and the resonances of the SCFT, such that virtual exchange of resonances induces flavor violating effects. Despite this violation, partial compositeness contains a built-in GIM-like mechanism suppressing flavor changing neutral currents (FCNC) <cit.>. This setup is known as the anarchic approach to the flavor problem.Although the GIM-like mechanism successfully suppresses most of the contributions to FCNC, the very stringent constraints from Kaon physics, in particular ϵ_K, push the scale of the composite resonances to m_cp∼ 10÷ 30 TeV, worsening the fine-tuning problem in the EW sector <cit.>. Some alternatives to avoid these constraints have been considered in the literature, such as the presence of flavor symmetries <cit.> and hierarchies in the SCFT <cit.>, as well as the presence of different dynamical scales <cit.>. These scenarios relax the constraints allowing m_cp to be lowered to a few TeV. In the context of anarchic flavor, the authors of Ref. <cit.> have shown that enhancing the global symmetry of the SCFT from SU(3)_c to SU(3)_L×SU(3)_R, where the indices can be associated with the chiralities of the quarks, the scale can be lowered to m_cp∼ 2 TeV. In that scenario, masses for the SM quarks require an extension of the scalar sector of the theory, with new scalar fields charged under the extended symmetry. The masses of these states are expected to be of the same order as m_cp, introducing a sizable breaking of the enlarged symmetry and spoiling the mechanism protecting ϵ_K, unless a fine-tuning of several orders of magnitude is introduced <cit.>.In the present paper we will elaborate on the possibility of protecting ϵ_K from large corrections by considering an extension of SU(3)_c, within the context of an anarchic SCFT. We will work along the lines of Ref. <cit.>, but making the new scalar a pNGB, based on a global symmetry SU(6) spontaneously broken by the strong dynamics to a subgroup containing SU(3)_L×SU(3)_R. Besides this extension of the color symmetry group, we add the usual SO(5)×U(1)_X global symmetry that allows us to describe also the Higgs field as a pNGB. The interactions between the SCFT and the SM gauge and fermion fields generate a potential for both scalars at one-loop. The potential can trigger electroweak symmetry breaking (EWSB) and it can also spontaneously break SU(3)_L×SU(3)_R to the diagonal subgroup that is aligned with SU(3)_c. The size of the spontaneous breaking can be described in terms of the parameters ϵ_5 and ϵ_6, that take values 0≤ϵ_5,6≤ 1, and measure the misalignment of the vacuum in the direction of SO(5) and SU(6), respectively. As in the usual Minimal Composite Higgs Model (MCHM): v_SM=ϵ_5 f_5, with f_5 the decay constant of the pNGB Higgs <cit.>. After spontaneous breaking of both symmetries fermion masses are successfully generated, leading to a realistic model. Quark masses, requiring breaking of both symmetries, are proportional to ϵ_5ϵ_6. The dominant effect in ϵ_K is given by a contribution to the Wilson coefficient C_4 that, compared with the case without extended symmetry, is suppressed by ϵ_6^2(1-ϵ_6^2). In general, factors of ϵ_6√(1-ϵ_6^2) are expected in operators with LR chiral structure, as for example magnetic dipole moments.The values of ϵ_5 and ϵ_6 depend on the parameters of the theory, as well as on the representations of the fermions under SO(5) and SU(6). The representations of SU(6) are fixed to cancel the dominant contributions to ϵ_K, according to <cit.>. We discuss the embedding into SO(5) representations, and show that a realistic case is obtained for q_L and u_R embedded in the representations 14 and 1 of SO(5) respectively. With this choice the potential can easily trigger spontaneous breaking of both symmetries, generating masses for the SM fields. We use a very simple description of the above dynamics by considering a 2-site model; that is, a four dimensional theory containing the first level of resonances of the SCFT <cit.>. Within this framework we are able to calculate the one-loop Coleman-Weinberg potential that is dominated by the top and gauge contributions. We compute ϵ_5 and ϵ_6, as well as the masses and mixing angle of the light scalars: the Higgs and the new neutral state. We find that, taking f≃ 1 TeV, the Higgs is usually somewhat heavier than 125 GeV, as expected in models with fermions in the 14. The other scalar is usually lighter, with a mass of order 50÷ 100 GeV due to a suppressed quartic coupling, whose size is determined by the embedding of the quarks into the SU(6) symmetry. The paper is organized as follows: In sec. <ref> we describe the symmetries of the theory assuming that the Higgs and the new scalar are pNGBs of a SCFT; we show the effective theory that is obtained after the heavy resonances of the SCFT are integrated out, codifying their effects in terms of a set of form factors. In sec. <ref> we compute the one-loop potential by using the form factors of the effective theory, we analyse the conditions for spontaneous symmetry breaking and the spectrum of light scalar states; we also discuss the dependence of the potential with the SO(5) embedding of the fermions. In sec. <ref> we describe a 2-site model that provides a calculable framework for the above dynamics and we compute the contributions to some flavor observables. In sec. <ref> we show numerical results for several interesting physical quantities, namely the spectrum of light states and the regions of the parameter space with spontaneous breaking of the symmetries. In sec. <ref> we discuss some important issues, such as the neutron dipole moments, the degree of tuning of the model and the phenomenology of the new states at the LHC. § A SYMMETRY FOR Ε_KWe consider a new strongly interacting sector beyond the SM. It has a global symmetry SO(5)×U(1)_X×SU(6). [As is well known, the extra U(1)_X factor is required to obtain the proper normalization of hypercharge in the fermionic sector.] The interactions produce bound states with masses of order TeV and spontaneously break the global symmetry to SO(4)×U(1)_X×SU(3)_L×SU(3)_R×U(1). A set of NGBs Π_5 and Π_6 transforming non-linearly under the global SO(5) and SU(6), respectively, emerge from this breaking. The field Π_5 has the proper quantum numbers to be identified with the Higgs multiplet. The field Π_6 results in a new scalar field, with some components charged under the color interactions of the SM. The SM fermions and gauge bosons are elementary fields, and can be thought of as external sources probing the strong dynamics. Depending on the orientation of the vacuum compared to the SM gauge symmetry SU(2)_L×U(1)_Y×SU(3)_c, the EW and color symmetries can be broken or not. This direction is determined dynamically by the potential generated at loop level from the interactions between the SM fields and the strongly interacting sector that explicitly violate the global symmetry. The presence of this potential turns Π_5 and Π_6 into pNGBs.The interactions between the SM gauge fields and the strongly interacting sector are introduced by the gauging of a subgroup of the global symmetry; in this way the elementary gauge fields are coupled to a subset of the conserved currents of the composite sector. The elementary fermions are coupled linearly with the strongly interacting sector, realizing partial compositeness.To shorten the notation we define G_6≡SU(6) and G_5≡SO(5)×U(1)_X, and for the subgroups H_6≡SU(3)_L×SU(3)_R×U(1) and H_5≡SO(4)×U(1)_X.A possible orientation of the vacuum in the direction of G_6 can be described by the following matrix:Ω_0=([I_30;0 -I_3 ])where I_3 is the identity matrix in three dimensions. Under G_6 this vacuum transforms as: g_6Ω_0g_6^†, with g_6 an element of G_6 in the fundamental representation. This vacuum preserves an H_6 subgroup. We identify SU(3)_c with the diagonal subgroup SU(3)_V contained in H_6. Therefore, expanding around this vacuum one can classify the fields according to their representation properties under the SM group of strong interactions. The NGBs Π_6 are given by a local element of G_6 of the form:U_6=e^i√(2)Π_6/f_6 ,Π_6=Π_6^b̂T_6^b̂ ,with T_6^b̂ the broken generators of G_6. We will use the subindex 6 for elements related with this group, to distinguish them from the elements associated to G_5. Only elements of the form U_6 transform the vacuum Ω_0. U_6 transforms non-linearly under G_6, as: g_6 U_6 h_6^†(g_6,Π_6), with h_6∈H_6. Π_6 transforms linearly under H_6, as a complex (3̅,3)_1/√(3). We find it useful to define the matrix Φ_6=Π_6|_ fund, where the subindex indicates that the generators T_6^b̂ are in the fundamental representation. For the suitable basis defined in Ap. <ref>, Φ_6 can be written as:Φ_6=([0 Φ_(3̅,3); Φ_(3̅,3)^†0 ])with Φ_(3̅,3) a 3× 3 complex matrix with eighteen real degrees of freedom, describing the NGB fields.A representative orientation of the vacuum in the direction of G_5 can be parametrized by the vector <cit.>Σ_0=(0,0,0,0,1).The vacuum Σ_0 transforms as: g_5Σ_0, with g_5 an element of SO(5) in the fundamental representation, Σ_0 is invariant under an H_5 subgroup. We identify the generators of the EW symmetry of the SM with a subset of the generators of H_5. Expanding around this vacuum all the fields can be classified according to their representations under the EW gauge symmetry. The NGB of G_5/H_5 are given by:U_5=e^i√(2)Π_5/f_5 ,Π_5=Π_5^âT_5^âwith T_5^â the broken generators of G_5. In this case we use the subindex 5 for elements related with G_5. Only elements of the form U_5 transform the vacuum. U_5 transforms non-linearly under G_5, as: g_5 U_5 h_5^†(g_5,Π_5), with h_5∈H_5, whereas Π_5 transforms with the fundamental representation (2,2)_0 of H_5. Similar to the case of Φ_6, we define the matrix Φ_5=Π_5|_ fund, with the generators T_5^â in the fundamental representation.As we will show in sec. <ref>, virtual exchange of elementary fields can misalign the vacuum at loop level, breaking H_5 to K_5≡SO(3) and H_6 to K_6≡SU(3)_V. Under K_5 the NGB Π_5 decomposes as: (2,2)∼ 1⊕ 3, whereas under K_6 the NGB Π_6 decomposes as: (3̅,3)∼ 1⊕ 8, with the singlet and octet being complex fields. The singlets of Π_5 and Π_6 under K_5 and K_6 will be called h_5 and h_6 e^iθ_6, respectively, and we define:s_5= sin(h_5/f_5),s_6 = sin(h_6/√(6)f_6),ϵ_5=sin(v_5/f_5),ϵ_6 =sin(v_6/√(6)f_6),with v_5=⟨ h_5⟩ and v_6=⟨ h_6⟩ the vacuum expectation values of the singlets. By using these definitions the vacuum can be characterized by the following objects:Σ_v=(0,0,0,ϵ_5,√(1-ϵ_5^2)), Ω_v=([(1-2ϵ_6^2) I_3 i2ϵ_6√(1-ϵ_6^2) I_3; i2ϵ_6√(1-ϵ_6^2) I_3 -(1-2ϵ_6^2) I_3 ]).Thus ϵ_5 and ϵ_6 measure the misalignment of the vacuum, with ϵ_5=ϵ_6=0 in the case of neither EWSB nor H_6 breaking, and ϵ_5=ϵ_6=1 in the case of maximal symmetry breaking of both groups. §.§ Effective theoryIntegrating out the heavy degrees of freedom of the strongly interacting sector one can obtain an effective theory for the elementary degrees of freedom and the NGBs. We find it useful to add spurious degrees of freedom to the elementary sector to extend its symmetry to G_5×G_6. Therefore one can write an effective Lagrangian that is formally invariant under this extended symmetry. In this procedure one has to choose the representations of the elementary fermions. We choose the fundamental representation of G_6 for the quarks, which under H_6 decomposes as: 6∼ (3,1)_1/2√(3)⊕(1,3)_-1/2√(3). We choose (3,1) for q_L and (1,3) for q_R to suppress the main contribution to ϵ_K <cit.>. The representations of G_5 are not fixed, the only constraint being that they have to contain the proper doublets and singlets of SU(2)_L to match with the SM fields. As is well known, the choice of the representations of G_5 has important consequences for the phenomenology; we will discuss their impact in the present model in sec. <ref>.As usual, due to the non-linear transformation properties of the NGB, a G_5×G_6 invariant Lagrangian can be obtained by dressing the elementary fields with the NGB matrices U_5 and U_6, and building with them a Lagrangian that superficially looks invariant under H_5×H_6. Calling R a representation of G_5, R decomposes under H_5 as R∼⊕_j r_j. Similarly, calling S a representation of G_6, S decomposes under H_6 as S∼⊕_k s_k. Therefore an elementary fermion ψ in the representation R of G_5 and S of G_6, decomposes under H_5×H_6 as: ψ∼⊕_r,sψ_rs. Dressing ψ with the NGB matrices usually leads to fields that transform with reducible representations, therefore the building blocks for the effective theory are the projections of the dressed fields into the irreducible representations of the unbroken subgroup: ψ̃_rs=(U_5^† U_6^†ψ)_rs, where the generators contained in U_5 are in the representation R of G_5 and the generators contained in U_6 are in the fundamental representation of G_6, as discussed previously. G_5×G_6 invariant operators can be obtained by considering products of factors ψ̃_rs, and selecting from these products the invariants of H_5×H_6. Using these objects, at the quadratic level in the elementary fields, the effective Lagrangian for the quarks is:L_ eff⊃ ∑_f=q,u,dZ_fψ̅_f ψ_f+ ∑_f=q,u,d∑_r,s(U_5^† U_6^†ψ_f)_rsΠ^f_rs(p^2) (U_5^† U_6^†ψ_f)_rs+ ∑_f=u,d∑_r,s(U_5^† U_6^†ψ_q)_rs M^f_rs(p^2) (U_5^† U_6^†ψ_f)_rs +h.c. ,where Z_f are the factors of the elementary kinetic terms. Π^f_rs(p^2) and M^f_rs(p^2) are momentum dependent form factors that contain the information on the composite degrees of freedom; they are independent of the NGB fields. The components of the different factors of the dressed fermions ψ̃_rs in Eq. (<ref>) are contracted in the usual way to obtain H_5×H_6 invariants. Generation indices are understood in Eq. (<ref>), and in equations below.For the gauge sector, it is useful to notice that the adjoint representation of SO(5) decomposes under SO(4) as: 10∼⊕_i t_i = (3,1)⊕(1,3)⊕(2,2), whereas the adjoint of G_6 decomposes under H_6 as: 35∼⊕_j u _j = (8,1)_0⊕(1,8)_0⊕(3̅,3)_1/√(3)⊕(1,1)_0. Calling x_μ, a_μ and g_μ the fields of U(1)_x, SO(5) and G_6, respectively, the quadratic effective Lagrangian for the elementary gauge fields is:L_ eff⊃1/2P^μν[Π^X(p^2)x_μ x_ν + ∑_tΠ^A_t(p^2)(U_5^† a_μ)_t (U_5^† a_ν)_t + ∑_uΠ^G_u(p^2)(U_6^† g_μ)_u (U_6^† g_ν)_u ] ,with P_μν=η_μν-p_μ p_ν/p^2, U_5 and U_6 in the adjoint representation, and t and u running over the decompositions of each adjoint under the subgroups as presented in the previous paragraph. There are also elementary kinetic terms for the gauge fields that were not written in Eq. (<ref>). The form factors Π(p^2) encode the effects of the strongly coupled sector and are independent of the NGB fields. There are also kinetic terms for the NGB fields that can be constructed, as usual, by making use of the Maurer-Cartan form d_μ defined from: iU^† D_μ U=e_μ^aT^a+d_μâT^â≡ e_μ+d_μ. There is a d_μ^5 for the NGBs Π_5 and a d_μ^6 for the NGBs Π_6. The NGBs kinetic terms are given by:L_ eff⊃f_5^2/4d^â_5,μ d_5^â,μ + f_6^2/4d^b̂_6,μ d_6^b̂,μ .Keeping only the dynamical fermions: q_L=(u_L,d_L) contained in ψ_q, as well as u_R and d_R contained in ψ_u and ψ_d, Eq. (<ref>) leads to:L_ eff⊃q̅_LΠ_q(Π_5,Π_6,p^2) q_L+u̅_R Π_u(Π_5,Π_6,p^2) u_R+d̅_R Π_d(Π_5,Π_6,p^2) d_R+q̅_LM_u(Π_5,Π_6,p^2)u_R+q̅_LM_d(Π_5,Π_6,p^2)d_R +h.c. .Since Π(Π_5,Π_6,p^2) and M(Π_5,Π_6,p^2) depend on momentum and on the NGBs, they have non-trivial indices under SU(3)_c×SU(2)_L. Using the form factors defined in Eq. (<ref>), they can be written as:Π_f(Π_5,Π_6,p^2)=Z_f+∑_r,sΠ^f_rs(p^2)F^f_rs(Π_5,Π_6), f=q,u,d ,M_f(Π_5,Π_6,p^2)=∑_r,sM^f_rs(p^2)G^f_rs(Π_5,Π_6),f=u,d,where all the momentum dependence is codified in the form factors defined in Eq. (<ref>), and the NGB dependence is contained in the functions F^f_rs and G^f_rs. These functions can be obtained from the invariants built with the dressed quarks:[F^q_rs(Π_5,Π_6)]_αβ ij=∂_q̅_L^α i∂_q_L^β j[(U_5^† U_6^†ψ_q)_rs (U_5^† U_6^†ψ_q)_rs], [F^f_rs(Π_5,Π_6)]_αβ=∂_f̅_R^α∂_f_R^β[(U_5^† U_6^†ψ_f)_rs (U_5^† U_6^†ψ_f)_rs],f=u,d, [G^f_rs(Π_5,Π_6)]_α ij=∂_q̅_L^α i∂_f_R^β[(U_5^† U_6^†ψ_q)_rs (U_5^† U_6^†ψ_f)_rs],f=u,d,where α,β=1,2,3 are color indices, and i,j=1,2 are SU(2)_L indices. It is straightforward to show that these functions factorize in one factor containing the Π_5-dependence and another factor containing the Π_6-dependence: F^f_rs(Π_5,Π_6)=F^f_r(Π_5)F^f_s(Π_6) and similarly for G.The dynamical gauge fields of QCD are associated to the generators of the diagonal subgroup T_6^V,a≡ T_6^L,a+T_6^R,a, for a=1,… 8. In this case Eq. (<ref>) leads to:L_ eff⊃1/2P^μνg^V,a_μΠ_G^ab(Π_6,p^2)g^V,b_ν ,where Π_G(Π_6,p^2) is given by:Π_G(Π_6,p^2)=Z_6+∑_sΠ^G_s(p^2)F^G_s(Π_6),with Z_6=g_6,0^-2 being the coefficient of the elementary kinetic term, and[F^G_s(Π_6)]_ab=∂_g^V,a∂_g^V,b[(U_6^† g^V)_s (U_6^† g^V)_s]. A similar description can be done for the EW sector <cit.>.In order to obtain the effective Lagrangian at quadratic level, it is useful to evaluate the scalar fields to their vev in Eq.(<ref>). In this case the functions F^f_rs of Eq. (<ref>) are proportional to the identity in color space [Π_q splits into four 3× 3 blocks, with those in the diagonal proportional to the identity and the others vanishing.], thus they are color independent, as are the form factors. The functions G^f_rs of Eq. (<ref>) split into two 3× 3 blocks, one proportional to the identity and the other vanishing. Moreover, the NGB matrices U_5 and U_6 can be resummed, leading to matrices that can be expressed as functions of ϵ_5 and ϵ_6. Taking into account the representations of the quarks under G_6 and H_6, and for generic representations of G_5 and H_5, the form factors can be written as:Π_f_L(p^2)=Z_q+∑_rΠ^q_r31(p^2) F^f_L_r(ϵ_5)+ϵ_6^2∑_r[Π^q_r13(p^2)-Π^q_r31(p^2)] F^f_L_r(ϵ_5), Π_f_R(p^2)=Z_f+∑_rΠ^f_r13(p^2) F^f_R_r(ϵ_5)+ϵ_6^2∑_r[Π^f_r31(p^2)-Π^f_r13(p^2)] F^f_R_r(ϵ_5), Π_f_Lf_R(p^2)=-iϵ_6√(1-ϵ_6^2)∑_r[M^f_r31(p^2)-M^f_r13(p^2)] G^f_Lf_R_r(ϵ_5),f=u,d ,where 31 and 13 label the H_6 representations (3,1) and (1,3), respectively, omitting the parenthesis to simplify the notation. The functions F_r(s_5) can be obtained from Eqs. (<ref>) and (<ref>), and have been computed for several representations in the literature, see for example Refs. <cit.> and Ap. <ref>.As usual, the spectrum of fermions that mix with the SM ones corresponds to the zeroes of the following function:Zeroes[Π_f_L(p^2)Π_f_R(p^2)-|Π_f_Lf_R(p^2)|^2].The zero of each function that is closest to the origin gives the mass of the corresponding SM fermion. §.§ Partial compositeness and flavorBesides the group theoretical structure of the correlators, another useful piece of information arises from the hypothesis of partial compositeness <cit.>. We define ϵ_f as the degree of compositeness of the fermions, with ϵ_f≪ 1 for mostly elementary fermions and ϵ_f∼ 1 for partially composite fermions <cit.>. Thus, to leading order in ϵ_f: Π^f∝ϵ_f^2 and M^f∝ϵ_qϵ_f. There are two sources of flavor structure: the structure of the SCFT and the structure of partial compositeness contained in ϵ_f. Flavor anarchy corresponds to the assumption of anarchic flavor structure of the SCFT, meaning that there are no hierarchies in the couplings and masses of this sector. As is well known, in this case the hierarchies between the quark masses can be obtained by hierarchical degrees of compositeness ϵ_f. In addition, this mechanism can also lead to a realistic V_CKM. The hierarchy in ϵ_f can be naturally obtained in five dimensional models <cit.> and in cases where the SCFT has an approximate conformal symmetry at high energies <cit.>. We will show more details in sec. <ref>.A useful approximate expression for the masses of the SM fermions can be derived from the effective Lagrangian <cit.>. By canonically normalizing the kinetic term of the fermions and evaluating the scalars to their vev, we obtain:m_f≃|Π_f_Lf_R(0)|/[Π_f_L(0)Π_f_R(0)-2|Π_f_Lf_R(0)|.d |Π_f_Lf_R(p^2)|/d p^2|_p^2=0]^1/2 ,where Π_f_L, Π_f_R and Π_f_Lf_R have been defined in Eq. (<ref>). Notice that, as expected, besides EWSB fermion masses require breaking of H_6. Also, for maximal breaking of H_6: ϵ_6=1, the fermion masses vanish. Partial compositeness shows us that, to leading order in the mixing, the masses are of orderm_f∼ϵ_6√(1-ϵ_6^2)G^f_Lf_R_r(ϵ_5) ϵ_qm_r ϵ_f,with m_r the mass scale of the fermionic resonances of the SCFT. All the group structure is contained in the first three factors, whereas the flavor structure is contained in ϵ_qϵ_f m_r. Comparing with the usual result in composite Higgs models, there is an extra factor ϵ_6√(1-ϵ_6^2) that arises from the extended symmetry G_6.The Yukawa couplings with the scalars h_5 and h_6 can be computed by taking the derivative of m_f, as approximated in Eq. (<ref>), with respect to v_5 and v_6, respectively:y_f,5≃∂ m_f/∂ v_5 , y_f,6≃∂ m_f/∂ v_6 .The dependence of the Yukawa couplings on the parameters of the theory is encoded in the form factors, but the dependence on the vev's of the scalar fields is simple, it is encoded in ϵ_6 and in the functions F_r(ϵ_5) and G_r(ϵ_5) of Eq. (<ref>). These functions are straightforward to compute once the embedding of the fermions on G_5 is chosen, see Ap. <ref>. A good estimate of the size of the Yukawa couplings can be obtained by expanding Eq. (<ref>) to leading order in powers of 1/Z_f, similar to an expansion in powers of the fermion mixing ϵ_f:y_f,5≃ m_f √(1-ϵ_5^2)/f_5 d/dϵ_5log∑_r[M^f_r31(0)-M^f_r13(0)]G_r^f_Lf_R(ϵ_5), y_f,6≃ m_f1/√(6)f_61-2ϵ_6^2/ϵ_6√(1-ϵ_6^2) .A non-vanishing y_f,5 requires 0<ϵ_6<1. In the cases with just one non-trivial invariant function G_r, the dependence of the derivative on the form factors cancels out, and y_f,5 depends on the microscopic parameters of the theory only through ϵ_5 and the combination m_f/f_5. The predictions for y_f,5 in the case of representations with custodial symmetry of Zb_Lb̅_L have been reported in Refs. <cit.>. On the other hand, a non-vanishing y_f,6 requires ϵ_5>0. Also, y_f,6 adopts a very simple form in this expansion, since all the dependence on the microscopic parameters of the theory has been absorbed in m_f/f_6, and the dependence on ϵ_6 is shown explicitly. This dependence is determined by the embedding of the quarks in the extended symmetry G_6.We are also interested in contributions to the Wilson coefficients of four-fermion operators, in particular to the operator O_4=(ψ̅_qψ_d)(ψ̅_dψ_q).However, instead of keeping track of the full NGB dependence, we are interested only in the coefficient evaluated in the vacuum of Eq. (<ref>), and keeping the dynamical fermions only. We obtain: L_ eff⊃ϵ_6^2 (1-ϵ_6^2)Π_f_Lf_R^(4)(p^2) (f̅_L f_R)(f̅_R f_L),where Π_f_Lf_R^(4)(p^2) is a form factor independent of ϵ_5 and ϵ_6, and where flavor indices are understood. The Wilson coefficient C_4 isC_4=ϵ_6^2 (1 - ϵ_6^2)Π_f_Lf_R^(4)(0).Partial compositeness implies that Π^(4)_f_Lf_R, and consequently also C_4, is proportional to (ϵ_f_Lϵ_f_R)^2. Eq. (<ref>) must be compared with the usual result in composite Higgs models, without the extended symmetry G_6, where the factor ϵ_6^2 (1 - ϵ_6^2) is absent. In that case the bounds on the scale of the masses of the composite resonances arising from flavor physics are m_cp≳ 10÷ 30TeV <cit.>. In the present case, the extended symmetry of the SCFT gives an extra suppression factor to the contribution to C_4, that can alleviate this bound. Up to corrections of numerical factors of O(1), that depend on the specific realization of the SCFT, for ϵ_6√(1 - ϵ_6^2)≃ 0.1÷ 0.3, the scale can be as low as m_cp∼ 3 TeV. Moreover, as we will show in sec. <ref>, in 2-site models there is an extra factor 1/2 that relaxes ϵ_6√(1 - ϵ_6^2)≲ 0.2÷ 0.5 for m_cp∼ 3 TeV. Notice that not only a small ϵ_6 can suppress the contribution to C_4, also for large ϵ_6, near one, there can be a supression. The same factor involved in C_4: ϵ_6√(1 - ϵ_6^2), suppresses the fermion masses, see Eq. (<ref>). Thus there is a tension between the large top mass and a small C_4. We will elaborate more on this topic in sec. <ref>. § RADIATIVE POTENTIALThe interactions between the fermions of the SM and the strongly interacting sector explicitly break G_5×G_6 to the subgroup SU(2)_L×U(1)_Y×H_6, generating a potential for the NGB fields at the one-loop level. This potential can misalign the vacuum and induce EW and H_6 spontaneous breaking. The interactions between the color gauge fields of the SM and the SCFT explicitly break G_6 to SU(3)_V, whereas the interactions with the EW gauge fields break G_5 to SU(2)_L×U(1)_Y. In general the interactions with the gauge fields do not induce misalignment of the vacuum. Below we discuss the contributions of the fermions, as well as the contributions of the gauge sector with color. The contributions of the EW gauge fields have been discussed extensively; we refer the reader to the original articles on SO(5)/SO(4) <cit.>.Let us start with the fermion contribution. We will consider only the effect of q_L and u_R of the third generation, since, having the largest mixing, they give the dominant contribution. We obtain:V_f(Π_5,Π_6)=-2∫d^4p/(2π)^4log( [ Π_q M_u; M_u^†Π_ u ]),where, as shown in Eqs. (<ref>-<ref>), the SU(3)_c and SU(2)_L indices of Π_q, Π_u and M_u, allow us to express them as 6× 6, 3× 3 and 6× 3 matrices, respectively.Since the SM fermions are in full representations of SU(3)_L×SU(3)_R×SU(2)_L×U(1)_Y, the one-loop potential can spontaneously break this symmetry. Expanding to fourth-order in powers of the NGB fields Π_5 and Π_6, the most general potential is:V_f(Π_5,Π_6)=1/2m_f, 5^2Π_5^2+1/2m_f, 6^2Π_6^2+λ_f, 5Π_5^4+λ_f, 6(Π_6^4)+λ'_f, 6(Π_6^4)'+λ_f, 56Π_5^2Π_6^2 + … ,where the ellipsis stands for higher order terms. The indices of the NGB fields in the different products are contracted to form all the invariants allowed by the group structure, namely:Π_5^2=[Φ_5^2]=∑_a(Π_5^â)^2 ,Π_5^4=^2[Φ_5^2]=[∑_a(Π_5^â)^2]^2 , Π_6^2=[Φ_6Φ_6^†]=1/2∑_b(Π_6^b̂)^2 , (Π_6^4)=[Φ_6Φ_6^†Φ_6Φ_6^†] , (Π_6^4)'=^2[Φ_6Φ_6^†] .The quadratic and quartic coefficients can be expressed as integrals of the correlators of the effective theory. They depend on the representation chosen for the fermions.For negative m^2_f,5 and m^2_f,6 and suitable quartic couplings, this potential can trigger spontaneous symmetry breaking of H_5 and H_6.Since the gauge interactions explicitly break G_6 to SU(3)_V, it is necessary to decompose the representation of Π_6 under this subgroup. Given that (3̅,3)∼ 8+8+1+1, we define: Φ_(3̅,3)∼ϕ_8+I_3ϕ_1, where ϕ_8 and ϕ_1 are complex fields, and ϕ_1 can be identified with h_6e^iθ_6. In terms of these fields, the potential generated by virtual exchange of elementary gluons at one-loop is:V_G(Π_6)=∫d^4p/(2π)^4logΠ_G,where Π_G is an 8× 8 matrix, as defined in Eq. (<ref>).Expanding to fourth order in Π_6:V_G(Π_6)= m_g,1^2ϕ_1ϕ_1^†+m_g,8^2(ϕ_8ϕ_8^†)+λ_g,1(ϕ_8ϕ_8^†ϕ_8ϕ_8^†)+λ_g,2^2(ϕ_8ϕ_8^†)+λ_g,3(ϕ_8^†ϕ_8^†)(ϕ_8ϕ_8)+ λ_g,4(ϕ_8ϕ_8^†)ϕ_1ϕ_1^†+λ_g,5(ϕ_8^†ϕ_8^†)(ϕ_1)^2+λ_g,6(ϕ_8ϕ_8)(ϕ_1^†)^2+λ_g,7(ϕ_1ϕ_1^†)^2 + … ,where the ellipsis stands for higher order terms. Matching the quadratic and quartic coefficients with the expansion of Eq. (<ref>) we obtain:m_g,1^2=0, λ_g,7=0, m_g,8^2=3/f_6^2∫_p2Π^G_33-Π^G_81-Π^G_18/2Z_6+Π^G_81+Π^G_18 , λ_g,1=-3/2f_6^4∫_p1/(2Z_6+Π^G_81+Π^G_18)^2[(Π^G_18)^2+3(Π^G_33)^2+(Π^G_81)^2+(Π^G_33-2Π^G_11)(Π^G_18+Π^G_81).6.5cm . -Π^G_18Π^G_81+2Z_6(Π^G_81+Π^G_18+2Π^G_11-4Π^G_33)], λ_g,2=-1/2f_6^4∫_p1/(2Z_6+Π^G_81+Π^G_18)^2[(21Π^G_33-2Π^G_11)(Π^G_81+Π^G_18)-33(Π^G_33)^2-5Π^G_18Π^G_81.6.5cm . +2Z_6(7Π^G_81+7Π^G_18-2Π^G_11-12Π^G_33)], λ_g,3=-1/4f_6^4∫_p1/(2Z_6+Π^G_81+Π^G_18)^2[6(Π^G_18)^2+6(Π^G_33)^2+6(Π^G_81)^2+(2Π^G_11-33Π^G_33)(Π^G_18+Π^G_81).6.5cm . +17Π^G_18Π^G_81+2Z_6(Π^G_81+Π^G_18-2Π^G_11)], λ_g,4=-2λ_g,5=-2λ_g,6=6/f_6^4∫_pΠ^G_81+Π^G_18-2Π^G_33/2Z_6+Π^G_81+Π^G_18 .Similar to the case of the fermions, we have shortened the notation of the representations by omitting the parenthesis. A few comments are in order. As expected, there is no contribution to the quadratic and quartic terms involving only the singlet. Besides, Eqs. (<ref>) and (<ref>) are independent of θ_6, which at this order remains as a NGB. In the specific realization of a two-site model (as well as simple extra dimensional models) m_g,8^2≥ 0, and the gauge potential does not misalign the vacuum. For m_g,8^2>|m_f,6|^2, only ϕ_1 has a negative mass term and can spontaneously break H_6 to SU(3)_V. §.§ Symmetry breakingWe analyse in this section the potential for the singlets of SU(2)_V and SU(3)_V: h_5 and h_6. For small s_5 and s_6 the potential can be expanded in powers of these variables; to fourth order it results in:V(s_5,s_6)≃α_5 s_5^2+α_6 s_6^2+β_5 s_5^4+β_6 s_6^4+β_56 s_5^2s_6^2,where the coefficients α and β scale as f^4, and are given by integrals of the correlators. As we will show in sec. <ref>, in general β_6 is suppressed compared with the other quartic couplings. This suppression will have deep implications for model building, as well as for the phenomenology of the scalars.Minimizing for a non-trivial vacuum we obtain:ϵ_5=(α_6β_56-2α_5β_6/4β_5β_6-β_56^2)^1/2 , ϵ_6=(α_5β_56-2α_6β_5/4β_5β_6-β_56^2)^1/2 .For β_56≫β_5,β_6 the minimum is not stable and in general one of the symmetries is not spontaneously broken. As we discuss in the next subsection, this behavior has important consequences for model building. §.§ Spectrum of neutral scalar statesFrom Eqs. (<ref>) and (<ref>), trading α_5 and α_6 for ϵ_5 and ϵ_6, one can obtain the mass matrix of the excitations of h_5 and h_6. In the basis (h_5,h_6) the squared mass matrix is:M^2=([8β_5/f_5^2(1 - ϵ_5)^2 ϵ_5^2 4/√(6)β_56/f_5f_6ϵ_5^2√(1 - ϵ_5^2) ϵ_6^2√(1 - ϵ_6^2); 4/√(6)β_56/f_5f_6ϵ_5^2√(1 - ϵ_5^2) ϵ_6^2√(1 - ϵ_6^2)4/3β_6/f_6^2(1 - ϵ_6)^2 ϵ_6^2 ]).There are different interesting limits to study: * ϵ_5^2≪ 1: the masses and mixing angles of the physical states are given by m_1^2≃ 2ϵ^2_5/f_5^2(4β_5-β_56^2/β_6),m_2^2≃4/3ϵ^2_6(1-ϵ^2_6)/f_6^2β_6 ,θ≃√(3/2(1-ϵ^2_6))ϵ_5 f_6/ϵ_6 f_5β_56/β_6 .In this case m_1 and the mixing are suppressed by powers of the small parameter ϵ_5. The usual Higgs can be approximately identified with h_5, and there is a new scalar state which is expected to be rather light also, since its mass is suppressed by ϵ^2_6(1-ϵ^2_6) and by the smaller quartic β_6.* β_6≪β_5,β_56: this case is particularly relevant because it is realized in some interesting models, according to the discussion of sec. <ref>. The masses and mixing angles of the physical states are given bym_1^2≃ 4β_5ϵ^2_5(1-ϵ^2_5)/f_5^2(1+√(1+c)),m_2^2≃ 4β_5ϵ^2_5(1-ϵ^2_5)/f_5^2(1-√(1+c))+2/3β_6ϵ^2_6(1-ϵ^2_6)/f_6^21+√(1+c)/√(1+c) , θ≃1/√(2)√(c)/√(1+c)-√(1+c) ,c=1/6 ϵ^2_6(1-ϵ^2_6)β_56^2/ϵ^2_5(1-ϵ^2_5)β_5^2 f_5^2/f_6^2 .It is worth analysing this result in some detail. Since the first term of m_2^2 is negative, the second term hast to be greater that the first one to obtain a stable minimum. However the second term is subdominant compared to the first one, in the expansion of small β_6. In this way one can see the tension introduced by the presence of a small β_6 and the requirement of spontaneous breaking of H_5 and H_6 by the potential. This scenario in general leads to m_2 smaller than m_1, due to the suppression of β_6 compared with β_5 and due to the tuning required to obtain a suitable minimum. * If in the previous case we consider β_56≪β_5, we obtainm_1^2≃ 8β_5ϵ^2_5(1-ϵ^2_5)/f_5^2 , m_2^2≃4/3β_6ϵ^2_6(1-ϵ^2_6)/f_6^2 ,θ≃1/2√(6) √(1-ϵ^2_6/1-ϵ^2_5) β_56/β_5 ϵ_6 f_5/ϵ_5 f_6 .This case is interesting because it is realized in several regions of the parameter space of the model studied in sec. <ref>. Since the mixing angle is small, the physical states can be approximated by h_5 and h_6. The mass of h_5 is controlled by the quartic β_5 and suppressed by ϵ^2_5, whereas the mass of h_6 is suppressed by β_6, that was assumed to be smaller than the other quartic couplings, and by ϵ^2_6(1-ϵ^2_6). The previous examples show that in general one expects the presence of two light scalar states. In the case of small mixing angle, there is a state with quantum numbers similar to the SM Higgs boson, whose mass is suppressed by ϵ^2_5, similar to the MCHM. Compared with that scenario, there is a new state, whose mass is in general suppressed by the small quartic β_6 and by ϵ^2_6(1-ϵ^2_6). §.§ Model buildingIn this subsection we discuss the impact of the fermion representations under G_5 on the potential V(s_5,s_6).Using the form factors of Eq. (<ref>), and considering only the singlet scalar fields, the potential generated by exchange of virtual fermions at one-loop is:V_f(s_5,s_6)=-2N_c∫_p[log(p^2Π_u_LΠ_u_R-|Π_u_Lu_R|^2)+log p^2Π_d_L]. By expanding V_f of Eq. (<ref>) in powers of s_5 and s_6 to fourth order it is possible to express the quadratic and quartic coefficients of Eq. (<ref>) in terms of integrals of the form factors. A simplified description of V_f can be obtained by expanding also in powers of 1/Z_f, or equivalently in powers of the fermion mixing ϵ_L and ϵ_R. As discussed in Ref. <cit.>, although for the third generation the mixing can be large, it can still provide a good estimate. Moreover, by performing numerical calculations to all orders in these parameters, we have checked that this expansion correctly describes the size of the coefficients.Expanding in powers of s_5, s_6 and 1/Z_f, we have analysed models with fermions in representations 1, 4, 5, 10, 14 and 16 of G_5. We find that: in all the cases α_5,α_6 and β_56 are of O(Z_f^-1), whereas β_6 is of O(Z_f^-2). For models involving only representations 1 and 5 β_5 is of O(Z_f^-2), for example ψ_q∼ 5 and ψ_u∼ 5 or 1. For models where at least one of the fermions, ψ_q or ψ_u, is in the representation 4, 10, 14 or 16, β_5 is of O(Z_f^-1). If P_LR-symmetry is added to protect the Zb_Lb̅_L coupling, representation 10 also gives β_5 of O(Z_f^-2). As shown in sec. <ref>, for β_56≫β_5,β_6 it is not possible to break both symmetries simultaneously, thus models involving only representations 1, 5 and 10 with P_LR-symmetry are not favored. By considering the 2-site models of sec. <ref> we have computed the potential to all orders in s_5, s_6 and Z_f, and we have checked that this is indeed the situation. Therefore, although the mixings of the quarks of the third generation are large, the simplified analysis allows for a correct selection of the best representations. As is well known, the choice of the representation also affects flavor physics. In theories with a pNGB Higgs, fermion representations allowing more than one invariant Yukawa coupling induce Higgs mediated flavor violating processes that in general are incompatible with the current bounds from flavor physics for f∼TeV <cit.>. In order to proceed with our analysis, we will assume that, for a given type of fermion ψ_f, all the generations are in the same representation. Once the representations for ψ_q and ψ_u are chosen, it is straightforward to count the number of invariants. As done in Eq. (<ref>), a simple procedure is to decompose the representations of G_5×G_6 in irreducible representations of H_5×H_6: ψ_f∼⊕_rs(ψ_f)_rs, and count how many representations rs in the decomposition of ψ_q coincide with the ones in ψ_u. This number gives the number of independent invariants corresponding to Yukawa structures. There is one exception to this rule: if ψ_q and ψ_u are in the same representation of the group, there is one linear combination of the invariants that is independent of the pNGB, thus in this case one has to subtract one unit to the previous counting <cit.>. A similar situation holds for the down sector and leptons. Taking into account these results, as well as the results from the previous paragraph, an interesting set of representations is: ψ_q∼ 14, ψ_u∼ 1 and ψ_d∼ 10 of SO(5). In this case there is just one non-trivial invariant for the Yukawa of the up-sector and one for the down-sector. Calling the invariants I_rs, where r labels the representation of SO(4) and s the representation of SU(3)_L×SU(3)_R, the invariant for the up-sector can be symbolically written as: I_(1,1),(3,1)-I_(1,1),(1,3), and the one for the down-sector as: I_(2,2),(3,1)-I_(2,2),(1,3). In this counting we have taken into account that, since q_L and q_R are in the same representation of G_6, there is only one linear combination of invariants of H_6, that we have chosen as: I_r(3,1)-I_r(1,3).Model 14-1According to the discussion of the previous paragraph, we consider in detail the potential generated by fermions in the following representations of SO(5): ψ_q∼ 14 and ψ_u∼ 1, we will refer to this case as 14-1. Under H_5 ψ_q decomposes as: 14∼(3,3)⊕(2,2)⊕(1,1), whereas ψ_u is a singlet (1,1). These decompositions indicate the values that the subindex r can take in the expressions for the form factors defined in Eqs. (<ref>) and (<ref>). For the representations of H_6: s=(3,1),(1,3). As done previously, to shorten the notation we will omit the parenthesis, thus we will write Π^f_ijkℓ, with i,j labeling the representation of H_5, and k,ℓ that of H_6, and similar for M^f_ijkℓ. We obtain: m_f,5^2=-2/f_5^2∫_p 3/45Π^q_1131-14Π^q_2231+9Π^q_3331/Z_q+Π^q_2231 , m_f,6^2=-2/f_6^2∫_p [Π^q_2231-Π^q_2213/Z_q+Π^q_2231+Π^u_1131-Π^u_1113/Z_u+Π^q_1113],λ_f,5=-2/f_5^4∫_p -3/16(Z_q+Π^q_2231)^2[75 (Π^q_1131)^2-140 Π^q_1131Π^q_2231+150 Π^q_1131Π^q_3331+44 (Π^q_2231)^23cm-252 Π^q_2231Π^q_3331+123 (Π^q_3331)^2+16 Z_q (10 Π^q_1131-19 Π^q_2231+9Π^q_3331)],λ_f,6=-2/f_6^4∫_p [(Π^q_2231-Π^q_2213)(2Z_q+3Π^q_2213-Π^q_2231)/12(Z_q+Π^q_2231)^2+Π^u_1113-Π^u_1131/12(Z_u+Π^u_1113)-(Π^u_1113-Π^u_1131)^2/8(Z_u+Π^u_1113)^2],λ'_f,6= 0, λ_f,56=-2/f_5^2f_6^2∫_p 1/32(Z_q+Π^q_2231)^2[5Π^q_1113Π^q_2231-5 Π^q_1131Π^q_2213-9 Π^q_2213Π^q_3331+9 Π^q_2231Π^q_33134cm+Z_q (5 Π^q_1113-5 Π^q_1131-14 Π^q_2213+14 Π^q_2231+9 Π^q_3313-9 Π^q_3331)]. By keeping only the scalar fields h_5 and h_6, and expanding in powers of s_5 and s_6, one can also compute the potential V(s_5,s_6) defined in Eq. (<ref>). For large Z_f, the leading non-trivial contribution to the coefficients of the potential are:α_5≃1/2Z_q∫_p(-9Π^q_3331+14Π^q_2231-5Π^q_1131),α_6≃∫_p[4/Z_q(Π^q_2231-Π^q_2213)-2/Z_u(Π^u_1131-Π^u_1113)],β_5≃1/2Z_q∫_p(3Π^q_3331-8Π^q_2231+5Π^q_1131),β_6≃∫_p[-1/Z_q^2(Π^q_2231-Π^q_2213)^2+1/2Z_u^2(Π^u_1131-Π^u_1113)^2],β_56≃1/2Z_q∫_p(9Π^q_3331-9Π^q_3313-14Π^q_2231+14Π^q_2213+5Π^q_1131-5Π^q_1113).A few observations can be made: as already discussed, in this model all the coefficients are of O(Z_f^-1), except for β_6 that is of O(Z_f^-2); the sign of the quadratic coefficients is not fixed, thus for suitable regions of the parameter space it is possible to break both symmetries. § A 2-SITE MODELMoose models can provide an effective description of a strongly coupled sector weakly coupled to external sources <cit.>, they can also describe the lower level of resonances of extra-dimensional theories <cit.>. In this section we consider the simplest case given by 2-site models; by using this framework one can explicitly compute the potential as well as some relevant observables. We follow the approach proposed in Ref. <cit.> to describe NGB fields arising from the SCFT (see also <cit.>). The setup is given by two sites containing different sets of fields, and connected by a set of non-linear sigma model fields. The first site, also called site-0, contains the same field content and symmetries as the SM but without scalar fields. For convenience, as already described in sec. <ref>, we add spurious fields to extend the gauge symmetry of the SM to G_5,0×G_6,0, with G_5=SO(5)× U(1)_X and G_6=SU(6), the subindex labelling the site-0. The corresponding gauge couplings are: g_X,0, g_5,0 and g_6,0. Following the discussion of sec. <ref>, we put ψ_q, ψ_u and ψ_d in the representations 14_2/3, 1_2/3 and 10_2/3 of G_5, respectively. We take the fundamental representation of G_6 for the quarks, with q_L in (3,1)_1/2√(3) and q_R in (1,3)_-1/2√(3) of the subgroup H_6, defined in the next paragraph. In the second site, also called site-1, we put the fields that describe the first level of resonances of the SCFT. The spin one resonances can be described by the fields of the gauge symmetry G_5,1×G_6,1, with gauge couplings g_X,1, g_5,1 and g_6,1. We add two sets of NGB scalar fields, Π_5^1 and Π_6^1, [The index 1 indicates that these fields belong to site-1, and allows us to distinguish them from the non-linear sigma model fields connecting both sites that will be described below.] respectively describing the spontaneous breaking G_5,1/H_5,1 and G_6,1/H_6,1, by the strong dynamics, with H_5=SO(4)×U(1)_X and H_6=SU(3)_L×SU(3)_R×U(1). Each spontaneous breaking is characterized by a scale f_5,1 and f_6,1; both scales are assumed to be of the same order. Thus we include in the Lagrangian of site-1 the kinetic terms of the gauge fields and the NGB scalars, that as usual can be expressed in terms of the Cartan-Maurer form.In addition, there are some fermion multiplets: Ψ_q, Ψ_u and Ψ_d, that are in correspondence with the elementary ones, and are in the same representations of G_5×G_6 as the elementary fermions. The fermions of site-1 are vector-like, with masses arising from the strong dynamics. The non-linear transformation properties of the NGB fields allow us to write gauge invariant Yukawa interactions with the fermions on site-1, as shown below:L_1⊃ ∑_f=q,u,dΨ̅_f (i-m_f,1)Ψ_f+f_1 ∑_f=u,d∑_rs y^f_rs(U_5^† U_6^†Ψ_qL)_rs (U_5^† U_6^†Ψ_fR)_rs +h.c. ,where y^f_rs are dimensionless Yukawa couplings, and f_1 is a dimensionful parameter of the same order as the NGB decay constants.The second term of Eq. (<ref>) generates Yukawa couplings for the elementary fermions after interactions between both sites. The representations chosen for the fermions allow for just one G_5 invariant for the Yukawa operator of the up sector, obtained by taking r=(1,1). Since a 10_2/3 of G_5 decomposes under H_5 as: 10_2/3∼ (2,2)_2/3⊕(3,1)_2/3⊕(1,3)_2/3, there is also just one G_5 invariant for the down sector, obtained by taking r=(2,2). The index s takes the values (3,1) and (1,3). Since for y^f_r31=y^f_r13 the Yukawa is independent of Π_6, the linear combination y^f_r31-y^f_r13 measures the Yukawa coupling involving Π_6, whereas y^f_r31+y^f_r13 leads to a mixing between Ψ_q and Ψ_f. Thus in the present 2-site model the Yukawa couplings are proportional to y^f_r31-y^f_r13. These results are in agreement with the arguments of sec. <ref> for the low energy effective theory.In Eq. (<ref>) we have included only one chiral structure, avoiding terms with the opposite chiral structures: (U^†Ψ_qR)_rs (U^†Ψ_fL)_rs. As discussed in Ref. <cit.>, there is no fundamental reason for this, the motivation being that in this way one can obtain a finite one-loop potential. All the couplings of site-1, generically called g_1, are assumed to be larger than the SM ones but still perturbative: g_SM≪ g_1≪ 4π. All the dimensionful parameters are assumed to be of order TeV, the masses of the fermions being m_f,1∼ g_1 f_1.The two sites are connected by a set of non-linear sigma models: Π_X,0, Π_5,0 and Π_6,0, associated to the gauge symmetry groups. These fields parametrize the cosets (G_5,0×G_5,1)/G_5,0+1 and (G_6,0×G_6,1)/G_6,0+1, where G_5,0+1 and G_6,0+1 are the diagonal subgroups.The field U_5,0=e^i√(2)Π_5,0/f_5,0 transforms linearly under G_5,0×G_5,1, as: U_5,0→ĝ_5,0U_5,0ĝ_5,1^†, with ĝ_5,0∈G_5,0 and ĝ_5,1∈G_5,1. Similar properties apply to U_6,0=e^i√(2)Π_6,0/f_6,0. Besides, these fields play an important role in the realization of partial compositeness, that is obtained by linear interactions between the fermions on site-0 and site-1, through the following terms:L_mix⊃ f_0 ∑_f=q,u,dλ_f ψ̅_f U_X,0^2/3U_5,0U_6,0Ψ_f + h.c.with f_0 a dimensionful constant and λ_f a dimensionless coupling that controls the size of the mixing between the fermions. All the dimensionful parameters, namely: the decay constants and f_0, are taken of order TeV, whereas the couplings λ_f can be hierarchical. Let us briefly count the number of NGB fields. One can choose a gauge where the fields Π_X,0, Π_5,0 and Π_6,0 are removed. In this gauge it is straightforward to see that the gauge fields of the corresponding cosets become massive, with masses m_1∼ g_1 f_0/√(2), whereas the gauge fields of the diagonal cosets remain massless. The NGBs Π_5,1 and Π_6,1 remain in the spectrum; Π_5,1 can be associated with the Higgs multiplet and Π_6,1 gives a new scalar multiplet. The previous gauge is not the unitary one, that can be obtained by demanding the absence of mixing terms between the gauge and the NGB fields, as described in Refs. <cit.> and <cit.>. The decay constants of the physical NGB fields Π_5 and Π_6 are given by: f^-2_5=f^-2_5,0+f^-2_5,1 and f^-2_6=f^-2_6,0+f^-2_6,1, respectively.The degree of compositeness of the light fermions is given by ϵ_f=λ_f/g_1, where we have considered f_0∼ f_1. The light fermions are mostly elementary if ϵ_f≪ 1, leading to small mixing, whereas for ϵ_f∼ 1 the mixing between the fermions on site-0 and site-1 is large, leading to partially composite states. We are interested in the anarchic scenario, that in the 2-site model is obtained by taking structureless Yukawa couplings, with all the elements of the matrices being of the same order: (y^f_rs)_jk∼ g_1, where j and k are flavor indices. Keeping the NGB dependence, the mass matrices of the SM fermions are:(m_f)_jk∼√(5)/2ϵ_qjϵ_fk s_5 c_5 s_6 c_6 f_1(y^f_rs)_jk ,thus a light fermion requires at least the mixing of one of the chiralities to be small, whereas the top requires both mixings of order one to be able to obtain a Yukawa coupling y_t∼ 1. Besides the masses, it is also necessary to reproduce the mixing angles of the CKM matrix. Taking hierarchical mixings ϵ_f leads to the hierarchies between the masses and CKM angles. [Although hierarchical λ_fj may look arbitrary, they appear naturally when considering the running of the couplings between the elementary fermions and the SCFT <cit.>. A simple realization can be obtained in extra dimensional models <cit.>.]Following Refs. <cit.> we take:ϵ_q1/ϵ_q2∼λ_C , ϵ_q2/ϵ_q3∼λ_C^2, ϵ_q1/ϵ_q3∼λ_C^3.where λ_C is the Cabibbo parameter. By using v_SM=ϵ_5 f_5, m_t∼ v_SM/√(2) and y^u_rs∼ g_1, the following relations arise from anarchy:ϵ_6 √(1 - ϵ_6^2) ϵ_q3ϵ_u3g_1∼1,ϵ_d3∼ϵ_u3y^u/y^dm_b/m_t , ϵ_u2∼ϵ_u3/λ_C^2m_c/m_t ,ϵ_d2∼ϵ_u3/λ_C^2y^u/y^dm_s/m_t , ϵ_u1∼ϵ_u3/λ_C^3m_u/m_t ,ϵ_d1∼ϵ_u3/λ_C^3y^u/y^dm_d/m_t .The factor y^u/y^d in the second column is of O(1) if all couplings on site-1 are of the same order, but differs from this estimate if the Yukawa couplings of the down- and up-sectors on site-1 are taken different. The upper-left equation of (<ref>) shows that ϵ_6 √(1 - ϵ_6^2) cannot be too small if g_1 is required to be perturbative since in the most favorable case, for ϵ_q3∼ϵ_u3∼ 1, it leads to: g_1∼(ϵ_6 √(1 - ϵ_6^2))^-1.Integrating out the spin-one fields and fermions on site-1, one obtains the effective theory described in sec. <ref>. The correlators of Eqs. (<ref>) and (<ref>) can be computed explicitly in terms of the parameters of the 2-site theory. We show them in Ap. <ref>.§.§ Dominant contributions to ϵ_KIntegrating out the heavy resonances, new contributions to the Wilson coefficients C_i of dimension-6 operators contributing to K-K̅ mixing are generated. The most dangerous contributions are those generated by exchange of massive gluons, that give a contribution to C_4:C_4=C_4^SU(3)ϵ^2_6 (1 - ϵ^2_6)f_6,1^2/2(f_6,0^2+f_6,1^2) ,where C_4^SU(3) is the contribution in the usual case with SU(3) global symmetry only.Bounds on C_4^sd from ϵ_K usually require m_cp≳ 10÷ 30 TeV. In the present case, taking f_6,0≃ f_6,1, for ϵ_6 √(1-ϵ_6^2)∼ 0.2÷ 0.5 the scale of the masses of the composite gluons can be lowered to ∼ 2.5 TeV. Since 0<ϵ_6<1, ϵ_6 √(1-ϵ_6^2) is always smaller than (at most equal to) 0.5.As mentioned in sec. <ref>, the decomposition of the adjoint of G_6 under H_6 contains a (1,1)_0 - the generator of the U(1) factor in H_6. The presence of this state induces an extra contribution to C_5, that is not suppressed by ϵ_6: C_5=C_4^SU(3)1/6 .The contribution of C_5^sd to ϵ_K is suppressed by N_c, and in the present case by an extra factor 1/6 compared with the usual case. Therefore the contribution of this state is not critical for ϵ_K. §.§ Estimation of the top massThe mass of the top can be approximated by the coefficient j=k=3 of Eq. (<ref>). Taking ϵ_q3∼ϵ_u3∼ 1, f_1≃ 1 TeV and the values for ϵ_6 that saturate C_4^sd, we obtainy^u_rs∼ 2.4÷ 0.6,where the largest value corresponds to ϵ_5∼ 0.3 and ϵ_6∼ 0.2 and the smallest one to ϵ_5∼ 0.5 and ϵ_6 √(1 - ϵ^2_6)∼ 0.5. Although there is an extra suppression from the factor ϵ_6 √(1 - ϵ^2_6) compared with the usual case, O(1) Yukawa can lead to the top mass in the present model.§ NUMERICAL RESULTSIn this section we present some numerical results that arise from the 2-site model described in sec. <ref>. We focus first on the issue of spontaneous symmetry breaking of the extended symmetries: H_5 that contains the EW symmetry group, and H_6 that contains the color symmetry group. By performing a random scan over the parameters of the 2-site models, we have checked the properties of the potential discussed in sec. <ref>. In particular we have checked the predictions of sec. <ref> for different combinations of the representations 1, 5, 10 and 14 of the fermions under SO(5), and their impact on the possibility to obtain successful breaking of both symmetries. We have found that, in models with fermions in combinations of the representations 1, 5, and 10, in general there is no spontaneous breaking at all, and in some cases only H_5 is broken. [We imposed Left-Right parity for the 10 of SO(5).] We have also checked that when the 14 is included there are suitable regions of the parameter space where the symmetry is spontaneously broken to K_5×K_6.In the rest of this section we will present the results for model 14-1 that, as discussed in sec. <ref>, is one of the most interesting models. We start by explaining our scan of the parameter space. It is useful to define mixing angles for the fermions by the following relation: tanθ_f=λ_f f_1/m_f, with m_f the mass of the vector-like fermions on site-1. We will use s_f and c_f for sinθ_f and cosθ_f respectively; notice that with this parametrization the absence of mixing is given by s_f=0, whereas large mixing corresponds to s_f≃ 1. We have fixed the masses m_f and the Yukawa couplings y^u_rs, and we have varied s_q and s_u. For the bosonic sector we have fixed the ratio between the couplings g_0/g_1 and we have matched the coupling of the diagonal subgroups with the SM gauge couplings. We have also fixed the masses of the spin-one resonances, m_1. In Fig. <ref>we show the results obtained when scanning over s_q and s_u for a point with parameters g_0/g_1 = 0.26, m_q = m_u = 2 TeV, m_1 = 2.5 TeV, y^u_1131 = 0.4, y^u_1113 = -0.82 and f_5 = f_6 = 1 TeV (upper panels, labeled 1) and another with g_0/g_1 = 0.25, m_q = 3 TeV, m_u = 1.4 TeV, m_1 = 3 TeV, y^u_1131 = -0.7, y^u_1113 = 0.5 and f_5 = f_6 = 1 TeV (lower panels, labeled 2). The panels on the left (labeled A) show the expectation values obtained for s_5 (red dotted) and s_6(blue solid), with the regions where ϵ_6 = 0 colored in light blue, those were ϵ_5 = 0 in red and the ones with maximal H_6 breaking (ϵ_6 = 1) in gray. The white, uncolored regions in each A panel are the regions where we found spontaneous symmetry breaking to K_5×K_6, and thus the most interesting phenomenologically. The panels labeled with a B show the masses of the Higgs boson (red dotted) and the h_6 scalar (blue solid). We have verified that the mixing angle between these scalars is very small (≤ 0.1 rad) and so their identification with the mass eigenstates is a reasonable approximation. It is worth noting that the mass of h_6 was always found to be smaller than that of the Higgs within the phenomenologically interesting region; this is a general feature of this model and a consequence of the discussion presented in sec. <ref>. The phenomenological implications of having this additional light scalar will be analysed in the following section. In the B panels we also show the mass of the SM top quark (black dashed), which is calculated as the mass of the lightest up-type quark. Even though the vev's and Higgs masses shown in these plots are not completely realistic, more realistic values could be found with a more extensive scan. All sharp angles and jagged lines in these plots are only a consequence of the limitations of the numerical calculations employed and do not carry any physical meaning.§ DISCUSSIONS §.§ Neutron dipole momentsBesides the constraints from K-K̅ mixing, the electromagnetic (EDM) and chromomagnetic dipole moments (CDM) of the neutron give the most important set of constraints in models with anarchic flavor <cit.>. In theories where the SCFT has a global symmetry SU(3), gauged by the color interactions, assuming that the dipole operators are induced at loop level the following bound is obtained: f≳ 4.5 TeV <cit.>. As expected, in the presence of the larger SU(6) symmetry, given the representations chosen for the fermions, the contribution to the Wilson coefficients of operators with LR chiral structures require insertions of the scalar Π_6. In particular, the group structure associated to the SU(6) symmetry of the EDM operators is identical to the structure of the mass operators, therefore at least one insertion of the scalar field Π_6 is required for the EDM operator. Resumming these insertions and evaluating in the vacuum, the bound is relaxed as: f/(ϵ_6√(1-ϵ_6^2))≳ 4.5 TeV. As an example, f=2 TeV is compatible with the bounds for ϵ_6≲ 0.2 or ϵ_6≳ 0.98.§.§ Phenomenology of the scalar statesAn interesting and distinctive phenomenology of the present model is given by the presence of the new scalar states. Let us briefly characterize this sector of the theory. As discussed in sec. <ref>, there are two light scalar states, corresponding to the excitations of the fields that acquire a vev at loop level, h_5 and h_6. As mentioned in the previous section, we find that the mixing between these fields is small, and therefore, to leading order the mass eigenstates can be identified with h_5 and h_6. Since h_5 arises from SO(5)/SO(4), its interactions with the EW gauge bosons are similar to those of the Higgs in the MCHM. On the other hand, h_6 does not interact with the gauge bosons at tree level. The Yukawa couplings at low energies are shown in Eqs. (<ref>) and (<ref>). In the full theory the scalars also interact with the resonances.Let us characterize the creation and decay of these scalars at the LHC. As for the usual Higgs, the main creation channel of the scalars at the LHC is gluon fusion, with much smaller contributions from creation in association with tt̅, as well as creation in association with EW gauge bosons for the case of h_5. Gluon fusion is induced at loop level by virtual exchange of fermions f^(n). In the limit of heavy fermions, 2m_f^(n)≫ m_j, this coupling is proportional to ∑_n≥0 y_f,j^(n)/m_f^(n), where j=5,6 labels the scalars and quantities with n = 0 pertain to the would be zero mode corresponding to the SM fermion. By standard algebraic manipulations it is possible to rewrite this expression as tr(Y_f,jℳ_f^-1), with Y_f,j=∂_h_jℳ_f and ℳ_f the fermion mass matrix of the full theory. As usual in models with NGB scalars, this trace leads to a compact expression:tr(Y_u,5ℳ_u^-1)=1/v_SM1-2ϵ_5^2/√(1-ϵ_5^2), tr(Y_u,6ℳ_u^-1)=1-2ϵ_6^2/√(6)f_6ϵ_6√(1-ϵ_6^2).The case of h_5 has been extensively studied, as well as its dependence on the fermion representations <cit.>. The coupling to h_6 has a similar behavior, but in terms of the parameters associated to G_6. By comparing with Eq. (<ref>), one can see that the sum over states is saturated by the lightest fermion, in this case corresponding to the top. [The light quarks and the leptons do not contribute in the present setup, see Ref. <cit.> for discussions.] Given these results, creation of h_j by gluon fusion and tt̅ h_j processes is similar to Higgs creation in the MCHM.Since the couplings of h_5 are similar to the couplings of the Higgs in the MCHM, its decay width and branching ratios are similar to that case. On the other hand h_6 can decay at tree level to SM fermion-antifermion, with Yukawa coupling given by Eq. (<ref>). This Yukawa coupling can be written in terms of the usual SM Yukawa with the Higgs, up to a factor depending on ϵ_6 and f_6: y_f,6≃ y_f^SM δ , δ=v_SM/f_61-2ϵ_6^2/√(6)ϵ_6√(1-ϵ_6^2) .At loop level, for a light h_6, the most important decay channels are pairs of gluons and photons. The amplitude of the former is modulated by Eq. (<ref>). The amplitude of the later is modulated by a similar factor, multiplied by the corresponding factors of electric charge. Compared with h_5, in the one-loop level contribution to h_6γγ there is no contribution from virtual EW gauge bosons. For light h_6 we expect diphoton and bb̅ final states to be the most relevant ones, whereas for a heavier scalar the final state tt̅ will become important, and massive diboson states mediated by loops of fermions also will become available. According to the results of Eqs. (<ref>) and (<ref>), the creation cross-section by gluon fusion and the partial widths of h_6 are similar to the ones of the SM Higgs, with a rescaling factor δ^2, and changing the Higgs mass by the mass of h_6. The only qualitative difference being that decays to pairs of massive gauge bosons are not present at tree level for h_6. In Refs. <cit.> and <cit.>, the ATLAS and CMS Collaborations have presented limits in the production cross-section of new scalar states decaying to photon pairs. In order to compare with those bounds, we consider a reference mass for h_6 of 80 GeV. In this case the production cross-section of h_6 is: σ≃ 45pb×δ^2, and the BR to photon pair is of order 10^-3, leading to a bound δ≲ 1. For f_6= 1 TeV, this bound can be satisfied as long as 0.1≲ϵ_6≲ 0.99. Therefore a new scalar state, somewhat lighter than the Higgs, could have evaded detection at the LHC.Besides the neutral scalar fields already discussed, the present model contains an axion-like state, associated to the spontaneous breaking of the U(1) subgroup of H_6. Indeed the vacuum expectation value v_6 spontaneously breaks this global U(1), and the scalar phase θ_6 becomes the associated NGB. In order to understand the phenomenology of this state, let us briefly discuss its interactions.Due to the embedding of the quarks into the fundamental representation of G_6, the Left- and Right-handed quarks of the SM have opposite charges under U(1). The axion-like scalar has interactions with the fermions of the form: θ_6 h_5 q̅_Lγ^5 q_R, that at one-loop level induce interactions θ_6 F_μνF̃^μν and θ_6 G_μνG̃^μν with photons and gluons. It is usually assumed that the axions acquire mass only through non-perturbative effects, however it has been shown recently that they can also obtain a mass at perturbative level <cit.>. The simplest scenario is obtained by considering the presence of a 3-form field, that does not propagate since its equation of motion fixes its field strength to be a constant. It has been shown that the interactions of this form with the axion induce a potential, with a mass for the axion. If the dimensional coupling is of order TeV, the axion mass can be of order several hundred GeV to a few TeV. This scalar state could mix with the other scalars. It could also be produced by gluon and photon fusion, and decay to pairs of photons, gluons and fermion-antifermions. A precise description of its phenomenology requires assumptions on the specific realization, since there can be several 3-forms involved. We will not consider those details in the present work.The new scalar sector also contains octets of SU(3)_c. The mass of these multiplets can be estimated from m_g,8^2 in Eq. (<ref>), leading to m^2≃α_s/(4π) g_1^2 f_6^2 ∼ O( TeV^2). Being color octets, these states can be copiously produced by QCD interactions, with a large branching fraction to dijets. ATLAS and CMS have searched for narrow resonances in dijet final states <cit.>, giving bounds on the mass of the scalar octets of order 3 TeV.§.§ TuningThe tuning can be roughly estimated as (ϵ_5ϵ_6)^-2. The factor ϵ_5^-2 is the usual tuning of the MCHM, that could be increased depending on the representations of the fermions <cit.>. EW precision tests usually require ϵ_5^2≲ 0.1÷ 0.3, leading to a tuning of order 4÷ 10. Numerical calculations using different representations for the fermions show that the tuning varies between 5 and 10^3 in the MCHM <cit.>. The need in the present paper for the spontaneous breaking of another symmetry is expected to worsen the tuning, introducing an extra factor ϵ_6^-2. As discussed in sec. <ref>, by demanding the masses of the resonances to be of order 2.5 TeV, constraints from the Kaon system give the bounds: ϵ_6≲ 0.2÷ 0.98. In the most stringent case of ϵ_6≲ 0.2, we expect to increase the tuning by a factor of order 25, whereas in the most favorable case we expect no sizable rise of the tuning compared with the MCHM. Going beyond this general estimate requires considering the details of the model, in particular the embedding of the fermions, as well as numerical calculations.Following the definition for the sensitivity parameter of Refs. <cit.>, we have computed the tuning in our model with the fermions q_L and u_R in the representations 14 and 1 of SO(5). For the regions of the parameter space of Fig. <ref> we find a tuning of order 50-300, with a few points where it rises to 400. One can compare this tuning with the results of Ref. <cit.>, that has reported a tuning of order 80÷ 300 for the MCHM_14-1. However in that paper the Higgs, top and W masses were restricted to their physical values, selecting a region of the parameter space with non-natural cancellations in the scalar potential and thus increasing the tuning. In Fig. <ref>, the masses of the Higgs, top and EW gauge bosons are not fixed to their physical values. Constraining these masses requires non-trivial cancellations also in the model with the extended symmetry G_6, and we expect a larger tuning in that case. §.§ Other representations and symmetry groupsThe need for a larger quartic coefficient β_6 suggests changing the representation of the quarks under G_6. A larger β_6 could improve the tuning of the model by allowing a larger region in the parameter space with the right spontaneous breaking of symmetries. Cancellation of C_4 requires q_L∼(3,1) and q_R∼(1,3) of SU(3)_L×SU(3)_R, thus the quark representations under SU(6) must contain these representations of the subgroup SU(3)_L×SU(3)_R. However we have not been able to find other small representations of SU(6) that satisfy those conditions.We have also explored the possibility of embedding the SO(4) as well as the extended SU(3)_L×SU(3)_R symmetry groups into a single unified group. We have found several examples of groups that, after spontaneous breaking to SO(4)×SU(3)_L×SU(3)_R, deliver a NGB field transforming as a (2,2,3,3) of the unbroken subgroup, playing the role of the Higgs field in the extended quark sector <cit.>, the most interesting one being the exceptional group E_8. However we have not been able to obtain suitable representations for the quarks. For example, for q_L, adding custodial symmetry to protect the Zb_Lb̅_L coupling, one would require a representation containing a (2,2,3,1) of the unbroken subgroup, whereas for u_R one would require (1,1,1,3) or (1,3,1,3). By exploring the lowest dimensional representations of the unified groups, we have not been able to find the proper representations for the quarks. § CONCLUSIONSFlavor anarchic composite Higgs models with partial compositeness of the SM fermions offer a rationale to understand the hierarchies in the flavor of the quarks and leptons. The most stringent constraint from flavor physics, arising from ϵ_K in K-K̅ system, pushes the scale of compositeness to f∼ O(10)TeV, introducing a little hierarchy problem for a light Higgs. Ref. <cit.> showed that if the composite sector has an SU(3)_L×SU(3)_R global symmetry, with SU(3)_c=SU(3)_L+R, the main contribution to ϵ_K, given by the Wilson coefficient C_4^sd, can be suppressed. However quark masses require the presence of a new scalar field with a vacuum expectation value, whose presence can destabilize the cancellation protecting C_4. We have embedded the global symmetry into a larger SU(6) group, showing that a proper spontaneous breaking of the symmetries can occur dynamically. We have made an analysis of the potential, showing which are the representations of the fermions that can trigger this breaking. We have also shown that fermion masses can be reproduced, and the Wilson coefficient C_4 can be successfully suppressed, with a compositeness scale f∼TeV. We have also briefly discussed the phenomenology at the LHC of the new light scalar states.§.§ AcknowledgementsL. D. thanks Kaustubh Agashe for discussions that triggered the realization of this work. We thank Luis Ibáñez for pointing out references on the physics of axions and Eduardo Andrés for useful discussions as well as help with group theoretical issues. We thank Giuliano Panico for questions on magnetic dipole operators. This work was partially supported by ANPCyT PICT 2013-2266.§ FUNDAMENTAL REPRESENTATION OF SU(6)In this appendix we provide some basic ingredients of the group SU(6). The generators of SU(6) in the fundamental representation consist of thirty five 6× 6 matrices, which are easily described as a combination of three by three blocks, i.e.:T^i = [[ A^i_3 × 3 B^i_3 × 3; C^i_3 × 3 D^i_3 × 3 ]], for the ith generator of SU(6). After the dynamics of the SCFT break SU(6) down, the generators can be organized into those of the preserved subgroup, SU(3)_L ×SU(3)_R×U(1) and those of the broken coset.For the generators of SU(3)_L ×SU(3)_R, which we will consider to be the first 16 generators of the basis, B^i = C^i = 0_3× 3, i = 1, ... , 16. Then, for those of SU(3)_L, A^i =1/2λ_i, with the λ matrices being the usual Gell-Mann matrices, and D^i = 0_3× 3, i = 1,...,8. On the other hand, for SU(3)_R, A^i =0_3× 3 and D^i =1/2λ_i-8, i = 9,...,16. The generator of U(1) is given by B^35 = C^35 = 0_3 × 3, A^35 = -D^35 = 1/2 √(3)I_3× 3.For the coset, the generators arrange into a complex (3,3). In this case only blocks B and C will be populated by non-zero elements. Furthermore, the 18 generators of the complex (3,3) can be organized into two octets and two singlets under the diagonal subgroup of SU(3)_L ×SU(3)_R, SU(3)_V. For the first octet one can choose A^i = C^i = D^i = 0_3× 3 and B^i = K^i - 16, i = 17,...,24; for the other A^i = B^i = D^i = 0_3× 3 and C^i = K^i - 24, i = 25,...,32. The matrix elements for the 8 three by three K^i blocks are as follows:K^1_(1,3) = K^2_(2,3) = K^3_(3,1) = K^4_(2,1) = -K^5_(3,2) = -K^6_(1,2) = i/√(2), K^7_(1,1) = K^7_(3,3) = i/2 √(3),K^7_(2,2) = -i/√(3),K^8_(1,1) = - K^8_(3,3) = - i/2,with all other elements being zeroes. The two singlets can be described by A^33 = D^33 = A^34 = D^34 = 0_3 × 3, B^33 = -C^33 = i B^34 = i C^34 = -i/2 √(3)I_3 × 3, with I_3 × 3 the three by three identity matrix.§ CORRELATORS IN THE SYMMETRIC VACUUM FOR THE 2-SITE MODEL Let's consider a 2-site model like the one used in this paper or the ones described in <cit.>, where the composite sector has a global symmetry group G that is broken down spontaneously to an H subgroup and fermions transform under full irreducible representations (irreps) of G. If we focus solely on the top quark sector of the model, its fermion content consist of two chiral elementary multiplets, one containing the electroweak doublet, q_L, and the other containing the electroweak singlet, u_R, and two vector-like composite multiplets corresponding to the composite counterparts of the aforementioned elementary multiplets, whose chiral components we call Ψ_qL/R and Ψ_uL/R respectively. The composite multiplets Ψ_q and Ψ_u are in irreps of G, which we shall call α and β respectively. Assuming the partial compositeness scheme, we add spurious fieldsto the elementary multiplets q or u and embed them in the same irreducible representations of G as the composite ones; these elementary multiplets are called ψ_q and ψ_u. Let's define α_H as the set of irreps of H with a non-zero multiplicity in the decomposition of α, and similarly define β_H. For the model to contain Yukawa interactions, it is necessary that Γ := α_H ⋂β_H ≠∅. Assuming that to be the case, the Lagrangian density for the fermionic sector of this model in the H-symmetric vacuum can be described as follows:ℒ = ℒ_kin + ℒ_mass + ℒ_mix + ℒ_Yuk + ℒ_0ℒ_kin =ψ_qLψ_qL + ψ_uRψ_uR + Ψ_qLΨ_qL +Ψ_qRΨ_qR + Ψ_uLΨ_uL + Ψ_uRΨ_uRℒ_mass = -m_q,1 Ψ_qLΨ_qR - m_u,1 Ψ_uLΨ_uR + h.c.ℒ_mix = f_0 λ_qψ_qLΨ_qR + f_0 λ_uψ_uRΨ_uL + h.c.ℒ_Yuk = f_1 ∑_r ∈Γy_rP_r (Ψ_qL) P_r (Ψ_uR) + h.c.ℒ_0 = - δ_αβm_yΨ_qRΨ_uL + h.c. where P_r is the projector from the representation space of G to the subspace associated to the irrep r of H. Proceeding in a similar way it is straightforward to include the d_R sector, as well as the light generations of fermions.We now integrate the composite degrees of freedom using their tree-level equations of motion and calculate the correlators for the elementary fields that transform according to each of the irreps inα_H or β_H (both dynamical and spurious). After integration, the effective fermionic Lagrangian density for the elementary fields in this vacuum takes the form:ℒ_eff = ∑_r ∈α_Hψ_qL^r (1 + Π_L^r) ψ_qL^r + ∑_s ∈β_Hψ_uR^s (1 + Π_R^s) ψ_uR^s + ∑_t ∈Γ (ψ_qL^tΠ_LR^tψ_uR^t + ψ_uR^tΠ_RL^tψ_qL^t) with the different correlators given by the following expressions:Π_L^r= -f_0^2 λ_q^2p^2 -I_Γ^r (m_u,1^2 + |f_1 y_r|^2)/δ_αβ [|m_y|^2 |f_1 y_r|^2 - p^2 (|f_1 y_r|^2 + |m_y|^2) + f_1 (y_r^* m_y + y_r m_y^*) m_q,1 m_u,1]+ (m_q,1^2 - p^2) (I_Γ^r m_u,1^2 - p^2) Π_R^r= -f_0^2 λ_u^2p^2 -I_Γ^r (m_q,1^2 + |f_1 y_r|^2)/δ_αβ [|m_y|^2 |f_1 y_r|^2 - p^2 (|f_1 y_r|^2 + |m_y|^2) + f_1 (y_r^* m_y + y_r m_y^*) m_q,1 m_u,1]+ (I_Γ^r m_q,1^2 - p^2) (m_u,1^2 - p^2) Π_LR^r= f_0^2 λ_qλ_uδ_αβm_y (|f_1 y_r|^2 - p^2) +I_Γ^r m_q,1m_u,1f_1 y_r/δ_αβ [|m_y|^2 |f_1 y_r|^2 - p^2 (|f_1 y_r|^2 + |m_y|^2) + f_1 (y_r^* m_y + y_r m_y^*) m_q,1 m_u,1]+ (m_q,1^2 - p^2) (m_u,1^2 - p^2) Π_RL^r= f_0^2 λ_qλ_uδ_αβm_y^* (|f_1 y_r|^2 - p^2) +I_Γ^r m_q,1m_u,1f_1 y_r^*/δ_αβ [|m_y|^2 |f_1 y_r|^2 - p^2 (|f_1 y_r|^2 + |m_y|^2) + f_1 (y_r^* m_y + y_r m_y^*) m_q,1 m_u,1]+ (m_q,1^2 - p^2) (m_u,1^2 - p^2)where I_Γ^r is the characteristic or indicator function of the set Γ which takes the value 1 if r ∈Γ and is 0 otherwise.Note that in the case α≠β and r ∉Γ the correlators reduce to:Π_L^r = -f_0^2 λ_q^2/(p^2 - m_q,1^2) ,Π_R^r = -f_0^2 λ_u^2/(p^2 - m_u,1^2) ,Π_LR^r = Π_RL^r = 0.which are independent of r and thus the same for all irreps of H that don't belong to Γ. Proceeding in a similar way for the gauge fields we obtain the following correlators in the SU(6)-sector:Π^G_81=Π^G_18=Π^G_11=p^2f_6,0^2/2p^2-f_6,0^2g_6,1^2 ,Π^G_33=f_6,0^2(2p^2-f_6,1^2g_6,1^2)/2(2p^2-f_6,0^2g_6,1^2-f_6,1^2g_6,1^2) .The case of SO(5)×U(1)_X can be found, for example, in Ref. <cit.>.§ INVARIANTS FOR THE IRREPS OF SO(5) In this Appendix, we present the explicit form of the F^f_r(s_5) and G^ff'_r(s5) functions of Eq. (<ref>). Most of these had been calculated previously in the literature, see for example: <cit.>.For the singlet of SO(5) we have: 0.7 1_2/3 of SO(5)×U(1)_X u_L d_L u_R (1,1)_2/3 of SU(2)_L ×SU(2)_R F^u_L_11(s_5) = 0 F^d_L_11(s_5) = 0 F^u_R_11(s_5) = 1 Now, keeping the same layout for the tables but simplifying the notation, the rest of the F^f_r functions are: 0.7 4_1/6u_L d_L u_R (2,1)_1/6 1 + c_5/2 1 + c_5/2 1 - c_5/2(1,2)_1/6 1 - c_5/2 1 - c_5/2 1 + c_5/25_2/3u_L d_L u_R (2,2)_2/3 1 + c_5^2/2 1 s_5^2(1,1)_2/3 s_5^2/2 0 c_5^210_2/3u_L d_L u_R (2,2)_2/3 1 + c_5^2/2 c_5^2 s_5^2/2(3,1)_2/3 s_5^2/4 s_5^2/2 (c_5^2 - 1)^2/4(1,3)_2/3 s_5^2/4 s_5^2/2 (c_5^2 + 1)^2/4 0.7 14_2/3u_L d_L u_R (3,3)_2/3 1/4 (2 + 3 c_5^2) s_5^2 s_5^2 15/64 (1 - c_5^2 + s_5^2)^2(2,2)_2/3 1/2(c_ 5^4 + s_5^4 + c_5^2 (1 - 2 s_5^2)) c_5^2 5 c_5^2 s_5^2/2(1,1)_2/3 5 c_5^2 s_5^2/4 0 1/64 (3 + 5 c_5^2 - 5 s_5^2)^2 0.716^(q_L ∈ (2,1))_1/6u_L d_L u_R (3,2)_1/6 -15/32 (c_5 - 1)(c_5 + 1)^2 -15/32 (c_5 - 1)(c_5 + 1)^2 15/32 (c_5 - 1)^2(c_5 + 1)(2,3)_1/6 15/32 (c_5 - 1)^2(c_5 + 1) 15/32 (c_5 - 1)^2(c_5 + 1) -15/32 (c_5 - 1)(c_5 + 1)^2(2,1)_1/6 1/32 (5 c_5 - 1)^2 (c_5 + 1) 1/32 (5 c_5 - 1)^2 (c_5 + 1) -1/32 (c_5 - 1) (5 c_5 + 1)^2(1,2)_1/6 -1/32 (c_5 - 1) (5 c_5 + 1)^2 -1/32 (c_5 - 1) (5 c_5 + 1)^2 1/32 (5 c_5 - 1)^2 (c_5 + 1) 0.7 16^(q_L ∈ (2,3))_1/6u_L d_L u_R (3,2)_1/6 1/32 (11 - 13 c_5 + 5 c_5^2 - 3 c_5^3) 1/32 (11 - 13 c_5 + 5 c_5^2 - 3 c_5^3) 15/32 (c_5 - 1)^2 (c_5 + 1)(2,3)_1/6 1/32 (11 + 13 c_5 + 5 c_5^2 + 3 c_5^3) 1/32 (11 + 13 c_5 + 5 c_5^2 + 3 c_5^3) -15/32 (c_5 - 1) (c_5 + 1)^2(2,1)_1/6 5/32 (c_5 - 1)^2 (c_5 + 1) 5/32 (c_5 - 1)^2 (c_5 + 1) -1/32 (c_5 - 1) (5 c_5 + 1)^2(1,2)_1/6 -5/32 (c_5 - 1) (c_5 + 1)^2 -5/32 (c_5 - 1) (c_5 + 1)^2 1/32 (5 c_5 - 1)^2 (c_5 + 1)where we have used the shorthand notation s_5 = sin(h_5f_5) and c_5 = cos(h_5f_5)In a similar fashion, but with a different layout, the expressions for the G^u_L u_R_r are shown in the following table: 0.7 4-9 6 c| u_R in 3-9 1 |c|r ↓ 1_2/3 4_1/6 5_2/3 10_2/3 14_2/3 16_1/6 1-9 1 |c18*[origin=c]90u_L in1 |c2*4_1/61 |c| (2,1)_1/6 - i/2 s_5 - - - -1/8 (5 c_5 + 1) s_5 3-9 1 |c 1 |c1 |c| (1,2)_1/6 - -i/2 s_5 - - - -i/8 (5 c_5 - 1) s_5 2-9 1 |c 1 |c2*5_2/31 |c| (2,2)_2/3 - - s_5 c_5/√(2) i s_5/2 i √(5)/2 c_5 s_5 - 3-9 1 |c 1 |c 1 |c| (1,1)_2/3 -s_5/√(2) - -s_5 c_5/√(2) - 5 s_5^3 - 4 s_5/4√(2) -2-9 1 |c 1 |c3*10_2/31 |c| (2,2)_2/3 - - s_5/√(2) i c_5 s_5/2 i √(5)/2 c_5^2 s_5 -3-9 1 |c 1 |c 1 |c| (3,1)_2/3 - - - -i/4 s_5 (c_5 - 1) - -3-9 1 |c 1 |c 1 |c| (1,3)_2/3 - - - -i/4 s_5 (c_5 + 1) - -2-9 1 |c 1 |c3*14_2/31 |c| (3,3)_2/3 - - - - i 3 √(5)/8 c_5 s_5^3 -3-9 1 |c 1 |c 1 |c| (2,2)_2/3 - - c_5 s_5/√(2) i/2 (s_5 - 2 s_5^3) i √(5)/2 c_5 s_5 (1-2 s_5^2) -3-9 1 |c 1 |c 1 |c| (1,1)_2/3 -i √(5)/2 c_5 s_5 - -i √(5)/2 c_5^2 s_5 - -i √(5)/8 c_5 s_5 (4 - 5 s_5^2) -2-9 1 |c 1 |c4*16^(q_L ∈ (2,3))_1/61 |c| (3,2)_1/6 - - - - - -√(5)/32(c_5-1)(3 c_5-1) s_53-9 1 |c 1 |c 1 |c| (2,3)_1/6 - - - - - √(5)/32(c_5 + 1)(3 c_5 + 1) s_53-9 1 |c 1 |c 1 |c| (2,1)_1/6 - -i √(5)/8(c_5 - 1) s_5 - - - -√(5)/32(c_5-1)(5 c_5 + 1) s_5 3-9 1 |c 1 |c 1 |c| (1,2)_1/6 - -√(5)/8(c_5 + 1) s_5 - - - -√(5)/32(c_5 + 1)(5 c_5 - 1) s_52-9 1 |c 1 |c4*16^(q_L ∈ (2,1))_1/61 |c| (3,2)_1/6 - - - - - -15/32 s_5^33-9 1 |c 1 |c 1 |c| (2,3)_1/6 - - - - - 15/32 s_5^33-9 1 |c 1 |c 1 |c| (2,1)_1/6 - i/8 (5 c_5 - 1) s_5 - - - 1/32(1 - 25 c_5^2) s_53-9 1 |c 1 |c 1 |c| (1,2)_1/6 - 1/8 (5 c_5 + 1) s_5 - - - -1/32(1 - 25 c_5^2) s_51-9 If a d_R quark was to be introduced in the model, the functions F^d_R_r and G^d_L d_R_r would also be needed. 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B 347, 300 (1995) [hep-ph/9409419]. | http://arxiv.org/abs/1704.08704v2 | {
"authors": [
"Leandro Da Rold",
"Iván A. Davidovich"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170427180143",
"title": "A symmetry for $ε_K$"
} |
Departamento de Astronomía y Astrofísica, Universitat de València, Dr. Moliner 50, 46100, Burjassot (València), SpainDepartamento de Astronomía y Astrofísica, Universitat de València, Dr. Moliner 50, 46100, Burjassot (València), Spain Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apdo. Postal 48-3, 62251, Cuernavaca, Morelos, México. Departamento de Astronomía y Astrofísica, Universitat de València, Dr. Moliner 50, 46100, Burjassot (València), Spain Observatori Astronòmic, Universitat de València, C/ CatedráticoJosé Beltrán 2, 46980, Paterna (València), Spain There is increasing numerical evidence that scalar fields can formlong-lived quasi-bound states around black holes. Recent perturbative andnumerical relativity calculations have provided further confirmation in avariety of physical systems, including both static and accreting black holes,and collapsing fermionic stars. In this work we investigate this issue yet againin the context of gravitationally unstable boson stars leading to black holeformation. We build a large sample of spherically symmetric initial models, bothstable and unstable, incorporating a self-interaction potential with a quarticterm. The three different outcomes of unstable models, namely migration to thestable branch, total dispersion, and collapse to a black hole,are also presentfor self-interacting boson stars. Our simulations show that forblack-hole-forming models, a scalar-field remnant is found outside the black-holehorizon, oscillating at a different frequency than that of the originalboson star. This result is in good agreement with recent spherically symmetricsimulations of unstable Proca stars collapsing to blackholes <cit.>.95.30.Sf 04.70.Bw04.25.dgQuasistationary solutions of scalar fields around collapsing self-interacting boson stars José A. Font December 30, 2023 ========================================================================================= § INTRODUCTIONThe development of stable numerical relativity codes based on hyperbolic formulations of Einstein's equations, accompanied with suitable gauge conditions, has beencritical for recent advances in our understanding of astrophysical systemsinvolving strong gravity. In particular, those technical developments have allowed accuratenumerical evolutions of highly dynamical spacetimes up to, and well beyond, the formationof black holes. Further steps have also been taken with the incorporation of mattercontent in black-hole spacetimes, specifically in the form of scalar fields, a typeof matter that has found recurrent use in numerical relativity. Within such aframework, recent studies of the Einstein-Klein-Gordon (EKG) system, both inthe linear and nonlinear regime, have shown that massive scalar fields surroundingblack holes can accommodate a type of oscillatory mode which only decays atinfinity <cit.>. Thesequasi-bound states may thus linger around the black hole in the form of a long-livedremnant (a wig) of scalar field. For both, scalar fields around supermassiveblack holes and axion-like scalar fieldsaroundprimordialblack holes,ithas beenfoundthat thefields can indeed survive for cosmological timescales <cit.>. Moreover, for spinning black holes, quasi-bound states can yield exponentially growing modes <cit.> and hairy-black-holesolutions <cit.>.On the other hand, scalar fields are also known to allow for soliton-likesolutions, i.e. static,sphericallysymmetricsolutionsoftheEKG systemfor a massive andcomplex field <cit.>,whichare commonly knownas boson stars (see <cit.> for a review). Thedynamical fate of boson stars has been thoroughly investigated numerically, bothusing perturbation theory <cit.> and fully nonlinearnumericalsimulations <cit.>.Ref. <cit.> in particular first showed that the fate of unstableboson-star solutions was either the formation of a black hole or the migration of the starto the stable branch, regardless of the sign of the binding energy. A third outcome forunstable boson stars is their totaldispersion <cit.> a situationwhich only happens for boson stars with negative binding energy.In this work we build a comprehensive sample of initial models of boson stars,incorporating a self-interaction potential with a quartic term. The inclusion ofsuch self-interaction provides extra pressure support against gravitationalcollapse and increases the range of possible maximum masses of boson stars,allowing to encompass models with astrophysical significance. Here, we revisit the stability of the solutions for different values of theself-interaction coupling constant λ, incorporating values as large asΛ≡λ/4π G μ^2=100, not previously accounted for (hereμ is the bare mass of the scalar field). Our findings are consistent withthe three different outcomes for unstable models, namely migration to the stablebranch, total dispersion, and collapse to a black hole, reported previously for both theλ=0 (mini-)boson star case and for self-interacting bosonstars (see <cit.>).We, however, focus on a particular subset of collapsing models.Making use of thespecific techniques developed by <cit.> to evolve black-holespacetimes in spherical symmetry using spherical-polar coordinates, we are able tofollow the dynamics of the system for very long periods of time, well beyond black holeformation and in an entirely stable manner. Using these techniques we showed recently that quasi-bound states can form in the vicinity of a black hole born dynamicallyfrom the collapse of a neutron star surrounded by a scalar field <cit.>.Here we show that long-lived quasi-bound states can also form after the collapse of aself-interacting boson star. Similar results have also recently been obtained inspherical simulations of unstable Proca stars collapsing to blackholes <cit.> as well as for axion stars <cit.>.This paper is organized as follows: Section <ref> briefly describes themathematical formulation of the EKG system. Section <ref> discusses the constructionof the initial data while Section <ref> gives a brief account of numericalaspects of the simulations. Our main findings and results are discussed inSection <ref>. Finally, Section <ref> summarizesour conclusions. Greek indices run over spacetime indices whileLatin indices run over spatial indices only. We use geometrized units, c=G=1. § BASIC EQUATIONS We investigate the dynamics of a self-interacting scalar field configuration around a black hole by solving numerically the coupled EKG system R_αβ-1/2g_αβR=8π T_αβ ,with the scalar field matter content given by the stress energy tensorT_αβ = 1/2(D_αΦ )^*(D_βΦ )+1/2(D_αΦ)(D_βΦ )^*- 1/2g_αβ(D^σΦ)^*(D_σΦ) -μ^2/2g_αβ|ΦΦ^*|- 1/4 λ g_αβ|ΦΦ^*|^2.We consider the following potential for the scalar fieldV(Φ^2)=μ^2|Φ|^2 +λ/2|Φ|^4, whereV_int:=1/4 λ |Φ|^4 is a quartic self-interactionpotential with coupling λ. We also introduce the dimensionless quantityΛ≡λ/4πμ^2. The scalar field obeys the Klein-Gordonequation( -dV/d|Φ|^2)Φ=0,where the D'Alambertian operator is defined by := (1/√(-g))∂_α(√(-g)g^αβ∂_β). Φ is dimensionless and μ has dimensions of (length)^-1. In spherical symmetry, the spatial line element can be written asdl^2 = e^4χ (a(t,r)dr^2+ r^2 b(t,r)dΩ^2),where dΩ^2 = dθ^2 + sin^2θ dφ^2 is the solid angle element and a(t,r) and b(t,r) are two non-vanishing metric functions. Moreover, χ is related to the conformal factor ψ as ψ = e^χ = (γ/γ̂)^1/12, with γ and γ̂ being the determinants of the physical and conformal 3-metrics, respectively. Theyare conformally related by γ_ij = e^4 χγ̂_ij.In this work we employ the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism of Einstein's equations <cit.> where the evolved fields are the conformally related 3-dimensional metric, the conformal exponent χ, the trace of the extrinsic curvature K, the independent component of the traceless part of the conformal extrinsic curvature, A_a≡ A^r_r, A_b≡ A^θ_θ=A^φ_φ, and the radial component of the conformal connection functions. The interested reader is addressed to <cit.> for further details. In particular, the explicit form of the evolution equations for the gravitational field that we use in this work are given by Eqs. (9)-(11) and (13)-(15) in Ref. <cit.>.As in our previous work <cit.>, in order to solve the Klein-Gordon equation we use two first-order variables defined as Π :=n^α∂_αΦ=1/α(∂_tΦ-β^r∂_rΦ), Ψ := ∂_rΦ .Therefore, from Eq. (<ref>) we obtain the following system of first-order equations: ∂_tΦ = β^r∂_rΦ+αΠ , ∂_tΨ = β^r∂_rΨ+Ψ∂_rβ^r+∂_r(αΠ), ∂_tΠ = β^r∂_rΠ+α/ae^4χ[∂_rΨ+ Ψ(2/r-∂_ra/2a+∂rb/b+2∂_rχ)]+ Ψ/ae^4χ∂_rα+α KΠ - α(μ^2+λΦ^2)Φ .The right-hand-sides of the gravitational field evolution equations contain matter source terms (see Eqs. (9)-(11) and (13)-(15) in <cit.>), denoted by ℰ, S_a, S_b and j_r. These terms are components of the energy-momentum tensor, Eq. (<ref>), or suitable projections thereof, and are given byℰ := n^αn^βT_αβ=1/2(|Π|^2+|Ψ|^2/ae^4χ)+1/2μ^2|Φ|^2+1/4λ |Φ|^4, j_r := -γ^α_rn^βT_αβ=-1/2(Π^*Ψ+Ψ^*Π) ,S_a := T^r_r=1/2(|Π|^2+|Ψ|^2/ae^4χ)-1/2μ^2|Φ|^2-1/4λ |Φ|^4 ,S_b := T^θ_θ=1/2(|Π|^2-|Ψ|^2/ae^4χ) -1/2μ^2|Φ|^2-1/4λ |Φ|^4 . The Hamiltonian and momentum constrains are given by the following two equations: ℋ ≡R-(A^2_a+2A_b^2)+2/3K^2-16πℰ=0, ℳ_r ≡ ∂_rA_a-2/3∂_rK+6A_a∂_rχ+ (A_a-A_b)(2/r+∂_rb/b)-8π j_r=0.The latter two equations are only computed in our code to monitor the accuracy of the numerical evolutions. § INITIAL DATA Spherical boson stars are described by the radial function Φ(r,t) = Φ_0(r) e^iω t where ω is the oscillation frequency of the field. Following <cit.> we obtainthe initial data for a boson star in polar-areal coordinates, for which the line element is given byds^2 = -α^2(r')dt^2 + a^2(r')dr^2+r'^2dΩ^2 ,and r' is the radial coordinate. The EKG system for a boson star reads:∂_r' a/a = 1-a^2/2r'+2π r'[ ω^2Φ_0^2 a^2/α^2+Ψ_0^2 +a^2Φ_0^2(μ^2+1/2λΦ_0^2)] , ∂_r'α/α = a^2 - 1/r' +∂_r' a/a-4πr' a^2Φ_0^2(μ^2+1/2λΦ_0^2) , ∂_r'Φ_0= Ψ_0 , ∂_r'Ψ_0=-Ψ_0(2/r' + ∂_r'α/α- ∂_r' a/a)-ω^2Φ_0^2a^2/α^2 + a^2(μ^2 + λΦ_0^2)Φ_0 . By solving these equations we obtain the spacetime metric potentials,g_tt=-α^2, g_rr=a^2, and theradial distribution of the scalarfield, Φ_0. The mass of a boson star is computed usingthe definition ofthe Misner-Sharp mass function,M_MS = r'_max/2( 1 -1/a^2(r'_max)) ,where r'_max is the radial coordinate at the outer boundary of our computational grid. The total mass of a boson star can also be computed using the Komar integral <cit.>,M_BS = ∫_Σ(2T_t^t-T_α^α)α√(γ) dr dθ dφ ,where Σ is a spacelike slice extending from a horizon, in case one exists,up to spatial infinity. To study the stability properties of the constructedequilibrium models we need to compute the Noether charge associated with the total bosonic number N, which is defined asN = ∫ g^0νj_να√(γ) dr dθ dφ ,where j_ν = i/2(Φ^*∂_νΦ - Φ∂_νΦ^*)is the conserved current associated with the transformation of the U(1) group.Finally, the sign of the binding energyE_b = M_ MS-Nμ,will determine the outcome of unstable models.Representative sequences of equilibrium models of boson stars are plotted inFigure <ref>. This figure shows four different mass profiles as afunction of the central scalar field value for four different values of theself-interaction coupling constant, namely Λ ={0,10,40,100}.For any given Λ two important points are explicitly indicated in eachcurve, the maximum mass, marked with a purple square, and the point at whichE_b = 0, marked with a cyan inverted triangle. For each sequence, thelocation of the maximum mass indicates the critical point separating the stableand the unstable branches. Boson stars situated at the left of the point ofmaximum mass are stable, while those on the right are unstable. The maximum massincreases monotically with Λ^1/2 <cit.>. Forsufficiently large values of Λ the self-interaction term allows forsignificantly larger masses than for non-self-interacting (mini-)boson stars. We study the stability of these equilibrium models through numerical timeevolutions. These are triggered by adding suitable small-amplitude perturbationsto the initial data profiles. We consider two types of perturbations, eitherthose associated with the intrinsic truncation error of the finite-differencerepresentation of the PDEs we solve or those associated with a functionalmodification of the actual radial distributions. While the evolutions of theboson stars may seem a priori easily predictable, telling from their locationwith respect to the maximum in the M_ BS vs. Φ(r=0) diagram,there are other aspects to consider which may affect the actual evolutions. Infact, depending on the sign of the binding energy, the point at whichE_b = 0, and on the perturbation, the stars will undergo differentfates. On the one hand, as we show below, an unstable boson star with positivebinding energy which is perturbed only with the discretization numerical error,migrates to the model with the same mass in the corresponding stable branch.However, if it is perturbed by slightly increasing its mass, it can collapsegravitationally and form a black hole. On the other hand, a boson star with anexcess energy, i.e. placed at the right of the zero binding energy point, is nolonger bounded and it will disperse away with time.The specific boson-star models that we generate and evolve numerically are indicated by the empty circles in Fig. <ref>. Quantitative details of the main model parameters are reported in Table <ref>. The boson star initial configurations are built in polar-areal coordinates but the time evolutions are performed in our code using isotropic coordinates, Eq. (<ref>). To solve this problem, we have to take two steps, see <cit.>. First, we perform a change of coordinates from polar-areal to isotropic coordinates withr_max = [ (1+√(a(r'_max))/2)^2r'_max/a(r'_max)],dr/dr' =ar/r' ,where Eq. (<ref>) is used as the initial value to integrate Eq. (<ref>) backwards. Then, we obtain the conformal factor usingψ = √(r'/r) .With this procedure our initial solution is described in isotropic coordinates and we can write the initial values of the other scalar field quantities as:Φ(r, t = 0)= Φ_0 , Ψ(r, t = 0)= Ψ_0 ,Π(r, t = 0) =iω/αΦ_0 .Finally, we interpolate the solution in an isotropic grid employing acubic-spline interpolation <cit.> that guarantees thecontinuity of the second derivative to minimize high-frequency noise associatedwith the interpolation.§ NUMERICAL FRAMEWORK As in our previous papers, the BSSN evolution equations for the geometry andthe evolution equations for the scalar field are solved numerically using asecond-order PIRK scheme <cit.>. This scheme canhandle in a satisfactory way the singular terms that appear in the evolution equations due to our choice of curvilinear coordinates.Explicit details about our numerical implementation have been reportede.g. in <cit.>. We also notethat the convergence properties of our numerical code have been extensivelytested before in various physical systems, including the EKG equations withself-interaction, seee.g. <cit.>.In the simulations reported in this work we consider two computational grids,namely a grid to obtain the equilibrium models of boson stars in polar-arealcoordinates, and another one to evolve those models in isotropic coordinates.Our polar-areal grid is an equidistant grid with spatial resolution Δ r' =0.001 spanning the interval r' ∈ [0.0,150.0]. On the other hand, ourisotropic grid is composed of two patches, a geometrical progression in theinterior part up to a given radius and a hyperbolic cosine outside. Using theinner grid alone would require too many grid points to place the outer boundarysufficiently far from the origin (and hence prevent the effects of possiblespurious reflections), while using only the hyperbolic cosine patch wouldproduce very small grid spacings in the inner region of the domain, leading toprohibitively small timesteps due to the Courant-Friedrichs-Lewy (CFL)condition. Details about the computational grid can be found in<cit.>. In our work the minimum resolution Δ rwe choose for the isotropic logarithmic grid is Δ r=0.025. With thischoice the inner boundary is then set to r_min = 0.0125 and the outerboundary is placed at r_max = 6000 at the nearest (in some models itis placed even further away, at r_max = 10000). The time step isgiven by Δ t = 0.3 Δ r in order to obtain long-term stablesimulations. § RESULTS§.§ Stable models Models A, D, G, and J in Fig. <ref> are all stable models.Therefore, the time evolution of the physical quantities that characterize them,as e.g. the central value of the scalar field, should remain constant. However,due to the grid discretization error, any of those quantities will insteadoscillate around the equilibrium value. This is shown in Fig. <ref>where we plot the central value of the scalar field for model A for twodifferent resolutions of the polar-areal grid employed to build the initialdata. In this figure we can also observe how when the resolution of the initialdata is reduced from Δ r'=0.01 to Δ r'=0.001 the amplitude of theoscillation is significantly reduced. All of our stable initial models areindeed seen to oscillate around the central equilibrium values. As an example oftheir stability we plot in Fig. <ref> the time evolution of thecentral scalar field for models A (Λ =0) and J (Λ =100).Note that the (purely numerical) secular drift of the initial central value of the scalar field ofmodel Aapparent in Fig. <ref> and hardly visible in Fig. <ref> (compare the two blue curves, both corresponding to the model with Λ=0) issimply a consequence of the change of scalein the vertical axes of both figures. By Fourier-transforming the time evolutionof the central value of the scalar field we obtain the corresponding frequency ofoscillation ω of the models. Those values are reported in the last column ofTable <ref>. The stable models oscillate with a single fundamentalfrequency whose value decreases with increasing Λ. §.§ Unstable models Another possible outcome of the evolution of our initial data is the totaldispersion of the boson star or its gravitational collapse. Let us startconsidering the first possibility. Such unstable situation will happen when thebinding energy is positive since due to the energy excess thestar will no longer remain bounded. The subset of initial models that can followthis trend are boson stars C, F, I, L, and M in Fig. <ref>. As anexample we plot in Fig. <ref> the radial profiles of the scalarfield at selected times of the evolution corresponding to model C (indicated inthe legend). In this case the central value of the scalar field rapidlydecreases with time, the boson star suffers a drastic radial expansion anddisperses away. All other unstable models (F, I, L and M) with a positivebinding energy display the same fate.Let us now consider the evolution of initial models that are located in theunstable branch, i.e. between the critical point (maximum mass of theconfiguration) and the E_b= 0 point. These are models B, E, H, and Kin Fig. <ref>. Previous numerical work <cit.> hasshown that the fate of these models is to collapse gravitationally to form ablack hole. However, we find that these models can also migrate to thestable branch of equilibrium configurations,depending on the perturbation (seealso earlier work by <cit.>). If the only perturbation of theinitial data is the one due to the discretization error, the outcome is amigration to the stable branch. However, if we include a slightly largerperturbation in the initial data, the models collapse to form black holes. Thefirst type of evolution, while mathematically plausible but unlikely on physicalgrounds, has been previously observed in the case of neutron stars(see <cit.> for details and arguments against this evolution), bosonstars <cit.> and, recently, also in the case of unstable Procastars <cit.>. A migrating boson star will result in adifferent boson star. It will have the same mass but it will be located in thestable branch of the equilibrium configurations (and hence the central scalarfield will have a smaller value). As an example, Fig. <ref> showsthe migration of boson star models E and K. For model E the star moves from acentral value of the scalar field Φ(r=0, t=0) = 0.09 towards a final value ofΦ(r=0) ∼ 0.04. This is precisely the value for which we obtain a stableboson star with the same mass (cf. Fig. <ref>). In the bottom panelof Fig. <ref> one can also observe that, besides the overall migration,the evolution of model K excites more frequencies of oscillation than that ofmodel E. This behaviour is consistent with the fact that the nonlinear term inthe potential induces nonlinear couplings among the frequencies, which aremore apparent the larger the value of the self-interaction coupling constant. Let us now consider the evolution of truly perturbed unstable models.To perturb these models we add an extra 2% value to the initial scalar fieldby multiplying Φ by 1.02 after solving equations (<ref>)-(<ref>). We have checked by computing the binding energy that the perturbation does not changethe sign. We then compute the auxiliary variables given by Eqs. (<ref>)-(<ref>)using the perturbed scalar field. For simplicity, after adding the perturbation we do notrecompute the spacetime variables a and α. This produces a slight violationof the constraints and leads to a small difference between the masses computed with Eq. (<ref>) andEq. (<ref>) in polar-areal coordinates.However, since the perturbation is fairly small (yet still larger than that associated with thediscretization error) it does not substantially alter our original solution.To diagnose the appearance of a black hole in the evolution we compute themass of the BH through the apparent horizon (AH) area 𝒜, usingM_BH=√(𝒜/16π). The timeevolution of both the scalarfield energy (mass) and the BH mass for all unstable models is shown inFig. <ref>. The mass of the boson star is computed using theKomar integral, Eq. (<ref>). Contrary to the migrating case, addinga 2% perturbation on the initial data triggers the collapse of the solutionsand at some point in the evolution an AH forms. This time is indicated inFig. <ref> by the sudden change that is observed in theevolution of the energy of the scalar field, which is associated with the suddenincrease of the black hole mass from a zero value.Fig. <ref> shows that, as expected, there is a small differencebetween the boson star masses computed with Eq. (<ref>)and the black hole masses computed through the apparent horizon. This isbecause some part of the scalar field is released after the collapse.For all models, the black hole mass is consistently, and slightly, smaller (seeTable <ref>). This means that during the collapseto a black hole, a remnant of the initial scalar field is not swallowed by thehole but instead lingers around in the form of a spherical shell or cloud.Figs. <ref> and <ref> show the time evolution of the amplitudeof the central value of the scalar potential for the four unstable models. Thistime series is extracted at an observation point with a fixed radiusr_obs= 10.The scalar field does not disappear after the formation ofthe AH (which takes place for all models (well) before t=100;cf. Fig. <ref>), forming instead long-lived quasi-bound states.Forallofthemodelsthe fieldisseentobe clearly oscillating, asit is best visualized in the insets of the two figures. To identify thefrequencies at which thefield oscillates we perform a Fourier transform of thetime series and obtain the power spectrum. This power spectrum shows a set ofdistinct frequencies, as indicated in Table <ref>. Moreover, modelsB, E, and H, show a distinctive beating pattern due to the presence of overtonesof the fundamental frequency. In order to compare the frequencies of the scalar clouds resulting from the collapse of boson stars with the known frequencies of quasi-bound states around Schwarzschildblack holes, we consider next an scenario in which initially the black hole is alreadyformed and has the same mass than that formed after the collapse of a boson star. This initial Schwarzschild black hole is surrounded by a Gaussian spherical shell ofscalar field which is evolved maintaining the background metric fixed. In this setupwe find that after a short initial transient the field settles down into a long-lived modeakin to the ones showed in Figs. <ref> and <ref>. In order tocharacterize this field we Fourier-transform its amplitude to obtain the oscillationfrequencies ω_qb^(2). The results are shown in the last column ofTable <ref>. The excellent agreement between the frequenciesω_qb^(1) computed after black hole formation andthefrequencies ω_qb^(2) computed from the Gaussian pulse,is a clear indication that the configurations formed after the collapse of bosonstars are indeed nonlinear quasi-bound states. Finally, in order to study the effect of Λ on the frequencies and on thetime decay of the quasi-bound states, we perform the same scattering experiment butkeeping the mass M_BH fixed. We find that for sufficiently long times t∼10^5 M_BH the effect of Λ on the scalar field becomesnegligible. This result is expected because the field decays exponentially andthe dominating term is the the scalar-field mass. Therefore, despite the presenceof nonlinear terms in the potential, the frequencies of all quasi-bound states willeventually tend to that of the quasi-bound state with Λ=0 (for the sameblack hole mass). The timescale to reach that situation depends on the value ofΛ because the frequency is different for each value of the coupling constant(both, the real and imaginary parts). Note that if we rescale the frequencies reported inTable <ref> with the BH mass, they do not coincide. § CONCLUSIONS We have presented a new numerical study of the Einstein-Klein-Gordon system inspherical symmetry. In particular we have discussed numerical relativitysimulations of a large number of initial models of boson stars, both stable andunstable, and which incorporate a self-interaction potential with a quarticterm.Self-interaction provides extra pressure support against gravitationalcollapse, increasing the range of possible maximum masses of boson stars,allowing to encompass models with possibly larger astrophysical significance. Wehave revisited the stability of the initial solutions for different values ofthe self-interaction coupling constant Λ, as large as Λ=100, notpreviously considered in the literature (to the best of our knowledge). Oursimulations have shown that the three different outcomes for unstable models,namely migration to the stable branch, total dispersion, and collapse to a blackhole, reported before for the Λ=0 (mini-)boson starcase <cit.>, are also present forself-interacting boson stars. We have focused our investigation on a subset ofcollapsing models, studying the effects the self-interaction potential may have in thepresence of quasi-bound states. The existence of such long-lived quasi-boundstates around black holes is supported by increasing numerical evidence, bothbased on perturbative calculations as on fully numerical relativity <cit.>. Moreover,they have been confirmed in a variety of physical systems, including both staticand accreting black holes, and collapsing fermionic stars. In this work we haverevisited this issue in the context of gravitationally unstable boson starsleading to black hole formation. We have found that for black-hole-formingmodels, a scalar field remnant can indeed be found outside the black holehorizon, oscillating at a different frequency than that of the original bosonstar. This result is in good agreement with recent spherically symmetricsimulations of unstable Proca stars collapsing to blackholes <cit.>. § ACKNOWLEDGEMENTSWe thank Carlos Herdeiro and João G. Rosa for useful discussions and comments on the manuscript. Work supported by the Spanish MINECO (AYA2015-66899-C2-1-P), by theGeneralitat Valenciana (PROMETEOII-2014-069, ACIF/2015/216), by the CONACYTNetwork Project 280908 “Agujeros Negros y Ondas Gravitatorias”, and byDGAPA-UNAM through grant IA103616. The computations have been performed at theServei d'Informàtica de la Universitat de València. | http://arxiv.org/abs/1704.08023v1 | {
"authors": [
"Alejandro Escorihuela-Tomàs",
"Nicolas Sanchis-Gual",
"Juan Carlos Degollado",
"José A. Font"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170426091149",
"title": "Quasistationary solutions of scalar fields around collapsing self-interacting boson stars"
} |
This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 714955-POPSTAR), as well as the ANR projects JCJC VIP ANR-11-JS02-006 and Sequoia ANR-14-CE28-0030-01.D. Baelde, S. Delaune, and L. Hirschi]David Baeldea ]Stéphanie Delauneb ]Lucca Hirschic a,cLSV, ENS Cachan & CNRS, Université Paris-Saclay, France {baelde,hirschi}@lsv.ens-cachan.fr bCNRS & IRISA, France [email protected] privacy-type properties of security protocols can be modelledusing trace equivalence properties in suitable process algebras.It has been shown that such properties can be decided for interesting classes of finite processes (without replication) by means ofsymbolic execution and constraint solving. However, this does not suffice to obtain practical tools. Current prototypes suffer from a classical combinatorial explosion problem caused by the exploration of many interleavings in the behaviour of processes. Mödersheim et al. <cit.> have tackled this problem for reachability properties using partial order reduction techniques. We revisit their work, generalize it and adapt it for equivalence checking.We obtain an optimisation in the form of a reduced symbolic semantics that eliminates redundant interleavings on the fly. The obtained partial order reduction technique has been integrated in a tool called . We conducted complete benchmarks showing dramatic improvements. A reduced semantics for deciding trace equivalence [ December 30, 2023 ==================================================§ INTRODUCTION Security protocols are widely used today to secure transactions that rely onpublic channels like the Internet, where malicious agents may listen tocommunications and interfere with them.Security has a different meaning depending on the underlyingapplication. It ranges from the confidentiality of data (medical files,secret keys, etc.) to, verifiability in electronic voting systems. Anotherexample is the notion of privacy that appears in many contexts such asvote-privacy in electronic voting or untraceability in RFID technologies.To achieve their security goals, security protocols rely on various cryptographic primitives such as symmetric and asymmetric encryptions, signatures, and hashes. Protocols also involve a high level of concurrency and are difficult to analyse by hand.Actually,many protocols have been shown to be flawed several years after their publication (and deployment). For example, a flaw has been discovered in the Single-Sign-Onprotocol used, by Google Apps. It has been shown that a maliciousapplication could very easily get access to any other application(Gmail or Google Calendar) of their users <cit.>. This flaw hasbeen found when analysing the protocol using formal methods,abstracting messages by a term algebra and using the Avantssarvalidation platform <cit.>. Another example is a flaw on vote-privacydiscovered during the formal and manual analysis of an electronicvoting protocol <cit.>. Formal symbolic methods have proved their usefulness for precisely analysing thesecurity of protocols. Moreover, it allows one to benefit from machinesupport through the use of various existing techniques, ranging from model-checking to resolution and rewriting techniques. Nowdays, several verification tools are available, <cit.>. A synthesis of decidability and undecidability results for equivalence-based security properties, and an overview of existing verification toolsthat may be used to verify equivalence-based security properties can be found in <cit.>. In order to design decision procedures, a reasonable assumption is to bound the number of protocol sessions, thereby limiting the length of execution traces.Under such an hypothesis, a wide variety of model-checking approaches have beendeveloped (<cit.>), and several tools are now available to automatically verify cryptographicprotocols, <cit.>. A major challenge faced here is that one has to account for infinitely many behaviours of the attacker, who can generate arbitrary messages. In order to cope with this prolific attacker problem and obtain decision procedures, approaches based on symbolic semantics and constraint resolution have been proposed <cit.>. This has lead to tools for verifying reachability-based security propertiessuch as confidentiality <cit.> or, more recently, equivalence-based properties such as privacy <cit.>.Unfortunately, the resulting tools, especially those for checking equivalence ( <cit.>, <cit.>, <cit.>) have a very limited practical impact because they scale badly.This is not surprising since they treat concurrency in a very naive way, exploring all possible symbolic interleavings of concurrent actions.*Related work. In standard model-checking approaches for concurrent systems, theinterleaving problem is handled using partial order reduction (POR) techniques <cit.>. In a nutshell, these techniques aim to effectively exploit the fact that the order of execution of twoindependent (parallel) actions is irrelevant when checking reachability. The theory of partial order reduction is well developed in the context of reactive systems verification ( <cit.>). However, as pointed out by Clarke et al. in <cit.>, POR techniques from traditional model-checking cannot be directly applied in the context of security protocol verification. Indeed, the application to security requires one to keep track of the knowledge of the attacker, and to refer to this knowledge in a meaningful way (in particular to know which messages can be forged at some point to feed some input). Furthermore, security protocol analysis does not rely on the internal reduction of a protocol, but has to consider arbitrary execution contexts (representing interactions with arbitrary, active attackers). Thus, any input may depend on any output, since the attacker has the liberty of constructing arbitrary messages from past outputs. This results in a dependency relation which is a priori very large, rendering traditional POR arguments suboptimal, and calling for domain-specific techniques.In order to improve existing verification tools for security protocols, one has to design POR techniques that integrate nicely with symbolic execution. This is necessary to precisely deal with infinite, structured data. In this task, we get some inspiration from Mödersheim et al. <cit.>, who design a partial order reduction technique that blends well with symbolic execution in the context of security protocols verification. However, we shall see that their key insight is not fully exploited, and yields only a quite limited partial order reduction. Moreover, they only consider reachability properties (like all previous work on POR for security protocol verification) while we seek an approach that is adequate for model-checking equivalence properties.*Contributions. In this paper, we revisit the work of <cit.> to obtain apartial order reduction technique for the verification of equivalenceproperties. Among the several definitions of equivalence that have been proposed, we consider trace equivalence in this paper: two processes are trace equivalent when they have the same sets of observable traces and, for each such trace, sequences of messages outputted by the two processesare statically equivalent, indistinguishable for the attacker. This notion is well-studied and several algorithms and tools support it <cit.>. Contrary to what happens for reachability-based properties, trace equivalencecannot be decided relying only on the reachable states. The sequence ofactions that leads to this state plays a role. Hence, extra precautions have to be taken before discarding a particular interleaving: we have toensure that this is done in both sides of the equivalence in a similar fashion. Our main contribution is an optimised form of equivalence that discards a lot of interleavings, and a proof that this reduced equivalence coincides with trace equivalence.Furthermore, our study brings an improvement of the original technique <cit.> that would apply equally well for reachability checking. On the practical side, we explain how we integrated our partial order reduction into the state-of-the arttool <cit.>, prove the correctness of this integration, and provide experimental results showing dramatic improvements. We believe that our presentation is generic enough to be easily adapted forother tools (provided that they are based on a forward symbolic exploration of traces combined with a constraint solving procedure). A big picture of the whole approach along with the new results is given in Figure <ref>. Vertically, it goes from the regular semantics, to symbolic semantics and 's semantics. Those semantics have variants when our optimisations are applied or not: no optimisation, only compression or compression plus reduction. This paper essentially subsumes the conference paper that has been published in 2014 <cit.>. However, we consider here a generalization of the semantics used in <cit.>. This generalization notably allows us to capture the semantics usedin , which allows us to formally prove the integration of our optimisations in that tool. In addition, this paper incorporates proofs of all the results, additional examples, and an extensive related work section. Finally, it comes with a solid implementation in the tool <cit.>.*Outline. In Section <ref>, we introduce our model for security processes.We then consider the class of simple processes introduced in <cit.>, with else branches and no replication. Then we present two successive optimisations in the form of refined semantics and associated trace equivalences. Section <ref> presentsa compressed semantics that limits interleavings by executing blocks of actions.Then, by adapting well-known argument, this is lifted to a symbolic semantics inSection <ref>. Section <ref> presents the reduced semantics which makes use of dependency constraints to remove more interleavings. In Section <ref>, we explain how this reduced semantics has been integrated in the tool , prove its correcteness, and give some benchmarks obtained on several case studies. Finally, Section <ref> is devoted to related work, and concluding remarks are given in Section <ref>. An overview of the different semantics we will define and the results relating them is depicted in Figure <ref>. A table of symbols can be found in Appendix <ref>. § MODEL FOR SECURITY PROTOCOLSIn this section, we introduce the cryptographic process calculus that we will use to describe security protocols. This calculus is close to the applied pi calculus <cit.>.We consider a semantics in the spirit of the one used in <cit.> but we also allow to block some actions depending on a validity predicate. This predicate can be chosen in such a way that no action is blocked, making the semanticsas in <cit.>. It can also be chosen as in as we eventually do in order to prove the integration of our optimisations into this tool.§.§ SyntaxA protocol consists of some agents communicating on a network. Messages sent by agents are modeled using a term algebra. We assume two infinite and disjoint sets of variables,and . Members ofare denoted x, y, z, whereas members of are denoted w and usedas handles for previously output terms. We also assume a set of names, which are used for representing keys or nonces[Note that wedo not have an explicit set of restricted (private) names.Actually, allnames are restricted and public ones will be explicitly given tothe attacker.], and a signature Σ consisting of a finite set of function symbols. Terms are generated inductively from names, variables, and functionsymbols applied to other terms. For S ⊆∪∪, the set of terms built from S by applying function symbols in Σ is denoted by (Σ, S).We write (t) for the set of syntactic subterms of a term t. Terms in (Σ, ∪) are denoted by u, v, etc. while terms in (Σ,) represent recipes (describing how the attacker built a term from the available outputs) and are written M, N, R.We write (t) for the set of variables (fromor ) occurring in a term t. A term is ground if it does not contain any variable, it belongs to (Σ,). One may rely on a sort system for terms, but its details are unimportant for this paper.To model algebraic properties of cryptographic primitives, we consider an equational theory . The theory will usually be generated from a finite set of axioms enjoying nice properties (convergence) but these aspects are irrelevant for the present work. In order to model asymmetric encryption and pairing, we consider:Σ = {··, ··,·,··,·, ·}.To take into account the properties of these operators, we consider the equational theory _𝖺𝖾𝗇𝖼 generated by the three following equations:xyy =x, x_1x_2 = x_1, x_1x_2 = x_2.For instance, we have nskaskbskb =__𝖺𝖾𝗇𝖼ska.Our model is parameterized by a notion of message, intuitively meant to represent terms that can actually be communicated by processes. Formally, we assume a special subset of ground terms , only requiring that it contains at least one public constant. Then, we say that a ground term u is valid, denoted _(u), whenever for any v ∈(u), we have that there exists v' ∈ such that v =_ v'. This notion of validity will be imposed on communicated terms. As we shall see,can be chosen in such a way that the validity constraint allows us to discard some terms for which the computation of some parts fail. Note thatcan also be chosen to be the set of all ground terms, yielding a trivial validity predicate that holds for all ground terms. The following developments are parametrized by . The signature used in is Σ=Σ_c∪Σ_d where:[ Σ_c = Σ_0∪{𝖺𝖾𝗇𝖼(·,·),𝗉𝗄(·),𝖾𝗇𝖼(·,·), 𝗁𝖺𝗌𝗁(·), 𝗌𝗂𝗀𝗇(·,·),𝗏𝗄(·), ⟨·,·⟩}; Σ_d ={𝖺𝖽𝖾𝖼(·,·), 𝖽𝖾𝖼(·,·), 𝖼𝗁𝖾𝖼𝗄(·,·), π_1(·),π_2(·)} ]where Σ_0 may contain some additional user-defined function symbols.The equational theory _ of is an extension of the theory _𝖺𝖾𝗇𝖼 generated by adding the following equations:xyy =x 𝖼𝗁𝖾𝖼𝗄(𝗌𝗂𝗀𝗇(x,y),𝗏𝗄(y)) = x The validity predicate used in the semantics of is obtained by taking = (Σ_c,), the ground terms built using constructor symbols. This choice allows us to discard terms for which a failure will happen during the computation and which therefore do not correspond to a message: ⟨, akk'⟩ is not valid since akk' is not equal modulo _ to a term in (Σ_c,). We do not need the full applied pi calculus <cit.> to represent security protocols.Here,we only consider public channels and we assume that each process communicates on a dedicated channel. Formally, we assume a setof channels and we consider the fragment of simple processes without replication built on basic processes as defined in <cit.>. A basic process represents a party in a protocol, which may sequentially performactions such as waiting for a message, checking that a message has acertain form, or outputting a message. Then, a simple process is a parallel composition of such basic processes playing on distinct channels.The setof basic processeson c ∈ is defined using the following grammar (where u,v∈(Σ,∪) and x ∈):[ P,Q:= 0; | u=vPQ; | (c,x).P; | (c,u).P; ] A simple process = { P_1, …, P_n }is a multiset of basic processes P_i on pairwisedistinct channels c_i. We assume that null processes are removed. Intuitively, a multiset of basic processes denotes a parallel composition. For conciseness, we often omit brackets, null processes, and even “ 0”. Basic processes are denoted by the letters P and Q, whereas simple processes are denoted using and .During an execution, the attacker learns the messages that have been sent on the different public channels. Those messages are organized into a frame.A frame Φ is a substitution whose domain is included inand image is included in (Σ, ∪). It is written { wu, …}. A frame is ground when its image only contains ground terms.In the remainder of this paper, we will actually only consider ground frames that are made of valid terms.An extended simple process (denoted A or B) is a pair made of a simple process and a frame.Similarly, we define extended basic processes.When the context makes it clear, we may omit “extended” and simply call them simple processes and basic processes.We consider the protocol given in <cit.> designed for authenticating an agent with another one without revealing their identities to other participants. In this protocol, A is willing to engage in communication with B and wants to be sure that she is indeed talking to B and not to an attacker who is trying to impersonate B. However, A does not want to compromise her privacy by revealing her identity or the identity of B more broadly. The participants A and B proceed as follows:[ A → B : {N_a,_A}__B; B → A : {N_a,N_b,_B}__A ]First A sends to B a nonce N_a and her public key encrypted with the public key of B. If the message is of the expected form then B sends to A the nonce N_a,a freshly generated nonce N_b and his public key, all of this being encrypted with the public key of A.Moreover, if the message received by B is not of the expected form then B sends out a “decoy” message: {N_b}__B. This message should basically look like B's othermessage from the point of view of an outsider.Relying on the signature and equational theory introduced in Example <ref>, a session of role A played by agent a (with private key ska) with b (with public key pkb) can be modeled as follows:[ P(, pkb) (c_A, n_askapkb).;(c_A,x).;xskaxska = n_apkb0 ]Here, we are onlyconsidering the authentication protocol. A more comprehensive model should include the access to an application in case of a success.Similarly, a session of role B played by agent b with a can be modeled by the following basic process, where N = yskb.[Q(skb, pka)(,y) .;𝚒𝚏 N = pka 𝚝𝚑𝚎𝚗 (, Nn_bskbpka);𝚒𝚏 N = pka 𝚎𝚕𝚜𝚎 (, n_bskb) ] To model a scenario with one session of each role (played by the agents a and b), we may consider the extended process Φ_0 where: * {P(ska, skb), Q(skb, ska)}, and* Φ_0 {w_0 ska', w_1 ska,w_2 skb}. The purpose of ' will be clear later on. It allows us toconsider the existence of another agent a' whose public key ska' is known by the attacker. §.§ Semantics We first define a standard concrete semantics using a relation overground extended simple processes, extended simple processes Φ such that() = ∅ (as said above, we also assume that Φ contains only valid ground terms). The semantics of a ground extended simple process Φ is induced by the relation a over ground extended simple processes as defined in Figure <ref>.A process may input any valid term that an attacker can build (rule In): {x ↦ u} is a substitution that replaces any occurrence of x with u.Once a recipe M is fixed, we may note that there are still different instances of the rule, but only in the sense that u is chosen modulo the equational theory . In practice, of course, not all such u are enumerated. How this is achieved in practice is orthogonal to the theoretical development carried out here.In the Out rule, we enrich the attacker's knowledge by adding the newlyoutput term u, with a fresh handle w, to the frame. The two remaining rules are unobservable (τ action) from the point ofview of the attacker.When ℳ contains all the ground terms, (u) is true for any term u and this semantics coincides with the one defined in <cit.>. However, this parameter gives us enough flexibility to obtain a semantics similar to the one used in , and therefore formally prove in Section <ref> how to integrate our techniques in . The relation A a_1 ·…· a_k Bbetween extended simple processes,where k≥ 0 andeach a_i is an observable or a τ action, is defined in the usual way. We also consider the relationdefined as follows: A B if, and only if, there exists a_1 ·…· a_k such that A a_1 ·…· a_k B, andis obtained from a_1 ·…· a_k by erasing all occurrences of τ. Consider the simple process Φ_0 introduced in Example <ref> (withequal to (Σ_c,) as in ). We have:Φ_0(c_A,w_3) ·(,w_3) ·τ·(,w_4) ·(c_A,w_4) ·τ∅Φ. This trace corresponds to the normal execution of one instance of the protocol. The two silent actions have been triggered using the Then rule. The resulting frame Φ is as follows:Φ_0 ⊎{w_3 n_askaskb,w_4 n_an_bskbska}. §.§ Trace equivalenceMany interesting security properties, such as privacy-type properties studied, in <cit.>, are formalized using the notion of trace equivalence.Before defining trace equivalence,we first introduce the notion of static equivalencethat compares sequences of messages. Two frames Φ and Φ' are in static equivalence, ΦΦ', when we have that (Φ) = (Φ'), and: * (MΦ)⇔(MΦ') for any term M ∈(Σ,(Φ)); and* MΦ =_ NΦ⇔ MΦ' =_ NΦ' for any terms M, N ∈(Σ,(Φ)) such that (MΦ) and (NΦ).Intuitively, two frames are equivalent if an attacker cannot see the difference between the two situations they represent, they satisfy the same equalities and failures.Consider the frame Φ given in Example <ref> and the frame Φ' below:Φ' Φ_0 ⊎{w_3 n_aska'skb, w_4 n_bskb }.We have that ΦΦ'. This is a non-trivial equivalence.Intuitively, it holds since the attacker is not able to decrypt any of the ciphertexts, and each ciphertext contains a nonce that prevents him to build it from its components. Now, if we decide to give access to n_a to the attacker, consideringΦ_+ = Φ⊎{w_5n_a} and Φ'_+ = Φ' ⊎{w_5n_a}, then the two frames Φ_+ and Φ'_+ are not in static equivalence anymore as witnessed by M = w_5w_1w_2 and N = w_3. Indeed, we have that MΦ_+ =__𝖺𝖾𝗇𝖼 NΦ_+ whereas MΦ'_+ ≠__𝖺𝖾𝗇𝖼 NΦ'_+, and all thesewitnesses are valid. Let A and B be two extended simple processes. We have that AB if, for every sequence of actionssuch that AΦ, there exists 'Φ' such that B'Φ' and ΦΦ'.The processes A and B aretrace equivalent, denoted by AB, if AB and BA.Intuitively, the private authentication protocol presented inExample <ref>preserves anonymity if an attacker cannotdistinguish whether b is willing to talk to a (represented by theprocess Q(,)) or willing to talk to a'(represented by theprocess Q(,')), provided a, a' and b are honest participants. This can be expressed relying on the following equivalence:Q(,)Φ_0?Q(,')Φ_0. For illustration purposes,we also consider a variant of the process Q,denoted Q_0, where its 𝚎𝚕𝚜𝚎 branch has been replaced by 0 (the null process). We will see that the “decoy”message plays a crucial role to ensure privacy.We have that:Q_0(,)Φ_0(,w_1w_1w_2) ·τ·(,w_3)∅Φwhere Φ = Φ_0 ⊎{w_3 skan_bskbska}. We may note that this trace does not correspond to a normal execution of the protocol. Still, the first input is fed with the message skaskaskb which is a message of the expected format from the point of view of the process Q_0(skb,ska). Therefore, once conditionals are positively evaluated, the output (,w_3) can be triggered.This trace has no counterpart in Q_0(,')Φ_0. Indeed, we have that:Q_0(,')Φ_0(,w_1w_1w_2) ·τ∅Φ_0.Hence, we have that Q_0(,)Φ_0Q_0(,')Φ_0.However, it is the case that Q(,)Φ_0Q(,')Φ_0. This equivalence can be checked using the tool <cit.> within few seconds for a simple scenario as the one considered here, and that takes few minutes/days as soon as we want to consider 2/3 sessions of each role. § COMPRESSION BASED ON GROUPING ACTIONSOur first refinement of the semantics, which we call compression, is closely related to focusing from proof theory <cit.>: we will assign a polarity to processes and constrain the shape of executed traces based on thosepolarities. This will provide a first significant reduction of the number of traces to consider when checking equivalence-based properties between simple processes. Moreover,compression can easily be used as a replacement for the usual semantics in verification algorithms. The key idea is to force processes to perform all enabled output actions as soon as possible. In our setting, we can even safely force them to perform a complete block of input actions followed by ouput actions. Consider the process Φ with𝒫 = {(c_1,x).P_1, (c_2,b).P_2}. In order to reach {P_1{x ↦ u}, P_2}Φ∪{wb}, we have to execute the action (c_1,x) (using a recipe M that allows one to deduce u) and the action (c_2,b) (giving us a label of the form (c_2,w)).In case of reachability properties, the execution order of these actions onlymatters if M uses w.Thus we can safely perform the outputs in priority.The situation is more complex when considering trace equivalence. In that case, we are concerned not only with reachable states, but also with how those states are reached. Quite simply, traces matter. Thus, if we want to discard the trace (c_1,M).(c_2,w) when studying process 𝒫 and consider only its permutation (c_2,w). (c_1,M), we have tomake sure that the same permutation is available on the other process. The key to ensure that identical permutations will be available on both sides of the equivalence is our restriction to the class of simple processes. §.§ Compressed semantics We now introduce the compressed semantics. Compression is anoptimisation, since it removes some interleavings. But it also gives rise toconvenient “macro-actions”, called blocks, that combine a sequence ofinputs followed by some outputs, potentially hiding silent actions. Manipulating those blocks rather than indiviual actions makes it easier to define our second optimisation.For sake of simplicity, we consider initial simple processes. A simple process A = Φ is initial if for any P ∈, we have that P = 0, P = (c,x).P' or P = (c,u).P' for some term u such that (u). In other words, each basic process composing A starts with an input unless it is blocked due to an unfeasible output. Continuing Example <ref>, {P(, ),Q(,)}Φ_0 is not initial.Instead, we may consider {P_, Q(,)}Φ_0 whereP_(c_A,z). z = P(,)assuming thatis a (public) constant in our signature.The main idea of the compressed semantics is to ensure that when a basic process starts executing some actions, it actually executes a maximal block of actions. In analogy with focusing in sequent calculus, we say that the basic process takes the focus, and can only release it under particular conditions. We define in Figure <ref> how blocks can be executed by extended basic processes. In that semantics, the label ℓ denotes the stage of the execution, starting with i^+, then i^* after the first input and o^* after the first output. Going back to Example <ref>, we have that:Q_0(,)Φ_0(,w_1w_1w_2)·(,w_3) i^+0Φwhere Φ is as given in Example <ref>. As illustrated by the proof tree below, we also have Q_0(,')Φ_0i^+Φ_0 with = (,w_1w_1w_2). Q_0(,')Φ_0Q'Φ_0Q'Φ_0τ0Φ_00Φ_0ϵi^*Φ_0Q'Φ_0ϵi^*Φ_0Q_0(,')Φ_0i^+Φ_0where Q'= '(c_B,u) for some message u. Then we define the relationbetween extended simple processes as the least reflexive transitive relation satisfying the rules given in Figure <ref>.A basic process is allowed to properly end a block execution when ithas performed outputs and it cannot perform any more output or unobservable action (τ).Accordingly, we call proper block a non-emptysequence of inputs followed by a non-empty sequence of outputs, all on thesame channel. For completeness, we also allow blocks to be terminated improperly, when the process that is executing has performed inputs but no output, and has reached the null process 0 or an output which is blocked. Accordingly, we call improper block a non-empty sequence of inputs on the same channel. Continuing Example <ref>,using the rule block, we can derive{P_, Q_0(,)}Φ_0(,w_1w_1w_2)·(,w_3) P_Φwhere P_ is defined in Example <ref>. We can also derive {P_, Q_0(,')}Φ_0(,w_1w_1w_2)∅Φ_0 using the rule Failure. Note that the resulting simple process is reduced to ∅ even though P_ has never been executed. At first sight, killing the whole process when applying the rule Failure may seem too strong.However, even if this kind of scenario is observable by the attacker, it does not bring him any new knowledge, hence it plays only a limited role in traceequivalence:it is in fact sufficient to consider such improper blocks only at the end of traces. Consider = { (c,x).(c,y), (c',x')}.Its compressed traces are of the form (c,M).(c,N) and (c',M').The concatenation of those two improper traces cannot be executed in the compressed semantics. Intuitively, we do not loose anything fortrace equivalence, because if a process can exhibit those two improperblocks they must be in parallel and hence considering their combination is redundant.We now define the notions of compressed trace equivalence (denoted ) and compressed trace inclusion (denoted ), similarly toand ⊑ but relying oninstead of .Let A and B be two extended simple processes. We have that AB if, for every sequence of actionssuch that AΦ, there exists 'Φ' such that B'Φ' and ΦΦ'.The processes A and B arecompressed trace equivalent, denoted by AB, if A B and BA. We have that {P_,Q_0(,)}Φ_0{P_,Q_0(,')}Φ_0. The trace (,w_1w_1w_2)·(,w_3) exhibited in Example <ref> is executable from {P_,Q_0(,)}Φ_0. However, this trace has no counterpart when starting with {P_,Q_0(,')}Φ_0. §.§ Soundness and completeness We shall now establish soundness andcompleteness of the compressed semantics. More precisely, we show thatthe two relations and coincide on initial simpleprocesses (Theorem <ref>). All theproofs of this section are given in Appendix <ref>.Intuitively, we can always permute output(resp. input) actions occurring on distinct channels, and we can also permute an output with an input if the outputted message is not used to build the inputted term. More formally,we define an independence relation _a over actions as the least symmetric relation satisfying: * (c_i,w_i) _a (c_j,w_j) and (c_i,M_i) _a (c_j,M_j) as soon as c_i ≠ c_j, * (c_i,w_i) _a (c_j,M_j) when in addition w_i ∉(M_j).Then, we consider =__a to be the least congruence (w.r.t. concatenation) satisfying:·' =__a'· for alland ' with _a ',and we show that processes are equally able to execute equivalent (w.r.t. =__a) traces. lemmalempermuteLet A, A' be two extended simple processes and , ' be such that =__a'. We have that A A' if, and only if, A' A'.Now, considering traces that are only made of proper blocks, a strong relationship can be established between the two semantics. propositionprocompsoundnessLet A, A' be two simple extended processes, andbe a trace made of proper blockssuch that AA'. Then we have thatA A'. Actually, the result stated in Proposition <ref>immediately follows from the observation thatis included infor traces made of proper blocks since for them Failure cannot arise. propositionprocompcompletenessLet A, A' be two initial simple processes, andbe a trace made of proper blocks such thatA A'. Then, we have that AA'.This result is more involved and relies on the additional hypothesis that A and A' have to be initial to ensure that no Failure will arise. theoremtheocompsoundnesscompletenessLet A and B be two initial simple processes. We have thatAB ⟺ AB. The main difficultyis that Proposition <ref> only considers traces composed of proper blocks whereas we have to consider all traces. To prove the ⇒ implication, we have to pay attention to the last block of the compressed trace that can be an improper one (composed of several inputs on a channel c).The ⇐ implication is more difficult since we have to consider a traceof a process A that is an interleaving of some prefix of proper and improper blocks. We will first complete it with ^+ to obtain an interleaving of proper and improper blocks. We then reorder theactions to obtain a trace ' such that ·^+=__a' and ' = _io·_in where _io is made of proper blocks while _in is made of improper blocks. For each improper block b of _in, we show by applying Lemma <ref> and Proposition <ref> that A is able to perform _io in the compressed semantics and the resulting extended process can execute the improper block b. We thus have that A is able to perform_io· b in the compressed semantics and thus B as well. Finally, we show that the executions of all those (concurrent) blocks b can be put together, obtaining that B can perform ', and thusas well. Note that, as illustrated by the following example, the two underlying notions of traceinclusion do not coincide. Let = {(c,x)} and = {(c,x).(c,n)} accompanied with an arbitrary frame Φ.We have ΦΦ but ΦΦ since in the compressed semantics Φ is not allowed to stop its execution after its first input. § DECIDING TRACE EQUIVALENCE VIA CONSTRAINT SOLVINGIn this section, we propose a symbolic semantics for our compressed semantics following, <cit.>.Such a semantics avoids potentially infinite branching of our compressed semantics due to inputs from the environment.Correctness is maintained by associating with each process a set of constraints on terms.§.§ Constraint systemsFollowing the notations of <cit.>, we consider a new set ^2 ofsecond-order variables, denoted by X, Y, etc.We shall use those variables to abstract over recipes. We denote by (o) the set of free second-order variables of an object o, typically a constraint system. To prevent ambiguities, we shall useinstead offor free first-order variables.A constraint system = Φ consists of a frameΦ, and a set of constraints .We consider three kinds of constraints:DXx uvuv where D⊆, X∈^2, x∈ and u,v∈(Σ, ∪). The first kind of constraint expresses that a second-order variable X has to be instantiated by a recipe that uses onlyvariables from a certain set D, and that the obtained term should be x. The handles in Drepresent terms that have been previously outputted by the process. We are not interested in general constraint systems, but only consider constraint systems that are well-formed. Given a constraint system , we define a dependency order on first-order variables in () ∩ by declaring that x depends on y if, and only if,contains a deduction constraint DXx with y∈(Φ(D)). A constraint systemis well-formed if:* the dependency relationship is acyclic, and* for every x∈() ∩ (resp. X∈()) there is a unique constraint DXx in .For X∈(), we write D_(X) for the domain D⊆ of the deduction constraint DXx associated to X in .Continuing Example <ref>, let Φ = Φ_0 ⊎{w_3 Nn_bskbska} with N = yskb, andbe a set containing two constraints:{w_0,w_1,w_2}YyNska. We have that = Φ is a well-formed constraint system. There is only one first-order variable y ∈() ∩, and it does not occur in (Φ({w_0,w_1,w_2})), which is empty. Moreover, there is indeed a unique constraint that introduces y. Our notion of well-formed constraint systems is in line with what is used, in <cit.>. We use a simpler variant here that is sufficient for our purpose. A solution of a constraint system = Φ is a substitution θ such that (θ) = (), and Xθ∈(Σ, D_(X)) for any X ∈(θ). Moreover, we require that there exists a ground substitution λ with (λ) = () such that: * for every DXx in , we have (Xθ)(Φλ) =_ xλ,((Xθ)(Φλ)), and (xλ);* for every uv in , we have uλ =_ vλ, (uλ), and (vλ); and* for every uv in , we have uλ≠_ vλ,or (uλ), or (vλ).Moreover, we require that all the terms occurring in Φλ are valid. The set of solutions of a constraint systemis denoted (). Since we consider constraint systems that are well-formed, the substitution λ is unique modulogivenθ∈(). We denote it by λ_θ whenis clear from the context. Note that the validity constraints in the notion of solution of symbolic processes reflect the validity constraints of the concrete semantics (outputted and inputted terms must be valid and the equality between terms requires the two terms to be valid). Since we consider well-formed constraint systems, we may note that the substitution λ above is not obtained through unification. This substitution is entirely determined (modulo ) from θ by considering the deducibility constraints only. Consider again the constraint systemgiven in Example <ref>. We have that θ = {Y ↦w_1w_1w_2} is a solution of . Its associated first-order solution is λ_θ = {y ↦skaskaskb}.§.§ Symbolic processes: syntax and semanticsGiven an extended simple process Φ,we compute the constraint systems capturing its possible executions, starting from the symbolic process Φ∅. Note that we are now manipulating processes that are not ground anymore, but may contain free variables.A symbolic process is a tuple Φ where Φ is a constraint system and ()⊆(() ∩). We give in Figure <ref>a standard symbolic semantics for symbolic basic processes. From this semantics given on symbolic basic processes only, we derive a semantics on simple symbolic processes in a natural way:[] {P}⊎Φα{P'}⊎Φ''PΦαP'Φ'' We can also deriveour compressed symbolicsemanticsfollowingthe same pattern as for the concrete semantics (see Figure <ref>).We consider interleavings that execute maximal blocks of actions, and we allow improper termination of a block only at the end of a trace. Note that the ¬ valid(u) conditions of the third Proper rule and the second Improper rule are replaced by u≠^? u constraints in theirsymbolic counterparts.Continuing Example <ref>, we have that{Q_0(,)}Φ_0∅∅Φ where: * = (,Y )·(,w_3), and* = Φ is the constraint system defined in Example <ref>.We are now able to define the notion of equivalence associated to these two semantics, namely symbolic trace equivalence (denoted ) and symbolic compressed trace equivalence (denoted ). For a trace , we note () the trace obtained fromby removing all τ actions.Let A = Φ and B=Ψ be two simple processes. We have that AB when, for every tracesuch that Φ∅ 'Φ'_A, for every θ∈(Φ';_A), we have that: * Ψ∅ ''Ψ'_B where (')=() with θ∈(Ψ';_B), and * Φ'λ^A_θΨ'λ^B_θ where λ^A_θ (resp. λ^B_θ) is the substitution associated to θ w.r.t. Φ'_A (resp. Ψ'_B).We have that A and B are in trace equivalence w.r.t. ,denoted AB, if AB and B A.We derive similarly the notion of trace equivalence induced by . We do not have to take care of the τ actions since they are performed implicitly in the compressed semantics.Let A = Φ and B=Ψ be two extended simple processes. We have that AB when, for every tracesuch that Φ∅ 'Φ'_A, for every θ∈(Φ';_A), we have that: * Ψ∅ 'Ψ'_B with θ∈(Ψ';_B), and * Φ'λ^A_θΨ'λ^B_θ where λ^A_θ (resp. λ^B_θ) is the substitution associated to θ w.r.t. Φ'_A (resp. Ψ'_B).We have that A and B are in trace equivalence w.r.t. , denoted AB, if AB and BA.We have that {Q_0(,)}Φ_0{Q_0(,')}Φ_0. Continuing Example <ref>, we have seen that: * {Q_0(,)}Φ_0∅∅Φ (see Example <ref>), and * θ = {Y ↦w_1w_1w_2} is a solution of = (Φ;) (see Example <ref>). The only symbolic process that is reachable from {Q_0(,')}Φ_0∅usingis ∅Φ'' with: * Φ' = Φ_0 ⊎{w_3 Nn_bskbska'}, and* ' = {{w_0,w_1,w_2}Yy;Nska'}. One can check that θ is not a solution of Φ''. For processes without replication, the symbolic transition system induced by(resp ) is essentially finite. Indeed, the choice of fresh names for handles and second-order variables does not matter, and therefore the relationsandare essentially finitely branching. Moreover, the length of traces of a simple process is obviously bounded. Thus, deciding (symbolic) trace equivalence between processes boils down to the problem of deciding a notion of equivalence between sets of constraint systems. This problem is well-studied and several procedures already exist <cit.>, <cit.> (see Section <ref>). §.§ Soundness and completenessIt is well-known that the symbolic semanticsis sound and complete w.r.t. , and therefore that the two underlying notions of equivalence, namelyand , coincide. This has been proved for instance in <cit.>. Using the same approach, we can show soundness and completeness of our symbolic compressed semantics w.r.t. our concrete compressed semantics. We have: * Soundness: each transition in the compressed symbolic semantics represents a set of transitions that can be done in the concrete compressed semantics.* Completeness: each transition in the compressed semantics can be matched by a transition in the compressed symbolic semantics.These results are formally expressed in Proposition <ref> and Proposition <ref> below. These propositions are simple consequences of similar propositions that link the (small-step) symbolic semantics and the (small-step) standard semantics. Lifting these results to the compressed semantics is straightforward since both semantics are built using exactly the same scheme (see Figures <ref> and <ref>).Let Φ be an extended simple process such that Φ∅'Φ'', and θ∈(Φ';'). We have that Φθ'λΦ'λ where λ is the first-order solution of 'Φ'' associated to θ. Let Φ be an extended simple process such that Φ'Φ'. There exists a symbolic process _sΦ_s, a solution θ∈(Φ_s;), and a sequence _s such that: * Φ∅_s_sΦ_s;* 'Φ' = _sλΦ_sλ; and* = _sθ where λ is the first-order solution of _sΦ_s associated to θ. Finally, relying on these two results, we can establish that symbolic trace equivalence () exactly captures compressed trace equivalence ().Actually, both inclusions can be established separately. theoremthrmsintcssymv For any extended simple processes A and B, we have that:ABAB. As an immediate consequence of Theorem <ref> and Theorem <ref>, we obtain that the relations and coincide.For any initial simple processes A and B, we have that:ABAB. § REDUCTION USING DEPENDENCY CONSTRAINTSUnlike compression,which is essentially based on the input/output nature of actions, our second optimisation takes into account the exchanged messages. Let us first illustrate one simple instance of our optimisation and how dependency constraints <cit.> may be used to incorporate it into symbolic semantics. Let P_i = (c_i,x_i).(c_i,u_i).P'_i with i ∈{1,2}, and Φ_0 = {w_0n} be a ground frame. We consider the simple process A = {P_1,P_2}Φ_0, and thetwo symbolic interleavings depicted in Figure <ref>.The two resulting symbolic processes are of the form {P'_1,P'_2}Φ_i where Φ = Φ_0 ⊎{w_1u_1, w_2u_2}, _1 = {w_0X_1x_1;w_0, w_1X_2x_2}, _2 = {w_0X_2x_2; w_0, w_2X_1x_1}.The sets of concrete processes that these two symbolic processes represent are different, which means that we cannot discard any of those interleavings. However, these sets have a significant overlap corresponding to concrete instances of the interleaved blocks that are actually independent, where the output of one block is not necessary to obtain the input of the next block. In order to avoid considering such concrete processes twice, we may add a dependency constraint X_1w_2 in _2, whose purpose is to discard all solutions θ such that the message x_1λ_θ can be derived without using w_2u_2λ_θ. For instance, the concrete trace(c_2, w_0)·(c_2,w_2)·(c_1,w_0)·(c_1,w_1) would be discarded thanks to this new constraint. The idea of <cit.> is to accumulate dependency constraints generated whenever such a pattern is detected in an execution, and use an adapted constraint resolution procedure to narrow and eventually discard the constrained symbolic states. We seek to exploit similar ideas for optimising the verification of trace equivalence rather than reachability. This requires extra care, since pruning traces as described above may break completeness when considering trace equivalence. Asbefore, the key to obtain a valid optimisation will be to discard traces in asimilar way on the two processes being compared. In addition to handling this necessary subtlety, we also propose a new proof technique for justifying dependency constraints. The generality of thattechnique allows us to add more dependency constraints, taking into account more patterns than the simple one from the previous example.§.§ Reduced semanticsWe start by introducing dependency constraints.A dependency constraint is a constraint of the form X w where X is a vector of second-order variables in ^2,and w is a vector of handles, variables in .Given a constraint system = Φ, a set _D of dependency constraints, and θ∈(). We write θ_Φ_D when θ also satisfies the dependency constraints in _D, whenfor eachX w∈_Dthere is some X_i ∈ X such that for all recipes M∈(Σ, D_(X_i)) satisfying M(Φλ_θ) =_ (X_iθ)(Φλ_θ) and (M(Φλ_θ)), we have that (M) ∩ w ≠∅where λ_θ is the substitution associated to θ w.r.t. Φ. Intuitively, a dependency constraintX w is satisfied as soon as at least one message among those in ( Xθ)(Φλ_θ) canonly be deduced by using a message stored in w. Continuing Example <ref>, assume that u_1 = u_2 = n and let θ = {X_1 ↦ w_2; X_2 ↦ w_0}. We have that θ∈(_2) and the substitution associated to θ w.r.t. _2 is λ^2_θ = {x_1 ↦ n; x_2 ↦ n}. However, θ does not satisfy the dependency constraint X_1w_2. Indeed, we have that w_0(Φλ^2_θ) =_ (X_1θ)(Φλ^2_θ) whereas {w_0}∩{w_2} = ∅. Intuitively, this means that there is no good reason to postpone the execution of the block on channel c_1 if the output on c_2 is not useful to build the message used in input on c_1.We shall now define formally how dependency constraints will be added to our constraint systems. For this, we fix an arbitrary total orderon channels. Intuitively, this order expresses which executions should be favored, and which should be allowed only under dependency constraints. To simplify the presentation, we use the notation cXw as a shortcut for (c,X_1)·…·(c,X_ℓ)·(c,w_1) ·…·(c,w_k) assuming that X = (X_1,…,X_ℓ) and w = (w_1,…,w_k). Note that X and/or w may be empty.Let c be a channel, and= c_1X_1w_1·…·c_nX_nw_nbe a trace. If there exists a rank k≤ n such thatc c_k and c_ic for all k < i ≤ n, then c ={w | w ∈w_i}.Otherwise, we have that c = ∅.Then, given a trace , we defineby ϵ = ∅ and·cXw = ∪{ Xc} if c≠∅otherwise Intuitively,corresponds to the accumulation of the dependency constraints generated for all prefixes of .Let a, b, and c be channels insuch that abc. The dependency constraints generated during the symbolic execution of a simple process of the form {(a,x_a).(a,u_a),(b,x_b).(b,u_b), (c,x_c).(c,u_c)}Φ are depicted below. [inner sep=0pt] (4,3) node (racine) ∙;(1,2) node (a) ∙; (4,2) node (b) ∙; (7,2) node (c) ∙;(0,1) node (ab) ∙; (2,1) node (ac) ∙; (3,1) node (ba) ∙; (5,1) node (bc) ∙; (6,1) node (ca) ∙; (8,1) node (cb) ∙;(0,0) node (abc) ∙; (2,0) node (acb) ∙; (3,0) node (bac) ∙; (5,0) node (bca) ∙; (6,0) node (cab) ∙; (8,0) node (cba) ∙; (racine) edgenode[left]𝚒𝚘_a (a); (racine) edgenode[left]𝚒𝚘_b (b); (racine) edgenode[right]𝚒𝚘_c (c);(a) edge node[left]𝚒𝚘_b (ab); (a) edge node[right]𝚒𝚘_c (ac); (b) edge node[left]𝚒𝚘_a (ba); (b) edge node[right]𝚒𝚘_c (bc); (c) edge node[left,above]𝚒𝚘_a (ca); (c) edge node[left,below]𝚒𝚘_b (cb); (ab) edge node[left]𝚒𝚘_c (abc); (ac) edge node[left]𝚒𝚘_b (acb); (ba) edge node[left]𝚒𝚘_c (bac); (bc) edge node[left]𝚒𝚘_a (bca); (ca) edge node[left,pos=0.6]𝚒𝚘_b (cab); (cb) edge node[left]𝚒𝚘_a (cba);[->,>=latex,color=blue, line width=1pt] (acb) to[bend right] (ac); [->,>=latex,color=blue, line width=1pt] (ba) to[bend right] (b); [->,>=latex,color=blue, line width=1pt] (bca) to[bend right] (bc); [->,>=latex,color=blue, line width=1pt] (ca) to[out=20,in=250] (c); [->,>=latex,color=blue, line width=1pt] (cb) to[bend right] (c); [->,>=latex,color=blue, line width=1pt] (cba) to[bend right] (cb);[-latex,color=red,dashed,line width=1pt] (cab) .. controls +(20:8mm) and +(290:4mm) .. (c); [-latex,color=red,dashed,line width=1pt] (cab) .. controls +(20:4mm) .. (ca); We use 𝗂𝗈_i as a shortcut for (i,X_i)·(i,w_i) and we representdependency constraints using arrows. For instance, on the trace 𝗂𝗈_a·𝗂𝗈_c·𝗂𝗈_b, a dependency constraint of the form X_bw_c (represented by the left-most arrow) is generated.Now, on the trace 𝗂𝗈_c·𝗂𝗈_a·𝗂𝗈_b we add X_aw_c after the second transition, and X_b{w_c,w_a} (represented by thedashed 2-arrow) after the third transition. Intuitively, the latter constraint expresses that 𝗂𝗈_b is only allowed to come after 𝗂𝗈_c if itdepends on it, possibly indirectly through 𝗂𝗈_a.Dependency constraints give rise to a new notion of trace equivalence, which further refines the previous ones. Let A = Φ and B= Ψ be two extended simple processes. We have that A B when, for every sequencesuch that Φ∅ 'Φ'_A, for every θ∈(Φ';_A) such that θ_Φ'_A, we have that: * Ψ∅ 'Ψ'_B with θ∈(Ψ';_B), and θ_Ψ'_B;* Φ'λ^A_θΨ'λ^B_θ where λ^A_θ (resp. λ^B_θ) is the substitution associated to θ w.r.t. Φ'_A (resp. Ψ'_B).We have that A and B are in reduced trace equivalence, denoted A B, if AB and BA.§.§ Soundness and completenessIn order to establish thatandcoincide, we shall study more carefully concrete traces, consisting of proper blocks possibly followed by a single improper block. We will then define a precise characterization of executions whose associated solution satisfies dependency constraints. We denote by the set of blocks cMw such that c∈𝒞, M_i∈(Σ, ) for each M_i∈ M, and w_j ∈ for each w_j ∈w. In this section, a concrete trace is seen as a sequence of blocks, it belongs to ^*.Twoblocks b_1 = c_1M_1w_1 and b_2 = c_2M_2w_2are independent, written b_1 || b_2, when c_1 ≠ c_2 and none of the variables of w_2 occurs in M_1, and none of the variables of w_1 occurs in M_2. Otherwise the blocks are dependent.It is easy to see that independent blocks that are proper can be permuted in a compressed trace without affecting the executability and the result of executing that trace. It is not the case for improper blocks, which can only be performed at the very end of a compressed execution.However, this notion of independence based on recipes is too restrictive: it may introduce spurious dependencies. Indeed, it is often possible to make two blocks dependent by slightly modifying recipes without altering the inputted messages. For instance, w' does not occur in recipe M = w but does in M'=π_1(⟨ w, w' ⟩) while M' induces the same message as M. We thus define a more permissive notion of equivalence over traces, which allows permutations of independent blocks but also changes of recipes that preserve messages. During these permutations, we require that (concrete) traces remain plausible. A traceis plausible if for any input (c,M) such that = _0 ·(c,M) ·_2, we have M ∈(Σ,_0) where _0 is the set of handles occurring in _0. Given two blocks b_1 = c_1M_1w_1 and b_2 =c_2M_2w_2,we note(b_1 =_ b_2)Φ when M_1Φ =_M_2Φ, (M_1Φ), (M_2Φ), and w_1 = w_2. Intuitively, the two blocks only differ by a change of recipes such that the underlying messages are kept unchanged. We lift this notion to sequences of blocks, ( =_')Φ, in the natural way.Given a frame Φ, the relation ≡_Φ is the smallest equivalence over plausible traces (made of blocks) such that: * · b_1 · b_2 ·' ≡_Φ· b_2 · b_1·' when b_1 || b_2; and * · b_1 ·' ≡_Φ· b_2 ·' when (b_1 =_ b_2)Φ. LetA Φ withbe a trace made of proper blocks. We have that A 'Φ for any ' ≡_Φ. This result is easily proved, following from the fact that proper compressed executions are preserved by the two generators of ≡_Φ.The first case is given by Lemma <ref>. The second one follows from a simple observation of the transition rules: only the derived messages matter, while the recipes that are used to derive them are irrelevant (as long as validity is ensured). We established that compressed executions are preserved by changes of traces within ≡_Φ-equivalence classes. We shall now prove that, by keeping only executions satisfying dependencyconstraints, we actually select exactly one representative in this class.We lift the ordering on channels to blocks: cMw≺c'M'w' if and only if c ≺ c'. Finally, we define ≺ on concrete traces as the lexicographic extension of the order on blocks.Given a frame Φ, we say that a plausible traceis Φ-minimal if it is minimal in its equivalence class modulo ≡_Φ.Let A Φ and θ∈Φ. We have that θ is Φλ_θ-minimal if, and only if, θ_Φ.Let A and Φ be such that A Φ and θ∈Φ. Let λ_θ be the substitution associated to θ w.r.t. Φ. (⇒) We first show that if θ is Φλ_θ-minimal then θ_Φ, by induction on the length of the trace . The base case, =ϵ, is straightforward since = ∅. Now, assume that =_0 · b for some block b and A _0_0Φ_0_0bΦ. Let θ_0 be the substitution θ restricted to variables occurring in Φ_0_0, and λ_θ_0 be the associated first-order substitution. We have that θ_0 ∈Φ_0_0, λ_θ_0 coincides with λ_θ on variables occurring in Φ_0_0, and Φ_0λ_θ_0 coincides with Φλ_θ on the domain of Φ_0λ_θ_0. As a prefix of θ, we have that _0θ is Φλ_θ-minimal.We can thus apply our induction hypothesis onA _0_0Φ_0_0 and θ_0 ∈Φ_0_0. Assume that b = cXw. If _0c = ∅, we immediately conclude. Otherwise, it only remains to show that θ_Φ X_0c. By definition of the generation of dependency constraints,we know that _0 is of the form '_0 · b_c_0· b_c_1·…· b_c_n where: * ∀ 0≤ i≤ n,b_c_i=_c_i(X_i,w_i),* c≺ c_0 and c_i ≺ c for all 0 < i ≤ n; and* c = { w | w ∈w_i 0 ≤ i ≤ n}. Assume that the dependency constraint is not satisfied, this means that for some M such that ( Xθ)(Φλ_θ) =_ ( M)(Φλ_θ) and (( M)(Φλ_θ)), we have that ^1( M) ∩{w | w ∈w_i 0 ≤ i ≤ n} = ∅. Therefore, we have that[θ=_0θ· bθ; = '_0θ· b_c_0θ· b_c_1θ·…·b_c_nθ·cXθw;≡_Φλ_θ '_0θ·cMw·b_c_0θ· b_c_1θ·…· b_c_nθ. ]Since c ≺ c_0, this would contradict the Φλ_θ-minimality of θ. Hence the result.(⇐) Now, assuming that θ is not Φλ_θ-minimal, we shall establish that there is a dependency constraint X w∈ that is not satisfied by θ. Let _m be a Φλ_θ-minimal trace of the equivalence class of θ. We have in particular _m ≡_Φλ_θθ and _m ≺θ.Let _0 (resp. _m^0) be the longest prefix of(resp. _m) such that (_0θ =__m^0)Φλ_θ. We have that = _0· b·_1and _m = _m^0· b_m·_m^1 with c_m ≺ c where c_m (resp. c)is the channel used in block b_m (resp. b).By definition of ≡_Φλ_θ, block b_m must have a counterpart inand, more precisely, in _1. We thus have a more precise decomposition of : = _0· b·_11· b'_m·_12 such that (b'_mθ =_ b_m)Φλ_θ.Let b'_m = c_mXw. We now show that the constraintX_0· b·_11c_m is inand is not satisfied by θ, implyingθ_Φ. We have seen that(b'_mθ =_ b_m)Φλ_θ and c_m ≺ c. Since c_m ≺ c, by definition of ·· we deduce that∅≠_0· b·_11c_m⊆{w| (d,w)b·_11 for somed,w}. But, since we also know that (b'_mθ =_ b_m)Φλ_θ and _m = ^0_m· b_m·_m^1 is a plausible trace, we have that b_m = c_mMw_m for some recipes M such that MΦλ_θ =_ (Xθ)Φλ_θ, (MΦλ_θ), and ^1(M) ∩_0· b·_11c_m = ∅. This allows us to conclude that θ_Φ. We are now able to show that the notion of trace equivalence based on this reduced semantics coincides with the compressed one (as well as its symbolic counterpart as given in Definition <ref>). Even though the reduced semantics is based on the symbolic compressed semantics, it is more natural to establish the theorem by going back to the concrete compressed semantics, because we have to consider a concrete execution to check whether dependency constraints are satisfied or not in our reduced semantics anyway. For any extended simple processes A and B, we have that: AB if and only if AB.Let A = Φ and B = Ψ be two extended simple processes. (⇒) Consider an execution of the form Φ∅_s_sΦ_s_A and a substitution θ∈Φ_s_A such that θ_Φ_s_A_s. Thanks to Proposition <ref>, we have that Φ_sθ_sλ^A_θΦ_sλ^A_θ whereλ^A_θ is the substitution associated to θ w.r.t. Φ_s_A.Since AB, we deduce that there exists 'Ψ' such that:B 𝖽𝖾𝖿=Ψ_sθ'Ψ'Φ_sλ^A_θΨ'Relying on Proposition <ref>, we deduce that there exists _sΨ_s_B such that:Ψ∅_s_sΨ_s_B, θ∈Ψ_s_B_sλ^B_θΨ_sλ^B_θ = 'Ψ'where λ^B_θ is the substitution associated to θ w.r.t. Ψ_s_B. The fact that we get the same symbolic trace _s and same solution θ comes from the third point of Proposition <ref>and the flexibility of the symbolic semantics · that allows us to choose second order variables of our choice(as long as they are fresh).Lemma <ref> tells us that θ is Φ_sλ^A_θ-minimal.Since Φ_sλ^A_θΨ_sλ^B_θ, we easily deduce that _sθ is also Ψ_sλ^B_θ-minimal, and thusLemma <ref> tells us that θ_Ψ_s_B. This allows us to conclude.(⇐) Consider an execution of the form Φ'Φ'. We prove the result by induction on the number of blocks involved in , and we distinguish two cases depending on whether ends with an improper block or not.Case whereis made of proper blocks. Let _m be a Φ'-minimal trace in the equivalence class of . Lemma <ref> tells us that Φ_m'Φ'. Thanks to Proposition <ref>, we know that there exist _sΦ_s_A, ^s_m, and θ such that:Φ∅^s_m_sΦ_s_A, θ∈Φ_s_A,_sλ^A_θΦ_sλ^A_θ = 'Φ', ^s_mθ =_m.Using Lemma <ref>, we deduce that θ_Φ_s_A^s_m. By hypothesis, AB, hence: * Ψ∅ ^s_m_sΨ_s_B with θ∈(Ψ_s;_B), and θ_Ψ_s_B^s_m;* Φ_sλ^A_θΨ_sλ^B_θ where λ^B_θ is the substitution associated to θ w.r.t. Φ_s_BThanks to Proposition <ref>, we deduce that Ψ^s_mθ_sλ^B_θΨ_sλ^B_θ.Moreover, since Φ' = Φ_sλ^A_θΨ_sλ^B_θ, we get _m ≡_Ψ_sλ^B_θ from the fact that _m ≡_Φ'. Applying Lemma <ref>, we conclude thatΨ_sλ^B_θΨ_sλ^B_θΦ' Ψ_sλ^B_θ. Case whereis of the form _0· b where b is an improper block. We have that:Φ_0'Φ'b∅Φ'Let _m be a Φ'-minimal trace in the equivalence class of . By definition of the relation ≡,block b must have a counterpart in _m. We thus have that _m is of the form _m = _1· b_m·_2 where b_m is the improper block corresponding to b. We do not necessarily have that b = b_m but we know that (b =_ b_m)Φ'. If _2 is an empty trace, b_m is at the end of _m, the reasoning from the previous case applies. Otherwise, we have that _1·_2 ≡_Φ'_0,_1·_2 is Φ'-minimal, and _2 is non empty. Thus, thanks toLemma <ref>, we have that:Φ_1_1Φ_1_2'Φ'Since _1·_2 is made of proper blocks, we can apply our previous reasoning, and conclude that there exist _1Ψ_1 and 'Ψ' such that: Ψ_1_1Ψ_1_2'Ψ'Φ' Ψ' (b =_ b_m)Ψ' Since we know that 'Φ'b∅Φ', and _0· b ≡_Φ'_1· b_m·_2, we deduce that _1Φ_1b_m∅Φ_1. We have that Φ_1· b_m∅Φ_1, and _1· b_m is a Φ_1-minimal trace (note that the improper block is at the end). Thus, applying our induction hypothesis, we have that:Ψ_1_1Ψ_1b_m∅Ψ_1. Since the channel used in b_m does not occur in _2, we deduce thatΨ_1_1Ψ_1_2'Ψ'b_m∅Ψ'Relying on Lemma <ref> and the fact that (b_m =_ b)Ψ', we deduce that Ψ_0'Ψ'b∅Ψ'.*Putting together Theorem <ref> and Theorem <ref>, we are now able to state our main result: our notion of reduced trace equivalence actually coincides with the usual notion of trace equivalence. This result is generic and holds for an arbitrary equational theory, as well as for an arbitrary notion of validity (as defined in Section <ref>). For any initial simple processes A and B, we have that: AB if and only if AB.§ INTEGRATION IN We validate our approach by integrating our refined semantics in the tool. As we shall see, the compressed semantics can easily be used as a replacement for the usual semantics in verificationalgorithms. However, exploiting the reduced semantics is not trivial, and requires to adapt the constraint resolution procedure. It is beyond the scope of this paper to provide a detailed summary of how the verification tool actually works. A 50 pages paper describing solely the constraint resolution procedure of is available <cit.>. This procedure manipulates matrices of constraint systems, with additional kinds of constraints necessary for its inner workings. Proofs of the soundness, completeness and termination of the algorithm are available in a long and technical appendix (more than 100 pages).In order to show how our reduced semantics have been integrated in the constraint solving procedure of , we choose to provide a high-level axiomatic presentation of 's algorithm. This allows us to prove that our integration is correct without having to enter into complex, unnecessary details of 's algorithm. Our axioms are consequences of results stated and proved in <cit.> and have been written in concertation with Vincent Cheval. However, due tosome changes in the presentation, proving them will require to adapt most of the proofs. It is therefore beyond the scope of this paper to formally prove that our axioms are satisfied by the concrete procedure. We start this section with a high-level axiomatic presentation of 's algorithm, following the original procedure <cit.> but assuming public channels only (sections <ref>, <ref>). The purpose of thispresentation is to provide enough details about to explain how our optimisations have been integrated, leaving out unimportant details. Next, we show that this axiomatization is sufficient to prove soundness and completeness of w.r.t. trace equivalence (Section <ref>). Then we explain the simplifications induced by the restriction to simple processes, and how compressed semantics can be used to enhance the procedure and prove the correctness of this integration (Section <ref>). We finally describe how our reduction technique can be integrated, and prove the correctness of this integration(Section <ref>).We present some benchmarks in Section <ref>, showing that our integration allows to effectively benefit from both of our partial order reduction techniques. §.§ in a nutshell has been designed for a fixed equational theory_ (formally defined in Example <ref>) containing standard cryptographic primitives. It relies on a notion of message which requires that only constructors are used, and a semantics in which actions are blocked unless they are performed on such messages. This fits in our framework, described in Section <ref>, by taking = (Σ_c,). We now give a high-level description of the algorithm that is implemented in .The main idea is to perform all possible symbolic executions of the processes, keeping together the processes that can be reached using the same sequence of symbolic action. Then, at each step of this symbolic execution, the procedure checks that for every solution of every process on one side, there is a corresponding solution for some process on the other side so that the resulting frames are in static equivalence. This check for symbolic equivalence is not obviously decidable. To achieve it, 's procedure relies on a set of rules for simplifying sets of constraint systems. These rules are used to put constraint systems in a solved form that enables the efficient verification of symbolic equivalence.The symbolic execution used in is the same as described inSection <ref>. However, 's constraint resolution procedure introduces new kinds of constraints. Fortunately, we do not need to enter into the details of those constraints and how they are manipulated. Instead, we treat them axiomatically.An extended constraint system ^+ = Φ^+ consists of a constraint system = Φ together with an additional set ^+ of extended constraints. We treat this latter set abstractly, only assuming an associated satisfaction relation, written θ^+, such that θ∅ always holds, and θ^+_1 implies θ^+_2 when ^+_2 ⊆^+_1. We define the set of solutions of ^+ as ^+(^+) = { θ∈()| θ^+}. An extended symbolic processΦ^+ is a symbolic process with an additional set of extended constraints ^+. We shall denote extended constraint systems by ^+, ^+_1, etc. Extended symbolic processes will be denoted by A^+, B^+, etc. Sets of extended symbolic processes will simply be denoted by , , etc. For convenience, we extendand ^+ to symbolic processes and extended symbolic processes in the natural way:Φ = Φ^+Φ^+ = ^+Φ^+.We may also use the following notation to translate back and forth between symbolic processes and extended symbolic processes:Φ=Φ∅Φ^+=Φ . We can now introduce the key notion of symbolic equivalence between sets of extended symbolic processes, or more precisely between their underlying extended constraint systems.Given two sets of extended symbolic processes and , we have thatif for every A^+ = _AΦ_A_A^+_A∈, for every θ∈^+(A^+), there exists B^+ = _BΦ_B_B^+_B∈ such that θ∈^+(B^+) and Φ_Aλ^A_θΦ_Bλ^B_θ where λ^A_θ (resp. λ^B_θ) is the substitution associated to θ w.r.t. Φ_A_A (resp. Φ_B_B).We say thatandare in symbolic equivalence, denoted by , if and .The whole trace equivalence procedure can finally be abstractly described by means of a transition systemon pairs of sets of extended symbolic processes, labelled by observable symbolic actions. Informally, the intent is that a pair of processes is in trace equivalence iff only symbolically equivalent pairs may be reached from the initial pair using .We now defineformally. A transition (;) can take place iffandare in symbolic equivalence[ This definition yields infinite executions forif no inequivalent pair is met. Each such execution eventually reaches (∅;∅) while, in practice, executions are obviously not explored past empty pairs. We chose to introduce this minor gap to make the theory more uniform. ].Each transition for some observable action α consists of two steps, (;)α(”;”) iff (;)α(';') and (';')(”;”), where the latter transitions are described below: * The first part of the transition consists in performing an observable symbolic action α (either (c,X) or (c,w)) followed by all available unobservable (τ) actions. This is done for each extended symbolic process that occurs in the pair of sets, and each possible transition of one such process generates a new element in the target set. Formally, we have (;) α (';') if' = ⋃_(;Φ;;^+)∈{(';Φ';';^+)|(;Φ;) α·τ^*(';Φ';') τ },and correspondingly for '. Note that elements of (;) that cannot perform α are simply discarded, and that the constraint systems of individual processes are enriched according to their own transitions whereas the extended part of constraint systems are left unchanged. For a fixed symbolic action α, the α transition is deterministic. The choice of names for handles and second-order variables does not matter, and therefore therelationis also finitely branching.* The second part consists in simplifying the constraint systems of (';') until reaching solved forms. This part of the transition is non-deterministic, several different (”;”) may be reached depending on various choices, whether a message is derived by using a function symbol or one of the available handles. Although branching, this part of the transition is finitely branching. Moreover, only extended constraints may change: for any (;Φ;;^+_1)∈” there must bea ^+_0 such that (;Φ;;^+_0)∈', and similarly for ”. An important invariant of this construction is that all the processes occurring in any of the two sets of processes have constraint systems that share a common structure.More precisely the transitions maintain that for any (_1;Φ_1;_1;^+_1),(_2;Φ_2;_2;^+_2)∈∪, (_1)=(_2) and DXx occurs in _1 iff it occurs in _2. Consider the simple basic processesR_i=(c_i,x_i).x_i=(c_i,n_i) for i∈ℕ, x_i∈, n_i∈,a public constant. We illustrate the roles ofandon the pair ({Q_0};{Q_0}) where Q_0 = {R_1,R_2}∅∅∅. We have that({Q_0};{Q_0})(c_2,X_2)({Q_0^t,Q_0^e};{Q_0^t,Q_0^e})where Q_0^t and Q_0^e are the two symbolic processes one may obtain by executing the observable action (c_2,X_2), depending on the conditional after that input. Specifically, we have: * Q_0^t={R_1,(c_2,n_2)}∅{X_2∅x_2,x_2}∅* Q_0^e={R_1}∅{X_2∅x_2,x_2}∅After this first step,is going to non-deterministicallysolve the constraint systems. From the latter pair, it will produce only two alternatives. Indeed, if x_2 holds then infers that the only recipe that it needs to consider is the recipe R=. In that case, the only considered solution is {X_2↦}. Otherwise, x_2 holds but, at this point, no more information is inferred on X_2. Formally,[ ({Q_0^t,Q_0^e};{Q_0^t,Q_0^e})({Q_1^t};{Q_1^t}); ({Q_0^t,Q_0^e};{Q_0^t,Q_0^e})({Q_1^e};{Q_1^e}) ]where * Q_1^t={R_1,(c_2,n_2)}∅{X_2∅x_2,x_2}_1^t and ^+(Q_1^t)={Θ^t_1}where Θ^t_1={X_2↦};* Q_1^e={R_1}∅{X_2∅x_2,x_2}_1^e.The content of _1^t and _1^e is not important. Note that after , only one alternative remains (there is only one extended symbolic process on each side of the resulting pair) because only one of the two processes Q_0^t,Q_0^e complies with the choices made in each branch.Let A = _AΦ_A and B = _BΦ_B be two processes. We say that AB whenfor any pair (;) such that(_AΦ_A∅∅;_BΦ_B∅∅) (;). As announced above, we expectto coincide with trace equivalence. We shall actually prove it (see Section <ref>),after having introduced a few axioms (Section <ref>). We note, however, that this can only hold under some minor assumptions on processes. In practice, does not need those assumptions but they allow for a more concise presentation. A simple process (resp. symbolic process) A is said to be quiescent when Aτ (resp. Aτ). An extended symbolic process A^+ is quiescent when A^+τ. In α transitions, processes must start by executing an observable action α and possibly some τ actions after that. Hence, it does not make sense to considertransitionson processes that can still perform τ actions. We shall thus establish thatandcoincide only on quiescent processes, which is not a significant restriction since it is always possible to pre-execute all available τ-actions before testing equivalences.§.§ Specification of the procedureWe now list and comment the specification satisfied by the exploration performed by . These statements are consequences of results stated and proved in <cit.> but it is beyond the scope of this paper to prove them.§.§.§ Soundness and completeness of constraint resolutionThestep, corresponding to 's constraint resolutionprocedure, only makes sense under some assumptions on the (common) structure of the processes that are part of the pairs of sets under consideration. Rather thanprecisely formulating these conditions (which would be at odds with theabstract treatment of extended constraint systems) we start bydefining an under-approximation of the set of pairs on which we may applyat some point. We choose this under-approximation sufficiently large to cover pairs produced by the compressed semantics, and we then formulate our specifications in that domain. More precisely, the under-approximation has to cover two things: * we have to consider additional disequalities of the form u ≠^? u in constraint systems since they are eventually added by our compressed symbolic semantics (see Figure <ref>);* we have to allow the removal of some extended symbolic process from the original sets since they are eventually discarded by our compressed (resp. reduced) symbolic semantics.Given an extended symbolic process A^+ = Φ^+, we denote add(A^+)the set of extended symbolic processes obtained from A^+ by adding intoa number of disequalities of the form u≠^?u with (u) ⊆().This is then extended to sets of extended symbolic processes as follows: add({A^+_1,…,A^+_n}) = {{ B^+_1, …, B^+_n } | B^+_i ∈add(A^+_i) }.The set of valid pairs is the least set such that: * For all quiescent, symbolic processes A=(;Φ;∅) and B=(;Ψ;∅), ({A};{B}) is valid.* If (;) is valid and , (;)α(_1;_1), _2⊆_1, _2⊆_1, _3 ∈add(_2), _3 ∈add(_2), and (_3;_3)(';') then (';') is valid. In that case, the pair (_3;_3) is called an intermediate valid pair. It immediately follows that ({A};{B})(;) implies that (;) is valid and only made of quiescent, extended symbolic processes. But the notion of validity accomodates more pairs: it will cover pairs accessible under refinements ofbased on subset restrictions of . We may note that these pairs are actually pairs that would have been explored bywhen starting with another pair of processes (a process that makes explicit the use of trivial conditionals of the form u=uPQ ). Therefore, those pairsdo not cause any trouble when they have to be handled by . [soundness of constraint resolution]Let (';') be an intermediate valid pair such that (';')(”;”). Then, for all A”∈” (resp. B”∈”) there exists some A'∈' (resp. B'∈') such that A'=A” (resp. B'=B”) and ^+(A”) ⊆^+(A') (resp. ^+(B”) ⊆^+(B')).treats almost symmetrically the twocomponents of the pair of sets on which transitions take place. This is reflected by the fact that axioms concern both sides and are completelysymmetric, like Axiom <ref>.In order to make the following specifications more concise and readable, we stateproperties only for one of the two sets and consider the other “symmetrically” as well. The completeness specification is in two parts: it first states that nofirst-order solution is lost in the constraint resolution process, and then that the branching ofcorresponds to different second-order solutions. [first-order completeness of constraint resolution]Let (;) be an intermediate valid pair. For all A^+∈ and θ∈(A^+) there exists (;)(_2;_2), A_2^+∈_2 and θ^+∈^+(A^+_2) such that A^+_2=A^+ and λ_θ^A =_λ_θ^+^A, where λ^A_θ (resp. λ^A_θ^+) is the substitution associated to θ (resp. to θ^+) w.r.t. A^+.Symmetrically for B^+∈. [second-order consistency of constraint resolution]Let (;) be an intermediate valid pair such that (;)(_2;_2), θ∈^+(A^+) for some A^+∈ and θ∈^+(C^+_2) for some C^+_2∈_2∪_2. Then there exists some A^+_2∈_2 such that A^+=A^+_2 and θ∈^+(A^+_2). Symmetrically for B^+∈. §.§.§ Partial solutionIn order to avoid performing some explorations when dependency constraints of our reduced semantics are not satisfied, we shall be interested in knowing when all solutions of a given constraint system assign a given recipe to some variable. Such information is generally available in the solved forms computed by , but not always in a complete fashion. We reflect this by introducing an abstract function that represents the information that can effectively be inferred by the procedure.We assume a partial solution[We use the notation σ_1⊔σ_2 to emphasize the fact that the two substitutions do not interact together. They havedisjoint domain, i.e. (σ_1) ∩(σ_2) = ∅, andno variable of (σ_i) occurs in (σ_j) with {i,j} = {1,2}.] functionwhich maps sets of extended constraints ^+ to a substitution, such that for any θ∈^+Φ^+, there exists θ' such that θ = (^+) ⊔θ'. We extendto extended symbolic processes: Φ^+ = (^+). Intuitively, given an extended constraint system, the functionreturns the value of some of its second-order variables (those for which their instantiation is already completely determined). Our specification of the partial solution shall postulate that the partial solution returned by is the same for each extended symbolic process occurring in a pair (; ) reached during the exploration. Moreover, there is a monotonicity property that ensures that this partial solution becomes more precise along the exploration.We assume the following about the partial solution: * For any valid pair (;), we have that (A) = (B) for any A,B ∈∪. This allows us to simply write (;) when ∪≠∅.* For any intermediate valid pair (;) such that (;)(';') and ' ∪' ≠∅, we have (';') = (;) ⊔θ for some θ.Continuing Example <ref>, we first note that ({Q_0};{Q_0}) is a valid pair. Second, the exploration ({Q_0};{Q_0})(c_2,X_2)({Q_1^t};{Q_1^t}) covers all executions of the formQ_0(c_2,X_2).τQ_1^t going to the 𝚝𝚑𝚎𝚗 branch even though the only solution of Q_1^t is Θ_1^t. Indeed, if Θ∈(Q_1^t) then the message computed by X_2Θ should be equal toand thus no first-order solution is lost as stated by Axiom 2. Moreover, because the value of X_2 is already known in Q_1^t, we may have (Q_1^t)=(^+_1)={X_2↦}.§.§ Proof of the original procedure The procedure, axiomatized as above, can be proved correct w.r.tthe regular symbolic semanticsand its induced traceequivalence as defined in Section <ref>. Of course, Axiom <ref> is unused in this first result. It will be used later on when implementing our reduced semantics. We first start by establishing that all the explorations performed bycorrespond to symbolic executions.This result is not new and has been established from scratch (without relying on the axioms stated in the previous section) in <cit.>. Nevertheless, we found it useful to establish that our axioms are sufficient to prove correctness of the original procedure.The proofs provided in the following sections to establish correctness of our optimised procedure follow the same lines as the onespresented below. Let (;) be a valid pair such that (;)(';'). Then, for all A'∈' there is some A∈ such that A'A' for some ' with (')=. Symmetrically for B'∈'. We proceed by induction on .Whenis empty, we have that (;)=(';'), and the result trivially holds. Otherwise we have that:(;) α (_1;_1) (_2;_2) _0 (';') with = α·_0. Let A' be a process of '. By induction hypothesis we have some A_2 ∈_2 such that A_2'_0A' with ('_0)=_0. By spec:sound there is some A_1 ∈_1 such that A_1 = A_2, and by definition ofwe finally find some A ∈ such that Aα·τ^kA_1. To sum up, we have A ∈ such that AA' with (') =. We now turn to completeness results. Assuming that processes under study are in equivalence(so that will not stop its exploration prematurely), we are able to show that any valid symbolic execution (a symbolic execution with a solution in its resulting constraint system) is captured by an exploration performed by .Actually, since discards some second-order solution during itsexploration, we can only assume that another second-order solutionwith the same associated first-order solution will be found. Let A=Φ∅, B=Ψ∅ andA'='Φ'' be three quiescent, symbolic processes such that ΦΨ, AA', and θ∈(A'). Then there exists an exploration ({A};{B}) _o (';') and some A^+ ∈', θ^+ ∈^+(A^+) such that ()=_o, A^+ = A' and λ_θ =_λ_θ^+, where λ_θ (resp. λ_θ^+) is the substitution associated to θ (resp. to θ^+) with respect to Φ''.Symmetrically for B B'.By hypothesis, we have that AA'. We will first reorganize this derivation to ensure that τ actions are always performed as soon as possible. Then,we proceed by induction on (). When () is empty, we have that A'=A since A isquiescent. Let (';') = ({A};{B}), A^+ = A, θ^+ = θ. We have that θ∈(A) and therefore θ∈^+(A), θ^+ = θ∈^+(A^+). We easily conclude. Otherwise, consider A _0 A_1 α·τ^k A' with θ∈(A'). Let A' = 'Φ'' and A_1 = _1Φ_1_1. We have that _1 ⊆'.Since θ∈(A'), we also have θ|_V∈(A_1) where V = ^2(_1). Therefore, we apply our induction hypothesis and we obtain that there exists an exploration ({A};{B})'_0(_1;_1) and some A_1^+∈_1,θ_1^+∈^+(A_1^+) such that (_0) = '_0, A_1^+=A_1, and the first-order substitutions associated to θ|_V and θ_1^+ with respect to (Φ_1; _1) are identical. By hypothesis we have ΦΨ, thus _1 _1. Hence atransition can take place on that pair. By definition ofand since A_1^+ = A_1α·τ^k A' with A' quiescent, there must be some (_1;_1) α (_2;_2) with A_2^+∈_2, A_2^+=A'. Thus θ∈(A^+_2) and we can apply spec:complete. There exists (_2;_2)(';'),A^+∈', A^+=A_2^+ and θ^+∈^+(A^+) such that A^+_2 = A^+, and the substitutions associated to θ (resp. θ^+) w.r.t. (Φ';') coincide. To sum up, the exploration({A};{B})'_0(_1;_1) α (_2;_2) (';') together with A^+ ∈', and θ^+ ∈^+(A^+) satisfy all the hypotheses. Let A,B,A' be quiescent symbolic processes such that AA' = 'Φ'', θ∈(A') and ({A};{B})_o(';') with ()=_o and θ∈^+(C) for some C ∈'∪'. Then there exists some A^+ ∈' such that A^+ = A' and θ∈^+(A^+). Symmetrically for BB'.We proceed by induction on _o.When _o is empty, we have that A' = A (because Ais quiescent),' = {A}, and ' = {B}. Let A^+ beA = A'. We deduce that θ∈^+(A^+) from the fact that θ∈(A) and A^+ = A. We consider now the case of a non-empty execution:({A};{B})_o(_1;_1)α(_2;_2)(_3;_3)AA_1 α·τ^k A_3. Note that, by reordering τ actions, we can assume A_1 to be quiescent. By assumption we have θ∈(A_3), ()=_o and θ∈^+(C_3) for some C_3 ∈_3∪_3. By spec:sound, there exists some C_2 ∈_2∪_2 such that θ∈^+(C_2). By definition ofwe obtain C_1∈_1∪_1 such that C_1α·τ^kC_2 and ^+(C_1) = ^+(C_2) (the sets of extended constraints of C_1 and C_2 coincide). The first fact implies θ|_V∈(C_1) by monotonicity (where V = ^2((C_1)), second-order variables that occur in the set of non-extended constraints of C_1), and the second allows us to conclude more strongly that θ|_V∈^+(C_1). Since we also have θ|_V∈(A_1) by monotonicity, the induction hypothesis applies and we obtain some A^+_1 ∈_1 with A^+_1 = A_1 and θ|_V∈^+(A^+_1).By definition of , and since A^+_1α·τ^k A_3τ (A_3 is quiescent by hypothesis), we have A^+_2∈_2 such that A^+_2 = A_3 and ^+(A^+_1)=^+(A^+_2). Therefore, we have that θ∈(A^+_2), andthe fact that^+(A^+_1)=^+(A^+_2) allows us to say that θ∈^+(A^+_2) We can finally apply spec:consistent to obtain some A^+_3 such that A^+_3 = A^+_2 = A_3 and θ∈^+(A^+_3).For any quiescent extended simple processes, we have that:AB if, and only if, A B. Let A_0 = Φ, B_0 = 'Φ', _0 = {(;Φ;∅;∅)} and _0 = {(';Φ';∅;∅)}. We prove the two directions separately.(⇒) Assume A_0B_0 and consider some exploration (_0;_0)_o(;). We shall establish that .Let A = _AΦ_A_A be inand θ∈^+(A). By Lemma <ref>, we have Φ∅A such that ()=_o. By hypothesis, there exists B = _BΦ_B_B such that'Φ'∅' B, (')=()=_o, θ∈(B) and Φ_Bλ_θ^B Φ_Aλ_θ^A.We can finally apply Lemma <ref>, which tells us that there must be some B^+ ∈ such that B^+ = B and θ∈^+(B^+).(⇐) We now establish A_0B_0 assuming A_0B_0. Consider Φ∅ A and θ∈(A).If A is not quiescent, it is easy to complete the latter execution into Φ∅·τ^k A' = _AΦ_A_A and θ∈(A') such that A' is quiescent. By Lemma <ref> we know that (_0;_0)_o(;) with ()=_o, A^+ ∈, θ^+∈^+(A^+) with A' = A^+ and λ_θ =_λ_θ^+ where λ_θ (resp. λ_θ^+) is the substitutionassociated to θ (resp. θ^+) w.r.t. (Φ_A;_A).By assumption we haveand thus there exists some B = (_B; Φ_B; _B; ^+_B) ∈ with θ^+∈^+(B),and Φ_Bλ_θ^+^BΦ_Aλ_θ^+ where λ_θ^+^Bis the substitution associated to θ^+ w.r.t. (Φ_B;_B).By Lemma <ref> we have 'Φ'∅'B with (')=_o=(). To conclude the proof, it remains to show that θ∈(B)and that Φ_Aλ_θΦ_Bλ^B_θ whereλ^B_θ is the substitution associated to θ w.r.t. (Φ_B; _B).For any X ∈(_B) = (_A), we have((Xθ)(Φ_Aλ_θ^+)), ((Xθ^+)(Φ_Aλ_θ^+)), and(Xθ)(Φ_Aλ_θ^+) =_ (Xθ)(Φ_Aλ_θ) =_ x_Aλ_θ =_ x_Aλ_θ^+ =_ (Xθ^+)(Φ_Aλ_θ^+)where x_A is the first-order variable associated to X in _A. SinceΦ_Aλ_θ^+∼Φ_Bλ_θ^+^B, we deduce that(Xθ)(Φ_Bλ_θ^+^B) =_ (Xθ^+)(Φ_Bλ_θ^+^B),((Xθ)(Φ_Bλ_θ^+^B)) and thereforeθ∈(B), and its associated substitution λ_θ^Bw.r.t. (Φ_B;_B)coincides with λ_θ^+^B, and therefore Φ_Aλ_θΦ_Bλ^B_θ is a direct consequence ofΦ_Bλ_θ^+^BΦ_Aλ_θ^+ and λ_θ =_λ_θ^+. §.§ Integrating compressionWe now discuss the integration of the compressed semantics ofSection <ref> as a replacement for the regular symbolicsemantics in . Although our compressed semanticshas been defined as executingblocks rather than elementary actions, we allow ourselves to viewit in a slightly different way in this section: we shall assume that thesymbolic compressed semantics deals with elementary actions and enforces that those actions, when put together, form a prefix of a sequence of blocks that can actually be executed (for the process under consideration) in the compressed semantics of Section <ref>.This can easily be obtained by means of extra annotations at the level of processes, and we will not detail that modification. This slight change makes it simpler to integrate compression into , both in the theory presented here and in the implementation. Given two sets of extended symbolic processes ,,and an observable action α, we write (;)α(';')when' = ⋃_(;Φ;;^+)∈{(';Φ';';^+)|(;Φ;) α(';Φ';')τ },and similarly for '. We say that (;)α(”;”) when (;)α(';') and (';')(”;”).Finally, given two simple extended processes A = (_A;Φ_A) and B = (_B; Φ_B), we say that AB when ≈^+ for any ({(_A;Φ_A;∅)}; {(_B;Φ_B;∅)})(; ). As expected,allows to consider much fewer explorations than with the original . It inherits the features of compression, prioritizing outputs, not considering interleavings of outputs, executing inputs only under focus, and preventing executions beyond improper blocks. These constraints apply to individual processes in ∪, but we remark that they also have a global effect in , all processes of ∪ must start a new block simultaneously: recall that the beginning of a block corresponds to an input after some outputs, and such inputs can only be executed if no more outputs are available.Continuing Example <ref>, there is only one non-trivial[ We dismiss here the (infinitely many) transitions obtained for infeasible actions, which yield (∅;∅). ] compressed exploration of one action from the valid pair ({Q_1^t};{Q_1^t}). It corresponds to the output on channel c_2: ({Q_1^t};{Q_1^t})(c_2,w_2)({Q_2},{Q_2}) for Q_2={R_1}{w_2 n_2}{X_2∅x_2,x_2}_2^+. In particular, for any i ∈{1,2}, we have ({Q_1^t};{Q_1^t})(c_i,X_i)(∅;∅).Observe that, becauseis obtained fromby a subset restriction in up to some disequalities, we have that (';') is a valid pair when ({A};{B})(';') for some quiescent, symbolic processes A,B having empty sets of constraints. Following the same reasoning as the one performed in Section <ref>,we can establish thatcoincides with . The main difference is thatalready ignores τ-actions, and therefore we do not need to apply the(·) operator. Let (;) be a valid pair such that (;)(';'). Then, for all A'∈' there is some A∈ such that AA'. Symmetrically for B'∈'. Let A=Φ∅, B=Ψ∅, and A'='Φ'' be three quiescent, symbolic processes such that ΦΨ, AA' and θ∈(A'). Then there exists an exploration ({A};{B})(';') and some A^+ ∈', θ^+ ∈^+(A^+) such that A^+ = A' and λ_θ =_λ_θ^+, where λ_θ (resp. λ_θ^+) is the substitution associated to θ (resp. to θ^+) with respect to Φ''.Symmetrically for BB'. Let A,B and A' be quiescent, simple symbolic processes such that AA' = 'Φ'', θ∈(A'), and ({A};{B})(';') with θ∈^+(C) for some C ∈'∪'. Then there exists some A^+ ∈' such that A^+ = A' and θ∈^+(A^+). Symmetrically for BB'.For any quiescent extended simple processes, we have that:AB if, and only if, AB.§.§ Integrating dependency constraints inWe now define a final variant of explorations, which integrates the ideas of Section <ref> to further reduce redundant explorations. We can obviously generate dependency constraints in , just like we did in Section <ref>, but the real difficulty is to exploit them in constraint resolution to prune some branches of the exploration performed by . Roughly, we shall simply stop the exploration when reaching a state for which we know that all of its solutions violate dependency constraints. To do that, we rely on the notion of partial solution introduced in Section <ref>. In other words, we do not modify 's constraint resolution, but simply rely on information that it already provides to know when dependency constraints become unsatisfiable. As we shall see, this simple strategy is very satisfying in practice. We defineas the greatest relation contained inand such that, for any symbolic processes A and B with empty constraint sets, ({A};{B})(';') implies that there is no X w∈ such thatfor all X_i∈ X we have X_i∈((';')), and w∩(X_i(';'))=∅.Finally, given two simple extended process A = (_A; Φ_A) and B = (_B; Φ_B), we say that AB whenfor any pair (; ) such that ((_A; Φ_A; ∅;∅);(_B;Φ_B;∅;∅))(;). Continuing Example <ref>, consider the following compressed exploration, where Q_3 containsthe constraints X_2∅x_2, X_1{w_2 n_2}x_1, x_2 and x_1:({Q_0};{Q_0})(c_2,X_2)(c_2,w_2) ({Q_2};{Q_2})(c_1,X_1)…(c_1,w_1) ({Q_3};{Q_3}).Assuming that (Q_3)={X_2↦,X_1↦} (which is the case in the actual procedure) this compressed exploration is not explored bybecauseX_1w_2∈c_2X_2w_2·c_1X_1w_1,X_1(Q_3) = {w_2}∩()=∅.Let A=Φ∅, B=Ψ∅ and A'='Φ'' be quiescent, simple symbolic processes such that ΦΨ, AA', θ∈(A') and θ_(Φ';'). Then there exists an exploration ({A};{B})(';') and some A^+ ∈', θ^+ ∈^+(A^+) such that A^+ = A' and λ_θ =_λ_θ^+, where λ_θ (resp. λ_θ^+) is the substitution associated to θ (resp. to θ^+) with respect to Φ''.Symmetrically for BB'.We proceed by induction on . The empty case is easy.Otherwise, consider AA_1 α A_3 = _3Φ_3_3 withθ∈(A_3), A_1,A_3 quiescent, and θ_(Φ_3; _3)·α. Let A_1 = _1Φ_1_1 and V_1 = (_1). We also have θ|_V_1∈(A_1) and θ|_V_1_(Φ_1; _1), so the induction hypothesis applies and we obtain ({A};{B})(_1;_1) with A_1^+∈_1, A_1^+=A_1 and θ_1^+∈^+(A_1^+)such that the first-order substitutions associated to θ|_V_1 and θ_1^+ w.r.t. (Φ_1;_1) coincide. By hypothesis we have AB, thus _1 _1. Hence atransition can take place on that pair. By definition ofand since A_1^+ = A_1α A_3, there must be some (_1;_1) α (_2;_2) with A_2^+∈_2, A_2^+=A_3. Thus θ∈(A^+_2) and we can apply spec:complete to obtain (_2;_2)(_3;_3) with A^+_3∈_3, A^+_3=A^+_2 and θ^+_3∈^+(A^+_3) such that the subsitutions associated to θ and θ^+_3 w.r.t. (Φ_3;_3) coincide.It only remains to show that this extra execution step inis also present in , that (_3;_3) does not violate ·α in the sense of Definition <ref>. This is because, by definition of the partial solution, we have that θ_3^+ = (_3;_3) ⊔τ for some τ, so that if (_3;_3) violated ·α then we would have θ_3^+_(Φ_3;_3)·α. Since θ_3^+ and θ induce the same first-order substitutions with respect to (Φ_3;_3), we would finally have θ_(Φ_3;_3)·α, contradicting the hypothesis on θ. For any quiescent initial simple processes A and B, we have that:AB if, and only if, AB.Let A=Φ and B=Ψ be two quiescent, initial simple processes. Thanks to our previous results, we have that AB implies AB. Then, we obviously have AB: for any ({Φ∅};{Ψ∅})(';') we have ({Φ∅};{Ψ∅})(';') by definition of , and thus ' ' by hypothesis.For the other direction, it suffices to show that AB implies AB. Let Φ∅ A' = (';Φ';') with θ∈(A') and θ_(Φ';'). By Lemma <ref> we have({Φ∅};{Ψ∅})(';') with A^+∈', θ^+∈^+(A^+) such that A^+=A' and λ_θ^A' =_λ_θ^+^A' where λ_θ^A' (resp. λ_θ^+^A') is the substitution associated to θ (resp. θ^+) w.r.t. (Φ';').Since AB, we have '': there must be some B^+ = (_B';Φ_B';_B';_B^+) ∈' such that θ^+∈^+(B^+)and Φ'λ_θ^+^A'Φ_B'λ_θ^+^B' where λ_θ^+^B' is the substitution associated to θ^+ w.r.t. (Φ_B';_B'). By Lemma <ref> we have Ψ∅B^+. Furthermore, we can show as before (see the end of the proof of Theorem <ref>) that θ∈(B^+) and Φ'λ_θ^A'Φ_B'λ_θ^B', where λ_θ^B' is the substitution associated to θ w.r.t. (Φ_B';_B'). Finally, by θ_(Φ';'), D_(Φ';')=D_(Φ_B';_B') (sets of handlesthat second-order variables may use coincide), and Φ'λ_θ^A'Φ_B'λ_θ^B', we obtain that θ_(Φ_B';_B'). §.§ BenchmarksThe optimisations developed in the present paper have been implemented, following the above approach, in the official version of <cit.>.In practice, many processes enjoy a nice property that allows one to ensure that non-blocking outputs will never occur: it is often the case that enough tests have been performed before outputting a termto ensure its validity.Consider the following process, where k' is assumed to be valid (because it is a pure constructor term):(c,x).xk =𝗁𝖺𝗌𝗁(u)(c,xkk')The term outputted during an execution is necessarily valid thanks to the test that is performed just before this output.We exploit this property in order to avoid adding additional disequalities when integrating compression in . Therefore, in this section, we will restrict ourselves to simple processes that are non-blocking as defined below.Let (;Φ) be a simple process. We say that (;Φ) is non-blocking ifu is valid for any , c, u, Q', , Ψ such that(;Φ) ({(c,u).Q'}∪;Ψ).This condition may be hard to check in general, butit is actually quiteeasy to see that it is satisfied on all of our examples. Roughly, enough tests are performed before any output action, and this ensures the validity of the term when the output actionbecomes reachable, as in Example <ref>.Our modified version of can verify our optimised equivalences in addition to the original trace equivalence. It has been integrated into the main development line of the tool.The modifications of the code (≈ 2kloc) are summarized at <https://github.com/lutcheti/APTE/compare/ref...APTE:POR> For reference, the version of that we are using in the benchmarks below is available at<https://github.com/APTE/APTE/releases/tag/bench-POR-LMCS> together with all benchmark files, in subdirectory bench/protocols. More details, including instructions for reproducing our benchmarks are available at <http://www.lsv.fr/ hirschi/apte_por>. We ran the tool (compiled with OCaml 3.12.1) on a single 2.67GHz Xeon core (memory is not relevant) and compared three different versions: * reference: the reference version without our optimisations ();* compression: using only the compression optimisation ();* reduction: using both compression and reduction (). We first show examples in which equivalence holds. They are the most significant, because the time spent on inequivalent processes is too sensitive to the order in which the (depth-first) exploration is performed.Toy example.Weconsider a parallel composition of n roles R_i as defined in Example <ref>: P_n := Π_i=1^n R_i. When executed in the regular symbolic semantics , the 2 n actions of P_n may be interleaved in (2n)!/2^n ways in a trace containing all actions. In the compressed symbolic semantics , the actions of individual R_i processes must be bundled in blocks, so there are only n! interleavings containing all actions. In the reduced symbolic semantics , only one interleaving of that length remains: the trace cannot deviate from the priority order, since the only way to satisfy a dependency constraint would be to feed an input with a message that cannot be derived without some previously output nonce n_i, but in that case the message will not beand the trace won't be explored further. Note that there is still an exponential number ofsymbolic traces in the reduced semantics when one takes into account traces with less than 2n actions. We show in Figure <ref> the time needed to verify P_n ≈ P_n for n = 1 to 22 in the three versions of described above: reference, compression andreduction. The results, in logarithmic scale, show that each of our optimisations brings an exponential speedup, as predicted by our theoretical analysis. Similar improvements are observed if one compares the numbers of explored pairs rather than execution times.Denning-Sacco protocol. We ran a similar benchmark, checking that Denning-Sacco ensures strong secrecy in various scenarios. The protocol has three roles and we added processes playing those roles in turn, starting with three processes in parallel.Srong secrecy is expressed by considering, after one of the roles B, the output of a message encrypted with the established key on one side of the equivalence, and with a fresh key on the other side.The results are plotted in Figure <ref>. The fact that we add one role out of three at each step explains the irregular growth in verification time.We still observe an exponential speedup for each optimisation. Practical impact. Finally, we illustrate how our optimisations make much more useful in practice for investigating interesting scenarios. Verifying a single session of a protocol brings little assurance into its security. In order to detect replay attacks and to allow the attacker to compare messages that are exchanged, at least two sessions should be considered. This means having at least four parallel processes for two-party protocols, and six when a trusted third party is involved. This is actually beyondwhat the unoptimised can handle in a reasonable amount of time. We show in Figure <ref> how many parallel processes could be handled in 20 hours by on various use cases of protocols, for the same three variants of as before, reference, compression and reduction. We verify an anonymity property for the Passive Authentication protocol of e-passports. For other protocols, we analyse strong secrecy of established keys:for one of the roles we add, on one side of the equivalence, an output encrypted by the established key and, on the other side, an output encrypted by a fresh key. We finally present the benefits of our optimisations for discoveringattacks. We performed some experiments on flawed variants of protocols,shown in Figure <ref>, corresponding to example files in subdirectory bench/protocols/attacks/ of the above mentioned release. The scenario Denning-Sacco A expresses strong secrecy of the (3-party) Denning-Sacco protocol, but this time on two instances of roles at the same time (instead of one as in Figure <ref>). In Denning-Sacco B, we consider again a form of strong secrecy expressed by outputting encrypted messagesbut this time at the end of role B.The Needham-Schroeder pub scenario corresponds to strong secrecy of the public-key Needham-Schroder protocol. The E-Passport PA exposed experiments show that anonymity is(obviously) lost with the Passive Authentication protocol when the secret key is made public. Similarly, the Yahalom exposed experiment shows that strong secrecy of Yahalom is lost when secrets keys are revealed.Since stops its exploration as soon as an attack is found, the time needed for to find the attack highly depends on the order in which the depth-firstexploration is performed. However,as shown in Figure <ref>, we always observein practice dramatic improvements brought by our optimisations compared to the reference version of .In some cases, ouroptimisations are even mandatory for to find the attack using reasonable resources.§ RELATED WORKThe techniques we have presented borrow from standard ideas from concurrency theory and trace theory.Blending all these ingredients, and adapting them to the demanding framework of security protocols, we have come up with partial order reduction techniques that can effectively be used in symbolic verification algorithms for equivalence properties of security protocols. We now discuss related work, and there is a lot of it given the huge success of POR techniques in various application areas. We shall focus on the novel aspects of our approach, and explain why such techniques have not been needed outside of security protocol analysis. These observations are not new: as pointed out by Baier and Katoen <cit.>, “[POR] is mainly appropriate to control-intensive applications and less suited for data-intensive applications”; Clarke et al. <cit.> also remark that “In the domain of model checking of reactive systems, there are numerous techniques for reducing the state space of the system. One such technique is partial-order reduction. This technique does not directly apply to [security protocol analysis]because we explicitly keep track of knowledge of various agents, and our logic can refer to this knowledge in a meaningful way.”We first compare our work with classical POR techniques, andthen comment on previous work in the domain of security protocol analysis. §.§ Classical POR Partial order reduction techniques have proved very useful in the domain of model checking concurrent programs. Given a Labelled Transition System (LTS) and some property to check (a Linear Temporal Logic formula),the basic idea of POR <cit.> is to only consider a reduced version of the given LTS whose transitions of some states might be not exhaustive but are such that this transformation does not affect the property. POR techniques can be categorized in two groups <cit.>. First, the persistent set techniques (stubborn sets, ample sets) where only a sufficiently representative subset of available transitions is explored. Second, sleep set techniques memoize past exploration and use this information along with available transitions to disable some provably redundant transitions. Note that these two kinds of techniques are compatible, and are indeed often combined to obtain better reductions. Theoretical POR techniques apply to transition systems which may not be explicitly available in practice, or whose explicit computation may be too costly. In such cases, POR is often applied to an approximation of the LTS that is obtained through static analysis. Another, more recent approach is to use dynamicPOR <cit.> where the POR arguments are applied based on information that is obtained during the execution of the system.Clearly, classical POR techniques would apply to our concrete LTS, but that would not be practically useful since this LTS is wildly infinite, taking into account all recipes that the attacker could build. Applying most classical POR techniques to the LTS from which data would have been abstracted away would be ineffective: any input would be dependent on any output (since the attacker's knowledge, increased by the output, may enable new input messages). Our compression technique lies between these two extremes. It exploits a semi-commutation property: outputs can be permuted before inputs, but not the converse in general. Further, it exploits the fact that inputs do not increase the attacker's knowledge, and can thus be executed in a chained fashion, under focus. The semi-commutation is reminiscent of the asymmetrical dependency analysis enabled by the conditional stubborn set technique <cit.>, and the execution of inputs under focus may be explained by means of sleep sets. While it may be possible to formally derive our compressed semantics by instantiating abstract POR techniques to our setting, we have not explored this possibility in detail[ Although this would be an interesting question, we do not expect that any improvement of compression would come out of it. Indeed, compression can be argued to be maximal in terms of eliminating redundant traces without analysing data: for any compressed trace there is a way to choose messages and modify tests to obtain a concrete execution which does not belong to the equivalence class of any other compressed trace. ].Concerning our reduced semantics, it may be seen as an application of the sleep set technique (or even as a reformulation of Anisimov's and Knuth's characterization of lexicographic normal forms) but the real contribution with this technique is to have formulated it in such a way(see Definition <ref>) that it can be implemented without requiring an a priori knowledge of data dependencies: it allows us to eliminate redundant traces on-the-fly as data (in)dependency is discovered by the constraint resolution procedure (as explained in Section <ref>)—in this sense, it may be viewed as a case of dynamic POR.Narrowing the discussion a bit more, we now focus on the fact that our techniques are designed for the verification of equivalence properties.This requirement turns several seemingly trivial observations into subtle technical problems. For instance, ideas akin to compression are often applied without justification (in <cit.>) because they are obvious when one does reachability rather than equivalence checking.To understand this, it is important to distinguish between two very different ways of applying POR to equivalence checking (independently of the precise equivalence under consideration). The first approach is to reduce a system such that the reduced system and the original systems are equivalent. In the second approach, one only requires that two reduced systems are equivalent iff the original systems are equivalent. The first approach seems to be more common in the PORliterature (where one finds, reductions that preserve LTL-satisfiability <cit.> or bisimilarity <cit.>) though there are instances of the second approach (for Petri nets <cit.>). In the present work, we follow the second approach: neither of our two reduction techniques preserves trace equivalence. This allows stronger reductions but requires extra care: one has to ensure that the independencies used in the reduction of one process are also meaningful for the other processes; in other words, reduction has to be symmetrical. We come back to these two different approaches later, when discussing specific POR techniques for security.§.§ Security applications The idea of applying POR to the verification of security protocols dates back, at least, to the work of Clarke et al. <cit.>. In this work, the authors remark that traditional POR techniques cannot be directly applied to security mainly because “[they] must keep track of knowledge of various agents” and “[their] logic can refer to this knowledge in a meaningful way”. This led them to define a notion of semi-invisible actions (output actions, that cannot be swapped after inputs but only before them) and design a reduction that prioritizes outputs and performs them in a fixed order. Compared to our work, this reduction is much weaker (even weaker than compression only), only handles a finite set of messages, and only focuses on reachability properties checking.In <cit.>, the authors develop “state space reduction” techniques for the Maude-NRLProtocol Analyzer (Maude-NPA). This tool proceeds by backwards reachability analysis and treats at the same level the exploration of protocol executions and attacker's deductions. Several reductions techniques are specific to this setting, and most are unrelated to partial order reduction in general, and to our work in particular. We note that the lazy intruder techniques from <cit.> should be compared to what is done in constraint resolution procedures (the one used in ) rather than to our work. A simple POR technique used in Maude-NPA is based on the observation that inputs can be executed in priority in the backwards exploration, which corresponds to the fact that we can execute outputs first in forward explorations. We note again that this is only one aspect of the focused strategy, and that it is not trivial to lift this observation from reachability to trace equivalence. Finally, a “transition subsumption” technique is described for Maude-NPA. While highly non-trivial due to the technicalities of the model, this is essentially a tabling technique rather than a partial order reduction. Though it does yield a significant state space reduction (as shown in the experiments <cit.>) it falls short of exploiting independencies fully, and has a potentially high computational cost (which is not evaluated in the benchmarks of<cit.>).In <cit.>, Fokkink et al. model security protocols as labeled transition systems whose states contain the control points of different agents as well as previously outputted messages.They devise some POR technique for these transition systems, where output actions are prioritized and performed in a fixed order. In their work, the original and reduced systems are trace equivalent modulo outputs (the same traces can be found after removing output actions). The justification for their reduction would fail in our setting, where we consider standard trace equivalence with observable outputs. More importantly, their requirement that a reduced system should be equivalent to the original one makes it impossible to swap input actions, and thus reductions such as the execution under focus of our compressed semantics cannot be used. The authors leave as future work the problem of combining their algorithm with symbolic executions, in order to be able to lift the restriction to a finite number of messages.Cremers and Mauw proposed <cit.> a reduction technique for checking secrecy in security protocols. Their method allows to perform outputs eagerly, as in our compressed semantics. It also uses a form of sleep set technique to avoid redundant interleavings of input actions. In addition to being applicable only for reachability property, the algorithm of <cit.> works under the assumption that for each input only finitely many input messages need to be considered. The authors identify as important future work the need to lift their method to the symbolic setting.Earlier work by Mödersheim et al. has shown how to combine POR techniques with symbolic semantics <cit.> in the context of reachability properties for security protocols. This has led to high efficiency gains in the OFMC tool of the AVISPA platform <cit.>. While their reduction is very limited, it brings some key insight on how POR may be combined with symbolic execution. For instance, their reduction imposes a dependency constraint (called differentiation constraint in their work) on the interleavings of {(c,x).(c,m), (d,y).(d,m')}.Assuming that priority is given to the process working on channel c, this constraint enforces that any symbolic interleaving of the form (d,M').(d,w').(c,M).(c,w) would only be explored for instances of M that depend on w'. Our reduced semantics constrains patterns of arbitrary size (instead of just size 2 diamond patterns as above) by means of dependency constraints. Going back to Example <ref>, their technique willonly be able (at most) to exploit the dependencies depicted in plain blue arrows, and they will not consider the one represented by the dashed 2-arrow.Moreover, while we generate dependency constraints on the fly, they implement their technique by looking for such a pattern afterwards.This causes a tradeoff between reduction and the cost of redundancydetection: their technique fails to detect all patterns of this kind. Besides these differences, we note that Mödersheim et al. use dependency constraints to guide a dedicated constraint resolution procedure,while we chose to treat constraint resolution (almost) as a black box, and leave it unchanged. Finally, we recall that our POR technique has been designed tobe sound and complete for trace equivalence checking as well as reachabilitychecking. Finally, in <cit.>, the authors of the present paper extend some of the results presented here. Instead of considering the syntactic fragment of simple processes, we work under the more general semantical assumption of action-determinism. We show that compression and reduction can be extended to that case, preserving the main result: the induced equivalences coincide. However, that work is completely carried out in concrete rather than symbolic semantics. Thus, this development should be viewed as being orthogonal to the one carried out in the present paper. The main ideas behind the integration in symbolic semantics and would apply to the action-deterministic case as well. The line of research followed in <cit.>, that consists in extending the supported fragment for our POR techniques, is still open: it would be interesting to support processes that are not action-deterministic, which are commonplace when analysing anonymity or unlinkability scenarios. § CONCLUSIONWe have developed two POR techniques that are adequate for verifying trace equivalence properties between simple processes.The first refinement groups actions in blocks, while the second one uses dependency constraints to restrict to minimalinterleavings among a class of permutations. In both cases, the refined semantics has less traces, yet we show that the associated trace equivalence coincides with the standard one.We have effectively implemented these refinements in , and shown that they yield the expected, significant benefit.We claim that our POR techniques – at least compression – and the significant optimisations they allow are generic enough to be applicable to other verification methods as long as they perform forward symbolic executions. In addition to the integration in Apte we have extensively discussed, we also have successfully done so in Spec <cit.>. Furthermore, parts of our POR techniques have been independently integrated and implemented in the distributed version of [See <https://github.com/akiss/akiss>.].We are considering several directions for future work.Regarding the theoretical results presented here, it is actually possible to slightly relax the syntactic condition we imposed on processes by an action-determinism hypothesis and apply our reduction techniques on replicated processes <cit.>. The question of whether the action-determinism condition can be removed without degrading the reductions too much is left open. Another interesting direction would be to adapt our techniques for verification methods based on backward search instead of forward search asis the case in this paper. We also believe that stronger reductions can be achieved: for instance, exploiting symmetries should be very useful for dealing with multiple sessions.Regarding the practical application of our results, we can certainly go further. We could investigate the role of the particular choice of the order ≺, to determine heuristics for maximising the practical impact of reduction. *Acknowledgements We would like to thank Vincent Cheval for interesting discussions and comments, especially on Section <ref>.abbrv § NOTATIONS Symbol Description Referencetransition for concrete processes [fig:concrete-semantics]Figure <ref> up-to non-observable actions [subsec:semantics]Section <ref>static equivalence [def:statequiv]Definition <ref>trace inclusion for concrete processes [def:concrete-equivalence]Definition <ref>trace equivalence for concrete processes [def:concrete-equivalence]Definition <ref>focused semantics [fig:sintf]Figure <ref>compressed semantics [fig:sintc]Figure <ref> trace inclusion induced by [def:compressed-equivalence]Definition <ref>trace equivalence induced by[def:compressed-equivalence]Definition <ref>symbolic semantics [figure:symbolic-sem]Figure <ref>focused symbolic semantics [figure:compressed-symbolic-sem]Figure <ref>compressed symbolic semantics [figure:compressed-symbolic-sem]Figure <ref>trace inclusion induced by[def:equiv-symb]Definition <ref>trace inclusion induced by[def:equiv-comp-symb]Definition <ref>trace equivalence induced by[def:equiv-symb]Definition <ref>trace equivalence induced by[def:equiv-comp-symb]Definition <ref> X w dependency constraint [def:dep-constr]Definition <ref>dependency constraints induced by a trace [def:alldep]Definition <ref>trace inclusion up-to dependency constraints [def:reduced-equivalence]Definition <ref>trace equivalence up-to dependency constraints[def:reduced-equivalence]Definition <ref> || independence of blocks [def:independence]Definition <ref> ≡_Φ equivalence of two traces [def:equiv-phi]Definition <ref> Φ associated extented symbolicprocess [subsec:nutshell]Section <ref> Φ^+ associated symbolic process [subsec:nutshell]Section <ref>symbolic inclusion of sets of extended symbolic processes [def:symbeqset]Definition <ref>symbolic equivalence of sets of extended symbolic processes [def:symbeqset]Definition <ref>exploration step [subsec:nutshell]Section <ref>first part of exploration step [subsec:nutshell]Section <ref>second part of exploration step [subsec:nutshell]Section <ref>compressed version of[subsec:apte-compression]Section <ref>reduced version of[subsec:apte-dependency]Section <ref>equivalence induced by[def:eqapte]Definition <ref>equivalence induced by[def:subsec:apte-compression]Section <ref>equivalence induced by[subsec:apte-dependency]Section <ref>§ PROOFS OF SECTION <REF>*It suffices to establish that A α·α'A' implies A α'·αA' for any α _a α'. * Assume that we have AA_iA' with c_i ≠ c_j. Because we are considering simple processes, the two actions must be concurrent. More specifically, our process A must be of the form {P_i, P_j}⊎_rΦ with P_i (resp. P_j) being a basic process on channel c_i (resp. c_j).We assume that in our sequence of reductions, τ actions pertaining to P_i are all executed before reaching A_i, and that τ actions pertaining to _r are executed last. This is without loss of generality, because a τ action on a given basic process can easily be permuted with actions taking place on another basic process, since it does not depend on the context and has no effect on the frame.Thus we have that A_i = {P'_i,P_j}⊎_rΦ⊎{w_i m_i}, A'= {P'_i;P'_j}⊎'_rΦ⊎{w_i m_i, w_jm_j}. Since the τ actions taking place on '_r rely neither on the frame nor on the first two basic processes, we easily obtain the permuted execution:[A{P_i,P'_j}⊎_rΦ⊎{w_j m_j}; {P'_i,P'_j}⊎'_rΦ⊎{w_i m_i,w_j m_j} ]* The permutation of two input actions on distinct channels is very similar. In this case, the frame does not change at all, and the order in which messages are derived from the frame does not matter. Moreover, the instantiation of the input variable on one basic process has no impact on the other ones. * Assume that we have AA_iA' with c_i ≠ c_j andw_i∉(M). Again, the two actions are concurrent, and we can assume that τ actions are organized conveniently so that A is of the form {P_i,P_j}⊎_rΦ with P_i (resp. P_j) a basic process on c_i (resp. c_j); A_i is of the form {P'_i,P_j}⊎_rΦ⊎{w_im_i}; and A' is of the form {P'_i,P'_j}⊎'_rΦ⊎{w_i m_i}. As before, the τ actions from _r to '_r are easily movedaround. Additionally, w_i ∉(M) implies (M)⊆(Φ) and thus we have:{P_i,P_j}⊎_rΦ{P_i,P'_j}⊎_rΦThe next step is trivial:{P_i,P'_j}⊎_rΦ{P'_i,P'_j}⊎'_rΦ⊎{w_i m_i}* We also have to perform the reverse permutation, but we shall not detail it; this time we are delaying the derivation of M from theframe, and it only gets easier.* We first observe that AA' implies A o^* A' if A' is initial andis a (possibly empty) sequence of output actions on the same channel. We prove this by induction on the sequence of actions. If it is empty, we can conclude using one of the Proper rules because A = A' is initial. Otherwise, we have:A (c,w) A”A'.We obtain A”o^* A' by induction hypothesis, and conclude using rules Tau and Out.The next step is to show that A A' implies A i^* A', if A' is initial andis the concatenation of a (possibly empty) sequence of inputs and a non-empty sequence of outputs, all on the same channel. This is easily shown by induction on the number of input actions. If there are none we use the previous result, otherwise we conclude by induction hypothesis and using rules Tau and In. Otherwise, the first output action allows us to conclude from the previous result and rules Tau and Out.We can now show that A A' implies A i^+ A' if A' is initial andis a proper block. Indeed, we must haveA (c,M) A”'A'which allows us to conclude using the previous result and rules Tau and In.We finally obtain that AA' implies AA' when A and A' are initial simple processes andis a sequence of proper blocks. This is done by induction on the number of blocks. The base case is trivial. Because A is initial, the execution of its basic processes can only start with observable actions, thus only one basic process is involved in the execution of the first block. Moreover, we can assume without loss of generality that the execution of this first block results in another initial process: indeed the basic process resulting from that execution is either in the final process A', which is initial, or it will perform another block, it can perform τ actions followed by an input, in which case we can force those τ actions to take place as early as possible. Thus we haveA 𝖻 A”' A'where 𝖻 is a proper block, and we conclude using the previous result and the induction hypothesis.Before proving Theorem <ref>, we establish the following result. Letbe a trace of observable actions such that, for any channel c occurring in the trace, it appears first in an input action. There exists a sequence of proper blocks _io and a sequence of improper blocks _i such that =__a_io·_i.We proceed by induction on the length of , and distinguish two cases: * Ifhas no output action then, by swapping input actions on distinct channels, we reorderso as to obtain _i = ^c_1·…·^c_n=__a where the c_i's are pairwise distinct and ^c_i is an improper block on channel c_i.* Otherwise, there must be a decomposition = _1·cw·_2 such that _1 does not contain any output. We can perform swaps involving input actions of _1 on all channel c'≠ c, so that they are delayed after the first output on c. We obtain =__acM_1·…·cM_n·cw·_1'·_2 with n≥ 1. Next, we swap output actions on channel c from _1'·_2 that are not preceded by another input on c, so as to obtain=__acM_1…cM_n·cw·cw_1…cw_m·_2'such that either _2' does not contain any action on channel c or the first one is an input action. We have thus isolated a first proper block, and we can conclude by induction hypothesis on _2'. Note that the above result does not exploit all the richness of _a. In particular, it never relies on the possibility to swap an input action before an output when the input message does not use the output handled. Indeed, the idea behind compression does not rely on messages. This is no longer the case in Section <ref> where we use _a more fully. We finally prove the main result about the compressed semantics relying on Proposition <ref> stated and proved above. Given two simple process A = Φ and A' = 'Φ', we shall write Φ(A) Φ(A') (or even AA') instead of ΦΦ'.* We prove the two directions separately.(⇒)Let A be an initial simple process such that A ≈ B and AA'. One can easily see that the tracemust be of the form _io·_i where _io is made of proper blocks and _i is a (possibly empty) sequence of inputs on the same channel c_j. We have:A _io A”_i A' Using Proposition <ref>, we obtain that A _io A”. We also claim that A”_i A^+ for some A^+ having the same frame as A' which is itself equal to the one of A”. This is obvious when _i is empty—in that case we can simply choose A^+ = A' = A”. Otherwise, the execution of the improper block _i results from theapplication of rule Improper. Except for the fact that this rule “kills” the resulting process, its subderivation simply packages a sequence of inputs, and so we have a suitable A^+. We thus have:A_ioA”_iA^+ By hypothesis, it implies that B _ioB” and B_io·_iB^+ with A” B” and A^+B^+. Relying on the fact that B is a simple process, we have:B _io B”_i B^+It remains to establish that BB' such that B' A'. We can assume that B” does not have any basic process starting with a test, without loss of generality since forcing τ actions cannot break static equivalence. Further, we observe that B” is initial. Otherwise, it would mean that a basic process of B is not initial (absurd) or that one of the blocks of _io, which are maximal for A, is not maximal for B (absurd again, because it contradicts A ≈ B). This allows us to apply Proposition <ref> to obtainB _io B”.This concludes when _i is empty, because B' = B” A” = A'. Otherwise, we note that A^+ cannot perform any action on channel c_j, because the execution of _i in the compressed semantics must be maximal. Since A ≈ B, it must be that B^+ cannot perform any observable action on the channel c_j either. Thus B” can complete an improper step:B”_iB'B' = ∅Φ(B^+).We can finally conclude that BB' with Φ(B') = Φ(B^+) Φ(A^+) = Φ(A').(⇐) Let A be an initial simple process such that AB and AA'. We “complete” this execution as follows: * We force τ actions whenever possible.* If the last action on c inis an input, we trigger available inputs on c using a valid public constant as a recipe. * We trigger all the outputs that are available and not blocked. We obtain a trace of the form ·^+. Let A^+ be the process obtained from this trace:AA' ^+ A^+We observe that A^+ is initial. Indeed, for each basic process that performs actions in ·^+, we have that:* either the last action on its channel is an output and the basic process is of the form (c,_).P or (c,u).P with (u), * or the last action is an input and the basic process is reduced to 0 and disappears, or it is an output which is blocked. Next, we apply Proposition <ref> to obtain traces _sio (resp. _i) made of proper (resp. improper) blocks, such that ·^+ =__a_io·_i. By Lemma <ref> we know that this permuted trace can also lead to A^+:A _io A_io_i A^+As before, we can assume that A_io cannot perform any τ action. Under this condition, since A^+ is initial, A_io must also be initial.By Proposition <ref> we have that A _io A_io, and AB implies that:B _io B_ioΦ(A_io) Φ(B_io).A simple inspection of the Proper rules shows that a basic process resulting from the execution of a proper block must be initial. Thus, since the whole simple process B is initial, B_io is initial too.Thanks to Proposition <ref>, we have that B _io B_io. Our goal is now to prove that we can complete this execution with _i. This trace is of the form ^c_1·^c_2…^c_n where ^c_i contains only inputs on channel c_i and the c_i are pairwise disjoint. Now, we easily see that for each i,A_io^c_i A_iand A_i has no more atomic process on channel c_i. Thus we have A_io^c_i A_i^0 with A_i^0 = ∅Φ(A_i). Since A ≈_c B, we must have some B_i^0 such that:B _io B_io^c_i B_i^0We can translate this back to the regular semantics, obtaining B _io B_io^c_i B_i. We can now execute all these inputs to obtain an execution of _io·_i towards some process B^+:B _io B_io_i B^+Permuting those actions, we obtain thanks to Lemma <ref>:BB' ^+ B^+We observe that Φ(B^+) = Φ(B_io) Φ(A_io) Φ(A^+), and it follows that A'B' because those frames have the same domain, which is a subset of that of Φ(A^+) Φ(B^+). | http://arxiv.org/abs/1704.08540v5 | {
"authors": [
"David Baelde",
"Stéphanie Delaune",
"Lucca Hirschi"
],
"categories": [
"cs.LO",
"C.2.2; D.2.4; F.3.1"
],
"primary_category": "cs.LO",
"published": "20170427124618",
"title": "A Reduced Semantics for Deciding Trace Equivalence"
} |
Threshold-activated transport stabilizes chaotic populations to steady states Chandrakala Meena, Pranay Deep Rungta, Sudeshna Sinha* Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, SAS Nagar, Sector 81, Manauli PO 140306, Punjab, [email protected] § ABSTRACT We explore Random Scale-Free networks of populations, modelled by chaotic Ricker maps, connected by transport that is triggered when population density in a patch is in excess of a critical threshold level. Our central result is that threshold-activated dispersal leads to stable fixed populations, for a wide range of threshold levels. Further, suppression of chaos is facilitated when the threshold-activated migration is more rapid than the intrinsic population dynamics of a patch. Additionally, networks with large number of nodes open to the environment, readily yield stable steady states. Lastly we demonstrate that in networks with very few open nodes, the degree and betweeness centrality of the node open to the environment has a pronounced influence on control. All qualitative trends are corroborated by quantitative measures, reflecting the efficiency of control, and the width of the steady state window. § INTRODUCTION Nonlinear systems, describing both natural phenomena as well as human-engineered devices, can give rise to a rich gamut of patterns ranging from fixed points to cycles and chaos. So mechanisms that enable a chaotic system to maintain a fixed desired activity (the “goal”) has witnessed enormous research attention <cit.>. In early years the focus was on controlling low-dimensional chaotic systems, and guiding chaotic states to desired target states <cit.>. Efforts then moved on to the arena of lattices modelling extended systems, and the control of spatiotemporal patterns in such systems <cit.>. With the advent of network science to describe connections between complex sub-systems, the new challenge is to find mechanisms or strategies that are capable of stabilizing these large interactive systems <cit.>. In this work we consider a network of population patches <cit.>, or “a population of populations” <cit.>. Now, most models of metapopulation dynamics consider density dependent dispersal, analogous to reaction-diffusion processes <cit.>. Here we consider a different scenario, namely one where the inter-patch connection is triggered by the excess of population density in a patch <cit.>. This describes a system comprising of many spatially discrete sub-populations connected by threshold-activated dispersal. Our principal question will be the following: can threshold-activated coupling serve to control networks of intrinsically chaotic populations on to regular behaviour? In the sections below, we will first discuss details of the nodal dynamics, as well as the salient features of pulsatile transport triggered by threshold mechanisms. We will then go on to demonstrate, through qualitative and quantitative measures, that such threshold-activated connections manage to stabilize chaotic populations to steady states. Further we will explore how the critical threshold that triggers the migration, and the timescales of the nodal dynamics vis-a-vis transport, influences the emergent dynamics. § MODEL Consider a network of N sub-systems, characterized by variable x_n(i) at each node/site i (i=1,… N) at time instant n. Specifically, we study a prototypical map, the Ricker (Exponential) Map, at the local nodes. Such a map models population growth of species with non-overlapping generations, and is given by the functional form: x_n+1 (i) =f(x_n (i))= x_n(i) exp(r(1-x_n(i))) where r is interpreted as an intrinsic growth rate and (dimensionless) x_n(i) is the population scaled by the carrying capacity at generation n at node/site i. We consider r=4 in this work, namely, an isolated uncoupled population patch displays chaotic behaviour. The coupling in the system is triggered by a threshold mechanisms <cit.>. Namely, the dynamics of node i is such that if x_n+1 (i) > x_c, the variable is adjusted back to x_c and the “excess” x_n+1-x_c is distributed to the neighbouring patches. The threshold parameter x_c is the critical value the state variable has to exceed in order to initiate threshold-activated coupling. So this class of coupling is pulsatile, rather than the more usual continuous coupling forms, as it is triggered only when a node exceeds threshold. Specifically, we study such population patches coupled in a Random Scale-Free network, where the network of underlying connections is constructed via the Barabasi-Albert preferential attachment algorithm, with the number of links of each new node denoted by parameter m <cit.>. The resultant network is characterized by a fat-tailed degree distribution, found widely in nature. The underlying web of connections determines the “neighbours” to which the excess is equi-distributed. Further, certain nodes in the network may be open to the environment, and the excess from such nodes is transported out of the system. Such a scenario will model an open system, and such nodes are analogous to the “open edge of the system”. We denote the fraction of open nodes in the network, that is the number of open nodes scaled by system size N, by f^open. In this work we also consider closed systems with no nodes open to the environment, where nothing is transported out of the system, i.e. f^open=0. The threshold-activated migration from an over-critical patch can trigger subsequent transport, as the redistribution of excess can cause neighbouring sites to become over-critical, thus initiating a domino effect, much like an “avalanche” in models of self-organized criticality <cit.>. All transport within patches stop when all patches are under the critical value, i.e. all x (i) < x_c. So there are two natural time-scales here. One time-scale characterizes the chaotic update of the populations at node i. The other time scale involves the redistribution of population densities arising from threshold-activated transport. We denote the time interval between chaotic updates, namely the time available for redistribution of excess resulting from threshold-activated transport processes, by T_R. This is analogous to the relaxation time in models of self-organized criticality, such as the influential sandpile model <cit.>. T_R then indicates the comparative time-scales of the threshold-activated migration and the intrinsic population dynamics of a patch. § RESULTS We have simulated this threshold-coupled scale-free network of populations, under varying threshold levels x_c (0 ≤ x_c ≤ 2). We considered networks with varying number of open nodes, namely systems that have different nodes/sites open to the environment from where the excess population can migrate out of the system. Further, we have studied a range of redistribution times T_R, capturing different timescales of migration vis-a-vis population change <cit.>. With no loss of generality, in the sections below, we will present salient results for Random Scale-Free networks with m=1, and specifically demonstrate, both qualitatively and quantitatively, the stabilization of networks of chaotic populations to steady-states under threshold-activated coupling. §.§ Emergence of Steady States First, we consider the case of large T_R, where the transport processes are fast compared to the population dynamics, or equivalently, the population dynamics of the patch is slow compared to inter-patch migrations. Namely, since the chaotic update is much slower than the transport between nodes, the situation is analogous to the slow driving limit <cit.>. In such a case, the system has time for many transport events to occur between chaotic updates, and avalanches can die down, i.e. the system is “relaxed” or “under-critical” between the chaotic updates. So when the transport/migration is significantly faster than the population update (namely the time between generations), the system tends to reach a stationary state where all nodal populations are less than critical. An illustrative case of the state of the nodes in the network is shown in Fig. <ref>. Without much loss of generality, we display results for a network of size N=100, for a representative large value of redistribution time T_R=5000. It is clear that all the nodes in the network gets stabilized to a fixed point, namely all population patches evolve to a stable steady state. The next natural question is the influence of the critical threshold x_c on the emergent dynamics. This dependence is demonstrated in bifurcation diagrams displayed in Fig. <ref>. It is clearly evident from these that a large window of threshold values (0 ≤ x_c < 1) yield spatiotemporal steady states in the network <cit.>. It is also apparent that the degree of the open node does not affect the emergence of steady states here. Further, for threshold values beyond the window of control to fixed states, one obtains cycles of period 2. Namely for threshold levels 1 < x_c < 2 the populations evolve in regular cycles, where low population densities alternate with a high population densities. This behaviour is reminiscent of the field experiment conducted by Scheffer et al <cit.> which showed the existence of self-perpetuating stable states alternating between blue-green algae and green algae. We discuss the underlying reason for this behaviour in the Appendix, and offer analytical reasons for the range of period-1 and 2 behaviour considering a single threshold-limited map. So our first result can be summarized as follows: when redistribution time T_R is large and the critical threshold x_c is small, we have very efficient control of networks of chaotic populations to steady states. This suppression of chaos and quick evolution to a stable steady states occurs irrespective of the number of open nodes. §.§ Influence of the redistribution time and the number of opennodes on the suppression of chaos Now we focus on the network dynamics when T_R is small, and the time-scales of the nodal population dynamics and the inter-patch transport are comparable. So now there will be nodes that may remain over-critical at the time of the subsequent chaotic update, as the system does not have sufficient time to “relax” between population updates. The network is then akin to a rapidly driven system, with the de-stabilizing effect of the chaotic population dynamics competing with the stabilizing influence of the threshold-activated coupling. So for small T_R, the system does not get enough time to relax to under-critical states and so perfect control to steady states may not be achieved. Importantly now, the fraction of open nodes f^open is crucial to chaos suppression. In general, a larger fraction of open nodes facilitates control of the intrinsic chaos of the nodal population dynamics, as the de-stabilizing “excess” is transported out of the system more efficiently. We investigate this dependence, through space-time plots of representative networks with varying number of open nodes and redistribution times (cf. Fig. <ref>), and through bifurcation diagrams of this system with respect to critical threshold x_c (cf. Fig. <ref>). It is apparent from Fig. <ref>, that when there are enough open nodes, the network relaxes to the steady state even for low redistribution times. Also notice from Fig. <ref>(d) that the system reaches the steady state very rapidly, namely within a few time steps, from the random initial state. So more open nodes yields better control of the intrinsic chaos of the nodal population dynamics to fixed populations. This is also corroborated in the bifurcation diagrams displayed in Fig. <ref>, where control to steady states is seen even for low T_R, when there are large number of open nodes, vis-a-vis networks with few open nodes. Further contrast this with the dynamics of a system with large T_R, shown earlier in Fig. <ref>, where even a single open node leads to stable steady states for a large range of threshold values. Similar qualitative trends are also borne out in Random Scale-Free network with m=2, where again more open nodes and longer redistribution times result in better control to fixed population densities. As a limiting case, we also studied the spatiotemporal behaviour of threshold-coupled networks without open nodes. Here the network of coupled population patches is a closed system. Again the intrinsic chaos of the populations is suppressed to regular behaviour, for large ranges of threshold values. However, rather than steady states, one now obtains period-2 cycles. This is evident through the bifurcation diagram of a closed network (cf. Fig. <ref>) vis-a-vis networks with at least one open node (cf. Fig. <ref>). Also, note the similarity of the bifurcation diagram of the closed system with that of a system with low T_R and few open nodes. This similarity stems from the underlying fact that in both cases the network cannot relax to completely under-critical states by redistribution of excess between the population updates, either due to paucity of time for redistribution (namely low T_R) or due to the absence of open nodes to transport excess out of the system. Lastly, we explore the case of networks with very few (typically 1 or 2) open nodes, and study the effect of the degree and betweenness centrality <cit.> of these open nodes on the control to steady states. We observe that when there are very few open nodes, the degreeand betweenness centrality of the open node is important, with the region of control being large when the open node has the high degree/betweenness centrality, and vice versa. This interesting behaviour is clearly seen in the bifurcation diagrams shown in Figs. <ref>a-d, which demonstrate that the degree and betweeness centrality of the open node has a pronounced influence on control. §.§ Quantitative Measures of the Efficiency of Chaos Suppression We now investigate a couple of quantitative measures that provide indicators of the efficiency and robustness of the suppression of chaos in the network. The first quantity is the average redistribution time ⟨ T ⟩, defined as the time taken for all nodes in a system to be under-critical (i.e. x_i < x_c for all i), averaged over a large sample of random initial states and network configurations. So ⟨ T ⟩ provides a measure of the efficiency of stabilizing the system, and reflects the rate at which the de-stabilizing “excess” is transported out of the network. Fig. <ref> shows the dependence of ⟨ T ⟩ on system size N. Clearly, while larger networks need longer redistribution times in order to reach steady states, this increase is only logarithmic. This is further corroborated by calculating the average fraction of nodes in the network that go to steady states with respect to the redistribution time T_R, for networks of different sizes, with varying number of open nodes (cf. Fig. <ref>).Clearly for small systems, with sufficiently high f^open, very low T_R can lead to stabilization of all nodes. Importantly, when the fraction of open nodes is very small, the average redistribution time ⟨ T ⟩ depends sensitively on the betweenness centrality of the open node, and to a lesser extent its degree. Figs. <ref>a-b present illustrative results demonstrating this observation. Next we calculate the range of threshold values yielding steady states, averaged over a large sample of network configurations and initial states, denoted by ⟨ R ⟩. Larger ⟨ R ⟩ implies that steady states will be obtained in a larger window in x_c space, thereby signalling a more robust control. We have explored the dependence of this quantity on redistribution time T_R, and also on the fraction of open nodes in the network, denoted by f^open. From Fig. <ref> we see that the steady-state window in x_c rapidly converges to ∼ 1 (namely, the range 0 ≤ x_c < 1), as the number of open nodes increases. So the window yielding suppression of chaos is almost independent of the number of open nodes, after a critical fraction of open nodes f_c^open. We observe that f_c^open tends to zero as the redistribution times increases and system size decreases, implying that very few open nodes are necessary in order to lead the network to a steady state. Lastly we explore the scenario of very few open nodes (f^open <<f_c^open) in greater depth, through the quantitative measures⟨ R ⟩ and ⟨ T ⟩. In particular, weinvestigate the limiting case of a single open node. Ourattempt will be to understand the influence of the degree k andbetweeness centrality b of the open node on the capacity tosuppress chaos. We have already observed the significant effect ofthe betweeness centrality of the open node on the efficiency ofcontrol to steady states through bifurcation diagrams inFig. <ref>. This is now further corroborated quantitatively bythe dependence of ⟨ R ⟩ and ⟨ T ⟩,displayed in Figs. <ref> and <ref>(b). Theeffect of the degree of the open node is less pronounced, though italso does have a discernable effect on the suppression of chaos. Asevident from Fig. <ref>(a), when the open node has ahigher degree, it has a higher ⟨ R ⟩, indicating thatopen nodes with higher degree yield larger steady state windows. § CONCLUSIONS We have explored Random Scale-Free networks of populations under threshold-activated transport. Namely we have a system comprising of many spatially distributed sub-populations connected by migrations triggered by excess population density in a patch. We have simulated this threshold-coupled Random Scale-Free network of populations, under varying threshold levels x_c.We considered networks with varying number of open nodes, namely systems that have different nodes/sites open to the environment from where the excess population can migrate out of the system. Further, we have studied a range of redistribution times T_R, capturing different timescales of migration vis-a-vis population change. Our first important observation is as follows: when redistribution time T_Ris large and the critical threshold x_c is small (0 ≤ x_c < 1), we havevery efficient control of networks of chaotic populations to steady states.This suppression of chaos and quick evolution to a stable steady states occursirrespective of the number of open nodes. Further, for threshold values beyondthe window of control to fixed states, one obtains cycles of period 2. Namelyfor threshold levels 1 < x_c < 2 the populations evolve in regular cycles,where low population densities alternate with a high population densities. Thisbehaviour is reminiscent of field experiments <cit.> thatshow the existence of alternating states. We offer an underlying reason forthis behaviour through the analysis of a single threshold-limited map.For small redistribution time T_R, the system does not get enough time torelax to under-critical states and so perfect control to steady states may notbe achieved. Importantly, now the number of open nodes is crucial to chaossuppression. We clearly demonstrate that when there are enough open nodes, thenetwork relaxes to the steady state even for low redistribution times. So moreopen nodes yields better control of the intrinsic chaos of the nodal populationdynamics to fixed populations. We corroborate all qualitative observations byquantitative measures such as average redistribution time, defined as the timetaken for all nodes in a system to be under-critical, and the range ofthreshold values yielding steady states.We also explored the case of networks with very few (typically 1 or 2) opennodes in detail, in order to gauge the effect of the degree and betweennesscentrality of these open nodes on the control to steady states. We observedthat the degree of the open node does not have significant influence on chaossuppression. However, betweenness centrality of the open node is important,with the region of control being large when the open node has the highbetweenness centrality, and vice versa.In summary, threshold-activated transport yields a very potent coupling form ina network of populations, leading to robust suppression of the intrinsic chaosof the nodal populations on to regular steady states or periodic cycles. Sothis suggests a mechanism by which chaotic populations can be stabilizedrapidly through migrations or dispersals triggered by excess population densityin a patch.§ APPENDIX : ANALYSIS OF A SINGLE POPULATION PATCH UNDER THRESHOLD-ACTIVATED TRANSPORTWe will now analyze the dynamics of a single Ricker map, modelling a single population patch, under threshold-activated transport. Specifically then we have the following scenario: in the dynamical evolution of the system, if the updated state exceeds a critical threshold x_c, it transports the excess out of the system and“re-sets” to level x_c. So the effective map of the dynamics is: x_n+1=f(x_n) if f(x_n) < x_cx_n+1=x_c if f(x_n) ≥ x_cThis is effectively a “beheaded” or “flat-top” map, with the curve lying above x_n+1>x_c in the usual Ricker map being “sliced” to x_c (cf. Fig. <ref>a). The level at which the map is chopped off depends on the threshold x_c. The fixed point solution x^⋆ occurs at the intersection of this f(x) curve and the 45^0 line, namely x^⋆ = x_c. Remarkably, this fixed point is super-stable if the intersection occurs at the “flat top”, since f^' (x^⋆) = 0 there.Clearly, as the threshold increases the intersection of the effective map andthe 45^0 line is no longer located at the “flat-top”. This is clear for theeffective maps for x_c=0.5 vis-a-vis that for x_c = 1.5 inFig. <ref>a. So x^⋆ for sufficiently high x_c will nolonger be stable (eg. x_c = 1.5 will not yield a stable fixed point). So wego on to inspect the second iterate of the effective map, in order to ascertainif a stable period-2 cycle is obtained (cf. Fig. <ref>b). Now theperiod-2 cycle solutions occur at the intersection of the f^2(x) curve andthe 45^0 line, and again this cycle is stable if and only if the intersectionoccurs at the “flat top”, namely where f^' (x) = 0. In theillustrative example displayed in Fig. <ref>b it is clear that forx_c=0.5, where the fixed point is super-stable, the period-2 is alsonaturally super-stable. Interestingly now, for x_c=1.5, which had an unstablefixed point solution, the period-2 solution is super-stable. So higher x_calso controls the intrinsic chaos. However, instead of a stable steady state,it yields stable periodic behaviour.Alternately, one can understand the emergence of stable cycles under thresholdcontrol as follows: The ergodicity of the system ensures that the system willexplore the available phase space fully, and the state variable is thusguaranteed to exceed threshold at some point in time. So one can analyse thedynamics of the effective map starting with the initial state at x_c. Nowstarting from x_c the dynamics will run as in the usual Ricker population mapuntil x_n+1>x_c, at which point it is re-set back to x_c and the cyclestarts again. So once it exceeds the critical value it is trapped immediatelyin a stable cycle whose periodicity is determined by the value of thethreshold. Further, this allows us to exactly obtain the values of thresholdx_c that yield stable fixed points x^* (namely period-1). This is simplythe range of x_c for which the first iterate of the Ricker map lies abovex_c. In this range f (x_c) > x_c. So starting from an initial state x_c,we will be updated in the next iterate to a state greater than x_c, leadingto the transport of the excess f(x)-x_c out of the system and the“relaxation” of the system to x_c.The curves f_n (x_c) as a function of threshold x_c aredisplayed in Fig. <ref>. For n=0, f_0(x_c)= x_c; forn=1, f_0(x_c)=x_cexp(r(1- x_c)), and in generalf_n(x_c) = f ∘ f_n-1(x_c) = f ∘ f ∘…f(x_c). From the figure it can be clearly seen that in therange of x_c ∈ [0:1], f(x_c) > x_c. So if the threshold isin this range, the system will evolve quickly to a steady stateat x^*=x_c, and transport the excess, namely f(x_c)-x_c, outof the system after every update of the population in the patch.Similarly, it can be seen that f_2 (x_c) = f (f (x_c)) is larger than x_c (while f (x_c) < x_c) in the range of threshold x_c ∈ (1,2]. So in this range of threshold, we obtain a stable period 2 cycle. Namely, the population at x_c evolves to f(x_c)< x_c which then evolves to f^2 (x_c). Sincef^2 (x_c)> x_c, it is mapped back to x_c. Hence a cycle of period 2 arises, with the values of the two points in the cycle beingx_c and f (x_c). It can be seen from Fig. <ref> that this range is from x_c ∼ 1 to x_c ∼ 2. This also corroborates the analysis using effective “flat-top” maps (cf. Fig. <ref>). When there is enough time to relax between chaotic updates (namely T_R is large and/or the number of open nodes is sufficiently high), the collective excess of the network is transported out of the system. This implies that the individual nodes behave essentially like the “flat-top” map analysed here. This explains why the range of threshold values yielding fixed points and period-2 cycles obtained in networks of threshold-coupled chaotic systems (cf. Fig. <ref>) matches so well with that obtained here(cf. Fig. <ref>). 20 control A few representative examples: A. Garfinkel, M. Spano, W. 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Pastor-Satorras and A Vespignani, Nature Physics 3,276-282 (2007). dispersal2 B Kerr, C Neuhauser, B J M Bohannan & A M Dean, Nature, 442, 75 (2006). dispersal4 JH Brown & A Kodric-Brown, Ecology, 58, 445 (1977). dispersal5 T Reichenbach, M Mobilia & E Frey, Nature, 448, 1046 (2007). dispersal6 B Cazelles, S Bottani & L Stone, Proc. R. Soc. B, 268, 2595 (2001). ssprl S. Sinha and D. Biswas, Phys. Rev. Letts. 71 (1993)2010. sspre S. Sinha, Phys. Rev. E, 49 (1994) 4832; Phys. Letts. A, 199 (1995) 365; Int. Jour. Mod. Phys. B 9 (1995) 875.kazu T. Morie, D. Atuti, K. Ifuku, Y. Horio, K. Aihara (2011) 20th European Conference on Circuit Theory and Design (ECCTD), 126-129;G. He, M.D. Shrimali, K. Aihara (2007) International Joint Conference onNeural Networks, pp. 350-354. scalefree Barabasi, A.-L. and R. Albert, Science 286, 509 (1999). bak P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Letts. 59 (1987) 381. relax A. Mondal and S. Sinha, Phys. Rev. E 73 (2006). thresh S. Sinha, Phys. Rev. E, 63 (2001) 036212; ibid, 69 (2004) 066209; K. Murali and S. Sinha, Phys. Rev. E, 68 (2003) 016210. self-perpetuatingM. Scheffer, S. Rinaldi,A. Gragnani, L. R. Mur, & E. H. van Nes, Ecology, 78, 272 (1997).betweeness Betweenness centrality of a node is given asb(i)=∑_s,t∈ Iσ(s,t|i)/σ(s,t), where I is the setof all nodes, σ(s,t) is the number of shortest paths between nodes sand t and σ(s,t|i) is the number of shortest paths passing through thenode i. | http://arxiv.org/abs/1704.08506v1 | {
"authors": [
"Chandrakala Meena",
"Pranay Deep Rungta",
"Sudeshna Sinha"
],
"categories": [
"nlin.CD"
],
"primary_category": "nlin.CD",
"published": "20170427110938",
"title": "Threshold-activated transport stabilizes chaotic populations to steady states"
} |
Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA 1Mani L. Bhaumik Institute for Theoretical Physics, University of California, Los Angeles, CA 90095, USA [email protected] dwarfs have atmospheres that are expected to consist nearly entirely of hydrogen and helium, since heavier elements will sink out of sight on short timescales. However, observations have revealed atmospheric pollution by heavier elements in about a quarter to a half of all white dwarfs. While most of the pollution can be accounted for with asteroidal or dwarf planetary material, recent observations indicate that larger planetary bodies, as well as icy and volatile material from Kuiper belt analog objects, are also viable sources of pollution. The commonly accepted pollution mechanisms, namely scattering interactions between planetary bodies orbiting the white dwarfs, can hardly account for pollution by objects with large masses or long-period orbits. Here we report on a mechanism that naturally leads to the emergence of massive body and icy and volatile material pollution. This mechanism occurs in wide binary stellar systems, where the mass loss of the planets' host stars during post main sequence stellar evolution can trigger the Eccentric Kozai-Lidov mechanism. This mechanism leads to large eccentricity excitations, which can bring massive and long-period objects close enough to the white dwarfs to be accreted. We find that this mechanism readily explains and is consistent with observations.Throwing Icebergs at White Dwarfs Alexander P. Stephan, Smadar Naoz1, B. ZuckermanApril 27, 2017 ===================================================== § INTRODUCTION Most stars (≤ 8 M_⊙) that have exhausted their nuclear fuel end up as white dwarfs (WDs). Over the last few decades, many WDs have been observed with spectra that show the presence of significant amounts of elements heavier than hydrogen or helium in their atmospheres <cit.>. These heavy elements are expected to sink rapidly to the core of WDs, which implies a recent replenishment <cit.>. The commonly expected source for this replenishment is material from asteroidal or minor planet size bodies <cit.>, some of which have been observed in the process of tidal break-up <cit.>. However, some observations have suggested that planetary bodies, with sizes of the order of Mars or larger, can also contribute to WD pollution <cit.>. Furthermore, for the first time a WD (WD 1425+540) has shown signs of pollution by icy and volatile material from a Kuiper belt analog object that was initially on a very wide orbit <cit.>.Here we report on a mechanism that naturally leads to the emergence of both of these observed features, large planetary mass and icy and volatile material pollution. In this mechanism we utilize the observed high binary fraction of stars <cit.> that facilitates the accretion of long-period planets and Kuiper belt analog objects onto WDs. Secular (i.e., long term coherent) gravitational perturbations exerted by a stellar companion on such objects can excite their eccentricities to extreme values, and even lead to accretion onto the host star. If these extreme eccentricity excitations take place after the host star has evolved to the WD phase, the WD can be polluted by a planet or a Kuiper belt analog object. Most efforts to explain the aforementioned polluted WD observations have focused on mechanisms to bring rocky asteroids onto a WD, either through scattering or secular effects <cit.>. These can explain small mass object accretion by material that orbits the WD on relatively close orbits, as well as Fe, Mg, O, and Si signatures in the WD atmosphere <cit.>. However, they cannot readily account for, or have not been used to explain, accretion of planetary size objects <cit.> or icy and volatile material <cit.> from wide orbits. Here, we consider a potential polluter (PP), whose mass can be anywhere between several times the mass of Jupiter to the mass of a large asteroid, orbiting its host star on a relatively wide orbit that is being gravitationally perturbed by a much more distant stellar companion through the Eccentric Kozai-Lidov mechanism.In this paper, planetary material refers to rocky and metallic material in large amounts (i.e. from Mars-size, or larger, bodies), while icy material refers to water ice, and volatile material refers to volatile chemicals based largely on nitrogen, carbon, or sulfur. Neptune-like ice giants and Kuiper belt analog objects contain both icy and volatile material, while ice-giants also have large rocky and metallic cores. Volatile material, and thus nitrogen, is difficult to bring onto a WD as its snow line is much further from a star than for planetary or icy material. While the accretion of volatile material most noticably enriches a WD's atmosphere in carbon and nitrogen <cit.>, the accretion of icy material will over time accumulate hydrogen in the atmosphere (especially noticeable for helium-dominated WDs), since hydrogen always remains in the atmosphere <cit.>. The majority of WD progenitors have a binary companion <cit.> that can excite the eccentricities of PPs to extreme values <cit.>. This can cause some of these PPs to accrete onto the primary star during its main sequence lifetime or to be engulfed by the star as it becomes a red giant. During the Asymptotic Giant Branch (AGB) stage, stars lose mass, which causes the orbits of surviving PPs and distant stellar companions to expand. After this stage, if a companion can trigger extreme eccentricity excitations, accretion onto and pollution of WDs can occur. The paper is organized as follows: We begin by describing the numerical setup of our calculations and Monte Carlo simulations (Section <ref>), followed by a description of the orbital parameters that lead to accretion (Section <ref>). We end the paper by discussing the implications of our results and our conclusions (Section <ref>).§ INITIAL CONDITIONS AND NUMERICAL SETUP§.§ Monte Carlo Simulations We perform large Monte Carlo simulations of two representative example scenarios covering different mass scales, Neptune-like planets (denoted Neptune-runs) <cit.> and Kuiper belt analogs (denoted Kuiper-runs), and give a proof of concept for the proposed pollution mechanism. The initial parameters for these systems are chosen to be consistent with observed main sequence binary stars (see Table <ref> for an overview of the parameters). As such we choose the primary stellar massfrom a Salpeter distribution <cit.>, limited between 1 and 8 M_⊙. More massive stars are not expected to evolve into WDs, while less massive stars have not had enough time to evolve to become WDs over the age of the Galaxy. The mass of the companion star is chosen from a normal distribution of mass ratios consistent with observations of field binaries <cit.>.The masses and radii of Neptune-like planets are set equal to Neptune's, while the mass and radius of Kuiper belt dwarf planet Eris is used for the Kuiper belt analog objects.c rrr Initial Parameters before applying stability criteria4 Parameter Neptune-runs Kuiper-runs WD 1425+540 runs # of runs 3000 1500 1500 a_PP,i [AU] 20-50 40-100 120-300 a_c,i [AU] ≳ 200^D&M≳ 400^D&M 1120 m_⋆,i [M_⊙] 1-8^S 1-8^S 2 m_PP,i [M_⊙] 5.149 × 10^-5 8.345 × 10^-98.345 × 10^-9 m_c,i [M_⊙] 0.1-8^D&M 0.1-8^D&M0.75 e_pp,i 0.01 0-0.15 0-0.15 D&MBinary separation and companion mass distributions are taken from <cit.>, determined from binary observations. SPrimary stellar mass distribution taken from the Salpeter Initial Mass Function <cit.>. Listed are the initial semi-major axis (a), mass (m), and eccentricity (e) distributions for Potential Polluters (subscript PP), primary stars (subscript ⋆), and companion stars (subscript c). Unless stated differently, all parameters distributions are uniform within the given ranges. The eccentricity of the companion star's orbit is picked from a uniform distribution between 0 and 1, while i_tot is chosen isotropically, for all three runs.The semi-major axis (SMA) values (a_pp) of the Neptune-like planets are chosen from a uniform distribution between 20 and 50 AU and set with initially very low eccentricities (e_pp=0.01), while Kuiper belt analog objects' SMA values are chosen from a uniform distribution between 40 and 100 AU with eccentricities chosen from a uniform distribution between 0 and 0.15 <cit.>. Long term stability requires: ϵ=a_pp/a_ce_c/1-e_c^2<0.1and a_pp/a_c<0.1 <cit.> so that the orbits of the PP and outer companion do not ever lead to dynamical instability and strong short-term interactions, such as, for example, scattering. This restricts the potential binary companions to wide orbits (a_c≳200 AU), as shown in Figure <ref>. We still use observational estimates from field binaries for the orbital separation <cit.>. Note that we also reject systems with initial a_c values of ≳10000 AU, as galactic tides become strong enough to efficiently dissolve binaries of such separations. Due to the wide binary restriction, our simulations describe only about 20% of the entire binary population parameter space. The companion's orbital eccentricity, e_c, is picked from a uniform distribution between 0 and 1, as long as the orbits still fulfill the long-term stability criteria. Overall we performed 3000 Neptune-runs and 1500 Kuiper-runs, about 15% of which started with an initial companion star more massive than the initial primary star and which served as test systems. For these test systems no WD pollution is expected, as explained in Subsection <ref>.We also perform a separate Monte Carlo simulation for the example system of WD 1425+540, which shows signs of volatile pollution suspected to stem from an accreting Kuiper belt analog object <cit.>. This WD has a known K-dwarf companion star <cit.>, and we use the simulation to determine the likelihood that our mechanism is producing the observed pollution. The WD progenitor mass is estimated to be around 2 M_⊙ <cit.>, and the K-dwarf companion has a B-V color of about 1.29 <cit.>, from which we estimate its mass to be approximately 0.75 M_⊙. A 2 M_⊙ star loses about 2/3 of its mass before becoming a WD <cit.>, meaning that the total system lost about half its mass. Given that the current visual separation of the binary is 40 arcseconds <cit.>, which corresponds to ∼2240 AU, the separation before mass loss would have been ∼1120 AU. We use this value as the minimum possible apoapsis value for the initial binary orbital parameters, which restricts the minimum possible SMA and eccentricity value for the stellar companion. The Kuiper belt analog's SMA is uniformly chosen between 120 and 300 AU.The lower limit of 120 AU is the roughly estimated closest distance for which a Kuiper belt analog object can retain a large amount of volatiles around a 2 M_⊙ star <cit.>. The objects' initial eccentricities are uniformly chosen between 0 and 0.15. The inclination between the Kuiper belt analog's orbit and the stellar companion's orbit is chosen isotropically. After mass loss has occurred, the Kuiper belt analog objects' SMA values increase by a factor of ∼3, while the SMA value of the companion star only increases by a factor of ∼2, which leads to the increasing value of ϵ. We perform 1500 runs of possible system configurations. §.§ Numerical Methods and Triggering EKLWe solve the hierarchical three-body Hamiltonian up to the octupole order of approximation and average over the orbits to obtain the hierarchical secular dynamical evolution equations, also called the Eccentric Kozai-Lidov (EKL) mechanism <cit.>. In this framework the three-body system consists of an inner binary formed by the host star and a potential polluter (PP), with an initial SMA of a_pp, and which is orbitied by the stellar binary companion on a much wider orbit with SMA a_c, forming an outer binary. We consider as a proof of concept two representative examples, which vary in the mass of the PPs; one is Neptune-size planets and the other is Kuiper belt analogs. We include equilibrium tidal models for the inner binary following <cit.> and <cit.>, see <cit.> for complete equations. We also implement general relativity precessions for the inner and outer binary <cit.>. Finally, we include radial expansion, contraction, structure changes, and mass loss due to stellar evolution for the two stars following the stellar evolution code SSE <cit.>[We have tested the inclusion of post-main sequence evolution to the secular code in the past and showed that it played an important role in three-body dynamical evolution <cit.>], where we follow <cit.> and <cit.> for the magnetic braking coefficients. We adopt the nominal tidal coefficent parameters for our calculations. The tidal Love number of Kuiper belt analog objects is set to 5×10^-5, since they are icy solid objects <cit.>, for stars it is set to 0.014 and for gas giants to 0.25 <cit.>. The tidal viscous evolution timescale, t_V, for stars, Neptune-like planets, and Kuiper belt analog objects is set to 1.5 years. However, we tested different t_V values over a range of several orders of magnitudes and found that they have no measurable influence on our results. In particular, once a star evolves to become a WD, its small radius suppresses tidal effects on the WD, unless the orbiting planet reaches extremely close separations, at which point the planet itself will already be tidally disrupted. The inclusion of the octupole level of approximation leads to qualitatively different behaviors from the quadrupole level, including extreme eccentricity spikes and inclination flips from prograde to retrograde, and a generally more chaotic evolution (seefor review).During the evolution of the system, the companion induces eccentricity and inclination oscillations on the orbit of a PP.In many cases the PP's eccentricity increases until it is accreted onto the evolving primary star during or before stellar expansion. However, in most cases the eccentricity excitations are not large enough to plunge the PPonto the star before the WD phase, since the octupole effects are not strong enough. The strength of the octupole oscillations are estimated by the pre-factor of the octupole level of the Hamiltonian, ϵ, which is given by Equation (<ref>),see <cit.> for a detailed explanation. The onset of this behavior can be estimated in the ϵ-inclination phase space <cit.>. This parameter space is depicted in Figure <ref>. where the solid black line marks the predicted onset of octupole induced inclination flips <cit.>. Adiabatic (slow and uniform) mass loss in gravitationally bound two body systems of total mass m leads to the expansion of the SMA a according to a_f=m/m_fa,where f subscripts denote post mass loss values. In hierarchical three body systems, the mass loss in one of the inner binary members will lead to SMA expansion for both the inner binary and the outer companion. However, the total masses to consider for each SMA are different. In our case, the PP's SMA changes toa_pp,f=m_⋆+m_pp/m_⋆,f+m_ppa_pp∼m_⋆/m_⋆,fa_pp,while the companion's star SMA changes toa_c,f=m_⋆+m_pp+m_c/m_⋆,f+m_pp+m_ca_c∼m_⋆+m_c/m_⋆,f+m_ca_c. To simplify the description of the mechanism, we present a case for which the companion does not lose mass. However, we note that, throughout the rest of the calculation, we do account for any mass lost by the companion.With these expressions we calculate the final value of ϵ, which isϵ_f=m_⋆/m_⋆,fm_⋆,f+m_c/m_⋆+m_cϵ.As m_⋆,f is always going to be smaller than m_⋆ and m_c is larger than zero, the value of m_⋆/m_⋆,f is always going to increase faster than the value of (m_⋆,f+m_c)/(m_⋆+m_c) will decrease, due to which ϵ_f will always be larger than ϵ. This can move systems that were in the quadrupole dominated regime before mass loss occurred into the part of the parameter space that is octupole dominated, as shown in Figure <ref>. The red-shaded area in the left panel shows the estimated area of the ϵ-inclination parameter space that shifts from quadrupole to octupole dominated behavior for an example case (see figure caption for details). The possibility of increasing the strength of the octupole behavior through mass loss during stellar evolution has been shown before in the context of triple stars <cit.>, as well as WD pollution by non-volatile material of objects smaller than Mars <cit.>. If, however, the companion star is more massive than the host star and evolves first beyond the main sequence phase, the effect will be opposite. ϵ will become smaller and the strength of octupole level perturbations decreases. Likewise, once a companion star to a WD becomes a WD itself, the mass loss might move the system from the octupole-dominated regime back to the quadrupole-dominated one. In Figures <ref> and <ref>, we show two example evolutions where the ϵ values increase by a factor of 2, leading to large eccentricity excitations during the WD phase (see the far right side of the lower parts of the plots). Note that periapsis distance and inclination oscillations (both in black) are fairly regular during the main sequence phase of the host stars (see blue- and red-shaded parts of the plots). The periapsis distance also does not reach extreme values, and the PPs never cross the stellar Roche limit (in red) or the stellar surface (in purple) to be destroyed or accreted. However, is shown in the parts of the plots with white background, this behavior changes after the stars have lost most of their mass during post main sequence evolution, as marked by the expanding semi-major axes of the PPs and companion stars (in blue and green, respectively, with the companion stars' periapsis distances in cyan). The eccentricities can now reach extreme peak values, at which the periapsis distances can become small enough such that the PPs cross their Roche limits (in grey, see red circles) and disintegrate around the WD, forming rings of material, which can be accreted <cit.>. The stellar Roche limit is R_Roche,⋆ = 1.66 × r_⋆(m_⋆+m_PP/m_⋆)^1/3,while the PP's Roche limit is R_Roche,PP = 1.66 × r_PP(m_⋆+m_PP/m_PP)^1/3.Here, r_⋆ and r_PP are the stellar and PP radius, respectively. When a PP crosses either Roche limit we halt the simulation. We assume that the PP is lost if it crosses the stellar Roche limit during the main sequence and red giant phases. If it crosses its own Roche limit during the WD phase, we assume that the PP will be accreted onto the WD.§ EKL INDUCED WD POLLUTIONFrom the Kuiper and Neptune set of Monte Carlo runs we predict the orbital properties of binary stars that can lead to WD pollution through planets or Kuiper belt analog objects. Stellar companions that facilitate this mechanism are likely to have a semi-major axis on the order of a few thousand AU and are fairly eccentric (>0.2), as depicted in Figure <ref>, top and middle left panels. Furthermore, the mass of the companion is most likely slightly less than a solar mass (Figure <ref> bottom left). Such lower mass stars have a long main sequence lifetime, which is beneficial for the EKL mechanism. Once the companion stars evolve past the main sequence phase, they lose mass, increasing the semi-major axis of the outer binary, a_c. As can be seen from Equation (<ref>), ϵ will decrease, and further EKL evolution suppressed. We also find that accretion can take place over a large range of WD cooling ages with a higher accretion likelihood during the first few 100 Myrs to first few Gyrs (Figure <ref> bottom right). The distribution of the initial masses of host stars that lead to WD pollution was not found to differ substantially from the initial Salpeter distribution and was therefore omitted in Figure <ref>.We note that nearly all Neptune-like planets that were later accreted onto the WD reached periapsis distances, during their host stars' AGB phase, within a few AU of the expanding host star, as seen in the example in Figure <ref>. During this phase the planet can potentially lose part of its gaseous envelope due to the strong radiation from the evolving star, which heats up the planet. This may inflate the planet's atmosphere and remove gas until only the planet's rocky core and a diminished atmosphere remain <cit.>. The body that will accrete onto the WD will therefore consist mostly of this core, not of the gaseous envelope. Furthermore, if these planets were able to retain most of their envelopes during this phase, once they plunge inwards of their Roche limit around the WD the remaining gaseous envelope will be the first part to be tidally stripped off of the planets.In contrast to Neptune-like planets, because of their wider initial SMA, Kuiper belt analog objects that accrete during the WD phase do not typically reach periapsis distances closer than a few dozen AU during the main sequence and red giant phase, as seen in the example in Figure <ref>. It is therefore highly likely that these objects will retain most or at least a significant part of their volatile material. In some cases the closest approach distance, combined with extreme radiation during the red giant phase, might trigger a comet-like behavior of the Kuiper belt analog object. For a more detailed discussion of the thermal evolution of these objects, see <cit.> and <cit.>. We estimate the likelihoods of accretion using binary configurations that are otherwise consistent with field binaries <cit.>, disregarding the test systems with more massive companion than host stars, for which no WD pollution is expected and for which none occurred(see Section <ref>). We find that 5% of our Neptune-runs result in accretion of the planet onto the WD, as well as 6.5% of our Kuiper-runs. However, our Kuiper-runs do not actually reflect the probability of WDs accreting volatile material, as they do not account for different Kuiper belt analog orbital configurations per system. A given Kuiper belt analog can be expected to contain thousands of objects with different orbital configurations, such as the solar system's Kuiper belt. For any given Kuiper belt analog in a wide binary system, at least some objects of the belt can be expected to accrete onto the WD, as long as the initial inclination of the belt to the stellar companion lies in the favorable regime (inclinations between ∼40^∘ and ∼140^∘, see also Figure <ref>). Thus, we conducted a proof-of-concept simulation for a single example system that is in the favorable part of the parameter space[We chose a fixed inclination of 70^∘, a WD initial progenitor mass of 1.5 M_⊙, a companion star mass of 0.75 M_⊙, a companion orbit SMA of 2000 AU and orbital eccentricity of 0.5, and SMAs and eccentricities for the Kuiper belt analog objects from uniform distributions between 40 and 100 AU and 0 and 0.15, respectively. 1000 objects were tested, with about equal contributions to object survival, destruction during main sequence or red giant phases, and WD accretion.] and found that up to a third of the objects of a given Kuiper belt analog can over time accrete onto the WD. We estimate that about 75 % of wide binary systems systems lie in this favorable initial inclination regime, based on an inclination distribution initially uniform in cosine. Our simulations represent about 20% of the stellar binary population, since we are restricted to wide binaries (Figure <ref>), and we adopt a 50% binary[37]l[+]12.5cm < g r a p h i c s > Orbital parameter space for polluted WDs. We show the orbital parameter likelihood distributions for systems associated with WDs polluted by the Eccentric Kozai-Lidov mechanism, after the host star has gone through mass loss and become a WD. Shown are the parameter distributions for inducing pollution by Neptune-like planets (in blue) and Kuiper belt analog objects (KBAOs, in red). Shown are (left to right, top to bottom): SMA of the companion star and of the PP; eccentricity of the companion star and mutual inclination between inner and outer orbits' angular momenta; mass of the companion star and WD cooling age at time of accretion. The histograms are normalized such that the integral of each one is unity. The black arrows mark the known and estimated parameters for the WD 1425+540 system (see Figure <ref> for tighter estimation of the parameter space of this system). We note that the position of our Neptune polluters are consistent with the HR 8799 <cit.> planetary system. fraction, consistent with observations for main sequence stars <cit.>. <cit.> show that the binary fraction of WDs with main sequence stellar companions is lower than 50%; they suggest that this is mostly due to stellar mergers of tight binaries, which should not influence our results. Based on these assumptions, our simulations are applicable to about 10 % of the entire WD population. Assuming that an average star system starts out with a Neptune-like planet (the solar system, for example, has two, Uranus and Neptune), we estimate from our results that about 1% of all WDs should accrete such Neptune-like planets. Assuming that an average star system possesses a Kuiper belt analog, our results indicate that up to ∼7.5 % of all WDs should accrete Kuiper belt analog objects. This may be consistent with observations of volatile and planetary material pollution.The total number of confirmed polluted WDs is on the order of ∼200, while the overall pollution rate for all WDs is about 25 to 50% <cit.>[Note that there are ∼1000 polluted WD candidates known from SDSS data <cit.>.]. However, there have been detailed abundance measurements for only about 15 polluted WDs so far <cit.>, two of which show signs of planetary pollution in terms of composition and amount (see Section <ref> for planetary pollution) <cit.>, and one with signs of volatile material pollution in the form of nitrogen <cit.>. While these are so far very small number statistics, these current results could imply that the planetary and volatile pollution rates are relatively high, on the order of ∼10 %, roughly consistent with our findings that ∼ 7.5 % of all WDs could be polluted by Kuiper belt analog objects, while about 1 % could be polluted by Neptune-like planets. More detailed abundance measurements are needed for a larger number of WDs to determine the exact occurance rates of volatile and planetary pollution. We note here that our pollution mechanism generally operates on any objects that are large enough to avoid major orbital changes by non-gravitational forces <cit.> and are smaller than a significant fraction of the host star's mass (up to a few Jupiter masses). If any such objects commonly exist on long-period orbits in wide binary systems, they would also accrete onto WDs with a chance of ∼1 %, and could potentially be sources of planetary material pollution. Given the large uncertainty in observed planetary and volatile pollution rates, our results are consistent with the observations.Our additional Monte Carlo simulation for the example system of WD 1425+540 leads to accretion of icy material onto the WD for about 12.5% of tested configurations. From our results we estimate that the current periapsis distance of the K-dwarf companion should be about 3500 AU, and that the mutual inclination between the Kuiper belt analog and companion star can be anywhere between ∼30-150^∘ (see also Figure <ref>). Within these loose parameter constraints our mechanism can efficiently deliver Kuiper belt analog objects onto the WD. WD 1425+540 has an age estimate of a few 100 Myrs <cit.>, consistent with our estimated pollution likelihood over time (Figure <ref>, lower right panel).§ DISCUSSION AND CONCLUSIONS We have shown that white dwarf (WD) pollution by icy and volatile material from Neptune-like planets and Kuiper belt analog objects can naturally be explained through the Eccetric Kozai-Lidov (EKL) evolution in wide binary systems. The mass loss during post main sequence stellar evolution enhances the strength of EKL eccentricity excitations, which can lead to the accretion of potential polluters (PP) onto WDs (see Figures <ref> and <ref> as examples).Systems that are more likely to facilitate this type of pollution have had an increase in ϵ value (where ϵ describes the strength of the octupole level perturbations, see Equation (<ref>)) due to stellar mass loss during the Asymptotic Giant Branch phase. These systems occupy a specific part in the ϵ-inclination parameter space (where inclination is defined as the angle between the inner and outer orbits' angular momenta), as shown in Figure <ref>.We note that we find some PPs that were initially in the octupole favored regime were able to avoid accretion onto the WD progenitor during the main sequence and red giant phases. Accretion only took place in the WD phase for these PPs, after mass loss had occured (Figure <ref>).<cit.> studied WD pollution in binary systems and showed that pollution should preferentely occur if the binary companion was on a wide pre-mass loss orbit of at least about 1000 AU, which indicates that binary separations closer than that either suppress planetary system formation or prohibit long-term stability and survival into the WD phase. This is consistent with our results, where we find that the bulk of polluted WDs had initial binary separations of about 1000 AU, as shown in Figure <ref>.This is related to the previously discussed stability criterion, which constrained our initial system architectures to be long-term stable. However, this criterion still allows for systems that are in the octupole favored regime (as depicted in Figures <ref> and <ref>). In these systems, the eccentricity excitations will drive the PPs to be accreted during the main sequence or red giant phases, before the star can become a WD. As mentioned before, only systems with mildeccentricity excitations before mass loss can occur are able to result in WD pollution. These systems correspond to the a_c distribution shown in Figure <ref>, upper left panel.We note here also that we ignored the effects of galactic tides on the orbital parameters of the binaries. It has been shown that galactic tides can excite eccentricities of wide stellar binaries such that their pericenter distance can get sufficiently close to scatter planetesimals onto a WD <cit.>. However, this galactic tides mechanism acts on extremely long timescales, on the order of a few to ten Gyrs, while the EKL mechanism is most efficient for the first one or two Gyrs. In our case galactic tides would change the companion stars' eccentricity, e_c, which would change ϵ (see Equation (<ref>)), and thus the strength of the octupole level of approximation. This could lead to both suppression and enhancement of EKL oscillations over time. Tides might thereby increase the efficiency of our pollution mechanism, as binaries that are outside of the appropriate inclination-ϵ parameter space could be moved inside of it. Furthermore, comparing our mechanism directly to the one in <cit.>, it appears that galactic tides will be most relevant at polluting WDs in very wide binaries (a_c ≳ 3000 AU), while our mechanism is also effective at modestly wide separations (a_c ≳ 500 AU), for which galactic tides are weak and slow. Given that the number of binary systems drops rapidly with separation <cit.>, our mechanism appears to be sufficient to describe the general WD population.The implications of our results for multi-planet systems broadly fall into two categories. In general, if a multi-planet system is packed tightly enough with massive enough planets, we expect EKL and large eccentricity excitations to be mostly suppressed by the planets' mutual gravitational interactions <cit.>. However, if the system is not tightly packed or if the objects are not very massive, such as, for example, in a debris disk or a Kuiper belt analog, EKL excitations should still occur for each object in the disk or belt. In such systems the eccentricity and inclination changes can lead to the crossing of orbits, and potentially planet collisions or strong scattering. WD pollution should still occur for those systems. As we have discussed in Section <ref>, the presence of multiple objects exhibiting a range of orbital parameters indead increases the chance of WD accretion for a given system in the favorable regime. WD 1425+540, which exhibits volatile material pollution signatures <cit.>, has a low-mass stellar companion on a wide orbit <cit.>, consistent with our proposed pollution model. We are able to make additional predictions for its orbital parameters, as shown in Figure <ref>.We predict that WDs polluted by planets or volatile material are more likely to have a binary companion with orbital parameters consistent with the distributions in Figure <ref>, in particular large companion separations and low companion masses. Future observations of these systems and their pollution signatures can be used to gain insights into the outer planetary and Kuiper belt analog architectures of wide stellar binary systems. By using upcoming GAIA data releases it should be possible to find more WDs with wide binary companions due to detailed proper motion measurements. These WDs will be excellent observational targets to find more volatile pollution signatures and to investigate the long-period planetary architectures of wide binary systems. The authors would like to dedicate this paper to the late UCLA Professor Michael Jura for his numerous contributions to polluted white dwarf science and for inspiration. We thank the anonymous referee for fast responses and helpful comments. We also thank Siyi Xu, Dimitri Veras, and Boris Gänsike for useful discussions. SN acknowledges partial support from a Sloan Foundation Fellowship. BZ acknowledges support from a NASA grant to UCLA. Calculations for this project were performed on the UCLA cluster Hoffman2. apj | http://arxiv.org/abs/1704.08701v3 | {
"authors": [
"Alexander P. Stephan",
"Smadar Naoz",
"B. Zuckerman"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170427180014",
"title": "Throwing Icebergs at White Dwarfs"
} |
SIT: A Lightweight Encryption Algorithm for Secure Internet of Things Muhammad Usman Faculty of Engineering Science and Technology,Iqra University, Defence View,Shaheed-e-Millat Road (Ext.), Karachi 75500, Pakistan.Email: [email protected] Irfan Ahmed and M. Imran Aslam Department of Electronic Engineering,NED University of Engineering and Technology,University Road, Karachi 75270, Pakistan.Email: [email protected], [email protected] Shujaat Khan Faculty of Engineering Science and Technology,Iqra University, Defence View,Shaheed-e-Millat Road (Ext.), Karachi 75500, Pakistan.Email: [email protected] S.M Usman Ali Department of Electronic Engineering,NED University of Engineering and Technology,University Road, Karachi 75270, Pakistan.Email: [email protected] December 30, 2023 =============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== The Internet of Things (IoT) being a promising technology of the future is expected to connect billions of devices.The increased number of communication is expected to generate mountains of data and the security of data can be a threat.The devices in the architecture are essentially smaller in size and low powered.Conventional encryption algorithms are generally computationally expensive due to their complexity and requires many rounds to encrypt, essentially wasting the constrained energy of the gadgets.Less complex algorithm, however, may compromise the desired integrity.In this paper we propose a lightweight encryption algorithm named as Secure IoT (SIT).It is a 64-bit block cipher and requires 64-bit key to encrypt the data.The architecture of the algorithm is a mixture of feistel and a uniform substitution-permutation network.Simulations result shows the algorithm provides substantial security in just five encryption rounds.The hardware implementation of the algorithm is done on a low cost 8-bit micro-controller and the results of code size, memory utilization and encryption/decryption execution cycles are compared with benchmark encryption algorithms. The MATLAB code for relevant simulations is available online at https://goo.gl/Uw7E0W.IoT; Security; Encryption; Wireless Sensor Network WSN; Khazad § INTRODUCTION The Internet of Things (IoT) is turning out to be an emerging discussion in the field of research and practical implementation in the recent years.IoT is a model that includes ordinary entities with the capability to sense and communicate with fellow devices using Internet <cit.>.As the broadband Internet is now generally accessible and its cost of connectivity is also reduced, more gadgets and sensors are getting connected to it <cit.>.Such conditions are providing suitable ground for the growth of IoT.There is great deal of complexities around the IoT, since we wish to approach every object from anywhere in the world <cit.>.The sophisticated chips and sensors are embedded in the physical things that surround us, each transmitting valuable data.The process of sharing such large amount of data begins with the devices themselves which must securely communicate with the IoT platform.This platform integrates the data from many devices and apply analytics to share the most valuable data with the applications.The IoT is taking the conventional internet, sensor network and mobile network to another level as every ‘thing’ will be connected to the internet.A matter of concern that must be kept under consideration is to ensure the issues related to confidentiality, data integrity and authenticity that will emerge on account of security and privacy <cit.>.§.§ Applications of IoT: With the passage of time, more and more devices are getting connected to the Internet.The houses are soon to be equipped with smart locks <cit.>, the personal computer, laptops, tablets, smart phones, smart TVs, video game consoles even the refrigerators and air conditioners have the capability to communicate over Internet.This trend is extending outwards andit is estimated that by the year 2020 there will be over 50 billion objects connected to the Internet <cit.>.This estimates that for each person on earth there will be 6.6 objects online.The earth will be blanketed with millions of sensors gathering information from physical objects and will upload it to the Internet. It is suggestedthat application of IoT is yet in the early stage but is beginning to evolve rapidly <cit.>.An overview of IoT in building automation system is given in <cit.>.It is suggested in <cit.> that various industries have a growing interest towards use of IoT. Various applications of IoT in health care industries are discussed in<cit.> and the improvement opportunities in health care brought in by IoT will be enormous <cit.>.It has been predicted that IoT will contribute in the making the mining production safer <cit.> and the forecasting of disaster will be made possible.It is expected that IoT will transform the automobile services and transportation systems <cit.>.As more physical objects will be equipped with sensors and RFID tags transportation companies will be able to track and monitor the object movement from origin to destination <cit.>, thus IoT shows promising behavior in the logistics industry as well.With so many applications eying to adapt the technology with the intentions to contribute in the growth of economy, health care facility, transportation and a better life style for the public, IoT must offer adequate security to their data to encourage the adaptation process.§.§ Security Challenges in IoT: To adopt the IoT technology it is necessary to build the confidence among the users about its security and privacy that it will not cause any serious threat to their data integrity, confidentiality and authority.Intrinsically IoT is vulnerable to various types of security threats,if necessary security measures are not taken there will be a threat of information leakage or could prove a damage to economy<cit.>.Such threats may be considered as one of the major hindrance in IoT <cit.>.IoT is extremely open to attacks <cit.>, for the reasons that there is a fair chance of physical attack on its components as they remain unsupervised for long time.Secondly, due to the wireless communication medium, the eavesdropping is extremely simple.Lastly the constituents of IoT bear low competency in terms of energy with which they are operated and also in terms of computational capability.The implementation of conventional computationally expensive security algorithms will result in the hindrance on the performance of the energy constrained devices.It is predicted that substantial amount of data is expected to be generated while IoT is used for monitoring purposes and it is vital to preserve unification of data <cit.>.Precisely, data integrity and authentication are the matters of concern. From a high level perspective, IoT is composed of three components namely, Hardware, Middleware and Presentation <cit.>.Hardware consists of sensors and actuators, the Middleware provides storage and computing tools and the presentation provides the interpretation tools accessible on different platforms.It is not feasible to process the data collected from billions of sensors, context-aware Middleware solutions are proposed to help a sensor decide the most important data for processing <cit.>.Inherently the architecture of IoT does not offer sufficient margin to accomplish the necessary actions involved in the process of authentication and data integrity.The devices in the IoT such as RFID are questionable to achieve the fundamental requirements of authentication process that includes constant communication with the servers and exchange messages with nodes. In secure systems the confidentiality of the data is maintained and it is made sure that during the process of message exchange the data retains its originality and no alteration is unseen by the system.The IoT is composed of many small devices such as RFIDs which remain unattended for extended times, it is easier for the adversary to access the data stored in the memory <cit.>.To provide the immunity against Sybil attacks in RFID tags, received signal strength indication (RSSI) based methodologies are used in <cit.>, <cit.>, <cit.> and <cit.>.Many solutions have been proposed for the wireless sensor networks which consider the sensor as a part of Internet connected via nodes <cit.>.However, in IoT the sensor nodes themselves are considered as the Internet nodes making the authentication process even more significant.The integrity of the data also becomes vital and requires special attention towards retaining its reliability. §.§ Motivation And Organization of PaperRecently a study by HP reveals that 70% of the devices in IoT are vulnerable to attacks <cit.>.An attack can be performed by sensing the communication between two nodes which is known as a man-in-the-middle attack. No reliable solution has been proposed to cater such attacks.Encryption however could lead to minimize the amount of damage done to the data integrity.To assure data unification while it is stored on the middle ware and also during the transmission it is necessary to have a security mechanism.Various cryptographic algorithms have been developed that addresses the said matter, but their utilization in IoT is questionable as the hardware we deal in the IoT are not suitable for the implementation of computationally expensive encryption algorithms.A trade-off must be done to fulfil the requirement of security with low computational cost.In this paper, we proposed a lightweight cryptographic algorithm for IoT named as Secure IoT (SIT).The proposed algorithm is designed for IoT to deal with the security and resource utilization challenges mentioned in section <ref>.The rest of the paper is organized as follows, in section <ref>, a short literature review is provided for the past and contemporary lightweight cryptographic algorithms, in section <ref>, the detail architecture and functioning of the proposed algorithm is presented.Evaluation of SIT and experimental setup is discussed in section <ref>.Conclusion of the paper is presented in section <ref>.§ CRYPTOGRAPHIC ALGORITHMS FOR IOT: The need for the lightweight cryptography have been widely discussed <cit.>, also the shortcomings of the IoT in terms of constrained devices are highlighted.There in fact exist some lightweight cryptography algorithms that does not always exploit security-efficiency trade-offs.Amongst the block cipher, stream cipher and hash functions, the block ciphers have shown considerably better performances.A new block cipher named mCrypton is proposed <cit.>. The cipher comes with the options of 64 bits, 96 bits and 128 bits key size.The architecture of this algorithm is followed by Crypton <cit.> however functions of each component is simplified to enhance its performance for the constrained hardware.In <cit.> the successor of Hummingbird-1 <cit.> is proposed as Hummingbird-2(HB-2).With 128 bits of key and a 64 bit initialization vector Hummingbird-2 is tested to stay unaffected by all of the previously known attacks.However the cryptanalysis of HB-2 <cit.> highlights the weaknesses of the algorithm and that the initial key can be recovered. <cit.> studied different legacy encryption algorithms including RC4, IDEA and RC5 and measured their energy consumption.They computed the computational cost of the RC4 <cit.>, IDEA <cit.> and RC5 ciphers on different platforms.However, various existing algorithms were omitted during the study.TEA <cit.>, Skipjack <cit.> and RC5 algorithms have been implemented on Mica2 hardware platform <cit.>.To measure the energy consumption and memory utilization of the ciphers Mica2 was configured in single mote. Several block ciphers including AES <cit.>, XXTEA <cit.>, Skipjack and RC5 have been implemented <cit.>, the energy consumption and execution time is measured.The results show that in the AES algorithm the size of the key has great impact on the phases of encryption, decryption and key setup i-e the longer key size results in extended execution process.RC5 offers diversified parameters i-e size of the key, number of rounds and word size can be altered.Authors have performed variety of combinations to find out that it took longer time to execute if the word size is increased.Since key setup phase is not involved in XXTEA and Skipjack, they drew less energy but their security strength is not as much as AES and RC5.<cit.> proposed lightweight block cipher Simon and Speck to show optimal results in hardware and software respectively.Both ciphers offer a range of key size and width, but atleast 22 numbers of round require to perform sufficient encryption.Although the Simon is based on low multiplication complexity but the total number of required mathematical operation is quite high <cit.>§ PROPOSED ALGORITHM The architecture of the proposed algorithm provides a simple structure suitable for implementing in IoT environment.Some well known block cipher including AES (Rijndael) <cit.>, 3-Way <cit.>, Grasshopper <cit.>, PRESENT <cit.>, SAFER <cit.>, SHARK <cit.>, and Square <cit.> use Substitution-Permutation (SP) network.Several alternating rounds of substitution and transposition satisfies the Shannon's confusion and diffusion properties that ensues that the cipher text is changed in a pseudo random manner.Other popular ciphers including SF <cit.>, Blowfish <cit.>, Camelia <cit.> and DES <cit.>, use the feistel architecture.One of the major advantage of using feistel architecture is that the encryption and decryption operations are almost same.The proposed algorithm is a hybrid approach based on feistel and SP networks.Thus making use of the properties of both approaches to develop a lightweight algorithm that presents substantial security in IoT environment while keeping the computational complexity at moderate level.SIT is a symmetric key block cipher that constitutes of 64-bit key and plain-text.In symmetric key algorithm the encryption process consists of encryption rounds, each round is based on some mathematical functions to create confusion and diffusion.Increase in number of rounds ensures better security but eventually results in increase in the consumption of constrained energy <cit.>.The cryptographic algorithms are usually designed to take on an average 10 to 20 rounds to keep the encryption process strong enough that suits the requirement of the system.However the proposed algorithm is restricted to just five rounds only, to further improve the energy efficiency, each encryption round includes mathematical operations that operate on 4 bits of data.To create sufficient confusion and diffusion of data in order to confront the attacks, the algorithm utilizes the feistel network of substitution diffusion functions.The details of SIT design is discussed in section <ref> and <ref>.Another vital process in symmetric key algorithms is the generation of key.The key generation process involves complex mathematical operations.In WSN environment these operations can be performed wholly on decoder<cit.>,<cit.>, on the contrary in IoT the node themselves happens to serve as the Internet node, therefore, computations involved in the process of key generation must also be reduced to the extent that it ensures necessary security.In the sub-sections the process of key expansion and encryption are discussed in detail.Some notations used in the explanation are shown in Table <ref>§.§ Key Expansion The most fundamental component in the processes of encryption and decryption is the key.It is this key on which entire security of the data is dependent, should this key be known to an attacker, the secrecy of the data is lost.Therefore necessary measures must be taken into account to make the revelation of the key as difficult as possible.The feistel based encryption algorithms are composed of several rounds, each round requiring a separate key.The encryption/decryption of the proposed algorithm is composed of five rounds, therefore, we require five unique keys for the said purpose.To do so, we introduce a key expansion block which is described in this section.To maintain the security against exhaustive search attack the length of the true key k_t must be large so that it becomes beyond the capability of the enemy to perform 2^k_t-1 encryptions for key searching attacks.The proposed algorithm is a 64-bit block cipher, which means it requires 64-bit key to encrypt 64-bits of data.A cipher key (Kc) of 64-bits is taken as an input from the user.This key shall serve as the input to the key expansion block.The block upon performing substantial operations to create confusion and diffusion in the input key will generate five unique keys.These keys shall be used in the encryption/decryption process and are strong enough to remain indistinct during attack.The architecture of the key expansion block is shown in Fig. <ref>.The block uses an f-function which is influenced by tweaked Khazad block cipher <cit.>.Khazad is not a feistel cipher and it follows wide trial strategy.The wide trial strategy is composed of several linear and non-linear transformations that ensures the dependency of output bits on input bits in a complex manner <cit.>. Detailed explanation of the components of key expansion are discussed below: * In the first step the 64-bit cipher key (Kc) is divided into the segments of 4-bits. * The f-function operates on 16-bits data. Therefore four f-function blocks are used. These 16-bits for each f-function are obtained after performing an initial substitution of segments of cipher key (Kc) as shown in equation (<ref>). Kb_if=∥_j=1^4 Kc_4(j-1)+iwhere i = 1 to 4 for first 4 round keys as shown in Fig. <ref>. * The next step is to get Ka_if by passing the 16-bits of Kb_if to the f-function as shown in equation (<ref>). Ka_if= f(Kb_if)* f-function is comprised of P and Q tables.These tables perform linear and non-linear transformations resulting in confusion and diffusion as illustrated in Fig. <ref>.* The transformations made by P and Qare shown in the tables <ref> and <ref>. * The output of each f-function is arranged in 4 × 4 matrix named Km shown below:Km_1=[Ka_1f_1Ka_1f_2Ka_1f_3Ka_1f_4;Ka_1f_5Ka_1f_6Ka_1f_7Ka_1f_8;Ka_1f_9 Ka_1f_10 Ka_1f_11 Ka_1f_12; Ka_1f_13 Ka_1f_14 Ka_1f_15 Ka_1f_16 ] Km_2= [Ka_2f_1Ka_2f_2Ka_2f_3Ka_2f_4;Ka_2f_5Ka_2f_6Ka_2f_7Ka_2f_8;Ka_2f_9 Ka_2f_10 Ka_2f_11 Ka_2f_12; Ka_2f_13 Ka_2f_14 Ka_2f_15 Ka_2f_16 ]Km_3= [Ka_3f_1Ka_3f_2Ka_3f_3Ka_3f_4;Ka_3f_5Ka_3f_6Ka_3f_7Ka_3f_8;Ka_3f_9 Ka_3f_10 Ka_3f_11 Ka_3f_12; Ka_3f_13 Ka_3f_14 Ka_3f_15 Ka_3f_16 ]Km_4= [Ka_4f_1Ka_4f_2Ka_4f_3Ka_4f_4;Ka_4f_5Ka_4f_6Ka_4f_7Ka_4f_8;Ka_4f_9 Ka_4f_10 Ka_4f_11 Ka_4f_12; Ka_4f_13 Ka_4f_14 Ka_4f_15 Ka_4f_16 ] * To obtain round keys, K1, K2, K3 and K4 the matrices are transformed into four arrays of 16 bits that we call round keys (Kr). The arrangement of these bits are shown in equations (<ref>), (<ref>), (<ref>) and (<ref>).K1 = a_4 a_3 a_2 a_1 a_5 a_6 a_7 a_8a_12 a_11 a_10 a_9 a_13 a_14 a_15 a_16K2 =b_1 b_5 b_9 b_13 b_14 b_10 b_6 b_2b_3 b_7 b_11 b_15 b_16 b_12 b_8 b_4K3 =c_1 c_2 c_3 c_4 c_8 c_7 c_6 c_5c_9 c_10 c_11 c_12 c_16 c_15 c_14 c_13 K4 =d_13 d_9 d_5 d_1 d_2 d_6 d_10 d_14d_15 d_11 d_7 d_3 d_4 d_8 d_12 d_16* An XOR operation is performed among the four round keys to obtain the fifth key as shown in equation (<ref>).K5 = ⊕_i=1^4 Ki§.§ EncryptionAfter the generation of round keys the encryption process can be started.For the purpose of creating confusion and diffusion this process is composed of some logical operations, left shifting, swapping and substitution.The process of encryption is illustrated in Fig. <ref>.For the first round an array of 64 bit plain text (Pt) is first furcated into four segments of 16 bits Px_0-15, Px_16-31, Px_32-47 and Px_48-63.As the bits progresses in each round the swapping operation is applied so as to diminish the data originality by altering the order of bits, essentially increasing confusion in cipher text.Bitwise XNOR operation is performed between the respective round key K_i obtained earlier from key expansion process and Px_0-15 and the same is applied between K_i and Px_48-63 resulting in Ro_11and Ro_14 respectively.The output of XNOR operation is then fed to the f-function generating the result Ef_l1 and Ef_r1 as shown in Fig. <ref>. The f-function used in encryption is the same as of key expansion, comprised of swapping and substitution operations the details of which are discussed earlier in section <ref>.Bitwise XOR function is applied between Ef_l1 & Px_32-47 to obtain Ro_12 and Ef_r1 & Px_16-31 to obtain Ro_13. Ro_i,j ={ Px_i,jK_i;j=1 & 4Px_i,j+1⊕ Ef_li ;j=2Px_i,j-1⊕ Ef_ri ;j=3.Finally a round transformation is made in such a way that for succeeding round Ro_11 will become Px_16-31, Ro_12 will become Px_0-15, Ro_13 will become Px_48-63 and Ro_13 will become Px_32-47 as shown in Fig. <ref>.Same steps are repeated for the remaining rounds using equation (<ref>).The results of final round are concatenated to obtain Cipher Text (Ct) as shown in equation (<ref>). Ct= R_51 R_52 R_53 R_54 § SECURITY ANALYSISThe purpose of a cipher is to provide protection to the plaintext.The attacker intercepts the ciphertext and tries to recover the plain text.A cipher is considered to be broken if the enemy is able to determine the secret key.If the attacker can frequently decrypt the ciphertext without determining the secret key, the cipher is said to be partially broken.We assume that the enemy has complete access of what is being transmitted through the channel.The attacker may have some additional information as well but to assess the security of a cipher, the computation capability of the attacker must also be considered.Since the proposed algorithm is a combination of feistel and uniform substitution -combination network, it benefits from existing security analysis.In the following a the existing security analysis of these two primitives are recalled and their relevancy with the proposed algorithm is discussed. §.§ Linear and Differential CryptanalysisThe f-function is inspired by <cit.> whose cryptanalysis shows that differential and linear attacks does not have the succeed for complete cipher.The input and output correlation is very large if the linear approximation is done for two rounds.Also the round transformation is kept uniform which treats every bit in a similar manner and provides opposition to differential attacks.§.§ Weak KeysThe ciphers in which the non-linear operations depend on the actual key value maps the block cipher with detectable weakness.Such case occurs in <cit.>.However proposed algorithm does not use the actual key in the cipher, instead the is first XORed and then fed to the f-function.In the f-function all the non-linearity is fixed and there is no limitation on the selection of key. §.§ Related KeysAn attack can be made by performing cipher operations using unknown or partially known keys.The related key attack mostly relies upon either slow diffusion or having symmetry in key expansion block.The key expansion process of proposed algorithm is designed for fast and non-linear diffusion of cipher key difference to that of round keys. §.§ Interpolation AttacksThese attacks are dependent upon the simple structures of the cipher components that may yield arational expression witha handy complexity.The expression of the S-box of the proposed algorithm along with the diffusion layer makes such type of attack impracticable. §.§ SQUARE AttackThis attack was presented by <cit.> to realize how efficiently the algorithm performs against it.The attack is able to recover one byte of the last key and the rest of keys can be recovered by repeating the attack eight times.However to be able to do so, the attack requires 2^8 key guesses by2^8 plaintexts which is equal to 2^16 S-box lookups. § EXPERIMENTAL SETUP §.§ Evaluation Parameters To test the security strength of the proposed algorithm, the algorithm is evaluated on the basis of the following criterion.Key sensitivity, effect of cipher on the entropy, histogram and correlation of the image.We further tested the algorithm for computational resource utilization and computational complexity.For this we observe the memory utilization and total computational time utilized by the algorithm for the key generation, encryption and decryption.§.§.§ Key Sensitivity An encryption algorithm must be sensitive to the key. It means that the algorithm must not retrieve the original data if the key has even a minute difference from the original key.Avalanche test is used to evaluate the amount of alterations occurred in the cipher text by changing one bit of the key or plain text.According to Strict Avalanche Criterion SAC <cit.> if 50% of the bits are changed due to one bit change, the test is considered to be perfect.To visually observe this effect, we decrypt the image with a key that has a difference of only one bit from the correct key.§.§.§ Execution Time One of the fundamental parameter for the evaluation of the algorithm is the amount of time it takes to encode and decode a particular data. The proposed algorithm is designed for the IoT environment must consume minimal time and offer considerable security.§.§.§ Memory Utilization Memory utilization is a major concern in resource constrain IoT devices. An encryption algorithm is composed of several computational rounds that may occupy significant memory making it unsuitable to be utilized in IoT.Therefore the proposed algorithm is evaluated in terms of its memory utilization.Smaller amount of memory engagement will be favourable for its deployment in IoT.§.§.§ Image Histogram A method to observe visual effect of the cipher is to encrypt an image with the proposed algorithm and observe the randomness it produces in the image.To evaluate the generated randomness, histogram of the image is calculated.A uniform histogram after encryption depicts appreciable security.§.§.§ Image Entropy The encryption algorithm adds extra information to the data so as to make it difficult for the intruder to differentiate between the original information and the one added by the algorithm.We measure the amount of information in terms of entropy, therefore it can be said that higher the entropy better is the performance of security algorithm.To measure the entropy (H) for an image, equation (<ref>) is applied on the intensity (I) values P(I_i) being the probability of intensity value I_i. H(I)=-∑_i=1^2^8P(I_i) log_b P(I_i)§.§.§ CorrelationThe correlation between two values is a statistical relationship that depicts the dependency of one value on another.Data points that hold substantial dependency has a significant correlation value.A good cipher is expected to remove the dependency of the cipher text from the original message.Therefore no information can be extracted from the cipher alone and no relationship can be drawn between the plain text and cipher text.This criterion is best explained by Shannon in his communication theory of secrecy systems <cit.>.In this experiment we calculated the correlation coefficient for original and encrypted images.The correlation coefficient γ is calculated using equation (<ref>).For ideal cipher case γ should be equal to 0 and for the worst case γ will be equal to 1. γ_x,y=cov(x,y)/√(D(x)√(D(y))), withD(x) where cov(x,y), D(x) and D(y) are covariance and variances of variable x and y respectively.The spread of values or variance of any single dimension random variable can be calculated using equation (<ref>).Where D(x) is the variance of variable x. D(x)=1/N∑_i=1^N(x_i-E(x))^2, For the covariance between two random variables the equation (<ref>) can be transformed into equation (<ref>).Where cov(x,y) is the covariance between two random variables x and y. cov(x,y)=1/N∑_i=1^N(x_i-E(x))(y_i-E(y)), In equation (<ref>) and (<ref>) E(x) and E(y) are the expected values of variable x and y.The expectation can be calculated using equation (<ref>).E(x)=1/N∑_i=1^Nx_i,where N is the total pixels of the image, N=row× col, x is a vector of length N and x_i is the ith intensity values of the original image. §.§ ResultsThe simulation of the algorithm is done to perform the standard tests including Avalanche and image entropy and histogram on Intel Core [email protected] GHz processor using MATLAB®.To evaluate the performance in the real IoT environment we implemented the algorithm on ATmega 328 based Ardinuo Uni board as well.The memory utilization and execution time of the proposed algorithm is observed.The execution time is found to be 0.188 milliseconds and 0.187 milliseconds for encryption and decryption respectively, the proposed algorithm utilizes the 22 bytes of memory on ATmega 328 platform.We compare our algorithm with other algorithms being implemented on hardware as shown in table <ref>.Block and key size is in bits while code and RAM is in bytes.The cycles include key expansions along with encryption and decryption. The Avalanche test of the algorithm shows that a single bit change in key or plain text brings around 49% change in the cipher bits, which is close to the ideal 50% change.The results in Fig. <ref> show that the accurate decryption is possible only if the correct key is used to decrypt image, else the image remains non recognizable.For a visual demonstration of avalanche test, the wrong key has a difference of just bit from the original key, the strength of the algorithm can be perceived from this result.To perform entropy and histogram tests we have chosen five popular 8-bits grey scale images.Further in the results of histogram in Fig. <ref> for the original and encrypted image, the uniform distribution of intensities after the encryption is an indication of desired security.An 8-bits grey scale image can achieve a maximum entropy of 8 bits.From the results in table <ref>, it can be seen that the entropy of all encrypted images is close to maximum, depicting an attribute of the algorithm. Finally the correlation comparison in Fig. <ref> illustrates the contrast between original and encrypted data. Original data, which in our case is an image can be seen to be highly correlated and detaining a high value for correlation coefficient.Whereas the encrypted image does not seem to have any correlation giving strength to our clause in section <ref>§ FUTURE WORKFor future research, the implementation of the algorithm on hardware and software in various computation and network environment is under consideration. Moreover, the algorithm can be optimized in order to enhance the performance according to different hardware platforms. Hardware like FPGA performs the parallel execution of the code, the implementation of the proposed algorithm on an FPGA is expected to provide high throughput. The scalability of algorithm can be exploited for better security and performance by changing the number of rounds or the architecture to support different key length. § CONCLUSION In the near future Internet of Things will be an essential element of our daily lives. Numerous energy constrained devices and sensors will continuously be communicating with each other the security of which must not be compromised. For this purpose a lightweight security algorithm is proposed in this paper named as SIT. The implementation show promising results making the algorithm a suitable candidate to be adopted in IoT applications.In the near future we are interested in the detail performance evaluation and cryptanalysis of this algorithm on different hardware and software platforms for possible attacks.IEEEtran | http://arxiv.org/abs/1704.08688v2 | {
"authors": [
"Muhammad Usman",
"Irfan Ahmed",
"M. Imran Aslam",
"Shujaat Khan",
"Usman Ali Shah"
],
"categories": [
"cs.CR",
"cs.IT",
"math.IT"
],
"primary_category": "cs.CR",
"published": "20170427175808",
"title": "SIT: A Lightweight Encryption Algorithm for Secure Internet of Things"
} |
SIT: A Lightweight Encryption Algorithm for Secure Internet of Things Muhammad Usman Faculty of Engineering Science and Technology,Iqra University, Defence View,Shaheed-e-Millat Road (Ext.), Karachi 75500, Pakistan.Email: [email protected] Irfan Ahmed and M. Imran Aslam Department of Electronic Engineering,NED University of Engineering and Technology,University Road, Karachi 75270, Pakistan.Email: [email protected], [email protected] Shujaat Khan Faculty of Engineering Science and Technology,Iqra University, Defence View,Shaheed-e-Millat Road (Ext.), Karachi 75500, Pakistan.Email: [email protected] S.M Usman Ali Department of Electronic Engineering,NED University of Engineering and Technology,University Road, Karachi 75270, Pakistan.Email: [email protected] December 30, 2023 =============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but little was known for non-convex problems. We propose a novel analytical framework and show that under certain dual conditions, the optimal solution of the matrix factorization program is the same as its bi-dual and thus the global optimality of the non-convex program can be achieved by solving its bi-dual which is convex. These dual conditions are satisfied by a wide class of matrix factorization problems, although matrix factorization problems are hard to solve in full generality. This analytical framework may be of independent interest to non-convex optimization more broadly.We apply our framework to two prototypical matrix factorization problems: matrix completion and robust Principal Component Analysis (PCA). These are examples of efficiently recovering a hidden matrix given limited reliable observations of it. Our framework shows that exact recoverability and strong duality hold with nearly-optimal sample complexity guarantees for matrix completion and robust PCA. empty § INTRODUCTIONNon-convex matrix factorization problems have been an emerging object of study in theoretical computer science <cit.>, optimization <cit.>, machine learning <cit.>, and many other domains. In theoretical computer science and optimization, the study of such models has led to significant advances in provable algorithms that converge to local minima in linear time <cit.>. In machine learning, matrix factorization serves as a building block for large-scale prediction and recommendation systems, e.g., the winning submission for the Netflix prize <cit.>. Two prototypical examples are matrix completion and robust Principal Component Analysis (PCA).This work develops a novel framework to analyze a class of non-convex matrix factorization problems with strong duality, which leads to exact recoverability for matrix completion and robust Principal Component Analysis (PCA) via the solution to a convex problem.The matrix factorization problems can be stated as finding a target matrix ^* in the form of ^*=, by minimizing the objective function H()+1/2_F^2 over factor matrices ∈^n_1× r and ∈^r× n_2 with a known value of r≪min{n_1,n_2}, where H(·) is some function that characterizes the desired properties of ^*.Our work is motivated by several promising areas where our analytical framework for non-convex matrix factorizations is applicable. The first area is low-rank matrix completion, where it has been shown that a low-rank matrix can be exactly recovered by finding a solution of the formthat is consistent with the observed entries (assuming that it is incoherent) <cit.>. This problem has received a tremendous amount of attention due to its important role in optimization and its wide applicability in many areas such as quantum information theory and collaborative filtering <cit.>. The second area is robust PCA, a fundamental problem of interest in data processing that aims at recovering both the low-rank and the sparse components exactly from their superposition <cit.>, where the low-rank component corresponds to the product ofandwhile the sparse component is captured by a proper choice of function H(·), e.g., the ℓ_1 norm <cit.>. We believe our analytical framework can be potentially applied to other non-convex problems more broadly, e.g., matrix sensing <cit.>, dictionary learning <cit.>, weighted low-rank approximation <cit.>, and deep linear neural network <cit.>, which may be of independent interest. Without assumptions on the structure of the objective function, direct formulations of matrix factorization problems are NP-hard to optimize in general <cit.>. With standard assumptions on the structure of the problem and with sufficiently many samples, these optimization problems can be solved efficiently, e.g., by convex relaxation <cit.>. Some other methods run local search algorithms given an initialization close enough to the global solution in the basin of attraction <cit.>. However, these methods have sample complexity significantly larger than the information-theoretic lower bound; see Table <ref> for a comparison. The problem becomes more challenging when the number of samples is small enough that the sample-based initialization is far from the desired solution, in which case the algorithm can run into a local minimum or a saddle point. Another line of work has focused on studying the loss surface of matrix factorization problems, providing positive results for approximately achieving global optimality. One nice property in this line of research is that there is no spurious local minima for specific applications such as matrix completion <cit.>, matrix sensing <cit.>, dictionary learning <cit.>, phase retrieval <cit.>, linear deep neural networks <cit.>, etc. However, these results are based on concrete forms of objective functions. Also, even when any local minimum is guaranteed to be globally optimal, in general it remains NP-hard to escape high-order saddle points <cit.>, and additional arguments are needed to show the achievement of a local minimum. Most importantly, all existing results rely on strong assumptions on the sample size. §.§ Our ResultsOur work studies the exact recoverability problem for a variety of non-convex matrix factorization problems. The goal is to provide a unified framework to analyze a large class of matrix factorization problems, and to achieve efficient algorithms. Our main results show that although matrix factorization problems are hard to optimize in general, under certain dual conditions the duality gap is zero, and thus the problem can be converted to an equivalent convex program. The main theorem of our framework is the following.Theorems <ref> (Strong Duality. Informal). Under certain dual conditions, strong duality holds for the non-convex optimization problem(,)=_∈^n_1× r,∈^r× n_2 F(,)=H()+1/2_F^2, H(·) is convex and closed,where “the function H(·) is closed” means that for each α∈, the sub-level set {∈^n_1× n_2:H()≤α} is a closed set. In other words, problem (<ref>) and its bi-dual problem=_∈^n_1× n_2 H()+_r*,have exactly the same optimal solutions in the sense that =, where _r* is a convex function defined by _r*=max_⟨,⟩-1/2_r^2 and _r^2=∑_i=1^rσ_i^2() is the sum of the first r largest squared singular values. Theorem <ref> connects the non-convex program (<ref>) to its convex counterpart via strong duality; see Figure <ref>. We mention that strong duality rarely happens in the non-convex optimization region: low-rank matrix approximation <cit.> and quadratic optimization with two quadratic constraints <cit.> are among the few paradigms that enjoy such a nice property.Given strong duality, the computational issues of the original problem can be overcome by solving the convex bi-dual problem (<ref>).The positive result of our framework is complemented by a lower bound to formalize the hardness of the above problem in general. Assuming that the random 4-SAT problem is hard (see Conjecture <ref>) <cit.>, we give a strong negative result for deterministic algorithms. If also BPP = P (see Section <ref> for a discussion), then the same conclusion holds for randomized algorithms succeeding with probability at least 2/3. Theorem <ref> (Hardness Statement. Informal). Assuming that random 4-SAT is hard on average, there is a problem in the form of (<ref>) such that any deterministic algorithm achieving (1+ϵ) in the objective function value with ϵ≤ϵ_0 requires 2^Ω(n_1+n_2) time, whereis the optimum and ϵ_0>0 is an absolute constant. If BPP = P, then the same conclusion holds for randomized algorithms succeeding with probability at least 2/3. Our framework only requires the dual conditions in Theorem <ref> to be verified. We will show that two prototypical problems, matrix completion and robust PCA, obey the conditions. They belong to the linear inverse problems of form (<ref>) with a proper choice of function H(·), which aim at exactly recovering a hidden matrix ^* with (^*)≤ r given a limited number of linear observations of it.For matrix completion, the linear measurements are of the form {_ij^*: (i,j) ∈Ω}, where Ω is the support set which is uniformly distributed among all subsets of [n_1] × [n_2] of cardinality m. With strong duality, we can either study the exact recoverability of the primal problem (<ref>), or investigate the validity of its convex dual (or bi-dual) problem (<ref>). Here we study the former with tools from geometric analysis. Recall that in the analysis of matrix completion, one typically requires an μ-incoherence condition for a given rank-r matrix ^* with skinny SVD Σ^T <cit.>:^T_i_2≤ √(μ r/n_1),^T_i_2≤√(μ r/n_2),for all iwhere _i's are vectors with i-th entry equal to 1 and other entries equal to 0. The incoherence condition claims that information spreads throughout the left and right singular vectors and is quite standard in the matrix completion literature. Under this standard condition, we have the following results.Theorems <ref>, <ref>, and <ref> (Matrix Completion. Informal). ^*∈^n_1× n_2 is the unique matrix of rank at most r that is consistent with the m measurements with minimum Frobenius norm by a high probability, provided that m=(κ^2μ(n_1+n_2)rlog (n_1+n_2)log_2κ(n_1+n_2)) and ^* satisfies incoherence (<ref>). In addition, there exists a convex optimization for matrix completion in the form of (<ref>) that exactly recovers ^* with high probability, provided that m=(κ^2μ(n_1+n_2)rlog (n_1+n_2)log_2κ(n_1+n_2)), where κ is the condition number of ^*.To the best of our knowledge, our result is the first to connect convex matrix completion to non-convex matrix completion, two parallel lines of research that have received significant attention in the past few years. Table <ref> compares our result with prior results. For robust PCA, instead of studying exact recoverability of problem (<ref>) as for matrix completion, we investigate problem (<ref>) directly. The robust PCA problem is to decompose a given matrix =^*+^* into the sum of a low-rank component ^* and a sparse component ^* <cit.>. We obtain the following theorem for robust PCA.Theorems <ref> (Robust PCA. Informal). There exists a convex optimization formulation for robust PCA in the form of problem (<ref>) that exactly recovers the incoherent matrix ^*∈^n_1× n_2 and ^*∈^n_1× n_2 with high probability, even if (^*)=Θ(min{n_1,n_2}/μlog^2 max{n_1,n_2}) and the size of the support of ^* is m=Θ(n_1n_2), where the support set of ^* is uniformly distributed among all sets of cardinality m, and the incoherence parameter μ satisfies constraints (<ref>) and ^*_∞≤√(μ r/n_1n_2)σ_r(^*). The bounds in Theorem <ref> match the best known results in the robust PCA literature when the supports of ^* are uniformly sampled <cit.>, while our assumption is arguably more intuitive; see Section <ref>. Note that our results hold even when ^* is close to full rank and a constant fraction of the entries have noise. Independently of our work, Ge et al. <cit.> developed a framework to analyze the loss surface of low-rank problems, and applied the framework to matrix completion and robust PCA. Their bounds are: for matrix completion, the sample complexity is (κ^6μ^4r^6(n_1+n_2)log (n_1+n_2)); for robust PCA, the outlier entries are deterministic and the number that the method can tolerate is (n_1n_2/μ rκ^5). Zhang et al. <cit.> also studied the robust PCA problem using non-convex optimization, where the outlier entries are deterministic and the number of outliers that their algorithm can tolerate is (n_1n_2/rκ).The strong duality approach is unique to our work.§.§ Our Techniques Reduction to Low-Rank Approximation. Our results are inspired by the low-rank approximation problem:min_∈^n_1× r,∈^r× n_21/2--_F^2.We know that all local solutions of (<ref>) are globally optimal (see Lemma <ref>) and that strong duality holds for any given matrix -∈^n_1× n_2 <cit.>. To extend this property to our more general problem (<ref>), our main insight is to reduce problem (<ref>) to the form of (<ref>) using the ℓ_2-regularization term. While some prior work attempted to apply a similar reduction, their conclusions either depended on unrealistic conditions on local solutions, e.g., all local solutions are rank-deficient <cit.>, or their conclusions relied on strong assumptions on the objective functions, e.g., that the objective functions are twice-differentiable <cit.>. Instead, our general results formulate strong duality via the existence of a dual certificate . For concrete applications, the existence of a dual certificate is then converted to mild assumptions, e.g., that the number of measurements is sufficiently large and the positions of measurements are randomly distributed. We will illustrate the importance of randomness below. More specifically, denote by (,) the optimal solution to (<ref>). Define ={Ł+: Ł∈^r× n_2,∈^n_1× r}, ^⊥ the complement of , and _ the orthogonal projection onto subspace . Let ∂ H()={∈^n_1× n_2: H()≥ H()+⟨,-⟩} be the sub-differential of function H evaluated at . To perform the reduction from problem (<ref>) to (<ref>), we study the Lagrangian L(,,) of (<ref>), which is equivalent to problem (<ref>) if we fix =. We show that, for a fixed Lagrangian multiplier ∈∂ H(), minimizing the primal problem (<ref>) reduces to minimizing the Lagrangian function L(,,), (i.e., problem (<ref>), thus strong duality holds), if (,) remains globally optimal to L(,,) (i.e., problem (<ref>)). This can be translated to: a) ∃∈∂ H(), b) (,) is a stationary point of the Lagrangian L(,,) so that c) =_r(-), where _r(-)=_:,1:rΣ_1:r,1:r_:,1:r^T if Σ^T is the SVD of -, and _:,1:r and Σ_1:r,1:r are the first r columns ofand top left r× r submatrix of Σ, respectively. We note that conditions b) and c) can be rephrased as _(-)= and σ_1(_^⊥)<σ_r(), respectively. To satisfy conditions a), b) and c) simultaneously, one may want to find a certificatesuch that among all matrices ∈∂ H() (i.e., condition a)) with _(-)= (i.e., condition b)),is the one with minimum Frobenius norm, so that condition c) is easier to satisfy. Following this principle, we build our dual certificateby -_∂ H_(__∂ H_)^-1(). It can be easily checked that conditions a) and b) hold for our construction. Thus the remainder is to prove condition c) for specific applications. We observe that ∂ H=Ω for matrix completion, where Ω is the linear space that characterizes the sample support, i.e., Ω={:_ij=0if(i,j)is unsampled}.[Without confusion, we denote by Ω both the linear subspace and the index set concerning the sampled positions.] This nice property serves as a bridge, connecting our analytical framework to the concrete application of matrix completion. We will then state how to prove condition c) by randomness as follows.The Blessing of Randomness. The desired dual certificatemay not exist in the deterministic world. A hardness result <cit.> shows that for the problem of weighted low-rank approximation, which can be cast in the form of (<ref>), without some randomization in the measurements made on the underlying low rank matrix, it is NP-hard to achieve a good objective value, not to mention to achieve strong duality. A similar phenomenon was observed for deterministic matrix completion <cit.>. Thus we should utilize such randomness to analyze the existence of a dual certificate. For matrix completion, the assumption that the measurements are random is standard, under which, the angle between the space Ω (the space of matrices which are consistent with observations) and the space(the space of matrices which are low-rank) is small with high probability, namely, ^* is almost the unique low-rank matrix that is consistent with the measurements. Thus, our dual certificate can be represented as another form of a convergent Neumann series concerning the projection operators on the spaces Ω and . The remainder of the proof is to show that such a construction obeys the dual conditions.To prove the dual conditions for matrix completion, we use the fact that the subspace Ω and the complement space ^⊥ are almost orthogonal when the sample size is sufficiently large. This implies the projection of our dual certificate on the space ^⊥ has a very small norm, which exactly matches the dual conditions.R5cm< g r a p h i c s >Feasibility. Non-Convex Geometric Analysis. Strong duality implies that the primal problem (<ref>) and its bi-dual problem (<ref>) have exactly the same solutions in the sense that =. Thus, to show exact recoverability of linear inverse problems such as matrix completion and robust PCA, it suffices to study either the non-convex primal problem (<ref>) or its convex counterpart (<ref>). Here we do the former analysis for matrix completion. We mention that traditional techniques <cit.> for convex optimization break down for our non-convex problem, since the subgradient of a non-convex objective function may not even exist <cit.>. Instead, we apply tools from geometric analysis <cit.> to analyze the geometry of problem (<ref>). Our non-convex geometric analysis is in stark contrast to prior techniques of convex geometric analysis <cit.> where convex combinations of non-convex constraints were used to define the Minkowski functional (e.g., in the definition of atomic norm) while our method uses the non-convex constraint itself.For matrix completion, problem (<ref>) has two hard constraints: a) the rank of the output matrix should be no larger than r, as implied by the form of ; b) the output matrix should be consistent with the sampled measurements, i.e., _Ω()=_Ω(^*). We study the feasibility condition of problem (<ref>) from a geometric perspective: =^* is the unique optimal solution to problem (<ref>) if and only if starting from ^*, either the rank of ^*+ or ^*+_F increases for all directions 's in the constraint set Ω^⊥={∈^n_1× n_2:_Ω(^*+)=_Ω(^*)}. This can be geometrically interpreted as the requirement that the set _(^*)={-^*∈^n_1× n_2:()≤ r, _F ≤^*_F} and the constraint set Ω^⊥ must intersect uniquely at(see Figure <ref>). This can then be shown by a dual certificate argument. Bypassing the Golfing Scheme to Obtain Optimal Bounds. Compared to the nuclear norm method <cit.>, our non-convex geometric analysis leads to a multiplicative log n_(1) factor improvement in the sample complexity for matrix completion. The main reason is that we can avoid using the technique in the so-called golfing scheme, which is loose but inevitable in the analysis of the nuclear norm method. The golfing scheme is a sequential way of building a certificate vector: suppose we would like to build a dual certificate that is close to _*. Starting from an initialization _0, in the i-th step the golfing scheme calculates the distance between our current guess _i and the target _*, and takes fresh i.i.d. samples the entries of _i-_* projected onto subspacein each step, using them for our next guess of _i+1-_*. In expectation, the distance between _i and _* becomes smaller and after 2⌈log n_(1)⌉ steps, _2⌈log n_(1)⌉ is very close to _* with high probability. However, the golfing scheme re-samples matrices in each of 2⌈log n_(1)⌉ iterative steps, and so the analysis incurs an additional multiplicative log n_(1) factor in the sample complexity. Although it might be possible to modify the golfing scheme to use the same samples in each iteration or to reuse many samples, it is unknown how to do so and this remains an important open question. In contrast, our analysis completely avoids the necessity of resampling by going through a very different geometric approach.Putting Things Together. We summarize our new analytical framework with the following figure.Other Techniques. An alternative method is to investigate the exact recoverability of problem (<ref>) via standard convex analysis. We find that the sub-differential of our induced function ·_r* is very similar to that of the nuclear norm. With this observation, we prove the validity of robust PCA in the form of (<ref>) by combining this property of ·_r* with standard techniques from <cit.>.§ PRELIMINARIES We will use calligraphy to represent a set, bold capital letters to represent a matrix, bold lower-case letters to represent a vector, and lower-case letters to represent scalars. Specifically, we denote by ^*∈^n_1× n_2 the underlying matrix. We use _:t∈^n_1× 1 (_t:∈^1× n_2) to indicate the t-th column (row) of . The entry in the i-th row, j-th column ofis represented by _ij. The condition number ofis κ=σ_1()/σ_r(). We let n_(1)=max{n_1, n_2} and n_(2)=min{n_1, n_2}. For a function H() on an input matrix , its conjugate function H^* is defined by H^*()=max_⟨,⟩-H(). Furthermore, let H^** denote the conjugate function of H^*.We will frequently use ()≤ r to constrain the rank of . This can be equivalently represented as =, by restricting the number of columns ofand rows ofto be r. For norms, we denote by _F=√(∑_ij_ij^2) the Frobenius norm of matrix . Let σ_1()≥σ_2()≥...≥σ_r() be the non-zero singular values of . The nuclear norm (a.k.a. trace norm) ofis defined by _*=∑_i=1^rσ_i(), and the operator norm ofis =σ_1(). Denote by _∞=max_ij|_ij|. For two matricesandof equal dimensions, we denote by ⟨,⟩=∑_ij_ij_ij. We denote by ∂ H()={∈^n_1× n_2: H()≥ H()+⟨,-⟩} the sub-differential of function H evaluated at . We define the indicator function of convex setby _()=0,∈;+∞,. For any non-empty set , denote by ()={t:∈, t≥ 0}.We denote by Ω the set of indices of observed entries, and Ω^⊥ its complement. Without confusion, Ω also indicates the linear subspace formed by matrices with entries in Ω^⊥ being 0. We denote by _Ω: ^n_1× n_2→^n_1× n_2 the orthogonal projector of subspace Ω. We will consider a single norm for these operators, namely, the operator norm denoted byand defined by =sup__F=1()_F. For any orthogonal projection operator _ to any subspace , we know that _=1 whenever ()≠0. For distributions, denote by (0,1) a standard Gaussian random variable, (m) the uniform distribution of cardinality m, and (p) the Bernoulli distribution with success probability p. § ℓ_2-REGULARIZED MATRIX FACTORIZATIONS: A NEW ANALYTICAL FRAMEWORK In this section, we develop a novel framework to analyze a general class of ℓ_2-regularized matrix factorization problems. Our framework can be applied to different specific problems and leads to nearly optimal sample complexity guarantees. In particular, we study the ℓ_2-regularized matrix factorization problem(P)min_∈^n_1× r,∈^r× n_2 F(,)=H()+1/2_F^2, H(·) is convex and closed.We show that under suitable conditions the duality gap between (P) and its dual (bi-dual) problem is zero, so problem (P) can be converted to an equivalent convex problem. Examples. There are many examples in which the data under study can be modelled in the form of (P) and fit our framework. To shed light on the nature of our results, we give two examples inspired by contemporary challenges in recommendation system, statistics and machine learning. * Matrix Completion. Setting H()=_{:_Ω()=_Ω(^*)}(), we obtain the models for low-rank matrix completion: min_∈^n_1× r,∈^r× n_21/2_F^2,_Ω()=_Ω(^*), where Ω∼(m). As shown in Section <ref>, the solution to this optimization problem exactly recovers the incoherent matrix ^* with optimal sample complexity up to a constant factor.* Matrix Sensing. Setting H()=_{:()=(^*)}(), we obtain the models for low-rank matrix sensing: min_∈^n_1× r,∈^r× n_21/2_F^2,()=(^*), where (·)={⟨_i,·⟩}_i=1^m is the linear operator with standard Gaussian matrix {_i}_i=1^m. As shown in Section <ref>, the solution to this optimization problem performs as well as the best known results, being able to exactly recover the underlying matrix ^* with sample complexity as small as the information-theoretic limit. §.§ Strong Duality We first consider an easy case where H()=1/2_F^2-⟨,⟩ for a fixed , leading to the objective function 1/2-_F^2. For this case, we establish the following lemma. For any given matrix ∈^n_1 × n_2, any local minimum of f(,)=1/2-_F^2 over ∈^n_1× r and ∈^r× n_2 (r ≤min{n_1, n_2}) is globally optimal, given by _r(). The objective function f(,) around any saddle point has a negative second-order directional curvature. Moreover, f(,) has no local maximum.[Prior work studying the loss surface of low-rank matrix approximation assumes that the matrixis of full rank and does not have the same singular values <cit.>. In this work, we generalize this result by removing these two assumptions.]The proof of Lemma <ref> is basically to calculate the gradient of f(,) and let it equal to zero; see Appendix <ref> for details. Given this lemma, we can reduce F(,) to the form 1/2-_F^2 for someplus an extra term:F(,) =1/2_F^2+H()=1/2_F^2+H^**()=max_1/2_F^2+⟨,⟩-H^*()=max_1/2--_F^2-1/2_F^2-H^*()≜max_ L(,,),where we define L(,,)≜1/2--_F^2-1/2_F^2-H^*() as the Lagrangian of problem (P),[One can easily check that L(,,)=min_ L'(,,,), where L'(,,,) is the Lagrangian of the constraint optimization problem min_,,1/2_F^2+H(),=. With a little abuse of notation, we call L(,,) the Lagrangian of the unconstrained problem (P) as well.] and the second equality holds because H is closed and convex w.r.t. the argument . For any fixed value of , by Lemma <ref>, any local minimum of L(,,) is globally optimal, because minimizing L(,,) is equivalent to minimizing 1/2--_F^2 for a fixed .The remaining part of our analysis is to choose a propersuch that (,,) is a primal-dual saddle point of L(,,), so that min_, L(,,) and problem (P) have the same optimal solution (,). For this, we introduce the following condition, and later we will show that the condition holds with high probability.For a solution (, ) to problem , there exists an ∈∂_ H()|_= such that -^T=^T^T(-)=^T.Explanation of Condition <ref>. We note that ∇_ L(,,)=^T+^T ∇_ L(,,)=^T+^T for a fixed . In particular, if we setto be thein (<ref>), then ∇_ L(,,)|_== and ∇_ L(,,)|_==. So Condition <ref> implies that (,) is either a saddle point or a local minimizer of L(,,) as a function of (,) for the fixed .The following lemma states that if it is a local minimizer, then strong duality holds. Let (,) be a global minimizer of F(,). If there exists a dual certificatesatisfying Condition <ref> and the pair (,) is a local minimizer of L(,,) for the fixed , then strong duality holds. Moreover, we have the relation =_r(-).By the assumption of the lemma, we can show that (,,) is a primal-dual saddle point to the Lagrangian L(,,); see Appendix <ref>. To show strong duality, by the fact that F(,)=max_ L(,,) and that =_ L(,,), we have F(,)=L(,,)≤ L(,,), for any ,, where the inequality holds because (,,) is a primal-dual saddle point of L. So on the one hand, min_,max_ L(,,)=F(,)≤min_, L(,,)≤max_min_, L(,,). On the other hand, by weak duality, we have min_,max_ L(,,)≥max_min_, L(,,). Therefore, min_,max_ L(,,)=max_min_, L(,,), i.e., strong duality holds. Therefore, =_ L(,,)=_1/2_F^2+⟨,⟩-H^*()=_1/2--_F^2=_r(-), as desired. This lemma then leads to the following theorem. Denote by (,) the optimal solution of problem . Define a matrix space≜{^T+, ∈^n_2× r, ∈^n_1× r}.Then strong duality holds for problem , provided that there existssuch that∈∂H()≜Ψ,_(-)=, _^⊥< σ_r().The proof idea is to construct a dual certificateso that the conditions in Lemma <ref> hold.should satisfy the following:∈∂H(), (+)^T=^T(+)=, =_r(-). It turns out that for any matrix ∈^n_1× n_2, _^⊥=(-^†)(-^†) and so _^⊥≤, a fact that we will frequently use in the sequel. Denote bythe left singular space ofandthe right singular space. Then the linear spacecan be equivalently represented as =+. Therefore, ^⊥=(+)^⊥=^⊥∩^⊥. With this, we note that: (b) (+)^T= and ^T(+)= imply +∈(^T)=()^⊥ and +∈()^⊥ (so +∈^⊥), and vice versa. And (c) =_r(-) implies that for an orthogonal decomposition -=+, ∈, ∈^⊥, we have <σ_r(). Conversely, <σ_r() and condition (b) imply =_r(-). Therefore, the dual conditions in (<ref>) are equivalent to (1) ∈∂ H()≜Ψ; (2) _(-)=; (3) _^⊥< σ_r(). To show the dual condition in Theorem <ref>, intuitively, we need to show that the angle θ between subspaceand Ψ is small (see Figure <ref>) for a specific function H(·). In the following (see Section <ref>), we will demonstrate applications that, with randomness, obey this dual condition with high probability.§ MATRIX COMPLETIONIn matrix completion, there is a hidden matrix ^*∈^n_1× n_2 with rank r. We are given measurements {_ij^*: (i,j) ∈Ω}, where Ω∼(m), i.e., Ω is sampled uniformly at random from all subsets of [n_1] × [n_2] of cardinality m. The goal is to exactly recover ^* with high probability. Here we apply our unified framework in Section <ref> to matrix completion, by setting H(·)=_{:_Ω()=_Ω(^*)}(·).A quantity governing the difficulties of matrix completion is the incoherence parameter μ. Intuitively, matrix completion is possible only if the information spreads evenly throughout the low-rank matrix. This intuition is captured by the incoherence conditions. Formally, denote by Σ^T the skinny SVD of a fixed n_1× n_2 matrixof rank r. Candès et al. <cit.> introduced the μ-incoherence condition (<ref>) to the low-rank matrix . For conditions (<ref>), it can be shown that 1≤μ≤n_(1)/r. The condition holds for many random matrices with incoherence parameter μ about √(rlog n_(1)) <cit.>.We first propose a non-convex optimization problem whose unique solution is indeed the ground truth ^*, and then apply our framework to show that strong duality holds for this non-convex optimization and its bi-dual optimization problem. Let Ω∼(m) be the support set uniformly distributed among all sets of cardinality m. Suppose that m≥ cκ^2μ n_(1)rlog n_(1)log_2κ n_(1) for an absolute constant c and ^* obeys μ-incoherence (<ref>). Then ^* is the unique solution of non-convex optimizationmin_,1/2_F^2,s.t._Ω()=_Ω(^*),with probability at least 1-n_(1)^-10. Here we sketch the proof and defer the details to Appendix <ref>. We consider the feasibility of the matrix completion problem:_Ω()=_Ω(^*),()≤ r, _F ≤^*_F .Our proof first identifies a feasibility condition for problem (<ref>), and then shows that ^* is the only matrix which obeys this feasibility condition when the sample size is large enough. More specifically, we note that ^* obeys the conditions in problem (<ref>). Therefore, ^* is the only matrix which obeys condition (<ref>) if and only if ^*+ does not follow the condition for all , i.e., _(^*)∩Ω^⊥={}, where _(^*) is defined as_(^*)={-^*∈^n_1× n_2:()≤ r, _F≤^*_F}. This can be shown by combining the satisfiability of the dual conditions in Theorem , and the well known fact that ∩Ω^⊥={} when the sample size is large. Given the non-convex problem, we are ready to state our main theorem for matrix completion. Let Ω∼(m) be the support set uniformly distributed among all sets of cardinality m. Suppose ^* has condition number κ=σ_1(^*)/σ_r(^*). Then there are absolute constants c and c_0 such that with probability at least 1-c_0n_(1)^-10, the output of the convex problem=__r*,_Ω()=_Ω(^*),is unique and exact, i.e., =^*, provided that m≥ cκ^2μ rn_(1)log_2κ (n_(1))log (n_(1)) and ^* obeys μ-incoherence (<ref>). Namely, strong duality holds for problem (<ref>).[In addition to our main results on strong duality, in a previous version of this paper we also claimed a tight information-theoretic bound on the number of samples required for matrix completion; the proof of that latter claim was problematic as stated, and so we have removed that claim in this version.]We have shown in Theorem <ref> that the problem (,)=_,1/2_F^2,_Ω()=_Ω(^*), exactly recovers ^*, i.e., =^*, with small sample complexity. So if strong duality holds, this non-convex optimization problem can be equivalently converted to the convex program (<ref>). Then Theorem <ref> is straightforward from strong duality.It now suffices to apply our unified framework in Section <ref> to prove the strong duality. We show that the dual condition in Theorem <ref> holds with high probability by the following arguments. Let (,) be a global solution to problem (<ref>). For H()=_{∈^n_1× n_2: _Ω=_Ω^*}(), we haveΨ = ∂ H() ={∈^n_1× n_2: ⟨,⟩≥⟨,⟩, ∈^n_1× n_2_Ω=_Ω^*}={∈^n_1× n_2: ⟨,^*⟩≥⟨,⟩, ∈^n_1× n_2_Ω=_Ω^*} =Ω,where the third equality holds since =^*. Then we only need to show ∈Ω,_(-)=,_^⊥<2/3σ_r().It is interesting to see that dual condition (<ref>) can be satisfied if the angle θ between subspace Ω and subspaceis very small; see Figure <ref>. When the sample size |Ω| becomes larger and larger, the angle θ becomes smaller and smaller (e.g., when |Ω|=n_1n_2, the angle θ is zero as Ω=^n_1× n_2). We show that the sample size m= Ω(κ^2μ rn_(1)log_2κ (n_(1))log (n_(1))) is a sufficient condition for condition (<ref>) to hold. This positive result matches a lower bound from prior work up to a logarithmic factor, which shows that the sample complexity in Theorem <ref> is nearly optimal. Denote by Ω∼(m) the support set uniformly distributed among all sets of cardinality m. Suppose that m≤ cμ n_(1)rlog n_(1) for an absolute constant c. Then there exist infinitely many n_1× n_2 matrices ' of rank at most r obeying μ-incoherence (<ref>) such that _Ω(')=_Ω(^*), with probability at least 1-n_(1)^-10. § ROBUST PRINCIPAL COMPONENT ANALYSISIn this section, we develop our theory for robust PCA based on our framework. In the problem of robust PCA, we are given an observed matrix of the form =^*+^*, where ^* is the ground-truth matrix and ^* is the corruption matrix which is sparse. The goal is to recover the hidden matrices ^* and ^* from the observation . We set H()=λ-_1.To make the information spreads evenly throughout the matrix, the matrix cannot have one entry whose absolute value is significantly larger than other entries. For the robust PCA problem, Candès et al. <cit.> introduced an extra incoherence condition (Recall that ^*=Σ^T is the skinny SVD of ^*)^T_∞≤√(%s/%s)μ rn_1n_2.In this work, we make the following incoherence assumption for robust PCA instead of (<ref>):^*_∞≤√(μ r/n_1n_2)σ_r(^*).Note that condition (<ref>) is very similar to the incoherence condition (<ref>) for the robust PCA problem, but the two notions are incomparable. Note that condition (<ref>) has an intuitive explanation, namely, that the entries must scatter almost uniformly across the low-rank matrix.We have the following results for robust PCA.Suppose ^* is an n_1× n_2 matrix of rank r, and obeys incoherence (<ref>) and (<ref>). Assume that the support set Ω of ^* is uniformly distributed among all sets of cardinality m. Then with probability at least 1-cn_(1)^-10, the output of the optimization problem(,)=_,_r*+λ_1,=+,with λ=σ_r(^*)/√(n_(1)) is exact, namely, =^* and =^*, provided that (^*)≤ρ_rn_(2)/μlog^2 n_(1)m≤ρ_sn_1n_2, where c, ρ_r, and ρ_s are all positive absolute constants, and function ·_r* is given by (<ref>).The bounds on the rank of ^* and the sparsity of ^* in Theorem <ref> match the best known results for robust PCA in prior work when we assume the support set of ^* is sampled uniformly <cit.>. § COMPUTATIONAL ASPECTS Computational Efficiency. We discuss our computational efficiency given that we have strong duality. We note that the dual and bi-dual of primal problem (P) are given by (see Appendix <ref>)()max_∈^n_1× n_2 -H^*()-1/2_r^2,_r^2=∑_i=1^rσ_i^2(),()min_∈^n_1× n_2 H()+_r*,_r*=max_⟨,⟩-1/2_r^2.Problems (D1) and (D2) can be solved efficiently due to their convexity. In particular, Grussler et al. <cit.> provided a computationally efficient algorithm to compute the proximal operators of functions 1/2·_r^2 and ·_r*. Hence, the Douglas-Rachford algorithm can find global minimum up to an ϵ error in function value in time (1/ϵ) <cit.>.Computational Lower Bounds. Unfortunately, strong duality does not always hold for general non-convex problems (P). Here we present a very strong lower bound based on the random 4-SAT hypothesis. This is by now a fairly standard conjecture in complexity theory <cit.> and gives us constant factor inapproximability of problem (P) for deterministic algorithms, even those running in exponential time.If we additionally assume that BPP =P, where BPP is the class of problems which can be solved in probabilistic polynomial time, and P is the class of problems which can be solved in deterministic polynomial time, then the same conclusion holds for randomized algorithms. This is also a standard conjecture in complexity theory, as it is implied by the existence of certain strong pseudorandom generators or if any problem in deterministic exponential time has exponential size circuits <cit.>. Therefore, any subexponential time algorithm achieving a sufficiently small constant factor approximation to problem (P) in general would imply a major breakthrough in complexity theory.The lower bound is proved by a reduction from the Maximum Edge Biclique problem <cit.>. The details are presented in Appendix <ref>.Assume Conjecture <ref> (the hardness of Random 4-SAT). Then there exists an absolute constant ϵ_0 > 0 for which any deterministic algorithm achieving (1+ϵ) in the objective function value for problemwith ϵ≤ϵ_0, requires 2^Ω(n_1 + n_2) time, whereis the optimum. If in addition, BPP = P, then the same conclusion holds for randomized algorithms succeeding with probability at least 2/3.Theorem <ref> is proved by using the hypothesis that random 4-SAT is hard to show hardness of the Maximum Edge Biclique problem for deterministic algorithms. We then do a reduction from the Maximum Edge Biclique problem to our problem.The complete proofs of other theorems/lemmas and related work can be found in the appendices.Acknowledgments. We thank Rina Foygel Barber, Rong Ge, Jason D. Lee, Zhouchen Lin, Guangcan Liu, Tengyu Ma, Benjamin Recht, Xingyu Xie, and Tuo Zhao for useful discussions. This work was supported in part by NSF grants NSF CCF-1422910, NSF CCF-1535967, NSF CCF-1451177, NSF IIS-1618714, NSF CCF-1527371, a Sloan Research Fellowship, a Microsoft Research Faculty Fellowship, DMS-1317308, Simons Investigator Award, Simons Collaboration Grant, and ONR-N00014-16-1-2329. § OTHER RELATED WORK Non-convex matrix factorization is a popular topic studied in theoretical computer science <cit.>, machine learning <cit.>, and optimization <cit.>. We review several lines of research on studying the global optimality of such optimization problems.Global Optimality of Matrix Factorization. While lots of matrix factorization problems have been shown to have no spurious local minima, they either require additional conditions on the local minima, or are based on particular forms of the objective function. Specifically, Burer and Monteiro <cit.> showed that one can minimize F(^T) for any convex function F by solving fordirectly without introducing any local minima, provided that the rank of the outputis larger than the rank of the true minimizer _true. However, such a condition is often impossible to check as (_true) is typically unknown a priori. To resolve the issue, Bach et al. <cit.> and Journée et al. <cit.> proved that =^T is a global minimizer of F(), ifis a rank-deficient local minimizer of F(^T) and F() is a twice differentiable convex function. Haeffele and Vidal <cit.> further extended this result by allowing a more general form of objective function F()=G()+H(), where G is a twice differentiable convex function with compact level set and H is a proper convex function such that F is lower semi-continuous. However, a major drawback of this line of research is that these result fails when the local minimizer is of full rank.Matrix Completion. Matrix completion is a prototypical example of matrix factorization. One line of work on matrix completion builds on convex relaxation (e.g., <cit.>). Recently, Ge et al. <cit.> showed that matrix completion has no spurious local optimum, when |Ω| is sufficiently large and the matrixis incoherent. The result is only for positive semi-definite matrices and their sample complexity is not nearly optimal.Another line of work is built upon good initialization for global convergence. Recent attempts showed that one can first compute some form of initialization (e.g., by singular value decomposition) that is close to the global minimizer and then use non-convex approaches to reach global optimality, such as alternating minimization, block coordinate descent, and gradient descent <cit.>. In our result, in contrast, we can reformulate non-convex matrix completion problems as equivalent convex programs, which guarantees global convergence from any initialization. Robust PCA. Robust PCA is also a prototypical example of matrix factorization. The goal is to recover both the low-rank and the sparse components exactly from their superposition <cit.>. It has been widely applied to various tasks, such as video denoising, background modeling, image alignment, photometric stereo, texture representation, subspace clustering, and spectral clustering.There are typically two settings in the robust PCA literature: a) the support set of the sparse matrix is uniformly sampled <cit.>; b) the support set of the sparse matrix is deterministic, but the non-zero entries in each row or column of the matrix cannot be too large <cit.>. In this work, we discuss the first case. Our framework provides results that match the best known work in setting (b) <cit.>. Other Matrix Factorization Problems. Matrix sensing is another typical matrix factorization problem <cit.>. Bhojanapalli et al. <cit.> and Tu et al. <cit.> showed that the matrix recovery model min_,1/2(-)_F^2, achieves optimality for every local minimum, if the operatorsatisfies the restricted isometry property. They further gave a lower bound and showed that the unstructured operatormay easily lead to a local minimum which is not globally optimal.Some other matrix factorization problems are also shown to have nice geometric properties such as the property that all local minima are global minima. Examples include dictionary learning <cit.>, phase retrieval <cit.>, and linear deep neural networks <cit.>. In multi-layer linear neural networks where the goal is to learn a multi-linear projection ^*=∏_i _i, each _i represents the weight matrix that connects the hidden units in the i-th and (i+1)-th layers. The study of such linear models is central to the theoretical understanding of the loss surface of deep neural networks with non-linear activation functions <cit.>. In dictionary learning, we aim to recover a complete (i.e., square and invertible) dictionary matrixfrom a given signalin the form of =, provided that the representation coefficientis sufficiently sparse. This problem centers around solving a non-convex matrix factorization problem with a sparsity constraint on the representation coefficient <cit.>. Other high-impact examples of matrix factorization models range from the classic unsupervised learning problems like PCA, independent component analysis, and clustering, to the more recent problems such as non-negative matrix factorization, weighted low-rank matrix approximation, sparse coding, tensor decomposition <cit.>, subspace clustering <cit.>, etc. Applying our framework to these other problems is left for future work.Atomic Norms. The atomic norm is a recently proposed function for linear inverse problems <cit.>. Many well-known norms, e.g., the ℓ_1 norm and the nuclear norm, serve as special cases of atomic norms. It has been widely applied to the problems of compressed sensing <cit.>, low-rank matrix recovery <cit.>, blind deconvolution <cit.>, etc. The norm is defined by the Minkowski functional associated with the convex hull of a set : _=inf{t>0: ∈ t}. In particular, if we setto be the convex hull of the infinite set of unit-ℓ_2-norm rank-one matrices, then ·_ equals to the nuclear norm. We mention that our objective term _F in problem (<ref>) is similar to the atomic norm, but with slight differences: unlike the atomic norm, we setto be the infinite set of unit-ℓ_2-norm rank-r matrices for ()≤ r. With this, we achieve better sample complexity guarantees than the atomic-norm based methods. § PROOF OF LEMMA <REF>Lemma <ref> (Restated). For any given matrix ∈^n_1 × n_2, any local minimum of f(,)=1/2-_F^2 over ∈^n_1× r and ∈^r× n_2 (r ≤min{n_1, n_2}) is globally optimal, given by _r(). The objective function f(,) around any saddle point has a negative second-order directional curvature. Moreover, f(,) has no local maximum.(,) is a critical point of f(,) if and only if ∇_ f(,)= and ∇_ f(,)=, or equivalently,^T=^T^T=^T.Note that for any fixed matrix(resp. ), the function f(,) is convex in the coefficients of(resp. ).To prove the desired lemma, we have the following claim.If two matricesanddefine a critical point of f(,), then the global mapping = is of the form=_,withsatisfying^†^T=^†^T^†=^T^†. Ifanddefine a critical point of f(,), then (<ref>) holds and the general solution to (<ref>) satisfies=(^T)^†^T+(-^†)Ł,for some matrix Ł. So ==(^T)^†^T=^†=_ by the property of the Moore-Penrose pseudo-inverse: ^†=(^T)^†^T.By (<ref>), we also have^T^T=^T^T^T=^T.Plugging in the relation =^†, (<ref>) can be rewritten as^†^T^†=^T^†.Note that the matrix ^†^T^† is symmetric. Thus^†^T^† = ^†^T,as desired. To prove Lemma <ref>, we also need the following claim.Denote by ={i_1,i_2,...,i_r} any ordered r-index set (ordered by λ_i_j, j∈[r] from the largest to the smallest) and λ_i, i∈[n_1], the ordered eigenvalues of ^T∈^n_1× n_1 with p distinct values. Let =[_̆1,_̆2,...,_̆n_1] denote the matrix formed by the orthonormal eigenvectors of ^T∈^n_1× n_1 associated with the ordered p eigenvalues, whose multiplicities are m_1,m_2,...,m_p (m_1+m_2+...+m_p=n_1). For any matrix , let _: denote the submatrix [_i_1,_i_2,...,_i_r] associated with the index set .Then two matricesanddefine a critical point of f(,) if and only if there exists an ordered r-index set , an invertible matrix , and an r× n matrix Ł such that=()_:=^†+(-^†)Ł,whereis a p-block-diagonal matrix with each block corresponding to the eigenspace of an eigenvalue. For such a critical point, we have=_, f(,)=1/2((^T)-∑_i∈λ_i)=1/2∑_i∉λ_i. Note that ^T is a real symmetric covariance matrix. So it can always be represented as ^T, where ∈^n_1× n_1 is an orthonormal matrix consisting of eigenvectors of ^T and ∈^n_1× n_1 is a diagonal matrix with non-increasing eigenvalues of ^T.Ifandsatisfy (<ref>) for some , Ł, and , then^T=^T^T=^T,which is (<ref>). Soanddefine a critical point of f(,).For the converse, notice that_^T=^T(^T)^†=^T^†=^T_,or equivalently, _=_^T^T. Thus (<ref>) yields_^T^T^T=^T_^T^T,or equivalently, _^T=_^T. Notice that ∈^n_1× n_1 is a diagonal matrix with p distinct eigenvalues of ^T, and _^T is an orthogonal projector of rank r. So _^T is a rank-r restriction of the block-diagonal PSD matrix :=^T with p blocks, each of which is an orthogonal projector of dimension m_i, corresponding to the eigenvalues λ_i, i∈[p]. Therefore, there exists an r-index setand a block-diagonal matrixsuch that _^T=_:_:^T, where =. It follows that_=_^T^T=_:_:^T^T=()_:()_:^T.Since the column space ofcoincides with the column space of ()_:,is of the form =()_:, andis given by (<ref>). Thus =(^T)^†^T+(-^†)Ł=_ andf(,) =1/2-_F^2=1/2-__F^2=1/2_^⊥_F^2=1/2∑_i∉λ_i.The claim is proved. So the local minimizer of f(,) is given by (<ref>) withsuch that (λ_i_1, λ_i_2, …, λ_i_r) = Φ, where Φ is the sequence of the r largest eigenvalues of ^T. Such a local minimizer is globally optimal according to (<ref>).We then show that whenconsists of other combinations of indices of eigenvalues, i.e., (λ_i_1, λ_i_2, …, λ_i_r)≠Φ, the corresponding pair (,) given by (<ref>) is a strict saddle point. Ifis such that (λ_i_1, λ_i_2, …, λ_i_r) ≠Φ, then the pair (,) given by (<ref>) is a strict saddle point.If (λ_i_1, λ_i_2, …, λ_i_r) ≠Φ, then there exists a i such that λ_i_i does not equal the i-th element λ_i of Φ.Denote by =𝐑. It is enough to slightly perturb the column space oftowards the direction of an eigenvector of λ_i. More precisely, let j is the largest index in . For any ϵ, let 𝐑 be the matrix such that 𝐑_:k = 𝐑_:k(k≠ j) and 𝐑_:j=(1+ϵ^2)^-1/2(𝐑_:j+ϵ𝐑_:i).Let =𝐑_ and =^†+(-^†)Ł. A direct calculation shows thatf(,)=f(,)-ϵ^2(λ_i-λ_j)/(2+2ϵ^2).Hence,lim_ϵ→ 0f(,)-f(,)/ϵ^2=-1/2(λ_i-λ_j)<0,and thus the pair (,) is a strict saddle point.Note that all critical points of f(,) are in the form of (<ref>), and if ≠Φ, the pair (,) given by (<ref>) is a strict saddle point, while if =Φ, then the pair (,) given by (<ref>) is a local minimum. We conclude that f(,) has no local maximum. The proof is completed. § PROOF OF LEMMA <REF> Lemma <ref> (Restated). Let (,) be a global minimizer of F(,). If there exists a dual certificateas in Condition <ref> such that the pair (,) is a local minimizer of L(,,) for the fixed , then strong duality holds. Moreover, we have the relation =_r(-). By the assumption of the lemma, (,) is a local minimizer of L(,,)=1/2--_F^2+c(), where c() is a function that is independent ofand . So according to Lemma <ref>, (,)=_, L(,,), namely, (,) globally minimizes L(,,) whenis fixed to . Furthermore, ∈∂_ H()|_= implies that ∈∂_ H^*()|_= by the convexity of function H, meaning that ∈∂_ L(,,). So =_ L(,,) due to the concavity of L(,,) w.r.t. variable . Thus (,,) is a primal-dual saddle point of L(,,).We now prove the strong duality. By the fact that F(,)=max_ L(,,) and that =_ L(,,), we haveF(,)=L(,,)≤ L(,,),∀,.where the inequality holds because (,,) is a primal-dual saddle point of L. So on the one hand, we havemin_,max_ L(,,)=F(,)≤min_, L(,,)≤max_min_, L(,,).On the other hand, by weak duality,min_,max_ L(,,)≥max_min_, L(,,).Therefore, min_,max_ L(,,)=max_min_, L(,,), i.e., strong duality holds. Hence,=_ L(,,)=_1/2--_F^2-1/2_F^2-H^*()=_1/2--_F^2=_r(-),as desired. § EXISTENCE OF DUAL CERTIFICATE FOR MATRIX COMPLETIONLet ∈^n_1× r and ∈^r × n_2 such that =^*. Then we have the following lemma.Let Ω∼(m) be the support set uniformly distributed among all sets of cardinality m. Suppose that m≥ cκ^2μ n_(1)rlog n_(1)log_2κ n_(1) for an absolute constant c and ^* obeys μ-incoherence (<ref>). Then there existssuch that ∈Ω, _(-)=, _^⊥<2/3σ_r().with probability at least 1-n_(1)^-10. The rest of the section is denoted to the proof of Lemma <ref>. We begin with the following lemma.If we can construct ansuch that ∈Ω, _(-)-_F≤√(r/3n_(1)^2)σ_r(), _^⊥<1/3σ_r(),then we can construct ansuch that Eqn. (<ref>) holds with probability at least 1 - n_(1)^-10. To prove the lemma, we first claim the following theorem. Assume that Ω is sampled according to the Bernoulli model with success probability p=Θ(m/n_1n_2), and incoherence condition (<ref>) holds. Then there is an absolute constant C_R such that for β>1, we havep^-1__Ω_-_≤ C_R√(βμ n_(1)rlog n_(1)/m)≜ϵ,with probability at least 1-3n^-β provided that C_R√(βμ n_(1)rlog n_(1)/m) < 1.Suppose that Condition (<ref>) holds. Let =-∈Ω be the perturbation matrix betweenandsuch that _(-)=. Such aexists by setting =_Ω_(__Ω_)^-1(_(-)-). So __F≤√(r/3n_(1)^2)σ_r(). We now prove Condition (3) in Eqn. (<ref>). Observe that_^⊥ ≤_^⊥+_^⊥≤1/3σ_r()+_^⊥.So we only need to show _^⊥≤1/3σ_r().Before proceeding, we begin by introducing a normalized version _Ω:^n_1× n_2→^n_1× n_2 of _Ω:_Ω=p^-1_Ω-.With this, we have__Ω_=p_(+_Ω)_.Note that for any operator :→, we have^-1=∑_k≥ 0 (_-)^k _-<1.So according to Theorem <ref>, the operator p(__Ω_)^-1 can be represented as a convergent Neumann seriesp(__Ω_)^-1=∑_k≥ 0(-1)^k(__Ω_)^k,because __Ω_≤ϵ<1/2 once m≥ Cμ n_(1)rlog n_(1) for a sufficiently large absolute constant C. We also note thatp(_^⊥_Ω_)=_^⊥_Ω_,because _^⊥_=0. Thus_^⊥ =_^⊥_Ω_(__Ω_)^-1(_(-)-))=_^⊥_Ω_ p(__Ω_)^-1((_(-)-))=∑_k≥ 0 (-1)^k_^⊥_Ω(__Ω_)^k((_(-)-))≤∑_k≥ 0 (-1)^k_^⊥_Ω(__Ω_)^k((_(-)-))_F≤_Ω∑_k≥ 0__Ω_^k_(-)-))_F≤4/p_(-)-)_F≤Θ(n_1n_2/m) √(r/3 n_(1)^2)σ_r()≤1/3σ_r()with high probability. The proof is completed. It thus suffices to construct a dual certificatesuch that all conditions in (<ref>) hold. To this end, partition Ω=Ω_1∪Ω_2∪...∪Ω_b into b partitions of size q. By assumption, we may chooseq≥128/3 C βκ^2μ rn_(1)log n_(1) b≥1/2log_2κ(24^2 n_(1)^2κ^2)for a sufficiently large constant C. Let Ω_j∼𝖡𝖾𝗋(q) denote the set of indices corresponding to the j-th partitions. Define _0= and set _k=n_1n_2/q∑_j=1^k_Ω_j(_j-1), _k=-_(_k) for k=1,2,...,b. Then by Theorem <ref>,_k_F =_k-1-n_1n_2/q__Ω_k(_k-1)_F=(_-n_1n_2/q__Ω_k_)(_k-1)_F≤1/2κ_k-1_F.So it follows that -_(_b)_F=_b_F≤ (2κ)^-b_0_F≤ (2κ)^-b√(r)σ_1()≤√(r/24^2n_(1)^2)σ_r().The following lemma together implies the strong duality of (<ref>) straightforwardly. Under the assumptions of Theorem <ref>, the dual certification _b obeys the dual condition (<ref>) with probability at least 1-n_(1)^-10. It is well known that for matrix completion, the Uniform model Ω∼(m) is equivalent to the Bernoulli model Ω∼(p), where each element in [n_1] × [n_2] is included with probability p = Θ(m/(n_1 n_2)) independently; see Section <ref> for a brief justification. By the equivalence, we can suppose Ω∼(p).To prove Lemma <ref>, as a preliminary, we need the following lemmas. Supposeis a fixed matrix. Suppose Ω∼(p). Then with high probability,(-p^-1_Ω)≤ C_0'(log n_(1)/p_∞+√(log n_(1)/p)_∞,2),where C_0'>0 is an absolute constant and_∞,2=max{max_i√(∑_b _ib^2),max_j√(∑_a_aj^2)}.Suppose Ω∼(p) andis a fixed matrix. Then with high probability,-p^-1__Ω_∞≤ϵ_∞,provided that p≥ C_0ϵ^-2(μ rlog n_(1))/n_(2) for some absolute constant C_0>0. Suppose thatis a fixed matrix and Ω∼(p). If p≥ c_0 μ rlog n_(1)/n_(2) for some c_0 sufficiently large, then with high probability,(p^-1__Ω-_)_∞,2≤1/2√(n_(1)/μ r)_∞+1/2_∞,2.Observe that by Lemma <ref>,_j_∞≤(1/2)^j_∞,and by Lemma <ref>,_j_∞,2≤1/2√(n_(1)/μ r)_j-1_∞+1/2_j-1_∞,2.So_j_∞,2≤(1/2)^j√(n_(1)/μ r)_∞+1/2_j-1_∞,2≤ j(1/2)^j√(n_(1)/μ r)_∞+(1/2)^j_∞,2.Therefore,_^⊥_b≤∑_j=1^bn_1n_2/q_^⊥_Ω_j_j-1=∑_j=1^b_^⊥(n_1n_2/q_Ω_j_j-1-_j-1)≤∑_j=1^b(n_1n_2/q_Ω_j-)(_j-1).Let p denote Θ(q/n_1n_2). By Lemma <ref>,_^⊥_b≤ C_0'log n_(1)/p∑_j=1^b _j-1_∞+C_0'√(log n_(1)/p)∑_j=1^b _j-1_∞,2≤ C_0'log n_(1)/p∑_j=1^b (1/2)^j_∞+C_0'√(log n_(1)/p)∑_j=1^b [j(1/2)^j√(n_(1)/μ r)_∞+(1/2)^j_∞,2]≤ C_0'log n_(1)/p_∞+2C_0'√(log n_(1)/p)√(n_(1)/μ r)_∞+C_0'√(log n_(1)/p)_∞,2. Setting =^*, we note the facts that (we assume WLOG n_2≥ n_1)^*_∞,2=max_i _i^TΣ^T_2≤max_i_i^Tσ_1(^*)≤√(μ r/n_1)σ_1(^*) ≤√(μ r/n_1)κσ_r(^*),and that^*_∞ =max_ij⟨^*,_i_j^T⟩=max_ij⟨Σ^T,_i_j^T⟩=max_ij⟨_i^TΣ,_j^T⟩≤max_ij_i^TΣ^T_2_j^T_2≤max_j^*_∞,2_j^T_2≤μ rκ/√(n_1n_2)σ_r(^*).Substituting p=Θ(κ^2μ r n_(1)log (n_(1))log_2κ(n_(1))/n_1n_2), we obtain _^⊥_b<1/3σ_r(^*). The proof is completed. § SUBGRADIENT OF THE R* FUNCTION Let Σ^T be the skinny SVD of matrix ^* of rank r. The subdifferential of ·_r* evaluated at ^* is given by∂^*_r*={^*+: ^T=,=,≤σ_r(^*)}. Note that for any fixed function f(·), the set of all optimal solutions of the problemf^*(^*)=max_⟨^*,⟩-f()form the subdifferential of the conjugate function f^*(·) evaluated at ^*. Set f(·) to be 1/2·_r^2 and notice that the function 1/2·_r^2 is unitarily invariant. By Von Neumann's trace inequality, the optimal solutions to problem (<ref>) are given by [,^⊥]([σ_1(),...,σ_r(),σ_r+1(),...,σ_n_(2)()])[,^⊥]^T, where {σ_i()}_i=r+1^n_(2) can be any value no larger than σ_r() and {σ_i()}_i=1^r are given by the optimal solution to the problemmax_{σ_i()}_i=1^r∑_i=1^rσ_i(^*)σ_i()-1/2∑_i=1^rσ_i^2().The solution is unique such that σ_i()=σ_i(^*), i=1,2,...,r. The proof is complete. § PROOF OF THEOREM <REF>Theorem <ref> (Uniqueness of Solution. Restated).Let Ω∼(m) be the support set uniformly distributed among all sets of cardinality m. Suppose that m≥ cκ^2μ n_(1)rlog n_(1)log_2κ n_(1) for an absolute constant c and ^* obeys μ-incoherence (<ref>). Then ^* is the unique solution of non-convex optimizationmin_,1/2_F^2,s.t._Ω()=_Ω(^*),with probability at least 1-n_(1)^-10.We note that a recovery result under the Bernoulli model automatically implies a corresponding result for the uniform model <cit.>; see Section <ref> for the details. So in the following, we assume the Bernoulli model.Consider the feasibility of the matrix completion problem:_Ω()=_Ω(^*),_F≤^*_F, ()≤ r.Note that if ^* is the unique solution of (<ref>), then ^* is the unique solution of (<ref>). We now show the former. Our proof first identifies a feasibility condition for problem (<ref>), and then shows that ^* is the only matrix that obeys this feasibility condition when the sample size is large enough. We denote by_(^*)={-^*∈^n_1× n_2:()≤ r, _F≤^*_F},and= {^T+^T, ∈^n_2× r, ∈^n_1× r},where Σ^T is the skinny SVD of ^*.We have the following proposition for the feasibility of problem (<ref>). ^* is the unique feasible solution to problem (<ref>) if _(^*)∩Ω^⊥={}. Notice that problem (<ref>) is equivalent to another feasibility problem(^*+)≤ r,^*+_F≤^*_F,∈Ω^⊥.Suppose that _(^*)∩Ω^⊥={}. Since (^*+)≤ r and ^*+_F≤^*_F are equivalent to ∈_(^*), and note that ∈Ω^⊥, we have =, which means ^* is the unique feasible solution to problem (<ref>). The remainder of the proof is to show _(^*)∩Ω^⊥={}. To proceed, we note that_(^*) ={ - ^* ∈^n_1× n_2:()≤ r, 1/2_F^2≤1/2^*_F^2}⊆{- ^*∈^n_1× n_2:_r*≤^*_r*}(since 1/2_F^2=_r* for any rank-r matrix)≜__*(^*).We now show that__*(^*)∩Ω^⊥={},when m≥ cκ^2μ rn_(1)log_2κ (n_(1))log (n_(1)), which will prove _(^*)∩Ω^⊥={} as desired.By Lemma <ref>, there exists asuch that ∈Ω, _(-)=^*, _^⊥<2/3σ_r(^*). Consider any ∈Ω^⊥ such that ≠. By Lemma <ref>, for any ∈^⊥ and ≤σ_r(^*),^*+_r* ≥^*_r*+⟨^*+,⟩.Since ⟨,⟩ = ⟨_^⊥,⟩ = ⟨, _^⊥⟩, we can choosesuch that ⟨,⟩=σ_r(^*)_^⊥_*. Then^*+_r* ≥^*_r*+σ_r(^*)𝒫_^⊥_*+⟨^*,⟩=^*_r*+σ_r(^*)𝒫_^⊥_*+⟨^*+,⟩(since ∈Ω and ∈Ω^⊥)=^*_r*+σ_r(^*)𝒫_^⊥_*+⟨^*+_,⟩+⟨_^⊥,⟩=^*_r*+σ_r(^*)𝒫_^⊥_*+⟨_^⊥,⟩(by condition (2))=^*_r*+σ_r(^*)𝒫_^⊥_*+⟨_^⊥_^⊥,⟩=^*_r*+σ_r(^*)𝒫_^⊥_*+⟨_^⊥,_^⊥⟩≥^*_r*+σ_r(^*)𝒫_^⊥_*-_^⊥_^⊥_*(by Hölder's inequality)≥^*_r*+1/3σ_r(^*)_^⊥_*(by condition (3)).So if ∩Ω^⊥={}, since ∈Ω^⊥ and ≠, we have ∉. Therefore,^*+_r*>^*_r*which then leads to __*(^*)∩Ω^⊥={}.The rest of proof is to show that ∩Ω^⊥={}. We have the following lemma.Assume that Ω∼(p) and the incoherence condition (<ref>) holds. Then with probability at least 1-n_(1)^-10, we have _Ω^⊥_≤√(1-p+ϵ p), provided that p≥ C_0ϵ^-2(μ rlog n_(1))/n_(2), where C_0 is an absolute constant. If Ω∼𝖡𝖾𝗋(p), we have, by Theorem <ref>, that with high probability_-p^-1__Ω_≤ϵ,provided that p≥ C_0ϵ^-2μ rlog n_(1)/n_(2). Note, however, that since =_Ω+_Ω^⊥,_-p^-1__Ω_=p^-1(__Ω^⊥_-(1-p)_)and, therefore, by the triangle inequality__Ω^⊥_≤ϵ p+(1-p).Since _Ω^⊥_^2 ≤__Ω^⊥_, the proof is completed. We note that _Ω^⊥_<1 implies Ω^⊥∩={}. The proof is completed.§ PROOF OF THEOREM <REF> We have shown in Theorem <ref> that the problem (,)=_,1/2_F^2,_Ω()=_Ω(^*), exactly recovers ^*, i.e., =^*, with nearly optimal sample complexity. So if strong duality holds, this non-convex optimization problem can be equivalently converted to the convex program (<ref>). Then Theorem <ref> is straightforward from strong duality.It now suffices to apply our unified framework in Section <ref> to prove the strong duality. LetH()=_{∈^n_1× n_2: _Ω=_Ω^*}()in Problem , and let (,) be a global solution to the problem. Then by Theorem <ref>, =^*. For Problemwith this special H(), we haveΨ = ∂ H() ={∈^n_1× n_2: ⟨,⟩≥⟨,⟩, ∈^n_1× n_2_Ω=_Ω^*}={∈^n_1× n_2: ⟨,^*⟩≥⟨,⟩, ∈^n_1× n_2_Ω=_Ω^*} =Ω,where the third equality holds since =^*. Combining with Lemma <ref> shows that the dual condition in Theorem <ref> holds with high probability, which leads to strong duality and thus proving Theorem <ref>. § PROOF OF THEOREM <REF> Theorem <ref> (Robust PCA. Restated). Suppose ^* is an n_1× n_2 matrix of rank r, and obeys incoherence (<ref>) and (<ref>). Assume that the support set Ω of ^* is uniformly distributed among all sets of cardinality m. Then with probability at least 1-cn_(1)^-10, the output of the optimization problem(,)=_,_r*+λ_1,=+,with λ=σ_r(^*)/√(n_(1)) is exact, namely, =^* and =^*, provided that (^*)≤ρ_rn_(2)/μlog^2 n_(1)m≤ρ_sn_1n_2, where c, ρ_r, and ρ_s are all positive absolute constants, and function ·_r* is given by (<ref>). §.§ Dual CertificatesAssume that _Ω_≤ 1/2 and λ<σ_r(^*). Then (^*,^*) is the unique solution to problem (<ref>) if there exists (,,𝐊) for which^*+=λ((^*)++_Ω𝐊),where ∈^⊥, ≤σ_r(^*)/2, ∈Ω^⊥, _∞≤1/2, and _Ω𝐊_F≤1/4.Let (^*+,̋^*-)̋ be any optimal solution to problem (<ref>). Denote by ^*+^* an arbitrary subgradient of the r* function at ^* (see Lemma <ref>), and (^*)+^* an arbitrary subgradient of the ℓ_1 norm at ^*. By the definition of the subgradient, the inequality follows^*+_r*+λ ^*-_1≥^*_r*+λ^*_1+⟨^*+^*,⟩̋-λ⟨(^*)+^*,⟩̋=^*_r*+λ^*_1+⟨^*-λ(^*),⟩̋+⟨^*,⟩̋-λ⟨^*,⟩̋=^*_r*+λ^*_1+⟨^*-λ(^*),⟩̋+σ_r(^*)_^⊥_*+λ_Ω^⊥_1 =^*_r*+λ^*_1+⟨λ+λ_Ω𝐊-,⟩̋+σ_r(^*)_^⊥_*+λ_Ω^⊥_1≥^*_r*+λ^*_1+σ_r(^*)/2_^⊥_*+λ/2_Ω^⊥_1-λ/4_Ω_F,where the third line holds by picking ^* such that ⟨^*,⟩̋=σ_r(^*)_^⊥_* and ⟨^*,⟩̋=-_Ω^⊥_1.[For instance, ^*=-(_Ω^⊥)̋ is such as matrix. Also, by the duality between the nuclear norm and the operator norm, there is a matrix obeying =σ_r(^*) such that ⟨,_^⊥⟩̋=σ_r(^*)_^⊥_*. We pick ^*=_^⊥ here.] We note that_Ω_F ≤_Ω__F+_Ω_^⊥_F≤1/2_F+_^⊥_F≤1/2_Ω_F+1/2_Ω^⊥_F+_^⊥_F,which implies that λ/4_Ω_F≤λ/4_Ω^⊥_F+λ/2_^⊥_F≤λ/4_Ω^⊥_1+λ/2_^⊥_*. Therefore,^*+_r*+λ ^*-_1≥^*_r*+λ^*_1+σ_r(^*)-λ/2_^⊥_*+λ/4_Ω^⊥_1≥^*+_r*+λ^*-_1+σ_r(^*)-λ/2_^⊥_*+λ/4_Ω^⊥_1,where the second inequality holds because (^*+,̋^*-)̋ is optimal. Thus ∈̋∩Ω. Note that _Ω_<1 implies ∩Ω={0} and thus =̋0. This completes the proof. According to Lemma <ref>, to show the exact recoverability of problem (<ref>), it is sufficient to find an appropriatefor which∈^⊥, ≤σ_r(^*)/2, _Ω(^*+-λ(S^*))_F≤λ/4, _Ω^⊥(^*+)_∞≤λ/2,under the assumptions that _Ω_≤ 1/2 and λ<σ_r(^*). We note that λ=σ_r(^*)/√(n_(1))<σ_r(^*). To see _Ω_≤ 1/2, we have the following lemma. Suppose that Ω∼(p) and incoherence (<ref>) holds. Then with probability at least 1-n_(1)^-10, _Ω_^2≤ p+ϵ, provided that 1-p≥ C_0ϵ^-2μ rlog n_(1)/n_(2) for an absolute constant C_0. Setting p and ϵ as small constants in Lemma <ref>, we have _Ω_≤ 1/2 with high probability. §.§ Dual Certification by Least Squares and the Golfing SchemeThe remainder of the proof is to constructsuch that the dual condition (<ref>) holds true. Before introducing our construction, we assume Ω∼(p), or equivalently Ω^⊥∼(1-p), where p is allowed be as large as an absolute constant. Note that Ω^⊥ has the same distribution as that of Ω_1∪Ω_2∪...∪Ω_j_0, where the Ω_j's are drawn independently with replacement from (q), j_0=⌈log n_(1)⌉, and q obeys p=(1-q)^j_0 (q=Ω(1/log n_(1)) implies p=(1)). We constructbased on such a distribution.Our construction separatesinto two terms: =^L+^S. To construct ^L, we apply the golfing scheme introduced by <cit.>. Specifically, ^L is constructed by an inductive procedure:_j=_j-1 +q^-1_Ω_j_(^*-_j-1), _0=,^L=_^⊥_j_0.To construct ^S, we apply the method of least squares by <cit.>, which is^S=λ_^⊥∑_k≥ 0(_Ω__Ω)^k(^*).Note that _Ω_≤ 1/2. Thus _Ω__Ω≤ 1/4 and the Neumann series in (<ref>) is well-defined. Observe that _Ω^S=λ(_Ω-_Ω__Ω)(_Ω-_Ω__Ω)^-1(^*)=λ(^*). So to prove the dual condition (<ref>), it suffices to show that^L≤σ_r(^*)/4,_Ω(^*+^L)_F≤λ/4,_Ω^⊥(^*+^L)_∞≤λ/4, ^S≤σ_r(^*)/4,_Ω^⊥^S_∞≤λ/4.§.§ Proof of Dual ConditionsSince we have constructed the dual certificate , the remainder is to show thatobeys dual conditions (<ref>) and (<ref>) with high probability. We have the following. Assume Ω_j∼(q), j=1,2,...,j_0, and j_0=2⌈log n_(1)⌉. Then under the other assumptions of Theorem <ref>, ^L given by (<ref>) obeys dual condition (<ref>).Let _j=_(^*-_j)∈. Then we have_j=__j-1-q^-1__Ω_j__j-1=(_-q^-1__Ω_j_)_j-1,and _j=∑_k=1^j q^-1_Ω_k_k-1∈Ω^⊥. We set q=Ω(ϵ^-2μ rlog n_(1)/n_(2)) with a small constant ϵ.Proof of (a). It holds that^L =_^⊥_j_0≤∑_k=1^j_0q^-1_^⊥_Ω_k_k-1=∑_k=1^j_0_^⊥(q^-1_Ω_k_k-1-_k-1)≤∑_k=1^j_0q^-1_Ω_k_k-1-_k-1≤ C_0'(log n_(1)/q∑_k=1^j_0_k-1_∞+√(log n_(1)/q)∑_k=1^j_0_k-1_∞,2).We note that by Lemma <ref>,_k-1_∞≤(1/2)^k-1_0_∞,and by Lemma <ref>,_k-1_∞,2≤1/2√(n_(1)/μ r)_k-2_∞+1/2_k-2_∞,2.Therefore,_k-1_∞,2 ≤(1/2)^k-1√(n_(1)/μ r)_0_∞+1/2_k-2_∞,2≤ (k-1)(1/2)^k-1√(n_(1)/μ r)_0_∞+(1/2)^k-1_0_∞,2,and so we have^L≤ C_0'[log n_(1)/q∑_k=1^j_0(1/2)^k-1_0_∞+√(log n_(1)/q)∑_k=1^j_0((k-1)(1/2)^k-1√(n_(1)/μ r)_0_∞+(1/2)^k-1_0_∞,2)]≤ 2C_0'[log n_(1)/q^*_∞+√(n_(1)log n_(1)/qμ r)^*_∞+√(log n_(1)/q)^*_∞,2]≤1/16[n_(2)/μ r^*_∞+√(n_(1)n_(2))/μ r^*_∞+√(n_(2)/μ r)^*_∞,2]≤σ_r(^*)/4,where we have used the fact that^*_∞,2≤√(n_(1))^*_∞≤√(μ r/n_(2))σ_r(^*). Proof of (b). Because _j_0∈Ω^⊥, we have _Ω(^*+_^⊥_j_0)=_Ω(^*-__j_0)=_Ω_j_0. It then follows from Theorem <ref> that for a properly chosen t,_j_0_F ≤ t^j_0^*_F≤ t^j_0√(n_1n_2)^*_∞≤ t^j_0√(n_1n_2)√(μ r/n_1n_2)σ_r(^*)≤λ/8. (t^j_0≤ e^-2log n_(1)≤ n_(1)^-2) Proof of (c). By definition, we know that ^*+^L=_j_0+_j_0. Since we have shown _j_0_F≤λ/8, it suffices to prove _j_0_∞≤λ/8. We have_j_0_∞ ≤ q^-1∑_k=1^j_0_Ω_k_k-1_∞≤ q^-1∑_k=1^j_0ϵ^k-1^*_∞≤n_(2)ϵ^2/C_0μ rlog n_(1)√(μ r/n_(1)n_(2))σ_r(^*)≤λ/8,if we choose ϵ= C(μ r(log n_(1))^2/n_(2))^1/4 for an absolute constant C. This can be true once the constant ρ_r is sufficiently small. We now prove that ^S given by (<ref>) obeys dual condition (<ref>). We have the following.Assume Ω∼(p). Then under the other assumptions of Theorem <ref>, ^S given by (<ref>) obeys dual condition (<ref>).According to the standard de-randomization argument <cit.>, it is equivalent to studying the case when the signs δ_ij of _ij^* are independently distributed asδ_ij=1, p/2,0, 1-p,-1, p/2. Proof of (d). Recall that^S =λ_^⊥∑_k≥ 0(_Ω__Ω)^k(^*)=λ_^⊥(^*)+λ_^⊥∑_k≥ 1(_Ω__Ω)^k(^*).To bound the first term, we have (^*)≤ 4√(n_(1)p) <cit.>. So λ_^⊥(^*)≤λ(^*)≤ 4√(p)σ_r(^*)≤σ_r(^*)/8.We now bound the second term. Let =∑_k≥ 1(_Ω__Ω)^k, which is self-adjoint, and denote by N_n_1 and N_n_2 the 1/2-nets of 𝕊^n_1-1 and 𝕊^n_1-1 of sizes at most 6^n_1 and 6^n_2, respectively <cit.>. We know that [<cit.>, Lemma 5.4]((^*)) =sup_∈𝕊^n_2-1,∈𝕊^n_1-1⟨(^T),(^*)⟩≤ 4sup_∈ N_n_2,∈ N_n_1⟨(^T),(^*)⟩.Consider the random variable X(,)=⟨(^T),(^*)⟩ which has zero expectation. By Hoeffding's inequality, we have(|X(,)|>t)≤ 2exp(-t^2/2(^T)_F^2)≤ 2exp(-t^2/2^2).Therefore, by a union bound,(((^*))>t)≤ 2×6^n_1+n_2exp(-t^2/8^2).Note that conditioned on the event {_Ω_≤σ}, we have = ∑_k≥ 1(_Ω__Ω)^k≤σ^2/1-σ^2. So(λ((^*))>t)≤ 2×6^n_1+n_2exp(-t^2/8λ^2(1-σ^2/σ^2)^2)(_Ω_≤σ)+(_Ω_>σ).Lemma <ref> guarantees that event {_Ω_≤σ} holds with high probability for a very small absolute constant σ. Setting t=σ_r(^*)/8, this completes the proof of (d). Proof of (e). Recall that ^S=λ_^⊥∑_k≥ 0(_Ω__Ω)^k(^*) and so_Ω^⊥^S =λ_Ω^⊥(-_)∑_k≥ 0(_Ω__Ω)^k(^*)=-λ_Ω^⊥_∑_k≥ 0(_Ω__Ω)^k(^*).Then for any (i,j)∈Ω^⊥, we have_ij^S=⟨^S,_i_j^T⟩=⟨λ(^*),-∑_k≥ 0(_Ω__Ω)^k_Ω_(_i_j^T)⟩.Let X(i,j)=-∑_k≥ 0(_Ω__Ω)^k_Ω_(_i_j^T). By Hoeffding's inequality and a union bound,(sup_ij|_ij^S|>t)≤ 2∑_ijexp(-2t^2/λ^2X(i,j)_F^2).We note that conditioned on the event {_Ω_≤σ}, for any (i,j)∈Ω^⊥,X(i,j)_F ≤1/1-σ^2σ_(_i_j^T)_F≤1/1-σ^2σ√(1-_^⊥(_i_j^T)_F^2)= 1/1-σ^2σ√(1-(-^T)_i_2^2(-^T)_j_2^2)≤1/1-σ^2σ√(1-(1-μ r/n_(1))(1-μ r/n_(2)))≤1/1-σ^2σ√(μ r/n_(1)+μ r/n_(2)).Then unconditionally,(sup_ij|_ij^S|>t)≤ 2n_(1)n_(2)exp(-2t^2/λ^2(1-σ^2)^2n_(1)n_(2)/σ^2μ r(n_(1)+n_(2)))(_Ω_≤σ)+(_Ω_>σ).By Lemma <ref> and setting t=λ/4, the proof of (e) is completed.§ PROOF OF THEOREM <REF>Our computational lower bound for problem (P) assumes the hardness of random 4-SAT. Let c > ln 2 be a constant. Consider a random 4-SAT formula on n variables in which each clause has 4 literals, and in which each of the 16n^4 clauses is picked independently with probability c/n^3. Then any algorithm which always outputs 1 when the random formula is satisfiable, and outputs 0 with probability at least 1/2 when the random formula is unsatisfiable, must run in 2^c' n time on some input, where c' > 0 is an absolute constant. Based on Conjecture <ref>, we have the following computational lower bound for problem (P). We show that problem (P) is in general hard for deterministic algorithms. If we additionally assume BPP = P, then the same conclusion holds for randomized algorithms with high probability.Theorem <ref> (Computational Lower Bound. Restated). Assume Conjecture <ref>. Then there exists an absolute constant ϵ_0 > 0 for which any algorithm that achieves (1+ϵ) in objective function value for problemwith ϵ≤ϵ_0, and with constant probability, requires 2^Ω(n_1 + n_2) time, whereis the optimum. If in addition, BPP = P, then the same conclusion holds for randomized algorithms succeeding with probability at least 2/3. Theorem <ref> is proved by using the hypothesis that random 4-SAT is hard to show hardness of the Maximum Edge Biclique problem for deterministic algorithms. The problem is Input: An n-by-n bipartite graph G.Output: A k_1-by-k_2 complete bipartite subgraph of G, such that k_1 · k_2 is maximized.<cit.> showed that under the random 4-SAT assumption there exist two constants ϵ_1 > ϵ_2 > 0 such that no efficient deterministic algorithm is able to distinguish between bipartite graphs G(U, V,E) with |U| = |V | = n which have a clique of size ≥ (n/16)^2 (1+ϵ_1) and those in which all bipartite cliques are of size ≤ (n/16)^2(1 + ϵ_2). The reduction uses a bipartite graph G with at least t n^2 edges with large probability, for a constant t.Given a given bipartite graph G(U, V,E), define H(·) as follows. Define the matrixand : _ij = 1 if edge (U_i, V_j) ∈ E, _ij = 0 if edge (U_i, V_j) ∉E; _ij = 1 if edge (U_i, V_j) ∈ E, and _ij = poly(n) if edge (U_i, V_j) ∉E. Choose a large enough constant β>0 and let H() = β∑_ij_ij^2 (_ij - ()_ij)^2. Now, if there exists a biclique in G with at least (n/16)^2(1 + ϵ_2) edges, then the number of remaining edges is at most tn^2 - (n/16)^2(1 + ϵ_1), and so the solution to min H() + 1/2_F^2 has cost at most β[t n^2 - (n/16)^2(1 +ϵ_1) ] + n^2. On the other hand, if there does not exist a biclique that has more than (n/16)^2(1 + ϵ_2) edges, then the number of remaining edges is at least (n/16)^2(1 + ϵ_2), and so any solution to min H() + 1/2_F^2 has cost at least β[tn^2 - (n/16)^2 (1 + ϵ_2)]. Choose β large enough so that β[tn^2 - (n/16)^2 (1 + ϵ_2)] > β[t n^2 - (n/16)^2(1 +ϵ_1) ] + n^2. This combined with the result in <cit.> completes the proof for deterministic algorithms.To rule out randomized algorithms running in time 2^α(n_1 + n_2) for some function α of n_1, n_2 for which α = o(1), observe that we can define a new problem which is the same as problemexcept the input description of H is padded with a string of 1s of length 2^(α/2)(n_1 + n_2). This string is irrelevant for solving problembut changes the input size to N = (n_1, n_2) + 2^(α/2)(n_1+n_2). By the argument in the previous paragraph, any deterministic algorithm still requires 2^Ω(n) = N^ω(1) time to solve this problem, which is super-polynomial in the new input size N. However, if a randomized algorithm can solve it in 2^α(n_1+n_2) time, then it runs in (N) time. This contradicts the assumption that BPP = P. This completes the proof.§ DUAL AND BI-DUAL PROBLEMS In this section, we derive the dual and bi-dual problems of non-convex program (P). According to (<ref>), the primal problem (P) is equivalent tomin_,max_1/2--_F^2-1/2_F^2-H^*().Therefore, the dual problem is given bymax_min_,1/2--_F^2-1/2_F^2-H^*()=max_1/2∑_i=r+1^n_(2)σ_i^2(-)-1/2_F^2-H^*()=max_ -1/2_r^2-H^*(), (D1)where _r^2=∑_i=1^rσ_i^2(). The bi-dual problem is derived bymin_max_,' -1/2_r^2-H^*(')+⟨,'-⟩=min_max_-[⟨,-⟩-1/2-_r^2]+max_'[⟨,'⟩-H^*(')]=min__r*+H(), (D2)where _r*=max_⟨,⟩-1/2_r^2 is a convex function, and H()=max_'[⟨,'⟩-H^*(')] holds by the definition of conjugate function.Problems (D1) and (D2) can be solved efficiently due to their convexity. In particular, <cit.> provided a computationally efficient algorithm to compute the proximal operators of functions 1/2·_r^2 and ·_r*. Hence, the Douglas-Rachford algorithm can find the global minimum up to an ϵ error in function value in time (1/ϵ) <cit.>.§ RECOVERY UNDER BERNOULLI AND UNIFORM SAMPLING MODELSWe begin by arguing that a recovery result under the Bernoulli model with some probability automatically implies a corresponding result for the uniform model with at least the same probability. The argument follows Section 7.1 of <cit.>. For completeness, we provide the proof here.Denote by _𝖴𝗇𝗂𝖿(m) and _𝖡𝖾𝗋(p) probabilities calculated under the uniform and Bernoulli models and let “Success” be the event that the algorithm succeeds. We have_𝖡𝖾𝗋(p)(Success) =∑_k=0^n_1n_2_𝖡𝖾𝗋(p)(Success||Ω|=k)_𝖡𝖾𝗋(p)(|Ω|=k)≤∑_k=0^m_𝖴𝗇𝗂𝖿(k)(Success| |Ω|=k)_𝖡𝖾𝗋(p)(|Ω|=k)+∑_k=m+1^n_1n_2_𝖡𝖾𝗋(p)(|Ω|=k)≤_𝖴𝗇𝗂𝖿(m)(Success)+_𝖡𝖾𝗋(p)(|Ω|>m),where we have used the fact that for k≤ m, _𝖴𝗇𝗂𝖿(k)(Success)≤_𝖴𝗇𝗂𝖿(m)(Success), and that the conditional distribution of |Ω| is uniform. Thus_𝖴𝗇𝗂𝖿(m)(Success)≥_𝖡𝖾𝗋(p)(Success)-_𝖡𝖾𝗋(p)(|Ω|>m).Take p=m/(n_1n_2)-ϵ, where ϵ>0. The conclusion follows from _𝖡𝖾𝗋(p)(|Ω|>m)≤ e^-ϵ^2 n_1n_2/2p. Let n be the number of Bernoulli trials and suppose that Ω∼(m/n). Then with probability at least 1-n^-10, |Ω|=Θ(m), provided that m≥ clog n for an absolute constant c. By the scalar Chernoff bound, with ϵ>0 we have(|Ω|≤ m-nϵ)≤exp(-ϵ^2n^2/(2m)),and(|Ω|≥ m+nϵ)≤exp(-ϵ^2n^2/(3m)).Taking ϵ=m/(2n) and m≥ c_1log n in (<ref>) for an appropriate absolute constant c_1, we have(|Ω|≤ m/2)≤exp(-m/4)≤n^-10/2.Taking ϵ=m/n and m≥ c_2log n in (<ref>) for an appropriate absolute constant c_2, we have(|Ω|≥ 2m)≤exp(-m/3)≤n^-10/2.Given (<ref>) and (<ref>), we conclude that m/2<|Ω|<2m with probability at least 1-n^-10, provided that m≥ clog n for an absolute constant c. alpha | http://arxiv.org/abs/1704.08683v5 | {
"authors": [
"Maria-Florina Balcan",
"Yingyu Liang",
"David P. Woodruff",
"Hongyang Zhang"
],
"categories": [
"cs.DS",
"cs.LG",
"stat.ML"
],
"primary_category": "cs.DS",
"published": "20170427175446",
"title": "Matrix Completion and Related Problems via Strong Duality"
} |
http://arxiv.org/abs/1704.08305v1 | {
"authors": [
"Tobias Glasmachers"
],
"categories": [
"cs.LG",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170426191237",
"title": "Limits of End-to-End Learning"
} |
|
Institute of Natural Sciences and Mathematics,Ural Federal University, Ekaterinburg, Russia [email protected]@cs.uni.wroc.pl Institute of Computer Science,University of Wrocław, Wrocław, [email protected] Institute of Computer Science,University of Wrocław, Wrocław, Poland Given a subset of states S of a deterministic finite automaton and a word w, the preimage is the subset of all states mapped to a state in S by the action of w. We study three natural problems concerning words giving certain preimages. The first problem is whether, for a given subset, there exists a word extending the subset (giving a larger preimage). The second problem is whether there exists a totally extending word (giving the whole set of states as a preimage)—equivalently, whether there exists an avoiding word for the complementary subset. The third problem is whether there exists a resizing word. We also consider variants where the length of the word is upper bounded, where the size of the given subset is restricted, and where the automaton is strongly connected, synchronizing, or binary. We conclude with a summary of the complexities in all combinations of the cases.Keywords: avoiding word, extending word, extensible subset, reset word, synchronizing automatonPreimage problems for deterministic finite automata Marek Szykuła=================================================== § INTRODUCTION A deterministic finite complete (semi)automaton A is a triple (Q,Σ,δ), where Q is the set of states, Σ is the input alphabet, and δ Q ×Σ→ Q is the transition function. We extend δ to a function Q ×Σ^* → Q in the usual way. Throughout the paper, by n we always denote the number of states |Q|.When the context is clear, given a state q ∈ Q and a word w ∈Σ^*, we write shortly q· w for δ(q,w). Given a subset S ⊆ Q, the image of S under the action of a word w ∈Σ^* is S· w = δ(S,w) = {q· w | q ∈ S}. The preimage is S· w^-1 = δ^-1(S,w) = {q ∈ Q | q· w ∈ S}. If S={q}, then we usually simply write q· w^-1.We say that a word w compresses a subset S if |S· w| < |S|, avoids S if (Q· w) ∩ S = ∅, extends S if |S· w^-1| > |S|, and totally extends S if S· w^-1 = Q. A subset S is compressible, avoidable, extensible, and totally extensible, if there is a word that, respectively, compresses, avoids, extends and totally extends it. A word w ∈Σ^* is avoiding for S ⊆ Q if and only if w is totally extending for Q ∖ S. Fig. <ref> shows an example automaton. For S={2,3}, the shortest compressing word is aab, and we have {2,3}· aab = {1}, while the shortest extending word is ba, and we have {2,3}· (ba)^-1 = {1,2}· b^-1 = {1,2,4}.Note that the preimage of a subset under the action of a word can be smaller than the subset. In this case, we say that a word shrinks the subset (not to be confused with compressing when the image is considered). For example, in Fig. <ref>, subset {3,4} is shrank by b to subset {4}.Note that shrinking a subset is equivalent to extending its complement. Similarly, a word totally extending a subset also shrinks its complement to the empty set.|S· w^-1| > |S| if and only if |(Q ∖ S)· w^-1| < |Q ∖ S|, and S · w^-1 = Q if and only if (Q ∖ S)· w^-1 = ∅.Therefore, avoiding a subset is equivalent to shrinking it to the empty set.The rank of a word w is the cardinality of the image Q · w. A word of rank 1 is called reset or synchronizing, and an automaton that admits a reset word is called synchronizing. Also, for a subset S ⊆ Q, we say that a word w ∈Σ^* such that |S· w|=1 synchronizes S.Synchronizing automata serve as transparent and natural models of various systems in many applications in different fields, e.g., in coding theory <cit.>, model testing of reactive systems <cit.>, robotics <cit.>, and biocomputing <cit.>. They also reveal interesting connections with many parts of mathematics. For example, some of the recent works involve group theory <cit.>, representation theory <cit.>, computational complexity <cit.>, optimization and convex geometry <cit.>, regular languages and universality <cit.>, approximability <cit.>, primitive sets of matrices <cit.>, and graph theory <cit.>. For a brief introduction to the theory of synchronizing automata we refer the reader to an excellent, though quite outdated, survey <cit.>.The famous Černý conjecture <cit.>, which was formally stated in 1969 during a conference <cit.>, is one of the most longstanding open problems in automata theory. It states that a synchronizing automaton has a reset word of length at most (n-1)^2. The currently best upper bound is cubic and has been improved recently <cit.> (cf. <cit.>). Besides the conjecture, algorithmic issues are also important. Unfortunately, the problem of finding a shortest reset word is computationally hard <cit.>, and also its length approximation remains hard <cit.>. We also refer to surveys <cit.> dealing with algorithmic issues and the Černý conjecture.Compressing and extending a subset in general play a crucial role in the synchronization of automata and related areas. In fact, all known algorithms finding a reset word use finding words that either compresses or extends a subset as subprocedures (e.g. <cit.>). Moreover, probably all proofs of upper bounds on the length of the shortest reset words use bounds on the length of words that compress (e.g. <cit.>) or extend (e.g. <cit.>) some subsets.In this paper, we study several problems about finding a word yielding a certain preimage. We provide a systematic view of their computational complexity in various combinations of cases. §.§ Compressing a subsetThe complexities of problems related to images of a subset have been well studied. It is known that given an automaton A and a subset S ⊆ Q, determining whether there is a word that synchronizes it is PSPACE-complete <cit.>. The same holds even for strongly connected binary automata <cit.>.On the other hand, checking whether the automaton is synchronizing, i.e. whether there is a word that synchronizes Q, can be solved in Ø(|Σ|n^2) time and space <cit.> and in Ø(n) average time and space when the automaton is randomly chosen <cit.>. To this end, we verify whether all pairs of states are compressible. Using the same algorithm, we can determine whether a given subset is compressible.Deciding whether there exists a synchronizing word of a given length is NP-complete <cit.> (cf. <cit.> for the complexity of the corresponding functional problems), even if the given automaton is binary. The NP-completeness holds even when the automaton is Eulerian and binary <cit.>, which immediately implies that for the class of strongly connected automata the complexity is the same.However, deciding whether there exists a word of a given length that only compresses a subset still can be solved in Ø(|Σ|n^2) time, as for every pair of states we can compute a shortest word that compresses the pair.The problems related to images have been also studied in other settings for both complexity and the bounds on the length of the shortest words, for example, in the case of a nondeterministic automaton <cit.>, in the case of a partial deterministic finite automaton <cit.>, in the partial observability setting for various kinds of automata <cit.>, and for the reachability of a given subset in the case of a deterministic finite automaton <cit.>. §.§ Extending a subset and our contributions In contrast to the problems related to images (compression), the complexity of the problems related to preimages has not been thoroughly studied in the literature. In the paper, we fill this gap and give a comprehensive analysis of all basic cases. We study three families of problems. As noted before, extending is equivalent to shrinking the complementary subset, hence we need to deal only with the extending word problems. Similarly, totally extending words are equivalent to avoiding the complement, thus we do not need to consider avoiding a set of states separately.Extending words: Our first family of problems is the question whether there exists an extending word (Problems <ref>, <ref>, <ref>, <ref>, <ref>, <ref> in this paper).This is motivated by the fact that finding such a word is the basic step of the so-called extension method of finding a reset word, which is used in many proofs and also some algorithms. The extension method of finding a reset word is as follows: we start from some singleton S_0 ={q} and iteratively find extending words w_1,…,w_k such that |S_0 · w_1^-1⋯ w_i^-1| > |S_0 · w_1^-1⋯ w_i-1^-1| for 1 ≤ i ≤ k, and where S_0 · w_1^-1⋯ w_k^-1 = Q. For finding a short reset word one needs to bound the lengths of the extending words. For instance, in the case of synchronizing Eulerian automata, the fact that there always exists an extending word of length at most n-1 implies the upper bound (n-2)(n-1)+1 on the length of the shortest reset words for this class <cit.> (the first extending step requires just one letter, as we can choose an arbitrary singleton). In this case, a polynomial algorithm for finding extending words has been proposed <cit.>.Totally extending words and avoiding: We study the problem whether there exists a totally extending word (Problems <ref>, <ref>, <ref>, <ref>, <ref>, <ref> in this paper). The question of the existence of a totally extending word is equivalent to the question of the existence of an avoiding word for the complementary subset.Totally extending words themselves can be viewed as a generalization of reset words: a word totally extending a singleton to the whole set of states Q is a reset word. If we are not interested in bringing the automaton into one particular state but want it to be in any of the states from a specified subset, then it is exactly the question about totally extending word for our subset. In view of applications of synchronization, this can be particularly useful when we deal with non-synchronizing automata, where reset words cannot be applied.Avoiding word problem is a recent concept that is dual to synchronization: instead of being in some states, we want not to be in them. A quadratic upper bound on the length of the shortest avoiding words of a single state has been established <cit.>, which led to an improvement of the best known upper bound on the length of the shortest reset words (see also <cit.> for a very recent improvement of that improvement of the upper bound). Furthermore, better upper bounds on the length of the shortest avoiding words would lead to further improvements; in particular, a subquadratic upper bound implies the upper bound on the reset threshold equal to 7n^3/48+o(n^3) <cit.>. There is a precise conjecture that the shortest avoiding words have length at most 2n-2 <cit.>. The computational complexity of the problems related to avoiding, both a single state or a subset, has not been established before. We give a special attention to the problem of avoiding one state and a small subset of states (totally extending a large subset), as since they seem to be most important in view of their applications (and as we show, the complexity grows with the size of the subset to avoid).Resizing: Shrinking a subset is dual to extending, i.e. shrinking a subset means extending its complement. Therefore, the complexity immediately transfers from the previous results. However, in Section <ref> we consider the problem of determining whether there is a word whose inverse action results in a subset having a different size, that is, either extends the subset or shrinks it (Problems <ref>, <ref>).Interestingly, in contrast with the computationally difficult problems of finding a word that extends the subset and finding a word that shrinks the subset, for this variant there exists a polynomial algorithm finding a shortest resizing word in all cases.We can mention that in some cases extending and shrinking words are related, and it may be enough to find either one. For instance, this is used in the so-called averaging trick, which appears in several proofs <cit.>.Summary: For all the problems we consider the subclasses of strongly connected, synchronizing, and binary automata. Also, we consider the problems where an upper bound on the length of the word is additionally given in a binary form in the input. Since, in most cases, the problems are computationally hard, in Section <ref> and Section <ref>, we consider the complexity parameterized by the size of the given subset.Table <ref> and Table <ref> summarize our results together with known results about compressing words. For the cases where a polynomial algorithm exists, we put the time complexity of the best one known. All the hardness results hold also in the case of a binary alphabet. § EXTENDING A SUBSET IN GENERAL§.§ Unbounded word length In the first studied case, we do not have any restriction on the given subset S neither on the length of the extending word. We deal with the following problems: [Extensible subset] Given A=(Q,Σ,δ) and a subset S ⊆ Q, is S extensible?[Totally extensible subset] Given A=(Q,Σ,δ) and a subset S ⊆ Q, is S totally extensible?Problem <ref> and Problem <ref> are PSPACE-complete, even if A is strongly connected. To solve one of the problems in NPSPACE, we guess the length of a word w with the required property, and then guess the letters of w from the end. Of course, we do not store w, which may have exponential length, but just keep the subset S· u^-1, where u is the current suffix of w. The current subset can be stored in Ø(n), and since there are 2^n different subsets, |w| ≤ 2^n and the current length also can be stored in Ø(n). By Savitch's theorem, the problems are in PSPACE.For PSPACE-hardness, we construct a reduction from the problem of determining whether an intersection of regular languages given as DFAs is non-empty. We create one instance for both problems that consists of a strongly connected automaton and a subset S extensible if and only if it is also totally extensible, which is simultaneously equivalent to the non-emptiness of the intersection of the given regular languages.Let (𝒟_i)_i ∈{1,…,m} be the given sequence of DFAs with an i-th automaton 𝒟_i=(Q_i,Σ,δ_i,s_i,F_i) recognizing a language L_i, where Q_i is the set of states, Σ is the common alphabet, δ_i is the transition function, s_i is the initial state, and F_i is the set of final states. The problem whether there exists a word accepted by all 𝒟_1,…,𝒟_m (i.e. the intersection of L_i is non-empty) is a well known PSPACE-complete problem, called Finite Automata Intersection <cit.>. We can assume that the DFAs are minimal; in particular, they do not have unreachable states from the initial state, otherwise, we may easily remove them in polynomial time.For each 𝒟_i we choose an arbitrary f_i ∈ F_i. Let M = ∑_i=1^m |Q_i|. We construct the (semi)automaton 𝒟'=(Q',Σ',δ')and define S ⊆ Q' as an instance of our both problems. The scheme of the automaton is shown in Fig. <ref>.* For i ∈{0,1,…,m}, let Γ_i = {f_i}×{0,…,2M-1} be fresh states and let Q_i' = (Q_i ∖{f_i}) ∪Γ_i. Let Q_0'={s_0,t_0}∪Γ_0, where s_0 and t_0 are fresh states. Then Q' = ⋃_i=0^m Q_i'.* Σ' = Σ∪{α,β}, where α and β are fresh letters. * δ' is defined by: * For q ∈ Q_i ∖{f_i} and a ∈Σ, we haveδ'(q,a)=δ_i(q,a)if δ_i(q,a) ≠ f_i,(f_i,0)otherwise. * For a ∈Σ, we haveδ'(t_0,a)=t_0,δ'(s_0,a)=s_0. * For k ∈{0,…,2M-1}, i ∈{1,…,m}, and a ∈Σ, we haveδ'((f_0,k),a) = t_0,δ'((f_i,k),a) = δ_i(f_i,a)if δ_i( f_i,a) ≠ f_i,(f_i,0)otherwise. * For q ∈ Q'_i, we haveδ'(q,α)=s_(i+1)(m+1). * For i ∈{0,…,m} and k ∈{0,…,2M-1}, we haveδ'((f_i,k),β) = (f_i,k+12M). * We haveδ'(s_0,β) = (f_0,0). * For the remaining states q ∈ Q' ∖ (⋃_i=0^mΓ_i ∪{s_0}), we haveδ'(q,β)=q.* The subset S ⊆ Q' is defined asS=(⋃_i=1^m F_i∩ Q') ∪⋃_i=0^mΓ_i ∪{s_0}.It is easy to observe that 𝒟' is strongly connected. Take any i,j ∈{0,…,m}. We show how to reach any state q ∈ Q_j' from a state p ∈ Q_i'. First, we can reach s_j by α^(m+1+j-i)(m+1). For j ≥ 1, each state q ∈ Q_j' ∖(Γ_j ∖{(f_j,0)}) is reachable from s_j, since δ' restricted to Σ acts on Q_j' as δ_j on Q_j (with f_j replaced by (f_j,0)) and 𝒟_j is minimal.For j=0, states (f_0,0) and t_0 are reachable from s_0 by the transformations of β and β a respectively, for any a∈Σ. States q ∈Γ_j can be reached from (f_j,0) using δ_β.We will show the following statements: (1) If S is extensible in 𝒟', then the intersection of the languages L_i is non-empty.(2) If the intersection of the languages L_i is non-empty, then S is extensible to Q' in 𝒟'.This will prove that the intersection of the languages L_i is non-empty if and only if S is extensible, which is also equivalent to that S is extensible toQ'.(1): Observe that, for each i ∈{0,…,m}, if (S · w^-1) ∩Γ_i ≠∅, then (S · w^-1) ∩Γ_i = Γ_i. This follows by induction: the empty word possesses this property; the transformation δ_a of a ∈Σ∖{β} maps every state from Γ_i to the same state, so it preserves the property; δ_β acts cyclically on Γ_i so also preserves the property.Suppose that S is extensible by a word w. Notice that, M is an upper bound on the number of states in Q' ∖⋃_i=0^m Γ_i (for m ≥ 2). We also have |S| ≥ 1+(m+1)· 2M. We conclude that Γ_i ⊆ S · w^-1 for each i ∈{0,…,m}, since|Q' ∖Γ_i| ≤ m · 2M + M ≤ (m+1) · 2M < |S|,so (S · w^-1) ∩Γ_i ≠∅ and then our previous observation Γ_i ⊆ S · w^-1.Now, the extending word w must contain the letter α. For a contradiction, if w ∈ (Σ' ∖{α})^*, then if it contains a letter a ∈Σ, then S· w^-1 does not contain any state from Γ_0 ∪{t_0}, as the only outgoing edges from this subset are labeled by α, t_0 ∉ S, Γ_0 ·β^-1 = Γ_0, and Γ_0 · a^-1 = ∅. This contradicts the previous paragraph. Also, w cannot be of the form β^k, for k ∈ℕ, since S ·β^k = S. Hence, w = w_p α w_s, where w_p ∈ (Σ')^* and w_s ∈ (Σ' ∖{α}). Note that if T is a subset of Q' such that T ∩ Q'_i = ∅ for some i, then also (T· u^-1) ∩ Q'_i' = ∅ for every word u and some i'; because only α maps states Q_i outside Q_i, and it acts cyclically on these sets.Hence, in this case, every preimage of T does not contain some Γ_i' set. So {s_i | i ∈{0,⋯,m}}⊆ S · (w_s)^-1, since in the opposite case (S · (α w_s)^-1) ∩ Q'_i = ∅ for some i.Let w'_s be the word obtained by removing all β letters from w_s. Note that, for every i ∈{1,…,m} and every suffix u of w_s, we have (S · u^-1) ∩ Q'_i = (S · (β u)^-1) ∩ Q'_i. Hence, (S · w_s^-1) ∩ (Q' ∖ Q'_0) = S · (w'_s)^-1∩ (Q' ∖ Q'_0).Now, the word w'_s is in Σ^*, and S · w_s^-1 contains s_i for all i ∈{1,…,m}. Hence, the action of w'_s maps s_i to either a state in F_i ∖{f_i} or (f_i,0), which means that w'_s maps s_i to F_i in 𝒟_i. Therefore, w'_s is in the intersection of the languages L_i.(2): Suppose that the intersection of the languages L_i is non-empty, so there exists a word w ∈Σ^* such that s_i · w ∈ F_i for every i. Then we have S · (α w)^-1 = Q', thus S is extensible to Q'. We ensure that both problems remain PSPACE-complete in the case of a binary alphabet, which follows from the following theorem.Given an automaton A=(Q,Σ,δ) and a subset S ⊆ Q, we can construct in polynomial time a binary automaton A'=(Q',{a',b'},δ') and a subset S' ⊆ Q' such that: (1) A is strongly connected if and only if A' is strongly connected;(2) S' is extensible in A' if and only if S is extensible in A;(3) S' is totally extensible in A' if and only if S is totally extensible in A.Let Σ = {a_0,…,a_k-1}. The idea is as follows: We reduce A to a binary automaton A' that consists of k copies of A. The first letter a acts in an i-th copy as the letter a_i in A. The second letter b acts cyclically on these copies. Then we define S' to contain states from S in the first copy and all states from the other copies. The construction is shown in Fig. <ref>.We construct A'=(Q',{a',b'},δ') with Q' = Q ×Σ and δ' defined as follows: δ'((q,a_i),a') = (δ(q,a_i),a_i), and δ'((q,a_i),b') = (q,a_(i+1)k). Clearly, A' can be constructed in Ø(nk) time, where k=|Σ|.(1): Suppose that A is strongly connected; we will show that A' is also strongly connected. Let (q_1,a_i) and (q_2,a_j) be any two states of A'. In A, there is a word w such that q_1· w=q_2. Let w' be the word obtained from w by replacing every letter a_h by the word (b')^ha'(b')^k-h. Note that in A' we have(p,a_0)· (b')^ha'(b')^k-h = (p· a_h,a_0),hence (q_1,a_0)· w' = (q_1 · w,a_0). Then the action of the word (b')^k-i w' (b')^j maps (q_1,a_i) to (q_2,a_j).Conversely, suppose that A' is strongly connected, so every (q_1,a_i) can be mapped to every (q_2,a_j) by the action of a word w'. Thenw' = (b')^h_1a' … (b')^h_m-1a' (b')^h_m,for some m ≥ 1 and h_1,…,h_m ≥ 0. We construct w of length m-1, where the s-th letter is a_r with r=(i+Σ_j=1^s h_j)k.Then w maps q_1 to q_2 in A.(2) and (3): For i ∈{0,…,k-1} we define U_i = (Q ×{Σ∖{a_i}}). Observe that for any word u' ∈{a',b'}^*, we have U_i· (u')^-1 = U_j for some j, which depends on i and the number of letters b' in u'.We defineS' = (S ×{a_0}) ∪ U_0.Suppose that S is extensible in A by a word w, and let w' be the word obtained from w as in (1). Then (w')^-1 maps U_0 to U_0, and (S ×{a_0}) to (S · w^-1) ×{a_0}). We have:S' (w')^-1 = ((S· w^-1) ×{a_0}) ∪ U_0,and since |S· w^-1|>|S|, this means that w' extends S'. By the same argument, if w extends S to Q, then w' extends S' to Q'.Conversely, suppose that S' is extensible in A' by a word w', and let w be the word obtained from w' as in (1). Then, for some i, we haveS'· (w')^-1 = ((S· w^-1) ×{a_i}) ∪ U_i,and since |U_0|=|U_i| it must be that |S· w^-1|>|S|. Also, if S'· (w')^-1 = Q' then S· w^-1=Q. Now, we consider the subclass of synchronizing automata. We show that synchronizability does not change the complexity of the first problem, whereas the second problem becomes much easier. When the automaton is binary and synchronizing, Problem <ref> remains PSPACE-complete. From Theorem <ref>, Problem <ref> is in PSPACE, as the algorithm works the same in the restricted case.Problem <ref> for binary and synchronizing automata is PSPACE-hard, as any general instance with a binary automaton A=(Q,{a,b},δ) can be reduced to an equivalent instance with a binary synchronizing automaton A'. For this, we just add a sink state s and a letter which synchronizes Q to s. Additionally, a standard tree-like binarization is suitably used to obtain a binary automaton A'.Formally, we construct a synchronizing binary automaton A' from the binary automaton A as follows. We can assume that Q = {q_1,…,q_n}. Let s be a fresh state. Let Q' = Q ∪{q^a_1,…,q^a_n}. We construct A'=(Q' ∪{s},{a,b},δ'), where δ' for all i is defined as follows: δ'(q_i,a)=q^a_i, δ'(q_i,b)=s, δ'(q^a_i,a)=δ(q,a), and δ'(q^a_i,b) = δ(q,b). Then bb is a synchronizing word for A', and each S ⊆ Q is extensible in A' if and only if it is extensible in A.When the automaton is synchronizing, Problem <ref> can be solved in Ø(|Σ|n) time and it is NL-complete. Since A is synchronizing, Problem <ref> reduces to checking whether there is a state q ∈ S reachable from every state: It is well known that a synchronizing automaton has precisely one strongly connected sink component that is reachable from every state. If w is a reset word that synchronizes Q to p, and u is such that p · u = q, then wu extends {q} to Q. If S does not contain a state from the sink component, then every preimage of S also does not contain these states.The problem can be solved in Ø(|Σ|n) time, since the states of the sink component can be determined in linear time by Tarjan's algorithm <cit.>.It is also easy to see that the problem is in NL: Guess a state q ∈ S and verify in logarithmic space that it is reachable from every state.For NL-hardness, we reduce from ST-connectivity: Given a graph G=(V,E) and vertices s,t, check whether there is a path from s to t. We will output a synchronizing automaton A=(V,Σ,δ) and S ⊆ Q such that S is extensible to Q if and only if there is a path from s to t in G.First, we compute the maximum out-degree of G, and set Σ = Σ' ∪{α}, where |Σ'| is equal to the maximum out-degree. We output A such that for every q ∈ V, every edge (q,p) ∈ E is colored by a different letter from Σ'. If there is no outgoing edge from q, then we set the transitions of all letters from Σ' to be loops. If the out-degree is smaller than |Σ'|, then we simply repeat the transition of the last letter. Next, we define δ(q,α) = s for every q ∈ V. Finally, let S = {t}. The reduction uses logarithmic space since it requires only counting and enumerating through V and Σ'. The produced automaton A is synchronizing just by α.Suppose that there is a path from s to t. Then there is a word w such that δ(s,w)=t, and so {t}· (α w)^-1 = Q.Suppose that {t} is extensible to Q by some word w. Let w' be the longest suffix of w that does not contain α. Since α^-1 results in ∅ for any subset not containing s, it must be that s ∈{t} (w')^-1. Hence δ(s,w') = t, and the path labeled by w' is the path from s to t in G. Note that in the case of strongly connected synchronizing automaton, both problems have a trivial solution, since every non-empty proper subset of Q is totally extensible (by a suitable reset word); thus they can be solved in constant time, assuming that we can check the size of the given subset and the number of states in constant time. §.§ Bounded word length We turn our attention to the variants in which an upper bound on the length of word w is also given.[Extensible subset by short word] Given A=(Q,Σ,δ), a subset S ⊆ Q, and an integer ℓ given in binary representation, is S extensible by a word of length at most ℓ? [Totally extensible subset by short word] Given A=(Q,Σ,δ), a subset S ⊆ Q, and an integer ℓ given in binary representation, is S totally extensible by a word of length at most ℓ? Obviously, these problems remain PSPACE-complete (also when the automaton is strongly connected and binary), as we can set ℓ=2^n, which bounds the number of different subsets of Q. In this case, both the problems are reduced respectively to Problem <ref> and Problem <ref>.When the automaton is synchronizing, Problem <ref> is NP-complete, which will be shown in Corollary <ref>. Of course, Problem <ref> remains PSPACE-complete for a synchronizing automaton by the same argument as in the general case. § EXTENDING SMALL SUBSETS The complexity of the extending problems is caused by an unbounded size of the given subset. Note that in the proof of PSPACE-hardness in Theorem <ref> the used subsets and simultaneously their complements may grow with an instance of the reduced problem, and it is known that the problem of the emptiness of intersection can be solved in polynomial time if the number of given DFAs is fixed. Here, we study the computational complexity of the extending problems when the size of the subset is not larger than a fixed k. §.§ Unbounded word length [Extensible small subset] For a fixed k ∈ℕ∖{0}, given A=(Q,Σ,δ) and a subset S ⊆ Q with |S| ≤ k, is S extensible?Problem <ref> can be solved in Ø(|Σ|n^k) time. We build the k-subsets automaton A^≤ k=(Q^≤ k,Σ,δ^≤ k, S_0, F), where Q^≤ k={A ⊆ Q|A| ≤ k} and δ^≤ k is naturally defined by the image of δ on a subset. Let the set of initial states be I={A ∈ Q^≤ k |A · a^-1| > |S|for somea ∈Σ}, and the set of final states be the set of all subsets of S. A final state can be reached from an initial state if and only if S is extensible in A. We can simply check this condition by a BFS algorithm.Note that we can compute whether a subset A of size at most k is in I in Ø(|Σ|), by summing the sizes |q· a^-1| for all q ∈ A, where |q· a^-1| are computed during a preprocessing, which takes O(n) time for a single a ∈Σ. Also, for a given subset A of size at most k, we can compute T· a in constant time (which depends only k). Hence, the BFS works in linear time in the size of A^≤ k, so in O(|Σ|n^k) time. [Totally extensible small subset] For a fixed k ∈ℕ∖{0}, given A=(Q,Σ,δ) and a subset S ⊆ Q with |S| ≤ k, is S totally extensible? For k=1, Problem <ref> is equivalent to checking if the automaton is synchronizing to the given state, thus can be solved in Ø(|Σ|n^2) time. For larger k we have the following: Problem <ref> can be solved in Ø(|Σ|n^k + n^3) time. Let u be a word of the minimal rank in A. We can find such a word and compute the image Q· u in Ø(n^3+|Σ|n^2) time, using the well-known algorithm <cit.> generalized to non-synchronizing automata. The algorithm just stops when there are no more compressible pairs of states contained in the current subset, and since the subset cannot be further compressed, the found word has the minimal rank.For each w ∈Σ^* we have S· w^-1=Q if and only if Q · w ⊆ S. We can meet the required condition for w if and only if (Q· u) · w ⊆ S. Surely |(Q· u) · w|=|Q· u|. The desired word does not exist if the minimal rank is larger than |S|=k. Otherwise, we can build the subset automaton A^≤ |Q · u| (similarly as in the proof of Proposition <ref>). The initial subset is Q · u. If some subset of S is reachable by a word w, then the word uw totally extends S in A. Otherwise, S is not totally extensible. The reachability can be checked in at most Ø(|Σ| n^k) time. However, if the rank r of u is less than k, the algorithm takes only Ø(|Σ|n^r) time.§.§ Bounded word length We also have the two variants of the above problems when an upper bound on the length of the word is additionally given.[Extensible small subset by short word] For a fixed k ∈ℕ∖{0}, given A=(Q,Σ,δ), a subset S ⊆ Q with |S| ≤ k, and an integer ℓ given in binary representation, is S extensible by a word of length at most ℓ? Problem <ref> can be solved by the same algorithm in a Proposition <ref>, since the procedure can find a shortest extending word.[Totally extensible small subset by short word] For a fixed k ∈ℕ∖{0}, given A=(Q,Σ,δ), a subset S ⊆ Q with |S| ≤ k, and an integer ℓ given in binary representation, is S totally extensible by a word of length at most ℓ?For every k, Problem <ref> is NP-complete, even if the automaton is simultaneously strongly connected, synchronizing, and binary. The problem is in NP, as the shortest extending words have length at most Ø(n^3+n^k) (since words of this length can be found by the procedure from Proposition <ref>).When we choose S of size 1, the problem is equivalent to finding a reset word that maps every state to the state in S. In <cit.> it has been shown that for Eulerian automata that are simultaneously strongly connected, synchronizing, and binary, deciding whether there is a reset word of length at most ℓ is NP-complete. Moreover, in this construction, if there exists a reset word of this length, then it maps every state to one particular state s_2 (see <cit.>). Therefore, we can set S={s_2}, and thus Problem <ref> is NP-complete.§ EXTENDING LARGE SUBSETS In this section, we consider the case where the subset S contains all except at most a fixed number of states k. §.§ Unbounded word length [Extensible large subset] For a fixed k ∈ℕ∖{0}, given A=(Q,Σ,δ) and a subset S ⊆ Q with |Q∖ S| ≤ k, is S extensible? [Totally extensible large subset] For a fixed k ∈ℕ∖{0}, given A=(Q,Σ,δ) and a subset S ⊆ Q with |Q∖ S| ≤ k, is S totally extensible? Problem <ref> is equivalent to deciding the existence of an avoiding word for a subset S of size ≤ k. Note that Problem <ref> and Problem <ref> are equivalent for k=1, when they become the problem of avoiding a single given state. Its properties will also turn out to be different than in the case of k ≥ 2. We give a special attention to this problem, defined as follows, and study it separately.[Avoidable state] Given A=(Q,Σ,δ) and a state q ∈ Q, is {q} avoidable? The following result may be a bit surprising, in view of that it is the only case where a general problem (i.e., Problems <ref> and <ref>) remains equally hard when the subset size is additionally bounded. We show that Problem <ref> is PSPACE-complete for all k ≥ 2, although the question about its complexity remains open for the class of strongly connected automata. Problem <ref> is PSPACE-complete for every fixed k ≥ 2, even if the given automaton is synchronizing and binary. Problem <ref> is in PSPACE as a special case of Problem <ref>, which is PSPACE-complete (Thm. <ref>).Now, we show a reduction from Problem <ref>. The idea is as follows. We construct an automaton A' from the automaton A=(Q,Σ,δ) given for Problem <ref>. We add two new states, e and s, and let the initial set S' contain all the original states of A. State s is a sink state ensuring that the automaton is synchronizing; it cannot be reached from S' by inverse transitions. Hence, to extend S', one needs to get e, which is doable only by a new special letter α. This letter has the transition that shrinks all states Q to the initial subset S for the totally extensible problem. This is done through an arbitrary selected state f ∈ Q. Then we can reach Q ∪{e} only by a totally extending word for A. The overall construction is presented in Fig. <ref>.Let A=(Q,Σ,δ) and S⊆ Q be an instance of Problem <ref>. We construct an automaton A'=(Q'=Q ∪{e,s},Σ'=Σ∪{α},δ'), where e,s are fresh states and α is a fresh letter. Let f be an arbitrary state from Q ∖ S (if S=Q then the problem is trivial). We define δ' as follows: * δ'(q,a)=δ(q,a) for q ∈ Q, a ∈Σ;* δ'(q,a)=q for q ∈{e,s}, a ∈Σ;* δ'(q,α)=f for q ∈ S ∪{e};* δ'(q,α)=s for q ∈ (Q ∪{s}) ∖ S.We define S'=Q. Note that |Q'∖ S'|=2, and hence automaton A' with S' is an instance of Problem <ref> for k=2. We will show that S' is extensible in A' if and only if S is totally extensible in A.If S is totally extensible in A by a word w ∈Σ^*, we have S' ·(wα)^-1=Q∖{s}, which means that S' is extensible in A'.Conversely, if S' is extensible in A', then there is some extending word of the form wα for some w ∈Σ^*, because S'· a^-1=S' for a∈Σ, (Q' ∖{s}) ·α^-1⊆ S'·α^-1, and each reachable set (as a preimage) is a subset of Q' ∖{s}. We know that S' · (wα)^-1=(S ∪{e}) · w^-1 = (S · w^-1) ∪{e}. From the fact that |S' · (wα)^-1| > |S'|, we conclude that S · w^-1 = Q, so S is totally extensible in A.Note that A' is synchronizing, since Q' ·α^2={f,s}·α = {s}.Now, we show that we can reduce the alphabet to two letters. Consider the application of the Theorem <ref> to Problem <ref>.Note that the reduction in the proof keeps the size of complement set the same (i.e. |Q' ∖ S'| = |Q”∖ S”|, where Q” and S” are the set and the subset of states in the constructed binary automaton), so we can apply it.Furthermore, we identify all the states of the form (s,a) for a ∈Σ in the obtained binary automaton to one sink state s”. In this way, we get a synchronizing binary automaton (since A' is synchronizing). The extending words remain the same, since the identified state s” is not reversely reachable from S”, and s” is not contained in the subset S”.Finally, we conclude that the proof generalizes to the case of any k ≥ 2 since we can add an arbitrary number of states with the same transitions as e. Now, we focus on totally extending words for large subsets, which we study in terms of avoiding small subsets. First we provide a complete characterization of single states that are avoidable:Let A=(Q,Σ,δ) be a strongly connected automaton. For every q ∈ Q, state q is avoidable if and only if there exists p ∈ Q ∖{q} and w ∈Σ^* such that q· w=p· w. First, for a given q ∈ Q, let p ∈ Q ∖{q} and w ∈Σ^* be such that q· w = p· w. Since the automaton is strongly connected, there is a word w' such that (p· w)· w'=(q· w)· w'=p. For each subset S ⊆ Q such that p ∈ S we have p ∈ S· ww'.Moreover, if q ∈ S then |S· ww'| < |S|, because {q,p}· ww'={p}.If q is not avoidable, then all subsets Q· (ww'),Q· (ww')^2,… contain q and they form an infinite sequence of subsets of decreasing cardinality, which is a contradiction.Now, consider the other direction. Suppose for a contradiction that a state q ∈ Q is avoidable, but there is no state p ∈ Q ∖{q} such that {q,p} can be compressed. Let u be a word of the minimal rank in A, and v be a word that avoids q. Then w=uv has the same rank and also avoids q. Let ∼ be the equivalence relation on Q defined with a word w as follows:p_1∼ p_2p_1 · w = p_2 · w.The equivalence class [p]_∼ for p ∈ Q is (p· w)· w^-1. There are |Q/∼|=|Q· w| equivalence classes and one of them is {q}, since q does not belong to a compressible pair of states. For every state p ∈ Q, we know that |(Q· w) ∩ [p]_∼| ≤ 1, because [p]_∼ is compressed by w to a singleton and Q· w cannot be compressed by any word. Note that every state r ∈ Q· w belongs to some class [p]_∼. From the equality |Q/∼|=|Q· w| we conclude that for every class [p]_∼ there is a state r ∈ (Q· w) ∩ [p]_∼, thus |(Q· w) ∩ [p]_∼| = 1. In particular, 1=|(Q· w) ∩ [q]_∼|=|(Q· w) ∩{q}|. This contradicts that w avoids q. Note that if A is not strongly connected, then every state from a strongly connected component that is not a sink can be avoided. If a state belongs to a sink component, then we can consider the sub-automaton of this sink component, and by Theorem <ref> we know that given q ∈ Q, it is sufficient to check whether q belongs to a compressible pair of states. Hence, Problem <ref> can be solved using the well-known algorithm (stage 1 in the proof of <cit.>) computing the pair automaton and performing a breadth-first search with inverse edges on the pairs of states. It works in Ø(|Σ|n^2) time and Ø(n^2+|Σ|n) space.We note that in a synchronizing automaton all states are avoidable except a sink state, which is a state q such that q· a=q for all a ∈Σ. We can check this condition and hence verify if a state is avoidable in a synchronizing automaton in Ø(|Σ|) time.The above algorithm does not find an avoiding word but checks avoidability indirectly. For larger subsets than singletons, we construct another algorithm finding a word avoiding the subset, which also generalizes the idea from Theorem <ref>. From the following theorem, we obtain that Problem <ref> for a constant k ≥ 2 can be solved in polynomial time.Let A=(Q,Σ,δ), let r be the minimum rank in A over all words, and let S ⊆ Q be a subset of size ≤ k. We can find a word w such that (Q· w) ∩ S = ∅ or verify that it does not exist in Ø(|Σ|(n^min(r,k)+n^2)+n^3) time and Ø(n^min(r,k)+n^2+|Σ|n) space. Moreover the length of w is bounded by Ø(n^min(r,k)+n^3)). Similarly to the proof of Theorem <ref>, let u be a word of the minimal rank r in A and let ∼ be the equivalence relation on Q defined by word u as follows:p_1 ∼ p_2p_1 · u = p_2 · u.The equivalence class [p]_∼ for p ∈ Q is the set (p· u)· u^-1. There are |Q/∼|=|Q· u| equivalence classes.First, we prove a key observation that the image of each word starting with prefix u has exactly one state in each equivalence class of ∼ relation. Let w=uw'. Then the word w has rank r and its image is not compressible. For every state p ∈ Q, we know that |(Q· w) ∩ [p]_∼| ≤ 1, because [p]_∼ is compressed by u to a singleton and Q· w cannot be compressed by any word. Note that every state q ∈ Q· w belongs to some class [p]_∼. From the equality |Q/∼|=|Q· u|=|Q· w| we conclude that for every class [p]_∼ there is an unique state q_[p]_∼∈ (Q· w) ∩ [p]_∼. This proves the mentioned observation.Now, we are going to show the following characterization: S is avoidable if and only if there exist a subset Q' ⊆ Q· u of size |S/∼| and a word w' such that (Q' · w') ∩ ([s]_∼∖ S) ≠∅ for each s ∈ S. The idea of the characterization is illustrated in Fig. <ref>.Suppose that S is avoidable, and let w' be an avoiding word for S. Then the word w=uw' also avoids S. Observe that Q · w has an unique state q_[p]_∼∈ (Q· w) ∩ [p]_∼ for each class [p]_∼. Then for every state s ∈ S, we have q_[s]_∼∈ [s]_∼∖ S, because w avoids S and q_[s]_∼∈ Q· w. Notice that [s]_∼∩ S can contain more than one state, so the set {q_[s]_∼| s ∈ S} has size |S/∼|, which is not always equal to |S|. Therefore, there exists a subset Q' ⊆ Q· u of size |S/∼| such that Q'· w' = {q_[s]_∼| s ∈ S}. Now, we know that for every s ∈ S we have q_[s]_∼∈ Q' · w' and q_[s]_∼∈ [s]_∼∖ S. We conclude that, if S is avoidable, then there exist a subset Q' ⊆ Q· u of size |S/∼| and a word w' such that (Q' · w') ∩ ([s]_∼∖ S) ≠∅ for every s ∈ S. Conversely, suppose that there is a subset Q' ⊆ Q· u of size |S/∼| and a word w' such that (Q' · w') ∩ ([s]_∼∖ S) ≠∅ for every s ∈ S. Since in the image Q · uw' there is exactly one state in each equivalence class, we have ((Q· u) ∖ Q') · w' ⊆ Q ∖⋃_s ∈ S([s]_∼) ⊆ Q ∖ S, and by the assumption, (Q' · w') ∩ S = ∅. Therefore, we get that uw' is an avoiding word for S.This characterization gives us Alg. <ref> to find w or verify that S cannot be avoided.Alg. <ref> first finds a word u of the minimal rank. This can be done by in Ø(n^3+|Σ|n^2) time and Ø(n^2+|Σ|n) space by the well-known algorithm <cit.> generalized to non-synchronizing automata (cf. the proof of Proposition <ref>. For every subset Q' ⊆ Q · u of size z=|S/∼| the algorithm checks whether there is a word w' mapping Q' to avoid S, but using its ∼-classes. This can be done by constructing the automaton A^z(Q^z,Σ,δ^z), where δ^z is δ naturally extended to z-tuples of states, and checking whether there is a path from Q' to a subset containing a state from each class [s]_∼ but avoiding the states from S. Note that since Q' cannot be compressed, every reachable subset from Q' has also size |Q'|. The number of states in this automaton is nz∈Ø(n^z). Also, note that we have to visit every z-tuple only once during a run of the algorithm, and we can store it in Ø(n^z+|Σ|n) space. Therefore, the algorithm works in Ø(n^3+|Σ|(n^2+n^z)) time and Ø(n^2+n^z+|Σ|n) space. The length of u is bounded by Ø(n^3), and the length of w' is at most Ø(n^z). Note that z=|S/∼| ≤min(r,|S|), where r is the minimal rank in the automaton.§.§ Bounded word length We now turn our attention to the variants of Problem <ref>, Problem <ref>, and Problem <ref> where an upper bound on the length of the word is additionally given.[Extensible large subset by short word] For a fixed k ∈ℕ∖{0}, given A=(Q,Σ,δ), a subset S ⊆ Q with |Q ∖ S| ≤ k, and an integer ℓ given in binary representation, is S extensible by a word of length at most ℓ? [Totally extensible large subset by short word] For a fixed k ∈ℕ∖{0}, given A=(Q,Σ,δ), a subset S ⊆ Q with |Q∖ S| ≤ k, and an integer ℓ given in binary representation, is S totally extensible by a word of length at most ℓ? As before, both problems for k=1 are equivalent to the following:[Avoidable state by short word] Given A=(Q,Σ,δ), a state q ∈ Q, and an integer ℓ given in binary representation, is {q} avoidable by a word of length at most ℓ? Problem <ref> for k ≥ 2 obviously remains PSPACE-complete. By the following theorem, we show that Problem <ref> is NP-complete, which then implies NP-completeness of Problem <ref> for every k ≥ 1 (by Corollary <ref>). Problem <ref> is NP-complete, even if the automaton is simultaneously strongly connected, synchronizing, and binary. The problem is in NP, because we can non-deterministically guess a word w as a certificate, and verify q ∉ Q· w in Ø(|Σ|n) time. If the state q is avoidable, then the length of the shortest avoiding words is at most Ø(n^2) <cit.>. Then we can guess an avoiding word w of at most quadratic length and compute Q· w in Ø(n^3) time.In order to prove NP-hardness, we present a polynomial-time reduction from the problem of determining the reset threshold in a specific subclass of automata, which is known to be NP-complete <cit.>. The reduction has two steps. First, we construct a strongly connected synchronizing ternary automaton A' for which deciding about the length of an avoiding word is equivalent to determining the existence of a bounded length reset word in the original automaton. Then, based on the ideas from <cit.>, we turn the automaton into a binary automaton A, which still has the desired properties.Let us have an instance of this problem from the Eppstein's proof of <cit.>. Namely, for a given synchronizing automaton B = (Q_B, {α_0,α_1}, δ_B) and an integer m>0, we are to decide whether there is a reset word w of length at most m. We do not want to reproduce here the whole construction from the Eppstein proof but we need some ingredients of it. Specifically, B is an automaton with a sink state z ∈ Q_B, and there are two subsets S={s_1,…,s_d} and F ⊆ Q_B with the following properties: * Each state q ∈ Q_B∖ S is reachable from a state s ∈ S through a (directed) path in the underlying digraph of B.* For each state s ∈ S and each word w of length m, we have δ_B(s,w) ∈ F ∪{z}.* For each f ∈ F we have δ_B(f,α_0) = δ_B(f,α_1) = z.* For each state s ∈ S and a non-empty word w ∈{α_0,α_1}^<m, we have δ_B(s,w) ∉ (F ∪ S).In particular, it follows that each word of length m+1 is reset. Deciding whether B has a reset word of length m is NP-hard.We transform the automaton B into A' as follows. First, we add the subset R = {r_0,r_1, …, r_m} of states to provide that z is not avoidable by words of length less than m+1. The transitions of both letters are δ_A'(r_i,α_0) = δ_A'(r_i,α_1) = r_i+1 for i=0,…,m-1, and δ_A'(r_m,α_0) = δ_A'(r_m,α_1) = z.Secondly, we add a set of states S' = { s'_1, …, s'_d} of size d=|S| and a letter α_2 to make the automaton strongly connected. Letters α_0 and α_1 map S' to the corresponding states from S, that is, δ_A'(s'_i,α_0) = δ_A'(s'_i,α_1) = s_i∈ S. Letter α_2 connects states r_0, s'_1,s'_2 …, s'_d into one cycle, i.e.δ_A'(r_0,α_2) = s'_1,δ_A'(s'_1,α_2) = s'_2,…,δ_A'(s'_d-1,α_2) = s'_d, δ_A'(s'_d,α_2) = r_0.We also set δ_A'(s_d,α_2) = r_1, δ_A'(z,α_2) = r_0, and all the other transitions of α_2 we define equal to the transitions of α_0.Finally, we transform A' to the final automaton A=(Q,{a,b},δ). We encode letters α_0,α_1,α_2 by 2-letter words over {a,b} alike it was done in <cit.>. Namely, for each state q ∈ Q_A'∖ (F ∪{z}), we add two new states q^a, q^b and define their transitions as follows:δ(q,a) = q^a,δ(q^a,a) = δ(q^a,b) = δ_A'(q,α_0),δ(q,b) = q^b,δ(q^b,a) = δ_A'(q,α_1),δ(q^b,b) = δ_A'(q,α_2).Then, aa,ab correspond to applying letter α_0, ba corresponds to applying letter α_1, and bb corresponds to applying letter α_2. Denote this encoding function by ϕ, i.e. ϕ(α_0) = aa, ϕ(α_1) = ba, and ϕ(α_2) = bb. We also extend ϕ to words over {α_0,α_1,α_2}^* as usual. For simplicity, we denote also ϕ(q) = {q, q^a, q^b}, and extend to subsets of Q_A' as usual.It remains to define the transitions for F ∪{z}. We set δ(z,a) = z, δ(z,b) = r_0, and δ(f,a) = δ(f,b) = z for each f ∈ F. Automaton A is shown in Fig. <ref>.Observe that A' is strongly connected: z is reachable from each state, from z we can reach r_0 by α_2, from r_0 we can reach every state from S' by applying a power of letter α_2, and we can reach every state of S from the corresponding state from S'. Then every state from Q_B is reachable from a state from S by Property 1. It follows that A is also strongly connected, since for every q ∈ Q_A', every state from ϕ(q) is reachable from q, and since for F ∪{z} the outgoing edges correspond to those in 𝒜.Observe that A is synchronizing: We claim that a^4m+6 is a reset word for A. Indeed, aa does not map any state into ϕ(S'). Every word of length m+1 is reset for B and synchronizes to z, in particular, α_0^m+1. Since ϕ(α_0^m+1)=a^2m+2 does not contain bbb, state z cannot go to S' by a factor of this word. Hence, we haveδ(Q,a^2m+4) ⊆{z}∪ϕ(R).Then, finally, a^2(m+1) compresses {z}∪ϕ(R) to z.Now, we claim that the original problem of checking whether B has a reset word of length m is equivalent to determining whether z can be avoided in A by a word of length at most 2m+3.Suppose that B has a reset word w of length m, and consider u=ϕ(α_0w)b. Note that ϕ(α_0)=aa does not map any state into ϕ(S') nor into ϕ(r_0). Hence, we haveδ(Q,ϕ(α_0)) ⊆ϕ(Q_B) ∪ϕ(R ∖{r_0}).Due to the definition of ϕ, factor bbb cannot appear in the image of words from {α_0,α_1}^* by ϕ. Henceforth, z cannot go to S' by a factor of ϕ(w). Since |ϕ(w)|=2m and to map z into ϕ(r_m)we require a word of length 2m+1, the factors of ϕ(w) do not map z into ϕ(r_m). Since also w is a reset word for B that maps every state from Q_B to z, we haveδ(ϕ(Q_B),ϕ(w)) ⊆{z}∪ϕ(R ∖{r_m}).By the definition of the transitions on R ∪{z} (only ϕ(α_2) maps r_0 outside), and since |ϕ(w)|=2m, we also haveδ(ϕ(R ∖{r_0}),ϕ(w)) ⊆{z}∪ϕ(R ∖{r_m}).Finally, we get that δ({z}∪ϕ(R ∖{r_m}),b) ⊂ R, thus u avoids z.Now, we prove the opposite direction. Suppose that state z can be avoided by a word u of length at most 2m+3. Then, by the definition of the transitions on R, |u|=2m+3 because z ∈δ(R,w) for each w of length at most 2(m+1). Let u = u' u” u”' with |u'|=2, |u”|=2m, and |u”'|=1.For words w ∈{a,b}^* of even length, we denote by ϕ̃^-1(w) the inverse image of encoding ϕ with respect to the definition on A', that is, ϕ̃^-1(aa) = ϕ̃^-1(ab) = α_0, ϕ̃^-1(ba) = α_1, ϕ̃^-1(bb) = α_2, which is extended to words of even length by concatenation.First notice that ϕ̃^-1(u') ≠α_2. Otherwise {z,r_0,r_1,r_2,…,r_m}⊆δ(S' ∪ R ∪{z}, ϕ̃^-1(u')) whence by the definition of R the word u”u”' of length 2m+1 cannot avoid z. Therefore ϕ̃^-1(u') ≠α_2 and S ⊆δ(S ∪ S',u').If α_2 is the second letter of ϕ̃^-1(u), then s_d goes to r_1 and we get {r_1,r_2,…,r_m,z} in the image of the prefix of u of length 4. Then, due to the definition of R, no word of length at most 2m can avoid z.Hence, the first two letters of ϕ̃^-1(u) are either α_0 or α_1.By Property 2 of B, every zero-one word of length m maps s ∈ S into {z}∪ F. Since the letter α_2 acts like α_0 on Q_B∖ S in A' and ϕ̃^-1(u”) starts with α_0 or α_1, u” maps S into {z}∪ F. If u” maps some state to F, then by Property 3 u cannot avoid z. Hence, ϕ̃^-1(u”) with all α_2 replaced with α_0 must be a reset word for B. By a corollary from Theorem <ref> and Theorem <ref>, we complete our results about extending subsets. Problem <ref> is NP-complete, Problem <ref> is NP-complete when the automaton is synchronizing, and Problem <ref> is NP-complete when the automaton is strongly connected and synchronizing. They remain NP-complete when the automaton is simultaneously strongly connected, synchronizing, and binary. NP-hardness for all the problems follows from Theorem <ref>, since we can set S = Q ∖{q}.Problem <ref> is solvable in NP as follows. By Theorem <ref> if there exists a totally extending word, then there exists such a word of polynomial length. Thus we first run this algorithm, and if there is no totally extending word then we answer negatively. Otherwise, we know that the length of the shortest totally extending words is polynomially bounded, so we can nondeterministically guess such a word of length at most ℓ and verify whether it is totally extending.Similarly, Problem <ref> is solvable in NP for synchronizing automata. For a synchronizing automaton there exists a reset word w of length at most n^3 <cit.>. Furthermore, if S is totally extensible, then there must exist a reset word w such that Q· w = {q}⊆ S, which has length at most n^3+n-1. Therefore, if the given ℓ is larger than this bound, we answer positively. Otherwise, we nondeterministically guess a word of length at most ℓ and verify whether it totally extends S.By the same argument for Problem <ref>, if the automaton is strongly connected and synchronizing, then for a non-empty proper subset of Q using a reset word we can always find an extending word of length at most n^3+n-1, thus the problem is solvable in NP. § RESIZING A SUBSET In this section we deal with the following two problems:[Resizable subset] Given an automaton A=(Q,Σ,δ) and a subset S ⊆ Q, is S resizeable? [Resizable subset by short word] Given an automaton A=(Q,Σ,δ), a subset S ⊆ Q, and an integer ℓ given in binary representation, is S resizeable by a word of length at most ℓ? In contrast to the cases |S· w^-1|>|S| and |S· w^-1|<|S|, there exists a polynomial-time algorithm for both these problems. Furthermore, we prove that if S is resizeable, then the length of the shortest resizing words is at most n-1.To obtain a polynomial-time algorithm, one could reduce Problem <ref> to the multiplicity equivalence of NFAs, which is the problem whether two given NFAs have the same number of accepting paths for every word. It can be solved in Ø(|Σ| n^4) time by a Tzeng's algorithm <cit.>, assuming that arithmetic operations on real numbers have a unitary cost; this algorithm relies on linear algebra methods. Alternatively, it can be solved in Ø(|Σ|^2 n^3) time by an algorithm of Archangelsky <cit.>. It was noted by Diekert that the Tzeng's algorithm could be improved to Ø(|Σ| n^3) time <cit.> (unpublished).However, to obtain the tight upper bound n-1 on the length we need to design and analyze a specialized algorithm for our problem. It is also based on the Tzeng's linear algebraic method. Assuming that in our computational model every arithmetic operation has a unitary cost, there is an algorithm with Ø(|Σ| n^3) time and Ø(|Σ|n+n^2) space complexity, which, given an n-state automaton A=(Q,Σ,δ) and a subset S ⊆ Q, returns the minimum length ℓ such that |S· w^-1| ≠ |S| for some word w ∈Σ^≤ℓ if it exists or reports that there is no such a word. Furthermore, we always have 1 ≤ℓ≤ n-1. The idea of the algorithm is based on the ascending chain condition, often used for automata (e.g. <cit.>). We need to introduce a few definitions from linear algebra. We associate a natural linear structure with automaton A. By ℝ^n we denote the real n-dimensional linear space of row vectors. The value at an i-th entry of a vector v ∈ℝ^n we denote by v(i). Without loss of generality, we assume that Q={1,2,…,n} and then assign to each subset K⊆ Q its characteristic vector [K] ∈ℝ^n, whose i-th entry v(i)=1 if i ∈ K, and v(i)=0, otherwise. By (S) we denote the linear span of S ⊆ℝ^n. The dimension of a linear subspace L is denoted by (L).Each word w ∈Σ^* corresponds to a linear transformation of ℝ^n. By [w] we denote the matrix of this transformation in the standard basis [1],…,[n] of ℝ^n. For example, if A is the automaton from Fig. <ref>, then[a]=( [ 0 1 0 0; 0 0 1 0; 0 0 0 1; 1 0 0 0 ]), [b]=( [ 1 0 0 0; 0 1 0 0; 0 0 1 0; 1 0 0 0 ]), [ba]=( [ 0 1 0 0; 0 0 1 0; 0 0 0 1; 0 1 0 0 ]).Clearly, as the automaton is deterministic, the matrix [w] has exactly one non-zero entry in each row. In particular, [w] is row stochastic, which means that the sum of entries in each row is equal to 1. For every words u,v ∈Σ^*, we have [uv]=[u][v].By [w]^T we denote the transpose of the matrix [w]. The transpose corresponds to the preimage by the action of a word; one verifies that [S · w^-1]=[S][w]^T. For two vectors v_1,v_2 ∈ℝ^n, we denote their usual inner (scalar) product by v_1v_2.Algorithm description. Now, we design the algorithm, which consists of two parts.First, consider the auxiliary Filter function shown in Algorithm <ref>. Its goal is to filter a stream of vectors g ∈ℝ^n, keeping only a subset of those vectors that are linearly independent. To perform this subroutine efficiently, we maintain a sequence of vectors G (basis) and a sequence of indices I, which are empty at the beginning. Every time, we use the Gaussian approach to reduce the matrix of vectors from G to a pseudo-triangular form. The sequence of (column) indices I = (i_1,i_2,…,i_k) and vectors G = (g_1,…,g_k) have the property that for each j, 1 ≤ j ≤ k, there is exactly one vector from {g_1,…,g_k} with non-zero i_j-th entry, which contains 1.We begin with the first non-zero vector g_1 and put its smallest index i of a non-zero entry to I, and the vector itself is normalized to have 1 in the i-th entry. Now, suppose we are given a vector g and we have already built G = (g_1,…,g_k) and I = (i_1,i_2,…,i_k) with aforementioned properties. Then, we just compute g' = g - ∑_r=1^kg(i_r)· g_r. Due to the construction, all the entries at the coordinates from I in g' are zero. If there is a non-zero coordinate left in g', then we need to normalize g', and it to G, and update the previous vectors. So we take the smallest coordinate i' whose entry is non-zero in g', normalize g' to have 1 in the i'-th entry, and add g' to G. To update the previous vectors, for each r, 1 ≤ r ≤ k, we set g_rg_r - g_r(i')· g', which results in that g_r has now zero in the i'-th entry, and finally we add i' to I. In the opposite case, if g'=0, then g belongs to (G) and thus should not be added.Note that at any point, the set G is a basis of the linear span of all the processed vectors, which is a straightforward corollary from using the Gaussian approach.We now turn to the main procedure of our algorithm, which is shown in Algorithm <ref>. Our goal is to find the minimum length of a word w such that |S· w^-1| ≠ |S|. This is equivalent to [S][Q][w] ≠ |S|. We do this by using a wave approach as in breadth-first search. We start by feeding [Q] to Filter and let W_0 = {[Q]}. Then in each iteration 1 ≤ i ≤ n-1, we consider the set of vectors D = { g[a] | g ∈ W_i-1, a∈Σ} and build a new subset of independent vectors W_i as follows. For each vector z from D, we first check whether [S]z = |S|. If this is not the case, we claim that i is the length of a shortest word which changes the size of the preimage of S. Otherwise, we feed z to Filter and add it to (initially empty) W_i if the corresponding basis vector was added to G. Note that the current G after the i-th iteration is equal to ⋃_j=0^i W_i. We stop if either W_i = ∅ or the last (n-1)-th iteration ends, which means that there is no resizing word.Correctness. To prove the correctness, note that by the construction all vectors from W_i can be written as [Q][w] for some word w of length i. Thus, if we have found a vector z ∈ D such that [S]z ≠ |S|, this means there is a word w of length i such that[S][Q][w] = [S · w^-1][Q] = |S · w^-1| ≠ |S|. It remains to show that if we get to an i-th iteration, then there is no word w of length less than i which violates [S][Q][w] = |S|. For r ≥ 0, denote U_r = ⋃_i=0^r W_i. We prove by induction that for each word w of length r < i, [Q][w] ∈([Q][U_r]). For r=0 this is trivial. If r>0, then w=w' a for some a ∈Σ and by induction [Q][w'] ∈([Q][U_r-1]), that is,[Q][w'] = ∑_j=0^r-1∑_u ∈ W_jλ_u [Q][u], for some values λ_u ∈ℝ. It follows that [Q][w' a] = [Q][w'][a] = ∑_j=0^r-1∑_u ∈ W_jλ_u [Q][u][a] = g_v + ∑_u ∈ W_r-1λ_u [Q][u][a], where g_v ∈([Q][U_r-1]). By the construction, we feed all vectors of the form [Q][u][a] for u ∈ W_r-1 and a ∈Σ to Filter function. Since the added vectors to G, and so to W_r, are a linear basis of the linear span of all the processed vectors, every vector [Q][u][a] belongs to ([Q][U_r]), which proves the induction step.Thus, if we had a word of length w of length less than i with [S][Q][w] ≠ |S|, we would have [Q][w] = ∑_u ∈ U_i-1λ_u [Q][u] for some λ_u ∈ℝ. Now, on the one hand we have n = [Q][w][Q] = ∑_u ∈ U_i-1λ_u ([Q][u][Q]) = n∑_u ∈ U_i-1λ_u,while on the other hand we have|S| ≠ [Q][w][S] = ∑_u ∈ U_i-1λ_u [Q][u][S] = ∑_u ∈ U_i-1λ_u |S|contradicting (<ref>).On the other hand, if W_i is empty for an i<n, this means that ([Q][Σ^≤ i]) = ([Q][Σ^≤ i-1]) and by the linear extending argument we know that the same holds for all j ≥ i, hence there cannot be a word that violates [S][Q][w] = |S|. Note that if there is no resizing word, then we always have this case for some i<n, because (([Q][w]| w ∈Σ^*)) ≤ n-1 and the vectors from all W_j are a basis.We also conclude that i cannot exceed n-1, which proves that the shortest resizing words have length at most n-1. Note that the upper bound n-1 is the best possible, at least in the cases |S| ∈{1,n-1}, which can be observed in the Černý automata (see Fig. <ref> with S={3}).Complexity. Assume that in our computational model every arithmetic operation has a unitary cost. Then clearly a k-th call of Feed can be performed in Ø(kn)-time. However, note that, if an exact computation is performed using rational numbers, then we may require to handle values of exponential order, and the total complexity would depend on the algorithms used for particular arithmetic operations.Notice that at an i-th iteration, we call Feed at most |Σ| |W_i| times, since, by the construction, sets W_i are disjoint because the corresponding vectors are independent. Since the complexity of Feed is in Ø(n^2), all calls work in Ø(|Σ|n^3)-time. The other operations took amortized time at most Ø(|Σ| n^2), which is the cost of computing sets D (at most n vectors in sets W_i; note that one g[a] can be computed in Ø(n) time, because the automaton is deterministic). Thus, the whole algorithm works in Ø(|Σ|n^3) time.The space complexity is at most Ø(|Σ|n+n^2), which is caused by storing the automaton and at most Ø(n^2) vectors in the sets W_i, G, and I. The running time Ø(|Σ|n^3) of the algorithm is quite large (and may require large arithmetic as discussed in the proof), and it is an interesting open question whether there is a faster algorithm for Problems <ref> and <ref>.We note that Problem <ref> becomes trivial when the automaton is synchronizing: A word resizing the subset exists if and only if S ∅ and SQ, because if w is a reset word and {q} = Q· w, then S· w^-1 is either Q when q ∈ S or ∅ when q ∉ S. This implies that there exists a faster algorithm in the sense of expected running time when the automaton over at least a binary alphabet is drawn uniformly at random: The algorithm from <cit.> checks in expected Ø(n) time (regardless of the alphabet size, which is not fixed) whether a random automaton is synchronizing, and it is synchronizing with probability 1-(1/n^0.5|Σ|) (for |Σ|≥ 2). Then only if it is not synchronizing we have to use the algorithm from Theorem <ref>. Thus, Problem <ref> can be solved for a random automaton in the expected timeØ(|Σ|n^3)·(1/n^0.5|Σ|) + Ø(n) = Ø(|Σ|n^3-0.5|Σ|) ≤Ø(n^2).Note that the bound is independent on the alphabet size, and this is because a random automaton with a growing alphabet is more likely to be synchronizing, so less likely we need to use Theorem <ref>.§ CONCLUSIONS We have established the computational complexity of problems related to extending words. Indirectly, our results about the complexity imply also the bounds on the length of the shortest compressing/extending words, which are of separate interest. In particular, PSPACE-hardness implies that the shortest words can be exponentially long in this case, and polynomial deterministic or nondeterministic algorithms in our proofs imply polynomial upper bounds. For example, the question about the length of the shortest totally extending words (in the equivalent terms of compressing Q to a subset included in S) was recently considered <cit.>, and from our results (PSPACE-completeness) we could infer an answer that the tight upper bound is exponential. The algorithm from Theorem <ref> implies also a bound on the length of the shortest avoiding words for a subset. That length is at least cubic, which is useless in the case of synchronizing automata, since reset words can be used as avoiding words and there exists a cubic upper bound on the length of the shortest reset words <cit.>.Some problems are left open. In Tables <ref> and <ref> there is a gap. The complexity of the existence of an extending word when the subset is large (Problem <ref>) and the automaton is strongly connected is unknown. The same holds in the case when the length of the extending word is bounded (Problem <ref>); now, we can only conclude that it is NP-hard, which follows from Corollary <ref>. The proof of Theorem <ref> relies on the automaton being not strongly connected.Further questions may concern other complexity classes like NL (cf. Theorem <ref>). Also, one could try improving the complexity of algorithms, in particular, those from Theorems <ref> and <ref> for avoiding words, and also that from Theorem <ref> for resizing words. § ACKNOWLEDGEMENTS We thank the anonymous referee for careful reading and detailed comments. This work was supported by the Competitiveness Enhancement Program of Ural Federal University (Mikhail Berlinkov), and by the National Science Centre, Poland under project number 2014/15/B/ST6/00615 (Robert Ferens) and 2017/25/B/ST6/01920 (Marek Szykuła).plainurl | http://arxiv.org/abs/1704.08233v4 | {
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Max Planck Institute for Mathematics in SciencesKreutzstrasse 21, 04287 Leipzig, GermanyBernstein Center for Computational Neuroscience, GöttingenAm Fassberg 12, 37077 Göttingen, [email protected], School of Informatics, University of Edinburgh10 Crichton St, Edinburgh, EH8 9AB, [email protected] define the Abelian distribution and study its basic properties. Abelian distributions arisein the context of neural modeling and describe the size of neural avalanches in fully-connectedintegrate-and-fire models of self-organized criticality in neural systems. THE ABELIAN DISTRIBUTION J. MICHAEL HERRMANN December 30, 2023 ======================== § INTRODUCTIONIn the present manuscript we introduce Abelian distributions.We have called the distribution Abelian because of a number of identities that arisein analysis and that resemble the Abel identity(x+y)^n=∑_i=0^n ni x (x-iz)^i-1(y+iz)^n-i <cit.>.This distribution appeared in 2002 in the study of a fully connected neural network <cit.> as a distribution of sizes of “avalanches” ofneural activity. Apart from Ref. <cit.>, so far there no systematic and accessible study of the distribution has been published.The related results that were reported in the context of Cayley's theorem <cit.> are also based on Ref. <cit.>.Here we will discuss the basic properties of this probability mass distribution and describe its importance for the applications in theoretical physics and biology.§ DEFINITIONLet N∈ℕ, α∈(0,1). The Abelian distribution is defined for 0≤ L≤ N byP_α,N(L) = C_α,NN L(Lα/N)^L-1(1-Lα/N)^N-L-1,where C_α,N= 1-α/N-(N-1)α, is a normalizing constant. Because P_α,N(0) =0 we will in the following often assume that L>0.The Abelian distribution defined by (<ref>),(<ref>) is a probability distribution.We have to show that ∑_L=1^N C_α,NNL(Lα/N)^L-1(1-Lα/N)^N-L-1=1.Introducing a new continuous variable x instead of α/N, we get ∑_L=1^N NL(L x)^L-1(1-Lx)^N-L-1=1/C_α,N,which is equivalent to∑_L=1^N-1NL(L x)^L-1(1-L x)^N-L-1= 1/C_α,N-(Nx)^N-1/1-Nx.We can expand the sum on the left side of (<ref>) and obtain ∑_L=1^N-1NL(L x)^L-1∑_m=0^N-L-1(-1)^m N-L-1m(Lx)^m. Introducing k=L we can rewrite the sum in the previous expression as a polynomial in x∑_i=0^N-2x^i∑_k=0^i (-1)^i-kNkN-k-1i-k(k)^i=∑_i=0^N-2P_i(N) x^i,where P_i(N) is a polynomial in N of degree i. For every N we have P_0(N)=1. Consider now i>0. To identify uniquelythe polynomial P_i(N) it is sufficient to find its values in i+1 different points that we select to be N=1,…,i+1. Because N-1k=0 for k>N-1, we have also N-k-2i- k=0 for N<i+2 for any k<N-1. Hence,P_i(N)=(-1)^i-kNN-1i- kN^i =N^i for N=1,…,i+1and i>0. This means that P_i(N)=N^i-1 for any N and i>0. Therefore the left side of (<ref>) is1+∑_i=1^N-2x^i N^i-1=1+x1-(Nx)^N-2/1-Nx.Inserting (<ref>) and (<ref>) into (<ref>) we arrive at 1+x1-(Nx)^N-2/1-Nx=N-(N-1)α/N(1-α)-(Nx)^N-1/N(1-Nx),which holds for any N and α<1.The authors of Ref. <cit.> mention that the theorem can also be proved by using a generalized binomial theorem.An Abelian-distributed probability mass function is shown in Fig. <ref> for several values of the parameter α.For small values of parameter α<0.9distribution is monotone and is dominated by approximately exponential decay, for α⪅ 1 distribution isnon-monotonous. For some small interval of parameter valuesα≈ 0.9 the distribution closely resemblesa power-law (with exponential cutoff at large L), see the double logarithmic plot in the inset. If a sample of data-points of size 10^5 is drawn from this distribution, the hypothesis of an underlying power-law distribution cannot be rejected <cit.>. The shape of the distribution varies in a similar way for all N, although for large N the non-monotonous regime ispresent only for α∈(α_crit(N),1) where the value of α_crit(N)has been numerically found to behave roughly as 1-1/√(N).§ EXPECTED VALUE We will now consider the moments of the Abelian distribution. Suppose ξ has an Abelian distribution with parameters α and N, then Eξ =N/N-(N-1)α.From (<ref>) and Lemma <ref> we have Eξ =∑_L=1^N L^L-1N-1L-1(α/N)^L-1(1-Lα/N)^N-L-1N(1-α)/N-(N-1)α.We have to prove that ∑_L=1^N L^L-1N-1L-1(α/N)^L-1(1-Lα/N)^N-L-1=1/1-α.Using again x=α/N we can rewrite this equation as ∑_L=1^NN-1L-1(Lx)^L-1(1-Lx )^N-L-1=1/1-N x.Transforming the sum in (<ref>) we obtain ∑_L=1^N-1N-1L-1(Lx)^L-1(1-Lx )^N-L-1+(Nx)^N-1(1-Nx)^-1=1/1-Nx.which is equivalent to ∑_L=1^N-1N-1L-1(Lx)^L-1(1-Lx )^N-L-1=∑_i=0^N-2(Nx)^i. Both the left and the right side of the equation (<ref>) are polynomials in x of degree N-2. Hence in order to provethat equation (<ref>) is an identity it is sufficient to show that the coefficients of x^i on the both sides are equal for every i. In other words, we have to show that ∑_k=0^i(-1)^i-kN-1kN-k-2i- k(k+1)^i =N^i. Again, both sides of (<ref>) are polynomials of N of the degree i. It is sufficient to prove that both sides of (<ref>) are equal for i+1 different points. We can select these points to be N=1, …, i+1. Obviously, if k>N-1, then N-1k=0, but also N-k-2i- k=0 for k<N-1 because N<i+2.Hence the only non-zero item of the sum is the one corresponding to k=N-1, in this case we have(-1)^i-kN-1N-1-1i- kN^i =N^i.§ MOTIVATION Power-law distributions have been studied in the sciences for a long time, the most prominent examplebeing the Gutenberg-Richter law which describes the energy distribution in earthquakes <cit.>.Other examples <cit.> include forest fires, migratory patterns, infectious diseases,solar flares, sandpiles <cit.> andneural activity dynamics <cit.>. Some of these examples can be related to critical branching processes <cit.> whichare known to produce power-law event distributions <cit.>. The relation between power-laws and branching processes usually requires a limit of large systems size <cit.> which is, however, not relevant when a comparison to numerical computations or mesoscopic experiments is desired. Nevertheless, the Abelian distribution converges to a power-law (asymptotically for large event sizesL→∞ or as an event density) in theexchangeable limits N→∞ and α→ 1. The exponent of the power-law γ = - 3/2is closely obeyed even for small L. Criticality being defined as the divergence of certain physical quantities (such as the mean event size)cannot occur in finite systems.Therefore it is tempting to use the Abelian distribution to define ananalogon of criticality also for finite systems.Depending on the parameters the Abelian distribution has monotonic or non-monotonic behavior, the latter being characterizedby a relative dominance of events with a size near the size of the system. The two behaviors, the sub- and the supercriticalregime are separated by a “critical” distribution, which is, however, unambiguously defined in terms of a power-law only for large systems. Avoiding the dependence of the critical parameters on the sample size that may arise when using atest (e.g. Kolmogorov-Smirnov) in order to determine the likelihood of criticality, we propose instead to define criticalityby qualitative criteria implied by the local similarity to a power-law. Consider the set A(N) of parameters α forwhich the equation d^2 log P_α,N(L)/d(log L)^2=0 has a solution L∈{1,…,N} for fixed N<∞.Expecting A(N) to contract into {1} for N→∞, we can define A(N) as the critical region for a finite system. Another possibility is to define a single critical value α_crit as an sup_α<1{α: d log P_α,N(L)/d L<0, ∀ L<N }. This definition uses the property of a critical state to stay between strictly monotonous and non-monotonous regimes.For all our numerical evaluations we found α_crit∈ A(N).Thus, the Abelian distribution is one of the few cases wherethe emergence of criticality in an infinite system can be studied explicitly as alimit of finite systems which enables a direct comparison with numerical computations or mesoscopic experiments. The Abelian distribution has been studied mainly in the context of neural avalanche dynamics <cit.>, where it not only turned out be successful in predicting an experimental result fromneuroscience <cit.>,but also allowed for an explicit and exact study of finite size effects. It is interpreted in this context asthe conditional probability of L-1 other neurons being activated given that one neuron just became spontaneouslyactive, thus forming an avalanche of L neural action potentials. From Theorem <ref> follows that the expectation exists also in the limit of large N ifα<1 as required by Definition <ref>. Correspondingly, in the neural system,a single nonterminating avalanche is observed at α=1. The application of the Abelian distribution as an event size distribution may require an appropriate definition of events. Although neurons produce quasi-discrete action potentials, in the experiments <cit.> events have been defined by threshold crossings, where an invariance of the distribution of the choice of the threshold is required for justification. In other time series, events can be defined either in a similar way. While the parameter N has usually a natural interpretation as the size of the system, for e.g. financial time series its meaning is less obvious. If N can be found by maximum likelihood, it can be interpreted as an effective system size. The parameter α describes in all cases the strength of the interaction between the elements in the system. If the elements are not all connected or if the system is heterogeneous, it seems reasonable touse, respectively, connectivity-rescaled parameters or an average interaction strength to determine estimates of this parameter.§ OPEN QUESTIONSA large number of questions related to the Abelian distribution are left for future investigation. Most important among them are the higher moments, characteristic function, stability and properties related to parameter estimation. Especially interesting for the application to critical system would be a scaling law for the critical value α_crit and relation between different possibilities to define ctiticality for finite system. § ACKNOWLEDGMENTSThe authors wish to thank Zakhar Kablutschko and Theo Geisel for helpful discussions andManfred Denker for valuable comments, help and support. Supported by the Federal Ministry of Education and Research (BMBF) Germany under grant number 01GQ1005B.10 Bakb P. Bak. How Nature Works: The Science of Self-Organized Criticality. Springer Verlag, 1999. Bak1987 P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality: An explanation of 1/f noise. Phys. Rev. Lett., 59:381–384, 1987. Beggs2003 J. Beggs and D. Plenz. Neuronal avalanches in neocortical circuits. J. Neurosci, 23:11167–11177, 2003. Denker2011 M. Denker and A. Rodrigues. The combinatorics of avalanche dynamics. arXiv preprint arXiv:1111.5071, 2011. Eurich2002 C. W. Eurich, M. Herrmann, and U. Ernst. Finite-size effects of avalanche dynamics. Phys. Rev. E, 66:066137–1–15, 2002. Gutenberg1954 B. Gutenberg and C.F. Richter. Seismicity of the Earth and Associated Phenomena. Princeton University Press, Princeton, N.J., 2nd edition, 1954. kolmogorov1947branching A. N. Kolmogorov and N. A. Dmitriev. Branching stochastic processes. Doklady Akademii Nauk SSSR, 56(1):5–8, 1947. LevinaDiss A. Levina. A mathematical approach to self-organized criticality in neural networks. Nieders. Staats- u. Universitätsbibliothek Göttingen, 2008. Dissertation (Ph.D. thesis), webdoc.sub.gwdg.de/diss/2008/levina/levina.pdf. Levina2008 A. Levina, J. M. Herrmann, and M. Denker. Critical branching processes in neural networks. PAMM, 7(1):1030701–1030702, 2008. Levina2007 A. Levina, J. M. Herrmann, and T. Geisel. Dynamical synapses causing self-organized criticality in neural networks. Nat. Phys., 3:857–860, 2007. Levina2009 A. Levina, J. M. Herrmann, and T. Geisel. Phase transitions towards criticality in a neural system with adaptive interactions. Phys. Rev. Lett., 102(11):118110, 2009. Otter1949 R. Otter. The multiplicative process. Ann. Math. Statist., 20:248–263, 1949. Saslaw1989 W. C. Saslaw. Some properties of a statistical distribution function for galaxy clustering. Astrophys. J., 341:588–598, 1989. | http://arxiv.org/abs/1704.08496v1 | {
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"title": "The Abelian distribution"
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[e-mail: ][email protected] Univ. Grenoble Alpes, F-38000 Grenoble, France CEA, LETI, MINATEC Campus, F-38054 Grenoble, France Univ. Grenoble Alpes, F-38000 Grenoble, France CEA, LETI, MINATEC Campus, F-38054 Grenoble, France Univ. Grenoble Alpes, F-38000 Grenoble, France CEA, LETI, MINATEC Campus, F-38054 Grenoble, France Univ. Grenoble Alpes, F-38000 Grenoble, France CEA, LETI, MINATEC Campus, F-38054 Grenoble, France Univ. Grenoble Alpes, F-38000 Grenoble, France CEA, LETI, MINATEC Campus, F-38054 Grenoble, France Univ. Grenoble Alpes, F-38000 Grenoble, France CEA, LETI, MINATEC Campus, F-38054 Grenoble, France Univ. Grenoble Alpes, F-38000 Grenoble, France CEA, LETI, MINATEC Campus, F-38054 Grenoble, France [e-mail: ][email protected] Univ. Grenoble Alpes, F-38000 Grenoble, France CEA, LETI, MINATEC Campus, F-38054 Grenoble, France Due to their wide band gaps, III-N materials can exhibit behaviors ranging from the semiconductor class to the dielectric class. Through an analogy between a Metal/AlGaN/AlN/GaN diode and a MOS contact, we make use of this dual nature and show a direct path to capture the energy band diagram of the nitride system. We then apply transparency calculations to describe the forward conduction regime of a III-N heterojunction diode and demonstrate it realizes a tunnel diode, in contrast to its regular Schottky Barrier Diode designation. Thermionic emission is ruled out and instead, a coherent electron tunneling scenario allows to account for transport at room temperature and higher. Coherent tunneling in an AlGaN/AlN/GaN heterojunction captured through an analogy with a MOS contact Marc Plissonnier December 30, 2023 ====================================================================================================§ INTRODUCTIONIII-N materials are today at the heart of continuous academic and industrial efforts worldwide. With applications in lighting, radio-frequency or power technologies, Gallium, Indium and Aluminum Nitride alloys offer versatile and outstanding platforms that enable high performance electronics and enhanced solutions in multiple sectors.Due to its intrinsic properties, GaN is nowadays drawing a lot of attention in the power electronics area as it leads to lower conversion losses at higher frequency compared to Silicon or Silicon Carbide devices. Multiple leading companies have already demonstrated GaN power technologies with extremely promising results <cit.>,thereby paving the way to new and greener energy converters with a market expected to ramp up in the forthcoming years. At the core of most of these GaN power devices lies the AlGaN/GaN heterojunction due to its exceptional properties, in particular the existence of a two-dimensional electron gas. Finding its origin in spontaneous and piezoelectric polarization <cit.>, it gives rise to a high electron concentration combined with a high electron mobility. In this work, we propose to draw parallels between an AlGaN/AlN/GaN heterojunction diode and a Metal/Oxide/Semiconductor (MOS) contact by using the duality found in III-N wide band gap materials: semiconductors on the one hand <cit.>, dielectrics on the other hand <cit.>, which can depend on their doping and on their electromagnetic environment. With the use of this analogy, we are able to map the energy band diagram at the rectifying contact vicinity in a direct way in order to address the underlying transport mechanisms. By combining the Transfer Matrix formalism <cit.> to compute the system's transparency and the Tsu-Esaki current formula <cit.>, we describe the forward conduction regime of the heterojunction diode with respect to applied voltage and operating temperature.We show thereby that the III-N heterojunction diode realizes a tunnel diode and rule out thermionic emission, too often incorrectly used to capture such architectures. Through this approach we propose an alternative path of understanding the physics of III-N heterostructures and devices.§ MOS ANALOGYWe start by considering an AlGaN/AlN/GaN double heterojunction, a widely used design <cit.> where the AlGaN layer is often referred to as the barrier layer, the AlN as the spacer layer and the GaN as the channel layer which contains a Two-Dimensional Electron Gas (2DEG) at the AlN/GaN interface. The system is completed with a gate metal on top of the AlGaN, more specifically Titanium Nitride (TiN). A common approach to studying this system is the use of a Poisson-Schrödinger simulator which allows the derivation of the energy band diagram of the contact and captures the existence of the 2DEG <cit.>. We propose here an alternative approach to the problem.To allow an equivalence between the TiN/AlGaN/AlN/GaN contact and a MOS contact we make two main assumptions. First we assume that the AlGaN and AlN layers are depleted from free carriers, thus allowing to consider purely their dielectric nature and equate them to the gate oxide found within a MOS. Then we assume the GaN channel layer to be a n type semiconductor which will be the stage of an electron accumulation layer as found at the semiconductor/oxide interface of a MOS contact. Figure <ref> illustrates this analogy by representing schematically an AlGaN/AlN/GaN diode (a) where the rectifying contact part or anode (b, dotted box) can be mapped on an effective MOS contact (c).This analogy allows us to employ the main equation used to describe MOS capacitors with a first focus being the derivation of the energy band diagram. We start by considering only the rectifying contact part and express the potential equilibrium as found in a MOS capacitor:V=V_fb+V_ox+Ψ_sWhere V is the metal potential (referenced to the bulk GaN supposed grounded), V_fb the flat band potential, V_ox the potential drop across the oxide and Ψ_s the surface potential of the semiconductor which translates the amount of band bending in the accumulation layer. We detail now the different terms of this equation when applied to the Metal/AlGaN/AlN/GaN contact. The flat band potential expresses the difference between the Fermi levels in the metal (TiN) and in the semiconductor (bulk GaN).V_fb=W-χ ^GaN-(E_C-E_F)^GaNWhere W is the gate metal work function, χ the electron affinity of GaN and (E_C-E_F) the energy difference between the bottom of the conduction band and the Fermi energy in the bulk GaN. As far as V_ox is concerned, we recall that III-N materials have polar bonds. Within the bulk, the dipoles created by each chemical bond are canceled by the neighboring ones, two by two. However, at a hetero-interface this balance is no longer present and polarization surface charges develop at them. Figure <ref> (a) illustrates the different interface charges and their polarity present in the case of a Ga-face oriented heterojunction. The effective oxide (AlGaN+AlN) under consideration is therefore natively charged. By taking into account spontaneous and piezoelectric charges and expressing the continuity of the normal component of the electric field in the structure, the voltage drop across the AlN and AlGaN layers reads:V_ox=-σ_s+σ_10/C_AlN-σ_s+σ_10+σ_21/C_AlGaNWhere σ_s denotes the accumulation charge (mobile electrons) per unit area in the GaN accumulation layer and C_AlN and C_AlGaN denote the equivalent AlN and AlGaN capacitances per unit area respectively. σ_10=σ_0^-+σ_1^+ and σ_21=σ_1^-+σ_2^+ express the net polarization charges found at the GaN/AlN and AlN/AlGaN interfaces respectively (refer to supplementary information). For each material, the spontaneous contribution is determined following <cit.> and the piezoelectric part is determined following <cit.> by assuming the AlN and the AlGaN to be fully strained on a relaxed GaN layer.It can be shown that in the case of a 2D accumulation layer, the surface charge density σ_s can be related to the surface potential Ψ_s by solving the Poisson equation <cit.>:σ_s=±ϵ k_BT/qL_D√(N_v/p_0F^-+qΨ_s/k_BT+n_0/p_0(N_c/n_0F^+-qΨ_s/k_BT))Where σ_s is taken to be negative when Ψ_s is positive and vice versa. n_0,p_0 correspond to the electron and hole concentrations in the bulk GaN, N_C and N_V to the GaN conduction and valence band effective density of states and L_D to the Debye length written as:L_D=√(ϵ k_BT/2q^2p_0)The F^- and F^+ functions are defined as:F^-=F_3/2(E_V-E_F-qΨ_s/k_BT)-F_3/2(E_V-E_F/k_BT) F^+=F_3/2(E_F-E_C+qΨ_s/k_BT)-F_3/2(E_F-E_C/k_BT)With F_3/2 corresponding to the normalized Fermi-Dirac integral F_j(x) with order j=3/2 <cit.>:F_j(x)=1/Γ (j+1)∫_0^∞ξ^j/1+exp(ξ-x)dξCombining (<ref>) and (<ref>) and injecting the outcome with (<ref>) into (<ref>), one can solve numerically the surface potential Ψ_s at the GaN/AlN interface, which in turn determines σ_s and finally V_ox. To illustrate the procedure, we apply it to the computation of the band diagram of a reference structure and more precisely we focus on the conduction band profile. The main operating and structural parameters used are summarized in the following table: Figure <ref> (b) illustrates the obtained conduction band profile for the reference TiN/AlGaN/AlN/GaN contact where the conduction band offsets were accounted for following <cit.>. Note that the profile of the conduction band from the bulk GaN (at z=-50 A) to the GaN/AlN interface (z=0 A) is shown for the sake of clarity and was not calculated explicitly. Indeed the model used assumes a 2D accumulation layer and returns the amount of band bending at the surface of the GaN referenced to the bulk, where the position of the Fermi energy is known through n_0.We observe that the existence of the 2D electron gas, or the electron accumulation layer, naturally arises from the calculation and is induced by the strong polarization charges found within the hetero-structure. The extracted sheet carrier density within the accumulation layer is n_s=1e13 cm^-2 which agrees well with experimental values reported for similar heterojunctions <cit.>.§ PRACTICAL CONSIDERATIONS §.§ Transfer lengthIII-N heterojunction diodes have been fabricated at CEA-LETI using 200 mm GaN on Silicon wafers grown on site that were processed through a CMOS compatible integration flow. Figure <ref> shows a close view of the TiN/AlGaN/AlN/GaN contact found on aprocessed diode which in this case contains a recessed Al_0.25Ga_0.75N barrier with 6 nm left above the AlN spacer. For the purpose of this study, the fabricated diodes were designed with anode length, cathode length and anode to cathode distance all of 15 μm. Before going into the details of the transport mechanisms entering the on state of the presented diode, we recall a common feature found in contact physics: the transfer length notion. This parameter, often characterized through Transmission Line Measurements (TLM), expresses the length over which a contact, ohmic in the TLM case, is effectively operating. The concept can be extended to nonlinear contacts and corresponding nonlinear transmission lines <cit.>. Understand thereby that for heterojunction diodes with sufficiently long anode contacts, the majority of the current is emitted at the anode periphery in the forward regime. To account for this feature and avoid 2D calculations to reproduce the on state current, we propose a compromise between the regular 1D surface independent current emission (figure <ref>, a) and the more complex 2D surface dependent current (figure <ref>, b). Figure <ref> (c) represents the intermediate case we will consider in the following which makes use of an effective anode contact length L_eff over which the current is assumed constant. Through this simplification a 1D approach is possible with a reduced contact surface S=2L_effω, where ω stands for the anode width. §.§ Anode recess impactAs detailed in the previous section, the rectifying contact of the diode under study is realized through a partial recess of the Al_0.25Ga_0.75N barrier with a targeted thickness of 6 nm left above the AlN spacer. This strategy is common in such architectures and offers the benefit of reducing the turn-on voltage of the diode <cit.> allowing more current to be delivered in the on state, a critical feature for end users.We derive the band diagram of the recessed diode using the same set of parameters as in the reference structure (table <ref>) and adjust the AlGaN thickness to 6 nm. The calculated resulting sheet carrier density under the contact is n_s=7.3e12 cm^-2. C(V) measurements performed on representative test structures reveal an experimentally smaller value falling in the range 1e12 cm^-2 to 3e12 cm^-2.To explain this difference, we turn back to the etching process steps which allow the anode contact recess. The process sequence uses Reactive Ion Etching and comprises the Si_3N_4 passivation layer opening using Fluorine species (F^-) followed by the Al_0.25Ga_0.75N etching using Chlorine species (Cl^-). Fluorine implantation through a similar process was already reported in the AlGaN barrier by coworkers <cit.> and we believe that within the etching condition used, negatively charged ions contaminate the active layers, more precisely the AlGaN left under the TiN metal.In order to take into consideration ion trapping in the AlGaN, we introduce an effective surface charge at the AlN/AlGaN interface. We modify σ_21 in equation (<ref>) to σ_21=σ_1^-+σ_2^++σ_etch where σ_etch=-0.013 C.m^-2 represents a negative surface charge density linked to Fluorine/Chlorine ion implantation. The resulting sheet carrier density becomes n_s=1.8e12 cm^-2 which falls into the experimental range. § TUNNELING CURRENTOnce the conduction band profile is known, the question of the current flowing through the contact can be addressed through the transmission probability of an electron moving from the accumulation layer to the metal. In order to determine this probability through an arbitrary potential barrier, in our case the conduction band profile of the AlN and AlGaN layers, we employ the Transfer Matrix formalism <cit.> which goes beyond the WKB approximation <cit.>.The procedure to follow is to discretize the potential barrier into N regions of constant potential (figure <ref>, a) where locally the shape of the electron wavefunction Φ_i in region i can be assigned as:Φ_i(z)=A_iexp(jk_iz)+B_iexp(-jk_iz)Where the wavevector as a function of energy E, potential U_i and effective mass m^* reads:k_i=√([2m_i^*(E-U_i)])/ħThrough the continuity of the wavefunction from the i to the i+1 region, one can express how it transforms through a local transfer matrix M_i:M_i= [ (1+S_i)e^-j(k_i+1-k_i)z_i (1-S_i)e^-j(k_i+1+k_i)z_i;(1-S_i)e^j(k_i+1+k_i)z_i(1+S_i)e^j(k_i+1-k_i)z_i; ]S_i=m_i+1^*k_i/m_i^*k_i+1By iteration, a product of N matrices leads to the global transfer matrix through the potential barrier. From it, the electron transmission probability T(E) from region i=0 (accumulation layer) to region i=N+1 (metal) can be derived as a function of energy:M_i=∏_i=0^i=NM_i= [ M_11 M_12; M_21 M_22;]A_N+1 =m_N+1^*k_0/m_0^*k_N+11/M_22 T(E) =m_0^*k_N+1/m_N+1^*k_0|A_N+1|^2 We show in figure <ref> (b) the calculation of the transmission probability T(E) for the recessed TiN/AlGaN/AlN/GaN contact. The computation is carried out from the bottom of the conduction band in the accumulation layer to a cut-off energy of 2 eV, using a voltage bias of 200 mV at 300 K. In order to derive the current flow through the contact, we follow the Landauer vision of transport in a mesoscopic system where its quantum resistance is expressed in terms of the transmission and reflection properties of the structure <cit.>. In the coherent tunneling limit and none-interacting electrons, the current density reads: J(V)=2e/h∫ T(E,V)[f_L(E)-f_R(E)]dEWhere f_L and f_R represent the equilibrium distribution functions in both reservoirs, indexed L and R, connected to the quantum conductor and T(E,V) denotes the conductor's transmission probability.The use of such an approach to mesoscopic transport has proved to be versatile and a powerful tool which has been applied to various systems such as molecular junctions and transistors <cit.>, carbon nanotubes <cit.>, or quantum dots <cit.>.In the case of parallel wavevector conservation and effective masses for each parabolic band of interest, equation (<ref>) can be reduced to the so called Tsu-Esaki current formula <cit.>: J(V)=qm_0^*k_BT/2π^2ħ^3∫_E_0^∞T(E_⊥,V)ln[1+e^E_F-E_⊥/k_BT/1+e^E_F-qV-E_⊥/k_BT]dE_⊥ Where T(E_⊥,V) is the transmission probability at transverse energy, that is to say in the transport direction, and which will be evaluated through the Transfer Matrix formalism. The logarithmic term arises from the assumption that the occupancy functions f_L(R) follow the Fermi-Dirac distribution in both leads. The metal (TiN) and the electron accumulation layer (2DEG) are treated here as incoherent reservoirs where temperature dictates the precise shape of these distributions. The summation from the bottom of the conduction band (E_0) at the GaN/AlN interface to a high energy cut-off allows to account for electron emission from the GaN accumulation layer at different energies. Therefore multiple elastic processes, from direct tunneling in the forbidden band gap of the effective oxide (AlN and AlGaN) to thermionic emission over the potential barrier (above the AlN), are taken into consideration which is represented schematically in figure <ref> (c). Within this framework, an electron emitted at energy E from the 2DEG (left Fermi reservoir) tunnels coherently to the contact Metal (right Fermi reservoir) where it is absorbed. Understand thereby that it losses its energy and previous state memory through incoherent relaxation.§ APPLICATION TO FORWARD CONDUCTIONWe focused in the last sections on the rectifying contact part of the lateral diode (figure <ref> b, dotted box). In order to reproduce the I(V) characteristics of the entire device, we also need to take into consideration the access resistance (figure <ref>, b dashed box), that is to say the 2DEG resistance in the anode to cathode region in series with the ohmic contact's resistance. We suppose this resistance R to be constant at a given temperature within the voltage bias range we are about to explore. The effect of the access resistance is accounted for by setting the effective voltage drop across the rectifying contact in the right hand side of equation (<ref>) to: V_eff=V-RI, V being now the Anode-Cathode voltage bias. The diode's current density then becomes implicit and needs to be solved at each applied bias. Finally the total current is determined by recalling that I(V)=2ω L_eff J(V) (figure <ref>, c), where ω=26 mm by design.Before presenting the results of our calculations we recall a widely used empirical formula to reproduce the on state current of several different types of diodes including the studied architecture, the thermionic formula:I_TE(V)=SA^*T^2exp(-qΦ_b/k_BT))exp(q(V-RI)/η k_BT) Where S denotes the rectifying contact surface, A^* denotes the modified Richardson constant, Φ_b the barrier height and η the ideality factor. Figure <ref> shows the forward conduction state of the fabricated diode at room temperature fitted via equation (<ref>) at 300 K (a) and 425 K (b). The adjustable parameters extracted from each fit are (Φ_b=0.76 eV, η=2.39, R=0.21 Ω) at room temperature and (Φ_b=0.94 eV, η=1.91, R=0.4 Ω) at 425 K. Although the agreement seems very good, such fits miss completely the physics of the device which we are about to show. On the one hand high ideality factors are synonymous of underlying tunnel events <cit.> which thermionic emission does not account for. On the other handΦ_b and η need to have strong temperature dependencies in order to capture the evolution of the current, another sign of the irrelevance of such a physical picture in the present case.Figure <ref> (a) presents the results of the calculations compared to the raw data acquired at 300 K and 425 K. As can be observed, an excellent agreement is met within temperature and over the entire operating voltage range with the current spanning over more than 10 orders of magnitude. Note that the material parameters used are close to the experimental measured and targeted values (table <ref>). We emphasize that the only parameter changed with temperature is the access resistance, where the 2DEG mobility is known to be a strong function of temperature due to electron-phonon interaction in particular <cit.>. In the present case, R is adjusted to fit the ohmic regime of the diode above the turn-on voltage and is the single free parameter once the calculation is calibrated at a given temperature.Figure <ref> (b) allows us to go deeper into the transport mechanisms involved under and above the turn-on voltage of the recessed diode by providing the current spectroscopy. As can be seen, the majority of the current at all voltages is emitted in an energy band of a few hundred of meV centered near the Fermi level of the GaN layer. By correlating this observation with band diagrams obtained at different voltages (figure <ref>, c to e), different conclusions can be drawn. First, within the voltage range explored, electrons never overcome the AlN potential barrier due to its important height, they tunnel directly through it. Second, below the turn-on voltage (figure <ref>, c and d) where the current grows exponentially, hardly any electrons are emitted above the AlGaN conduction band near the AlN/AlGaN interface. At best, they tunnel through its band gap over a certain distance before reaching its conduction band minimum further in the layer. Such a mechanism could be approached to Nordheim-Fowler tunneling <cit.>. Overall, within the operating voltage and temperature range, no thermionic processes participate to the transport in the forward conduction regime.We infer that the use of the thermionic equation for similar devices as it is commonly done, is bound to return incorrect parameters and lead to misinterpretations. The derived electronic transport scenario reveals that the recessed TiN/AlGaN/AlN/GaN realizes a tunnel diodewith large output currents. More precisely, the formalism employed suggests that this diode is the stage of coherent electron tunneling events at the vicinity of the anode at room temperature and even higher, a property that would be worth investigating more deeply. From a wider perspective than conventional power conversion, we believe that ultra fast operation could be possible using more compact and fine tuned designs in order to achieve rectification at very high frequencies such as in the infrared spectrum <cit.>.Beyond the studied diode design, the sensitivity of electron tunneling to potential barrier heights and thickness raises different questions. First, the removal of the AlN layer would allow higher transparency as far as electrons propagating from the 2DEG to the metal, though it would also imply lower carrier concentration and higher access resistance. Indeed the AlN layer is known to lead to improved sheet resistance in the electron channel due to stronger polarization and enhanced confinement <cit.>. A comparable argument could be made as far as the Al content in the AlGaN barrier and in both cases a compromise could be possible. Moreover, the reduction of the AlGaN thickness under the TiN metal would also lead to higher transparency. Through further recessing, the impact on the Two-Dimensional Electron Gas arises. For the extreme case of full recess, the 2DEG is removed below the metal gate and a different type of contact is realized: a lateral Metal/GaN contact <cit.> for which the transport mechanisms would be worth investigating thoroughly. Finally, for thicker barriers (or alternatively shallower recesses) the coherent tunneling picture might break down due to the loss of phase coherence. Indeed, one cannot rule out incoherent scattering processes over longer tunneling distances and more a general tunneling approach might be required to describe a wider variety of heterostructure designs.§ CONCLUSION AND PERSPECTIVESVia an analogy with a MOS contact and current calculations based on the Tsu-Esaki formula, a coherent tunneling transport picture could be provided as far as forward conduction is concerned in a III-N heterojunction diode. This result contrasts the common approach involving thermionic emission which can lead to invalid and misleading physical interpretations. We believe that, in essence, the proposed approach which highlights the importance of field effect and takes into account natively electron emission at multiple energies, can be applied to other architectures and materials. As far as III-N heterojunctionapplications are concerned, we may cite RF or power GaN High Electron Mobility Transistors for which the gate electrostatic control and leakage could be tested on a solid basis. We may also mention III-N LED which involve multiple quantum wells and where transport in the forward regime could be explored more thoroughly.We emphasize that within the coherent tunneling framework used, questions related to trap assisted events or inelastic tunneling may arise <cit.>. The good agreement between experimental and calculated data proves that such events do not play an essential role within the device and associated conduction regime explored. Nonetheless pushing the reasoning further to incorporate such effects would provide a larger frame of investigation. Other refinements could be further implemented such as describing the access resistance in better details to capture its temperature variation for example. One may also consider self-heating effects which mainly occur for high current densities and that would open broader voltage ranges to be analyzed. Finally, the energy quantization of the Two-Dimensional Electron Gas, which was treated as a classical incoherent electron reservoir, would be worth evaluating.§ METHODSIII-N on Silicon growth was realized via MOCVD using a single chamber closed coupled showerhead equipment manufactured by AIXTRON capable of fully automated handling of 200 mm wafers. Au free process integration was carried out using CEA-LETI's CMOS facilities on the basis of a process flow comprised of approximately 80 technological steps. Electrical characterization was performed by the use of a Tesla 300 mm semi-automatic prober and an Agilent B1505A power device analyzer. All numerical calculations were performed on specifically developed scripts and were double checked using various commercial and open source coding environments.§ ACKNOWLEDGMENTSThe authors thank CEA-LETI's clean room department and technical staff for carrying out multiple process integration steps and in-line control measurements. They also thank Stéphane Bécu for fruitful discussions and comments on the manuscript. This work was performed in the frame of the TOURS 2015 project supported by the french "Programme de l'économie numérique des Investissements d'Avenir". | http://arxiv.org/abs/1704.08505v3 | {
"authors": [
"Yannick Baines",
"Julien Buckley",
"Jérôme Biscarrat",
"Gennie Garnier",
"Matthew Charles",
"William Vandendaele",
"Charlotte Gillot",
"Marc Plissonnier"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170427110555",
"title": "Coherent tunneling in an AlGaN/AlN/GaN heterojunction captured through an analogy with a MOS contact"
} |
Hardware for Dynamic Quantum Computing Thomas A. Ohki December 30, 2023 ====================================== Without resorting to complex numbers <cit.> or advanced topological arguments <cit.>, we show that any real polynomial of degree >2 has a real quadratic factor, which is equivalent to the seminal version of the Fundamental Theorem of Algebra (FTA) <cit.>. Thus it is established that basic real analysis suffices as the minimal platform to prove FTA. Keywords: Fundamental Theorem of Algebra, Polynomial Interlacing.AMS subject classifications: 12D05, 12D10, 26C10.§ INTRODUCTIONLet us consider polynomials of the field of real numbers ℝ, which is the minimal field containing the rational number field ℚ as a subfield and satisfying the least upper-bound property <cit.>.f(c̅_N, x) =∑_n=0^N c_n x^n∈ℝwhere c̅_N=(c_0 , c_1,c_2,..., c_N, 0,0,0,...) ∈ℝ^∞ is fixed, with c_00 (otherwise divide by x and start redifining c̅_N from c_1), c_N0, c_k=0 ∀ k>N. x ∈ℝ is independent. Such a description is useful for representing any polynomial as a unique power series. The seminal version of FTA only stated factorizability of any such polynomial into a product of linear and quadratic polynomials <cit.>, which can be rephrased to an even more compact statement:For N>2, given any c̅_N, ∃ A, B ∈ℝ and d̅_N-2 such that f(c̅_N, x) = (x^2-Ax-B) f(d̅_N-2, x) We will prove this by looking at the remainder of the polynomial long division of f(c̅_N, x) by (x^2-ax-b), where a,b∈ℝ. Given any c̅_N and a,b ∈ℝ, ∃ unique c̅'̅_N-2, P(c̅_N, a,b)∈ℝ and Q(c̅_N, a,b)∈ℝ that satisfyf(c̅_N, x) = (x^2-ax-b) f(c̅'̅_N-2, x)+ P(c̅_N, a,b) x + Q(c̅_N, a,b)(x^2-ax-b)=f(c̅_2, x), where c̅_2=(-b, -a, 1, 0,0,0,...). Both P(c̅_N, a,b) and Q(c̅_N,a,b) are bivariate polynomials in variables a and b. By studying their interlacing, we will show the existence of A,B ∈ℝ that satisfy P(c̅_N, A,B)=0 and Q(c̅_N, A,B)=0. The crucial steps will be to show the following: * ∃ b̅<0 such that for any fixed b<b̅, P(c̅_N, a,b) and Q(c̅_N, a,b) have maximum possible number of (real) roots in the variable a which are interlacing each other. For brevity, this is abbreviated as interleaving. * For some b>b̅, interleaving fails either due to the failure in interlacing of the roots, or due to existence of smaller number of (real) roots of P(c̅_N, a,b) and Q(c̅_N, a,b) than their individual degrees in a.Figure <ref> illustrates these conditions for a particular example polynomial. In order to show such interleaving, we will reduce P(c̅_N, a,b) and Q(c̅_N, a,b) to simpler formulae (equation <ref>), using the following lemmas:Linearity P(c̅_N, a,b) = ∑_n=0^N c_n P_n(a,b) and Q(c̅_N, a,b) = ∑_n=0^N c_n Q_n(a,b)where f(ϵ̅_n, x)=x^n, P_n(a,b)=P(ϵ̅_n,a,b) and Q_n(a,b)=Q(ϵ̅_n,a,b). Proof: Given c̅_N and d̅_M, by definition ∃c̅'̅_N-2 and ∃d̅'̅_M-2 such thatf(c̅_N, x)= (x^2-ax-b) f(c̅'̅_N-2, x)+ P(c̅_N, a,b) x + Q(c̅_N, a,b) f(d̅_M, x)= (x^2-ax-b) f(d̅'̅_M-2, x)+ P(d̅_M, a,b) x + Q(d̅_M, a,b) α_1 f(c̅_N, x) + α_2 f(d̅_M, x)= (x^2-ax-b) [ α_ 1 f(c̅'̅_N-2, x) + α_2 f(d̅'̅_M-2, x) ]+ [ α_ 1 P(c̅_N, a,b) + α_2 P(d̅_M, a,b) ] x+ [ α_ 1 Q(c̅_N, a,b) + α_2 Q(d̅_M, a,b) ]Butα_1 f(c̅_N, x) + α_2 f(d̅_M, x) =α_1 ∑_n=0^N c_n x^n+ α_2 ∑_m=0^M d_m x^m =∑_k=0^max(M,N)[ α_1 c_k + α_2 d_k] x^k = f(α_1 c̅_N + α_2 d̅_M, x). ∃s̅'_max{N,M}-2 such thatf(α_1 c̅_N + α_2 d̅_M, x)= (x^2-ax-b) f(s̅'_max{N,M}-2, x) + P(α_1 c̅_N + α_2 d̅_M, a,b) x + Q(α_1 c̅_N + α_2 d̅_M, a,b)Since polynomial division gives unique remainder, the coefficients of the linear residues in equations <ref> and <ref> must be equal. Finally the linear representation is written in terms of P(ϵ̅_n,a,b) and P(ϵ̅_n,a,b) to obtain equation <ref>. □ Recursion P_n+2(a,b) = a P_n+1(a,b) + b P_n(a,b) Q_n+2(a,b) = a Q_n+1(a,b) + b Q_n(a,b)Proof: For any (x^2-ax-b) f(c̅_N) = f(c̅^*_N+2), by the definition of residues in equation <ref>, P(c̅^*_N+2,a,b)=0 and Q(c̅^*_N+2,a,b)=0. Applying lemma <ref> onx^n+2 = (x^2-ax-b) x^n + a x^n+1 + b x^nP_n+2(a,b)= 0 + a P_n+1(a,b) + b P_n(a,b)and Q_n+2(a,b)= 0 + a Q_n+1(a,b) + b Q_n(a,b) □ Interrelation Q_n+1(a,b)=bP_n(a,b) Proof: By induction. The relation explicitly holds for n=0,1,2.Assume that equation <ref> holds until some n>2. For n+1, lemma <ref> givesQ_n+2(a,b) = a Q_n+1(a,b) + b Q_n(a,b) = a b P_n(a,b) + b^2 P_n-1(a,b) = b [ a P_n(a,b) + b P_n-1(a,b) ] = b P_n+1(a,b)□ § INTERLEAVING USING SIMPLER REPRESENTATION Using the lemmas, we arrive at a simpler representation of equation <ref>:f(c̅_N, x)= (x^2-ax-b) f(c̅'̅_N-2, x) + x ∑_n=1^N c_n P_n(a,b) + b ∑_n=1^N c_n P_n-1(a,b) + c_0Therefore we only need to look at the interleaving of ∑_n=1^N c_n P_n(a,b) and b ∑_n=1^N c_n P_n-1(a,b) + c_0. Since the extra variable b and constant c_0 renders direct comparison of their roots intractable, we resort to comparing roots of the following polynomials similar in structure to the famous Sturm chains <cit.>:h_m(a,b)=∑_n=m^N c_n P_n-m(a,b)h_m(a,b) inherits the following recursion formula from lemma 2:h_m(a,b) = a h_m+1(a,b) + b h_m+2(a,b) + c_m+1∑_n=1^N c_n P_n(a,b) = h_0(a,b) and b ∑_n=1^N c_n P_n-1(a,b) + c_0 = b h_1(a,b) + c_0. For any given c̅_N, ∃ b_c̅_N<0such that for any fixed b<b_c̅_N , and for all 0≤ m ≤ N-2, ( h_m(a,b), h_m+1(a,b) ) are interleaving. Recapitulating for the sake of clarity, for any fixed b<b_c̅_N, * h_m(a,b) has N-m-1 distinct (real) roots in the variable a, say α_1^(m)(b)< α_2^(m)(b)<...<α_N-m-1^(m)(b) in ascending order. * All pairs ( h_m(a,b), h_m+1(a,b) ) have interlacing roots α_1^(m)(b)<α_1^(m+1)(b) < α_2^(m)(b) < α_2^(m+1)(b) <...<α_N-m-2^(m+1)(b) <α_N-m-1^(m)(b). Proof: By induction and contradiction. We can explicitly verify for m=N-3 and m=N-2 h_N-3(a,b)= c_N P_3(a,b)+c_N-1P_2(a,b) + c_N-2P_1(a,b) = c_N (a^2 + b) +c_N-1a+c_N-2 h_N-2(a,b)= c_N P_2(a,b) +c_N-1 P_1(a,b) =c_Na + c_N-1 h_N-2(a,b)=0only at α^(N-2)_1(b) = - c_N-1/c_N. For any fixed b< b_N-3=- c_N-2/c_N, h_N-3(a,b) has distinct (real) roots. When c_N-1 0, these roots are -c_N-1/2 c_N (1 ±√(1 - D) ), where D=4 c_N c_N-2/ c_N-1^2+4 c_N^2 / c_N-1^2b. When c_N-1 = 0, the roots are ±√(-c_N-2/c_N-b).Either α^(N-3)_1(b)=-c_N-1/2 c_N (1 + √(1 - D) ) when c_N-1/c_N>0, or α^(N-3)_1(b)=-c_N-1/2 c_N (1 - √(1 - D) ) when c_N-1/c_N<0. By directly comparing the formulae of roots, we get interleaving of ( h_N-3(a,b), h_N-2(a,b) ) ∀ b<b_N-3 : α^(N-3)_1(b) <α^(N-2)_1(b) < α^(N-3)_2(b)Now let us assume that for the given c̅_N, ∃ b_m+1<0 such that for any fixed b<b_m+1, the pairs ( h_k(a,b), h_k+1(a,b) ) interleave for m+1 ≤ k ≤ N-3.Interleaving implies that h_m+2(α^(m+1)_l(b),b )h_m+2( α^(m+1)_l+1(b),b ) <0, ∀ l. In the particular case of c_m+1=0, equation <ref> implies that for any such b, h_m(α^(m+1)_l(b),b)h_m(α^(m+1)_l+1(b),b) =( 0 + b h_m+2(α^(m+1)_l(b),b) ) ( 0 + b h_m+2(α^(m+1)_l+1(b),b) ) <0Applying intermediate value theorem establishes the existence of α^(m)_l(b) ∈(α^(m+1)_l-1(b), α^(m+1)_l(b) ), ∀ 2 ≤ l ≤ N-m-2 such that h_m(α^(m)_l(b),b)=0. Since h_m(a,b) and h_m+2(a,b) both have the same leading coefficient c_N and differ by degree 2 in the highest power of a, they take same signs at ±∞: lim_a →±∞ h_m(a,b) h_m+2(a,b) = ∞. In addition, due to the negative sign of b<b_m+1<0, we have at the extremal roots of h_m+1(a,b):h_m( α^(m+1)_1(b),b )h_m+2( α^(m+1)_1(b),b ) = b ( h_m+2( α^(m+1)_1(b),b ) )^2<0h_m( α^(m+1)_N-m-2(b),b )h_m+2( α^(m+1)_N-m-2(b),b ) = b ( h_m+2( α^(m+1)_N-m-2(b),b ) )^2 <0 Due to the location of all its roots in I(b<b_m+1)=( α^(m+1)_1(b), α^(m+1)_N-m-2(b)), h_m+2(a,b) cannot change sign outside I(b). Therefore h_m(a,b) must change sign outside I(b). By the intermediate value theorem (real) roots must exist in both ( -∞, α^(m+1)_1(b) ) and ( α^(m+1)_N-m-2(b), ∞). Thus in the particular case of c_m+1=0 interleaving of the pair ( h_m(a,b), h_m+1(a,b) ) is directly established. For the general case of c_m+1 0, intermediate value theorem is not applicable using equation <ref> without b h_m+2(α^(m+1)_l(b),b) somehow dominating c_m+1. Let us now derive such a sufficient growth property for large negative b:Growth property For any 1 ≤ l ≤ N-m-2 lim_b → -∞| h_m+2(α^(m+1)_l(b), b) |→∞Proof: By contradiction. Let us fix an 1 ≤ l ≤ N-m-2. Assuming that lemma <ref> is false (equivalent tolim inf_b → -∞| h_m+2(α^(m+1)_ l(b)) | < ∞) implies that ∃ϵ >0 such that we can find at least one unbounded sequence { b_i: i ∈ℕ, b_i+1<b_i<0 }that satisfies | h_m+2(α^(m+1)_l(b_i), b_i) | < ϵ. Consequently from equation <ref> h_m+1 ( α^(m+1)_l(b_i), b_i ) = α^(m+1)_l(b_i) h_m+2(α_m+1(b_i), b_i) +b_i h_m+3(α^(m+1)_l(b_i), b_i) + c_m+2 0 =h_m+3(α^(m+1)_l(b_i), b_i)+ α^(m+1)_l(b_i)/b_i h_m+2(α_m+1(b_i), b_i) + c_m+2/b_i| h_m+3(α^(m+1)_l(b_i), b_i) |≤|α^(m+1)_ l(b_i)/b_i|| h_m+2(α_m+1(b_i), b_i) |+|c_m+2/b_i|| h_m+3(α^(m+1)_l(b_i), b_i) |≤|α^(m+1)_ l(b_i)/b_i| ϵ +|c_m+2/b_1|Writing h_m+1(a,b) as a polynomial in a,h_m+1(a,b)=∑_k=0^N'g_N'-k(b)a^k with N'=N-m-2, and then applying triangle inequality we get ∀ a0,| h_m+1(a,b) |≥| c_N|| a|^N-m-2( 1- ∑_k=1^N-m-2| g_k(b) |/| c_N|| a | ^k) | h_m+1(a,b) | >0 ∀| a | > max_1 ≤ k ≤ N-m-2[ (| g_k(b) |/N | c_N|) ^1/k] |α^(m+1)_l (b_i) | < max_1 ≤ k ≤ N-m-2[ (| g_k(b) |/N | c_N|) ^1/k] |α^(m+1)_l(b_i) /b_i| < max_1 ≤ k ≤ N-m-2[ (| g_k(b_i) |/N | c_N|| b_i|^k) ^1/k ] Since the appearance of one b while deriving the formula for h_m(a,b) from ∑_n=0^N-mc_N-m-n x^n is equivalent to continually replacing x^2 with ax+b until only linear residue is left is x, it can be seen that appearance of one b in such replacement is equivalent to losing x^2. This gives *k/2 as the upper limit of the power of b in g_k(b), where *y gives the largest integer ≤ y ∈ℝ. Thuslim_b_i→ -∞max_1 ≤ k ≤ N-m-2[( | g_k(b_i ) |/N | c_N || b_i|^k) ^1/k ] ≤lim_b_i→ -∞max_1 ≤ k ≤ N-m-2 K [( | b_i|^*k/2/N | c_N || b_i|^k) ^1/k ]for someK>0 ≤lim_b_i→ -∞K/(N | c_N |) ^1/N | b_i|^1/2 = 0 lim_b_i→ -∞|α^(m+1)_l(b_i) /b_i| = 0Thus we get a strong limiting behavior over { b_i: i ∈ℕ, b_i+1<b_i<0 }: lim_b_i → -∞| h_m+3(α^(m+1)_l(b_i), b_i) |≤lim_b_i→ -∞[ ϵ|α^(m+1)_l(b_i) /b_i| +|c_m+2/b_i|] = 0Applying limit <ref> to equation <ref> for m+2, we get similar limiting behaviorlim_b_i → -∞| h_m+4(α^(m+1)_l(b_i), b_i) |≤lim_b_i → -∞|α^(m+1)_ l(b_i)/b_i| lim_b_i → -∞| h_m+3 (α_l^(m+1)(b_i), b_i) | +lim_b_i → -∞|h_m+2( α_l^(m+1)(b_i), b_i) /b_i|+ lim_b_i → -∞|c_m+3/b_i| = 0 Continuing sequentially for larger m, we deduce for m=N-3 and m=N-2: lim_b_i → -∞ h_N-3(α^(m+1)_l(b_i), b_i) =0lim_b_i → -∞ h_N-2(α^(m+1)_l(b_i), b_i) =0 lim_b_i → -∞α^(m+1)_l(b_i) = -c_N-1/c_N If equations <ref>and and <ref> hold true simultaneously, then by equations <ref> lim_b_i → -∞ h_N-3(α^(m+1)_l(b_i), b_i) = c_N (c_N-1/c_N)^2 + lim_b_i → -∞ c_N b_i - c_N-1c_N-1/c_N + c_N-2= lim_b_i → -∞ c_N b_i + c_N-2= - ∞c_N/| c_N |To avoid contradiction between equations <ref> and <ref>, lemma <ref> must hold. □ Let us define b_m+1<0 such that ∀ b<b_m+1, | b h_m+2(α^(m+1)_l(b), b) | > | c_m+1| holds, thereby making intermediate value theorem applicable. Continuing this procedure sequentially over m, we get the set { b_m: N-3 ≤ m ≤ 0 }.b_c̅_N = min_N-3 ≤ m ≤ 0{ b_m} proves theorem <ref>. □ For m=0, lemma <ref> implies that lim_b_i → -∞| h_1(α^(0)_l(b_i), b_i) | = ∞. Thus ∃ b̅<0 such that |h_1(α^(0)_l(b), b) | > |c_0/b| holds ∀ b< b̅. Applying the intermediate value theorem proves interleaving of ( h_0(a,b), h_1(a,b) + c_0/b) for any fixed b<b̅. Since for fixed b the roots of h_1(a,b) + c_0/b and b h_1(a,b) + c_0 are the same, ( h_0(a,b), b h_1(a,b) + c_0) also interleave for any fixed b<b̅.§ EXISTENCE OF A,B SATISFYING P(C̅_N,A,B)=0 AND Q(C̅_N,A,B)=0 FROM INTERLEAVINGLet us consider values of b for which ( h_0(a,b), b h_1(a,b) + c_0) does not interleave. Clearly this non-interleaving set is bounded below by b̅. For small | b |, h_1(a,b)=-c_0/b can only be attained either at two values of a or none, due to the boundedness of the coefficients of h_1(a,b). Particularly at b=0, bh_1(a,b) + c_0=c_0 is only a constant and therefore has no roots. Thus the non-interleaving set contains some interval around b=0. Since the non-interleaving set is non-empty and bounded below, by the least upper-bound property <cit.> it has an infimum B ∈ℝ satisfying b̅≤ B <0. Since the coefficients of both h_0(a,b) and b h_1(a,b) + c_0, when written as polynomials in a, are bounded for b̅≤ b ≤ B<0, the roots of both h_0(a,b) and b h_1(a,b) + c_0 in variable a are bounded for these values of b. Thus by Bolzano-Weierstrass theorem, roots can be defined at b=B as subsequential limits of roots taken over any sequence { b_k: k ∈ℕ, b̅≤ b_k < B, b_k < b_k+1, lim_k →∞ b_k =B }. Writing the roots of b h_1(a,b) + c_0 as β_l(b) ∀ b < B, we can define limits over any converging subsequences of β_l(b_k) and α_l^(1)(b_k) indexed by σ_k and σ'_k.β_j=lim_j →∞β_l(b_σ_k),1 ≤ i ≤ N-2α_i=lim_j →∞α_l^(1)(b_σ'_k),1 ≤ i ≤ N-1α_1 ≤β_1 ≤α_2 ≤ ... ≤β_N-2 ≤α_N-1∀ b ≤ B If all the inequalities are strict, then B belongs to the interleaving set. Then ∃ t>0 such that (α_i-t, α_i+t) and (β_j-t, β_j+t) are disjoint. This implies h_0(α_i-t,B) h(α_i+t,B) <0 ( B h_1(β_j-t,B)+c_0)( B h_1(β_j+t,B)+c_0)<0 Due to continuity of bivariate polynomials, ∃ 0 <δ <-B/2 such that h_0(a,b) and b h_1(a,b) + c_0 do not change sign in inside circles of radius δ centered around (α_i ± t,B) and (β_j ± t,B), respectively. Then ∀ B ≤ b ≤ B+δh_0(α_i-t,b) h_0(α_i+t,b) <0 ∀ B ≤ b ≤ B+δ h_0(a,b)has at least one root in(α_i- t, α_i+t) ( b h_1(β_j- t,b)+c_0) ( b h_1(β_j+t,b)+c_0) <0∀ B ≤ b ≤ B+δ b h_1(a,b)+c_0 has at least one root in(β_j- t, β_j+t) But then interleaving holds ∀ B ≤ b ≤ B+δ, which contradicts the definition of B as the infimum of thenon-interleaving set. Therefor at least two neighboring quantities in equation <ref> must be equal. The value of A is given by these equal quantities at b=B, thereby proving theorem <ref>. ▪Additionally, we have proven that we can always find B<0 satisfying theorem <ref> for any c̅_N, N>2. To obtain complete factorization of f(c̅_N, x), factorization can be continued onward from f(d̅_N-2, x) until it halts due to lack in powers of x. 9 basu S. Basu and D. J. Velleman, https://www.tandfonline.com/doi/abs/10.4169/amer.math.monthly.124.8.688On Gauss's First Proof of the Fundamental Theorem of Algebra, The American Mathematical Monthly, vol 124, number 8, pages 688-694. Taylor & Francis. 2017. Pukhlikov A. V. Pukhlikov, http://mi.mathnet.ru/mp6Mat. Pros., Ser. 3, vol 1, pages 85–89. Moscow Center for Continuous Mathematical Education. Moscow 1997. Pushkar P.E. Pushkar, http://mi.mathnet.ru/mp6Mat. Pros., Ser. 3, vol 1, pages 90–95.Moscow Center for Continuous Mathematical Education. Moscow 1997. Gauss C. F. Gauss, http://www.paultaylor.eu/misc/gauss.htmlDemonstratio nova theorematis omnem functionem http://www.paultaylor.eu/misc/gauss.htmlalgebraicam rationalem integram unius variabilis in factores reales primi http://www.paultaylor.eu/misc/gauss.htmlvel secundi gradus resolvi posse. PhD thesis. Helmstedt 1799. Rudin W. Rudin, Principles of mathematical analysis (third edition), Section 1.10. McGraw-Hill. New York 1976. Sturm J.C.F. Sturm, Mémoire sur la résolution des équations numériques, Bulletin des Sciences de Férussac 11, no. 271,pages 419-425. 1829. | http://arxiv.org/abs/1704.08312v6 | {
"authors": [
"Soham Basu"
],
"categories": [
"math.CA",
"26C10, 65H04, 26B35"
],
"primary_category": "math.CA",
"published": "20170426193324",
"title": "Strictly Real Fundamental Theorem of Algebra"
} |
Trevor Dorn-Wallenstein [email protected]]Trevor Dorn-Wallenstein University of Washington Astronomy Department Physics and Astronomy Building, 3910 15th Ave NESeattle, WA 98105, USA 0000-0003-2184-1581]Emily M. Levesque University of Washington Astronomy Department Physics and Astronomy Building, 3910 15th Ave NESeattle, WA 98105, USA0000-0001-8665-5523]John J. Ruan University of Washington Astronomy Department Physics and Astronomy Building, 3910 15th Ave NESeattle, WA 98105, USAWe identify an object previously thought to be a star in the disk of M31, J0045+41, as a background z≈0.215 AGN seen through a low-absorption region of M31. We present moderate resolution spectroscopy of J0045+41 obtained using GMOS at Gemini-North. The spectrum contains features attributable to the host galaxy. We model the spectrum to estimate the AGN contribution, from which we estimate the luminosity and virial mass of the central engine. Residuals to our fit reveal a blue-shifted component to the broad Hα and Hβ at a relative velocity of ∼4800 km s^-1. We also detect Na1 absorption in the Milky Way restframe. We search for evidence of periodicity using g-band photometry from the Palomar Transient Factory and find evidence for multiple periodicities ranging from ∼80-350 days. Two of the detected periods are in a 1:4 ratio, which is identical to the predictions of hydrodynamical simulations of binary supermassive black hole systems. If these signals arise due to such a system, J0045+41 is well within the gravitational wave regime. We calculate the time until inspiral due to gravitational radiation, assuming reasonable values of the mass ratio of the two black holes. We discuss the implications of our findings and forthcoming work to identify other such interlopers in the light of upcoming photometric surveys such as the Zwicky Transient Facility (ZTF) or the Large Synoptic Survey Telescope (LSST) projects. § INTRODUCTIONActive Galactic Nuclei (AGN) are among the most luminous persistent sources of radiation in the Universe, capable of outshining their host galaxies when in a quasar state. They are hosts to supermassive black holes (SMBHs) and are found throughout the history of the universe from redshift z∼7 onward <cit.>. With the advent of surveys like the Sloan Digital Sky Survey (SDSS, ), the number of cataloged AGN has increased by many orders of magnitude. As incredibly powerful sources of ionizing radiation, AGN drive and regulate the evolution of the stars, gas, and dust of their host galaxies. The major merger of two gas-rich galaxies can trigger intense dust production and star formation, while the increased accretion onto the central black hole of one or both galaxies can increase its luminosity, triggering outflows and regulating star formation <cit.>, leaving behind a massive, gas-poor elliptical remnant. Such mergers appear to be not only frequent, but the primary means by which both SMBHs and galaxies are grown <cit.>. If both galaxies in a merger contain SMBHs, simulations indicate that the black holes themselves can merge over ∼Gyr timescales <cit.>. At early times, the SMBHs in a merger will appear as dual or offset AGN (depending on the accretion rate of both black holes, ). As their orbits decay, the black holes can form a supermassive black hole binary (SMBHB), which could be observed as an apparently single AGN that displays periodic variability. We present here spectroscopic and time-domain analyses of an AGN behind M31 that has been previously misidentified as a red supergiant, a globular cluster, and an eclipsing binary. We find evidence for the periodic variability of the AGN and discuss the implications of its misidentification in light of forthcoming large photometric surveys. §.§ J0045+41 As a part of a search for red supergiant X-ray binaries — a still-theoretical class of exotic stellar binary system — we used the single-epoch photometry of the Local Group Galaxy Survey (LGGS, ), which covers M31, M33, the Magellanic Clouds and 7 dwarf galaxies in the Local Group, to assemble a statistical sample of Local Group red supergiants (RSGs). We used the method of <cit.> to reduce contamination from the far more prevalent foreground M-dwarfs by taking advantage of the separation of the two populations in B-V vs. V-R space. After creating our sample (and ensuring our results agreed within M31, where we find 437 candidate RSGs), we searched the Chandra Source Catalog (CSC, ) for X-ray sources within 10” of the LGGS RSGs. This search yielded one close match. LGGS J004527.30+413254.3 (α = 00^ h45^ m27^ s.30, δ = +41^o32'54”.31, Figure <ref>), which we will refer to as J0045+41 hereafter, is a bright (V ≈ 19.9) object of previously-unknown nature in the disk of M31. <cit.> classified J0045+41 as an eclipsing binary with a period of ∼76 days. While the observed variability is of order 1 magnitude in B and V, their data are poorly sampled in phase. On the other hand, <cit.> included J0045+41 in a catalog of candidate globular clusters, and it has also been included in catalogs of M31 globular clusters as recently as 2014 <cit.>. The LGGS photometry was consistent with the color and brightness of a typical 12-15 M_⊙ RSG in M31, with an inferred effective temperature of ∼3500 K and bolometric magnitude of -6.67 (following ). However, the best SED fit to photometry from the Panchromatic Hubble Andromeda Treasury (PHAT, ) using the Bayesian Extinction And Stellar Tool (BEAST, ) is a 300 M_⊙, 10^ 5 K “star”, extincted by A_V∼4 magnitudes, which we exclude as being unphysical. This discrepancy is likely due to the broader wavelength coverage of the PHAT dataset, as well as the fact that the BEAST performs a complete SED fit, whereas our RSG selection criteria are purely based on color and magnitude cuts to select for bright, red objects roughly consistent with the photometric properties of RSGs. Furthermore, the object appears extended in the PHAT images (though its radial profile appears similar to that of other nearby stars; see Figure <ref>), implying that J0045+41 may be a background AGN or quasar. Given the angular size of M31 at optical wavelengths (∼10 deg^2) and the typical surface density of bright quasars on the sky (∼18 deg^-2, ), we expect ∼180 sources in the entirety of M31 to actually be background AGN.J0045+41 is separated by ∼1.18^'' (4.45 pc at the distance of M31) from an X-ray source in the <cit.> catalog. This source, CXO J004527.3+413255 (α = 00^ h45^ m27^ s.30, δ = +41^o32'55”.46), is bright (F_X = 1.98×10^-13 erg s^-1 cm^-2) and has hardness ratios from <cit.> that are consistent with an unabsorbed X-ray binary or AGN. To confirm this, we fit a spectrum from the publicly available ChandraPHAT dataset (Obs. ID 17010, ) with an absorbed power law model (xstbabs * powlaw1d) in Sherpa <cit.>. We use the atomic cross-sections from <cit.>, and abundances from <cit.>. The spectrum is binned to ensure each bin has a minimum of five counts, and we fit the background-subtracted spectrum from 0.3 to 8 keV. The best-fit (χ^2_red = 0.34) model has a neutral H column density N_H = 1.7×10^21 cm^-2 and a power-law slope Γ = 1.5. The spectrum and fit are shown in Figure <ref>. The value of N_H derived from the fit corresponds to an extinction of A_V < 1, which would be surprising if CXO J004527.3+413255 was a background AGN or quasar seen through the disk of M31, as we would expect a significantly higher column density. In addition, using the ChandraPHAT data, Williams et al. (in prep) derive improved source locations and positional errors, resulting in a much better alignment between CXO J004527.3+413255 and J0045+41 (see Figure <ref>)To conclusively determine the nature of J0045+41, we decided to obtain optical spectrophotometry. We discuss our observations and data reduction in <ref>. We present the spectrum, use it to classify J0045+41 as an AGN, identify key features, and analyze it in <ref>, and search for evidence of periodicity using archival data in <ref>. We conclude with a discussion of our results and their implications in <ref>.§ OBSERVATIONS AND DATA REDUCTIONWe obtained a longslit spectrum of J0045+41 using the Gemini Multi-Object Spectrograph (GMOS) on Gemini-North <cit.>. Four 875 second exposures were taken 2016 July 5 using thegrating centered on 5000 Å, and four 600 second exposures were taken 2016 July 9 using thegrating centered on 7000 Å, with a blocking filter to remove 2^ nd-order diffraction. Two of each set of exposures were offset by +50 Å to fill in the gaps between the three CCDs in GMOS. We followed the standard GMOS-N reduction pipeline using thepackage in<cit.>. Flux calibration was performed using HZ 44 <cit.> as a standard star for both sets of observations. The final reduced spectrum is continuous from ∼4000 to ∼9100 Å at a resolution of R ∼ 1688 ( blue)/1918 ( red). § SPECTRUM AND ANALYSIS The optical spectrum is shown in Figure <ref>. It shows the broad emission lines characteristic of an AGN. We use Ca2 H & K, the Fe1/Hγ/[O3] G band, [O3] λ5007, Mg1 λλ5192,5197, Na1 D, and He1 λ7067 to determine that J0045+41 is at z ≈ 0.215. We also detect Na1 D doublet absorption in the rest frame of the Local Group; however our data are not of sufficient resolution to distinguish Milky Way from M31 absorption. Both Hα and Hβ are broad, with full widths at half maximum of ∼10^4 km s^-1. The centers of broad Hα and Hβ are slightly blueshifted (z ≈ 0.21) relative to the rest of the spectrum, which may be indicative of an outflow or motion of the central engine relative to the host galaxy.Mistaking a blue AGN for a red star might seem unsurprising given that it is seen through the disk of M31. However, the low amount of extinction implied from the fit to the X-ray spectrum seems inconsistent with an object seen through an entire galactic disk. In Figure <ref>, we show our spectrum of J0045+41 compared with the composite Sloan Digital Sky Survey (SDSS, ) quasar template spectrum from <cit.> as well as a template Seyfert 2 spectrum from PySynphot (a Python implementation of Synphot destributed by Space Telescope Science Institute, ), both redshifted to z=0.215 and reddened by 1 (top) and 2 (bottom) magnitudes of extinction in V using a standard <cit.> R_V = 3.1 extinction law. While hardly a robust fit, this comparison serves to illustrate that either a larger value of extinction is required to reproduce the overall spectral shape of J0045+41 with a pure QSO template or that many of the spectral features — e.g., the apparent break in spectral slope at ∼5500 Å and the presence of strong absorption lines in the spectrum – are intrinsic to the host galaxy. To decompose the spectrum into host and AGN spectra, we follow <cit.>. We use the first five galaxy eigenspectra and the first ten QSO eigenspectra derived from a Principal Component Analyses (PCA) of SDSS galaxy and quasar samples <cit.> as a set of basis spectra, which we redden using the <cit.> extinction law, redshift to z=0.215, and fit to our spectrum of J0045+41 as follows. If the measured fluxes are represented by a column vector, f, then the residuals between the data and the basis spectra fit is simplyE = f - G· cwhere G is a matrix whose columns are the redshifted and reddened basis spectra interpolated to the values of the observed wavelengths in our spectrum and c is a column vector containing the coefficients for each basis spectrum. Taking the errors on each point into account, the scaled residual at each point can be represented by the scalarR = E^TΣ^-1Ewhere Σ is the covariance matrix and E^T denotes the matrix transpose. It can be shown that the coefficients that minimize R are given byc = (G^TΣ^-1G)^-1(G^TΣ^-1)· fIn order to estimate a suitable value of A_V to use when reddening the basis spectra, we redden the spectra with integer values of 0≤ A_V≤ 10 mag. Some of these fits are shown in Figure <ref>. While the basis spectra sufficiently fit the spectrum for 0 ≤ A_V ≤ 2 mag, at higher values, the basis spectra are unable to reproduce the spectral shape, especially in the blue. Going forward, we adopt A_V = 1 mag. <cit.> mapped the dust extinction in M31 at a resolution of 25 pc using the PHAT dataset. They model the probability distribution of A_V in each pixel with a log-normal distribution, parametrized by the median extinction, Ã_V and the dimensionless width, σ, such that the mean extinction ⟨ A_V⟩ is ⟨ A_V⟩ = Ã_Ve^σ^2/2 and the variance in the extinction σ_A^2 isσ_A^2 = Ã_V^2e^σ^2(e^σ^2 - 1)<cit.> also include the fraction of stars in each pixel that are reddened, f_red. In the pixel containing J0045+41, f_red = 0.206, Ã_V = 0.72, and σ = 0.28. The latter two values correspond to ⟨ A_V⟩ = 0.75, σ_A = 0.21, consistent with our estimate of A_V. Spectral modeling at higher resolution would further constrain the extinction along the particular line of sight towards J0045+41.The galaxy and AGN components of this fit are shown in the top panel of Figure <ref>. The bottom panel shows the dereddened rest-frame luminosity spectrum of each component. The luminosity of the underlying AGN component is L_λ = 3.46× 10^39 erg s-1 Å^-1 at 5100 Å. The derived host galaxy spectrum appears similar to an early type galaxy. This is unsurprising as the hosts of low-luminosity AGN (like J0045+41) tend to be early type <cit.>. If the periodicity (discussed in Section <ref>) arises from a SMBH binary formed through the major merger of two late type AGN hosts, it would also be unsurprising that the resulting host is an early type galaxy.With the underlying contribution to the spectrum from the central engine now known, it is possible to estimate the mass of the SMBH <cit.>. We use the full width at half maximum of Hβ (1.11× 10^4 km s^-1), the continuum rest frame luminosity from the quasar at 5100 Å, and the Hβ virial mass estimator coefficients from <cit.> to calculate log(M/M_⊙) = 8.30. We use the bolometric correction from <cit.> to calculate the bolometric luminosity, from which we determine the Eddington ratio Γ≡ L_bol / L_Edd = 0.007. This small value for Γ may indicate that the accretion flow is radiatively inefficient <cit.>. § POTENTIAL PERIODICITY §.§ Searching for Periodicity Using the Supersmoother Algorithm Though the light curve in <cit.> is sparsely sampled, the suggestion of a ∼76 day period in J0045+41 prompted further investigation. While continuum emission from AGN is well-known to be stochastically variable due to a variety of phenomena associated with the central engine and surrounding environment, periodicities in the variability have long been predicted as a signature of SMBHBs (e.g., ). A short-period SMBH system would be well within the gravitational wave regime. We investigated the reported periodicity using data from the Palomar Transient Factory (PTF, ). PTF observed J0045+41 in both g and r, though the g-band data cover a broader range in time, and thus we focus our analyses solely on those data. These data are shown in Figure <ref>.AGN continuum variability is well fit by a damped random walk (DRW) process <cit.>, described by a characteristic timescale (τ) and long-term rms variability (σ or SF_∞ = √(2)σ). The power spectral distribution (PSD) of a DRW process <cit.> isPSD(T) = 4σ^2τ/1 + 4π(τ/T)^2and the covariance function isS(Δ t) = σ^2e^-|Δ t|/τwhere Δ t is the time between two observations.Previous searches for periodicities in AGN lightcurves commonly use Lomb-Scargle periodograms (). Lomb-Scargle periodograms detect periodicities in irregularly-sampled lightcurves by fitting sinusoids to the data <cit.>. It is important to note that sinusoidal variability is expected if the periodicity arises due to the relativistic Doppler boost of the emission of the secondary component of a steadily-accreting binary (see ). However, the predicted periodicity from SMBHBs is not necessarily sinusoidal if caused by periodic episodes of accretion (e.g., ). Furthermore, <cit.> show that the behavior generated by red noise processes can be well fit by a sinusoid over a few `cycles'. Therefore the statistical significance of previously-reported detections using Lomb-Scargle periodogram analysis may be overestimated. To provide a robust assessment of periodicities in the lightcurve of J0045+41, we utilize the Supersmoother algorithm <cit.>, which uses a non-parametric periodic model to test the strength of signals at various periods. Using the implementation in the gatspy Python package <cit.>, we calculate the periodogram of the g-band data on a linearly spaced grid of 2000 periods between 60 and 1000 days — we are unlikely to see periods shorter than 60 days (see ), and our data do not cover more than two cycles of a signal with more than a 1000 day period. The periodogram is shown in Figure <ref>. As expected by a DRW signal, the power appears to rise to a constant level at long periods. However, there do appear to be real peaks superimposed onto the expected DRW behavior.§.§ Estimating the Significance of Measured Peaks To check that the measured power of the true signal (P_S(T)) is not attributable to a DRW process, we generate simulated DRW lightcurves, following the prescription of <cit.>, and compare the distribution of the periodograms of the simulated lightcurves to P_S(T). While it is possible to calculate the DRW parameters, σ and τ, from the estimated mass of J0045+41, we choose to instead estimate those parameters by fitting the lightcurve directly, thus incorporating the distribution of possible values. We implement (<ref>) as a kernel function in celerite <cit.>, a Python package for Gaussian process computations, which calculates the likelihood, ℒ, of a DRW with given σ and τ:lnℒ = -1/2r^TK^-1r - 1/2ln |K| - Cwhere r is a vector of the observed data minus the mean, K is the covariance matrix incorporating the photometric errors and the DRW covariance function, and C is a constant proportional to the number of measurements (for a discussion of Gaussian processes and the derivation of this likelihood function, see ). We then use emcee <cit.>, an affine-invariant MCMC Python package, to fit for σ, τ, and the mean magnitude ⟨ g⟩ by sampling the posterior distribution. We use 32 walkers, and, after discarding 500 burn-in steps, record 3000 samples per walker for a total of 96,000 samples. A corner plot of these samples is shown in Figure <ref>.Drawing the value of σ, τ and ⟨ g⟩ from the posterior distribution of samples, we generate 96,000 DRW lightcurves. The lightcurves are sampled at the same times as the PTF observations and have identical photometric errors. The final points in the simulated lightcurve are then drawn from a Gaussian distribution with the magnitude of the raw point as the mean, and standard deviation equal to the photometric error. We then calculate periodograms for each simulated DRW lightcurve on the same grid of periods as P_S. The mean (P_DRW) and standard deviation (P_σ) of the simulated periodograms are plotted as P_DRW± P_σ along with P_S and the theoretical DRW PSD with σ = 0.2, τ = 200 days (scaled to match the values returned by Supersmoother) for comparison in the left panel of Figure <ref>. Much of the structure in the true periodogram is matched by the simulated periodograms, but not in the theoretical PSD. This is likely due to the irregular sampling of the PTF lightcurve, which is reflected in the simulated lightcurves. However, some of the peaks in the true periodogram do not appear in the DRW noise.To identify periods with power in excess of the DRW noise, we search for peaks in σ = (P_S - P_DRW)/P_σ. σ(T) is plotted in the right panel of Figure <ref>, with the ten peaks with largest σ indicated by blue triangles. As Supersmoother only returns values between 0 and 1 when it calculates the periodogram — and thus the values are not normally distributed — σ as a statistic is meaningless by itself. We instead want to estimate the false-alarm probability (FAP) of each peak. Traditional estimates of significance (see , for example) assume that the null hypothesis is pure white noise. Because the background noise is dependent on the period, we split the grid of periods into N_trial = 100 bins with 20 periods each. In each bin, we find the period T associated with the largest value of P_S. We then calculate the number of simulated periodograms that have at least one point with power greater than P_S(T) (N_DRW(>P_S(T))) within the period bin. The FAP is thus N_DRW(>P_S(T)) divided by the number of simulated DRW periodograms (N_DRW = 96,000) times N_trial,which accounts for the fact that there are N_trial× N_DRW `chances' to randomly generate a peak with more power than the true peak (the look-elsewhere effect).§.§ Distinguishing Periodicity from Systematics The above process results in a number of periods that correspond to local minima in FAP vs. T, shown as blue triangles in the right side of Figure <ref>. Between the sampling of the lightcurve and the algorithm used to generate the DRW lightcurves, it is possible that some of these detections are only arising due to artificial suppression of the DRW noise. To determine this, we use the same algorithm to simulate white noise lightcurves (τ→ 0, with σ and ⟨ g⟩ drawn from the DRW samples in Figure <ref>), and calculate the average (P_WN) and standard deviation (P_σ,WN) of the periodograms. P_WN± P_σ,WN is shown in purple on the left panel of Figure <ref>. It is clear that P_WN and P_σ,WN are roughly constant over the range of tested periods, and thus that none of the detected periodicities arise due to suppression of the DRW noise.It is also possible that the period detected at T = 354.8 days is due to approximately yearly systematic variations in observing conditions — e.g., airmass, observability, weather, etc. — at Palomar Observatory, and that the period at T = 708.5 ≈ 2×354.8 is an alias of the same effects. This appears to be reflected in Figure <ref>, where the phase-sampling of both the g and r band data is nearly identical at these periods. Because J0045+41 is nearly at the detection limit of PTF, it is certainly possible that those systematics can masquerade as real effects; our discussion of these results comes with the major caveat that the ∼yearly periodicity may not be real. However, even discounting the 354.8-day period, there is a secondary peak at 328 days that is unlikely to be a result of these yearly systematics. Finally, if these periods are real, they should be detectable by other means. We add a sinusoidal mean model to our implementation of the DRW kernel within celerite, and simultaneously sample the posterior distribution of the model parameters — mean, amplitude, period, and phase — and the DRW parameters as described above using emcee, using double the number of walkers, and restricting the period of the sinusoid to lie between 60 and 1000 days. As discussed above, a sinusoidal model is not necessarily an accurate one; however, the periods revealed by this analysis should be similar to the periods found above. A histogram of the posterior distribution of the period is shown in Figure <ref>, with the periods with local minima in FAP indicated by blue triangles. It is clear that at least some of the peaks found — namely at T = 82.1,117.8,202.0,328.0,354.8,and,708.3 days — are retrieved. The phase-folded, mean-subtracted data and the best-fit Supersmoother model at the six periods detected with celerite, along with the phase-folded r-band are shown in Figure <ref>. Table <ref> contains the period T, the value of P_S(T), the bounds of the period bin containing T (T_min and T_max), the estimated FAP, and whether a strong peak in the celerite posterior appears at a similar period.lccccResults from <ref>. T is the period, P_S(T) is as described in the text, T_min and T_max are the bounds of the period bin in which the FAP is calculated. The last column shows whether the period is detected using a DRW + sinusoidal mean model in celerite. T P_S(T)(T_min,T_max) FAP Detected with daysdays celerite?82.10 0.120592(78.809,79.280) 6.98469 × 10^-3 Yes117.84 0.139525(116.428,116.898) 7.14281 × 10^-3 Yes162.04 0.148967(154.047,154.517) 7.78917 × 10^-3 No202.01 0.212229(201.071,201.541) 4.72885 × 10^-3 Yes328.03 0.233829(323.332,323.802) 3.30188 × 10^-3 Yes354.84 0.270498(351.546,352.016) 1.01854 × 10^-3 Yes409.85 0.248934(407.974,408.444) 4.27479 × 10^-3 No702.34 0.281859(699.520,699.990) 4.59042 × 10^-3 Yes867.86 0.300183(859.400,859.870) 5.84198 × 10^-3 NoThe period of ∼82.1 days (FAP∼0.007) is similar to <cit.> who find a period of ∼76 days. We plot the PTF data, the historical data from <cit.> (offset by a constant for clarity), and the best-fit Supersmoother model folded on the period found by <cit.> in Figure <ref>. None of the structure in the <cit.> data is seen in the PTF data or the Supersmoother fit; however, with so few observations, it is possible that the true period detected by <cit.> is closer to that detected in the PTF data. Unfortunately, the historical data are only available phase-folded, and we are unable to include them in our analysis of other periods.§ DISCUSSION AND CONCLUSIONOne possible interpretation of a periodic signal in an AGN is that it is due to the orbital motions of a SMBHB, formed through a major galaxy merger. Though small, the number of z < 1 candidate SMBHBs discovered is consistent with this model <cit.>. The detected periodicities of J0045+41 are thus quite interesting. Most intriguingly, the ∼82.1 day period is almost exactly in a 1:4 ratio with the ∼328-day period. It is possible that either of these peaks is an alias of the other, as the observed periodogram is the convolution of the true periodogram with the Fourier transform of the sampling function <cit.>. However, multiple periodicities beyond the orbital period are predicted to occur in SMBHBs at similar period ratios as a result of interactions in the circumbinary disk <cit.>. In particular, <cit.> found that the periodogram of the accretion rate in their simulation displayed significant peaks at frequencies approximately generated by the formula ω = 2/9KΩ_bin, where Ω_bin is the binary orbital angular frequency, and K = 1,2,6,7,8,9,10. We search for the orbital period, T_bin, that generates a set of periods closest to the first five observed periods (discounting the 354.8 and 708.5 day periods). We find that T_bin = 169.29 (Ω_bin = 3.7×10^-2 day^-1) creates periods that match quite well with the two shortest periods, though it underpredicts the 202 day period by ∼75 days, and overpredicts the 328 day period by ∼50 days. Finally, <cit.> find that, for varying mass ratios and simulation setups, periodic variations in the accretion rate onto one or both black holes can arise at frequencies with the same 1:4 correspondence as the 82 and 328 day periods. These occur at 1/4Ω_bin and Ω_bin. This points to the 82.1 day period being the orbital period of the binary. <cit.> also find frequencies arising at Ω_bin and 2Ω_bin. Interestingly, we do detect a period with FAP∼0.008 at 162≈2×82≈1/2×328 days. While we do not detect a strong peak in the celerite posterior around this period, this hints that the orbital period may also be 162 or 328 days.If we assume that any of these three periods is the orbital period of a SMBHB in a circular Keplerian orbit, and that the virial mass derived in <ref> is the total mass of the two black holes M_tot, then the semimajor axis of the orbit ranges from 216 to 544 AU (or 0.3 to 1 microarcseconds at the angular diameter distance of J0045+41, which is unresolvable using current radio interferometric arrays). Such a separation would be well within the regime where loss due to gravitational radiation is significant. We can approximate the time for two circularly orbiting black holes to inspiral due to gravitational radiation using equations (5.9) and (5.10) from <cit.>:[t_GW = 5/256c^5/G^3R^4/(M_1 + M_2)(M_1M_2); =5/256c^5/G^3R^4/M_tot^3(1+q)^2/q ]where R is the semimajor axis of the orbit, M_1, M_2 are the masses of the individual black holes and q ≡ M_2/M_1. t_GW ranges between ∼350 yr (for the shortest period, with q = 1) to 360 kyr (for the longest period, with q = 0.01). The gravitational waves produced by SMBHBs are expected to be detectable at the nHz frequencies probed by pulsar timing arrays (PTAs, ). The amplitude of the dimensionless gravitational strain (h_0) of a SMBHB with mass ratio q at redshift z, assuming a circular orbit with period T can be expressed ash_0 = 4G/c^2qM_tot/(1+q)^2D_L(z)(2π GM_tot/c^3 T)^2/3where D_L(z) is the luminosity distance <cit.>. The expected strain of a SMBHB with the derived mass and orbital period of J0045+41 would range from ∼10^-16 (for the shortest detected period, with q = 1) to ∼10^-18 (for the longest period, with q = 0.01). These results, in addition to the expected orbital velocity of the secondary black hole (see below) are summarized in Table <ref>. While the latter strain would be orders of magnitude below the stochastic background of gravitational radiation from all SMBHBs at that period (h ≈ 10^15 at T = 1 yr, ), the background falls off at higher frequencies as fewer sources are expected to be inspiraling at shorter and shorter periods, and the signal from a ∼80 day SMBHB would be detectable above the background <cit.>. Indeed, the signal would be just shy of the anticipated sensitivity — ∼6×10^-16 <cit.> — of the Square Kilometer Array (SKA, ). While this is an exciting finding, it is important to note that there are a number of other possible interpretations of a periodic signal, e.g: a long-lived or periodically-generated hot spot in the accretion disk, geodetic precession, and self-warping of the disk (seefor a concise review). lccccOrbital and gravitational properties of proposed orbital periods T R/θ v_orb t_GW h_0 days AU/μarcsec 10^3 km s^-1yr(q = 1/0.01) (q = 1/0.01) (q = 1/0.01)82.10 216.02/0.30 14.312/28.341 3.522 × 10^2/8.982 × 10^3 9.252 × 10^-17/3.628 × 10^-18 162.04 339.90/0.47 11.410/22.594 2.159 × 10^3/5.505 × 10^4 5.880 × 10^-17/2.306 × 10^-18 328.03 543.93/0.75 9.020/17.860 1.416 × 10^4/3.610 × 10^5 3.674 × 10^-17/1.441 × 10^-18 Even if it is not a SMBHB, J0045+41 is an interesting object. For one, it appears to be probing a relatively extinction-free region of the ISM in M31. The detection of the Na1 D doublet is promising, and follow-up optical and infrared observations at higher spectral resolution may disentangle absorption from M31 and from the Milky Way, and reveal more about the dynamics of the ISM along the line of sight towards J0045+41. The spectrum is well fit by a mixture of the galaxy and quasar eigenspectra from <cit.> redshifted to z = 0.215 and reddened by an A_V = 1.0±1.0 mag <cit.> extinction law. However, Hα and Hβ both have a blueshifted broad component. Indeed, the residuals to the fit shown in Figure <ref> appear to be Gaussian. Fitting these residuals with a Gaussian profile shows that this component is at z = 0.196, a ∼4800 km s^-1 difference from the host redshift. This shift may be due to an outflow from the central engine, a hot spot in the blueshifted side of the accretion disk, or the blending of the broad lines of each SMBH component; as the less massive SMBH moves towards us, we would see its broad lines blueshifted, which would explain the excess of blue flux in the broad lines <cit.>. Indeed, a similar binary model has been used to explain SDSS J092712.65+294344.0, which also appears to have blueshifted broad lines relative to the narrow lines in the spectrum <cit.>. At the short periods found in <ref>, orbital velocities are expected to be ∼10^4 km s^-1 (depending on the assumed mass ratio), so this blueshift would be consistent with the orbital velocities for all of the periods in Table <ref>, for any value of the mass ratio. Follow-up spectroscopy on a cadence of a few months would be able to search for or exclude periodic changes of the Hα and Hβ profiles relative to the narrow lines over time, which would help point to an explanation.To search for any objects similar to J0045+41 in color space, we used PySynphot (a Python implementation of Synphot distributed by Space Telescope Science Institute, ) to generate synthetic photometry from our spectrum in g, r, i, and z — there was not enough signal in u to synthesize a magnitude. We then downloaded photometry of all low-redshift (z<1) SDSS quasars from Data Release 13 <cit.> within 0.1 magnitudes of J0045+41 in g-r vs. r-i vs. i-z color space. These quasars are shown in color space in Figure <ref>. Each point is colored by the assumed value of the extinction in g. Of these 446 objects, only 197 of them have redshifts that are positive — implying the remaining objects are not plausibly quasars. Indeed, the spectra of many of the `quasars' in this sample are quite clearly cool stars. Some of these objects are simply misidentified; however, many are flagged with a Z_WARNING: NOT_QSO by the SDSS pipeline. While this is helpful for reducing contamination of the quasar sample, it illustrates than many objects of interest fall through the cracks of classification algorithms (seefor further discussion). Of the true quasars in the sample, none are extincted by more than 1.5 magnitudes in g. It is likely that these quasars (and the AGN component of J0045+41) are intrinsically red as described by <cit.>. These quasars may have been reddened by dust intrinsic to the host galaxy, or have excess red flux due to synchrotron emission with an optical turnover. Higher resolution spectroscopic follow-up would allow for more detailed fitting of J0045+41 to determine if a red quasar template yields a better fit.The confusion of stars and quasars represents a unique problem for purely photometric surveys, such as the upcoming LSST project <cit.>. Stars and higher redshift (z>2.2) quasars are well separated in color space. However, at lower redshifts, the two color loci appear closer and closer. The difference between the two populations is most apparent in u-band flux and u-g color; indeed the u filter was designed in part to leverage the difference between power-law spectra and spectra with strong Balmer decrements <cit.>. Thus, in any single-visit catalog, the colors of the lower-redshift, low-luminosity, and intrinsically red AGN are the hardest population to distinguish from stars. LSST will visit most of its survey area ∼50-180 times in each filter over 10 years. <cit.> demonstrated that it is possible to use variability in addition to colors to distinguish stars from AGN with high (>90%) completeness. However, the accuracy of classifications in the lowest redshift bins studied drops to ∼80%. While the number of quasars at low redshift is small, this highlights the importance of developing accurate classification algorithms for objects similar to J0045+41. Forthcoming work will focus on distinguishing between stars and quasars in the low-redshift, low-luminosity, red regime.J0045+41 is an exciting and unique object. It represents an extreme end of color space in which photometric classification methods fail. Both the simple selection methods (described in <ref>) and more sophisticated machine learning algorithms are unable to correctly classify objects in this regime. Finding these intrinsically red AGN is important, as they are still poorly understood. The evidence of multiple periodic signals in the photometric lightcurve of J0045+41 is compelling, and warrants more dedicated spectroscopic observations at higher spectral resolution and deeper photometric observations sampled at a higher rate. Such observations would be crucial to confirm the presence of a SMBHB in J0045+41. They would also allow for the confirmation of the periods that we detected. The photometric data will soon be attainable in the form of the Zwicky Transient Facility (ZTF, ), a next-generation transient survey that will see first light this year. The authors thank Jessica Werk, Julianne Dalcanton, Ben Williams, Jake VanderPlas and Scott Anderson for their valuable advice and feedback on this work. We wish to thank the anonymous referee for their extremely helpful comments.Based on observations (Program ID GN-2016A-FT-30) obtained at the Gemini Observatory (processed using the Gemini IRAF package), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil). The authors thank the Gemini-North support staff. The scientific results reported in this article are based in part on data obtained from the Chandra Data Archive and the SDSS archive. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org.SDSS-IV is managed by the Astrophysical Research Consortium for theParticipating Institutions of the SDSS Collaboration including theBrazilian Participation Group, the Carnegie Institution for Science,Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics,Instituto de Astrofísica de Canarias, The Johns Hopkins University,Kavli Institute for the Physics and Mathematics of the Universe (IPMU) /University of Tokyo, Lawrence Berkeley National Laboratory,Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg),Max-Planck-Institut für Astrophysik (MPA Garching),Max-Planck-Institut für Extraterrestrische Physik (MPE),National Astronomical Observatories of China, New Mexico State University,New York University, University of Notre Dame,Observatário Nacional / MCTI, The Ohio State University,Pennsylvania State University, Shanghai Astronomical Observatory,United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona,University of Colorado Boulder, University of Oxford, University of Portsmouth,University of Utah, University of Virginia, University of Washington, University of Wisconsin,Vanderbilt University, and Yale University. 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"authors": [
"Trevor Dorn-Wallenstein",
"Emily M. Levesque",
"John J. Ruan"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170427180001",
"title": "A Mote in Andromeda's Disk: a Misidentified Periodic AGN Behind M31"
} |
[pages=1-last]Arxiv_finite_time_ratio_consensus_with_delay_final_v2.pdf | http://arxiv.org/abs/1704.08297v1 | {
"authors": [
"Mangal Prakash",
"Saurav Talukdar",
"Sandeep Attree",
"Sourav Patel",
"Murti V. Salapaka"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170426185629",
"title": "Distributed Finite Time Termination of Ratio Consensus for Averaging in the presence of Delays"
} |
#1#1 equationsection definDefinition theoremTheorem noticeNotice lemmaLemma corCorollary exampleExample remarkRemark conjConjecture Proof:height5pt width5pt depth0pt÷∇· ∇× sign arsinh arcosh diag const ḍε ϕφ θϑℕ ℝ .24em.1exheight1.3ex width.9ptC I.16em P .24em.1exheight1.3ex width.9ptQ I.16emIM Z.20em ZPART:#1#2∂ #1/∂ #2#1#2#3∂^#3 #1/∂ #2^#3#1D #1 Dt∇_x∇_v∇ #1 #1#10=#1-.025em0-0 -.05em0-0 -.025em.0433em0 #1(<ref>)1/2#11/#1#1⟨ #1 ⟩#1#2#̣1/#̣2Δḍℕℝ𝕋ε𝔼Å𝒜u̅^*ξ_A_F_D_ w_u^_∞ A B x yα·Åℬℋ̋Łℒ#1 [#1]Well posedness and approximation for quantitative traits]Well posedness and Maximum Entropy Approximation for the Dynamics of Quantitative TraitsBoďová]Katarína Boďová[Katarína Boová] Institute of Science and Technology Austria (IST Austria), Klosterneuburg A-3400, [email protected]]Jan Haskovec[Jan Haskovec] Computer, Electrical and Mathematical Sciences & EngineeringKing Abdullah University of Science and Technology, 23955 Thuwal, [email protected]]Peter Markowich[Peter Markowich] Computer, Electrical and Mathematical Sciences & EngineeringKing Abdullah University of Science and Technology, 23955 Thuwal, [email protected][] We study the Fokker Planck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The Fokker-Planck equation is posed on a bounded domain and its transport and diffusion coefficients vanish on the domain's boundary. We first argue that, despite this degeneracy, the standard no-flux boundary condition is valid. We derive the weak formulation of the problem and prove the existence and uniqueness of its solutions by constructing the corresponding contraction semigroup on a suitable function space. Then, we prove that for the parameter regime with high enough mutation rate the problem exhibits a positive spectral gap, which implies exponential convergence to equilibrium.Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy (DynMaxEnt) method for approximation of moments of the Fokker-Planck solution, which can be interpreted as a nonlinear Galerkin approximation. The limited applicability of the DynMaxEnt method inspires us to introduce its modified version that is valid for the whole range of admissible parameters. Finally, we present several numerical experiments to demonstrate the performance of both the original and modified DynMaxEnt methods. We observe that in the parameter regimes where both methods are valid, the modified one exhibits slightly better approximation properties compared to the original one. [ [ December 30, 2023 ===================== § INTRODUCTIONThe dynamics of allele frequencies = (x_1, …, x_L),where L is the number of loci that contribute to the trait, can be described by a diffusion process using a deterministic forward Kolmogorov equation. The evolution of the joint probability density u = u(t,) of allele frequencies for a population of N diploid individuals satisfies the linear Fokker-Planck equation PART:ut = - 1/2∑_i=1^L PART:x_i( ξ_i PART:()x_i u ) + 1/4N∑_i=1^L x_i2 (ξ_i u), on Ω_ := (0,1)^L, where we denoted ξ_i := ξ(x_i) = x_i(1-x_i) for i=1,…, L. The diffusion term captures the stochasticity of the allele frequencies arising from random sampling.Here we assume that linkage disequilibria are negligible, otherwise this term would be of cross-diffusion type, reflecting correlations between loci <cit.>. The drift term captures deterministic effects on allele frequencies that are described by a vector of coefficientsand a vector of complementary quantities . We consider directional selection and dominance with symmetrical mutation, which, using the notation of <cit.>, corresponds to the choice= (ξ'_1, …, ξ'_L, ξ_1,…,ξ_L, lnξ_1,…,lnξ_L)and = - β∑_i=1^L γ_i ξ_i' + 2h ∑_i=1^L η_i ξ_i + 2μ∑_i=1^L lnξ_i, where the nondimensional parameters β, h, γ_i, η_i ∈ represent the effects of loci on the traits, μ>0 is the mutation rate, and ξ_i' := ξ'(x_i) = 1-2x_i. For notational simplicity and without loss of generality, we set β = h = 1 in the sequel, so that= (-γ_1, …, -γ_L, 2η_1,…,2η_L, 2μ,…,2μ)∈^3L . This drift-diffusion process (<ref>) is known to be an accurate continuous-time approximation to a wide range of specific population genetics models <cit.>.In order to represent the population in terms of allele frequencies, we must assume that linkage disequilibria are negligible, which will be accurate if recombination is sufficiently fast. For simplicity, we also assume two alleles per locus.The main difficulty for analysis of the Fokker-Planck equation (<ref>) is the degeneracy of the diffusion coefficients ξ_i = x_i(1-x_i) at the boundary of Ω_. Consequently, the task of prescribing boundary conditions that lead to a well-posed problem is far from obvious; see also <cit.> for related issues in population genetics problems. As noted above, we aim at interpreting the solution u as a time-dependent probability density, which calls for a no-flux boundary condition. In Section <ref> we argue that the standard no-flux boundary condition is indeed appropriate for (<ref>). In Section <ref> we derive the weak formulation of (<ref>) subject to the no-flux boundary condition and prove the existence and uniqueness of its solutions by constructing the corresponding contraction semigroup. Then, in Section <ref> we prove that for the parameter regime with high enough mutation rate the problem exhibits a positive spectral gap, which implies exponential convergence to equilibrium.In typical applications in quantitative genetics the solution of the Fokker-Planck equation (<ref>) is not the main object of interest. One is rather interested in the evolution of its certain moments that correspond to the macroscopic dynamics of observable quantitative traits. Therefore, Section <ref> is devoted to the study of the so-called Dynamic Maximum Entropy (DynMaxEnt) method for approximation of moments of the Fokker-Planck solution. We first show in Section <ref> that a related constrained entropy maximization is equivalent to a moment-matching problem, which we solve in a simple case. Then, in Section <ref> we provide a simple and straightforward derivation of the DynMaxEnt method by adopting a quasi-stationary approximation, which results in a nonlinear system of ordinary differential equations. It can be interpreted as a nonlinear Galerkin approximation of the Fokker-Planck equation (<ref>). However, this "original" DynMaxEnt method cannot be applied in the regime of small mutations, i.e., when 4Nμ≤ 1. This inspires us to introduce a modified version, which is valid for the whole range of admissible parameters, Section <ref>. Finally, in Section <ref> we present several numerical experiments to demonstrate the performance of both the original and modified DynMaxEnt methods. We observe that in the parameter regimes where both methods are valid, the modified one exhibits slightly better approximation properties compared to the original one.The surprisingly good approximation properties of the DynMaxEnt method, as documented by the numerical results in <cit.> and Section <ref> of this paper, suggest that the infinitely-dimensional dynamics of the Fokker-Planck equation (<ref>) can be well approximated by suitable finitely-dimensional dynamical systems. This is reminiscent of the recent series of works of E. Titi and collaborators <cit.> where a data assimilation (downscaling) approach to fluid flow problems is developed, inspired by ideas applied for designing finite-parameters feedback control for dissipative systems. The goal of a data assimilation algorithm is to obtain (numerical) approximation of a solution of an infinitely-dimensional dynamical system corresponding to given measurements of a finite number of observables. In particular, in <cit.>, it has been shown that solutionsof the two-dimensional Navier-Stokes equations can be well reconstructed from a relatively low number of low Fourier modes or local averages over finite volume elements. In <cit.>, continuous data assimilation (CPA) algorithm was proposed and analyzed for a two-dimensional Bénard convection problem, where the observables were incorporated as a feedback (nudging) term in the evolution equation of the horizontal velocity. In <cit.> CPA was applied for downscaling a coarse resolution configuration of the 2D Bénard convection equations into a finer grid, while in <cit.> the CPA method is studied for a three-dimensional Brinkman-Forchheimer-extended Darcy model of porous media, and in <cit.> for the three-dimensional Navier-Stokes–α model. Finally, in <cit.> numerical performance of the CPA algorithm in the context of the two-dimensional incompressible Navier–Stokes equations was studied. It was shown that the numerical method is computationally efficient and performs far better than the analytical estimates suggest. This is similar to our numerical observations showing very good approximation properties of the DynMaxEnt method applied to the Fokker-Planck equation (<ref>).§ BOUNDARY CONDITIONS FOR THE STATIONARY PROBLEMThe stationary solution of the Fokker-Planck equation (<ref>) is of the form u_ = 1/ℤ_exp(2N)/∏_i=1^L ξ_i, where ℤ is a normalization constant (partition function). We aim at interpreting the solution u_ as a probability density, therefore, we set ℤ_ := ∫_Ω_exp(2N)/∏_i=1^L ξ_ix̣. Observe that the above integral is finite for μ>0, which we assumed. Let us rewrite (<ref>) in the form PART:ut = _·( D u__(u/u_)) with u_ defined in (<ref>), and the diagonal diffusion matrix D = D(), D_ij = 1/4Nξ_iδ_ij for i,j=1,…,L.To provide an insight into the problem of prescribing valid boundary conditions for (<ref>), we consider the related stationary problem in the spatially one-dimensional setting, ∂_x ( D u_∂_x (u/u_)) = f for x∈ (0,1), where f∈ L^1(0,1) is a prescribed function with ∫_0^1 f(s) ṣ = 0, ξ = ξ(x) = x(1-x), D = 1/4Nξ andu_ = ℤ_^-1ξ^-1exp (2N) =ℤ_^-1ξ^4Nμ -1exp (2Nγξ' + 4Nηξ),with ℤ_ defined in (<ref>). We recall that ℤ_ is finite and u_ is integrable for the relevant range of parameters. Moreover, note that the product D u_ behaves like ξ^4Nμ close to x = 0 and x=1, so that it vanishes at the boundary and leads to a degeneracy in the formal no-flux boundary condition D u_∂_x (u/u_) = 0x∈{0,1}. To avoid possible difficulties due to this degeneracy, we integrate (<ref>) for a fixed x∈(0,1) on the interval (1/2,x),D u_∂_x (u/u_) = ∫_1/2^x f(s) ṣ + C_1,where C_1 is an integration constant. We see that imposing the formal no-flux boundary condition (<ref>) at, say, x=0 is equivalent to setting C_1 to the particular valueC_1 = ∫_0^1/2 f(s) ṣ.The assumption ∫_0^1 f(s) ṣ = 0 then implies that (<ref>) is verified at x=1. Integrating once again yields u = C_2 u_ + C_1 u_∫_1/2^x ṣ/D(s) u_ (s) + u_∫_1/2^x F(s)ṣ/D(s) u_(s), with F(s) := ∫_1/2^s f(r) ṛ. Observe that∫_1/2^xṣ/D(s) u_ (s)≈ξ^-4Nμ + 1 x∈{0,1},so that the second term in (<ref>) is bounded on [0,1] and thus integrable. Due to the boundedness of F(s), the same holds also for the third term in (<ref>). Consequently, the solution u constructed in (<ref>) is integrable on (0,1).We conclude that, for the aforementioned range of parameter values, the Fokker-Planck equation (<ref>) has to be supplemented with the standard no-flux boundary condition (<ref>) regardless of the degeneracy of D u_ at the boundary. Although the above argument only applies to the spatially one-dimensional setting, it provides a strong heuristic hint that the conclusion also holds in the multidimensional case.§ EXISTENCE AND UNIQUENESS OF SOLUTIONSIn this section we construct solutions of the Fokker-Planck equation (<ref>), supplemented with the boundary condition D u__(u/u_) ·ν = 0 ∂Ω_, where ν=ν(x) denotes the unit normal vector to the boundary of Ω_. Moreover, we prescribe the initial condition u(t=0) = u_0 Ω_. Our strategy is to convert the problem to the Hamiltonian form (- + V) for a suitable potential V and construct the corresponding semigroup. In order to obtain some intuition, we first carry out the transform formally.§.§ Formal calculationsSettingÅ() := ∑_i=1^L lnξ(x_i) - 2N(), (<ref>) is written in the formPART:ut = _·( D(_ u + u_Å ))with the boundary condition D(_ u + u_Å )·ν = 0. For i=1,…,N we introduce the coordinate transform y_i := y(x_i) := 2√(N)∫_0^x_iṣ/√(ξ(s)) =4√(N)arcsin√(x_i), and denote := (y_1,…,y_L). Note that ↦ maps Ω_ = (0,1)^L onto Ω_ := (0, Y_N)^L with Y_N:=2 π√(N). Introducing the new variable u̅() := J(()) u(()),J():= (2√(N))^-L∏_j=1^L ξ^1/2(x_i) transforms (<ref>) to the form PART:u̅t = _·( _u̅ + u̅_( Å̅- lnJ̅) ), with Å̅() := Å(()) and J̅() := J(()). By () we denote the componentwise inverse transform x_i=x(y_i). The no-flux boundary condition (<ref>) transforms as1/2√(N)∑_i=1^L J̅^-1√(ξ_i)( ∂_y_iu̅ + u̅∂_y_i( Å̅- lnJ̅) ) ν_i = 0 ∂Ω_,where we use the shorthand notation ξ_i = ξ(x_i(y_i)). Note that the product J̅^-1√(ξ_i) is constant in y_i and positive on the set {∈∂Ω_; y_i∈{0,Y_N }, 0 < y_j < Y_Nj≠ i}. Consequently, the transformed boundary condition is equivalent to the nondegenerate expression ( _u̅ + u̅_( Å̅- lnJ̅) ) ·ν = 0 ∂Ω_, which can be also written as ν·_ln (u̅/u̅_) = 0 a.e. on ∂Ω_. The steady state for (<ref>)–(<ref>) isu̅_ := ℤ_^-1exp(-(Å̅- lnJ̅)), ℤ_ := ∫_Ω_exp(-(Å̅- lnJ̅)) .Finally, settingz():=u̅() / √(u̅_()),the Fokker-Planck equation (<ref>) transforms to the Hamiltonian form PART:zt = _ z - V()z, withV() = _√(u̅_)/√(u̅_) = 1/2_u̅_/u̅_ - 1/4|_u̅_|^2/u̅_^2,which can be further expressed as V() = -1/2_ (Å̅- lnJ̅) + 1/4 |_ (Å̅- lnJ̅))|^2. The boundary condition (<ref>) transforms to √(u̅_)( _ z + 1/2 z _( Å̅- lnJ̅) )·ν = 0 ∂Ω_. Let us remark that with (<ref>), u̅_ behaves like (∏_i=1^L ξ_i)^4Nμ-1/2 close to the boundary, so that for 4Nμ-1/2 > 0 the boundary condition (<ref>) is degenerate.Inserting the expression (<ref>) forinto (<ref>) gives the explicit expression for the potential V = 1/16N(4Nμ - 1/2)(4Nμ - 3/2) (ξ'_i)^2/ξ_i + , where (bounded terms) are expressions involvingPART:ξ_iy_i = √(ξ_i)/2√(N)ξ_i',ξ_iy_i2 =-1/2Nξ_i + 1/4N (ξ_i')^2, PART:ξ'_iy_i = - 1/√(N)√(ξ_i),ξ'_iy_i2 = 1/4Nξ'_i,that are uniformly bounded on Ω_. The unbounded term in V is(ξ'_i)^2/ξ_i = (1-2x_i)^2/x_i(1-x_i),so for the potential to be bounded below, we need 4Nμ≥ 3/2. §.§ Construction of solutions for the case 4Nμ≥ 1/2In this Section we shall construct weak solutions of the Fokker-Planck equation (<ref>) with 4Nμ≥ 1/2, subject to the no-flux boundary condition (<ref>) and the initial datum (<ref>). However, since the equivalent form (<ref>) is more suitable to study the asymptotic behavior of the solution for large times, we shall work with this formulation. Due to the issues caused by the degeneracy of the boundary condition, we shall start from a weak formulation of (<ref>) and carry out the coordinate transform as in previous Section in order to arrive at a weak formulation of (<ref>).To obtain a symmetric form, we multiply (<ref>) by ϕ/u_, with a test function ϕ∈ C^∞(Ω_), and integrate by parts, taking into account the no-flux boundary condition (<ref>). We arrive att∫_Ω_u/u_ϕ/u_ u_= - ∫_Ω_ D _(u/u_) ·_(ϕ/u_) u_. ̣̅u_() Carrying out the coordinate transform ↦(<ref>), with the Jacobian J given by (<ref>), yieldst∫_Ω_u̅/u̅_ϕ̅/u̅_ =- ∫_Ω__(u̅/u̅_) ·_(ϕ̅/u̅_) ,with u̅ given by (<ref>), ϕ̅() := J(()) ϕ(()) and := u̅_. Finally, defining z:=u̅/√(u̅_) and ψ:=ϕ̅/√(u̅_), we arrive at t∫_Ω_z/√(u̅_)ψ/√(u̅_) =- ∫_Ω__(z/√(u̅_)) ·_(ψ/√(u̅_)) . We thus define the space _̋ := { z∈ L^2(Ω_); ∫_Ω_| _z/√(u̅_)|^2< +∞} with the scalar product(z, ψ)__̋ :=∫_Ω_z/√(u̅_)ψ/√(u̅_)+ ∫_Ω__(z/√(u̅_)) ·_(ψ/√(u̅_))and the induced norm z__̋^2 := (z,z)__̋. Central for our analysis is the following result. Let 4Nμ≥ 1/2. Then for every z∈_̋ the inequality holds∫_Ω_| z/√(u̅_)|^2 ≥∫_Ω_ | z|^2 + Vz^2 ,with V defined in (<ref>).We have∫_Ω_| z/√(u̅_)|^2= ∫_Ω_( | z|^2 + 1/4|_u̅_|^2/u̅_^2z^2 - _ z ·_u̅_/u̅_ z ) .We integrate by parts in the last term of the right-hand side,- ∫_Ω__ z ·_u̅_/u̅_ z=-1/2∫_Ω__u̅_/u̅_·_ z^2 = 1/2∫_Ω_ z^2 _·( _u̅_/u̅_) - 1/2∫_∂Ω_ z^2 _u̅_/u̅_·νṢ_.With (<ref>) we have_y_iu̅_/u̅_ = _y_i (lnJ̅ - Å̅) = √(ξ_i)/2√(N)[ (4Nμ-1/2) ξ_i'/ξ_i - 4Nγ_i + 4Nηξ_i' ].Since ξ_i vanishes for y_i∈{0, Y_N} and ξ_i' is bounded on [0,Y_N], we have- 1/2∫_∂Ω_ z^2 _u̅_/u̅_·νṢ_=- 1/4√(N)(4Nμ-1/2) ∑_i=1^L ∫_∂Ω_z^2 ξ_i'/√(ξ_i)ν_i Ṣ_.We write the boundary of the hypercube Ω_ as an union of the pairs of faces,∂Ω_ = ⋃_i=1^L F_i, F_i:={∈∂Ω_,y_j ∈{0,Y_N}},then we have∑_i=1^L ∫_∂Ω_z^2 ξ_i'/√(ξ_i)ν_i Ṣ_ = ∑_i=1^L ∫_F_i[ z^2 ξ_i'/√(ξ_i)]_y_i=0^Y_NṢ_F_i,where Ṣ_F_i denotes the (L-1)-dimensional Lebesgue measure on F_i. Since x_i'(x(Y_N)) = x_i'(1) = -1 and x_i'(x(0)) = x_i'(0) = 1, we have[ z^2 ξ_i'/√(ξ_i)]_y_i=0^Y_N≤ 0.Therefore, if 4Nμ-1/2≥ 0,- 1/2∫_∂Ω_ z^2 _u̅_/u̅_·νṢ_≥ 0.Consequently,∫_Ω_| z/√(u̅_)|^2≥ ∫_Ω_( | z|^2 + 1/4|_u̅_|^2/u̅_^2z^2 + 1/2z^2 _·( _u̅_/u̅_) ) = ∫_Ω_| z|^2 + Vz^2 .Finally, the above formal calculation are made rigorous by replacing u̅_ by u̅_^ := u̅_ + for >0 and subsequently passing to the limit → 0.Let 4Nμ≥ 1/2. Then the space _̋ defined in (<ref>) with the scalar product (·,·)__̋ is a Hilbert space, and is densely embedded into L^2(Ω_).Completeness follows from the fact that if z_k is a Cauchy sequence in _̋, then due to Lemma <ref> it is also a Cauchy sequence in L^2. The density of the embedding into L^2(Ω_) is due to the fact that the set of smooth functions with compact support is dense in _̋.We call z∈ L^2((0,T); _̋) ∩ C([0,T]; L^2(Ω_)) a weak solution of (<ref>) on [0,T) subject to the boundary condition (<ref>) if (<ref>) holds for every ψ∈_̋ and almost all t∈(0,T), and the initial condition is satisfied by continuity in C([0,T]; L^2(Ω_)). We remark that a formal integration by parts in the right-hand side of (<ref>) gives- ∫_Ω__(z/√(u̅_)) ·_(ψ/√(u̅_)) = - ∫_∂Ω_[ √(u̅_)( _ z + 1/2 z _ (Å̅-lnJ̅))·ν] ψ/√(u̅_)Ṣ_+ ∫_Ω_ [ _ z - V z ] ψ/√(u̅_).This justifies the interpretation of (<ref>) as the weak formulation of(<ref>) subject to the boundary condition (<ref>).We now define the operator Ł: D(Ł) ⊂_̋→ L^2(Ω_) by its action ⟨Ł z, ψ⟩ := - ∫_Ω__(z/√(u̅_)) ·_(ψ/√(u̅_)) for all z, ψ∈_̋. We shall prove that the closure Ł of Ł generates a contraction semigroup on L^2(Ω_). For this sake, we study the resolvent problem (-Ł + λ) z = f for (some) λ>0 and f ∈ L^2(Ω_). Let 4Nμ≥ 1/2. Then for every f∈ L^2(Ω_) the resolvent problem (<ref>) has a unique solution z∈_̋. For a fixed λ>0 we define the bilinear form a: _̋×_̋→,a_λ(z,ψ) := ⟨ -Ł z,ψ⟩ + λ (z,ψ),where (z,ψ) denotes the standard scalar product on L^2(Ω_). The resolvent problem (<ref>) with the no-flux boundary conditions is equivalent toa_λ (z,ψ) = (f, ψ) ψ∈_̋.A straightforward application of the Hölder inequality gives the continuity of a_λ,a_λ(z,ψ) ≤ C z__̋ψ__̋for a suitable constant C>0; coercivity is straightforward. Finally, the mapping ψ↦ (f,ψ) with f∈ L^2(Ω_) is an element of thedual space (_̋)'. Consequently, an application of the Lax-Milgram theorem yields the existence and uniqueness of the solution z∈_̋.Let 4Nμ≥ 1/2. Then the closure Ł of Ł generates a contraction semigroup on L^2(Ω_). Since _̋ is densely embedded into L^2(Ω_), the operator Ł is densely defined, and dissipative. Moreover, due to Lemma <ref>, the range of -Ł + λ is L^2(Ω_) for all λ > 0. The claim then follows by an application of the Lumer-Phillips theorem <cit.>. The contraction semigroup constructed in Theorem <ref> provides the announced existence and uniqueness of weak solutions z∈ L^2((0,T); _̋) ∩ C([0,T]; L^2(Ω_)) of (<ref>) subject to the no-flux boundary condition (<ref>) in the sense of Definition <ref>. The solutions are formally written as z(t) = e^Ł tz_0, where z_0∈ L^2(Ω_) is the initial datum; see, e.g., <cit.>. By the inverse coordinate transform to (<ref>) we obtain weak solutions of the original Fokker-Planck equation (<ref>) subject to the no-flux boundary condition (<ref>).§ SPECTRAL GAP - EXPONENTIAL CONVERGENCE TO EQUILIBRIUMIn this Section we shall perform a spectral analysis of the operator (-Ł) and prove that boundedness below of the potential V(<ref>) implies exponential convergence to equilibrium for (<ref>). From the explicit expression (<ref>) for V we see that V is bounded below if 4Nμ≥ 3/2. Let 4Nμ≥ 3/2. Then the operator (-Ł) defined in (<ref>) has compact resolvent.We need to show that for some λ>0 the operator (-Ł + λ)^-1 is compact as a mapping from L^2(Ω_) into itself. Let f∈ L^2(Ω_) and z=(-Ł + λ)^-1 f, constructed in Lemma <ref>. From Lemma <ref> we have((-Ł + λ)z, z) ≥∫_Ω_ |_ z|^2 + (V + λ) z^2 ≥ C z^2_H^1(Ω_)for some constant C>0 and λ chosen such that min_∈Ω_ (V()+λ) > 0. On the other hand, the Cauchy-Schwartz inequality gives((-Ł + λ)z, z) = (f,z) ≤1/2f_L^2(Ω_)^2 + /2z_L^2(Ω_)^2,so for sufficiently small >0 we conclude(-Ł + λ)^-1 f_H^1(Ω_) = z_H^1(Ω_)≤ C f_L^2(Ω_)and the claim follows by the compact embedding of the Sobolev space H^1 into L^2. Together with the obvious self-adjointness of (-Ł), Lemma <ref> implies that(-Ł) has a discrete spectrum without finite accumulation points. Moreover, all its eigenvalues are nonnegative. This implies the existence of a positive spectral gap and, consequently, exponential convergence to equilibrium as t→∞, see, e.g., <cit.>.§ THE DYNAMICAL MAXIMUM ENTROPY APPROXIMATIONIn typical applications in quantitative genetics the solution of the Fokker-Planck equation (<ref>) is not the main object of interest. One is rather interested in the evolution of its certain moments that correspond to the macroscopic dynamics of observable quantitative traits. This naturally leads to the question whether one can derive a finite-dimensional system of differential equations that approximates the evolution of the moments of interest, avoiding the need of solving (<ref>). This question has been studied previously by analogy with statistical mechanics: the allele frequency distribution is approximated by the stationary form, which maximizes the logarithmic relative entropy. Called Maximum Entropy Method, it has been applied to broad spectrum of problems ranging from the statistics of neural spiking <cit.>, bird flocking <cit.>, protein structure <cit.>, immunology <cit.> and more. For transient problems described by known dynamical equations (e.g., Fokker-Planck equation), the Dynamical Maximum Entropy (DynMaxEnt) method assumes quasi-stationarity at each time point. It has been applied, e.g., to modeling of cosmic ray transport <cit.>, general Fokker-Planck equation <cit.>, analysis of genetic algorithms <cit.>, and population genetics <cit.>. In <cit.> it is observed that the "classical" DynMaxEnt method cannot be applied in the regime of small mutations, and the theory is extended for this regime to account for changes in mutation strength. Surprisingly, systematic numerical simulations document superb approximation properties of the method even far from the quasi-stationary regime. However, derivation of analytic error estimates remains an open problem. In this section we discuss several aspects of the DynMaxEnt method. First, in Section <ref> we show that constrained maximization of a logarithmic entropy functional leads to a moment-matching condition.Then, in Section <ref> we provide a simple and straightforward derivation of the DynMaxEnt method by adopting a quasi-stationary approximation. To our best knowledge, this derivation has not been known before. Finally, in Section <ref> we consider the scalar case and derive a modified version of the DynMaxEnt method, which is valid for the whole range of admissible parameters. §.§ Constrained entropy maximizationWe shall call the vector ∈^dadmissible if the corresponding normalization factor ℤ_(<ref>) is finite. For any integrable function u∈ L^1(Ω_) with ∫_Ω_ u() = 1 and any admissible ∈^d we define the logarithmic relative entropy H(u | u_) := ∫_Ω_ ulnu/u_, where u_ is the normalized stationary solution of the Fokker-Planck equation (<ref>), given by formula (<ref>). Note that this is a different approach compared with <cit.>, where the logarithmic entropy is taken relative to the neutral distribution of allele frequencies in the absence of mutation or selection, ∏_i=1^L ξ_i^-1, and the variational problem is complemented with normalization and moment constraints.For a fixed u∈ L^1(Ω_) with finite -moments, let us consider the maximization of the relative entropy (<ref>) in terms of admissible ∈^d, i.e., the task of maximizing the function ↦ H(u | u_). If a critical point exists, then for i=1,…,d,α_i H(u | u_)=- ∫_Ω_u/u_α_i u_= A_i_u_ - A_i_u = 0.Consequently, if a maximizer ^* exists, then the -moments corresponding to u_^* must be matching the same moments of u. This naturally leads to the question of solvability of the nonlinear system of equations_u_=_uin terms of the admissible parameter vector ∈^d, for a given, normalized u∈ L^1(Ω_) with finite -moments. To address this question seems to be a very difficult task that we leave open. We merely remark that the Hessian matrix of ↦ H(u | u_),^̣2/α̣_i α̣_j H(u | u_) = A_i A_j_u_ - A_i_u_A_j_u_,is equal to the covariance matrix of the random variableswith the probability density u_. Thus, the Hessian matrix is positive semidefinite. In the scalar case, solvability of the moment equation A_u_α=A_u can be studied for particular choices of A. We shall give an example below in Section <ref>.§.§ Derivation of the DynMaxEnt methodLet us consider u=u(t) a solution of the Fokker-Planck equation (<ref>) with admissible parameter vector , subject to the initial datum u(t=0) = u_^0 for some admissible ^0. The DynMaxEnt method is derived in two steps: First, we multiply the equation in its form (<ref>) by the vectorand integrate,t_u(t) = ∫_Ω__( D u__( u/u_) ) =- ∫_Ω__D u__( u/u_) ,where we assumed that the boundary term in the integration by parts vanishes (note that, in general, this does not necessarily follow from (<ref>)). In the second step, we substitute u(t) in the above expression by u_^*(t) with some time-dependent parameter vector ^*=^*(t), which leads tot_u_^*(t) = - ∫_Ω__D u__( u_^*(t)/u_) + 𝐑,where 𝐑 is a vector-valued residuum term. We now introduce an approximation by neglecting the residuum 𝐑. Expanding the derivatives on both sides of the above equation leads then to ( ⊗_u_^*(t)- _u_^*(t)⊗_u_^*(t)) ^*(t)t == 1/2ξ_ : __u_^*(t) ( - ^*(t)), where _ : _ is the symmetric d× d matrix with the (i,k)-component ∑_j=1^d ∂_x_j A_i ∂_x_j A_k. The nonlinear ODE system for ^*=^*(t) is called the DynMaxEnt method for approximation of the moments of (<ref>). However, two comments have to be made: First, the matrix on the left-hand side,( ⊗_u_^*(t)- _u_^*(t)⊗_u_^*(t)),is positive semidefinite, since it is the covariance matrix of the observablesof the probability distribution u_^*(t). However, in order (<ref>) to be globally solvable, the covariance matrix must be uniformly (positive) definite, which in general may not be the case. Furthermore, the matrix ξ_ : __u_^*(t) may have infinite entries even for some admissible ^*(t), and if this is the case, then again the ODE system is not solvable. Since these two issues are very hard to resolve in general, we shall below resort to a simple case whereis a scalar.§.§ Scalar caseTo gain some more insight into the ODE (<ref>), we consider the single locus case x∈(0,1) withbeing a scalar function A=A(x) and α∈. The DynMaxEnt method (<ref>) simplifies to the following ODE for α^* = α^*(t), ( A^2_u_α^*(t)- A^2_u_α^*(t)) α^*(t)t =1/2ξ (∂_x A)^2_u_α^*(t) (α - α^*(t)). An application of the Cauchy-Schwartz inequality implies thatA^2_u_α^*(t) - A^2_u_α^*(t)≥ 0,and, moreover, equality holds if and only if A is a constant function. Consequently, for every nonconstant A the ODE (<ref>) can be rewritten as α^*(t)t =1/2( A^2_u_α^*(t)- A^2_u_α^*(t))^-1ξ (∂_x A)^2_u_α^*(t) (α - α^*(t)). However, the question of finiteness of the moment ξ (∂_x A)^2_u_α^*(t) can be only answered by making a particular choice for A=A(x).As a toy model, let us choose A=A(x) to be the scalar function ln(ξ(x)). This corresponds to a population of individuals in a neutral environment (β = h = 0 in (<ref>)) with the nonzero mutation rate α = 2μ. With the singularities at x∈{0, 1}, the function A(x)=ln(ξ(x)) well represents the issues that one encounters with the generic choice (<ref>). It is easily checked that the set of admissible values of α is the interval (0,∞). Moreover, the moment ξ (∂_x A)^2_u_α^* is only finite for α^* > 1. Consequently, the DynMaxEnt method (<ref>) is only applicable if both the initial value α^*(0) = α^0 and α are strictly larger than 1. Then, since obviously the solution α^*(t) of (<ref>) is a monotone function of time, it will stay strictly larger than 1 for all t≥ 0 and asymptotically converge to α.The issue of non-finiteness of the term ξ (∂_x A)^2_u_α^*(t) was addressed in <cit.> by introducing a special treatment near the boundary (see Appendix E, equations E.10-E.13 of <cit.> for details of the derivation of the modified method). Here we propose an alternative way that treats the problem at least in the case A(x) := ln(ξ(x)). It is based on the idea of multiplying the Fokker-Planck equation by a suitable function B=B(x), instead of A=A(x), and integrating on Ω_. In the second step, one again approximates u(t) by u_^*(t) and neglects the residuum. This leads, in the scalar case, to the ODE ( AB_u_α^*(t)- A_u_α^*(t)B_u_α^*(t)) α^*(t)t =1/2ξ (∂_x B)^2_u_α^*(t) (α - α^*(t)). Choosing B(x) := ξ(x) leads then to finite ξ (∂_x B)^2_u_α^* for all α^* > 0, i.e., for all admissible values of α^*.Thus, our strategy is to obtain α^*(t) by solving (<ref>) for t≥ 0 and then calculate the moment ln(ξ)_u_α^*(t), which is expected to be a good approximation of the true moment ln(ξ)_u(t). Clearly, one can use this strategy to obtain an approximation of any other moment of u(t).However, the method (<ref>) suffers from a serious drawback, namely, it is only solvable if the covarianceAB_u_α^*(t) - A_u_α^*(t)B_u_α^*(t)is nonvanishing for all t≥ 0, which is not clear. Nonetheless, for the particular choice A(x) = ln(ξ(x)) and B(x) = ξ(x) this seems to be the case, as is documented by our numerical calculation in Fig. <ref>. Analytically we are only able to calculate the limits ξln(ξ)_u_α^* - ln(ξ)_u_α^*ξ_u_α^*→ 0 α^*→ 0, +∞, which is based on the following Lemma. For σ>0 and x∈(0,1) denote ν_σ(x) := ξ(x)^σ-1/∫_0^1 ξ(s)^σ-1ṣ. with ξ(x) = x(1-x). Then, in the sense of distributions,ν_σ → δ(· - 1/2) σ→∞, ν_σ → 1/2δ(·) + 1/2δ(· - 1) σ→ 0,where δ(· - x) denotes the Dirac-delta distribution concentrated at x.Obviously, ν_σ is a probability measure on (0,1). Let ϕ∈ C_c^∞(0,1) be any test function on the interval (0,1). We shall show thatlim_σ→∞∫_0^1 ϕ(x) ν̣_σ(x) = ϕ(1/2).The mean-value theorem gives| ϕ(1/2) - ∫_0^1 ϕ(x) ν̣_σ(x) | ≤∫_0^1 | ϕ'(η(x)) | |x-1/2| ν̣_σ(x)≤ C_ϕ∫_0^1 |x-1/2| ν̣_σ(x).Thus, our goal is to show that ∫_0^1 |x-1/2| ν̣_σ(x) vanishes as σ→∞. For the numerator, we have∫_0^1 (4ξ(x))^σ-1 |x-1/2| x̣ = 2 ∫_0^1/2 (4ξ(x))^σ-1 (1/2 - x) x̣,and using the identity 1/2 - x = ξ'(x)/2, we calculate∫_0^1 (4ξ(x))^σ-1 |x-1/2| x̣ = 1/σ.The denominator is estimated from below using the elementary inequalities4ξ(x)≥3x x∈[0,1/4], ≥x+1/2x∈[1/4,1/2],which give∫_0^1 (4ξ(x))^σ-1x̣≥2/σ[ (3/2)^σ - 2/3(3/4)^σ].Thus,∫_0^1 |x-1/2| ν̣_σ(x) ≤1/2[ (3/2)^σ - 2/3(3/4)^σ]^-1→ 0σ→∞,which proves the first claim.To calculate the limit σ→ 0, due to the symmetry of ξ(x)=x(1-x) with respect to x=1/2, it is sufficient to prove thatξ(x)^σ-1/∫_0^1/2ξ(s)^σ-1ṣ→δ(·) σ→ 0.Again, picking a test function ϕ∈ C_c^∞[0,1/2) and using the mean-value theorem, we have to show that| ϕ(0) - ∫_0^1/2ϕ(x) ξ(x)^σ-1x̣/∫_0^1/2ξ(s)^σ-1ṣ| ≤ C_ϕ∫_0^1/2| ϕ'(η(x)) | x ξ(x)^σ-1x̣/∫_0^1/2ξ(s)^σ-1ṣtends to zero as σ→ 0.However, this follows directly from the fact that the numerator is uniformly bounded for, say, 0 ≤σ < 1, and that, obviously, the denominator tends to +∞ as σ→ 0. The statement (<ref>) follows directly from the fact that for A(x) = ln(ξ(x)) we readily have u_α^* = ν_α^* with ν_α^* given by (<ref>).Consequently, the "modified" DynMaxEnt method (<ref>) can be safely used with A(x) = ln(ξ(x)) and B(x) = ξ(x). It even seems to provide better approximation results than the "original" method (<ref>), as is documented by our numerical experiments in Section <ref>. §.§.§ Solvability of the moment equationFinally, we study the solvability with respect to α>0 of the moment equation A_u_α=A_u with A(x) = ln(ξ(x)), assuming that the right-hand side is finite. First of all, we note that the mapping α↦A_u_α is strictly increasing for α > 0. Indeed,αA_u_α = A^2_u_α - A^2_u_α > 0,where the strict positivity follows as before by the Cauchy-Schwartz inequality. Consequently, if a solution to the moment equation (<ref>) exists, it is unique. Next we claim that for any u≥ 0 with ∫_0^1 u(x) x̣ = 1 we have ln(ξ)_u∈ [-∞,ln(1/4)). Indeed, since ξ(x) < 1/4 on (0,1)∖{1/2},ln(ξ)_u < ln( 1/4) ∫_0^1 u(x) x̣ = ln( 1/4).Thus, it remains to prove that the range of α↦A_u_α is the interval (-∞,ln(1/4)). Since for A(x) = ln(ξ(x)) we readily have u_α = ν_α with ν_α given by (<ref>), Lemma <ref> givesln(ξ)_u_α = ∫_0^1 ln(ξ(x)) ν̣_α(x)→ ln(1/4) α→+∞,→-∞α→ 0+.Indeed, the range of the mapping α↦ln(ξ)_u_α is the interval (-∞,ln(1/4)) and, therefore, the moment equation (<ref>) is uniquely solvable for every normalized u∈ L^1(0,1) with finite ln(ξ)_u-moment. § NUMERICAL EXPERIMENTS§.§ Scalar caseWe present results of several numerical experiments that aim to demonstrate the performance of the original (<ref>) and modified (<ref>) DynMaxEnt methods for the scalar (single locus) case A(x) = ln(ξ(x)), as discussed in Section <ref>. Let us recall that this case corresponds to a population of individuals in a neutral environment (β = h = 0 in (<ref>)) with the nonzero mutation rate α = 2Nμ. For the modified method (<ref>) we again choose B(x) = ξ(x).In all simulations we set N=1 and start from the initial condition α^*(t=0) = α^0 := 2 for the ODEs (<ref>), (<ref>), and the initial datum u(t=0) = u_α^0 for the Fokker-Planck equation (<ref>). The ODEs (<ref>), (<ref>) are solved with simple forward Euler discretization on the time interval [0,T] for different values of T>0. We use B(x) = ξ(x) for the modified DynMaxEnt method (<ref>). The Fokker-Planck equation is discretized in space using the Chang-Cooper scheme <cit.> and forward Euler in time.In Fig. <ref> we plot the time evolution of the ln(ξ)-moment of the Fokker-Planck solution u(t) and its approximation obtained by the DynMaxEnt methods (<ref>), (<ref>) for the parameter values α∈{1.1, 1.5, 2.5, 3}. Note that since α > 1, both the methods (<ref>), (<ref>) are applicable. However, we observe that the modified method (<ref>) gives better approximation results. To quantify the approximation error, we calculate the indicator e := ∫_0^T ( ln(ξ)_u(t) - ln(ξ)_u_α^*(t))^2 ṭ/∫_0^T ln(ξ)_u(t)^2 ṭ, where ln(ξ)_u_α^*(t) is the moment calculated by one of the DynMaxEnt methods(<ref>), (<ref>). The results for the values α∈{1.1, 1.5, 2.5, 3} given in Table <ref> indeed suggest that the modified method (<ref>) provides better approximation of the moment ln(ξ)_u(t). Moreover, we observe that with increasing value of α the approximation properties of both methods seem to improve. In Fig. <ref> we plot the time evolution of the ln(ξ)-moment of the Fokker-Planck solution u(t) and its approximation obtained by the modified DynMaxEnt method (<ref>) for the parameter values α∈{0.7, 0.5, 0.3, 0.2}. Note that the original method (<ref>) is no longer applicable since for α^* < 1 the term ξ(∂_x ln(ξ))^2_u_α^* is not finite. Again, we calculate the approximation error (<ref>) for the above mentioned valued of α in Table <ref>. We observe that the approximation worsens for smaller values of α. This is presumably a consequence of the singularity of u_α at x∈{0,1} becoming stronger when α approaches zero. In fact, numerical solution of the Fokker-Planck equation (<ref>) also becomes more difficult for small values of α. For α<0.2 our discrete scheme ceases to provide reliable results. That is why α=0.2 is the smallest value that we take into account.§.§ Vector caseFinally, we consider the more general case with the function =() being vector-valued, : Ω_→^k with some k∈. It has been observed in <cit.> that using more moments (i.e., higher k) in general improves the approximation properties of the DynMaxEnt method. Inspired by the success of the modified DynMaxEnt method (<ref>) demonstrated in Section <ref>, we consider an analogous approach also in the vector case. For this, we employ the idea of deriving a modified DynMaxEnt method as in Section <ref>: We multiply the Fokker-Planck equation (<ref>) by a vector-valued function : Ω_→^k to be chosen later and integrate by parts, assuming the boundary terms to vanish. Then, we approximate u(t) by u_^*(t) with the time-dependent vector ^* = ^*(t) and neglect the residual term. This gives ( ⊗_u_^*(t)- _u_^*(t)⊗_u_^*(t)) ^*(t)t=1/2ξ_ : __u_^*(t) ( - ^*(t)), where ⊗ is the d× d matrix with the (i,k)-component B_i A_k and _ : _ is the d× d matrix with the (i,k)-component ∑_j=1^d ∂_x_j B_i ∂_x_j A_k. Clearly, uniform invertibility of the matrix( ⊗_u_^*(t)- _u_^*(t)⊗_u_^*(t)) is necessary for global solvability of the ODE system (<ref>). This condition is satisfied in our numerical experiments below.The goal of this Section is to illustrate the performance of the original (<ref>) and modified (<ref>) DynMaxEnt methods for the generic choice =(ξ',ξ,lnξ). For simplicity, we shall still stick to the 1D (single locus) setting x∈(0,1). Choosing again β = h = 1 in (<ref>), we have= - γξ' + 2 ηξ + 2μlnξ,with = (-γ,2η,2μ). The parameters γ, η∈ represent the effects of loci on the traits and μ>0 is the mutation rate. For the modified DynMaxEnt method (<ref>) we choose =(ξ',ξ,ξ^2). Note that this choice prevents the issue of non-finitness of the moment ξ_⊗__u_^*(t) for 4Nμ < 1.We carry out two numerical experiments. In both simulations we set N=1 and the initial condition for the Fokker-Planck equation (<ref>) to be the stationary distribution (<ref>) with the parameters 4μ_0 = 2, η_0=-1, γ_0=2. As before, the Fokker-Planck equation is discretized in space using the Chang-Cooper scheme <cit.> and forward Euler method in time. The ODE systems (<ref>), (<ref>) are discretized in time using the forward Euler scheme.For the first experiment we use the parameter values 4μ=1.1, η=1, γ=0. This corresponds to the abrupt change of parameters (evolutionary forces)4μ: 2 ↦ 1.1,η: -1 ↦ 1,γ: 2 ↦ 0.In Fig. <ref> we plot the time evolution of the moments ln(ξ), ξ and ξ' of the Fokker-Planck solution u(t) and its approximation obtained by the original and, resp., modified DynMaxEnt methods (<ref>), resp., (<ref>). Note that in this case both methods (<ref>), (<ref>) are applicable since the moment ξ_⊗__u_^*(t) is finite for all t≥ 0. Calculating the error of approximation (<ref>) for the three moments, Table <ref>, we observe that the modified method (<ref>) provides slightly more accurate results. For our second experiment we use the parameter values 4μ=0.5, η=1, γ=0. This corresponds to the rapid change of evolutionary forces4μ: 2 ↦ 0.5,η: -1 ↦ 1,γ: 2 ↦ 0.Note that in this case the original method (<ref>) is not applicable any more since the moment ξ_⊗__u_^*(t) is not defined for 4μ < 1. In Fig. <ref> we plot the time evolution of the moments ln(ξ), ξ and ξ' of the Fokker-Planck solution u(t) and its approximation obtained by the modified DynMaxEnt method (<ref>). On the other hand, the results presented in Fig. <ref> and Table <ref> indicate that the modified DynMaxEnt method (<ref>) provides a reasonably good approximation of the three moments.Acknowledgments. We thank Nicholas Barton (IST Austria) for his useful comments and suggestions. JH and PM are funded by KAUST baseline funds and grant no. 1000000193.10ALT D. Albanez, H. Nussenzveig Lopes, and E. Titi: Continuous data assimilation for the three-dimensional Navier-Stokes–α model. Asymptotic Analysis 97 (2016), 139–164.ATKZ M. Altaf, E. Titi, O. Knio, L. Zhao, M. McCabe, and I. Hoteit: Downscaling the 2D Bénard Convection Equations Using Continuous Data Assimilation. Computational Geosciences (to appear, 2017).AMTU A. Arnold, P. A. Markowich, G. Toscani, and A. 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Biol. 10 (2014), e1003408.Weigt M. Weigt, R. A. White, H. Szurmant, J. A. Hoch, and T. Hwa: Identification of direct residue contacts in protein-protein interaction by message passing. Proc. Natl. Acad. Sci. USA 106 (2009), 67–72.§.§ Derivation of the DynMaxEnt approximation (simplified case)We consider x∈ (0,1) and A: (0,1) ↦, so that (<ref>) reduces to PART:ut = - PART:x[ ξ(x) PART:(α A(x))x u ] + x2[ ξ(x) u ], with a given parameter α∈ and ξ(x) := x(1-x). We prescribe the boundary datum - ξ(x) PART:(α A(x))x u + PART:x(ξ(x) u)= 0x∈{0,1}, which guarantees mass conservation for (<ref>) if the mass is finite initially. Note that, due to possible singularities of u and A, the first term is not necessarily zero, even though ξ(x) has zeros at x∈{0,1}. With the boundary condition (<ref>), the stationary solution of (<ref>) isu̅_α(x) = K_α e^α A(x)ξ(x)^-1,with the normalization constant K_α > 0. Its integrability depends in general on the choice of A=A(x) and α. Let us assume that A=A(x) is chosen such that e^α Aξ^-1∈ L^1(0,1) for all α>0; this is, for instance, guaranteed with the choice A(x) = ln(ξ(x)).The dynamical maxEnt method can be derived by inserting the ansatz u(t,x) = u̅_α^*(t)(x) into (<ref>), which gives ∂_t u̅_α^* = PART:x(ξPART:Ax (α^*-α) u̅_α^*) + R, where R=R(t,x) is the residuum. Due to mass conservation, we have to choose K_α^* in u̅_α^* so that∫_0^1 u̅_α^*(x) x̣ = K_α^*∫_0^1e^α^* A(x)ξ(x)^-1x̣≡ M,where M>0 is the mass of the initial datum. Taking a time derivative leads to tln K_α^*(t) = - A_u̅_α^*(t)/Mα^*(t)t. Using the identity ∂_t u̅_α^*(t) = u̅_α^*(t)tln K_α^* + A u̅_α^*(t)tα^*, taking the A-moment and integrating by parts in (<ref>) gives then( A^2_u̅_α^*(t) - A^2_u̅_α^*(t)/M) α^*(t)t = ξ(∂_x A)^2_u̅_α^*(t) (α-α^*) - (α-α^*)[ Aξ∂_x A u̅_α^*(t)]_x=0^1 + RA_u̅_α^*(t).Unfortunately, the boundary term does not vanish in general. For the choice A(x)=lnξ(x), it reads[ Aξ∂_x A u̅_α^*(t)]_x=0^1 = [ ξ^α^*-1∂_x ξlnξ]_x=0^1,which vanishes if α^* > 1. Assuming that the boundary term indeed vanishes, the residuum term RA_u̅_α^*(t) is zero if( A^2_u̅_α^*(t) - A^2_u̅_α^*(t)/M) α^*(t)t = ξ(∂_x A)^2_u̅_α^*(t) (α-α^*).An application of the Cauchy-Schwartz inequality gives( A^2_u̅_α^*(t) - A^2_u̅_α^*(t)/M) ≥ 0,and, moreover, the inequality is sharp whenever A is not constant. In this case we can write α^*(t)t = ( A^2_u̅_α^*(t) - A^2_u̅_α^*(t)/M)^-1 B^*(t) (α - α^*(t)) with B^*(t) = :ξ(∂_x A)^2_u̅_α^*(t). Note that for nonconstant A, the steady state α^* = α for (<ref>) is asymptotically linearly stable.It is interesting to calculate the evolution of the A-moment, which is the quantity we are interested in. For the case 4Nμ > 1 the boundary term in the integration by parts vanishes and the result is tA_u̅_α^*(t) =B^*(t) (α - α^*(t)) = ξ(∂_x A)^2_u̅_α^*(t) (α - α^*(t)).§.§.§ Alternative - taking a different moment?We can also take a different moment in (<ref>), say multiply by F=F(x), integrate by parts and set the residuum (the term RF_u̅_α^*(t)) to zero. Assuming that F and 4Nμ are such that the boundary term vanishes, this leads to ( AF_u̅_α^*(t) - A_u̅_α^*(t)F_u̅_α^*(t)/M)α^*(t)t = ξ(∂_x A)(∂_x F)_u̅_α^*(t) (α-α^*) Now, with different choices of F one obtains different approximations of the Fokker-Planck equation. Is there a way to tell which choice is the best for the given purpose? In Figure below the evolution of the moment A is plotted, calculated from the solution of the Fokker-Planck equation (blue curves), and compared to the result provided by the "original" DynMaxEnt method (<ref>), green curves, and with its "generalized" version (<ref>) with the choice F(x): = ξ(x), red curves. Cleraly, the approximation provided by the modified method (<ref>) is better then the one produced by the "original" one. Why is it so?Moreover, note that for F:=lnξ, the term ξ(∂_x A)(∂_x F)_u̅_α^*(t) is defined only for 4Nμ > 1, while with F:=ξ, it is defined for all 4Nμ>0. Consequently, the applicability of the method is extended. In general, any quasistationary approximation can be written asα^*(t)t = G[u̅_α^*(t)] (α-α^*).Is there a way to determine the optimal functional G that gives the best approximation of the A-moment of the Fokker-Planck solution?Let 4Nμ≥ 3/2. Then the space _̋ defined in (<ref>) with the scalar product (·,·)__̋ is compactly embedded into L^2(Ω_).The claim follows directly from Lemma <ref>, the below boundedness of V and the compact embedding of the Sobolev space H^1 into L^2. §.§ Spectral gap - exponential convergence to equilibrium §.§.§ Weak formulationThe weak formulation of (<ref>) with the no-flux boundary condition is obtained by multiplying ∂_t u = _x· (Du__x (u/u_)) by the test function ϕ/u_,∫∂_t u ϕ/u_x̣ =∫ Du__xu/u_·_xϕ/u_.As before, we introduce the coordinate transform y_i := y(x_i) := 2√(N)∫_0^x_iṣ/√(ξ(s)), the new variable u̅(y) := J(x(y)) u(x(y)), the Jacobian J(x):= (2√(N))^-L∏_j=1^L ξ^1/2(x_i) and z:=u/√(u_).§.§ Derivation of the DynMaxEnt approximation (simplified case)We consider x∈ (0,1) and A: (0,1) ↦, so that (<ref>) reduces to PART:ut = - PART:x[ ξ(x) PART:(α A(x))x u ] + x2[ ξ(x) u ], with a given parameter α∈ and ξ(x) := x(1-x). We prescribe the boundary datum - ξ(x) PART:(α A(x))x u + PART:x(ξ(x) u)= 0x∈{0,1}, which guarantees mass conservation for (<ref>) if the mass is finite initially. Note that, due to possible singularities of u and A, the first term is not necessarily zero, even though ξ(x) has zeros at x∈{0,1}. With the boundary condition (<ref>), the stationary solution of (<ref>) isu̅_α(x) = K_α e^α A(x)ξ(x)^-1,with the normalization constant K_α > 0. Its integrability depends in general on the choice of A=A(x) and α. Let us assume that A=A(x) is chosen such that e^α Aξ^-1∈ L^1(0,1) for all α>0; this is, for instance, guaranteed with the choice A(x) = ln(ξ(x)).The dynamical maxEnt method can be derived by inserting the ansatz u(t,x) = u̅_α^*(t)(x) into (<ref>), which gives ∂_t u̅_α^* = PART:x(ξPART:Ax (α^*-α) u̅_α^*) + R, where R=R(t,x) is the residuum. Due to mass conservation, we have to choose K_α^* in u̅_α^* so that∫_0^1 u̅_α^*(x) x̣ = K_α^*∫_0^1e^α^* A(x)ξ(x)^-1x̣≡ M,where M>0 is the mass of the initial datum. Taking a time derivative leads to tln K_α^*(t) = - A_u̅_α^*(t)/Mα^*(t)t. Using the identity ∂_t u̅_α^*(t) = u̅_α^*(t)tln K_α^* + A u̅_α^*(t)tα^*, taking the A-moment and integrating by parts in (<ref>) gives then( A^2_u̅_α^*(t) - A^2_u̅_α^*(t)/M) α^*(t)t = ξ(∂_x A)^2_u̅_α^*(t) (α-α^*) - (α-α^*)[ Aξ∂_x A u̅_α^*(t)]_x=0^1 + RA_u̅_α^*(t).Unfortunately, the boundary term does not vanish in general. For the choice A(x)=lnξ(x), it reads[ Aξ∂_x A u̅_α^*(t)]_x=0^1 = [ ξ^α^*-1∂_x ξlnξ]_x=0^1,which vanishes if α^* > 1. Assuming that the boundary term indeed vanishes, the residuum term RA_u̅_α^*(t) is zero if( A^2_u̅_α^*(t) - A^2_u̅_α^*(t)/M) α^*(t)t = ξ(∂_x A)^2_u̅_α^*(t) (α-α^*).An application of the Cauchy-Schwartz inequality gives( A^2_u̅_α^*(t) - A^2_u̅_α^*(t)/M) ≥ 0,and, moreover, the inequality is sharp whenever A is not constant. In this case we can write α^*(t)t = ( A^2_u̅_α^*(t) - A^2_u̅_α^*(t)/M)^-1 B^*(t) (α - α^*(t)) with B^*(t) = :ξ(∂_x A)^2_u̅_α^*(t). Note that for nonconstant A, the steady state α^* = α for (<ref>) is asymptotically linearly stable.It is interesting to calculate the evolution of the A-moment, which is the quantity we are interested in; the result is tA_u̅_α^*(t) =B^*(t) (α - α^*(t)) = ξ(∂_x A)^2_u̅_α^*(t) (α - α^*(t)). §.§ Error estimates in terms of relative entropyLet us now define the entropy of u relative to u̅_α^*, S^*(u|α^*) := ∫ ulnu/u̅_α^*x̣ = ∫ ulnu/u̅_αx̣ + ∫ ulnu̅_α/u̅_α^*x̣. We havet∫ ulnu/u̅_αx̣ = [ ξ u PART:x( -α A + ln(ξ u) ) lnu/u̅_α]_x=0^1- ∫[ PART:x(α A - ln(ξ u)) ]^2 ξ u x̣,where the boundary term vanishes due to (<ref>) if the term lnu/u̅_α is not too singular at the boundary. Moreover, noting that lnu̅_α/u̅_α^* = (α - α^*)A + ln(K_α/K_α^*),t∫ ulnu̅_α/u̅_α^*x̣ = ∫∂_t u (α - α^*) A x̣ + A_u ∂_t (α - α^*) - M tln K_α^*= (α - α^*)[ A ξ u PART:x( -α A + ln(ξ u) ) ]_x=0^1 +(α - α^*) ( α(PART:Ax)^2 ξ_u + Ax2ξ_u ),where we used (<ref>) in the second line. Combined with the terms from above, and assuming that both boundary terms vanish, we arrive att S^*(u|α^*) = - α^*α(PART:Ax)^2 ξ_u - (α^* + α) Ax2ξ_u- ∫ [∂_x (ξ u)]^2 ξ u x̣.Only the middle term of the right-hand side does not have a sign, but it is not clear how to control it. §.§ Hilbert (or Chapman-Enskog?) expansionWe introduce the time scaling PART:ut = - PART:x[ ξ(x) PART:(α A(x))x u ] + x2[ ξ(x) u ], with a (small) parameter >0, and assume the solution can be expanded asu = u_0 +u_1 + ^2 u_2 + O(^3).Comparing terms of zeroth order in , we obtain u_0 = u̅_α^* for some α^*=α^*(t). We normalize u̅_α^* such that it has the mass M=∫ ux̣, which implies that ∫ u_1 x̣ = ∫ u_2 x̣ = 0. Comparison of the first-order terms gives PART:u̅_α^*t = - PART:x[ ξ(x) PART:(α A(x))x u_1 ] + x2[ ξ(x) u_1 ]. The mass normalization of u̅_α^* implies∂_t u̅_α^*(t) = u̅_α^*(t)(-A_u̅_α^*(t)/M + A) α^*t,so the solvability condition t∫u̅_α^*x̣ = 0 is satisfied. Multiplication of (<ref>) by A and integration gives( A^2_u̅_α^*(t) - A^2_u̅_α^*(t)/M) α^*(t)t = ( α(PART:Ax)^2 ξ_u_1 + ξAx2_u_1),where we again assumed that the boundary term coming from integration by parts vanishes.Comparison of terms of order ^2 gives PART:u_1t = - PART:x[ ξ(x) PART:(α A(x))x u_2 ] + x2[ ξ(x) u_2 ], and the solvability condition t∫ u_1 x̣ = 0 is satisfied since ∫ u_1 x̣ = 0. §.§.§ Estimate in Wasserstein distanceIn the spatially 1D case, the p-Wasserstein distance W_p(u,u̅) of the probability measures u and u̅ is equal to the L^p-difference of the corresponding pseudoinverse functions,W_p(u,u̅)^p = ∫ |V - V̅|^p,whereV(y) := inf{x;U(x)>y} U(x) = ∫_0^x u(s) ṣ,and analogously for V̅; we can also write V=U^-1 with U being the cumulative distribution function of u.Let us express the equation (<ref>) in terms of V. Integration with respect to x givesPART:Ut =- ξ(x) PART:(α A(x))x u + PART:x[ ξ(x) u ] =- ξ(x) PART:(α A(x))x∂_x U + PART:x[ ξ(x) ∂_x U ].Taking derivatives with respect to t and x in the identityV(t,U(t,x)) = x,which holds if U is strictly increasing (i.e., u>0 everywhere), gives the relations∂_t V(t,U(t,x)) + ∂_y V(t,U(t,x))∂_t U(t,x)=0,∂_y V(t,U(t,x)) ∂_x U(t,x)=1.To express the second-order derivative, we take another x-derivative in the second identity, which leads to∂_xx^2 U(t,x) = -.(∂_yy^2 V(t,y)/(∂_y V(t,y))^3)|_y = U(t,x).Using these identities, we rewrite (<ref>) as ∂_t V = ξ(V) ∂_V (α A(V)) - ∂_V ξ(V) - ξ(V) ∂_y (1/∂_y V). This can be used for calculating the steady state. Indeed, division by ξ(V) gives∂_V (α A(V) - lnξ(V) = ∂_y (1/∂_y V),and integration leads toα A - lnξ + C = ∫(ỵ/Ṿ) Ṿ/ỵ = ln(1/∂_y V) = ln (∂_x U) = ln u.Now, one can calculate for the quadratic Wasserstein distance (let us write just V̅ for the pseudoinverse corresponding to u̅_α^*)t1/2 W_2(u,u̅_α^*)^2 = ∫ (V - V̅) ∂_t (V - V̅) ỵ.Taking into account the above identities, we have∂_t V̅(t,y) = -∂_yV̅(t,y)∂_t U̅(t,V) = -∂_tα/∂_xU̅(t,V)∫_0^V A(s)u̅(s)ṣ.§.§ Error estimatesWe approximate the true solution u=u(t,x) by u^*(t,x) = u̅(x,α^*(t)), where α^*(t) solves (<ref>). The goal is to evaluate the error u-u^* in a suitable norm and bound it in terms of |α_0 - α|, where α_0∈^K is a given vector and we solve (<ref>) for u subject to the initial datum u(t=0) = u̅(·,α_0).§.§.§ Two Fokker-Planck equationsWe consider the case of two scalar Fokker-Planck equations, ∂_t u= ∂_x (α u) + d/2∂_xx^2 u,∂_t v= ∂_x (β v) + d/2∂_xx^2 v, where α, β and d are fixed parameters. We prescribe the same initial datum for both equations, so that u(t=0)=v(t=0). Integration by parts (assuming vanishing boundary terms) gives1/2t∫ |u-v|^2 x̣ = - ∫∂_x (u-v) (α u - β v) x̣ - d/2∫ |∂_x (u-v)|^2 x̣.We write the first term of the right-hand as∫∂_x (u-v) (α u - β v) x̣ = ∫∂_x (u-v) (α(u-v) + (α-β) v) x̣= α/2∫∂_x |u-v|^2 x̣ + ∫ (α - β) v ∂_x (u-v) x̣.The first term of the last line vanishes, while the second is estimated by∫ (α - β) v ∂_x (u-v) x̣≤/2∫ |∂_x (u-v)|^2 x̣ + (α-β)^2/2∫ |v|^2 x̣.Thus, we have1/2t∫ |u-v|^2 x̣≤( /2 - d/2) ∫ |∂_x (u-v)|^2 x̣+ (α-β)^2/2∫ |v|^2 x̣.As can be easily checked, for constant β (and suitable boundary data), the term ∫ |v|^2 x̣ is nonincreasing in time. Thus, choosing := d, we havet∫ |u-v|^2 x̣≤C_0 (α-β)^2/dwith C_0 :=∫ |v(t=0)|^2 x̣. § FOR GIACOMOWe need to solve for x∈Ω:=(0,1) the linear FP equation PART:ut = - PART:x[ ξ(x) PART:(α A(x))x u ] + x2[ ξ(x) u ], with a given parameter α∈,a given function A: (0,1) ↦, and ξ(x) := x(1-x). We prescribe the boundary datum - ξ(x) PART:(α A(x))x u + PART:x(ξ(x) u)= 0x∈{0,1}, which guarantees mass conservation for (<ref>) if the mass is finite initially. Note that, due to possible singularities of u and A, the first term is not necessarily zero, even though ξ(x) has zeros at x∈{0,1}. With the boundary condition (<ref>), the stationary solution of (<ref>) is u̅_α(x) = K_α e^α A(x)ξ(x)^-1, with the normalization constant K_α > 0, i.e., K_α^-1 = ∫_0^1 e^α A(x)ξ(x)^-1x̣. The integrability depends in general on the choice of A=A(x) and α. Let us choose A(x) = ln(ξ(x)), then the above expression is integrable for all α>0.Your mission: Solve (<ref>) for the generic choice α=1 and A(x) = ln(ξ(x)). As the initial datum choose (<ref>) with a different α - try, e.g., α∈{ 0.01, 0.5, 2, 100}. The solution should converge to u̅_1 as t→∞. We need to see how fast does it converge, i.e., plot, say, the L^1, L^2 and L^∞ norms of the difference u(t)-u̅_1 as a function of t and see whether the convergence is exponential or slower. Thanks! :-) 10Simon B. Simon: Schrödinger operators with purely discrete spectrum. Methods of Functional Analysis and Topology.Thus, denoting A^*(t):= A_(t), the assumption ∂_t A^*(t) = 0 implies∑_k=1^K α^*_k(t) B_j,k[(t)] + D_j[(t)] = 0. But (t) does not satisfy (<ref>), so neither it does (<ref>).. so why does the above hold? Let us introduce the shorthand notation B_j,k^*:=B_j,k[(t)] and D_j^*(t) := D_j[(t)], then ∑_k=1^K α^*_k B_j,k^* + D_j^* = 0. We now approximate the dynamics of A(t) as follows:PART:tA(t)≈∑_k=1^K α_k B_j,k(t) + D_j(t) §.§ Entropies and co.At this point just a remark: The usual definition of entropy relative to a given equilibrium u̅ isS[u] = ∫u/u̅lnu/u̅ - u x̣.Then the global minimizer of S is equal to u̅. I believe this should replace the formula (7) in the notes "II"; the calculation of the entropy dissipation t S[u(t)] remains essentially the same, due to the mass conservation t∫ u = 0.§.§.§ Comparison with the "old" methodIt is interesting to compare the A-moment evolution (<ref>) with the corresponding result for the "old" method (<ref>) for L=1, which we rewrite asα^*(t)t = A^2_u̅_α^*(t)^-1ξ(∂_x A)^2_u̅_α^*(t)(α - α^*(t)).Note that u̅_α^*(t) is normalized such that ∫u̅_α^*(t)x̣ = M. The evolution of the A-moment is then governed by the equation tA_u̅_α^*(t) = ( A^2_u̅_α^*(t) - A^2_u̅_α^*(t)/M)A^2_u̅_α^*(t)^-1ξ(∂_x A)^2_u̅_α^*(t) (α - α^*(t)). Now, we see that the "old" and "new" methods give different results, since at t=0, they start from the same initial value of the A-moment, but the right-hand sides in (<ref>) and (<ref>) have different values. Moreover, it seems that the "new" method is indeed more consistent, judging so solely due to the fact that (<ref>) is so much neater than (<ref>). §.§ RegularizationWe define (x):= + x(1-x) on (0,1) and consider the regularized problem for u^=u^(t,x)∂_tu = - 1/2∂_x [ (x) ∂_x(α(x)) u ] + 1/4N∂_xx^2 [ (x) u ] where (x) = ln(x) and we skip the indexin u^. The equation is considered subject to the no-flux boundary condition = 0x=0,1, where the fluxcan be written as := ∂_x u/,= 1/4N, with the steady state= K_^-1exp(2Nα) = K_exp(2Nα - ln).Sinceandare bounded on [0,1] for every >0, the boundary condition (<ref>) is equivalent to ∂_x u/ = 0x=0,1. In particular, it implies that u/ must be bounded at the boundary.Since (<ref>) is uniformly parabolic, the standard theory provides the existence of a unique nonnegative weak solution for any nonnegative initial datum u(0,x) = u_0(x)∈ L^2(0,1). Due to the no-flux boundary condition (<ref>), the solution conserves mass and we assume, without loss of regularity, ∫_0^1 u(t,x) x̣≡ 1 for all t≥ 0.We define the relative entropy with the generating function ψ, H_ψ(u|) := ∫_0^1 ψ(u/) x̣, where ψ satisfies the conditions (2.12) in <cit.>. In particular, we will consider the logarithmic relative entropy H generated by ψ(s) = sln s - s + 1 and the regularized entropies H_β generated by ψ_β(s) = (s+β)lns+β/1+β - (s-1) for β>0. We have then the entropy dissipationt H_β(u|)= ∫_0^1 ψ_β'(u/) ∂_t u x̣= ∫_0^1 ψ_β'(u/) ∂_x x̣= [ ψ_β'(u/) ]_x=0^1 - ∫_0^1 ψ_β”(u/) (^T ) x̣,where we used (<ref>) and denoted:=∂_x (u/).The boundary term vanishes sinceψ_β'(s) = lns+β/1+βis bounded on bounded subintervals of [0,∞) and (<ref>) implies that u/ is bounded at the boundary.Thus, for any β>0 the entropy H_β(u(t)|) is bounded by H_β(u^0|) for all t>0. Then, since the logarithmic entropy is the limit of the β-entropy as β→ 0, we haveH(u|)= ∫_0^1 u ln(u/) x̣= ∫_0^1 uln u - u ( ln K_ + (2Nα-1)ln) x̣≤ H(u^0|)for all t>0. From this, we shall deduce the uniform boundedness of the term ∫_0^1 uln u x̣ in . Denote V(x) := -( ln K_ + (2Nα-1)ln), i.e., = e^-V. Then there exists a constant C>0 independent of >0 such that ∫_0^1 V u x̣≤ C. By assumption, we have∫_0^1 uln u + V u x̣≤ Kfor some K>0, so that, with ψ(s) = sln s -s +1,K≥ ∫_0^1 ψ(u) + V u = ∫_0^1 ψ(u) + λ V u + (1-λ) V u x̣= ∫_0^1 ψ(u) - ψ'(u_∞,λ) u + (1-λ) V u x̣,where u_∞,λ = ( K_ξ^2Nα-1)^λ satisfies - ψ'(u_∞,λ) = λ V for some 0 < λ < 1. Due to the convexity of the function ψ, we construct the supporting hyperplane to ψ at u_∞,λ, i.e.,ψ(u) ≥ψ(u_∞,λ) + ψ'(u_∞,λ)(u-u_∞,λ),so thatK≥ ∫_0^1 ψ(u_∞,λ) - ψ'(u_∞,λ) u_∞,λ + (1-λ) V u x̣= ∫_0^1 ψ(u_∞,λ) + λ V u_∞,λ + (1-λ) V u x̣= ∫_0^1 ψ(u_∞,λ) + λ V u_∞,λx̣ + (1-λ) ∫_0^1 V u x̣.It can be easily checked that the first integral is finite for 0 < λ < 1, and denoting its value by I_λ, we have(1-λ) ∫_0^1 V u x̣≤ K - I_λ,so that it suffices to choose C:=K-I_λ/1-λ in (<ref>) There exists a constant C>0 independent of >0 such that ∫_0^1 u ln^+ u x̣≤ C, where ln^+ x = max{0,ln x}.We have∫_0^1 u |ln e^V u| x̣ = ∫_0^1 u ln (e^V u) x̣ + 2 ∫_u≤ e^-V u ln(1/e^V u) x̣=C + 2/e∫_0^1 e^-V≤ C,where we used the inequality x ln(1/x) ≤1/e for 0<x<1, the uniform boundedness of ∫_0^1 u ln (e^V u) x̣ = ∫_0^1 u ln u + V u x̣ and the identity = e^-V. Moreover,∫_0^1 u ( ln^+ u + V ) x̣ = ∫_u≥ 1 u ( ln^+ u + V ) x̣ + ∫_u<1 u V x̣≤ ∫_0^1 u ( |ln u| + V) x̣ + C ≤ C,We call u a weak solution of the Fokker-Planck equation (<ref>) if for every test function ψ∈ C^∞([0,1]) t∫_0^1 uψx̣ = 1/2∫_0^1 ∂_x (α A) u ∂_x ψx̣ + 1/4N∫_0^1u ∂_xxψx̣.Inserting into the formula, we obtain-2V = ∑_i=1^L [ ξ_i”/4N - 3/4N(ξ_i')^2/ξ_i - 1/2(α· A)x_i2+ ξ_i'/2( 1 - 1/2ξ_i^1/2) PART:(α· A)x_i-N/2(PART:(α· A)x_i)^2 ξ_i + 1/4N(ξ_i')^2/ξ_i^3/2],where we denoted ξ_i:=ξ(x_i(y)). Spectral gap exists if V is bounded below. The "standard" choice of A isA(x) = ( ∑_j=1^L γ_i (2x_i-1), ∑_j=1^L 2x_i(1-x_i), ∑_j=1^L 2ln x_i, ∑_j=1^L 2ln(1-x_i)).Let us set L=1 for simplicity. We will insert and inspect the terms one-by-one. Choosing A(x)=ln x, we obtain-2V = - 1/2N - 3/4N(1-2x)^2/x(1-x) + 1/2α/x^2+ 1-2x/2( 1 - 1/2√(x(1-x))) α/x-N/2α^2 (1-x)/x + 1/4N(1-2x)^2/(x(1-x))^3/2.Now, spectral gap exists if the right-hand side is bounded above. However, this is never the case for α>0, since at x=1 it is unbounded. Indeed, for x→ 1-,- 3/4N(1-2x)^2/x(1-x) + 1/4N(1-2x)^2/(x(1-x))^3/2≈1/1-x( -3 + 1/√(1-x)) → +∞and1-2x/2( 1 - 1/2√(x(1-x))) α/x→ +∞,while the other terms are bounded at x=1. Choosing A(x)=ln(1-x) leads to the same issue at x=0. With A(x)=2x-1, one gets-2V = - 1/2N - 3/4N(1-2x)^2/x(1-x) + 1/2α/x^2+ α(1-2x)( 1 - 1/2√(x(1-x)))-N α^2 x (1-x) + 1/4N(1-2x)^2/(x(1-x))^3/2,and, again, for α>0 the right-hand side tends to +∞ at x=1-. §.§ Spectral gap for the 1D caseWe consider the Fokker-Planckequation ∂_tu = - 1/2∂_x [ ξ(x) ∂_x(α· A(x)) u ] + 1/4N∂_xx^2 [ ξ(x) u ] on the interval x ∈ (0,1), with ξ(x):=x(1-x), α∈^K a constant and A: (0,1)→^K a given vector-valued function.SettingD(x) := 1/4Nξ(x),Å(x) := lnξ(x) - 2Nα· A(x),the equation is rewritten as ∂_tu = ∂_x ( D(∂_x u + u∂_xÅ )). Then the condition for the existence of the spectral gap (A1) in Lemma 2.13 of <cit.> for (<ref>) reads-1/4N - 1/16N(ξ')^2/ξ - 1/4ξ' ∂_x (α· A) - 1/2ξ∂_xx^2 (α· A) ≥λ_1on (0,1) for some λ_1>0.We make the standard choiceα· A = 2μlnξ + βξ - γξ'and inserting into the above condition gives4Nμ-1/16N( 4 + (ξ')^2/ξ) + β(ξ - 1/4 (ξ')^2 ) - γ/2ξ' ≥λ_1.We observe that the terms β(ξ - 1/4 (ξ')^2 ) - γ/2ξ' are bounded on [0,1], while (ξ')^2/ξ→ +∞ as x→ 0, 1. Consequently, the condition can only be satisfied if 4Nμ-1≥ 0. If 4Nμ-1 > 0, then positive spectral gap exists if |β| and |γ| are sufficiently small. Now, let us consider u=u(t) being a solution of the Fokker-Planck equation (<ref>) with parameters , subject to the initial datum u(t=0) = u_^0 for some admissible ^0. Assuming that there exists ^* = ^*(t) such that the identity _u(t) = _u_^*(t) holds for all t≥ 0, we take its derivative with respect to time. Using the formula (<ref>) for u_^*(t), we have for the right-hand sidet_u_^*(t) = ( ⊗_u_^*(t)- _u_^*(t)⊗_u_^*(t)) ^*t.For the left-hand side, we insert for ∂_t u from (<ref>) and integrate by parts, using the no-flux boundary condition (<ref>). This yields for i=1,…,d,tA_i_u(t) = - ∫_Ω__ A_i ·(D u__( u/u_) ) .The DynMaxEnt method is derived by exchanging (approximating) u=u(t) in the above expression by u_^*(t). A simple calculation gives then the expressiontA_i_u(t)≈1/2ξ_ : __u_^*(t) ((t) - ^*(t)),where _ : _ is the symmetric d× d matrix with the (i,k)-component ∑_j=1^d ∂_x_j A_i ∂_x_j A_k. Thus, the DynMaxEnt method for (<ref>) reads( ⊗_u_^*(t)- _u_^*(t)⊗_u_^*(t)) ^*(t)t= 1/2ξ_ : __u_^*(t) ((t) - ^*(t)).Therefore,ξ(x)^σ-1/∫_0^1 ξ(s)^σ-1ṣ≤ (4x(1-x))^σ-1σ →_σ→∞+∞ x=1/2,→_σ→∞0x∈(0,1)∖{1/2}.Consequently, ν_σ→δ(· - 1/2) in the sense of distributions as σ→∞.For the second claim, we calculate∫_σ^1-σξ(x)^σ-1x̣ = ∫_σ^1-σ (x(1-x))^σ-1x̣≤(1-2σ) σ^2(σ-1)→ 1 σ→ 1.Since ∫_0^1 ξ(x)^σx̣… | http://arxiv.org/abs/1704.08757v1 | {
"authors": [
"Katarina Bodova",
"Jan Haskovec",
"Peter Markowich"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20170427214614",
"title": "Well posedness and Maximum Entropy Approximation for the Dynamics of Quantitative Traits"
} |
[email protected] Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 USA Materials composed of two dimensional layers bonded to one another through weak van der Waals interactions often exhibit strongly anisotropic behaviors and can be cleaved into very thin specimens and sometimes into monolayer crystals. Interest in such materials is driven by the study of low dimensional physics and the design of functional heterostructures. Binary compounds with the compositionsandwhere M is a metal cation and X is a halogen anion often form such structures. Magnetism can be incorporated by choosing a transition metal with a partially filled d-shell for M, enabling ferroic responses for enhanced functionality. Here a brief overview of binary transition metal dihalides and trihalides is given, summarizing their crystallographic properties and long-range-ordered magnetic structures, focusing on those materials with layered crystal structures and partially filled d-shells required for combining low dimensionality and cleavability with magnetism. Crystal and Magnetic Structures in Layered, Transition Metal Dihalides and Trihalides Michael A. McGuire December 30, 2023 =====================================================================================§ INTRODUCTIONBinary transition metal halides MX_y (M= metal cation, X= halogen anion) provide a rich family of materials in which low dimensional magnetism can be examined, and such studies were carried out through much of the last century <cit.>. The dihalides contain triangular nets of transition metal cations, and geometrical frustration is expected when the magnetic interactions are antiferromagnetic (AFM) <cit.>. Several of thecompounds form helimagnetic structures and display multiferroic behavior <cit.>. In the trihalides, on the other hand, the transition metal cations form honeycomb nets. This lattice is not frustrated for simple AFM nearest neighbor interactions, but in the case of RuCl more complex magnetic interactions and spin-orbit coupling are expected to result in a spin-liquid ground state that is currently of much interest <cit.>. For many of the materials considered here, the in plane interactions are ferromagnetic (FM). Chromium trihalides were identified as some of the earliest ferromagnetic semiconductors, and CrX_3 compounds in general have received recent attention as candidate materials for the study of magnetic monolayers and for use in van der Waals heterostructures, in which their magnetism can be coupled to electronic and optical materials via proximity effects <cit.>. Developing cleavable ferroic materials, both magnetic and electric, is key to expanding the toolbox available for designing and creating custom, functional heterostructures and devices <cit.>. Although FM and ferroelectric materials play the most clear role in such applications, the development of spintronics employing antiferromagnetic materials may open the door to a much larger set of layered transition metal halides <cit.>.It is generally observed that binary halides often form low dimensional crystal structures, comprising either molecular units, one dimensional chains, or two dimensional layers. This holds true especially for the chlorides, bromides, and iodides, and can be attributed to the low ionic charge X^- and relatively large ionic radii (1.8–2.2 Å) of these anions. This results in multiple large anions for each metal cation assuming oxidation states typical for transition metals. Thus the cations are usually found in six-fold coordination and are well separated into structural units that are joined to one another in the crystal by van der Waals bonding between halogen anions. Fluoride has a much smaller ionic radius (1.3 Å) and forms three dimensional crystal structures with the divalent and trivalent cations that are the focus of most of this work. Note that this simple ionic picture is not appropriate when metal-metal bonding is present, which is often the case in the early, heavy transition metals. Typically larger anion to cation ratios result in lower dimensional structures, but often polymorphs of different dimensionality occur for a single composition. For example, TiCl forms with 1D chains of face sharing octahedra or 2D layers of edge sharing octahedra <cit.>. Dihalides () and trihalides () represent the majority of layered binary transition metal halides. There are, however, other examples, including Nb_3Cl_8 <cit.>, in which interesting magnetic behavior has been noted recently <cit.>.Here, a brief review ofandcompounds with partially filled d-shells is presented, with a focus on crystal and magnetic structures. A general overview of the crystallographic properties and magnetic behavior including static magnetic order in these two families is given. This short survey is not meant to be exhaustive, but rather to give a general introduction to these materials and a broad overview of the trends observed in their crystallographic and magnetic properties, with references to the literature where more detailed discussion can be found.§ CRYSTAL STRUCTURES OF LAYERED, BINARY, TRANSITION METAL HALIDES§.§ MX_2 Compounds Crystal structure information forcompounds with partially filled d-shells is collected in Table <ref>. Non-magnetic, layered dihalides of Zn, Cd, and Hg, with valence electronic configurations 3d^10, 4d^10, and 5d^10, respectively, are also known <cit.>, but these are not considered here. It can be seen that most of the compounds in Table <ref> adopt either the trigonal CdI structure type or the rhombohedral CdCl structure type. These structures are shown in Figure <ref>. Both contain triangular nets of cations in edge sharing octahedral coordination forming layers of compositionseparated by van der Waals gaps between the X anions. The structures differ in how the layers are stacked. The CdI structure type has AA stacking with one layer per unit cell, and the X anions adopt a hexagonal close packed arrangement. The CdCl structure has ABC stacking with three layers per unit cell, and the X anions adopt a cubic close packed arrangement.Figure <ref> also shows sections of the periodic table highlighting the transition metals for which thecompounds listed in Table <ref> form. Note that compounds with stoichiometryare known for other M, for example Cr, Mo, and Pd, but they form molecular (cluster) compounds or 1D chain structures. Among the dichlorides, the CdCl structure is found only for the later transition metals, and only for Ni in the dibromides and diiodides. However, Schneider et al. have shown that MnBr may undergo a crystallographic phase transition from the CdI structure type to the CdCl structure type at high temperature <cit.>. NiI undergoes a crystallographic phase transition at 60 K <cit.>. It is monoclinic below this temperature, resulting from a slight distortion (β = 90.2^∘) from the C-centered orthorhombic description of the hexagonal lattice. In addition, diffraction measurements on FeCl under pressure have shown a transition from the CdCl structure to the the CdI structure near 0.6 GPa <cit.>.Interatomic distances between M cations within the layers and the spacing of the layers are shown in Table <ref>. For the CdCl and CdI structure types the in-plane M-M distance is equal to the length of the crystallographic a axis (hexagonal settings). The layer spacing, defined as the distance between the midpoints of neighboring layers measured along the stacking direction, is equal to the length of the c axis in the CdI structure and c/3 in the CdCl structure. Moving across the series from Mn to Ni, both of these distances generally decrease, while less systematic behavior is seen for TiX_2 and VX_2.The layered phases are restricted to the first row of the transition metals, with the exception of the 4d element Zr. Both ZrCl and ZrI are reported, but not the dibromide. The zirconium compounds are found to have different structures than the layered 3d transition metal dihalides. As shown in Figure <ref>, ZrCl adopts the MoS structure type <cit.>, which has the same triangular nets of metal cations and ABC stacking found in CdCl. However, in ZrCl the Zr atoms are in trigonal prismatic coordination rather than octahedral coordination. As a result the Cl anions do not form a cubic close packed arrangement in ZrClbut instead an AABBCC stacking sequence. ZrI is reported to adopt both the MoTe and WTe structure types <cit.>. The closely related structures are shown in Figure <ref>. The regular triangular net of M cations found in the compounds described previously is disrupted in ZrI, which has zigzag chains of Zr atoms (see M-M in-plane distances in Table <ref>). This points to the tendency of heavier (4d and 5d) transition metals to form metal-metal bonds. Indeed, in addition to the layered MoS structure described above for ZrCl, a molecular crystal structure with Zr_6 clusters is also known <cit.>. Further examples of this tendency will be noted later in discussion ofcompounds.§.§ MX_3 Compounds Crystal structure information for layeredcompounds is collected in Table <ref>. Only compounds with transition metals containing partially filled d-shells are included, since those are the materials in which magnetism may be expected. Non-magnetic, layered trihalides of Sc and Y, with valence electronic configurations 3d^0 and 4d^0, respectively, are also known <cit.>.Note that several of these materials also form in 1D chain structures, but those polymorphs are outside of the scope of the present work. All of the layered compounds have been reported to adopt either the monoclinic AlCl structure type or the rhombohedral BiI structure type, and this is indicated for each element on the periodic table sections shown in Figure <ref>. In these materials the common structural motif is a honeycomb net of M cations that are in edge sharing octahedral coordination, as shown in Figure <ref>. In the BiI structure the layer stacking sequence is strictly ABC, and the stacking in AlCl is approximately ABC. In the former case subsequent layers are shifted along one of the M-M “bonds” (to the right in the BiI structure shown in Figure <ref>), while in the latter case the layers are shifted perpendicular to this direction (into the page in the AlCl structure shown in Figure <ref>).In the BiI structure the honeycomb net is regular due to the three-fold symmetry. In the AlCl structure the honeycomb net can be distorted, and the y-coordinate of the M site determines the degree of distortion. This results in two unique in-plane M-M distances (Table <ref>). In most of the compounds these two distances are seen to be quite similar, that is the honeycomb nets are nearly undistorted. The two exceptions are the heavier transition metal compounds MoCl and TcCl, in which the net is broken into dimers that are separated from one another by a distance about one Angstrom longer than their intradimer distance. Metal-metal bonding in Tc halides including layered (β) TcCl is discussed in []. Three elements, Ti, Fe, and Ru, are reported to form multiple layered crystal structures with stoichiometry MCl_3 (Table <ref>). This is indicated by the crosshatching on the table in Figure <ref>. As noted in Table <ref>,TiCl is also reported to form in the trigonal Ti_3O structure type. This is similar to the BiI structure shown in Figure <ref>, but with an ABB stacking sequence. This same structure type is also found for one of the FeCl polymorphs, which also forms in a third structure type (trigonal, P3) with twelve honeycomb layers per unit cell and a c axis length of 70 Å. TiCl undergoes a structural phase transition at low temperature <cit.>. Troyanov et al. demonstrated that the distortion upon cooling corresponds to a dimerization similar to that noted above in MoCl and TcCl <cit.>. Below 220 K a monoclinic structure was reported. The space group, C2/m is the same as the AlCl structure type, but the structure is different, with three layers per unit cell. The dimerization is not as extreme in TiCl as it is in MoCl and TcCl. At 160 K the Ti-Ti distances within the distorted honeycomb net are 3.36 and 3.59 Å <cit.>, so the dimerization is not as strong at this temperature, 60 K below the transition, as it is in MoCl and TcCl (Table <ref>) at room temperature. A structural phase transition is also reported for TiBr, with a triclinic low temperature structure (P1) <cit.>, and this same triclinic structure was also later reported for TiCl <cit.>.All three of the layered chromium trihalides are known to undergo temperature induced crystallographic phase transitions between the AlCl and BiI structure types <cit.>. At high temperatures all three adopt the AlCl structure and transition to the BiI structure upon cooling. This happens near 240, 420, and 210 K in the chloride, bromide, and iodide, respectively. The phase transition is first order, displaying thermal hysteresis and a temperature range over which both phases coexist. Interestingly, it is the lower symmetry monoclinic phase that is preferred at higher temperatures. The transition must be driven by interlayer interactions, since the layers themselves are changed little between the two phases. As expected, twinning and stacking faults develops during the transition upon cooling as the layers rearrange themselves into the BiI stacking, which can complicate interpretation of diffraction data <cit.>. Multiple structure types have been assigned to the layered form of RuCl, known as α-RuCl. Early reports assigned the trigonal space group P3_112 <cit.> (known as the CrCl_3 structure type, although it has been shown that CrCl_3 does not actually adopt it) and the AlCl type <cit.>, and a tendency to form stacking defects has been noted <cit.>. The Ti_3O type was also reported <cit.>. More recently an X-ray and neutron diffraction study reported the monoclinic AlCl structure for small single crystals at and below room temperature, and a phase transition in large single crystals from a trigonal structure at room temperature to the monoclinic AlCl structure type below about 155 K <cit.>. A recent report finds high quality crystals undergo a crystallographic phase transition upon cooling from the AlCl-type at room temperature to the BiI-type below about 60 K <cit.>, the same transition described above for CrX_3. Note that even in the monoclinic form the honeycomb net of Ru has little or no distortion (Table <ref>).Finally, layered IrCl has the AlCl structure with a nearly regular honeycomb net (Table <ref>), but it is also known to adopt a less stable orthorhombic polymorph (Fddd). The orthorhombic structure is made up of edge sharing octahedra like the layered structure, but the connectivity extends the structure in three dimensions <cit.>. It is interesting to note that the structure of orthorhombic IrCl is made up of fragments of honeycomb nets like those found in the layered structures shown in Figure <ref>.Clearly there are many variants on the stacking sequence in these layered materials due to the weak van der Waals interactions between layers that results in small energy differences between arrangements with different stacking sequences. This is apparent from the crystallographic results from the Ti, Cr, Fe, and Ru trichlorides discussed above. This has been demonstrated using first principles calculations for RuClwhere multiple structures are found to be very close in energy, and the ground state can depend on the fine details of spin-orbit coupling and electron correlations <cit.>. The possibility of mechanically separating these materials into thin specimens or even monolayers is of great interest from the point of view of low dimensional magnetism and potential applications and is greatly facilitated by the weakness of the interlayer interactions. The cleavability of several of these compounds has been studied with first principles calculations, using density functionals that incorporate the weak interlayer dispersion forces that are missing from many conventional functionals. For the Ti, V, and Cr trihalides, cleavage energies are reported to be near 0.3 J/m^2, which is smaller than that of graphite <cit.>. Stable monolayer crystals of CrI have recently been demonstrated experimentally <cit.>. § MAGNETIC STRUCTURES OF LAYERED, BINARY, TRANSITION METAL HALIDES The magnetic order in layered MX_2 and MX_3 compounds is described below, and some description of the high temperature paramagnetic behavior is given as well. Magnetic excitations and magnetic correlations that develop above the long range ordering temperature are not considered here. Magnetism in these insulating transition metal halide compounds arises from the angular momentum associated with partially filled d orbitals. In octahedral coordination, interaction with the coordinating anions split the five d orbitals into a set of three levels at lower energy, the t_2g levels (d_xy, d_xz, d_yz), and two levels at higher energy, the e_g levels (d_x^2-y^2, d_z^2). According to Hund's rules, the d electrons first fill these states singly with their spins parallel, unless the energy cost of putting electrons in the higher energy e_g states overcomes the cost of doubly occupying a single state. In addition to their spin, the electrons in these levels also have orbital angular momentum. In ideal octahedral coordination, the total orbital angular momentum can be shown to be zero for certain electronic configurations. This arises due to rotational symmetry of the system, and when this occurs the orbital angular momentum is said to be “quenched”. For octahedral coordination the orbital angular momentum is quenched when there is exactly one electron in each of the t_2g orbitals, and when there are two electrons in each of the t_2g orbitals. Otherwise there is an orbital moment that must be considered. There is no orbital angular momentum associated with the e_g orbitals. Of course, distortions of the octahedral environments can affect the details of the magnetism that are based on symmetry and degeneracy of electronic states. Despite the partially filled d-orbitals, the materials considered here are electrically insulating under ambient conditions. This can be attributed to a Mott-Hubbard type mechanism by which electron-electron interactions produce a band gap related to the Coulomb repulsion among the well-localized electrons (see for example <cit.>). Magnetism in a material is often first characterized by measurements of magnetization (M) as functions of applied magnetic field (H) and temperature (T). Considering magnetic interactions between localized magnetic moments, the temperature dependence of the magnetic susceptibility (χ = M/H) can often be described by the Curie-Weiss formula, χ(T) =C/(T-θ). The Curie constant (C) is a measure of the size of the magnetic moment and is given by C = N_A/3k_Bμ_B^2g^2S(S+1), where N_A is Avogadro's number, k_B is the Boltzmann constant, μ_B is the Bohr magneton, S is the total spin, and g = 2.00 is the electron gyromagnetic ratio. The “effective moment” (μ_eff) is also often quoted, μ_eff = g√(S(S+1))μ_B. In cgs units, μ_eff≈√(8C). Fully ordered magnetic moments are expected to be equal to gS in units of Bohr magnetons. When both orbital and spin moments are present, the total angular momentum and associated g-factor must be used. The Weiss temperature (θ) is a measure of the strength of the magnetic interactions. Considering magnetic interactions between nearest neighbors of the form H_ij = -J S_i·S_j, it can be shown that the Weiss temperature depends on the spin S, the magnetic exchange interaction strength J, and the number of nearest neighbors z according to θ = 2zJ/3k_BS(S+1). Positive values of θ indicate positive values of J, which indicate ferromagnetic interactions. Negative values of θ indicate antiferromagnetic interactions. In a simple mean field model, the Weiss temperature corresponds to the ordering temperature (T_C,N = |θ|). Note that the presence of multiple types of interactions, for example FM intralayer interactions and AFM interlayer interactions, complicates the interpretation of Weiss temperatures.In the materials considered here, the in-plane magnetic interactions between transition metal cations are expected to arise mainly from superexchange through shared coordinating halogen anions. The sign of the superexchange interaction depends upon many factors, including the orbital occupations and the M-X-M angle (see, for example, the discussion in <cit.>).It is often AFM and strong when the angle is 180^∘. When this angle is 90^∘, as it is in the edge sharing octahedral coordination found in layeredandcompounds, superexchange can be either FM or AFM. There are also direct M-M exchange interactions, which tend to be AFM, but this is expected to be relatively weak in these materials due to the relatively large M-M distances. The in-plane magnetic order in most of the compounds described below either is ferromagnetic, contains ferromagnetic stripes, or has a helimagnetic arrangement. The later two scenarios are expected to arise from competing magnetic interactions. The exceptions are VX_2 in which the interactions are predominantly AFM <cit.>, and perhaps TiCl.Note that in the figures below showing the magnetic structures ofandcompounds, only the M sublattices are shown. The magnetic moments directions are indicated by red arrows. In addition, to make the magnetic structures easier to view, different colored balls are used to represent atoms with moments along different directions, except for in the more complex helimagnetic structures. §.§ MX_2 Compounds §.§.§ TiX_2 and ZrX_2 These compounds have Ti and Zr in the unusual formal oxidation state of 2+. However, as noted above, metal-metal bonds are present in ZrI, so this simple electron counting is invalid in this case. Divalent Ti and Zr in TiX_2 and ZrCl have electron configurations of 3d^2 and 4d^2, respectively, with an expected spin of S = 1. There have been very few magnetic studies of these materials, likely due in part to their instability and reactivity. Magnetic susceptibility measurements on TiCl down to 80 K have revealed a cusp near 85 K <cit.>. The authors suggest that this may indicate antiferromagnetic ordering at this temperature, although they note that previous measurements showed smoothly increasing susceptibility upon cooling from 300 to 20 K <cit.>, but this was based on only six temperature points and significant features could have been overlooked. Magnetic susceptibility versus temperature curves have somewhat unusual shapes, and effective moments of 1.1 and 2.0 μ_B perTi have been reported <cit.>. A Weiss temperature of -702 K was determined by Starr et al. <cit.>, which would indicate strong antiferromagnetic interactions. Frustration of these interactions by the triangular Ti lattice may be responsible for the relatively low ordering temperature of 85 K proposed in Ref. <cit.>.ZrCl is reported to have a reduced magnetic moment at room temperature <cit.>, but no temperature dependent data were reported. The authors suggest that this may indicate strong antiferromagnetic interactions between Zr magnetic moments. No magnetic structure determinations for TiCl were located in the literature and no magnetic information was found for TiBr or TiI. §.§.§ VX_2 These materials contain divalent V with an electronic configuration 3d^3, S = 3/2. An early report on VCl found it to be paramagnetic with a large negative Weiss temperature (-565 K) indicating strong antiferromagnetic interactions <cit.>. Niel et al. later reported Weiss temperatures of -437, -335, and -143 K for VCl, VBr, and VI, respectively, with effective moments close to the expected value of 3.9 μ_B, andexplained their behavior in terms of a 2D Heisenberg model <cit.>.A neutron powder diffraction study showed that all three of the vanadium dihalides order antiferromagnetically with Néel temperatures of 36.0 K for VCl, 29.5 K for VBr and 16.3 K for VI_2 <cit.>. The strong suppression of these ordering temperatures relative to the Weiss temperatures is a result of geometrical frustration. Both temperatures trend to lower values as the halogen is changed from Cl to Br to I. Further neutron scattering experiments revealed that the magnetic order in VCl develops in two steps, with phase transition temperatures separated by about 0.1 K, and found the magnetic structure at low temperature to be a 120^∘ Néel state shown in Figure <ref>, where each moment in the triangular lattice is rotated by this angle with respect to its neighbors, with moments in the ac-plane <cit.>. The ordered moment corresponded to a spin of 1.2. In that study, three types of critical behavior were observed, corresponding to 2D Heisenberg, 3D Heisenberg, and 3D Ising models. A similar magnetic structure was found for VBr with moments of about 83% of the expected value <cit.>. The magnetic order in VI develops in two steps with T_N1 = 16.3 K and T_N2 near 15 K, but the low temperature magnetic structure of this compound was not resolved with any certainty <cit.>.Recently Abdul Wasey et al. proposed VX_2 materials as promising candidates for extending 2D materials beyond graphene and dichalcogenides <cit.>. They report results of first principles calculations of the magnetic order in these systems in both bulk and monolayer forms. In the bulk crystal the experimental spin structure was reproduced. A similar structure is predicted for the monolayer, and the authors suggest that magnetic order in the monolayer may occur at much higher temperature than in the bulk. §.§.§ MnX_2 Divalent Mn has a 3d^5 electronic configuration, with S=5/2. Magnetization measurements for MnCl indicate weak antiferromagnetic interactions (θ = -3.3 K) and an effective moment of 5.7 μ_B, close to the expected value of 5.9 μ_B <cit.>. Heat capacity measurements indicate magnetic phase transitions at 1.96 and 1.81 K <cit.>. The magnetic structures of MnCl below these two transitions have not been completely determined. Neutron diffraction from single crystals were analyzed assuming a collinear structure and complex orderings with stripes of ferromagnetically aligned spins in the plane were proposed <cit.>. A more recent investigation of MnCl-graphite intercalation compounds found that the magnetic order within isolated MnCl layers could be described by an incommensurate helimagnetic arrangement, and it was suggested that this may also hold for the magnetic structure of the bulk crystal <cit.>.A heat capacity anomaly was reported at 2.16 K in MnBr, and neutron diffraction showed that antiferromagnetic order is present below this temperature <cit.>. The magnetic structure has ferromagnetic stripes within the layers with antiferromagnetic coupling between neighboring stripes, as depicted in Figure <ref>. The moments are along the a axis of the hexagonal cell of the crystal structure. There is antiferromagnetic order between the layers. Later, an incommensurate magnetic phase was identified between this phase and the paramagnetic state, persisting up to about 2.3 K <cit.>.MnI adopts a complicated helical magnetic structure below 3.4 K <cit.>. The moments lie in the (307) planes, and are ferromagnetically aligned within each of these planes. The variation of the moment direction upon moving between (307) planes was originally reported to be a rotation by 2π/16 <cit.>. Furthermeasurements resolved multiple phase transitions as the magnetic order develops and find the helical ordering to be incommensurate, but with a wave vector close to that reported in the earlier work <cit.>.It was recently noted that a ferroelectric polarization develops in the magnetically ordered state of MnI, spurring interest in this compound as a multiferroic material <cit.>. Density functional theory calculations suggest that spin-orbit coupling on the iodine ions is the main source of the ferroelectric polarization in MnI <cit.>, which has been measured to exceed 120 C/m^2 <cit.>. While spin-orbit coupling is required to accurately describe the polarization, is was found to have little influence on the magnetic interactions determined by fitting density functional theory results to a Heisenberg model <cit.>. In that study it was found that the observed helimagnetic order arises from competing magnetic interactions on the triangular Mn lattice, that electronic correlations, which weaken AFM superexchange, must be considered to accurately reproduce the experimental magnetic structures, and that the details of the spiral structure are sensitive to relatively strong interplane magnetic interactions. The ferroelectric polarization responds to applied magnetic fields in multiferroic MnI. Magnetic fields affect the polarization by modifying the helimagnetic domain structure at low fields, and by changing the magnetic order at higher fields <cit.>. Ferroelectric distortions onsetting at the magnetic ordering temperatures and associated multiferroictiy is a common occurrence in MX_2 compounds that adopt non-collinear magnetic structures (see CoI, NiBr, and NiI below); however, the details of the coupling between the spin and electric polarization in these and related triangular lattice multiferroics is not well understood <cit.>.§.§.§ FeX_2 The divalent iron, 3d^6, in these compounds is expected to be in the high spin state with S=2. The partially filled t_2g levels means that orbital angular momentum is not quenched, and an orbital moment may be expected, as discussed in []and references therein. Significant anisotropy is observed in the paramagnetic state in all three of the iron dihalides, with a larger effective moment measured along the c axis <cit.>. Moments in the magnetically ordered states are also along this direction. Weiss temperatures determined from measurements with the field in the plane (||) and out of the plane (⊥) are9 K (||) and 21 K (⊥) for FeCl, -3.0 K (||) and 3.5 K (⊥) for FeBr, and 24 K (||) and 21.5 K (⊥) for FeI <cit.>.Although the crystallographic structures of FeCl and FeBr differ (Table <ref>), they have the same ordered arrangement of spins at low temperature. This magnetic structure is shown in Figure <ref>, and contains ferromagnetic intralayer order and antiferromagnetic stacking. The chloride orders below 24 K and has an ordered moment of 4.5 μ_B <cit.>, and the bromide orders below 14 K and has an ordered moment of 3.9 μ_B <cit.>. The iodide adopts a different low temperature structure below its Néel temperature of 9 K, with two-atom-wide ferromagnetic stipes in the plane (ordered moment of 3.7 μ_B) that are aligned antiferromagnetically with neighboring stripes <cit.>. The moment arrangement is shifted from layer to layer so that the magnetic unit cell contains four layers. This is similar to the magnetic structure of MnBr shown in Figure <ref>, but the stripes run in different directions in the plane. There is no apparent correlation between the Weiss temperatures and magnetic ordering temperatures in the FeX_2 series. This is likely related to the presence of both FM and AFM interaction in these materials, which complicates interpretation of the fitted Weiss temperatures.As noted above, FeCl undergoes a transition from the CdCl structure to the CdI structure at a pressure of 0.6 GPa. At higher pressures two additional phase transitions occur, with pronounced effects on the magnetic behavior. Above 32 GPa the orbital moment is quenched and the magnetic moments cant away from the c axis. A further increase in pressure results in the collapse of the magnetization and an insulator-metal transition that is attributed to delocalization of the Fe d electrons <cit.>. Similar behavior is reported for FeI <cit.>. In both materials the Néel temperature in increased with applied pressure, and reaches room temperature before collapsing into the non-magnetic state.In the antiferromagnetic state, magnetic field induced phase transitions, or metamagnetic transitions, occur in FeCl, FeBr, and FeI at applied fields near 11, 29, and 46 kOe, 0.95[0.95]respectively <cit.>. This arises from stronger ferromagnetic coupling within the layers compared to the weak antiferromagnetic coupling between them, and led to much of the early interest in these materials, as summarized in []and references therein. The most complex behavior is seen in FeI_2 <cit.>. From magnetization and heat capacity measurements, Katsumata et al. identified four different field induced phases, in addition to the antiferromagnetic ground state, and proposed ferrimagnetic structures for them <cit.>. In addition, Binek et al. have proposed the emergence of a Griffith's phase in FeCl_2 <cit.>, and neutron diffraction has been used to construct the temperature-field magnetic phase diagram of FeBr <cit.>. §.§.§ CoX_2 Cobalt dihalides have cobalt in electronic configuration 3d^7, which can have a high (S = 3/2) or low (S = 1/2) spin state. Orbital magnetic moments may be expected in either state. It is apparent from neutron diffraction results that the high spin state is preferred, at least for CoCl and CoBr. The ordered moment on Co in CoClwhich orders below 25 K <cit.>, is 3.0 μ_B <cit.>, and it is 2.8 μ_B in CoBr_2 <cit.>, which orders at 19 K <cit.>. These are close to the expected value of gS for S = 3/2 for high-spin only. However, magnetization measurements on CoCl_2 <cit.> indicate an enhanced effective moment in the paramagnetic state (5.3 μ_B), which suggests an orbital contribution, and a Weiss temperature of 38 K.Below their ordering temperatures, both of these compounds adopt the magnetic structure shown in Figure <ref>, with ferromagnetic alignment within each layer and antiferromagnetic stacking. The moments are known to be parallel or antiparallel to the hexagonal [210] direction for CoCl <cit.>, as shown in the Figure. The moments in CoBr are only known to lie within the ab plane <cit.>. The magnetic behavior in CoI is more complex. CoI is a helimagnet with a spiral spin structure, and anisotropic magnetic susceptibility in the paramagnetic state arising from spin-orbit coupling <cit.>. Powder neutron diffraction analysis indicated a cycloidal structure with moments in the plane and planes stacked antiferromagnetically <cit.>, which is supported by Mössbauer spectroscopy <cit.>. The corresponding in-plane spin arrangement is shown Figure <ref>.Mekata et al. used single crystal neutron diffraction to examine the magnetic order in CoI and found evidence of a more complicated magnetic structure that requires an additional propagation vector to describe. The same study identified a first order magnetic phase transition at 9.4 K, just below the magnetic ordering transition at 11.0 K, and suggested that these successive transitions may arise due to in-plane magnetic frustration, but no change in the magnetic structure was observed at 9.4 K <cit.>.An electric polarization of about 10 μC/m^2 that varies with applied magnetic field is induced below the magnetic ordering transition in CoI indicating multiferroic behavior <cit.> (see MnI above, NiBr, NiI below).§.§.§ NiX_2 The octahedrally coordinated, divalent nickel in these compounds has a 3d^8 electronic configuration, with filled t_2g and half-filled e_g orbitals. Magnetic moments are expected to be spin only, as orbital angular momentum is quenched in this configuration. Magnetization data for NiCl indicate an effective moment of 3.3 μ_B, somewhat larger than the spin only value of 2.8 μ_B expected for S = 1, and Weiss temperature of 68 K, suggesting predominantly ferromagnetic interactions <cit.>. The Néel temperatures of NiCl and NiBr are quite similar; upon cooling, both develop long range antiferromagnetic order below 52 K <cit.>. Their fully ordered moments are 2.11 and 2.0 μ_B, respectively <cit.>, as expected for S = 1. The resulting magnetic structure is shown in Figure <ref>a. The moments lie within the ab plane and are ferromagnetically aligned within each layer, with antiferromagnetic stacking. The moment directions were determined from Mössbauer spectroscopy to be parallel and antiparallel to the [210] direction, as depicted in the figure <cit.>.While the magnetic structure shown in Figure <ref> describes NiCl at all temperatures below T_N, NiBr undergoes a second phase transition, to a more complicated magnetic structure below 23 K <cit.>. Below this first order transition the magnetic moments adopts an incommensurate helimagnetic structure with a periodicity that varies with temperature. As described by Adam et al., the magnetic moments still lie within the basal plane, but vary in direction at 4.2 K by 9.72^∘ from site to site along both the hexagonal a and b axes <cit.>, as depicted in Figure <ref>. This results in a periodicity of about 37 crystallographic unit cells along each in-plane direction. The stacking remains antiferromagnetic.Heat capacity data show that NiI undergoes two phase transitions upon cooling, at 75 and 60 K <cit.>. Helimagnetic order develops at 75 K, and the phase transition at 60 K is crystallographic <cit.>. The helimagnetic structure of NiI is incommensurate with the nuclear structure and the moments rotate in a plane that makes a 55^∘ angle with the c axis, as depicted in <cit.>.The ordered moment at 4.2 K was determined to be 1.6 μ_B.Like helimagnetic MnI and CoI described above, NiBr and NiI also develop a ferroelectric polarization in their helimagnetic states <cit.>. Polarizations of 20-25 μC/m^2 are observed in the bromide, and polarizations exceeding 120 μC/m^2 are reported for the iodide. As in MnI and CoI, the polarization can be controlled by applied magnetic fields through their influence on the helimagnetic domain structure <cit.>. §.§ MX_3 Compounds Several of the MX_3 compounds listed in Table <ref> are not known to form magnetically ordered states. These include TiX_3, MoCl, TcCl, RhX_3, and IrX_3. The later two materials have electron configuration 4d^6, and are expected to have non-magnetic ground states with all electrons paired. A clue to the non-magnetic nature of MoCl <cit.> is found in the magnetic behavior of TiCl. Although neutron diffraction shows no magnetic ordering in layered TiCl at low temperature, magnetic susceptibility shows a dramatic and sharp decrease near 217 K. This corresponds to the structural distortion noted above in the discussion of TiCl and described in [] and [].Ogawa had earlier observed a lattice response coincident with the magnetic anomaly, and proposed that the formation of covalently bonded Ti-Ti dimers that pair the d electrons on each Ti as the reason for the collapse of the magnetic moment <cit.>. Thus the strong dimerization in MoCl (Table <ref>) is expected to be responsible for its non-magnetic nature. Dimerized TcCl is also expected to be non-magnetic <cit.>. §.§.§ VX_3 Little information about magnetic order in VCl or VBr is available. These compounds are expected to be magnetic due to their electron configuration 3d^2 (S = 1) and the undistorted honeycomb net of the BiI structure type reported for these materials (Table <ref>). Magnetic susceptibility data <cit.> for VCl give an effective moment of 2.85 μ_B, close to the expected value for S = 1 (2.82 μ_B), and a Weiss temperature of -30 K, indicating antiferromagnetic interactions. The maximum displayed near 20 K in the temperature dependence of the susceptibility suggests antiferromagnetic order at lower temperatures. First principles calculations have been done to examine the electronic and magnetic properties of monolayers of VCl and (hypothetical) VI <cit.>. Both are predicted to be ferromagnetic. §.§.§ CrX_3 In these compounds Cr is expected to be in a 3d^3 electronic configuration, and effective moments determined from high temperature magnetic susceptibility range from 3.7 to 3.9 μ_B per Cr as expected for S=3/2 <cit.>. Weiss temperatures determined from these measurements are 27, 47, and 70 K for CrCl, CrBr, and CrI, respectively, indicating predominantly ferromagnetic interactions. In fact, among the layeredandmaterials, the chromium trihalide family contains the only compounds in which long-range, 3D ferromagnetic ground states are observed. The magnetic structures are shown in Figure <ref>. Below 61 K for CrI and 37 K for CrBr, moments directed out of the plane order ferromagnetically <cit.>. In CrCl below about 17 K, ferromagnetic order is also observed within the layers, but the layers stack antiferromagnetically <cit.>. Also unlike the tribromide and triiodide, the moments in CrCl lie within the planes. In this series, the ordering temperatures scale nicely with the Weiss temperatures. The ordered moments determined by neutron diffraction and magnetic saturation are all close to 3μ_B as expected for the 3d^3 electronic configuration of Cr^3+. Reported values are 2.7–3.2 μ_Bfor CrCl <cit.>, 3 μ_B for CrBr <cit.>, and 3.1 μ_Bfor CrI <cit.>. Significant magnetic anisotropy is observed in the ferromagnetic state of CrI; the anisotropy field, the field required to rotate the ordered moments away from the c-axis and into the ab-plane, is found to be near 30 kOe near 2 K <cit.>. Ferromagnetic CrBr has a significantly lower anisotropy field of about 5 kOe <cit.>. With moments in the plane and antiferromagnetic stacking of the layers CrCl is unique among the chromium trihalides. It also has weak magnetic anisotropy. A magnetic field of only a few kOe is sufficient to overcome the antiferromagnetic order and fully polarize the magnetization in any direction <cit.>. Using the optical technique of Faraday rotation, Kuhlow followed closely the evolution of the magnetization in CrCl with changing temperature and applied magnetic field <cit.>. It was noted that the magnetic order appears to onset in two stages upon cooling, first developing ferromagnetic correlations and 16.7 K with long range antiferromagnetic order as shown in Figure <ref> below 15.5 K.The ferromagnetism in these CrX_3 compounds makes them particulary interesting for incorporating magnetism into functional van der Waals heterostructures. Several relevant theoretical studies have been reported that suggest ferromagnetic order may persist into monolayer 0.95[0.95]specimens <cit.>. Recently ferromagnetic monolayers of CrI_3 were demonstrated experimentally <cit.>. Ferromagnetic CrI was also recently incorporated into a van der Waals heterostructure in which an exchange field effect equivalent to a 13 T applied magnetic field was observed in the electronic properties of monolayer WSe when the heterostructure was cooled through the Curie temperature of CrI <cit.>. Although CrCl has an antiferromagnetic structure in the bulk, each layer is ferromagnetically ordered. If this proves to be independent of sample thickness then ferromagnetic monolayers may be realized in all three of the chromium trihalides, with a range of magnetic anisotropy that may allow easy tuning of the magnetization direction in the chloride or more robust moment orientation in the iodide.§.§.§ FeX_3FeCl and FeBr have iron in the 3d^5 configuration. There has been considerable study of the magnetism in the chloride, but very little for the bromide. Early magnetization measurements on FeCl found an effective moment of 5.7 μ_B, close to the expected spin-only value of 5.9 μ_B, and a Weiss temperature of -11.5 K, indicating antiferromagnetic interactions <cit.>. A neutron diffraction study found a helimagnetic structure for FeCl below about 15 K, with an ordered moment on 4.3 μ_B per iron at 4.2 K <cit.>. The reduction from the expected value of 5 μ_B may be due to some disorder still present at 4.2 K or could arise from a slight distortion from the periodic model used to describe the magnetic order. Later magnetization measurements place the Néel temperature at 9–10 K <cit.>.The magnetic structure of FeCl is shown in Figure <ref>. The figure shows one layer of Fe atoms, with dashed lines denoting (140) planes. Sites in this layer on a common (140) plane have parallel moments with their orientation indicated at the left of the Figure. Note that the moments all have the same magnitude, but their projections on to the plane of the page vary as their orientations rotate about the [140] direction by 2π/15 between neighboring planes <cit.>. The layers stack antiferromagnetically. A field induced magnetic phase transition was noted in FeCl by Stampfel et al. and Johnson et al. with the magnetic structure evolving with field up to about 15 kOe and experiencing a spin-flop 0.95[0.95]near 40 kOe <cit.>. A Mössbauer spectroscopy study of FeBr found magnetic order below 15.7 K, and the authors proposed the order below this temperature to be antiferromagnetic in analogy with FeCl <cit.>. §.§.§ RuX_3 Recent interest in RuCl began with Plumb et al. identifying it as a spin-orbit assisted Mott insulator in which a small band gap arises from a combination of spin-orbit interactions and strong electron-electron correlations <cit.>. In this compound, Ru has electron configuration 3d^5. Considering spin only gives an option of a high spin configuration (S=5/2) with all t_2g and e_g levels half filled, or a lower spin configurations with some levels doubly occupied. In this case the crystal field splitting is large enough so that all five of the d electrons go into the lower, t_2g set leaving only one unpaired spin (S=1/2). However, the orbital angular momentum is not quenched and cannot be neglected. In addition, the spin orbit coupling interaction, which increases in strength as Z^4 where Z is the atomic number of the nucleus,must also be considered for this heavy, 4d transition metal. In RuCl, spin orbit coupling, along with significant electron-electron correlations, splits the otherwise degenerate t_2g states into states with effective angular momentum j_eff = 3/2 and j_eff = 1/2 <cit.>. The j_eff = 3/2 states are lower in energy, and hold four of the five d electrons, leaving one for the higher energy level, and giving Ru in this compound an angular momentum of j_eff = 1/2. Magnetization measurements in the paramagnetic state have been reported for both powder and single crystals. Powder measurements give effective moments of 2.2–2.3 μ_B and Weiss temperatures of 23–40 K <cit.>. Single crystal measurements show strong paramagnetic anisotropy, and give μ_eff = 2.1μ_B and θ = 37 K with the field applied in the plane, and μ_eff = 2.7μ_B and θ = -150 K with the field applied perpendicular to the layers <cit.>.Two magnetic phase transitions have been observed in RuCl, at 14 K and 7 K. It is believed that the difference depends upon the details of the stacking sequence of the RuCl layers and the density of stacking faults <cit.>. Some crystals show only one transition or the other, while others samples show both. In crystals which undergo no crystallographic phase transition upon cooling (see above) and remain monoclinic at all temperatures magnetic order occurs below 14 K, while crystals that undergo a structural transition upon cooling show only the 7 K transition <cit.>. Pristine crystals that have shown a phase transition at 7 K can be transformed into crystals with only the 14 K transition through mechanical deformation <cit.>. Although all of the details of the magnetic structures of the two phases are have not been settled, there is consensus that the in-plane magnetic structures are of the so-called zig-zag type <cit.> shown in Figure <ref>. Determinations of the size of the ordered moment include ≤0.4 μ_B<cit.>,≤0.45 μ_B <cit.>, ≥0.64 μ_B <cit.>, and 0.73 μ_B<cit.>.The moment direction is reported to lie in the monoclinic ac-plane, with components both in and out of the plane of the RuCl layers <cit.>. The layers stack antiferromagnetically with a different stacking sequence associated with the different transition temperatures. AB magnetic stacking is seen in crystals with a 14 K transition, ABC stacking is seen in crystals with a 7 K transition, and both types of stacking onsetting at the appropriate temperatures are seen in samples with both transitions <cit.>. With j_eff = 1/2 and strong spin orbit coupling on a honeycomb lattice, RuCl is identified as a promising system for studying the Kitaev model <cit.>. In this model, anisotropic interactions result in a type of magnetic frustration. This can give rise to a quantum spin liquid ground state, in which fluctuations prevent magnetic order even at very low temperature, and in which particulary exotic magnetic excitations are predicted <cit.>. This is, in fact, the motivation for much of the current interest in RuCl. § SUMMARY AND CONCLUSIONS The binary transition-metal halidesandreviewed here have simple layered crystal structures containing triangular and honeycomb transition metal nets, yet they display a wide variety of interesting crystallographic and magnetic behaviors. Several compounds display polymorphism, with multiple layered and non-layered structures reported. Temperature and pressure induced crystallographic phase transitions are observed in some. Dimerization of transition metal cations results in a quenching of the magnetic moment in materials like TiX_3 and MoCl. All of the materials which maintain a local magnetic moment are observed to order magnetically, although the magnetic order in TiCl is not definitively confirmed. This compound and the vanadium dihalides clearly show evidence of geometrical frustration of strong antiferromagnetic interactions on their triangular lattices, with ordering temperatures an order of magnitude smaller than their Weiss temperatures. Effects of a different kind of frustration, due to competing anisotropic exchange interactions, is observed in RuCl, making it a promising candidate for the realization of a Kitaev spin liquid. It appears that the in-plane magnetic interactions are at least partly ferromagnetic in most of the other magneticandcompounds, and field induced phase transitions that may arise from competing magnetic interactions and multiple low energy magnetic configurations are observed in several of cases. Several dihalides adopt helimagnetic structures and develop electric polarization at their magnetic ordering temperature, providing an interesting class of multiferroic materials. Finally, interest is growing in producing monolayer magnetic materials from several of these compounds, in particular the chromium trihalides, which will enable exciting advances in functional van der Waals heterostructures. Particularly interesting for this application is the wide variety of in-plane magnetic structures that occur inandcompounds. Although several of these materials have been studied for many decades, it is likely that layered, binary, transition-metal halides will continue to provide a fruitful playground for solid state chemists, physicists, and materials scientists seeking to further our understanding of low dimensional magnetism and to develop new functional materials.§ ACKNOWLEDGMENTS This work is supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.146 fxundefined [1]ifx#1fnum [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[de Jongh(1990)]layered-TM-book author author L. 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^1 Theoretische Physik III, Ruhr-Universität Bochum, D-44801 Bochum, Germany The charge transport in a dirty 2-dimensionalelectron system biasedin the presence of a lateral potential barrier under magnetic field is theoretically studied. The quantum tunnelling across the barrierprovides the quantum interference of the edge states localized on its both sides that resultsin giant oscillations of the charge current flowing perpendicular to the lateral junction.Our theoretical analysis is in a good agreementwith the experimental observations presentedin Ref. <cit.>.In particular, positions of the conductance maximacoincide with the Landau levels while the conductance itself is essentially suppressed even at the energies at which the resonant tunnelling occurs and hence these puzzling observations can be resolvedwithout taking into account the electron-electron interaction.75.47.-m,03.65.Ge,05.60.Gg,75.45.+j Giant oscillations of the current in a dirty 2D electron system flowing perpendicular to a lateral barrier under magnetic field. A. M. Kadigrobov December 30, 2023 ================================================================================================================================§ INTRODUCTION.Investigations of low dimensional electronic structureshave opened new fields in condense matter physics such as Berezinskii-Kosterlitz-Thouless phase transition <cit.>, the quantum Hall effect <cit.>,the macroscopic quantum tunnelling<cit.>, the conductance quantization inQPCs <cit.>, to name a few.Energy gaps inelectronic spectra in semiconductors and insulators play a crucial role in their kineticand optic properties, and one of the fascinating features of low dimensional structures is a possibility to get energy gaps in electronic spectra which are gapless in the three dimensional case. One such example is a two dimensional electron gas (2DEG) with a lateral barrier under an external magnetic field. In this case the spectrum of electrons skipping along the barrier is an alternating series of extremely narrowbands and gaps <cit.>, the widths of which being ∼ħω_H for the barrier transparency D∼ 1. Such a drastic change of the spectrum is due to the quantum interference of the edge states located on the opposite sides of the barrier, the spectra ofthe latter beinggapless in the absence of the tunnelling. Such a spectra ofalternating narrow bands and gaps arisesif the quantum interference of the electron wave functions with semiclassically large phases takes place. The most prominent and seminal phenomenon of this type is the magnetic breakdownphenomenon<cit.> in which large semiclassical orbits of electrons under magnetic field are coupled by quantum tunnelling through very small areas in the momentum space. Other systemswith analogous quantum interference are those with multichannel reflection of electrons from sample boundaries <cit.>, samples with grain <cit.> or twin boundaries <cit.>. Common to all these systems are analogous dispersion equations which are sums of 2 π periodic trigonometric functions of semiclassically large phases of the interfering wave functions. Dynamics and kinetics of electrons in 2DEG in the presence of a lateral barrier under magnetic field H was experimentally and theoretically investigated in the situation that the current flows along<cit.> and perpendicular <cit.> to the barrier in ballistic samples. In all these cases giant conductance oscillations have been shown to arise. In paper <cit.> a ballistic sample with a lateral barrier in the quantum Hall regime was considered. It has been analytically shown that the lateral junction placed perpendicular to the current serves as a unique quantum-mechanical scatter for propagating magnetic edge states, and thatthis quantum "anti-resonant" scatter provides an essential increase of the transverse conductance as soon as the Fermi energy is inside one of the energy gaps in the spectrum of electrons skipping along the junction.The object of the present paper is to investigate transport properties of a dirty biased 2DEG with a lateral barrier under semiclassical magnetic field, the barrier being placed perpendicular to the current. In contrast to the ballistic situation considered in paper <cit.>the contribution of the conventional edge statesto the conductanceis neglected assuming the following inequalities being satisfied: ω_H τ≫ 1; L_y ≫ l_0 (ω_H τ) where ω_H = e H/mc and l_0=v_F τ are the cyclotron frequency andtheelectron free path length, respectively, while τ isthe free path time and v_F is the Fermi velocity.It is shown that the above-mentioned bands and gaps manifest themselves by giant oscillations of the transverse conductance with a change of the magnetic field of the gate voltage. Detailed analysis of the phases of the tunnelling matrix and the gap positions shows that the conductance peaks coincide with Landau levels in agreement with observations presentedin paper <cit.>.§ FORMULATION AND SOLUTION OF THE PROBLEM.Let us consider a 2D dimensional electron system in the presence of a lateral barrier subject to an externalmagnetic field H appliedperpendicular to its plane as is shown in Fig.<ref>.In this paper electron dynamics and kinetics are considered in the semiclassical approximation that is ħω_H ≪ε_F whereω_H =e H/m c is the cyclotron frequency and ε_F is the Fermi energy. It is also assumed that the electron free path lengthl_0 ≫ R_h where R_H = c p_F/eH is the Larmour radius and p_F is the Fermi momentum; the width of the sample L_y ≫ l_0^2/R_H and hence the contribution of the conventional edge states (which are localized at the external boundaries) to the sample conductance is negligible. The x-axis is parallel to the current flowing along the biased sample while the y-axis is along the lateral barrier.§.§ Dynamics of quasi-particles skipping along the lateral junction under magnetic field.As is seen in Fig.<ref> electrons are in three qualitatively different states: a) there areelectrons in the Landau states moving along closed orbits, b) those in the conventional edge states skipping along the external boundaries of the sample, and c) electrons in peculiar field-dependent quasi-particle states highly localized around the lateral barrier. The quantum interference between the left and right edge states results inpeculiar one-dimensional spectrum arises.As shown in Appendix<ref>, at low transparency of the barrier D=|t|^2 ≪ 1 the dispersion equation which determines the energy E_n(P_y) of quasi-particles skipping along the barrier iscosθ̅_1(E,P_y)cosθ̅_2(E,P_y)= |t|^2/4cos(θ̅_-(E,P_y))where θ̅_1,2=θ_1,2 -π/4 and θ̅_+=θ̅_1+θ̅_2 whileP_y is the conserving projection of the generalized electron momentum on the direction of the lateral barrier whileθ_1=c/e H ħ∫_-p_E^P_y√(p_E^2-p̅_y^2 )d p̅_y; θ_2=c/e H ħ∫^p_E_P_y√(p_E^2-p̅_y^2 )d p̅_y;Here p_E=√(2 m E). As one easily sees θ̅_+(E) does not depend on P_y.One sees fromEq.(<ref>) that at t=0 there are a large number of crossing points of the spectra of the left and right independent edge states (see Appendix <ref>). At final barrier transparency the degeneracy is lifted that opens gaps Δ∼ |t| ħω in the quasi-particle spectrum (see Fig.<ref>).As follows from Eq.(<ref>) degeneration takes place at |t|=0 if two equations are satisfied:cosθ̅_1(E,P_y)=cosθ̅_2(E,P_y)=0Hence positions of the degeneration points in the (E,P_y) plane are determined by the conditions: θ̅_1(E)=π/2(2 k +1) and θ̅_2(E)=π/2(2 l +1) where k,l are integer. Summing and subtracting them with the use of Eq.(<ref>) one gets{θ_+=E_n/ħω_H =π (n+1/2); θ_-(E_n,P_y)=π(2 k -n+1);where θ_±=θ_2(E,P_y) ±θ_1(E,P_y and n, k =0,1, ....Therefore, the degeneration pointsare in line withdiscrete Landau levels E_nbeing situated in discrete points P_y =P_k inside the each Landau level n and hence their positions may be uniquely classified with two discrete indexesas P_y=P_k^(n).One easily sees that the distances between neighboring points areδ P^(n)_k =|P^(n)_k-P^(n)_k± 1|∼ħ/R_H≪ p_F Therefore, if Q(P^(n)_k , E_n) is a slow varying function of the momentumon the ħ/R_H scale one may changes the summationwith respect to k to the integration as follows:∑_k Q(E_n, P^(n)_k ) = - c/π e ħ H∫ Q(E_n, P_y)√(2 m E_n -P_y^2)dP_ySumming up, the spectrum of electrons skipping along the junction is an alternating sequence of narrow energy bands E_n (P_y) (the widths of which are ∼√(1-|t|^2)ħω_H) and energy gaps Δ_n ∼ |t|ħω_H (see Ref.<cit.>), the latter lining discrete Landau levels (n is theLandau number).§.§ Current flowingalong dirty sample and perpendicular to lateral junction under magnetic field.As in the case of the magnetic breakdown phenomenon, dynamic and kinetic properties of quasi-particles skipping along the junction under magnetic field are of the fundamentally quantum mechanical nature due to the quantum interference of their wave functions with semiclassically large phases. Thus, in order to analyze the transport properties of thequasi-particles in the presence of impuritiesit is convenient to start withthe equation for the density matrix ρ̂ in the τ-approximation: i/ħ[ρ̂,Ĥ]+ρ̂ - f_0(Ĥ)/τ=-i e/ħ[ρ̂,V(x̂)]; Here, Ĥ is the Hamiltonian of the system under consideration in the absence of the bias voltage V(x), f_0 is the Fermi distribution function, τ is the electron scattering time.In this paper we assume that the barrier transparency |t|^2 ≪ 1 is so low that the main drop of the voltage applied to the sample takes place on the lateral barrier, that is it may be written asV(x)= V_0 Θ(-x)where V_0 is the voltage drop on the barrier and Θis the unit step function.Writing the density matrix in the form ρ̂= f_0(Ĥ)+ ρ̂^(1) and linearizing Eq.(<ref>) with respect to the bias potentialone gets [ρ̂^(1),Ĥ]- i ħ/τρ̂^(1) =-e[f_0(Ĥ),V(x̂)];In terms of the density matrix thethe current density at a point r_0 is written as follows: J=2 e Tr{v̂ρ̂} where v̂ is theoperator of the quasi-particle velocity.Taking matrix elements ofequation Eq.(<ref>) with respectto the proper functions of the Hamiltonian Ĥ written in the Dirac notations Ĥ|n, P_y⟩=E_n(P_y) |n,P_y⟩;one finds the density matrix. Inserting the found solution in Eq.(<ref>) one obtains the current Jflowing perpendicularto the barrier as follows: J=-i 2 e^2ħν_0 L_y/L_x∑_n ≠ n^'∫dP_y/2πħ V_n,n^'(P_y)v^(x)_n^',n(P_y)/[E_n^' (P_y)-E_n (P_y)]^2 ×[ f_0(E_n (P_y)) - f_0(E_n^' (P_y))];where O_n,n^'(P_y)=<n, P_y|Ô|n^', P_y> and ν_0 =1/τ is the electron-impurity relaxation frequency. The equation is written under assumption that ν_0 ≪ |t| ω_H. Matrix elements of the applied voltage andthe velocity operator are presented in Appendix (see Eqs.(<ref>,<ref>)).Such a peculiar dynamics as isdescribed in Subsection <ref> manifests itself inthe resonant properties of the matrix elements in Eq.(<ref>) that is especially pronounced at |t|≪ 1. Consider, e.g., V_n,n^', Eq.(<ref>). At |t|^2=0 the wave functions Ψ_2 in Eq.(<ref>) are orthogonal edge state functions andhence V_n,n^'(P_y)=0 at n ≠ n^'. Therefore, one has V_n,n^'(P_y)≠ 0 exclusively due to a final barrier transparency. Using Eq.(<ref>) one easily finds that far from the degenerate points V_n,n^'∝ |t|^2 while in the vicinity of them the resonance tunnellingtakes place and V_n,n^'∝ 1/2 and hence the main contribution to the integral in Eq.(<ref>)is from P_y in the vicinity of the degenerate points P^(n)_k(see Eq.(<ref>). Therefore, when calculating the current Eq.(<ref>) one may use Eqs.(<ref>,<ref>) and get it as follows:J=-2e^2 ν_0V_0 L_y/L_x∑_n ≠ n^'∫dP_y/2πħ T_1(κ) C_1^*(κ) C_2(κ^') ∑_α=1^2C_α^*(κ^') C_α(κ)X_α (κ)f_0(E_n (P_y)) - f_0(E_n^' (P_y))/E_n (P_y)-E_n^' (P_y);where κ ={n, P_y} and κ^' ={n^', P_y} while X_α = ∫_0^T_αx(t)dt and x(t) is the x-coordinate of the electron which is defined by the classical Hamilton equation Eq.(<ref>).Expanding the integrandin the vicinities of degeneration points P^(n)_k (where the resonant tunnelling takes place) with the use of Eq.(<ref>) one re-writesthe current, Eq.(<ref>), as follows:J=L_y/L_x2 e^2 ν_0 V_0∑_n,k( c P^(n)_y(k) / e H T_1) ×∫_-∞^∞ d P_y |t|^2 /(( v_- P_y/2 ħω_12)^2 +|t|^2×f_0(E_n+v_1 P_y))-f_0(E_n+v_2 P_y))/(v_2-v_1)P_yHere the summation is over all the degeneration points;τ =ν^-1 is the electron-impurity relaxation time,ω_12 =1/ √(T_1 T_2) andv_1,2 arethevelocitiesof the left (1) and right (2) edge states at |t|=0 while T_1(E,P_y) and T_1(E,P_y) are the times ofelectron motionbetween points y_a and y_b along the left and right classical orbitsshown in Fig.<ref>, respectively:v_1,2 = d E_1,2^(0)/d P_y =-∂θ_1,2/∂ P_y/∂θ_1,2/∂E; T_1,2 =ħ∂θ_1,2/∂E=1/ω_H(π/2±arcsinP_y/p_E)All the above-mentioned quantities are taken at E=E_n,P_y=P_y^(n)(k). The first resonantterm of the integrand is due to the resonanttransmission ofelectrons between left and right edge statesskipping along the lateral junction:at the degenerate pointsP_y=P_k^(n) the widths of the left and right wells (which arecreated by the magnetic field) are of suchvalues thatthe electron energies in them (at |t|=0)coincidecausing resonant transmissions between the wells (see Eq.<ref>and the text below it).For the case that the temperature satisfies the inequality kT ≳ |t|ħω_H one may expand the Fermi distribution functions with respect to v_1,2P_y and obtain the current as follows:J =|t|L_y/L_xσ_0V_0/ (ω_H τ)^2× ħω_H /4 π^2 T∑_ncosh^-2[ħω_H(n+1/2) -ε_F/2 T]where σ_0 n_F e^2 τ/m is the Drude conductivity , n_F =p_F^2/ħ^2 is the electron density. While writing this equation the summation with respect to k was changed to integration according to Eq.(<ref>). As one sees from Eq.(<ref>), at T ≪ħω the current oscillates with a giant amplitude under a change of the magnetic field or the gate voltage(see Fig.<ref>). If T ≫ħω the summation with respect to n may be changed to integration,∑_n ... →∫ dn ..., and thecurrent oscillations are smoothed out and the current becomes of the conventional form. In conclusion, it is shown that kinetic properties of a dirty 2DEG system with lateral junction under magnetic field is extremely sensitive to actions ofexternal fields. In particular, even a rather weakvariation of the magnetic filed or thegate voltage causes giant oscillations of the charge current flowing perpendicular to the junction. The theoretical analysis based on quantum resonance tunnelling of quasi-particles skipping along the junction is in a good agreement with experimental data: the period of the conductance oscillations,the position and the value of the conductance maxima correspond to the observations presented in Rwf.<ref>.§ WAVE FUNCTIONS AND DISPERSION EQUATION FOR QUASI-PARTICLES SKIPPING ALONG LATERAL JUNCTION. Quantum dynamics of electronswith a lateral junction under magnetic field is described by the wave function Ψ(x,y) satisfying the two-dimensional Schrödinger equation: - ħ^2/2 m∂^2 Ψ/∂ x^2+[1/2 m(-i ħ∂/∂ y+e H x/c)^2 + m ω_1^2/2y^2 + V (x)-E]Ψ =0, where thegauge is used for which the vector-potentialA = (0,Hx,0), The semiclassical solutions of the above equation on the left and right sides of the junction (x_l<x<0 and 0 <x<x_r, respectively) are Ψ_1=C_1(n,P_y)/√(p(x)/m)[ exp{i/ħ∫_x_1^x p(x^')dx^')-π/4} + h.c.] Ψ_2=C_2(n,P_y)/√(p(x)/m)[ exp{i/ħ∫_x^x_2 p(x^')dx^')-π/4} + h.c.]where quantum numbers n and P_y are the band number and the conserving projection momentum, respectivelywhile x_1,2=c/eH(∓√(2 m E)-P_y); p(x)=√(2 m E -(P_y+eH/cx)^2) Here x_1,2 are the turning points. The dependence of the constant factors C_1, C_2 andthe quasi-particle dispersion law E_n=E_n(P_y) are found by matching the above wave functions at the lateral barrier and theirnormalizationas it is shown below.In the vicinity of the junction |x|≪ R_H,one may expand the phasesof the wave functions Eq.(<ref>) in |x| and see that they areincoming and outgoing plane waves exp{(± i px/ħ)} the constant factors at whichA_1,2 and B_1,2 are A_1=C_1exp{i (θ_1-π/4)};B_1=C_1exp{-i (θ_1-π/4)};A_2=C_2exp{i (θ_2-π/4)};B_2=C_2 exp{-i (θ_2-π/4)};where θ_1=∫_x_1^0 p(x^')dx^'; θ_2=∫^x_2_0 p(x^')dx^';Changing variables in the integrals here one finds Eq.(<ref>) of the main text. The found plane wavesundergotwo-channel scatteringatthe junctionandtheconstantfactorsat the outgoing functions are coupled with the incoming oneswitha 2× 2 scatteringunitarymatrix whichis writteninthegeneral case as follows: ( B_1B_2 )=e^iΦ( rt-t^∗r^∗)( A_1A_2 ),where t and r are the probability amplitudes for the incoming electron to pass through and to be scattered back at the junction, respectively, |t|^2+|r|^2=1.Using Eqs.(<ref>,<ref>) one finds the set of equations that couples the constant factors in wave functions Eq.(<ref>):(e^-i(θ̅_1+Φ)- re^iθ̅_1)C_1 -te^+iθ_2C_2=0; t^∗e^iθ̅_1C_1 +(e^-i(θ̅_2+Φ)- r^∗e^iθ̅_2)C_2=0;where θ̅_1,2=θ_1,2-π/4. Equating the determinant of equation Eq.<ref> to zero and using the inequality |t|≪ 1 onefindsdispersion equation Eq.(<ref>) of the main text.Using equations Eq.(<ref>) one easily finds the quasi-particledispersion law in the vicinity ofpoints of degeneration, P_y=P_k^n, as follows:δ E_±=1/2(δ P_y v_+±√((δ P_y v_-)^2 +4(|t| ħω_1,2)^2))where v_± = v_2-v_1 and ω_1,2=1/√(T_1 T_2) while definitions of the velocities v_1,2 and times T_1,2 atdegeneration points are given by Eq.(<ref>). Normalization ofthe wave function Eq.(<ref>)to unity gives the second independent equation for constants C_1,2:T_1 |C_1|^2+T_2 |C_2|^2=1where T_1,2(E,P_y) =ħ ∂θ_1,2/∂ E are times of electron motion alongclassical orbits 1 and 2.For the case under considerations |t|^2≪ 1, using Eqs.(<ref>,<ref>) one finally finds|C_1(E_n,P_y)|^2 = |t|^2/4 T_2 cos^2θ̅_1 + |t|^2 T_1|C_2(E_n,P_y)|^2 =4 cos^2θ̅_1 /4 T_2 cos^2θ̅_1 + |t|^2T_1where functions θ̅_1,2(E_n(P_y),P_y) are determined by Eq.(<ref>).Using Eq.(<ref>) one easily finds cos^2θ̅_1 ≈ |t|^2T_1/4 T_2at degeneracy point P_y= P_k^(n) and hence |C_1,2(E_n,P_y)|^2=1/2 T_1,2 that is a resonant tunnelling takes place at these points. § MATRIX ELEMENTS OFAPPLIED VOLTAGE AND QUASI-PARTICLE VELOCITY. 1. Matrix elements of the applied voltage Eq.(<ref>) areV_n,n^'(P_y)=∫_x_1^0Ψ^∗_1, κ^'(x)Ψ_1,κ(x)dxwhere κ =(n, P_y) and κ^' =(n^', P_y). Using Eq.(<ref>)one findsV_n,n^'(P_y)= V_0 C_1^∗(κ) C_1(κ^') ×sin{T_1[E_n(P_y)-E_n^'(P_y)]/ħ)}/E_n(P_y)-E_n^'(P_y),n ≠ n^' In the vicinity of degeneration points one has |E_n(P_y)-E_n^'(P_y)|≪ħω_H and hence Eq.(<ref>) may be written as follows:V_n,n^'(P_y)=ħ V_0 T_1 |C_1(n,P_y)|^22. Matrix elements of the quasi-particle velocity arev^(x)_n,n' = {∫_x_1^0 Ψ_1,κ^∗i/ħ[Ĥ,x]Ψ_1,κ^'dx + ∫^x_2_0 Ψ_2,κ^∗i/ħ[Ĥ,x]Ψ_2,κ^'dx} Usingthe explicit form of the semiclassical wave functions Eq.(<ref>) one finds thevelocity matrix elements as followsv^(x)_n,n'=i E_n(P_y)-E_n^'(P_y)/ħ∑_α=1^2 C_α^∗(κ)C_α(κ^')×∫_0^T_αx^(α)(t)exp{i [E_n^'(P_y)-E_n(P_y)](T_α-t)/ħ}d twhere x(t) is defined by the classical Hamilton equation:d p/d t=e/c[vH]while x^(1)(t) and x^(2)(t) are coordinates of semiclassical packets moving along the left and right sections of the closed orbit, respectively (see Fig.<ref>). They are defined in such a way thattheir motion starts at the beginning of the corresponding section: e.g., for the motion along the closed orbit in Fig.<ref> x^(1)(0)=0, y^(1)(0)=y_b and x^(2)(0)=0, y^(2)(0)=y_a. Usinginequality |E_n(P_y)-E_n^'(P_y)|≪ħω_H, near degeneration pointsone may write Eq.(<ref>) in the form:v_n,n^'= i E_n(P_y)-E_n^'(P_y)/ħ ×∑_α=1^2 C_α^∗(κ)C_α(κ^')∫_0^T_αx^(α)(t)dt99 Berezinskii V. L. Berezinskii, Sov. Phys. JETP, 32, 493 (1971). KT J. M. Kosterlitz and D. J. Thouless,Journal of Physics C: Solid State Physics, 6, 1181 (1973). Heinzel Th. Heinzel,Mesoscopic Electronics in Solid State Nanostructures, Wiley-VCH (2003). Dittrich W. Zwerger, Theory of Coherent Transport in Th. Dittrich, G-L. Ingold, G. Schön, P. Hänggi, B. Kramer, and W. Zwerger,Quantum Transport and Dissipation, Wiley-VCH (1998). QH B. Kramer, 1998 Quantization of Transport,in Th. Dittrich, G-L. Ingold, G. Schön, P. Hänggi, B. Kramer, and W. Zwerger, Quantum Transport and Dissipation, Wiley-VCH (1998). Devoret M. H. Devoret , J. M. Martinis, and J. Clarke, 1985 Phys. Rev. Lett. 55 (1908); J. M. Martinis, M. H. Devoret, and J. ClarkePhys. Rev. B 35, 4682 (1987). Beenakker C. W. J. Beenakker and H. van Houten, Solid State Physics, 441 (1991). kang W. Kang, H.L. Stormer, L.N. Pfeifer, K.W. Baldwin, and K.W. West, Letters to Nature, 403, 59 (2000). barrier A.M. Kadigrobov, M.V. Fistul, and K.B. Efetov,Phys. Rev. B 73, 235313 (2006). graphene A.M. Kadigrobov, arXiv:1609.06648, to be published in Low Temperature Physics 43, No 1 (2017). Cohen Morrel H. Cohen and L.M. Falikov, Pfys. Rev. Lett. 6, 231 (231). KaganovSlutskin M.I. Kaganov and A.A. Slutskin, Physics Reports 98, 189 (1983). FTT A.A. Slutskin and A.M. Kadigrobov, Soviet Physics - Solid State, 9, 138 (1967). Slutskin A.A. Slutskin,Sov. Phys. JETP Lett. 26, 474 (1968). reflection A.A. Slutskin and A.M. Kadigrobov, JETP Lett. 32, 338 (1980) physica A.A. Slutskin and A.M. Kadigrobov, Physica B & C 108, 877 (1981).Peschanski Y.A. Kolesnichenko, V.G. Peschanski, Fizika Nizkikh Temperatur 10, 1141(1984) KoshkinA.M. Kadigrobov and I.V. Koshkin, Sov. J. Low Temp. Phys. 12, 249 (1986). barrierperp A.M. Kadigrobov andM.V. Fistul, J. Phys.: Condens. Matter 28, 255301 (2016). | http://arxiv.org/abs/1704.08093v1 | {
"authors": [
"A. M. Kadigrobov"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170426131215",
"title": "Giant oscillations of the current in a dirty 2D electron system flowing perpendicular to a lateral barrier under magnetic field"
} |
22.5cm 17.0cm - -1.0in8pt5pt-25pt12pt30pt24pt Higher-order radiative corrections for bb̅→ H^- W^+Nikolaos Kidonakis Department of Physics, Kennesaw State University,Kennesaw, GA 30144, USAI present higher-order radiative corrections from collinear and soft gluonemission for the associated production of a charged Higgs boson with a Wboson. The calculation uses expressions from resummation atnext-to-leading-logarithm accuracy. From the resummed cross section I derive analytical formulas atapproximate NNLO and N^3LO. Total cross sections are presented for the process bb̅→ H^- W^+ at various LHC energies.The transverse-momentum and rapidity distributions of the charged Higgsboson are also calculated. Introduction Higgs bosons play a central role in both the Standard Model and in searches for new physics. Two-Higgs-doublet models in new physics scenarios, such as the Minimal Supersymmetric Standard Model, involve charged Higgs bosons in addition to neutral ones. One of the Higgs doublets gives mass to up-type fermions while the other to down-type fermions, with the ratio of the vacuum expectation values for the two doublets denoted by tanβ. Two charged Higgs bosons,H^+ and H^-, appear in such models.An important charged Higgs production process at LHC energies is the associated production of a charged Higgs boson with a W boson, which may proceed via the partonic process bb̅→ H^- W^+ or bb̅→ H^+ W^-. This process was studied in Refs. <cit.>, and various kinds of radiative corrections were calculated in those works. There is good potential for the LHC to discover charged Higgs bosons via this process, so it is useful to calculate higher-order corrections that may enhance the cross section.An important set of higher-order corrections is due to soft-gluon emission, dominant near partonic threshold; another is due to collinear gluon emission. These corrections can in principle be resummed, and the resummation formalism can be used to construct approximate higher-order results.In this paper I present a first study of collinear and soft-gluon resummation for the associated production of a charged Higgs boson with a W boson via b-quark annihilation. Since the charged Higgs is presumably very massive, its possible production at the LHC would be a near-threshold process.I employ the resummation formalism that has been used for several related processes, including charged Higgs production in association with a top quark <cit.>, neutral Higgs production via bb̅ annihilation <cit.>, W or Z production at large transverse momentum <cit.>, top-quark production in association with a W boson <cit.>, and top-antitop pair production <cit.>.In the next section we discuss collinear and soft-gluon corrections and present their resummation. Using the expansion of the resummed cross section at next-to-leading order (NLO), next-to-next-to-leading order (NNLO), and next-to-next-to-next-to-leading order (N^3LO), we derive approximate NLO (aNLO), approximate NNLO (aNNLO), and approximate N^3LO (aN^3LO), cross sections. In Section 3 we present results for H^- W^+ total cross sections at LHC energies. In Section 4 we present results for the charged Higgs transverse momentum and rapidity distribution in this process. We conclude in Section 5.Collinear and soft-gluon resummation for bb̅→ H^- W^+For the process bb̅→ H^- W^+, involving bottom quarks inthe initial state, we assign the momentab(p_1)+b̅(p_2) → H^-(p_3)+ W^+(p_4) , and define the kinematical variabless=(p_1+p_2)^2, t=(p_1-p_3)^2, t_1=t-m_H^2, t_2=t-m_W^2,u=(p_2-p_3)^2, u_1=u-m_H^2, and u_2=u-m_W^2, where m_H is thecharged Higgs mass andm_W is the W-boson mass while the b-quarkmass is taken to be 0. We note that we work in the five-flavor scheme where the b-quark is treated as a parton in the proton. We also define the variable s_4=s+t_1+u_2, whichmeasures distance from partonic threshold where there is no energy foradditional emission; however, even when s_4=0 the charged Higgs boson andthe W boson are not constrained to be produced at rest.We note that identical considerations apply to H^+ W^- production.Radiative corrections, including collinear and soft-gluon corrections,appear at each order in the perturbative expansion of the cross section.The resummation of these corrections in our formalism is performed for the double-differential cross section in single-particle-inclusive (1PI) kinematics, in terms of the variable s_4. We note that while resummation for colorless final states is well established, previous studies have not been done in 1PI kinematics but have instead used the more inclusive variable z=M^2/s, where M is the invariant mass of the final state. Therefore, the present work is distinct from other work on Higgs or other electroweak final states. Using the s_4 resummation introduces several additional new terms in the expressions for the higher-order corrections, as we will discuss later. Furthermore, our 1PI resummation formalism allows the calculation of higher-order soft-gluon contributions to the Higgs transverse-momentum and rapidity distributions, something which is not possible with the resummation in invariant mass.The soft-gluon terms are plus distributions of logarithms of s_4,[ln^k(s_4/m_H^2)/s_4]_+, with k an integer ranging from 0to 2n-1 for the nth order corrections in the strong coupling, α_s. The plus distributions are defined by their integrals with functions f,which in our case involve perturbative coefficients and parton distributionfunctions (pdf) as discussed later, via the expression∫_0^s_4^max ds_4[ln^k(s_4/m_H^2)/s_4]_+ f(s_4)= ∫_0^s_4^max ds_4 ln^k(s_4/m_H^2)/s_4 [f(s_4) - f(0)]+1/k+1ln^k+1(s_4^max/m_H^2) f(0) . In addition, further logarithmic terms of the form (1/m_H^2) ln^k(s_4/m_H^2), of collinear origin, also appear in the perturbative expansion. These collinear terms are fully known only at leading logarithmic accuracy. In this paper we provide the first analytical and numerical study of such terms in 1PI kinematics with the s_4 variable.Resummation of collinear and soft-gluon contributions follows fromthe factorization of the cross section into various functions that describe collinear and soft emission in the partonic process. Taking moments of the partonic scattering cross section, σ̂(N)=∫ (ds_4/s)e^-N s_4/sσ̂(s_4), with N the moment variable, we write a factorized expression in 4-ϵ dimensions: σ̂^H^- W^+(N,ϵ)=( ∏_i=b,b̅ J_i (N,μ,ϵ) ) H^H^- W^+(α_s(μ))S^H^- W^+(m_H/N μ,α_s(μ) ) where μ is the scale, J_i are jet functions that describesoft and collinear emission from the incoming b and b̅ quarks,H^H^- W^+ is the hard-scattering function, andS^H^- W^+ is the soft-gluon function for non-collinear soft-gluon emission. The lowest-order cross section is given by the product of the lowest-orderhard and soft functions.The soft function S^H^- W^+ requires renormalization, and its N-dependencecan be resummed via renormalization group evolution.Thus, S^H^- W^+ satisfies the renormalization group equation (μ∂/∂μ +β(g_s, ϵ)∂/∂ g_s) S^H^- W^+ =-2 S^H^- W^+ Γ_S^H^- W^+ where g_s^2=4πα_s;β(g_s, ϵ)=-g_s ϵ/2 + β(g_s)with β(g_s) the QCD beta function; andΓ_S^H^- W^+ is the soft anomalous dimension that controls the evolution of the soft-gluon function S^H^- W^+.The evolution of the soft and jet functions provides resummed expressionsfor the cross section <cit.>. For H^- W^+production the resummed partonic cross section in moment spaceis given by σ̂_ res^H^- W^+(N)= exp[∑_i=b,b̅ E_i(N_i)] H^H^- W^+(α_s(√(s))) S^H^- W^+(α_s(√(s)/Ñ') ) ×exp[2∫_√(s)^√(s)/Ñ'dμ/μ Γ_S^H^- W^+(α_s(μ))].The first exponent <cit.> in Eq. (<ref>) resums soft andcollinear corrections from the incoming b and b̅ quarks and iswell known (see <cit.> for details). Since the resummation is performed in 1PI kinematics, we have N_b=N(-u_2/m_H^2) andN_b̅=N(-t_2/m_H^2), and this generateslogarithms involving t_2 and u_2 in the fixed-order expansions. This is animportant point, as no such terms appear in invariant-mass resummations,for which N_b=N_b̅=N.The specific forms of the expressions for the individual termsin Eq. (<ref>) depend on the gauge, although the overall resultfor the resummed cross section of course does not. In Feynman gauge the one-loop soft anomalous dimension forbb̅→ H^- W^+ vanishes; in axial gauge it is(α_s/π) C_F, where C_F=(N_c^2-1)/(2N_c) with N_c=3 the number ofcolors.We calculate the soft-gluon corrections at next-to-leading-logarithmaccuracy. However, as mentioned previously, only the leading collinearcorrections are fully known.We expand the resummed cross section, Eq. (<ref>), in α_s,and then we invert to momentum space. We provide explicit analytical resultsthrough third order for the collinear and soft-gluon corrections. The NLO collinear and soft-gluon corrections from the resummation are d^2σ̂^(1)/dt du = πα^2 m_t^4^2β/48 sin^4θ_W m_W^4 s^2 t_1^2(m_W^2 s +t_2 u_2) α_s(μ_R)/π C_F {-4/m_H^2ln(s_4/m_H^2) . +4 [ln(s_4/m_H^2)/s_4]_+ -2 [ln(t_2 u_2/m_H^4) +ln(μ_F^2/s)] [1/s_4]_+. +[ln(t_2 u_2/m_H^4)-3/2] ln(μ_F^2/m_H^2) δ(s_4) } where α=e^2/(4π), θ_W is the weak mixing angle, μ_R is the renormalization scale, and μ_F is the factorization scale. We note that the logarithmic terms involving the variables t_2 and u_2in the above expression arise from the 1PI nature of our resummation andwould not appear in an invariant-mass resummation.The NNLO collinear and soft-gluon corrections from the resummation are d^2σ̂^(2)/dt du = πα^2 m_t^4^2β/48 sin^4θ_W m_W^4 s^2 t_1^2(m_W^2 s +t_2 u_2)α_s^2(μ_R)/π^2 C_F×{-8 C_F 1/m_H^2ln^3(s_4/m_H^2)+8 C_F [ln^3(s_4/m_H^2)/s_4]_+ .+[-12 C_F (ln(t_2 u_2/m_H^4) +ln(μ_F^2/s)) -11/3 C_A +2/3 n_f] [ln^2(s_4/m_H^2)/s_4]_+ +[4C_F ln^2(μ_F^2/m_H^2) +C_F (12 ln(t_2 u_2/m_H^4) +8ln(m_H^2/s)-6) ln(μ_F^2/m_H^2).. +(11/3 C_A -2/3 n_f) ln(μ_R^2/m_H^2)] [ln(s_4/m_H^2)/s_4]_+ +[(-2 C_Fln(t_2 u_2/m_H^4) +3 C_F+11/12 C_A -n_f/6) . ln^2(μ_F^2/m_H^2). . -(11/6 C_A -n_f/3) ln(μ_F^2/m_H^2) ln(μ_R^2/m_H^2)][1/s_4]_+ } where C_A=N_c, and n_f=5 is the number of light-quark flavors. Again, the logarithmic terms involving the variables t_2 and u_2in the above expression arise from the 1PI nature of the resummation.Equation (<ref>) can be written more compactly asd^2σ̂^(2)/dt du =F_LOα_s^2/π^2{-C_3^(2)1/m_H^2ln^3(s_4/m_H^2)+∑_k=0^3 C_k^(2)[ln^k(s_4/m_H^2)/s_4]_+ } where F_LO denotes the overall leading-order factor and the C_k^(2) arecoefficients of the logarithms, and they can be read off by comparingEq. (<ref>) with Eq. (<ref>), e.g. C_3^(2)=8 C_F^2.This compact form for the aNNLO corrections will be useful in the next section.Finally, one can consider the contribution of even higher-order corrections although not all logarithms can be determined. The N^3LO collinear and soft-gluon corrections from the resummation are d^2σ̂^(3)/dt du =F_LOα_s^3/π^3{-C_5^(3)1/m_H^2ln^5(s_4/m_H^2)+∑_k=0^5 C_k^(3)[ln^k(s_4/m_H^2)/s_4]_+ } where the C_k^(3) are coefficients of the logarithms. We haveC_5^(3)=8 C_F^3, C_4^(3)=-20 C_F^3[ln(t_2 u_2/m_H^4) +ln(μ_F^2/s)] -10/3β_0 C_F^2 ,C_3^(3) = -64 C_F^3 ζ_2 +8 C_F^3 ln(μ_F^2/m_H^2) [2ln(μ_F^2/m_H^2)+4ln(m_H^2/s) +5 ln(t_2 u_2/m_H^4)-3/2]+4C_F^2 β_0 [2/3ln(μ_F^2/m_H^2) +ln(μ_R^2/m_H^2)]C_2^(3) = 160 C_F^3 ζ_3-4C_F^3 ln^3(μ_F^2/m_H^2) -12 C_F^3 ln^2(μ_F^2/m_H^2) [2ln(t_2 u_2/m_H^4)+ln(m_H^2/s) -3/2]+96C_F^3 ζ_2 [ln(t_2 u_2/m_H^4) +ln(μ_F^2/s)] -6β_0 C_F^2 ln(μ_F^2/m_H^2) ln(μ_R^2/m_H^2) +3/2β_0 C_F^2 ln^2(μ_F^2/m_H^2) C_1^(3) = -160 C_F^3 ζ_3 ln(μ_F^2/m_H^2) +4 C_F^3 ln^3(μ_F^2/m_H^2) [ln(t_2 u_2/m_H^4)-3/2]+C_F^2 β_0 ln^2(μ_F^2/m_H^2) [2ln(μ_R^2/m_H^2) -ln(μ_F^2/m_H^2)] -40 C_F^3 ζ_2 ln^2(μ_F^2/m_H^2)-24 C_F^3 ζ_2 ln^2(μ_F^2/m_H^2) [4ln(t_2 u_2/m_H^4) +10/3ln(m_H^2/s)-1] .In the above expressions, β_0=(11 C_A-2 n_f)/3. Once again, the logarithmic terms involving the variables t_2 and u_2in the above expression arise from the details of the 1PI resummation. Total cross sections for H^- W^+ productionWe consider proton-proton collisions with momentap(p_A)+p(p_B) → H^-(p_3)+W^+(p_4).In analogy to the partonic variables defined in Section 2, we define thehadronic kinematical variables S=(p_A+p_B)^2, T=(p_A-p_3)^2, T_1=T-m_H^2, T_2=T-m_W^2,U=(p_B-p_3)^2, and U_1=U-m_H^2. The hadronic variables are related to thepartonic variables via p_1=x_1 p_A and p_2=x_2 p_B, where x_1 and x_2are the fractions of the momentum carried by the partons in protonsA and B, respectively.The hadronic total cross section can be written asσ^H^- W^+ = ∫_T^min^T^max dT ∫_U^min^U^max dU ∫_x_2^min^1 dx_2 ∫_0^s_4^max ds_4x_1 x_2/x_2 S+T_1 ϕ(x_1)ϕ(x_2)d^2σ̂/dt duwhere the ϕ denote the pdf; x_1=(s_4-m_H^2+m_W^2-x_2U_1)/(x_2 S+T_1);T^^max_min=-(1/2)(S-m_H^2-m_W^2) ± (1/2) [(S-m_H^2-m_W^2)^2-4m_H^2m_W^2]^1/2;U^max=m_H^2+S m_H^2/T_1 and U^min=-S-T_1+m_W^2; x_2^min=-T_2/(S+U_1); and s_4^max=x_2(S+U_1)+T_2.Specifically, using the properties of plus distributions, Eq. (<ref>),and the compact form of Eq. (<ref>),the aNNLO corrections to the total cross section, Eq. (<ref>),can be written asσ^(2)_H^- W^+ = α_s^2/π^2∫_T^min^T^max dT ∫_U^min^U^max dU ∫_x_2^min^1 dx_2ϕ(x_2) x_2/x_2 S+T_1 ×{-∫_0^s_4^max ds_4 1/m_H^2ln^3(s_4/m_H^2) F_LOC_3^(2)x_1ϕ(x_1) . +∑_k=0^3 [∫_0^s_4^max ds_41/s_4ln^k(s_4/m_H^2)(F_ LOC_k^(2)x_1ϕ(x_1) -F_ LO^ elC_k^(2)elx_1^ el ϕ(x_1^ el)) .. .+1/k+1ln^k+1(s_4^max/m_H^2)F_ LO^ elC_k^(2)elx_1^ el ϕ(x_1^ el)] } where x_1^ el, F_ LO^ el, and C_k^(2)el denote theelastic variables, i.e. these quantities with s_4=0. Analogous results can be written for the aNLO and aN^3LO corrections.We now present results for the total H^- W^+cross section at LHC energies using MMHT2014 NNLO pdf <cit.>.For convenience we set tanβ=1 but it is easy to rescale the resultsfor any value of tanβ.In Fig. <ref> we plot the aNLO cross sections for bb̅→ H^- W^+ in proton-proton collisions at the LHC versus charged Higgs mass for energies of 7, 8, 13, and 14 TeV.The cross sections vary greatly with charged Higgs mass, falling by threeorders of magnitude over the mass range at each energy. We also observe an order of magnitude or so increase in the cross section at 13 and 14 TeV relative to 7 and 8 TeV.The inset plot of Fig. <ref> shows theK-factors, i.e. the ratios of cross sections at various orders.The four lines at the top of the inset plot show the aNLO/LO ratiosfor the four LHC energies. The corrections are clearly very significantfor all LHC energies. We also note that the K-factors at different energiesare rather similar, and are slightly higher for smaller energies. It is also important to determine how much of the full NLO corrections<cit.> are accounted for by the soft and collinear contributions.The lower line in the inset plot of Fig. <ref> shows the aNLO/NLOratio at 14 TeV energy. We see that the ratio is close to 1 for smallercharged-Higgs masses and it remains above 0.9 up to a mass of 500 GeV,indicating that the soft and collinear gluon corrections are dominant andprovide numerically the majority of the NLO corrections. The ratio remainswell above 0.8 through 1000 GeV, showing that the collinear and soft-gluoncorrections are still large and significant.In Fig. <ref> we plot the aNNLO cross sections for bb̅→ H^- W^+ versus charged Higgs massfor LHC energies of 7, 8, 13, and 14 TeV.Again, we observe a large increase in the cross section at 13 and 14 TeVrelative to 7 and 8 TeV, and a large dependence of the cross section on themass of the charged Higgs between 200 and 1000 GeV at each energy.The inset plot shows the aNNLO/LO K-factors.We note that the leading collinear terms by themselves make a significant contribution to the total collinear plus soft corrections. For example, for 200 GeV charged Higgs mass at 13 TeV energy, they amount to 20% of the aNNLO corrections.Theoretical uncertainties arise from scale variation as well as from pdf uncertainties. Scale variation by a factor of 2 around the central scale μ=m_H produces a moderate uncertainty, ± 15% at 13 TeV LHC energy for a 500 GeV charged Higgs, with similar numbers at other energies. The uncertainties from the pdf are smaller, ± 5% at 13 TeV for a 500 GeV charged Higgs.We find that results using other pdf sets are very similar. If one usesthe CT14 NNLO pdf <cit.> the results are essentially the same.We note that the aN^3LO corrections are incomplete and their numerical contribution typically small relative to the aNLO and aNNLO corrections. For example, for 300 GeV charged-Higgs mass at 13 TeV energy, the aNLO corrections contribute a 23% enhancement, the aNNLO corrections an additional 14% enhancement, and the aN^3LO corrections a further 2% enhancement. The fact that the aN^3LO corrections are much smaller than the corrections at previous orders is an indication of perturbative convergence, and is also in line with related results for Higgs production and top-quark production (see e.g. <cit.>). Since the uncertainty due to uknown terms at aN^3LO can be of the order of the size of these corrections, we do not study them further. We also note that there are no pdf available at N^3LO for such calculations, and the effect of such pdf may also be nonnegligible.Charged Higgs p_T and rapidity distributionsWe continue with the charged Higgs p_T and rapidity distributions. The charged Higgs p_T distribution is given bydσ/dp_T =2 p_T ∫_Y^min^Y^max dY ∫_x_2^min^1 dx_2 ∫_0^s_4^max ds_4x_1 x_2 S/x_2 S+T_1 ϕ(x_1)ϕ(x_2)d^2σ̂/dt du where T_1=-√(S)(m_H^2+p_T^2)^1/2e^-Y, U_1=-√(S)(m_H^2+p_T^2)^1/2e^Y, Y^^max_min=± (1/2) ln[(1+β_T)/(1-β_T)] withβ_T=[1-4(m_H^2+p_T^2)S/(S+m_H^2-m_W^2)^2]^1/2, and the other quantitiesare defined in Section 3. We note that the total cross section can also becalculated by integrating the p_T distribution, dσ/dp_T,over p_T from 0 to p_T^max=[(S-m_H^2-m_W^2)^2-4m_H^2m_W^2]^1/2/(2√(S)),and we have checked for consistency that we get the same numerical resultsas in Section 3.In Fig. <ref> we plot the aNNLO p_T distributions, dσ/dp_T, of the charged Higgs boson with mass 200 GeV for LHC energies of 7, 8, 13, and 14 TeV. The inset plot shows the aNNLO/LO K-factors. The corrections are large, around 50%, for much of the p_T range shown. The distributions peak at a p_T value of around 65 GeV for this mass choice.In Fig. <ref> we plot the corresponding aNNLO p_T distributions of the charged Higgs boson with mass 500 GeV. The inset plot shows the aNNLO/LO K-factors and, again, the corrections are large. The distributions now peak at a higher p_T value of around 110 GeV.It is useful to also study normalized distributions since normalization removes the dependence on tanβ and it minimizes the dependence on the choice of pdf. Such normalized distributions are also often favored in experimental studies and comparisons with theory.In Fig. <ref> we plot the aNNLO normalized p_T distributions, (1/σ) dσ/dp_T, of the charged Higgs boson with mass 200 GeV (left plot) and 500 GeV (right plot) for LHC energies of 7, 8, 13, and 14 TeV. The shape of the normalized p_T distributions depends on the energy, as expected, with higher peaks at lower energies. We also observe that the peaks are lower for a 500 GeV mass than for 200 GeV. The charged-Higgs rapidity, Y, distribution is given bydσ/dY = ∫_0^p_T^max 2 p_T dp_T ∫_x_2^min^1 dx_2 ∫_0^s_4^max ds_4x_1 x_2 S/x_2 S+T_1 ϕ(x_1)ϕ(x_2)d^2σ̂/dt du where p_T^max=((S+m_H^2-m_W^2)^2/(4Scosh^2Y)-m_H^2)^1/2 and the rest of the quantities are defined as before. We again note that the total cross section can also be obtained by integrating the rapidity distribution, dσ/dY, over rapidity with limits Y^^max_min=± (1/2) ln[(1+β)/(1-β)] where β=(1-4m_H^2/S)^1/2, and again we have checked for consistency that we get the same numerical results as in Section 3.In Fig. <ref> we plot the aNNLO rapidity distributions,dσ/d|Y|, of the charged Higgs boson with mass 200 GeV for LHC energiesof 7, 8, 13, and 14 TeV. The inset plot shows the aNNLO/LO K-factors. The corrections are quite large, especially at lower LHC energies, andthey grow at larger values of charged Higgs rapidity.In Fig. <ref> we plot the corresponding aNNLO rapidity distributions of the charged Higgs boson with mass 500 GeV. The aNNLO/LO K-factors are again shown in the inset plot. We observe that the 7 and 8 TeV K-factors increase rapidly at larger values of rapidity.Finally, in Fig. <ref> we plot the aNNLO normalized rapidity distributions, (1/σ) dσ/d|Y|, of the charged Higgs boson with mass 200 GeV (left plot) and 500 GeV (right plot) for LHC energiesof 7, 8, 13, and 14 TeV. For a given charged Higgs mass the normalized rapidity distributions at lower energies have higher peaks at central rapidity with corresponding smaller values at large |Y|, as expected. The fall of the distributions with increasing |Y| is sharper for m=500 GeV than for 200 GeV at all LHC energies.ConclusionsThe cross sections for the associated production of a charged Higgs boson witha W boson, via bb̅→ H^- W^+, receive sizable contributionsfrom collinear and soft gluon corrections.These radiative contributions have been resummed, and approximate double-differential cross sections have been derived at NLO, NNLO, and N^3LO.Numerical predictions have been provided for the total cross section forH^- W^+ production at LHC energies as well as for the p_T and rapiditydistributions of the charged Higgs boson.The higher-order corrections are significant and they enhance thetotal cross section and differential distributions for H^- W^+ productionat the LHC.*Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. PHY 1519606.99DHKR D.A. Dicus, J.L. Hewett, C. Kao, and T.G. Rizzo, Phys. Rev. D40, 787 (1989).DK D.A. 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"authors": [
"Nikolaos Kidonakis"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170427131651",
"title": "Higher-order radiative corrections for $b{\\bar b} \\rightarrow H^- W^+$"
} |
[email protected] School of Physics, CRANN and AMBER, Trinity College Dublin, Dublin 2, Ireland Stephenson Institute for Renewable Energy andDepartment of Chemistry, The University of Liverpool, L69 3BX Liverpool, United Kingdom Beijing Computational Science Research Center, Beijing 100094, China School of Physics, CRANN and AMBER, Trinity College Dublin, Dublin 2, Ireland In electronic structure methods based on the correction of approximate density-functional theory (DFT) for systematic inaccuracies, Hubbard U parameters may be usedto quantify and amend the self-interactionerrors ascribed to selected subspaces.Here, in order to enable the accurate, computationally convenientcalculation of Uby means of DFT algorithms that locate the ground-state by direct total-energyminimization, we introduce areformulation of the successfullinear-response method for U in terms of the fully-relaxed constrained ground-state density.Defining U as an implicit functional of the ground-state density implies the comparability ofDFT + Hubbard U (DFT+U)total-energies, and related properties, as external parameters such as ionic positions are varied together with theircorresponding first-principlesU values.Our approachprovides a framework in whichto address the partially unresolvedquestion of self-consistency over U, for whichplausibleschemes have been proposed, and to precisely define theenergy associated with subspace many-body self-interaction error.We demonstrate that DFT+Uprecisely corrects the total energy for self-interaction error under ideal conditions,but only if a simple self-consistency condition is applied.Such parameters also promote to first-principlesarecently proposed DFT+U based method for enforcing Koopmans' theorem. A self-consistent ground-stateformulation of the first-principles Hubbard Uparameter validated on one-electron self-interaction error David D. O'Regan December 30, 2023 =========================================================================================================================================§ INTRODUCTIONApproximate density-functional theory(DFT) <cit.> is a central element in the simulation ofmany-body atomistic systems and an indispensable partner toexperiment <cit.>.DFT is prone, however, within its commonplacelocal-density (LDA) <cit.>, generalized-gradient (GGA) <cit.>, and hybrid <cit.>exchange-correlation (xc) approximations, to significant systematic errors <cit.>.The most widely encountered of these is the many-electron self-interaction error(SIE) <cit.>, or delocalization <cit.>error, which is manifested as a spurious curvaturein the total-energy profile of a system with respect to its total electronnumber <cit.>. The SIE contributes to inaccuracies ininsulating band gaps <cit.>,charge-transfer energies <cit.>, activation barriers <cit.> ,binding and formation energies, as well as in spin-densities and their moments.While the nature of SIE is well understood, it remains persistently challenging to reliably avoid its introduction usingapproximate xc functionals of a computationally tractable, explicit analytical form, even if exact exchange is incorporated (see, e.g.,the B3LYP curve inFig. 2 of Ref. doi:10.1021/cr200107z). DFT+U(DFT + Hubbard U) <cit.> is a computationally efficient <cit.> and formally straightforward methodthat has matured as a corrective approach for SIE in systems where it may be reasonably attributed to particular selected subspaces <cit.>.Originally designed to capture Mott-Hubbardphysics in transition-metal oxides <cit.>,it now sees very diverseapplications <cit.>.It has gained a transparent interpretation as an efficient corrective method forSIE particularly since the work of Kulik, Cococcioni, and co-workers inRef. PhysRevLett.97.103001.The DFT+U corrective energy term is often invoked in itsrotationally-invariant,simplified form <cit.>, given byE_U [ n̂^Iσ]=∑_I,σU^I/2Tr[ n̂^I σ - n̂^I σn̂^I σ],where the density-matrices n̂^I σ = P̂^I ρ̂^σP̂^I are those for the subspaces I over which the SIE is to becorrected. Here, the Kohn-Sham density-matrix ρ̂^σ corresponds to thespin indexed by σ, which we hereafter suppress for simplicity,and the idempotent subspace projection operatorsP̂^I=∑_m|φ^I_m⟩⟨φ^I m|are usually built from fixed, spin-independent,orthonormal, localized orbitals, which may also be nonorthogonal <cit.>andself-consistent <cit.>.The quadratic term of Eq. <ref>alters the intra-subspace self-interaction, on a one-electron basis in the frame of the individual orthonormal eigenstates of n̂^I,which may generallybe expected to change as a result.The linear term then imposes the conditionthat the correction to the total-energy should vanishfor each subspace eigenstate as its correspondingeigenvalue n^I_i approaches zero or one,implying that the xc functional isassumed to be correct for such eigenstates.This mirrors the well-known resultthat the total-energy of open systems at integer fillingis reasonably well described by conventionalapproximate xc functionals <cit.>.While the linear term does notdirectly affect the SIE explicitly, it represents an important boundary condition on the SIE correction.Simultaneously , the corresponding correction to the potentialv̂_U^I = U^I(1̂-2n̂^I)/2 vanishes at eigenvalues of one-half and,when a Kohn-Sham gap is symmetry-allowed, the occupancy-dependence of the potentialacts to energetically split stateslesser and greater in occupancy than one-half by an energy intervalon the order of U <cit.>. §.§ A one-electron litmus test:how DFT+U affects H_2^+ As it exhibits nomulti-reference or static correlation error effects, by definition,but a straightforwardly variable bonding regime, the dihydrogen cation H_2^+ is perhaps the ideal system for the study of pure SIE, also known as delocalization error <cit.>.It serves as a convenient test bed for the exploration ofsystem-specificadditive corrections, such as DFT+U, and more generally for density-functionals which are, at least in part, implicitly defined via parametersto be calculated, such as the self-consistent Hubbard U. Particularly subject tothe idealpopulation analysisand non-overlapping subspace conditions availablein the dissociated limit, H_2^+ willallow us to draw firm conclusions regardingthe numerous plausible butdifferent strategiescurrently in use fordefining self-consistency over the Hubbard U. The action of the DFT+U functional under varying bonding conditions may be observedin the dissociation curves of H_2^+ depicted inFig. <ref>.Here, the total-energy error in approximate DFT, specifically the PBE xc-functional <cit.>, is seen to grow significantly with bond-length asthe electron count on each atomapproaches one-half.The result of the exact xc-functional,in which the Hartree and xc energies and potentials cancel,is indicated by the solid line,and the results ofDFT(PBE)+U = 0, 4, 8 eV,are indicated by thedashedlines [Calculations were performedusing the DFT+Ufunctionality <cit.> available in the ONETEP linear-scaling DFT package <cit.>with a hard (0.65 a_0 cutoff)norm-conserving pseudopotential <cit.>,10 a_0 Wannier function cutoff radii, and openboundary conditions <cit.>. DFT+U was applied simultaneously to each an atom, using a separate 1sorbital subspacecentred on each, defined using the occupied Kohn-Sham state of the pseudopotentialfor neutral hydrogen. The correct symmetry of H_2^+was maintained for all values of Ugiven a symmetric initial guess, i.e., we observedno tendency for the charge to localize on a single ion.].The Hubbard U parameter required to correct the PBE total-energy to the exact valuevaries over approximately 8 eV from the fully bonded todissociated limits, highlighting theimportance of chemical environment dependent, and not just species-dependent,U parameters, as previouslyshown,e.g., in Refs. doi:10.1063/1.3660353, doi:10.1063/1.4865831,PhysRevB.90.115105.A critical and perhaps defining characteristicof an SIE-free system isits compliance with Koopmans'condition <cit.>,and in a one-electron system such asH_2^+ this implies that the total-energy and theoccupied Kohn-Sham eigenvalue ε should differ only by the ion-ion energy. Thus, the dissociation curve of H_2^+should be equivalently accessible by calculating the total energies for the dimer and its constituent atoms directly,or by using total energiesderived from the occupied eigenvalue and the expression E = ε + E_ion-ion. Fig. <ref> illustrates thestrikingly poor results of DFT+U when combinedwith this latterprocedure [In all dissociation curves presented,the fully dissociated reference energy was fixed to the total-energyof a single exact-functional hydrogen atom placed at the midpoint of the dimer, which is equal tothe occupied Kohn-Sham eigenvalue of the same system, or two half-charged exact hydrogen atoms. In this way, only the SIE specific to the PBE dimer is analysed,without the SIE present inisolated,half-charged PBE hydrogen atoms.].A U ≳ 4 eV is required for the eigenvalue-derived dissociation curve to exhibit a local minimum.We observe that the non-compliance with Koopmans' condition (disagreement between Figs. <ref> and <ref>) broadly decreases both withbond-length and with the Hubbard U,but that the effect of DFT+U on the eigenvalue is lost entirely in the dissociated limitsince both factors drive thesubspace occupancy to 1/2.Across the dissociation curve, the Hubbard Urequired to enforce compliance with Koopmans' condition, and that needed to attain theexact result typically differ substantially. The facile correction of the total-energy of this systemat each bond length with a varying but reasonableU value sharply contrastswith the inefficacy of DFT+U for fixing the occupied eigenvalue.At dissociation, the latter is not significantly affected by DFT+U, suggesting an intrinsic limitation in the linear term of that correction.We have previously introduced a generalized DFT+U functional in Ref. PhysRevB.94.220104, in whichthe linear term was amended to enforce Koopmans condition. We will return to put this approach on a first-principles footing using self-consistent variational Hubbard Uparameters in Section <ref>.§.§ The Hubbard U as a first-principles response property: motivations for seeking a variational formalism TheHubbard U^Iare external parameters that define the SIE correction strength applied to each subspace in DFT+U.They may be thought of as subspace-averaged SIEs quantified in situ <cit.>. Historically and to this day,the Hubbard U has frequently been determinedvia the empirical fitting of calculatedDFT+U observables to experimental data, typically spectral <cit.>;structural <cit.> or energetic <cit.>.This approach is pragmatic and in many cases very successful,but it is clearly inapplicable when the necessary experimental data is unavailable or difficult to measure.If the Hubbard U parameters are instead themselves calculated as properties of the electronic structure,however, becoming no longer free parameters but auxiliary variables,in effect, then DFT+U is restored to a first-principles status.If the Hubbard U are calculated strictly as variational ground-state density-functional properties, even implicitly,i.e., one of the central developments of the present work,then DFT+U as whole becomes afully self-contained, variational first-principles method. Only under the latter condition would we expect thefully rigorous direct comparability of the total energies,and their derived thermodynamic observables,calculated from different DFT+U calculations with different system-specificU parameters.In this work, we build upon the very widely-used <cit.>andsuccessful linear-response method proposed inRef. PhysRevB.71.035105, in which Cococcioni and de Gironcolidemonstrated that a small number of DFT calculations is sufficient to calculate first-principles Hubbard U parametersby finite-differences, as well as uponthe earlier linear-response scheme proposed byPickett and co-workers <cit.>,and aspects of the modified constrained LDA scheme of Aryasetiawan and co-workers <cit.>.In this linear-response DFT+U method <cit.>,a small, external uniform perturbation of strength αis appliedto the subspace of interest andthe interacting response function χ, and its non-interacting Kohn-Sham equivalent χ_0,are computed, respectively, from the first derivativesof the self-consistent and non-self-consistentsubspace total occupancies N^I=Tr[n̂^I]with respect to α.The scalarDyson equationU^I = (χ_0^-1- χ^-1)^IyieldsUfor a single subspace model, which may be further improvedunder self-consistency <cit.>.The Hubbard U parameters appropriate to a generalized model in which inter-subspace parametersV are included <cit.>,as well as the inter-subspace V themselves,may also be calculated <cit.>by treating the Dyson equation as a site-indexedmatrix equation [In this work, since we find it necessary to use only single-site DFT+U with no +V term,we treat the two 1s atomic subspaces as decoupled, each comprising the majority of the screening bath for the other. Hence we usea pair of scalar Dyson equations(identical by symmetry, i.e., only one is treated numerically) to calculate the Hubbard U,rather than selecting the diagonalofthe 2 × 2 site-indexedHubbard U.].To date, based on our extensive literature search,the linear-response method has only been used in conjunction with the self-consistent field (SCF) algorithms very typically used to solve the Kohn-Sham equations for smaller isolated and periodic systems. In the SCF case, it is convenient to calculatethe non-interacting response function χ_0following the first iteration of the SCF cycle as prescribed in Ref. PhysRevB.71.035105, i.e.,following the initial charge re-organizationinduced by the external potential v̂_ext = α^I P̂^I, but before anyupdate of the remaining terms in the Kohn-Sham potentialis carried out.This techniqueis impractical to implement, however,in codes that utilize a direct minimization ofthe total-energy with respect to the density, Kohn-Sham orbitals, or density-matrixto locate the ground-state, since there it is not efficient or customary to nest the density andpotential update processes.These codes comprise a growing number of linear-scalingor large-system adapted packages, where explicit Hamiltonian diagonalization is typically avoidedaltogether where possible,such as ONETEP <cit.>,CONQUEST <cit.>, Siesta <cit.>,BigDFT <cit.>,OpenMX <cit.>,and CP2K <cit.>, among others,albeit that the SCF techniquemay also be available in some of these.We are therefore motivated to seek a linear-response formalism for the Hubbard U that is readily compatible with direct-minimization DFT and large systems,particularly for linear-scaling DFT+U <cit.>. In this work, we develop and presenta minimal revision of the established `SCF linear-response' approach(terminology specific to this article,introduced for the avoidance of ambiguity)for the Hubbard Uparameters, one based on the response of the fully relaxed ground-state density subject to a varying perturbation.We have implemented our `variational linear-response'method for the U in thelinear-scaling direct-minimization DFT packageONETEP <cit.>,where the cost of the method itself scales with the number oftargeted subspaces to be assessed, multiplied by the totalnumber of atoms present. It is thus readily applicable to systemsthat are both spatially disorderedand electronically challenging.More generally, the variational linear-response method is equally applicable todirect-minimization and SCF DFT codes irrespective of the basis set used, and itmay prove helpful in cases where the SCF non-interacting response χ_0 is numerically problematic <cit.>. We find that it provides a convenientframeworkin which to analyzea number of different criteria that have beenproposed for defining the self-consistentHubbard U and with it, for the particular case of the variational linear-response Hubbard U at least, we identify awell-defined best choice ofself-consistency criterion supported by numerical results. §.§ Article outlineIn Section <ref>, we investigatethe conditions that must holdfor a first-principles Hubbard U parameterto correct SIE subspace-locally by means of Eq. <ref>.Arriving at a simple, variational linear-responseformulation in terms offully-relaxed constrained density and its resulting properties,we make the calculation of Hubbard U parameters accessible to direct-minimization DFT codes.In Section <ref>,we address, for specific case of variationallinear-response, the question: which of thepreviously-proposedand availableHubbard U self-consistency criteria, if any is necessary,is suitable for correcting the SIE-affectedtotal energy by means of DFT+U?In Section <ref>, we further analyse our results by means of numerically stringent DFT+U calculations along the dissociation curve of H_2^+,an ideal system for studying one-electron SIE <cit.>.Finally, in our concluding Section <ref>, we discuss the theoretical relationship betweenthe SCF and variational linear-response formalisms,the relevanceto the comparability of total-energies and other thermodynamics quantities from DFT+U calculations with system-specific first-principles Hubbard U parameters, and our outlook on thepracticability of such parameters.§ A VARIATIONAL GROUND-STATE APPROACH TO THELINEAR-RESPONSE HUBBARD U PARAMETERIn order to calculate the Hubbard Uparameter required to subtractthe many-electron SIEattributed to a particular subspace,by means of Eq. <ref>, we may define the parameter for each subspace as the net average electronic interaction acting within it.More specifically, we seek only the interactions at leading order in the subspace density-matrices,that is those coupling to ( n_i^I )^2,in order to comply with Eq. <ref>, Thus, for a particular site,wedefine the U on the basis of the interaction kernelf̂_int =δ^2 E_int / δρ̂^2 only,i.e., not ĝ_int =δ^3 E_int / δρ̂^3 etc., where E_int is the interactingcontribution to thetotal-energy.Furthermore, we require only the components ofthe interaction for each subspace that arise due to density variations within it, so that f̂_int must be appropriately projected.In order to illustrate the requirements of such a projection, let us consider some candidate formulae for HubbardU parameters which do not meet them. The many-electron SIE of an approximate xc functional, applied to an an open quantum systemthat does not interact with its bath for particle exchange, is characterized by the spurious non-zero second total-energyderivative with respect to the total occupancy.We may apply this definition to the individual DFT+U subspaces,with occupancies given by N = Tr [ n̂](suppressing subspace indices), under the reasonable assumption that the subspace-bath interactions are negligible compared to the interactions within the subspace.By defining the Hubbard U for each subspaceas the net value of the latter interaction,in a precise sense yet to be determined, the DFT+U functional should act to correct the many-body SIE by subtracting the individual one-electron SIEof each eigenstate of the subspace density-matrix.Immediately, we may rule outas a Hubbard U parameter thestraightforward fully interacting curvature d^2 E / d N^2, discussed in Ref. PhysRevB.58.1201,which may be calculated as - d α / d N.Here, v̂_ext = αP̂is the external potential inducing the occupancy change.As discussedin Ref. PhysRevB.71.035105,this term comprises a substantial non-interacting contribution,which, in accordance with Dyson equations quite generally, is superfluous to the definition of an interaction and must besubtracted. On the other hand, one may suggest the direct subspace projection of the interaction kernel(Hartree, xc, any other electronic interaction terms),denoted here for a single site by P̂( δ^2 E_int /δn̂^ 2 ) P̂= P̂f̂_intP̂.Any bare interaction of this kindneglects the potentially substantial screening effectsof density-matrix variations outside the subspace.Thus, it is also an unsuitable starting point for measuringmany-body SIE,ruling it out.TheU must be bath-screened, yet remain bare ofintra-subspace screening. More interesting is the curvature of the interaction term in the total-energy, d^2 E_int / dN^2,and the reasons for itsnon-suitability are perhaps more subtle.Since the Hellman-Feynman theoremcannot be applied to E_int alone, its first total derivative with respect to N, i.e., dE_int/dN = ∂ E_int/∂ N + Tr[ δ E_int/δρ̂d ρ̂/ d N]yields not only the partial derivative(vanishing due to no explicit N-dependence in E_int, and only an implicitdependence via the changes to the total density-matrix ρ̂),but it alsocomprises a term proportional totheinteraction potential δ E_int / δρ̂,bath-screened since d ρ̂ / d N couples to the external potential v̂_ext = αP̂.The second total derivative d^2 E_int / d N^2incorporates screening again,and the resulting twice-screened objects areunphysical. This problem here is the opposite, in a sense, to that of P̂f̂_intP̂,from which one may surmise thecorrect definition is an intermediate case,where screening effects due to the complementof the subspace at hand should be incorporated, but only once.This motivates us to work not from the energy, butfrom the potential, i.e., from the unscreened functional derivative of the energy with respect to the density-matrix, and todifferentiate by N. As a functional derivative, i.e., a generalized partial derivative, the interaction termin the Kohn-Sham potentialv̂_int = δ E_int/ δρ̂is bare of screening, as is its subspace projection P̂(δ E_int/ δn̂ ) P̂ =P̂v̂_intP̂. The quantity then given by Tr [ P̂( d v̂_int / d n̂) P̂] / Tr [P̂ ]^2seems to fulfil many of the requirements for a valid Hubbard parameter, namely, that it is a subspace-averaged, once-screened interaction that is non-extensive,i.e. it does not scale extensively with the subspaceeigenvalue count Tr [ P̂ ].In practice, however,the screened kernel d v̂_int / d n̂is cumbersome to calculate, even in orbital-free density-functional theory, and, more importantly, it includes screening effectsdue to density-matrix rearrangementswithin the subspace, which make it unsuitable as a quantifier for the subspace-bare interactionto be explicitly corrected by DFT+U.Instead, the object that we required isthe average, net, subspace-barebut bath (i.e., environment) screenedself-interaction of the subspace.We may meet these specifications by taking the total derivativewith respect to the total subspace occupancy N, and,finally, by defining U ≡d v_int/dN, v_int≡Tr [v̂_intP̂] /Tr [P̂ ] is the conveniently calculated,non-extensive, subspace-averaged interaction potential (comprising Hartree, xc, etc.).Here, since the uniform potential αused in the linear-response methodinduces to first-order no microscopicdensity-variations within the subspace except for the uniform shift,the screening processes withinthe subspaces are effectively suppressed,much as in the constrained random phase approximation <cit.>.In practice,as we return to discuss around Eq. <ref>. the proposed variational linear-response U for asingle-site model may still be computedusing the Dyson equation, but with the response functionsχ = d N / d α and χ_0 = d N / d v_KS,where v_KS≡Tr[v̂_KSP̂] / Tr [P̂ ]. Here, both χ and χ_0 are to calculated at the endof the minimization procedurefrom the same set ofconstrained ground-state densities defined by α.Thus, while χ is identical to that used in the SCF linear-responseintroduced inRef. PhysRevB.71.035105, our χ_0 formula is somewhat different (at least formally, the numerical differences remain unclear).The SCF and variational linear-responseformalisms are equally compatible with SCF and direct-minimization DFT,as well as with the matrix Dyson equation required, e.g., for calculating longer-rangedinter-subspace parameters V <cit.>and their corresponding Hubbard U values. §.§ The subspace contribution to total-energy curvature The subspace contributionto the interacting part of the total-energy SIE,specifically that corresponding to the variational linear-response Hubbard U, is theintegral of thethe interacting part of the Kohn-Shampotential over the subspace occupancyup to its ground-state value. To the same effect, we may use the negative of the integral overthe external potential α needed to fully deplete that occupancy back to zero, as in E^SIE_int( N ) = ∫_0^N v_int(N') dN' =- ∫^∞_0 v_int(N'(α)) dN'/dα dα= ∫^∞_0 v_int(N'(α)) ( . d^2 E_total/d N”^2|_N”( α))^-1 dα In the final line,we make the connection to the occupancy curvature of the total-energy E_total using the result for the constrained DFT system,d E_total / d N = - α.Although E^SIE_int does notappear anywhere in our DFT+U implementation in practice, we emphasise that for a single-site model, in the variational linear-responseformalismat least, it isd^2 E_SIE / d N^2 = U, and not the total-energycurvature d^2E_total / d N^2 = - χ^-1, which yields theparameter required for DFT+U. Considering the difference of theenergy curvatures ascribed tothe bath-screened subspace and the overall global system,both as a function of subspace occupancy, we find that d^2 ( E_total - E_SIE) / d N^2 = - χ^-1 - U = - χ_0^-1≥ 0, where the latter inequality was proven in Ref. PhysRevB.94.035159.This result is reminiscent of the findingsof Kulik et al. in Ref :/content/aip/journal/jcp/145/5/10.1063/1.4959882,to wit, that while the application ofDFT+U can only be expectedto mitigate subspace SIE,and at the very least it cannot disimprove the global SIE, albeitfor a different sense of global pertaining to total occupancy.The quantity E^SIE_int differs from the full subspace contribution to the total-energy SIE by a non-interacting contribution required to restore Koopmans'condition.It is interesting to assume, for a moment, that the SIE kernel U = d v_int / d N is Hartree-dominated and hence approximately constant,so that v_int ( N ) ≈ U N and E^SIE_int≈ U N^2 / 2. If we further assume that the subspace is SIE-free at the nearest integer occupancy,N_0, as well as at N_0 ± 1, with a linear (i.e., non-interacting) interpolation term being requiredbetween these points, then we may make the curvature-preserving modificationE^SIE≈ ( U / 2 ) [ ( N - N_0)^2 -| N - N_0 | ]. Taking the negative of this energy to estimate a total-energy correction, and consideringsingle-orbital, single-spin sites, we effectively re-derive the E_U of Eq. <ref>.Even non-self-consistently,this turns out to be an acceptable energy correction for H_2^+ in the dissociated limit, with two subspaces ofU ≈ 8 eV and N ≈ 1/2, yielding- E_SIE≈ 2 eV ≈ E_exact - E_PBE. § SELF-CONSISTENCY OVER THE HUBBARD U Beginning with the work ofKulik and co-workers in Ref PhysRevLett.97.103001, and in later works <cit.>,it has been demonstrated thata self-consistently calculated U can be requiredfor certain systems wherethe nature of the electronic states (and corresponding response properties)in the DFT+U ground-state differqualitatively from those of the DFT ground-state <cit.>.In self-consistency schemesgenerally,incremental values of U_in are applied to the subspace at hand, with varying ground-state orbitals and densities as a result, and a new first-principles U_out iscomputed for each U_in.Thenumerical relationshipU_out ( U_in ) is then used to select the self-consistent U, using a pre-definedcriterion.Its clear conceptual elegance aside, a self-consistent Uhas been shown to provide improvements in transition-metalchemistry <cit.>,biological systems <cit.>, photovoltaics <cit.>,and high-density energy storage <cit.>.While many researchers have used an original,linear-extrapolation type U_scf in their studies <cit.>,others have used the equality between U_in and U_outas an alternative self-consistency condition <cit.>.The majority of published first-principles U calculationsinvolve no self-consistency over theparameter at all, and there mayeven be a case to be made that none is ordinarily warranted. The resolution of this ambiguity is, in itself,an intriguing open challenge in abstract DFT, butit particularly demands investigation in the present context of the variational linear-response U since, ideally, the optimal schemeto match that method should be established from the outset.On the basis of this study, however, weemphasise that we cannot drawconclusions concerning theself-consistency schemes forU parameters calculated bymeans of any other methods.In order to compute variational linear-responseU_outfor a single subspace already subject to a DFT+U term of strength U_in,the subspace-averaged interaction v_int must incorporate the DFT+U potential v̂_U, as well as the usual Hartree + xc term v̂_Hxc.Each component in the subspace-averaged interaction potentialv_KS - v_ext≡v_int=v_Hxc+v_U_in must be defined in such a manner thatdoes not scale extensively with the orbital count ofthe subspace, Tr [ P̂ ]. For Hartree + xc, the appropriate average isv_Hxc=Tr[v̂_HxcP̂]/Tr[P̂] (the operator v̂_Hxc may approximatelyscale with N but the averaging scheme does not),while the average differential to the external potential is, similarly, d v_ext=Tr[d̂ v_extP̂]/Tr[P̂]=Tr[d αP̂P̂]/Tr[P̂] = d α by theidempotency of P̂.Unlike v̂_Hxc, which acts on one state but is generated by all occupied states, theDFT+U potentialv̂_U_in = U_in (P̂- 2P̂ρ̂P̂)/2 is intrinsically both specific toand due toeach subspace occupancy matrix eigenvector individually.Thus, we find that the simple tracev_U_in = Tr [ v̂_U_in ]= U_in ( Tr [ P̂ ] - 2 N ) / 2 is that which scales appropriatelywith N or, put another way, v_U_in would be the average DFT+U potential acting on asubspace eigenvector were there Tr [P̂] copies of that eigenvector, and thus is comparable with v_Hxc. The factor Tr [ P̂ ] separatingthe definitions of v_Hxc and v̂_U_inis consistent with DFT+U correcting the Hartree + xc generatedmany-body subspace SIE, which is assumed to be proportional to N^2≈ ( Tr [ P̂ ] ⟨ n_i ⟩ ) ^2, by only Tr [ P̂ ] one-electron SIE corrector terms on the order of ⟨ n_i ⟩^2.Finally we may write, for the single-site variational linear-response Hubbard U_out in the presence of a non-zeroU_in,that U_out= χ_0^-1- χ^-1=d v_KS-d v_ext/d N=d v_int/d N= d v_Hxc/d N-U_in =f^P̂_Hxc(U_in)-U_in,where f^P̂_Hxc (U_in) ≡ dv_Hxc/d N is the subspace-averaged, subspace-bare but bath-screened Hxc interaction calculated at the fully-relaxed DFT+U_in ground-state.From Eq. <ref>we may readily identifythree unique self-consistency criteria.The first is a very plausible self-consistency criterion,first proposed in Ref. campo2010extendedand later utilized in Refs. PhysRevB.84.115108,PhysRevB.93.085135,which requires that U_out=U_in and thus gives U_in= f^P̂_Hxc(U_in)/2. This U_in, denoted here as U^(1),appears to account for, i.e., cancel away one-halfof the subspace SIE that remains at that DFT+U_in.The second criterion is given by U_out=0,denoted hereby U^(2), which dictates that U_in= f^P̂_Hxc(U_in),implying that U_infully cancelsthe subspace-related SIEcomputed at the same DFT+_in ground-state.The third condition, denoted by U^(3),matches (albeit with a different underling linear-response procedure) the originalself-consistency scheme <cit.> where it is denoted U_scf.Here, the U_out ( 0 ) of the DFT+U electronic structure is calculated by a linear-extrapolation of U_out ( U_in ) for sufficiently large U_in to obtain a good fit, back to U_in = 0 eV. For our present purposes, it is reasonable to assume thata DFT+U corrected electronic structure has been well-obtainedat U^(2), and thus performing thelinear extrapolation for U^(3) aroundU^(2),we find thatU^(3)=U^(2)(1-. d f^P̂_Hxc/ dU_in|_U^(2)).From this, a clear interpretation of U^(3) as screened version of U^(2) emerges, in a generalized sense of screening in which, instead of an externally applied potentialbeing attenuated by relaxation of the electronic structure,it is instead the externally applied interaction correction which is attenuated.A normal dielectric screening operatormeasures the rate of change ofthe potential with respect to an external perturbation, takingthe formϵ̂^-1= dv̂_KS / d v̂_ext = 1̂+ f̂_Hxcχ̂.A generalized screening function here instead measures the rate of reduction insubspace-averaged SIE with respect to U_in, and is given by ϵ_U^-1= - dU_out / dU_in = 1-df^P̂_Hxc/dU_in.Therefore, while werequire a DFT+U correction with parameterU^(2) tocancel the subspace-averaged SIEincluding allself-consistent response effects in the electronic structure,when we have done so we havein fact removed an SIE (with respect to DFT)of magnitude U^(3)= ϵ_U^-1 U^(2),which is typically smaller in magnitude than U^(2).There is a numerically relevant distinction between the external `bare'U_inthat weapply using DFT+U,and the `screened' SIE quantifierU_out that we then measure. The SIE measureU^(3), calculated around the U^(2) ground-state, isof particular interest, e.g., for quantifying the change in SIE in a subspace in response to an external parameter such as atomic position, or if comparing the SIE of an atom in two different charge states.We also expect U^(3) to be suitable as an input Hubbard U parameter for non-self-consistent protocols such as a post-processing DFT+U band-structure correction based on the DFT density,or a DFT + dynamical mean-field theory (DMFT) calculation with no density self-consistency. U^(3) linearly accounts for the resistance to SIE reduction that would be met were density self-consistency in response to U allowed.On the basis of the above analysis, however, we conclude that the criterion U^(2) represents the appropriateself-consistency scheme for the variational linear-response method, wherever the standard self-consistent response of the density occurs upon application of DFT+U.The value of U^(2) may be efficiently obtained, e.g.,by the bisection method.The three self-consistency conditions aresummarizedin Table <ref>.§ NUMERICAL RESULTS §.§ Self-consistent U schemesapplied to dissociating H_2^+ In order toassess the potency ofDFT+U for correcting SIE under varying bonding conditions without the complicating effects of static correlation error,we calculatedself-consistent Hubbard U values and the resulting DFT+U electronic structure along the binding curve of the dissociatingone-electron dimer H_2^+.A further advantage of the one-electron system is thatthe PBE and exact (i.e., for one electron, simply no Hartree or xc)functionals are availableusing the same first-principles code and pseudopotential,which ensures the accurate comparabilityof energies across the parameter space.Stringent numerical conditions were applied, with an extremelyaccurate small-core norm-conserving PBE pseudopotential and a plane-wave equivalent kinetic energycutoff of approximately 2650eV, yielding deviations from 0.5 Ha within w (x) and y (z) on the isolated-atom total-energy and occupied Kohn-Shameigenvalue, respectively, for the exact (PBE in parentheses) functional.The dissociated limit is of particular interest for confirming the relative appositeness of Hubbard U self-consistency schemes that may yieldnumerically similar results since, in that limit, the neutral-atom PBE 1s orbitals used to define each of the twoDFT+U subspacesspatially overlap (i.e., double-count) and spill the total charge minimally, and the DFT+U population analysis for the PBE dimer becomesideal.Furthermore, as we approach the dissociated limit,the assumption that each of the two DFT+U subspaces interacts relatively weakly with its bath (in each case, the other atom) becomes increasingly realistic, represening the best availableperformance of DFT+U using an fixed atomicpopulation analysis (i.e., one that is not dependent on the charge, applied U, or other details of the electronic structure, as Wannier functionsare <cit.>).While conserving the overall charge,the external potential α was varied within the range± 0.05 eV and applied to one atom.DFT+U was applied to both atoms equally, with U_insampledfrom 0 eV up to the valuethat yielded U_out=0 eV.A typical calculation ofU_out is shown in the left inset of Fig. <ref>.For each bond-length,a U_in versus U_outprofile was calculatedaccording to Eq. <ref>, as illustrated in the right inset of Fig <ref> with due care to error accumulation.These profiles were found to remain highly linearacross all bond-lengthsfor this particular system and linear-response methodology,and we note that the slope remainedgreater than -1, signifying d f_Hxc^P̂ / d U_in > 0 and a `resistance' to SIE reduction, for all but the small bond-lengths ≲ 1.3 a_0 strongly affected by subspace double-counting. The linear fit to U_out ( U_in ) was then used to evaluateU^(1), U^(2), U^(3), according to Table <ref>,and their values are depicted bydashed, dotted and dot-dashed lines, respectively,in Fig. <ref>.For each bond-length,we also estimated, by interpolation, the U_int (solid line) required to recover the exact total-energy. The U^(2) and U^(3)schemes, and particularly the former, closely approximatethe U_int requiredto correct the SIE in the total-energyin the dissociated limit, whereas U^(1) clearly representsan underestimation by a factor of2,as indicated by Table <ref>.The numerical situation is reversed within the equilibriumbond-length of approximately 2 a_0, where U^(1) appears to perform better than the alternatives.We emphasise that the latterresult ismisleading, however,since U^(1) performs better at short bond lengths only due to the cancellation of its factor-of-two magnitude reductionwith the double-counting effects of spatially overlapping DFT+U subspaces, as well as the breakdown, in the strong-bonding regime, of the subspace-bath separationunderpinningDFT+U.This highlights a risk when assessing the relative merits of correction formulae of this kind solely on the basisof numerical results gathered under equilibrium conditions,where bonding or overlap effects complicate the analysis.The total-energy baseddissociation curves of H_2^+ were recalculatedusing the bond-length dependentU^(1) (dashed),U^(2) (dotted),and U^(3) (dot-dashed), for comparison with the exact total-energy (solid) in Fig. <ref>.We note that any attempt to extend ourbond length interval beyond 8.5 a_0 resulted in numerical instabilitiesdue to the near-degeneracy ofthe Kohn-Sham σ and σ^*eigenstates, and present here arethe results only of well-converging calculations. As already suggested by Fig. <ref>,U^(1) fails to correctthe SIE in the total-energy at bond-lengths further from equilibrium,whereas U^(2) and U^(3) provide a more universal correction of the total-energy,becomingacceptable in the dissociation limit.The inset of Fig. <ref> illustrates, however,that the PBE+U^(3) scheme, which is numerically equivalent to noHubbardU self-consistency in this particular system,begins to under-perform with respect toPBE+U^(2) in the dissociated limit.The PBE+U^(2) total-energy,meanwhile,seems to converge uponthe exact total-energy asymptotically. Our results confirm that DFT+Uiscapable of precisely correctingthe total-energy SIE of a one-electron systemunder ideal population-analysis conditions but only, it seems, when usingthe simplest self-consistency scheme, U^(2).It is clear, notwithstanding, that DFT+Uis an efficient and effective corrector for the SIEmanifested in the total-energy,as discussed in detail inRefs. PhysRevLett.97.103001,:/content/aip/journal/jcp/133/11/10.1063/1.3489110,:/content/aip/journal/jcp/145/5/10.1063/1.4959882.§.§ Restoration of Koopmans' condition: DFT+U_1+U_2Despite the success of DFT+Uin SIE-correctingthe total-energy using a suitably calculated U value,the fact remains thatit is incapable of simultaneously correcting the highestoccupied Kohn-Sham eigenvalue to minus the ionization potential in compliance with Koopmans' condition, as indicated in Fig. <ref>. This issue has previously beenexplored in Ref. :/content/aip/journal/jcp/145/5/10.1063/1.4959882,and by us in Ref. PhysRevB.94.220104where we constructed a generalized,two-parameter DFT+U functional,comprising separate parameters for the linear (U_1)and quadratic (U_2) terms.In fact, Eq. 9 of Ref. PhysRevB.94.220104 indicates that if a symmetric system of two one-orbital subspaces (a very good approximation for H_2^+, with approximately constant subspace occupancies N) is Koopmans' compliant (so that the Koopmans' U_K = 0), and it is then correctedusing DFT+U for the SIE in the total-energy (it is possible for the interaction strength to be inaccurate, but for the system still to comply with Koopmans'condition), then DFT+U will act to spoil that condition unless U_1 = 2 U_2 ( N - N^2) / ( 1 - 2 N ).More pragmatically, we expect the extra degree of freedomfurnished by U_1 to bebeneficial in cases wherethe quadratic approximation to the subspace-averaged self-interaction does not remain valid all the way down to the ionizedstate, which is particularly relevant for H_2^+ since there that state corresponds to the low-density limit.For compliance with Koopmans' condition, it seemsunavoidable that data must be collected from both the approximate neutral and ionized(the total energy of which may be sufficient) systems, in order to calculate U_1 and U_2.We carried outdensity non-self-consistentDFT+U_1+U_2 calculations on the basis of the PBEtotal energy and occupied Kohn-Sham eigenvalue,following theformulae given in Ref. PhysRevB.94.220104.To put the method on a first-principles footing,we used the self-consistent value U^(2) to calculate U_1 and U_2,resulting in a densitycorrection to thetotal-energysumming to Δ E = U^(2) (N - N^2).The corresponding modificationto the subspace potentials is given byΔ v_U =U^(2) (N -N^2) - U_K/2 where,for this system, U_K/2 = E_ion-ion - E_PBE +ε_PBE.Noting that the correction to the Kohn-Sham eigenvalue ε_PBE tends toΔ v_U in the dissociated limit wherechanges to the occupied Kohn-Shamorbital are negligible, there we find that ε_PBE+U_1+U_2≡ε_PBE + Δ v_U = ε_PBE + Δ E - ( E_ion-ion - E_PBE + ε_PBE) = - (E_ion-ion -E_PBE+U_1+U_2 ) ≡ - IP, if IP is the ionization potential, i.e., thatKoopmans' compliance is restored for aSIE correction strength of U^(2).Fig. <ref> illustratesthe result of this simple technique,whichsimultaneously reconciles thetotal energy E and eigenvalue ε derived dissociation curves with the PBE+U^(2) dissociation curve of Fig. <ref>, albeit impreciselyas this is a non-self-consistent post-processing step.Our results highlightsthe potential of the DFT+U_1+U_2 approach and its immediate compatibility with self-consistently calculated Hubbard U parameters.To our knowledge, the SIE of approximate DFThas not previously beensimultaneously addressed for the total-energy and the occupied Kohn-Sham eigenvalue using a first-principles correction method of DFT+U type, even for aone-electron system such as this. In the manner in which we have performed ithere,non-self-consistent DFT+U_1+U_2requires only one total-energy calculation,at the ionized state, on top of the usual apparatus of a linear-response DFT+U calculation, in order to simultaneously, albeit approximately,correct the total energy and the highest occupied Kohn-Sham eigenvalue for SIE.Interesting avenues for the development of this methodinclude its extension to multi-electron, heterogeneous, and non-trivially spin-polarized systems, as well asto perform self-consistency over the density and to lift the fixed-occupancy approximation,as outlined in Ref. PhysRevB.94.220104.In principle, a further refinement of the method might entail the self-consistent linear-response calculation of U_1 and U_2 separately for the neutral and ionized states. §.§ Binding Curve ParametersIn order to further quantify the results of the various Hubbard U self-consistency schemes tested,we determined the equilibriumbond-length R_e,dissociation energy E_D,harmonic frequency ω_eand anharmonicity ω_eχ_e,corresponding toeach, as shown inTable. <ref>,by fitting a polynomial about the energyminima. As compared with theexperimental data ofRef. HERZBERG1972425,the exact calculations perform well in determiningthe bond-length and harmonicityin particular, with errors that reflect theinaccuracies due to our fitting scheme, finitecomputational basis set size, core pseudization, and absent physical effects, as well as experimental factors.The PBE functional overestimates theequilibrium bond-lengthand dissociation energy,while underestimatingthe harmonic frequency and anharmonicity.The various DFT+U schemes tested generally preserve the PBE dissociation energy(N.B.,calculated with respect to the exact-functional one-atom total-energy, rather than to the PBE local maximum at 6-7 a_0) but they reduce the bond-lengthand increase the frequencyand anharmonictiy.The U^(2),U^(3) and U_1+U_2schemes over-correct the latter three and remain as poorly predictive of the experimental data as the uncorrected PBE is.We attribute this toimperfect DFT+U population analysis at shorter bond-lengths, featuring bothdouble-counting across the two subspaces and spillage,as well as the breakdown of the subspace-bath separation.The double-counting, in particular, isnot properly compensated forby self-consistently calculatedU parameters,since the formula for U_out features no quantification of this effect.The U^(1) scheme performs well here,as reflected also in Fig. <ref>,approximately recovering the exactbond-length, harmonic frequency and anharmonicity.We emphasise that this is due entirely to the U^(1) parameter simply being smallerbe definition, so that the over-correction due to double-counting is less extreme.It thereforecoincides with the exact regime serendipitously,and not bydesign.§ THE COMPARABILITY OF DFT+U TOTAL ENERGIES AND THE CONNECTIONS BETWEEN THE SCF AND VARIATIONAL LINEAR-RESPONSEIn this section, we explore the relevance of first-principles calculated Hubbard U values, particularly atself-consistency, to the comparabilityof the DFT+U total energies across different calculations in whichthe Hubbard U is separately calculated due to the variation of external parameterssuch as stoichiometry and crystallographic geometry.We alsoclarify the technical similarities and differences betweenSCF linear-response <cit.>and its derived variational linear-response method.The open question of the rigorous comparability ofDFT+U total-energies that are generated by calculations with different U values, which ordinarily represent external parameters with the same status as ionic positions,is of considerable contemporary relevance.This is demonstrated by recent progress in calculating thermodynamic quantities <cit.>, in high-throughput materials informatics <cit.>, catalysis <cit.>, and in the study of ion-migration in battery materials <cit.>by means of DFT+Uand its related methods. The variational linear-response definition U = d v_int / d N, so-called as it is based on the variational response of theground-state density,demands that the subspace-averagednon-interacting responseχ_0 = d N / d v_KS is calculated using the same set of ground-state densities, parameterized by the external perturbation strength α, as that used for calculatingthe interacting response χ = d N / d α.We may therefore write, taking the limit ofsmall perturbations, that U_out= χ_0^-1 - χ^-1 = . d v_KS[ ρ̂] /d N [ ρ̂] |_ρ̂_0- . d v_ext/d N [ ρ̂] |_ρ̂_0 ,where ρ̂_0 is the unperturbedKohn-Sham density-matrix.From this, it is clear that U_out[ ρ̂_0 ]is a ground-state density-functional, albeit not one of an explicit algebraic form.This definition is readily adaptable to orbital-free DFT, in which there is no Kohn-Sham Hamiltonian to diagonalize but only a density (rather than a Kohn-Sham density-matrix) to optimize, andwhere v̂_KS is replaced by thetotal potential. If we perform a variational linear-response calculationfor a given U_in, notwithstanding, then the resulting U_out maythought of as a ground-state density functional parameterisedby U_in.If we can then uniquely determine U_in by applying a self-consistency criterion such asU_out = 0 eV,we will thereby uniquely determine the self-consistent ground-stateDFT+U density-matrix (up tounitary transformations) ρ̂_0 ( U_in ), as well as its derived properties such as the total-energy,in termsof the remainingparameters, e.g.,ionic positions.The comparability of total-energies betweenvarious crystallographicor molecular structures withdifferingself-consistent Hubbard U values, and the validity of thermodynamic calculations based onDFT+U, directly follows.In this way, given the underlying explicit algebraicxc functional such as PBE, together with the choice of a set ofsubspaces to target for SIE correction,DFT+U is elevated to the status of aself-containedorbital-dependentdensity-functional in its own right, incorporating the Hubbard U as a non-algebraicbut readily computable auxiliary ground-state variable.The SCF <cit.>and variational linear-responsemethods are identical in terms of their applied external perturbation,the use of the Dyson equation for multi-site models, and issues of DFT+Upopulation analysis choice and convergence. While they areequally convenient for usewith SCF-type DFT solvers,the variational approach is likely to be more convenientforuse with direct-minimization solvers.They are also perfectly identical insofar as the calculation ofχ is concerned.They differ only in the definition and set of densities used to calculate the subspace-averaged non-interacting response χ_0.A calculation of the non-interacting response χ_0 following the first step of the SCF cycle, as instate-of-the-art linear-response calculations,relies, by construction,upon the density (or Kohn-Sham orbitals,or density-matrix, as the case may be) not being converged to the variationalground-state for each given external parameter α. A non-optimized density of this type will typically not correspondto theground-statefor any value of α, although its subspace-total N will be.As a result, thefinite-differencedata points for α 0 that build the SCF linear-response χ_0are individually not properties of the ground-states for their corresponding external potentials, noting that linear-response does not imply the sufficiency of first-orderperturbation theory or first-order screening.Put another way,the SCFχ_0 (and hence the derived U) is not a ground-stateproperty of the unperturbed ground-state density, but instead an excited-state property(in the simple sense of non-ground-state, rather than of aresonance) dependent on the eigensystemof the non-interacting Kohn-Sham Hamiltonian. The comparison of the resulting DFT+U total-energiesthus remainswell defined in terms of ground-state densities,since the Kohn-Sham eigenspectra are themselves ground-state properties.However, by virtue of the SCF Hubbard U not itself being a purely ground-state density-functional property,in general,the total energies are also not necessarily so. The precise effects of the departure from the ground-stateenergy surface in the calculation of SCF χ_0have not beenquantified to date, to the best of our knowledge.Therefore, while the self-consistency scheme dubbed U^(2) seems to be optimal for use with the variational linear-response scheme, as we have shown,this result does not automatically extend to its SCF progenitor.Nonetheless, we may expect that the two inequivalent approachesfor χ_0 should yield similar numericalresults in practice.§ CONCLUSIONS We have developed a simple, variational adaptation of the widely-used linear-response method for directly calculatingthe Hubbard U of DFT+U, in which the U incorporates only quantities calculated from ground-state densities.This method puts DFT+U on afirst-principles footing within the context of direct-minimization DFT solvers, even the linear-scaling solversof the type now routinely used to simulate systems which are simultaneously spatially and electronicallycomplex <cit.> Our formalism simplifies the analysis of parameter self-consistency schemes considerably and, at least forthis specific method, there emerges a clear best choice of self-consistency criterion, U^(2), whichhas been explored relatively little in the literature to date. We recommendthe use of a more complicatedcriterion, the previously proposed U^(3),particularly for density-non-self-consistent methods such aspost-processing DFT+DMFT.In stringent calculations of the dissociated limit of the purely SIE-afflicted system PBE H_2^+,where DFT+U operates under ideal conditions,we are able to directly observe that the method corrects the SIE in the total-energyvery precisely, as foreseen in Ref. PhysRevLett.97.103001.It does so entirely from first-principleswhen the U^(2) scheme is used.Our analysis also shows that the comparison ofthermodynamically relevant DFT+U quantities such as the total-energybetween dissimilar systemsdemanding differentfirst-principles U parameters is, at least, well defined.This comparison evidently becomes one between purely ground-state properties, moreover, in the case where the variational linear-response method is applied together with parameter self-consistency, but there may well be other circumstances in which this also holds true.TheDFT+U_1+U_2 method <cit.>,put here on a first-principles basis,extends the DFT+U^(2) full SIE correction of theH_2^+ total-energy to the highest occupied eigenvalue,approximately enforcing Koopman's condition.Finally, we note that to properly account for SIE in the total-energy across the bond-length range,one would need to fully take into accountthe effects of subspace charge spillage, overlap and double-counting,possibly through the use ofWannier functions <cit.>generated self-consistently with the DFT+U electronic structure <cit.>.At least as important for correcting SIE in thebonding regime, perhaps, is the necessityto overcome the breakdown of thesingle-site approximation.For this, the account of inter-subspace SIEoffered by the multi-site methodDFT+U+V <cit.> is a promising avenue forinvestigation.This work was enabled by the Royal Irish Academy – Royal SocietyInternational Exchange Cost Share Programme (IE131505). 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B volume 49, pages 6736–6740 (year 1994)NoStop [Note1()]Note1 note Calculations were performed using the DFT+U functionality <cit.> available in the ONETEP linear-scaling DFT package <cit.> with a hard (0.65 a_0 cutoff) norm-conserving pseudopotential <cit.>, 10 a_0 Wannier function cutoff radii, and open boundary conditions <cit.>. DFT+U was applied simultaneously to each an atom, using a separate 1s orbital subspace centred on each, defined using the occupied Kohn-Sham state of the pseudopotential for neutral hydrogen. The correct symmetry of H_2^+ was maintained for all values of U given a symmetric initial guess, i.e., we observed no tendency for the charge to localize on a single ion.Stop [Kulik and Marzari(2011a)]doi:10.1063/1.3660353 author author H. J. Kulik and author N. Marzari, title title Accurate potential energy surfaces with a DFT+U(R) approach, 10.1063/1.3660353 journal journal The Journal of Chemical Physics volume 135, pages 194105 (year 2011a)NoStop [Lu and Liu(2014)]doi:10.1063/1.4865831 author author D. Lu and author P. Liu,title title Rationalization of the hubbard u parameter in ceox from first principles: Unveiling the role of local structure in screening, 10.1063/1.4865831 journal journal The Journal of Chemical Physicsvolume 140, pages 084101 (year 2014), http://arxiv.org/abs/http://dx.doi.org/10.1063/1.4865831 http://dx.doi.org/10.1063/1.4865831 NoStop [Aykol and Wolverton(2014)]PhysRevB.90.115105 author author M. Aykol and author C. Wolverton, title title Local environment dependent GGA+U method for accurate thermochemistry of transition metal compounds, 10.1103/PhysRevB.90.115105 journal journal Phys. Rev. B volume 90, pages 115105 (year 2014)NoStop [Dabo et al.(2010)Dabo, Ferretti, Poilvert, Li, Marzari, and Cococcioni]PhysRevB.82.115121 author author I. Dabo, author A. Ferretti, author N. Poilvert, author Y. Li, author N. Marzari,and author M. Cococcioni, title title Koopmans' condition for density-functional theory, 10.1103/PhysRevB.82.115121 journal journal Phys. Rev. B volume 82,pages 115121 (year 2010)NoStop [Borghi et al.(2014)Borghi, Ferretti, Nguyen, Dabo, andMarzari]PhysRevB.90.075135 author author G. Borghi, author A. Ferretti, author N. L. Nguyen, author I. Dabo,and author N. Marzari, title title Koopmans'-compliant functionals and their performance against reference molecular data, 10.1103/PhysRevB.90.075135 journal journal Phys. Rev. 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Cococcioni, title title Extended DFT + U + V method with on-site and inter-site electronic interactions,http://stacks.iop.org/0953-8984/22/i=5/a=055602 journal journal Journal of Physics: Condensed Mattervolume 22, pages 055602 (year 2010)NoStop [Note3()]Note3 note In this work, since we find it necessary to use only single-site DFT+U with no +V term, we treat the two 1s atomic subspaces as decoupled, each comprising the majority of the screening bath for the other. Hence we use a pair of scalar Dyson equations (identical by symmetry, i.e., only one is treated numerically) to calculate the Hubbard U, rather than selecting the diagonal of the 2 × 2 site-indexed Hubbard U.Stop [Skylaris et al.(2005)Skylaris, Haynes, Mostofi, andPayne]:/content/aip/journal/jcp/122/8/10.1063/1.1839852 author author C.-K. Skylaris, author P. D. Haynes, author A. A. Mostofi,and author M. C. 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"authors": [
"Glenn Moynihan",
"Gilberto Teobaldi",
"David D. O'Regan"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170426124113",
"title": "A self-consistent ground-state formulation of the first-principles Hubbard U parameter validated on one-electron self-interaction error"
} |
Bipartite entanglement infermion systems N. Gigena, R. Rossignoli – Dedicated to John Butcher, on the occasion of his 84-th birthday – ========================================================================§ INTRODUCTIONBy the end of 20th century it was discovered that the Universe is expanding at an accelerating rate<cit.>.The current cosmic acceleration can be explained by the existence of a positive cosmologicalconstant in the Einstein field equations <cit.>. However, the cosmological constant presents a hugediscrepancy between its observed and its theoretical value <cit.>.Modifications of gravitytheory <cit.> and exotic forms of fields <cit.>are some alternatives to the cosmological constant toexplain the cosmic acceleration. However, the information about the cosmological parametersobtained from these alternative scenarios largely depends on the model under consideration. Cosmokinetics (or cosmography) <cit.>is the least model-dependent method to get information aboutUniverse expansion history. The basic assumption of cosmokinetics is thecosmological principle. No assumptions about sources or gravity theory are made. Therefore, itis expected that the results obtained from this kinematic approach remainvalid regardless of the underlying cosmology. This feature may be an efficient weaponto probe the viability of several cosmological models proposed to describe the currentphase of accelerated expansion of the Universe. For instance, since j(z)=1 for the ΛCDM model, we can rule out this model if we find that j≠1. Cosmography methodology consists of expanding cosmological observables such as the Hubble parameter and theluminosity distance in power series. However, to obtain some information aboutthe kinematic state of the Universe, these series should be stopped. In such anapproximate process issues arise concerning the series convergence and theseries truncation order. The series convergence problem can be circumvented bychoosing a suitable expansion variable such as the so-called y-redshift,y=z/(1+z) <cit.>, or the scale factor,a=(1+z)^-1=1-y <cit.>, instead of the z redshift. The seriestruncation problem can be alleviated by performing the so-called F-test<cit.> to find which truncation orderprovides the more statistically significant fit to a given data set.In this paper we follow the procedure adopted in <cit.> andperform the series expansion of the luminosity distance, d_L(a), and of the Hubbleparameter H(a) around an arbitrary scale factor ã. The F- test indicates that the most statistically significant truncation order for both seriesis the third. Since the third order approximation of d_L has one parameter less thanthe third order truncation of H, we also consider, for sake of completeness, the fourth order d_L approximation. We use some of the mostrecent Type Ia Supernovae (SNe Ia) and H(z) data sets toconstrain the Hubble (H), deceleration (q), jerk (j) and snap (s)kinematic parameters at z0.For the third order expansions, the results obtained from both SNe Ia and H(z) data are incompatible with the ΛCDM model at 2σ confidence level, but also incompatible with each other. When the fourth order expansion of d_L is taken into account, the results obtained from SNe Ia data are compatible with the ΛCDM modelat 2σ confidence level, but still remains incompatible with results obtained from H(z) data.The constraints on j and s are conflicting and indicate adiscrepancy between SNe Ia and H(z) measurements. § COSMOKINETICS Cosmokinetics relies on the assumption that at large scales the Universe ishomogeneous and isotropic. Mathematically, this assumption is translated by theRobertson-Walker (RW) metric ds^2=-c^2dt^2+a^2(t)[dr^2/1-kr^2+r^2(dθ^2+sin^2θ dϕ^2)], where a(t) is the scale factor of the Universe and k is theUniverse spatial curvature. In agreement with recent results of the CMB powerspectrum <cit.>, we restrict our attention to a spatially flat Universe(k = 0) in this paper. For a flat RW line element, the luminosity distancetakes the form, d_L=c 1/a ∫_t^t_0dt^'/a(t^')=c (1+z) ∫_0^zdz^'/H(z^'), where the subscript 0 denotes the value of a variable at the present epoch, H≡ a^-1(da/dt) is the Hubble parameter, which provides the expansionrate of the Universe, and we have used the convention a_0=1. Cosmokinetics works at time domains where a complete knowledge of the a(t)function is not necessary. The standard approach consists in performing a Taylorexpansion of a cosmological observable in terms of the redshift, keeping theexpansion center fixed at z=0 <cit.>. By focusingon the Hubble parameter and the luminosity distance, such a procedure leads to H(z) = H_0[1+(1+q_0)z+1/2(j_0-q_0^2)z^2+1/6(3q_0^3+3q_0^2-3j_0+4q_0j_0-s_0)z^3++ 1/24(l_0+8s_0+7q_0s_0+12j_0+32q_0j_0+25q_0^2j_0-4j_0^2-12q_0^2-24q_0^3-- 15q_0^4)z^4+⋯] and d_L(z) = c/H_0{z+1/2(1-q_0) z^2 +1/6(3q_0^2+q_0-1-j_0)z^3+ 1/24(2-2q_0-15q_0^2-- 15q_0^3+5j_0+10q_0j_0+s_0)z^4- 1/120[6(1-q_0)-3q_0^2(27+55q_0-35q_0^2)++ 5q_0j_0(21q_0+22)+15q_0s_0-10j_0^2+27j_0+11s_0+l_0]z^5+⋯},whereq≡-ä/H^2a,j≡⃛a/H^3a, s≡⃜a/H^4al≡a^(5)/H^5aare, respectively, the deceleration, the jerk, the snap and the lerk parameters, and thedot denotes time derivatives. These parameters provide information about thekinematic state of the Universe. Physically, q specifies if the Universe isexpanding at an accelerated (q<0), decelerated(q>0) or constant (q=0)rate; j shows whether the Universe's acceleration is increasing (j>0), decreasing(j<0) or constant (j=0); s tells us if d^3a/dt^3 is increasing (s>0),decreasing (s<0) or constant (s=0) and l tell us if d^4a/dt^4 is increasing (l>0),decreasing (l<0) or constant (l=0).Thus, the kinematic approach allow usto investigate the cosmic acceleration without assuming modifications of the gravitytheory or dark energy models. The truncation of the expansions(<ref>) and (<ref>) at the first two or three termsshould be good approximations if z does not lie outside the convergenceradius of these series, z<1 <cit.>. However,currently we have measurements of H and d_L at z>1. Applying low orderapproximations to cover such a redshift range may result in artificiallystrong constraints on the free parameters, while taking higher order terms, and consequentlyincreasing the number of free parameters, can make the analysis more laborious than necessary. Therefore, we need to find a way tocover the higher redshift range using the lowest number of parameters possible.This problem can be handled if we work with the y-redshift, y=z/(1+z) <cit.> which maps the redshift domainz∈[0,∞[ into y∈[0,1[or, equivalently, if we work with the scalefactor a (a=1-y) <cit.> as expansion variables. Here we choose the scale factor as the expansion variable. Note that an expansion around z=0 is translated to anexpansion around a=1 when the scale factor is used as the expansion variable. Thestandard approach consists in taking the expansion center at z=0 (a=1). However,nothing prevents us from changing the expansion center to an arbitrary redshift orscale factor. By assuming that the Hubble parameter and the luminosity distanceare analytical functions in the range ]ã-ϵ,ã+ϵ[,where ã is expansion center, we get H(a) = H̃{1+(1+q̃)(1-a/ã)+1/2(2+2q̃-q̃^2+j̃)(1-a/ã)^2-1/6(s̃-3j̃+3q̃^2-3q̃^3-- 6q̃+4q̃j̃-6)(1-a/ã)^3+ 1/24[l̃+7q̃s̃+5q̃^2(5j̃-3q̃^2)-4j̃^2- 4(s̃-3j̃+3q̃^2-- 3q̃^3-6q̃+4q̃j̃-6)](1-a/ã)^4+⋯} and H̃d_L(a)/c = 1/a{ãH̃d̃_L/c+1/ã(1-a/ã)[1 + 1/2(1-q̃) (1-a/ã)+ 1/6(2-2q̃+3q̃^2-j̃)(1- a/ã)^2++ 1/24(s̃-3j̃+9q̃^2-15q̃ ^3-6q̃+10q̃j̃+6) (1-a/ã)^3-1/120(l̃+15q̃s̃+105q̃^2j̃-- 10j̃^2- 105q̃^4-4s̃+12j̃-36q̃^2+60q̃ ^3+24q̃- 40q̃j̃-24)(1-a/ã)^4]⋯}, where a tilde denotes a function evaluated at ã. The mainadvantage of this procedure is thatwe can estimate the value of thecosmographicparameters at z0 and so, changing the expansion center,discover how these parameters evolve in a completely model-independent way. Notethatd̃_L=d_L(ã) is also a free parameter in our cosmographicanalysis. Since d_L=0 at a=1, we can write d̃_L in terms ofH̃, q̃, j̃ and so on.Thus, by expanding H and d_L around an arbitrary scale factor ã it ispossible to obtain the cosmographic parameters as a function of a independent of theunderlying cosmological model. Also, it is worth mentioning that the lower thevalue of ϵ the better the approximation that describes the real H and d_Lfunctions.§ OBSERVATIONAL CONSTRAINTS§.§ Data In order to constrain the cosmographic parameters we use separately the 580 SNe Iadistance measurements of the Union 2.1 compilation <cit.> and the 30measurements of the Hubble parameter compiled in <cit.>, plus themeasurement of the Hubble constant H_0=73.24±1.74 Km· s^-1· Mpc^-1provided by <cit.>. The SNe Ia data are distributed in the redshift interval0.015≤ z ≤ 1.414 (0.414≤ a≤ 0.985), corresponding to a maximumϵ of ∼ 0.571, while the Hubble parameter data cover the redshift range 0≤ z ≤ 1.965 (0.337≤ a≤ 1), corresponding to a maximumϵ of ∼ 0.663. For SNe Ia data, the statistical analysis is performed using the distance modulus definition:μ(z|{θ_i}) = 5log_10 d_L(z|{θ_i})+25= 5log_10(H̃d_L/c)-5log_10(h̃/3)+40,where {θ_i}={H̃, q̃, j̃, …}is the set of parameters to be fitted andh̃=H̃/(100Km· s^-1· Mpc^-1).The best fit parameters are obtained by minimizing the quantityχ^2_ SN({θ_i})=∑_i=1^580[μ(z|{θ_i})- μ^ obs(z_i)]^2/σ_μ,i^2,where μ^ obs(z_i) is the observed value of the distance moduli atredshift z_i and σ_μ,i^2 is the error of μ^ obs(z_i).For the Hubble parameter data, the best fit parameters are obtained by minimizingthe quantityχ^2_H({θ_i})=∑_i=1^31[H(z|{θ_i})- H^ obs(z_i)]^2/σ_H,i^2,where H^ obs(z_i) is the observed value of the Hubble parameter atz_i and σ_H,i^2 is the error associated with the H^ obs(z_i)measurement. §.§ F-test In order to decide the order in which the series should be stopped,we perform the so-called F-test, defined asF_kl=χ_k^2-χ_l^2/n_l-n_kχ_l^2/N-n_l,where χ_i^2 and n_i are, respectively, the minimum chi-squaredfunction and the number of parameters of the ith model and N isthe number of data points. This test compares two models, identifying theone that provides the best fit to the data, with the null hypothesis implying thecorrectness of the first model. In the following we compare successivetruncations of the Taylor series (<ref>) and (<ref>)to decide the number of parameters that we need to take into account in our analysis.Table <ref> displays the constraints on the cosmographicparameters at the present time for successive approximations of H and d_L. It iseasy to see that for both expansions the last relevant term is the third,F_34≈0.2 for H and F_34=0 for d_L. However, the third orderapproximation of H contains four parameters while the third order approximationof d_L contains three parameters. Therefore, for sake of completeness,we will also work with one term beyond than necessary inthe d_L series approximation. In what follows we take the third order approximation of H and the third and fourth order approximation of d_L and compare the constraints onH, q, j,and s obtained from H(z) and SNe Ia data. §.§ Results The evolution of the cosmographic parameters H, q, jands is obtainedfollowing the algorithm: 1. fix the expansion center ã_i=(1+z̃_i)^-1 in eqs. (<ref>) and (<ref>);2. perform the statistical analysis with H and SNe Ia data to constrain H, q, jands at z̃_i;3. set z̃_i+1=z̃_i+Δz̃ and repeat the previous step to constrain H, q, jands at z̃_i+1.Here we take a step of Δz̃=0.1 and cover the interval 0≤z̃_i≤1.4for both data sets used in our analysis.Table <ref> contain the results obtained from H data. Tables <ref> and <ref> containthe results obtained from SNe Ia data for the third and fourth order approximations, respectively. The errors correspond to a 2σ(Δχ^2=4) confidence interval for each parameter.In all cases the reduced chi-square values (χ^2_ν=χ^2_min/ NDoF)remain unchanged when the expansion center is shifted.In order to make the comparison between these results clearer, a graphical representation of the results contained in Tables <ref>and <ref> is given in Figs. <ref> and <ref> and a graphical representation of the results contained in Tables <ref>and <ref> is given in Figs. <ref> and <ref>. Figures <ref> and <ref> shows, respectively, the constraints onH (left panel of Figure <ref>), q (right panel of Figure <ref>)and j (Figure <ref>) for the third order approximations of H and d_L at 15 points equallyspaced in the redshift range 0≤ z≤1.4.Figure <ref> shows the constraints on H (left panel) and q (right panel)and Figure <ref> shows the constraints on j (top panel) and s (bottom panel)when a fourth order approximation of d_L isconsidered.The blue boxes standfor 2σ confidence intervals obtained from H data whilethe orange boxes stand for 2σ confidence intervals obtained fromSNe Ia data. The gray region in the q plots, the dashed line in the j plots and the gray region in the s plotscorrespond, respectively, to the ΛCDM bounds:q(a) = -1+3H_0^2Ω_m,0/2H^2a^3>-1, j(a) = 1 and s(a) = 1-9H_0^2Ω_m,0/2H^2a^3<1,where Ω_m,0 is matter density parameter at the present time. When we stop the d_L expansion in the third term (Figs. <ref> and <ref>), a general feature is that SNe Ia constraints are tighter than the constraints obtained from H(z) measurements in all redshift range covered. Particularly, the constraints on j obtained from SNe Ia data are significantly stronger than the constraints obtained from H(z) data. The constraints on H and q obtained from SNe Ia and from H(z) data are compatible with the ΛCDM model and compatible with each other. For SNe Ia, values of q>0 are allowed for z≥0.5 and values of q<0 are allowed for z≤0.8,indicating that the transition between the decelerated to accelerated phases should stay in the range 0.5<z_t<0.8. In turn, for H data, positive values of q are allowed for z≥0.3 showing that in this case thetransition redshift, z_t, is greater than 0.3. The constraints on j obtained from both, SNe Ia and H(z) data are incompatible with the ΛCDM model. For SNe Ia data, j begin to depart from the ΛCDM model at z>0.3. For H(z) data j is above the ΛCDM value, j=1, at z=0.3 and z=0.4 and below this value for z≥ 1.2. The constraints on j reveals yet that the results obtained from H(z) and SNe Ia data are incompatible with each other. For the fourth order expansion of d_L, the constraints from SNe Ia data, as expected, becomes weaker (Figs. <ref> and <ref>). For z<1 the constraints on H obtained from SNe Ia data are tighter than the constraints obtained from H measurements, reversing the roles for z≥ 1. A similar behavior is observed for q, with SNe Ia providing tighter constraints for z≤0.5. For j the constraints obtained from H(z) data are tighter than the constraints provided by the SNe Ia data for z≥1, while the constraints on s obtained from H data are tighter than those obtained from SNe Ia data for z≥0.8. The constraints on j obtained from H data begin to depart from those from SN e Ia data for z>0.8, going to negative values. For the snap, the difference between the results obtained from SNe Ia and H(z) data begins at z>0.5. As we can see, for this case, the results obtained from SNe Ia data are in agreement with the ΛCDMbounds, but still remains incompatible with the results obtained from H data. Therefore, the inclusion of the fourth order term in the expansion of d_L does not alleviate the tension between the data sets observed early. These results indicate a discrepancy between the H and SNe Ia data sets. Such a discrepancy cannot be seen when we restrict our analysis to the neighborhood ofz=0. At z=0, the constraints on the parameters j and s are completely without statistical significance.Therefore, the standard cosmographicapproach, which consists in expanding the Taylor series of H and d_L around z=0, does not seem a useful toolfor testing models designed to explain the cosmic acceleration. This result is in agreementwith the findings of <cit.>. However, since their resultsremain valid regardless of the underlying cosmology, performing the series expansion around an arbitraryã≠1 cosmography can still be an efficient way to rule out cosmological models. For instance,a single value of j≠1 for some z≠0 should be considered as evidence against the ΛCDM model.It is important to note that, when we consider the fourth order expansion of d_L, at z=0, both, SNe Ia and H(z) results do not exclude a decelerated Universe, q_0>0.However, it is an observational fact that, at the present time, the Universe is expanding at an accelerated rate<cit.>, i.e., q_0<0. So, how can we explain such a result?For SNe Ia data, this result can be explained by the fact that we are working with more termsin the d_L expansion than necessary. When the expansion of d_L is truncated at the most statisticallysignificant term, we have q_0<0 at 2σ (see Table <ref>). Since, for H data,we are already using the most relevant approximation, we suspect that this result may be due the low numberof H measurements or to the lack of precision of these measurements, or both.Also note that, for the case of a fourth order expansion of d_L, values ofq<0 are allowed in the entire redshift interval considered, i. e., both SNe Ia and H data setsare compatible with an early time accelerated Universe. In this case, for SNe Ia, values of q>0 are allowed for z≥0.5,indicating that the transition between the decelerated to accelerated phases should occur for redshifts greaterthan 0.5.Also, we observe that from z≥0.6 onwards the constraints on the snap obtained from SNe Ia data begin tobecome incompatible with the constraints coming from H data. This confirms that we cannot combine the two data setsto reconstruct the time-dependence of the cosmographic parameters.Finally, it should be mentioned that, even working with more parameters than necessary(which can be seem as a conservative analysis), the constraints obtained from SNe Ia data barely touch the ΛCDMdiagnostic line j=1. That is, although compatible with the results, the ΛCDM is not the model most consistent with the data.§.§ Transition redshift Although our results allow us to estimate the transition redshift z_t by mere inspection of right panels of Figures <ref> and <ref>, we want make it more precise. In ref. <cit.> it was noticed that the function f(z)≡ H(z)/(1+z) has an absolute minima at z_t. Then, building f(z) from H(z) data and fitting it with a piecewise linear function composed of two intervals (one for acceleration and one for deceleration), the authors were able to obtain a model-independent determination of z_t. By following this approach, we use the estimates of H contained in the Tables <ref>, <ref> and <ref> to estimate z_t. For the sake of comparison, we also fit the open ΛCDM model H^2=H_0^2[Ω_m,0(1+z)^3+Ω_k,0(1+z)^2+Ω_Λ,0],for which z_t=[2Ω_Λ,0/Ω_m,0]^1/3-1. Since the oΛCDM model has two parameters less than the piecewise linear function, we use the corrected Akaike Information Criterion (AIC_ C) <cit.>, and the Bayesian Information Criterion (BIC) <cit.> to provide a fair comparison of the fits. These informations criteria are defined, respectively, as: AIC_C≡-2lnℒ_max +2kN/N-k-1andBIC≡-2lnℒ_max +kln N, where k is the number of parameters of a given model and N the Number of data point. Table <ref> contain the constraints on z_t at 2σ confidence level. The sets 1, 2, and 3 refers to estimates of f(z) obtained from Tables <ref>, <ref> and <ref>, respectively. Our results are compatible with the findings of <cit.> that constrain the transition redshift at 1σ confidence level to 0.3≤ z_t≤0.5 for a piecewise linear function fit and 0.58≤ z_t≤74 for the oΛCDM model. AIC_ C and BIC estimators reveal that H(z) data (set 1) provides strong evidence in favor of the oΛCDM model (Δ AIC_C, Δ BIC>5) while SNe Ia data (sets 2 and 3) do not favor any of the models considered[in fact, the set 3 provides Δ AIC_C>2.5, which is a significant evidence in favor of oΛCDM model, but Δ BIC<2.5 which is a weak evidence].Now, instead of use f(z), we can constrain z_t with our estimates of q by building the function g(z)≡ f' /f= q(z)/(1+z). Since g(z_t)=0, it is natural try to adjust g by a second order expansion, i. e., g(z)=g'(z_t)(z-zt)+1/2g”(z_t)(z-z_t)^2. Since our estimates of g(z) are cosmology-independent, we should presume that the estimate of z_t achieved in this way it is also cosmology-independent. Table <ref> contain the constraints on z_t at 2σ confidence level for this case. AIC_ C and BIC estimators reveal that SNe Ia data (sets 2 and 3) favor the oΛCDM model (Δ AIC_C, Δ BIC>2.5) while H(z) data (set 1) do not favor any of the fitting functions considered. These results confirms what we have already noticed. Note that the weak constraints on z_t from set 3 can be due the unnecessary term include in the d_L approximation.Figure <ref> shows the functions f (top panels) and g=f'/f (bottom panels) obtained from Tables <ref> (left), <ref> (center) and <ref> (right). The solid curve corresponds to the best fit of piecewise linear function (top panels) and g(z) function given by (<ref>). The dashed curve corresponds to the oΛCDM model. The vertical grey region is the constraint on z_t for the piecewise linear function and for the polynomial fit (<ref>). The vertical dashed lines denotes the constraint on z_t for the oΛCDM model. The horizontal solid line in the bottom panels marks the transition from the decelerated to the accelerated phase.By following the vertical stripes we can see that the constraints on z_t for the oΛCDM model from both f and g estimates are entirely compatible with each other. Also, the oΛCDM constraints are entirely compatible with the model-independent constraints on z_t provide for the polynomial fit (<ref>), but are not in good agree with the piecewise bounds on z_t.§ FINAL REMARKS In this paper we have used the cosmographic approach to constrain the Hubble (H), deceleration (q), jerk (j) and snap (s) parameters at z0 from SNe Ia and Hubble parameter data. These constraints are obtained from data by changing the expansion center of the H and d_L Taylor series at small intervals. Such simple implementation allows us to map the time evolution of the cosmographic parameters without assuming a specific gravity model or making assumptions about the sources. This approach can be a useful tool to decide between modified gravity or dark energy models designed to explain the current accelerated expansion of the Universe. For instance, for the main candidate used to explain the present cosmic acceleration, the ΛCDM model, j=1. In the usual approach, where the expansion center is fixed at z=0, evidence against the ΛCDM model is possible only if we find j_0≠1 with some statistical significance. However, many cosmographic analyses performed with multiple data sets have shown that the constraints on j_0 are too weak and do not allow us to decide either for or against ΛCDM (or many other competing models). On the other hand, in the method used in this paper, it is enough to find a single value of j≠1 with some statistical significance to rule out the ΛCDM model.For both, SNe Ia and H(z) data, we show that the value j=1 is rule out at 2σ confidence level when we stop the series of d_L and H at the last relevant term. This result put difficulties on the ΛCDM model. Our results also indicates that SNe Ia and H(z) data are incompatible with each other. When we take a fourth order expansion for d_L expansion, the SNe Ia data accommodate the ΛCDM model. In this case,the constraints on the cosmographic parameters obtained from SNe Ia data are weaker than they should be. Even so, the 2σ bounds do not overlap and the results obtained from SNe Ia data remains incompatible with results obtained from H(z) data.These conflicting results may indicates a tension between SNe Ia and H(z) data, which is masked at z=0. Such a discrepancy indicates that we cannot combine these two data sets to reconstruct the time evolution of the kinematic parameters. In fact, the Taylor series of H and d_L cannot be treated on equal footing since we need to include more terms than necessary in the d_L approximation to make a combination possible. If we look at the results of SNe Ia and H(z) data separately, we will conclude that the ΛCDM model is excluded. However we cannot make such an extreme statement since both, the results of SNe Ia and H(z) data, are not in agreement with each other.We believe that future analyses with a larger and more accurate H data can help us to clarify this problem.CRF acknowledge the financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). The authors acknowledge Thomas Dumelow and Jailson Alcaniz for useful comments.99Riess A. Riess et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astron.J. 116 (1998) 1009Perlmutter S. Perlmutter et al., Measurements of Ω and Λ from 42 High-Redshift Supernovae, Astrophys.J. 517 (1999) 565LCDM T. Padmanabhan, Cosmological Constant - the Weight of the Vacuum, Phys. Rept. 380 (2003) 235CC_problem S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys. 61 (1989) 1fr1 S. 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C. Busti,A. Cruz-Dombriz, P. K. S. Dunsby and D. Sáez-Gómez, Is cosmography a useful tool for testing cosmology?, Phys. Rev D 92 (2015) 123512AICc N. Sugiura, Further analysis of the data by Akaikes information criterion and the finite corrections, Communications in Statistics - Theory and Methods A7 (1978) 13.BIC G. Schwarz, Estimating the dimension of a model, Ann. Statist. 6 (1978) 461. | http://arxiv.org/abs/1704.08089v2 | {
"authors": [
"C. Rodrigues Filho",
"Edésio M. Barboza Jr"
],
"categories": [
"astro-ph.CO",
"gr-qc"
],
"primary_category": "astro-ph.CO",
"published": "20170426130626",
"title": "Constraints on kinematic parameters at $z\\ne0$"
} |
shapes.geometric, arrows startstop = [rectangle, rounded corners, minimum width=3cm, minimum height=1cm,text centered, draw=black, fill=red!30]env=[circle,ball color = green!20, minimum size= 80mm] central=[circle, ball color = red!100, minimum size=8mm] bath=[circle, ball color =blue!75, minimum size=4mm] plain thmTheorem cor[thm]Corollary[ |⟩⟩⟨⟨| [email protected] Research Institute, Allahabad 211019, Indiaand Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400 085, [email protected] Research Institute, Allahabad 211019, Indiaand Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400 085, [email protected] and Quantum Information Group, The Institute of Mathematical Sciences, H.B.N.I., C.I.T. campus, Taramani, Chennai 600113, [email protected] Research Institute, Allahabad 211019, Indiaand Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400 085, India. An exact reduced dynamical map along with its operator sum representation is derived for a central spin interacting with a thermal spin environment. The dynamics of the central spin shows high sustainability of quantum traits like coherence and entanglement in the low temperature regime. However, for sufficiently high temperature and when the number of bath particles approaches the thermodynamic limit, this feature vanishes and the dynamics closely mimicsMarkovian evolution. The properties of the long time averaged state and the trapped information of the initial state for the central qubit are also investigated in detail, confirming that the non-ergodicity of the dynamics can be attributed to the finite temperature and finite size of the bath. It is shown that if a certain stringent resonance condition is satisfied, the long time averaged state retains quantum coherence, which can have far reaching technological implications in engineering quantum devices. An exact time local master equation of the canonical form is derived . With the help of this master equation, the non-equilibrium properties of the central spin system are studied by investigating the detailed balance condition and irreversible entropy production rate. The result reveals that the central qubit thermalizes only in the limit of very high temperature and large number of bath spins. 03.65.Yz, 42.50.Lc, 03.65.Ud, 05.30.RtDynamics and thermodynamics of a central spin immmersed in a spin bath Arun Kumar Pati Received: date / Accepted: date ======================================================================§ INTRODUCTIONIn the microscopic world, physical systems are rarely isolated from environmental influence. Systems relevant for implementation of quantum information theoretic and computationaltasks like ion traps <cit.>,quantum dots <cit.>, NMR qubits <cit.>, polarized photons <cit.>, Josephson junction qubits <cit.> or NV centres <cit.>all interact with their respective environments to some extent. Therefore it is necessary to study the properties of open system dynamics for such quantum systems immersed in baths. For quantum systems exposed to usual Markovian baths, their quantumness gradually fades over time, thus negating any advantage gained through the use of quantum protocols over classical ones. Even in thermodynamics, the presence of quantum coherence <cit.> or entanglement <cit.> enhances the performance of quantum heat machines. Thus, it is imperative to engineer baths in such a way so as to retain nonclassicalfeatures of the system for large durations.Baths can be broadly classified into two different classes, namely Bosonic and Fermionic. Paradigmatic examples for Bosonic baths include the Caldeira-Leggett model <cit.> or the spin Boson model <cit.>. Lindblad type master equations for these models can be found in the literature <cit.>. However, in the Fermionic case, where one models the bath as a collection of a large number of spin-1/2 particles,the situation is generally trickier and one often has to rely on perturbative techniques or time nonlocal master equations <cit.>. Far from from being a theoretical curiosity, the solution of such systems is of paramount importance in physical situations such as magnetic systems <cit.>, quantum spin glasses <cit.> or superconducting systems <cit.>. One specific example of a qubit immersed in aFermionic bath is the Non-Markovian spin star model (schematic diagram in Fig. <ref>) <cit.>, whichis relevant for quantum computing with NV centre <cit.> defects within a diamond lattice. We show that it is possible to preserve coherence and entanglement in this system for quite a long time by choosing bath parameter values appropriately. Even more interestingly we confirm the presence of quantum coherence in the system for the long time averaged state for certain resonance conditions, which is an utter impossibility for the usual Markovian thermal baths. Such strict and fragile resonance conditions underlie our emphasis on the need for ultra-precise engineering of the bath. We also investigate theamount of information trapped <cit.> in the central spin system and draw a connection of the same with the process of equilibration. A time nonlocal integrodifferential master equation was set up for the central spin model using the correlated projection operator technique in Ref. <cit.>. An exact time local master equation for this system was derived in the limit of infinite bath temperature in Ref. <cit.> from the corresponding reduced dynamical map.In this paper, we considerably extend the scope of previous results by deriving the exact reduced dynamics and the exact Lindblad type master equation for arbitrary bath temperature and system bath coupling strength. Our formalism allows us to study the approach towards equilibration in sufficient detail. Thepaper is organised as follows. In Section <ref> we introduce the central spin model and find the exact reduced dynamics for the system and the corresponding Kraus operator representation. We use the solution for the exact reduced dynamics to study the evolution of quantum coherence and entanglement. In Section <ref> we study the long time averaged state and its properties. We analyse the resonance condition for the existence of quantum coherence even in the long time averaged state and the phenomenon of information trapping in the central qubit. In Section <ref>, we begin with the derivation of the exact time-local master equation for this system and use this master equation to investigate the non-equilibrium nature of the dynamics through a thorough study of the deviation from the detailed balance condition as well as the temporal dependence of irreversible entropy production rate. We finally conclude in Section <ref>. § CENTRAL SPIN MODEL AND ITS REDUCED DYNAMICSIn this section we present the model for the qubit coupled centrally to a thermal spin bath. Then we derive the exact dynamical map for the qubit. We also derive the Kraus operators for the reduced dynamics.§.§ The modelWe consider a spin-1/2 particle interacting uniformly with N other mutually non-interacting spin-1/2 particles constituting the bath. The total Hamiltonian for this spin bath model is given by H = H_S+H_B+H_SB= ħ/2ω_0σ_z^0+ħω/2N∑_i=1^Nσ_z^i+ħϵ/2√(N)∑_i=1^N (σ_x^0σ_x^i+σ_y^0σ_y^i),with σ_k^i (k=x,y,z) as the Pauli matrices of the i-th spin of the bath and σ_k^0 (k=x,y,z) as the same for the central spin and ϵ is the system-bath interaction parameter. Here H_S, H_B and H_SB are the system, bath and interaction Hamiltonian respectively. N is the number of bath atoms directly interacting with the central spin. The bath frequency and the system-bath interaction strength are both rescaled as ω/N and ϵ/√(N) respectively. By the use of collective angular momentum operators for the bath spins J_l=∑_i=1^N σ_l^i(where l=x,y,z,+,-), we rewrite the bath and interaction Hamiltonians as[ H_B=ħω/2NJ_z,; H_SB=ħϵ/2√(N)(σ_x^0J_x+σ_y^0J_y). ] We then use the Holstein-Primakoff transformation <cit.> to redefine the collective bath angular momentum operators asJ_+=√(N)b^†(1-b^†b/2N)^1/2 , J_-=√(N)(1-b^†b/2N)^1/2b,where b and b^† are the bosonic annihilation and creation operators with the property [b,b^†]=1. Then the Hamiltonians of Eq. (<ref>) can be rewritten as[H_B =-ħω/2(1-b^†b/N),; H_SB=ħϵ[σ_0^+(1-b^†b/2N)^1/2b+σ_0^-b^†(1-b^†b/2N)^1/2]. ] §.§ Dynamical map of the central spinIn the following, we derive the exact reduced dynamical map of the central spin after performing the Schrödinger evolution for the total system and bath and then tracing over the bath degrees of freedom. It is assumed that the initial system bath joint state is a product state ρ_SB(0)=ρ_S(0)⊗ρ_B(0), which ensures the complete positivity of the reduced dynamics <cit.>. The initial bath state is considered as a thermal state ρ_B(0)=e^-H_B/KT/Z, where K, T and Z are the Boltzman constant, temperature of the bath and the partition function respectively. Consider the evolution of the state |ψ(0)|=⟩|1||⟩x|$⟩, where|1|$⟩ is the system excited state and |x|$⟩ is an arbitrary bath state. After the unitary evolutionU(t)=exp(-iHt/ħ), let the state is|ψ(t)|=⟩γ_1(t)|1||⟩x'|+⟩γ_2(t)|0||⟩x”|$⟩. let us now define two operators Â(t) and B̂(t) corresponding to the bath Hilbert space such that Â(t)|x|=⟩γ_1(t)|x'|$⟩ andB̂(t)|x|=⟩γ_2(t)|x”|$⟩. Then we have |ψ(t)|=⟩Â(t)|1||⟩x'|+⟩B̂(t)|0||⟩x”|$⟩. Now from the Schrödinger equationd/dt|ψ(t)|=⟩-i/ħH|ψ(t)|$⟩, we have[dÂ(t)/dt=-i(ω_0/2-ω(1-b^†b/2N))Â(t)-iϵ(1-b^†b/2N)^1/2bB̂(t),; dB̂(t)/dt=i(ω_0/2+ω(1-b^†b/2N))B̂(t)-iϵ b^†(1-b^†b/2N)^1/2Â(t). ] By substituting Â(t)=Â_1(t) and B̂(t)=b^†B̂_1(t), we have[ dÂ_1(t)/dt=-i(ω_0/2-ω(1-n̂/2N))Â_1(t)-iϵ(1-n̂/2N)^1/2(n̂+1)B̂_1(t),; dB̂_1(t)/dt=i(ω_0/2+ω(1-n̂+1/2N))B̂_1(t)-iϵ(1-n̂/2N)^1/2Â_1(t), ] where n̂=b^†b is the number operator. The operator equations (<ref>) can be straight forwardly solved and the solutions will be functions of n̂ and t. Then Â_1(t)|n|=⟩A_1(n,t)|n|$⟩, wheren̂|n|=⟩n|n|$⟩. Therefore the evolution of the reduced state of the qubit (|1|⟨%s|⟩1|) can now be found by tracing over the bath modes as [ϕ(|1|⟨%s|⟩1|)=Tr_B[|ψ(t|⟨%s|⟩ψ(t)|)]=; 1/Z∑_n=0^N (|A_1(n,t)|^2|1|⟨%s|⟩1|+(n+1)|B_1(n,t)|^2|0|⟨%s|⟩0|)e^-ħω/KT(n/2N-1/2), ] where from the solution of (<ref>), we have |B_1(n,t)|^2=4ϵ^2(1-n/2N)sin^2(η t/2)/η and |A_1(n,t)|^2=1-(n+1)|B_1(n,t)|^2.Similarly we define χ(0)=|0||⟩x|$⟩ andχ(t)=Ĉ(t)|0||⟩x|+⟩D̂(t)|1||⟩x|$⟩. Following the similar procedure and with the substitution Ĉ(t)=Ĉ_1(t), D̂(t)=bD̂_1(t), we find [ dĈ_1(t)/dt=i(ω_0/2+ω(1-n̂/2N))Ĉ_1(t)-iϵn̂(1-n̂-1/2N)^1/2D̂_1(t),; dD̂_1(t)/dt=-i(ω_0/2-ω(1-n̂-1/2N))D̂_1(t)-iϵ(1-n̂-1/2N)^1/2Ĉ_1(t), ]From the solution of (<ref>), we find[ϕ(|0|⟨%s|⟩0|)=Tr_B[|χ(t|⟨%s|⟩χ(t)|)]; =1/Z∑_n=0^N (n|D_1(n,t)|^2|1|⟨%s|⟩1|+|C_1(n,t)|^2|0|⟨%s|⟩0|)e^-ħω/KT(n/2N-1/2), ] with |D_1(n,t)|^2=4ϵ^2(1-(n-1)/2N)sin^2(η' t/2)/η' and |C_1(n,t)|^2=1-n|D_1(n,t)|^2. For the off-diagonal component of the reduced density matrix, we have[ϕ(|1|⟨%s|⟩0|)=Tr_B[|ψ(t|⟨%s|⟩χ(t)|)]; =1/Z∑_n=0^N (A_1(n,t)C_1^*(n,t)|1|⟨%s|⟩0|)e^-ħω/KT(n/2N-1/2), ] with A_1(n,t)C_1^*(n,t)=Δ(t). Therefore the reduced state of the system after the unitary evolution of the joint system-bath state, can be expressed as[ ρ_S(t)=_B[e^-iHt/ħρ_S(0)⊗ρ_B(0)e^iHt/ħ],; =(ρ_11(t)ρ_12(t) ρ_21(t)ρ_22(t) ), ] where the components of the density matrix are given by[ ρ_11(t)=ρ_11(0)(1-α(t))+ρ_22(0)β(t),; ρ_12(t)=ρ_12(0)Δ(t), ] with [ α(t)=1/Z∑_n=0^N 4(n+1)ϵ^2(1-n/2N)sin^2(η t/2)/η^2e^-ħω/KT(n/2N-1/2),; ; β(t)=1/Z∑_n=0^N 4nϵ^2(1-n-1/2N)sin^2(η' t/2)/η'^2e^-ħω/KT(n/2N-1/2),; ;Δ(t)=1/Z∑_n=0^N e^-iω t/2N(cos(η t/2)-i(ω_0-ω/2N)sin(η t/2))×;(cos(η' t/2)+i(ω_0-ω/2N)sin(η' t/2))e^-ħω/KT(n/2N-1/2), ] and [ η = √((ω_0-ω/2N)^2+4ϵ^2(n+1)(1-n/2N)),; ; η' = √((ω_0-ω/2N)^2+4ϵ^2 n(1-n-1/2N)), ] where the partition function is Z=∑_n=0^N e^-ħω/KT(n/2N-1/2). §.§ Operator sum representationA very important aspect of general quantum evolution, represented by completely positive trace preserving operation is the Kraus operator sum representation, given as ρ(t)=∑_i K_i(t)ρ(0)K_i^†(t). The Kraus operators can be constructed <cit.> from the eigenvalues and eigenvectors of the corresponding Choi-Jamiolkowski (CJ) state <cit.>. The CJ state for a dynamical map Φ[ρ] acting on a d dimensional system is given by (𝕀_d ⊗Φ)[Φ_+], with Φ_+=|Φ_+⟩⟨Φ_+| being the maximally entangled state in d× d dimension. For the particular evolution considered here, we find the CJ state to be( 1-α(t)/2 00 Δ(t)/20 α(t)/2 0000 β(t)/2 0 Δ^*(t)/2 00 1-β(t)/2).From the eigensystem of the CJ state given in (<ref>), we derive the Kraus operators as [ K_1(t)=√(β(t))( 0100 ),;; K_2(t)=√(α(t))( 0010 ),;; K_3(t)=√(X_1/1+Y_1^2)( Y_1 e^iθ(t) 001 ),;; K_4(t)=√(X_2/1+Y_2^2)( Y_2 e^iθ(t) 001 ),; ] where θ(t)=arctan[Δ_I(t)/Δ_R(t)] and X_1,2=(1-α(t)+β(t)/2)±1/2√((α(t)-β(t))^2+4|Δ(t)|^2),Y_1,2=√((α(t)-β(t))^2+4|Δ(t)|^2)∓(α(t)-β(t))/2|Δ(t)|.One can check that the Kraus operators satisfy the condition ∑_i K_i^†(t)K_i(t)=𝕀.§.§ Coherence and Entanglement dynamics of the central spinHaving obtained the exact reduced dynamics of the central spin, in the following we study the temporal variation of non-classical properties, viz. quantum coherence and entanglement of the system. It is well known that for usual Markovian systems, such non-classical quantities decay monotonically over time and eventually disappear <cit.>. However, the central spin system is strongly non-Markovian in nature and therefore, a natural and pertinent question is to ask whether it is possible to preserve quantum features for long periods of time for this system. The following subsections are devoted to answering that question for various parameter regimes of the spin bath model. Quantum Coherence: In this article we consider l_1-norm of coherence as a quantifier of quantum coherence. For a qubit system, the l_1-norm of coherence <cit.>C_l_1 is simply given by twice the absolute value of any off-diagonal element, i.e., 2 |ρ_12(t) |. The evolution of coherence is then given byC_l_1(t) = C_l_1(0) |Δ(t)|.This is a straightforward scaling of the initial quantum coherence. One immediate consequence is that we cannot create coherence over and above the coherence present in the system initially, even though this is a strongly non-Markovian system. In subsequent analysis, we can thus take the initial coherence to be unity, i.e. the maximally coherent state without loss of generality. Quantum Entanglement: Operationally, quantum entanglement is the most useful resource in quantum information theory <cit.>. However, it is also a fragile one <cit.> and decays quite quickly for Markovian evolution <cit.>. We suppose a scenario in which the central spin qubit is initially entangled to an ancilla qubit A in addition to the spin bath. There is no subsequent interaction between the ancilla qubit and the central spin. Our goal is to investigate the entanglement dynamics of the joint two-qubit state ρ_SA. From the factorization theorem for quantum entanglement <cit.>, we haveE(ρ_SA (t)) = E(ρ_SA (0))E(χ_SA(t)),where χ_SA(t) is the CJ State in (<ref>) and the entanglement measure E isconcurrence <cit.>. Concurrence of a two qubit system is given as E(ρ_AB) = max{0,λ_1-λ_2-λ_3-λ_4}, where λ_1,…,λ_4 are the square roots of the eigenvalues of ρ_ABρ̃_AB indecreasing order, ρ̃_AB= (σ_y ⊗σ_y)ρ_AB^*(σ_y ⊗σ_y).Herethe complex conjugation ρ_AB^* is taken in the computational basis, and σ_y is the Pauli spin matrix. From now on, we mean concurrence by entanglement throughout the paper. Then the entanglement of the CJ state can be written asE (χ_SA (t))= max ( 0, |Δ (t) | - √(α(t) β(t))). Since the initial entanglement E(ρ_SA (0)) is simply a constant scaling term, we take this to be unity, i.e. consider a maximally entangled initial ρ_SA (0) state without loss of generality and study the subsequent dynamics.We now present the results for time evolution of quantumcoherence and entanglement with the bath temperature T, the strength of system-bath interaction ϵ and number of spins (N) in the spin bath attached to the central spin. If the spin bath is in a very high temperature, we expect the thermal noise to swamp signatures of quantumness, which is broadly confirmed in Fig. <ref>and <ref>. However, small fluctuations in quantum coherence continue to occur testifying to the non-Markovianityof the dynamics. On the contrary, for low bath temperature, as demonstrated in Fig. <ref>, quantum coherence does not decay noticeably and for the timespan we considered, it does not dip below a certain value that is in itself quite high.For intermediate temperatures, coherence broadly decays with increasing decay rate as we increase the bath temperature, but along with small fluctuations due to non-Markovianity. The dynamics of entanglementas shown in Fig. <ref>, is quite similar to that of coherence.At the high temperature limit, the difference with dynamics for quantum coherence lies in the fact that entanglement encounters a sudden death and never revives. This is entirely consistent with the usual observation for many physical systems where quantum coherence turns out to be more robust against noise than entanglement <cit.>. In the opposite regime, for low enough temperatures, entanglement dynamics is very much similar to that of coherence.Another parameter we can tune is the system-bath interaction strength ϵ, which depending upon the species of the central spin as well as the bath spins, may differ. In case the interaction parameter is too small, the system evolves almost independently from the bath and therefore the coherence and entanglement of the system decay quite slowly as shown inFig. <ref> and<ref>. In the opposite limit, if the system-bath interaction is comparable to the energy difference of the spin levels of the central spin, we observe a rapid decay in quantum coherence with the presence of usual non-Markovian fluctuations. Whereas, entanglement decays to zero almost immediately with no revival detected in the time span considered in Fig. <ref>. Eq. (<ref>) also allows us to study the dynamics of coherence for varying number of bath spins. If the number of spins in the bath is large, we observe from Fig. <ref>, that the coherence rapidly decays and only small fluctuations are subsequently detected. In case the number of spins in the bath is not very large, the evolution of coherence undergoes periodic revivals. The magnitude of such revivals decreases with increasing bath size, eventually reducing to being indistinguishable with smaller fluctuations for large enough number of spins in the bath. As seen in Fig. <ref>,revivals themselves occur in periodic packets, magnitudes of which decrease steadily with time. On the other hand, if the number of bath particles is quite large, entanglement decays very quickly to zero. However for smaller number of spins in the bath, the entanglement dynamics depicted in Fig. <ref> is quite similar to the corresponding dynamics of coherence captured earlier in Fig. <ref>. § ANALYSIS OF TIME AVERAGED DYNAMICAL MAPIn this section we probe the behaviour of long time averaged state of the central spin qubit. We study under what condition the long time averaged state is coherent. We further investigate whether or under what conditions the long time averaged state is a true fixed point of the dynamical map, i.e. independent of initial condition. In connection to that we further study what role the finite size of the environment plays in this context. The long time averaged state of the central spin qubit is given byρ = lim_τ→∞∫^τ_0ρ (t) dt /τ.Following this definition, we find [ ρ_11 = lim_τ→∞∫^τ_0ρ_11(t) dt /τ=ρ_11 (0) (1 - α)+ ρ_22(0) β,; ρ_12 = ρ_12(0) Δ, ]where α, βand Δ are long time averages of α(t), β(t) and Δ(t) respectively. When we integrate a bounded periodic function over a long timeand divide by the total time elapsed, we can consider the integral being over a large integer number of periods without loss of generality. Now,[ α = ∑_n=0^N 2(n+1)ϵ^2(1-n/2N)1/η^2e^-ħω/KT(n/2N-1/2)/Z, ] where the result follows from the fact that average of sin^2 (θ (t)) over any integer number of time periods = 1/2. Similarly we getβ=∑_n=0^N 2nϵ^2(1-n-1/2N)1/η'^2e^-ħω/KT(n/2N-1/2)/ZThe equation for population dynamics shows Eq (<ref>) that even the very long time averaged state retains the memory of the initial state, which is a signature of the system being strongly non-Markovian. This initial state dependence is captured in Fig. <ref>. It is observed that the parameter (ρ_11/ρ_22) which captures the population distribution for long time averaged state is heavily dependent on the initial ground state population. If the initial population of the ground state increases, so does the population of the ground state for long time averaged state. However, in case the bath is very large, the population statistics for the long time averaged state is markedly less sensitive to the initial population. This leads us to posit that the only true fixed point independent of the initial conditions for this system exists only in the limit N →∞. We also observe that in the limit ρ_11 (0) = ρ_22 (0) = 1/2, (ρ_11/ρ_22) tends towards 1 regardless of bath size N indicating the dynamics is almost unital. Also we should mention that in the thermodynamic limit (N→∞), when the temperature of the bath is infinite, the state ρ̅_11=ρ̅_22=1/2 is not only the fixed point of the dynamics but the canonical equilibrium state also. Thus we can conclude that in the limit N→∞ and T→∞, the present open system dynamics is ergodic. Moreover, we see that the system-bath coupling strength not only affects the timescale of evolution but also plays significant role in the population statistics of the time averaged state. This we can see fromEq.s (<ref>) and (<ref>), which is also depicted in Fig. <ref>. Also for most of the cases, we have Δ=0. It is interesting to note that the long-time averaged state ρ is incoherent in general. This implies, even though quantum coherence or entanglement persists for quite a long time if the bath temperature is very low, as depicted in Fig. <ref> or Fig. <ref> respectively, they must eventually decay. It is important to mention that there are specific resonance conditions under which Δ can have finite value, which will be analysed in the following section. §.§ Resonance Condition for long lived quantum coherenceWe have mentioned previously that the long time averaged state is in general diagonal, but for very specific choices of parameter values, this is not true and there indeed is long lived quantum coherence even in the long time averaged state. This can be of significant interest for theoretical and experimental purposes. For the off-diagonal component, the real and imaginary parts of Δ(t), defined as Δ_R(t) and Δ_I(t) respectively equals to [ Δ_R(t) =; ∑_ncosω t/2N[cosη t/2cosη' t/2+ ( ω_0 - ω/2N) ^2/ηη'sinη t/2sinη' t/2]e^-ħω/KT(n/2N-1/2)/Z; + ∑_n( ω_0 - ω/2N)[sinω t/2Ncosη t/2sinη' t/2/η' - sinω t/2Nsinη t/2cosη' t/2/η] e^-ħω/KT(n/2N-1/2)/Z,; Δ_I(t) =;-∑_nsinω t/2N[ cosη t/2cosη' t/2+( ω_0 - ω/2N) ^2/ηη'sinη t/2sinη' t/2]e^-ħω/KT(n/2N-1/2)/Z; +∑_n( ω_0 - ω/2N)[ cosω t/2Ncosη t/2sinη' t/2/η' - cosω t/2Nsinη t/2cosη' t/2/η] e^-ħω/KT(n/2N-1/2)/Z. ] We always have sinθ_1(t) sinθ_2 (t) sinθ_3(t) = sinθ_1(t) cosθ_2(t)cosθ_3(t) = 0.For each of the rest of the terms, it can be shown that the criteria for non-zero time averaged coherence reads ω/2N = |η±η'/2| .For the condition ω/2N = | η + η'/2| to hold, it is easily shown thatN ≤ω/ω_0.This, given that ω and ω_0 are usually of the same order of magntitude,we feel is a rather unrealistic demand on N, since we are concerned with a heat bath, albeit finite sized. We thus concentrate on the other condition ω/2N = ( η- η'/2). The equation ω/2N = ( η- η'/2) can be explicitly expanded out and the following quadratic equation in n isobtained[ ( ϵ^4/N^2 + ϵ^2ω^2/2 N^3) n^2 - (2 ϵ^4/N + ϵ^2ω^2/N^2) n +;(ω_0ω^3/4 N^3 - ω^2ω_0^2/4 N^2 - ϵ^2ω^2/2 N^2 + ϵ^4) = 0. ]By solving this quadratic equation and noting that the value of n must be an integer, we reach the following equation, which is the resonance condition.N ±ϵω/2√(q_1/8 N^3 + q_2/16 N^4 + q_3/32 N^5 - q_4/64 N^6)/ϵ^4/4 N^2 + ϵ^2ω^2/8 N^3∈ℤ_+,with [q_1=ϵ^4, q_2=( ϵ^2ω^2 + ϵ^2ω_0^2 + 2 ϵ^4),; q_3=( ω^2ω_0^2 + 2 ϵ^2ω^2 - 2 ϵ^2ωω_0), q_4=2ω_0ω^3, ]where ℤ^+ is the set of positive integers ∈ [0,N]. Taking ω = ω_0 = 1 and in the limit N ≫ 1, we have the resonance condition as [ N ±√(N)/ϵ√(2)∈ℤ^+, ]Thus, if we are interested in obtaining non zero amount of quantum coherence in the long time averaged state, we have to tune the interaction parameter exactly in such a way that N ±√(N)/ϵ√(2) is a positive integer. This is a nice example where precise bath engineering can help us achieve long sustained coherence.§.§ Information trapping in the Central Spin System Let us now investigate whether or under what condition the dynamical map considered here does have a true fixed point; i.e. the existence of a state which is invariant under the particular dynamics. In order to do that, define the time-averaging map Λ as the map which takes any initial state ρ tothe corresponding time averaged state ρ as given by Eq. (<ref>). Now suppose the system is initially in a state ρ. Then a natural question to ask is the following - “Is the corresponding time averaged state ρ invariant under the map Λ ?" This can only happen when the map Λ is an idempotent one, i.e. Λ^2 = Λ. Clearly, if the time averaged state did not retain the memory of the initial state, this would be the case. Therefore the deviation from idempotence of the map Λ can serve as a useful measure of the initial state dependence of the system in the long run, which is termed as Information Trapping<cit.> and defined by𝒯( Λ) = max_ρ∈ℋ_S D [ Λ^2ρ, Λρ],where D[.,.] is a suitable distance measure on the Hilbert space of the system. Choosing the trace norm as our distance measure, the expression for 𝒯 in the central spin model is computed as𝒯( Λ)= |β̅ - α̅|.We immediately note that this quantity vanishes iff β̅ = α̅, which is the case only in the limit N →∞, T →∞,i.e. the thermodynamic and high temperature limit. The above statement is confirmed in Fig. <ref>. As we increase the temperature of the bath, the trapped information 𝒯 asymptotically vanishes. It is also observed that at any given temperature, the amount of information trapped is greater for a smaller sized bath. This is consistent with the observation that a very large bath is required for 𝒯to vanish.Fig. <ref> and <ref> lead to the observation that as the system-bath coupling gets stronger, the amount of information trapping, i.e. the dependence of the time averaged state on the initial state, also increases. § CANONICAL MASTER EQUATION AND THE PROCESS OF EQUILIBRATIONFinding the generator of a general dynamical evolution of a quantum system is one of the fundamental problems in the theory of open quantum systems, which leads to a better understanding of the actual nature of decoherence. It is our aim here to derive a canonical master equation without resorting to weak coupling and Born-Markov approximation for the reduced dynamics presented in Eq. (<ref>), by virtue of which we will later analyse various thermodynamic aspects of the qubit system. Using the formalism of <cit.>, we obtain the following exact time local master equation for the central spin in the Lindblad form.[ ρ̇(t)=i/ħδ(t)[ρ(t),σ_z]+Γ_deph(t)[σ_zρ(t)σ_z-ρ(t)]; +Γ_dis(t)[σ_-ρ(t)σ_+-1/2{σ_+σ_-,ρ(t)}];+Γ_abs(t)[σ_+ρ(t)σ_--1/2{σ_-σ_+,ρ(t)}], ] where σ_±=σ_x ± iσ_y/2, and Γ_dis(t), Γ_abs(t), Γ_deph(t) are the rates of dissipation, absorption and dephasing processes respectively, andδ(t) corresponds to the unitary evolution, respectively, given as [Γ_dis(t)=[d/dt(α(t)-β(t))/2-(α(t)-β(t)+1)/2d/dtln(1-α(t)-β(t))],;; Γ_abs(t)=-[d/dt(α(t)-β(t))/2-(α(t)-β(t)-1)/2d/dtln(1-α(t)-β(t))],;;Γ_deph(t)=1/4d/dt[ln(1-α(t)-β(t)/|Δ(t)|^2)],;; δ(t)=-1/2d/dt[ln(1+(Δ_R(t)/Δ_I(t))^2)]. ] For the detailed derivation of the master equation, one can look into the Refs. <cit.>. Note that the system environment interaction generates a time dependent Hamiltonian evolution in the form of δ(t). This is analogous to the Lamb-shift correction in the unitary part of the evolution. Complete positivity <cit.> is one of the important properties of a general quantum evolution, following the argument that for any valid quantum dynamical map, the positivity must be preserved if the map is acting on a system which is correlated to an ancilla of any possible dimension. For a Lindblad type evolution, this is guaranteed by the condition ∫_0^t Γ_i(s)ds ≥ 0<cit.>, which can be easily verified for the specific decay rates given in (<ref>). However since the dynamical map here is derived starting from an initial product state, complete positivity is always guaranteed <cit.>.§.§ The principle of detailed balanceHere we investigate the process of approach towards steady state for the open system dynamics considered in this paper. There are various different approaches to explore the process of equilibration in an open system dynamics, each of which has their own merit <cit.>. In this work we carry out this investigation for the specific system considered here from a few different aspects, one of which is the quantum detailed balance first introduced by Boltzmann, who used it to prove the famous H-theorem <cit.>. When two or more irreversible processes occur simultaneously, they naturally interfere with each other. If due to the interplay between those different processes, over a sufficient period of evolution time, a certain balance condition between them is reached, then the system reaches a steady state. Consider the Pauli master equation for the atom undergoing such processes <cit.> given byṖ_n=∑_m γ_nmP_m -∑_m γ_mnP_n,where P_n is the diagonal matrix element of the density operator and γ_mn is the transition probability for the process |m|→⟩|n|$⟩. The well known detailed balance condition <cit.> for Pauli master equation is given asγ_mnP_n^(s)=γ_nmP_m^(s), whereP_n^(s)is diagonal density matrix element at the steady state. We first derive a rate equation of the form of Eq.(<ref>) from the master equation (<ref>) in order to study the detailed balance for our particular system <cit.>. Let us consider the unitary matrixU(t), which diagonalizes the system density matrix (ρ(t)) asρ_D(t)=U(t)ρ(t)U^†(t). Then we can straightforwardly derive the equation of motion for the diagonalized density matrix as[ ρ̇_D(t)=i/ħδ(t)[ρ_D(t),σ̅_z(t)]; +Γ_deph(t)[σ̅_z(t)ρ_D(t)σ̅_z(t)-ρ_D(t)];+Γ_dis(t)[σ̅_-(t)ρ_D(t)σ̅_+(t)-1/2{σ̅_+(t)σ̅_-(t),ρ_D(t)}]; +Γ_abs(t)[σ̅_+(t)ρ_D(t)σ̅_-(t)-1/2{σ̅_-(t)σ̅_+(t),ρ_D(t)}], ] whereA̅_j(t)=U(t)A_jU^†(t). ConsideringP_a(t)=⟨a||ρ_D(t)|a|$⟩, we get the rate equation similar to the Pauli equation asṖ_a(t)=∑_i∑_b |⟨a||A̅_i(t)|b||⟩^2 P_b(t)-∑_i ⟨a||A̅_i^†(t)A̅_i(t)|a|P⟩_a(t),where A̅_i(t)s are all the Lindblad operators in the diagonal basis as given in Eq. (<ref>). For the instantaneous steady state we must have Ṗ_a(t)=0, for all a. Thus, we have the detailed balance condition∑_i Γ_i(t_s)⟨a||A̅_i^†(t_s)A̅_i(t_s)|a|P⟩_a(t_s)/∑_i∑_b Γ_i(t_s)|⟨a||A̅_i(t_s)|b||⟩^2 P_b(t_s)=1,where t_s is the time at which the system comes to the steady state. From Eq. (<ref>) and (<ref>), we arrive at the following condition D(t_s)=Γ_dis(t_s)P_a(t_s)/Γ_abs(t_s)P_b(t_s)=1,where P_a,b(t)=1/2(1±√((ρ_11(t)-ρ_22(t))^2+4|ρ_12(t)|^2)) are the eigenvalues of the system density matrix. Any deviation of D(t) from its steady state value, implies that the system has not attained a steady state at that instant of time. The magnitude of such deviations may be regarded as a measure of how far away the system is from equilibrating. In the following we study the time dynamics of deviations from the detailed balance condition Eq. (<ref>). From Fig. <ref>, we observe that the deviations from detailed balance condition are quite persistent in the low temperature limit. In the opposite limit, as we go on increasing the bath temperature, Fig. <ref> shows that the fluctuations in deviation from the detailed balance condition increasingly tend to damp down. In the limit of a completely unpolarized bath, the detailed balance condition is met if the system size is large enough. For an initially coherent central qubit, any study of approach towards steady state has to also take the coherence dynamics into account. In the very low temperature limit, the value of quantum coherence (Fig. <ref>) is encapsulated within a narrow band whose width does not decay much over time. The persistence of coherence in this case implies the deviations are further away from D(t)=1 than in Fig. <ref>. In the opposite limit of a high temperature bath, quantum coherence dies down very quickly, as seen in Fig. <ref>. This explains why, just like Fig. <ref>, D(t) again approaches 1 in Fig. <ref>. In the intermediate regime, as we increase the temperature, the approach towards D(t) =1 becomes faster. If the system-bath coupling strength is very weak, we see from Fig. <ref> that the deviation of D(t) from unity is very small. This is understandable because as the system-bath interaction gets weaker, the change in the state of the system due to the exposure of bath interaction becomes slower and the process becomes more and more quasi-static. Hence, the system remains close to its steady state. As we go on increasing the strength of the interaction, the fluctuations in population levels increase, implying that the deviation from detailed balance condition also increases which is confirmed in Fig. <ref>. With increasing the bath size, we see from Fig. <ref> that deviations from detailed balance condition becomes smaller and smaller. This is fully consistent with the observation for many physical systems thatenergy exchange and consequent thermalization of a system isbetterfacilitated by having a large bath rather than a small ancilla attached to it. §.§ Irreversible Entropy productionHere we investigate how this system approaches towards a steady state from another thermodynamic perspective, i.e. the phenomenon of irreversible entropy production (IEP). The entropy production rate is formally defined as the negative rate of change of relative entropy between the instantaneous state and the steady state, i.e., Σ(t)=-d/dtS(ρ(t)||ρ_st). For an ideal Markovian evolution, Σ(t) is always positive<cit.>. This happens for few ideal situations and in general is not satisfied. The rate equation (<ref>) can be compactly represented as Ṗ_a(t)=∑_b ℒ_abP_b(t), withℒ=( -Γ_dis(t)Γ_abs(t)Γ_dis(t)-Γ_abs(t)).The entropy of the system is defined as S(t)=-∑_b P_b(t)ln P_b(t). By differentiating S(t) with respect to time, it can be easily shown that[ Ṡ(t)=∑_abℒ_abP_b(t) ln(ℒ_abP_b(t)/ℒ_baP_a(t)) -∑_abℒ_abP_b(t) ln(ℒ_ab/ℒ_ba),;=Σ(t)+Φ(t). ] The first term in the right hand side can be identified as the entropy production rate Σ(t) and the second term Φ(t) defines the effective rate at which entropy is transferred from the environment to the system. For the particular central spin system considered in this paper, the IEP rate is given by Σ(t)=(Γ_dis(t)P_a(t)-Γ_abs(t)P_b(t))ln(Γ_dis(t)P_a(t)/Γ_abs(t)P_b(t)).We see from (<ref>) that IEP rate is related to D(t) and at the time (t_s) when system obeys the detailed balance condition,we have Σ(t_s)=0. We also see from the expression of IEP rate that for Markovian situation (i.e. Γ_dis(t),Γ_abs(t)≥ 0), it will always be non-negative. This behaviour is illustrated in Fig. <ref>. Whenever the irreversible entropy production rate Σ (t) is negative, the absorption and dissipation rates are also negative and vice versa in the time span we probed. Since negativity of at least one Lindblad coefficient Γ(t) is a necessary and sufficient condition <cit.> for non-Markovianity, this leads us to conclude that whenever this system is non-Markovian, a negative IEP rate Σ (t) is obtained. While the negativity of IEP rate at any point in the dynamics necessarily implies that the dynamics is non-Markovian, the opposite is not true in general. However, in this illustration we note that the opposite is also true.If the bath temperature is very low, we have already seen from Fig. <ref> that the quantum coherence of the central spin qubitpersists for a long time, resulting in persistent deviations from thesteady state detailed balance condition as depicted in Fig. <ref>. Therefore, it is expected that the IEP rate will also fluctuate and not show any sign of dying down to zero. This is indeed captured inFig. <ref>. In the opposite limit, as we go on increasing the bath temperature, as seenFig. <ref>, the approach towards a steady state becomes quicker. This is again confirmed in Fig. <ref>, where the fluctuations in IEP rate die down more and more quickly for higher temperatures.As we have already observed in Fig. <ref>, the approach towards a steady state through exchange of energy between the system and the bath is quicker for a larger bath. This is again confirmed in Fig. <ref> which shows the IEP rate becoming smaller and smaller as we increase the bath size. The period of fluctuations also diminish with increasing bath size.§ CONCLUSIONIn this paper we explore various aspects of a central qubit system in the presence of anon-interacting thermal spin environment. We solve the Schrödinger dynamics of the total state and derive the exact reduced dynamical map for the central qubit. We compute the corresponding Kraus decomposition and evaluate the time evolution of quantum coherence (quantified through the l_1 norm) for the qubit in various parameter regions in section <ref>. We note that as the number of bath spins and the temperature increases, quantum coherence decays steadily with very small fluctuations thus enabling us to conclude that in the thermodynamic limit (N→∞) and for sufficiently high temperature, the decay of coherence closely mimics the corresponding behaviour in Markovian systems. We observe quite similar phenomena for quantum entanglement in the same limit, where we see the usual entanglement sudden death. On the contrary, for low temperature both coherence and entanglement, sustain steadily in a band for a very long period of time. This is an important observation having potential practical applications in quantum information processing. For the sake of concretenss, assuming typical order of magnitude values of various parameters governing the dynamics of quantum coherence, we are able to estimate the timescale for which coherence is sustained. Supposing the coupling strength ϵ∼ 1 MHz <cit.>, and assuming the spins having intrinsic energies ∼ 100 MHz <cit.>, we can conclude that at room temperature (T = 300 K) and for N=100, the value of coherence is guaranteed to be at least 80 percent of the initial coherence for at least ∼ 100 μ s. Interestingly, this timescale for guaranteeing at least 80 percent of the initial coherence is not too sensitive on the bath temperature in practice. For example, if we assume the bath to be in a very low temp, say 10^-4 K, then this time increases to only around ∼ 300μ s. It implies that for the open system considered in this paper, the environment can be designed in such particular ways that quantum signatures like coherence or entanglement can be preserved for along period of time. For diminishing number of bath spins, steady oscillations of both coherence and entanglement increases both in magnitude and frequency, which can be attributed to the finite size effect. We can contrast the situation with the the extreme case where only one auxillary spin is coupled to the central spin. In that extremal case, the coherence merely oscillates steadily, which is to be expected. But as the number of bath spin increases, the coherence suppression also increases.In the second part of our work, we derive the exact canonical master equation for the central qubit, without weak coupling approximation or Born-Markov approximation to study under what condition the central qubit thermalizes with its environment and if not, whether it at all comes to any steady state other than the corresponding thermal state. Probing the quantum detailed balance relation and IEP rate, we conclude that as the completely unpolarized (T→∞) spin bath reaches thermodynamic limit, the system equilibrates faster. We see that in the non-Markovian region (Γ_i(t)<0) of the dynamics, the IEP rate is negative, which is a signature of a system driven away from the equilibrium. However with the increasing number of bath spins and the temperature, we observe that this effect vanishes and the IEP rate remains very close to zero. In fact from further study of long time averaged state and information trapping, we also see that in the mentioned limit, the system actually equilibrates to the corresponding canonical state at infinite temperature. Hence, one may naturally infer that in the limit of N→∞,T→∞, the dynamics is ergodic and the bath does not retain the memory of the initial state. However as we deviate from this limit, ergodicity breaks down. In those cases, we observe finite amount of information trapping in the central spin system, which demonstrates that then the bath does hold the memory of the initial state. Perhaps the most important result of the present work is the finding of the existence of coherence in thelong time averaged state of the central spin. We have shown that for specific choices of the system-bath interaction parameter, a resonance condition is satisfied and as a result the long time averaged state retains a finite amount of coherence. Here no external coherent driving is required to preserve this coherence. Our result shows that through precise bath engineering, a spin environment can be manipulated in such a way that it acts as a quantum resource to preserve coherence and potentially entanglement. The presence of such long time quantumness can have potentially far reaching consequence for the construction of quantum thermal machines whose performances are augmented by coherence.§ ACKNOWLEDGEMENTSSB thanks Anindita Bera of HRI for careful reading of the manuscript. Authors acknowledge financial support from the Department of Atomic Energy, Govt. of India.apsrev4-1IntroductionIn many body problemsthe dynamics of microscopic (e.g. spin systems) or mesoscopic (e.g. SQUIDs) systems always gets complicated owing to its interaction with a background environment. To have the reduced dynamics of the quantum system that we are interested in, it is a general custom to model the environment as a collection of oscillators or spin half particles <cit.> which is often abbreviated as bath. They constitute two different universal classes of quantum environment <cit.>. In the oscillator bath model, the environment is described as a set of uncoupled harmonic oscillators. Paradigmatic examples of this kind of baths are spin-boson <cit.> and the Caldeira-Leggett model <cit.> originating from a scheme proposed by Feynman and Vernon <cit.>. These oscillator models have been widely studied in the context of various physical phenomena under Markovian approximation <cit.>. On the other hand, the spin bath models remain relatively less explored. However, the spin bath models play a pivotal role in the quantum theory of magnetism <cit.>, quantum spin glasses <cit.>, theory of conductors and superconductors <cit.>. To get the exact dynamics of a quantum systems under this spin bath model is of paramount importance yet a difficult task. Indeed, in most of the cases the dynamics cannot be described exactly and several approximation techniques, both local and nonlocal in time, have been employed <cit.>. In this work, we will focus on the dynamical behavior of a central spin interacting uniformly with a spin bath and derive an exact time-local master equation of the Lindblad type. Moreover, the Kraus representation of the dynamical map is also derived. Reduced dynamics of this particular spin bath model has been considered before <cit.> where correlated projection operator technique has been used to approximate the master equation of the central spin. However, the given master equation is time nonlocal and not of the standard canonical form. In contrast, we start from the exact reduced state of the central spin at an arbitrary given time <cit.> to derive the canonical master equation without considering any approximations. The thrust of our result is not only that the master equation is exact but the method used here allows us to unravel the less explored but far reaching consequences of the strong coupling regimes which can be instrumental in performing information theoretic, quantum thermodynamic and several other quantum technological tasks.Moreover, the relaxation rates in the canonical master equation are insightful to understand several physical processes such as the dissipation, absorption and dephasing and thus the nature of decohrence.One of the characteristics of the spin bath models is to exhibit the non-Markovian features <cit.>. The non-Markovianity has been identified as a key resource in information theoretic <cit.>, thermodynamic <cit.> and precision measurement protocols <cit.>. We study the non-Markovian features of the reduced dynamics and it is shown that the non-Markovianity increases with the interaction strength.Irreversible increase of entropy due to dissipation of energy and work into the environment is inevitable for systems out of equilibrium. The analysis of irreversible or nonequilibrium entropy production and its rate have been instrumental to understand nonequilibrium phenomena in different branches of physics <cit.>. According to the Spohn's theorem <cit.>, the irreversible entropy production rate is always non-negative under the Markovian dynamics.Whereas non-Markovianity of the dynamics allows negative irreversible entropy production rate and thereby this partial reversibility of the work and entropy influences the performance of quantum heat engines, refrigerators and memory devices. As our study enables us to probe the strong coupling regime, it can be far reaching to unravel the hitherto unexplored consequences of the non-Markovian dynamics in the strong coupling regime for more efficient thermodynamic protocols. Here, we investigate the entropy production rate and shown that the non-Markovianity of the dynamics is always associated with a negative entropy production rate of the central spin for a certain initial state. We also investigate the non-Markovianity in terms of the rate of change of the purity of the central qubit and it is observed that the rate of change of the purityof the qubit is positive for the same aforesaid initial state, whenever the dynamics is non-Markovian. Experimental detection of the non-Markovianity and the entropy production rates for quantum systems are of paramount interest in current research. As purity can be measured in the laboratory, the study of this article can pave novel avenues to experimentally demonstrate non-Markovian features and negative entropy production rate in spin bath models. The organization of the paper is as follows. In Sec. <ref>, we derive the proposed canonical master equation of Lindblad type. The non-Markovian features of the dynamics of the central qubit are demonstrated explaining the indivisibility of the dynamical map and non-monotonicity of the trace distance fidelity. In this section, we also derive the Kraus operators for the dynamical evolution. The nonequilibrium entropy production rate and dynamics of purity of the qubit are studied in Sec. <ref>. Finally we conclude in Sec. <ref>. ] | http://arxiv.org/abs/1704.08291v2 | {
"authors": [
"Chiranjib Mukhopadhyay",
"Samyadeb Bhattacharya",
"Avijit Misra",
"Arun Kumar Pati"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170426184438",
"title": "Dynamics and thermodynamics of a central spin immersed in a spin bath"
} |
^1Vienna University of Technology, Atominstitut, Stadionallee 2, 1020 Vienna, Austria^2National Institute of Standards and Technology, Time and Frequency Division, Boulder, Colorado 80305, USA We propose to build a bad cavity laser using forbidden transitions in large ensembles of cold ions that form a Coulomb crystal in a linear Paul trap. This laser might realize an active optical frequency standard able to serve as a local oscillator in next-generation optical clock schemes. In passive optical clocks, large ensembles of ions appear less promising, as they suffer from inhomogeneous broadening due to quadrupole interactions and micromotion-relates shifts. In bad cavity lasers however, the radiating dipoles can synchronize and generate stable and narrow-linewidth radiation. Furthermore, for specific ions, micromotion-induced shifts can be largely suppressed by operating the ion trap at a magic frequency. We discuss the output radiation properties and perform quantitative estimations for lasing on the ^3D_2→^1S_0 transition in ^176Lu^+ ions in a spherically-symmetric trap. 42.50.Gy, 42.50.HzProspects for a bad cavity laser using a large ion crystal Georgy A. Kazakov^1E–mail: [email protected], Justin Bohnet^2, Thorsten Schumm^1 December 30, 2023 ===========================================================================================§ INTRODUCTION Optical frequency standards are the most stable clocks to date. The most advanced implementations reach a short-term stability at the 3.4× 10^-16/√(τ) level <cit.>, and systematic uncertainty of 3.2× 10^-18 <cit.>. Further improvement of optical frequency standards would allow a multitude of new applications in fundamental and applied science, such as study of fundamental constant variations <cit.> and relativistic geodesy <cit.>. Modern optical clocks are passive clocks, where the frequency of a local oscillator, i.e., some stable narrow-band laser, is feedback-stabilized to a narrow and robust etalon transition in trapped atoms or ions. This etalon transition may be extremely narrow, down to a nHz level in some species <cit.>, but the real spectroscopic linewidth is limited by the short-term stability of the local oscillator and usually does not surpass the sub-Hz level. Also, on a timescale shorter than the interrogation time of the etalon transition, the stability of the local oscillator entirely determines the stability of the whole frequency standard. Fluctuations of the local oscillator frequency may also contribute to the instability of the frequency standard on longer timescales via the Dick effect <cit.>. Therefore, improving the local oscillators is one of the key tasks for the development of more precise optical clocks. The best modern local oscillators are lasers that are prestabilized to an ultrastable macroscopic cavity. Their stability is usually limited by mechanical and thermal noise <cit.>, and may attain a level of 8× 10^-17 on the timescale up to 10^3 s at room temperature <cit.>, and even 4× 10^-17 on the timescale up to 10^2 s in cryogenic environments <cit.>, but the progress in this direction is slow.One possible alternative approach is to create an active optical frequency standard, i.e., a laser where atoms with a narrow and robust lasing transition play the role of the gain medium. Such a laser would operate in the so-called bad cavity regime, where the linewidth of the cavity mode is much broader than the gain profile. The output frequency of such a laser is determined primarily by the gain medium, which makes it robust to fluctuations of the cavity length. Such standards have been proposed by several authors recently <cit.>, and a series of proof-of-principle experiments have been performed <cit.>. Active atoms that constitute the gain for an active optical frequency standard must be confined to the Lamb Dicke regime to avoid Doppler and recoil shifts. Such a confinement may be realized with an optical lattice potential at a so-called magic frequency, where the upper and the lower lasing states experience the same ac Stark shift <cit.>. These shifts depend on the polarization of the trapping fields and can be controlled to the necessary level of precision only for ^3P_0→^1S_0 transitions in Sr and other alkali-earth atoms, Zn, Cd, Hg and Yb. A first proof-of-principle experiment with such a transition in trapped Sr atoms has been recently performed in a pulsed regime <cit.>.The optical lattice potential trapping neutral atoms is relatively shallow, of order of a few tens of μK <cit.>. This leads to a short trap lifetime, therefore some method of compensating for atom losses must be implemented to practically realize an active optical frequency standard <cit.>. The implementation of such methods is rather complicated, although certain efforts in this direction are being made <cit.>.In contrast to neutral atoms, charged ions may be trapped in much deeper Paul or Penning traps, which leads to much longer trap lifetimes. Trapped ions may also be cooled via co-trapped ions of another species (sympathetic cooling) <cit.>. A bad cavity laser utilizing trapped ions may operate continuously over hours, even days, without the need to compensate for ion losses. On the other hand, micromotion of the ions and their interactions with trapping fields and with each other causes shifts and inhomogeneous broadening of the etalon transition; these effects are especially pronounced in large ion ensembles. Thus, ion optical clocks have been built primarily with single ions <cit.> or with few-ion ensembles <cit.>. Inhomogeneous broadening may be considerably reduced for ions with negative differential polarizability of the clock states in RF Paul traps at a specially chosen magic frequency of the trapping field <cit.>. Also, bad cavity lasers with inhomogeneously broadened gain may produce synchronous and stable output radiation, if the total homogeneous broadening of the lasing transition exceeds the inhomogeneous broadening by at least a few times <cit.>. In lasers based on a 3-level scheme, such homogeneous broadening will be dominated by repumping, which opens the possibility to build a bad cavity laser with ions <cit.>.In this paper, we present a detailed discussion of the bad cavity laser based on Coulomb crystals in Paul traps. In Section <ref> we consider a generic model of a harmonic co-axial Paul trap formed by static and radio-frequency (rf) harmonic potentials, obtain general expressions for micromotion-induced Doppler and Stark shifts in a cold Coulomb crystal, and introduce the “magic” frequency, which allows one to compensate these shifts in leading order. In Section <ref> we consider residual terms of the shifts, and specify the trap geometry. In Section <ref> we derive the equation for the intracavity field, taking into account standing-wave periodicity and Gaussian shape of the cavity mode. In Section <ref> we present some quantitative estimations for a bad cavity laser with trapped ^176Lu^+ ions. In Section <ref> we discuss the results, envisaged difficulties, and possible ways to overcome them.§ MAGIC FREQUENCY Here we consider micromotion-induced second-order Doppler and dc Stark shifts for ions forming a cold Coulomb crystal in a harmonic rf Paul trap. Such a many-ion crystal has been considered in <cit.>, although some higher-order terms have been omitted there. These terms, however, can be easily calculated, if we note that the only macroscopic force acting on the ion in the Paul trap is proportional to the same local electric field that causes the Stark shift.We consider a Paul trap formed by the potential ϕ (,̊ t) =m Ωω_z/2 q ^̊T ·[ cos (Ω t)+ϵ/2] ·,̊ where =̊x _x+y _y+ z_z≡ x_1 _x+x_2 _y+ x_3 _z is the position vector, m and q are the mass and the charge of the ion, Ω is the rf drive frequency, ω_z the frequency characterizing the trap confinement, ϵ=2 ω_z/Ω, andandare traceless dimensionless symmetric matrices determining curvatures of the potentials. We use a single bar to denote column vectors with 3 spatial components, a double bar to denote 3× 3 matrices, a dot (·) for the inner product, and the superscript T for the transposition (we will often omit this superscript for vectors, for the sake of brevity). Also we suppose that ω_z ≪Ω, i.e., such that ϵ may be considered as a small parameter. Scaling time and length (in Gaussian units) by tΩ/2→ t, /ℓ→,̊ whereℓ=(q^2/m ω_z^2)^1/3, we write the equation of motion (e.o.m.) of i-th ion as _i+ϵ^2(·_̊i)+2 ϵ (·_̊i) cos(2 t)=ϵ^2 ∑_j≠ i_̊ij/r^3_ij, where _̊ij=_̊i-_̊j, r=||̊. Following <cit.> we assume the existence of a stable π-periodic solution of the e.o.m. (<ref>), which may be expressed as _̊i(t) = _0,i+2 ∑_n=1^∞_2n,icos(2 n t). We suppose that all the motions of the ions except the micromotion (<ref>) are frozen out (cold Coulomb crystal).The main trap-induced corrections to the frequency ν=2 πω of the clock transition of the ith ion are the micromotion-induced second-order Doppler shift ^i, and the Stark shift ^i caused by the time-dependent local electric field of the trap and nearby ions acting on the ith ion at its instantaneous position. The second-order fractional Doppler shift averaged over the period of the micromotion is ^i/ν= -Ω^2 ℓ^2/4⟨^2 ⟩/2 c^2=-Ω^2 ℓ^2/c^2∑_n=1^∞n^2 _2n,i^2, where c is the speed of light, and the prefactor comes from the scaling (<ref>).We first consider the Stark shift of the clock transition caused by the local electric field acting on the ion. Following <cit.>, we suppose that the ion trap is placed into a homogeneous external magnetic fieldcausing the Zeeman splitting to be much larger than the tensor component of the Stark shift (see estimations in the end of Sec. <ref>). Then the Stark shift of some Zeeman sublevel can be written asΔ = -α_0/2^2 - α_2/4 (3 E_z^2-^2), where the axis z is oriented along , α_0 is the scalar polarizability, and α_2 isα_2 = α_2(η,J,F,m_F)= α_tens (η,J) 3m_F^2-F (F+1)/3F^2-F (F+1) (-1)^I+J+F{[ F J I; J F 2 ]} ×(F (2F-1) (2F+1)(2J+3)(2J+1)(J+1)/(2F+3)(F+1)J(2J-1))^1/2.Here η, J, F, m_F are the principal quantum number, the angular momentum of the electronic shell, the total angular momentum and its projection onto the direction of the magnetic field respectively, and α_tens (η,J) is the tensor polarizability characterizing the state η, J of the electronic shell of the atom <cit.>. Denoting the differential polarizabilities Δα_k = α_k^u - α_k^l (k=0 or 2) of the upper (u) and lower (l) clock states, and taking into account the relation between the local electric field and the instantaneous acceleration of the ion, we can write the time-averaged scalar and tensor Stark shifts ^i and ^i of the clock transition of the ith ion as^i= -Δα_0/4 πħ⟨^2 ⟩= -Δα_0/4 πħΩ^4 ℓ^2 m^2/16 q^2⟨^2 ⟩= - Δα_0/2 πħΩ^4 ℓ^2 m^2/q^2∑_n=1^∞ n^4 ^2_2n,i,^i=Δα_2/8 πħ (⟨^2 ⟩ - 3 ⟨_z^2 ⟩)= Δα_2/4 πħΩ^4 ℓ^2 m^2/q^2∑_n=1^∞ n^4 (^2_2n,i-3 Z^2_2n,i).where Z_2n,i is the z-projection of R_2n,i. Combining (<ref>), (<ref>) and (<ref>), we can express the sum of the Doppler and Stark shifts in the formΔν^i =^i+^i = -Ω^2 ℓ^2 ν/c^2[ 1 + ( Δα_0-Δα_2/2) (m Ω c)^2/2 πħν q^2] ∑_n=1^∞ n^4 ^2_2n,i+ Ω^2 ℓ^2 ν/c^2∑_n=1^∞(n^4-n^2) ^2_2n,i- 3 Δα_2 νΩ^4 m^2 ℓ^2/4 πħν q^2∑_n=1^∞ Z^2_2n,i n^4.It is easy to see that if Δα_2 > 2 Δα_0, the first term in (<ref>) will be zero at so-called magic value Ω_0 of the radio frequency Ω: Ω_0=q/mc√(4 πħν/Δα_2-2 Δα_0). In the next section we consider remaining terms of (<ref>).§ RESIDUAL MICROMOTION-RELATED SHIFTSTo estimate residual terms in (<ref>), we expand ^2_2n,i and Z^2_2n,i by the small parameter ϵ=2ω_z/Ω. Also we suppose that the oscillating terms _2n,i (n≠ 0) are small in comparison with the time-independent components R_0,i. Then we can decompose the right part of the e.o.m. (<ref>) as _̊ij/r_ij^3=_0,ij/R_0,ij^3+_ij·_̊ij^'+..., where _̊ij^'=_̊ij-_0,ij, and_ij=-3 _0,ij⊗_0,ij -R_0,ij^2/R_0,ij^5. Here the symbol ⊗ denotes the outer product, andis the identity matrix. Substituting (<ref>) and (<ref>) into (<ref>), we obtain: _2,i= ϵ/4·_0,i+ϵ^3/16[(+^2/16) ··_0,i. .-∑_j≠ i_ij··_0,ij]+O(ϵ^5), _4,i= ϵ^2/64^2 ·_0,i+O(ϵ^4), _6,i= ϵ^3/2304^3 ·_0,i+O(ϵ^5).Consider the two residual terms in (<ref>). The second term contains only the summands with n≥2. It may be estimated as Ω^2 ℓ^2/c^2∑_n=1^∞(n^4-n^2) ^2_2n,i≈ 12 Ω^2 ℓ^2/c^2_4,i^2≈3 ϵ^4/1024Ω^2 ℓ^2/c^2_0,i·^4 ·_0,i. The last term of (<ref>) - 3 Δα_2Ω^4 m^2 ℓ^2/4 πħν q^2∑_n=1^∞ Z^2_2n,i n^4 contains also the summand with n=1, which is proportional to ϵ^2. However, it can be substantially reduced by a proper choice of the trap geometry. Namely, if the radio frequency component of the trap field is orthogonal to the z axis (see Fig. <ref>), i.e., if = [ [a00;0 -a0;000 ]], then Z_6,i=O(ϵ^4), Z_4,i=O(ϵ^4), and Z_2,i=O(ϵ^3). Therefore, the whole last term of (<ref>) is of order of ϵ^6, and it is dominated by the second term of (<ref>). We can neglect it, and approximate Δν^i as Δν^i ≈3 νϵ^4/1024Ω^2 ℓ^2/c^2 a^4 (X_0,i^2+Y_0,i^2).For our future estimations we consider the particular case of a spherical trap. Namely, we take a=√(3) in (<ref>), and= [ [ -1/200;0 -1/20;001 ]], which corresponds to the pseudopotentialV()̊=m ω_z^2/2 ·̊(+^2/2)·=̊m ω_z^2 ^̊2/2. It is easy to show that a large Coulomb crystal of N ions in such a trap will have an approximate spherical shape with the radius R≈ N^1/3 in the units of ℓ (i.e., ℓ is the Wigner-Seitz radius), and the density of the crystal will be homogeneous on the scales exceeding ℓ. The micromotion-related shift (<ref>) goes to zero at the center of the crystal and reaches its maximum valueΔ_max = 2 π Δν^i_max≈(3/4)^32 πν N^2/3 ω_z^8/3q^4/3/Ω_0^2 c^2m^2/3 at X_0,i^2 + Y_0,i^2 = N^2/3. To illustrate the dependence of Δ_max on ω_z, we present in Fig. <ref> the maximal micromotion-related shifts Δ_max(ω_z) for different |^3D_2,F_u,m_F=0⟩→ |^1S_0,F_l,m_F=1⟩ transitions in ^176Lu^+ ions, supposing that the spherical Coulomb crystal contains N=10^5 ions, and that the radio-frequencies Ω are equal to the magic frequencies Ω_0 for the corresponding transitions, see Section <ref> for details.We should note that the expressions (<ref>) – (<ref>) for the micromotion-related shift as well as for the magic frequency Ω_0 (<ref>) of the radio-frequency field have been obtained in the leading order (ϵ^4) of the small parameter ϵ, and in this leading order the individual shift Δ_i ∝ X_i^2+Y_i^2. However, it is easy to see that the first term in (<ref>) (which turns to zero at Ω=Ω_0) is also proportional to X^2+Y^2 in the leading order of ϵ^2. Therefore, fine tuning of Ω near Ω_0 may be used for further compensation of the micromotion-related shifts down to the order of ϵ^6; the respective correction of the magic frequency has been considered in <cit.>. Also, this tuning may be used for compensation of the light shifts caused by the pumping and cooling fields, or for suppression of the sensitivity of the frequency of the lasing transition to fluctuations of non-perfectly controlled parameters of the trap, such as amplitudes of the trapping and pumping fields. Detailed investigations of these possibilities are beyond the scope of this paper.§ CAVITY FIELD In this section we estimate the output power of the bad cavity laser based on a spherical Coulomb crystal with radius R_c=N^1/3ℓ coupled with the cavity field. We neglect here the micromotion-induced and quadrupole shifts of the lasing transitions; this assumption is acceptable, if the inhomogeneous broadening caused by these shifts is small in comparison with the homogeneous one <cit.>. These assumptions will be proven in the end of Section <ref>. Instead, we take into account that the cavity mode is a standing-wave Gaussian mode with waist w_0. We start from the mean-field equations (see Appendix <ref> for details of the derivation), where we neglect detunings and suppose equivalence of the cavity eigenfrequency ω_c with the transition frequencies ω and ω_ul^j of the laser field and lasing transitions: d/dt=- κ/2 -i/2∑_j , d /dt=i[ - ]-γ_∥ + w - γ, d /dt=i/2 - γ_⊥. Here , ^+ are the cavity field operators, _αβ^j=|α^j⟩⟨β^j | (|α^j⟩ and |β^j⟩ are the generic notations for the levels of jth atom), =-,w is the incoherent pumping rate, γ is the spontaneous rate of the lasing transition, γ_∥=w+γ, γ_⊥ = (γ+w)/2 + γ_R, γ_R is the incoherent dephasing rate (limited from the bottom by the value ξ w/2, where ξ=Γ_1/Γ_2 is the ratio of decay rates Γ_1 and Γ_2 of the intermediate pumping state into the lower and the upper lasing states |l⟩ and |u⟩ respectively).is the coupling coefficient of the cavity field with the lasing transition in jth ion. As is shown in Appendix <ref>,= g()̊= g_0e^ - ^2/^2 cos(·̨)̊.One may obtain the steady-state (cw) solution of equations (<ref>) – (<ref>) setting time derivatives to zero. Then, from (<ref>) follows _cw=- i/κ 3 N/4 π R_c^3 ×∫_-R_c^R_c∫_0^√(R_c^2-z^2)()̊⟨_lu(()̊) ⟩_cw 2 π d dz, where R_c=N^1/3ℓ is the radius of the Coulomb crystal. In turn, ⟨_lu((r)) ⟩_cw=⟨_lu^j ⟩_cw may be expressed via _cw with the help of (<ref>) and (<ref>) as ⟨_lu(()̊)⟩_cw= 1/2i()̊_cw (w-γ)/γ_⊥γ_∥+|_cw|^2^2. Substituting (<ref>) into (<ref>) and (<ref>) into (<ref>) and reducing _cw, we obtain the equation 1=(w-γ)/2 κ3 N/4 π R_c^3∫_-R_c^R_c∫_0^√(R_c^2-z^2) 2 π×g_0^2 cos^2(k z) e^-2 ^2/^2/γ_⊥γ_∥+|_cw|^2 g_0^2 cos^2(k z)e^-2 ^2/^2d dz. The integral overmay be taken analytically. Then (<ref>) transforms into 1=(w-γ)/2 κ3 N/4 R_c^3w_0^2/2 |_cw|^2 ×∫_-R_c^R_clog[γ_⊥γ_∥+|_cw|^2 g_0^2 cos^2(k z)/γ_⊥γ_∥+|_cw|^2 g_0^2 cos^2(k z) e^-2 R_c^2 - z^2/^2] dz. Because the cavity waistand the radius of the crystal R_c are large in comparison with 1/k, we can average (<ref>) on the scale of 2π/k with the help of relation 2/π∫_0^π/2log (1+b cos^2 z) d z= 2 log[√(1+b)+1/2]. It allows to represent (<ref>) in the form 1=3 N ζ^2 (w-γ)/4 κ |_cw|^2[ log(1+√(1+A)) - F(A, ζ) ], whereF(A, ζ) = ∫_0^1log(1+√(1+Aexp2(x^2-1)/ζ^2) )dx, A= |_cw|^2 g_0^2/γ_∥γ_⊥,ζ =/R_c. To find the steady-state intracavity field, one has to solve (<ref>) numerically. § PROSPECT FOR BAD-CAVITY LASER WITH ^176LU^+ IONSIn this section we consider the implementation of a bad-cavity laser using the ^3D_2→^1S_0 transition in ^176Lu^+ (I=7) ion. Briefly this possibility has been mentioned in <cit.>, here we present more detailed quantitative analysis.A possible pumping scheme is shown in Figure <ref>: a 350.84 nm pumping laser populates the ^3P_1^o state which decays with a 42 % probability into the ^3D_2 upper lasing state, and with a 37.6 % probability back into the lower lasing state <cit.>.The decay of the ^3P_1^o state will populate also the long-living ^3D_1 state, which can be depopulated via the ^3P_2^o state with the help of a 484.10 nm laser. Two additional 661.37 nm and 547.82 nm lasers should be applied to pump the atoms out of the^1D_2 and^3D_3 states populated by the decay of the ^3P_2^o state. 484.10 nm, 547.82 nm and 661.37 nm lasers may be detuned to the red side and be used also for cooling of the ion ensemble; sympathetic cooling with an additional ion species is also possible.An important point is that all involved states except the ground state have a hyperfine structure, therefore the pumping lasers should have several frequency components to effectively repump the atoms. Finally, a 5-component 499.55 nm laser should be employed to pump the populations into the upper lasing state, for example, with specific F=F_u and m_F=0. This can be realized if one component of this laser is tuned in resonance with the |^3D_2, F_e⟩→ |^3P_2^o, F_e ⟩ transition and polarized along the z axis of the trap, coinciding with the direction of the auxiliary magnetic field.For our calculations, we use values from <cit.>, where the spontaneous rate of the ^3D_2→^1S_0 lasing transition is γ= 4.19 × 10^-2 s^-1, the differential scalar polarizability is Δα_0 = -0.9 a_0^3, and the tensor polarizability of the upper state is α_tens(^3D_2)=-5.6 a_0^3, where a_0 is the Bohr radius.Let the lower indices l,u and e correspond to the lower lasing ^1S_0, upper lasing ^3D_2, and auxillary ^3P_2^o levels. With the help of the 499.55 nm laser, the populations may be pumped either into one of the |^3D_2,F_u,m_F=± F_u ⟩ states (both m_F=± F_u states may also be populated simultaneously, if F_u>5), or into some of the |^3D_2,F_u,m_F=0 ⟩ states. Quadrupole transitions may, generally speaking, be accompanied by Δ m = 0, ± 1, ± 2. In this paper we restrict our consideration to the geometry shown in Fig. <ref>. In such a configuration, micromotion of the ions takes place primarily in the plane orthogonal to the cavity axis (up to the terms of order of ϵ^3, see (<ref>) – (<ref>)), and the ions will be confined on length scales significantly smaller than the mode wavelength in axial direction, i.e., in the Lamb-Dicke regime. As shown in Appendix <ref>, only the transitions with Δ m = ± 1 will be coupled with the cavity modes in such a configuration.Note that two modes (σ^+- andσ^--polarized) with the same eigenfrequency may be excited simultaneously in the cavity. These modes couple the upper |^3D_2, F_u, m_F=0⟩ lasing state with two |^1S_0, F_l, m_l=± 1 ⟩ lower states, forming a “Λ-system”. Generally speaking, lasing in such a system can not be represented as a simple superposition of 2 independent lasers, because the modes will be coupled via the coherence between two lower lasing levels. However, a detailed investigation of such a system lies beyond the scope of the present paper, and we will consider only a single circularly-polarized mode coupling |u⟩ = |^3D_2, F_u, m_F=0⟩ and |l⟩ = |^1S_0, F_l=I, m_F=1⟩ states. Note also, that a selective excitation of a single mode may be performed, if the mirrors of the optical cavity will have slightly different transparencies for left- and right-polarized modes, and the lasing threshold will be more easily attainable for one of them. Using the method presented in Sec. <ref>, we find that the magic frequency Ω_0 exists for F_u=5, 8 and 9 (Ω_0=2π× 25.3, 45.3 and 22.3 MHz respectively). Also, one may find the values of the “magic magnetic field” B_m, at which the sensitivity of the lasing transition frequency to the fluctuation of this field vanishes in the first order: B_m= 0.388, -1.035 and -1.040 G for F_u=5,8 and 9 respectively.The maximal value g_0 of the coupling coefficient of the Gaussian standing-wave cavity mode with the lasing transition may be estimated as (see Appendix <ref> for details) g_0 =Θ_ul√(5 π c^3 γ/ω^2 ). where =π w_0^2 L is the effective mode volume, L is the cavity length, w_0 is the mode waist, Θ_ul^2=8(2 F_l+1) (2J_u+1) ×{[ J_l I F_l; F_u 2 J_u ]}^2 (C^F_u m_u_F_lm_l2-1)^2, and m_u=m_l-1. It is easy to see that the ratio g_0^2/κ does not depend on the cavity length L, but only on the cavity finesse , because κ may be expressed via L andas κ=π c/( L).To study the dependence of the output power on the mode waist , it is convenient to expressvia the radius R_c of the Coulomb crystal as =ζ R_c, like in Section <ref>. The output power P may be estimated as P=ħωκ |_cw|^2, where _cw is the steady-state intracavity field which may be found from the numeric solution of equation (<ref>). Note also that _cw appears in (<ref>) – (<ref>) and (<ref>) either as |_cw|^2 g^2 or as |_cw|^2 κ, therefore, the output power depends on the cavity finesse , not on the length L.For a quantitative estimation of the output power, we consider a spherical Coulomb crystal containing N=10^5 ^176Lu^+ ions, where the lasing transition is one of the |^3D_2, F_u, m_F=0⟩→⟩ |^1S_0, F_l=I, m_F=1⟩ quadrupole transitions with F_u=5, 8 or 9. The cavity finesse is =10^5. Also we suppose the repumping efficiency ξ=0.6 (i.e., 40 % of the atoms pumped into the ^3P^o_1 state from the lower lasing state decays back, see Appendix <ref> for details). Fig. <ref> presents the output power P (<ref>) for these 3 transitions for 3 different values of the confinement frequency ω_z (ω_z = 2π× 200 kHz, 500 kHz and 1 MHz), and different values of ζ. In figure <ref> we show the dependence of the maximum output power P_ max, corresponding to the optimized pumping rate w, on the parameter ζ. One can see that the optimal values of ζ are about 0.8. Let us discuss the linewidth Δω of such a bad-cavity laser. Our semiclassical mean-field model can not predict the linewidth; one needs to construct at least some “second-order theory” keeping second-order cumulants of the operators related to different ions, as it has been done in <cit.>, but with a larger amount of groups of ions with different coupling strengths (and shifts, generally speaking). Such a task claims additional attention. However, as an order-of-magnitude estimation, we can take the formula Δω≈ g^2/κ from <cit.>, and substitute g_0 for g. For =100 μ m (which corresponds to ζ=0.8 for spherical Coulomb crystal with N=10^5 ions at ω_z=2π× 1 MHz) this estimation gives Δω∼ 2π× 3 - 4 mHz for the transitions considered in this section; weaker confinement results in an even narrower linewidth.Concerning the validity of negligence of the micromotion-induced frequency shift: The optimal values of the repumping rate w, at which the maximum output powers are attained, are about 15, 50 and 150 s^-1 for ω_z=2π× 0.2 MHz, 0.5 MHz and 1 MHz respectively. These values of w significantly exceed the maximum micromotion-induced shifts Δ_max corresponding to the respective confinement frequencies, see Fig. <ref>. The repumping rate determines the homogeneous broadening, therefore, near the optimal regime we may neglect the inhomogeneous broadening related to the micromotion-induced shifts, at least for the parameters considered above.A few words about the quadrupole shift. This shift has been investigated in <cit.> for the ^1S_0→^3D_1 transition in Lu^+ ion. It has been shown that for ω_z=2π× 200 kHz and a spherical Coulomb crystal with more than a thousand ions, the distribution of the quadrupole shift is symmetric and has a dispersion below 0.1 Hz. The quadrupole shift scales as ℓ^-3, or as ω_z^2. Because the quadrupole moment of the ^3D_1 and ^3D_2 states are similar, we estimate that for ω_z=2π× 1 MHz, the dispersion of the quadrupole shifts does not exceed a few Hz, which is much less than the optimal repumping rate. Finally, let us compare the Zeeman splitting with the tensor Stark shift. One can easily calculate the Zeeman shifts between the upper lasing state |F_u,m_F=0⟩ and the nearby Zeeman state |F_u,m_F=1⟩ at the respective “magic” value of the magnetic field B_z: they are 211, 258, and 377 kHz respectively. At the same time, at ω_z=2 π× 1 MHz and N=10^5, the tensor Stark shift will be only about 1 kHz on the edge of the Coulomb crystal. Therefore, the tensor Stark shift is small in comparison with the Zeeman shift, and the theory in ref. <cit.> may be applied. § DISCUSSION AND CONCLUSION In the present paper, we studied the possibility to create a bad-cavity laser on forbidden transitions in cold ions trapped in a linear Paul trap and forming a large Coulomb crystal. We considered the particular case of a spherical Coulomb crystal of ^176Lu^+ ions, where the |^3D_2,F_u, m_F=0⟩→ |^1S_2, F_l=I, m_F=1⟩ transition is coupled to the circularly polarized mode of the high-finesse (=10^5) optical cavity, whose axis coincides with the trap axis and with the direction of external magnetic field. We showed that 10^5 ions in the trap with ω_z=2π× 1 MHz may provide about 0.5 picowatts of output power with a 150 s^-1 repumping rate if the mode waistis about 80% of the crystal radius, i.e., ∼ 100 μm.To increase this power, one could increase the number of ions, increase the frequency ω_z of the radial confinement, or use an elongated trap. Here we consider the main advantages and disadvantages of these measures in some detail.First, the ion crystal radius R_c scales as R_c ∝ N^1/3. To keep the value ζ=R_c/ near the optimum (about 0.8), we have to increase the cavity waist ∝ N^1/3, so the coupling coefficient g ∝ N^-1/3. The total output power P ∝ N^2 g^2 <cit.>, which gives the scaling law P ∝ N^4/3. Also, in larger ion ensembles, the maximum micromotion-related shifts grow with N, particularly, Δ_max∝ N^2/3 at Ω=Ω_0 (<ref>). At the same time, a decrease of g may lead to a decrease of the linewidth δω, as shown in (<ref>). We should note, however, that both controlling a large number of ions and fabricating a high-finesse resonator with a large mode waist may be technically challenging.Second, increasing the confinement frequency ω_z will lead to a scaling R_c∝ω_z^-2/3, which allows one to increase the coupling coefficient g ∝ω_z^2/3, keeping the same ζ, so that the output power scales as P ∝ω_z^4/3. At the same time, the maximal micromotion-induced shift Δ_max at Ω=Ω_0 will scale as Δ_max∝ω_z^8/3, as shown in (<ref>). Using the semiclassical model with equal coupling g for all the atoms, but with non-zero frequency shifts (<ref>), we found that the maximum output power might be attained at ω_z ∼ 2 π× 10-20 MHz (for different transitions) if the other parameters are the same as considered in Section <ref>. However, such a large value of ω_z is much higher than typical ion trap axial frequencies <cit.>. Moreover, the parameter ϵ is no longer a small parameter at such large ω_z, and the theory presented in Sections <ref> and <ref> is not valid. Finally, the increasing g may lead to a drastic increase of the linewidth Δω.Third, the Coulomb crystal in the elongated trap will be less regular than in the spherical one, and the quadrupole shifts may play a more significant, non-negligible role. On the other hand, such a method allows one to pack more ions into the same cavity mode. Such a setup should be designed carefully.In the present paper, we neglect the excitation of the second circularly polarized mode in the cavity. If such a mode will be excited, it will lead to a reduction of the output power. The picture will become more complex because of interactions of these modes via the coherence between the lower lasing states. A detailed investigation of their interaction will be presented in future work. Here we can note that both the output fields will have different polarizations and frequencies, and the frequency difference will be of the order of a few hundreds of Hz (for “magic” magnetic fields). The beat signal between two modes may be used for a stabilization of the magnetic field near its “magic” value.In addition to ^176Lu^+, some other ions with metastable states and negative differential polarizabilities may also be considered as candidates, although it seems to be less straightforward to find a proper repumping scheme. Instead, one may implement a “passive” scheme with cavity-enhanced non-linear spectroscopy, similar to the one proposed in <cit.>. Such a scheme may also be used for locking the frequency of some slave laser to the optical transition, and this approach does not require pumping of the atoms into the upper lasing state. We considered the “collinear” configuration, where trap axis, cavity axis and external magnetic field are co-aligned. In such a configuration, the cavity mode will be coupled only with Δ m = ± 1 quadrupole transitions. To allow the coupling with Δ m = 0 and/or Δ m = ± 2 transitions, some non-zero angle between the cavity axis and the magnetic field should be introduced. It may be attained, for example, by tilting the cavity axis with respect to magnetic field, co-aligned with the trap axis, or by tilting the magnetic field with respect to the cavity axis coinciding with the trap axis.In the first case (tilted cavity) the broadening related to the tensor Stark shift (the third term in eq. (<ref>)) will be suppressed, as well as in the collinear configuration considered in the main text. However, such a scheme does not allow the use of an elongated trap geometry, and will cause additional problems connected with confinement of the atoms to the Lamb-Dicke regime along the cavity axis. Particularly, our estimations shows that for the parameters of the ions considered in the paper, the amplitude R_2,iℓ of the micromotion on the edges of the crystal exceeds the wavelength of the mode. Note that this micromotion-related issue may be of less importance for some long-wavelength transitions, such as ^2D_3/2→^2S_1/2 and ^2D_5/2→^2S_1/2 transitions in Ba^+ ions. Tilting the magnetic field instead of the cavity axis allows the use elongated traps and to keep the ions in the Lamb-Dicke regime, but requires special measures to suppress the tensor Stark shift. As it has been shown in <cit.>, compensation of this shift may be performed with a special choice of the magnetic field.In summary, we have shown that a bad-cavity laser may be realized on a Coulomb crystal composed of ions with negative differential polarizability of the clock transition. As an example, we considered the ^3D_2→ ^1S_0 lasing transition in ^176Lu^+, with the ions forming a spherical Coulomb crystal in a linear Paul trap. Such a crystal may provide a route to truly steady-state lasing in the bad cavity regime if the proper continuous cooling and pumping is performed. § ACKNOWLEDGMENTS We are grateful to Murray Douglas Barrett and John Bollinger for fruitful discussion and valuable comments, and Athreya Shankar for useful remarks. The study has been supported by the EU–FET-Open project 664732 NuClock. § SEMICLASSICAL EQUATIONS FOR THREE-LEVEL MODEL AND ADIABATIC ELIMINATION OF THE INTERMEDIATE STATE.Consider the system of N 3-level trapped atoms (ions) with states |l⟩, |u⟩ and |i⟩ , whose |u⟩→ |l⟩ transition is coupled with the cavity mode , and |l⟩→ |i⟩ transition is pumped by external field with Rabi frequency V, see Fig. <ref>. Neglecting the dipole-dipole interaction between different atoms and their collective coupling to the bath modes (the role of these effects have been considered in Refs. <cit.>), we can write the master equation for such a system in the formdρ/dt=-i/ħ[,]+ _c[] +∑_j _j[], where the Lindbladian _c desctribes the relaxation of the cavity field: _c[] =-κ/2[ ^++ ^+-2 ^+ ], the Lindbladians _j desribes the relaxations of individual atoms _j[]=γ/2[2 ^j_lu^j_ul-^j_uu - ^j_uu] +Γ_1/2[2 ^j_li^j_il-^j_ii - ^j_ii] +Γ_2/2[2 ^j_ui^j_iu-^j_ii - ^j_ii], and the Hamiltonian in the respective rotating frame has the form=ħδ^++ ħ∑_j( (Δ_j+δ) _uu^j + Δ^p_j _ii^j) +ħ/2∑_j(^+_lu^j + _ul^j )+ħ V/2∑_j(_il^j + _li^j). Here and below we use the notation ^j_rq = |r^j⟩⟨ q^j|, whereis the coupling strength between the lth lower and uth upper lasing states and the cavity field, V is Rabi frequency of the pumping field.In the semiclassical (mean-field) approximation, where all the correlators are decoupled, the set of equations for atomic and field expectation values is:d/dt =-i[ κ/2 + iδ]-i/2∑_j , d/dt =-i/2[- ]-iV/2[-]+ γ + Γ_1 ,d/dt = ig/2[- ] - γ + Γ_2 ,d/dt = iV/2[-] - Γ d/dt = -(γ/2 + i (Δ_j+δ) ) - i/2[-]+ iV/2,d/dt = -( Γ/2 + i Δ_j^p )+i/2 - iV/2[ - ],d/dt =-(γ+Γ/2 +i(Δ_j^p-Δ_j-δ) ) +i/2 -iV/2,plus respective equations for ,,. Here Γ=Γ_1+Γ_2 is the total decay rate of the intermediate state |i⟩.Supposing that Γ≫ (V, Δ_j^p) ≫ (, Δ_j, δ, γ), we can adiabatically eliminate the intermediate level |i⟩ usingEqs. (<ref>) – (<ref>) and corresponding conjugated equations. It gives=^* =-iV/Γ+2i Δ_j^p, =V^2/Γ^2+4Δ_j^p ^2, = ^*=-i V/Γ+2iΔ_j^p .Substituting (<ref>) – (<ref>) into (<ref>) – (<ref>), we obtain: d /dt =i[ - ]+w [1-] - γ[1+],d /dt =i/2- (γ+w/2+γ_R+i (Δ_j + δ + Δ_j^LS)).Here =-,w=Γ_2 V^2/Γ^2+4Δ^p_j^2 andΔ_j^LS = w (ξ+1)/ΓΔ_j^p are the incoherent pumping rate and the light shift,γ_R=ξ w/2 is the rate of incoherent dephasing caused by the repumping, and ξ=Γ_1/Γ_2.Let us discuss briefly the light shifts Δ_j^LS. First, they are proportional to individual detunings Δ_j^p of the |l⟩→ |i⟩ pumping transition in the jth ion from the pumping field. These detunings depends on the micromotion-related second-order Doppler and Stark shifts. If the pumping field is tuned into resonance with the pumping transition of the ion in the center of the trap, they occurs to be proportional to X_j,0^2+Y_j,0^2, as well as the micromotion-related shifts of the lasing transition(<ref>), which may be compensated with the help of the fine tuning of the radio frequency Ω near its “magic” value Ω_0, as mentioned in the end of Section <ref>. Third, light shifts are suppressed, in comparison with the detunings, by a factor of (ξ+1)w/Γ. For example, in the scheme with ^176Lu^+ ion considered in the paper, Γ=2.8 × 10^7 s^-1 and ξ=0.6, which for w ≈ 100 s^-1 gives (ξ+1)w/Γ≈ 6× 10^-6. In the present paper, we neglect this shift.§ COUPLING OF ELECTRIC QUADRUPOLE TRANSITION WITH THE CAVITY FIELD Here we suppose that the cavity mode is a Gaussian standing wave. For the sake of simplicity, we will neglect the beam divergence and the Gouy phase; this simplification is valid, because we only need to calculate the coupling of the cavity mode with ions localized near the cavity waist. Then the electric field of the cavity mode is ()̊= ω/c√(8 πħ c^2/V_ eff ω)e^ - ^2/^2 sin(·̨)̊[ + ^+ ], whereis the (complex) polarization unit vector, ω is the mode frequency, $̨ is the wave vector,is the cavity waist,=π^2 Lis the effective mode volume,Lis the cavity length,is the (time-dependent) field operator, andr_⊥=|-̊(̨·̨)̊/k^2|is the projection of$̊ on the plane orthogonal to $̨. The origin is on the axis of the cavity. If≫1/k, then the interaction of this electric field with quadrupole momentumof some ion localized in the position$̊ may be approximately written as _̋int= 1/6∂Ê_α/∂ x_β_αβ≈e_α k_β/6_αβ[ +^+ ] ×√(8 πħω/V_ eff) exp[ - ^2/^2]cos(·̨)̊, where the summations over twice appearing Cartesian indices α, β, ... are implied here and below, for the sake of brevity. Cartesian components of the quadrupole momentum operator are _αβ = ∫(3 x_α x_β - ^̊2 δ_αβ) ρ̂()̊ d^3 x, where ρ̂()̊ is the operator of the charge density. Let us consider some lasing transition between the upper and lower lasing states |u⟩ and |l⟩. Supposing that the ion is placed into the origin, we express the absolute value of the coupling strength g()̊ of the ion situated in $̊ as g()̊=2/ħ| ⟨ l, 1| _̋int|u, 0 ⟩| = g_0e^- ^2/^2 cos(·̨)̊,where g_0=√(8 πω/ħ )|e_α k_β/3⟨ l |_αβ|u⟩|.is the coupling coefficient for the ion placed on the cavity axis in the antinode of the mode.The quadrupole momentumis the symmetric traceless 2nd rank tensor, and its Cartesian components may be expressed via the spherical components_2q = √(4 π/5)∫ r^2ρ̂()̊Y_2q(/r) d^3 xas _xx= √(3/2)(_22+_2-2)-_20; _yy= -√(3/2)(_22+_2-2)-_20; _zz= 2 _20; _xy= -i √(3/2)(_22-_2-2); _zx= √(3/2)(_21-_2-1); _zy= -i √(3/2)(_21+_2-1). Now we should express the matrix elements of_2qvia the rateγof spontaneous transition. The states|a⟩=|η_a J_a I F_a m_a⟩(a=uorl) are characterized by principal quantum numbersη_a, the electronic shell angular momentaJ_a, the nuclear angular momentumI, the total angular momentaF_aand its projectionsm_a. Then, according to the well known expression for the electric multipole spontaneous transition rate <cit.> we can write γ = ω^5/15 ħ c^5∑_F_l,m_l|⟨η_u J_u I F_u m_u|_2q|η_l J_l I F_l m_l ⟩|^2.Using the Wigner-Eckart theorem <cit.>, we can express the matrix element as ⟨η_u J_u I F_u m_u|_2q|η_l J_l I F_l m_l ⟩ =(-1)^F_l+J_u+I-2√(2 F_l+1)C_F_lm_l 2 q^F_u m_u ×{[ J_l I F_l; F_u 2 J_u ]}⟨η_u J_u ||_2||η_l J_l ⟩where⟨η_u J_u ||_2||η_l J_l ⟩is a reduced matrix element. Using the properties of the Clebsch-Gordan coefficients and6J-symbols <cit.>∑_m_l,q(C_F_l m_l 2q^F_um_u)^2=1, ∑_F_l (2 F_l +1) {[ J_l I F_l; F_u 2 J_u ]}^2 = 1/2 J_u+1,we obtain⟨η_u J_u ||_2||η_l J_l ⟩^2= 15 ħ c^5 γ (2 J_u+1)/ω^5,what gives ⟨η_u J_u I F_u m_u|_2q|η_l J_l I F_l m_l ⟩^2 =(2 F_l+1)(2 J_u+1) (C_F_lm_l 2 q^F_u m_u)^2×{[ J_l I F_l; F_u 2 J_u ]}^215 ħ c^5 γ/ω^5. Let us calculate the coupling strengthsgfor circularly polarized cavity mode in configuration shown on the Figure <ref>, i.e., when=(i _y ±_x)/√(2)and=̨_z ω/c. Then e_α k_β_αβ= ω/c √(2)(i _yz±_xz)=ω√(3)/ c_2 ± 1.Substituting (<ref>) into (<ref>), we obtain g_0 = ω/c √(3)√(8 πω/ħ )⟨ l |_2± 1|u⟩.One can see that transitions withm_u-m_l=±1can be coupled with the cavity mode in such a configuration.Let us suppose, for the sake of definiteness, thatm_u=m_l. Then g_0^2 = 40 π c^3 γ/ ω^2 (2 F_l+1) (2 J_u+1)×{[ J_l I F_l; F_u 2 J_u ]}^2 (C_F_lm_l 2 ± 1^F_u m_u)^2 =5 π c^3 γ/ ω^2Θ_ul^2,whereΘ_ul^2is given by (<ref>).§ HYPERFINE STRUCTURE OF ^176LU^+^3D_2 STATE AND SECOND-ORDER ZEEMAN SHIFT The hyperfine structure of low-lying levels of^175Lu^+ion (I=7/2) has been measured in <cit.>. Particularly, it was found that the energies of the hyperfine sublevels of^3D_2state grows asFincreases from 3/2 to 11/2. The distances between adjacent levels are 0.139, 0.210, 0.288 and 0.382cm^-1. Using the standard expression for the hyperfine energy levels E_ hfs(F)/ħ =A_ hfsK/2+B_ hfs3/2K (K+1)-2 I (I+1) J (J+1)/4 I (2 I-1) J (2 J-1)whereK = F (F+1)-J(J+1)-I(I+1),we can fit the hyperfine constants as:A_hfs,175=2π×1935MHz,B_hfs,175=2π×1388MHz.To estimate the hyperfine constantsA_hfs,176andB_hfs,176for^3D_2state of^176Lu^+ion, we can use the fact thatB_hfsis proportional to the nuclear quadrupole moment, andA_hfsis proportional to the nuclear g-factorg_I=-μ/(Iμ_B), whereμis the nuclear magnetic moment,μ_Bis the Bohr magneton. According <cit.>, nuclear quadrupole moments of^175Lu^+and^176Lu^+are 3415 and 4818 mbarn respectively, which givesB_hfs,176=2 π×1963MHz. In turn, the magnetic moments areμ_175 = 2.2327μ_Nandμ_176 = 3.162μ_Naccording <cit.>, which givesA_hfs,176=2 π×1370MHz.In the presence of external magnetic field, the hyperfine energy levels experience a Zeeman splitting. Magnitudes of the Zeeman shift may be found from the diagonalization of the hyperfine-Zeeman Hamiltonian Ĥ/ħ=μ_B(g_I + g_J )· + A_ hfs·+ 3 B_ hfs/4 J (2 J -1) I (2 I -1) ×[ 2 (·)^2+·-2/3^2 ^2 ],where g_J=g_L J(J+1)-S(S+1)+L(L+1)/2 J (J+1)+g_S J(J+1)+S(S+1)-L(L+1)/2 J (J+1)is the electronic Lande g-factor. Takingg_L=1,g_S=2.002319043617,L=2,S=1andJ=2, we findg_J≈1.16705. Also,g_I≈-0.000246for^176Lu^+.In weak magnetic fieldthe Zeeman shiftsΔ_Z|F,m_F=0⟩(B)of the states|F,m_F=0⟩are quadratic in||. We find Δ_Z|5,0⟩(B)/2 π ||^2=-440.0 Hz/G^2, Δ_Z|6,0⟩(B)/2 π ||^2=-15.19 Hz/G^2 Δ_Z|7,0⟩(B)/2 π ||^2=127.3 Hz/G^2, Δ_Z|8,0⟩(B)/2 π ||^2= 166.3 Hz/G^2 Δ_Z|9,0⟩(B)/2 π ||^2=165.5 Hz/G^2. In turn, Zeeman shiftΔ_Z,0of the state|^1S_0, m=1⟩(playing the role of the lower lasing state in the scheme considered in the main text) is determined by the nuclear gyromagnetic ratio and is linear in the magnetic field:Δ_Z,0/B_z=-2 π×344.3 Hz/G. 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"authors": [
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"Justin Bohnet",
"Thorsten Schumm"
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"published": "20170426193748",
"title": "Prospects for a bad cavity laser using a large ion crystal"
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-0.5in 0.9in 0.3in 8.3in 0.3in -0.1in -0.1in 6.6in | http://arxiv.org/abs/1704.08161v3 | {
"authors": [
"Adam B. Barrett"
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"primary_category": "q-fin.EC",
"published": "20170426152713",
"title": "Stability of zero-growth economics analysed with a Minskyan model"
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State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences & Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, [email protected] Experimental Two-dimensional Quantum Walk on a Photonic Chip Hao Tang^1,2, Xiao-Feng Lin^1,2, Zhen Feng^1,2, Jing-Yuan Chen^1, Jun Gao^1,2, Ke Sun^1, Chao-Yue Wang^1, Peng-Cheng Lai^1, Xiao-Yun Xu^1,2, Yao Wang^1,2, Lu-Feng Qiao^1,2, Ai-Lin Yang^1,2 and Xian-Min Jin^* December 30, 2023 ===================================================================================================================================================================================================================Quantum walks, in virtue of the coherent superposition and quantum interference, possess exponential superiority over its classical counterpart in applications of quantum searching and quantum simulation. The quantum enhanced power is highly related to the state space of quantum walks, which can be expanded by enlarging the photon number and/or the dimensions of the evolution network, but the former is considerably challenging due to probabilistic generation of single photons and multiplicative loss. Here we demonstrate a two-dimensional continuous-time quantum walk by using the external geometry of photonic waveguide arrays, rather than the inner degree of freedoms of photons. Using femtosecond laser direct writing, we construct a large-scale three-dimensional structure which forms a two-dimensional lattice with up to 49×49 nodes on a photonic chip. We demonstrate spatial two-dimensional quantum walks using heralded single photons and single-photon-level imaging. We analyze the quantum transport properties via observing the ballistic evolution pattern and the variance profile, which agree well with simulation results. We further reveal the transient nature that is the unique feature for quantum walks of beyond one dimension. An architecture that allows a walk to freely evolve in all directions and a large scale, combining with defect and disorder control, may bring up powerful and versatile quantum walk machines for classically intractable problems.§ INTRODUCTIONQuantum walks (QWs), the quantum analogue of classical random walks <cit.>, demonstrate remarkably different behaviours comparing to classical random walks, due to the superposition of the quantum walker in its paths. This very distinct feature leads the quantum walks to be a stunningly powerful approach to quantum information algorithms<cit.>, and quantum simulation for various systems <cit.>. For instance, theoretical research has revealed that QWs propagating in one dimension (1D) possess superior transport properties to 1D classical random walks<cit.>, and the coherence in QWs is crucial in simulating energy transport in the photosynthetic process<cit.>. The potential of applying QWs in machine learning algorithms such as artificial neural network <cit.> also draw wide attention from multidisciplinary researchers. Inspired by the prospects of QWs, many endeavours have been made to realize QWs in different physics systems, including nuclear magnetic resonance<cit.>, trapped neutral atoms<cit.>, trapped ions<cit.>, and photonic systems<cit.>.However, these experimental implementations reveal a very evident limitation, that the realized quantum walk is normally of only one dimension, andthe evolving scale of QWs remains very small. Simple demonstration of 1D QW could not suffice the ever growing demand for further speed-up of certain quantum algorithms, or the simulation of quantum systems of a much higher complexity<cit.>. In the spatial search algorithm, a quantum walk outperforms its classical counterparts only when the dimension is higher than one<cit.>; In the simulation of graphene, photosynthesis, or neural network systems, these complex networks always intuitively have high dimensions. Experimental research on quantum walks of beyond 1D becomes indispensable, and a few attempts having covered 2D QWs in experiments are worth noted. A discrete-time 2D quantum walk was achieved in the fibre network system by dynamically controlling the time interval of two walkers<cit.>, in the so-called delayed-choice scheme <cit.>, or using two walkers sharing coins <cit.>. They ingeniously use either time-polarization dimension or the analog from two walkers acting on 1D graph to represent one walker on a 2D lattice, and the 2D lattice does not physically occur.A quasi-2D continuous-time quantum walk was explored in the waveguide coupled in a `Swiss cross' arrangement<cit.>, but this is not, strictly speaking, a 2D quantum walk, because photons could not freely propagate in the diagonal and many other directions as they suppose to do in the 2D array. In this paper, we for the first time experimentally observe the evolution of 2D continuous-time quantum walks with single photons on the 2D waveguide array. We set up the heralded single-photon source and measure the evolution results that agree well with theoretical simulation using an ultra-low-noise single-photon-level imaging technique. We further analyze the transport and recurrent properties, measured from the variance and the probability from the initial waveguide, respectively. We experimentally verify the unique features for two-dimensional quantum walks that differ from both classical random walks and quantum walks of one-dimension. § MAINPhotons propagating through the coupled waveguide arrays can be described by the Hamiltonian:H=∑_i^N β_i a_i^† a_i + ∑_i ≠ j^N C_i,j a_i^† a_j(1)where β_i is propagating constant in waveguide i, C_i,j is the coupling strength between waveguide i and j. For a uniform array, all β_i is regarded equal to β, and C_i,j that mainly depends on waveguide spacing can be obtained via a coupled mode approach <cit.>. In our implementation, we fabricate two-dimensional waveguide arrays using femtosecond laser writing techniques <cit.> (Fig.1.a). The waveguides are written in different depths of the borosilicate glass to form a two-dimensional array<cit.> from the cross-section view (Fig.1.b). The centre-to-centre spacing between two nearest waveguides is set as a spacing unit that is 15 μm in the vertical direction (Δ d_V) and 13.5 μm in the horizontal direction (Δ d_H). In such a two-dimensional array, each waveguide is involved into comprehensive coupling with surrounded waveguides, e.g., Waveguide O has different waveguide spacings to Waveguide P, Q, M and N as marked in Fig.1.c, namely, Δ d_V, Δ d_H, √(Δ d_H^2+Δ d_V^2) and √((2Δ d_H)^2+Δ d_V^2) for Δ d_PO, Δ d_QO, Δ d_MO and Δ d_NO respectively.Such differences in waveguide spacings and waveguide-pair orientations affect the coupling coefficient significantly, as shown in Fig.1.d. Through the measured value of C following the standard method <cit.>, we observe the exponential decay as waveguide spacing increases and some discrepancy of C in different directions. We hence select Δ d_H and Δ d_V to ensure uniform coupling coefficients for nearest waveguide pairs in the horizontal and vertical directions. For other waveguide pairs in inclined directions, such as Pair M-O and Pair N-O in Fig.1.c, as the directional discrepancy of C gets smaller when waveguide spacing increases, we use the average of the horizontal and vertical value at the corresponding spacing for their coupling coefficient. For a quantum walk that evolves along the waveguide, the propagation length z is proportional to the propagation time by z = ct, where c is the speed of light in the waveguide, and hence all terms that are a function of t would use z instead in this paper for simplicity. The wavefunction that evolves from an initial wavefunction satisfies: |Ψ(z)⟩=e^-iHz|Ψ(0)⟩(2)where |Ψ(z)⟩=∑_ja_j(z)|j⟩, and |a_j(z)|^2=|⟨j|Ψ(z)||⟩^2=P_j(z) respectively. |a_j(z)|^2 and P_j(z) is the probability of the walker<cit.> being found at waveguide j at the propagation length z. As is shown in Fig.1.e, we observe the dynamics by injecting a vertically polarized heralded single photon source (810 nm) into the central waveguide<cit.> and measuring the evolution patterns using an ICCD camera. More details about our single-photon source and the ultra-low-noise single-photon-level imaging can be found in the Method section. These two-dimensional patterns of different propagation lengths from both experimental evolution of heralded single photons and theoretical simulations are then collected (Fig.2). Clearly, the intensity peaks always emerge at the diagonal positions, and they move further in these directions when the propagation length z increases. The similarity between two probability distributions Γ_i,j and Γ_i,j' can be defined by<cit.>: S=(∑_i,j(Γ_i,jΓ_i,j')^1/2)^2/∑_i,jΓ_i,j∑_i,jΓ_i,j'. For the five pairs in Fig. 2, the similarities are calculated as 0.961(a & f), 0.957 (b & g), 0.920 (c & h), 0.917 (d & i) and 0.913 (e & j), respectively. Therefore, there is a good match between experimental evolution patterns and the theoretical results of two-dimensional quantum walks. §.§ The transport properties of quantum walks We know quantum walks have unique transport properties, which could be examined from the variance against the propagation length, as defined in Eq.(3): σ(z)^2=∑_i=1^NΔ l_i^2P_i(z)/∑_i=1^NP_i(z)(3)where Δ l_i is the normalized spacing between waveguide i and the central waveguide where the single photons are injected into. Plotting the variance-propagation length relationship with double-logarithmic axes, the ballistic 1D quantum walk is known for yielding a straight line with slope 2, while the diffusive 1D classical random walk results in a straight line with slope 1, i.e., QW transports quadratically faster than the classical random walk <cit.>.The variance for both one-dimensional quantum walks in theory, and two-dimensional quantum walks in theory and in experiments are presented in Fig.3.a. All quantum walks have the same coupling coefficient for waveguide pairs of the nearest spacing, and the walks evolve in a lattice large enough to ignore boundary effects. For two-dimensional quantum walk, the experimental results agree well with the theoretical ones. The variance from one-dimensional quantum walk in theory goes all the way below the two-dimensional case, as a walker can move in more directions in the latter. However, the variance for all these quantum walks follows the trend of slope 2 rather than slope 1, suggesting the universal ballistic spreading for both one and two-dimensional quantum walks, which distinguishes them from diffusive classical random walks. Projecting the evolution patterns of a 2D quantum walk and a 2D classical random walk onto x axis and y axis (Fig 3.b and c), the random walk in a two-dimensional Gaussian distribution <cit.> has the projection profiles of a 1D Gaussian distribution, while the projection profiles for the quantum case show a ballistic shape similar to the 1D quantum walks. It indicates that the intensity peaks in random walks always remain in the centre, but those in quantum walks always move to all frontiers, causing a larger variance for the latter. §.§ The recurrent properties of quantum walks We further investigate the difference between quantum walks of different dimensions, which can be gauged by P_0(z) and Pólya number, two indices that concern the recurrent properties of a walker in a network<cit.>.P_0(z), the probability of a walker being found at the initial waveguide after a propagation length z, is plotted in Fig.4.a. All quantum walks have a decreasing P_0(z) as z increases, but follow different asymptotic lines. A walker in a 2D lattice evolves away from the original site much faster through many additional paths and is less likely to move back (with a smaller oscillation) comparing to the 1D scenario.A system can be judged to be recurrent or transient depending on the Pólya number, through the definition<cit.>: P=1-∏_m=1^∞[1-P_0(z_m)](4) where z_m is a set of propagation lengths sampled periodically <cit.>. When the Pólya number is 1, a system is recurrent, because P_0(z_m) can always be a large value to make ∏_i=1^∞[1-P_0(z_m)] close to zero, while for a transient system, P_0(z_m) quickly drops to a very marginal value so the Pólya number would be smaller than one <cit.>.Two-dimensional QWs in experiment and in theory, and one-dimensional QWs in theory show a Pólya number approaching 0.887, 0.912 and 0.998, respectively (Fig.4.b). Clearly, the 2D QW is much less inclined to be recurrent than 1D QW. Further interpretation<cit.> comes from the asymptotic features z^-d. It has been pointed out that transient systems tend to have a value of d larger than 1, while d for recurrent systems would be equal to or go below 1. From Fig.4.a, the 2D quantum walks in experiment and in theory both follow an asymptotic line z^-2, revealing the transient nature for these 2D continuous-time quantum walks in our implementation. We for the first time measure the transient nature of a 2D quantum walk in experiment, which makes it different from all experimentally realized quantum walks that were either in 1D or in 2D with limited scales. § DISCUSSION Here, we have demonstrated strong capacity in achieving large-scale three-dimensional photonic chips and ultra-low-noise single-photon-level imaging techniques that are crucial for the implementation and measurement of our spatial two-dimensional continuous-time quantum walks. The first and large-scale realization of real spatial 2D quantum walk may not only be fundamentally interesting but also provide a powerful platform for quantum simulation and quantum computing. Since we increase the dimensions by the evolution network geometry, even with single walker, photon evolution on lattices up to 49×49 nodes may lead to a huge state space being large enough to explore new physics in entirely new regimes. Quantum advantage/supremacy may also be explored in such platform using analog quantum computing protocols, such as 2D Boson sampling<cit.>, fast hitting<cit.> and even universal quantum computing protocols<cit.>, instead of using circuit-model protocols of universal quantum computing.The spatial structure itself can also be freely fabricated with special geometric arrangement, defect, disorder, topological structure in a programmable way, which may offer a new approach of Hamiltonian engineering to enable designing and building quantum simulators on demand on a photonic chip. Such a Hamiltonian engineering can be realized by adding waveguide curvature, variation of the fabrication power or dynamic waveguide spacings, etc. Through these we could potentially extend the issue of localization in quantum walks to higher dimensions<cit.>, as well as exploring topological photonics and the simulation of quantum open systems in photonic lattices<cit.>. Further, we would go beyond two dimensions through various ways. Quantum simulations in (2+1) dimensions are possible and their dynamic properties can be explored if we introduce time-varying Hamiltonian along the propagating axis. For issues such as quantum walks in bosonic and fermionic behaviours <cit.>, multi-particle entanglement and evolution, etc., the multi-photon source interfaced to the robust and precise photonic chips could give the research of high-dimension quantum systems an instant boost, and demonstrate its strong potential for quantum simulation in a highly complex regime.§ METHODSPhotonic lattice preparation: Waveguide arrays were prepared by steering a femtosecond laser (10W, 1026 nm, 290 fs pulse duration, 1 MHz repetition rate and 513 nm working frequency) into an spatial light modulator (SLM) to create burst trains onto a borosilicate substrate with a 50X objective lens (numerical aperture: 0.55) at a constant velocity of 10 mm/s. Power and SLM compensation were processed to ensure the waveguides to be uniform and depth independent<cit.>. The borosilicate glass wafer is of a size 1×20×20 mm, and consists of 20 set of lattices of different evolution lengths from 0.31 mm to 9.81 mm. Each lattices has 49×49 waveguides in a size of 0.72 mm×0.648 mm in the cross section view.Single-photon source and imaging: A 405nm diode laser pumped a PPKTP crystal to generate pairs of 810nm via type II spontaneous parametric downconversion. The resulted single-channel count rate and two-channel coincidence count rate reach 510000 and 120000, respectively. The generated photon pairs then pass a 810nm band-pass filter and a polarized beam splitter to be divided to two purified components of horizontal and vertical polarization. The vertically polarized photon was coupled into a single-mode optical fiber and then injected into the photonic chips, while the horizontally polarized photon is connected to a single photon detector that sets a trigger for heralding the vertically polarized photons on ICCD camera with a time slot of 10 ns. If without such an external trigger, the measured patterns would come from light in thermal states rather than single-photon states. ICCD camera captures each evolution pattern from the photon output end of the photonic chip, after accumulating single photon injections in the `external' mode for around an hour. Simulation of light field evolution: Solving Eq.(2) requires a matrix exponential method and this yields the light evolving pattern that contains the probability matrix for all waveguides. The Pade approximation function <cit.> in Matlab is used in the simulation. 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M. titleInvisibility Cloak Printed on a Photonic Chip. journalSci. Rep. volume6, pages28527 (year2016).Moler1978 authorMoler, C., & Van Loan, C. titleNineteen dubious ways to compute the exponential of a matrix. journalSIAM Review volume20, pages801-836 (year1978).Acknowledgements.The authors thank J.-W. Pan for helpful discussions. This research is supported by the National Key Research and Development Program of China (2017YFA0303700), National Natural Science Foundation of China (Grant No. 61734005, 11761141014, 11690033, 11374211), the Innovation Program of Shanghai Municipal Education Commission, Shanghai Science and Technology Development Funds, and the open fund from HPCL (No. 201511-01), X.-M.J. acknowledges support from the National Young 1000 Talents Plan. | http://arxiv.org/abs/1704.08242v2 | {
"authors": [
"Hao Tang",
"Xiao-Feng Lin",
"Zhen Feng",
"Jing-Yuan Chen",
"Jun Gao",
"Ke Sun",
"Chao-Yue Wang",
"Peng-Cheng Lai",
"Xiao-Yun Xu",
"Yao Wang",
"Lu-Feng Qiao",
"Ai-Lin Yang",
"Xian-Min Jin"
],
"categories": [
"quant-ph",
"cond-mat.dis-nn"
],
"primary_category": "quant-ph",
"published": "20170426175709",
"title": "Experimental Two-dimensional Quantum Walk on a Photonic Chip"
} |
Chains of type I radio bursts ... M. Karlický, [email protected] Astronomical Institute of the Czech Academy of Sciences, Fričova 258, CZ – 251 65 Ondřejov, Czech Republic Owing to similarities of chains of type I radio bursts and drifting pulsation structures a question arisesif both these radio bursts are generated by similar processes. Characteristics and parameters of both these radio bursts are compared. We present examples of the both types of bursts and showtheir similarities and differences.Then for chains of type I bursts a similar model as for drifting pulsationstructures (DPSs) is proposed.We show that similarly as in the DPS model, the chains of type I bursts can be generated by the fragmented magnetic reconnectionassociated with plasmoids interactions. To support this new model of chains of type I bursts,we present an effect of mergingof two plasmoids to one larger plasmoid on the radio spectrum of DPS. This process can also explain the "wavy" appearance of somechains of type I bursts. Then we show that the chains of type Ibursts with the "wavy" appearance can be used for estimation of the magnetic field strength in their sources. We think thatdifferences of chains of type I bursts and DPSs are mainly owing to different regimesof the magnetic field reconnection.While in the case of chains of type I bursts the magnetic reconnection and plasmoidinteractions are in the quasi-separatrix layer of the active region in more or less quasi-saturated regime,in the case of DPSs, observed in the impulsive phase of eruptive flares,the magnetic reconnection and plasmoids interactions are in the current sheet formed under the flare magneticrope, which moves upwards and forces this magnetic reconnection.Are chains of type I radio bursts generated by similar processes as drifting pulsation structures observed during solar flares? M. Karlický Received ; accepted=============================================================================================================================== § INTRODUCTION Solar radio bursts are generally divided into five groups designated as type I, II, III, IV and V <cit.>. Most of them show fine structures, especially type IV bursts: fibers, zebra patterns, spikes and pulsations <cit.>.In the present paper we study two types of bursts: type I bursts <cit.> and special type of the pulsations called drifting pulsation structures (DPSs) <cit.>.Type I radio bursts (noise storms) are quite common phenomenon observed in metric wavelength range up to about 400 MHz. They last from a few hours to several days and they are associated with the active region passage over solar disc <cit.>. They appear in clouds of short duration and narrow bandwidth bursts (0.1 - 3 s; several MHz), superimposed on a broadband continuum. The polarization of type I bursts is in most cases the same as the background continuum polarization. While the polarization up to 100% is observed for the noise storms located close to the solar disc center, at the solar limb their polarization is lower <cit.>. The type I bursts are structured, they form chains of type I bursts (in short type I chains) <cit.>. These chains preferentially drift towards lower frequencies and their drift is used for estimations of the magnetic field in their sources <cit.>. Sometimes the type I chains even oscillate in frequencies ("wavy" appearance on dynamic spectrum) <cit.>.Recently, <cit.> have found that noise storms consist of an extended halo and several compact cores which intensity is changing over a few seconds. Regions, where storms were originated, were much denser than the ambient corona and their vertical extent was smaller than estimated from hydrostatic equilibrium.Moreover, <cit.> have proposed that persistent magnetic reconnection along quasi-separatrix layers of the active region is responsible for the continuous metric noise storm.Several models of type I bursts were proposed, see the book of <cit.> and <cit.>. Among them the most promising models are those based on the plasma emission mechanism, e.g., the model by <cit.> connecting the type I chains with weakly super-Alfvenic shocks generated in the front of emerging magnetic flux.In solar flares, in the decimetric range, pulsations are quite common <cit.>. Among them a special type of the pulsations called now drifting pulsation structures (DPSs) have been recognized and interpreted <cit.>. They are relatively narrowband and drifting mostly towards lower frequencies. They are usually observed during the impulsive phase of eruptive flares in the 0.6 - 3 GHz frequency range in connection with the plasmoid ejection.Nice example of the plasmoid ejection, observed in soft X-rays during the 5 October 1992 event,is described in the paper by <cit.>. It is is shown that the plasmoid is a small part of the 3-D loop (Figures 2, 5a, and 10 in <cit.>). Further analysis showed that the plasmoid is a small part of the 3-D current-carrying loop which is embedded in the current sheet, where the flare magnetic reconnection takes place (see Figure 2 in <cit.>). The magnetic reconnection accelerates superthermal electrons <cit.> that are then trapped in a denser O-type magnetic structure, which thus becomes "visible" in the soft X-rays <cit.>, EUV <cit.> or at 17 GHz radio waves <cit.> as the plasmoid.A limited extent of the plasmoid along the loop (i.e. trapping of superthermal electrons along the loop) can be explained by a complexity of magnetic field lines at the plasmoid as shown in the 3-D kinetic simulation of the magnetic reconnection <cit.> or by the distribution of superthermal electrons with the high-pitch angles only. Further possibilities of this trapping are discussed in the paper by <cit.>.The plasmoid in 3-D has a cylindrical form, which in its 2D models (invariant in the third coordinate) corresponds to circular magnetic structure having the plasma density greater than that in the surrounding plasma, see <cit.> and the following Figure <ref>.The model of DPSs was developed in papers by <cit.>. In the 5 October 1992 event the frequency of DPS was found to be close to the plasma frequency derived from the plasmoid density, see Figure 1 in the paper by <cit.>. Therefore in the DPS model the plasma emission mechanism is considered. In the flare current sheet, during the magnetic reconnection plasmoids are generated due to the tearing mode instability. At X-points of the magnetic reconnection superthermal electrons are accelerated <cit.>. Then these electrons are trapped in a nearby plasmoid (O-type magnetic field structure), where they generate plasma waves that after their conversion to electromagnetic waves produce DPSs (see Figure 9 in <cit.>). The narrow bandwidth of DPSs is given by the limited interval of the plasma densities (plasma frequencies) inside the plasmoid. In the flare current sheet plasmoids preferentially move upwards in the solar atmosphere (due to a tension of the surrounding magnetic field lines <cit.>), i.e. in the direction, where the electron plasma density decreases, that is why DPSs mostly drift to low frequencies. The velocity of plasmoids is in the range from zero to the local Alfvén speed. The acceleration of electrons at X-points of the magnetic reconnection is quasi-periodic, which causes quasi-periodic pulsations of DPSs. The typical period of the pulses is about one second. In some cases these pulses have the frequency drift which is caused by propagation of the superthermal electrons inside the plasmoid.In the paper we compare chains of type I bursts and drifting pulsation structures observed in the impulsive phase of eruptive flares. Based on this comparison we propose a new model of the chains of type I bursts. The chains of type I bursts are considered to be radio signatures of processes that heat the solar corona. Therefore, a correct model of type I bursts can contribute to a solutionof the problem of the hot corona.The paper is structured as follows: In Section 2 we compare chains of type I bursts and DPSs and show their similarities and differences. Then in section 3, for chains of type I bursts we propose a model similar to that of DPSs and then we discuss processes which could explain their differences. Conclusions are in Section 4. § COMPARISON OF CHAINS OF TYPE I RADIO BURSTS AND DRIFTING PULSATING STRUCTURES Examples of type I drifting chains and drifting pulsation structures (DPSs) are shown in Figures <ref> and <ref>. Their typical parameters are summarized in Table <ref>. The parameters of the type I chains were taken combining the results presented by <cit.>. On the other hand, the parameters of DPSs are taken from papers by <cit.>. Remark: For DPSs there is only one polarization measurement (P ∼ 30 %), presented in the paper by <cit.>. The brightness temperature of DPS was calculated for the 5 October 1992 event <cit.> assuming that the DPS source size is equal to the plasmoid size. While for type I chains it is commonly believed that their polarization is consistent with the O-mode <cit.>, for DPS it is unclear. Periods of repetition of type I bursts in chains and pulsations in DPSs are similar, from fractions of second to several seconds.As shown in Table <ref> some parameters of type I chains and DPSs are comparable (duration and brightness temperature), other parameters differ. However, type I chains are observed in metric range and DPSs in decimetric range, i.e. in different altitudes of the solar atmosphere with different densities, different density gradients and magnetic field strengths. Moreover, while DPSs are usually observed during the impulsive phase of eruptive flares <cit.>, type I chains are a part of noise storms connected with the reconnection activity at the quasi-separatrix layer in active region <cit.>.As concerns an appearance of the chains of type I bursts and DPSs (Figures <ref> and <ref>) they look to be similar. The both types of bursts preferentially drift towards lower frequencies. In rare cases both reveal "wavy" appearance, see the chain of type I bursts observed in July 2, 2012 (Figure <ref>, the upper right part) and DPS observed in June 30, 2002 (Figure <ref>, the upper right part). § NEW MODEL OF TYPE I CHAINS Considering all observational parameters of the both burst types and differences in conditions in their generation we propose that the type I chain is produced by very similar processes as DPS; they differ only in plasma parameters and initial conditions.Therefore the model of the drifting chain of type I bursts can be explained as follows:During the magnetic reconnection in the current sheet formed in the quasi-separatrix layer of the active region the current-carrying loops are generated. In the magnetic reconnection between interacting current-carrying loops there are the X-magnetic points, where electrons are accelerated. These electrons penetrate into interacting loops at their interaction region. This region (plasmoid) is simultaneously rapidly heated. For the acceleration process and penetration of electrons to interacting current-carrying loops, see Figure 3 in the paper by <cit.>.As seen in Figures2, 5a, 10 in the paper by <cit.>, the observed plasmoid in the DPS case is a small part of the loop. We assume that a similar spatially limited plasmoid is also formed in the case of the drifting chain of type I bursts. The limited extent of the plasmoid means that the hot plasma (observed in soft X-rays in the DPS case) is trapped by some processes in this plasmoid also in direction along its axis. Some processes explaining this trapping were proposed in the paper by <cit.>. But based on the papers by <cit.> and <cit.> we think that this trapping is mainly due a complexity of magnetic field lines in the plasmoid. Considering the observational evidence about the hot plasma confinement in the plasmoid, we assume that also superthermal electrons (accelerated during the interaction of the current-carrying loops) are trapped in the plasmoid. (Note that in the perpendicular direction to the plasmoid axis the hot plasma as well as superthermal electrons are kept by the magnetic field in the plasmoid.)These trapped superthermal electrons generate in the plasmoid the Langmuir (electrostatic) waves, which are then transformed into the electromagnetic (radio) waves, observed as type I chains. These processes are shown in details in the paper by <cit.>. There, using a 2.5-D particle-in-cell model, we self-consistently described not only the interaction of plasmoids, but also how the electrostatic (Langmuir) and electromagnetic (radio) waves are generated. We found that the distribution of superthermal electrons, penetrating the plasmoid, is unstable for the Langmuir waves (due to the bump-on-tail instability) and then these electrostatic waves are transformed to the electromagnetic (radio) waves, see Figure 9 in this paper. In agreement with the models of type I bursts based on the plasma wave theories <cit.>, we assume that the emission of type I bursts is dominant on the fundamental frequency and the processes (e.g. the coalescence of two Langmuir waves) giving the emission on the harmonic frequency are not effective. The observed polarization of type I bursts is up to 100 percent. To reach such high polarization, in agreement with <cit.>, and <cit.>, we assume that the type I burst emission is in O-mode generated in the region where the emission frequency ω is greater than the plasma frequency ω_pe and smaller than the cutoff frequency for the X-mode ω_x. This O-mode (electromagnetic one) is generated from Langmuir waves by scattering on thermal ions and in such a case the frequency of the resulting emission is essentially equal to that of Langmuir waves (ω_l = ω_pe (1 + 3 v_te^2/v_ϕ^2)^1/2, where v_te is the thermal plasma velocity and v_ϕ is the phase velocity). Now using the commonly accepted assumption that in the type I burst source the electron gyro-frequency ω_ce is much smaller that the plasma frequency, we have ω_x ≈ ω_pe + 1/2 ω_ce. Then the above mention condition ω≅ω_l ≤ω_xgives a lower limit for the phase velocities of generated Langmuir waves, expressed as v_ϕ≥ v_te (3 ω_pe/ω_ce)^1/2. Because the Langmuir waves with the phase velocity v_ϕ are produced by electrons having velocities greater than v_ϕ, it also gives a lower limit for energies of energetic electrons.Due to a limited interval of densities inside the plasmoid, the trapped superthermal electrons generate Langmuir waves in the limited interval of plasma frequencies and thus the instantaneous bandwidth of type I chain is limited. Because the plasmoid expands or moves upwards in the solar atmosphere, plasma densities inside the plasmoid decrease, and therefore most of type I chains drift towards lower frequencies. Similarly as in DPSs, the acceleration of superthermal electrons is quasi-periodic which leads to quasi-periodic repetition of type I bursts (which form the type I chain) with the typical period of about one second. If accelerated electrons are trapped in several plasmoids simultaneously then several type I chains are simultaneously generated. Thus, type I chains can be mutually superimposed in the radio spectrum (if there are similar densities in the plasmoids) or they are separated in frequencies in the radio spectrum, if densities inside plasmoids are different.As already mentioned, in some cases of type I chains and DPSs there is also similarity in their "wavy" appearances. Up to now this feature was not explained in any model of type I chains.In the DPS model, during the flare magnetic reconnection below the rising magnetic rope, plasmoids are formed due to the tearing mode instability <cit.>. The plasmoids can merge to larger plasmoids which after this merging (coalescence) process start to oscillate with the period P ∼ L/v_ A, where L is the characteristic length in the merging process and v_ A is the local Alfvén speed <cit.>. Oscillations of the plasmoid (compression and expansion) periodically change densities inside the plasmoid and thus periodically change the plasma frequencies and frequencies of DPS. Just this process was proposed for explaining of quasi-periodic variations of frequencies ("wavy appearance") of DPSs <cit.>. Here the same process is proposed for explanation of the "wavy" appearance of type I chains. To illustrate it, we made similar numerical simulations as in the paper by <cit.>, see Figure <ref>, where two plasmoids (P_1 and P_2) merge into one larger plasmoid P. After merging the resulting plasmoid starts to oscillate. Figure <ref> shows time evolution of this merging and oscillating process expressed by positions of the selected magnetic field lines (with fixed magnetic vector potential) at the top and bottom of the plasmoids at the axis of the vertical and gravitationally stratified current sheet (x = 0 in Figure <ref>). In our case the period of the oscillation is about 22 s. From computations we know the maximal and minimal densities inside plasmoids (which are delimited by the selected magnetic field lines) and then assuming the emission based on the plasma emission mechanism we computed the artificial radio spectrum shown in Figure <ref>. The vertical lines in this DPS spectrum mimic pulses with the typical period of about one second. Namely, the pulses are generated on the kinetic level of the plasma description and thus their generation cannot be included into the used magnetohydrodynamic simulation.As seen in Figure <ref>, the artificial drifting pulsation structure has the "wavy" appearance. When the oscillating plasmoid is compressed then the frequency band of the drifting pulsation structure is shifted to higher frequencies and vice versa. Compare this artificial drifting pulsation structure with that, observed in June 30, 2002, shown in Figure <ref> (the upper right part). For details of computations, see <cit.>.Because type I chains and DPSs with the "wavy" appearance are relatively rare, it indicates that the full merging of two plasmoids with comparable sizes into one larger and oscillating plasmoid is also relatively rare.In previous models the frequency drift of type I chains is connected with the the Alfvén speed at the source of type I chain, e.g., in the model by <cit.> and thus used for the magnetic field estimations <cit.>. However, in the new model the speed of the plasmoid can be in the range from zero up to the local Alfvén speed, see <cit.>.Type I chainsare limited from the high-frequency side at about 400 MHz. It is commonly believed that it is due to the collisional optical depth of the emission increases with the frequency to the square <cit.>. On the other hand, the range of type I chains (below ∼ 400 MHz) shows that these chains are generated at upper parts of the active-region loops located close to the quasi-separatrix layers of the active region as proposed by <cit.>.DPSs are observed on higher frequencies than type I chains, at which in the "quiet" conditions of the solar atmosphere the plasma emission is fully absorbed. However, during solar flares the atmosphere becomes highly inhomogeneous and thus transparent for the plasma emission even in the decimetric range.If the size of the resulting plasmoid (L), formed during a merging process, is estimated (e.g. from the instantaneous frequency bandwidth of the chain and some density model of the solar atmosphere), then the period (P_W) of the chains with the "wavy" appearance can be used for further estimation of the magnetic field strength B in the chain radio source (the Alfvén speed is v_ A = B/√(μ_0 ρ) ∼ L/P_w, where μ_0 is the magnetic permeability of free space and ρ is the plasma density, that can be determined from the frequency of the chain, see also <cit.>).There is an important difference in processes generating type I chains and DPSs. While, in DPSs the magnetic reconnection is forced by the positive feedback between the magnetic reconnection and plasma inflow, given by the ejection of the whole flare structure upwards, in the processes generating type I chains this positive feedback in the magnetic reconnection is missing.These two regimes (without and with the positive feedback) of the magnetic reconnection together with plasmoids generating type I chains and drifting pulsation structures can be seen, e.g., in the 2012 July 12 flare <cit.>. Before the flare (at 15:00 – 16:16 UT) chains of type I bursts (noise storm) and then during the flare impulsive phase (at 16:16 UT), when the flare magnetic rope was ejected, the drifting pulsation structures were observed.§ CONCLUSIONSAs shown in previous section some parameters of type I chains and DPSs are similar (duration, repetition period of type I bursts in the type I chain and that of pulses in DPS, preference of the frequency drift towards lower frequencies, brightness temperature and "wavy" appearance in some cases) and other parameters like frequency range, bandwidth, frequency drift, source size and polarization are different. However, the differences can be caused by different conditions where type I chains and DPSs are generated (different altitude in the solar atmosphere, different densities, different density gradients and different magnetic field strengths).Therefore, considering all these similarities and differences we propose a new model of chains of type I bursts that is very similar to the DPS model. Although the magnetic reconnection was already proposed for explanation of noise storms <cit.>, this new model is more specific about the processes generating the chains of type I bursts, which are a part of noise storms. The chains of type I bursts are considered to be radio signatures of processes that heat the solar corona. Therefore, a correct model of these processes can contribute to a solution of the problem of the hot solar corona.We show that the chains of type I bursts can be generated by the magnetic reconnection associated with plasmoids (parts of current-carrying loops). While a trapping of accelerated superthermal electrons in a single plasmoid leads to normal type I chain (without the "wavy" appearance), the trapping of superthermal electrons in an oscillating plasmoid, which can be the result of merging of two smaller plasmoids, produces the type I chain with the "wavy" appearance.Similarly as in DPSs, individual type I bursts (forming the type I chain) are generated by quasi-periodic acceleration of superthermal electrons and their plasma emission. The frequency drift of these individual type I bursts can be caused by propagation of these superthermal electrons inside the plasmoid.We think that differences of both these types of bursts are also owing to different regimes of the magnetic reconnection. While in the case of type I chains the magnetic reconnection and plasmoid interactions are in the quasi-separatrix layers of the active region in more or less quasi-saturated regime, in the case of DPSs the magnetic reconnection and plasmoids formation and their interactions are forced by the upward motion of the flare magnetic rope.The new model can explain the "wavy" appearance of some chains of type I bursts by the merging of two plasmoids into one larger and oscillating one. This feature was not up to now explained.We showed that the chains of type I bursts with the "wavy" appearance can be used for estimation of the magnetic field strength in their sources. Unfortunately, examples of the chains of type I bursts with the "wavy" appearance are rare.DPSs are generated in deeper and denser layers of the solar atmosphere than chains of type I bursts. In the "quiet" coronal conditions the plasma radio emission from these deep and dense layers are absorbed. However, during the impulsive phase of solar flares these deep and dense layers are strongly disturbed and thus they become transparent also for the DPS emission. This new model can also explain the finding that the vertical extent of the noise storm is smaller than estimated from hydrostatic equilibrium <cit.>. Namely, complicated magnetic field structure in the region with plasmoids can shorten the density scale-height similarly as was proposed for microwave type III pair bursts by <cit.>. Furthermore, this model explains an enhanced density in the noise storm source comparing to the ambient corona <cit.>. As was shown, the plasmoids, where the type I chains are generated, are denser than the surrounding plasma.In this new model the plasmoid velocity, which is assumed to be connected with the frequency drift of the type I chain, is in the range from zero to the local Alfvén speed; contrary to previous models, where the velocity of the agent producing the frequency drift were strictly the Alfvén speed. It should be taken into consideration when the frequency drift of type I chains is used for magnetic field estimations.There are still many questions, especially about the plasmoid formation, its magnetic structure and evolution of the superthermal electrons in real 3-dimensional configuration. For their answers new simulations in extended 3-D kinetic models are necessary. The author thanks the referee for constructive comments that improved the paper. 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"authors": [
"Marian Karlicky"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170427122938",
"title": "Are chains of type I radio bursts generated by similar processes as drifting pulsation structures observed during solar flares?"
} |
Rogue periodic waves of the mKdV equation Jinbing Chen^1,2 and Dmitry E. Pelinovsky^2,3 ^1 School of Mathematics, Southeast University, Nanjing, Jiangsu 210096, P.R. China ^2 Department of Mathematics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1^3 Department of Applied Mathematics, Nizhny Novgorod State Technical University 24 Minin street, 603950 Nizhny Novgorod, RussiaDecember 30, 2023 ====================================================================================================================================================================================================================================================================================================================================================================== Giao Ky Duong ^† , [ G. K. Duong is fully funded by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 665850.], Van Tien Nguyen^∗ and Hatem Zaag^† , [ H. Zaag is supported by theANRprojet ANAÉref. ANR -13-BS01-0010-03] ^† Université Paris 13, Sorbonne Paris Cité, LAGA,CNRS (UMR 7539), F-93430, Villetaneuse, France. ^∗ New York University in Abu Dhabi, P.O. Box 129188, Abu Dhabi, United Arab Emirates. We considerthe semilinear heat equation∂_t u =Δ u + |u|^p-1 u ln ^α( u^2+2),in the whole space ℝ^n, where p > 1 and α∈ℝ. Unlike the standard case α = 0, this equation isnot scaling invariant. We construct for this equation a solution which blows up in finitetime T only at one blowup point a, according to the followingasymptotic dynamics:u(x,t) ∼ψ(t) (1 + (p-1)|x-a|^2/4p(T -t)|ln(T -t)|)^-1/p-1 ast → T, where ψ(t) is the uniquepositivesolutionof the ODEψ' = ψ^p ln^α(ψ^2+2), lim_t→ Tψ(t) = + ∞.The constructionrelies on the reduction of theproblemto afinitedimensional oneand a topological argument basedon the index theory to get the conclusion. By the interpretation of the parametersof the finite dimensionalproblemin terms of the blowup time and the blowup point, we showthe stabilityof the constructedsolutionwith respectto perturbations ininitial data. To our knowledge, this is the first successful construction for a genuinelynon-scale invariantPDE of a stableblowup solution with the derivation of the blowup profile. From this point of view, we consider our result as a breakthrough. Rogue periodic waves of the mKdV equation Jinbing Chen^1,2 and Dmitry E. Pelinovsky^2,3 ^1 School of Mathematics, Southeast University, Nanjing, Jiangsu 210096, P.R. China ^2 Department of Mathematics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1^3 Department of Applied Mathematics, Nizhny Novgorod State Technical University 24 Minin street, 603950 Nizhny Novgorod, RussiaDecember 30, 2023 ======================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTION. We are interested in the semilinear heat equation{[ ∂_t u = Δ u + F(u),;u(0) = u_0 ∈ L^∞(ℝ^n), ].where u(t): ℝ^n →ℝ,Δ standsforthe Laplacianin ℝ^n andF(u)= |u|^p-1 u ln^α(u^2+2),p > 1, α∈ℝ.By standardresultsthe model(<ref>) is wellposedin L^∞(ℝ^n) thanks to a fixed-point argument. More precisely, there is aunique maximalsolution on [0, T), with T ≤ +∞. If T < +∞,then the solution of (<ref>)may developsingularitiesinfinite time T, inthe sense thatu(t)_L^∞→+ ∞ ast → T. In this case, T is called the blowup time of u. Given a ∈ℝ^n, we say that a is a blowup point of u if and only ifthere exists (a_j,t_j ) → (a,T) as j → +∞ such that |u(a_j,t_j)| → +∞ as j → +∞. In the special caseα = 0, the equation (<ref>) becomes the standard semilinear heat equation ∂_t u = Δ u + |u|^p-1u.This equation is invariant under the following scaling transformationu ↦ u_λ(x,t):= λ^2/p-1u(λ x, λ^2 t).An extensive literature is devoted to equation (<ref>)and no rewiew can be exhaustive. Given our interest in the constructionquestion with aprescribed blowup behavior, weonly mention previous work in this direction.In <cit.>, Bricmont and Kupiainen showed the existence of a solution of (<ref>)such that (T - t)^1/p-1 u(a + z √((T -t) |ln(T -t)|), t) - φ_0(z)_L^∞(ℝ^n)→ 0,ast → T,whereφ_0 (z) = (p-1 + (p-1)^2 z^2/4p )^-1/p-1,(note that Herrero and Velázquez <cit.> proved the same result with a different method; note also that Bressan <cit.> made a similar construction in the case of an exponential nonlinearity).Later, Merle and Zaag <cit.> (seealso the note <cit.>) simplifiedthe proof of <cit.> and proved the stability of the constructed solution verifying the behavior (<ref>). Their method relies on the linearization of the similarity variables version around the expected profile. In that setting, the linearized operator has two positive eigenvalues, then a non-negative spectrum. Then, they proceed in two steps: - Reduction of an infinite dimensional problem to finite dimensional one: they show that controlling the similarity variable version around the profile reduces to the control of the components corresponding to the two positive eigenvalues.- Then, they solve the finite dimensional problem thanks to a topological argument based on index theory. The method of Merle and Zaag <cit.> has been proved to be successful in various situations. This was the case of the complexGinzgburg-Landau equation by Masmoudi and Zaag <cit.> (see also Zaag <cit.> for an ealier work) and also for thecase of a complex semilinear heat equation with no variational structure by Nouaili and Zaag <cit.>. We also mention the work of Tayachi and Zaag <cit.> (see also the note <cit.>) and the work of Ghoul, Nguyen and Zaag <cit.> dealing with a nonlinear heat equation with a double source depending on the solution and its gradient in a critical way. In <cit.>,Ghoul, Nguyen and Zaag successfully adapted the method to construct a stable blowup solution for a non variational semilinear parabolic system.In other words, the method of <cit.> was proved to be efficient even for the case of systems with non variational structure. However, all the previous examples enjoy a common scaling invariant property like (<ref>), which seemed at first to be a strong requirement for the method.In fact, this was proved to be untrue.As matter of fact, Ebde and Zaag <cit.> were able to adapt the method to construct blowup solutions for the following non scaling invariant equation∂_t u = Δ u + |u|^p-1 u+ f(u,∇ u),where|f(u,∇ u)| ≤ C(1+ |u|^q +|∇ u|^q'),withq < p, q' < 2p/p + 1.These conditions ensure that the perturbation f(u, ∇ u) turns out to exponentially small coefficients in the similarity variables. Later, Nguyen and Zaag <cit.> did a more spectacular achievement by addressing the case of stronger perturbation of (<ref>), namely∂_t u = Δ u + |u|^p-1 u + μ |u|^p-1 u/ln^a( 2 + u^2),where μ∈ℝ and a > 0. When moving to the similarity variables, the perturbation turns out to have a polynomial decay. Hence, when a > 0 is small, we are almost in the case of a critical perturbation.In both cases addressed in <cit.> and <cit.>, the equations are indeed non-scaling invariant, which shows the robustness of the method. However, since both papers proceed by perturbations around the standard case (<ref>), it is as if we are still in the scaling invariant case. In this paper, we aim at trying the approach on a genuinely non-scaling invariant case, namely equation (<ref>). This is our main result.There exists aninitial data u_0 ∈ L^∞(ℝ^n) such that the correspondingsolution to equation (<ref>) blows up in finite time T=T(u_0) > 0,only at the origin. Moreover, we have (i) For all t ∈ [0,T), there exists a positive constant C_0 such thatψ^-1(t) u(x,t)- f_0(x/√((T-t)|ln(T-t)|))_L^∞(ℝ^n)≤C_0/√(|ln (T -t)|),where ψ(t) is the uniquepositive solution of the following ODEψ'(t)= ψ^p(t) ln^α(ψ^2(t) +2), lim_t → Tψ(t) = + ∞,(see Lemma <ref> for the existence and uniqueness of ψ), and the profile f_0 is defined by f_0(z) = ( 1 + (p-1)/4p|z|^2 )^-1/p-1. (ii) There exits u^*(x) ∈ C^2(ℝ^n\{0}) such that u(x,t) → u^*(x)ast → Tuniformly on compact sets of ℝ^n ∖{0}, where u^*(x) ∼[ (p - 1)^2 |x|^2/ 8 p |ln|x||]^-1/p- 1( 4 |ln|x||/p - 1)^-α/ p -1 asx → 0, From (i), we see that u(0,t) ∼ψ(t) → +∞ as t → T, which means that the solution blows up in finite time T at x = 0. From (ii), we deduce that the solution blows up only at the origin.Note that the behavior in (<ref>) is almost the same as in the standard case α = 0 treated in <cit.> and <cit.>. However, the final profile u^* has a difference comingfrom the extra multiplication of the size |ln|x||^-α/p-1, which shows that the nonlinear source in equation (<ref>) has a strong effect to the dynamic of the solution in comparing with the standard case α = 0.Item(ii) is in facta consequence of(<ref>) and Lemma <ref>. Therefore, the main goal of this paper is to construct for equation (<ref>) a solution blowing up in finite time and verifying the behavior (<ref>).By the parabolic regularity, one can show that if the initial data u_0 ∈ W^2,∞(ℝ^n), then we have for i = 0, 1, 2,ψ^-1 (t) (T- t)^i/2∇^i_x u (x,t) - (T -t)^i/2∇^i_x f_0(x/√((T -t)|ln(T-t)|)) _L^∞≤C/√( |ln (T - t)|),where f_0 is defined by (<ref>).From the technique of Merle <cit.>, we can prove the following result.For arbitrary given set of m points x_1,...,x_m. There exists initial data u_0 such that the solution u of (<ref>) with initial data u_0 blows up exactly atm points x_1,...,x_m.Moreover, the local behavior at each blowup point x_i is also given by (<ref>)by replacing x byx - x_i.As a consequence of our technique, we prove the stability of the solution constructed in Theorem <ref>under the perturbations of initial data. In particular, we have the followingresult. Consider û the solution constructed in Theorem <ref> and denote by T̂ its blowup time.Then there exists𝒰_0 ⊂ L^∞(ℝ^n) a neighborhood of û(0)such that for allu_0 ∈𝒰_0,equation (<ref>) with the initial datau_0 has aunique solution u(t)blowing up in finite time T(u_0) at a single point a(u_0). Moreover, the statements (i) and (ii) in Theorem <ref> are satisfiedby u(x-a(u_0),t), and( T(u_0), a(u_0)) → (T̂, 0) as u_0 - û_0_L^∞(ℝ^n)→ 0.We will not give the proof of Theorem <ref> because the stability result follows from the reduction to a finite-dimensional case as in <cit.> with the same proof. Here we only prove the existence and refer to <cit.> for the stability. § FORMULATION OF THE PROBLEM.In this section, we first use the matched asymptotic technique to formally derive the behavior (<ref>). Then, we give the formulation of the problem in order to justify the formal result.§.§A formal approach. In this part, we follow the approach of Tayachi and Zaag <cit.> to formally explain how to derive the asymptotic behavior (<ref>). To do so, we introduce the following self-similarity variablesu(x,t) = ψ(t) w(y,s), y = x/√(T -t), s= -ln (T -t),where ψ(t) is the unique positive solutionof equation (<ref>) and ψ(t) → +∞ as t → T. Then, we see from (<ref>) that w(y,s) solves the following equation: for all (y,s) ∈ℝ^n × [-ln T, +∞) ∂_s w = Δw - 1/2 y. ∇ w - h(s) w+ h(s) |w|^p-1w ln ^α( ψ^2_1w^2+ 2)/ln^α(ψ^2_1 +2), whereh(s)=e^-sψ^p-1_1(s) ln^α(ψ^2_1(s)+2),andψ_1(s) = ψ(T - e^-s).Note that h(s) admits the following asymptotic behavior as s → +∞,h(s) =1/ p-1( 1 - α/ s- α^2 ln s/s^2) + O(1/s^2),(seeii) of Lemma <ref> for the proof of (<ref>)). From (<ref>), we see that the study of the asymptotic behavior of u(x,t) as t → T is equivalent to the study of the long time behavior of w(y,s) as s → +∞. In other words, the construction of the solution u(x,t), which blows up in finite time T and verifies the behavior (<ref>), reduces to the construction of a global solution w(y,s) for equation (<ref>)satisfying0 < ϵ_0 ≤lim sup_s → + ∞w(s)_L^∞(ℝ^n)≤1/ϵ_0, ϵ_0 > 0,andw(y,s)- ( 1 + (p-1)y^2/4p s)^-1/p-1_L^∞(ℝ^n)→ 0ass → +∞.In the following, we will formally explain how to derive the behavior (<ref>). §.§.§ Inner expansionWe remark that0 , ± 1 are the trivialconstantsolutionsto equation (<ref>). Sinceweare lookingfor a non zero solution, let usconsider the case whenw→ 1 ass → +∞. We nowintroduce w = 1+ w̅,then from equation (<ref>),we see that w̅ satisfies ∂_s w̅ = ℒ(w̅) + N(w̅,s),where ℒ=Δ - 1/2y . ∇ + ,N(w̅, s) =h(s) |w̅ +1|^p-1(w̅+1) ln^α(ψ^2_1 (w̅+1)^2 +2)/ln^α( ψ^2_1 +2) - h(s)(w̅+1) - w̅,ψ_1(s) is defined in (<ref>)and h(s) behaves asin (<ref>).Note that N admits the following asymptotic behavior,N(w̅, s) = p w̅^2/2 + O(|w̅|ln s /s^2) + O(|w̅|^2/s) + O(|w̅|^3)as(w̅ ,s)→ (0 ,+∞),(see Lemma <ref> for the proof of this statement). Since w̅(s) → 0 as s → +∞ and the nonlinear term N is quadratic in w̅, we see from equation (<ref>) that the linear part will play the main role in the analysis of our solution. Let us recall some properties of ℒ.The linear operator ℒ is self-adjoint in L^2_ρ(ℝ^n), where L^2_ρ is the weighted space associated with the weight ρ defined byρ(y) =e^- |y|^2/4/(4 π)^n/2,and (ℒ) = {1 - m/2, m ∈ℕ}.More precisely, we have * When n = 1, all the eigenvalues of ℒ are simple and the eigenfunction corresponding to the eigenvalue 1 - m/2 is the Hermite polynomial defined byh_m(y) = ∑_j=0^[ m/2](-1)^jm! y^m - 2j/j! ( m -2j)!.In particular, we have the following orthogonality ∫_ℝ h_i h_jρ dy=i! 2^i δ_i,j, ∀ (i,j) ∈ℕ^2.* When n ≥ 2, the eigenspace corresponding to the eigenvalue 1 - m/2 is defined as followsℰ_m = { h_β = h_β_1⋯ h_β_n,for all β∈ℕ^n, |β|= m , |β| = β_1 + ⋯ +β_n}. Since the eigenfunctions of ℒis a basic of L^2_ρ, we can expand w̅ in this basic as followsw̅(y,s)= ∑_β∈ℕ^nw̅_β(s)h_β(y) .For simplicity, let us assume that w̅ is radially symmetric in y.Since h_β with |β| ≥ 3 corresponds to negative eigenvalues of ℒ, we may consider the solution w̅ taking the formw̅ =w̅_0+ w̅_2(s)(|y|^2- 2n), where |w̅_0(s)| and |w̅_2(s)| go to 0 as s → +∞. Injecting (<ref>) and(<ref>) into (<ref>), thenprojecting equation (<ref>) ontheeigenspace ℰ_m with m=0 and m =2,{[ w̅_0' = w̅_0+ p/2( w̅_0^2 + 8 n w̅_2^2) + O( ( |w̅_0|+|w̅_2| )ln s/s^2);+ O( |w̅_0|^2 + |w̅_2|^2/s) + O(|w̅_0|^3+ |w̅_2|^3),; w̅_2'=4p w̅_2^2 + p w̅_0 w̅_2+ O( (|w̅_0|+|w̅_2| )ln s/s^2);+ O( |w̅_0|^2 + |w̅_2|^2/s) + O(|w̅_0|^3+ |w̅_2|^3), ].as s → + ∞. we now assume that|w̅_0(s)| ≪ |w̅_2(s)| as s → +∞, then (<ref>) becomes{[w̅_0' = w̅_0+ O(|w̅_2|^2) +O(|w̅_2| ln s/s^2),; w̅_2' = 4p w̅_2^2+ o(|w̅_2|^2) +O(|w̅_2| ln s/s^2), ] ass → + ∞. .We consider the following cases:- Case 1: Either|w̅_2| = O( ln s/s^2) or |w̅_2| ≪ln s/sas s → + ∞, then the second equation in (<ref>) becomes w̅_2' = O( |w̅_2| ln s/s^2)ass→+∞,which yieldsln|w̅_2|= O( ln s/s)ass → +∞,whichcontradicts with the condition w̅_2(s) → 0 as s → +∞.- Case 2: |w̅_2| ≫ln s/s^2 as s → + ∞, then (<ref>) becomes{[ w̅_0' = w̅_0+ O(|w̅_2|^2) ,; w̅_2' = 4p w̅_2^2+ o(|w̅_2|^2), ] ass → + ∞. . This yields {[ w̅_0 = O( 1/s^2) ,; w̅_2 =- 1/4ps+ o(1/s), ] ass → + ∞. .Substituting(<ref>) into (<ref>) yields{[ w̅_0' = O( 1/s^2) ,; w̅_2' =4p w̅^2_2+O(ln s/s^3), ] ass → + ∞, .from which we improve the error for w̅_2 as follows{[w̅_0 =O(1/s^2),; w̅_2 =- 1/4p s + O( ln ^2s/s^2), ] ass → +∞ ..Hence, from (<ref>), (<ref>) and (<ref>), we derive w(y,s)= 1- y^2/4 ps + n/2ps+ O(ln^2s/s^2),in L^2_ρ (ℝ^n) as s → + ∞. Note that the asymptotic expansion (<ref>) also holds for all |y| ≤ K, K is an arbitrary positive number. §.§.§ Outer expansion.Theasymptotic behaviorof (<ref>) suggests that the blowup profile depends on the variable z = y/√(s),From(<ref>), let us try tosearcha regular solution of equation (<ref>) of the formw(y,s) = ϕ_0(z) + n/2 ps + o( 1/s)inL^∞_loc ass → +∞,where ϕ_0 is abounded, smooth function to be determined. From (<ref>), we impose the condition ϕ_0(0) = 1.Since w(y,s) is supposed to be bounded, we obtain from Lemma <ref> that | h(s) | w|^p-1 w ln^α(ψ_1^2 w^2+2)/ln^α(ψ_1^2+2) -|w|^p-1w/p-1| =O(1/s),Note also that| |ϕ_0(z) +O( 1/s)|^p-1(ϕ_0(z) +O( 1/s)) - |ϕ_0(z)|^p-1ϕ_0(z)|=O(1/s).Hence, injecting (<ref>) into equation (<ref>) and comparing terms of order O( 1/s^i) for j = 0, 1, ⋯, we derive the following equation for j = 0,- 1/2 z .∇ϕ_0(z) - ϕ_0(z)/p-1 + |ϕ_0|^p-1ϕ_0(z)/p-1 = 0, ∀ z ∈ℝ^n.Solving (<ref>) with condition (<ref>), we obtainϕ_0(z)= ( 1+ c_0 |z|^2 )^-1/p-1,for some constantc_0 ≥ 0 (since we want ϕ_0 to be bounded for all z ∈ℝ^n). From (<ref>), (<ref>) and a Taylor expansion, we obtainw(y,s) = 1 - c_0 y^2/(p-1)s + n/2ps+ o(1/s), ∀ |y| ≤ Kass → + ∞ ,from which and the asymptotic behavior (<ref>), we find that c_0 = p-1/4p.In conclusion,we havejust derivedthe following asymptotic profilew(y,s) ∼φ(y,s)ass → +∞,whereφ(y,s) = ( 1+ (p-1)y^2/4ps)^-1/p-1+ n/2ps. §.§ Formulation of the problem.In this subsection, we set up the problemin order tojustifythe formal approachpresentedin the Section <ref>. In particular, we give a formulation to prove item (i) of Theorem <ref>. We aim at constructing for equation (<ref>) a solution blowing up in finite time T only at the origin and verifying the behavior (<ref>). In the similarity variables (<ref>), this is equivalent to the construction of a solution w(y,s) for equation (<ref>)defined for all (y,s) ∈ℝ^n × [s_0, +∞) and satisfying (<ref>). The formal approach given in subsection <ref> (see (<ref>)) suggests to linearize w around the profile function φ defined by (<ref>). Let us introduceq(y,s)= w(y,s) - φ(y,s),where φ is defined by (<ref>). From (<ref>), we see that q satisfies the equation∂_s q = ℒ q+ Vq+ B(q)+ R(y,s) + D(q,s),where ℒ is the linear operator defined by(<ref>) and V= p/p-1[ φ^p-1-1], B(q) = |q + φ|^p-1 (q + φ) - φ^p - p φ^p-1 q/p-1,R(y,s) = Δφ - 1/2 y ∇φ- φ/p-1+ φ^p/p-1 - ∂_s φ,D(q,s) = (q + φ)( ( h(s)-1/p-1) ( |q + φ|^p-1 - 1 )+ h(s) |q + φ|^p-1 (q + φ) L(q + φ, s)),L(v, s) = 2 αψ^2_1/ln(ψ_1^2+ 2)(ψ_1^2 +2)(v -1)+ 1/ln^α(ψ_1^2 +2)∫_1^vf”(u) (v -u)du,with h, ψ_1(s) andφ being defined by(<ref>), (<ref>) and(<ref>)respectively, andf(z)= ln^α(ψ_1^2 z^2+ 2), z ∈ℝ.Hence, proving (<ref>) now reduces to construct for equation (<ref>) a solution q such that lim_s → +∞q(s)_L^∞→ 0.Since we construct for equation (<ref>) a solution q verifying q(s)_L^∞→ 0” as s → +∞, and the fact that|B(q)| ≤ C|q|^min(2,p), R(s)_L^∞+D(q,s)_L^∞≤C/s,(see Lemmas <ref>, <ref> and <ref> for these estimates), we see that the linear part of equation (<ref>) will play an important role in the analysis of the solution. The property of the linear operator ℒ has been studied in previous section (see page Hermite),and the potential V has the following properties:i) Perturbation of effect of ℒ inside the blowup region {|y| ≤ K√(s)}:V(s)_L^2_ρ→ 0 as s→ +∞. ii) For eachϵ > 0, there exist K_ϵ >0 and s_ϵ >0 such thatsup_y/√(s)≥ K_ϵ, s ≥ s_ϵ| V(y,s)+ p/p-1| ≤ϵ. Since 1 is the biggest eigenvalue of ℒ, the operator ℒ+ Vbehaves as one with with a fully negative spectrumoutside blowup region {|y| ≥ K√(s)}, which makes the control of the solution in this region easily.Since the behavior of the potential V inside and outside the blowup region is different,we will consider the dynamics of the solution for |y| ≤ 2K√(s) and for |y| ≥ K√(s) separately for some K to be fixed large. We introduce the following functionχ(y,s)= χ_0(|y|/K √(s)),where χ_0 ∈ C^∞_0[0,+∞), χ_0_L^∞≤ 1 and χ_0(x) = {[1forx≤ 1,; 0for x≥ 2, ].andK is a positive constant to be fixed large later. We now decompose q byq = χ q + (1- χ) q =q_b+ q_e. (Note that (q_b) ⊂{|y| ≤ 2 K √(s)} and (q_e) ⊂{|y| ≥K √(s)}). Since the eigenfunctions of ℒ span the whole space L^2_ρ, let us writeq_b(y,s)=q_0(s) + q_1(s) · y + 1/2 y^T · q_2(s) · y - (q_2(s))+ q_-(y,s),where q_m(s) = (q_β(s))_β∈ℕ^n, |β| = m and∀β∈ℕ^n,q_β(s) = ∫_ℝ^nq_b(y,s) h̃_β(y) ρ dy, h̃_β =h_β/h_β^2_L^2_β,and q_-(y,s) = ∑_β∈ℕ^n, |β| ≥ 3 q_β(s) h_β(y).In particular, we denoteq_1= (q_1,i)_1 ≤ i ≤ n and q_2(s) is a n × n symmetric matrix defined explicitly byq_2(s)= ∫ q_b ℳ(y) ρ dy = (q_2,i,j)_1 ≤ i,j ≤ n,with ℳ = {1/4 y_i y_j- δ_i,j/2}_1 ≤ i,j ≤ n.Hence, by (<ref>) and (<ref>), we can writeq(y,s)= q_0(s) + q_1(s) · y + 1/2 y^T · q_2(s) · y - (q_2(s))+ q_-(y,s) + q_e(y,s).Note that q_m(m=0,1,2) and q_- are the components of q_b, and not those of q. § PROOF OF THE EXISTENCE ASSUMING SOME TECHNICAL RESULTS.In this section, we shall describe the main argument behind the proof of Theorem <ref>. To avoid winding up with details, we shall postpone most of the technicalities involved to the next section.According to the transformations (<ref>) and (<ref>), proving (i) of Theorem <ref> is equivalent to showing that there exists an initial dataq_0(y) at the time s_0such that the corresponding solution q ofequation (<ref>) satisfiesq(s)_L^∞(ℝ^n)→0ass → + ∞.In particular, we consider the following functionψ_d_0,d_1(y) = A/s_0^2( d_0+ d_1 .y)χ (2 y , s_0),as the initial data for equation (<ref>), where (d_0, d_1) ∈ℝ^1 + n are the parameters to be determined, s_0 > 1 and A > 1 are constants to be fixed large enough, andχ is the function defined by (<ref>). We aim at proving that there exists (d_0,d_1) ∈ℝ×ℝ^n such that the solution q(y,s) = q_d_0, d_1(y,s) of (<ref>) withinitial data ψ_d_0,d_1(y)satisfiesq_d_0,d_1(s)_L^∞→0ass → + ∞.More precisely, we will show that there exists (d_0,d_1) ∈ℝ×ℝ^n such that the solution q_d_0,d_1(y,s) belongs to the shrinking set S_A defined as follows:For all A ≥ 1, s ≥ 1 we define S_A(s) being the set of all functionsq ∈ L^∞(ℝ^n) such that|q_0| ≤A/s^2,|q_ 1,i| ≤A/s^2,|q_2, i,j| ≤ A^2 ln^2 s/s^2, ∀1 ≤ i,j ≤ n,q_-(y)/1+|y|^3_L^∞(ℝ^n)≤A/s^2, q_e(y)_L^∞(ℝ^n)≤A^2/√(s),where q_0, q_1 = (q_1,i)_1≤ i ≤ n, q_2 = (q_2,i,j)_1 ≤ i,j≤ n, q_- and q_e are defined as in (<ref>).We also denote by Ŝ_A(s) being the set Ŝ_A(s) = [ - A/s^2,A/ s^2] ×[ - A/s^2,A/ s^2]^n.For each A ≥ 1, s ≥ 1, we have the following estimates for all q(s) ∈ S_A(s):|q(y,s)| ≤C A^2 ln^2 s /s^2(1 + |y|^3), ∀ y ∈ℝ^n, q(s)_L^∞({|y| ≤ 2 K √(s)})≤CA/√(s), q(s)_L^∞(ℝ^n)≤CA^2/√(s). We aim at proving the following central proposition which implies Theorem <ref>.There exists A_1 ≥ 1 such that for all A ≥ A_1 there exists s_1(A) ≥ 1 such that for all s_0 ≥ s_1(A), there exists (d_0,d_1) ∈ℝ^1 + n such that the solution q(y,s) = q_d_0,d_1(y,s) of (<ref>) with the initial data at the time s_0 given by q(y,s_0) = ψ_d_0,d_1(y), where ψ_d_0,d_1 is defined as in (<ref>), satisfiesq(s)∈ S_A(s), ∀ s ∈ [s_0,+∞). From (<ref>), wesee that once Proposition <ref> is proved, item (i) of Theorem <ref> directly follows. In the following, we shall give all the main arguments for the proof of this proposition assuming some technical results which are left to the next section. As for the initial data at time s_0 defined as in (<ref>), we have the following properties.For each A ≥ 1,there exists s_2(A) > 1 such that for all s_0 ≥ s_2(A) we have the following properties:i) There exists 𝔻_A,s_0⊂ [-2;2] × [-2;2]^nsuch that the mapping Φ_1 : ℝ^1 + n → ℝ^1+n,(d_0,d_1)↦ (ψ_0, ψ_1) is linear, one to one from 𝔻_A,s_0 onto Ŝ_A(s_0). Moreover Φ_1 ( ∂𝔻_A, s_0) ⊂∂Ŝ_A(s_0).ii) For all (d_0,d_1) ∈𝔻_A, s_0 we haveψ_d_0,d_1∈ S_A(s_0) with strict inequalities in the sense that|ψ_0|≤A/s_0^2,|ψ_1,i| ≤A/s_0^2,|ψ_2,i,j|<A ln^2 s_0/s^2_0, ∀ 1 ≤ i,j ≤ n, ψ_-/1 + |y|^3_L^∞(ℝ)<A/s^2_0, ψ_e ≡ 0. where χ(y,s_0) is defined in(<ref>), ψ_0, (ψ_1,i)_1 ≤ i ≤ n, (ψ_2,i,j)_1 ≤ i,j ≤ 2, ψ_-, ψ_e are the components of ψ_d_0,d_1 defined as in (<ref>), ψ_d_0,d_1 and Ŝ_A(s) are defined by (<ref>) and (<ref>). See Propositon 4.5 of Tayachi and Zaag<cit.> for a similar proof of this proposition. From now on, we denote by C the universal constant which only depends on K, where K is introduced in (<ref>). Let us now give the proof of Proposition <ref> to complete the proof of item (i) of Theorem <ref>. We proceed into two steps to prove Proposition <ref>:- In the first step, we reduce the problem of controlling q(s) in S_A(s) to the control of (q_0,q_1)(s) in Ŝ_A(s), where q_0 and q_1 are the component of q corresponding to the positive modes defined as in (<ref>) and Ŝ_A is defined by (<ref>). This means that we reduce the problem to a finite dimensional one.- In the second step, we argue by contradiction to solve the finite dimensional problem thanks to a topological argument.Step 1: Reduction to a finite dimensional problem.In this step, we show through a priori estimate that the control of q(s) in S_A(s) reduces to the control of (q_0,q_1)(s) in Ŝ_A(s). This mainly follows from a good understanding of the properties of the linear part ℒ + V of equation (<ref>). In particular, we claim the following which is the heart of our analysis. There exists A_3 ≥ 1 such that for all A ≥ A_3, there exists s_3(A) ≥ 1 such that for all s_0 ≥ s_3(A), the following holds:If q(y,s) is the solution of equation (<ref>) with the initial data at time s_0 given by (<ref>) with (d_0,d_1) ∈𝔻_A,s_0, and q(s) ∈ S_A(s) for all s ∈ [s_0, s_1] for some s_1 ≥ s_0 and q(s_1) ∈∂ S_A(s_1), then:(i) (Reduction to a finite dimensional problem) We have (q_0,q_1)(s_1) ∈∂Ŝ_A(s_1).(ii) (Transverse outgoing crossing) There exists δ_0 > 0 such that∀δ∈ (0, δ_0),(q_0, q_1)(s_1 + δ) ∉Ŝ_A(s_1 + δ),hence, q(s_1 + δ)∉S_A(s_1 + δ), whereŜ_A is defined in(<ref>) and 𝔻_A,s_0 is introduced in Proposition <ref>.Let us suppose for the moment that Proposition <ref> holds. Then we can take advantage of a topological argument quite similar to that already used in <cit.>.Step 2: A basic topological argument.From Proposition <ref>, we claim that there exists (d_0,d_1) ∈𝔻_A,s_0 such that equation (<ref>) with initial data (<ref>) has a solution q_d_0,d_1(s) ∈ S_A(s), ∀ s ∈ [s_0, +∞),for suitable choice of the parameters A, K, s_0. Since the argument is analogous as in <cit.>, we only give the main ideas. Let us consider s_0, K, A such that Propositions <ref>and <ref>hold. From Proposition <ref>, we have ∀ (d_0, d_1) ∈𝔻_A,s_0,q_d_0,d_1(y,s_0):=ψ_d_0,d_1∈ S_A(s_0),where ψ_d_0,d_1 is defined by (<ref>). Since the initial data belongs to L^∞, we then deduce from the local existence theory for the Cauchy problem of (<ref>) in L^∞ that we can define for each (d_0,d_1) ∈𝔻_A,s_0 a maximum time s_*(d_0,d_1) ∈ [s_0, +∞) such that q_d_0,d_1(s) ∈ S_A(s), ∀ s ∈ [s_0, s_*).If s_*(d_0,d_1) = +∞ for some (d_0, d_1) ∈𝔻_A,s_0, then we are done. Otherwise, we argue by contradiction and assume that s_*(d_0, d_1) < +∞ for all (d_0, d_1) ∈𝔻_A,s_0. By continuity and the definition of s_*, we deduce that q_d_0,d_1(s_*) is on the boundary of S_A(s_*). From item (i) of Proposition <ref>, we have (q_0,q_1)(s_*) ∈∂Ŝ_A(s_*).Hence, we may define the rescaled functionΓ:𝔻_A,s_0 ↦∂([-1,1]^1 + n) (d_0,d_1)→s_*^2/A(q_0, q_1)(s_*). From item (i) of Proposition <ref>, we see that if (d_0,d_1) ∈∂𝔻_A,s_0, then q(s_0) ∈ S_A(s_0),(q_0,q_1)(s_0) ∈∂Ŝ_A(s_0).From item (ii) of Proposition <ref>,we see that q(s) must leave S_A(s) at s = s_0,thus, s_*(d_0,d_1) = s_0. Therefore, the restriction ofΓto ∂𝔻_A,s_0 ishomeomorphicto theidentity mapping, which is impossible thanks to index theorem, and the contradiction is obtained. This concludes the proof of Proposition <ref> as well as item (i) of Theorem <ref>, assuming that Proposition <ref> holds. We now give the proof of item (ii) of Theorem <ref>.The existence of u^* in C^2(ℝ^n ∖{0}) follows from the technique of Merle <cit.>. Here, we want tofind an equivalent formation for u^* neartheblowup point x = 0. The case α = 0 was treated in <cit.>. When α≠ 0, we follow the method of <cit.>, and no new idea is needed. Therefore, we just sketch the main steps for the sake of completeness.We consider K_0 > 0 the constant to be fixed large enough, and |x_0| ≠0 small enough. Then, we introduce the following function υ (x_0, ξ, τ)= ψ^-1 ( t_0(x_0))u(x,t),where (ξ, τ) ∈ℝ^n ×[ - t_0(x_0)/T - t_0(x_0), 1 ), and (x,t) = (x_0 + ξ√(T - t_0(x_0)),t_0(x_0) + τ (T - t_0(x_0))),with t_0(x_0)being uniquely determinedby |x_0| = K_0 √((T - t_0(x_0)) |ln(T -t_0(x_0))|).From (<ref>), (<ref>) , (<ref>) and (<ref>) we derive thatsup_|ξ| < 2 |ln(T - t_0(x_0))|^1/4| v (x_0, ξ, 0) - φ_0(K_0)| ≤C/ 1 + (|ln(T - t_0(x_0))|^1/4)→ 0asx_0 → 0,where φ_0 (x)= ( 1 + (p - 1)x^2/4p)^1/p-1. As in <cit.>, we use the continuity with respect to initial data for equation (<ref>) associated to a space-localization in the ball B(0, |ξ| < |ln(T - t_0(x_0))|^1/4) to derivesup_|ξ| <|ln(T - t_0(x_0))|^1/4, τ∈ [0,1)| v (x_0, ξ, τ) - v̂_K_0 (τ) | ≤ϵ(x_0) → 0, asx_0 → 0,where v̂_K_0 (τ)= ( (1 - τ) + (p-1) K_0^2/4 p)^-1/p-1.From (<ref>) and (<ref>), we deduceu^* (x_0) = lim_t → T u(x_0, t) = ψ(t_0(x_0)) lim_τ→ 1 v (x_0, 0, τ) ∼ψ(t_0(x_0)) (p -1/ 4 p)^- 1/p-1. Using the relation (<ref>), we find thatT- t_0 ∼|x_0|^2/ 2 K_0 |ln |x_0|| andln(T - t_0(x_0)) ∼ 2 ln (|x_0|),asx_0 → 0,The formula (<ref>) then follows from Lemma<ref>, (<ref>) and (<ref>). This concludes the proof of Theorem <ref>, assuming that Proposition <ref> holds.§ PROOF OF PROPOSITION <REF>.This section is devoted to the proof of Proposition <ref>,which is the heart of our analysis. We proceed into two parts. In the first part, we derive a priori estimates on q(s) in S_A(s). In the second part, we show that the new bounds are better than those defined in S_A(s), except for the first two components (q_0, q_1). This means that the problem is reduced to the control of a finite dimensional function (q_0,q_1), which is the conclusion of item (i) of Proposition <ref>. Item (ii) of Proposition <ref> is just direct consequence of the dynamics of the modes q_0 and q_1. Let us start the first part. §.§ A priori estimates on q(s) in S_A(s).In this part we derive the a priori estimates on the componentsq_2, q_-, q_e which implies the conclusion of Proposition <ref>.Firstly, let us give some dynamics of q_0, q_1 = (q_1,i)_1 ≤ i ≤ n and q_2 = (q_2,i,j)_1 ≤ i,j ≤ n. More precisely, we claim the following.There exists A_4≥ 1, such that ∀ A ≥ A_4 there exists s_4(A)≥ 1, such that the following holds for all s_0 ≥ s_4(A): Assume that for all s ∈ [s_0,s_1] for some s_1≥s_0, q(s) ∈ S_A(s), then the following holds for all s ∈ [s_0,s_1]:(i) (ODE satisfied by the positive and null modes)m = 0, 1, |q_m' (s) - (1 - m/2)q_m(s)| ≤C/s^2,and |q_2'(s)+ 2/s q_2(s)| ≤Cln s/s^3.(ii) (Control of the negative and outer parts) q_-(y,s)/1 + |y|^3_L^∞ ≤C/s^2((s - σ) + e^-s - σ/2A + e^-(s-σ)^2A^2),q_e(s)_L^∞ ≤C/√(s)((s - σ) + A^2e^-s - σ/p + Ae^s - σ).We proceed in two parts: - In the first part we project equation (<ref>) to write ODEs satisfied by q_m for m = 0, 1,2.- In the second part we use the integral form of equation (<ref>) and the dynamics of the linear operator ℒ + V to derive a priori estimates on q_- and q_e.-Part 1: ODEs satisfying by the positive and null modes. We give the proof of (<ref>) and (<ref>) in this part. We only deal with the proof of (<ref>) because the same proof holds for (<ref>). By formula (<ref>) and equation (<ref>), we write for each 1 ≤ i,j ≤ n, |q_2,i,j'(s)-∫[ ℒ q + Vq + B(q) + R(y,s) + D(q,s) ] χ( y_i y_j /4 - δ_i,j/2) ρ dy|≤ C e^-s.Using the assumption q(s) ∈ S_A(s) for alls ∈ [s_0, s_1], we derive the following estimates for all s ∈ [s_0,s_1]: |∫ℒ(q)χ( y_i y_j /4 - δ_i,j/2)ρ dy| ≤C/s^3,from Lemmas <ref>, <ref> and <ref>| ∫ V q χ( y_i y_j /4 - δ_i,j/2) ρdy+ 2 /s q_2,i,j(s)|≤ C A/s^3, |∫ B(q) χ( y_i y_j /4 - δ_i,j/2) ρ dy |≤ C /s^3, |∫R χ( y_i y_j /4 - δ_i,j/2) ρdy |≤ C/s^3,|∫ D(q, s) χ( y_i y_j /4 - δ_i,j/2) ρ dy|≤ Cln s/s^3.Gathering all these above estimates to (<ref>) yields |q'_2,i,j + 2/sq_2,i,j| ≤Cln s/s^3,which concludes the proof of (<ref>). -Part 2: Control of the negative and outer parts. We give the proof of (<ref>) and (<ref>) in this part. the control of q_- and q_e is mainly based on the dynamics of the linear operator ℒ + V. In particular,we use the following integral form of equation (<ref>): for each s ≥σ≥ s_0,q(s)= 𝒦(s,σ) q(σ) + ∫_σ^s 𝒦(s,τ) [ B(q)(τ)+ R(τ) + D(q,τ)] d τ = ∑_i=1^4 ϑ_i(s,σ),where {𝒦(s,σ)}_s ≥σ is defined by{[ ∂_s 𝒦(s,σ)= (ℒ + V)𝒦(s,σ),s > σ,; 𝒦(σ,σ) = Id, ].and ϑ_1(s,σ)= 𝒦(s,σ) q(σ), ϑ_2(s,σ) = ∫_σ^s 𝒦(s,τ) B(q)(τ) d τ, ϑ_3(s,σ)= ∫_σ^s 𝒦(s,τ) R(.,τ) d τ, ϑ_4(s,σ) = ∫_σ^s 𝒦(s,τ) D(q,τ) d τ. From (<ref>), it is clear to see the strong influence of the kernel 𝒦 in this formula. It is therefore convenient to recall the following result which the dynamics of the linear operator 𝒦 = ℒ + V. (A priori estimates of the linearized operator in thedecomposition in (<ref>)). For all ρ^* ≥ 0, there exists s_5(ρ^*)≥ 1, such that if σ≥ s_5(ρ^*) and v ∈ L^2_ρ satisfying∑_m=0^2 |v_m| + v_-/1 + |y|^3_L^∞ +v_e_L^∞ < ∞. Then, ∀ s ∈ [σ, σ + ρ^*]the function θ(s) = 𝒦(s,σ) v satisfies [ θ_-(y,s)/1 + |y|^3_L^∞≤C e^s -σ( (s - σ)^2 +1 )/s(|v_0|+ |v_1|+ √(s) |v_2|);+ C e^-(s-σ)/2v_-/1 + |y|^3_L^∞ + C e^-(s-σ)^2/s^3/2v_e_L^∞, ]andθ_e(y,s)_L^∞≤C e^s -σ(∑_l=0^2 s^l/2 |v_l|+s^3/2v_-/1+|y|^3_L^∞) + C e^-s -σ/pv_e_L^∞.The proof of this result was given by Bricmont and Kupiainen <cit.> in one dimensional case. It was then extended in higher dimensional case in <cit.>. We kindly refer interested readers to Lemma 2.9 in <cit.> for a detail of the proof. In view of formula (<ref>), we see that Lemma (<ref>) plays an important role in deriving the new bounds on the components q_- and q_e. Indeed, given bounds on the components of q, B(q), D(q) and R, we directly apply Lemma <ref> with 𝒦(s, σ) replaced by 𝒦(s, τ) and then integrating over τ to obtain estimates on q_- and q_e. In particular, we claim the following which immediately follows (<ref>) and (<ref>) by addition.For all Ã≥ 1, A ≥ 1, ρ^* ≥ 0,there exists s_6(A, ρ^*) ≥ 1 such that ∀ s_0 ≥ s_6(A,ρ^*) and q(s) ∈ S_A(s), ∀ s ∈ [σ, σ + ρ^*]where σ≥ s_0. Then, we have the following properties: a)Case σ≥ s_0: for all s ∈ [σ, σ + ρ^*], i) (The linear term ϑ_1(s,σ))(ϑ_1(s,σ))_-/1 + |y|^3_L^∞ ≤C( 1 +e^-s-σ/2 A + e^-(s -σ)^2 A^2)/s^2,(ϑ_1(s,σ))_e_L^∞ ≤C A^2 e^- s -σ/p + A e^s -σ/s^1/2. ii) (The quadratic term ϑ_2(s,σ))(ϑ_2(s,σ))_-/1 + |y|^3_L^∞ ≤ C(s - σ)/ s^2 + ϵ,(ϑ_2(s,σ))_e_L^∞≤C (s - σ)/ s^1/2+ϵ.where ϵ = ϵ(p) > 0. iii) (The correction term ϑ_3(s,σ) )(ϑ_3(s,σ))_-/1 + |y|^3_L^∞ ≤ C (s- σ) / s^2,(ϑ_3(s,σ))_e_L^∞≤C (s-σ ) / s^3/4. iv) (The nonlinear term ϑ_4(s,σ))(ϑ_4(s,σ))_-/1 + |y|^3_L^∞≤C (s - σ)/ s^2 ,(ϑ_4(s,σ))_e_L^∞≤C (s -σ ) / s^3/4. b) Case σ=s_0, we assume in addition|q_m(s_0)|≤Ã/s_0^2,|q_2(s_0)| ≤Ãln^2 s_0/s_0^2, q_-(y,s_0)/1 + |y|^3_L^∞≤Ã/s_0^2, q_e(s_0)_L^∞≤Ã/√(s_0).Then, forall s ∈ [s_0,s_0 + ρ^*] we have a) and the following properties:(ϑ_1(s,s_0))_-/1+ |y|^3_L^∞ ≤ C Ã/s^2, (ϑ_1(s,s_0))_e_L^∞≤C à (1 + e^s - s_0)/√(s). The proof simply follows from definition of the set S_A and Lemma <ref>.In particular, we make use Lemmas <ref> , <ref> and <ref>to derive the bounds on the components of the term B, D and R as follows:∑_m ∈ℕ^n, |m| = 0^2| B(q)_m(s)| ≤C/s^3, B(q)_-(s)/1 + |y|^3_L^∞≤C/s^2 + ϵ,B(q)_e(s) _L^∞≤C /s^1/2 + ϵ,and∑_m ∈ℕ^n, |m| = 0^2| R_m(s)| ≤C/s^2, R_-(s)/1 + |y|^3_L^∞≤C/s^2 + 1/2,R_e(s) _L^∞≤C/s^3/4,and∑_m ∈ℕ^n, |m| = 0^2| D(q)_m(s)| + D(q)_-(s)/1 + |y|^3_L^∞≤Cln s/s^3,D(q)_e(s) _L^∞≤C/s^3/4,where ϵ = ϵ (p) > 0.We simply inject these bounds to the a priori estimates given inLemma <ref> to obtain the bounds on (ϑ_m)_- and (ϑ_m)_e for m = 2, 3, 4. The estimate on ϑ_1 directly follows from Lemma <ref> and the assumption q(s) ∈ S_A(s). This ends the proof of Lemma <ref>.From the formula (<ref>), the estimates (<ref>) and (<ref>) simply follows from Lemma <ref> by addition. This concludes the proof of Proposition <ref>.§.§ Conclusion of Proposition <ref>.In this part, wegive the proof of Proposition <ref> whichis a consequence of the dynamics of equation (<ref>) given inProposition <ref>. Indeed, the item (i) of Proposition <ref> directly follows from the following result. There exists A_7≥ 1 such that ∀ A ≥ A_7, there exists s_7(A)≥ 1 such that for all s_0 ≥ s_7(A), we have the following properties:a) q(s_0) = ψ_d_0,d_1,s_0(y), where(d_0,d_1) ∈𝔻_A,s_0,b) For all s ∈ [s_0,s_1], q(s) ∈ S_A(s).Then for all s ∈ [s_0,s_1], we have∀ i,j ∈{1, ⋯, n},|q_2,i,j(s)|<A^2 ln^2 s/ s^2,q_-(y,s)/1 + |y|^3_L^∞≤A/2 s^2, q_e(s)_L^∞≤A^2/2 √(s),where 𝔻_A,s_0 is introduced in Proposition <ref> and ψ_d_0, d_1 is defined as in (<ref>). Since the proof of (<ref>) is similar to the one written in <cit.>, we only deal with the proof of (<ref>) and refer to Proposition 3.7 in <cit.> for the proof of (<ref>). We argue by contradiction to prove (<ref>). Let i, j ∈{1, ⋯, n} and assume that there is s_* ∈ [s_0, s_1] such that ∀ s ∈ [s_0, s_*),|q_2,i,j(s)| < A^2 ln^2(s)/s^2and |q_2,i,j(s_*)| = A^2 ln^2(s_*)/s_*^2.Assuming that q_2,i,j(s_*) > 0 (the negative case is similar), we have on the one handq'_2,i,j(s_*) ≥d/ds(A^2 ln^2 s/s^2)_s = s_* = 2A^2 ln s_*/s_*^3 - 2A^2 ln^2 s_*/s_*^3.On the other hand, we have from (<ref>), q'_2,i,j(s_*) ≤ - 2A^2 ln^2 s_*/s_*^3 + C ln s_*/s_*^3.The contradiction then follows if 2A^2 > C. This concludes the proof of Proposition <ref>. From Proposition <ref>, we see that if q(s) ∈∂ S_A(s_1),the first two components (q_0,q_1)(s_1) must be in ∂Ŝ_A(s_1), which is the conclusion of item (i) of Proposition <ref>.The proof of item (ii) of Proposition <ref> follows from (<ref>). Indeed, it is easy to see from (<ref>) that for alli ∈{ 1,...,n} and for each ε_0, ε_i= ± 1, then if q_0 (s_1) = ε_0 As_1^2 and q_1,i (s_1) = ε_i As^2_1, it follows that the sign of dq_0 ds( s_1 ) anddq_1,ids( s_1 )are opposite the sign of dds(ε_0 As^2 )( s_1 ) and dds( ε_i As^2 )( s_1 ) respectively. Hence, (q_0,q_1)(s) will actually leave Ŝ_A (s) at s_1 ≥ s_0 for s_0 large enough. This concludes the proof of Proposition <ref>. § SOME ELEMENTARY LEMMAS. For each T > 0, there exists only one positive solution of equation (<ref>). Moreover, the solution ψ satisfies the following asymptotic: ψ (t) ∼κ_α(T - t)^-1/p-1 |ln(T - t)|^-α/p-1,ast → T, where κ_α = (p-1)^-1/p-1( p-1/2)^α/p-1. Consider the ODEψ' = ψ^p ln^α(ψ^2+2), ψ(0) > 0.The uniqueness and local existence are derived by the Cauchy-Lipschitz property. Let T_max, T_min be the maximum and minimumtimeof the existence of the positive solution, i.e.ψ(t) exists for all t ∈ (T_min, T_max). We now prove that T_max < + ∞ and T_min = - ∞. By contradiction, we suppose thatthe solution exists on [0, + ∞), we havelim_t_1 → +∞∫_0^t_1ψ'/ψ^pln^α(ψ^2+2) dt = lim_t_1 → +∞∫_0^t_1dt = +∞.Since ∫_0^t_1ψ'/ψ^pln^α(ψ^2+ 2) dt is bounded, the contradiction then follows.Witha similarargument we can prove that T_min = - ∞.Let us now prove (<ref>). We deduce from (<ref>) thatT - t =∫_ψ(t)^+∞du/ u^p ln^α(u^2+2).Thus, for all δ∈ (0, p-1), there exist t_δ such that for all t ∈ (t_δ, T), we have∫_ψ(t)^+∞du/u^p + δ≤ T -t ≤∫_ψ(t)^+∞du/u^p-δ.This follows for all t ∈ (t_δ, T):(p-1 + δ)^-1/p - 1 + δ (T - t)^-1/ p - 1 + δ≤ψ(t) ≤ (p-1 - δ)^-1/p - 1 - δ (T - t)^-1/ p - 1 - δ,from which we havelnψ (t)∼ - 1/p-1ln(T-t)ast → T,andln (ψ^2 + 2) ∼ -2/p-1ln(T-t)ast → T.Hence, we obtainψ' = ψ^p ln(ψ^2 +2) ∼ψ^p [ - 2/p-1ln(T - t)]^α ast → T,which yieldsψ'/ψ^p∼( 2/p-1)^α |ln(T - t)|^α ast → T.This implies 1/p-1ψ^1-p∼( 2/p-1)^α∫^T_t |ln(T - v)|^α dv ∼( 2/p-1)^α (T - t) |ln(T - t)|^α as t → T,which concludes the proof of (<ref>).For all α∈ (0,1), θ > 0 and 0 < h < 1, the integralI (h)= ∫_h^1(s - h)^- α s^-θ dssatisfies: i) if α+ θ > 1, then I(h) ≤( 1/1 - α + 1/α+θ- 1) h^1 - α- θ. ii) If α+ θ = 1, thenI(h)≤1/ 1 - α+ |ln h|. iii) If α + θ < 1, thenI(h) ≤1/ 1 - α - θ. See Lemma 2.2 of Giga and Kohn <cit.>If y(t), r(t) and q(t)are continuous functions definedon [t_0, t_1] such that y(t) ≤y_0+ ∫_t_0^t y(s) r(s) ds++ ∫_t_0^t h(s) ds, ∀ t ∈ [t_0, t_1].Then,y(t) ≤ e^∫_t_0^t r(s)ds [ y_0+ ∫_t_0^t h(s) e^ -∫_t_0^s r(τ) dτ ds].See Lemma 2.3 of Giga and Kohn <cit.>.For each T_2 < T, δ > 0. There exists ϵ= ϵ (T,T_2, δ, n, p) > 0 such that for each v(x,t) satisfying | ∂_tv - Δ v| ≤C |v|^p ln^α(v^2 + 2), ∀ |x| ≤δ,t ∈ (T_2, T), δ > 0, and| v(x,t)| ≤ϵψ(t), ∀ |x| ≤δ,t ∈ (T_2, T),whereψ (t) isthe unique positive solution of (<ref>). Then, v(x,t) does not blow up at (0,T).Since the argument is almost the same as in <cit.> treated for the case α = 0, we only sketch the main step for the sake of completeness. Let ϕ∈ C^∞ (ℝ^n), ϕ = 1if|x| ≤δ/2, ϕ = 0if|x| ≥δ, and consider ω= ϕ v satisfying∂_t ω - Δω = f ϕ + g,wheref = ∂_t v - Δ vandg = v Δϕ -2 ∇ . (v ∇ϕ).By using the Duhamel's formula, we writeω(t) = e^(t - T_2)Δ (ω(T_2)) + ∫_T_2^t(e^(t - τ)Δ (ϕ f) +e^(t - τ)Δ(g) ) dτ, ∀ t ∈ [T_2, T),where e^tΔ is the heat semigroup satisfying the following properties: for all h ∈ L^∞,e^t Δ h _L^∞≤h _L^∞ ande^t Δ∇ h _L^∞≤C t^- 1/2h _L^∞, ∀ t >0.The formula (<ref>) then yields ω(t) _L^∞ ≤C + C∫_T_2^t ω(τ)_L^∞|v|^p-1ln^α( v^2+2)(τ)_L^∞(|x| ≤δ) + C∫_T_2^t (t - τ)^-1/2v(τ)_L^∞(|x| ≤δ) dτ,for some constant C = C(n, p, ϕ, T, T_2,δ) > 0.From (<ref>), (<ref>) and Lemma (<ref>), we find thatfor all |x| ≤δ, and τ∈ [T_2, T), |v(τ)|^p-1ln^α( v^2(τ)+2) ≤ C ψ^p-1(τ) ln^α(ψ^2(τ) + 2 ) ≤ C (T - τ)^-1,and|v(τ) | ≤ C (T -τ)^-1/p-1 |ln(T -τ)|^- α/p-1.The estimate (<ref>) becomesω(t) _L^∞ ≤C + Cϵ^p-1∫_T_2^t( T-τ)^-1ω(τ)_L^∞dτ + Cϵ∫_T_2^t (t - τ)^-1/2(T -τ)^-1/p-1 |ln(T -τ)|^-α/p-1 dτ.In particular, we now consider 0 < λ≪1/2 fixed, thenwe have:(T - τ )^- 1/p-1 |ln(T-τ)|^-α/p-1≤ C(α, λ) (T - τ)^- ( 1/p-1 + λ), ∀τ∈ (T_2, T).Hence, we rewrite (<ref>) as followsω(t) _L^∞ ≤C + C ϵ^p-1∫_T_2^t( T-τ)^-1ω(τ)_L^∞dτ + Cϵ∫_T_2^t (t - τ)^-1/2(T -τ)^- (1/p-1+ λ)dτ,whereC(n,p, ϕ, α, ϵ, λ,p). Beside that, by changing variabless =T - τ, h = T-t we have∫_T_2^t (t - τ)^-1/2(T -τ)^- θ(p, λ)dτ = ∫^T - T_2_h (s - h)^-1/2 (s)^- θ(p,λ) ds,where θ(p,λ) =(1/p-1+ λ).Case 1: If θ(p,λ) < 1/2, by usingiii) of Lemma <ref> we deduce from (<ref>), (<ref>) that ω (t)_L^∞≤C + C ϵ^p-1∫_T_2 ^t(T- s)^-1ω (s)_L^∞ds,Therefore, by Lemma<ref>, ω (t)_L^∞≤C (T - t)^ - Cϵ^p-1,Choosing ϵ small enough such that C ϵ^p-1≤1/2 (p-1). Then, we conclude from (<ref>) that |v (x, t)| ≤ C (T - t) ^- 1/2 (p-1),for|x| ≤1/2, t ≤ T.By using parabolic regularitytheory and the same argument as in Lemma 3.3 of <cit.>, we can prove that (<ref>) actually prevents blowup.Case 2: θ (λ, p) = 1/2, it is similar to the first case, by using ii) of Lemma <ref>,(<ref>) and(<ref>) we yieldω (t)_L^∞≤C (1 + |ln (T -t)|) + C ϵ^p-1∫_T_2 ^t(T- s)^-1ω (s)_L^∞ds,However, we derive from Lemma<ref> thatω (t)_L^∞≤ C (T -t)^-Kϵ^p-1,where C =C(n,p,ϕ, T,T_2, δ). We now take ϵ is small enough such that C ϵ^p-1≤1/2 (p-1), which follows (<ref>).Case 3: θ (λ, p) > 1/2, by using Lemmas <ref>,<ref> and arguments similar to obtain |v(x,t)| ≤ C (T -t)^1/2 - θ(p,λ), ∀ |x| ≤δ,t ∈ [T_2, T),Repeating the step in finite steps would end up with (<ref>). This concludes the proof of Lemma <ref>. The following lemma gives the asymptotic behavior of h(s) ans ψ_1(s) defined in (<ref>) and (<ref>). Leth(s) and ψ_1(s) be defined as in (<ref>) and (<ref>)respectively.Then we havei) 1/ln ( ψ^2_1(s) + 2)= p-1/2s+ α (p-1) ln s/ 2 s^2+ O( 1/s^2),ass → +∞. ii) h(s) =1/ p-1[1 - α/ s- α^2 ln s/s^2]+ O(1/s^2),ass → +∞. i) Consider ψ(t) the unique positive solution of (<ref>). We have T -t= ∫_ψ(t)^+∞dx/x^p ln^α(x^2+2).An integration by parts yieldsT -t= 1/ψ^p-1(t) ln^α(ψ^2(t)+2)[1/p-1 -2 α/(p-1)^2ln( ψ^2(t)+2) + O( 1/(ln^2( ψ^2(t)+2))) ].Let us write ψ (t)= ψ_1(s) where s = -log(T-t), then we haveln ( ψ_1 (s)) = s/p-1 - α/(p-1)ln( ln ( ψ_1( s)) ) +O( 1),ass → + ∞,from which,we deduce that ln ( ψ_1(s)) = s/p-1- αln( s ) /p-1 +O(1),ass → + ∞, which is the conclusion (i).ii) From (<ref>) and (<ref>), we have h(s)= 1/p-1- 2 α/ (p-1)^2 ln ( ψ^2_1(s) + 2)+ O ( 1/ln^2( ψ^2_1(s) +2)),Using (<ref>) we conclude the proof of (<ref>) as well as Lemma (<ref>).Let N be defined as in (<ref>), we have N(w̅, s) = p w̅^2/2 + O(|w̅|ln s /s^2) + O(|w̅|^2/s) + O(|w̅|^3)as(w̅ ,s)→ (0 ,+∞). From the definition (<ref>) of N, let us write N(w̅, s) = N_1(w̅, s) + N_2(w̅, s), where N_1(w̅, s)= h(s) ( |w̅+1|^p-1 (w̅+1) -(w̅+1) ) - w̅,N_2(w̅, s)= h(s) |w̅ +1|^p-1(w̅+1) ( ln^α(ψ_1^2 (w̅+1)^2 +2)/ln^α( ψ_1^2 +2) -1 ).From (<ref>) and a Taylor expansion,we find that N_1(w̅, s) =p w̅^2/2- αw̅/s + O(|w̅| ln s /s^2) + O( |w̅|^2/s) + O(|w̅|^3)as(w̅ ,s)→ (0 ,+∞). We now claim the followingN_2(w̅, s) = αw̅/s+ O(|w̅| ln s /s^2) + O(|w̅|^2/s)as(w̅ ,s)→ (0 ,+∞), then, the proof of (<ref>) simply follows by addition.Let us now give the proof of (<ref>) to complete the proof of Lemma <ref> .We set f(w̅)= ln^α ( ψ_1^2 (w̅ +1)^2+ 2),| w̅|≤1/2. We applyTaylor expansion tof(w̅) at w̅ = 0 to find thatf(w̅) = ln^α(ψ_1^2+ 2) + 2 αln^α -1(ψ_1^2+2) ψ_1^2 /ψ_1^2+2w̅ + f”(θ)/2 (w̅)^2,where θ is between 0 and w̅, and f”( θ) = α (α -1) ln^α -2 ( ψ_1^2 (θ+1)^2 +2 ) (2 (θ +1) ψ_1^2/ψ_1^2 (θ +1)^2+2)^2 +αln^α -1(ψ_1^2(θ +1)^2+2)( 4 ψ_1 - 2ψ_1^4 (θ +1)^2)/(ψ_1^2 (θ +1)^2+2)^2. Since |θ| ≤1/2, one can show that |f”(θ)| ≤C ln^α - 1(ψ_1^2 + 2), ∀ |θ| ≤1/2.Thus, we have f(w̅) = ln^α(ψ_1^2+2 )+ 2 αln^α -1(ψ_1^2+2)w̅+ O (|w̅|^2 ln^α -1(ψ_1^2+2)) + O(|w̅| ln^α -1(ψ_1^2+ 2)/ψ_1^2 ),as s → +∞.Thisyieldsln^α(ψ_1^2(w̅+1)^2+ 2)/ln^α(ψ_1^2 +2) = 1 +2 αw̅/ln(ψ_1^2+2)+ O(|w̅|^2/ln(ψ_1^2 +2))+O(|w̅|/ln(ψ_1^2 + 2)ψ_1^2 ),as (w̅, s)→ (0,+ ∞), from which and(<ref>)we derive ln^α(ψ_1^2(w̅+1)^2+ 2)/ln^α(ψ_1^2(s)+2)- 1 =α (p-1) w̅/s + O(ln s |w̅|/s^2)+ O(|w̅|^2/s).From the definition of N_2, (<ref>), (<ref>) and the fact that|w̅+1|^p-1 (w̅+1) = 1 +p w̅ + O(|w̅|^2) as w̅→ 0,we conclude the proof of (<ref>)as well as Lemma <ref>.For all |z| ≤K_1, then there exists C(K_1) such that∀ s ≥ 1 we have| h(s) |z|^p-1z ln^α(ψ_1^2 z^2 +2)/ln^α(ψ_1^2 +2) - |z|^p-1 z/p-1| ≤ C(K_1)/s, where h(s) satisfies the asymptotic(<ref>). We consider f(z) = ln^α(ψ_1^2z^2 + 2) ∀ z ∈ℝ, then we writeln^α(ψ_1^2 z^2 +2) = ln^α(ψ_1^2+ 2) + ∫_1^|z|f'(v)dv.Recall from (<ref>) that h(s) = 1/p-1 + O(1/s), we have| h(s)|z|^p-1zln^α(ψ_1^2 z^2+ 2)/ln^α(ψ_1^2 +2)- |z|^p-1z/p-1|≤C|z|^p/ln^α(ψ_1^2+2)∫_1^|z||f'(v)|dv +C |z|^p/s,From i) of Lemma <ref> we have 1/ln(ψ_1^2 + 2)≤C/s, it issufficient to show that A(z):= |z|^p/ln^α -1(ψ_1^2 +2)∫_1^|z| |f'(v)|dv ≤ C(K_1), ∀ |z| ≤ K_1,where f'(v) = αln^α -1 (ψ_1^2 v^2 +2) 2 v ψ_1^2/ψ_1^2 v^2 + 2.For1 ≤ |z| ≤ K_1, it is trivial to see that |A(z)| ≤ C(K_1). For |z| <1, we consider two cases:- Case 1: α - 1 ≥ 0, thenA(z) ≤2|α| |z|^p ∫_|z|^1 1/v dv ≤ C(K_1). - Case 2: α-1 < 0, then A(z) ≤2 |α| |z|^p ln^α -1 (ψ_1^2 z^2+2)/ln^α -1 (ψ_1+2 )∫_|z|^1 1/v dv. + ifψ_1 z^2 ≥ 1 then A(z) ≤ 2 |α|ln^1-α(ψ_1^2+ 2)/ln^1 - α (ψ_1+ 2) |z|^p ∫_|z|^1 1/v dv≤ C(K_1). + if ψ_1 z^2 ≤ 1 then |z| ≤ v≤ψ_1^-1/2 we deduce that|A(z)| ≤ 2 |α|ψ_1^1-p/2ln^1 - α(ψ_1^2 +2)/ln^1-α(2)|z| ∫_|z|^1 ≤ C(K_1).This concludesthe proof ofLemma <ref>. For all A ≥ 1, there exists σ_3(A) ≥ 1such that for all s ≥σ_3(A), q(s) ∈ S_A(s) implies∀ |y| ≤ 2 K √(s),| D(q,s) | ≤ C(K) ln s (1+|y|)^4/s^3, and D(q,s)_L^∞(ℝ^n)≤C/s.From the definition (<ref>)of D, let us decomposeD(q, s)= D_1 (q ,s)+D_2(q,s),whereD_1(q,s) = ( h(s)- 1/p-1) ( |q + φ|^p-1(q+φ)-(q + φ) ), D_2(q ,s) = h(s)|q + φ|^p-1(q + φ) L(q + φ,s),and h(s)admits the asymptotic behavior(<ref>),L is definedin (<ref>). The proof of (<ref>) will follow once the following is proved: for all | y |≤ 2K √(s)|D_1 - ( α (| y|^2 - 2n)/4ps^2- α/sq)| ≤ C (1 + |y|^4)ln s/s^3,and| D_2+ ( α (| y|^2 - 2n)/4ps^2- α/s q ) | ≤ C (1+|y|^4)ln s/s^3.Let us give a proof of (<ref>). From the definition of S_A(s), we note that if q(s) ∈ S_A(s), then∀ y ∈ℝ^n, |q(y,s)|≤ C A^2ln^2 s(1 + |y|^3)/s^2,q(s)_L^∞(ℝ^n) ≤C A^2/√(s).From the definition (<ref>) of φ and (<ref>), we see thatfor all |y| ≤ 2K√(s), there exists a positive constant C(K) such that 0 < 1/C(K)≤(q + φ )(y,s) ≤C(K).Using Taylor expansion and the asymptotic (<ref>), we write D_1 (q,s) = ( -α/(p-1)s+ O( ln s/s^2)) ( φ^p- φ+ ( p φ^p-1 -1 )q ) + O( q^2).Using again the definition of φ and a Taylorexpansion, we derive φ^p= 1- (| y|^2 - 2n)/4s+ O( 1 + |y|^4/s^2),φ = 1 - (| y|^2 - 2n)/4ps+ O( 1 + |y|^4/s^2),p φ^p-1-1= p-1- (p-1)(| y|^2 - 2n)/4ps+ O( 1 + |y|^4/s^2),as s → +∞. Inserting (<ref>) and these estimates into (<ref>) yields (<ref>).We now turn to the proof of (<ref>). Recall from (<ref>) the definition of L,L(q + φ, s)= 2 αψ_1^2/ln(ψ_1^2+ 2)(ψ_1^2 +2)( q + φ -1)+ 1/ln^α(ψ_1^2 +2)∫_1^q + φf”(v) (q + φ -v)dt,where f(v) = ln^α(ψ_1^2 v^2+2), v ∈ℝ. From(<ref>) and a direct computation, we estimate|1/ln^α(ψ_1^2+ 2)∫_1^q + φ f”(v) (q+ φ- v) dv| ≤ C(K)|q + φ -1|^2/s,which yields|L(q + φ, s) - 2 αψ_1^2 (q + φ- 1)/ln(ψ_1^2 +2)(ψ_1^2+2)|≤ C(K) |q + φ -1|^2/s.From (<ref>) and (<ref>), we then have|L(q + φ, s) - α (p-1)(q + φ- 1)/s| ≤ C(K) ( |q + φ -1|^2/s+ ln s | q + φ -1 |/s^2),and beside that we have| q + φ - 1 |≤C ( 1 + | y |^2 /s,imply that|L(q + φ, s) - α (p-1)(q + φ- 1)/s| ≤ C(K) ln s(1 + | y|^4)/s^3,Moreover, from definition of D_2 and (<ref>) we deduce that| D_2(q,s) - α/s(φ^p+1 - φ^p+ ((p+1)φ^p- pφ^p-1 )q) | ≤ C(1+|y|^4) ln s/s^3,andφ^p+1 - φ^p = - (| y|^2 - 2)/4ps + O ( 1 + |y|^4/s^2),as ,(p+1) φ^p - pφ^p-1 = 1 - (| y|^2 - 2)/2s+ O( 1 + |y|^4/s^2),as ,as s → +∞ which yield (<ref>).We now prove for (<ref>).From (<ref>) and the boundedness of q and φ, we have |D_1(q,s)| ≤C/s.It is sufficient to prove that for all y ∈ℝ^n,|D_2(q,s)| ≤C(K) /s,Indeed, from definition (<ref>) of L we deduce thatD_2(q,s) = h(s) |q + φ|^p-1(q + φ) ln^α(ψ_1^2 z^2 +2)/ln^α(ψ^2 +2) - h(s)|q + φ|^p-1(q + φ).Using Lemma <ref> we deduce|D_2(q,s)| ≤C(K)/s.This completes the proof of Lemma <ref>. When s large enough, then wehave for all y ∈ℝ^n: i) (Estimates on V): | V (y,s)| ≤C (1 + |y|^2)/s, ∀ y ∈ℝ^n,andV = -(|y|^2- 2n)/4s + ṼwithṼ = O ( 1 + |y|^4/s^2), ∀ |y| ≤ K √(s). ii) (Estimates on R ) | R(y,s) | ≤C/s, ∀ y ∈ℝ^n,and R(y,s) = c_p/s^2+R̃(y,s) withR̃= O (1 + |y|^4/s^3), ∀ |y| ≤ K √(s).The proof simply follows from Taylor expansion. We refer to Lemmas B.1 and B.5 in <cit.> for a similar proof. For all A > 0there exists σ_5(A) > 0 such that for all s ≥σ_5 (A), q(s) ∈ S_A(s) implies| B (q (y,s))|≤ C |q|^2, and |B(q)|≤ C |q|^p̅, with p̅= min (p,2). See Lemma 3.6 in <cit.> for the proof of this lemma.' 19 urlstyle[Bressan(1992)]Brejde92 A. Bressan. Stable blow-up patterns. J. Differential Equations, 980 (1):0 57–75, 1992. ISSN 0022-0396. 10.1016/0022-0396(92)90104-U. URL <http://dx.doi.org/10.1016/0022-0396(92)90104-U>.[Bricmont and Kupiainen(1994)]BKnon94 J. Bricmont and A. Kupiainen. Universality in blow-up for nonlinear heat equations. Nonlinearity, 70 (2):0 539–575, 1994. 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Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term. arXiv:1506.08306, 2016.[Zaag(1998)]ZAAihn98 H. Zaag. Blow-up results for vector-valued nonlinear heat equations with no gradient structure. Ann. Inst. H. Poincaré Anal. Non Linéaire, 150 (5):0 581–622, 1998. ISSN 0294-1449. 10.1016/S0294-1449(98)80002-4. URL <http://dx.doi.org/10.1016/S0294-1449(98)80002-4>. | http://arxiv.org/abs/1704.08580v1 | {
"authors": [
"G. K. Duong",
"V. T. Nguyen",
"H. Zaag"
],
"categories": [
"math.AP",
"35K50, 35B40"
],
"primary_category": "math.AP",
"published": "20170427140513",
"title": "Construction of a stable blowup solution with a prescribed behavior for a non-scaling invariant semilinear heat equation"
} |
[vega]This work has been supported under grant VEGANo. 1/0234/14 and APVV-14-0073dtf]V. Štubňa [email protected] dtf]M. Jaščurcor1 [email protected] [cor1]Corresponding author [dtf]Department of Theoretical Physics and Astrophysics, Institute of Physics, P.J.Šafárik Universityin Košice, Park Angelinum 9, 040 01 Košice, Slovakia A mixed spin-1/2 and spin-3/2 Ising model on a decoratedsquare lattice with a nearest-neighbor interaction, next-nearest-neighbor bilinear interaction, three-site four-spin interactionand single-ion anisotropy is exactly investigated using a generalizeddecoration-iteration transformation, Callen-Suzuki identity and differential operator technique.The ground-state and finite-temperature phase boundaries are obtained by identifying all relevantphases corresponding to minimum internal or free energy of the system. The thermal dependencies of magnetization, correlation functions, entropy and specific heat are also calculated exactly and the most interesting cases are discussed in detail.mixed-spinIsing model many-body interactionsexact resultsdecorated latticephase transitions.§ INTRODUCTION The investigation of higher-order spin couplingshas been initiated many decades ago by Anderson <cit.> and Kittel <cit.> whohavestudiedthe role of biquadratic exchange interactions of the form S_i^2 S_j^2, in connection with the superexchangeinteraction and elastic properties of magnetic materials. Later, thehigher-order spin interactions have been experimentally found inmagneticcompounds MnO and NiO<cit.>-<cit.>.Since then, various types of multi-spin interactionshave been intensively studiedindifferent physical systems, in order to explaindiverse physical phenomena (see forexample <cit.>and references therein).It has been found that these interactions are usually much weaker than thestandard pair Heisenberg exchange coupling, however, due to their non-conventional symmetries they may significantly modify many physical quantities in the systems under investigation.In this work we will study a special kind of higher-order spin interactions, that are usually calledas three-sitefour-spin interactions due to their geometry. These interactions have been originally introducedby Iwashita and Uryu<cit.>, in order to describe magnetic properties of some clustered complex systems.In general these interaction take the form of (S_iS_j)(S_jS_k) andas a rule,they principially modify physical properties of various localized magnetic system <cit.>-<cit.>.Among recent works in this researchfield one should also notice ourstudy ofa decorated exactly solvable mixedspin-1/2 and spin-1 Ising modelwiththree-site fours-spin interaction<cit.>. In that work we have found that the such a decorated planar system may exhibit unusually interesting and richmagneticbehavior, among others includingalso the existence of phases with non-zero ground state entropy. Owing to many interesting phenomena found inour previous study <cit.> it is of interestto understand the role of varying spin value in the three-site four-spin exchange interaction and to clarify which magnetic properties will be significantly changed due to varying spin of decoratingatoms.For this purpose,wewill studyin this work abond-decoratedsquare lattice consisting of nodal atoms withspin 1/2and decorating atoms with spin 3/2.The Hamiltonian of the systems will include, except of standardbilinear interactionsterms, also an unconventional three-site four-spin interactions.The role of single-ion anisotropy will be also taken into account.The outline of this paper is as follows. In Section 2 we briefly summarize the application of decoration-iteration transformation toobtain exact relations for all relevant physical quantities. In Section 3 we discuss the most interesting numerical results and finally mainconclusions are summarized in the last section.§ FORMULATIONThe subject of our study is a mixed spin-1/2 and spin-3/2 Ising model withbilinear,three-site four-spin interactions and single ion anisotropy on a decorated square lattice as it is depicted in Fig 1.As one can see from the figure, the system consists of N spin-1/2 atoms located on the sites of square latticeand 2N decorating spin-3/2 atoms that occupy all bonds of the original square lattice.Thus,as a whole the system can be treated as a two-sublattice mixed-spin system with unequal number ofsub-lattice atoms which can bedescribed by the HamiltonianH=∑_k=1^2N H_k,where the summation runs over all bonds of the original square lattice andℋ_k represents the Hamiltonian ofk-th bond which takes the followingexplicit formℋ_k= -J'μ_k1μ_k2-JS_k(μ_k1+μ_k2) -J_4S_k^2μ_k1μ_k2-DS_k^2.Here the parameter J denotes the bilinear exchange interaction between nearest neighbors (n.n) μ_ki-S_k, parameter J'>0 represents the bilinear interaction between next-nearest neighbors (n.n.n) μ_k1-μ_k2, the parameter J_4 stands for the three-site four-spin interaction between μ_k1-S_k^2-μ_k2 spins andD denotes a single-ion anisotropy. The interaction parameters J, J_4 and D are allowed to takearbitrary positive or negative valuesand the spin variablestakeobviously the following values: μ_ki=± 1/2 and S_k=±1/2,± 3/2. One should notice here that the summation in (<ref>) must be performed in a way that accounts for different spinterms only once.Using Eq. (<ref>) the partition function for the present system can be written as𝒵=_{μ_ki}∑_{ S_k}∑exp(-β2N∑_k=1ℋ_k) =_{μ_ki}∑∏_k^2N∑_S_k=±1/2,±3/2exp(-βℋ_k),where β=1/k_BT, k_B is the Boltzmann constant, T is absolute temperature and lastly ∑_{μ_ki} and ∑_{ S_k} mean summation over degrees of freedom of μ_ki and S_k spins, respectively.Now introducing the following generalized decoration-iteration transformation <cit.>-<cit.>.∑_ S_k=±1/2,±3/2exp (-βℋ_k ) = Ae^β Rμ_k1μ_k2,one may recast the partition function of the system in the form𝒵=A^2N𝒵_0(β R)where 𝒵_0 represents the partition function of the original (undecorated)square lattice with N spin-1/2 atoms that interact via nearest-neighboreffective exchange interaction R. Here one should recall thatexact analytical expresion of 𝒵_0 is well known from the Onsager seminal work <cit.>.The expression A^2N represents the contribution of the decorating spins to the total partition function.Both unknown parameters A, R entering Eq. (<ref>) can be straightforwardly evaluated performing thesummation on the l.h.s. in Eq. (<ref>) and substituting μ_k1=±1/2 and μ_k2=± 1/2into resulting expression. In this way one easily gets the following relationsA =(V_1V_2)^1/2,β R = 2lnV_1/V_2,where V_1= 2e^1/4β J'e^9/16β J_4e^9/4β DK_1, V_2= 2e^-1/4β J'e^-9/16β J_4e^9/4β DK_2,withK_1=cosh(3/2β J)+ ^-1/2β J_4e^-2β Dcosh(1/2β J) K_2=1+e^1/2β J_4^-2β D.Having obtained exact mapping relations (<ref>)-(<ref>), and exact expression for the partition functionof the system (<ref>), we are now ableto gain exact equations forphase boundaries and exact analytical relations forall physical quantities of interest.At first, after substituting the value of inverse critical temperature of the square lattice, β_cR=2ln(1+√(2)),into l.h.s of (<ref>), we obtain formula for finite-temperature phase diagrams of the decorated system in the form1 + √(2) = V_1c/V_2c,where V_1c = V_1(β_c),V_2c = V_2(β_c) andβ_c = 1/k_B T_c.Next, using the relation F = -k_BT ln𝒵 one simply obtains from Eq- (<ref>) the following relation for the Helmholtz free energy of the entire systemF( β, J,J_4,J',D)=-2N β^-1ln A ( β, J,J_4,J',D)+F_0(β, R),where parametersA,R are given by Eq. (<ref>)-(<ref>) and F_0(β, R) represents the Helmholtz free energyof the original undeciratedIsingsquare lattice <cit.>F_0=-N/β[ln(2coshβ R/2)+1/2π_0^πζ(ϕ)dϕ],withζ(ϕ)=ln[1/2(1+√(1-κ^2sin^2ϕ))],andκ = 2sinh (β R/2) /cosh^2 ( β R/2 ).Now, theentropy Sand specific heatCcan calculated from Eq. (<ref>) using the following thermodynamic relationsS=-(∂ F/∂ T)_V,C_V= -T(∂^2 F/∂ T^2)_V,and consequently the internal energy can be also easily obtained using equationU = F + TS.In addition to the above mentioned thermodynamic quantities, we will also investigate the spin-orderingin all possible phases of the system. For this purpose it is necessaryto analyze the total and sub-lattice magnetization along with various spin-correlation functions.The total magnetization per one site of the decorated lattice is given by m=(m_A+2m_B)/3,wherethe sub-lattice magnetizationm_A and m_B are respectively given by m_A=⟨μ_ki⟩= 1/𝒵_{μ_ki}∑_{ S_k}∑μ_kiexp(-β2N∑_k=1ℋ_k) m_B=⟨ S_k⟩= 1/𝒵_{μ_ki}∑_{ S_k}∑S_kexp(-β2N∑_k=1ℋ_k)The calculation of m_A is a particularly simple task, since using Eqs. (<ref>) and (<ref>) one obtains from (<ref>) the following relation <cit.>,<cit.>, <cit.>⟨ f(μ_k1, μ_k2, …, μ_ki)⟩= ⟨ f(μ_k1, μ_k2, …, μ_ki)⟩_0.Here f represents an arbitrary function depending exclusively on the spin variables ofA sublattice. Thus, setting f(μ_k1, μ_k2, …, μ_ki) = μ_ki one obtains ⟨μ_ki⟩ = ⟨μ_ki⟩_0 = m_0, wherem_0 represents themagnetization per one lattice site of the original square lattice which has been exactly calculated by Yang <cit.> and in our case it takes the form m _0=1/2(1-16 e^-2β R/(1-e^-β R)^4)^1/8.On the other hand,for the calculation of m_B i.e., the mean value of ⟨ S_k⟩, one can use the exact Callen-Suzuki identity <cit.>, <cit.>, <cit.> as a starting point.⟨ S_k⟩ =⟨∑_S_kS_k^-βℋ_k/∑_S_k^-βℋ_k⟩ ,where ℋ_k is defined in Eq. (<ref>). After performing the summation over S_k in previous equation one obtains ⟨ S_k⟩ =1/2⟨3sinh(3 β/2 h)+ ^-2β(h_4 +D)sinh(β/2h)/cosh(3 β/2 h) + ^-2β(h_4 +D)cosh(β/2h)⟩ ,where we have denoted effective fields acting on the k-th lattice site ash = J(μ_k1 + μ_k2), h_4 = J_4 μ_k1μ_k2.In order to calculate the ensemble average in last equation it is very comfortable to utilize thedifferential operator method which is based on the following relations f(x +λ_x, y + λ_y) =^(λ_x∇_x+λ_y∇_y)f(x,y) ^a μ=cosh(a/2)+2μsinh(a/2), μ = ±1/2where ∇_x=∂/∂ x, ∇_y=∂/∂ y are standard differential operators anda stands for an arbitrary parameter.Now, with the help of(<ref>)and (<ref>)one obtains for the sublattice magnetization m_B the following simple expression ⟨ S_k⟩ =⟨μ_k1⟩ A_1,where A_1 = 3sinh(3/2β J)+^-1/2β J_4^-2β Dsinh(1/2β J)/cosh(3/2β J)+ ^-1/2β J_4^-2β Dcosh(1/2β J).Subsequently, the total magnetization per one lattice site (<ref>) for the present system can be explicitly written in the formm=1/3⟨μ_k1⟩(1+2A_1).Hawing obtained the total reduced magnetization, one can now calculate the compensation temperatureT_k from the condition m=0 ∧m_A≠0 ∧m_B≠0 <cit.>. By means of this definition, we derive the condition for T_k in the following form7w^3/2α-5w^-3/2α+3w^γ_--w^γ_+=0where we have defined the following termsw = e^β_kJ_4 γ_+= -1/2(4d+1+α) γ_- = -1/2(4d+1-α).Of course, the equation (<ref>) has to be used in line with inequality T_k<T_c. In a similar way, one also obtains equations for the quadrupolar moment and various spin-correlation functions:q_B = ⟨ S_k^2⟩=1/8(A_2+A_3)+1/2⟨μ_k1μ_k2⟩(A_2-A_3), ⟨ S_kμ_k1⟩=(1/8+1/2⟨μ_k1μ_k2⟩)A_1,⟨ S_k^2μ_k⟩=1/4⟨μ_k⟩ A_2,⟨ S_k^2μ_k1μ_k2⟩=1/32(A_2-A_3)+1/8⟨μ_k1μ_k2⟩(A_2+A_3),where the coefficientsA_2 and A_3 are defined asA_2=9cosh(3/2β J)+e^-1/2β J_4e^-2β Dcosh(1/2β J)/cosh(3/2β J)+e^-1/2β J_4e^-2β Dcosh(1/2β J), A_3=9+e^1/2β J_4e^-2β D/1+e^1/2β J_4e^-2β D.We recall here thataverage valuesentering r.h.s. of (<ref>), (<ref>)-(<ref>) can be simplyevaluted,since on the basis of Eq. (<ref>) one obtains ⟨μ_k1⟩ = ⟨μ_k1⟩_0 and ⟨μ_k1μ_k2⟩ = ⟨μ_k1μ_k2⟩_0.§ NUMERICAL RESULTSIn this part we will present the most interestingresults obtained numerically from equations presented in the previous section.For the sake of simplicity,we introduce the following dimensionless parametersα= J/J_4,d = D/J_4,λ = J^'/J_4.§.§ Ground-statephase diagrams The ground-state phase diagram of the present system has been obtained by investigating the internal energy of all relevant spin configurations at T=0. Our findings are summarized in Fig. <ref> where we have depicted the phase diagramin α-d plane which is valid for arbitrary λ. As one can see,thewhole parameter space is divided into fourregions in whichdifferent ordered magnetic phases can be found.Namely:I. the ferrimagnetic phase withm_A=1/2, m_B =-3/2and q_B = 9/4,for α < 0 and d > α/2 - 1/4.II.the ferromagnetic phase withm_A=1/2, m_B =3/2and q_B = 9/4,for α > 0 and d >-α/2 - 1/4.III.the ferrimagnetic phase withm_A=1/2, m_B =-1/2/2and q_B = 1/4, for α < 0 and d <α/2 - 1/4.IV. the ferromagnetic phase withm_A=1/2, m_B =1/2and q_B = 1/4,for α > 0 and d < -α/2 - 1/4. The boundariesseparatingthese regions represent the lines of first-order phase transitions along whichrelevant couples of phases co-exist.Therefore, for d = α/2 - 1/4 and α<0,we have found thephase with m_A=1/2, m_B =-1and q_B = 5/4,while ford = -α/2 - 1/4 and α>0,onefinds m_A=1/2, m_B =1and q_B = 5/4.Similarly,for the case of pure three-site four spin interaction, (i.e. α=0),we have againthe co-existence of relevant phases, but now the resultingphase will be onlypartially ordered, since each decorating atom occupiesequally likely ± 3/2 or ± 1/2 spin states, respectively.Consequently, for α=0 and d>-1/4 one gets m_A=1/2, m_B =0and q_B = 9/4, while forα=0 and d<1/4one finds m_A=1/2, m_B =0and q_B = 1/4. Finally, thepoint withco-ordinates (α, d) = (0, -1/4) represents a special point in which coexist all four ordered phases, so that the minimum of the internal energy now corresponds to the partially ordered phase with m_A=1/2, m_B =0 and q_B = 5/4, since now all the statesS_k =± 1/2,± 3/2 on B sublattice are occupied equally likely.Here one should notice that the disorderappearing alongthe phase boundary α=0 willnaturally generate non-zero values of the entropy at the ground state. This interesting phenomenon will bediscussed in detail in Subsection 3.2. §.§ Finite-temperature phase diagrams and thermodynamic propertiesIn order to investigatethermal properties of the system, we have at firstcalculatedcritical and compensation temperaturesusing Eq. (<ref>) and Eq. (<ref>). In Figs. <ref>-<ref>wehave depictedrepresentative results by selecting various combinations of model parameters. Atfirst,in Fig. 3we have shown the results in the α - T_c space for λ = 1.0and somecharacteristic values of the single-ion anisotropy parameter d. In the figure, the solid and doted curves correspondtocritical and compensation temperatures, respectively.In agreement with the ground-state analysis,onefindsthe standardferri- or ferromagnetic phases to bestable at low temperatures for α<0 or α>0,respectively.On the other hand, the standard paramagnetic phase exists above each phase boundary.It is therefore clearthat the normalized spontaneous magnetization of these phases will take itssaturation value at T=0 and thenwill gradually decrease with increasing temperature, until itcontinuouslyvanishes at thecorresponding critical temperature.Moreover one can see that for fixed values of λand d one can observe the compensation effect by choosingappropriate negative values of the parameterα. We have also investigatedother cases and we have foundthat independently of the values of d and λall phase boundaries exhibit a symmetric U-shape form with minimum values at α= 0. Here one should recallthat the phase boundary for α = 0 representsthe critical temperatures of partially ordered phase.Intuitively one can simplyunderstand the minimum value of T_c, since in the partially ordered phase (i.e. forα = 0) only the sublattice Aexhibits a non-zero magnetization and therefore in this caseit iseasier todestroy the long-range order than that one appearing inthe fully orderedferrimagnetic or ferromagneticphases. Next, in order to demonstrate the influence of the n.n.n. bilinear interaction on physical behavior of our system, wehave studiedcritical boundaries andcompensation temperatures inλ - T space.Our findings are summarized in Fig. <ref> for|α| = 0.1 and characteristic values of the parameter d. As expected, all phase boundaries exhibit almost a perfect linear dependence with the increasing strength of theparameter λ and the compensation temperatures are again clearly visible for α = -0.1 andappropriate combinations of the parameters α and d.Finally, we have studied the critical and compensation temperatures in d - T_c space. As one can see from Figs. 5 the phase boundaries have very similar shapes as a phase diagram ofthe standard spin-3/2 Blume-Capelmodel <cit.>. This typical behavior of the system is mainly driven by the variation of the crystal-fieldparameter d and it is clear that on the sublattice B positive values ofd promotethe spin states ± 3/2, while the negative values prefer the occupation of ±1/2 spin states.One should emphasize here that such suppressing or favoring of the relevant spin states on the B sublattice has a principal influence on the three-sitefour-spin interaction term which which takes the form of- J_4S_k^2μ_k1μ_k2.As a matter of fact, one easily identifies that at the ground statethis three-site term reduces to- 9J_4μ_k1μ_k2/4 or - J_4μ_k1μ_k2/4 for d →∞ or d → -∞, respectively. Thus it is clear that for strong values of the crystal field the three-site four-spininteractionacts as an effective n.n.n pair interaction which substantially reinforces the effect of parameter λ and consequently keeps high values of the sublattice magnetization m_A even at higher temperature region. Due to this interesting effect,the compensation temperatures may exist in very large regions of d and they can exhibit very interesting behavior. In fact, we have found that for a non-zero λ and some appropriate values of α thecompensation temperature alwaystakes its saturation value for d → -∞ and it very slowly increases withincreasing the crystal-field parameter. On the other hand, the change of relevant compensation curves becomesmoredramatic in the region of d > -2.0, where for α < -0.211 each compensation temperature terminatesat the relevant critical boundary, while for α > -0.211 the existence of compensation temperatures extends up to infinite values of the crystal-field parameter.Numerical analysis of the critical and compensation temperatureshas clearly revealed several interestingphysical phenomena that appear in our system due to the presence of unconventional three-site four-spin interaction. To put further insight on thermodynamic properties of the system, let us discuss the mostinteresting thermal variation of the magnetization, entropy, specific heat and Helmholtz free energy. As far as concerns the magnetization, we restrict ourselvesto the ferrimagnetic case, in order to illustrateexistence of compensation points at finite temperatures.For this purpose,in Fig. <ref>,we have depictedseveral temperature dependencies of the absolute value of total magnetization per one atom by selecting suitablecombinations of allrelevant parameters. The presented curvesare in perfect agreement with our results presented in Figs. 2-5.Hereone should mention that the detailed theoretical investigation ofmagneticsystems exhibiting compensationtemperaturesis also of interest in connection with development of new recording media.Nowlet us turn our attention to the analysis of the thermal variations of entropy and specific heat.Here we willpreferablydiscuss thepartially ordered phase i.e. α = 0 and alsothe phasesthat are stable along the phase boundaries given byequations d = ± 0.5 α - 0.25.As we have already mentioned above, in the case of α = 0 the sublattice B remainsstrongly disordered down to zero temperature, while the sublatice Aexhibits the standard long-range order at T=0, thus the non-zeroentropy appears at the ground state. The situation is shown in Fig. <ref>, where we havepresented the thermal variationsof the entropy for α = 0 and λ = 1.0 and several typical negative andpositive values of d. As one can see from the figure, for d=-0.25 we have obtained S/Nk_B = ln (2^4) = 2.7726 whilefor all other values of d ≠ -0.25 one finds S/Nk_B = ln (2^2) = 1.3863.Here one should notice that for α =0remain the ground-state values of the entropyunchanged even for arbitrary non-negative λ.Next,in order to demonstrate the role of n.n, pair interaction,we have in Fig. <ref> depicted temperature dependenciesof the entropy for α = ± 1, λ =1.0 and some generic values of d. In this case only one non zerovalue of the entropy appears for d = -0.75, which corresponds to the ground-state phase point located exactly onthe line given by the equation d = ± 0.5 α - 0.25 and again, this situation does not change for non-negative values of λ. Moreover,our results also indicate thatthe entropy does not depend on the sign of parameter α, thus for arbitraryfixed values of λ and d,the system takesexactly the same values of entropy for theferromagneticas well asfor the ferrimagnetic equilibrium thermodynamic states.Of course, the describedbehavior of entropy is in a full agreement with our previous discussion. To complete our investigation of thermal properties of our system we have calculated the magnetic part of the specificheat. Our main finding are illustrated in Fig.<ref> andFig. <ref> for several representative combinations of parameters. As onecan see, all curvesgo to zero value for T → 0 in agreement with the Third Law of Thermodynamics and also eachcurve exhibits Onsager type singularity at the corresponding critical temperature. We can also observe that severalthermal dependencies of the specific heat exhibit a very clear local maximum at low-temperature region. This phenomenon isobservable whenever the relevant set of parameters is selected from the close neighborhoodof ground-state phase boundaries. In such a case there appears a strong mixing of different spin stateson the B sublattice, since therelevant stets are very easily thermally excited.§ CONCLUSIONIn this paper we have investigated phase diagrams and thermal properties of the complex mixed spin-1/2 andspin-3/2 Ising model on decorated square lattice. We have mainly concentrated on understanding of the influence ofthree-site four-spin interaction on magnetic properties of the system. Applyingthe generalized decoration-iterationtransformation we have obtained exact results for phase diagrams and all relevant thermodynamic quantities of themodel. We have also clearly demonstrated that due to multi-spin interactions the model exhibitsseveral unexpected features,for example, the existence of a partially ordered phase or non-zero ground state entropy. Comparing the presentresults with those obtained in our previous work for the decorating spin value of S_B =1,one can then formulate the following general physical statements:1. On the contrary to four-site four-spin coupling,the three-site four spin interaction is able to generate a partiallyordered phase even in the magnetic systems without bilinear interactions. This partially ordered phase can be stablein a wide temperature region and it exhibits a second-order phase transition at some critical temperature, which can depend on other physical parameters of the model, such as the crystal field.2. Due to its special symmetry with respect to spatial reversal of spins,the three-site four-spin interaction alwayssuppressesthe long-range ordering in arbitrary magnetic phase.3. In the mixed-spinmagnetic systems with three-site four-spin interactions the paramagnetic phase can by stable atT=0 for negative values of the crystal field, whenever atoms of the one sublattice are integer (i.e. S_B = 1,2,...). This behavior is impossible to observe in the systems with a half-integer values of S_B.In general,the theoretical investigation of the systems with many-body interactions is extraordinarily complicated task, however, the localized-spin models represent an excellentbasis for deep understanding of variousmany-body interactions going beyond the standard pair-wisepicture.For that reason, we hope that the presentstudy may initiate a wider interest in investigation of magnetic systems with multi-spin interactions.model1-num-names 90 Anderson1959P. W. Anderson, Phys. Rev. B 115,2 (1959).Kittel1960C. Kittel, Phys. Rev. B 120, 335 (1960). Harris1963E.A. Harris and J. Owen, Phys. Rev. Lett.11,9 (1963).Rodbell1963D. S. Rodbell, I.S. Jacobson, J. Owen and E.A. 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Jaščur, Physica A 186, 495 (1992). | http://arxiv.org/abs/1704.08565v1 | {
"authors": [
"Viliam Štubňa",
"Michal Jaščur"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170427133712",
"title": "Mixed spin-1/2 and 3/2 Ising model with multi-spin interactions on a decorated square lattice"
} |
Evolution of Activity-Dependent Adaptive Boolean Networks towards Criticality: An Analytic Approach Taichi Haruna^ 1 ^ 1 Department of Information and Sciences, Tokyo Woman's Christian University 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan E-mail: [email protected] ============================================================================================================================================================================================= We propose new activity-dependent adaptive Boolean networks inspired by the cis-regulatory mechanism in gene regulatory networks. We analytically show that our model can be solved for stationary in-degree distribution for a wide class of update rules by employing the annealed approximation of Boolean network dynamics and that evolved Boolean networks have a preassigned average sensitivity that can be set independently of update rules if certain conditions are satisfied. In particular, when it is set to 1, our theory predicts that the proposed network rewiring algorithm drives Boolean networks towards criticality. We verify that these analytic results agree well with numerical simulations for four representative update rules. We also discuss the relationship between sensitivity of update rules and stationary in-degree distributions and compare it with that in real-world gene regulatory networks.§ INTRODUCTION Boolean networks (BNs) <cit.> were originally proposed as a model of gene regulatory networks (GRNs) by S. Kauffman in 1969 <cit.>. Since then, it also has been used as a useful representation for modeling other complex systems such as neuronal networks <cit.> and social networks <cit.>. Although Boolean abstraction of real-world complex systems ignores fine details of them, it enables us to study some important aspects of their generic features. One such feature of BNs is the phase transition between ordered phase and disordered phase <cit.>. It has often been argued, but still is controversial, that real-world living systems such as GRNs and neuronal networks adjust their dynamical behavior towards the boundary between the two phases, criticality <cit.>. The advantages of criticality also have been studied: Optimal computational ability <cit.>, maximal sensitivity to external stimuli <cit.>, maximal memory capacity <cit.> and so on. So far, many plausible theoretical models of network self-organization towards criticality have been proposed although the exact mechanisms in gene regulatory or neuronal networks have not yet been known. For example, the following literatures discuss biologically inspired mechanisms: Hebbian learning <cit.>, spike-timing dependent plasticity <cit.>, dynamical synapses <cit.> and homeostatic plasticity <cit.> for neuronal networks and local control of feedback loops <cit.> and adaptation towards both adaptability and stability <cit.> for gene regulatory networks. Such models have been collectively called adaptive networks, in which network structure and network state coevolve, and have been paid much attention recently <cit.>. The study of adaptive networks originates from the work by Bornholdt and Rohlf <cit.>, which is also motivated by a preliminary work on the relationship between network structure and network state <cit.>. They showed that a simple activity-dependent rewiring rule based on measurement of local dynamics drives random threshold networks towards criticality by numerical simulation. The model has been extended to different situations: Liu and Bassler <cit.> reported that the activity-dependent rewiring rule drives random Boolean networks towards criticality by numerical simulation. Recently, this model was extended to networks with modular structure <cit.>. In spite of the structural constraint, self-organization towards criticality was shown to be preserved. Rohlf introduced an activity-dependent threshold change into the original Bornholdt-Rohlf model <cit.>. In his model, the threshold change and rewiring are switched stochastically. It was shown that the adaptive thresholds yield a new class of self-organized networks. However, it was confirmed numerically that networks still evolve towards criticality in the large size limit. In summary, these previous works based on numerical simulation suggest that the activity dependent rewiring rule robustly drives networks towards criticality in different conditions. However, in these adaptive Boolean network models, no analytic approach has been reported so far to the best of the author's knowledge. One reason for this would be the fact that the definition of activity is dependent on attractors which are usually avoided to discuss the phase transition of Boolean networks in the limit of large system size <cit.>. In the activity-dependent rewiring rule of Bornholdt and Rohlf <cit.>, a node is defined to be active if it does not change its state on the attractor reached from a random initial condition. Otherwise, the node is said to be static. The rewiring rule is as follows: The active node loses one of its incoming link randomly and the static node acquires a new incoming link randomly. Indeed, it seems that the activity on attractors is crucial for self-organization towards criticality. Bornholdt and Rohlf <cit.> numerically identified a first-order-like transition of the frozen component defined as the fraction of static nodes and argued that this transition is the main mechanism of robust self-organization of networks towards criticality. In this paper, we propose a new activity-dependent adaptive Boolean network model inspired by the cis-regulatory mechanism of real-world GRNs in which activity does not dependent on attractors but is defined by typical states that will be defined in Sec. <ref>. By this change of the definition of activity, we expect that our model admits analysis based on a mean-field theory called the annealed approximation in the limit of large system size. In the following, we show that our model can be solved for stationary in-degree distribution for a wide class of update rules to which the annealed approximation of the Boolean dynamics is applicable. At first sight, one would suspect that our network rewiring rule is designed towards a desired result, namely, criticality. However, it turns out that whether our model can self-organize towards criticality depends on a parameter of our network rewiring rule independent of update rules. We analytically show that the average sensitivity of stationary BN dynamics is equal to the parameter if certain conditions are satisfied. Thus, only when the value of the parameter is set to 1, we expect that BNs evolve towards criticality. The analytic result is verified by numerical simulation in four representative update rules. We also discuss the relationship between sensitivity of update rules and the tail of stationary in-degree distributions and compare it with that in real-world GRNs.§ MODEL Boolean networks (BNs) consist of a directed network with N nodes that can take two states 0 and 1. The state of node i at time step t is denoted by x_i(t) and is updated by a rule f_i selected from a given ensemble of Boolean functions ℰ_i: x_i(t+1)=f_i( x_i(t)),where x_i(t)=(x_j_1(t),…,x_j_k_i(t)) and j_1,…,j_k_i are nodes from which node i receives inputs. The number of inputs k_i is called in-degree of i. In this paper, all nodes are updated simultaneously. We also assume that the ensemble of Boolean functions ℰ_i associated with node i only depends on its in-degree k_i. Our activity-dependent rewiring rule for network evolution is different from those proposed in previous work <cit.> in the following two respects. First, both nodes and arcs can be selected at each time step of network evolution, in contrast to the previous models where only nodes are assumed to be selected. Second, we consider activity of arcs rather than that of nodes. In the previous models, activity of a selected node is measured by time-averaging its state value along a reached attractor and the decision whether the selected node gets a new incoming arc or loses an existing arc is made depending on the value of activity. In our model, when a node is selected, the node gets a new incoming arc. On the other hand, when an arc is selected, it is deleted when it is active. Here, activity of the arc is evaluated by the response of the target node i to perturbations on the arc given a typical state. That is, given an input x_i=(x_j_1,…,x_j_k_i) sampled randomly from a collection of states after sufficiently long time steps starting from a random initial condition, the arc is said to be active if f_i( x_i) ≠ f_i(x̃_i) where x̃_i=(x̃_j_1,…,x̃_j_k_i) is given by x̃_j_l=1-x_j_l if j_l is the source of the selected arc and x̃_j_l=x_j_l otherwise. These modifications are motivated by the following biological consideration: Deletion of an arc in a GRN of an organism can be caused by mutations in cis-regulatory elements (CREs) <cit.> of a gene that are nearby non-coding regions of DNA where a number of proteins called transcription factors (TFs) that are themselves products of other genes can bind. TFs regulate expression of the gene by increasing or decreasing the frequency of transcription initiation. If mutations in existing CREs of a gene change the binding pattern of TFs and the expression level of the gene, it could result in undesirable behavior of the organism and the corresponding arcs in its gene regulatory network are deleted in an evolutionary time scale <cit.>. On the other hand, mutations in a non-coding region of DNA within functional interaction range that is not involved in existing CREs could give rise to binding of a new TF. This means addition of a new incoming arc to the node representing the gene. Thus, nodes in a GRN can be conceived as carrying capacity to accept new incoming arcs incarnated by non-coding regions of DNA rather than coding DNA. In summary, when considering rewiring of a GRN, it is natural to treat nodes and arcs on the same footing because the physical basis of them is the same. In detail, our algorithm for network evolution in this paper is as follows:(i) An initial BN with a given ensemble of Boolean functions is generated. The in-degree of each node is sampled from a Poisson distribution with mean k_0 and the source of each arc is chosen randomly. (ii) The state of the BN is evolved from a random initial state for sufficiently long time steps to find a typical state. For any BN of finite size N, its state trajectory eventually falls onto an attractor. Hence, it is ideal to choose a state randomly from the attractor. However, when numerically simulating the model, it is difficult to find an attractor in a reasonable time if the BN is in the disordered phase. For efficient numerical simulation, we limit the maximum length of attractors to be detected as T. If no attractor is found within 2T+T' time steps, the last T steps are stored and a state is chosen randomly from the T states. In this paper, we set T=1000 and T'=100. We expect that this way of sampling a state approximates that of sampling from true typical states in the limit of large N because correlations between nodes are negligible for N ≫ 1 if the underlying network is locally tree-like and thus whether a state is on an attractor or not does not matter if it is reached after many time steps from a random initial state <cit.>. Indeed, this expectation accommodates to the assumptions of the mean-field theory used in Sec. <ref> and we will see that the numerically obtained in-degree distributions by this network rewiring algorithm agree well with the theoretical predictions based on the mean-field theory. (iii) A particular node or a particular arc is chosen with probability π_n or π_a, respectively. Here, we fix the ratio σ:=π_n/π_a throughout the network evolution. If a node is chosen, then a new incoming arc is added to the node. The source of the new arc is chosen uniformly at random. If an arc is chosen, then its activity in the state chosen in step (ii) is assessed. If the arc is active, then it is deleted. Otherwise, do nothing. (iv) The Boolean function on the chosen node or the target of the chosen arc in step (iii) is re-assigned following the given ensemble of Boolean functions. (v) Go back to step (ii). The steps (ii)-(v) constitute time unit of network evolution. We call it epoch after <cit.>. Note that π_n N + π_a z(e) N=1 should hold for all epoch e where z(e) is the average in-degree of the underlying directed network of BN at epoch e. Thus, π_n=σ/[(σ+z(e))N] and π_a=1/[(σ+z(e))N]. In each epoch, the network topology and Boolean functions assigned are fixed as in typical applications of BNs for modeling real-world complex systems. Thus, in the above model, the time scale separation between BN dynamics and network evolution is taken for granted.§ ANALYTIC RESULTS In this section, first we develop a general mean-field theory of network evolution that can be applied to any update rule which satisfies certain conditions mentioned below. Second, we apply the analytic result derived from the mean-field theory to four update rules that have been paid attention in the literature.§.§ Mean-field theory If the large system size limit N →∞ is taken and the underlying directed network is random networks with a specified degree distribution P(k,l) <cit.>, where P(k,l) is the probability that a randomly chosen node has in-degree k and out-degree l, the stability of BN dynamics can be analyzed by a mean-field theory so-called annealed approximation <cit.>. In the annealed approximation, correlations between nodes are neglected. This is manifested as the following ansatz taken in the mean-field calculation of BN dynamics <cit.>: The sources of incoming arcs to a node are chosen randomly at each time step and the Boolean functions are also re-assigned randomly at each time step. We apply the annealed approximation to BN dynamics in each epoch and assess its stability. For this purpose, we need to calculate sensitivity of Boolean functions selected from a given ensemble for each input <cit.>. Let λ_k,j be the probability that the output of an assigned Boolean function with k inputs changes when j-th input is flipped for 1 ≤ j ≤ k. We put λ_k:=∑_j=1^k λ_k,j. In general, λ_k,j depends on the fraction b_t of nodes with state 1 at time step t. b_t evolves by the following equation b_t+1=∑_k β_k(b_t) P_ in(k),where β_k(b_t) is the probability that the output of a node with k inputs is 1 at time step t+1 and P_ in(k)=∑_l P(k,l) is the in-degree distribution. Although Eq. (<ref>) can have periodic or chaotic solutions depending on update rules <cit.>, we only consider the case that Eq. (<ref>) has a unique stable stationary solution b^* in the following. Now let us suppose that the dynamics of a BN settle down to the stationary regime and apply a small perturbation. Let d̃_t be the fraction of damaged inputs at time step t. That is, d̃_t is the probability that the source node of a randomly chosen arc is flipped. Neglecting the higher order terms of d̃_t, we obtain d̃_t+1= λd̃_tfor the time evolution of d̃_t by a similar reasoning with previous work <cit.>, where λ=∑_k,ll P(k,l)/zλ_k which we call average sensitivity, z=∑_k k P_ in(k) is the average in-degree and λ_k is evaluated at b^*. Let d_t be the fraction of damaged nodes at time step t. Since d_t+1=λ̅d̃_t where λ̅=∑_k P_ in(k)λ_k, d_t also follows Eq. (<ref>). When in-degree and out-degree are independent as we expect for networks evolved by the proposed network rewiring algorithm, we have λ=λ̅=∑_k P_ in(k)λ_k.When λ < 1, d_t dies out eventually and the dynamics are said to be ordered or stable. If λ >1, d_t grows exponentially at first and the dynamics are said to be disordered or unstable. λ=1 is the boundary between the two cases and the dynamics are said to be critical. Now let us write down the equation for the time evolution of in-degree distribution by assuming the annealed approximation for the dynamics of BN at each epoch. Let P_ in(e,k) be the in-degree distribution at epoch e. According to the proposed network rewiring algorithm, we have P_ in(e+1,k)= ( 1 - π_n -π_a λ_k ) P_ in(e,k) + π_n P_ in(e,k-1) + π_a λ_k+1 P_ in(e,k+1)for k ≥ 1 and P_ in(e+1,0)=( 1 - π_n ) P_ in(e,0) + π_a λ_1 P_ in(e,1).In order to iteratively solve Eqs. (<ref>) and (<ref>), in each iteration one must calculate λ_k which is in general a function of b^*, which in turn depends on the entire in-degree distribution at epoch e through Eq. (<ref>). In addition, π_n and π_a are functions of average in-degree z(e). A stationary solution P_ in^s(k) of Eqs. (<ref>) and (<ref>) should satisfy π_n P_ in^s(k)= π_a λ_k+1 P_ in^s(k+1)for k ≥ 0 if it exists. When the stationary solution exists, we obtain λ=π_n/π_aby substituting Eq. (<ref>) into Eq. (<ref>). Thus, we predict that we can control the stability of evolved BNs by adjusting the ratio σ=π_n/π_a which we call target average sensitivity (TAS) hereafter. Note that σ can be given independently of update rules. In particular, when σ=1, that is, when a node or an arc is selected uniformly at random, the proposed network rewiring algorithm is expected to drive BNs towards criticality. The limitation of our mean-field theory arises from the normalization condition for the stationary in-degree distribution. If Eq. (<ref>) has a solution, it is solved by P_ in^s(k)=P_ in^s(0) σ^k ( ∏_l=1^k λ_l )^-1.Hence the infinite series ∑_k=0^∞ r_k must be convergent, where r_k=σ^k ( ∏_l=1^k λ_l )^-1. Since r_k+1/r_k=σ/λ_k+1, this is always the case when λ_k diverges as k →∞ by d'Alembert's ratio test. However, when λ_k converges to a number α as k →∞, it must hold that σ≤α. When b^* is independent of P_ in^s, we can give the condition for the existence of P_ in^s as follows: (i) If λ_k →∞ as k →∞, then P_ in^s exists. (ii) If λ_k →α < ∞ as k →∞, then P_ in^s exists if σ<α. If σ>α, then P_ in^s does not exist. If σ=α, then the existence of P_ in^s depends on the precise form of λ_k. Even when P_ in^s does not exist in the mean-field theory, we can formally obtain P_ in^s by truncating Eq. (<ref>) at k=N for BNs of finite size N. However, it is not guaranteed that the truncated P_ in^s can reproduce the stationary in-degree distribution of the evolved finite size BNs. This is because the assumption of the absence of correlations between nodes in the annealed approximation of BN dynamics will be violated in such case due to the existence of non-negligible amount of nodes with in-degree proportional to system size N.§.§ Examples In this subsection, we apply the analytic result presented in Sec. <ref> to four ensembles of Boolean functions: (a) Biased functions (BF) <cit.>: All Boolean functions with k_i inputs are weighted with bias p. The value of output of f_i is assigned to be 1 with probability p or 0 with probability 1-p for each input x_i. (b) Threshold functions (TF) <cit.>: Only threshold functions are considered. f_i( x_i)=1 if ∑_l=1^k_i w_j_l i(2x_j_l-1)+h_i ≥ 0 or 0 otherwise, where x_i=(x_j_1,…,x_j_k_i) ∈{0,1}^k_i and w_j_l i=± 1 with equal probability. In the following, we only consider the case h_i=0 for all i. (c) Heterogeneous biased functions (HBF) <cit.>: In this update rule, we allow the bias of BFs to depend on in-degree. That is, a BF with bias p_k_i is selected for node i with in-degree k_i. (d) Nested Canalizing functions (NCF) <cit.>: A nested canalizing function is given by f( x_i)=s_1if x_j_1=c_1s_2if x_j_1≠ c_1 and x_j_2=c_2s_3if x_j_1≠ c_1 and x_j_2≠ c_2 and x_j_3=c_3 ⋮ s_k_i if x_j_1≠ c_1 and … and x_j_k_i=c_k_is_dotherwisefor x_i=(x_j_1,…,x_j_k_i) ∈{0,1}^k_i, where c_l ∈{0,1} is the canalizing value for input from node j_l and s_l ∈{0,1} is the corresponding output value for l=1,…,k_i. Here, we consider a weight on NCFs defined by the following parameters <cit.>: s_l=1 with probability a and c_l=1 with probability c for l=1,…,k_i, and s_d=1 with probability d. The formula of λ_k for BFs, TFs and HBFs are given by λ_k=2p(1-p)k, λ_k=k2^-(k-1)k-1⌊ k/2 ⌋∼√(2/π)√(k) <cit.> and λ_k=2p_k(1-p_k)k, respectively. For these three rules, λ_k is independent of b^*. However, λ_k of NCFs depends on b^*. We have β_k(b_t)=a+(d-a)(1-γ(b_t))^k in Eq. (<ref>) where γ(b_t)=b_t c + (1-b_t)(1-c) is the probability that a randomly chosen input is at its canalizing value <cit.>. λ_k of NCFs is shown to be λ_k=(1-η)(1-(1-γ(b^*))^k)/γ(b^*)+k(1-γ(b^*))^k-1(η-η_0) ∼ (1-η)/γ(b^*) when 0<γ(b^*)<1, where η=a^2+(1-a)^2 and η_0=ad+(1-a)(1-d) at stationarity <cit.>. By substituting λ_k into the right-hand side of Eq. (<ref>), we obtain stationary in-degree distributions. For BFs, we get a Poisson stationary in-degree distribution P_ in^s(k)=e^-z_sz_s^k/k! with the stationary average in-degree z_s=σ/[2p(1-p)]. The tail of the stationary in-degree distribution for TFs decays slower than that of any Poisson distribution but does faster than that of any exponential distribution. HBFs have different stationary in-degree distributions depending on the functional form of p_k if it exists. For NCFs, the stationary in-degree distribution exists and is asymptotically equal to an exponential distribution provided that 0<γ(b^*)<1 and σ < (1-η)/γ(b^*) where b^* satisfies b^*=∑_k β_k(b^*)P_ in^s(k). In next section, we test these analytic predictions for TAS σ close to 1 since our primary interest is evolution towards criticality. The behavior of our model for a wider range of σ is investigated in Appendix where we also present an example in which our mean-field theory fails. § NUMERICAL RESULTS We compared analytic results with numerical simulations for the above four ensembles of Boolean functions. We simulated evolution of BNs with N=200 for three different values of TAS: σ=0.95, 1.00 and 1.05. Parameters used are p=0.7 for BFs, p_k=(1+√(1-2q_k))/2 with q_k=1/2 if 1 ≤ k ≤ 3 and q_k=2/k if k ≥ 4 for HBFs (thus, we have λ_k=2 for k ≥ 4) and a=1/3, c=0.95 and d=0 for NCFs. The condition for the existence of the stationary in-degree distribution for HBFs is σ < 2 and is satisfied in the numerical simulation here. For NCFs, we numerically checked that 0<γ(b^*)<1 and σ < (1-η)/γ(b^*) hold for the above parameter values. Fig. <ref> shows time evolution of the average sensitivities for each update rule from five different initial average in-degree 1 ≤ k_0 ≤ 5. For each pair of values of σ and k_0, 100 realizations were averaged. In Fig. <ref>, the average sensitivity of a BN at epoch e was calculated by Eq. (<ref>) with a numerical in-degree distribution at epoch e and analytic values of λ_k. We can clearly see that the average sensitivities approach to given values of σ independent of k_0. The numerical stationary in-degree distributions agree well with the theoretical predictions (Eq. (<ref>)) for all three values of TAS σ (Fig. <ref>). Here, they were obtained by averaging numerical in-degree distributions over last 10000 epochs in Fig. <ref> of 100 realizations for each k_0. Finally, we verified numerically that Eq. (<ref>) (with replacing d̃_t and d̃_t+1 by d_t and d_t+1, respectively) holds in evolved BNs for all three values of TAS σ by constructing so-called Derrida plots (Fig. <ref>) <cit.>. Derrida plots show the fraction of damaged nodes d_t+1 at time step t+1 as a function of the fraction of damaged nodes d_t at time step t. In Fig. <ref>, the value of d_t+1 was averaged over 200 states of 500 realizations of evolved BNs (those at the last step in Fig. <ref>) for each value of d_t. We can see that for all three values of TAS, the slope at the origin agrees well between numerical calculations and theoretical predictions. In constructing Derrida plots numerically, a subtlety arises when λ_k depends on b^* as in case of NCFs. For BFs, TFs and HBFs, we can choose a random state and randomly flip its fraction of d_t nodes to compute d_t+1 because λ_k is independent of b^* in these update rules. On the other hand, for NCFs, we must choose a typical state and then randomly flip its fraction of d_t nodes. It was predicted that this procedure produces the correct slope at the origin of Derrida plots <cit.>. However, in order for a Derrida plot to be correct for larger values of d_t, the perturbed state must also be a random sample of typical states (This does not guarantee that the Derrida plot is correct over all the range of d_t as shown in <cit.>). Here, we are interested in only the slope of the Derrida plots at the origin. Hence, it suffices for our purpose to adopt the above procedure. § DISCUSSION In this paper, we proposed a new activity-dependent adaptive Boolean network model and presented its analytic solutions for stationary in-degree distribution by employing the annealed approximation of Boolean dynamics. We showed analytically that stationary BNs evolved by the proposed network rewiring algorithm have in-degree distributions whose average sensitivity is equal to TAS if certain conditions are satisfied and verified the analytic solutions agree well with numerical simulations for four representative update rules. We emphasize that TAS can be given independently of update rules. In particular, if it is set to 1, our mean-field theory predicts that BNs evolve towards criticality. In previous work <cit.>, network self-organization towards criticality has been explained by the self-organized criticality picture <cit.>. That is, criticality is achieved by slowly adding links in the subcritical phase and rapidly deleting links in the supercritical phase of an absorbing transition of network activity. In particular, Droste et al. <cit.> analytically demonstrated this mechanism based on the pair-approximation of the network activity dynamics. They showed that two different time-scale separations are necessary to realize self-organization towards criticality: one is that between state dynamics on networks and topological changes of networks and the other is that between deletion of links and addition of links. In our model, the former time-scale separation is incorporated. However, the latter does not hold because the ratio of the probability of link addition to that of link deletion is finite. Thus, the self-organized criticality picture seems not to hold. In our model, the criticality is realized by stochastically balancing the mutually opposed processes, addition and deletion of links. In previous work on activity-dependent adaptive Boolean networks, influence of the update rule on the structure of evolved networks is assessed by only numerical simulations <cit.>. In our model, we have a simple relationship between the sensitivity of update rules represented by λ_k and the stationary in-degree distribution as shown above. Although our model is parsimonious, it is worth to compare our result with real-world GRNs. The in-degree distribution of the prokaryote Escherichia coli is best fitted by a Poisson distribution, whereas that of the eukaryote Saccharomyces cerevisiae is best fitted by an exponential distribution <cit.>. As for update rules, NCFs were introduced to model the yeast GRN <cit.> because NCFs are found abundantly in eukaryotic GRNs by an extensive literature study <cit.>. On the other hand, the analysis by Balleza et al. <cit.> suggested that BFs are enough to model the GRN of E. coli. They modeled several real-world GRNs including the bacterium GRN by biased functions to reveal whether they operate close to criticality or not and showed that changes in the fraction of canalizing functions for genes with at least 4 inputs do not affect the near critical dynamical behavior of the bacterium GRN. On the other hand, most of genes in the bacterium GRN have at most 3 inputs and canalizing functions are abundant just by chance for such genes <cit.>. Thus, there is no need for the bacterium to bias the sampling strategy of update rules towards canalizing functions even if they have an evolutionary advantage. Our model predicts Poisson and exponential stationary in-degree distributions for BFs and NCFs, respectively, and thus is consistent with the real-world GRNs. We are almost ignorant of out-degree distributions in this paper. Under the proposed network rewiring algorithm, the stationary out-degree distribution becomes a Poisson distribution independent of update rules. This disagrees with real-world GRNs because they have heavy-tailed out-degree distributions <cit.>. However, we can control the shape of stationary out-degree distribution by modifying step (iii) of the algorithm without changing the value of average sensitivity: selecting the source of a new arc following an appropriate weight depending on the out-degree of each node <cit.>. Finally, we note that it is an interesting open question whether our model can be extended to the network ensembles to which the semi-annealed approximation of Boolean dynamics <cit.> is applicable.§ ACKNOWLEDGMENTSThis work was partially supported by JSPS KAKENHI Grant Number 25280091. The author thanks the anonymous reviewers for their helpful comments to improve the manuscript.§ APPENDIX In this appendix, we compare our theoretical results with numerical simulation for BFs and HBFs for TAS σ apart from criticality. The parameters of the update rules are the same as those in Sec. <ref>. 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"authors": [
"Taichi Haruna"
],
"categories": [
"nlin.AO",
"q-bio.MN"
],
"primary_category": "nlin.AO",
"published": "20170427141341",
"title": "Evolution of Activity-Dependent Adaptive Boolean Networks towards Criticality: An Analytic Approach"
} |
alph empty< g r a p h i c s >High-Dimensional Function Approximation: Breaking the Cursewith Monte Carlo MethodsD I S S E R T A T I O Nzur Erlangung des akademischen Gradesdoctor rerum naturalium (Dr. rer. nat.) vorgelegt dem Rat derFakultät für Mathematik und Informatikder Friedrich-Schiller-Universität Jena von M.Sc. Robert Kunschgeboren am 19. Februar 1990 in GothaGutachter * ... * ... * ... Tag der öffentlichen Verteidigung: ... CHAPTER: ACKNOWLEDGEMENTS First of all, I wish to express my deepest gratitude to my supervisor Prof. Dr. Erich Novak. His sincere advice, not only on scientific matters, carried me forward in so many ways during my time as his student. He helped me to realize when attempts to solve a problem might be unsuccesful. Even more often his encouragement led me to pursue new ideas. Not to forget, the numerous hints on the written text allowed the dissertation to evolve.I am also grateful to my many colleagues for all their support. Especially David Krieg, who meticulously read large portions of the manuskript. Hopefully, his remarks led to a more comprehensive text. I would also like to thank Prof. Dr. Stefan Heinrich, Prof. Dr. Aicke Hinrichs, Prof. Dr. Winfried Sickel, Prof. Dr. Henryk Woźniakowski, Dr. Daniel Rudolf, Dr. Mario Ullrich, Dr. Tino Ullrich, Dr. Jan Vybíral, and my fellow students Glenn Byrenheid and Dr. Van Kien Nguyen, for interesting and encouraging scientific discussions during conferences and seminars. If I were to include all the friendly associates at mathematical meetings, this list would extend considerably.Finally, I wish to thank my friends Nico Kennedy and Rafael Silveira for reading the english introduction for linguistic aspects, never deterred from the formulas, even as non-mathematicians. I almost forgot to mention Kendra Horner who helped polish this very text. empty empty page emptyscrheadings roman CHAPTER: ZUSAMMENFASSUNGtocchapterZusammenfassung ZusammenfassungZusammenfassungFür viele Probleme, die in wissenschaftlich-technischen Anwendungen auftreten, ist es praktisch unmöglich, exakte Lösungen zu finden. Stattdessen sucht man Näherungslösungen mittels Verfahren, die in endlich vielen Schritten umsetzbar sind. Insbesondere muss man mit unvollständiger Information über die jeweilige Probleminstanz arbeiten; abgesehen von Strukturannahmen (dem sogenannten -Wissen) können wir nur endlich viel Information sammeln, typischerweise in Form von n reellen Zahlen aus Messungen oder vom Nutzer bereitzustellenden Unterprogrammen. Es besteht wachsendes Interesse an der Lösung hochdimensionaler Probleme, das sind Probleme mit Funktionen, die auf d-dimensionalen Gebieten definiert sind. Wir untersuchen die Informationskomplexität , die minimal benötigte Anzahl von Informationen, um ein Problem bis auf einen Fehler zu lösen. Tractability ist die Frage nach dem Verhalten dieser Funktion , also nach der Durchführbarkeit einer Aufgabe. In vielen Fällen wächst die Komplexität exponentiell in d für festgehaltenes , wir sprechen dann vom Fluch der Dimension. Es gibt im Grunde zwei Wege, dem Fluch der Dimension zu begegnen – so er denn auftritt. Die eine Variante ist, mehr a priori-Wissen einzubeziehen und so die Menge der denkbaren Eingabegrößen einzugrenzen. Die andere Möglichkeit besteht in der Erweiterung der Klasse zulässiger Algorithmen. In dieser Dissertation liegt der Fokus auf dem zweiten Ansatz, und zwar untersuchen wir das Potential von Randomisierung für die Approximation von Funktionen.Ein d-dimensionales Approximationsproblem ist eine Identitätsabbildung: F^d ↪ G^d,f ↦ f,mit einer Inputmenge F^d, die d-variate reellwertige Funktionen enthält, und einem normierten Raum G^d. Funktionen aus F^d sind in dieser Arbeit für gewöhnlich auf dem d-dimensionalen Einheitswürfel definiert, der Zielraum G^d ist dann beispielsweiseoder . Deterministische Algorithmen sind Abbildungen , wobei die InformationsabbildungN : F^d →^n ,f ↦ (L_1(f),…,L_n(f)),mittels n linearer Funktionale L_i Information über die Probleminstanz f sammelt. Der Fehler des Verfahrens ist durch den ungünstigsten Fall bestimmt,e(A_n,F^d) := sup_f ∈ F^df - A_n(f)_G^d .Randomisierte Verfahren sind Familien von Abbildungen obiger Struktur, indiziert durch ein Zufallselement aus einem Wahrscheinlichkeitsraum . Der Fehler eines solchen Monte-Carlo-Algorithmus ist der erwartete Fehler für den schlechtesten Input,e((A_n^ω)_ω,F^d) := sup_f ∈ F^df - A_n^ω(f)_G^d .In beiden Fällen wird eine signifikante Verkleinerung des Anfangsfehlers angestrebt,e(0,F^d) := inf_g ∈ G^dsup_f ∈ F^df - g_G^d ,welcher bereits ohne Information erreichbar ist. Insbesondere interessieren wir uns für den Vergleich der Komplexität im deterministischen und randomisierten Fall,n^(,d):= inf{n ∈_0 |∃ A_n: e(A_n,F^d) ≤} , n^(,d):= inf{n ∈_0 |∃ (A_n^ω)_ω: e((A_n^ω)_ω,F^d) ≤} ,wobei . Sämtliche in dieser Arbeit angegebene Algorithmen sind nichtadaptiv mit der einfachen Struktur von N siehe oben. Untere Fehlerschranken werden für wesentlich allgemeinere Verfahren gezeigt, welche die Information auch adaptiv sammeln oder eine veränderliche Kardinalität aufweisen. Zu diesen Begriffen und einer ausführlichen Einführung in das Themengebiet der Informationskomplexität, siehe chap:basics.Neue Resultate sind in den Kapiteln 2–4 enthalten, welche mehr oder weniger für sich stehende Themen behandeln. chap:Bernstein befasst sich mit unteren Schranken für randomisierte Verfahren, mittels derer sich für verschiedene Beispiele zeigen lässt, dass Monte-Carlo-Methoden nicht viel besser als deterministische Algorithmen sein können. Im Gegensatz dazu ist chap:Hilbert der Suche nach Problemen gewidmet, wo deterministische Algorithmen unter dem Fluch der Dimension leiden, Randomisierung diesen jedoch auf recht eindrucksvolle Weise zu brechen vermag. chap:monotone bespricht ein konkretes Problem für welches Zufallsalgorithmen zwar den Fluch aufheben, das Problem aber trotzdem noch sehr schwer ist. §.§.§ Zu chap:Bernstein: Untere Schranken für lineare Probleme mittels Bernstein-Zahlen Das Hauptergebnis dieses Kapitels stellen untere Schranken für den Fehler von Monte-Carlo-Algorithmen für allgemeine lineare ProblemeS: F → Gdar, d.h. S ist ein linearer Operator zwischen normierten Räumen F und G, zudem ist die Inputmenge F die Einheitskugel in F. Es wird gezeigt, dass für jede (adaptive) Monte-Carlo-Methode , welche n beliebige stetige lineare Funktionale L_i zur Informationsgewinnung einsetzt, die Abschätzunge((A_n^ω)_ω,F) ≥1/30b_2n(S)gilt, wobei b_m(S) die m-te Bernstein-Zahl des Operators S ist, siehe thm:BernsteinMCada. Der Beweis basiert auf einem Ergebnis von Heinrich <cit.>, welches Normerwartungswerte von Gauß-Maßen in Beziehung zum Monte-Carlo-Fehler setzt. Die Neuerung besteht in der Anwendung des Theorems von Lewis für die Wahl eines optimalen Gauß-Maßes. Dieses Ergebnis wurde in <cit.> angekündigt und ein kurzer Beweis ohne explizite Konstanten aufgeführt.In sec:Cinf->Linf wird dieses allgemeine Werkzeug für die L_∞-Approximation bestimmter Klassen von C^∞-Funktionen angewandt. Wir betrachten das Problem: F_p^d ↪ L_∞([0,1]^d)mit der InputmengeF_p^d := {f ∈ C^∞([0,1]^d) |∇__k⋯∇__1 f_∞≤ |_1|_p ⋯ |_k|_pfürk ∈_0,_1,…,_k ∈^d} ,wobei die Richtungsableitung entlang eines Vektors bezeichnet unddie p-Norm von , . Über die dazugehörigen Bernstein-Zahlen erhalten wir die untere Schranken^(,d,p) > 2^⌊d^1/p/3⌋ - 1für 0 < ≤1/30,siehe cor:LinfAppLB. Für ergibt sich daraus der Fluch der Dimension, auch bei Randomisierung. Die Beweistechnik zur Bestimmung der Bernstein-Zahlen ist von Novak und Woźniakoswki <cit.> bekannt, welche den Fluch der Dimension für den Fall im deterministischen Szenario gezeigt haben.Eine einfache Taylor-Approximation liefert obere Schranken für die Komplexität mittels deterministischer Verfahren,n^(,d,p) ≤n^(,d,p) ≤ exp( log(d+1) max{log1/, d^1/p} ),siehe thm:Cinf->LinfUB. Die algorithmische Idee stammt von Vybíral <cit.>, welcher ein Problem untersucht hat, das dem Fallnahekommt.Für dieses Beispiel beobachten wir grob gesprochen eine exponentielle Abhängigkeit der Komplexität von d^1/p. Dies kann durch Randomisierung nicht verbessert werden. Das betrachtete Problem ist zudem ein Beispiel dafür, wie eine Einschränkung der Input-Menge die Durchführbarkeit der Approximation beeinflusst. §.§.§ Zu chap:Hilbert: Gleichmäßige Approximation von Funktionen aus einem Hilbert-Raum Dieses Kapitel enthält einen neuen Monte-Carlo-Ansatz für die L_∞-Approximation von Funktionen aus einem Hilbert-Raum mit reproduzierendem Kern. Die Menge F der Eingabegrößen ist die Einheitskugel in , d.h. für eine Orthonormalbasis von haben wirF := {∑_k=1^∞ a_kψ_k | a_k ∈, ∑_k=1^∞ a_k^2 ≤ 1 } .Die Idee für den neuen Algorithmus basiert auf einer fundamentalen Monte-Carlo-Approximationsmethode nach Mathé <cit.>, siehe auch sec:HilbertFundamental. Jene wurde in der Originalarbeit zur Rekonstruktion in endlichdimensionalen Folgenräumen , , angewandt und diente in Verbindung mit Diskretisierungstechniken für Funktionenräume der Bestimmung der Konvergenzordnung für Einbettungsoperatoren. In sec:HilbertPlainMCUB verfolgen wir einen direkteren Ansatz über die lineare Monte-Carlo-MethodeA_n^ω(f) := 1/n∑_i=1^n L_i^ω(f) g_i^ω ,wobeiL_i^ω(f) := ∑_k=1^∞ X_ik ⟨ψ_k,f ⟩_, und g_i^ω := ∑_k=1^∞ X_ik ψ_k,mit unabhängigen standardnormalverteilten Zufallsvariablen X_ik. Die Funktionen g_i^ω sind unabhängige Realisierungen des mit assoziierten Gauß-Feldes Ψ, die Kovarianzfunktion von Ψ ist der reproduzierende Kern von . Für dieses Verfahren gilt die Fehlerabschätzunge((A_n^ω)_ω,F) ≤2Ψ_∞/√(n) .Zugegebenermaßen sind die zufälligen Funktionale L_i^ω unstetig mit Wahrscheinlichkeit 1, für festes jedoch sind die Werte zentrierte Gauß-Variablen mit Varianz und somit fast sicher endlich. Das Verfahren (A_n^ω)_ω kann allerdings auch als Grenzwert von Methoden gesehen werden, die fast sicher stetige Funktionale verwenden, siehe lem:stdMCapp.Mit Werkzeugen aus der Stochastik, siehe sec:E|Psi|_sup für eine Zusammenstellung, können wir den Wert abschätzen, sofern die zufällige Funktion Ψ beschränkt ist. Insbesondere mit der Technik majorisierender Maße nach Fernique lässt sich der Fall periodischer Funktionen auf dem d-dimensionalen Torus ^d angehen, siehe sec:HilbertPeriodic. Hierbei istdas Intervall [0,1] mit identifizierten Randpunkten. Im eindimensionalen Fall bezeichnen wir mitden Raum mit Orthonormalbasis{λ_0, λ_ksin(2πk·), λ_kcos(2πk·) }_k ∈ ,wobei . Der d-variate Fall ist über das Tensorprodukt definiert,_(^d) := ⊗_j=1^d_().Wir nehmenan, sodass der Anfangsfehler konstant 1 ist. Unter diesen Voraussetzungen gilt der Fluch der Dimension für deterministische Verfahren, siehe thm:curseperiodic. Die deterministische untere Schranke basiert auf einer Beweistechnik von Kuo, Wasilkowski und Woźniakowski <cit.>, ebenso Cobos, Kühn und Sickel <cit.>, siehe sec:HilbertWorLB. Im randomisierten Fall leiten wir Bedingungen an den reproduzierenden Kern periodischer Hilbert-Räume ab, für die der assoziierte Gauß-Prozess beschränkt ist. Im Speziellen betrachten wir Korobov-Räumemit für . Hierbei istso gewählt, dass der Anfangsfehler immer noch über die Wahl vonangepasst werden kann. Für Glattheit lässt sich zeigen, dass das Approximationsproblem: H_r^(^d) ↪ L_∞(^d)eine polynomiell beschränkte Monte-Carlo-Komplexität besitzt,n^(,d,r) ≤ C_r d (1 + log d)^-2 ,wobei . Für weniger Glattheit, konkret , können wir immer noch Durchführbarkeit der Approximation mit polynomiell beschränktem Aufwand (polynomial tractability) zeigen, wobei die Schranken für die Komplexität schlechter werden. Hierbei wird die fundamentale Monte-Carlo-Methode nur noch auf endlichdimensionale Teilräume von angewandt, siehe thm:Korobov. Auf diese Weise bricht Monte Carlo den Fluch. §.§.§ Zu chap:monotone: Approximation monotoner Funktionen Wir untersuchen das Problem der L_1-Approximation für die Klasse beschränkter, monotoner Funktionen,F_^d := {f : [0,1]^d → [0,1] | ≤⇒ f() ≤ f() } ,unter Nutzung von Funktionswerten als Information. Dies ist kein lineares Problem, weil die Inputmenge asymmetrisch ist. Hinrichs, Novak und Woźniakowski <cit.> zeigten, dass das Problem im deterministischen Fall dem Fluch der Dimension unterliegt. Dies ist bei Randomisierung nicht mehr der Fall, dennoch bleibt das Problem sehr schwer zu lösen.Aus einem Ergebnis von Blum, Burch und Langford <cit.> für Boole'sche monotone Funktionen kann man ableiten, dass für festes die Komplexität mindestens exponentiell von abhängt. sec:monoMCLBs enthält einen modifizierten Beweis, dank dem wir eine untere Schranke mit aussagekräftiger -Abhängigkeit bekommen,n^(,d) > νexp(c√(d) ^-1) für _0√(d_0/d)≤≤_0 ,wobei und , siehe thm:monotonLB. Insbesondere wenn wir eine gemäßigt abfallende Folge von Fehlertoleranzenwählen, lässt sich beobachten, dass die Komplexität exponentiell in d wächst. Man sagt, das Problem sei nicht "`weakly tractable"', siehe rem:monMCLBintractable.In sec:monoUBs werden obere Schranken bewiesen, die zeigen, dass die Komplexität für festes tatsächlich "`nur"' exponentiell von √(d) modulo logarithmischer Terme abhängt. Die algorithmische Idee wurde bereits von Bshouty und Tamon <cit.> für Boole'sche monotone Funktionen umgesetzt, siehe sec:BooleanUBs. Ein vergleichbarer Ansatz für reellwertige, auf definierte, monotone Funktionen wird nun in sec:monoRealUBs verfolgt. Darin beschreiben und analysieren wir einen neuen Monte-Carlo-Algorithmus (A_r,k,n^ω)_ω mit wünschenswerten Fehlerschranken, hierbei . Im Wesentlichen basiert dieser auf einer Standard Monte-Carlo-Näherung für die wichtigsten Wavelet-Koeffizienten der Haar-Basis in , wobei die zu approximierende Funktion an n zufällig gewählten Stellen ausgewertet wird. Die ausgegebene Funktion ist konstant auf Teilwürfeln der Seitenlänge 2^-r, d.h. nur Wavelet-Koeffizienten bis zu einer bestimmten Auflösung kommen in Betracht. Außerdem sind nur solche Wavelet-Koeffizienten von Interesse, die – für eine Input-Funktion f – die gleichzeitige Abhängigkeit von bis zu k Variablen messen. Für festes hat dieser Parameter das asymptotische Verhalten . Es gibt eine lineare Version des Algorithmus, siehe thm:monoUBsreal, sowie eine nichtlineare mit verbesserter -Abhängigkeit der Komplexität, siehe rem:monoMCUBeps.CHAPTER: INTRODUCTION AND RESULTStocchapterIntroduction and Results Introduction and ResultsIntroduction and ResultsFor many problems arising in technical and scientific applications it is practically impossible to give exact solutions. Instead, one is interested in approximate solutions that are to be found with methods that perform a finite number of steps. In particular, we need to cope with incomplete information about a problem instance; apart from structural assumptions (the so-called a priori knowledge), we may collect only a finite amount of information, let us say n real numbers originating from measurements or from subprograms provided by the user. There is a growing interest in solving high-dimensional problems that involve functions defined on a d-dimensional domain. We study the so-called information-based complexity , that is the minimal number of information needed in order to solve the problem within a given error tolerance . Tractability studies in general are concerned with the behaviour of this function . In many cases the complexity increases exponentially in d for some fixed , this phenomenon is called the curse of dimensionality. If a problem suffers from the curse of dimensionality, there are basically two ways to deal with it. One way is to include more a priori knowledge, thus narrowing the set of possible inputs. The other way is to widen the class of admissible algorithms. In this dissertation we focus on the second approach, namely, we study the potential of randomization for function approximation problems.A d-dimensional function approximation problem is an identity mapping: F^d ↪ G^d,f ↦ f,with an input set F^d which contains d-variate real-valued functions, and a normed space G^d. In this study, functions from F^d are usually defined on the d-dimensional unit cube , the output space G^d could be or . Deterministic algorithms are mappings , where the information mappingN : F^d →^n ,f ↦ (L_1(f),…,L_n(f)),uses n linear functionals L_i as information about the problem instance f. The error is then defined by the worst case,e(A_n,F^d) := sup_f ∈ F^df - A_n(f)_G^d .Randomized methods are modelled as a family of mappings as before, where is a random element from a probability space . The error of such a Monte Carlo algorithm is defined as the expected error for the worst input,e((A_n^ω)_ω,F^d) := sup_f ∈ F^df - A_n^ω(f)_G^d .In both cases, the aim is to significantly reduce the initial errore(0,F^d) := inf_g ∈ G^dsup_f ∈ F^df - g_G^d ,which is achievable without any information. We are interested in the comparison of the complexity in the deterministic and the randomized setting, n^(,d):= inf{n ∈_0 |∃ A_n: e(A_n,F^d) ≤} , n^(,d):= inf{n ∈_0 |∃ (A_n^ω)_ω: e((A_n^ω)_ω,F^d) ≤} ,where . All algorithms presented in this thesisfor upper bounds on these quantities are non-adaptive algorithms with the simple structure of N as indicated above. The lower bounds are proven for more general adaptive algorithms, even varying cardinality is considered. For these notions and a detailed introduction to information-based complexity see chap:basics.New results are contained in Chapters 2–4, which treat more or less stand-alone topics. chap:Bernstein is concerned with lower bounds for randomized methods, by means of which in some cases one can show that Monte Carlo methods are not much better than optimal deterministic algorithms. In contrast to this, in chap:Hilbert we find settings where deterministic algorithms suffer from the curse of dimensionality but randomization can break the curse in a very impressive way. chap:monotone deals with a problem for which randomization breaks the curse of dimensionality, yet the problem is quite difficult. §.§.§ On chap:Bernstein: Lower Bounds for Linear Problems via Bernstein Numbers The main result of this chapteris a lower bound for Monte Carlo algorithms for general linear problemsS: F → G,that is, S is a linear operator between normed spaces F and G, and the input set F is the unit ball in F. We show that for any adaptive Monte Carlo method using n arbitrary continuous linear functionals L_i as information, we havee((A_n^ω)_ω,F) ≥1/30b_2n(S),where b_m(S) is the m-th Bernstein number of the operator S, see thm:BernsteinMCada. The proof is based on a result due to Heinrich <cit.> which connects norm expectations for Gaussian measures with the Monte Carlo error. The innovation is that we use Lewis' theorem for choosing optimal Gaussian measures. This result has been announced in <cit.>, a short proof without the explicit constant has been included there.In sec:Cinf->Linf we apply this general tool to the L_∞-approximation of certain classes of C^∞-functions,: F_p^d ↪ L_∞([0,1]^d).Here, the input set is defined asF_p^d := {f ∈ C^∞([0,1]^d) |∇__k⋯∇__1 f_∞≤ |_1|_p ⋯ |_k|_pfork ∈_0,_1,…,_k ∈^d} ,where denotes the directional derivative along a vector , and we write ||_p for the p-norm of , . Via the corresponding Bernstein numbers we obtain the lower boundn^(,d,p) > 2^⌊d^1/p/3⌋ - 1for 0 < ≤1/30,see cor:LinfAppLB. For this implies the curse of dimensionality even in the randomized setting. The technique for determining the Bernstein numbers is known from Novak and Woźniakoswki <cit.>, where the curse of dimensionality for the case was shown in the deterministic setting.A simple Taylor approximation provides upper bounds for the complexity with deterministic methods,n^(,d,p) ≤n^(,d,p) ≤ exp( log(d+1) max{log1/, d^1/p} ),see thm:Cinf->LinfUB. The algorithmic idea goes back to Vybíral <cit.> who considered a setting similar to the case .For this example we observe an exponential dependency of the complexity on d^1/p, roughly, which cannot be removed with randomization. It is also an example which shows how narrowing the input set may affect tractability. §.§.§ On chap:Hilbert: Uniform Approximation of Functions from a Hilbert Space In this chapterwe study a new Monte Carlo approach to the L_∞-approximation of functions from a reproducing kernel Hilbert space . The input set F is the unit ball of , that is, for an orthonormal basis of we haveF := {∑_k=1^∞ a_kψ_k | a_k ∈, ∑_k=1^∞ a_k^2 ≤ 1 } .The idea for the new algorithm is based on a fundamental Monte Carlo approximation methodwhich is due to Mathé <cit.>, see also sec:HilbertFundamental. In the original paper it has been applied to finite dimensional sequence recovery , , it was then used in combination with discretization techniques for function space embeddings in order to determine the order of convergence. In sec:HilbertPlainMCUB we take a more direct approach, proposing the linear Monte Carlo methodA_n^ω(f) := 1/n∑_i=1^n L_i^ω(f) g_i^ωwhereL_i^ω(f) := ∑_k=1^∞ X_ik ⟨ψ_k,f ⟩_, and g_i^ω := ∑_k=1^∞ X_ik ψ_k,with the X_ik being independent standard Gaussian random variables. The functions g_i^ω are independent copies of the Gaussian field Ψ associated to , the covariance function of Ψ is the reproducing kernel of . We have the error estimatee((A_n^ω)_ω,F) ≤2Ψ_∞/√(n) .Admittedly, the random functionals L_i^ω are discontinuous with probability 1, but for any fixed the value is a zero-mean Gaussian random variable with variance , hence it is almost surely finite. This method, however, can be seen as the limiting case of methods that use continuous random functionals, see lem:stdMCapp.Using tools from stochastics, see sec:E|Psi|_sup for a summary, we can estimate the value , provided that the random function Ψ is bounded. Namely, via the technique of majorizing measures due to Fernique, we tackle the case of periodic functions on the d-dimensional torus ^d, see sec:HilbertPeriodic. Here, is the interval [0,1] where the endpoints are identified. In the univariate case, we denote bythe space with orthonormal basis{λ_0, λ_ksin(2πk·), λ_kcos(2πk·) }_k ∈ ,where . The d-variate case is defined by the tensor product,_(^d) := ⊗_j=1^d_().We assume , and then the initial error is constant 1.For this situation we obtain the curse of dimensionality in the deterministic setting, see thm:curseperiodic. The deterministic lower bound is based on a technique due to Kuo, Wasilkowski, and Woźniakowski <cit.>, also Cobos, Kühn, and Sickel <cit.>, see sec:HilbertWorLB. For the randomized setting, we derive conditions on the reproducing kernel of periodic Hilbert spaces such that the associated Gaussian process is bounded. Specifically, we consider Korobov spaces with for , heresuch that the initial error may still be adjusted with . For smoothness we can show that the approximation problem: H_r^(^d) ↪ L_∞(^d)possesses the Monte Carlo complexityn^(,d,r) ≤ C_r d (1 + log d)^-2 ,where . Hence this problem is polynomially tractable. For smaller smoothness , we can still prove polynomial tractability with a worse complexity bound, in that case the fundamental Monte Carlo method is only applied to a finite dimensional subspace of , see thm:Korobov. By this, Monte Carlo breaks the curse. §.§.§ On chap:monotone: Approximation of Monotone Functions We study the L_1-approximation for the class of bounded monotone functions,F_^d := {f : [0,1]^d → [0,1] | ≤⇒ f() ≤ f() } ,based on function values as information. This problem is not linear since the input set is unbalanced. Hinrichs, Novak, and Woźniakowski <cit.> showed that the problem suffers from the curse of dimensionality in the deterministic setting. This is not the case in the randomized setting anymore, still the problem is very difficult.From a result by Blum, Burch, and Langford <cit.> for monotone Boolean functions , we can conclude that for fixed the complexity depends exponentially on at least. In sec:monoMCLBs a modified proof is given by what we obtain a lower bound that includes a meaningful -dependency,n^(,d) > νexp(c√(d) ^-1) for _0√(d_0/d)≤≤_0 ,where and , see thm:monotonLB. In particular, choosing a moderately decaying sequence of error tolerances , we observe that the complexity grows exponentially in d. This implies that the problem is not weakly tractable, see rem:monMCLBintractable.In sec:monoUBs we prove upper bounds which show that, for fixed , the complexity indeed depends exponentially on √(d) times some logarithmic terms only. The algorithmic idea has been performed for monotone Boolean functions in Bshouty and Tamon <cit.>, see sec:BooleanUBs. Inspired by this, in sec:monoRealUBs a new Monte Carlo algorithm (A_r,k,n^ω)_ω with desirable error bounds for real-valued monotone functions defined on is proposed and studied, here . Essentially, we use standard Monte Carlo approximation for the most important wavelet coefficients of the Haar basis in , using n random samples. The output will be constant on subcubes of sidelength 2^-r, so only wavelet coefficients up to a certain resolution come into consideration. Further, only those wavelet coefficients are of interest that – for an input function f – measure the simultanious dependency on at most k variables. For fixed , this parameter has the asymptotic behaviour . There is a linear version of the algorithm, see thm:monoUBsreal, and a non-linear version with improved -dependency of the complexity, see rem:monoMCUBeps. scrplain arabicscrheadings [section]chapterCHAPTER: BASIC NOTIONS IN INFORMATION-BASED COMPLEXITYIn sec:IBC e(n),N the basic notions for the model of computation and approximation in information-based complexity (IBC) are introduced. In sec:tractability on tractability we provide the notions for a classification of multi-dimensional problems by the difficulty of solving them. After these two sections the reader may immediately go forward to one of the three main chapters (Chapters 2–4) that cover different topics. sec:VaryCard on algorithms with varying cardinality is an extension of the computational model, we collect tools that help to extend lower bounds to this broader class of algorithms. sec:measurable is a comment on the computational model, especially on measurability assumptions, it has no further connection to the rest of the thesis.§ TYPES OF ERRORS AND INFORMATIONWe collect all notions we need for a basic understanding of information-based complexity (IBC). For an elaborate introduction to this field, refer to the book of Traub, Wasilkowski, and Woźniakowski <cit.>.Let S: F→ G be the so-called solution mapping between the input space F, and the target space G which is a metric space. We aim to approximate S for inputs from an input set with respect to the metric _G of the target space G, using algorithms that collect only a limited amount of information on the input by evaluating finitely many functionals from a given class Λ.A very common example are linear problems where * S is a linear operator between Banach spaces, * the input set F is the unit ball in F, or – more generally – a centrally symmetric convex set, and * the class Λ of all admissible functionals is a subclass of the class of all continuous linear functionals.Chapters <ref> and <ref> deal with linear problems. In chap:monotone, however, we will consider an input set F consisting of monotone functions which is not centrally symmetric. Within this research we mainly examine approximation problems , , that is, F is identified with a subset of G. Another typical example for problems is the computation of the definite integral , , with F being a class of integrable functions ; here, algorithms may use function values, also called standard information .Let (Ω,Σ,ℙ) be a suitable probability space. Further, let (G) denote the Borelof G, and ℱ be a suitableon F, e.g. the Borelif F is a metric space. By randomized algorithms, also called Monte Carlo algorithms, we understand -measurable mappings . This means that the output for an input f is random, depending on . We consider algorithms of cardinality n that use at most n pieces of information,[ See sec:VaryCard for the extention of the computational model to algorithms with varying cardinality.] i.e. whereis the so-called information mapping. The mappinggenerates an output as a compromise for all possible inputs that lead to the same information .[ Some authors call ϕ^ω an algorithm and a method. In this dissertation, “method” and “algorithm” are used synonymously, both referring to . ] If, for any information vector , we take the output as the solution for an elementfrom the input set which interpolates the data, that means , then the algorithm is called interpolatory. The combinatory cost for the computation of ϕ (arithmetic operations, comparison of real numbers, operations in G) is usually neglected.[ We make one exception in rem:monoMCUBphicost, where we compare two different outputs ϕ.]There are different types of information mappings. In this research the information is obtained by computing n functionals from the class Λ for the particular input. This could be function values , or arbitrary continuous linear functionals . We do not care about how these functionals are evaluated – they could be obtained by some measuring device or by a subroutine provided by the user – to us, evaluating an information functional is an oracle call. An information mapping is called non-adaptive, ifN^ω(f) = [L_1^ω(f),…,L_n^ω(f)] = (y_1,…,y_n) =,where all functionals are chosen independently from f. In that case, N^ω is a linear mapping for fixed . For adaptive information N^ω the choice of the functionals may depend on previously obtained information, we assume that the choice of the k-th functional is a measurable mappinginto the space of functionals, here,for . Further, the mapping as a whole shall be -measurable. By we denote the class of all Monte Carlo algorithms that use n pieces of adaptively obtained information, for the subclass of non-adaptive algorithms we write .If the solution operator S is a linear operator that maps between Banach spaces, we consider two more special types of algorithms.Linear algorithms comprise non-adaptive algorithms where not only N^ω, but also ϕ^ω, and therefore , is linear for every . For linear algorithms we usually say rank instead of cardinality.As another special class we consider homogeneous algorithms . The information mapping may still be adaptive, however with the special constraintL_k,_[k-1]^ω = L_k,λ _[k-1]^ωfor information vectors = N^ω(f) and . In particular, this implies homogeneity for the info mapping,for all and . For the mapping ϕ we assume the same, , thus inducing .We regard the class of deterministic algorithms as a subclass of algorithms that are independent from ,[ This means in particular that we assume deterministic algorithms to be measurable. For a deeper discussion on measurability see sec:measurable.] for a particular algorithm we write , omitting the random element . For a deterministic algorithm A_n, the (absolute) error at f is defined as the distance between output and exact solution,e(A_n,S,f) := _G(A_n(f),S(f)).For randomized algorithms A_n = (A_n^ω(·))_ω∈Ω, this can be generalized as the expected error at f,e(A_n,S,f) := 𝔼_G(A_n^ω(f),S(f)),however, some authors prefer the root mean square errore_2(A_n,S,f) := √(𝔼_G(A_n^ω(f),S(f))^2) .(The expectation is written for the integration over all ω∈Ω with respect to .) Note that .Another criterion for rating Monte Carlo methods is the margin of error[ This is a common notion in statistics.] for some preferably small uncertainty level ,e_δ(A_n,S,f) := inf{ > 0 |(_G(A_n^ω(f),S(f)) > ) ≤δ} ,in other words, we have a confidence level for the error . This criterion is more difficult to analyse than the other two definitions of a Monte Carlo error, however, a basic understanding of the power of randomization can already be gained with a simple mean error criterion.[ In sec:BooleanUBs we cite an algorithm proposed by Bshouty and Tamon <cit.>. They studied the margin of error, but we only reproduce the analysis for the expected error.]If the input space F is a normed space, one can also consider the normalized error criterion where for deterministic algorithms theis defined ase_(A_n,S,f) := S(f) - A_n(f)_G/f_F .The normalized error for randomized algorithms is defined analogously.The global error of an algorithm A_n is defined as the error for the worst input from the input set F ⊂F, we writee(A_n,S,F) := sup_f ∈ F e(A_n,S,f).For technical purposes, we also need the μ-average error, which is defined for any (sub-)probability measure μ (the so-called input distribution) on the input space F,e(A_n,S,μ) := ∫ e(A_n,S,f) μ(f).(A sub-probability measure μ on F is a positive measure with .)The difficulty of a problem within a particular setting refers to the error of optimal algorithms, we define the n-th minimal errore^♢,⋆(n,S,F,Λ) := inf_A_n ∈𝒜_n^♢,⋆(Λ) e(A_n,S,F) and e^♢,⋆(n,S,μ,Λ) := inf_A_n ∈𝒜_n^♢,⋆(Λ) e(A_n,S,μ),where and . These quantities are inherent properties of the problem S with proper names. So, given an input set F, the worst input error for optimal randomized algorithmsis called the Monte Carlo error, the worst input error for deterministic algorithmsis called the worst case error of the problem S. Given an input distribution μ, we only consider deterministic algorithms, and – for better distinction from the other two settings – we introduce a new labelling , calling it the μ-average (case) error of the problem S. For n=0 we obtain the initial error, that is the minimal error that we achieve if we have to generate an output without collecting any information about the actual input. The inverse notion is the -complexity[ More precisely, we should call this quantity information-based -complexity. In the book on IBC by Traub et al. <cit.> it is called -cardinality, whereas the notion complexity is associated to the total computational cost taking combinatory operations such as addition, multiplication, comparisons and evaluation of certain elementary functions into account.] for a given error tolerance ,n^♢,⋆(,S,∙,Λ) := inf{n ∈_0 |∃_A_n ∈𝒜_n^♢,⋆(Λ) , e(A_n,S,∙) ≤} ,where ∙ either stands for an input set F⊂F, or for an input distribution μ. Obviously, in any setting, the error is monotonously decreasing for growing n. Similarly, the inverse notion of complexity is growing for . By definition, the error (or the complexity, respectively) is smaller or equal for smaller input sets , e^♢,⋆(n,S,F',Λ) ≤ e^♢,⋆(n,S,F,Λ). In general, a broader class of algorithms can only lead to a smaller error (and complexity), so, since adaption and randomization are additional features for algorithms, we have e^,⋆(n,S,∙,Λ) ≤ e^,⋆(n,S,∙,Λ) and e^♢,(n,S,∙,Λ) ≤ e^♢,(n,S,∙,Λ). For the same reason, more general classes of information functionals will diminish the error (and the complexity), e^♢,⋆(n,S,∙,Λ') ≤ e^♢,⋆(n,S,∙,Λ). If for a particular problem function evaluations are continuous, then arbitrary continuous functionals are a generalization, so in that case we have . Another important relationship connects average errors and the Monte Carlo error. It has already been used by Bakhvalov <cit.>. Let μ be an arbitrary (sub-)probability measure supported on the input set . Then e^,⋆(n,S,F,Λ) ≥e^,⋆(n,S,μ,Λ). Let be a Monte Carlo algorithm. We find e(A_n,S,F) = sup_f ∈F e(A_n^ω,S,f) ≥∫e(A_n^ω,S,f) μ(f) [Fubini] = ∫e(A_n^ω,S,f) μ(f) = 𝔼 e(A_n^ω,S,μ) ≥inf_ω e(A_n^ω,S,μ) ≥inf_A_n^' ∈𝒜_n^,⋆(Λ) e(A_n^',S,μ). Here, we used that for any fixed elementary event the realization A_n^ω can be seen as a deterministic algorithm. The proof of the above relation also showse^,⋆(n,S,μ,Λ) = e^,⋆(n,S,μ,Λ) ≡ e^,⋆(n,S,μ,Λ),so there is no need for randomized algorithms in an average case setting. Bakhvalov's technique provides the standard tool for proving lower bounds for the Monte Carlo error by considering particular average case situations. This has the advantage that we have to deal only with deterministic algorithms. We have some freedom to choose a suitable distribution μ.[ There are only few situations where lower bounds for the Monte Carlo error have been proven directly without switching to the average case setting, see for example the non-adaptive Monte Carlo setting for the integration of univariate monotone functions in Novak <cit.>, or an estimate for small errors for the approximation of monotone Boolean functions in Bshouty and Tamon <cit.>, see also rem:monMCLBintractable.] The proof of upper bounds basically relies on the analysis of proposed algorithms. Mathé <cit.> showed that in several cases one can theoretically find input distributions μ supported on F such that the μ-average error matches the Monte Carlo error. Lower (upper) bounds for the n-th minimal error correspond to lower (upper) bounds for the -complexity, in detail, e^♢,⋆(n_0,S,∙,Λ) > _0 ⇒ n^♢,⋆(_0,S,∙,Λ) > n_0. Consequently, Bakhvalov's technique can also be written down in the notion of -complexity, n^,⋆(,S,F,Λ) ≥ n^,⋆(,S,μ,Λ) .If no confusion is possible, in the future we will use a reduced notation, e.g. writinginstead of if the class Λ of information functionals is known from the context, the input set F is the unit ball of the input space F in the setting of a linear operator S between Banach spaces, and taking into account that adaptive algorithms are the most general type of algorithms we consider. The same applies for the complexity . In any case, the notation should be compact, yet include all aspects needed to distinguish different settings within the context. § TRACTABILITYWe give a short overview over different notions used in tractability theory. For a more detailed introduction refer to the book by Novak and Woźniakowski <cit.>.In tractability analysis we do not just consider a single solution operator but an entire family of solution operators(S^d : F^d → G^d)_d ∈,with d being a dimensional parameter. This could mean, for example, that F^d and G^d are classes of d-variate functions defined on the unit cube .In classical numerical analysis, however, the dimension d is typically considered a fixed parameter – along with smoothness parameters etc. – so within a complexity setting[ The notion complexity setting comprises all features of algorithms like adaptivity or non-adaptivity, randomization, the class of information functionals Λ, as well as the error criterion, be it the absolute or the normalized error.] the error is perceived as a function in n,e^♢,⋆(n,S^d,F^d,Λ) = e(n).For solvable problems this function is monotonously decreasing and converging to 0 for . Problems are then classified by their speed of convergence: * a function e(n) converges faster than a function e'(n) iff , we write , * a function e(n) converges at least as fast as a function e'(n) iff there exists a constant and such that for , we write , * two functions e(n) and e'(n) have the same speed of convergence iff and , we write .[ In chap:Bernstein we will encounter relations like . It is worth thinking about an alternative definition of equal speed, which holds if there exist constants andsuch that ce(k_1 n) ≤ e'(k_2 n) ≤ C e(k_3 n) for sufficiently large n. For polynomial rates this will not make any difference, but if exponential functions are involved, two functions exp(-n^p) and exp(-2 n^p) would be classified the same speed of decay only for the new notion.]A widely used classification is done by the comparison to polynomial decay, a problem has the order of convergence at least p iff , where . Determining the optimal order of convergence means finding constants c,C > 0 such that for large n we havec n^-p≤ e(n) ≤ C n^-p.A common phenomenon when determining the optimal order p_d for d-variate problems is that the corresponding constants c_d and C_d deviate widely, and even worse, “large n” meansand n_0(d) can be huge for growing dimension d. In sec:MonoOrder we find an example where difficulties become apparent as soon as we consider the inverse notion of -complexity. Last but not least, for discrete problems such as the approximation of Boolean functions, see chap:monotone, the concept of order of convergence is meaningless since discrete problems may be solved with a finite amount of information.For tractability analysis now, we regard the complexity as a function depending on > 0 and d ∈,n^♢,⋆(,S^d,F^d,Λ) = n(,d). A first approach to this complexity function is to fix > 0 and to consider the growth in d, see for example the results on lower bounds in cor:LinfAppLB, thm:curseperiodic, or thm:monotonLB. It is unpleasant if the complexity depends exponentially on d, we say that a problem suffers from the curse of dimensionality[ This notion goes back to Bellman 1957 <cit.>.] iff there exist and such thatn(,d) ≥ c (1+γ)^d for d ≥ d_0.There are problems that have arbitrarily high order of convergence but suffer from the curse of dimensionality, see for example the case in sec:Cinf->Linf.For positive results, we do not only want the dependency on d to be moderate, but also the dependency on . A problem is polynomially tractable iff there exist constantssuch thatn(,d) ≤ C^-pd^q.If we can even choose q = 0, that is if the complexity is essentially independent from the dimension d, we have strong polynomial tractability.In contrast to the curse of dimensionality, problems for that the complexity does not depend exponentially on d or ^-1, in detail, wherelim_^-1 + d →∞log n(,d)/^-1 + d = 0,are called weakly tractable. This notion is fairly new and has been studied first around the time where the book on tractability, Novak and Woźniakowski 2008 <cit.>, has been written. A problem which is not weakly tractable is called intractable.[ The notion of “intractability” as it is used within the IBC community since the book on the tractability of multivariate problems, Novak and Woźniakowski <cit.>, is different from definitions of “intractability” in other scientific communities. In computer science, see for example the book on NP-completeness by Garey and Johnson <cit.>, all problems that, for solving a problem exactly, need a running time which is superpolynomial in the size m of the input, are called “intractable”. Thus even m^log m would fall into that category. In tractability studies for IBC, instead of the input size we consider the dimension d and the error tolerance , so automatically new notions arose. But also the observation that many problems have a sub-exponential yet superpolynomial running time motivated the introduction of new notions like weak tractability.] Note that there are intractable problems that do not suffer from the curse of dimensionality, for example the randomized approximation of monotone functions, see chap:monotone.More recently, the refined notion of -weak tractability has been promoted in Siedlecki and Weimar <cit.>. It is fulfilled ifflim_^-1 + d →∞log n(,d)/^-s + d^t = 0with . This notion coincides with weak tractability for .Last but not least, Gnewuch and Woźniakowski <cit.> promoted the notion quasi-polynomial tractability. It holds iff there exist constants such thatn(,d) ≤ Cexp[p (1+log^-1)(1+log d)].In this case the complexity behaves almost polynomially in d with an exponent that grows very slowly in ^-1, and vice versa.For an example of quasi-polynomial and -weak tractability, see thm:Cinf->LinfUB.Whether or not a problem falls into one of the tractability classes above, highly depends on the particular choice of the d-dependent setting . One criterion of a natural d-dependent problem could be that the input set F^d can be identified with a subset of F^d+1, and therefore we can consider S^d to be a restriction of S^d+1. Another possible criterion is whether the initial error is properly normalized, that is, the initial error should be a constant,e(0,S^d,F^d) = c > 0 for all d ∈.Typically c = 1, see for example the problem in sec:Cinf->Linf; however, in chap:monotone we have , see rem:monoInit. § ALGORITHMS WITH VARYING CARDINALITYFor some problems it might be convenient to allow algorithms that collect a varying amount of information, but in average they do not use more than n pieces of information. In Ritter <cit.> one can find examples of average case settings where varying cardinality does help. Anyways, for upper bounds we try to find algorithms that are as simple as possible, whereas for lower bounds it is desirable that they hold for as general classes of algorithms as possible, that is, we allow for randomization, adaption, or even varying cardinality.[ Similarly, it is good to find lower bounds for very small input sets, but upper bounds that hold for very general and large input sets.] In the end one might see what features are really making a big difference.We need to adjust our model of algorithms where the number of information we collect may depend on the random element ω and (adaptively) on the input. Now, the information mapping shall be a mapping yielding an information sequence , and for possible information sequences we need to define an output via a mapping . As before, the k-th piece of information is obtained by evaluating an adaptively chosen functional from a given class Λ, or the zero functional,[ Considering for example , in general the zero functional is not a function evaluation.]y_k := L_k,_[k-1]^ω(f). At some point we need to stop collecting further information. Within the model, this means that for some index we choose L_k,_[k-1]^ω to be the zero functional for all , so the actual amount of information for a particular algorithm is a functionn(ω,) := inf{n ∈_0 | L_k,_[k-1]^ω = 0 for all k > n } ,with being a proper information sequence.[ For fixed ω∈Ω, not all sequences ∈^ can be the outcome of the information mapping N^ω.] (For non-adaptive algorithms this function is independent from the input, , for deterministic algorithms it is a function .) For convenience, we will also write instead of . Then the worst input cardinality of the algorithm is defined as(A) = (A,F) := sup_f ∈ F n(ω,f).For any (sub)-probability measure μ the μ-average cardinality is(A,μ) := ∫ n(ω,f)μ( f).The μ-average cardinality is usually defined for deterministic algorithms.As before, we define different classes of algorithmswhere and . The definition of the error for a particular algorithm does not change, however, the new concept of cardinality brings about new error and complexity notions associated to a problem . For we havee̅^♢,⋆(n̅,S,∙,Λ) := inf_A ∈𝒜^♢,⋆(Λ)(A,∙) ≤n̅ e(A,S,∙),and for a given error tolerance we definen̅^♢,⋆(,S,∙,Λ) := inf{n̅≥ 0 |∃ A ∈𝒜^♢,⋆(Λ) : (A,∙)≤n̅ , e(A,S,∙) ≤} ,where for ∙ we may insert an input set , or an input distribution μ. Be aware that the cardinality may be a real number now.Note that algorithms from classes of fixed cardinalitycan be identified with methods from , so for n̅≥ 0 we have the general estimatee̅^♢,⋆(n̅,S,∙,Λ) ≤ e^♢,⋆(⌊n̅⌋,S,∙,Λ).For the worst case setting it is easy to see that the new notion even coincides with the old notion of fixed cardinality, that is, for we havee̅^,⋆(n,S,F,Λ) = e^,⋆(n,S,F,Λ). For Monte Carlo methods with non-adaptively varying cardinality n(ω), there is a direct relation to the fixed cardinality setting. This relation is well known, see Heinrich <cit.>. For n ∈ we have e̅^,(n,S,F,Λ) ≥1/2e^,(2n, S, F, Λ). The proof also works for classes of adaptive algorithms as long as the actual cardinality does not depend on the input. In this sense, let be a Monte Carlo algorithm with non-adaptively varying cardinality such that . Then we have e(A,F) = sup_f ∈ F e(A^ω,S,f) ≥sup_f ∈ F e(A^ω,S,f) _{n(ω) ≤ 2n}= {n(ω) ≤ 2n} sup_f ∈ F' e(A^ω,S,f)Here, ' denotes the expectation for the conditional probability space where we integrate over with respect to the conditional measure . We can regard as a Monte Carlo algorithm from the class with another underlying probability space than for A. Together with (by Markov's inequality), this gives the lower bound e(A,F) ≥1/2e^,⋆(2n,S,F,Λ).For Monte Carlo methods with adaptively varying cardinality we need special versions of Bakhvalov's technique. For any (sub-)probability measure μ on F, and , the Monte Carlo error and the average error in the setting of (adaptively) varying cardinality are related by e̅^,(n̅,S,F,Λ) ≥1/2 e̅^,(2n̅, S, μ, Λ). If we have an estimate with a convex and decaying function (n̅) for , the lower bound can be improved to e̅^,(n̅,S,F,Λ) ≥(n̅). Let be a Monte Carlo algorithm with adaptively varying cardinality such that n̅ ≥(A,F) := sup_f ∈ F n(ω,f).We can relate this to the average cardinality with respect to μ regarding A^ω as a deterministic algorithm for fixed ω, ≥∫ n(ω,f) μ( f)[Fubini] = ∫ n(ω,f)μ( f) _= (A^ω, μ) [Markov's ineq.] ≥ 2n̅ {(A^ω,μ) > 2n̅} . This gives us the estimate {(A^ω,μ) ≤ 2n̅}≥1/2 . Now, considering the error, we find e(A,F) = sup_f ∈ F e(A^ω,f) ≥∫ e(A^ω,f)μ( f)[Fubini] = ∫ e(A^ω,f)μ( f) =e(A^ω,μ) ≥e̅^,⋆((A^ω,μ), S, μ, Λ). A rough estimate via (<ref>) will give e(A,F) ≥{(A^ω,μ) ≤ 2n̅}_= 1/2 e̅^,⋆(2n̅, S, μ, Λ). If we have a specially structured estimate for the average error, we can proceed in a better way, e(A,F) ≥((A^ω,μ))[convexity] ≥((A^ω,μ))[monotonicity] ≥(n̅). This finishes the proof. A similar convexity argument will help to find good bounds for average case settings with varying cardinality. Let μ be a probability measure on F. Assume that for any deterministic algorithm with varying cardinality there exists a version of the conditional measure μ_ such that ∫_G(ϕ(),S(f))μ_( f) ≥(n()), where is convex and decaying for . Then the average error for algorithms with varying cardinality is bounded by this function, e̅^,(n̅,S,μ,Λ) ≥(n̅). If μ is supported on F, by lem:n(om,f)Bakhvalov the very lower bound holds for the Monte Carlo error as well. Let be a deterministic algorithm with adaptively varying cardinality n(f) such that . By definition we have e(A,μ) = ∫ e(A,f)μ( f). We split the integral into the integration over ∈ N^ω(F), with an appropriate conditional distributions μ_ on , fulfilling the assumptions of the lemma, and obtain = ∫[∫_G(ϕ(),S(f)) μ_( f) ] μ∘ N^-1() ≥∫(n()) μ∘ N^-1()[convexity] ≥(∫ n(f)μ( f)) [monotonicity] ≥(n̅) .By lem:n(om)MC we see that non-adaptively varying cardinality does not help a lot when trying to find better Monte Carlo algorithms. For adaptively varying cardinality the situation is slightly more complicated; however, lem:n(om,f)Bakhvalov gives us a tool to prove lower bounds that are similar to those that we can obtain for the fixed cardinality setting, see sec:n(om,f)Bernstein for an application of this lemma. In many more cases even better, lem:n(om,f)avgspecial applies to the average setting so that we obtain lower bounds which coincide with the computed estimates for the fixed cardinality setting. In this dissertation, we have this nice situationfor the lower bounds in thm:BernsteinMChom (homogeneous algorithms and Bernstein numbers), and in thm:MonAppOrderConv and thm:monotonLB (approximation of monotone functions).This justifies that in the main parts of this thesis we focus on algorithms with fixed cardinality. § ON THE MEASURABILITY OF ALGORITHMSIt seems natural to assume measurability for algorithms since real computers can only deal with a finite amount of states. In the IBC setting, however, it is convenient to assume that we can operate with real numbers, otherwise the concept of linear algorithms for real-valued functions would not make sense. Further justification for why we work with the real number model is gathered in Novak and Woźniakowski <cit.>. Unfortunately, the real number model tails the problem of measurability. Heinrich and Milla <cit.> presented a simple Monte Carlo sampling algorithm for indefinite integration that at first view appears natural but, in fact, is not measurable.[ I would like to thank Mario Hefter for interesting discussions on measurability of algorithms during our stay at Brown University's ICERM in fall 2014. I would also like to thank Prof. Dr. Klaus Ritter for pointing me to the paper of Heinrich and Milla <cit.>.] We will comment on that.Consider the indefinite integrationS^d : L_p([0,1]^d) → L_∞([0,1]^d), [S^d(f)]() := ∫_, f λ^d,where . The simple Monte Carlo sampling algorithm A_n is given by[A_n(f)]() := 1/n∑_i=1^n [_i ≤] f(_i),with iid random variables . As discussed in <cit.>, this algorithm is not measurable since the method is not separably valued. Indeed, considering the constant function f_1 = 1, for two realizations A_n^ω and A_n^ω' with distinct sample points and modulo ordering, we have . Still, the error mappingis measurable and an error analysis makes sense.[ In detail, Heinrich and Milla <cit.> showed polynomial tractability in the randomized setting. Note that in the deterministic setting the problem is unsolvable because we may only use function values of L_p-functions as information. By this, indefinite integration is an example of a problem where the output space consists of functions and where randomization does help. Heinrich and Milla also note that only few polynomially tractable problems with unweighted dimensions have been known so far. Their example of indefinite integration is such an unweighted problem with polynomial tractability. In sec:HilbertPeriodic of this dissertation we add another example: The L_∞-approximation of Hilbert space functions from unweighted periodic Korobov spaces with standard information is polynomially tractable in the Monte Carlo setting.] In detail, Heinrich and Milla show that it suffices to consider the pointwise difference for points from a regular grid with mesh size 1/m, see <cit.>.This motivates a very natural measurable modification of the sample algorithm. A computer can only store finitely many digits of the coordinates of , in general, for fixed ω, the function is not exactly implementable. Therefore, let be the largest rational number representable with r binary digits after the radix point, . The algorithm [A_n,r(f)]() := 1/n∑_i=1^n [_i^(r)≤] f(_i)is composed of measurable mappings, and thereby measurable itself. Indeed, the mappingΩ→ L_∞([0,1]^d),ω↦[(_i^(r))^ω≤· ]is measurable since it only has discrete values. Furthermore, the pointwise error of A_n and A_n,r coincides on the grid Γ_2^r. With increasing r we can get arbitrarily close to the error of A_n, compare Heinrich and Milla <cit.>. The original publication contains another modification with continuous outputs. The modification given here, however, nourishes the belief in measurability of implementable algorithms.This was an example of non-measurability of Monte Carlo algorithms. Typically, measurability is an assumption in the randomized and in the average setting, but we do not need it for the worst case setting. If we assume measurability only for randomized algorithms, but allow non-measurability for deterministic algorithms in the worst case setting, for linear problems S with the input set F being the unit ball in F and with general linear information , one can still statee̅^(n,S,F,) ≤ 4 e^(n,S,F,),see Heinrich <cit.>.As we have seen in the example above, we do not really need measurability for the algorithm as long as the error mapping is measurable. Measurability, however, is a convenient assumption, especially for the average case analysis in chap:Bernstein, where we need to establish the conditional measure for given information , see sec:GaussCond. As long as we do not find meaningful non-measurable algorithms that could not be replaced by equally successful measurable algorithms, measurability is a justifiable assumption for lower bound studies. For the lower Monte Carlo bounds in chap:monotone, however, measurability is unproblematic since we consider average settings with discrete measures, see thm:MonAppOrderConv and thm:monotonLB. In that case, relaxed measurability assumptions would suffice, but we do not go into details.We finish with a final remark on an alternative approximation concept, aside from the IBC setting with the real number model. Given a numerical problem , we define entropy numbers for n ∈_0,e_n(S,F) := inf{ > 0| ∃ g_1,…,g_2^n∈ G : S(F) ⊆⋃_i=1^2^n B_G(g_i,) } ,where denotes the closed -ball around , see Carl <cit.>, alternatively Pisier <cit.>. One interpretation of this concept is the question on how well we can approximate the problem S if we are only allowed to use n bits to represent 2^n different outputs.[ The given definition contains an index shift compared to the definition to be found in Carl <cit.>. This is a matter of taste. Here,coincides with the initial error from the IBC setting. According to Carl's notation, we would start with n=1, and the initial error would match . Similar index shifts compared to related notions from IBC are commonly found for s-numbers, following the axiomatic scheme of Pietsch <cit.>. Contrarily, Hutton, Morrell, and Retherford <cit.> use a definition of approximation numbers which happens to fit the IBC notion. Heinrich <cit.>, in turn, in his paper on lower bounds, on which chap:Bernstein is based on, and Mathé <cit.> in his fundamental research on random approximation by , which inspired chap:Hilbert, both kept consitency with the s-number conventions, even for the definition of the Monte Carlo error. In this thesis, however, we strictly follow IBC conventions for error quantities. In contrast, for the definition of Bernstein numbers in sec:BernsteinSetting, we use a definition which fits to the s-number scheme, see the footnote given there for additional justification. See also lem:H->,lin=opt,singular (b) for the link between singular values and the worst case error in the Hilbert space setting.] Carl studies linear problems and establishes a lower bound for certain s-numbers based on entropy numbers. Some of the s-numbers are closely related to the error quantities for deterministic algorithms with , approximation numbers correspond to the error of linear methods, Gelfand numbers are linked to the error of general deterministic methods.[ See the book on IBC by Traub et al. <cit.>.]Within this dissertation, in rem:monMCLBintractable we cite a lower bound for the Monte Carlo error for the approximation of monotone functions which is due to Bshouty and Tamon <cit.>. Their proof uses an entropy argument.In chap:Hilbert, in the context of estimates on the expected maximum of zero-mean Gaussian fields, we will step across the inverse concept metric entropy , that is the logarithm of the minimal number of -balls needed to cover a set, see prop:Dudley (Dudley).CHAPTER: LOWER BOUNDS FOR LINEAR PROBLEMS VIA BERNSTEIN NUMBERSWe consider adaptive Monte Carlo methods for linear problems and establish a lower bound via Bernstein numbers, see sec:BernsteinSetting for the definition and an overview of already known relations. The abstract main result and the proof is contained in sec:BernsteinAda. It is based on a technique due to Heinrich <cit.> that relates the Monte Carlo error to norm expectations of Gaussian measures. The innovation is the application of Lewis' theorem in order to find optimal Gaussian measures, see sec:OptGauss. Within the supplementary sec:BernsteinSpecial we present versions of the main result for two interesting special settings: varying cardinality, and homogeneous algorithms. A major application is the L_∞-approximation of certain classes of C^∞-functions, see sec:Cinf->Linf. With the new technique we obtain lower bounds via Bernstein numbers, which show that in these cases randomization cannot give us better tractability than that what we already have with deterministic methods. § THE SETTING AND BERNSTEIN NUMBERSLet be a compact linear operator between Banach spaces over the reals. Throughout this chapter the input set is the unit ball of F. We consider algorithms that may use arbitrary continuous linear functionals as information.The operator S can be analysed in terms of Bernstein numbersb_m(S) := sup_X_m ⊆Finf_f ∈ X_m f = 1S(f)_G,where the supremum is taken over m-dimensional[ Some authors take the supremum over -dimensional spaces <cit.>, which might be motivated by relations like (<ref>). The present version, however, is also in common use <cit.>, besides it looks quite natural, and in view of the sharp estimate (<ref>), any index shift would appear like a disimprovement. Although Bernstein numbers are not s-numbers, according to the definition in Pietsch <cit.>, in some cases they coincide with certain s-numbers, in particular for operators between Hilbert spaces where Bernstein numbers match the singular values. ] linear subspaces . These quantities are closely related to the Bernstein widths[ I wish to thank Prof. Dr. Stefan Heinrich for making me aware of the non-equivalence of both notions.] of the image within G,b_m(S(F),G) := sup_Y_m ⊆ Gsup{r ≥ 0| B_r(0) ∩ Y_m ⊆ S(F) } ,where the first supremum is taken over m-dimensional linear subspaces . By we denote the (closed) ball around with radius r. In general, Bernstein widths are greater than Bernstein numbers, however, for injective operators (like embeddings) both notions coincide (consider Y_m = S(X_m)). In the case of Hilbert spaces F and G, Bernstein numbers and widths match the singular values σ_m(S).For deterministic algorithms it can be easily seen thate^(n,S) ≥ b_n+1(S(F),G) ≥ b_n+1(S),since for any information mapping and any , there always exists an with and , i.e. f cannot be distinguished from . If both F and G are Hilbert spaces, lower bounds for the (root mean square) Monte Carlo error have been found by Novak <cit.>,e_2^(n,S) ≥√(2)/2 σ_2n(S).For operators between arbitrary Banach spaces the estimate reads quite similar, see thm:BernsteinMCada,e^,(n,S) > 1/30b_2n(S).The constant can be improved for extremely large n, see rem:1/6*b_2m, or when imposing further assumptions. The following lower bound for non-adaptive algorithms has been proven first within the author's master thesis and published later in <cit.>,e^,(n,S) ≥1/2b_2n+1(S).For homogeneous algorithms, possibly adaptive, and even with varying cardinality, one can prove an estimate with optimal constant, see thm:BernsteinMChom,e^,(n,S) ≥1/2b_2n(S).Within <cit.> the results of (<ref>) and (<ref>) have been mentioned, for the adaptive setting a proof for a result with slightly worse constants based on results from Heinrich <cit.> has been given. In this chapter now one can find a self-contained proof following the lines of Heinrich <cit.> but with optimized constants and slight simplifications that are possible when relying on Bernstein numbers. § ADAPTIVE MONTE CARLO METHODSthm:BernsteinMCada below is the main result of this chapter. The proof needs several results that are provided in the subsequent subsections.The proof is based on the idea of Heinrich <cit.> to use truncated Gaussian measures in order to obtain lower bounds for the Monte Carlo error. Considering Gaussian measures is quite convenient as there is an easy representation for the conditional distribution, even when collecting adaptive information, see sec:GaussCond. The key tool for Heinrich's technique is a deviation result for zero-mean Gaussian measures μ̃ on a normed space F,μ̃{f: f_F > λ ^μ̃f_F}≤exp(- (λ-1)^2/π),see cor:deviationGauss. This shows how far the norm may deviate from its expected value, that way enabling us to estimate how much we lose when truncating a Gaussian measure. The expected norm for a truncated Gaussian measure is estimated in sec:E|JX|trunc, in the case of Bernstein numbers a simplified result with slightly better constants is feasible.The new idea now is to apply Lewis' theorem in order to find optimal Gaussian measures that are “well spread” into all directions within the input space F, see sec:OptGauss.[ This idea has already been published in <cit.>. I wish to thank Prof. Dr. Aicke Hinrichs and my doctoral advisor Prof. Dr. Erich Novak for pointing me to the book of Pisier <cit.> in search of optimal Gaussian measures.]This dissertation includes a self-contained proof of the theorem. That way we are able to adapt for simplifications that are possible in the particular situation. Furthermore, we work on the improvement of constants. For being a compact linear operator between Banach spaces, and the input set F being the unit ball in F, we have e^,(n,S,F,) > 1/15 m-n/mb_m(S) for m > n. For all > 0 there exists an m-dimensional subspace X_m ⊆F such that S(f)_G ≥f_F (b_m(S) - ) for f ∈ X_m. Note that for the restricted operator we have , and in general . Hence it suffices to show the theorem for S|_X_m, so without loss of generality we assume , and therefore holds for all . Below,andare used to describe probabilities and expectations for an average case setting whenever it seems convenient. This is not to be confused with the probability space used to define Monte Carlo algorithms within chap:basics. Let be a standard Gaussian vector within ^m = ℓ_2^m. We choose a matrix in order to define a Gaussian measure μ̃ on F as the distribution of . The restricted measure μ(E) := μ̃|_F (E) = {J ∈ E ∩ F} , for measurable E ⊆F, is a sub-probability measure supported on the unit ball . By Bakhvalov's technique, see prop:Bakh, we know e^,(n,S,F,) ≥ e^,(n,S,μ,). Let ϕ∘ N : F→ G be an adaptive deterministic algorithm using n pieces of information. Let denote the distribution of the information . Without loss of generality, is surjective, so by lem:condGauss, for all we have an orthogonal projection P_ and an element m_∈F to describe the conditional distribution μ̃_ as the distribution of . We write the error e(ϕ∘ N,μ) = ∫_^n∫_F ∩ N^-1()S(f) - ϕ()_G μ̃_( f) ν̃(). Defining , we can continue using the representation of the conditional measure μ̃_, further cutting off parts of the integral, ≥∫_^n[ S J P_ - g__G _{J P__F ≤ 1 - m__F }] ν̃(). Due to symmetry, the two versions are identically distributed (), so we can rewrite = ∫_^n[1/2∑_σ = ± 1S J P_ + σ g__G _[Δ-ineq.] ≥S J P__G _{J P__F ≤ 1 - m__F }] ν̃(). Applying the triangle inequality, and further truncating the integral, we get ≥∫_{m__F ≤ 1 - r}[S J P__G _{J P__F ≤ r }] ν̃(). Using the definition of the Bernstein numbers, and replacing the projections P_ by a general estimate for orthogonal rank-(m-n) projections, we end up with e(ϕ∘ N,μ) ≥ ν̃{ : m__F ≤ 1-r} inf_P orth. Proj.P = m-n[J P _F _{J P__F ≤ r}] b_m(S). From now on we write . First, we need an estimate for the probability of a small m_, this is done in lem:|m_y|<r with , ν̃{ : m__F ≤ 1-r}≥ 1 - 2exp(- (1-r/2α-1)^2 /π) =: ν(r,α). This estimate is meaningful for . Second, for orthogonal projections P on ℓ_2^m and t>0, the truncated expectation can be estimated by [J P _F _{J P _F ≤λ J _F }] ≥[J P _F _{J P _F ≤λ J P _F }] ≥β(λ)J P _F. Here, within the first step we used , see lem:E|JPX|<=E|JX|. The second inequality is the application of lem:E|JX|trunc with the operator and the constant defined there. In our situation . By cor:optGauss we know that J can be chosen in a way such that for any rank (m-n) projection P on ^m we have[ This idea is new and special for the situation of Bernstein numbers. However, similar properties have been known to Heinrich <cit.> for the special case of the standard Gaussian distribution in sequence spaces ℓ_p^m, compare also rem:UniqueGauss. Heinrich used a symmetry argument that can be found in Mathé <cit.>.] J P _F ≥m-n/m J _F. In detail, we choose , with J being the optimal operator from cor:optGauss, and . Putting all this together, we obtain the estimate e(ϕ∘ N,μ) ≥ cm-n/mb_m(S) with . Note that iff . On the other hand,gives meaningful results for only, so we need in order to obtain positive estimates. Combining this, we have the constraint 0 < α < 1/5.0513 + 2√(πlog 2) < 0.125 = 1/8 . With and , we find a constant . §.§ The Conditional MeasureThe following lemma gives the conditional measure for adaptive information mappings applied in an Gaussian average case setting. The conditional measure is well known since the study of average errors, see the book on IBC, Traub et al. <cit.>. This reference has also been given in Heinrich <cit.>. The proof given here is intended to be self-contained and uses a slightly different notation. Let be a standard Gaussian vector in ^m, and an injective linear operator defining a measure μ̃ on as the distribution of . Furthermore, let be a non-wasteful[ That means, , so if F is m-dimensional, N shall be surjective.] adaptive deterministic information mapping. Then the conditional measure μ̃_, given the information , can be described as the distribution of , with P_ being a suitable rank- orthogonal projection within , and a suitable vector . That is, for all measurable we have μ̃(E) = {J ∈ E} = ∫_^n{J P_ + m_∈ E}_= μ̃_(E) μ̃∘ N^-1 (). Before going into the details of the proof, we want to clarify that expressions containing are random variables, whereas the information vector , and everything depending on it, is fixed. In particular, is the information as a random vector, is an event fixing the information. We denote the partial information for . We will show by induction that the conditional measure , knowing the first k information values _[k], can be represented as the distribution of , with P_k,_[k-1] being a suitable rank- orthogonal projection within , and a suitable vector . Moreover, there exists a vector such that and . For convenience, we also show that the information mapping can be chosen in a way such that the distribution of the partial information is thestandard Gaussian distribution which we denote by γ_k. Starting with means that we have no information ,[ ^0 = {0} is the zero vector space.] the conditional distribution is described by where and , or . The partial information is distributed according to the “zero-dimensional standard Gaussian distribution”, that is . Now for . Given the partial information _[k-1], the (adaptively chosen) information functional L_k,_[k-1] actually gives us information about the random vector . In detail, is a functional in ℓ_2^m, so there exists a representing vector such that the k-th information value (as a random variable for fixed _[k-1]) is . Not waisting any information actually means that is not the zero functional, therefore we may assume . This is equivalent to , and by induction it further implies the orthogonality . In addition, we may assume that such that is a standard Gaussian random variable. We now set P_k,_[k-1] := P_k-1,_[k-2] - ⟨_k,_[k-1], ⟩ _k,_[k-1] , and__[k] := __[k-1] + y_k_k,_[k-1] , thus defining μ̃__[k]. Note that by construction . Then for any measurable set we have μ̃__[k-1](E) = {J P_k-1,_[k-2] + m__[k-1]∈ E}= {J (P_k,_[k-1] + ⟨_k,_[k-1], ⟩ _k,_[k-1]) + J __[k-1]∈ E} (∗)=∫_{J P_k,_[k-1] + J (__[k-1] + y_k_k,_[k-1]) _= J __[k] = m__[k]∈ E } γ_1( y_k) def.=∫_μ̃__[k](E)γ_1( y_k). The step of splitting the integral into two integrations was possible because * the Gaussian random vector is stochastically independent from the Gaussian random variable due to orthogonality, and * the span of is not inside the image of , which provides that for any there is a unique representation with and a vector . Now, by induction, we have μ̃(E) = ∫_^k-1μ̃__[k-1](E) γ_k-1(_[k-1]),which by the above results and the product structure of the standard Gaussian measure may be continued as = ∫_^k-1∫_μ̃__[k](E) γ_1( y_k) γ_k-1(_[k-1]) = ∫_^kμ̃__[k](E)γ_k(_[k]). The lemma is obtained for with and . Having the representation of the conditional measure for the untruncated Gaussian measure, it is of interest to know the probability of obtaining information such that the mass of the conditional measure is concentrated inside the unit ball and therefore truncation is not a great loss. In the situation of lem:condGauss, with , and the image measure , for we have the estimate ν̃{ : m__F ≤ 2κ J _F} ≥ 1 - 2exp(- (κ - 1)^2ρ^2/2) ≥ 1 - 2exp(- (κ-1)^2/π). Writing , basic estimates give ν̃{ : m__F ≤ 2 t} ≥μ̃{f: f_F ≤ t} - μ̃{f: m__F > 2 t with = N(f) and f_F ≤ t}≥ 1 - μ̃{f: f_F > t} - sup_μ̃_{f: f - m__F > t}≥ 1 - {J_F > t} - sup_P orth. proj.P = m-n{J P _F ≥ t} . Applying cor:deviationGauss with gives us ≥ 1 - exp(- (t - J _F)^2 /2J_2 → F^2) - exp(- (t - J _F)^2 /2J P_2 → F^2), which by , see lem:E|JPX|<=E|JX|, and , reduces to ≥ 1 - 2exp(- (t - J _F)^2 /2J_2 → F^2) = 1 - 2exp(- (κ-1)^2ρ^2/2) . The first lower bound is meaningful for , otherwise it is not positive and should be replaced by the trivial lower bound 0. lem:E|JX|>c|J| gives us the general estimate , which leads to the second lower bound. This now is meaningful for . [Why m__F ≤ r is a complicated constraint] We considerfor m ≥ 2 and . The center m_ = _ of the conditional distribution on the affine subspace of inputs with the same information is orthogonal to that subspace, i.e. for all . Consider the situation m_ = _ = (1,1/√(m)+1,…,1/√(m)+1) ∈^m. If, for example, then f = = (2/√(m)+1,…,2/√(m)+1) ∈^m lies within because ( - _) _ . In this situation we have m__∞ = 1 andf_∞ = 2/√(m)+1 0 . §.§ Norm Expectation of Truncated Gaussian MeasuresThe following lemma is a simplification of a result in Heinrich <cit.>. This simplification is only feasible in the situation of Bernstein numbers. The more complicated version is given in lem:E|SJX|trunc. For and being the standard Gaussian vector in , set . Then for λ>1 we have [J _F _{J _F ≤λ J _F }] ≥[1 - (λ+1/(λ-1)ρ^2) exp(- (λ-1)^2ρ^2/2) ]_+_=: β(λ,ρ) J _F≥[1 - (λ+π/2 (λ-1)) exp(- (λ-1)^2/π) ]_+_=: β(λ) J _F. (Note that vanishes for , but it is positive for and monotonically increasing with limit .) With we have [J _F _{J _F ≤ t}] = J _F - t{J _F ≥ t } - ∫_t^∞{J _F ≥ s} s. By the deviation result cor:deviationGauss, and substituting , we can bound this by ≥[1 - λ exp(- (1-λ)^2ρ^2/2) - ∫_λ^∞exp(- (1-κ)^2ρ^2/2) κ] J _F. Using the estimate ∫_λ^∞exp(- (1-κ)^2ρ^2/2) κ ≤∫_λ^∞κ-1/λ-1 exp(- (κ-1)^2ρ^2/2)s = 1/(λ-1)ρ^2 exp(- (λ-1)^2ρ^2/2), we obtain the final lower bound. The factor is monotonically increasing in ρ, so taking the general bound , see lem:E|JX|>c|J|, we obtain the second estimate. Of course, the truncated expectation is non-negative, so we take the positive part of the prefactor.For comparison, we cite the original lemma from Heinrich <cit.> concerning the truncated norm expectation when dealing with two different norms at once. Consider a similar situation to lem:E|JX|trunc above with the ratios and . Then for we have [S J _G _{J _F ≤κ J _F }] ≥[β(λ,σ) - λexp(-(κ-1)^2ρ^2/2) ]_+_=: β̃(κ,λ,ρ,σ) S J _F≥[β(λ) - λexp(-(κ-1)^2/π) ]_+_=: β̃(κ,λ) S J _F. The trick is that we replace the truncation with respect to the F-norm by a truncation with respect to the G-norm, for that purpose introducing an auxiliary parameter λ. We estimate the difference between both truncation variants, [S J _G _{J _F ≤κ J _F }]≥[S J _G _{S J _G ≤λ S J _G }] - (λ S J _G) {J _F > κ J _F} . The first term may be estimated by applying lem:E|JX|trunc to the operator , for the second term we can directly use the deviation result cor:deviationGauss. Heinrich's result <cit.> originally provides lower bounds for the Monte Carlo error via norm expectations of Gaussian measures. In detail, there exists a constant such that for and any injective linear operator J: ℓ_2^m →F we have e^,(n,S,F,) ≥ c'inf_P orth. Proj.P = m-nS J P _G/J _F , where is a standard Gaussian random vector in ^m = ℓ_2^m. The proof works similarly to the proof of thm:BernsteinMCada. In detail, for any one may rescale the operator J such that , we truncate the rescaled measure. The constant then is determined as , now applying lem:E|SJX|trunc instead of lem:E|JX|trunc. With , , and , we obtain which is not much worse than the constant c in thm:BernsteinMCada. How should J be chosen? When applying cor:optGauss from the next section to find an optimal , that is, the image measure shall be “well spread” within G, we may get rid of the infimum within (<ref>) and write e^,(n,S,F,) ≥ c'm-n/m S J _G/J _F . Especially for the identity mapping between sequence spaces , the optimal Gaussian measure will be the standard Gaussian measure, see rem:UniqueGauss. §.§ Optimal Gaussian Measures§.§.§ Lewis' Theorem and Application to Gaussian Measures We want to find optimal Gaussian measures with respect to the F-norm in ^m. We therefore apply Lewis' Theorem, originally <cit.>. The proof given here is taken from Pisier <cit.>. It is included for completeness. Let α be an arbitrary norm on the space of automorphisms . Letmaximize the determinant subject to . Then for any operator we have (J^-1 T) ≤ mα(T). Since any invertible operator J can be rescaled such that , there are admissible operators that fulfil . The constraint defines a compact subset within the finite dimensional space , that is, the supremum sup_α(J)=1(J) > 0 is attained. Let J be a maximizer. Then for any and >0 we have (J + T /α(J + T)) ≤(J). By homogeneity, and after dividing by , (1 +J^-1 T) ≤(α(J + T))^m Δ-ineq.≤ (1 +α(T))^m. Finally, (J^-1 T) = lim_→ 0(1 +J^-1 T) - 1 /≤lim_→ 0(1 +α(T))^m - 1 / = mα(T). For any norm on ^m there is an operator with and J P _F ≥ P/m for any projection P ∈(^m). Here, is a standard Gaussian vector in ^m. First note thatdefines a norm on the space of linear operators J:^m →^m. Indeed, because the expectation operator is linear and is a semi-norm for any linear operator J, and if . We then may apply prop:Lewis withand for projections P. §.§.§ Properties and Examples of Optimal Gaussian Measures The remaining part of this subsection collects some results that expand our knowledge on optimal Gaussian measures but are not necessary for the basic version of the chapter's main result, thm:BernsteinMCada. For any orthogonal matrix , the distribution of is identical to that of , of course, . Consequently, J and J A define the same distribution and are equivalent maximizers of the absolute value of the determinant , subject to . Moreover, J is unique modulo orthogonal transformations, i.e. for all similarly optimal operators there exists an orthogonal matrix A with , see Pisier <cit.> for a proof. In particular, all similarly optimal operators have the same operator norm , and there is one unique optimal Gaussian measure μ associated with F. Let us now consider sequence spaces ℓ_p^m with , and letbe a random vector distributed according to the corresponding optimal Gaussian measure. Due to the symmetry of sequence spaces, for any operator Q permuting the coordinates of a vector in ^m and possibly changing some of their signs, clearly, the distribution of will be optimal as well. By the uniqueness we conclude that the covariance matrix must be a multiple of the identity, so the optimal Gaussian measure for sequence spaces is a scaled standard Gaussian vector. In other words, the optimal operator J may be chosen as a multiple of the identity, . Compare Mathé <cit.> for similar symmetry arguments in a more general setting. This has been used by Heinrich <cit.> to prove properties for standard Gaussian measures on sequences spaces ℓ_p^m which we obtain by optimality according to Lewis' theorem.Next, we find bounds for the operator norm of an optimal operator corresponding to the optimal Gaussian measure on . It is not known to the author whether this particular upper bound has already been proven before, however, its implication together with the deviation result cor:deviationGauss in high dimensions is not surprising, and similar results are known under names such as concentration phenomenon and thin shell estimates. There exists a constant such that for any norm on ^m, , the operator J defined as in cor:optGauss is bounded by J_2 → F≤ c (log m)^-1/2 . On the other hand, we have the lower bound √(π/2)m^-1≤J_2 → F . The lower bound is rather simple. Let denote the projections onto the i-th coordinate. Then, using , we have 1= J_F Δ-ineq.≤∑_i=1^m J P_i _F ≤ m√(2/π) J_2 → F . Now for the upper bounds. There exists an orthogonalprojection P_1 on such that L := J_2 → F = J P_1_2 → F . For the complementaryprojection we have J P_2_2 → F≤J_2 → F = L. Due to orthogonality, we can split into two independent zero-mean random vectors. Letdenote the expectation of the norm of the second part. Clearly, by cor:optGauss we have 1-1/m≤β≤ 1. Note that is a real random variable with probability density p_1(t) := √(2/π L^2) exp(- t^2/2 L^2) [t ≥ 0]. For the cumulative distribution function of the real random variable we write , and by we denote the corresponding density function for . prop:dev2 directly implies that for we have F_2(s) ≥ 1 - exp(-(s-β)^2/2 L^2). Now, by symmetry and independence we have 1 = J_F = 1/2[ J P_1+ J P_2 _F + J P_1- J P_2 _F ] Δ-ineq.≥max(J P_1 _F, J P_2 _F). It follows 1/m(<ref>)≥ 1 - β = J_F - J P_2 _F (<ref>)≥[ max(J P_1 _F, J P_2 _F ) - J P_2 _F ] =[(J P_1 _F - J P_2 _F _=: r) _{J P_1 _F ≥J P_2 _F} ] = ∫_0^∞ r∫_0^∞ p_1(r+s) p_2(s)s r part. int.=∫_0^∞ r[ p_1(r+s) F_2(s) |_0^∞_ = 0 - ∫_0^∞p_1'(r+s)_<0 F_2(s)s ]r (<ref>)≥∫_0^∞ r[ - ∫_0^∞ p_1'(r+t+β) (1-exp(- t^2/2 L^2)) t ]r part. int.=∫_0^∞ r[ - p_1(r+t+β) (1-exp(- t^2/2 L^2)) |_0^∞_ = 0 + L^-2∫_0^∞ t p_1(r+t+β) exp(- t^2/2 L^2)t ]r = √(2/π)L^-3∫_0^∞ rexp(- (r+β)^2/4 L^2) ∫_0^∞ texp(- (t+r+β/2)^2/L^2) tr. Note that for . After the substitution , the inequality can be continued as ≥√(2/π) 1/2L^-3∫_0^∞ rexp(- (r+β)^2/4 L^2) ∫_r+β^∞τ exp(- τ^2/L^2) τ r = √(2/π) 1/4L^-1∫_0^∞ rexp(-5(r+β)^2/4 L^2) r. Usingfor , and with the substitution , we go on to ≥√(2/π) 1/8L^-1∫_2 β^∞ρ exp(-5ρ^2/4 L^2) ρβ≤ 1≥√(2/π) 1/20Lexp(- 5/L^2). For we have L ≥√(π/2)L^2 > √(π/2)20 1/1 + 20/L^2 > √(π/2)20exp(- 20/L^2). By this we finally obtain 1/m≥exp(- 25/L^2). Inverting the inequality, we get L ≤ 5 (log m)^-1/2 . This is the proposition with . As shown before in rem:UniqueGauss, for sequence spaces ℓ_p^m, due to symmetry, the operator J for the optimal Gaussian measure is a multiple of the identity. Considering ℓ_1^m, we observe _1 = ∑_i=1^m |_i| = √(2/π)m . Therefore, the choice is optimal. The norm of matches the lower bound in prop:|J|<bound. Furthermore, for projections P onto coordinates, that is, with an index set , we have and J P _1 =P/m . Therefore, the lower bound in cor:optGauss is sharp. Now consider ℓ_∞^m. Clearly, 𝕀_^m_2 →∞ = 1. Furthermore, we can show _∞≤ C √(1 + log m) , see lem:gaussqnormvector. We rescale with , i.e. generates the optimal Gaussian measure, and the order of is determined precisely thanks to the upper bound from prop:|J|<bound. Using prop:|J|<bound, the constant 1/15 in thm:BernsteinMCada can be improved for (extremely) large n and m. In detail, we have constants such that for m ≥ 2n we can state e^,(n,S,F,) ≥ c_mm-n/mb_m(S). Following the proof of thm:BernsteinMCada, thereby setting and with , we attain an estimate with a factor ν̃{m_≤2/3} inf_P orth. Proj.P = m-n[J P _F _{J P _F ≤1/3}] _≥β(λ, ρ/2) for m ≥ 2n α . Using lem:|m_y|<r and lem:E|JX|trunc with , we can choose a ρ-dependent constant c = c(ρ) = [1 - 2 exp(- δ^2ρ^2/2)] [1 - (1+ δ + 4/δ ρ^2) exp(- δ^2ρ^2/8) ] 1/3 (1+δ) . This expression is monotonically growing in ρ, and converging to for , if we choose . For example, with we have c(ρ) = [1 - 2 exp(- δ_0^2ρ/2)] [1 - (1 + δ_0/√(ρ) + 4/δ_0ρ^3/2) exp(- δ_0^2ρ/8) ] 1/3 (1+δ_0/√(ρ)) . Since by prop:|J|<bound we know , this shows that we can choose . However, this convergence is extremely slow. If one is really interested in better constants, it is recommendable to include best knowledge about ρ directly into the estimate. § SPECIAL SETTINGSWe study two interesting modifications of the main result thm:BernsteinMCada. They are non-essential for the applications in sec:BernsteinAppl. First, in sec:n(om,f)Bernstein we widen the class of admissible algorithms, now allowing varying cardinality n(ω,f). Still, we can show a similar inequality, however with worse constants. Second, in sec:BernsteinHomo we restrict to homogeneous algorithms and obtain an estimate with sharp constants, even for varying cardinality.§.§ Varying CardinalityUp to this point we ignored the additional feature of varying cardinality because it does not change the big point but gives us unpleasant constants that detract from the main relation. However, for lower bounds it is of interest to assume the most general shape for algorithms.In Heinrich <cit.> results were actually given for algorithms with non-adaptively varying cardinality n(ω). In this setting, by lem:n(om)MC and thm:BernsteinMCada, for we directly obtain an estimate likee̅^(n,S) ≥1/2e^(2n,S) > 1/60b_4n(S) We can even consider adaptively varying cardinality , and still get similar bounds. Let be a compact linear operator between Banach spaces, and the input set F be the unit ball in F. Considering algorithms with adaptively varying cardinality , forwe have e̅^,(n,S,F,) > 1/63b_4n(S). More generally, for any injective linear operator J: ℓ_2^8n→F, we can estimate e^,(n,S,F,) > 1/64 inf_P orth. Proj.P = 4nS J P _G/J _F , where is a standard Gaussian random vector in ^8n = ℓ_2^8n. The proof works similar to that of thm:BernsteinMCada. Again, we assume . As before, we define a measure μ̃ as the distribution of with being a standard Gaussian random vector in , and set μ to be the restriction of μ̃ to the unit ball . We write . In view of lem:n(om,f)Bakhvalov (Bakhvalov's technique), we aim to bound the μ-average error for a deterministic algorithm with varying cardinality such that n̅ ≥(ϕ∘ N, μ) = ∫ n(f)μ( f)[Markov's ineq.] ≥ 2n̅ μ{n(f) > 2n̅} . Thus, using the definition of the truncation, we can estimate μ̃{n(f) ≤ 2n̅} ≥μ̃{n(f) ≤ 2n̅ andf ∈ F}= μ{n(f) ≤ 2n̅}≥μ(F) - 1/2 [cor:deviationGauss] ≥1/2 - exp(- (1/α - 1)^2 /π). The conditional measure μ̃_ can be represented as the distribution of , with a suitable orthogonal projection P_ of rank , and a vector , not very different from the case of fixed cardinality, compare lem:condGauss. Again, we write for the distribution of the information. Following the same arguments as in the proof of thm:BernsteinMCada, and in addition restricting the integral to the case , we obtain the estimate for e(ϕ∘ N, μ) ≥ ν̃{ : n() ≤ 2n̅ and m__F ≤ 1 - r }inf_P orth. Proj.P ≥ m-2n̅[J P _F _{J P _F ≤ r}] b_m(S). The first factor can be estimated using inequality (<ref>) and a slight adjustment of lem:|m_y|<r, ν̃{ : m__F ≤ 1-r} ≥1/2 - exp(- (1/α - 1)^2 /π) - 2exp(- (1-r/2α - 1)^2 /π) =: ν̅(r,α). The second factor can be bounded from below by , if the operator J is chosen optimally. This estimate is exactly the same as in the proof of thm:BernsteinMCada, here with instead of m-n/m, so e̅^,(n̅,S,μ) ≥(m-2n̅/m)_+ ν̅(r,α) β(r/α) αb_m(S) =: (n̅). This lower bound exhibits convexity, so by the subtle version of lem:n(om,f)Bakhvalov (Bakhvalov's technique), taking , we obtain e̅^,(n,S,F) ≥1/2 ν̅(r,α) β(r/α) α_=: c̅b_4n(S). With and , this gives us a constant . For the more general estimate with Gaussian measures, we take and the rough version of lem:n(om,f)Bakhvalov (Bakhvalov's technique), inserting in the adjusted version of the above estimates, and obtain a constant . Choosing , , and , we get the numerical value . In regard of the estimate for fixed cardinality,e^(2n,S) > 1/30b_4n(S),we see that taking twice as much information as in the varying cardinality setting gives us bigger lower bounds by roughly a factor two only.[ For the general estimate for Gaussian measures we lose roughly a factor 4, for both the error and the cardinality.]However, several estimates involved in this proof seem to be far from optimal. For homogeneous algorithms, see sec:BernsteinHomo, using lem:n(om,f)avgspecial for the analysis of the average setting, we will obtain sharp estimates that equally hold for algorithms with fixed and varying cardinality. Here, we could not apply lem:n(om,f)avgspecial because of the much more complicated situation arising from truncation. Anyways, even without the homogeneity assumption, we can state: If upper bounds achieved by Monte Carlo algorithms with fixed cardinality are close to the lower bounds obtained using Bernstein widths (or directly, Gaussian measures), varying cardinality does not help a lot. §.§ Homogeneous Monte Carlo MethodsFor linear problems (as considered within this chapter), common algorithms are homogeneous (and also non-adaptive).[ For example for the identity on sequence spaces , the basic structure of asymptotically best known algorithms is described in sec:An:lp->lq. ] There is a close and very basic connection to the normalized error. Concerning the approximation of a compact linear operator between Banach spaces over the reals using homogeneous algorithms, the absolute error criterion with the input set being the unit ball of F coincides with the normalized error criterion, e^⋆,(n,S,F,Λ) = e_^⋆,(n,S,F∖{0},Λ) , where . If A_n is a homogeneous algorithm that is defined for , it is naturally extended to such that . Indeed, this is unproblematic since the information mapping as well is homogeneous. Then for we have e_(A_n,f) = S(f) - A_n^ω(f)_G/f_F= S(f/f_F) - A_n^ω(f/f_F) _G = e(A_n,f/f_F)≤ e(A_n,F). This proves “≥”. Now, for any algorithm A_n, and any non-zero input from the unit ball, we have e(A_n,f) = S(f) - A_n^ω(f)_G ≤[S(f) - A_n^ω(f)_G/f_F] = e_(A_n,f) ≤ e_(A_n,F∖{0}). Trivially, , so this proves “≤”. For being a compact linear operator between Banach spaces, and the input set F being the unit ball within F, we have e^,(n,S,F,) ≥m-n/mb_m(S) for m > n. In general, for any injective linear operator J: ℓ_2^m →F we have e^,(n,S,F,) ≥inf_P orth. Proj.P = m-nS J P _G/J _F , where is a standard Gaussian random vector in ^m = ℓ_2^m. Actually, the same Bernstein estimate holds for homogeneous algorithms with varying cardinality as well, forwe can state e̅^,(n̅,S,F,) ≥(m-n̅/m)_+ b_m(S). Similarly to the proof of thm:BernsteinMCada, we choose a Gaussian measure μ̃ described as the distribution of . Here, however, we take the scaling , so for measurable sets , by μ(E) := ∫_E f_Fμ̃( f) = J _F_{J ∈ E} , a probability measure on F is defined. By lem:hom=normal, and Bakhvalov's technique in the normalized error criterion setting (see prop:Bakh for a proof in the absolute error criterion setting), we have e^,(n,F) = e_^,(n,F∖{0}) ≥ e_^,(n,μ). Now, for any homogeneous deterministic algorithm , first with fixed cardinality, we have e_(A_n,μ) = ∫S(f) - A_n(f)_G/f_F μ( f) = ∫S(f) - A_n(f)_Gμ̃( f) = ∫_^n[∫S(f) - ϕ()_G μ̃_( f) ] μ̃∘ N^-1 (). Using the representation of the conditional measure, see lem:condGauss, and with the same symmetrization argument as in the proof of thm:BernsteinMCada, we continue ≥inf_P orth. Proj.P = m-nS J P _G. By the definition of Bernstein numbers for F = ^m, choosing an optimal J as it is found in cor:optGauss, we end up with e_(A_n,μ)≥m-n/mb_m(S). Observe that the lower bound for the conditional error, given , exhibits the special structure of lem:n(om,f)avgspecial with a convex function . However, switching airily between different error criterions and measures, it is not immediate that this already implies lower bounds for homogeneous algorithms with varying cardinality, so we need to think about a modification of the proof of lem:n(om,f)Bakhvalov that fits to the present situation. Let be a homogeneous Monte Carlo method with varying cardinality. Due to homogeneity, for and , hence n̅ := sup_f ∈ F n(ω,f) = sup_f ∈F n(ω,f) [Fubini] ≥∫ n(ω,f)μ̃( f). The key insight is that the error can be related to the μ̃-average setting, here we use that lem:hom=normal holds for algorithms with varying cardinality as well, e(A,F) = e_(A,F∖{0}) [Fubini] ≥∫ e_(A^ω,f)μ( f) = ∫ e(A^ω,f)μ̃( f). For the inner integral we apply lem:n(om,f)avgspecial to the μ̃-average setting, for which we have the lower bound for the conditional average error, that is, with respect to μ̃_. We obtain ≥((A^ω,μ̃)). From here we can proceed as in the last inequality chain within the proof of lem:n(om,f)Bakhvalov, e(A,F) ≥(n̅). Hence the lower bound based on Bernstein numbers does even hold for algorithms with varying cardinality. For the direct estimate via general Gaussian measures, it depends on the particular situation how we can generalize the lower bound. The above theorem is optimal. Indeed, consider for example the identity with Bernstein number . Let be a randomly chosen index set such that , where , and define the linear (homogeneous) Monte Carlo method A_n̅^ω () := ∑_i ∈ I(ω) x_i_i, ∈ℓ_1^m , where _i are the vectors of the standard basis. The cardinality is (A) = # I(ω) = ∑_i=1^m {i ∈ I} = n̅ , the error e(A_n̅,𝕀_ℓ_1^m,) = 𝔼 - A_n̅^ω()_1 = ∑_i=1^mℙ{i ∉ I(ω)}|x_i| = m-n̅/m _1, so . On the other hand, by thm:BernsteinMChom we have the lower bound . This shows that is optimal. If , we can find a fixed-cardinality version for .§ APPLICATIONSThe first application on the recovery of sequences, sec:Bernstein:lp->lq, is meant to give a feeling for the potentials and limitations of Bernstein numbers when it comes to lower bounds for quantities from IBC. We also discuss other techniques for lower bounds, as well as general problems arising. The main application is the L_∞-approximation of C^∞-functions, see sec:Cinf->Linf. There we show that in the particular situation randomization does not help in terms of tractability classifications. §.§ Recovery of SequencesWe consider the approximation of the identity between finite dimensional sequence spaces,: ℓ_p^M ↪ℓ_q^M,with and . This example problem is also of interest when dealing with embeddings between function spaces, compare sec:ranApp:e(n).§.§.§ The Case M = 4n The following result on Bernstein numbers is well known and has been used e.g. in <cit.> in order to determine the order of decay of Bernstein numbers in different function space settings,b_m(ℓ_p^2m↪ℓ_q^2m) ≍ m^1/q-1/p if1 ≤ p ≤ q ≤∞ or1 ≤ q ≤ p ≤ 2, m^1/q-1/2 if1 ≤ q ≤ 2 ≤ p ≤∞ , 1 if2 ≤ q ≤ p ≤∞ .Here, the hidden constants may depend on p and q. Applying thm:BernsteinMCada with , this implies the estimatee^(n,ℓ_p^4n↪ℓ_q^4n) ≽ n^1/q-1/p if1 ≤ p ≤ q ≤∞ or1 ≤ q ≤ p ≤ 2, n^1/q-1/2 if1 ≤ q ≤ 2 ≤ p ≤∞ , 1 if2 ≤ q ≤ p ≤∞ . Since the lower bounds with Bernstein numbers were obtained using Gaussian measures, it is not surprising that in some parameter settings we will get significantly better estimates when directly working with Gaussian measures, as it has been done in Heinrich <cit.>, see rem:GaussMCada. In detail, with being a standard Gaussian vector on ^4n, by (<ref>) we have[ We chose m = 4n for better comparison with the results that were obtained via Bernstein numbers. In this case however,would give the same asymptotics.]e^(n, ℓ_p^4n↪ℓ_q^4n) ≥ c_q/_p≽ n^1/q-1/p if1 ≤ p,q < ∞ , n^-1/p(log n)^1/2 if1 ≤ p < q = ∞ , n^1/q(log n)^-1/2 if1 ≤ q < p = ∞ , 1 ifp = q = ∞ ,where c>0 is a universal constant, and the hidden constant for the second relation may depend on p and q. For this result we only need to knowfor , and , for a standard Gaussian vector in , see lem:gaussqnormvector. In Heinrich <cit.> we also find upper bounds, which are achieved by non-adaptive and homogeneous methods, see also sec:An:lp->lq,e^(n, ℓ_p^4n↪ℓ_q^4n) ≼ n^1/q-1/p if1 ≤ p,q < ∞ , n^-1/p(log n)^1/2 if1 ≤ p < q = ∞ , n^1/q if1 ≤ q < p = ∞ , 1 ifp = q = ∞ .We see that in most cases the lower bounds (<ref>) obtained by Gaussian measures match the upper bounds (<ref>), but for there is a logarithmic gap. For non-adaptive algorithms this gap can be closed. Within the authors master's thesis, see also <cit.>, Bernstein numbers have been related to the error of non-adaptive Monte Carlo methods by means of volume ratios.[ Instead of truncated Gaussian measures, in <cit.> the average case for the uniform distribution on finite-dimensional sub-balls of the input set F was considered.] In the general linear setting of this chapter, for we havee^,(n,S) ≥m-n/m+1 sup_X_m inf_Y_m-n(_m-n(S(F) ∩ Y_m-n)/_m-n(B_G ∩ Y_m-n))^1/(m-n) ,whereandare subspaces with dimensionand , furthermore, B_G denotes the unit ball in G, and for each choice of Y_m-n the volume measure _m-n may be any (m-n)-dimensional Lebesgue-like measure since we are only interested in the ratio of volumes. In the case of sequence spaces , the volume ratios could actually be computed in <cit.> (based on results from Meyer and Pajor <cit.>), and by that we obtainede^,(n,S) ≽ n^1/q-1/pfor the whole parameter range , thus closing the gap for . For , in turn, Gaussian measures do a better job. §.§.§ The Case of Little Information n ≪ M Up to this point the dimension of the vector spaces in consideration was a constant times the number of information. That setting is good enough when aiming for the order of convergence for function space embeddings. But what if the information is much smaller than the size of the sequence spaces?For an example of rather disappointing lower bounds we restrict to the caseℓ_1^M ↪ℓ_2^M.There is a well known result on deterministic algorithms,e^(n,ℓ_1^M ↪ℓ_2^M) ≍min{1, √(1 + logM/n/n)}for n<M.A proof based on compressive sensing can be found e.g. in Foucart and Rauhut <cit.>, the idea goes back to Kashin 1977 <cit.> and Garnaev and Gluskin 1984 <cit.>. These errors are obtained with homogeneous and non-adaptive algorithms, see sec:An:lp->lq.What lower bounds for the Monte Carlo error do we know? We could use the Bernstein numbers for ,b_m(ℓ_1^M ↪ℓ_2^M) = 1/√(m) ,see Pinkus <cit.>. Then by thm:BernsteinMCada, for we obtaine^(n,ℓ_1^M ↪ℓ_2^M) ≥1/15 max_n<m ≤ Mm-n/m 1/√(m)m = 3n=1/15 2/3√(3) 1/√(n) .It is not reasonable that the size M of the problem did not contribute at all to the error quantity, so this lower bound for the Monte Carlo error is seemingly not quite optimal. Directly considering Gaussian measures does not change the big point, as the next lemma shows. Let be an injective linear operator and be a standard Gaussian random vector in ℓ_2^m. Then for we have inf_P orth. Proj.P = m - nJ P _2/J _1≤√(π/2) √(m-n)/m≤√(π/2) 1/2 1/√(n) . Let denote the rows, andthe columns of J. Without loss of generality, the columns of J are orthogonal, compare rem:UniqueGauss. Obviously, the term inf_P orth. Proj.P = m - nJ P _2 does not change when transforming J into , with Q_M being an orthogonal matrix operating on ℓ_2^M, and Q_m respectively on ℓ_2^m. For the denominator we have J _1 = √(2/π) ∑_i=1^M _i_2. What happens to this, when we rotate rows (thus performing a transformation that contributes to Q_M)? Consider the rotation of the i_1-th and the i_2-th rows, with and , defined by _i_1 ↦_i_1' := √(ξ) _i_1 + √(1-ξ) _i_2 ,_i_2 ↦_i_2' := -√(1-ξ) _i_1 + √(ξ) _i_2 with ξ∈ [0,1]. By construction, _i_1_2^2 + _i_2_2^2 = _i_1'_2^2 + _i_2'_2^2. Now, for the special choice ξ := 1/2 (1 + ⟨_i_1, _i_2⟩/√(1+ 4 ⟨_i_1, _i_2⟩^2 /_i_1_2^2 - _i_2_2^2)), one can check that ⟨_i_1, _i_2⟩ = 0, and_i_1'_2 > _i_1_2 ≥_i_2_2 > _i_2'_2. Together with (<ref>), one can easily prove that _i_1'_2 + _i_2'_2 < _i_1_2 + _i_2_2. This means that by such transformations performed on J, the expression (<ref>) will be reduced. Now, repeatedly performing such transformations, and permuting rows of J, one can find a matrix with orthogonal rows of descending norm, in particular , moreover, . Since the columns of J are orthogonal, so are the columns of J'. Hence we can find an orthogonal matrix Q_m such that the only non-zero entries of the matrix lie on the diagonal, , and . Without loss of generality, ∑_k=1^m λ_k = 1, so J' _1 = J_1 = √(2/π) . Now, consider the projection P_n onto the last coordinates, i.e. the mapping with _k being the standard basis in ℓ_2^m. Then J P_n _2 ≤√(J P_n _2^2) = √(∑_k = n+1^m λ_k^2)≤1/√(m-n) ∑_k = n+1^m λ_k ≤√(m-n)/m . Here, we used the general inequality for with , and (<ref>) together with the monotonicity of the coefficients λ_k. Altogether we have inf_P orth. Proj.P = m - nJ P _2/J _1≤J P_n _2 /J_1≤√(π/2) √(m-n)/m . This is maximized for . The situation for volume ratios (<ref>) is the same, see <cit.> for further hints. This disappointing phenomenon is not new, Vybíral <cit.> showed that Gaussian measures, as well as uniform distributions, are not suitable in this and many other cases to obtain lower bounds that – up to a constant – match the upper bounds. In his paper on best m-term approximation[ The concept of best-m-term approximation is not covered by our framework of information-based complexity, but results on that topic often have implications on the performance of some types of algorithms. For instance, in the case , typical algorithms usually return vectors with only n non-zero entries, compare sec:An:lp->lq.] he basically shows that in those particular average case settings the initial error is already too small. He proposes other average case settings which work perfectly for best m-term approximation but are hard to use in the information based complexity framework. There are also situations where the lower bounds perfectly reflect the situation for , considerℓ_p^M ↪ℓ_q^Mfor the parameter range . The initial error ise(0,ℓ_p^M ↪ℓ_q^M) = M^1/q-1/p≥ 1,whereas the lower bounds by Gaussian measures (<ref>) give use^,(n,ℓ_p^M ↪ℓ_q^M) ≽ M^1/q-1/pforin the parameter rangeor ,[ Bernstein numbers only give comparably satisfying lower bounds for or .] where the hidden constant may depend on p and q. In the case , the lower bounds by Gaussian measures are worse by a logarithmic factor , but in the non-adaptive setting (<ref>) we get rid of that term. What does this mean for our strategies to approximate this embedding for ? Basically, we have the two alternatives of either * taking no information and accepting the initial error, or * taking full information , thus having no error at all.This is a reasonable approach to the problem, because any choice will be insufficient if we want to reduce the initial error by a significant factor, and taking at most four times as much information than really necessary is no big deal.§.§ L_∞-Approximation for C^∞-FunctionsWe consider the L_∞-approximation for subclasses of C^∞-functions defined on the d-dimensional unit cube ,: C^∞([0,1]^d) ↪ L_∞[0,1]^d.The space has no natural norm and it will be crucial for tractability what input sets we choose. Novak and Woźniakoswki <cit.> considered the input setF^d := {f ∈ C^∞([0,1]^d) | D^ f_∞≤ 1 for ∈_0^d} .Here,denotes the partial derivative of f belonging to a multi-index .In their study, Novak and Woźniakowski showed that with this input set the problem suffers from the curse of dimensionality for deterministic algorithms. Since the proof was based on the Bernstein numbers, thanks to thm:BernsteinMCada, the curse of dimensionality extends to randomized algorithms.We will cover this case within a slightly more general setting, considering the input setsF_p^d := {f ∈ C^∞([0,1]^d) |∇__k⋯∇__1 f_∞≤ |_1|_p ⋯ |_k|_pfor allk ∈_0,_1,…,_k ∈^d} ,where denotes the directional derivative along a vector , and we write ||_p for the p-norm of , . Note that, indeed, this is a generalization of the original problem since . The set F_p^d can be seen as the unit ball of the spaceF_p^d := {f ∈ C^∞([0,1]^d) | f_F_p < ∞} ,equipped with the normf_F_p := sup_k ∈_0_1,…,_k ∈^d |_1|_p^-1⋯ |_k|_p^-1 ∇__k⋯∇__1 f_∞ . First, we aim for lower bounds, to this end starting with the Bernstein numbers of the restricted operator . The proof follows the lines of Novak and Woźniakowski <cit.>. For 1 ≤ p < ∞ we have b_m(F_p^d ↪ L_∞) = 1 for n ≤ 2^⌊d^1/p/3⌋. In the case , this even holds for . Note that , and therefore for all . We set and . Consider the following linear subspace of F_p^d, V_p^d:= { f | f() = ∑_∈{0,1}^s a_ (x_1+ … + x_r)^i_1 ⋯ (x_r(s-1)+1 + … + x_rs)^i_s, a_∈} with . For and , we will show . Besides, , so it then easily follows that for . Therefore, with and the subspace , we obtain . Since the sequence of Bernstein numbers is decreasing, we know the first 2^s Bernstein numbers. In order to estimate , we first consider partial derivatives. For an index , where , for and we have |∂_i f()| = 1/r |f(x_1,…,x_(k-1)r,1,…,1,x_kr+1,…,x_d)-f(x_1,…,x_(k-1)r,0,…,0,x_kr+1,…,x_d)| ≤1/r |f(x_1,…,x_(k-1)r,1,…,1,x_kr+1,…,x_d)|+ |f(x_1,…,x_(k-1)r,0,…,0,x_kr+1,…,x_d)| ≤2/r f_∞≤ d^-1 + 1/p f_∞ . For the directional derivative , this gives us ∇_ f_∞ ≤∑_i=1^d |v_i|∂_i f_∞≤ ||_1 d^-1 + 1/p f_∞≤ ||_pf_∞ .By thm:BernsteinMCada (or thm:BernsteinMChom, respectively) we directly obtain the following result on the Monte Carlo complexity. Consider the approximation problem with parameter . Then the Monte Carlo complexity for achieving an error smaller than is bounded from below by n^,(,,F_p^d) > 2^⌊d^1/p/3⌋ - 1 . For homogeneous algorithms we have the same complexity for already, n^,(1/2,,F_p^d) ≥ 2^⌊d^1/p/3⌋ - 1 . In the case , the problem suffers from the curse of dimensionality. In general, for small , the -complexity depends exponentially on d^1/p. Note that the initial error is , hence properly normalized. Furthermore, functions from F_p^d can be identified with functions in F_p^d+1 that are independent from x_d+1.The upper bounds actually get close to the lower bound in terms of d-dependency. The idea originates from Vybíral <cit.>, where it has been used for slightly different settings, but included a case similar to the casehere. For the L_∞-approximation of smooth functions from the classes F_p^d, , for , we obtain the following upper bounds on the -complexity achieved by linear deterministic algorithms, n^,(,,F_p^d,) ≤exp( log(d+1) max{log1//log 2, d^1/p} ). In particular, in the case , the problem is -weakly tractable for all and , but not for . In the case of , the problem is quasi-polynomially tractable. As an algorithm we consider the k-th Taylor polynomial, , at the point , then the output function is defined for as [A_k(f)]() := ∑_j = 0^k [∇_^j f](_0)/j! = ∑_∈_0^d ||_1 ≤ k[D^ f](_0)/ ! ^ . For this deterministic algorithm we need n(k) := (A_k) = d+kd≤ (d+1)^k partial derivatives of the input f at _0 as information.[ Vybíral <cit.> even shows that the same amount of function values is actually sufficient to approximate the partial derivatives at the point _0 with arbitrarily high accuracy. The problem of counting the number of partial derivatives up to the order , , is equivalent to choosing d numbers from the set by the transformation , where . For our purpose, it is sufficient to know that we do not need more than partial derivatives (like deciding k times in which coordinate direction to derive – or not to derive – the function) since k is very small.] The error estimate then is |f() - [A_k(f)]()| ≤∇_^k+1 f_∞/(k+1)! , and with , for the input set F_p^d we obtain the error bound e(A_k,F_p^d) ≤1/(k+1)! (d^1/p/2)^k+1 [Stirling's formula] ≤1/√(2π(k+1)) (d^1/p/2 (k+1))^k+1 . What should we choose in order to guarantee an error smaller or equal a given tolerance ? This is ensured for (d^1/p/2 (k+1))^k+1≤⇔ k+1 ≥d^1/p/2 ^-1/k+1 . Note that for , so choosing k = k() = ⌊max{log1//log 2, d^1/p}⌋ will give us the guarantee we aim for. By this and (<ref>), we obtain the theorem on the -complexity. This upper bound is not optimal in terms of the speed of convergence, which is superpolynomial (as it has already been mentioned in Novak and Woźniakowski <cit.>). However, together with cor:LinfAppLB, it shows that in these cases randomization does not help to improve the tractability classification of the problems. Here, only narrowing the input set affects tractability.There are several other publications worth mentioning that study the tractability of the approximation of smooth functions in the worst case setting. Weimar <cit.> discusses several settings with weighted Banach spaces. Xu <cit.> considers the L_p-approximation for and the same input set F^d as in Novak and Woźniakowski <cit.>.CHAPTER: UNIFORM APPROXIMATION OF FUNCTIONS FROM A HILBERT SPACEWe study the L_∞-approximation of functions from Hilbert spaces with linear functionals as information. Based on a fundamental Monte Carlo approximation method (originally for finite dimensional input spaces) which goes back to Mathé <cit.>, see sec:HilbertFundamental, we propose a function approximation analogue to standard Monte Carlo integration, now using “Gaussian linear functionals” as random information, see sec:HilbertPlainMCUB. This method is intended to break the curse of dimensionality where it holds in the deterministic setting. The analysis relies on the theory of Gaussian fields, see sec:E|Psi|_sup, some theory of reproducing kernel Hilbert spaces is needed as well, see sec:RKHS. Using a known proof technique for lower bounds in the worst case setting, see sec:HilbertWorLB, we can prove the curse of dimensionality for the deterministic approximation of functions from unweighted periodic tensor product Hilbert spaces, whereas for the randomized approximation we can show polynomial tractability under certain assumptions, see sec:HilbertPeriodic for this particular application. A specific example are Korobov spaces, see thm:Korobov.§ MOTIVATION AND THE GENERAL SETTING For the integration problem with standard information , it is known since more than half a century that randomization can speed-up the order of convergence. For example, for r-times continuously differentiable functionsF_r^d := {f ∈ C^r([0,1]^d) | D^ f_∞≤ 1 for ∈_0^d with ||_1 ≤ r} ,one can showe^(n,,F_r^d) ≍ n^-r/d-1/2≺ e^(n,,F_r^d) ≍ n^-r/d ,where the hidden constants depend on r and d, see for instance the lecture notes of Novak <cit.>, the original result is due to Bakhvalov 1959 <cit.>. We see that for fixed smoothness and high dimensions the deterministic rate gets arbitrarily bad, whereas for Monte Carlo methods we have a guaranteed rate of convergence of n^-1/2. Even worse, in the deterministic setting upper bounds are achieved by product rules that use function values on a regular grid as information, which for high dimensions is of no practical use. The classical approach to lower bounds – when proving the rate of convergence, for both settings – involved constants that are exponentially small in d. It is only recent that Hinrichs, Novak, Ullrich, and Woźniakowski <cit.> proved the curse of dimensionality for these classes of C^r-functions in the deterministic setting,n^(,,F_r^d) ≥ c_r^d (d/)^d/rfor all and ,moreover, they proved that product rules are really the best what we can do. In contrast to this, the problem is strongly polynomially tractable in the randomized setting by simple means of the classical Monte Carlo methodM_n(f) := 1/n∑_i = 1^n f(_i),where the _i are iid uniformly distributed on the domain . Here, we have the bound , and thereforen^(,F_r^d) ≤⌈^-2⌉for > 0.When aiming for the optimal rate n^-r/d-1/2, a proof would usually consider algorithms that use exponentially in d many function values, so in many high-dimensional cases the standard Monte Carlo method might be the best approach to a practical solution.This observation raises the question whether there are approximation problems where randomization significantly reduces the complexity. Even more, can we find a comparably simple Monte Carlo method that breaks the curse of dimensionality?The short answer is: Yes, we can – at least for some problems.Throughout this chapter we consider linear problemsS: F↪ Gwith the input set F being the unit ball of F, allowing algorithms to use arbitrary continuous linear functionals for information. The latter is a crucial assumption for the new upper bounds based on a fundamental Monte Carlo approximation method, see prop:Ma91_l2G in sec:HilbertFundamental below. Whilst for the introductory part sec:HilbertIntro the input space F is not necessarily a Hilbert space, this will be the case for Sections <ref> and <ref>. The examples we present towards the end of the chapter are all with the output space , but it should also be possible to consider embeddings into classical smoothness spaces . § INTRODUCTION TO RANDOMIZED APPROXIMATIONThe most important part of this introduction is sec:HilbertFundamental with the fundamental Monte Carlo approximation method prop:Ma91_l2G, an idea which is due to Mathé <cit.>. Sections <ref>, <ref>, and <ref>, give an overview of different settings where this method can be applied. The reader can decide to skip these sections and go directly to sec:HilbertTools where we collect tools for the tractability analysis of function approximation in both, the deterministic and the randomized setting. These methods are applied to exemplary problems in sec:HilbertExamples.Still, those last three sections of the present introductory part (Sections <ref>–<ref>) may be helpful to gain some deeper insight into the potential of randomized approximation and the historical background. Within sec:An:lp->lq we consider finite dimensional sequence spaces and summarize what is known about this topic. Recovery of sequences is a keystone for the understanding of how the speed of convergence can be enhanced by randomization for function space embeddings, a short overview on that issue is to be found in sec:ranApp:e(n). Finally, in sec:l2->linf,curse we discuss a sequence space model for d-dependent problems where randomization breaks the curse of dimensionality. This example will give us strong indication that we should restrict to Hilbert spaces ^d of d-variate functions as input spaces, and L_∞ as the target space, in search for examples where randomization helps to break the curse.§.§ A Fundamental Monte Carlo Approximation MethodThe following result originates from Mathé <cit.> and is a key component for the Monte Carlo approximation of Hilbert space functions. Here we keep it a little more general than in the original paper, where the output space was a sequence space ℓ_q^m with . Letbe a linear operator between normed spaces and consider the unit ball as the input set. Let the information mapping N^ω = N be a random -Matrix with entries , where the X_ij are independent standard Gaussian random variables. Then defines a linear rank-n Monte Carlo method () and its error is bounded from above by e(A_n,S:ℓ_2^m → G) ≤2/√(n) S _G whereis a standard Gaussian vector in ^m. Note that A_n is an unbiased linear Monte Carlo algorithm. To see this, let ∈^m, then (N^⊤ N )(i) = 1/n∑_j=1^n ∑_k=1^m X_ji X_jk_= δ_ikx_k = x_i, i.e. , and by linearity of S we have . We start from the definition of the error for an input ∈ℓ_2^m, e(A_n,) = S-S N^⊤ N _G. Now, let M be an independent copy of N. We write ' for expectations with respect to M, andwith respect to N. Using and , we can write = ^' S(M^⊤M - M^⊤N + N^⊤ M - N^⊤ N) _G Δ-ineq.≤ 2 ^' S (M+N/√(2))^⊤(M-N/√(2)) _G. The distribution of (M,N) is identical to that of , therefore = 2 ^'S N^⊤ M _G. Here, M is a Gaussian vector distributed like _2/√(n) with being a standard Gaussian vector on ^n. So we continue, ^' now denoting the expectation with respect to , = 2 _2/√(n)^'S N^⊤_G. For fixed , the distribution of is identical to that of with being a standard Gaussian vector on ^m. Letdenote the expectation with respect to . By Fubini's theorem we get = 2 _2/n ^' [_2S _G]. Using , we finally obtain e(A_n,)≤2 _2/√(n) S _G. As mentioned within the proof of the error bound, the algorithm is unbiased, that is, for . However, in general the method is non-interpolatory since for non-trivial problems S with positive probability the output will be outside the image of the input set , which is the unit ball. If the solution operator S is injective, then the output is the solution for , and one can show √(N^⊤ N _2^2) = √(1 + m+1/n) _2. We will put this to an extreme in sec:HilbertPlainMCUB. Applied to function approximation problems, the method will produce an output function for which the Hilbert norm is almost surely infinite. This means that the output does not only lie outside of the input set, but it actually drops out of the input space , see rem:Lstoch for the general phenomenon, and rem:SmoothnessLost on the loss of smoothness in the particular context of Korobov spaces. §.§ Methods for the Recovery of Sequences We consider again the identity operator between sequence spaces,: ℓ_p^M ↪ℓ_q^M,whereand , compare sec:Bernstein:lp->lq where lower bounds have been discussed. Now, we summarize what is known on upper bounds. In addition to well-known deterministic bounds, we owe linear Monte Carlo results to Mathé 1991 <cit.>, and non-linear Monte Carlo estimates to Heinrich 1992 <cit.>.What is the basic structure of deterministic and randomized algorithms, depending on the parameters p and q? In what cases does randomization help?§.§.§ The Case 1 ≤ q ≤ p ≤∞ – Practically Complete Information Needed The simplest case is where the initial error by simple norm estimates ise(0,ℓ_p^M ↪ℓ_q^M) = M^1/q-1/p .If we allow to use n information functionals, it is optimal to simply compute the first n entries of the input vector and set the other entries to 0 for the approximant, which gives use(n,ℓ_p^M ↪ℓ_q^M) = (M-n)^1/q-1/p ,see Pietsch <cit.> for a proof. This is a fairly small reduction of the initial error. At best, we gain a factor at most 2 for n ≤M/2, in case any will be useless. Randomization and adaption does not help a lot, see page para:lp->lq,q<p,LB for a deeper discussion. So basically, in this case we can rely on deterministic linear methods. Ghost line.We now discuss several cases for , where the initial error ise(0,ℓ_p^M ↪ℓ_q^M) = 1. §.§.§ The Case 1 ≤ p < q ≤ 2 – Non-Linear Deterministic Methods In the case and , optimal deterministic error bounds are obtained by non-linear methods with a subtly chosen non-adaptive (that is linear) information mapping N. For an information obtained for an input , one then finds an output by ℓ_1-minimization,ϕ() := _∈^M N = _1.This definition of the outputsimply guarantees that the algorithm is interpolatory. Indeed, by construction, , and for inputs from the input set being the unit ball, which is the input set, the output will also be from that input set. Furthermore, it gives the same information. The structure of this method reflects that for linear problemsin the deterministic setting interpolatory algorithms based on non-adaptive information are optimal up to a factor 2, see the book onfor further details. In this particular case linear algorithms are far worse than interpolatory algorithms.The crucial point is to find a good information mapping N. Several non-constructive ways are known, e.g. taking an -Matrix with independent standard Gaussian entries, then with positive probability it will have the properties that ensure the up to a constant optimal error boundse(ϕ∘ N, ℓ_1^M ↪ℓ_2^M) ≤ Cmin{1,√(1+logM/n/n)} ,where is a numerical constant, see Foucart and Rauhut <cit.> for a proof. Almost surely, the matrix N will be such that the ℓ_1-minimization is solved by a unique . Computing , in fact, is a linear optimization problem, see Foucart and Rauhut <cit.>. Even more generally, for the parameter range and , the same algorithms give up to a constant optimal error ratese^(n, ℓ_1^M ↪ℓ_q^M) ≍min{1, (1+logM/n/n)^1-1/q} . On the other hand, the lower bounds known for the Monte Carlo setting actually statee^(n, ℓ_1^M ↪ℓ_q^M) ≽ n^-(1-1/q)for n ≤M/2,compare sec:Bernstein:lp->lq. As discussed there, it seems odd for the Monte Carlo error to be independent from the size M of the problem, so we conjecture that randomization may not help significantly in this setting. The gap between the lower and the upper bound is logarithmic in M/n.In Foucart and Rauhut <cit.> one can also find a summary on the worst case error for, it is based on results from Kashin 1981 <cit.>. The basic structure of algorithms in that case again is that the information will be non-adaptive and the output interpolatory, which can be achieved by ℓ_p-minimization (instead of ℓ_1-minimization). For simplicity, we only cite the error for ,e^(n,ℓ_p^M ↪ℓ_2^M) ≍min{1, M^1-1/p/√(n)}for n < M,where the hidden constants may depend on p. This stands in contrast to the best known lower bounds on the Monte Carlo error from sec:Bernstein:lp->lq,e^(n, ℓ_p^M ↪ℓ_2^M) ≽ n^-(1/p-1/2)for n ≤M/2,which exhibit a polynomial gap of a factor . Still, we conjecture that randomization will not help a lot in the parameter range .§.§.§ The Case 2 ≤ p < q ≤∞ – Linear Monte Carlo Approximation In the case and , since Smolyak 1965 <cit.> it is well known thate^(n, ℓ_2^M ↪ℓ_∞^M) = √(M-n/M) ,see ex:DiagOps for more details and references. The optimal algorithm is an orthogonal rank-n projection. In particular for , the deterministic error cannot go below . By norm estimates[ With 2 ≤ p, for the input set being the unit ball, we have . On the other hand, for the error measuring norms are related by , making the problem even more difficult for . ] we obtain that this lower bound holds in general for ,e^(n,ℓ_p^M ↪ℓ_q^M) ≥√(2)/2for n ≤M/2.Since this is no significant reduction of the initial error, practically, for the deterministic setting we have the choice between full information , and accepting the initial error.In this parameter range, however, it is helpful to apply the fundamental linear Monte Carlo method from Mathé <cit.>, see prop:Ma91_l2G, as long as n is big enough for that the method's error does not exceed the initial error 1. The Monte Carlo algorithm , where N is anwith independent zero-mean Gaussian entries of variance 1/n, has the errore(A_n, ℓ_p^M ↪ℓ_q^M) ≤ M^1/2-1/pe(A_n, ℓ_2^M ↪ℓ_q^M) ≤ 2 M^1/2-1/p _q/√(n),where is a standard Gaussian vector in ^M. Using the norm estimates for Gaussian vectors, see lem:gaussqnormvector, we obtaine^(n, ℓ_p^M ↪ℓ_q^M) ≼min{1, M^1/2 - 1/p + 1/q / √(n)} for 2 ≤ p < q < ∞, min{1, M^1/2 - 1/p √(1 + log M/n)} for 2 ≤ p < q = ∞.Here, the hidden constant may depend on q. Comparing this to the best known lower bounds, see sec:Bernstein:lp->lq, where for we havee^(n, ℓ_p^M ↪ℓ_q^M) ≽ n^-(1/p-1/q) for 1 ≤ p < q < ∞, n^-1/p √(1+log n) for 1 ≤ p < q = ∞,we observe a gap which is at least logarithmic in M (for and ), and can grow up to a factor of almost order (the limiting case is ). Once more, this gap seems to be a deficiency of the lower bounds and not of the algorithms proposed, especially in the case and .§.§.§ The Case – Non-Linear Monte Carlo Approximation It is easier to approximate with the error measured in an ℓ_q-norm for than with respect to the ℓ_2-norm, soe^(n,ℓ_p^M ↪ℓ_q^M) ≤ e^(n,ℓ_p^M ↪ℓ_2^M).In view of the preceding paragraph it is not surprising that for the order of the worst case error cannot be improved. Forwe have the estimatee^(n,ℓ_p^M ↪ℓ_q^M) ≍min{1, M^1-1/p/√(n)}for n ≤M/2,see Foucart and Rauhut <cit.>. All in all, we can use the same algorithms that we used for . Only for and , best known lower bounds, see Foucart and Rauhut <cit.>, do not match the upper bounds we obtain by (<ref>). In this case, for we havemin{1,(1+logM/n/n)^1-1/q}≼ e^(n,ℓ_1^M ↪ℓ_q^M) ≼min{1,√(1+logM/n/n)} . Randomization, however, enables us to exploit the advantage of measuring the error in an ℓ_q-norm. Namely, we combine the non-linear deterministic algorithms that we have for with the linear Monte Carlo approximation for , this idea is contained in Heinrich <cit.>. In detail, split the cost n = n_1+n_2, collecting informationfor , where N_1 is an -matrix as we would choose it for ℓ_p ↪ℓ_2, and N_2^ω is a random -matrix with iid zero-mean Gaussian entries of variance . The output is generated in two steps. In the first step we compute a rough deterministic approximant_1 = ϕ_1(_1) := _∈^m N_1= _1_p,in the second step we generate the refined outputϕ^ω(_1,_2) := _1 + [N_2^ω]^⊤(_2 - N_2^ω_1).The error for can be estimated ase((ϕ^ω∘ N^ω)_ω∈Ω, ℓ_p^M ↪ℓ_q^M, ) =- _1 - (N_2^ω)^⊤N_2^ω ( - _1) _q ≤ e(([N_2^ω]^⊤ N_2^ω))_ω∈Ω, ℓ_2^M ↪ℓ_q^M) - _1_2 ≤ e(([N_2^ω]^⊤ N_2^ω)_ω∈Ω, ℓ_2^M ↪ℓ_q^M) e(ϕ_1 ∘ N_1, ℓ_p^M ↪ℓ_2^M).One could go with , then using (<ref>) or (<ref>), respectively, together with (<ref>), we obtain[ Heinrich <cit.> contains only the case p=1 since the other cases were not needed for the application to Sobolev embeddings.]e^(n,ℓ_p^M ↪ℓ_q^M) ≼1/n √((1 +log M)(1+logM/n)) for 1 = p and q = ∞, M^1-1/p √(1 +log M) for 1 < p ≤ 2 and q = ∞, M^1/q √(1 + logM/n) for 1 = p and 2 < q < ∞, M^1-1/p+1/q for 1 < p ≤ 2 < q < ∞,where the hidden constant may depend on q. However, if n is too small, it might be better to omit the second step and choose , , thus simply taking as a deterministic algorithm that achieves the bound from (<ref>). Also note that for or , respectively, this estimate is not optimal, and one should rather use the relation between the case p<2 and the case of the input space being ℓ_2^M,e^(n,ℓ_p^M ↪ℓ_q^M) ≤ e^(n,ℓ_2^M ↪ℓ_q^M) (<ref>)≼1/√(n) M^1/q for q < ∞, √(1 + log M) for q = ∞.Again, known lower bounds (<ref>) do not reflect the size M of the problem.§.§ Speeding up the Convergence for Function ApproximationSeveral examples are known where the order of convergence can be improved by randomization. Heinrich <cit.> considered Sobolev embeddings: W_p^r([0,1]^d) ↪ L_q([0,1]^d),where , and , with the compactness condition . Here, denotes the Sobolev space of smoothness r, that is the space of all functions such that for all , with , the partial derivatives D^f exist in a weak sense and belong to L_p. We consider the norm[ Concerning the order of convergence, any equivalent norm will give the same results.]f_W_p^r :=(∑_∈_0^d ||_1 ≤ rD^ f_p^p )^1/p for 1 ≤ p < ∞, max_∈_0^d ||_1 ≤ rD^ f_∞ for p = ∞,see for example Evans <cit.>, or Triebel <cit.>.For the deterministic setting we refer to Vybíral <cit.>. For simplicity we only cite the result for smoothness ,e^(n,W_p^r([0,1]^d) ↪ L_q([0,1]^d)) ≍ n^-r/d for 1 ≤ q ≤ p ≤∞,or 1 ≤ p < q ≤ 2, n^-r/d + 1/2 - 1/q for 1 ≤ p < 2 < q ≤∞, n^-r/d + 1/p - 1/q for 2 ≤ p ≤ q ≤∞.For the one-dimensional case see also Pinkus <cit.>. From Heinrich <cit.> we know the randomized setting for smoothness ,e^(n,W_p^r([0,1]^d) ↪ L_q([0,1]^d)) ≼ n^-r/d for 1 ≤ p ≤∞and 1 ≤ q < ∞,or p = q = ∞, n^-r/d√(1 + log n) for 1 ≤ p < q = ∞.Heinrich also proved lower bounds for the adaptive Monte Carlo setting that match the rate of the upper bounds – except for the case , where a logarithmic gap of a factor occurrs.[ This gap can actually be closed in the non-adaptive Monte Carlo setting since the lower bounds of the Sobolev embeddings are based on estimates for the sequence space embedding . For this, in the case , we have a better lower bound (<ref>) when restricting to non-adaptive methods.] Note that in all cases the hidden constants may depend on r, d, p, and q. In comparison of these two results, Heinrich could show that, for and , randomized algorithms can improve the rate of convergence by a factor that can reach almost the order , the most prominent case is and . This phenomenon was already known to Mathé <cit.> in the 1-dimensional case.Similar gaps between the Monte Carlo and the worst case error have been found in Fang and Duan <cit.> for multi-variate periodic Sobolev spaces with bounded mixed derivative.[ In Fang and Duan <cit.>, while lower bounds hold for adaptive Monte Carlo methods, upper bounds are obtained with non-adaptive but in some cases non-linear methods.] Again, gaps occur in parameter settings where we also know that randomization can help for the sequence space embedding . This is not surprising since the estimates for function space embeddings heavily rely on results for sequence space embeddings. In order to illustrate the connection to sequence spaces, let us outline the methods of discretization.For lower bounds one usually finds m-dimensional subspaces with such that the restriction resembles the sequence space embedding . Upper bounds are based on Maiorov's discretization technique <cit.>, where the solution operatorS: F→ Gis split into finite rank operators, so-called blocks,S = ∑_i = 1^∞ S_i,S_i = h_i ∈ ,that can be related to sequence space embeddings by estimatese^(n,S_i) ≤γ_i e^(n,ℓ_p^h_i↪ℓ_q^h_i)with . Now, with , , we havee^(n,S) ≤S_-k_F → G + ∑_i=1^k e^(n_i,S_i) ≤S_-k_F → G + ∑_i=1^kγ_i e^(n_i,ℓ_p^h_i↪ℓ_q^h_i),where . A common shape of the block operators could be that for a Schauder basisof the input space , we have disjoint index sets_i = 1^∞ J_i =of cardinality , such that for we may writeS_i(f) := ∑_j ∈ J_i a_j S(ψ_j).This structure can be found in Fang and Duan <cit.>, however, in the case of Heinrich <cit.> the discretization is based on another decomposition that is described in the book of König <cit.>.As mentioned before, the hidden constants for the results on the order of convergence may depend on the problem parameters. Indeed, the upper and the lower bounds may differ largely, even exponentially in d. In particular, for Heinrich's result, the upper bounds are obtained with n being exponential in d.We want to point out another drawback of splitting the operator, especially in the randomized setting. Let the input space be a Hilbert spacewith orthonormal basis , and split the operator S into block operators S_i of the structure (<ref>). Now, performing Maiorov's technique, we approximate the first k block operators by known methods, using a part n_i of the total information, respectively, where . Assume that for each of the blocks, using n_i pieces of information, the fundamental Monte Carlo approximation method from prop:Ma91_l2G is the best method we know. Then we obtaine^(n,S_1 + … + S_k) ≤∑_i=1^k e^(n_i,S_i) ≤∑_i=1^k 2∑_j ∈ J_i X_j S (ψ_j) _G /√(n_i) ,where the X_j are independent standard Gaussian random variables. However, we could apply the fundamental Monte Carlo approximation method directly to the cluster , and obtain the far better estimatee^(n,S_1 + … + S_k) ≤2∑_i=1^k ∑_j ∈ J_i X_j S (ψ_j) _G /√(n) .(Apply the triangle inequality for comparison to the Maiorov type upper bound.) For this reason, for our analysis on breaking the curse, we will take a more direct approach to the problem, see sec:HilbertPlainMCUB.Let us add one final remark on the type of information. In this chapter we aim for examples of d-dependent problems where randomized approximation using information from arbitrary linear functionals can break the curse of dimensionality. The examples of enhanced speed of convergence were also based on general information . However, randomization can also help in some cases where only function values are available to the algorithms. This was shown by Heinrich in a series of papers <cit.>, where he studied the randomized approximation of Sobolev embeddings, and discovered cases of low smoothness where randomization can give a speedup over deterministic methods. It is an interesting task for future research to find examples of function approximation problems based on standard information where randomization can break the curse of dimensionality, or significantly improve the d-dependency of a problem.§.§ Breaking the Curse - a Sequence Space ModellConsider the following example with a rather artificial[ One could regard ℓ_2^2^d as an L_2-space on a Boolean domain equipped with the counting measure #.] dimensional parameter d ∈,: ℓ_2^2^d↪ℓ_∞^2^d .The initial error is 1, hence properly normalized. By (<ref>) we haven^(, ℓ_2^2^d↪ℓ_∞^2^d) ≥ 2^d-1for 0 < ≤√(2)/2,which clearly is the curse of dimensionality. Now, in the randomized setting, by (<ref>) we haven^(, ℓ_2^2^d↪ℓ_∞^2^d) ≤ C d^-2for > 0,where C>0 is a numerical constant. This means that the problem is polynomially tractable for Monte Carlo methods.We want to discuss briefly what happens for the problem: ℓ_p^2^d↪ℓ_q^2^din other parameter settings where Monte Carlo methods are known to improve the error significantly, that is forand , see sec:An:lp->lq.In what cases do we have the curse of dimensionality for the deterministic setting in the first place?It turns out that, if , then by (<ref>) for we haven^(,ℓ_1^2^d↪ℓ_q^2^d) ≼ d^-2 ,which implies polynomial tractability in the deterministic setting already.[ By (<ref>) we have polynomial tractability for as well, yet with a worse -dependency .]For , in turn, the problem is more difficult than the problem (<ref>) and we obviously inherit the curse of dimensionality.Furthermore, in the case , by (<ref>) we have the estimaten^(,ℓ_p^2^d↪ℓ_q^2^d) ≽ 2^(2-2/p) dfor 0 << _0,so in this case deterministic methods suffer from the curse of dimensionality, too.Now, what do we know about randomized approximation for ?If the target space is altered compared to the original example (<ref>), i.e. , or if the input set[ Recall that the input set is the unit ball B_p^2^d of ℓ_p^2^d.] is extended, that is the case for , then by (<ref>), or by (<ref>) and (<ref>), respectively, we only know the upper bounds for ,[ The constant _q > 0 is actually the hidden constant from the respective error estimates (<ref>), (<ref>) and (<ref>), the hidden constant for the complexity estimates is then _q^2, or _q for .]n^(, ℓ_p^2^d↪ℓ_q^2^d) ≼ 2^(1-2/p+2/q) d ^-2 for 2 ≤ p < q < ∞, d 2^(1 - 2/p) d ^-2 for 2 < p < q = ∞, min{2^(1-1/p+1/q) d ^-1, 2^2d/q ^-2} for 1 < p < 2 < q < ∞,which is still exponential in d. We do not know whether the curse of dimensionality actually holds in the randomized setting because the lower bounds we obtain from (<ref>) will be independent from d.The case left over isand . Here, we can break the curse of dimensionality similarly to the original example (<ref>). Indeed, the ℓ_p-ball is contained in the ℓ_2 ball, hence the problem is easier.To summarize, the ℓ_∞ approximation of finite ℓ_p-sequences with is the case where we know that randomization can break the curse of dimensionality. The most prominent case is where best known Monte Carlo methods are linear, yet linear methods would suffice to break the curse for as well.This sequence space example motivates the restriction to the L_∞-approximation of Hilbert space functions in search of function approximation problems where the curse of dimensionality holds in the worst case setting but polynomial tractability can be found in the Monte Carlo setting. § TOOLS FOR FUNCTION APPROXIMATIONIn sec:HilbertPlainMCUB we put the fundamental Monte Carlo method from prop:Ma91_l2G to an extreme and obtain a function approximation analogue to standard Monte Carlo integration (<ref>). Here, we restrict the input set to functions from Hilbert spaces. lem:stdMCapp stated below is still quite general, its specification to L_∞-approximation of functions is the starting point for the study of Gaussian random fields and their expected maximum. In preparation for this, in sec:RKHS we sketch major elements of the theory of reproducing kernel Hilbert spaces (RKHS). sec:E|Psi|_sup then outlines the well established theory of Gaussian fields associated to a RKHS, in particular the technique of majorizing measures due to Fernique, and Dudley's entropy-based estimates. The theory of RKHSs is also useful for the analysis of the worst case setting, for which we need lower bounds in order to show the superiority of Monte Carlo approximation. sec:HilbertWorLB addresses a general approach to deterministicof Hilbert space functions, that approach has been taken by several authors before <cit.>. §.§ A Plain Monte Carlo Upper Bound Consider the linear problem with a compact solution operator S: → G from a separable Hilbert space into a Banach space G, the input set being the unit ball. Assume that we have an orthonormal basis for such that the sum , with independent standard Gaussian random variables , converges almost surely in G. Then for we have e^(n,S,) ≤2/√(n) ∑_j=1^∞ X_j S(ψ_j) _G , or equivalently, for , n^(,S,) ≤⌈ 4 (∑_j=1^∞ X_j S(ψ_j) _G /)^2 ⌉ . For we define the linear Monte Carlo method which, for an input , returns the output g = A_n,m^ω(f) := 1/n ∑_i=1^n L_i,m^ω(f) g_i,m^ω , based on the information y_i = L_i,m^ω(f) =∑_j=1^m X_ij ⟨ψ_j, f ⟩_ , and with elements from the output space g_i,m^ω := ∑_j=1^m X_ijS(ψ_j). Here, the X_ij are independent standard Gaussian random variables. This algorithm is actually the fundamental Monte Carlo method from prop:Ma91_l2G when restricting S to the subspace that can be identified with ℓ_2^m. Let P_m denote the orthogonal projection onto _m. Since S is compact, we have S (𝕀_ - P_m) _→ G 0. Then, for elements f from the input set, , we have e((A_n,m^ω)_ω,f) ≤S (𝕀_ - P_m) f _G + e((A_n,m^ω)_ω, P_m f) ≤S(𝕀_ - P_m) _→ G + 2/√(n) ∑_j=1^m X_j S(ψ_j) _G2/√(n) ∑_j=1^∞ X_j S(ψ_j) _G . Note that is monotonously increasing for since the X_j are independent, see lem:E|Y|<E|Y+Z|. With the above lemma it seems natural to consider the idealized method , A_n(f) := 1/n ∑_i=1^n L_i^ω(f) g_i^ω , with information y_i = L_i^ω(f) := ∑_j=1^∞ X_ij ⟨ψ_j,f ⟩_ , and elements from the output space g_i^ω := ∑_j=1^∞ X_ijS(ψ_j). Observe the similarities with standard Monte Carlo integration (<ref>). Observe also the important difference that the present approximation method depends on the particular norm of the input space, whereas standard Monte Carlo integration is defined independently from the input set. Note that, almost surely, L_i^ω is an unbounded functional, so . To see this, for fixed ω, consider the sequence of normalized Hilbert space elements f_ik := 1/√(∑_j=1^k X_ij^2) ∑_j=1^k X_ijS(ψ_j) ∈ , where L_i^ω(f_ik) = √(∑_j=1^k X_ij^2)∞ . Indeed, from lem:gaussqnormvector we have . The deviation result prop:dev2 can be applied, similarly to cor:deviationGauss, to bound the probability for . The Borel-Cantelli lemma implies that the monotone sequence almost surely exceeds any for sufficiently large k. Specifically for embedding problems , the functions g_i^ω, and therefore the output as well, are functions defined on the same domain as the input functions, but they are not from the original Hilbert space (for the same reasons that caused the functionals to be discontinuous). This underlines the non-interpolatory nature of the fundamental Monte Carlo approximation method, compare rem:FundMC. Yet the functions g_i^ω correspond to the information functionals L_i^ω, similar to Hilbert space elements representing continuous linear functionals according to the Riesz representation theorem. Although the functionals L_i^ω are almost surely discontinuous, for any fixed input the random information is a standard Gaussian random variable with variance f_^2, hence almost surely finite. Since, by assumption, the g_i^ω are almost surely defined, we have a method that is almost surely defined for any fixed f. Even more, for any fixed , the idealized algorithm A_n can be approximated with almost sure convergence, A_n,m^ω(f) A_n^ω(f). These considerations motivate to extend the class of admissible information functionals to some class of “stochastically bounded” functionals Λ^ stoch. Actually, this kind of stochastically defined functionals is quite common. For example, the problem of integrating L_p-functions by function values is only solvable in the randomized setting since in that case standard information is discontinuous. Compare also the example from Heinrich and Milla <cit.> which has been discussed in sec:measurable. §.§ Reproducing Kernel Hilbert SpacesWe summarize several facts about reproducing kernel Hilbert spaces (RKHS) that are necessary for the numerical analysis of approximation problems: ↪ L_∞(D),with being a separable Hilbert space defined on a domain . For a general introduction to reproducing kernels, refer to Aronszajn <cit.>. For an introduction with focus on the associated Gaussian field, see Adler <cit.>.The theory of RKHSs is a powerful concept for the analysis of many other numerical settings, e.g. for certain average case problems (see for instance Ritter <cit.>), or when standard information is considered (see Novak and Woźniakowski <cit.> for a bunch of examples), it also proves useful for statistical problems (see Wahba <cit.>).§.§.§ Definition of Reproducing Kernels We assume function evaluations to be continuous on . Then by the Riesz representation theorem, for each there exists a unique function such that for we havef() = ⟨ K_,f ⟩_ .For we define a symmetric functionK(,) := K_() = ⟨ K_,K_⟩_ = ⟨ K_,K_⟩_ = K_() = K(,).This function is called the reproducing kernel of . The reproducing kernel K is positive-semidefinite, that is, for and we have∑_i,j=1^m a_i a_j K(_i,_j) = ⟨∑_i=1^m a_i K__i, ∑_i=1^m a_iK__i⟩_≥ 0.Reversely, any symmetric and positive-semidefinite function defines an inner product on the linear spaceby⟨∑_i=1^m a_i K__i, ∑_j=1^n b_j K__j⟩_K := ∑_i=1^m ∑_j=1^n a_i b_j K(_i,_j),for points and . Its completion with respect to the corresponding norm uniquely defines a Hilbert space which is then called the reproducing kernel Hilbert space with kernel K. This is the space we started with.§.§.§ Comparison to the sup-Norm Knowing the kernel, it is easy to estimate the sup-norm of normalized functions . Indeed, withwe getf_sup = sup_∈ D |f()| = sup_∈ D⟨ K_, f ⟩_ ≤sup_∈ DK__ = sup_∈ D√(⟨ K_, K_⟩_) = sup_∈ D√(K(,)) .This is the initial error for the sup-norm approximation. By the Cauchy-Schwarz inequality, the sup-norm of K is determined by values on the diagonal of ,sup_,∈ D |K(,)| ≤sup_∈ D K(,). Therefore from now on we assume K to be bounded.§.§.§ Decomposition of Reproducing Kernels and a Worst Case Upper Bound Let be an orthonormal basis of , then we can writeK(,) = ∑_i = 1^∞ψ_i()ψ_i().That way it is easy to determine the reproducing kernel of a subspace spanned by , it isK'(,) := ∑_i = n + 1^∞ψ_i()ψ_i() = K(,) - ∑_i=1^n ψ_i()ψ_i() .Therefore, via the linear algorithmA_n(f) := ∑_j=1^n ⟨ψ_j, f ⟩_ ψ_j,analogously to the initial error (<ref>),we can estimate the worst case error,e^,(n,(K) ↪ L_∞(D),) ≤sup_∈ D√(K(,) - ∑_i=1^n ψ_i()ψ_i()) .Actually, the optimal error can be achieved that way, see sec:HilbertWorLB for optimality and lower bounds.§.§.§ Tensor Product of Reproducing Kernel Hilbert Spaces Tensor products are a common way in IBC to define multivariate problems, compare for instance Novak and Woźniakowski <cit.>, or Ritter <cit.>, find many more examples in Novak and Woźniakowski <cit.>.Let andbe reproducing kernel Hilbert spaces defined on D_1 and D_2, respectively. Let and be corresponding orthonormal bases. Then the tensor product space is the Hilbert space with orthonormal basis . Here,denotes the tensor product of functions and ,[f_1 ⊗ f_2](_1,_2) := f_1(_1) f_2(_2) , defined for (_1,_2) ∈ D_1 × D_2.With another tensor product function of this sort, one easily obtains⟨ f_1 ⊗ f_2, g_1 ⊗ g_2 ⟩_1 ⊗_2 = ⟨ f_1,g_1 ⟩__1 ⟨ f_2,g_2 ⟩__2 .Using the representation (<ref>) for the reproducing kernel, it is easy to see that the reproducing kernel K of the new space is the tensor product of the kernels of the original spaces,K((_1,_2),(_1,_2)) := K_1(_1,_1) K_2(_2,_2),where .§.§.§ Canonical Metric and Continuity We consider the canonical metric[ If d_K(,) = 0 for some distinct ≠, we only have a semimetric. Then we still obtain a metric for the set of equivalence classes of points that are at distance 0.] ,d_K(,) := K_ - K__ = √(K(,) - 2 K(,) + K(,)) .Functions are Lipschitz continuous with Lipschitz constant with respect to the canonical metric,|f() - f()| = |⟨ K_ - K_, f ⟩_| ≤K_ - K__ f_ = f_d_K(,).Hence functions from are continuous with respect to any metric δ on D that is topologically equivalent to the canonical metric d_K.Since we assume K to be bounded, the domain D is bounded with respect to d_K,(D) = sup_,∈ D d_K(,) ≤ 2sup_∈ D√(K(,)) .Towards the end of the next section on the boundedness (and continuity) of associated Gaussian fields, we will need the stronger assumption that D is totally bounded with respect to d_K. That is, for any the set D can be covered by finitely many balls with radius r. In particular, if D is complete with respect to d_K, this implies compactness of D. Recall that compactness of a set in a metric space implies total boundedness.§.§ Expected Maximum of Zero-Mean Gaussian FieldsWe discuss zero-mean Gaussian fields and their connection to reproducing kernel Hilbert spaces. All results presented here can be found in the notes by Adler <cit.>. For some results, Lifshits <cit.> or Ledoux and Talagrand <cit.> will also be good references. The Gaussian field associated with a reproducing kernel Hilbert space is a random function such that, for any finite collection of points , the vector is distributed according to a zero-mean Gaussian distribution in ^m, and (Ψ_,Ψ_) = K(,).§.§.§ Series Representation Letbe an orthonormal basis of , .[ The interesting case of course is M = ∞. For M < ∞ almost sure boundedness is obvious, still, good upper bounds for the expected maximum are of interest.] Then the pointwise definitionΨ_ := ∑_i=1^M X_iψ_i(),with X_i being iid standard Gaussian random variables, produces a version of the Gaussian field associated with . Indeed, the covariance function of Ψ defined that way is the kernel K,(Ψ_,Ψ_) = ∑_i,j=1^∞ ( X_i X_j) ψ_i()ψ_j() = ∑_i = 1^∞ψ_i()ψ_i() = K(,). §.§.§ Continuity Note that for the canonical metric of the reproducing kernel Hilbert space we have the alternative representationd_K(,) = √( (Ψ_ - Ψ_)^2) .The question of the boundedness of Ψ is closely related to continuity with respect to the canonical metric . We say that the Gaussian field with covariance function K is continuous, if there exists a version of Ψ with almost surely continuous sample paths.The classical approach for continuity starts with a countable dense subset (we therefore assume D to be separable with respect to d_K):If is continuous on T with respect to the canonical metric d_K, it can be uniquely extended to a continuous function on D by the limitΨ_ := lim_T ∋→Ψ_ ,see Lifshits <cit.>. Working with a countable subset T enables us to determine the probability of Ψ being continuous on T. This probability is either 0 or 1, see e.g. Adler <cit.>. If Ψ is continuous on T with probability 1, indeed, (<ref>) defines an almost surely continuous version of the Gaussian field associated with according to def:Gauss-H(K).If the Gaussian field associated with has continuous sample paths and the domain D is totally bounded, one can show that the series representation (<ref>) converges uniformly[ That is, the series converges in the L_∞-norm.] on D with probability 1, see Adler <cit.>. In the sequel, when talking about Ψ, we always mean the series representation, for which uniform convergence implies continuity with respect to d_K.§.§.§ Boundedness From now on we assume that the domain D is totally bounded.[ If D is not totally bounded, the estimate in prop:Dudley (Dudley) will be infinite. Actually, total boundedness of D is a necessary condition for boundedness of the process Ψ.] In this case, for Gaussian fields, continuity with respect to the canonical metric d_K is equivalent to boundedness, see Adler <cit.>.Let denote the closed d_K-ball around with radius .The following result can be found in Adler <cit.>, it is originally due to Fernique 1975 <cit.>. Let μ be any probability measure on D, then sup_∈ DΨ_≤ C_Fernique sup_∈ D∫_0^∞√(log(1/μ(B_K(,r)))) r, where is a universal constant.From Adler <cit.> one can extract a value. This constant is not optimal. From the book of Ledoux and Talagrand <cit.> (via the Young function ) we gain the much better estimate .A measure for which the right hand side of the above proposition is finite, is called a majorizing measure for the metric space . Majorizing measures must be – in a certain way – “well spread” because the integral will be infinite if for some and . Yet it may be discrete, see the construction for the proof of prop:Dudley below. Furthermore, the integral vanishes for r exceeding the diameter of D with respect to d_K.Sometimes it is inconvenient to work with majorizing measures. An alternative way of estimating the maximum of a Gaussian field is based on metric entropy. For , let denote the minimal number of d_K-balls with radius r needed to cover D. The function is called the (metric) entropy of D. The following inequality is based on this quantity, it goes back to Dudley 1973 <cit.>. There exists a universal constant such that sup_∈ DΨ_≤ C_Dudley ∫_0^∞√(log N(r)) r.For a direct proof with explicit numerical bound , see Lifshits <cit.>. In the book of Adler <cit.> Dudley's inequality was derived from Fernique's estimate. By scaling, without loss of generality, the diameter of D is 1. For , let be a minimal collection of points such that D is covered by balls of radius , D = ⋃_j=1^N(2^-k) B_K(_k,j,2^-k). Then, defining μ(E) := 1/2∑_k=0^∞ 2^-k [1/N(2^-k)∑_j=1^N(2^-k)[_k,j∈ E] ] for E ⊆ D, we obtain a majorizing measure and may apply prop:Fernique (Fernique), see Adler <cit.> for more details.[ In Adler <cit.> the construction of the measure is less explicit. Check the proof with the construction given here.] Since we are interested in the expected sup-norm of Ψ, we also need the following easy lemma, compare Adler <cit.>. For the Gaussian field Ψ with covariance function K, we have Ψ_∞≤√(2/π) inf_∈ D√(K(,)) + 2sup_∈ DΨ_ . With _0 ∈ D, by the triangle inequality we obtain Ψ_∞≤ |Ψ__0| + Ψ - Ψ__0_∞ . Since Ψ__0 is a zero-mean Gaussian random variable with variance , we get |Ψ__0| = √(2/π) √(K(_0,_0)) . For the random field , we have , so by symmetry Φ__∞ = max{sup_∈ DΦ_, sup_∈ D (-Φ_) }≤sup_∈ DΦ_ + sup_∈ D (-Φ_) = 2sup_∈ DΦ_ . Finally, observe that sup_∈ DΦ_ = sup_∈ DΨ_ . The lemma is obtained taking the infimum over . §.§ A Lower Bound in the Worst Case SettingAs mentioned in sec:ranApp:e(n), commonly used discretization techniques are not feasible for tractability analysis. Osipenko and Parfenov 1995 <cit.>, Kuo, Wasilkowski, and Woźniakowski 2008 <cit.>, and Cobos, Kühn, and Sickel 2016 <cit.>, independently from each other found similar approaches to relate the error of L_∞-approximation to L_2-approximation. The formulation of prop:H->L_inf/L_2 follows <cit.>, giving a lower bound for the L_∞-approximation in terms of singular values of some -approximation. Parts of the proof in the original papers are based on the theory of absolutely summing operators. For this thesis, however, a proof that is based on tools from IBC appears more natural. Namely, in Kuo et al. <cit.> the worst case L_∞ error was compared to the average error with respect to the Gaussian field associated with the reproducing kernel Hilbert space .We start with a well known fact on the optimality of linear algorithms for the approximation of Hilbert space functions. The proofs are given for completeness. Consider a linear problem S: → G with the input set F being the unit ball of a Hilbert space . * Linear algorithms are optimal, more precisely, optimal algorithms have the structure where P is an orthogonal rank-n projection on . Hence we can write e^(n,S,) = inf_P Proj.P = nS (𝕀_ - P) _→ G . * If G = _2 is another Hilbert space and S is compact, we have a singular value decomposition. That is, there is an orthonormal system in such that is orthogonal in , and S can be written S f = ∑_k=1^M⟨ψ_k , f ⟩_S ψ_k, where . Furthermore, the sequence of the singular values σ_k := S ψ_k_G > 0 for k < M+1, 0for k > M, is ordered, . Then [ Following the axiomatic scheme of Pietsch <cit.>, singular values of linear operators between Hilbert spaces are a special case of s-numbers, . All kinds of s-numbers – which may differ for operators between arbitrary Banach spaces – coincide with the singular values in the Hilbert space setting. The correspondence to the error of deterministic methods exhibits the usual index shift we encounter when relating quantities from IBC to s-numbers, see also the discussion towards the end of sec:measurable, and the definition of Bernstein numbers in sec:BernsteinSetting. ] e^(n,S,) = σ_n+1 . (a) Let be any deterministic information mapping using adaptively chosen functionals L_k,_[k-1]. We define the non-adaptive information N^(f) := (L_1(f),L_2,0(f),L_3,_[2](f),…,L_n,_[n-1](f)), which uses the functionals that N would choose in the case of . Let P be the orthogonal projection onto , i.e. is the orthogonal projection onto , and consider the linear rank-n algorithm (it is indeed based on the information N^ since ). The error of this algorithm is e(A_n,S) = sup_f ∈ FS(𝕀_ - P) f_G = sup_f ∈⊷(𝕀_ - P)f_≤ 1S f _G = sup_f ∈ F N(f) = S f _G ≤ inf_ϕsup_f ∈ F N(f) = S f - [ϕ∘ N](f)_G ≤ inf_ϕ e(ϕ∘ N, S) . This shows that the error of A_n is maximal for zero information, a case which also occurs for the adaptive information mapping. (b) The singular value decomposition is a standard result from spectral theory. For n ≥ M, the statement is trivial since S itself, with rank , can be seen as a suitable algorithm. Now, for , take the rank-n algorithm A_n(f) = ∑_k=1^n ⟨ψ_k , f ⟩_ψ_k, with the error e(A_n,f) = ∑_k=n+1^M⟨ψ_k , f ⟩_ ψ_k _G = √(∑_k=n+1^M⟨ψ_k , f ⟩_^2 σ_k^2) ≤ σ_n+1 √(∑_k=n+1^M⟨ψ_k , f ⟩_^2) ≤ σ_n+1 f_ , where equality is attained for . This gives us the upper bound. This upper bound is optimal. Indeed, for any rank-n algorithm , there exists an element with f_ = 1 and , wherefore e(A_n,f) = f_G = √(∑_k=1^n+1σ_k^2 ⟨ψ_k , f ⟩_^2)≥σ_n+1f_ . This implies the matching lower bound e^(n,S,) ≥σ_n+1 .Let ρ be a measure on D (defined for Borel sets in D, with respect to the canonical metric d_K). Recall that denotes the space of (equivalence classes of) measurable functions defined on D and bounded in the normf_L_p(ρ) := (∫_D f^p ρ)^1/p for 1 ≤ p < ∞, _D,ρ |f| for p = ∞,where . For continuous functions it makes sense to consider the supremum normf_sup := sup_∈ D |f()| ≥f_L_∞(ρ) ,later, when the supremum norm and the L_∞-norm coincide, we will only write .The following version of a worst case lower bound is close to the formulation of Osipenko and Parfenov <cit.>, also Cobos et al. <cit.>, however, it is essentially contained in Kuo et al. <cit.> as well.[ Kuo et al. in their research work with eigenvalues of an integral operator defined via the kernel function K. These eigenvalues are the squared singular values, which in turn we prefer to use here.] The first part of the proof follows Kuo et al. <cit.>. Let ρ be a probability measure on the domain D, and let the separable reproducing kernel Hilbert space be compactly embedded into . The embedding is compact as well, and a singular value decomposition exists. This means, there is an orthonormal basis (with ) of which is orthogonal in as well, and the corresponding singular values , for , are in decaying order . Then we have e^(n,↪ L_∞(ρ),) ≥√(∑_k=n+1^∞σ_k^2 ) . Without loss of generality . By lem:H->,lin=opt,singular we know that optimal algorithms with cardinality n for the approximation problem can be built with orthonormal , A_n(f) = ∑_j=1^n ⟨φ_j, f ⟩_ φ_j. The orthonormal system can be completed to an orthonormal basis of . Then the worst case error is e(A_n,↪ L_∞(ρ)) = _∈ D√(∑_j=n+1^Mφ_j^2())≥√(∫∑_j=n+1^Mφ_j^2() ρ()) , compare (<ref>).[ The proof is simpler once knowing that the optimal algorithm, in fact, is built of an orthogonal projection within in composition with the solution operator. In Kuo et al. <cit.> a little more work is needed because it was only used that optimal algorithms are linear.] Consider the Gaussian field Ψ associated to , Ψ_ = ∑_j=1^M X_jφ_j(), where the X_j are independent standard Gaussian random variables, and let μ denote the distribution of Ψ. The intention is to study the algorithm A_n for the -approximation in the μ-average setting. For however, almost surely, but A_n uses functionals that are defined for functions from . So instead, consider the random functions Ψ^(m) for , Ψ_^(m) := ∑_j=1^m ∧ M X_jφ_j(), and let μ^(m) denote the corresponding distribution in . Clearly, Ψ_^(m) - [A_nΨ^(m)]() = ∑_j=n+1^m ∧ M X_jφ_j(), and for the root mean square average L_2(ρ)-error we have e_2(A_n,L_2(ρ),μ^(m)) = √(∫(∑_j=n+1^m ∧ M X_jφ_j() )^2 ρ()) [Fubini, X_j iid] = √(∫∑_j=n+1^m ∧ Mφ_j^2()ρ()) . In comparison with (<ref>), this shows e(A_n,↪ L_∞(ρ)) ≥lim_m →∞ e_2(A_n,L_2(ρ),μ^(m)), where the limit is approached monotonously from below. The RHS of (<ref>) can be expressed by means of singular values of the compact mapping . In detail, there exists an orthonormal system in such that is orthogonal in and [𝕀 - A_n] f = ∑_i=1^M'⟨χ_i , f ⟩_[𝕀 - A_n] χ_i, with the singular values , for , as always in decaying order . (For we have .) With Z_i^(m) := ⟨χ_i , Ψ^(m)⟩_ = ∑_j=1^m ∧ M⟨χ_i,φ_j ⟩_X_j, we can write e_2(A_n,L_2(ρ),μ^(m)) = √([𝕀 - A_n] Ψ^(m)_L_2(ρ)^2) = √( ∑_i=1^M' Z_i^(m)[𝕀 - A_n]χ_i _L_2(ρ)^2)= √(∑_i=1^M'τ_i^2 (Z_i^(m))^2) . Note that (Z_i^(m))^2 [X_j iid]= ∑_j = 1^m ∧ M⟨χ_i, φ_j ⟩_^2 χ_i_^2 = 1, where the sequence is monotonically increasing in m, so we have lim_m →∞ e_2(A_n,L_2(ρ),μ^(m)) = √(∑_i = 1^∞τ_i^2) . It remains to show that , and by (<ref>) we are done. Consider any algorithm A'_m for the approximation of . Then is an algorithm of cardinality for the approximation of . By this observation and lem:H->,lin=opt,singular (<ref>), we obtain τ_i = e^(i-1,[𝕀 - A_n] : → L_2(ρ)) = inf_A'_i-1sup_f_≤ 1[𝕀 - A_n]f - A'_i-1 f _L_2(ρ)≥inf_A”_n+i-1sup_f_≤ 1f - A”_n+i-1f= e^(n+i-1,𝕀 : ↪ L_2(ρ)) = σ_n+i .[Diagonal operators] The proposition above can be used to prove lower bounds for the approximation of diagonal operators on sequence spaces. This has already been pointed out by Osipenko and Parfenov <cit.>. We repeat the example for its connection to the RKHS framework. Consider a compact matrix operator A: ℓ_2 →ℓ_∞, (x_j)_j ∈↦(∑_j a_ijx_j)_i ∈ . This problem is equivalent to the embedding operator 𝕀 : (K) ↪ℓ_∞ , where we have the reproducing kernel K: ×→, K(i,j) = (A A^⊤)(i,j). Let ρ be a probability measure on , with denoting the probability of , we have . Then ℓ_2(ρ) is the space of sequences that are bounded in the norm _ℓ_2(ρ) := √(∑_i=1^∞ρ_i x_i^2) . Now, let A be a diagonal operator [A ](i) = λ_i x_ifor i ∈, where and . The reproducing kernel for the Hilbert space of the corresponding embedding problem is K(i,j) = δ_ij λ_i^2. Hence is an orthonormal basis of (K), here . It is also orthogonal in ℓ_2(ρ) for any measure ρ on . Taking ρ_i := λ_i^-2/∑_j=1^m λ_j^-2 [i ≤ m], with , for we have the singular values σ_1 = … = σ_m = (∑_j=1^m λ_j^-2)^-1/2 > 0 = σ_m+1 = σ_m+2 = … By prop:H->L_inf/L_2 we obtain e^(n,A: ℓ_2 →ℓ_∞,) ≥√(m-n/∑_j=1^m λ_j^-2) , taking the supremum over m, , this gives sharp lower bounds. The proof for the lower bound does not reveal any information about the structure of optimal methods. A construction of methods can be found in the book of Osipenko <cit.>. The finite dimensional case goes back to Smolyak 1965 <cit.>, the general case to Hutton, Morrell, and Retherford <cit.>.§ BREAKING THE CURSE FOR FUNCTION APPROXIMATIONWe study two examples of tensor product Hilbert spaces where for L_∞-Approximationwe can show the curse of dimensionality for the worst case, but in the randomized setting we have polynomial tractability with . The first example on the approximation with the Brownian sheet, see sec:BrownianSheet, is more or less a toy example intended to demonstrate the new techniques in a setting that is easy to visualize, read rem:BrownianToy for further comments on this rating of the example. For the second example on unweighted periodic tensor product Hilbert spaces, see sec:HilbertPeriodic, we find some general conditions that are sufficient for Monte Carlo to break the curse. These conditions are specified for Korobov spaces, see thm:Korobov. The latter constitutes the main application of the present chapter. We will close the chapter with some final hints that one should bear in mind when searching for further examples, see sec:HilbertFinalRemarks. §.§ Approximation with the Brownian SheetWe consider the Hilbert space with the Wiener sheet kernel K_d on , the associated continuous Gaussian field W is also called Brownian sheet, see e.g. Adler <cit.> for basic properties of this space and the associated Gaussian random field.[ In the lecture notes of Adler, sometimes a set-indexed Brownian sheet is dealt with. We do not need that concept here.]For we take the covariance kernel of the “two-armed” Brownian motion,K_1(x,z) := |x+z| - |x-z|/2= [ x =z]min{|x|,|z|} .This space consists of once weakly differentiable functions,(K_1) = { f:[-1,+1] →| f(0) = 0, f_(K_1) := f'_2 } ,the inner product is . For we take the tensor product kernel,K_d(,) := ∏_j=1^d K_1(x_j,z_j) = [ = ]∏_j=1^d min{|x_j|,|z_j|} ,where is the vector-valued signum function. Thus(K_d) = ⊗_j=1^d (K_1) = {f:[-1,+1]^d →|f() = 0 if ∃_j x_j = 0, and f_ := D^ f_L_2 < ∞} ,where .Note that the initial error is 1 (thus properly normalized) since the kernel takes its maximum 1 in the corners .Furthermore, functions can be identified with functions , f̃(x_1,…,x_d+1) := f(x_1,…,x_d) [x_d+1]_+,then , and , where for f̃ the maximal absolute value is attained with . Deterministic L_∞-approximation of functions from the Wiener sheet space on suffers from the curse of dimensionality, in detail, n^(,(K_d) ↪ L_∞,) ≥ 2^d-1 , for 0 < ≤√(2)/2. We consider the 2^d-dimensional subspace of spanned by , where K_() :=K_d(,) = [ = ]∏_j=1^d |x_j|. These functions are orthonormal in = (K_d) since ⟨ K_, K_'⟩_ =K_d(,') = [ = ']. Besides, they have essentially disjoint supports (which are the subcubes of with constant sign in each coordinate), and take their supremum in , which is . For we have f_(K_d) = √(∑_∈{± 1}^d a_^2) , andf_L_∞ = max_∈{± 1}^d |a_|. Hence we can estimate the error from below by comparison to a sequence space embedding, e^(n,(K_d) ↪ L_∞) ≥ e^(n, ℓ_2^2^d↪ℓ_∞^2^d) ≥√(1 - n 2^-d) , see (<ref>). This implies the stated lower bound for the complexity, compare sec:l2->linf,curse. Randomized L_∞-approximation of functions from the Wiener sheet space, using linear functionals for information, is polynomially tractable. In detail, n^(,(K_d) ↪ L_∞,) ≤ Cd (1+ log d)/^2 , with a numerical constant C > 0. We want to apply lem:stdMCapp, so we need to estimate W_∞ for the Brownian sheet W on defined by the covariance kernel K_d. This will be done by an entropy estimate and prop:Dudley (Dudley).[ The same approach for estimating the expected maximum of W is taken in Adler <cit.> though for the domain [0,1]^d instead, which admittedly is a minor change. Steps that have been left to the reader there are explicated here.] The canonical metric of the Wiener sheet kernel is d_K(,)^2 = ∏_j=1^d |x_j| + ∏_j=1^d |z_j| - 2 [ = ] (∏_j=1^d min{|x_j|,|z_j|}). If there exists an index such that , hence d_K(,)^2 = ∏_j=1^d |x_j| + ∏_j=1^d |z_j| ≤ |x_j| + |z_j| = |x_j - z_j| ≤ | - |_∞ . If , we obtain d_K(,)^2 = ∏_j=1^d |x_j| + ∏_j=1^d |z_j| - 2 ∏_j=1^d min{|x_j|,|z_j|}≤ 2(∏_j=1^d max{|x_j|,|z_j|} - ∏_j=1^d min{|x_j|,|z_j|})[telescoping sum] ≤ 2∑_k=1^d [ (∏_j=1^k-1min{|x_j|,|z_j|}) _≤ 1(max{|x_k|,|z_k|} - min{|x_k|,|z_k|}) _=|x_k-z_k|(∏_j=k+1^d max{|x_j|,|z_j|}) _≤ 1]≤ 2 | - |_1. This shows d_K(,)^2 ≤ 2 d | - |_∞ , and for ∈ [-1,1]^d and r>0 we have the inclusion B_∞(,r^2/2 d) ⊆ B_K(,r), where B_∞ denotes the ball in the ℓ_∞^d-metric. Since one can cover by balls with ℓ_∞^d-radius r^2/2 d, for the metric entropy with respect to d_K we obtain H(r) = log(N(r)) ≤ d log(1 + 2 d/r^2) ≤ C_1 d (1 + log d) (1 - log r). Note that with the ball around . Since the Brownian sheet W is zero on the coordinate hyperplanes, we do not need the additional term when applying lem:E|X|<init+EsupX, W_∞ ≤ 2sup_∈ [-1,1]^d W_ [prop:Dudley] ≤ 2 C_DudleyC_1 √(d (1 + log d))∫_0^1 √(1 - log r) r[subst. s^2/2 = 1 - log r] = C_2√(d (1 + log d)) /√(2) ∫_1^∞ s^2exp(-s^2/2) s]= C_3√(d (1 + log d)) . Hence by lem:stdMCapp, n^(,(K_d) ↪ L_∞,) ≤ 4 C_3^2d (1 + log d)/^2 . This finishes the proof. This is a first example of a function approximation problem where Monte Carlo methods can break the curse of dimensionality. As mentioned before, the initial error is properly normalized, and the problem for lower dimensions is contained in the problem for higher dimensions. This example, however, has a downside: Actually, we treat the simultaneous approximation of 2^d entirely independent functions (that only need to be bounded in a common Euclidean norm). This view is justified by the fact that functions from the space are zero at the coordinate hyperplanes, that way the domain is split into subcubes of constant sign. Observe the similarities with the sequence space example in sec:l2->linf,curse where we only lack the logarithmic term in the Monte Carlo upper bound. Adding some proper “function space nature” by this Wiener sheet example, honestly, serves as a fig-leaf for the artificiality of the sequence space example. The next section treats much more natural problems. Including this example in this study, however, was motivated by the fact that the Brownian sheet is widely known, and that the loss of smoothness becomes palpable. In addition, this gives us a non-periodic example where it was convenient to use entropy methods for the estimate (in contrast to the next section where we will rely on the technique of majorizing measures). The Wiener sheet – usually only defined on – is a common example for many topics in IBC, see for example Novak and Woźniakowski <cit.> or Ritter <cit.>. §.§ Tensor Product Spaces of Periodic Functions§.§.§ The General Setting We consider the L_∞-approximation of Hilbert space functions defined on the d-dimensional torus ^d, compare the notation in Cobos et al. <cit.> (with slight modifications). The Hilbert spaces we consider will be unweighted tensor product spaces.A few words on the domain. The one-dimensional torus can be identified with the unit interval tying the endpoints together. A natural way to define a metric on isd_(x,z) := min_k ∈{-1,0,1} |x-z+k|, for x,z ∈ [0,1).This is the length of the shortest connection between two points along a closed curve of length 1. For the d-dimensional torus we take the summing metricd_^d(,) := ∑_j=1^d d_(x_j,z_j).Smoothness and continuity are to be defined with respect to this metric.We start with a basis representation of spaces under consideration. First, for d=1, the Fourier system{φ_0 := 1, φ_-k := √(2) sin(2πk ·), φ_k := √(2) cos(2 pi k ·) }_k ∈is an orthonormal basis for . We consider Hilbert spaces where these functions are still orthogonal. Namely, letdenote the Hilbert space for which the system{ψ_0 := λ_0, ψ_-k := λ_ksin(2πk ·), ψ_k := λ_kcos(2πk ·) }_k ∈∖{0} ,is an orthonormal basis.[ If λ_k = 0 for certain k, of course, the corresponding zero-functions cannot be part of the orthonormal basis. In the proof of thm:Korobov we consider finite-dimensional subspaces where the corresponding orthonormal basis will be . The same holds for the basis of the d-dimensional space.] Here,indicates the importance of the different frequencies. Now, for general , we consider the unweighted[ This means that every coordinate is equally important. For weighted tensor product spaces one would take different values for for different dimensions , compare Cobos et al. <cit.>, or Kuo et al. <cit.>.] tensor product space with the tensor product orthonormal basis ,ψ_() := ∏_j=1^d ψ_k_j(x_j).Analogously, we write for the Fourier basis of , once more at the risk of some confusion from using the same letter for the index as in the one-dimensional case, merely with a different font style.For a suitable choice of the λ_k, we have the one-dimensional reproducing kernelK_(x,z) := λ_0^2 + ∑_k=1^∞λ_k^2 [cos(2 π k x)cos (2 π k z) + sin (2 π k x)sin (2 π k z)] = ∑_k=0^∞λ_k^2cos (2 π k (x-z)) ,for general dimensions we obtain the product kernelK_^d(,) := ∏_j=1^d K_(x_j,z_j).In particular, the initial error ise(0,_(^d) ↪ L_∞(^d)) = sup_∈^d√(K_^d(,)) = ( ∑_k=0^∞λ_k^2 )^d/2 .The condition is necessary and sufficient for the existence of a reproducing kernel and for the embeddingto be compact, see Cobos et al. <cit.> with an extended list of equivalent properties. We will assumefor that the initial error be constant 1.Note that under this last assumption, functions can be identified with functions for ,f̃(x_1,…,x_d̃) := f(x_1,…,x_d) K(0,x_d+1),the _- and the L_∞-norms coincide, the maximum values of the function being attained for . So indeed, the problems of lower dimensions are contained in the problems of higher dimensions, yet f̃ is a bit lopsided in the redundant variable. Suppose that and for non-negative λ_k. Then the approximation problem : _(^d) ↪ L_∞(^d) suffers from the curse of dimensionality in the deterministic setting. In detail, while the initial error is constant 1, we have e^(n,_(^d) ↪ L_∞(^d)) ≥√((1 - nβ^d)_+) , where . In other words, for we have the complexity bound n^(,_(^d) ↪ L_∞(^d)) ≥β^-d(1-)^2. Following prop:H->L_inf/L_2, we study the singular values of . Essentially, this can be traced back to the one dimensional case, ψ_k = σ_k φ_k for k ∈, where andfor k ∈ denote the unordered singular values of . In the multi-dimensional case we have ψ_ = σ_φ_for ∈^d, with the unordered singular values , in particular σ_^2 ≤(sup_k' ∈σ_k'^2)^d = (sup{λ_0^2,λ_k'^2 / 2}_k' ∈)^d = β^d. On the other hand, ∑_∈^dσ_^2 = (∑_k' ∈σ_k'^2 )^d = (∑_k' = 0^∞λ_k'^2 )^d = 1. So for any index set of size , we have √(∑_∈^d ∖ Iσ_^2)≥√((1 - n β^d)_+) . By prop:H->L_inf/L_2, this proves the lower bound. Within the above proof we applied prop:H->L_inf/L_2 with ρ being the uniform distribution on ^d. If we consider complex-valued Hilbert spaces, this approach will always give sharp lower bounds, see Cobos et al. <cit.>. In the real-valued setting we obtain sharp error results at least for those n where the optimal index set contains all indices belonging to the same frequency, that is, = (k_1,…,k_d) ∈ I ⇔ := (|k_1|, …, |k_d|) ∈ I. Still, in most cases it is hard to estimate the number of singular values within a certain range. The following abstract result relies on estimates for the shape of the kernel function . Consider the uniform approximation problem : (K_d) ↪ L_∞(^d) where (K_d) is a reproducing kernel Hilbert space on the d-dimensional torus ^d with the following properties: * K_d is the unweighted product kernel built from the one-dimensional case, this means for . * for all x ∈. (Consequently, for all , in particular the initial error is constant 1.) * The kernel function can be locally estimated from below with an exponential decay, that is, there existand such that K_1(x,z) ≥exp(-αd_(x,z)) forwith . (Hence for .)[ If one is interested in a version of this theorem with better constants for particularly nice kernels, one could start with a stronger assumption which is a comparison to a bell-shaped curve. The asymptotics of the complexity result, however, will not change. See also rem:Korobov-small r with a proposal for a modified version of this theorem. ] Then the problem is polynomially tractable in the randomized setting with general linear information , in detail, n^(,(K_d) ↪ L_∞(^d),) ≤ C (1 + α^2 - log 2 R_0)d (1 + log d)/^2 , with a universal constant . We are going to apply the method of majorizing measures in order to estimate the expected maximum norm of the Gaussian field Ψ associated with the reproducing kernel K_d. The majorizing measure μ we choose shall be the uniform distribution on ^d, this is the Lebesgue measure on . Supposed that , for the canonical metric in the d-dimensional case we have d_K(,)^2 = K_d(,) + K_d(,) - 2 K_d(,) ≤ 2 (1 - exp(-αd_^d(,))) ≤ 2αd_^d(,). By this, we have the inclusion B_(√(r/2 α), ) ⊆ B_K(r,), where denotes the d_^d-ball of radius R around ∈^d, andis the ball of radius r in the canonical metric associated with K_d. Hence μ(B_K(r,)) ≥μ(B_(√(r/2 α), ) ). We distinguish three cases: * For , the μ-volumeof the torus metric ball is the volume of an ℓ_1-ball in ^d with radius R, so with Stirling's formula, log(1/μ(B_(R,))) = log(1/(R B_1^d)) = logΓ(d + 1) - dlog 2 R ≤ C_1 d (1 + log d) (1 - log 2 R) ≤ C_1 d (1 + log d) (1 - log (2 R)^2). Such an estimate can be used for . * For , the μ-volume of can be estimated from below with the μ-volume of an ℓ_1-ball with radius . We will use this in the case . * For , we know with μ-volume 1 since . In this case the term log(1/μ(B_K(r,))) vanishes. Combining these cases, we can estimate ∫_0^∞√(log(1/μ(B_K(,r)))) r ≤∫_0^α/2√(log(1 / (√(r/2 α)B_1^d ) ) )r+ (2 - 2αR_0^2)_+ √(log(1 / (R_0 B_1^d ) ) )(<ref>)≤√(C_1 d (1 + log d))(∫_0^α/2√(1 - log2 r/α) r +2√(1 - log 2 R_0)) = C_2 (1 + α + √(- log 2 R_0))√(d (1 + log d)) . Here, the last integral can be transformed into a familiar integral by the substitution , ∫_0^α/2√(1 - log2 r/α) r =α/2 √(2) ∫_1^∞ s^2exp(-s^2/2) s. Now, consider the Gaussian field Ψ associated with the reproducing kernel K_d. Putting the above calculation into prop:Fernique (Fernique), with lem:E|X|<init+EsupX we obtain Ψ_∞ ≤√(2/π) + 2 C_FerniqueC_2 (1 + α + √(- log 2 R_0)) √(d (1 + log d))≤ C_3(1 + α + √(- log 2 R_0))√(d (1 + log d)) . By lem:stdMCapp, this gives us a final upper bound on the complexity. We finish the general part of the periodic setting with sufficient conditions for the parameters of the kernels K_ for that we can apply thm:MCUBperiodic. Given a kernel K_1(x,z) := K_(x,z) = ∑_k=0^∞λ_k^2cos 2 π k (x-z) with , and tensor product kernels K_^d as before, it is sufficient for polynomial tractability in the randomized setting that the following holds: * ∑_k=0^∞λ_k^2 = 1, * σ_ := ∑_k=1^∞ kλ_k^2 < ∞. In detail, there exists a universal constant C'> 0 such that n^(,(K_^d) ↪ L_∞(^d),) ≤ C' (1 + σ_^2) d (1 + log d)/^2 . Condition (<ref>) is for the normalization of the initial error, see (<ref>) in thm:MCUBperiodic. We first check, when assumption (<ref>) of thm:MCUBperiodic holds with , that is, the inequality is valid for all . It suffices to show that h(x) := exp(αx) K_(x,0) is monotonously increasing for , noting by (<ref>). Condition (<ref>) guarantees differentiability of with absolute convergence of the resulting series of sine functions. Moreover, for the derivative of h we obtain h'(x) = exp(αx) [α ∑_k=0^∞λ_k^2cos 2 π k x - 2 π∑_k=1^∞ kλ_k^2sin 2 π k x ] ≥exp(αx) [α (λ_0^2 - ∑_k=1^∞λ_k^2 ) - 2 π∑_k=1^∞ kλ_k^2 ]. Positivity of the left-hand term in is ensured if holds in addition to (<ref>). The right-hand term in is finite thanks to condition (<ref>), hence we can choose α :=2πσ_/2λ_0^2 - 1 to guarantee non-negativity of . This gives us as intended. Restricting to the case we have a better control over the constants, in that case getting . Hence by thm:MCUBperiodic, n^(,(K_^d) ↪ L_∞(^d), ) ≤ C (1 + (6π)^2σ_^2) d (1 + log d)^-2≤ C_0 (1 + σ_^2) d (1 + log d)^-2 . If ,[ In the case we need a local estimate because we can not a priori exlude negative or vanishing values for the kernel function for far apart points and . In the case a localized view will make better constants possible, in the end we aim for a universal constant C'. ] we compare K_ to a kernel K_ with and for , where is a scaling factor such that (<ref>) holds for as well. Besides, (<ref>) is inherited from . This shows the existence of a constant such that for . Note that K_(x,0) = λ_0^2 + c(K_(x,0) - 2/3) ≥ cexp(-αx) - (2/3 c - λ_0^2) = 3 (1-λ_0^2)exp(-αx) - (2 - 3 λ_0^2). If we choose , for we can finish with the estimate K_(x,0) ≥exp(- βx). Here we used that withand , for the following inequality holds, exp(- β x) ≤ (1+γ)exp(- α x) - γ , in our case and . (A proof for this inequality can be done by observing that / as a function in x is monotonously growing for small x>0.) By this, from thm:MCUBperiodic we obtain the complexity bound n^(,(K_^d) ↪ L_∞(^d), ) ≤ C (1 + β^2 - log 2 R_0) d (1 + log d)^-2≤ C_1 (1 + α^2 + logα) d (1 + log d)^-2 ≤ C_2 (1 + σ_^2) d (1 + log d)^-2 . Finally, the constant in the corollary is . §.§.§ Example: Korobov Spaces We apply the above results to unweighted Korobov spaces. In the framework of this section, these are spaces with and for , where β_0,β_1 > 0. For integers , the Korobov space norm can be given in a natural way in terms of weak partial derivatives (instead of Fourier coefficients), in the one-dimensional case we havef__r^()^2 = β_0^-1| ∫_ f(x)x |^2 + β_1^-1(2π)^-2r f^(r)_2^2.The d-dimensional case is a bit more complicated, in a squeezed way, the norm isf__r^(^d)^2 = ∑_J ⊆ [d]β_0^-(d - #J)(β_1^-1(2π)^-2r)^#J ∫_^[d] ∖ J(∏_j ∈ J∂_j^r) f()_[d] ∖ J_L_2(^J)^2,see Novak and Woźniakowski <cit.> for details on the derivation of this representation of the norm. There one can also find some information on the historical background concerning these spaces. It should be pointed out that in the same book tractability for L_2-approximation of Korobov functions based on has been studied <cit.>, in that case randomization does not help a lot. The condition is necessary and sufficient for the existence of a reproducing kernel (and the embedding to be compact), then∑_k=1^∞λ_k^2 = β_1∑_k=1^∞ k^-2 r = β_1ζ(2 r)with the Riemann zeta function ζ. Assumingβ_0 + β_1ζ(2 r) = 1,the initial error will be constant 1 in all dimensions. Furthermore, withwe have the curse of dimensionality for the deterministic setting, see thm:curseperiodic. Consider unweighted Korobov spaces as described above. For smoothness , fixing such that the initial error is constant 1 for all dimensions, we have polynomial tractability for the uniform approximation with Monte Carlo, in detail, n^(,_r^(^d) ↪ L_∞(^d), )≼ d (1 + log d)^-2 for r > 1, d (1 + (log d)^3)^-1(1 + (log^-1)^2) for r = 1, d^1/(r-1/2) - 1(1 + log d)^-1/(r-1/2) for 1/2 < r < 1. The hidden constant may depend on r. We start with the easiest case . By (<ref>) we have , thus we satisfy (<ref>) in cor:periodic-sufficient with σ_ = ∑_k=1^∞ kλ_k^2 = β_1∑_k=1^∞ k^-(2 r - 1)≤ζ(2 r - 1), and obtain (with the constant C' from the corollary) n^(,_r^(^d) ↪ L_∞(^d),) ≤ C' (1 + ζ(2 r - 1)^2) d (1 + log d)/^2 . For , the quantity σ_ is infinite. Therefore we apply the fundamental Monte Carlo method from prop:Ma91_l2G to a finite dimensional subspace of finite Fourier sums up to frequencies in each dimension. With the orthonormal basis of _(^d), see (<ref>), for f ∈_(^d) we define f_m := ∑_∈^d ||_∞≤ m⟨ψ_,f⟩__ ψ_ . Taking a Monte Carlo method with , we can estimate the error for by e((A_n^ω)_ω, f) ≤f - f_m_∞ + e((A_n^ω)_ω, f_m). For this term to be bounded from above by , we desire both summands to be bounded from above by /2. By the worst case error formula (<ref>), together with the kernel representation (<ref>), for it easily follows f - f_m_∞^2 ≤1 - ∑_∈^d ||_∞≤ mλ_^2 =1 - (∑_k=0^m λ_k^2 )^d. In our particular situation with for , we can estimate ∑_k = m+1^∞λ_k^2 ≤β_1 ∫_m^∞ t^-2s t = β_1/2r - 1m^-(2r - 1) . Hence, together with , we obtain f - f_m__^2 ≤ 1 - (1 - β_1/2r - 1m^-(2r - 1))^d[for β_1/2r - 1m^-(2r - 1) < 1/2] (∗)≤ 1 - exp(- log 2β_1/r - 1/2d m^-(2r - 1)) ≤log 2β_1/r - 1/2d m^-(2r - 1) . Here we used for . Choosing m := ⌈(4 (log 2)β_1/r - 1/2d^-1)^1/(2r-1)⌉ , step (∗) is actually valid, and we bound . For the error analysis of , we need to understand the restricted approximation problem : _'(^d) ↪ L_∞(^d), where . The method A_n shall be the fundamental Monte Carlo approximation method from prop:Ma91_l2G applied to this problem. The initial error is smaller than 1, so we cannot apply cor:periodic-sufficient directly to this problem. Therefore, consider another space with for and . The initial error of the approximation problem : _(^d) ↪ L_∞(^d) is then properly normalized by construction. Applying prop:Ma91_l2G to and means determining the expected L_∞-norm of the corresponding Gaussian processes Ψ^(',d) and Ψ^(,d), respectively. For better comparison it is useful to represent these via the Fourier basis of , Ψ^(',d) = ∑_∈^d ||_∞≤ m 2^-||_0/2 λ_X_ φ_ , andΨ^(,d) = ∑_∈^d ||_∞≤ m 2^-||_0/2 κ_X_ φ_ , where the X_ are iid standard Gaussian random variables. Note that, by construction, for , so we have , see lem:E|sum aXf|. Consequently, complexity bounds from applying cor:periodic-sufficient to also hold for . This gives n^(,_(^d) ↪ L_∞(^d))≤ n^(/2,_'(^d) ↪ L_∞(^d)) ≤ n^(/2,_(^d) ↪ L_∞(^d)) ≤ 4 C' (1 + σ_^2) d (1 + log d)/^2 . It remains to estimate σ_: σ_ = ∑_k=1^∞ kλ_k'^2 = β_1 ∑_k=1^m k^-(2r - 1) [neglect β_1 < 1] ≤ 1 + ∫_1^m t^-(2r - 1) t ≤ 1 + log m for r=1, 1/2(1-r)m^2(1-r) for 1/2 < r < 1. By the choice of m, see (<ref>), putting this into (<ref>), we obtain the final upper bound with constants that depend on r. In the case r>1, with the simple approach from sec:HilbertPlainMCUB we loose smoothness 1/2. This stresses the non-interpolatory nature of the method. In detail, check that the Gaussian process Ψ associated to _r^ (for any equivalent norm) lies almost surely in _s^ for , and it is almost surely not in _s^ for . The argument is similar to that in rem:FundMC. We had some difficulties with the case of smaller smoothness . However, there is some indication that nevertheless the associated Gaussian process Ψ is bounded. Then we could apply the simple approach from sec:HilbertPlainMCUB, and that way obtain better complexity bounds than in thm:Korobov. Sufficient and necessary conditions on for boundedness of the univariate Gaussian Fourier series Ψ associated to are known, see Adler <cit.>. The book of Marcus and Pisier <cit.> contains a lot more information on random Fourier series and could serve as a starting point for further research. There is a second hint. Plots of the one-dimensional kernel K_1 for nourish the conjecture that for we can find an estimate (iii)' K_1(x,z) ≥exp(-αd_(x,z)^p) for with , with and . An adapted version of thm:MCUBperiodic could give the desired upper bounds. Anyways, we were mainly interested in showing the superiority of Monte Carlo approximation over deterministic approximation in terms of tractability, and we have been successful for the whole range of continuous Korobov functions. The upper bounds in thm:Korobov do not give the optimal order of convergence for the approximation of Korobov functions. The rate of convergence we can guarantee is only for , and it can be arbitrarily bad for low smoothness r close to 1/2. From results on similar settings, see Fang and Duan <cit.>, one may conjecture a rate of something like .[ This topic is part of ongoing cooperation with Glenn Byrenheid and Dr. Van Kien Nguyen.] Proofs in that direction still rely on Maiorov's discretization technique. Maybe one can find bounds with better constants via more direct methods. In detail, propose an explicit Monte Carlo approximation method A_n with the following properties: * The most relevant Fourier coefficients, belonging to the indices , are approximated exactly at the cost of evaluatingfunctionals. * Fourier coefficients of medium importance, belonging to are approximated altogether now using the fundamental Monte Carlo approximation method from prop:Ma91_l2G with random functionals, the total information cost is . * The remaining Fourier coefficients, for indices , are ignored. The effect of truncation (<ref>) should be estimated with methods from sec:HilbertWorLB, see also Cobos et al. <cit.> or Kuo et al. <cit.>. The Monte Carlo part (<ref>) should be treated with methods on Gaussian fields, see sec:E|Psi|_sup and Adler <cit.>, or maybe Marcus and Pisier <cit.>. Note that by the deterministic part (<ref>) we will likely have no tensor product structure for the analysis of the Monte Carlo part (<ref>). §.§ Final Remarks on the Initial ErrorFor unweighted tensor product problems as in the above two subsections, the assumption of a normalized initial error is crucial for the new approach to work. If not, that is, ife(0,(K_1) ↪ L_∞(D_1)) = sup_x ∈ D_1√(K_1(x,x)) =: 1 + γ > 1,we havee(0,(K_d) ↪ L_∞(D_d)) = (1 + γ)^d,where K_d is the product kernel and . Then for the Gaussian field Ψ with covariance function K_d we haveΨ_∞≥sup_∈ D_d |Ψ_| = √(2/π)(1 + γ)^d.In this situation lem:stdMCapp can only give an impractical upper complexity bound that grows exponentially in d for fixed . However, if the constant function is normalized in , butis non-trivial and contains more than just constant functions, then , contrary to our requirements.Therefore we must accept that the constant function cannot be a normalized function in (K_d) if we want to break the curse for the L_∞-approximation with the present tools. This stands in contrast to many other problems: * Take the L_2 approximation of periodic Korobov spaces, see e.g. Novak and Woźniakowski <cit.>. The initial error is 1 iff the largest singular value is 1, so it is natural to let the constant function be normalized within the input space , hence it lies on the boundary of the input set. * Consider multivariate integration over Korobov spaces, see for example Novak and Woźniakowski <cit.>. The integral is actually the Fourier coefficient belonging to the constant function. The initial error is properly normalized iff the constant function has norm 1.I wish to thank my colleagues David Krieg, and Van Kien Nguyen (meanwhile Dr. rer. nat.), for making me aware of the difference to these particular two situations.CHAPTER: L_1-APPROXIMATION OF MONOTONE FUNCTIONSConcerning the L_1 approximation of d-variate monotone functions (and also monotone Boolean functions) by function values, in the deterministic setting the curse of dimensionality holds, see Hinrichs, Novak, and Woźniakowski <cit.>, and sec:monoCurse. For the randomized setting we still have intractability, i.e. the problem is not weakly tractable, see sec:monoMCLBs where we improve known lower bounds, the new bounds now exhibit a meaningful -dependency. Yet randomization may reduce the complexity significantly, for any fixed tolerance the complexity depends exponentially on √(d), roughly, see sec:monoUBs for the analysis of a known algorithm for Boolean functions, and a new extension to real-valued monotone functions that is based on a Haar wavelet decomposition. § THE SETTING AND BACKGROUNDWithin this chapter we mainly consider the L_1-approximation of d-variate monotone functions using function values as information,[ The input set contains different functions that belong to the same equivalence class in . Actually, we do not care about these equivalence classes but only need the L_1-norm as a seminorm. Furthermore, function evaluations are discontinuous functionals, but monotonicity provides a regularization to this type of information in some other useful ways such that deterministic approximation is actually possible.]: F_^d ↪ L_1([0,1]^d),where the input setF_^d := { f: [0,1]^d → [0,1] |≤⇒ f() ≤ f()} consists of monotonously increasing functions with respect to the partial order on the domain. For , the partial order is defined by≤ :⇔ x_j ≤x̃_jfor allj=1,…,d. This problem is closely related to the approximation of Boolean monotone functionsF_^d := { f: {0,1}^d →{0,1}|≤⇒ f() ≤ f()} .One can identify F_^d with a subclass if we split into 2^d subcubes indexed by ,C_ := _j = 1^d I_i_j , where I_0 := [0, 1/2) andI_1 := [1/2,1].Then, for any Boolean function , we obtain a subcubewise constant function by setting . If f is monotone then so is f̃. The corresponding distance between two Boolean functions is(f_1,f_2) := 1/2^d #{∈{0,1}^d | f_1() ≠ f_2() } .The metric space of Boolean functions shall be named G_^d, we consider the approximation problem: F_^d ↪ G_^d.Note that the metric on G_^d corresponds to the L_1-distance of the associated subcubewise constant functions defined on . Another way to think of the metric on G_^d is as the induced metric for G_^d as a subset of the Banach space L_1({0,1}^d).[ This property is the reason why it is convenient to consider real-valued monotone functions with range [0,1]. It is only in sec:monoUBs that we switch to the range because there we use linear approximation methods.]Approximation of monotone functions is not a linear problem becasue the set F_^dis not symmetric: For non-constant functions f ∈ F_^d, the negative -f is not contained in F_^d as it will be monotonously decreasing. The monotonicity assumption is very different from common smoothness assumptions, yet it implies many other nice properties, see for example Alberti and Ambrosio <cit.>. Integration and Approximation of monotone functions has been studied in several papers <cit.>. Monotonicity can also be an assumption for statistical problems <cit.>. Similarly, a structural assumption could be convexity (more generally: k-monotonicity), numerical problems with such properties have been studied for example in <cit.>.Within this research, Boolean monotone functions are considered in order to obtain lower bounds for the Monte Carlo approximation of real-valued monotone functions, see sec:monoMCLBs. We will show that the approximation of monotone (Boolean) functions is not weakly tractable, i.e. intractable in the IBC sense.[ In learning theory there exist many similar sounding notions like weak learnability, which, however, have different meanings.] General Boolean functions are of interest for logical networks and cryptographic applications. Monotone Boolean functions in particular constitute a widely studied topic in computer science and discrete mathematics with connections to graph theory amongst others. Much research has been done on different effective ways of exact representation of Boolean functions, see the survey paper of Korshunov <cit.>. On the other hand, the approximation of monotone Boolean functions (or subclasses thereof) is a good example for learning theory <cit.>. In cryptography one is interested in finding classes of easily representable Boolean functions (not only monotone functions) that are hard to learn from examples <cit.>, so the motivation is different from the motivation for tractability studies in IBC. Different perspectives on similar problems explain the sometimes unfamiliar way of presenting results in other scientific communities, see for instance sec:monoMCLBs.Finally, we give some examples of naturally occurring monotone functions.[Application of monotone functions] Think of a complex technical system with d components. Some of these components could be damaged but the system as a whole would still work. However, if more critical components fail, the system will fail as well. It is a natural assumption that, once the system stopped working, it will not come back to life after more components break – this, in fact, is monotonicity. We are interested in predicting when the machine will cease to function. Our framework fits to a test environment where we can manually deactivate components and check whether the system still does its job. We are interested in a good (randomized) strategy to test the system. This approach could work for the testing of uncritical applications where a test environment can be set up. For rather critical systems such as aircrafts or running systems, we need to learn from bad experience (that we actually wish to avoid). Different components will have different probabilities of failing, and it is with these probabilities that we obtain samples from which we can learn. On the other hand, not every case is equally important, so we judge the approximation of the Boolean function by the probability of an event occurring where the prediction fails. This situation fits better to the framework that we have for example in Bshouty and Tamon <cit.>. (From that paper we know the method presented in sec:BooleanUBs.) This picture can be extended to real-valued monotone functions . Now, each of the d components of a system can work at a different level, we are interested in how much this affects the performance of the entire system. When thinking of a home computer, the question could be how much the PC slows down in different situations. § FIRST SIMPLE ESTIMATES In view of the order of convergence for the approximation of real-valued monotone functions, see sec:MonoOrder, randomization does not help. Actually, for small errors the very simple deterministic algorithm to be found in this section is the best method we know, the randomized methods from sec:monoRealUBs will only help for larger .In sec:monoCurse we cite a result of Hinrichs, Novak, and Woźniakowski <cit.>, which states that the approximation of real-valued monotone functions suffers from the curse of dimensionality in the deterministic setting. A similar statement holds for the approximation of Boolean monotone functions as well.§.§ The Classical Approach – Order of ConvergenceThe integration problem for monotone functions,: F_^d →, f ↦∫_[0,1]^d f,based on standard information , is an interesting numerical problem, where in the randomized setting adaption makes a difference for the order of convergence (at least for ), but non-adaptive randomization helps only for to improve the convergence compared to deterministic methods. In the univariate case Novak <cit.> showede^,(n,,F_^1) ≍ n^-3/2 ≺ e^,(n,,F_^1) ≍ e^(n,,F_^1) ≍ n^-1 .Papageorgiou <cit.> examined the integration for d-variate monotone functions, for dimensions we havee^,(n,,F_^d) ≍ n^-1/d - 1/2 ≼ e^,(n,,F_^d) ≼ n^- 1/2d - 1/2 ≺ e^(n,,F_^d) ≍ n^-1/d ,where the hidden constants depend on d. It is an open problem to find lower bounds for the non-adaptive Monte Carlo error that actually show that adaption is better for as well, but from the one-dimensional case we conjecture it to be like that.For the L_1-approximation, the order of convergence does not reveal any differences between the various algorithmic settings. Applying Papageorgiou's proof technique to the problem of L_1-approximation, we obtain the following theorem. For the L_1-approximation of monotone functions, for fixed dimension d and , we have the following asymptotic behaviour, e^(n,,F_^d) ≍ e^(n,,F_^d) ≍ n^-1/d . This holds also for varying cardinality. We split into m^d subcubes indexed by : C_ := _j = 1^d I_i_j where for and . For the lower bounds, we consider fooling functions that are constant on each of the subcubes, in detail, f|_C_ = ||_1+δ_/d(m-1)+1 with δ_∈{0,1} and . Obviously, such functions are monotonously increasing. For a deterministic algorithm using n < m^d function values, when applied to such a function, we do not know the function on at least subcubes. There exist a maximal function f_+ and a minimal function f_- fitting to the computed function values, and the diameter of information is at least f_+ - f_-_1 ≥(1 - n/m^d)1/d(m-1)+1 . Hence we have the error bound e^(n,,F_^d) ≥1/2 (1 - n/m^d)1/d(m-1)+1 . For Monte Carlo lower bounds we switch to the average case setting, the measure μ may be described by such functions where the δ_ are independent Bernoulli variables with . For any information , let be the set of indices where we do not know anything about the function on the corresponding subcube C_. Again, , and for any output we have the following estimate on the local average error with respect to the conditional distribution μ_: ∫f - g_1μ_( f) ≥∑_∈ I^1/2∫_C_(|||_1/d(m-1)+1 - g() | +|||_1 + 1/d(m-1)+1 - g() | ) _≥1/d (m-1)+1 ≥1/2 (1 - n/m^d)1/d(m-1)+1 . Averaging over the information , we obtain the same lower bound in this average setting as in the worst case setting, by virtue of prop:Bakh (Bakhvalov's technique), this is a lower bound for the Monte Carlo error. Note that this lower bound is an estimate for the conditional error by a convex function , notably it holds for methods with varying cardinality , putting . Hence by lem:n(om,f)avgspecial we reason the alike error bounds for Monte Carlo methods with varying cardinality. Choosing , we obtain the general lower bound e^(n,,F_^d) ≥ e^(n,,F_^d) ≥1/4 (d√(2n) + 1)≥1/12 dn^-1/d . For the upper bounds, we give a deterministic, non-adaptive algorithm with cardinality , i.e. when allowed to use n function values, we choose . We split the domain into (m+1)^d subcubes as above, and define the output by g|_C_ := 1/2 [f(/m+1) + f(+/m+1) ], here := (1,…,1) ∈^d. Without loss of generality, we assume that on the boundary of the domain we have f|_[0,1)^d ∖ (0,1)^d = 0 and f|_[0,1]^d ∖ [0,1)^d = 1. This means that we only need to compute function values on a grid in the interior of the domain. For each subcube we take the medium possible value based on our knowledge on the function f in the lower and upper corners of that particular subcube. When analysing this algorithm, we group the subcubes into diagonals collected by index sets D_ := { + k∈{0,…,m}^d | k ∈} . There are such diagonals, each of them can be tagged by exactly one index . By monotonicity we have the following estimate for the error: e(A_m^d,f) = f - g_1≤1/(m+1)^d ∑_∈{0,…,m}^d ∖{1,…,m}^d ∑_∈ D_1/2 [f(+/m+1) - f(/m+1) ] _= 1/2 ≤d/2(m+1) = d/2(⌊√(n)⌋ + 1) ≤ d/2n^-1/d . The above proof yields the explicit estimate 1/12 dn^-1/d≤ e^(n,,F_^d) ≤ e^(n,,F_^d) ≤d/2n^-1/d . At first glance, this estimate looks quite nice, with constants differing only polynomially in d. This optimistic view, however, collapses dramatically when switching to the notion of -complexity for : (1/12 d)^d^-d≤ n^(,,F_^d) ≤ n^(,,F_^d) ≤(d/2)^d^-d . Here, the constants differ superexponentially in d. Of course, lower bounds for low dimensions also hold for higher dimensions, so given the dimension d_0, one can optimize over . Still, the upper bound is impractical for high dimensions since it is based on algorithms that use exponentially (in d) many function values. In fact, for the deterministic setting we cannot avoid a bad d-dependency, as the improved lower bounds of sec:monoCurse below show. For the randomized setting, however, we can significantly reduce the d-dependency (which is still high), at least as long as is fixed,[ For smallhowever, the best known method is the deterministic method from thm:MonAppOrderConv above, see rem:monoMCUBeps.] see sec:monoRealUBs. To summarize, if we only consider the order of convergence, we might think that randomization does not help, but for high dimensions randomization actually does help. §.§ Curse of Dimensionality in the Deterministic SettingHinrichs, Novak, and Woźniakowski <cit.> have shown that the integration (and hence also the L_p-approximation, ) of monotone functions suffers from the curse of dimensionality in the deterministic setting. We want to recap their result for our particular situation. The L_1-approximation of monotone functions suffers from the curse of dimensionality in the worst case setting. In detail, e^(n,,F_^d,) ≥1/2 (1 - n 2^-d), so for we have n^(,,F_^d,) ≥ 2^d-1 . Let N be any adaptive information mapping. We consider functions f for which we obtain the same information as for the diagonal split function . Consequently, such functions will be evaluated at the same points , and f(_i) = 0if |_i|_1 < d/2, 1if |_i|_1 ≥d/2, for . Having this information, there are two areas, D_0:= {∈ [0,1]^d |∃ _i ≥, f(_i) = 0 } = ⋃_i:f(_i) = 0,_i , and D_1:= {∈ [0,1]^d |∃ _i ≤, f(_i) = 1 } = ⋃_i:f(_i) = 1_i, , where we know the function for sure: and . The minimal monotone function fitting to that information is , the maximal function is . Their L_1-distance is f_+ - f_-_1 = λ^d([0,1]^d ∖ (D_0 ∪ D_1)), so we have the error bound e(ϕ∘ N,F_^d) ≥1/2 λ^d([0,1]^d ∖ (D_0 ∪ D_1)) , no matter which output ϕ we choose. For we have λ^d(,_i) = ∏_j=1^d _i(j) AM-GM ineq.≤(|_i|_1/d)^d < 2^-d . For we have a similar estimate, λ^d(_i,) = ∏_j=1^d (1-_i(j)) AM-GM ineq.≤(| - _i|_1/d)^d ≤ 2^-d . Consequently, λ^d([0,1]^d ∖ (D_0 ∪ D_1)) ≥ 1 - n 2^-d , which together with (<ref>) finishes the proof. For the integration the proof follows exactly the same lines. While for integration the standard Monte Carlo method easily achieves strong polynomial tractability, for the approximation we still have intractability in the randomized setting, see sec:monoMCLBs, yet the curse of dimensionality is broken, see sec:monoRealUBs.With slight modifications an analogue lower bound on the deterministic approximation of Boolean monotone functions is found. For the approximation of monotone Boolean functions we have e^(n,,F_^d,) ≥1/2 (1 - n 2^-⌊ d/2 ⌋). Hence, for , the -complexity is bounded from below by n^(,,F_^d,) ≥ 2^⌊ d/2 ⌋ - 1 . In particular, the approximation of monotone Boolean functions suffers from the curse of dimensionality in the worst case setting. Similarly to the proof of thm:HNW11, we consider fooling functions with f(_i) = 0if |_i|_1 < d/2, 1if |_i|_1 ≥d/2. For we have the estimate , for it holds . The remaining steps are analogous. For the class F_^d of real-valued monotone functions, it is easy to see that the initial error is 1/2 with the initial guess being the constant 1/2-function. For Boolean functions, since there is no 1/2-function, we need to exploit monotonicity properties in order to show that the initial error is 1/2 nevertheless. Let the initial guess be , indeed . For any we have #{∈{0,1}^d | f() ≠ x_1} = #{| f() = 1, x_1 = 0} + #{| f() = 0, x_1 = 1} [monotonicity] ≤#{| f() = 1, x_1 = 1} + #{| f() = 0, x_1 = 1}= 2^d-1 , from which we conclude . In fact, there are several functions that are equally suitable for an initial guess, a more canonical function without bias to one single coordinate could be the diagonal split , for Boolean monotone functions, however, this only works properly for odd d. The monotonicity assumption is crucial for us to prove the initial error to be independent from the algorithmic setting. In contrast to that, the approximation of general (non-monotone) Boolean functions with Boolean approximants is an interesting problem where it makes a difference whether we consider the randomized or the deterministic initial error. In the randomized setting, the algorithm that uses no information can return the constant 0 and the constant 1 functions with probability 1/2 each, that way obtaining the initial error 1/2. In the deterministic setting, however, for any initial guess g, the opposite function has a distance 1, the initial error is 1. Anyways, with the initial error being constant independently from the dimension d, the problem of the approximation of monotone (Boolean) functions is properly normalized. In addition, problems of lower dimensions are canonically contained in the problem of higher dimensions, F_^d can be seen as a subset of F_^d+1 consisting of all functions that do not depend on the coordinate x_d+1. These two properties make the class of multivariate monotone functions particularly interesting for tractability studies. While the lower bound of thm:MonAppOrderConv contains bad d-dependent constants, the bound from thm:HNW11 does not work for small . One could combine the ideas of both proofs in the following way: Given ∈^d, split the domain into sub-cuboids C_, each of side length m_j^-1 in the j-th dimension, and indexed by , . We consider those monotone functions f ∈ F_^d with different ranges on different sub-cuboids, f() ∈[||_1/||_1 - d + 1, ||_1 + 1/||_1 - d + 1] for ∈ C_. Note that, by construction, on each of the sub-cuboids, can be chosen independently from the values of the function on other sub-cuboids. If an algorithm computes n_ function values of f on C_, we can apply thm:HNW11 to the approximation of by scaling. Altogether, with , we obtain the lower bound e^(n,,F_^d,) ≥1/||_1 - d + 1 [1/∏∑_1/2 (1 - n_2^-d) ] = 1/2(||_1 - d + 1) (1 - n/2^d∏). For , choose , splitting the domain only for the first k coordinates, . For , split the domain into m^d subcubes, that is, with . By this we obtain the complexity bound n^(,,F_^d,) ≥ 2^d+⌊ 1/(4) ⌋-2 for ∈ [1/4(d+1), 1/4], 2^d (1 + log_2 ⌊ 1 / (4d)⌋) - 1 for ∈ (0,1/4(d+1)]. Still, the upper bounds in thm:MonAppOrderConv for the complexity are superexponential in d, from rem:MonAppOrderConv we have n^(,,F_^d,) ≤exp(dlogd/2 ), there is a logarithmic gap in the exponent. This idea of a combined lower bound can also be done for the randomized setting, see rem:combiLBran.§ INTRACTABILITY FOR RANDOMIZED APPROXIMATION §.§ The Result – A Monte Carlo Lower BoundAs we will show in sec:monoUBs, for the L_1-approximation of monotone functions, the curse of dimensionality does not hold anymore in the randomized setting. Within this section, however, we show that, for fixed , the -complexity depends at least exponentially on √(d) in the randomized setting. Yet worse, the problem is not weakly tractable.For the proof we switch to an average case setting for Boolean functions, an idea that has already been used by Blum, Burch, and Langford <cit.>.[ I wish to thank Dr. Mario Ullrich for pointing me to this paper, after learning about my first version of a lower bound proof.] They stated that for , and sufficiently large d, we havee^(n,,F_^d,) ≥(1-exp(-n/4)) (1/2 - Clog(6 d n)/√(d)),with a numerical constant . (From their proof the constant can be extracted.) Assuming this result to hold for all n and d, one could conclude that for we have a lower error bound of roughly[ That is, ignoring the prefactor (1-exp(-n/4)).] , where we may choose for meaningful bounds. Consequently, such an estimate gives us results only for . With the given value of C, this means that is the smallest dimension for which a positive error bound with n=1 is possible in the first place.[ One more example: d=39 168 is the first dimension for which a positive lower bound with n≥ d is possible.] Actually, interpreting “sufficiently large” as , one can check their proof and will find out that all proof steps work in those cases where we have non-trivial error bounds, see rem:1/2-...BBL98 for more information on the original proof. Note that1/d exp(σ √(d)) ≻exp(σ'√(d)) for d →∞, if 0 < σ' < σ.Therefore, by (<ref>), Blum et al. were the first to show that for fixed the Monte Carlo complexity for the approximation of monotone Boolean functions depends exponentially on √(d) at least.From the IBC point of view the structure of (<ref>) appears unfamiliar. Blum et al., however, wanted to show that if we allow to use algorithms with cardinality n growing only polynomially in the dimension d, the error approaches the initial error with a rate of almost . The interest for such a result is not motivated from practically solving a problem, but from proving that a problem is hard to solve, and could therefore be used in cryptographic procedures.[ Think of two parties A and B communicating. Both have a key, say, they know a high-dimensional Boolean function f. In order to check the identity of the other party they ask for some values of f. That is, A sends a list of points to B, and B answers with a list of correct values in order to approve a message. A third party C with bad intentions could intercept this communication and try to learn the function f from sample pairs (_i,f(_i)). This should be as hard as possible for that there is little chance that one day C could answer correctly to . Monotone Boolean functions are just one model of such a problem. Besides hardness of learning, another criterion is simplicity of representation (keeping the key), and probably other problems are better for that purpose. The big issue for complexity theory: With technical progress hackers have more capacities for cracking a code, with more intense communication they can collect more information. But technical progress also gives opportunities to make cryptography more complex in order to rule out cyber attacks.]One objective in this study was to extract the best constants possible whenever a constant error tolerance is given. Moreover, using some different inequalities than Blum et al.,[ Differences within the proof will be indicated in place by footnotes. ] it is possible to find a lower complexity bound that includes the error tolerance in a way that we can prove intractability for the Monte Carlo approximation of monotone functions, see rem:monMCLBintractable. A detailed comment on modifications of the proof needed for the case of getting close to the initial error, that is the case Blum et al. <cit.> studied, is made in rem:1/2-...BBL98. Consider the randomized approximation of monotone Boolean functions. There exist constants and such that for we have n^(_0,,F_^d,) > ν exp(σ_0 √(d)), and moreover, for and we have n^(,,F_^d,) > ν exp(c√(d)/), with c = σ_0_0. In particular, for andwe have n^(1/30,,F_^d,) > 108 ·exp(√(d)-√(100)), for this means . For and we have n^(,,F_^d,) > 108 ·exp(√(d)/30 -√(100)). All these lower bounds hold for varying cardinality as well. Before we give the proof in sec:monoMCLBs-Proof,we discuss some consequences of the theorem. The above theorem shows that the approximation of monotone Boolean functions is not weakly tractable. Indeed, consider the sequence of error tolerances . Then lim__d^-1 + d →∞log n(_d,d)/_d^-1 + d≥lim_d →∞σ_0 d /√(d_0) + logν/_0^-1√(d/d_0) + d = σ_0/√(d_0) > 0. This contradicts the definition of weak tractability, see sec:tractability. Actually, this behaviour has already been known since the paper of Bshouty and Tamon 1996 <cit.>, however, research on weak tractability has not yet been started at that time.[ For the historical background of tractability notions, see Novak and Woźniakowski 2008 <cit.>.] Their lower bound can be summarized as follows: For moderately decaying error tolerances and sufficiently large d, we have n^(_d,,F_^d,) ≥ c 2^d / √(d) , with some numerical constant c > 0. Interestingly, the proof is based on purely combinatorial arguments, without applying Bakhvalov's technique (prop:Bakh) and average case settings. Since a function value for Boolean functions may only be 0 or 1, after n oracle calls we have at most 2^n different possible information outcomes. Any Boolean function can approximate at most k(,d) := ∑_l = 0^⌊2^d ⌋2^dl[lem:BinomSum]Lem <ref>lem:BinomSum≤ 2^ 2^d log_2 ( / ) Boolean functions up to a distance . On the other hand, the total number of monotone Boolean functions is known to be # F_^d ≥ 2^d⌊ d/2 ⌋≥ 2^2^d-1 / √(d) , see Korshunov <cit.> and the references therein for the first inequality, and lem:(d d/2) for the second inequality. Then for any realization of a Monte Carlo method, that is, fixing ω, a portion of at least Boolean monotone functions is at distance more than to all of the output functions. This can be used to show the existence of poorly approximated functions, sup_f ∈ F_^d{(A_n^ω,f) > } ≥1/# F_^d∑_f ∈ F_^d{(A_n^ω,f) > } [Fubini] = #{f ∈ F_^d |(A_n^ω,f) > }/# F_^d≥(1 - 2^nk(,d)/#F_^d). Hence we get the error bound e^(n,,F_^d,) ≥ (1 - 2^nk(,d)/#F_^d), which implies the complexity bound n^(/2,,F_^d,) ≥log_2 (# F_^d/2 k(,d)) ≥2^d-1/√(d) - 2^d log_2 (/) - 1. This lower bound only makes sense for as , then it will give a complexity bound that is exponential in d. The lower bounds of thm:monotonLB also hold for the problem , which therefore is intractable as well. Indeed, the proof of the theorem is done by switching to a μ-average case setting on the set of monotone Boolean functions F_^d (Bakhvalov's technique, see prop:Bakh). Any measure μ on F_^d can be associated with a measure μ̃ on the set of subcubewise constant functions that only take the values 0 and 1. Since for the Boolean setting the output function must be Boolean as well, in order to prove the equivalence of both problems, it remains to show that optimal outputs with respect to μ̃ are constant 0 or 1 on each of the 2^d subcubes of the domain . For the μ̃-average setting on F_^d, take any deterministic information mapping using n sample points (possibly adaptively chosen). By we denote the conditional measure, and is the preimage of the information . We only need to consider the cases of information with non-vanishing probability . Taking any output mapping ϕ̃, the definition of the average error and the law of total probability lead to the following representation of the error: e(ϕ̃∘N, μ̃) Fubini=∑_μ̃(F_) ∫_F_[ϕ̃()]() - f()_L_1μ̃_( f) Fubini=∑_μ̃(F_) ∫_[0,1]^d[ ∫_F_|[ϕ̃()]() - f()| μ̃_( f) ] Since , for the integrand we have […] =μ̃_{f() = 0} | [ϕ̃()]() | + μ̃_{f() = 1} |1 - [ϕ̃()]() |, which is minimized for . This, of course, is a function that is constant 0 or 1 on each of the 2^d subcubes. Note that, since the set of Boolean functions is finite, by minimax principles there exists a measure μ^∗ on F_^d such that the μ^∗-average error coincides with the Monte Carlo error, see Mathé <cit.>. By this we have that the problem of approximating Boolean functions is strictly easier than the problem of L_1-approximation of real-valued functions, e^(n,,F_^d,) = e^(n,,μ^∗,) ≤ e^(n,,F_^d,). §.§ The Proof of the Monte Carlo Lower BoundWe start the proof with two preparatory lemmas.The calculation (<ref>) in rem:monoMCLBs-Realvalued actually brings us to a direct representation of the best error possible with the information N,inf_ϕ̃ e(ϕ̃∘N, μ̃) = ∑_μ̃(F_) ∫_[0,1]^dmin{μ̃_{f() = 0}, μ̃_{f() = 1}} .We summarize a similar identity for Boolean functions in the following Lemma. Let μ be a probability measure on the set F_^d, and let be any deterministic information mapping. Then inf_ϕ e(ϕ∘ N, μ) = ∑_μ(F_) 2^-d∑_∈{0,1}^dmin{μ_{f() = 0}, μ_{f() = 1}} , where is the conditional measure, and is the preimage of the information . By lem:01outputavg, for a given measure μ on F_^d, the average case analysis reduces to understanding the conditional measure μ_. The concept of augmented information allows us to simplify the conditional measure as long as we are concerned about lower bounds. Consider the general problem . Let be an arbitrary measurable information mapping and be an augmented information mapping, that is, for all possible information representers there exists an information representer such that . With this property, the augmented information allows for smaller errors, that means inf_ϕ̃ e(ϕ̃∘N,μ) ≤inf_ϕ e(ϕ∘ N,μ). There exists a mapping such that . For any mappingwe can define such that , and so r^(N,μ) :=inf_ϕ̃ e(ϕ̃∘N,μ) ≤inf_ϕ e(ϕ∘ N, μ) =: r^(N,μ). (The quantity is called μ-average radius of the information N.) We are now ready to proof the theorem. We will use Bakhvalov's technique (prop:Bakh) for lower bounds and switch to an average case setting on the set of monotone Boolean functions. The proof is organized in seven steps. proof:monoLB1: The general structure of the measure μ on F_^d. proof:monoLB2: Introduce the augmented information. proof:monoLB3: Estimate the number of points ∈{0,1}^d for which f() is still – to some extend – undetermined, even after knowing the augmented information. proof:monoLB4: Further specify the measure μ, and give estimates on the conditional probability for the event for the set of still fairly uncertain from the step before. proof:monoLB5: A general formula for the lower bound. proof:monoLB6: Connect estimates for _0 and d_0 with estimates for smaller and larger d. proof:monoLB7: Explicit numerical values. proof:monoLB1 General structure of the measure μ. We define a measure μ on the set of functions that can be represented by a randomly drawn set , withbeing a suitable parameter, and a boundary value , . We define by f_U() := [(||_1 > b) or (∃∈ U: ≤) ]. The boundary value will facilitate considerations in connection with the augmented information.[ The proof of Blum et al. <cit.> worked without a boundary value b when defining functions f_U for the measure μ. Then in proof:monoLB2 one needs to replace the first inequality of (<ref>). They used the Chernoff bound in order to control the size of the set V_0 of the augmented information ỹ with high probability, μ{# V_0 ≤2n/p}≥ 1 - exp(-n/4). The error bound, see proof:monoLB5, then needs to be multiplied with this factor, which for large n is neglectable, though. In proof:monoLB4 we specify , so we could use the estimate . The estimate (<ref>) we take instead can be compared to this using (<ref>), we have . For high dimensions we get the rough estimate . With the numerical values listed in proof:monoLB7 we have versus . In this case our version is slightly better and avoids the usage of Chernoff bounds. For the original result, however, Chernoff bounds prove to be useful, see rem:1/2-...BBL98. In turn, the case of varying cardinality becomes more difficult because the situation of lem:n(om,f)avgspecial does not apply anymore, compare proof:monoLB5. ] We draw U such that the are independent Bernoulli random variables with . The parameter will be specified in proof:monoLB4. proof:monoLB2 Augmented information. Now, for any (possibly adaptively obtained) info with , we define the augmented information ỹ := (V_0,V_1), where and represent knowledge about the instance f that implies the information . We know , and . In detail, let ≤_L be the lexicographic order[ Any other total order will be applicable as well.] of the elements of W, thendenotes the first element of a set V ⊆ W with respect to this order. For a single sample f() the augmented oracle reveals the sets V_0^ := ∅ if ||_1 > b, {∈ W |≤} if f() = 0, {∈ W | ≤ and<_Lmin_L{∈ U| ≤}} if f() = 1and ||_1 ≤ b, V_1^ := ∅ if ||_1 > b or f() = 0, {min_L{∈ U| ≤}} if f() = 1 and ||_1 ≤ b, and altogether the augmented information is ỹ = (V_0,V_1) :=(⋃_i=1^n V_0^_i,⋃_i=1^n V_1^_i). Note that computing f() for is a waste of information, so no algorithm designer would decide to compute such samples. Since for , and , we have the estimates[fn:f_U with b] # V_0 ≤ nbt , and# V_1 ≤ n. proof:monoLB3 Number of points where is still fairly uncertain. For any point we define the set W_ := {∈ W |≤} of points that are “relevant” to . Given an augmented information , we are interested in points where it is not yet clear whether or . In detail, these are points where , for thatbe still possible. Furthermore,shall be big enough, saywith , so that the conditional probability is not too small. For our estimates it will be necessary to restrict to points , we suppose . The set of all these points will be denoted by B:= {∈ D_ab| W_∩ V_1 = ∅ , #(W_∖ V_0) ≥ M } , where D_ab := {∈{0,1}^d | a ≤ ||_1 ≤ b} . We aim to find a lower bound for the cardinality of B. proof:monoLB3.1 Bounding # D_a,b. Let and with , then by the Berry-Esseen inequality (on the speed of convergence of the Central Limit Theorem), see prop:BerryEsseen and in particular cor:BerryEsseenBinom, we have # D_ab/#{0,1}^d = 1/2^d∑_k=a^bdk ≥ Φ(β) - Φ(α) _=: C_αβ - 2 C_0/√(d)=: r_0(α,β,d). Here, Φ is the cumulative distribution function of standard Gaussian variables.[ The original proof of Blum et al. <cit.> uses Hoeffding bounds #D_ab/#{0,1}^d≥ 1 - exp(-α^2) - exp(-β^2). The Berry-Esseen inequality enables us to obtain better constants. Moreover, for the considerations on small in proof:monoLB6, we need to take α,β→ 0, but this cannot be done with the Hoeffding bound, where for we would have trivial negative bounds.] proof:monoLB3.2 The influence of ∈ W (in particular ∈ V_1).[ This step helps to get better constants, but it becomes essential for small in proof:monoLB6. For the focus of Blum et al. <cit.> withbeing close to the initial error, the estimate will be sufficient.] Now, let with , and for define Q_ := {∈{0,1}^d |≤} , this is the set of all points inside the area of influence of . Applying cor:BerryEsseenBinom once more, we obtain # (Q_∩ D_ab)/# Q_ =#{∈{0,1}^d-t| a-t ≤ ||_1 ≤ b-t}/2^d-t=1/2^d-t∑_k=a-t^b-td-tk [[cor:BerryEsseenBinom]Cor <ref>cor:BerryEsseenBinom] ≤ Φ(2b - t/√(d-t)) - Φ(2a - t/√(d-t)) +2 C_0/√(d-t) [(<ref>), (<ref>)] ≤ [Φ(β-τ) - Φ(α - τ) _=: C_αβτ +(1/√(2 π) + 2 C_0) _=: C_11/√(d)] 1/√(1 - t/d) , where for , and this factor converges, . Within the above calculation, we exploited that the density of the Gaussian distribution is decreasing with growing distance to 0, in detail, for t_0<t_1 and κ≥ 1 we have Φ(κt_1) - Φ(κt_0) = 1/√(2π)∫_κt_0^κt_1exp(-t^2/2)t = κ/√(2π)∫_t_0^t_1exp(-κ^2 s^2/2)s ≤ κ/√(2π)∫_t_0^t_1exp(-s^2/2)s = κ[Φ(t_1) - Φ(t_0)] . Namely, we took which comes from replacing by . Furthermore, we shifted the , knowing its derivative being bounded between 0 and , so for andwe have |[Φ(t_1 + δ) - Φ(t_0 + δ)] -[Φ(t_1) - Φ(t_0)] | ≤|δ|/√(2π) , in our case . proof:monoLB3.3 The influence of V_0. Markov's inequality gives us ∑_∈ V_0# (Q_∩ D_ab) = ∑_∈ D_ab# (W_∩ V_0) ≥N#{∈ D_ab|#(W_∩ V_0) ≥ N} , with . Using this, we can carry out the estimate #{∈ D_ab|#(W_∖ V_0) ≥ M }= #{∈ D_ab|#(W_∩ V_0) ≤# W_ - M } ≥ #{∈ D_ab|#(W_∩ V_0) ≤at - M } = # D_ab - #{∈ D_ab|#(W_∩ V_0) > at - M } [(<ref>)]≥ # D_ab - 1/at - M + 1 ∑_∈ V_0# (Q_∩ D_ab). proof:monoLB3.4 Final estimates on # B. Putting all this together, we estimate the cardinality of B: # B/#{0,1}^d = #({∈ D_ab|#(W_∖ V_0) ≥ M }∖⋃_∈ V_1 Q_) /#{0,1}^d [(<ref>), any ∈ W] ≥# D_ab/#{0,1}^d - # Q_/#{0,1}^d (# V_0/at - M + 1 + #V_1) # (Q_∩ D_ab)/# Q_ [(<ref>), (<ref>), (<ref>)] ≥ C_αβ - 2 C_0/√(d) - n 2^-t (bt/at - M + 1 + 1) [C_αβτ +C_1/√(d)] κ_τ(d). We set with , and provided , which can be guaranteed for and , we estimate the ratio bt/at ≤(a+1/a-t+1)^b-a≤(d/2 + α√(d)/2 + 1 /d/2 + (α - 2τ) √(d)/2)^(β-α) √(d)/2 ≤exp((β - α)τ(1 + α - 2 τ/√(d))^-1_=: κ_ατ(d) +β-α/√(d) + α - 2τ_=: K_αβτ(d)) =: σ_αβτ(d), where we have and . (Note that the above estimate is asymptotically optimal, .) We finally choose the information cardinality , and obtain the estimate # B/#{0,1}^d ≥[C_αβ - 2 C_0/√(d)] - ν (σ_αβτ(d) /1-λ + 1 ) [C_αβτ +C_1/√(d)] κ_τ(d) =: r_0(α,β,d) - νr_1(α,β,τ,λ,d) =: r_B(α,β,τ,λ,ν,d) . With all the other conditions on the parameters imposed before, for sufficiently large d we will have . Furthermore, we always have , so choosing will guarantee r_B(…) to be positive. proof:monoLB4 Specification of μ and bounding of conditional probabilities. We specify the measure μ on the set of functions defined as in (<ref>) with . Remember that the (for ) shall be independent Bernoulli random variables with probability . Having the augmented information , the values are still independent random variables with probabilities μ_ỹ{f() = 1} = 0if ∈ V_0, 1if ∈ V_1, pif ∈ W ∖ (V_0 ∪ V_1). Then for we have the estimate μ_ỹ{f() = 0}≤ (1 - p)^M ≤exp( - p λat) = exp( - λϱ), where we write with . The other estimate is μ_ỹ{f() = 0}≥ (1 - p)^bt = exp(log(1 - p)bt)≥exp( - ϱ σ_αβτ(d)( 1/2 + 1/2 (1- ϱ/γ_ατ(d))_=: κ_ϱγ(d)) ) =: q_0(α,β,τ,ϱ,d)exp(-ϱ exp((β-α) τ) ) , Here we used that, for 0 ≤ p < 1, 0 ≥log(1-p) = - (p+∑_k=2^∞p^k/k) ≥ - (p+∑_k=2^∞p^k/2) = -p (1/2 + 1/2 (1-p)), together with the estimates pbt≤ϱ σ_αβτ(d), and, provided , at≥(a/t)^t ≥(√(d)+α/2(τ + 1 /√(d)))^τ√(d) =: γ_ατ(d). Note that implies . It follows that for , min{μ_ỹ{f() = 1}, μ_ỹ{f() = 0}} ≥min{1- exp( - ϱ λ), q_0(α,β,τ,ϱ,d) } =: q(α,β,τ,λ,ϱ,d). proof:monoLB5 The final error bound. By lem:01outputavg and Bakhvalov's technique (prop:Bakh) we obtain the final estimate for , where , e^(n,,F_^d,) ≥ e^(n,,μ,) [any valid ỹ] ≥# B/#{0,1}^d min{μ_ỹ{f() = 0}, μ_ỹ{f() = 1}|∈ B } [(<ref>) and (<ref>)] ≥ [r_0(α,β,τ) - νr_1(α,β,τ,λ,d)] · q(α,β,τ,λ,ϱ,d) =: (α,β,τ,λ,ν,ϱ,d) . Fixing , and with appropriate values for the other parameters, we can provide . The value of ϱ should be adapted for that q(…) is big (and positive in the first place). The functionis monotonously increasing in d, so an error bound for implies error bounds for while keeping in particular ν and τ. Clearly, for any , this gives lower bounds for the -complexity, n^(_0,,F_^d,) > ν exp(σ√(d)). Note that the definition of the measure does not depend on n. Moreover, by the above calculations, we have a general lower bound which holds for the conditional error in the case of varying cardinality as well. This estimate can be seen as a convex function in , indeed, fixing all parameters but , we have (n̅) = [r_0 - n̅2^-τ√(d)r_1]_+ q. By lem:n(om,f)avgspecial the lower bounds extend to methods with varying cardinality. proof:monoLB6 Smaller and bigger exponent τ for higher dimensions.[ These considerations give results of a new quality compared to Blum et al. <cit.>.] More sophisticated, if we have a lower bound , then for and we obtain the lower bound (α(τ),β(τ),τ,λ,ν,ϱ,d) > τ_0/τ _0 =: with and , supposed that in addition we fulfil the conditions and . This gives us a valid estimate n^(,,F_^d,) ≥ν2^τ√(d) = ν2^τ_0_0√(d) / under the restriction . In detail, the constraint is needed to contain several correcting terms that occur because a, b, and t can only take integer values. For example, from (<ref>) we have the correcting factor , for which holds 1 ≤κ_τ(d) = 1/√(1-τ/√(d)-1/d)≤ 1/√(1-τ_0/√(d_0)-1/d_0) = κ_τ_0(d_0). Furthermore, with the choice of and , the product is kept constant. This is the key element for the estimate σ_αβτ(d) ≤σ_α_0,β_0,τ_0(d_0), see its definition (<ref>). For the d-dependent correcting terms that occur therein, we have , and , where the assumption comes into play. For bounding , we also need . Having under control, one can easily show[ This effectively means examining , see (<ref>), where in particular we need to show , which relies on and . ] q(α(τ),β(τ),τ,λ,ϱ,d) ≥ q(α_0,β_0,τ_0,λ,ϱ,d_0), and, more complicated, r_B(α(τ),β(τ),τ,λ,ν,d) ≥τ_0/τr_B(α_0,β_0,τ_0,λ,ν,d_0). For the latter we need in particular the inequalities C_αβ≥τ_0/τC_α_0, β_0 , and C_αβτ≤τ_0/τC_α_0, β_0, τ_0 . We start with the first inequality, C_αβ = 1/√(2π)∫_α^βexp(-x^2/2) x [x = τ_0/τu] = τ_0/τ √(2π)∫_α_0^β_0exp(- (τ_0/τ)^2 u^2/2) _[τ≥τ_0]≥exp( - u^2/2) u≥τ_0/τC_α_0, β_0 . The second inequality is a bit trickier, C_αβτ = 1/√(2π)∫_α-τ^β-τexp(-x^2/2) x [subst. x + τ = τ_0/τ(u + τ_0)] = τ_0/τ √(2π)∫_α_0-τ_0^β_0-τ_0exp(- 1/2 (τ_0/τ(u + τ_0) - τ)^2 ) _≥exp( - u^2/2) u [τ_0/τ(u + τ_0) - τ≤ u ≤ 0] ≤τ_0/τC_α_0, β_0, τ_0 . Here,followed from the the upper integral boundary and the assumption . The other constraint, , followed from the monotonous decay of for , taking from the lower integral boundary into account, and recalling the assumption . proof:monoLB7 Example for numerical values. The stated numerical values result from the setting , , and . We adapt , and for starting dimension and (choosing ν appropriately) we obtain the lower error bound .§.§ Remarks on the Proof Fixing d_0 and n_0, one may vary α, β, τ, and λ so that the error bound is maximized (meanwhile adjustingand ϱ). Numerical calculations indicate that is likely to be the first dimension where with n_0 = 1 we can obtain a positive error bound (which is at around ). Choosing we first obtain a positive error bound for . That way it is also possible to find the maximal value of n_0 such that for a given dimension d_0 the error bound exceeds a given value _0. In doing so, we find a result with a particular and an estimate for the _0-complexity for : n^(_0,d) ≥ n_0 2^τ(√(d)-√(d_0)) = n_0exp(σ(√(d)-√(d_0))) . The following tabular lists the maximal n_0 for given d_0 such that we still find a lower bound that exceeds . In addition, we give the maximal possible value for τ such that we still obtain the error bound _0 with the same n_0. [ d_0 n_0 τ σ = τ log 2; 051 000 0011.06960.7414; 100 000 1081.47951.0255; 200 498 0981.97961.3721; ] As observable in the examples, the value for τ is increasing for growing dimension, so if we aim to find a good lower bound for the _0-complexity for a particular dimension d, it is preferable to use an estimate based on a big value . For example, for we obtain * n^(_0,d) > 000 179, based on d_0 = 51 and τ = 1.0696, * n^(_0,d) > 007 554, based on d_0 = 100 and τ = 1.4795, * n^(_0,d) > 498 098, computed directly for d_0 = 200. The question on how big the exponent can get is answered within the next remark. We have results of the type n^(,d) ≥ν exp(c √(d)/) that hold for “large d” and . In the asymptotics of , any estimate with a larger exponent will outstrip an estimate with a smaller exponent, so it is preferable to have a big constant c in the exponent, but ν can be arbitrarily small. In order to find the maximal value for c, we consider the limiting case for the detailed error bounds of thm:monotonLB, see proof:monoLB5 of the proof. First, we ask the question what error bounds are possible for a given τ, lim_λ→ 1lim_d →∞ ν→ 0(…) = (Φ(β)-Φ(α)) max_ϱ>0min{1- exp(-ϱ), exp(-ϱ exp((β-α) τ)) } . The maximal value for optimal α, β, and ϱ gives us a limiting value . Amongst all settings with constant difference , the factor is maximized (and hence also the asymptotic lower bound) for the symmetrical choice . Writing , we obtain (τ) := max_ϑ>0(2 Φ(ϑ/τ) - 1)max_ϱ>0min{1- exp(-ϱ), exp(-ϱ exp(2ϑ)) }_=:(τ,ϑ) , so the second factor is formally independent from τ now.[ Compare with the choice of α(τ) and β(τ) in proof:monoLB5 within the proof of thm:monotonLB.] The product is growing with τ when ϑ is fixed. Therefore, the product is maximized in the limit , lim_τ→∞τ (τ) = max_ϑ>0√(2/π) ϑmax_ϱ>0min{1- exp(-ϱ), exp(-ϱ exp(2ϑ)) }≈ 0.1586 . In other words, switching the basis of the exponential expression, now considering , the constant in thm:monotonLB cannot exceed when relying on the given proof technique. For comparison, in the numerical example of the theorem with , , and , we have . Compared to the upper bounds for Boolean functions, see thm:BooleanUBs in sec:monoUBs, there is a significant gap in the exponent that can reach arbitrarily high factors if (the growth, however, is only logarithmic). We discuss necessary modifications to the proof of thm:monotonLB in order to reproduce the result of Blum, Burch, and Langford <cit.>, which states that for sufficiently large dimensions d, and , we have e^(n,,F_^d,) ≥1/2 - Clog(d n)/√(d) , whereis a numerical constant. We start with direct modifications for a weaker version of (<ref>). Several parameters are chosen with regard to the estimates of proof:monoLB3. First, taking , by Hoeffding bounds we have #D_ab/#{0,1}^d≥ 1 - 2/√(d) , compare proof:monoLB3.1. The calculations of proof:monoLB3.2 can be replaced by the trivial estimate # (Q_∩ D_ab)/# Q_≤ 1. We choose ν = 1/d and , thus the final estimate (<ref>) of proof:monoLB3.4 reduces to # B/#{0,1}^d≥ 1 - 1/d - 2 + σ_αβτ(d)/√(d) . The choice of ν implies that, for given n, we need to take τ := log_2(d n)/√(d) , thus . Note thatcan be bounded by a constant as long as , which is equivalent to for some c>0. This log-term in the exponent is unpleasant when trying to reproduce the result of Blum et al. The term occurs from estimating , see (<ref>) in proof:monoLB2. In the original paper, for this purpose, Chernoff bounds are used, and we do not need to include the boundary value b in the definition of the measure, see proof:monoLB1. Then the cardinality of # B can be estimated by terms that only depend on τ and d, the termcan be replaced by a constant. More effort than before has to be put in estimating the conditional distribution of for , compare proof:monoLB4. For detailed calculations refer to the original proof of Blum et al. <cit.>. The parameter p of the distribution is determined by the equation (1-p)^d/2t!=1/2 , thus, for , the function values f() are 0 and 1 with equal probability under μ. Then we obtain μ_ỹ{f() = 0} ≤ 2^λat/d/2t≤1/2 + (β+1) t/√(d) , and μ_ỹ{f() = 0} = 2^- bt/d/2t≥1/2 - (β + 1) t/√(d) . Note that these two inequalities become trivial if exceeds 1/2. In particular, is an assumption needed for the proof of the second inequality when following the steps in Blum et al. Combining (<ref>) and (<ref>), and inserting the values for β and t, we obtain the estimate e(n,d) ≥1/2 - C(1 + √(log d))log d n/√(d) , with C > 0 being a numerical constant. This is a weaker version of the result (<ref>) by a logarithmic factor in d. Blum et al. proved a stronger version without this logarithmic factor by integrating over β from 1 to √(d). The weaker version with constant has also been mentioned in the original paper already. The integration over β is only possible if we estimate # V_1 by Chernoff bounds, the boundary value b may not be part of the definition of the functions for the measure μ. Interestingly, this integral runs also over such β where the estimates on the conditional measure (<ref>) give negative values, thereby weakening the lower bounds. Still, this refined proof technique gives an improvement by a logarithmic term. Similarly to rem:combiLBdeter, which was about the deterministic setting, we can find lower bounds for the Monte Carlo approximation of real-valued monotone functions that include arbitrarily small for small dimensions already, but still reflect the d-dependency known from thm:monotonLB. With the notation from rem:combiLBdeter, given , we split the domain into sub-cuboids C_, and consider monotone functions f ∈ F_^d that on each cuboid only have function values within an interval of length such that monotonicity is guaranteed whenever the function is monotone on each of the sub-cuboids. Having lower bounds from thm:monotonLB, e^(n = ν2^τ√(d), , F_^d , ) ≥ (r_0 - νr_1) q =: _1, see also (<ref>) for the inner structure of the lower bound, we can estimate the error we make on each of the sub-cuboids using function values only, e^(n = ν2^τ√(d), , F_^d , ) ≥1/||_1 - d + 1 [1/∏∑_ (r_0 - ν_r_1) q ] = 1/||_1 - d + 1 ( r_0 - ν/∏) q, where . For , choose an appropriate splitting parameter , and obtain the complexity bound n^(,,F_^d,) ≥ν2^τ√(d)+⌊_1 / ⌋-2 for ∈ [_1/d+1, _1], ν2^τ√(d) + dlog_2 ⌊_1 / (d)⌋) - 1 for ∈ (0,_1/d+1]. Knowing n^(,,F_^d,) > ν2^c√(d)/ , for and , we can take (<ref>) with the d-dependent values and in order to get enhanced error bounds for .§ BREAKING THE CURSE WITH MONTE CARLOA new algorithm for the approximation of real-valued monotone functions on the unit cube is presented and analysed in sec:monoRealUBs. It is the first algorithm to show that for this problem the curse of dimensionality does not hold in the randomized setting. The idea is directly inspired by a method for Boolean monotone functions due to Bshouty and Tamon <cit.>. We start with the presentation of the less complicated Boolean case in sec:BooleanUBs. The structure of the proofs in each of the two sections is analogous to the greatest possible extend so that one can always find the counterpart within the other setting (if there is one). §.§ A Known Method for Boolean FunctionsWe present a method known from Bshouty and Tamon <cit.> for the randomized approximation of Boolean functions that comes close to the lower bounds from thm:monotonLB. Actually, they considered a slightly more general setting, allowing product weights for the importance of different entries of a Boolean function f, we only study the special case that fits to our setting. The analysis of the original paper was done for the margin of error setting, but it can be easily converted into results on the Monte Carlo error as we prefer to define it by means of expectation.[ In the margin of error setting we want to determine such that for any input function f the actual error of the randomized algorithm exceeds only with probability δ. Since for Boolean functions the error cannot exceed 1, the corresponding expected error is bounded from above by . Conversely, for any that exceeds the Monte Carlo error e^, we obtain for the uncertainty level by Chebyshev's inequality. Practically, if we aim for a small δ, we lose a lot in this direction, and it is advisable to analyse the margin of error setting directly whenever it is of interest.]For the formulation and the analysis of the algorithm it is convenient to redefine the notion of Boolean functions and to consider the class of functionsG_±^d := {f: {-1,+1}^d →{-1,+1}} ,the input set of Boolean functions is renamed . We keep the distance that we had for the old version of G_^d, for we have(f_1,f_2) := 1/2^d #{∈{-1,1}^d | f_1() ≠ f_2() } ,compare (<ref>), thus the diameter of G_±^d is still 1. This metric differs by a factor 2 from the induced metric that we obtain when regarding G_±^d as a subset of the Euclidean space . For this space we choose the orthonormal basis ,ψ_() := ^ = ∏_j=1^d x_j^α_jfor ∈{-1,+1}^d.Every Boolean function can be written as the Fourier decompositionf = ∑_∈{0,1}^df̂()ψ_with the Fourier coefficientsf̂() := ⟨ψ_, f ⟩ = ^f(),where is uniformly distributed on . The idea of the algorithm is to use random samples , with , in order to approximate the low-degree Fourier coefficients for , k ∈,f̂() ≈ĥ() := 1/n∑_i=1^n _i^f(_i).Based on the Fourier approximationh := ∑_∈{0,1}^d ||_1 ≤ kĥ()ψ_ ,we return the output g := A_n,k^ω(f) withg() :=h() = +1if h() ≥ 0, -1if h() < 0. We will give a complete analysis of the above algorithm which is based on L_2-approximation. In fact, fromf() ≠ g() ⇔ f() ≠ h() ⇒ (f() - h())^2 ≥ 1we obtain(f,g) ≤f - h_L_2^2. Note that every Boolean function has norm 1 in the L_2-norm, so by Parseval's equation,∑_∈{0,1}^df̂^2() = f_L_2 = 1. A key result for the analysis of the above algorithm is the following fact about the Fourier coefficients that are dropped. For any monotone Boolean function f we have ∑_∈{0,1}^d ||_1 > kf̂^2() ≤√(d)/k+1 . Within the first step, we consider the special Fourier coefficients , which measure the sensitivity of f with respect to a single variable x_j. These are the only Fourier coefficients where monotonicity guarantees a non-negative value. For we consider the restricted functions f_-j() = f(), with z_j' = x_j' for j' ≠ j, and z_j = -1, f_+j() = f(), with z_j' = x_j' for j' ≠ j, and z_j = +1. Due to the monotonicity of f, we have , using this and Parseval's equation, we obtain f̂(_j) = <ψ__j,f> = 1/2^d ∑_∈{-1,+1}^d x_j f() = 1/2^d ∑_∈{-1,+1}^df_+j()-f_-j()/2_∈{0,+1}= f_+j-f_-j/2_L_2^2 = ∑_∈{0,1}^d⟨ψ_, f_+j-f_-j/2⟩^2. Since the functions f_-j and f_+j are independent from x_j, the summands with vanish. For all the other summands, with , we have ⟨ψ_, f_+j-f_-j/2⟩ = 1/2^d ∑_∈{-1,+1}^d^ f_+j()-f_-j()/2= 1/2^d ∑_∈{-1,+1}^dx_j^_= ^'f()= ⟨ψ_', f ⟩ = f̂('), where for , and . This leads to the identity f̂(_j) = ∑_∈{0,1}^d α_j = 1f̂^2(). Summing up over all dimensions, we obtain ∑_j=1^d f̂(_j) = ∑_j=1^d ∑_∈{0,1}^d α_j = 1f̂^2() = ∑_∈{0,1}^d ||_1f̂^2() ≥ (k+1)∑_∈{0,1}^d ||_1 > kf̂^2(). Finally, 1 = ∑_∈{0,1}^df̂^2() ≥∑_j=1^d f̂^2(_j) ≥1/d(∑_j=1^d f̂(_j))^2, which, combined with the inequality above, proves the lemma. This helps us to obtain the following error and complexity bound, which is a simplification of Bshouty and Tamon <cit.>, where the setting was more general, and the more demanding margin of error was considered. For the algorithm , , we have the error bound e(A_n,k,F_±^d ↪ G_±^d) ≤√(d)/k+1 + exp(k(1 + logd/k)) /n . In particular, given , the -complexity of the Monte Carlo approximation of monotone Boolean functions is bounded by n^(,F_±^d ↪ G_±^d,) ≤min{exp(C√(d)/ (1 + [log√(d) ]_+ ) ), 2^d } , whereis some constant. Hence the curse of dimensionality does not hold. We first compute the accuracy at which we approximate each of the Fourier coefficients, [f̂() - ĥ()]^2 = 1/n^2 [∑_i=1^n (f̂() - ψ_(_i) f(_i)) ]^2[independent mean 0 variables] = 1/n^2 ∑_i=1^n [f̂() - ψ_(_i) f(_i) _∈{-1,+1} ]^2 ≤1/n . This estimate is needed only for the set , we obtain the general bound (f,g) (<ref>)≤f - h_L_2^2≤ ∑_∈{0,1}^d ||_1 > kf̂^2() + ∑_∈{0,1}^d ||_1 ≤ k[f̂() - ĥ()]^2[[lem:monBooleSmallFourier]Lem <ref>lem:monBooleSmallFourier and (<ref>)]≤ √(d)/k+1 + # A/n . By lem:BinomSum, for , we estimate #A = ∑_l=0^k dl≤(d/k)^k = exp(k(1 + logd/k) ). This gives us the error bound for the Monte Carlo method . Choosing guarantees . The second term can be bounded by if we choose n := ⌈2# A/⌉≤⌈2/ exp(2√(d)/ (1 + [log√(d) ]_+ ) ) ⌉ . For , however, according to the error estimate of A_n,k, we would need to take . In this case , and n should be even larger. But then deterministically collected complete information is the best solution with already exact approximation. The present upper bounds fit the lower bounds from thm:monotonLB up to a factor in the exponent which is logarithmic in and d, but if we consider a sequence , there is no logarithmic gap at all. There is a natural transition to complete information as n approaches 2^d. Indeed, take for some natural number and sample f from function values computed for independent _i chosen uniformly from 2^k disjoint subsets in of equal size. For instance, let the first k entries within the random vector _i be given by the binary representation of i, and let the remaining entries be independent Bernoulli random variables. The calculation (<ref>) will still work essentially the same. The algorithm A_n,k is not always consistent with the knowledge we actually have on the function, and it does not even preserve monotonicity in general, so it is non-interpolatory. Take, for example,and k = 1, and the constant function . Assume that – for bad luck – all the sample points happened to be . Of course, the information already implies , thanks to monotonicity. But with the Fourier based algorithm, h() = 1 - ∑_i=1^d x_i and for the output we have g(1,…,1) = -1 < g(-1,…,-1) = +1, which violates monotonicity. We could modify the output, making the algorithm interpolatory, actually this is possible without affecting the error bounds. Obviously, it is an improvement to replace the original output g by g'() := +1if ∃ i:_i ≤ and f(_i) = +1, -1if ∃ i:_i ≥ and f(_i) = -1, g()else. Restoring monotonicity for the output is a bit more complicated since one needs to survey the output g as a whole.[ Usually we would not store all values of an approximant g in a computer but only the coefficients that are necessary for a computation of on demand. This process should be significantly cheaper than asking the oracle for a value f(), compare rem:monoMCUBphicost.] The general idea is to find pairs of points _1 ≤_2 with . At least one of these values is a misprediction of the input f. If we flip these values, that is, we create a new output g'() := -1for = _1, +1for = _2, g()else, then at least one of these values predicts f correctly, so g' approximates f not worse than g does. We could proceed like this until we obtain an interpolatory output, if needed. §.§ Real-Valued Monotone FunctionsWe present a generalization of the method from the above section to the situation of real-valued monotone functions. The method is based on Haar wavelets. For convenience, we change the range and now consider monotone functionsf: [0,1]^d → [-1,+1],the altered input set shall be named F_±^d.[ By the bijection , we can transfer results for F_±^d to results for , which comes along with a reduction of the error quantities by a factor 1/2. For proofs on lower bounds it was more convenient to have functions , because then the distance of Boolean functions coincides with the L_1-distance of the corresponding subcubewise constant functions.]We define dyadic cuboids on indexed by , or equivalently by an index vector pair with and , , such that for :C_ = C_, := _j=1^d I_α_j ,whereI_α_j = I_λ_j,κ_j := [κ_j 2^-λ_j , (κ_j+1) 2^-λ_j) for κ_j = 0,…,2^λ_j-2, [1 - 2^-λ_j , 1] for κ_j = 2^λ_j-1.Note that for fixed λ_j we have a decomposition of the unit interval into 2^λ_j disjoint intervals of length 2^-λ_j. One-dimensional Haar wavelets are defined for (if , we set and ),h_α_j :=_[0,1] if α_j = 0 (i.e. λ_j = -∞ and k=0), 2^λ_j/2(_I_λ_j+1, 2κ_j+1 - _I_λ_j+1, 2κ_j) if α_j ≥ 1 (i.e. λ_j ≥ 0).In we have the orthonormal basiswithψ_() := ∏_j=1^d h_α_j(x_j).The volume of the support of ψ_ is with . The basis function ψ_ only takes discrete values , hence it is normalized indeed.We can write any monotone function f as the Haar wavelet decompositionf = ∑_∈_0^df̃()ψ_with the wavelet coefficientsf̃() := ⟨ψ_, f ⟩ = ψ_() f(),where is uniformly distributed on . For the algorithm we will use random samples , with , in order to approximate the most important wavelet coefficientsf̃() ≈g̃() := 1/n∑_i=1^n ψ_(_i) f(_i).In particular, we choose a resolution , and a parameter , and only consider indices with and . The Monte Carlo method will give the outputg := A_n,k,r^ω(f) := ∑_∈_0^d ||_0 ≤ k < rg̃()ψ_ . We start with an analogue of lem:monBooleSmallFourier. For any monotone function f ∈ F_±^d we have ∑_∈_0^d ||_0 > k < rf̃()^2 ≤√(d r)/k+1 . Within the first step, we consider special wavelet coefficients that measure the average growth of f for the j-th coordinate within the interval . We will frequently use the alternative indexing with , where and . We define the two functions f_-α j() := 0 if x_j ∉ I_α, 2^λ+1 ∫_I_(λ+1,2κ) f()|_z_j' = x_j' for j ≠ j' z_j if x_j ∈ I_α, f_+α j() := 0 if x_j ∉ I_α, 2^λ+1 ∫_I_(λ+1,2κ+1) f()|_z_j' = x_j' for j ≠ j' z_j if x_j ∈ I_α. Due to monotonicity of f, we have . Using this and Parseval's equation, we obtain f̃(α _j) = ⟨ψ_α _j , f ⟩ = 2^λ/2 [⟨_I_λ+1,2κ+1, f ⟩ - ⟨_I_λ+1,2κ, f ⟩] = 2^λ/2 f_+α j - f_-α j/2_∈ [0,1]_L_1≥ 2^λ/2 f_+α j - f_-α j/2_L_2^2 = 2^λ/2 ∑_' ∈_0^d⟨ψ_', f_+α j - f_-α j/2⟩^2. Since the functions f_-α j and f_+α j are constant in x_j on I_α_j and vanish outside, we only need to consider summands with coarser resolution in that coordinate, and where the support of ψ_ contains the support of f_±α j. That is the case for with . For these indices we have ⟨ψ_', f_+α j - f_-α j/2⟩^2 = 2^max{0,λ_j'} - λ ⟨ψ_”,f ⟩^2 = 2^max{0,λ_j'} - λ f̃^2(”), where for , and . Hence we obtain f̃(α _j) ≥ 2^λ/2 (2^-λ + ∑_l = 0^λ-1 2^l-λ) _= 1 ∑_”∈_0^d α_j” = αf̃^2(”). Based on this relation between the wavelet coefficients, we can estimate 1 = f_L_2^2 = ∑_∈_0^df̃^2() ≥ ∑_j=1^d ∑_λ = 0^r-1∑_κ = 0^2^λ-1f̃^2((2^λ+κ)_j) ≥ ∑_j=1^d ∑_λ = 0^r-1(2^-λ/2 ∑_κ = 0^2^λ-1f̃((2^λ+κ)_j) )^2 (<ref>)≥∑_j=1^d ∑_λ = 0^r-1(∑_∈_0^d λ_j = λf̃^2() )^2. Taking the square root, and using the norm estimate for , here with , we get 1≥1/√(d r) ∑_j=1^d ∑_λ = 0^r-1∑_∈_0^d λ_j = λf̃^2() ≥1/√(d r) ∑_∈_0^d< r ||_0f̃^2() ≥k+1/√(d r) ∑_∈_0^d ||_0 > k < rf̃^2(). This proves the lemma. For the algorithm we have the error bound e(A_n,k,r,F_±^d ↪ L_1([0,1]^d)) ≤d/2^r+1 + √(√(d r)/k+1 +exp[k(1+logd/k + (log 2) r)]/n) . Given , the -complexity for the Monte Carlo approximation of monotone functions is bounded by n^(,F_±^d ↪ L_1([0,1]^d),) ≤ min{ exp[C √(d)/^2 ( 1 + (logd/)^3/2) ],exp[dlogd/2 ] } , with some numerical constant . In particular, the curse of dimensionality does not hold for the randomized L_1-approximation of monotone functions. Since we only take certain wavelet coefficients until a resolution r into account, the output will be a function that is constant on each of 2^rd subcubes C_r , where . The algorithm actually approximates f_r := ∑_∈_0^d< rf̃()ψ_ . Since on the one hand, the Haar wavelets are constant on each of the 2^rd subcubes, and on the other hand, we have 2^rd wavelets up to this resolution, the function f_r averages the function f on each of the subcubes. That is, for we have |f() - f_r()| = |f() - ' f(')| ≤1/2 [sup_∈ C_r , f() -inf_∈ C_r , f() ]. Following the same arguments as in the proof of thm:MonAppOrderConv, we group the subcubes into diagonals, each diagonal being uniquely represented by a with at least one 0-entry. By monotonicity, summing up (<ref>) for all cubes of a diagonal, we obtain the upper bound 1/2. Now that there are diagonals, and the volume of each subcube is 2^-rd, we obtain the estimate f - f_r_L_1≤d/2^r + 1 . Surprisingly, the fact that the wavelet basis functions have a small support, actually helps to keep the error for estimating the wavelet coefficients small. Exploiting independence and unbiasedness (compare (<ref>)), for we have [f̃() - g̃()]^2 = 1/n^2 ∑_i=1^n [f̃() - ψ_(_i) f(_i) ]^2 = 1/n([ψ_(_i) f(_i)]^2 - f̃^2() ) ≤1/n {_1 ∈ C_}_= 2^-λ [(ψ_(_1) f(_1))^2 _∈ [0,2^λ]|_1 ∈ C_] ≤1/n . Then by (<ref>), lem:monSmallWavelet, and (<ref>), the expected distance between input f and output g is f - g_L_1 ≤f - f_r_L_1 + f_r - g_L_2≤f - f_r_L_1 + (∑_∈_0^d ||_0 > k < rf̃()^2 + ∑_∈_0^d ||_0 ≤ k < r [f̃() - g̃()]^2 )^1/2≤d/2^r + 1 + √(√(d r)/k+1 + # A/n) , where A := {∈_0^d | ||_0 ≤ k and< r} . Using lem:BinomSum, we can estimate the size of A for , # A = ∑_l=0^k dl(2^r - 1)^l ≤ 2^r k (d/k)^k. This gives us the error bound for the Monte Carlo method A_n,k,r. Choosing the resolution will bound the first term . Taking then guarantees . Finally,can be bounded from above by if we choose n := ⌈8/^2 exp( 8√(d (1 + log_2 d/))/^2( 1 + logd/k + log2d/) ) ⌉ . By this choice we obtain the error bound we aimed for. Note that if is too small, we can only choose for the algorithm A_n,k,r. In this case, for the approximation of f_r, we would take 2^rd wavelet coefficients into account, n would become much bigger in order to achieve the accuracy we aim for. Instead, one can approximate f directly via the deterministic algorithm A_m^d from thm:MonAppOrderConv, which is based on function values on a regular grid. The worst case error is bounded by . Taking , this gives the same bound that we have for the accuracy at which f_r approximates f, see (<ref>). So for small we take the deterministic upper bound n^(,,F_^d,) ≤exp(dlogd/2 ), compare rem:MonAppOrderConv. It is rather unpleasant that the estimate in thm:monoUBsreal depends exponentially on ^-2, at least for . In other words, for d-dependent error tolerances , the cardinality of A_n,k,r (with appropriately chosen parameters) depends exponentially on d. However, there is a way to improve the -dependency at the price of losing the linearity of the algorithm. For the subclass of sign-valued monotone functions F_{±}^d := {f:[0,1]^d →{-1,+1}| f ∈ F_±^d} , we can modify the algorithm in a way similar to the Boolean setting, the new version Ã_n,k,r now returning an output . For we can estimate f - g̃_L1 ≤f - f_r _L_1 + 2f_r - g_L_2^2 ≤d/2^r+1 + 2√(d r)/k+1 + 2# A/n , and for the restricted input set we obtain the complexity bound n^(,F_{±}^d↪ L_1[0,1]^d,) ≤exp[C'√(d)/ ( 1 + (logd/)^3/2) ], with a numerical constant C' > 0. This complexity bound holds actually for the whole class F_±^d. Indeed, any bounded monotone function can be written as an integral composition of sign-valued functions , f = 1/2∫_-1^1 f_tt. We define a new algorithm by A̅_n,k,r^ω(f) := 1/2∫_-1^1 Ã_n,k,r^ω(f_t)t. Note that the information needed for the computation of can be derived from the same information mapping applied to f directly since the algorithm is non-adaptive. We can write in contrast to . By the triangle inequality we get e(A̅_n,k,r,f) = f - A̅_n,k,r^ω(f)≤1/2∫_-1^1 f_t - Ã_n,k,r^ω(f_t) t ≤ e(Ã_n,k,r,F_{±}^d). The resulting algorithm A̅_n,k,r is not linear anymore. It is an interesting question how much the combinatory cost for ϕ̅_n,k,r differs from the cost for ϕ_n,k,r. For the model of computation we refer to the book on IBC of Traub et al. <cit.>, and to Novak and Woźniakowski <cit.>. The cost for computing the linear representation ϕ_n,k,r is dominated by the following operations: * Each sample f(_i) contributes to wavelet coefficients. The relevant indices ∈_0^d can be determined effectively based on the binary representation of _i. * Compute the linear combination of wavelets. The first part is the most costly part with more than n operations needed. If the parameters are chosen according to thm:monoUBsreal, the second part only needs about operations in . We can roughly summarize the cost as . For the non-linear representation ϕ̅_n,k,r, proceed as follows: * Sort the information such that . * Define and , and use the representation ϕ̅_n,k,r^ω() = 1/2∑_i=0^n (y_i+1-y_i) ϕ_n,k,r^ω(-1,…,-1 ^i times, 1,…,1 ^(n-i) times) _=: g_i . Here, the cost for computing g_0 is the cost for computing . For , we obtain g_i from modifying g_i-1, indeed, by linearity of ϕ_n,k,r we have g_i := g_i-1 - 2ϕ_n,k,r^ω(_i). Since for we only need to take wavelet coefficients into account, doing this n times, the cost for computing is only twice the cost of ϕ_n,k,r. (Here, however, we need more operations in , and less operations in , but we assumed them to have the same cost, no matter how realistic that is.) The signum operator and the final sum contribute to the cost only linearly in n. Ordering the information has an expected cost of , which is likely to be dominated by the cost of . Assuming this, we obtain ϕ_n,k,r≍ϕ̅_n,k,r . Heinrich and Milla <cit.> pointed out that for problems with functions as output, the interesting question is not about a complete picture of the output , but about effective computation of approximate function values on demand. In our situation it makes sense to distinguish between pre-processing operations and operations on demand. For the linear representation ϕ_n,k,r we have Pre-processing: Compute and store the wavelet coefficients that are needed for the output. On demand: Compute . (Only wavelet coefficients are relevant to .) The pre-processing is approximately as expensive as the cost with the above computational model, computation on demand is significantly cheaper. For the non-linear representation ϕ̅_n,k,r we have Pre-processing: Rearrange the information. Store the wavelet coefficients needed for g_0. On demand: Compute . (For this, wavelet coefficients are relevant.) In order to compute , we need in particular the values for . These can be determined in a very effective way.[ Observe that n [ϕ_n,k,r^ω(_i)]() = ∑_∈_0^d ||_0 ≤ k ||_∞ < 2^rψ_(_i)ψ_() = ∑_∈_0^d ||_0 ≤ k ||_∞ < 2^r∏_j=1^d h_α_j(_i(j)) h_α_j(x_j) = ∑_∈{0,1}^d |β|_1 ≤ k^ , where . It is readily checked that Z_j = 2^r - 1 if ⌊ 2^r_i(j) ⌋ = ⌊ 2^r x_j ⌋, -1 else, so a comparison of the first r digits of the binary representation of _i(j) and x_j is actually enough for determining Z_j. In the end, we only need the number b of coordinates where , and obtain n [ϕ_n,k,r^ω(_i)]() = ∑_l = 0^b ∧ kbl(2^r - 1)^l∑_m = 0^(d-b)∧(k-l)d-bm(-1)^m =: χ(b) ∈ . These values are needed for . Since they only depend on parameters of the algorithm, they can be prepared before any information was collected. ] For the pre-processing there is no big difference from the setting before. The part “on demand” it is a little cheaper than the pre-processing part, however, we need more than n operations. Hence a linear algorithm with the same information cardinality on the one hand is less costly, on the other hand the error is larger. We conclude that it depends on and the ratio of information cost versus combinatory cost whether the linear or the non-linear algorithm should be preferred. If we are allowed asking the oracle for wavelet coefficients directly, the same algorithmic idea is implementable as a deterministic method and less information is needed. (This reduction of the complexity is by a factor or .) In particular, the curse of dimensionality does not hold in the deterministic setting with . It is an open problem whether similar lower bounds to those from sec:monoMCLBs can be found for . The proof technique of thm:monotonLB, however, will not work for , because one could choose a functional that injectively maps all possible Boolean functions onto the real line, and thus identify any given function by just one measurement. Indeed, one measurement is sufficient. Let be the binary representation of an integer, b() := ∑_j=1^d x_j 2^j-1∈_0. Define the functional L_1 for Boolean functions by L_1(f) := ∑_∈{0,1}^d f() 2^b() . This functional maps G_^d bijectively onto the set of 2^d-bit representable integers. This shows that is an inappropriate model for the approximation of Boolean functions. [chapter]chapterCHAPTER: ON GAUSSIAN MEASURES Let Φ(x) = 1/√(2π) ∫_-∞^x exp(- t^2/2)t denote the cumulative distribution function of a standard Gaussian variable.Let be a standard Gaussian vector in , i.e. the X_i are iid standard Gaussian variables. For any linear mapping with F being a Banach space, then is a zero-mean Gaussian vector in F. For the norm of a vector ∈ℓ_2^m we write . The operator norm of J shall be denoted .§ COMPARISON OF GAUSSIAN MEASURES Let andbe independent zero-mean Gaussian vectors in a normed space F. Then _F ≤ + _F. Due to symmetry of zero-mean Gaussian measures, the distributions ofand are identical. Hence, by the triangle inequality, _F = 1/2∑_σ = ± 1 ( + σ ) _F ≤1/2∑_σ = ± 1 + σ_F = + _F. Let be independent standard Gaussian random variables, be elements in a normed vector space F, and be real numbers. Then ∑_k=1^m a_k X_k f_k _F ≤∑_k=1^m a_k' X_k f_k _F. Take another m independent standard Gaussian random variables . Then the are identically distributed to . Thus, applying lem:E|Y|<E|Y+Z|, we obtain ∑_k=1^m a_k X_k f_k _F ≤∑_k=1^m a_k X_k f_k + ∑_k=1^m √(a_k'^2 - a_k^2)X_k' f_k _F = ∑_k=1^m a_k' X_k f_k _F.There are several other known comparison principles for Gaussian vectors, mainly based on covariance comparisons, and concerning the expected maximum component instead of arbitrary norms, see for example Lifshits <cit.>. For Gaussian fields one important result is Slepian's inequality, see e.g. Adler <cit.>.§ RESTRICTIONS OF GAUSSIAN MEASURES With P being an orthogonal projection in ℓ_2^m, the random vector can be interpreted as a restriction of the Gaussian measure to the subspace . We start with a rather simple comparison of the expected norms. For orthogonal projections P on ℓ_2^m we have J P _F ≤J _F. Due to orthogonality,and are stochastically independent. The claim follows from lem:E|Y|<E|Y+Z|. As an application we have a bound for the operator norm of J. For any linear operator J : ℓ_2^m →F we have J _F ≥√(2/π) J_2 → F . Applying lem:E|JPX|<=E|JX| to rank-1 orthogonal projections, we obtain J _F ≥sup_P orth. Proj. P=1JP _F = J_2 → F|X_1| = √(2/π) J_2 → F .§ DEVIATION ESTIMATES FOR GAUSSIAN MEASURES cor:deviationGauss is a deviation result for norms of Gaussian vectors known from Pisier <cit.>. It has been used in this form by Heinrich <cit.> for his proof on lower bounds for the Monte Carlo error via Gaussian measures, which is reproduced in Chapter <ref>. The deviation result for the norms of Gaussian vectors is a consequence of the following proposition. Let f : ℓ_2^m → be a Lipschitz function with Lipschitz constant L. Then for t>0 we have {f() -f() > t}≤exp(- t^2/ 2 L^2).For this result several proofs are known. One way uses stochastic integration, a rather short description of this approach can be found in Pisier <cit.>. A more direct proof is contained in Adler and Taylor <cit.>. Both methods require higher smoothness for f first, an assumption that can be removed in the end with Fatou's inequality. For a simpler proof with a slightly worse constant in the exponent, see Pisier <cit.>. Forand with we have {J _F > λ J _F}≤exp(- (λ-1)^2ρ^2/2) ≤exp(- (λ-1)^2/π). We define by f() := J _F. This function is Lipschitz continuous with Lipschitz constant L = sup_,∈ℓ_2^m≠|f()-f()|/-_2Δ-ineq.≤sup_,∈ℓ_2^m≠J(-)_F/-_2 = J_2 → F , so that we can apply prop:dev2 with . Actually, we have equality , to see this, consider the LHS with the supremum for . For this kind of estimate it is sufficient to bound from above, or the ratio from below. In the most general way, , see lem:E|JX|>c|J|.§ GAUSSIAN VECTORS IN SEQUENCE SPACESThe next two lemmas are needed for the proof of lem:gaussqnormvector which follows the lines of Pisier <cit.>. In there the inequalities of lem:gausstail and <ref> have been mentioned without giving a proof or detailed information about the constants. There exists a constant such that {|X|>α√(log x)}≥1/xfor x≥, where X is a standard Gaussian random variable. The value is optimal, providing equality for . Here, Φ denotes the cumulative distribution function of the standard normal distribution. Having chosen α as above, we consider the left-hand side of the inequality to be shown, = {|X|>α√(log x)} = √(2/π)∫_α√(log x)^∞exp(-t^2/2)t [t = α√(log s)] = α/√(2π)∫_x^∞s^-(α^2/2+1)/√(log s) s Its derivative is / x = - α/2π x^-(α^2/2 + 1)/√(log x) , whereas for the right-hand side we have / x = - 1/x^2 . The derivatives of both sides agree iff α^2/2π x^2-α^2 = log x. Since , we have that x^2-α^2 is convex, whereasis concave for , so there are no more than two points that solve (<ref>). Comparing both sides of (<ref>), we obtain: α/√(2π) > 0 for x = 1, α^2/2π^2-α^2α<1</2π < 1 for x =, and α^2/2π x^2-α^2 > log x for sufficiently large x, say x > x_2. Therefore 1<x_1<<x_2. Now for (<ref>), by choice of α, we have = for x=, lim_x→∞ = lim_x →∞ = 0. With / x|_x= = - α/√(2π) ^-(α^2/2+1)≈ -0.088107 > / x|_x= = - ^-2≈ -0.135335, and only having one point where the difference is locally extreme, we obtain thatfor ≤ x < ∞. There exists a constant such that for all we have ( |X|^q)^1/q≤ K √(q) , where X is a standard Gaussian variable. We can write γ(q) := ( |X|^q)^1/q = (√(2/π) ∫_0^∞ x^qexp(-x^2/2)x )^1/q . Given , one can find such that x^qexp(-x^2/2) ≤ c xexp(-α x^2/2) , in detail, is optimal with equality holding in . We obtain ∫_0^∞ x^qexp(-x^2/2)x ≤ c∫_0^∞ xexp(-α x^2/2)x = c/α = 1/α (q-1/(1-α))^(q-1)/2 . This estimate can be minimized by choosing , so γ(q) ≤1/√() ((q+1)/2π)^1/(2q) √(1 + 1/q)_ =: K(q) √(q) . We can carry out a final estimate (finding the extreme point by ), K := sup_q≥ 1 K(q) = K(2π/ - 1) = √(2π/(2π - ))≈ 0.805228. This value for K is not optimal because for our estimate is rough. But it is close to optimal since , i.e. if one could show that the lemma is true with , one would have a sharp estimate with equality holding for . Numerical calculations strongly encourage this conjecture. Now we can state and prove the norm estimates for Gaussian vectors, the proof (without explicit constants) is found in Pisier <cit.>. Let be a standard Gaussian vector on ^m. * For we have √(2/π)m^1/q≤_q ≤ K √(q)m^1/q . * There exist constants c,C>0 such that c√(1+log m)≤_∞≤ C√(1+log m) . (a) The upper bound follows from _q = (∑_j=1^m |X_j|^q )^1/qJensen≤(∑_j=1^m|X_j|^q)^1/q , and lem:gaussqnorm. For the lower bound we use the triangle inequality with the vector of absolute values , _q ≥_q = (∑_j=1^m (|X_j|)^q )^1/q = √(2/π)m^1/q . (b) The upper bound is obtained by a comparison to some q-norm for , from (a) we have _∞≤_q Jensen≤(_q^q)^1/q≤ K√(q)m^1/q . For the choice we have , so that we obtain the desired upper bound with . Finally the lower bound. Using lem:gausstail with , we get {|X_j|>α√(1 + log m)}≥1/m . By this, {max_j=1,…,m |X_j| ≤α√(1+log m)}≤(1 - 1/m)^m ≤exp(-^-1). We conclude _∞ = max_j=1,…,m |X_j| ≥α√(1+log m) (1 - {max_j=1,…,m |X_j| ≤α√(1+log m)}) ≥α(1 - exp(-^-1)) √(1 + log m) . This gives us .CHAPTER: USEFUL INEQUALITIES § COMBINATORICS The following two lemmas are well known. For k=1,…,d we have ∑_l=0^k dl < (d/k)^k. For k=1 we have = 1 + d < d =. For k=d we have = 2^d < ^d =. This completes the cases d=1,2. Assume that the lemma holds for d and . We show that it holds for d+1 and as well. ∑_l=0^k d+1l= ∑_l=0^k dl + ∑_l=0^k-1dl < (d/k-1)^k-1 + (d/k)^k = (/k)^k ((1 + 1/k-1)^k-1/_< 1k d^k-1 + d^k ) < ((d+1)/k)^k. By induction, this completes the proof.For d > 0 we have1/2 2^d/√(d)≤d⌊ d/2 ⌋≤√(2/π) 2^d/√(d) . For odd , , by induction we see that is monotonously increasing in k. Similarly, for even , , we observe that is increasing. Hence the lower bound is done by considering . For the upper bound we need the limit as , this can be obtained by Stirling's formula. § QUANTITATIVE CENTRAL LIMIT THEOREM Letbe iid random variables with zero mean, unit variance and finite third absolute moment β_3. Then there exists a universal constant C_0 such that |{1/√(d)∑_j=1^d X_j ≤ x } - Φ(x)| ≤C_0β_3/√(d) , where Φ(·) is the cumulative distribution function of the univariate standard normal distribution. The best known estimates on C_0 are C_E := √(10)+3/6√(2π) = 0.409732…≤ C_0 < 0.4748 see Shevtsova <cit.>. Let and with real numbers . Then we have 1/2^d∑_k=a^bdk ≥ Φ(β) - Φ(α) - 2 C_0/√(d) . Let be Bernoulli random variables and the corresponding Rademacher random variables. Note that the Z_i have zero mean, unit variance, and third absolute moment . Applying prop:BerryEsseen twice to the Z_j, we obtain 1/2^d∑_k=a^bdk = {d/2 + α√(d)/2≤∑_j=1^d X_j ≤d/2 + β√(d)/2}= {α≤1/√(d) ∑_j=1^d Z_j ≤β}≥Φ(β) - Φ(α) - 2 C_0/√(d) .CHAPTER: ABBREVIATIONS AND SYMBOLStocchapterAbbreviations and Symbols scrplain 2X Abbreviations a.s. almost surely IBC information-based complexity iid independent and identically distributed (random variables) / left-hand side/right-hand side (of an equation or inequality referred to) RKHS reproducing kernel Hilbert space void2XOn Functions and Real Numbers a_+ := max{a,0} positive part of a real number a ∈ ⌊ a ⌋ a ∈ rounded down to an integer ⌈ a ⌉ a ∈ rounded up to an integer [Statement] characteristic function, yielding 1 if Statement is true, and taking the value 0 if Statement is false δ_ij = [i=j] Kronecker symbol log natural logarithm f ≼ g f and g non-negative functions on a common domain andfor a constant f ≍ g f ≼ g and g ≼ f, that is, with c,C>0 a_n ≺ b_n for n →∞ with a_n ≼ b_n for n →∞ there existand such that for we have void2XVectors and Normed Spaces = (x_1,…,x_m) vector in ^m (or ℓ_p) with entries (i) = x_i _A = (x_i)_i ∈ A∈^A sub-vector for A ⊆{1,…,m} [k] := {1,…,k} first k natural numbers := (0,…,0) vector with all the entries set to 0 := (1,…,1) vector with all the entries set to 1 , := _j=1^m [x_j,z_j] closed cuboid with vector-valued interval boundaries ⟨, ⟩ scalar product for vectors ,∈^m = ℓ_2^m ℓ_p^m ^m with the norm for , orfor , analogously for ||_p in some contextes instead of _p for the ℓ_p^d-norm ||_0 number of non-zero entries of ∈^d B_p^m unit ball within ℓ_p^m _i standard basis vector in ^m = ℓ_p^m, or ^ L_p(ρ) for being a measure space, is the space of measureable functions that are bounded in the norm for , or for ; more precicely it is the space of equivalence classes of functions that are indistinguishable with respect to the metric induced by the norm L_p(Q) L_p on a domain Q ⊆^d with the d-dimensional Lebesgue measure void2XOperators (^m) set of linear operators ^m →^m J_2 → F operator norm of a linear operator between normed spaces T rank of a linear operator T P trace of a linear operator P ∈(^m) ⊷ J image J(^m) of an operator J : ^m →F F↪ G identity mapping, F is identified with a subset of Gvoid2XAnalysis and Topology ∇_ f directional derivative of a d-variate function f ∂_i f partial derivative of a d-variate function f, that is , into the direction _i of the i-th coordinate void2XStochastics and Measure Theory (Ω,Σ,) suitable probability space expectation, i.e. the integration over with respect to Φ(x) cummulative distribution function of a standard normal variable = (X_1,…,X_m) random vector (A) uniform distribution on a finite set A # A number of elements of a set A 99. plain tocchapterBibliographyAdl90 R. 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Journal of Complexity, 28(1):59–75, 2012.Xu15 G. Xu. On weak tractability of the Smolyak algorithm for approximation problems. Journal of Approximation Theory, 192:347–361, 2015. CHAPTER: EHRENWÖRTLICHE ERKLÄRUNGemptyHiermit erkläre ich, * dass mir die Promotionsordnung der Fakultät bekannt ist, * dass ich die Dissertation selbst angefertigt habe, keine Textabschnitte oder Ergebnisse eines Dritten oder eigenen Prüfungsarbeiten ohne Kennzeichnung übernommen und alle von mir benutzten Hilfsmittel, persönliche Mitteilungen und Quellen in meiner Arbeit angegeben habe, * dass ich die Hilfe eines Promotionsberaters nicht in Anspruch genommen habe und dass Dritte weder unmittelbar noch mittelbar geldwerte Leistungen von mir für Arbeiten erhalten haben, die im Zusammenhang mit dem Inhalt der vorgelegten Dissertation stehen, * dass ich die Dissertation noch nicht als Prüfungsarbeit für eine staatliche oder andere wissenschaftliche Prüfung eingereicht habe. Bei der Auswahl und Auswertung des Materials sowie bei der Herstellung des Manuskripts wurde ich durch Prof. Dr. Erich Novak unterstützt.Ich habe weder die gleiche, noch eine in wesentlichen Teilen ähnliche oder andere Abhandlung bei einer anderen Hochschule als Dissertation eingereicht. Jena, 31. März 2017 Robert Kunsch | http://arxiv.org/abs/1704.08213v1 | {
"authors": [
"Robert J. Kunsch"
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"title": "High-Dimensional Function Approximation: Breaking the Curse with Monte Carlo Methods"
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[pages=1-last]HPCCidle.pdf | http://arxiv.org/abs/1704.08244v1 | {
"authors": [
"Ivy Bo Peng",
"Stefano Markidis",
"Erwin Laure",
"Gokcen Kestor",
"Roberto Gioiosa"
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A Large-Scale Study on the Usage of Testing Patterns that Address Maintainability Attributes Patterns for Ease of Modification, Diagnoses, and ComprehensionDanielle Gonzalez 1, Joanna C.S. Santos1, Andrew Popovich1, Mehdi Mirakhorli1, Mei Nagappan2 1Software Engineering Department, Rochester Institute of Technology, USA 2David R. Cheriton School of Computer Science, University of Waterloo, Canada{dng2551,jds5109,ajp7560, mxmvse}@rit.edu, [email protected] 30, 2023 ===========================================================================================================================================================================================================================================================================================================================================Test case maintainability is an important concern, especially in open source and distributed development environments where projects typically have high contributor turn-over with varying backgrounds and experience, and where code ownership changes often.Similar to design patterns, patterns for unit testing promote maintainability quality attributes such as ease of diagnoses, modifiability, and comprehension. In this paper, we report the results of a large-scale study on the usage of four xUnit testing patterns which can be used to satisfy these maintainability attributes.This is a first-of-its-kind study which developed automated techniques to investigate these issues across 82,447 open source projects, and the findings provide more insight into testing practices in open source projects. Our results indicate that only 17% of projects had test cases, and from the 251 testing frameworks we studied, 93 of them were being used. We found 24% of projects with test files implemented patterns that could help with maintainability, while the remaining did not use these patterns. Multiple qualitative analyses indicate that usage of patterns was an ad-hoc decision by individual developers, rather than motivated by the characteristics of the project, and that developers sometimes used alternative techniques to address maintainability concerns. Unit Testing, Maintenance, Open Source, Mining Software Repositories, Unit Test Patterns, Unit Test Frameworks § INTRODUCTIONThe unit test serves as an important facet of software testing because it allows individual “units” of code to be tested in isolation. With the development of automated testing frameworks, such as the xUnit collection, writing and executing unit tests has become more convenient. However, writing quality unit tests is a non-trivial <cit.> task. Similar to production code, unit tests often need to be read and understood by different people. The moment a developer writes a unit test, it becomes legacy code that needs to be maintained and evolve with changes in production code.Particularly, writing quality unit tests that encourage maintainability and understandability is a very important consideration for open source projects and the distributed development environment in general, where projects have high contributor-turn over, contributors have varying backgrounds and experience, and code ownership changes often <cit.>.Existing research and best practices for unit testing focus on the quality of the test's design in terms of optimizing the coverage metric or defects detected. However, recent studies <cit.> emphasize that the maintainability and readability attributes of the unit tests directly affect the number of defects detected in the production code. Furthermore, achieving high code coverage requires evolving and maintaining a growing number of unit tests. Like source code, poorly organized or hard-to-read test code makes test maintenance and modification difficult, impacting defect identification effectiveness.In “xUnit Test Patterns: Refactoring Test Code” George Meszaro <cit.> presents a set of automated unit testing patterns, and promotes the idea of applying patterns to test code as a means of mitigating this risk, similar to how design patterns are applied to source code. Previous studies <cit.> have looked at test case quality in open source and industry, studying the effects of test smells and bad practices, and detecting quality problems in test suites. However, these and other testing-related works <cit.> are limited to a small number of projects and have not taken into account use of testing frameworks and patterns that can be used to address maintainability attributes.In this paper, we aim to assess the satisfaction of test maintainability quality attributes in open source projects by studying the use of automated unit testing frameworks and the adoption of four of Meszaros' testing patterns within a large number of projects. First, we conducted a large-scale empirical study to measure the application of software testing in the open source community, providing insights into whether open source projects contain unit tests and which testing frameworks are more common. Second, to evaluate the maintainability characteristics of unit tests written in the open source community, we have defined three quality attributes: Ease of Modification, Ease of Diagnoses, and Ease of Comprehension. To assess the maintainability characteristics of the tests with regard to these quality attributes, we detect the adoption of four unit testing patterns that, when applied to unit tests within open source projects, help satisfy these attributes. Our novel approach includes a data set ofopen source projects written in 48 languages, and our approach for identifying pattern use is language-agnostic, increasing the generalization of our results to open source development as a whole. We also conduct a qualitative study to reveal project factors which may have influenced a developer's decision whether or not to apply patterns to their tests. More precisely, the following motivating question and sub-questions have been explored and answered in this study: Motivating Question: Do Test Cases in Open Source Projects Implement Unit Testing Best Practices to Satisfy Test Maintainability Quality Attributes? To accurately answer this question, we investigate the following sub-questions:* RQ1: What percentage of projects with tests applied patterns that address Ease of Modification?* RQ2: What percentage of projects with tests applied patterns that address Ease of Diagnoses?* RQ3:Do automatically generated test plans and checklists focused on testing breaking points improve the security and reliability of a software product? * RQ3:Despite investigating the use of only four testing patterns, our novel approach to measuring test quality will highlight important facts about the quality of test cases present in open source projects.The reminder of this paper is organized as follows. A description of related works is provided in section <ref>, and section <ref> defines important terms used throughout the paper. Section <ref> describes in detail the methodology for answering our three research questions. Results are presented in section <ref>. Qualitative analyses and discussion of our pattern results is presented in section <ref>. Threats to validity are acknowledged in section <ref>, and we end with our conclusions and proposed future work. § RELATED WORKIn this section we identify relevant work studying unit testing in open source software, maintainability and quality of unit test cases, and the relationship between production and test code.The studies most closely related to the work presented in this paper were presented by Kochar et. al. This group was the first to conduct a large-scale study on the state of testing in open source software <cit.> using 20,817 projects mined from Github. They studied the relationship between the presence and volume of test cases and relevant software project demographics such as the number of contributing developers, the project size in lines of code (LOC), the number of bugs and bug reporters. They also measured the relationship between programming languages and number of test cases in a project. A year later, they conducted a quality-focused study <cit.> using code coverage as a measure of test adequacy in 300 open source projects. Other small-scale studies have evaluated the quality or effectiveness of unit tests in software projects. Badri et. al's used two open source Java projects to study levels of testing effort to measure the effectiveness of their novel quality assurance metric <cit.>. Bavota et. al <cit.> performed two empirical analyses to study the presence of poorly designed tests within industrial and open source Java projects using the JUnit framework, and the effects these bad practices have on comprehension during maintenance phases. Evaluations of the effectiveness of automated unit test generation tools show the importance of the usability and readability quality attribute, such as a study by Rojas et. al <cit.> on automatic unit tests generated byEvosuite. To improve understandability of test cases Panichella et. al created a summarization technique called Test Describer. <cit.>. Athanasiou et. al <cit.> created a model to measure test code quality based on attributes such as maintainability found to positively correlate with issue handling.Some works have taken an approach opposite to the one presented in this paper to measure the quality of unit tests by measuring the presence of `test smells' and bad practices within projects. These works uses the `test smells' defined by Van Deursen et. al <cit.> and broadened by Meszaros <cit.>.Van Rompaey and Demeyer <cit.> defined a metric-based approach to detect two test smells, and later Brugelmans and Rompaey <cit.> developed TestQ to identify test smells in test code. To reduce erosion of test maintainability over time due to fixture refactoring, Greiler et. al <cit.>, developed TestHound, which detects test smells.Neukirchen et. al <cit.> have used the presence of test smells to identify maintenance and reusability problems in test suites which use TCCN-3.A number of works have been published which study the relationships between production code and unit tests. Using graphs obtained from static and dynamic analysis, Tahir and MacDonell <cit.> applied centrality measures to quantify the distribution of unit tests across five open source projects in order to identify the parts of those projects where the most testing effort was focused. Zaidman et. al <cit.> use coverage and size metrics along with version control metadata to explore how tests are created and evolve alongside production code at the commit level on two open source projects. Van Rompaey and Demeyer <cit.> have established conventions for improving the traceability of test code such as naming conventions, static call graphs, and lexical analysis. These methods have been used in other papers to <cit.> establish heuristics for identifying tests in source code. In this study we expand the scope of existing explorations of testing in open source by increasing the number of projects and languages being examined. The novel contributions presented in this paper are the use of three quality attributes, measured via pattern and test framework adoption in projects, as a measure of unit test maintainability in the open source community. § BACKGROUND AND DEFINITIONSWe begin by defining the important concepts and terms required to understand our work. In this section we define the metrics, three maintenance quality attributes for unit tests, and the testing patterns used throughout the paper. §.§ Project Metrics §.§.§ Testing RatioTo better understand the quantity of test code present in open source projects, we used the TPRatio metric, defined as the Production Code to Unit Test Code Ratio:TPRatio = Unit Test LOCProduction LOCTPRatio measures the ratio of unit test lines of code to production (non-test) lines of code. Other works have used code coverage to measure test volume in projects <cit.>, but we chose the TPRatio metric as a reasonable alternative because we do not have a reliable means of collecting coverage data for all the languages in our dataset. Ward Cunningham provides a discussion of this metric on his website <cit.> which mentions it as a `quick check' before measuring coverage. This metric has also been used in a previous study by Kochar et. al <cit.>.Project SizeProject size within our dataset varies widely, which can impact how certain results, such as TPRatio, are presented. To prevent this, we divide projects in our dataset into four sizes when presenting results:* Very Small: Project Size < 1 kLOC* Small: 1 kLOC ≤ Project Size < 10 kLOC * Medium: 10 kLOC ≤Project Size < 100 kLOC* Large: Project Size ≥ 100 kLOC§.§ Unit Test Quality Attributes To evaluate the maintainability characteristics of unit tests written in the open source community, we have defined the following quality attributes, derived from literature <cit.> that can impact the quality of unit tests. These attributes, summarized below, were used as driving requirements to identify patterns that improve the quality of unit tests. * QA#1 - Ease of Modification: The ability of unit tests in a system to undergo changes with a degree of ease.* QA#2 - Ease of Diagnoses: The measure of difficulty when attempting to track and inspect a test failure.* QA#3 - Ease of Comprehension: The degree of understandability of a unit test being read by a developer not originally involved in its writing.§.§ Testing Patterns`Test smells', introduced by Van Deursen et. al <cit.>, negatively affect the maintainability of unit tests <cit.>. Guidelines for refactoring test code to remove these smells have been proposed <cit.> but in order to prevent smells and promote high quality tests, Meszaros has detailed 87 goals, principles, and patterns for designing, writing and organizing tests using the xUnit family of unit testing frameworks <cit.>Therefore, in order to evaluate the maintainability of unit tests in open source projects, we chose to measure the adoption of some of these patterns. We developed 3 criteria when choosing patterns; first, a pattern must address at least one of the quality attributes defined in section <ref>. Next, a pattern must be optional for a developer to implement, i.e. not enforced by the framework. This will help us to understand the decisions open source developers make. Finally, it must be possible to detect the use of the pattern or principle using a programming-language-agnostic heuristic, because we needed to be able to detect their use in test files written in numerous languages.After careful review, we chose four patterns that met the selection criteria. Table <ref> provides a brief mapping between each pattern and the quality attributes they address as well as rationale for satisfying the quality attributes based on existing works. Patterns are described in brief Coplin form <cit.> in the context of test smells/quality goals below. Our detection heuristics are described in section <ref>. Simple Tests:Context:A test with multiple assertions.Problem: Obscure Test(Eager Tests) <cit.> If a test verifies multiple conditions and one fails, any conditions after it will not be executed. After a failure, a test with multiple assertions is more difficult to diagnose, and any checks appearing after the failing assertion would not be executed.Solution: A simple test is a test method which only checks one code path of the subject under test. Meszaros categorizes simple tests as a goal of test automation, which can be realized by following the Verify One Condition Per Test principle.Implicit Teardown: Context: Objects and fixtures are created within a test class or method to simulate the right test environment.Problem: Conditional Test Logic (Complex Teardown) <cit.>.It is difficult to ensure that test fixtures have been removed properly if the teardown code is difficult to comprehend, and leftover fixtures can affect other tests. While most objects can be automatically destroyed in languages with garbage-collection, some objects need to be destroyed manually such as connections or database records.Solution: In this pattern, amethod containing removal code for items that are not automatically removed is run after each test method execution, no matter if it passes or fails. In a class with many tests, this pattern eases maintenance by ensuring all persisting items are removed, and promotes readability by making it easier for developers to understand which test items are being destroyed, and when. Assertion Message: Context: Tests written with xUnit based frameworks verify conditions using Assertion Methods. By default, these methods do not return any descriptive information when they fail.Problem: Assertion Roulette <cit.> In a test with multiple assertions, or when many tests are run simultaneously, this lack of output can make it difficult to diagnose which assertion failed. Solution:This problem can be mitigated by adding an Assertion Message to each assertion method. While this might seem like a trivial pattern, its presence in an assertion shows a conscious choice to improve the readability of the test. This pattern can also be used as an alternative solution to the problem discussed in Simple Test. Testcase Class Per Class (TCCPC) Context: a project with a large number of classes. Problem: Poor Test Organization. Where do I put a new test? There are a number of options, with the worst-case being ALL project test cases in ONE test class. Solution: Developers who take this into consideration early in a project may wish to create one Testcase Class per (Source) Class being tested. This improves readability and maintenance while making organization more straightforward. This pattern can also be applied to a mature project with existing tests, but if it contains a very large test suite, this refactoring can be time consuming. §.§ Testing FrameworksInstead of manually writing and executing complex custom tests, developers can use a unit testing framework, which provides syntax conventions and utility methods. Presence of a test framework is one criteria we use to identify test files, described in section <ref>. Mezaros' <cit.> collection of test patterns are based on the xUnit family of unit test frameworks (JUnit, CUnit, etc.). However, since there are usually multiple test frameworks for each programming language, we look for both xUnit and other frameworks. § METHODOLOGY To answer our motivating question, our methodology consists of three steps: (i) retrieve a large number of open source projects from online repositories, (ii) identify test files across these projects, and (iii) detect testing patterns across test files. We discuss each of these steps in the following subsections.The data used in this study includeopen source projects,unit testing frameworks, and 4 unit testing patterns. A full copy of this data can be found in our project website [All the collected data and the tools developed in this paper can be found at: <https://goo.gl/Mc7tHk>]. To detect test files and pattern use in the projects, we developed automated, heuristic-based techniques. §.§ Gathering Open Source ProjectsOur first step was to extract a large-scale collection of software projects from open source repositories. Our current collection containsprojects extracted from GitHub. To retrieve projects, we used GHTorrent <cit.>, which provides Github's public data and events in the form of MongoDB data dumps containing project metadata such as users, commit messages, programming languages, pull requests, and follower-following relations. We also used Sourcerer <cit.>, which provided versioned source code from multiple releases, documentation (if available), and a coarse-grained structural analysis.After extracting projects from these resources, we performed a data cleanse in which we removed all the empty projects and duplicate forks. Table <ref> shows the number of projects collected per programming language.§.§ Extracting and Analyzing Test Cases Next, we developed a heuristical approach to identify test files among other source files and identify which automated testing frameworks were used to implement the tests. Our automated test case extraction and analysis tool contains the following components: Catalog of Automated Testing Frameworks This catalog contains a list of existing testing frameworks and regular expressions which represent the import statements for each framework. To establish this catalog, we started with a pre-compiled list of unit testing frameworks sorted by language from Wikipedia <cit.>. For each framework in this list we then manually collected additional information such as applicable languages. Most importantly, we searched source code, official documentation, and tutorials to find the framework's API and code annotations in order to determine how each framework is imported into a project. With this information we developed heuristics for determining which (if any) framework a project was using. Frameworks which required payment for use, without source code or sufficient documentation, or for which we could not develop a technique to identify its import statement were then excluded from the list. Our final list containedtesting frameworks fordifferent languages. Test Case Detection and Analysis This component used two heuristical conditions to analyze each project in order to determine they contained test files. Our analysis used a voting mechanism to determine if each condition had been met. The conditions were as follows:For each test framework we used the list of specific import statement(s) to create regular expressions, and pattern matching was used to identify the import statement within the source code. The accuracy of this approach was evaluated using a peer review process including 50 randomly selected projects. With 100% accuracy, this approach identifies various forms of imported testing libraries.We also curated a list of common testing keywords and constructs (such asand ) found in each programming language (Table <ref>.). This list was established by manually analyzing at least 10 sample test cases per language, and collecting framework-specific terms from the documentation of each testing framework.We processed every file in our collection and calculated a binary value for both condition. A file received a vote of 0 for a condition if it did not meet that particular condition, or a 1 if the file did meet the condition. A file was considered a test file if both conditions received a vote of 1.This heuristical approach to test case identification differs from most existing methods in the literature <cit.> such as selecting files which include the word `test' in the file name. We chose not to include this condition to avoid relying on file name conventions because we believe the use of frameworks and test-keywords are stronger indications of test files and will not exclude files that do not follow the `testX' naming convention. We also excluded other conditions such as Fixture Element Types or Static Call Graphs <cit.> in order to develop language-agnostic heuristics which could be applied to all the languages in our dataset. §.§ Detecting Patterns in Test Files To detect the selected patterns within test files we developed another set of heuristical techniques. For each pattern, we were able to identify unique characteristics such as code structure, test method signature, and reserved keywords which indicate the presence of said pattern. The Implicit Teardown and Assertion Message patterns are built-in but optional components of the xUnit framework, and can be best identified in source code by verifying the presence of certain keywords. The heuristical approach used to detect each pattern is described in the following:* Assertion Messages are passed as arguments into anmethod in a test. Common forms for an assertion with a message include: , , and . By developing regular expressions for these three method signatures we can detect if an assert function contains a message. We totaled the number of matches for each signature variation in every file, and considered a test case if there was at least one match. * Simple Test is detected by tracking the number of asserts in a test case. If there is only one assert, we consider it a simple test. We adopted the previous heuristic to detect this pattern. * Implicit Teardown required a semantic heuristic and a relatively more complex search to verify its presence. Our heuristic first finds amethod, a built-in signature in testing frameworks used to implement the teardown pattern. After finding this method signature in a test method, our heuristic searches for `remove' or `destroy' methods to indicate that items are being deleted. In our heuristic, the method name `teardown' must be found, along with at least least one of the two other methods.* Testcase Class Per Class is identified by first creating a call graph between all classes in the project, including test classes. This call graph is used to check if there is one test class which depends on a source class in production code. We used an existing commercial tool <cit.> to generate the call graph. This tool can only perform static analysis on a limited number of languages. Therefore, we were only able to detect this pattern in 11 of thelanguages present in our dataset: C, C++, C#, FORTRAN, Java, Pascal, VHDL, HTML, CSS, PHP, and Javascript. §.§ Validation of Data Extraction TechniquesTo validate the correctness of the data collected, we performed the following checks: §.§.§ Test FilesTo verify the correctness of the test files, we randomly selected 20 projects and manually identified any test files and any test framework the projects imported. We then ran our test case analysis (conditions 1 & 2) on these files and verified that the results matched. Our approach achieved an accuracy of 100%. §.§.§ Pattern FilesTo verify the correctness of the pattern files, we randomly selected and manually verified files which were identified by our heuristic to contain a pattern, 15 per pattern. Our pattern detection approach had 100% accuracy. § RESULTSThis section presents the results for the main motivating question and the three consequent research questions.§.§ The State of Software Testing in Open Source The following data provides general insights into the state of software testing in open source, specifically project size, test-to-code ratios, and framework usage across the projects studied. This provides better context and background for the interpretation of our research question results.Distribution of Project Sizes: As mentioned in section <ref>, we grouped theprojects in our collection into four size categories (very small, small, medium and large) to better understand how testing varies with project size. After grouping the projects, we found that our collection was 46.55% very small projects,30.82% small projects, 16.16% medium projects and 6.45% large projects.Test Code to Production Code Ratio: We observed that(17.17%) of projects included test files. We computed the ratio of lines of code from test cases to total number of line in the production code (TPRatio <ref>) and report this metric for each project size category in Table <ref>. The average TPRatio of all the projects in our collection was 0.06. We also note that small and very small projects included fewer test files. Out of 38,379 very small projects and25,425 small projects, only 3,207 (8.4%) and 4,203 (16.5%) of projects had at least one test file, respectively. Medium and large projects in our dataset contained more tests (31% and 67.2%, respectively).Testing In Open Source: 17.17% of projects studied contained test files. The average ratio of test code to production code was very low, 0.06. Smaller projects had a lower ratio than larger projects. One hypothesis for this result is that medium and large projects are more complex and, therefore, introduce automated testing practices more to ensure code correctness. Test Framework Adoption in the Open Source Community: Of theunit testing frameworks we searched for in 48 programming languages, we detected usage offrameworks in 9 languages (C, C++, C#, Java, JavaScript, Perl, PHP, Python and Ruby). Table <ref> presents the three most-used frameworks per language. In the case of the Perl language, we only detected the usage of two testing frameworks. Relying only on the total number of projects that adopted a framework to identify the most commonly used ones would result in a bias. As presented in Table <ref>, the distribution of the programming languages over the projects in our repository is not uniform. Therefore, the frameworks that are used in the most frequent languages in our repository would more likely be placed at the top of the ranking. To overcome this, we normalized the frequency of the frameworks by dividing the framework frequency by the total number of projects in the framework's programming language(s). Table <ref> enumerates the ten most used frameworks sorted by this normalization value. From this table we observe that , a testing framework for JavaScript projects, is the most used framework.Test Framework Usage:From 251 testing frameworks included in our study, only 93 were imported by developers in open source projects. Mocha (Javascript) was the most commonly used framework.§.§ Satisfaction of Quality Attributes through Pattern AdoptionWe identifiedprojects that contain at least one pattern (23.76%) andprojects which contained all four patterns (0.08%).Table <ref> shows the number of test files and projects which implement each pattern. Assertion Message was the most commonly adopted pattern, occurring in 49,552 test files among the 399,940 test files found in projects in our collection. The other patterns were not as widely adopted. Testcase Class Per Class was adopted in 4,565 test files, Simple Test was only applied in 1,614 test files, and Implicit Teardown was used in 920 test files. Therefore, few test cases applied patterns which can impact maintainability attributes. All further analysis focuses on projects with tests containing patterns. Pattern Adoption Ratio: In order to better understand how often the patterns were adopted in projects, we computed the adoption ratio of the patterns as the number of test case files with the pattern divided by the total number of test files in the project. This ratio was computed per pattern for the projects that adopted each pattern.The adoption ratios of patterns are presented in Figure <ref>. From this figure, we find that both Assertion Message and TCCPC had the highest adoption rates, which means that these patterns were consistently applied throughout the projects' test cases. In fact, the median adoption rates for Assertion Message and TCCPC were 41.2% and 33.3%, respectively.Conversely, the adoption ratios for Simple Test and Implicit Teardown were both 5%. Pattern Adoption And Number of Test Cases: In addition to the analysis of the raw results, we wanted to examine if patterns are typically found in projects containing a lot of test cases or not. Therefore, we built a linear regression model to examine the relationship between the adoption ratio of each pattern (independent variables) to the number of test cases (dependent variable) in the corresponding project. However, since the number of test cases could just be a function of the size of a project, we include the size of the project (in KLOC) as a control variable in the regression model.From Table <ref>, we find that as expected there is a strong positive relationship between size of the project and the number of test cases. However, surprisingly, we notice that the relationship between adoption ratio and the number of test cases is negative. This implies that smaller projects are more disciplined in using test case patterns.Relationship Between Use of a Framework and PatternsAdoption: In order to study the relationship between the choice of framework and the adoption ratio we built a random forest model. The dependent variable was the type of framework used and the independent variables were the the adoption ratios for each pattern and the total number of test files (which is a control variable that might be correlated with the choice of framework). We did not use a linear regression model here since choice of framework is a categorical variable and not a linear variable like the number of test files. Once, we built the model, we determined the importance of each independent variable. The mean “decrease in Gini coefficient” values indicate the importance of the variable in helping classify the framework. Hence, the higher the values indicates a stronger relationship between the variable and the choice of framework. We see that while the total number of test cases is an important variable (2nd most important), the most important variable is the adoption ratio for the assert pattern. This implies that there is a strong relationship between the choice of a framework and the adoption ratio for the assert pattern. Some frameworks could have very high assert pattern adoption ratios and some other frameworks have very little. For instance Danielle mentions here in which frameworks the developers used Assert more oftenTCCPC pattern was the third most important variable, while Simple Test and Automated Teardown were fourth and fifth most important variables.Pattern Adoption: Open source projects do not frequently adopt the studied testing patterns. Smaller projects have fewer test files and test-to-production code ratios, but a higher frequency of pattern adoption compared to larger projects. To answer our three research questions, we used a QA-Pattern mapping (See Table <ref>) to relate the patterns we chose to study with the quality attributes they best represent. From theprojects which included patterns, for each quality attribute we calculated the percentage of projects that implemented each mapped pattern as well as all mapped patterns. This data is shown in Table <ref>. Note that percentages are based only on the total number of projects with patterns. §.§ RQ1: What percentage of projects with patterns addressed the Ease of Modification Quality Attribute? The testing patterns which can satisfy the Ease of Modification quality attribute for tests are Implicit Teardown and Testcase Class Per Class. 143 (4.32%) projects applied Implicit Teardown, improving modifiability of internal test contents. 815 (23.02%) projects implemented Testcase Class Per Class (TCCPC), improving the modifiability of test structure in relation to production code. 41 (1.16%) of projects implemented both patterns, addressing both internal and structural test modifiability. Ease of Modification: Approximately one-fourth of the projects with patterns implemented the Testcase Class per Class pattern, providing a maintainable design for test classes and their associated production classes. Fewer projects adopted Implicit Teardown, which helps structure the destruction of test fixtures.§.§ RQ2: What percentage of projects with patterns addressed the Ease of Diagnosis Quality Attribute?To satisfy Ease of Diagnoses, we looked for adoption of Simple Test and Assertion Message patterns. Both patterns improve defect traceability and test readability. From all projects that implement patterns, 576 projects (16.27%) implemented Simple Test, and 3,048 projects (86.02%) implemented Assertion Message. 473 (13.36%) projects implemented both, strongly improving defect traceability and test readability. Although these two patterns are intuitive to enforce, a large number of projects have not implemented them.Ease of Diagnoses: Despite benefits in diagnoses of the Simple Test and Assertion Message patterns, less than one fourth of projects adopted both or Simple Test without Assertion Message. However, over three fourths of projects used Assertion Message without Simple Test. §.§ RQ3: What percentage of projects with patterns addressed Ease of Comprehension Quality Attribute?Comprehension improvement involves a combination of test modifiability and diagnosis improvements. The testing patterns which can satisfy Ease of Comprehension in our study were Implicit Teardown, Assertion Message, and Simple Test.As individual usage of these patterns has been reported above, we will now discuss combinations of these patterns. 422 (11.92%) projects implemented Assertion Message and Simple Test, 170 (4.80%) projects implemented Assertion Message and Implicit Teardown, and 1 (0.03%) project implemented Simple Test and Implicit Teardown. 51 (1.44%) implemented all three patterns.Ease of Comprehension: Comprehension can be addressed with three of the patterns studied, but less than 2% of projects with patterns implemented all three. The most implemented combination was Assertion Message and Simple Test (11.92%), which are more intuitive to implement than Implicit Teardown. From the results reported for our three RQs, we can see that the quality attribute addressed in the highest number of projects is Ease of Diagnoses. This indicates that developers most often applied the patterns which could help them identify why a test has failed. Relatively, Ease of Modification and Ease of Comprehension were not satisfied across many open source projects.To further investigate these findings, we report three qualitative analyses on projects with different pattern usage in the following section. § QUALITATIVE ANALYSES In addition to the quantitative results reported in the previous section, we have conducted three qualitative studies to understand differences between projects with different levels of pattern adoption. First we investigated whether there are any additional project characteristics which influenced pattern adoption. Next, we studied the contributions of developers to the test cases with pattern within projects which applied all four patterns. Finally, we examined test files from projects with no patterns to determine if they used other approaches to address test maintainability. §.§ Analysis #1: Pattern Adoption and Project CharacteristicsTo identify potential correlations between test pattern adoption and other project characteristics, we collected project artifacts such as source code, online repositories, and documentation for 29 randomly selected projects from 3 categories: 9 projects that applied all 4 patterns, 10 projects that applied 1, 2, or 3 (Some) patterns, and 10 that applied 0 (No) patterns. We looked at test documentation, organization, `coverage', industry sponsorship, and number of contributors per file. Project Selection: All projects in our dataset were sorted by randomly generated numbers. 10 projects from each category (All, Some, and No patterns) were then randomly chosen from the sorted list. To ensure that projects were suitable for our study, we developed exclusion criteria. Selected projects were verified that: (1) there was at least one online artifact available such as a Github page, (2) contributor information was available, and (3) the project contained original source code. For projects with Some patterns, we also tried to choose projects with different combinations of patterns. If a project failed to meet these criteria, it was excluded from the study and a new project was randomly selected using the process above. Of the 11 total projects in our dataset which applied all 4 patterns, 2 had to be excluded, so we were only able to study 9.Analysis Results: Project Demographics: For all 29 projects selected, the number of contributors ranges from 2 to 620, and the number of forks ranges from 0 to 3,108. 16 unit testing frameworks were used, and projects were written in Java, Javascript, C#, C, C++, Python, Scala, and PHP. Test Documentation & Rules: Our first question was whether the presence of test-specific documentation affected pattern adoption. Therefore, we manually searched available artifacts to see if projects had any guidelines (documentation) or templates specifically for unit testing code. We also wanted to know if projects had rules for new contributions, specifically that existing tests must pass, and new code requires new unit tests.Projects with All patterns required the developers to add new tests alongside source code contributions more frequently than the projects with Some or No patterns. Projects with Some patterns most often required that existing tests run and pass (6) and also included more guidelines for writing tests (4). We found instances of unit test template code in 2 projects, one with All patterns and the other with No patterns.Test Organization & `Coverage': We examined if projects with well-organized tests were more likely to adopt patterns. To measure how well tests were organized in a project, one author searched its directories to find where all test files were located. Project test organization was ranked Great, Good, OK, or Poor based on how easily all tests in the project could be located and if they were organized by component. Due to the range of languages used, we were unable to quantitatively measure coverage, so we considered `coverage' in terms of how many project components had any tests. For test organization, only 1 project (All-pattern) received a Great rating, but projects with Some and No patterns had equal amounts of Great(0), Good(7), and Poor(1) ratings. Surprisingly, projects with All patterns had the highest amount of Poorly organized projects (2). For `coverage', more projects with Some patterns had tests for all components (7), and projects with No tested more than half of components (4).Industry Contribution: Company sponsorship of a project is another characteristic which could potentially affect pattern adoption. We also noted if the `core contributors' of a project (identified through acknowledgments and Github metrics) worked for the sponsoring company.Of the 12 projects with industry sponsors, 4 applied All patterns, 2 applied Some patterns, and 6 applied No patterns. 8 projects had rules for unit testing (2 All, 2 Some, and 4 No) and 6 included test specific guidelines or templates (1 All, 1 Some, and 4 No). 2 projects had Poor test organization (1 All, 1 No), 9 had Great or Good organization (4 All, 1 Some, 4 No), and 1 (No patterns) had OK organization. The majority of industry projects had high `coverage', with only 1 (Some) project testing less than half of its components.# Contributors Per Test File: Finally, we were interested in a possible correlation between pattern adoption and the number of contributors who work on an individual test file. Across all projects, the number of contributors to a single test file ranged from 1 to 10. Projects with All or Some pattern usage did not have higher contributors per test file than projects with no pattern usage. The average for all projects was 1 to 2 contributors per file.§.§ Analysis Results Project Demographics: For all 29 projects selected, the number of contributors ranges from 2 to 620, and the number of forks ranges from 0 to 3,108. 12 projects were sponsored by companies, but all accepted outside contributions. Only 1 of the 29 projects did not accept contributions from outside the founding group. 16 unit testing frameworks were used, and projects were written in Java, Javascript, C#, C, C++, Python, Scala, and PHP.Test Documentation & Rules: Table <ref> shows the percentage of projects in each category that had rules, guidelines (documentation), or templates specifically for unit testing. Interestingly, no test-specific artifacts mentioned any of the patterns studied.Test Organization & `Coverage': Table <ref> shows the percentage of projects per category which received each possible rating for test organization:`Coverage', measured as how many components have any tests, for 2 projects (1 All, 1 No patterns) could not be precisely determined due to the Poor organization of test files in those projects. Table <ref> shows the percentage of remaining projects with each `coverage' rating:Industry Contribution: Of the 12 projects with industry sponsors, 4 applied All patterns, 2 applied Some patterns, and 6 applied No patterns. 8 projects had rules for unit testing (2 All, 2 Some, and 4 No) and 6 included test specific guidelines or templates (1 All, 1 Some, and 4 No). 2 projects had Poor test organization (1 All, 1 No), 9 had Great or Good organization (4 All, 1 Some, 4 No), and 1 (No patterns) had OK organization. The majority of industry projects had high `coverage', with only 1 (Some) project testing less than half of its components. # Contributors Per Test File: Across all projects, the number of contributors to a single test file ranged from 1 to 10. Projects with All or Some pattern usage did not have higher contributors per test file than projects with no pattern usage. The average for all projects was 1 to 2 contributors per file. Analysis Conclusions: The only characteristic found most frequently in projects with All patterns was requiring new tests to be added to contributions, and one instance of Great test organization. Projects with Some and No patterns were more similar than projects with All and Some patterns in their inclusion of these characteristics as well. Interestingly, the differences in occurrences of these characteristics across all project categories was very small. It does appear, however, that projects with industry sponsors often addressed test organization, testing rules & guidelines, and coverage. However, only 6 of these projects implemented All or Some patterns. Because we were unable to identify a strong influence on pattern adoption in this study, we performed another qualitative analysis on the contributors to projects using All patterns.§.§ Analysis #2: Contributors to Projects Using All PatternsNext, we investigated the influence of developers on the adoption of testing patterns. For this analysis, we identified the developers that contributed to test files in the 9 projects with All patterns. To do this, we retrieved the commit logs of these projects and identified the developers that committed code to each test file with a pattern and computed the total number of testing files with patterns that each developer contributed to. Analysis Results:By observing the total number of test pattern files each developer contributed to, we found that a fewer number of developers were contributing to most of the testing files with patterns.Figure shows the proportion of contributions to testing files with patterns made by each developer in the projects. In this figure, each color represents one developer that contributed to a testing pattern within the project. From this, we observe that in almost all projects between 2 to 3 developers are contributing the most to test files with patterns. For KbEngine, there was only one developer that was contributing to testing patterns, who is also the creator and maintainer of the project. For Ember Cli, only 9 of 100 total developers who wrote test cases were contributing most of the test files.Analysis Conclusions: We observed that fewer developers were contributing to the majority of test files with patterns. This suggests that testing patterns adoption in a project is an ad-hoc personal decision made by individual developers rather than a project-wide effort to improve test quality. §.§ Analysis #3: Tests in Projects Using No PatternsOur final analysis was an investigation into test cases from projects which did not use patterns. Since these files contain no patterns, our goal was to see what, if any, other techniques were used to satisfy ease of modifiability, diagnosis, and comprehension. 5 test files from each of the 10 projects without patterns used in the first qualitative study were randomly selected (50 total files) containing a total of 219 test methods. Our search was focused on solutions which implemented test failure feedback (Assertion Message) for ease of diagnoses and fixture destruction (Implicit Teardown) for ease of modifiability, but we also looked for characteristics which would improve comprehension of the tests such as naming conventions and the presence of comments.Analysis Results:The first general observation from this study was that within a project, files were usually consistent in their use of naming conventions and comment usage, which helped with comprehension of the test cases. This was not the case in only 3 projects.Only 34% of files used comments to describe the behavior or purpose of a unit test. 54% of projects used clear, natural language naming conventions for file, method and variable names. 38% of files (written in Javascript) used an alternative naming convention called , a special method where the functionality of the test is passed as a parameter. The remaining 8% of files used sequential numbering. As an alternative to Implicit Teardown, 74% of test files relied solely on garbage collection and the remaining 26% of projects used custom methods (such as ). 82% of the files used an alternative to Assertion Message: Most Javascript files studied used an alternative assertion library called `Should', which uses anmethod syntax rather than an assertion (38%). On failure, this message is displayed. 26% of files used exception throwing for errors, 6% used try/catch blocks, 4% used custom logging objects, and the remaining 26% used no failure feedback at all.Analysis Conclusions: From these observations it is clear that the projects with no patterns use other techniques besides the XUnit patterns to address ease of maintainability and diagnoses, such as readable naming conventions and alternative failure feedback techniques. However, the data also shows that these projects seldom use comments to describe test behavior, which would improve comprehension. A future study to investigate these alternative techniques may reveal interesting results.Qualitative Conclusions: Pattern adoption is independent of project characteristics, and it is dependent on individual developers. Project without patterns instead used existing libraries and proprietary and primitive techniques to satisfy quality attributes.§ THREATS TO VALIDITY The following threats to the validity are identified. §.§ Construct ValidityWe assumed that all test cases written for a project were included in the project's source code repository. Thus, if zero test files were detected, we say the project does not contain tests. Our approach detected the presence of test files in the project but not how or if they were run, so we do not know if the tests are still relevant. However, this is of low concern because we are more interested in the quality of the tests present in the project.§.§ Internal ValidityWhen mining software repositories, it is important to consider that some repositories may not contain projects, or may be a `fork' of another project. To prevent repeated data in our results, we removed all empty and duplicate project forks. For duplicates, we removed all but the original project or the oldest fork if we did not have the original project in our dataset. §.§ External ValidityTheprojects studied do not provide a complete perspective of the entire open source community. Projects may have custom tests, or use a testing framework which which was not included in this study. However, the large volume of projects and frameworks included allow new insights into the current state and quality of open source testing. Also, while the use of only four testing patterns was investigated, we believe their use is a good indication that developers took quality into consideration when designing their tests. We do not claim that a project without the patterns studied does not address quality as it is possible that other patterns or techniques were applied. §.§ Reliability ValidityWe used heuristical methods (defined in section <ref>) to detect test files and patterns. This approach is similar to those used in other studies (see section <ref>), but there are related risks. It is possible that some test frameworks were not included in our detection process because there are multiple ways to import a framework into a project. To reduce this risk, we maually searched all available documentation for each framework to find import statements. We also used a list of reserved testing keywords in our detection tool. Further, while we searched for all possible ways of implementing the patterns within the frameworks considered, it is possible that we missed instances of pattern use if the implementation did not match our heuristics. Finally, the static analysis tool <cit.> used to generate the file dependency data for detecting the Testcase Class Per Class pattern does not recognize all of the languages in our dataset. Therefore, our results for this pattern only included the projects which we were compatible with Understand. In the future we would like to detect use of this pattern in all the languages in our dataset.§ CONCLUSIONIn this paper we have performed a large-scale empirical analysis on unit tests inopen source software projects written inlanguages.We found(17%) projects containing test files written in 9 languages usingtest frameworks.(24%) projects applied at least one of the 4 patterns, andprojects applied all 4 patterns.Ease of Diagnoses was the most commonly addressed quality attribute, and Assertion Message was the most adopted pattern. We also found that although smaller projects contain fewer test files than larger projects, they apply patterns more frequently.Through three qualitative analyses of the projects with and without patterns, we found that pattern adoption is an ad-hoc decision by individual project contributors.In summary we find that open source projects often do not adopt testing patterns that can help with maintainability attributes. More research is needed to understand why and how we can help developers write better test cases that can be maintained and evolved easily.§ ACKNOWLEDGMENTSThis work was partially funded by the US National Science Foundation under grant numbers CCF-1543176. abbrv | http://arxiv.org/abs/1704.08412v1 | {
"authors": [
"Danielle Gonzalez",
"Joanna C. S. Santos",
"Andrew Popovich",
"Mehdi Mirakhorli",
"Mei Nagappan"
],
"categories": [
"cs.SE"
],
"primary_category": "cs.SE",
"published": "20170427022226",
"title": "A Large-Scale Study on the Usage of Testing Patterns that Address Maintainability Attributes (Patterns for Ease of Modification, Diagnoses, and Comprehension)"
} |
http://arxiv.org/abs/1704.08925v2 | {
"authors": [
"Georgios Kofinas",
"Nelson A. Lima"
],
"categories": [
"gr-qc",
"astro-ph.CO",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170427125946",
"title": "Dynamics of cosmological perturbations in modified Brans-Dicke cosmology with matter-scalar field interaction"
} |
|
An Open-Source Framework for N-Electron Dynamics: II. Hybrid Density Functional Theory/Configuration Interaction Methodology Gunter Hermann corresponding author Institut für Chemieund Biochemie, Freie Universität Berlin, Takustraße 3, 14195 Berlin, GermanyThese authors contributed equally to this work., Vincent Pohl[2] [3],and Jean Christophe Tremblay[2] December 30, 2023 =================================================================================================================================================================================================================================================§ ABSTRACT In this contribution, we extend our framework for analyzing and visualizingcorrelated many-electron dynamics to non-variational, highly scalable electronic structure method. Specifically, an explicitly time-dependent electronic wave packet is written as a linear combination ofN-electron wave functions at the configuration interaction singles (CIS) level, which are obtained from a reference time-dependent density functional theory (TDDFT) calculation. The procedure is implemented in the open-source Python program detCI@ORBKIT, which extends the capabilities of our recently published post-processing toolbox [J. Comput. Chem. 37 (2016) 1511]. From the output of standard quantum chemistry packages using atom-centered Gaussian-type basis functions,the framework exploits the multi-determinental structure of the hybrid TDDFT/CIS wave packetto compute fundamental one-electron quantities such asdifference electronic densities, transient electronic flux densities, and transition dipole moments. The hybrid scheme is benchmarked against wave function data for the laser-driven state selective excitation in LiH. It is shown that all features of the electron dynamics are in good quantitative agreement with the higher-level method provided a judicious choice of functional is made. Broadband excitation of a medium-sized organic chromophore further demonstrates the scalability of the method. In addition, the time-dependent flux densities unravel the mechanistic details of the simulated charge migration process at a glance.§ INTRODUCTION Unraveling the flow of electrons inside a molecule out of equilibrium is key to understand its reactivity. Since the pioneering laser experiments by Zewail and co-workers<cit.>, the development of new light sourceshas now granted access to the indirect observation of electron dynamics on its natural timescale. To shed light on the mechanistic details of this attosecond dynamics, accurate theoretical methods are required that capture the subtle details of the transient electronic structure evolution. Various approaches based on explicitlytime-dependent density functional theory (TDDFT) and wave function ansatz have been developed over the years and enjoyed mixed degrees of success. While TDDFT appears as more intuitive and scalable, it was shown to suffer from problems for ultrafast dynamics instrong laser fields. On the other hand, the advantages of wave function-based methods in terms of convergence become rapidly compensated by their unfavorable computational cost. Further, the intuitive picture of electrons flowing on a molecular skeleton can become blurred by correlation effects between the N particles.This contribution is motivated by the need for a robust, scalable wave function method to investigate ultrafast N-electron dynamics in systems of large dimension. The method we advocate is based on a combination of linear-response time-dependent functional theory (LR-TDDFT) and configuration interaction singles (CIS) methodologies, as was introduced recently <cit.>. In principle, the method is similar to the well-established CIS ansatz, with the exception that the energies and the pseudo-CIS eigenvectors are obtained from a reference LR-TDDFT calculation. This allows to improve the energetic properties of the states while keeping a simple electron-particle picture to describe the transient N-electron wave packet. This TDDFT/CIS hybrid formalism inherits the qualities of both underlying methods and ensures the N-representability of all reduced density matrices, at all times and under all laser conditions. The ensuing N-electron dynamics remains marred by the non-intuitive interpretation of quantities beyond the density itself.Recently, we demonstrated that correlated electron dynamics can be accurately described by means of the electronic flux density operator and derived one-electron properties. We introduced an open-source framework <cit.> to post-process multi-determinantal configuration interaction wave functions directly from the output of standard quantum chemistry packages. It thus becomes possible to reconstruct the transient N-representative one-electron density and current density (flux density) using a library of transition moments calculated from the multi-determinantal configuration interaction wave functions, yielding an intuitive tool for visualizing and analyzing the correlated electron dynamics. A wide variety of established wave function-based methods are covered, ranging from configuration interaction singles to Full CI viarestricted active space CI and multi-configuration self-consistent-field methods. It is the purpose of this work to extend the formalism to the TDDFT/CIS hybrid formalism mentioned above,which should retain the qualities of the wave function ansatz and the scalability of DFT-based schemes with respect to the system size.In the next section, the hybrid TDDFT/CIS methodology is first introduced, followed by the description of the analysis toolset based on the flux density. The application section reports on benchmark calculations on the LiH molecule and the demonstration of the scalability of the scheme by investigation of the broadband excitation in an organic chromophore. The findings are summarizedin the conclusion section. Unless otherwise stated, atomic units are used throughout the manuscript (ħ = m_e = e = 4πε_0 = 1).§ THEORY§.§ Hybrid TDDFT/CI Methodology The evolution of the electronic state of a molecular system obeys the time-dependent Schrödinger equation<cit.>, which can be written in the clamped nuclei approximationi∂/∂ tΨ_ el( t) = (Ĥ - μ̂·F⃗(t)) Ψ_ el( t).The interaction of the molecular dipole μ̂ with an external laser field F(t) is treated here semi-classically. For a system consisting of N electrons and N_A nuclei, the field-free electronic Hamiltonian readsĤ = - 1/2∑_i=1^N∇_i^2 + ∑_i=1^N∑_j>i^N1/r_ij - ∑_i=1^N∑_A=1^N_AZ_A/r_Ai,where 1/r_ij=1/|r⃗_i - r⃗_j| is inter-electronic Coulomb repulsion, and r_Ai is the distance between the ith electron and nucleus A of charge Z_A. In this work, an electronic wave packet Ψ_ el( t)satisfying Eq. (<ref>) is expressed as a linear superposition of stationary electronic states Φ_λΨ_ el(t) = ∑_λ B_λ( t) Φ_λ .Here, B_λ( t) are the expansion coefficients of state λ, which describe the time-evolution of the wave packet. For molecules in strong laser fields, a large number of stationary electronic states is required to offer a proper description of the N-electron dynamics. The equations of motion for the coefficients in Eq. (<ref>), associated with the basis set expansion Eq. (<ref>), can be integrated numerically.In the time-dependent configuration interaction methodology, the stationary electronic states are chosen aslinear combinations of excited configuration state functionsΦ_λ^ CI = C_ref^(λ)ϕ_ref+∑_arC_a^r(λ)ϕ_a^r +∑_abrsC_ab^rs(λ)ϕ_ab^rs + … .The expansion parameters C^(λ) are associated with the formal excitationof a reference configuration, ϕ_ref, from occupied orbitals { a,b,c }to virtual orbitals { r,s,t }. Including all possible excitations leads to the exact Full CI limit. The reference and excited configurations are defined as Slater determinants, which builds antisymmetrized products of one-electron spin orbitals φ_a. Note that, in the time-dependent configuration interaction (TDCI) methodology in the form presented above, the field-free electronic Hamiltonian is considered to be diagonal in the basis of CI eigenstates at a given level of theory. The matrix elements of the dipole operator can be computed from the knowledge of these eigenfunctions, which serve as a basis for the variational representation of the molecule-field interaction.For large molecules, it is customary to truncate the CI expansion to a chosen maximum rank of excitations (e.g., CI Singles or CI Singles Doubles) in order to reduce the number of possible excited configurations. Unfortunately, this often compromises the energetic description of the excitedstates. To circumvent this limitation while keeping the problem computationally tractable, Sonk and Schlegel <cit.> first recognized that only excitation energies and transition dipole moments are required to perform TDCI simulations. These can be obtained from linear-responsetime-dependent density functional theory (LR-TDDFT). To generalize this approach, it was proposed to use the solutions of the LR-TDDFT calculation to generate a basis of pseudo-CI eigenstates <cit.>.All required information for a TDCI simulation is thus availablefrom the output of standard quantum chemistry programs, provided excitations are performed from the ground state.According to the Runge-Gross theorem, it is possible to recast the N-electron Schrödinger equation and calculate all observables from the sole knowledge of the one-electron density. Using the Kohn-Sham ansatz for the density, the N-electron time-dependentSchrödinger equation can be mapped onto a one-electron equation for the orbitalsi∂φ_a(𝐫,t)/dt = (-∇^2/2 + v_KS(𝐫,t))φ_a(𝐫,t).The time-dependent Kohn-Sham potential v_KS(𝐫,𝐭) contains the classical electrostatic interaction (v_Hartree(𝐫,t)), an external potential (v_ext(𝐫,t)), and an exchange-correlation contribution (v_xc(𝐫,t)), i.e.,v_KS(𝐫,𝐭)=v_Hartree(𝐫,t)+v_ext(𝐫,t)+v_xc(𝐫,t).In explicit TDDFT, the Kohn-Sham potential is usually assumed to be local in time. A celebrated success of TDDFT comes from its linear-response formulation, which allows to accurately compute spectral properties of large molecules. For this endeavor, the response kernel of the electron density to an external weak, long wavelength perturbation can be evaluated from the electric susceptibility of the ground state. The search for the poles of the response function can be recast as an eigenvalue problem of the form[([ 𝐀 𝐁; 𝐁^† 𝐀^† ])-ω([ -𝐈0;0𝐈 ])] ([ 𝐗; 𝐘 ])=-([ δ𝐯; δ𝐯^† ]),where δ𝐯 is the response of the system state to the perturbation, δ v_ext(𝐫,t). The elements of matrices 𝐀 and 𝐁 are obtained from the orbital energies and integrals over theexchange-correlation kernel, see Eq. (<ref>). At the resonance frequencies ω, where the response vanishes (δ𝐯=0), the solution of the Casida Eq. (<ref>) yields simultaneously the excitations andde-excitations amplitudes, 𝐗 and 𝐘.In the present work, we make use of the fact that these are usually given in the output of standard quantum chemistry programs, together with the excitation energies and the oscillator strengths.From Eq. (<ref>), it is possible to define pseudo-CI Singles eigenvectors in the Tamm-Dancoff approximation, which consistsin neglecting the off-diagonal blocks 𝐁. This procedure can alter the quality of the energetic properties of the excited states. On the other hand, the dominant characters present in the pseudo-CI eigenvectors are often not strongly affected by this approximation. In the TDDFT/CI procedure, we thus advocate using directly the transition energies and amplitudes obtained from a LR-TDDFT calculation to take advantage of the full solution of Eq. (<ref>) and to obtain a good energetic description of the excited states. A separate Tamm-Dancoff calculation may be used to confirm the character of the excited states. The LR-TDDFT excitation amplitudes are then re-orthonormalized using a modified Gram-Schmidt procedure to define a pseudo-CI basis for the TDCIS dynamics. All properties not directly deriving from the energies can be subsequently calculated at the CIS level of theory using theorbitals and the pseudo-CI eigenstates, which are treated as configuration interaction singles expansions. Note that the Slater determinants {ϕ_ref, ϕ_a^r} are constructed from Kohn-Sham orbitals. Importantly, all the information required to reconstruct these KS-orbitals and the N-electron pseudo-eigenfunctions are directly accessible from the output of standard quantum chemistry packages. As a consequence, only the evaluation of one-electron integrals is required to generate a library of molecular properties andtransition moments of various one-electron operators, which can be used to characterize the properties of transient wave packets, as explained below.§.§ Analysis Tools for Electron Dynamics For the analysis of the N-electron dynamics, we propose using a set of tools composed from the one-electron density, ρ(𝐫,t ), and the associated electronic flux density, 𝐣(𝐫,t). These are related by the electronic continuity equation∂/∂ tρ(𝐫,t ) = - ∇⃗·𝐣(𝐫,t).Whereas the electron density gives information about the probability distribution of the electron, the flux density yields complementary information about the phase of the electronic wave packet. This in turn reveals the mechanistic aspects of the time-evolution of the one-electron density. The one-electron density can be used to define the electron flow, ∂/∂ tρ(𝐫,t ), as the left-hand-side of the continuity equation. The difference density, 𝐲(𝐫,t ), is a widespread quantity used for visualization purposes, and it can be obtained by integrating the electron flow from a chosen initial condition ρ(𝐫,0 ), i.e.,𝐲(𝐫,t )= ∫_0^t d t' ∂ρ(𝐫,t' )/∂ t' = ρ(𝐫,t ) - ρ(𝐫,0 ).We will resort to both quantities in later analyses.In operator form, the one-electron density and the electronic flux density respectively readρ̂( 𝐫)= ∑_k^Nδ( 𝐫 - 𝐫_k)= ∑_k^Nδ_k(𝐫) ĵ( 𝐫)= 1/2∑_k^N(δ_k(𝐫)p̂_k + p̂^†_kδ_k(𝐫)),where 𝐫 is an observation point, δ( 𝐫 - 𝐫_k)= δ_k(𝐫) is the Dirac delta distribution at the position 𝐫_k of electron k, and p̂_k=-i∇⃗_k is the associated momentum operator. In general, the expectation value of any one-electron operator F̂can be expressed using Eqs. (<ref>) and (<ref>) as F̂(t) =Ψ_ el(t) | F̂| Ψ_ el(t) =∑_λν B_λ^†( t) B_ν( t) Φ_λ| F̂| Φ_ν.Evaluation of the matrix elements Φ_λ| F̂| Φ_ν can be done by exploiting the structureof the functions {Φ_λ,Φ_ν}. In the hybrid TDDFT/CIS methodology, these take the form of singly excitedconfigurations, i.e., the truncation of Eq. (<ref>) at the singles level. The matrix elements in the basisof singly excited configurations readΦ_λ| F̂| Φ_ν=C_ref^(λ)C_ref^(ν)ϕ_ref| F̂| ϕ_ref + ∑_bsC_ref^(λ) C_b^s(ν)ϕ_ref| F̂| ϕ_b^s+ ∑_arC_a^r(λ)C_ref^(ν)ϕ_a^r| F̂| ϕ_ref + ∑_abrsC_a^r(λ) C_b^s(ν)ϕ_a^r| F̂| ϕ_b^s. = C_ref^(λ)C_ref^(ν)∑_aφ_a| F̂| φ_a+ ∑_arC_a^r(λ)C_a^r(ν)∑_aφ_a| F̂| φ_a+ ∑_bsC_ref^(λ) C_b^s(ν)φ_b| F̂| φ_s+ ∑_arC_a^r(λ)C_ref^(ν)φ_a| F̂| φ_r + ∑_ar sC_a^r(λ) C_a^s(ν)φ_r| F̂| φ_s + ∑_a brC_a^r(λ) C_b^r(ν)φ_a| F̂| φ_b.where a∈{1,2,…,a-1,r,a+1,…} denote the occupied spin orbitals of the configuration state function ϕ_a^r. Note that we make use of the Slater-Condon rules<cit.> to resolve Eq. (<ref>) in terms of one-electron integrals in the basis of the spin orbitals, φ_a(𝐫).The transition moments between spin orbitals are usually computed in the spin-free representation by first integrating over the spin coordinates. Specifically, the expectation value for the electron density requires the following integralsφ_a| ρ̂| φ_b = ρ_ab(𝐫)=φ_a(𝐫)φ_b(𝐫).The electronic flux density for a wave packet of the form Eq. (<ref>) can be formulated as𝐣(𝐫,t) = 2 i∑_λ<ν Im[B^†_λ(t)B_ν(t)] 𝐉_λν(𝐫,t),which can be calculated by exploiting the CIS structure of the eigenfunctions. The transition electronic flux density from state λ to state ν is denoted 𝐉_λν(𝐫,t), which simplifies using the Slater-Condon rules to 𝐉_λν(𝐫,t)= ∑_ar(C_ref^(λ) C_a^r(ν) + C_a^r(λ)C_ref^(ν))𝐣_ar + ∑_a,r sC_a^r(λ) C_a^s(ν)𝐣_rs+ ∑_a brC_a^r(λ) C_b^r(ν)𝐣_abwhere 𝐣_ab = -i/2(φ_a(𝐫)∇⃗φ_b(𝐫) - φ_b(𝐫)∇⃗φ_a(𝐫))are molecular orbital (MO) electronic transition flux densities from MO φ_a(𝐫) to MO φ_b(𝐫).As one of the most widespread bases used in quantum chemistry, we specialize here to spatial MO definedas linear combination of atom-centeredorbitals (MO-LCAO for “Molecular Orbital - Linear Combination of Atomic Orbitals”)φ_a(𝐫) = ∑_A=1^N_A∑_i_A=1^n_ AO(A) D^(a)_i_Aχ_i_A(𝐫-𝐑_A),where D^(a)_i_A is the i_Ath expansion coefficient for MO a. The atomic orbitals χ_i_A are expressed as a function of the Cartesian coordinates of one electron 𝐫 and the spatial coordinates 𝐑_A of nucleus A. N_A labels the number of atoms and n_ AO(A) is the number of atomic orbitals on atom A. Using the MO-LCAO ansatz, the transition moments between spin orbitals readφ_a| F̂| φ_b =∑_A,B^N_A∑_i_A=1^n_ AO(A)∑_j_B=1^n_ AO(B) D^(a)_i_A D^(b)_j_Bχ_i_A| F̂| χ_j_B.The MO-LCAO coefficients D^(a)_iA and the definition of the atomic orbitals can be read directly from the output of standard quantum chemistry program packages. All required derivatives and integrals in the atomic orbital basis are computed analytically using our Python post-processing toolbox ORBKIT<cit.>, with which the molecular orbital density (cf. Eq. (<ref>)) and the molecular orbital electronic flux density (cf. Eq. (<ref>)) can then be projected on an arbitrary grid. Combining the information in this list with the occupation patterns of the quasi-CI eigenvectors associated with the excited states obtained at the LR-TDDFT level of theory, it is possible to create a library of transition moments between CI-states to be used in the dynamics.Note that the transition dipole moments are also computed using the same information and exploiting the multi-determinantalstructure of the N-electron basis functions, cf. Eqs. (<ref>) and (<ref>). The analysis tools for the hybrid TDDFT/CIS methodology are implemented, along with various other one-electron quantities, in a recently introduced open-source Python framework detCI@ORBKIT, available at . The program requires a preliminary LR-TDDFT calculation usingGaussian-type atom-centered orbitals. There is no restriction for the choice of functional. Currently, the code supports the GAMESS<cit.> and TURBOMOLE<cit.> formats. Our program then computes matrix elements of one-electron operators, projects them on an arbitrary grid, and stores them in a library to be used for analyzing the N-electron dynamics.The framework detCI@ORBKIT is written in Python, simplifying its portability on different platforms and offering efficient standard libraries for visualization purposes. Implementation details are given elsewhere <cit.>.The dynamics program is not part of the standard implementation and can be performed using either a user-written code or, e.g., the Matlab WAVEPACKET package<cit.>.In the present work, all dynamical simulations were performed using GLOCT, an in-house implementation of a propagator for the reduced-density matrix and related quantities <cit.>. § RESULTS AND DISCUSSION To demonstrate the capabilities of detCI@ORBKIT, we perform the analysis of correlated electron dynamics in two selected molecular systems. First, the charge transfer process in lithium hydride is studied to benchmark the quality of the TDDFT/CIS description against Full CI results. The electron migration in an alizarin dye induced by broadband laser excitation is then used as an example to demonstrate the scalability of the method. §.§ Benchmark: Charge Transfer in LiH In lithium hydride, charge migration can be initiated, e.g., by laser excitation from the molecular ground state Ψ_ g to the first excited state Ψ_ e. The charge transfer mechanism can be understood from the analysis of the superposition stateΨ_ el(t) = 1/√(2)(Ψ_ g e^-iE_ g t/ħ +Ψ_ e e^-i (E_ e t/ħ + η))which leads to the time-dependent one-electron densityρ(𝐫,t) = 1/2(ρ_ g(𝐫) + ρ_ e(𝐫)) + ρ_ ge(𝐫)cos(Δ E t/ħ + η),where Δ E=E_ e -E_ g, and η is the relative phase. It is set to η=π in the present example. Similarly, the electronic flux density takes the form𝐣(𝐫,t) =[𝐉_ ge(𝐫)] sin(Δ Et/ħ + η),where the transition electronic flux density 𝐉_ ge(𝐫) is obtained from Eq. (<ref>). The charge transfer dynamics can be thus rationalized in terms of the static transition moment between the two states involved. The electron densities, ρ_ g(𝐫) and ρ_ e(𝐫), of the ground state X^1 Σ^+ and the charge transfer state A^1 Σ^+ and the transition density between both, ρ_ ge(𝐫), are computed by combining the MO contributions obtained from Eq. (<ref>).The Kohn-Sham orbitals, the pseudo-CI eigenvectors, and the associated LR-TDDFT excitation energies are computed using an aug-cc-pVTZ at the B2-PLYP level of theory, as implemented in TURBOMOLE.<cit.> The character of the charge transfer state is found to be dominated by the HOMO-LUMO transition (see Fig. <ref> (right side)). This is in good agreement with the character determined from CIS and Full CI calculations,both performed with PSI4<cit.> using the identical basis set. The corresponding frontier orbitals from the Hartree-Fock reference are shown in the left side of Fig. <ref>. It can be seen that the HOMO is similar in both cases, while the LUMO is more delocalized at the B2-PLYP level of theory. We will show below that this difference has only a marginal influence on the electron dynamics.A great advantage of LR-TDDFT over CIS is the improved energetic description of the excited states at virtually the same computational cost. This is demonstrated by the good agreement of the excitation energies at the B2-PLYP level of theory (Δ E=3.51 eV) with the Full CI reference (Δ E=3.56 eV), as compared to the rather poor value for the truncated CIS expansion (Δ E=4.04 eV). Since the excitation is dominated by single excitations in all cases, the transition energies will affect mostly the timescale of the dynamics. Further, considering the similarities between the MOs involved in the dominant transition, a similar dynamical behavior is to be expected. This is indeed what is observed in Fig. <ref>, where the left panels report the flux density 𝐣(𝐫,t)at time t=τ/4 computed using Eq. (<ref>) and the right panels show the difference density 𝐲(𝐫,t) computed from the one-electron density, Eq. (<ref>). In the top right panel, the benchmark Full CI calculation reveals that the charge is transferred from the hydrogen atom (electron density depletion region in blue) to the lithium ion (red region). Some p-like regions of increasing density can also be recognized around the lithium atom.The same features are also observed for both the CIS (central panels) and hybrid (lower panels) method, while the magnitude of the difference density is larger around the hydrogen for the two single electron approaches.The electronic flux density maps are depicted as streamlines in the left panels of Fig. <ref>, where the blue shades indicate its magnitude. The charge is seen to flow from the polarizable hydrogen around and towards the back of the hard lithium. The same large vortex around the lithium ion is observed forthe Full CI benchmark, the CIS, and the hybrid TDDFT/CIS approach.The main quantitative differences between the methods are located in the low-density regions, e.g., at x>5 a_0, where the Full CI benchmark predictsa flux almost parallel to the molecular axis. The critical point (at x∼ 2 a_0) between the lithium and hydrogen atoms also appears to be slightly shifted to the right at the CIS and TDDFT/CIS level of theory. In general, it can be said that both single electron excitation ansatz yield a very similar picture of the electron difference density and the associated flux density.This conclusion is likely to hold for all dynamical processes involving N-electron eigenstates dominated by a single excitation character.§.§ Scalability: Electron Migration in Alizarin In this second example, we demonstrate the computational scalability of the hybrid TDDFT/CIS approach to analyzethe correlated electron dynamics for more extended molecular systems. The necessity of such a method is due to the fact that high-level electronic structuretheory methods, such as MCSCF, are often not applicable for larger molecules. The CI scheme truncated at the singles excitation represents a simple, intuitive,and computationally cheap approach to compute qualitatively correct excited electronic states.<cit.>However, it yields inaccurate vertical excitation energies from the ground state.<cit.> As advocated in the theory section above, a suitable alternative is LR-TDDFT<cit.>, which usually provides better energetic description than CIS whileretaining the same quality for the wave function. It can be inferred from the example in the previous subsection, that this will provide an adequate description for a large number of photochemical processes dominated by a single excitation character. In addition, it benefits from the versatility and continuous improvement of density functionals. Proper treatment of the excited states strongly depends on the appropriate choice of a functional, which can be chosen to correctly describe electronic correlations, the dispersive nature, or the charge-transfer character of a given excitation. <cit.> Fortunately, extensive experience has been accumulated over the years concerning the applicatibility of each functional in specific chemical contexts. For example, it is known that non-local exchange improves greatly the description of charge transfer states <cit.>.To show the scalability of the hybrid TDDFT/CIS formalism for the analysis of correlatedelectron dynamics, we initiate a photoinduced ultrafast charge migration process in alizarin. This organic chromophore is used as a π-conjugated photosensitizer in dye-sensitized solar cells. Prior to the dynamical simulation, the electronic and optical properties of alizarin are determined by means of LR-TDDFT. Therefore, we perform a TURBOMOLE<cit.> calculation with the B3LYP<cit.> hybrid functional and the def2-SVP basis set<cit.>at the equilibrium geometry of alizarin. This setup has been previously proven to yield accurate results for the electronicand spectroscopic properties of such systems.<cit.> The computed absorption spectrum is depicted in Fig. <ref>(a) along withthe experimentally observed absorption band of free alizarin (dashed black line). The good agreement for the first absorption band between theory (437 nm) and experiment(431 nm) underlines the suitability of TDDFT to model the electronic spectra of medium-sized organicmolecules dominated by single excitation character. It is important to recognize that the second absorption band is composed of a multitude of excited states in the UV/VIS range.In order to simulate an ultrafast charge migration process in alizarin, we proceed to the broadband excitation of all excited states in the energy range between 200 nm and 500 nm(cf. Fig. <ref>(a) yellow filling). For the promotion of these states from the ground state, a superposition of state-to-state sin^2-shaped pulses with a duration 19 fs is constructed. The pulse is adjusted to the parameters of a realistic experimental laser field used in similar investigations to initiate, e.g., ultrafast photoinduced processes in alizarin-TiO_2 solar cells.<cit.> The resulting electric field is shown in the inset of Fig. <ref>(b). The laser excitation is followed by a 20 fs field-free propagation. The time-evolution of the N-electron wave packet (cf. Eqs. (<ref>) and (<ref>)) is accomplished using an adaptive Runge-Kutta algorithm in the interaction picture. The methodology and implementation details are described elsewhere.<cit.> Fig. <ref>(b) shows the evolution of the state populations and the applied laser field in the inset. As it is often the case for molecules in strong fields, the population dynamics is very intricate while the laser is on, in part due to important polarization effects and in part due to the number of states that are excited by the broadband laser. To account for the electronic response of the system to the laser field, 25 eigenstates are incorporated in the simulation. After the laser excitation, only twelve states are significantly populated (P_λ>0.01).To unravel the mechanistic pathways and give an intuitive picture of the electron dynamics, we advocate using the time-dependent electron density, electron flux density, and electronic flow, which are reconstructed from the N-electron wave packet in the pseudo-CI eigenvector basis. This can become computationally tedious, since the number of Slater determinantsin the wave function expansion increases with the number of occupied and unoccupied orbitals. For alizarin in the current basis set, the 62 occupied MOs times 248 virtual MOs correspond to 15 376 determinants for eachof the excited states. Recalling that reconstruction of the flux density, Eq. (<ref>), requirescombining one-electron integrals of all orbitals pairwise, for each pair of eigenstates, this amounts to a tremendous computational task. However, inclusion of the complete set of determinants is not necessary, since theexpansion coefficient of many of these is either zero or negligibly small. Moreover, the pseudo-CI eigenvectors are usually dominated by a few determinant contributions. Applying the maximum correspondence principle between determinants and exploiting the Slater-Condon rules lead to significant numerical savings, which are automatized in our implementation and can be controlled by a user-defined convergence threshold.In order to examine the influence of truncating the complete set of determinants to aphysically meaningful subset in the wavefunction ansatz, we choose to use a threshold for the expansion coefficients of the quasi-CI wave functions, Eq. (<ref>). All determinants with a contribution under this value are neglected in the evaluation of the one-electron density and the associated flux density. We define three different thresholds, |C_a^r,(λ)| > {10^-3, 10^-2, 10^-1}. To illustrate the computational savings that can be expected, the thirteenth excited state is selected as an example,since it is the most populated state after the broadband laser excitation. The three chosen threshold values retain 926, 31, and 7 Slater determinants in the wave function expansion, respectively. For consistency check, the pruned expansions recover 99.94 %, 99.31 %, and 95.92 % of the norm of the thirteenth excited state, respectively. These numbers are similar for other excited states. After excluding the contributions lying under the respective threshold, the remaining coefficients are renormalized using a modified Gram-Schmidt algorithm. Regarding the computational effort, the reduction of determinants means a drastic decrease of computational steps. For example, the calculation of the transition electronic flux density 𝐣_5 13(𝐫) between the fifth and thirteenth excited state requires N_ SD^5× N_ SD^13 =573× 926 = 530 598determinant combinations for the more strict threshold of |C_a^r,λ| >10^-3. This number reduces to N_ SD^5× N_ SD^13 = 3×7=21 determinant combinations for the lenient threshold |C_a^r,λ| >10^-1. This simple constraint thus confers great scalability to the method presented here.To assess the quality of this approximation, the electronic flux densities𝐣(𝐫,t) reconstructed using the different thresholds are illustrated in Fig. <ref>(a),(c),(d) at a characteristic point in time after the laser-pulse excitation (here, t=25.3 fs). This analysis could be performed during the laser-field at the zeros of the pulse function to avoid simply representing the contributionof the electric field to the flux density. On the other hand, after the pulse, the flux is solely caused by the coherences between the electronic states, simplifying its analysis. In Fig. <ref>(b), the corresponding electron flow, ∂ρ(𝐫,t) / ∂ t, is displayed additionally for the tight threshold, |C_a^r,(λ)| > 10^-3, to facilitate the interpretation of the flux densities. For all three wave packet expansions, 𝐣(𝐫,t) shows nearly identical qualitative features. These include: (1) an electron flow along the bonds mediated by the π-system,(2) an anti-clockwise flux at the outer right ring,and (3) a charge migration from the outer right ring and the top part of the left ring to the lower hydroxyl group. Due to the cyclic nature of the field-free evolution of the N-electron wave packet, the features (2)-(3) exhibits a Rabi-type change of direction during the dynamics. Since the qualitative picture is the same even at all levels of analysis, the very important numerical savings at the crudest level of approximation confer great scalability to the proposed methodology to analyze N-electron dynamics in large molecules. Despite this success, one striking difference can be noticed, i.e., the electron migrationfrom the lower hydroxyl group to the neighboring carboxyl group is not fully reproducedwith the two larger thresholds (cf. Fig. <ref>(c) and (d)). This corresponds to a through-space charge transfer and is mediated by a large number of small contributions in the pseudo-CI eigenstate basis. While the major phenomenological characteristics of the correlated electron dynamics are still captured, some minor mechanistic information is lost by reducing the wave functions to their dominant determinant contributions.To understand the electron dynamics, we extend its analysis to the time-dependent electronic yield. It is defined as the difference between the electron density at a given time and the electron density at t=0fs, integrated over a given volume,y_ A(t)= ∫_ A d𝐫(ρ(𝐫,t) - ρ(𝐫,0) )In Fig. <ref>, the difference densities projected on right and left aromatic ring of alizarin are reported for times after the laser has been switched off. These reveal intricate synchronous fluctuations of electron density between both rings. The slight asymmetry of the electron redistributioncorrelates with the asymmetric substituents on the two outer rings: the fluctuation amplitudes in the left ring are larger than in the unsubstituted, rightmost aromatic ring. In Fig. <ref>, the electronic flux density is plotted for selected characteristic points in order to unveil the mechanism of the charge migration between both rings. The times associated with these snapshots are marked as vertical gray lines in Fig. <ref>. In panel (a), where the electronic yield is largest on the rightmost aromatic ring(t=24.5 fs), the electronic flux density is seen to rotate clockwise, which will create a temporary local magnetic field. In the next snapshot (panel (b), t=25.3 fs), theyields in the left and right aromatic rings are equivalent. The electrons are seen to flow laminarly from the right to left aromatic moieties. Starting from the lowest carbon atom of the right ring, electrons move anticlockwise along the bonds of the right ring, passing by the bottom carbonyl group to bottom hydroxyl group below the left aromatic ring, and back to the central carbonyl unit via a through space mechanism. This is a strong indication of the presence of a hydrogen bond, which will lead toa rapid tautomeric hydrogen transfer.At the third time step (t=25.9 fs), the electronic yield is maximal in the right ring and the electron flux density rotates clockwise. Contrary to panel (a), the electron flow is not as strongly localized inside the ring, probably due to the asymmetry of the ring substituents. In the last snapshot (t=26.4 fs), the density migrates back from the leftmost to the rightmost aromatic ring along a different path: the electrons retain a clockwise orientation in the left ring, mostly flowing from the bottom hydroxyl to the top central carbonyl group. A marginal amount flows from the bottom lefthydroxy group over the hydrogen bond to the bottom carbonyl group and into rightmost aromatic ring. For the complete picture of the dynamics after the laser excitation, a short film of the charge migration process is made available online in the Supporting Information.§ CONCLUSIONS In this paper, we have introduced a novel procedure to analyze and visualize many-electron dynamics from a hybrid time-dependent density functional theory (TDDFT)/ configuration interaction singles (CIS) formalism. The method resorts to a linear-response TDDFT calculation to generate a basis of pseudo-CI eigenvectors and associated energies, which are then used as a basis to describe an N-electron dynamics at the CIS level of theory. The time-dependent CIS wave function retains the simple character of coupled electron-hole pairs, which facilitates its interpretation in terms of configuration states while keeping the size of the basis relatively small. This renders the hybrid method amenable to large systems – in fact, any system that can betackled and described accurately using linear-response TDDFT.From the evolution of a TDDFT/CIS wave packet, we showed how it is possible to rationalize the N-electron dynamics of a system in terms of transition moments of various one-electron operators. These include difference electronic densities, transient electronic flux density maps, and transition dipole moments, which we have implemented ina Python toolbox for postprocessing multi-determinantal wave function data. This new module of our open-source project ORBKIT creates a library of transition moments of user-specified one-electron operators, which are projected on an arbitrary grid. The required information (molecular orbitals, structure, Gaussian atomic basis, pseudo-CI coefficients) can be directly extracted from the output of various quantum chemistry program packages. These are then used to reconstruct quantities that help understanding the flow of electrons in molecules out of equilibrium.We first applied the hybrid TDDFT/CIS scheme to a test system, the charge transfer in lithium hydride, in order to benchmark the quality of the ansatz against standard CIS and Full CI reference simulations. The results demonstrated that a good choice of the functional improves mostly the energetic description of the charge transfer state, which can be brought close to the Full CI benchmark. On the other hand, the pseudo-CI basis retains a single excitation character, which is found to be similar to the standard CIS reference. Both ansatz agree semi-quantitatively with the higher level wave function description, with the discrepancies mostly found in the regions of low density.In a second example, the application to the broadband excitation of a prototypical chromophore for dye-sensitized solar cells demonstrated the scalability of the method and the versatility of the new toolkit. In particular, it was found that the main features of the electron flow mechanism can be recovered using a stringent basis pruning strategy, in which each pseudo-CI eigenstate is representedusing only a few dominant configurations. By doing so, marginal features of the many-electron dynamics involving, e.g., electron flow through hydrogen bonds, may be lost. Because of the favorablescaling of the method even at tighter convergence Thresholds, it is expected to be applicable to a large number of medium-sized molecules. 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"authors": [
"Gunter Hermann",
"Vincent Pohl",
"Jean Christophe Tremblay"
],
"categories": [
"physics.chem-ph"
],
"primary_category": "physics.chem-ph",
"published": "20170426142507",
"title": "An Open-Source Framework for $N$-Electron Dynamics: II. Hybrid Density Functional Theory/Configuration Interaction Methodology"
} |
Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, People's Republic of China [Corresponding author: ][email protected] Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, People's Republic of ChinaKey Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, People's Republic of ChinaKey Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, People's Republic of ChinaKey Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, People's Republic of ChinaKey Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, People's Republic of ChinaKey Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, People's Republic of China [Corresponding author: ][email protected] Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, People's Republic of China High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, People's Republic of China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, People's Republic of China The electronic and magnetic properties of ZrS_2 nanoribbons (NRs) are investigated based on the first-principles calculations. It is found that the ZrS_2 NRs with armchair edges are all indirect-band-gap semiconductors without magnetism and the band-gap exhibits odd-even oscillation behavior with the increase of the ribbon width. For the NRs with zigzag edges, those with both edges S-terminated are nonmagnetic direct-band-gap semiconductors and the gap decreases monotonically as a function of the ribbon width. However, the NRs with one edge S-terminated and the other edge Zr-terminated are ferromagnetic half-metal, while those with both edges Zr-terminated tend to be ferromagnetic half-metal when the width N≥9. The magnetism of both systems mainlyoriginates from the unsaturated edge Zr atoms. Depending on the different edge configurations and ribbon widths, the ZrS_2 NRs exhibit versatile electronic and magnetic properties, making them promising candidates for the applications of electronics and spintronics. Edge-controlled half-metallic ferromagnetism and direct gap in ZrS_2 nanoribbons Y. P. Sun================================================================================ The electronic and magnetic properties of nanoscale materials have been the subject of extensive research due to their potential applications in electronics and spintronics. Carbon-based systems, graphene for example, are among the mostly studied low-dimensional materials. Graphene has a high mobility at room temperature, making it a promising candidate for the future information technology. However, the lack of intrinsic band gap has limited its practical applications. Cutting the two-dimensional (2D) graphene into one-dimensional (1D) nanoribbon (NR) can open the band gap in graphene, which was firstly predicted theoretically<cit.> and then verified by the experiments.<cit.> Because of the additional edge states unique to the dimensionality, 1D NRs could exhibit rich properties, which can be further tuned by modifying their edges. It was found that the narrow zigzag graphene NR is semiconducting with the two edges antiferromagnetically coupled to each other.<cit.> In addition, the graphene NR can be converted into half-metal in different ways, such as by applying a homogeneous electrical field<cit.> or by chemical decoration at the edges.<cit.>Besides graphene, low-dimensional transition-metal dichalcogenides TMDs (with the formula MX_2, M=transition metal, X=S, Se, or Te) are another important materials that received considerable attention. Different from graphene, single layer MX_2 has three atomic layers, with an M-layer sandwiched between two-X layers. As a typical representative of TMDs, single layer MoS_2 is a direct-band-gap semiconductor.<cit.> Intrinsic 2D MoS_2 monolayer is nonmagnetic. However, distinct electronic and magnetic properties were reported for 1D MoS_2 NRs, that is, armchair MoS_2 NRs are nonmagnetic semiconductors, while the zigzag NRs have ferromagnetic (FM) and metallic ground states.<cit.> The WS_2 NRs exhibit similar properties as to MoS_2 NRs.<cit.> The zigzag-edge related ferromagnetism was then observed experimentally in MoS_2<cit.> and WS_2<cit.> nanosheets. Furthermore, the electronic and magnetic properties of MoS_2 NRs can be modified by edge passivation<cit.> or by applying external strain and/or electric field.<cit.> It was theoretically predicted that by applying a transverse electrical field, the insulator-metal transition occurs and ferromagnetism emerges beyond a critical electrical field in the armchair MoS_2 NRs.<cit.>Both MoS_2 and WS_2 crystallize in the honeycomb (H) structure. ZrS_2 is another kind of TMDs, which crystallizes in the centered honeycomb (T) structure. Different from MoS_2 monolayer, single layer ZrS_2 is an indirect-band-gap semiconductor.<cit.> In experiment, 1D ZrS_2 NRs have been synthesized by the process of chemical vapor transport and vacuum pyrolysis.<cit.> Then what different properties the 1D NRs may present? In the present work, we investigate the electronic and magnetic properties of a series of ZrS_2 NRs. The results show that depending onthe different edge configurations, the ZrS_2 NRs can be indirect-band-gap semiconductor, direct-band-gap semiconductor, antiferromagnetic (AFM) metal, or FM half-metal, exhibiting versatile electronic and magnetic properties. Our calculations were performed within the framework of the density functional theory (DFT), <cit.> as implemented in the QUANTUM ESPRESSO code.<cit.> The exchange correlation energy was in the form of Perdew-Burke-Ernzerhof (PBE)<cit.> with generalized gradient approximation (GGA). The Brillouin zones were sampled with 12×1×1 and 20×1×1 Monkhorst-Pack k meshes for the armchair and zigzag NRs, respectively. The cutoff energy for the plane-wave expansion was set to be 40 Ry. The distance between each NR and its periodic image is set to be larger than 15 Å so that they can be treated as independent entities.The ZrS_2 NRs can be obtained by cutting the monolayer, and there are mainly two kinds of ZrS_2 NRs, i.e., armchair and zigzag NRs, according to the cutting directions with respect to the monolayer. As for the zigzag ZrS_2 NRs, the edges can be terminated by Zr or S atoms, thus three cases exist. We denote the case that two edges are both S atoms terminated as 1S-1S zigzag NR, and the other two cases are 1Zr-1S (one edge is Zr and the other edge is S atoms) and 1Zr-1Zr (both edges are Zr atoms) zigzag NRs. The relaxed structures of these four kinds of ZrS_2 NRs are demonstrated in Fig. 1 and the left and right panels are top and side views, respectively. The width of the NR is defined according to the number of the Zr atoms across the ribbon width, as shown in Figs. 1(a) and (c). Compared with the initial structures, the fully relaxed structures of the NRs change a little, mainly coming from the edge atoms, that is, the edge S atoms tend to leave away from the inner atoms, while the Zr atoms prefer to approach the inner side.To investigate the stability of the ZrS_2 NRs, we calculated the binding energy E_b, which is defined as E_b=(E_Zr_nS_m-nE_Zr-mE_S)/(n+m), where E(Zr_nS_m), E(Zr), and E(S) are the total energies of the ZrS_2 NRs, the Zr and S atoms, respectively. The larger the binding energy, the more stable the corresponding structure. The NRs with the width N=4-14 are considered in this work. The calculated binding energies of the four kinds of ZrS_2 NRs are in the range of 6.19-6.66 eV per atom, indicating that all the investigated NRs are very stable. Moreover, as shown in Fig. 2, the binding energies of the 1S-1S zigzag NRs are the highest among the four kinds of NRs, indicating that this type of NR is the most stable one, followed by the armchair NRs, and the 1S-1Zr and 1Zr-1Zr zigzag NRs are relatively less stable. For each kind of NRs, the system becomes more and more stable as the ribbon width increases.To check the possibility of magnetism in ZrS_2 NRs, both the spin-unpolarized and spin-polarized calculations were carried out. For the 1S-1Zr and 1Zr-1Zr zigzag NRs, the total energies of the FM states are lower than those of the nonmagnetic states, indicating that these two kinds of NRs have a magnetic state. However, the other two cases, i.e, armchair and 1S-1S zigzag NRs, are nonmagnetic. First, we focus on the electronic properties of the nonmagnetic armchair and 1S-1S zigzag ZrS_2 NRs. The results of the band structures show that all the investigated armchair ZrS_2 NRs are indirect-band-gap semiconductors (see Fig. 3(a)). The band gap exhibits interesting odd-even oscillation as increasing the ribbon width (Fig. 3(d)), that is, the even-numbered NRs have relatively larger band gaps than the neighboring odd-numbered NRs. Moreover, the band gap of the even-numbered NRs decreases as a function of the width, while it changes slightly for the odd-numbered NRs. On the other hand, all the 1S-1S zigzag NRs are semiconducting as well but have direct band gaps (Fig. 3(b)). The band gap decreases monotonically as the ribbon width increases (see the inset of Fig. 3(d)). Although the ZrS_2 monolayer is an indirect-band-gap semiconductor,<cit.> which makes it less investigated compared with MoS_2, the indirect-direct band gap transition can be obtained by cutting it into 1D zigzag NR, with both edges terminated by S atoms .To investigate the origin of the odd-even oscillation of the band gap, we plot in Fig. 3 (c) the band-decomposed charge densities for armchair ZrS_2 NRs with widths N=6 and 7. We can see from Fig. 3(a) that for the armchair NRs, the valence band maximum (VBM) locates at the Γ point for both the even- and odd-numbered NRs, while the conduction band minimum (CBM) locates at the X points for the odd-numbered NRs and at the position between the Γ and X points for the even-numbered ones. The corresponding charge densities for the CBM and VBM are plotted in the upper and lower panels in Fig. 3(c), respectively. We can see that the CBM are mainly determined by the Zr atoms located in the inner part of the NRs and there are much smaller contributions coming from the edge atoms. Therefore, the edge configurations have little impact on the electronic properties of the CBM. In contrast, the VBM are mainly controlled by the S atoms at the edge of the NRs. The structures of the odd-numbered NRs are symmetric with respect to the central line, thus the two nearest-neighboring S atoms at the two edges (which are circled by the green lines) are just opposite to each other, making their distance shortest for the two edge atoms. However, for the even-numbered NRs, the two nearest-neighboring S atoms at the two opposite edges are staggered. Therefore, their distance is enlarged, making the interactions between the two edges relatively smaller. As a result, the VBM is lowered for the even-numbered NRs and their band gaps are enlarged accordingly compared with those of the neighboring odd-numbered NRs.As discussed above, the other two kinds of ZrS_2 NRs, i.e., 1S-1Zr and 1Zr-1Zr zigzag NRs are magnetic. The results of the spin densities (Fig. 4(b)) demonstrate that the magnetism of both systems mainly originate from the unsaturated edge Zr atoms. This is different from the H-structured zigzag MoS_2 NRs, whose magnetic moments concentrate on both the edge Mo and S atoms.<cit.> For narrow graphene NRs, the spins of the two edges are antiparallel to each other, i.e., the two edges are antiferromagnetically coupled.<cit.> Since the 1Zr-1Zr ZrS_2 NRs have both edges terminated by the Zr atoms, we also check the case when the Zr atoms at the two edges are antiferromagnetically coupled. When the width N=4-8, the energies of the AFM states are relatively smaller than those of the FM states and thus the ground states of the NRs are AFM. When the width increases up to 9, the system tends to be FM. The similar property was also observed in zigzag graphene NR, for which the switching from AFM to FM states may occur when the ribbon width is larger than 7 nm, due to the FM inter-edge coupling.<cit.> For both the 1S-1Zr and 1Zr-1Zr zigzag NRs, the magnetic moments for the FM states are nearly width-independent, as shown in Fig. 4(a). Since both edges of 1Zr-1Zr NR are terminated by Zr atoms while for 1S-1Zr NR, there is only one edge of Zr atoms, the total magnetic moments of 1Zr-1Zr NRs are twice those of the 1S-1Zr systems.In Fig. 5, we plot the band structures and partial density of states (PDOS) for the 1S-1Zr and 1Zr-1Zr zigzag NRs with ribbon width N=6 and 10. The spin-up and spin-down channels are drawn in the black and red lines, respectively. For the PDOS, the results of four kinds of atoms (denoted in Figs. 1(c) and (d)), including the edge S atom (S1), edge Zr atom (Zr1) and two inner Zr atoms near the edge (Zr2 and Zr3), are shown. For 1S-1Zr ZrS_2 NRs (Figs. 5(a) and (b)), the spin-up channels are metallic while the spin-down channels are semiconducting, thus the NRs are 100% spin-polarized and the systems are FM half-metal. The results of the PDOS show that for S1, the spin-up and spin-down channels are symmetry, so it has no contribution to the magnetic moment. For Zr1, Zr2, and Zr3, the spin-up and spin-down channels are asymmetry. The edge Zr atom (Zr1) contributes most to the magnetic moment and from edge to inner sites, the contribution becomes more and more smaller. Therefore, the magnetic moments mainly originate from the unsaturated Zr atoms located at the edge, which is consistent with the results of spin densities (Fig. 4(b)). For the 1Zr-1Zr ZrS_2 NRs, both the spin-up and spin-down states are metallic when N≤8 (Fig. 5(c)), thus the systems are AFM metals. However, the system tends to be FM half-metal when N increases up to 9 and the magnetic moments mainly come from the Zr atoms at the two edges.Compared with the H-structured MoS_2 NRs, the T-structured MX_2 NRs are less investigated.Reyes-Retana et al. reported that the zigzag NiSe_2 NRs are metallic and the armchair systems are semiconducting, but both of which are nonmagnetic.<cit.> Our results demonstrate that for the 1S-1Zr and 1Zr-1Zr zigzag ZrS_2 NRs, 1D electrical current with completely spin polarization can be realized along the Zr edges of the systems. The intrinsic half-metallicity predicted in the ZrS_2 NRs is highly desirable for applications in spintronics.In conclusion, we have investigated the electronic and magnetic properties of ZrS_2 NRs. The armchair ZrS_2 NRs are indirect-band-gap semiconductors and the gap exhibits odd-even oscillation as increasing the ribbon width. For the zigzag ZrS_2 NRs, the 1S-1S NRs are direct-band-gap semiconductors and the gap decreases with the increase of the width. Both the armchair and 1S-1S zigzag ZrS_2 NRs are nonmagnetic. However, the 1S-1Zr and 1Zr-1Zr (N≥9) zigzag NRs are found to be FM half-metal and the magnetism is mainly contributed by the edge Zr atoms. Our results indicate that by tuning the edge configurations, the ZrS_2 NRs could exhibit rich electronic and magnetic properties, which is desirable for the future applications in electronics and spintronics. This work was supported by the National Key Research and Development Program under Contract No. 2016YFA0300404, National Natural Science Foundation of China under Contract Nos. 11404340, 11674326, and U1232139, the Key Research Program of Frontier Sciences of CAS (QYZDB-SSW-SLH015) and Hefei Science Center of CAS (2016HSC-IU011), the Anhui Provincial Natural Science Foundation under Contract No. 1708085QA18. 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section.equation equationsection ℙ ℚ ℝ ℕ 𝔻 𝔼 1𝔼 ℙ ℝ ℚ ℕ 𝕎 ∂̣ ℤE F - _ = + theoremTheorem[section] lemme[theorem]Lemma corollary[theorem]Corollary definition[theorem]Definition proposition[theorem]Proposition property[theorem]Propertyremark remarkRemark exampleExample #1#21#1_ #2 #1#2 1#1_ #2 #1#2 1#1_ #2 #1#2 1#1_ #2 #1#2 #1#2#1 height .81 depth .851_ #2 | http://arxiv.org/abs/1704.08199v1 | {
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http://arxiv.org/abs/1704.07999v1 | {
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IEEEexample:BSTcontrolgobble Multiscale Analysis for Higher-order Tensors Alp Ozdemir, Ali Zare, Mark A. Iwen, and Selin Aviyente A. Ozdemir ([email protected]) and S. Aviyente ([email protected]) are with the Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824, USA. Ali Zare ([email protected]) is with the Department of Computational Mathematics, Science, and Engineering (CMSE), Michigan State University, East Lansing, MI, 48824, USA.Mark A. Iwen ([email protected]) is with the Department of Mathematics, and the Department of Computational Mathematics, Science, and Engineering (CMSE), Michigan State University, East Lansing, MI, 48824, USA.This work was in part supported by NSF DMS-1416752 and NSF CCF-1615489.December 30, 2023 ==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================0 ptThe widespread use of multisensor technology and the emergence of big datasets have created the need to develop tools to reduce, approximate, and classify large and multimodal data such as higher-order tensors. While early approaches focused on matrix and vector based methods to represent these higher-order data, more recently it has been shown that tensor decomposition methods are better equipped to capture couplings across their different modes. For these reasons, tensor decomposition methods have found applications in many different signal processing problems including dimensionality reduction, signal separation, linear regression, feature extraction, and classification. However, most of the existing tensor decomposition methods are based on the principle of finding a low-rank approximation in a linear subspace structure, where the definition of the rank may change depending on the particular decomposition. Since many datasets are not necessarily low-rank in a linear subspace, this often results in high approximation errors or low compression rates. In this paper, we introduce a new adaptive, multi-scale tensor decomposition method for higher order data inspired by hybrid linear modeling and subspace clustering techniques. In particular, we develop a multi-scale higher-order singular value decomposition (MS-HoSVD) approach where a given tensor is first permuted and then partitioned into several sub-tensors each of which can be represented as a low-rank tensor with increased representational efficiency. The proposed approach is evaluated for dimensionality reduction and classification for several different real-life tensor signals with promising results.Higher-order singular value decomposition, tensor decomposition, multi-scale decomposition, data reduction, big data applications.§ INTRODUCTION Data in the form of multidimensional arrays, also referred to as tensors, arise in a variety of applications including chemometrics, hyperspectral imaging, high resolution videos, neuroimaging, biometrics and social network analysis <cit.>. These applications produce massive amounts of data collected in various forms with multiple aspects and high dimensionality. Tensors, which are multi-dimensional generalizations of matrices, provide a useful representation for such data. A crucial step in many applications involving higher-orders tensors is multiway reduction of the data to ensure that the reduced representation of the tensor retains certain characteristics. Early multiway data analysis approaches reformatted the tensor data as a matrix and resorted to methods developed for classical two-way analysis. However, one cannot discover hidden components within multiway data using conventional matrix decomposition methods as matrix based representations cannot capture multiway couplings focusing on standard pairwise interactions. To this end, many different types of tensor decomposition methods have been proposed inliterature <cit.>.In contrast to the matrix case where data reduction is often accomplished via low-rank representations such as singular value decomposition (SVD), the notion of rank for higher order tensors is not uniquely defined. The CANDECOMP/PARAFAC (CP) and Tucker decompositions are two of the most widely used tensor decomposition methods for data reduction <cit.>. For CP, the goal is to approximate the given tensor as a weighted sum of rank-1 tensors, where a rank-1 tensor refers to the outer product of n vectors with n being equal to the order of the tensor.The Tucker model allows for interactions between the factors from different modes resulting in a typically dense, but small, core tensor. This model also introduces the notion of Tucker rank or n-rank, which refers to the n-tuple of ranks corresponding to the tensor's unfoldings along each of its modes. Therefore, low rank approximation with the Tucker model can be obtained by projections onto low-rank factor matrices. Unlike the CP decomposition, the Tucker decomposition is in general non-unique. To help obtain meaningful and unique representations by the Tucker decomposition, orthogonality, sparsity, and non-negativity constraints are often imposed on the factors yielding, e.g., the Non-Negative Tensor Factorization (NTF) and the Sparse Non-Negative Tucker Decomposition <cit.>. The Tucker decomposition with orthogonality constraints on the factors is known as the Higher-Order Singular Value Decomposition (HoSVD), or Multilinear SVD <cit.>.The HoSVD can be computed by simply flattening the tensor in each mode and calculating the n-mode singular vectors corresponding to that mode. With the emergence of multidimensional big data, classical tensor representation and decomposition methods have become inadequate since the size of these tensors exceeds available working memory and the processing time is very long. In order to address the problem of large-scale tensor decomposition, several block-wise tensor decomposition methods have been proposed <cit.>. The basic idea is to partition a big data tensor into smaller blocks and perform tensor related operations block-wise using a suitable tensor format. Preliminary approaches relied on a hierarchical tree structure and reduced the storage of d-dimensional arrays to the storage of auxiliary three-dimensional ones such as the tensor-train decomposition (T-Train), also known as the matrix product state (MPS) decomposition, <cit.> and the Hierarchical Tucker Decomposition (H-Tucker) <cit.>. In particular, in the area of large volumetric data visualization, tensor based multiresolution hierarchical methods such as TAMRESH have attracted attention <cit.>. However, all of these methods are interested in fitting a low-rank model to data which lies near a linear subspace, thus being limited to learning linear structure. Similar to the research efforts in tensor reduction, low-dimensional subspace and manifold learning methods have also been extended for higher order data clustering and classification applications. In early work in the area, Vasilescu and Terzopoulos <cit.> extended the eigenface concept to the tensorface by using higher order SVD and taking different modes such as expression, illumination and pose into account. Similarly,2D-PCA for matrices has been used for feature extraction from face images without converting the images into vectors <cit.>. He et al. <cit.> extended locality preserving projections <cit.> to second order tensors for face recognition. Dai and Yeung <cit.> presented generalized tensor embedding methods such as the extensions of local discriminant embedding methods <cit.>, neighborhood preserving embedding methods <cit.>, and locality preserving projection methods <cit.> to tensors.Li et al. <cit.> proposed a supervised manifold learning method for vector type data which preserves local structures in each class of samples, and then extended the algorithm totensors to provide improved performance for face and gait recognition. Similar to vector-type manifold learning algorithms, the aim of these methods is to find an optimal linear transformation for the tensor-type training data samples without vectorizing them and mapping these samples to a low dimensional subspace while preserving the neighborhood information. In this paper, we propose a novel multi-scale analysis technique to efficiently approximate tensor type data using locally linear low-rank approximations. The proposed method consists of two major steps: 1) Constructing a tree structure by partitioning the tensor into a collection of permuted subtensors, followed by 2) Constructing multiscale dictionaries by applying HoSVD to each subtensor.The contributions of the proposed framework and the novelty in the proposed approach with respect to previously published work in <cit.> are manifold.They include: 1) The introduction of a more flexible multi-scale tensor decomposition method which allows the user to approximate a given tensor within given memory and processing power constraints; 2) the introduction of theoretical error bounds for the proposed decomposition;3) the introduction of adaptive pruning to achieve a better trade-off between compression rate and reconstruction error for the developed factorizations; 4) the extensive evaluation of the method for both data reduction and classification applications; and 5) a detailed comparison of the proposed method to state-of-the-art tensor decomposition methods including the HoSVD, T-Train, and H-Tucker decompositions. The remainder of the paper is organized as follows. In Section <ref>, basic notation and tensor operations are reviewed.The proposed multiscale tensor decomposition method along with theoretical error bounds are then introduced in Section <ref>. Sections <ref> and <ref> illustrate theresults of applying the proposed framework to data reduction and classification problems, respectively. § BACKGROUND §.§ Tensor Notation and AlgebraA multidimensional array with N modes 𝒳∈ℝ^I_1× I_2× ... × I_N is called a tensor, where x_i_1,i_2,..i_N denotes the (i_1,i_2,..i_N)^th element of the tensor 𝒳. The vectors in ℝ^I_n obtained by fixing all of the indices of such a tensor 𝒳 except for the one that corresponds to its nth mode are called its mode-n fibers. Let [N] := {1, …, N } for all N ∈ℕ.Basic tensor operations are reviewed below (see, e.g., <cit.>, <cit.>, <cit.>).Tensor addition and multiplication by a scalar: Two tensors 𝒳, 𝒴∈ℝ^I_1× I_2× ... × I_N can be added using component-wise tensor addition.The resulting tensor 𝒳 + 𝒴∈ℝ^I_1× I_2× ... × I_N has its entries given by ( 𝒳 + 𝒴)_i_1,i_2,..i_N = x_i_1,i_2,..i_N + y_i_1,i_2,..i_N.Similarly, given a scalar α∈ℝ and a tensor 𝒳∈ℝ^I_1× I_2× ... × I_N the rescaled tensor α𝒳∈ℝ^I_1× I_2× ... × I_N has its entries given by ( α𝒳)_i_1,i_2,..i_N = α x_i_1,i_2,..i_N.Mode-n products: The mode-n product of a tensor 𝒳∈ℝ^I_1× ... I_n× ...× I_N and a matrix U∈ℝ^J× I_n is denoted as 𝒴=𝒳×_nU, (𝒴)_i_1,i_2,…,i_n-1,j,i_n+1,…,i_N=∑_i_n=1^I_nx_i_1,…,i_n,…,i_Nu_j,i_n.It is of size I_1× ...× I_n-1× J × I_n+1× ...× I_N.The following facts about mode-n products are useful (see, e.g., <cit.>,<cit.>). Let 𝒳, 𝒴∈ℝ^I_1× I_2× ... × I_N, α, β∈ℝ, and U^(n),V^(n)∈ℝ^J_n × I_n for all n ∈ [N].The following are true: (a) ( α𝒳 + β𝒴) ×_nU^(n) = α( 𝒳×_nU^(n)) + β( 𝒴×_nU^(n)). (b) 𝒳×_n( α U^(n) + β V^(n)) = α( 𝒳×_nU^(n)) + β( 𝒳×_nV^(n)).(c) If n ≠ m then 𝒳×_nU^(n)×_mV^(m) = ( 𝒳×_nU^(n)) ×_mV^(m) = ( 𝒳×_mV^(m)) ×_nU^(n) = 𝒳×_mV^(m)×_nU^(n) . (d) If W∈ℂ^P × J_n then 𝒳×_nU^(n)×_nW = ( 𝒳×_nU^(n)) ×_nW = 𝒳×_n (WU^(n)) = 𝒳×_nWU^(n).Tensor matricization: The process of reordering the elements of the tensor into a matrix is known as matricization or unfolding. The mode-n matricization of a tensor 𝒴∈ℝ^I_1× I_2× ... × I_N is denoted as Y_(n)∈ℝ^I_n ×∏_m ≠ n I_m and is obtained by arranging 𝒴's mode-n fibers to be the columns of the resulting matrix. Unfolding the tensor 𝒴= 𝒳×_1U^(1)×_2U^(2)...×_NU^(N) =: 𝒳_n=1^NU^(n) along mode-n is equivalent to Y_(n) =U^(n) X_(n)( U^(N)⊗...U^(n+1)⊗ U^(n-1)...⊗ U^(1) )^⊤,where ⊗ is the matrix Kronecker product.In particular, (<ref>) implies that the matricization ( 𝒳×_nU^(n))_(n) =U^(n) X_(n).[Simply set U^(m) =I (the identity) for all m ≠ n in (<ref>).This fact also easily follows directly from the definition of the mode-n product.]It is worth noting that trivial inner product preserving isomorphisms exist between a tensor space ℝ^I_1× I_2× ... × I_N and any of its matricized versions (i.e., mode-n matricization can be viewed as an isomorphism between the original tensor vector space ℝ^I_1× I_2× ... × I_N and its mode-n matricized target vector space ℝ^I_n ×∏_m ≠ n I_m).In particular, the process of matricizing tensors is linear.If, for example, 𝒳, 𝒴∈ℝ^I_1× I_2× ... × I_N then one can see that the mode-n matricization of 𝒳 + 𝒴∈ℝ^I_1× I_2× ... × I_N is ( 𝒳 + 𝒴)_(n) =X_(n) +Y_(n) for all modes n ∈ [N].Tensor Rank:Unlike matrices, which have a unique definition of rank, there are multiple rank definitions for tensors including tensor rank and tensor n-rank. The rank of a tensor 𝒳∈ℝ^I_1× ... I_n× ...× I_N is the smallest number of rank-one tensors that form 𝒳 as their sum. The n-rank of 𝒳 is the collection of ranks of unfoldings X_(n) and is denoted as: n(𝒳) = ( ( X_(1)), (X_(2)),..., (X_(N)) ). Tensor inner product: The inner product of two same sized tensors 𝒳, 𝒴∈ℝ^I_1× I_2× ... × I_N is the sum of the products of their elements.⟨𝒳,𝒴⟩ = ∑_i_1=1^I_1∑_i_2=1^I_2 ...∑_i_N=1^I_N x_i_1,i_2,...,i_Ny_i_1,i_2,...,i_N.It is not too difficult to see that matricization preserves Hilbert-Schmidt/Frobenius matrix inner products, i.e., ⟨𝒳,𝒴⟩ = ⟨ X_(n),Y_(n)⟩_ F =Trace(X_(n)^⊤ Y_(n)) holds for all n ∈ [N].If ⟨𝒳,𝒴⟩ = 0, 𝒳 and 𝒴 are orthogonal.Tensor norm: Norm of a tensor 𝒳∈ℝ^I_1× I_2× ... × I_N is the square root of the sum of the squares of all its elements.∥𝒳∥ = √(⟨𝒳,𝒳⟩)= √(∑_i_1=1^I_1∑_i_2=1^I_2 ...∑_i_N=1^I_N x_i_1,i_2,...,i_N^2).The fact that matricization preserves Frobenius matrix inner products also means that it preserves Frobenius matrix norms.As a result we have that 𝒳 =X_(n)_ F holds for all n ∈ [N].If 𝒳 and 𝒴 are orthogonal and also have unit norm (i.e., have 𝒳 = 𝒴 = 1) we will say that they are an orthonormal pair. §.§ Some Useful Facts Concerning Mode-n Products and Orthogonality Let I∈ℝ^I_n × I_n be the identity matrix.Given a (low rank) orthogonal projection matrix P∈ℝ^I_n × I_n one can decompose any given tensor 𝒳∈ℝ^I_1× I_2× ... × I_N into two orthogonal tensors using Lemma <ref> (b)𝒳 = 𝒳×_nI = 𝒳×_n (( I -P) +P) = 𝒳×_n ( I -P) + 𝒳×_nP.To check that the last two summands are orthogonal one can use (<ref>) to compute that⟨𝒳×_n ( I -P),𝒳×_nP⟩ = ⟨ ( I -P)X_(n), P X_(n)⟩_ F =Trace(X_(n)^⊤ ( I -P)P X_(n)) = 0.As a result one can also verify that the Pythagorean theorem holds, i.e., that 𝒳^2 = 𝒳×_nP^2 + 𝒳×_n ( I -P) ^2.If we now regard 𝒳×_nP as a low rank approximation to 𝒳 then we can see that its approximation error𝒳 - 𝒳×_nP = 𝒳×_n ( I -P)is orthogonal to the low rank approximation 𝒳×_nP, as one would expect.Furthermore, the norm of its approximation error satisfies 𝒳×_n ( I -P) ^2 = 𝒳^2 - 𝒳×_nP^2.By continuing to use similar ideas in combination with lemma <ref> for all modes one can prove the following more general Pythagorean result (see, e.g., theorem 5.1 in <cit.>). Let 𝒳∈ℝ^I_1× I_2× ... × I_N and U^(n)∈ℝ^I_n× I_n be an orthogonal projection matrix for all n ∈ [N].Then,𝒳 - 𝒳×_1U^(1)×_2U^(2)...×_NU^(N)^2 =: 𝒳 - 𝒳_n=1^NU^(n)^2 = ∑^N_n=1𝒳_h=1^n-1 U^(h)×_n( I -U^(n)) ^2. §.§ The Higher Order Singular Value Decomposition (HoSVD) Any tensor 𝒳∈ℝ^I_1× I_2× ...× I_N can be decomposed as mode products of a core tensor 𝒞∈ℝ^I_1× I_2× ...× I_N with N orthogonal matrices U^(n)∈ℝ^I_n× I_n each of which is composed of the left singular vectors of X_(n) <cit.>:𝒳 = 𝒞×_1U^(1)×_2U^(2)...×_N U^(N)=𝒞_n=1^NU^(n)where 𝒞 is computed as 𝒞 = 𝒳×_1 ( U^(1))^⊤×_2 ( U^(2))^⊤ ...×_N ( U^(N))^⊤ . Let 𝒞_i_n=α be a subtensor of 𝒞 obtained by fixing the nth index to α. This subtensor satisfies the following properties:* all-orthogonality: 𝒞_i_n=α and 𝒞_i_n=β are orthogonal for all possible values of n, α and β subject to α≠β.⟨𝒞_i_n=α, 𝒞_i_n=β⟩ = 0 when α≠β. * ordering: ∥𝒞_i_n=1∥≥∥𝒞_i_n=2∥≥ ... ≥∥𝒞_i_n=I_n∥≥ 0 for n∈[N].§ MULTISCALE ANALYSIS OF HIGHER-ORDER DATASETSIn this section, we present a new tensor decomposition method named Multiscale HoSVD (MS-HoSVD) for an Nth order tensor, 𝒳∈ℝ^I_1× I_2× ...× I_N. The proposed method recursively applies the following two-step approach: (i) Low-rank tensor approximation, followed by (ii) Partitioning the residual (original minus low-rank) tensor into subtensors. A tensor 𝒳 is first decomposed using HoSVD as follows:𝒳= 𝒞×_1U^(1)×_2U^(2)...×_N U^(N),where the U^(n)'s are the left singular vectors of the unfoldings X_(n). The low-rank approximation of 𝒳 is obtained by𝒳̂_0= 𝒞_0×_1 Û^(1)×_2 Û^(2)...×_N Û^(N)where Û^(n)∈ℝ^I_n× r_nsare the truncated matrices obtained by keeping the first r_n columns of U^(n) and 𝒞_0=𝒳×_1 ( Û^(1))^⊤×_2 ( Û^(2))^⊤...×_N ( Û^(N))^⊤. The multilinear-rank of 𝒳̂_0, { r_1,..., r_N }, can either be given a priori, or an energy criterion can be used to determine the minimum number of singular values to keep along each mode as:r_n= argmin_i ∑_l=1^i σ^(n)_l s.t.∑_l=1^i σ^(n)_l ∑_l=1^I_nσ^(n)_l≥τ,where σ^(n)_l is the lth singular value of the matrix obtained from the SVD of the unfolding X_(n),and τ is an energy threshold. Once 𝒳̂_0 is obtained, the tensor 𝒳 can be written as 𝒳= 𝒳̂_0 + 𝒲_0,where 𝒲_0 is the residual tensor.For the first-scale analysis, to better encode the details of 𝒳, we adapted an idea similar to the one presented in <cit.>. The 0^ th-scale residual tensor, 𝒲_0 is first decomposed into subtensors as follows. 𝒲_0∈ℝ^I_1× I_2× ...× I_N is unfolded across each mode yielding W_0,(n)∈ℝ^I_n×∏_j≠ nI_j whose columns are the mode-n fibers of 𝒲_0.For each mode, rows of W_0,(n) are partitioned into c_n non-overlapping clusters using a clustering algorithm such as local subspace analysis (LSA) <cit.> in order to encourage the formation of new subtensors which are intrinsically lower rank, and therefore better approximated via a smaller HoSVD at the next scale. The Cartesian product of the partitioning labels coming from the N modes yields K=∏_i=1^Nc_i disjoint subtensors 𝒳_1,k where k∈ [K]. Let J_0^n be the index set corresponding tothe nth mode of𝒲_0 with J_0^n=[I_n],and let J_1,k^n be the index set of the subtensor 𝒳_1,k for the nth mode, where J_1,k^n⊂ J_0^n for all k ∈ [K] and n∈ [N]. Index sets of subtensors for the nth mode satisfy ⋃_k=1^K J_1,k^n= J_0^nfor all n ∈ [N]. The kth subtensor 𝒳_1,k∈ℝ^| J_1,k^1 | ×| J_1,k^2 | ×…×| J_1,k^N | is obtained by [ 𝒳_1,k(i_1, i_2,..., i_N)= 𝒲_0(J_1,k^1(i_1),J_1,k^2(i_2), ...,J_1,k^N(i_N)),;𝒳_1,k= 𝒲_0(J_1,k^1× J_1,k^2 × ...× J_1,k^N), ] where i_n ∈[ | J_1,k^n | ]. Low-rank approximation for each subtensor is obtained by applying HoSVD as: 𝒳̂_1,k = 𝒞_1,k×_1 Û^(1)_1,k×_2 Û^(2)_1,k...×_N Û^(N)_1,k, where 𝒞_1,k and Û^(n)_1,k∈ℝ^|J^n_1,k|× r^(n)_1,ks correspond to the core tensor and low-rank projection basis matrices of 𝒳_1,k, respectively. We can then define 𝒳̂_1 as the 1^ st-scale approximation of 𝒳 formed bymapping all of the subtensors onto 𝒳̂_1,k as follows:𝒳̂_1(J_1,k^1× J_1,k^2 × ...× J_1,k^n)=𝒳̂_1,k.Similarly, 1^ st scale residual tensor is obtained by𝒲_1(J_1,k^1× J_1,k^2 × ...× J_1,k^n)=𝒲_1,k,where 𝒲_1,k = 𝒳_1,k -𝒳̂_1,k. Therefore, 𝒳 can be rewritten as:𝒳=𝒳̂_0+𝒲_0= 𝒳̂_0+ 𝒳̂_1 + 𝒲_1. Continuing in this fashion the j^th scale approximation of 𝒳 is obtained by partitioning 𝒲_j-1,ks into subtensors 𝒳_j,ks and fitting a low-rank model to each one of them in a similar fashion. Finally, the j^th scale decomposition of 𝒳 can be written as: 𝒳= ∑_i=0^j𝒳̂_i + 𝒲_j.Algorithm <ref> describes the pseudo code for this approach and Fig. <ref> illustrates 1-scale MS-HoSVD.§.§ Memory Cost of the First Scale Decomposition Let 𝒳∈ℝ^I_1× I_2 × .... × I_N be an Nth order tensor. To simplify the notation, assume that the dimension of each mode is the same, i.e. I_1 = I_2 = .... = I_N=I. Assume𝒳is approximated by HoSVD as:𝒳̂ = 𝒞_H×_1U^(1)_H ×_2U^(2)_H...×_N U^(N)_H,by fixing the rank of each mode matrix as ( U^(i)_H) = r_H for i∈{ 1, 2,...,N}. Let 𝔽(·) be a function that quantifies the memory cost,then the storage cost of 𝒳 decomposed by HoSVD is 𝔽(𝒞_H) + ∑_i=1^N(𝔽( U^(i)_H))≈ r_H^N + N I r_H. For multiscale analysis at scale 1, 𝒳̂= 𝒳̂_0 + 𝒳̂_1. The cost of storing 𝒳̂_0 is 𝔽(𝒞_0) + ∑_i=1^N(𝔽(Û^(i))) ≈ r_0^N + N I r_0 where the rank of each mode matrix is fixed at ( U^(i)) = r_0 for i∈{ 1, 2,...,N}. The cost of storing 𝒳̂_1 is the sum of the storage costs for each of the K=∏ _i=1^N c(i) subtensors 𝒳̂_1,k. Assume c(i)=c for all i∈{ 1, 2,...,N} yielding c^N equally sized subtensors, and that each 𝒳̂_1,k is decomposed using the HoSVD as 𝒳̂_1,k = 𝒞_1,k×_1 Û^(1)_1,k×_2 Û^(2)_1,k...×_N Û^(N)_1,k. Let the rank of each mode matrix be fixed as (Û^(i)_1,k) = r_1 for all i∈{ 1, 2,...,N} and k∈{ 1, 2,...,K}. Then, the memory cost for the first scale is ∑_k=1^K (𝔽(𝒞_1,k)+∑_i=1^N𝔽(Û^(i)_1,k) ) ≈ c^N (r^N_1+ N I r_1c). Choosing r_1 ≲r_0c^(N-1) ensures that the storage cost does not grow exponentially so that 𝔽(𝒳̂_1)<𝔽(𝒳̂_0) since the total cost becomes approximately equal to r_0^N (1+1c^N^2-2N)+2N Ir_0. Thus, picking r_0 ≈ r_H/2 can now provide lower storage cost for the first scale analysis than for HoSVD. §.§ Computational ComplexityThe computational complexity of MS-HoSVD at the first scale is equal to the sum of computational complexity of computing HoSVD at the parent node, partitioning into subtensors and computing HoSVD for each one of the subtensors. Computational complexity of HoSVD of an N-way tensor 𝒳∈ℝ^I_1× I_2× ...× I_N where I_1= I_2= ...= I_N =I is 𝒪(N I^(N+1)) <cit.>.By assuming that the partitioning is performed using K-means (via Lloyd's algorithm) with c_i=c along each mode, the complexity partitioning along each mode is 𝒪( N I^Nci ), where i is the number of iterations used in Lloyd's algorithm.Finally, the total complexity of applying the HoSVD to c^N equally sized subtensors is 𝒪(c^NN (I/c )^(N+1)).Therefore, first scale MS-HoSVD has a total computational complexity of 𝒪(NI^(N+1) + N I^Nci + c^N N (I/c )^(N+1)).Note that this complexity is similar to that of the HoSVD whenever ci is small compared to I. The runtime complexity of these multiscale methods can be reduced even further bycomputing the HoSVDs for different subtensors in parallel whenever possible, as well as by utilizing distributed and parallel SVD algorithms such as <cit.> when computing all the required HoSVD decompositions. §.§ A Linear Algebraic Representation of the Proposed Multiscale HoSVD Approach Though the tree-based representation of the proposed MS-HoSVD approach used above in Algorithm <ref> is useful for algorithmic development, it is somewhat less useful for theoretical error analysis.In this subsection we will develop formulas for the proposed MS-HoSVD approach which are more amenable to error analysis.In the process, we will also formulate a criterion which, when satisfied, guarantees that the proposed fist scale MS-HoSVD approach produces an accurate multiscale approximation to a given tensor.Preliminaries:We can construct full size first-scale subtensors of the residual tensor 𝒲_0 ∈ℝ^ I_1 × I_2 × ... I_N from (<ref>), 𝒳|_k ∈ℝ^ I_1 × I_2 × ... I_N for all k ∈ [K], using the index sets J_1,k^n from (<ref>) along with diagonal restriction matrices.Let R_k^(n)∈{ 0, 1 }^I_n×I_n be the diagonal matrix with entries given by R_k^(n)(i,j)=1, ifi=j,j ∈ J_1,k^n0,otherwisefor all k ∈ [K], and n ∈ [N].We then define𝒳|_k := 𝒲_0 _n=1^NR_k^(n) = 𝒲_0 ×_1R_k^(1)×_2R_k^(2)...×_NR_k^(N). Thus, the kth subtensor 𝒳|_k will only have nonzero entries, given by 𝒲_0(J_1,k^1×...× J_1,k^N), in the locations indexed by the sets J_1,k^n from above.The properties of the index sets J_1,k^n furthermore guarantee that these subtensors all have disjoint support.As a result both𝒲_0 = ∑^K_k = 1𝒳|_kand⟨𝒳|_k,𝒳|_j ⟩ = 0 for all j,k ∈ [K] with j ≠ kwill always hold.Recall that we want to compute the HoSVD of the subtensors we form at each scale in order to create low-rank projection basis matrices along the lines of those in (<ref>).Toward this end we compute the top r^(n)_k≤ rank( R_k^(n)) = |J^n_1,k| left singular vectors of the mode-n matricization of each 𝒳|_k, X|_k_(n)∈ℝ^I_n ×∏_m ≠ n I_m, for all n ∈ [N].Note that X|_k_(n) =R_k^(n) X|_k_(n) always holds for these matricizations since R_k^(n) is a projection matrix.[Here we are implicitly using (<ref>).]Thus, the top r^(n)_k left singular vectors of X|_k_(n) will only have nonzero entries in locations indexed by J^n_1,k.Let Û^(n)_k∈ℝ^I_n× r^(n)_k be the matrix whose columns are these top singular vectors.As a result of the preceding discussion we can see that Û^(n)_k =R_k^(n)Û^(n)_k will hold for all n ∈ [N] and k ∈ [K]. Our low-rank projection matrices Q^(n)_k∈ℝ^I_n×I_n used to produce low-rank approximations of each subtensor 𝒳|_k can now be defined as Q^(n)_k := Û^(n)_k( Û^(n)_k)^⊤. As a consequence of Û^(n)_k =R_k^(n)Û^(n)_k holding, combined with the fact that (R_k^(n))^⊤ =R_k^(n) since each R_k^(n) matrix is diagonal, we have thatQ^(n)_k := Û^(n)_k( Û^(n)_k)^⊤ = R_k^(n)Û^(n)_k(R_k^(n)Û^(n)_k)^⊤ =R_k^(n)Û^(n)_k( Û^(n)_k)^⊤(R_k^(n))^⊤ =R_k^(n) Q^(n)_kR_k^(n)holds for all n ∈ [N] and k ∈ [K].Using (<ref>) combined with the fact that R_k^(n) is a projection matrix, we can further see thatR_k^(n) Q^(n)_k =R_k^(n)(R_k^(n) Q^(n)_kR_k^(n)) = R_k^(n) Q^(n)_kR_k^(n)=Q^(n)_k =R_k^(n) Q^(n)_kR_k^(n) = (R_k^(n) Q^(n)_kR_k^(n))R_k^(n) =Q^(n)_kR_k^(n)also holds for all n ∈ [N] and k ∈ [K].1-scale Analysis of MS-HoSVD: Using this linear algebraic formulation we are now able to re-express the the 1^ st scale approximation of 𝒳∈ℝ^ I_1 × I_2 × ... I_N, 𝒳̂_1 ∈ℝ^ I_1 × I_2 × ... I_N, as well as the 1^ st scale residual tensor tensor, 𝒲_1 ∈ℝ^ I_1 × I_2 × ... I_N, as follows (see (<ref>) – (<ref>)).We have that𝒳̂_1 = ∑^K_k = 1( 𝒳|_k _n=1^NQ^(n)_k ) = ∑^K_k = 1( 𝒲_0 _n=1^NQ^(n)_kR_k^(n))(Using Lemma <ref> and (<ref>))= ∑^K_k = 1( 𝒲_0 _n=1^NQ^(n)_k )(Using the properties in (<ref>))= ∑^K_k = 1( ( 𝒳 - 𝒳̂_0 ) _n=1^NQ^(n)_k )(Using (<ref>))holds.Thus, we see that the residual error 𝒲_1 from (<ref>) satisfies𝒳 = 𝒳̂_0+ ∑^K_k = 1( ( 𝒳 - 𝒳̂_0 ) _n=1^NQ^(n)_k ) + 𝒲_1. Having derived (<ref>), it behooves us to consider when using such a first-scale approximation of 𝒳 is actually better than, e.g., just using a standard HoSVD-based 0^ th-scale approximation of 𝒳 along the lines of (<ref>).As one might expect, this depends entirely on (i) how well the 1^ st-scale partitions (i.e., the restriction matrices utilized in (<ref>)) are chosen, as well as on (ii) how well restriction matrices of the type used in (<ref>) interact with the projection matrices used to create the standard HoSVD-based approximation in question.Toward understanding these two conditions better, recall that 𝒳̂_0 ∈ℝ^ I_1 × I_2 × ... I_N in (<ref>) is defined as 𝒳̂_0 = 𝒳_n=1^NP^(n) = 𝒳×_1P^(1)×_2P^(2)…×_NP^(N)where the orthogonal projection matrices P^(n)∈ℝ^I_n × I_n are given by P^(n) = Û^(n)( Û^(n))^⊤ for the matrices Û^(n)∈ℝ^I_n× r_n used in (<ref>).For simplicity let the ranks of the P^(n) projection matrices momentarily satisfy r_1 = r_2 = … = r_N =: r_0 (i.e., let them all be rank r_0 < max_n { rank( X_(n)) }).Similarly, let allthe ranks, r^(n)_k, of the 1^ st scale projection matrices Q^(n)_k in (<ref>) be r_1 for the time being.Motivated by, e.g., the memory cost analysis of Section <ref> above, one can now ask when the multiscale approximation error, 𝒲_1, resulting from (<ref>) will be less than a standard HoSVD-based approximation error, 𝒳 - 𝒳̅_0, where𝒳̅_0 := 𝒳_n=1^N P̅^(n) = 𝒳×_1 P̅^(1)×_2 P̅^(2)…×_N P̅^(N),and each orthogonal projection matrix P̅^(n) is of rank r̅_n = r_H ≥ 2 r_0 ≥ r_0 + c^N-1r_1 (i.e., where each P̅^(n) projects onto the top r_H left singular vectors of X_(n)).In this situation having both 𝒲_1< 𝒳 - 𝒳̅_0 and r_H ≥ 2 r_0 ≥ r_0 + c^N-1r_1 hold at the same time would imply that one could achieve smaller approximation error using MS-HoSVD than using HoSVD while simultaneously achieving better compression (recall Section <ref>).In order to help facilitate such analysis we prove error bounds in Appendix <ref> that are implied by the choice of a good partitioning scheme for the residual tensor 𝒲_0 in (<ref>) – (<ref>).In particular, with respect to the question concerning how well the 1^ st-scale approximation error, 𝒲_1, from (<ref>) might compare to the HoSVD-based approximation error 𝒳 - 𝒳̅_0, we can use the following notion of an effective partition of 𝒲_0. The partition of 𝒲_0 formed by the restriction matrices R_k^(n) in (<ref>) – (<ref>) will be called effective if there exists another pessimistic partitioning of 𝒲_0 via (potentially different) restriction matrices {R̃_k^(n)}^K_k=1 together with a bijection f:[K]→[K] such that∑_n=1^N 𝒳|_k×_n ( I -Q^(n)_k ) ^2≤∑_n=1^N 𝒲_0 ×_n R̃_f(k)^(n)( I-P̃^(n))_h≠ n^NR̃_f(k)^(h)^2holds for each k∈ [K].In (<ref>) the {P̃^(n)} are the orthogonal projection matrices obtained from the HoSVD of 𝒲_0 with ranks r̃_n = r_H (i.e., where each P̃^(n) projects onto the top r̃_n = r_H left singular vectors of the matricization W_0,(n)). In Appendix <ref>, weshow that (<ref>) holding for 𝒲_0 implies that the error 𝒲_1 resulting from our 1^ st-scale approximation in (<ref>) is less than an upper bound of the type often used for HoSVD-based approximation errors of the form 𝒳 - 𝒳̅_0 (see, e.g., <cit.>).In particular, we prove the following result. Suppose that (<ref>) holds.Then, the first scale approximation error given by MS-HoSVD in (<ref>) is bounded by𝒲_1 ^2 = 𝒳-𝒳̂_0-𝒳̂_1 ^2 ≤∑_n=1^N 𝒳×_n (I-P̅^(n)) ^2,where {P̅^(n)} are low-rankprojection matrices of rank r̅_n = r̃_n = r_H obtained from the truncated HoSVD of 𝒳 as per (<ref>). See Appendix <ref>. Theorem <ref> implies that 𝒲_1 may be less than 𝒳 - 𝒳̅_0 when (<ref>) holds.It does not, however, actually prove that 𝒲_1 ≤𝒳 - 𝒳̅_0 holds whenever (<ref>) does.In fact, directly proving that 𝒲_1 ≤𝒳 - 𝒳̅_0 whenever (<ref>) holds does not appear to be easy.It also does not appear to be easy to prove the error bound in theorem <ref> without an assumption along the lines of (<ref>) which simultaneously controls both (i) how well the restriction matrices utilized to partition 𝒲_0 in (<ref>) are chosen, as well as (ii) how poorly (worst case) restriction matrices interact with the projection matrices used to create standard HoSVD-based approximations of 𝒲_0 and/or 𝒳.The development of simpler and/or weaker conditions than (<ref>) which still yield meaningful error guarantees along the lines of theorem <ref> is left as future work.See Appendix <ref> for additional details and comments, and Appendix <ref> below for an example illustrating Theorem <ref> on an idealized tensor..§.§ Adaptive Pruning in Multiscale HoSVD for Improved Performance In order to better capture the local structure of the tensor, it is important to look at higher scale decompositions. However, as the scale increases, the storage cost and computational complexity will increase making any gain in reconstruction error potentially not worth the additional memory cost. For this reason, it is important to carefully select the subtensors adaptively at higher scales. To help avoid the redundancy in decomposition structure we propose an adaptive pruning method across scales.In adaptive pruning, the tree is pruned by minimizing the following cost function ℍ = Error + λ· Compression similar to the rate-distortion criterion commonly used by compression algorithms where λ is the trade-off parameter <cit.>. To minimize this function we employ a greedy procedure similar to sequential forward selection <cit.>. First, the root node which stores 𝒳̂_0 is createdand scale-1 subtensors 𝒳̂_1,k are obtained from the 0th order residual tensor 𝒲̂_0 as discussed in Section <ref>. These subtensors are stored in a list and the subtensor which decreases the cost function the most is then added to the tree structure under its parent node. Next, scale-2 subtensors belonging to the added node are created and added to the list. All of the scale-1 and scale-2 subtensorsin the list are again evaluated to find the subtensor that minimizes the cost function. This procedure is repeated until the cost function ℍ converges or the decrease is minimal.A pseudocode of the algorithm is given in Algorithm <ref>. It is important to note that this algorithm is suboptimal similar to other greedy search methods.§ DATA REDUCTIONIn this section we demonstrate the performance of MS-HoSVD for tensor type data reduction on several real 3-mode and 4-mode datasets as compared with three other tensor decompositions: HoSVD, H-Tucker, and T-Train. The performance of tensor decomposition methods are evaluated in terms of reconstruction error and compression rate. In the tables and figures below the error rate refers to the normalized tensor approximation error 𝒳 - 𝒳̂_F/𝒳_F and the compression rate is computed as # total bits to store 𝒳̂/# total bits to store 𝒳. Moreover, we show the performance of the proposed adaptive tree pruning strategy for data reduction.§.§ Datasets §.§.§ PIE dataset A 3-mode tensor 𝒳∈ℝ^ 244× 320× 138 is created from PIE dataset <cit.>. The tensor contains 138 images from 6 different yaw angles and varying illumination conditions collected from a subject where each image isconverted to gray scale. Fig. <ref> illustrates the images from different frames of the PIE dataset.§.§.§ COIL-100 dataset The COIL-100 database contains 7200 images collected from 100 objects where the images of each object were taken at pose intervals of 5^∘.A 4-mode tensor 𝒳∈ℝ^ 128× 128× 72 × 100 is created from COIL-100 dataset <cit.>. The constructed 4-mode tensor contains 72 images of size 128× 128 from 100 objects where each image is converted to gray scale. In Fig. <ref>, sample images of four objects taken from different angles can be seen. §.§.§ The Cambridge Hand Gesture DatasetThe Cambridge hand gesture database consists of900 image sequences of nine gesture classes of three primitive hand shapes and three primitive motions where each class contains 100 image sequences (5 different illuminations × 10 arbitrary motions × 2 subjects). In Fig. <ref>, sample image sequences collected for nine hand gestures can be seen. The created 4-mode tensor 𝒳∈ℝ^ 60× 80× 30 × 900 contains 900 image sequences of size 60× 80 × 30 where each image is converted to gray scale. §.§.§ Hyperspectral ImageIn this experiment, we used a hyperspectral image from <cit.> to create a 3-mode tensor 𝒳∈ℝ^ 201× 250× 148 where the modes are rows, columns and spectral bands, respectively. Fig. <ref> illustrates the images from different spectral bands of the hyperspectral image. §.§ Data Reduction with Fixed Rank In the following experiments, clustering is performed by LSA and the cluster number along each mode is chosen as c_i=2. The rank used in HoSVD is selected based on the size of the datasets and gradually increased to illustrate the relationship between reconstruction error and compresion rate.In MS-HoSVD with 1-scale, rank of each scale is selected according to the criterion R_1 ≤R_0c^(N-1) derived in Section <ref>. As seen in Fig. <ref>, MS-HoSVD provides better compression performance for all datasets. Moreover, selecting a smaller multilinear rank yielding lower compression rate increases the normalized reconstruction error for both MS-HoSVD and HoSVD as expected. Therefore, as the compression rate goes down the performance of HoSVD and MS-HoSVD become comparable to each other. §.§ Data Reduction Experiments In this section, we evaluate the performance of MS-HoSVD for 1 and 2-scale decompositions compared to HoSVD, H-Tucker and T-Train decompositions. In the following experiments, tensor partitioning is performed by LSAand the cluster number along each mode is chosen as c_i=2. The rank used in HoSVD isselected adaptively using the energy criterion as per Section <ref>'s (<ref>). In our experiments, we performed MS-HoSVD with τ= 0.7 and τ= 0.75 and we kept τ the same for each scale. For the same compression rates as the MS-HoSVD, the reconstruction error of HoSVD, H-Tucker and T-Train models are computed.Fig. <ref> explores the interplay between compression rate and approximation error for MS-HoSVD in comparison to HoSVD, H-Tucker and T-Train for PIE, COIL-100 and hand gesture datasets. Starting from the left in Figs.<ref>(a), <ref>(b) and <ref>(c), the first two compression rates correspond to 1-scale MS-HoSVD with τ= 0.7 and τ= 0.75, respectively while the last two are obtained from 2-scale approximation with τ= 0.7 and τ= 0.75, respectively. As seen in Fig.<ref>,MS-HoSVD outperforms other approaches with respect to reducingPIE, COIL-100 and hand gesture tensors at varying compression rates. Moreover, adding the 2^ nd scaleincreases the storage requirements while decreasing the error of MS-HoSVD. Fig. <ref> illustrates the influence of scale on the visual quality of the reconstructed images.As expected, introducing additional finer scales into a multiscale approximation of video data improves image detail in each frame. Moreover, the data reduction performance of T-Train is seen to be slightly better than H-Tucker in most of the experiments. §.§ Data Reduction with Adaptive Tree Pruning In this section, we evaluate the performance of adaptive tree pruning multiscale decompositions. In the pruning experiments, clustering is performed by LSA and the cluster number along each mode is chosen as c_i=2. The rank used in HoSVD is selected adaptively based on the energy threshold τ = 0.7. A pruned version of 2-scale MS-HoSVD that greedily minimizes thecost function ℍ= Error + λ· Compression for is implemented for PIE, COIL-100 and Hand Gesture datasets with varying λ values as reported in Tables<ref>, <ref> and <ref>.As λ increases, reducing the compression rate becomes more important and the algorithm prunes the leaf nodes more.For example, a choice ofλ = 0.75 prunes all of the nodes corresponding to thesecond scale subtensors for PIE data (see Table <ref>).As can be seen from Tables <ref>, <ref>, and <ref>, the best tradeoffs achieved between reconstruction error and compression rate occur at different λ values for different datasets. For example, for PIE data, increasing λ value does not provide much change in reconstruction error while increasing the compression. On the other hand, for COIL-100, λ=0.75 provides a good tradeoff between reconstruction error and compression rate. Small changes in λ yield significant effects on pruning the subtensors of 2-scale decomposition of hand gesture data.Fig. <ref> illustrates the performance of the pruning algorithm on the PIE dataset. Applying pruning with λ=0.25 increases the reconstruction error from 0.0276 to 0.0506 while reducing the compression rate by a factor of 4 (Table <ref>). As seen in Fig. <ref>, the 2^ nd-scale approximation obtained by the adaptive pruning algorithm preserves most of the facial details in the image.Performance of the pruning algorithm reported in Tables <ref>, <ref> and <ref> is also compared with HoSVD, H-Tucker and T-Train decompositions in Fig. <ref>. As seen in Fig.<ref> (b) and (c),MS-HoSVD outperforms other approaches for compressing COIL-100 and Hand Gesture datasets at varying compression rates. However, for PIE data, the performance of MS-HoSVD and HoSVD are very close to each other while both approaches outperform H-Tucker and T-Train, as can be seen in Fig.<ref> (a). In Fig. <ref>, sampleframes of PIE data reconstructed by T-Train (top-left), H-Tucker (top-right), HoSVD (bottom-left) and pruned MS-HoSVD with 2-scales (bottom-right) are shown. It can be easily seen that the reconstructed images by H-Tucker and T-Train are more blurred than the ones obtained by HoSVD and MS-HoSVD. One can also see the facial details captured by MS-HoSVD are clearer than HoSVD although the performances of both algorithms are very similar to each other. The reason for capturing facial details better by MS-HoSVD is that the higher-scale subtensors encode facial details.§ FEATURE EXTRACTION AND CLASSIFICATIONIn this section, we evaluate the features extracted from MS-HoSVD for classification of 2-mode and 3-mode tensors containing object images and hand gesture videos. The classification accuracy of MS-HoSVD features are compared to the features extracted by HoSVD and T-Train using three different classifiers: 1-NN, Adaboost and Naive Bayes.§.§ COIL-100 Image DatasetFor computational efficiency, each image was downsampled to a gray-scale image of 32 × 32 pixels.Number of images per object used for training data was gradually increased from 18 to 54 and selected randomly. A 3-mode tensor 𝒳^tr∈ℝ^32× 32× I_tr is constructed from training imageswhere I_tr∈ 100×{18,36, 54 } and the rest of the images are used to create the testing tensor 𝒳^te∈ℝ^32× 32× I_tewhere I_te=7200-I_tr. §.§ The Cambridge Hand Gesture DatasetFor computational efficiency, each imagewas downsampled to a gray-scale image of 30 × 40 pixels. Number of image sequences used for training data gradually increased from 25 to 75 per gesture and selected randomly. A 4-mode tensor 𝒳^tr∈ℝ^30× 40×30 × I_tr is constructed from training image sequenceswhere I_tr∈ 9×{ 25,50,75 } and the rest of the image sequences are used to create the testing tensor 𝒳^te∈ℝ^30× 40×15× I_tewhere I_te=900-I_tr. §.§ Classification Experiments §.§.§ TrainingFor MS-HoSVD, the training tensor 𝒳^tr is decomposed using 1-scale MS-HoSVD as follows. Tensor partitioning is performed by LSAand the cluster number along each mode is chosen as c={ 2, 3,1} yielding 6 subtensors for COIL-100 dataset and c={ 2, 2, 3,1} yielding 12 subtensors for hand gesture dataset. We did not partition the tensor along the last mode that corresponds to the classes to make the comparison with other methods fair. The rank used in 0th scale is selected based on the energy criterion with τ=0.7, while the full rank decomposition is used for the 1st scale. The 0th scale approximation𝒳̂_0^tr= 𝒞_0^tr×_1 Û^tr,(1)×_2 Û^tr,(2)...×_N Û^tr,(N)provides the 0^ th-scale core tensor 𝒞_0^tr, factor matrices Û^tr,(i) and residual tensor 𝒲_0^tr= 𝒳^tr- 𝒳̂_0^tr. Next, the 0^ th-scale feature tensor 𝒮^tr_0 for the training data is created by projecting 𝒳^trs onto the first N-1 factor matrices U^tr,(i) as:𝒮^tr_0= 𝒳^tr×_1 (Û^tr,(1))^⊤×_2 (Û^tr,(2))^⊤...×_N-1(Û^tr,(N-1))^⊤.Subtensors of𝒲_0^tr obtained by 𝒳^tr_1,k= 𝒲_0^tr(J^tr,1_1,k× J^tr,2_1,k× ...× J^tr,N_1,k) are used to extract 1^st-order core tensors 𝒞_1,k and factor matrices U^tr,(i)_1,k as: 𝒳_1,k^tr = 𝒞_1,k^tr×_1U^tr,(1)_1,k×_2U^tr(2)_1,k...×_NU^tr(N)_1,k. 1st-order feature tensors are then created by projecting 𝒳^tr_1,ks onto the first N-1 factor matrices U^tr,(i)_1,k as:𝒮^tr_1,k= 𝒳^tr_1,k×_1 ( U^tr,(1)_1,k)^⊤×_2 ( U^tr,(2)_1,k)^⊤...×_N-1( U^tr,(N-1)_1,k)^⊤.Unfolding the feature tensors 𝒮^tr_0 and 𝒮^tr_1,k along the sample mode Nand concatenating them to each other yields a high dimensional feature vector for each of the training samples. From these vectors, N_f features with the highestFisher Score <cit.> are selected to form the lower-dimensional feature vectors x^tr∈ℝ^N_f× 1 for each training sample where the number of features (N_f) is determined 100 for COIL-100 and 200 for hand gesture dataset emprically. For HoSVD and T-Train, full rank decompositions are computed and feature vectors are created by selecting N_ffeatures with the highest Fisher Score from the core tensors as described above. For T-Train, the procedure described in <cit.> is used without reducing the dimensionality. §.§.§ Testing To create the 0^ th-order feature tensor 𝒮^te_0 for testing samples, first, the testing tensor 𝒳_te is projected ontoÛ^tr,(i) where i∈[N-1] as:𝒮^te_0= 𝒳^te×_1 (Û^tr,(1))^⊤×_2 (Û^tr,(2))^⊤...×_N-1(Û^tr,(N-1))^⊤. The 0^ th-order residual tensor 𝒲_0^te of testing data is computed as 𝒲_0^te=𝒳^te- 𝒳^te_n=1^n=N(Û^tr,(n))^⊤. Then 1^ st-order subtensors are created from 𝒲_0^te using the same partitioning as the 0th order training residual tensor 𝒲_0^tr as 𝒳^te_1,k= 𝒲_0^te(J^tr,1_1,k× J^tr,2_1,k× ...× J^tr,N_1,k). The 1^ st-order feature tensors 𝒮^te_1,k for the testing samples are then obtained by𝒮^te_1,k= 𝒳^te_1,k×_1 ( U^tr,(1)_1,k)^⊤×_2 ( U^tr,(2)_1,k)^⊤...×_N-1( U^tr,(N-1)_1,k)^⊤. Similar to the training step, unfolding the feature tensors 𝒮^te_0 and 𝒮^te_1,k along the sample mode Nand concatenating them with each other yields high dimensional feature vectors for the testing samples. Thefeaturescorresponding to the features selected from the training step are used to form the feature vectors for testing samples x^te∈ℝ^N_f× 1.A similar two-step procedure, i.e projecting the testing tensor onto training factor matricesfollowed by selecting N_f features, is used to create testing feature vectorsfor HoSVD and T-Train. Discrimination performance of thefeature vectors are evaluated using different classifiers including 1-NN, Adaboost and Naive Bayes.Tables <ref> and <ref> summarize the classification accuracy for the three methods using three different classifiers for COIL-100 and Hand gesture data sets, respectively. As it can be seen from these Tables, for both data sets and all classifiers MS-HoSVD performs the best except for a Naive Bayes Classifier trained by 25% of the data to classify hand gesture dataset. As seen in Tables <ref> and <ref>, the performance of HoSVD, T-Train and MS-HoSVD become close to each other asthe size of the training dataset increases, as expected. The reason for the superior performance of MS-HoSVD is that MS-HoSVD captures the variations and nonlinearities across the modes such as rotation or translation better than the other methods. In both of the datasets used in this section, the images are rotated across different frames. Since these nonlinearities are encoded in the higher-scale (1^ st-scale) featureswhile the average characteristics, which are the same as HoSVD, are captured by the lower scale (0^ th-scale) MS-HoSVD features, the classification performance of the MS-HoSVD is slightly better than HoSVD. It is also seen that T-Train features are not as good as MS-HoSVD and HoSVD features for capturing rotations and translations in the data and requires larger training set to reach the performance of MS-HoSVD and HoSVD. § CONCLUSIONS In this paper, we proposed a new multi-scale tensor decomposition technique for better approximating the local nonlinearities in generic tensor data. The proposed approach constructs a tree structure by considering similarities along different fibers of the tensor and decomposes the tensor into lower dimensional subtensors hierarchically. A low-rank approximation of each subtensor is then obtained by HoSVD. We also introduced a pruning strategy to find the optimum tree structure by keeping the important nodes andeliminating redundancy in the data. The proposed approach is applied to a set of 3-way and 4-way tensors to evaluate its performance on both data reduction and classification applications. As it is illustrated in sections <ref> and <ref>, any application involving tensor data reduction and classification would benefit from the proposed method. Some examples include hyper-spectral image compression, high-dimensional video clustering and functional connectivity network analysis in neuroscience.Although this paper focused on the integration of a single existing tensor factorization technique (i.e., the HoSVD) into a clustering-enhanced multiscale approximation framework, we would like to emphasize that the ideas presented herein are significantly more general.In principal, for example, there is nothing impeding the development of multiscale variants of other tensor factorization approaches (e.g., PARAFAC, T-Train, H-Tucker, etc.) in essentially the same way.In this paper it is demonstrated that the use of the HoSVD as part of a multiscale approximation approach leads to improved compression and classification performance over standard HoSVD approaches.However, this paper should additionally be considered as evidence that similar improvements are also likely possible for other tensor factorization-based compression and classification schemes, as well as for other related applications.Future work will consider automatic selection of parameters such as the number of clustersand the appropriate rank along each mode. The computational efficiency of the proposed method can also be improved through parallelization of the algorithm by, e.g., constructing the disjoint subtensors at each scale in parallel, as well as by utilizing distributed and parallel SVD algorithms such as <cit.> when computing their required HoSVD decompositions (see also, e.g., <cit.> for other related parallel implementations). Such efficient implementations will enable the computation of finer-scale decompositions for higher-order and higher-dimensional tensors.IEEEtran §.§ Effective Partitioning, and Error AnalysisIn order to facilitate error analysis for the 1-scale MS-HoSVD that is similar to the types of error analysis available for various HoSVD-based low-rank approximation strategies (see, e.g., <cit.>), we will engage in a more in depth discussion of condition (<ref>) herein.Recall that the partition of 𝒲_0 formed by the restriction matrices R_k^(n) in (<ref>) – (<ref>) is called effective if there exists another pessimistic partitioning of 𝒲_0 via restriction matrices {R̃_k^(n)}^K_k=1 together with a bijection f:[K]→[K] such that∑_n=1^N 𝒳|_k×_n ( I -Q^(n)_k ) ^2≤∑_n=1^N 𝒲_0 ×_n R̃_f(k)^(n)( I-P̃^(n))_h≠ n^NR̃_f(k)^(h)^2holds for each k∈ [K]. In (<ref>) the {P̃^(n)} are the orthogonal projection matrices obtained from the HoSVD of 𝒲_0 with ranks r̃_n ≥r̅_n ≥ r_n (i.e., where each P̃^(n) projects onto the top r̃_n left singular vectors of the matricization W_0,(n)). Below we will show that (<ref>) holding for 𝒲_0 implies that the error 𝒲_1 resulting from our 1^ st-scale approximation in (<ref>) is less than an upper bound of the type given for a high-rank standard HoSVD-based approximation (<ref>) in <cit.>.Considering condition (<ref>) above, we note that experiments show that it is regularly satisfied on real datasets when (i) the effective restriction matrices { R_k^(n)}^K_k=1 in (<ref>) – (<ref>) are first formed by clustering the rows of each unfolding of 𝒲_0 using, e.g., local subspace analysis (LSA), after which (ii) pessimistic restriction matrices {R̃_k^(n)}^K_k=1 are randomly generated in order to create another (random) partition of 𝒲_0 into K different disjoint subtensors for comparison.The bijection f can then be created by, e.g., (i) sorting the left-hand side errors in (<ref>) for each k ∈ [k], (ii) sorting the right-hand side errors in (<ref>) for each k ∈ [K], and then (iii) matching the largest left-hand and right-hand errors for comparision, the second largest left-hand and right-hand errors for comparision, etc..When checked in this way the sorted right-hand side errors often dominate (entrywise) the sorted left-hand side errors for various reasonable ranks r̅_n = r̃_n = r_H = r_0 + c^N-1r_1 (as a function of r_0 and r_1 with, e.g., c = 2) on every dataset considered in Section <ref> above, thereby verifying that (<ref>) does indeed regularly hold.We will now begin to prove Theorem <ref> with a lemma that shows our subtensor-based approximation of 𝒲_0 is accurate whenever (<ref>) is satisfied. Let 𝒲_0 = 𝒳 - 𝒳̂_0 ∈ℝ^ I_1 × I_2 × ... I_N.Suppose that { R_k^(n)} is a collection of effective restriction matrices that form an effective partition of 𝒲_0 with respect to a pessimistic partition formed via pessimistic restriction matrices {R̃_k^(n)} as per (<ref>) above. Similarly, let P̃^(n) be the rank r̃_n ≥r̅_n ∀ n orthogonal projection matrices from (<ref>) obtained via the truncated HoSVD of 𝒲_0 as above. Then, 𝒲_0 -𝒳̂_1 ^2 = ( 𝒳 - 𝒳̂_0 ) - ∑^K_k = 1( ( 𝒳 - 𝒳̂_0 ) _n=1^NQ^(n)_k ) ^2≤∑_n=1^N ( 𝒳 - 𝒳̂_0 ) ×_n (I-P̃^(n))^2. We have that 𝒲_0 -𝒳̂_1 ^2= 𝒲_0 -∑^K_k = 1𝒲_0 _n=1^NQ^(n)_k ^2 (Using (<ref>) and (<ref>)) = ∑_k=1^K 𝒲_0 _n=1^NR_k^(n) -∑_k=1^K 𝒲_0 _n=1^NQ^(n)_kR_k^(n)^2(Using (<ref>), (<ref>), and (<ref>)) = ∑_k=1^K 𝒲_0 _n=1^N (R_k^(n) -Q^(n)_kR_k^(n))^2(Using Lemma <ref>) = ∑_k=1^K 𝒳|_k _n=1^N (I -Q^(n)_k ) ^2.(Using Lemma <ref>, (<ref>), (<ref>), and support disjointness) Applying lemmas <ref> and <ref> to (<ref>) we can now see that 𝒲_0 -𝒳̂_1 ^2 = ∑_k=1^K ∑_n=1^N 𝒳|_k _h=1^n-1 Q^(h)_k ×_n( I -Q^(n)_k ) ^2 ≤ ∑_k=1^K ∑_n=1^N 𝒳|_k×_n ( I -Q^(n)_k ) ^2 since the Q^(n)_k matrices are orthogonal projections.Using assumption (<ref>) we now get that 𝒲_0 -𝒳̂_1 ^2 ≤ ∑_k=1^K ∑_n=1^N 𝒲_0 ×_n R̃_k^(n)( I-P̃^(n)) _h≠ n^NR̃_k^(h)^2 = ∑_n=1^N 𝒲_0 ×_n ( I-P̃^(n)) ^2 where we have used the fact that the pessimistic restriction matrices R̃_k^(n) partition 𝒲_0 in the last line. Lemma <ref> indicates that the error in approximating 𝒲_0 via low-rank approximations of its effective subtensors is potentially smaller than the error obtained by approximating 𝒲_0 via (higher-rank) truncated HoSVDs whenever (<ref>) holds.[That is, the upper bound on the error provided by Lemma <ref> is less than or equal to the upper bound on the error for truncated HoSVDs provided by, e.g., <cit.> when/if (<ref>) holds.]The following theorem shows that this good error behavior extends to the entire 1^ st-scale approximation provided by (<ref>) whenever (<ref>) holds. Let 𝒳∈ℝ^I_1× I_2...× I_N.Suppose that (<ref>) holds.Then, the first-scale approximation error given by MS-HoSVD (<ref>) is bounded by 𝒲_1 ^2 = 𝒳-𝒳̂_0-𝒳̂_1 ^2 ≤ ∑_n=1^N 𝒳×_n (I-P̅^(n)) ^2 where {P̅^(n)} are low-rankprojection matrices of rank r̅_n ≥ r_n obtained from the truncated HoSVD of 𝒳 as per (<ref>). Using (<ref>) and (<ref>) together with lemma <ref> we can see that 𝒲_1 ^2 = 𝒳-𝒳̂_0-𝒳̂_1 ^2 = 𝒲_0 -𝒳̂_1 ^2≤∑_n=1^N ( 𝒳 - 𝒳̂_0 ) ×_n (I-P̃^(n))^2 ≤∑_n=1^N ( 𝒳 - 𝒳̂_0 ) ×_n ( I-Q̅^(n)) ^2 where Q̅^(n)∈ℝ^I_n × I_n is the orthogonal projection matrix of rank r̃_n which projects onto the subspace spanned by the top r̃_n left singular vectors of X_(n).Here (<ref>) holds because the orthogonal projection matrices P̃^(n) are chosen in (<ref>) so that P̃^(n) W_0,(n) is a best possible rank r̃_n approximation to W_0,(n).As a result, we have that ( 𝒳 - 𝒳̂_0 ) ×_n (I-P̃^(n))^2 = (I-P̃^(n))W_0,(n)^2_ F≤(I-Q̅^(n))W_0,(n)^2_ F = ( 𝒳 - 𝒳̂_0 ) ×_n ( I-Q̅^(n)) ^2 must hold for each n ∈ [N]. Continuing from (<ref>) we can use the definition of 𝒳̂_0 in (<ref>) to see that 𝒲_1 ^2≤∑_n=1^N ( 𝒳 - 𝒳_h=1^NP^(h)) ×_n ( I-Q̅^(n)) ^2 = ∑_n=1^N 𝒳×_n ( I-Q̅^(n))- 𝒳_h=1^NP^(h)×_n ( I-Q̅^(n)) ^2 by lemma <ref>.Due to the definition of Q̅^(n) together with the fact that its rank is r̃_n ≥ r_n we can see that ( I-Q̅^(n))P^(n) =0.As a consequence, lemma <ref> implies that 𝒳_h=1^NP^(h)×_n ( I-Q̅^(n)) =0 for all n ∈ [N].Continuing from (<ref>) we now have that 𝒲_1 ^2 ≤∑_n=1^N 𝒳×_n ( I-Q̅^(n)) ^2. Again, appealing to the definition of both Q̅^(n) and P̅^(n) in (<ref>), combined with the fact that r̃_n ≥r̅_n, finally yields the desired result. We refer the reader to the strong empirical performance of MS-HoSVD in Section <ref> for additional evidence supporting the utility of (<ref>) as a means of improving the compression performance of standard HoSVD-based compression techniques.In addition, we further refer the reader to Section <ref> where it is empirically demonstrated that MS-HoSVD is also capable of selecting more informative features than HoSVD-based methods for the purposes of classification.These two facts together provide strong evidence that combining the use of clustering-enhanced multiscale approximation with existing tensor factorization techniques can lead to improved performance in multiple application domains.§.§ Experiment for Error Analysis In this experiment we evaluate the error obtained by the 1^ st-scale MS-HoSVD analysis of a tensor along the lines of the model described in Section <ref>. Herein we consider a three-way tensor 𝒳∈ℝ^20×20×20 that is the sum of two tensors as 𝒳=𝒳_0 + 𝒳_1 where𝒳_0∈ℝ^20×20×20 has n-rank ( 2, 2, 2 ), and 𝒳_1∈ℝ^20×20×20 is formed by concatenating 8 subtensors 𝒳_k∈ℝ^10×10×10 each also with n-rank ( 2, 2, 2 ). Low-rank approximations for 𝒳 and its subtensors are always obtained via the truncated HoSVD.The 1-scale MS-HoSVD is applied with the ground truth partitions R_k^(n), partitions provided by Local Subspace Analysis (LSA) clustering R̂_k^(n), and also with randomly chosen partitions R̃_k^(n) of the 0^ th-scale residual error 𝒲_0 into 8 different 10×10×10 subtensors. For the LSA clustering, the cluster numbers are selected as 2 along each mode also yielding 8 subtensors.The 0^ th-scale n-rank for MS-HoSVD is selected as ( 2, 2, 2 ), and the 1^ st-scale ranks are varied in the experiments as shown in Table <ref>. The normalized reconstruction error computed for these varying 1^ st-scale n-ranks can also be seen in Table <ref>. As seen there, using ground truth partition provides lower-rank subtensors, and using clustering as part of the 1-scale MS-HoSVD leads to much better approximations than HoSVD does in general.In addition, the left (LHS) and right (RHS) sides in Theorem <ref> are computed where the first-scale projections Q^(n)_k each have rank r_1=2 and are obtained via both ground truth partitioning and clustering after the 0^ th-scale P^(n) are obtained from the truncated HoSVD of 𝒳 with r_0 = 2.For comparison the P̅^(n) are also computed from the truncated HoSVD of 𝒳 with varying ranks r̅_n=κ r_0 for κ∈{ 2 ,3 ,4}. In Table <ref>, we report the mean value and standard deviation of both the right-hand side (RHS) and left-hand side (LHS) of the error bound in Theorem <ref>. As seen in Table <ref>, Theorem <ref> holds for the 1^ st-scale approximation of 𝒳 via MS-HoSVD since the RHS errors based on the P̅^(n) projections are larger than the LHS errors no matter whether the Q^(n)_k are obtained via ground truth partitioning or LSA clustering. | http://arxiv.org/abs/1704.08578v3 | {
"authors": [
"Alp Ozdemir",
"Ali Zare",
"Mark A. Iwen",
"Selin Aviyente"
],
"categories": [
"cs.NA"
],
"primary_category": "cs.NA",
"published": "20170427140246",
"title": "Multiscale Analysis for Higher-order Tensors"
} |
Polarization-based Tests of Gravity with the Stochastic Gravitational-Wave Background Eric Thrane December 30, 2023 ===================================================================================== In this manuscript we present a fast gpu implementation for tomographic reconstruction of large datasets using data obtained at the Brazilian synchrotron light source. The algorithm is distributed in a cluster with 4 gpu's through a fast pipeline implemented in c programming language.Our algorithm is theoretically based on a recently discovered low complexity formula, computing the total volume within O(N^3log N) floating point operations; much less than traditional algorithms that operates within O(N^4) flops over an input data of size O(N^3). The results obtained with real data indicate that a reconstruction can be achieved within 1 second provided the data is transferred completely to the memory.§ INTRODUCTION In this manuscript, we present a fast implementation of the well-known filtered-backprojection algorithm (fbp) <cit.>, which has the ability to reproduce reliable image reconstructions in a reasonable amount of time, before taking further decisions. The fbp is easy to implement and can be used to take fast decisions about the quality of the measurement, i.e., sample environment, beam-line conditions, among others. Figure <ref> shows the fluxogram for an ideal tomographic experiment.A synchrotron facility able to measure a three-dimensional dataset Y within few seconds, needs a fast reconstruction algorithm able to provide a fast "preview" of the tomography within the same amount of time (Fig.<ref>.(a.1)). If the experimental conditions are not satisfactory, the quality of the reconstruction will decrease, and the researcher can decide either to make another scan, or to process later the data using advanced reconstruction algorithms or even high quality segmentation methods (Fig.<ref>.(d)). The difficulty here is that the fbp algorithm consist basically in two mathematical operators, which are filtering and backprojection. Filtering is fft based since is a low-pass convolution operation <cit.>. Backprojection, on the other hand, is defined as an average through all the x-rays passing at a given pixel; therefore presenting high computational complexity of O(N^3) for an image of N^2 pixels. The brute-force approach to compute the backprojection operator can be made extremely slow, even using a gpu implementation. Sophisticated ray-tracing strategies can also be used to make the running time faster and others analytical strategies reduce the backprojection complexity to O(N^2log N), see <cit.>. Our approach to compute the backprojection is called bst - as an acronym to backprojection-slice theorem <cit.> - having the same low complexityof O(N^2 log N) although easier to implement than his competitors, producing less numerical artifacts and following a more traditional "gridding strategy" similar as <cit.>. There are several others reconstruction softwares reproducing quasi-real time reconstructions, <cit.>. The computational gain of bst over the brute-force approach for computing the backprojection operator is presented using a fast pipeline for the data access. Our results indicatethat reconstructions through bst can be performed within 1 second using the ibm/minsky (4 nvidia P100) for datasets of size 2048× 2048 × 2048. A comparison is made with asgi/tesla using 4 nvidia K80. To make a reliable comparison, all codes were implemented without taking advantages of the nvidia-nvlink. Also, we have tested our algorithm at small dimensions using only one gpu. Devices like jetson TX1, indicates that a fast reconstruction is possible for images of size 512× 512 × 512, a conventional gpu like gt740M usually coupled with domestic notebooks can handle datasets of size 1024× 1024 × 1024 and finally a titan-X coupled with a standard pc handle volumes of size 2048× 2048× 2048. In this sense, the reconstruction package supply three different aspects of a tomographic experiment: i) live reconstruction at the beamline assisting fast decisions by the researcher, ii) advanced reconstructions after the measurement using the beamline gpu power iii) reconstructions of the dataset without the beamline gpu power, at conventional desktops/notebooks. Description of the problem: We consider the transmission tomographic problem using x-rays generated at synchrotron facility based on parallel x-rays. The Radon transform has been used extensively as the mathematical object modelling the inverse problem. A typical tomographic measure provides a three-dimensional dataset Y as presented in Figure <ref>.a. Dataset Y is such that each plane s × t determines a radiography of the sample, for a constant projection angle θ_j varying on a discrete mesh with V points from 0 to π. The plane s× t is a discretization of a ccd camera with N× N pixels. The pair (N,V) is a characteristic of the imaging beamline, typically N ∼ 2048 and V > 1001, which means that Y is a large dataset with approximately 9GiB. The tomographic problem for parallel rays is posed in the following manner: find a reconstructed three-dimensional dataset X from Y in such a way that a given slice (s_k constant) x(u)= x(u_1,u_2) of the cube Xis related to the same slice y(t,θ) from volume Y through the linear operationy(t,θ) =x(t,θ) = ∫_^2 x(u) δ(t- u·ξ_θ) d uwith ξ = (cosθ,sinθ). Equation (<ref>) is the Radon transform <cit.> from the two-dimensional function x.Inverting the operatoris the mathematical core of most tomographic problems. There are several numerical algorithms for this task. One of the most celebrated algorithms is known as filtered-backprojection, given by the following formula x =[ℱ y] whereis thebackprojection operator - the adjoint of - defined as the following integral operatorb(u) = [ y](u) = ∫_0^π y(u·ξ_θ, θ) dθThe operator ℱ is a convolution, acting only on the first variable t, i.e., h(t,θ) = ℱ y(t,θ) =y(t,θ) ⋆ℓ(t) with ℓ̂(σ) = |σ|. Figure <ref> illustratethe filtered-backprojection action, together with ,andℱ on a point source function x(u) = δ(u - a) (for a random point a∈^2). The function y=x is often referred to as a sinogram because the Radon transform of an off-center point source is a sinusoid. The backprojection operation simply propagates the measured sinogram back into the image space along the projection paths.Under physical assumptions that are beyond the scope of this manuscript, the photon propagation through a sample obeys the Lambert-Beer law I(η) =I_0(η) e^- y(η) , η=(t,θ) with I, I_0 standing for the transmitted and incidentphoton counting on the pixel camera, parameterized by the point(t,θ) and over the slice s_k. The filtered backprojection algorithm applied over the sinogram y (fbp) offers a good reconstruction under strictly severe conditions on the measured data { I, I_0}, almost never satisfied at real measurements. There are three main processing steps on the data, before reconstruction, they are: (a) Normalization: the sinogram y is obtained theoretically using the logarithmic function. In practice, a dark/flat field correction is used y = -log[ ( I -D)/( I_0 -D) ], where D is the dark-field measurement (i.e., without the sample). (b) Centering: In practice, the tomographic device suffers from several mechanical imprecisions, most of them, intrinsic to the problem. In this case, we obtain an experimental sinogram which is a not an exact realization of a Radon transform. This is case of a sample rotating above a precision stage, which is not perfectly aligned to the camera vertical axis. Here, the center of rotation of the sample is unknown, and the originated sinogram is slightly shifted from the correct center of rotation. As a consequence, the measured sinogram y is a noisy and shifted version of the theoretical one y(t - β, θ) where β is the unknown shift. (c) Ring Filtering: Dead camera pixels or imperfections on the scintillator remain constant as the angle θ varies within the interval [0,π], giving origin to constant artifacts on the gathered frames I. Therefore, strong stripes arise in the sinogram y, producing concentric rings on the final reconstructed image. There are also several ring artifacts correction algorithms, producing anapproximation of y as discussed in <cit.>.§ FAST BACKPROJECTIONThe backprojection corresponds to an average 'smear' of all projections passing through a single pixel x. Since an average is described asa convolution, it is natural to regard formula (<ref>) as an average in the frequency domain. It was shown recently <cit.> that the operator y can be computed within O(N^2 log N) flops per slice, assuming that y is a sinogram having dimension of order O(N^2). This is not the only fast approach to compute , many others can be found in <cit.> while other algorithms as <cit.> propose to reconstruct x using y as an input, through the well known Slice Theorem or Fourier-Slice Theorem (fst)relating the values of y and y in the reciprocal space.Our approach, the so-called Backprojection-slice Theorem (bst) isbased on the following theorem: Let y be a given sinogram and · denotes the Fourier transform operation. The backprojection b =y can be computed through b(σcosθ, σsinθ) = ŷ(σ,θ) / σ with σ>0 ∈ and θ∈[0,2π]. The bst approach is the dual of the fst, in the sense thaty=y and b =y are related in the reciprocal space within a polar grid. If y= x the computation of b= x follows easily convolving the feature image x with the point-spread function 1/u_2 <cit.>, but x is unknown for pratical experiments. Also, fst provide a direct link between { x, y}and { b, y}. If y≠ x the link between { x, y} is no longer valid through fst but we can still provide a connection between { b,y}. This is the main goal of the bst approach. The straight usage of the Fourier transform for the implementation of (<ref>) produce big artifacts near the origin, caused by the fact that the values on sinogram on the line t=0 are not equal to 0. This problem can be solved with usage of short-time Fourier transform near the origin, with window w, e.g. Kaiser-Bessel window function.The bst strategy applied over a sinogram image y is obtained after 7 processing stages {P_k}, i.e. b = P_6 P_5 P_4 P_3 P_2 P_1 P_0 ( y). Each processing step P_k can be implemented in a parallel form with cuda. Step P_0 and P_1 indicates an interpolation to polar coordinates with the multiplication of the Kaiser-Bessel window function, respectively. This is an easy process, computed with complexity O(N V). Step P_2 is the zero padding of the polar sinogram - equivalent to an oversampling in the frequency domain - with the same complexity of P_0. Even though P_1 and P_2 can be merged into one single step, they were considered disjoint operations in our customized implementation. Step P_3 is a one-dimensional Fourier Transform of the data from step P_2. Step P_4 is the convolution of the polar sinogram with kernel 1/σ. This part was divided in m parallel fft's, each computed at an individual thread using advanced strategies with complexity O(N log N). Step P_5 is an interpolation from polar to cartesian coordinates in the frequency domain. The bigger the zero padding at step P_2, the better this part will behave, preventing aliasing artifacts. Step P_6 is a two-dimensional inverse Fourier transform of the data from step P_5. This is an operation with low computational complexity and obtained with order O(N^2 log N). Step P_7 is the fft shifting from previous step. We could add a P_8 step, an optional two-dimensional interpolation of the resulting image to the correct feature domain [-1,1]×[-1,1]. A theoretical fluxogram for bst, describing processing steps {P_0, …, P_7} is presented in Figure <ref>. Computing the backprojection operatordirectly from equation (<ref>) produce a high-computational complexity algorithm which is easy to implement in a parallel structure, either in cpu or gpu. Such approach has a computational complexity of O(N^3) per slice, which in turns implies a total cost of O(N^4) for a set of N sinogram images N. We denote this as a Slant-stack approach (SS) since summation is performed over straight lines with slope according to the input angle θ <cit.>. A block of Q sinograms is usually processed at once due to the fact that the backprojection kernel can be processed simultaneously for a tomographic scan using parallel rays. The fluxogram presented in Figure <ref> indicates that each step of bst has computational complexity bounded by O(N^2 log N).§ PIPELINE FOR DISTRIBUTIONThe full implementation is composed of multiple different stages: sinogram normalization (N), low-pass filtering (F), backprojection (B), ring filtering (R), centering of sinograms (R) and saving reconstructed data to storage (S). A typical sequence for these operations is presented in Figure <ref>. Although all stages operate on the same data, each has different characteristics regarding resource usage. Loading the input data and storing the output results are I/O bound operations, other stages are cpu bound and some are gpu bound. Even among the gpu bound stages, some are more compute intensive while others are more memory intensive. Having such a heterogeneous set of process stages, it is desirable to optimize their execution in order to maximize resource utilization and consequently minimize execution time. Such optimization becomes non-trivial as the available resources increase and the scheduling possibilities are amplified. Although the stages must be executed in a given order for a given set of input data, they can be executed in parallel with different sets of input data, as long as each data set is processed by each stage in the expected order. To help with the scheduling of the available jobs (the execution of one of the stages on one of the input data sets), a helper framework was implemented. This framework consists of an abstract representation of a pipeline of interconnected stages. Data enters the pipeline and is processed by each stage sequentially until it reaches the end of the pipeline, producing the resulting output data for the given input data. Each stage is represented as a processing function that receives a generic data blob and returns a resulting generic data blob. The function therefore processes a data block into another data block. The contents of the data block is specific to each stage, and it is a requirement that the resulting output block is in the expected format for the input block of the stage that follows. With the implementation modeled as a pipeline, some resource optimizations can be applied generically. The first is the execution of each stage in a separate cpu thread. This is not an optimization to primarily use different cpu cores to execute stages in parallel, although it does contribute to the usage of more resources. Instead, the main advantage is that it allows the operating system to schedule the different stages depending on their resource usage patterns. This means that I/O bound stages can start, and when an I/O operation halts the execution, a stage that is gpu bound can be executed, and when that stage is waiting on the results from the gpu another stage can be executed or the idle stages can be resumed if the data they're waiting for is ready.With each stage executing in different threads, the jobs are executed concurrently. An optimization opportunity arises in managing the number of jobs in flight. In one extremity, the number of jobs in flight can be limited to one, which effectively makes the pipeline run sequentially with no concurrency. On the other extremity, the number can be unlimited. While this might appear to be the optimal solution, because generally stages don't have the same execution time, some stages end up being bottlenecks. When this occurs, many jobs remain queued before the slowest stage, and they can potentially be holding resources that they are not using while idle. An example of this was that between two gpu specific stages, the data transferred between them resided in the gpu memory. Because the second stage was slower than the first stage, the gpu ran out of memory due to the amount of queued jobs for the second stage. The solution to this problem was to allow limiting the number of jobs in queue before each stage. Each stage can have its maximum queue length individually tuned, to allow limiting only the jobs in flight that may strain resources while idle.The final generic optimization is to allow running the same stage in a pool of threads. If the execution isn't fully using a specific resource, the stage that uses it most can be executed in more than one thread, which means they use that resource concurrently, potentially increasing its utilisation. An example is cpu core usage. If not all of the systems core are in use, have the same stage execute in more than one thread allows the operating system to schedule more threads on the cpu, attempting to maximize its usage. This also makes it easier when using multiple gpus, since each thread can be assigned to work with a single gpu. Even having more than one thread per gpu can increase throughput, because even if the task is the same, the gpu driver can schedule data transfers between the main memory and the gpu memory while the gpu is actively running another job (this is allowed in cuda as long as the program is configured to have a per-thread default stream).Such abstract implementation of a work pipeline allows the algorithm to be broken up into stages - as shown in Figure <ref> - and makes it easier to manage the execution of these stages in order to maximize resource usage. After adapting the implementation to the pipeline, some execution parameters can be tweaked to increase throughput. These include the number of threads for each stage, and the maximum queue size before each stage. This allows a simple way to experiment with different implementations by adapting the work done in each stage, and also quick experimentation with job scheduling to validate how different allocations result in different resource usage and performance. § PERFORMANCE Our pipeline, as presented in Section <ref>, is dependent on the time for reading the input data Y - see definition on Figure <ref>.The hierarchical data format hdf5 is becoming very popular for the storage of measured data at synchrotron facilities. In this sense, our algorithm is evaluated in two ways: (a) Extracting - without mpi capabilities - a block of Q images. Here, we have used the standard call of hdf to read the given number of images. The block of images is then processed following the pipeline of Figures <ref> and <ref>, except that we are not concerned with output. Hence, the flux of reconstructed data to storage is not considered in our analysis. Executions using the pipeline of Figure <ref> are refered as e(cpu/gpu) as they have an extra processing of parallel kernels R and C at cpu. Executions using the pipeline of Figure <ref> are refereed as e(gpu). The pipeline was executed using W=8 work items, each using T = 2 × G threads, being G the number of availables gpus.Needless to say that Q is bounded by the gpu memory and therefore by G and W, i.e., A_(Q W T) + A_(Q W T)+ A_(QWT) ≤ with A_ the amount of memory allocation for kernel N and M the global memory of a single gpu. (b) Measuring the elapsed time of the pipeline just reading the input data Y, and turning off all subsequent kernels. In average, we understand that the execution time of the reconstruction is bounded by this step of the pipeline. We denote this execution as r(cpu/gpu).Table <ref> presents the elapsed times for the pipeline e(gpu) using only one gpu and standard tomographic dimensions[We have adopted a three-dimensional dataset described by an ellipsoid. The feature image is defined as x(u) = ρ if (u_1/A)^2 + (u_2/B)^2 ≤ 1 - (s/C)^2 and zero otherwise. Here, A and B are parameters defining the shape of the semi-major axis u_1,u_2 of the ellipse and C defining the length of slice axis |s| ≤ C. The Radon transform of x can be obtained analytically <cit.>. A discretization of the sinogram using {N,V} points for t and θ axis respectively, gives us a three-dimensional dataset Y.]. Using bst with a powerful gpu like titan x we can achieve a feasible reconstruction of a dataset of dimension 2048^3 within 3 minutes. At the same dimension and after 10 executions of the pipeline r(cpu/gpu) with a blocksize of Q=10, weobtain an average of 107 seconds wasted on reading the input data - see Table <ref>. Hence, the reconstruction can be achieved within one minute. Using the same reasoning for the other devices,the reconstruction with jetson runs within 6 seconds for a dimension of 512^2 andwithin 1 second for dimensions 512^2 × 256. The standard gpu-gt740 runs a reconstruction within 46 seconds for a dimension 1024^3. A large number of executions (at two different clusters) of the pipeline e(gpu) using real data with dimensions (N,V)=(2048,2000)is presented in Figures <ref> and <ref>. Plots (a.k) therein represents 10 elapsed times (in seconds) versus the blocksize number Q - with k gpus for code distribution running at a SGI/C2108-GP5 server coupled with 4 nvidia K80. Plots (b.k) represents 10 elapsed times (in seconds) versus the blocksize number Q - with k gpus for code distribution running at a Minsky/IBM server coupled with 4 nvidia P100. Times for bst (Fig.<ref>) are much lesser than ss (Fig.<ref>), as predicted by theory. The average time expended reading the input data for these servers using a non-optimized hdf call, is presented in Table <ref> for different values ofblocksize Q.The processing time expected for bst is near 1 second using 1,2,3 or 4 gpus at Minsky cluster while 30 seconds for the sgi cluster. The average time for a complete reconstruction using the high-complexity kernel ss depends on the number of gpus used for distribution. The elapsed times for the hybrid pipeline e(cpu/gpu) - see Figure <ref> - running on the same data is presented in Figure <ref>. Now, it is clear that the time expended processing the data on the cpu affect performance dramatically. In fact, the cpu kernel R process a single slice using a Conjugate gradient method for stripe suppression <cit.>. Processing a block of Q images, although easy to implement, does not present the same quality of sinogram restoration because correction is slice dependent. Hence, even with a O(N) computational complexity, the ring suppression plus data transfer from cpu/gpu should be avoided in this pipeline. There is certainly a huge space for improvements on the ring suppression algorithm, handling a block of sinograms. The cpu kernel C - for centering sinograms - is optional and easy to implement for a block of images not affecting the final performance. § CONCLUSIONS Fast reconstructions of large tomographic datasets is feasible using the bst formula (<ref>). The superscalar pipeline - running only gpu kernels with computational complexity bounded above by O(N^2 log N) - is able to complete a 3d reconstruction within 1 second of running time provided the input dataset is transferred completely to the memory. New imaging detectors capable to store the data in a local buffer could benefit from this reconstruction approach. We emphasize thatother low-complexity algorithms <cit.> are also capable to provide fast results. Nonetheless, a fast backprojector (bst) and a fast projector (fst) implemented in the same pipeline presented here, give us the chance to implement advanced reconstruction strategies <cit.> (mostly iterative) with the same low computational complexity. Further improvements on the code[Download: ] are under progress which include: usage of mpi strategies to read the data, ring-filtering kernel to process a block of sinograms, thread optimization among others.§.§ Acknowledgments We would like to thank the ibm team for providing access to thePoughkeepsie Benchmarking Technical Computing Cloud: Douglas M. Dreyer, Victoria Nwobodo, James Kuchler, Khajistha Fattu, Antonio C.Navarro and Leonardo A.G. Garcia. Thanks also to Harry Westfahl Jr. for many valuable suggestions. The titan x used for this research was donated by the NVIDIA Corporation.unsrt | http://arxiv.org/abs/1704.08364v1 | {
"authors": [
"Gilberto Martinez Jr.",
"Janito V. Ferreira Filho",
"Eduardo X. Miqueles"
],
"categories": [
"cs.DC"
],
"primary_category": "cs.DC",
"published": "20170426221834",
"title": "Low-complexity Distributed Tomographic Backprojection for large datasets"
} |
roman EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)pdflatexCERN-EP-2017-062LHCb-PAPER-2017-008April 26, 2017 Resonances and violation inand → K^+K^- decays in the mass region above the ϕ(1020)The LHCb collaboration[Authors are listed at the end of this paper.] The decays of and mesons into the K^+K^- final state are studied in the K^+K^- mass region above the ϕ(1020) meson in order to determine the resonant substructure and measure the -violating phase, ,the decay width, , and the width difference between light and heavy mass eigenstates, .A decay-time dependent amplitude analysis is employed. The data sample corresponds to an integrated luminosity of 3produced in 7 and 8pp collisions at the LHC, collected by the experiment. The measurement determines ϕ_s = 119±107±34 mrad. A combination with previous LHCb measurements using similar decays into theπ^+π^- and ϕ(1020) final states gives =1±37 mrad, consistent with the Standard Model prediction.Published in JHEP 08 (2017) 037 CERN on behalf of the collaboration, licence http://creativecommons.org/licenses/by/4.0/CC-BY-4.0. plain arabic§ INTRODUCTIONMeasurements of violation through the interference of mixing and decay amplitudes are particularly sensitive to the presence of unseen particles or forces. The Standard Model (SM) prediction of the -violating phase in quark-level b→ ccs transitions is very small, ϕ_s^ SM≡ -2 arg(-V_tsV_tb^*/V_csV_cb^*)=-36.5_-1.2^+1.3 mrad <cit.>.Although subleading corrections from penguin amplitudes are ignored inthis estimate, the interpretation of the current measurements is not affected,since those subleading terms are known to be small <cit.> compared to the experimental precision. Initial measurements of ϕ_s were performed at the Tevatron <cit.>, followed by measurements using both and decays[Whenever a flavour-specific decay is mentioned it also implies use of the charge-conjugate decay except when dealing with -violating quantities or other explicitly mentioned cases.] into π^+π^- and K^+K^-, with K^+K^- invariant masses[Natural units are used where ħ=c=1.]<1.05,from 3of integrated luminosity.The measurements were found to be consistent with the SM value <cit.>, as are more recent and somewhat less accurate results from the CMS <cit.> and ATLAS <cit.>collaborationsusing ϕ(1020) final states.[The final states <cit.> and ψ(2S)ϕ(1020) <cit.> are also used by , but the precisions are not comparable due to lower statistics.] The average of all of the above mentioned measurements is =-30±33 mrad <cit.>.Previously, using a data sample corresponding to 1integrated luminosity, the LHCb collaboration studied the resonant structures in the → K^+K^- decay <cit.> revealing a rich resonance spectrum in the K^+K^- mass distribution. In addition to the ϕ(1020) meson, there are significant contributionsfrom the f_2'(1525) resonance <cit.> and nonresonant S-wave, which are large enough to allow further studies of violation. This paper presents the first measurement of ϕ_s using → K^+ K^- decays with above the ϕ(1020) region, using data corresponding to an integrated luminosity of 3, obtained from pp collisions at the LHC. One third of the data was collected at a centre-of-mass energy of 7, and the remainder at 8. An amplitude analysis as a function of theproper decay time <cit.> is performed to determine the -violating phase , by measuring simultaneously the -even and -odd decay amplitudes for each contributing resonance (and nonresonant S-wave), allowingthe improvement of theϕ_s accuracy and, in addition, further studies of the resonancecomposition in the decay. These → K^+K^- decays are separated into two K^+K^- mass intervals. Those with <1.05 GeV are called low-mass and correspond to the region of the ϕ(1020) resonance, while those with >1.05 GeV are called high-mass. The high-mass region has not been analyzed for violation before, allowing the measurement of violation in several decay modes, including a vector-tensor final state,f_2^'(1525). In the SM the phase ϕ_s is expected to be the same in all such modes. One important differencefrom the previous low-mass analysis <cit.> is that modelling of thedistribution is included to distinguish different resonance and nonresonance contributions. In the previous low mass -violation analysis only the ϕ(1020) resonance and an S-wave amplitude were considered. This analysis follows very closely the analyses of violation in →π^+π^- decays <cit.> and in ^0→π^+π^- decays <cit.>, and only significant changes with respect to those measurements are described in this paper. The analysis strategy is to fit the -even and -odd components in the decay width probability density functions that describe the interfering amplitudes in the particle and antiparticle decays. These fits are done as functions of the proper decay time and in a four-dimensional phase space including the three helicity angles characterizing the decay and .Flavour tagging, described below, allows us to distinguish between initialandstates.This paper is organized as follows.Section <ref> describes theproper-time dependent decay widths. Section <ref> gives a description of the detector and the associated simulations. Section <ref> contains the event selection procedure and the extracted signal yields. Section <ref> shows the measurement of the proper-time resolution and efficiencies for the final state in the four-dimensional phase space. Section <ref> summarizes the identification of the initial flavour of the state, a process called flavour tagging. Section <ref> gives the masses and widths of resonant states that decay into K^+K^-, and the description of a model-independent S-wave parameterization. Section <ref> describes the unbinned likelihood fit procedure used to determine the physics parameters, and presents the results of the fit, while Section <ref> discusses the systematic uncertainties. Finally, the results are summarized and combined with other measurements in Section <ref>. § DECAY RATES FORAND → K^+ K^- The total decay amplitude for a () meson at decay time equal to zero is taken to be the sum over individual K^+K^- resonant transversity amplitudes <cit.>, and one nonresonant amplitude, with each component labelled as A_i (A_i).Because of the spin-1in the final state, the three possible polarizations of thegenerate longitudinal (0), parallel (∥) and perpendicular (⊥) transversity amplitudes. When the K^+K^- forms a spin-0 state the final system only has a longitudinal component. Each of these amplitudes is apure eigenstate. By introducing the parameter λ_i ≡q/p_i/A_i, relating violation in the interference between mixing and decay associated with the state i, the total amplitudes A andcan be expressed as the sums of the individualamplitudes,A=∑ A_i and =∑q/p_i =∑λ_i A_i= ∑η_i |λ_i| e^-iϕ_s^i A_i. The quantities q and p relate the mass to the flavour eigenstates <cit.>. For each transversity state i the -violating phase ϕ_s^i≡ -(η_iλ_i) <cit.>, with η_i being the eigenvalue of the state. Assuming that any possible violation in the decay is the same for all amplitudes, then λ≡η_iλ_i and ϕ_s≡ -(λ) are common.The decay rates into the K^+K^- final state are[|p/q|=1 is used. The latest LHCb measurement determined |p/q|^2=1.0039±0.0033 <cit.>.]Γ(t) ∝e^- t{||^2+||^2/2+ ||^2-||^2/2.- .(^*)-(^*)}, Γ(t) ∝e^- t{||^2+||^2/2- ||^2-||^2/2.- .(^*)+(^*)},where ≡- is the decay width difference between the light and the heavy mass eigenstates, ≡ m_ H-m_ L is the mass difference, and ≡ (+)/2 is the average width.The sensitivity to the phase ϕ_s is driven by the terms containing ^*. Fordecays to μ^+μ^- final states, these amplitudes are themselves functions of four variables: theinvariant mass , andthree angular variables Ω≡ (cos, cos, χ), defined in the helicity basis. These consist of the anglebetween the K^+ direction in the K^+K^- rest frame with respect to the K^+K^- direction in therest frame, the anglebetween the μ^+ direction in therest frame with respect to thedirection in therest frame, and the angle χ between theand K^+K^- decay planes in therest frame <cit.>.These angles are shown pictorially in Fig. <ref>. These definitions are the same forand , namely, using μ^+ and K^+ to define the angles for bothanddecays. The explicit forms of | A(,Ω)|^2,|(,Ω)|^2, and ^*(,Ω)(,Ω) in Eqs. (<ref>) and (<ref>) are given in Ref. <cit.>. § DETECTOR AND SIMULATIONThe detector <cit.> is a single-arm forward spectrometer covering therange 2<η <5, designed for the study of particles containing or quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet. The tracking system provides a measurement of momentum, , of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200. The minimum distance of a track to a primary vertex (PV), the impact parameter (IP),is measured with a resolution of (15+29/), where is the component of the momentum transverse to the beam, in . Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov (RICH) detectors.Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The online event selection is performed by a trigger,which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. The software trigger is composed of two stages, the first of which performs a partial reconstruction and requires either a pair of well-reconstructed, oppositely charged muons having an invariant mass above 2.7, or a single well-reconstructed muon with high and large IP. The second stageapplies a full event reconstruction and for this analysis requires two opposite-sign muons to form a good-quality vertex that is well separated from all of the PVs, and to have an invariant mass within ±120of the known mass <cit.>. In the simulation, pp collisions are generated using 8 <cit.>. Decays of hadronic particles are described by <cit.>, in which final-state radiation is generated using <cit.>. The interaction of the generated particles with the detector, and its response, are implemented using the toolkit <cit.> as described in Ref. <cit.>. The simulation covers the full K^+K^- mass range.§ EVENT SELECTION AND SIGNAL YIELD EXTRACTION Acandidate is reconstructed by combining a →μ^+μ^- candidate with two kaons of opposite charge.The offline selection uses a loose preselection, followed by a multivariate classifier based on a Gradient Boosted Decision Tree (BDTG) <cit.>.In the preselection, the candidates are formed from two oppositely charged particles withgreater than 550, identified as muons and consistent with originating from a common vertex but inconsistent with originating from any PV. The invariant mass of thepair is required to be within [-48, +43] MeV of the known mass <cit.>, corresponding to a window of about ±3 times the mass resolution. The asymmetry in the cut values is due to the radiative tail. The two muons are subsequently kinematically constrained to the knownmass. Kaon candidates are required to be positively identified in the RICH detectors, to havegreater than 250, and the scalar sum of the two transverse momenta, (K^+)+(K^-), must be larger than 900.The four tracks from acandidate decay must originate from a common vertex with a good fit χ^2 and have a decay time greater than 0.3.Eachcandidate is assigned to a PV for which it has the smallest ,defined as the difference in theof the vertex fit for a given PV reconstructed with and without the considered particle. The angle between the momentum vector of thedecay candidates and the vector formed from the positions of the PV and the decay vertex (pointing angle) is required to be less than 2.5^∘.Events are filtered with a BDTG to further suppress the combinatorial background. The BDTG uses six variables: (K^+)+(K^-); the vertex-fit χ^2, pointing angle, , andof thecandidates; and the smaller of the DLL(μ-π) for the two muons, where DLL(μ-π) is the difference in the logarithms of the likelihood values from the particle identification systems <cit.> for the muon and pion hypotheses. The BDTG is trained on a simulated sample of 0.7 million reconstructed signal events, with the final-state particles generated uniformly in phase space assuming unpolarized J/ψ→μ^+μ^- decays, and a background data sample from the sideband 5516<m(J/ψ K^+K^-)< 5616 . Separate samples are used to train and test the BDTG. TheBDTG and particle identification (PID) requirements for the kaons are chosen to maximize the signal significance multiplied by the square root of the purity, S/√(S+B)×√(S/(S+B)), for candidates with >1.05, where S and B are the numbers ofsignal andbackground candidate combinations, respectively. This figure of merit optimizes the total uncertainty including both statistical and background systematic errors. In addition to the expected combinatorial background, studies of the data in sidebands of the m() spectrum show contributions from approximately8700 (430)and10 700(800)decays atgreater (less) than 1.05, where thein the former or p in the latter is misidentified as a . In order to avoid dealing with correlations between the angular variables and m( K^+K^-), the contributions from these reflection backgrounds are statistically subtracted by adding to the data simulated events of these decays with negative weights.These weights are chosen so that the distributions of the relevant variables used in the overall fit (see below) describe the background distributions both in normalization and shapes. The simulation uses amplitude models derived from data for → <cit.> and → p decays <cit.>. The invariant mass of the selected K^+K^- combinations, separated into samples forbelow or above 1.05, are shown in Fig. <ref>, where the expected reflection backgrounds are subtracted using simulation. The combinatorial background is modelled with an exponential function and thesignal shape is parameterized by a double-sided Hypatia function <cit.>, where the signal radiative tail parameters are fixed to values obtained from simulation.In total, 53 440±240 and 33 200±240 signal candidates are found for the low and highintervals, respectively.Figure <ref> shows the Dalitz plot distribution ofm^2_K^+K^-versusm^2_ K^+ for → K^+K^- candidates within ±15of themass peak. Clear resonant contributions from ϕ(1020) and f_2^'(1525) mesons are seen, but no exotic K^+ resonanceis observed.§ DETECTOR RESOLUTION AND EFFICIENCYThe resolution on the decay time is determined with the same method as described in Ref. <cit.> by using a large sample ofpromptcombinations produced directly in the pp interactions. These events are selected using→μ^+μ^- decays via a prescaled trigger that does not impose any requirements on the separation of the from the PV. The candidates are combined with two oppositely charged tracks that are identified as kaons, using a similar selection as for the signal decay, without adecay-time requirement. The resolution function, T(t-t̂ | δ_t), where t̂ is the true decay time, is a sum of three Gaussian functions with a common mean, and separate widths. To implement the resolution model each of the three widths aregiven by S_i·(δ_t+σ_t^0), where S_i is scale factor for the ith Gaussian, δ_t is an estimated per-candidate decay-time error and σ_t^0 is a constant parameter. The parameters of the resolution modelare determined by using a maximum likelihood fit to the unbinned decay time and δ_t distributions of the promptcombinations, using a δ function to represent the prompt component summed with two exponential functions for long-lived backgrounds; these are convolved with the resolution function. Taking into account the δ_t distribution of thesignal, the average effective resolution is found to be 44.7.The reconstruction efficiency is not constant as a function of decay time due to displacement requirements made on thecandidates in the trigger and offline selections. The efficiency is determined using the control channel → K^*(892)^0, with K^*(892)^0 →π^-, which is known to have a purely exponential decay-time distribution with τ_ = 1.520 ± 0.004 <cit.>. The selection efficiency is calculated asε_ data^(t) = ε_ data^(t) ×ε_ sim^(t)/ε_ sim^(t),where ε_ data^(t) is the efficiency of the control channel and ε_ sim^(t)/ε_ sim^(t) is the ratio of efficiencies of the simulated signal and control mode after the full trigger and selection chain has been applied. This correction accounts for the small differences in the kinematics between the signal and control mode. The details of the method are explained in Ref. <cit.>. The decay-time efficiencies for the twointervals are shown in Fig. <ref>.The efficiency as a function of the → K^+K^- helicity angles and the K^+K^- invariant mass is not uniform due to the forward geometry of the LHCb detector and the requirements imposed on the final-state particle momenta. The four-dimensional efficiency, ε(,Ω), is determined using simulated events that are subjected to the same trigger and selection criteria as the data.The efficiency is parameterized by ϵ(,Ω) = ∑_a,b,c,dϵ^abcdP_a(cos)Y_bc(,χ)P_d(2-^ min/^ max-^ min-1),where P_a and P_d are Legendre polynomials, Y_bc are spherical harmonics, and ^ min=2 m_K^+ and ^ max=m_-m_ are the minimum and maximum allowed values for , respectively. The Y_bc are complex functions. To ensure that the efficiency function is real, we set ϵ^abcd=-ϵ^ab(-c)d. The values of ϵ^abcd are determined by summing overthe fully simulated phase-space events ϵ^abcd=1/∑_i w_i∑_i w_i 2a+1/22d+1/2P_a(cos_,i)Y^*_bc(_,i,χ_i)P_d(2_,i-^ min/^ max-^ min-1)1/g_i,where the weights w_i account for corrections of PID and tracking efficiencies, and g_i=P^i_RP^i_B is the value of the phase-space probability density for event i with P_R being the momentum of either of the two hadrons in the dihadron rest frame and P_B the momentum of thein therest frame. This approach allows the description of multidimensional correlations without assuming factorization. In practice, the sum is over a finite number of terms (a≤10, b≤8, -2≤ c≤2, d≤8) and only coefficients with a statistical significance larger than three standard deviations (σ) from zero are retained. The number of events in thesimulated signal sample is about 20 times of that observed in data.Since a symmetric K^+ and K^- efficiency is used, a and b+c must be even numbers. Projections of the efficiency integrated over other variables are shown in Fig. <ref>. The modelling functions describe well the simulated data.Since is not used as a variable in the selection for the two hadrons, the efficiency is quite uniform over all the four variables varying only by about ±10%. (A dedicated simulation of ϕ(1020) decays is used to determine the efficiency in the region of <1.05, in order to have a large enough sample for an accurate determination.) § FLAVOUR TAGGINGThe candidate flavour at production is determined using two independent classes of flavour-tagging algorithms, the opposite-side (OS) tagger <cit.> and the same-side kaon (SSK) tagger <cit.>, which exploit specific features of the production ofquark pairs incollisions, and their subsequent hadronisation. Each tagging algorithm provides a tag decision and a mistag probability. The tag decision, 𝔮, is +1, -1, or 0, if the signal meson is tagged as , , or is untagged, respectively. The fraction of candidates in the sample with a nonzero tagging decision gives the efficiency of the tagger, ε_ tag. The mistag probability, η, is estimated event by event, and represents the probability that the algorithm assigns a wrong tag decision to the candidate; it is calibrated using data samples of several flavour-specific , ,and B_s2^*0 <cit.> decays to obtain the corrected mistag probability, ω, for an initialmeson, and separately obtain ω for an initial meson. A linear relationship between η and ω is used for the calibration. When candidates are tagged by both the OS and the SSK algorithms, a combined tag decision and a wrong-tag probability are given by the algorithm defined in Ref. <cit.> and extended to include SSK tags. This combined algorithm is implemented in the overall fit. The effective tagging power is given by ε_ tag(1-2ω)^2 and for the combined taggers in thesignal sample is (3.82 ± 0.13 ± 0.12)%. Whenever two uncertainties are quoted in this paper, the first is statistical and the second is systematic.§ RESONANCE CONTRIBUTIONS The entire K^+K^- mass spectrum is fitted by including the resonance contributions previously found in the time-integrated amplitude analysis using 1of integrated luminosity <cit.>, except for the unconfirmed f_2(1640) state. They are shown in Table <ref> and are described by Breit-Wigner amplitudes. The S-wave amplitude S()=c()+i s() is described in a model-independent way, makingno assumptions aboutits f_0 meson composition, or about the form of any S-wave nonresonant terms.Explicitly, two real parameters c^k=c(^k) and s^k=s(^k) are introduced to define the total S-wave amplitude at each of a set of invariant mass values =^k (k=1,..,N_s). Third-order spline interpolations are used to define c() and s() between these points of ^k. The c^k and s^k values are treated as model-independent parameters, and are determined by a fit to the data. In total N_s=13 knots are chosen at =(1.01, 1.03, 1.05, 1.10, 1.40, 1.50, 1.65, 1.70, 1.75, 1.80, 1.90, 2.1, 2.269) . The S-wave amplitude is proportional to momentum P_B <cit.>; at the last point since P_B=0, the amplitude is zero <cit.>.To describe thedependence for each resonance R, the formula of Eq. (18) in Ref. <cit.> is modified by changing (P_R/m_KK)^L_R to (P_R/m_0)^L_R, whereP_R is the momentum of either of the two hadrons in the dihadron rest frame, m_0 is the mass of resonance R, and L_R the orbital angular momentum in thedecay, and thus corresponds to theresonance's spin.This change modifies the lineshape of resonances with spin greater than zero. The original formula followed the convention from the Belle collaboration <cit.> and was used in two LHCb publications <cit.>, while the new one follows the convention of PDG/, and was used in analyzing → p K^- decays <cit.>.§ MAXIMUM LIKELIHOOD FITThe physics parameters are determined from a weighted maximum likelihood fit of a signal-only probability density function (PDF) to the five-dimensional distributions ofand decay time,and helicity angles. The negative log-likelihood function to be minimized is given by-ln L = - α∑_i W_i ln ( PDF),where i runs over all event candidates,W_i is the computed using m(J/ψ K^+K^-) as the discriminating variable <cit.> and the factor α≡∑_i W_i / ∑_i W_i^2 is a constant factor accounting for the effect of the background subtraction on the statistical uncertainty.The s are determined by separate fits in four |cos| bins for the event candidates.The PDF is given by PDF= F/∫ F dtddΩ, where F isF(t,,Ω,𝔮 | η,δ_t)= [ R(t̂,,Ω,𝔮 | η) ⊗ T(t-t̂ | δ_t)] ·ε_ data^(t)·ε(,Ω),with R(t̂,,Ω,𝔮 | η) = 1/1+|𝔮|[[1+𝔮(1-2ω(η))]Γ(t̂,,Ω).. +[1-𝔮(1-2ω̅(η))]1+A_ P/1-A_ PΓ̅(t̂,,Ω)],where t̂ is the true decay time, 0.18em-0.05em Γis defined in Eqs. (<ref>) and (<ref>), andA_ P=(1.09±2.69)% is the LHCb measured production asymmetry of and mesons <cit.>.To obtain a measurement that is independent of the previous publication that used mainly ϕ(1020) decays <cit.>, two different sets of fit parameters (ϕ_s, |λ|, , )^ L,H are used to account for the low (L) and high (H)regions. Simulated pseudoexperiments show that this configuration removes the correlation for these parameters between the two regions. A simultaneous fit to the two samples is performed by constructing the log-likelihood as the sum of that computed from the L and H events. The shared parameters are all the resonance amplitudes and phases, and, which is freely varied in the fit.In the nominal fit configuration, violation is assumed to be the same for all the transversity states. In total 69 free parameters are used in the nominal fit. The decay observables resulting from the fit for the highregion are listed in Table <ref>.The measurements for these parameters andin the ϕ(1020) region are consistent with the reported values in Ref. <cit.> within 1.4σ, taking into account the overlap between the two samples used. In addition, good agreement is also found for the S-wave phase. The fit gives =17.783±0.049from the fullregion, which is consistent with the most precise measurement 17.768±0.023±0.006from LHCb in →π^+ decays <cit.>. The value of |λ| is consistent with unity, thus giving no indication of any direct violation in the decay amplitude.While a complete description of the → K^+K^- decay is given in terms of the fitted amplitudes and phases, knowledge of the contribution of each component can be summarized by the fit fraction, FF_i, defined as the integral of the squared amplitude of each resonance over the phase space divided by the integral of the entire signal function over the same area, as given in Eq. <ref> FF_i = ∫ |A_i|^2 d dΩ/∫| A|^2 d dΩ.The sum of the fit fractions is not necessarily unity due to the potential presence of interference between two resonances.The fit fractions are reported in Table <ref> and resonance phases in Table <ref>. Fit projections are shown in Fig. <ref> for the ϕ(1020) region and above.The fit reproduces the data in each of the projected variables. Each contributing component is shown in Fig. <ref> as a function of . To check the fit quality in the highregion, χ^2 tests are performed. ForandΩ, χ^2=1401 for 1125 bins (25 for , 5 for cos, 3 for cos and 3 for χ); for the two variablesand cos, χ^2=380 for 310 bins. The fit describes the data well. Note, adding the f_2(1640) into the fit improves the -2ln L by 0.4 with an additional 6 degrees of freedom, showing that this state is not observed. As a check a fit is performed allowing independent sets of -violating parameters (|λ_i|,ϕ_s^i): three sets for the three corresponding ϕ(1020) transversity states, one for the K^+K^- S-wave, one common to all three transversity states ofthe f_2(1270), one for thef_2^'(1525), one for the ϕ(1680), and one for the combination of the two high-mass f_2(1750) and f_2(1950) resonances.In total, eight sets of -violating parameters are used instead of two sets in the nominal fit. The -2ln L value is improved by 16 units with 12 additional parameters compared to the nominal fit, corresponding to the fact that all states have consistent violation within 1.3 σ. All valuesof |λ| are consistent with unity and ϕ_s differences ofthe longitudinal ϕ(1020) component are consistent with zero, showing no dependence of violation for the different states. § SYSTEMATIC UNCERTAINTIESThe systematic uncertainties are summarized for the physics parameters in Table <ref> and for the fit fractions in Table <ref>. They are small compared to the statistical ones for the -violating parameters. Generally, the largest contribution results from the resonance fit model.The fit model uncertainties are determined by doubling the number of S-wave knots in the highregion, allowing the centrifugal barrier factors, of nominal value 1.5^-1 for K^+K^- resonances and 5.0^-1 for themeson <cit.>, to vary within 0.5–2 times of these values <cit.>. Additional systematic uncertainties are evaluated by increasing the orbital angular momentum between the and the K^+K^- system from the lowest allowed one, which is taken as the nominal value, and varying the masses and widths of contributing resonances by their uncertainties. The largest variation among those changes is assigned as the systematic uncertainty for resonance modelling.The effect of using the m_0in the fit, rather than following the Belle approach usingis evaluated by redoing the fit. This change worsens the -2ln L by more than 100 units, which clearly shows the variation doesn't give a good fit; as a consequence, no systematic uncertainty is assessed.Differences resulting from the two conventions are comparable to the quoted modelling uncertainty for the -violating parameters, but generally are larger than the quoted systematic uncertainties for the fit fractions of nonscalar resonances.The sources of uncertainty for the modelling of the efficiency variation of the three angles andinclude the statistical uncertainty from simulation, and the efficiency correction due to the differences in kinematic distributions between data and simulation fordecays. The former is estimated by repeating the fit to the data 100 times. In each fit, the efficiency parameters are resampled according to the corresponding covariance matrix determined from simulation. For the latter, the efficiency used by the nominal fit is obtained by weighting the distributions of p andof the kaon pair andmeson to match the data. Such weighting is removed to assign the corresponding systematic uncertainty. The uncertainties due to thelifetime and decay time efficiency determination are estimated.Each source is evaluated by adding to the nominal fit an external correlated multidimensional Gaussian constraint, either given by the fit to the → sample with varying τ_=1.520±0.004 <cit.>, or given by the fit to simulation for the decay time efficiency correction, ε_ sim^(t)/ε_ sim^(t) in Eq. (<ref>). A systematic uncertainty is given by the difference in quadrature of the statistical uncertainties for each physics parameter between the nominal fit and the alternative fit with each of these constraints. The uncertainties due to the decay time acceptance are found to be negligible for the fit fraction results. The sample of promptmesons combined with two kaon candidates is used to calibrate the per-candidate decay-time error. This method is validated by simulation. Since the detached selection, pointing angle and BDTG requirements cannot be applied to the calibration sample,the simulations show that the calibration overestimates the resolution fordecays after final selection by about 4.5%.Therefore, a 5% variation of the widths, and the uncertainty of the mean value are used to estimate uncertainty of the time resolution modelling. The average angular resolution is 6 mrad for all three decay angles. This is small enough to have only negligible effects on the analysis.A large number of pseudoexperiments is used to validate the fitter and check potential biases in the fit outputs. Biases onand , 20% of their statistical uncertainties, are taken as systematic uncertainties. Calibration parameters of the flavour-tagging algorithm and the – production asymmetry A_ P=(1.09±2.69)% <cit.> are fixed. The systematic uncertainties due to the calibration of the tagging parameters or the value of A_ P are given by the difference in quadrature between the statistical uncertainty for each physics parameter between the nominal fit and an alternative fit where the tagging parameters or A_ P are Gaussian-constrained by the corresponding uncertainties.Background sources are tested by varying the decay-time acceptance of the injected reflection backgrounds,changing these background yields by 5%, and also varying thelifetime.To evaluate the uncertainty of the method that requires the fit observables being uncorrelated with the variable m( K^+K^-) used to obtain the s, two variations are performed to obtain new s, and the fit is repeated. The first consists of changing the number of |cos| bins.In the nominal fit, the s are determined by separate fits in four |cos| bins for the event candidates, as significant variations of signal invariant mass resolution are seen as a function of the variable. In another variation of the analysisstarting with the nominal number of |cos| bins the decay time dependence is explored, since the combinatorial background may have a possible variation as a function of m( K^+K^-). Herethe decay time is further divided into three intervals. The larger change on the physics parameter of interest is taken as a systematic uncertainty. About 0.8% of the signal sample is expected from the decays ofmesons <cit.>. Neglecting thecontribution in the nominal fit leads to a negligible bias of 0.0005for <cit.>. The correlation matrix with both statistical and systematic uncertainties is shown in Table <ref>. § CONCLUSIONSWe have studiedand decays into thefinal state using a time-dependent amplitude analysis. In the >1.05region we determineϕ_s= 119±107±34 mrad,|λ|= 0.994±0.018±0.006,= 0.650±0.006±0.004 ,=0.066±0.018±0.010 . Many resonances and a S-wave structure have been found.Besides the ϕ(1020) meson these include the f_2(1270), the f_2'(1525), the ϕ(1680), the f_2(1750), and the f_2(1950) mesons.The presence of the f_2(1640) resonance is not confirmed. The measured -violating parameters of the individual resonances are consistent.The f_2^'(1525) mass and width are determined as 1522.2±1.3±1.1and 78.0±3.0±3.7, respectively. The fit fractions of the resonances in → are also determined, and shown in Table <ref>.These results supersede our previous measurements <cit.>. The combination with the previous results fromdecays in the ϕ(1020) region <cit.> givesϕ_s= -25±45±8 mrad,|λ|= 0.978±0.013±0.003,= 0.6588±0.0022±0.0015 ,=0.0813±0.0073±0.0036 .The two results are consistent within 1.1σ. A further combination is performed by including theand |λ| measurements fromanddecays into <cit.>, which results in = 1±37 mrad and |λ|=0.973±0.013, whereandare unchanged. The correlation matrix is shown in Table <ref>. The measurement of the -violating phaseis in agreement with the SM prediction -36.5_-1.2^+1.3 mrad <cit.>. These new combined results supersede our combination reported in Ref. <cit.>.§ ACKNOWLEDGEMENTS We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID (Poland) and OSC (USA). We are indebted to the communities behind the multiple open source software packages on which we depend. Individual groups or members have received support from AvH Foundation (Germany), EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust (United Kingdom). Appendix § ANGULAR MOMENTSθ_KK AWe define the moments⟨ Y_ℓ^0⟩, as the efficiency-corrected and background-subtractedinvariant mass distributions, weighted by the ℓth spherical harmonic functions of the cosine of the helicity angle . The moment distributions provide an additional way of visualizing the presence of different resonances and their interferences, similar to a partial wave analysis. Figures <ref> and <ref> show the distributions of the even angular moments for the events around ± 30 MeV of ϕ(1020) mass peak and those above the ϕ(1020), respectively. The general interpretation of the even moments is that⟨ Y^0_0⟩ is the efficiency-corrected and background-subtracted event distribution,⟨ Y^0_2⟩reflects the sum of P-wave, D-wave and the interference of S-wave and D-wave amplitudes,and ⟨ Y^0_4⟩the D-wave. The average ofanddecays cancels the interference terms that involve P-wave amplitudes. This causes the odd moments to sum to zero. The fit results reproduce the moment distributions relatively well. For the region near the ϕ(1020), the p-values are 3%, 3%, 48% for the ℓ=0, 2, 4 moments, respectively. For the high mass region, the p-values are 37%, 0.2% 0.5% for the ℓ=0, 2, 4 moments, respectively.tocsectionReferences inbibliographytrue LHCbLHCb collaboration R. Aaij^40, B. Adeva^39, M. Adinolfi^48, Z. Ajaltouni^5, S. Akar^59, J. Albrecht^10, F. Alessio^40, M. Alexander^53, S. Ali^43, G. Alkhazov^31, P. Alvarez Cartelle^55, A.A. Alves Jr^59, S. Amato^2, S. Amerio^23, Y. Amhis^7, L. An^3, L. Anderlini^18, G. Andreassi^41, M. 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Sadykhov^32, N. Sagidova^31, B. Saitta^16,f, V. Salustino Guimaraes^1, D. Sanchez Gonzalo^38, C. Sanchez Mayordomo^69, B. Sanmartin Sedes^39, R. Santacesaria^26, C. Santamarina Rios^39, M. Santimaria^19, E. Santovetti^25,j, A. Sarti^19,k, C. Satriano^26,s, A. Satta^25, D.M. Saunders^48, D. Savrina^32,33, S. Schael^9, M. Schellenberg^10, M. Schiller^53, H. Schindler^40, M. Schlupp^10, M. Schmelling^11, T. Schmelzer^10, B. Schmidt^40, O. Schneider^41, A. Schopper^40, H.F. Schreiner^59, K. Schubert^10, M. Schubiger^41, M.-H. Schune^7, R. Schwemmer^40, B. Sciascia^19, A. Sciubba^26,k, A. Semennikov^32, A. Sergi^47, N. Serra^42, J. Serrano^6, L. Sestini^23, P. Seyfert^21, M. Shapkin^37, I. Shapoval^45, Y. Shcheglov^31, T. Shears^54, L. Shekhtman^36,w, V. Shevchenko^68, B.G. Siddi^17,40, R. Silva Coutinho^42, L. Silva de Oliveira^2, G. Simi^23,o, S. Simone^14,d, M. Sirendi^49, N. Skidmore^48, T. Skwarnicki^61, E. Smith^55, I.T. Smith^52, J. Smith^49, M. Smith^55, l. Soares Lavra^1, M.D. Sokoloff^59, F.J.P. Soler^53, B. Souza De Paula^2, B. Spaan^10, P. Spradlin^53, S. Sridharan^40, F. Stagni^40, M. Stahl^12, S. Stahl^40, P. Stefko^41, S. Stefkova^55, O. Steinkamp^42, S. Stemmle^12, O. Stenyakin^37, H. Stevens^10, S. Stoica^30, S. Stone^61, B. Storaci^42, S. Stracka^24,p, M.E. Stramaglia^41, M. Straticiuc^30, U. Straumann^42, L. Sun^64, W. Sutcliffe^55, K. Swientek^28, V. Syropoulos^44, M. Szczekowski^29, T. Szumlak^28, S. T'Jampens^4, A. Tayduganov^6, T. Tekampe^10, G. Tellarini^17,g, F. Teubert^40, E. Thomas^40, J. van Tilburg^43, M.J. Tilley^55, V. Tisserand^4, M. Tobin^41, S. Tolk^49, L. Tomassetti^17,g, D. Tonelli^24, S. Topp-Joergensen^57, F. Toriello^61, R. Tourinho Jadallah Aoude^1, E. Tournefier^4, S. Tourneur^41, K. Trabelsi^41, M. Traill^53, M.T. Tran^41, M. Tresch^42, A. Trisovic^40, A. Tsaregorodtsev^6, P. Tsopelas^43, A. Tully^49, N. Tuning^43, A. Ukleja^29, A. Ustyuzhanin^35, U. Uwer^12, C. Vacca^16,f, V. Vagnoni^15,40, A. Valassi^40, S. Valat^40, G. Valenti^15, R. Vazquez Gomez^19, P. Vazquez Regueiro^39, S. Vecchi^17, M. van Veghel^43, J.J. Velthuis^48, M. Veltri^18,r, G. Veneziano^57, A. Venkateswaran^61, T.A. Verlage^9, M. Vernet^5, M. Vesterinen^12, J.V. Viana Barbosa^40, B. Viaud^7, D. Vieira^63, M. Vieites Diaz^39, H. Viemann^67, X. Vilasis-Cardona^38,m, M. Vitti^49, V. Volkov^33, A. Vollhardt^42, B. Voneki^40, A. Vorobyev^31, V. Vorobyev^36,w, C. Voß^9, J.A. de Vries^43, C. Vázquez Sierra^39, R. Waldi^67, C. Wallace^50, R. Wallace^13, J. Walsh^24, J. Wang^61, D.R. Ward^49, H.M. Wark^54, N.K. Watson^47, D. Websdale^55, A. Weiden^42, M. Whitehead^40, J. Wicht^50, G. Wilkinson^57,40, M. Wilkinson^61, M. Williams^40, M.P. Williams^47, M. Williams^58, T. Williams^47, F.F. Wilson^51, J. Wimberley^60, M.A. Winn^7, J. Wishahi^10, W. Wislicki^29, M. Witek^27, G. Wormser^7, S.A. Wotton^49, K. Wraight^53, K. Wyllie^40, Y. Xie^65, Z. Xing^61, Z. Xu^4, Z. Yang^3, Z. Yang^60, Y. Yao^61, H. Yin^65, J. Yu^65, X. Yuan^61, O. Yushchenko^37, K.A. Zarebski^47, M. Zavertyaev^11,c, L. Zhang^3, Y. Zhang^7, A. Zhelezov^12, Y. Zheng^63, X. Zhu^3, V. Zhukov^33, S. Zucchelli^15.^1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil^2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil^3Center for High Energy Physics, Tsinghua University, Beijing, China^4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France^5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-Ferrand, France^6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France^7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France^8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France^9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany^10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany^11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany^12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany^13School of Physics, University College Dublin, Dublin, Ireland^14Sezione INFN di Bari, Bari, Italy^15Sezione INFN di Bologna, Bologna, Italy^16Sezione INFN di Cagliari, Cagliari, Italy^17Sezione INFN di Ferrara, Ferrara, Italy^18Sezione INFN di Firenze, Firenze, Italy^19Laboratori Nazionali dell'INFN di Frascati, Frascati, Italy^20Sezione INFN di Genova, Genova, Italy^21Sezione INFN di Milano Bicocca, Milano, Italy^22Sezione INFN di Milano, Milano, Italy^23Sezione INFN di Padova, Padova, Italy^24Sezione INFN di Pisa, Pisa, Italy^25Sezione INFN di Roma Tor Vergata, Roma, Italy^26Sezione INFN di Roma La Sapienza, Roma, Italy^27Henryk Niewodniczanski Institute of Nuclear PhysicsPolish Academy of Sciences, Kraków, Poland^28AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland^29National Center for Nuclear Research (NCBJ), Warsaw, Poland^30Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania^31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia^32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia^33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia^34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia^35Yandex School of Data Analysis, Moscow, Russia^36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia^37Institute for High Energy Physics (IHEP), Protvino, Russia^38ICCUB, Universitat de Barcelona, Barcelona, Spain^39Universidad de Santiago de Compostela, Santiago de Compostela, Spain^40European Organization for Nuclear Research (CERN), Geneva, Switzerland^41Institute of Physics, Ecole PolytechniqueFédérale de Lausanne (EPFL), Lausanne, Switzerland^42Physik-Institut, Universität Zürich, Zürich, Switzerland^43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands^44Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, The Netherlands^45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine^46Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine^47University of Birmingham, Birmingham, United Kingdom^48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom^49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom^50Department of Physics, University of Warwick, Coventry, United Kingdom^51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom^52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom^53School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom^54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom^55Imperial College London, London, United Kingdom^56School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom^57Department of Physics, University of Oxford, Oxford, United Kingdom^58Massachusetts Institute of Technology, Cambridge, MA, United States^59University of Cincinnati, Cincinnati, OH, United States^60University of Maryland, College Park, MD, United States^61Syracuse University, Syracuse, NY, United States^62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to ^2^63University of Chinese Academy of Sciences, Beijing, China, associated to ^3^64School of Physics and Technology, Wuhan University, Wuhan, China, associated to ^3^65Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China, associated to ^3^66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia, associated to ^8^67Institut für Physik, Universität Rostock, Rostock, Germany, associated to ^12^68National Research Centre Kurchatov Institute, Moscow, Russia, associated to ^32^69Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia - CSIC, Valencia, Spain, associated to ^38^70Van Swinderen Institute, University of Groningen, Groningen, The Netherlands, associated to ^43^aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil^bLaboratoire Leprince-Ringuet, Palaiseau, France^cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia^dUniversità di Bari, Bari, Italy^eUniversità di Bologna, Bologna, Italy^fUniversità di Cagliari, Cagliari, Italy^gUniversità di Ferrara, Ferrara, Italy^hUniversità di Genova, Genova, Italy^iUniversità di Milano Bicocca, Milano, Italy^jUniversità di Roma Tor Vergata, Roma, Italy^kUniversità di Roma La Sapienza, Roma, Italy^lAGH - University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Kraków, Poland^mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain^nHanoi University of Science, Hanoi, Viet Nam^oUniversità di Padova, Padova, Italy^pUniversità di Pisa, Pisa, Italy^qUniversità degli Studi di Milano, Milano, Italy^rUniversità di Urbino, Urbino, Italy^sUniversità della Basilicata, Potenza, Italy^tScuola Normale Superiore, Pisa, Italy^uUniversità di Modena e Reggio Emilia, Modena, Italy^vIligan Institute of Technology (IIT), Iligan, Philippines^wNovosibirsk State University, Novosibirsk, Russia^†Deceased | http://arxiv.org/abs/1704.08217v2 | {
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],
"categories": [
"hep-ex"
],
"primary_category": "hep-ex",
"published": "20170426170620",
"title": "Resonances and $CP$ violation in $B_s^0$ and $\\overline{B}_s^0 \\to J/ψK^+K^-$ decays in the mass region above the $φ(1020)$"
} |
http://arxiv.org/abs/1704.08728v1 | {
"authors": [
"C. S. Alves",
"T. A. Silva",
"C. J. A. P. Martins",
"A. C. O. Leite"
],
"categories": [
"astro-ph.CO",
"gr-qc",
"hep-ph",
"hep-th"
],
"primary_category": "astro-ph.CO",
"published": "20170427193857",
"title": "Fisher matrix forecasts for astrophysical tests of the stability of the fine-structure constant"
} |
|
Rogue periodic waves of the mKdV equation Jinbing Chen^1,2 and Dmitry E. Pelinovsky^2,3 ^1 School of Mathematics, Southeast University, Nanjing, Jiangsu 210096, P.R. China ^2 Department of Mathematics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1^3 Department of Applied Mathematics, Nizhny Novgorod State Technical University 24 Minin street, 603950 Nizhny Novgorod, RussiaDecember 30, 2023 ====================================================================================================================================================================================================================================================================================================================================================================== Rogue periodic waves stand for rogue waves on the periodic background. Two families of traveling periodic waves of the modified Korteweg–de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn. By using one-fold and two-fold Darboux transformations, we construct explicitly the rogue periodic waves of the mKdV equation. Since the dn-periodic wave is modulationally stable with respect to long-wave perturbations, the “rogue" dn-periodic solution is not a proper rogue wave on the periodic background but rather a nonlinear superposition of an algebraically decaying soliton and the dn-periodic wave. On the other hand, since the cn-periodic wave is modulationally unstable with respect to long-wave perturbations, the rogue cn-periodic wave is a proper rogue wave on the periodic background, which generalizes the classical rogue wave (the so-called Peregrine's breather) of the nonlinear Schrödinger (NLS) equation. We compute the magnification factor for the rogue cn-periodic wave of the mKdV equation and show that it remainsfor all amplitudes the same as in the small-amplitude NLS limit. As a by-product of our work, we find explicit expressions for the periodic eigenfunctions of the AKNS spectral problem associated with the dn and cn periodic waves of the mKdV equation.§ INTRODUCTION Simplest models for nonlinear waves in fluids such as the nonlinear Schrödinger equation (NLS), the Korteweg–de Vries equation (KdV), and the modified Korteweg–de Vries equation (mKdV) have many things in common. First, they appear to be integrable by using the inverse scattering transform method for the same AKNS (Ablowitz–Kaup–Newell–Segur) spectral problem <cit.>. Second, there exist asymptotic transformations of one nonlinear evolution equation to another nonlinear evolution equation, e.g. from defocusing NLS to KdV and from KdV and focusing mKdV to the defocusing and focusing NLS respectively <cit.>.Modulation instability of the constant-wave background in the focusing NLS equation has been a paramount concept in the modern nonlinear physics <cit.>. More recently, spectral instability of the periodic waves expressed by the elliptic functions dn and cn has been investigated in the focusing NLS <cit.> (see also <cit.>). Regarding periodic waves in the focusing mKdV equation, it was found that the dn-periodic waves are modulationally stable with respect to the long-wave perturbations, whereas the cn-periodic waves are modulationally unstable <cit.> (see also <cit.>).The outcome of the modulation instability in the focusing NLS equation is the emergence of the localizedspatially-temporal patterns on the background of the unstable periodic or quasi-periodic waves (see review in <cit.>). Such spatially-temporal patterns are known under the generic name of rogue waves <cit.>.In the simplest setting of the constant wave background, the rogue waves are expressed as rational solutions of the NLS equation. Explicit expressions for such rational solutions have been obtained by using available algebraic constructions such as applications of the multi-fold Darboux transformations <cit.>. For example, if the focusing NLS equation is set in the formi ψ_t + ψ_xx + 2 (|ψ|^2 - 1) ψ = 0,then the classical rogue wave up to the translations in (x,t) is given byψ(x,t) = 1 - 4 (1+4it)/1 + 4 x^2 + 16 t^2.As |t| + |x| →∞, the rogue wave (<ref>) approaches the constant wave background ψ_0(x,t) = 1. On the other hand, at (x,t) = (0,0), the rogue wave reaches the maximum at |ψ(0,0)| = 3, from which we define the magnification factor of the constant wave background to be M_0 = 3. The rogue wave (<ref>) was derived by Peregrine <cit.> as an outcome of the modulational instability of the constant-wave background and is sometimes referred to as Peregrine's breather.Rogue waves over nonconstant backgrounds (e.g., the periodic waves or the two-phase solutions) were addressed only recently in the context of the focusing NLS equation (<ref>). Computations of such rogue waves rely on the numerical implementation of the Bäcklund transformation to the periodic waves <cit.> or the two-phase solutions <cit.> of the NLS. Further analytical work to characterize the general two-phase solutions of the NLS can be found in <cit.> and in <cit.>.The purpose of this work is to obtain exact solutions for the rogue waves on the periodic background, which we name here as rogue periodic waves. Computations of such rogue waves are developed by an analytical algorithm with precise characterization of the periodic and non-periodic eigenfunctions of the AKNS spectral problem at the periodic wave. Although our computations are reported in the context of the focusing mKdV equation, the algorithm can be applied to other nonlinear evolution equations associated to the AKNS spectral problem such as the NLS.Hence we consider the focusing mKdV equation written in the normalized formu_t+6u^2u_x+u_xxx=0.Some particular rational and trigonometric solutions of the mKdV were recently constructed in <cit.> and discussed in connection to rogue waves of the NLS. In comparison with <cit.>, the novelty of our work is to obtain the rogue periodic waves expressed by the Jacobian elliptic functions and to investigate how these rogue periodic waves generalize in the small-amplitude limit the classical rogue wave (<ref>). In particular, we shall compute explicitly the magnification factor for the rogue periodic waves that depends on elliptic modulus of the Jacobian elliptic functions.There are two particular periodic wave solutions of the mKdV. One solution is strictly positive and is given by the dn elliptic function. The other solution is sign-indefinite and is given by the cn elliptic function. Up to the translations in (x,t) as well as a scaling transformation, the positive solution is given byu_ dn(x,t) =dn(x-ct;k),c = c_ dn(k) := 2 - k^2,whereas the sign-indefinite solution is given byu_ cn(x,t) = kcn(x-ct;k),c = c_ cn(k) := 2k^2 - 1.In both cases, k ∈ (0,1) is elliptic modulus, which defines two different asymptotic limits. In the limit k → 0, we obtainu_ dn(x,t) ∼ 1 - 1/2 k^2sin^2(x - 2t)andu_ cn(x,t) ∼ k cos(x + t)which are understood in the sense of the Stokes expansion of the periodic waves. As is well-known <cit.>, the mKdV equation can be reduced asymptotically to the NLS equation in the small-amplitude limit. The cn-periodic wave of the mKdV in the limit k → 0 is reduced to the constant wave background ψ_0 of the NLS equation (<ref>), which is modulationally unstable with respect to the long-wave perturbations. Hence, the cn-periodic wave for the mKdV generalizes the constant wave background of the NLS and inherits modulational instability with respect to long-wave perturbations. The dn-periodic wave has nonzero mean value, which is large enough to make the dn-periodic wave modulationally stable with respect to long-wave perturbations <cit.>.In the limit k → 1, both Jacobian elliptic functions (<ref>) and (<ref>) converges to the normalized mKdV solitonu_ dn(x,t), u_ cn(x,t) → u_ soliton(x,t) =sech(x-t).Very recently, the rogue waves of the mKdV built from a superposition of slowly interacting nearly identical solitons were considered numerically <cit.> and analytically <cit.>. It was found in these studies that the magnification factor of the rogue waves built from N nearly identical solitons is exactly N.In our work, we compute the rogue periodic waves for the dn and cn Jacobian elliptic functions with the following algorithm. First, by using the algebraic technique based on the nonlinearization of the Lax pair <cit.>, we obtain the explicit expressions for the eigenvalues λ with Re(λ) > 0 and the associated periodic eigenfunctions in the AKNS spectral problem associated with the Jacobian elliptic functions. These eigenvalues correspond to the branch points of the continuous bands, when the AKNS spectral problem with the periodic potentials is considered on the real line with the help of the Floquet–Bloch transform <cit.>. For each periodic eigenfunction, we construct the second, linearly independent solution of the AKNS spectral problem, which is not periodic but linearly growing in (x,t). The latter eigenfunction is expressed in terms of integrals of the Jacobian elliptic functions and hence it is not explicit. Finally, by using the one-fold and two-fold Darboux transformations <cit.> with the nonperiodic eigenfunctions of the AKNS spectral problem, we obtain the rogue periodic waves. Although the resulting solutions are not explicit, we prove that these solutions approach the dn and cnperiodic waves as |x|+|t| →∞ almost everywhere and that they have maximum at the origin (x,t) = (0,0), where the magnification factors can be computed in the explicit form.Figure <ref> shows the “rogue" dn-periodic wave for k = 0.5 (left) and k = 0.99 (right). We write the name of “rogue wave" in commas, because the solution is not a proper rogue wave, the latter one is supposed to appear from nowhere and to disappear without a trace as time evolves <cit.>. Instead, we obtain a solution that corresponds to a nonlinear superposition of the algebraically decaying soliton of the mKdV <cit.> and the dn-periodic wave, hence the maximal amplitude is brought by the algebraic soliton from infinity. This outcome of our algorithm is related to the fact that the dn-periodic wave in the mKdV is modulationally stable with respect to the long-wave perturbations <cit.>. Indeed, it is argued in <cit.> on several examples involving the constant wave background that the rogue wave solutions exist only in the parameter regions where the constant wave background is modulationally unstable.Figure <ref> shows the rogue cn-periodic wave for k = 0.5 (left) and k = 0.99 (right). This solution is a proper rogue wave on the periodic background because it appears from nowhere and disappears without a trace as time evolves. The existence of such rogue periodic wave is related to the fact that the cn-periodic wave in the mKdV is modulationally unstable with respect to the long-wave perturbations <cit.>.The magnification factors of the rogue periodic waves can be computed in the explicit form:M_ dn(k) = 2 + √(1 - k^2),M_ cn(k) = 3,k ∈ [0,1].It is remarkable that the magnification factor M_ cn(k) = 3 is independent of the wave amplitude in agreement with M_0 = 3 for the classical rogue wave (<ref>) thanks to the small-amplitude asymptotic limit (<ref>). At the same time M_ dn(k) ∈ [2,3] and M_ dn(k) → 3 as k → 0 due to the fact that the limit (<ref>) gives the same potential to the AKNS spectral problem as the constant wave background ψ_0(x,t) = 1 of the NLS equation (<ref>).In the soliton limit (<ref>), M_ dn(k) → 2 as k → 1 in agreement with the recent results in <cit.>. Indeed, the “rogue" dn-periodic wave degenerates as k → 1 to the two-soliton solutions constructed of two nearly identical solitons. Such solutions are constructed by the one-fold Darboux transformation from the one-soliton solutions, when the eigenfunction of the AKNS spectral problem is nondecaying (exponentially growing) <cit.>. Therefore, the magnification factor M_ dn(1) = 2 is explained by the weak interaction between two nearly identical solitons. On the other hand, M_ cn(1) = 3 is explained by the fact that the rogue cn-periodic wave is built from the two-fold Darboux transformation, hence it degenerates as k → 1 to the three-soliton solutions constructed of three nearly identical solitons <cit.>.The paper is organized as follows. Section 2 gives details of the periodic eigenfunctions of the AKNS spectral problem associated with the dn and cn Jacobian elliptic functions. The non-periodic functions of the AKNS spectral problem are computed in Section 3. Section 4 presents the general N-fold Darboux transformation for the mKdV equation and the explicit formulas for the one-fold and two-fold Darboux transformations. The rogue dn-periodic and cn-periodic waves of the mKdV are constructed in Sections 5 and 6 respectively. Appendix A gives a proof of the N-fold Darboux transformations in the explicit form. Acknowledgement. The authors thank E.N. Pelinovsky for suggesting the problem of computing the rogue periodic waves in the mKdV. J. Chen is grateful to the Department of Mathematics of McMaster University for the generous hospitality during his visit. J. Chen is supported by the National Natural Science Foundation of China (No. 11471072) and the JiangsuOverseas Research & Training Programme for University Prominent Young & Middle-aged Teachers and Presidents (No. 1160690028). D.E. Pelinovsky is supported by the state task of Russian Federation in the sphere of scientific activity (Task No. 5.5176.2017/8.9).§ PERIODIC EIGENFUNCTIONS OF THE AKNS SPECTRAL PROBLEM The mKdV equation (<ref>) is obtained as a compatibility condition of the following Lax pair of two linear equations for the vector φ = (φ_1,φ_2)^t:φ_x = U(λ,u) φ,U(λ,u) =([λu; -u -λ;]),andφ_t = V(λ,u) φ,V(λ,u) = ([-4λ^3-2λ u^2 -4λ^2u-2λ u_x-2u^3-u_xx;4λ^2u-2λ u_x+2u^3+u_xx 4λ^3+2λ u^2; ]).The first linear equation (<ref>) is referred to as the AKNS spectral problem as it defines the spectral parameter λ for a given potential u(x,t) at a frozen time t, e.g. at t = 0. By using the Pauli matricesσ_1 = ( [ 0 1; 1 0 ]), σ_2 = ( [0 -i;i0 ]), σ_3 = ( [10;0 -1 ]),we can rewrite U(λ,u) and V(λ,u) in (<ref>) and (<ref>) in the formU(λ,u) =λσ_3 + u σ_3 σ_1, V(λ,u) = -(4 λ^3 + 2 λ u^2) σ_3 - 4 λ^2 u σ_3 σ_1 - 2 λ u_x σ_1 - (2 u^3 + u_xx) σ_3 σ_1. If u is either dn or cn Jacobian elliptic functions (<ref>) and (<ref>), the potentials are L-periodic in x with the period L = 2K(k) for dn-functions and L = 4 K(k) for cn-functions, where K(k) is the complete elliptic integral. If the AKNS spectral problem (<ref>) is considered in the space of L-periodic functions, then the admissible set for the spectral parameter λ is discrete as the AKNS spectral problem has a purely point spectrum.In the case of periodic or quasi-periodic potentials u, the algebraic technique based on the nonlinearization of the Lax pair <cit.> (see also applications in <cit.>) can be used to obtain explicit solutions for the eigenfunctions of the AKNS spectral problem related to the particular eigenvalues λ with Re(λ) > 0. Below we simplify the general method in order to obtain particular solutions of the AKNS spectral problem for the periodic waves in the focusing mKdV equation (<ref>). The following two propositions represent the explicit expressions for eigenvalues and periodic eigenfunctions of system (<ref>) and (<ref>) related to the travelling periodic wave solution of the mKdV.Let u be a travelling wave solution of the mKdV equation (<ref>) satisfyingd^2 u/d x^2 + 2 u^3 = c u, (du/dx)^2 + u^4 = c u^2 + d,where c and d are real constants parameterized byc = 4 λ_1^2 + 2 E_0,d = - E_0^2with possibly complex λ_1 and E_0. Then, there exists a solution φ = (φ_1,φ_2)^t of the AKNS spectral problem (<ref>) with λ = λ_1 such thatφ_1^2 + φ_2^2 = u, φ_1^2 - φ_2^2 = 1/2λ_1du/dx,4 λ_1 φ_1 φ_2 = E_0 - u^2.In particular, if u is periodic in x, then φ is periodic in x.Following <cit.>, we set u = φ_1^2 + φ_2^2 and consider a nonlinearization of the AKNS spectral problem (<ref>) given by the Hamiltonian systemd φ_1/dx = ∂ H/∂φ_2, d φ_2/dx = -∂ H/∂φ_1,which is related to the Hamiltonian functionH(φ_1,φ_2) = 1/4 (φ_1^2+φ_2^2)^2 + λ_1 φ_1 φ_2 = 1/4 E_0,where E_0 is constant in x. It follows from (<ref>) that 4 λ_1 φ_1 φ_2 = E_0 - u^2. Also note thatdu/dx = 2 ( φ_1 d φ_1/d x + φ_2 d φ_2/dx) = 2 λ_1 (φ_1^2 - φ_2^2),so that all three equations in (<ref>) are satisfied by the construction.Let us introduceQ(λ) = ([λφ_1^2 + φ_2^2; -φ_1^2 - φ_2^2 -λ ]),W(λ) = ([ W_11(λ) W_12(λ);W_12(-λ) -W_11(-λ) ]),withW_11(λ) = 1 - φ_1 φ_2/λ - λ_1 + φ_1 φ_2/λ + λ_1, W_12(λ) =φ_1^2/λ-λ_1 + φ_2^2/λ + λ_1.Then, one can check directly that the Lax equationd/dx W(λ) = Q(λ) W(λ) - W(λ) Q(λ),is satisfied for every λ∈ℂ if and only if (φ_1,φ_2) satisfies (<ref>). In particular, the (1,2)-entry in the above relations yields the equationd/dx W_12(λ) = 2 λ W_12(λ) - 2 (φ_1^2+φ_2^2) W_11(λ)and the representationW_12(λ) = 1/λ^2 - λ_1^2[ λ (φ_1^2 + φ_2^2) + λ_1 (φ_1^2 - φ_2^2) ] =: (λ-μ) (φ_1^2 + φ_2^2)/a(λ),witha(λ) := λ^2 - λ_1^2 μ := -λ_1 φ_1^2 - φ_2^2/φ_1^2 + φ_2^2 = -1/2udu/dx.In addition, we note that[W(λ)] = -[W_11(λ)]^2 - W_12(λ) W_21(λ) = -b(λ)/a(λ)withb(λ) := λ^2 - λ_1^2 - 4 λ_1 φ_1 φ_2 - (φ_1^2+φ_2^2)^2 = λ^2 - λ_1^2 - E_0.Since W_12(λ) has a simple zero at λ = μ, then[W_11(μ)]^2 = b(μ)/a(μ).By substituting (<ref>), (<ref>), andd/dx W_12(μ) = -(φ_1^2+φ_2^2)/a(μ)dμ/dxto equation (<ref>) and squaring it, we obtain the closed equation on μ:1/4( dμ/dx)^2 = a(μ) b(μ) = (μ^2 - λ_1^2)(μ^2 - λ_1^2 - E_0).Substituting the representation (<ref>) yieldsu^2 ( d^2u/dx^2)^2 - 2 u ( du/dx)^2 [ d^2u/dx^2 - 2(E_0+2λ_1^2) u ] = 16 λ_1^2 (E_0+λ_1^2) u^4.Let u be a solution of the differential equations (<ref>) with parameters c and d. Substituting (<ref>) to (<ref>) yields the relations (<ref>) between (c,d) and (λ_1,E_0). Hence, the constraint (<ref>) is fulfilledif u satisfies (<ref>) with parameters (c,d) satisfying (<ref>). Let u, φ = (φ_1,φ_2)^t, and λ_1 be the same as in Proposition <ref>. Then φ(x-ct) satisfies the linear system (<ref>) with λ = λ_1 and u(x-ct).By using (<ref>), we rewrite the first equation of system (<ref>) with λ = λ_1 as∂_t φ_1 = - (4 λ_1^3 + 2 λ_1 u^2) φ_1 - (4 λ_1^2 u + 2 λ_1 u_x + c u) φ_2.By using (<ref>), we note that(4 λ_1^2 + 2 u^2) φ_1 + (4 λ_1 u + 2 u_x) φ_2 = (4 λ_1^2 + 2 u^2 + 8 λ_1 φ_1 φ_2) φ_1 = (4 λ_1^2 + 2 E_0) φ_1.By using (<ref>) and the first equation in system (<ref>), equation (<ref>) becomes∂_t φ_1 = - λ_1 c φ_1 - c u φ_2 = -c ∂_x φ_1,hence φ_1(x-ct) is a solution of system (<ref>) and (<ref>) with λ = λ_1 and u(x-ct). Similar computations hold for φ_2by symmetry from the second equations in systems (<ref>) and (<ref>). For the dn Jacobian elliptic functions (<ref>), we have c = 2 - k^2 and d = k^2 - 1 ≤ 0. Since u(x) > 0 for every x ∈ℝ, the periodic eigenfunction φ = (φ_1,φ_2)^t in Proposition <ref> is real with parameters E_0 = ±√(1 - k^2) andλ_1^2 = 1/4[ 2 - k^2 ∓ 2 √(1-k^2)].Taking the positive square root of (<ref>), we obtain two particular real pointsλ_±(k) := 1/2( 1 ±√(1-k^2)),such that 0 < λ_-(k) < λ_+(k) < 1 for every k ∈ (0,1). As k → 0, we have λ_-(k) → 0 and λ_+(k) → 1, whereas as k → 1, we have λ_-(k), λ_+(k) → 1/2.For the cn Jacobian elliptic functions (<ref>), we have c = 2k^2-1 and d = k^2(1-k^2) ≥ 0. Since u(x) is sign-indefinite, the periodic eigenfunction φ = (φ_1,φ_2)^t in Proposition <ref> is complex-valued with parameters E_0 = ± i k √(1 - k^2) andλ_1^2 = 1/4[ 2 k^2 - 1 ∓ 2 i k √(1-k^2)].Defining the square root of (<ref>) in the first quadrant of the complex plane, we obtainλ_I(k) := 1/2( k + i √(1-k^2)).As k → 0, we have λ_I(k) → i/2, whereas as k → 1, we have λ_I(k) → 1/2.Figure <ref> shows the spectral plane of λ with the schematic representation of the Floquet–Bloch spectrum for the dn-periodic wave with k = 0.75 (left) and the cn-periodic wave with k = 0.75 (right). The branch points λ_±(k) and λ_I(k) obtained in (<ref>) and (<ref>) are marked explicitly as the end points of the Floquet–Bloch spectral bands away from the imaginary axis.§ NON-PERIODIC SOLUTIONS OF THE AKNS SPECTRAL PROBLEM Here we construct the second linearly independent solution to the AKNS spectral problem (<ref>) with λ = λ_1 and extend it to satisfy the linear system (<ref>). The second solution is no longer periodic in variables (x,t). The following result represents the corresponding solution.Let u, λ_1, E_0, and φ = (φ_1,φ_2)^t be the same as in Proposition <ref>. Assume that u(x)^2 - E_0 ≠ 0 for every x. The second linearly independent solution of the AKNS spectral problem (<ref>) with λ = λ_1 is given by ψ = (ψ_1,ψ_2)^t, whereψ_1 = θ - 1/φ_2, ψ_2 = θ + 1/φ_1,andθ(x) = -4 λ_1 (u(x)^2-E_0) ∫_0^xu(y)^2/(u(y)^2-E_0)^2dy.In particular, if u is periodic in x, then θ grows linearly in x as |x| →∞, so that ψ_1 and ψ_2 are not periodic in x.Since the AKNS spectral problem (<ref>) is related to the traceless matrix, the Wronskian of the two linearly independent solutions φ = (φ_1,φ_2)^t and ψ = (ψ_1,ψ_2)^t is independent of x. Normalizing it by 2, we write the relationφ_1 ψ_2 - φ_2 ψ_1 = 2,from which the representation (<ref>) follows with arbitrary θ. If u(x)^2 - E_0 ≠ 0 for every x, then φ_1(x) ≠ 0 and φ_2(x) ≠ 0 for every x. Substituting (<ref>) to (<ref>), we obtain the following scalar linear differential equation for θ:d θ/d x = u θφ_2^2-φ_1^2/φ_1 φ_2 + u φ_1^2+φ_2^2/φ_1 φ_2.By using relations (<ref>), we rewrite it in the equivalent forms:d θ/d x = θ2 u u'/u^2 - E_0 - 4 λ_1 u^2/u^2 - E_0⇒d/dx[ θ/u^2 - E_0] = - 4 λ_1 u^2/(u^2 - E_0)^2.Integrating the last equation with the boundary condition θ(0) = 0, we obtain (<ref>). Let u, λ_1, E_0, φ = (φ_1,φ_2)^t, and ψ = (ψ_1,ψ_2)^t be the same as in Proposition <ref>. Then, ψ = (ψ_1,ψ_2)^t expressed by (<ref>) satisfies the linear system (<ref>) with λ = λ_1 and u(x-ct) if θ is expressed byθ(x,t) = -4 λ_1 (u(x-ct)^2-E_0) [ ∫_0^x-ctu(y)^2/(u(y)^2-E_0)^2 dy - t]. By using (<ref>), we rewrite the first equation of system (<ref>) with λ = λ_1 as∂_t ψ_1 = - (4 λ_1^3 + 2 λ_1 u^2) ψ_1 - (4 λ_1^2 u + 2 λ_1 u_x + c u) ψ_2.By using (<ref>), (<ref>), and expressing ∂_t φ_2 from the second equation of system (<ref>), we obtain from (<ref>):∂_t θ=(4 λ_1^2 u - 2 λ_1 u_x + c u) φ_1 (θ - 1)/φ_2 - (4 λ_1^2 u + 2 λ_1 u_x + c u) φ_2 (θ + 1)/φ_1 = - 16 λ_1^2 φ_1 φ_2 - c u/φ_1 φ_2[ θ (φ_2^2-φ_1^2) + φ_1^2 + φ_2^2 ]= 4 λ_1 (u^2 - E_0) - c ∂_x θ.Let us represent θ = -4 λ_1 (u^2 - E_0) χ so that χ satisfies∂_t χ =- c ∂_x χ - 1.Hence χ(x,t) = - t + f(x-ct), where f is obtained from (<ref>) in the formf(x) = ∫_0^xu(y)^2/(u(y)^2-E_0)^2 dyto yield the representation (<ref>). Similar computations hold for ψ_2 by symmetry from the second equations in systems (<ref>) and (<ref>). Note that a more general solution for ψ = (ψ_1,ψ_2)^t is defined arbitrary up to an addition to the first solution φ = (φ_1,φ_2)^t. However, this addition is equivalent to the arbitrary choice of the lower limit in the integral (<ref>), which is then equivalent to the translation in time t. Thus, the second linearly independent solution in the form (<ref>) and (<ref>) is unique up to the translation in x and t. § ONE-FOLD, TWO-FOLD, AND N-FOLD DARBOUX TRANSFORMATIONS Here we give the explicit formulas for the one-fold and two-fold Darboux transformations for the focusing mKdV equation (<ref>), as well as the general formula for the N-fold Darboux transformation. Although the formal derivation of the N-fold Darboux transformation can be found in several sources, e.g. in book <cit.> or original papers <cit.>, we find it useful to derive the explicit transformation formulas by using purely algebraic calculations.By definition, we say that T(λ) is a Darboux transformation ifφ = T(λ) φ,where φ satisfies (<ref>)–(<ref>) for a particular potential u and φ satisfies (<ref>)–(<ref>) for a new potential u, which is related to u. The transformation formulas between φ and φ follow from the Darboux equations∂_x T(λ) + T(λ) U(λ,u) = U(λ,u) T(λ)and∂_t T(λ) + T(λ) V(λ,u) = V(λ,u) T(λ). In many derivations, e.g. in <cit.>, the N-fold Darboux transformation is deduced formally from a linear system of equations imposed on the coefficients of the polynomial representation of T(λ) without checking all the constraints arising from the Darboux equations (<ref>) and (<ref>). In order to avoid such formal computations, we give in Appendix A a rigorous derivation of the N-fold Darboux transformation in the explicit form and show how the Darboux equations (<ref>) and (<ref>) are satisfied. Our derivation relies on a particular implementation of the dressing method <cit.> which was recently reviewed in the context of the cubic NLS equation in <cit.>.The general N-fold Darboux transformation is given by the following theorem.Let u be a smooth solution of the mKdV equation (<ref>). Let φ^(k) = (p_k,q_k)^t, 1 ≤ k ≤ N be a particular smooth nonzero solution of system (<ref>) and (<ref>) with fixed λ = λ_k ∈ℂ\{0} and potential u. Assume that λ_k ≠±λ_j for every k ≠ j. Let {φ^(k)}_1 ≤ k ≤ N be a solution of the linear algebraic systemσ_3 σ_1 φ^(j) = ∑_k=1^N ⟨φ^(j), φ^(k)⟩/λ_j + λ_kφ^(k),1 ≤ j ≤ N,where ⟨φ^(j), φ^(k)⟩ := p_j p_k + q_j q_k is the inner vector product. Assume that the linear system (<ref>) has a unique solution. Then, φ^(k) = (p_k,q_k)^t, 1 ≤ k ≤ N is a particular solution of system (<ref>) and (<ref>) with λ = λ_k and the new potential u given byu = u + 2 ∑_j=1^N p_j p_j = u - 2 ∑_j=1^N q_j q_j.Consequently, u is a new solution of the mKdV equation (<ref>). The proof of Theorem <ref> is given in Appendix A. The following two propositions represent the one-fold and two-fold Darboux transformation formulas deduced from Theorem <ref> for N = 1 and N = 2 respectively. Let u be a smooth solution of the mKdV equation (<ref>). Let φ = (p,q)^t be a particular smooth nonzerosolution of system (<ref>) and (<ref>) with fixed λ = λ_1 ∈ℂ\{0}. Then,u = u + 4 λ_1 p q/p^2+q^2is a new solution of the mKdV equation (<ref>).Solving the linear system (<ref>) for φ = (p,q)^t yieldsp = 2 λ_1 q/p^2+q^2, q = -2 λ_1 p/p^2+q^2.Substituting (<ref>) into (<ref>) for N = 1 results in the transformation formula (<ref>).Let u be a smooth solution of the mKdV equation (<ref>). Let φ^(k) = (p_k,q_k)^t be a particular smooth nonzero solution of system (<ref>) and (<ref>) with fixed λ = λ_k ∈ℂ\{0} for k = 1,2 such that λ_1 ≠±λ_2. Then,ũ = u + 4 (λ_1^2-λ_2^2) [λ_1 p_1 q_1 (p_2^2 + q_2^2) - λ_2 p_2 q_2 (p_1^2 + q_1^2) ]/ (λ_1^2 + λ_2^2) (p_1^2+q_1^2) (p_2^2+q_2^2) - 2 λ_1 λ_2 [4 p_1 q_1 p_2 q_2 + (p_1^2-q_1^2)(p_2^2-q_2^2) ]is a new solution of the mKdV equation (<ref>).The linear system (<ref>) is generated by the matrix A with the entriesA_jk = ⟨φ^(j), φ^(k)⟩/λ_j + λ_k,1 ≤ j,k ≤ N.For N = 2, we compute the determinant of this matrix asdet(A) =1/4 λ_1 λ_2 (λ_1+λ_2)^2[ (λ_1 + λ_2)^2 (p_1^2+q_1^2)(p_2^2+q_2^2) - 4 λ_1 λ_2 (p_1p_2 +q_1q_2)^2]=1/4 λ_1 λ_2 (λ_1+λ_2)^2[ (λ_1^2 + λ_2^2) (p_1^2+q_1^2)(p_2^2+q_2^2) - 2 λ_1 λ_2 ( 4 p_1 p_2 q_1 q_2 + (p_1^2-q_1^2)(p_2^2 - q_2^2) ) ].Solving the linear system (<ref>) with Cramer's rule yields the componentsp_1 = (λ_1 + λ_2) q_1 (p_2^2+q_2^2)-2 λ_2 q_2 (p_1p_2+q_1q_2)/2 λ_2 (λ_1+λ_2)det(A)andp_2 = (λ_1 + λ_2) q_2 (p_1^2+q_1^2)-2 λ_1 q_1 (p_1p_2+q_1q_2)/2 λ_1 (λ_1+λ_2)det(A).Substituting these formulas to the representation (<ref>) with N = 2 and reordering the similar terms result in the transformation formula (<ref>). § THE “ROGUE" DN-PERIODIC WAVE Here we apply the one-fold Darboux transformation (<ref>) to the Jacobian elliptic function dn in (<ref>) in order to obtain the “rogue" dn-periodic wave. We write the “rogue" wave in commas, because the corresponding solution is a nonlinear superposition of an algebraically decaying soliton and the dn-periodic wave, hence the maximal amplitude is brought by the algebraic soliton from infinity. The proper rogue dn-periodic wave does not exist in the mKdV equation (<ref>) because the dn-periodic wave is modulationally stable. We note however that very similar solutions to the NLS equation define a proper rogue dn-periodic wave, as is shown numerically in <cit.>.Let u be the dn periodic wave (<ref>), whereas φ = (φ_1,φ_2)^t be the periodic eigenfunction of the linear system (<ref>) and (<ref>) with λ = λ_1 defined by Propositions <ref> and <ref>. Since the connection formulas (<ref>) are satisfied for every t ∈ℝ, substituting p = φ_1 and q = φ_2 into the one-fold Darboux transformation (<ref>) yields another solution of the mKdV equation in the formũ = u + 4 λ_1 φ_1 φ_2/φ_1^2 + φ_2^2 = E_0/u,where E_0 = ±√(1 - k^2). However, sincedn(x + K(k);k) = √(1-k^2)/ dn(x;k),the new solution ũ to the mKdV equation (<ref>) is obtained trivially by the spatial translation of the dn periodic wave on the half-period 1/2 L = K(k). This computation explains why we need to use the second non-periodic solution ψ instead of the periodic eigenfunction φ. Let u be the dn periodic wave (<ref>), whereas ψ = (ψ_1,ψ_2)^t be the non-periodic solution to the linear system (<ref>) and (<ref>) with λ = λ_1 defined by Propositions <ref> and <ref>. Recall that there exist two choices for λ_1 in (<ref>). However, for the choice λ_1 = λ_-(k), we have E_0 = √(1-k^2) and u(x)^2 - E_0 = 0 for some values of x in [-K(k),K(k)], therefore, the assumption of Proposition <ref> is not satisfied. For the choice λ_1 = λ_+(k), we have E_0 = -√(1-k^2) and u(x)^2 - E_0 > 0 for every x, therefore, the assumption of Proposition <ref> is satisfied. Substituting p = ψ_1 and q = ψ_2 given by (<ref>) into the one-fold Darboux transformation (<ref>) with λ_1 = λ_+(k) and E_0 = -√(1-k^2) yields another solution of the mKdV equation in the formũ = u + 4 λ_1 ψ_1 ψ_2/ψ_1^2 + ψ_2^2 = u + 4 λ_1 φ_1 φ_2 (θ^2 -1)/(φ_1^2+φ_2^2) (1 + θ^2) - 2 (φ_1^2-φ_2^2) θ.By using relations (<ref>) again, we finally write the new solution in the formu_ dn-rogue = u_ dn + (1-θ_ dn^2) (u_ dn^2+√(1-k^2))/ (1+θ_ dn^2)u_ dn - λ_1^-1θ_ dn u_ dn'whereθ_ dn(x,t) = -4 λ_1 (u_ dn(x-ct)^2+√(1-k^2)) [ ∫_0^x-ctu_ dn(y)^2/(u_ dn(y)^2+ √(1-k^2))^2dy - t ].We refer to the exact solution (<ref>)–(<ref>) as the “rogue" dn periodic wave of the mKdV equation.If k = 0, then u_ dn(x,t) = 1, λ_1 = 1, c = 2, θ_ dn(x,t) = -2(x-6t), andk = 0 :u_ dn-rogue(x,t) = -1 + 4/1 + 4 (x-6t)^2.Although this expression is an analogue of the rogue wave of the NLS on the constant wave background <cit.>, it corresponds to the algebraically decaying soliton of the mKdV <cit.>.If k = 1, then u_ dn(x,t) =sech(x-t), λ_1 = 1/2, c = 1,θ_ dn(x,t) = -(x - 3t)sech^2(x-t) - tanh(x-t),andk = 1 :u_ dn-rogue(x,t) = 2sech(x-t) 1 - (x-3t) tanh(x-t)/1 + (x-3t)^2sech^2(x-t).in agreement with the two-soliton solutions of the mKdV for two nearly identical solitons <cit.>.Next, we show that for every k ∈ [0,1), there exists a particular line x = c_* t with c_* > c such that θ_ dn(x,t) given by (<ref>) remains bounded as |x| + |t| →∞. This value of c_* gives the speed of the algebraically decaying soliton propagating on the dn-periodic wave background. For instance, if k = 0, then c_* = 6 > 2 = c.In order to show the claim above, we inspect the expression∫_0^x-ctu_ dn(y)^2/(u_ dn(y)^2+ √(1-k^2))^2dy - t.Since the integrand is a positive L=2K(k)-periodic function with a positive mean value denoted by I(k), then the expression can be written asI(k) (x-ct) - t + .Therefore, θ_ dn(x,t) is bounded at x = c_* t, where c_* = c + [I(k)]^-1 > c.Except for the line x = c_* t, the function θ_ dn(x,t) given by (<ref>) grows linearly in x and t as |x| + |t| →∞ for every k ∈ [0,1). Hence the representation (<ref>) yields asymptotic behavioru_ dn-rogue(x,t) ∼ - √(1-k^2)/ dn(x-ct;k) = - dn(x-ct+K(k);k) = -u_ dn(x-ct+K(k)).The maximal value of u_ dn-rogue(x,t) as |x| + |t| →∞ except for the line x = c_* t coincides with the maximal value of u_ dn(x,t) =dn(x-ct;k).For t = 0, u_ dn(x,0) is even in x, θ_ dn(x,0) is odd in x, hence u_ dn-rogue(x,0) is even in x. The maximal value of u_ dn(x,0) occurs at u_ dn(0,0) = 1. Since u_ dn-rogue(x,0) is even in x, then x = 0 is an extremal point of u_ dn-rogue(x,0). Moreover, ∂_x^2 u_ dn-rogue(0,0) < 0, which follows from the expansions u_ dn(x,0) = 1 - 1/2 k^2 x^2 + 𝒪(x^4), θ_ dn(x,0) = -4 λ_1 (1+√(1-k^2))^-1 x + 𝒪(x^3), andu_ dn-rogue(x,0) = 2 + √(1-k^2) - [ 8 - 3 k^2 + 8 √(1-k^2) - 1/2 k^2 √(1-k^2)] x^2 + 𝒪(x^4).Hence x = 0 is the point of maximum of u_ dn-rogue(x,0). Defining the magnification number asM_ dn(k) = u_ dn-rogue(0,0)/max_x ∈ [-K(k),K(k)] u_ dn(x,0) = 2 + √(1-k^2),we obtain the expression in (<ref>). The value M_ dn(k) corresponds to the amplitude of the algebraically decaying soliton propagating on the background of the dn-periodic wave.§ THE ROGUE CN-PERIODIC WAVE Here we apply the one-fold and two-fold Darboux transformations (<ref>) and (<ref>) to the Jacobian elliptic function cn in (<ref>) in order to obtain the rogue cn-periodic wave. This is a proper rogue cn-periodic wave because the cn-periodic wave is modulationally unstable.Let u be the cn periodic wave (<ref>), whereas φ = (φ_1,φ_2)^t be the periodic solution to the linear system (<ref>) and (<ref>) with λ = λ_1 defined by Propositions <ref> and <ref>. Without loss of generality, we choose λ_1 = λ_I(k), where λ_I(k) is given by (<ref>), so that E_0 = - i k √(1-k^2). Since the periodic solution φ is complex, the one-fold Darboux transformation (<ref>) produces a complex-valued solution to the mKdV, hence we should use the two-fold Darboux transformation (<ref>).By virtue of relations (<ref>), substituting (p_1,q_1) = (φ_1,φ_2) with λ_1 = λ_I and (p_2,q_2) = (φ_1,φ_2) with λ_2 = λ_I to the two-fold Darboux transformation (<ref>) yields another solution of the mKdV equation in the formũ = u + 4 k^2 (1-k^2) u/(2k^2-1) u^2 - u^4 - k^2(1-k^2) - (u')^2 = -u,where the first-order invariant in (<ref>) is used in the second identity with c = 2k^2-1 and d = k^2(1-k^2). Thus, the new solution ũ in the two-fold transformation (<ref>) is trivially related to the previous solution u if the functions (p_1,q_1) and (p_2,q_2) are periodic.Let us now consider the non-periodic solution ψ = (ψ_1,ψ_2)^t to the linear system (<ref>) and (<ref>) with λ = λ_I. The assumption of Proposition <ref> is satisfied because E_0 = - i k √(1-k^2)≠ 0 for k ∈ (0,1) and u(x)^2 - E_0 ≠ 0 for every x. Therefore, the non-periodic solution ψ inPropositions <ref> and <ref> is well-defined. Substituting (p_1,q_1) = (ψ_1,ψ_2) with λ_1 = λ_I and (p_2,q_2) = (ψ_1,ψ_2) with λ_2 = λ_I into the two-fold Darboux transformation (<ref>) yields another solution of the mKdV in the formũ = u + 4 (λ_I^2-λ_I^2) [ λ_I ψ_1 ψ_2 (ψ_1^2 + ψ_2^2) - λ_I ψ_1 ψ_2 (ψ_1^2 + ψ_2^2) ]/ (λ_I^2 + λ_I^2) |ψ_1^2+ψ_2^2|^2 - 2 |λ_I|^2 [4 |ψ_1|^2 |ψ_2|^2 + |ψ_1^2-ψ_2^2|^2] = u + F_1/F_2,whereF_1 = 8Im(λ_I^2)Im[ λ_I φ_1 φ_2 (1-θ^2) [(1+θ^2)(φ_1^2+φ_2^2) - 2 θ (φ_1^2-φ_2^2)] ], F_2 = Re(λ_I^2) |(1+θ^2)(φ_1^2+φ_2^2) - 2 θ (φ_1^2-φ_2^2)|^2 t- |λ_I|^2 ( 4 |1-θ^2|^2 |φ_1|^2 |φ_2|^2 + |(1+θ^2)(φ_1^2-φ_2^2) - 2 θ (φ_1^2+φ_2^2)|^2 ).By using relations (<ref>) and (<ref>), we finally write the new solution in the formu_ cn-rogue = u_ cn + G_1/G_2,whereG_1 = 4 k √(1-k^2) Im[ (u_ cn^2+i k √(1-k^2)) (1-θ_ cn^2) [(1+θ_ cn^2) u_ cn - λ_I^-1θ_ cn u_ cn'] ], G_2 = (1-2k^2) |(1+θ_ cn^2) u_ cn - λ_I^-1θ_ cn u_ cn'|^2 t+ |1-θ_ cn^2|^2 [ u_ cn^4 + k^2(1-k^2) ] + |(1+θ_ cn^2) (2λ_I)^-1 u_ cn' - 2 θ_ cn u_ cn|^2,andθ_ cn(x,t) = -4 λ_I (u_ cn(x-ct)^2+i k √(1-k^2)) [ ∫_0^x-ctu_ cn(y)^2/(u_ cn(y)^2+ i k√(1-k^2))^2dy - t].We refer to the exact solution (<ref>)–(<ref>) as the rogue cn periodic wave of the mKdV equation.As k → 0, then u_ cn(x,t) → 0, λ_I → i/2, θ_ cn(x,t) → 0, and u_ cn-rogue(x,t) → 0. Although the limit is zero, one can derive asymptotic expansions at the order of 𝒪(k) which recovers the rogue wave of the NLS equation (<ref>), according to the asymptotic transformation of the focusing mKdV to the focusing NLS in the small-amplitude limit <cit.>. The rouge cn-periodic wave generalizes the rogue wave (<ref>) on the constant wave background.As k → 1, then u_ cn(x,t) → sech(x-t), λ_I → 1/2, and it may first seem that the second term in (<ref>) vanishes. However, G_1 = 𝒪(1-k^2) and G_2 = 𝒪(1-k^2), hence a non-trivial limit exists to yield a three-soliton solution to the mKdV with three nearly identical solitons <cit.>.Let us inspect the expression∫_0^x-ctu_ cn(y)^2/(u_ cn(y)^2+ i k√(1-k^2))^2dy - t = ∫_0^x-ctu_ cn(y)^2 (u_ cn(y)^2- i k √(1-k^2))^2/(u_ cn(y)^4+ k^2 (1-k^2))^2dy - t.For every k ∈ (0,1), the imaginary part in the integrand is a negative L=4K(k)-periodic function with a negative mean value. It is only bounded on the line x = ct, however, the real part of the last term in the expression grows linearly in t. Therefore, for every k ∈ (0,1), |θ_ cn(x,t)| grows linearly in x and t as |x| + |t| →∞ everywhere on the (x,t) plane. Hence the representation (<ref>) yields the asymptotic behavioru_ cn-rogue(x,t)∼u_ cn(x,t) + 4 k^2 (1-k^2) u_ cn(x,t)/ (2k^2-1) u_ cn(x,t)^2 - (∂_x u_ cn(x,t))^2 - u_ cn(x,t)^4-k^2(1-k^2)= -u_ cn(x,t),where the first-order invariant in (<ref>) is used for the last identity with c = 2k^2-1 and d = k^2(1-k^2). The maximal value of u_ cn-rogue(x,t) as |x| + |t| →∞ coincides with the maximal value of u_ cn(x,t) = kcn(x-ct;k).For t = 0, u_ cn(x,0) is even in x, θ_ cn(x,0) is odd in x, hence u_ cn-rogue(x,0) is even in x. The maximal value of u_ cn(x,0) occurs at u_ cn(0,0) = k. Since u_ cn-rogue(x,0) is even in x, then x = 0 is an extremal point of u_ cn-rogue(x,0). Moreover, ∂_x^2 u_ cn-rogue(0,0) < 0, which follows from the expansions u_ cn(x,0) = k - 1/2 k x^2 + 𝒪(x^4), θ_ cn(x,0) = -4 λ_I (k^2 - i k √(1-k^2)) x + 𝒪(x^3), andu_ cn-rogue(x,0) = 3k - [ 3/2 k + 16 k^3 ] x^2 + 𝒪(x^4).Hence x = 0 is the point of maximum of u_ cn-rogue(x,0). Defining the magnification number asM_ cn(k) = u_ cn-rogue(0,0)/max_x ∈ [-2K(k),2K(k)] |u_ cn(x,0)| = 3,we obtain the expression in (<ref>). The magnification factor is independent of the amplitude of the cn-periodic wave. § PROOF OF N-FOLD DARBOUX TRANSFORMATIONHere we prove Theorem <ref> with explicit algebraic computations. The Darboux transformation matrix T(λ) in (<ref>) is sought in the following explicit form:T(λ) = I + ∑_k=1^N 1/λ - λ_k T_k,T_k = φ^(k)⊗ (φ^(k))^t σ_1 σ_3,where the sign ⊗ denotes the outer vector product and I denotes an identity 2 × 2 matrix.We note that φ^(k)∈ ker(T_k) and φ^(k)∈ ran(T_k). It is assumed in Theorem <ref> that φ^(k) = (p_k,q_k)^t, 1 ≤ k ≤ N is a particular smooth nonzero solution to system (<ref>) and (<ref>) with fixed λ = λ_k ∈ℂ\{0} satisfying λ_k ≠±λ_j for every k ≠ j, whereas {φ^(k)}_1 ≤ k ≤ N is a unique solution of the linear algebraic system (<ref>). Deeper in the proof, we will be able to show that φ^(k) = (p_k,q_k)^t, 1 ≤ k ≤ N is a particular solution to system (<ref>) and (<ref>) with λ = λ_k and new potential u given by the transformation formula (<ref>).First, let us show that the two lines in the definition (<ref>) are identical. Let us define entries of the matrix A by (<ref>). Each entry is finite, moreover, A_jk = A_kj. The linear system (<ref>) can be split into two parts as follows∑_k=1^N ⟨φ^(j), φ^(k)⟩/λ_j + λ_kp_k = q_j, ∑_k=1^N ⟨φ^(j), φ^(k)⟩/λ_j + λ_kq_k = -p_j.Thanks to the symmetry of A, we obtain from (<ref>):∑_j=1^N q_j q_j =∑_j=1^N∑_k=1^N ⟨φ^(j), φ^(k)⟩/λ_j + λ_kp_k q_j = ∑_j=1^N∑_k=1^N ⟨φ^(j), φ^(k)⟩/λ_j + λ_kp_j q_k = -∑_j=1^N p_j p_j.This proves that the two lines in the definition (<ref>) are identical. For further use, let us also derive another relation from the system (<ref>):∑_j=1^N λ_j q_j q_j - ∑_j=1^N λ_j p_j p_j =∑_j=1^N∑_k=1^N ⟨φ^(j), φ^(k)⟩p_k q_j=( ∑_j=1^N p_j p_j ) ( ∑_k=1^N p_k q_k ) + ( ∑_j=1^N q_j q_j ) ( ∑_k=1^N p_k q_k )=1/2 (u-u) ( ∑_k=1^N p_k q_k- ∑_k=1^N p_k q_k ). Next, we show validity of the Darboux equation (<ref>) under the transformation formula (<ref>). Substituting (<ref>) to (<ref>) yields the following equations at the simple poles∂_x T_k + T_k U(λ_k,u) = U(λ_k,u) T_k,1 ≤ k ≤ N,and the following equation at the constant termuσ_3 σ_1 = u σ_3 σ_1+ ∑_k=1^N T_k σ_3 - σ_3 T_k.Equation (<ref>) yields (<ref>) due to representation (<ref>).Let us show that equations (<ref>) are satisfied if φ^(k) solves (<ref>) with λ = λ_k and u, whereas φ^(k) solves (<ref>) with λ = λ_k and u. Recall that σ_1 σ_3 = -σ_3 σ_1 and σ_1 σ_1 = σ_3 σ_3 = I. Substituting (<ref>) to both sides of (<ref>) yieldst [ ∂_x φ^(k)] ⊗ (φ^(k))^t σ_1 σ_3 + φ^(k)⊗[ ∂_x (φ^(k))^t] σ_1 σ_3 + φ^(k)⊗ (φ^(k))^t σ_1 σ_3 U(λ_1,u) t= [ ∂_x φ^(k)] ⊗ (φ^(k))^t σ_1 σ_3 + φ^(k)⊗[ ∂_x (φ^(k))^t] σ_1 σ_3 - φ^(k)⊗ (φ^(k))^t U(λ_1,u)^t σ_1 σ_3 t= [ ∂_x φ^(k)] ⊗ (φ^(k))^t σ_1 σ_3andU(λ_1,u) φ^(k)⊗ (φ^(k))^t σ_1 σ_3 = [ ∂_x φ^(k)] ⊗ (φ^(k))^t σ_1 σ_3,hence equation (<ref>) is satisfied.We show now that if {φ^(k)}_1 ≤ k ≤ N solve (<ref>) with {λ_k}_1 ≤ k ≤ N and u and {φ^(k)}_1 ≤ k ≤ N are obtained from the linear algebraic system (<ref>), then {φ^(k)}_1 ≤ k ≤ N solve (<ref>)with {λ_k}_1 ≤ k ≤ Nand u. We note from the linear system (<ref>) that∂_x ⟨φ^(j), φ^(k)⟩ = (λ_j + λ_k) ⟨φ^(j), σ_3 φ^(k)⟩.Differentiating (<ref>) in x and substituting (<ref>) and (<ref>) yieldt ∑_k=1^N ⟨φ^(j), φ^(k)⟩/λ_j + λ_k[ ∂_x φ^(k) - λ_k σ_3 φ^(k) - uσ_3 σ_1 φ^(k)] = (u-u) φ^(j) - ∑_k=1^N [ ⟨φ^(j), σ_3 φ^(k)⟩φ^(k) + ⟨φ^(j), φ^(k)⟩σ_3 φ^(k)] = 0,where the last equality is due to the transformation formula (<ref>). Thus, if the linear system (<ref>) is assumed to admit a unique solution, then φ^(k) solves (<ref>)with λ = λ_kand u.It remains to show validity of the Darboux equation (<ref>) under the transformation formula (<ref>). Substituting (<ref>) to (<ref>) yields the following equations at the simple poles∂_t T_k + T_k V(λ_k,u) = V(λ_k,u) T_k,1 ≤ k ≤ N,the same equation (<ref>) at λ^2 and the following two equations at λ^1 and λ^0 respectively:u^2 σ_3 + u_x σ_1 + 2 u∑_k=1^N σ_3 σ_1 T_k = u^2 σ_3 + u_x σ_1 + 2 u ∑_k=1^N T_k σ_3 σ_1 + 2 ∑_k=1^N λ_k ( T_k σ_3 - σ_3 T_k )and(2 u^3 + u_xx) σ_3 σ_1 + 2 u^2 ∑_k=1^N σ_3 T_k + 2 u_x ∑_k=1^N σ_1 T_k + 4 u∑_k=1^N λ_k σ_3 σ_1 T_k + 4 ∑_k=1^N λ_k^2 σ_3 T_kt = (2 u^3 + u_xx) σ_3 σ_1 + 2 u^2 ∑_k=1^N T_k σ_3 + 2 u_x ∑_k=1^N T_k σ_1 + 4 u ∑_k=1^N λ_k T_k σ_3 σ_1 + 4 ∑_k=1^N λ_k^2 T_k σ_3.t Let us show that equations (<ref>) are satisfied if φ^(k) solves (<ref>) with λ = λ_k and u, whereas φ^(k) solves (<ref>) with λ = λ_k and u. Substituting (<ref>) to both sides of (<ref>) yieldst [ ∂_t φ^(k)] ⊗ (φ^(k))^t σ_1 σ_3 + φ^(k)⊗[ ∂_t (φ^(k))^t] σ_1 σ_3 + φ^(k)⊗ (φ^(k))^t σ_1 σ_3 V(λ_1,u) t= [ ∂_t φ^(k)] ⊗ (φ^(k))^t σ_1 σ_3 + φ^(k)⊗[ ∂_t (φ^(k))^t] σ_1 σ_3 - φ^(k)⊗ (φ^(k))^t V(λ_1,u)^t σ_1 σ_3 t= [ ∂_t φ^(k)] ⊗ (φ^(k))^t σ_1 σ_3andV(λ_1,u) φ^(k)⊗ (φ^(k))^t σ_1 σ_3 = [ ∂_t φ^(k)] ⊗ (φ^(k))^t σ_1 σ_3,hence equation (<ref>) is satisfied.In order to show the validity of equation (<ref>), we differentiate (<ref>) in x and substitute (<ref>) to obtain(u_x-u_x)σ_1 = 2∑_k=1^Nλ_k(T_kσ_3-σ_3T_k) + u∑_k=1^N(σ_1T_kσ_3-σ_3σ_1T_k) + u ∑_k=1^N(σ_3T_kσ_1+T_kσ_3σ_1).Substituting (<ref>) into (<ref>) yields a simplified form of the equation:(u^2 - u^2) σ_3 + u∑_k=1^N σ_1 T_k σ_3 + σ_3 σ_1 T_k + u ∑_k=1^N σ_3 T_k σ_1 - T_k σ_3 σ_1 = 0.Further substituting (<ref>) into (<ref>) yields∑_k=1^N(σ_1T_kσ_3+σ_3σ_1T_k+T_kσ_3σ_1-σ_3T_kσ_1) = 0.The validity of equation (<ref>) is satisfied thanks again to equation (<ref>):(u-u) σ_3 = ∑_k=1^N (T_k σ_3 σ_1 - σ_3 T_k σ_1 ),(u-u) σ_3 = ∑_k=1^N (σ_1 T_k σ_3 + σ_3 σ_1 T_k).Hence, equation (<ref>) is satisfied.In order to show the validity of equation (<ref>), we differentiate (<ref>) in x and substitute (<ref>) to obtain(ũ_xx-u_xx)σ_3σ_1 = 4∑_k=1^Nλ_k^2(T_kσ_3-σ_3T_k)+ 2ũ∑_k=1^Nλ_k(σ_1T_kσ_3-σ_3σ_1T_k) +2u∑_k=1^Nλ_k(σ_3T_kσ_1+T_kσ_3σ_1)+ũ_x∑_k=1^N(σ_3σ_1T_kσ_3-σ_1T_k)+u_x∑_k=1^N(T_kσ_1+σ_3T_kσ_3σ_1)+(ũ^2+u^2)∑_k=1^N(σ_3T_k-T_kσ_3)+2uũ∑_k=1^N(σ_3σ_1T_kσ_1+σ_1T_kσ_3σ_1).Substituting (<ref>) and (<ref>) into (<ref>) yields a simplified form of the equation:t2u∑_k=1^Nλ_k(σ_1T_kσ_3+σ_3σ_1T_k)+ 2u∑_k=1^Nλ_k(σ_3T_kσ_1-T_kσ_3σ_1) t+ u_x∑_k=1^N(σ_3σ_1T_kσ_3+σ_1T_k) +u_x∑_k=1^N(σ_3T_kσ_3σ_1-T_kσ_1) +(u^2-u^2) ∑_k=1^N(T_kσ_3+σ_3T_k)t+2 u u ∑_k=1^N (σ_3 σ_1 T_k σ_1 + σ_1 T_k σ_3 σ_1 + T_k σ_3 - σ_3 T_k) = 0.The last term in the left-hand side of (<ref>) is identically zero thanks to equation (<ref>) after multiplication by σ_1 on the right. Multiplication of equation (<ref>) by σ_3 on the right allows us to group the terms containing u_x and u_x. As a result, we rewrite (<ref>) in the equivalent formt2u∑_k=1^Nλ_k(σ_1T_kσ_3+σ_3σ_1T_k)+2u∑_k=1^Nλ_k(σ_3T_kσ_1-T_kσ_3σ_1) t+ (u_x-u_x) ∑_k=1^N(σ_3T_kσ_3σ_1-T_kσ_1) +(u^2-u^2) ∑_k=1^N(T_kσ_3+σ_3T_k) = 0.Multiplying (<ref>) by σ_1 from the left and from the right, we obtain(u_x-u_x) I = 2∑_k=1^Nλ_k ( σ_1 T_kσ_3 + σ_3 σ_1 T_k) + u∑_k=1^N (T_k σ_3 + σ_3 T_k) + u ∑_k=1^N (σ_1 σ_3T_kσ_1 + σ_1 T_kσ_3σ_1)and(u_x-u_x) I =2∑_k=1^Nλ_k ( T_k σ_3 σ_1 - σ_3 T_k σ_1 ) + u∑_k=1^N(σ_1 T_k σ_3 σ_1 -σ_3σ_1T_k σ_1 ) + u ∑_k=1^N(σ_3T_k+T_kσ_3),from which one can rewrite (<ref>) in the equivalent form(u_x - u_x) (u - u) I - (u_x - u_x) ∑_k=1^N(σ_3T_kσ_3σ_1-T_kσ_1) = 0,which is satisfied thanks to equation (<ref>). Hence, equation (<ref>) is satisfied.Finally, we show that if {φ^(k)}_1 ≤ k ≤ N solve (<ref>) with {λ_k}_1 ≤ k ≤ N and u and {φ^(k)}_1 ≤ k ≤ N are obtained from the linear algebraic system (<ref>), then {φ^(k)}_1 ≤ k ≤ N solve (<ref>)with {λ_k}_1 ≤ k ≤ Nand u. We note from the linear system (<ref>) that∂_t ⟨φ^(j), φ^(k)⟩= -(λ_j + λ_k) [ 4 (λ_j^2 - λ_j λ_k + λ_k^2) + 2 u^2 ] ⟨φ^(j), σ_3 φ^(k)⟩t+ 4 (λ_j^2 - λ_k^2) u ⟨φ^(j), σ_3 σ_1 φ^(k)⟩ - 2 (λ_j + λ_k) u_x ⟨φ^(j), σ_1 φ^(k)⟩.Differentiating (<ref>) in t and substituting (<ref>) and (<ref>) yieldt ∑_k=1^N ⟨φ^(j), φ^(k)⟩/λ_j + λ_k[ ∂_t φ^(k) + (4 λ_k^3 + 2 λ_k u^2) σ_3 φ^(k) + 4 λ_k^2 uσ_3 σ_1 φ^(k) + 2 λ_k u_x σ_1 φ^(k) + (2 u^3 + u_xx) σ_3 σ_1 φ^(k)] =∑_k=1^N [ (4 λ_j^2 - 4 λ_j λ_k + 4 λ_k^2 + 2 u^2 ) ⟨φ^(j), σ_3 φ^(k)⟩ + 4 (λ_k - λ_j) u ⟨φ^(j), σ_3 σ_1 φ^(k)⟩ + 2 u_x ⟨φ^(j), σ_1 φ^(k)⟩] φ^(k) +∑_k=1^N ⟨φ^(j), φ^(k)⟩[ (4 λ_k^2 - 4 λ_k λ_j + 4 λ_j^2 + 2 u^2) σ_3 φ^(k) + 4 (λ_k - λ_j) uσ_3 σ_1 φ^(k) + 2 u_x σ_1 φ^(k)]+2 λ_j (u^2 - u^2) σ_1 φ^(j) + 4 λ_j^2 (u-u) φ^(j) - 2 λ_j (u_x - u_x) σ_3 φ^(j)+ (2 u^3 + u_xx - 2 u^3 - u_xx) φ^(j).The terms proportional to 4 λ_j^2 cancel out due to the same relation (<ref>). The terms proportional to 2 λ_j cancel out if the following relation is true:t(u^2 - u^2) σ_1 φ^(j) + (u_x - u_x) σ_3 φ^(j) = 2 ∑_k=1^N λ_k [ ⟨φ^(j), σ_3 φ^(k)⟩φ^(k) + ⟨φ^(j), φ^(k)⟩σ_3 φ^(k)] t+ 2 u∑_k=1^N ⟨φ^(j), φ^(k)⟩σ_3 σ_1 φ^(k) + 2 u ∑_k=1^N ⟨φ^(j), σ_3 σ_1 φ^(k)⟩φ^(k).The other λ_j-independent terms cancel out if the following relation is true:t(2 u^3 + u_xx - 2 u^3 - u_xx) φ^(j)= 4 ∑_k=1^N λ_k^2 [ ⟨φ^(j), σ_3 φ^(k)⟩φ^(k) + ⟨φ^(j), φ^(k)⟩σ_3 φ^(k)] t+ 2 u^2 ∑_k=1^N ⟨φ^(j), φ^(k)⟩σ_3 φ^(k) + 2 u^2 ∑_k=1^N ⟨φ^(j), σ_3 φ^(k)⟩φ^(k)t+ 4 u∑_k=1^N λ_k ⟨φ^(j), φ^(k)⟩σ_3 σ_1 φ^(k) + 4 u ∑_k=1^N λ_k ⟨φ^(j), σ_3 σ_1 φ^(k)⟩φ^(k)t+ 2 u_x ∑_k=1^N ⟨φ^(j), φ^(k)⟩σ_1 φ^(k) + 2 u_x ∑_k=1^N ⟨φ^(j), σ_1 φ^(k)⟩φ^(k).Provided equations (<ref>) and (<ref>) are satisfied, the right-hand side of equation (<ref>) is zero. If the linear system (<ref>) is assumed to admit a unique solution, then φ^(k) solves (<ref>)with {λ_k}_1 ≤ k ≤ Nand u.Finally, we show validity of equations (<ref>) and (<ref>). In order to show (<ref>), we first obtain the relation∂_x ⟨φ^(j), σ_3 φ^(k)⟩ = (λ_j + λ_k) ⟨φ^(j), φ^(k)⟩ + 2u ⟨φ^(j), σ_1 φ^(k)⟩,in addition to the relation (<ref>). Then, we differentiate (<ref>) in x, substitute (<ref>), (<ref>), and (<ref>), and obtaint(u_x-u_x) φ^(j) + (u-u) u σ_3 σ_1 φ^(j) = 2 ∑_k=1^N λ_k [ ⟨φ^(j), σ_3 φ^(k)⟩σ_3 φ^(k) + ⟨φ^(j), φ^(k)⟩φ^(k)] t+ u∑_k=1^N [ ⟨φ^(j), σ_3 φ^(k)⟩σ_3 σ_1 φ^(k) + ⟨φ^(j), φ^(k)⟩σ_1 φ^(k)] + 2 u ∑_k=1^N ⟨φ^(j), σ_1 φ^(k)⟩φ^(k),where the relation (<ref>) was used to cancel the λ_j term. By using the transformation formulas (<ref>), we verify that(u - u) σ_1 φ^(j) =∑_k=1^N [ ⟨φ^(j), σ_1 φ^(k)⟩σ_3 φ^(k) -⟨φ^(j), σ_3 σ_1 φ^(k)⟩φ^(k)].This allows us to simplify (<ref>) to the form(u_x-u_x) φ^(j)= 2 ∑_k=1^N λ_k [ ⟨φ^(j), σ_3 φ^(k)⟩σ_3 φ^(k) + ⟨φ^(j), φ^(k)⟩φ^(k)] t+ u∑_k=1^N [ ⟨φ^(j), σ_3 φ^(k)⟩σ_3 σ_1 φ^(k) + ⟨φ^(j), φ^(k)⟩σ_1 φ^(k)] t+ u ∑_k=1^N [ ⟨φ^(j), σ_1 φ^(k)⟩φ^(k) +⟨φ^(j), σ_3 σ_1 φ^(k)⟩σ_3 φ^(k)].Substituting (<ref>) to (<ref>) yields the following equation(u^2 - u^2) σ_1 φ^(j)=u∑_k=1^N [ ⟨φ^(j), φ^(k)⟩σ_3 σ_1 φ^(k) -⟨φ^(j), σ_3 φ^(k)⟩σ_1 φ^(k)] t+ u ∑_k=1^N [⟨φ^(j), σ_3 σ_1 φ^(k)⟩φ^(k) -⟨φ^(j), σ_1 φ^(k)⟩σ_3 φ^(k)].Thanks to the relations (<ref>) and (<ref>), equation (<ref>) is satisfied, and so is equation (<ref>).In order to show (<ref>), we first obtain the relations∂_x ⟨φ^(j), σ_1 φ^(k)⟩ = (λ_j - λ_k) ⟨φ^(j), σ_3 σ_1 φ^(k)⟩ - 2u ⟨φ^(j), σ_3 φ^(k)⟩and∂_x ⟨φ^(j), σ_3 σ_1 φ^(k)⟩ = (λ_j - λ_k) ⟨φ^(j), σ_1 φ^(k)⟩.Then, we differentiate (<ref>) in x, substitute (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>), and obtaint(u_xx - u_xx) φ^(j) + u (u_x - u_x) σ_3 σ_1 φ^(j) = 4 ∑_k=1^N λ_k^2 [ ⟨φ^(j), σ_3 φ^(k)⟩φ^(k) + ⟨φ^(j), φ^(k)⟩σ_3φ^(k)] t+ 2 u∑_k=1^N λ_k [ ⟨φ^(j), σ_3 φ^(k)⟩σ_1 φ^(k) + ⟨φ^(j), φ^(k)⟩σ_3 σ_1 φ^(k)] + 4 u ∑_k=1^N λ_k ⟨φ^(j), σ_1 φ^(k)⟩σ_3 φ^(k)t+ u_x ∑_k=1^N [ ⟨φ^(j), σ_3 φ^(k)⟩σ_3 σ_1 φ^(k) + ⟨φ^(j), φ^(k)⟩σ_1 φ^(k)] t+ u_x ∑_k=1^N [ ⟨φ^(j), σ_1 φ^(k)⟩φ^(k) +⟨φ^(j), σ_3 σ_1 φ^(k)⟩σ_3 φ^(k)] t+ u u ∑_k=1^N [ 3 ⟨φ^(j), σ_1 φ^(k)⟩σ_3 σ_1 φ^(k) +⟨φ^(j), σ_3 σ_1 φ^(k)⟩σ_1 φ^(k)] t- u^2 ∑_k=1^N [ ⟨φ^(j), σ_3 φ^(k)⟩φ^(k) +⟨φ^(j), φ^(k)⟩σ_3 φ^(k)] - 2 u^2 ∑_k=1^N ⟨φ^(j), σ_3 φ^(k)⟩φ^(k),where the relation (<ref>) was used to cancel the λ_j term. Substituting (<ref>) into (<ref>) and using (<ref>) and (<ref>) yieldt2 (u^3 - u^3) φ^(j)= (2u - u) (u_x - u_x) σ_3 σ_1 φ^(j) + u^2 ∑_k=1^N [ 3 ⟨φ^(j), φ^(k)⟩σ_3 φ^(k) +⟨φ^(j), σ_3 φ^(k)⟩φ^(k)] t+ 4 u^2 ∑_k=1^N ⟨φ^(j), σ_3 φ^(k)⟩φ^(k) - u u ∑_k=1^N [ 3 ⟨φ^(j), σ_1 φ^(k)⟩σ_3 σ_1 φ^(k) +⟨φ^(j), σ_3 σ_1 φ^(k)⟩σ_1 φ^(k)] t+ 2 u∑_k=1^N λ_k [ ⟨φ^(j), φ^(k)⟩σ_3 σ_1 φ^(k) - ⟨φ^(j), σ_3 φ^(k)⟩σ_1 φ^(k)]t+ 4 u ∑_k=1^N λ_k [ ⟨φ^(j), σ_3 σ_1 φ^(k)⟩φ^(k) - ⟨φ^(j), σ_1 φ^(k)⟩σ_3 φ^(k)].Substituting (<ref>) to (<ref>) yieldst2 (u - u) (u^2 + u u + u^2) φ^(j)= 2 u^2 ∑_k=1^N [ ⟨φ^(j), φ^(k)⟩σ_3 φ^(k) +⟨φ^(j), σ_3 φ^(k)⟩φ^(k)] t+ 2 u u ∑_k=1^N [ ⟨φ^(j), φ^(k)⟩σ_3 φ^(k) - ⟨φ^(j), σ_3 φ^(k)⟩φ^(k) - 2 ⟨φ^(j), σ_1 φ^(k)⟩σ_3 σ_1 φ^(k)] t+ 2 u^2 ∑_k=1^N [ 2 ⟨φ^(j), σ_3 φ^(k)⟩φ^(k) + ⟨φ^(j), σ_1 φ^(k)⟩σ_3 σ_1 φ^(k) - ⟨φ^(j), σ_3 σ_1 φ^(k)⟩σ_1 φ^(k)] t+ 4 u ∑_k=1^N λ_k [ ⟨φ^(j), σ_3 σ_1 φ^(k)⟩φ^(k) + ⟨φ^(j), φ^(k)⟩σ_3 σ_1φ^(k). t texttexttext. - ⟨φ^(j), σ_1 φ^(k)⟩σ_3 φ^(k) - ⟨φ^(j), σ_3 φ^(k)⟩σ_1 φ^(k)].By using the relations (<ref>) and explicit computations, we obtaint ∑_k=1^N [ ⟨φ^(j), φ^(k)⟩σ_3 φ^(k) - ⟨φ^(j), σ_3 φ^(k)⟩φ^(k) - 2 ⟨φ^(j), σ_1 φ^(k)⟩σ_3 σ_1 φ^(k)]= (u-u) φ^(j) - 2 ( ∑_k=1^N p_k q_k - ∑_k=1^N p_k q_k ) σ_1 φ^(j), t ∑_k=1^N [ 2 ⟨φ^(j), σ_3 φ^(k)⟩φ^(k) + ⟨φ^(j), σ_1 φ^(k)⟩σ_3 σ_1 φ^(k) - ⟨φ^(j), σ_3 σ_1 φ^(k)⟩σ_1 φ^(k)]= (u-u) φ^(j) + 2 ( ∑_k=1^N p_k q_k - ∑_k=1^N p_k q_k ) σ_1 φ^(j)andt ∑_k=1^N λ_k [ ⟨φ^(j), σ_3 σ_1 φ^(k)⟩φ^(k) + ⟨φ^(j), φ^(k)⟩σ_3 σ_1φ^(k). t . texttext - ⟨φ^(j), σ_1 φ^(k)⟩σ_3 φ^(k) - ⟨φ^(j), σ_3 φ^(k)⟩σ_1 φ^(k)] =- 2 ( ∑_k=1^N λ_k p_k p_k- ∑_k=1^N λ_k q_k q_k ) σ_1 φ^(j).Substituting (<ref>), (<ref>), and (<ref>) to (<ref>) cancel all terms thanks to the relations (<ref>) and (<ref>). 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"authors": [
"Jinbing Chen",
"Dmitry E. Pelinovsky"
],
"categories": [
"nlin.SI",
"math-ph",
"math.AP",
"math.MP",
"nlin.PS"
],
"primary_category": "nlin.SI",
"published": "20170427140801",
"title": "Rogue periodic waves of the mKdV equation"
} |
Qatar Foundation, P.O. Box 5825, Doha, Qatar ISMO, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay cedex, France [email protected] ISMO, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay cedex, France A semiclassical model supporting the destructive interference interpretation of zero-width resonances (ZWR) is extended to wavelengths inducing c^--type curve crossing situations in Na_2 strong field dissociation. This opens the possibility to get critical couples of wavelengths λ and field intensities I to reach ZWRs associated with the ground vibrationless level v=0, that, contrary to other vibrational states (v>0), is not attainable for the commonly referred c^+-type crossings. The morphology of such ZWRs in the laser (I, λ) parameter plane and their usefulness in filtration strategies aiming at molecular cooling down to the ground v=0 state are examined within the frame of an adiabatic transport scheme.33.80.Gj, 42.50.Hz, 37.10.Mn, 37.10.Pq Vibrational-ground-state zero-width resonances for laser filtration: An extended semiclassical analysis. Osman Atabek December 30, 2023 ======================================================================================================== § INTRODUCTION Laser filtration based on zero-width resonances (ZWR) has already been referred to as a selective and robust technique that, starting from a given vibrational distribution, aims at shaping a chirped laser pulse such as to efficiently photodissociate all vibrational states, at the exception of but one <cit.>. The basic mechanism is an adiabatic transport of the vibrational state to be filtrated (i.e., protected against dissociation) on its associated infinitely long-lived ZWR. Such control strategies have already been worked out for vibrational cooling purpose on the specific example of Na_2, prepared by photoassociation in some excited vibrational levels. The theoretical observation is that for certain critical field parameters (wavelength λ and intensity I) the photodissociation rate vanishes, resulting into a ZWR. The objective is to produce ZWRs at will and in a controllable way, continuously tuning laser parameters. A laser pulse is shaped in such a way to adiabatically transport a given vibrational state v on its parent ZWR and to track it all along the pulse referring to an effective phase strategy <cit.>. For long enough durations, when the pulse is switched off, only the single vibrational state v remains populated, all others v'≠ v having decayed, leading thus to a robust vibrational cooling control strategy.It has been shown that ZWRs result from destructive interference between two outgoing wave componentsaccommodated by laser-induced adiabatic potentials of a semiclassical two-channel description <cit.>. Roughly speaking, the critical phase matching of the interference scheme relies on the degeneracy of two energy levels: One which originates, in field-free conditions, from vand the other v_+, supported by the field-dressed upper bound adiabatic potential. Depending on the wavelength, field dressing is such that a given v could be brought in energy coincidence with any v_+=0,1,..., resulting into several ZWRs. But it has recently been argued that the ground vibrational level v=0 constitutes an exception (in a generic sense and at least for Na_2), without the possibility to merge into any ZWR <cit.>. This does not however prevent a vibrational cooling objective, still achievable through filtration aiming at a single level protection v'≠ 0, although this is not the ground one. In a second step, a STIRAP process could then bring the v' population onto v=0 <cit.>.The purpose of the present work is to extend the semiclassical analysis having in mind the specific goal of preparing the vibrationless state directly from a Boltzmann-type thermal distribution. This is done by referring to some wavelength regime inducing curve crossing schemes at internuclear distances less than the equilibrium geometry of v=0. Such situations are labeled as c^- crossings, as opposite tothe c^+ onesof previous investigations <cit.>.The paper is organized in the following way:In section <ref>, ZWRs are introducedwithin a two-state photodissociation model of Na_2 in a time-independent close-coupled Floquet Hamiltonian formalism. Their interpretation is based on a semiclassical model with an original extension to c^- avoided crossings. The filtration strategy based both on the ground and first excited vibrational states (v=0,1) are presented in Section <ref>.§ PHOTODISSOCIATION DYNAMICS §.§ Zero-width resonances: Quantum descriptionRotationless Na_2 multiphoton dissociation is described within a two electronic states Born-Oppenheimer approximation. These are labeled 1 ⟩ for the bound a^3Σ_u^+ (3^2S+3^2S) and 2 ⟩ for the dissociative excited state (1)^3Π_g (3^2S+3^2P). R being the internuclear distance, the nuclear components ϕ_1,2(R,t) of the time-dependent wave function:Φ (R,t) ⟩ = |ϕ_1(R,t) ⟩ 1 ⟩ + |ϕ_2 (R,t) ⟩ 2 ⟩.are solutions of the Time Dependent Schrödinger Equation (TDSE) :iħ∂/∂ t[[ ϕ_1 (R,t); ϕ_2 (R,t) ]] = ( T_N + [[ V_1(R)0;0 V_2(R) ]] . - . μ_12(R)ℰ(t)[[ 0 1; 1 0 ]] )[[ ϕ_1 (R,t); ϕ_2 (R,t) ]] where T_N represents the nuclear kinetic energy. V_1(R) and V_2(R) are the Born-Oppenheimer potentials and μ_12(R) is the transition dipole between 1 ⟩ and 2 ⟩.The linearly polarized electric field ℰ(t) is given, for a continuous wave (cw) laser by: ℰ(t) = Ecos(ω t)The intensity and the wavelength are given by I∝ E^2 and λ= 2 π c /ω, c being the speed of light. Due to time-periodicity, the Floquet ansatz leads to <cit.>:[ [ ϕ_1 (R,t); ϕ_2 (R,t) ]] = e^-iE_v t/ħ[ [ χ_1 (R,t); χ_2 (R,t) ]].where Fourier expanded χ_k(R,t) (k=1,2):χ_k(R,t)=∑_n=-∞^∞ e^inω tφ_k,n(R)involve components satisfying a set of coupled differential equations, for any n, which for moderate field intensities (retaining only n=0,1) reduce to: [ T_N + V_1(R)+ħω-E_v ]φ_1,1(R)-1/2 E μ_12(R) φ_2,0(R)=0 [ T_N + V_2(R)-E_v ]φ_2,0(R)-1/2 E μ_12(R) φ_1,1(R)=0Resonances are quantized solutions with Siegert type outgoing-wave boundary conditions <cit.> and have complex quasi-energies of the form(E_v)-iΓ_v/2, where Γ_v is the resonance width related to its decay rate. In the following, label v designates both the field-free vibrational level and the laser-induced resonance originating from this vibrational state.Going beyond the cw laser assumption, we considera chirped laser pulse with parameters ϵ(t)≡{E(t), ω(t)} involving slowly varying envelope and frequency.The purpose of optimizing laser parameters such that the survival probability of a resonance state originating in field-free conditions from a given vibrational state v be maximized, while all other resonances (originating from v'≠ v) are decaying fast, is conducted within the frame of the adiabatic Floquet formalism <cit.>. The full control strategy consists in trapping the system into a single eigenvector of the adiabatic Floquet Hamiltonian, in a so called extended Hilbert space and shaping a pulse with field parameters such that this eigenstate presents the lowest (zero, if possible) dissociation rate. We have recently shown <cit.> that this is achieved by an optimal choice for the field parameters:ϵ^*(t)≡{E^ZWR(t), ω_eff^ZWR(t)}ω_eff being an effective frequency in this extended Hilbert space and such that:[E_v{ϵ^ZWR(t)}]=0 ∀ twhere (E_v) is the imaginary part of the energies of these field-induced resonances. Eq.(<ref>) is nothing but a ZWR path (originating from |v ⟩) in the amplitude, frequency parameter plane (or equivalently, intensity I, wavelength λ) as a function of t, that is:ϵ^ZWR(t)≡{[ E^ZWR(t) = [I^ZWR(t)]^1/2,; ω_eff^ZWR(t) = 2π c/λ^ZWR(t) ]}Finally, the optimal laser pulse acting in the original Hilbert space where the evolution is monitored by the TDSE displayed in Eq.(<ref>) is given by <cit.>:ℰ^*(t)= [I^ZWR(t)]^1/2·cos(∫_0^t 2π c/λ^ZWR(t') dt')It has in particular been shown that ZWRs are good candidates for a full adiabatic Floquet treatment as initially derived for pure bound states <cit.>. The molecule, initially in a particular field-free vibrational state v, is supposed to be adiabatically driven by such a pulse. Adiabaticity means here that a single resonance Φ_v(t), labeled v according to its field-free parent bound state, is followed during the whole dynamics. This resonance wave function involves, through its complete basis set expansion, a combination of both bound and continuum eigenstates of the field-free molecular Hamiltonian. But the important issue is that, at the end of the pulse, the molecule is again on its initial single vibrational state v (adiabaticity condition). For such open systems, contrary to dynamics involvingbound states only, there is unavoidably an irreversibledecay process, precisely due to the fact that vibrational continuum states are temporarily populated under the effect of the pulse, even though this is minimized by a ZWR path tracking. A quantitative measure of such a decay is given in terms of the overall fraction of non-dissociated molecules, assuming a perfect adiabatic following of the selected resonance<cit.>:P_v(t) =exp[- ħ^-1∫_0^tΓ_v(ϵ (t')) dt']where the decay rate Γ_v(ϵ(t)) is associated with the relevant Floquet resonance quasi-energy E_v(ϵ(t)) using the instantaneous field parameters ϵ(t) ≡{E(t), ω(t)} at time t.The control issue consists in investigating how rates are changing with the field parameters and in particular find optimal combinations ϵ^ZWR(t) for which these rates are small enough (or even ideally zero) to insure the survival of the vibrational state v to the laser excitation, that is P_v(τ) ≈ 1, τ being the total pulse duration. §.§ Zero-width resonances: Semiclassical modelA full destructive interference interpretation of ZWRs is provided by a semiclassical theory based on a two-channel scattering model, involving adiabatic potentials V_±(R) resulting from the diagonalization ofmolecule-field interaction matrix <cit.>. Dissociation quenching related to a null value of the outgoing scattering amplitude in the lower (open) adiabatic channel V_- occurs when the following two conditions are simultaneously fulfilled <cit.>:∫_R_+^R_c dR' k_+(R')+∫_R_c^R_t dR' k_+(R')+ χ=(ṽ_++1/2)πand∫_R_-^R_c dR' k_-(R')+∫_R_c^R_tdR' k_+(R') =(ṽ+1/2)πThe wavenumbers are given by: k_±(R)=ħ^-1[2m(ε-V_±(R)]^1/2.m is the reduced nuclear mass, R_± are the left turning points of V_± potentials, R_t is the right turning point of V_+ and R_c is the diabatic crossing point resulting from field-dressing. With integer quantum numbers ṽ_+ and ṽ these conditions are nothing but the requirements of Bohr-Sommerfeld quantization involving a coincidence between two energies; namely one ε=ε_ṽ^+ of the upper adiabatic potential V_+(R), with a phase correction χ, which in weak coupling is -π/4 <cit.>, and another ε=ε_ṽ of a potential Ṽ(R) made of two branches, namely, V_-(R) for R ≤ R_c, and V_+(R) otherwise. For a weak coupling, this is practically the field-free diabatic attractive potential V_1(R). An analytical expression of the resonance width Γ_v is <cit.>:Γ_v= 2π/ħe^2πν (e^2πν-1) ω_d ω_+/[ω_++(e^2πν-1)ω_d]^3 (ε_ṽ-ε_ṽ_+)^2This expression clearly displays the role played by such energy coincidences, in terms of the square of their differences. In Eq.(<ref>)ω_d and ω_+ are the local energy spacings of the modified diabatic and adiabatic potentials respectively. ν is the Landau-Zener coupling parameter:ν=μ_12^2(R_c)E^2/ħv̅|Δ F|where v̅ and Δ F are the classical velocity and slope difference of the diabatic potentials at R_c. Clearly, the two energies ε=ε_ṽ and ε=ε_ṽ^+, and therefore the width Γ_v are dependent on field parameters, i.e., both frequency (or wavelength) and amplitude (or intensity). This is in particular due to the (λ, I)-dependence of the corresponding field-dressed adiabatic potentials V_±(R). As a consequence, ZWRs in photodissociation can be produced at will by a fine tuning of the wavelength and intensity.Moreover, for a wavelength λ which roughly brings into coincidence the levels ṽ (corresponding to the field-free vibrational level v in consideration) and ṽ_+=0, a fine tuning of the intensity I will result in an accurate determination of a ZWR, that is Γ_v(λ , I)=0.In some cases, a stronger field (higher I) may also bring into coincidence ṽ with ṽ_+=1, producing thus a second ZWR, for the same wavelength, and so on for ṽ_+=2,3... But, one can also envisage slightly different wavelengths which build energetically close enough ṽ and ṽ_+ levels in a field-dressed picture, such that a subsequent fine tuning of the intensity brings them into precise coincidence. This flexibility offered by the field parameters that, in principle, can be continuously modified, is at the origin of not only quasi-zero width photodissociation resonances, but also for their multiple occurrence in the (λ, I)-parameter plane <cit.>. We emphasize that the semiclassical description of photodissociation is based on field-dressed adiabatic potential energy curves with avoided crossings mainly controlled by frequency, whereas couplings are intensity dependent. In addition, according to Child's diagrammatic approach we are following here, these crossings should be reached classically for both channels, their turning points being at the left of the crossing point. Such a situation is the one which is valid for the most commonly refereed c^+-type crossings. But, as will be discussed hereafter, the semiclassical analysis could still be useful, for somec^--type crossings, occurring at higher frequencies, compatible with the classically allowed picture and leading to the previously discarded possibility to reach a ZWR associated with v=0. Finally, it is worthwhile noting that some extensions, at even higher frequencies, to classically non-reachable crossing situations have already been worked out using complex crossing points <cit.>. We now examine in more detail some generic properties of ZWR behaviors in the (λ, I) parameter plane, by distinguishing the c^+ and c^- cases. §.§.§ Low frequency regime: c^+-type crossing. Such cases correspond to low frequency field-dressing with a diabatic curve crossing at the right of the equilibrium distance, R_c > R_e. A typical situation is the one illustrated on the left panel of the schematic view of Fig.<ref> with the lowest possible wavelength λ_0 leading to R_c = R_e.Semiclassical rationalization of ZWRs generic behaviors, according to the energy coincidences between ε_ṽ and ε_ṽ_+ involved in Eq.<ref>, can be conducted in three steps: (i) field-dressing with λ (intensity being taken as negligible), which is the major shifting effect on v_+ levels not affecting ṽ=v; (ii) introduction of the additional phase χ (taken as -π/4, for low enough intensities) affecting v_+ which becomes ṽ_+; (iii) consideration of the role played by the field intensity in locally changing the adiabatic potentials supporting both ṽ and ṽ_+.The argument discarding the possibility of a ZWR associated with v=0 is based on the fact that in a field-dressed picture the upper adiabatic potential V_+(R) accommodating level v_+, presents a localcurvature (close to R_c) higher than the one of the bound diabatic state V_1(R) supporting level v, at least for low intensities. This is depicted in Fig.<ref> for the lowest possible wavelength λ = λ_0, most favorable candidate for an energy coincidence. The consequence is thatε_ṽ_+=0 > ε_ṽ=0.Moreover, the phase χ produces an additional energy increase on ε_ṽ_+=0. Finally, the effect of the field intensity is such that it will affect the system by increasing ε_ṽ_+=0, while slightly decreasing ε_ṽ=0. Obviously, all other wavelengths (λ > λ_0) of this low frequency c^+ regime will shift ε_ṽ_+=0at even higher energies. The coincidence condition for v=0, required by Eq.<ref>, can never be fulfilled for wavelengths inducing a c^+-type crossing.For all other levels v > 0, semiclassical expectations are different. As is clear from Fig.<ref>, for λ = λ_0 and v=1,ε_ṽ_+=0 < ε_ṽ=1.Both the neglected additional phase χ and changes in field control parameters (increase of wavelength λ > λ_0 and intensity I) result in increasing ε_ṽ_+=0. It is important to note that ε_ṽ=1 is in turn affected by the increase of the field intensity, but much less than ε_ṽ_+=0. Specific laser parameters (λ^ZWR, I^ZWR) could then be found to achieve the semiclassical energy coincidence of Eq.<ref>, leading to ZWR(v=1, v_+=0) originating from v=1. To follow a typical ZWR map in the laser parameter plane, we suppose that a first coincidence (ε_ṽ = ε_ṽ_+) has been obtained for some critical (I, λ) parameters. When the wavelength is progressively increased, ε_ṽ_+ is blue shifted, whereas ε_ṽ is only slightly affected. The coincidence required for a ZWR is no more achieved. In order to compensate the increase of ε_ṽ_+ we have to lower the field intensity. As a consequence, the ZWR path in the (I, λ)-plane is of negative slope, as illustrated in Fig.<ref>.§.§.§ High frequency regime: c^--type crossing. At higher frequencies, for λ < λ_0, curve crossings occur on the left of the equilibrium geometry, that is R_c < R_e. The right panels of Fig.<ref> illustrate such a typical situation. As already mentioned, there is still a wavelength window for which, even in this regime, the semiclassical model is still valid, with left turning points of Eq. <ref> satisfying the following condition: R_- < R_+ < R_c < R_0. At negligible field intensities, the two laser-dressed potentials V_1(R) and V_+(R) are very similar, at least in the deeper part of their common well accommodating the lowest vibrational levels, and in particular v=0. The consequence is that the coincidence condition could be reached, at least approximately. It is important to note that, even if for v > 0 such c^--type crossings are merely an extension of the wavelength regime for which ZWRs are expected, the situation is completely different for v=0. Concerning semiclassical energy coincidence arguments, this actually appears to be the only regime where, at least approximately, one can expect ε_ṽ_+=0≃ε_ṽ=0, that is to get a ZWR originating from the vibrationless ground state v=0 at field free conditions.Finally, similarities between adiabatic potentials Ṽ(R) and V_+(R) regarding their common potential well around equilibrium geometry, are better obtained by: (i) decreasing the wavelength λ < λ_0 in order to enhance the potential well extension; (ii) decreasing the intensity I in order to reduce energy separations of the avoided crossing area. The conclusion is that ZWRs paths in the (I, λ)-plane should behave with a positive slope, as opposite to the c^+-type crossing situation.§ RESULTSTransitionally and rotationally cold, tightly bound and vibrationally hot Na_2 species are experimentally produced by photoassociation in the metastable bound state ^3Σ_u^+ (3^2S+3^2S), considered as an initial ground state (referred to as state 1) radiatively coupled with the repulsive, thus dissociating excited (1)^3Π_g (3^2S+3^2P) electronic state (referred to as state 2). The corresponding Born-Oppenheimer potential energy curves V_1,2(R) and the electronic transition dipole moment μ_12(R) between states 1 and 2 are taken from the literature <cit.>. Finally, Na_2 reduced mass is taken as 20963.2195 au. As depicted in Fig.<ref>, the equilibrium geometry corresponds to R_e = 9.79 au. The critical wavelength for c^0 crossing (for R_c=R_e) is actually λ_0=552 nm, such that c^+-type crossings are obtained for λ > λ_0, whereas wavelengths λ < λ_0 lead to c^--type. Moreover, the lowest possible wavelength still fulfilling the requirement of Eq. <ref>, for v=0, turns out to be λ=550 nm. This means that the c^- extension of the semiclassical model within its diagrammatic presentation would only concern a moderate range of wavelengths, namely 550 nm < λ < 552 nm. Photoassociation typically prepares vibrational levels with quantum numbers v ≥ 8. We have previously used a filtration strategy using ZWR tracking in the (I,λ) parameter plane by adiabatically transporting v=8 level on its associated ZWR <cit.>. This leads, after the laser pulse is over, to vibrational population left only on v=8 and therefor achieves efficient and robust cooling by preparing a single vibrational level, although not the ground one v=0. In other experimental situations, with an initial thermal distribution of vibrational states, the filtration targeting the vibrationless ground state v=0 would require the generalization of a similar strategy but now based on a ZWR associated with v=0. In the following, we start with the more common case of filtration referring to ZWR(v=1) to illustrate both c^+ and c^--type crossing behaviors, in conformity with the previous semiclassical model. In a second attempt, we analyze the case of ZWR(v=0), showing that, with specific range of wavelengths (roughly inducing c^--type crossings), efficient filtration still remains possible.§.§ Filtering using ZWR(v=1). Solving time-independent coupled equations Eq.(<ref>) withSiegert boundary conditions for a set of continuous wave cw laser parameters{I,λ}, gives rise to resonances with complex eigenvalues E_vcorrelating, in field-free conditions, with the real vibrational eigenenergies.We are actually interested in finding specific couples of fieldparameters for which the imaginary part of resonance eigenvalues are close tozero. More specifically, we analyze the wavelength regime 549 nm < λ <λ_0 corresponding to the semiclassical extended c^--type crossingregion for v=1.Results of exploratory calculations illustrating the behavior of resonances originating from v=1 are displayed in panel (a) of Figure <ref>. We have selected three wavelengths within the semiclassical extension window and intensities up to I = 2 GW/cm^2. The overall tendency is a smooth regular decrease of the widths for increasing field strengths, in agreement with the generic behavior of Feshbach-type resonances (due to decreasing non-adiabatic couplings <cit.>). But, more interestingly, for specific intensities, we obtain sharp dips corresponding to resonance widths typically less than 10^-3 cm^-1, clear signatures of ZWRs. Within numerical inaccuracies inherent to the evaluation of such very small width resonances(less than 10^-6 cm^-1), we observe that there are several couples of critical wavelengths and intensities producing ZWRs originating from a single vibrational level v. Figure <ref>, panel b, displays in the (I, λ) laser parameter plane, ZWRs path originating from (v=1) for both c^- and c^+ regimes. As expected from the extended semiclassical analysis summarized in Fig.<ref>, the c^--type crossing region reached for 549 nm < λ < λ_0, roughly corresponds to a ZWR path with a positive slope. This is to be contrasted with the behavior in the c^+ crossing region λ > λ_0, where the slope is negative, once again in conformity with the semiclassical analysis of Fig.<ref>.The last step for the filtration control is to shape frequency chirped laser pulses resulting from the effective phase adiabatic transport strategy of Eq.(<ref>), where λ^ZWR and I^ZWR are those depicted in panel (b), exclusively for the c^- region. A wavepacket evolution based on TDSE solved by a third-order split-operator technique <cit.>, gives the vibrational population dynamics under the effect of such a pulse acting either on v=1 as an initial state, or neighboring v=0,2 levels. The results are displayed in Fig.<ref> in panels c and d. The efficiency of filtration strategy is well proven. The vibrational population of level v=1 is well protected against dissociation (up to 80%, on panel c), whereas the neighboring levels populations are decaying fast (panel d). We emphasize that similar observations have previously been discussed for c^+-type crossings, the originality of the present work is to show their possible extension to c^--type crossings. We however notice that when referring to ZWRs in the c^- semiclassical extension regime, the filtration process is slightly less selective (remaining v=0 population being not less than 30%). This is due to the fact that ZWRs originating from v=1 and v=0 are close to each other. As is clear from Figure <ref>, laser parameters inducing the energy coincidence which is looked for v=0 approximately correspond to the ones valid for a similar coincidence for v=1.§.§ Filtering using ZWR(v=0).Having shown the validity of a possible extension of the semiclassical approach to c^--type crossings, we are now in a position to examine the most challenging case of a ZWR originating from the vibrationless ground (v=0) state together with its potentiality to support robust filtration control.The results are gathered in Fig.<ref> following a graphical illustration similar to the above discussed case of v=1. As previously, exploratory calculations are carried out for c^--type crossing regime, covered by 550 nm < λ <λ_0. A selection of three such wavelengths is illustrated in panel a for the resonance widths originating from v=0 as a function of intensity. This clearly shows the possibility to reach ZWRs as sharp dips (widths typically less than 10^-4 cm^-1) superimposed to a smoothly decreasing background. More unexpectedly, some smaller wavelengths are also producing ZWRs. One of them corresponding to λ = 549 nm is shown in Fig.<ref>. Panel b displays the ZWR(v=0) path in the (I, λ) parameter plane. A few observations deserve interest: (i) No ZWR is obtained in the low frequency regime, for wavelengths λ > λ_0 leading to c^+-type crossing, in conformity with the semiclassical analysis; (ii) Unexpectedly, for wavelengths λ < 550 nm, ZWR are still observed, although the semiclassical model is no more valid, due to the fact that the crossing is not within the classically allowed region (or even no crossing at all); (iii) In the intermediate wavelength regime 550 nm < λ < λ_0 fully supported by the extended semiclassical c^--type crossing, the slope of ZWRs path is positive in a region well on the left of R_e (550 nm < λ < 550.5 nm) as expected from the analysis of Fig.<ref>. But, when R_c becomes closer to R_e, the slope changes to be negative, presumably due to a competition between decreasing energy separations (ε_ṽ_+=0 - ε_ṽ=0) on the one hand, and increasing additional phase χ on the other hand, when the field strength is decreasing. Panel c shows the robustness of v=0 population efficiently protected against dissociation (up to 95%), whereas panel d displays populations of neighboring states (v=1,2) which are decaying fast, but with still 27% remaining v=1 population at the end of the pulse, for reasons similar to those already discussed in the previous paragraph.§ CONCLUSION The diagrammatic semiclassical model, of crucial importance in the destructive interference interpretation of ZWRs and in their localization in the laser (I, λ) parameter plane, is extended to wavelengths inducingc^--type crossings in adiabatic potentials description. Such an extension remains however limited to wavelengths windows of moderate size, as additional requirements of classically reachable crossings within vibrational wavefunctions spatial stretching should be fulfilled. With this extension, the validity of which is first checked on the standard case of v=1, the semiclassical model acquires the capacity of a possible depiction of ZWRs originating, in field-free conditions, from the vibrationless ground state v=0, by approximately defining a couple of (I, λ) parameters. Actually, quantum Floquet photodissociation theory confirms these ZWR(v=0) parameters by refining their values. But more unexpectedly, additional ZWRs(v=0) are obtained in the c^- regime, even though classical conditions are no more fulfilled.A full quantum wave packet propagation shows that an adiabatic transport of population from v=0 to its associated ZWRs(v=0) tracked all along an appropriately shaped laser pulse duration results in efficient vibrational population protection against photodissociation. When the pulse is over, the v=0 population is almost unchanged, pointing thus to the robustness of the mechanism. At the same time, all other (v > 0) vibrational populations of the initial thermal distribution are decaying, pointing to the selectivity of the filtration process, even though this is less than the one of the c^+ case.As a conclusion, a laser controlled filtration strategy based on (v=0) ZWR tracking is shown to be robust and selective enough for molecular vibrational cooling aiming at obtaining the ground vibrational state in a single step laser excitation. O. A. acknowledges support from the European Union (Project No. ITN-2010-264951, CORINF). R. L. thanks Pr. F. Leyvraz for his hospitality at the "Centro Internacional de Ciencias", Cuernavaca, Mexico.apsrev 30 catherine O. Atabek, R. Lefebvre, C. Lefebvre and T. T. Nguyen-Dang, Phys. Rev. A 77, 043413 (2008)AtabekRC O. Atabek, R. Lefebvre, A. Jaouadi and M. Desouter-Lecomte, Phys. Rev. A 87, 031403(R) (2013).Lecl2016 A. Leclerc, D. Viennot, G. Jolicard, R. Lefebvre and O. Atabek, Phys. Rev. A 94, 043409 (2016).Bandrauk A. D. 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Phys. 98, 7113 (1993).aymar M. Aymar and O. Dulieu, J. Chem. Phys. 122, 204302 (2005).AtabekPRL O. Atabek, R. Lefebvre, M. Lepers, A. Jaouadi, O. Dulieu and V. Kokoouline, Phys. Rev. Lett., 106, 173002 (2011).chrysos O. Atabek, M. Chrysos and R. Lefebvre, Phys. Rev. A 49, R8 (1994)feit M.D. Feit J.A. Fleck and A. Steiger, J. Comp. Phys. 47, 412 (1982) | http://arxiv.org/abs/1704.08211v1 | {
"authors": [
"Amine Jaouadi",
"Roland Lefebvre",
"Osman Atabek"
],
"categories": [
"quant-ph",
"J.2"
],
"primary_category": "quant-ph",
"published": "20170426165326",
"title": "Vibrational-ground-state zero-width resonances for laser filtration: An extended semiclassical analysis"
} |
abbrvnat[mytitlenote]Supported by the program of High-end Foreign Experts of the SAFEA (No. GDW20163200216) and by FCT and CIDMAwithin project UID/MAT/04106/2013.mymainaddress]Wei Liu [email protected]]Guoju Ye [email protected],mysecondaryaddress]Dafang Zhao [email protected],myfourthaddress]Delfim F. M. Torresmycorrespondingauthor [mycorrespondingauthor]Corresponding author ([email protected]). [email protected], [email protected][mymainaddress]College of Science, Hohai University, Nanjing 210098, P. R. China[mysecondaryaddress]School of Mathematics and Statistics,Hubei Normal University, Huangshi 435002, P. R. China[mythirdaddress]Center for Research and Developmentin Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro,3810-193 Aveiro, Portugal[myfourthaddress]African Institute for Mathematical Sciences (AIMS-Cameroon), P. O. Box 608 Limbe, Cameroon In this work, we are concerned with existence of solutions fora nonlinear second-order distributional differential equation,which contains measure differential equations and stochasticdifferential equations as special cases. The proof is basedon the Leray–Schauder nonlinear alternative andKurzweil–Henstock–Stieltjes integrals. Meanwhile,examples are worked out to demonstratethat the main results are sharp.distributional differential equation measure differential equation stochastic differential equation regulated function Kurzweil–Henstock–Stieltjes integral Leray–Schauder nonlinear alternative.[2010] 26A39 34B15 46G12. § INTRODUCTIONThe first-order distributional differential equation (DDE) in the formDx=f(t,x)+g(t,x)Du,where Dx and Du stand, respectively,for the distributional derivative of function x and u in the sense of Schwartz, has been studied as a perturbed system of the ordinary differential equation (ODE)x'=f(t,x)(':=d/dt).The DDE (<ref>) provides a good model for many physical processes, biological neural nets, pulse frequency modulation systems and automatic control problems <cit.>. Particularly, when u is an absolute continuous function, then (<ref>) reduces to an ODE. However, in physical systems, one cannot always expect the perturbations to be well-behaved. For example, if u is a function of boundary variation, Du can be identified with a Stieltjes measure and will have the effect of suddenly changing the state of the system at the points of discontinuity of u, that is, the system could be controlled by some impulsive force. In this case, (<ref>) is also called a measure differential equation (MDE), see <cit.>. Results concerning existence, uniqueness, and stability of solutions, were obtained in those papers. However, this situation is not the worst,because it is well-known that the solutions of a MDE, if exist,are still functions of bounded variation. The case when u isa continuous function has also been considered in <cit.>.The integral there is understood as a Kurzweil–Henstock integral <cit.> (or Kurzweil–Henstock–Stieltjes integral, or distributional Kurzweil–Henstock integral), which is a generalization of the Lebesgue integral. Especially, if u denotes a Wiener process(or Brownian motion), then (<ref>) becomes a stochastic differential equation (SDE), see, for example, <cit.>.In this case, u is continuous but pointwise differentiable nowhere, and the Itô integral plays an important role there. As for the relationship between the Kurzweil–Henstock integral and the Itô integral, we refer the interested readers to <cit.> and references therein.It is well-known that regulated functions (that is, a function whose one-side limits exist at every point of its domain) contain continuous functions and functions of bounded variation as special cases <cit.>. Therefore, it is natural to consider the situation when u is a regulated function, see <cit.>. Denote by G[0,1] the space of all real regulated functions on [0,1], endowed with the supremum norm ·. Since the DDE allows both the inputs and outputs of the systems to be discontinuous, most conventional methods for ODEs are inapplicable,and thus the study of DDEs becomes very interesting and important.The purpose of our paper is to apply the Leray–Schauder nonlinear alternative and Kurzweil–Henstock–Stieltjes integrals to establish existence of a solution to the second order DDE of type-D^2x=f(t,x)+g(t,x)Du, t∈ [0,1],subject to the three-point boundary condition (cf. <cit.>)x(0)=β Dx(0), Dx(1)+Dx(η)=0,where D^2 x stands for the second order distributional derivative of the real function x ∈ G[0,1], u∈ G[0,1], β is a constant, and η∈ [0,1]. This approach is well-motivated sincethis topic has not yet been addressed in the literature, and by thefact that the Kurzweil–Henstock–Stieltjes integral is a powerfultool for the study of DDEs. We assume that f and g satisfythe following assumptions: (H_1)f(t,x) is Kurzweil–Henstock integrablewith respect to t for all x∈ G[0,1]; (H_2) f(t,x) is continuous with respect to xfor all t∈ [0,1]; (H_3) there exist nonnegative Kurzweil–Henstockintegrable functions k and h such that - k x-h≤ f(·,x) ≤ k x+h∀ x ∈ B_r, where B_r={x∈ G[0,1] : x≤ r}, r>0; (H_4) g(t,x) is a function with bounded variationon [0,1] and g(0,x)=0 for all x∈ G[0,1]; (H_5) g(t,x) is continuous with respectto x for all t∈ [0,1]; (H_6) there exists M>0 such that sup_x ∈ B_r_[0,1]g≤ M, where _[0,1]g=sup∑_n |g(s_n,x(s_n))-g(t_n,x(t_n))|, the supremum taken over every sequence {(t_n, s_n)}of disjoint intervals in [0,1],is called the total variation of g on [0,1]. Now, we state our main result.Suppose assumptions (H_1)–(H_6) hold. If(|β|+2)max_t∈[0,1]|∫_0^t k(s)ds|<1,then problem (<ref>)–(<ref>) has at least one solution. If k(t)≡ 0 on [0,1], then (H_3) can be reduced to (H_3') there exists a nonnegativeKurzweil–Henstock function h such that -h≤ f(·,x)≤ h ∀ x ∈ B_r. Thus, the following result holds as a direct consequence.Assume that (H_1), (H_2), (H_3') and (H_4)–(H_6)are fulfilled. Then, problem (<ref>)–(<ref>)has at least one solution. It is worth to mention that condition (H_3'), togetherwith (H_1) and (H_2), was firstly proposed by<cit.>, to deal with first-order Cauchy problems.The paper is organized as follows. In Section <ref>, we give two useful lemmas: we prove that under our hypotheses problem (<ref>)–(<ref>) can be rewritten in anequivalent integral form (Lemma <ref>) and we recall the Leray–Schauder theorem (Lemma <ref>). Then, in Section <ref>, we prove our existence result (Theorem <ref>). We end with Section <ref>,providing two illustrative examples. Along all the manuscript,and unless stated otherwise, we always assume thatx, u∈ G[0,1]. Moreover, we use the symbol ∫_a^bto mean ∫_[a,b].§ AUXILIARY LEMMASBy (H_1) and (H_4), we defineF(t,x) =∫_0^t f(s,x(s))ds,G_u(t,x) =∫_0^t g(s,x(s))du(s),for all t∈[0,1].Under the assumptions (H_1)–(H_6), problem (<ref>)–(<ref>)is equivalent to the integral equationx(t)= t+β/2(F(1,x) +F(η,x)+G_u(1,x)+G_u(η,x))-∫_0^t F(s,x)ds-∫_0^t G_u(s,x)dson [0,1], where F and G_u are given in (<ref>),u∈ G[0,1], and β and η are constantswith 0≤η≤ 1.For all t∈ [0,1], s∈[0,1], and x∈ G[0,1], we have∫_0^ts D^2x(s)ds=∫_0^ts d(Dx(s))=tDx(t)-x(t)+ x(0)by the properties of the distributional derivative. Integrating (<ref>) once over [0,t], we obtain thatDx(t)=Dx(0)-F(t,x)- G_u(t,x).Combining with the boundary conditions (<ref>), one hasDx(0)=1/2( F(1,x)+F(η,x)+G_u(1,x)+G_u(η,x))andx(0)=β/2( F(1,x)+F(η,x)+G_u(1,x)+G_u(η,x)).It follows from (<ref>) and (<ref>) thatx(t)= tDx(0)+x(0)-∫_0^t (t-s)f(s,x(s))ds-∫_0^t (t-s)g(s,x(s))du(s).Therefore, by (<ref>)–(<ref>) and thesubstitution formula <cit.>, one hasx(t)=t+β/2( F(1,x)+F(η,x)+G_u(1,x)+G_u(η,x))-∫_0^t F(s,x)ds-∫_0^t G_u(s,x)ds, t∈ [0,1].It is not difficult to calculate that(<ref>)–(<ref>) holds by taking the derivative both sides of (<ref>). This completes the proof. Now, we present the well-known Leray–Schaudernonlinear alternative theorem.Let E be a Banach space, Ω a bounded open subset of E,0∈Ω, and T: Ω→ E be acompletely continuous operator. Then, either there existsx∈∂Ω such that T(x)=λ x with λ>1,or there exists a fixed point x^*∈Ω. We prove existence of a solution to problem(<ref>)–(<ref>) with the helpof the preceding two lemmas.§ PROOF OF THEOREM <REF>LetH(t) =∫_0^t h(s)ds,K(t) =∫_0^t k(s)ds,t∈ [0,1]. Then, by (H_3),H and K are continuous functions. According to (<ref>) and (H_1), function F is continuous on [0,1], andF=max_t∈[0,1]|∫_0^t f(s,x(s))ds| ≤Kx+H.On the other hand, by <cit.> and (H_4), G_u is regulated on [0,1]. Further, from (H_6) and the Hölder inequality <cit.>, it follows thatG_u ≤(|g(0,x(0))|+ |g(1,x(1))|+ _[0,1] g )u≤ 2Mu.Letr=(|β|+2)(H+2Mu)/1-(|β|+2)K>0.For each x∈ B_r and t∈ [0,1], define the operator𝒯:G[0,1]→ G[0,1] by 𝒯x(t):= t+β/2( F(1,x)+F(η,x)+G_u(1,x)+G_u(η,x))-∫_0^t F(s,x)ds-∫_0^t G_u(s,x)ds.We prove that 𝒯 is completely continuous in three steps. Step 1: we show that 𝒯:B_r→ B_r. Indeed,for all x∈ B_r, one has𝒯x ≤(|β|+2)(F+G_u)≤ (|β|+2)(rK+H+2Mu)= rby (<ref>) and (<ref>). Hence, 𝒯(B_r)⊆ B_r. Step 2: we show that 𝒯(B_r) is equiregulated(see the definition in <cit.>). For t_0∈[0, 1)and x∈ B_r, we have|𝒯x(t)-𝒯x(t_0+)|=|t-(t_0+)/2(F(1,x)+F(η,x)+G_u(1,x)+G_u(η,x))..-∫_t_0+^tF(s,x)+G_u(s,x)ds|≤ 2|t-(t_0+)|(rK+H+2Mu) ⟶ 0as t→ t_0+. Similarly, we can prove that |𝒯 x(t_0-) -𝒯x(t)|→ 0ast→ t_0- for each t_0∈(0,1]. Therefore, 𝒯(B_r)is equiregulated on [0,1]. In view of Steps 1 and 2and an Ascoli–Arzelà type theorem <cit.>, we conclude that 𝒯(B_r)is relatively compact. Step 3: we prove that 𝒯is a continuous mapping. Let x∈ B_r and {x_n}_n∈ℕbe a sequence in B_r with x_n → x as n→∞.By (H_2) and (H_4), one hasf(·,x_n)→ f(·,x) and g(·,x_n)→ g(·,x) as n→∞. According to the assumption (H_3) and the convergence Theorem 4.3 of <cit.>, we havelim _n →∞∫_0^t f(s,x_n(s))ds= ∫_0^t f(s,x(s))ds, t∈ [0,1]. Moreover, (H_6), together with the convergenceTheorem 1.7 of <cit.>, yields thatlim _n →∞∫_0^t g(s,x_n(s))du(s)= ∫_0^t g(s,x(s))du(s),t∈ [0,1]. Hence,𝒯x_n(t)-𝒯x(t)= β+t/2[(F(1,x_n) +F(η,x_n)+G_u(1,x_n)+G_u(η,x_n)).. -(F(1,x)+F(η,x)+G_u(1,x)+G_u(η,x))] -∫_0^t F(s,x_n(s))-F(s,x(s))ds -∫_0^t G_u(s,x_n)-G_u(s,x)ds, t∈[0,1].Therefore, lim_n→∞𝒯x_n(·)=𝒯x(·), and thus 𝒯 is a completely continuous operator. Finally, let Ω={x∈ G[0,1] : x<r} and assume that x∈∂Ωsuch that 𝒯x=λ x for λ>1. Then, by (<ref>), one hasλ r =λx=𝒯x≤ r,which implies that λ≤ 1. This is a contradiction. Therefore, by Lemma <ref>, there exists a fixed point of 𝒯, which is a solutionof problem (<ref>)-(<ref>). The proof of Theorem <ref> is complete.§ ILLUSTRATIVE EXAMPLESWe now give two examples to illustrate Theorem <ref>and Corollary <ref>, respectively. Let g^*(t,x(t))=0if t=0 and g^*(t,x(t))=1 if t∈ (0,1] for all x∈ B_r.Then, it is easy to see that g^* satisfies hypotheses(H_4)–(H_6) with M=1.Consider the boundary value problem{[ -D^2 x=xsin(x)/3√(5+t)+g^*(t,x)Dℋ(t-1/2) ,t∈ [0,1],; x(0)=4Dx(0), Dx(1)+Dx(1/4)=0, ].where ℋ is the Heaviside function, i.e.,ℋ(t)=0 if t<0 and ℋ(t)=1 if t>0. It is easy to see that ℋ is of bounded variation,but not continuous. Let f(t,x)=xsin(x)/2√(4+t), g(t,x)=g^*(t,x), and u(t)=ℋ(t-1/2). Then, (H_1), (H_2), and (H_4)–(H_6) hold. Moreover, there exist HK integrable functions k(t)=1/3√(5+t) and h(t)=1 such that-kx-h≤f(·,x)≤ kx+h ∀ x∈ G[0,1],i.e., (H_3) holds. Further, by (<ref>),K=2/3(√(6)-√(5)), H=1, u=ℋ=1. Let β=4 and η=1/4.From (<ref>), we haver=(|β|+2)(H+2Mu)/1-(|β|+2)K =18/1-4(√(6)-√(5)).Therefore, by Theorem <ref>, problem (<ref>)has at least one solution x^* withx^*≤18/1-4(√(6)-√(5)).Consider the boundary value problem{[ -D^2x=sin(x)+ 2tsin( t^-2)-2/tcos(t^-2); +g^*(t,x)D𝒲,t∈ [0,1],;x(0)=-1/6Dx(0), Dx(1)+Dx(2/3)=0, ].where 𝒲 is the Weierstrass function 𝒲(t)=∑_n=1^∞sin 7^nπ t/2^n in <cit.>. It is well-known that 𝒲(t)is continuous but pointwise differentiable nowhere on [0,1],so 𝒲(t) is not of bounded variation. Letf(t,x) =sin(x)+ 2tsin( t^-2) -2/tcos(t^-2),g(t,x) =g^*(t,x), u =𝒲.Then, (H_1), (H_2) and (H_4)–(H_6) hold.Moreover, letk(t)=0, h(t)=1+2tsin( t^-2) -2/tcos(t^-2).Obviously, the highly oscillating function h(t) is Kurzweil–Henstock integrable but not Lebesgue integrable, andH(t)=∫_0^t h(s)ds= {[ t+t^2sin(t^-2),t∈(0,1],;0,t=0. ].Moreover, we have-h≤f(·,x)≤ h ∀ x∈ G[0,1],that is, (H_3) holds. Let β=-1/6and η=2/3. Since 0≤u=𝒲≤∑_n=1^∞1/2^n=1, H_∞=1+sin(1),we have by (<ref>) that3.9899≈13/6(sin(1)+1)≤r = (|β|+2)(H+2Mu)/1-(|β|+2)K_∞≤13/6(sin(1)+3)≈ 8.3232.Therefore, by Corollary <ref>, problem (<ref>)has at least one solution x^* with x^*≤13/6(sin(1)+3).§ ACKNOWLEDGMENTS The authors are grateful to two referees for their comments and suggestions. 30urlstyle[Antunes Monteiro and Slavík(2016)]MS16 G. Antunes Monteiro and A. Slavík. Extremal solutions of measure differential equations. J. Math. Anal. Appl.,4440 (1):0 568–597, 2016. [Boon and Lam(2011/12)]BL11 T. S. Boon and T. T. Lam. The Itô-Henstock stochastic differential equations. Real Anal. Exchange,370 (2):0 411–424, 2011/12. [Chew and Flordeliza(1991)]TSC T. S. Chew and F. Flordeliza. On x'=f(t,x) and Henstock-Kurzweil integrals. Differential Integral Equations,40 (4):0 861–868, 1991. [Chew et al.(2001/02)Chew, Tay, and Toh]CTT01 T.-S. Chew, J.-Y. Tay, and T.-L. Toh. The non-uniform Riemann approach to Itô's integral. Real Anal. 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Well-posedness results for abstract generalized differential equations and measure functional differential equations. J. Differential Equations,2590 (2):0 666–707, 2015. [Sun and Zhao(2015)]SZ15 Y. Sun and M. Zhao. Existence of positive pseudo-symmetric solution for second-order three-point boundary value problems. J. Appl. Math. Comput.,470 (1-2):0 211–224, 2015. [Talvila(2008)]Eri08 E. Talvila. The distributional Denjoy integral. Real Anal. Exchange,330 (1):0 51–82, 2008. [Toh and Chew(2012)]TC12 T.-L. Toh and T.-S. Chew. The Kurzweil-Henstock theory of stochastic integration. Czechoslovak Math. J.,62(137)0 (3):0 829–848, 2012. [Tvrdý(1994)]MT94 M. Tvrdý. Linear distributional differential equations of the second order. Math. Bohem., 1190 (4):0 415–436, 1994. [Tvrdý(2002)]TVR02 M. Tvrdý. Differential and integral equationsin the space of regulated functions. Mem. Differential Equations Math. Phys.,25:0 1–104, 2002. [Ye and Liu(2016)]YL15 G. Ye and W. Liu. The distributional Henstock-Kurzweil integral and applications. Monatsh. Math., 1810 (4):0 975–989, 2016. [Zhou et al.(2015)Zhou, Ye, Liu, and Wang]ZY15 H. Zhou, G. Ye, W. Liu, and O. Wang. The distributional Henstock-Kurzweilintegral and measure differential equations. Bull. Iranian Math. Soc.,410 (2):0 363–374, 2015. | http://arxiv.org/abs/1704.08009v1 | {
"authors": [
"Wei Liu",
"Guoju Ye",
"Dafang Zhao",
"Delfim F. M. Torres"
],
"categories": [
"math.CA",
"26A39, 34B15, 46G12"
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"primary_category": "math.CA",
"published": "20170426082859",
"title": "Existence theorems for a nonlinear second-order distributional differential equation"
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Linking axionlike dark matter to neutrino massesO. Zapata December 30, 2023 ================================================== INTRODUCTIONIn 1956, Selberg expressedthe trace of an invariant kernel actingon a locally symmetric space Z=Γ\ G/K asa sum of certain integrals on the orbits of Γ in G,the so called “orbital integrals", andhe gave a geometric expression for such orbital integralsfor the heat kernel when G= _2(), andthe corresponding locally symmetric space isa compactRiemann surface of constant negative curvature. In this case, the orbital integrals are one to one correspondence with the closed geodesics in Z.In the general case, Harish-Chandra worked on the evaluation oforbital integrals from the 1950s until the 1970s. He couldgive an algorithm to reduce the computation ofan orbital integral to lower dimensional Lie groups bythe discrete series method. Given a reductive Lie group,in a finite number of steps, there is a formula for such orbital integrals. See Section <ref> for a brief description of Harish-Chandra's Plancherel theory.It is important to understandthe different properties oforbital integrals even without knowing their explicit values. The orbital integrals appear naturally in Langlands program.About 15 years ago, Bismut gave a natural construction of a Hodge theory whose corresponding Laplacianis a hypoelliptic operator acting on the total spaceof the cotangent bundle of a Riemannian manifold. This operator interpolates formally between the classical elliptic Laplacian on the base and the generator of the geodesic flow. We will describe recent developments in the theory of the hypoelliptic Laplacian, andwe will explain two consequences of this program,the explicit formula obtained by Bismut fororbital integrals, and the recent solution by Shen of Fried's conjecture (dating back to 1986)for locally symmetric spaces. The conjecture predicts the equality of the analytic torsion andof the value at 0 of the Ruelle dynamicalzeta function associated with the geodesic flow.We will describe in more detail these two last results.Let G be a connected reductive Lie group, let be its Lie algebra, let θ∈ Aut(G) be the Cartan involution of G.Let K⊂ G be the maximal compact subgroup of G given by the fixed-points ofθ, and letbe its Lie algebra. Let =⊕ be the corresponding Cartan decomposition of .Let B be a nondegenerate bilinear symmetric form onwhich is invariant under the adjoint action of G on and also under θ.We assume B is positive onand negative on . Then ⟨·, ·⟩=-B(·,θ·) is a K-invariant scalar product onthat issuch that the Cartan decomposition is an orthogonal splitting.Let C^∈ U() be the Casimir element of G. If {e_i}_i=1^m is an orthonormal basis of and {e_i}_i=m+1^m+n is an orthonormal basis of , set B^*()= -1/2∑_1≤ i,j≤ m|[e_i,e_j]|^2-1/6∑_m+1≤ i,j≤ m+n|[e_i,e_j]|^2, =1/2C^ +1/8 B^*().Let E be a finite dimensional Hermitian vector space,let ρ^E:K→ U(E) be a unitary representation of K.Let F=G×_K E be the corresponding vector bundle overthe symmetric space X=G/K. Thendescends toa second order differential operator^X actingon C^∞(X,F). For t>0, let e^-t^X(x,x') be the smooth kernel ofthe heat operator e^-t^X.Assume γ∈ G is semisimple.Then up to conjugation, there exist a∈, k∈ Ksuch that γ= e^a k^-1 and Ad(k)a=a. Let ^[γ][e^-t^X] denotethe corresponding orbital integral ofe^-t^X (cf. (<ref>), (<ref>)). If γ=1, then the orbital integral associated with 1∈ Gis given by ^[γ=1][e^-t^X] =^F[e^-t^X(x,x)]which does not depend on x∈ X.Let Z(γ)⊂ G be the centralizer of γ,and let (γ) be its Lie algebra. Set(γ)= (γ)∩,(γ)= (γ)∩. Then (γ)= (γ)⊕(γ).Set _0=((a)), _0=_0∩.Let _0^ be the orthogonal space to_0 in . Let _0^(γ) be theorthogonal space to (γ) in _0, and_0^(γ) be the orthogonal space to(γ) in _0, so that _0^(γ) = _0^(γ)⊕_0^(γ). For a self-adjoint matrix Θ, set A(Θ)= ^1/2[Θ/2/sinh(Θ/2)]. For Y∈(γ), set J_γ(Y)= | (1-(γ)) |__0^|^-1/2A(i(Y)|_(γ))/A(i(Y)|_(γ)) ×[1/(1-(k^-1))|_^_0(γ)(1-e^-i(Y)(k^-1) )|_^_0(γ)/(1-e^-i(Y)(k^-1) )|_^_0(γ)]^1/2.If γ=1, then the above equation reduces to J_1(Y)=A(i(Y)|_)/A(i(Y)|_) for Y∈=(1).(Bismut's orbital integral formula<cit.>) Assume γ∈ G issemisimple.Then for any t>0, we have ^[γ][e^-t^X] = (2π t)^-(γ)/2 e^-|a|^2/2t ∫_(γ) J_γ(Y) ^E[ρ^E(k^-1) e^-iρ^E(Y)] e^-|Y|^2/2tdY/(2π t)^(γ)/2.There are some striking similarities of Equation (<ref>)with the Atiyah-Singer index formula, wherethe A-genus of the tangent bundle appears.Here the A-function of bothandparts(with different roles) appear naturally in the integral (<ref>). A more refined versionof Theorem <ref> for the orbital integral associated with the wave operator is given in<cit.> (cf. Theorem <ref>).Let Γ⊂ G be a discrete cocompact torsion freesubgroup. The above objets constructedon X descend tothe locally symmetric space Z=Γ\ X and π_1(Z)=Γ. We denote by ^Z the corresponding differentialoperator on Z. Let [Γ] be the set of conjugacy classes in Γ.The Selberg trace formula (cf. (<ref>), (<ref>)) forthe heat kernel of the Casimir operator on Z says that[e^-t^Z]=∑_[γ]∈ [Γ](Γ∩ Z(γ)\Z(γ) )^[γ][e^-t ^X].Each term ^[γ][·] in (<ref>) is evaluated in(<ref>). Assume m=X is odd now. Let ρ:Γ→ U( q) be a unitary representation.Then F=X×_Γ^ q is a flat Hermitian vector bundleon Z=Γ\ X.Let T(F) be the analytic torsionassociated with F on Z (cf. Definition <ref>),which is a regularized determinant of the Hodge Laplacianfor the de Rham complex associated with F. In 1986, Fried discovered a surprising relation ofthe analytic torsion to dynamical systems. In particular,for a compact orientable hyperbolic manifold, he identifiedthe value at zero of the Ruelle dynamical zeta function associated with the closed geodesics in Z and with ρ,to the corresponding analytic torsion, and he conjectured thata similar result should hold for general compactlocally homogenous manifolds.In 1991, Moscovici-Stanton <cit.> made an importantprogress in the proof ofFried's conjecture for locally symmetric spaces.The following recent result of Shen establishes Fried's conjecture for arbitrarylocally symmetric spaces, and Theorem <ref> is one important ingredient in Shen's proof. Given [γ]∈ [Γ]\{1}, letB_[γ]be the space of closed geodesics in Z which lie in the homotopy class [γ],and let l_[γ] be the length of the geodesic associated with γ in Z. The group 𝕊^1 acts onB_[γ] by rotations. This action is locally free.Denote by χ_orb(𝕊^1\B_[γ])∈ℚ the orbifold Euler characteristic number for the quotient orbifold𝕊^1\ B_[γ]. Letn_[γ]=|(𝕊^1→Diff(B_[γ]))|be the generic multiplicity of B_[γ].<cit.>For any unitary representation ρ:Γ→ U( q), R_ρ(σ)=exp( ∑_[γ]∈ [Γ]\{1}[ρ(γ)]χ_orb(𝕊^1\ B_[γ])/n_[γ]e^-σ l_[γ])is a well-defined meromorphic function on . IfH^∙(Z,F)=0, then R_ρ(σ) is holomorphic at σ=0 and R_ρ(0)=T(F)^2. This article is organized as follows. In Section <ref>, we describeBismut's program on the geometric hypoelliptic Laplacian in de Rham theory, and we give its applications.In Section <ref>, we introduce the heat kernel onsmooth manifolds and the basic ideas in the heat equation proofof the Lefschetz fixed-point formulas, which will serve as a model for the proof of Theorem <ref>. In Section <ref>, we revieworbital integrals, their relation toSelberg trace formula, andwe state Theorem <ref>. In Section <ref>, we give the basic ideas in howto adapt theconstruction of the hypoelliptic Laplacian ofSection <ref> in the context of locally symmetric spaces in order toestablish Theorem <ref>.In Section <ref>,we concentrate on Shen's solution of Fried's conjecture. Notation: If A is a _2-graded algebra, if a,b∈ A, the supercommutator [a,b] is given by[a,b]= ab -(-1)^ a· b ba.If B is another _2-graded algebra, we denote byA⊗ B the _2-graded tensor product, such that the _2-degreeof a⊗ b is given bya + b, and where the product is given by(a⊗ b)· (c⊗ d) = (-1)^ b· c ac ⊗bd.If E=E^+⊕ E^- is a _2-graded vector space, and τ=± 1 on E^±,for u∈(E), the supertrace _s[u] is given by_s[u]= [τ u].In what follows, we will often add a superscript to indicate where the trace or supertrace is taken. For β a number or a matrix, we denotesinh(β)=1/2(e^β-e^-β), cosh(β) =1/2(e^β+e^-β), tanh(β) =sinh(β)/cosh(β).Acknowledgments. I thank Professor Jean-Michel Bismut very heartily for his helpand advice during the preparation of this manuscript.It is a pleasure to thank Laurent Clozel, Bingxiao Liu,George Marinescu and Shu Shen for their help and remarks. § FROM HYPOELLIPTICLAPLACIANS TO THE TRACE FORMULAIn this section, we describe some basic ideastaken from Bismut's programon the geometric hypoelliptic Laplacian and its applicationsto geometry and dynamical systems.A differential operator P is hypoelliptic iffor every distribution u defined on an open set Usuch that Pu is smooth, then u is smooth on U. Elliptic operators are hypoelliptic, but there arehypoelliptic differential operators which are not elliptic. Classical examples are Kolmogorov operator∂ ^2/∂ y^2- y ∂/∂ x on ^2 <cit.> and Hörmander's generalization ∑_j=1^k X_j^2 + X_0 on Euclidean spaces <cit.>. Along this line, see for exampleHelffer-Nier's <cit.> recent book and Lebeau's work <cit.> on the hypoelliptic estimates and Fokker-Planck operators.In 1978, Malliavin <cit.>introduced the so-called`Malliavin calculus' to reprove Hörmander's regularity result<cit.> from a probabilisticpoint of view. Malliavin calculus was furtherdeveloped by Bismut <cit.> and Stroock <cit.>. About 15 years ago, Bismut initiated a program whose purposeis to study the applications of hypoelliptic second order differential operators to differential geometry.In <cit.>, Bismut constructed a (geometric)hypoelliptic Laplacian onthe total space of the cotangent bundleT^*M of a compact Riemannian manifold M, that depends on a parameter b>0. This hypoelliptic Laplacianis a deformation of the usual Laplacian on M. More precisely, when b→ 0, itconverges to the Laplacian on M in a suitable sense, and when b→ +∞, it converges to the generator ofthe geodesic flow.In this way,properties of the geodesic flow on Mare potentially related to the spectral properties of the Laplacian on M.We now explain brieflyBismut'shypoelliptic Laplacian in de Rham theory. Let (M, g^ TM) be a compact Riemannian manifoldof dimension m. Let (Ω^∙ (M), d) be the de Rham complex of M, let d^* be the formal L_2 adjoint of d, andlet □^M= (d+d^*)^2 be the Hodge Laplacianacting on Ω^∙ (M). Let π: → M be the total space of the cotangent bundle T^*M.Let Δ^V be theLaplacian along the fibersT^*M, and letbe the function ondefined by(x,p)= 1/2|p|^2 forp∈ T_x^*M, x∈ M.Let Y^ be the Hamiltonian vector field on associated withandwith the canonical symplectic form on . Then Y^ is the generator of the geodesic flow.Let L_Y^ denote the correspondingLie derivative operator acting on Ω^∙ (). For b>0, the Bismut hypoelliptic Laplacian onis given by _b =1/b^2α + 1/bβ + ϑ,with α = 1/2 (-Δ^V + |p|^2 -m +⋯),β = -L_Y^ + ⋯,where the dots andϑ are geometric termswhich we will not be made explicit. The operator _b is essentially the weighted sum ofthe harmonic oscillator along the fiber,minus the generator of the geodesic flow - L_Y^ along the horizontal direction.[On Euclidean spaces, all geometric termsvanish and the operator _b acting on functionsreduces tothe Fokker-Planck operator.] The vector space (α) is spanned by the functionexp(-|p|^2/2). We identify Ω^∙ (M) to (α) by the maps →π^* s exp(-|p|^2/2)/ π^m/4.Let P be the standard L_2-projector fromΩ^∙ () on (α). Thenby <cit.>, P(ϑ - βα^-1β)P = 1/2□^M.In <cit.>, equation (<ref>)is used to prove thatas b→ 0, we have the formal convergence of resolvents (λ - _b)^-1→ P (λ - 1/2□^M)^-1P. Bismut-Lebeau <cit.> set up the proper analysis foundation for the study of the hypoelliptic Laplacian _b. They not only proved a corresponding version ofthe Hodge theorem, but they also studied the precise properties of its resolventand of the corresponding heat kernel.Sinceis noncompact, they needed to refinethe hypoelliptic estimates ofHörmander in order to control hypoellipticity at infinity.They developed the adequate theoryof semiclassical pseudodifferential operators with parameter ħ= b and obtained the proper version of the convergenceof resolvents in (<ref>). They developed also a hypoelliptic local index theorywhich is itself a deformation of classical elliptic local index theory. In<cit.>, Bismut-Lebeau defined a hypoelliptic version ofthe analytic torsion ofRay-Singer <cit.> associated withthe elliptic Hodge Laplacianin (<ref>).The main result in<cit.> is the proof of the equality of the hypoelliptictorsion with the Ray-Singer analytic torsion.In his thesis <cit.>, Shen studied the Witten deformationof the hypoelliptic Laplacian for a Morse functionon the base manifold, andidentified the hypoelliptic torsion to the combinatory torsion. Shen's work gives a new proofof Bismut-Lebeau's result on the equality of the hypoelliptic torsion and the Ray-Singer analytictorsion. This article concentrates on applications of the hypoelliptic Laplacian to orbital integrals. We will briefly summarize other applications. Aversion of Theorem <ref> for compact Lie groupscan be found in <cit.>. In <cit.>, as a test of his ideas,Bismut gave a new proof ofthe classical explicit formula for the scalar heatkernel in terms of the coroots lattice <cit.> for a simple simply connected compact Lie group,by using the hypoelliptic Laplacian on the totalspace of the cotangent bundle of the group. In <cit.>, Bismut also constructed a hypoellipticDirac operator which is a hypoelliptic deformation of the usualDirac operator.In <cit.>, Bismut established aGrothendieck-Riemann-Roch theorem for a proper holomorphicsubmersion π: M→ B of complex manifolds inBott-Chern cohomology.For compact Kähler manifolds, Bott-Chern cohomology coincides with de Rham cohomology. In the general situation considered in <cit.>,the elliptic methods of <cit.>, <cit.> are known to fail, and hypoelliptic methods seem to bethe only way to obtain this result. As in the case of the Dirac operator, there does notexist a universalhypoelliptic Laplacian which works for all situations,there are several hypoelliptic Laplacians. To attack a specific(geometric) problem, we need to constructthe corresponding hypoelliptic Laplacian. Still all the hypoelliptic Laplacians have naturally the samestructure, but the geometric terms depend on the situation.Probability theory plays an important role, bothformally and technically in its construction and in its use.In this article,we will not touch the analytic andprobabilistic aspects of the proofs. We will explain how to give a natural construction of the hypoelliptic Laplacian which is needed in order to establishTheorem <ref>.The method consists in giving a cohomological interpretation to orbital integrals, so as to reducetheir evaluation to methods related to the proof ofLefschetz fixed-point formulas. Theorem <ref> gives a direct link ofthe trace formula toindex theory.We hope this article can be used as an invitation to the original papers <cit.> and to several surveys on this topic <cit.> and <cit.>. § HEAT KERNEL AND LEFSCHETZ FIXED-POINT FORMULA This section is organized as follows. In Section <ref>,we explain some basic facts about heat kernels.In Section <ref>, we review the heatequationproof of the Lefschetz fixed-point formula.This proof will beused as a model for the proof of the main theorem of this article. §.§ A brief introduction to the heat kernelLet M be a compact manifold of dimension m.Let TM be the tangent bundle, T^*M be the cotangentbundle, and let g^TM be a Riemannian metric on M. Let F be a complex vector bundle over M, andlet h^F be a Hermitian metric on F.Let C^∞(M,F) be the space of smooth sectionsof F on M. Let ⟨· ,·⟩ be the L_2-Hermitian product on C^∞(M,F)defined by the integral of the pointwise productwith respect to the Riemannian volume form dx. We denote byL_2(M,F) the vector space of L_2-integrable sectionsof F on M.Let ∇^F: C^∞(M,F)→ C^∞(M,T^*M⊗ F) bea Hermitian connection on (F, h^F) and let ∇^F,*be its formal adjoint.Then the (negative) Bochner Laplacian Δ^Facting on C^∞(M,F), is defined by-Δ^F=∇^F,*∇^F. The operator -Δ^F isan essentially self-adjoint second order elliptic operator.Let ∇^TM be the Levi-Civita connection on(TM,g^TM). We can rewrite it as-Δ^F=-∑^m_i=1((∇_e_i^F)^2 -∇^F_∇^TM_e_ie_i), where {e_i}^m_i=1 is a local smooth orthonormal frame of (TM, g^TM). For a self-adjoint section Φ∈ C^∞(M,End(F)) (for any x∈ MthatΦ_x∈End(F_x)is self-adjoint), set -Δ^F_Φ=-Δ^F-Φ. Then the heat operator e^t Δ^F_Φ: L_2(M,F) → L_2(M,F) for t>0 of -Δ^F_Φis the unique solution of { (∂/∂ t-Δ^F_Φ ) e^tΔ^F_Φ=0 lim_t→ 0e^tΔ^F_Φs=s ∈ L_2(M,F)for anys∈ L_2(M,F). .For x,x'∈ M, let e^tΔ^F_Φ(x,x')∈ F_x⊗ F^*_x'be the Schwartz kernel of the operator e^tΔ^F_Φ with respect to the Riemannian volume element dx'.Classically, e^tΔ^F_Φ is smooth in x,x'∈ M, t>0. Since M is compact, the operator -Δ^F_Φ hasdiscrete spectrum, consisting of eigenvalues λ_1≤λ_2≤⋯≤λ_k≤⋯ counted with multiplicities,with λ_k→ +∞ as k→ +∞. Let {φ_j}^+∞_j=1be a systemof orthonormal eigenfunctions such that-Δ^F_Φφ_j=λ_j φ_j. Then {φ_j}_j=1^+∞ is anorthonormal basisof L_2(M,F). The heat kernel can also be written as(cf. <cit.>, <cit.>)e^tΔ^F_Φ(x,x')=∑^+∞_j=1 e^-tλ_jφ_j(x)⊗φ_j(x')^*where φ_j(x')^*∈ F^*_x' isthe metric dual of φ_j(x')∈ F_x'.The trace of the heat operator is given by[e^tΔ^F_Φ]=∑^+∞_j=1 e^-tλ_j.The (heat) trace [e^tΔ^F_Φ] involvesthe full spectrum information ofoperator Δ^F_Φand has many applications. In general, it is difficult to evaluate explicitly[e^tΔ^F_Φ] for t>0.However, we will explain the explicit formula obtained byBismut for locally symmetric spaces and its connection with Selberg trace formula.Let π: M→ M be the universal cover of M with fiber π_1(M), the fundamental group of M.Then geometric data on M lift toM, and we willadd a to denote the corresponding objets on M.It's well-known(see for instance <cit.>) that if x, x'∈M are such that π (x)=x, π (x')=x', we havee^tΔ^F_Φ(x,x')=∑_γ∈π_1(M)γe^tΔ^F_Φ (γ^-1x,x'), where the right-hand side is uniformly convergent. §.§ The Lefschetz fixed-point formulasLet Ω^∙(M)=⊕_j Ω^j(M) =⊕_j C^∞(M, Λ^j(T^*M)) be the vector spaceof smooth differential forms on M (with values in ), which is-graded by degree.Let d:Ω^j(M)→Ω^j+1(M)be the exterior differential operator. Then d^2=0 so that (Ω^∙(M),d) formsthe de Rham complex. Thede Rham cohomology groups of M are defined by H^j(M,)=(d|_Ω^j(M))/ (d|_Ω^j-1(M)), H^∙ (M,)= ⊕_j=0^m H^j(M,).They are canonically isomorphic to the singular cohomologyof M. Let d^*: Ω^∙(M)→Ω^∙-1(M) be theformal adjoint of d with respect to the scalar product⟨· , ·⟩ on Ω^∙(M), i.e., for all s,s'∈Ω^∙(M),⟨ d^*s,s'⟩ :=⟨ s, ds'⟩.Set D=d+d^*.Then D is a first order elliptic differential operator, and we haveD^2=dd^*+d^*d.The operator D^2 is called the Hodge Laplacian,it is an operator of the type (<ref>)for F=Λ^∙(T^*M), which preserves the -gradingon Ω^∙(M). By Hodge theory, we havethe isomorphism,(D|_Ω^j(M))= (D^2|_Ω^j(M)) ≃ H^j(M,), forj=0,1,⋯,m. We give here a baby example to explainthe heat equation proof of the Atiyah-Singer index theorem (cf. <cit.>).Let H be a compact Lie group acting on M on the left.Since the exterior differential commutes with theaction of H on Ω^∙(M),H acts naturally on H^j(M,) for any j. The Lefschetz number for h∈ H is given byχ_h(M) = ∑^m_j=0 (-1)^j [h|_H^j(M,)] = _s[h|_H^∙(M,)].The Lefschetz fixed-point formula computesχ_h(M) in term ofgeometric data on the fixed-point set of h.Instead of working on H^j(M,), we will work onthe much larger space Ω^∙(M) to establishthe Lefschetz fixed-point formulas.Since H is compact, by an averaging argument on H,we can assume that the metric g^TM is H-invariant.Then the operator D defined above is also H-invariant. We have the following result(cf. <cit.>), [McKean-Singer formula]For any t>0, χ_h(M)=_s [he^-tD^2].For any t>0, we have∂/∂ t_s[he^-tD^2] = -_s[hD^2 e^-t D^2]=-1/2_s[[D,hDe^-tD^2]]=0.Here [·,·] is a supercommutator defined as in (<ref>), and as in the case of matrices, the supertrace ofa supercommutator vanishes by a simple algebraic argument. By (<ref>) and (<ref>), we havelim_t→ +∞_s[he^-tD^2]=χ_h(M).Combining (<ref>) and (<ref>), we get (<ref>). A simpleanalysis shows that only thefixed-points of h contribute to the limit of _s[he^-tD^2] ast→ 0.Further simple work then leads tothe Lefschetz fixed-point formulas.Even though we will work on a more refined object the trace of a heat operator,the above philosophy still applies. § BISMUT'S EXPLICIT FORMULA FOR THE ORBITAL INTEGRALS In this section, we give an introduction to orbital integrals and to Selberg trace formula, and we present the main result of thisarticle: Bismut's explicit evaluation of the orbital integrals. Also, we compare Harish-Chandra's Plancherel theory withBismut's explicit formula for the orbital integrals.This section is organized as follows. In Section <ref>, we recall some basic facts on symmetric spaces, and we explain how the Casimir operator for a reductive Lie group induces a Bochner Laplacian on the associated symmetric space.In Section <ref>, we give an introduction to orbital integrals and to Selberg trace formula, and in Section <ref>,we describe the geometric definition oforbital integrals given by Bismut. In Section <ref>,we present the main result of this article,Bismut's explicit evaluation of the orbital integrals,and give some examples. Finally in Section <ref>, we present briefly Harish-Chandra'sPlancherel theory for comparison with Bismut's result.§.§ Casimir operator and Bochner Laplacian Let G be a connected real reductive Lie group with Lie algebra and Lie bracket [·,·].Let θ∈Aut(G) be its Cartan involution.Let K be the subgroup of G fixed by θ,with Lie algebra . Then K is a maximal compact subgroup of G, and K is connected. The Cartan involution θ acts naturally as a Lie algebraautomorphism of . Then the Cartan decomposition ofis given by=⊕,with ={a∈ : θ a=-a}, ={a∈ : θ a=a}. From (<ref>), we get[,]⊂,[,] ⊂, [,] ⊂.Put n=, m=.Then =m+n.If g,h ∈ G,u ∈, let (g)h=ghg^-1 be the adjoint action of g on h, and let Ad(g)u∈ denote the action of g on u via the adjoint representation. If u,v∈, set ad(u)v=[u,v],then ad is the derivative of the mapg∈ G→Ad(g)∈Aut().Let B be a real-valued nondegenerate symmetric bilinearform onwhich is invariant under the adjoint actionof G on , and also under the action of θ.Then (<ref>) is an orthogonal splitting of with respect to B.We assume that B is positive onand negative on . Put⟨·,·⟩=-B(·,θ·)the associated scalar product on ,which is invariant under the adjoint action of K. Let |·| be the corresponding norm on .The splitting (<ref>) is also orthogonalwith respect to ⟨·,·⟩.For G=GL^+( q,)={A∈GL( q,),A>0},the Cartan involution is given by θ(g)=^tg^-1, where ^t· denotes the transpose of a matrix. Then K=SO( q), the special orthogonal group, andis the vector space of anti-symmetric matrices andis the vector space ofsymmetric matrices. We can take B(u,v)=2^^ q[uv] foru,v∈=𝔤𝔩( q,)=End(^ q). Let U() be the enveloping algebra ofwhich willbe identified with the algebra of left-invariant differential operators on G. Let C^∈ U() be the Casimir element. If {e_i}^m_i=1 is an orthonormal basis of(,⟨·,·⟩) andif {e_i}^m+n_i=m+1 is an orthonormal basis of(,⟨·,·⟩), then C^= C^+C^,with C^=-∑^m_i=1e_i^2,C^=∑^m+n_i=m+1 e_i ^2.Then C^ is the Casimir element ofwith respect to the bilinear form induced by B on .Note that C^ lies in the center of U().Let ρ^V: K→Aut(V) be an orthogonalor unitary representation of K on a finite dimensional Euclideanor Hermitian vector space V.We denote by C^,V∈End(V)the corresponding Casimir operator acting on V, given byC^,V=∑^m+n_i=m+1ρ^V,2(e_i). Let p: G→ X=G/Kbe the quotient space.Then X is contractible.More precisely, X is a symmetric space and the exponential mapexp: → G/K,a↦p e^a isa diffeomorphism. We have a natural identification of vector bundles on X:TX=G×_K ,where K acts on via the adjoint representation. The scalar product ofdescends to a Riemannian metricg^TX on TX.Let ω^ be the canonical left-invariant 1-form on Gwith values in , and let ω^ be the-component of ω^. Then ω^defines a connection on the K-principal bundle G→ G/K. The connection ∇^TX on TX induced by ω^ and by (<ref>) is preciselythe Levi-Civita connection on (TX, g^TX).Note since the adjoint representation of K preserves and ,we obtain C^,∈End(),C^,∈End(). In fact, ^[C^,]is the scalar curvature of X, and-14^[C^,] is the scalar curvature of Kfor the Riemannian structure induced by B (cf. <cit.>).Let ρ^E: K→Aut(E) be a unitary representation of K. Then the vector space Edescends toa Hermitian vector bundle F=G×_K E on X, andω^ induces a Hermitian connection ∇^F on F.Then C^∞(X,F) can be identified to C^∞(G,E)^K, the K-invariant part of C^∞(G,E).The Casimir operator C^,acting on C^∞(G,E), descends to an operatoracting on C^∞(X,F), which will still be denoted byC^.Let A be a self-adjoint endomorphism of E which isK-invariant. Then A descends to a parallel self-adjoint sectionof End(F) over X. Let ^X, ^X_A act on C^∞(X,F) by the formulas,^X=1/2C^ +1/16^ [C^,] +1/48^[C^, ];^X_A=^X+A.From (<ref>), -C^ descends tothe Bochner Laplacian Δ^F on C^∞(X,F), the operator C^descends to a parallel section C^,F of (F) on X.If the representationρ^E above is irreducible, then C^,F acts asc Id_F, where c is a constant function on X.Thus from (<ref>) and (<ref>), we have^X=-1/2Δ^F_ϕwithϕ=-C^,F - 1/8^ [C^,] -1/24^ [C^,]. The group G acts on X on the left. This action lifts toF. More precisely, for any h∈ G and [g,v]∈ F,the left action of h is given byh.[g,v]=[hg,v]∈ G×_K E=F.Then the operators ^X, ^X_Acommute with G. Let Γ⊂ G be a discrete subgroup of Gsuch that the quotient space Γ\ G is compact.SetZ=Γ\ X=Γ\ G/ K.Then Z is a compact locally symmetric space.In general Z is an orbifold. If Γ is torsion-free(i.e.,if γ∈Γ, k∈^*, then γ^k=1 implies γ=1),then Z is a smooth manifold.From now on, we assume that Γ is torsion free, so thatΓ=π_1(Z) and X is just the universal cover of Z.A vector bundle like F on X descends to a vector bundleon Z, which we still denote by F.Then the operators ^X, ^X_Adescend to operators ^Z, ^Z_Aacting on C^∞(Z,F). For t>0,let e^-t ^X_A(x,x') (x,x'∈ X),e^-t ^Z_A(z,z') (z,z'∈ Z)be the smooth kernels of the heat operators e^-t ^X_A, e^-t ^Z_A with respect to the Riemannian volumeforms dx', dz' respectively. By (<ref>), we get[e^-t ^Z_A] = ∫_Z [e^-t ^Z_A(z,z)]dz= ∫_Γ\ X∑_γ∈Γ[γ e^-t ^X_A(γ^-1z, z)]dz.§.§ Orbital integrals and Selberg trace formulaLet C^b(X,F) be the vector space of continuous bounded sectionsof F over X. Let Q be an operator acting on C^b(X,F)with a continuous kernel q(x,x') with respect to the volume form dx'. It is convenient to view q as a continuous function q(g,g') defined on G× Gwith values in (E) which satisfiesforany k,k'∈ K,q(gk,g'k')=ρ^E(k^-1)q(g,g')ρ^E(k'). Now we assume that the operator Q commutes withthe left action of G on C^b(X,F) defined in (<ref>).This is equivalent to q(gx,gx')=gq(x,x')g^-1 for any x,x'∈ X,g∈ G, wherethe action of g^-1 mapsF_gx' to F_x', the action of g maps F_x to F_gx.If we consider instead the kernel q(g,g'), then this impliesthat for all g”∈ G,q(g”g,g”g')=q(g,g')∈(E).Thus the kernel q is determinedby q(1,g). Setq(g)=q(1,g).Then we obtain from (<ref>) and (<ref>)that for g∈ G, k∈ K,q(k^-1gk)=ρ^E(k^-1)q(g)ρ^E(k).This implies that ^E[q(g)] is invariant when replacing g by k^-1gk. In the sequel, we will use the same notation q for the variousversions of the corresponding kernel Q.The element γ∈ G is said to be elliptic if it is conjugate inG to an element of K. We say that γ is hyperbolicif it is conjugate in G to e^a,a∈. For γ∈ G, γ is semisimple if there existg∈ G, a∈, k∈ K such that (k)a=a, γ=(g)(e^ak^-1).By <cit.>, if γ∈ G is a semisimple element, (g)e^a and (g)k^-1 areuniquely determined by γ (i.e., they do not depend ong∈ G such that (<ref>) holds), and Z(γ)=Z((g)e^a)∩ Z((g)k^-1),where Z(γ)⊂ G is the centralizer of γ in G. Let dk be the Haar measure on K that givesvolume 1 to K. Let dg be measure on G(as a K-principal bundle on X=G/K) given bydg=dx dk.Then dg is a left-invariant Haar measure on G.Since G is unimodular, it is also a right-invariant Haar measure. For γ∈ G semisimple, Z(γ) is reductive andK(γ), the fixed-points set of (g) θ(g)^-1 in Z(γ) (cf. (<ref>)), is a maximal compact subgroup. Let dy be the volume element on the symmetric space X(γ)=Z(γ)/K(γ)induced by B. Let dk' be the Haar measure onK(γ) that gives volume 1 toK(γ). Then dz=dydk' is a left and right Haar measureon Z(γ). Let dv be the canonical measure onZ(γ)\ G that is canonically associated with dgand dz so that dg=dzdv. [Orbital integral] For γ∈ G semisimple, we define the orbital integral associated with Q and γby^[γ][Q]=∫_Z(γ)\ G^E[q(v^-1γv)]dv, once the integral converges.Note that the mapZ(γ) \ G→𝒪_γ=_Gγgiven by v→ v^-1γ videntifies Z(γ)\ G as the orbit𝒪_γ of γ with the adjoint actionof G on G. This justifies the name “orbital integral"for (<ref>).Let Γ⊂ G be a discrete torsion freecocompact subgroup as in Section <ref>. Since the operator Q commutes with the left action of G, Q descends to an operator Q^Z acting on C^∞(Z,F).We assume that the sum∑_γ∈Γ q(g^-1γ g') is uniformly and absolutely convergent on G× G. Let [Γ] be the set of conjugacy classes in Γ.If [γ]∈ [Γ], set q^X,[γ](g,g')=∑_γ'∈ [γ] q(g^-1γ' g').Then from (<ref>)–(<ref>), we get q^Z(z,z')=∑_[γ]∈ [Γ] q^X,[γ](g,g'),with g,g'∈ G fixed liftof z,z'∈ Z. Thus as in (<ref>),[Q^Z]=∑_[γ]∈ [Γ][Q^Z,[γ]] with[Q^Z,[γ]] =∫_Z [q^X,[γ](z,z)]dz. From (<ref>), (<ref>), (<ref>), and the fact that[γ]≃Γ∩ Z(γ)\Γ,we have[Q^Z,[γ]] =∫_Γ∩ Z(γ) \ G^E[q(g^-1γ g)]dg=(Γ∩ Z(γ)\ Z(γ)) ^[γ][Q]=(Γ∩ Z(γ)\ X(γ)) ^[γ][Q].From (<ref>) and(<ref>),we get[Selberg trace formula][Q^Z]=∑_[γ]∈ [Γ](Γ∩ Z(γ)\ X(γ)) ^[γ][Q].Selberg <cit.>was the first to give a closed formula forthe trace of the heat operator on a compact hyperbolic Riemann surface via (<ref>),which is the original Selberg trace formula.Harish-Chandra's Plancherel theory, developed from the 1950s until the 1970s, isan algorithm to reduce the computation ofan orbital integral to a lower dimensional group by the discrete series method, cf. Section <ref>. To understand better the structure of each integral in (<ref>), we first reformulate it in more geometric terms. §.§ Geometric orbital integrals Let d(·,·) be the Riemannian distance on X. If γ∈ G, the displacement function d_γis given by for x∈ X,d_γ(x)=d(x,γ x).By <cit.>, the function d_γ isconvex on X, i.e., for any geodesic t∈→ x_t∈ X withconstant speed, the function d_γ(x_t) is convex on t∈. Recall that p: G→ X=G/K is the natural projection in (<ref>). We have the following geometricdescription on the semisimple elements in G.<cit.>. The elementγ∈ G is semisimple if and only ifthe function d_γattains its minimum in X. If γ∈ G is semisimple, and X(γ)={x∈ X: d_γ(x)=m_γ: =inf_y∈ Xd_γ(y)},for g∈ G, x=pg∈ X, then x∈ X(γ) if and only if there exist a∈,k∈ Ksuch that γ=(g)(e^ak^-1) and(k)a=a.If g_t=ge^ta,then t∈ [0,1]→ x_t=pg_t is the unique geodesicconnecting x∈ X(γ) and γ x in X.Moreover, we have m_γ=|a|.Since the integral (<ref>) depends only on the conjugacy class of γ, from Theorem <ref> or (<ref>), we may and we will assume that γ=e^ak^-1, (k)a=a, a∈, k∈ K. Furthermore,by (<ref>), we haveZ(γ)=Z(e^a)∩ Z(k),(γ) =(e^a)∩(k),where we use the symbol 𝔷 to denote thecorresponding Lie algebras of the centralizers.Put (γ)=(γ)∩,(γ)=(γ)∩. From (<ref>) and (<ref>), we get(γ)=(γ)⊕(γ).Thus the restriction of B to (γ) is non-degenerate. Let ^⊥(γ)be the orthogonal space to(γ) inwith respect to B.Then ^⊥(γ) splits as ^⊥(γ)=^⊥(γ)⊕^⊥(γ), where ^⊥(γ)⊂, ^⊥(γ)⊂ are the orthogonal spaces to(γ), (γ) in , with respect tothe scalar product induced by B.Set K(γ)=K∩ Z(γ),then from (<ref>) and (<ref>), (γ) isjust the Lie algebra of K(γ). <cit.>The set X(γ) is a submanifold of X. In the geodesic coordinate system centered at p1,we have the identificationX(γ)=(γ).The action of Z(γ) on X(γ) is transitive andwe have the identification of Z(γ)-manifolds,X(γ)≃ Z(γ)/K(γ). The map ρ_γ: (g,f,k')∈ Z(γ)×_K(γ) (^⊥(γ)× K)→ ge^fk'∈ G is a diffeomorphism of left Z(γ)-spaces, and of right K-spaces. The map (g,f,k')↦ (g,f) corresponds to the projectionp: G→ X=G/K. In particular,the mapρ_γ: (g,f)∈ Z(γ)×_K(γ)^⊥(γ)→ p(ge^f)∈ Xis a diffeomorphism.Moreover, under the diffeomorphism (<ref>),we have the identity of right K-spaces,^⊥(γ)_K(γ)× K = Z(γ)\ G.Finally, there exists C_γ>0 such that iff∈^(γ), |f|>1,d_γ(ρ_γ(1,f))≥ |a|+C_γ |f|. The map ρ_γ in (<ref>) is the normal coordinate system on X based at X(γ).Recall that dy is the volume element on X(γ)(cf. Section <ref>).Let df be the volume element on ^⊥(γ).Then dydf is a volume form onZ(γ)×_K(γ)^⊥(γ)that is Z(γ)-invariant.Let r(f) be the smooth functionon ^⊥(γ) that is K(γ)-invariantsuch that we have the identity of volume element on X via (<ref>),dx= r(f)dydf,withr(0)=1. In view of (<ref>), (<ref>), Bismut could reformulate geometrically the orbital integral (<ref>)as an integral along the normal direction of X(γ) in X. The orbital integral for the operator Q inSection <ref> and a semisimple element γ∈ Gis given by ^[γ][Q]=∫_^⊥(γ)^E[q(e^-fγ e^f)]r(f)df.Equation(<ref>)gives a geometric interpretationfor orbital integrals. It is remarkable that even before its explicit computation, the variational problem connected with the minimization of thedisplacement function d_γ is used in(<ref>).We need the following criterion forthe semisimplicity of an element.(Selberg <cit.>) If Γ⊂ G is a discrete cocompact subgroup,then for any γ∈Γ, γ is semisimple, and Γ∩ Z(γ) is cocompact in Z(γ). Let U be a compact subset of G such that G=Γ· U.Let γ∈Γ. Let {x_k}_k∈ be afamily of points in X such thatd(x_k,γ x_k)→ m_γ =inf_x∈ Xd(x,γ x) as k→ +∞. Then there exists γ_k∈Γ,x'_k∈ U such thatγ_k x'_k=x_k. Since U is compact,there is a subsequence{x'_k_j}_j∈ of {x'_k}_k∈ such thatas j→ +∞, x'_k_j→ y∈ U. Thend(y,γ^-1_k_jγγ_k_jy) ≤ d(x'_k_j,y) +d(x'_k_j, γ^-1_k_jγγ_k_jx'_k_j) +d(γ^-1_k_jγγ_k_jx'_k_j,γ^-1_k_jγγ_k_jy)=2d(x'_k_j,y)+d(x_k_j, γ x_k_j), where the right side tends to m_γas j→ +∞. Since Γ is discrete and eachγ^-1_k_jγγ_k_j∈Γ, the set of such γ^-1_k_jγγ_k_jis bounded, so that there exist infinitely many j such thatγ^-1_k_jγγ_k_j=γ'∈Γ.Then m_γ=d(y,γ'y) =d(γ_k_jy,γγ_k_jy).This means that d_γ reaches its minimum in X.Therefore γ is semisimple.Since Γ is discrete, [γ] is closed in G, thus Γ· Z(γ) as the inverse image of [γ] of the continuous map g∈ G→ g γ g^-1∈ G, is closed in G. This implies Γ∩ Z(γ)\ Z(γ) = Γ\Γ· Z(γ)is a closed subset of the compact quotient Γ\ G. ThusΓ∩ Z(γ) is cocompact in Z(γ).Let Γ⊂ G be a discrete torsion free cocompact subgroup as in Section <ref>. From (<ref>), (<ref>) and (<ref>),we get for γ∈Γ,[Q^Z,[γ]]=(Γ∩ Z(γ) \ X(γ)) ^[γ][Q].Combing (<ref>) and (<ref>), we can reformulateTheorem <ref> as [Selberg trace formula] [Q^Z]=∑_[γ]∈ [Γ](Γ∩ Z(γ)\ X(γ))^[γ][Q]. Set Z= Γ\ X, then Γ=π_1(Z). For x∈ X(γ),the unique geodesic from x to γ x descends tothe closed geodesic in Zin the homotopy class γ∈Γwhich has the shortest lengthm_γ.Thus the Selberg trace formula (<ref>)relates the trace of an operatorQ to the dynamicalproperties of the geodesic flow on Z viaorbital integrals. §.§ Bismut's explicit formula for orbital integralsBy the standard heat kernel estimate,for the heat operator e^-t_A^X on X, there exist c>0,λ, C>0, M>0 such that for anyt>0, x,x'∈ X, we have(cf. for instance <cit.>)|e^-t_A^X(x,x')|≤ C t^-M e^λt-c d^2(x,x') /t.Note also that by Rauch's comparison theorem, there exist C_0, C_1>0 such that for all f∈^(γ),|r(f)|≤ C_0 e^C_1 |f|.From (<ref>), (<ref>) and (<ref>),the orbital integral ^[γ][e^-t_A^X] is well-defined for any semisimple element γ∈ G.Let γ∈ G be the semisimple element asin (<ref>). Set_0=(a)∩, _0=(a)∩, _0=(a)=_0⊕_0.Let ^⊥_0 be the orthogonal space to _0 inwith respect to B. Let ^⊥_0(γ) be the orthogonal to (γ) in _0, and let ^⊥_0(γ) bethe orthogonal space to (γ) in _0.Then the orthogonal space to (γ) in _0 is^⊥_0(γ)=^⊥_0(γ)⊕^⊥_0(γ). For Y_0^𝔨∈(γ), we claim that (1-exp(-iθ(Y_0^)) (k^-1))|_^⊥_0(γ)(1-(k^-1))|_^⊥_0(γ)has a natural square root, which depends analytically on Y_0^.Indeed, (Y_0^) commutes with(k^-1), and no eigenvalue of (k)acting on ^⊥_0(γ) is equal to 1.If ^⊥_0(γ) is 1-dimensional, then(k)|__0^⊥(γ)=-1 and (Y_0^)|__0^⊥(γ)=0,the square root is just 2. If ^⊥_0(γ) is2-dimensional, if (k)|__0^⊥(γ)is a rotation of angle ϕ and θ(Y_0^)|__0^⊥(γ)acts by an infinitesimal rotation of angle ϕ', such a square root is given by (cf. <cit.>)4 sin(ϕ/2)sin(ϕ+iϕ'/2).If V is a finite dimensional Hermitian vector space and ifΘ∈End(V) is self-adjoint, then Θ/2sinh(Θ/2) is a self-adjoint positiveendomorphism.SetA(Θ)=det^1/2 [Θ/2/sinh(Θ/2) ].In (<ref>), the square root is taken to be the positive square root.For Y^_0∈(γ), setJ_γ(Y^_0) =1/| (1-(γ))|_^⊥_0|^1/2·A(i(Y^_0)|_(γ))/A(i(Y^_0)|_(γ))·[1/ (1-(k^-1)) |__0^⊥(γ)(1-exp(-i (Y_0^)) (k^-1))|_^⊥_0(γ)/(1-exp(-i(Y_0^)) (k^-1))|_^⊥_0(γ)]^1/2. From (<ref>), we know that (<ref>) is well-defined.Moreover, there exist c_γ,C_γ>0 such that for any Y^_0∈(γ)|J_γ(Y^_0)|≤ c_γe^C_γ|Y^_0|.We note that p=(γ),q=(γ)and r=(γ)=p+q. Now we can restate Theorem <ref> as follows.<cit.> For any t>0, we have^[γ][e^-t ^X_A]=e^-|a|^2/2t/(2π t)^p/2 ∫_(γ) J_γ(Y^_0)^E[ρ^E(k^-1)e^-i ρ^E(Y^_0)-tA]e^-|Y^_0|^2/2tdY^_0/(2π t)^q/2.For γ=1, we have (1)=, (1)=, and forY^_0∈, by (<ref>),J_1(Y^_0)=A(i(Y^_0)|_)/A(i(Y^_0)|_).Let 𝒮() be the Schwartz space of .Let ^[γ][cos(s√(_A^X))] be the evendistribution ondetermined by the condition that for any even function μ∈𝒮()with compactly supported Fourier transformation μ,we have^[γ][μ(√(^X_A))] =∫_μ(s)^[γ][cos(2π s√(_A^X))] ds.The wave operator cos( √(2)π s√(_A^X)) definesa distribution on × X× X. Let Δ^(γ) be the standard Laplacian on(γ) with respect to the scalar product⟨·,·⟩=-B(·,θ·). Now we can state the followingmicrolocal version of Theorem <ref> for the wave operator. <cit.>. We have the following identity of even distributionsonsupported on{|s|≥√(2)|a|} and with singular support in ±√(2)|a|,^[γ][cos(s√(_A^X))] =∫_H^γ^E[cos(s√(-1/2Δ^(γ)+A)) J_γ(Y^_0)ρ^E(k^-1)e^-iρ^E(Y_0^)], where H^γ={0}× (a,(γ))⊂(γ)×(γ). We assume that the semisimple element γ is nonelliptic,i.e., a≠ 0. We also assume that [(γ), _0]=0.Then for Y^_0∈(γ),ad(Y^_0)|_(γ)=0, ad(Y^_0)|__0^⊥(γ)=0.Now from (<ref>), we have<cit.>: for t>0,^[γ][e^-t^X_A]= e^-|a|^2/2t/|(1-Ad(γ)) |__0^⊥|^1/21/(1-Ad(k^-1))|__0^⊥(γ)1/(2π t)^p/2·^E[ ρ^E(k^-1) exp(-t (A+1/48^_0[C^_0,_0] +1/2C^_0,E))]. Note that if G is of real rank 1, then _0 is thevector subspace generated by a, so that (<ref>) holds.Thus (<ref>) recovers the result of Sally-Warner <cit.> where they assume that the real rank of G is 1.From (<ref>) and (<ref>), we obtaina refined version of the Selberg trace formulafor the Casimir operator :[e^-t^Z_A]=∑_[γ]∈ [Γ](Γ∩ Z(γ)\X(γ))^[γ][e^-t ^X_A],and each term ^[γ][·] isgiven by the closed formula (<ref>).We give two examples here to explain theexplicit version ofthe Selberg trace formula (<ref>). [Poisson summation formula] Take G= and A=0. Then K={0}.We have X= and ^X_A= - 12Δ^ =-12∂^2∂ x^2, where x is the coordinate on .Let p_t(x,x') be the heat kernel associated withe^tΔ^/2.For a∈, we have Z(a)=, (a)={0}. By (<ref>) or (<ref>), we have^[a][e^-t^X_A]= p_t(0,a).From (<ref>), we get^[a][e^-t^X_A]=1/√(2π t) e^-a ^2/2t.Thus(<ref>) gives simply an evaluationof the heat kernel onwhich is well-known thatp_t(x,x')=1/√(2π t)e^-(x-x')^2/2t.Take Γ=⊂, then Z=\=𝕊^1. For any γ∈Γ,X(γ)=Z(γ)/K(γ)=Z(γ)=.Thus Γ∩ Z(γ)\ Z(γ) =\=𝕊^1 and(𝕊^1)=1. The Selberg trace formula(<ref>) reducesto the Poisson summation formula: ∑_k∈ e^-2π^2 k^2 t=∑_k∈1/√(2π t)e^-k^2/2t for any t>0.Let G=SL_2() be the 2× 2 realspecial linear group with Lie algebra =𝔰𝔩_2().The Cartan involution is given byθ:G→ G,g↦^tg^-1.Then K=SO(2)={[cosβsinβ; -sinβcosβ ]: β∈}≃𝕊^1 is the corresponding maximal compact subgroup and X=G/K is the Poincaré upper half-planeℍ={z=x+iy∈ :y>0, x∈}.Precisely, an element g=[ a b; c d ]∈SL_2() acts on ℍ byg z=az+b/cz+d∈ℍ for z∈ℍ. The Cartan decomposition of 𝔰𝔩_2() is=⊕,whereis the set of real antisymmetric matrices,andis the set of traceless symmetric matrices. Let B be the bilinear form ondefined for u,v∈ byB(u,v)=2^^2[uv]. Set e_1=[1/20;0 -1/2 ],e_2=[ 0 1/2; 1/2 0 ],e_3=[01/2; -1/20 ].Then {e_1,e_2} is a basis of , and e_3 is a basisof . They together form an orthonormal basis ofthe Euclidean space(,⟨·,·⟩=-B(·,θ·)). Moreover,we have the relations,[e_1,e_2]=e_3,[e_2,e_3]=-e_1,[e_3,e_1]=-e_2. The metric on X is given by 1/y^2(dx^2+ dy^2). The scalar curvature of X is^[C^,]=-2|[e_1,e_2]|^2=-2.Let Δ^X be the Bochner Laplacian acting on C^∞(X,).Then Δ^X= y^2(∂^2/∂ x^2 + ∂^2/∂ y^2). Since ^[C^,]=0 here, we have onC^∞(X,),^X=1/2C^ +1/16^ [C^,] +1/48^[C^, ] =-1/2Δ^X-1/8. From (<ref>), we see that a semisimplenonelliptic element γ∈ G is hyperbolic.Thus such γ is conjugate to e^a e_1with some a ∈\{0}.Note that the orbital integral depends only onthe conjugacy class of γ in G If γ=e^ae_1 with a ∈\{0},then by (<ref>), (γ)=0, _0=(γ)= e_1, and we have(1-Ad(γ))|_^⊥_0 =-(e^a /2-e^-a /2)^2.From Theorem <ref>, (<ref>) and (<ref>),we get^[γ][e^tΔ^X/2] =1/√(2π t)exp(-a ^2/2t-t/8)/2sinh(|a |/2). For Y^_0=y_0e_3∈, the relations (<ref>) imply thatA(iad(Y^_0)|_) =y_0/2/sinh(y_0/2).From Theorem <ref>, (<ref>) and (<ref>),we get^[1][e^tΔ^X/2]= e^-t/8/2π t∫_ e^-y_0^2/2ty_0/2/sinh(y_0/2)dy_0/√(2π t).By taking the derivative with respect to y_0 in both sides of1/√(2π t)e^-y^2_0/2t =1/2π∫_ e^-tρ^2/2-iρ y_0dρ, we get1/√(2π t) e^-y^2_0/2ty_0/t =1/2π∫_ e^-tρ^2/2ρsin(ρ y_0)dρ.Thus1/ t∫_ e^-y_0^2/2ty_0/2/sinh(y_0/2)dy_0/√(2π t) =1/4π∫_ e^-tρ^2/2ρ( ∫_-∞^+∞sin(ρ y_0)/sinh(y_0/2)dy_0)dρ=1/2∫_ e^-tρ^2/2ρtanh(πρ)dρ,where we use the identity∫_-∞^+∞sin(ρ y_0)/sinh(y_0/2)dy_0 =2πtanh(πρ). Let Γ⊂SL_2() bea discrete torsion-freecocompact subgroup.Then Z=Γ\ X is a compact Riemann surface.We say that γ∈Γ is primitive if there does not exist β∈Γ and k∈, k≥ 2 such that γ=β^k.If γ=e^ae_1∈Γ is primitive,then |a | is the length of the corresponding closed geodesic in Z and for any k∈,k≠ 0,Z(γ^k)=Z(γ)= e^ e_1, and moreover,(Z(γ^k)∩Γ\ Z(γ^k)) =|a |. Thus by (<ref>), (<ref>), (<ref>),(<ref>) and (<ref>), we get[e^tΔ^Z/2]=∑_γ∈Γ primitive, [γ]=[e^ae_1], a ≠ 0|a |∑_k ∈, k ≠ 0^[e^k ae_1] [e^tΔ^X/2]+ (Z)^[1][e^tΔ^X/2]=∑_γ∈Γ primitive,[γ]=[e^ae_1], a ≠ 0 |a | ∑_k ∈, k ≠ 01/√(2π t)1/2sinh(k |a |/2) e^-k ^2a ^2/2t-t/8+ (Z)/4πe^-t/8∫_ e^-tρ^2/2ρtanh(πρ)dρ.Formula (<ref>) is exactly the original Selberg traceformula in <cit.>(cf. also <cit.>).§.§ Harish-Chandra's Plancherel TheoryIn this subsection, we briefly describe Harish-Chandra's approach to orbital integrals.This approach can be used to evaluate the orbital integrals of arbitrary test function,for sufficiently regular semisimple elements.This formulacontainscomplicated expressions involving infinite sums which do not converge absolutely, and have no obvious closed form exceptfor some special groups. An useful reference onHarish-Chandra's work on orbital integrals is Varadarajan's book <cit.>.Recall that G is a connected reductive group.Denote by G'⊂ G the space of regular elements. Let C^∞_c(G) be the vector space of smooth functions with compact support on G. For f∈ C^∞_c(G), attached to each θ-invariantCartan subgroupH of G, Harish-Chandra introduce a smooth function ^' F_f^H (cf. <cit.>),asan orbital integralof f in a certain sense, defined onH∩ G', which has reasonable limiting behavior onthe singular set in H. Let γ be a semisimple element such that (<ref>)holds. If γ is regular, then up to conjugation there existsa unique θ-invariant Cartan subgroup H which containsγ.In this case, ^' F_f^H(γ) is equal to a product of ^[γ][f] andan explicit Lefschetz likedenominator of γ. Now if γ is a singular semisimpleelement,let H be the unique (up to conjugation)θ-invariantCartan subgroup with maximal compactdimension, which contains γ.Following Harish-Chandra <cit.>, there is an explicit differential operatorD defined onH such that ^[γ][f]= lim_γ'∈ H∩ G'→γD ^' F_f^H(γ').Thus, to determine the orbital integral^[γ][f], it is enough tocalculate ^' F_f^H on the regular setH∩ G'.Take γ∈ H∩ G' a regular element in H.Harish-Chandra developed certain techniquesto calculate^' F_f^H, obtaining formulas which are known asFourier inverse formula.Indeed, f∈ C_c^∞(G)→^' F_f^H(γ) defines an invariant distribution on G. The idea is to write^'F_f^H(γ) as a combination of invariant eigendistributions (i.e.,a distribution on G which is invariant under the adjoint action ofG, and which is an eigenvector of the center of U()),like the global character of the discrete series representationsand the unitary principal series representations of G,as well as certain singular invariant eigendistributions.More precisely, let H=H_IH_R be Cartan decomposition ofH (cf. <cit.>), where H_I is a compactAbelian group and H_R is a vector space. Denote by H, H_I,H_R the set of irreducible unitary representationsof H,H_I,H_R.Then H= H_I ×H_R.Following <cit.>, fora^*=(a_I^*,a_R^*)∈H, we can associateaninvariant eigendistribution Θ^H_a^* on G.Note that if H is compact and if a^*_I is regular,then Θ^H_a^* is the globalcharacter ofthe discrete series representations of G, and that if H is noncompact and if a^*_I is regular, thenΘ^H_a^* is the globalcharacter ofthe unitary principal series representations of G.When a_I^* is singular, Θ^H_a^* is much morecomplicated. It is an alternating sum of some unitary characters,which in general arereducible.In <cit.>, Harish-Chandra announced the following Theorem. <cit.>. Let {H_1,⋯, H_l} be the complete set of non conjugatedθ-invariant Cartan subgroups of G.Then there exist computable continuous functions Φ_ijon H_i×H_j such that for any regular elementγ∈ H_i∩ G',^'F_f^H_i(γ)=∑_j=1^l∑_a^*_I∈H_jI∫_a^*_R∈H_jRΦ_ij(γ,a_I^*,a_R^*)Θ^H_j_a^*(f)da_R^*. In <cit.>, Harish-Chandra only explained the idea of a proof by induction on G.A more explicit version is obtained bySally-Warner <cit.> when G is of real rank one(cf. Remark <ref>), and by Herb <cit.>(cf. also Bouaziz <cit.>) for general G.However, Herb's formula only holds for γ in an opendense subset of H_i∩ G' and involvescertain infinite sumof integrals which converges, but cannot be directly differentiated,term by term. In particular, the orbital integral of singularsemisimple elements could not be obtained from Herb's formulaby applying term by term the differential operator D in (<ref>).When γ=1, much more is known: [Harish-Chandra <cit.>] There exists computable real analytic elementary functionsp^H_j(a^*) defined on H_j such that for f∈ C^∞_c(G),we have ^[1][f]=f(1)=∑_j=1^l∑_a^*_I∈H_jI,regular∫_a^*_R∈H_jRΘ^H_j_a^*(f)p^H_j(a_I^*,a_R^*)da_R^*. Theorem <ref> can be applied to more general functionssuch asHarish-Chandra Schwartz functions, e.g., the trace of the heat kernel q_t∈ C^∞(G,(E)) ofe^-tℒ_A^X.Thus,^[1][e^-tℒ_A^X] =∑_j=1^l∑_a^*_I ∈H_jI,regular∫_a^*_R∈H_jRΘ^H_j_a^*(^E[q_t]) p^H_j(a_I^*,a_R^*)da_R^*.For H_j, we can associate a cuspidal parabolic subgroup P_jwith Langlands decomposition P_j=M_jH_jRN_j such thatH_jI⊂ M_j is a compact Cartan subgroup of M_j. For a^*=(a^*_I, a^*_R )∈H_j with a^*_I regular,denote by (ς_a^*_I,V_a^*_I) the discrete seriesrepresentations of M_j associated to a^*_I, and denote by (π_a^*,V_a^*) the associated principal seriesrepresentations of G associated to ς_a^*_I anda^*_R. We have Θ^H_j_a^*(^E[q_t]) =^V_a^*⊗ E[π_a^*(q_t)] with π_a^*(q_t)= ∫_G q_t(g) π_a^*(g) dg. It is not difficult to see thatthe image of the operatorπ_a^*(q_t) is(V_a^*⊗ E)^K ≃ (V_a^*_I⊗ E)^K∩ M_j, andπ_a^*(q_t) acts ase^-t(1/2C^,π_a^* +1/16^[C^,]+1/48^[C^, ]+A) on its image. We get ^V_a^*⊗ E[π_a^*(q_t)] = e^-t(1/2C^,π_a^* +1/16^[C^,]+1/48^[C^, ])^(V_a^*_I⊗ E)^K∩ M_j[e^-tA].Thus,^[1][e^-tℒ_A^X] =∑_j=1^l∑_a^*_I ∈H_jI,regular∫_a^*_R∈H_jRe^-t(1/2C^,π_a^* +1/16^[C^,]+1/48^[C^, ]) ^(V_a^*_I⊗ E)^K∩ M_j [e^-tA]p^H_j(a^*_I,a_R^*)da_R^*.Equation (<ref>) is not as explicitas (<ref>),because in general it is not easy to determine all parabolic subgroups,all the discrete series of M, andthe Plancherel densitiesp^H_j(a^*).We hope that from these descriptions, the readers got an idea onHarish-Chandra's Plancherel theory as an algorithm to compute orbital integrals.These resultsuse the full forceof the unitary representation theory (harmonic analysis) of reductive Lie groups, both at the technicaland the representation level. Bismut's explicit formula of the orbital integrals associated with the Casimir operator gives a closedformula in full generality for any semisimple element and any reductive Lie group.Bismut avoided completely the use of the harmonic analysis on reductive Lie groups. The hypoelliptic deformation allows him to localize the orbital integral for γ to any neighborhood of the family of shortest geodesicassociated with γ, i.e., X(γ).There is a mysterious connection between Harish-Chandra's Plancherel theoryand Theorem <ref>:in Harish-Chandra's Plancherel theory, the integral are takenon thepart, but in Theorem <ref>, the integral is on thepart. In particular, in Example <ref> for G= _2(), we obtain the contribution ∫_e^-tρ^2/2ρtanh (πρ) dρ from the Plancherel theory for γ=1. This coincide with (<ref>) by using a Fourier transformation argument as explained in(<ref>).Assume G=SO^0(m,1) with m odd.There exists only one Cartan subgroup H, and p^H(a^*_I,·) is an explicit polynomial.In this case, (<ref>) becomes completely explicit. § GEOMETRIC HYPOELLIPTIC OPERATOR ANDDYNAMICALSYSTEMS In this section, we explain how to construct geometrically thehypoelliptic Laplacians for a symmetric space,with the goal to proveTheorem <ref> in the spirit of the heat kernel proof of theLefschetz fixed-point formula (cf. Section <ref>). We introduce a hypoelliptic version of the orbital integral that depends on b.The analogue of the methods of local index theory are needed to evaluate the limit.Theorem <ref> identifies the orbital integral associatedwith the Casimir operator to the hypoelliptic orbital integral for the parameter b>0. As b→ +∞, the hypoelliptic orbital integral localizes near X(γ).This section is organized as follows. In Section <ref>,we explain how to computethe cohomology of a vector space by using algebraic de Rham complex and its Bargmann transformation, whose Hodge Laplacian is a harmonic oscillator. In Section <ref>,we recall the construction of the Dirac operator of Kostant, and in Sections <ref>, we construct the geometric hypoelliptic Laplacianby combining the constructions in Sections <ref> and <ref>. In Section <ref>, we introduce the hypoelliptic orbital integralsand a hypoelliptic version of the McKean-Singer formula for these orbital integrals. In Section <ref>, we describe the limitof the hypoelliptic orbital integrals as b→ +∞. Finally, in Section <ref>, we explainsome relations of the hypoelliptic heat equationto the wave equation on the base manifold, whichplays an important role in the proof of uniform Gaussian-like estimates for the hypoelliptic heat kernel.§.§ Cohomology of a vector space and harmonic oscillator Let V be a real vector space of dimension n, and let V^* be its dual. Let Y be the tautological section of Vover V. Then Y can be identified with the corresponding radialvector field. Let d^V denote the de Rham operator. Let L_Y be the Lie derivative associated with Y, and let i_Y be the contraction of Y. By Cartan's formula, we have the identityL_Y=[d^V, i_Y].Let S^∙(V^*)=⊕_j=0^∞ S^j(V^*) be the symmetric algebra of V^*, which can be canonically identifiedwith the polynomial algebra of V.Then Λ^∙(V^*)⊗ S^∙(V^*) is the vector spaceof polynomial forms on V. Let N^S^∙(V^*), N^Λ^∙(V^*) be the number operators onS^∙(V^*), Λ^∙(V^*),which act by multiplication byk on S^k(V^*), Λ^k(V^*). ThenL_Y|_Λ^∙(V^*)⊗ S^∙(V^*) =N^S^∙(V^*)+N^Λ^∙(V^*).By (<ref>) and(<ref>),the cohomology ofthe polynomial forms(Λ^∙(V^*)⊗ S^∙(V^*), d^V) on V is equal to1.Assume that V is equipped with a scalar product. ThenΛ^∙(V^*), S^∙(V^*) inheritassociated scalar products. For instance, if V=, then 1^⊗ j^2=j!.With respect to this scalar product onΛ^∙(V^*)⊗ S^∙(V^*), i_Y is the adjoint of d^V. Therefore L_Y is the associated Hodge Laplacian on Λ^∙(V^*)⊗ S^∙(V^*). Remarkably enough, it does not depend on g^V.By (<ref>), we get(L_Y)= 1.We have given a Hodge theoretic interpretation to the proof that the cohomology of the complex of polynomial formsis concentrated in degree 0.Let Δ^V denote the (negative) Laplacian on V.Let L_2(V) be the corresponding Hilbert space ofsquare integrable real-valued functions on V.Let T: S^∙(V^*)→ L_2(V) be the map such that given P∈ S^∙(V^*), then(TP)(Y)=π^-n/4 e^-Y^2/2(e^-Δ^V/2P) (√(2)Y). Since P is a polynomial, e^-Δ^V/2Pis defined by taking the obvious formal expansion ofe^-Δ^V/2. Its inverse, the Bargmann kernel, is given by (Bf)(Y)=π^n/4 e^Δ^V/2( e^Y^2/4 f(Y/√(2))).Here the operator e^Δ^V/2 is defined viathe standard heat kernel of V.Setd=T d^V B,d^*=T i_Y B: Λ^∙(V^*)⊗ L_2(V)→Λ^∙(V^*)⊗ L_2(V).Then by (<ref>) and (<ref>), we getd=1/√(2)(d^V+Y^*∧) ,d^*=1/√(2)(d^V*+i_Y).Here Y^* is the metric dual of Y in V^*,and d^V* is the usual formal L_2 adjoint of d^V.Let {e_j} be an orthonormal basis of V and let{e^j} be its dual basis. For U∈ V, let ∇_U be the usual differential along the vector U.Put Y=∑^n_j=1 Y_j e_j, thend^V=∑^n_j=1 e^j∧∇_e_j,d^V*=-∑^n_j=1 i_e_j∇_e_j;Y^*∧=∑^n_j=1Y_j e^j∧,i_Y=∑^n_j=1 Y_j i_e_j.From (<ref>) and (<ref>), we getd^2=(d^*)^2=0, T L_Y T^-1 = [d,d^*] =1/2(-Δ^V+Y^2-n) +N^Λ^∙(V^*).Note that 1/2(-Δ^V+Y^2-n)is the harmonic oscillator on V already appeared in (<ref>).In (<ref>), we saw that the kernel of [d^V,i_Y] in Λ^∙(V^*)⊗ S^∙(V^*) is generated by1 and so it is 1-dimensional and is concentratedin total degree 0. Equivalently the kernel ofthe unbounded operator [d,d^*]acting on Λ^∙(V^*)⊗ L_2(V)is 1-dimensional and is generated by the functione^-Y^2/2/π^n/4.§.§ The Dirac operator of Kostant Let V be a finite dimensional real vector space of dimension nand let B be a real valued symmetric bilinear form on V.Let c(V) be the Clifford algebra associated to (V,B). Namely,c(V) is the algebra generated overby 1,u∈ V and the commutation relations for u,v∈ V,uv+vu=-2B(u,v).We denote by c(V) the Clifford algebra associated to-B. Then c(V),c(V) are filtered by length,and their corresponding Gr^· is justΛ^∙ (V). Alsothey are _2-graded by length.In the sequel, we assume that B is nondegenerate.Let φ: V→ V^* be the isomorphism such thatif u,v∈ V,(φ u,v)=B(u,v). If u∈ V, let c(u),c(u) act onΛ^∙(V^*) by c(u)=φ u∧ - i_u,c(u)=φ u∧+ i_u.Here i_u is the contraction operator by u. Using supercommutators as in (<ref>),from (<ref>), we find that for u,v∈ V,[c(u),c(v)]=-2B(u,v), [c(u),c(v)]=2B(u,v),[c(u),c(v)]=0.Equation (<ref>) shows that c(·), c(·) are representations of the Clifford algebrasc(V), c(V) on Λ^∙(V^*).We will apply now the above constructions to the vector space(, B) of Section <ref>.If {e_i}^m+n_i=1 is a basis of , we denote by {e^*_i}^m+n_i=1 its dual basis ofwith respectto B (i.e., B(e_i,e_j^*)=δ_ij),and by {e^i}^m+n_i=1 the dual basis of^*.Let κ^∈Λ^3(^*) be such that ifa,b,c∈,κ^(a,b,c)=B([a,b],c).Let c(κ^)∈c(V) correspond toκ^∈Λ^3(^*) defined byc(κ^)=1/6κ^(e_i^*,e_j^*,e_k^*)c(e_i) c(e_j) c(e_k).Let D^∈c()⊗ U() be theDirac operatorD^=∑^m+n_i=1c(e_i^*) e_i -1/2c(κ^). Note that c(κ^), D^ areG-invariant. The operator D^ acts naturally on C^∞(G,Λ^∙(^*)).[Kostant formula, <cit.>, <cit.>] D^,2=-C^-1/8^[C^,]- 1/24^[C^,]. §.§ Construction of geometric hypoelliptic operators The operator D^ acts naturally onC^∞(G,Λ^∙(^*)) and also on C^∞(G,Λ^∙(^*)⊗ S^∙(^*)).As we saw in Section <ref>,from a cohomological point of view,Λ^∙(^*)⊗ S^∙(^*)≃. This is how ultimately C^∞(G,) (andC^∞(X,))will reappear. We denote by Δ^⊕ the standard EuclideanLaplacian on the Euclidean vector space =⊕.If Y∈, we split Y in the form Y= Y^ +Y^ withY^∈, Y^∈.If U∈, we use the notationc(ad(U)) =-1/4B([U,e^*_i],e^*_j) c(e_i) c(e_j), c(ad(U)) =1/4B([U,e^*_i],e^*_j) c(e_i) c(e_j).Here is theoperator 𝔇_b appeared in <cit.> which acts on C^∞(G,Λ^∙(^*)⊗ S^∙(^*) ) “≃” C^∞(G×,Λ^∙(^*)). Set𝔇_b=D^+ic([Y^,Y^]) +√(2)/b(d^ -id^+d^*+id^*). The introduction of i in the third term in the right-hand sideof (<ref>) is made so that its principal symbolanticommutes with the principal symbol of D^. Let {e_j}_j=1^m be an orthonormal basis of , and let{e_j}_j=m+1^m+n be an orthonormal basis of . If U∈, ad(U)|_ acts as an antisymmetricendomorphism ofand by (<ref>), we havec(ad(U)|_)=1/4∑_1≤ i,j≤ m⟨ [U,e_i],e_j⟩ c(e_i)c(e_j).Finally, if v∈, ad(v) exchangesand and is antisymmetric with respect to B, i.e., it is symmetricwith respect to the scalar product on .Moreover, by (<ref>)c(ad(v))= - 1/2∑_m+1≤ i≤ m+n 1≤ j≤ m⟨ [v,e_i], e_j⟩ c(e_i)c(e_j). If v∈, we denote by ∇^V_v the correspondingdifferential operator along . In particular, ∇^V_[Y^,Y^] denotes the differentiation operator in the direction [Y^,Y^]∈. If Y∈, we denote byY^+ iY^the section of U()⊗_ associated with Y^+i Y^∈⊗_.<cit.> The following identity holds:𝔇^2_b/2=D^,2/2 +1/2|[Y^,Y^]|^2 +1/2b^2(-Δ^⊕+Y^2-m-n) +N^Λ^∙(^*)/b^2+1/b( Y^ +iY^-i∇^V_[Y^,Y^]+c(ad(Y^+iY^))+2ic(ad(Y^)|_) -c(ad(Y^))). By (<ref>),G×_K=TX⊕ N, withN=G×_K. Let 𝒳 be the total space of TX⊕ Nover X, and letπ:𝒳→ X bethe natural projection.Let Y=Y^TX+Y^N, Y^TX∈ TX, Y^N∈ N be the canonicalsections of π^*(TX⊕ N), π^*(TX),π^*(N) over 𝒳. Note that the natural action of K onC^∞(,Λ^∙(^*)⊗ E) is given by (k·ϕ)(Y)=ρ^Λ^∙(^*)⊗ E(k) ϕ(Ad(k^-1)Y), for ϕ∈ C^∞(,Λ^∙(^*)⊗ E).Therefore S^∙(T^*X⊕ N^*) ⊗Λ^∙(T^*X⊕ N^*)⊗ F = G×_K (S^∙(^*)⊗Λ^∙(^*)⊗ E),and the bundleG×_K C^∞(,Λ^∙(^*)⊗ E) over X is justC^∞(TX⊕ N,π^* (Λ^∙(T^*X⊕ N^*)⊗ F)).By (<ref>), the K action on C^∞(G×,Λ^∙(^*)⊗ E) is givenby (k· s) (g,Y)=ρ^Λ^∙(^*⊕^*)⊗ E(k) s(gk,(k^-1)Y).If a vector space W is a K-representation, we denote by W^K its K-invariant subspace. Then C^∞(G,S^∙(^*)⊗Λ^∙(^*)⊗ E)^K= C^∞(X, S^∙(T^*X⊕ N^*) ⊗Λ^∙(T^*X⊕ N^*)⊗ F)“≃”C^∞(X, C^∞(TX⊕ N,π^* (Λ^∙(T^*X⊕ N^*)⊗ F)))= C^∞(𝒳, π^* (Λ^∙(T^*X⊕ N^*)⊗ F)).As we saw in Section <ref>, the connection form ω^ on K-principal bundle p: G→ X=G/K alsoinduces a connection on C^∞(TX⊕ N,π^* (Λ^∙(T^*X⊕ N^*)⊗ F)) over X, which is denoted by ∇^C^∞(TX⊕ N,π^* (Λ^∙(T^*X⊕ N^*)⊗ F)). In particular, for the canonical section Y^TX of π^*(TX) over 𝒳,the covariant differentiation with respect to the givencanonical connection in the horizontal direction corresponding toY^TX is ∇_Y^TX^C^∞(TX⊕ N,π^* (Λ^∙(T^*X⊕ N^*)⊗ F)). Since the operator 𝔇_b is K-invariant, by (<ref>), itdescends to an operator 𝔇^X_b acting onC^∞(𝒳, π^* (Λ^∙(T^*X⊕ N^*)⊗ F)).It is the same for the operator D^,which descends to anoperator D^,X over X.Recall that A is the self-adjoint K-invariant endomorphism of E in Section <ref>. For b>0, let ^X_b, ^X_A,b act onC^∞(𝒳, π^* (Λ^∙(T^*X⊕ N^*)⊗ F)) by _b^X=-1/2D^,X,2 +1/2𝔇_b^X,2,^X_A,b= ^X_b +A. Let ⟨·,·⟩ be theusual L_2 Hermitian producton the vector space of smooth compactly supported sections ofπ^*(Λ^∙(T^*X⊕ N^*)⊗ F) over𝒳. Setα= 1/2(-Δ^TX⊕ N +Y^2-m-n)+N^Λ^∙(T^*X⊕ N^*), β= ∇^C^∞(TX⊕ N,π^* (Λ^∙(T^*X⊕ N^*)⊗ F))_Y^TX + c(ad(Y^TX))-c(ad(Y^TX)+iθad(Y^N)) -iρ^E(Y^N), ϑ = 1/2|[Y^N,Y^TX]|^2.<cit.> We have_b^X=α/b^2+β/b+ϑ.The operator ∂/∂ t+^X_b ishypoelliptic. Also 1b∇^C^∞(TX⊕ N,π^* (Λ^∙(T^*X⊕ N^*)⊗ F))_Y^TXis formally skew-adjoint with respect to⟨·,·⟩ and ^X_b-1b∇^C^∞(TX⊕ N,π^* (Λ^∙(T^*X⊕ N^*)⊗ F))_Y^TXis formally self-adjoint with respect to ⟨·,·⟩.We will now explain the presence of the termic([Y^,Y^]) in the right-hand side of (<ref>). Instead of 𝔇_b, we could consider the operatorD_b= D^ + 1/b(d^ -id^+d^*+i d^ *).From (<ref>), (<ref>), (<ref>)and (<ref>), we getD^2_b=D^,2 + 1/2b^2(-Δ^⊕)+√(2)/b (Y^+ iY^)+ zero orderterms. If e∈, let ∇_e,l be the differentiation operatorwith respect to the left invariant vector field e,by (<ref>), fors∈ C^∞(G, C^∞(,Λ^∙(^*)⊗ E))^K,∇_e,ls= (L^V_[e,Y]-ρ^E(e))s.Here [e,Y] is a Killing vector field on =⊕and the corresponding Lie derivative L^V_[e,Y]acts on C^∞(,Λ^∙(^*)). By <cit.>, we have the formula L^V_[e,Y]=∇^V_[e,Y]-(c+c)(ad(e)).When we use the identification (<ref>), the operator iY^ contributes the first order differential operator i∇^V_[Y^N,Y^TX] along TX. This term is very difficult to control analytically.The miraculous fact is that after addingic([Y^,Y^]) to D_b, in theoperator ^X_b, we have eliminated i∇^V_[Y^N,Y^TX] and we add instead the termϑ = 1/2[Y^N,Y^TX]^2, which is nonnegative.This ensures that the operatorα/b^2+ϑ isbounded below. The operator ^X_b is a nice operator. <cit.> We have the identity[𝔇^X_b, ℒ^X_A,b]=0. The classical Bianchi identity say that [𝔇^X_b,𝔇^X,2_b]=0.By (<ref>), D^,X,2 is the Casimir operator (up to a constant), so that[𝔇^X_b,D^,X,2]=0. We have the trivial [𝔇^X_b, A]=0. From (<ref>), (<ref>) and (<ref>), we get (<ref>).By analogy with (<ref>), we will need to showthat as b→ 0, in a certain sense,e^-t ℒ^X_A,b→ e^-t ℒ^X_A.We explain here an algebraic argumentwhich gives evidence for (<ref>).This will bethe analogue of (<ref>).We denote by H the fiberwise kernel of α,so that H=e^-|Y|^2/2⊗ F.Let H^ be the orthogonal to H inL_2(𝒳, π^*(Λ^∙(T^*X⊕ N^*)⊗ F)). Note that β maps H to H^. Let α^-1 be the inverse of α restricted to H^.Let P, P^ be the orthogonal projections onH and H^ respectively. We embed L_2(X,F) intoL_2(𝒳, π^*(Λ^∙(T^*X⊕ N^*)⊗ F)) isometrically vias→π^*s e^-|Y|^2/2/π^(m+n)/4. <cit.>The following identify holds:P(ϑ-βα^-1β)P=^X. From (<ref>), we can write1/√(2)𝔇^X_b =E_1+F_1/b, with E_1 =1/√(2)(D^,X +ic([Y^N, Y^TX])). Then comparing (<ref>), (<ref>) and (<ref>), we getα=F_1^2,β=[E_1,F_1],ϑ =E_1^2-1/2D^,X,2. Since H is the kernel of F_1, we havePF_1=F_1P=0.We obtain thus P(ϑ-βα^-1β)P=P(E_1^2-E_1P^ E_1 -1/2D^,X,2)P=(PE_1P)^2-1/2PD^,X,2P. But H is of degree 0 in Λ^∙(^*), D^+ic([Y^, Y^]) is of odd degree, we know that PE_1P=0. Thus, (<ref>) holds. §.§ Hypoelliptic orbital integralsUnder the formalism of Section <ref>, we replace nowthe finite dimensional vector space E by the infinite dimensionalvector spaceℰ=Λ^∙(^*⊕^*)⊗ S^∙(^*⊕^*)⊗ E.Using (<ref>), from now on, wewill work systematically on C^∞(𝒳, π^* (Λ^∙(T^*X⊕ N^*)⊗ F)).Let dY be the volume element of =⊕with respect to the scalar product⟨· ,·⟩=-B(·,θ·). It defines a fiberwise volume elementon the fiberTX⊕ N, which we still denote by dY.Our kernel q(g) now acts as an endomorphism ofℰ and verifies (<ref>) and (<ref>). In what follows, the operator q(g) is given by continuous kernels q(g,Y,Y'), Y,Y'∈.Let q((x,Y),(x',Y')), (x,Y),(x',Y')∈𝒳 be the corresponding kernel on 𝒳. <cit.> Fora semisimple element γ∈ G, we define_s^[γ][Q] as in (<ref>), _s^[γ][Q]=∫_^(γ)×_s^Λ^∙(^*⊕^*)⊗ E[q(e^-fγ e^f,Y,Y)]r(f)dfdYonce it is well-defined. Note here_s^Λ^∙(^*⊕^*)⊗ E[·] =^Λ^∙(^*⊕^*)⊗ E [(-1)^N^Λ^∙(^*⊕^*)·], i.e., we use the natural ℤ_2-grading onΛ^∙(^*⊕^*).Let P be the projection from Λ^∙(T^*X⊕ N^*)⊗ F on Λ^0(T^*X⊕ N^*)⊗ F.Recall that e^-t^X_A(x,x') is the heat kernel of ^X_A in Section <ref>. For t>0, (x,Y), (x', Y')∈𝒳, put q^X_0,t((x,Y), (x', Y'))= P e^-t^X_A(x,x')π^-(m+n)/2 e^-1/2(|Y|^2+|Y'|^2) P. Let e^-tℒ_A,b^X be the heat operator ofℒ_A,b^X andq^X_b,t((x,Y), (x', Y')) be the kernel of theheat operator e^-tℒ_A,b^Xassociated with the volume form dx' dY'. In <cit.>, Bismut studied in detailthe smoothness of q^X_b,t((x,Y), (x', Y')) for t>0, b>0, (x,Y), (x', Y')∈𝒳. In particular,he showed thatit is rapidly decreasing in the variables Y,Y'.Now we state an important result<cit.> whose proof was given in<cit.> where Theorem <ref>plays an important role. It ensures that the hypoelliptic orbital integral is well-defined fore^-tℒ_A,b^X and that the analogue ofTheorem <ref> holds for h= e^-tℒ^X_Aand 𝔇_b^X. Given 0<ϵ≤ M, there exist C,C'>0 such that for 0<b≤ M, ϵ≤ t ≤ M, (x,Y), (x', Y')∈𝒳, |q^X_b,t((x,Y), (x', Y'))|≤ Cexp(-C' (d^2(x,x')+|Y|^2+|Y'|^2)). Moreover, as b→ 0,q^X_b,t((x,Y), (x', Y'))→ q^X_0,t((x,Y), (x', Y')).The formal analogueof Theorem <ref> is as follows.<cit.>For any b>0, t>0, we have^[γ][e^-tℒ_A^X]=_s^[γ][e^-tℒ_A,b^X] .In <cit.>, Bismut showedthatthe hypoelliptic orbital integral (<ref>)is a trace on certain algebras of operators given by smooth kernels which exhibit a Gaussian decay like in (<ref>). By Theorem <ref>,the kernel function q^X_b,t is in this algebra.As in (<ref>),by Proposition <ref>,∂/∂ b_s^[γ][e^-tℒ_A,b^X] = _s^[γ][-t (∂/∂ bℒ_A,b^X) e^-tℒ_A,b^X] = -t _s^[γ][ 1/2[𝔇^X_b, ∂/∂ b𝔇^X_b] e^-tℒ_A,b^X] =-t/2_s^[γ][ [𝔇^X_b, (∂/∂ b𝔇^X_b) e^-tℒ_A,b^X]] =0.By Theorem <ref>, we havelim_b→ 0_s^[γ][ e^-tℒ_A,b^X]=^[γ][e^-tℒ_A^X].From(<ref>), (<ref>), we get (<ref>).§.§ Proof of Theorem <ref>For b>0, s(x,Y)∈ C^∞(𝒳, π^*(Λ^∙(T^*X⊕ N^*)⊗ F)),set F_bs(x,Y)=s(x,-bY).Put ℒ^X_A,b=F_bℒ_A,b^XF_b^-1.Let q^X_b,t((x,Y),(x',Y')) be the kernel associated withe^-tℒ_A,b^X. When t=1,we will write q^X_b instead of q^X_b,1.Then from (<ref>), we haveq^X_b,t((x,Y),(x',Y')) =(-b)^m+n q^X_b,t((x,-bY),(x',-bY')).Let a^TX be the vector field on X associated witha in (<ref>) induced by the left action of Gon X (cf. (<ref>)). Let d(·,X(γ))be the distance function to X(γ). <cit.>Given 0<ϵ≤ M, there exist C,C'>0, such that for any b≥ 1, ϵ≤ t≤ M, (x,Y),(x',Y')∈𝒳,|q^X_b,t((x,Y),(x',Y'))|≤ Cb^4m+2nexp(-C(d^2(x,x')+|Y|^2+|Y'|^2)). Given δ>1, β>0, 0<ϵ≤ M, there existC,C'>0, such that for any b≥ 1, ϵ≤ t≤ M, (x,Y)∈𝒳, if d(x,X(γ))≥β, |q^X_b,t((x,Y),γ(x,Y))|≤ Cb^-δexp(-C'(d^2_γ(x)+|Y|^2)). Given δ> 1, β>0, μ>0, there exist C,C'>0 such that for any b≥ 1, (x,Y)∈𝒳,if d(x,X(γ))≤β, and |Y^TX-a^TX(x)|≥μ,|q^X_b,t((x,Y),γ(x,Y))|≤ Cb^-δe^-C'|Y|^2. In view of Theorem <ref>, the proof of Theorem <ref>consists in obtainingthe asymptotics of_s^[γ][e^-tℒ^X_A,b]as b→ +∞. By <cit.>,the operator in (<ref>) associated with B/t isup to conjugation, tℒ^X_√(t)b. Observe thatJ_γ(Y^_0) is unchanged when replacing the bilinear form B by B/t, t>0. Thus we only need toestablish the corresponding result for t=1. When f∈^(γ),we identify e^f with e^fp1. For f∈^(γ),Y∈ (TX⊕ N)_e^f, set Q^X_b(e^f,Y)= _s^Λ^∙(T^*X⊕ N^*)⊗ F[γq_b^X((e^f,Y),γ(e^f,Y))].Then _s^[γ][e^-ℒ^X_A,b] =∫_(e^f,Y)∈π^-1^(γ)Q^X_b(e^f,Y)r(f)dfdY.Take β∈]0,1]. By Theorem <ref>, as b→ +∞,∫_(e^f,Y)∈π^-1^(γ), |f| ≥βQ^X_b(e^f,Y)r(f)dfdY→ 0,∫_(e^f,Y)∈π^-1^(γ),|f|< β,|Y^TX-a^TX(e^f)|≥μQ^X_b(e^f,Y) r(f)dfdY→ 0.We need to understand the integral on the domain|f|<β, |Y^TX-a^TX(e^f)|<μ, when b→ +∞. Let π: 𝒳→ X be the total space ofthe tangent bundle TX to X. Let φ_t|_t∈ bethe group of diffeomorphisms of 𝒳 induced bythe geodesic flow. By <cit.>,φ_1(x, Y^TX)= γ·(x, Y^TX)is equivalent tox∈ X(γ) and Y^TX= a^TX(x). Equation (<ref>) shows that as b→ +∞, the right-hand side of (<ref>) localizes nearthe minimizing geodesic x_t connecting x and γ x so that ẋ = a^TX.Let N(γ) be the vector bundle on X(γ) which is the analogue of the vector bundleN on X in (<ref>).Then N(γ)⊂ N|_X(γ).Let N^(γ) be the orthogonal to N(γ) inN|_X(γ). Clearly,N^(γ)=Z(γ)×_K(γ)^(γ).Let p_γ: X→ X(γ) be the projectiondefined by (<ref>) and (<ref>). We trivialize the vector bundles TX,N by parallel transport along the geodesics orthogonal to X(γ)with respect to the connection ∇^TX, ∇^N,so that TX, N can be identified withp^*_γ TX|_X(γ), p^*_γ N|_X(γ).At x=p1, we haveN(γ)=(γ),N^(γ) =^(γ).Therefore at ρ_γ(1,f), we may write Y^N∈ N in the form Y^N=Y_0^+Y^N,, withY_0^∈(γ),Y^N,∈^(γ).Let dY_0^, dY^N, be the volume elements on(γ), ^(γ), so that dY^N=dY_0^dY^N,.To evaluate the limit of (<ref>) as b→ +∞ for β>0, we may by (<ref>), as well consider the integral∫_|f|< β,|Y^TX-a^TX(e^f)|< μQ^X_b(e^f,Y)r(f)dfdY^TXdY_0^ dY^N,=b^-4m-2n+2r∫_|f|< b^2β,|Y^TX|< b^2μQ^X_b(e^f/b^2,Y^TX/b^2+a^TX(e^f/b^2),Y_0^+Y^N,/b^2) r(f/b^2)dfdY^TXdY_0^ dY^N,. Let (γ) be the another copy of (γ),and let (γ)^* be the corresponding copyof the dual of (γ). Also, for u∈(γ)^*,we denote by u the corresponding element in(γ)^*. Let e_1,⋯,e_r be a basisof (γ), let e^1,⋯,e^r bethe corresponding dual basis of (γ)^*. Put =(Λ^∙(^*))⊗Λ^∙((γ)^*).Let e_r+1,⋯,e_m+n be a basis of^(γ), and lete^*_r+1,⋯,e^*_m+n be the dual basis toe_r+1,⋯,e_m+nwith respect toB|_^(γ).Thenis generated by all the monomials in c(e_i),c(e_i), 1≤ i≤ m+n, e^j, 1≤ j≤ r.Let _s be the linear map fromintothat, up to permutation, vanishes on all monomials exceptthose of the following form:_s[c(e_1)e^1⋯ c(e_r)e^rc(e^*_r+1)c(e_r+1)⋯ c(e^*_m+n) c(e_m+n)]=(-1)^r(-2)^m+n-r.For u∈, v∈(E), we define _s[uv]=_s[u]^E[v].Set α=∑_i=1^r c(e_i)e^i∈ c((γ)) ⊗Λ^∙((γ)^*).Let _b^X((x,Y),(x',Y')) denote the smooth kernel associated with e^-ℒ_A,b^X-α, andQ_b^X(x,Y)=γ_b^X((x,Y),γ(x,Y)). Since _A,b^X+α can be obtained from _A,b^X by a conjugation, bya simple argument on Clifford algebras, we get:<cit.>. For b>0, the following identity holds: Q^X_b(x,Y)=b^-2r_s[Q_b^X(x,Y)].Now we define a limit operator acting onC^∞(×,Λ^∙(^*) ⊗Λ^∙((γ)^*) ⊗ E). We denote by y the tautologicalsection of the first component ofin ×,and by Y=Y^+Y^ the tautological sectionof =⊕. Let dy the volume form on and let dY the volume form on =⊕. Given Y_0^∈(γ), set_a,Y^_0=1/2|[Y^,a]+[Y_0^,Y^]|^2-1/2Δ^⊕+ ∑_i=1^r c(e_i)e^i-∇^H_Y^-∇^V_[a+Y_0^,[a,y]] -c((a))+c((a)+iθ(Y_0^)) acting on C^∞(×,Λ^∙(^*)⊗Λ^∙((γ)^*)⊗ E). Let R_Y_0^((y,Y),(y',Y^'))be the smooth kernel of e^-_a,Y^_0with respect to the volume form dydY on ×.Then R_Y_0^((y,Y),(y',Y^'))∈(Λ^∙(^(γ)^*)) ⊗c((γ))⊗Λ^∙((γ)^*). The following result gives an estimate and pointwise asymptoticsof Q_b^X. <cit.> Given β>0, there exist C,C_γ'>0 such that for b≥1, f∈^(γ), |f|≤β b^2, and |Y^TX|≤β b^2,b^-4m-2n|Q^X_b(e^f/b^2,a^TX(e^f/b^2) +Y^TX/b^2,Y_0^+Y^N,/b^2)| ≤ Cexp(-C'|Y_0^|^2 -C_γ'(|f|^2+|Y^TX|^2 +|((k^-1)-1)Y^N,| +|[a,Y^N,]|)).As b→ +∞,b^-4m-2nQ^X_b(e^f/b^2,a^TX(e^f/b^2) +Y^TX/b^2,Y_0^+Y^N,/b^2) → e^-(|a|^2+|Y_0^|^2)/2(k^-1)R_Y_0^((f,Y), (k^-1)(f,Y))ρ^E(k^-1) e^-iρ^E(Y_0^)-A. A crucial computation in <cit.> gives the following key result.For Y_0^∈(γ), we have the identity(2π)^r/2∫_^(γ)×(⊕^(γ))_s [(k^-1) R_Y_0^((y,Y),(k^-1)(y,Y)) ]dydY= J_γ(Y_0^). From Theorems <ref>, <ref>,(<ref>)-(<ref>),and (<ref>),we obtain Theorem <ref>. <cit.>, <cit.>.InExample <ref>, we haveN=0,𝒳=TX⊕ N=T=⊕.Using the coordinates (x,y)∈⊕, we get^X_b=M_b+N^Λ^∙()/b^2,withM_b=1/2b^2(-∂^2/∂ y^2+y^2-1) +y/b∂/∂ x. The heat kernel p_b,t((x,y),(x',y')) associated withe^-tM_b depends only on x'-x,y,y'.The heat kernel of theoperator-∂^2/∂ y^2+y∂/∂ x was first calculatedby Kolmogorov <cit.>, and p_b,t((x,y),(x',y')) has been computed explicitly in<cit.>.Let a∈=. Then a acts astranslation by a on the first componentof ⊕. From (<ref>) and (<ref>), we deduce that_s^[a][e^-t_b^X] =(1-e^-t/b^2)∫_p_b,t((0,Y),(a,Y))dY.Theorem <ref> can be stated in the special case of this example as follows.For any t>0, b>0, we have ^[a][e^tΔ^/2] =_s^[a][e^-t_b^X].We give a simple direct proof which can be ultimately easily justified.Note thatM_b=1/2b^2(-∂^2/∂ y^2 +(y+b∂/∂ x)^2-1) -1/2∂^2/∂ x^2.By (<ref>), we get e^-b∂^2/∂ x∂ yM_b e^b∂^2/∂ x∂ y =1/2b^2(-∂^2/∂ y^2 +y^2-1)-1/2∂^2/∂ x^2.Using the fact that p_b,t((x,y),(x',y')) only depends onx'-x, y, y', we deduce from (<ref>) that ∫_p_b,t((0,y),(a,y))dy=[e^-t/2b(-∂^2/∂ y^2+y^2-1)] ^[a][e^tΔ^/2]= 1/1-e^-t/b^2^[a][e^tΔ^/2],since the spectrum of the harmonic oscillator1/2(-∂^2/∂ y^2+y^2-1) is . By (<ref>) and(<ref>), we get (<ref>).By (<ref>), we can compute the limitas b→ +∞of the right-hand side of (<ref>) from the explicit formula of p_b,t((x,y),(x',y')), and in this way we get (<ref>). In other words, we interpret(<ref>) as a consequence of a local index theorem.§.§ A brief idea on the proof of Theorems <ref>, <ref>, <ref> The wave operator forthe elliptic Laplacian has the property of finite propagation speed,which explain the Gaussian decayof the elliptic heat kernel.The hypoelliptic Laplacian does not have a wave equation.One difficult point in Theorems <ref>, <ref> and <ref> is to get the uniform Gaussian-likeestimate.Let us give an argument back to <cit.> which explains some heuristic relations of the hypoelliptic heat equation to the wave equation on X.Here q^X_b,t will denote scalar hypoelliptic heat kernel on the total space 𝒳 of the tangent bundle TX. Putσ_b,t((x,Y), x') = ∫_Y'∈T_x'X q^X_b,t((x,Y),(x',Y'))dY',M_b,t((x,Y), x') =1/σ_b,t((x,Y), x') ∫_Y'∈ T_x'X q^X_b,t((x,Y),(x',Y')) ( Y'⊗ Y') dY'.Then M_b,t((x,Y), x') takes its values in symmetric positiveendomorphisms of T_x'X. We can associate to M_b,t the second order elliptic operator acting on C^∞ (X,),M_b,t(x,Y) g(x')= ⟨∇^TX_·∇_·, M_b,t((x,Y), x')g(x')⟩,where the operator ∇^TX_·∇_· acts on the variable x'. Then we have <cit.>(b^2∂^2/∂ t^2 + ∂/∂ t - M_b,t(x,Y) ) σ_b,t((x,Y), ·)=0.This is a hyperbolic equation. As b→ 0, it converges in the proper sense to the standard parabolic heat operator(∂/∂ t - 1/2Δ)p_t(x,·)=0. The above consideration plays an important role in the proof given in <cit.> of the estimates(<ref>), (<ref>), (<ref>)and (<ref>).§ ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTIONRecall that a flat vector bundle (F,∇) with flat connection∇ over a smooth manifold Mcomes from a representationρ: π_1(M)→ GL( q,) so thatifM is the universal cover of M,thenF= M×_ρ^ q.The analytic torsion associated witha flat vector bundle on a smooth compactRiemannian manifold M is a classical spectral invariantdefined by Ray and Singer <cit.> in 1971.It is a regularized determinant of the Hodge Laplacian forthe de Rham complex associated with this flat vector bundle.For Γ⊂ G a discrete cocompact torsion freesubgroup of a connected reductive Lie group G,if Z= Γ\ G/K is the locally symmetric spaceas in (<ref>), then Γ = π_1(Z).By the superrigidity theorem of Margulis <cit.>,if the real rank of G is ≥ 2, a generalrepresentation of Γ is not too far froma unitary representation of Γ or the restriction to Γ of a representation of G_, the complexification of G.See <cit.> for more details.Assume that the difference of the complex ranks ofG and Kis different from 1. For a flat vector bundle induced by a G_-representation,as an application of Theorem <ref>,we obtain a vanishing result of individualorbital integrals that appearin the supertrace of the heat kernelfrom which the analytic torsion can be obtained. In particular, this implies that the associated analytic torsion is equal to 1 (cf. Theorem <ref>). We explain finally Shen's recent solution on Fried's conjecturefor locally symmetric spaces:for any unitary representation of Γ such that the cohomology of the associated flat vector bundle on Z=Γ\ G/K vanishes, the value at zero of a Ruelle dynamical zeta functionidentifies to the associated analytic torsion. This section is organized as follows. In Section <ref>,we introducethe Ray-Singer analytic torsion. In Section <ref>,we study the analytic torsion on locally symmetric spaces for flat vector bundlesinduced bya representation ofG_.Finally in Section <ref>,we describe Shen's solution ofFried's conjecture in the case of locally symmetric spaces.In Section <ref>, we make some remarks onrelated research directions.§.§ Analytic torsionLet M be a compact manifold of dimension m. Let (F,∇) be a flat complex vector bundle on Mwith flat connection ∇(i.e., its curvature (∇)^2=0). The flat connection ∇ inducesan exterior differential operator d on Ω^∙(M,F), the vector space of differential forms on M with values in F, and d^2=0. Let H^∙ (M,F) be the cohomology group ofthe complex (Ω^∙(M,F),d) as in (<ref>).Let h^F be a Hermitian metric on F. Then as explained inSection <ref>,g^TM and h^F induce naturally a Hermitian product onΩ^∙(M,F).Let D be as in (<ref>). We introduce here a refined spectral invariant of D^2 which isparticularly interesting. Let P be the orthogonal projection from Ω^∙(M,F)onto (D) and let P^⊥=1-P.Let N be the number operator acting on Ω^∙(M,F), i.e., multiplication by j on Ω^j(M,F). For s∈and (s)>m2, setθ(s) = - ∑^m_j=0(-1)^jj|_Ω^j(M,F) [(D^2)^-sP^⊥]=-1/Γ(s)∫_0^+∞_s[Ne^-tD^2P^⊥]t^sdt/t,where Γ(·) is the Gamma function. From the small time heat kernel expansion (cf. <cit.>), we know thatθ(s) is well-defined for (s)>m2 andextends holomorphically near s=0.<cit.> The (Ray-Singer) analytic torsion is defined as T(g^TM,h^F)=exp(12∂θ∂ s(0)). We have the formal identity, T(g^TM,h^F) =∏^m_j=0 (D^2|_Ω^j(M,F))^(-1)^jj/2.a) If h^F is parallel with respect to ∇, thenF is induced by a unitary representation of π_1(M), and we say that (F,∇,h^F) is a unitary flat vector bundle.In this case, if m is even and M is orientable,by a Poincaré duality argument, we have T(g^TM,h^F)=1.b) If m is odd, and H^∙(M,F)=0, then T(g^TM,h^F)does not depend on the choiceof g^TM,h^F, thus it is atopological invariant (cf. <cit.>). §.§ Analytic torsion for locally symmetric spacesWe use the same notation and assumptions as in Section <ref>.Recall that ρ^E: K→U(E) is a finite dimensionalunitary representation of K, and F=G×_K E is the induced Hermitian vector bundle on the symmetric space X=G/K.Assume form now on that the complexification G_ of G exists, and the representation ρ^E is induced by a holomorphicrepresentation of G_→Aut(E), that is stilldenoted by ρ^E.We have the canonical identification ofG×_K E as a trivial bundle E on X:F= G×_K E→ X× E,(g,v)→ρ^E(g)v.This induces a canonical flat connection ∇ on F such that∇=∇^F+ρ^Eω^.Let U be a maximal compact subgroup of G_. Then U is the compact form of G and 𝔲=i⊕ is its Lie algebra.By Weyl's unitary trick <cit.>, ifU is simply connected, it is equivalent to considerrepresentations of G, of U on E, or holomorphicrepresentations of the complexification G_ of G on E,or representations of , or 𝔲 on E. We fix a U-invariant Hermitian metric on E. This implies in particular it isK-invariant,and ρ^E(v)∈(E) is symmetric for v∈.This induces a Hermitian metric h^F on F. As in Section <ref>, we consider now the operatorD acting on Ω^∙(X,F) induced by g^TX,h^F.Let C^,X be the Casimir operator of G acting onC^∞(X,Λ^∙(T^*X)⊗ F) as in (<ref>).Then by <cit.> and <cit.>,we have D^2=C^,X-C^,E. Let T bea maximal torus in K and let ⊂be its Lie algebra. Set ={v∈: [v,]=0}.Put 𝔥=⊕.By <cit.>, we know that 𝔥 isa Cartan subalgebra ofand thatisthe complex rank of K and 𝔥 isthe complex rank of G. Also, m and have the same parity.For γ= e ^a k^-1∈ G a semisimple element as in (<ref>), let K^0(γ)⊂ K(γ) bethe connected component of the identity.Let T(γ)⊂ K^0(γ) be a maximal torus inK^0(γ), and let (γ)⊂(γ) be its Lie algebra. By (<ref>) and (<ref>),k commutes with T(γ), thus by <cit.>, there exists k_1∈ K such that k_1T(γ)k_1^-1⊂ T,k_1k k_1^-1⊂ T. By working onk_1γ k_1^-1=e ^(k_1) a ((k_1) k)^-1 instead of γ, we may and we will assume that T(γ)⊂ T, k∈ T. In particular (γ)⊂. Set(γ)={v∈: [v,(γ)]=0,(k) v=v}.Then ⊂(γ) and(1)= .Recall that N^Λ^∙(T^*X) is the number operator on Λ^∙(T^*X).<cit.>, <cit.>, <cit.>.For any semisimple element γ∈ G,if m is even, or if m is odd and (γ)≥ 2,then for any t>0, we have _s^[γ][(N^Λ^∙(T^*X) -m/2)e^-t/2D^2]=0.By Theorem <ref>, (<ref>) and (<ref>),for any t>0 and any semisimple element γ∈ G,_s^[γ][(N^Λ^∙(T^*X) -m/2)e^-t/2D^2] =e^-|a|^2/2t/(2π t)^p/2 e^t/16^[C^,] +t/48^[C^,] ∫_(γ) J_γ(Y^_0)_s^Λ^∙(^*)⊗ E[(N^Λ^∙(^*)-m/2) ρ^Λ^∙(^*)⊗ E(k^-1) e^-i ρ^Λ^∙(^*)⊗ E(Y^_0) +t/2C^,E]e^-|Y^_0|^2/2tdY^_0/(2π t)^q/2.But _s^Λ^∙(^*)⊗ E[(N^Λ^∙(^*)-m/2) ρ^Λ^∙(^*)⊗ E(k^-1) e^-i ρ^Λ^∙(^*)⊗ E(Y^_0) +t/2C^,E]=_s^Λ^∙(^*)[(N^Λ^∙(^*)-m/2) ρ^Λ^∙(^*)(k^-1) e^-i ρ^Λ^∙(^*)(Y^_0)] ^E[ρ^E(k^-1)e^-i ρ^E(Y^_0) +t/2C^,E]. If u is an isometry of , we have _s^Λ^∙(^*)[u]=(1-u^-1),_s^Λ^∙(^*)[N^Λ^∙(^*)u] =∂/∂ s(1-u^-1e^s)(0).If the eigenspace associated with the eigenvalue 1 is of dimension ≥1, the fist quantity in (<ref>)vanishes. If it is of dimension ≥ 2, the second expressionin (<ref>) also vanishes. Also, if m is even andu preserves the orientation, then _s^Λ^∙(^*)[(N^Λ^∙(^*)-m/2)u]=0. From (<ref>), (<ref>) and (<ref>),we get (<ref>). Now let Γ be a discrete torsion free cocompact subgroup of G.Set Z=Γ\ X. Then π_1(Z)=Γ and the flat vector bundle F descents as a flat vector bundle F over Z.<cit.>. For a flat vector bundle F on Z=Γ\ Xinduced by a holomorphic representation of G_, if m is even, or if m is odd and 𝔟≥3,then T(g^TZ,h^F)=1. Under the condition of Theorem <ref>,from Theorem <ref>,(<ref>)and (<ref>), we get _s[(N^Λ^∙(T^*Z)-m/2) e^-t/2D^Z,2]=0.Now Theorem <ref> is a direct consequence of(<ref>) for h=1, (<ref>) and (<ref>). a) If F is trivial, i.e., it is induced by the trivialrepresentation of G, then Theorem <ref> under the condition of Theorem<ref> was firstobtained by Moscovici-Stanton <cit.>.b) Assume G is semisimple, then the induced metric h^F on F is unimodular, i.e., the metric h^ Fon F:= Λ^maxFinduced by h^F is parallel with respect to the flat connection onF.In this case,Theorem <ref> for γ=1 was first obtained by Bergeron-Venkatesh <cit.>,and Müller and Pfaff <cit.> gave a new proof of Theorem <ref>. c) We can drop the condition on torsion freeness ofΓ in (<ref>). For p,q∈, let SO^0(p,q)be the connected component of the identity in the real groupSO(p,q). By <cit.> and <cit.>,among the noncompact simple connected complex groupssuch that m is odd and =1, there is onlySL_2(), and among the noncompact simple realconnected groups, there are only SL_3(),SL_4(), SL_2(ℍ),and SO^0(p,q) with pq odd >1.Also, by <cit.>,𝔰𝔩_2()=𝔰𝔬(3,1),𝔰𝔩_4()=𝔰𝔬(3,3),𝔰𝔩_2(ℍ)=𝔰𝔬(5,1).Therefore the abovelist can be reduced toSL_3() and SO^0(p,q) with pq odd >1.Assume from now on that ρ:Γ→ U( q) be a unitary representation.Then F=X×_Γ^ q is a flat vector bundle on Z=Γ\ X with metric h^F induced bythe canonical metric on ^ q,i.e.,F is a unitary flat vector bundle on Zwith holonomy ρ.By Remark <ref>, if m is even, thenT(g^TZ,h^F)=1. Thus we can simply assume that m is odd.Observe that the pull back of (F,h^F) over X is ^ q with canonical metric, thusthe heat kernel on X is given bye^-tD^2(x,x')=e^-tD_0^2(x,x')⊗Id_^ qwhere e^-tD_0^2(x,x')∈Λ^∙(T_x^*X) ⊗Λ^∙(T_x'^*X)^* is the heat kernel on Xfor the trivial representationG→Aut().Thus,_s[(N^Λ^∙(T^*X)-m/2) γ e^-tD^2(γ^-1z, z)]=[ρ(γ)]_s[(N^Λ^∙(T^*X) -m/2)γ e^-tD^2_0 (γ^-1z,z)].Note that for γ∈Γ, [ρ(γ)] depends only on the conjugacy class of γ,thus form (<ref>), (<ref>) and (<ref>),we get the analogue of Theorem <ref>,_s[(N^Λ^∙(T^*Z)-m/2) e^-tD^Z,2]=∑_[γ]∈ [Γ](Γ∩ Z(γ) \ X(γ)) [ρ(γ)]_s^[γ][(N^Λ^∙(T^*X)-m/2) e^-tD^2_0]. Since the metric h^F is given by the unitary representation ρand g^TZ is induced by the bilinear form B on ,we denote the analytic torsion in Section <ref> by T(F). By Theorem <ref> for the trivial representationG→Aut() and (<ref>), we geta result similar to Theorem <ref>. <cit.>. For a unitary flat vector bundle F onZ=Γ\ X,if m is odd and ≥ 3, then for t>0, we have _s[(N^Λ^∙(T^*Z)-m/2) e^-t D^Z,2/2]=0.In particular, T(F)=1. §.§ Fried's conjecture for locally symmetricspaces The possible relation of the topological torsion tothe dynamical systems was first observed by Milnor<cit.> in 1968.A quantitative description of their relation was formulated byFried <cit.> when Z is a closed oriented hyperbolicmanifold. Namely, he showed that for an acyclic unitary flat vector bundle F, the value at zero of the Ruelle dynamical zeta function,constructed using the closed geodesics in Zand the holonomy of F, is equal to the associated analytic torsion. In <cit.>, Fried's conjectured that a similar resultstill holds for general closed locally homogenous manifolds. In 1991, Moscovici-Stanton <cit.> made an importantcontribution to the proof of Fried's conjecture for locally symmetric spaces. Let Γ⊂ G be a discrete cocompact torsion freesubgroup of a connected reductive Lie group G. Then we get the symmetric space X= G/Kand the locally symmetric spaceZ= Γ\ X. By Remark <ref>, we may assumethat Z=m is odd. Recall that [Γ] is the set of conjugacy classes of Γ. For [γ]∈ [Γ]\{1}, denote byB_[γ]the space of closed geodesics in [γ]. As a subset of the loop space LZ, we equippedB_[γ] the induced topology and smooth structure.By Proposition <ref>,B_[γ]≃Γ∩ Z(γ)\X(γ) is a compact locally symmetric space,and the elements of B_[γ] have the same lengthl_[γ]>0.The group 𝕊^1 acts onB_[γ] by rotation. This action is locally free.Denote by χ_orb(𝕊^1\B_[γ])∈ℚ the orbifold Eulercharacteristic number for the quotient orbifold 𝕊^1\ B_[γ].Recall that if e(𝕊^1\ B_[γ], ∇^T(𝕊^1\ B_[γ])) ∈Ω^∙(𝕊^1\ B_[γ], o(T(𝕊^1\ B_[γ])))is the Euler formdefined using Chern-Weil theory for the Levi-Civita connection∇^T(𝕊^1\ B_[γ]), then χ_orb(𝕊^1\B_[γ])=∫_𝕊^1\B_[γ]e(𝕊^1\ B_[γ], ∇^T(𝕊^1\ B_[γ])).Let n_[γ]=|(𝕊^1→Diff(B_[γ]))|be the generic multiplicity of B_[γ]. Given a representation ρ:Γ→ U( q),we say that the dynamical zeta function R_ρ(σ)is well-defined if the following properties hold: * For σ∈, Re(σ)≫1, the sumΞ_ρ(σ)=∑_[γ]∈ [Γ]\{1}[ρ(γ)]χ_orb(𝕊^1\ B_[γ])/n_[γ]e^-σ l_[γ]. defines to a holomorphic function.* The function R_ρ(σ)=exp(Ξ_ρ(σ)) has a meromorphic extension to σ∈. Note that γ∈Γ is primitive meansif γ=β^k,β∈Γ, k∈^*, then γ=β. If Z is a compact oriented hyperbolic manifold, then𝕊^1\ B_[γ] is a point.Moreover, if ρ is the trivial representation, then R_ρ(σ)=exp(∑_[γ]∈[Γ]\{1}1/n_[γ]e^-σ l_[γ]) =∏_[γ] primitive, γ≠1(1-e^-σ l_[γ])^-1.<cit.>For any unitary flat vector bundle F on Z with holonomy ρ,the dynamical zeta function R_ρ(σ) is a well-defined meromorphic function onwhich is holomorphic for Re(σ)≫1. Moreover, there exist explicit constants C_ρ∈ and r_ρ∈such that, whenσ→ 0, we haveR_ρ(σ)=C_ρ T(F)^2σ^r_ρ +𝒪(σ^r_ρ+1). If H^∙ (Z,F)=0,thenC_ρ=1, r_ρ=0,so that R_ρ(0)=T(F)^2. The most difficult part of the proof is toexpress the R_ρ(σ) as a product of determinant of shifted Casimir operators, [The gap in Moscovici-Stanton's paper comes from using an operator Δ^j,l_ϕ <cit.> which could not be defined on Z.] in fact being the analytic torsion. Shen's idea is to interpret the right-hand side of (<ref>)as the Selberg trace formula (by eliminating the term ^[1])of the heat kernel for some representations of K by usingTheorems <ref> and <ref>.By Theorem <ref>, we can concentrate on the proof inthe case =1.From now on, we assume =1. Up to conjugation, there existsa unique standard parabolic subgroup Q⊂ G withLanglands decomposition Q=M_QA_QN_Q such thatA_Q=1. Let 𝔪, , bethe respective Lie algebras of M_Q, A_Q, N_Q. Let M be the connected component of identity of M_Q. Then M is a connected reductive group with maximal compact subgroup K_M=M∩ K andwith Cartan decomposition𝔪=_𝔪⊕_𝔪,and K_M acts on _𝔪, M acts on and M acts trivially on . We have an identity of real K_M-representations≃_𝔪⊕⊕,andis even. Moreover there exists ν∈^* such that (cf.<cit.>) [a,f]=⟨ν,a⟩ ffor anyf∈,a∈.For γ∈ G semisimple, Shen<cit.>observes thatγcan be conjugated into H:=exp()T(cf. (<ref>))iff (γ)=1.Let R(K,), R(K_M,) be the real representation ringsofK and K_M. We can prove that the restrictionR(K,)→ R(K_M,) is injective. The key result <cit.> is thatthe K_M-representation onhas a unique lift in R(K,).Asis a K-representation, K_M-action on _𝔪 also lifts to K by liftingas a trivial K-representation in (<ref>). For a real finite dimensional representation ς ofM on the vector space E_ς, such thatς|_K_Mcan be lifted into R(K,), this implies that there existsa real finite dimensional _2-representationς=ς^+ - ς^- of K onE_ς =E_ς^+ -E_ς^- such that we have the equality in R(K_M,),E_ς|_K_M =∑_i=0^_𝔪 (-1)^iΛ^i(^*_𝔪)⊗ (E_ς|_K_M).Let ℰ_ς =G×_K E_ςbe the induced _2-graded vector bundle on X, andℱ_ς =Γ\ℰ_ς.Let C^,Z,ς,ρ bethe Casimir element of G acting onC^∞(Z,ℱ_ς⊗ F).Modulo some technical conditions,with the help of Theorem <ref> and (<ref>), Shen<cit.> obtains the identity_s[e^-tC^,Z,ς,ρ/2] = q(Z)^[1][e^-tC^,X,ς/2]+1/√(2π t)e^-c_ςt∑_[γ]∈ [Γ]\{1}[ρ(γ)]χ_orb (𝕊^1\ B_[γ])/n_[γ]^E_ς[ς(k^-1)]/|(1-(γ))|__0^|^1/2 |a|e^-|a|^2/2t,where c_ς is some explicit constant, and k^-1 is defined in (<ref>).We do not write here the exact formula for^[1][e^-tC^,X,ς/2]. By (<ref>), if we setΞ_ς,ρ(σ) =∑_[γ]∈ [Γ]\{1}[ρ(γ)]χ_orb(𝕊^1 \ B_[γ])/n_[γ]^E_ς[ς(k^-1)]/|(1-(γ))|__0^|^1/2 e^-|a|σ,we need to eliminate the denominator|(1-(γ))|__0^|^1/2to relate Ξ_ς,ρ to Ξ_ρ, the logarithm ofthe dynamical zeta function R_ρ(σ). Set 2l =. The following observation is crucial. <cit.> Forγ= e^ak^-1∈ H:=exp()T,a≠0, with ν∈^* in (<ref>),we have|(1-(γ))|__0^|^1/2 =∑_j=0^2l(-1)^j^Λ^j(^*)[(k^-1)]e^(l-j)|ν||a|, Let ς_j be the representation of M onΛ^j(^*). By (<ref>) and (<ref>), we haveΞ_ρ(σ)=∑_j=0^2l(-1)^jΞ_ς_j,ρ (σ+(j-l)|ν|).On the other hand, since =1, from (<ref>), we have the following identity in R(K_M,),∑_i=0^(-1)^i-1iΛ^i(^*) =∑_j=0^(-1)^j (∑^_𝔪_i=0(-1)^iΛ^i(_𝔪^*))⊗Λ^j(^*). Note that ∑_i=0^(-1)^i-1iΛ^i(^*) ∈ R(K,) is used to define the analytic torsion. From (<ref>) and (<ref>),Shen obtains a very interesting expression for R_ρ(σ) in term of determinants of shifted Casimir operators, from which he could obtain equation (<ref>).For a representation ς of M in (<ref>), set r_ς,ρ =_ (C^,Z,ς^+,ρ) -_ (C^,Z,ς^-,ρ). Then Shen <cit.> obtains the formulaC_ρ=∏_j=0^l-1(-4|l-j|^2|ν|^2)^(-1)^j-1 r_ς_j,ρ,r_ρ=2∑_j=0^l(-1)^j-1r_ς_j,ρ. Shen showsthat if H^∙(Z,F)=0,then r_ς_j,ρ=0for any 0≤ j≤ l by usingthe spectral aspect of the Selberg trace formula (Theorem <ref> and (<ref>)),and some deep results on the representation theoryof reductive groups.For a G-representation π: G→ Aut(V) and v∈ V, recall that v is said to be differentiable if c_v: G→ V, c_v(g)= π(g)v is C^∞, that v is said to be K-finite if it is contained in a finitedimensional subspace stable under K. We denote by V_(K) the subspace of differentiable and K-finite vectors in V. Let H^∙ (,K;V_(K)) be the (,K)-cohomologyof V_(K). We denote by G_u the unitary dual of G,that is the set of equivalence classes of complex irreducibleunitary representation π of G on the Hilbert space V_π.Let χ_π be the corresponding infinitesimal character. Let p:Γ\ G →Z bethe natural projection. The group G acts unitarily on the right onL_2(Γ\ G, p^*F), then L_2(Γ\ G, p^*F) decomposes into a discrete Hilbert direct sum withfinite multiplicities of irreducible unitary representations of G,L_2(Γ\ G, p^*F) = ⊕_π∈G_u n_ρ(π) V_π with n_ρ(π) <+∞,heremeans the Hilbert completion.Set W= ⊕_π∈G_u, χ_π is trivial n_ρ(π) V_π,then W isthe closureinL_2(Γ\ G, p^*F) of W^∞,the subspace of C^∞(Γ\ G,p^*F)on which the center of U(𝔤) acts by the same scalaras in the trivial representation of 𝔤. By standard arguments<cit.>,the cohomology H^∙ (Z,F) is canonically isomorphic tothe (,K)-cohomology H^∙ (,K;W_(K)) of W_(K), the vector space of differentiable and K-finite vectors of W, i.e., H^∙ (Z,F) = ⊕_π∈G_u, χ_π is trivial n_ρ(π)H^∙ (,K;V_π, (K)).Vogan-Zuckerman <cit.> and Vogan <cit.> classified all irreducible unitary representations of G with nonzero(,K)-cohomology. On the other hand, in <cit.>, Salamanca-Riba showed that any irreducibleunitary representation of G with trivial infinitesimal character is in the class specified byVogan and Zuckerman, which means that it possesses nonzero (,K)-cohomology. In summary, if (π,V_π)∈G_u, thenχ_π is non-trivial if and only ifH^∙ (,K; V_π,(K))=0.By the above considerations,H^∙(Z,F)=0 is equivalentto W=0.This is the main algebraic ingredient inthe proof of (<ref>). Shen's contribution <cit.> is to give a formulafor r_ς_j,ρ using Hecht-Schmid's work<cit.>on the -homologyof W, Theorem <ref> and (<ref>).From this formula,we see immediately that W=0 implies r_ς_j,ρ=0 for 0≤ j≤ l. §.§ Final remarks1. Theorem <ref> gives an explicit formula forthe orbital integrals for the heat kernel of the Casimir operator and it holds for any semisimple element γ∈ G.A natural question is how to evaluate or definethe weighted orbital integrals that appear in Selberg trace formula fora discrete subgroup Γ⊂ G such thatΓ\ G has a finite volume. 2. Bismut-Goette <cit.> introduced a local topologicalinvariant for compact manifolds with a compact Lie group action:the V-invariant. It appears as an exotic term in the differencebetween two natural versions of equivariant analytic torsion.The V-invariant shares formally many similaritieswith the analytic torsion, and if we apply formally the constructionof the V-invariant to the associated loop space equipped with its natural𝕊^1 action, then we get the analytic torsion.In Shen's proof of Fried's conjecture for locally symmetric spaces,Shen observed that the V-invariant for the 𝕊^1-actionon B_[γ] is exactly -χ_orb(𝕊^1\ B_[γ])/2n_[γ]. This suggests a general definition of the Ruelle dynamical zeta function for any compact manifold by replacing-χ_orb(𝕊^1\ B_[γ])/2n_[γ] in Definition <ref> by the associated V-invariant.One could then compare it with the analytic torsion,and obtain a generalized version of Fried's conjecture forany manifold with non positive curvature .Note that for a strictly negative curvature manifold, B_[γ]is a circle and the V-invariant is -1/2n_[γ]. 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Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 51–90. | http://arxiv.org/abs/1704.08337v2 | {
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"title": "Geometric hypoelliptic Laplacian and orbital integrals (after Bismut, Lebeau and Shen)"
} |
Inclusive Breakup Theory of Three-Body Halos T. Frederico Received: date / Revised version: date ============================================The final publication is available at Springer via<http://doi.org/10.1007/s10958-017-3407-3>The paper is concerned with the computational complexity of the initial value problem (IVP) for a system of ordinary dynamical equations. Formal problem statement is given, containing a Turing machine with an oracle for getting the initial values as real numbers. It is proven that the computational complexity of the IVP for the three body problem is not bounded by a polynomial. The proof is based on the analysis of oscillatory solutions of the Sitnikov problem that have complex dynamical behavior. These solutions contradict the existence of an algorithm that solves the IVP in polynomial time. § INTRODUCTIONThe problem of numerical integration of ODE systems is undoubtedly one of the most popular problems in applied mathematics. There exists a huge number of algorithms and program packages for obtaining numerical solutions of systems of differential equations originating from math, physics, celestial mechanics and engineering. However, there is little available research in the area of computational complexity of the initial value problem itself (some results are obtained in <cit.> and <cit.>). In most other works, complexity of particular algorithms is analyzed, in terms of either the number of basic arithmetical operations performed on each step, or the number of calls to first or higher-order derivatives.In this work, a formal statement is presented of the IVP for a system of ODEs. In that statement, the input data for a problem will be: the initial conditions, the point t in time, and a precision ε. An algorithm is supposed to consume the input and produce the output (an approximate state of the system at time t) that matches the actual state of the system up to ε. It must be noted that since the said statement includes real numbers, we can not work with just data of finite length. While it is sufficient to treat t and ε as rationals, initial conditions are a different story: there is no prior knowledge of how many digits in them will be sufficient to ensure that the solution at t>0 will be obtained with precision ε.There are several approaches to work around that difficulty. The first is to consider an infinite input tape (or several infinite tapes) whose cells contain the digits of the initial conditions. A Turing machine for the given IVP can read the digits on demand. The second approach is to have a Turing machine use an oracle that gives the needed digits on demand. The third approach is to use a secondary Turing machine that prints out the digits into the tape by request from the main Turing machine.The third approach, as opposed to the first two, is that it would limit us to just the constructive real numbers. In this work, the second approach (with an oracle) is used. It differs from the first one in the conventions of complexity analysis: the calls to an oracle account for the time complexity of algorithm as a function from the (finite) input length, while in the first approach makes the input infinite, rendering the complexity analysis difficult.Another obstacle in the formal statement of the problem is the following: even if the derivatives in the system of ODEs are known Lipschitz-continuous functions, the problem of existence of the ODE solution at point t can be undecidable.We will show that even in the provably decidable case, the complexity of the IVP can be non-polynomial. The proof is based on the investigation of systems with complex dynamical behavior. As a basic example, we will use the classical Sitnikov problem for a three-body gravitational system, where two bodies follow elliptic orbits on a plane, and the third body stays on the line perpendicular to that plane. In the general case, the third body does unending oscillations with arbitrary amplitudes.Instead of the oscillating solution of Sitnikov problem, we could use other dynamical systems exhibiting complex behavior, like, for instance, a neighborhood of some homoclinic solution. Their computational complexity would have turned non-polynomial, too. In this work, we do not use a natural representation of such solution in the terms of symbolic dynamics <cit.>. Rather, to prove the absence of a polynomial algorithm for our formal IVP statement, it is sufficient to show that for a certain neighborhood of initial conditions in phase space, the number of algorithmically distinguishable trajectories is exponential in t.§ TURING MACHINE FOR THE INITIAL VALUE PROBLEM LENGTH 𝐱 𝐟 ℝ ℚ ℕ ℤ ḍ 𝐪 𝐩 𝐯 𝒫We estimate the computational complexity of the initial value problem for the dynamical system [ =(); (0) =_0 ]where ∈ D, _0 ∈ D is a real vector, and : D →^n is a computable real vector-valued function (open set D⊆^n is the phase space of the system).This work deals with the case when the solution ^*(t) : → D: * exists on the whole ;* is unique;* is a computable real vector-valued function. Solutions that do not extend toare called singular. The problem of determining the singularity of a solution is undecidable (see section <ref>). Uniqueness of a solution, it it exists, is guaranteed given that the functionis locally Lipschitz-continuous in every point in D. (The proof of that fact can be found e.g. in <cit.>.) However, the local Lipschitz-continuity does not imply the existence of the solution on .Ifis defined on D when D=^n and is (globally) Lipschitz-continuous, then the solution ondoes exist for all _0 and is unique due to the Cauchy-Lipschitz theorem.Ifis continuous at every point in D, then every unique solution is computable by a (non-practical) combinatorial algorithm <cit.>. In particular, that holds for any computable , since every computable function is continuous.In <cit.>, it is proven that the solution of an IVP is computable with a modification of Picard–Lindelöf method, ifis Lipschitz-continuous on D. This important fact is quite non-trivial, despite the existence of hundreds numerical integrators for ODE. The vast majority of these integrators suffer from saturation: the step size being small enough, the error grows upon further decrease of the step size. Therefore, these integrators can not in principle obtain a solution up to an arbitrary precision <cit.>.To summarize: with D=^n and Lipschitz-continuous , the solution of (<ref>) with any _0 exists on , is unique and computable. It follows independently from <cit.> and <cit.>. In both sources, the computability is proven for the solution being the function of _0 and t, rather than just t.In this work, we limit ourselves with the study of a particular instance of the three-body problem (see Section <ref>). The subject for study is the asymptotic dependence of the computational complexity of the solution ^*(t) on the value of t; the dependence on the precision of t is not considered. In the text that follows, t in the IVP is treated as rational, while _0 is a real vector. The complexity analysis of another special case of IVP, where t ∈, is given in <cit.>. The solution function of an initial value problem (<ref>) is the function S(_0, t): D ×→ D, where S|_=_0: → D is a computable real vector-valued function, whose closure on the real axis is the solution of (<ref>).Turing machine that computes the solution function of an IVP is a Turing machine that accepts rational t and ε as input; has an oracle φ that instruments _0 as a computable real vector; and produces the value of the solution (t) corresponding to given _0 and t, with the precision ε. It should be noted that in terms of complexity theory, the IVP belongs to the class of function problems, as opposed to more studied decision problems. The job of the oracle in the Turing machineis to write into its tape the representation of _0 up to an arbitrary precision, specified by the machine itself. It is obvious that the time required by the Turing machine includes the time to read the oracle tape. The IVP (<ref>) has polynomial complexity if there exists aTuring machine from the definition <ref> that computes its solution function in time bounded by ((t), (ε)), whereis an arbitrary polynomial. Remark. Without loss of generality, it can be assumed that ε = 2^-l, hence (ε) = l. Suppose A and B are two IVPs. A is called polynomially reducible to B if there exist the following functions, computable in polynomial time: G : D^(A)→ D^(B) and H : D^(B)→ D^(A), so that for any initial state _0^(A)∈ D^(A) and a corresponding solution ^*(A)(t) the following holds: ^*(A)(t) = H(^*(B)(t)), where ^*(B)(t) is a solution of B with initial state _0^(B) = G(_0^(A)). Statement. If IVP A is polynomially reducible to IVP B, and B has polynomial complexity, then A has polynomial complexity as well.§ ANALYSIS OF THE COMPUTATIONAL COMPLEXITY OF THE IVP FOR THE THREE-BODY PROBLEM §.§ N-body problemGravitational N-body problem is concerned with the Newtonian motion of N point-masses in three dimensions. The system of ODEs for this problem is the following::. [ _i=_i,i = 1..N; _i= ∑_j=1 j≠ i^Nμ_j _j-_i/|_j-_i|^3,i = 1..N ]}where μ_i∈, μ_i ≥ 0, _i ∈^3, _i ∈^3.With N=3, the initial state of the system is given by a 21-vector _0=(μ_1,μ_2,μ_3,p_1,1,…,p_3,3,v_1,1,…,v_3,3), while the system (<ref>) defines a computable real vector-valued function = 𝐟(). (The first three variables do not depend onor t.) §.§ Known resultsThe classical two-body problem (N=2) has a solution in algebraic functions of initial state and t. Depending on the configuration of the system, the two bodies follow either a Keplerian orbit (a parabola, hyperbola, or ellipse) or move along a line. The detailed description of the solutions can be found in multiple sources. Given those algebraic solutions, it is not difficult to show that the IVP for a nonsingular two-body problem has polynomial complexity.With N=3 the problem does not have a generic algebraic solution, as proven by Poincaré. However, Sundman in 1912 derived a solution in the form of converging series. Unfortunately, the estimate of the number of terms required to calculate the series at point t with a sensible precision is exponential in t <cit.>. Merman improved Sundman's result and found other series <cit.>, though still exponential in t.In practical tasks related to the N-body problem (in particular, in ephemeris astronomy) algorithms of numerical integration are used to obtain approximate solutions. The time complexity of such algorithms has a fundamental lower bound of O(t), hence it can not be upper-bounded by a polynomial of (t).The bottom line is that the known algorithms for the IVP for the three-body problem are non-polynomial.However, that does not disprove the polynomial complexity of the problem.On a different note, let us show that there is a singular solution of the N-body problem that has a nonsingular one in any neighborhood. Let N=2. Two bodies collide if they are thrown upon each other along a straight line, while a smallest deviation from the straight line will prevent the collision (if the velocity is big enough). This implies the undecidability of the problem of determination of singularity with computable real _0: it requires the solution of equality relation of real numbers which does not exist. §.§ Sitnikov problemFrom now on, we will focus on a special case of the three-body problem, where two of the bodies are of equal positive mass, while the third body is massless and lies on a line, perpendicular to the plane of the motion of the first two bodies and passing through their center of mass(Fig. <ref>). Hence, the two bodies follow the unperturbed (Keplerian) orbit; in this problem, the elliptic orbit is the case.Let us place the center of mass at the origin, and the Z axis along the line where the third body is. Let us denote r(t) the distance from the first body (and the second, as their trajectories are symmetric) to the origin.Following Newtonian laws (<ref>), the coordinate of the third body, denoted as z, obeys the following differential equation:z̈ = -2 μ z/√(z^2 + r(t)^2)^3, where μ is the gravitational constant of the first and second bodies. Periodic function r(t) comes from the solution of the two-body problem: [r(t) = a(1-e cos E(t)); E(t) - e sin E(t) = √(%s/%s)μa^3(t - t_0) ] a (semimajor axis), e (eccentricity) and t_0 (epoch) are constants that can be calculated from the initial state of the two bodies. E(t) is the eccentric anomaly angle. The period of r(t) is P = 2π√(%s/%s)a^3μ.The initial values in the Sitnikov problem are:* a > 0, e ∈ (0..1), μ > 0 — parameters of the orbit of the two bodies;* z_0 = z(0) — initial position of the third body in the Z axis.* v_0 = ż(0) – initial velocity of the third body in the Z axis.* ϕ = E(0), 0 ≤ϕ < 2 π— initial value of the eccentric anomaly of the orbit of the two bodies.The state vector of the system is accordingly = (a,e,μ,z,v,E). a, e and μ do not depend on time; ż = v; v̇ = z̈ from (<ref>); Ė follows from (<ref>):[ = () = (0, 0, 0, v, z̈, Ė); z̈= -2μ z/√(z^2+a^2(1 - ecos E)^2)^3;Ė= √(μ a)/1 - e cos E ] Statement. IVP for the Sitnikov problem (<ref>) is polynomially reducible to the IVP for the three-body problem (<ref>).The study of the trajectories of z(t) in this system was started by Kolmogorov, while Sitnikov was the first to prove the existence of the oscillatory motions in this system <cit.>. His proof was also the first proof of this kind for three-body systems in general. In the Sitnikov problem, there are no singularities, and the functionis Lipschitz-continuous on the whole domain. From Eqs. (<ref>) and (<ref>), along with the fact that r(t) > 0, instantly follows thatis defined and continuous with any z, v, E ∈.Let us prove the Lipschitz-continuity ofby showing that all its partial derivatives w.r.t.are bounded. We write down those derivatives, skipping the zero ones:∂ v / ∂ v=1∂z̈ / ∂ z=-2μ(1/w^3-3z^2/w^5) ∂z̈ / ∂ E=-3μ z2a^2(1-ecos E)sin E/w^3 ∂Ė / ∂ E=-√(μ a)e sin E/(1 - e cos E)^2(Notion w=√(z^2+a^2(1 - ecos E)^2) is used for brevity.)It is evident that all those functions are defined and continuous for any z,v,E∈ (for (<ref>) it is important that 0<e<1). The boundedness of (<ref>) and (<ref>) is trivial. The boundedness of (<ref>) follows from the fact that it approaches zero as z→±∞: 1/w^3→ 0 and z^2/w^5→ 0. Similarly, (<ref>) is bounded because z/w^3→ 0 at z→±∞. Existence, uniqueness, and computability of the solution of the IVP for the Sitnikov problem follow from Theorem <ref> and the references given in Section <ref>.For the rest of the article, we consider the Sitnikov problem with z_0 = 0, omitting the solutions where the third body never crosses the plane. §.§ Combinatorial properties of the solutions of the Sitnikov problemSitnikov's result about the oscillatory motion was significantly extended by Alexeyev, who not only discovered the existence of all the classes of final motions in this problem, but also proved the following <cit.>:For any sufficiently small eccentricity e > 0 there exists an m(e) such that for any double-infinite sequence {s_n}_n∈, s_n ≥ m there exists a solution z(t) of the equation (<ref>) whose roots satisfy the equation⌊τ_k+1-τ_k/P⌋ = s_k, ∀ k ∈.The shortened version of the original theorem is given, excluding the finite and semi-infinite sequences. Alexeyev also proved a generalization of his theorem to the case when the third body has a nonzero mass. A simpler proof was later obtained by Moser <cit.>.In what follows, we restrict our analysis to t≥ 0, k≥ 0 (τ_0 = 0). Let C(T) be the set of (finite) sequences of the form (s_1,…, s_k), s_i ≥ m > 1, s_i 2 = 0, m2 = 0, for each of which any sequence (τ_0,…,τ_k+1) satisfying (<ref>) lies in the interval [0, T] (i.e. τ_k+1≤ T). |C(T)| has an asymptotic lower bound exponential in T.Obviously, C((m+1)P)=1. For some T≥(m+1)P, let us consider the interval [T, T + (m+1)P]. Any sequence (s_1,… s_k)∈ C(T) can be extended to a sequence from C(T+(m+1)P) by the following ways: * (s_1,…, s_k,m)∈ C(T+(m+1)P)* (s_1,…, s_k+2i)∈ C(T+(m+1)P), ∀ 0<i≤ m/2Consequently, |C(T+(m+1)P)| ≥ (m/2+1)|C(T)|, and that implies |C(T)|≥ (m/2+1)^T/(m+1)P for sufficiently large T. If m>0, this bound is exponential in T.§.§ Computational complexity of the IVP for the Sitnikov problemWe give two lemmas that describe important properties of z(t). The first lemma gives a lower bound of |z(t)| between two roots separated by a certain distance. In the proof of the lemma, the Sturm's comparison theorem is used:[Sturm's comparison theorem] Consider two equations:ẍ = - q(t) xandẍ = - Q(t) x,where q and Q are continuous functions. Let a nonzero solution of (<ref>) x(t) has roots a and b, and Q(t) > q(t) on t∈[a,b]. Then any solution of (<ref>) has a root on (a,b).Let z^*(t) be a solution of the Sitnikov problem (<ref>) with initial values a, e, μ, ϕ, v_0. According to the previous assumptions, let z^*(0) = 0. To be specific, we consider v_0>0 (the case of negative v_0 is a mirroring of that). Let τ be the smallest positive root ofz^*. Then ∃ t ∈ (0, τ) : z^*(t) ≥ h, whereh = H(τ) = √((2μτ^2/π^2)^2/3-a^2)Suppose z^*(t)<h, 0≤ t ≤τ. Since z^* is the solution of (<ref>), then it is also the solution of the following equation:z̈ = -2 μ z/√(z^*(t)^2 + r(t)^2)^3,where the factor of z depends only on t, but not on z. Let us denote this factor Q(t):z̈ = - Q(t) z.Since z^*(t)<h by the assumption, and r(t)≤ a, thenQ(t) > 2 μ/√(h^2 + a^2)^3Denotingq=2 μ / √(h^2 + a^2)^3,we write a differential equationz̈ = - q z.Since q>0 the equation (<ref>) is the equation of a harmonic oscillator. We examine its solution z^** for initial conditions z(0) = 0, ż(0) = v_0:z^**(t) = v_0 sin(√(q) t)By the Sturm's comparison theorem, between two roots of z^**—0 and π/√(q)—there exist roots of any solution of(<ref>), including z^*. Since τ was chosen as the smallest positive root of z^*, it must be that τ<π/√(q). However, by construction of q (<ref>) and h (<ref>) it follows τ=π/√(q), hence the contradiction. Consider a nonnegative function z(t), continuous and convex on [t_1, t_2]; let z(t_1) = z(t_2) = 0; let at some t∈[t_1,t_2] z(t)>h>0. Then ∃ t_a, t_b ∈ [t_1, t_2] : (t_b - t_a) > 3/4(t_2 - t_1), ∀ t∈ (t_a, t_b) z(t) > h/4. z(t) has one (strict) maximum at (t_1,t_2), let us say that t_3 is the point where the maximum is reached. Let us place points (Fig. <ref>): A(t_1, 0), B(t_3, z(t_3)), C(t_2, 0). Let the line z=h/4 cross AB at point D and BC at point E. Similarly, let the same line cross the z(t) curve at F and G.Since z(t) is convex, it lies above ABC, with the exception of A, B and C themselves (Fig. <ref>). Consequently, |DE| < |FG|. At the same time, from the similarity of triangles it follows that|DE|/|AC| = 1-h/4/z(t_3). Since z(t_3) > h and |AC| = (t_2 - t_1), we get |FG| > 3/4(t_2 - t_1). The horizontal coordinates of F and G are the desired t_a and t_b.The time complexity of an initial value problem for the Sitnikov problem with any fixed value of eccentricity does not have a polynomial upper bound. Suppose that there exists a Turing machine M that calculates the solution function of the IVP for the Sitnikov problem in time ((t), (ε)), whereis arbitrary polynomial.We examine the solutions at the interval t∈[0, T], T∈. From Lemma <ref> and Alexeyev's theorem, the number C(T) of different solutions z(t), forming different sequences(s_1,…,s_k) with s_k2 = 0, s_k ≥ m (m2 = 0), has a lower bound of (m/2+1)^T/(m+1)P, where m depends only on e. (The Alexeyev's theorem allows zero and odd m, but we can round the m up to be a nonzero even number, without trouble to the theorem.We build an algorithm for recovery of the sequence (s_1,…,s_k) that corresponds to a solution z(t) for some initial values, using our supposedly existing Turing machine M. We choose the parameters δ∈, δ < mP/2 and ε = 2^-l (l∈), ε < h/4, where h = H(mP). (Note that P is a computable real number.) Let us build on [0, T] a uniform grid with a step δ; on each node {t_i = iδ, 0 < i ≤⌊ T/δ⌋} we can compute the state of the system up to the precision ε. The grid has the following important properties: * If |z(t_i)| > h/4, then from Lemma <ref> follows that the closest root to t_i lies no farther thanmP/4.* From above it follows that two neighbor nodes can not both have |z|<h/4* Calculated z(t_i) can be divided into three classes: positive (z > 0 for sure), negative (z < 0 for sure) and undefined (the sign of z is not determined within the given precision).* Positive and negative nodes can go any number in a row, while there can be only one undefined node in a row.* From the estimate of the distance between roots, it is evident that if there are no nodes between a positive node and a negative node, or if there is (one) undefined node, then z has exactly one root in between.Given that the s_k are even, it is easily seen thatp nodes in a row of the same sign correspond to s_k = ⌈ (p+1)/2 ⌉; undefined nodes do not correspond to any s_k.It is not important how long it took to recover the sequence of s_k. What matters is that all the “calls” to out Turing machine M have used the same oracle for the computation of the (same) initial state. But, as we supposed, M did not have a chance to read more than ((t_i), (ε)) digits from the oracle tape for any t_i, which is no more than P(log_2 T, l); hence, basing on what it had read, it can possibly generate no more than 2^(log_2 T, l) different outcomes. At the same time, we proved that our algorithm recovers any of at least(m/2+1)^T/((m+1)P) sequences, which (as m>0) is not bounded by the said polynomial.§ CONCLUSION AND FUTURE WORKIn this work we examined the theoretical complexity of the initial value problem. We have shown that the lower time bound of that complexity can not be polynomial for the three-body problem (instantly meaning the absence of such a bound for the N-body problem). The choice of the three-body problem and oscillatory trajectories is not principal. We believe that similar results can be obtained in other systems, where, with the help of methods of symbolic dynamics, complex dynamical behavior can be shown and analyzed. We already mentioned homoclinic trajectories, discovered by Poincaré for the three body problem. It seems appropriate to quote his work “New methods of celestial mechanics” <cit.> here:“One is struck by the complexity of this figure I am not even attempting to draw. Nothing can give us a better idea of the complexity of the three-body problem and of all problems of dynamics where there is no holomorphic integral and Bohlin’s series diverge.”On a different note, for the integrable dynamical systems—those who have computable integrals of motion with good complexity bounds in t and ε— it is possible to derive complexity bounds for the initial value problem in our formal statement. Those bounds will be polynomial by log(t) and log(1/ε). That can point to a link between computational complexity of the IVP and integrability.On another different note, in this work the computational complexity of the IVP is examined at the “macro level” (rational t→∞), but what is left aside is the “micro level” (real t), where the precision of t plays an important role <cit.>. Another work is planned devoted to that case.99kawamura Akitoshi Kawamura, Hiroyuki Ota, Carsten Rösnick, Martin Ziegler. Computational Complexity of Smooth Differential Equations. In: Branislav Rovan, Vladimiro Sassone, Peter Widmayer (Eds.)Lecture Notes in Computer Science 7464: Mathematical Foundations of Computer Science, Springer-Verlag, 2012, 578–589.reif J. H. Reif, S. R. Tate. The Complexity of N-body Simulation. In: Proceedings of the 20th International Colloquium on Automata, Languages and Programming (ICALP '93), Springer-Verlag, London, 1993, 162–176.alexeyev1 V. M. Alekseev. Quasirandom dynamical systems. I. Quasirandom diffeomorphisms. Mathematics of the USSR-Sbornik(1968), 5(1):73.alexeyev2 V. M. Alekseev. Quasirandom dynamical systems. II. One-dimensional nonlinear oscillations in a field with periodic perturbation. Mathematics of the USSR-Sbornik(1968),6(4):505.alexeyev3 V. M. Alekseev. Quasirandom dynamical systems. III.Quasirandom oscillations of one-dimensional oscillators. Mathematics of the USSR-Sbornik(1969),7(1):1.alexeyev V. M. Alexeyev. Final motions in the three-body problem and symbolic dynamics. Russian Mathematical Surveys, Volume 36, Number 4, 1981, 181–200.burke James V. Burke, Ordinary Differential Equations. Existence and Uniqueness Theory. In: Math 555 Course Notes (Linear Analysis), University of Washington, 2015. URL: <www.math.washington.edu/ burke/crs/555/555_notes/exist.pdf>.collins Peter Collins, Daniel S. Graça. Effective Computability of Solutions of Ordinary Differential Equations. The Thousand Monkeys Approach. Electronic Notes in Theoretical Computer Science 221(25), 2008, 103–114.repin Svetlana Matculevich, Pekka Neittaanmäki, Sergey Repin.Guaranteed Error Bounds for a Class of Picard-Lindelöf Iteration Methods. In: Sergey Repin, Timo Tiihonen, Tero Tuovinen (Eds.) Computational Methods in Applied Sciences Vol. 27: Numerical Methods for Differential Equations, Optimization, and Technological Problems, Springer Netherlands, 2013, 175–189.babenko К. И. Бабенко. Основы численного анализа. Москва-Ижевск: НИЦ <<Регулярная и хаотическая динамика>>, 2002.beloriszky M. D. Belorizky. Application pratique des méthodes de M. Sundman à un cas particulier du problème des trois corps. Bulletin Astronomique 6 (2), 1930, 417–434.merman Г. А. Мерман. Бюллетень Института теоретической астрономии АН СССР. № 10(83), 713, 1958.sitnikov1960 K. A. Sitnikov. The Existence of Oscillatory Motions in the Three-Body Problem. Soviet Physics Doklady, Vol. 5, p. 647 (1961)moser Jürgen Moser. Stable and Random Motions in Dynamical Systems with Special Emphasis on Celestial Mechanics. Princeton University Press, 1973.poincare H. Poincaré. Les méthodes nouvelles de la mécanique céleste, volume 2. Paris: Gauthier-Villars, 1892. | http://arxiv.org/abs/1704.08762v3 | {
"authors": [
"N. N. Vasiliev",
"D. A. Pavlov"
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"categories": [
"cs.CC",
"math.DS"
],
"primary_category": "cs.CC",
"published": "20170427220743",
"title": "The computational complexity of the initial value problem for the three body problem"
} |
Three-dimensional structure of the magnetic field in the disk of the Milky Way A. Ordog1 J.C. Brown1R. Kothes2T.L. Landecker2 =============================================================================================== We give some new formulas about factorizations of K-k-Schur functions , analogous to the k-rectangle factorization formula R_t∪=R_t of k-Schur functions, whereis any k-bounded partition and R_t denotes the partition (t^k+1-t) called k-rectangle.Although a formula of the same form does not hold for K-k-Schur functions, we can prove thatdivides , and in fact more generally that P divides P∪ for any multiple k-rectangles P= and any k-bounded partition . We give the factorization formula of such P and the explicit formulas of P∪/P in some cases,including the case whereis a partition with a single part as the easiest example. § INTRODUCTION Let k be a positive integer. K-k-Schur functionsare inhomogeneous symmetric functions parametrized by k-bounded partitions ,namely by the weakly decreasing strictly positive integer sequences =(_1,…,_l), l∈_≥ 0, whose terms are all bounded by k. They are K-theoretic analogues of another family of symmetric functions called k-Schur functions, which are homogeneous and also parametrized by k-bounded partitions.Historically,k-Schur functions were first introduced by Lascoux, Lapointe and Morse <cit.>, and subsequent studies led to several (conjectually equivalent) characterizations ofsuch asthe Pieri-like formula due to Lapointe and Morse <cit.>, and Lam proved that k-Schur functions correspond to the Schubert basis of homology of the affine Grassmannian <cit.>.Moreover it was shown by Lam and Shimozono that k-Schur functions play a central role in the explicit description of the Peterson isomorphism between quantum cohomology of the Grassmannian and homology of the affine Grassmannian up to suitable localizations <cit.>.These developments have analogues in K-theory. Lam, Schilling and Shimozono <cit.> characterized the K-theoretick-Schur functions as the Schubert basis of the K-homology of the affine Grassmannian, and Morse <cit.> investigated them from a conbinatorial viewpoint, giving various properties including the Pieri-like formulas using affine set-valued strips (the form using cyclically decreasing words are also given in <cit.>). In this paperwe start from this combinatorial characterization (see Definition <ref>) and show certain new factorization formulas of K-k-Schur functions.Among the k-bounded partitions,those of the form (t^k+1-t)=(t,…,t_k+1-t)=:R_t, 1≤ t ≤ k, called k-rectangle, play a special role. In particular,if a k-bounded partition has the form R_t∪, where the symbol ∪ denotes the operation of concatenating the two sequences and reordering the terms in the weakly decreasing order, then the corresponding k-Schur function has the following factorization property <cit.>:R_t∪ = R_t.Note that, under the bijection between the set of k-bounded partitions and the set of affine Grassmannian elements in the affine symmetric group, the correspondent of the k-rectangle R_i is congruent,in the extended affine Weyl group,to the translation t_-ϖ_i^∨by the negative of a fundamental coweight, modulo left multiplication by the length zero elements. It is suggested in <cit.> thatthe K-k-Schur functions should also possess similar properties, including the divisibility of R_t∪ by R_t.The present work is an attempt to materialize this suggestion. We do show in Proposition <ref> thatR_t divides R_t∪in the ring =ℤ[h_1,…,h_k], where h_i denotes the complete homogeneous symmetric functions of degree i, of which the K-k-Schur functions form a basis. However, unlike the case of k-Schur functions, the quotient R_t∪/R_t is not a single termbut, in general, a linear combination of K-k-Schur functions with leading term , namely in whichis the only highest degree term. Even the simplest case whereconsists of a single part (r), 1≤ r ≤ k, displays this phenomenon: we show in Theorem <ref> thatR_t∪(r) =R_t·(r)( if t<r),R_t·((r) + (r-1) + ⋯ + ∅) ( if t≥ r)(actually we have (s) = h_s for 1 ≤ s ≤ k, and ∅=h_0=1). So we may ask:Question 1. Which μ, besides , appear in the quotient /? With what cofficients? A k-bounded partition can always be written in the form R_t_1∪…∪ R_t_m∪ with not having so many repetitions of any part as to form a k-rectangle. In such an expression we temporarily callthe remainder, although this term is only used in the Introduction. Proceeding in the direction of Question 1, one ultimate goal may be to give a factorization formula in terms of the k-rectangles and the remainder. In the case of k-Schur functions, the straightforward factorization in (<ref>) above leads to the formula ∪=R_t_1…R_t_m. On the contrary, with K-k-Schur functions, the simplest case having a multiple k-rectangle, to be shown in the author's following paper <cit.>, givesR_t∪ R_t = ∑_⊂ R_t.Hence we cannot expect R_t∪ R_t to be divisible bytwice. Instead, upon organizing the part consisting of k-rectangles in the form with t_1 < … < t_m and a_i ≥ 1 (1≤ i ≤ m), with R_t^a = R_t∪…∪ R_t_a, actually we show in Proposition <ref> that∪ is divisible by , which actually holds whether or notis the remainder. Then we can subdivide our goal as follows:Question 1'. Which μ, besides , appear in the quotient P∪/P where P=, and with what coefficients? Question 2. How canbe factorized? In this (and author's following) paper, we give a reasonably complete answer to Question 2, and partial answers to Question 1'. For Question 2, we first show in Theorem <ref> that multiple k-rectangles of different sizes entirely split, namely that we have =R_t_1^a_1…R_t_m^a_m. Then in the following paper <cit.>, we show that for each 1≤ t ≤ k and a>1, we have a nice factorizationR_t^a=(∑_⊂ R_t)^a-1, generalizing the formula (<ref>). Thus, we have= R_t_1(∑_^(1)⊂ R_t_1^(1))^a_1-1…R_t_m(∑_^(m)⊂ R_t_m^(m))^a_m-1. For Question 1', unfortunately we cannot give a complete answer yet. Still we obtain some nice explicit formulas, including the case (<ref>).We first show an auxiliary result that, being given P= and putting Q= without multiplicities, we have P∪/P = Q∪/Q for any . Thus we can reduce Question 1' to the case where the k-rectangles are of all different sizes.We then derive explicit formulas in some limited cases where, writing =(_1,…,_l) and =(_1,…,_l-1), the parts ofexcept for _l are all larger than the widths of the k-rectangles,andis contained in a k-rectangle.An easiest case is where =(r) consists of a single part,which generalizes the case (<ref>).Namely we show that if P= with t_1<…<t_m and a_1,…,a_m>0 and 0≤ r ≤ k, then P∪(r) decomposes as P·∑_s=0^rr+s-1s(r-s), where u = #{i| 1≤ i ≤ m, t_i≥ u}.Considering the case m=1 and a_1=1, we obtain the formula (<ref>).Generalizing this case,we show in Theorem <ref> thatif P and u are the same as above and =(_1,…,_l) satisfies _l-1>t_m and =(_1,…,_l-1) is contained in a k-rectangle, then P∪ decomposes as P·∑_s=0^_l_l+s-1s∪(_l-s). In particular, if t_n < _l, the summation on the right-hand side consists of a single term .[scale=0.12] (0,0) -| (13,1) -| (11,3) -| (0,0);(10,1.5) to [out=45,in=180] (13,3) node [right] ; (0,3) rectangle (9,6); (8,4.5) to [out=45,in=180] (11,6.5) node [right] R_t_m;(0,8) rectangle (7,12);(6,10) to [out=45,in=180] (9,12) node [right] R_t_p+1;(0,13) rectangle (5,17);(4,15) to [out=45,in=180] (7,17) node [right] R_t_p;(0,20) rectangle (3,25); (2,22.5) to [out=45,in=180] (5,24.5) node [right] R_t_1;[pattern=north east lines] (0,12) rectangle (6,13);(0,12) to [out=-20,in=-160] node[inner sep=0pt] (la)(6,12);[<-] (la) to [out=-135,in=0] (-2,10) node [left] _l; (0,6) – (0,8);[loosely dotted, thick] (4.5,6.2) – (3.5,7.8);(0,17) – (0,20);[loosely dotted, thick] (2.5,17.2) – (1.5,19.8); [right,text width=5cm] at (23,12.5)In this figure p=m-_l and a_i=1 for all i. ; (0,5) rectangle (7,11);at (3.5,8) R_t_n;(0,11) rectangle (5,18);at (2.5,14.5) R_t_1; [pattern=north east lines] (0,18) rectangle (5,19);[pattern=north east lines] (5,11) rectangle (6,12); (0,19) to [out=15,in=165] node[above] _l (6,19);[loosely dotted, thick] (6,19) – (6,12); It is worth noting that, in all cases we have seen,P∪/P is a linear combinationof K-k-Schur functionswith positive coefficients. Moreover,if P=R_t, it seems that each coefficient is 0 or 1 and the set of μ such that the coefficient of μ in P∪/P is 1 is an interval (with respect to the strong order. See Conjecture <ref>). Anyway, it should be interesting to study the geometric meaning of these results and conjectures.This paper is organized as follows. In Section <ref>, we review some basic notations and facts about combinatorial backgrounds of K-k-Schur functions.In Section <ref>, we show some auxiliary results which provide a basis for our work.In Section <ref>, we give explicit factorization formulas in an easiest case where the remainder consists of a single part.In Section <ref>, we generalize the result of the previous section and give a “straightforward factorization” formula fora multiple k-rectangles of different sizes. In Section <ref>, we state some observations and conjectures.Acknowledgement.The author would like to express his gratitude to T. Ikeda for suggesting the problem to the author and helping him with many fruitful discussions.He is grateful to H. Hosaka and I. Teradafor many valuable comments and pointing out mistakes and typos in the draft version of this paper. He is also grateful to the committee of the 29th international conference on Formal Power Series and Algebraic Combinatorics (FPSAC) for many valuable comments for the extended abstract version of this paper.This work was supported by the Program for Leading Graduate Schools, MEXT, Japan.The contents of this paper is the first half of the author's master-thesis <cit.>.§ PRELIMINARIES In this section we review some requisite combinatorial backgrounds. For detailed definitions, see for instance <cit.> or <cit.>. §.§ Partitions Let 𝒫 denote the set of partitions.A partition =(_1 ≥_2 ≥…)∈𝒫 is identified with its Young diagram (or shape), for which we use the French notation here. quadrant[We use the French notation] of Cartesian plane so that there are _i boxes arranged in left justified way in the i-th row from the bottom.[scale=0.25] [->] (-0.5,0) – (6,0); [->] (0,-0.5) – (0,3); (0,0) rectangle (1,1); (0,1) rectangle (1,2); (1,0) rectangle (2,1); (1,1) rectangle (2,2); (2,0) rectangle (3,1); (3,0) rectangle (4,1); [below] at(3,-1) the Young diagram of (4,2);We denote the size of a partitionby ||, the length by l(), and the conjugate by '.For partitions , μ we say ⊂μ if _i≤μ_i for all i. The dominance orderon 𝒫 is defined by saying that μ if ||=|μ| and ∑_i=1^r_i≤∑_i=1^rμ_i for all r≥ 1.Sometimes we abbreviate horizontal strip (resp. vertical strip) (of size r) to (r-)h.s. (resp. (r-)v.s.). For a partitionand a cell c=(i,j) in ,we denote the hook length of c inby c=_i+'_j-i-j+1. For a partition , a removable corner of(or -removable corner) is a cell (i,j)∈ with (i,j+1),(i+1,j)∉. (i,j)∈(_>0)^2 is said to be an addable corner of(or -addable corner) if (i,j-1),(i-1,j)∈ with the understanding that (0,j),(j,0)∈.In order to avoid making equations too wide, we may denote removable corner (resp. addable corner) briefly byrem. cor. (resp. add. cor.).A cell (i,j)∈ is called extremal if (i+1,j+1)∉.For partitions =(_1,…,_l()), μ=(μ_1,…,μ_l(μ)),we write ⊕μ = (_1+μ_1,_2+μ_1,…,_l()+μ_1,μ_1,…,μ_l(μ)).For partitions ^(1),…,^(n), we define ^(1)⊕⋯⊕^(n) = (^(1)⊕⋯⊕^(n-1))⊕^(n), recursively.[scale=0.16](0,15) – (1,15) – (1,14) – (3,14) – (3,13) – (4,13) – (4,12) – (0,12) – cycle;(2,13) to [out=45,in=180] (6,14.5) node[right] ^(n); (4,12) – (6,12) – (6,10) – (8,10) – (8,8) – (4,8) – cycle;(6,9) to [out=45,in=180] (9,10.5) node[right] ^(n-1); (11,5) – (13,5) – (13,4) – (16,4) – (16,2) – (17,2) – (17,0) – (11,0) – cycle;(15,1.5) to [out=45,in=180] (19,3) node[right] ^(1);[loosely dotted,very thick] (8,8) – (11,5);(0,12) – (0,0) – (11,0);the shape of ^(1)⊕…⊕^(n) §.§ Bounded partitions, cores, affine Grassmannian elements, and k-rectangles R_t A partition λ is called k-bounded if λ_1 ≤ k. Letbe the set of all k-bounded partitions. An r-core (or simply a core if no confusion can arise) is a partition none of whose cells have a hook length equal to r. We denote by 𝒞_r the set of all r-core partitions. When we consider a partition as a core, the notion of size differs from the usual one: the length (or size) of an r-coreis the number of cells inwhose hook length is smaller than r, and denoted by ||_r. The affine symmetric group S_k+1 is given bygenerators { s_0,s_1,…,s_k} and relations s_i^2=1, s_i s_i+1 s_i = s_i+1 s_i s_i+1, s_i s_j = s_j s_i for i-j ≢0,1,k(k+1), with all indices are considered mod (k+1).Note that the symmetric group S_k+1 generated by {s_1,…,s_k} is a subgroup of S_k+1.We identify the left cosets of S_k+1/S_k+1 with their minimal length representatives, which we call affine Grassmannian elements. Namely, the set of affine Grassmannian elements is { w∈ S_k+1| l(w s_i)>l(w) (∀ i≠ 0)}.Hereafter we fix a positive integer k. For a cell c=(i,j), the content of c is j-i and the residue of c is res(c)=j-i(k+1) ∈ℤ/(k+1).For a set X of cells, we write Res(X) = { (c) | c∈ X }. We will write a -removable corner of residue i simply a -removable i-corner.For simplicity of notation, we may use an integer to represent a residue, omitting “mod (k+1)”. We denote by R_t the partition(t^k+1-t)=(t,t,…,t) ∈ for 1≤ t ≤ k, which is called a k-rectangle. Naturally a k-rectangle is a (k+1)-core.Now we recall the bijection between the k-bounded partitions in , the (k+1)-cores in , and the affine Grassmannian elements in :@<0.5ex>[rr]^[rd]_by taking “word”@<0.5ex>[ll]^[ru]_The mapsand :The map ; ↦ is defined by _i=#{j| (i,j)∈, (i,j)≤ k}.The map ; ↦ is defined by the following procedure: given a k-bounded partitionthen work from the smallest part to the largest. For each row, calculate the hook lengths of all its cells. If there is a cell with hook length greater than k, slide this row to the right until all its cells have hook length not greater than k. In the end this process produces a skew shape μ/ν, where in fact μ is a (k+1)-core. Then letbe this μ.Then in factandare bijective and =^-1. See <cit.> for the proof. The next lemma givesa more explicit description for , which follows from the argument given just before <cit.>:For ∈ and j≥ 1, ()_j = ()_j+k+1-_j+_j.Note that ifis contained in a k-rectangle then ∈ and ∈, and besides ()==().The mapand the inverse: For ∈ and i=0,1,…,k, we define s_i· as follows: * if there is a -addable i-corner, then let s_i· be with all -addable i-corners added, * if there is a -removable i-corner, then let s_i· be with all -removable i-corners removed, * otherwise, let s_i· be . In fact first and second case never occur simultaneously and s_i·∈ and then we have a well-defined S_k+1-action onand it induces a bijection: ⟶ ; w ↦ w·∅. The inverse map is given by; ↦() ↦ w_(),where w_ is the affine permutation s_i_1 s_i_2… s_i_l, where (i_1,i_2,…,i_l) is the sequence obtained by reading the residues of the cells in , from the shortest row to the largest, and within each row from right to left.See <cit.> for the proof. §.§ Weak order and weak strips In this subsection we review the weak order on ≃≃.For a k-bounded partition , its k-conjugateis also a k-bounded partition given by =(()'). The weak order ≺ onis defined by the following covering relation:wv :∃ i such that s_i w=v, l(w)+1=l(v). It is transferred to andby the bijection described aboveas follows:on : μ ⊂μ, ⊂μ, ||+1=|μ|. on : τ ∃ i such that s_i τ=, |τ|_k+1+1=||_k+1. (<ref>)(<ref>): see <cit.>. (<ref>)(<ref>): see <cit.>. For (k+1)-cores τ⊂∈, /τ is called a weak strip of size r (or a weak r-strip) if the following equivalent conditions hold:* /τ is horizontal strip and τ∃τ^(1)…∃τ^(r) =.* /τ is horizontal strip and =τ+r and #Res(/τ)=r.* ()/(τ) is a horizontal strip and (')/(τ') is a vertical strip and = τ+r.* = s_i_1… s_i_rτ for some cyclically decreasing element s_i_1… s_i_r. (Here, an affine permutation w=s_i_1… s_i_r (a reduced expression) is called cyclically decreasing if i_1,…,i_r are distinct and j never precedes j+1 (taken modulo k+1) in the sequence i_1 i_2 … i_r. This definition is in fact independent of which reduced expression we choose. )(1)(3): see <cit.>. (3)(1): see <cit.>. (3)(2): see <cit.>.(4)(1), (1)(4):see Appendix <ref>.(2)(1): omitted since (2) is not used in this paper.§.§ Symmetric functions Let =[h_1,h_2,…] be the ring of symmetric functions, generated by thecomplete symmetric functions h_r = ∑_i_1≤ i_2≤…≤ i_r x_i_1… x_i_r. For a partitionwe seth_ = h__1h__2… h__l(). Then {h_}_∈𝒫 forms a -basis of . The Schur functions {s_}_∈𝒫 are the family of symmetric functions satisfying the Pieri rule:h_r s_ = ∑_μ/:horizontal r-strip s_μ. Note that h_r s_ = s_∪(r) + ∑_μ∪(r) a_μs_μ for some a_μ. Using this repeatedly, we can write h_ = s_ + ∑_μ K_μ s_μ for some coefficients K_μ. Thus Schur functions {s_}_∈𝒫 form a basis ofsince the transformation matrix between {s_}_ and {h_μ}_μ is unitriangular.§.§ k-Schur functions We recall a characterization of k-Schur functions given in <cit.>, since it is a model for and has a relationship with K-k-Schur functions.[k-Schur function via “weak Pieri rule”] k-Schur functions {}_∈ are the family of symmetric functions such that ∅ =1, h_r= ∑_μμfor r≤ k and μ∈,summed over μ∈ such that (μ)/() is a weak strip of size r. According to the fact that if (ν)/(η) is a weak strip then ν/η is a horizontal strip, we can writeh_ =+ ∑_μ∈ K^(k)_μμ for ∈ by the same argument as the case of Schur functions, which ensures the well-definedness ofand shows that {}_∈ forms a basis of =[h_1,…,h_k]⊂. In additionis homogeneous of degree ||. Note that (r)=h_r for 1≤ r ≤ k since ()/∅ is a weak r-strip if and only if =(r). In <cit.> it is proved that if _1+l()≤ k+1 (in other words ⊂ R_t for some t) then =s_.It is proved in <cit.> thatFor 1≤ t ≤ k and ∈, we have R_t∪ = R_t (=s_R_t). §.§ K-k-Schur functions In <cit.> a combinatorial characterization of K-k-Schur functions is given via an analogue of the Pieri rule, using some kind of strips called affine set-valued strips.For a partition , (i,j)∈(_>0)^2 is called -blocked if (i+1,j)∈. [affine set-valued strip] For r≤ k, (/β,ρ) is called an affine set-valued strip of size r (or an affine set-valued r-strip) if ρ is a partition and β⊂ are cores both containing ρ such that * /β is a weak (r-m)-strip where we put m=#Res(β/ρ), * β/ρ is a subset of β-removable corners, * /ρ is a horizontal strip, * For all i∈(β/ρ), all β-removable i-corners which are not -blocked are in β/ρ. In this paper we employ the following characterization <cit.> of the K-k-Schur function as its definition. [K-k-Schur function via an “affine set-valued” Pieri rule] K-k-Schur functions {}_∈ are the family of symmetric functions such that∅=1 andfor ∈ and 0 ≤ r ≤ k,h_r · = ∑_(μ,ρ) (-1)^||+r-|μ|μ,summed over (μ,ρ) such that ((μ)/(),ρ) is an affine set-valued strip of size r.Notice that,given a weak strip /β, taking a ρ such that (/β,ρ) becomes an affine set-valued strip is equivalent to choosing a subset of the set of residues i∈ℤ/(k+1) where there is at least one -nonblocked β-removable i-corner.Now we introduce a notation for the convinience: For partitions ,μ, we denote by r_μ the number of distinct residues of -nonblocked μ-removable corners.Then for a fixed weak (r-m)-strip /β, the number of ρ such that (/β,ρ) is an affine set-valued r-strip is equal to r_βm. Notice that /β with all -nonblocked β-removable corners added is a horizontal strip. Therefore we can rewrite Definition <ref>: For ∈ and 0 ≤ r ≤ k,h_r · = ∑_s=0^r (-1)^r-s∑_μ (μ)/():weak s-stripr_(μ)()r-sμ. We can prove similarly that is uniquely determined by (<ref>): for ∈ and 1≤ r≤ k, we have h_r= ∪(r) + ∑_μ a_μμ with μ∈ satisfying |μ|<|∪(r)| or μ∪(r). Thus, for ∈ we can write h_ =+ ∑_μ𝒦^(k)_μμ, summed over μ∈ satifying |μ|<|| or μ. Henceis well-defined and {}_∈ forms a basis of . Note that (r)=h_r for 1≤ r ≤ k since if ((μ)/∅,ρ) is an affine set-valued r-strip then (μ,ρ)=((r),∅). Moreover, thoughis an inhomogeneous symmetric function in general, from the form of (<ref>) we can deduce that the degree ofis || and its homogeneous part of highest degree is equal toby using induction. §.§ Some properties of bounded partitions and cores In this section we review some properties which show that the k-rectangles R_t=(t^k+1-t) are important and thus it can be expected that there are some good properties of 's wherecan be written in the form =R_t∪μ.Recall the weak order ≺ of Definition <ref>. For μ,∈ and P=R_t_1^a_1∪⋯∪ R_t_m^a_m (1≤ t_1< ⋯ < t_m ≤ k and a_1,…,a_m∈_>0), ∪ P ≼μ∃ν∈, μ = ν∪ P,≼ν. See <cit.> for the case where m=1 and a_1=1 (i.e. P=R_t_1). The general case follows by using this case repeatedly.For ν,∈ and P=R_t_1^a_1∪⋯∪ R_t_m^a_m (1≤ t_1< ⋯ < t_m ≤ k and a_1,…,a_m∈_>0), (ν)/() is a weak strip(ν∪ P) / (∪ P) is a weak strip The case where m=1 and a_1=1 (i.e. P=R_t_1) is proved in the proof of <cit.>. The general case follows by using this case repeatedly. For η,∈ and P=R_t_1^a_1∪⋯∪ R_t_m^a_m (1≤ t_1< ⋯ < t_m ≤ k and a_1,…,a_m∈_>0), (μ)/(∪ P) is a weak strip∃ν∈, μ = ν∪ P,(ν)/() is a weak strip. : We have ∪ P≼μ by the definition of weak strips. Thus we can write μ=∃ν∪ P by Proposition <ref>. Then we have that (ν)/() is a weak strip by Proposition <ref>. ⟹: By Proposition <ref>. Let ∈, 1≤ t≤ k, and let r∈_≥ 0 such that _r ≥ t ≥_r+1, where we regard _0=∞. Put = ∪ R_t. Then()_i =()_i+t (if i≤ r+(k+1-t)),()_i-(k+1-t)(if i≥ (r+1)+(k+1-t)).The latter case is obvious since _i+(k+1-t)=_i for i≥ r+1.For i = r+(k+1-t),…,r+1,()_i =()_i+k+1-_i + _i (by Lemma <ref>)=()_i+k+1-t+t (since _i=t)=()_i + t. (by the latter case)Then for i=r,r-1,…,1, by descending induction on i,()_i = ()_i+k+1-_i+_i(by Lemma <ref>)= ()_i+k+1-_i_≤ r+k+1-t + _i(since i≤ r)= ()_i+k+1-_i + t + _i(induction hypothesis)= ()_i + t.(by Lemma <ref>)Remark. There are more than one candidates for r ifhas a part equal to t, thus in such situations both equalities of the above lemma may hold for some i.§ POSSIBILITY OF FACTORING OUTAND SOME OTHER GENERAL RESULTSRecall how to prove the formula R_t ∪=R_t in <cit.>: first consider a linear map Θ extending ↦R_t∪ for all ∈. Then from the weak Pieri rule it was shown that it commutes with the multiplication by h_r, and thus that Θ coincides with the multiplication by R_t. In the case of K-k-Schur functions, a similar map Θ does not commute with the multiplication of h_r since the Pieri rule is different in lower terms. However, it holds that R_t divides R_t ∪. We prove it in a slightly more general form.The following notation is often referred later: Let 1≤ t_1,…,t_m ≤ k be distinct integers and a_i∈_>0 (1≤ i≤ m), where m∈_>0. Then we put P =, u = #{t_v| 1≤ v ≤ m, t_v≥ u}for each u∈_>0.Let P be as in the above . Then, for λ=(λ_1,⋯,λ_l) ∈,we have P | λ∪ P in the ring . Remark.Note thatmay still have the form =R_t∪μ. Hereafter we will not repeat the same remark in similar statements. we prove it by induction on λ, with respect to the order ≤ defined by μ≤ |μ|<|| or (|μ|=|| and μ).The statement is obvious when λ = ∅.Assume ≠∅ and put λ̂ = (λ_1,⋯,λ_l-1). ThenP∪λ̂·(λ_l) =P∪λ + ∑_μ a_λμP∪μfor some coefficients a_λμ, since adding a weak strip to P∪ yields a k-bounded partition in the form of P∪μ for some μ∈, by Proposition <ref>. Here μ in the summation runs under the condition |μ| < |λ| or μλ. By induction hypothesis P∪λ̂ and P∪μ are divisible by P if |μ|<|λ| or μλ. This completes the proof. Since the homogeneous part of highest degree ofis equal tofor any , it follows from Propositions <ref> and <ref> that Let P be as in . Then, for any ∈, we can write P ∪ = P( + ∑_μ a_μμ), summing over μ∈ such that |μ|<||, for some coefficients a_μ (depending on P).Now we are interested in finding a explicit description of P∪/P. Let us consider the case P=R_t for simplicity.As noted above,a linear map Θ extending ↦ (∀∈) does not coincide with the multiplication ofbecause it does not commute with the multiplication by h_r in the first place.However, in the remaining part of this section, we can prove that the restriction of Θ to the subspace spanned by {R_t∪μ}_μ∈ (in fact this is the principal ideal generated by ) commutes with the multiplication by h_r, and thus it coincides with the multiplication of Θ()/ = R_t∪ R_t/ on that ideal (Proposition <ref>). Thus it is of interest to describe the value of R_t∪ R_t/, which is shown to be ∑_ν⊂ R_tν in the author's following paper <cit.>.Now let us begin with seeing how Θ and the multiplication by h_r do not commute.Recall the K-k-Schur version of the Pieri rule (<ref>)h_r · = ∑_s=0^r (-1)^r-s∑_ν (ν)/():weak s-stripr_(ν)()r-sν,and compare with the formula obtained by replacingwith R_t ∪:h_r ·R_t∪ = ∑_s=0^r (-1)^r-s∑_η (η)/(R_t∪):weak s-stripr_(η)(R_t∪)r-sη.By Corollary <ref>, the summation in (<ref>) is formed for all η having the form η=R_t∪ν such that (ν)/() is a weak s-strip. Hence the right-hand side of (<ref>) differs from what is obtained by replacing each ν in the right-hand side of (<ref>) by R_t∪ν according to the difference betweenr_(ν)() andr_(R_t∪ν)(R_t∪).The next lemma says r_(R_t∪ν)(R_t∪)=r_(ν)() holds ifhas a part equal to t. For ν,∈ such that has a part equal to t and (ν)/() is a weak strip, we have r_(ν),() = r_(ν∪ R_t),(∪ R_t). We write = ∪ R_t and ν̃ = ν∪ R_t. We take r such that _r = t > _r+1 (then ν_r ≥ t = _r ≥ν_r+1 since ν/ is a horizontal strip). Then we have _r = _r+1 = ⋯ = _r+k+1-t = tν̃_r+1 = ⋯ = ν̃_r+k+1-t = t,therefore by Lemma <ref> ()_i = ()_i + t (i ≤ r+(k+1-t))()_i = ()_i-(k+1-t)(i ≥ r+(k+1-t)) (here we applied Lemma <ref> toand r-1 for the lower equation) and (ν)_i = (ν)_i + t (i ≤ r+(k+1-t))(ν)_i = (ν)_i-(k+1-t)(i > r+(k+1-t)). [scale=0.26][red,very thick] (5,3) |- (8,2) |- (12,1) – (12,0);[red,very thick] (5,6) |- (8,5) |- (12,4) – (12,3);[red,very thick] (18,3) |- (21,2) |- (25,1) – (25,0);(0,0) |- (3,4) |- (5,3) |- (8,2) |- (12,1) |- (18,0) |- (26,-1) – (26,-2);[loosely dotted,thick] (0,0) – (0,-2);[loosely dotted,thick] (26,-2) – (26,-3);[pattern=north east lines] (0,4) rectangle (2,5);[pattern=north east lines] (5,2) rectangle (6,3);[pattern=north east lines] (8,1) rectangle (10,2);[pattern=north east lines] (12,0) rectangle (14,1); [pattern=dots] (-0.1,7) rectangle (2,8);[pattern=dots] (5,5) rectangle (6,6);[pattern=dots] (8,4) rectangle (10,5);[pattern=dots] (12,3) rectangle (14,4); [pattern=dots] (18,2) rectangle (19,3);[pattern=dots] (21,1) rectangle (23,2);[pattern=dots] (25,0) rectangle (27,1); [pattern=north east lines] (18,-1) rectangle (22,0);[pattern=dots] (31,-1) rectangle (35,0);[blue,decorate,decoration=zigzag,segment length=2mm,amplitude=.3mm] (-0.1,-0.1) |- (3,7) |- (5,6) |- (8,5) |- (12,4) |- (18,3) |- (21,2) |- (25,1) |- (31,0) |- (39,-1) – (39,-2);[blue,loosely dotted,thick] (0,0) – (0,-2);[blue,loosely dotted,thick] (39,-2) – (39,-3); [red,thick,->,decorate,decoration=snake,amplitude=.4mm] (7,2) to (7,5);[<-,red] (7,3.5) to [out=10, in=260] (10,6) node[above] k+1-t;[red,thick,->,decorate,decoration=snake,amplitude=.4mm] (8,1.5) to node[above]t (21,1.5); [loosely dotted,thick] (0,0) – (12,0);[loosely dotted,thick] (0,3) – (3,3); [loosely dotted,thick] (6,3) – (12,3); [left] at (0,-0.5) r;[left] at (0,0.5) r+1;[left] at (0,2.5) r+k+1-t; ( Here [scale=0.33] (0,0)–(2,0);: outline of () [scale=0.33] [blue,decorate,decoration=zigzag,segment length=2mm,amplitude=.3mm] (0,0)–(2,0);: outline of () [scale=0.33] [pattern=north east lines] (0,0) rectangle (2,1);: (ν)/() [scale=0.33] [pattern=dots] (0,0) rectangle (2,1);: (ν)/()) Then, * if i < r+(k+1-t), (i,()_i) is a ()-removable corner(i,()_i) is a ()-removable corner, * if i ≥ r+(k+1-t), (i,()_i) is a ()-removable corner(i-(k+1-t),()_i-(k+1-t)) is a ()-removable corner. Moreover, when (i,()_i) is a ()-removable corner (of residue a), we consider two cases: * if i < r+(k+1-t). Then (i,()_i) is (ν)-blocked ()_i ≤(ν)_i+1()_i+t ≤(ν)_i+1+t (i,()_i) is (ν)-blocked, and the residue of (i,()_i) is a-t. * if i ≥ r+(k+1-t). Then (i,()_i) is (ν)-blocked ()_i ≤(ν)_i+1()_i-(k+1-t)≤(ν)_i+1-(k+1-t)(i-(k+1-t),()_i-(k+1-t)) is (ν)-blocked, and the residue of (i-(k+1-t),()_i-(k+1-t)) is a-t. Hence, for each a ∈/(k+1), there exists a non-(ν)-blocked ()-removable a-corner if and only if there exists a non-(ν)-blocked ()-removable (a-t)-corner. Therefore we have r_(ν)() = r_(ν)(). As a corollary of the proof of the above lemma, we have For any ,ν∈ and 1≤ t≤ k we have r_(R_t∪ν)(R_t∪)=r_(ν)() or r_(ν)()+1. Take r such that _r≥ t>_r+1 and do a same argument as the above lemma. Then we have that, if i≠ r+(k+1-t), there exists a (ν)-nonblocked ()-removable a-corner in i-th row if and only if there exists a (ν)-nonblocked ()-removable (a-t)-corner in i'-th row. (Here we put i'=i if i<r+(k+1-t) and i'=i-(k+1-t) if i>r+(k+1-t)) Hence we have r_(ν)()≤ r_(ν)()≤ r_(ν)() + 1.For λ∈ and 1 ≤ t ≤ k, we have λ∪ R_t ∪ R_t = λ∪ R_t·R_t∪ R_t/R_t.Write μ̃= μ∪ R_t for μ∈.Define a linear map Θ : by μμ̃ for all μ∈ and put X = span{|λ∈}. Then X is an ideal of because h_r· can be written as a linear combination of {ν̃|ν∈}, by (<ref>) and Proposition <ref>.Next we claimΘ|_X ∘ (h_r·) = (h_r·) ∘Θ|_X:X Xfor 1≤ r ≤ k, where h_r· denotes the multiplication by h_r.Proof of claim.It suffices to show h_r·μ∪ R_t = Θ(h_r·μ̃) for μ∈. More generally, we can show h_r·μ∪ (t) = Θ(h_r·μ∪ (t)) for μ∈: h_r·μ∪ (t) = ∑_s=0^r (-1)^r-s∑_η (η)/(μ∪(t)) is a weak s-stripr_(η),(μ∪(t))r-sη= ∑_s=0^r (-1)^r-s∑_ν (ν)/(μ∪(t)) is a weak s-stripr_(ν̃),(μ∪(t))r-sν̃= ∑_s=0^r (-1)^r-s∑_ν (ν)/(μ∪(t)) is a weak s-stripr_(ν),(μ∪(t))r-sν̃= Θ(∑_s=0^r (-1)^r-s∑_ν (ν)/(μ∪(t)) is a weak s-stripr_(ν),(μ∪(t))r-sν) = Θ(h_r·μ∪(t)).Here the second equality uses Proposition <ref>, and the third equality uses Lemma <ref>. Hence the claim is proved.Since h_1,…,h_k generate , the claim implies that Θ|_X is a -module homomorphism. Hence for any x ∈ X, x ·Θ(R_t) = Θ(xR_t) = Θ(x) ·R_t,which implies Θ(x) = x ·R_t∪ R_t/R_t for any x ∈ X. Setting x=R_t∪ gives the proposition.Let P= be as in , and put Q=R_t_1∪⋯∪ R_t_m. Then, for ∈ we haveP∪/P=Q∪/Q. Induction on ∑_i (a_i-1). If ∑_i (a_i-1)=0 then it is obvious since P=Q. Otherwise, we can assume a_1>1 without loss of generality. Write P=R_t_1∪ R_t_1∪ P'. By Proposition <ref> we have P'∪∪ R_t_1∪ R_t_1/P'∪∪ R_t_1 = R_t_1∪ R_t_1/R_t_1 = P'∪ R_t_1∪ R_t_1/P'∪ R_t_1, thus we conclude P'∪∪ R_t_1∪ R_t_1/P'∪ R_t_1∪ R_t_1 = P'∪∪ R_t_1/P'∪ R_t_1 = Q∪/Q. Here we used induction hypothesis for the second equality.§ A FACTORIZATION OF ∪(R) In this section we will give an explicit formula for ∪/ when =(r).Roughly speaking, K-k-Schur functions can be calculated by “solving” the system of Pieri rule formulas (<ref>). To solve such a system, it is important to understand concretely what weak strips (ν)/(μ) are.If μ is a union of k-rectangles P= the situation is simple:if (ν)/(P) is a weak strip then ν has the form P∪(s) for some s, as we will see in the proof of the following proposition. Thus the Pieri rule also has a simple explicit expression as follows: Let P and u (u ∈_>0) be as in in Section <ref>, before Proposition <ref>.Then, for 1 ≤ r ≤ k, we haveP· h_r = ∑_s=0^r(-1)^r-ss+1r-sP∪(s).We have (P) = R_t_m⊕…⊕ R_t_m_a_m⊕⋯⊕R_t_1⊕…⊕ R_t_1_a_1 and all addable corners of (P) has the same residue, say i. Moreover, (P) has a total of ∑_j a_j removable corners, a_j of which are derived from the removable corner of R_t_j and having the residue i+t_j for each j.Next we claim that if γ/(P) is a weak s-strip then γ = s_i+s-1⋯ s_i+1s_i((P)).Proof of the claim. We prove it by induction on s. If s=1, it is obvious because all addable corners of (P) have the same residue i.Let s>1 and γ/(P) be a weak s-strip. Then we can write γ = s_j_s⋯ s_j_2s_j_1((P)),where (j_s,⋯,j_1) is cyclically decreasing (see Definition-Proposition <ref>(4)).Since s_j_s-1⋯ s_j_2s_j_1((P))/(P) is a weak (s-1)-strip, we have (j_s-1,⋯,j_1)=(i+s-2,⋯,i+1,i) by the induction hypothesis. Since (j_s,i+s-2,⋯,i+1,i) is cyclycally decreasing, we have j_s ∉{i-1,i,i+1,⋯,i+s-2}.If j_s ≠ i+s-1, then s_j_s commutes with s_i,s_i+1,⋯,s_i+s-2 andγ = s_j_ss_i+s-2⋯ s_i+1s_i((P)) = s_i+s-2⋯ s_i+1s_is_j_s((P)).However, |s_j_s((P))|_k ≤ |(P)|_k because (P) doesn't have an addable corner of residue j_s. Hence |γ|_k ≤ |(P)|_k+s-1, violating the assumption that γ/(P) is a weak s-strip. Hence we have j_s = i+s-1, completing the proof of the claim.Since s_i+s-1⋯ s_i+1s_i((P)) has the form below on the right, we can see that the corresponding k-bounded partition has the form P∪(s). Now we get back to the proof of the proposition. Let γ = s_i+s-1⋯ s_i+1s_i((P)). Then the removable corner of (P) corresponding to the removable corner of R_t_a is γ-blocked if and only if s≥ t_a. Then the number of residues of γ-nonblocked removable corners of (P) is exactly s+1.The above proposition gives an expression for P h_r as a linear combination of {P∪(s)}_s. To solve this linear equation, we need the following lemma of binomial coefficients. Let l be a positive integer and β_1,β_2,⋯,β_l+1 be integers such that β_i≥β_i+1≥β_i-1 for each i.Let C=((-1)^r-sβ_s+1r-s)_r,s=0^l. Then C^-1 = (β_r + r-s-1r-s)_r,s=0^l.Here ab is considered to be 0 if b<0. Put D=(β_r + r-s-1r-s)_r,s=0^l. The (p,q) element of the matrix DC is(DC)_pq = ∑_i=0^lβ_p + p-i-1p-i· (-1)^i-qβ_q+1i-q= ∑_i=q^pβ_p + p-i-1p-i· (-1)^i-qβ_q+1i-q= ∑_j=0^p-qβ_p + p-q-j-1p-q-j· (-1)^jβ_q+1j,which is 0 unless p≥ q.Let us consider the case p≥ q.By applying the next lemma for a=β_q+1, b=β_p, c=p-q≥ 0, we have(DC)_pq = (-1)^p-qβ_q+1-β_pp-q= 0 ( if p>q), 1 ( if p=q),where the last equality follows from β_q+1-β_p∈{0,1,⋯,p-q-1} (if q+1≤ p).For integers a,b and a nonnegative integer c,∑_i=0^c (-1)^i aib-1+c-ic-i = (-1)^c a-bc. Since mn is the coefficient of X^n in (1+X)^m∈[[X]] for m∈ and n∈_≥ 0, we have (LHS)= ∑_i=0^c (-1)^c ai-bc-i= (-1)^c (the coefficient of X^c in (1+X)^a(1+X)^-b∈[[X]])= (-1)^c a-bc.Now we can deduce the formula showing the goal of this section. If P, u and r are as in Proposition <ref>, then we haveP∪(r)/P = ∑_s=0^rr+r-s-1r-sh_s. In particular, if t_m < r, which means r=0, we haveP∪(r)/P = h_r = (r) On the other hand, when m=1,R_t∪(r)/R_t = h_r ( if r>t), h_r+h_r-1+⋯+h_0 ( if r≤ t). Apply Lemma <ref> for Proposition <ref>. § A FACTORIZATION OF G^(K)_∪ WITH SMALLAND SPLITTINGINTO R_T_1^A_1…R_T_M^A_M §.§ StatementsOur goal in this section is to show the equalityR_t_1^a_1∪⋯∪ R_t_m^a_m = R_t_1^a_1⋯R_t_m^a_mfor1≤ t_1 < ⋯ < t_m ≤ k and a_i>0 (see Theorem <ref>).The essential part is to prove R_t_1∪…∪ R_t_m = R_t_1∪…∪ R_t_m-1R_t_m, and the remainging part follows from the results from Section <ref> and induction (on n).This is a simple statement, but our proof involves an induction on the shape of partitions, thus we have to prove a more general statement (see the case t_n < r of part (2) of Theorem <ref>): Let P=⋃_i=1^m R_t_i^a_i be as in , Section <ref>, before Proposition <ref>, andas follows: Let (∅≠)∈ with satisfying ⊂, where we write =(_1,_2,…,_l()-1) and = l() = l() - 1. (Here we consider R_t to be ∅ unless 1≤ t≤ k) (Note: when l()=1, we have = 0 and = ∅ = thussatisfies . When l() > k+1, we have > k and ≠∅ = thusdoes not satisfy . ) Then, P∪ = Pwhen _l()>max_i{t_i}. §.§ Proofs We will prove a slightly even more general formula than (<ref>) (see part (2) of Theorem <ref>) in the following procedure. * Step (A): First we writeas a linear combination of products of h_i's and μ's with l(μ)<l(): putting =(_1,…,_l()-1), we have = ∑_μ s.t. ⊂μ⊂ R_k-l()+1 μ/ :vertical strip (-1)^|μ/|μ∑_i≥ 0(|μ/|+r_μ'')+i-1i h__l()-|μ/|-i if _1+l()≤ k+1 (Lemmas <ref> (1), <ref> (1), <ref> (1) and <ref>). * Step (B): Derive a similar expression for P∪ (parts Lemmas <ref> (2)-(3), <ref> (2)-(3), <ref> (2)-(3), <ref>). * Step (C): Compare (B) with the equality obtained by multiplying the formula in Step (A) by P, noticing Pμ=P∪μ by induction. Step (A) consists of two substeps: * Step (A-1): Write down the Pieri rule for μ h_r explicitly. * Step (A-2): Solve the system of Pieri rule formulas to give expressions foras a linear combination of {μ h_r}_μ,r. Obtaining an expression for P∪ in Step (B) follows from similar steps (B-1) and (B-2). * Step (B-1): Write down the Pieri rule for P∪μ h_r explicitly. * Step (B-2): Solve the system of Pieri rule formulas to give expressions for P∪ as a linear combination of {P∪μ h_r}_μ,r.§.§ Steps (A-1) and (B-1)Toward Step (A-1) and (B-1), let us begin with describing weak strips ()/(μ) where μ is contained in a k-rectangle. Assume μ⊂ and μ_l>0. Let 0 ≤ u ≤μ_l be an integer. (1) For ∈, ()/(μ) is a weak u-strip ⟺/μ is a horizontal u-strip,_1≤ k-l+1, ⟺=ν∪(s) , ⟺whereν⊂,ν/μ is a horizontal strip of size ≤ u, s = u-|ν/μ|. [scale=0.18](0,0)–(12,0)–(12,2)–(9,2)–(9,4)–(6,4)–(6,7)–(0,7)–cycle; [loosely dotted,thick] (0,7)-|(14,0); [pattern=dots] (0,7) rectangle (2,8); [pattern=dots] (6,4) rectangle (7,5);[pattern=dots] (12,0) rectangle (14,1);at (3,3) μ;(0,0) to [out=-20,in=-160] node[below]k+1-l (14,0);(14,0) to [out=70,in=-70] node[right]l (14,7); (2) For ∈, ()/(P∪μ) is a weak u-strip=P∪, where ()/(μ) is a weak u-strip=P∪ν∪(x), whereν⊂,ν/μ: horizontal strip, |ν/μ|+x = u. (1): The second equivalence is obvious. The “if” part of the first equivalence is easy: since _1≤ k+1-l and l()≤ l+1, we have ()= or ()=(_1+_l+1,_2,…) and hence /μ is a vertical strip. Hence it suffices to prove the “only if” part of the first equivalence: let μ⊂ R_k+1-l, μ_l>0, and ()/(μ) be a weak strip of size ≤μ_l, and we shall prove _1 ≤ k+1-l.[scale=0.18](0,0)–(11,0)–(11,2)–(8,2)–(8,4)–(5,4)–(5,7)–(0,7)–cycle; [loosely dotted,thick] (0,7)-|(13,0); [pattern=dots] (0,7) rectangle (3,8); [pattern=dots] (5,4) rectangle (7,5); [pattern=dots] (8,2) rectangle (11,3); [pattern=dots] (11,0) rectangle (16,1);at (3,2.5) μ;(0,0) to [out=-10,in=-170] node[below]k+1-l (13,0);(13,0) to [out=-30,in=-150] node[below]u (16,0);(16,0) to [out=-20,in=-160] node[below]_l+1-u (23,0);[loosely dotted,thick] (0,4)–(5,4); [loosely dotted,thick] (0,5)–(5,5); [loosely dotted,thick] (7,0)–(7,4);(16,0) rectangle (23,1);[pattern=north west lines] (0,0) rectangle (7,1);[left] at (0,4.5) l+1-u;[left] at (0,7.5) l+1; [<-] (3,0.5) to [out=-135,in=90] (1,-4) node[below] cells with hook lengths >k; Assume, on the contrary, that _1 > k+1-l. Write =(_2,_3,…)⊂. Then by Lemma <ref> we have ()_i = ()_i-1 = _i-1=_i (for i>1), and ()_1 = _1 + ()_1+k+1-_1_< l+1_≥_l ≥μ_l≥_1+μ_l > k+1-l+μ_l. Hence, the hook lengths of (1,1),⋯, (1,μ_l) in () are all greater than k because h_(1,j)(())= ()_1 + ()'_j-1-j+1 > k+1-l+μ_l + ()'_j_≥'_j≥μ'_j ≥ l- j_≥ -μ_l≥ k+1 for 1≤ j ≤μ_l. On the other hand, those of (2,1),⋯, (2,μ_l) in () are less than or equal to k because ⊂μ⊂. Hence ^ω_k_j = ()'_j - 1 = '_j - 1 for 1≤ j ≤μ_l. Since ()/(μ) is a weak strip, /μ is a vertical strip. Hence '_j - 1 = _j ≥μ_j = μ'_j = l for 1 ≤ j ≤μ_l, which implies _l+1≥μ_l. Then we have |/μ| ≥ (_l+1-μ_l+1_=0) + (_1-μ_1_>0) > μ_l since _1>k+1-l≥μ_1. This is a contradiction. (2): The first equivalence follows from Corollary <ref>. The second equivalence follows from (1). Next let us explicitly describe the weak Pieri rule (<ref>), after we prepare a notation for convenience. Let P and u (u∈_>0) be as in . For ν⊂ R_k+1-l(ν), 0≤ u ≤ν_l(ν), p ∈ℤ, we set T_ν,u,p := ∑_s=0^u (-1)^s psν∪ (u-s), T'_P,ν,u,p := ∑_s=0^u (-1)^s p+u+1-ssP∪ν∪ (u-s). Let P and u (u∈_>0) be as in . Assume μ⊂, μ_l>0, μ_l≥ r ≥ 0. Then we have(1)μ h_r= ∑_ν⊂ R_k+1-l ν/μ:h.s.∑_s=0^r-|ν/μ| (-1)^s r_νμsν∪ (r-|ν/μ|-s) ( = ∑_ν⊂ R_k+1-l ν/μ:h.s. T_ν,r-|ν/μ|,r_ν,μ). (2) If <μ_l,P∪μ h_r= ∑_ν⊂ R_k+1-l ν/μ:h.s.∑_s=0^r-|ν/μ| (-1)^s r_νμ+r-|ν/μ|+1-ssP∪ν∪ (r-|ν/μ|-s) ( = ∑_ν⊂ R_k+1-l ν/μ:h.s. T'_P,ν,r-|ν/μ|,r_ν,μ). (3) If =μ_l,P∪μ h_r= ∑_ν⊂ R_k+1-l ν/μ:h.s.∑_s=0^r-|ν/μ| (-1)^s r_νμ+r-|ν/μ|+1-s-1sP∪ν∪ (r-|ν/μ|-s) ( = ∑_ν⊂ R_k+1-l ν/μ:h.s. T'_P,ν,r-|ν/μ|,r_ν,μ-1). (1) We transform the right-hand side of Eq. (<ref>), Proposition <ref>, into the right-hand side of part (1) of the Lemma as follows:μ h_r= ∑_u=0^r (-1)^r-u∑_ ()/(μ):weak u-stripr_()(μ)r-u(i)=∑_u=0^r (-1)^r-u∑_ν s.t. ν⊂ ν/μ: h.s. of size≤ ur_(ν∪(u-|ν/μ|)),(μ)r-uν∪(u-|ν/μ|)= ∑_ν s.t. ν⊂ ν/μ: h.s.∑_u=|ν/μ|^r (-1)^r-ur_(ν∪(u-|ν/μ|)),(μ)r-uν∪ (u-|ν/μ|)(ii)=∑_ν s.t. ν⊂ ν/μ: h.s.∑_s=0^r-|ν/μ| (-1)^sr_ν,μsν∪ (r-s-|ν/μ|). Here, the equality (i) uses Lemma <ref> (1) in order to change the summation variable fromto ν according to =ν∪(u-|ν/μ|). For the equality (ii) we use (1) of the following Lemma <ref> and put s=r-u. Note that u-|ν/μ| ≥μ_l occurs only if u-|ν/μ| = u = r = μ_l since u ≤ r ≤μ_l, in which case we have r_ν∪(u-|ν/μ|),μr-u = 1 = r_ν,μr-u. We can prove (2) and (3) almost the same as (1), using Lemma <ref> (2) for (i), and (2) of the following Lemma <ref> for (ii). Note that, in the same way as (1), the case u-|ν/μ|≥μ_l and <μ_l appears in the expression for P∪ corresponding to (<ref>) only in the form r_νμ+μ_l+1-10, which is equal to r_νμ+μ_l+10. Let μ be as in Lemma <ref>. Let μ⊂ν⊂ and assume ν/μ is a horizontal strip. Let 0 ≤ x ≤ν_l. Then we have (1) r_(ν∪(x)),(μ) = r_νμ - [x≥μ_l]. (2) Let P and u (u ∈_>0) be as in Lemma <ref> and assume ≤μ_l. Then r_(P∪ν∪(x)),(P∪μ) = r_ν,μ + x+1 - [x≥μ_l](if <μ_l), r_ν,μ + x+1 - 1(if =μ_l). [scale=0.125] (0,17) rectangle (5,29);at (2.5,23) R_t_1;(5,7) rectangle (12,17);at (8,12) R_t_m;(12,0)–(27,0)–(27,2)–(24,2)–(24,5)–(21,5)–(21,7)–(12,7)–cycle;at (17,3) μ;[loosely dotted,very thick] (0,17) |- (12,0); [below] at(12,0) (P∪μ);(0,52) rectangle (5,64);at (2.5,58) R_t_1;(0,42) rectangle (7,52);at (3.5,47) R_t_m;(0,35)–(15,35)–(15,37)–(12,37)–(12,40)–(9,40)–(9,42)–(0,42)–cycle;at (5,38) μ; [below] at (9,35) P∪μ;(35,17) rectangle (40,29);at (37.5,23) R_t_1;(40,7) rectangle (47,17);at (43.5,12) R_t_m;(47,0) -| (62,2) -| (59,5) -| (56,7) -| (47,0);at (52,3) μ;[loosely dotted,very thick] (35,17) |- (47,0); [below] at (47,0) (P∪ν∪(x));[pattern=dots] (35,29) rectangle (40,30); [pattern=dots] (40,17) rectangle (46,18); [pattern=dots] (47,7) rectangle (53,8);[pattern=north west lines] (56,5) rectangle (59,6); [pattern=north west lines] (59,2) rectangle (60,3); [pattern=north west lines] (62,0) rectangle (64,1); [pattern=dots] (64,0) rectangle (70,1); (47,8) to [out=20,in=160] node[above]x (53,8);(64,0) to [out=-20,in=-160] node[below]x (70,0);[red,decorate,decoration=snake,segment length=.2mm,amplitude=.1mm] (46.9,-0.2) – (64.1,-0.2) – (64.1,1.0) – (62.1,1.0) – (62.1,1.95) –(60.1,1.95) – (60.1,2.95) –(59.0,2.95) – (59.0,5.95) – (56.0,5.95) – (56.0,6.95) – (46.9,6.95) – (46.9,-0.2);[red] (59.2,6.2) to [out=45,in=190] (64,8) node[right]ν;(35,52) rectangle (40,64);at (37.5,58) R_t_1;(35,42) rectangle (42,52);at (38.5,46) R_t_m;(35,35) -| (50,37) -| (47,40) -| (44,42) -| (35,35);[red,decorate,decoration=snake,segment length=.2mm,amplitude=.1mm] (34.9,34.8) -| (52.1,36.10) -| (50.1,37.1) -| (48.1,38.1) -| (47.1,41.1) -| (44.15,42.10) -| (34.9,34.8);[red] (47.2,41.2) to [out=45,in=190] (52,43) node[right]ν;at (40,38) μ;[below] at (47,35) P∪ν∪(x);[pattern=dots] (35,64) rectangle (40,65); [pattern=dots] (40,52) rectangle (41,53);[pattern=north west lines] (44,40) rectangle (47,41); [pattern=north west lines] (47,37) rectangle (48,38); [pattern=north west lines] (50,35) rectangle (52,36); (35,52) to [out=-20,in=-160] node[below] x (41,52); (1): Since μ⊂ we have (μ) = μ. Since ν⊂ and x ≤ν_l, we have (ν∪(x))_i = (ν∪(x))_i for i≠ 1. Thus r_(ν∪(x)),(μ) = r_ν∪(x),μ. Moreover, r_ν∪(x),μ≠ r_ν,μ happens only if the (l+1)-th part of ν∪(x) blocks the μ-removable corner in the l-th row, i.e. x ≥μ_l, in which case r_ν∪(x),μ = r_ν,μ - 1. (2): Assume t_1 < … < t_m without loss of generality and thus =t_m. We put T=(P)_1=∑_j t_j. Since (P∪μ) = μ⊕(P), a removable corner (r,c) of (P∪μ) satisfies one of the following: * (type 1) r≥ l+1, and (r-l,c) is a removable corner of (P), * (type 2) c≥ T+1, and (r,c-T) is a removable corner of μ. We put X_j:= Res{removable corners of (P∪μ) of type j} Y_j:= Res{(P∪ν∪(x))-nonblocked removable corners of (P∪μ) of type j} for j=1,2. We denote by i the residue of top addable corner of (P∪μ). Then we have X_1 = {i+t_1,i+t_2,…,i+t_m} and i+μ_l ∈ X_2 ⊂ [i+μ_l, i+k-1]. Note that {i+t_1,…,i+t_m}∩ [i+μ_l, i+k-1] = ∅ (if t_m<μ_l),{i+μ_l} (if t_m=μ_l). Next we show that, for i≥ 1, (P∪ν∪(x))_l+i =(P∪(x))_i,(P∪ν∪(x))'_T + i = (ν∪(x))'_i. (<ref>) is obvious since the smallest part of ν, which is ν_l, is greater than or equal to the largest part of P∪(x), which is max{x,t_m}. For (<ref>), first we note that, by (<ref>) and Figure <ref> in the proof of Proposition <ref>, we have (P∪ν∪(x))_l+1 =(P∪(x))_1 = T+x,(P∪ν∪(x))_l+2 =(P∪(x))_2 = T, ⋮ (P∪ν∪(x))_l+k+1-μ_l =(P∪(x))_k+1-μ_l = T. Then by Lemma <ref> we have, for 1≤ i ≤ l, (P∪ν∪(x))_i = (P∪ν∪(x))_i+(k+1-ν_i) + ν_i = (P∪ν∪(x))_l+1 + ν_1 = T+x+ν_1(if i=1 and ν_1=k+1-l), T + ν_i(otherwise), where we used (P∪ν∪(x))_i = ν_i for 1≤ i ≤ l for the first equality and l+1≤ i+(k+1-ν_i) ≤ l+(k+1-μ_n) (the first equality holds if and only if i=1 and ν_1=k+1-l) for the second equality. Thus we have (P∪ν∪(x))_i=(ν∪(x))_i+T for 1≤ i≤ l+1 and (P∪ν∪(x))_i≤ T for i> l+1, which implies (<ref>). Hence, |Y_1| = r_(P∪(x)),(P) = x+1, and |Y_2| = r_(ν∪(x)),μ = r_νμ-[x≥μ_l]. Moreover Y_1∩ Y_2 = {i+μ_l} if x < t_m=μ_l, and Y_1∩ Y_2=∅ otherwise. Then r_(P∪ν∪(x)),(P∪μ) = |Y_1|+|Y_2|-|Y_1∩ Y_2| =x+1 + r_νμ - [x≥μ_l](if t_m<μ_l),x+1 + r_νμ- [x≥μ_l] - [x<t_m]_=-1 (if t_m=μ_l). Thus Steps (A-1) and (B-1) have been achieved. §.§ Steps (A-2) and (B-2)The next lemma is technically important to perform the instructions in Step (A-2) and (B-2). Let ν, u, p be as in the assumptions in Definition <ref> and n be an integer. Then we have the following equalities. In particular, in either case, the left-hand side does not depend on p.(1) ∑_i=0^up+n+i-1i T_ν,u-i,p = ∑_s=0^un+s-1sν∪(u-s).(2) ∑_i=0^up+n+i-1i T'_P,ν,u-i,p = ∑_s=0^un-u+1-s+s-1sP∪ν∪ (u-s). Since both equality can be proved in a parallel manner, we prove (2) here. By the definition of T'_P,ν,u-i,p, we have (LHS)= ∑_i=0^up+n+i-1i∑_s=0^u-i (-1)^s p+u-i+1-ssP∪ν∪(u-i-s),then putting t=i+s, = ∑_t = 0^u( ∑_s=0^tp+n-1+t-st-s (-1)^s p+u+1-ts) P∪ν∪(u-t),then using Lemma <ref>, = ∑_t=0^u (-1)^t -n+u+1-ttP∪ν∪(u-t)= ∑_t=0^un-u+1-t+t-1tP∪ν∪(u-t). Now we can express(resp. P∪) as a linear combination of μh_r (resp. P∪μh_r) as proposed in the description of Step (A-2) (resp. (B-2)). Let P and u (u ∈_>0) be as in . Let ,, be as in in Section <ref>. Write r=_l(). Assume that ≥ 1 and ≤_.(1) We have = ∑_μ s.t.⊂μ⊂ ∑_q∈ℤ A_μ,,q μ ∑_i≥0 q+i-1i h_r-|μ/|-i. (2) If <_, we have P∪= ∑_μ s.t.⊂μ⊂ ∑_q∈ℤ A_μ,,q P∪μ ∑_i≥0 q+i+r-1i h_r-|μ/|-i. (3) If =_, we have P∪= ∑_μ s.t.⊂μ⊂ ∑_q∈ℤ A_μ,,q P∪μ ∑_i≥0 q+i+r-2+[μ_≠_]i h_r-|μ/|-i. Here, in all of the three expressions, the number A_μ,,q is defined by the following recursion formula:A_,,q = δ_q,r_, A_μ,,q = -∑_μ/: h.s. ⊂⊊μ A_,,q-(r_μμ-r_μ)for ⊊μ⊂.Notice that for each μ, A_μ,,q = 0 except for finitely many q. The explicit value of A_μ,,q will be given in Lemma <ref> below. Remark. In the above recursion formula, r_μμ - r_μ≥ 0 always holds because there must be a μ-removable corner in every row in which there is a μ-nonblocked -removable corner since μ/ is a horizontal strip. (1) By Lemma <ref>(1), (RHS)= ∑_μ s.t. ⊂μ⊂∑_q A_μ,,q∑_μ⊂ν⊂ ν/μ: h.s.∑_i≥ 0q+i-1i T_ν,r-|μ/|-i-|ν/μ|,r_νμ,then splitting the third summation according to whether μ=ν or μ⊊ν, = ∑_μ s.t. ⊂μ⊂∑_q A_μ,,q∑_i≥ 0q+i-1i T_μ,r-|μ/|-i,r_μμ=+ ∑_μ,ν s.t. ⊂μ⊊ν⊂ ν/μ: h.s.∑_q A_μ,,q∑_i≥ 0q+i-1i T_ν,r-|ν/|-i,r_νμ,then replacing the variable μ for the first summation by ν, and splitting it again according to whether =ν or ⊊ν, and rearranging the summands, = ∑_q A_,,q∑_i≥ 0q+i-1i T_,r-i,r_=+∑_ν s.t. ⊊ν⊂(∑_qA_ν,,q∑_i≥ 0q+i-1i T_ν,r-|ν/|-i,r_νν_(X)+∑_ν s.t. ⊊ν⊂ + ∑_q∑_μ s.t. ⊂μ⊊ν ν/μ: h.s. A_μ,,q∑_i≥ 0q+i-1i T_ν,r-|ν/|-i,r_νμ_(Y)).Then, by the definition of A_ν,,q, noting that ⊊ν, (X)= - ∑_q∑_μ s.t. ν/μ: h.s. ⊂μ⊊ν A_μ,,q-(r_νν-r_νμ)∑_i ≥ 0q+i-1i T_ν,r-|ν/|-i,r_νν,then replacing q by q+(r_νν-r_νμ), = -∑_q∑_μ s.t. ν/μ: h.s. ⊂μ⊊ν A_μ,,q∑_i ≥ 0q+r_νν-r_νμ+i-1i T_ν,r-|ν/|-i,r_νν,then using the independence of the LHS on p of Lemma <ref>(1) (note that the range of i can be limited to 0≤ i ≤ r-|ν/| since i originally occurs in h_r-|μ/|-i in the statement of part (1) of the Lemma), = -∑_q∑_μ s.t. ν/μ: h.s. ⊂μ⊊ν A_μ,,q∑_i ≥ 0q+i-1i T_ν,r-|ν/|-i,r_νμ= -(Y).Hence,(RHS) = ∑_q A_,,q∑_i≥ 0q+i-1i T_,r-i,r_= ∑_i≥ 0r_+i-1i T_,r-i,r_,again by Lemma <ref>(1), noting that 0+s-1s vanishes unless s=0, = . (3) is proved almost parallel to (1): By Lemma <ref>(2) and (3),(RHS)= ∑_μ s.t. ⊂μ⊂∑_q A_μ,,q∑_μ⊂ν⊂ ν/μ: h.s.∑_i≥ 0q+i+r-2+[μ_l≠_l]i T'_P,ν,r-|μ/|-i-|ν/μ|,r_νμ-[μ_l=_l],then, by Lemma <ref>(2), shifting p by μ_l=_ and noting that μ_l≠_ + μ_l=_ = 1, = ∑_μ s.t. ⊂μ⊂∑_q A_μ,,q∑_μ⊂ν⊂ ν/μ: h.s.∑_i≥ 0q+i+r-1i T'_P,ν,r-|ν/|-i,r_νμ.Note that the following deformation is also valid for the case <_. Applying the same argument as (1), = ∑_i≥ 0r_+r-1+ii T'_P,,r-i,r_,then by Lemma <ref>(2), = ∑_s≥ 0-r+1-s+r-1+ssP∪∪(r-s)= P∪.Here the last equality follows from -r+1-s+r-1+ss=(-1)^sr+1-s-rs and0≤r+1-s-r≤ s-1 for s≥ 1.For (2), we have(RHS)= ∑_μ s.t. ⊂μ⊂∑_q A_μ,,q∑_μ⊂ν⊂ ν/μ: h.s.∑_i≥ 0q+i+r-1i T'_P,ν,r-|μ/|-i-|ν/μ|,r_νμ,which is equal to P∪ since this sum has exactly the same form as appeared in the proof of (3). In fact we can explicitly solve the recursion formula of A_μ,,q appeared in the previous proposition. This result is needed in the author's following paper <cit.> and included in Appendix <ref>.Now Step (A) and (B) have been accomplished. §.§ Step (C)We multiplyby P, and express it as a linear combination of K-k-Schur functions, and solve it: Let P and u (for u∈_>0) be as in in Section <ref>, before Proposition <ref>. Let ,, be as in in Section <ref>. Write r=_l(). Assume max_i{t_i} < _. Then we have (1) P= ∑_s=0^r (-1)^s r+1-ss P∪∪(r-s). (2) P∪= P ∑_s=0^r r+s-1s ∪(r-s). In particular, if t_n < r then r=0 and P∪ = P. (2) follows from (1) and Lemma <ref>. We prove (1) by induction on ≥ 0. The case =0 was proved in Proposition <ref> and Theorem <ref>. Assume ≥ 1. From Lemma <ref>, (LHS)= P∑_μ s.t. ⊂μ⊂∑_q A_μ,,qμ∑_i≥ 0q+i-1i h_r-|μ/|-i,by the induction hypothesis, we have Pμ=P∪μ in the above summation. Hence = ∑_μ s.t. ⊂μ⊂∑_q A_μ,,qP∪μ∑_i≥ 0q+i-1i h_r-|μ/|-i,then by Lemma <ref>(2), (notice that μ_≥_>) =∑_μ s.t. ⊂μ⊂∑_q A_μ,,q∑_μ⊂ν⊂ ν/μ: h.s.∑_i≥ 0q+i-1i T'_P,ν,r-|ν/|-i,r_ν,μ,then by doing the same argument as Lemma <ref>(1), (formally replacing T_… by T'_P,… and using Lemma <ref>(2) instead of (1), the proof works) =∑_i≥ 0r_+i-1i T'_P,,r-i,r_,then by Lemma <ref>(2), = ∑_s=0^r-r+1-s+s-1sP∪∪(r-s)= ∑_s=0^r (-1)^s r+1-ssP∪∪(r-s).Now we can achieve our goal in this section. For 1≤ t_1 < ⋯ < t_m ≤ k and a_1,…,a_m>0, R_t_1^a_1∪⋯∪ R_t_m^a_m = R_t_1^a_1⋯R_t_m^a_m. Use induction on m>0.The base case m=1 is obvious. Assume m>1.Applying Proposition <ref> for=R_t_m^i and t=t_m, we haveR_t_m^i+2 = R_t_m^i+1R_t_m∪ R_t_m/R_t_m.Multiplying this for i=0,…,a_m-2, we haveR_t_m^a_m = R_t_m(R_t_m∪ R_t_m/R_t_m)^a_m-1. Put P = R_t_1^a_1∪⋯∪ R_t_m-1^a_m-1.Similarly applying Proposition <ref> for=P∪ R_t_m^i and t=t_m,then multiplying this for i=0,…,a_m-2, we have P∪ R_t_m^a_m = P∪ R_t_m(R_t_m∪ R_t_m/R_t_m)^a_m-1. On the other hand, applying the previous theorem for P, = R_t_m, we have P ∪ R_t_m = PR_t_m.Hence we have P∪ R_t_m^a_m = PR_t_m(R_t_m∪ R_t_m/R_t_m)^a_m-1 = PR_t_m^a_m= R_t_1^a_1…R_t_m^a_m,where the last equality follows by the induction hypothesis. § DISCUSSIONS In this section we state some conjectures, that are consistent with our results in previous sections. For all ∈ and P=, write P∪ = P∑_μ a_P,,μμ. Then a_P,,μ≥ 0 for any μ. In the case P=R_t, it is observed that a_R_t,,μ=0 or 1. Moreover, the set of μ such that a_R_t,,μ=1 is expected to be an “interval”, but we have to consider the strong order on ≃≃, which can be seen as just inclusion as shapes in the poset of cores, or strong Bruhat order on the affine symmetric group. Namely, the strong order ≤μ onis defined by ()⊂(μ). Notice that ≼μ⊂μ≤μ for ,μ∈. Then, For all ∈ and 1≤ t ≤ k, there exists μ∈ such that R_t∪ = R_t∑_μ≤ν≤ν. Assuming this conjecture, we shall write (,t) = μ.We can make some conjectures about the behavior of minindex: * It is expected that ifgets “bigger” with respect to inclusion, then minindex gets weakly bigger in the strong order. Namely, For any two elements μ⊂ of , we have (μ,t) ≤(,t). * If a bounded partition has a form R_s∪ for s≠ t, its minindex still be expected to contains R_s, and the “remaining part” is bigger or equal to (,t) in the strong order: For all ∈ and 1≤ s ≠ t ≤ k, (R_s∪,t) has the form R_s∪μ and (,t) ≤μ. * If a bounded partition has a form R_s∪ R_s∪ for s≠ t, its minindex would be equal to the union of R_s and (R_s∪): For all ∈ and 1≤ s≠ t≤ k, we have R_s∪(R_s∪,t) = (R_s∪ R_s ∪, t).Next, consider a bounded partition that has a form R_t∪. We wroteR_t∪ R_t∪/R_t = ∑_(R_t∪,t) ≤≤ R_t∪.On the other hand, by Proposition <ref>R_t∪ R_t∪/ = /R_t∪ R_t/= ∑_(,t)≤μ≤μ∑_ν⊂ R_tν.Now we can expect thatfor any μ∈, μ∑_ν⊂ R_tν = ∑_∈ I_μ,t,where I_μ,t is an order filter of the interval [∅, R_t∪μ] (inwith the strong order) such that _μ∈[(,t),] I_μ,t = [(R_t∪,t), R_t∪]. § EXAMPLESIn this section we sometimes abbreviate ∑_ a_[3] as ∑_ a_ for ease to see.Example A. [3]1,1,1,2 = [3]1,1,1[3]2 [3]2,2,2 = [3]2,2( [3]2 + [3]1 + [3]∅)[3]2,3 = [3]3( [3]2 + [3]1 + [3]∅)[3]1,1,1,2,2,2 = [3]1,1,1[3]2,2( [3]2 + [3]1 + [3]∅) = [3]1,1,1[3]2,2,2= [3]2,2( [3]1,1,1,2 + [3]1,1,1,1)[3]1,1,1,2,3 = [3]1,1,1[3]3( [3]2 + [3]1 + [3]∅) = [3]1,1,1[3]2,3= [3]3( [3]1,1,1,2 + [3]1,1,1)[3]2,2,2,3 = [3]2,2[3]3( [3]2 + 2 [3]1 + 3 [3]∅) = [3]2,2( [3]2,3 + [3]1,3 + [3]3) = [3]3( [3]2,2,2 + [3]1,2,2 + [3]2,2)[3]1,1,1,2,2,2,3 = [3]1,1,1[3]2,2[3]3( [3]1,1 + 2 [3]1 + 3 [3]∅) = [3]1,1,1[3]2,2( [3]2,3 + [3]1,3 + [3]3) = [3]1,1,1[3]3( [3]2,2,2 + [3]1,2,2 + [3]2,2) = [3]2,2[3]3( [3]1,1,1,2 + 2 [3]1,1,1,1 + [3]1,1,1) = [3]1,1,1[3]2,2,2,3= [3]2,2( [3]1,1,1,2,3 + [3]1,1,1,1,3) = [3]3( [3]1,1,1,2,2,2 + [3]1,1,1,1,2,2) Example B. R_2∪ R_3∪ R_4 ∪ (1) = R_2∪ R_3 ∪ R_4( (1) + 3 ∅)(= R_2R_3R_4( (1) + 3 ∅) )R_2∪ R_3∪ R_4 ∪ (2) = R_2∪ R_3 ∪ R_4( (2) + 3 (1) + 6 ∅)R_2∪ R_3∪ R_4 ∪ (3) = R_2∪ R_3 ∪ R_4( (3) + 2 (2) + 3 (1) + 4 ∅)R_2∪ R_3∪ R_4 ∪ (4) = R_2∪ R_3 ∪ R_4( (4) +(3) +(2) +(1) +∅)R_2∪ R_3∪ R_4 ∪ (5) = R_2∪ R_3 ∪ R_4·(5)Example C.Assume k ≥ 7.R_2∪ R_3∪ R_4 ∪ (6,5,1) = R_2∪ R_3 ∪ R_4( (6,5,1) + 3 (6,5))R_2∪ R_3∪ R_4 ∪ (6,5,2) = R_2∪ R_3 ∪ R_4( (6,5,2) + 3 (6,5,1) + 6 (6,5))R_2∪ R_3∪ R_4 ∪ (6,5,3) = R_2∪ R_3 ∪ R_4( (6,5,3) + 2 (6,5,2) + 3 (6,5,1) + 4 (6,5))R_2∪ R_3∪ R_4 ∪ (6,5,4) = R_2∪ R_3 ∪ R_4( (6,5,4) +(6,5,3) +(6,5,2) +(6,5,1) +(6,5))R_2∪ R_3∪ R_4 ∪ (6,5,5) = R_2∪ R_3 ∪ R_4·(6,5,5)Example D. Assume k ≥ 8. R_2∪ R_5 ∪ (7,6,4) = R_2·R_5 ∪ (7,6,4)= R_5( R_2∪ (7,6,4) + R_2∪ (7,6,3) + R_2∪ (7,6,2)). § PROOF OF PROPOSITION <REF>(4)(1): The latter condition τ∃τ^(1)…∃τ^(r) = is obvious.If /τ is not a horizontal strip,then (a,b),(a+1,b)∈/τ for ∃ a,b. Write their residues i=b-a, i-1=b-(a+1). Then a s_i-1-action should be performed after a s_i-action.Then the representation of (4) has the form = … s_i-1… s_i …τ, which contradicts (4).(1)(4): Assume= … s_i … s_i+1…τ, = τ + r, and /τ is a horizontal strip.Consider the moment just before performing the action of s_i+1. At that time the situation around each extremal cell of residue i+1 is one of the following:(1): -4ex [scale=0.5] (0,1) |- (1,0); (0,0) circle (2pt); [loosely dotted, thick] (0,0) – (1.0,1.0) node [anchor=south west] i+1;(2): -6ex [scale=0.5] (0,1) – (0,-1); (0,0) circle (2pt); [loosely dotted, thick] (0,0) – (1.0,1.0) node [anchor=south west] i+1;(3): -4ex [scale=0.5] (-1,0) – (1,0); (0,0) circle (2pt); [loosely dotted, thick] (0,0) – (1.0,1.0) node [anchor=south west] i+1;(4): -6ex [scale=0.5] (-1,0) -| (0,-1); (0,0) circle (2pt); [loosely dotted, thick] (0,0) – (1.0,1.0) node [anchor=south west] i+1;In the case (1), furthermore it should be -6ex [scale=0.5] (-1,1) – (0,1) |- (1,0); (0,0) circle (2pt); (0,1) circle (2pt); [loosely dotted, thick] (0,0) – (1.0,1.0) node [anchor=south west] i+1; [loosely dotted, thick] (0,1) – (1.0,2.0) node [anchor=south west] i; since /τ is a horizontal strip. Besides, the case (1) should happen because the action of s_i+1 must add more than or equal to one box.In fact the case (4) never happens since the action of s_i+1 does not remove boxes.The case (3) is divided to (3-1): -4ex [scale=0.5] (-1,1) – (-1,0) – (1,0); (0,0) circle (2pt); (-1,0) circle (2pt); [loosely dotted, thick] (0,0) – (1.0,1.0) node [anchor=south west] i+1; [loosely dotted, thick] (-1,0) – (0,1) node [anchor=south west] i; and (3-2): -4ex [scale=0.5] (-2,0) – (-1,0) – (1,0); (0,0) circle (2pt); (-1,0) circle (2pt); [loosely dotted, thick] (0,0) – (1.0,1.0) node [anchor=south west] i+1; [loosely dotted, thick] (-1,0) – (0,1) node [anchor=south west] i; .The case (3-1) should happen since later the action of s_i must add more than or equal to one box.Thus we have a contradiction that there are both addable corners and removable corners of residue i in this moment.§ EXPLICIT DESCRIPTION OF A_Μ,,Q In the setting of Lemma <ref>,A_μ,,q = (-1)^|μ/|(if μ/: vertical strip and q=|μ/|+r_μ''), 0 (otherwise). We fix , and set f_μ(t) := ∑_q A_μ,,qt^q ∈[t]. Then the definition of A_μ,,q (in the statement of Lemma <ref>) is transformed into the recursion formulaf_(t)= t^r_, f_μ(t)= -∑_μ/: h.s. ⊂⊊μ t^r_μμ-r_μf_(t) for μ≠,and the desired result becomes the conditionf_μ(t) = (-1)^|μ/| t^|μ/|+r_μ'' (if μ/: vertical strip) 0(otherwise).We prove it by induction on |μ| (for μ satisfying ⊂μ⊂). The base case μ= is obvious by definition.Then we assume ⊊μ⊂. First we consider the case where μ/ is a vertical strip. In this case we put{x_1,…,x_s} := {x|'_x<μ'_x} (x_1 < ⋯ < x_s), a_i:= μ'_x_i - '_x_i, b_i:= '_x_i-1 - '_x_i (≥ a_i) (if x_1=1 set b_1=∞),[scale=0.22](0,0) -| (10,3) -| (7,5) -| (4,9) -| (0,0);at (4,2.5) ;[loosely dotted,thick] (8,5.5) – (11,3);(0,9) rectangle (1,14);(1,9) to [out=60, in=-60] node[right=1pt]a_1 (1,14); (4,5) rectangle (5,8);(5,5) to [out=60, in=-60] node[right=1pt]a_2 (5,8);(4,5) to [out=120, in=-120] node[left=1pt]b_2 (4,9);(10,0) rectangle (11,2);(11,0) to [out=60, in=-60] node[right=1pt]a_s (11,2);(10,0) to [out=120, in=-120] node[left=1pt]b_s (10,3);and we denote by (c_1,…,c_s) the partition defined by(c_1,…,c_s)'_x ='_x + c_iif x=x_i for some i '_xotherwisefor 0≤ c_i ≤ a_i (1≤ i≤ s). In particular (0,…,0)= and (a_1,…,a_s)=μ.Since |(c_1,…,c_s)/| = ∑_i c_i and r_(c_1,…,c_s)'' = r_ - #{i| c_i=b_i}, we havef_(c_1,…,c_s)(t)= (-1)^|(c_1,…,c_s)/| t^|(c_1,…,c_s)/|+r_(c_1,…,c_s)''= (-1)^∑_i c_i t^r_+∑_i (c_i-c_i=b_i),for 0≤ c_i ≤ a_i and (c_1,…,c_s)≠(a_1,…,a_s), by the induction hypothesis. For S⊂{1,…,s}, we set(S) = (a_1-[1∈ S],…,a_s-[s∈ S]). Then ⊂⊊μ μ/: horizontal strip = (S)for ∅≠∃ S ⊂{1,…,s}.Therefore,f_μ(t)= -∑_μ/: h.s. ⊂⊊μ t^r_μμ-r_μf_(t) = - ∑_∅≠ S ⊂{1,2,…,s} t^r_μμ-r_μ(S)f_(S)(t)=-t^r_μμ-r_μ({1})f_({1})(t) = -∑_∅≠ T⊂{2,…,s}(t^r_μμ-r_μ(T)f_(T)(t) + t^r_μμ-r_μ({1}∪ T)f_({1}∪ T)(t)_(X)). In fact it can be proved that (X)=0 by the following Claim 1 and Claim 2.Claim 1.r_μ,({1}∪ T) = r_μ,(T) - a_1<b_1 (for all T ⊂{2,…,s}). Proof of Claim 1: Reduced to next lemma: Let γ⊂β and y=(r,c) be an addable corner of γ. Put =∪{y}. Assume that _1+l()≤ k+1 and y is β-nonblocked. Then r_βγ-r_βγ= 0 (if (r,c-1) is a β-nonblocked removable corner of ), 1 (otherwise).Note that r_β=#{β-nonblocked -removable corners} since ⊂, and the same equality holds for r_β.If z is a β-blocked (resp. nonblocked) removable corner ofother than (r-1,c) or (r,c-1),then z is also β-blocked (resp. nonblocked) removable corner of , and vice versa. Note that * y=(r,c) is a β-nonblocked removable corner of , and not in . * (r,c-1) is not a removable corner of . * (r-1,c) is not a removable corner of . Even if (r-1,c) is a removable corner of , it is β-blocked.Hence we concluder_β,-r_β,= 0(if (r,c-1) is a β-nonblocked removable corner of ), 1(otherwise). 5mm .21 boundary of γ̃ around y=(r,c) [scale=0.25] (-1,1) -| (1,-1); (0,1) |- (1,0); (0.5,0.5) to [out=90,in=190] (2,2) node[right]y; [dotted,thick] (-1,1)–(-2,2); [dotted,thick] (1,-1)–(2,-2);[scale=0.25] (-1,1) -| (1,0); (0,1) |- (2,0); (0.5,0.5) to [out=90,in=190] (2,2) node[right]y; [dotted,thick] (-1,1)–(-2,2); [dotted,thick] (2,0)–(3,-1);[scale=0.25] (0,1) -| (1,-1); (0,2) |- (1,0); (0.5,0.5) to [out=90,in=190] (2,2) node[right]y; [dotted,thick] (0,2)–(-1,3); [dotted,thick] (1,-1)–(2,-2);[scale=0.25] (0,1) -| (1,0); (0,2) |- (2,0); (0.5,0.5) to [out=90,in=190] (2,2) node[right]y; [dotted,thick] (0,2)–(-1,3); [dotted,thick] (2,0)–(3,-1);r_β-r_β (r+1,c-1)∉β (r+1,c-1)∉β 0 0 Claim 2.f_({1}∪ T)(t) = - f_(T)(t)· t^-a_1<b_1for ∅≠ T⊂{2,…,s}Proof of Claim 2: Put a'_i=a_i-[i∈ T], thenf_({1}∪ T)(t)= (-1)^|μ/|-|T|-1 t^r_+(a_1-1)+∑_i>1 (a'_i-a'_i=b_i)= - (-1)^|μ/|-|T| t^r_+(a_1-a_1=b_1)-a_1<b_1+∑_i>1 (a'_i-a'_i=b_i)= - f_(T)(t)· t^-a_1<b_1.(End of the proof of Claim 2) Hence,f_μ(t)= -t^r_μμ-r_μ({1})f_({1})(t) = -t^a_1<b_1· (-1)^|μ/|-1t^r_+∑_i(a_i-a_i=b_i)-a_1<b_1= (-1)^|μ/| t^r_+∑_i(a_i-a_i=b_i)= (-1)^|μ/| t^|μ/|+r_μ''.This completes the proof in the case where μ/ is a vertical strip.Next we consider the case where μ/ is not a vertical strip.We take the same x_i, a_i, b_i (1≤ i≤ s) as above (in this case we have a_i>b_i for some i), and (c_1,…,c_s) (0≤ c_i ≤ a_i, 1≤ i ≤ s, so long as adding c_i cells on top of the x_i-th column of , for all i,yields a Young diagram of a partition) and (S) (S⊂{1,2,…,s} but bound by the same restriction). Notice that (c_1,…,c_s)/ is a vertical strip if and only if c_i≤ b_i for all i.Now we havef_μ(t)= -∑_μ/: h.s. ⊂ ≠μ t^r_μμ-r_μf_(t). By the induction hypothesis, we have f_(t)=0 unless / is a vertical strip. Since μ/ must be a horizontal strip,must have the form (S). Therefore= - ∑_∅≠ S ⊂{1,2,…,s}a_i-i∈ S≤ b_i(∀ i) t^r_μμ-r_μ(S)f_(S)(t) If there exists some i such that a_i>b_i+1, then f_μ(t)=0 since it is equal to an empty sum. So we assume that there is no such i. We set U := {i∈{1,…,s}| a_i = b_i+1}≠∅. It is easily seen that 1∉ U. Then= - ∑_U⊂ S ⊂{1,2,…,s} t^r_μμ-r_μ(S)f_(S)(t)= -∑_U⊂ T⊂{2,…,s}(t^r_μμ-r_μ(T)f_(T)(t) + t^r_μμ-r_μ({1}∪ T)f_({1}∪ T)(t)_(X))= 0 (since (X)=0 by the same reason as the above case).Remark. By Lemma <ref>, Lemma <ref>(1), say, can be rewritten as:= ∑_μ s.t. ⊂μ⊂ μ/: v.s. (-1)^|μ/|μ∑_i≥ 0(|μ/|+r_μ'')+i-1i h_r-|μ/|-i. Lam08article author=Lam, Thomas,title=Schubert polynomials for the affine Grassmannian,journal=J. Amer. Math. Soc.,volume=21,date=2008,number=1,pages=259–281,MR1950481articleauthor=Lapointe, Luc,author=Lascoux, Alain,author=Morse, Jennifer,title=Tableau atoms and a new Macdonald positivity conjecture,journal=Duke Math. J.,volume=116,date=2003,number=1,pages=103–146, MR1851953articleauthor=Lascoux, Alain,title=Ordering the affine symmetric group,conference= title=Algebraic combinatorics and applications (Gößweinstein, 1999),,book= publisher=Springer, Berlin,,date=2001,pages=219–231, MR3379711collection author=Lam, Thomas,author=Lapointe, Luc,author=Morse, Jennifer,author=Schilling, Anne,author=Shimozono, Mark,author=Zabrocki, Mike,title=k-Schur functions and affine Schubert calculus,series=Fields Institute Monographs,volume=33,publisher=Springer, New York; Fields Institute for Research inMathematical Sciences, Toronto, ON,date=2014,pages=viii+219, MR2741963article author=Lam, Thomas,author=Lapointe, Luc,author=Morse, Jennifer,author=Shimozono, Mark,title=Affine insertion and Pieri rules for the affine Grassmannian,journal=Mem. Amer. Math. Soc.,volume=208,date=2010,number=977,pages=xii+82, isbn=978-0-8218-4658-2,MR2079931article author=Lapointe, Luc,author=Morse, Jenniger,title=Order ideals in weak subposets of Young's lattice and associatedunimodality conjectures,journal=Ann. Comb.,volume=8,date=2004,number=2,pages=197–219, MR2167475article author=Lapointe, Luc,author=Morse, Jennifer,title=Tableaux on k+1-cores, reduced words for affine permutations,and k-Schur expansions,journal=J. Combin. Theory Ser. A,volume=112,date=2005,number=1,pages=44–81, MR2331242article author=Lapointe, Luc,author=Morse, Jennifer,title=A k-tableau characterization of k-Schur functions,journal=Adv. Math.,volume=213,date=2007,number=1,pages=183–204, MR2923177articleauthor=Lam, Thomas,author=Shimozono, Mark,title=From quantum Schubert polynomials to k-Schur functions via theToda lattice,journal=Math. Res. Lett.,volume=19,date=2012,number=1,pages=81–93,MR2660675article author=Lam, Thomas,author=Schilling, Anne,author=Shimozono, Mark,title=K-theory Schubert calculus of the affine Grassmannian,journal=Compos. Math.,volume=146,date=2010,number=4,pages=811–852, MR1354144bookauthor=Macdonald, Ian G.,title=Symmetric functions and Hall polynomials,series=Oxford Mathematical Monographs,edition=2,publisher=The Clarendon Press, Oxford University Press, New York,date=1995,Morse12article author=Morse, Jennifer,title=Combinatorics of the K-theory of affine Grassmannians,journal=Adv. Math.,volume=229,date=2012,number=5,pages=2950–2984, Takigikuarticle author=Takigiku, Motoki,title=On some factorization formulas of K-k-Schur functions II,MasterThesisarticleauthor=Takigiku, Motoki,title=On some factorization formulas of K-k-Schur functions,journal=Master's thesis at University of Tokyo, | http://arxiv.org/abs/1704.08643v1 | {
"authors": [
"Motoki Takigiku"
],
"categories": [
"math.CO"
],
"primary_category": "math.CO",
"published": "20170427162726",
"title": "Factorization formulas of $K$-$k$-Schur functions I"
} |
Large-scale Feature Selection of Risk Genetic FactorsQ. Li et al.^1School of Computing, Informatics, and Decision Systems Engineering, Arizona State Univ.,Tempe, AZ; ^2Imaging Genetics Center, Institute for Neuroimaging and Informatics, Univ.of Southern California, Marina del Rey, CA; ^3Dept. of Computational Medicine and Bioinformatics, Univ. of Michigan, Ann Arbor, MILarge-scale Feature Selection of Risk Genetic Factors Authors' Instructions Large-scale Feature Selection of Risk Genetic Factors for Alzheimer's Disease via Distributed Group Lasso Regression Qingyang Li^1, Dajiang Zhu^2, Jie Zhang^1, Derrek Paul Hibar^2, Neda Jahanshad^2, Yalin Wang^1, Jieping Ye^3, Paul M. Thompson^2, Jie Wang^3 December 30, 2023 ================================================================================================================================================ Genome-wide association studies (GWAS) have achieved great success in the genetic study of Alzheimer's disease (AD). Collaborative imaging genetics studies across different research institutions show the effectiveness of detecting genetic risk factors. However, the high dimensionality of GWAS data poses significant challenges in detecting risk SNPs for AD. Selecting relevant features is crucial in predicting the response variable. In this study, we propose a novel Distributed Feature Selection Framework (DFSF) to conduct the large-scale imaging genetics studies across multiple institutions. To speed up the learning process, we propose a family of distributed group Lasso screening rules to identify irrelevant features and remove them from the optimization. Then we select the relevant group features by performing the group Lasso feature selection process in a sequence of parameters. Finally, we employ the stability selection to rank the top risk SNPs that might help detect the early stage of AD. To the best of our knowledge, this is the first distributed feature selection model integrated with group Lasso feature selection as well as detecting the risk genetic factors across multiple research institutions system. Empirical studies are conducted on 809 subjects with 5.9 million SNPs which are distributed across several individual institutions, demonstrating the efficiency and effectiveness of the proposed method.§ INTRODUCTION Alzheimer’s disease (AD) is known as the most common type of dementia. Genome-Wide Association Studies (GWAS) <cit.> achieved great success in finding single nucleotide polymorphisms (SNPs) associated with AD. Some large-scale collaborative network such as ENIGMA <cit.> Consortium consists of 185 research institutions around the world, analyzing genomic data from over 33,000 subjects, from 35 countries. However, processing and integrating genetic data across different institutions is challenging. The first issue is the data privacy since each participating institution wishes to collaborate with others without revealing its own data set. The second issue is how to conduct the learning process across different institutions. Local Query Model (LQM) <cit.> is proposed to perform the distributed Lasso regression for large-scale collaborative imaging genetics studies across different institutions while preserving the data privacy for each of them. However, in some imaging genetics studies<cit.>, we are more interested in finding important explanatory factors in predicting responses, where each explanatory factor is represented by a group of features since lots of AD genes are continuous or relative with some other features, not individual features. In such cases, the selection of important features corresponds to the selection of groups of features. As an extension of Lasso, group Lasso <cit.> has been proposed for feature selection in a group leveland quite a few efficient algorithms <cit.> have been proposed for efficient optimization. However, integrating group Lasso with imaging genetics studies across multiple institutions has not been studied well. In this study, we propose a novel Distributed Feature Selection Framework (DFSF) to conduct the large-scale imaging genetics studies analysis across multiple research institutions. Our framework has three components. In the first stage, we proposed a family of distributed group lasso screening rules (DSR and DDPP_GL) to identify inactive features and remove them from the optimization. The second stage is to perform the group lasso feature selection process in a distributed manner, selecting the top relevant group features for all the institutions. Finally, each institution obtains the learnt model and perform the stability selection to rank the top risk genes for AD. The experiment is conducted on the Alzheimer's Disease Neuroimaging Initiative (ADNI) GWAS data set, including approximately 809 subjects with 5.9 million loci. Empirical studies demonstrate that proposed method the proposed method achieved a 35-fold speedup compared to state-of-the-art distributed solvers like ADMM. Stability selection results show that the proposed DFSF detects APOE, GRM8, GPC6 and LOC100506272 as top risk SNPs associated with AD, demonstrating a superior result compared to Lasso regression methods <cit.>. The proposed method offers a powerful feature selection tool to study AD and its early symptom.§ PROBLEM STATEMENT§.§ Problem Formulation Group Lasso <cit.> is a highly efficient feature selection and regression technique used in the model construction. Group Lasso takes the form of the equation: min_x∈ℝ^N F(x)=1/2||y-∑_g=1^G[A]_g [x]_g||_2^ 2+ λ∑_g=1^G w_g ||[x]_g||_2,where A represents the feature matrix where A∈ℝ^N× P and y denotes the N dimensional response vector. λ is a positive regularization parameter. Different from Lasso regression <cit.>, group Lasso partitions the original feature matrix A into G non-overlapping groups [A]_1, [A]_2,......,[A]_G and w_g denotes the weight for the g-th group. After solving the group Lasso problem, we get the corresponding G solution vector [x]_1, [x]_2,......,[x]_G and the dimension of [x]_g is the same as the feature space in [A]_g. §.§ ADNI GWAS data The ADNI GWAS dataset contains genotype information of 809 ADNI participants. To store statistically relevant SNPs called using Illumina’s CASAVA SNP Caller, the ADNI WGS SNP data is stored in variant call format (VCF) for storing gene sequence variations. SNPs at approximately 5.9 million specific loci are recorded for each participant. We encode SNPs using the coding scheme in <cit.> and apply Minor Allele Frequency (MAF) <0.05 and Genotype Quality (GQ) <45 as two quality control criteria to filter high quality SNPs features. We follow the same SNP genotype coding and quality control scheme in <cit.>. We have m institutions to conduct the collaborative learning. The ith institution maintains its own data set (A_i,y_i) where A_i∈ℝ^n_i× P, n_i is the sample number, P is the feature number and y_i∈ℝ^n_i is the response and N=∑_i^m n_i. We assume P is the same across m institutions. We aim at conducting the feature selection process of group lasso on the distributed datasets (A_i,y_i), i=1,2,...,m.§ PROPOSED FRAMEWORK In this section, we present the streamline of proposed DFSF framework. The DFSF framework is composed of three main procedures: * Identify the inactive features by the distributed group Lasso screening rules and remove inactive features from optimization.* Solve the group Lasso problem on the reduced feature matrix along a sequence of parameter values and select the most relevant features for each participating institution.* Perform the stability selection to rank SNPs that may collectively affect AD.§.§ Screening Rules for Group Lasso Strong rule <cit.> is an efficient screening method for fitting lasso-like problems by pre-identifying the features which have zero coefficients in the solution and removing these features from optimization, significantly cutting down on the computation required for optimization.For the group lasso problem <cit.>, the gth group of x—[x]_g— will be discarded by strong rules if the following rule holds:||[A]_g^T y||_2≤ w_g(2λ-λ_max)The calculation of λ_max follows λ_max=max_g||[A]_g^T y||_2/w_g. [x]_g could be discarded in the optimization without sacrificing the accuracy since all the elements of [x]_g are zero in the optimal solution vector. Let J denote the index set of groups in the feature space and J={1, 2,......,G }. Suppose that there are G remaining groups after employing screening rules, we use J to represent the index set of remaining groups and J={1, 2,......,G}. As a result, the optimization of group lasso problem (<ref>) can be reformulated as:min_x∈ℝ^NF(x)=1/2||y-∑_g=1^G[A]_g [x]_g||_2^ 2+ λ∑_g=1^G w_g ||[x]_g||_2,where N is the dimension of reduced feature space and x∈ℝ^N. §.§ Distributed Screening Rules for Group LassoAs the data set are distributed among multiple research institutions, it is necessary to conduct a distributed learning process without compromising the data privacy for each institution. LQM <cit.> is proposed to optimize the lasso regression while preserving the data privacy for each participating institution. In this study, we aim at selecting the group features to detect the top risk genetic factors for the entire GWAS data set. Since each institution maintains its own data pair (A_i, y_i), we develop a family of distributed group Lasso screening to identify and discard the inactive features in a distributed environment. We summarize the Distributed Strong Rules (DSR) as follows:* For the ith institution, compute Q_i by Q_i=A_i^T y_i.* Update Q = ∑_i^m Q_i by LQM, then send Q back to all the institutions.* In each institution, calculate λ_max by: λ_max=max_g ||[Q]_g||_2/w_g where [Q]_g is the elements of gth group in Q and it is similar as the definition of [A]_g. * For each gth group in the problem (<ref>), we will discard it and remove from the optimization when the following rule holds:||[Q]_g||_2≤ w_g(2λ-λ_max). In many real word applications, the optimal value of regularization parameter λ is unknown. To tune the value of λ, commonly used methods such as cross validation needs to solve the Lasso problem along a sequence of parameter values λ_0>λ_1>...>λ_κ ,which can be very time-consuming. A sequential version of strong rules was proposed in EDPP <cit.> by utilizing the information of optimal solutions in the previous parameter, achieving about 200x speedups for real-world applications. The implementation details of EDPP is available on the GitHub: http://dpc-screening.github.io/glasso.html. We omit the introduction of EPDD for brevity. We propose a distributed safe screening rules for group Lasso, known as the Distributed Dual Polytope Projection Group Lasso (DDPP_GL), to quickly identify and discard inactive features along a sequence of parameters in a distributed manner. We summarize DDPP_GL in algorithm <ref>.§.§ Distributed Block Coordinate Descent for Group Lasso After we apply DDPP_GL to discard the inactive features, the feature space shrank from P to P and there are remaining G groups. The problem of group Lasso (<ref>) could be reduced as (<ref>). We need to optimize (<ref>) in a distributed manner. The block coordinate descent (BCD) <cit.> is one of the most efficient solvers in the big data optimization. BCD optimize the problem by updating one or a few blocks of variables at a time, rather than updating all the block together. The order of update can be deterministic or stochastic. For the group lasso problem, we can randomly pick up a group of variables to optimize and keeps other groups fixed. Following this idea, we propose a Distributed Block Coordinate Descent (DBCD) to solve the group Lasso problem in algorithm <ref>.In algorithm <ref>, we use a variable R_i to store the result of A_ix-y_i. R_i is initialized as -y_i since x is initialized to be zero at the beginning. In DBCD, the update of gradient can be divided as three steps: * Compute the gradient: ∇ F([x]_g)_i=[A_i]_g^T R_i and get ∇ F([x]_g) by LQM. * Get Δ [x]_g by the gradient information ∇ F([x]_g). * Update R_i: R_i=R_i+Δ [x]_g [A_i]_g^T The update of [x]_g follow the equations in 7rd line of algorithm <ref>. We update [x]_g if ||[x]_g||_2 is larger than λ w_g/L_g, otherwise all the elements of [x]_g are set to be zero. L_g denotes the Lipschitz constant in gth group. For the group Lasso problem, L_g is set to be ||[A]_g||_2^2. DBCD updates R_i at the end of each iteration to make sure R_i stores the correct information of A_ix-y_i in each iteration.§.§ Feature selection by Group LassoGiven a sequence of parameter values: λ_0>...>λ_κ, we can obtain a sequence of learnt models {x^*(λ_0),...,x^*(λ_κ)} by employing DDPP_GL+DBCD. For each group g in the feature space G, we count the frequency of nonzero entries in the learnt model and rank the frequency by descent to get the top relevant features. We summarize the top K feature selection process as follows: * For each group g in the feature space G, I_g = I_g+1, If [x^*(λ_k)]_g is not equal to zero where k∈ (0, κ) and I ∈ℝ^G. * Rank I by descent and select the top K relevant features from A_i to construct the feature matrix A̅_̅i̅. After obtaining the relevant features, we perform the stability selection <cit.> to rank the top genetic factors that are associated with the disease AD.§ EXPERIMENTAL RESULTSIn this section, we conduct several experiments to evaluate the efficiency and effectiveness of our methods. The proposed framework is implemented across three institutions with thirty computation nodes on Apache Spark: http://spark.apach-e.org, a state-of-the-art distributed computing platform. We perform DDPP_GL+ DBCD on a sequence of parameter values and employ stability selection with our methods to determine top risk SNPs related to AD. §.§ Performance Comparison In this experiment, we choose the volume of lateral ventricle as variables being predicted whichcontaining 717 subjects by removing subjects without labels. The volumes of brain regions were extracted from each subject's T1 MRI scan using Freesurfer: http://freesurfer.net. The distributed platform is built across three research institutions that maintain 326, 215, and 176 subjects, respectively and each institution has ten computation nodes. We perform the DDPP_GL+DBCD along a sequence of 100 parameter values equally spaced on the linear scale of λ/λ_maxfrom 1.00 to 0.1. As a comparison, we run the state-of-the-art distributed solver ADMM <cit.> with the same experiment setup. The group size is set to be 20 and we vary the number of features by randomly selecting 0.5 million to 5.9 million from GWAS dataset and report the result in Fig <ref>.The proposed method achieved a 38-fold speedup compared to ADMM. §.§ Stability selection for top risk genetic factorsWe employ stability selection <cit.> with DDPP_GL+DBCD to select top risk SNPs from the entire GWAS data set with 5,906,152 features. We conduct two different groups of trials by choosing the volume of hippocampus and entorhinal cortex at baseline as the response variable for each group, respectively. In each trial, DDPP_GL+DBCD is carried out along a 100 linear-scale sequence of parameter values from 1 to 0.05, respectively. Then we select the top 10000 features and perform stability selection <cit.> to rank the top risk SNPs for AD. As a comparison, we perform D_EDPP+F_LQM<cit.> with the same environment setup and report the result in Table <ref>. In both of trials, APOE is ranked 1st while DDPP_GL+DBCD detects more risk genes like GRM8, GPC6, PIK3C2G and LOC100506272 that are associated with the disease AD in GWAS <cit.>. splncs03 | http://arxiv.org/abs/1704.08383v1 | {
"authors": [
"Qingyang Li",
"Dajiang Zhu",
"Jie Zhang",
"Derrek Paul Hibar",
"Neda Jahanshad",
"Yalin Wang",
"Jieping Ye",
"Paul M. Thompson",
"Jie Wang"
],
"categories": [
"cs.LG",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170427000234",
"title": "Large-scale Feature Selection of Risk Genetic Factors for Alzheimer's Disease via Distributed Group Lasso Regression"
} |
=-0.6in =-0.80in =-0.3in =0.00in=210mm =170mm =0.1in ReIm =12pt plain =16pt An initial-boundary value problem of the general three-componentnonlinear Schrödinger equation with a 4× 4 Lax pair on a finite interval Zhenya Yan[ Email address: [email protected]]Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China =15pt=15pt We investigate the initial-boundary value problem for the general three-component nonlinear Schrödinger (gtc-NLS) equation with a 4× 4 Lax pair on a finite interval by extending the Fokas unified approach. The solutions of the gtc-NLS equation can be expressed in terms of the solutions of a 4× 4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Moreover, the relevant jump matrices of the RH problem can be explicitly found via the three spectral functions arising from the initial data, the Dirichlet-Neumann boundary data. The global relation is also established to deduce two distinct but equivalent types of representations (i.e., one by using the large k of asymptotics of the eigenfunctions and another one in terms of the Gelfand-Levitan-Marchenko (GLM) method) for the Dirichlet and Neumann boundary value problems. Moreover,the relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-lineas the length of the interval approaches to infinity. Finally, we also give the linearizable boundary conditions for the GLM representation.Keywords: Riemann-Hilbert problem; General three-component nonlinear Schrödinger equation; Initial-boundary value problem;Global relation;Maps between Dirichlet and Neumann problems; Gelfand-Levitan-Marchenko representation § INTRODUCTIONIn the theory of integrable systems, the powerful inverse scattering transform (IST) <cit.> (also called nonlinear Fourier transform) was presented to analytically study the initial value problems of the integrable nonlinear waveequations starting from the spectral analysis of their associated systems of linear eigenvalue equations (also known as the Lax pair <cit.>). After that, some significant extensions of the IST were gradually developed. For instance, Deift and Zhou <cit.> developed the IST to present the nonlinear steepest descent method to explicitly explore the long-time asymptotics of the Cauchy problems of (1+1)-dimensional integrable nonlinear evolution equations in terms of RH problems. Fokas <cit.> extended the idea of the IST to put forward a unified method studyingboundary value problems of both linear and integrable nonlinear PDEs with Lax pairs <cit.>. Especially, the Fokas' method can be used to study integrable nonlinear PDEs in terms ofthe simultaneous spectral analysis of both parts of the Lax pairs and the global relations among spectral functions. This approach obviously differs from the standard IST in which the spectral analysis of only one part of the Lax pairs was considered <cit.>.The Fokas' unified method has been used to explore boundary value problems of some physically significant integrable nonlinear evolution equations (NLEEs) with 2× 2 Lax pairs on the half-line and the finite interval (e.g., the nonlinear Schrödinger equation <cit.>, the sine-Gordon equation <cit.>, the KdV equation <cit.>, the mKdV equation <cit.>, the derivative nonlinear Schrödinger equation <cit.>, Ernst equations <cit.>, and etc. <cit.>) and ones with 3× 3 Lax pairs on the half-line and the finite interval (e.g., <cit.>, the Degasperis-Procesi equation <cit.>, the Sasa-Satsuma equation <cit.>, the coupled nonlinear Schrödinger equations <cit.>, and the Ostrovsky-Vakhnenko equation <cit.>). To the best of our knowledge, there was no report on the initial-boundary value (IBV) problems of integrable NLEEs with 4× 4 Lax pairs on the half-line or the finite interval before. The aim of this paper is to develop a methodology for analyzing the IBV problems for integrable NLEEs with 4× 4 Lax pairs on a finite interval. The extension will contain some novelties from 2× 2 and 3× 3 to 4× 4 matrix Lax pairs, but the two key steps of this method <cit.> keep invariant: (i) Finding an integral representation of the solution in terms of a matrix RH problem formulated in the complex k-plane (k is a spectral parameter of the associated Lax pair). The integral representation in general contains the unknown boundary data such that this expression of the solution is not effective yet; (ii) Applying a global relation to consider the unknown boundary values. The representation of the unknown boundary values in general involves the solution of a nonlinear problem. But, this problem for the linearizable boundary conditions can be ignored since the unknown boundary values can be avoided in terms of only algebraic operations.In this paper, we will exhibit how steps (i) and (ii) can be actualized for the integrable general three-component nonlinear Schrödinger (gtc-NLS) equation with a 4× 4 Lax pair <cit.>{[ ị q_1t+ q_1xx-2[α_11|q_1|^2+α_22|q_2|^2+α_33|q_3|^2+2 (α_12q̅_1q_2+α_13q̅_1q_3+α_23q̅_2q_3)]q_1=0,; ị q_2t+ q_2xx-2[α_11|q_1|^2+α_22|q_2|^2+α_33|q_3|^2+2 (α_12q̅_1q_2+α_13q̅_1q_3+α_23q̅_2q_3)]q_2=0,; ị q_3t+ q_3xx-2[α_11|q_1|^2+α_22|q_2|^2+α_33|q_3|^2+2 (α_12q̅_1q_2+α_13q̅_1q_3+α_23q̅_2q_3)]q_3=0, ].where the complex-valued vector fields q_j=q_j(x,t),j=1,2,3 are the sufficiently smooth functions defined in the finite regionΩ={(x,t)| x∈ [0, L],t∈ [0, T]}, with L>0 being the length of the interval and T>0 being the fixed finite time, the overbar denotes the complex conjugate, (·) denotes the real part, and the six coefficients α_ij's (1≤ i≤ j≤ 3) combine a 3× 3 Hermitian-unitary matrix ℳ=([α_11α_12α_13; α̅_12α_22α_23; α̅_13 α̅_23α_33 ]),ℳ=ℳ^†, ℳ^2=𝕀.The gtc-NLS equations contain the group velocity dispersion (GVD, i.e., q_jxx), self-phase modulation (SPM, e.g., |q_j|^2q_j), cross-phase modulation (XPM, e.g., |q_j|^2q_s,j≠s), pair-tunnelingmodulation (PTM, e.g., q_j^2q̅_s,j≠s), and three-tunnelingmodulation (TTM, e.g., q_1q̅_2q_3). System (<ref>) admits the distinct cases for the six parameters α_ij,(1≤ i≤ j≤ 3) such as the three-component focusing NLS equation for α_jj=-1 and α_ij=0 with i<j, the three-component defocusing NLS equation for α_jj=1 and α_ij=0 with i<j, the three-component mixed NLS equation for (α_11=-1, α_22=α_33=1) or (α_11=1, α_22=α_33=-1) and α_ij=0 with i<j, and other general three-component NLS equation. Recently, the three-component defocusing NLS equation with nonzero boundary conditions was studied via the IST <cit.>.We would like to investigate the gtc-NLS equation (<ref>) with the initial-boundary value problems[Initial conditions: q_j(x, t=0)=q_0j(x), j=1,2,3,; Dirichletboundaryconditions: q_j(x=0, t)=u_0j(t),q_j(x=L, t)=v_0j(t), j=1,2,3,; Neumannboundaryconditions:q_jx(x=0, t)=u_1j(t), q_jx(x=L, t)=v_1j(t), j=1,2,3, ] where the initial data q_0j(x), (j=1,2,3), and Dirichlet and Neumann boundary data u_0j(t), v_0j(t) and u_1j(t),v_1j(t),j=1,2,3 are sufficiently smooth and compatible at points (x,t)=(0, 0),(L, 0), respectively.The rest of this paper is organized as follows. In Sec. 2, we investigate thespectral analysis of the associated 4× 4 Lax pair of Eq. (<ref>), such as the eigenfunctions, the jump matrices, and the global relation. Sec. 3 gives the corresponding 4× 4 matrix RH problemby means of the jump matrices obtained in Sec. 2. The global relation is used to establish the map between the Dirichlet and Neumann boundary values in Sec. 4. Particularly, the relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-lineas the length of the interval approaches to infinity. In Sec. 5, we present the Gelfand-Levitan-Marchenko (GLM) representation of the eigenfunctions in terms of the global relation. Moreover, we also show that the GLM representation is equivalent to one in Sec. 4. Finally, we also give the linearizable boundary conditions for the GLM representation. § THE SPECTRAL ANALYSIS OF A 4× 4 LAX PAIR§.§2.1. The exact one-formThe gtc-NLS system (<ref>) can be regarded as the compatible condition of a 4 × 4 Lax pair <cit.>{[ψ_x+ikσ_4ψ=U(x,t)ψ,; ψ_t+2ik^2σ_4ψ=V(x,t,k)ψ, ].where ψ=ψ(x,t,k) is a complex 4×4 matrix-valued or 4× 1 column vector-valuedeigenfunction, k∈ℂ is an iso-spectral parameter, σ_4= diag(1,1,1,-1), and the 4 × 4 matrices U and V are defined byU(x,t)=([000 q_1(x,t);000 q_2(x,t);000 q_3(x,t); p_1(x,t) p_2(x,t) p_3(x,t)0 ]),V(x,t,k)=2kU(x,t)+V_0(x,t),with p_1(x,t)=α_11q̅_1+α̅_12q̅_2+α̅_13q̅_3,p_2(x,t)=α_12q̅_1+α_22q̅_2+α̅_23q̅_3, p_3(x,t)=α_13q̅_1+α_23q̅_2+α_33q̅_3, andV_0(x,t)=-i(U_x+U^2)σ_4=-i([q_1p_1q_1p_2q_1p_3 -q_1x;q_2p_1q_2p_2q_2p_3 -q_2x;q_3p_1q_3p_2q_3p_3 -q_3x;p_1xp_2xp_3x -(q_1p_1+q_2p_2+q_3p_3) ]), Define a new eigenfunction μ(x,t,k) byμ(x,t,k)=ψ(x,t,k)e^i(kx+2k^2t)σ_4, such that the Lax pair (<ref>) becomes the equivalent form for μ(x,t,k){[ μ_x+ik[σ_4,μ]= U(x,t)μ,; μ_t+2ik^2[σ_4,μ]=V(x,t,k)μ, ].where [σ_4, μ]≡σ_4μ-μσ_4. Let σ̂_4 denote the commutator with respect to σ_4 and the operator acting on a 4× 4 matrix X by σ̂_4X=[σ_4, X] such that e^σ̂_4X=e^σ_4Xe^-σ_4, then the Lax pair (<ref>) can be written as a full derivative formd[e^i(kx+2k^2t)σ̂_4μ(x,t,k)]=W(x,t,k),where the exact one-form W(x,t,k) is of the form W(x,t,k)=e^i(kx+2k^2t)σ̂_4[U(x,t)dx+V(x,t,k)dt]μ(x,t,k). §.§2.2. The definition and boundedness of eigenfunctions μ_j'sFor any point (x,t) in the region Ω={(x,t)| x∈ [0, L],t∈ [0, T]},let {γ_j}_1^4 be four contours connecting fours vertexes (x_1, t_1)=(0, T),(x_2, t_2)=(0, 0),(x_3, t_3)=(L, 0),(x_4, t_4)=(L, T) to (x,t), respectively (see Fig. <ref>). Therefore we get the following inequalities on these contours: [ γ_1: (0, T)→ (x,t), x-x' ≥ 0, t-τ≤ 0,; γ_2: (0, 0)→ (x,t), x-x' ≥ 0, t-τ≥ 0,; γ_3: (L, 0)→ (x,t), x-x' ≤ 0, t-τ≥ 0,; γ_4: (L, T)→ (x,t), x-x' ≤ 0, t-τ≤ 0, ]By means of the Volterra integral equations, it follows from Eqs. (<ref>) and (<ref>) that we introduce the four eigenfunctions {μ_j}_1^4 on the four contours {γ_j}_1^4[ μ_j(x,t,k)=I+∫_(x_j, t_j)^(x,t)e^-i(kx+2k^2t)σ̂_4W_j(x',τ,k), ]where 𝕀= diag(1,1,1,1), the integral is over a piecewise smooth curve from (x_j, t_j) to (x,t), and W_j(x,t,k) is given by Eq. (<ref>) with μ(x,t,k) replaced by μ_j(x,t,k). Since the one-form W_j are closed, thus μ_jare independent of the path of integration. If we take the paths of integration to be parallel to the x and t axes, then the integral Eq. (<ref>) reduces to [ μ_j(x,t,k)=I+∫_x_j^x e^-ik(x-x')σ̂_4(Uμ_j)(x',t,k)dx'+e^-ik(x-x_j)σ̂_4∫_t_j^te^-2ik^2(t-τ)σ̂_4 (Vμ_j)(x_j,τ,k)dτ, ]It follows from Eq. (<ref>) that the four columns of the matrix μ_j(x,t,k) contain the following exponentials [μ_j]_s:e^2ik(x-x')+4ik^2(t-τ), j=1,2,3,4; s=1,2,3,[μ_j]_4:e^-2ik(x-x')-4ik^2(t-τ),e^-2ik(x-x')-4ik^2(t-τ), e^-2ik(x-x')-4ik^2(t-τ),j=1,2,3,4 To analyze the bounded regions of the eigenfunctions {μ_j}_1^4, we need to use the curve {k∈ℂ |( f(k))( g(k))=0,f(k)=ik,g(k)=ik^2} to separate the complex k-plane into four regions (see Fig. <ref>): [ D_1={k∈ℂ |f(k)<0and g(k)<0},; D_2={k∈ℂ |f(k)<0and g(k)>0},; D_3={k∈ℂ |f(k)>0and g(k)<0},;D_4={k∈ℂ |f(k)>0 and g(k)>0}, ] which implies that D_1 and D_3 (D_2 and D_4) are symmetric about the origin. Thus it follows from Eqs. (<ref>), (<ref>) and (<ref>) that the regions, where the different columns of eigenfunctions {μ_j}_1^4 are bounded and analytic in the complex k-plane, are presented below: {[ μ_1: (f_- ∩ g_+,f_- ∩ g_+,f_- ∩ g_+,f_+ ∩ g_-)=: (D_2, D_2, D_2, D_3),; μ_2: (f_- ∩ g_-,f_- ∩ g_-,f_- ∩ g_-,f_+ ∩ g_+)=: (D_1, D_1, D_1, D_4),; μ_3: (f_+ ∩ g_-,f_+ ∩ g_-,f_+ ∩ g_-,f_- ∩ g_+)=: (D_3, D_3, D_3, D_2),; μ_4: (f_+ ∩ g_+,f_+ ∩ g_+,f_+ ∩ g_+,f_- ∩ g_-)=: (D_4, D_4, D_4, D_1), ].where f_+=:f(k)>0,f_-=: f(k)<0,g_+=:g(k)>0, and g_-=: g(k)<0. §.§2.3. The definition of the new matrix-valued functions M_n'sTo construct the jump matrix in a RH problem, we introduce the solutions M_n(x,t,k) of Eq. (<ref>)(M_n)_sj(x,t,k)=δ_sj+∫_(γ^n)_sj(e^-i(kx+2k^2t)σ̂_4W_n(x',τ,k))_sj,k∈ D_n,s,j=1,2,3,4,where W_n(x,t,k) isdefined by Eq. (<ref>) with μ(x,t,k) replaced with M_n(x,t,k), and the contours (γ^n)_sj's are given by(γ^n)_sj={[γ_1,if f_s(k)>f_j(k)and g_s(k) ≤ g_j(k),; γ_2,if f_s(k)>f_j(k)and g_s(k) >g_j(k),; γ_ 3,if f_s(k) ≤ f_j(k)and g_s(k) ≥ g_j(k),;γ_4, if f_s(k)≤ f_j(k) and g_s(k)≤ g_j(k), ].for k∈ D_n, where f_1,2,3(k)=-f_4(k)=-ik,g_1,2,3(k)=-g_4(k)=-2ik^2.Notice that to distinguish (γ^n)_sj's to be the contour γ_3 or γ_4 for the special cases, f_s(k)= f_j(k) and g_s(k)= g_j(k), we choose them in these cases as γ_3 (or γ_4) which must appear in the matrix γ^n, otherwise, we choose them in all these cases as the same γ_3 (or γ_4).The definition (<ref>) of (γ^n)_sjimplies that γ^n(n=1,2,3,4) are explicitly given by [ γ^1=( [ γ_4 γ_4 γ_4 γ_2; γ_4 γ_4 γ_4 γ_2; γ_4 γ_4 γ_4 γ_2; γ_4 γ_4 γ_4 γ_4 ]), γ^2=( [ γ_3 γ_3 γ_3 γ_1; γ_3 γ_3 γ_3 γ_1; γ_3 γ_3 γ_3 γ_1; γ_3 γ_3 γ_3 γ_3 ]),; γ^3=( [ γ_3 γ_3 γ_3 γ_3; γ_3 γ_3 γ_3 γ_3; γ_3 γ_3 γ_3 γ_3; γ_1 γ_1 γ_1 γ_3 ]), γ^4=( [ γ_4 γ_4 γ_4 γ_4; γ_4 γ_4 γ_4 γ_4; γ_4 γ_4 γ_4 γ_4; γ_2 γ_2 γ_2 γ_4 ]), ]Proposition 2.1. For the matrix-valued functions M_n(x,t,k)(n=1,2,3,4) defined by Eq. (<ref>) for k∈D̅_n and (x,t)∈Ω, and any fixed point (x,t), M_n(x,t,k)'s are the bounded and analytic functions of k∈ D_n away from a possible discrete set of singularity {k_j} at which the Fredholm determinants vanish. Moreover, M_n(x,t,k)'s admit the bounded and continuous extensions to D̅_n and M_n(x,t,k)=𝕀+O(1/k),k∈ D_n,k→∞,n=1,2,3,4.Proof. Similar to the proof for the 3× 3 Lax pair in <cit.>, we can also proof the bounedness and analyticity of M_n. The substitution of μ(x,t,k)=M_n(x,t,k)=M_n^(0)(x,t,k)+∑_j=1^∞M_n^(j)(x,t,k)/k^j, k →∞,into the x-part of the Lax pair (<ref>) yields Eq. (<ref>). □ The above-defined matrix-valued functions M_n's can be used to formulate a 4× 4 matrix Riemann-Hilbert problem. §.§2.4.Thespectral functions and jump matricesWe introduce the spectral functions S_n(k)(n=1,2,3, 4) by S_n(k)=M_n(x=0,t=0,k),k∈ D_n, n=1,2,3,4.Let M(x,t,k) denote the sectionally analytic function on the Riemann k-spere which is equivalent to M_n(x,t,k) for k∈ D_n. Then M(x,t,k) solves the jump equations M_n(x,t,k)=M_m(x,t,k)J_mn(x,t,k), k∈D̅_n∩D̅_m,n,m=1,2,3,4, n≠ m,with the jump matrices J_mn(x,t,k) defined byJ_mn(x,t,k)=e^-i(kx+2k^2t)σ̂_4(S_m^-1(k)S_n(k)).§.§2.5.The minors or the transpose of the adjugates of eigenfunctionsTo conveniently calculate the spectral functions S_n(k) in the following sections, we need to use the cofactor matrix X^A (or the transpose of the adjugate) of a 4× 4 matrix X defined asadj(X)^T=X^A=([m_11(X) -m_12(X)m_13(X) -m_14(X); -m_21(X)m_22(X) -m_23(X)m_24(X);m_31(X) -m_32(X)m_33(X) -m_34(X); -m_41(X)m_42(X) -m_43(X)m_44(X) ]),where m_ij(X) denotes the (ij)th minor of X and (X^A)^TX = adj(X) X= X.It follows from the Lax pair (<ref>) that the eigenfunction {μ_j^A}_1^4 of the matrices {μ_j(x,t,k)}_1^4 satisfy the Lax equation {[ μ_x^A-ik[σ_4,μ^A]= -U^T(x,t)μ^A,; μ_t^A-2ik^2[σ_4,μ^A]=-V^T(x,t,k)μ^A, ]. whose solutions can be written as the form μ_j^A(x,t,k)=𝕀-∫_γ_je^i[k(x-x')+2k^2(t-τ)]σ̂_4[U^T(x', τ)dx'+V^T(x', τ, k)dτ] μ_j^A(x', τ, k), j=1,2,3,4,in terms of the Volterra integral equations.It is easy to check that the regions of boundedness of μ_j^A: {[ μ_1^A(x,t,k)isbounded for k∈ (D_3, D_3, D_3, D_2),; μ_2^A(x,t,k)isbounded for k∈ (D_4, D_4, D_4, D_1),; μ_3^A(x,t,k)isbounded for k∈ (D_2, D_2, D_2, D_3),; μ_4^A(x,t,k)isbounded for k∈ (D_1, D_1, D_1, D_4). ].which are symmetric ones of μ_j about the k-axis (cf. Eq. (<ref>)). §.§2.6.Symmetries of eigenfunctionsLet Ǔ(x,t, k)=-ikσ_4+U(x,t),V̌(x,t, k)=-2ik^2σ_4+V(x,t,k).in the Lax pair (<ref>). Then we havePǓ(x,t, k̅)P=-Ǔ(x,t,k)^T,PV̌(x,t, k̅)P=-V̌(x,t,k)^T,where the symmetric matrix P is taken asP=([α_11 α̅_12 α̅_13 0;α_12α_22 α̅_23 0;α_13α_23α_33 0; 0 0 0-1 ]), P^2=𝕀, P=P^†, Notice that the symmetric matrix P used herediffers from the diag ones used in 3× 3 Lax pairs <cit.>.Similar to the proof in Ref. <cit.>, based on Eq. (<ref>) and (<ref>) we have the following proposition:Proposition 2.2. The matrix-valued eigenfunctions ψ(x,t,k) of the Lax pair (<ref>) and μ_j(x,t,k) of the Lax pair (<ref>) both possess the same symmetric relations [ ψ^-1(x,t,k)=Pψ(x,t,k̅)^TP,μ_j^-1(x,t,k)=Pμ_j(x,t,k̅)^TP, j=1,2,3,4, ]Moreover, In the domains where μ_j is bounded, we have μ_j(x,t,k)=𝕀+O(1/k), k→∞,j=1,2,3, 4anddet [μ_j(x,t,k)]=1,j=1,2,3, 4since the traces of the matricesU(x,t, k) and V(x,t,k) are zero. §.§2.7. The relations between spectral functions and jump matrices J_mnSince these functions μ_j are dependent, thus we can define three 4× 4 matrix-valued functions S(k),s(k) and 𝕊(k) between μ_2 and μ_j,j=1,3,4 in the form (cf. Fig. <ref>) {[ μ_1(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4S(k),; μ_3(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4s(k),; μ_4(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4𝕊(k), ]. Evaluatingsystem (<ref>) at (x,t)=(0,0) and the three equations in system (<ref>) at (x, t)=(0, T), (L, 0), (L, T), respectively, we have {[S(k)=μ_1(0,0,k)=e^2ik^2Tσ̂_4μ_2^-1(0,T,k),; s(k)=μ_3(0,0,k)=e^ikLσ̂_4μ_2^-1(L,0,k),; 𝕊(k)=μ_4(0,0,k)=e^i(kL+2k^2T)σ̂_4μ_2^-1(L,T,k), ]. Except for the defined three relations, it follows from Eqs. (<ref>) and (<ref>) that we can find other threerelations:(i) the relation between μ_3(x,t,k) and μ_4(x,t,k)[ μ_4(x,t,k)= μ_3(x,t,k)e^-i[k(x-L)+2k^2(t-T)]σ̂_4μ_3^-1(L, T, k)=μ_3(x,t,k)e^-i[k(x-L)+2k^2t]σ̂_4S_L(k), ] withS_L(k)=μ_4(L, 0,k)=e^2ik^2Tσ̂_4μ_3^-1(L, T, k),(ii) the relation between μ_1(x,t,k) and μ_4(x,t,k)[ μ_3(x,t,k)= μ_1(x,t,k)e^-i(kx+2k^2t)σ̂_4𝔖(k), ] with 𝔖(k)=S^-1(k)s(k),and (iii) the relation between μ_1(x,t,k) and μ_4(x,t,k)[ μ_4(x,t,k)= μ_1(x,t,k)e^-i(kx+2k^2t)σ̂_4s_T(k), ] withs_T(k)=S^-1(k)𝕊(k),It follows from Eqs. (<ref>) and (<ref>) that we have the relation 𝕊(k)=s(k)e^ikLσ̂_4S_L(k),The map of these relations among μ_j is exhibited in Fig. <ref>.According to the definition (<ref>) of μ_j, Eq. (<ref>) and (<ref>) imply that [ s(k)=I-∫_0^L e^ikx'σ̂_4(Uμ_3)(x',0,k)dx' =[𝕀+∫_0^L e^ikx'σ̂_4(Uμ_2)(x', 0,k)dx']^-1 ,; S(k)=I-∫_0^T e^2ik^2τσ̂_4(Vμ_1)(0, τ,k)dx' =[𝕀+∫_0^T e^2ik^2τσ̂_4(Vμ_2)(0,τ,k)dτ]^-1,; S_L(k)= I-∫_0^T e^2ik^2τσ̂_4(Vμ_4)(L,τ,k)dτ=[𝕀+∫_0^T e^2ik^2τσ̂_4(Vμ_3)(L,τ,k)dτ]^-1,; 𝕊(k)= I-∫_0^L e^ikx'σ̂_4(Uμ_4)(x',0,k)dx' -e^ikLσ̂_4∫_0^T e^2ik^2τσ̂_4(Vμ_4)(L,τ,k)dτ; = [𝕀+e^2ik^2Tσ̂_4∫_0^L e^ikx'σ̂_4(Uμ_2)(x',T,k)dx'+∫_0^T e^2ik^2τσ̂_4(Vμ_2)(0,τ,k)dτ]^-1, ] which leads to 𝔖(k) and s_T(k) in terms of Eqs. (<ref>) and (<ref>), where μ_j_2(0,t,k), j_2=1,2, μ_j_3(L, t, k),j_3=3,4, μ_j_1(x,0,k),j_1=2,3,4, μ_2(x,T,k),0<x<L,0<t<T are defined by the integral equations [μ_1(0,t,k)=I+∫_T^te^-2ik^2(t-τ)σ̂_4 (Vμ_1)(0,τ,k)dτ,;μ_2(0,t,k)=I+∫_0^te^-2ik^2(t-τ)σ̂_4 (Vμ_2)(0,τ,k)dτ,;μ_3(L,t,k)=I+∫_0^te^-2ik^2(t-τ)σ̂_4 (Vμ_3)(L,τ,k)dτ,;μ_4(L,t,k)=I+∫_T^te^-2ik^2(t-τ)σ̂_4 (Vμ_4)(L,τ,k)dτ,; μ_2(x,0,k)=I+∫_0^x e^ikx'σ̂_4(Uμ_2)(x',0,k)dx',; μ_3(x,0,k)=I+∫_L^x e^ikx'σ̂_4(Uμ_3)(x',0,k)dx',; μ_4(x,0,k)=I+∫_L^x e^ikx'σ̂_4(Uμ_4)(x',0,k)dx'-e^-ik(x-L)σ̂_4∫_0^Te^2ik^2τσ̂_4 (Vμ_4)(L,τ,k)dτ,; μ_2(x,T,k)=I+∫_0^x e^-ik(x-x')σ̂_4(Uμ_2)(x',T,k)dx'+e^-ikxσ̂_4∫_0^Te^-2ik^2(T-τ)σ̂_4 (Vμ_2)(0,τ,k)dτ, ] It follows from the properties of μ_j and μ_j^A that the functions{S(k),s(k), 𝕊(k),S_L(k)} and{S^A(k),s^A(k), 𝕊^A(k),S_L^A(k)} have the following boundedness:{[ S(k) isbounded for k∈(D_2∪ D_4, D_2∪ D_4, D_2∪ D_4, D_1∪ D_3),; s(k) isbounded for k∈(D_3∪ D_4, D_3∪ D_4, D_3∪ D_4, D_1∪ D_2),; 𝕊(k) isbounded for k∈(D_4, D_4, D_4, D_1),; S_L(k) isbounded for k∈(D_2∪ D_4, D_2∪ D_4, D_2∪ D_4, D_1∪ D_3),;S^A(k) isbounded for k∈(D_1∪ D_3, D_1∪ D_3, D_1∪ D_3, D_21∪ D_4),; s^A(k) isbounded for k∈(D_1∪ D_2, D_1∪ D_2, D_1∪ D_2, D_3∪ D_4),; 𝕊^A(k) isbounded for k∈(D_2, D_2, D_2, D_3),; S_L^A(k) isbounded for k∈(D_2∪ D_4, D_2∪ D_4, D_2∪ D_4, D_1∪ D_3), ]. Proposition 2.3. The matrix-valued functions S_n(x,t,k)(n=1,2,3,4) defined by M_n(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4S_n(k),k∈ D_n, with M_n given by Eq. (<ref>) can be determined by the entries of the data S(k)=(S_ij)_4× 4,s(k)=(s_ij)_4× 4, and 𝕊(k)=(𝕊_ij)_4× 4given by Eq. (<ref>) as follows: [ S_1(k)=([ 𝕊_11 𝕊_12 𝕊_130; 𝕊_21 𝕊_22 𝕊_230; 𝕊_31 𝕊_32 𝕊_330; 𝕊_41 𝕊_42 𝕊_43 1m_44(𝕊) ]),S_2(k)=([s_11s_12s_13 S_14(S^Ts^A)_44;s_21s_22s_23 S_24(S^Ts^A)_44;s_31s_32s_33 S_34(S^Ts^A)_44;s_41s_42s_43 S_44(S^Ts^A)_44 ]),;S_3(k)=([ S_3^(11) S_3^(12) S_3^(13) s_14; S_3^(21) S_3^(22) S_3^(23) s_24; S_3^(31) S_3^(32) S_3^(33) s_34; S_3^(41) S_3^(42) S_3^(43) s_44 ]), S_4(k)=([ n_11,44(𝕊)𝕊_44 n_12,44(𝕊)𝕊_44 n_13,44(𝕊)𝕊_44 𝕊_14; n_21,44(𝕊)𝕊_44 n_22,44(𝕊)𝕊_44 n_23,44(𝕊)𝕊_44 𝕊_24; n_31,44(𝕊)𝕊_44 n_32,44(𝕊)𝕊_44 n_33,44(𝕊)𝕊_44 𝕊_34;000 𝕊_44 ]), ] where n_i_1j_1,i_2j_2(X) denotes the determinant of the sub-matrix generated by choosing the cross elements of i_1,2th rows and j_1,2th columns of X, and {[ S_3^(1l)=%̣ṣ/̣%̣ṣm_24(S)n_1l,24(s)-m_34(S)n_1l,34(s)+m_44(S)n_1l,44(s)(s^TS^A)_44,; S_3^(2l)=%̣ṣ/̣%̣ṣm_14(S)n_2l,14(s)-m_34(S)n_2l,34(s)+m_44(S)n_2l,44(s)(s^TS^A)_44,; S_3^(3l)=%̣ṣ/̣%̣ṣm_14(S)n_3l,14(s)-m_24(S)n_3l,24(s)+m_44(S)n_3l,44(s)(s^TS^A)_44,; S_3^(4l)=%̣ṣ/̣%̣ṣm_14(S)n_4l,14(s)-m_24(S)n_4l,24(s)+m_34(S)n_4l,34(s)(s^TS^A)_44, ].l=1,2,3,Proof.We introduce the matrix-valued functions R_n(k), S_n(k), T_n(k), and P_n(k),n=1,2,3,4) by M_n(x,t,k) and μ_j(x,t,k){[ M_n(x,t,k)=μ_1(x,t,k)e^-i(kx+2k^2t)σ̂_4R_n(k),; M_n(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4S_n(k),; M_n(x,t,k)=μ_3(x,t,k)e^-i(kx+2k^2t)σ̂_4T_n(k),; M_n(x,t,k)=μ_4(x,t,k)e^-i(kx+2k^2t)σ̂_4P_n(k), ]. It follows from Eq. (<ref>) that we have the relations {[R_n(k)=e^2ik^2Tσ̂_4M_n(0,T,k),;S_n(k)=M_n(0,0,k),; T_n(k)=e^ikLσ̂_4M_n(L,0,k),; P_n(k)=e^i(kL+2k^2T)σ̂_4M_n(L,T,k), ].and {[ S(k)=μ_1(0,0,k)=S_n(k)R_n^-1(k),; s(k)=μ_3(0,0,k)=S_n(k)T_n^-1(k),; 𝕊(k)=μ_4(0,0,k)=S_n(k)P_n^-1(k), ].which can in general obtain the functions {R_n, S_n, T_n, P_n} for the given functions {s(k), S(k), 𝕊(k)}.Moreover, we can also determine some entries of {R_n, S_n, T_n, P_n} in terms of Eqs. (<ref>) and (<ref>) {[(R_n(k))_ij=0, if(γ^n)_ij=γ_1,;(S_n(k))_ij=0,if (γ^n)_ij=γ_2,;(T_n(k))_ij=δ_ij, if (γ^n)_ij=γ_3,; (P_n(k))_ij=δ_ij,if (γ^n)_ij=γ_4, ]. Thus it follows from systems (<ref>) and(<ref>) that we can find Eq. (<ref>). □ §.§2.8.The residue conditions for M_nSince μ_2(x,t,k) is an entire function, it follows from Eq. (<ref>) that M_n(x,t,k) only have singularities at the points where the S_n(k)'s have singularities. We find from the expressions of S_n(k) given byEq. (<ref>) that the possible singularities of M_n are as follows: * [M]_4 could admit poles in D_1 at the zeros of m_44(𝕊)(k);* [M]_4 could have poles in D_2 at the zeros of (S^Ts^A)_44(k);* [M]_l,l=1,2,3 could be of poles in D_3 at the zeros of(s^TS^A)_44(k);* [M]_l,l=1,2,3 could have poles in D_4 at the zeros of 𝕊_44(k). We introduce the above possible zeros by {k_j}_1^N and suppose that they satisfy the following assumption.Assumption 2.4. We assume that* m_44(𝕊)(k) has n_1 possible simple zeros in D_1 denoted by {k_j}_1^n_1; * (S^Ts^A)_44(k) has n_2-n_1 possible simple zeros in D_2 denoted by {k_j}_n_1+1^n_2; * (s^TS^A)_44(k) has n_3-n_2 possible simple zeros in D_3 denoted by {k_j}_n_2+1^n_3; * 𝕊_44(k) has N-n_3 possible simple zeros in D_4 denoted by {k_j}_n_3+1^N;and that none of these zeros coincide. Moreover, none of these functions are assumed to have zeros on the boundaries od the D_n's (n=1,2,3,4). We can deduce the residue conditions at these zeros in the following expressions:Proposition 2.5.Let {M_n}_1^4 be the eigenfunctions given by Eq. (<ref>) and suppose that the set {k_j}_1^N of singularities is as the above-mentioned Assumption 2.4. Then we have the following residue conditions for M_n: [ Ṛẹṣ_k=k_j [M_1]_4= %̣ṣ/̣%̣ṣn_12,23(𝕊)(k_j)[M_1(k_j)]_1 - n_11,23(𝕊)(k_j)[M_1(k_j)]_2 +n_11,22(𝕊)(k_j)[M_1(k_j)]_3ṁ_44(𝕊)(k_j)m_34(𝕊)(k_j)e^2θ(k_j),;for 1≤ j≤ n_1, k∈ D_1, ][Ṛẹṣ_k=k_j[M_2]_4= [M_2(k_j)]_1[S_14(k_j)n_22,43(s)(k_j)-S_24(k_j)n_12,43(s)(k_j)+S_44(k_j)n_12,23(s)(k_j)](̇Ṡ^̇Ṫṡ^̇Ȧ)̇_44(k_j)m_34(s)(k_j)e^-2θ(k_j);-̣[M_2(k_j)]_2[S_14(k_j)n_21,43(s)(k_j)-S_24(k_j)n_11,43(s)(k_j)+S_44(k_j)n_11,23(s)(k_j)](̇Ṡ^̇Ṫṡ^̇Ȧ)̇_44(k_j)m_34(s)(k_j)e^-2θ(k_j);+[M_2(k_j)]_3[S_14(k_j)n_21,42(s)(k_j)-S_24(k_j)n_11,42(s)(k_j)+S_44(k_j)n_11,22(s)(k_j)](̇Ṡ^̇Ṫṡ^̇Ȧ)̇_44(k_j)m_34(s)(k_j)e^-2θ(k_j),;for n_1+1≤ j≤ n_2, k∈ D_2, ][ Res_k=k_j[M_3]_l= m_14(S)(k_j)n_4l,14(s)(k_j)-m_24(S)(k_j)n_4l,24(s)(k_j)+m_34(S)(k_j)n_4l,34(s)(k_j)(̇ṡ^̇ṪṠ^̇Ȧ)̇_44(k_j)s_44(k_j)e^2θ(k_j); ×[M_3k_j)]_4, for n_2+1≤ j≤ n_3, k∈ D_3, l=1,2,3, ] Res_k=k_j[M_4]_l= -𝕊_4l(k_j)𝕊̇_44(k_j)[M_4(k_j)]_4e^-2θ(k_j), for n_3+1≤ j≤ N, k∈ D_4,l=1,2,3, where the overdot stands for the derivative with resect to the parameter k and θ=θ(k)=-i(kx+2k^2t).Proof.It follows from Eqs. (<ref>) and (<ref>) that the four columns of M_1 are given bythe matrices μ_2 and S_1(k) [M_1]_1=[μ_2]_1𝕊_11+[μ_2]_2𝕊_21+[μ_2]_3𝕊_31+[μ_2]_4𝕊_41e^-2θ,[M_1]_2=[μ_2]_1𝕊_12+[μ_2]_2𝕊_22+[μ_2]_3𝕊_32+[μ_2]_4𝕊_42e^-2θ,[M_1]_3=[μ_2]_1𝕊_13+[μ_2]_2𝕊_23+[μ_2]_3𝕊_33+[μ_2]_4𝕊_43e^-2θ,[M_1]_4=%̣ṣ/̣%̣ṣ[μ_2]_4m_44(𝕊),the four columns of M_2 are given bythe matrices μ_2 and S_2(k) [M_2]_1=[μ_2]_1 s_11 +[μ_2]_2s_21+[μ_2]_3s_31+[μ_2]_4s_41e^-2θ,[M_2]_2=[μ_2]_1 s_12 +[μ_2]_2s_22+[μ_2]_3s_32+[μ_2]_4s_42e^-2θ,[M_2]_3=[μ_2]_1 s_13 +[μ_2]_2s_23+[μ_2]_3s_33+[μ_2]_4s_43e^-2θ,[M_2]_4=%̣ṣ/̣%̣ṣ[μ_2]_1S_14(S^Ts^A)_44e^2θ+[μ_2]_2S_24/(S^Ts^A)_44e^2θ +[μ_2]_3S_34/(S^Ts^A)_44e^2θ+[μ_2]_4S_44/(S^Ts^A)_44,the four columns of M_3 are given bythe matrices μ_2 and S_3(k) [M_3]_1=[μ_2]_1 S_3^(11) +[μ_2]_2S_3^(21)+[μ_2]_3S_3^(31)+[μ_2]_4S_3^(41)e^-2θ,[M_3]_2=[μ_2]_1 S_3^(12) +[μ_2]_2S_3^(22)+[μ_2]_3S_3^(32)+[μ_2]_4S_3^(42)e^-2θ,[M_3]_3=[μ_2]_1 S_3^(13) +[μ_2]_2S_3^(23)+[μ_2]_3S_3^(33)+[μ_2]_4S_3^(43)e^-2θ, [M_3]_4=[μ_2]_1s_14e^2θ+[μ_2]_2s_24e^2θ+[μ_2]_3s_34e^2θ+[μ_2]_4s_44, and the four columns of M_4 are given bythe matrices μ_2 and S_4(k) [M_4]_1=[̣μ_2]_1 n_11,44(𝕊)/𝕊_44+[μ_2]_2 n_21,44(𝕊)/𝕊_44 +[μ_2]_3 n_31,44(𝕊)/𝕊_44,[M_4]_2=[̣μ_2]_1 n_12,44(𝕊)/𝕊_44+[μ_2]_2 n_22,44(𝕊)/𝕊_44 +[μ_2]_3 n_32,44(𝕊)/𝕊_44,[M_4]_3=[̣μ_2]_1 n_13,44(𝕊)/𝕊_44+[μ_2]_2 n_23,44(𝕊)/𝕊_44 +[μ_2]_3 n_33,44(𝕊)/𝕊_44,[M_4]_4=[μ_2]_1𝕊_14e^2θ+[μ_2]_2𝕊_24e^2θ+[μ_2]_3𝕊_34e^2θ+[μ_2]_4𝕊_44, For the case that k_j∈ D_1 is a simple zero of m_44(𝕊)(k), it follows from Eqs. (<ref>)-(<ref>) that we have [μ_2]_j,j=1,2,4 and then substitute them into Eq. (<ref>) to yield[M_1]_4=%̣ṣ/̣%̣ṣn_12,23(𝕊)[M_1]_1-n_11,23(𝕊)[M_1]_2+n_11,22(𝕊)[M_1]_3m_34(𝕊)m_44(𝕊)e^2θ-[μ_2]_3/m_34(𝕊)e^2θ,whose residue at k_j yields Eq. (<ref>)for k_j∈ D_1, respectively.Similarly, we solve Eqs (<ref>)-(<ref>) for [μ_2]_j,j=1,2,4 and then substitute them into Eq (<ref>) to yield [[M_2]_4= [M_2]_1[S_14n_22,43(s)-S_24n_12,43(s)+S_44n_12,23(s)](S^Ts^A)_44m_34(s)e^2θ;-[M_2]_2[S_14n_21,43(s)-S_24n_11,43(s)+S_44n_11,23(s)](S^Ts^A)_44m_34(s)e^2θ; +̣[M_2]_3[S_14n_21,42(s)-S_24n_11,42(s)+S_44n_11,22(s)](S^Ts^A)_44m_34(s)e^2θ -[μ_2]_3/m_34(s)e^2θ, ] whose residues at k_j yields Eq. (<ref>) for k_j∈ D_2, respectively. Similarly, we can show Eq. (<ref>) for k_j∈ D_3 and Eq. (<ref>) for k_j∈ D_4 by analyzing Eqs. (<ref>)-(<ref>).□ §.§2.9.The global relationThe definitions of the above-mentioned spectral functions S(k), s(k), S_L(k), and 𝕊(k) imply that they are dependent. It follows from Eqs. (<ref>) and (<ref>) that [ μ_4(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4𝕊(k); = μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4[s(k)e^ikLσ̂_4S_L(k)]; = μ_1(x,t,k)e^-i(kx+2k^2t)σ̂_4[S^-1(k)s(k)e^ikLσ̂_4S_L(k)], ] which leads to theglobal relationc(T,k)=μ_4(0, T, k)=e^-2ik^2Tσ̂_4[S^-1(k)s(k)e^ikLσ̂_4S_L(k)],by evaluating Eq. (<ref>) at the point (x,t)=(0, T) and using μ_1(0, T, k)=𝕀.§ THE 4× 4 MATRIX RIEMANN-HILBERT PROBLEMBy using the district contours γ_j(j=1,2,3,4), the integral solutions of the revised Lax pair (<ref>), and S_n due to {S(k), s(k), 𝕊(k), S_L(k)}, we have defined the sectionally analytic function M_n(x,t,k)(n=1,2,3,4), which solves a 4× 4 matrix Riemann-Hilbert (RH) problem. This RH problem can be formulated on basis of the initial and boundary data of the functions q_1(x,t), q_2(x,t) and q_3(x,t). Thus the solution of Eq. (<ref>) for all values of x,t can be refound by solving the RH problem.Theorem 3.1.Let (q_1(x,t), q_2(x,t), q_3(x,t)) be a solution of Eq. (<ref>) in the interval domainΩ={(x,t)| x∈ [0, L],t∈ [0, T]}. Then it can be reconstructed from the initial data defined by q_j(x, t=0)=q_0j(x), j=1,2,3,and Dirichlet and Neumann boundary values defined by[ Dirichletboundarydata: q_j(x=0, t)=u_0j(t),q_j(x=L, t)=v_0j(t), j=1,2,3,; Neumannboundarydata:q_jx(x=0, t)=u_1j(t), q_jx(x=L, t)=v_1j(t), j=1,2,3, ] We can use the initial and boundary data to define the jump matrices J_mn(x, t, k),(n, m = 1,..., 4) given by Eq. (<ref>) as well as the spectral functions S(k), s(k) and𝕊(k) defined by Eq. (<ref>). Assume that the possible zeros {k_j}^N_1 of the functions m_44(𝕊)(k), (S^Ts^A)_44(k), (s^TS^A)_44(k), and 𝕊_44(k) are as in Assumption 2.4. Then the solution (q_1(x,t),q_2(x,t), q_3(x,t)) of Eq. (<ref>) is given by M(x,t,k) in the formq_j(x,t)=2̣ilim_k→∞(kM(x,t,k))_j4, j=1,2,3,where M(x,t,k) satisfies the following 4× 4 matrix Riemann-Hilbert problem: * M(x,t,k) is sectionally meromorphic on the Riemann k-sphere with jumpsacross the contours D̅_n∪D̅_m, (n, m = 1,..., 4) (see Fig. <ref>). *Across the contours D̅_n∪D̅_m(n, m = 1,..., 4), M(x, t, k) satisfies thejump condition (<ref>). *The residue conditions of M(x,t,k) are satisfied in Proposition 2.5. * M(x, t, k) = I+O(1/k) as k→∞. Proof.System (<ref>) can be deduced from the large k asymptotics of the eigenfunctions. We can follow the similar one in Refs. <cit.>to show the rest proof of the Theorem. □§ THE NONLINEARIZABLE BOUNDARY CONDITIONSThe key difficulty of initial-boundary value problems is to find the boundary values for a well-posed problem.All boundary value conditions are required for the definition of S(k) and S_L(k), and hence for the formulate theRH problem. Our main conclusion exhibits the unknown boundary condition on basis of the prescribed boundary condition and the initial conditionin terms of the solution of a system of nonlinear integral equations.§.§4.1.The generalized global relationBy evaluating Eqs. (<ref>) and (<ref>) at the point (x,t)=(0, t), we have c(t,k)=μ_2(0,t,k)e^-2ik^2tσ̂_4[s(k)e^ikLσ̂_4S_L(k)], which and Eq. (<ref>) lead to [c(t,k)= μ_2(0,t,k)e^-2ik^2tσ̂_4[s(k)e^ikLσ̂_4e^2ik^2tσ̂_4μ_3^-1(L,t,k)],;= μ_2(0,t,k)[e^-2ik^2tσ̂_4s(k)][e^ikLσ̂_4μ_3^-1(L,t,k)], ] Thus, the column vectors [c(t,k)]_j,j=1,2,3 are analytic and bounded in D_4 away from the possible zeros of 𝕊_44(k) and of order O(1+e^-2ikL/k) as k→∞, and the column vector [c(t,k)]_4 is analytic and bounded in D_1 away from the possible zeros of m_44(𝕊)(k) and of order O(1+e^2ikL/k) as k→∞, §.§4.2. Asymptotic behaviors of eigenfunctionsIt follows from the Lax pair (<ref>) that the eigenfunctions {μ_j}_1^4 possess the following asymptotics as k→∞[ μ_j(x,t,k)= I+∑_i=1^2 1/k^i([ μ_j,11^(i) μ_j,12^(i) μ_j,13^(i) μ_j,14^(i); μ_j,21^(i) μ_j,22^(i) μ_j,23^(i) μ_j,24^(i); μ_j,31^(i) μ_j,32^(i) μ_j,33^(i) μ_j,34^(i); μ_j,41^(i) μ_j,42^(i) μ_j,43^(i) μ_j,44^(i) ]) +O(1/k^3); = I+1/k([ ∫_(x_j, t_j)^(x,t)Δ_11^(1) ∫_(x_j, t_j)^(x,t)Δ_12^(1) ∫_(x_j, t_j)^(x,t)Δ_13^(1) -̣i/2q_1; ∫_(x_j, t_j)^(x,t)Δ_21^(1) ∫_(x_j, t_j)^(x,t)Δ_22^(1) ∫_(x_j, t_j)^(x,t)Δ_23^(1) -̣i/2q_2; ∫_(x_j, t_j)^(x,t)Δ_31^(1) ∫_(x_j, t_j)^(x,t)Δ_32^(1) ∫_(x_j, t_j)^(x,t)Δ_33^(1) -̣i/2q_3;%̣ṣ/̣%̣ṣi2p_1%̣ṣ/̣%̣ṣi2p_2%̣ṣ/̣%̣ṣi2p_3 ∫_(x_j, t_j)^(x,t)Δ_44^(1) ]); +%̣ṣ/̣%̣ṣ1k^2([ ∫_(x_j, t_j)^(x,t)Δ_11^(2) ∫_(x_j, t_j)^(x,t)Δ_12^(2) ∫_(x_j, t_j)^(x,t)Δ_13^(2) μ_j,14^(2); ∫_(x_j, t_j)^(x,t)Δ_21^(2) ∫_(x_j, t_j)^(x,t)Δ_22^(2) ∫_(x_j, t_j)^(x,t)Δ_23^(2) μ_j,24^(2); ∫_(x_j, t_j)^(x,t)Δ_31^(2) ∫_(x_j, t_j)^(x,t)Δ_32^(2) ∫_(x_j, t_j)^(x,t)Δ_33^(2) μ_j,34^(2); μ_j,41^(2) μ_j,42^(2) μ_j,43^(2) ∫_(x_j, t_j)^(x,t)Δ_44^(2) ])+O(1/k^3), ] where we have introduced the following functions {[Δ_jl^(1)= %̣ṣ/̣%̣ṣi2q_jp_ldx+1/2(q_jp_lx-q_jxp_l)dt,j,l=1,2,3,;Δ_44^(1)=-̣i/2∑_j=1^3q_jp_jdx+1/2∑_j=1^3(p_jq_jx-p_jxq_j)dt, ].and {[ μ_j,l4^(2)=%̣ṣ/̣%̣ṣ14q_lx+1/2iq_l∫_(x_j,t_j)^(x,t)Δ_44^(1),l=1,2,3,; μ_j,4l^(2)=%̣ṣ/̣%̣ṣ14p_lx+i/2∑_s=1^3p_s∫_(x_j,t_j)^(x,t)Δ_sl^(1),l=1,2,3,; Δ_sl^(2)=[1/4q_sp_lx+i/2q_s∑_n=1^3p_n∫_(x_j,t_j)^(x,t)Δ_nl^(1)]dx; +̣{1/4[q_sp_lx+iq_sxp_lx-iq_sp_l∑_j=1^3q_jp_j]+1/2∑_n=1^3(q_sp_nx-q_sxp_n)∫_(x_j,t_j)^(x,t)Δ_nl^(1)}dt,s,l=1,2,3,; Δ_44^(2)= [1/4∑_l=1^3p_lq_lx-i/2∑_l=1^3p_lq_l∫_(x_j,t_j)^(x,t)Δ_44^(1)]dx;+{1/4[∑_l=1^3(p_lq_lx-ip_lxq_lx)+i(∑_l=1^3p_lq_l)^2]+ 1/2∑_l=1^3(p_lq_lx-p_lxq_l)∫_(x_j,t_j)^(x,t)Δ_44^(1)}dt, ].The functions {μ^(i)_jl=μ^(i)_jl(x,t)}_1^4,i=1, 2 are independent of k.We define the function {Ψ_ij(t,k)}_i,j=1^4as [ μ_2(0, t,k)=(Ψ_sj(t, k))_4× 4=I+∑_l=1^21/k^l([ Ψ_11^(l)(t) Ψ_12^(l)(t) Ψ_13^(l)(t) Ψ_14^(l)(t); Ψ_21^(l)(t) Ψ_22^(l)(t) Ψ_23^(l)(t) Ψ_24^(l)(t); Ψ_31^(l)(t) Ψ_32^(l)(t) Ψ_33^(l)(t) Ψ_34^(l)(t); Ψ_41^(l)(t) Ψ_42^(l)(t) Ψ_43^(l)(t) Ψ_44^(l)(t) ]) +O(1/k^3), ]Based on the asymptotic of Eq. (<ref>) and the boundary data at x=0, we find {[ Ψ_14^(1)(t)=-%̣ṣ/̣%̣ṣi2u_01(t), Ψ_24^(1)(t)=-i/2u_02(t), Ψ_34^(1)(t)=-i/2u_03(t),; Ψ_41^(1)(t)=%̣ṣ/̣%̣ṣi2[α_11u̅_01(t)+α̅_12u̅_02(t)+α̅_13u̅_03(t)],;Ψ_42^(1)(t)=%̣ṣ/̣%̣ṣi2[α_12u̅_01(t)+α_22u̅_02(t)+α̅_23u̅_03(t)],; Ψ_43^(1)(t)=%̣ṣ/̣%̣ṣi2[α_13u̅_01(t)+α_23u̅_02(t)+α_33u̅_03(t)],; Ψ_14^(2)=%̣ṣ/̣%̣ṣ14u_11+1/2iu_01Ψ_44^(1),Ψ_24^(2)=%̣ṣ/̣%̣ṣ14u_12+1/2iu_02Ψ_44^(1),Ψ_34^(2)=%̣ṣ/̣%̣ṣ14u_13+1/2iu_03Ψ_44^(1),; Ψ_44^(1)=%̣ṣ/̣%̣ṣ12∫^t_0{u_11[α_11u̅_01(t)+α̅_12u̅_02(t)+α̅_13u̅_03(t)]+u_12[α_12u̅_01(t)+α_22u̅_02(t)+α̅_23u̅_03(t)];+u_13[α_13u̅_01(t)+α_23u̅_02(t)+α_33u̅_03(t)]-u_01[α_11u̅_11(t)+α̅_12u̅_12(t)+α̅_13u̅_13(t)]; -u_02[α_12u̅_11(t)+α_22u̅_12(t)+α̅_23u̅_13(t)]-u_03[α_13u̅_11(t)+α_23u̅_12(t)+α_33u̅_13(t)]}dt, ]. Thus we have the the boundary data at x=0: {[ u_01(t)=2iΨ_14^(1)(t),u_02(t)=2iΨ_24^(1)(t), u_03(t)=2iΨ_34^(1)(t),; u_11(t)= 4Ψ_14^(2)(t)+2iu_01(t)Ψ_44^(1)(t),; u_12(t)= 4Ψ_24^(2)(t)+2iu_02(t)Ψ_44^(1)(t),; u_13(t)= 4Ψ_34^(2)(t)+2iu_03(t)Ψ_44^(1)(t), ]. Similarly, we assume that the asymptotic formula of μ_3(L, t, k)={ϕ_ij(t,k)}_i,j=1^4 is of the from [ μ̣_3(L, t,k)=(ϕ_sj(t, k))_4× 4= 𝕀+∑_l=1^21/k^l([ ϕ_11^(l)(t) ϕ_12^(l)(t) ϕ_13^(l)(t) ϕ_14^(l)(t); ϕ_21^(l)(t) ϕ_22^(l)(t) ϕ_23^(l)(t) ϕ_24^(l)(t); ϕ_31^(l)(t) ϕ_32^(l)(t) ϕ_33^(l)(t) ϕ_34^(l)(t); ϕ_41^(l)(t) ϕ_42^(l)(t) ϕ_43^(l)(t) ϕ_44^(l)(t) ]) +O(1/k^3), ]By using the asymptotic of Eq. (<ref>) and the boundary data at x=L, we find {[ ϕ_14^(1)(t)=-%̣ṣ/̣%̣ṣi2v_01(t), ϕ_24^(1)(t)=-i/2v_02(t), ϕ_34^(1)(t)=-i/2v_03(t),; ϕ_41^(1)(t)=%̣ṣ/̣%̣ṣi2[α_11v̅_01(t)+α̅_12v̅_02(t)+α̅_13v̅_03(t)],;ϕ_42^(1)(t)=%̣ṣ/̣%̣ṣi2[α_12v̅_01(t)+α_22v̅_02(t)+α̅_23v̅_03(t)],; ϕ_43^(1)(t)=%̣ṣ/̣%̣ṣi2[α_13v̅_01(t)+α_23v̅_02(t)+α_33v̅_03(t)],; ϕ_14^(2)=%̣ṣ/̣%̣ṣ14v_11+1/2iv_01ϕ_44^(1),ϕ_24^(2)=%̣ṣ/̣%̣ṣ14v_12+1/2iv_02ϕ_44^(1),ϕ_34^(2)=%̣ṣ/̣%̣ṣ14v_12+1/2iv_03ϕ_44^(1),; ϕ_44^(1)=%̣ṣ/̣%̣ṣ12∫^t_0{v_11[α_11v̅_01(t)+α̅_12v̅_02(t)+α̅_13v̅_03(t)]+v_12[α_12v̅_01(t)+α_22v̅_02(t)+α̅_23v̅_03(t)];+v_13[α_13v̅_01(t)+α_23v̅_02(t)+α_33v̅_03(t)]-v_01[α_11v̅_11(t)+α̅_12v̅_12(t)+α̅_13v̅_13(t)]; -v_02[α_12u̅_11(t)+α_22v̅_12(t)+α̅_23v̅_13(t)]-v_03[α_13v̅_11(t)+α_23v̅_12(t)+α_33v̅_13(t)]}dt, ].which generates the following expressions for the boundary values at x=L{[ v_01(t)=2iϕ_14^(1)(t),v_02(t)=2iϕ_24^(1)(t), v_03(t)=2iϕ_34^(1)(t),; v_11(t)= 4ϕ_14^(2)(t)+2iv_01(t)ϕ_44^(1)(t),; v_12(t)= 4ϕ_24^(2)(t)+2iv_02(t)ϕ_44^(1)(t),; v_13(t)= 4ϕ_34^(2)(t)+2iv_03(t)ϕ_44^(1)(t), ]. For the vanishing initial values, it follows from Eq. (<ref>) that we have the following asymptotic of the global relation c_j4(t,k) and c_4j(t,k), j=1,2,3.Proposition 4.1. Let the initial and Dirichlet boundary conditions be compatible at points x=0, L (i.e., q_0j(0)=u_0j(0) at x=0 andq_0j(L)=v_0j(0) at x=L, j=1,2,3). Then, the global relation (<ref>) with the vanishing initial data implies that the large k behaviors ofc_j4(t,k) and c_4j(t,k), j=1,2,3 are of the form[ c_14(t,k)= %̣ṣ/̣%̣ṣΨ_14^(1)k+Ψ_14^(2)+Ψ_14^(1)ϕ̅_44^(1)/k^2 +O(1/k^3); -̣{α_11ϕ̅_41^(1)+α̅_12ϕ̅_42^(1)+α̅_13ϕ̅_43^(1)/k +1/k^2[α_11ϕ̅_41^(2)+α̅_12ϕ̅_42^(2)+α̅_13ϕ̅_43^(2)..;+̣Ψ_11^(1)(α_11ϕ̅_41^(1)+α̅_12ϕ̅_42^(1)+α̅_13ϕ̅_43^(1)) +Ψ_12^(1)(α_12ϕ̅_41^(1)+α_22ϕ̅_42^(1) +α̅_23ϕ̅_43^(1));+̣.. Ψ_13^(1)(α_13ϕ̅_41^(1)+α_23ϕ̅_42^(1)+α_33ϕ̅_43^(1))] +O(1/k^3)}e^2ikL, k→∞, ][ c_24(t,k)= %̣ṣ/̣%̣ṣΨ_24^(1)k+Ψ_24^(2)+Ψ_24^(1)ϕ̅_44^(1)/k^2 +O(1/k^3); -̣{α_12ϕ̅_41^(1)+α_22ϕ̅_42^(1)+α̅_23ϕ̅_43^(1)/k +1/k^2[α_12ϕ̅_41^(2)+α_22ϕ̅_42^(2)+α̅_23ϕ̅_43^(2)..;+̣Ψ_21^(1)(α_11ϕ̅_41^(1)+α̅_12ϕ̅_42^(1)+α̅_13ϕ̅_43^(1)) +Ψ_22^(1)(α_12ϕ̅_41^(1)+α_22ϕ̅_42^(1) +α̅_23ϕ̅_43^(1));+̣.. Ψ_23^(1)(α_13ϕ̅_41^(1)+α_23ϕ̅_42^(1)+α_33ϕ̅_43^(1))] +O(1/k^3)}e^2ikL, k→∞, ][ c_34(t,k)= %̣ṣ/̣%̣ṣΨ_34^(1)k+Ψ_34^(2)+Ψ_34^(1)ϕ̅_44^(1)/k^2 +O(1/k^3); -̣{α_13ϕ̅_41^(1)+α_23ϕ̅_42^(1)+α_33ϕ̅_43^(1)/k +1/k^2[α_13ϕ̅_41^(2)+α_23ϕ̅_42^(2)+α_33ϕ̅_43^(2)..;+̣Ψ_31^(1)(α_11ϕ̅_41^(1)+α̅_12ϕ̅_42^(1)+α̅_13ϕ̅_43^(1)) +Ψ_32^(1)(α_12ϕ̅_41^(1)+α_22ϕ̅_42^(1) +α̅_23ϕ̅_43^(1));+̣.. Ψ_33^(1)(α_13ϕ̅_41^(1)+α_23ϕ̅_42^(1)+α_33ϕ̅_43^(1))] +O(1/k^3)}e^2ikL, k→∞, ][ c_41(t,k)=-̣{α_11ϕ̅_14^(1)+α_12ϕ̅_24^(1)+α_13ϕ̅_34^(1)/k +1/k^2[α_11ϕ̅_14^(2)+α_12ϕ̅_24^(2)+α_13ϕ̅_34^(2)..; ..+Ψ_44^(1)(α_11ϕ̅_14^(1)+α_12ϕ̅_24^(1)+α_13ϕ̅_34^(1))] +O(1/k^3)}e^-2ikL; +̣1/k[(α_11^2+|α_12|^2+|α_13|^2)Ψ_41^(1)+(α_11α_12+α_12α_22+α_13α̅_23)Ψ_42^(1).; . +(α_11α_13+α_12α_23+α_13α_33)Ψ_43^(1)] +1/k^2[(α_11^2+|α_12|^2+|α_13|^2)Ψ_41^(2).; +(α_11α_12+α_12α_22+α_13α̅_23)Ψ_42^(2)+(α_11α_13+α_12α_23+α_13α_33)Ψ_43^(2);+̣Ψ_41^(1)[α_11(α_11ϕ̅_11^(1)+α̅_12ϕ̅_12^(1) +α̅_13ϕ̅_13^(1))+α_12(α_11ϕ̅_21^(1)+α̅_12ϕ̅_22^(1)+α̅_13ϕ̅_23^(1)).; .+α_13(α_11ϕ̅_31^(1)+α̅_12ϕ̅_32^(1)+α̅_13ϕ̅_33^(1))]+Ψ_42^(1)[α_11(α_12ϕ̅_11^(1)+α_22ϕ̅_12^(1) +α̅_23ϕ̅_13^(1)) .; . +α_12(α_12ϕ̅_21^(1)+α_22ϕ̅_22^(1)+α̅_23ϕ̅_23^(1))+α_13(α_12ϕ̅_31^(1)+α_22ϕ̅_32^(1)+α̅_23ϕ̅_33^(1))];+Ψ_43^(1)[α_11(α_13ϕ̅_11^(1)+α_23ϕ̅_12^(1)+α_33ϕ̅_13^(1))+α_12(α_13ϕ̅_21^(1)+α_23ϕ̅_22^(1)+α_33ϕ̅_23^(1)).;. +α_13(α_13ϕ̅_31^(1)+α_23ϕ̅_32^(1)+α_33ϕ̅_33^(1))]+O(1/k^3), k→∞, ][c_42(t,k)= -̣{α̅_12ϕ̅_14^(1)+α_22ϕ̅_24^(1)+α_23ϕ̅_34^(1)/k +1/k^2[α̅_12ϕ̅_14^(2)+α_22ϕ̅_24^(2)+α_23ϕ̅_34^(2)..; ..+Ψ_44^(1)(α̅_12ϕ̅_14^(1)+α_22ϕ̅_24^(1)+α_23ϕ̅_34^(1))]+O(1/k^3)}e^-2ikL;+̣1/k{(|α_12|^2+α_22^2+|α_23|^2)Ψ_42^(1)+(α_11α̅_12+α̅_12α̅_22+α̅_13α_23)Ψ_41^(1).;+̣(α̅_12α_13+α_22α_23+α_23α_33)Ψ_43^(1) +1/k^2[(|α_12|^2+α_22^2+|α_23|^2)Ψ_42^(2).; +(α_11α̅_12+α̅_12α̅_22+α̅_13α_23)Ψ_41^(2)+(α̅_12α_13+α_22α_23+α_23α_33)Ψ_43^(2); +̣Ψ_41^(1)[α̅_12(α_11ϕ̅_11^(1)+α̅_12ϕ̅_12^(1) +α̅_13ϕ̅_13^(1))+α_22(α_11ϕ̅_21^(1)+α̅_12ϕ̅_22^(1)+α̅_13ϕ̅_23^(1)).; .+α_23(α_11ϕ̅_31^(1)+α̅_12ϕ̅_32^(1) +α̅_13ϕ̅_33^(1))]+Ψ_42^(1)[α̅_12(α_12ϕ̅_11^(1)+α_22ϕ̅_12^(1) +α̅_23ϕ̅_13^(1)) .; . +α_22(α_12ϕ̅_21^(1)+α_22ϕ̅_22^(1)+α̅_23ϕ̅_23^(1))+α_23(α_12ϕ̅_31^(1)+α_22ϕ̅_32^(1)+α̅_23ϕ̅_33^(1))]; +Ψ_43^(1)[α̅_12(α_13ϕ̅_11^(1)+α_23ϕ̅_12^(1)+α_33ϕ̅_13^(1))+α_22(α_13ϕ̅_21^(1)+α_23ϕ̅_22^(1)+α_33ϕ̅_23^(1)).;. +α_23(α_13ϕ̅_31^(1)+α_23ϕ̅_32^(1)+α_33ϕ̅_33^(1))]+O(1/k^3), k→∞, ][ c_43(t,k)=-̣{α̅_13ϕ̅_14^(1)+α̅_23ϕ̅_24^(1)+α_33ϕ̅_34^(1)/k +1/k^2[α̅_13ϕ̅_14^(2)+α̅_23ϕ̅_24^(2)+α_33ϕ̅_34^(2)..;..+Ψ_44^(1)(α̅_13ϕ̅_14^(1)+α̅_23ϕ̅_24^(1)+α_33ϕ̅_34^(1))]+O(1/k^3)}e^-2ikL;+̣1/k{(|α_13|^2+|α_23|^2+α_33^2)Ψ_43^(1)+(α_11α̅_13+α̅_12α̅_23+α̅_13α_33)Ψ_41^(1).;+̣(α̅_13α_12+α_22α̅_23+α_13α̅_23)Ψ_42^(1) +1/k^2[(|α_13|^2+|α_23|^2+α_33^2)Ψ_43^(2).; +(α_11α̅_12+α̅_12α̅_22+α̅_13α_23)Ψ_41^(2)+(α̅_12α_13+α_22α_23+α_23α_33)Ψ_43^(2);+̣Ψ_41^(1)[α̅_13(α_11ϕ̅_11^(1)+α̅_12ϕ̅_12^(1) +α̅_13ϕ̅_13^(1))+α̅_23(α_11ϕ̅_21^(1)+α̅_12ϕ̅_22^(1)+α̅_13ϕ̅_23^(1)).; .+α_33(α_11ϕ̅_31^(1)+α̅_12ϕ̅_32^(1) +α̅_13ϕ̅_33^(1))]+Ψ_42^(1)[α̅_13(α_12ϕ̅_11^(1)+α_22ϕ̅_12^(1) +α̅_23ϕ̅_13^(1)) .;. +α̅_23(α_12ϕ̅_21^(1)+α_22ϕ̅_22^(1)+α̅_23ϕ̅_23^(1))+α_33(α_12ϕ̅_31^(1)+α_22ϕ̅_32^(1)+α̅_23ϕ̅_33^(1))];+Ψ_43^(1)[α̅_13(α_13ϕ̅_11^(1)+α_23ϕ̅_12^(1)+α_33ϕ̅_13^(1))+α̅_23(α_13ϕ̅_21^(1)+α_23ϕ̅_22^(1)+α_33ϕ̅_23^(1)).;. +α_33(α_13ϕ̅_31^(1)+α_23ϕ̅_32^(1)+α_33ϕ̅_33^(1))]+O(1/k^3), k→∞, ]Proof.The global relation (<ref>) under the vanishing initial data can be simplified as[c_14(t,k)= Ψ_14(t,k)ϕ̅_44(t,k̅)-e^2ikL[Ψ_11(t,k)(α_11ϕ̅_41(t,k̅)+α̅_12ϕ̅_42(t,k̅) +α̅_13ϕ̅_43(t,k̅)); + Ψ_12(t,k)(α_12ϕ̅_41(t,k̅)+α_22ϕ̅_42(t,k̅)+α̅_23ϕ̅_43(t,k̅));+ Ψ_13(t,k)(α_13ϕ̅_41(t,k̅)+α_23ϕ̅_42(t,k̅)+α_33ϕ̅_43(t,k̅))], ] [c_24(t,k)= Ψ_24(t,k)ϕ̅_44(t,k̅)-e^2ikL[Ψ_21(t,k)(α_11ϕ̅_41(t,k̅)+α̅_12ϕ̅_42(t,k̅) +α̅_13ϕ̅_43(t,k̅)); + Ψ_22(t,k)(α_12ϕ̅_41(t,k̅)+α_22ϕ̅_42(t,k̅)+α̅_23ϕ̅_43(t,k̅));+ Ψ_23(t,k)(α_13ϕ̅_41(t,k̅)+α_23ϕ̅_42(t,k̅)+α_33ϕ̅_43(t,k̅))], ] [c_34(t,k)= Ψ_34(t,k)ϕ̅_44(t,k̅)-e^2ikL[Ψ_31(t,k)(α_11ϕ̅_41(t,k̅)+α̅_12ϕ̅_42(t,k̅) +α̅_13ϕ̅_43(t,k̅)); + Ψ_32(t,k)(α_12ϕ̅_41(t,k̅)+α_22ϕ̅_42(t,k̅)+α̅_23ϕ̅_43(t,k̅));+ Ψ_33(t,k)(α_13ϕ̅_41(t,k̅)+α_23ϕ̅_42(t,k̅)+α_33ϕ̅_43(t,k̅))], ] where ϕ̅_ij(t,k̅)=ϕ_ij(t,k̅).Recalling the time-part of the Lax pair (<ref>) μ_t+2ik^2[σ_4, μ]=V(x,t,k)μ, It follows from the first column of Eq. (<ref>) with μ=μ_2 that we have {[Ψ_11,t(t,k)=(2ku_01+iu_11)Ψ_41-iΨ_11(α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03); -iΨ_21(α_12|u_01|^2+α_22u_01u̅_02+α̅_23u_01u̅_03); -iΨ_31(α_13|u_01|^2+α_23u_01u̅_02+α_33u_01u̅_03),;Ψ_21,t(t,k)=(2ku_02+iu_12)Ψ_41-iΨ_11(α_11u_02u̅_01+α̅_12|u_02|^2+α̅_13u_02u̅_03); -iΨ_21(α_12u_02u̅_01+α_22|u_02|^2+α̅_23u_02u̅_03); -iΨ_31(α_13u_02u̅_01+α_23|u_02|^2+α_33u_02u̅_03),;Ψ_31,t(t,k)= (2ku_03+iu_13)Ψ_41 -iΨ_11(α_11u_03u̅_01+α̅_12u_03u̅_02+α̅_13|u_03|^2); -iΨ_21(α_12u_03u̅_01+α_22u_03u̅_02+α̅_23|u_03|^2); -iΨ_31(α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2),;Ψ_41,t(t,k)=Ψ_11[α_11(2ku̅_01-iu̅_11)+α̅_12(2ku̅_02-iu̅_12)+α̅_13(2ku̅_03-iu̅_13)]; +Ψ_21[α_12(2ku̅_01-iu̅_11)+α_22(2ku̅_02-iu̅_12) +α̅_23(2ku̅_03-iu̅_13)];+Ψ_31[α_13(2ku̅_01-iu̅_11)+α_23(2ku̅_02-iu̅_12) +α_33(2ku̅_03-iu̅_13)];+iΨ_41[4k^2+α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03+α_12u_02u̅_01; +α_22|u_02|^2+α̅_23u_02u̅_03+α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2], ]. The second column of Eq. (<ref>) with μ=μ_2 yields {[Ψ_12,t(t,k)=(2ku_01+iu_11)Ψ_42-iΨ_12(α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03); -iΨ_22(α_12|u_01|^2+α_22u_01u̅_02+α̅_23u_01u̅_03); -iΨ_32(α_13|u_01|^2+α_23u_01u̅_02+α_33u_01u̅_03),;Ψ_22,t(t,k)=(2ku_02+iu_12)Ψ_42-iΨ_12(α_11u_02u̅_01+α̅_12|u_02|^2+α̅_13u_02u̅_03); -iΨ_22(α_12u_02u̅_01+α_22|u_02|^2+α̅_23u_02u̅_03); -iΨ_32(α_13u_02u̅_01+α_23|u_02|^2+α_33u_02u̅_03),;Ψ_32,t(t,k)= (2ku_03+iu_13)Ψ_42 -iΨ_12(α_11u_03u̅_01+α̅_12u_03u̅_02+α̅_13|u_03|^2); -iΨ_22(α_12u_03u̅_01+α_22u_03u̅_02+α̅_23|u_03|^2); -iΨ_32(α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2),;Ψ_42,t(t,k)=Ψ_12[α_11(2ku̅_01-iu̅_11)+α̅_12(2ku̅_02-iu̅_12)+α̅_13(2ku̅_03-iu̅_13)]; +Ψ_22[α_12(2ku̅_01-iu̅_11)+α_22(2ku̅_02-iu̅_12) +α̅_23(2ku̅_03-iu̅_13)];+Ψ_32[α_13(2ku̅_01-iu̅_11)+α_23(2ku̅_02-iu̅_12) +α_33(2ku̅_03-iu̅_13)];+iΨ_42[4k^2+α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03+α_12u_02u̅_01; +α_22|u_02|^2+α̅_23u_02u̅_03+α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2], ].The third column of Eq. (<ref>) with μ=μ_2 yields {[Ψ_13,t(t,k)=(2ku_01+iu_11)Ψ_43-iΨ_13(α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03); -iΨ_23(α_12|u_01|^2+α_22u_01u̅_02+α̅_23u_01u̅_03); -iΨ_33(α_13|u_01|^2+α_23u_01u̅_02+α_33u_01u̅_03),;Ψ_23,t(t,k)=(2ku_02+iu_12)Ψ_43-iΨ_13(α_11u_02u̅_01+α̅_12|u_02|^2+α̅_13u_02u̅_03); -iΨ_23(α_12u_02u̅_01+α_22|u_02|^2+α̅_23u_02u̅_03); -iΨ_33(α_13u_02u̅_01+α_23|u_02|^2+α_33u_02u̅_03),;Ψ_33,t(t,k)= (2ku_03+iu_13)Ψ_43 -iΨ_13(α_11u_03u̅_01+α̅_12u_03u̅_02+α̅_13|u_03|^2); -iΨ_23(α_12u_03u̅_01+α_22u_03u̅_02+α̅_23|u_03|^2); -iΨ_33(α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2),;Ψ_43,t(t,k)=Ψ_13[α_11(2ku̅_01-iu̅_11)+α̅_12(2ku̅_02-iu̅_12)+α̅_13(2ku̅_03-iu̅_13)]; +Ψ_23[α_12(2ku̅_01-iu̅_11)+α_22(2ku̅_02-iu̅_12) +α̅_23(2ku̅_03-iu̅_13)];+Ψ_33[α_13(2ku̅_01-iu̅_11)+α_23(2ku̅_02-iu̅_12) +α_33(2ku̅_03-iu̅_13)];+iΨ_43[4k^2+α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03+α_12u_02u̅_01; +α_22|u_02|^2+α̅_23u_02u̅_03+α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2], ]. The fourth column of Eq. (<ref>) with μ=μ_2 yields {[Ψ_14,t(t,k)= (2ku_01+iu_11)Ψ_44-iΨ_14(4k^2+α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03); -iΨ_24(α_12|u_01|^2+α_22u_01u̅_02+α̅_23u_01u̅_03); -iΨ_34(α_13|u_01|^2+α_23u_01u̅_02+α_33u_01u̅_03),;Ψ_24,t(t,k)=(2ku_02+iu_12)Ψ_44-iΨ_14(α_11u_02u̅_01+α̅_12|u_02|^2+α̅_13u_02u̅_03);-iΨ_24(4k^2+α_12u_02u̅_01+α_22|u_02|^2+α̅_23u_02u̅_03); -iΨ_34(α_13u_02u̅_01+α_23|u_02|^2+α_33u_02u̅_03),;Ψ_34,t(t,k)= (2ku_03+iu_13)Ψ_44 -iΨ_14(α_11u_03u̅_01+α̅_12u_03u̅_02+α̅_13|u_03|^2); -iΨ_24(α_12u_03u̅_01+α_22u_03u̅_02+α̅_23|u_03|^2);-iΨ_34(4k^2+α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2),;Ψ_44,t(t,k)=Ψ_14[α_11(2ku̅_01-iu̅_11)+α̅_12(2ku̅_02-iu̅_12)+α̅_13(2ku̅_03-iu̅_13)]; +Ψ_24[α_12(2ku̅_01-iu̅_11)+α_22(2ku̅_02-iu̅_12) +α̅_23(2ku̅_03-iu̅_13)];+Ψ_34[α_13(2ku̅_01-iu̅_11)+α_23(2ku̅_02-iu̅_12) +α_33(2ku̅_03-iu̅_13)];+iΨ_44[α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03+α_12u_02u̅_01+α_22|u_02|^2;+α̅_23u_02u̅_03+α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2], ].Suppose that Ψ_j1's, j=1,2,3,4 are of the form ([ Ψ_11; Ψ_21; Ψ_31; Ψ_41 ]) =(a_10(t)+a_11(t)/k+a_12(t)/k^2+⋯)+(b_10(t)+b_11(t)/k+b_12(t)/k^2+⋯)e^4ik^2t,where the 4× 1 column vector functions a_1j(t),b_1j(t)(j=0,1,...,) are independent of k.By substituting Eq. (<ref>) into Eq.(<ref>) and using the initial conditionsa_10(0)+b_10(0)=(1, 0,0,0)^T,a_11(0)+b_11(0)=(0, 0, 0, 0)^T,we have ([ Ψ_11; Ψ_21; Ψ_31; Ψ_41 ]) =([ 1; 0; 0; 0 ]) +∑_s=1^21/k^s([ Ψ_11^(s); Ψ_21^(s); Ψ_31^(s); Ψ_41^(s) ]) +O(1/k^3) +[1/2ik([0;0;0; α_11u̅_01(0)+α̅_12u̅_02(0)+α̅_13u̅_03(0) ])+O(1/k^2)]e^4ik^2t,Similarly,it follows from Eqs. (<ref>)-(<ref>) that we have the asymptotic formulae for Ψ_ij,i=1,2,3,4; j=2,3,4 in the form ([ Ψ_12; Ψ_22; Ψ_32; Ψ_42 ]) =([ 1; 0; 0; 0 ]) +∑_s=1^21/k^s([ Ψ_12^(s); Ψ_22^(s); Ψ_32^(s); Ψ_42^(s) ]) +O(1/k^3) +[1/2ik([ 0; 0; 0; α_12u̅_01(0)+α_22u̅_02(0)+α̅_23u̅_03(0) ])+O(1/k^2)]e^4ik^2t, ([ Ψ_13; Ψ_23; Ψ_33; Ψ_43 ]) =([ 1; 0; 0; 0 ]) +∑_s=1^21/k^s([ Ψ_13^(s); Ψ_23^(s); Ψ_33^(s); Ψ_43^(s) ]) +O(1/k^3) +[1/2ik([0;0;0; α_13u̅_01(0)+α_23u̅_02(0)+α_33u̅_03(0) ])+O(1/k^2)]e^4ik^2t,and ([ Ψ_14; Ψ_24; Ψ_34; Ψ_44 ]) =([ 1; 0; 0; 0 ]) +∑_s=1^21/k^s([ Ψ_14^(s); Ψ_24^(s); Ψ_34^(s); Ψ_44^(s) ]) +O(1/k^3) +[i/2k([ u_01(0); u_02(0); u_03(0); 0 ])+O(1/k^2) ]e^-4ik^2t, Similar to Eqs. (<ref>)-(<ref>) for μ_2(0,t,k), we also know that the function μ(x,t,k)=μ_3(L, t,k) at x=L satisfy the t-part of Lax pair (<ref>).The first column of Eq. (<ref>) with μ=μ_3 yields {[ϕ_11,t(t,k)=(2kv_01+iv_11)ϕ_41-iϕ_11(α_11|v_01|^2+α̅_12v_01v̅_02+α̅_13v_01v̅_03); -iϕ_21(α_12|v_01|^2+α_22v_01v̅_02+α̅_23v_01v̅_03); -iϕ_31(α_13|v_01|^2+α_23v_01v̅_02+α_33v_01v̅_03),;ϕ_21,t(t,k)=(2kv_02+iv_12)ϕ_41-iϕ_11(α_11v_02v̅_01+α̅_12|v_02|^2+α̅_13v_02v̅_03); -iϕ_21(α_12v_02v̅_01+α_22|v_02|^2+α̅_23v_02v̅_03); -iϕ_31(α_13v_02v̅_01+α_23|v_02|^2+α_33v_02v̅_03),;ϕ_31,t(t,k)= (2kv_03+iv_13)ϕ_41 -iϕ_11(α_11v_03v̅_01+α̅_12v_03v̅_02+α̅_13|v_03|^2); -iϕ_21(α_12v_03v̅_01+α_22v_03v̅_02+α̅_23|v_03|^2); -iϕ_31(α_13v_03v̅_01+α_23v_03v̅_02+α_33|v_03|^2),;ϕ_41,t(t,k)=ϕ_11[α_11(2kv̅_01-iv̅_11)+α̅_12(2kv̅_02-iv̅_12)+α̅_13(2kv̅_03-iv̅_13)]; +ϕ_21[α_12(2kv̅_01-iv̅_11)+α_22(2kv̅_02-iv̅_12) +α̅_23(2kv̅_03-iv̅_13)];+ϕ_31[α_13(2kv̅_01-iv̅_11)+α_23(2kv̅_02-iv̅_12) +α_33(2kv̅_03-iv̅_13)];+iϕ_41[4k^2+α_11|v_01|^2+α̅_12v_01v̅_02+α̅_13v_01v̅_03+α_12v_02v̅_01; +α_22|v_02|^2+α̅_23v_02v̅_03+α_13v_03v̅_01+α_23v_03v̅_02+α_33|v_03|^2], ].The second column of Eq. (<ref>) with μ=μ_3yields {[ϕ_12,t(t,k)=(2kv_01+iv_11)ϕ_42-iϕ_12(α_11|v_01|^2+α̅_12v_01v̅_02+α̅_13v_01v̅_03); -iϕ_22(α_12|v_01|^2+α_22v_01v̅_02+α̅_23v_01v̅_03); -iϕ_32(α_13|v_01|^2+α_23v_01v̅_02+α_33v_01v̅_03),;ϕ_22,t(t,k)=(2kv_02+iv_12)ϕ_42-iϕ_12(α_11v_02v̅_01+α̅_12|v_02|^2+α̅_13v_02v̅_03); -iϕ_22(α_12v_02v̅_01+α_22|v_02|^2+α̅_23v_02v̅_03); -iϕ_32(α_13v_02v̅_01+α_23|v_02|^2+α_33v_02v̅_03),;ϕ_32,t(t,k)= (2kv_03+iv_13)ϕ_42 -iϕ_12(α_11v_03v̅_01+α̅_12v_03v̅_02+α̅_13|v_03|^2); -iϕ_22(α_12v_03v̅_01+α_22v_03v̅_02+α̅_23|v_03|^2); -iϕ_32(α_13v_03v̅_01+α_23v_03v̅_02+α_33|v_03|^2),;ϕ_42,t(t,k)=ϕ_12[α_11(2kv̅_01-iv̅_11)+α̅_12(2kv̅_02-iv̅_12)+α̅_13(2kv̅_03-iv̅_13)]; +ϕ_22[α_12(2kv̅_01-iv̅_11)+α_22(2kv̅_02-iv̅_12) +α̅_23(2kv̅_03-iv̅_13)];+ϕ_32[α_13(2kv̅_01-iv̅_11)+α_23(2kv̅_02-iv̅_12) +α_33(2kv̅_03-iv̅_13)];+iϕ_42[4k^2+α_11|v_01|^2+α̅_12v_01v̅_02+α̅_13v_01v̅_03+α_12v_02v̅_01; +α_22|v_02|^2+α̅_23v_02v̅_03+α_13v_03v̅_01+α_23v_03v̅_02+α_33|v_03|^2], ].The third column of Eq. (<ref>) with μ=μ_3yields {[ϕ_13,t(t,k)=(2kv_01+iv_11)ϕ_43-iϕ_13(α_11|v_01|^2+α̅_12v_01v̅_02+α̅_13v_01v̅_03); -iϕ_23(α_12|v_01|^2+α_22v_01v̅_02+α̅_23v_01v̅_03); -iϕ_33(α_13|v_01|^2+α_23v_01v̅_02+α_33v_01v̅_03),;ϕ_23,t(t,k)=(2kv_02+iv_12)ϕ_43-iϕ_13(α_11v_02v̅_01+α̅_12|v_02|^2+α̅_13v_02v̅_03); -iϕ_23(α_12v_02v̅_01+α_22|v_02|^2+α̅_23v_02v̅_03); -iϕ_33(α_13v_02v̅_01+α_23|v_02|^2+α_33v_02v̅_03),;ϕ_33,t(t,k)= (2kv_03+iv_13)ϕ_43 -iϕ_13(α_11v_03v̅_01+α̅_12v_03v̅_02+α̅_13|v_03|^2); -iϕ_23(α_12v_03v̅_01+α_22v_03v̅_02+α̅_23|v_03|^2); -iϕ_33(α_13v_03v̅_01+α_23v_03v̅_02+α_33|v_03|^2),;ϕ_43,t(t,k)=ϕ_13[α_11(2kv̅_01-iv̅_11)+α̅_12(2kv̅_02-iv̅_12)+α̅_13(2kv̅_03-iv̅_13)]; +ϕ_23[α_12(2kv̅_01-iv̅_11)+α_22(2kv̅_02-iv̅_12) +α̅_23(2kv̅_03-iv̅_13)];+ϕ_33[α_13(2kv̅_01-iv̅_11)+α_23(2kv̅_02-iv̅_12) +α_33(2kv̅_03-iv̅_13)];+iϕ_43[4k^2+α_11|v_01|^2+α̅_12v_01v̅_02+α̅_13v_01v̅_03+α_12v_02v̅_01; +α_22|v_02|^2+α̅_23v_02v̅_03+α_13v_03v̅_01+α_23v_03v̅_02+α_33|v_03|^2], ].The fourth column of Eq. (<ref>) with μ=μ_3yields {[ϕ_14,t(t,k)= (2kv_01+iv_11)ϕ_44-iϕ_14(4k^2+α_11|v_01|^2+α̅_12v_01v̅_02+α̅_13v_01v̅_03); -iϕ_24(α_12|v_01|^2+α_22v_01v̅_02+α̅_23v_01v̅_03); -iϕ_34(α_13|v_01|^2+α_23v_01v̅_02+α_33v_01v̅_03),;ϕ_24,t(t,k)=(2kv_02+iv_12)ϕ_44-iϕ_14(α_11v_02v̅_01+α̅_12|v_02|^2+α̅_13v_02v̅_03);-iϕ_24(4k^2+α_12v_02v̅_01+α_22|v_02|^2+α̅_23v_02v̅_03); -iϕ_34(α_13v_02v̅_01+α_23|v_02|^2+α_33v_02v̅_03),;ϕ_34,t(t,k)= (2kv_03+iv_13)ϕ_44 -iϕ_14(α_11v_03v̅_01+α̅_12v_03v̅_02+α̅_13|v_03|^2); -iϕ_24(α_12v_03v̅_01+α_22v_03v̅_02+α̅_23|v_03|^2);-iϕ_34(4k^2+α_13v_03v̅_01+α_23v_03v̅_02+α_33|v_03|^2),;ϕ_44,t(t,k)=ϕ_14[α_11(2kv̅_01-iv̅_11)+α̅_12(2kv̅_02-iv̅_12)+α̅_13(2kv̅_03-iv̅_13)]; +ϕ_24[α_12(2kv̅_01-iv̅_11)+α_22(2kv̅_02-iv̅_12) +α̅_23(2kv̅_03-iv̅_13)];+ϕ_34[α_13(2kv̅_01-iv̅_11)+α_23(2kv̅_02-iv̅_12) +α_33(2kv̅_03-iv̅_13)]; +iϕ_44[α_11|v_01|^2+α̅_12v_01v̅_02+α̅_13v_01v̅_03+α_12v_02v̅_01; +α_22|v_02|^2+α̅_23v_02v̅_03+α_13v_03v̅_01+α_23v_03v̅_02+α_33|v_03|^2], ]. Similarly, we can also obtain the asymptotic formulae for ϕ_ij,i,j=1,2,3,4. The substitution of these formulae into Eq. (<ref>) and using the assumption that the initial and boundary data are compatible at x=0 and x=L, we find the asymptotic result (<ref>)of c_14(t,k) for k→∞. Similarly we can also show Eqs. (<ref>) and (<ref>) for c_24(t,k) and c_34(t,k) as k→∞. Similarly, we have the global relation (<ref>) under the vanishing initial data as [ c_41(t,k)= -Ψ_44(t,k)(α_11ϕ̅_14+α_12ϕ̅_24+α_13ϕ̅_34)e^-2ikL;+Ψ_41(t,k)[α_11(α_11ϕ̅_11+α̅_12ϕ̅_12+α̅_13ϕ̅_13)+α_12(α_11ϕ̅_21+α̅_12ϕ̅_22+α̅_13ϕ̅_23);+α_13(α_11ϕ̅_31+α̅_12ϕ̅_32+α̅_13ϕ̅_33)]+Ψ_42(t,k)[α_11(α_12ϕ̅_11+α_22ϕ̅_12+α̅_23ϕ̅_13); +α_12(α_12ϕ̅_21+α_22ϕ̅_22+α̅_23ϕ̅_23)+α_13(α_12ϕ̅_31+α_22ϕ̅_32+α̅_23ϕ̅_33)];+Ψ_43(t,k)[α_11(α_13ϕ̅_11+α_23ϕ̅_12+α_33ϕ̅_13)+α_12(α_13ϕ̅_21+α_23ϕ̅_22+α_33ϕ̅_23);+α_13(α_13ϕ̅_31+α_23ϕ̅_32+α_33ϕ̅_33)], ][c_42(t,k)= -Ψ_44(t,k)(α̅_12ϕ̅_14+α_22ϕ̅_24+α_23ϕ̅_34)e^-2ikL; +Ψ_41(t,k)[α̅_12(α_11ϕ̅_11+α̅_12ϕ̅_12+α̅_13ϕ̅_13)+α_22(α_11ϕ̅_21+α̅_12ϕ̅_22+α̅_13ϕ̅_23); +α_23(α_11ϕ̅_31+α̅_12ϕ̅_32+α̅_13ϕ̅_33)]+Ψ_42(t,k)[α̅_12(α_12ϕ̅_11+α_22ϕ̅_12+α̅_23ϕ̅_13); +α_22(α_12ϕ̅_21+α_22ϕ̅_22+α̅_23ϕ̅_23)+α_23(α_12ϕ̅_31+α_22ϕ̅_32+α̅_23ϕ̅_33)]; +Ψ_43(t,k)[α̅_12(α_13ϕ̅_11+α_23ϕ̅_12+α_33ϕ̅_13)+α_22(α_13ϕ̅_21+α_23ϕ̅_22+α_33ϕ̅_23);+α_23(α_13ϕ̅_31+α_23ϕ̅_32+α_33ϕ̅_33)], ][ c_43(t,k)= -Ψ_44(t,k)(α̅_13ϕ̅_14+α̅_23ϕ̅_24+α_33ϕ̅_34)e^-2ikL;+Ψ_41(t,k)[α̅_13(α_11ϕ̅_11+α̅_12ϕ̅_12+α̅_13ϕ̅_13)+α̅_23(α_11ϕ̅_21+α̅_12ϕ̅_22+α̅_13ϕ̅_23); +α_33(α_11ϕ̅_31+α̅_12ϕ̅_32+α̅_13ϕ̅_33)]+Ψ_42(t,k)[α̅_13(α_12ϕ̅_11+α_22ϕ̅_12+α̅_23ϕ̅_13);+α̅_23(α_12ϕ̅_21+α_22ϕ̅_22+α̅_23ϕ̅_23)+α_13(α_12ϕ̅_31+α_22ϕ̅_32+α̅_23ϕ̅_33)];+Ψ_43(t,k)[α̅_13(α_13ϕ̅_11+α_23ϕ̅_12+α_33ϕ̅_13)+α̅_23(α_13ϕ̅_21+α_23ϕ̅_22+α_33ϕ̅_23);+α_33(α_13ϕ̅_31+α_23ϕ̅_32+α_33ϕ̅_33)], ] where ϕ̅_ij=ϕ̅_ij(t,k̅)=ϕ_ij(t,k̅), such that we can show Eqs. (<ref>)-(<ref>) for c_4j(t,k),j=1,2,3 as k→∞.□ §.§4.3.The relation between Dirichlet and Neumann boundary value problemsIn what follows we show that the spectral functions S(k) and S_L(k) can be expressed in terms of the prescribed Dirichlet and Neumann boundary data and the initial data using the solution of a system of integral equations. Introduce the new notations as F_± (t,k)=F(t,k)± F(t, -k), Σ_±(k)=e^2ikL± e^-2ikL.The sign ∂ D_j, j=1,2,3,4 stands for the boundary of the jth quadrant D_j, oriented so that D_j lies to the left of ∂ D_j. ∂ D_3^0 denotes the boundary contour which has not contain the zeros of Σ_-(k) and ∂ D_3^0=-∂ D_1^0. Theorem 4.2. Let q_0j(x)=q_j(x,t=0)=0,j=1,2,3 be the initial data of Eq. (<ref>) on the interval x∈ [0, L] andT<∞. (i) For the Dirichlet problem, the boundary data u_0j(t) and v_0j(t)(j=1,2,3) on the interval t∈ [0, T) are sufficiently smooth and compatible with the initial data q_0j(x),(j=1,2,3) at points (x_2, t_2)=(0, 0) and (x_3, t_3)=(L, 0), respectively, i.e., u_0j(0)=q_0j(0),v_0j(0)=q_0j(L), j=1,2,3; (ii) For the Neumann problem, the boundary data u_1j(t) and v_0j(t)(j=1,2,3) on the interval t∈ [0, T) are sufficiently smooth and compatible with the initial data q_0j(x),(j=1,2,3) at the origin (x_2, t_2)=(0, 0) and (x_3, t_3)=(L, 0), respectively. For simplicity, let n_33,44(𝕊)(k) have no zeros in the domain D_1. Then the spectral functions S(k) and S_L(k) are defined byS(k)=e^2ik^2Tσ̂_4[P([ Ψ_11(T,k̅) Ψ_21(T,k̅) Ψ_31(T,k̅) Ψ_41(T,k̅); Ψ_12(T,k̅) Ψ_22(T,k̅) Ψ_32(T,k̅) Ψ_42(T,k̅); Ψ_13(T,k̅) Ψ_23(T,k̅) Ψ_33(T,k̅) Ψ_43(T,k̅); Ψ_14(T,k̅) Ψ_24(T,k̅) Ψ_34(T,k̅) Ψ_44(T,k̅) ])P], S_L(k)=e^2ik^2Tσ̂_4[P([ ϕ_11(T,k̅) ϕ_21(T,k̅) ϕ_31(T,k̅) ϕ_41(T,k̅); ϕ_12(T,k̅) ϕ_22(T,k̅) ϕ_32(T,k̅) ϕ_42(T,k̅); ϕ_13(T,k̅) ϕ_23(T,k̅) ϕ_33(T,k̅) ϕ_43(T,k̅); ϕ_14(T,k̅) ϕ_24(T,k̅) ϕ_34(T,k̅) ϕ_44(T,k̅) ])P], where the matrix P is given by Eq. (<ref>), and the complex-valued functions {Ψ_ij(t,k)}_i,j=1^4 have the following system of integral equations {[Ψ_11,t(t,k)=1̣+∫_0^t[(2ku_01+iu_11)Ψ_41-iΨ_11(α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03);-iΨ_21(α_12|u_01|^2+α_22u_01u̅_02+α̅_23u_01u̅_03) -iΨ_31(α_13|u_01|^2+α_23u_01u̅_02+α_33u_01u̅_03)](t',k)dt',; Ψ_21,t(t,k)=∫̣_0^t[ (2ku_02+iu_12)Ψ_41-iΨ_11(α_11u_02u̅_01+α̅_12|u_02|^2+α̅_13u_02u̅_03); -iΨ_21(α_12u_02u̅_01+α_22|u_02|^2+α̅_23u_02u̅_03)-iΨ_31(α_13u_02u̅_01+α_23|u_02|^2+α_33u_02u̅_03)](t',k)dt',;Ψ_31,t(t,k)=∫̣_0^t[ (2ku_03+iu_13)Ψ_41 -iΨ_11(α_11u_03u̅_01+α̅_12u_03u̅_02+α̅_13|u_03|^2); -iΨ_21(α_12u_03u̅_01+α_22u_03u̅_02+α̅_23|u_03|^2)-iΨ_31(α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2)](t',k)dt',; Ψ_41,t(t,k)=∫̣_0^te^4ik^2(t-t'){Ψ_11[α_11(2ku̅_01-iu̅_11)+α̅_12(2ku̅_02-iu̅_12)+α̅_13(2ku̅_03-iu̅_13)];+Ψ_21[α_12(2ku̅_01-iu̅_11)+α_22(2ku̅_02-iu̅_12) +α̅_23(2ku̅_03-iu̅_13)]; +Ψ_31[α_13(2ku̅_01-iu̅_11)+α_23(2ku̅_02-iu̅_12) +α_33(2ku̅_03-iu̅_13)];+iΨ_41[α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03+α_12u_02u̅_01;+α_22|u_02|^2+α̅_23u_02u̅_03+α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2]}(t',k)dt', ]. {[ Ψ_12,t(t,k)=∫̣_0^t[(2ku_01+iu_11)Ψ_42 -iΨ_12(α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03);-iΨ_22(α_12|u_01|^2+α_22u_01u̅_02+α̅_23u_01u̅_03) -iΨ_32(α_13|u_01|^2+α_23u_01u̅_02+α_33u_01u̅_03)](t',k)dt',; Ψ_22,t(t,k)=1̣+∫_0^t[ (2ku_02+iu_12)Ψ_42-iΨ_12(α_11u_02u̅_01+α̅_12|u_02|^2+α̅_13u_02u̅_03);-iΨ_22(α_12u_02u̅_01+α_22|u_02|^2+α̅_23u_02u̅_03) -iΨ_32(α_13u_02u̅_01+α_23|u_02|^2+α_33u_02u̅_03)](t',k)dt',;Ψ_32,t(t,k)=∫̣_0^t[ (2ku_03+iu_13)Ψ_42 -iΨ_12(α_11u_03u̅_01+α̅_12u_03u̅_02+α̅_13|u_03|^2);-iΨ_22(α_12u_03u̅_01+α_22u_03u̅_02+α̅_23|u_03|^2) -iΨ_32(α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2)](t',k)dt',; Ψ_42,t(t,k)=∫̣_0^te^4ik^2(t-t'){Ψ_12[α_11(2ku̅_01-iu̅_11)+α̅_12(2ku̅_02-iu̅_12)+α̅_13(2ku̅_03-iu̅_13)];+Ψ_22[α_12(2ku̅_01-iu̅_11)+α_22(2ku̅_02-iu̅_12) +α̅_23(2ku̅_03-iu̅_13)]; +Ψ_32[α_13(2ku̅_01-iu̅_11)+α_23(2ku̅_02-iu̅_12) +α_33(2ku̅_03-iu̅_13)];+iΨ_42[α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03+α_12u_02u̅_01;+α_22|u_02|^2+α̅_23u_02u̅_03+α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2]}(t',k)dt', ]. {[Ψ_13,t(t,k)=∫̣_0^t[(2ku_01+iu_11)Ψ_43-iΨ_13(α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03);-iΨ_22(α_12|u_01|^2+α_22u_01u̅_02+α̅_23u_01u̅_03) -iΨ_32(α_13|u_01|^2+α_23u_01u̅_02+α_33u_01u̅_03)](t',k)dt',; Ψ_23,t(t,k)=∫̣_0^t[ (2ku_02+iu_12)Ψ_42-iΨ_13(α_11u_02u̅_01+α̅_12|u_02|^2+α̅_13u_02u̅_03); -iΨ_23(α_12u_02u̅_01+α_22|u_02|^2+α̅_23u_02u̅_03)-iΨ_33(α_13u_02u̅_01+α_23|u_02|^2+α_33u_02u̅_03)](t',k)dt',;Ψ_33,t(t,k)=1̣+∫_0^t[ (2ku_03+iu_13)Ψ_43 -iΨ_13(α_11u_03u̅_01+α̅_12u_03u̅_02+α̅_13|u_03|^2); -iΨ_23(α_12u_03u̅_01+α_22u_03u̅_02+α̅_23|u_03|^2)-iΨ_33(α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2)](t',k)dt',; Ψ_43,t(t,k)=∫̣_0^te^4ik^2(t-t'){Ψ_13[α_11(2ku̅_01-iu̅_11)+α̅_12(2ku̅_02-iu̅_12)+α̅_13(2ku̅_03-iu̅_13)];+Ψ_23[α_12(2ku̅_01-iu̅_11)+α_22(2ku̅_02-iu̅_12) +α̅_23(2ku̅_03-iu̅_13)]; +Ψ_33[α_13(2ku̅_01-iu̅_11)+α_23(2ku̅_02-iu̅_12) +α_33(2ku̅_03-iu̅_13)];+iΨ_43[α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03+α_12u_02u̅_01;+α_22|u_02|^2+α̅_23u_02u̅_03+α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2]}(t',k)dt', ].and {[ Ψ_14,t(t,k)=∫̣_0^te^-4ik^2(t-t')[(2ku_01+iu_11)Ψ_44-iΨ_14(α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03);-iΨ_24(α_12|u_01|^2+α_22u_01u̅_02+α̅_23u_01u̅_03)-iΨ_34(α_13|u_01|^2+α_23u_01u̅_02+α_33u_01u̅_03)](t',k)dt',;Ψ_24,t(t,k)=∫̣_0^te^-4ik^2(t-t')[ (2ku_02+iu_12)Ψ_44-iΨ_14(α_11u_02u̅_01+α̅_12|u_02|^2+α̅_13u_02u̅_03);-iΨ_24(α_12u_02u̅_01+α_22|u_02|^2+α̅_23u_02u̅_03)-iΨ_34(α_13u_02u̅_01+α_23|u_02|^2+α_33u_02u̅_03)](t',k)dt',; Ψ_34,t(t,k)=∫̣_0^te^-4ik^2(t-t')[ (2ku_03+iu_13)Ψ_44 -iΨ_14(α_11u_03u̅_01+α̅_12u_03u̅_02+α̅_13|u_03|^2);-iΨ_24(α_12u_03u̅_01+α_22u_03u̅_02+α̅_23|u_03|^2)-iΨ_34(α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2)](t',k)dt',; Ψ_44,t(t,k)=1̣+∫_0^t{Ψ_14[α_11(2ku̅_01-iu̅_11)+α̅_12(2ku̅_02-iu̅_12)+α̅_13(2ku̅_03-iu̅_13)]; +Ψ_24[α_12(2ku̅_01-iu̅_11)+α_22(2ku̅_02-iu̅_12) +α̅_23(2ku̅_03-iu̅_13)];+Ψ_34[α_13(2ku̅_01-iu̅_11)+α_23(2ku̅_02-iu̅_12) +α_33(2ku̅_03-iu̅_13)]; +iΨ_44[α_11|u_01|^2+α̅_12u_01u̅_02+α̅_13u_01u̅_03+α_12u_02u̅_01; +α_22|u_02|^2+α̅_23u_02u̅_03+α_13u_03u̅_01+α_23u_03u̅_02+α_33|u_03|^2]}(t',k)dt', ].The functions {ϕ_ij(t,k)}_i,j=1^4 are of the same integral equations (<ref>)-(<ref>) by replacing the functions {u_0j,u_1j} with {v_0j,v_1j},(j=1,2,3), that is, ϕ_ij(t,k)=Φ_ij(t,k)|_{u_0l(t)=v_0l(t),u_1l(t)=v_1l(t)},(i,j=1,2,3,4; l=1,2,3)(i) For the given Dirichlet problem, the unknown Neumann boundary data {u_1j(t)}_j=1^3 and {v_1j(t)}_j=1^3, 0<t<T can be given by [u_11(t)= ∫̣_∂ D_3^0[2Σ_+/iπΣ_-(kΨ_14-+iu_01)+u_01(2Ψ_44- -ϕ̅_44-)]dk; +∫̣_∂ D_3^04/πΣ_-[α_11(-ikϕ̅_41-+α_11v_01+α_12v_02+α_13v_03).; +̣α̅_12(-ikϕ̅_42-+α̅_12v_01+α_22v_02 +α_23v_03); .+α̅_13(-ikϕ̅_43-+α̅_13v_01 +α̅_23v_02+α_33v_03)]dk; +∫̣_∂ D_3^04k/iπΣ_-[Ψ_14(ϕ̅_44-1)e^-2ikL-(Ψ_11-1)(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .; .- Ψ_12(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)- Ψ_13(α_13ϕ̅_41 +α_23ϕ̅_42+α_33ϕ̅_43)]_-dk, ][ u_12(t)=∫̣_∂ D_3^0[2Σ_+/iπΣ_-(kΨ_24-+iu_02) +u_02(2Ψ_44- -ϕ̅_44-) ]dk; +∫̣_∂ D_3^04/πΣ_-[α_12(-ikϕ̅_41-+α_11v_01+α_12v_02+α_13v_03).;+̣α_22(-ikϕ̅_42-+α̅_12v_01+α_22v_02 +α_23v_03); .+α̅_23(-ikϕ̅_43-+α̅_13v_01 +α̅_23v_02+α_33v_03)]dk; +∫̣_∂ D_3^04k/iπΣ_-[Ψ_24(ϕ̅_44-1)e^-2ikL-Ψ_21(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .;.- (Ψ_22-1)(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)-Ψ_23(α_13ϕ̅_41 +α_23ϕ̅_42+α_33ϕ̅_43)]_-dk, ][ u_13(t)= ∫̣_∂ D_3^0[2Σ_+/iπΣ_-(kΨ_34-+iu_03)+u_03(2Ψ_44- -ϕ̅_44-) ]dk; +∫̣_∂ D_3^04/πΣ_-[α_13(-ikϕ̅_41-+α_11v_01+α_12v_02+α_13v_03).;+̣α_23(-ikϕ̅_42-+α̅_12v_01+α_22v_02 +α_23v_03);.+α_33(-ikϕ̅_43-+α̅_13v_01 +α̅_23v_02+α_33v_03)]dk; +∫̣_∂ D_3^04k/iπΣ_-[Ψ_34(ϕ̅_44-1)e^-2ikL-Ψ_31(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .;.- Ψ_32(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)- (Ψ_33-1)(α_13ϕ̅_41+α_23ϕ̅_42+α_33ϕ̅_43)]_-dk, ] andv_1j(t)=4ϕ_j4^(2)+2/π∫_∂ D_3^0v_0jϕ_44-dk,j=1,2,3, where [ ([ ϕ_14^(2); ϕ_24^(2); ϕ_34^(2) ]) =1/4π∫_∂ D_3^0[2iΣ_+/Σ_-([ kϕ_14-+iv_01; kϕ_24-+iv_02; kϕ_34-+iv_03 ])-Ψ_44-^(1)([ v_01; v_02; v_03 ])]dk +ℳ^T/2iπ([ I_1(t); I_2(t); I_3(t) ]), ] with [ I_1(t)=∫̣_∂ D_3^0{2/Σ_-[(α_11^2+|α_12|^2+|α_13|^2) (kΨ̅_41-+i(α̅_11u_01+α_12u_02+α_13u_03)) .;+̣(α_11α̅_12+α̅_12α_22+α̅_13α_23) (kΨ̅_42-+i(α̅_12u_01+α_22u_02+α_23u_03));+̣(α_11α̅_13+α̅_12α̅_23+α̅_13α_33) (kΨ̅_43-+i(α̅_13u_01+α̅_23u_02+α_33u_03))]}dk;+∫̣_∂ D_3^02k/Σ_-{(1-Ψ̅_44)(α_11ϕ_14+α̅_12ϕ_24+α̅_13ϕ_34)e^2ikL.; +Ψ̅_41[α_11(α_11(ϕ_11-1)+α_12ϕ_12+α_13ϕ_13)+α̅_12(α_11ϕ_21+α_12(ϕ_22-1)+α_13ϕ_23);+α̅_13(α_11ϕ_31+α_12ϕ_32+α_13(ϕ_33-1))] +Ψ̅_42[α_11(α̅_12(ϕ_11-1)+α_22ϕ_12+α_23ϕ_13); +α̅_12(α̅_12ϕ_21+α_22(ϕ_22-1)+α_23ϕ_23)+α̅_13(α̅_12ϕ_31+α_22ϕ_32+α_23(ϕ_33-1))]; +Ψ̅_43[α_11(α̅_13(ϕ_11-1)+α̅_23ϕ_12+α_33ϕ_13)+α̅_12(α̅_13ϕ_21+α̅_23(ϕ_22-1)+α_33ϕ_23); +α̅_13(α̅_13ϕ_31+α̅_23ϕ_32+α_33(ϕ_33-1))]}_-dk,; ][I_2(t)= ∫̣_∂ D_3^0{2/Σ_-[(α_12^2+α_22^2+|α_23|^2) (kΨ̅_42-+i(α̅_12u_01+α_22u_02+α_23u_03)) .; +̣(α_11α_12+α̅_23α_22+α̅_33α_33) (kΨ̅_41-+i(α̅_11u_01+α_12u_02+α_13u_03));+̣(α_12α̅_13+α_22α̅_23+α̅_23α_33)(kΨ̅_43-+i(α̅_13u_01+α̅_23u_02+α_33u_03))]}dk; +∫̣_∂ D_3^02k/Σ_-{(1-Ψ̅_44)(α_12ϕ_14+α_22ϕ_24+α̅_23ϕ_34)e^2ikL.; +Ψ̅_41[α_12(α_11(ϕ_11-1)+α_12ϕ_12+α_13ϕ_13) +α_22(α_11ϕ_21+α_12(ϕ_22-1)+α_13ϕ_23);+α̅_23(α_11ϕ_31+α_12ϕ_32+α_13(ϕ_33-1))] +Ψ̅_42[α_12(α̅_12(ϕ_11-1)+α_22ϕ_12+α_23ϕ_13); +α_22(α̅_12ϕ_21+α_22(ϕ_22-1)+α_23ϕ_23) +α̅_23(α̅_12ϕ_31+α_22ϕ_32+α_23(ϕ_33-1))];+Ψ̅_43[α_12(α̅_13(ϕ_11-1)+α̅_23ϕ_12+α_33ϕ_13)+α_22(α̅_13ϕ_21+α̅_23(ϕ_22-1)+α_33ϕ_23); +α̅_23(α̅_13ϕ_31+α̅_23ϕ_32+α_33(ϕ_33-1))]}_-dk,;][I_3(t)= ∫̣_∂ D_3^0{2/Σ_-[(α_12^2+α_22^2+|α_23|^2) (kΨ̅_43-+i(α̅_13u_01+α̅_23u_02+α_33u_03)).; +̣(α_11α_13+α_12α_23+α̅_13α_23)(kΨ̅_41-+i(α̅_11u_01+α_12u_02+α_13u_03)); +̣(α_13α̅_12+α_22α_23+α̅_13α_23) (kΨ̅_42-+i(α̅_12u_01+α_22u_02+α_23u_03))]}dk;+∫̣_∂ D_3^02k/Σ_-{(1-Ψ̅_44)(α_13ϕ_14+α_23ϕ_24+α_33ϕ_34)e^2ikL.; +Ψ̅_41[α_13(α_11(ϕ_11-1)+α_12ϕ_12+α_13ϕ_13) +α_23(α_11ϕ_21+α_12(ϕ_22-1)+α_13ϕ_23); +α_33(α_11ϕ_31+α_12ϕ_32+α_13(ϕ_33-1))] +Ψ̅_42[α_13(α̅_12(ϕ_11-1)+α_22ϕ_12+α_23ϕ_13);+α_23(α̅_12ϕ_21+α_22(ϕ_22-1)+α_23ϕ_23) +α_33(α̅_12ϕ_31+α_22ϕ_32+α_23(ϕ_33-1))];+Ψ̅_43[α_13(α̅_13(ϕ_11-1)+α̅_23ϕ_12+α_33ϕ_13)+α_23(α̅_13ϕ_21+α̅_23(ϕ_22-1)+α_33ϕ_23);+α_33(α̅_13ϕ_31+α̅_23ϕ_32+α_33(ϕ_33-1))]}_-dk,;](ii) For the known Neumannproblem, the unknown Dirichlet boundary data {u_0j(t)}_j=1^3 and {v_0j(t)}_j=1^3,0<t<T can be given by[ u_01(t)=∫̣_∂ D_3^01/πΣ_-[Σ_+Ψ_14+-2 (α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43)_+]dk;+∫̣_∂ D_3^02/πΣ_-{Ψ_14(ϕ̅_44-1)e^-2ikL-[(Ψ_11-1)(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .;. + Ψ_12(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)+ Ψ_13(α_13ϕ̅_41+α_23ϕ̅_42+α_33ϕ̅_43)]}_+dk, ] [ u_02(t)= ∫̣_∂ D_3^01/πΣ_-[Σ_+Ψ_24+-2 (α_12ϕ̅_41+α_22ϕ̅_42 +α̅_23ϕ̅_43)_+]dk;+∫̣_∂ D_3^02/πΣ_-{Ψ_24(ϕ̅_44-1)e^-2ikL-[Ψ_21(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .;. + (Ψ_22-1)(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)+ Ψ_23(α_13ϕ̅_41+α_23ϕ̅_42+α_33ϕ̅_43)]}_+dk, ] [ u_03(t)=∫̣_∂ D_3^01/πΣ_-[Σ_+Ψ_34+-2 (α_13ϕ̅_41+α_23ϕ̅_42 +α_33ϕ̅_43)_+]dk;+∫̣_∂ D_3^02/πΣ_-{Ψ_34(ϕ̅_44-1)e^-2ikL-[Ψ_31(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .;. + Ψ_32(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)+ (Ψ_33-1)(α_13ϕ̅_41+α_23ϕ̅_42+α_33ϕ̅_43)]}_+dk, ] andv_01(t)=2iϕ_14^(1), v_02(t)=2iϕ_24^(1), v_03(t)=2iϕ_34^(1),where [ ([ ϕ_14^(1); ϕ_24^(1); ϕ_34^(1) ]) =-1/2iπ∫_∂ D_3^0Σ_+/Σ_-([ ϕ_14+; ϕ_24+; ϕ_34+ ])dk+ℳ^T/2iπ([ J_1(t); J_2(t); J_3(t) ]), ] with [J_1(t)= ∫̣_∂ D_3^02/Σ_-[(α_11^2+|α_12|^2+|α_13|^2) Ψ̅_41++(α_11α̅_12+α̅_12α_22+α̅_13α_23) Ψ̅_42+;+̣(α_11α̅_13+α̅_12α̅_23+α̅_13α_33) Ψ̅_43+]dk; +∫̣_∂ D_3^02/Σ_-{(1-Ψ̅_44)(α_11ϕ_14+α̅_12ϕ_24+α̅_13ϕ_34)e^2ikL.; +Ψ̅_41[α_11(α_11(ϕ_11-1)+α_12ϕ_12+α_13ϕ_13)+α̅_12(α_11ϕ_21+α_12(ϕ_22-1)+α_13ϕ_23);+α̅_13(α_11ϕ_31+α_12ϕ_32+α_13(ϕ_33-1))] +Ψ̅_42[α_11(α̅_12(ϕ_11-1)+α_22ϕ_12+α_23ϕ_13); +α̅_12(α̅_12ϕ_21+α_22(ϕ_22-1)+α_23ϕ_23)+α̅_13(α̅_12ϕ_31+α_22ϕ_32+α_23(ϕ_33-1))]; +Ψ̅_43[α_11(α̅_13(ϕ_11-1)+α̅_23ϕ_12+α_33ϕ_13)+α̅_12(α̅_13ϕ_21+α̅_23(ϕ_22-1)+α_33ϕ_23); +α̅_13(α̅_13ϕ_31+α̅_23ϕ_32+α_33(ϕ_33-1))]}_+dk,;][J_2(t)=∫̣_∂ D_3^02/Σ_-[(α_12^2+α_22^2+|α_23|^2) Ψ̅_42++(α_11α_12+α̅_23α_22+α̅_33α_33) Ψ̅_41+;+̣(α_12α̅_13+α_22α̅_23+α̅_23α_33)Ψ̅_43+]dk;+∫̣_∂ D_3^02/Σ_-{(1-Ψ̅_44)(α_12ϕ_14+α_22ϕ_24+α̅_23ϕ_34)e^2ikL.; +Ψ̅_41[α_12(α_11(ϕ_11-1)+α_12ϕ_12+α_13ϕ_13) +α_22(α_11ϕ_21+α_12(ϕ_22-1)+α_13ϕ_23);+α̅_23(α_11ϕ_31+α_12ϕ_32+α_13(ϕ_33-1))] +Ψ̅_42[α_12(α̅_12(ϕ_11-1)+α_22ϕ_12+α_23ϕ_13); +α_22(α̅_12ϕ_21+α_22(ϕ_22-1)+α_23ϕ_23) +α̅_23(α̅_12ϕ_31+α_22ϕ_32+α_23(ϕ_33-1))];+Ψ̅_43[α_12(α̅_13(ϕ_11-1)+α̅_23ϕ_12+α_33ϕ_13)+α_22(α̅_13ϕ_21+α̅_23(ϕ_22-1)+α_33ϕ_23); +α̅_23(α̅_13ϕ_31+α̅_23ϕ_32+α_33(ϕ_33-1))]}_+dk,;][J_3(t)= ∫̣_∂ D_3^02/Σ_-[(α_12^2+α_22^2+|α_23|^2) Ψ̅_43++(α_11α_13+α_12α_23+α̅_13α_23) Ψ̅_41+;+̣(α_13α̅_12+α_22α_23+α̅_13α_23) Ψ̅_42+]dk; +∫̣_∂ D_3^02/Σ_-{(1-Ψ̅_44)(α_13ϕ_14+α_23ϕ_24+α_33ϕ_34)e^2ikL.; +Ψ̅_41[α_13(α_11(ϕ_11-1)+α_12ϕ_12+α_13ϕ_13) +α_23(α_11ϕ_21+α_12(ϕ_22-1)+α_13ϕ_23); +α_33(α_11ϕ_31+α_12ϕ_32+α_13(ϕ_33-1))] +Ψ̅_42[α_13(α̅_12(ϕ_11-1)+α_22ϕ_12+α_23ϕ_13);+α_23(α̅_12ϕ_21+α_22(ϕ_22-1)+α_23ϕ_23) +α_33(α̅_12ϕ_31+α_22ϕ_32+α_23(ϕ_33-1))];+Ψ̅_43[α_13(α̅_13(ϕ_11-1)+α̅_23ϕ_12+α_33ϕ_13)+α_23(α̅_13ϕ_21+α̅_23(ϕ_22-1)+α_33ϕ_23);+α_33(α̅_13ϕ_31+α̅_23ϕ_32+α_33(ϕ_33-1))]}_+dk,;] where Ψ_14=Ψ_14(t,k), ϕ̅_44=ϕ_44(t, k̅)=ϕ̅_44(t, k̅) and other functions have the similar expressions.Proof. We can show that Eqs. (<ref>) and (<ref>) hold by means of Eqs. (<ref>) and (<ref>) with replacing T by t, that is,S(k)=e^-2ik^2tσ̂_4μ_2^-1(0,t,k) and S_L(k)=e^-2ik^2tσ̂_4μ_3^-1(L,t,k) and the symmetry relation (<ref>). Moreover, Eqs. (<ref>)-(<ref>) for Ψ_ij(t,k),i,j=1,2,3,4 can be obtained by using the Volteral integral equations of μ_2(0,t,k). Similarly, the expressions of ϕ_ij(t,k),(i,j=1,2,3,4) can be found by means of the Volteral integral equations of μ_3(L,t,k). In what follows we show Eqs. (<ref>)-(<ref>), that is the map between Dirichlet and Neumann boundary conditions.(i) The Cauchy's theorem is employed to study Eq. (<ref>) togenerate [iπΨ_44^(1)(t)= -̣(∫_∂ D_2+∫_∂ D_4)[Ψ_44(t,k)-1]dk = (∫_∂ D_1+∫_∂ D_3)[Ψ_44(t,k)-1]dk; = ∫̣_∂ D_3[Ψ_44(t,k)-1]dk-∫_∂ D_3[Ψ_44(t,-k)-1]dk =∫_∂ D_3Ψ_44-(t,k)dk, ] and [ iπΨ_14^(2)(t)=(∫_∂ D_1+∫_∂ D_3)[kΨ_14(t,k)+i/2u_01(t)]dk, =∫̣_∂ D_3[kΨ_14(t,k)+i/2u_01(t)]_-dk,;= ∫̣_∂ D_3^0{kΨ_14(t,k)+i/2u_01(t)+2e^-2ikL/Σ_-(k)[kΨ_14(t,k)+i/2u_01(t)]}_-dk+C_1(t),;=∫̣_∂ D_3^0Σ_+/Σ_-(kΨ_14-+iu_01)dk+C_1(t), ] where we have introduced the function C_1(t) asC_1(t)=-∫̣_∂ D_3^0{2e^-2ikL/Σ_-[kΨ_14(t,k)+i/2u_01(t)]}_-dk, We use the global relation (<ref>) to further reduce C_1(t) in the form [ C_1(t)=-∫̣_∂ D_3^0{2e^-2ikL/Σ_-[kΨ_14(t,k)+i/2u_01(t)]}_-dk; = ∫̣_∂ D_3^0{2e^-2ikL/Σ_-[-kc_14+Ψ_14^(1)+ Ψ_14^(1)ϕ̅_44^(1)/k-(α_11ϕ̅_41^(1)+α̅_12ϕ̅_42^(1) +α̅_13ϕ̅_43^(1))e^2ikL]}_-dk;-∫̣_∂ D_3^0{2e^-2ikL/Σ_-[ Ψ_14^(1)ϕ̅_44^(1)/k+ (α_11(kϕ̅_41-ϕ̅_41^(1)) +α̅_12(kϕ̅_42-ϕ̅_42^(1)) ..; .. +α̅_13(kϕ̅_43-ϕ̅_43^(1))) e^2ikL]}_-dk; +∫̣_∂ D_3^0{2ke^-2ikL/Σ_-[ Ψ_14(ϕ̅_44-1)-[(Ψ_11-1)(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) ..;.. + Ψ_12(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)+ Ψ_13(α_13ϕ̅_41+α_23ϕ̅_42+α_33ϕ̅_43)]e^2ikL]}_-dk, ]By applying the Cauchy's theorem and asymptotic (<ref>) to Eq. (<ref>), we find that C_1(t) can reduce to [ C_1(t)=-iπΨ_14^(2)-∫̣_∂ D_3^0{i/2u_01ϕ̅_44- +2i/Σ_-[α_11(-ikϕ̅_41-+α_11v_01 +α_12v_02+α_13v_03)..; +̣α̅_12(-ikϕ̅_42-+α̅_12v_01+α_22v_02+α_23v_03) ..+α̅_13(-ikϕ̅_43-+α̅_13v_01+α̅_23v_02+α_33v_03)]}dk; +∫̣_∂ D_3^02k/Σ_-[Ψ_14(ϕ̅_44-1)e^-2ikL-(Ψ_11-1)(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .;.- Ψ_12(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)- Ψ_13(α_13ϕ̅_41+α_23ϕ̅_42+α_33ϕ̅_43)]_-dk, ]It follows from Eqs. (<ref>) and (<ref>) that we have [2iπΨ_13^(2)(t)=∫̣_∂ D_3^0[Σ_+/Σ_-(kΨ_14-+iu_01)-i/2u_01ϕ̅_44-]dk;+∫̣_∂ D_3^02i/Σ_-[α_11(-ikϕ̅_41-+α_11v_01 +α̅_12v_02+α̅_13v_03).;+̣α̅_12(-ikϕ̅_42-+α_12v_01+α_22v_02+α̅_23v_03);..+α̅_13(-ikϕ̅_43-+α_13v_01+α_23v_02+α_33v_03)]}dk; +∫̣_∂ D_3^02k/Σ_-[Ψ_14(ϕ̅_44-1)e^-2ikL-(Ψ_11-1)(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .;.- Ψ_12(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)- Ψ_13(α_13ϕ̅_41+α_23ϕ̅_42+α_33ϕ̅_43)]_-dk, ] Thus substituting Eqs. (<ref>) and (<ref>) into the third one of system (<ref>), we can get Eq. (<ref>).Similarly, we can also showEqs. (<ref>) and (<ref>).To use Eq. (<ref>) to show Eq. (<ref>) for v_11(t) we need to find these functions ϕ_44^(1)(t,k) and ϕ_14^(2)(t,k).Applying the Cauchy's theorem to Eq. (<ref>), we have [iπ [α_11ϕ_14^(2) +α̅_12ϕ_24^(2)+α̅_13ϕ_34^(2)]; = ∫̣_∂ D_3[α_11(kϕ_14(t,k)-ϕ_14^(1))+α̅_12(kϕ_24(t,k)-ϕ_24^(1)) +α̅_13(kϕ_34(t,k)-ϕ_34^(1))]_-dk,; =∫̣_∂ D_3^0{α_11[kϕ_14(t,k)-ϕ_14^(1)-2e^2ikL/Σ_-(kϕ_14-ϕ_14^(1))] .; .+α̅_12[kϕ_24(t,k)-ϕ_24^(1)-2e^2ikL/Σ_-(kϕ_24-ϕ_24^(1))].; .+α̅_13[kϕ_34(t,k)-ϕ_34^(1)-2e^2ikL/Σ_-(kϕ_34-ϕ_34^(1))]}_-dk+C_2(t),; = ∫̣_∂ D_3^0-Σ_+/Σ_-[α_11(kϕ_14--2ϕ_14^(1))+α̅_12(kϕ_24--2ϕ_24^(1)) +α̅_13(kϕ_34--2ϕ_34^(1))]dk+C_2(t), ] where we have introduced the function C_2(t) asC_2(t)=∫̣_∂ D_3^0{2e^2ikL/Σ_-[α_11(kϕ_14-ϕ_14^(1))+α̅_12(kϕ_24-ϕ_24^(1)) +α̅_13(kϕ_34-ϕ_34^(1))]}_-dk, We use the global relation (<ref>) to further reduce C_2(t) in the form [ C_2(t)= ∫̣_∂ D_3^0{2/Σ_-[-kc̅_41(t,k̅)-(α_11ϕ_14^(1)+α̅_12ϕ_24^(1)+α̅_13ϕ_34^(1))e^2ikL..; -̣Ψ̅_44^(1)(α_11ϕ_14^(1)+α̅_12ϕ_24^(1)+α̅_13ϕ_34^(1))e^2ikL/k+(α_11^2+|α_12|^2+|α_13|^2)Ψ̅_41^(1);+(α_11α̅_12+α̅_12α_22+α̅_13α_23)Ψ̅_42^(1)..+(α_11α̅_13+α̅_12α̅_23+α̅_13α_33)Ψ_43^(1)]}_-dk; +∫̣_∂ D_3^0{2/Σ_-[Ψ̅_44^(1)(α_11ϕ_14^(1)+α̅_12ϕ_24^(1)+α̅_13ϕ_34^(1))e^2ikL/k..; +(α_11^2+|α_12|^2+|α_13|^2)(kΨ̅_41-Ψ̅_41^(1))+(α_11α̅_12+α̅_12α_22+α̅_13α_23)(kΨ̅_42-Ψ̅_42^(1)); ..+(α_11α̅_13+α̅_12α̅_23+α̅_13α_33)(kΨ̅_43-Ψ̅_43^(1))]}_-dk;+∫̣_∂ D_3^02k/Σ_-{(1-Ψ̅_44)(α_11ϕ_14+α̅_12ϕ_24+α̅_13ϕ_34)e^2ikL.; +Ψ̅_41[α_11(α_11(ϕ_11-1)+α_12ϕ_12+α_13ϕ_13)+α̅_12(α_11ϕ_21+α_12(ϕ_22-1)+α_13ϕ_23);+α̅_13(α_11ϕ_31+α_12ϕ_32+α_13(ϕ_33-1))] +Ψ̅_42[α_11(α̅_12(ϕ_11-1)+α_22ϕ_12+α_23ϕ_13); +α̅_12(α̅_12ϕ_21+α_22(ϕ_22-1)+α_23ϕ_23)+α̅_13(α̅_12ϕ_31+α_22ϕ_32+α_23(ϕ_33-1))];+Ψ̅_43[α_11(α̅_13(ϕ_11-1)+α̅_23ϕ_12+α̅_33ϕ_13)+α̅_12(α̅_13ϕ_21+α̅_23(ϕ_22-1)+α_33ϕ_23); . +α̅_13(α̅_13ϕ_31+α̅_23ϕ_32+α_33(ϕ_33-1))]}_-dk,; ] We need to further reduce C_2(t) by using the asymptotic (<ref>) and the Cauchy's theorem such that we have C_2(t) in the formC_2(t)=-̣iπ [α_11ϕ_14^(2)+α̅_12ϕ_24^(2)+α̅_13ϕ_34^(2)] -∫̣_∂ D_3^0i/2Ψ̅_44-^(1)(α_11v_01+α̅_12v_02+α̅_13v_03)dk +I_1(t), It follows from Eqs. (<ref>) and (<ref>)that we have [ 2iπ [α_11ϕ_14^(2) +α̅_12ϕ_24^(2)+α̅_13ϕ_34^(2)]; = -∫̣_∂ D_3^0Σ_+(k)/Σ_-(k)[α_11(kϕ_14-+iv_01)+α̅_12(kϕ_24-+iv_02+α̅_13(kϕ_34-+iv_03)]dk; -∫̣_∂ D_3^0i/2Ψ̅_44-^(1)(α_11v_01+α̅_12v_02+α̅_13v_03)dk +I_1(t), ] where I(t) is given by Eq. (<ref>). Similarly, we can also the expressions of iπ [α_12ϕ_14^(2)+α_22ϕ_24^(2)+α̅_23ϕ_34^(2)] and 2iπ [α_13ϕ_14^(2)+α_23ϕ_24^(2)+α_33ϕ_34^(2)] such that we can show that Eq. (<ref>) holds. (ii) We now deduce the Dirichlet boundary value problems (<ref>)-(<ref>) at x=0 from the Neumann boundary value problems. It follows from the first one of Eq. (<ref>) that u_01(t) can be expressed by means of Ψ_14^(1). Applying the Cauchy's theorem to Eq. (<ref>) yields [iπΨ_14^(1)(t)= (∫_∂ D_1+∫_∂ D_3)Ψ_14(t,k)dk =∫_∂ D_3Ψ_14-(t,k)dk; = ∫̣_∂ D_3^0[Ψ_14-(t,k)+2/Σ_-(k)(e^-2ikLΨ_14)_+]dk+C_3(t); =∫̣_∂ D_3^0Σ_+(k)/Σ_-(k)Ψ_14+dk+C_3(t), ] where C_3(t) is defined byC_3(t)=-∫̣_∂ D_3^02/Σ_-(k)(e^-2ikLΨ_14)_+dk,By applying the global relation (<ref>), the Cauchy's theorem and asymptotics (<ref>) to Eq. (<ref>), we find [C_3(t)=-∫̣_∂ D_3^02/Σ_-(e^-2ikLΨ_14)_+dk;= ∫̣_∂ D_3^02/Σ_-[-c_14e^-2ikL-(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43)]_+dk;+∫̣_∂ D_3^02/Σ_-[ Ψ_14(ϕ̅_44-1)e^-2ikL-[(Ψ_11-1)(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .;. + Ψ_12(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)+ Ψ_13(α_13ϕ̅_41+α_23ϕ̅_42+α_33ϕ̅_43)]]_+dk,;=-iπΨ_14^(1)-∫̣_∂ D_3^02/Σ_-(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43)_+dk; +∫̣_∂ D_3^02/Σ_-{Ψ_14(ϕ̅_44-1)e^-2ikL-[(Ψ_11-1)(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .;. + Ψ_12(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)+ Ψ_13(α_13ϕ̅_41+α_23ϕ̅_42+α_33ϕ̅_43)]}_+dk,;]Eqs. (<ref>) and (<ref>) imply that [2iπΨ_14^(1)(t)= ∫̣_∂ D_3^0[Σ_+(k)/Σ_-(k)Ψ_14+-2/Σ_-(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43)_+]dk; +∫̣_∂ D_3^02/Σ_-{Ψ_14(ϕ̅_44-1)e^-2ikL-[(Ψ_11-1)(α_11ϕ̅_41+α̅_12ϕ̅_42 +α̅_13ϕ̅_43) .;. + Ψ_12(α_12ϕ̅_41+α_22ϕ̅_42+α̅_23ϕ̅_43)+ Ψ_13(α_13ϕ̅_41+α_23ϕ̅_42+α_33ϕ̅_43)]}_+dk, ] Thus, substituting Eq. (<ref>) into the first one of Eq. (<ref>) yields Eq. (<ref>). Similarly, by applying the expressions of Ψ_24^(1)(t) and Ψ_34^(1)(t)to the second one of Eq. (<ref>), we can find Eqs. (<ref>) and (<ref>).Similarly we also can show that the Dirichlet boundary value problems (<ref>) at x=L hold from the Neumann boundary value problems.□ §.§4.3.The effective characterizations Substituting the perturbated expressions for eigenfunctions and initial boundary conditions {[ Ψ_ij(t,k)=Ψ_ij^[0](t,k)+ϵΨ_ij^[1](t,k)+ϵ^2Ψ_ij^[2](t,k)+⋯, i,j=1,2,3,4,; ϕ_ij(t,k)=ϕ_ij^[0](t,k)+ϵϕ_ij^[1](t,k)+ϵ^2ϕ_ij^[2](t,k)+⋯, i,j=1,2,3,4,;u_sj(t)=ϵ u_sj^[1](t)+ϵ^2 u_sj^[2](t)+⋯, s=0,1; j=1,2,3,;v_sj(t)=ϵ v_sj^[1](t)+ϵ^2 v_sj^[2](t)+⋯, s=0,1; j=1,2,3, ].into Eqs. (<ref>)-(<ref>), where ϵ>0 is a small parameter, we have these terms of O(1), and O(ϵ)asO(1): {[ Ψ_jj^[0]=1, j=1,2,3,4,; Ψ_ij^[0]=0, i,j=1,2,3,4,i≠j, ].O(ϵ): {[ Ψ_js^[1]=Ψ_44^[1]=0,j,s,=1,2,3,;Ψ_j4^[1]=∫̣_0^te^-4ik^2(t-t')(2ku_0j^[1]+iu_1j^[1])(t')dt', j=1,2,3,; Ψ_41^[1]=∫̣_0^te^4ik^2(t-t')[α_11(2ku̅_01^[1]-iu̅_11^[1])+α̅_12(2ku̅_02^[1]-iu̅_12^[1])+α̅_13(2ku̅_03^[1]-iu̅_13^[1])] (t')dt',;Ψ_42^[1]=∫̣_0^te^4ik^2(t-t')[α_12(2ku̅_01^[1]-iu̅_11^[1])+α_22(2ku̅_02^[1]-iu̅_12^[1])+α̅_23(2ku̅_03^[1]-iu̅_13^[1])] (t')dt',; Ψ_43^[1]=∫̣_0^te^4ik^2(t-t')[α_13(2ku̅_01^[1]-iu̅_11^[1])+α_23(2ku̅_02^[1]-iu̅_12^[1])+α_33(2ku̅_03^[1]-iu̅_13^[1])] (t')dt',; ]. Similarly, we can also obtain the analogous expressions for {ϕ_ij^[l]}_i,j=1^4, l=0,1 by means of the boundary values at x=L, that is, {v_ij^[l]}, i=0,1; j=1,2,3; l=0,1.If we assume that m_44(𝕊) has no zeros, then we expand Eqs. (<ref>)-(<ref>) to have[ u_11^[n](t)= ∫̣_∂ D_3^0{2Σ_+/iπΣ_-(kΨ_14-^[n]+iu_01^[n]) dk+4/πΣ_-[α_11(-ikϕ̅_41-^[n]+α_11v_01^[n]+α_12v_02^[n]+α_13v_03^[n])..; +̣α̅_12(-ikϕ̅_42-^[n]+α̅_12v_01^[n]+α_22v_02^[n] +α_23v_03^[n]);..+α̅_13(-ikϕ̅_43-^[n]+α̅_13v_01^[n] +α̅_23v_02^[n]+α_33v_03^[n])]}dk+lowerorderterms,;][u_12^[n](t)= ∫̣_∂ D_3^0{2Σ_+/iπΣ_-(kΨ_24-^[n]+iu_02^[n])dk+4/πΣ_-[α_12(-ikϕ̅_41-^[n]+α_11v_01^[n]+α_12v_02^[n]+α_13v_03^[n])..; +̣α_22(-ikϕ̅_42-^[n]+α̅_12v_01^[n]+α_22v_02^[n]+α_23v_03^[n]);..+α̅_23(-ikϕ̅_43-^[n]+α̅_13v_01^[n] +α̅_23v_02^[n]+α_33v_03^[n])]}dk+lowerorderterms,; ][u_13^[n](t)= ∫̣_∂ D_3^0{2Σ_+/iπΣ_-(kΨ_34-^[n]+iu_03^[n])dk+4/πΣ_-[α_13(-ikϕ̅_41-^[n]+α_11v_01^[n]+α_12v_02^[n]+α_13v_03^[n])..; +̣α_23(-ikϕ̅_42-^[n]+α̅_12v_01^[n]+α_22v_02^[n]+α_23v_03^[n]); ..+α_33(-ikϕ̅_43-^[1]+α̅_13v_01^[n] +α̅_23v_02^[n]+α_33v_03^[n])]}dk+lowerorderterms,; ] where `lower order terms' stands for the result involving known terms of lower order.The terms of O(ϵ^n) in Eqs. (<ref>)-(<ref>) and the similar equations for ϕ_ij yield {[Ψ_j4^[n](t,k)= ∫_0^t e^-4ik^2(t-t')(2ku_0j^[n]+iu_1j^[n])(t')dt'+lowerorderterms, j=1,2,3,;ϕ̅_41^[n](t,k̅)= ∫̣_0^te^-4ik^2(t-t')[α_11(2kv_01^[n]+iv_11^[n])+α_12(2kv_02^[n]+iv_12^[n])+α_13(2kv_03^[n]+iv_13^[n])](t')dt'; +lowerorderterms,;ϕ̅_42^[n](t,k̅)=∫̣_0^te^-4ik^2(t-t')[α̅_12(2kv_01^[n]+iv_11^[n])+α_22(2kv_02^[n]+iv_12^[n])+α_23(2kv_03^[n]+iv_13^[n])](t')dt'; +lowerorderterms,;ϕ̅_43^[n](t,k̅)= ∫̣_0^te^-4ik^2(t-t')[α̅_13(2kv_01^[n]+iv_11^[n])+α̅_23(2kv_02^[n]+iv_12^[n])+α_33(2kv_03^[n]+iv_13^[n])](t')dt'; +lowerorderterms, ].which leads to [ Ψ_j4-^[n](t,k)= 4k∫_0^t e^-4ik^2(t-t')u_0j^[n](t')dt'+lowerorderterms, j=1,2,3,; ϕ̅_41-^[n](t,k̅)=4̣k∫_0^te^-4ik^2(t-t')(α_11v_01^[n]+α_12v_02^[n]+α_13v_03^[n])(t')dt'+lowerorderterms,; ϕ̅_42-^[n](t,k̅)= 4̣k∫_0^te^-4ik^2(t-t')(α̅_12v_01^[n]+α_22v_02^[n]+α_23v_03^[n])(t')dt'+lowerorderterms,; ϕ̅_43-^[n](t,k̅)= 4̣k ∫_0^te^-4ik^2(t-t')(α̅_13v_01^[n]+α̅_23v_02^[n]+α_33v_03^[n])(t')dt'+lowerorderterms, ] It follows from system (<ref>) that Ψ_1j-^[n] and ϕ_4j-^[n],j=1,2,3 can be generated at each step from the known Dirichlet boundary data u_0j^[n] and v_0j^[n] such that we know that the Neumann boundary data u_1j^[n] can then be given by Eqs. (<ref>)-(<ref>). Similarly, we also show that the Neumann boundary data v_1j^[n] can then be determined by the known Dirichlet boundary data u_0j^[n] and v_0j^[n] .Similarly, the substitution of Eq. (<ref>) into Eqs. (<ref>) and (<ref>) yields the terms of O(ϵ^n) as u_01^[n](t)=∫̣_∂ D_3^0[Σ_+/πΣ_-Ψ_14+^[n]-2/πΣ_-(α_11ϕ̅_41^[n] +α̅_12ϕ̅_42^[n] +α̅_13ϕ̅_43^[n])]dk+lowerorderterms, u_02^[n](t)=∫̣_∂ D_3^0[Σ_+/πΣ_-Ψ_24+^[n] -2/πΣ_-(α_12ϕ̅_41^[n]+α_22ϕ̅_42^[n] +α̅_23ϕ̅_43^[n])]dk+lowerorderterms, u_03^[n](t)=∫̣_∂ D_3^0[Σ_+/πΣ_-Ψ_34+^[n]-2/πΣ_-(α_13ϕ̅_41^[n] +α_23ϕ̅_42^[n] +α_33ϕ̅_43^[n])]dk+lowerorderterms,Eq. (<ref>) implies that [Ψ_j4+^[n](t,k)= 2̣i∫_0^t e^-4ik^2(t-t')u_1j^[n](t')dt'+lowerorderterms, j=1,2,3,;ϕ̅_41+^[n](t,k̅)=∫̣_0^te^-4ik^2(t-t')(α_11v_11^[n]+α_12v_12^[n]+α_13v_13^[n])(t')dt' +lowerorderterms,;ϕ̅_42+^[n](t,k̅)= ∫̣_0^te^-4ik^2(t-t')(α̅_12v_11^[n]+α_22v_12^[n]+α_23v_13^[n])(t')dt' +lowerorderterms,;ϕ̅_43+^[n](t,k̅)= ∫̣_0^te^-4ik^2(t-t')(α̅_13v_11^[n])+α̅_23v_12^[n]+α_33v_13^[n])(t')dt' +lowerorderterms, ]It follows from system (<ref>) that Ψ_j4+^[n] and ϕ_4j+^[n],j=1,2,3 can be generated at each step from the known Neumann boundary data u_1j^[n] and v_1j^[n] such that we know that the Dirichlet boundary data u_0j^[n] can then be given by Eqs. (<ref>)-(<ref>). Similarly, we also show that the Dirichlet boundary data v_0j^[n] can then be determined by the known Neumann boundary data u_1j^[n] and v_1j^[n]. §.§4.4. The large L limit from the interval to the half-lineThe formulae for the initial and boundary value conditions u_0j(t) and u_1j(t),j=1,2,2 of Theorem 4.2 in the limit L→∞ can reduce to the corresponding ones on the half-line. Since when L→∞, [ v_0j→ 0, v_1j→ 0, j=1,2,3, ϕ_ij→δ_ij, %̣ṣ/̣%̣ṣΣ_+(k)Σ_-(k)→ 1as k→∞ in D_3, ]Thus, according to Eq. (<ref>), the L→∞ limits of Eqs. (<ref>), (<ref>), (<ref>), and (<ref>) yield the unknown Neumann boundary data u_1j(t)=%̣ṣ/̣%̣ṣ2π∫_∂ D_3^0[u_0j(Ψ_44-+1)-ikΨ_j4-]dk, j=1,2,3,for the given Dirichlet boundary problem, and the unknown Dirichlet boundary data u_0j(t)=%̣ṣ/̣%̣ṣ1π∫_∂ D_3^0Ψ_j4+dk, j=1,2,3,for the given Neumann boundary problem.§ THE GLM REPRESENTATION AND EQUIVALENCEIn this section we deduce the eigenfunctions Ψ(t,k) and ϕ(t,k) in terms of the Gel'fand-Levitan-Marchenko (GLM) approach <cit.>. Moreover, the global relation can be used to find the unknown Neumann (Dirichlet) boundary values from the given Dirichlet (Neumann) boundary values by means of the GLM representations. Moreover, the GLM representations are shown to be equivalent to the ones obtained in Sec. 4. Finally, the linearizable boundary conditions are presented fortheGLM representations.§.§5.1. The GLM representation Proposition 5.1. The eigenfunctions Ψ(t,k) and ϕ(t,k) possess the GLM representationΨ(t,k)=𝕀+∫̣_-t^t[L(t,s)+(k+i/2U^(0)σ_4)G(t,s)]e^-2ik^2(s-t)σ_4ds,ϕ(t,k)=𝕀+∫̣_-t^t[ℒ(t,s)+(k+i/2𝒰^(L)σ_4)𝒢(t,s)]e^-2ik^2(s-t)σ_4ds,where the 4× 4 matrix-valued functions L(t, s)=(L_ij)_4× 4 and G(t, s)=(G_ij)_4× 4,-t≤ s≤ t satisfy a Goursat system {[ Ḷ_t(t, s)+σ_4L_s(t,s)σ_4= iσ_4U_x^(0)L(t,s) -1/2[(U^(0))^3+iU̇^(0)σ_4+[U_x^(0), U^(0)]]G(t,s),;G_t(t, s)+σ_4G_s(t,s)σ_4=2U^(0)L(t,s)+iσ_4U_x^(0) G(t,s), ].with the initial conditions {[L_lj(t,-t)=L_44(t,-t)=0,l,j=1,2,3,;G_lj(t,-t)=G_44(t,-t)=0,l,j=1,2,3,; G_14(t,t)= u_01(t), G_24(t,t)=u_02(t), G_34(t,t)=u_03(t),; G_41(t,t)=α_11u̅_01(t)+α̅_12u̅_02(t)+α̅_13u̅_03(t),; G_42(t,t)= α_12u̅_01(t)+α_22u̅_02(t)+α̅_23u̅_03(t),; G_43(t,t)=α_13u̅_01(t)+α_23u̅_02(t)+α_33u̅_03(t),; L_14(t,t)= i/2u_11(t),L_24(t,t)=i/2u_12(t), L_34(t,t)=i/2u_13(t),; L_41(t,t)=-i/2(α_11u̅_11(t)+α̅_12u̅_12(t)+α̅_13u̅_13(t)),; L_42(t,t)= -i/2(α_12u̅_11(t)+α_22u̅_12(t)+α̅_23u̅_13(t)),; L_43(t,t)=-i/2(α_13u̅_11(t)+α_23u̅_12(t)+α_33u̅_13(t)), ].U^(0)=([ 0 0 0 u_01(t); 0 0 0 u_02(t); 0 0 0 u_03(t); p_01(t) p_02(t) p_03(t) 0 ]),U_x^(0)=([ 0 0 0 u_11(t); 0 0 0 u_12(t); 0 0 0 u_13(t); p_11(t) p_12(t) p_13(t) 0 ]),with [ p_01=α_11u̅_01+α̅_12u̅_02+α̅_13u̅_03, p_02=α_12u̅_01+α_22u̅_02+α̅_23u̅_03, p_03=α_13u̅_01+α_23u̅_02+α_33u̅_03,; p_11=α_11u̅_11+α̅_12u̅_12+α̅_13u̅_13, p_12=α_12u̅_11+α_22u̅_12+α̅_23u̅_13, p_13=α_13u̅_11+α_23u̅_12+α_33u̅_13, ]Similarly, ℒ(t,s), 𝒢(t,s) satisfy the similar Eqs. (<ref>) and (<ref>) with u_0j→ v_0j,u_1j→ v_1j, U^(0)→𝒰^(L)=U^(0)|_u_0j→ v_0j,U_x^(0)→𝒰_x^(L)=U_x^(0)|_u_1j→ v_1j.Proof. We assume thatthe function ψ(t,k)=e^-2ik^2tσ_4+∫̣_-t^t[L_0(t,s)+kG(t,s)]e^-2ik^2sσ_4ds,satisfies the time-part of Lax pair (<ref>) with the boundary data ψ(0,k)=𝕀 at x=0, where L_0(t,s) and G(t,s) are the unknown 4× 4 matrix-valued functions. We substitute Eq. (<ref>) into the time-part of Lax pair (<ref>) with the boundary data (<ref>) and use the identity ∫̣_-t^tF(t,s)e^-2ik^2sσ_4ds=i/2k^2[F(t,t)e^-2ik^2tσ_4-F(t,-t)e^2ik^2tσ_4- ∫̣_-t^tF_s(t,s)e^-2ik^2sσ_4ds]σ_4,where the function F(t,s) is a 4× 4 matrix-valued function. As a consequence, we find {[L_0(t, -t)+σ_4L_0(t,-t)σ_4=-iU^(0)G(t,-t)σ_4,;G(t, -t)+σ_4G(t,-t)σ_4=0,;L_0(t, t)-σ_4L_0(t,t)σ_4=iU^(0)G(t,t)σ_4+V_0^(0),; G(t, t)-σ_4G(t,t)σ_4=2U^(0),; L_0t(t, s)+σ_4L_0s(t,s)σ_4=-iU^(0)G_s(t,s)σ_4+V_0^(0)L_0(t,s),; G_t(t, s)+σ_4G_s(t,s)σ_4=2U^(0)L_0(t,s)+V_0^(0)G(t,s), ].where U^(0) is given by Eq. (<ref>) andV_0^(0)=-i(U_x^(0)+U^(0)2)σ_4=-i([u_01p_01u_01p_02u_01p_03 -u_11;u_02p_01u_02p_02u_02p_03 -u_21;u_03p_01u_03p_02u_03p_03 -u_31;p_11p_21p_31 -(u_01p_01+u_02p_02+u_03p_03) ]),To reduce system (<ref>) we further introduce the new matrix L(t,s) byL(t,s)=L_0(t,s)-i/2U^(0)σ_4G(t,s),such that the first four equations of system (<ref>) become {[ L(t, -t)+σ_4L(t,-t)σ_4=0,; G(t, -t)+σ_4G(t,-t)σ_4=0,; L(t, t)-σ_4L(t,t)σ_4=V_0^(0),;G(t, t)-σ_4G(t,t)σ_4=2U^(0), ].which leads to Eq. (<ref>), and from the last two equations of system (<ref>) we have Eq. (<ref>). By means oftransformation (<ref>), that is, μ_2(0, t, k)=Ψ(t,k)=ψ(t,k)e^2ik^2tσ_4, we know that Ψ(t,k) is given by Eq. (<ref>). Similarly, we can also show that Eq. (<ref>) holds. □ For convenience, we rewrite a 4× 4 matrix C=(C_ij)_4× 4 asC=(C_ij)_4× 4=([ C̃_3× 3 C̃_j4 C̃_4jC_44 ]), C̃_3× 3=(C_ij)_3× 3, C̃_j4=(C_14, C_24, C_34)^T,C̃_4j=(C_41, C_42, C_43), The Dirichlet and Neumann boundary values at x=0, L are simply written as [u_j(t)=(u_j1(t), u_j2(t), u_j3(t)),v_j(t)=(v_j1(t), v_j2(t), v_j3(t)), j=1,2,3,; w_j0(t)=(p_j1(t), p_j2(t), p_j3(t)), w_jL(t)=(p_j1(t), p_j2(t), p_j3(t))|_u_sj→ v_sj,s=0,1; j=1,2,3, ]For a matrix-valued function F(t,s), we introduce the F̂(t,k) by F̂(t,k)=∫̣_-t^tF(t,s)e^2ik^2(s-t)ds,Thus, the GLM expressions (<ref>) and (<ref>) of {Ψ_ij, ϕ_ij} can be rewritten as{[ ̣̃Ψ_3× 3(t,k)=𝕀+L̂̃̂_3× 3-i/2u_0^T(t)Ĝ̃̂_4j+kĜ̃̂_3× 3,; ̣̃Ψ_j4(t,k)=L̂̃̂_j4-i/2u_0^T(t)Ĝ̃̂_44+kĜ̃̂_j4,j=1,2,3,; ̣̃Ψ_4j(t,k)=L̂̃̂_4j+i/2u̅_0(t)ℳĜ̃̂_3× 3+kĜ̃̂_4j,j=1,2,3,; ̣̃Ψ_44(t,k)=1+L̂̃̂_44+i/2u̅_0(t)ℳĜ̃̂_j4+kĜ̃̂_44, ].{[̣̃ϕ_3× 3(t,k) =𝕀+ℒ̂̃̂_3× 3-i/2v_0^T(t)𝒢̂̃̂_4j+k𝒢̂̃̂_3× 3,;̣̃ϕ_j4(t,k)=ℒ̂̃̂_j4-i/2v_0^T(t)𝒢̂̃̂_44+k𝒢̂̃̂_j4, j=1,2,3,; ̣̃ϕ_4j(t,k)=ℒ̂̃̂_4j+i/2v̅_0(t)ℳ𝒢̂̃̂_3× 3+k𝒢̂̃̂_4j,j=1,2,3,;̣̃ϕ_44(t,k)=1+ℒ̂̃̂_44+i/2v̅_0(t)ℳ𝒢̂̃̂_j4 +k𝒢̂̃̂_44, ]. For the given Eqs. (<ref>) and (<ref>) we have the following proposition.Proposition 5.2. ḷịṃ_t'→ t∫̣_∂ D_1^0ke^4ik^2(t-t')/Σ_-(F̃_j4e^-2ikL)_-dk=∫̣_∂ D_1^0[i k/2u_0^T(Ĝ̃̂_44-𝒢̅̂̅̃̅̂̅_44)+k/Σ_-(F̃_j4e^-2ikL)_-]dk, ḷịṃ_t'→ t∫̣_∂ D_1^0ke^4ik^2(t-t')/Σ_-F̃_4j-dk= ∫̣_∂ D_1^0[ik/2ℳ^T v_0^T(𝒢̂̃̂_44-Ĝ̅̃̅̂̅_44)+k/Σ_-F̃_4j-]dk,ḷịṃ_t'→ t∫̣_∂ D_1^0e^4ik^2(t-t')/Σ_-(F̃_j4e^-2ikL)_+dk= ∫̣_∂ D_1^01/Σ_-(F̃_j4e^-2ikL)_+dk,ḷịṃ_t'→ t∫̣_∂ D_1^0e^4ik^2(t-t')/Σ_-F̃_4j+dk= ∫̣_∂ D_1^01/Σ_-F̃_4j+dk,where the vector-valued functions F̃_j4(t,k) and F̃_4j(t,k)(j=1,2,3) are defined by[ F̃_j4(t,k)=-̣i/2u_0^TĜ̃̂_44+i/2ℳ^T𝒢̅̂̅̃̅̂̅_3× 3^Tℳv_0^Te^2ikL; +̣(L̂̃̂_j4-i/2u_0^TĜ̃̂_44+kĜ̃̂_j4)(ℒ̅̂̅̃̅̂̅_44-i/2𝒢̅̂̅̃̅̂̅_j4^Tℳv_0^T+k𝒢̅̂̅̃̅̂̅_44); -̣(L̂̃̂_3× 3-i/2u_0^TĜ̃̂_4j+kĜ̃̂_3× 3)ℳ^T (ℒ̅̂̅̃̅̂̅_4j^T-i/2𝒢̅̂̅̃̅̂̅_3× 3^Tℳv_0^T +k𝒢̅̂̅̃̅̂̅_4j^T)e^2ikL, ][F̃_4j(t,k)=-̣i/2Ĝ̅̃̅̂̅^T_3× 3ℳu_0^T+i/2ℳ^Tv_0^T𝒢̂̃̂_44e^2ikL;+̣ℳ^T(ℒ̂̃̂_3× 3-i/2v_0^T𝒢̂̃̂_4j+k𝒢̂̃̂_3× 3)ℳ^T (L̅̂̅̃̅̂̅^T_4j-i/2Ĝ̅̃̅̂̅^T_3× 3ℳu_0^T+kĜ̅̃̅̂̅^T_4j); -̣ℳ^T (ℒ̂̃̂_j4-i/2v_0^T𝒢̂̃̂_44+k𝒢̂̃̂_j4)(L̅̂̅̃̅̂̅_44-i/2Ĝ̅̃̅̂̅^T_j4ℳu_0^T+kĜ̅̃̅̂̅_44) e^2ikL, ]Proof. Similar to the proof of Lemma 4.3 in Ref. <cit.>, we here show Eq. (<ref>) in detail. We multiply Eq. (<ref>) by k/Σ_-e^4ik^2(t-t') with 0<t'<t and integrate along along ∂ D_1^0 with respect to dk to yield [ ∫̣_∂ D_1^0k/Σ_-e^4ik^2(t-t')(F̃_j4e^-2ikL)_-dk= ∫̣_∂ D_1^0i k/2e^4ik^2(t-t')u_0^TĜ̃̂_44dk -∫̣_∂ D_1^0 k^3e^4ik^2(t-t')Ĝ̃̂_j4𝒢̅̂̅̃̅̂̅_44dk;-∫̣_∂ D_1^0 k e^4ik^2(t-t')(L̂̃̂_j4-i/2u_0^TĜ̃̂_44) (ℒ̅̂̅̃̅̂̅_44-i/2𝒢̅̂̅̃̅̂̅_j4^Tℳv_0^T)dk; +̣∫̣_∂ D_1^0k^2Σ_+/Σ_-e^4ik^2(t-t')[(L̂̃̂_j4 -i/2u_0^TĜ̃̂_44)𝒢̅̂̅̃̅̂̅_44 +Ĝ̃̂_j4(ℒ̅̂̅̃̅̂̅_44-i/2𝒢̅̂̅̃̅̂̅_j4ℳv_0^T)]dk; -∫̣_∂ D_1^02 k^2/Σ_-e^4ik^2(t-t')[(L̂̃̂_3× 3-i/2u_0^TĜ̃̂_4j)ℳ^T𝒢̅̂̅̃̅̂̅_4j^T+ Ĝ̃̂_3× 3ℳ^T(ℒ̅̂̅̃̅̂̅_4j^T-i/2𝒢̅̂̅̃̅̂̅_3× 3^Tℳv_0^T)]dk, ]To further analyse the above equation, the following identities are introduced ∫̣_∂ D_1ke^4ik^2(t-t')F̂(t,k)dk={[ %̣ṣ/̣%̣ṣπ2 F(t, 2t'-t), 0<t'<t,; %̣ṣ/̣%̣ṣπ4F(t, t),0<t'=t, ].and ∫̣_∂ D_1^0k^2/Σ_-e^4ik^2(t-t')F̂(t,k)dk= 2∫̣_∂ D_1^0k^2/Σ_-[∫_0^t'e^4ik^2(t-t')F̂(t,2τ-t)dτ-F(t, 2t'-t)/4ik^2]dk,which also holds for the cases that k^2/Σ_- is taken place by k^2Σ_+/Σ_- or k^2.It follows from the first integral on the RHS of Eq. (<ref>) and Eq. (<ref>) that we haveḷịṃ_t'→ t∫̣_∂ D_1^0i k/2e^4ik^2(t-t')u_0^TĜ̃̂_44dk =lim_t'→ tiπ/2 u_0^TG̃_22(t, 2t'-t) =iπ/4u_0^TG̃_44(t, t),ḷịṃ_t'→ t∫̣_∂ D_1^0i k/2e^4ik^2(t-t')u_0^TĜ̃̂_44dk =∫̣_∂ D_1^0i k/2u_0^TĜ̃̂_44dk =iπ/8u_0^TG̃_44(t,t), Therefore, we know that the first integral on the RHS of Eq. (<ref>) yields the following two terms ḷịṃ_t'→ t∫_∂ D_1^0i k/2e^4ik^2(t-t')u_0^TĜ̃̂_44dk =∫_∂ D_1^0i k/2u_0^TĜ̃̂_44dk|_(<ref>) +∫_∂ D_1^0i k/2u_0^TĜ̃̂_44dk|_(<ref>), Nowadays we study the second integral on the RHS of Eq. (<ref>). It follows from the second integral on the RHS of Eq. (<ref>) and Eq. (<ref>) that we have [ -∫̣_∂ D_1^0 k^3e^4ik^2(t-t')Ĝ̃̂_j4𝒢̅̂̅̃̅̂̅_44dk =-2∫̣_∂ D_1^0 k^3∫_0^te^4ik^2(τ-t')G̃_j4(t, 2τ-t)𝒢̅̂̅̃̅̂̅_44dτ dk;=-2∫̣_∂ D_1^0 k^3[∫_0^t'e^4ik^2(τ-t')G̃_j4(t, 2τ-t)dτ-G̃_j4(t, 2t'-t)/4ik^2]𝒢̅̂̅̃̅̂̅_44dk, ] Thus we take the limit t'→ t of Eq. (<ref>) to have-lim_t'→ t∫̣_∂ D_1^0 k^3e^4ik^2(t-t')Ĝ̃̂_j4𝒢̅̂̅̃̅̂̅_44dk =-∫̣_∂ D_1^0 k^3Ĝ̃̂_j4𝒢̅̂̅̃̅̂̅_44dk +∫̣_∂ D_1^0k/2iu_0^T𝒢̅̂̅̃̅̂̅_44dk Finally, following the proof in Ref. <cit.> we can show the limits t'→ t of the rest three integrals (i.e., the third, fourth and fifth integrals) of Eq. (<ref>) can be deduced by simply making the limit t'→ t inside the every integral, that is, no additional terms arise in these integrals. For example, lim_t'→ t∫̣_∂ D_1^0 k e^4ik^2(t-t')(L̂̃̂_j4-i/2u_0^TĜ̃̂_44) (ℒ̅̂̅̃̅̂̅_44-i/2𝒢̅̂̅̃̅̂̅_j4^Tℳv_0^T)dk =∫̣_∂ D_1^0 k (L̂̃̂_j4-i/2u_0^TĜ̃̂_44) (ℒ̅̂̅̃̅̂̅_44-i/2𝒢̅̂̅̃̅̂̅_j4^Tℳv_0^T)dk.Thus we complete the proof of Eq. (<ref>). Similarly, we can show that Eqs. (<ref>), (<ref>) and (<ref>) also hold. □Theorem5.3. Let q_0j(x)=q_j(x,t=0)=0,j=1,2,3 be the initial data of Eq. (<ref>) on the interval x∈ [0, L] andT<∞. For the Dirichlet problem, the boundary data u_0j(t) and v_0j(t)(j=1,2,3) on the interval t∈ [0, T) are sufficiently smooth and compatible with the initial data q_j0(x)(j=1,2,3) at the points (x_2, t_2)=(0, 0) and(x_3, t_3)=(L, 0), respectively. For the Neumann problem, the boundary data u_1j(t) and v_1j(t)(j=1,2,3) on the interval t∈ [0, T) are sufficiently smooth and compatible with the initial data q_0j(x)(j=1,2,3) at the points (x_2, t_2)=(0, 0) and(x_3, t_3)=(L, 0), respectively. For simplicity, let n_33,44(𝕊)(k) have no zeros in the domain D_1. Then the spectral functions S(k) and S_L(k) are defined by Eqs. (<ref>) and (<ref>) with Ψ(t,k) and ϕ(t,k) given by Eq. (<ref>) and (<ref>).(i) For the given Dirichlet boundary values u_0(t) and v_0(t), the unknown Neumann boundary values u_1(t) and v_1(t) are given by[u_1^T(t)= %̣ṣ/̣%̣ṣ4iπ∫_∂ D_1^0{Σ_+/Σ_-[k^2Ĝ̃̂_j4(t,t)+ i/2u_0^T(t)] -%̣ṣ/̣%̣ṣ2ℳ^TΣ_-[k^2𝒢̅̂̅̃̅̂̅_4j^T(t, t)+i/2ℳv_0^T(t)].; .+%̣ṣ/̣%̣ṣi k2u_0^T(Ĝ̃̂_44-𝒢̅̂̅̃̅̂̅_44)+k/Σ_-[F̃_j4e^-2ikL]_-}dk, ] [ v_1^T(t)= %̣ṣ/̣%̣ṣ4iπ∫_∂ D_1^0{-Σ_+/Σ_-[k^2𝒢̂̃̂_j4(t,t)+i/2v_0^T(t)] +%̣ṣ/̣%̣ṣ2ℳ^TΣ_-[k^2Ĝ̅̃̅̂̅_4j^T(t, t)+i/2ℳu_0^T(t)].;.+%̣ṣ/̣%̣ṣi k2v_0^T(𝒢̂̃̂_44-Ĝ̅̃̅̂̅_44) +k/Σ_-ℳ^TF̃_4j-}dk, ](ii) For the given Neumann boundary values u_1(t) and v_1(t), the unknown Dirichlet boundary values u_0(t) and v_0(t) are given by[u_0^T(t)= %̣ṣ/̣%̣ṣ2π∫̣_∂ D_1^0[Σ_+/Σ_-L̂̃̂_j4-2ℳ^T/Σ_-ℒ̅̂̅̃̅̂̅_4j^T +1/Σ_-(F̃_j4e^-2ikL)_+]dk, ] [ v_0^T(t) = %̣ṣ/̣%̣ṣ2π∫_∂ D_1^0[ 2ℳ^T/Σ_-L̅̂̅̃̅̂̅^T_j4-1/Σ_-ℒ̂̃̂_4j +ℳ^T/Σ_-F̃_4j+]dk, ] where F̃_j4(t,k) and F̃_4j(t,k) are defined by Eqs. (<ref>) and (<ref>).Proof. By means of the global relation (<ref>) and Proposition 5.1, we can show that the spectral functions S(k) and S_L(k) are defined by Eqs. (<ref>) and (<ref>) with Ψ(t,k) and ϕ(t,k) given by Eq. (<ref>) and (<ref>). (i) we firstly consider the Dirichlet problem. It follows from the global relation (<ref>) with the vanishing initial data c(t,k)=μ_2(0, t,k)e^ikLσ̂_4μ_3^-1(L, t,k),that we findc̃_j4(t,k)=-Ψ̃_3× 3ℳ^Tϕ̅̃̅_4j^T(t,k̅)e^2ikL+Ψ̃_j4ϕ̅̃̅_44(t,k̅), c̃_4j(t,k)=Ψ̃_4jℳ^Tϕ̅̃̅_3× 3^T(t,k̅)ℳ^T-Ψ̃_44ϕ̅̃̅_j4^T(t,k̅)ℳ^Te^-2ikL,Substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>) yields ℳ^Tℒ̅̂̅̃̅̂̅_4j^Te^2ikL-L̂̃̂_j4=kĜ̃̂_j4 -kℳ^T𝒢̅̂̅̃̅̂̅_4j^Te^2ikL+F̃_j4(t,k)-c̃_j4(t,k),where F̃_j4(t,k) is given by Eq. (<ref>). Eq. (<ref>) with k→ -k further yields ℳ^Tℒ̅̂̅̃̅̂̅_4j^Te^-2ikL-L̂̃̂_j4=-kĜ̃̂_j4 +kℳ^T𝒢̅̂̅̃̅̂̅_4j^Te^-2ikL+F̃_j4(t,-k)-c̃_j4(t,-k), It follows from Eqs (<ref>) and (<ref>) that weget ̣̂L̃_j4=kΣ_+/Σ_-Ĝ̃̂_j4 -2 k/Σ_-ℳ^T𝒢̅̂̅̃̅̂̅_4j^T+1/Σ_-{[F̃_j4(t, k)-c̃_j4(t, k)]e^-2ikL}_-. We multiply Eq. (<ref>)by k e^4ik^2(t-t') with 0<t'<t and integrate them along ∂ D_1^0 with respect to dk, respectively to yield [ ∫̣_∂ D_1^0k e^4ik^2(t-t')L̂̃̂_j4dk= ∫̣_∂ D_1^0 e^4ik^2(t-t')k^2Σ_+/Σ_-Ĝ̃̂_j4dk -∫̣_∂ D_1^0e^4ik^2(t-t')2k^2/Σ_-ℳ^T𝒢̅̂̅̃̅̂̅_4j^Tdk; +∫̣_∂ D_1^0ke^4ik^2(t-t')/Σ_-[F̃_j4(t,k)e^-2ikL]_-dk, ] where we have used ∫̣_∂ D_1^0 k e^4ik^2(t-t')c̃_j4-(t, k)dk =∫̣_∂ D_1^0k e^4ik^2(t-t')(c̃_j4(t, k)e^-2ikL)_-dk=0in terms of their analytical properties in D_1^0.Based on these conditions given by Eqs. (<ref>) and (<ref>), Eq. (<ref>) can become [ %̣ṣ/̣%̣ṣπ2L̃_j4(t, 2t'-t)=2∫̣_∂ D_1^0k^2Σ_+/Σ_-[∫_0^t'e^4ik^2(t-t')G̃_j4(t,2τ-t)dτ-G̃_j4(t, 2t'-t)/4ik^2]dk;-4∫̣_∂ D_1^0k^2ℳ^T/Σ_-[∫_0^t'e^4ik^2(t-t')𝒢̅̃̅_4j^T(t,2τ-t)dτ-𝒢̅̃̅_4j^T(t, 2t'-t)/4ik^2]dk;+∫̣_∂ D_1^0k/Σ_-e^4ik^2(t-t')[F̃_j4(t, k)e^-2ikL]_-dk, ]We choose the limit t'→ t of Eq. (<ref>) with the initial data (<ref>) and Proposition 5.2 to find [ %̣ṣ/̣%̣ṣπ2L̃_j4(t, t)=2ḷịṃ_t'→ t∫_∂ D_1^0k^2Σ_+/Σ_-[∫_0^t'e^4ik^2(t-t')G̃_j4(t,2τ-t)dτ-G̃_j4(t, 2t'-t)/4ik^2]dk;-4ḷịṃ_t'→ t∫_∂ D_1^0k^2ℳ^T/Σ_-[∫_0^t'e^4ik^2(t-t')𝒢̅̃̅_4j^T(t,2τ-t)dτ-𝒢̅̃̅_4j^T(t, 2t'-t)/4ik^2]dk; +ḷịṃ_t'→ t∫_∂ D_1^0k/Σ_-e^4ik^2(t-t')[F̃_j4(t, k)e^-2ikL]_-dk;= ∫̣_∂ D_1^0{Σ_+/Σ_-[k^2Ĝ̃̂_j4(t,t)+i/2G̃_j4(t, t)] -%̣ṣ/̣%̣ṣ2ℳ^TΣ_-[k^2𝒢̅̂̅̃̅̂̅_4j^T(t, t)+i/2𝒢̅̃̅_4j^T(t, t)].;. +%̣ṣ/̣%̣ṣi k2u_0^T(Ĝ̃̂_44-𝒢̅̂̅̃̅̂̅_44)+k/Σ_-(F̃_j4e^-2ikL)_-}dk, ] Since the initial data (<ref>) are of the form L̃_j4(t, t)=i/2u_1^T(t)=i/2(u_11(t), u_12(t), u_13(t))^T,then we know that Eq. (<ref>) holds ny means of Eqs. (<ref>) and (<ref>).To show Eq. (<ref>) we rewrite Eq. (<ref>) in the form c̅̃̅^T_4j(t,k̅)=ℳ^Tϕ̃_3× 3ℳ^TΨ̅̃̅^T_4j(t,k̅)-ℳ^Tϕ̃_j4Ψ̅̃̅_44^T(t,k̅)e^2ikL, We substitute Eqs. (<ref>) and (<ref>) into Eq. (<ref>) to have-L̅̂̅̃̅̂̅^T_4j+ℳ^Tℒ̂̃̂_j4e^2ikL=kĜ̅̃̅̂̅^T_4j -kℳ^T𝒢̂̃̂_j4e^2ikL+F̃_4j(t,k)-c̅̃̅^T_4j(t,k̅),where F̃_4j(t,k) is givenby Eq. (<ref>). Eq. (<ref>) with k→ -k yields-L̅̂̅̃̅̂̅^T_4j+ℳ^Tℒ̂̃̂_j4e^-2ikL=-kĜ̅̃̅̂̅^T_4j +kℳ^T𝒢̂̃̂_j4e^-2ikL+F̃_4j(t,-k)-c̅̃̅^T_4j(t,-k̅), It follows from Eqs. (<ref>) and (<ref>) that we have M^Tℒ̂̃̂_j4=2k/Σ_-Ĝ̅̃̅̂̅^T_4j -kΣ_+/Σ_-ℳ^T𝒢̂̃̂_j4+1/Σ_-[F̃_4j(t, k)-c̅̃̅_4j^T(t, k̅)]_- We multiply Eq. (<ref>)by k e^4ik^2(t-t') with 0<t'<t, integrate them along ∂ D_1^0 with respect to dk, and use these conditions given by Eqs. (<ref>) and (<ref>) to yield [%̣ṣ/̣%̣ṣπ2ℳ^Tℒ̃_j4(t, 2t'-t)= -2 ∫̣_∂ D_1^0k^2Σ_+/Σ_-ℳ^T[∫_0^t'e^4ik^2(t-t')𝒢̃_j4(t,2τ-t)dτ-𝒢̃_j4(t, 2t'-t)/4ik^2]dk;+4∫̣_∂ D_1^0 k^2/Σ_-[∫_0^t'e^4ik^2(t-t')G̅̃̅_4j^T(t,2τ-t)dτ-G̅̃̅_4j^T(t, 2t'-t)/4ik^2]dk;+∫̣_∂ D_1^0k/Σ_-e^4ik^2(t-t')F̃_4j-(t, k)dk, ] where we have used the relation ∫̣_∂ D_1^0k/Σ_-e^4ik^2(t-t')c̅̃̅^T_4j-(t, k̅)dk=0due to the analytical property of the integrand in D_1^0. We consider the limit t'→ t of Eq. (<ref>) with the initial data (<ref>) and Proposition 5.2 to have [%̣ṣ/̣%̣ṣπ2ℳ^Tℒ̃_j4(t, t)= ∫̣_∂ D_1^0{-Σ_+/Σ_-ℳ^T[k^2𝒢̂̃̂_j4(t,t)+i/2𝒢̃_j4(t, t)] +%̣ṣ/̣%̣ṣ2Σ_-[k^2Ĝ̅̃̅̂̅_4j^T(t, t)+i/2G̅̃̅_4j^T(t, t)].; .+%̣ṣ/̣%̣ṣik2ℳ^Tv_0^T(𝒢̂̃̂_44-Ĝ̅̃̅̂̅_44)+k/Σ_-F̃_4j-(t,k)}dk, ] Since the initial conditions are of the form ℒ̃_j4(t, t)=i/2v_1^T(t)=i/2(v_11(t), v_12(t), v_13(t))^T,then we have Eq. (<ref>) by combining Eqs. (<ref>) and (<ref>).(ii) We now turn to consider the Neumann problem. It follows from Eqs (<ref>), (<ref>), (<ref>) and (<ref>) that we havệG̃_j4=1/kΣ_-{Σ_+L̂̃̂_j4-2ℳ^T ℒ̅̂̅̃̅̂̅_4j^T+[(F̃_j4(t, k)-c̃_j4(t, k))e^-2ikL]_+}, ̣̂𝒢̃_j4=1/kΣ_-{2ℳ^TL̅̂̅̃̅̂̅^T_j4-Σ_+ℒ̂̃̂_j4+ℳ^T[F̃_4j(t, k)-c̅̃̅^T_44(t, k̅)]_+}.We multiply Eqs. (<ref>) and (<ref>) by k e^4ik^2(t-t') with 0<t'<t, integrate them along ∂ D_1^0 with respect to dk, and use these conditions given by Eqs. (<ref>) and (<ref>) to yield[ %̣ṣ/̣%̣ṣπ2G̃_j4(t, 2t'-t)= ∫̣_∂ D_1^02Σ_+/Σ_-[∫_0^t'e^4ik^2(t-t')L̃_j4(t,2τ-t)dτ-L̃_j4(t, 2t'-t)/4ik^2]dk;-∫̣_∂ D_1^04ℳ^T/Σ_-[∫_0^t'e^4ik^2(t-t')ℒ̅̃̅_4j^T(t,2τ-t)dτ -ℒ̅̃̅_4j^T(t, 2t'-t)/4ik^2]dk; +∫̣_∂ D_1^0e^4ik^2(t-t')/Σ_-(F̃_j4e^-2ikL)_+dk, ][ %̣ṣ/̣%̣ṣπ2𝒢̃_j4(t, 2t'-t)= ∫̣_∂ D_1^04ℳ^T/Σ_-[∫_0^t'e^4ik^2(t-t')L̅̃̅^T_j4(t,2τ-t)dτ-L̅̃̅^T_j4(t, 2t'-t)/4ik^2]dk; -∫̣_∂ D_1^02 /Σ_-[∫_0^t'e^4ik^2(t-t')ℒ̃_4j(t,2τ-t)dτ-ℒ̃_4j(t, 2t'-t)/4ik^2]dk;+∫̣_∂ D_1^0ℳ^T/Σ_-e^4ik^2(t-t')F̃_4j+dk, ] where we have used the analytical property of the matrix-valued functions ∫̣_∂ D_1^01/Σ_-e^4ik^2(t-t')(c̃_j4(t, k)e^-2ikL)_+dk=∫̣_∂ D_1^01/Σ_-e^4ik^2(t-t')c̅̃̅^T_4j+(t, k̅)dk=0.We consider the limits t'→ t of Eqs. (<ref>) and (<ref>) with the initial data (<ref>) and Proposition 5.2 to find[ %̣ṣ/̣%̣ṣπ2G̃_j4(t, t)= ∫̣_∂ D_1^0[Σ_+/Σ_-L̂̃̂_j4-2ℳ^T/Σ_-ℒ̅̂̅̃̅̂̅_4j^T +1/Σ_-(F̃_j4e^-2ikL)_+]dk, ][ %̣ṣ/̣%̣ṣπ2𝒢̃_j4(t, t)= ∫̣_∂ D_1^0(2ℳ^T/Σ_-L̅̂̅̃̅̂̅^T_j4 -1/Σ_-ℒ̂̃̂_4j+ℳ^T/Σ_-F̃_4j+)dk, ] Since the initial conditions are of the form [ G̃_j4(t, t)=u_0^T(t)=(u_01(t), u_02(t), u_03(t))^T,𝒢̃_j4(t, t)=v_0^T(t)=(v_01(t), v_02(t), v_03(t))^T, ] then we have Eqs. (<ref>) and (<ref>) by using Eqs. (<ref>) and (<ref>). This completes the proof of the Theorem. □§.§5.2. Equivalence of the two distinct representations We now show that the above-mentionedGLMrepresentation for the Dirichlet and Neumann boundary data in Theorem 5.3 is equivalent to one in Theorem 4.2.Case i. From the Dirichlet boundary conditions to the Neumann boundary ones It followsfrom Eqs. (<ref>) and (<ref>) that we obtain Ĝ̃̂_j4=1/2kΨ̃_j4-,𝒢̂̃̂_j4=1/2kϕ̃_j4-,Ĝ̃̂_44=1/2kΨ̃_44-,𝒢̂̃̂_44=1/2kϕ̃_44-, Substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>) yields[ u_1^T(t)=%̣ṣ/̣%̣ṣ4iπ∫_∂ D_1^0{Σ_+/Σ_-[k^2Ĝ̃̂_j4(t,t)+ i/2u_0^T(t)] -%̣ṣ/̣%̣ṣ2ℳ^TΣ_-[k^2𝒢̅̂̅̃̅̂̅_4j^T(t, t)+i/2ℳv_0^T(t)].; .+ịk u_0^TĜ̃̂_44+ k/2iu_0^T𝒢̅̂̅̃̅̂̅_44+k/Σ_-[Ψ̃_j4(ϕ̅̃̅_44-1)e^-2ikL -(Ψ̃_3× 3-𝕀)ℳ^Tϕ̅̃̅_4j^T]_-}dk; = ∫̣_∂ D_1^0{2Σ_+/iπΣ_-[kΨ̃_j4-+iu_0^T(t)] +%̣ṣ/̣%̣ṣ4iℳ^TπΣ_-[ kϕ̅̃̅^T_4j-+iℳ v_0^T(t)]+1/πu_0^T(2Ψ̃_44--ϕ̅̃̅_44-).;.+̣4k/iπΣ_-[Ψ̃_j4(ϕ̅̃̅_44-1)e^-2ikL -(Ψ̃_3× 3-𝕀)ℳ^Tϕ̅̃̅_4j^T]_- }dk, ] Since the integrand in Eq. (<ref>) is an odd function about k, which makes sure that the contour ∂ D_1^0 can be replaced by ∂ D_3^0, thus we can find the same Neumann boundary data u_1j(t)(j=1,2,3) at x=0 given by Eqs. (<ref>)-(<ref>) from Eq. (<ref>). Similarly, we can also find the Neumann boundary data v_1j(t)(j=1,2,3) at x=L given by Eq. (<ref>) from Eq. (<ref>).Case ii. From theNeumannboundary conditions to the Dirichlet boundary ones Eqs. (<ref>) and (<ref>) imply that L̂̃̂_j4=1/2Ψ̃_j4+(t,k)+i/2u_0^TĜ̃̂_44,ℒ̅̂̅̃̅̂̅^T_4j=1/2ϕ̅̃̅^T_4j+(t,k)+i/2𝒢̅̂̅̃̅̂̅^T_3× 3ℳv_0^T,The substitution of Eqs. (<ref>) and (<ref>) into Eq. (<ref>) yields [u_0^T(t)=%̣ṣ/̣%̣ṣ2π∫̣_∂ D_1^0[Σ_+/Σ_-L̂̃̂_j4-2ℳ^T/Σ_-ℒ̅̂̅̃̅̂̅_4j^T +1/Σ_-(F̃_j4e^-2ikL)_+]dk;= ̣̣∫_∂ D_1^0{Σ_+/πΣ_-Ψ̃_j4+-2ℳ^T/πΣ_-ϕ̅̃̅_4j+^T . .+2/πΣ_-[ Ψ̃_j4(ϕ̅̃̅_44(t,k̅)-1)e^-2ikL-(Ψ̃_3× 3-𝕀)ℳ^Tϕ̅̃̅_4j^T ]_+}dk, ]Since the integrand in Eq. (<ref>) is an odd function about k, which makes sure that the contour ∂ D_1^0 can be replaced by ∂ D_3^0, thus Eq. (<ref>) yields the Dirichlet boundary values u_0j(t),j=1,2,3 again. Similarly, we can also deduce the Dirichlet boundary values v_0j(t),j=1,2,3 from Eq. (<ref>).§.§5.3. Linearizable boundary conditions for the GLM representationIn what follows we further explore the linearizable boundary conditions for the GLM representation given in Theorem 5.3.Proposition 5.4. Let q_j(x, t=0)=q_0j(x),j=1,2,3 be the initial conditions of the gtc-NLS equation (<ref>) on the interval x∈ [0, L], and one of the following boundary conditions, either(i) the Dirichlet boundary conditions at x=0, L, q_j(x=0,t)=u_0j(t)=0 and q_j(x=L,t)=v_0j(t)=0,j=1,2,3,or(ii) the Robin boundary conditions x=0, L, q_jx(x=0,t)-χ q_j(x=0,t)=u_1j(t)-χ u_0j(t)=0,j=1,2,3 and q_jx(x=L,t)-ϑ q_j(x=L,t)=v_1j(t)-ϑ v_0j(t)=0,j=1,2, where χ and ϑ are both real parameters.Then the eigenfunctions Ψ(t,k) and ϕ(t,k) can be expressed as(i)Ψ(t,k)=𝕀+([ L̂̃̂_3× 3 L̂̃̂_j4; L̂̃̂_4j L̂̃̂_44 ]), ϕ(t,k)=𝕀+([ ℒ̂̃̂_3× 3 ℒ̂̃̂_j4; ℒ̂̃̂_4j ℒ̂̃̂_44 ]),where the 4× 4 matrix-valued function L(t, s)=(L_ij)_4× 4satisfies a reduced Goursat system {[ L̃_3× 3t+L̃_3× 3s=iu_1^T(t)L̃_4j,; L̃_j4t-L̃_j4s=iu_1^T(t)L̃_44,j=1,2,3,; L̃_4jt-L̃_4js=-i u̅_1(t)ℳL̃_3× 3,j=1,2,3,;L̃_44t+L̃_44s=-iu̅_1(t)ℳL̃_j4, ].with the initial data (cf. Eq. (<ref>)) L̃_3× 3(t, -t)=0_3× 3,L̃_44(t, -t)=0, L̃_j4(t, t)=i/2u_1^T(t),L̃_4j(t, t)=-i/2u̅_1(t)ℳ,Similarly, the 4× 4 matrix-valued function ℒ(t, s)=(ℒ_ij)_4× 4satisfies the analogous system (<ref>) with u_1(t) replaced by v_1(t). (ii)Ψ(t,k)=I+([ L̂̃̂_3× 3 L̂̃̂_j4; L̂̃̂_4j L̂̃̂_44 ])+([ -%̣ṣ/̣%̣ṣi2u_0^T(t)Ĝ̃̂_4j kĜ̃̂_j4; kĜ̃̂_4j%̣ṣ/̣%̣ṣi2u̅_0(t)ℳĜ̃̂_j4 ]),ϕ(t,k)=I+([ ℒ̂̃̂_3× 3 ℒ̂̃̂_j4; ℒ̂̃̂_4j ℒ̂̃̂_44 ])+([ -%̣ṣ/̣%̣ṣi2v_0^T(t)𝒢̂̃̂_4j k𝒢̂̃̂_j4; k𝒢̂̃̂_4j%̣ṣ/̣%̣ṣi2v̅_0(t)ℳ𝒢̂̃̂_j4 ]),where the 4× 4 matrix-valued functions L(t, s)=(L_ij)_4× 4 and G(t, s)=(G_ij)_4× 4satisfy the reduced nonlinear Goursat system {[ L̃_3× 3t+L̃_3× 3s=iχ u_0^T(t)L̃_4j +%̣ṣ/̣%̣ṣ12[iu̇_0^T(t)-u_0^T(t)u̅_0(t)ℳu_0^T(t)]G̃_4j,;L̃_44t+L̃_44s=-iχu̅_0(t)ℳL̃_j4 -%̣ṣ/̣%̣ṣ12[iu̇̅̇_0(t)ℳ+u̅_0(t)ℳu^T_0(t)u̅_0(t)ℳ]G̃_j4,; L̃_j4t-L̃_j4s=iχ u_0^T(t)L̃_44,; L̃_4jt-L̃_4js=-iχu̅_0(t)ℳL̃_3× 3,; G̃_j4t-G̃_j4s=2u_0^T(t)L̃_44,;G̃_4jt-G̃_4js=2 u̅_0(t)ℳL̃_3× 3, ].with the initial data (cf. Eq. (<ref>)) {[L̃_3× 3(t, -t)=0_3× 3,; L̃_44(t, -t)=0,; L̃_j4(t, t)=%̣ṣ/̣%̣ṣi2χ u_0^T(t),; L̃_4j(t, t)=-%̣ṣ/̣%̣ṣi2χu̅_0(t)ℳ,; G̃_j4(t, t)=u_0^T(t),; G̃_4j(t, t)=u̅_0(t)ℳ, ].Similarly, the 4× 4 matrix-valued functions ℒ(t, s)=(ℒ_ij)_4× 4 and 𝒢(t, s)=(𝒢_ij)_4× 4satisfy the similar system (<ref>) with u_0(t) and χ replaced by v_0(t) and ϑ, respectively.Proof. Let us show that the linearizable boundary data correspond to the special cases of Proposition 5.1.Case (i) The Dirichlet zero boundary data q_j(x=0,t)=u_0j(t)=0. It follows from the second one of system (<ref>) that G̃_ij(t,s) satisfy {[ G̃_3× 3t+G̃_3× 3s=iu_1^T(t)G̃_4j,; G̃_j4t-G̃_j4s=iu_1^T(t)G̃_44,; G̃_4jt-G̃_4js=-i u̅_1(t)ℳG̃_3× 3,; G̃_44t+G̃_44s=-i u̅_1(t)ℳG̃_j4, ].with the initial data (cf. Eq. (<ref>)) G̃_3× 3(t, -t)=0_3× 3,G̃_44(t, -t)=0, G̃_j4(t, t)=0_j4,G̃_4j(t, t)=0_4j, Thus the unique solution of Eq. (<ref>) is trivial, that is, G̃_3× 3(t,s)=0, G̃_4j(t,s)=0, G̃_j4(t,s)=0, G̃_44(t,s)=0 such that Eq. (<ref>) reduces to Eq. (<ref>) and the condition (<ref>) with (<ref>) becomes(<ref>) with (<ref>). Similarly, for the Dirichlet zero boundary data q_j(x=L,t)=v_0j(t)=0,j=1,2,3, we can also showEq. (<ref>).(ii) Consider the Robin boundary dataq_jx(x=0,t)-χ q_j(x=0,t)=u_1j(t)-χ u_0j(t)=0, (j=1,2,3), that is, the Dirichlet and Neumann boundary data have the linear relation u_1(t)=χ u_0(t).We introduce a 4× 4matrix Q(t, s)=2L(t, s)-iχσ_4G(t, s) by the linear combinations of L and G such that we have {[ ̣̃Q_3× 3(t,s)=2L̃_3× 3(t,s)-iχG̃_3× 3(t,s),; ̣̃Q_j4(t,s)=2L̃_j4(t,s)-iχG̃_j4(t,s),; ̣̃Q_4j(t,s)=2L̃_4j(t,s)+iχG̃_4j(t,s),; ̣̃Q_44(t,s)=2L̃_44(t,s)+iχG̃_44(t,s), ]. It follows from Eq. (<ref>) and (<ref>) with Eq. (<ref>) that Q̃_ij(t,s), G̃_ij(t,s),i,j=1,2 satisfy {[ Q̃_3× 3t+Q̃_3× 3s =[iu̇_0^T(t)-u_0^T(t)u̅_0(t)ℳu_0^T(t)+χ^2u_0^T(t)]G̃_4j,;Q̃_j4t-Q̃_j4s=[iu̇_0^T(t)-u_0^T(t)u̅_0(t)ℳu_0^T(t)+χ^2u_0^T(t)]G̃_44,; Q̃_4jt-Q̃_4js=[-u̅_0(t)ℳu_0^T(t)u̅_0(t)ℳ-iu̇̅̇_0(t)ℳ +χ^2u̅_0(t)ℳ]G̃_3× 3,; Q̃_44t+Q̃_44s=[-u̅_0(t)ℳu_0^T(t)u̅_0(t)ℳ-iu̇̅̇_0(t)ℳ +χ^2u̅_0(t)ℳ]G̃_j4,;G̃_3× 3t+G̃_3× 3s=u_0^T(t) Q̃_4j,;G̃_j4t-G̃_j4s=u_0^T(t) Q̃_44,;G̃_4jt-G̃_4js= u̅_0(t)ℳQ̃_3× 3,;G̃_44t+G̃_44s= u̅_0(t)ℳQ̃_j4, ].with the initial data (cf. Eq. (<ref>)) {[ G̃_3× 3(t, -t)=0_3× 3,G̃_44(t, -t)=0,G̃_j4(t, t)=u_0^T(t),G̃_4j(t, t)=u̅_0(t)ℳ,;Q̃_3× 3(t, -t)=0_3× 3,Q̃_44(t, -t)=0,Q̃_j4(t, t)=0_j4, Q̃_4j(t, t)=0_4j, ]. Thus the unique solution of Eq. (<ref>) is trivial, that is, Q̃_j4(t,s)=Q̃_4j(t,s) =G̃_3× 3(t,s)=G̃_44(t,s)=0such that Eq. (<ref>) reduces to Eq. (<ref>) and the condition (<ref>) with Eq. (<ref>) becomes Eq. (<ref>) with Eq. (<ref>). Similarly, for the Robin boundary dataq_jx(x=L,t)-ϑ q_j(x=L,t)=v_1j(t)-ϑ v_0j(t)=0,j=1,2,3, that is, v_1(t)=ϑ v_0(t), we can also showEq. (<ref>). □Based on the Theorem 5.3 and Proposition 5.4, we have the following Proposition.Proposition 5.5 For the linearizable Dirichlet boundary data u_0(t)=v_0(t)=0, we have the Neumann boundary data u_1(t) and v_1(t): u_1^T(t)=%̣ṣ/̣%̣ṣ4iπ∫̣_∂ D_1^0kΨ̃_j4(ϕ̅̃̅_44-𝕀)dk, v_1^T(t)=%̣ṣ/̣%̣ṣ4iπ∫̣_∂ D_1^0kϕ̃_j4(Ψ̅̃̅_44-𝕀)dk,where {[ Ψ̃_j4t+4ik^2Ψ̃_j4=iu_1^T(t)(Ψ̃_44+𝕀),; Ψ̃_44t=-iu̅_1(t)ℳΨ̃_j4,; ϕ̃_j4t+4ik^2ϕ̃_j4=iv_1^T(t)(ϕ̃_44+𝕀),; ϕ̃_44t=-iv̅_1(t)ℳϕ̃_j4. ].Remark 5.6. The analogous analysis of the Fokas unified method will use also to explore the IBV problems for other integrable nonlinear evolution PDEs with 4× 4 Lax pairs both on the the half-line and the finite interval, such as the three-component derivative-NLS equation andthe three-component higher-order NLS equation, which will be considered in other papers.AcknowledgmentsThis work was partially supported by the NSFC under Grant No.11571346 and the Youth Innovation Promotion Association, CAS.99ist C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Methods for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967) 1095-1097.ist2 M. J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, Society for Industrial and Applied Mathematics, 1981.soliton M.J. Ablowitz, P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, Cambridge, 1991. lax P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968) 467-490. rh P. Deift, Z. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, Bull. Am. Math. Soc. 20 (1992) 119-123.f1 A.S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. Ser. A 453 (1997) 1411-1443.f2 A.S. Fokas, On the integrability of linear and nonlinear partial differential equations, J. Math. Phys. 41 (2000) 4188-4237.f3 A.S. Fokas, Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys. 230 (2002) 1-39.f4 A.S. Fokas, A unified approach to boundary value problems, in: CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2008.f5 B. Pelloni, Advances in the study of boundary value problems for nonlinear integrable PDEs, Nonlinearity 28 (2015) R1-R38.nls1 A.S. Fokas, A.R. 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"authors": [
"Zhenya Yan"
],
"categories": [
"nlin.SI",
"math-ph",
"math.AP",
"math.MP",
"quant-ph"
],
"primary_category": "nlin.SI",
"published": "20170427133120",
"title": "An initial-boundary value problem of the general three-component nonlinear Schrodinger equation with a 4x4 Lax pair on a finite interval"
} |
Chemical Enhancements in Shock-accelerated Particles: Ab-initio Simulations Anatoly Spitkovsky December 30, 2023 =========================================================================== We consider relative error low rank approximation of tensors with respect to the Frobenius norm. Namely, given an order-q tensor A ∈ℝ^∏_i=1^q n_i, output a rank-k tensor B for which A-B_F^2 ≤ (1+ϵ), where = inf_rank-k A'A-A'_F^2. Despite much success on obtaining relative error low rank approximations for matrices, no such results were known for tensors for arbitrary (1+ϵ)-approximations.One structural issue is that there may be no rank-k tensor A_k achieving the above infinum. Another,computational issue, is that an efficient relative error low rank approximation algorithm for tensors would allow one to compute the rank of a tensor, which is NP-hard. We bypass these two issues via (1) bicriteria and (2) parameterized complexity solutions: * We give an algorithmwhich outputs a rank k' = O((k/ϵ)^q-1) tensor B for which A-B_F^2 ≤ (1+ϵ) in (A) + n ·(k/ϵ) time in the realmodel, whenever either A_k exists or > 0. Here (A) denotes the number of non-zero entries in A. If both A_k does not exist and = 0, then B instead satisfies A-B_F^2 < γ, where γ is any positive, arbitrarily small function of n. * We give an algorithm for any δ >0 which outputs a rank k tensor B for which A-B_F^2 ≤ (1+ϵ) and runs in ( (A) + n (k/ϵ) + exp(k^2/ϵ) ) · n^δ time in the unit costmodel, whenever > 2^- O(n^δ) and there is a rank-k tensor B = ∑_i=1^k u_i ⊗ v_i ⊗ w_i for which A-B_F^2 ≤ (1+ϵ/2) and u_i_2, v_i_2, w_i_2 ≤ 2^O(n^δ). If ≤ 2^- Ω(n^δ), then B instead satisfies A-B_F^2 ≤ 2^- Ω(n^δ).Our first result is polynomial time, and in fact input sparsity time, in n, k, and 1/ϵ, for any k≥ 1 and any 0 < ϵ <1, while our second result is fixed parameter tractable in k and 1/ϵ. For outputting a rank-k tensor, or even a bicriteria solution with rank-Ck for a certain constant C > 1, we show a 2^Ω(k^1-o(1)) time lower bound under the Exponential Time Hypothesis. Our results are based on an “iterative existential argument”, and also give the first relative error low rank approximations for tensors for a large number of error measures for which nothing was known. In particular, we give the first relative error approximation algorithms on tensors for: column row and tube subset selection, entrywise ℓ_p-low rank approximation for 1 ≤ p < 2, low rank approximation with respect to sum of Euclidean norms of faces or tubes, weighted low rank approximation, and low rank approximation in distributed and streaming models. We also obtain several new results for matrices, such as (A)-time CUR decompositions, improving the previous (A)log n-time CUR decompositions, which may be of independent interest.empty § INTRODUCTIONLow rank approximation of matrices is one of the most well-studied problems in randomized numerical linear algebra. Given an n × d matrix A with real-valued entries, we want to output a rank-k matrix B for which A-B is small, under a given norm. While this problem can be solved exactly using the singular value decomposition for some norms like the spectral and Frobenius norms, the time complexity is still min(nd^ω-1, dn^ω-1), where ω≈ 2.376 is the exponent of matrix multiplication <cit.>. This time complexity is prohibitive when n and d are large. By now there are a number of approximation algorithms for this problem, with the Frobenius norm [Recall the Frobenius norm A_F of a matrix A is (∑_i = 1^n ∑_j=1^d A_i,j^2)^1/2.] being one of the most common error measures.Initial solutions <cit.> to this problem were based on sampling and achieved additive error in terms of ϵA_F, where ϵ > 0 is an approximation parameter, which can be arbitrarily larger than the optimal cost = min_rank-kBA-B_F^2. Since then a number of solutions based on the technique of oblivious sketching <cit.> as well as sampling based on non-uniform distributions <cit.>, have been proposed which achieve the stronger notion of relative error, namely, which output a rank-k matrix B for which A-B_F^2 ≤ (1+ϵ) with high probability. It is now known how to output a factorization of such a B = U · V, where U is n × k and V is k × d, in (A) + (n+d) (k/ϵ) time <cit.>. Such an algorithm is optimal, up to the (k/ϵ) factor, as any algorithm achieving relative error must read almost all of the entries.Tensors are often more useful than matrices for capturing higher order relations in data. Computing low rank factorizations of approximations of tensors is the primary task of interest in a number of applications, such as in psychology<cit.>, chemometrics <cit.>, neuroscience <cit.>, computational biology <cit.>, natural language processing <cit.>, computer vision <cit.>, computer graphics <cit.>, security <cit.>, cryptography <cit.> data mining <cit.>, machine learning applications such as learning hidden Markov models, reinforcement learning, community detection, multi-armed bandit, ranking models, neural network, Gaussian mixture models and Latent Dirichlet allocation <cit.>, programming languages <cit.>, signal processing <cit.>, and other applications <cit.>.Despite the success for matrices, the situation for order-q tensors for q > 2 is much less understood. There are a number of works based on alternating minimization <cit.> gradient descent or Newton methods <cit.>, methods based on the Higher-order SVD (HOSVD) <cit.> which provably incur Ω(√(n))-inapproximability for Frobenius norm error <cit.>, the power method or orthogonal iteration method <cit.>, additive error guarantees in terms of the flattened (unfolded) tensor rather than the original tensor <cit.>, tensor trains <cit.>, the tree Tucker decomposition <cit.>, or methods specialized to orthogonal tensors <cit.>. There are also a number of works on the problem of tensor completion, that is, recovering a low rank tensor from missing entries <cit.>. There is also another line of work using the sum of squares (SOS) technique to study tensor problems <cit.>, other recent work on tensor PCA <cit.>, and work applying smoothed analysis to tensor decomposition <cit.>. Several previous works also consider more robust norms than the Frobenius norm for tensors, e.g., the R_1 norm (ℓ_1-ℓ_2-ℓ_2 norm in our work) <cit.>, ℓ_1-PCA <cit.>, entry-wise ℓ_1 regularization <cit.>, M-estimator loss <cit.>, weighted approximation <cit.>, tensor-CUR <cit.>, or robust tensor PCA <cit.>. Some of the above works, such as ones based on the tensor power method or alternating minimization, require incoherence or orthogonality assumptions. Others, such as those based on the simultaneous SVD, require an assumption on the minimum singular value. See the monograph of Moitra <cit.> for further discussion. Unlike the situation for matrices, there is no work for tensors that is able to achieve the following natural relative error guarantee: given a q-th order tensor A ∈ℝ^n^⊗ q and an arbitrary accuracy parameter ϵ > 0,output a rank-k tensor B for which A-B_F^2 ≤ (1+ϵ),where = inf_rank-kB'A-B'_F^2, and where recall the rank of a tensor B is the minimal integer k for which B can be expressed as ∑_i=1^k u_i ⊗ v_i ⊗ w_i. A third order tensor, for example, has rank which is aninteger in {0, 1, 2, …, n^2}. We note that <cit.> is able to achievea relative error 5-approximation for third order tensors, and an O(q)-approximation for q-th order tensors, though it cannot achieve a (1+ϵ)-approximation.We compare our work to <cit.> in Section <ref> below.For notational simplicity, we will start by assuming third order tensors with all dimensions of equal size, but we extend all of our main theorems below to tensors of any constant order q > 3 and dimensions of different sizes.The first caveat regarding (<ref>) for tensors is that an optimal rank-k solution may not even exist! This is a well-known problem for tensors (see, e.g., <cit.> and more details in section 4 of <cit.>), for which for any rank-k tensor B, there always exists another rank-k tensor B' for which A-B'_F^2 < A-B_F^2. If = 0, then in this case for any rank-k tensor B, necessarily A-B_F^2 > 0, and so (<ref>) cannot be satisfied. This fact was known to algebraic geometers as early as the 19th century, which they refer to as the fact that the locus of r-th secant planes to a Segre variety may not define a (closed) algebraic variety <cit.>. It is also known as the phenomenon underlying the concept of border rank[<https://en.wikipedia.org/wiki/Tensor_rank_decomposition#Border_rank>]<cit.>.In this case it is natural to allow the algorithm to output an arbitrarily small γ > 0 amount of additive error. Note that unlike several additive error algorithms for matrices, the additive error here can in fact be an arbitrarily small positive function of n. If, however, > 0, then for any ϵ > 0, there exists a rank-k tensor B for which A-B_F^2 ≤ (1+ϵ), and in this case we should still require the algorithm to output a relative-error solution. If an optimal rank-k solution B exists, then as for matrices, it is natural to require the algorithm to output a relative-error solution.Besides the above definitional issue, a central reason that (<ref>) has not been achieved is that computing the rank of a third order tensor is well-known to be NP-hard <cit.>. Thus, if one had such a polynomial time procedure for solving the problem above, one could determine the rank of A by running the procedure on each k ∈{0, 1, 2, …, n^2}, and check for the first value of k for which A-B_F^2 = 0, thus determining the rank of A. However, it is unclear if approximating the tensor rank is hard. This question will also be answered in this work. The main question which we address is how to define a meaningful notion of (<ref>) for the case of tensors and whether it is possible to obtain provably efficient algorithms which achieve this guarantee, without any assumptions on the tensor itself. Besides (<ref>), there are many other notions of relative error for low rank approximation of matrices for which provable guarantees for tensors are unknown, such as tensor CURT, R_1 norm, and the weighted and ℓ_1 norms mentioned above. Our goal is to provide a general technique to obtain algorithms for many of these variants as well. §.§ Our Results To state our results, we first consider the case when a rank-k solution A_k exists, that is, there exists a rank-k tensor A_k for which A-A_k_F^2 =.We first give a poly(n, k, 1/ϵ)-time (1+ϵ)-relative error approximation algorithm for any 0 < ϵ <1 and any k≥ 1, but allow the output tensor B to be of rank O((k/ϵ)^2) (for general q-order tensors, the output rank is O((k/ϵ)^q-1), whereas we measure the cost of B with respect to rank-k tensors. Formally, A-B_F^2 ≤ (1+ϵ) A-A_k_F^2. In fact, our algorithm can be implemented in (A) + n ·(k/ϵ) time in the real- model, where (A) is the number of non-zero entries of A. Such an algorithm is optimal for any relative error algorithm, even bicriteria ones.If A_k does not exist, then our output B instead satisfies A-B_F^2 ≤ (1+ϵ) + γ, where γ is an arbitrarily small additive error. Since γ is arbitrarily small, (1+ϵ) + γ is still a relative error whenever > 0.Our theorem is as follows. Given a 3rd order tensor A ∈ℝ^n× n× n, if A_k exists then there is a randomized algorithm running in (A) + n ·(k/ϵ) time which outputs a (factorization of a) rank-O(k^2/ϵ^2) tensor B for which A-B_F^2 ≤ (1+ϵ)A-A_k_F^2. If A_k does not exist, then the algorithm outputs a rank-O(k^2/ϵ^2) tensor B for which A-B_F^2 ≤ (1+ϵ) + γ, where γ > 0 is an arbitrarily small positive function of n. In both cases, the success probability is at least 2/3. One of the main applications of matrix low rank approximation is parameter reduction, as one can store the matrix using fewer parameters in factored form or more quickly multiply by the matrix if given in factored form, as well as remove directions that correspond to noise.In such applications, it is not essential that the low rank approximation have rank exactly k, since one still has a significant parameter reduction with a matrix of slightly larger rank. This same motivation applies to tensor low rank approximation; we obtain both space and time savings by representing a tensor in factored form, and in such applications bicriteria applications suffice. Moreover, the extremely efficient (A) + n ·(k/ϵ) time algorithm we obtain may outweigh the need for outputting a tensor of rank exactly k. Bicriteria algorithms are common for coping with hardness;see e.g., results on robust low rank approximation of matrices <cit.>, sparse recovery <cit.>, clustering <cit.>, and approximation algorithms more generally. We note thatthere are other applications, such as unique tensor decomposition in the method of moments, see, e.g., <cit.>, where one may have a hard rank constraint of k for the output. However, in such applications the so-called Tucker decomposition is still a useful dimensionality-reduction analogue of the SVD and our techniques for provingTheorem <ref> can also be used for obtaining Tucker decompositions, see Section <ref>. We next consider the case when the rank parameter k is small, and we try to obtain rank-k solutions which are efficient for small values of k. As before, we first suppose that A_k exists.If A_k = ∑_i=1^k u_i ⊗ v_i ⊗ w_i and the norms u_i_2, v_i_2, and w_i_2 are bounded by 2^(n), we can return a rank-k solution B for which A-B_F^2 ≤ (1+)A-A_k_F^2 + 2^-(n), in f(k,1/ϵ) ·(n) time in the standard unit costmodel with words of size O(log n) bits. Thus, our algorithm is fixed parameter tractable in k and 1/ϵ, and in fact remains polynomial time for any values of k and 1/ϵ for which k^2/ϵ = O(log n). This is motivated by a number of low rank approximation applications in which k is typically small. The additive error of 2^-(n) is only needed in order to write down our solution B in the unit costmodel, since in general the entries of B may be irrational, even if the entries of A are specified by (n) bits. If instead we only want to output an approximation to the value A-A_k_F^2, then we can output a number Z for which ≤ Z ≤ (1+ϵ), that is, we do not incur additive error.When A_k does not exist, there still exists a rank-k tensor à for whichA-Ã_F^2 ≤ + γ. We require there exists such a à for which if à = ∑_i=1^k u_i ⊗ v_i ⊗ w_i, then the norms u_i_2, v_i_2, and w_i_2 are bounded by 2^(n).The assumption in the previous two paragraphs that the factors of A_k and of à have norm bounded by 2^(n) is necessary in certain cases, e.g., if = 0 and we are to write down the factors in (n) time. An abridged version of our theorem is as follows. Given a 3rd order tensor A ∈ℝ^n× n× n, for any δ >0, if A_k = ∑_i=1^k u_i ⊗ v_i ⊗ w_i exists and each of u_i_2, v_i_2, and w_i_2 is bounded by 2^O(n^δ), then there is a randomized algorithm running in O( (A) + n(k,1/ϵ) + 2^O(k^2/ϵ) ) · n^δ time in the unit costmodel with words of size O(log n) bits[The entries of A are assumed to fit in n^δ words.], which outputs a (factorization of a) rank-k tensor B for which A-B_F^2 ≤ (1+ϵ)A-A_k_F^2 + 2^- O(n^δ). Further, we can output a number Z for which ≤ Z ≤ (1+ϵ) in the same amount of time. When A_k does not exist, if there exists a rank-k tensor à for which A-Ã_F^2 ≤ + 2^-O(n^δ) and à = ∑_i=1^k u_i ⊗ v_i ⊗ w_i is such that the norms u_i_2, v_i_2, and w_i_2 are bounded by 2^O(n^δ), then we can output a (factorization of a) rank-k tensor à for which A-Ã_F^2 ≤ (1+ϵ) + 2^-O(n^δ). Our techniques for proving Theorem <ref> and Theorem <ref> open up avenues for many other problems in linear algebra on tensors. We now define the problems and state our results for them. There is a long line of research on matrix column subset selection and CUR decomposition <cit.> under operator, Frobenius, and entry-wise ℓ_1 norm. It is natural to consider tensor column subset selection or tensor-CURT[T denotes the tube which is the column in 3rd dimension of tensor.], however most previous works either give error bounds in terms of the tensor flattenings <cit.>, assume the original tensor has certain properties <cit.>, consider the exact case which assumes the tensor has low rank <cit.>, or only fit a high dimensional cross-shape to the tensor rather than to all of its entries <cit.>. Such works are not able to provide a (1+ϵ)-approximation guarantee as in the matrix case without assumptions.We consider tensor column, row, and tube subset selection, with the goal being to find three matrices: a subset C ∈ℝ^n × c of columns of A, a subset R ∈ℝ^n × r of rows of A, and a subset T∈ℝ^n× t of tubes of A, such that there exists a tensor U∈ℝ^c× r× t for which U(C,R,T) -A _ξ≤α A_k - A _ξ + γ,where γ=0 if A_k exists and γ=2^-(n) otherwise, α>1 is the approximation ratio, ξ is either Frobenius norm or Entry-wise ℓ_1 norm, and U(C,R,T) = ∑_i=1^c ∑_j=1^r ∑_l=1^t U_i,j,l· C_i ⊗ R_j ⊗ T_l. In tensor CURT decomposition, we also want to output U.We provide a (nearly) input sparsity time algorithm for this, together with an alternative input sparsity time algorithm which chooses slightly larger factors C,R, and T. To do this, we combine Theorem <ref> with the following theorem which, given a factorization of a rank-k tensor B, obtains C, U, R, and T in terms of it: Given a 3rd order tensor A∈ℝ^n× n × n, let k≥ 1, and let U_B,V_B,W_B∈ℝ^n× k be given. There is an algorithm running in O((A) log n) + O( n^2 )( k, 1/ϵ) time (respectively, O((A)) + n (k,1/ϵ) time) which outputs a subset C∈ℝ^n× c of columns of A, a subset R∈ℝ^n× r of rows of A, a subset T∈ℝ^n× t of tubes of A, together with a tensor U∈ℝ^c× r× t with (U)=k such that c=r=t=O(k/ϵ) (respectively, c = r = t = O(klog k+k/ϵ)), and U(C,R,T) - A _F^2 ≤ (1+ϵ) U_B⊗ V_B ⊗ W_B -A _F^2 holds with probability at least 9/10. Combining Theorems <ref> and <ref> (with B being a (1+O(ϵ))-approximation to A) we achieve Equation (<ref>) with α=(1+ϵ) and ξ=F with the optimal number of columns, rows, tubes, and rank of U (we mention our matching lower bound later), though the running time has an 2^O(k^2/ϵ) term in it.We note that insteadcombining Theorem <ref> and Theorem <ref> gives a bicriteria result for CURT without a 2^O(k^2/ϵ) term in the running time, though it is suboptimal in the number of columns, rows, tubes, and rank of U. We also obtain several algorithms for tensor entry-wise ℓ_p norm low-rank approximation,as well as results for asymmetric tensor norms, which are natural extensions of the matrix ℓ_1-ℓ_2 norm. Here, for a tensor A,A_v = ∑_i ( ∑_j,k (A_i,j,k )^2 )^1/2 and A_u = ∑_i,j ( ∑_k (A_i,j,k )^2 )^1/2.Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, let r=O(k^2). If A_k exists then there is an algorithm which runs in (A) · t + O(n) (k) time andoutputs a (factorization of a) rank-r tensor B for which B - A _ξ≤(k,log n)· A_k - A _ξ holds. If A_k does not exist, we have B-A_ξ≤(k,log n)· + γ, where γ is an arbitrarily small positive function of n. The success probability is at least 9/10. For ξ = 1 or p, t=O(k); for ξ=v, t=O(1); for ξ=u, t=O(n). As in the case of Frobenius norm, we can get rank-k and CURT algorithms for the above norms. Our results for asymmetric norms can be extended to ℓ_p-ℓ_2-ℓ_2, ℓ_p-ℓ_p-ℓ_2, and families of M-estimators. We also obtain the following result for weighted tensor low-rank approximation.Suppose we are given a third order tensor A∈ℝ^n× n× n, as well as a tensor W∈ℝ^n× n × n with r distinct rows and r distinct columns. Suppose there is a rank-k tensor A'∈ℝ^n× n× n for which W∘ (A'-A)_F^2 = and one can write A'=∑_i=1^k u_i ⊗ v_i ⊗ w_i for u_i_2, v_i_2, and w_i_2 bounded by 2^n^δ. Then there is an algorithm running in ( (A)+(W)+ n 2^O(r^2k^2/ϵ) ) · n^δ time and outputting n× k matrices U_1,U_2,U_3 for whichW∘( U_1 ⊗ U_2 ⊗ U_3 - A ) _F^2 ≤ (1+ϵ) with probability at least 2/3. We next strengthen Håstad's NP-hardnessto show that even approximating tensor rank is hard (we note at the time of Håstad's NP-hardness, there was no PCP theorem available; nevertheless we need to do additional work here): Let q ≥ 3. Unless the Exponential Time Hypothesis (𝖤𝖳𝖧) fails, there is an absolute constant c_0>1 for which distinguishing if a tensor in ℝ^n^q has rank at most k, or at least c_0 · k, requires 2^δ k^1-o(1) time, for a constant δ>0.Under random-<cit.>, an average case hardness assumption for , we can replace the k^1-o(1) in the exponent above with a k. We also obtain hardness in terms of ϵ: Let q ≥ 3. Unless fails, there is no algorithm running in 2^o(1/ϵ^1/4) time which, given a tensor A ∈ℝ^n^q, outputs a rank-1 tensor B for which A-B_F^2 ≤ (1+ϵ). As a side result worth stating, our analysis improves the best matrix CUR decomposition algorithm under Frobenius norm <cit.>, providing the first optimal (A)-time algorithm: There is an algorithm, which given a matrix A∈ℝ^n× d and an integer k ≥ 1, runs in O((A)) + (n+d)(k,1/ϵ) time and outputs three matrices: C∈ℝ^n × c containing c columns of A, R∈ℝ^r× d containing r rows of A, and U∈ℝ^c× r with (U)=k for which r=c=O(k/ϵ) and CUR - A _F^2 ≤ (1+ϵ) min_rank-k A_k A_k - A _F^2, holds with probability at least 9/10. §.§ Our Techniques Many of our proofs, in particular those for Theorem <ref> and Theorem <ref>, are based on what we call an “iterative existential proof”, which we then turn into an algorithm in two different ways depending if we are proving Theorem <ref> or Theorem <ref>.Henceforth, we assume A_k exists; otherwise replace A_k with a suitably good tensor à in what follows. Since A_k = ∑_i=1^k U_i^* ⊗ V_i^* ⊗ W_i^*[For simplicity, we define U⊗ V⊗ W=∑_i=1^k U_i ⊗ V_i ⊗ W_i, where U_i is the i-th column of U.], we can create three n × k matrices U^*, V^*, and W^* whose columns are the vectors U_i^*, V_i^*, and W_i^*, respectively. Now we consider the three different flattenings (or unfoldings) of A_k, which express A_k as an n × n^2 matrix. Namely, by thinking of A_k as the sum of outer products, we can write the three flattenings of A_k as U^* · Z_1, V^* · Z_2, and W^* · Z_3, where the rows of Z_1 are ( V^*_i ⊗ W^*_i) [(V^*_i ⊗ W^*_i) denotes a row vector that has length n_1 n_2 where V^*_i has length n_1 and W^*_i has length n_2.] ( For simplicity, we write Z_1 = (V^*⊤⊙ W^*⊤). [(V^*⊤⊙ W^*⊤ ) denotes a k× n_1n_2 matrix where the i-th row is ( V^*_i ⊗ W^*_i), where length n_1 vector V_i^* is the i-th column of n_1× k matrix V^*, and length n_2vector W^*_i is the i-th column of n_2 × k matrix W^*, ∀ i∈ [k].] ), the rows of Z_2 are (U^*_i ⊗ W^*_i), and the rows of Z_3 are ( U^*_i ⊗ V^*_i), for i ∈ [k] {1, 2, …, k}. Letting the three corresponding flattenings of the input tensor A be A_1, A_2, and A_3, by the symmetry of the Frobenius norm, we haveA-B_F^2 = A_1-U^*Z_1_F^2 = A_2-V^*Z_2_F^2 = A_3-W^*Z_3_F^2. Let us consider the hypothetical regression problem min_U A_1 - UZ_1_F^2. Note that we do not know Z_1, but we will not need to. Let r = O(k/ϵ), and suppose S_1 is an n^2 × r matrix of i.i.d. normal random variables with mean 0 and variance 1/r, denoted N(0, 1/r). Then by standard results for regression (see, e.g., <cit.> for a survey), if Û is the minimizer to the smaller regression problem Û = argmin_U UZ_1S_1 - A_1 S_1_F^2, then A_1 -ÛZ_1_F^2 ≤ (1+ϵ) min_U A_1 - UZ_1_F^2.Moreover, Û = A_1 S_1 (Z_1 S_1)^†. Although we do not know know Z_1, this implies Û is in the column span of A_1 S_1, which we do know, since we can flatten A to compute A_1 and then compute A_1 S_1. Thus, this hypothetical regression argument gives us an existential statement - there exists a good rank-k matrix Û in the column span of A_1 S_1. We could similarly define V̂ = A_2 S_2 (Z_2 S_2)^† and Ŵ = A_3 S_3 (Z_3 S_3)^† as solutions to the analogous regression problems for the other two flattenings of A, which are in the column spans of A_2 S_2 and A_3 S_3, respectively. Given A_1 S_1, A_2 S_2, and A_3 S_3, which we know, we could hope there is a good rank-k tensor in the span of the rank-1 tensors {( A_1 S_1)_a ⊗ ( A_2 S_2 )_b ⊗ ( A_3 S_3)_c}_a,b,c ∈ [r].However, an immediate issue arises. First, note that our hypothetical regression problem guarantees that A_1 - ÛZ_1_F^2 ≤ (1+ϵ)A-A_k_F^2, and therefore since the rows of Z_1 are of the special form ( V^*_i ⊗ W^*_i ), we can perform a “retensorization” to create a rank-k tensor B = ∑_i Û_i ⊗ V^*_i ⊗ W^*_i from the matrix ÛZ_1 for which A-B_F^2 ≤ (1+ϵ)A-A_k_F^2. While we do not know Û, since it is in the column span of A_1 S_1, it implies that B is in the span of the rank-1 tensors {( A_1 S_1 )_a ⊗ V^*_b ⊗ W^*_c}_a ∈ [r], b,c ∈ [k]. Analogously, we have that there is a good rank-k tensor B in the span of the rank-1 tensors {U^*_a ⊗ (A_2 S_2)_b ⊗ W^*_c}_a, c ∈ [k], b ∈ [r], and a good rank-k tensor B in the span of the rank-1 tensors {U^*_a ⊗ V^*_b ⊗ (A_3 S_3 )_c}_a,b ∈ [k], c ∈ [r]. However, we do not know U^* or V^*, and it is not clear there is a rank-k tensor B for which simultaneously its first factors are in the column span of A_1 S_1, its second factors are in the column span of A_2 S_2, and its third factors are in the column span of A_3 S_3, i.e., whether there is a good rank-k tensor B in the span of rank-1 tensors in (<ref>).We fix this by an iterative argument. Namely, we first compute A_1 S_1, and write Û = A_1 S_1 (Z_1 S_1)^†. We now redefine Z_2 with respect to Û, so the rows of Z_2 are (Û_i ⊗ W^*_i) for i ∈ [k], and consider the regression problem min_V A_2 - VZ_2_F^2. While we do not know Z_2, if S_2 is an n^2 × r matrix of i.i.d. Gaussians, we again have the statement that V̂ = A_2 S_2 (Z_2 S_2)^† satisfies A_2 - V̂Z_2_F^2≤(1+ϵ) min_V A_2 - VZ_2_F^2by the regression guarantee with Gaussians ≤(1+ϵ)A_2 - V^*Z_2_F^2 since V^*is no better than the minimizer V = (1+ϵ)A_1-ÛZ_1_F^2by retensorizing and flattening along a different dimension≤(1+ϵ)^2 min_U A_1 - UZ_1_F^2by (<ref>) = (1+ϵ)^2 A - A_k_F^2by definition of Z_1.Now we can retensorize V̂ Z_2 to obtain a rank-k tensor B for which A-B_F^2 = A_2 - V̂ Z_2_F^2 ≤ (1+ϵ)^2 A-A_k_F^2. Note that since the columns of V̂ are in the span of A_2 S_2, and the rows of Z_2 are (Û_i ⊗ W^*_i) for i ∈ [k], where the columns of Û are in the span of A_1 S_1, it follows that B is in the span of rank-1 tensors{(A_1 S_1)_a ⊗ ( A_2 S_2 )_b ⊗V̂_c}_a,b ∈ [r], c ∈ [k]. Suppose we now redefine Z_3 so that it is now an r^2 × n^2 matrix with rows ( ( A_1 S_1)_a ⊗ ( A_2 S_2)_b ) for all pairs a,b ∈ [r], and consider the regression problem min_W A_3 - WZ_3_F^2. Now observe that since we knowZ_3, and since we can form A_3 by flattening A, we can solve for W ∈ℝ^n × r^2 in polynomial time by solving a regression problem. Retensorizing WZ_3 to a tensor B, it follows that we have found a rank-r^2 = O(k^2/ϵ^2) tensor B for which A-B_F^2 ≤ (1+ϵ)^2 A-A_k_F^2 = (1+O(ϵ))A-A_k_F^2, and the result follows by adjusting ϵ by a constant factor.To obtain the (A) + n (k/ϵ) running time guarantee of Theorem <ref>, while we can replace S_1 and S_2 with compositions of a sparse CountSketch matrix and a Gaussian matrix (see chapter 2 of <cit.> for a survey), enabling us to compute A_1 S_1 and A_2 S_2 in (A) + n (k/ϵ) time, we still need to solve the regression problem min_W A_3 - WZ_3_F^2 quickly, and note that we cannot even write down Z_3 without spending r^2 n^2 time. Here we use a different random matrix S_3 called TensorSketch, which was introduced in <cit.>, but for which we will need the stronger properties of a subspace embedding and approximate matrix product shown to hold for it in <cit.>. Given the latter properties, we can instead solve the regression problem min_W A_3S_3 - WZ_3S_3_F^2, and importantly A_3S_3 and Z_3 S_3 can be computed in (A) + n (k/ϵ) time. Finally, this small problem can be solved in n (k/ϵ) time.If we want to output a rank-k solution as in Theorem <ref>, then we need to introduce indeterminates at several places in the preceding argument and run a generic polynomial optimization procedure which runs in time exponential in the number of indeterminates. Namely, we write Û as A_1 S_1 X_1, where X_1 is an r × k matrix of indeterminates, we write V̂ as A_2 S_2 X_2, where X_2 is an r × k matrix of indeterminates, and we write Ŵ as A_3 S_3 X_3, where X_3 is an r × k matrix of indeterminates. When executing the above iterative argument, we let the rows of Z_1 be the vectors (V^*_i ⊗ W^*_i), the rows of Z_2 be the vectors (Û_i ⊗ W^*_i), and the rows of Z_3 be the vectors (Û_i ⊗ V_i). Then Û is a (1+ϵ)-approximate minimizer to min_U A_1 - UZ_1_F, while V̂ is a (1+ϵ)-approximate minimizer to min_V A_2 - VZ_2_F, while Ŵ is a (1+ϵ)-approximate minimizer to min_W A_3 - WZ_3_F. Note that by assigning X_1 = (Z_1 S_1)^†, X_2 = (Z_2 S_2)^†, and X_3 = (Z_3 S_3)^†, it follows that the rank-k tensor B = ∑_i=1^k (A_1 S_1 X_1)_i ⊗ (A_2 S_2 X_2)_i ⊗ (A_3 S_3 X_3)_i satisfies A-B_F^2 ≤ (1+ϵ)^3 A-A_k_F^2, as desired. Note that here the rows of Z_2 are a function of X_1, while the rows of Z_3 are a function of both X_1 and X_2. What is important for us though is that it suffices to minimize the degree-6 polynomial ∑_a,b,c ∈ [n] (∑_i=1^k (A_1 S_1 X_1)_a,i· (A_2 S_2 X_2)_b,i· (A_3 S_3 X_3)_c,i - A_a,b,c)^2,over the 3rk = O(k^2/ϵ) indeterminates X_1, X_2, X_3, since we know there exists an assignment to X_1, X_2, and X_3 providing a (1+O(ϵ))-approximate solution, and any solution X_1, X_2, and X_3 found by minimizing the above polynomial will be no worse than that solution. This polynomial can be minimized up to additive 2^-(n) additive error in (n) time <cit.> assuming the entries of U^*, V^*, and W^* are bounded by 2^(n), as assumed in Theorem <ref>. Similar arguments can be made for obtaining a relative error approximation to the actual valueas well as handling the case when A_k does not exist.To optimize the running time to (A), we can choose CountSketch matrices T_1, T_2, T_3 of t=(k,1/ϵ) × n dimensions and reapply the above iterative argument. Then it suffices to minimize this small size degree-6 polynomial∑_a,b,c ∈ [t] (∑_i=1^k (T_1 A_1 S_1 X_1)_a,i· (T_2 A_2 S_2 X_2)_b,i· (T_3 A_3 S_3 X_3)_c,i - (A(T_1,T_2,T_3))_a,b,c)^2,over the 3rk = O(k^2/ϵ) indeterminates X_1, X_2, X_3. Outputting A_1S_1X_1, A_2S_2X_2, A_3S_3X_3 then provides a (1+ϵ)-approximate solution.Our iterative existential argument provides a general framework for obtaining low rank approximation results for tensors for many other error measures as well. §.§ Other Low Rank Approximation Algorithms Following Our Framework.Column, row, tube subset selection, and CURT decomposition. In tensor column, row, tube subset selection, the goal is to find three matrices: a subset C of columns of A, a subset R of rows of A, and a subset T of tubes of A, such that there exists a small tensor U for which U(C,R,T) - A _F^2 ≤ (1+ϵ). We first choose two Gaussian matrices S_1 and S_2 with s_1=s_2=O(k/ϵ) columns, and form a matrix Z_3' ∈ℝ^(s_1s_2) × n^2 with (i,j)-th row equal to the vectorization of (A_1 S_1)_i ⊗ (A_2 S_2)_j. Motivated by the regression problem min_W A_3-WZ_3'_F, we sample d_3= O(s_1 s_2 /ϵ) columns from A_3 and let D_3 denote this selection matrix. There are a few ways to do the sampling depending on the tradeoff between the number of columns and running time, which we describe below. Proceeding iteratively, we write down Z_2' by setting its (i,j)-th row to the vectorization of ( A_1 S_1)_i ⊗ ( A_3 D_3)_j. We then sample d_2 =O(s_1d_3/ϵ) columns from A_2 and let D_2 denote that selection matrix. Finally, we define Z_1' by setting its (i,j)-th row to be the vectorization of (A_2 D_2)_i ⊗ (A_3 D_3)_j. We obtain C=A_1 D_1, R=A_2 D_2 and T= A_3 D_3. For the sampling steps, we can use a generalized matrix column subset selection technique, which extends a column subset selection technique of <cit.> in the context of CUR decompositions to the case when C is not necessarily a subset of the input.This gives O((A)log n) + O(n^2) (k,1/ϵ) time. Alternatively, we can use a technique we develop called tensor leverage score sampling described below, yielding O((A)) + n (k,1/ϵ) time.A body of work in the matrix case has focused on finding the best possible number of columns and rows of a CUR decomposition, and we can ask the same question for tensors. It turns out that if one is given the factorization ∑_i=1^k (U_B)_i ⊗ (V_B)_i ⊗ (W_B)_i of a rank-k tensor B ∈ℝ^n × n × n with U_B , V_B , W_B ∈ℝ^n × k, then one can find a set C of O(k/ϵ) columns, a set R of O(k/ϵ) rows, and a set T of O(k/ϵ) tubes of A, together with a rank-k tensor U for which U(C,R,T) - A_F^2 ≤ (1+ϵ)A-B_F^2. This is based on an iterative argument, where the initial sampling (which needs to be our generalized matrix column subset selection rather than tensor leverage score sampling to achieve optimal bounds) is done with respect to V_B^⊤⊙ W_B^⊤, and then an iterative argument is carried out. Since we show a matching lower bound on the number of columns, rows, tubes and rank of U, these parameters are tight. The algorithm is efficient if one is given a rank-k tensor B which is a (1+O(ϵ))-approximation to A; if not then one can use Theorem <ref> and and this step will be exponential time in k. If one just wants O(klog k + k/ϵ) columns, rows, and tubes, then one can achieve O((A)) + n (k,1/ϵ) time, if one is given B. Column-row, row-tube, tube-column face subset selection, and CURT decomposition.In tensor column-row, row-tube, tube-column face subset selection, the goal is to find three tensors: a subset C∈ℝ^c× n × n of row-tube faces of A, a subset R∈ℝ^n× r × n of tube-column faces of A, and a subset T∈ℝ^n× n× t of column-row faces of A, such that there exists a tensor U∈ℝ^tn × cn × rn with small rank for which U(T_1,C_2,R_3) - A_F^2 ≤ (1+ϵ), where T_1∈ℝ^n× tn denotes the matrix obtained by flattening the tensor T along the first dimension, C_2∈ℝ^n× cn denotes the matrix obtained by flattening the tensor C along the second dimension, and R_3∈ℝ^n× rn denotes the matrix obtained by flattening the tensor T along the third dimension.We solve this problem by first choosing two Gaussian matrices S_1 and S_2 with s_1=s_2=O(k/ϵ) columns, and then forming matrix U_3∈ℝ^n× s_1s_2 with (i,j)-th column equal to (A_1S_1)_i, as well as matrix V_3∈ℝ^n× s_1 s_2 with (i,j)-th column equal to (A_2S_2)_j.Inspiredby the regression problem min_W∈ℝ^n× s_1s_2 V_3 · (W^⊤⊙ U_3^⊤) - A_2_F, we sample d_3= O(s_1 s_2 /ϵ) rows from A_2 and let D_3 ∈ℝ^n× n denote this selection matrix. In other words, D_3 selects d_3 tube-column faces from the original tensor A. Thus, we obtain a small regression problem: min_WD_3 V_3 · (W^⊤⊙ U_3^⊤) - D_3 A_2_F. By retensorizing the objective function, we obtain the problem min_WU_3 ⊗ (D_3 V_3) ⊗ W - A(I,D_3,I) _F. Flattening the objective function along the third dimension, we obtain min_WW · (U_3^⊤⊙ (D_3 V_3)^⊤) -(A(I,D_3,I))_3 _F which has optimal solution (A(I,D_3,I))_3 (U_3^⊤⊙ (D_3 V_3)^⊤)^†. Let W' denote A(I,D_3,I))_3. In the next step, we fix W_2= W'(U_3^⊤⊙ (D_3 V_3)^⊤)^† and U_2=U_3, and consider the objective function min_V U_2 · ( V^⊤⊙ W_2^⊤) - A_1 _F. Applying a similar argument, we obtain V'= (A(D_2,I,I))_2 and U'=( A(I,I,D_1)_1). Let C denote A(D_2,I,I), R denote A(I,D_3,I), and T denote A(I,I,D_1). Overall, this algorithm selects (k,1/ϵ) faces from each dimension.Similar to our column-based CURT decomposition, our face-based CURT decomposition has the property that if one is given the factorization ∑_i=1^k (U_B)_i⊗ (V_B)_i ⊗ (W_B)_i of a rank-k tensor B∈ℝ^n× n× n with U_B,V_B,W_B∈ℝ^n× k which is a (1+O(ϵ))-approximation to A, then one can find a set C of O(k/ϵ) row-tube faces, a set R of O(k/ϵ) tube-column faces, and a set T of O(k/ϵ) column-row faces of A, together with a -k tensor U for which U(T_1,C_2,R_3) -A _F^2 ≤ (1+ϵ).Tensor multiple regression and tensor leverage score sampling. In the above we need to consider standard problems for matrices in the context of tensors. Suppose we are given a matrix A∈ℝ^n_1 × n_2 n_3 and a matrix B= (V^⊤⊙ W^⊤) ∈ℝ^k × n_2n_3 with rows (V_i ⊗ W_i) for an n_2 × k matrix V and n_3 × k matrix W. Using TensorSketch <cit.> one can solve multiple regression min_U U B - A_F without forming B in O(n_2 + n_3) (k,1/ϵ) time, rather than the naïve O(n_2n_3)(k,1/ϵ) time. However, this does not immediately help us if we would like to sample columns of such a matrix B proportional to its leverage scores. Even if we apply TensorSketch to compute a k × k change of basis matrix R in O(n_2 + n_3)(k, log(n_2 n_3)) time, for which the leverage scores of B are (up to a constant factor) the squared column norms of R^-1 B, there are still n_2n_3 leverage scores and we cannot write them all down! Nevertheless, we show we can still sample by them by using that the matrix of interest is formed via a tensor product, which can be rewritten as a matrix multiplication which we never need to explicily materialize.In more detail, for the i-th row e_iR^-1 of R^-1, we create a matrix V^'i by scaling each of the columns of V^⊤ entrywise by the entries of z. The squared norms of e_iR^-1B are exactly the squared entries of (V^'i)W^⊤. We cannot compute this matrix product, but we can first sample a column of it proportional to its squared norm and then sample an entry in that column proportional to its square. To sample a column, we compute G (V^'i)W^⊤ for a Gaussian matrix G with O(log n_3) rows by computing G · V^'i, then computing (G · V^'i) · W^⊤, which is O(n_2 + n_3)(k, log(n_2 n_3)) total time. After sampling a column, we compute the column exactly and sample a squared entry. We do this for each i ∈ [k], first sampling an i proportional to GV^'iW^⊤_F^2, then running the above scheme on that i. The (log n) factor in the running time can be replaced by (k) if one wants to avoid a (log n) dependence in the running time.Entry-wise ℓ_1 low-rank approximation. We consider the problem of entrywise ℓ_1-low rank approximation of an n × n × n tensor A, namely, the problem of finding a rank-k tensor B for which A-B_1 ≤(k, log n), where = inf_rank-kBA-B_1, and where for a tensor A, A_1 = ∑_i,j,k |A_i,j,k|. Our iterative existential argument can be applied in much the same way as for the Frobenius norm. We iteratively flatten A along each of its three dimensions, obtaining A_1, A_2, and A_3 as above, and iteratively build a good rank-k solution B of the form (A_1S_1X_1) ⊗ (A_2 S_2 X_2) ⊗ (A_3 S_3 X_3), where now the S_i are matrices of i.i.d. Cauchy random variables or sparse matrices of Cauchy random variables and the X_i are O(k log k) × k matrices of indeterminates. For a matrix C and a matrix S of i.i.d. Cauchy random variables with k columns, it is known <cit.> that the column span of CS contains a (k log n)-approximate rank-k space with respect to the entrywise ℓ_1-norm for C. In the case of tensors, we must perform an iterative flattening and retensorizing argument to guarantee there exists a tensor B of the form above. Also, if we insist on outputting a rank-k solution as opposed to a bicriteria solution, (A_1 S_1 X_1) ⊗ (A_2 S_2 X_2) ⊗ (A_3 S_3 X_3) -A_1 is not a polynomial of the X_i, and if we introduce sign variables for the n^3 absolute values, the running time of the polynomial solver will be 2^# of variables = 2^Ω(n^3). We perform additional dimensionality reduction by Lewis weight sampling <cit.> from the flattenings to reduce the problem size to (k). This small problem still has Õ(k^3) sign variables, and to obtain a 2^Õ(k^2) running time we relax the reduced problem to a Frobenius norm problem, mildly increasing the approximation factor by another (k) factor.Combining the iterative existential argument with techniques in <cit.>, we also obtain an ℓ_1 CURT decomposition algorithm (which is similar to the Frobenius norm result in Theorem <ref>), which can find O(k) columns, O(k) rows, O(k) tubes, and a tensor U.Our algorithm starts from a given factorization of a rank-k tensor B = U_B ⊗ V_B ⊗ W_B found above. We compute a sampling and rescaling diagonal matrix D_1 according to the Lewis weights of matrix B_1=(V_B^⊤⊙ W_B^⊤), where D_1 has O(k) nonzero entries. Then we iteratively construct B_2, D_2, B_3 and D_3. Finally we have C=A_1 D_1 (selecting O(k) columns from A), R=A_2 D_2 (selecting O(k) rows from A), T=A_3 D_3 (selecting O(k) tubes from A) and tensor U = ( (B_1 D_1)^†) ⊗ ( (B_2 D_2)^† ) ⊗ ( (B_3 D_3)^† ).We have similar results for entry-wise ℓ_p, 1 ≤ p < 2, via analogous techniques.ℓ_1-ℓ_2-ℓ_2 low-rank approximation (sum of Euclidean norms of faces). For an n × n × n tensor A, in ℓ_1-ℓ_2-ℓ_2 low rank approximation we seek a rank-k tensor B for which A - B _v ≤(k,log n), where =inf_rank-k B A - B _v and where A_v = ∑_i ( ∑_j,k (A_i,j,k )^2 )^1/2 for a tensor A. This norm is asymmetric, i.e., not invariant under permutations to its coordinates, and we cannot flatten the tensor along each of its dimensions while preserving its cost. Instead, we embed the problem to a new problem with a symmetric norm. Once we have a symmetric norm, we apply an iterative existential argument. We choose an oblivious sketching matrix (the M-Sketch in <cit.>) S∈ℝ^s× n with s=(k,log n), and reduce the original problem to S (A-B) _v,by losing a small approximation factor. Because s is small, we can then turn the ℓ_1 part of the problem to ℓ_2 by losing another √(s) in the approximation, so that now the problem is a Frobenius norm problem. We then apply our iterative existential argument to the problem S( ∑_i=1^k U^*_i ⊗ (A_2 S_2 X_2)_i ⊗ (A_3 S_3 X_3)_i - A )_F where U^* is a fixed matrix and A = SA, and output a bicriteria solution.ℓ_1-ℓ_1-ℓ_2 low-rank approximation (sum of Euclidean norms of tubes). For an n × n × n tensor A, in the ℓ_1-ℓ_1-ℓ_2 low rank approximation problem we seek a rank-k tensor B for which A - B _u ≤(k,log n), where =inf_rank-k B A - B _u and A_u = ∑_i,j ( ∑_k (A_i,j,k )^2 )^1/2. The main difficulty in this problem is that the norm is asymmetric, and we cannot flatten the tensor along all three dimensions. To reduce the problem to a problem with a symmetric norm, we choose random Gaussian matrices S∈ℝ^n × s with s=O(n). By Dvoretzky's theorem <cit.>, for all tensors A, A S_1 ≈A_u, which reduces our problem to min_rank-k B (A - B)S _1. Via an iterative existential argument, we obtain a generalized version of entrywise ℓ_1 low rank approximation, ((A_1 S_1 X_1) ⊗ (A_2 S_2 X_2) ⊗ (A_3S_3X_3) -A) S_1, where A = AS is an n× n × s size tensor. Finally, we can either use a polynomial system solver to obtain a rank-k solution, or output a bicriteria solution.Weighted low-rank approximation. We also consider weighted low rank approximation. Given an n × n × n tensor A and an n× n × n tensor W of weights, we want to find a rank-k tensor B for which W∘(A-B)_F^2 ≤ (1+ϵ), where = inf_rank-kB W∘(A-B)_F^2 and where for a tensor A, W∘ A _F = (∑_i,j,k W_i,j,k^2 A_i,j,k^2)^1/2. We provide two algorithms based on different assumptions on the weight tensor W. The first algorithm assumes that W has r distinct faces on each of its three dimensions. We flatten A and W along each of its three dimensions, obtaining A_1,A_2,A_3 and W_1,W_2,W_3. Because each W_i has r distinct rows, combining the “guess a sketch” technique from <cit.> with our iterative argument, we can create matrices U_1, U_2, and U_3 in terms of O(rk^2/ϵ) total indeterminates and for which a solution to the objective function W ∘( ∑_i=1^k (U_1)_i ⊗ (U_2)_i ⊗ (U_3)_i - A) _F^2, together with O(r) side constraints, gives a (1+ε)-approximation. We can solve the latter problem in (n) · 2^O(rk^2/ϵ) time. Our second algorithm assumes W has r distinct faces in two dimensions. Via a pigeonhole argument, the third dimension will have at most 2^O(r) distinct faces. We again use O(rk^2/ϵ) variables to express U_1 and U_2, but now express U_3 in terms of these variables, which is necessary since W_3 could have an exponential number of distinct rows, ultimately causing too many variables needed to express U_3 directly. We again arrive at the objective function W ∘( ∑_i=1^k (U_1)_i ⊗ (U_2)_i ⊗ (U_3)_i - A) _F^2, but now have 2^O(r) side constraints, coming from the fact that U_3 is a rational function of the variables created for U_1 and U_2 and we need to clear denominators. Ultimately, the running time is 2^O(r^2k^2/ϵ).Computational Hardness. Our 2^δ k^1-o(1) time hardness for c-approximation in Theorem <ref> is shown via a reduction from approximating MAX-3SAT to approximating MAX-E3SAT, where the latter problem has the property that each clause in the satisfiability instance has exactly 3 literals (in MAX-3SAT some clauses may have 2 literals). Then, a reduction <cit.> from approximating MAX-E3SAT to approximating MAX-E3SAT(B) is performed, for a constant B which provides an upper bound on the number of clauses each literal can occur in. Given an instance ϕ to MAX-E3SAT(B), we create a 3rd order tensor T as Håstad does using ϕ <cit.>. While Håstad's reduction guarantees that the rank of T is at most r if ϕ is satisfiable, and at least r+1 otherwise, we can show that if ϕ is not satisfiable then its rank is at least the minimal size of a set of variables which is guaranteed to intersect every unsatisfied clause in any unsatisfiable assignment. Since if ϕ is not satisfiable, there are at least a linear fraction of clauses in ϕ that are unsatisfied under any assignment by the inapproximability of MAX-E3SAT(B), and since each literal occurs in at most B clauses for a constant B, it follows that the rank of T when ϕ is not satisfiable is at least c_0r for a constant c_0 > 1. Further, under , our reduction implies one cannot approximate MAX-E3SAT(B), and thus approximate the rank of a tensor up to a factor c_0, in less than 2^δ k^1-o(1) time. We need the near-linear size reduction of -to -of <cit.> to get our strongest result.The 2^Ω(1/ϵ^1/4) time hardness for (1+ϵ)-approximation for rank-1 tensors in Theorem <ref> strengthens the NP-hardness for rank-1 tensor computation in Section 7 of <cit.>, where instead of assuming the NP-hardness of the Clique problem, we assume . Also, the proof in <cit.> did not explicitly bound the approximation error; we do this for a (1/ϵ)-sized tensor (which can be padded with 0s to a (n)-sized tensor) to rule out (1+ϵ)-approximation in 2^o(1/ϵ^1/4) time.The same hard instance above shows, assuming , that 2^Ω(1/ϵ^1/2) time is necessary for (1+ε)-approximation to the spectral norm of a symmetric rank-1 tensor (see Section <ref> and Section <ref>).Assuming , the 2^1/ϵ^1-o(1)-hardness <cit.> for matrix ℓ_1-low rank approximation gives the same hardness for tensor entry-wise ℓ_1 and ℓ_1-ℓ_1-ℓ_2 low rank approximation. Also, under , we strengthen the NP-hardness in <cit.> to a 2^1/ϵ^Ω(1)-hardness for ℓ_1-ℓ_2-low rank approximation of a matrix, which gives the same hardness for tensor ℓ_1-ℓ_2-ℓ_2 low rank approximation.Hard Instance. We extend the previous matrix CUR hard instance <cit.> to 3rd order tensors by planting multiple rotations of the hard instance for matrices into a tensor. We show C must select Ω(k/ϵ) columns from A, R must select Ω(k/ϵ) rows from A, and T must select Ω(k/ϵ) tubes from A. Also the tensor U must have rank at least k. This generalizes to q-th order tensors.Optimal matrix CUR decomposition. We also improve the (A)log n + (n+d) (log n, k, 1/ϵ) running time of <cit.> for CUR decomposition of A ∈ℝ^n × d to (A) + (n+d)(k,1/ϵ), while selecting the optimal number of columns, rows, and a rank-k matrix U. Using <cit.>, we find a matrix U with k orthonormal columns in (A)+n(k/ε) time for which min_VUV-A_F^2≤ (1+ε)A-A_k_F^2. Let s_1=O(k/ε^2) and S_1∈ℝ^s_1× n be a sampling/rescaling matrix by the leverage scores of U. By strengthening the affine embedding analysis of <cit.> to leverage score sampling (the analysis of <cit.> gives a weaker analysis for affine embeddings using leverage scores which does not allow approximation in the sketch space to translate to approximation in the original space), with probability at least 0.99, for all X' which satisfy S_1UX'-S_1A_F^2≤ (1+ε')min_XS_1UX-S_1A_F^2, we have UX'-A_F^2≤ (1+ε)min_XUX-A_F^2, where ε'=0.0001ε. Applying our generalized row subset selection procedure, we can find Y,R for which S_1UYR-S_1A_F^2≤ (1+ε')min_XS_1UX-S_1A_F^2, where R contains O(k/ε')=O(k/ε) rescaled rows of S_1A. A key point is that rescaled rows of S_1A are also rescaled rows of A. Then, UYR-A_F^2≤ (1+ε)min_XUX-A_F^2. Finding Y,R can be done in d(s_1/ε)=d(k/ε) time. Now set V=YR. We can choose S_2 to be a sampling/rescaling matrix, and thenfind C,Z for which CZVS_2-AS_2_F^2≤ (1+ε')min_X XVS_2-AS_2_F^2 in a similar way, where C contains O(k/ε) rescaled columns of AS_2, and thus also of A. We thus haveCZYR-A_F^2≤ (1+O(ε))A-A_k_F^2.Distributed and streaming settings. Since our algorithms use linear sketches, they are implementable in distributed and streaming models. We use random variables with limited independence to succinctly store the sketching matrices <cit.>. Extension to other notions of tensor rank. This paper focuses on the standard CP rank, or canonical rank, of a tensor. As mentioned, due to border rank issues, the best rank-k solution does not exist in certain cases. There are other notions of tensor rank considered in some applications which do not suffer from this problem, e.g., the tucker rank <cit.>, and the train rank <cit.>). We also show observe that our techniques can be applied to these notions of rank. §.§ Comparison to <cit.>In <cit.>, the authors show for a third order n_1 × n_2 × n_3 tensor Ahow to find a rank-k tensor B for whichA-B_F^2 ≤ 5 in (n_1 n_2 n_3) exp((k)) time. They generalize this to q-th order tensors to find a rank-k tensor B for which A-B_F^2 = O(q) in (n_1 n_2 ⋯ n_q) exp((qk)) time. In contrast, we obtain a rank-k tensor B for which A-B_F^2 ≤ (1+ϵ) in (A) + n ·(k/ϵ) + exp((k^2/ϵ) (q)) time for every order q. Thus, we obtain a (1+ϵ) instead of an O(q) approximation. The O(q) approximation in <cit.> seems inherent since the authors apply triangle inequality q times, each time losing a constant factor. This seems necessary since their argument is based on the span of the top k principal components in the SVD in each flattening separately containing a good space to project onto for a given mode. In contrast,our iterative existential argument chooses the space to project onto in successive modes adaptively as a function of spaces chosen for previous modes, and thus we obtain a (1+ϵ)^O(q) = (1+O(ϵ q))-approximation, which becomes a (1+ϵ)-approximation after replacing ϵ with ϵ/q.Also, importantly, our algorithm runs in (A) + n ·(k/ϵ) + exp((k^2/ϵ) (q))time and there are multiple hurdles we overcome to achieve this, as described in Section <ref> above.§.§ An Algorithm and a RoadmapRoadmap Section <ref> introduces notation and definitions. Section <ref> includes several useful tools. We provide our Frobenius norm low rank approximation algorithms in Section <ref>. Section <ref> extends our results to general q-th order tensors. Section <ref> has our results for entry-wise ℓ_1 norm low rank approximation. Section <ref> has our results for entry-wise ℓ_p norm low rank approximation. Section <ref> has our results for weighted low rank approximation.Section <ref> has our results for asymmetric norm low rank approximation algorithms. We present our hardness results in Section <ref> and Section <ref>. Section <ref> and Section <ref> extend the results to distributed and streaming settings. Section <ref> extends our techniques from tensor rank to other notions of tensor rank including tensor tucker rank and tensor train rank. § NOTATION For an n∈ℕ_+, let [n] denote the set {1,2,⋯,n}.For any function f, we define O(f) to be f·log^O(1)(f). In addition to O(·) notation, for two functions f,g, we use the shorthand f≲ g (resp. ≳) to indicate that f≤ C g (resp. ≥) for an absolute constant C. We use f g to mean cf≤ g≤ Cf for constants c,C. For a matrix A, we use A_2 to denote the spectral norm of A. For a tensor A, let A and A_2 (which we sometimes use interchangeably) denote the spectral norm of tensor A,A= sup_x,y,z ≠ 0|A(x,y,z)|/ x · y · z .Let A_F denote the Frobenius norm of a matrix/tensor A, i.e., A_F is the square root of sum of squares of all the entries of A. For 1≤ p<2, we use A _p to denote the entry-wise ℓ_p-norm of a matrix/tensor A, i.e., A_p is the p-th root of the sum of p-th powers of the absolute values of the entries of A. A_1 will be an important special case of A_p, which corresponds to the sum of absolute values of all of the entries. Let (A) denote the number of nonzero entries of A. Let (A) denote the determinant of a square matrix A. Let A^⊤ denote the transpose of A. Let A^† denote the Moore-Penrose pseudoinverse of A. Let A^-1 denote the inverse of a full rank square matrix.For a 3rd order tensor A ∈ℝ^n× n × n, its x-mode fibers are called column fibers (x=1), row fibers (x=2) and tube fibers (x=3). For tensor A, we use A_*,j,l to denote its (j,l)-th column, we use A_i,*,l to denote its (i,l)-th row, and we use A_i,j,* to denote its (i,j)-th tube.A tensor A is symmetric if and only if for any i,j,k, A_i,j,k = A_i,k,j = A_j,i,k = A_j,k,i = A_k,i,j = A_k,j,i.For a tensor A∈ℝ^n_1 × n_2 × n_3, we use ⊤ to denote rotation (3 dimensional transpose) so that A^⊤∈ℝ^n_3× n_1 × n_2.For a tensor A∈ℝ^n_1 × n_2 × n_3 and matrix B∈ℝ^n_3 × k, we define the tensor-matrix dot product to be A · B ∈ℝ^n_1× n_2 × k.We use ⊗ to denote outer product, ∘ to denote entrywise product, and · to denote dot product. Given two column vectors u,v ∈ℝ^n, let u⊗ v∈ℝ^n× n and (u ⊗ v)_i,j = u_i · v_j, u^⊤ v= ∑_i=1^n u_i v_i ∈ℝ and (u ∘ v )_i = u_i v_i.Given q vectors u_1∈ℝ^n_1, u_2 ∈ℝ^n_2, ⋯, u_q∈ℝ^n_q, we use u_1⊗ u_2 ⊗⋯⊗ u_q to denote an n_1 × n_2 ×⋯× n_q tensor such that, for each (j_1,j_2,⋯,j_q)∈ [n_1]× [n_2]×⋯× [n_q],( u_1⊗ u_2 ⊗⋯⊗ u_q )_j_1,j_2,⋯,j_q = (u_1)_j_1 (u_2)_j_2⋯ (u_q)_j_q,where (u_i)_j_i denotes the j_i-th entry of vector u_i.Given a tensor A∈ℝ^n_1 × n_2 ×⋯× n_q, let (A)∈ℝ^1×∏_i=1^q n_i be a row vector, such that the t-th entry of (A) is A_j_1,j_2,⋯,j_q where t=(j_1-1)∏_i=2^q n_i+(j_2-1)∏_i=3^q n_i+⋯+(j_q-1-1) n_q+j_q.For example if u=[ 1; 2 ],v=[ 3; 4; 5 ] then (u⊗ v)=[34568 10 ].Given q matrices U_1∈ℝ^n_1 × k, U_2 ∈ℝ^n_2 × k, ⋯, U_q∈ℝ^n_q× k, we use U_1 ⊗ U_2 ⊗⋯⊗ U_q to denote an n_1 × n_2 ×⋯× n_q tensor which can be written as,U_1 ⊗ U_2 ⊗⋯⊗ U_q = ∑_i=1^k (U_1)_i ⊗ (U_2)_i ⊗⋯⊗ (U_q)_i ∈ℝ^n_1 × n_2 ×⋯× n_q,where (U_j)_i denotes the i-th column of matrix U_j ∈ℝ^n_j × k.Given q matrices U_1 ∈ℝ^k× n_1, U_2∈ℝ^k× n_2, ⋯, U_q ∈ℝ^k× n_q, we use U_1 ⊙ U_2 ⊙⋯⊙ U_q to denote a k ×∏_j=1^q n_j matrix where the i-th row of U_1 ⊙ U_2 ⊙⋯⊙ U_q is the vectorization of (U_1)^i⊗ (U_2)^i ⊗⋯⊗ (U_q)^i, i.e.,U_1 ⊙ U_2 ⊙⋯⊙ U_q = [ ( (U_1)^1 ⊗ (U_2)^1 ⊗⋯⊗ (U_q)^1 ); ( (U_1)^2 ⊗ (U_2)^2 ⊗⋯⊗ (U_q)^2 ); ⋯; ( (U_1)^k ⊗ (U_2)^k ⊗⋯⊗ (U_q)^k ) ]∈ℝ^k×∏_j=1^q n_j.where (U_j)^i∈ℝ^n_j denotes the i-th row of matrix U_j∈ℝ^k× n_j.Suppose we are given three matrices U∈ℝ^n_1 × k, V∈ℝ_n_2 × k, W∈ℝ^n_3 × k. Let tensor A∈ℝ^n_1 × n_2 × n_3 denote U⊗ V ⊗ W. Let A_1 ∈ℝ^n_1 × n_2 n_3 denote a matrix obtained by flattening tensor A along the 1st dimension. Then A_1 = U · B, where B = V^⊤⊙ W^⊤∈ℝ^k× n_2 n_3 denotes the matrix for which the i-th row is ( V_i ⊗ W_i ), ∀ i ∈ [k]. We let the “flattening” be the operation that obtains A_1 by A. Given A_1 = U · B, we can obtain tensor A by unflattening/retensorizing A_1. We let “retensorization” be the operation that obtains A from A_1. Similarly, let A_2∈ℝ^n_2 × n_1 n_3 denote a matrix obtained by flattening tensor A along the 2nd dimension, so A_2 = V· C, where C= W^⊤⊙ U^⊤∈ℝ^k× n_1 n_3 denotes the matrix for which the i-th row is (W_i ⊗ U_i), ∀ i∈ [k]. Let A_3 ∈ℝ^n_3 × n_1 n_2 denote a matrix obtained by flattening tensor A along the 3rd dimension. Then, A_3 = W · D, where D= U^⊤⊙ V^⊤∈ℝ^k× n_1 n_2 denotes the matrix for which the i-th row is (U_i ⊗ V_i), ∀ i∈ [k].Given tensor A∈ℝ^n_1 × n_2 × n_3 and three matrices B_1∈ℝ^n_1× d_1, B_2 ∈ℝ^n_2× d_2, B_3∈ℝ^n_3 × d_3, we define tensors A(B_1,I,I)∈ℝ^d_1× n_2× n_3, A(I,B_2,I)∈ℝ^n_1 × d_2 × n_3, A(I,I,B_3)∈ℝ^n_1 × n_2 × d_3, A(B_1,B_2,I)∈ℝ^d_1 × d_2 × n_3, A(B_1,B_2,B_3)∈ℝ^d_1 × d_2 × d_3 as follows,A(B_1,I,I)_i,j,l = ∑_i'=1^n_1 A_i',j,l (B_1)_i',i,∀ (i,j,l) ∈ [d_1]× [n_2] × [n_3] A(I,B_2,I)_i,j,l = ∑_j'=1^n_2 A_i,j',l (B_2)_j',j,∀ (i,j,l) ∈ [n_1]× [d_2] × [n_3] A(I,I,B_3)_i,j,l = ∑_l'=1^n_3 A_i,j,l' (B_3)_l',l,∀ (i,j,l) ∈ [n_1]× [n_2] × [d_3] A(B_1,B_2,I)_i,j,l = ∑_i'=1^n_1∑_j'=1^n_2 A_i',j',l (B_1)_i',i (B_2)_j',j,∀ (i,j,l) ∈ [d_1]× [d_2] × [n_3] A(B_1,B_2,B_3)_i,j,l = ∑_i'=1^n_1∑_j'=1^n_2∑_l'=1^n_3 A_i',j',l' (B_1)_i',i (B_2)_j',j (B_3)_l',l,∀ (i,j,l) ∈ [d_1]× [d_2] × [d_3] Note that B_1^⊤ A = A(B_1,I,I), A B_3 = A(I,I,B_3) and B_1^⊤ A B_3 = A(B_1,I,B_3). In our paper, if ∀ i∈[3], B_i is either a rectangular matrix or a symmetric matrix, then we sometimes use A(B_1,B_2,B_3) to denote A(B_1^⊤,B_2^⊤,B_3^⊤) for simplicity. Similar to the (·,·,·) operator on 3rd order tensors, we can define the (·,·,⋯,·) operator on higher order tensors.For the matrix case, -k A'minA-A'_F^2 always exists. However, this is not true for tensors <cit.>. For convenience, we redefine the notation ofand min.Given tensor A∈ℝ^n_1× n_2× n_3,k>0, if -k A'minA-A'_F^2 does not exist, then we define =-k A'infA-A'_F^2+γ for sufficiently small γ>0, which can be an arbitrarily small positive function of n. We let -k A'minA-A'_F^2 be the value of , and we let -k A'minA-A'_F^2 be a -k tensor A_k∈ℝ^n_1× n_2× n_3 which satisfies A-A_k_F^2=. § PRELIMINARIESSection <ref> provides the definitions for Subspace Embeddings and Approximate Matrix Product. We introduce the definition for Tensor-CURT decomposition in Section <ref>. Section <ref> presents a tool which we call a “polynomial system verifier”. Section <ref> introduces a tool which is able to determine the minimum nonzero value of the absolute value of a polynomial evaluated on a set, provided the polynomial is never equal to 0 on that set. Section <ref> shows how to relax an ℓ_p problem to an ℓ_2 problem. We provide definitions for CountSketch and Gaussian transforms in Section <ref>. We present Cauchy and p-stable transforms in Section <ref>. We introduce leverage scores and Lewis weights in Section <ref> and Section <ref>. Finally, we explain an extension of CountSketch, which is called TensorSketch in Section <ref>.§.§ Subspace Embeddings and Approximate Matrix ProductA (1±ϵ) ℓ_2-subspace embedding for the column space of an n× d matrix A is a matrix S for which for all x∈ℝ^d, SA x_2^2 = (1±ϵ)A x_2^2. Let 0<ϵ<1 be a given approximation parameter. Given matrices A and B, where A and B each have n rows, the goal is to output a matrix C so that A^⊤ B - C_F ≤ϵ A _FB _F. Typically C has the form A^⊤ S^⊤ S B, for a random matrix S with a small number of rows. See, e.g., Lemma 32 of <cit.> for a number of example matrices S with O(ϵ^-2) rows for which this property holds. §.§ Tensor CURT decompositionWe first review matrix CUR decompositions:Given a matrix A∈ℝ^n× d, we choose C∈ℝ^n× c to be a subset of columns of A and R∈ℝ^r × n to be a subset of rows of A. If there exists a matrix U∈ℝ^c× r such that A can be written as,CUR = A,then we say C,U,R is matrix A's CUR decomposition.Given a matrix A∈ℝ^n× d, a parameter k≥ 1, an approximation ratio α >1, and a norm _ξ, we choose C∈ℝ^n× c to be a subset of columns of A and R∈ℝ^r × n to be a subset of rows of A. Then if there exists a matrix U∈ℝ^c× r such that,CUR - A _ξ≤αmin_-k A_k A_k - A_ξ,where _ξ can be operator norm, Frobenius norm or Entry-wise ℓ_1 norm, we say that C, U, R is matrix A's approximate CUR decomposition, and sometimes just refer to this as a CUR decomposition.Given matrix A∈ℝ^m× n, integer k, and matrix C∈ℝ^m× r with r>k, we define the matrix Π_C,k^ξ(A) ∈ℝ^m× n to be the best approximation to A (under the ξ-norm) within the column space of C of rank at most k; so, Π_C,k^ξ(A) ∈ℝ^m× n minimizes the residual A - A_ξ, over all A∈ℝ^m× n in the column space of C of rank at most k. We define the following notion of tensor-CURT decomposition. Given a tensor A∈ℝ^ n_1 × n_2 × n_3, we choose three sets of pair of coordinates S_1 ⊆ [n_2] × [n_3], S_2 ⊆ [n_1] × [n_3], S_3 ⊆ [n_1] × [n_2]. We define c=|S_1|, r=|S_2| and t=|S_3|. Let C ∈ℝ^ n_1 × c denote a subset of columns of A, R ∈ℝ^ n_2 × r denote a subset of rows of A, and T ∈ℝ^ n_3 × t denote a subset of tubes of A. If there exists a tensor U∈ℝ^c × r × t such that A can be written as( ( ( U · T^⊤ )^⊤· R^⊤ )^⊤· C^⊤ )^⊤ = A,or equivalently,U(C,R,T) = A,or equivalently,∀ (i,j,l) ∈ [n_1] × [n_2] × [n_3], A_i,j,l = ∑_u_1=1^c∑_u_2=1^r∑_u_3=1^t U_u_1,u_2,u_3 C_i,u_1 R_j,u_2 T_l,u_3,then we say C,U,R, T is tensor A's CURT decomposition. Given a tensor A∈ℝ^ n_1 × n_2 × n_3, for some k≥ 1, for some approximation α>1, for some norm _ξ, we choose three sets of pair of coordinates S_1 ⊆ [n_2] × [n_3], S_2 ⊆ [n_1] × [n_3], S_3 ⊆ [n_1] × [n_2]. We define c=|S_1|, r=|S_2| and t=|S_3|. Let C ∈ℝ^ n_1 × c denote a subset of columns of A, R ∈ℝ^ n_2 × r denote a subset of rows of A, and T ∈ℝ^ n_3 × t denote a subset of tubes of A. If there exists a tensor U∈ℝ^c × r × t such thatU(C, R, T ) - A _ξ≤αmin_-k A_k A_k - A _ξ,where _ξ is operator norm, Frobenius norm or Entry-wise ℓ_1 norm, then we refer to C, U, R, T as an approximate CUR decomposition of A, and sometimes just refer to this as a CURT decomposition of A. Recently, <cit.> studied a very different face-based tensor-CUR decomposition, which selects faces from tensors rather than columns. To achieve their results, <cit.> need to make several incoherence assumptions on the original tensor. Their sample complexity depends on log n, and they only sample two of the three dimensions. We will provide more general face-based tensor CURT decompositions. Given a tensor A∈ℝ^ n_1 × n_2 × n_3, we choose three sets of coordinates S_1 ⊆ [n_1], S_2 ⊆ [n_2], S_3 ⊆ [n_3]. We define c=|S_1|, r=|S_2| and t=|S_3|. Let C ∈ℝ^ c× n_2 × n_3 denote a subset of row-tube faces of A, R ∈ℝ^ n_1 × r × n_3 denote a subset of column-tube faces of A, and T ∈ℝ^ n_1 × n_2 × t denote a subset of column-row faces of A. Let C_2∈ℝ^n_2 × cn_3 denote the matrix obtained by flattening the tensor C along the second dimension. Let R_3∈ℝ^n_3 × rn_1 denote the matrix obtained by flattening the tensor R along the third dimension. Let T_1 ∈ℝ^n_1 × tn_2 denote the matrix obtained by flattening the tensor T along the first dimension. If there exists a tensor U∈ℝ^t n_2 × cn_3 × r n_1 such that A can be written as∑_i=1^t n_2∑_j=1^cn_3∑_l=1^rn_1 U_i,j,l (T_1)_l⊗ (C_2)_i⊗ (R_3)_j = A, U(T_1,C_2,R_3) = A,or equivalently,∀ (i',j',l') ∈ [n_1] × [n_2] × [n_3], A_i,j,l =∑_i=1^t n_1∑_j=1^cn_3∑_l=1^rn_2U_i,j,l (T_1)_i',i (C_2)_j',j (R_3)_l',l ,then we say C,U,R, T is tensor A's (face-based) CURT decomposition. Given a tensor A∈ℝ^ n_1 × n_2 × n_3, for some k≥ 1, for some approximation α>1, for some norm _ξ,we choose three sets of coordinates S_1 ⊆ [n_1], S_2 ⊆ [n_2], S_3 ⊆ [n_3]. We define c=|S_1|, r=|S_2| and t=|S_3|. Let C ∈ℝ^ c× n_2 × n_3 denote a subset of row-tube faces of A, R ∈ℝ^ n_1 × r × n_3 denote a subset of column-tube faces of A, and T ∈ℝ^ n_1 × n_2 × t denote a subset of column-row faces of A. Let C_2∈ℝ^n_2 × cn_3 denote the matrix obtained by flattening the tensor C along the second dimension. Let R_3∈ℝ^n_3 × rn_1 denote the matrix obtained by flattening the tensor R along the third dimension. Let T_1 ∈ℝ^n_1 × tn_2 denote the matrix obtained by flattening the tensor T along the first dimension.If there exists a tensor U∈ℝ^t n_2 × cn_3 × r n_1 such thatU(T_1,C_2,R_3 ) - A _ξ≤αmin_-k A_k A_k - A _ξ,where _ξ is operator norm, Frobenius norm or Entry-wise ℓ_1 norm, then we refer to C, U, R, T as an approximate CUR decomposition of A, and sometimes just refer to this as a (face-based) CURT decomposition of A. §.§ Polynomial system verifier We use the polynomial system verifiers independently developed by Renegar <cit.> and Basu et al. <cit.>.Given a real polynomial system P(x_1, x_2, ⋯, x_v) having v variables and m polynomial constraints f_i (x_1, x_2, ⋯, x_v) Δ_i 0, ∀ i ∈ [m], where Δ_i is any of the “standard relations”: { >, ≥, =, ≠, ≤, < }, let d denote the maximum degree of all the polynomial constraints and let H denote the maximum bitsize of the coefficients of all the polynomial constraints. Then in(m d)^O(v)(H),time one can determine if there exists a solution to the polynomial system P.Recently, this technique has been used to solve a number of low-rank approximation and matrix factorization problems <cit.>. §.§ Lower bound on the cost of a polynomial systemAn important result we use is the following lower bound on the minimum value attained by a polynomial restricted to a compact connected component of a basic closed semi-algebraic subset of ℝ^v. Let T={ x∈ℝ^v | f_1(x) ≥ 0, ⋯, f_ℓ(x) ≥ 0,f_ℓ+1(x) = 0, ⋯, f_m(x) = 0 } be defined by polynomials f_1, ⋯, f_m ∈ℤ[x_1, ⋯, x_v ] with n ≥ 2, degrees bounded by an even integer d, and coefficients of absolute value at most H, and let C be a compact connected (in the topological sense) component of T. Let g ∈ℤ[x_1, ⋯, x_v] be a polynomial of degree at most d and coefficients of absolute value bounded by H. Then, the minimum value that g takes over C satisfies that if it is not zero, then its absolute value is greater than or equal to(2^4-v/2H d^v)^-v 2^v d^v,where H = max{ H, 2v+2m}.While the above theorem involves notions from topology, we shall apply it in an elementary way. Namely, in our setting T will be bounded and so every connected component, which is by definition closed, will also be bounded and therefore compact. As the connected components partition T the theorem will just be applied to give a global minimum value of g on T provided that it is non-zero. §.§ Frobenius norm and ℓ_2 relaxation Given matrices A∈ℝ^n× d, B∈ℝ^n× p, and C∈ℝ^q× d, let the SVD of B be B=U_BΣ_B V_B^⊤ and the SVD of C be C=U_CΣ_C V_C^⊤. Then,B^† ( U_B U_B^⊤ A V_C C_C^⊤ )_k C^† = -k X∈ℝ^p× qmin A - B X C _F,where (U_B U_B^⊤ A V_C V_C^⊤)_k ∈ℝ^p× q is of rank at most k and denotes the best rank-k approximation to U_B U_B^⊤ A V_C V_C^⊤∈ℝ^p× d in Frobenius norm. [ℓ_2 relaxation of ℓ_p-regression] Let p∈ [1,2). For any A∈ℝ^n× d and b∈ℝ^n, define x^* =x∈ℝ^dmin A x - b_p and x'=x ∈ℝ^dmin A x - b_2. Then,Ax^* -b_p ≤ A x' - b_p ≤ n^1/p-1/2· A x^* - b_p.[(Matrix) Frobenius norm relaxation of ℓ_p-low rank approximation] Let p∈ [1,2) and for any matrix A∈ℝ^n× d, define A^* = -k B∈ℝ^n× dmin B - A_p and A' =-k B ∈ℝ^n× dmin B - A _F. ThenA^* - A _p ≤ A'- A _p ≤ (nd)^1/p-1/2 A^* - A _p.[(Tensor) Frobenius norm relaxation of ℓ_p-low rank approximation] Let p∈ [1,2) and for any matrix A∈ℝ^n_1 × n_2 × n_3, defineA^* = -k B∈ℝ^n_1× n_2 × n_3min B - A_p andA' =-k B ∈ℝ^n_1× n_2 × n_3min B - A _F.ThenA^* - A _p ≤ A'- A _p ≤ (n_1 n_2 n_3)^1/p-1/2 A^* - A _p. §.§ CountSketch and Gaussian transforms A CountSketch transform is defined to be Π=σ·Φ D∈ℝ^m× n. Here, σ is a scalar, D is an n× n random diagonal matrix with each diagonal entry independently chosen to be +1 or -1 with equal probability, and Φ∈{0,1}^m× n is an m× n binary matrix with Φ_h(i),i=1 and all remaining entries 0, where h:[n]→ [m] is a random map such that for each i∈ [n], h(i) = j with probability 1/m for each j ∈ [m]. For any matrix A∈ℝ^n× d, Π A can be computed in O((A)) time. For any tensor A∈ℝ^n× d_1 × d_2, Π A can be computed in O((A)) time. Let Π_1, Π_2, Π_3 denote three CountSktech transforms. For any tensor A∈ℝ^n_1 × n_2 × n_3, A(Π_1,Π_2,Π_3) can be computed in O((A)) time.If the above scalar σ is not specified in the context, we assume the scalar σ to be 1. Let S=σ· G ∈ℝ^m× n where σ is a scalar, and each entry of G∈ℝ^m× n is chosen independently from the standard Gaussian distribution. For any matrix A∈ℝ^n× d, SA can be computed in O(m ·(A)) time. For any tensor A∈ℝ^n × d_1 × d_2, SA can be computed in O(m ·(A)) time.If the above scalar σ is not specified in the context, we assume the scalar σ to be 1/√(m). In most places, we can combine CountSketch and Gaussian transforms to achieve the following: Let S' = S Π, where Π∈ℝ^t× n is the CountSketch transform (defined in Definition <ref>) and S∈ℝ^m × t is the Gaussian transform (defined in Definition <ref>). For any matrix A∈ℝ^n× d, S'A can be computed in O((A) + dtm^ω-2) time, where ω is the matrix multiplication exponent.Given matrices A∈ℝ^n× r,B∈ℝ^n× d, and (A)=k, let m=(k/ε), S∈ℝ^m× n be a sparse embedding matrix (Definition <ref>) with scalar σ=1. Then with probability at least 0.999, ∀ X∈ℝ^r× d, we have(1-ε)·AX-B_F^2≤S(AX-B)_F^2≤ (1+ε)AX-B_F^2. Let m=Ω(k/ε), S=1/√(m)· G, where G∈ℝ^m× n is a random matrix where each entry is an i.i.d Gaussian N(0,1). Then with probability at least 0.998, S satisfies (1±1/8) Subspace Embedding (Definition <ref>) for any fixed matrix C∈ℝ^n× k, and it also satisfies O(√(ε/k)) Approximate Matrix Product (Definition <ref>) for any fixed matrix A and B which has the same number of rows.Let m=Ω(k^2+k/ε), Π∈ℝ^m× n, where Π is a sparse embedding matrix (Definition <ref>) with scalar σ=1, then with probability at least 0.998,S satisfies (1±1/8) Subspace Embedding (Definition <ref>) for any fixed matrix C∈ℝ^n× k, and it also satisfies O(√(ε/k)) Approximate Matrix Product (Definition <ref>) for any fixed matrix A and B which has the same number of rows.Let m_2=Ω(k^2+k/ε), Π∈ℝ^m_2× n, where Π is a sparse embedding matrix (Definition <ref>) with scalar σ=1. Let m_1=Ω(k/ε), S=1/√(m_1)· G, where G∈ℝ^m_1× m_2 is a random matrix where each entry is an i.i.d Gaussian N(0,1). Let S'=SΠ. Then with probability at least 0.99,S' is a (1±1/3) Subspace Embedding (Definition <ref>) for any fixed matrix C∈ℝ^n× k, and it also satisfies O(√(ε/k)) Approximate Matrix Product (Definition <ref>) for any fixed matrix A and B which have the same number of rows.Given A∈ℝ^n× k,B∈ℝ^n× d, suppose S∈ℝ^m× n is such that S is a (1±1/√(2)) Subspace Embedding for A, and satisfies O(√(ε/k)) Approximate Matrix Product for matrices A and C where C with n rows, where C depends on A and B. IfX=min_X∈ℝ^k× dSAX-SB_F^2,thenAX-B_F^2≤ (1+ε)min_X∈ℝ^k× dAX-B_F^2. §.§ Cauchy and p-stable transformsLet S = σ· C ∈ℝ^m× n where σ is a scalar, and each entry of C∈ℝ^m× n is chosen independently from the standard Cauchy distribution. For any matrix A∈ℝ^n× d, SA can be computed in O(m·(A)) time. Let Π = σ· S C∈ℝ^m × n, where σ is a scalar, S∈ℝ^m× n has each column chosen independently and uniformly from the m standard basis vectors of ℝ^m, and C∈ℝ^n× n is a diagonal matrix with diagonals chosen independently from the standard Cauchy distribution.For any matrix A∈ℝ^n× d, Π A can be computed in O((A)) time. For any tensor A∈ℝ^n× d_1 × d_2, Π A can be computed in O((A)) time. Let Π_1 ∈ℝ^m_1 × n_1,Π_2∈ℝ^m_2 × n_2,Π_3∈ℝ^m_3 × n_3 denote three sparse Cauchy transforms. For any tensor A∈ℝ^n_1× n_2× n_3, A(Π_1,Π_2,Π_3)∈ℝ^m_1 × m_2 × m_3 can be computed in O((A)) time.Let p∈ (1,2). Let S = σ· C ∈ℝ^m× n, where σ is a scalar, and each entry of C∈ℝ^m× n is chosen independently from the standard p-stable distribution. For any matrix A∈ℝ^n× d, SA can be computed in O(m (A) ) time.Let p∈ (1,2). Let Π = σ· S C∈ℝ^m× n, where σ is a scalar, S∈ℝ^m× n has each column chosen independently and uniformly from the m standard basis vectors of ℝ^m, and C∈ℝ^n× n is a diagonal matrix with diagonals chosen independently from the standard p-stable distribution. For any matrix A∈ℝ^n× d, Π A can be computed in O((A)) time. For any tensor A∈ℝ^n× d_1 × d_2, Π A can be computed in O((A)) time. Let Π_1 ∈ℝ^m_1 × n_1,Π_2∈ℝ^m_2 × n_2,Π_3∈ℝ^m_3 × n_3 denote three sparse p-stable transforms. For any tensor A∈ℝ^n_1× n_2× n_3, A(Π_1,Π_2,Π_3)∈ℝ^m_1 × m_2 × m_3 can be computed in O((A)) time. §.§ Leverage scores Let U∈ℝ^n× k have orthonormal columns, and let p_i = u_i^2 / k, where u_i^2 =e_i^⊤ U _2^2 is the i-th leverage score of U. Given A∈ℝ^n× d with rank k, let U∈ℝ^n× k be an orthonormal basis of the column space of A, and for each i let p_i be the squared row norm of the i-th row of U, i.e., the i-th leverage score. Let k· p_i denote the i-th leverage score of U scaled by k. Let β>0 be a constant and q=(q_1, ⋯,q_n) denote a distribution such that, for each i∈ [n], q_i ≥β p_i. Let s be a parameter. Construct an n× s sampling matrix B and an s× s rescaling matrix D as follows. Initially, B = 0^n× s and D = 0^s× s. For each column j of B, D, independently, and with replacement, pick a row index i∈ [n] with probability q_i, and set B_i,j=1 and D_j,j=1/√(q_i s). We denote this procedure Leverage score sampling according to the matrix A.§.§ Lewis weightsWe follow the exposition of Lewis weights from <cit.>.For a matrix A, let a_i denote the i^th row of A, where a_i(=(A^i)^⊤ ) is a column vector. The statistical leverage score of a row a_i isτ_i(A) def= a_i^⊤ (A^⊤ A)^-1 a_i =(A^⊤ A )^-1/2 a_i _2^2.For a matrix A and norm p, the ℓ_p Lewis weights w are the unique weights such that for each row i we havew_i = τ_i ( W^1/2-1/p A ).or equivalently,a_i^⊤ (A^⊤ W^1-2/p A)^-1 a_i = w_i^2/p. Given a matrix A∈ℝ^n× d, n≥ d, for any constant C>0,4>p≥ 1, there is an algorithm which can compute C-approximate ℓ_p Lewis weights for every row i of A in O((nnz(A)+d^ωlog d)log n) time, where ω < 2.373 is the matrix multiplication exponent<cit.>.Given matrix A∈ℝ^n× d (n≥ d) with ℓ_p (4>p≥ 1) Lewis weights w, for any set of sampling probabilities p_i, ∑_i p_i=N,p_i≥ f(d,p)w_i,if S∈ℝ^N× n has each row chosen independently as the i^th standard basis vector, multiplied by 1/p_i^1/p, with probability p_i/N. Then, overall with probability at least 0.999,∀ x∈ℝ^d, 1/2Ax_p^p≤SAx_p^p≤ 2Ax_p^p.Furthermore, if p=1, N=O(dlog d). If 1<p<2, N=O(dlog dloglog d). If 2≤ p<4, N=O(d^p/2log d). Given matrix A∈ℝ^n× d (n≥ d), there is an algorithm to compute a diagonal matrix D=SS_1 with N nonzero entries in O(n(d)) time such that, with probability at least 0.999, for all x∈ℝ^d1/10DAx_p^p≤Ax_p^p≤ 10DAx_p^p,where S,S_1 are two sampling/rescaling matrices. Furthermore, if p=1, then N=O(dlog d). If 1<p<2, then N=O(dlog dloglog d). If 2≤ p<4, then N=O(d^p/2log d). Given a matrix A∈ℝ^n× d (n≥ d), by Lemma <ref> and Lemma <ref>, we can compute a sampling/rescaling matrix S in O((nnz(A)+d^ωlog d)log n) time with O(d) nonzero entries such that∀ x∈ℝ^d, 1/2Ax_p^p≤SAx_p^p≤ 2Ax_p^p.Sometimes, (d) is much smaller than log n. In this case, we are also able to compute such a sampling/rescaling matrix S in n(d) time in an alternative way.To do so, we run one of the input sparsity ℓ_p embedding algorithms (see e.g., <cit.>) to compute a well conditioned basis U of the column span of A in n(d/ε) time. By sampling according to the well conditioned basis (see e.g. <cit.>), we can compute a sampling/rescaling matrix S_1 such that (1-ε)Ax_p^p≤S_1Ax_p^p≤ (1+ε)Ax_p^p where ε∈(0,1) is an arbitrary constant. Notice that S_1 has (d/ε) nonzero entries, and thus S_1A has size (d/ε). Next, we apply Lewis weight sampling according to S_1A, and we obtain a sampling/rescaling matrix S for which∀ x∈ℝ^d, (1-1/3)S_1Ax_p^p≤SS_1Ax_p^p≤ (1+1/3)S_1Ax_p^p.This implies that∀ x∈ℝ^d, 1/2Ax_p^p≤SS_1Ax_p^p≤ 2Ax_p^p.Note that SS_1 is still a sampling/rescaling matrix according to A, and the number of non-zero entries is O(d). The total running time is thus n(d/ϵ), as desired. §.§ Let ϕ(v_1,v_2,⋯,v_q) denote the function that maps q vectors(u_i∈ℝ^n_i) to the ∏_i=1^q n_i-dimensional vector formed by v_1⊗ v_2 ⊗⋯⊗ u_q.We first give the definition of TensorSketch. Similar definitions can be found in previous work <cit.>.Given q points v_1,v_2,⋯, v_q where for each i∈ [q], v_i ∈ℝ^n_i, let m be the target dimension. The TensorSketch transform is specified using q 3-wise independent hash functions, h_1,⋯,h_q, where for each i∈ [q], h_i : [n_i]→ [m], as well as q 4-wise independent sign functions s_1, ⋯, s_q, where for each i∈ [q], s_i : [n_i] →{-1,+1}.TensorSketch applied to v_1,⋯,v_q is then CountSketch applied to ϕ(v_1,⋯,v_q) with hash function H: [∏_i=1^q n_i] → [m] and sign functions S : [∏_i=1^q n_i] →{-1,+1} defined as follows:H(i_1,⋯,i_q) = h_1(i_1) + h_2(s_2) + ⋯ + h_q(i_q)m,andS(i_1,⋯,i_q) = s_1(i_1) · s_2(i_2) ·⋯· s_q(i_q).Using the Fast Fourier Transform, TensorSketch(v_1,⋯,v_q) can be computed in O(∑_i=1^q ((v_i) + mlog m) ) time.Note that Theorem 1 in <cit.> only defines ϕ(v) = v⊗ v⊗⋯⊗ v. Here we state a stronger version of Theorem 1 than in <cit.>, though the proofs are identical; a formal derivation can be found in <cit.>.Let S be the (∏_i=1^q n_i) × m matrix such that TensorSketch (v_1,v_2,⋯,v_q) is ϕ(v_1,v_2,⋯,v_q) S for a randomly selected TensorSketch. The matrix S satisfies the following two properties.Property 1 (Approximate Matrix Product). Let A and B be matrices with ∏_i=1^q n_i rows. For m≥ (2+3^q) / (ϵ^2 δ), we have[ A^⊤ S S^⊤ B - A^⊤ B _F^2 ≤ϵ^2A _F^2B _F^2 ] ≥ 1-δ. Property 2 (Subspace Embedding). Consider a fixed k-dimensional subspace V. If m ≥ k^2 (2+3^q)/ (ϵ^2 δ), then with probability at least 1-δ, x S _2 = (1±ϵ)x _2 simultaneously for all x∈ V.§ FROBENIUS NORM FOR ARBITRARY TENSORS Section <ref> presents a Frobenius norm tensor low-rank approximation algorithm with (1+ϵ)-approximation ratio. Section <ref> introduces a tool which is able to reduce the size of the objective function from n^3 to (k,1/ϵ). Section <ref> introduces a new problem called tensor multiple regression. Section <ref> presents several bicriteria algorithms. Section <ref> introduces a powerful tool which we call generalized matrix row subset selection. Section <ref> presents an algorithm that is able to select a batch of columns, rows and tubes from a given tensor, and those samples are also able to form a low-rank solution. Section <ref> presents several useful tools for tensor problems, and also two (1+ϵ)-approximation CURT decomposition algorithms: one has the optimal sample complexity, and the other has the optimal running time. Section <ref> shows how to solve the problem if the size of the objective function is small. Section <ref> extends several techniques from 3rd order tensors to general q-th order tensors, for any q≥ 3. Finally, in Section <ref> we also provide a new matrix CUR decomposition algorithm, which is faster than <cit.>. For simplicity of presentation, we assume A_k exists in theorems (e.g., Theorem <ref>) which concern outputting a -k solution, as well as the theorems (e.g., Theorem <ref>, Theorem <ref>, Theorem <ref>) which concern outputting a bicriteria solution (the output rank is larger than k). For each of the bicriteria theorems, we can obtain a more detailed version when A_k does not exist, like Theorem <ref> in Section <ref> (by instead considering a tensor sufficiently close to A_k in objective function value). Note that the theorems for column, row, tube subset selection Theorem <ref> and Theorem <ref> also belong to this first category. In the second category, for each of the rank-k theorems we can obtain a more detailed version handling all cases, even when A_k does not exist, like Theorem <ref> in Section <ref> (by instead considering a tensor sufficiently close to A_k in objective function value).Several other tensor results or tools (e.g., Theorem <ref>, Lemma <ref>, Theorem <ref>, Theorem <ref>, Theorem <ref>, Theorem <ref>) that we build in this section do not belong to the above two categories. It means those results do not depend on whether A_k exists or not and whetheris zero or not.Given a 3rd order tensor A ∈ℝ^n× n× n, for any δ>0, if A_k = ∑_i=1^k u_i ⊗ v_i ⊗ w_i exists and each of u_i_2, v_i_2, and w_i_2 is bounded by 2^(n), then there is a randomized algorithm running in O( (A) + n (k,1/ϵ) + 2^O(k^2/ϵ^2) ) n^δ time in the unit costmodel with words of size O(log n) bits, which outputs a rank-k tensor B for which A-B_F^2 ≤ (1+ϵ)A-A_k_F^2 + 2^-n^δ. Further, we can output a number Z for which ≤ Z ≤ (1+ϵ) in the same amount of time. When A_k does not exist, if there exists a rank-k tensor à for which A-Ã_F^2 ≤ + 2^-n^δ and à = ∑_i=1^k u_i ⊗ v_i ⊗ w_i is such that the norms u_i_2, v_i_2, and w_i_2 are bounded by 2^n^δ, then we can output a rank-k tensor à for which A-Ã_F^2 ≤ (1+ϵ) + 2^-n^δ.§.§ (1+ϵ)-approximate low-rank approximationGiven a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1,ε∈(0,1), there exists an algorithm which takes O((A)) + n (k,1/ϵ) + 2^O(k^2/ϵ) time and outputs three matrices U∈ℝ^n× k, V∈ℝ^n× k, W∈ℝ^n× k such that∑_i=1^k U_i ⊗ V_i ⊗ W_i -A _F^2 ≤ (1+ϵ) -k A_kmin A_k -A _F^2holds with probability 9/10.Given any tensor A∈ℝ^n_1× n_2 × n_3, we define three matrices A_1 ∈ℝ^n_1 × n_2 n_3, A_2 ∈ℝ^n_2 × n_3 n_1, A_3 ∈ℝ^n_3 × n_1 n_2 such that, for any i∈ [n_1], j ∈ [n_2], l ∈ [n_3],A_i,j,l = ( A_1)_i, (j-1) · n_3 + l = ( A_2 )_ j, (l-1) · n_1 + i= ( A_3)_l, (i-1) · n_2 + j . We defineas=-k A'min A' -A _F^2. Suppose the optimal A_k=U^*⊗ V^*⊗ W^*. We fix V^* ∈ℝ^n× k and W^* ∈ℝ^n× k. We use V_1^*, V_2^*, ⋯, V_k^* to denote the columns of V^* and W_1^*, W_2^*, ⋯, W_k^* to denote the columns of W^*.We consider the following optimization problem,min_U_1, ⋯, U_k ∈ℝ^n ∑_i=1^k U_i ⊗ V_i^* ⊗ W_i^* - A _F^2,which is equivalent tomin_U_1, ⋯, U_k ∈ℝ^n [ U_1 U_2 ⋯ U_k ][ V_1^* ⊗ W_1^*; V_2^* ⊗ W_2^*; ⋯; V_k^* ⊗ W_k^* ] - A _F^2. We use matrix Z_1 to denote [ vec(V_1^* ⊗ W_1^* ); vec(V_2^* ⊗ W_2^* ); ⋯; vec(V_k^* ⊗ W_k^* ) ]∈ℝ^k× n^2 and matrix U to denote [ U_1 U_2 ⋯ U_k ]. Then we can obtain the following equivalent objective function,min_U ∈ℝ^n× k U Z_1- A_1 _F^2.Notice that min_U ∈ℝ^n× k U Z_1- A_1 _F^2=, since A_k=U^*Z_1.Let S_1^⊤∈ℝ^s_1× n^2 be a sketching matrix defined in Definition <ref>, where s_1=O(k/ε). We obtain the following optimization problem,min_U ∈ℝ^n× k U Z_1 S_1 - A_1 S_1 _F^2.Let U∈ℝ^n× k denote the optimal solution to the above optimization problem. Then U = A_1 S_1 (Z_1 S_1)^†. By Lemma <ref> and Theorem <ref>, we have U Z_1- A_1_F^2 ≤ (1+ϵ) U∈ℝ^n× kmin U Z_1 - A_1 _F^2 = (1+ϵ) , which implies∑_i=1^k U_i ⊗ V_i^* ⊗ W_i^* - A _F^2 ≤ (1+ϵ) .To write down U_1, ⋯, U_k, we use the given matrix A_1, and we create s_1 × k variables for matrix (Z_1 S_1)^†. As our second step, we fix U∈ℝ^n× k and W^* ∈ℝ^n× k, and we convert tensor A into matrix A_2. Let matrix Z_2 denote [ vec ( U_1 ⊗ W_1^* ); vec ( U_2 ⊗ W_2^* ); ⋯; vec ( U_k ⊗ W_k^* ) ]. We consider the following objective function,min_V ∈ℝ^n× k V Z_2 -A_2_F^2,for which the optimal cost is at most (1+ϵ).Let S_2^⊤∈ℝ^s_2× n^2 be a sketching matrix defined in Definition <ref>, where s_2=O(k/ε). We sketch S_2 on the right of the objective function to obtain the new objective function,V∈ℝ^n× kmin V Z_2 S_2 - A_2 S_2 _F^2.Let V∈ℝ^n× k denote the optimal solution of the above problem. Then V = A_2 S_2 (Z_2 S_2)^†. By Lemma <ref> and Theorem <ref>, we have,V Z_2 - A_2 _F^2 ≤ (1+ϵ ) V∈ℝ^n× kmin V Z_2- A_2 _F^2 ≤(1+ϵ)^2 ,which implies∑_i=1^k U_i ⊗V_i ⊗ W_i^* - A _F^2 ≤ (1+ϵ )^2 .To write down V_1, ⋯, V_k, we need to use the given matrix A_2 ∈ℝ^n^2 × n, and we need to create s_2× k variables for matrix (Z_2 S_2)^†.As our third step, we fix the matrices U∈ℝ^n× k and V∈ℝ^n × k. We convert tensor A∈ℝ^n× n × n into matrix A_3 ∈ℝ^n^2 × n. Let matrix Z_3 denote [ vec ( U_1 ⊗V_1 ); vec ( U_2 ⊗V_2 );⋯; vec ( U_k ⊗V_k ) ]. We consider the following objective function,W∈ℝ^n× kmin W Z_3 - A_3 _F^2,which has optimal cost at most (1+ϵ)^2.Let S_3^⊤∈ℝ^s_3× n^2 be a sketching matrix defined in Definition <ref>, where s_3=O(k/ε). We sketch S_3 on the right of the objective function to obtain a new objective function,W ∈ℝ^n× kmin W Z_3 S_3 - A_3 S_3 _F^2.Let W∈ℝ^n× k denote the optimal solution of the above problem. Then W = A_3 S_3 (Z_3 S_3)^†. By Lemma <ref> and Theorem <ref>, we have,W Z_3 - A_3 _F^2 ≤ (1+ϵ) W∈ℝ^n× kmin W Z_3 - A_3 _F^2 ≤ (1+ϵ)^3 .Thus, we havemin_X_1,X_2,X_3∑_i=1^k (A_1 S_1 X_1)_i ⊗ (A_2S_2 X_2)_i ⊗ (A_3S_3 X_3)_i - A _F^2 ≤ (1+ϵ)^3 .Let V_1=A_1S_1,V_2=A_2S_2,V_3=A_3S_3, we then apply Lemma <ref>, and we obtain V_1,V_2,V_3,C. We then apply Theorem <ref>. Correctness follows by rescaling ε by a constant factor. Running time. Due to Definition <ref>, the running time of line <ref> is O((A))+n(k).The running time of line <ref> is shown by Lemma <ref>, and the running time of line <ref> is shown by Theorem <ref>. Suppose we are given a 3rd order n× n× n tensor A such that each entry can be written using n^δ bits, where δ > 0 is a given, value which can be arbitrarily small (e.g., we could have n^δ being O(log n)). Define =inf_-k A_kA_k - A _F^2. For any k≥ 1, and for any 0<ϵ<1, define n^δ' = O( n^δ 2^O(k^2/ϵ) ). (1) If >0, and there exists a rank-k A_k=U^*⊗ V^* ⊗ W^* tensor, with size n× n× n, such that A_k-A _F^2 =, and max (U^*_F,V^*_F,W^*_F) ≤ 2^O(n^δ'), then there exists an algorithm that takes ((A)+ n(k,1/ϵ) + 2^O(k^2/ϵ) ) n^δ time in the unit costmodel with word size O(log n) bits[The entries of A are assumed to fit in n^δ words.] and outputs three n × k matrices U,V,W such thatU ⊗ V ⊗ W - A_F^2 ≤ (1+ϵ)holds with probability 9/10, and each entry of each of U,V,W fits in n^δ' bits.(2) If >0, and A_k does not exist, and there exist three n× k matrices U',V',W' for which max (U'_F,V'_F,W'_F) ≤ 2^O(n^δ') and U' ⊗ V'⊗ W' - A_F^2 ≤ (1+ϵ/2), then we can find U,V,W such that (<ref>) holds.(3) If =0 and A_k does exist, and there exists a solution U^*,V^*,W^* such that each entry can be written by n^δ' bits, then we can obtain (<ref>).(4) If =0, and there exist three n× k matrices U,V,W such that max (U_F,V_F,W_F)≤ 2^O(n^δ') and U ⊗ V ⊗ W - A _F^2 ≤ (1+ϵ)+ 2^-Ω(n^δ') = 2^-Ω(n^δ') ,then we can output U,V,W such that (<ref>) holds.Further if A_k exists, we can output a number Z for which ≤ Z ≤ (1+ϵ). For all the cases above, the algorithm runs in the same time as (1) and succeeds with probability at least 9/10. This follows by the discussion in Section <ref>, Theorem <ref> and Theorem <ref> in Section <ref>.Part (1) Suppose δ>0 and A_k = U^*⊗ V^*⊗ W^* exists and each of U^*_F, V^*_F, and W^* _F is bounded by 2^O(n^δ'). We assume the computation model is the unit costmodel with words of size O(log n) bits, and allow each number of the input tensor A to be written using n^δ bits. For the case whenis nonzero, using the proof of Theorem <ref> and Theorems <ref>, <ref>, there exists a lower bound on the cost , which is at least 2^-O(n^δ) 2^O(k^2/ϵ). We can round each entry of matrices U^*,V^*,W^* to be an integer expressed using O(n^δ') bits to obtain U',V',W'.Using the triangle inequality and our lower bound on , it follows that U',V',W' provide a (1+ϵ)-approximation. Thus, applying Theorem <ref> by fixing U',V',W' and using Theorem <ref> at the end, we can output three matrices U,V, W, where each entry can be written using n^δ' bits, so that we satisfy U ⊗ V ⊗ W - A_F^2 ≤ (1+ϵ).For the running time, since each entry of the input is bounded by n^δ bits, due to Theorem <ref>, we need ((A)+n(k/ε))· n^δ time to reduce the size of the problem to (k/ε) size (with each number represented using O(n^δ) bits). According to Theorem <ref>, the running time of using a polynomial system verifier to get the solution is 2^O(k^2/ϵ) n^O(δ')=2^O(k^2/ϵ) n^O(δ) time. Thus the total running time is ((A)+n(k/ε))n^δ+2^O(k^2/ϵ)· n^O(δ). Part (2) is similar to Part (1). Part (3) is trivial to prove since there exists a solution which can be written down in the bit model, so we obtain a (1+ϵ)-approximation. Part (4) is also very similar to Part (2). §.§ Input sparsity reductionLet (k,1/ϵ) ≥ b_1b_2b_3≥ k. Given a tensor A∈ℝ^n× n× n and three matrices V_1∈ℝ^n× b_1, V_2 ∈ℝ^n× b_2, and V_3 ∈ℝ^n× b_3, there exists an algorithm that takes O((A) + (V_1) + (V_2) + (V_3) )=O((A)+n(k/ε)) time and outputs a tensor C∈ℝ^c_1× c_2× c_3 and three matrices V_1∈ℝ^c_1× b_1, V_2 ∈ℝ^c_2× b_2 and V_3 ∈ℝ^c_3 × b_3 with c_1=c_2=c_3=(k,1/ϵ), such that with probability at least 0.99, for all α>0,X_1,X'_1∈ℝ^b_1× k, X_2,X'_2∈ℝ^b_2× k, X_3,X'_3∈ℝ^b_3× k satisfy that, ∑_i=1^k (V_1 X_1')_i ⊗ (V_2 X_2')_i ⊗ (V_3 X_3')_i - C _F^2 ≤α∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - C _F^2,then, ∑_i=1^k (V_1 X_1')_i ⊗ (V_2 X_2')_i ⊗ ( V_3 X_3')_i - A _F^2 ≤ (1+ϵ) α∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _F^2.Let X_1∈ℝ^b_1× k, X_2∈ℝ^b_2× k, X_3∈ℝ^b_3× k. First, we define Z_1 = ( (V_2 X_2)^⊤⊙ (V_3 X_3)^⊤ )∈ℝ^k× n^2. (Note that, for each i∈ [k], the i-th row of matrix Z_1 is ( (V_2 X_2)_i ⊗ (V_3 X_3)_i ).)Then, by flattening we have∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _F^2=V_1 X_1 · Z_1 - A_1_F^2.We choose a sparse embedding matrix (Definition <ref>) T_1∈ℝ^c_1× n with c_1=(k,1/ϵ) rows. Since V_1 has b_1≤(k/ε) columns, according to Lemma <ref> with probability 0.999, for all X_1 ∈ℝ^b_1 × k,Z∈ℝ^k× n^2,(1-ϵ) V_1 X_1 Z - A_1_F^2 ≤ T_1 V_1 X_1 Z - T_1 A_1_F^2 ≤ (1+ϵ) V_1 X_1 Z - A_1_F^2.Therefore, we have T_1V_1 X_1 · Z_1 - T_1 A_1 _F^2=(1±ε)∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _F^2.Second, we unflatten matrix T_1A_1∈ℝ^c_1 × n^2 to obtain a tensor A'∈ℝ^c_1× n× n. Then we flatten A' along the second direction to obtain A_2 ∈ℝ^n× c_1 n. We define Z_2 = (T_1 V_1 X_1)^⊤⊙ (V_3 X_3)^⊤∈ℝ^k× c_1n. Then, by flattening, V_2 X_2 · Z_2 - A_2 _F^2 = T_1V_1 X_1 · Z_1 - T_1 A_1 _F^2= (1±ε)∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _F^2.We choose a sparse embedding matrix (Definition <ref>) T_2∈ℝ^c_2 × n with c_2 = (k,1/ϵ) rows. Then according to Lemma <ref> with probability 0.999, for all X_2∈ℝ^b_2× k, Z∈ℝ^k× c_1 n,(1-ϵ)V_2 X_2 Z - A_2 _F^2 ≤ T_2 V_2 X_2 Z - T_2 A_2 _F^2 ≤ (1+ϵ)V_2 X_2 Z - A_2 _F^2.Therefore, we haveT_2 V_2 X_2 · Z_2 - T_2 A_2 _F^2= (1±ε)V_2 X_2 · Z_2 - A_2 _F^2 = (1±ε)^2∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _F^2. Third, we unflatten matrix T_2 A_2∈ℝ^c_2 × c_1 n to obtain a tensor A”(=A(T_1,T_2,I))∈ℝ^c_1 × c_2 × n. Then we flatten tensor A” along the last direction (the third direction) to obtain matrix A_3∈ℝ^n × c_1 c_2. We define Z_3 = (T_1 V_1 X_1)^⊤⊙ (T_2 V_2 X_2)^⊤∈ℝ^k × c_1 c_2. Then, by flattening, we have V_3 X_3 · Z_3 - A_3 _F^2 = T_2 V_2 X_2 · Z_2 - T_2 A_2 _F^2 = (1±ε)^2∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _F^2.We choose a sparse embedding matrix (Definition <ref>) T_3 ∈ℝ^c_3 × n with c_3= (k,1/ϵ) rows. Then according to Lemma <ref> with probability 0.999, for all X_3∈ℝ^b_3 × k, Z ∈ℝ^k× c_1 c_2,(1-ϵ)V_3 X_3 Z - A_3 _F^2 ≤ T_3 V_3 X_3 Z - T_3 A_3 _F^2 ≤ (1+ϵ)V_3 X_3 Z - A_3 _F^2.Therefore, we haveT_3 V_3 X_3 · Z_3 - T_3 A_3 _F^2=(1±ε)^3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _F^2.Note thatT_3 V_3 X_3 · Z_3 - T_3 A_3 _F^2=∑_i=1^k (T_1 V_1 X_1)_i ⊗ (T_2 V_2 X_2)_i ⊗ (T_3 V_3 X_3)_i- A(T_1,T_2,T_3) _F^2, and thus, we have ∀ X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× k∑_i=1^k (T_1 V_1 X_1)_i ⊗ (T_2 V_2 X_2)_i ⊗ (T_3 V_3 X_3)_i- A(T_1,T_2,T_3) _F^2=(1±ε)^3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _F^2.Let V_i denote T_i V_i, for each i∈ [3]. Let C∈ℝ^c_1 × c_2 × c_3 denote A(T_1,T_2,T_3). For α>1, if∑_i=1^k (V_1 X_1')_i ⊗ (V_2 X_2')_i ⊗ (V_3 X_3')_i - C _F^2 ≤α∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - C _F^2,then∑_i=1^k (V_1 X_1')_i ⊗ (V_2 X_2')_i ⊗ (V_3 X_3')_i - C _F^2 ≤ 1/(1-ε)^3∑_i=1^k (V_1 X_1')_i ⊗ (V_2 X_2')_i ⊗ (V_3 X_3')_i - C _F^2 ≤ 1/(1-ε)^3α∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - C _F^2 ≤ (1+ε)^3/(1-ε)^3α∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - C _F^2 By rescaling ε by a constant, we complete the proof of correctness. Running time. According to Section <ref>, for each i∈ [3], T_i V_i can be computed in O((V_i)) time, and A(T_1,T_2,T_3) can be computed in O((A)) time.By the analysis above, the proof is complete. §.§ Tensor multiple regression Given matrices A∈ℝ^d× n^2, U,V∈ℝ^n× k, let B∈ℝ^k× n^2 denote U^⊤⊙ V^⊤. There exists an algorithm that takes O( (A) + (U) + (V) + d(k,1/ϵ) ) time and outputs a matrix W'∈ℝ^d× k such that,W' B - A _F^2 ≤ (1+ϵ) min_W∈ℝ^d× k WB - A _F^2.We choose a TensorSketch (Definition <ref>) S∈ℝ^n^2 × s to reduce the problem to a smaller problem,min_W∈ℝ^d× k W B S - A S_F^2.Let W' denote the optimal solution to the above problem. Following a similar proof to that in Section <ref>, if S is a (1± 1/2)-subspace embedding and satisfies √(ϵ/k)-approximate matrix product, then W' provides a (1+ϵ)-approximation to the original problem. By Theorem <ref>, we have s=O(k^2+k/ϵ). Running time. According to Definition <ref>, BS can be computed in O((U)+(V))+(k/ε) time. Notice that each row of S has exactly 1 nonzero entry, thus AS can be computed in O((A)) time. Since BS∈ℝ^k× s and AS∈ℝ^d× s, min_W∈ℝ^d× k W B S - A S_F^2 can be solved in d(sk)=d(k/ε) time.§.§ Bicriteria algorithms§.§.§ Solving a small regression problemGiven tensor A∈ℝ^n× n × n and three matrices U∈ℝ^n× s_1, V∈ℝ^n× s_2 and W∈ℝ^n× s_3, there exists an algorithm that takes O((A) + n (s_1,s_2,s_3,1/ε)) time and outputs α'∈ℝ^s_1 × s_2 × s_3 such that∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3α'_i,j,l· U_i ⊗ V_j ⊗ W_l - A _F^2 ≤ (1+ϵ) min_α∈ℝ^s_1 × s_2 × s_3∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3α_i,j,l· U_i ⊗ V_j ⊗ W_l - A _F^2.holds with probability at least .99.We define b∈ℝ^n^3 to be the vector where the i+(j-1)n+(l-1)n^2-th entry of b is A_i,j,l. We define A∈ℝ^n^3 × s_1 s_2 s_3 to be the matrix where the (i+(j-1)n+(l-1)n^2,i'+(j'-1)s_2+(l'-1)s_2s_3) entry is U_i',i· V_j',j· W_l',l. This problem is equivalent to a linear regression problem,min_x ∈ℝ^s_1 s_2 s_3A x - b_2^2,where A∈ℝ^n^3 × s_1 s_2 s_3, b∈ℝ^n^3. Thus, it can be solved fairly quickly using recent work <cit.>. However, the running time of this naïvely is Ω(n^3), since we have to write down each entry of A. In the next few paragraphs, we show how to improve the running time to (A) + n(s_1,s_2,s_3).Since α∈ℝ^s_1× s_2× s_3, α can be always written as α=X_1⊗ X_2⊗ X_3, where X_1∈ℝ^s_1× s_1s_2s_3,X_2∈ℝ^s_2× s_1s_2s_3,X_3∈ℝ^s_3× s_1s_2s_3, we havemin_α∈ℝ^s_1 × s_2 × s_3∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3α_i,j,l· U_i ⊗ V_j ⊗ W_l - A _F^2=X_3∈ℝ^s_3× s_1s_2s_3X_2∈ℝ^s_2× s_1s_2s_3X_1∈ℝ^s_1× s_1s_2s_3min(UX_1)⊗(VX_2)⊗ (WX_3)-A_F^2.By Lemma <ref>, we can reduce the problem size n× n× n to a smaller problem that has size t_1× t_2 × t_3, min_X_1,X_2,X_3∑_i=1^s_1s_2s_3( T_1 U X_1)_i ⊗ ( T_2 V X_2)_i ⊗ (T_3 WX_3)_i - A(T_1,T_2,T_3) _F^2where T_1∈ℝ^t_1 × n, T_2∈ℝ^t_2 × n, T_3∈ℝ^t_3 × n,t_1=t_2=t_3=(s_1s_2s_3/ε). Notice that min_X_1,X_2,X_3∑_i=1^s_1s_2s_3( T_1 U X_1)_i ⊗ ( T_2 V X_2)_i ⊗ (T_3 WX_3)_i - A(T_1,T_2,T_3) _F^2 = min_α∈ℝ^s_1 × s_2 × s_3∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3α_i,j,l· (T_1U)_i ⊗ (T_2V)_j ⊗ (T_3W)_l - A(T_1,T_2,T_3) _F^2.Letα'=α∈ℝ^s_1 × s_2 × s_3min∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3α_i,j,l· (T_1U)_i ⊗ (T_2V)_j ⊗ (T_3W)_l - A(T_1,T_2,T_3) _F^2,then we have∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3α'_i,j,l· U_i ⊗ V_j ⊗ W_l - A _F^2 ≤ (1+ϵ) min_α∈ℝ^s_1 × s_2 × s_3∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3α_i,j,l· U_i ⊗ V_j ⊗ W_l - A _F^2.Again, according to Lemma <ref>, the total running time is then O((A) + n (s_1,s_2,s_3,1/ε)).Note that, by Theorem <ref>, there exist three matrices X_1,X_2,X_3 such thatmin_X_1,X_2,X_2∑_i=1^k( A_1 S_1 X_1)_i ⊗ (A_2 S_2 X_2)_i ⊗ (A_3S_3X_3)_i - A _F^2 ≤(1+ϵ) min_-k A' A' - A _F^2.By Lemma <ref>, we reduce the size n× n× n problem to a smaller problem that has size t_1× t_2 × t_3, min_X_1,X_2,X_2∑_i=1^k( T_1 A_1 S_1 X_1)_i ⊗ ( T_2 A_2 S_2 X_2)_i ⊗ (T_3 A_3S_3X_3)_i - A(T_1,T_2,T_3) _F^2≤ (1+O(ϵ))min_X_1,X_2,X_2∑_i=1^k( A_1 S_1 X_1)_i ⊗ (A_2 S_2 X_2)_i ⊗ (A_3S_3X_3)_i - A _F^2≤ (1+O(ϵ)) min_-k A' A' - A _F^2,where T_1∈ℝ^t_1 × n, T_2∈ℝ^t_2 × n, T_3∈ℝ^t_3 × n.Given tensor A∈ℝ^n× n× n, and two matrices U∈ℝ^n× s, V∈ℝ^n× s with (U)=r_1,(V)=r_2, let T_1∈ℝ^t_1× n,T_2∈ℝ^t_2× n be two sparse embedding matrices (Definition <ref>) with t_1=(r_1/ε),t_2=(r_2/ε). Then with probability at least 0.99, ∀ X∈ℝ^n× s,(1-)U⊗ V⊗ X-A_F^2≤T_1U⊗ T_2V⊗ X-A(T_1,T_2,I)_F^2≤ (1+)U⊗ V⊗ X-A_F^2.Let X∈ℝ^n× s. We define Z_1 = ( V^⊤⊙ X^⊤ )∈ℝ^s× n^2. We choose a sparse embedding matrix (Definition <ref>) T_1∈ℝ^t_1× n with t_1=(r_1/ϵ) rows. According to Lemma <ref> with probability 0.999, for all Z∈ℝ^s× n^2,(1-ϵ) U Z - A_1_F^2 ≤ T_1 UZ - T_1 A_1_F^2 ≤ (1+ϵ) T_1U Z - A_1_F^2.It means that(1-ϵ) U Z_1 - A_1_F^2 ≤ T_1 UZ_1 - T_1 A_1_F^2 ≤ (1+ϵ) T_1U Z_1 - A_1_F^2.Second, we unflatten matrix T_1A_1∈ℝ^t_1 × n^2 to obtain a tensor A'∈ℝ^t_1× n× n. Then we flatten A' along the second direction to obtain A'_2 ∈ℝ^n× t_1 n. We define Z_2 = ((T_1 U)^⊤⊙ X^⊤) ∈ℝ^s× t_1n. Then, by flattening, V · Z_2 - A'_2 _F^2=T_1U · Z_1 - T_1 A_1 _F^2=(1±ε)U⊗ V⊗ X-A_F^2.We choose a sparse embedding matrix (Definition <ref>) T_2∈ℝ^t_2 × n with t_2 = (r_2/ϵ) rows. Then according to Lemma <ref> with probability 0.999, for all Z∈ℝ^s× t_1n,(1-ϵ)VZ - A'_2 _F^2 ≤ T_2 VZ - T_2 A'_2 _F^2 ≤ (1+ϵ)V Z - A'_2 _F^2.Thus,T_2V · Z_2 - T_2A'_2 _F^2=(1±ε)^2U⊗ V⊗ X-A_F^2.After rescaling ε by a constant, with probability at least 0.99, ∀ X∈ℝ^n× s,(1-)U⊗ V⊗ X-A_F^2≤T_1U⊗ T_2V⊗ X-A(T_1,T_2,I)_F^2≤ (1+)U⊗ V⊗ X-A_F^2.§.§.§ Algorithm 1We start with a slightly unoptimized bicriteria low rank approximation algorithm. Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, ϵ∈ (0,1), let r=O(k^3/ϵ^3). There exists an algorithm that takes O((A) + n(k,1/ϵ)) time and outputs three matrices U ∈ℝ^n× r, V∈ℝ^n× r, W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _F^2 ≤ (1+ϵ) -k A_kmin A_k - A _F^2holds with probability 9/10. At the end of Theorem <ref>, we need to run a polynomial system verifier. This is why we obtain exponential in k running time. Instead of running the polynomial system verifier, we can use Lemma <ref>. This reduces the running time to be polynomial in all parameters: n,k,1/ϵ. However, the output tensor has rank (k/ϵ)^3 (Here we mean that we do not obtain a better decomposition than (k/ϵ)^3 components). According to Section <ref>, for each i, A_iS_i can be computed in O((A))+n(k/ε) time. Then T_i(A_iS_i) can be computed in n(k,1/ϵ) time and A(T_1,T_2,T_3) also can be computed in O((A)) time. The running time for the regression is (k/ε). Now we present an optimized bicriteria algorithm.Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, ϵ∈ (0,1), let r=O(k^2/ϵ^2). There exists an algorithm that takes O((A)+ n(k,1/ϵ)) time and outputs three matrices U ∈ℝ^n× r, V∈ℝ^n× r, W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _F^2 ≤ (1+ϵ) -k A_kmin A_k - A _F^2holds with probability 9/10.Note that there are two different ways to implement algorithm FTensorLowRankBicriteriaQuadraticRank. We present the proofs for both of them here.Approach 1.Let =-k A_kminA_k-A_F^2. According to Theorem <ref>, we know that there exists a sketching matrix S_3∈ℝ^n^2× s_3 where s_3=O(k/ε), such thatmin_X_1∈ℝ^s_1× k,X_2∈ℝ^s_2× k,X_3∈ℝ^s_3× k∑_l=1^k (A_1 S_1 X_1)_l ⊗ (A_2S_2 X_2)_l ⊗ (A_3S_3 X_3)_l - A _F^2 ≤ (1+ϵ)Now we fix an l and we have:(A_1 S_1 X_1)_l ⊗ (A_2S_2 X_2)_l ⊗ (A_3S_3 X_3)_l= (∑_i=1^s_1 (A_1S_1)_i (X_1)_i,l)⊗(∑_j=1^s_2 (A_2S_2)_j (X_2)_j,l)⊗ (A_3S_3 X_3)_l= ∑_i=1^s_1∑_j=1^s_2 (A_1S_1)_i ⊗ (A_2S_2)_j ⊗ (A_3S_3 X_3)_l(X_1)_i,l(X_2)_j,lThus, we havemin_X_1,X_2,X_3∑_i=1^s_1∑_j=1^s_2 (A_1S_1)_i ⊗ (A_2S_2)_j ⊗( ∑_l=1^k(A_3S_3 X_3)_l(X_1)_i,l(X_2)_j,l) - A _F^2 ≤ (1+ϵ) .We use matrices A_1 S_1 ∈ℝ^n× s_1 and A_2 S_2 ∈ℝ^n× s_2 to construct a matrix B ∈ℝ^s_1s_2 × n^2 in the following way: each row of B is the vector corresponding to the matrix generated by the ⊗ product between one column vector in A_1 S_1 and the other column vector in A_2 S_2, i.e.,B^i+ (j-1)s_1 = vec( (A_1 S_1)_i ⊗ (A_2 S_2)_j), ∀ i∈ [s_1], j∈ [s_2] ,where (A_1 S_1)_i denotes the i-th column of A_1S_1 and (A_2 S_2)_j denote the j-th column of A_2 S_2.We create matrix U∈ℝ^n× s_1 s_2 by copying matrix A_1 S_1 s_2 times, i.e.,U = [ A_1 S_1 A_1 S_1 ⋯ A_1 S_1 ].We create matrix V∈ℝ^n× s_1 s_2 by copying the i-th column of A_2 S_2 a total of s_1 times, into columns (i-1)s_1, ⋯, i s_1 of V, for each i∈ [s_2], i.e.,V = [ (A_2 S_2)_1 ⋯ (A_2 S_2)_1 (A_2 S_2)_2 ⋯ (A_2 S_2)_2 ⋯ (A_2 S_2)_s_2 ⋯ (A_2 S_2)_s_2 ].Thus, we can use U and V to represent B,B = ( U^⊤⊙V^⊤ ) ∈ℝ^s_1 s_2 × n^2. According to Equation (<ref>), we have:W∈ℝ^n× s_1 s_2min WB - A_3 _F^2≤ (1+ε). Next, we want to find matrix W∈ℝ^n× s_1 s_2 by solving the following optimization problem,W∈ℝ^n× s_1 s_2min WB - A_3 _F^2.Note that B has size s_1s_2× n^2. Naïvely writing down B already requires Ω(n^2) time. In order to achieve nearly linear time in n, we cannot write down B. We choose S_3 ∈ℝ^n_1 n_2 × s_3 to be a TensorSketch (Definition <ref>). In order to solve multiple regression, we need to set s_3 = O( (s_1 s_2)^2+ (s_1s_2)/ϵ). Let W denote the optimal solution to WB S_3 - A_3 S_3 _F^2. Then W = (A_3 S_3) (BS_3)^†. Since each row of S_3 has exactly 1 nonzero entry, A_3S_3 can be computed in O((A)) time. Since B = ( U^⊤⊙V^⊤ ), according to Definition <ref>, BS_3 can be computed in n (s_1s_2/ϵ)=n(k/ε) time. By Theorem <ref>, we haveW B - A_3 _F^2 ≤ (1+ϵ) min_W∈ℝ^n× s_1 s_2 W B - A_3 _F^2.Thus, we haveU⊗V⊗W-A_F^2≤ (1+ε).According to Definition <ref>, A_1S_1,A_2S_2 can be computed in O((A)+(k/ε)) time. Te total running time is thus O((A)+(k/ε)). Approach 2.Let =-k A_kminA_k-A_F^2. Choose sketching matrices (Definition <ref>) S_1∈ℝ^n^2× s_1, S_2∈ℝ^n^2× s_2, S_3 ∈ℝ^n^2× s_3, and sketching matrices (Definition <ref>) T_1∈ℝ^t_1 × n and T_2 ∈ℝ^t_2 × n with s_1 =s_2 = s_3=O(k/ϵ),t_1 =t_2=(k/ε). We create matrix U∈ℝ^n× s_1 s_2 by copying matrix A_1 S_1 s_2 times, i.e.,U = [ A_1 S_1 A_1 S_1 ⋯ A_1 S_1 ].We create matrix V∈ℝ^n× s_1 s_2 by copying the i-th column of A_2 S_2 a total of s_1 times, into columns (i-1)s_1, ⋯, i s_1 of V, for each i∈ [s_2], i.e.,V = [ (A_2 S_2)_1 ⋯ (A_2 S_2)_1 (A_2 S_2)_2 ⋯ (A_2 S_2)_2 ⋯ (A_2 S_2)_s_2 ⋯ (A_2 S_2)_s_2 ]. As we proved in Approach 1, we havemin_X∈ℝ^n× s_1s_2U⊗V⊗ X-A_F^2≤ (1+ε).Let B=( (T_1U)^⊤⊙ (T_2 V)^⊤)∈ℝ^s_1s_2× t_1t_2, and flatten A(T_1,T_2,I) along the third direction to obtain C_3∈ℝ^n× t_1t_2. LetW=X∈ℝ^n× s_1s_2minT_1U⊗ T_2V⊗ X-A(T_1,T_2,I)_F^2=X∈ℝ^n× s_1s_2minXB-C_3_F^2.LetW^*=X∈ℝ^n× s_1s_2minU⊗V⊗ X-A_F^2.According to Lemma <ref>,U⊗V⊗W-A_F^2 ≤ 1/1-εT_1U⊗ T_2V⊗W-A(T_1,T_2,I)_F^2 ≤ 1/1-εT_1U⊗ T_2V⊗ W^*-A(T_1,T_2,I)_F^2 ≤ 1+ε/1-εU⊗V⊗ W^*-A_F^2 ≤ (1+ε)^2/1-ε. According to Definition <ref>, A_1S_1,A_2S_2 can be computed in O((A)+(k/ε)) time. The total running time is thus O((A)+(k/ε)). Since T_1,T_2 are sparse embedding matrices, T_1U,T_2V can be computed in O((A)+(k/ε)) time. The total running time is in O((A)+(k/ε)). Given a 3rd order tensor A ∈ℝ^n× n× n, for any k≥ 1 and any 0 < ϵ <1, if A_k exists then there is a randomized algorithm running in (A) + n ·(k/ϵ) time which outputs a rank-O(k^2/ϵ^2) tensor B for which A-B_F^2 ≤ (1+ϵ)A-A_k_F^2. If A_k does not exist, then the algorithm outputs a rank-O(k^2/ϵ^2) tensor B for which A-B_F^2 ≤ (1+ϵ) + γ, where γ is an arbitrarily small positive function of n. In both cases, the algorithm succeeds with probability at least 9/10.If A_k exists, then the proof directly follows the proof of Theorem <ref> and Theorem <ref>. If A_k does not exist, then for any γ>0, there exist U^*∈ℝ^n× k,V^*∈ℝ^n× k,W^*∈ℝ^n× k such thatU^*⊗ V^*⊗ W^*-A_F^2≤inf_-k A'A-A'_F^2+1/10γ.Then we just regard U^*⊗ V^*⊗ W^* as the “best” k approximation to A, and follow the same argument as in the proof of Theorem <ref> and the proof of Theorem <ref>. We can finally output a tensor B∈ℝ^n× n× n with rank-O(k^2/ϵ^2) such thatB-A_F^2≤ (1+ε)U^*⊗ V^*⊗ W^*-A_F^2≤ (1+ε)( inf_-k A'A-A'_F^2+1/10γ)≤ (1+ε) inf_-k A'A-A'_F^2+γwhere the first inequality follows by the proof of Theorem <ref> and the proof of theorem <ref>. The second inequality follows by our choice of U^*,V^*,W^*. The third inequality follows since 1+ε<2 and γ>0. §.§.§ (k)-approximation to multiple regressionLet s≥ k. Let U∈ℝ^n × k denote a matrix that has orthonormal columns, and S ∈ℝ^s× n denote an i.i.d. N(0,1/s) Gaussian matrix. Then SU is also an s × k i.i.d. Gaussian matrix with each entry draw from N(0,1/s), and furthermore, we have with arbitrarily large constant probability,σ_max(SU) = O(1) and σ_min(SU) = Ω(1/√(s)).Note that √(s) - √(k-1)=s-k-1 /√(s) + √(k-1) = Ω(1/√(s)). Given matrices A∈ℝ^n× k, B∈ℝ^n× d, let S∈ℝ^s× n denote a standard Gaussian N(0,1) matrix with s=k. Let X^*=X∈ℝ^k× dmin A X - B _F. Let X'=X∈ℝ^k× dmin SA X - SB_F. Then, we have thatA X' - B _F ≤ O(√(k))A X^* - B _F,holds with probability at least 0.99. Let X^* ∈ℝ^k× d denote the optimal solution such thatA X^* - B _F = X ∈ℝ^k× dmin AX - B _F. Consider a standard Gaussian matrix S ∈ℝ^k× n scaled by 1/√(k) with exactly k rows. Then for any X ∈ℝ^k× d, by the triangle inequality, we haveSAX - SB _F ≤ SAX - SA X^* _F +SA X^* - SB _F,andSAX - SB _F≥ SAX - SA X^* _F -SA X^* - SB _F. We first show how to bound SAX - SA X^* _F, and then show how to bound SA X^* - SB _F. Note that Lemma <ref> implies the following result,For any X ∈ℝ^k× d, with probability 0.999, we have1 /√(k) A X - AX^* _F ≲ SA X - SA X^* _F ≲ A X - A X^* _F.First, we can write A= UR ∈ℝ^n × k where U∈ℝ^n× k has orthonormal columns and R ∈ℝ^k× k. It gives,SAX - SAX^* _F =SU ( R X - R X^* ) _F.Second, applying Lemma <ref> to SU ∈ℝ^s× k completes the proof.Using Markov's inequality, for any fixed matrix AX^* - B, choosing a Gaussian matrix S, we have thatS A X^* -S B _F^2 = O(A X^* - B_F^2)holds with probability at least 0.999. This is equivalent toS A X^* - S B_F = O( A X^* - B _F), holding with probability at least 0.999.LetX' = X ∈ℝ^k× dmin S A X - S B _F. Putting it all together, we haveAX'-B _F ≤ AX'- AX^* _F +AX^* - B _F by triangle inequality ≤ O(√(k))S AX'- SA X^* _F + A X^* - B_F by Claim <ref> ≤ O(√(k))S AX'- S B _F + O(√(k))SA X^* - SB _F+ A X^* - B_F by triangle inequality ≤ O(√(k))S AX^*- S B _F + O(√(k))SA X^* - SB _F+ A X^* - B_F by definition of X'≤ O(√(k))A X^* - B _F. by Equation (<ref>) §.§.§ Algorithm 2Given a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, let r=k^2. There exists an algorithm which takes O((A) k) + n (k) time and outputs three matrices U,V,W∈ℝ^n× r such that,∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _F ≤(k) min_-k A' A' - A _Fholds with probability 9/10. Let =-k A'min A' - A _F, we fix V^*∈ℝ^n × k, W^*∈ℝ^n× k to be the optimal solution of the original problem. We use Z_1 = (V^*⊤⊙ W^*⊤)∈ℝ^k× n^2 to denote the matrix where the i-th row is the vectorization of V_i^* ⊗ W_i^*. Let A_1 ∈ℝ^n× n^2 denote the matrix obtained by flattening tensor A∈ℝ^n× n × n along the first direction. Then, we havemin_U U Z_1 -A_1 _F ≤.Choosing an N(0,1/k) Gaussian sketching matrix S_1∈ℝ^n^2× s_1 with s_1=k, we can obtain the smaller problem,min_U∈ℝ^n× k U Z_1 S_1 - A_1 S_1 _F.Define U = A_1S_1 (Z_1 S_1)^†. Define α=O(√(k)). By Lemma <ref>, we haveU Z_1 - A_1 _F ≤α.Second, we fix U and W^*. Define Z_2,A_2 similarly as above. Choosing an N(0,1/k) Gaussian sketching matrix S_2∈ℝ^n^2× s_2 with s_2=k, we can obtain another smaller problem,min_V∈ℝ^n× k V Z_2 S_2 - A_2 S_2 _F.Define V = A_2 S_2 (Z_2 S_2)^†. By Lemma <ref> again, we haveV Z_2 - A_2 _F ≤α^2 .Thus, we now havemin_X_1,X_2,WA_1S_1X_1⊗ A_2S_2X_2 ⊗ W -A_F≤α^2 We use a similar idea as in the proof of Theorem <ref>. We create matrix U∈ℝ^n× s_1 s_2 by copying matrix A_1 S_1 s_2 times, i.e.,U = [ A_1 S_1 A_1 S_1 ⋯ A_1 S_1 ].We create matrix V∈ℝ^n× s_1 s_2 by copying the i-th column of A_2 S_2 a total of s_1 times, into columns (i-1)s_1, ⋯, i s_1 of V, for each i∈ [s_2], i.e.,V = [ (A_2 S_2)_1 ⋯ (A_2 S_2)_1 (A_2 S_2)_2 ⋯ (A_2 S_2)_2 ⋯ (A_2 S_2)_s_2 ⋯ (A_2 S_2)_s_2 ].We havemin_X∈ℝ^n× s_1s_2U⊗V⊗ X-A_F≤α^2. Choose T_i ∈ℝ^t_i× n to be a sparse embedding matrix (Definition <ref>) with t_i = (k/ε), for each i∈ [2]. By applying Lemma <ref>, we have, if W' satisfies,T_1U⊗ T_2V⊗ W'-A(T_1,T_2,I)_F= min_X∈ℝ^n× s_1s_2T_1U⊗ T_2V⊗ X-A(T_1,T_2,I)_Fthen,U⊗V⊗ W'-A_F≤(1+ϵ) min_X∈ℝ^n× s_1s_2U⊗V⊗ X-A_F≤ (1+ε)α^2 .Thus, we only need to solvemin_X∈ℝ^n× s_1s_2T_1U⊗ T_2V⊗ X-A(T_1,T_2,I)_F.which is similar to the proof of Theorem <ref>. Therefore, we complete the proof of correctness. For the running time, A_1S_1,A_2S_2 can be computed in O((A)k) time, T_1U,T_2V can be computed in n(k) time. The final regression problem can be computed in n(k) running time.§.§ Generalized matrix row subset selectionNote that in this section, the notation Π_C,k^ξ is given in Definition <ref>. Given matrices A∈ℝ^n× m and C∈ℝ^n× k, there exists an algorithm which takes O((A) log n ) + (m+n)(k,1/ϵ) time and outputs a diagonal matrix D ∈ℝ^n× n with d=O(k/ϵ) nonzeros (or equivalently a matrix R that contains d=O(k/ϵ) rescaled rows of A) and a matrix U∈ℝ^k × d such thatC U D A - A _F^2 ≤ (1+ϵ) min_X∈ℝ^k× m C X - A _F^2holds with probability .99. This follows by combining Lemma <ref> and <ref>. Let U,R denote the output of procedure GeneralizedMatrixRowSubsetSelection,A - CUR _F^2 ≤ (1+ϵ)A - Z_2Z_2^⊤ A R^† R_F^2≤ (1+ϵ) (1+60ϵ)A - Π_C,k^F(A) _F^2 ≤ (1+130ϵ)A - Π_C,k^F(A) _F^2.Because R is a subset of rows of A and R has size O(k/ϵ)× m, there must exist a diagonal matrix D∈ℝ^n× n with O(k/ϵ) nonzeros such that R = D A. This completes the proof.Given matrices A∈ℝ^n× m and C∈ℝ^n× k, if min(m,n)=(k,1/ϵ), then there exists an algorithm which takes O((A)) + (m+n)(k,1/ϵ) time and outputs a diagonal matrix D ∈ℝ^n× n with d=O(k/ϵ) nonzeros (or equivalently a matrix R that contains d=O(k/ϵ) rescaled rows of A) and a matrix U∈ℝ^k × d such thatC U D A - A _F^2 ≤ (1+ϵ) min_X∈ℝ^k× m C X - A _F^2holds with probability .99. The log n factor comes from the adaptive sampling where we need to choose a Gaussian matrix with O(log n) rows and compute SA. If A has (k,1/ϵ) columns, it is sufficient to choose S to be a CountSketch matrix with (k,1/ϵ) rows. Then, we do not need a log n factor in the running time. If S has (k,1/ϵ) rows, then we no longer need the matrix S.Given matrices A∈ℝ^m× n and C∈ℝ^m× c, let Y ∈ℝ^m× c,Φ∈ℝ^c× c and Δ∈ℝ^c× k denote the output of procedure ApproxSubspaceSVD(A,C,k,ϵ). Then with probability .99, we have,A - Y ΔΔ^⊤ Y^⊤ A _F^2 ≤ (1+30ϵ) A - Π_C,k^F (A) _F^2.This follows by Lemma 3.12 in <cit.>.The matrices R and Z_2 in procedure GeneralizedMatrixRowSubsetSelection (Algorithm <ref>) satisfy with probability at least 0.17-2/n,A- Z_2 Z_2^⊤ A R^† R _F^2 ≤ A - Π_C,k^F(A) _F^2 + 60ϵ A - Π_C,k^F(A) _F^2. We can show,A - Z_2 Z_2^⊤ A _F^2 + 30ϵ/4820 A - A R_1^† R_1 _F^2= A -BB^† A_F^2 + 30ϵ/4820 A - A R_1^† R_1 _F^2≤ A -BB^† A_F^2 + 30ϵ A - A_k _F^2≤ A - Y ΔΔ^⊤ Y A _F^2 + 30ϵ A - Π_C,k^F(A) _F^2≤ (1+30ϵ) A - Π_C,k^F(A) _F^2 + 30ϵ A - Π_C,k^F(A) _F^2,where the first step follows by the fact that Z_2 Z_2^⊤ = Z_2 D D^-1 Z_2^⊤ = (Z_2 D) (Z_2 D)^† =BB^†, the second step follows by A - A R_1^† R_1_F^2 ≤ 4820A - A_k_F^2, the third step follows by B=Y Δ and B^† = (Y Δ)^† = Δ^† Y^† = Δ^⊤ Y^⊤, and the last step follows by Claim <ref>. The matrices C,U and R in procedure GeneralizedMatrixRowSubsetSelection (Algorithm <ref>) satisfy thatA- CUR _F^2 ≤ (1+ϵ)A - Z_2 Z_2^⊤ A R^† R _F^2with probability at least .99. Let U_R,Σ_R,V_R denote the SVD of R. Then V_R V_R^⊤ = R^† R.We define Y^* to be the optimal solution ofmin_X∈ℝ^k× r W A V_R V_R^⊤ - WC Φ^-1Δ D^-1 Y R _F^2.We define X^* to be Y^* R ∈ℝ^k × n, which is also equivalent to defining X^* to be the optimal solution ofmin_X∈ℝ^k× n W A V_R V_R^⊤ - W C Φ^-1Δ D^-1 X _F^2.Furthermore, it implies X^* = (WC Φ^-1Δ D^-1)^† WA V_R V_R^†.We also define X^* to be the optimal solution ofmin_X∈ℝ^k× n A V_R V_R^† - C Φ^-1Δ D^-1 X _F^2,which implies that,X^* = (CΦ^-1Δ D^-1)^† A V_R V_R^⊤ = Z_2^⊤ A V_R V_R^⊤.Now, we start to prove an upper bound on A - CUR _F^2,A - CUR _F^2 = A - C Φ^-1Δ D^-1 Y^* R_F^2 by definition of U = A - C Φ^-1Δ D^-1X^*_F^2 by X^* = Y^*R = A V_R V_R^⊤ - CΦ^-1Δ D^-1X^* + A - A V_R V_R^⊤_F^2 = A V_R V_R^⊤ - CΦ^-1Δ D^-1X^* _F^2 _α + A - A V_R V_R^⊤_F^2_β,where the last step follows by X^*=MV_R^⊤, A-AV_RV_R^⊤ = A(I-V_RV_R^⊤) and the Pythagorean theorem. We show how to upper bound the term α,α≤ (1+ϵ)AV_R V_R^⊤ -C Φ^-1Δ D^-1 X^*_F^2 by Lemma <ref>= ϵ AV_R V_R^⊤ -C Φ^-1Δ D^-1 X^*_F^2 +AV_R V_R^⊤ -C Φ^-1Δ D^-1 X^*_F^2 = ϵ AV_R V_R^⊤ -C Φ^-1Δ D^-1 X^*_F^2 +AV_R V_R^⊤ -C Φ^-1Δ D^-1 (Z_2^⊤ A R^† R)_F^2.By the Pythagorean theorem and the definition of Z_2 (which means Z_2 = C Φ^-1Δ D^-1), we have,AV_R V_R^⊤ -C Φ^-1Δ D^-1 (Z_2^⊤ A R^† R)_F^2 + β= AV_R V_R^⊤ -C Φ^-1Δ D^-1 (Z_2^⊤ A R^† R)_F^2 +A - A V_R V_R^⊤_F^2 = A - C Φ^-1Δ D^-1 (Z_2^⊤ A R^† R) _F^2 = A - Z_2 Z_2^⊤ A R^† R _F^2.Combining Equations (<ref>), (<ref>) and (<ref>) together, we obtain,A - CUR _F^2 ≤ϵ A V_RV_R^⊤ - CΦ^-1Δ D^-1 X^*_F^2 +A -Z_2 Z_2^⊤ A R^† R _F^2.We want to show A V_RV_R^⊤ - CΦ^-1Δ D^-1 X^*_F^2≤ A -Z_2 Z_2^⊤ A R^† R_F^2,A V_RV_R^⊤ - CΦ^-1Δ D^-1 X^*_F^2= A V_RV_R^⊤ - CΦ^-1Δ D^-1 Z_2^⊤ A V_R V_R^⊤_F^2 by X^* = Z_2^⊤ A V_R V_R^⊤ ≤ A- CΦ^-1Δ D^-1 Z_2^⊤ A _F^2 by properties of projections ≤ A- CΦ^-1Δ D^-1 Z_2^⊤ A R^† R_F^2 by properties of projections= A - Z_2 Z_2^⊤ A R^† R_F^2. by Z_2 = C Φ^-1Δ D^-1This completes the proof.Let A∈ℝ^n× d have rank ρ and B∈ℝ^n × r. Let W∈ℝ^r× n be a randomly chosen sparse subspace embedding with r= Ω(ρ^2 ϵ^-2). Let X^* = X∈ℝ^d× rmin WA X -W B _F^2 and let X^*=X∈ℝ^d× rmin A X - B _F^2. Then with probability at least .99,A X^* - B _F^2 ≤ (1+ϵ)AX^* - B _F^2. §.§ Column, row, and tube subset selection, (1+ϵ)-approximation We provide two bicriteria CURT results in this Section. We first present a warm-up result. That result (Theorem <ref>) does not output tensor U and only guarantees that there is a -(k/ϵ) tensor U. Then we show the second result (Theorem <ref>), our second result is able to output tensor U. The U has rank (k/ϵ), but not k. Given a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm which takes O((A))+ n (k,1/ϵ) time and outputs three matrices: C∈ℝ^n× c, a subset of columns of A, R∈ℝ^n× r a subset of rows of A, and T∈ℝ^n× t, a subset of tubes of A where c=r=t= (k,1/ϵ), andthere exists a tensor U∈ℝ^c× r× t such that( ( ( U · T^⊤)^⊤· R^⊤ )^⊤· C^⊤ )^⊤- A _F^2≤ (1+ϵ) -k A_kmin A_k - A _F^2,or equivalently,∑_i=1^c ∑_j=1^r ∑_l=1^t U_i,j,l· C_i ⊗ R_j ⊗ T_l -A _F^2 ≤ (1+ϵ) -k A_kmin A_k - A _F^2holds with probability 9/10. We mainly analyze Algorithm <ref> and it is easy to extend to Algorithm <ref>.We fix V^*∈ℝ^n× k and W^*∈ℝ^n× k. We define Z_1∈ℝ^k× n^2 where the i-th row of Z_1 is the vector V_i ⊗ W_i. Choose sketching (Gaussian) matrix S_1 ∈ℝ^n^2 × s_1 (Definition <ref>), and let U = A_1 S_1 (Z_1 S_1)^†∈ℝ^n× k. Following a similar argument as in the previous theorem, we haveU Z_1 - A_1 _F^2 ≤ (1+ϵ) .We fix U and W^*. We define Z_2∈ℝ^k× n^2 where the i-th row of Z_2 is the vector U_i ⊗ W^*_i. Choose sketching (Gaussian) matrix S_2∈ℝ^n^2× s_2 (Definition <ref>), and let V = A_2 S_2 (Z_2 S_2)^†∈ℝ^n× k. Following a similar argument as in the previous theorem, we haveV Z_2 - A_2 _F^2 ≤ (1+ϵ)^2 . We fix U and V. Note that U=A_1 S_1 (Z_1 S_1)^† and V=A_2 S_2 (Z_2 S_2)^†. We define Z_3∈ℝ^k× n^2 such that the i-th row of Z_3 is the vector U_i ⊗V_i. Let z_3 = s_1 · s_2. We define Z'_3∈ℝ^z_3× n^2 such that, ∀ i∈ [s_1], ∀ j∈ [s_2], the i+(j-1) s_1-th row of Z'_3 is the vector (A_1 S_1)_i ⊗ (A_2 S_2)_j. We consider the following objective function,min_W ∈ℝ^n× k, X∈ℝ^k × z_3 W X Z_3' - A_3 _F^2 ≤min_W ∈ℝ^n× k W Z_3 - A_3 _F^2≤ (1+ϵ)^2 . Using Theorem <ref>, we can find a diagonal matrix D_3∈ℝ^n^2× n^2 with d_3 = O(z_3/ϵ) = O(k^2/ϵ^3) nonzero entries such thatmin_X∈ℝ^d_3× z_3A_3 D_3 XZ_3' - A_3_F^2≤ (1+ϵ)^3 .In the following, we abuse notation and let A_3 D_3 ∈ℝ^n× d_3 by deleting zero columns.Let W' denote A_3 D_3 ∈ℝ^n× d_3. Then,min_X ∈ℝ^d_3× z_3 W' X Z_3' - A_3 _F^2 ≤ (1+ϵ)^3 . We fix U and W'. Let z_2 = s_1 · d_3. We define Z'_2 ∈ℝ^z_2× n^2 such that, ∀ i∈ [s_1], ∀ j ∈ [d_3],the i+(j-1)s_1-th row of Z'_2 is the vector (A_1 S_1)_i⊗ (A_3 D_3)_j.Using Theorem <ref>, we can find a diagonal matrix D_2 ∈ℝ^n^2 × n^2 with d_2= O(z_2/ϵ) = O(s_1 d_3/ϵ) = O(k^3 /ϵ^5) nonzero entries such thatmin_X∈ℝ^d_2× z_2A_2 D_2 XZ_2' - A_2 _F^2≤ (1+ϵ)^4 .Let V' denote A_2 D_2. Then,min_X∈ℝ^d_2× z_2V' X Z'_2 - A_2 _F^2 ≤ (1+ϵ)^4 . We fix V' and W'. Let z_1 = d_2 · d_3. We define Z'_1 ∈ℝ^z_1× n^2 such that, ∀ i∈ [d_2], ∀ j ∈ [d_3],the i+(j-1)s_1-th row of Z'_1 is the vector (A_2 D_2)_i⊗ (A_3 D_3)_j.Using Theorem <ref>, we can find a diagonal matrix D_1 ∈ℝ^n^2 × n^2 with d_1= O(z_1/ϵ) = O(d_2 d_3/ϵ) = O(k^5/ϵ^9) nonzero entries such thatmin_X∈ℝ^d_1× z_1A_1 D_1 XZ_1' - A_1 _F^2≤ (1+ϵ)^5 .Let U' denote A_1 D_1. Then,min_X∈ℝ^d_1× z_1U' X Z'_1 - A_1 _F^2 ≤ (1+ϵ)^5 . Putting U',V',W' all together, we complete the proof.All the above analysis gives the running time O((A)) log n + n^2 (log n, k, 1/ϵ). To improve the running time, we need to use Algorithm <ref>, the similar analysis will go through, the running time will be improved to O((A) + n(k,1/ϵ)), but the sample complexity of c,r,k will be slightly worse (log factors).Given a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm which takes O((A) + n(k,1/ϵ)) time and outputs three matrices: C∈ℝ^n× c, a subset of columns of A, R∈ℝ^n× r a subset of rows of A, and T∈ℝ^n× t, a subset of tubes of A, together with a tensor U∈ℝ^c× r× t with (U)=k' where c=r=t=(k,1/ϵ) and k'=(k,1/ϵ) such thatU(C,R,T)- A _F^2≤ (1+ϵ) -k A_kmin A_k - A _F^2,or equivalently,∑_i=1^c ∑_j=1^r ∑_l=1^t U_i,j,l· C_i ⊗ R_j ⊗ T_l -A _F^2 ≤ (1+ϵ) -k A_kmin A_k - A _F^2holds with probability 9/10. The proof follows by combining Theorem <ref> and Theorem <ref> directly.Given a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm which takes (n) 2^(k,1/ϵ) time and outputs three matrices: C∈ℝ^n× c, a subset of columns of A, R∈ℝ^n× r a subset of rows of A, and T∈ℝ^n× t, a subset of tubes of A, as well as a tensor U∈ℝ^c× r× t where c=r=t=(k,1/ϵ) such that∑_i=1^c ∑_j=1^r ∑_l=1^t U_i,j,l· C_i ⊗ R_j ⊗ T_l -A_F^2 ≤ (1+ϵ) -k A_kmin A_k - A _F^2holds with probability at least 9/10. This follows by combining Theorem <ref> and the polynomial system verifier.§.§ CURT decomposition, (1+ϵ)-approximation§.§.§ Properties of leverage score sampling and BSS sampling Notice that, the BSS algorithm is a deterministic procedure developed in <cit.> for selecting rows from a matrix A∈ℝ^n× d (with A_2≤ 1 and A _F^2≤ k) using a selection matrix S so thatA^⊤ S^⊤ S A - A^⊤ A _2 ≤ϵ.The algorithm runs in (n,d,1/ϵ) time.Using the ideas from <cit.> and <cit.>, we are able to reduce the number of nonzero entries from O(ϵ^-2 klog k) to O( ϵ^-2k ), and also improve the running time to input sparsity.Given a -k matrix A∈ℝ^n× d, via leverage score sampling, we can obtain a diagonal matrix D with m nonzero entries such that, letting B = DA, if m=O(ϵ^-2 klog k), then, with probability at least 0.999, for all x∈ℝ^d,(1-ϵ)A x_2≤ B x_2 ≤ (1+ϵ)A x_2Given a -k matrix A∈ℝ^n× d, there exists an algorithm that runs in O((A) + n ( k,1/ϵ)) time and outputs a matrix B containing O(ϵ^-2 klog k ) re-weighted rows of A, such that with probability at least 0.999, for all x∈ℝ^d,(1-ϵ)A x_2≤ B x_2 ≤ (1+ϵ)A x_2 We choose a sparse embedding matrix (Definition <ref>) Π∈ℝ^d× s with s=(k/ε). With probability at least 0.999, Π^⊤ is a subspace embedding of A^⊤. Thus, (AΠ)=(A). Also, the leverage scores of AΠ are the same as those of A. Thus, we can compute the leverage scores of AΠ. The running time of computing AΠ is O((A)). Thus the total running time is O((A) + n ( k,1/ϵ)).Let B denote a matrix which contains O(ϵ^-2 klog k) rows of A∈ℝ^n× d. Choosing Π to be a sparse subspace embedding matrix of size d× O( ϵ^-6 (klog k)^2 ), with probability at least 0.999,B ΠΠ^⊤ B^⊤ - B B^⊤_2 ≤ϵ B _2^2. Combining Lemma <ref>, <ref> and the BSS algorithm, we obtain:Given a -k matrix A∈ℝ^n× d, there exists an algorithm that runs in O((A)+ n (k,1/ϵ)) time and outputs a sampling and rescaling diagonal matrix S that selects O(ϵ^-2 k) re-weighted rows of A, such that, with probability at least 0.999,A^⊤ S^⊤ S A - A^⊤ A _2 ≤ϵ A_2^2.or equivalently, for all x∈ℝ^d,(1-ϵ)A x_2 ≤SA x_2 ≤ (1+ϵ)A x_2.Using Lemma <ref>, we can obtain B. Then we apply a sparse subspace embedding matrix Π on the right of B. At the end, we run the BSS algorithm on BΠ and we are able to output O(ϵ^-2 k) re-weighted rows of BΠ. Using these rows, we are able to determine O(ϵ^-2 k) re-weighted rows of A.§.§.§ Row sampling for linear regression We are given A∈ℝ^n× d with A_2^2 ≤ 1 and A_F^2 ≤ k, and an ϵ∈ (0,1). There exists a diagonal matrix S with O(k/ϵ^2) nonzero entries such that(SA)^⊤ SA - A^⊤ A _2 ≤ϵ.Given a rank-k matrix A∈ℝ^n× d, vector b∈ℝ^n, and parameter ϵ>0, let U∈ℝ^n× (k+1) denote an orthonormal basis of [A,b].Let S∈ℝ^n× n denote a sampling and rescaling diagonal matrix according to Leverage score sampling and sparse BSS sampling of U with m nonzero entries. If m=O(k), then S is a (1± 1/2) subspace embedding for U; if m=O(k/ϵ), then S satisfies √(ϵ)-operator norm approximate matrix product for U. This follows by Lemma <ref>, Lemma <ref> and Theorem <ref>.Given A∈ℝ^n× d and b∈ℝ^n, let S∈ℝ^n× n denote a sampling and rescaling diagonal matrix. Let x^* denote min_xAx-b_2^2 and x' denote min_xS A x - Sb_2^2. If S is a (1± 1/2) subspace embedding for the column span of A, and ϵ' (=√(ϵ))-operator norm approximate matrix product for U adjoined with b-Ax^*, then, with probability at least .999,A x' - b _2^2 ≤ (1+ϵ )A x^* - b _2^2. We define = xmin A x - b_2. We define x' = xmin SA x - S b _2^2 and x^* = xmin A x - b _2^2. Let w = b-Ax^*. Let U denote an orthonormal basis of A. We can write Ax'-Ax^*=U β. Then, we have,Ax'-b_2^2 = Ax' - A x^* + AA^† b - b_2^2 by x^*=A^† b= U β + (UU^⊤ - I) b _2^2= Ax^*-Ax' _2^2 +Ax^*-b _2^2 by Pythagorean Theorem= U β_2^2 + ^2 = β_2^2 + ^2.If S is a (1±1/2) subspace embedding for U, then we can showβ_2 -U^⊤ S^⊤ S U β_2≤ β- U^⊤ S^⊤ S U β_2 by triangle inequality= ( I - U^⊤ S^⊤ S U ) β_2≤ I - U^⊤ S^⊤ S U _2 ·β_2≤ 1/2β_2. Thus, we obtainU^⊤ S^⊤ S U β_2 ≥β_2/2. Next, we can showU^⊤ S^⊤ S U β_2 = U^⊤ S^⊤ S (Ax'-Ax^*) _2^2 = U^⊤ S^⊤ S (A (SA)^† Sb -Ax^*) _2 by x'= (SA)^† Sb= U^⊤ S^⊤ S (b -Ax^*) _2 by SA (SA)^† = I= U^⊤ S^⊤ S w_2. by w= b- Ax^*We define U' = [U w/ w_2 ]. We define X and y to satisfy U=U'X and w=U'y. Then, we haveU^⊤ S^⊤ Sw _2 = U^⊤ S^⊤ Sw - U^⊤ w_2 by U^⊤ w= 0= X^⊤ U'^⊤ S^⊤ S U' y - X^⊤ U'^⊤ U' y _2 = X^⊤(U'^⊤ S^⊤ S U' - I) y _2≤ X _2 · U'^⊤ S^⊤ S U' - I _2 · y _2≤ ϵ' X_2 y_2 = ϵ'U_2 w _2= ϵ' , by U _2=1 and w_2 =where the fifth inequality follows since S satisfies ϵ'-operator norm approximate matrix product for the column span of U adjoined with w.Putting it all together, we haveAx'-b_2^2 = A x^* -b _2^2 +Ax^* - A x'_2^2 = ^2 + β_2^2≤ ^2 + 4 U^⊤ S^⊤ S w _2^2≤ ^2 + 4 (ϵ' )^2≤ (1+ϵ) ^2. by ϵ'= 1/2√(ϵ).Finally, note that S satisfies ϵ'-operator norm approximate matrix product for U adjoined with w if it is a (1±ϵ')-subspace embedding for U adjoined with w, which holds using BSS sampling by Theorem 5 of <cit.> with O(d/ϵ) samples. §.§.§ Leverage scores for multiple regressionGiven matrix A∈ℝ^n× d with orthonormal columns, and parameter ϵ>0, if S∈ℝ^n× n is a sampling and rescaling diagonal matrix according to the leverage scores of A where the number of nonzero entries is t=O(1/ϵ^2), then, for any B ∈ℝ^n × m, we haveA^⊤ S^⊤ S B - A^⊤ B _F^2 < ϵ^2A _F^2B _F^2 ,holds with probability at least 0.9999.Given matrix A∈ℝ^n× d with orthonormal columns, and parameter ϵ>0, if S∈ℝ^n× n is a sampling and rescaling diagonal matrix according to the leverage scores of A with m nonzero entries, then if m=O(d log d), then S is a (1± 1/2) subspace embedding for A. If m=O(d/ϵ), then S satisfies √(ϵ/d)-Frobenius norm approximate matrix product for A. This follows by Lemma <ref> and Lemma <ref>.Given A∈ℝ^n× d and B∈ℝ^n× m, let S∈ℝ^n× n denote a sampling and rescaling matrix according to A. Let X^* denote min_X A X - B_F^2 and X' denote min_X S A X - S B _F^2. Let U denote an orthonormal basis for A. If S is a (1± 1/2) subspace embedding for U, and satisfies ϵ'(=√(ϵ/d))-Frobenius norm approximate matrix product for U, then, we have thatA X' - B _F^2 ≤(1+ϵ)A X^* - B _F^2holds with probability at least 0.999.We define = min_X A X - B_F. Let A = U Σ V^⊤ denote the SVD of A. Since A has rank k, U and V have k columns. We can write A (X'-X^*)=U β. Then, we haveA X'-B _F^2 = A X'- AX^* + AA^† B - B _F^2 by X^* = A^† B = U β + (UU^⊤ - I) B _F^2= AX^* - AX'_F^2 + AX^* - B_F^2 by Pythagorean Theorem= U β_F^2 + ^2 = β_F^2 + ^2. If S is a (1± 1/2) subspace embedding for U, then we can show,β_F -U^⊤ S^⊤ S SU β_F ≤ β - U^⊤ S^⊤ S U β_F by triangle inequality= (I - U^⊤ S^⊤ S U) β_F≤ (I - U^⊤ S^⊤ S U) _2 ·β_F by A B _F ≤A_2 B_F≤ 1/2β_F. by (I - U^⊤ S^⊤ S U) _2≤ 1/2Thus, we obtainU^⊤ S^⊤ S U β_F ≥β_F/2.Next, we can showU^⊤ S^⊤ S U β_F = U^⊤ S^⊤ S (AX'- AX^*) _F = U^⊤ S^⊤ S (A(SA)^† Sb - AX^* ) _F by X'=(SA)^† SB = U^⊤ S^⊤ S (B - A X^* ) _F. by SA (SA)^† = I Then, we can showU^⊤ S^⊤ S (B-AX^*) _F ≤ ϵ'U^⊤_FB- AX^* _F by Lemma <ref>= ϵ' √(d). by U _F=√(d) and B-AX^*_F =Putting it all together, we haveAX'-B_F^2 = A X^* -B _F^2 +AX^* - A X'_F^2 = ^2 + β_F^2 by Equation (<ref>) ≤ ^2 + 4 U^⊤ S^⊤ S w _F^2 by Equation (<ref>) ≤ ^2 + 4 (ϵ' √(d))^2 by Equation (<ref>) ≤ (1+ϵ) ^2. by ϵ'= 1/2√(ϵ/d)§.§.§ Sampling columns according to leverage scores implicitly, improving polynomial running time to nearly linear running timeThis section explains an algorithm that is able to sample from the leverage scores from the ⊙ product of two matrices U,V without explicitly writing down U⊙ V. To build this algorithm we combine TensorSketch, some ideas from <cit.> and some ideas from <cit.>. Finally, we are able to improve the running time of sampling columns according to leverage scores from Ω(n^2) to O(n). Given two matrices U, V∈ℝ^k× n, we define A∈ℝ^k × n_1 n_2 to be the matrix where the i-th row of A is the vectorization of U^i ⊗ V^i, ∀ i ∈ [k]. Naïvely, in order to sample O((k,1/ϵ)) rows from A^⊤ according to leverage scores, we need to write down n^2 leverage scores. This approach will take at least Ω(n^2) running time. In the rest of this section, we will explain how to do it in O(n ·(log n,k,1/ϵ) ) time. In Section <ref>, we will explain how to extend this idea from 3rd order tensors to general q-th order tensors and remove the (log n) factor from running time, i.e., obtain O(n ·(k,1/ϵ) ) time. Given two matrices U∈ℝ^k× n_1 and V∈ℝ^k× n_2, there exists an algorithm that takes O( (n_1 +n_2) ·(log(n_1 n_2), k) · R_samples) time and samples R_samples columns of U ⊙ V ∈ℝ^k × n_1 n_2 according to the leverage scores of R^-1 (U ⊙ V), where R is the R of a QR factorization. We choose Π∈ℝ^n_1 n_2 × s_1 to be a TensorSketch. Then, according to Section <ref>, we can compute R^-1 in n ·(log n, k, 1/ϵ) time, where R is the R in a QR-factorization. We want to sample columns from U ⊙ V according to the square of the ℓ_2-norms of each column of R^-1 (U ⊙ V). However, explicitly writing down the matrix R^-1 (U ⊙ V) takes kn_1n_2 time, and the number of columns is already n_1 n_2. The goal is to sample columns from R^-1 (U ⊙ V) without explicitly computing the square of the ℓ_2-norm of each column.The first simple observation is that the following two sampling procedures are equivalent in terms of the column samples of a matrix that they take. (1) We sample a single entry from the matrix R^-1 (U ⊙ V) proportional to its squared value. (2) We sample a column from the matrix R^-1 (U ⊙ V) proportional to its squared ℓ_2-norm. Let the (i,j_1,j_2)-th entry denote the entry in the i-th row and the (j_1-1)n_2 +j_2-th column. We can show, for a particular column (j_1-1)n_2 +j_2,[sample an entry from the (j_1-1)n_2 +j_2 th column of a matrix]= ∑_i=1^k [sample the (i,j_1,j_2)-th entry of matrix]= ∑_i=1^k| (R^-1(U⊙ V))_i,(j_1-1)n_2 +j_2 |^2 / R^-1 (U ⊙ V) _F^2 = (R^-1(U⊙ V))_(j_1-1)n_2 +j_2^2 / R^-1 (U ⊙ V) _F^2 = [sample the (j_1-1)n_2 +j_2 th column of matrix].Thus, it is sufficient to show how to sample a single entry from matrix R^-1(U⊙ V) proportional to its squared value without writing down all of the entries of a k× n_1 n_2 matrix.We choose a Gaussian matrix G_1∈ℝ^g_1× k with g_1 = O(ϵ^-2log(n_1 n_2)). By Claim <ref> we can reduce the length of each column vector of matrix R^-1 U ⊙ V from k to g_1 while preserving the squared ℓ_2-norm of all columns simultaneously. Thus, we obtain a new matrix GR^-1 (U ⊙ V) ∈ℝ^g_1 × n_1 n_2, and sampling from this new matrix is equivalent to sampling from the original matrix R^-1 (U ⊙ V). In the following paragraphs, we explain a sampling procedure (also described in Procedure FastTensorLeverageScore in Algorithm <ref>) which contains three sampling steps. The first step is sampling i from [g_1], the second step is sampling j_2 from [n_2], and the last step is sampling j_1 from [n_1]. For each j_1∈[n_1], let U_j_1 denote the j_1-th column of U. For each i∈ [g_1], let G_1^i denote the i-th row of matrix G_1∈ℝ^g_1 × k, let U'^i∈ℝ^k× n_1 denote a matrix where the j_1-th column is (G^i R^-1)^⊤∘ U_j_1∈ℝ^k,∀ j∈ [n_1]. Then, using Claim <ref>, we have that (G^i R^-1) · (U ⊙ V) ∈ℝ^n_1 n_2 is a row vector where the entry in the (j_1-1)n_2 + j_2-th coordinate is the entry in the j_1-th row and j_2-th column of matrix (U'^i⊤ V)∈ℝ^n_1 × n_2. Further, the squared ℓ_2-norm of vector (G^i R^-1) · (U ⊙ V) is equal to the squared Frobenius norm of matrix (U'^i⊤ V). Thus, sampling i proportional to the squared ℓ_2-norm of vector (G^i R^-1) · (U ⊙ V) is equivalent to sampling i proportional to the squared Frobenius norm of matrix (U'^i⊤ V). Naïvely, computing the Frobenius norm of an n_1 × n_2 matrix requires O(n_1 n_2) time. However, we can choose a Gaussian matrix G_2,i∈ℝ^g_2 × n_1 to sample according to the value (G_2,i U'^i⊤) V _F^2, which can be computed in O((n_1+n_2)g_2k) time. By claim <ref>, (G_2,i U'^i⊤) V _F^2 ≈ ( U'^i⊤) V _F^2 with high probability. So far, we have finished the first step of the sampling procedure.For the second step of the sampling procedure, we need to sample j_2 from [n_2]. To do that, we need to compute the squared ℓ_2-norm of each column of U'^i⊤ V∈ℝ^n_1 × n_2. This can be done by choosing another Gaussian matrix G_3,i∈ℝ^g_3 × n_1. For all j_2∈ [n_2], by Claim <ref>, we have G_3,i U'^i⊤ V_j_2_2^2 ≈U'^i⊤ V_j_2_2^2. Also, for j_2∈ [n_2], G_3,i U'^i⊤ V_j_2_2^2 can be computed in nearly linear in n_1+n_2 time.For the third step of the sampling procedure, we need to sample j_1 from [n_1]. Since we already have i and j_2 from the previous two steps, we can directly compute | (U'^i⊤)^j_1 V_j_2 |^2, for all j_1. This only takes O(n_1 k) time.Overall, the running time is O( (n_1 +n_2) ·(log (n_1 n_2), k, 1/ϵ) ). Because our estimates are accurate enough, our sampling probabilities are also good approximations to the leverage score sampling probabilities. Putting it all together, we complete the proof.Given matrix R^-1 (U ⊙ V) ∈ℝ^k× n_1n_2, let G_1 ∈ℝ^g_1 × k denote a Gaussian matrix with g_1 = (ϵ^-2log(n_1 n_2)). Then with probability at least 1-1/(n_1 n_2), we have: for all j∈ [n_1n_2],(1-ϵ)R^-1 (U ⊙ V)_j _2^2 ≤ G_1 R^-1 (U ⊙ V)_j _2^2 ≤ (1+ϵ)R^-1 (U ⊙ V)_j _2^2.This follows by the Johnson-Lindenstrauss Lemma.For a fixed i∈ [g_1], let G_2,i∈ℝ^g_2× n_1 denote a Gaussian matrix with g_2 = O(ϵ^-2log(n_1 n_2)). Then with probability at least 1-1/(n_1n_2), we have: for all j_2 ∈ [n_2],(1-ϵ) U'^i⊤ V_j_2_2^2 ≤ (G_2,i U'^i⊤) V_j_2_2 ≤ (1+ϵ)U'^i⊤ V_j_2_2^2. By taking the union bound over all i∈ [g_1], we obtain a stronger claim,With probability at least 1-1/(n_1n_2), we have : for all i∈ [g_1], for all j_2 ∈ [n_2],(1-ϵ) U'^i⊤ V_j_2_2^2 ≤ (G_2,i U'^i⊤) V_j_2_2 ≤ (1+ϵ)U'^i⊤ V_j_2_2^2. Similarly, if we choose G_3,i to be a Gaussian matrix, we can obtain the same result as for G_2,i:With probability at least 1-1/(n_1n_2), we have : for all i∈ [g_1], for all j_2 ∈ [n_2],(1-ϵ) U'^i⊤ V_j_2_2^2 ≤ (G_3,i U'^i⊤) V_j_2_2 ≤ (1+ϵ)U'^i⊤ V_j_2_2^2. For any i∈ [g_1], j_1∈ [n_1], j_2∈ [n_2], let G_1^i denote the i-th row of matrix G_1∈ℝ^g_1 × k. Let (U ⊙ V)_(j_1-1)n_2 +j_2 denote the (j_1-1)n_2 + j_2-th column of matrix ℝ^k× n_1 n_2. Let (U'^i⊤)^j_1 denote the j_1-th row of matrix (U'^i⊤)∈ℝ^n_1 × k. Let V_j_2 denote the j_2-th column of matrix V∈ℝ^k× n_2. Then, we haveG_1^i R^-1 (U ⊙ V)_(j_1-1)n_2 +j_2 = (U'^i⊤)^j_1 V_j_2.This follows by,G_1^i R^-1 (U ⊙ V)_(j_1-1)n_2 +j_2 = G_1^i R^-1 (U_j_1∘ V_j_2) = (G_1^i R^-1∘ (U_j_1)^⊤ ) V_j_2 =(U'^i⊤)^j_1 V_j_2. Given A∈ℝ^n× n^2, V,W∈ℝ^k× n, for any ϵ>0, there exists an algorithm that runs in O(n ·(k,1/ϵ)) time and outputs a diagonal matrix D∈ℝ^n^2 × n^2 with m=O(klog k + k/ϵ) nonzero entries such that,U (V⊙ W) - A_F^2 ≤ (1+ϵ) min_U∈ℝ^n× k U (V⊙ W) - A_F^2,holds with probability at least 0.999, where U denotes the optimal solution to min_UU (V ⊙ W) D - A D _F^2. This follows by combining Theorem <ref>, Corollary <ref>, and Lemma <ref>.Replacing Theorem <ref> (Algorithm <ref>) by Lemma <ref> (Algorithm <ref>), we can obtain a slightly different version of Lemma <ref> with n(log n,k,1/ϵ) running time, where the dependence on k is better. §.§.§ Input sparsity time algorithmGiven a 3rd order tensor A∈ℝ^n× n × n, let k≥ 1, and let U_B,V_B,W_B∈ℝ^n× k denote a rank-k, α-approximation to A. Then there exists an algorithm which takes O((A) +n ( k, 1/ϵ) ) time and outputs three matrices C∈ℝ^n× c with columns from A, R∈ℝ^n× r with rows from A, T∈ℝ^n× t with tubes from A, and a tensor U∈ℝ^c× r× t with (U)=k such that c=r=t=O(klog k +k/ϵ), and∑_i=1^c ∑_j=1^r ∑_l=1^t U_i,j,l· C_i ⊗ R_j ⊗ T_l - A _F^2 ≤(1+ϵ) αmin_-k A' A' - A_F^2holds with probability 9/10.We define: = min_-k A' A' - A_F^2.We already have three matrices U_B∈ℝ^n× k, V_B∈ℝ^n× k and W_B∈ℝ^n× k and these three matrices provide a -k, α-approximation to A, i.e.,∑_i=1^k ( U_B )_i ⊗ (V_B)_i ⊗ (W_B)_i - A _F^2 ≤α.Let B_1 = V_B^⊤⊙ W_B^⊤∈ℝ^k× n^2 denote the matrix where the i-th row is the vectorization of (V_B)_i ⊗ (W_B)_i. Let D_1 ∈ℝ^n^2 × n^2 be a sampling and rescaling matrix corresponding to sampling by the leverage scores of B_1^⊤; there are d_1 nonzero entries on the diagonal of D_1. Let A_i∈ℝ^n× n^2 denote the matrix obtained by flattening A along the i-th direction, for each i∈ [3].Define U^*∈ℝ^n× k to be the optimal solution to U∈ℝ^n× kmin U B_1 - A_1_F^2, U = A_1 D_1 (B_1 D_1)^†∈ℝ^n× k, and V_0 ∈ℝ^n× k to be the optimal solution to V∈ℝ^n× kmin V ·(U^⊤⊙ W_B^⊤) - A_2 _F^2. Due to Lemma <ref>, if d_1=O(klog k+k/ϵ) then with constant probability, we haveU B_1 - A _1 _F^2 ≤α_D_1 U^* B_1 - A_1 _F^2. Recall that ( U^⊤⊙ W_B^⊤) ∈ℝ^k× n^2 denotes the matrix where the i-th row is the vectorization of U_i ⊗ (W_B)_i, ∀ i∈ [k]. Now, we can show,V_0 · ( U^⊤⊙ W_B^⊤ ) - A_2 _F^2 ≤ U B_1 - A_1 _F^2 by V_0 = V∈ℝ^n× kmin V · ( U^⊤⊙ W_B^⊤ ) - A_2_F^2≤ α_D_1 U^* B_1 - A_1 _F^2 by Equation (<ref>) ≤ α_D_1 U_B B_1 - A_1 _F^2 by U^* = U∈ℝ^n× kmin U B_1 - A_1 _F^2≤ α_D_1α. by Equation (<ref>) We define B_2= U^⊤⊙ W_B^⊤. Let D_2∈ℝ^n^2 × n^2 be a sampling and rescaling matrix corresponding to the leverage scores of B_2^⊤. Suppose there are d_2 nonzero entries on the diagonal of D_2.Define V^*∈ℝ^n× k to be the optimal solution to min_V∈ℝ^n× k V B_2 - A_2 _F^2, V= A_2 D_2 (B_2 D_2)^†∈ℝ^n× k, W_0∈ℝ^n× k to be the optimal solution to W∈ℝ^n× kmin W· ( U^⊤⊙V^⊤ ) - A_3 _F^2, and V' to be the optimal solution to V∈ℝ^n× kminV B_2 D_2 - A_2 D_2_F^2.Due to Lemma <ref>, with constant probability, we haveV B_2 - A_2 _F^2 ≤α_D_2 V^* B_2 - A_2 _F^2.Recall that (U^⊤⊙V^⊤) ∈ℝ^k× n^2 denotes the matrix where the i-th row is the vectorization of U_i ⊗V_i, ∀ i∈ [k]. Now, we can show,W_0 · (U^⊤⊙V^⊤ ) - A_3 _F^2 ≤ V B_2 - A_2 _F^2 by W_0 = W∈ℝ^n× kmin W · ( U^⊤⊙V^⊤ ) - A_3 _F^2≤ α_D_2 V^* B_2 - A_2 _F^2 by Equation (<ref>) ≤ α_D_2 V_0 B_2 - A_2 _F^2 by V^* =V∈ℝ^n× kmin V B_2 - A_2 _F^2≤ α_D_2α_D_1α. by Equation (<ref>)We define B_3= U^⊤⊙V^⊤. Let D_3∈ℝ^n^2 × n^2 denote a sampling and rescaling matrix corresponding to sampling by the leverage scores of B_3^⊤. Suppose there are d_3 nonzero entries on the diagonal of D_3.Define W^*∈ℝ^n× k to be the optimal solution to min_W∈ℝ^n× k W B_3 - A_3 _F^2, W= A_3 D_3 (B_3 D_3)^†∈ℝ^n× k, and W' to be the optimal solution to W∈ℝ^n× kminW B_3 D_3 - A_3 D_3_F^2. Due to Lemma <ref> with constant probability, we haveW B_3 - A_3 _F^2 ≤α_D_3 W^* B_3 - A_3 _F^2. Now we can show,W B_3 - A_3 _F^2 ≤ α_D_3 W^* B_3 - A_3 _F^2, by Equation (<ref>) ≤ α_D_3 W_0 B_3 - A_3 _F^2, by W^* = W∈ℝ^n× kmin W B_3 - A_3 _F^2≤ α_D_3α_D_2α_D_1α. by Equation (<ref>)This implies,∑_i=1^k U_i ⊗V_i ⊗W_i - A _F^2 ≤ O(1) α^2.where U = A_1 D_1 (B_1 D_1)^†, V = A_2D_2 (B_2 D_2)^†, W=A_3D_3 (B_3 D_3)^†.By Lemma <ref>, we need to set d_1=d_2=d_3=O(klog k+k/ϵ). Note that B_1= (V_B^⊤⊙ W_B^⊤). Thus D_1 can be found in n·(k,1/ϵ) time. Because D_1 has a small number of nonzero entries on the diagonal, we can compute B_1 D_1 quickly without explicitly writing down B_1. Also A_1 D_1 can be computed in (A) time. Using (A_1D_1) and (B_1D_1), we can compute U in n(k,1/ϵ) time. In a similar way, we can compute B_2, D_2, B_3, and D_3. Since tensor U is constructed based on three (k,1/ϵ) size matrices, (B_1D_1)^†, (B_2D_2)^†, and (B_3 D_3)^†, the overall running time is O((A) + n (k,1/ϵ) ) §.§.§ Optimal sample complexity algorithmGiven a 3rd order tensor A∈ℝ^n× n × n, let k≥ 1, and let U_B,V_B,W_B∈ℝ^n× k denote a rank-k, α-approximation to A. Then there exists an algorithm which takes O((A) log n +n^2 (log n, k, 1/ϵ) ) time and outputs three matrices: C∈ℝ^n× c with columns from A, R∈ℝ^n× r with rows from A, T∈ℝ^n× t with tubes from A, and a tensor U∈ℝ^c× r× t with (U)=k such that c=r=t=O(k/ϵ), and∑_i=1^c ∑_j=1^r ∑_l=1^t U_i,j,l· C_i ⊗ R_j ⊗ T_l - A _F^2 ≤(1+ϵ) αmin_-k A' A' - A_F^2holds with probability 9/10. The proof is almost the same as the proof of Theorem <ref>. The only difference is that instead of using Theorem <ref>, we use Theorem <ref>.§.§ Face-based selection and decompositionPreviously we provided column-based tensor CURT algorithms, which are algorithms that can select a subset of columns from each of the three dimensions. Here we provide two face-based tensor CURT decomposition algorithms. The first algorithm runs in polynomial time and is a bicriteria algorithm (the number of samples is (k/ϵ)). The second algorithm needs to start with a rank-k (1+O(ϵ))-approximate solution, which we then show how to combine with our previous algorithm. Both of our algorithms are able to select a subset of column-row faces, a subset of row-tube faces and a subset of column-tube faces. The second algorithm is able to output U, but the first algorithm is not.§.§.§ Column-row, column-tube, row-tube face subset selection Given a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm which takes O((A)) log n + n^2 (log n, k,1/ϵ) time and outputs three tensors : a subset C∈ℝ^c× n× n of row-tube faces of A, a subset R∈ℝ^n× r × n of column-tube faces of A, and a subset T∈ℝ^n× n × t of column-row faces of A, where c=r=t=(k,1/ϵ), and for which there exists a tensor U∈ℝ^tn × cn × rn for whichU(T_1,C_2,R_3)-A _F^2 ≤ (1+ϵ) min_-k A' A' - A _F^2,or equivalently,∑_i=1^tn∑_j=1^cn∑_l=1^rnU_i,j,l· (T_1)_i ⊗ (C_2)_j ⊗ (R_3)_l- A _F^2 ≤(1+ϵ)min_-k A' A' - A_F^2. We fix V^*∈ℝ^n× k and W^*∈ℝ^n× k. We define Z_1∈ℝ^k× n^2 where the i-th row of Z_1 is the vector V_i ⊗ W_i. Choose a sketching (Gaussian) matrix S_1 ∈ℝ^n^2 × s_1 (Definition <ref>), and let U = A_1 S_1 (Z_1 S_1)^†∈ℝ^n× k. Following a similar argument as in the previous theorem, we haveU Z_1 - A_1 _F^2 ≤ (1+ϵ) .We fix U and W^*. We define Z_2∈ℝ^k× n^2 where the i-th row of Z_2 is the vector U_i ⊗ W^*_i. Choose a sketching (Gaussian) matrix S_2∈ℝ^n^2× s_2 (Definition <ref>), and let V = A_2 S_2 (Z_2 S_2)^†∈ℝ^n× k. Following a similar argument as in the previous theorem, we haveV Z_2 - A_2 _F^2 ≤ (1+ϵ)^2 . We fix U and V. Note that U=A_1 S_1 (Z_1 S_1)^† and V=A_2 S_2 (Z_2 S_2)^†. We define Z_3∈ℝ^k× n^2 such that the i-th row of Z_3 is the vector U_i ⊗V_i. Let z_3 = s_1 · s_2. We define Z'_3∈ℝ^z_3× n^2 such that, ∀ i∈ [s_1], ∀ j∈ [s_2], the i+(j-1) s_1-th row of Z'_3 is the vector (A_1 S_1)_i ⊗ (A_2 S_2)_j.We define U_3∈ℝ^n× z_3 to be the matrix where the i+(j-1) s_1-th column is (A_1 S_1)_i and V_3 ∈ℝ^n× z_3 to be the matrix where the i+(j-1) s_1-th column is (A_2 S_2)_j. Then Z_3' = (U_3^⊤⊙ V_3^⊤). We first have,min_W ∈ℝ^n× k, X∈ℝ^k × z_3 W X Z_3' - A_3 _F^2 ≤min_W ∈ℝ^n× k W Z_3 - A_3 _F^2≤ (1+ϵ)^2 . Now consider the following objective function,min_W∈ℝ^n× z_3 V_3 ·( W^⊤⊙ U_3^⊤ ) - A_2 _F^2.Let D_3 denote a sampling and rescaling diagonal matrix according to V_1∈ℝ^n× z_3, let d_3 denote the number of nonzero entries of D_3. Then we havemin_W∈ℝ^n× z_3 D_3 V_3 ·( W^⊤⊙ U_3^⊤ ) - D_3 A_2 _F^2 = min_W∈ℝ^n× z_3U_3 ⊗ (D_3 V_3) ⊗ W- A(I,D_3,I) _F^2 = min_W∈ℝ^n× z_3 W· ( U_3^⊤⊙ (D_3 V_3)^⊤ )- (A(I,D_3,I))_3 _F^2,where the first equality follows by retensorizing the objective function, and the second equality follows by flattening the tensor along the third dimension.Let Z_3 denote ( U_3^⊤⊙ (D_3 V_3)^⊤ ) ∈ℝ^z_3 × nd_3 and W'=(A(I,D_3,I))_3∈ℝ^n × nd_3. Using Theorem <ref>, we can find a diagonal matrix D_3∈ℝ^n^2× n^2 with d_3 = O(z_3/ϵ) = O(k^2/ϵ^3) nonzero entries such thatU_3 ⊗ V_3 ⊗ (W'Z_3^†) - A _F^2 ≤ (1+ϵ)^3 . We define U_2=U_3∈ℝ^n× z_2 with z_2=z_3. We define W_2 = W' Z_3^†∈ℝ^n× z_2 with z_2=z_3. We consider,min_V∈ℝ^n× z_2 U_2 · (V^⊤⊙ W_2^⊤ ) -A_1_F^2.Let D_2 denote a sampling and rescaling matrix according to U_2, and let d_2 denote the number of nonzero entries of D_2. Then, we havemin_V∈ℝ^n× z_2 D_2 U_2 · (V^⊤⊙ W_2^⊤) - D_2 A_1_F^2 = min_V∈ℝ^n× z_2 D_2 U_2 ⊗ V ⊗ W_2- A(D_2,I,I)_F^2 = min_V∈ℝ^n× z_2 V· (W_2^⊤⊙(D_2 U_2)^⊤ ) - (A(D_2,I,I))_2_F^2,where the first equality follows by retensorizing the objective function, and the second equality follows by flattening the tensor along the second dimension.Let Z_2 denote (W_2^⊤⊙(D_2 U_2)^⊤ ) ∈ℝ^z_2 × nd_2 and V' = (A(D_2,I,I))_2 ∈ℝ^n× nd_2. Using Theorem <ref>, we can find a diagonal matrix D_2 ∈ℝ^n^2 × n^2 with d_2= O(z_2/ϵ)nonzero entries such thatU_2 ⊗ (V' Z_2^†) ⊗ W_2 -A _F^2 ≤ (1+ϵ)^4 . We define W_1=W_2∈ℝ^n× z_1 with z_1=z_2, and define V_1=(V' Z_2^†) ∈ℝ^n× z_1 with z_1 =z_2.Let D_1 denote a sampling and rescaling matrix according to W_1, and let d_1 denote the number of nonzero entries of D_1. Then we havemin_U∈ℝ^n× z_1 D_1 W_1 · (U^⊤⊙ V_1^⊤) - D_1 A_3 _F^2= min_U∈ℝ^n× z_1 U ⊗ V_1 ⊗(D_1 W_1) -A(I,I,D_1) _F^2= min_U∈ℝ^n× z_1 U ·( V_1^⊤⊙(D_1 W_1)^⊤) -A(I,I,D_1)_1 _F^2where the first equality follows by unflattening the objective function, and second equality follows by flattening the tensor along the first dimension.Let Z_1 denote ( V_1^⊤⊙(D_1 W_1)^⊤)∈ℝ^z_1× nd_1, and U'=A(I,I,D_1)_1∈ℝ^n× nd_1. Using Theorem <ref>, we can find a diagonal matrix D_1 ∈ℝ^n^2 × n^2 with d_1= O(z_1/ϵ)nonzero entries such that(U' Z_1^†) ⊗ (V_1) ⊗ W_1 -A _F^2 ≤ (1+ϵ)^5 ,which means,(U' Z_1^†) ⊗ (V'Z_2^†) ⊗ (W'Z_3^†) -A _F^2 ≤ (1+ϵ)^5 . Putting U',V',W' together completes the proof. Given a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm which takes O((A)) + n^2 ( k,1/ϵ) time and outputs three tensors : a subset C∈ℝ^c× n× n of row-tube faces of A, a subset R∈ℝ^n× r × n of column-tube faces of A, and a subset T∈ℝ^n× n × t of column-row faces of A, where c=r=t=(k,1/ϵ), so that there exists a tensor U∈ℝ^tn × cn × rn for whichU(T_1,C_2,R_3)-A _F^2 ≤ (1+ϵ) min_-k A' A' - A _F^2,or equivalently,∑_i=1^tn∑_j=1^cn∑_l=1^rnU_i,j,l· (T_1)_i ⊗ (C_2)_j ⊗ (R_3)_l- A _F^2 ≤(1+ϵ)min_-k A' A' - A_F^2If we allow a (k/ε) factor increase in running time and a (k/ε) factor increase in the number of faces selected, then instead of using generalized row subset selection, which has running time depending on log n, we can use the technique in Section <ref> to avoid the log n factor.§.§.§ CURT decompositionGiven a 3rd order tensor A∈ℝ^n× n × n, let k≥ 1, and let U_B,V_B,W_B∈ℝ^n× k denote a rank-k, α-approximation to A. Then there exists an algorithm which takes O((A)) log n +n^2 (log n, k, 1/ϵ) time and outputs three tensors: C∈ℝ^c× n × n with row-tube faces from A, R∈ℝ^n× r × n with colum-tube faces from A, T∈ℝ^n× n× t with column-row faces from A, and a (factorization of a) tensor U∈ℝ^tn× cn× rn with (U)=k for which c=r=t=O(k/ϵ) andU(T_1,C_2,R_3) - A _F^2 ≤ (1+ϵ) αmin_-k A' A' - A _F^2,or equivalently,∑_i=1^tn∑_j=1^cn∑_l=1^rnU_i,j,l· (T_1)_i ⊗ (C_2)_j ⊗ (R_3)_l- A _F^2 ≤(1+ϵ) αmin_-k A' A' - A_F^2holds with probability 9/10. We already have three matrices U_B∈ℝ^n× k, V_B∈ℝ^n× k and W_B∈ℝ^n× k and these three matrices provide a -k, α-approximation to A, i.e.,U_B ⊗ V_B ⊗ W_B - A _F^2 ≤αmin_-k A' A' - A _F^2 _. We can consider the following problem,min_U ∈ℝ^n× k W_B · (U^⊤⊙ V_B^⊤) - A_3 _F^2.Let D_1 denote a sampling and rescaling diagonal matrix according to W_B, and let d_1 denote the number of nonzero entries of D_1. Then we havemin_U ∈ℝ^n× k (D_1 W_B) ·(U^⊤⊙ V_B^⊤ ) - D_1 A_3 _F^2 =min_U ∈ℝ^n× k U ⊗ V_B ⊗ D_1 W_B - A(I,I,D_1) _F^2 =min_U ∈ℝ^n× k U · ( V_B^⊤⊙ (D_1 W_B)^⊤) - ( A(I,I,D_1) )_1 _F^2,where the first equality follows by retensorizing the objective function, and the second equality follows by flattening the tensor along the first dimension. Let Z_1 denote V_B^⊤⊙ (D_1 W_B)^⊤∈ℝ^k × n d_1, and define U = (A(I,I,D_1))_1 Z_1^†∈ℝ^n× k. Then we haveU⊗ V_B ⊗ W_B -A _F^2 ≤ (1+ϵ) α.In the second step, we fix U and W_B, and consider the following objective function,min_V∈ℝ^n× kU· (V^⊤⊙ W_B) - A_1 _F^2.Let D_2 denote a sampling and rescaling matrix according to U, and let d_2 denote the number of nonzero entries of D_2. Then we have,min_V∈ℝ^n× k (D_2 U) · (V^⊤⊙ W_B^⊤) - D_2 A_1 _F^2 = min_V∈ℝ^n× k (D_2 U) ⊗ V⊗ W_B - A(D_2,I,I) _F^2 = min_V∈ℝ^n× k V ·(W_B^⊤⊙ (D_2 U)^⊤ )- (A(D_2,I,I))_2 _F^2,where the first equality follows by unflattening the objective function, and the second equality follows by flattening the tensor along the second dimension. Let Z_2 denote ( W_B^⊤⊙ (D_2 U)^⊤ )∈ℝ^k × nd_2, and define V =(A(D_2,I,I))_2 (Z_2)^†∈ℝ^n× k. Then we have,U⊗V⊗ W_B - A _F^2 ≤ (1+ϵ)^2 α.In the third step, we fix U and V, and consider the following objective function,min_W∈ℝ^n× kV· ( W ⊙U ) - A_2 _F^2.Let D_3 denote a sampling and rescaling matrix according to V, and let d_3 denote the number of nonzero entries of D_3. Then we have,min_W∈ℝ^n× k (D_3 V) · (W^⊤⊙U^⊤) - D_3 A_2 _F^2 = min_W∈ℝ^n× kU⊗ (D_3 V) ⊗ W-A(I,D_3,I) _F^2 = min_W∈ℝ^n× k W · (U^⊤⊙ (D_3 V)^⊤ )-(A(I,D_3,I))_3 _F^2,where the first equality follows by retensorizing the objective function, and the second equality follows by flattening the tensor along the third dimension. Let Z_3 denote (U^⊤⊙ (D_3 V)^⊤ ) ∈ℝ^k× nd_3, and define W = (A(I,D_3,I))_3(Z_3)^†. Putting it all together, we have,U⊗V⊗W - A_F^2 ≤ (1+ϵ)^3 α.This implies(A(I,I,D_1))_1 Z_1^†⊗ (A(D_2,I,I))_2 Z_2^†⊗ (A(I,D_3,I))_3 Z_3^† - A_F^2 ≤ (1+ϵ)^3 α. §.§ Solving small problemsLet max_i{t_i, d_i}≤ n. Given a t_1 × t_2 × t_3 tensor A and three matrices: a t_1 × d_1 matrix T_1, a t_2 × d_2 matrix T_2, and a t_3 × d_3 matrix T_3, if for any δ > 0 there exists a solution tomin_X_1,X_2,X_3∑_i=1^k (T_1 X_1)_i ⊗ (T_2 X_2)_i ⊗ (T_3 X_3)_i - A _F^2 := ,and each entry of X_i can be expressed using O(n^δ) bits, then there exists an algorithm that takes n^O(δ)· 2^ O( d_1 k+d_2 k+d_3 k) time and outputs three matrices: X_1, X_2, and X_3 such that (T_1 X_1)⊗ (T_2 X_2) ⊗ (T_3X_3) - A_F^2 =.For each i∈ [3], we can create t_i× d_i variables to represent matrix X_i. Let x denote this list of variables. Let B denote tensor ∑_i=1^k (T_1 X_1)_i ⊗ (T_2X_2)_i⊗ (T_3X_3)_i and let B_i,j,l(x) denote an entry of tensor B (which can be thought of as a polynomial written in terms of x). Then we can write the following objective function,min_x∑_i=1^t_1∑_j=1^t_2∑_l=1^t_3 ( B_i,j,l(x) -A_i,j,l )^2.We slightly modify the above objective function to obtain a new objective function,min_x,σ ∑_i=1^t_1∑_j=1^t_2∑_l=1^t_3(B_i,j,l(x) -A_i,j,l)^2 ,s.t. x _2^2≤ 2^O(n^δ),where the last constraint is unharmful, because there exists a solution that can be written using O(n^δ) bits. Note that the number of inequality constraints in the above system is O(1), the degree is O(1), and the number of variables is v=(d_1k+d_2k+d_3k). Thus by Theorem <ref>, the minimum nonzero cost is at least(2^O(n^δ) )^-2^O ( v ) .It is clear that the upper bound on the cost is at most 2^O(n^δ). Thus the number of binary search steps is at most log (2^O(n^δ) ) 2^O(v). In each step of the binary search, we need to choose a cost C between the lower bound and the upper bound, and write down the polynomial system,∑_i=1^t_1∑_j=1^t_2∑_l=1^t_3(B_i,j,l(x) -A_i,j,l)^2 ≤ C, x _2^2≤ 2^O(n^δ).Using Theorem <ref>, we can determine if there exists a solution to the above polynomial system. Since the number of variables is v, and the degree is O(1), the number of inequality constraints is O(1). Thus, the running time is( bitsize) · (#·)^# = n^O(δ) 2^O(v). §.§ Extension to general q-th order tensors This section provides the details for our extensions from 3rd order tensors to general q-th order tensors. In most practical applications, the order q is a constant. Thus, to simplify the analysis, we use O_q(·) to hide dependencies on q.§.§.§ Fast sampling of columns according to leverage scores, implicitly This section explains an algorithm that is able to sample from the leverage scores from the ⊙ product of q matrices U_1,U_2,⋯,U_q without explicitly writing down U_1 ⊙ U_2 ⊙⋯ U_q. To build this algorithm we combine TensorSketch, some ideas from <cit.>, and some techniques from <cit.>. Finally, we improve the running time for sampling columns according to the leverage scores from (n) to O(n). Given q matrices U_1, U_2,⋯, U_q, with each such matrix U_i having size k × n_i, we define A∈ℝ^k×∏_i=1^q n_i to be the matrix where the i-th row of A is the vectorization of U_1^i ⊗ U_2^i ⊗⋯⊗ U_q^i, ∀ i ∈ [k]. Naïvely, in order to sample (k,1/ϵ) rows from A according to the leverage scores, we need to write down ∏_i=1^q n_i leverage scores. This approach will take at least ∏_i=1^q n_i running time. In the remainder of this section, we will explain how to do it in O_q(n ·(k,1/ϵ)) time for any constant p, and max_i∈[q] n_i ≤ n. Given q matrices U_1∈ℝ^k× n_1, U_2 ∈ℝ^k× n_2, ⋯, U_q ∈ℝ^k× n_q, let max_i n_i ≤ n. There exists an algorithm that takes O_q( n ·(k,1/ϵ) · R_samples) time and samples R_samples columns of U_1 ⊙ U_2 ⊙⋯⊙ U_q ∈ℝ^k ×∏_i=1^q n_i according to the leverage scores of U_1 ⊙ U_2 ⊙⋯⊙ U_q. Let max_i n_i ≤ n. First, choosing Π_0 to be a TensorSketch, we can compute R^-1 in O_q(n ( k,1/ϵ)) time, where R is the R in a QR-factorization. We want to sample columns from U_1 ⊙ U_2 ⊙⋯⊙ U_q according to the square of the ℓ_2-norm of each column of R^-1(U_1 ⊙ U_2 ⊙⋯ U_q ). The issue is the number of columns of this matrix is already ∏_i=1^q n_i. The goal is to sample columns from R^-1(U_1 ⊙ U_2 ⊙⋯ U_q ) without explicitly computing the square of the ℓ_2-norm of each column.Similarly as in the proof of Lemma <ref>, we have the observation that the following two sampling procedures are equivalent in terms of sampling a column of a matrix: (1) We sample a single entry from matrix R^-1 (U_1⊙ U_2 ⊙⋯⊙ U_q) proportional to its squared value, (2) We sample a column from matrix R^-1 (U_1 ⊙ U_2 ⊙⋯⊙ U_q) proportional to its squared ℓ_2-norm. Let the (i,j_1,j_2,⋯,j_q)-th entry denote the entry in the i-th row and the j-th column, wherej = ∑_l=1^q-1 (j_l - 1) ∏_t=l+1^q n_t + j_q.Similarly to Equation (<ref>), we can show, for a particular column j,[we sample an entry from the j-th column of matrix] = [we sample the j-th column of a matrix].Thus, it is sufficient to show how to sample a single entry from matrix R^-1 (U_1 ⊙ U_2 ⊙⋯⊙ U_q) proportional to its squared value without writing down all the entries of the k×∏_i=1^q n_i matrix.Let V_0 denote R^-1. Let n_0 denote the number of rows of V_0.In the next few paragraphs, we describe a sampling procedure (procedure FastTensorLeverageScoreGeneralOrder in Algorithm <ref>) which first samples j_0 from [n_0], then samples j_1 from [n_1], ⋯, and at the end samples j_q from [n_q].In the first step, we want to sample j_0 from [n_0] proportional to the squared ℓ_2-norm of that row. To do this efficiently, we choose Π_1∈ℝ^∏_i=1^q n_i × s_1 to be a TensorSketch to sketch on the right of V_0 ( U_1 ⊙ U_2 ⊙⋯⊙ U_q ). By Section <ref>, as long as s_1 = O_q((k,1/ϵ)), then Π_1 is a (1±ϵ)-subspace embedding matrix. Thus with probability 1-1/Ω(q), for all i∈ [n_0],(V_0)^i ( ( U_1 ⊙ U_2 ⊙⋯⊙ U_q ) Π_1)_2^2 = (1±ϵ)(V_0)^i ( ( U_1 ⊙ U_2 ⊙⋯⊙ U_q ))_2^2,which means we can sample j_0 from [n_0] in O_q( n(k,1/ϵ) ) time.In the second step, we have already obtained j_0. Using that row of V_0 with U_1, we can form a new matrix V_1∈ℝ^n_1× k in the following sense,(V_1)^i =(V_0)^j_0∘ (U_1)_i^⊤, ∀ i ∈ [n_1],where (V_1)^i denotes the i-th row of matrix V_1, (V_0)^j_0 denotes the j_0-th row of V_0 and (U_1)_i is the i-th column of U_1. Another important observation is, the entry in the (j_1,j_2,⋯,j_q)-th coordinate of vector (V_0)^j_0 ( U_1 ⊙ U_2 ⊙⋯⊙ U_q ) is the same as the entry in the j_1-th row and (j_2,⋯,j_q)-th column of matrix V_1 (U_2⊙ U_3 ⊙⋯⊙ U_q). Thus, sampling j_1 is equivalent to sampling j_1 from the new matrix V_1 (U_2⊙ U_3 ⊙⋯⊙ U_q) proportional to the squared ℓ_2-norm of that row. We still have the computational issue that the length of the row vector is very long. To deal with this, we can choose Π_2∈ℝ^∏_i=2^q n_i × s_2 to be a TensorSketch to multiply on the right of V_1 (U_2⊙ U_3 ⊙⋯⊙ U_q).By Section <ref>, as long as s_2 = O_q((k,1/ϵ)), then Π_2 is a (1±ϵ)-subspace embedding matrix. Thus with probability 1-1/Ω(q), for all i∈ [n_1],(V_1)^i ( (U_2 ⊙⋯⊙ U_q ) Π_2)_2^2 = (1±ϵ)(V_1)^i ( ( U_2 ⊙⋯⊙ U_q ))_2^2,which means we can sample j_1 from [n_1] in O_q( n( k,1/ϵ ) ) time.We repeat the above procedure until we obtain each of j_0, j_1, ⋯, j_q. Note that the last one, j_q, is easier, since the length of the vector is already small enough, and so we do not need to use TensorSketch for it.By Section <ref>, the time for multiplying by TensorSketch is O_q (n( k, 1 / ϵ ) ). Setting ϵ to be a small constant, and taking a union bound over O(q) events completes the proof.Given A∈ℝ^n_0×∏_i=1^q n_i, U_1,U_2,⋯, U_q∈ℝ^k× n, for any ϵ>0, there exists an algorithm that runs in O(n ·(k,1/ϵ)) time and outputs a diagonal matrix D∈ℝ^∏_i=1^q n_i ×∏_i=1^q n_i with m=O(klog k + k/ϵ) nonzero entries such that,U (U_1⊙ U_2 ⊙⋯⊙ U_q) - A_F^2 ≤ (1+ϵ) min_U∈ℝ^n× k U (U_1⊙ U_2 ⊙⋯⊙ U_q) - A_F^2,holds with probability at least 0.999, where U denotes the optimal solution ofmin_U∈ℝ^n_0 × k U (U_1⊙ U_2 ⊙⋯⊙ U_q) D - A D _F^2.This follows by combining Theorem <ref>, Corollary <ref>, and Lemma <ref>.§.§.§ General iterative existential proof Given a q-th order tensor A∈ℝ^n× n×⋯× n, we fix U_1^*, U_2^*, ⋯, U_q^* ∈ℝ^n× k to be the best rank-k solution (if it does not exist, then we replace it by a good approximation, as discussed). We define =U_1^* ⊗ U_2^* ⊗⋯⊗ U_q^* - A _F^2. Our iterative proof works as follows. We first obtain the objective function,min_U_1 ∈ℝ^n× k U_1 · Z_1 - A_1 _F^2 ≤,where A_1 is a matrix obtained by flattening tensor A along the first dimension, Z_1 = (U_2^*⊤⊙ U_3^*⊤⊙⋯⊙ U_q^*⊤ ) denotes a k× n^q-1 matrix. Choosing S_1∈ℝ^n^q-1× s_1 to be a Gaussian sketching matrix with s_1 = O(k/ϵ), we obtain a smaller problem,min_U_1 ∈ℝ^n× k U_1 · Z_1 S_1 - A_1 S_1 _F^2.We define U_1 to be A_1 S_1 (Z_1 S_1)^†∈ℝ^n× k, which gives,U_1 · Z_1 - A_1 _F^2 ≤ (1+ϵ) .After retensorizing the above, we have, U_1 ⊗ U_2^*⊗⋯⊗U_q^* - A _F^2 ≤ (1+ϵ) .In the second round, we fix U_1, U_3^*, ⋯, U_q^* ∈ℝ^n× k, and choose S_2 ∈ℝ^n^q-1× s_2 to be a Gaussian sketching matrix with s_2 = O(k/ϵ). We define Z_2 ∈ℝ^k× n^q-1 to be (U_1^⊤⊙ U_3^*⊤⊙⋯⊙ U_q^*⊤ ). We define U_2 to be A_2 S_2 (Z_2 S_2)^†∈ℝ^n× k. Then, we haveU_1 ⊗U_2⊗ U_3^*⊗⋯⊗ U_q^* - A _F^2 ≤ (1+ϵ)^2 .We repeat the above process, where in the i-th round we fix U_1, ⋯, U_i-1, U_i+1^*, ⋯, U_q^*∈ℝ^n× k, and choose S_i∈ℝ^n^q-1× s_i to be a Gaussian sketching matrix with s_i = O(k/ϵ). We define Z_i ∈ℝ^k× n^q-1 to be (U_1^⊤⊙⋯⊙U_i-1^⊤⊙ U_i+1^*⊤⊙⋯⊙ U_q^*⊤ ). We define U_i to be A_i S_i (Z_i S_i)^†∈ℝ^n× k. Then, we haveU_1 ⊗⋯⊗U_i-1⊗U_i ⊗ U_i+1^* ⊗⋯⊗ U_q^* - A _F^2 ≤ (1+ϵ)^2 .At the end of the q-th round, we haveU_1 ⊗⋯⊗U_q - A _F^2 ≤ (1+ϵ)^q . Replacing ϵ = ϵ'/ (2q), we obtainU_1 ⊗⋯⊗U_q - A _F^2 ≤ (1+ϵ') .where for all i∈[q], s_i = O( k q /ϵ' ) = O_q(k/ϵ') .§.§.§ General input sparsity reductionThis section shows how to extend the input sparsity reduction from third order tensors to general q-th order tensors. Given a tensor A∈ℝ^n× n ×⋯× n and q matrices, for each i∈ [q], matrix V_i has size V_i ∈ℝ^n× b_i, with b_i ≤(k,1/ϵ). We choose a batch of sparse embedding matrices T_i∈ℝ^t_i × n. Define V_i = T_i V_i, and C=A(T_1,T_2,⋯,T_q). Thus we have with probability 99/100, for any α≥ 0, for all { X_i,X_i'∈ℝ^b_i × k}_i∈ [q], ifV_1 X_1' ⊗V_2 X_2' ⊗⋯⊗V_q X_q' - C _F^2 ≤αV_1 X_1 ⊗V_2 X_2 ⊗⋯⊗V_q X_q -C_F^2,thenV_1 X_1' ⊗ V_2 X_2' ⊗⋯⊗ V_q X_q' -A_F^2 ≤ (1+ϵ)α V_1 X_1 ⊗ V_2 X_2 ⊗⋯⊗ V_q X_q -A_F^2,where t_i = O_q( (b_i,1/ϵ) ).§.§.§ Bicriteria algorithmThis section explains how to extend the bicriteria algorithm from third order tensors (Section <ref>) to general q-th order tensors. Given any q-th order tensor A∈ℝ^n× n×⋯× n, we can output a -r tensor (or equivalently q matrices U_1, U_2, ⋯, U_q ∈ℝ^n× r) such that,U_1 ⊗ U_2 ⊗⋯⊗ U_q - A _F^2 ≤ (1+ϵ) ,where r= O_q( (k/ϵ)^q-1 ) and the algorithm takes O_q( (A) + n ·(k,1/ϵ)). §.§.§ CURT decompositionThis section extends the tensor CURT algorithm from 3rd order tensors (Section <ref>) to general q-th order tensors. Given a q-th order tensor A∈ℝ^n× n×⋯× n and a batch of matrices U_1, U_2, ⋯, U_q ∈ℝ^n× k, we iteratively apply the proof in Theorem <ref> (or Theorem <ref>) q times. Then for each i∈ [q], we are able to select d_i columns from the i-th dimension of tensor A (let C_i denote those columns) and also find a tensor U∈ℝ^d_1 × d_2 ×⋯× d_q such that,U(C_1, C_2, ⋯, C_q) - A_F^2 ≤ (1+ϵ)U_1 ⊗ U_2 ⊗⋯⊗ U_q - A _F^2,where either d_i = O_q(klog k+k/ϵ) (similar to Theorem <ref>) or d_i = O_q(k/ϵ) (similar to Theorem <ref>). §.§ Matrix CUR decompositionThere is a long line of research on matrix CUR decomposition under operator, Frobenius or recently, entry-wise ℓ_1 norm <cit.>. We provide the first algorithm that runs in (A) time, which improves the previous best matrix CUR decomposition algorithm under Frobenius norm <cit.>.§.§.§ Algorithm Given matrix A∈ℝ^n× n, for any k≥ 1 and ϵ∈ (0,1), there exists an algorithm that takes O((A) + n(k,1/ϵ) ) time and outputs three matrices C∈ℝ^n× c with c columns from A, R∈ℝ^r× n with r rows from A, and U∈ℝ^c × r with (U)=k such that r=c=O(k/ϵ) and,CUR - A _F^2 ≤ (1+ϵ) min_-k A_k A_k - A _F^2,holds with probability at least 9/10. We define= min_-k A_k A_k - A _F^2.We first compute U∈ℝ^n× k by using the result of <cit.>, so that U satisfies:min_X ∈ℝ^k× nU X - A _F^2 ≤ (1+ϵ) .This step can be done in O((A) + n(k,1/ϵ)) time.We choose S_1∈ℝ^n× n to be a sampling and rescaling diagonal matrix according to the leverage scores of U, where here s_1 = O(ϵ^-2 klog k) is the number of samples. This step also can be done in O( n(k,1/ϵ)) time.We run GeneralizedMatrixRowSubsetSelection(Algorithm <ref>) on matrices S_1 A and S_1U. Then we obtain two new matrices R and Y, where R contains r=O(k/ϵ) rows of S_1A and Y has size k× r. According to Theorem <ref> and Corollary <ref>, this step takes n(k,1/ϵ) time.We construct V=YR, and choose S_2^⊤ to be another sampling and rescaling diagonal matrix according to the leverage scores of V^⊤ with s_2=O(ϵ^-2 k log k) nonzero entries. As in the case of constructing S_1, this step can be done in O( n(k,1/ϵ)) time.We run GeneralizedMatrixRowSubsetSelection(Algorithm <ref>) on matrices (A S_2)^⊤ and (VS_2)^⊤. Then we can obtain two new matrices C^⊤ and Y^⊤, where C^⊤ contains c=O(k/ϵ) rows of (AS_2)^⊤ and Z^⊤ has size k× c. According to Theorem <ref> and Corollary <ref>, this step takes n(k,1/ϵ) time.Thus, overall the running time is O((A) + n (k,1/ϵ)). Correctness. LetX^*=min_X∈ℝ^n× kXV-A_F^2.According to Corollary <ref>,CZVS_2-AS_2_F^2≤ (1+ε”)min_X∈ℝ^n× kXVS_2-AS_2_F^2≤ (1+ε”)X^*VS_2-AS_2_F^2.According to Theorem <ref>, ε”=0.001ε',CZV-A_F^2≤ (1+ε')X^*V-A_F^2. LetX=min_X∈ℝ^k× nUX-A_F^2.According to Corollary <ref>,S_1UYR-S_1A_F^2≤ (1+ε”)min_X∈ℝ^k× nS_1UX-S_1A_F^2≤ (1+ε”)S_1UX-S_1A_F^2.According to Theorem <ref>, since ε”=0.001ε',UYR-A_F^2≤ (1+ε')UX-A_F^2.Then, we can concludeCUR-A_F^2 = CZYR-A_F^2= CZV-A_F^2 ≤ (1+ε') min_X∈ℝ^n× kXV-A_F^2 ≤ (1+ε') UV-A_F^2 ≤ (1+ε')^2 min_X∈ℝ^k× nUX-A_F^2 ≤ (1+ε')^3≤ (1+) .The first equality follows since U=ZY. The second equality follows since YR=V. The first inequality follows by Equation (<ref>). The third inequality follows by Equation (<ref>). The fourth inequality follows by Equation (<ref>). The last inequality follows since '=0.1.Notice that C has O(k/ε) reweighted columns of AS_2, and AS_2 is a subset of reweighted columns of A, so C has O(k/ε) reweighted columns of A. Similarly, we can prove that R has O(k/ε)reweighted rows of A. Thus, CUR is a CUR decomposition of A. §.§.§ Stronger property achieved by leverage scores Given matrix A∈ℝ^n× m, for any distribution p=(p_1,p_2,⋯,p_n) define random variable X such that X=A_i _2^2 /p_i with probability p_i, where A_i is the i-th row of matrix A. Then take m independent samples X^1, X^2, ⋯, X^m, and let Y = 1/m∑_j=1^m X^j. We have[Y ≤ 100A_F^2 ] ≥ .99.We can compute the expectation of X^j, for any j∈ [m],[X^j ] = ∑_i=1^nA_i _2^2 /p_i· p_i =A_F^2.Then [Y] = 1/m∑_j=1^m [X^j] =A_F^2. Using Markov's inequality, we have[Y ≥ A_F^2 ] ≤ .01. Let A∈ℝ^n× k, B∈ℝ^n× d. Let S∈ℝ^n× n denote a sampling and rescaling diagonal matrix according to the leverage scores of A. If the event occurs that S satisfies (ε/√(k))-Frobenius norm approximate matrix product for A, and also S is a (1+ϵ)-subspace embedding for A, then let X^* be the optimal solution of min_XAX-B_F^2, and B≡ AX^*-B. Then, for all X∈ℝ^k× d,(1-2ε)AX-B_F^2≤S(AX-B)_F^2+B_F^2-SB_F^2≤ (1+2ε)AX-B_F^2.Furthermore, if S has m=O(ϵ^-2 klog(k) ) nonzero entries, the above event happens with probability at least 0.99.Note that Theorem 39 in <cit.> is stated in a way that holds for general sketching matrices. However, we are only interested in the case when S is a sampling and rescaling diagonal matrix according to the leverage scores. For completeness, we provide the full proof of the leverage score case with certain parameters.Suppose S is a sampling and rescaling diagonal matrix according to the leverage scores of A, and it has m=O( ϵ^-2 klog k ) nonzero entries. Then, according to Lemma <ref>, S is a (1+ε)-subspace embedding for A with probability at least 0.999, and according to Lemma <ref>, S satisfies (ϵ/√(k))-Frobenius norm approximate matrix product for A with probability at least 0.999.Let U∈ℝ^n× k denote an orthonormal basis of the column span of A. Then the leverage scores of U are the same as the leverage scores of A. Furthermore, for any X∈ℝ^k× d, there is a matrix Y such that AX=UY, and vice versa, so we can now assume A has k orthonormal columns.Then,S(AX-B)_F^2-SB_F^2= SA(X-X^*)+S(AX^*-B)_F^2-SB_F^2= SA(X-X^*)_F^2+S(AX^*-B)_F^2+2((X-X^*)^⊤ A^⊤ S^⊤ S (AX^*-B))-SB_F^2= SA(X-X^*)_F^2+2((X-X^*)^⊤ A^⊤ S^⊤ S B) _α.The second equality follows using C+D_F^2=C_F^2+D_F^2+2(C^⊤ D). The third equality follows from B=AX^*-B. Now, let us first upper bound the term α in Equation (<ref>):SA(X-X^*)_F^2+2((X-X^*)^⊤ A^⊤ S^⊤ S B) ≤ (1+ε)A(X-X^*)_F^2+2X-X^*_FA^⊤ S^⊤ S B_F ≤ (1+ε)A(X-X^*)_F^2+2(ε/√(k))·X-X^*_FA_F B_F ≤ (1+ε)A(X-X^*)_F^2+2εA(X-X^*)_FB_F.The first inequality follows since S is a (1+ε) subspace embedding of A, and (C^⊤ D)≤C_FD_F. The second inequality follows since S satisfies (ε/√(k))-Frobenius norm approximate matrix product for A. The last inequality follows using that A_F≤√(k) since A only has k orthonormal columns. Now, let us lower bound the term α in Equation (<ref>):SA(X-X^*)_F^2+2((X-X^*)^⊤ A^⊤ S^⊤ S B) ≥ (1-ε)A(X-X^*)_F^2-2X-X^*_FA^⊤ S^⊤ S B_F ≥ (1-ε)A(X-X^*)_F^2-2(ε/√(k))·X-X^*_FA_F B_F ≥ (1-ε)A(X-X^*)_F^2-2εA(X-X^*)_FB_F.The first inequality follows since S is a (1+ε) subspace embedding of A, and (C^⊤ D)≥ -C_FD_F. The second inequality follows since S satisfies (ε/√(k))-Frobenius norm approximate matrix product for A. The last inequality follows using that A_F≤√(k) since A only has k orthonormal columns.Therefore,(1-ε)A(X-X^*)_F^2-2εA(X-X^*)_FB_F ≤S(AX-B)_F^2-SB_F^2,and(1+ε)A(X-X^*)_F^2+2εA(X-X^*)_FB_F ≥S(AX-B)_F^2-SB_F^2.Notice that B=AX^*-B=AA^† B-B=(AA^†-I)B, so according to the Pythagorean theorem, we haveAX-B_F^2=A(X-X^*)_F^2+B_F^2,which means thatA(X-X^*)_F^2=AX-B_F^2-B_F^2.Using Equation (<ref>), we can rewrite and lower bound theof Equation (<ref>),(1-ε)A(X-X^*)_F^2-2εA(X-X^*)_FB_F = A(X-X^*)_F^2 - ε( A(X-X^*)_F^2+ 2 A(X-X^*)_FB_F ) = AX-B_F^2-B_F^2-ε(A(X-X^*)_F^2+2A(X-X^*)_FB_F)≥ AX-B_F^2-B_F^2-ε(A(X-X^*)_F+B_F)^2≥ AX-B_F^2-B_F^2-2ε(A(X-X^*)_F^2+B_F^2) = (1-2ε)AX-B_F^2-B_F^2.The second step follows by Equation (<ref>). The first inequality follows using a^2+2ab<(a+b)^2. The second inequality follows using (a+b)^2≤2(a^2+b^2). The last equality follows using A(X-X^*)_F^2+B_F^2=AX-B_F^2. Similarly, using Equation (<ref>), we can rewrite and upper bound theof Equation (<ref>)(1+ε)A(X-X^*)_F^2+2εA(X-X^*)_FB_F ≤(1+2ε)AX-B_F^2-B_F^2.Combining Equations (<ref>),(<ref>),(<ref>),(<ref>), we conclude that(1-2ε)AX-B_F^2-B_F^2≤S(AX-B)_F^2-SB_F^2≤ (1+2ε)AX-B_F^2-B_F^2.Let A∈ℝ^n× k, B∈ℝ^n× d, and 1/2>ε>0. Let X^* be the optimal solution to min_XAX-B_F^2, and B≡ AX^*-B. Let S∈ℝ^n× n denote a sketching matrix which satisfies the following: * SB_F^2≤ 100·B_F^2,* for all X∈ℝ^k× d,(1-ε)AX-B_F^2≤S(AX-B)_F^2+B_F^2-SB_F^2≤ (1+ε)AX-B_F^2. Then, for all X_1,X_2∈ℝ^k× d satisfyingSAX_1-SB_F^2≤(1+ε/100)·SAX_2-SB_F^2,we haveAX_1-B_F^2≤ (1+5ε)·AX_2-B_F^2.Let A,B,S,ε be the same as in the statement of the theorem, and suppose S satisfies those two conditions. Let X_1,X_2∈ℝ^k× d satisfySAX_1-SB_F^2≤(1+ε/100)SAX_2-SB_F^2.We haveAX_1-B_F^2 ≤ 1/1-ε(S(AX_1-B)_F^2+B_F^2-SB_F^2) ≤ 1/1-ε((1+ε/100)·S(AX_2-B)_F^2+B_F^2-SB_F^2)= 1/1-ε((1+ε/100)·(S(AX_2-B)_F^2+B_F^2-SB_F^2)-ε/100·(B_F^2-SB_F^2)) ≤ 1/1-ε·(1+ε/100) ·AX_2-B_F^2-1/1-ε·ε/100·(B_F^2-SB_F^2) ≤ (1+3ε)AX_2-B_F^2+1/1-ε·ε/100SB_F^2 ≤ (1+3ε)AX_2-B_F^2+2εB_F^2 ≤ (1+5ε)AX_2-B_F^2.The first inequality follows since S satisfies the second condition. The second inequality follows by the relationship between X_1 and X_2. The third inequality follows since S satisfies the second condition. The fifth inequality follows using that ε<1/2 and that S satisfies the first condition. The last inequality follows using that B_F^2=AX^*-B_F^2≤AX_2-B_F^2.Let A∈ℝ^n× k, B∈ℝ^n× d, and 1/2>ε>0. Let S∈ℝ^n× n denote a sampling and rescaling diagonal matrix according to the leverage scores of A. If S has at least m=O(klog(k)/ε^2) nonzero entries, then with probability at least 0.98, for all X_1,X_2∈ℝ^k× d satisfyingSAX_1-SB_F^2≤ (1+ε/500)·SAX_2-SB_F^2,we haveAX_1-B_F^2≤ (1+ε)·AX_2-B_F^2.The proof directly follows by Claim <ref>, Theorem <ref> and Theorem <ref>. Because of Claim <ref>, S satisfies the first condition in the statement of Theorem <ref> with probability at least 0.99. According to Theorem <ref>, S satisfies the second condition in the statement of Theorem <ref> with probability at least 0.99. Thus, with probability 0.98, by Theorem <ref>, we complete the proof.§ ENTRY-WISE ℓ_1 NORM FOR ARBITRARY TENSORSIn this section, we provide several different algorithms for tensor ℓ_1-low rank approximation. Section <ref> provides some useful facts and definitions. Section <ref> presents several existence results. Section <ref> describes a tool that is able to reduce the size of the objective function from (n) to (k). Section <ref> discusses the case when the problem size is small. Section <ref> provides several bicriteria algorithms. Section <ref> summarizes a batch of algorithms. Section <ref> provides an algorithm for ℓ_1 norm CURT decomposition.Notice that if the -k solution does not exist, then every bicriteria algorithm in Section <ref> can be stated in a form similar to Theorem <ref>, and every algorithm which can output a -k solution in Section <ref> can be stated in a form similar to Theorem <ref>. See Section <ref> for more details. §.§ FactsWe present a method that is able to reduce the entry-wise ℓ_1-norm objective function to the Frobenius norm objective function.Given a 3rd order tensor C∈ℝ^c_1 × c_2 × c_3, three matrices V_1∈ℝ^c_1 × b_1, V_2∈ℝ^c_2× b_2, V_3∈ℝ^c_3× b_3, for any k ∈ [1,min_i b_i], if X'_1∈ℝ^b_1× k,X'_2∈ℝ^b_2× k,X'_3∈ℝ^b_3× k satisfies that,(V_1 X'_1) ⊗ (V_2 X'_2) ⊗ (V_3 X'_3) - C _F ≤αmin_X_1,X_2,X_3 (V_1 X_1) ⊗ (V_2 X_2) ⊗ (V_3 X_3) - C _F,then(V_1 X'_1) ⊗ (V_2 X'_2) ⊗ (V_3 X'_3) - C _1 ≤α√(c_1 c_2 c_3)min_X_1,X_2,X_3 (V_1 X_1) ⊗ (V_2 X_2) ⊗ (V_3 X_3) - C _1.We extend Lemma C.15 in <cit.> to tensors:Given tensor A∈ℝ^n× n× n, let =-k A_kmin A - A_k _1. For any r≥ k, if -r tensor B∈ℝ^n× n× n is an f-approximation to A, i.e.,B -A _1 ≤ f ·,and U,V,W ∈ℝ^n× k is a g-approximation to B, i.e.,U ⊗ V ⊗ W - B _1 ≤ g ·-k B_kmin B_k - B _1,then,U ⊗ V ⊗ W - A _1 ≲ g f ·.We define U, V, W∈ℝ^n× k to be three matrices, such thatU⊗V⊗W - B _1 ≤ g -k B_kmin B_k - B _1,and also define,U, V, W = U,V,W∈ℝ^n× kmin U ⊗ V ⊗ W - B _1 and U^*,V^*,W^* = U,V,W∈ℝ^n× kmin U ⊗ V ⊗ W - A _1.It is obvious that,U⊗V⊗W - B _1 ≤ U^* ⊗ V^* ⊗ W^* - B _1.Then,U⊗V⊗W - A _1≤ U⊗V⊗W - B _1 +B - A _1 by the triangle inequality ≤ g U⊗V⊗W - B _1 +B - A _1 by definition ≤ gU^* ⊗ V^* ⊗ W^* - B _1 +B - A _1 by Equation (<ref>) ≤ gU^* ⊗ V^* ⊗ W^* - A _1 + gB-A_1 +B-A_1 by the triangle inequality= g+ (g+1)B - A _1 by definition of ≤ g+ (g+1 ) f · since B is an f-approximation to A ≲ g f .This completes the proof. Using the above fact, we are able to optimize our approximation ratio.§.§ Existence resultsGiven matrices U∈ℝ^n× r,A∈ℝ^n× d, let S∈ℝ^m× n. If ∀β≥ 1,V∈ℝ^r× d which satisfySUV-SA_1≤β·min_V∈ℝ^r× dSUV-SA_1,it holds thatUV-A_1≤β· c·min_V∈ℝ^r× dUV-A_1,then S provides a c-ℓ_1-multiple-regression-cost-preserving-sketch for (U,A).Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, there exist three matrices S_1∈ℝ^n^2× s_1, S_2∈ℝ^n^2× s_2, S_3 ∈ℝ^n^2 × s_3 such that min_X_1, X_2 , X_3 ∑_i=1^k (A_1S_1 X_1)_i ⊗ (A_2S_2X_2)_i ⊗ (A_3S_3X_3)_i -A _1 ≤α-k A_k ∈ℝ^n× n × nmin A_k -A _1,holds with probability 99/100.(1). Using a dense Cauchy transform,s_1=s_2=s_3=O(k), α = O(k^1.5) log^3 n. (2). Using a sparse Cauchy transform,s_1=s_2=s_3=O(k^5), α = O(k^13.5) log^3 n. (3). Guessing Lewis weights,s_1=s_2=s_3=O(k), α = O(k^1.5).We useto denote: = -k A_k ∈ℝ^n× n × nmin A_k - A _1. Given a tensor A∈ℝ^n_1× n_2 × n_3, we define three matrices A_1 ∈ℝ^n_1 × n_2 n_3, A_2 ∈ℝ^n_2 × n_3 n_1, A_3 ∈ℝ^n_3 × n_1 n_2 such that, for any i∈ [n_1], j ∈ [n_2], l ∈ [n_3],A_i,j,l = ( A_1)_i, (j-1) · n_3 + l = ( A_2 )_ j, (l-1) · n_1 + i= ( A_3)_l, (i-1) · n_2 + j . We fix V^* ∈ℝ^n× k and W^* ∈ℝ^n× k, and use V_1^*, V_2^*, ⋯, V_k^* to denote the columns of V^* and W_1^*, W_2^*, ⋯, W_k^* to denote the columns of W^*.We consider the following optimization problem,min_U_1, ⋯, U_k ∈ℝ^n ∑_i=1^k U_i ⊗ V_i^* ⊗ W_i^* - A _1,which is equivalent tomin_U_1, ⋯, U_k ∈ℝ^n [ U_1 U_2 ⋯ U_k ][ V_1^* ⊗ W_1^*; V_2^* ⊗ W_2^*; ⋯; V_k^* ⊗ W_k^* ] - A _1. We use matrix Z_1 to denote V^*⊤⊙ W^*⊤∈ℝ^k× n^2 and matrix U to denote [ U_1 U_2 ⋯ U_k ]. Then we can obtain the following equivalent objective function,min_U ∈ℝ^n× k U Z_1- A_1 _1. Choose an ℓ_1 multiple regression cost preserving sketch S_1 ∈ℝ^n^2 × s_1 for (Z_1^⊤,A_1^⊤). We can obtain the optimization problem,min_U ∈ℝ^n× k U Z_1 S_1 - A_1 S_1 _1 = min_U∈ℝ^n× k∑_i=1^nU^i Z_1 S_1 - (A_1 S_1)^i _1,where U^i denotes the i-th row of matrix U∈ℝ^n× k and (A_1 S_1)^i denotes the i-th row of matrix A_1 S_1. Instead of solving it under the ℓ_1-norm, we consider the ℓ_2-norm relaxation,U ∈ℝ^n× kmin U Z_1 S_1 - A_1 S_1 _F^2 = U∈ℝ^n× kmin∑_i=1^nU^i Z_1 S_1 - (A_1 S_1)^i _2^2.Let U∈ℝ^n× k denote the optimal solution of the above optimization problem. Then, U = A_1 S_1 (Z_1 S_1)^†. We plug U into the objective function under the ℓ_1-norm. According to Claim <ref>, we have,U Z_1 S_1 - A_1 S_1 _1= ∑_i=1^n U^i Z_1 S_1 - (A_1 S_1)^i _1≤√(s_1)min_U ∈ℝ^n× k U Z_1 S_1 - A_1 S_1 _1.Since S_1 ∈ℝ^n^2 × s_1 satisfies Definition <ref>, we haveU Z_1- A_1_1 ≤αU∈ℝ^n× kmin U Z_1 - A_1 _1 = α,where α=√(s_1)β and β (see Definition <ref>) is a parameter which depends on which kind of sketching matrix we actually choose. It impliesU⊗ V^* ⊗ W^* - A _1 ≤α.As a second step, we fix U∈ℝ^n× k and W^* ∈ℝ^n× k, and convert tensor A into matrix A_2. Let matrix Z_2 denote U^⊤⊙ W^*⊤. We consider the following objective function,min_V ∈ℝ^n× k V Z_2 -A_2_1,and the optimal cost of it is at most α.Choose an ℓ_1 multiple regression cost preserving sketch S_2 ∈ℝ^n^2 × s_2 for (Z_2^⊤,A_2^⊤), and sketch on the right of the objective function to obtain this new objective function,V∈ℝ^n× kmin V Z_2 S_2 - A_2 S_2 _1 = min_U∈ℝ^n× k∑_i=1^nV^i Z_2 S_2 - (A_2 S_2)^i _1,where V^i denotes the i-th row of matrix V and (A_2 S_2)^i denotes the i-th row of matrix A_2 S_2. Instead of solving this under the ℓ_1-norm, we consider the ℓ_2-norm relaxation,U ∈ℝ^n× kmin V Z_2 S_2 - A_2 S_2 _F^2 = V∈ℝ^n× kmin V^i (Z_2 S_2) - (A_2 S_2)^i_2^2. Let V∈ℝ^n× k denote the optimal solution of the above problem. Then V = A_2 S_2 (Z_2 S_2)^†. By properties of the sketching matrix S_2 ∈ℝ^n^2 × s_2, we have,V Z_2 - A_2 _1 ≤αV∈ℝ^n× kmin V Z_2- A_2 _1 ≤α^2 ,which impliesU⊗V⊗ W^* - A _1 ≤α^2 .As a third step, we fix the matrices U∈ℝ^n× k and V∈ℝ^n × k. We can convert tensor A∈ℝ^n× n × n into matrix A_3 ∈ℝ^n^2 × n. Let matrix Z_3 denote U^⊤⊙V^⊤∈ℝ^k× n^2. We consider the following objective function,W∈ℝ^n× kmin W Z_3 - A_3 _1,and the optimal cost of it is at most α^2.Choose an ℓ_1 multiple regression cost preserving sketch S_3 ∈ℝ^n^2 × s_3 for (Z_3^⊤,A_3^⊤) and sketch on the right of the objective function to obtain the new objective function,W ∈ℝ^n× kmin W Z_3 S_3 - A_3 S_3 _1.Let W∈ℝ^n× k denote the optimal solution of the above problem. Then W = A_3 S_3 (Z_3 S_3)^†. By properties of sketching matrix S_3∈ℝ^n^2 × s_3, we have,W Z_3 - A_3 _1 ≤αW∈ℝ^n× kmin W Z_3 - A_3 _1 ≤α^3 .Thus, we obtain,min_X_1 ∈ℝ^s_1× k, X_2 ∈ℝ^s_2 × k, X_3 ∈ℝ^s_3 × k∑_i=1^k (A_1S_1 X_1)_i ⊗ (A_2 S_2 X_2)_i ⊗ (A_3 S_3 X_3)_i - A _1 ≤α^3 . Proof of (1) By Theorem C.1 in <cit.>, we can use dense Cauchy transforms for S_1, S_2, S_3, and then s_1=s_2=s_3 = O(klog k) and α = O(√(klog k)log n).Proof of (2) By Theorem C.1 in <cit.>, we can use sparse Cauchy transforms for S_1, S_2, S_3, and then s_1=s_2=s_3 = O(k^5log^5 k) and α = O(k^4.5log^4.5 k log n).Proof of (3) By Theorem C.1 in <cit.>, we can sample by Lewis weights. Then S_1, S_2, S_3 ∈ℝ^n^2 × n^2 are diagonal matrices, and each of them has O(klog k) nonzero rows. This gives α = O(√(klog k) ). §.§ Polynomial in k size reductionGiven a matrix M∈ℝ^n× d, if matrix S∈ℝ^m× n satisfiesSM_1≤βM_1,then S has at most β dilation on M.Given a matrix U∈ℝ^n× k, if matrix S∈ℝ^m× n satisfies∀ x∈ℝ^k, SUx_1≥1/βUx_1,then S has at most β contraction on U.Given a tensor A∈ℝ^n_1× n_2× n_3 and three matrices V_1∈ℝ^n_1× b_1,V_2∈ℝ^n_2× b_2,V_3∈ℝ^n_3× b_3, let X_1^*∈ℝ^b_1× k,X_2^*∈ℝ^b_2× k,X_3^*∈ℝ^b_3× k satisfiesX_1^*,X_2^*,X_3^*=X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kminV_1X_1⊗ V_2X_2 ⊗ V_3X_3 - A_1.Let S∈ℝ^m× n have at most β_1≥ 1 dilation on V_1X_1^*· ((V_2X_2^*)^⊤⊙ (V_3X_3^*)^⊤)-A_1 and S have at most β_2≥ 1 contraction on V_1. If X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× k satisfiesSV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA _1 ≤βX_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kminSV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA_1,where β≥ 1, thenV_1X_1⊗ V_2X_2⊗ V_3X_3 - A _1 ≲β_1β_2βmin_X_1,X_2,X_3V_1X_1⊗ V_2X_2⊗ V_3X_3 - A_1. The proof idea is similar to <cit.>.Let A,V_1,V_2,V_3,S,X_1^*,X_2^*,X_3^*,β_1,β_2 be the same as stated in the theorem. Let X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× k satisfySV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA _1 ≤βX_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kminSV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA_1.We have,SV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA _1 ≥ SV_1X_1⊗ V_2X_2⊗ V_3X_3 - SV_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3 _1-SV_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-SA_1 ≥ 1/β_2V_1X_1⊗ V_2X_2⊗ V_3X_3 - V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3 _1-β_1V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_1 ≥ 1/β_2V_1X_1⊗ V_2X_2⊗ V_3X_3 - A _1-1/β_2V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_1 -β_1V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_1= 1/β_2V_1X_1⊗ V_2X_2⊗ V_3X_3 - A _1-(1/β_2+β_1)V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_1.The first and the third inequality follow by the triangle inequalities. The second inequality follows using thatSV_1X_1⊗ V_2X_2⊗ V_3X_3 - SV_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3 _1= SV_1(X_1-X^*_1)·((V_2(X_2-X^*_2))^⊤⊙(V_3(X_3-X^*_3))^⊤) _1 ≥ 1/β_2V_1(X_1-X^*_1)·((V_2(X_2-X^*_2))^⊤⊙(V_3(X_3-X^*_3))^⊤) _1 ≥ 1/β_2V_1X_1⊗ V_2X_2⊗ V_3X_3 - V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3 _1,andSV_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-SA_1= S(V_1X_1^*· ((V_2X_2^*)^⊤⊙ (V_3X_3^*)^⊤)-A_1)_1 ≤ V_1X_1^*· ((V_2X_2^*)^⊤⊙ (V_3X_3^*)^⊤)-A_1_1= β_1V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_1.Then, we haveV_1X_1⊗ V_2X_2⊗ V_3X_3 - A _1 ≤ β_2 SV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA _1+(1+β_1β_2) V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_1 ≤ β_2βSV_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3 - SA _1+(1+β_1β_2) V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_1 ≤ β_1β_2βV_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3 - A _1+(1+β_1β_2) V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_1 ≤ β(1+2β_1β_2)V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3 - A _1.The first inequality follows by Equation (<ref>). The second inequality follows bySV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA _1 ≤βX_1,X_2,X_3minSV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA_1. The third inequality follows by Equation (<ref>). The final inequality follows using that β≥ 1. Let min(b_1,b_2,b_3)≥ k. Given three matrices V_1∈ℝ^n× b_1, V_2 ∈ℝ^n× b_2, and V_3 ∈ℝ^n× b_3, there exists an algorithm that takes O((A)) + n (b_1,b_2,b_3) time and outputs a tensor C∈ℝ^c_1× c_2× c_3 and three matrices V_1∈ℝ^c_1× b_1, V_2 ∈ℝ^c_2× b_2 and V_3 ∈ℝ^c_3 × b_3 with c_1=c_2=c_3=(b_1,b_2,b_3), such that with probability 0.99, for any α≥ 1, if X'_1,X'_2,X'_3 satisfy that,∑_i=1^k (V_1 X_1')_i ⊗ (V_2 X_2')_i ⊗ (V_3 X_3')_i - C _1 ≤αX_1, X_2, X_3min∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - C _1,then,∑_i=1^k (V_1 X_1')_i ⊗ ( V_2 X_2')_i ⊗ (V_3 X_3')_i - A _1 ≲αmin_X_1, X_2, X_3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _1.For simplicity, we defineto bemin_X_1, X_2, X_3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _1.Let T_1∈ℝ^c_1× n sample according to the Lewis weights of V_1∈ℝ^n× b_1, where c_1=O(b_1). Let T_2∈ℝ^c_2× n sample according to the Lewis weights of V_2∈ℝ^n× b_2, where c_2=O(b_2). Let T_3∈ℝ^c_3× n sample according to the Lewis weights of V_3∈ℝ^n× b_3, where c_3=O(b_3).For any α≥ 1, let X'_1∈ℝ^b_1× k,X'_2∈ℝ^b_2× k,X'_3∈ℝ^b_3× k satisfyT_1V_1X'_1 ⊗ T_2V_2X'_2 ⊗ T_3V_3X'_3 - A(T_1,T_2,T_3)_1≤ αmin_X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kT_1V_1X_1 ⊗ T_2V_2X_2 ⊗ T_3V_3X_3 - A(T_1,T_2,T_3)_1.First, we regard T_1 as the sketching matrix for the remainder. Then by Lemma D.11 in <cit.> and Theorem <ref>, we haveV_1X'_1 ⊗ T_2V_2X'_2 ⊗ T_3V_3X'_3 - A(I,T_2,T_3)_1≲ αmin_X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kV_1X_1 ⊗ T_2V_2X_2 ⊗ T_3V_3X_3 - A(I,T_2,T_3)_1.Second, we regard T_2 as a sketching matrix for V_1X_1 ⊗ V_2X_2 ⊗ T_3V_3X_3 - A(I,I,T_3). Then by Lemma D.11 in <cit.> and Theorem <ref>, we haveV_1X'_1 ⊗ V_2X'_2 ⊗ T_3V_3X'_3 - A(I,I,T_3)_1≲ αmin_X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kV_1X_1 ⊗ V_2X_2 ⊗ T_3V_3X_3 - A(I,I,T_3)_1.Third, we regard T_3 as a sketching matrix for V_1X_1 ⊗ V_2X_2 ⊗ V_3X_3 - A. Then by Lemma D.11 in <cit.> and Theorem <ref>, we have∑_i=1^k (V_1 X_1')_i ⊗ ( V_2 X_2')_i ⊗ (V_3 X_3')_i - A _1 ≲αmin_X_1, X_2, X_3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _1. First, we define Z_1∈ℝ^k× n^2 to be the matrix where the i-th row is ( (V_2 X_2)_i ⊗ (V_3 X_3)_i ). Then, we can flatten the original problem tomin_X_1 ∈ℝ^b_1 × k, X_2 ∈ℝ^b_2× k, X_3 ∈ℝ^b_3× kV_1 X_1 · Z_1 - A_1_1.We choose a sampling and rescaling matrix T_1∈ℝ^c_1× n according to the Lewis weights of V_1∈ℝ^n× b_1, where the number of nonzero entries on the diagonal is c_1=O(b_1). Let X_1^1, X_2^1, X_3^1 denote the optimal solution to min_X_1, X_2, X_3T_1V_1 X_1 · Z_1 - T_1A_1 _1.Let Z_1^1 = (V_2 X_2^1)^⊤⊙ (V_3 X_3^1)^⊤∈ℝ^k× n^2 denote the matrix where the i-th row is ( (V_2 X_2^1)_i ⊗ (V_3 X_3^1)_i ). By Lemma D.11 in <cit.> and Theorem <ref>, with probability 0.999,V_1 X_1 Z_1^1 - A_1_1 ≲min_X_1,X_2,X_3 V_1 X_1 Z - A_1_1 ≲Therefore, we reduce the original problem to a new problem where one dimension is only (k/ϵ) size.Second, we unflatten matrix T_1A_1∈ℝ^c_1 × n^2 to obtain a tensor A'∈ℝ^c_1× n× n. Then we flatten A' along the second direction to obtain matrix A_2 ∈ℝ^n× c_1 n. We define Z_2∈ℝ^k× c_1n to be the matrix where the i-th row is ( (T_1 V_1 X_1)_i ⊗ (V_3 X_3)_i ). Then, we can flatten the problem,min_X_1, X_2, X_3V_2 X_2 · Z_2 - A_2 _1.We choose a sampling and rescaling matrix T_2∈ℝ^c_2 × n according to the Lewis weights of V_2 ∈ℝ^n× b_2, and the number of nonzero entries on the diagonal is c_2 = O(b_2). Let X_1^2, X_2^2, X_3^2 denote the optimal solution tomin_X_1,X_2,X_3 T_2 V_2 X_2 · Z_2 - T_2 A_2 _1.Let Z_2^2 = (V_1X_1^1)^⊤⊙ (V_3 X_3^2)^⊤∈ℝ^k× c_1 n denote the matrix where the i-th row is ( (V_1X_1^1)_i ⊗ (V_3 X_3^2)_i ). By Lemma D.11 in <cit.> and Theorem <ref>, with probability 0.999,V_2 X_2 Z_2^2 - A_2 _1 ≲min_X_1,X_2,X_3 V_2 X_2 Z_2 - A_2 _1 ≲.Therefore, we reduce the original problem to a problem where the two dimensions are only (k/ϵ) size.Third, we unflatten matrix S_2 A_2∈ℝ^c_2 × c_1 n to obtain a tensor A”∈ℝ^c_1× c_2 × n. Then we flatten A” along the second direction to obtain A_3 ∈ℝ^n× c_1 c_2. We define Z_3∈ℝ^k× c_1c_2 to be the matrix where the i-th row is ( (T_1 V_1 X_1)_i ⊗ (T_2 V_2 X_2)_i ). Then, we can flatten the problem,min_X_1, X_2, X_3V_3 X_3 · Z_3 - A_3 _1.We choose a sampling and rescaling matrix T_3∈ℝ^c_3 × n according to the Lewis weights of V_3∈ℝ^n× b_3, and the number of nonzero entries on the diagonal is c_3 = O(b_3). Let X_1^3, X_2^3, X_3^3 denote the optimal solution tomin_X_1,X_2,X_3 T_3 V_3 X_3 · Z_3 - T_3 A_3 _1.Let Z_3^3 = ( T_1 V_1X_1^3 )^⊤⊙ (T_2 V_2 X_2^3)^⊤∈ℝ^k× c_1 c_2 denote the matrix where the i-th row is ( (T_1 V_1X_1^3)_i ⊗ (T_2 V_2 X_2^3)_i ). By Section D in <cit.>, with probability 0.999,V_3 X_3^3 Z_3^3 - A_3 _1 ≲min_X_1,X_2,X_3 V_3 X_3 Z_3 - A_3 _1 ≲.Therefore, we reduce the problem tomin_X_1,X_2,X_3 T_3 V_3 X_3 · Z_3 - T_3 A_3 _1.Note that the above matrix (flattening) version objective function is equivalent to the tensor (unflattening) version objective function,min_X_1,X_2,X_3∑_i=1^k (T_1 V_1 X_1)_i ⊗ (T_2 V_2 X_2)_i ⊗ (T_3 V_3 X_3)_i- A(T_1,T_2,T_3) _1.Let V_i denote T_i V_i, for each i∈ [3]. Let C∈ℝ^c_1 × c_2 × c_3 denote A(T_1,T_2,T_3). Then we obtain,min_X_1,X_2,X_3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i- C _1.Let X'_1, X'_2, X'_3 denote an α-approximation to the above problem. Then combining the above analysis, we have,(V_1 X'_1) ⊗ (V_2 X'_2) ⊗ (V_3 X'_3) - C _1≤ αmin_X_1∈ℝ^s_1 × k,X_2∈ℝ^s_2 × k,X_3 ∈ℝ^s_3 × k(V_1 X_1) ⊗ (V_2 X_2) ⊗ (V_3 X_3)- C _1≲ α,which completes the proof.Given tensor A∈ℝ^n_1× n_2× n_3, and two matrices U∈ℝ^n_1× s, V∈ℝ^n_2× s with (U)=r, let T∈ℝ^t× n_1 be a sampling/rescaling matrix according to the Lewis weights of U with t=O(r). Then with probability at least 0.99, for all X'∈ℝ^n_3× s,α≥ 1 which satisfyT_1U⊗ V⊗ X'-T_1A_1≤α·min_X∈ℝ^n_3× sT_1U⊗ V⊗ X-T_1A_1,it holds thatU⊗ V⊗ X'-A_1≲α·min_X∈ℝ^n_3× sU⊗ V⊗ X-A_1. The proof is similar to the proof of Lemma <ref>.Let X^*=X∈ℝ^n_3× sminU⊗ V⊗ X-A_1. Then according to Lemma D.11 in <cit.>, T has at most constant dilation (Definition <ref>) on U·(V^⊤⊙ (X^*)^⊤) - A_1, and has at most constant contraction (Definition <ref>) on U. We first look atTU⊗ V⊗ X' - TA_1= TU·(V^⊤⊙ (X')^⊤) - TA_1_1 ≥ TU·( (V^⊤⊙ (X')^⊤) - (V^⊤⊙ (X^*)^⊤)) _1-TU·(V^⊤⊙ (X^*)^⊤) - TA_1_1 ≥ 1/β_2U·( (V^⊤⊙ (X')^⊤)-A_1_1-(1/β_2+β_1)U·(V^⊤⊙ (X^*)^⊤) - A_1_1,where β_1≥ 1,β_2≥ 1 are two constants. Then we have:U⊗ V⊗ X' - A_1 ≤ β_2 TU·(V^⊤⊙ (X')^⊤) - TA_1_1 + (1+β_1β_2)U·(V^⊤⊙ (X^*)^⊤) - A_1_1 ≤ αβ_2TU·(V^⊤⊙ (X^*)^⊤) - TA_1_1 + (1+β_1β_2)U·(V^⊤⊙ (X^*)^⊤) - A_1_1 ≤ αβ_1β_2U·(V^⊤⊙ (X^*)^⊤) - A_1_1 + (1+β_1β_2)U·(V^⊤⊙ (X^*)^⊤) - A_1_1 ≲ αU⊗ V⊗ X^*-A_1.Given tensor A∈ℝ^n× n× n, and two matrices U∈ℝ^n× s, V∈ℝ^n× s with (U)=r_1,(V)=r_2, let T_1∈ℝ^t_1× n be a sampling/rescaling matrix according to the Lewis weights of U, and let T_2∈ℝ^t_2× n be a sampling/rescaling matrix according to the Lewis weights of V with t_1=O(r_1),t_2=O(r_2). Then with probability at least 0.99, for all X'∈ℝ^n× s,α≥ 1 which satisfyT_1U⊗ T_2V⊗ X'-A(T_1,T_2,I)_1≤α·min_X∈ℝ^n× sT_1U⊗ T_2V⊗ X-A(T_1,T_2,I)_1,it holds thatU⊗ V⊗ X'-A_1≲α·min_X∈ℝ^n× sU⊗ V⊗ X-A_1.We apply Lemma <ref> twice: ifT_1U⊗ T_2V⊗ X'-A(T_1,T_2,I)_1≤α·min_X∈ℝ^n× sT_1U⊗ T_2V⊗ X-A(T_1,T_2,I)_1,thenU⊗ T_2V⊗ X'-A(I,T_2,I)_1≲α·min_X∈ℝ^n× sU⊗ T_2V⊗ X-A(I,T_2,I)_1.Then, we haveU⊗ V⊗ X'-A_1≲α·min_X∈ℝ^n× sU⊗ V⊗ X-A_1.Let min(b_1,b_2,b_3)≥ k. Given three matrices V_1∈ℝ^n× b_1, V_2 ∈ℝ^n× b_2, and V_3 ∈ℝ^n× b_3, there exists an algorithm that takes O((A)) + n (b_1,b_2,b_3) time and outputs a tensor C∈ℝ^c_1× c_2× n and three matrices V_1∈ℝ^c_1× b_1, and V_2 ∈ℝ^c_2× b_2 with c_1=c_2=(b_1,b_2,b_3), such that with probability 0.99, for any α≥ 1, if X'_1,X'_2,X'_3 satisfy that,∑_i=1^k (V_1 X_1')_i ⊗ (V_2 X_2')_i ⊗ (V_3 X_3')_i - C _1 ≤αX_1, X_2, X_3min∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - C _1,then,∑_i=1^k (V_1 X_1')_i ⊗ ( V_2 X_2')_i ⊗ (V_3 X_3')_i - A _1 ≲αmin_X_1, X_2, X_3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _1.§.§ Solving small problemsLet max_i{t_i, d_i}≤ n. Given a t_1 × t_2 × t_3 tensor A and three matrices: a t_1 × d_1 matrix T_1, a t_2 × d_2 matrix T_2, and a t_3 × d_3 matrix T_3, if for δ > 0 there exists a solution tomin_X_1,X_2,X_3∑_i=1^k (T_1 X_1)_i ⊗ (T_2 X_2)_i ⊗ (T_3 X_3)_i - A _1 := ,such that each entry of X_i can be expressed using O(n^δ) bits, then there exists an algorithm that takes n^O(δ)· 2^ O( d_1 k+d_2 k+d_3 k) time and outputs three matrices: X_1, X_2, and X_3 such that (T_1 X_1)⊗ (T_2 X_2) ⊗ (T_3X_3) - A_1 =. For each i∈ [3], we can create t_i× d_i variables to represent matrix X_i. Let x denote the list of these variables. Let B denote tensor ∑_i=1^k (T_1 X_1)_i ⊗ (T_2X_2)_i⊗ (T_3X_3)_i. Then we can write the following objective function,min_x∑_i=1^t_1∑_j=1^t_2∑_l=1^t_3 |B_i,j,l(x) -A_i,j,l|.To remove the |·|, we create t_1t_2t_3 extra variables σ_i,j,l. Then we obtain the objective function:min_x,σ ∑_i=1^t_1∑_j=1^t_2∑_l=1^t_3σ_i,j,l (B_i,j,l(x) -A_i,j,l)s.t. σ_i,j,l^2 =1, σ_i,j,l (B_i,j,l(x) -A_i,j,l) ≥ 0, x _2^2 + σ_2^2 ≤ 2^O(n^δ)where the last constraint is unharmful, because there exists a solution that can be written using O(n^δ) bits. Note that the number of inequality constraints in the above system is O(t_1t_2t_3), the degree is O(1), and the number of variables is v=(t_1t_2t_3+d_1k+d_2k+d_3k). Thus by Theorem <ref>, we know that the minimum nonzero cost is at least(2^O(n^δ) )^-2^O ( v ) .It is immediate that the upper bound on cost is at most 2^O(n^δ), and thus the number of binary search steps is at most log(2^O(n^δ)) 2^O(v). In each step of the binary search, we need to choose a cost C between the lower bound and the upper bound, and write down the polynomial system,∑_i=1^t_1∑_j=1^t_2∑_l=1^t_3σ_i,j,l (B_i,j,l(x) -A_i,j,l) ≤ C, σ_i,j,l^2 =1, σ_i,j,l (B_i,j,l(x) -A_i,j,l) ≥ 0, x _2^2 + σ_2^2 ≤ 2^O(n^δ).Using Theorem <ref>, we can determine if there exists a solution to the above polynomial system. Since the number of variables is v, and the degree is O(1), the number of inequality constraints is t_1 t_2 t_2. Thus, the running time is(bitsize) · (#·)^# = n^O(δ) 2^O(v)§.§ Bicriteria algorithms We present several bicriteria algorithms with different tradeoffs. We first present an algorithm that runs in nearly linear time and outputs a solution with rank O(k^3) in Theorem <ref>. Then we show an algorithm that runs in (A) time but outputs a solution with rank (k) in Theorem <ref>. Then we explain an idea which is able to decrease the cubic rank to quadratic rank, and thus we can obtain Theorem <ref> and Theorem <ref>. §.§.§ Input sparsity timeGiven a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, ϵ∈ (0,1), let r=O(k^3). There exists an algorithm which takes (A)·O(k) + O(n) (k) + (k) time and outputs three matrices U,V,W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _1 ≤O(k^3/2) log^3 n min_-k A_k A_k - A_1holds with probability 9/10.We first choose three dense Cauchy transforms S_i∈ℝ^n^2× s_i. According to Section <ref>, for each i∈[3], A_i S_i can be computed in (A) ·O(k) time. Then we apply Lemma <ref> (Algorithm <ref>). We obtain three matrices Y_1,Y_2,Y_3 and a tensor C. Note that for each i∈ [3], Y_i can be computed in n (k) time. Because C= A(T_1,T_2,T_3) and T_1,T_2,T_3 ∈ℝ^n×O(k) are three sampling and rescaling matrices, C can be computed in (A) + O(k^3) time. At the end, we just need to run an ℓ_1-regression solver to find the solution to the problem,min_X∈ℝ^s_1 × s_2 × s_3∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3 X_i,j,l (Y_1)_i ⊗ (Y_2)_j ⊗ (Y_3)_j_1,where (Y_1)_i denotes the i-th column of matrix Y_1. Since the size of the above problem is only (k), this can be solved in (k) time.Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, ϵ∈ (0,1), let r=O(k^15). There exists an algorithm that takes (A) + O(n) (k) + (k) time and outputs three matrices U,V,W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _1 ≤( k, log n ) min_-k A_k A_k - A_1holds with probability 9/10.We first choose three dense Cauchy transforms S_i∈ℝ^n^2× s_i. According to Section <ref>, for each i∈[3], A_i S_i can be computed in O((A)) time. Then we apply Lemma <ref> (Algorithm <ref>), and can obtain three matrices Y_1,Y_2,Y_3 and a tensor C. Note that for each i∈ [3], Y_i can be computed in O(n) (k) time. Because C= A(T_1,T_2,T_3) and T_1,T_2,T_3 ∈ℝ^n×O(k) are three sampling and rescaling matrices, C can be computed in (A) + O(k^3) time. At the end, we just need to run an ℓ_1-regression solver to find the solution to the problem,min_X∈ℝ^s_1 × s_2 × s_3∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3 X_i,j,l (Y_1)_i ⊗ (Y_2)_j ⊗ (Y_3)_l - C_1,where (Y_1)_i denotes the i-th column of matrix Y_1. Since the size of the above problem is only (k), it can be solved in (k) time. §.§.§ Improving cubic rank to quadratic rank Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, ϵ∈ (0,1), let r=O(k^2). There exists an algorithm which takes (A)·O(k) + O(n) (k) + (k) time and outputs three matrices U,V,W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _1 ≤O(k^3/2) log^3 n min_-k A_k A_k - A_1holds with probability 9/10. Let =A_k∈ℝ^n× n × nmin A_k -A _1. We first choose three dense Cauchy transforms S_i ∈ℝ^n^2 × s_i, ∀ i∈ [3]. According to Section <ref>, for each i∈ [3], A_iS_i can be computed in (A) ·O(k) time. Then we choose T_i to be a sampling and rescaling diagonal matrix according to the Lewis weights of A_iS_i, ∀ i∈ [2].According to Theorem <ref>, we havemin_X_1∈ℝ^s_1× k,X_2∈ℝ^s_2× k,X_3∈ℝ^s_3× k∑_l=1^k (A_1 S_1 X_1)_l ⊗ (A_2S_2 X_2)_l ⊗ (A_3S_3 X_3)_l - A _1 ≤O(k^1.5)log^3 nNow we fix an l and we have:(A_1 S_1 X_1)_l ⊗ (A_2S_2 X_2)_l ⊗ (A_3S_3 X_3)_l= (∑_i=1^s_1 (A_1S_1)_i (X_1)_i,l)⊗(∑_j=1^s_2 (A_2S_2)_j (X_2)_j,l)⊗ (A_3S_3 X_3)_l= ∑_i=1^s_1∑_j=1^s_2 (A_1S_1)_i ⊗ (A_2S_2)_j ⊗ (A_3S_3 X_3)_l(X_1)_i,l(X_2)_j,l Thus, we havemin_X_1,X_2,X_3∑_i=1^s_1∑_j=1^s_2 (A_1S_1)_i ⊗ (A_2S_2)_j ⊗( ∑_l=1^k(A_3S_3 X_3)_l(X_1)_i,l(X_2)_j,l) - A _1 ≤O(k^1.5)log^3 n .By applying Corollary <ref>, we have if X'_1,X'_2,X'_3 gives an α-approximation tomin_X_1,X_2,X_3 (T_1A_1S_1X_1)⊗ (T_2A_2S_2X_2) ⊗ (A_3 S_3X_3) - A(T_1,T_2,I) _1,then X'_1, X'_2,X'_3 is an O(α)-approximation tomin_X_1,X_2,X_3 (A_1S_1X_1)⊗ (A_2S_2X_2) ⊗ (A_3 S_3X_3) - A _1,which is at most O(k^3/2) log^3 n according to Theorem <ref>.We use (T_1 A_1 S_1)_i to denote the i-th column of matrix T_1 A_1 S_1, and use (T_2A_2S_2)_j to denote the j-th column of matrix T_2A_2 S_2. We construct an s_1s_2× t_1t_2 matrix in the following way,B^i+(j-1)s_1←( (T_1A_1 S_1)_i ⊗ (T_2A_2S_2)_j ), ∀ i∈[s_1],j∈[s_2],where B^i+(j-1)s_1 denotes the i+(j-1)s_1-th row of matrix B. We use C∈ℝ^t_1 × t_2 × n to denote A(T_1,T_2,I). Thus,min_W W B - C_1 ≤min_X_1,X_2,X_3 (T_1A_1S_1X_1)⊗ (T_2A_2S_2X_2) ⊗ (A_3 S_3X_3) - A(T_1,T_2,I) _1.Note that using an ℓ_1 regression solver, we can find a solution W to the above problem.We create matrix U∈ℝ^n× s_1 s_2 by copying matrix A_1S_1 s_2 times, i.e.,U = [ A_1 S_1 A_1 S_1 ⋯ A_1 S_1 ].We create matrix V∈ℝ^n× s_1 s_2 by copying the i-th column of A_2 S_2 a total of s_1 times into the columns (i-1)s_1, ⋯, is_1 of V, for each i∈ [s_2], i.e.,V = [ (A_2S_2)_1⋯(A_2 S_2)_1 (A_2S_2)_2⋯(A_2 S_2)_2 ⋯ (A_2S_2)_s_2⋯ (A_2 S_2)_s_2. ]According to Equation (<ref>), we have:W∈ℝ^n× s_1 s_2minU⊗V⊗ W - A _1≤O(k^1.5)log^3 n·. LetW=W∈ℝ^n× s_1 s_2min T_1U⊗ T_2V⊗ W - A(T_1,T_2,I) _1.Due to Corollary <ref>, we haveU⊗V⊗W-A_1≤O(k^1.5)log^3 n·.Putting it all together, we have that U,V,W gives a rank-O(k^2) bicriteria algorithm to the original problem. Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, ϵ∈ (0,1), let r=O(k^10). There exists an algorithm which takes (A) + O(n) (k) + (k) time and outputs three matrices U,V,W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _1 ≤( k, log n ) min_-k A_k A_k - A_1holds with probability 9/10. The proof is similar to the proof of Theorem <ref>. The only difference is that instead of choosing dense Cauchy matrices S_1,S_2, we choose sparse Cauchy matrices. Notice that if we firstly apply a sparse Cauchy transform, we can reduce the rank of the matrix to (k). Then we apply a dense Cauchy transform and can further reduce the dimension while only incurring another (k) factor in the approximation ratio. By combining a sparse Cauchy transform and a dense Cauchy transform, we can improve the running time from (A) ·O(k) to (A). Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, ϵ∈ (0,1), let r=O(k^2). There exists an algorithm which takes (A) + O(n) (k) + (k) time and outputs three matrices U,V,W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _1 ≤( k, log n ) min_-k A_k A_k - A_1holds with probability 9/10. §.§ AlgorithmsIn this section, we show two different algorithms by using different kind of sketches. One is shown in Theorem <ref> which gives a fast running time. Another one is shown in Theorem <ref> which gives the best approximation ratio. §.§.§ Input sparsity time algorithm Given a 3rd tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm that takes (A) ·O(k) + O(n) (k) + 2^O(k^2) time and outputs three matrices U,V,W∈ℝ^n× k such that,U ⊗ V ⊗ W - A _1 ≤(k,log n) min_-k A'A' - A _1.holds with probability at least 9/10. First, we apply part (2) of Theorem <ref>. Then A_i S_i can be computed in O((A)) time. Second, we use Lemma <ref> to reduce the size of the objective function from O(n^3) to (k) in n (k) time by only losing a constant factor in approximation ratio. Third, we use Claim <ref> to relax the objective function from entry-wise ℓ_1-norm to Frobenius norm, and this step causes us to lose some other (k) factors in approximation ratio. As a last step, we use Theorem <ref> to solve the Frobenius norm objective function. Notice again that if we first apply a sparse Cauchy transform, we can reduce the rank of the matrix to (k). Then as before we can apply a dense Cauchy transform to further reduce the dimension while only incurring another (k) factor in the approximation ratio. By combining a sparse Cauchy transform and a dense Cauchy transform, we can improve the running time from (A) ·O(k) to (A), while losing some additional (k) factors in approximation ratio.Given a 3rd tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm that takes (A) + O(n) (k) + 2^O(k^2) time and outputs three matrices U,V,W∈ℝ^n× k such that,U ⊗ V ⊗ W - A _1 ≤(k,log n) min_-k A'A' - A _1.holds with probability at least 9/10.§.§.§ O(k^3/2)-approximation algorithmGiven a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm that takes n^O(k) 2^O(k^3) time and output three matrices U,V,W∈ℝ^n× k such that,U ⊗ V ⊗ W - A _1 ≤O(k^3/2) min_-k A'A' - A _1.holds with probability at least 9/10.First, we apply part (3) of Theorem <ref>. Then, guessing S_i requires n^O(k) time. Second, we use Lemma <ref> to reduce the size of the objective from O(n^3) to (k) in polynomial time while only losing a constant factor in approximation ratio. Third, we use Theorem <ref> to solve the entry-wise ℓ_1-norm objective function directly.§.§ CURT decomposition Given a 3rd order tensor A∈ℝ^n× n × n, let k≥ 1, let U_B,V_B,W_B∈ℝ^n× k denote a rank-k, α-approximation to A. Then there exists an algorithm which takes O((A)) + O(n^2) (k) time and outputs three matrices: C∈ℝ^n× c with columns from A, R∈ℝ^n× r with rows from A, T∈ℝ^n× t with tubes from A, and a tensor U∈ℝ^c× r× t with (U)=k such that c=r=t=O(klog k), and∑_i=1^c ∑_j=1^r ∑_l=1^t U_i,j,l· C_i ⊗ R_j ⊗ T_l - A _1 ≤O(k^1.5) αmin_-k A' A' - A_1holds with probability 9/10.We define: = min_-k A' A' - A_1.We already have three matrices U_B∈ℝ^n× k, V_B∈ℝ^n× k and W_B∈ℝ^n× k and these three matrices provide a -k, α approximation to A, i.e.,∑_i=1^k ( U_B )_i ⊗ (V_B)_i ⊗ (W_B)_i - A _1 ≤αLet B_1 = V_B^⊤⊙ W_B^⊤∈ℝ^k× n^2 denote the matrix where the i-th row is the vectorization of (V_B)_i ⊗ (W_B)_i. By Section B.3, we can compute D_1 ∈ℝ^n^2 × n^2 which is a sampling and rescaling matrix corresponding to the Lewis weights of B_1^⊤ in O(n^2(k)) time, and there are d_1 = O(klog k) nonzero entries on the diagonal of D_1. Let A_i∈ℝ^n× n^2 denote the matrix obtained by flattening A along the i-th direction, for each i∈ [3].Define U^*∈ℝ^n× k to be the optimal solution to U∈ℝ^n× kmin U B_1 - A_1_1, U = A_1 D_1 (B_1 D_1)^†∈ℝ^n× k, V_0 ∈ℝ^n× k to be the optimal solution to V∈ℝ^n× kmin V ·(U^⊤⊙ W_B^⊤) - A_2 _1, and U' to be the optimal solution to U∈ℝ^n× kmin U B_1 D_1 - A_1 D_1 _1.By Claim <ref>, we haveU B_1 D_1 - A_1 D_1 _1 ≤√(d_1) U' B_1 D_1 - A_1 D_1_1Due to Lemma D.11 and Lemma D.8 (in <cit.>) with constant probability, we haveU B_1 - A _1 _1 ≤√(d_1)α_D_1 U^* B_1 - A_1 _1,where α_D_1 = O(1).Recall that ( U^⊤⊙ W_B^⊤) ∈ℝ^k× n^2 denotes the matrix where the i-th row is the vectorization of U_i ⊗ (W_B)_i, ∀ i∈ [k]. Now, we can show,V_0 · ( U^⊤⊙ W_B^⊤) - A_2 _1 ≤ U B_1 - A_1 _1 by V_0 = V∈ℝ^n× kmin V · ( U^⊤⊙ W_B^⊤) - A_2_1≲ √(d_1) U^* B_1 - A_1 _1 by Equation (<ref>) ≤ √(d_1) U_B B_1 - A_1 _1 by U^* = U∈ℝ^n× kmin U B_1 - A_1 _1≤ O(√(d_1)) α by Equation (<ref>) We define B_2= U^⊤⊙ W_B^⊤. We can compute D_2∈ℝ^n^2 × n^2 which is a sampling and rescaling matrix corresponding to the Lewis weights of B_2^⊤ in O(n^2 (k)) time, and there are d_2 = O(klog k) nonzero entries on the diagonal of D_2.Define V^*∈ℝ^n× k to be the optimal solution of min_V∈ℝ^n× k V B_2 - A_2 _1, V= A_2 D_2 (B_2 D_2)^†∈ℝ^n× k, W_0∈ℝ^n× k to be the optimal solution of W∈ℝ^n× kmin W· (U^⊤⊙V^⊤) - A_3 _1, and V' to be the optimal solution of V∈ℝ^n× kminV B_2 D_2 - A_2 D_2_1.By Claim <ref>, we haveV B_2 D_2 - A_2 D_2 _1 ≤√(d_2)V' B_2 D_2 - A_2 D_2 _1.Due to Lemma D.11 and Lemma D.8(in <cit.>) with constant probability, we haveV B_2 - A_2 _1 ≤√(d_2)α_D_2 V^* B_2 - A_2 _1,where α_D_2 = O(1).Recall that (U^⊤⊙V^⊤) ∈ℝ^k× n^2 denotes the matrix for which the i-th row is the vectorization of U_i ⊗V_i, ∀ i∈ [k]. Now, we can show,W_0 · (U^⊤⊙V^⊤ ) - A_3 _1 ≤ V B_2 - A_2 _1 by W_0 = W∈ℝ^n× kmin W · ( U^⊤⊙V^⊤ ) - A_3 _1≲ √(d_2) V^* B_2 - A_2 _1 by Equation (<ref>) ≤ √(d_2) V_0 B_2 - A_2 _1 by V^* =V∈ℝ^n× kmin V B_2 - A_2 _1≤ O(√(d_1 d_2)) α by Equation (<ref>)We define B_3= U^⊤⊙V^⊤. We can compute D_3∈ℝ^n^2 × n^2 which is a sampling and rescaling matrix corresponding to the Lewis weights of B_3^⊤ in O(n^2 (k)) time, and there are d_3 = O(klog k) nonzero entries on the diagonal of D_3.Define W^*∈ℝ^n× k to be the optimal solution to min_W∈ℝ^n× k W B_3 - A_3 _1, W= A_3 D_3 (B_3 D_3)^†∈ℝ^n× k, and W' to be the optimal solution to W∈ℝ^n× kminW B_3 D_3 - A_3 D_3_1.By Claim <ref>, we haveW B_3 D_3 - A_3 D_3 _1 ≤√(d_3)W' B_3 D_3 - A_3 D_3 _1.Due to Lemma D.11 and Lemma D.8(in <cit.>) with constant probability, we haveW B_3 - A_3 _1 ≤√(d_3)α_D_3 W^* B_3 - A_3 _1,where α_D_3 = O(1). Now we can show,W B_3 - A_3 _1 ≲ √(d_3) W^* B_3 - A_3 _1, by Equation (<ref>) ≤ √(d_3) W_0 B_3 - A_3 _1, by W^* = W∈ℝ^n× kmin W B_3 - A_3 _1≤ O(√(d_1d_2d_3)) α by Equation (<ref>)Thus, it implies,∑_i=1^k U_i ⊗V_i ⊗W_i - A _1 ≤(k,log n) .where U = A_1 D_1 (B_1 D_1)^†, V = A_2D_2 (B_2 D_2)^†, W=A_3D_3 (B_3 D_3)^†. Given a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm which takes O((A)) + O(n^2)(k) + 2^O(k^2) time and outputs three matrices C∈ℝ^n× c with columns from A, R∈ℝ^n× r with rows from A, T∈ℝ^n× t with tubes from A, and a tensor U∈ℝ^c× r × t with (U)=k such that c=r=t=O(klog k), and∑_i=1^c ∑_j=1^r ∑_l=1^t U_i,j,l· C_i ⊗ R_j ⊗ T_l - A _1 ≤(k,log n) -k A'min A' - A_1,holds with probability 9/10. This follows by combining Corollary <ref> and Theorem <ref>. § ENTRY-WISE ℓ_P NORM FOR ARBITRARY TENSORS, 1<P<2There is a long line of research dealing with ℓ_p norm-related problems <cit.>.In this section, we provide several different algorithms for tensor ℓ_p-low rank approximation. Section <ref> formally states the ℓ_p version of Theorem C.1 in <cit.>. Section <ref> presents several existence results. Section <ref> describes a tool that is able to reduce the size of the objective function from (n) to (k). Section <ref> discusses the case when the problem size is small. Section <ref> provides several bicriteria algorithms. Section <ref> summarizes a batch of algorithms. Section <ref> provides an algorithm for ℓ_p norm CURT decomposition.Notice that if the -k solution does not exist, then every bicriteria algorithm in Section <ref> can be stated in the form as Theorem <ref>, and every algorithm which can output a -k solution in Section <ref> can be stated in the form as Theorem <ref>. See Section <ref> for more details. §.§ Existence results for matrix caseLet 1≤ p<2. Given V∈ℝ^k× n,A∈ℝ^d× n. Let S∈ℝ^n× s be a proper random sketching matrix. LetU=min_U∈ℝ^d× kUVS-AS_F^2,i.e.,U=AS(VS)^†.Then with probability at least 0.999,UV-A_p^p≤α·min_U∈ℝ^d× kUV-A_p^p. (1). S denotes a dense p-stable transform,s=O(k), α = O(k^1-p/2) log d. (2). S denotes a sparse p-stable transform,s=O(k^5), α = O(k^5-5p/2+2/p) log d. (3). S^⊤ denotes a sampling/rescaling matrix according to the ℓ_p Lewis weights of V^⊤,s=O(k), α = O(k^1-p/2). We give the proof for completeness. Let S∈ℝ^n× s be a sketching matrix which satisfies the property (*): ∀ c≥ 1,U∈ℝ^d× k which satisfyUVS-AS_p^p≤ c·min_U∈ℝ^d× kUVS-AS_p^p,we haveUV-A_p^p ≤ cβ_S·min_U∈ℝ^d× kUV-A_p^p,where β_S≥ 1 only depends on the sketching matrix S. Let∀ i∈[d],(U^i)^⊤=min_x∈ℝ^kx^⊤ VS-A^iS_2^2,i.e.,U=AS(VS)^†.LetU=min_U∈ℝ^d× kUVS-AS_p^p.Then, we have:UVS-AS_p^p= ∑_i=1^d U^i VS-A^iS_p^p ≤ ∑_i=1^d (s^1/p-1/2U^i VS-A^iS_2)^p ≤ ∑_i=1^d (s^1/p-1/2U^i VS-A^iS_2)^p ≤ ∑_i=1^d (s^1/p-1/2U^i VS-A^iS_p)^p ≤ s^1-p/2U VS-AS_p^p.The first inequality follows using ∀ x∈ℝ^s,x_p≤ s^1/p-1/2x_2 since p<2. The third inequality follows using ∀ x∈ℝ^s,x_2≤x_p since p<2. Thus, according to the property (*) of S,UV-A_p^p ≤s^1-p/2β_S min_U∈ℝ^d× kUV-A_p^p.Due to Lemma E.8 and Lemma E.11 of <cit.>, we have:for (1), s=O(k),β_S=O(log d),α=s^1-p/2β_S=O(k^1-p/2)log d,for (2), s=O(k^5),β_S=O(k^2/plog d),α=s^1-p/2β_S=O(k^5-5p/2+2/p) log d,for (3), s=O(k),β_S=O(1),α=s^1-p/2β_S=O(k^1-p/2).§.§ Existence results Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, there exist three matrices S_1∈ℝ^n^2× s_1, S_2∈ℝ^n^2× s_2, S_3 ∈ℝ^n^2 × s_3 such that min_X_1, X_2 , X_3 ∑_i=1^k (A_1S_1 X_1)_i ⊗ (A_2S_2X_2)_i ⊗ (A_3S_3X_3)_i -A _p^p ≤α-k A_k ∈ℝ^n× n × nmin A_k -A _p^p,holds with probability 99/100.(1). Using a dense p-stable transform,s_1=s_2=s_3=O(k), α = O(k^3-1.5p) log^3 n. (2). Using a sparse p-stable transform,s_1=s_2=s_3=O(k^5), α = O(k^15-7.5p+6/p) log^3 n. (3). Guessing Lewis weights,s_1=s_2=s_3=O(k), α = O(k^3-1.5p). We useto denote: = -k A_k ∈ℝ^n× n × nmin A_k - A _p^p. Given a tensor A∈ℝ^n_1× n_2 × n_3, we define three matrices A_1 ∈ℝ^n_1 × n_2 n_3, A_2 ∈ℝ^n_2 × n_3 n_1, A_3 ∈ℝ^n_3 × n_1 n_2 such that, for any i∈ [n_1], j ∈ [n_2], l ∈ [n_3]A_i,j,l = ( A_1)_i, (j-1) · n_3 + l = ( A_2 )_ j, (l-1) · n_1 + i= ( A_3)_l, (i-1) · n_2 + j. We fix V^* ∈ℝ^n× k and W^* ∈ℝ^n× k, and use V_1^*, V_2^*, ⋯, V_k^* to denote the columns of V^* and W_1^*, W_2^*, ⋯, W_k^* to denote the columns of W^*.We consider the following optimization problem,min_U_1, ⋯, U_k ∈ℝ^n ∑_i=1^k U_i ⊗ V_i^* ⊗ W_i^* - A _p^p,which is equivalent tomin_U_1, ⋯, U_k ∈ℝ^n [ U_1 U_2 ⋯ U_k ][ V_1^* ⊗ W_1^*; V_2^* ⊗ W_2^*; ⋯; V_k^* ⊗ W_k^* ] - A _p^p. We use matrix Z_1 to denote V^*⊤⊙ W^*⊤∈ℝ^k× n^2 and matrix U to denote [ U_1 U_2 ⋯ U_k ]. Then we can obtain the following equivalent objective function,min_U ∈ℝ^n× k U Z_1- A_1 _p^p. Choose a sketching matrix (a dense p-stable, a sparse p-stable or an ℓ_p Lewis weight sampling/rescaling matrix to Z_1) S_1 ∈ℝ^n^2 × s_1. We can obtain the optimization problem,min_U ∈ℝ^n× k U Z_1 S_1 - A_1 S_1 _p^p = min_U∈ℝ^n× k∑_i=1^nU^i Z_1 S_1 - (A_1 S_1)^i _p^p,where U^i denotes the i-th row of matrix U∈ℝ^n× k and (A_1 S_1)^i denotes the i-th row of matrix A_1 S_1. Instead of solving it under the ℓ_p-norm, we consider the ℓ_2-norm relaxation,U ∈ℝ^n× kmin U Z_1 S_1 - A_1 S_1 _F^2 = U∈ℝ^n× kmin∑_i=1^nU^i Z_1 S_1 - (A_1 S_1)^i _2^2.Let U∈ℝ^n× k denote the optimal solution of the above optimization problem. Then, U = A_1 S_1 (Z_1 S_1)^†. We plug U into the objective function under the ℓ_p-norm. By choosing s_1 and by the properties of sketching matrices (a dense p-stable, a sparse p-stable or an ℓ_p Lewis weight sampling/rescaling matrix to Z_1) S_1 ∈ℝ^n^2 × s_1, we haveU Z_1- A_1_p^p ≤αU∈ℝ^n× kmin U Z_1 - A_1 _p^p = α.This impliesU⊗ V^* ⊗ W^* - A _p^p ≤α.As a second step, we fix U∈ℝ^n× k and W^* ∈ℝ^n× k, and convert tensor A into matrix A_2. Let matrix Z_2 denote U^⊤⊙ W^*⊤. We consider the following objective function,min_V ∈ℝ^n× k V Z_2 -A_2_p^p,and the optimal cost of it is at most α.We choose a sketching matrix (a dense p-stable, a sparse p-stable or an ℓ_p Lewis weight sampling/rescaling matrix to Z_2) S_2 ∈ℝ^n^2 × s_2 and sketch on the right of the objective function to obtain the new objective function,V∈ℝ^n× kmin V Z_2 S_2 - A_2 S_2 _p^p = min_V∈ℝ^n× k∑_i=1^nV^i Z_2 S_2 - (A_2 S_2)^i _p^p,where V^i denotes the i-th row of matrix V and (A_2 S_2)^i denotes the i-th row of matrix A_2 S_2. Instead of solving this under the ℓ_p-norm, we consider the ℓ_2-norm relaxation,V ∈ℝ^n× kmin V Z_2 S_2 - A_2 S_2 _F^2 = V∈ℝ^n× kmin∑_i=1^nV^i (Z_2 S_2) - (A_2 S_2)^i_2^2. Let V∈ℝ^n× k denote the optimal solution of the above problem. Then V = A_2 S_2 (Z_2 S_2)^†. By properties of sketching matrix S_2 ∈ℝ^n^2 × s_2, we have,V Z_2 - A_2 _p^p ≤αV∈ℝ^n× kmin V Z_2- A_2 _p^p ≤α^2 ,which impliesU⊗V⊗ W^* - A _p^p ≤α^2 ,As a third step, we fix the matrices U∈ℝ^n× k and V∈ℝ^n × k. We can convert tensor A∈ℝ^n× n × n into matrix A_3 ∈ℝ^n^2 × n. Let matrix Z_3 denote U^⊤⊙V^⊤∈ℝ^k× n^2. We consider the following objective function,W∈ℝ^n× kmin W Z_3 - A_3 _p^p,and the optimal cost of it is at most α^2.We choose sketching matrix (a dense p-stable, a sparse p-stable or an ℓ_p Lewis weight sampling/rescaling matrix to Z_3) S_3∈ℝ^n^2 × s_3 and sketch on the right of the objective function to obtain the new objective function,W ∈ℝ^n× kmin W Z_3 S_3 - A_3 S_3 _p^p.Instead of solving this under the ℓ_p-norm, we consider the ℓ_2-norm relaxation,W ∈ℝ^n× kmin W Z_3 S_3 - A_3 S_3 _F^2 = W∈ℝ^n× kmin∑_i=1^nW^i (Z_3 S_3) - (A_3 S_3)^i_2^2.Let W∈ℝ^n× k denote the optimal solution of the above problem. Then W = A_3 S_3 (Z_3 S_3)^†. By properties of sketching matrix S_3∈ℝ^n^2 × s_3, we have,W Z_3 - A_3 _p^p ≤αW∈ℝ^n× kmin W Z_3 - A_3 _p^p ≤α^3 .Thus, we obtain,min_X_1 ∈ℝ^s_1× k, X_2 ∈ℝ^s_2 × k, X_3 ∈ℝ^s_3 × k∑_i=1^k (A_1S_1 X_1)_i ⊗ (A_2 S_2 X_2)_i ⊗ (A_3 S_3 X_3)_i - A _p^p ≤α^3 . According to Theorem <ref>, we let s=s_1=s_2=s_3 and take the corresponding α. We can directly get the results for (1), (2) and (3).§.§ Polynomial in k size reduction Given a matrix M∈ℝ^n× d, if matrix S∈ℝ^m× n satisfiesSM_p^p≤βM_p^p,then S has at most β dilation on M in the ℓ_p case.Given a matrix U∈ℝ^n× k, if matrix S∈ℝ^m× n satisfies∀ x∈ℝ^k, SUx_p^p≥1/βUx_p^p,then S has at most β contraction on U in the ℓ_p case.Given a tensor A∈ℝ^n_1× n_2× n_3 and three matrices V_1∈ℝ^n_1× b_1,V_2∈ℝ^n_2× b_2,V_3∈ℝ^n_3× b_3, let X_1^*∈ℝ^b_1× k,X_2^*∈ℝ^b_2× k,X_3^*∈ℝ^b_3× k satisfyX_1^*,X_2^*,X_3^*=X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kminV_1X_1⊗ V_2X_2 ⊗ V_3X_3 - A_p^p.Let S∈ℝ^m× n have at most β_1≥ 1 dilation on V_1X_1^*· ((V_2X_2^*)^⊤⊙ (V_3X_3^*)^⊤)-A_1 and S have at most β_2≥ 1 contraction on V_1 in the ℓ_p case. If X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× k satisfySV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA _p^p ≤βX_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kminSV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA_p^p,where β≥ 1, thenV_1X_1⊗ V_2X_2⊗ V_3X_3 - A _p^p ≲β_1β_2βmin_X_1,X_2,X_3V_1X_1⊗ V_2X_2⊗ V_3X_3 - A_p^p. The proof is essentially the same as the proof of Theorem <ref>:Let A,V_1,V_2,V_3,S,X_1^*,X_2^*,X_3^*,β_1,β_2 be as stated in the theorem. Let X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× k satisfySV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA _p^p ≤βX_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kminSV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA_p^p.Similar to the proof of Theorem <ref>, we have,SV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA _p^p= 2^2-2p1/β_2V_1X_1⊗ V_2X_2⊗ V_3X_3 - A _p^p-(2^1-p1/β_2+β_1)V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_p^pThe only difference from the proof of Theorem <ref> is that instead of using triangle inequality, we actually use x+y_p^p≤ 2^p-1x_p^p+y_p^p. Then, we haveV_1X_1⊗ V_2X_2⊗ V_3X_3 - A _p^p ≤ 2^2p-2β_2 SV_1X_1⊗ V_2X_2⊗ V_3X_3 - SA _p^p+(2^p-1+2^2p-2β_1β_2) V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_p^p ≤ 2^2p-2β_2βSV_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3 - SA _p^p+(2^p-1+2^2p-2β_1β_2) V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_p^p ≤ 2^2p-2β_1β_2βV_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3 - A _p^p+(2^p-1+2^2p-2β_1β_2) V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3-A_p^p ≤ 2^p-1β(1+2β_1β_2)V_1X^*_1⊗ V_2X^*_2⊗ V_3X^*_3 - A _p^p. Let min(b_1,b_2,b_3)≥ k. Given three matrices V_1∈ℝ^n× b_1, V_2 ∈ℝ^n× b_2, and V_3 ∈ℝ^n× b_3, there exists an algorithm which takes O((A)) + n (b_1,b_2,b_3) time and outputs a tensor C∈ℝ^c_1× c_2× c_3 and three matrices V_1∈ℝ^c_1× b_1, V_2 ∈ℝ^c_2× b_2 and V_3 ∈ℝ^c_3 × b_3 with c_1=c_2=c_3=(b_1,b_2,b_3), such that with probability 0.99, for any α≥ 1, if X'_1,X'_2,X'_3 satisfy that,∑_i=1^k (V_1 X_1')_i ⊗ (V_2 X_2')_i ⊗ (V_3 X_3')_i - C _p^p ≤αX_1, X_2, X_3min∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - C _p^p,then,∑_i=1^k (V_1 X_1')_i ⊗ ( V_2 X_2')_i ⊗ (V_3 X_3')_i - A _p^p ≲αmin_X_1, X_2, X_3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _p^p. For simplicity, we defineto bemin_X_1, X_2, X_3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _p^p.Let T_1∈ℝ^c_1× n correspond to sampling according to the ℓ_p Lewis weights of V_1∈ℝ^n× b_1, where c_1=b_1. Let T_2∈ℝ^c_2× n be sampling according to the ℓ_p Lewis weights of V_2∈ℝ^n× b_2, where c_2=b_2. Let T_3∈ℝ^c_3× n be sampling according to the ℓ_p Lewis weights of V_3∈ℝ^n× b_3, where c_3=b_3.For any α≥ 1, let X'_1∈ℝ^b_1× k,X'_2∈ℝ^b_2× k,X'_3∈ℝ^b_3× k satisfyT_1V_1X'_1 ⊗ T_2V_2X'_2 ⊗ T_3V_3X'_3 - A(T_1,T_2,T_3)_p^p≤ αmin_X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kT_1V_1X_1 ⊗ T_2V_2X_2 ⊗ T_3V_3X_3 - A(T_1,T_2,T_3)_p^p.First, we regard T_1 as the sketching matrix for the remainder. Then by Lemma D.11 in <cit.> and Theorem <ref>, we haveV_1X'_1 ⊗ T_2V_2X'_2 ⊗ T_3V_3X'_3 - A(I,T_2,T_3)_p^p≲ αmin_X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kV_1X_1 ⊗ T_2V_2X_2 ⊗ T_3V_3X_3 - A(I,T_2,T_3)_p^p.Second, we regard T_2 as the sketching matrix for V_1X_1 ⊗ V_2X_2 ⊗ T_3V_3X_3 - A(I,I,T_3). Then by Lemma D.11 in <cit.> and Theorem <ref>, we haveV_1X'_1 ⊗ V_2X'_2 ⊗ T_3V_3X'_3 - A(I,I,T_3)_p^p≲ αmin_X_1∈ℝ^b_1× k,X_2∈ℝ^b_2× k,X_3∈ℝ^b_3× kV_1X_1 ⊗ V_2X_2 ⊗ T_3V_3X_3 - A(I,I,T_3)_p^p.Third, we regard T_3 as the sketching matrix for V_1X_1 ⊗ V_2X_2 ⊗ V_3X_3 - A. Then by Lemma D.11 in <cit.> and Theorem <ref>, we have∑_i=1^k (V_1 X_1')_i ⊗ ( V_2 X_2')_i ⊗ (V_3 X_3')_i - A _p^p ≲αmin_X_1, X_2, X_3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i - A _p^p. First, we define Z_1∈ℝ^k× n^2 to be the matrix where the i-th row is ( (V_2 X_2)_i ⊗ (V_3 X_3)_i ). Then, we can flatten the original problem tomin_X_1 ∈ℝ^b_1 × k, X_2 ∈ℝ^b_2× k, X_3 ∈ℝ^b_3× kV_1 X_1 · Z_1 - A_1_p^p.We choose a sampling and rescaling matrix T_1∈ℝ^c_1× n according to the ℓ_p Lewis weights of V_1∈ℝ^n× b_1, and the number of nonzero entries on the diagonal is c_1=O(b_1). Let X_1^1, X_2^1, X_3^1 denote the optimal solution to min_X_1, X_2, X_3T_1V_1 X_1 · Z_1 - T_1A_1 _p^p.Let Z_1^1 = (V_2 X_2^1)^⊤⊙ (V_3 X_3^1)^⊤∈ℝ^k× n^2 denote the matrix where the i-th row is ( (V_2 X_2^1)_i ⊗ (V_3 X_3^1)_i ). By Section E in <cit.>, with probability 0.999,V_1 X_1^1 Z_1^1 - A_1_p^p ≲min_X_1,X_2,X_3 V_1 X_1 Z - A_1_p^p ≲.Therefore, we reduce the original problem to a new problem where one dimension is only c_1=(k/ϵ) size.Second, we unflatten matrix T_1A_1∈ℝ^c_1 × n^2 to obtain a tensor A'∈ℝ^c_1× n× n. Then we flatten A' along the second direction to obtain matrix A_2 ∈ℝ^n× c_1 n. We define Z_2∈ℝ^k× c_1n to be the matrix where the i-th row is ( (T_1 V_1 X_1)_i ⊗ (V_3 X_3)_i ). Then, we can flatten the problem,min_X_1, X_2, X_3V_2 X_2 · Z_2 - A_2 _p^p.We choose a sampling and rescaling matrix T_2∈ℝ^c_2 × n according to the ℓ_p Lewis weights of V_2 ∈ℝ^n× b_2, and the number of nonzero entries on the diagonal is c_2 = O(b_2). Let X_1^2, X_2^2, X_3^2 denote the optimal solution tomin_X_1,X_2,X_3 T_2 V_2 X_2 · Z_2 - T_2 A_2 _p^p.Let Z_2^2 = (V_1X_1^1)^⊤⊙ (V_3 X_3^2)^⊤∈ℝ^k× c_1 n denote the matrix where the i-th row is ( (V_1X_1^1)_i ⊗ (V_3 X_3^2)_i ). By Section E in <cit.>, with probability 0.999,V_2 X_2 Z_2^2 - A_2 _1 ≲min_X_1,X_2,X_3 V_2 X_2 Z_2 - A_2 _p^p ≲.Therefore, we reduce the original problem to a problem where the two dimensions are only (k/ϵ) size (one dimension is c_1 and another is c_2).Third, we unflatten matrix S_2 A_2∈ℝ^c_2 × c_1 n to obtain a tensor A”∈ℝ^c_1× c_2 × n. Then we flatten A” along the second direction to obtain A_3 ∈ℝ^n× c_1 c_2. We define Z_3∈ℝ^k× c_1c_2 to be the matrix where the i-th row is ( (T_1 V_1 X_1)_i ⊗ (T_2 V_2 X_2)_i ). Then, we can flatten the problem,min_X_1, X_2, X_3V_3 X_3 · Z_3 - A_3 _p^p.We choose a sampling and rescaling matrix T_3∈ℝ^c_3 × n according to the ℓ_p Lewis weights of V_3∈ℝ^n× b_3, and the number of nonzero entries on the diagonal is c_3 = O(b_3). Let X_1^3, X_2^3, X_3^3 denote the optimal solution tomin_X_1,X_2,X_3 T_3 V_3 X_3 · Z_3 - T_3 A_3 _p^p.Let Z_3^3 = ( T_1 V_1X_1^3 )^⊤⊙ (T_2 V_2 X_2^3)^⊤∈ℝ^k× c_1 c_2 denote the matrix where the i-th row is ( (T_1 V_1X_1^3)_i ⊗ (T_2 V_2 X_2^3)_i ). By Section E in <cit.>, with probability 0.999,V_3 X_3^3 Z_3^3 - A_3 _p^p ≲min_X_1,X_2,X_3 V_3 X_3 Z_3 - A_3 _p^p ≲.Therefore, we reduce the problem tomin_X_1,X_2,X_3 T_3 V_3 X_3 · Z_3 - T_3 A_3 _p^p.Note that the above matrix (flattening) version of the objective function is equivalent to the tensor (unflattening) version the objective function,min_X_1,X_2,X_3∑_i=1^k (T_1 V_1 X_1)_i ⊗ (T_2 V_2 X_2)_i ⊗ (T_3 V_3 X_3)_i- A(T_1,T_2,T_3) _p^p.Let V_i denote T_i V_i, for each i∈ [3]. Let C∈ℝ^c_1 × c_2 × c_3 denote A(T_1,T_2,T_3). Then we obtain,min_X_1,X_2,X_3∑_i=1^k (V_1 X_1)_i ⊗ (V_2 X_2)_i ⊗ (V_3 X_3)_i- C _p^p.Let X'_1, X'_2,X'_3 denote an α-approximation to the above problem. Then combining the above analysis, we have,(V_1 X'_1) ⊗ (V_2 X'_2) ⊗ (V_3 X'_3) - C _p^p≤ αmin_X_1∈ℝ^s_1 × k,X_2∈ℝ^s_2 × k,X_3 ∈ℝ^s_3 × k(V_1 X_1) ⊗ (V_2 X_2) ⊗ (V_3 X_3)- C _p^p≲ α,which completes the proof.§.§ Solving small problemsCombining Section B.5 in <cit.> and the proof of Theorem <ref>, for any p=a/b with a,b are integers, we can obtain the ℓ_p version of Theorem <ref>. §.§ Bicriteria algorithm We present several bicriteria algorithms with different tradeoffs. We first present an algorithm that runs in nearly linear time and outputs a solution with rank O(k^3) in Theorem <ref>. Then we show an algorithm that runs in (A) time but outputs a solution with rank (k) in Theorem <ref>. Then we explain an idea which is able to decrease the cubic rank to quadratic, and thus we can obtain Theorem <ref>. Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, let r=O(k^3). There exists an algorithm which takes (A)·O(k) + n (k) + (k) time and outputs three matrices U,V,W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _p^p ≤O(k^3-p/2) log^3 n min_-k A_k A_k - A_p^pholds with probability 9/10.We first choose three dense Cauchy transforms S_i∈ℝ^n^2× s_i. According to Section <ref>, for each i∈[3], A_i S_i can be computed in (A) ·O(k) time. Then we apply Lemma <ref>. We obtain three matrices Y_1=T_1A_1S_1,Y_2=T_2A_2S_2,Y_3=T_3A_3S_3 and a tensor C=A(T_1,T_2,T_3). Note that for each i∈ [3], Y_i can be computed in n (k) time. Because C= A(T_1,T_2,T_3) and T_1,T_2,T_3 ∈ℝ^n×O(k) are three sampling and rescaling matrices, C can be computed in (A) + O(k^3) time. At the end, we just need to run an ℓ_p-regression solver to find the solution for the problem:min_X∈ℝ^s_1 × s_2 × s_3∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3 X_i,j,l (Y_1)_i ⊗ (Y_2)_j ⊗ (Y_3)_j_p^p,where (Y_1)_i denotes the i-th column of matrix Y_1. Since the size of the above problem is only (k), this can be solved in (k) time. Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, let r=O(k^15). There exists an algorithm that takes (A) + n (k) + (k) time and outputs three matrices U,V,W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _p^p ≤( k, log n ) min_-k A_k A_k - A_p^pholds with probability 9/10.We first choose three sparse p-stable transforms S_i∈ℝ^n^2× s_i. According to Section <ref>, for each i∈[3], A_i S_i can be computed in O((A)) time. Then we apply Lemma <ref>, and can obtain three matrices Y_1=T_1A_1S_1,Y_2=T_2A_2S_2,Y_3=T_3A_3S_3 and a tensor C=A(T_1,T_2,T_3). Note that for each i∈ [3], Y_i can be computed in n (k) time. Because C= A(T_1,T_2,T_3) and T_1,T_2,T_3 ∈ℝ^n×O(k) are three sampling and rescaling matrices, C can be computed in (A) + O(k^3) time. At the end, we just need to run an ℓ_p-regression solver to find the solution to the problem,min_X∈ℝ^s_1 × s_2 × s_3∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3 X_i,j,l (Y_1)_i ⊗ (Y_2)_j ⊗ (Y_3)_l - C_p^p,where (Y_1)_i denotes the i-th column of matrix Y_1. Since the size of the above problem is only (k), it can be solved in (k) time.Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, ϵ∈ (0,1), let r=O(k^2). There exists an algorithm which takes (A)·O(k) + n (k) + (k) time and outputs three matrices U,V,W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _p^p ≤O(k^3-1.5p) log^3 n min_-k A_k A_k - A_p^pholds with probability 9/10. The proof is similar to Theorem <ref>.As for ℓ_1, notice that if we first apply a sparse Cauchy transform, we can reduce the rank of the matrix to (k). Theyn we can apply a dense Cauchy transform and further reduce the dimension, while only incurring another (k) factor in the approximation ratio. By combining sparse p-stable and dense p-stable transforms, we can improve the running time from (A) ·O(k) to be (A) by losing some additional (k) factors in the approximation ratio. Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, ϵ∈ (0,1), let r=O(k^2). There exists an algorithm which takes (A)+ n (k) + (k) time and outputs three matrices U,V,W∈ℝ^n× r such that∑_i=1^r U_i ⊗ V_i ⊗ W_i - A _p^p ≤(k,log n) min_-k A_k A_k - A_p^pholds with probability 9/10.§.§ AlgorithmsIn this section, we show two different algorithms by using different kind of sketches. One is shown in Theorem <ref> which gives a fast running time. Another one is shown in Theorem <ref> which gives the best approximation ratio.Given a 3rd tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm which takes O((A))+ n (k) + 2^O(k^2) time and outputs three matrices U,V,W∈ℝ^n× k such that,U ⊗ V ⊗ W - A _p^p ≤(k,log n) min_-k A'A' - A _p^p.holds with probability at least 9/10. First, we apply part (2) of Theorem <ref>. Then A_i S_i can be computed in O((A)) time. Second, we use Lemma <ref> to reduce the size of the objective function from O(n^3) to (k) in n (k) time by only losing a constant factor in approximation ratio. Third, we use Claim <ref> to relax the objective function from entry-wise ℓ_p-norm to Frobenius norm, and this step causes us to lose some other (k) factors in approximation ratio. As a last step, we use Theorem <ref> to solve the Frobenius norm objective function.Given a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm that takes n^O(k) 2^O(k^3) time and output three matrices U,V,W∈ℝ^n× k such that,U ⊗ V ⊗ W - A _p^p ≤O(k^3-1.5p) min_-k A'A' - A _p^p.holds with probability at least 9/10.First, we apply part (3) of Theorem <ref>. Then, guessing S_i requires n^O(k) time. Second, we use Lemma <ref> to reduce the size of the objective from O(n^3) to (k) in polynomial time while only losing a constant factor in approximation ratio. Third, we solve the small optimization problem.§.§ CURT decomposition Given a 3rd order tensor A∈ℝ^n× n × n, let k≥ 1, and let U_B,V_B,W_B∈ℝ^n× k denote a rank-k, α-approximation to A. Then there exists an algorithm which takes O((A)) + O(n^2) (k) time and outputs three matrices C∈ℝ^n× c with columns from A, R∈ℝ^n× r with rows from A, T∈ℝ^n× t with tubes from A, and a tensor U∈ℝ^c× r× t with (U)=k such that c=r=t=O(klog kloglog k), and∑_i=1^c ∑_j=1^r ∑_l=1^t U_i,j,l· C_i ⊗ R_j ⊗ T_l - A _p^p ≤O(k^3-1.5p) αmin_-k A' A' - A_p^pholds with probability 9/10.We define: = min_-k A' A' - A_p^p.We already have three matrices U_B∈ℝ^n× k, V_B∈ℝ^n× k and W_B∈ℝ^n× k and these three matrices provide a -k, α approximation to A, i.e.,∑_i=1^k ( U_B )_i ⊗ (V_B)_i ⊗ (W_B)_i - A _p^p ≤α.Let B_1 = V_B^⊤⊙ W_B^⊤∈ℝ^k× n^2 denote the matrix where the i-th row is the vectorization of (V_B)_i ⊗ (W_B)_i. By Section B.3 in <cit.>, we can compute D_1 ∈ℝ^n^2 × n^2 which is a sampling and rescaling matrix corresponding to the Lewis weights of B_1^⊤ in O(n^2(k)) time, and there are d_1 = O(klog kloglog k) nonzero entries on the diagonal of D_1. Let A_i∈ℝ^n× n^2 denote the matrix obtained by flattening A along the i-th direction, for each i∈ [3].Define U^*∈ℝ^n× k to be the optimal solution to U∈ℝ^n× kmin U B_1 - A_1_p^p, U = A_1 D_1 (B_1 D_1)^†∈ℝ^n× k, V_0 ∈ℝ^n× k to be the optimal solution to V∈ℝ^n× kmin V ·(U^⊤⊙ W_B^⊤) - A_2 _p^p, and U' to be the optimal solution to U∈ℝ^n× kmin U B_1 D_1 - A_1 D_1 _p^p.By Claim <ref>, we haveU B_1 D_1 - A_1 D_1 _p^p ≤ d_1^1-p/2 U' B_1 D_1 - A_1 D_1_p^p.Due to Lemma E.11 and Lemma E.8 in <cit.>, with constant probability, we haveU B_1 - A _1 _p^p ≤ d_1^1-p/2α_D_1 U^* B_1 - A_1 _p^p,where α_D_1 = O(1).Recall that ( U^⊤⊙ W_B^⊤) ∈ℝ^k× n^2 denotes the matrix where the i-th row is the vectorization of U_i ⊗ (W_B)_i, ∀ i∈ [k]. Now, we can show,V_0 · ( U^⊤⊙ W_B^⊤) - A_2 _p^p ≤ U B_1 - A_1 _p^p by V_0 = V∈ℝ^n× kmin V · ( U^⊤⊙ W_B^⊤) - A_2_p^p≲ d_1^1-p/2 U^* B_1 - A_1 _p^p by Equation (<ref>) ≤ d_1^1-p/2 U_B B_1 - A_1 _p^p by U^* = U∈ℝ^n× kmin U B_1 - A_1 _p^p≤ O(d_1^1-p/2) α. by Equation (<ref>) We define B_2= U^⊤⊙ W_B^⊤. We can compute D_2∈ℝ^n^2 × n^2 which is a sampling and rescaling matrix corresponding to the ℓ_p Lewis weights of B_2^⊤ in O(n^2 (k)) time, and there are d_2 = O(klog kloglog k) nonzero entries on the diagonal of D_2.Define V^*∈ℝ^n× k to be the optimal solution of min_V∈ℝ^n× k V B_2 - A_2 _p^p, V= A_2 D_2 (B_2 D_2)^†∈ℝ^n× k, W_0∈ℝ^n× k to be the optimal solution of W∈ℝ^n× kmin W· (U^⊤⊙V^⊤) - A_3 _p^p, and V' to be the optimal solution of V∈ℝ^n× kminV B_2 D_2 - A_2 D_2_p^p.By Claim <ref>, we haveV B_2 D_2 - A_2 D_2 _p^p ≤ d_2^1-p/2V' B_2 D_2 - A_2 D_2 _p^p.Due to Lemma E.11 and Lemma E.8 in <cit.>, with constant probability, we haveV B_2 - A_2 _p^p ≤ d_2^1-p/2α_D_2 V^* B_2 - A_2 _p^p,where α_D_2 = O(1).Recall that (U^⊤⊙V^⊤) ∈ℝ^k× n^2 denotes the matrix for which the i-th row is the vectorization of U_i ⊗V_i, ∀ i∈ [k]. Now, we can show,W_0 · (U^⊤⊙V^⊤ ) - A_3 _p^p≤ V B_2 - A_2 _p^p by W_0 = W∈ℝ^n× kmin W · ( U^⊤⊙V^⊤ ) - A_3 _p^p≲ d_2^1-p/2 V^* B_2 - A_2 _p^p by Equation (<ref>) ≤ d_2^1-p/2 V_0 B_2 - A_2 _p^p by V^* =V∈ℝ^n× kmin V B_2 - A_2 _p^p≤ O((d_1d_2)^1-p/2) α. by Equation (<ref>)We define B_3= U^⊤⊙V^⊤. We can compute D_3∈ℝ^n^2 × n^2 which is a sampling and rescaling matrix corresponding to the ℓ_p Lewis weights of B_3^⊤ in O(n^2 (k)) time, and there are d_3 = O(klog kloglog k) nonzero entries on the diagonal of D_3.Define W^*∈ℝ^n× k to be the optimal solution to min_W∈ℝ^n× k W B_3 - A_3 _p^p, W= A_3 D_3 (B_3 D_3)^†∈ℝ^n× k, and W' to be the optimal solution to W∈ℝ^n× kminW B_3 D_3 - A_3 D_3_p^p.By Claim <ref>, we haveW B_3 D_3 - A_3 D_3 _p^p ≤ d_3^1-p/2W' B_3 D_3 - A_3 D_3 _p^p.Due to Lemma E.11 and Lemma E.8 in <cit.>, with constant probability, we haveW B_3 - A_3 _p^p ≤ d_3^1-p/2α_D_3 W^* B_3 - A_3 _p^p,where α_D_3 = O(1). Now we can show,W B_3 - A_3 _p^p ≲ d_3^1-p/2 W^* B_3 - A_3 _p^p, by Equation (<ref>) ≤ d_3^1-p/2 W_0 B_3 - A_3 _p^p, by W^* = W∈ℝ^n× kmin W B_3 - A_3 _p^p≤ O((d_1d_2d_3)^1-p/2) α. by Equation (<ref>)Thus, it implies,∑_i=1^k U_i ⊗V_i ⊗W_i - A _p^p ≤(k,log n) .where U = A_1 D_1 (B_1 D_1)^†, V = A_2D_2 (B_2 D_2)^†, W=A_3D_3 (B_3 D_3)^†.§ ROBUST SUBSPACE APPROXIMATION (ASYMMETRIC NORMS FOR ARBITRARY TENSORS)Recently, <cit.> and <cit.> study the linear regression problem and low-rank approximation problem under M-Estimator loss functions. In this section, we extend the matrix version of the low rank approximation problem to tensors, i.e., in particular focusing on tensor low-rank approximation under M-Estimator norms. Note that M-Estimators are very different from Frobenius norm and Entry-wise ℓ_1 norm, which are symmetric norms. Namely, flattening the tensor objective function along any of the dimensions does not change the cost if the norm is Frobenius or Entry-wise ℓ_1-norm. However, for M-Estimator norms, we cannot flatten the tensor along all three dimensions. This property makes the tensor low-rank approximation problem under M-Estimator norms more difficult. This section can be split into two independent parts. Section <ref> studies the ℓ_1-ℓ_2-ℓ_2 norm setting, and Section <ref> studies the ℓ_1-ℓ_1-ℓ_2 norm setting. §.§ Preliminaries The Huber norm <cit.>, for example, is specified by a parameter τ>0, and its measure function H us given byH(a) =a^2 / (2τ) if |a|≤τ,|a|-τ/2 otherwise ,combining an ℓ_2-like measure for small x with an ℓ_1-like measure for large x.For a vector z∈ℝ^n, the Huber “norm” isz _H = ∑_i=1^n H(z_i).We provide a tool called “Huber” inequality, we can relate the Huber norm to the ℓ_1 and ℓ_2 norms by using this inequality.For z∈ℝ^n,Θ(n^-1/2 ) min{ z _1,z_2^2 /(2τ) }≤ z _H ≤ z _1.We say an M-Estimator is nice if M(x) = M(-x), M(0)=0, M is non-decreasing in |x|, there is a constant C_M>0 and a constant p≥ 1 so that for all a,b∈ℝ_>0 with a≥ b, we haveC_m |a|/|b|≤ M(a) / M(b) ≤ (a/b)^p,and also that M(x)^1/p is subadditive, that is, M(x+y)^1/p≤ M(x)^1/p + M(y)^1/p.Let M_2 denote the set of such nice M-estimators, for p=2. Let L_p denote M-Estimators with M(x) = |x|^p and p∈ [1,2).§.§ ℓ_1-Frobenius (a.k.a ℓ_1-ℓ_2-ℓ_2) normSection <ref> presents basic definitions and facts for the ℓ_1-ℓ_2-ℓ_2 norm setting. Section <ref> introduces some useful tools. Section <ref> presents the “no dilation” and “no contraction” bounds, which are the key ideas for reducing the problem to a “generalized” Frobenius norm low rank approximation problem. Finally, we provide our algorithms in Section <ref>.§.§.§ DefinitionsWe first give the definition for the v-norm of a tensor, and then give the definition of the v-norm for a matrix and a weighted version of the v-norm for a matrix.For an n× n× n tensor A, we define the v-norm of A, denoted A _v, to be( ∑_i=1^n M(A_i,*,*_F ) ) ^1/p,where A_i,*,* is the i-th face of A (along the 1st direction), and p is a parameter associated with the function M(), which defines a nice M-Estimator.For an n× d matrix A, we define the v-norm of A, denoted A _v, to be∑_i=1^n M(A_i,*_2 )^1/p,where A_i,* is the i-th row of A, and p is a parameter associated with the function M(), which defines a nice M-Estimator.Given matrix A∈ℝ^n× d, let A_i,* denote the i-th row of A. Let T_S ⊂ [n] denote the indices i such that e_i is chosen for S. Using a probability vector q and a sampling and rescaling matrix S∈ℝ^n× n from q, we will estimate A _v using S and a re-weighted version, S ·_v,w' of ·_v, withSA _v,w' = ( ∑_i∈ T_S w_i' M( A_i,*_2 ) )^1/p,where w_i'= w_i/q_i. Since w' is generally understood, we will usually just write SA _v. Wewill also need an “entrywise row-weighted” version :|||SA||| = ( ∑_i∈ T_Sw_i/q_i A_i,*_M^p )^1/p = ( ∑_i∈ T_S, j ∈ [d]w_i/q_i M(A_i,j) )^1/p,where A_i,j denotes the entry in the i-th row and j-th column of A.For p=1, for any two matrices A and B, we have A+ B_v ≤ A _v + B_v. For any two tensors A and B, we have A + B _v ≤ A _v +B_v. §.§.§ Sampling and rescaling sketches Note that Lemmas 42 and 44 in <cit.> are stronger than stated. In particular, we do not need to assume X is a square matrix. For any m ≥ z, if X∈ℝ^d× m, then we have the same result.Let ρ>0 and integer z>0. For sampling matrix S, suppose for a given y∈ℝ^d with failure probability δ it holds that SA y_M = (1± 1/10)A y _M. There is K_1 = O(z^2 / C_M) so that with failure probability δ (K_ N / C_M)^(1+p)d, for a constant K_ N, any rank-z matrix X∈ℝ^d× m has the property that if A X_v ≥ K_1 ρ, then S A X _v≥ρ, and that if A X_v ≤ρ /K_1, then SAX_v ≤ρ.Let δ, ρ>0 and integer z>0. Given matrix A∈ℝ^n× d, there exists a sampling and rescaling matrix S∈ℝ^n× n with r=O(γ(A,M,w) ϵ^-2 dz^2 log(z/ϵ) log(1/δ) ) nonzero entries such that, with probability at least 1-δ, for any -z matrix X∈ℝ^d× m, we have eitherSA X _v ≥ρ,or(1-ϵ) AX _v -ϵρ≤ SAX _v ≤(1+ϵ)A X _v + ϵρ. For r>0, let r = r/γ(A,M,w), and let q∈ℝ^n haveq_i = min{ 1, rγ_i (A,M,w) }.Let S be a sampling and rescaling matrix generated using q, with weights as usual w_i' = w_i /q_i. Let W∈ℝ^d× z, and δ >0. There is an absolute constant C so that for r≥ C z log(1/δ) /ϵ^2, with probability at least 1-δ, we have(1-ϵ)A W _v,w≤ SA W_v,w'≤ (1+ϵ)A W _v,w.§.§.§ No dilation and no contraction Given matrices A∈ℝ^n× m, U∈ℝ^n× d, let V^* = -k V∈ℝ^d× mmin U V - A _v. If S∈ℝ^s× n has at most c_1-dilation on UV^*-A, i.e.,S (U V^* - A ) _v ≤ c_1UV^* - A _v,and it has at most c_2-contraction on U, i.e.,∀ x ∈ℝ^d,SU x _v ≥1/c_2 U x _v,then S has at most (c_2,c_1+1/c_2)-contraction on (U,A), i.e.,∀ -k V ∈ℝ^d× m,SUV -SA _v ≥1/c_2 UV - A_v - (c_1+1/c_2)UV^* -A _v.Let A∈ℝ^n× m, U∈ℝ^n× d and S∈ℝ^s× n be the same as that described in the lemma. Let (V-V^*)_j denote the j-th column of V-V^*. Then ∀ -k V∈ℝ^d× m,SU V - S A_v ≥ SUV - SUV^*_v -SU V^* - SA _v≥ SUV - SUV^* _v - c_1UV^* - A _v = SU (V- V^*) _v - c_1UV^* - A _v = ∑_j=1^mSU (V-V^*)_j _v- c_1UV^* - A _v≥ ∑_j=1^m 1/c_2 U (V - V^*)_j _v - c_1UV^* -A _v = 1/c_2 UV - UV^* _v -c_1U V^* - A _v≥ 1/c_2 UV - A _v - 1/c_2 UV^* -A _v - c_1UV^* -A _v = 1/c_2 U V - A _v - ((1/c_2 +c_2)UV^* -A _v ),where the first inequality follows by the triangle inequality, the second inequality follows since S has at most c_1 dilation on UV^* - A, the third inequality follows since S has at most c_2 contraction on U, and the fourth inequality follows by the triangle inequality.Given matrix A∈ℝ^n× m, for any distribution p=(p_1,p_2,⋯,p_n) define random variable X such that X=A_i _2 /p_i with probability p_i where A_i is the i-th row of matrix A. Then take m independent samples X^1, X^2, ⋯, X^m, and let Y = 1/m∑_j=1^m X^j. We have[Y ≤ 1000A_v ] ≥ .999.We can compute the expectation of X^j, for any j∈ [m],[X^j ] = ∑_i=1^nA_i _2 /p_i· p_i =A_v.Then [Y] = 1/m∑_j=1^m [X^j] =A_v. Using Markov's inequality, we have[Y ≥ A_v ] ≤ .001. For any fixed U^*∈ℝ^n× d and -k V^*∈ℝ^d× m with d=(k), there exists an algorithm that takes (n,d) time to compute a sampling and rescaling diagonal matrix S∈ℝ^n× n with s= (k) nonzero entries such that, with probability at least .999, we have: for all -k V∈ℝ^d× m,U^* V^* - U^* V _v ≲ S U^* V^* - S U^* V _v ≲ U^* V^* - U^* V _v. Given matrices A∈ℝ^n× m, U^*∈ℝ^n× d with d=(k), define V^*∈ℝ^d× m to be the optimal solution -k V∈ℝ^d× mmin U^* V - A _v. Choose a sampling and rescaling diagonal matrix S∈ℝ^n× n with s=(k) according to Lemma <ref>. Then with probability at least .99, we have: for all -k V∈ℝ^d× m,SU^* V - SA_v ≲ U^*V^* - U^* V _v + O(1)U^* V^* - A _v ≲U^* V -A _v.Using Claim <ref> and Lemma <ref>, we have with probability at least .99, for all -k V∈ℝ^d× m,SU^* V - SA _v≤ SU^* V - SU^* V^* _v +SU^* V^* - SA _v by triangle inequality ≲ SU^* V - SU^* V^* _v + O(1)U^* V^* - A _v by Claim <ref> ≲ U^* V - U^* V^*_v + O(1)U^* V^* -A _v by Lemma <ref> ≲ U^* V - A _v + U^* V^* -A _v + O(1)U^* V^* - A _v by triangle inequality ≲ U^* V - A _v. Given matrices A∈ℝ^n× m, U^*∈ℝ^n× d with d=(k), define V^*∈ℝ^d× m to be the optimal solution -k V∈ℝ^d× mmin U^* V - A _v. Choose a sampling and rescaling diagonal matrix S∈ℝ^n× n with s=(k) according to Lemma <ref>. Then with probability at least .99, we have: for all -k V∈ℝ^d× m,U^* V - A _v ≲ S U^* V - S A _v + O(1)U^* V^* - A _v.This follows by Lemma <ref>, Claim <ref> and Lemma <ref>. §.§.§ Oblivious sketches, MSketchIn this section, we recall a concept called M-sketches for M-estimators which is defined in <cit.>. M-sketch is an oblivious sketch for matrices. Letdenote min_x∈ℝ^d A x - b _G. There is an algorithm that in O((A)) + (dlog n) time, with constant probability finds x' such that Ax'-b_G ≤ O(1). Given parameters N,n,m,b>1, define h_max = ⌊log_b (n/m) ⌋, β = (b-b^-h_max) / (b-1) and s=N h_max. For each p∈[n], σ_p,g_p,h_p are generated (independently) in the following way,σ_p←± 1, chosen with equal probability, g_p∈ [N], chosen with equal probability, h_p ← t, chosen with probability 1/(β b^t) for t∈{0,1,⋯ h_max}.For each p∈[n], we define j_p = g_p + N h_p.Let w∈ℝ^s denote the scaling vector such that, for each j∈ [s],w_j = β b^h_p,if there exists p∈[n] s.t. j = j_p, 0 otherwise.Let S∈ℝ^N h_max× n be such that, for each j∈ [s],for each p∈ [n],S_j,p = σ_p, if j = g_p + N · h_p,0, otherwise.Let D_w denote the diagonal matrix where the i-th entry on the diagonal is the i-th entry of w. Let S=D_w S. We say (S,w) or S is an MSketch.For a tensor A∈ℝ^d× n_1 × n_2 and a vector w∈𝕕, we defineA _v,w = ∑_i=1^d w_iA_i,*,*_F. Let (S,w) denote an MSketch, and let S= D_wS. If v corresponds to a scale-invariant M-Estimator, then for any three matrices U,V,W, we have the following,(S U ) ⊗ V ⊗ W _v,w =(D_w S U ) ⊗ V ⊗ W _v =(S U ) ⊗ V ⊗ W _v.For a tensor A∈ℝ^n× n× n, let S∈ℝ^s× n denote an MSketch (defined in <ref>) with s=(k,log n). Then SA can be computed in O((A)) time.For any fixed U^*∈ℝ^n× d and -k V^*∈ℝ^d× m with d=(k), let S∈ℝ^s× n denote an MSketch (defined in Definition <ref>) with s= (k,log n) rows. Then with probability at least .999, we have: for all -k V∈ℝ^d× m,U^* V^* - U^* V _v ≲ S U^* V^* - S U^* V _v≲ U^* V^* - U^* V _v. Given matrices A∈ℝ^n× m, U^*∈ℝ^n× d with d=(k), define V^*∈ℝ^d× m to be the optimal solution to -k V∈ℝ^d× mmin U^* V - A _v. Choose an MSketch S∈ℝ^s× n with s=(k,log n) according to Definition <ref>. Then with probability at least .99, we have: for all -k V∈ℝ^d× m,SU^* V - SA_v≲ U^*V^* - U^* V _v + O(1)U^* V^* - A _v ≲U^* V -A _v. Given matrices A∈ℝ^n× m, U^*∈ℝ^n× d with d=(k), define V^*∈ℝ^d× m to be the optimal solution to -k V∈ℝ^d× mmin U^* V - A _v. Choose an MSketch S∈ℝ^s× nwith s=(k,log n) according to Definition <ref>. Then with probability at least .99, we have: for all -k V∈ℝ^d× m,U^* V - A _v ≲ S U^* V - S A _v + O(1)U^* V^* - A _v.§.§.§ Running time analysisGiven a tensor A∈ℝ^n× d× d, let S∈ℝ^s× n denote an MSketch with s rows. Let SA denote a tensor that has size s× d × d. For each i∈{2,3}, let (SA)_i∈ℝ^d × ds denote a matrix obtained by flattening tensor SA along the i-th dimension. For each i∈{2,3}, let S_i∈ℝ^ds× s_i denote a CountSketch transform with s_i columns. For each i∈{2,3}, let T_i∈ℝ^t_i× d denote a CountSketch transform with t_i rows. Then (1) For each i∈{2,3}, (SA)_i S_i can be computed in O((A)) time.(2) For each i∈{2,3}, T_i (SA)_i S_i can be computed in O((A)) time.Proof of Part (1). First note that (SA)_2 S_2 has size n× S_2. Thus for each i∈ [d], j∈ [s_2], we have,( (SA)_2 S_2 )_i,j = ∑_x'=1^ds ((SA)_2)_i,x' (S_2)_x',j by (SA)_2 ∈ℝ^d× ds, S_2∈ℝ^ds × s_2= ∑_y=1^d∑_z=1^s((SA)_2)_i,(y-1)s+z (S_2)_(y-1)s+z,j= ∑_y=1^d∑_z=1^s(SA)_z,i,y (S_2)_(y-1)s+z,j by unflattening= ∑_y=1^d∑_z=1^s( ∑_x=1^n S_z,x A_x,i,y)(S_2)_(y-1)s+z,j= ∑_y=1^d∑_z=1^s ∑_x=1^nS_z,x· A_x,i,y·(S_2)_(y-1)s+z,j.For each nonzero entry A_x,i,y, there is only one z such that S_z,x is nonzero. Thus there is only one j such that (S_2)_(y-1)s+z,j is nonzero. It means that A_x,i,y can only affect one entry of ( (SA)_2 S_2 )_i,j. Thus, (SA)_2 S_2 can be computed in O((A)) time. Similarly, we can compute (SA)_3S_3 in O((A)) time.Proof of Part (2). Note that T_2 (SA)_2 S_2 has size t_2 × s_2. Thus for each i∈ [t_2], j∈ [s_2], we have,( T_2(SA)_2 S_2)_i,j = ∑_x=1^d∑_y'=1^ds (T_2)_i,x ((SA)_2)_x,y' (S_2)_y',j by (SA)_2 ∈ℝ^d × ds= ∑_x=1^d∑_y=1^d∑_z=1^s (T_2)_i,x ((SA)_2)_x,(y-1)s+z(S_2)_(y-1)s+z,j= ∑_x=1^d∑_y=1^d∑_z=1^s (T_2)_i,x (SA)_z,x,y (S_2)_(y-1)s+z,j by unflattening= ∑_x=1^d∑_y=1^d∑_z=1^s (T_2)_i,x(∑_w=1^n S_z,w A_w,x,y) (S_2)_(y-1)s+z,j= ∑_x=1^d∑_y=1^d∑_z=1^s∑_w=1^n (T_2)_i,x· S_z,w· A_w,x,y· (S_2)_(y-1)s+z,j.For each nonzero entry A_w,x,y, there is only one z such that S_z,w is nonzero. There is only one i such that (T_2)_i,x is nonzero. Since there is only one z to make S_z,w nonzero, there is only one j, such that (S_2)_(y-1)s+z,j isnonzero. Thus, T_2(SA)_2 S_2 can be computed in O((A)) time. Similarly, we can compute T_3 (SA)_3 S_3 in O((A)) time. §.§.§ AlgorithmsWe first give a “warm-up”algorithm in Theorem <ref> by using a sampling and rescaling matrix. Then we improve the running time to be polynomial in all the parameters by using an oblivious sketch, and thus we obtain Theorem <ref>. Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, let r=O(k^2). There exists an algorithm which takes n^(k) time and outputs three matrices U,V,W∈ℝ^n× r such thatU ⊗ V ⊗ W - A_v ≤(k) -k A'min A' - A _v,holds with probability at least 9/10.We defineas follows,= U,V,W ∈ℝ^n× kminU ⊗ V⊗ W - A _v =U,V,W ∈ℝ^n× kmin∑_i=1^k U_i ⊗ V_i ⊗ W_i - A _v.Let A_1∈ℝ^n× n^2 denote the matrix obtained by flattening tensor A along the 1st dimension. Let U^*∈ℝ^n× k denote the optimal solution. We fix U^*∈ℝ^n× k, and consider this objective function,min_V,W ∈ℝ^n× kU^* ⊗ V ⊗ W - A _v ≡min_V,W ∈ℝ^n× kU^* · ( V^⊤⊙ W^⊤ )- A_1 _v,which has cost at most , and where V^⊤⊙ W^⊤∈ℝ^k× n^2 denotes the matrix for which the i-th row is a vectorization of V_i ⊗ W_i,∀ i∈ [k]. (Note that V_i∈ℝ^n is the i-th column of matrix V∈ℝ^n× k). Choose a sampling and rescaling diagonal matrix S∈ℝ^n× n according to U^*, which has s=(k) non-zero entries. Using S to sketch on the left of the objective function when U^* is fixed (Equation (<ref>)), we obtain a smaller problem,min_V,W ∈ℝ^n× k (SU^*) ⊗ V ⊗ W - SA _v ≡min_V,W ∈ℝ^n× kS U^* · ( V^⊤⊙ W^⊤ )- S A_1 _v.Let V',W' denote the optimal solution to the above problem, i.e.,V', W' = V,W∈ℝ^n× kmin (SU^*) ⊗ V ⊗ W -SA _v.Then using properties (no dilation Lemma <ref> and no contraction Lemma <ref>) of S, we haveU^* ⊗ V' ⊗ W' - A _v ≤α.where α is an approximation ratio determined by S.By definition of ·_v and ·_2 ≤·_1 ≤√(dim)·_2, we can rewrite Equation (<ref>) in the following way,(SU^*) ⊗ V ⊗ W - SA _v = ∑_i=1^s(∑_j=1^n ∑_l=1^n ( ( ( SU^*) ⊗V ⊗ W )_i,j,l - (SA)_i,j,l)^2)^1/2 ≤ √(s)( ∑_i=1^s ∑_j=1^n ∑_l=1^n ( ( ( SU^*) ⊗V ⊗ W )_i,j,l -(SA)_i,j,l)^2 )^1/2= √(s)(SU^*)⊗ V ⊗ W - SA _F.Given the above properties of S and Equation (<ref>), for any β≥ 1, let V”,W” denote a β-approximate solution of V,W∈ℝ^n× kmin (SU^*) ⊗ V ⊗ W -S A_F, i.e.,(SU^*) ⊗ V”⊗ W” - SA _F ≤β·V,W∈ℝ^n× kmin (SU^*) ⊗ V ⊗ W -S A_F.Then,U^* ⊗ V”⊗ W” - A _v ≤√(s)αβ·.In the next few paragraphs we will focus on solving Equation (<ref>). We start by fixing W^*∈ℝ^n× k to be the optimal solution ofmin_V,W∈ℝ^n× k(SU^*) ⊗ V ⊗ W - SA _F. We use (SA)_2 ∈ℝ^n× ns to denote the matrix obtained by flattening the tensor SA ∈ℝ^s× n × n along the second direction. We useZ_2 = (SU^*)^⊤⊙ (W^*)^⊤∈ℝ^k× ns to denote the matrix where the i-th row is the vectorization of (SU^*)_i ⊗ W_i^*. We can consider the following objective function,min_V∈ℝ^n× k V Z_2- (SA)_2 _F.Choosing a sketching matrix S_2 ∈ℝ^ns × s_2 with s_2=O(k/ϵ) gives a smaller problem,min_V∈ℝ^n× k V Z_2 S_2 - (SA)_2 S_2 _F.Letting V = (SA)_2 S_2 (Z_2 S_2)^†∈ℝ^n× k, thenV Z_2- (SA)_2_F ≤ (1+ϵ) min_V∈ℝ^n× k V Z_2- (SA)_2 _F = (1+ϵ) min_V∈ℝ^n× k V ( (SU^*)^⊤⊙ (W^*)^⊤ ) - (SA)_2 _F = (1+ϵ) min_V∈ℝ^n× k(SU^*) ⊗ V ⊗ W^*- SA _F by unflattening= (1+ϵ) min_V,W∈ℝ^n× k(SU^*) ⊗ V ⊗ W- SA _F. by definition of W^* We define D_2∈ℝ^n^2 × n^2 to be a diagonal matrix obrained by copying the n× n identity matrix s times on n diagonal blocks of D_2. Then it has ns nonzero entries. Thus, D_2 also can be thought of as a matrix that has size n^2 × ns.We can think of (SA)_2 S_2 ∈ℝ^n× s_2 as follows,(SA)_2 S_2 = (A(S,I,I))_2 S_2 = A_2_n× n^2· D_2_n^2 × n^2·S_2_ns × s_2 by D_2 can be thought of as having size n^2× ns = A_2 ·[ c_2,1 I_n; c_2,2 I_n; ⋱; c_2,n I_n ]· S_2where I_n is an n× n identity matrix, c_2,i≥ 0 for each i∈ [n], and the number of nonzero c_2,i is s. For the last step, we fix SU^* and V. We use (SA)_3 ∈ℝ^n× ns to denote the matrix obtained by flattening the tensor SA∈ℝ^s× n× n along the third direction. We use Z_3 = (SU^*)^⊤⊙V^⊤∈ℝ^k× ns to denote the matrix where the i-th row is the vectorization of (S U^*)_i ⊗V_i. We can consider the following objective function,min_W∈ℝ^n× k W Z_3 - (SA)_3 _F.Choosing a sketching matrix S_3∈ℝ^ns × s_3 with s_3=O(k/ϵ) gives a smaller problem,min_W∈ℝ^n× k W Z_3 S_3 - (SA)_3 S_3 _F.Let W = (SA)_3 S_3 (Z_3 S_3)^†∈ℝ^n× k. ThenW Z_3- (SA)_3_F ≤ (1+ϵ) min_W∈ℝ^n× k W Z_3- (SA)_3 _F by property of S_3 = (1+ϵ) min_W∈ℝ^n× k W ((SU^*)^⊤⊙V^⊤)- (SA)_3 _F by definition Z_3 = (1+ϵ) min_W∈ℝ^n× k (SU^*) ⊗V⊗ W- SA _F by unflattening ≤ (1+ϵ)^2 (SU^*) ⊗ V ⊗ W - SA _F. by Equation (<ref>) We define D_3∈ℝ^n^2 × n^2 to be a diagonal matrix formed by copying the n× n identity matrix s times on n diagonal blocks of D_3. Then it has ns nonzero entries. Thus, D_3 also can be thought of as a matrix that has size n^2 × ns and D_3 is uniquely determined by S.Similarly as to the 2nd dimension, for the 3rd dimension, we can think of (SA)_3 S_3 as follows,(SA)_3 S_3 = ( A(S,I,I) )_3S_3 = A_3_n× n^2·D_3_n^2× n^2·S_3_ns × s_3 by D_3 can be thought of as having size n^2× ns = A_3 ·[ c_3,1 I_n; c_3,2 I_n; ⋱; c_3,n I_n ]· S_3where I_n is an n× n identity matrix, c_3,i≥ 0 for each i∈ [n] and the number of nonzero c_3,i is s.Overall, we have proved that,min_X_2,X_3(SU^*) ⊗ (A_2 D_2 S_2 X_2) ⊗ (A_3 D_3 S_3 X_3 ) - S A _F ≤ (1+ϵ)^2 (SU^*) ⊗ V ⊗ W - SA _F,where diagonal matrix D_2∈ℝ^n^2 × n^2 (with ns nonzero entries) and D_3∈ℝ^n^2 × n^2 (with ns nonzero entries) are uniquely determined by diagonal matrix S∈ℝ^n× n (s nonzero entries). Let X'_2 and X_3' denote the optimal solution to the above problem (Equation (<ref>)). Let V”=(A_2 D_2 S_2 X_2')∈ℝ^n× k and W”=(A_3 D_3 S_3 X_3' ) ∈ℝ^n× k. Then we haveU^* ⊗ V”⊗ W” - A _v ≤√(s)αβ.We construct matrix V∈ℝ^n× s_2 s_3 by copying matrix (SA)_2 S_2∈ℝ^n× s_2 s_3 times,V = [(SA)_2 S_2(SA)_2 S_2 ⋯ (SA)_2 S_2. ]We construct matrix W∈ℝ^n× s_2 s_3 by copying the i-th column of matrix (SA)_3 S_3∈ℝ^n× s_3 into (i-1)s_2+1, ⋯, i s_2 columns of W,W = [ ( (SA)_3 S_3)_1⋯( (SA)_3 S_3)_1( (SA)_3 S_3)_2⋯ ( (SA)_3 S_3)_2 ⋯ ( (SA)_3 S_3)_s_3⋯ ( (SA)_3 S_3)_s_3. ] Although we don't know S, we can guess all of the possibilities. For each possibility, we can find a solution U∈ℝ^n× s_2 s_3 to the following problem,min_U∈ℝ^n× s_2 s_3∑_i=1^s_2∑_j=1^s_3 U_(i-1)s_3+j⊗ ( (SA)_2 S_2 )_i ⊗ ( (SA)_3 S_3 )_j - A _v= min_U∈ℝ^n× s_2 s_3∑_i=1^s_2∑_j=1^s_3 U_(i-1)s_3+j·( ( (SA)_2 S_2 )_i ⊗ ( (SA)_3 S_3 )_j ) - A_1 _v = min_U∈ℝ^n× s_2 s_3∑_i=1^s_2∑_j=1^s_3 U_(i-1)s_3+j· (V^⊤⊙W^⊤ )^(i-1)s_3+j - A_1 _v = min_U∈ℝ^n× s_2 s_3 U· (V^⊤⊙W^⊤ ) - A_1 _v = min_U∈ℝ^n× s_2 s_3 U Z - A_1 _v = min_U ∈ℝ^n× s_2 s_3∑_i=1^s_2 s_3U^i Z - A_1^i _2,where the first step follows by flattening the tensor along the 1st dimension, U_(i-1)s_3+j denotes the (i-1)s_3+j-th column of U∈ℝ^n× s_2 s_3, A_1∈ℝ^n× n^2 denotes the matrix obtained by flattening tensor A along the 1st dimension, the second step follows since V^⊤⊙W^⊤∈ℝ^s_2s_3 ∈ n^2 is defined to be the matrix where the (i-1)s_3 +j-th row is vectorization of ( (SA)_2 S_2 )_i ⊗ ( (SA)_3 S_3 )_j, the fourth step follows by defining Z to be V^⊤⊙W^⊤, and the last step follows by definition of ·_v norm. Thus, we obtain a multiple regression problem and it can be solved directly by using <cit.>.Finally, we take the best U, V, W over all the guesses. The entire running time is dominated by the number of guesses, which is n^(k). This completes the proof.Given a 3rd order tensor A∈ℝ^n× n × n, for any k≥ 1, let r=O(k^2). There exists an algorithm which takes O((A) ) + n (k,log n) time and outputs three matrices U,V,W∈ℝ^n× r such thatU ⊗ V ⊗ W - A_v ≤(k,log n) -k A'min A' - A _vholds with probability at least 9/10.We defineas follows,= U,V,W ∈ℝ^n× kminU ⊗ V⊗ W - A _v =U,V,W ∈ℝ^n× kmin∑_i=1^k U_i ⊗ V_i ⊗ W_i - A _v.Let A_1∈ℝ^n× n^2 denote the matrix obtained by flattening tensor A along the 1st dimension. Let U^*∈ℝ^n× k denote the optimal solution. We fix U^*∈ℝ^n× k, and consider the objective function,min_V,W ∈ℝ^n× kU^* ⊗ V ⊗ W - A _v ≡min_V,W ∈ℝ^n× kU^* · ( V^⊤⊙ W^⊤ )- A_1 _v,which has cost at most , and where V^⊤⊙ W^⊤∈ℝ^k× n^2 denotes the matrix for which the i-th row is a vectorization of V_i ⊗ W_i,∀ i∈ [k]. (Note that V_i∈ℝ^n is the i-th column of matrix V∈ℝ^n× k). Choose an (oblivious) MSketch S∈ℝ^s× nwith s=(k,log n) according to Definition <ref>. Using MSketch S,w to sketch on the left of the objective function when U^* is fixed (Equation (<ref>)), we obtain a smaller problem,min_V,W ∈ℝ^n× k (SU^*) ⊗ V ⊗ W - SA _v≡ min_V,W ∈ℝ^n× kS U^* · ( V^⊤⊙ W^⊤ )- S A_1 _v.Let V',W' denote the optimal solution to the above problem, i.e.,V', W' = V,W∈ℝ^n× kmin (SU^*) ⊗ V ⊗ W -SA _v.Then using properties (no dilation Lemma <ref> and no contraction Lemma <ref>) of S, we haveU^* ⊗ V' ⊗ W' - A _v ≤α.where α is an approximation ratio determined by S.By definition of ·_v and ·_2 ≤·_1 ≤√(dim)·_2, we can rewrite Equation (<ref>) in the following way,(SU^*) ⊗ V ⊗ W - SA _v = ∑_i=1^s(∑_j=1^n ∑_l=1^n ( ( ( SU^*) ⊗V ⊗ W )_i,j,l - (SA)_i,j,l)^2)^1/2 ≤ √(s)( ∑_i=1^s ∑_j=1^n ∑_l=1^n ( ( ( SU^*) ⊗V ⊗ W )_i,j,l -(SA)_i,j,l)^2 )^1/2= √(s)(SU^*)⊗ V ⊗ W - SA _FUsing the properties of S and Equation (<ref>), for any β≥ 1, let V”,W” denote a β-approximation solution of V,W∈ℝ^n× kmin (SU^*) ⊗ V ⊗ W -S A_F, i.e., (SU^*) ⊗ V”⊗ W” - SA _F ≤β·V,W∈ℝ^n× kmin (SU^*) ⊗ V ⊗ W -S A_F.Then,U^* ⊗ V”⊗ W” - A _v ≤√(s)αβ·. Let A denote SA. Choose S_i∈ℝ^ns × s_i to be Gaussian matrix with s_i=O(k/ϵ), ∀ i {2,3}. By a similar proof as in Theorem <ref>, we have if X_2',X_3' is a β-approximate solution tomin_X_2,X_3 (SU^*) ⊗ (A_2 S_2 X_2) ⊗ (A_3 S_3 X_3)-SA_F,then,U^* ⊗ (A_2 S_2 X_2) ⊗ (A_3 S_3 X_3) - A_v ≤√(s)αβ. To reduce the size of the objective function from (n) to (k/ϵ), we use perform an “input sparsity reduction” (in Lemma <ref>). Note that, we do not need to use this idea to optimize the running time in Theorem <ref>. The running time of Theorem <ref> is dominated by guessing sampling and rescaling matrices. (That running time is ≫(A).) Choose T_i ∈ℝ^t_i × n to be a sparse subspace embedding matrix (CountSketch transform) with t_i=(k,1/ϵ), ∀ i ∈{2,3}. Applying the proof of Lemma <ref> here, we obtain, if X_2',X_3' is a β-approximate solution tomin_X_2,X_3 (SU^*) ⊗ (T_2 (SA)_2 S_2 X_2) ⊗ (T_3 (SA)_3 S_3 X_3)-SA_F,then,U^* ⊗ ( (SA)_2 S_2 X_2) ⊗ ( (SA)_3 S_3 X_3) - A_v ≤√(s)αβ. Similar to the bicriteria results in Section <ref>, Equation (<ref>) indicates that we can construct a bicriteria solution by using two matrices (SA)_2 S_2 and(SA)_3 S_3. The next question is how to obtain the final results U,V,W. We first show how to obtain U. Then we show to construct V and W. To obtain U, we need to solve a regression problem related to two matrices V,W and a tensor A(I,T_2,T_3). We construct matrix V∈ℝ^t_2× s_2 s_3 by copying matrix T_2 (SA)_2 S_2∈ℝ^t_2 × s_2 s_3 times,V = [T_2(SA)_2 S_2T_2(SA)_2 S_2⋯ T_2 (SA)_2 S_2 ].We construct matrix W∈ℝ^t_3 × s_2 s_3 by copying the i-th column of matrix T_3(SA)_3 S_3∈ℝ^t_3 × s_3 into (i-1)s_2+1, ⋯, i s_2 columns of W,W = [F_1⋯F_1 F_2⋯ F_2⋯ F_s_3⋯ F_s_3 ],where F=T_3 (SA)_3 S_3.Thus, to obtain U∈ℝ^s_2 s_3, we just need to use a linear regression solver to solve a smaller problem,min_U ∈ℝ^s_2 s_3 U · ( V^⊤⊙W^⊤ )- A(I,T_2,T_3) _F,which can be solved in O((A))+ n (k,log n) time. We will show how to obtain V and W.We construct matrix V∈ℝ^n× s_2 s_3 by copying matrix (SA)_2 S_2∈ℝ^n× s_2 s_3 times,V = [(SA)_2 S_2(SA)_2 S_2 ⋯ (SA)_2 S_2. ]We construct matrix W∈ℝ^n× s_2 s_3 by copying the i-th column of matrix (SA)_3 S_3∈ℝ^n× s_3 into (i-1)s_2+1, ⋯, i s_2 columns of W,W = [F_1⋯F_1 F_2⋯ F_2⋯ F_s_3⋯ F_s_3 ],where F=(SA)_3 S_3. §.§ ℓ_1-ℓ_1-ℓ_2 normSection <ref> presents some definitions and useful facts for the tensor ℓ_1-ℓ_1-ℓ_2 norm. We provide some tools in Section <ref>. Section <ref> presents a key idea which shows we are able to reduce the original problem to a new problem under entry-wise ℓ_1 norm. Section <ref> presents several existence results. Finally, Section <ref> introduces several algorithms with different tradeoffs. §.§.§ Definitions(Tensor u-norm) For an n× n × n tensor A, we define the u-norm of A, denoted A _u, to be( ∑_i=1^n ∑_j=1^n M(A_i,j,*_2 ) )^1/p,where A_i,j,* is the (i,j)-th tube of A, and p is a parameter associated with the function M(), which defines a nice M-Estimator.(Matrix u-norm) For an n× n matrix A, we define u-norm of A, denoted A _u, to be( ∑_i=1^nM(A_i,*_2 ) )^1/p,where A_i,* is the i-th row of A, and p is a parameter associated with the function M(), which defines a nice M-Estimator.For p=1, for any two matrices A and B, we have A + B _u≤ A _u + B _u. For any two tensors A and B, we have A + B _u ≤ A _u +B _u.§.§.§ Projection via Gaussians Let p≥ 1. Let ℓ_p^ S^n-1 be an infinite dimensional ℓ_p metric which consists of a coordinate for each vector r in the unit sphere S^n-1. Define function f: S^n-1→ℝ. The ℓ_1-norm of any such f is defined as follows:f _1 =( ∫_r ∈ S^n-1 |f(r) |^pd r )^1/p.Let f_v(r) = ⟨ v, r⟩. There exists a universal constant α_p such thatf_v _p = α_pv _2.We have,f_v _p = ( ∫_r∈ S^n-1 | ⟨ v, r⟩ |^p d r)^1/p= (∫_θ∈ S^n-1 v_2^p · |cosθ|^p dθ)^1/p= v_2 ( ∫_θ∈ S^n-1 |cosθ|^p dθ)^1/p= α_pv _2.This completes the proof.Let G∈ℝ^k× n denote i.i.d. random Gaussian matrices with rescaling. Then for any v∈ℝ^n, we have[ (1-ϵ)v_2 ≤ G v _1 ≤ (1+ϵ)v _2] ≥ 1 - 2^-Ω(kϵ^2).For each i∈ [k], we define X_i = ⟨ v, g_i⟩, where g_i∈ℝ^n is the i-th row of G. Then X_i = ∑_j=1^n v_j g_i,j and [|X_i|] = α_pv_2. Define Y = ∑_i=1^k |X_i|. We have [Y] = kα_1v _2 = k α_1. We can show[Y ≥ (1+ϵ) α_1 k ] = [ e^sY≥ e^s(1+ϵ) α_1 k ] for all s>0≤ [e^sY] / e^s(1+ϵ) α_1 k by Markov's inequality= e^-s(1+ϵ) α_1 k·[ ∏_i=1^k e^s|X_i|] by Y =∑_i=1^k |X_i| = e^-s(1+ϵ) α_1 k· ([ e^s|X_1|])^kIt remains to bound [ e^s|X_1|]. Since X_1∼ N(0,1), we have that X_1 has density function e^-t^2/2. Thus, we have,[e^s |X_1| ] = 1/√(2π)∫_-∞^+∞ e^s|t|· e^-t^2/2d t= 1/√(2π)∫_-∞^+∞ e^s^2/2· e^- (|t|-s)^2/2d t = e^s^2/2 ( erf( s/√(2)) + 1)≤ e^s^2/2 ( (1-exp(-2s^2/π) )^1/2 +1) by 1-exp(-4x^2/π) ≥erf(x)^2≤ e^s^2/2 ( √(2/π) s +1). by 1-e^-x≤ xThus, we have[Y ≥ (1+ϵ) α_1 k ]≤ e^-s(1+ϵ)ke^ks^2/2 (1+s √(2/π))^k = e^-s(1+ϵ) α_1 ke^ks^2/2 e^ k·log (1+s √(2/π) )≤ e^-s(1+ϵ) α_1 k+ks^2/2 + k· s √(2/π) ≤ e^-Ω(kϵ^2). by α_1 ≥√(2/π) and setting s= ϵ For any ϵ∈ (0,1), let k=O(n/ϵ^2). Let G∈ℝ^k× n denote i.i.d. random Gaussian matrices with rescaling. Then for any v∈ℝ^n,with probability at least 1-2^-Ω(n/ϵ^2 ), we have : for all v∈ℝ^n,(1-ϵ)v_2 ≤ G v _1 ≤ (1+ϵ)v _2.Let S denote { y∈ℝ^n | y_2=1}. We construct a γ-net so that for all y∈ S, there exists a vector w∈ N for which y-w_2 ≤γ. We set γ = 1/2.For any unit vector y, we can writey=y^0 + y^1+y^2 +⋯,where y^i _2 ≤ 1/2^i and y^i is a scalar multiple of a vector in N. Thus, we haveG y _1 = G (y^0 + y^1 + y^2 +⋯) _1≤ ∑_i=0^∞ G y^i _1 by triangle inequality ≤ ∑_i=0^∞ (1+ϵ) y^i_2≤ ∑_i=0^∞ (1+ϵ) 1/2^i ≤ 1+ Θ(ϵ).Similarly, we can lower bound Gy_1 by 1-Θ(ϵ). By Lemma 2.2 in <cit.>, we know that for any γ∈ (0,1), there exists a γ-net N of S for which | N|≤ (1 +4/γ)^n. §.§.§ Reduction, projection to high dimension Given a 3rd order tensor A∈ℝ^n× n× n, let S∈ℝ^n× s denote a Gaussian matrix with s= O(n/ϵ^2) columns. With probability at least 1-2^-Ω(n/ϵ^2), for any U,V,W∈ℝ^n× k, we have(1-ϵ) U ⊗ V ⊗ W - A_u ≤ (U ⊗ V ⊗ W )S -AS _1 ≤ (1+ϵ) U ⊗ V ⊗ W - A _u.By definition of the ⊗ product between matrices and · product between a tensor and a matrix, we have (U⊗ V ⊗ W) S = U ⊗ V ⊗ (SW) ∈ℝ^n× n × s. We use A_i,j,*∈ℝ^n to denote the (i,j)-th tube (the column in the 3rd dimension) of tensor A. We first prove the upper bound,(U ⊗ V ⊗ W)S - AS _1 = ∑_i=1^n ∑_j=1^n( ( U ⊗ V ⊗ W )_i,j,* - A_i,j,*) S _1≤ ∑_i=1^n ∑_j=1^n(1+ϵ)( U ⊗ V ⊗ W )_i,j,* - A_i,j,*_2 = (1+ϵ)U ⊗ V ⊗ W - A _u,where the first step follows by definition of tensor ·_u norm, the second step follows by Lemma <ref>, and the last step follows by tensor entry-wise ℓ_1 norm. Similarly, we can prove the lower bound,(U ⊗ V ⊗ W)S - AS _1≥ ∑_i=1^n ∑_j=1^n(1-ϵ)( U ⊗ V ⊗ W )_i,j,* - A_i,j,*_2= (1-ϵ)U ⊗ V ⊗ W - A _u. This completes the proof.For any α≥ 1, if U',V',W' satisfy(U'⊗ V'⊗ W' -A) S _1 ≤γmin_-k A_k (A_k - A) S _1,thenU'⊗ V'⊗ W' -A _u ≤γ1+ϵ/1-ϵmin_-k A_k A_k - A _u.Let U, V, W denote the optimal solution to min_-k A_k (A_k - A) S _1. Let U^*,V^*,W^* denote the optimal solution to min_-k A_k A_k - A _u. Then,U'⊗ V'⊗ W' -A _u ≤ 1/1-ϵ ( U'⊗ V'⊗ W' -A ) S _1≤ γ1/1-ϵ ( U⊗V⊗W -A ) S _1≤ γ1/1-ϵ ( U^* ⊗ V^* ⊗ W^* -A ) S _1≤ γ1+ϵ/1-ϵU^* ⊗ V^* ⊗ W^* -A _u,which completes the proof. §.§.§ Existence resultsGiven a 3rd order tensor A∈ℝ^n× n× n and a matrix S∈ℝ^n×n, letdenote min_-k A_k∈ℝ^n× n× n (A_k - A)S _1, let A = AS∈ℝ^n× n×n. For any k≥ 1, there exist three matrices S_1∈ℝ^nn× s_1, S_2∈ℝ^nn× s_2, S_3 ∈ℝ^n^2 × s_3 such thatmin_X_1∈ℝ^s_1× k, X_2∈ℝ^s_2× k, X_3∈ℝ^s_3 × k (A_1 S_1 X_1) ⊗ (A_2 S_2 X_2) ⊗ ( A_3 S_3 X_3) - A_1 ≤α,or equivalently,min_X_1∈ℝ^s_1× k, X_2∈ℝ^s_2× k, X_3∈ℝ^s_3 × k( (A_1 S_1 X_1) ⊗ (A_2 S_2 X_2) ⊗ ( A_3 S_3 X_3) - A ) S _1 ≤α,holds with probability 99/100.(1). Using a dense Cauchy transform,s_1=s_2=s_3=O(k), α = O(k^1.5) log^3 n. (2). Using a sparse Cauchy transform,s_1=s_2=s_3=O(k^5), α = O(k^13.5) log^3 n. (3). Guessing Lewis weights,s_1=s_2=s_3=O(k), α = O(k^1.5).We useto denote the optimal cost,:= -k A_k ∈ℝ^n× n×nmin (A_k - A )S _1. We fix V^*∈ℝ^n× k and W^*∈ℝ^n× k to be the optimal solution tomin_U,V,W (U ⊗ V⊗ W - A) S _1. We define Z_1∈ℝ^k× nn to be the matrix where the i-th row is the vectorization of V_i^* ⊗ (S W_i^*). We define tensorA = AS ∈ℝ^n× n×n.Then we also have A= A(I,I,S) according to the definition of the · product between a tensor and a matrix.Let A_1∈ℝ^n× nn denote the matrix obtained by flattening tensor A along the first direction. We can consider the following optimization problem,min_U ∈ℝ^n× k U Z_1 - A_1 _1.Choosing S_1 to be one of the following sketching matrices:(1) a dense Cauchy transform,(2) a sparse Cauchy transform,(3) a sampling and rescaling diagonal matrix according to Lewis weights. Let α_S_1 denote the approximation ratio produced by the sketching matrix S_1. We use S_1∈ℝ^nn× s_1 to sketch on right of the above problem, and obtain the problem:min_U ∈ℝ^n× k U Z_1 S_1 - A_1 S_1 _1 = min_U∈ℝ^n× k∑_i=1^nU^i Z_1 S_1 - (A_1 S_1)^i _1,where U^i denotes the i-th row of matrix U∈ℝ^n× k and (A_1S_1)^i denotes the i-th row of matrix A_1 S_1. Instead of solving it under ℓ_1-norm, we consider the ℓ_2-norm relaxation,U ∈ℝ^n× kmin U Z_1 S_1 - A_1 S_1 _F^2 = U∈ℝ^n× kmin∑_i=1^nU^i Z_1 S_1 - ( A_1 S_1 )^i _2^2.Let U∈ℝ^n× k denote the optimal solution of the above optimization problem, so that U = A_1 S_1 (Z_1 S_1)^†. We plug U into the objective function under the ℓ_1-norm. By the property of sketching matrix S_1∈ℝ^nn× s_1, we have,U Z_1 - A_1 _1 ≤α_S_1min_U∈ℝ^n× k U Z_1 - A_1 _1 =α_S_1,which implies that,U⊗ V^* ⊗ (SW^*) - A_1 = (U⊗ V^* ⊗ W^*) S - A_1 ≤α_S_1. In the second step, we fix U∈ℝ^n× k and W^* ∈ℝ^n× k. Let A_2 ∈ℝ^n× n n denote the matrix obtained by flattening tensor A∈ℝ^n× n ×n along the second direction. We choose a sketching matrix S_2∈ℝ^n n× s_2. Let Z_2 = U^⊤⊙(SW^*)^⊤∈ℝ^k× nn denote the matrix where the i-th row is the vectorization of U_i ⊗ (SW_i^*). Define V = A_2 S_2 (Z_2 S_2)^†. By the properties of sketching matrix S_2, we haveV Z_2 - A_2 _1 ≤α_S_2α_S_1, In the third step, we fix U∈ℝ^n× k and V∈ℝ^n× k. Let A_3 ∈ℝ^n× n^2 denote the matrix obtained by flattening tensor A∈ℝ^n× n ×n along the third direction. We choose a sketching matrix S_3∈ℝ^n^2 × s_3. Let Z_3∈ℝ^k× n^2 denote the matrix where the i-th row is the vectorization of U_i ⊗V_i. Define W'=A_3 S_3 (Z_3 S_3)^†∈ℝ^n× k and W = A_3 S_3 (Z_3 S_3)^†∈ℝ^n × k. Then we have,W' = A_3 S_3 (Z_3 S_3)^† = ( A(I,I,S) )_3 S_3 (Z_3 S_3)^† = (S^⊤ A_3 ) S_3 (Z_3 S_3)^† = S^⊤W By properties of sketching matrix S_3, we haveW' Z_3 - A_3 _1 ≤α_S_3α_S_2α_S_1.Replacing W' by S^⊤W, we obtain,W' Z_3 - A_3 _1 =S^⊤W Z_3 - A_3 _1 =S^⊤W Z_3 - S^⊤ A_3 _1 =(U⊗V⊗W - A) S _1.Thus, we havemin_X_1∈ℝ^s_1× k, X_2∈ℝ^s_2× k, X_3∈ℝ^s_3 × k (A_1 S_1 X_1) ⊗ (A_2 S_2 X_2) ⊗ ( A_3 S_3 X_3) - A_1 ≤α_S_1α_S_2α_S_3.§.§.§ Running time analysisGiven tensor A∈ℝ^n× n × n and a matrix B∈ℝ^n× d with d=O(n), let AB denote an n× n× d size tensor, For each i∈ [3], let (AB)_i denote a matrix obtained by flattening tensor AB along the i-th dimension, then(AB)_1∈ℝ^n× nd, (AB)_2∈ℝ^n× nd, (AB)_3 ∈ℝ^d × n^2.For each i∈[3], let S_i ∈ℝ^n d × s_i denote a sparse Cauchy transform, T_i ∈ℝ^t_i× n.Then we have,(1) If T_1 denotes a sparse Cauchy transform or a sampling and rescaling matrix according to the Lewis weights, T_1 (A B)_1 S_1 can be computed in O((A)d) time. Otherwise, it can be computed in O((A)d+ns_1t_1).(2) If T_2 denotes a sparse Cauchy transform or a sampling and rescaling matrix according to the Lewis weights, T_2 (A B)_2 S_2 can be computed in O((A)d) time. Otherwise, it can be computed in O((A)d+ns_2t_2).(3) If T_3 denotes a sparse Cauchy transform or a sampling and rescaling matrix according to the Lewis weights, T_3 (A B)_3 S_3 can be computed in O((A)d) time. Otherwise, it can be computed in O((A)d+ds_3t_3).Part (1). Note that T_1 (A B)_1 S_1 ∈ℝ^t_1 × s_1 and (AB)_1 ∈ℝ^n× nd, for each i∈ [t_1], j∈ [s_1],(T_1 (A B)_1 S_1)_i,j= ∑_x=1^n ∑_y'=1^nd (T_1)_i,x ( (AB)_1 )_x,y' (S_1)_y',j= ∑_x=1^n ∑_y=1^n∑_z=1^d (T_1)_i,x ( (AB)_1 )_x,(y-1)d+z (S_1)_(y-1)d+z,j= ∑_x=1^n ∑_y=1^n∑_z=1^d (T_1)_i,x (AB)_x,y,z (S_1)_(y-1)d+z,j= ∑_x=1^n ∑_y=1^n∑_z=1^d (T_1)_i,x∑_w=1^n (A_x,y,w B_w,z) (S_1)_(y-1)d+z,j= ∑_x=1^n ∑_y=1^n (T_1)_i,x∑_w=1^n A_x,y,w∑_z=1^d B_w,z (S_1)_(y-1)d+z,j.We look at a non-zero entry A_x,y,w and the entry B_w,z. If T_1 denotes a sparse Cauchy transform or a sampling and rescaling matrix according to the Lewis weights, then there is at most one pair (i,j) such that (T_1)_i,xA_x,y,wB_w,z(S_1)_(y-1)d+z,j is non-zero. Therefore, computing T_1 (A B)_1 S_1 only needs (A)d time. If T_1 is not in the above case, since S_1 is sparse, we can compute (A B)_1 S_1 in (A)d time by a similar argument. Then, we can compute T_1(A B)_1 S_1 in nt_1s_1 time.Part (2). It is as the same as Part (1).Part (3). Note that T_3 (A B)_3 S_3 ∈ℝ^t_3 × s_3 and (AB)_3 ∈ℝ^d × n^2. For each i∈ [t_3], j∈ [s_3],(T_3 (A B)_3 S_3)_i,j= ∑_x=1^d ∑_y'=1^n^2 (T_3)_i,x ( (AB)_3 )_x,y' (S_3)_y',j = ∑_x=1^d ∑_y=1^n∑_z=1^n (T_3)_i,x ( (AB)_3 )_x,(y-1)n+z (S_3)_(y-1)n+z,j = ∑_x=1^d ∑_y=1^n∑_z=1^n (T_3)_i,x (AB)_y,z,x (S_3)_(y-1)n+z,j = ∑_x=1^d ∑_y=1^n∑_z=1^n (T_3)_i,x∑_w=1^n A_y,z,w B_w,x (S_3)_(y-1)n+z,jSimilar to Part (1), if T_1 denotes a sparse Cauchy transform or a sampling and rescaling matrix according to the Lewis weights, computing T_3 (A B)_3 S_3 only needs (A)d time. Otherwise, it needs dt_3s_3+(A)d running time.§.§.§ AlgorithmsGiven a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm which takes O((A)n) + O(n) (k) + n 2^O(k^2) time and outputs three matrices U,V,W∈ℝ^n× k such that,U⊗ V ⊗ W - A _u ≤(k,log n) min_-k A' A' - A _u,holds with probability at least 9/10. We first choose a Gaussian matrix S∈ℝ^n×n with n=O(n). By applying Corollary <ref>, we can reduce the original problem to a “generalized” ℓ_1 low rank approximation problem. Next, we use the existence results (Theorem <ref>) and polynomial in k size reduction (Lemma <ref>). At the end, we relax the ℓ_1-norm objective function to a Frobenius norm objective function (Fact <ref>).Given a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, there exists an algorithm which takes n^O(k) 2^O(k^3) time and outputs three matrices U,V,W∈ℝ^n× k such that,U⊗ V ⊗ W - A _u ≤ O(k^3/2) min_-k A' A' - A _u,holds with probability at least 9/10. We first choose a Gaussian matrix S∈ℝ^n×n with n=O(n). By applying Corollary <ref>, we can reduce the original problem to a “generalized” ℓ_1 low rank approximation problem. Next, we use the existence results (Theorem <ref>) and polynomial in k size reduction (Lemma <ref>). At the end, we solve an entry-wise ℓ_1 norm objective function directly.Given a 3rd order tensor A∈ℝ^n× n× n, for any k≥ 1, let r=O(k^2). There is an algorithm which takes O((A)n) + O(n) (k) time and outputs three matrices U, V, W∈ℝ^n× r such thatU ⊗ V ⊗ W - A _u ≤(log n, k) min_-k A_k A_k - A_u,holds with probability at least 9/10.We first choose a Gaussian matrix S∈ℝ^n×n with n=O(n). By applying Corollary <ref>, we can reduce the original problem to a “generalized” ℓ_1 low rank approximation problem. Next, we use the existence results (Theorem <ref>) and polynomial in k size reduction (Lemma <ref>). At the end, we solve an entry-wise ℓ_1 norm objective function directly.§ WEIGHTED FROBENIUS NORM FOR ARBITRARY TENSORS This section presents several tensor algorithms for the weighted case. For notational purposes, instead of using U,V,W to denote the ground truth factorization, we use U_1,U_2,U_3 to denote the ground truth factorization. We use A to denote the input tensor, and W to denote the tensor of weights. Combining our new tensor techniques with existing weighted low rank approximation algorithms <cit.> allows us to obtain several interesting new results. We provide some necessary definitions and facts in Section <ref>. Section <ref> provides an algorithm when W has at most r distinct faces in each dimension. Section <ref> studies relationships between r distinct faces and r distinct columns. Finally, we provides an algorithm with a similar running time but weaker assumption, where W has at most r distinct columns and r distinct rows in Section <ref>. The result in Theorem <ref> is fairly similar to Theorem <ref>, except for the running time. We only put a very detailed discussion in the statement of Theorem <ref>. Note that Theorem <ref> also has other versions which are similar to the Frobnius norm -k algorithms described in Section <ref>. For simplicity of presentation, we only present one clean and simple version (which assumes A_k exists and has factor norms which are not too large).§.§ Definitions and Facts For a matrix A∈ℝ^n× m and a weight matrix W∈ℝ^n× m, we define W ∘ A_F as follows,W ∘ A_F = ( ∑_i=1^n ∑_j=1^m W_i,j^2 A_i,j^2 )^1/2.For a tensor A∈ℝ^n× n× n and a weight tensor W∈ℝ^n× n× n, we define W ∘ A _F as follows,W ∘ A_F = ( ∑_i=1^n ∑_j=1^n ∑_l=1^n W_i,j,l^2 A_i,j,l^2 )^1/2.For three matrices A∈ℝ^n× m, U∈ℝ^n× k, V∈ℝ^k× m and a weight matrix W, from one perspective, we have(UV - A) ∘ W _F^2 = ∑_i=1^n( U^i V - A^i ) ∘ W^i _2^2 = ∑_i=1^n( U^i V - A^i ) D_W^i_2^2,where W^i denote the i-th row of matrix W, and D_W^i∈ℝ^m× m denotes a diagonal matrix where the j-th entry on diagonal is the j-th entry of vector W^i. From another perspective, we have(UV - A) ∘ W _F^2 = ∑_j=1^m( U V_j - A_j ) ∘ W_j _2^2 = ∑_j=1^m( U V_j - A_j ) D_W_j_2^2,where W_j denotes the j-th column of matrix W, and D_W_j∈ℝ^n× n denotes a diagonal matrix where the i-th entry on the diagonal is the i-th entry of vector W_j.One of the key tools we use in this section is,Let R be an n× n invertible matrix. Then, for each i∈ [n], j∈ [n],(R^-1)^j_i = (R_ j^ i ) /(R),where R_ j^ i is the matrix R with the i-th row and the j-th column removed. §.§ r distinct faces in each dimensionNotice that in the matrix case, it is sufficient to assume that A'_F is upper bounded <cit.>. Once we have that A'_F is bounded, without loss of generality, we can assume that U_1^* is an orthonormal basis<cit.>. If U_1^* is not an orthonormal basis, then let U_1'R denote a QR factorization of U_1^*, and then write U'_2 = RU_2^*. However, in the case of tensors we have to assume that each factor U^*_i_F is upper bounded due to border rank issues (see, e.g., <cit.>). Given a 3rd order n× n× n tensor A and an n× n × n tensor W of weights with r distinct faces in each of the three dimensions for which each entry can be written using O(n^δ) bits, for δ >0,define =inf_-k A_k W∘ (A_k - A )_F^2. Let k≥ 1 be an integer and let 0 < ϵ <1. If >0, and there exists a rank-k A_k=U_1^*⊗ U_2^* ⊗ U_3^* tensor (with size n× n× n) such that W∘ (A_k-A)_F^2 =, and max_i∈ [3]U_i^*_F ≤ 2^O(n^δ), then there exists an algorithm that takes ((A)+(W)+ n 2^O(rk^2/ϵ) ) n^O(δ) time in the unit costmodel with words of size O(log n) bits[The entries of A and W are assumed to fit in n^δ words.] and outputs three n × k matrices U_1,U_2,U_3 such thatW∘( U_1 ⊗ U_2 ⊗ U_3 - A ) _F^2 ≤ (1+ϵ)holds with probability 9/10.Note that W has r distinct columns, rows, and tubes. Hence, each of the matrices W_1,W_2,W_3 ∈ℝ^n× n^2 has at most r distinct columns, and at most r distinct rows. Let U_1^*,U_2^*,U_3^*∈ℝ^n× k denote the matrices satisfying W∘ (U_1^* ⊗ U_2^* ⊗ U_3^* -A )_F^2 =. We fix U_2^* and U_3^*, and consider a flattening of the tensor along the first dimension,min_U_1 ∈ℝ^n× k (U_1 Z_1 - A_1) ∘ W_1 _F^2 = ,where matrix Z_1= U_2^*⊤⊙ U_3^*⊤ has size k × n^2 and for each i∈ [k] the i-th row of Z_1 is ( (U_2^*)_i ⊗ (U_3^*)_i ). For each i∈ [n], let W_1^i denote the i-th row of n× n^2 matrix W_1. For each i∈ [n], let D_W^i_1 denote the diagonal matrix of size n^2 × n^2, where each diagonal entry is from the vector W^i_1 ∈ℝ^n^2. Without loss of generality, we can assume the first r rows of W_1 are distinct. We can rewrite the objective function along the first dimension as a sum of multiple regression problems. For any n× k matrix U_1,(U_1 Z_1 - A_1) ∘ W_1 _F^2 = ∑_i=1^nU_1^i Z_1 D_W_1^i - A_1^i D_W_1^i_2^2.Based on the observation that W_1 has r distinct rows, we can group the n rows of W^1 into r groups. We use g_1,1, g_1,2,⋯, g_1,r to denote r sets of indices such that, for each i ∈ g_1,j, W_1^i = W_1^j. Thus we can rewrite Equation (<ref>),(U_1 Z_1 - A_1) ∘ W_1 _F^2 = ∑_i=1^nU_1^i Z_1 D_W_1^i - A_1^i D_W_1^i_2^2 = ∑_j=1^r ∑_i∈ g_1,j U_1^i Z_1 D_W_1^i - A_1^i D_W_1^i_2^2.We can sketch the objective function by choosing Gaussian matrices S_1∈ℝ^n^2 × s_1 with s_1 = O(k/ϵ).∑_i=1^nU_1^i Z_1 D_W_1^i S_1 - A_1^i D_W_1^i S_1 _2^2.Let U_1 denote the optimal solution of the sketch problem,U_1 =U_1∈ℝ^n× kmin∑_i=1^nU_1^i Z_1 D_W_1^i S_1 - A_1^i D_W_1^i S_1 _2^2.By properties of S_1(<cit.>), plugging U∈ℝ^n× k into the original problem, we obtain,∑_i=1^n U_1^i Z_1 D_W_1^i - A_1^i D_W_1^i_2^2 ≤ (1+ϵ) .Note that U_1 ∈ℝ^n× k also has the following form. For each i∈ [n],U_1^i = A_1^i D_W_1^i S_1(Z_1 D_W_1^i S_1)^†= A_1^i D_W_1^i S_1(Z_1 D_W_1^i S_1)^⊤ ( (Z_1 D_W_1^i S_1) (Z_1 D_W_1^i S_1)^⊤ )^-1.Note that W_1 has r distinct rows. Thus, we only have r distinct D_W_1^i. This implies that there are r distinct matrices Z_1 D_W_1^i S_1 ∈ℝ^k× s_1. Using the definition of g_1,j, for j∈ [r], for each i∈ g_1,j⊂ [n], we haveU^i_1 = A_1^i D_W_1^i S_1(Z_1 D_W_1^i S_1)^†= A_1^i D_W_1^j S_1(Z_1 D_W_1^j S_1)^† by W_1^i = W_1^j,which means we only need to write down r different Z_1 D_W_1^j S_1.For each k× s_1 matrix Z_1 D_W_1^j S_1, we create k× s_1 variables to represent it. Thus, we need to create rks_1 variables to represent r matrices,{ Z_1 D_W_1^1 S_1, Z_1 D_W_1^2 S_1, ⋯, Z_1 D_W_1^r S_1 }.For simplicity, let P_1,i∈ℝ^k× s_1 denote Z_1 D_W_1^i S_1. Then we can rewrite U^i∈ℝ^k as follows,U_1^i = A_1^i D_W_1^i S_1P_1,i^⊤ ( P_1,i P_1,i^⊤ )^-1.If P_1,i P_1,i^⊤∈ℝ^k× k has rank k, then we can use Cramer's rule (Lemma <ref>) to write down the inverse of P_1,i P_1,i^⊤. However, vector W_1^i could have many zero entries. Then the rank of P_1,i P_1,i^⊤ can be smaller than k. There are two different ways to solve this issue. One way is by using the argument from <cit.>, which allows us to assume that P_1,i P_1,i^⊤∈ℝ^k× k has rank k. The other way is straightforward: we can guess the rank. There are k possibilities. Let t_i ≤ k denote the rank of P_1,i. Then we need to figure out a maximal linearly independent subset of rows of P_1,i. There are 2^O(k) possibilities. Next, we need to figure out a maximal linearly independent subset of columns of P_1,i. We can also guess all the possibilities, which is at most 2^O(k).Because we have r different P_1,i, the total number of guesses we have is at most 2^O(rk). Thus, we can write down ( P_1,i P_1,i^⊤ )^-1 according to Cramer's rule.After U_1 is obtained, we will fix U_1 and U_3^* in the next round. We consider the flattening of the tensor along the second direction,min_U_2 ∈ℝ^n× k (U_2 Z_2 - A_2 ) ∘ W_2 _F^2,where n× n^2 matrix A_2 is obtained by flattening tensor A along the second dimension, k× n^2 matrix Z_2 denotes U_1^⊤⊙ U_3^*⊤, and n× n^2 matrix W_2 is obtained by flattening tensor W along the second dimension. For each i∈ [n], let W_2^i denote the i-th row of n× n^2 matrix W_2. For each i∈ [n], let D_W_1^i denote the diagonal matrix which has size n^2 × n^2 and for which each entry is from vector W_2^i∈ℝ^n^2. Without loss of generality, we can assume the first r rows of W_2 are distinct. We can rewrite the objective function along the second dimension as a sum of multiple regression problems. For any n × k matrix U_2,(U_2 Z_2 - A_2) ∘ W_2 _F^2 = ∑_i=1^nU_2^i Z_2 D_W_2^i - A_2^i D_W_2^i_2^2.Based on the observation that W_2 has r distinct rows, we can group the n rows of W^2 into r groups. We use g_2,1, g_2,2, ⋯, g_2,r to denote r sets of indices such that, for each i∈ g_2,j, W_2^i = W_2^j. Thus we obtain,(U_2 Z_2 - A_2) ∘ W_2 _F^2 = ∑_i=1^nU_2^i Z_2 D_W_2^i - A_2^i D_W_2^i_2^2 = ∑_j=1^r ∑_i ∈ g_2,j U_2^i Z_2 D_W_2^i - A_2^i D_W_2^i_2^2.We can sketch the objective function by choosing a Gaussian sketch S_2∈ℝ^n^2 × s_2 with s_2 = O( k/ϵ ). Let U_2 denote the optimal solution to the sketch problem. Then U_2 has the form, for each i∈ [n],U_2^i = A_2^i D_W_2^i S_2 ( Z_2 D_W_2^i S_2)^†.Similarly as before, we only need to write down r different matrices Z_2 D_W_2^iS_1, and for each of them, create k× s_2 variables. Let P_2,i∈ℝ^k× s_2 denote Z_2 D_W_2^i S_2. By our guessing argument, we can obtain U_2.In the last round, we fix U_1 and U_2. We then write down U_3. Overall, by creating l=O(rk^2/ϵ) variables, we have rational polynomials U_1(x), U_2(x), U_3(x). Putting it all together, we can write this objective function, min_x∈ℝ^l ( U_1(x) ⊗U_2(x) ⊗U_3(x) - A ) ∘ W_F^2.s.t. h_1,i(x) ≠ 0, ∀ i ∈ [r]. h_2,i(x) ≠ 0, ∀ i ∈ [r]. h_3,i(x) ≠ 0, ∀ i ∈ [r].where h_1,i(x) denotes the denominator polynomial related to a full rank sub-block of P_1,i(x).By a perturbation argument in Section 4 in <cit.>, we know that the h_1,i(x) are nonzero. By a similar argument as in Section 5 in <cit.>, we can show a lower bound on the cost of the denominator polynomial h_1,i(x). Thus we can create new bounded variables x_l+1, ⋯, x_3r+l to rewrite the objective function, min_x∈ℝ^l+3r q(x) / p(x).s.t. h_1,i(x) x_l+i = 0, ∀ i ∈ [r]. h_2,i(x) x_l+r+i = 0, ∀ i ∈ [r]. h_3,i(x) x_l+2r+i = 0, ∀ i ∈ [r]. p(x) = ∏_i=1^r h_1,i^2(x) h_2,i^2(x) h_3,i^2(x)Note that the degree of the above system is (kr) and all the equality constraints can be merged into one single constraint. Thus, the number of constraints is O(1). The number of variables is O(rk^2/ϵ). Using Theorem <ref> and a similar argument from Section 5 of <cit.>, we have that the minimum nonzero cost is at least 2^-n^δ 2^O(rk^2/ϵ). Combining the binary search explained in Section <ref>(similar techniques also can be found in Section 6 of <cit.>) with the lower bound we obtained, we can find the solution for the original problem in time,((A) + (W) + n 2^O(rk^2/ϵ)) n^O(δ).§.§ r distinct columns, rows and tubesLet W∈ℝ^n× n× n denote a tensor that has r distinct columns and r distinct rows, then W has (1) r distinct column-tube faces.(2) r distinct row-tube faces.Proof of Part (1). Without loss of generality, we consider the first (which is the bottom one) column-row face. Assume it has r distinct rows and r distinct columns. We can re-order all the column-tube faces to make sure that all the n columns in the bottom face have been split into r continuous disjoint groups C_i, e.g., {C_1, C_2, ⋯, C_r } = [n]. Next, we can re-order all the row-tube faces to make sure that all the n rows in the bottom face have been split into r continuous disjoint groups R_i, e.g., {R_1, R_2, ⋯, R_r } = [n]. Thus, the new bottom face can be regarded as r× r groups, and the number in each position of the same group is the same.Suppose that the tensor has r+1 distinct column-tube faces. By the pigeonhole principle there exist two different column-tube faces belonging to the same group C_i, for some i∈ [r]. Note that these two column-tube faces are the same by looking at the bottom (column-row) face. Since they are distinct faces, there must exist one row vector v which is not in the bottom (column-row) face, and it has a different value in coordinates belong to group C_i. Note that, considering the bottom face, for each row vector, it has the same value over coordinates belonging to group C_i. But v has different values in coordinates belong to group C_i. Also, note that the bottom (column-row) face also has r distinct rows, and v is not one of them. This means there are at least r+1 distinct rows, which contradicts that there are r distinct rows in total. Thus, there are at most r distinct column-tube faces.Proof of Part (2). It is similar to Part (1).Let W∈ℝ^n× n× n denote a tensor that has r distinct columns, r distinct rows, and r distinct rubes. Then W has r distinct column-tube faces, r distinct row-tube faces, and r distinct column-row faces. This follows by applying Lemma <ref> twice. Thus, we obtain the same result as in Theorem <ref> by changing the assumption from r distinct faces in each dimension to r distinct columns, r distinct rows and r distinct tubes.§.§ r distinct columns and rowsThe main difference between Theorem <ref> and Theorem <ref> is the running time. The first one takes 2^O(rk^2/ϵ) time and the second one is slightly longer, 2^O(r^2k^2/ϵ). By Lemma <ref>, r distinct columns in two dimensions implies r distinct faces in two of the three kinds of faces. Thus, the following theorem also holds for r distinct columns in two dimensions.Given a 3rd order n× n× n tensor A and an n× n × n tensor W of weights with r distinct faces in two dimensions (out of three dimensions) such that each entry can be written using O(n^δ) bits for some δ >0, define =inf_-k A_k W∘ (A_k - A )_F^2. For any k≥ 1 and any 0 < ϵ <1. (1) If >0, and there exists a rank-k A_k=U_1^*⊗ U_2^* ⊗ U_3^* tensor (with size n× n× n) such that W∘ (A_k-A)_F^2 =, and max_i∈ [3]U_i^*_F ≤ 2^O(n^δ), then there exists an algorithm that takes ((A)+(W)+ n 2^O(r^2k^2/ϵ) ) n^O(δ) time in the unit costmodel with words of size O(log n) bits[The entries of A and W are assumed to fit in n^δ words.] and outputs three n × k matrices U_1,U_2,U_3 such thatW∘( U_1 ⊗ U_2 ⊗ U_3 - A ) _F^2 ≤ (1+ϵ)holds with probability 9/10. (2) If >0, A_k does not exist, and there exist three n× k matrices U_1',U_2',U_3' where each entry can be written using O(n^δ) bits and W ∘( U_1'⊗ U_2'⊗ U_3' - A) _F^2 ≤ (1+ϵ/2), then we can find U,V,W such that (<ref>) holds.(3) If =0, A_k exists, and there exists a solution U_1^*,U_2^*,U_3^* such that each entry of the matrix can be written using O(n^δ) bits, then we can obtain (<ref>). (4) If =0, and there exist three n× k matrices U_1,U_2,U_3 such that max_i∈ [3]U_i^*_F ≤ 2^O(n^δ) and W∘( U_1 ⊗ U_2 ⊗ U_3 - A ) _F^2 ≤ (1+ϵ)+ 2^-Ω(n^δ),then we can output U_1,U_2,U_3 such that (<ref>) holds.(5) Further if A_k exists, we can output a number Z for which ≤ Z ≤ (1+ϵ).For all the cases, the algorithm succeeds with probability at least 9/10.By Lemma <ref>, we have W has r distinct column-tube faces and r distinct row-tube faces. By Claim <ref>, we know that W has R=2^O(rlog r) distinct column-row faces.We use the same approach as in proof of Theorem <ref> (which is also similar to Section 8 of <cit.>) to create variables, write down the polynomial systems and add not equal constraints. Instead of having 3r distinct denominators as in the proof of Theorem <ref>, we have 2r+R.We create l=O(rk^2/) variables for { Z_1 D_W_1^1 S_1, Z_1 D_W_1^2 S_1, ⋯ , Z_1 D_W_1^r S_1}. Then we can write down U_1 with r distinct denominators g_i(x). Each g_i(x) is non-zero in an optimal solution using the perturbation argument in Section 4 in <cit.>. We create new variables x_2l+i to remove the denominators g_i(x), ∀ i∈ [r]. Then the entries of U_1 are polynomials as opposed to rational functions.We create l=O(rk^2/) variables for { Z_2 D_W_2^1 S_2, Z_2 D_W_2^2 S_2, ⋯ , Z_2 D_W_2^r S_2}. Then we can write down U_2 with r distinct denominators g_r+i(x). Each g_r+i(x) is non-zero in an optimal solution using the perturbation argument in Section 4 in <cit.>.We create new variables x_2l+r+i to remove the denominators g_r+i(x), ∀ i∈ [r]. Then the entries of U_2 are polynomials as opposed to rational functions. Using U_1 and U_2 we can express U_3 with R distinct denominators f_i(x), which are also non-zero by using the perturbation argument in Section 4 in <cit.>, and using that W_3 has at most this number of distinct rows. Finally we can write the following optimization problem,x∈ℝ^2l+2rmin p(x)/q(x)s.t.g_i(x) x_2l+i -1=0, ∀ i∈ [r] g_r+i(x) x_2l+r+i -1=0, ∀ i∈ [r]f_j^2(x) ≠ 0, ∀ j∈ [ R ] q(x) =∏_j=1^ Rf_j^2(x)We then determine if there exists a solution to the above semi-algebraic set in time((k,r) R)^O(rk^2/) = 2^O(r^2k^2/).Using similar techniques from Section 5 of <cit.>, we can show a lower bound on the cost similar to Section 8.3 of <cit.>, namely, the minimum nonzero cost is at least2^-n^δ 2^O(r^2 k^2/ϵ). Combining the binary search explained in Section <ref> (a similar techniques also can be found in Section 6 of <cit.>) with the lower bound we obtained, we can find a solution for the original problem in time((A) + (W)+ n2^O(r^2k^2/)) n^O(δ). Note that the running time for the Frobenius norm and for the ℓ_1 norm are of the form (n) + exp((k/ϵ)) rather than (n) ·exp(k/ϵ). The reason is, we can use an input sparsity reduction to reduce the size of the objective function from (n) to (k). Let W∈ℝ denote a third order tensor that has r distinct columns and r distinct rows. Then it has 2^O(rlog r) distinct column-row faces.By similar arguments as in the proof of Lemma <ref>, the bottom (column-row) face can be split into r groups C_1, C_2, ⋯, C_r based on r columns, and split into r groups R_1,R_2,⋯, R_r based on rows. Thus, the bottom (column-row) face can be regarded as having r× r groups, and the number in each position of the same group is the same. We can assume that all the r^2 blocks in the bottom column-row face have the same size. Otherwise, we can expand the tensor to the situation that all the r^2 blocks have the same size. Because this small tensor is a sub-tensor of the big tensor, if the big tensor has at most t distinct column-row faces, then the small tensor has at most t distinct column-row faces.By Lemma <ref>, we know that the tensor W has at most r distinct column-tube faces and row-tube faces. Because it has r distinct column-tube faces, then all the faces belonging to coordinates in C_r are the same. Thus, all the columns belonging to C_r and in the second column-row face are the same. Similarly, we have that all the rows belonging to R_r and in the second column-row face are the same. Thus we have that all the entries in block C_R ∪ R_r and in the second column-row faces are the same. Further, we can conclude, for every column-row face, for every C_i ∪ R_j block, all the entries in the same block are the same.The next observation is, if there exist r^2+1 different values in the tensor, then there exist either r distinct columns or r distinct rows. Indeed, otherwise since we have r distinct columns, each column has at most r distinct entries given our bound on the nunber of distinct rows. Thus, the r distinct columns could have at most r^2 distinct entries in total, a contradiction.For each column-row face, there are at most r^2 blocks, and the value in each block can have at most r^2 possibilities. Thus, overall we have at most (r^2)^r^2 = 2^O(r^2log r) column-row faces. By using different argument, we can improve the above bound. Note that we already show in each column-row face of a tensor, it has r^2 blocks, and all the values in each block have to be the same. Since we have r distinct rows, we can fix the those r distinct rows. If we copy row v into one row of R_i, then we have to copy row v into every row of R_i. This is because if R_i contains two distinct rows, then there must exist a block C_j for which the entries in block R_i ∪ C_j are not all the same. Thus, for each row group, all the rows in that group are the same. Now, for each column-row face, consider the leftmost r blocks, R_1 ∪ C_1, R_2 ∪ C_1, ⋯, R_r ∪ C_1. There are at most r possible values in each block, because we have r distinct rows in total. Overall the total number of possibilities for the leftmost r blocks is at most (r)^r = 2^O(rlog r). Once the leftmost r blocks are determined, the remaining r(r-1) are also determined. This completes the proof. Also, notice that there is an example that has 2^Ω(rlog r) distinct column-row faces. For the bottom column-row faces, there are r× r blocks for which all the blocks have the same size, the blocks on the diagonal have all 1s, and all the other blocks contain 0s everywhere. For the later column-row faces, we can arbitrarily permute this block diagonal matrix, and the total number of possibilities is Ω(r!) ≥ 2^Ω(rlog r). We redefine W' to be the input weight tensor. Then each entry of the input weight tensor W' is in {0,1,2, ⋯, (n)}. For each entryW'_i,j,l, we round it to the smallest power (1+)^x such that W'_i,j,l≤ (1+)^x, where x is an integer. Because W' is bounded, the total number of choices for the power x is O(log(n) /). Define W to be the matrix after rounding. Define OPT to be min_U,V W' ∘ (UV - A)_F^2. Then W has the following properties1.W has r distinct columns, 1.W has r distinct rows, 3.W has R:= ( log (n)/ )^O(r) distinct tubes,4.OPT≤min_U_1,U_2,U_3 ∈ℝ^n× k W ∘ (U_1 ⊗ U_2 ⊗ U_3 - A)_F^2 ≤ (1+)^2 OPT.We prove the above three properties one by one.The rounding is a deterministic procedure: if two values are the same in W', then they are the same in W. Hence, Property 1 and 2 holds.To prove Property 3, take the r distinct columns i_1, ..., i_r. Then every other column can be labeled j in {i_1, ..., i_r}. If one fixes the values on entries i_1, …, i_r in a row, this fixes the values on every other column. So the number of distinct rows is the number of fixings to the values i_1, …, i_r. Each entry has log_1+(n) = O((log n)/) possibilities, so there are O((log n)/)^r distinct rows. Because of the rounding procedure, each W'_i,j,l satisfies W'_i,j,l≤ W_i,j,l≤ (1+) W'_i,j,l, which implies Property 4.§ HARDNESSWe first provide definitions and results for some fundamental problems in Section <ref>. Section <ref> presents our hardness result for the symmetric tensor eigenvalue problem. Section <ref> presents our hardness results for symmetric tensor singular value problems, computing tensor spectral norm, and rank-1 approximation. We improve Håstad's NP-hardness<cit.> result for tensor rank in Section <ref>. We also show a better hardness result for robust subspace approximation in Section <ref>. Finally, we discuss several other tensor hardness results that are implied by matrix hardness results in Section <ref>. §.§ DefinitionsWe first provide the definitions for , , -, -and then state some fundamental results related to those definitions.Given n variables and m clauses in a conjunctive normal form formula with the size of each clause at most 3, the goal is to decide whether there exists an assignment to the n Boolean variables to make the formula be satisfied. [Exponential Time Hypothesis (𝖤𝖳𝖧) <cit.>] There is a δ>0 such that the problem defined in Definition <ref> cannot be solved in O(2^δ n) time.Given n variables and m clauses, a conjunctive normal form formula with the size of each clause at most 3, the goal is to find an assignment that satisfies the largest number of clauses. We use -to denote the version of -where each clause contains exactly 3 literals.For every δ > 0, it is -hard to distinguish a satisfiable instance of -from an instance where at most a 7/8+δ fraction of the clauses can be simultaneously satisfied.Assume holds. For every δ>0, there is no 2^o(n^1-o(1)) time algorithm to distinguish a satisfiable instance of -from an instance where at most a fraction 7/8+δ of the clauses can be simultaneously satisfied. We use -to denote the restricted special case of -where every variable occurs in at most B clauses. Håstad <cit.> proved that the problem is approximable to within a factor 7/8+1/(64B) in polynomial time, and that it is hard to approximate within a factor 7/8+1/(log B)^Ω(1). In 2001, Trevisan improved the hardness result,Unless =, there is no polynomial time (7/8+5/√(B))-approximate algorithm for -. Unless fails, there is no 2^o(n^1-o(1)) time (7/8+5/√(B))-approximate algorithm for -. Unless fails, there is no 2^o(n) time algorithm for the Independent Set problem.Given a positive integer c^* and an unweighted graph G=(V,E) where V is the set of vertices of G and E is the set of edges of G, the goal is to determine whether there is a cut of G that has at least c^* edges. Note that Feige's original assumption<cit.> states that there is no polynomial time algorithm for the problem in Assumption <ref>. We do not know of any better algorithm for the problem in Assumption <ref> and have consulted several experts[Personal communication with Russell Impagliazzo and Ryan Williams.] about the assumption who do not know a counterexample to it. [Random Exponential Time Hypothesis] Let c>ln 2 be a constant. Consider a random formula on n variables in which each clause has 3 literals, and in which each of the 8n^3 clauses is picked independently with probability c/n^2. Then any algorithm which always outputs 1 when the random formula is satisfiable, and outputs 0 with probability at least 1/2 when the random formula is unsatisfiable, must run in 2^c'n time on some input, where c'>0 is an absolute constant.The 4𝖲𝖠𝖳-version of the above random-assumption has been used in <cit.> and <cit.> (Assumption 1.3).§.§ Symmetric tensor eigenvalue An eigenvector of a tensor A∈ℝ^n× n× n is a nonzero vector x∈ℝ^n such that∑_i=1^n ∑_j=1^n A_i,j,k x_i x_j = λ x_k, ∀ k∈ [n]for some λ∈ℝ, which is called an eigenvalue of A.Let G=(V,E) on v vertices have stability number (the size of a maximum independent set) α(G). Let n = v + v(v-1)/2 and 𝕊^n-1 = { (x,y) ∈ℝ^v ×ℝ^v(v-1)/2 :x_2^2 +y_2^2 = 1 }. Then,√(1-1/α(G)) = 3 √(3/2)(x,y)∈𝕊^n-1max∑_i< j, (i,j)∉ E x_i x_j y_i,j.For any graph G(V,E), we can construct a symmetric tensor A∈ℝ^n× n × n. For any 1≤ i < j < k ≤ v, letA_i,j,k =1 1≤ i < j ≤ v, k = v+ϕ(i,j), (i,j) ∉ E, 0otherwise,where ϕ(i,j) = (i-1) v - i(i-1)/2 + j-i is a lexicographical enumeration of the v(v-1)/2 pairs i<j. For the other cases i<k<j, ⋯, k<j<i, we setA_i,j,k = A_i,k,j = A_j,i,k = A_j,k,i = A_k,i,j = A_k,j,i.If two or more indices are equal, we set A_i,j,k=0. Thus tensor T has the following property,A(z,z,z) = 6 ∑_i<j, (i,j)∉ E x_i x_j y_i,j,where z = (x,y) ∈ℝ^n.Thus, we haveλ=max_z ∈𝕊^n-1 A(z,z,z) = max_(x,y)∈𝕊^n-1 6 ∑_i<j, (i,j)∉ E x_i x_j y_i,j.Furthermore, λ is the maximum eigenvalue of A. Unless fails, there is no 2^o(√(n)) time to approximate the largest eigenvalue of an n-dimensional symmetric tensor within (1±Θ(1/n)) relative error. The additive error is at least√(1-1/v) - √(1-1/(v-1)) =1/(v-1) - 1/v /√(1-1/v) + √(1-1/(v-1))≳ 1/(v-1) - 1/v ≥ 1/v^2.Thus, the relative error is (1±Θ(1/v^2)). By the definition of n, we know n=Θ(v^2). Assuming , there is no 2^o(v) time algorithm to compute the clique number of G. Because the clique number of G is α(G), there is no 2^o(v) time algorithm to compute α(G). Furthermore, there is no 2^o(v) time algorithm to approximate the maximum eigenvalue within (1±Θ(1/v^2)) relative error. Thus, we complete the proof.Unless fails, there is no polynomial running time algorithm to approximate the largest eigenvalue of an n-dimensional tensor within (1±Θ(1/log^2+γ(n))) relative-error, where γ>0 is an arbitrarily small constant. We can apply a padding argument here. According to Theorem <ref>, there is a d-dimensional tensor such that there is no 2^o(√(d)) time algorithm that can give a (1+Θ(1/d)) relative error approximation. If we pad 0s everywhere to extend the size of the tensor to n=2^d^(1-γ')/2, where γ'>0 is a sufficiently small constant, then (n)=2^o(√(d)), so d=log^2+O(γ')(n). Thus, it means that there is no polynomial running time algorithm which can output a (1+1/(log^2+γ))-relative approximation to the tensor which has size n. §.§ Symmetric tensor singular value, spectral norm and rank-1 approximation <cit.> defines two kinds of singular values of a tensor. In this paper, we only consider the following kind:Given a 3rd order tensor A∈ℝ^n_1 × n_2 × n_3, the number σ∈ℝ is called a singular value and the nonzero u∈ℝ^n_1,v∈ℝ^n_2,w∈ℝ^n_3 are called singular vectors of A if∑_j=1^n_2∑_k=1^n_3 A_i,j,k v_j w_k = σ u_i, ∀ i ∈ [n_1]∑_i=1^n_1∑_k=1^n_3 A_i,j,k u_i w_k = σ v_j, ∀ j ∈ [n_2]∑_i=1^n_1∑_j=1^n_2 A_i,j,k u_i v_j = σ w_k, ∀ k ∈ [n_3]. The spectral norm of a tensor A is:A_2=x,y,z≠ 0sup|A(x,y,z)|/x_2 y_2 z_2Notice that the spectral norm is the absolute value of either the maximum value of A(x,y,z)/x_2y_2z_2 or the minimum value of it. Thus, it is an ℓ_2-singular value of A. Furthermore, it is the maximum ℓ_2-singular value of A. Let A∈ℝ^n× n× n be a symmetric 3rd order tensor. Then,A_2=x,y,z≠ 0supA(x,y,z)/x_2 y_2 z_2 = x≠ 0sup|A(x,x,x)|/ x_2^3 .It means that if a tensor is symmetric, then its largest eigenvalue is the same as its largest singular value and its spectral norm. Then, by combining with Theorem <ref>, we have the following corollary:Unless fails, * There is no 2^o(√(n)) time algorithm to approximate the largest singular value of an n-dimensional symmetric tensor within (1+Θ(1/n)) relative-error.* There is no 2^o(√(n)) time algorithm to approximate the spectral norm of an n-dimensional symmetric tensor within (1+Θ(1/n)) relative-error. By Corollary <ref>, we have:Unless fails, * There is no polynomial time algorithm to approximate the largest singular value of an n-dimensional tensor within (1+ Θ(1/log^2+γ(n))) relative-error, where γ>0 is an arbitrarily small constant.* There is no polynomial time algorithm to approximate the spectral norm of an n-dimensional tensor within (1+ Θ(1/log^2+γ(n))) relative-error, where γ>0 is an arbitrarily small constant.Now, let us consider Frobenius norm rank-1 approximation.Let A∈ℝ^n× n× n be a symmetric 3rd order tensor. Then,min_σ≥ 0,u_2=v_2=w_2=1A-σ u⊗ v⊗ w_F=min_λ≥ 0,v_2=1A-λ v⊗ v⊗ v_F.Furthermore, the optimal σ and λ may be chosen to be equal. Notice thatA-σ u⊗ v⊗ w_F^2=A_F^2-2σ A(u,v,w)+σ^2u⊗ v⊗ w_F^2.Then, if u_2=v_2=w_2=1, we have:A-σ u⊗ v⊗ w_F^2=A_F^2-2σ A(u,v,w)+σ^2.When A(u,v,w)=σ, then the above is minimized.Thus, we have:min_σ≥ 0,u_2=v_2=w_2=1A-σ u⊗ v⊗ w_F^2+A_2^2=A_F^2.It is sufficient to prove the following theorem:Given A∈ℝ^n× n× n, unless fails, there is no 2^o(√(n)) time algorithm to compute u',v',w'∈ℝ^n such thatA-u'⊗ v'⊗ w'_F^2≤ (1+ε)min_u,v,w∈ℝ^nA-u⊗ v⊗ w_F^2,where ε=O(1/n^2).Let A∈ℝ^n× n× n be the same hard instance mentioned in Theorem <ref>. Notice that each entry of A is either 0 or 1. Thus, min_u,v,w∈ℝ^nA-u⊗ v⊗ w_F^2≤A_F^2. Notice that Theorem <ref> also implies that it is hard to distinguish the two cases A_2≤ 2√(2/3)·√(1-1/c) or A_2≥ 2√(2/3)·√(1-1/(c+1)) where c is an integer which is no greater than √(n). So the difference between (2√(2/3)·√(1-1/c))^2 and (2√(2/3)·√(1-1/(c+1)))^2 is at least Θ(1/n). Since A_F^2 is at most n (see construction of A in the proof of Lemma <ref>), Θ(1/n) is an ε=O(1/n^2) fraction of min_u,v,w∈ℝ^nA-u⊗ v⊗ w_F^2. Becausemin_u,v,w∈ℝ^nA- u⊗ v⊗ w_F^2+A_2^2=A_F^2,if we have a 2^o(√(n)) time algorithm to compute u',v',w'∈ℝ^n such thatA-u'⊗ v'⊗ w'_F^2≤ (1+ε)min_u,v,w∈ℝ^nA-u⊗ v⊗ w_F^2for ε=O(1/n^2), it will contradict the fact that we cannot distinguish whether A_2≤ 2√(2/3)·√(1-1/c) or A_2≥ 2√(2/3)·√(1-1/(c+1)).Given A∈ℝ^n× n× n, unless fails, for any ε for which 1/2≥ε≥ c/n^2 where c is any constant, there is no 2^o(ε^-1/4) time algorithm to compute u',v',w'∈ℝ^n such thatA-u'⊗ v'⊗ w'_F^2≤ (1+ε)min_u,v,w∈ℝ^nA-u⊗ v⊗ w_F^2.If ε=Ω(1/n^2), it means that n=Ω(1/√(ε)). Then, we can construct a hard instance B with size m× m× m where m=Θ(1/√(ε)), and we can put B into A, and let A have zero entries elsewhere. Since B is hard, i.e., there is no 2^o(m^-1/2)=2^o(ε^-1/4) running time to compute a rank-1 approximation to B, this means there is no 2^o(ε^-1/4) running time algorithm to find an approximate rank-1 approximation to A. Unless fails, there is no polynomial time algorithm to approximate the best rank-1 approximation of an n-dimensional tensor within (1+ Θ(1/log^2+γ(n))) relative-error, where γ>0 is an arbitrarily small constant. We can apply a padding argument here. According to Theorem <ref>, there is a d-dimensional tensor such that there is no 2^o(√(d)) time algorithm which can give a (1+Θ(1/d^4)) relative approximation. Then, if we pad with 0s everywhere to extend the size of the tensor to n=2^d^(1-γ')/2 where γ'>0 is a sufficiently small constant, then (n)=2^o(√(d)), and d^4=log^2+O(γ')(n). Thus, it means that there is no polynomial time algorithm which can output a (1+1/(log^2+γ))-relative error approximation to the tensor which has size n.§.§ Tensor rank is hard to approximateThis section presents the hardness result for approximating tensor rank under .According to our new result, we notice that not only deciding the tensor rank is a hard problem, but also approximating the tensor rank is a hard problem. This therefore strengthens Håstad's NP-Hadness <cit.> for computing tensor rank. §.§.§ Cover numberBefore getting into the details of the reduction, we provide a definition of an important concept called the “cover number” and discuss the cover number for the -problem.For any instance S with n variables and m clauses, we are allowed to assign one of three values {0,1,*} to each variable. For each clause, if one of the literals outputs true, then the clause outputs true. For each clause, if the corresponding variable of one of the literals is assigned to *, then the clause outputs true. We say y∈{0,1}^n is a string, and z∈{0,1,*}^n is a star string. For an instance S, if there exists a string y∈{0,1}^n that causes all the clauses to be true, then we say that S is satisfiable, otherwise it is unsatisfiable. For an instance S, let Z_S denote the set of star strings which cause all of the clauses of S to be true. For each star string z∈{0,1,*}^n, let star(z) denote the number of *s in the star-string z. We define the “cover number” of instance S to becover-number(S) = min_z∈ Z_Sstar(z).Notice that for a satisfiable instance S, the cover number p is 0. Also, for any unsatisfiable instance S, the cover number p is at least 1. This is because for any input string, there exists at least one clause which cannot be satisfied. To fix that clause, we have to assign * to a variable belonging to that clause. (Assigning * to a variable can be regarded as assigning both 0 and 1 to a variable) Let S denote a -instance with n variables and m clauses and S suppose S is at most 7/8+A satisfiable, where A∈ (0,1/8). Then the cover number of S is at least (1/8-A)m/B. For any input string y∈{0,1}^n, there exists at least (1/8-A)m clauses which are not satisfied. Since each variable appears in at most B clauses, we need to assign * to at least (1/8-A)m/B variables. Thus, the cover number of S is at least (1/8-A)m/B. We say x_1, x_2, ⋯, x_n are variables and x_1,x_1, x_2, x_2, ⋯, x_n, x_n are literals. For a list of clauses C and a set of variables P, if for each clause, there exists at least one literal such that the corresponding variable of that literal belongs to P, then we say P covers L. §.§.§ Properties of instances For any instance S with n variables and m=Θ(n) clauses, let c>0 denote a constant. If S is (1-c)m satisfiable, then let y∈{0,1}^n denote a string for which S has the smallest number of unsatisfiable clauses. Let T denote the set of unsatisfiable clauses and let b denote the number of variables in T. Then Ω( (cm)^1/3 )≤ b ≤ O( cm ). Note that in S, there is no duplicate clause. Let T denote the set of unsatisfiable clauses by assigning string y to S. First, we can show that any two literals x_i, x_i cannot belong to T at the same time. If x_i and x_i belong to the same clause, then that clause must be an “always” satisfiable clause. If x_i and x_i belong to different clauses, then one of the clauses must be satisfiable. This contradicts the fact that that clause belongs to T. Thus, we can assume that literals x_1, x_2, ⋯, x_b belong to T.There are two extreme cases: one is that each clause only contains three literals and each literal appears in exactly one clause in T. Then b = 3cm. The other case is that each clause contains 3 literals, and each literal appears in as many clauses as possible. Then b 3 =cm, which gives b = Θ( (cm)^1/3). For a random instance, with probability 1-2^-Ω( log n loglog n) there is no literal appearing in at least log n clauses. By the property of random , for any literal x and any clause C, the probability that x appears in C is 3/2n, i.e., [x ∈ C] = 3/2n = Θ(1/n). Let p denote this probability. For any literal x, the probability of x appearing in at least log n clauses (out of m clauses) is[ x appearing in≥log n clauses ] = ∑_i=log n^m mi p^i (1-p)^m-i= ∑_i=log n^m/2mi p^i (1-p)^m-i + ∑_i=m/2^m mi p^i (1-p)^m-i ≤ ∑_i=log n^m/2 (e m/i)^i p^i+ ∑_i=m/2^m mi p^i by (1-p) ≤ 1, m i≤ (em/i)^i ≤ (Θ(1/log n))^log n + 2 · (2e)^m/2·Θ(1/n)^m/2 ≤ 2^- Ω( log n ·loglog n).Taking a union bound over all the literals, we complete the proof,[ ∄ x appearing in≥log n clauses ] ≥ 1-2^-Ω( log n loglog n ). For a sufficiently large constant c'>0 and a constant c>0, for any random instance which has n variables and m=c'n clauses, suppose it is (1-c)m satisfiable. Then with probability 1-2^-Ω(log n loglog n), for all input strings y, among the unsatisfied clauses, each literal appears in O(log n) places.This follows by Lemma <ref>. Next, we show how to reduce the O(log n) to O(1).For a sufficiently large constant c, for any random instance that has n variables and m=cn clauses, for any constant B ≥ 1,b∈ (0,1), with probability at least 1-9m/Bbn, there exist at least (1-b)m clauses such that each variable (in these (1-b)mclauses) only appears in at most B clauses (out of these (1-b)m clauses). For each i∈ [m], we use z_i to denote the indicator variablesuch that it is 1, if for each variable in the ith clause, it appears in at most a clauses. Let B ∈ [1,∞) denote a sufficiently large constant, which we will decide upon later.For each variable x, the probability of it appearing in the i-th clause is 3/n. Then we have[ # clauses that contain x ] = ∑_i=1^m [ i-th clause contains x ] = 3m/nBy Markov's inequality, [ # clauses that contain x ≥ a ] ≤[ # clauses that contain x ] /B = 3m/B n By a union bound, we can compute [z_i] ,[z_i]= [z_i=1]≥ 1 - 3 [ one variable in i-th clause appearing≥ B clauses ]≥ 1 - 9m/B n.Furthermore, we have[z] = [∑_i=1^m z_i] = ∑_i=1^m [z_i] ≥ (1-9m/B n) m.Note that z≤ m. Thus [z]≤ m. Let b∈ (0,1) denote a sufficiently small constant. We can show[ m - z ≥ b m ]≤ [m-z]/ bm= m-[z]/bm ≤ m - (1-9m/Bn)m /bm= 9m/Bbn.This implies that with probability at least 1- 9m/Bbn, we have m-z ≤ bm. Notice that in random-, m=cn for a constant c. Thus, by choosing a sufficiently large constant B (which is a function of c,b), we can obtain arbitrarily large constant success probability. §.§.§ Reduction For any instance S, for any string y∈{0,1}, let T_S,y denote the set that contains unsatisfiable clauses when plugging y into S, let P_S,y denote the set that contains the literals belong to unsatisfiable clauses when plugging y into S. We say y is the “best-clause” string for S if |T_S,y| is minimized. We say y is the “best-literal” string for S if |P_S,y| is minimized. We reduce to tensor rank by following the same construction in <cit.>. To obtain a stronger hardness result, we use the property that each variable only appears in at most B (some constant) clauses and that the cover number of an unsatisfiable instance is large. Note that both -instances and random-instances have that property. Also each -is also a instance. Thus if the reduction holds for , it also holds for -, and similarly for random-.Recall the definition of : is the problem of given a Boolean formula of n variables in form with at most 3 variables in each of the m clauses, is it possible to find a satisfying assignment to the formula? We say x_1,x_2,⋯,x_n are variables and x_1,x_1, x_2, x_2, ⋯, x_n, x_n are literals. We transform this to the problem of computing the rank of a tensor of size n_1 × n_2 × n_3 where n_1 = 2+n+2m, n_2=3n and n_3 = 3n+m. T has the following n_3 column-row faces, where each of the faces is an m_1× n_2 matrix, * n variable matrices V_i ∈ℝ^ n_1 × n_2. It has a 1 in positions (1,2i-1) and (2,2i) while all other elements are 0.* n help matrices S_i ∈ℝ^ n_1 × n_2. It has a 1 position in (1,2n+i) and is 0 otherwise.* n help matrices M_i ∈ℝ^ n_1 × n_2. It has a 1 in positions (1,2i-1), (2+i,2i) and (2+i,2n+i) and is 0 otherwise.* m clause matrices C_l ∈ℝ^ n_1 × n_2.Suppose the clause c_l contains the literals u_l,1, u_l,2 and u_l,3. For each j∈ [3], u_l,j∈{x_1,x_2,⋯, x_n, x_1, x_2,⋯, x_n}. Note that x_i, x_i are the literals of the formula. We can also think of x_i, x_i as length 3n vectors. Let x_i denote the vector that has a 1 in position 2i-1, i.e., x_i=e_2i-1. Let x_i denote the vector that has a 1 in positions 2i-1 and 2i, x_i = e_2i-1+e_2i. * Row 1 is the vector u_l,1∈ℝ^3n,* Row 2+n + 2l-1 is the vector u_l,1-u_l,2∈ℝ^3n,* Row 2+n + 2l is the vector u_l,1-u_l,3∈ℝ^3n.First, we can obtain Lemma <ref> which follows by Lemma 2 in <cit.>. For completeness, we provide a proof.If the formula is satisfiable, then the constructed tensor has rank at most 4n+2m. We will construct 4n+2m rank-1 matrices V_i^(1), V_i^(2), S_i^(1), M_i^(1), C_l^(1) and C_l^(2) . Then the goal is to show that for each matrix in the set{ V_1, V_2,⋯, V_n ,S_1, S_2,⋯, S_n ,M_1, M_2, ⋯, M_n, C_1, C_2, ⋯, C_m}, it can be written as a linear combination of these constructed matrices. * Matrices V_i^(1) and V_i^(2). V_i^(1) has the first row equal to x_i iff α_i=1 and otherwise x_i. All the other rows are 0. We set V_i^(2) = V_i - V_i^(1).* Matrices S_i^(1). S_i^(1) = S_i.* Matrices M_i^(1).M_i^(1)= M_i - V_i^(1) if α_i =1 M_i - V_i^(1) - S_i if α_i =0 * Matrices C_l^(1) and C_l^(2). Let x_i = α_i be the assignment that makes the clause c_l true. Then C_l - V_i^(1) has rank 2, since either it has just two nonzero rows (in the case where x_i is the first variable in the clause) or it has three nonzero rows of which two are equal. In both cases we just need two additional rank 1 matrices.Once the instance S is unsatisfiable, then its cover number is at least 1. For each unsatisfiable instance S with cover number p, we can show that the constructed tensor has rank at most 4n+2m+O(p) and also has rank at least 4n+2m+Ω(p). We first prove an upper bound,For a instance S, let y∈{0,1} denote a string such that S(y) has a set L that contains unsatisfiable clauses. Let p denote the smallest number of variables that cover all clauses in L. Then the constructed tensor T has rank at most 4n+2m+p. Let y denote a length-n Boolean string (α_1, α_2,⋯, α_n). Based on the assignment y, all the clauses of S can be split into two sets: L contains all the unsatisfied clauses and L contains all the satisfied clauses. We use set P to denote a set of variables that covers all the clauses in set L. Let p=|P|. We will construct 4n+2m+p rank-1 matrices V_i^(1), V_i^(2),S_i^(1), M_i^(1), ∀ i∈ [n], C_l^(1), C_l^(2), ∀ l∈ [m], and V_j^(3), ∀ j ∈ P. Then the goal is to show that the V_i,S_i,M_i and C_l can be written as linear combinations of these constructed matrices. * Matrices V_i^(1) and V_i^(2). V_i^(1) has first row equal to x_i iff α_i=1 and otherwise x_i. All the other rows are 0. We set V_i^(2) = V_i - V_i^(1).* Matrices V_j^(3). For each j∈ P, V_j^(3) has the first row equal to x_i iff α_i=0 and otherwise x_i.* Matrices S_i^(1). S_i^(1) = S_i.* Matrices M_i^(1).M_i^(1)= M_i - V_i^(1) if α_i =1 M_i - V_i^(1) - S_i if α_i =0 * Matrices C_l^(1) and C_l^(2). * For each l∉ L, clause c_l is satisfied according to assignment y. Let x_i = α_i be the assignment that makes the clause c_l true. Then C_l - V_i^(1) has rank 2, since either it has just two nonzero rows (in the case where x_i is the first variables in the clause) or it has three nonzero rows of which two are equal. In both cases we just need two additional rank 1 matrices.* For each l∈ L. It means clause c_l is unsatisfied according to assignment y. Let x_j_1 = α_j_1, x_j_2 = α_j_2, x_j_3 = α_j_3 be an assignment that makes the clause c_l false. In other words, one of j_1, j_2, j_3 must be P according to the definition that P covers L. Then matrix C_l - V_j_1^(3) has rank 2, since either it has just two nonzero rows (in the case where x_j_1 is the first variables in the clause) or it has three nonzero rows of which two are equal. In both cases we just need two additional rank 1 matrices. We finish the proof by taking the P that has the smallest size.Further, we have:For a instance S, let p denote the cover number of S, then the constructed tensor T has rank at most 4n+2m+p. This follows by applying Lemma <ref> to all the input strings and the definition of cover number (Definition <ref>). We can split the tensor T∈ℝ^(2+n+3m) × 3n × (3n+m) into two sub-tensors, one is T_1 ∈ℝ^2 × 3n × (3n+m) (that contains the first two row-tube faces of T and linear combination of the remaining 2m row-tube faces of T), and the other is T_2 ∈ℝ^(n+2m) × 3n × (3n+m) (that contains the next n+2m row-tube faces of T). We first analyze the rank of T_1 and then analyze the rank of T_2. The rank of T_2 is n+2m. According to Figure <ref>, the nonzero rows are distributed in n+m fully separated sub-tensors. It is obvious that the rank of each one of those n sub-tensors is 1, and the rank of each of those m sub-tensors is 2. Thus, overall, the rank T_2 is n+2m. To make sure (T) = (T_1) +(T_2), the T_1∈ℝ^2 × 3n × (3n+m) can be described as the following 3n+m column-row faces, and each of the faces is a 2× 3n matrix. * Matrices V_i, ∀ i∈[n]. The two rows are from the first two rows of V_i in Figure <ref>, i.e., the first row is e_2i-1 and the second row is e_2i.* Matrices S_i, ∀ i∈[n]. The two rows are from the first two rows of S_i in Figure <ref>, i.e., the first row is e_2n+i and the second row is zero everywhere else.* Matrices M_i, ∀ i∈[n]. The first row is e_2i-1 + β_i,1 (e_2i+e_2n+i), while the second row is β_i,2 (e_2i + e_2n+i).* Matrices C_l, ∀ i∈[m]. The first row is (1+γ_l,1+γ_l,2) u_l,1 - γ_l,1 u_l,2 - γ_l,2 u_l,3 and the second is ( γ_l,3 + γ_l,4 ) u_l,1 - γ_l,3 u_l,2- γ_l,4 u_l,3,where for each i∈ [3n], we use vector e_i to denote a length 3n vector such that it only has a 1 in position i and 0 otherwise. β,γ are variables. The goal is to show a lower bound for,β,γ (T_1).Let P denote the set { i | the second row of matrix M_i is nonzero,∀ i∈ [n]}. Then the rank of T_1 is at least 3n+|P|. We define p = |P|. Without loss of generality, we assume that for each i∈ [p], the second row of matrix M_i is nonzero.Notice that matrices V_i, S_i, M_i have size 2× 3n, but we only focus on the first 2n+p columns. Thus, we have n+p column-row faces (from the 3rd dimension) A_j ∈ℝ^2 × (2n+p), * A_j, 1 ≤ j ≤ n, A_j is the first 2n+p columns of V_j - ∑_i=1^n α_i,jS_i ∈ℝ^2 × 3n, where α_i,j are some coefficients.* A_n+j, 1≤ j ≤ p, A_j is the first 2n+p columns of M_j - ∑_i=1^n α_i,n+jS_i ∈ℝ^2× 3n, where α_i,j are some coefficients. Consider the first 2n+p column-tube faces (from 2nd dimension), B_j, ∀ j∈ [2n+p], of T_1. Notice that these matrices have size 2× (n+p). * B_2i-1, 1≤ i≤ p, it has a 1 in positions (1,i) and (1,n+i).* B_2i, 1≤ i ≤ p, it has β_i,1 in position (1,n+i), 1 in position (2,i) and β_i,2 in position (2,n+i).* B_2i-1, p+1≤ i≤ n, it has 1 in position (1,i).* B_2i, p+1≤ i≤ n, it has 1 in position (2,i).* B_2n+i, 1 ≤ i≤ p, the first row is unknown, the second row has β_i,2 in position in (2,n+i).It is obvious that the first 2n matrices are linearly independent, thus the rank is at least 2n. We choose the first 2n matrices as our basis. For B_2n+1, we try to write it as a linear combination of the first 2n matrices { B_i}_i∈ [2n]. Consider the second row of B_2n+1. The first n positions are all 0. The matrices B_2i all have disjoint support for the second row of the first n columns. Thus, the matrices B_2i should not be used. Consider the second row of B_2i-1, ∀ i∈ [n]. None of them has a nonzero value in position n+1. Thus B_2n+1 cannot be written as a linear combination of of the first 2n matrices. Thus, we can show for any i∈ [p], B_2n+i cannot be written as a linear combination of matrices { B_i}_i∈ [2n]. Consider the p matrices {B_2n+i}_i∈ [p]. Each of them has a different nonzero position in the second row. Thus these matrices are all linearly independent. Putting it all together, we know that the rank of matrices {B_i}_i∈ [2n+p] is at least 2n+p. Next, we consider another special case when β_i,2=0, for all i∈ [n]. If we subtract β_i,1 times S_i from M_i and leave the other column-row faces (from the 3rd dimension) as they are, and we make all column-tube faces(from the 2nd dimension) for j>2n identically 0, then all other choices do not change the first 2n column-tube faces (from the 2nd dimension) and make some other column-tube faces (from the 2nd dimension) nonzero. Such a choice could clearly only increase the rank of T. Thus, we obtain,(T) = 2n + 2m + min( T_3),where T_3 is a tensor of size 2× 2n × (2n+m) given by the following column-row faces (from 3rd dimension) A_i, ∀ i∈ [2n+m] and each matrix has size 2 × 2n (shown in Figure <ref>). * A_i, i ∈ [n], the first 2n columns of V_i.* A_n+i, i ∈ [n], the first 2n columns of M_i. The first row is e_2i-1 + β_i,1 e_2i, and the second row is 0.* A_2n+l, l ∈ [m], the first 2n columns of C_l. The first row is (1+γ_l,1 + γ_l,2) u_l,1 - γ_l,1 u_l,2 - γ_l,2 u_l,3, and the second row is (γ_l,3+γ_l,4) u_l,1 - γ_l,3 u_l,2 - γ_l,4 u_l,3.We can showLet p denote the cover number of the instance. T_3 has rank at least 2n+Ω(p). First, we can show that all matrices A_n+i-A_i and A_n+i (for all i∈ [n] ) are in the expansion of tensor T_3. Thus, the rank of T_3 is at least 2n.We need the following claim:For any l∈ [m], if A_2n+l can be written as a linear combination of { A_n+i-A_i }_i∈ [n] and { A_n+i}_i∈ [n], then the second row of A_2n+l is 0, and the first row of one of the A_n+i is u_i where u_i is one of the literals appearing in clause c_l. We prove this for the second row first. For each l∈ [m], we consider the possibility of using all matrices A_n+i-A_i and A_n+i to express matrix A_2n+l. If the second row of A_2n+l is nonzero, then it must have a nonzero entry in an odd position. But there is no nonzero in an odd position of the second row of any of matrices A_n+i-A_i and A_n+i. For the first row. It is obvious that the first row of A_2n+l must have at least one nonzero position, for any γ_l,1, γ_l,2. Let u_j be a literal belonging to the variable x_i which appears in the first row of A_2n+l with a nonzero coefficient. Since only A_n+i of all the other A_n+s, ∀ s∈ [n] matrices has nonzero elements in either of the positions (1,2i-1) or (1,2i), then A_n+i must be used to cancel these elements. Thus, the first row of A_n+i must be a multiple of u_j and since the element in position (1,2i-1) of A_n+i is 1, this multiple must be 1.Note that matrices A_i,∀ i∈ [n] have the property that, for any matrix in { A_n+1, ⋯, A_2n+m}, it cannot be written as the linear combination of matrices A_i,∀ i∈ [n]. Let A∈ℝ^(n+m) × 2n denote a matrix that consists of the first rows of { A_n+1, ⋯, A_2n+m}. According to the property of matrices A_i,∀ i∈ [n], and that the rank of a tensor is always greater than or equal to the rank of any sub-tensor, we know that(T_3) ≥ n + min(A).For a instance S, for any input string y∈{0,1}^n, set β_*,1 to be the entry-wise flipping of y, (1)if the clause l is satisfied, then the (n+l)-th row of A∈ℝ^(n+m)× 2n can be written as a linear combination of the first n rows of A. (2) if the clause l is unsatisfied, then the (n+l)-th row of A cannot be written as a linear combination of the first n rows of A. Part (1), consider a clause l which is satisfied with input string y. Then there must exist a variable x_i belonging to clause l (either literal x_i or literal x_i) and one of the following holds: if x_i belongs to clause l, then α_i = 1; if x_i belongs to clause l, then α_i=0. Suppose clause l contains literal x_i. The other case can be proved in a similar way. We consider the (n+l)-th row. One of the following assignments (0,0), (-1,0), (0,-1) to γ_l,1, γ_l,1 is going to set the (n+l)-th row of A to be vector e_2i-1. We consider the i-th row of A. Since we set α_i=1, then we set β_i,1=0, it follows that the i-th row of A becomes e_2i-1. Therefore, the (n+l)-th row of A can be written as a linear combination of A.Part (2), consider a clause l which is unsatisfied with input string y. Suppose that clause contains three literals x_i_1, x_i_2, x_i_3 (the other seven possibilities can be proved in a similar way). Then for input string y, we have α_i_1=0, α_i_2=0 and α_i_3=0, otherwise this clause l is satisfied. Consider i_1-th row of A. It becomes e_2i_1 -1 + e_2i_1. Similarly for the i_2-th row and i_3-th row. Consider the (n+l)-th row. We can observe that all of positions 2i_1, 2i_2,2i_3 must be 0. Any linear combination formed by the i_1,i_2,i_3-th row of A must have one nonzero in one of positions 2i_1, 2i_2,2i_3. However, if we consider the (n+l)-th row of A, one of the positions 2i_1, 2i_2,2i_3 must be 0. Also, the remaining n-3 of the first n rows of A also have 0 in positions2i_1, 2i_2,2i_3. Thus, we can show that the (n+l)-th row of A cannot be written as a linear combination of the first n rows. Similarly, for the other seven cases.Note that in order to make sure as many as possible rows in n+1, ⋯, n+m can be written as linear combinations of the first n rows of A, the β_i,1 should be set to either 0 or 1. Also each possibility of input string y is corresponding to a choice of β_i,1. According to the above Claim <ref>, let l_0 denote the smallest number of unsatisfied clauses over the choices of all the 2^n input strings. Then over all choices of β,γ, there must exist at least l_0 rows of A_n+1, ⋯A_n+m, such that each of those rows cannot be written as the linear combination of the first n rows.Let A∈ℝ^(n+m) × 2n denote a matrix that consists of the first rows of A_n+i,∀ i∈[n] and A_n+l,∀ l∈ [m]. Let p denote the cover number of instance. Then min(A) ≥ n+ Ω(p).For any choices of {β_i,1}_i∈ [n], there must exist a set of rows out of the next m rows such that, each of those rows cannot be written as a linear combination of the first n rows. Let L denote the set of those rows. Let t denote the maximum size set of disjoint rows from L. Since those t rows in L all have disjoint support, they are always linearly independent. Thus the rank is at least n+t. Note that each row corresponds to a unique clause and each clause corresponds to a unique row. We can just pick an arbitrary clause l in L, then remove the clauses that are using the same literal as clause l from L. Because each variable occurs in at most B clauses, we only need to remove at most 3B clauses from L. We repeat the procedure until there is no clause L. The corresponding rows of all the clauses we picked have disjoint supports, thus we can show a lower bound for t, t ≥ |L| / (3B) ≥ l_0/(3B) ≥ p / (9B) ≳ p,where the second step follows by |L|≥ l_0, the third step follows 3l_0≥ p, and the last step follows by B is some constant.Thus, putting it all together, we complete the proof. Now, we consider a general case when there are q different i∈ [n] satisfying that β_i,2≠ 0. Similar to tensor T_3, we can obtain T_4 such that,(T) = 2n + 2m + min (T_4)where T_4 is a tensor of size 2× 2n × (2n+m) given by the following column-row faces (from 3rd dimension) A_i, ∀ i∈ [2n+m] and each matrix has size 2× 2n (shown in Figure <ref>). * A_i, i ∈ [n], the first 2n columns of V_i.* A_n+i, i ∈ [q], the first 2n columns of M_i. The first row is e_2i-1 + β_i,1 e_2i, and the second row is β_i,2 e_2i.* A_n+i, i ∈{q+1, ⋯,n}, the first 2n columns of M_i. The first row is e_2i-1 + β_i,1 e_2i, and the second row is 0.* A_2n+l, l ∈ [m], the first 2n columns of C_l. The first row is (1+γ_l,1 + γ_l,2) u_l,1 - γ_l,1 u_l,2 - γ_l,2 u_l,3, and the second row is (γ_l,3+γ_l,4) u_l,1 - γ_l,3 u_l,2 - γ_l,4 u_l,3.Note that modifying q entries(from Figure <ref> to Figure <ref>) of a tensor can only decrease the rank by q, thus we obtainLet q denote the number of i such that β_i,2≠ 0, and let p denote the cover number of the instance. Then T_4 has rank at least 2n+Ω(p)-q. Combining the two perspectives we haveLet p denote the cover number of an unsatisfiable instance. Then the tensor has rank at least 4n+2m+Ω(p). Let q denote the q in Figure <ref>. From one perspective, we know that the tensor has rank at least 4n+2m+Ω(p)-q. From another perspective, we know that the tensor has rank at least 4n+2m+q. Combining them together, we obtain the rank is at least 4n+2m+Ω(p)/2, which is still 4n+2m+Ω(p). Unless fails, there is a δ>0 and an absolute constant c_0>1 such that the following holds. For the problem of deciding if the rank of a q-th order tensor, q≥ 3, with each dimension n, is at most k or at least c_0k, there is no 2^δ k^1-o(1) time algorithm. The reduction can be split into three parts.[The first two parts are accomplished by personal communication with Dana Moshkovitz and Govind Ramnarayan.] The first part reduces the -problem to the -problem by <cit.>. For each -instance with size n, the corresponding -instance has size n^1+o(1). The second part is by reducing the -problem to -by <cit.>. For each -instance with size n, the corresponding -instance has size Θ(n) when B is a constant. The third part is by reducing the -problem to the tensor problem. Combining Theorem <ref>, Lemma <ref> with this reduction, we complete the proof. Unless random-fails, there is an absolute constant c_0>1 for which any deterministic algorithm for deciding if the rank of a q-th order tensor is at most k or at least c_0k, requires 2^Ω(k) time. This follows by combining the reduction with random-and Lemma <ref>.Note that, if BPP = P then it also holds for randomized algorithms which succeed with probability 2/3.Indeed, we know that any deterministic algorithm requires 2^Ω(n) running time on tensors that have size n× n× n. Let g(n) denote a fixed function of n, and g(n)=o(n). We change the original tensor from size n× n× n to 2^g(n)× 2^g(n)× 2^g(n) by adding zero entries. Then the number of entries in the new tensor is 2^3 g(n) and the deterministic algorithm still requires 2^Ω(n) running time on this new tensor. Assume there is a randomized algorithm that runs in 2^c g(n) time, for some constant c>3. Then considering the size of this new tensor, the deterministic algorithm is a super-polynomial time algorithm, but the randomized algorithm is a polynomial time algorithm. Thus, by assuming BPP = P, we can rule out randomized algorithms, which means Theorem <ref> also holds for randomized algorithms which succeed with probability 2/3.We provide some some motivation for the BPP = P assumption: this is a standard conjecture in complexity theory, as it is implied by the existence of strong pseudorandom generators or if any problem in deterministic exponential time has exponential size circuits <cit.>. §.§ Hardness result for robust subspace approximation This section improves the previous hardness for subspace approximation <cit.> from 1± 1/(d)to 1± 1/(log d). (Note that, we provide the algorithmic results for this problem in Section <ref>.) This <cit.> gives a way to extend any graph to regular graph.<cit.> said about independent set problem in 3-regular graph.the book <cit.>, david cited it.David said, we need to cite this work <cit.>Some useful papers need to read <cit.>.Assume NP-hard, we know can get 1+1/(d) hardness. We want to show that, by assuming , we can get 1+1/(log d) hardness.If cliques or independent sets of logarithmic size could be found in polynomial time, the exponential time hypothesis wouldbe false.More generally, the exponential time hypothesis implies that it is not possible to find cliques or independent sets of size k in time n^o(k) <cit.>. There does not exist a k≥ 2, an ϵ >0, and a randomized algorithm that succeeds (with high probability) in solving in time O( n^⌈ k/2⌉ - ϵ ).Assumes conjecture, there does not exist a k≥ 2, an ϵ>0, and a randomized algorithm that succeeds (with high probability) in solving in time O( n^⌈ k/2⌉ - ϵ ). It follows by <cit.>. For any graph G with n nodes, m edges, for which the maximum degree in graph G is d, there exists a d-regular graph G' with 2nd-2m nodes such that the clique size of G' is the same as the clique size of G. First we create d copies of the original graph G. For each i∈[n], let v_i,1, v_i,2,⋯, v_i,d denote the set of nodes in G' that are corresponding to v_i in G. Let d_v_i denote the degree of node v_i in graph G. In graph G', we create d-d_v_i new nodes v_i,1', v_i,2', ⋯, v'_i,d_v_i and connect each of them to all of the v_1, v_2, ⋯, v_d. Therefore, 1. For each i∈ [n], j∈ [d_v_i], node v_i,j' has degree d. 2. For each i∈ [n], j∈ [d], node v_i,j has degree d_v_i (from the original graph), and d-d_v_i degree (from the edges to all the v_i,1', v_i,2', ⋯, v'_i,d_v_i). Thus, we proved the graph G is d-regular.The number of nodes in the new graph G' is,nd + ∑_i=1^n (d - d_v_i) = 2nd - ∑_i=1^n d_v_i = 2nd -2m. It remains to show the clique size is the same in graph G and G'. Since we can always reorder the indices for all the nodes, without loss of generality, let us assume the the first k nodes v_1, v_2,⋯,v_k forms a k-clique that has the largest size. It is obvious that the clique size k' in graph G' is at least k, since we make k copies of the original graph and do not delete any edges and nodes. Then we just need to show k' ≤ k. By the property of the construction, the node in one copy does not connect to a node in any other copy. Consider the new nodes we created. For each node v_i,j', consider the neighbors of this node. None of them share a edge. Combining the above two properties gives k'≤ k. Thus, we finish the proof.Any n variable m clauses instance can be reduced to a graph G with 24m vertices,which is an instance of 10m-independent set. Furthermore G is a 3-regular graph.We give the proof for completeness here.Define o_i to be the number of occurrences of {x_i,x_i} in the m clauses. For each variable x_i, we construct 2o_i vertices, namely v_i,1,v_i,2,⋯,v_i,2o_i. We make these 2o_i vertices be a circuit, i.e., there are 2o_i edges: (v_i,1,v_i,2),(v_i,2,v_i,3),⋯,(v_i,2o_i-1,v_i,2o_i),(v_i,2o_i,v_i,1). For each clause with 3 literals a,b,c, we create 3 vertices v_a,v_b,v_c where they form a triangle, i.e., there are edges (v_a,v_b),(v_b,v_c),(v_c,v_a). Furthermore, assume a is the j^th occurrence of x_i (occurrence of x_i means a=x_i or a=x_i). Then if a=x_i, we add edge (v_a,v_i,2j), otherwise we add edge (v_a,v_i,2j-1).Thus, we can see that every vertex in the triangle corresponding to a clause has degree 3, half of vertices of the circuit corresponding to variable x_i have degree 3 and the other half have degree 2. Notice that the maximum independent set of a 2o_i circuit is at most o_i, and the maximum independent set of a triangle is at most 1. Thus, the maximum independent set of the whole graph has size at most m+∑_i=1^n o_i=m+3m=4m. Another observation is that if there is a satisfiable assignment for the instance, then we can choose a 4m-independent set in the following way: if x_i is true, then we choose all the vertices in set { v_i,1, v_i,3, ⋯, v_i,2j-1, ⋯ v_i,2o_i-1}; otherwise, we choose all the vertices in set { v_i,2, v_i,4, ⋯, v_i,2j, ⋯ v_i,2o_i}. For a clause with literals a,b,c: if a is satisfied, it means that v_i,t which connected to v_a is not chosen in the independent set, thus we can pick v_a.The issue remaining is to reduce the above graph to a 3 regular graph. Notice that there are exactly ∑_i=1^n o_i=3m vertices which have degree 2. For each of this kind of vertex u, we construct 5 additional vertices u_1,u_2,u_3,u_4,u_5 and edges (u_1,u_2),(u_2,u_3),(u_3,u_4),(u_4,u_5),(u_5,u_1),(u_2,u_4),(u_3,u_5) and (u_1,u). Because we can always choose exactly two vertices among u_1,u_2,⋯,u_5 no matter we choose vertex u or not, the value of the maximum independent set will increase the size by exactly 2∑_i=1^n o_i=6m.To conclude, we construct a 3-regular graph reduced from a instance. The graph has exactly 24m vertices. Furthermore, if the instance is satisfiable, the graph has 10m-independent set. Otherwise, it does not have a 10m-independent set.There is a constant 0<c<1, such that for any ε>0, there is no O(2^n^1-ε) time algorithm which can solve k-clique for an n-vertex (n-3)-regular graph where k=cn unless fails.According to Theorem <ref>, for a given n variable m=O(n) clauses instance, we can reduce it to a 3-regular graph with 24m vertices which is a 10m-independent set instance. If there exists ε>0 such that we have an algorithm with running time O(2^(24m)^1-ε) which can solve 10m-clique for a 24m-3 regular graph with 24m vertices, then we can solve the problem in O(2^n^1-ε') time, where ε'=Θ(ε). Thus, it contradicts .Let V be a k-dimensional subspace of ℝ^d, represented as the column span of a d× k matrix with orthonormal columns. We abuse notation and let V be both the subspace and the corresponding matrix. For a set Q of points, letc(Q,V) = ∑_q ∈ Q d(q,V)^p = ∑_q∈ Q q^⊤ (I - VV^⊤) _2^p = ∑_q∈ Q ( q^2 -q^⊤ V ^2 )^p/2,be the sum of p-th powers of distances of points in Q, i.e., Q - QVV^⊤_v with associated M(x) = |x|^p.For any k∈ [d], the k-dimensional subspaces V which minimize c(E,V) are exactly the nk subspaces formed by taking the span of k distinct standard unit vectors e_i, i∈ [d]. The cost of any such V is d-k.Given a set Q of (d) points in ℝ^d, for a sufficiently small ϵ = 1/(d), it is NP-hard to output a k-dimensional subspace V of ℝ^d for which c(Q,V) ≤ (1+ϵ) c(Q,V^*), where V^* is the k-dimensional subspace minimizing the expression c(Q,V), that is c(Q,V) ≥ c(Q,V^*) for all k-dimensional subspaces V.For a sufficiently small ε=1/(log(d)), there exist 1≤ k≤ d, unless fails, there is no algorithm that can output a k-dimensional subspace V of ℝ^d for which c(Q,V) ≤ (1+ϵ) c(Q,V^*), where V^* is the k-dimensional subspace minimizing the expression c(Q,V), that is c(Q,V) ≥ c(Q,V^*) for all k-dimensional subspaces V.The reduction is from the clique problem of d-vertices (d-3)-regular graph. We construct the hard instance in the same way as in <cit.>. Given a d-vertes (d-3)-regular graph graph G, let B_1=d^α,B_2=d^β where β>α≥ 1 are two sufficiently large constants. Let c be such that(1-1/B_1)^2+c^2/B_1=1.We construct a d× d matrix A as the following: ∀ i∈[d], let A_i,i=1-1/B_1 and ∀ i≠j, A_i,j=A_j,i=c/√(B_1 r) if (i,j) is an edge in G, and A_i,j=A_j,i=0 otherwise. Let us construct A'∈ℝ^2d× d as follows:A'=[A; B_2· I_d ],where I_d∈ℝ^d is a d× d identity matrix. [In proof of Theorem 54 in <cit.>] Let V'∈ℝ^d× k satisfy that c(A',V')≤ (1+1/d^γ) c(A',V^*), where A' is constructed as the above corresponding to the given graph G, and γ>1 is a sufficiently large constant, V^* is the optimal solution which minimizes c(A',V). Then if G has a , given V', there is a (d) time algorithm which can find the clique which has size at least k. Now, to apply here, we only need to apply a padding argument. We can construct amatrix A”∈ℝ^N× d as follows:A”=[ A'; A';⋯; A' ].Basically, A” contains N/(2d) copies of A' where N=2^d^1-α, and 0<α is a constant which can be arbitrarily small. Notice that ∀ V∈ℝ^d× k,c(V,A”)=∑_q∈ A” d(q,V)^p=N/(2d)∑_q∈ A' d(q,V)^p=N/(2d)c(V,A').So if V” gives a (1+1/d^γ) approximation to A”, it also gives a (1+1/d^γ) approximation to A'. So if we can find V” in (N,d) time, we can output a of G in (N,d) time. But unless fails, for a sufficiently small constant α'>0 there is no (N,d)=O(2^d^1-α') time algorithm that can output a of G. It means that there is no (N,d) time algorithm that can compute a (1+1/d^γ)=(1+1/(log(N))) approximation to A”. To make A” be a square matrix, we can just pad with 0s to make the size of A” be N× N. Thus, we can conclude, unless fails, there is no polynomial algorithm that can compute a (1+1/(log(N))) rank-k subspace approximation to a point set with size N.§.§ Extending hardness from matrices to tensorsIn this section, we briefly state some hardness results which are implied by hardness for matrices. The intuition is that, if there is a hard instance for the matrix problem, then we can always construct a tensor hard instance for the tensor problem as follos: the first face of the tensor is the hard instance matrix and it has all 0s elsewhere. We can prove that the optimal tensor solution will always fit the first face and will have all 0s elsewhere. Then the optimal tensor solution gives an optimal matrix solution.§.§.§ Entry-wise ℓ_1 norm and ℓ_1-ℓ_1-ℓ_2 normIn the following we will show that the hardness for entry-wise ℓ_1 norm low rank matrix approximation implies the hardness for entry-wise ℓ_1 norm low rank tensor approximation and asymmetric tensor norm (ℓ_1-ℓ_1-ℓ_2) low rank tensor approximation problems.Unless fails, for an arbitrarily small constant γ>0, given some matrix A∈ℝ^n× n, there is no algorithm that can compute x,y∈ℝ^n s.t.A-xy^⊤_1≤(1+1/log^1+γ(n))min_x,y∈ℝ^nA-xy^⊤_1,in (n) time.We can get the hardness for tensors directly.Unless fails, for an arbitrarily small constant γ>0, given some tensor A∈ℝ^n× n × n, * there is no algorithm that can compute x,y,z∈ℝ^n s.t.A-x⊗y⊗z_1≤(1+1/log^1+γ(n))min_x,y,z∈ℝ^nA-x⊗ y⊗ z_1,in (n) time.* there is no algorithm can compute x,y,z∈ℝ^n s.t.A-x⊗y⊗z_u≤(1+1/log^1+γ(n))min_x,y,z∈ℝ^nA-x⊗ y⊗ z_u,in (n) time.Let matrix A∈ℝ^n× n be the hard instance in Theorem <ref>. We construct tensor A∈ℝ^n× n× n as follows: ∀ i,j,l∈ [n],l≠1 we let A_i,j,1=A_i,j,A_i,j,l=0.Suppose x,y,z∈ℝ^n satisfiesA-x⊗y⊗z_1≤(1+1/log^1+γ(n))min_x,y,z∈ℝ^nA-x⊗ y⊗ z_1.Then letting z'=(1,0,0,⋯,0)^⊤, we haveA-x⊗y⊗ z'_1 ≤A-x⊗y⊗z_1 ≤(1+1/log^1+γ(n))min_x,y,z∈ℝ^nA-x⊗ y⊗ z_1.The first inequality follows since ∀ i,j,l∈[n],l≠1, we have A_i,j,l=0. Letx^*,y^*=min_x,y∈ℝ^nA-xy^⊤_1.ThenA-x⊗y⊗ z'_1 ≤(1+1/log^1+γ(n))A-x⊗y⊗z_1 ≤(1+1/log^1+γ(n))A-x^*⊗ y^*⊗ z'_1.Thus, we haveA-xy^⊤_1≤(1+1/log^1+γ(n))A-x^*(y^*)^⊤_1.Combining with Theorem <ref>, we know that unless ETH fails, there is no (n) running time algorithm which can outputA-x⊗y⊗z_1≤(1+1/log^1+γ(n))min_x,y,z∈ℝ^nA-x⊗ y⊗ z_1. Similarly, we can prove that if x,y,z∈ℝ^n satisfies:A-x⊗y⊗z_u≤(1+1/log^1+γ(n))min_x,y,z∈ℝ^nA-x⊗ y⊗ z_u,thenA-xy^⊤_1≤(1+1/log^1+γ(n))A-x^*(y^*)^⊤_1.We complete the proof. Unless fails, for arbitrarily small constant γ>0, * there is no algorithm that can compute (1+ε) entry-wise ℓ_1 norm rank-1 tensor approximation in 2^O(1/ε^1-γ) running time. (·_1-norm is defined in Section <ref>)* there is no algorithm that can compute (1+ε) ℓ_u-norm rank-1 tensor approximation in 2^O(1/ε^1-γ) running time. (·_u-norm is defined in Section <ref>) §.§.§ ℓ_1-ℓ_2-ℓ_2 normUnless fails, for arbitrarily small constant γ>0, given some tensor A∈ℝ^n× n × n, there is no algorithm can compute U,V,W∈ℝ^n× k s.t.A-U⊗V⊗W_v≤(1+1/(log n))min_U,V,W∈ℝ^n× kA-U⊗ V⊗ W_v,in (n) running time. (·_v-norm is defined in Section <ref>) Let matrix A∈ℝ^n× n be the hard instance in Theorem <ref>. We construct tensor A∈ℝ^n× n× n as follows: ∀ i,j,l∈ [n],l≠1 we let A_i,j,1=A_i,j,A_i,j,l=0.Suppose U,V,W∈ℝ^n× k satisfiesA-U⊗V⊗W_v≤(1+1/(log n))min_U,V,W∈ℝ^n× kA-U⊗ V⊗ W_v.Let W'∈ℝ^n× k be the following:W'=[ 1 1 ⋯ 1; 0 0 ⋯ 0; 0 0 ⋯ 0; ⋯ ⋯ ⋯ ⋯; 0 0 ⋯ 0;],then we haveA-U⊗V⊗ W'_v ≤A-U⊗V⊗W_v ≤(1+1/(log n))min_U,V,W∈ℝ^n× kA-U⊗ V⊗ W_v.The first inequality follows since ∀ i,j,l∈[n],l≠1, we have A_i,j,l=0. LetU^*,V^*=min_U,V∈ℝ^n× kA-UV^⊤_v.ThenA-U⊗V⊗ W'_v ≤ (1+1/(log n))A-U⊗V⊗W_v≤ (1+1/(log n))A-U^*⊗ V^*⊗ W'_v.Thus, we haveA-UV^⊤_v≤(1+1/(log n))A-U^*(V^*)^⊤_v.Combining with Theorem <ref>, we know that unless fails, there is no (n) time algorithm which can outputA-U⊗V⊗W_v≤(1+1/(log n))min_U,V,W∈ℝ^n× kA-U⊗ V⊗ W_v.§ HARD INSTANCE This section provides some hard instances for tensor problems. §.§ Frobenius CURT decomposition for 3rd order tensorIn this section we will prove that a relative-error Tensor CURT is not possible unless C has Ω(k/ϵ) columns from A, R has Ω(k/ϵ) rows from A, T has Ω(k/ϵ) tubes from A and U has rank Ω(k).We use a similar construction from <cit.> and extend it to the tensor setting.There exists a tensor A∈ℝ^n× n × n with the following property. Consider a factorization CURT, with C∈ℝ^n× c containing c columns of A, R∈ℝ^n× r containing r rows of A, T∈ℝ^n× t containing r tubes of A, and U∈ℝ^c× r × t, such thatA - ∑_i=1^n ∑_j=1^n ∑_l=1^n U_i,j,l· C_i ⊗ R_j ⊗ T_l _F^2 ≤ (1+ϵ)A - A_k _F^2.Then, for any ϵ < 1 and any k≥ 1,c = Ω(k/ϵ), r = Ω(k/ϵ), t=Ω(k/ϵ) and (U) ≥ k/3.For any i∈ [d], let e_i∈ℝ^d denote the i-th standard basis vector. For α >0 and integer d>1, consider the matrix D∈ℝ^(d+1) × (d+1),D = [e_1 + α e_2e_1 + α e_3⋯ e_1 +α e_d+10 ]= [ 1 1 ⋯ 1 0; α 0; α 0; ⋱ ⋮; α 0 ]We construct matrix B∈ℝ^(d+1)k/3 × (d+1)k/3 by repeating matrix D k/3 times along its main diagonal,B= [ D; D; ⋱; D ]Let m=(d+1)k/3. We construct a tensor A∈ℝ^n× n× n with n = 3m by repeating matrix B three times in the following way,A_1,j,l = B_j,l, ∀ j,l ∈ [m] × [m]A_m+i,m+1,m+l = B_i,l, ∀ i,l ∈ [m] × [m]A_2m+i,2m+j,2m+1 = B_i,j, ∀ j,i ∈ [m] × [m]and 0 everywhere else. We first state some useful properties for matrix D,D^⊤ D = [ 1_d 1_d^⊤ + α^2I_d0;00 ]∈ℝ^(d+1)× (d+1)whereσ_1^2(D)= d+α^2,σ_i^2(D)= α^2 ,∀ i=2, ⋯,dσ_d+1^2(D)= 0. By definition of matrix B, we can obtain the following properties,σ_i^2(B) = d + α^2 ,∀ i =1, ⋯, k/3σ_i^2(B) = α^2 ,∀ i =k/3+1, ⋯, dk/3σ_i^2(B) = 0 ,∀ i = dk+1 ,⋯, dk/3+k/3By definition of A, we can copy B into three disjoint n× n× n sub-tensors on the main diagonal of tensor A. Thus, we haveσ_i^2(A) = d + α^2 ,∀ i =1, ⋯, kσ_i^2(A) = α^2 ,∀ i =k+1, ⋯, dkσ_i^2(A) = 0 ,∀ i = dk+1 ,⋯, dk+kLet A_(k) denote the best rank-k approximation to A, and let D_1 denote the best rank-1 approximation to D. Using the above properties, for any k≥ 1, we can compute A -A_(k)_F^2,A -A_k _F^2 = kD- D_1 _F^2 = k (d-1)α^2.Suppose we have a CUR decomposition with c' = o(k/ϵ) columns, r' = o(k/ϵ) rows or t'=o(k/ϵ) tubes. Since the tensor is equivalent by looking through any of the 3 dimensions/directions, we just need to show why the cost will be at least (1+ϵ)A-A_k_F^2 if we choose t = o(k/ϵ) columns and t=o(k/ϵ) rows.Let C∈ℝ^n× c denote the optimal solution. Then it should have the following form,C = [ C_1; C_2; C_3 ]where C_1∈ℝ^m× c_1 contains c_1 columns from A_1:m,1:m,1:m∈ℝ^m× m× m, C_2∈ℝ^m× c_2 contains c_2 columns from A_m+1:2m,m+1:2m,m+1:2m∈ℝ^m× m× m, C_3∈ℝ^m× c_3 contains c_3 columns from A_2m+1:3m,2m+1:3m,2m+1:3m∈ℝ^m× m× m. Let R∈ℝ^n× r denote the optimal solution. Then it should have the following form,R = [ R_1; R_2; R_3 ] A - A(CC^†,RR^†,I) _F^2 ≥ B - R_1 R_1^† B _F^2 +B - C_2 C_2^† B_F^2 +B^⊤ - C_3 C_3^† B^⊤_F^2. By the analysis in Proposition 4 of <cit.>, we haveB -R_1 R_1^† B _F^2 ≥ (k/3)(1+b ·α) D-D_(1)_F^2.andB -C_2 C_2^† B _F^2 ≥ (k/3)(1+b ·α) D-D_(1)_F^2.Let C_3∈ℝ^m× c_3 contain any c_3 columns from B^⊤. Note that C_3 contains c_3(≤ t) columns from B^⊤, equivalently C_2^⊤ contains c_2 rows from B. Recall that B contains k copies of D∈ℝ^(d+1)× (d+1) along its main diagonal.Even if we choose t columns of B^⊤, the cost is at leastB^⊤ - C_3 C_3^† B^⊤_F^2 ≥ (k/3)D-D_(t)_F^2 ≥ (k/3) (d-t) α^2. Combining Equations (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), α=ϵ gives, A - C C^† A_F^2/ A - A_(k)_F^2 ≥ B - R_1 R_1^† B _F^2 +B - C_2 C_2^† B_F^2 +B^⊤ - C_3 C_3^† B^⊤_F^2/ A - A_(k)_F^2 by Eq. (<ref>) ≥ B - R_1 R_1^† B _F^2 +B - C_2 C_2^† B_F^2 +B^⊤ - C_3 C_3^† B^⊤_F^2/k(d-1)α^2 by Eq. (<ref>) ≥ 2(k/3) (1+bϵ)(d-1)ϵ^2 + (k/3)(d-t)ϵ^2 / k (d-1)ϵ^2 by Eq. (<ref>),(<ref>),(<ref>) and α=ϵ= k(d-1)ϵ^2+ (k/3)(-t+1)ϵ^2 + 2(k/3)bϵ (d-1)ϵ^2/k(d-1)ϵ^2= 1 +(k/3)ϵ^2 (2bϵ(d-1) -t+1) /k(d-1)ϵ^2= 1 +2bϵ(d-1) -t +1 /3(d-1) ≥ 1 + (b/3)ϵ by 2t ≤ b ϵ(d-1)/2≥ 1 +ϵ. by b>3. which gives a contradiction.§.§ General Frobenius CURT decomposition for q-th order tensorIn this section, we extend the hard instance for 3rd order tensors to q-th order tensors.For any constant q≥ 1, there exists a tensor A∈ℝ^n× n ×⋯× n with the following property. Define=min_-k A_k∈ℝ^c_1 × c_2 ×⋯× c_q A - A_k _F^2.Consider a q-th order factorization CURT, with C_1 ∈ℝ^n× c_1 containing c columns from the 1st dimension of A, C_2 ∈ℝ^n× c_2 containing c_2 columns from the 2nd dimension of A, ⋯, C_q ∈ℝ^n× c_q containing c_q columns from the q-th dimension of A and a tensor U∈ℝ^c_1× c_2 ×⋯× c_q, such thatA - ∑_i_1=1^n ∑_i_2=1^n ⋯∑_i_q=1^nU_i_1,i_2,⋯,i_q· C_1,i_1⊗ C_2,i_2⊗⋯⊗ C_q,i_q_F^2 ≤ (1+ϵ) .There exists a constant c'<1 such that for any ϵ < c' and any k≥ 1,c_1 = Ω(k/ϵ), c_2 = Ω(k/ϵ), ⋯, c_q=Ω(k/ϵ) and (U) ≥ c' k.We use the same matrix D∈ℝ^(d+1)× (d+1) as the proof of Theorem <ref>. Then we can construct matrix B ∈ℝ^(d+1) k/q × (d+1)k/q by repeating matrix D k/q times along the its main diagonal,B = [ D; D; ⋱; D ]Let m=(d+1)/q. We construct a tensor A∈ℝ^n × n ×⋯× n with n= qm by repeating the matrix q times in the following way,A_[1:m],[1:m],1,1,1,⋯,1,1= B, A_m+1,[m+1:2m],[m+1:2m],m+1,m+1, ⋯,m+1, m+1 = B^⊤, A_2m+1,2m+1,[2m+1:3m],[2m+1:3m],2m+1, ⋯,2m+1, 2m+1 = B, A_3m+1,3m+1,3m+1,[3m+1:4m],[3m+1:4m], ⋯,2m+1, 3m+1 = B^⊤,⋯ ⋯⋯A_(q-2)m+1,(q-2)m+1,(q-2)m+1,(q-2)m+1,(q-2)m+1,⋯,[(q-2)m+1:(q-1)m], [(q-2)m+1:(q-1)m]= B, A_[(q-1)m+1:qm],(q-1)m+1,(q-1)m+1,(q-1)m+1,(q-1)m+1,⋯, (q-1)m+1, [(q-1)m+1:qm]= B^⊤,where there are q/2 Bs and q/2 B^⊤s on the right when q is even, and there are (q+1)/2 Bs and (q-1)/2 Bs on the right when q is odd. Note that this tensor A is equivalent if we look through any of the q dimensions/directions. Similarly as before, we haveA - A_(k)_F^2 = kD - D_(1)_F^2 = k(d-1) α^2.Suppose there is a general CURT decomposition (of this q-th order tensor), with c_1=c_2=⋯ c_q =o(k/ϵ) columns from each dimension. Let C_1∈ℝ^n× c_1, C_2∈ℝ^n× c_2, ⋯, C_q∈ℝ^n× c_q denote the optimal solution. Then the C_i should have the following form,C_1 = [ C_1,1; C_1,2; ⋱; C_1,q ] , C_2 = [ C_2,1; C_2,2; ⋱; C_2,q ] ,⋯, C_q = [ C_q,1; C_q,2; ⋱; C_q,q ](In the rest of the proof, we focus on the case when q is even. Similarly, we can show the same thing when q is odd.) We haveA - A (C_1 C_1^†, C_2C_2^†, ⋯, C_q C_q^†) _F^2≥ ∑_i=1^q/2 B - C_2i-1,2i-1 C_2i-1,2i-1^† B _F^2 + B^⊤ - C_2i,2i C_2i,2i^† B^⊤_F^2≥ (q/2) ((k/q)(1+ bα)D - D_(1)_F^2+ (k/q) (d-t) α^2 ) = (q/2) ((k/q)(1+ bα) (d-1)α^2+ (k/q) (d-t) α^2 )where the second inequality follows by Equations (<ref>) and (<ref>), and the third step follows by D- D_(1)_F^2 = (d-1)α^2.Putting it all together, we haveA - A (C_1 C_1^†, C_2C_2^†, ⋯, C_q C_q^†) _F^2/ A - A_(k)_F^2 ≥ (q/2) ((k/q)(1+ bα) (d-1)α^2+ (k/q) (d-t) α^2 )/k(d-1)α^2= k(d-1)α^2 + (k/2) bα (d-1)α^2 + (k/q) (-t+1)α^2 /k(d-1)α^2= 1 + (k/2) bα (d-1)α^2 + (k/q) (-t+1)α^2 /k(d-1)α^2 ≤ 1 + (k/3) bα (d-1)α^2/k(d-1)α^2 = 1 + (b/3) ϵ by ϵ=α> 1 + ϵ by b>3.which leads to a contradiction. Similarly we can show the rank is at least Ω(k).§ DISTRIBUTED SETTING Input data to large-scale machine learning and data mining tasks may be distributed across different machines. The communication cost becomes the major bottleneck of distributed protocols, and so there is a growing body of work on low rank matrix approximations in the distributed model <cit.> and also many other machine learning problems such as clustering, boosting, and column subset selection <cit.>. Thus, it is natural to ask whether our algorithm can be applied in the distributed setting. This section will discuss the distributed Frobenius norm low rank tensor approximation protocol in the so-called arbitrary-partition model (see, e.g. <cit.>). In the following, we extend the definition of the arbitrary-partition model <cit.> to fit our tensor setting.There are s machines, and the i^th machine holds a tensor A_i∈ℝ^n× n× n as its local data tensor. The global data tensor is implicit and is denoted as A=∑_i=1^s A_i. Then, we say that A is arbitrarily partitioned into s matrices distributed in the s machines. In addition, there is also a coordinator. In this model, the communication is only allowed between the machines and the coordinator. The total communication cost is the total number of words delivered between machines and the coordinator. Each word has O(log(sn)) bits.Now, let us introduce the distributed Frobenius norm low rank tensor approximation problem in the arbitrary partition model:Tensor A∈ℝ^n× n× n is arbitrarily partitioned into s matrices A_1,A_2,⋯,A_s distributed in s machines respectively, and ∀ i∈[s], each entry of A_i is at most O(log(sn)) bits. Given tensor A, k∈ℕ_+ and an error parameter 0<ε<1, the goal is to find a distributed protocol in the model of Definition <ref> such that * Upon termination, the protocol leaves three matrices U^*,V^*,W^*∈ℝ^n× k on the coordinator.* U^*,V^*,W^* satisfies that∑_i=1^k U_i^*⊗ V_i^*⊗ W_i^* -A _F^2≤ (1+ε) min_-k A'A'-A_F^2. * The communication cost is as small as possible. Suppose tensor A∈ℝ^n× n× n is distributed in the arbitrary partition model (See Definition <ref>). There is a protocol( in Algorithm <ref>) which solves the problem in Definition <ref> with constant success probability. In addition, the communication complexity of the protocol is s((k/ε)+O(kn)) words.Correctness. The correctness is implied by Algorithm <ref> and Algorithm <ref> (Theorem <ref>.) Notice that A_1=∑_i=1^s A_i,1,A_2=∑_i=1^s A_i,2,A_3=∑_i=1^s A_i,3, which means thatY_1=T_1A_1S_1,Y_2=T_2A_2S_2,Y_3=T_3A_3S_3,andC=A(T_1,T_2,T_3).According to line <ref>,X^*_1,X^*_2,X^*_3=X_1,X_2,X_3minkj=1∑(Y_1X_1)_j⊗ (Y_2X_2)_j⊗ (Y_3X_3)_j-C _F.According to Lemma <ref>, we havekj=1∑(T_1A_1S_1X^*_1)_j⊗ (T_2A_2S_2X^*_2)_j⊗ (T_3A_3S_3X^*_3)_j-A(T_1,T_2,T_3)_F^2 ≤(1+O(ε))X_1,X_2,X_3minkj=1∑(A_1S_1X_1)_j⊗ (A_2S_2X_2)_j⊗ (A_3Y_3X_3)_j-A _F^2 ≤ (1+O(ε))U,V,Wmin∑_i=1^k U_i⊗ V_i⊗ W_i -A_F^2,where the last inequality follows by the proof of Theorem <ref>. By scaling a constant of ε, we complete the proof of correctness.Communication complexity. Since S_1,S_2,S_3 are w_1-wise independent, and T_1,T_2,T_3 are w_2-wise independent, the communication cost of sending random seeds in line <ref> is O(s(w_1+w_2)) words, where w_1=O(k),w_2=O(1) (see <cit.>). The communication cost in line <ref> is s·(k/ε) words due to T_1 A_i,1S_1,T_2 A_i,2S_2,T_3 A_i,3S_3∈ℝ^(k/ε)× O(k/ε) and C_i=A_i(T_1,T_2,T_3)∈ℝ^(k/ε)×(k/ε)×(k/ε).Notice that, since ∀ i∈[s] each entry of A_i has at most O(log(sn)) bits, each entry of Y_1,Y_2,Y_3,C has at most O(log(sn)) bits. Due to Theorem <ref>, each entry of X_1^*,X_2^*,X_3^* has at most O(log(sn)) bits, and the sizes of X_1^*,X_2^*,X_3^* are (k/ε) words. Thus the communication cost in line <ref> is s·(k/ε) words.Finally, since ∀ i∈[s],U^*_i,V^*_i,W^*_i∈ℝ^n× k, the communication here is at most O(skn) words. The total communication cost is s((k/ε)+O(kn)) words.If we slightly change the goal in Definition <ref> to the following: the coordinator does not need to output U^*,V^*,W^*, but each machine i holds U_i^*,V_i^*,W_i^* such that U^*=∑_i=1^s U_i^*,V^*=∑_i=1^s V_i^*,W^*=∑_i=1^s W_i^*, then the protocol shown in Algorithm <ref> does not have to do the line <ref>. Thus the total communication cost is at most s·(k/ε) words in this setting.Algorithm <ref> needs exponential in (k/ε) running time since it solves a polynomial solver in line <ref>. Instead of solving line <ref>, we can solve the following optimization problem:α^*= α∈ℝ^s_1 × s_2× s_3min∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3α_i,j,l·(Y_1)_i⊗(Y_2)_j⊗(Y_3)_l-C_F.Since it is actually a regression problem, it only takes polynomial running time to get α^*. And according to Lemma <ref>, ∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3α^*_i,j,l·(Y_1)_i⊗(Y_2)_j⊗(Y_3)_l gives a rank-O(k^3/ε^3) bicriteria solution. Further, similar to Theorem <ref>, we can solvemin_U∈ℝ^n× s_2 s_3∑_i=1^s_1∑_j=1^s_2 U_i+s_1(j-1)⊗ (Y_2)_i ⊗ (Y_3)_j -C _F,where C= ∑_i A_i(I,T_2,T_3). Thus, we can obtain a -O(k^2/ϵ^2) in polynomial time.If we select sketching matrices S_1,S_2,S_3,T_1,T_2,T_3 to be random Cauchy matrices, then we are able to compute distributed entry-wise ℓ_1 norm rank-k tensor approximation (see Theorem <ref>). The communication cost is still s((k/ε)+O(kn)) words. If we only require a bicriteria solution, then it only needs polynomial running time.Using similar techniques as in the proof of Theorem <ref>, we can obtain: Let max_i{t_i, d_i}≤ n. Given a t_1 × t_2 × t_3 tensor A and three matrices: a t_1 × d_1 matrix T_1, a t_2 × d_2 matrix T_2, and a t_3 × d_3 matrix T_3. For any δ > 0, if there exists a solution tomin_X_1,X_2,X_3∑_i=1^k (T_1 X_1)_i ⊗ (T_2 X_2)_i ⊗ (T_3 X_3)_i - A _F^2 := ,and each entry of X_i can be expressed using O(log n) bits, then there exists an algorithm that takes (log n) · 2^ O( d_1 k+d_2 k+d_3 k) time and outputs three matrices: X_1, X_2, and X_3 such that (T_1 X_1)⊗ (T_2 X_2) ⊗ (T_3X_3) - A_F^2 =.Let ℝ denote {0,±δ, ⋯, ±Δ} and Δ/δ=2^O(log n s) for γ>0. Let max_i{t_i, d_i}≤ n. Given tensor A∈ℝ^t_1 × t_2 × t_3 and three matrices T_1∈ℝ^t_1 × d_1, T_2∈ℝ^t_2 × d_2, T_3 ∈ℝ^t_3 × d_3, there exists an algorithm that takes O( log(Δ/δϵ) · 2^O( d_1 k+d_2 k+d_3 k) ) time and outputs X'_1,X'_2, X'_3 such that each entry of X'_i has O(log (ns)) bits and it is an (1+ϵ)-approximation tomin_X_1,X_2,X_3∑_i=1^k (T_1 X_1)_i ⊗ (T_2 X_2)_i ⊗ (T_3 X_3)_i - A _F^2.For simplicity, we assume δ=1 in the proof. For each i∈ [3], we can create t_i× d_i variables to represent matrix X_i. Let x denote those variables. Let B denote tensor ∑_i=1^k (T_1 X_1)_i ⊗ (T_2X_2)_i⊗ (T_3X_3)_i and let B_i,j,l(x) denote an entry of tensor B (which can be thought of as a polynomial written in terms of x). Then we can write the following objective function,min_x∑_i=1^t_1∑_j=1^t_2∑_l=1^t_3 ( B_i,j,l(x) -A_i,j,l )^2.We slightly modify the above objective function to obtain this new objective function,min_x,σ ∑_i=1^t_1∑_j=1^t_2∑_l=1^t_3(B_i,j,l(x) -A_i,j,l)^2s.t. x _2^2≤ 2^O(log ns),where the last constraint is unharmful, because there exists a solution that can be written using O(log(ns)) bits. Note that the number of inequality constraints in the above system is O(1), the degree is O(1), and the number of variables is v=(d_1k+d_2k+d_3k). Thus by Theorem <ref>, we know that the minimum nonzero cost is at least(2^O(log ns) )^-2^O ( v ) .It is obvious that the upper bound on cost is at most 2^O(log ns). Thus, the number of binary search steps is at most log (2^O(log ns) /ϵ) 2^O(v). In each step of binary search, we need to pick up a cost C between the lower bound and upper bound, and write down this polynomial system,∑_i=1^t_1∑_j=1^t_2∑_l=1^t_3(B_i,j,l(x) -A_i,j,l)^2 ≤ C, x _2^2≤ 2^O(log ns).Using Theorem <ref>, we can determine if there exists a solution to the above polynomial system. Since the number of variables is v, and the degree is O(1), the number of inequality constraints is O(1). Thus, the running time is(#·)^# = 2^O(v).§ STREAMING SETTING One of the computation models which is closely related to the distributed model of computation is the streaming model. There is a growing line of work in the streaming model. Some problems are very fundamental in the streaming model such like Heavy Hitters <cit.>, and streaming numerical linear algebra problems <cit.>. Streaming low rank matrix approximation has been extensively studied by previous work like <cit.>. In this section, we show that there is a streaming algorithm which can compute a low rank tensor approximation.In the following, we introduce the turnstile streaming model and the turnstile streaming tensor Frobenius norm low rank approximation problem. The following gives a formal definition of the computation model we study.Initially, tensor A∈ℝ^n× n× n is an all zero tensor. In the turnstile streaming model, there is a stream of update operations, and the i^th update operation is in the form (x_i,y_i,z_i,δ_i) where x_i,y_i,z_i∈[n], and δ_i∈ℝ has O(log n) bits. Each (x_i,y_i,z_i,δ_i) means that A_x_i,y_i,z_i should be incremented by δ_i. And each entry of A has at most O(log n) bits at the end of the stream. An algorithm in this computation model is only allowed one pass over the stream. At the end of the stream, the algorithm stores a summary of A. The space complexity of the algorithm is the total number of words required to compute and store this summary while scanning the stream. Here, each word has at most O(log(n)) bits.The following is the formal definition of the problem.Given tensor A∈ℝ^n× n× n, k ∈ℕ_+ and an error parameter 1>ε>0, the goal is to design an algorithm in the streaming model of Definition <ref> such that * Upon termination, the algorithm outputs three matrices U^*,V^*,W^*∈ℝ^n× k.* U^*,V^*,W^* satisft that∑_i=1^k U_i^*⊗ V_i^*⊗ W_i^* - A _F^2≤ (1+ε) min_-k A'A'-A_F^2. * The space complexity of the algorithm is as small as possible. Suppose tensor A∈ℝ^n× n× n is given in the turnstile streaming model (see Definition <ref>), there is an streaming algorithm (in Algorithm <ref>) which solves the problem in Definition <ref> with constant success probability. In addition, the space complexity of the algorithm is (k/ε)+O(nk/ε) words.Correctness. Similar to the distributed protocol, the correctness of this streaming algorithm is also implied by Algorithm <ref> and Algorithm <ref> (Theorem <ref>.) Notice that at the end of the stream V_1=A_1S_1 ∈ℝ^n× s_1, V_2=A_2S_2 ∈ℝ^n× s_2,V_3=A_3S_3∈ℝ^n× s_3, C=A(T_1,T_2,T_3)∈ℝ^t_1 × t_2 × t_3. It also means thatY_1=T_1A_1S_1,Y_2=T_2A_2S_2,Y_3=T_3A_3S_3.According to line <ref> of procedure TurnstileStreaming,X^*_1,X^*_2,X^*_3=X_1∈ℝ^s_1 × k,X_2 ∈ℝ^s_2 × k,X_3 ∈ℝ^s_3 × kminkj=1∑(Y_1X_1)_j⊗ (Y_2X_2)_j⊗ (Y_3X_3)_j-C _FAccording to Lemma <ref>, we havekj=1∑(Y_1X_1)_j⊗ (Y_2X_2)_j⊗ (Y_3X_3)_j-C_F^2= kj=1∑(T_1A_1S_1X^*_1)_j⊗ (T_2A_2S_2X^*_2)_j⊗ (T_3A_3S_3X^*_3)_j-A(T_1,T_2,T_3) _F^2 ≤ (1+O(ε))X_1,X_2,X_3minkj=1∑(A_1S_1X_1)_j⊗ (A_2S_2X_2)_j⊗ (A_3Y_3X_3)_j-A _F^2 ≤ (1+O(ε))U,V,Wmin∑_i=1^k U_i⊗ V_i⊗ W_i-A _F^2,where the last inequality follows by the proof of Theorem <ref>. By scaling a constant of ε, we complete the proof of correctness.Space complexity. Since S_1,S_2,S_3 are w_1-wise independent, and T_1,T_2,T_3 are w_2-wise independent, the space needed to construct these sketching matrices in line <ref> and line <ref> of procedure TurnstileStreaming is O(w_1+w_2) words, where w_1=O(k),w_2=O(1) (see <cit.>). The cost to maintain V_1,V_2,V_3 is O(nk/ε) words, and the cost to maintain C is (k/ε) words.Notice that, since each entry of A has at most O(log(sn)) bits, each entry of Y_1,Y_2,Y_3,C has at most O(log(sn)) bits. Due to Theorem <ref>, each entry of X_1^*,X_2^*,X_3^* has at most O(log(sn)) bits, and the sizes of X_1^*,X_2^*,X_3^* are (k/ε) words. Thus the space cost in line <ref> is (k/ε) words.The total space cost is (k/ε)+O(nk/ε) words.In the Algorithm <ref>, for each update operation, we need O(k/ε) time to maintain matrices V_1,V_2,V_3, and we need (k/ε) time to maintain tensor C. Thus the update time is (k/ε). At the end of the stream, the time to computeX^*_1,X^*_2,X^*_3=X_1,X_2,X_3∈ℝ^O(k/ε)× kminkj=1∑(Y_1X_1)_j⊗ (Y_2X_2)_j⊗ (Y_3X_3)_j-C _F,is exponential in (k/ε) running time since it should use a polynomial system solver. Instead of computing the rank-k solution, we can solve the following:α^*=α∈ℝ^s_1× s_2 × s_3min∑_i=1^s_1∑_j=1^s_2∑_l=1^s_3α_i,j,l·(Y_1)_i⊗(Y_2)_j⊗(Y_3)_l-C_Fwhich will then give ∑_i=1^s_1 ∑_j=1^s_2∑_l=1^s_3α^*_i,j,l·(Y_1)_i⊗(Y_2)_j⊗(Y_3)_l to be a rank-O(k^3/ε^3) bicriteria solution.Further, similar to Theorem <ref>, we can solvemin_U∈ℝ^n× s_2 s_3∑_i=1^s_1∑_j=1^s_2 U_i+s_1(j-1)⊗ (Y_2)_i ⊗ (Y_3)_j -C _Fwhere C= ∑_i A_i(I,T_2,T_3). Thus, we can obtain a -O(k^2/ϵ^2) in polynomial time.If we choose S_1,S_2,S_3,T_1,T_2,T_3 to be random Cauchy matrices, then we are able to apply the entry-wise ℓ_1 norm low rank tensor approximation algorithm (see Theorem <ref>) in turnstile model. § EXTENSION TO OTHER TENSOR RANKSThe tensor rank studied in the previous sections is also called the CP rank or canonical rank. The tensor rank can be thought of as a direct extension of the matrix rank. We would like to point out that there are other definitions of tensor rank, e.g., the tucker rank and train rank. In this section we explain how to extend our proofs to other notions of tensor rank. Section <ref> provides the extension to tucker rank, and Section <ref> provides the extension to train rank. §.§ Tensor Tucker rank Tensor Tucker rank has been studied in a number of works <cit.>. We provide the formal definition here: §.§.§ Definitions Given a third order tensor A∈ℝ^n × n × n, we say A has tucker rank k if k is the smallest integer such that there exist three matrices U,V,W∈ℝ^n× k and a (small) tensor C∈ℝ^k × k × k satisfyingA_i,j,l = ∑_i'=1^k ∑_j'=1^k ∑_l'=1^k C_i',j',l' U_i,i' V_j,j' W_l,l', ∀ i,j,l ∈ [n] × [n] × [n],or equivalently,A = C(U,V,W).§.§.§ AlgorithmGiven a third order tensor A∈ℝ^n× n× n, for any k≥ 1 and ϵ∈ (0,1), there exists an algorithm which takes O((A)) + n (k,1/ϵ) + 2^O(k^2/ϵ +k^3) time and outputs three matrices U,V,W∈ℝ^n× k, and a tensor C∈ℝ^k× k × k for which C(U,V,W) - A _F^2 ≤ (1+ϵ) -k A_k min A_k - A_F^2holds with probability 9/10. We defineto be = -k A'min A' -A _F^2. Suppose the optimal A_k= C^* (U^*,V^*,W^*). We fix C^* ∈ℝ^k× k × k, V^* ∈ℝ^n× k and W^* ∈ℝ^n× k. We use V_1^*, V_2^*, ⋯, V_k^* to denote the columns of V^* and W_1^*, W_2^*, ⋯, W_k^* to denote the columns of W^*.We consider the following optimization problem,min_U_1, ⋯, U_k ∈ℝ^nC^*(U,V^*,W^*) - A _F^2,which is equivalent tomin_U_1, ⋯, U_k ∈ℝ^nU · C^*(I,V^*,W^*) -A _F^2,because C^*(U,V^*,W^*) = U · C^*(I,V^*,W^*) according to Definition <ref>.Recall that C^*(I,V^*,W^*) denotes a k× n× n tensor. Let ( C^*(I,V^*,W^*) )_1 denote the matrix obtained by flattening C^*(I,V^*,W^*) along the first dimension. We use matrix Z_1 to denote ( C^*(I,V^*,W^*) )_1∈ℝ^k× n^2. Then we can obtain the following equivalent objective function,min_U ∈ℝ^n× k U Z_1- A_1 _F^2.Notice that min_U ∈ℝ^n× k U Z_1- A_1 _F^2=, since A_k=U^*Z_1.Let S_1^⊤∈ℝ^s_1× n^2 be the sketching matrix defined in Definition <ref>, where s_1=O(k/ϵ). We obtain the following optimization problem,min_U ∈ℝ^n× k U Z_1 S_1 - A_1 S_1 _F^2.Let U∈ℝ^n× k denote the optimal solution to the above optimization problem. Then U = A_1 S_1 (Z_1 S_1)^†. By Lemma <ref> and Theorem <ref>, we have U Z_1- A_1_F^2 ≤ (1+ϵ) U∈ℝ^n× kmin U Z_1 - A_1 _F^2 = (1+ϵ) , which impliesC^*( U, V^* , W^*) - A _F^2 ≤ (1+ϵ) .To write down U_1, ⋯, U_k, we use the given matrix A_1, and we create s_1 × k variables for matrix (Z_1 S_1)^†. As our second step, we fix U∈ℝ^n× k and W^* ∈ℝ^n× k, and we convert tensor A into matrix A_2. Let matrix Z_2 denote (C^*(U,I,W^*))_2 ∈ℝ^k× n^2. We consider the following objective function,min_V ∈ℝ^n× k V Z_2 -A_2_F^2,for which the optimal cost is at most (1+ϵ).Let S_2^⊤∈ℝ^s_2× n^2 be a sketching matrix defined in Definition <ref>, where s_2=O(k/ε). We sketch S_2 on the right of the objective function to obtain a new objective function,V∈ℝ^n× kmin V Z_2 S_2 - A_2 S_2 _F^2.Let V∈ℝ^n× k denote the optimal solution to the above problem. Then V = A_2 S_2 (Z_2 S_2)^†. By Lemma <ref> and Theorem <ref>, we have,V Z_2 - A_2 _F^2 ≤ (1+ϵ ) V∈ℝ^n× kmin V Z_2- A_2 _F^2 ≤(1+ϵ)^2 ,which impliesC^*( U ,V, W^*) - A _F^2 ≤ (1+ϵ )^2 .To write down V_1, ⋯, V_k, we need to use the given matrix A_2 ∈ℝ^n^2 × n, and we need to create s_2× k variables for matrix (Z_2 S_2)^†.As our third step, we fix the matrices U∈ℝ^n× k and V∈ℝ^n × k. We convert tensor A∈ℝ^n× n × n into matrix A_3 ∈ℝ^n^2 × n. Let matrix Z_3 denote ( C^*(U,V,I) )_3 ∈ℝ^k× n^2. We consider the following objective function,W∈ℝ^n× kmin W Z_3 - A_3 _F^2,which has optimal cost at most (1+ϵ)^2.Let S_3^⊤∈ℝ^s_3× n^2 be a sketching matrix defined in Definition <ref>, where s_3=O(k/ε). We sketch S_3 on the right of the objective function to obtain a new objective function,W ∈ℝ^n× kmin W Z_3 S_3 - A_3 S_3 _F^2.Let W∈ℝ^n× k denote the optimal solution of the above problem. Then W = A_3 S_3 (Z_3 S_3)^†. By Lemma <ref> and Theorem <ref>, we have,W Z_3 - A_3 _F^2 ≤ (1+ϵ) W∈ℝ^n× kmin W Z_3 - A_3 _F^2 ≤ (1+ϵ)^3 .Thus, we havemin_X_1,X_2,X_3 C^* ( (A_1 S_1 X_1) , (A_2S_2 X_2) , (A_3S_3 X_3) ) - A _F^2 ≤ (1+ϵ)^3 .Let V_1=A_1S_1,V_2=A_2S_2, and V_3=A_3S_3. We then apply Lemma <ref>, and we obtain V_1,V_2,V_3,B. We then apply Theorem <ref>. Correctness follows by rescaling ε by a constant factor. Running time. Due to Definition <ref>, the running time of line <ref> (Algorithm <ref>) is O((A))+n(k,1/ϵ). Due to Lemma <ref>, line <ref> and <ref> can be executed in (A) + n(k,1/ϵ) time. The running time of line <ref> is given by Theorem <ref>. (For simplicity, we ignore the bit complexity in the running time.)§.§ Tensor Train rank §.§.§ Definitions The tensor train rank has been studied in several works <cit.>. We provide the formal definition here.Given a third order tensor A∈ℝ^n × n × n, we say A has train rank k if k is the smallest integer such that there exist three tensors U ∈ℝ^1 × n × k, V∈ℝ^k× n × k, W∈ℝ^k× n× 1 satisfying:A_i,j,l = ∑_i_1=1^1 ∑_i_2=1^k ∑_i_3=1^k ∑_i_4=1^1 U_i_1,i,i_2 V_i_2,j,i_3 W_i_3,l,i_4, ∀ i,j,l ∈ [n] × [n] × [n],or equivalently,A_i,j,l =∑_i_2=1^k ∑_i_3=1^k(U_2)_i,i_2 (V_2)_j, i_2 + k(i_3-1)(W_2)_l,i_3,where V_2∈ℝ^n× k^2 denotes the matrix obtained by flattening the tensor U along the second dimension, and (V_2)_i,i_1+k(i_2-1) denotes the entry in the i-th row and i_1+k(i_2-1)-th column of V_2. We similarly define U_2, W_2 ∈ℝ^n× k. §.§.§ Algorithm Given a third order tensor A∈ℝ^n× n× n, for any k≥ 1, ϵ∈ (0,1), there exists an algorithm which takes O((A)) + n (k,1/ϵ) + 2^O(k^4/ϵ ) time and outputs three tensors U∈ℝ^1× n × k, V∈ℝ^k× n× k, W∈ℝ^k× n× 1 such that∑_i=1^k ∑_j=1^k(U_2)_i⊗ (V_2)_ i + k(j -1) ⊗ (W_2)_j - A _F^2 ≤ (1+ϵ) -k A_k min A_k - A_F^2holds with probability 9/10. We defineas= -k A'min A' -A _F^2. Suppose the optimalA_k=∑_i=1^k ∑_j=1^kU^*_i⊗ V^*_ i + k(j-1)⊗ W^*_j.We fixV^* ∈ℝ^n× k^2 and W^* ∈ℝ^n× k. We use V_1^*, V_2^*, ⋯, V_k^2^* to denote the columns of V^*, and W_1^*, W_2^*, ⋯, W_k^* to denote the columns of W^*.We consider the following optimization problem,min_U ∈ℝ^n× k∑_i=1^k ∑_j=1^kU_i⊗ V^*_ i + k(j-1)⊗ W^*_j - A _F^2,which is equivalent tomin_U ∈ℝ^n× k U ·[ kj=1∑ V_1 + k(j-1)^* ⊗ W^*_j; kj=1∑ V_2 + k(j-1)^* ⊗ W^*_j;⋯; kj=1∑ V_k + k(j-1)^* ⊗ W^*_j;] -A _F^2. Let A_1 ∈ℝ^n× n^2 denote the matrix obtained by flattening the tensor A along the first dimension. We use matrix Z_1 ∈ℝ^k× n^2 to denote[ kj=1∑( V_1 + k(j-1)^* ⊗ W^*_j ); kj=1∑( V_2 + k(j-1)^* ⊗ W^*_j ); ⋯; kj=1∑( V_k + k(j-1)^* ⊗ W^*_j ) ]. Then we can obtain the following equivalent objective function,min_U ∈ℝ^n× k U Z_1- A_1 _F^2.Notice that min_U ∈ℝ^n× k U Z_1- A_1 _F^2=, since A_k=U^*Z_1.Let S_1^⊤∈ℝ^s_1× n^2 be a sketching matrix defined in Definition <ref>, where s_1=O(k/ϵ). We obtain the following optimization problem,min_U ∈ℝ^n× k U Z_1 S_1 - A_1 S_1 _F^2.Let U∈ℝ^n× k denote the optimal solution to the above optimization problem. Then U = A_1 S_1 (Z_1 S_1)^†. By Lemma <ref> and Theorem <ref>, we have U Z_1- A_1_F^2 ≤ (1+ϵ) U∈ℝ^n× kmin U Z_1 - A_1 _F^2 = (1+ϵ) , which implies∑_i=1^k ∑_j=1^k U_i ⊗ V^*_i+ k(j-1)⊗ W^*_j - A _F^2 ≤ (1+ϵ) .To write down U_1, ⋯, U_k, we use the given matrix A_1, and we create s_1 × k variables for matrix (Z_1 S_1)^†.As our second step, we fix U∈ℝ^n× k and W^* ∈ℝ^n× k, and we convert the tensor A into matrix A_2. Let matrix Z_2 ∈ℝ^k^2× n^2 denote the matrix where the (i,j)-th row is the vectorization of U_i ⊗ W^*_j.We consider the following objective function,min_V ∈ℝ^n× k V Z_2 -A_2_F^2,for which the optimal cost is at most (1+ϵ).Let S_2^⊤∈ℝ^s_2× n^2 be a sketching matrix defined in Definition <ref>, where s_2=O(k^2/ϵ). We sketch S_2 on the right of the objective function to obtain the new objective function,V∈ℝ^n× kmin V Z_2 S_2 - A_2 S_2 _F^2.Let V∈ℝ^n× k denote the optimal solution of the above problem. Then V = A_2 S_2 (Z_2 S_2)^†. By Lemma <ref> and Theorem <ref>, we have,V Z_2 - A_2 _F^2 ≤ (1+ϵ ) V∈ℝ^n× kmin V Z_2- A_2 _F^2 ≤(1+ϵ)^2 ,which implies∑_i=1^k ∑_j=1^k U_i ⊗V_i+k(j-1)⊗ W^* - A _F^2 ≤ (1+ϵ )^2 .To write down V_1, ⋯, V_k, we need to use the given matrix A_2 ∈ℝ^n^2 × n, and we need to create s_2× k variables for matrix (Z_2 S_2)^†.As our third step, we fix the matrices U∈ℝ^n× k and V∈ℝ^n × k. We convert tensor A∈ℝ^n× n × n into matrix A_3 ∈ℝ^n^2 × n. Let matrix Z_3∈ℝ^k× n^2 denote[∑_i=1^k (U_i ⊗V_i+k · 0);∑_i=1^k (U_i ⊗V_i+k · 1); ⋯; ∑_i=1^k (U_i ⊗V_i+k ·(k-1)) ] .We consider the following objective function,W∈ℝ^n× kmin W Z_3 - A_3 _F^2,which has optimal cost at most (1+ϵ)^2.Let S_3^⊤∈ℝ^s_3× n^2 be a sketching matrix defined in Definition <ref>, where s_3=O(k/ε). We sketch S_3 on the right of the objective function to obtain a new objective function,W ∈ℝ^n× kmin W Z_3 S_3 - A_3 S_3 _F^2.Let W∈ℝ^n× k denote the optimal solution of the above problem. Then W = A_3 S_3 (Z_3 S_3)^†. By Lemma <ref> and Theorem <ref>, we have,W Z_3 - A_3 _F^2 ≤ (1+ϵ) W∈ℝ^n× kmin W Z_3 - A_3 _F^2 ≤ (1+ϵ)^3 .Thus, we havemin_X_1,X_2,X_3∑_i=1^k ∑_j=1^k (A_1 S_1 X_1)_i ⊗ (A_2S_2 X_2)_i+k(j-1)⊗ ( A_3S_3 X_3 )_j - A _F^2 ≤ (1+ϵ)^3 .Let V_1=A_1S_1,V_2=A_2S_2, and V_3=A_3S_3. We then apply Lemma <ref>, and we obtain V_1,V_2,V_3,B. We then apply Theorem <ref>.Correctness follows by rescaling ϵ by a constant factor. Running time. Due to Definition <ref>, the running time of line <ref> (Algorithm <ref>) is O((A))+n(k,1/ϵ). Due to Lemma <ref>, lines <ref> and <ref> can be executed in (A) + n(k,1/ϵ) time. The running time of 2^O(k^4/ϵ) comes from running Theorem <ref> (For simplicity, we ignore the bit complexity in the running time.) § ACKNOWLEDGMENTSThe authors would like to thank Udit Agarwal, Alexandr Andoni, Arturs Backurs, Saugata Basu, Lijie Chen, Xi Chen, Thomas Dillig, Yu Feng, Rong Ge, Daniel Hsu, Chi Jin, Ravindran Kannan, J. M. Landsberg, Qi Lei, Fu Li, Syed Mohammad Meesum, Ankur Moitra, Dana Moshkovitz, Cameron Musco, Richard Peng, Eric Price, Govind Ramnarayan, Ilya Razenshteyn, James Renegar, Rocco Servedio, Tselil Schramm, Clifford Stein, Wen Sun, Yining Wang, Zhaoran Wang, Wei Ye, Huacheng Yu, Huan Zhang, Kai Zhong, David Zuckerman for useful discussions. tocsectionReferences alpha | http://arxiv.org/abs/1704.08246v2 | {
"authors": [
"Zhao Song",
"David P. Woodruff",
"Peilin Zhong"
],
"categories": [
"cs.DS",
"cs.CC",
"cs.LG"
],
"primary_category": "cs.DS",
"published": "20170426175911",
"title": "Relative Error Tensor Low Rank Approximation"
} |
Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland We study entanglement generated between a charge qubit and a bosonic bath due to their joint evolution which leads to pure dephasing of the qubit. We tune the parameters of the interaction, so that the decoherence is quantitatively independent of the number of bosonic modes taken into account and investigate, how the entanglement generated depends on the size of the environment. A second parameter of interest is the mixedness of the initial state of the environment which is controlled by temperature. We show analytically that for a pure initial state of the environment, entanglement does not depend on environment size. For mixed initial states of the environment, the generated entanglement decreases with the increase of environment size. This effect is stronger for larger temperatures, when the environment is initially more mixed, but in the limit of an infinitely large environment, no entanglement is created at any finite temperature. Entanglement generation between a charge qubit and its bosonic environment during pure dephasing - dependence on environment size Katarzyna Roszak December 30, 2023 =================================================================================================================================§ INTRODUCTION Decoherence due to the interaction of a qubit with its environment can often be modeled by classical noise with a good deal of accuracy. This is especially true with respect to pure dephasing, i. e. when decoherence does not disturb the occupations of the qubit, but affects only its coherence (the off-diagonal elements of the density matrix which are responsible for quantum behavior). Such loss of qubit coherence can always be mapped by a random unitary channel acting on the system, where the channel describes the interaction with a fictitious classical environment <cit.> (see Ref. <cit.> for examples of constructions of such fictitious classical environments for different types of open quantum systems). The existence of such a mapping does not invalidate the importance of qubit-environment entanglement in pure-dephasing scenarios, since the presence of entanglement in the system will influence the evolution of the system in general, e. g. it changes the state of the environment pre- and post-measurement <cit.>. Furthermore, if the study of a quantum system is not limited to its preparation, allowing it to evolve freely, and then measurement, but also involves some manipulation of the system, such as performing gates <cit.>, coherence maximizing schemes <cit.>, etc. then the presence of system-environment entanglement will influence the end result and can be higly relevant. Hence, it is at least in principle possible to have different qubit-environment setups, in which the pure dephasing of the qubit is qualitatively and quantitatively the same, but the origin of the dephasing is different, since it can, but does not have to be the result of entanglement generation. If the whole system is always in a pure state (for an evolution described by a Hamiltionian this is equivalent to the initial states of the qubit and environment being pure) pure dephasing is unambiguously related to entanglement <cit.>. In the case of mixed states, the relation between qubit coherence and qubit-environment entanglement is much more ambiguous and although entanglement not accompanied by dephasing is not possible, dephasing without entanglement is <cit.>. In fact, the latter situation is often realized in real systems, especially in the case of large environments, high temperatures, or noise resulting from e. g. fluctiating semi-classical fields <cit.>. The distinction between entangling and non-entangling evolutions is not trivial in itself, since the non-entangling case is not limited to random unitary evolutions <cit.>, and a straightforward criterion for the generation of qubit-environment entanglement during pure dephasing has only recently been found <cit.>. We study the amount of qubit-environment entanglement generated during the joint evolution of a charge qubit interacting with a bath of phonons as an example of a realistic system in which the qubit undergoes strictly pure dephasing which is always accompanied by the creation of entanglement (with the exception of only the infinite-temperature situation) <cit.>. The system is particularly convenient, because not only can the decoherence-curves be reproduced using an arbitrary number of phonon modes (for short enough times and large enough temperatures), but the results can be obtained in a semi-analytical fashion, which simplifies changing the number of phonon modes and later interpretation of the results. Furthermore, we control the initial level of mixedness of the environment by setting the temperature of the phonon bath. The correlation build-up between a system and its bosonic environment have thus far been studied in the context of its relation towards non-Markovian dynamics <cit.>, decoherence (especially for mixed initial qubit states) <cit.>, and two qubit correlation decay <cit.>. Furthermore, studies of system-environment entanglement for a boson system and a bosonic environment have also been reported <cit.>. In these studies, the focus was on the relation between the appearance of quantum correlations with a large environment with other quantum features of open system dynamics. In this paper, the quantity of highest importance is the size of the environment, which is vital for the amount of entanglement generated in the whole system. We find that the dependence of generated entanglement on the temperature shows monotonously decreasing behavior which is steep above a physically well motivated threshold temperature. The temperature dependence is non-trivial even above this temperature, and for reasonably low temperatures displays exponential decay, while for high temperatures the dependence becomes proportional to 1/T^3. On the other hand, the dependence on the number of phonon modes is much more complex. For pure states, the amount of entanglement generated does not depend on the number of phonon modes at all. Yet for any mixed environmental state (finite temperature) there is a pronounced dependence on the number of phonon modes (on the size of the environment) of 1/n character (where n is the number of modes). The maximum entanglement generated throughout the evolution always decreases when the environment becomes larger, even though the decoherence curve is unaffected in the studied scenario for high enough temperatures. Furthermore, this decrease is steeper when the temperature is higher (the initial state is more mixed), but when the number of phonon modes approaches infinity (the continuous case), all finite-temperature entanglement vanishes. The article is organized as follows. In Sec. <ref> we introduce the system under study, the Hamiltonian describing this system, and the full qubit-environment evolution resulting from this Hamiltonian. We furthermore describe, how the respective strengths of the bosonic modes are determined, so that the decoherence at short times and large enough temperatures is qualitatively and quantitatively the same independently of the size of the environment. Sec. <ref> contains a brief description of Negativity, which is the entanglement measure which is later used to quantify qubit-environment entanglement. In Sec. <ref> the dependence of the purity of the whole system (which is constant throughout the evolution) on the initial state of the environment is determined, especially on the temperature and size of the environment. Sec. <ref> contains the results pertaining to the dependence of the generated entanglement on temperature and consequently, on the degree of mixedness of the initial state of the environment, as well as the results concerning the effect of environment size on entanglement, when the characteristics of the resulting qubit decoherence remain unchanged. The conclusions are given in Sec. <ref>. § THE SYSTEM AND THE EVOLUTION The system under study consists of a charge qubit interacting with phonons. The Hamiltonian of this system is H=ϵ|1 1|+∑_kħω_kb_k^†b_k +|1 1|∑_k (f_k^*b_k+f_kb_k^†), where the first term describes the energy of the qubit (ϵ is the energy difference between the qubit states |0⟩ and |1⟩ in the absence of phonons), the second term is the Hamiltonian of the free phonon subsystem and the third term describes their interaction. Here, ω_k is the frequency of the phonon mode with the wave vector k and b_k^†, b_k are phonon creation and annihilation operators corresponding to mode k and f_k are coupling constants. The Hamiltonian (<ref>) can be diagonalized exactly using the Weyl operator method (see Ref. roszak06b for details; the same results can be obtained using a different approach <cit.>). For a product initial state of the system and the environment, σ(0)=|ψ⟩⟨ψ |⊗ R(0), where the qubit state is pure, |ψ⟩ = α |0⟩ +β |1⟩, and the environment is at thermal equilibrium, R(0)=e^-1/k_B T∑_kħω_kb_k^†b_k/Tr[ e^-1/k_B T∑_kħω_kb_k^†b_k], where k_B is the Bolzmann constant and T is the temperature, the joint qubit-environment density matrix evolves according to σ̂(t)=( [ |α|^2R̂(0) αβ^*e^-iϵ t/ħR̂(0)û^†(t); α^*β e^iϵ t/ħû(t)R̂(0) |β|^2û(t)R̂(0)û^†(t) ]). Here, the matrix is written in the basis of the qubit states |0⟩ and |1⟩, while the degrees of freedom of the environment are contained in the density matrix R̂(0) and time-evolution operators acting only on the environmnet, û(t). The evolution operators can be found following Ref. [roszak06b], and are given by û(t) = exp[∑_k( f_k/ħω_k(1-e^-iω_k t)b_k^† -f_k^*/ħω_k(1-e^iω_k t) b_k)] ×exp[i∑_k|f_k|^2/(ħω_k)^2sinω_k t]. In order to obtain the density matrix of the qubit alone, a trace over the degrees of freedom of the environment needs to be performed, ρ̂(t)=_E σ̂. This yields a density matrix with time-independent occupations and coherences which undergo decay governed by the function |⟨û(t)⟩| = exp(-∑_kf_k/ħω_k^2(1-cosω_kt)(2n_k+1)), where n_k=1/(e^ħω_k/k_BT-1) is the Bose-Einstein distribution. If, as in our case, the quantity of interest is qubit-environment entanglement and not just the coherence of the qubit, the time-evolution of the full system density matrix (<ref>) is needed. This can be found by acting with the evolution operator given by eq. (<ref>) on the initial density matrix of the environment. The density matrix of the whole system σ̂ can be divided into four parts with respect to the way that the evolution operator acts on the density matrix of the environment, which correspond to the |0⟩ qubit state occupation (for which the environment remains unaffected R̂_00(t)=R̂(0)), the |1⟩ qubit state occupation (for which R̂_11(t)=û(t)R̂(0)û^†(t)), and the two qubit coherences (with R̂_01(t)=R̂(0)û^†(t) when the qubit density matrix element corresponds to |0⟩⟨ 1| and R̂_10(t)=û(t)R̂(0) when the qubit density matrix element corresponds to |1⟩⟨ 0|). Since the initial density matrix of the environment is a product of density matrices for each boson mode R̂(0)=⊗_kR̂^k(0) and so is the evolution operator at all times û(t)=⊗_kû^k(t), each matrix R̂_ij(t) (i,j=0,1) also has product form with respect to the different boson modes at all times, so each boson mode can be treated separately. This is convenient, since every R̂_ij^k(t) matrix is in principle of infinite dimension (the number of phonons in each mode can be arbitrarily large; the actual distribution of states for a single phonon mode is governed by the temperature and the qubit-phonon coupling) and a reasonable cut-off needs to be implemented to keep the density matrix σ manageable without the loss of physical meaning. Note, that although the R̂_ii(t) matrices corresponding to the diagonal elements of the qubit density matrix are density matrices themselves, this is not always true for the R̂_ij(t) matrices with i≠ j (which is a first indicator of qubit-environment entanglement). It can be shown that the evolution of any state of m phonons in mode k is given by |m(t)⟩_k = û^k(t)|m⟩_k= ∑_p=-m^∞( f_k/ħω_k)^p√(m!/(m+p)!) × L_m^(p)(|f_k/ħω_k|^2) |m+p⟩_k, where L_m^(p)(x) is a generalized Laguerre polynomial. Given the initial state of the environment, eq. (<ref>) is sufficient to find the time evolution of the whole system-environment density matrix σ̂, since R̂_11(t) = û(t)R̂(0)û^†(t)=⊗_k(∑_m_k=0^∞ c_m_k|m(t)⟩_k k⟨ m(t)| ), R̂_10(t) = û(t)R̂(0)= ⊗_k(∑_m_k=0^∞ c_m_k|m(t)⟩_k k⟨ m|), R̂_01(t) = R̂(0)û^†(t)=R̂^†_10(t). Here, the initial occupations of each state |m⟩_k are found for a given temperature using eq. (<ref>), c_m_k=e^-ħω_k/k_BTm_k(1-e^-ħω_k/k_BT). §.§ Excitonic quantum dot qubits The exciton-phonon interaction constants used in the calculations correspond to excitonic qubits confined in quantum dots <cit.>, where qubit state |0⟩ corresponds to an empty dot, while state |1⟩ denotes an exciton in its ground state confined in the dot. They are given by f_k =( σ_e -σ_h ) √(ħ k/2ϱ V_N c)∫_-∞^∞ d^3rψ^*(r) e^-ik·rψ(r), describing the deformation potential coupling, which is the dominating decoherence mechanism for excitons <cit.>. Hence, ω_k=ck, where c is the speed of longitudinal sound and the phonon-bath is super-Ohmic. Here ϱ is the crystal density, V_N unit cell volume, and σ_e,h are deformation potential constants for electrons and holes respectively. The exciton wave function ψ(r) is modeled as a product of two identical single-particle wave functions ψ(r_e) and ψ(r_h), corresponding to the electron and hole, respectively. The parameters used in the calculations correspond to small self-assembled InAs/GaAs quantum dots, which are additionally assumed to be isotropic (for the sake of simplicity when limiting the number of phonon modes and with little loss of realism, when the evolution of coherence is found). The single particle wave functions ψ(r) are modeled by Gaussians with 3 nm width in all directions. The deformation potential difference is σ_e-σ_h=9.5 eV, the crystal density is ϱ=5300 kg/m^3, and the speed of longitudinal sound is c=5150 m/s. The unit cell volume for GaAs is V_N=0.18 nm^3 (note, that this volume does not enter into the decoherence function, but is relevant, when individual elements of the system-environment density matrix needed to evaluate entanglement are found). §.§ Discretization Typically for realistic systems, the number of phonon modes is very large and the summation over k can be substituted by integration, which in spherical coordinates yields ∑_k→V/(2π)^3∫_0^2πdϕ∫_0^πsinθ dθ∫_0^∞k^2dk. If the studied system has spherical symmetry, as do the quantum dots, the parameters of which are used in the calculations, the integration over the angles can be performed analytically. In the following, when we study qubit-environment entanglement and qubit decoherence due to the interaction with an environment which supports only a limited number of phonon modes, we do not differentiate the modes with respect to their direction, only with respect to the length of the wave vector. This means that we consider a simplified scenario, where phonon modes are averaged over all directions. The actual discretization of the phonon modes is done only on the level of the length of the wave vector k. A minimum and maximum wave vector length is, somewhat arbitrarily, chosen, so that a large enough range of k is considered to account for different phonon modes with values of the function |f_k/ħω_k|^2 which are large enough to be relevant for both pure dephasing and entanglement generation. In the following they are always set to k_min=0.001 nm^-1 and k_max=0.9 nm^-1. For a given number of phonon modes n, the range [k_min,k_max+k_min] is evenly divided, and only wave vectors of lengths k_i=(i-1)Δ k+k_min, with Δ k=k_max/(n-1) and i=1,2,...,n, are taken into account. The slight off-set by of the range of k by k_min allows for the decoherence of the qubit in the continuous case (when an infinite number of phonon modes is taken into account) to be well approximated by only a few phonon modes for a wide range of temperatures as seen below. The decay of qubit coherence for a quantum dot interacting with an environment for which only a few discrete lengths of phonon wave vectors are allowed (in which only a few phonon modes are present) is plotted in Fig. (<ref>) for T=6 K (note that the coherence on the plot is limited from below by 0.7 and not by the minimum value of the degree of coherence, 0, for clarity). The initial drop of coherence which is present in the continuous case, and which is in the continuous case followed by a slight rise in coherence, after which the coherence stabilizes at some finite value due to the super-Ohmic nature of the environment (hence the term “partial pure dephasing”; see Ref. <cit.> for details) is reproduced very well already when only three phonon modes are present in the environment. For larger boson mode numbers n, the longer-time features of decoherence are also reproduced for a finite time, and the refocusing of the qubit (which is the result of the whole qubit-environment density matrix returning to its initial state in the course of its unitary evolution) is further and further delayed in time with increasing number of phonon modes. In fact, it is easy to reproduce continuous short-time behavior of the coherence even with small phonon mode numbers as long as the initial temperature is above some threshold, which depends on the phonon energy spectrum, and is here around 3 K (this also obviously depends on the number of phonon modes and the threshold is higher for higher mode numbers). Below this temperature, the approximation gets progressively worse, if the choice of phonon modes remains unchanged and spans the whole energy range where the spectral density is reasonably large. This is because the temperature enters into the calculation only through the initial state of the environment (which is at thermal equilibrium) and, if only a few phonon modes with well separated energies are allowed, then below some temperature practically no phonons will be initially excited. This can be remedied by a redistribution of the phonon modes to lower energies, but since we wish to compare qubit-environment entanglement evolutions, when environments with different numbers of boson modes lead to the same dynamics of the qubit alone, but with a set system under study for a given number of modes, we simply restrict ourselves to higher temperatures. This means that for a set number o phonon modes the studied system is always qualitatively the same (and even though the initial state of the system depends on temperature, the types of phonons which can be excited remain unchanged). § NEGATIVITY The measure of entanglement which is most convenient (easiest to compute) in the context of quantifying entanglement between a qubit (a small quantum system) and its environment (a large quantum system) is Negavitity <cit.> (or equivalently logarithmic Negativity <cit.>). The measure is based on the PPT criterion of separability <cit.>, which does not detect bound entanglement <cit.>. Fortunately, in the case of the evolution of a qubit initially in a pure state interacting with an arbitrary environment due to an interaction which can only lead to pure dephasing of the qubit, bound entanglement is never formed <cit.>, so non-zero Negativity is a good criterion for the presence of entanglement in the system and the value of Negativity unambiguously indicates the amount of said entanglement. Negativity can be defined as the absolute sum of the negative eigenvalues of the density matrix of the whole system after a partial transposition with respect to one of the two potentially entangled subsystems has been performed (it does not depend on which of the subsystems is chosen for the partial transposition), N(σ̂)=∑_i|λ_i|-λ_i/2, where λ_i are the eigenvalues of σ̂^Γ_A, and Γ_A denotes partial transposition with respect to system A=Q,E (qubit or environment). In the case of the studied system, it is particularly simple to perform partial transposition with respect to the qubit, as it is sufficient to exchange the off-diagonal terms in eq. (<ref>) to get the desired partially transposed state. § PURITY An important factor for the amount of entanglement generated between the qubit and the environment is the initial purity of the state of the whole system. Note, that since the qubit-environment evolution is unitary, the purity does not change with time, since P(σ̂(t)) = σ̂^2(t)=[ Û(t)σ̂(0)Û^†(t) Û(t)σ̂(0)Û^†(t) ]= σ̂^2(0)=P(σ̂(0)). Taking into account that the initial state of the studied system is a product state and the state of the qubit subsystem is pure, we have P(σ̂(0))=P(ρ̂(0))P(R̂(0))=P(R̂(0)), so the purity of the system only depends on the initial purity of the density matrix of the environment. Furthermore, this initial density matrix is a product of the thermal-equilibrium density matrices for each phonon mode, so the purity is a product of the purities of the state of each mode, P(R̂(0))=∏_kP(R̂^k(0)). The purity of the initial state of mode k is easily found from eq. (<ref>) and is given by P(R̂^k(0))= (1-e^-ħω_k/k_BT)^2/1-e^-2ħω_k/k_BT. Since e^-ħω_k/k_BT tends to one with growing temperature more slowly than e^-2ħω_k/k_BT, the numerator in eq. (<ref>) tends to zero much faster than the denominator, and the purity of the ininial state of a given phonon mode is a decreasing function of temperature (which reaches zero for infinite temperature, since the dimension of the Hilbert space of each phonon mode is infinite). Consequently, the purity of the whole environment for a set choice and number of phonon modes is always a decreasing function of temperature as well. Less obviously, for the system under study the purity is also a decreasing function of the number of phonon modes. For a given number of modes n, n wave vectors are taken into account which are evenly distributed throughout a set wave vector lengths k where the coupling constants are most relevant as explained in Sec. <ref>, and the purity of their initial state is a product of the corresponding single-phonon-mode purities (<ref>). The energy of each phonon mode is proportional to its wave vector length, ω_k=ck, so the single-phonon-mode purity is an increasing function of k for any finite temperature. If the number of phonon modes is increased by one, each phonon mode is substituted by one with smaller wave vector length k (with the exception of the two phonon modes limiting the rangle of k), and hence, of lesser purity, and an additional phonon mode with a longer wave vector is taken into account (since now we are dealing with n+1 phonon modes evenly distributed over the same range). Consequently, the product of the new n+1 purities, which yields the purity of the initial state of the environment for an increased number of phonon modes, must be smaller for any finite temperature than the purity for n modes. The exception is the zero-temperature case, for which the purity of the initial environment is always equal to one, since the initial state is pure, and the infinite-temperature case, when the purity is always equal to zero. Hence, although for every number of phonon modes the purity is a decreasing function of temperature ranging from one to zero, the decrease is faster, if n is larger. § RESULTS: QUBIT-ENVIRONMENT ENTANGLEMENT GENERATION §.§ Temperature dependence The time-evolution of entanglement for the initial qubit state with α=β=1/√(2) (equal superposition) quantified by Negativity is plotted in Fig. (<ref>) for n=10 boson modes at three different temperatures (the temperature increases from top to bottom of the figure). The relatively large number of boson modes guarantees that the evolution of the qubit state due to the exciton-phonon interaction not only reproduces the fast initial decay of qubit coherence which occurs in the first two picoseconds after the creation of the excitonic superposition for a continuous environment, but also gives a reasonable approximation of the coherence plateau for the next four picoseconds (up to slight oscillations which are absent when an infinite number of phonon modes is taken into account). The temperatures in the plot start at 6 K, well above the threshold temperature, so phonon modes which are evenly distributed over the range of relevant coupling constants f_k reproduce the dynamics of decoherence for short times well. The plots inFig. (<ref>) capture a single cycle of qubit-environment evolution, so at the right end of the plots, the density matrix of the whole system returns to its initial state, and then the evolution is repeated (this is an unavoidable feature of systems with discrete spectra). As can be seen, changing the temperature does not change the qualitative features of the evolution of Negativity, but entanglement (at any given time) is a decreasing function of temperature. Contrarily, decoherence increases with temperature, so for higher temperatures the effect of the environment on the qubit is larger (leading to stronger pure dephasing), but this is due to a buildup of classical qubit-environment correlations, since the amount of entanglement generated between the two subsystems decreases with decreasing purity of the state. In Fig. (<ref>) the dependence of the maximum Negativity reached during the pure dephasing evolution (Negativity reached at the first maximum which corresponds to the initial strong loss of coherence) is plotted as a function of temperature for a choice of three different numbers of boson modes. Below around T=2 K, the maximum Negativity stabilizes at an almost fixed value. This is because the initial density matrix of the environment below this temperature becomes a very weak function of temperature, since only phonon modes with high energies compared to k_BT are taken into account (if there are only a few phonon modes allowed in the system). The density matrix of the environment is then almost in the pure state R̂(0)≈ |0⟩⟨ 0|, and the resulting qubit decoherence is no longer a good approximation of the continuous case. Note that in such situations, the plateau in Negativity is strictly related to the discrete nature of the phonon energy spectrum. An agreement between continuous and few-phonon-mode decoherence could be reached also for low temperatures, but this would require changing k_max and would qualitatively change the system under study (which we want to avoid). If the necessary redistribution of the phonon modes taken into account (to account for decoherence well) were made, the plateau in low-temperature negativity would not be observed. At higher temperatures, maximum Negativity decreases strongly with temperature, regardless of the number of phonon modes taken into account, although the actual amount of entanglement in the system depends strongly on n. The shapes of the Negativity curves plotted in Fig. (<ref>) roughly resemble the temperature dependence of the purity, which is found using eq. (<ref>) with appropriate values of wave vectors k, meaning that the dependence in the shown temperature range is predominantly exponential decay. At high temperatures (for nanostructures, meaning far outside the 20 K range of Fig. (<ref>)), the decay is dominated by terms proportional to 1/T^3. The fitted dependence of Negativity on temperature is presented at the end of Sec. (<ref>), since the dependence on temperature is convoluted with the dependence on environment size and cannot be considered separately. Note that the trade-off temperature behavior, which is characteristic for the build up of correlations in the studied system <cit.> and which results from the decrease of purity with temperature accompanied by an increase of the overall effect of the environment on the qubit, is not present here, as only the purity of the system state is relevant for the generation of entanglement, as long as the system-environment interaction is capable of entangling the two subsystems. Contrarily, this type of trade-off behavior has been reported for boson-boson system-environment ensembles <cit.>. §.§ Dependence on environment size - pure initial stateIn the case of a pure initial state of the environment (at zero temperature in the case of the studied system, so there are initially no phonons), the joint evolution of the system and the environment remains pure and entanglement at any time can be evaluated in a straightforward manner using the von Neumann entropy of one of the entangled subsystems (such von Neumann entropy is theunique entanglement measure for pure states). The measure is defined asE(|ψ(t)⟩)=-1/ln 2( ρ(t)lnρ(t) ),where |ψ(t)⟩ is the pure system-environment state and ρ(t)=_E|ψ(t)⟩⟨ψ(t)| is the density matrix of the qubit at time t (obtained by tracing out the environment). The entanglement measure in eq. (<ref>) is normalized to yield unity for maximally entangled states. The same result would be obtained when tracing out the qubit degrees of freedom instead of the environmental degrees of freedom, but the small dimensionality of the qubit makes this way much more convenient.Let us denote the pure initial state of the environment as |R_0⟩. Then qubit-environment state at time t is given by|ψ(t)⟩ = α|0⟩⊗|R_0⟩+β e^iϵ t/ħ|1⟩⊗û(t)|R_0⟩,and û(t) is given by eq. (<ref>). The density matrix of the qubit is now of the formρ(t)=( [ |α|^2 αβ^*e^-iϵ t/ħu^*(t);α^*β e^iϵ t/ħ u(t) |β|^2 ]),where u(t)=⟨ R_0|û(t)|R_0⟩ and the absolute value of the function u(t) constitutes the degree of coherence retained in the qubit system at a given time (it is given by eq. (<ref>) with T=0).The entanglement measure of eq. (<ref>) can be calculated using eq. (<ref>) which yieldsE(|ψ(t)⟩) = -1/ln 2[ 1+√(Δ(t))/2ln1+√(Δ(t))/2. +. 1-√(Δ(t))/2ln1-√(Δ(t))/2],with Δ(t)=1-4|α|^2|β|^2+|α|^2|β|^2|u(t)|^2. Note that the von Neumann entropy during pure dephasing depends only on the degree of coherence |u(t)|. This means that, if two qubits lose the same amount of coherence, they must be entangled with the environment to the same degree, regardless of the numer of boson modes which constitute theenvironment. Hence, for pure initial environmental states, the amount of entanglementgenerated during evolution does not depend on the size of the environment, as long as the degree of coherence at a given time does not depend on its size.A similar analysis can be performed using Negativity as the measure of pure state entanglement (Negativity does not converge to von Neumann entropy for pure states contrarily to most entanglement measures). It is then fairly straightforward to show that entanglement does not depend on the size of the environment, but only on the degree of coherence u(t), taking into account the fact that Negativity in the studied system does not depend on the phase relations between different components of the densitymatrix σ̂. What is not straightforward is obtaining the explicit relation between Negativity and decoherence, and therefore the von Neumann entropy was used in the analysis above.§.§ Dependence on environment sizeAt finite temperatures (for non-pure initial environmental states) the number of boson modes taken into account becomes very important. In Fig. (<ref>) the evolution of Negativity for the equal superposition initial state of the qubit (α=β=1/√(2)) is plotted at T=6 K for different numbers of boson modes n. Note that the temperature is high enough, so that the initial drop of qubit coherence is always the same as in the continuouscase, while the platou is reproduced for some short time after the drop up to small oscillations (and this time is longer for larger n). This means that the three curves inFig. (<ref>) correspond to the same qubit decoherence curves at short times (up to roughly 5 ps here). Obviously, the amount of entanglement generated during these decoherence processes is not the same, as both the maximum values and the values at the plateau decrease with increasing number of boson modes. As the temperature dependence, this is related to the purity of the whole system during the evolution, but the temperature also affects the decoherence curves, while the number of phonon modes (when they are chosen as outlined in Sec. <ref>) does not.The dependence of maximum Negativity as a function of the number of boson modes taken into account is plotted in Fig. (<ref>) with pointsfor three different temperatures (well above the threshold value). The decrease of Negativity with growing n israther steep for the temperatures shown and this steepness increases when the temperature grows. This corresponds to the fact that at zero temperature, entanglement does not depend on the size of the environment, while at infinite temperature no entanglement between the qubit and the environment is generated at all<cit.>. Furthermore, for any finite temperature entanglement approaches zero with growing n according to a function proportional to 1/n, and for a continuous environment, no entanglement is generated in the system. Although separability is reached more slowly at lower temperatures it is reached nonetheless for large enough values of n (technically, zero-Negativity is only obtained for n=∞, but for high enough n the values of Negativity will be so small that such entanglement will no longer be detectable and will have practically no effect on the properties of the system).Fitting of the curves displayed in Fig. (<ref>) for temperatures above the threshold temperature and points displayed in Fig. (<ref>) allows to find the dependence of maximum Negativity on temperature and environment size. The dependence on size exhibits good ∼1/n behavior. The temperature dependence, on the other hand, shows strong exponential decayfor low temperatures and, while increasing temperature, a 1/T^3 dependence becomes dominant.Furthermore, the temperature and size dependencies are convoluted, so they cannot be represented as a simple product of temperature-dependent and size-dependent functions. A reasonable fit is obtained using the functionN_max(n,T)≈ e^-α TA/T(n-BT+CT^2+D),where the fitting parameters are given by α=0.0857, A=3.51,B=0.4674, C=0.01865, and D=2.57. The curves obtained using eq. (<ref>) are also plotted on Fig. (<ref>), showing a very good n-dependence for different temperatures, especially for higher numbers of boson modes (n≥ 3).§ CONCLUSIONWe have studied the generation of entanglement quantified by Negativity between a charge qubit and its bosonic environment during evolution which leads to pure dephasing of the qubit.In particular, we studied and excitonic qubit confined in a quantum dot in the presence of a super-Ohmic phonon bath, but the results could be easily extended to other charge qubits undergoing similar decoherence processes. The quantity of interest was the dependence of the amount of generated entanglement on the size of the environment (the number of boson modes taken into account) in the situation, when the evolution of the qubit alone does not depend on environment size (for short enough times and high enough temperatures, such a situation iseasily obtained). We have found that although for pure states entanglement does not depend on the system size (and the amount of generated entanglement for pure dephasing is an explicit function of the degree of qubit coherence), for finite temperatures Negativity is a decreasing function of environment size proportional to 1/n and there is no entanglement generated for a continous bosonic environment regardless of the temperature (as long as T≠ 0).The temperature, which governs the initial mixedness of the environment, and consequently the mixedness of the whole system throughout its unitary evolution, similarly governs entanglement generated between the system and environment. 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"authors": [
"Tymoteusz Salamon",
"Katarzyna Roszak"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170426161738",
"title": "Entanglement generation between a charge qubit and its bosonic environment during pure dephasing - dependence on environment size"
} |
The Web SSO Standard OpenID Connect: In-Depth Formal Security Analysis and Security Guidelines Daniel Fett, Ralf Küsters, and Guido SchmitzUniversity of Stuttgart,Germany Email:December 30, 2023 =============================================================================================================================== We study the parameterized complexity of several positional games. Our main result is that is -complete parameterized by the number of moves. This solves an open problem from Downey and Fellows' influential list of open problems from 1999. Previously, the problem was thought of as a natural candidate for -completeness.Our main tool is a new fragment of first-order logic where universally quantified variables only occur in inequalities. We show that model-checking on arbitrary relational structures for a formula in this fragment is -complete when parameterized by formula size.We also consider a general framework where a positional game is represented as a hypergraph and two players alternately pick vertices. In a Maker-Maker game, the first player to have picked all the vertices of some hyperedge wins the game. In a Maker-Breaker game, the first player wins if she picks all the vertices of some hyperedge, and the second player wins otherwise. In an Enforcer-Avoider game, the first player wins if the second player picks all the vertices of some hyperedge, and the second player wins otherwise.is -complete, whereas is -complete and -complete parameterized by the number of moves. This suggests a rough parameterized complexity categorization into positional games that are complete for the first level of the -hierarchy when the winning configurations only depend on which vertices one player has been able to pick, but -completeness when the winning condition depends on which vertices both players have picked. However, some positional games where the board and the winning configurations are highly structured are fixed-parameter tractable. We give another example of such a game, , which is fixed-parameter tractable when parameterized by the number of moves.§ INTRODUCTION In a positional game <cit.>, two players alternately claim unoccupied elements of the board of the game. The goal of a player is to claim a set of elements that form a winning set, and/or to prevent the other player from doing so.Tic-Tac-Toe, its competitive variant played on a 15× 15 board, Gomoku, as well as Hex are the most well-known positional games. When the size of the board is not fixed, the decision problem, whether the first player has a winning strategy from a given position in the game is -complete for many such games. The first result was established for , a variant played on an arbitrary graph <cit.>.soon followed up with results for gomoku <cit.> and Hex played on a board <cit.>. More recently, -completeness was obtained for Havannah <cit.> and several variants of Connect(m, n, k, p, q) <cit.>, a framework that encompasses Tic-Tac-Toe and Gomoku. In a Maker-Maker game, also known as strong positional game, the winner is the first player to claim all the elements of some winning set. In a Maker-Breaker game, also known as weak positional game, the first player, Maker, wins by claiming all the elements of a winning set, and the second player, Breaker, wins by preventing Maker from doing so. In an Enforcer-Avoider game, the first player, Enforcer, wins if the second player claims all the vertices of a winning set, and the second player, Avoider, wins otherwise.In this paper, we consider the corresponding short games, of deciding whether the first player has a winning strategy in ℓ moves from a given position in the game, and parameterize them by ℓ. The parameterized complexity of short games is known for games such as generalized chess <cit.>, generalized geography <cit.>, and pursuit-evasion games <cit.>. For , played on a hexagonal grid, the short game is and for , played on an arbitrary graph, the short game is -hard and in .When winning sets are given as arbitrary hyperedges in a hypergraph, we refer to the three game variants as Maker-Maker, Maker-Breaker, and Enforcer-Avoider, respectively. Maker-Breaker was first shown -complete by <cit.> under the name G_pos(POS DNF). A simpler proof was later given by <cit.> who also showed -completeness of Maker-Maker. To the best of our knowledge, the classical complexity of Enforcer-Avoider has not been established yet. In this paper we will show that the short game for is -complete, solving an open problem stated numerous times <cit.>, we establish that the short game for a generalization of Tic-Tac-Toe is , and we determine the parameterized complexity of the short games for Maker-Maker, Maker-Breaker, and Enforcer-Avoider. One of our main tools is a new fragment of first-order logic where universally-quantified variables only occur in inequalities and no other relations. After giving some necessary definitions in the next section, we will state our results precisely, and discuss them. The rest of the paper is devoted to the proofs of our results, with some parts deferred to the appendix, due to space constraints.§ PRELIMINARIESFinite structures.A vocabulary τ is a finite set of relation symbols, each having an associated arity. A finite structure 𝒜 over τ consists of a finite set A, called the universe, and for each R in τ a relation over A of corresponding arity. An (undirected) graph is a finite structure G=(V,E), where E is a symmetric binary relation. A hypergraph is a finite structure G=(V∪ E,𝐼𝑁), where 𝐼𝑁⊆ V × E is the incidence relation between vertices and edges. Sometimes it is more convenient to denote a hypergraph instead by a tuple G=(V,E) where E is a set of subsets of V.First-order logic.We assume a countably infinite set of variables. Atomic formulas over vocabulary τ are of the form x_1=x_2 or R(x_1,…,x_k) where R∈τ and x_1,…,x_k are variables. The classof all first-order formulas over τ consists of formulas that are constructed from atomic formulas over τ using standard Boolean connectives , ∧, ∨ as well as quantifiers ∃,∀ followed by a variable. Let φ be a first-order formula. The size of (a reasonable encoding of) φ is denoted by φ. The variables of φ that are not in the scope of a quantifier are called free variables. We denote by φ(𝒜) the set of all assignments of elements of A to the free variables of φ such that φ is satisfied. We call 𝒜 a model of φ if φ(𝒜) is not empty. The class Σ_1 contains all first-order formulas of the form ∃ x_1, …, ∃ x_k φ where φ is a quantifier free first-order formula.Parameterized complexity.The class contains all parameterized problems that can be decided by an -algorithm. An -algorithm is an algorithm with running time f(k)· n^𝒪(1), where f(·) is an arbitrary computable function that only depends on the parameter k and n is the size of the problem instance. An -reduction of a parameterized problem Π to a parameterized problem Π' is an -algorithm that transforms an instance (I, k) of Π to an instance (I', k') of Π' such that: (i) (I, k) is a yes-instance of Π if and only if (I', k') is a yes-instance of Π', and (ii) k' = g(k), where g(·) is an arbitrary computable function that only depends on k. Hardness and completeness with respect to parameterized complexity classes is defined analogously to the concepts from classical complexity theory, using -reductions. The following parameterized classes will be needed in this paper: ⊆⊆. Many parameterized complexity classes can be defined via a version of the following model checking problem. MC(Φ) Finite structure 𝒜 and formula φ∈Φ. φ. Decide whether φ(𝒜) ≠∅.In particular, the problem MC(Σ_1) is -complete and the problem MC() is -complete (see for example <cit.>).Positional games.Positional games are played by two players on a hypergraph G=(V,E). The vertex set V indicates the set of available positions, while the each hyperedge e∈ E denotes a winning configuration. For some games, the hyperedges are implicitly defined, instead of being explicitly part of the input. The two players alternatively claim unclaimed vertices of V until either all elements are claimed or one player wins. A position in a positional game is an allocation of vertices to the players, who have already claimed these vertices. The empty position is the position where no vertex is allocated to a player. The notion of winning depends on the game type. In a Maker-Maker game, the first player to claim all vertices of some hyperedge e ∈ E wins. In a Maker-Breaker game, the first player (Maker) wins if she claims all vertices of some hyperedge e∈ E. If the game ends and player 1 has not won, then the second player (Breaker) wins. In an Enforcer-Avoider game, the first player (Enforcer) wins if the second player (Avoider) claims all vertices of some hyperedge e∈ E. If the game ends and player 1 has not won, then the second player wins. A positional game is called an l-move game, if the game ends either after a player wins or both players played l moves. A winning strategy for player 1 is a move for player 1 such that for all moves of player 2 there exists a move of player 1…such that player 1 wins.§ RESULTSThe first game we consider is a Maker-Maker game that generalizes well-known games Tic-Tac-Toe, Connect6, and Gomoku (also known as Five in a Row).In , the vertices are cells of an m × n grid, each set of k aligned cells (horizontally, vertically, or diagonally) is a winning set, the first move by player 1 is to claim q vertices, and then the players alternate claim p unclaimed vertices at each turn. Tic-Tac-Toe corresponds to (3,3,3,1,1), Connect6 to (19,19,6,2,1), and Gomoku to (19,19,5,1,1). Variations with different board sizes are also common. In the problem, the input is the set of m· n vertices, an assignment of some of these vertices to the two players, the integer p, and the parameter ℓ. The winning sets corresponding to the k aligned cells are implicitly defined. The question is whether player 1 has a winning strategy from this position in at most ℓ moves. We omit q from the problem definition of since we are modeling games that advanced already past the initial moves. Our first result (proved in Section <ref>) is that is fixed-parameter tractable for parameter ℓ. (In all our results, the parameter is the number of moves, ℓ.)theoremthmconnectis . The main reason for this tractability is the rather special structure of the winning sets. It helps reducing the problem to model checking for first-order logic on locally bounded treewidth structures, which is <cit.>. A similar strategy was recently used to show that is <cit.>. The game is played on a parallelogram board paved by hexagons, each player owns two opposite sides of the parallelogram. Players alternately claim an unclaimed cell, and the first player to connect their sides with a path of connected hexagons wins the game. Note that we may view as a Maker-Breaker game: if the second player manages to disconnect the first players sides, he has created a path connecting his sides. <cit.> also considered a well-known generalization to arbitrary graphs. Thegame is played on a graph with two specified vertices s and t. The two players alternately claim an unclaimed vertex of the graph, and player 1 wins if she can connect s and t by vertices claimed by her, and player 2 wins if he can prevent player 1 from doing so. The problem has as input a graph G, two vertices s and t in G, an allocation of some of the vertices to the players, and an integer ℓ. The parameter is ℓ, and the question is whether player 1 has a winning strategy to connect s and t in ℓ moves.The problem is known to be in and was conjectured to be -complete <cit.>. In fact, is thought of as the natural home for most short games <cit.>, playing a similar role in parameterized complexity as PSPACE in classical complexity for games with polynomial length. However, <cit.> only managed to show that is -hard, leaving a complexity gap between and . Our next result is to show that is in . Thus, is in fact -complete.theoremthmhexis -complete. Our main tool is a new fragment of first-order logic for which model-checking on arbitrary relational structures is -complete parameterized by the length of the formula. This fragment, which we call , is the fragment of first-order logic where universally-quantified variables appear only in inequalities.theoremthmcustomizedis -complete. This result is proved by reducing a formula in to a formula in . The logic makes it convenient to express short games where we can express that player 1 can reach a certain configuration without being blocked by player 2, no matter what configurations player 2 reaches. This is indeed the case for , where we are merely interested in knowing if player 1 can connect s and t without being blocked by player 2.More generally, this is the case for , where the input is a hypergraph G=(V,E), a position, and an integer ℓ, and the question is whether player 1 has a winning strategy to claim all the vertices of some hyperedge in ℓ moves.theoremthmmbis -complete. The fact that is -complete and -complete (and not -complete) may challenge the intuition one has on alternation. Looking at the classical complexity (-completeness), it seems that both players have comparable expressivity and impact over the game. As the game length is polynomially bounded, if the outcome could be determined by only guessing a sequence of moves from one player, then the problem would lie in .Now from the parameterized complexity standpoint, is equivalent under FPT reductions to guessing the k vertices of a clique (as in the seminal -complete k-Clique problem); no alternation there.Those considerations may explain why it was difficult to believe that is not -complete as conjectured repeatedly <cit.>.This is also in contrast to , where the input is a hypergraph G=(V,E), a position, and an integer ℓ, and the question is whether player 1 has a strategy to be the first player claiming all the vertices of some hyperedge in ℓ moves.theoremthmmmis -complete. For the remaining type of positional games, the problem has as input a hypergraph G=(V,E), a position, and an integer ℓ, and the question is whether player 1 has a strategy to claim ℓ vertices that forces player 2 to complete a hyperedge. Again, player 1 can only block some moves of player 2, and the winning condition for player 1 can be expressed in .theoremthmeais -complete. Our results suggest that a structured board may suggest that a positional game is , but otherwise, the complexity depends on how the winning condition for player 1 can be expressed. If it only depends on what positions player 1 has reached, our results suggest that the problem is -complete, but when the winning condition for player 1 also depends on the position player 2 has reached, the game is probably -complete.§ ISGraph G represents an m × n board in the following sense. Every board cell is represented by a vertex. Horizontal, vertical and diagonal neighbouring cells are connected via an edge. Vertex sets V_1 and V_2 represent the vertices already occupied by Player 1 and Player 2. While integer p, the number of stones to be placed during a move, is part of the input, we restrict it to values below constant k as games with p≥ k are trivial.A graph G=(V,E) representing an m × n board, occupied vertices V_1, V_2 ⊆ V, and integer p and l. l. Decide whether Player 1 has a winning strategy with at most l moves. *We reduce to first-order model checking MC() on a bounded local treewidth structure. Using a result by Frick and Grohe <cit.>, it follows that is . Let (G, V_1, V_2, p, l) be an instance of , where G = (V,E). We construct instance (𝒜,φ) of MC() as follows. Let 𝐸𝐷𝐺𝐸 be a binary relation symbol and let 𝑉1 and 𝑉2 be unary relation symbols. Then 𝒜 is the {𝐸𝐷𝐺𝐸,𝑉1,𝑉2}-structure (V,𝐸𝐷𝐺𝐸^𝒜,𝑉1^𝒜,𝑉2^𝒜) with 𝐸𝐷𝐺𝐸^𝒜 E, 𝑉1^𝒜 V_1, and 𝑉2^𝒜 V_2. FO-formula φ is defined as φ≡∃ x_1^1 ∃ x_1^2 …∃ x_1^p ∀ y_1^1 …∀ y_1^p ∃ x_2^1 …∃ x_2^p ∀ y_2^1 …∃ x_l^p ∃ u_1 ∃ u_2 …∃ u_k ∀ v_1 ∀ v_2 …∀ v_k ψ, ψ≡⋁_i = 0^l [𝑙𝑒𝑔𝑎𝑙𝑃1_i(x_1^1,…,x_1^p,y_1^1,…,x_l^p) ( 𝑙𝑒𝑔𝑎𝑙𝑃2_i(x_1^1,…,x_1^p,y_1^1,…,x_l^p) ∨( 𝑐𝑜𝑛𝑓𝑖𝑔𝑃1_i(x_1^1,…,x_l^p,u_1,…,u_k) ⋀_j = 1^k - 2u_j,u_j + 1,u_j + 2( 𝑐𝑜𝑛𝑓𝑖𝑔𝑃2_i(y_1^1,…,y_l^p,v_1,…,v_k) ⋀_j = 1^k - 2v_j,v_j + 1,v_j + 2) ) ) ] u,v,w≡𝐸𝐷𝐺𝐸(u,v) 𝐸𝐷𝐺𝐸(v,w), u,v,w≡∃ x ∃ yu,v,wu,x,wu,y,wx,v,y∀ z [ ( z ≠ vz ≠ xz ≠ y ) →u,z,w], u,v,w≡u,v,w∀ x [ x ≠ v u,x,w], u,v,w≡u,v,wu,v,w. Variables x_i^j represent the jth stone in Player 1's ith move and variables y_i^j represent the jth stone in Player 2's ith move. The sequences u_1 … u_k and v_1 … v_k represent possible winning configurations for Player 1 and Player 2. The overall structure of ψ is the following. The first disjunction ranging from i=0 to i=l represents the number of moves Player 1 needs to win the game. We then ensure that the x variables represent legal moves by Player 1. Further, either variables y do not represent legal moves by Player 2, or Player 1 achieved a winning configuration. For the latter, we assure that variables u represent aligned vertices occupied by Player 1. Finally, we check that Player 2 did not achieve a winning configuration before, that is vertices v do not represent aligned vertices occupied by Player 2.Formula u,v,w expresses that there is a path of length 2 between vertices u and w via v (𝑐𝑜𝑛𝑓𝑖𝑔𝑃1_i and 𝑐𝑜𝑛𝑓𝑖𝑔𝑃2_i ensure that the arguments are disjoint vertices). Formula u,v,w expresses that vertices u, v, and w are aligned horizontally or vertically in this order. A case analysis shows that u, v and w are horizontally or vertically aligned if and only if there are exactly three nodes at distance 1 of u and w, and that v is in the middle of the other two. In case u, v and w are located on one of the border lines of the board, there are exactly two nodes at distance 1. Formula u,v,w expresses that vertices u, v, and w are diagonally aligned in this order. This is the case if there exists no other length 2 path between u and w. Formula u,v,w expresses that vertices u, v, and w are aligned (in that order). Formula 𝑙𝑒𝑔𝑎𝑙𝑃1_i (see Appendix <ref>) ensures that variables x_i^j represent legal moves of Player 1, that is vertices not contained in V_1 or V_2 or previously played vertices. Analogously, 𝑙𝑒𝑔𝑎𝑙𝑃2_i ensures that variables y_i^j represent legal moves of Player 2. Formula, 𝑐𝑜𝑛𝑓𝑖𝑔𝑃1_i (see Appendix <ref>) expresses that variables u_1,…,u_k form a valid configuration of exactly k vertices out of the set of V_1 or vertices played by Player 1. Analogously, 𝑐𝑜𝑛𝑓𝑖𝑔𝑃2_i states that variables v_1,…,v_k form a valid configuration of exactly k vertices out of the set of V_2 or vertices played by Player 2.The size of φ is polynomial in l, k, and p. Since k is a constant and p is bounded by k, we have an FO formula polynomial in our parameter l. Graph G represents a grid with diagonals. Hence, G has maximum degree 8. It follows from Seese <cit.> that Short Connect is .§ IS -COMPLETEThe class contains all first-order formulas of the form Q_1 x_1 Q_2 x_2 Q_3 x_3 … Q_k x_k φ, with Q_i ∈{∀ , ∃} and φ being a quantifier free first-order formula such that every ∀-quantified variable x_i only occurs in inequalities, that is in relations of the form x_i ≠ x_j for some variable x_j. Furthermore, φ does not contain any other variables besides x_1,…,x_k.*Hardness: Every Σ_1 formula is contained in the class . Hence, -hardness follows from -completeness of . Membership: By reduction to . Let (𝒜,φ) be an instance of . If φ contains only existential quantifiers then (𝒜,φ) is already an instance of . Hence, let φ = Q_1 x_1 Q_2 x_2 … Q_i-1 x_i-1∀ x_i ∃ x_i+1∃ x_i+2…∃ x_k ψ with Q_j ∈{∀,∃} for 1 ≤ j < i, ψ is in negation normal form and φ=l. That is, x_i is the rightmost of the universal quantified variables.In order to reduce (𝒜,φ) to an instance of , we need a way to remove all universal quantifications. We will show how to eliminate the universal quantification of x_i. This technique can then be used to iteratively eliminate all the universal quantifiers. Let φ_1(x_1,…,x_i-1) be the subformula φ_1(x_1,…,x_i-1) = ∀ x_i∃ x_i+1…∃ x_k ψ. We will show that we can replace φ_1(x_1,…,x_i-1) byφ_2(x_1,…,x_i-1) =∃ y_i∃ y_i+1…∃ y_k( ψ[y_i/x_i,y_i+1/x_i+1, … ,y_k/x_k] ⋀_j = 1^i-1∃ y_i+1^j ∃ y_i+2^j …∃ y_k^jψ[x_j/x_i, y_i+1^j/x_i+1, y_i+2^j/x_i+2, … , y_k^j/x_k] ⋀_j = i + 1^k ∃ y_i+1^j ∃ y_i+2^j …∃ y_k^jψ[y_j/x_i, y_i+1^j/x_i+1, y_i+2^j/x_i+2, … , y_k^j/x_k] ). This reduction is an -reduction, since the size of formula φ_2 is a function of the size of formula φ_1. Let c_1,…,c_i-1 be arbitrary but fixed elements of the universe A of 𝒜. We will show that φ_1(x_1,…,x_i-1) ≡φ_2(x_1,…,x_i-1) by proving (a) φ_1(c_1,…,c_i-1) →φ_2(c_1,…,c_i-1) and (b) φ_2(c_1,…,c_i-1) →φ_1(c_1,…,c_i-1). For (a) assume that φ_1(c_1,…,c_i-1) is true. This means, φ_1[c_i/x_i] is true for all c_i∈ A, that is for all c_i∈ A there exists an assignment to x_i+1,…,x_k such that ψ is true. Part (1) of φ_2(c_1,…,c_i-1) asks for some c_i ∈ A such that there exists an assignment to x_i+1,…,x_k such that ψ is true. Part (2) asks for the existence of an assignment to x_i+1,…,x_k such that ψ is true for each of the cases where x_i is one of the elements c_1,…,c_i-1. Part (3) asks for the existence of an assignment to x_i+1,…,x_k such that ψ is true for each of the cases where x_i is one of the elements that are assigned to x_i+1,…,x_k in the model of Part (1). All these are special cases of the universal quantification over x_i, hence φ_2(c_1,…,c_i-1) is true. For direction (b) assume towards a contradiction that φ_1(c_1,…,c_i-1) is false and that φ_2(c_1,…,c_i-1) is true. Since φ_1 is false, there exists c_i ∈ A such that φ_1[c_i/x_i] is false. We perform a case distinction on the value c_i. First let c_i = c_j for some j ∈{1,…,i-1}. Then let c_i+1,…,c_k be the assignments to variables y_i+1^j,…,y_k^j in the model of φ_2. The jth conjunct of Part (2) of φ_2 states that ψ holds for x_i=x_j using the assignment c_i+1,…,c_k. Hence, assigning c_i+1,…,c_k to variables x_i+1,…,x_k in φ_1 is a model for φ_1[c_i/x_i], which contradicts our assumption. As the next case, let c_i+1,…,c_k be the assignment to variables y_i+1,…,y_k in the model of φ_2 and let c_i = c_j for some j∈{i+1,…,k}. Let c_i+1',…,c_k' be the assignments to variables y_i+1^j,…,y_k^j in the model of φ_2. The conjunct with index j of Part (3) of φ_2 states that ψ holds for x_i=x_j=c_j using the assignment c_i+1',…,c_k'. Hence, assigning c_i+1',…,c_k' to variables x_i+1,…,x_k in φ_1 is a model for φ_1[c_i/x_i], which contradicts our assumption. For the last case, let c_i be one of the remaining values. Let l_1,…,l_m be all the literals in ψ that contain x_i. All of them are inequalities of the form x_i ≠ x_j for j ≠ i. Let c_i' be the assignment to y_i in the model of φ_2. Let l_1',…,l_m' be the literals in ψ[y_i/x_i,y_i+1/x_i+1, … ,y_k/x_k] in Part (1) of φ_2 that correspond to l_1,…,l_m. We have no knowledge about the truth value of these literals l_j' with 1 ≤ j ≤ m, but all of the literals l_j in ψ evaluate to true when assigning c_i+1,…,c_k to the variables x_i+1,…,x_k. Since ψ is in negation normal form and the literals l_1,…,l_m never occur in unnegated form, that is as equalities, changing the truth value of these literal from false to true will never result in changing the truth value of the whole formula from true to false. But since c_i' together with c_i+1,…,c_k is a model of Part (1) of φ_2, it holds that for all values of c_i that we consider in this case, that φ_1[c_i/x_i] is true, which contradicts our assumption. This completes the case distinction and we have φ_1(x_1,…,x_i-1) ≡φ_2(x_1,…,x_2).§ IS -COMPLETEGraph G=(V,E), vertices s,t∈ V, vertex sets V_1,V_2⊆ V with V_1∩ V_2 = ∅, and integer l. l. Decide whether Player 1 has a winning strategy with at most l moves in the generalized Hex game (G,s,t,V_1,V_2).A generalized Hex game (G,s,t,V_1,V_2) is a positional game (V',E'), where the positions V' and the winning configurations E' are defined as follows. Set V' contains all vertices of G, that is V' = V. Set E' contains a set of vertices {v_1,…,v_k} if and only if {v_1,…,v_k}∪{s,t} form an s-t path in G. Additionally, vertices in V_1 and V_2 are already claimed by player 1 and player 2, respectively. Since the set of winning configurations of is only defined implicitly, the input size of can be exponential smaller than the number of winning configurations.*Hardness is already known <cit.>. For membership, we reduce to . Let (G,s,t,V_1,V_2,l) be an instance of , where G = (V,E). Claimed vertices V_1 and V_2 can be preprocessed: (i) every v∈ V_1 and its incident edges are removed from G and the neighbourhood of v is turned into a clique; (ii) every v∈ V_2 and its incident edges are removed from G. Hence, w.l.o.g. we assume that V_1=V_2=∅. We construct an instance (𝒜,φ) of as follows. Let 𝐸𝐷𝐺𝐸 be a binary relation symbol and let S and T be unary relation symbols. Then 𝒜 is the {𝐸𝐷𝐺𝐸,S,T}-structure (V,𝐸𝐷𝐺𝐸^𝒜,S^𝒜,T^𝒜) with 𝐸𝐷𝐺𝐸^𝒜 E, S^𝒜{s}, and T^𝒜{t}. The -formula φ is defined as φ = ∃ s ∃ t ∃ x_1 ∀ y_1 ∃ x_2 ∀ y_2 …∀ y_l-1∃ x_l ∃ z_1 ∃ z_2 …∃ z_l ψ, withψ≡S(s) ∧T(t) ∧( 𝐸𝐷𝐺𝐸(s, t) ⋁_i=1^l ⋁_j=1^i ( 𝐸𝐷𝐺𝐸(s,z_1) 𝐸𝐷𝐺𝐸(z_j,t)𝑝𝑎𝑡ℎ_i,j(x_1,…,x_i,z_1,…,z_j) 𝑑𝑖𝑓𝑓_i(x_1,y_1,…,y_i-1,x_i) ) ),𝑝𝑎𝑡ℎ_i,j(x_1,…,x_i,z_1,…,z_j) ≡⋀_h=1^j-1 𝐸𝐷𝐺𝐸(z_h,z_h+1) ⋀_h=1^j ⋁_k=1^i z_h = x_k, 𝑑𝑖𝑓𝑓_i(x_1,y_1,…,x_i-1,y_i-1,x_i) ≡⋀_1 ≤j < k ≤i x_j ≠x_k ⋀_1 ≤j < k ≤i y_j ≠x_k.The intuition of φ is the following. The variables x_i, y_i, and z_i represent the moves of Player 1, the moves of Player 2, and the ordered (s,t)-path induced by Player 1's moves, respectively. The variables s and t represent the vertices of the same name. Formula φ expresses that there is either a direct edge between s and t or a s-t path of length j was played. The main disjunctions (⋁) ensure that we consider wins that take up to l moves, and build s-t path of length up to l. Subformula 𝑝𝑎𝑡ℎ_i,j will be true if and only if the z variables form a path using only values of the selected values for the x variables. Subformula 𝑑𝑖𝑓𝑓_i ensures that all x variables are pairwise distinct and they are distinct from all y variables with smaller index.We have φ = 𝒪(l^4), so this is indeed an -reduction and -membership follows.§ IS -COMPLETE Hypergraph G=(V,E), vertex sets V_1,V_2⊆ V with V_1∩ V_2 = ∅, and integer l. l. Decide whether Player 1 has a winning strategy with at most l if vertices V_1 and V_2 are already claimed by Player 1 and Player 2, respectively.*For membership, we reduce to . Let (G,V_1,V_2,l) be an instance of , where G = (V,E) is a hypergraph. Claimed vertices V_1 and V_2 can be preprocessed: (i) every v∈ V_1 is removed from V and every hyperedge e∈ E; (ii) every v∈ V_2 is removed from V and every hyperedge e∈ E with v ∈ e is removed from E. Hence, w.l.o.g. we assume that V_1=V_2=∅. We construct an instance (𝒜,φ) of as follows. Let 𝐼𝑁 and 𝑆𝐼𝑍𝐸 be binary relation symbols. Then 𝒜 is the {𝐼𝑁,𝑆𝐼𝑍𝐸}-structure (V∪ E ∪{1,…,V},𝐼𝑁^𝒜,𝑆𝐼𝑍𝐸^𝒜) with 𝐼𝑁^𝒜{(x,e) | x ∈ V, e ∈ E, x ∈ e} and 𝑆𝐼𝑍𝐸^𝒜{(e,i) | e ∈ E, e=i }. Hence, the universe of 𝒜 consists of the vertices of G, an element for each hyperedge, and an element for some bounded number of integers. The -formula φ is defined as φ≡∃ x_1 ∀ y_1 …∀ y_l-1∃ x_l ∃ e ∃ z_1 ∃ z_2 …∃ z_l ψ, withψ≡⋁_1 ≤j ≤i ≤l (𝑑𝑖𝑓𝑓_i(x_1,y_1,…,x_i) 𝑆𝐼𝑍𝐸(e,j) ⋀_k = 1^j ⋁_h = 1^i z_k = x_h ⋀_1 ≤k < h ≤j z_k ≠z_h ⋀_k = 1^j 𝐼𝑁(z_k,e) ).The subformula 𝑑𝑖𝑓𝑓_i(x_1,y_1,…,x_i) refers to the subformula with same name used in the proof of Theorem <ref>. That is, it ensures that all x variables are pairwise distinct and that they are distinct from all y variables with smaller index. The intuition of φ is the following. The variables x_i and y_i represent the moves of Maker and the moves of Breaker, respectively. The variables z_i represent the vertices forming the winning configuration of Maker and e represents the hyperedge of this winning configuration. The first disjunction ensures that we consider wins that take up to l moves. The second disjunction ensures that we consider winning configurations that consist of up to i vertices. After checking that e has the correct size (𝑆𝐼𝑍𝐸(e,j)), we encode that the values of the z variables are contained in the hyperedge represented by e and that these variables are pairwise disjoint and selected among the moves of Maker (the x variables).We have φ = 𝒪(l^4), so this is indeed an -reduction and -membership follows.For hardness, we reduce to .The reduction is essentially the same as the reduction used for showing -hardness of <cit.>. The crucial observation is that the construction of <cit.> contains only a polynomial number of possible s-t paths. Hence, we can encode every such s-t-path as a unique hyperedge denoting a winning configuration in . § IS -COMPLETEHypergraph G=(V,E), vertex sets V_1,V_2⊆ V with V_1∩ V_2 = ∅, and integer l. l. Decide whether Player 1 has a winning strategy with at most l if vertices V_1 and V_2 are already claimed by Player 1 and Player 2.*For membership, we reduce to . Let (G,V_1,V_2,l) be an instance of , where G = (V,E) is a hypergraph. We construct an instance (𝒜,φ) of as follows. Let 𝑉1, 𝑉2, and 𝐸𝐷𝐺𝐸 be unary relation symbols. Let 𝐼𝑁 be a binary relation symbol. Then 𝒜 is the {𝑉1, 𝑉2, 𝐸𝐷𝐺𝐸, 𝐼𝑁}-structure (V∪ E, 𝑉1^𝒜, 𝑉2^𝒜, 𝐸𝐷𝐺𝐸^𝒜, 𝐼𝑁^𝒜) with 𝑉1^𝒜 V_1, 𝑉2^𝒜 V_2, 𝐸𝐷𝐺𝐸^𝒜 E, and 𝐼𝑁^𝒜{(x,e) | x ∈ V, e ∈ E, x ∈ e}. Hence, the universe of 𝒜 consists of the vertices and the hyperedges of G. The -formula φ is defined as φ≡∃ x_1 ∀ y_1 …∀ y_l-1∃ x_l ψ, withψ≡⋁_i = 0^l 𝑙𝑒𝑔𝑎𝑙𝑃1_i(x_1, y_1, …, x_l) ( 𝑙𝑒𝑔𝑎𝑙𝑃2_i-1(x_1, y_1, …, x_l) ∨ (𝑤𝑖𝑛𝑃1_i(x_1,y_1,…, x_l) ∧𝑤𝑖𝑛𝑃2_i-1(x_1,y_1,…, x_l) )). 𝑤𝑖𝑛𝑃1_i(x_1,y_1,…,x_l) ≡∃e ∀z 𝐸𝐷𝐺𝐸(e) ∧(𝐼𝑁(z, e) ∨𝑉1(z) ∨⋁_j = 1^i z = x_j),𝑤𝑖𝑛𝑃2_i(x_1,y_1,…,x_l) ≡∃e ∀z 𝐸𝐷𝐺𝐸(e) ∧(𝐼𝑁(z, e) ∨𝑉2(z) ∨⋁_j = 1^i z = y_j).Variable x_j represent Player 1's jth move and variable y_j represent Player 2's jth move. The first disjunction represents the number of moves i that Player 1 needs to win the game. Formula 𝑙𝑒𝑔𝑎𝑙𝑃1_i (see Appendix <ref>) ensures that variables (x_j)_1 ≤ j ≤ i represent legal moves of Player 1, that is vertices not contained in V_1 or V_2 or previously played vertices. Analogously, 𝑙𝑒𝑔𝑎𝑙𝑃2_i ensures that variables (y_j)_1 ≤ j ≤ i represent legal moves of Player 2. Formula 𝑤𝑖𝑛𝑃1_i ensures that Player 1 has won within the i first moves, that is, it has completed a hyperedge with V_1 and variables up to x_i. Analogously, 𝑤𝑖𝑛𝑃2_i ensures that Player 2 has won within the i first moves.We have φ = 𝒪(l^3) and 𝒜= 𝒪(G^2), so this is indeed an -reduction and -membership follows.For hardness, we reduce from the -complete problem on bipartite graphs. The reduction is deferred to the appendix. § IS -COMPLETE Hypergraph G=(V,E), vertex sets V_1,V_2⊆ V with V_1∩ V_2 = ∅, and integer l. l. Decide whether Player 1 has a winning strategy with at most l moves if vertices V_1 and V_2 are already claimed by Player 1 and Player 2, respectively.*We show that the co-problem of is -complete. The co-problem of is to decide whether for all strategies of Enforcer, there exists a strategy of Avoider such that during the first l moves, Avoider does not claim a hyperedge. Again, vertices V_1 and V_2 are already claimed by Enforcer and Avoider, respectively. We prove -hardness by a parameterized reduction from Independent Set and -membership by reduction to .In the -complete Independent Set problem <cit.>, the input is a graph G=(V,E) and an integer parameter k, and the question is whether G has an independent set of size k, i.e., a set of k pairwise non-adjacent vertices. We construct a positional game G'=(V',E') by replacing each vertex of G by a clique of size k+1. The vertex set V' has vertices v(1),…,v(k+1) for each vertex v∈ V, and hyperedges are E' = {{v(i),v(j)} : v∈ Vand1≤ i<j≤ k+1}∪{{u(i),v(j)} : uv∈ Eand1≤ i,j≤ k+1}. We claim that G has an independent set of size k if and only if Avoider does not claim a hyperedge in the first k moves in the positional game G' starting from the empty position, that is V_1 = V_2 = ∅. For the forward direction, suppose I={v_1,…,v_k} is an independent set of G of size k. Then, a winning strategy for Avoider is to claim an unclaimed vertex from {v_i(1),…,v_i(k+1)} at round i∈{1,…,k}. We note that Enforcer cannot claim all the vertices from {v_i(1),…,v_i(k+1)}, since there are not enough moves to do so, and Avoider does not complete a hyperedge with this strategy. On the other hand, suppose Avoider has a winning strategy in k moves. For an arbitrary play by Enforcer, let {v_1(i_1),…,v_k(i_k)} denote the vertices claimed by Player 1. Then, v_iv_j and v_iv_j∉ E for any 1≤ i<j≤ k, since Player 1 would otherwise claim all the vertices of a hyperedge. Therefore, {v_1,…,v_k} is an independent set of G of size k.For membership, we reduce to . Let (G,V_1,V_2,l) be an instance of the co-problem of where G = (V,E) is a hypergraph. First we do some preprocessing. We remove all vertices from G that are contained in V_2, that is the vertices already claimed by Avoider. If this results in an empty hyperedge, the instance is a no-instance. Otherwise, we remove all hyperedges that contain a vertex in V_1, that is the vertices already claimed by Enforcer, since Avoider will never lose via these edges anymore. Finally, we remove all vertices from G that are contained in V_1. Let G = (V,E) now refer to the outcome of this preprocessing. By construction all vertices of G are unoccupied and some vertices might not be contained in any hyperedge. If G contains less than 2 l vertices we can solve the problem via brute force in time. Hence, in what follows we assume that there are at least 2 l unoccupied vertices in G. We construct an instance (𝒜,φ) of as follows. Let 𝐸𝐷𝐺𝐸_𝑖 be a i-ary relation symbol for 1 ≤ i ≤ l. Then 𝒜 is the {𝐸𝐷𝐺𝐸_1,…,𝐸𝐷𝐺𝐸_𝑙}-structure (V,𝐸𝐷𝐺𝐸_1^𝒜,…,𝐸𝐷𝐺𝐸_𝑙^𝒜) with 𝐸𝐷𝐺𝐸_𝑖^𝒜{(v_1,…,v_i) | e ∈ E, e=i, e = {v_1,…,v_l}}, that is 𝐸𝐷𝐺𝐸_𝑖^𝒜 contains every permutation of all hyperedges of cardinality i. The -formula φ is defined asφ≡∀ y_1 ∃ x_1 ∀ y_2 ∃ x_2 …∃ x_l𝑑𝑖𝑓𝑓_l(y_1,x_1,…,x_l) ⋀_1 ≤ i ≤ l⋀_{z_1,…,z_i}⊆{x_1,…,x_l}𝐸𝐷𝐺𝐸_𝑖(z_1,…,z_i), where 𝑑𝑖𝑓𝑓_i(y_1,x_1,…,x_i) ≡⋀_1 ≤ j < k ≤ i x_j ≠ x_k ⋀_1 ≤ j ≤ k ≤ i y_j ≠ x_k.Subformula 𝑑𝑖𝑓𝑓_i(y_1,x_1,…,x_i) ensures that all x variables are pairwise distinct and they are distinct from all y variables with index less or equal theirs. The intuition of φ is the following. The variables x_i and y_i represent the moves of Avoider and the moves of Enforcer, respectively. Avoider wins if the x variables do not cover a whole hyperedge after l moves. We only have to check hyperedges of size up to l. Hence, for each cardinality i ≤ l, we check for all subsets z_1,…, z_l of the x variables that they do not form a hyperedge. Formula φ does not pose any restrictions on the y variables, that is we do not force Enforcer to pick unoccupied vertices. We call a move by Enforcer that picks an already occupied vertex cheating. To prove correctness, we need to show that whenever Enforcer has a winning strategy σ_E that involves cheating, Enforcer also has a winning strategy σ'_E without cheating. We construct σ'_E as follows. We follow strategy σ_E while σ_E does not perform a cheating move. If the next move would be a cheating move, we play a random unoccupied vertex instead and keep track of this vertex in a new set V_r. The next time we need to select a move, we construct a board state s by removing all vertices in V_r from the picks of Enforcer and query strategy σ_E on this state s. If the answer is an unoccupied vertex, we perform this move normally. If instead the answer is a previously played vertex (which might be in V_r), we play a random unoccupied vertex instead and add it to V_r. Since σ_E was a winning strategy, so is σ'_E. Hence, formula φ does not need to check if the y variables correspond to unoccupied vertices.The construction can be done by an algorithm since for each hyperedge e ∈ E of cardinality i, we create i!≤ l! entries in the 𝐸𝐷𝐺𝐸_𝑖 relation. We have φ = 𝒪(l^l), so this is indeed an reduction and -membership follows.§ CONCLUSIONWe have seen that the parameterized complexity of short positional games depends crucially on whether both players compete for achieving winning sets, or whether the game can be seen as one player aiming to achieve a winning set and the other player merely blocking the moves of the first player. Naturally, blocking moves correspond to inequalities in first-order logic, and our fragment of first-order logic therefore captures that the universal player can only block moves of the existential player. Our -completeness of has been used several times in this paper, but our transformation of formulas into formulas may have other uses. As a concrete example related to positional games, <cit.> established that is by expressing the problem as aformula, and making use of Frick and Grohe's meta-theorem <cit.>, similarly as we did in Section <ref>. This establishes that the problem is but the running time is non-elementary in l. However, we remark that theirformula is actually a formula of size polynomial in l. Our transformation gives an equivalent formula whose length is single-exponential in l, and the meta-theorem of <cit.> then gives a running time for solving that istriply-exponential in l.*Acknowledgments We thank anonymous reviewers for helpful comments and we thank Yijia Chen and Paul Hunter for bringing 's work to our attention. Serge Gaspers is the recipient of an Australian Research Council (ARC) Future Fellowship (FT140100048). Abdallah Saffidine is the recipient of an ARC DECRA Fellowship (DE150101351). 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Mathematical Structures in Computer Science, 6(6):505–526, 1996.§ SUBFORMULAS FOR THEOREM <REF> 𝑙𝑒𝑔𝑎𝑙𝑃1_i(x_1^1,…,x_1^p,y_1^1,…,x_l^p) ≡⋀_j = 1^i ⋀_t = 1^p [𝑉1(x_j^t) 𝑉2(x_j^t) ⋀_r = 1^j-1⋀_q = 1^t (x_j^t ≠ x_r^q) ⋀_q = 1^t - 1 (x_j^t ≠ x_j^q) ⋀_r = 1^j-1⋀_q = 1^t (x_j^t ≠ y_r^q) ], 𝑙𝑒𝑔𝑎𝑙𝑃2_i(x_1^1,…,x_1^p,y_1^1,…,x_l^p) ≡⋀_j = 1^i-1⋀_t = 1^p [𝑉1(y_j^t) 𝑉2(y_j^t) ⋀_r = 1^j-1⋀_q = 1^t (y_j^t ≠ y_r^q) ⋀_q = 1^t - 1 (y_j^t ≠ y_j^q) ⋀_r = 1^j-1⋀_q = 1^t (y_j^t ≠ x_r^q) ].𝑐𝑜𝑛𝑓𝑖𝑔𝑃1_i(x_1^1,…,x_l^p,u_1,…,u_k) ≡ ⋀_j = 1^k [ ( 𝑉1(u_j) ⋁_r = 1^i ⋁_q = 1^p u_j = x_r^q ) ⋀_r = 1^j-1 u_j ≠ u_r ], 𝑐𝑜𝑛𝑓𝑖𝑔𝑃2_i(y_1^1,…,y_l^p,v_1,…,v_k) ≡ ⋀_j = 1^k [ ( 𝑉2(v_j) ⋁_r = 1^i-1⋁_q = 1^p v_j = y_r^q ) ⋀_r = 1^j-1 v_j ≠ v_r ]. § SUBFORMULAS FOR THEOREM <REF> 𝑙𝑒𝑔𝑎𝑙𝑃1_i(x_1,y_1,…,x_l) ≡⋀_1 ≤j ≤i [ 𝑉1(x_j) 𝑉2(x_j) ] ∧⋀_1 ≤j < k ≤i [ x_j ≠x_k y_j ≠x_k ], 𝑙𝑒𝑔𝑎𝑙𝑃2_i(x_1,y_1,…,x_l) ≡⋀_1 ≤j ≤i [ 𝑉1(y_j) 𝑉2(y_j) ] ∧⋀_1 ≤j < k ≤i (y_j ≠y_k) ⋀_1 ≤j ≤k ≤i x_j ≠y_k. § -HARDNESS OFReduction from the -complete problem on bipartite graphs. is played by two players on a bipartite graph. Players alternate in picking a vertex that is a neighbour of the previously picked vertex of the opponent. A vertex can only be picked, if it has not already been picked during the game. A player loses if there is no legal move left for her.Bipartite graph (X ⊎ Y,F), start vertex v_0 ∈ X and integer k. k. Decide whether Player 1 has a winning strategy that needs at most k moves.From an instance B=(X ⊎ Y, F, v_0), k of , with v_0 ∈ X, we build a hypergraph G=(V, E), l of size polynomial in |B| which will be an equivalent instance.In our reduction, the hypergraph G mainly involves two distinguished vertices ∃, ∀∈ V and gadgets corresponding to vertices and edges of B. In the initial setup, the vertex ∃ is assumed to have already been claimed by Player 1 and the vertex ∀ to have already been claimed by Player 2.Our construction ensures that all the hyperedges of E contain exactly one vertex in {∃, ∀}. We thus partition the hyperedges between the ones that can make Player 1 win and the ones that can make Player 2 win.Formally, G is defined as indicated in Equations (<ref>) and (<ref>). It uses gadgets ·, ·, V^D_4(·), E^∃(·), E^∀(·), D_4^∃(·), D_4^∀(·) detailed in the rest of this section. The parameter is linearly preserved from the input parameter: l=9(k+1)+6.V = {∃, ∀}∪⋃_u ∈ Xu∪⋃_u ∈ Yu∪4∃∪4∀E = {{∀, a^v_0}}∪⋃_u ∈ X E^∃(u) ∪⋃_u ∈ Y E^∀(u) ∪4∃∪4∀ V_1 = {∃}, V_2 = {∀} §.§ TerminologyA useless 3-threat for Player 1 is a 3-threat that can be defended, and for which after the 3-threat and its defense, Player 1 has not achieved anything. Formally, the threat and its defense are two vertices which, once played, do not appear in any other hyperedges that could make one player or their opponent win. Note that those threats can be disregarded for Player 1 but not for Player 2. Indeed, Player 2 could use a series of useless 3-threats to win by delaying the game.A losing 3-threat for a player is a 3-threat that can be met with a counter-attack winning in a constant number of moves; more precisely in at most 6 moves.A living 3-threat is a non losing 3-threat; if it is for Player 1, it should in addition be non useless. §.§ Delay gadgetAs a building block of the forthcoming existential and universal gadgets, we introduce the following delay gadgets where ? ∈{∃, ∀}. If ?=∃ (resp. ?=∀), we say that the delay gadget belongs to Player 1 (resp. to Player 2). ?1S := {S ∪{?, x^S_1}, S ∪{?, x^S_2}, S ∪{?, x^S_3}} ?2S := ⋃_i,j ∈[3] {S ∪{?,x^S_i,y^S_j}} ={[ S ∪{?, x^S_1, y^S_1}, S ∪{?, x^S_2, y^S_1}, S ∪{?, x^S_3, y^S_1},; S ∪{?, x^S_1, y^S_2}, S ∪{?, x^S_2, y^S_2}, S ∪{?, x^S_3, y^S_2},;S ∪{?, x^S_1, y^S_3}, S ∪{?, x^S_2, y^S_3}, S ∪{?, x^S_3, y^S_3} ]} ?4S := ⋃_g,h,i,j ∈[3] {S ∪{?,x^S_g,y^S_h,z^S_i,t^S_j}} 1S := {x^S_1, x^S_2, x^S_3} 2S := ⋃_i ∈[3] {x^S_i,y^S_i} = {x^S_1, x^S_2, x^S_3, y^S_1, y^S_2, y^S_3}4S := ⋃_i ∈[3] {x^S_i,y^S_i,z^S_i,t^S_i} The elements x^S_i, y^S_i, z^S_i, and t^S_i (with i ∈ [3]) will only appear in the corresponding delay gadgets. For any set S, we will introduce at most one set among 1S, 1S, 2S, 2S, 4S, and 4S. This implies that existing x^S_i and y^S_i (with i ∈ [3]) are well-defined.Let δ∈{1,2,4} and S ⊆ V be a set of vertices such that δS⊆ E (resp. δS⊆ E). If all vertices of S have been claimed by Player 1 (resp. Player 2), and if no more than one vertex of δS has been claimed by the opponent, then she (resp. he) has an unstoppable δ-threat. The two statements have identical proofs by switching Player 1 and Player 2. We therefore only give a proof for a delay gadget δS. Assume that Player 1 has played all the vertices of S. Without loss of generality, assume that the vertex claimed by the opponent, if any, is x^S_1. Recall that we assume that ∃ has already been claimed by Player 1 and ∀ has been claimed by Player 2.For δ = 1, Player 1 has at least two 1-threats, playing in x^S_2 or x^S_3, and Player 2 cannot block them both. Thus, if Player 2 claims x^S_i (with i ∈ [3]), she claims x^S_j with j ≠ i ∈ [3] and wins. For δ = 2, Player 1 has several 2-threats. If Player 2 claims x^S_i (resp. y^S_i) for some i ∈ [3], Player 1 claims x^S_j (resp. y^S_j) for some j ≠ i ∈{2,3} and obtains an unstoppable 1-threat.For δ = 4, the reasoning is similar and omitted. Let δ∈{1,2,4} and S ⊆ V be a set of vertices such that δS⊆ E (resp. δS⊆ E). If Player 1 (resp. Player 2) claims all vertices in S and no more than one vertex of δS has been claimed by the opponent, then if it is that player's turn, they can force a win in δ moves unless the opponent has a δ-1-threat. If it is the opponent's turn, then Player 1 (resp. Player 2) can force a win in δ moves unless the opponent has a δ-threat.§.§ Existential vertex gadget For each vertex u ∈ X in the existential partition of the instance, we introduce in G the following hyperedges:E^∃(u) =2a^u, b^u∪2b^u, e^u∪2b^u, g^u∪⋃_v∈ N(u)1a^u, c^u_v, d^u_v∪1b^u, d^u_v, e^u∪1c^u_v, e^u, f^u∪1d^u_v, f^u, g^u∪1c^u_v, g^u, h^u∪2d^u_v, i^u_v∪2i^u_v, a^v In terms of vertices of G introduced by the gadget, each vertex u ∈ X gives rise to a set u that contains all the vertices needed by the delay sub-gadgets along with {a^u, b^u, e^u, f^u, g^u, h^u}∪⋃_v ∈ N(u){c^u_v, d^u_v, i^u_v}.Consider the gadget for an existential vertex u ∈ X such that no element of u∖{a^u} has been claimed yet. Assume that a^u has been played by Player 1 and that it is Player 2's turn. If Player 2 has no non-losing 3-threats in the whole board, then for each v ∈ N(u) such that a^v has not been claimed yet, Player 1 has a strategy σ^∃(u,v) that ensures either that Player 2 plays a^v after no more than 8 moves all of which belonging to u and that there are no non-losing 3-threats left for Player 2 in the gadget or that Player 1 wins in no more than 14 moves. We exhibit the strategy for Player 1 and show that Player 2's answers are forced to prevent Player 1 from winning. By assumption Player 2 has no non-losing 3-threats anywhere else on the board and no vertices claimed in u, so unless Player 2 play b^u, Player 1 wins in 6 moves by claiming b^u herself via Corollary <ref> applied to 2a^u, b^u. Although Player 2 has now sets of 3-threats which involve e^u and g^u, he does not have any 2-threats. Player 1 plays c^u_v which forces Player 2 to claim d^u_v by Corollary <ref> applied to 1a^u, c^u_v, d^u_v.Player 1 plays e^u which forces Player 2 to claim f^u. Player 1 plays g^u which forces Player 2 to claim h^u. Player 1 plays i^u_v. At this point, 8 moves have been played, Player 2 has no 3-threats left in the gadget, so Player 2 is forced to play a^v lest Player 1 plays a^v and wins in a total of 14 moves by Corollary <ref> applied to 2i^u_v, a^v.Since Player 1 has claimed e^u, g^u, and i^u_v, the only local hyperedges remaining for Player 2 are 2d^u_w, i^u_w for w≠ v, and none of them feature a 3-threat. Consider the gadget for an existential vertex u ∈ X such that no element of u∖{a^u} has been claimed yet. Assume that for any vertex v ∈ Y, a^v has not been claimed by Player 1. Assume that a^u has been played by Player 1 and that it is Player 2's turn. If Player 1 has no living 3-threats elsewhere on the board, then Player 2 has a strategy σ^∀(u) that ensures either 1) that after no less than 8 moves, all of which either belong to u or are not in any live existential hyperedge, Player 2 plays a^v for some v and there are no living 3-threats left for Player 1, or it is Player 2's turn and there is no living 3-threat for Player 1; or 2) that Player 2 wins. We exhibit a local strategy for Player 2, any move by Player 1 in a non-living 3-threat elsewhere on the board is responded to accordingly. Player 2 plays b^u creating sets of 3-threats in 2b^u, e^u and 2b^u, g^u. Playing either of e^u and g^u is losing for Player 1 because Player 2 can play in the other vertex. Therefore, Player 1 needs to play in a 3-threat to avoid losing. Notwithstanding the non-living 3-threats, the only 3-threats for Player 1 can be found in the gadgets 1a^u, c^u_v, d^u_v for v ∈ N(u).As long as Player 1 plays in d^u_w for some w, Player 2 replies in the corresponding c^u_w voiding the threat. As soon as Player 1 plays a move other than d^u_v in 1a^u, c^u_v, d^u_v for some v, Player 2 can answer d^u_v, voiding the threat, and play proceeds as follows. Player 1 has no 2-threats and so replying e^u is forced to avoid losing via Corollary <ref> applied to 1b^u,d^u_v,e^u. Player 2 plays f^u which forces Player 1 to claim g^u. Player 2 plays h^u threatening to play i^u_v. Therefore, Player 1 needs to either play in 3-threats via the gadgets 1a^u, c^u_w, d^u_w for some w ∈ N(u) such that d^u_w has not been claimed yet, or Player 1 has to play in i^u_v herself. As long as Player 1 plays in d^u_w for some w, Player 2 replies in the corresponding c^u_w voiding the threat.Eventually, Player 1 has to play in i^u_v. If a^v has already been claimed by Player 2, then Player 2 is left with no 3-threat to defend. Otherwise, Player 2 plays a^v. §.§ Universal vertex gadgetFor each u ∈ Y, we introduce in G the following hyperedges: E^∀(u) =2a^u,b^u∪2b^u, g^u∪2b^u, i^u∪⋃_v∈ N(u)1a^u, c^u_v, d^u_v∪1b^u, d^u_v, e^u_v∪1c^u_v, e^u_v, f^u∪1b^u, c^u_v, j^u_v∪1d^u_v, j^u_v, f^u∪1c^u_v, f^u, g^u∪1d^u_v, f^u, g^u∪1e^u_v, g^u, h^u∪1f^u, h^u, i^u∪2e^u_v, a^v In terms of vertices of G introduced by the gadget, each vertex u ∈ Y gives rise to a set u that contains all the vertices needed by the delay sub-gadgets along with {a^u, b^u, f^u, g^u, h^u, i^u}∪⋃_v ∈ N(u){c^u_v, d^u_v, e^u_v, j^u_v}.We observe that the only shared vertices between the different existential and universal gadgets are a^u for u ∈ X ∪ Y. For instance, in the universal gadget, each a^v with v ∈ N(u) is the “starting vertex” of the existential gadget encoding the vertex v ∈ X.Consider the gadget for a universal vertex u ∈ Y such that no element of u∖{a^u} has been claimed yet. Assume that for any vertex v ∈ X, a^v has not been claimed by Player 2. Assume that a^u has been played by Player 2 and that it is Player 1's turn.If Player 2 has no non-losing 3-threats elsewhere on the board, then Player 1 has a strategy σ^∃(u) that ensures either 1) that after no more than 8 moves, all of which belong to u, Player 1 plays a^v for some v and there are no non-losing 3-threats left for Player 2 or it is Player 1's turn and there are no non-losing 3-threats for Player 2; or 2) that Player 1 wins in no more than 14 moves. We exhibit a local strategy for Player 1. Player 1 plays b^u creating sets of 3-threats in 2b^u, g^u and 2b^u, i^u. Claiming either of g^u and i^u is losing for Player 2 because Player 1 can play in the other vertex. Therefore, Player 2 needs to play in a 3-threat to avoid losing. The only non-losing 3-threats for Player 2 can be found in the gadget 1a^u, c^u_v, d^u_v for v ∈ N(u).If Player 2 claims d^u_v, Player 1 plays c^u_v, forcing Player 2 to claim j^u_v. Player 1 plays f^u, forcing Player 2 to claim g^u. At this point, Player 1 can play i^u and win by Corollary <ref> applied to 2b^u,i^u.If instead of d^u_v Player 2 starts by claiming c^u_v, then Player 1 plays d^u_v, forcing Player 2 to claim e^u_v. Player 1 plays f^u, forcing Player 2 to claim g^u. Player 1 plays h^u, forcing Player 2 to claim i^u. If a^v has already been claimed by Player 1, then Player 1 is left with no 3-threat to defend. Otherwise, Player 1 plays a^v. Consider the gadget for a universal vertex u ∈ Y such that no element of u∖{a^u} has been claimed yet. Assume that a^u has been played by Player 2 and that it is Player 1's turn. If Player 1 has no living 3-threats on the whole board, then for each v ∈ N(u) such that a^v has not been claimed yet, Player 2 has a strategy σ^∀(u,v) that ensures either that Player 1 plays a^v after no less than 8 moves all of which either belong to u or are not in any live existential hyperedge and that there are no living 3-threats left for Player 1 in the gadget; or that Player 2 wins.We exhibit the strategy for Player 2 and show that Player 1's answers are forced to prevent Player 2 from winning. By assumption Player 1 has no living 3-threats anywhere else on the board and no vertices claimed in u, so unless Player 1 plays b^u, Player 2 wins in 6 moves by claiming b^u himself via Corollary <ref> applied to 2a^u, b^u. Although Player 1 has now sets of 3-threats which involve e^u and g^u, she does not have any 2-threats. Player 2 plays c^u_v which forces Player 1 to claim d^u_v by Corollary <ref> applied to 1a^u, c^u_v, d^u_v.Player 2 plays e^u which forces Player 1 to claim f^u. Player 2 plays g^u which forces Player 1 to claim h^u. Player 2 plays i^u. At this point, 8 moves have been played, Player 1 has no 3-threats left in the gadget, so Player 1 is forced to play a^v lest Player 2 plays a^v and wins in a total of 14 moves by Corollary <ref> applied to 2e^u_v, a^v.Since Player 2 has claimed g^u and i^u, the only local hyperedges remaining for Player 1 are in 2b^u, d^u_w, e^u_w and 2b^u, c^u_w, j^u_w for w≠ v, and none of them is feature a living 3-threat.§.§ Correctness of the reduction To show that YES instances are mapped to YES instances and that NO instances are mapped onto NO instances, we prove that any Player 1 winning strategy in gives rise to a winning strategy for Player 1 in the corresponding instance, and conversely for Player 2 winning/delaying strategies.Assume that Player 1 can ensure a win within k moves in with strategy τ, and let us construct a strategy σ ensuring a Player 1 win within l moves in . After Player 1 starts with move a^v_0, we use τ, Lemma <ref>, and Lemma <ref> to create σ such that whenever τ prescribes that the token moves from a vertex u ∈ X to v ∈ Y, we use σ^∃(u,v) to leave the u-gadget and enter the v-gadget. When the game enters a u-gadget with u ∈ Y, we use σ^∃(u) to select moves in the gadget until the u-gadget is left and enters a v-gadget with v ∈ X. If a^v is already claimed by Player 1, then Player 2 has no non-losing threats and Player 1 can enter the 4∃ gadget and win by Corollary <ref>. Otherwise, we then update the game with Player 2 moving the token to v. Eventually, the game reaches a vertex u ∈ Y such that all neighbors have been visited before and the game ends. In the instance, Player 1 follows σ^∃(u) and then wins by entering the 4∃ gadget. If τ guarantees that at most k' ≤ k moves are played before Player 1 wins, then σ guarantees that at most 9(k'+1)+6 ≤ l moves are played before Player 1 wins.In the case of a NO instance, Player 2 has a strategy τ such that either Player 2 wins, or the game goes for longer than k moves. A corresponding strategy σ can be derived such that either Player 2 wins in the game, or the game goes for longer than 9(k+1)+6 = l moves. The construction is dual to the one above and relies on Lemma <ref> and Lemma <ref>. | http://arxiv.org/abs/1704.08536v1 | {
"authors": [
"Édouard Bonnet",
"Serge Gaspers",
"Antonin Lambilliotte",
"Stefan Rümmele",
"Abdallah Saffidine"
],
"categories": [
"cs.CC",
"F.2.2"
],
"primary_category": "cs.CC",
"published": "20170427124254",
"title": "The Parameterized Complexity of Positional Games"
} |
Experimental Two-dimensional Quantum Walk on a Photonic Chip Hao Tang^1,2, Xiao-Feng Lin^1,2, Zhen Feng^1,2, Jing-Yuan Chen^1, Jun Gao^1,2, Ke Sun^1, Chao-Yue Wang^1, Peng-Cheng Lai^1, Xiao-Yun Xu^1,2, Yao Wang^1,2, Lu-Feng Qiao^1,2, Ai-Lin Yang^1,2 and Xian-Min Jin^* December 30, 2023 ===================================================================================================================================================================================================================empty § INTRODUCTIONThe possibility to retrieve the shape and constitutive parameters of amedium from its scattered field is known asdetection and/or imaging problem and it is related to proper solutions of an electromagnetic inverse scattering problem (ISP) <cit.>. In this respect, the opposite task is avoiding such detection by hiding scatterers from external observers: this is known in the literature as cloaking and can occur, for example, exploiting plasmonic and metamaterial coatings <cit.>.Negative, or less than unity, ε (ENG or ENZ) and μ (MNG or MNZ) materials <cit.> serve to induce cancellation effects on scattered fields as in Plasmonic Cloaking(PC) <cit.> or to reroute the impinging waves as in Transformation Optics (TO) <cit.>. The common idea is acting on the internal field within or through the hidden region without perturbing the surrounding space.However, PC shows two main limitations: it requires plasmonic materials and formulas are valid only under quasi-static regime. For these reasons, the method is not well suited for objects that are large or comparable with respect to the impinging wavelength, especially in the radiofrequency and microwave bands where ENG and ENZ materials are not seldom available. On the other hand, by construction, TO requires inhomogeneous and strongly anisotropic dielectric and magnetic tensors to practically manufacture the cloak. Indeed, the quest for an exact, object-independent, invisibility coating is paid by enormous complications (if not impossibility) of the desired cloak <cit.>.Moreover, the possibility to achieve cloaking via PC or TO in actual applications is severely limited by the difficulty of considering the influence of the impinging wave (coming from a scanned array for example) on devices with a low scattering response quite sensitive to the working frequency and direction of arrival (DoA) of the incident field. In this paper, we address the synthesis of dielectric covers able to achieve null or very low values of the scattered fields (cloaking effect) wherein only natural dielectric materials (i.e., without negative or near-zero values) without magnetic properties (μ=μ_0) are considered. In the relevant literature, non-magnetic cloaks have been investigated for near-perfect invisibility <cit.> via TO. The useof only natural dielectric materials have been addressed by means of topology optimization approaches <cit.> at optical frequencies in the case of metallic objects, requiring extremely variable covers to induce cloaking exclusively for a narrow angular range of DoAs. Other all-ordinary dielectric cloaks have been proposed via global optimization techniques for radially cylindrical and spherical cloaks of metallic targets <cit.>. These approaches, although very attractive in terms of simplicity of the geometrical architecture of the covers, require very large refractive indexes <cit.>, rarely available in nature, or even near-zero dielectric constants <cit.>, moreover they achieve only nearly optimal performance in terms of residual scattering radiation <cit.>. Another optimization approach based on the phase field method has been recently proposed <cit.> for cloaking metallic cylinder, by considering six angular directions of the incident wave. Finally, graded refractive index structures have been proposed for surface cloaking <cit.>.Differently from the above contributions mostly concerned with cloaking metallic objects and surface, in this paper, we investigate through recent analytic <cit.> and numerical methods <cit.>, the conditions to pursue non-magnetic volumetric dielectric cloaking with natural permittivity materials (ε > ε_0) to cloak dielectric objects. In this respect, we adopt a design procedure based on the solution of the inverse scattering problem <cit.> exploiting a gradient based local optimization approach which allows to easily handle many of the design constrains concerned with the cloaking problem.A physical insight into artificial-natural and natural-natural dielectric cloaking systems is also given. The presented methodology is mainly focused on investigations against volumetric dielectric scatterers, with a possible extension towards composite structures such as metal-dielectric metasurfaces <cit.>. § METHODS: ANALYTIC AND NUMERICAL CLOAK SYNTHESIS PROCEDURES We consider a 2D domain Ω embedding one (or more) penetrable non-magnetic homogeneous object(s) of arbitrary shape with support Σ_1, see Fig.<ref>. A cloak region Σ_2 such that Σ=Σ_1 ∪Σ_2 is considered, and, for the sake of simplicity, the intersection between Σ_1 and Σ_2 form a null set. However, the analysis developed in the following is valid even if this hypothesis is removed.Let us assume the Ω domain in the xy plane, and one (or more) plane wave(s)E_i( r,θ_ν, ω)=e^- j k (ω,θ_ν) ·r ẑ with unitary amplitude and electric field polarized along the ẑ axis (TM polarization) impinging towards the center of Ω, wherein r=(x,y) denotes the vector position in Ω, ω=2π f the angular frequency and θ_ν the DoA. Throughout the paper, the time harmonic factor e^jω t is assumed and dropped.For the sake of clarity, the dielectric properties of the overall region are grouped in:ε(r)= {[ ε_1(r),r∈Σ_1(object region); ε_2(r),r∈Σ_2 (cloak region);ε_b,r∈Γ (observation region) ] . with Γ≡Ω/Σ the observation region where no “visible” scattering effects by the cloaking system would be desired and where the constitutive parameters are homogeneous (i.e., ε_b does not depend on r in Γ).From now on, arrow signatures on fields are suppressed, tacitly assumed all vectors being directed along ẑ.The electromagnetic scattering from such a cylindrical (i.e., longitudinal invariant) structure is due to the equivalent volumetric sources with support Σ, defined as J_eq(r,θ_ν, ω)= j ωε_bJ(r,θ_ν, ω) where J(r,θ_ν, ω)= χ(r)[ E_i(r,θ_ν, ω) + E_s(r,θ_ν, ω) ] In Eq. (<ref>), J(·)is the so called contrast source given bythe product between total internal fields E_t(r,·)≡ E_i(r,·)+E_s(r,·) and χ(r) is the contrast function defined as: χ(r)=ε(r)-ε_b/ε_b. The total field E_t, expressed in the whole region Ω as the sum of the incident (or primary) and scattered (or secondary) field, can be conveniently expressed via integral formulation as <cit.>: E_t(r,θ_ν,ω)=E_i(r,θ_ν, ω) +k_b^2 ∬_Ω J(r',θ_ν, ω) G(r,r', ω)dr' where k_b=ω√(μ_bε_b) (recalling that everywhere μ_b=μ_0). Moreover G(r,r',ω) is the 2D Green's function of the homogeneous background which has exact analytic form in a homogeneous background, namely the Hankel function of zero order and second kind<cit.>. According to the partition in Eq.(<ref>), by definition, the contrast function is zero in the observation domain Γ≡Ω/Σ and it is mostly non-zero elsewhere. Let us also notice that, by definition, if vacuum is assumed as background medium (i.e., ε_b=ε_0), the contrast function is coincident with the electric susceptibility. Eq.(<ref>) states that the physical cause of the scattering phenomenon is the contrast source J induced in Σ. It may be convenientto express the scattered field through a more compact notation with respect to Σ and Γ domains, as: E_t(r,θ_ν, ω)-E_i(r,θ_ν, ω)= 𝒜_Σ[J]E_s(r,θ_ν, ω)=𝒜_Γ[J] Eqs. (<ref>) and (<ref>) can be identified as the object and data integral equations of the ISP, respectively. From a physical point of view, the operators 𝒜_Σ : L^2(Σ) → L^2(Σ)and 𝒜_Γ : L^2(Σ) →L^2(Γ) map the contrast source into the corresponding scattered field in Σ and in Γ, respectively.Adopting this formulation, the cloaking effect can be pursued by enforcingE_s(r, θ_ν, ω)=0 ∀.In the following, solutions of Eqs. (<ref>)-(<ref>) are pursued to achieve cloaking in and beyond the quasi-static regime, using both analytic and numerical approaches. §.§ Cloaking in quasi-static regimeWhen Eq. (<ref>) is assumed as desired specification of any cloaked system, the data equation (<ref>) can be explicited as: ∬_Σ J(r',θ_ν,ω) G(r,r',ω)dr' =0 . Considering the overall system enclosed in a circular cylinder of radius b, in quasi-static condition (i.e., k_b b → 0), Eq. (<ref>)can be solved in a straightforward and simple manner since the overall system is so extremely compact in terms of λ that the Green’s function of the homogeneous background becomes constant over the entire domain Σ, i.e., lim_k_b b → 0 G(r,r',ω)= lim_k_b b → 0-j4_0^(2)(k_b|r-r'|)= C since k_b|r-r'|≈ 2π b/λ in the quasi static-limit. Expliciting the contrast source according to Eq. (<ref>), also the total field can be considered to be constant in the Σ domain in the quasi-static approximation, i.e., the Rayleigh scattering regime, thus giving: ∬_Σχ(r')dr'=0. Splitting Eq.(<ref>), namely the Contrast Cloaking Equation (CCE) <cit.> overthe object region, χ_1 ∈Σ_1 and the cloak region χ_2 ∈Σ_2, a necessary and sufficient condition to achieve cloaking comes out as a proper mix of positive/negative values of the local contrast function, i.e., χ_1 Σ_1+χ_2Σ_2=0. It is interesting to notice that the CCE generalizes in a very compact fashion PC for scatterers of arbitrary shape <cit.> for a general background medium, even when ε_b ≠ε_0. Therefore, in the quasi-static limit, the designer can turn-off the effect of volumetric sources by locally compensating the positive-induced source associated to a positive contrast (e.g., the object) with the negative-induced one associated to a negative contrast (e.g., the cloak) or viceversa, regardless shape of the cloak system and DoAs of the incident field. As a matter of fact, since the total field does not play any role in this derivation (being factored out from the integral), such a kind of cloaking is expected to behave as an omni-directional cloaking, i.e., its performances do not depend from the DoA of the incident field. §.§ Cloaking beyond quasi-static regimeWhen the overall cloaking dimension is not in subwavelength condition, the two main hypothesis for deriving the CCE are no longer valid. Beyond quasi-static condition,distributed effects take place with two consequences: (i) the Green's function is no more constant and (ii) the total field changes from point to point, taking into account all the non-local contributions of the scatterers in the whole domain Σ <cit.>.However, thanks to these considerations, as also depicted in Fig. <ref>, we figure out an important finding: the cloaking mechanism can be achieved with all-positive values of the contrast provided that a proper spatial organization of the contrast layout is pursued. As a result, the cancellation effects occurring between positive-negative values of the contrast, that are very close at subwavelength scale, can be synthesized also for all-positive contrast values which are not so close in terms of wavelength (e.g., crest and through of the working wavelength). On the other hand, due to the need of specific spatial distribution of the contrast function, the cloaking effect is expected not to be broadband. As a matter of fact, when the ratio D/λ increases, D being the diameter of the minimum circle enclosing Σ, the architecture scheme of the coating plays a crucial role with the possibility to get rid of metamaterials in Σ_2through a proper arrangement of all-positive dielectric values ε_2(r)≥ε_0. On the other hand, as stressed above,according to PC <cit.>,just one homogeneous layer of metamaterial is sufficient to achieve cancellation effects in quasi-static regime.Under the above reasonings, the synthesis of cloaking profiles can be conveniently addressed as the solution of an ISP without any approximation on the mathematical model in Eqs. (<ref>)-(<ref>). This can be performed through the minimization of a proper cost function, under the constraint χ_2(r)>0. Obviously, solutions accounting for both positive and negative values of the contrast function can be anyway pursued, but their investigation is beyond the scope of the present paper. Once fixed χ_1 in Σ_1, the adopted formulation of the cost function depends on both contrast χ_2(r) ∀ r∈Σ_2 and contrast source J(r) ∀ r∈Σ. In particular, the cost function is obtained as joint minimization of the weighted objectand data equation (<ref>)-(<ref>). To this aim, it is convenient to modify theobject equation according to the contrast source formulation<cit.>, justmultiplying both members of Eq. (<ref>) by χ. Moreover, the part of functional (<ref>) relevant to the data equation has to be properly modified to take into account zero scattered field enforced in the optimization task. A possible way to address such a problem is given by the minimization of the following weighted cost function: Φ(χ,J)=∑_ν=1^N{||J^ν-χ E_i^ν-χ𝒜_Σ[J^ν]||^2||E_i^ν||^2+ ||𝒜_Γ_0 [J^ν]||^2||E_i^ν||^2} ν=1,2.., N_ν,where the apex ^ν stands for the ν-th impinging DoA and || · || denotes the usual L^2-norm. In particular, in Eq. (<ref>), the first addendum enforces, for a given set of plane waves, the solution (in the least square sense) of the scattering equation in Σ, while, interestingly, the second term stands for the minimization of the radar cross section (RCS), or the echo width <cit.> in a subset Γ_0 ∈Γ, see Fig. <ref>. In this respect, it is worth noting that the finite bandwidth of the scattered fields <cit.> allows to not enforce zero scattered field in the whole observation region Γ, since it is sufficient to enforce such a value only in a finite number of points M over a surface Γ_0. For circular cylindrical geometry (a circumference of radius R for the particular case at hand), according to <cit.>, the minimum non redundant number of sampling points M=2k_b R can be considered as angularly equispaced along the circumference Γ_0 enclosing the cloaking system.As a last comment, it is worth noting that the minimization problem in Eq. (<ref>) entails a non-quadratic form <cit.>, so that the minimization procedure may get stuck into “local minima” <cit.>. The possibility to incur in local minima is strictly related to the functional shape (depending on the data and constraints of the problem) as well as on the initial guess adopted to start the minimization procedure. In this respect, while the first set of parameters are fixed in advance by the designer, the initial guess is a “degree of freedom” of the synthesis problem, which can be conveniently exploited to obtain several equivalent (in terms of cloaking effect) solutions provided that a satisfactory weighted residual error is reached in the minimization of (<ref>). § NUMERICAL RESULTS AND ANALYSIS To keep things simple, we have considered the cloaking of a circular scatterer in free-space background (i.e., ε_b=ε_0), made up oflossless allumina (ε_s=10ε_0) with radius a=0.42 cm, see Fig. <ref>(a). However, it is worth noticing that such an approach can be easily adopted for any arbitrary (more sophisticated) shape of the scattering system (object and cover). The cloaking effect will be demonstrated, both in and beyond the quasi static-limit, adopting circular coatings with different radii. To achieve cloaking in the subwavelength condition, we set the cover's radius to b=0.6 cm (slightly larger than that of the bare object). In particular, the coating diameter 2b corresponds to 0.16λ at 4 GHz (quasi-static regime).In quasi-static condition the CCE is exploited and, following Eq. (<ref>), the contrast in Σ_2 is computed as χ_2=-χ_1 Σ_1/Σ_2, where Σ_1= π a^2 and Σ_2=π b^2-Σ_1. The resulting value is χ_2=-8.64 (i.e., ε_2=-7.64ε_0) and the overall cloaking system is reported in Fig. <ref>(b). Beyond subwavelength condition, the architecture for the natural dielectric cloak is obtained solving the optimization problem (<ref>) via conjugate gradient fast Fourier transform method (CG-FFT) <cit.> adopting a standard pixel representation for the discretization of the state and data equations, for both the scatterer and the cover ε_2(r) in Σ_2. Two different dimensions for the cloak have been considered in order to take into account the performance of the cloak with respect to its electrical dimension (compared to the operational wavelength). The discretization of the analysis domain has been fixed according to rules of the integral equation method (MoM) <cit.>. Specifically, the dimension of the pixel has been set to 0.5mm, which is slightly larger than λ/10 of the minimum wavelength (as referred to the allumina). This choice has been established as trade-off between the accuracy required by the numerical procedure and the need to deal with reasonable resolution in realizing the cloak trough solid printing techniques. In order to take into account the dependence of the cloak from the impinging directions of the incoming wave, we have considered in the optimization problem (<ref>), differentnumber of DoAs, i.e. N=2, N=4 and N=8 plane waves angularly evenly spaced on a arc of 360^∘. On the other hand, since the cloaking effects is required all around the cloak we have enforced the scattered field to be zero in M=24 or M=36 equispaced observation points located on a circumference placed in the close proximity of the cloaking system. The circumference's radius has been set to R ≃ 1.45 b, b being the different coating radius, with b=1.2 cm and b=2.4 cm (1λ and 2 λ at 25 GHz, respectively) adopted in the synthesis procedure. These different values have been considered in order to take into account the different number of degrees of freedom of the scattered field pertaining to scatterers of different extent <cit.>.According to the above reasons, we have considered a discretization grid of 48×48 pixels for the cloak with radius b=1.2 cm (1λ) and 96×96 pixels for the cloak with radius b=2.4 cm (2λ). Moreover, in the minimization procedure, constraints about natural permittivity (i.e., ε≥ε_0) and lossless (i.e., σ=0) nature of the cover have been enforced at each step of the minimization procedure. In this respect, the maximum value of the permittivity to be used in the cover region has been set to that of the allumina (i.e., 10 ε_0). It is worth noticing this constrain simplifies the fabrication of the cloak in terms of number of materials to be employed, as well as, not less important, allows to take under control, i.e. to not violate, the spatial discretization of the integral equations involved in the synthesis strategy. The synthesis procedures was stopped when the functional reached a value of Φ≤ 10^-6. This ensures that both data and state equation are solved with a satisfactory accuracy. On the other hand, when the threshold value was not reached, the procedure has been stopped when no significant changes arose in the functional value between two subsequent steps of the minimization procedure.The initial guess of the dielectric cover has been set as homogeneous in the whole Σ_2 with relative permittivity of ε_2 =3.5ε_0 or ε_2 =5.5ε_0, for N=2, 8 and N=4, respectively, in order to escape from local minima wherein the residual value of the functional (<ref>) is unsatisfactory for achieving cloaking effects. It is worth to note that these initial values have been chosen after some a posteriori checks. The results of the synthesis strategy are shown in Fig.<ref>(c)-(h), after a proper “post-quantization” procedure, which slightly adjusts the dielectric profile of the cover to permittivity's levelswith of ε_r=1, 2.5,3.5,4.5,5.5,6.5,7.5,8.5,10. Such procedure entails a simplification of the cloak manufacturing in terms of piecewise constant dielectric profile taking into account the availability of solid materials in the considered frequency range, while not substantially modifying the cloaking performance, which results only slightly worsened.Fig. <ref>(a) and <ref>(b) show the real part of the total field for the uncloaked case and the plasmonic coating at 4 GHz: in Fig. <ref>(c)-(h),the real part of the total field for all the natural dielectric cloaks at 25 GHz is reported. The analysis was performed in COMSOL importing the synthesized profile and meshing it by means of triangular shaped element.At the lower frequency, the plasmonic cover is in cloaking operation with the flat phase fronts well recognizable behind the object, whereas, as expected, at the same frequency, no scattering cancellations occur for the natural dielectric cover (not shown). At frequency of 25GHz, the dielectric cloaks are all able to achieve cloaking mechanism for the DoAs considered in the design procedure (for the sake of brevity the cloaking effect is shown only for one impinging directions). From these results it is possible to observe that the cloaking can been achieved both for the cloak with radius of 1λ and 2λ, and that the smaller the cloak and the larger the number of DoAs, the more complicated the architecture of the covers to be synthesized is. Moreover, note that not all the discrete permittivity's values reported above are required by the synthesized covers. For example, for the cloaks of Fig.<ref> (c) and (h) only four kind of dielectrics are required with a further simplification in the architecture of the cloak.In order to quantify the cloaking effect as a function of the frequency, we have calculated the scattering cross section (SCS) of the synthesized cloak, which is the average value of the RCS calculated at 72 equispaced points on a circle of radius R=1.45 bfor each given DoA (the same shown in Fig. <ref>) [The numerical values of the SCS when calculated for different DoAs do not substantially change.]. SCS(θ_ν,ω)= 2π R [ 1/M∑_m=1^M| E_s(θ_ν,ω)/E_i(θ_ν,ω)| ^2] The overall function SCS(θ_ν,ω) has been calculated for different frequency values at a fixed DoA and for different DoA values at a fixed (designed) frequency: this leads to quantify the bandwidth performance of the synthesized cloaks in terms of frequency and omnidirectionality issue. As shown in Fig. <ref>(a), in subwavelength condition, the plasmonic cover (dotted line) drastically reduces the scattering levels with respect to the uncloaked case (continuous line) and, at the same time, its cloaking response at 4 GHz is of omnidirectional kind, as shown in Fig. <ref>(b), i.e., completely flat with respect to change in the DoA. For what concerns the ordinary dielectric cloaks with radius b=1.2 cm, in Fig. <ref>(a), they all share the minimum around the designed frequency 25 GHz, but they possess very different angular responses as reported in Fig.<ref>(c), due to the fact that they have been designed for different incoming waves. In the case b=1λ, the cloaking effect is clearly affected by changing the DoA: for the 2-views ordinary dielectric cloak (upward-pointing triangle line), the performance becomes worse when non-optimal DoA is considered, whereas, as expected, two different minima (as the numbers of views for which it has been designed) appear at DoA=0,π. The 4-views ordinary dielectric cloak (dash-dot line) improves its omnidirectional performance, ensuring an overall response which is always below the uncloaked case around -14 dB (continuous line), especially when the 4 minima occur. The 8-views ordinary dielectric cloak (dashed line) shows the best omnidirectional performance, ensuring an overall response between -27 dB and -32 dB.In Fig.<ref> (b)-(d), the SCS(θ_ν,ω) is shown for the case b=2.4 cm (2λ at 25 GHz).As reported in Fig. <ref>(b), the scattering from the ordinary dielectric cloak is larger in the low frequency window with respect their compact counterpart in Fig. <ref>(a), due to the fact that now their size is increased: around 25 GHz, they show a minimum with slightly worse performance in terms of dB reduction compared to the cloaks with b=1λ in Fig. <ref>(a). Even the angular swing between optimal and worst DoA value, as reported in Fig. <ref>(d), is increased. In this b=2λ case, the cloaking effect is still affected by changing the DoA, but in a different way with respect to the previous case. The 2-views ordinary dielectric cloak (upward-pointing triangle line) has wide regions between the minima points DoA=0,π for which it scatters more than the bare object. In this case, also the behavior of the4-views ordinary dielectric cloak (dash-dot line) becomes worse, with several values of the SCS at 25 GHz above the uncloaked case (continuous line) of about +5 dB. Even the 8-views ordinary dielectric cloak (dashed line) loses its omnidirectionality for a degradation in its performances, whereas ensuring all its 8 minima around the -27 dB value. § CONCLUSION The synthesis of all dielectric cloaks has been tackled through the solution of an inverse scattering problem where zero scattered field is properly pursued with artificial and natural dielectric materials within and outside the quasi-static frequency regime, respectively. It has been found and discussed that it is not strictly necessary the use of ENZ or ENG materials for cloaking beyond quasi-static limit, completely changing the paradigm suggested by quasi-static formulas and exploring the potentiality of cloaking techniques via scattering cancellation. For this reason, natural dielectric cloaks can besynthesized in a relatively compact and easy fashion. Moreover, the behavior of the cloak is non-resonant and a non-negligible operational bandwidth (at 3 dB) can be achieved (about 15%). Interestingly, being made of only natural dielectrics substances, the cloak could be suitably manufactured by means of solid multi-filament printing techniques when the permittivity values can be possibly achieved using alternative dielectric mixtures with differing volume fractions and particle sizes <cit.>. 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"authors": [
"Loreto Di Donato",
"Tommaso Isernia",
"Giuseppe Labate",
"Ladislau Matekovits"
],
"categories": [
"physics.class-ph"
],
"primary_category": "physics.class-ph",
"published": "20170426175025",
"title": "Towards Printable Natural Dielectric Cloaks via Inverse Scattering Techniques"
} |
Crowdsensing in Opportunistic Mobile Social Networks: A Context-aware and Human-centric Approach Phuong Nguyen Department of Computer ScienceUniversity of Illinois at Urbana-Champaign [email protected] Klara Nahrstedt Department of Computer ScienceUniversity of Illinois at Urbana-Champaign [email protected] 30, 2023 ===================================================================================================================================================================================================================================== In recent years, there have been efforts to collect human contact traces during social events (e.g., conferences) using Bluetooth devices (e.g., mobile phones, iMotes).The results of these studies have enabled the ability to do the crowd-sourcing task from within the crowd, in order to answer questions, such as: what is the current density of the crowd, or how many people are attending the event?However, in those studies, the sensing devices are usually distributed and configured in a certain manner.For example, the number of devices is fixed, people register for the devices on a volunteering basis.In this paper, we treat the above problem as an optimization problem and draw the connection to the vertex cover problem in graph theory.Since finding the optimal solution for minimum vertex cover problem is NP-complete, approximation algorithms have to be used.However, we will show that the well-known approximation algorithms do not perform well with the crowd-sensing task.In this paper, we propose the notions of node observability and coverage utility score and design a new context-aware approximation algorithm to find vertex cover that is tailored for crowd-sensing task.In addition, we design human-centric bootstrapping strategies to make initial assignment of sensing devices based on meta information about the participants (e.g., interests, friendship).The motivation is to assign the sensing task to a more “socialized” device to obtain better sensing coverage. We perform comprehensive experiments on real-world data traces obtained from previous experimental studies in conference and academic social context.The results show that our proposed approach significantly outperforms the baseline approximation algorithms in terms of sensing coverage. § INTRODUCTIONCrowd-sourcing has changed the way people obtain needed services, new ideas, or content by soliciting contributions from a large group of people, and especially from an online community (e.g., Amazon Mechanical Turks).There are also efforts of using crowd-sourcing to monitor, obtain information about the crowd – which is referred to as crowd-sensing <cit.>.For example, during Olympic London 2012, organizers released a smartphone app that allowed users to upload their location information to help determine how to manage the crowds and the associated city resources.With the proliferation of mobile devices and communication technologies, it has become more possible to use technologies to understand the behavior of large crowds.For example, during social events, such as conferences, some attendants are given Bluetooth-enabled devices (e.g., mobile phones, iMotes) to collect opportunistic contacts (i.e., the Bluetooth scanning results of neighboring devices).The collected contacts, or traces, can be used to answer a variety of questions about the crowd.For example, what is the density of the crowd, how many people, or groups of people exist in the crowd (i.e., for crowd monitoring purpose), or what is the current opportunistic contact graph between devices (i.e., to understand the ability of data dissemination within the crowd).We refer to the tasks of answering these questions as the crowd-sensing tasks.There have been previous experiments on collecting opportunistic mobile traces, such as SIGCOMM'09 <cit.> or UIM <cit.>, where the collected sensing data are valuable for studying crowd-sensing tasks.However, in those experiments, the settings are usually configured in a certain manner.For example, the number of sensing devices (e.g., Bluetooth-enabled devices that periodically scan for neighboring devices) is fixed, devices are given to attendants on a volunteer basis, and the sensing interval (i.e., the interval during which opportunistic contacts are collected) is fixed.In addition, in those experiments, each sensing device does the sensing task in isolation and data from all sensing devices are only collected at the end of the experiment.As a result, such data could not help answer timely questions about the crowd.Besides, while crowd sensing task can be handled successfully, given all individual devices collecting sensing data all of the time, such approach is overly demanding of the device's resources, and not energy-efficient.In this paper, we propose a crowd-sensing model, in which the sensing nodes periodically connect to a centralized server to send the collected data and receive the instruction for the next sensing interval.In addition, we treat the number of sensing devices and the length of sensing interval as the given constraints, and try to optimize the assignment of sensing task to appropriate devices to maximize the sensing coverage.Under this scheme, we will show that the crowd-sensing task poses some similarities to sensor placement task <cit.>, where the problem is to find an optimal set of positions to place sensors in order to obtain the best coverage of the environment, given the constraint as the number of available sensors.However, crowd-sensing problem is more challenging due to the spatio-temporal and social nature of the interactions between people in the crowd.For example, the collected sensing data might be highly overlapped if people carrying sensing devices are nearby each other at the same location.In another example, since social relationships and personal interests might have influence on who people frequently meet or interact with, the centralized server should take such “out-of-band” information into its assignment of sensing task to devices. In this paper, we show the connection between the crowd-sensing problem with the vertex cover problem in graph theory.While finding the optimal solution in vertex cover is NP-complete, we show that the constrained versions of existing approximation algorithms, such as random-based approximation, or greedy approximation <cit.>, can be used to derive approximate solutions.However, since those algorithms are designed for generic graph, they do not take into account the spatio-temporal and human-centric characteristics of the crowd-sensing task and optimize the individual coverage of covering vertex instead of combined coverage, as in greedy approximation.To solve this problem, we propose a new context-aware approximation algorithm that is tailored for crowd-sensing task.Particularly, we propose the notion of node observability and coverage utility score to optimize the combined sensing coverage objective.In addition, we incorporate the out-of-band, human-centric information about the participants, such as personal interests, and social relationship, to improve the bootstrapping of the crowd-sensing task.The experimental results on real-world mobile traces show a significant improvement in sensing coverage while satisfying the optimization constraints.In summary, our contributions in this paper are as follow: * We model the crowd-sensing problem as an optimization problem and draw the connection to the vertex cover problem in graph theory.* We propose a general 2-stage framework for incorporating vertex cover approximation into crowd-sensing task.* We propose the notions of node observability and coverage utility score and design a new context-aware approximation algorithm to find vertex cover that is tailored for crowd-sensing task.* We perform comprehensive experiments on real-world data traces to verify the effectiveness of our proposed approach.The paper is organized as follow: in Section <ref>, we describe the sensing model and formally define the problem.In Section <ref> we show how to solve the crowd-sensing problem using minimum vertex cover approximation.In Section <ref>, we describe our proposal of a context-aware approximation algorithm for crowd-sensing task.We show the results of experiments on real-world datasets in Section <ref> and discuss the related work in Section <ref>.Finally, in Section <ref>, we conclude the paper and discuss some future work.§ CROWD-SENSING MODEL AND PROBLEM DEFINITION In this section, we first describe the crowd-sensing model before formally defining the problem. §.§ Crowd-sensing model Let us denote V as the set of devices (e.g., mobile phones) attending a crowd event (e.g., conference sessions, classes on campus, etc.).Among devices in V, there is a sub-set of devices V_in (represented as circles in Figure <ref>) that register as the participants of the crowd-sensing experiment – we call it as the set of internal devices.The remaining set of devices V_ex (represented as the green rectangles in Figure <ref>) does not participate in the experiment – we call it as the set of external devices (apparently, we have V = V_in∪V_ex).An sensing application is installed on each internal device that runs in background and periodically connects to a centralized server to i) send collected sensing data, and ii) receive sensing instructions.In this paper, we refer the sensing task to the task of collecting opportunistic Bluetooth contacts, or wireless contacts, in mobile ad hoc networks.As a result, we assume that every device in V has its Bluetooth in discoverable mode.In addition, for the sensing devices, they need to periodically do the scanning of neighboring Bluetooth-enabled devices and record the observed contacts.Besides, since the sensing devices also need to contact with the centralized server for sending the collected data or receiving instructions, they need to have connection to the server by longer range communications, such as Wifi, or 3G.Although, ideally, we would like to have all devices in V_in perform the sensing task, we argue that such an approach is not energy efficient and even not necessary.On one hand, performing the sensing task by scanning for neighboring Bluetooth devices is very energy-consuming.On the other hand, the sensing results of multiple sensing devices could be overlapped and thus, not all the data are helpful in covering the crowd.The overlapping situation is more serious when the people carrying the sensing devices are at the same place, or move together.As a result, we only require a subset of registered devices doing the sensing task at a time, and we consider the number of sensing devices as a given constraint.Particularly, at any point of time, an internal device can be either in sensing mode (during which, the device collects the wireless contacts of its neighboring devices), or non-sensing mode (during which, the device does not do the sensing task and waits for the sensing instruction from centralized server for the next sensing interval).We denote the devices that are in the sensing mode as V_in^a (the red circles in Figure <ref>), and the ones that are in the non-sensing mode as V_in^p (the orange circles in Figure <ref>).Apparently, we have V_in = V_in^a∪V_in^p.Each internal device operates under two different time windows (Figure <ref>).During sensing time window t_s, if an internal device is in the sensing mode, it periodically senses the neighboring environment for wireless contacts and stores the data locally before sending to the server at the end of sensing interval.The value of t_s is chosen as a multiplication[The reasons for not choosing t_s = τ are to reduce the number of times communicating with centralized server, and to increase the chances of observing neighboring contacts, which might be missing if just a single scan is used, due to weak signal or obstacles] of inquiry interval of wireless sensor on each device τ (i.e., the time gap between two consecutive scanning of neighboring devices): t_s = T ×τ, with T is a fixed integer.If an internal device is in non-sensing mode during t_s, it simply does nothing.During decision time window t_d, both sensing and non-sensing devices listen to the instructions from the centralized server to decide which ones would do the sensing task in the next round.Also during t_d, the sensing devices will send its locally stored data of contacts observed during previous t_s to the server.After t_d, all the devices go to a new t_s period with a new set of sensing devices decided by the server.§.§ Problem definition Let us denote E_t_s as the set of wireless contacts observed by sensing devices during the time interval t_s.For each sensing device v∈V_in^a, let us denote E_t_s(v) as the set of wireless contacts that v observes during t_s.The set of wireless contacts obtained from devices in V_in^a can be used to construct a contact graph G_t_s = (V_t_s, E_t_s) of devices during t_s, with V_t_s being the set of nodes seen during t_s (including the sensing nodes) and each edge in E_t_s represents a contact between two devices in V_t_s during t_s.From now on, we might use nodes/devices, and edges/contacts interchangeably as they refer to the same notion.Since V_in^a is only a subset of V_in (V_in^a⊆V_in), it is possible that the set of observed contacts during t_s might not be complete: E_t_s⊂E_t_s, where E_t_s is the set of contacts observed during t_s if V_in^a≡V_in.Therefore, our objective is to find a set of vertices V_in^a, whose size is limited to a predefined n, that maximizes the number of observed contacts during t_s.The crowd-sensing problem is thus formally defined as follow:Problem Definition: Given a set of internal devices V_in = V_in^a∪V_in^p, for each sensing time interval t_s, find an optimal set of sensing devices V_in^a*, whose size equals a predefined n (n ≤ |V_in|), that maximizes the number of wireless contacts |E_t_s| observed during t_s: V_in^a* = _V_in^a⊂V_in, |V_in^a| = n|E_t_s| In the above definition, the absolute number of observed contacts (i.e., |E_t_s|) represents the coverage ability and is used as the maximization objective.Equivalently, we can also use the ratio between the number of observed contacts by devices in V_in^a and that by all devices in V_in, i.e., |E_t_s| / |E_t_s|, to measure thecoverage capability.We refer this ratio as sensing coverage ratio.§ CROWD-SENSING BY VERTEX COVER OPTIMIZATION In this section, we show the connection between the crowd-sensing problem and the vertex cover problem <cit.>.We first revisit the vertex cover problem and then, show why finding the minimum cover set of a contact graph also gives us an efficient solution for crowd-sensing problem.After that, we describe the constrained versions of two approximation algorithms <cit.> for vertex cover problem and how they can be applied to crowd-sensing problem. §.§ Vertex cover of a graph A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex of the set.Formally, a vertex cover of an undirected graph G = (V, E) is a subset V' ⊂V such that if edge (u, v) is an edge of G, then u ∈V', v ∈V', or both. It is not difficult to see there might have more than one vertex cover of a graph. In Figure <ref>a, the red nodes represent one example of the graph's vertex cover.A minimum vertex cover is a vertex cover of smallest possible size. In Figure <ref>b, the red dots represent an example of the graph's minimum vertex cover. From here, we can see more clearly the connection between vertex cover problem and crowd-sensing problem.As every edge in a graph is incident to at least one vertex in vertex cover, the set of nodes in the vertex cover could essentially “observe” all contacts (i.e., edges) of the graph.In crowd-sensing scenario, the vertex cover is basically the set of sensing nodes that can observe all number of contacts between all nodes in the graph.In addition, since the number of sensing devices in crowd-sensing problem is constrained to a predefined number n (which is preferred to be small, due to the cost and high energy consumption of having too many sensing devices), it is also desirable to find the minimum vertex cover as the set of sensing devices.In case the size of minimum vertex cover is greater than n, we need to exclude vertices from the cover set in a way that minimizes the effect to the observable edges (i.e., to make fewest number of edges become non-observable by vertices in the vertex cover).One simple strategy is to exclude the vertices that have smallest number of edges connected to it. Finding minimum vertex cover of a graph, however, is a NP-complete problem <cit.>.There have been efforts to come up with approximate solutions <cit.>.In the next section, we discuss the constrained version of the two well-known approximation algorithms for minimum vertex cover problem with our modification to comply with the constraint on the maximum number of sensing nodes n. §.§ Approximation algorithms for vertex cover problem The first approximation algorithm is based on random selection of vertices into vertex cover set.This randomization strategy is similar to the 2-Approximation algorithm that finds a factor-2 approximation by repeatedly taking both endpoints of an randomly selected edge into the vertex cover [The 2-Approximation algorithm is not directly applicable for crowd-sensing task, because not always the two vertices of a chosen edge are the internal devices].In our random-based approximation algorithm (Algorithm <ref>), we needs to comply with the constraint on the number of vertices on the cover set as well as the fact that the selected vertices for the sensing task needs to be in V_in.Particularly, we repeatedly randomly select a vertex from the set of internal devices and add it to the cover set (while also removing all adjacent edges to the selected vertex), until the size of the cover set equals n.In the second approximation algorithm, i.e., Top-n Greedy Cover (Algorithm <ref>), we repeatedly select a vertex in V_in with highest node degree from the contact graph to add to the vertex cover.The motivation behinds this approximation algorithm is that the higher the node degree of a vertex is, the more incident edges it has, and thus the more likely the selected vertices in the vertex cover can fully cover all edges in the contact graph.Similar to Algorithm <ref>, this greedy algorithm also ends when the size of cover set reach the limiting number n, to comply with the constraint on the number of vertices on the cover set.In the following section, we will describe a framework to plug-in the above approximation algorithms to solve crowd-sensing problem. §.§ Crowd-sensing by vertex cover approximation At the beginning of crowd-sensing task, since the centralized server does not even have a contact graph to start with approximation algorithms, the assignment of sensing task to a set of vertices V_in^a∈V_in is done by a bootstrapping algorithm.Then, after each sensing interval t_s, using the sensing data collected by nodes in V_in^a, the centralized server can construct a contact graph between devices of the previous interval and use this as the input of the approximation algorithms (e.g., Top-n Random-based Approximation Vertex Cover or Top-n Greedy Vertex Cover – Algorithm <ref> and <ref>) to find the new vertex cover for the next iteration.The general framework of applying vertex cover approximation to crowd-sensing problem is described in the following 2-stage algorithm (Algorithm <ref>):In Algorithm <ref>, bootstrap(V_in) is the bootstrapping method that returns the initial assignment of sensing task to a subset of vertices V_in^a (sized n) in V_in. vertexCoverApprox(V_t_s, E_t_s, V_in, n) is the call to a vertex cover approximation algorithm (e.g., Top-n Greedy, or Top-n Random-based Approximation Vertex Cover algorithms) to find the vertex cover sized n given a contact graph (i.e., (V_t_s, E_t_s)) and the set of registered devices (i.e., V_in). The method getCoveredGraph(V_in^a) constructs a contact graph given a vertex cover set based on all the wireless contacts that vertices in the cover set observe during a sensing interval. rounds represents the number of rounds to run the crowd-sensing task.§ CONTEXT-AWARE APPROXIMATION AND HUMAN-CENTRIC BOOTSTRAPPING FOR CROWD-SENSING While both of the above approximation algorithms are simple and seemingly intuitive, they avoid important underlining information about the contact graph and how it is constructed.Particularly, since each node in the graph is actually a device carried by a human user, the opportunistic contacts between devices are influenced by the spatio-temporal interactions between human users carrying the devices and the social context of the crowd event.For example, the collected opportunistic contacts might be highly overlapped (and thus, is not very helpful in covering a large crowd) if the sensing devices move together, or stay at the same location.In another example, in the context of an academic conference, people tend to attend presentation sessions of the topics of their interests.As a result, a sensing node tends to observe surrounding devices of people sharing similar interests.These insights motivate us to design a new context-aware approximation algorithm and human-centric bootstrapping methods for the crowd-sensing task that take into account the spatio-temporal interactions between devices and the social context of the crowd.In this section, we describe in more details our proposals of approximation algorithm and bootstrapping strategies as parts of the two stages in the general crowd-sensing framework (Algorithm <ref>).We first describe our proposed human-centric bootstrapping strategies based on meta information of the participants (i.e., ∼ bootstrap(V_in) – the first stage of the framework).After that, we introduce the notions of node observability and coverage utility score that are the key components of the proposed context-aware approximation algorithm (i.e., ∼ vertexCoverApprox(V_t_s, E_t_s, V_in, n) - the second stage of the framework).§.§ Human-centric bootstrapping strategies As mentioned in the previous section, at the beginning of the crowd-sensing process, without a contact graph to start with, we need to bootstrap the process by assigning the sensing task to a subset of nodes sized n in V_in.While the most basic way of bootstrapping is to randomly select nodes as sensing nodes, we propose to use “out-of-band” meta information about the social relationships and interests of human users who carry the internal devices to initialize the selection of sensing nodes.Particularly, we consider two types of relationship between people: friendship (i.e., whether two or more people know each other in person) and interests (i.e., whether two or more people share the same personal interest).Our approach is motivated from the fact that human mobility and interaction are influenced by their interests and how they are connected socially.Specifically, people who are friends or share similar interests tend to get together more often than the ones who do not know each other, or do not have anything in common.As a result, if we know such relationships between people, we can appropriately assign the sensing task to the persons who have a lot of friends, or the ones who share common interest with a lot of people. The three bootstrapping strategies used in this paper are summarized as follow:* Random-based bootstrapping: Randomly select a subset of n devices from V_in as the sensing devices.* Friendship-based bootstrapping: Build a friendship social network of participants in V_in.Sort all participating people in descending order of their node degree in friendship social network.Select top-n people as the initial set of users carrying sensing devices.* Interest-based bootstrapping: Group people into groups of interests. Sort all the interest groups in descending order of its number of members.For each group in the top-n groups, select a member with the most diverse affiliation (i.e., the member with the most number of groups he/she belongs to). In this paper, the fact that the proposed bootstrapping strategies are limited to only using friendship and personal interests is due to the availability of such meta information in the experimenting datasets.However, we believe that these are the good examples of using out-of-band information in bootstrapping the selection of sensing nodes. §.§ Node observability and coverage utility score Before describe our proposed context-aware approximation algorithm for finding vertex cover, we introduce the notions of node observability and coverage utility score that are the key components of the proposed algorithm.An appropriate utility function is essential for any approximation algorithm.For example, in Top-n Greedy Vertex Cover algorithm, the utility function is actually the method that calculates the node degree of each internal vertex in the contact graph.The motivation of the greedy objective function is that the higher node degree is, the better coverage a node has.While such a motivation seems to be intuitive, it only tries to obtain the local optimal and does not capture the fact that the coverage by different nodes might be overlapped and some nodes are less visible than others.As a result, the combined coverage by multiple sensing nodes might not be good enough to cover all contacts in the crowd.To account for different levels of visibility between nodes, we propose the notions of node observability for non-sensing nodes that helps measure how “easy” a non-sensing node can be observed by other sensing nodes.In addition, to establish an objective function that can help optimize the overall sensing coverage (i.e., the global optimal), we propose the notion of coverage utility score for sensing nodes.This utility score accounts for how important the contacts that a sensing node observes are, in terms of contributing to a high overall sensing coverage. Specifically, node observability is defined as the number of sensing nodes that observe a given non-sensing node during a sensing interval.The higher the observability score is, the more visible a node is during that interval.Definition 1 (Node observability): During a sensing interval t_s, node observability σ_t_s(v) of a non-sensing node v ∉V_in^a is defined as: σ_t_s(v) = |{u|u ∈V_in^a, (u, v) ∈E_t_s}| Since the ultimate objective is to obtain the best combined coverage – to cover as many contacts between nodes as possible (even the contacts with less visible nodes), covering nodes with low observability is also very important.In addition, if a node already has a high observability, the fact that this node is observed by some other sensing nodes is not very important (as that means overlapped contacts, or the node is in an area with a redundant number of sensing nodes). Based on this, we propose a notion of coverage utility score that gives higher reward to a node that can observe less visible nodes in its sensing results.Definition 2 (Coverage utility score): Coverage utility score Δ_t_s(u) of a sensing node u ∈V_in^a during a sensing time interval t_s is defined as: Δ_t_s(u) = Σ_v | (u, v) ∈E_t_s, v ∉V_in^aσ^-1_t_s(v)§.§ Context-aware approximation algorithm In the second stage of the crowd-sensing algorithm (Algorithm <ref>), after receiving data from the sensing nodes, the centralized server update the set of sensing nodes by using an approximation algorithm (i.e., vertexCoverApprox(V_t_s, E_t_s, V_in, n)).We now describe our proposed context-aware approximation algorithm (Algorithm <ref>) based on the notions of node observability and coverage utility score introduced in the previous section.In our approximation algorithm, we first calculate the observability scores of all non-sensing nodes and coverage utility scores of the sensing nodes from the collected sensing data.These scores are used to assess the effectiveness of the sensing nodes (i.e., using coverage utility scores) and the sensing potential of the non-sensing nodes (i.e., using observability scores).Particularly, top k (k < n) nodes with highest coverage utility scores are kept in the set for the next sensing interval.The (n-k) nodes with lowest coverage utility scores are replaced by non-sensing nodes with highest observability scores.In the above algorithm, k, i.e., the number of sensing nodes to keep after each sensing interval, is set in advance.In our implementation and experiments, we set k equals 0.5 * n as it produces the best results.§ EXPERIMENTS In this section, we show the results of our experiments on real-world datasets to verify the effectiveness of our proposed approach.We starts with describing our experimental settings (Section <ref>.A, and then, move on to compare our proposed approximation algorithm and bootstrapping strategies to the baselines (Section <ref>.B and <ref>.C, respectively).After that, we measure the effect of key parameters on the performance of our proposed algorithm (Section <ref>.D and <ref>.E). §.§ Experimental settings In our experiments, we use a simulation-based approach using real-world datasets of opportunistic Bluetooth contacts.Particularly, we use previously published datasets including (Table <ref>): i) mobile traces collected during SIGCOMM'09 <cit.> conference experiment in Barcelona, Spain, and ii) traces of Bluetooth encounters of a set of users collected through UIM experiment <cit.> at the University of Illinois at Urbana-Champaign in 2010.For SIGCOMM'09 dataset, it was collected from 76 smartphones distributed to a set of volunteers during the first two days of the conference.The participants were recruited on-site in conjunction of the conference registration.Besides the logs of Bluetooth encounters, which represent all observed wireless contacts, each device was initialized with the social profile of the participant that included some basic information, such as list of friends and interests in the social profile.For UIM dataset, it is collected by 28 Android phone users , who are staff, faculties, grads, and undergrads at University of Illinois, for 3 weeks in March 2010.Since in the above datasets, all internal devices (i.e., 76 smartphones in SIGCOMM'09 and 28 Android phones in UIM) are doing the sensing task all the time (i.e., V_in^a≡V_in), we are able to have the ground-truth information about the best possible set of contacts we can collect (i.e., E_t_s).As a result, we are able to use the sensing coverage ratio (described in Section <ref>) as the evaluation metric.Basically, this ratio measures the ratio between the number of contacts that devices in V_in^a can cover, and that by all devices in V_in.In terms of algorithms, we compare three approximation algorithms discussed in this paper: Top-n Random-based Approximation Vertex Cover (denoted as RANDOM), Top-n Greedy Vertex Cover (denoted as GREEDY), and our proposal – Context-aware Approximation Algorithm (denoted as HCONTEXT).For each experimental scenario, the time is divided into multiple rounds, each round consists of a sensing interval (i.e., t_s) and a decision interval (i.e., t_d) – as describe in Section <ref>.Since the main objective is to measure the sensing coverage performance of the approximation algorithms and bootstrapping methods, to simplify the experiment, we assume that all internal nodes have the same decision time interval t_d, and are be able to successfully send sensing data, receive sensing assignment from centralized server within t_d.The experimental results are presented in graph with y-axis is the sensing coverage ratio and x-axis is the starting time of each sensing round. Since there are multiple parameters that could affect the performance of an algorithm (including sensing time interval t_s, number of active sensing nodes n, the selection of bootstrapping strategy bootstrap(V_in)), in the following, we use a step-by-step experimenting approach.Particularly, we start with comparing sensing coverage between different algorithms (Section <ref>.B) by fixing the configuration parameters (i.e., t_s, n, and bootstrap(V_in)).After that, we fix the algorithm, t_s, and n to compare between bootstrapping strategies (Section <ref>.C).Similarly, in Section <ref>.D and <ref>.E, we measure the affect of t_s and n respectively by fixing the remaining parameters.§.§ Coverage capability comparison In terms of coverage capability, we compare the three approximation algorithms on three different scenarios.The first scenario, keynote presentation session (Figure <ref>), is the duration when keynote presentations are undertaken and people gather into large conference room to listen to the presentations.This scenario presents a crowd, less dynamic situation.The second scenario, poster/demo and socialized session (Figure <ref>), is the duration when the poster/demo session of the conference is undertaken. During this time, people gather in an open space venue and walk around to listen to poster/demo presentations.This scenario represents a less crowd and more dynamic situation.For the third scenario, we evaluate algorithms with the Bluetooth encounters of university users in UIM dataset during a weekday afternoon, where typical activities on campus are happening, such as classes, meeting, lab sessions, etc.In terms of setting-up the paramters, for the length of sensing interval t_s, since the keynote presentation session is more crowd than the poster/demo session, we set the length of time interval shorter (i.e., 480s compared to 720s).For the third scenario, since the daily encounters between university users happen at a lower frequency, the length of the sensing time interval is set to 30 minutes.The effect of choosing different lengths of sensing time interval is studied in the next section.For the sensing coverage comparison, we fix the bootstrapping strategy as random selection, the number of sensing devices is limited to be 40% of total number of internal devices.As we can see in Figure <ref>, <ref>, and <ref>, HCONTEXT outperforms RANDOM and GREEDY algorithm in all scenarios.The RANDOM algorithm, although looks to be the simplest one, is still better than the more intuitive GREEDY algorithm.However, it is not a surprise, since random-based strategy has been considered as one of the most effective general-purpose approximationalgorithms for vertex cover problem <cit.>.The superior of HCONTEXT compared with two other algorithms is clearer in all scenarios, and this helps confirm our intuition of designing HCONTEXT that takes into account the contextual information in forms of node observability and coverage utility scores of sensing nodes. §.§ Bootstrapping methods comparison In this section, we compare three different bootstrapping methods for selecting initial set of sensing devices: i) Randomization, ii) Friendship-based, and iii) Interest-based bootstrapping.We test each method with the HCONTEXT approximation algorithm during the SIGCOMM '09 keynote presentation session and poster/demo session, and report the sensing coverage for comparison (Figure <ref> and <ref>).Interestingly, the results show that during the keynote session, randomization-based bootstrapping yields the best initialized sensing coverage, followed by friendship-based, and lastly interest-based selection.This is because, during such a crowd and less dynamic event, the wireless encounters tend to be more random (e.g., due to different arrival time of audience and seating arrangement), and are less influenced by the common interests or friendship.On the other hand, during the poster and demo session, friendship-based bootstrapping produces the best result, followed by interest-based, and lastly randomization.This result is reasonable, since during such a more socialized event, people tend to interact more with whom they share similar interests or they know in person (i.e., friends). In the next sections, we measure the effects of different parameters/constraints, including the length of sensing interval t_s and the number of sensing devices n, to the performance of HCONTEXT algorithm. §.§ Varying different lengths of t Different length of a time interval affects the collection of wireless contacts by sensing devices.Apparently, the longer the time interval, the more wireless contacts a node can observe.If the sensing interval is set too short, the collected contact graph might not be complete, and thus, negatively affect our understanding of the context (i.e., node observability and coverage utility).In our experiment, we measure the effect of the length of time interval to different scenarios.The results support our aforementioned motivations.For example, in university campus scenario, if the length of sensing time t_s is set too short (i.e., 15 min), the result is not as good as when t_s is set longer (i.e., 30 min or 60 min) (see Figure <ref>).This is because 30 min or 60 min interval lengths are closer to the realistic frequency of people's encounters in daily life (while 15 min is too short).This result suggests that, the length of sensing interval should be chosen carefully, based on the context of the environment that we want to sense.§.§ Varying different number of sensing devices n In our last experiment, we measure the effect of using different numbers of sensing devices n.Obviously, the more sensing devices we have, the better coverage we could achieve.However, as the number of sensing devices can be considered as a cost and energy constraints, we would like our method to still perform well when there are few sensing devices available.In our experiment, we test our proposed HCONTEXT algorithm while varying the percentage of sensing devices in all internal devices during poster/demo session and report the coverage.The result in Figure <ref> show that (not surprisingly) as the percentage increases, we are able to obtain better coverage.More importantly, the result also shows an interesting insight about the ability of HCONTEXT to withstand the limited number of sensing devices.At the lowest percentage (i.e., 20%), although HCONTEXT starts not very well, it quickly gains better coverage and reaches a reasonably high coverage level, compared with higher percentage of sensing devices (see Figure <ref>).With 40% of sensing devices, the algorithm can perform almost as good as higher percentage levels.This result is highly desirable as lower percentage of sensing devices means saving energy and cost.In summary, throughout the experiments, we have shown that: i) HCONTEXT is significantly better than other state-of-the-art approximation algorithms for crowd-sensing task; ii) Different bootstrapping strategies should be employed in different scenarios to obtain the best performance; iii) The length of time interval needs to be set appropriately depending on the sensing context; and iv) HCONTEXT still performs well with limited number of available sensing devices. § RELATED WORK The idea ofcrowd computing was first introduced by Murray et al. <cit.>, which aims to combine mobile devices and social interactions to achieve large-scale distributed computation.The paper, however,only proposed one realistic model for crowd computing: static task farming, which does not take into account the dynamic nature of the crowd.In addition, while <cit.> focus on computational resource, our optimization objective is in the coverage of the task.<cit.> presents CrowdWatch, a scalable, distributed and energy-efficient crowd-sourcing framework, based on a building a hierarchy of participants.<cit.> desribes a crowd sensing system that is developed in IBM for the smart cities domain that utilize heterogeneous types of data sources.<cit.> present a technique for estimating crowd density by using a mobile phone to scan the environment for discoverable Bluetooth devices.Mashhadi et al. <cit.> reasons on users mobility patterns and quality of their past contributions to estimate user's credibility under crowd-sensing scenario.Along the line with human-centric computation, <cit.> provides a comprehensive survey on human-centric sensing tasks.Our paper falls nicely into the category of humans as sensor operators (collection campaigns) and directly solve one of the challenges mentioned in <cit.>: to identify the appropriate set of individuals who would collect the data. Nicolai et al. present an interesting study <cit.> that show limited number of people with discoverable Bluetooth devices (only 6% devices in San Francisco are detectable).This limitation does not affect our experiments, since the datasets used in our experiment, in fact, show that the number of external devices with discoverable Bluetooth during crowd event is quite big.Sensor coverage problem <cit.> has been long studied before, but usually in static setting (e.g., sensor placement).Cardei et al. present a comprehensive survey <cit.> on related work addressing energy-efficient coverage problems in the context of static wireless sensor networks.In our paper, we are able to adapt with the mobility nature ofusers by doing context-aware adjustment for assignment of sensing devices overtime.Wu et al., <cit.> model the problem of routing in wireless ad hoc network as the connected dominating set problem and propose distributed approximation algorithm to find the connected dominating set.In our paper, we present a centralized sensing model for monitoring the crowd and do not require the sensing devices to be connected to each other.Vertex cover problem <cit.> is a well-know optimization problem in graph theory.As far as we concern, this is the first paper that draw the connection between vertex cover and crowd-sensing problem.Since finding minimum vertex cover is a NP-complete problem <cit.>, there have been efforts <cit.> to develop approximation algorithms.However, all of the proposed approximation algorithms are designed for generic graph.In this paper, we propose a new approximation algorithm designed for crowd-sensing scenario that take into account the dynamic nature of the contact graph.§ CONCLUSION AND FUTURE WORK In summary, in this paper, we have modeled the crowd-sensing problem as an optimization problem and draw the connection to the vertex cover problem in graph theory.We show that the current state-of-the-art approximation algorithms for vertex cover are not well-designed to deal with the dynamic nature of the crowd and its social and spatio-temporal characteristics.We thus propose the notions of node observability and coverage utility score and design a new context-aware approximation algorithm and human-centric bootstrapping strategies to find vertex cover that is tailored for crowd-sensing task.We have also verified the effectiveness of our proposed approach via comprehensive experiments on real-world datasets.For future work, we would like to explore to include location information of the devices that can be inferred from Wifi access points data in the UIM dataset <cit.> to improve the assignment of sensing devices.In addition, from the results on the varying the length of time intervals and percentage of sensing nodes, it is interesting to investigate new algorithms that can adaptively adjust the sensing interval length and decrease/increase the number of sensing nodes depending on the context of the crowd, while maintaining a desirable sensing coverage.hcontext | http://arxiv.org/abs/1704.08598v1 | {
"authors": [
"Phuong Nguyen",
"Klara Nahrstedt"
],
"categories": [
"cs.SI"
],
"primary_category": "cs.SI",
"published": "20170427142828",
"title": "Crowdsensing in Opportunistic Mobile Social Networks: A Context-aware and Human-centric Approach"
} |
[email protected] Research Group, INTEC, Ghent University-IMEC, Ghent B-9000, Belgium Center for Nano-and Biophotonics (NB-Photonics), Ghent University, Ghent B-9000, Belgium Institute of Physics, University of Augsburg, D-86135 Augsburg, GermanyWe present a degenerate four-wave mixing experiment on a silicon nitride (SiN) waveguide covered with gated graphene. We observe strong dependencies on signal-pump detuning and Fermi energy, i.e. the optical nonlinearity is demonstrated to be electrically tunable. In the vicinity of the interband absorption edge (2|E_F|≈ħω) a peak value of the waveguide nonlinear parameter of ≈ 6400 m^-1W^-1, corresponding to a graphene nonlinear sheet conductivity |σ_s^(3)|≈4.3· 10^-19 A m^2V^-3 is measured. Electrically Tunable Optical Nonlinearities in Graphene-Covered SiN Waveguides Characterized by Four-Wave Mixing N. A. Savostianova, S. A. Mikhailov December 30, 2023 ================================================================================================================In recent years, there has been increasing interest in the nonlinear optical properties of graphene. Both theoretical predictions <cit.> and experimental studies <cit.> have indicated that graphene has a very high third-order sheet conductivity σ_s^(3), which leads to a strong nonlinear optical response. Despite the consensus that nonlinearities in graphene are strong, very different values of the corresponding material parameters have been reported (see, e.g., discussions in Refs. <cit.>). Reasons for this can be found in the fact that in different experiments different nonlinear effects (harmonics generation, four-wave mixing, Kerr effect) are probed at different wavelengths, in samples with different carrier densities and in different dielectric environments. Moreover, in experimental studies pulses with vastly different durations and optical bandwidths have been used. All these factors can significantly influence the final result. Hence a detailed quantitative study of the nonlinear response of graphene, at different frequencies and in samples with different electron densities, is imperative.Another important research direction is the search for interband resonances in the nonlinear response function of graphene. It has been theoretically predicted <cit.> that the nonlinear parameters of graphene should have resonances at frequencies corresponding to the interband absorption edge, e.g. at ħω=2|E_F|/3 for third harmonic generation or at ħω=2|E_F| for self-phase modulation (SPM) (SPM in graphene with fixed E_F on a waveguide has been measured e.g. in Ref. <cit.>), etc., where ω is the incident photon frequency and E_F the Fermi energy. Fig. <ref>a shows the band diagram of graphene, along with the photon energy. SPM, which scales as Im[σ_s^(3)], has been predicted to peak at 2|E_F|≈ħω, after which it decreases sharply for further increasing |E_F| (see for example Fig. 3 in Ref. <cit.>). The majority of experiments have been performed in weakly doped graphene at high frequencies (near-IR, visible), where ħω≫ 2|E_F|. An experimental observation of the Fermi energy related resonances would not only be interesting for fundamental science but also for nonlinear devices since it would provide a way to control the nonlinear optical response of practical systems.In this Letter, we characterize the Fermi energy dependence of the third order nonlinear effects in graphene, tuning the graphene from intrinsic (2|E_F|≪ħω) to beyond the interband absorption edge (2|E_F|>ħω). We do this by means of four-wave mixing (FWM) in an integrated silicon nitride (SiN) waveguide, covered with a monolayer of graphene. Because of the coupling between the evanescent tail of the highly confined waveguide mode and the graphene over a relatively long length, significant light-matter interaction can be achieved. Studies of nonlinear effects in graphene-covered silicon waveguides and resonators have been published previously <cit.>. However, an intrinsic disadvantage of using a silicon platform for the characterization of graphene nonlinearities is that silicon has a relatively strong nonlinear response itself. The real part of the nonlinear parameter of a typical Si waveguide is about γ_Si≈ 300 m^-1W^-1 <cit.>, as opposed to about γ_SiN≈ 1.4 m^-1W^-1 <cit.> for a SiN waveguide, which is negligible compared to the nonlinear parameters of the graphene-covered waveguide measured in this work (γ P L is the nonlinear phase shift acquired over length L at power P, see Supplemental Material or Ref. <cit.>). Using SiN, we can thus avoid any ambiguity about the origin of the strong nonlinear effects. Furthermore, we have achieved electrical tuning of E_F (gating) by using a polymer electrolyte <cit.>. We have performed measurements for a varying signal-pump detuning and for a broad range of charge carrier densities and demonstrate, for the first time to our knowledge, a significant increase of the nonlinear response of graphene in the vicinity the interband absorption edge ħω≈2|E_F|. Moreover, we demonstrate a good qualitative agreement with theoretical calculations. Degenerate four-wave mixing (FWM) is a third order nonlinear optical process in which two pump photons at frequency ω_p are converted into two photons at different frequencies, typically denoted as the signal ω_s and idler ω_i. Energy conservation dictates that ω_s+ω_i=2ω_p, which is schematically shown in Fig. <ref>b. For a small signal-pump detuning, ω_p≈ω_s≈ω_i, the theorypredicts a vanishing FWM response beyond the interband absorption edge, 2|E_F|>ħω_p, in analogy with SPM described before. However, as opposed to SPM, the theory as it is published in Refs. <cit.> does not predict a sharp peak at 2|E_F|≈ħω_p. This is because the FWM response scales as the absolute value of the third order conductivity, |σ_s^(3)|, in which Re[σ_s^(3)] typically dominates Im[σ_s^(3)].To investigate this experimentally, a pump at a fixed wavelength and a signal with variable wavelength are injected into the graphene-covered SiN waveguide. Under the current experimental conditions, one can prove that the conversion efficiency η, defined as the ratio between the idler power to the signal power at the output, is quadratically dependent on the nonlinear parameter γ of the waveguide (see Supplemental Material),η≡P_i(L)/P_s(L)≈|γ(ω_i;ω_p,ω_p,-ω_s) |^2 (ω_i/ω_p)^2 P_p(0)^2L_eff^2,where L is the length of the graphene-covered waveguide section, L_eff≈1-e^-α L/α the effective length of the nonlinear process and α the linear waveguide loss. The effect of the phase mismatch is neglected since Lβ_2Δω^2 ≪ 1 in the presented experiment (L=100 μm, Δω<10^13 rad/s and β_2 of a SiN waveguide is on the order of 10^-25 s^2/m <cit.>; here β_2≡∂^2 β/∂ω ^2, with β(ω) the propagation constant of the optical mode). The nonlinear parameter γ of the waveguide is, to a good approximation, proportional to the nonlinear conductivity σ_s^(3) of graphene (see Supplemental Material),γ(ω_i;ω_p,ω_p,-ω_s)≈i 3 σ^(3)_s, xxxx(ω_i; ω_p, ω_p, -ω_s)/16𝒫_p^2∫_G|𝐞(ω_p)_∥×ê_z|^4dℓ ,where 𝐞(ω_p)_∥ is the electric field component tangential to the graphene sheet at the pump frequency, ê_z is the unit vector along the propagation direction and 𝒫_p is the power normalization constant of the optical mode.A set of straight waveguides was patterned in a 330 nm thick LPCVD SiN layer on top of a 3 μm burried oxide layer on a silicon handle wafer. The sample was then covered with LPCVD oxide and planarized using a combination of chemical mechanical polishing, reactive ion etching and wet etching. Subsequently, a CVD-grown graphene layer was transferred to the samples by Graphenea <cit.> and patterned using photolithography and oxygen plasma etching so that different waveguides were covered with different lengths of graphene. Metallic contacts (Ti/Au; ≈ 5 nm/300 nm) were applied at both sides of each waveguide, with a spacing of 12 μm. Fig. <ref>c shows a SEM image of the waveguide cross-section, note that ≈ 80 nm of oxide is left on the waveguide. Finally the structures were covered with a polymer electrolyte consisting of LiClO_4 and polyethylene oxide (PEO) in a weight ratio of 0.1:1. Fig. <ref>d shows a sketch of the cross-section (not to scale). The gate voltage V_GS can be used to gate the graphene layer <cit.>. The dependence of E_F on V_GS can be approximated by the following formula <cit.>:V_GS-V_D = sgn(E_F)e E_F^2/ħ^2 v_F^2 π C_EDL+E_F/e ,where e is the electron charge, v_F≈ 10^6 m/s the Fermi velocity, C_EDL the electric double layer capacitance and V_D the Dirac voltage. Based on measurements of the optical loss and the graphene sheet resistance versus the gate voltage we estimatedC_EDL≈1.8·10^-2 F m^-2 and V_D≈0.64 V, see Supplemental Material.The setup used for the FWM experiment is shown in Fig. <ref>e. A pump laser (Syntune S7500, λ_p= 1550.18 nm) is amplified using an Erbium-doped fiber amplifier (EDFA), a tunable band-pass filtersuppresses the Amplified Spontaneous Emission (ASE) of the EDFA. The signal is provided by a Santec Tunable Laser TSL-510. Pump and signal are coupled into the waveguide through a grating coupler. At the output a fiber Bragg grating (FBG) filters out the strong pump light and the signal and idler are visualized on an Anritsu MS9740A optical spectrum analyser (OSA).Fig. <ref> summarizes the experimental results obtained for a set of 1600 nm wide waveguides. By measuring the transmission of a set of waveguides with varying graphene lengths, as well as the transmission as a function of V_GS, the propagation loss as a function of V_GS was extrapolated, Fig. <ref>. Note that the absorption drops sharply for negative voltages, indicating that the Fermi level of the graphene gets tuned beyond the interband absorption edge. In Fig. <ref>, some of the measured spectrafor the FWM experiment are plotted (V_GS=-0.5 V), the pump laser light is filtered out by the FBG. The conversion efficiency η can be read as the ratio between the idler (λ_i) and signal (λ_s) peaks (we correct for variations in the transmission of the grating couplers with changing wavelength). The measured conversion efficiencies are plotted in Fig. <ref>, for a range of different voltages and signal wavelengths (λ_p=1550.18 nm), with an estimated on-chip pump power of P_p(0)=10.5 dBm and a graphene length L=100 μm. The FWM conversion efficiency is highly dependent on both detuning λ_s-λ_p and the applied voltage. Using Eqs. (<ref>) and (<ref>), the magnitude of the nonlinear parameter γ(ω_i;ω_p,ω_p,-ω_s) and of the third order conductivity σ^(3)_s(ω_i; ω_p, ω_p, -ω_s) can be calculated. For the calculation of L_eff the loss measurement in Fig. <ref> was used. The integral and power normalization constant 𝒫_p in Eq. (<ref>) are calculated using a COMSOL Multiphysics^-model of the cross-section in Fig. <ref>c. Figs. <ref>a and <ref>b show the results of these conversions. The measured values for |γ| (|σ^(3)_s|) have a sharp resonance as a function of detuning and a broad asymmetric resonance as a function of E_F. |γ| (|σ^(3)_s|) is about 2800 m^-1W^-1 (2·10^-19 Am^2/V^3) at small |E_F| (for minimum detuning) and about 6400 m^-1W^-1(4.3·10^-19 Am^2/V^3) at its absolute peak.We can compare these experimental results with a slightly modified version of the theory published in Refs. <cit.>. In these papers analytical expressions for the third order conductivity σ^(3)_s, αβγδ(ω_1+ω_2+ω_3;ω_1,ω_2,ω_3,E_F,Γ) were derived at T=0, where the relaxation rate Γ was assumed to be energy independent. However, under these assumptions the theory does not predict the measured increase of σ^(3)_s with increasing |E_F| (see the solid line on Fig. S5 in the Supplemental Material). To get a better correspondence between theory and experiment, we can however assume that Γ(E) is a function of the electron energy. Both theoretical (e.g. Ref. <cit.>) and experimental (e.g. Ref. <cit.>) studies indicate that the relaxation rate Γ(E)∝ |E|^-α is a power-law function of energy E, at |E|≳ E_0, with α being determined by the scattering mechanism. According to theory, α=1 for impurity scattering and E_0 is related to the density of impurities <cit.>. Experimental data confirmed the power-law dependence of Γ(E) but showed a slightly smaller value of α, 0.5≲α≲ 1 (see the inset in Fig. 2 in Ref. <cit.>). To be able to use a formula for Γ(E) at all energies including the limit E→ 0 we adopt the modelΓ(E)=Γ_0/(1+E^2/E_0^2)^α/2,which has a correct asymptote Γ(E)∝ |E|^-α at large energies |E|≫ E_0 and gives a constant relaxation rate at E→ 0; the quantities Γ_0, E_0 and α in Eq. (<ref>) are treated as fitting parameters. In addition, to take into account the effects of nonzero temperatures, we can use the formula (the frequency arguments are omitted for clarity) <cit.>:σ^(3)_s, αβγδ (E_F, Γ_0,E_0,α,T)=1/4T∫_-∞^+∞σ^(3)_s, αβγδ (E_F',Γ_0,E_0,α,T=0) /cosh^2(E_F-E_F'/2T) dE_F'.Figs. <ref>c and <ref>d show thus obtained theoretical dependencies of the absolute value of the third order conductivity |σ^(3)_s, xxxx(ω_i;ω_p,ω_p,-ω_s)| on the Fermi energy and the detuning λ_s-λ_p. The parameters ħΓ_0=2.5 meV, E_0=250 meV and α=0.8 have been chosen so that good qualitative agreement was obtained with the experimental plots shown in Figs. <ref>a and <ref>b. One can see that the theory indeed describes the most important features of the FWM response: a narrow resonance as a function of λ_s-λ_p and a broad strongly asymmetric shape as a function of E_F; the inflection point at E_F≈ -0.4 eV corresponds to ħω_p≈ 2|E_F|. Quantitatively, the theory predicts about one order of magnitude larger response than was experimentally observed (this contrasts to a number of previous publications, where the measured nonlinear response was claimed to be larger than the theoretically calculated one, see discussions in Refs. <cit.>). This discrepancy should be subject to further investigation. Experimental errors could have an influence, such as an overestimated pump power P_p(0) or effective length L_eff, errors on the exact dimensions of the waveguide cross-section, etc. Inhomogeneities in the doping level of the graphene could also have an influence, effectively creating inhomogeneous broadening of the measured response in the |E_F|-direction and diminishing the height of the peak. Finally, the difference could partly be due to imperfections in the graphene which the theory might fail to fully take into account.In conclusion, we have performed a degenerate four-wave mixing experiment on a graphene-covered SiN waveguide. A polymer electrolyte enabled us to gate the graphene over a relatively large window. The experiment shows that the nonlinear conductivity of graphene has a sharp resonance as a function of signal-pump detuning, also a broad asymmetric resonance shape in the vicinity of the absorption edge 2|E_F|=ħω_p is observed. Qualitative agreement was obtained between these experimental data and an adapted version of previously published theory <cit.>, in which we introduced an energy-dependent relaxation rate Γ(E). From an application perspective, it is important to note that the measured nonlinear parameter of the waveguide |γ| is tunable by applying a gate voltage, and that optimizing this voltage the parameter surpasses ≈2000 m^-1W^-1 over the full measured bandwidth of 20 nm, with peak values over ≈6000 m^-1W^-1. This is more than 3 orders of magnitude larger than the nonlinear parameter of a standard SiN waveguide. The most obvious trade-off is the strongly increased linear absorption (more than 2 orders of magnitude, though this absorption is also tunable with voltage). 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Supplemental Material 1217 § THEORYOF FOUR-WAVE MIXING IN GRAPHENE-COVERED SIN WAVEGUIDES In this section the expressions for the nonlinear parameters and the linear loss of a waveguide covered with graphene are derived. In this derivation, the nonlinearities and linear losses will be treated as perturbations to the ideal lossless, linear waveguide. In addition, an expression for the degenerate four-wave mixing conversion efficiency η is derived.The complex amplitude of an unperturbed waveguide mode at frequency ω_j can be written as:𝐄_0(ω_j,𝐫)=A_0𝐞(ω_j,𝐫_⊥)/√(𝒫_j)e^iβ_j z , 𝐇_0(ω_j,𝐫)=A_0𝐡(ω_j,𝐫_⊥)/√(𝒫_j)e^iβ_j z .Where 𝐞(ω_j,𝐫_⊥) and 𝐡(ω_j,𝐫_⊥) are the vectorial electric and magnetic mode profiles, in what follows, we will often omit the arguments 𝐫 and 𝐫_⊥ for brevity. A_0 is the complex amplitude of the mode. β_j is the mode propagation constant and 𝒫_j is the power normalization constant, defined so that the total power of the mode equals |A_0|^2:∬_A_∞1/2{|A_0|^2 𝐞(ω_j)/√(𝒫_j)×𝐡^*(ω_j)/√(𝒫_j)} ·ê_z dA ≡ |A_0|^2⇒𝒫_j= 1/4∬_A_∞{𝐞(ω_j) ×𝐡^*(ω_j) + 𝐞^*(ω_j) ×𝐡(ω_j))}·ê_z dA.A_∞ is the plane perpendicular to the waveguide propagation direction. ê_z is the unit vector in the propagation direction z. By definition, these modes obey the Maxwell curl equations,∇×𝐄_0(ω_j)=iω_j μ_0 𝐇_0(ω_j) , ∇×𝐇_0(ω_j)=-iω_j ϵ_0n^2 𝐄_0(ω_j) ,n(𝐫_⊥) is the index of the unperturbed waveguide cross-section. One can include the effect of perturbations, such as linear losses and nonlinearities, by introducing complex slowly varying amplitudes A_j(z). The perturbed waveguide modes are then written as: 𝐄(ω_j, 𝐫)=A_j(z) 𝐞(ω_j,𝐫_⊥)/√(𝒫_j)e^iβ_j z , 𝐇(ω_j,𝐫)=A_j(z) 𝐡(ω_j,𝐫_⊥)/√(𝒫_j)e^iβ_j z .In practice, we will consider the total field to be a superposition of a number of monochromatic waves:𝐄(𝐫,t)= ∑_j{ A_j(z) 𝐞(ω_j, 𝐫_⊥)/√(𝒫_j)e^-i(ω_j t - β_j z)} , 𝐇(𝐫,t)= ∑_j{ A_j(z) 𝐡(ω_j, 𝐫_⊥)/√(𝒫_j)e^-i(ω_j t - β_j z)} .These perturbed modes should also obey the Maxwell curl equations, where the influence of the graphene sheet can be incorporated as a current density, 𝐉(ω_j),∇×𝐄(ω_j)=iω_j μ_0 𝐇(ω_j) , ∇×𝐇(ω_j)=-iω_j ϵ_0n^2 𝐄(ω_j) +𝐉(ω_j) .This current density can be written as the sum of a linear and a nonlinear contribution,𝐉(ω_j) = 𝐉_L(ω_j)+𝐉_NL(ω_j)= σ^(1)(ω_j) 𝐄(ω_j) + 1/4∑_ω_j=ω_k+ω_l+ω_mσ^(3)(ω_j; ω_k, ω_l, ω_m)⋮𝐄(ω_k)𝐄(ω_l) 𝐄(ω_m) ,σ^(1) and σ^(3) are the first and third order conductivity tensors. To derive the coupled-wave equations, we can start from the conjugated form of the Lorentz reciprocity theorem <cit.>:∬_A_∞∇·𝐅 = ∂/∂ z∬_A_∞𝐅·ê_z dA.A_∞ is the surface perpendicular to the propagation direction. The 𝐅-field can be constructed from the perturbed and unperturbed waveguide mode fields as 𝐅≡𝐄_0^*(ω_j)×𝐇(ω_j) + 𝐄(ω_j)×𝐇_0^*(ω_j). Substituting this in Eq. (<ref>) yields:∬_A_∞{ (∇×𝐄_0^*(ω_j))·𝐇(ω_j)- 𝐄_0^*(ω_j) · (∇×𝐇(ω_j))+ (∇×𝐄(ω_j))·𝐇_0^*(ω_j)- 𝐄(ω_j) · (∇×𝐇_0^*(ω_j))}dA= ∂/∂ z∬_A_∞A_0^* A_j(z)/𝒫_j{𝐞(ω_j) ×𝐡^*(ω_j) + 𝐞^*(ω_j) ×𝐡(ω_j))}·ê_z dA.The left hand side of Eq. (<ref>) can be simplified by substituting Eqs. (<ref>)-(<ref>) and (<ref>)-(<ref>). The right hand side can be simplified by using the normalization condition (Eq. (<ref>)). Eventually this gives∂/∂ zA_j= -e^-iβ_j z/4√(𝒫_j)∬_A_∞𝐞^*(ω_j)·𝐉(ω_j) dA ,substituting Eqs. (<ref>) and (<ref>) in Eq. (<ref>), and subsequently in Eq. (<ref>), one gets a general coupled-wave equation for the set of slowly varying amplitudes:∂/∂ zA_j= -A_j/4𝒫_j∬_A_∞𝐞^*(ω_j) ·σ^(1)(ω_j)𝐞(ω_j) dA -∑_ω_j=ω_k +ω_l+ω_m A_k A_l A_m e^i(β_k+β_l+β_m-β_j)z/16√(𝒫_j𝒫_k𝒫_l𝒫_m)∬_A_∞𝐞^*(ω_j)·σ^(3)(ω_j; ω_k, ω_l, ω_m)⋮𝐞(ω_k)𝐞(ω_l)𝐞(ω_m)dA ,here the summation goes over all possible combinations of 3 frequencies that add up to ω_j, including the negative frequencies. Moreover, since the time-dependent electrical fields are real-valued, one can make use of the equality 𝐞(-ω)=𝐞^*(ω).In the case of degenerate four-wave mixing, there are 3 monochromatic waves involved, the pump, signal and idler, at equally spaced frequency intervals (Δω=ω_s-ω_p=ω_p-ω_i). For this specific case, the coupled wave equations (Eq. (<ref>)) can be simplified to:∂ A_p/∂ z =i{γ(ω_p;ω_p,ω_p,-ω_p)| A_p|^2 A_p + 2γ(ω_p;ω_p,ω_s,-ω_s)| A_s|^2 A_p + 2γ(ω_p;ω_p,ω_i,-ω_i)| A_i|^2 A_p+ 2γ(ω_p;ω_s,ω_i,-ω_p) A_sA_iA_p^* e^-iΔβ z}-α(ω_p)/2A_p, ∂ A_s/∂ z =iω_s/ω_p{γ(ω_s;ω_s,ω_s,-ω_s)| A_s|^2 A_s + 2γ(ω_s;ω_s,ω_p,-ω_p)| A_p|^2 A_s + 2γ(ω_s;ω_s,ω_i,-ω_i)| A_i|^2 A_s+ γ(ω_s;ω_p,ω_p,-ω_i) A_pA_pA_i^* e^iΔβ z}-α(ω_s)/2A_s, ∂ A_i/∂ z =iω_i/ω_p{γ(ω_i;ω_i,ω_i,-ω_i)| A_i|^2 A_i + 2γ(ω_i;ω_i,ω_p,-ω_p)| A_p|^2 A_i + 2γ(ω_i;ω_i,ω_s,-ω_s)| A_s|^2 A_i+ γ(ω_i;ω_p,ω_p,-ω_s) A_pA_pA_s^* e^iΔβ z}-α(ω_i)/2A_i,where A_p(z), A_s(z) and A_i(z) are the complex amplitudes of respectively the pump, signal and idler, normalized so that | A_p,s,i|^2 equals the total power in the respective mode. Δβ = 2β(ω_p)-β(ω_s)-β(ω_i) is the phase mismatch term. α(ω) represents the linear loss and γ(ω_p+ω_q+ω_r;ω_p,ω_q,ω_r) is the nonlinear parameter of the waveguide. Comparing with Eq. (<ref>), these parameters become:α(ω_j)= 1/2𝒫_j∬_A_∞𝐞^*(ω_j) ·σ^(1)(ω_j)𝐞(ω_j) dA ,γ(ω_j=ω_p+ω_q+ω_r;ω_p,ω_q,ω_r)= i3/N_(p,q,r)∑_k,l,m1/16√(𝒫_j𝒫_k𝒫_l𝒫_m) ∬_A_∞𝐞^*(ω_j)·σ^(3)(ω_j; ω_k, ω_l, ω_m)⋮𝐞(ω_k)𝐞(ω_l)𝐞(ω_m)dA .In Eq. (<ref>), the summation parameters (k,l,m) take all different permutations of the set (p,q,r). N_(p,q,r) is the number of permutations of the set (p,q,r). In the specific case of graphene, σ^(1) and σ^(3) are only present in a very thin layer. The effects can be very well described using first and third order sheet conductivities, σ^(1)_s and σ^(3)_s. The surface integrals then become line integrals over the graphene,α(ω_j)= 1/2𝒫_j∫_G𝐞^*(ω_j) ·σ^(1)_s(ω_j)𝐞(ω_j) dℓ ,γ(ω_j=ω_p+ω_q+ω_r;ω_p,ω_q,ω_r)= i 3/N_(p,q,r)∑_k,l,m1/16√(𝒫_j𝒫_k𝒫_l𝒫_m) ∫_G𝐞^*(ω_j)·σ^(3)_s(ω_j; ω_k, ω_l, ω_m)⋮𝐞(ω_k)𝐞(ω_l)𝐞(ω_m)dℓ .In the specific case of the four-wave mixing experiment described in this work, the coupled-wave equations in Eqs. (<ref>)-(<ref>) can be strongly simplified. Firstly, the pump carries a much higher power than the signal, moreover the idler will be orders of magnitude weaker (| A_p | > | A_s |≫| A_i |). Secondly, in Ref. <cit.> it was demonstrated for a similar waveguide platform that nonlinear absorption only starts affecting the overall power transmission significantly for power levels above 1 W. In the experiments presented here the on-chip power levels were kept on the order of 10 mW or lower. Hence self-phase/amplitude modulation is expected to be much weaker than linear absorption (|γ|| A_p |^2 ≪|α(ω)/2|). Thirdly, the phase mismatch is negligible (Lβ_2Δω^2≪ 1, L=100 μm, Δω<10^13 rad/s and β_2≡∂^2 β/∂ω^2 of a SiN waveguide is on the order of 10^-25 s^2/m <cit.>). All these assumptions lead to heavily simplified coupled-wave equations:∂ A_p/∂ z≈ -α(ω_p)/2A_p, ∂ A_s/∂ z≈ -α(ω_s)/2A_s, ∂ A_i/∂ z≈iω_i/ω_pγ(ω_i;ω_p,ω_p,-ω_s) A_pA_pA_s^* -α(ω_i)/2A_i.Under these conditions the conversion efficiency η, defined as the ratio of the idler power to the signal power, has a quadratic dependence on the nonlinear parameter γ <cit.>:η≡P_i(L)/P_s(L)=| A_i(L)|^2/| A_s(L)|^2≈|γ(ω_i;ω_p,ω_p,-ω_s) |^2 (ω_i/ω_p)^2 P_p(0)^2L_eff^2 e^{α(ω_s)-α(ω_i)}L≈|γ(ω_i;ω_p,ω_p,-ω_s) |^2 (ω_i/ω_p)^2 P_p(0)^2L_eff^2 ,where the effective interaction length is defined as:L_eff≡1-e^-{α(ω_p)+α(ω_s)/2-α(ω_i)/2} L/α(ω_p)+α(ω_s)/2-α(ω_i)/2≈1-e^-α L/α .The final expressions in Eqs. (<ref>) and (<ref>) are only valid when the approximation α(ω_p)≈α(ω_s)≈α(ω_i)≡α holds, i.e. when the frequencies detuning is small (Δω≪ω_p). For the specific experiments described in this work, the expressions for α(ω) and γ(ω_i;ω_p,ω_p,-ω_s) can be further simplified. It is assumed that a flat sheet of graphene lies in the xz plane (this can easily be generalized to arbitrary graphene shapes). Firstly, the linear conductivity has only two nonzero elements, which are equal: σ^(1)_xx=σ^(1)_zz. Now we can treat the linear conductivity as a scalar parameter and calculate the linear loss as:α(ω_j)=σ^(1)_s,xx(ω_j)/2𝒫_j∫_G|𝐞(ω_j)_∥|^2dℓ ,where 𝐞(ω_j)_∥ is the electric field component tangential to the graphene sheet. Using symmetry considerations one can prove that the third order conductivity tensor of graphene has only the following nonzero elements <cit.>:σ^(3)_s, xxxx =σ^(3)_s, zzzz , σ^(3)_s, xxzz =σ^(3)_s, zzxx , σ^(3)_s, xzxz =σ^(3)_s, zxzx , σ^(3)_s, xzzx =σ^(3)_s, zxxz , σ^(3)_s, xxxx =σ^(3)_s, xxzz+σ^(3)_s, xzxz+σ^(3)_s, xzzx .Moreover, simulations show that the modes in the SiN waveguides used in this work are quasi-transversal, meaning that e_x≫e_z. This implies that the term containing σ^(3)_s, xxxx in Eq. (<ref>) is about two orders of magnitude larger than any of all other terms, the expression for the nonlinear parameter can be simplified to:γ(ω_i;ω_p,ω_p,-ω_s)≈ i3 σ^(3)_s, xxxx(ω_i; ω_p, ω_p, -ω_s)/16𝒫_p√(𝒫_i𝒫_s)∫_Ge^*(ω_i)_xe(ω_p)_xe(ω_p)_xe^*(ω_s)_xdℓ≈ i 3 σ^(3)_s, xxxx(ω_i; ω_p, ω_p, -ω_s)/16𝒫_p^2∫_G|e(ω_p)_x|^4dℓ .To arrive to the second expression, we have used the assumption that e(ω_p)≈e(ω_s)≈e(ω_i), which is the case when one considers the same spatial modes and small detunings (Δω≪ω_p). We can further generalize this expression to arbitrary graphene shapes:γ(ω_i;ω_p,ω_p,-ω_s)≈ i 3 σ^(3)_s, xxxx(ω_i; ω_p, ω_p, -ω_s)/16𝒫_p^2∫_G|𝐞(ω_p)_∥×ê_z|^4dℓ .In this Letter, the electric field profile 𝐞 was calculated for the cross-section of the waveguide using COMSOL Multiphysics^. The above formula was then used to convert between the waveguide nonlinear parameter γ and the material parameter σ^(3)_s. § GRAPHENE GATING USING POLYMER ELECTROLYTEIn this work, a polymer electrolyte (LiClO_4 and polyethylene oxide in a weight ratio of 0.1:1) is used to gate the graphene, i.e. to electrostatically change the carrier density or Fermi energy of the graphene. Fig. <ref>a shows a sketch of the sample cross-section. Each waveguide is covered by patterned graphene, which is contacted at both sides, making a simple resistance measurement possible by applying a voltage V_DS. The whole sample is covered with the polymer electrolyte, by applying a voltage V_GS to a `gate' contact (in principle any isolated contact in the vicinity of the waveguide, in the Letter the contact on the adjacent waveguide is used) the carrier density in the graphene can be tuned. The dependence of the Fermi energy E_F on the gate voltage V_GS can be approximated by the following formula <cit.>:V_GS-V_D = sgn(E_F)e E_F^2/ħ^2 v_F^2 π C_EDL+E_F/e ,with e the electron charge, v_F≈ 10^6 m/s the Fermi velocity and C_EDL the electric double layer capacitance. V_D is the Dirac voltage, the voltage at which the graphene becomes intrinsic and at which the conductance reaches a minimum. Note that Eq. (<ref>) is derived at temperature T=0 K, but the difference with room temperature is negligibly small (see Fig. <ref>). To obtain an estimate of V_D, an electrical resistance measurement of the gated graphene is used. Fig. <ref> shows the total resistance between the source and drain electrode as a function of the V_GS voltage. Based on this measurement we estimate V_D≈ 0.64 V. To estimate the capacitance C_EDL, a measurement of the optical loss through the waveguide, as a function of V_GS can be used. Fig. <ref> shows the measured loss and corresponding fit, which is proportional to the real part of the linear conductivity σ^(1)_s, which was calculated using the Kubo formula <cit.>. The obtained value for C_EDL is 1.8·10^-2 F m^-2 (other used parameters for this fit were ħΓ=10 meV and T=293 K). The resulting relation between Fermi energy and gate voltage is shown in Fig. <ref>.Figure <ref>b shows an optical microscope image of an actual set of contacted graphene-covered waveguides. The SiN waveguides can be seen, as well as the grating couplers used to couple to the optical fiber. To provide enough space for the contact needles, only every other waveguide is covered with a section of graphene. The graphene is not visible on the optical microscope image, therefore its extent is shown by the dashed lines. The graphene is contacted at both sides. On top of this structure the polymer electrolyte is spin-coated (not in this image).§ EVALUATION OF Σ_S,XXXX^(3) We compare our experimental results with the theory of Refs. <cit.>. Analytical expressions for the third order conductivity σ^(3)_s,αβγδ(ω_i;ω_p,ω_p,-ω_s) have been derived in these papers at T=0 and at the relaxation rate Γ=Γ_0 independent of energy. The calculated |σ^(3)_s,xxxx(ω_p;ω_p,ω_p,-ω_p)| is shown in Fig. <ref> by the solid curve; one can see that the Fermi-energy dependence of the third order conductivity at T=0 and Γ=Γ_0 has a step-like shape (compare with Fig. 9(b) from Ref. <cit.>) which does not look similar to the experimental curves in Fig. 3(a) of the main text. However, as discussed in the main text, a more realistic model of the relaxation rate supposes an energy dependence of Γ(E) given by Eq. (4) of the main Letter. Using this model for Γ(E) we plot |σ^(3)_s,xxxx(ω_p;ω_p,ω_p, -ω_p)| by the dashed curve at T=0 and by the dotted curve at room temperature (kT=25 meV); a good qualitative agreement of the dotted curve with the experimental data is now evident. The parameters Γ_0, α and E_0 are chosen to quantitatively fit the characteristic features of the curves |σ^(3)(E_F)| and |σ^(3)(λ_s-λ_p)| (e.g. the linewidth, the maxima and minima) to the experimental ones. 10S_osgood2009engineering R. M. Osgood, N. C. Panoiu, J. I. Dadap, Xiaoping Liu, Xiaogang Chen, I-Wei Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov. Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires. 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Third order optical nonlinearity of graphene. New Journal of Physics, 16(5):053014, 2014.S_thareja2015electrically Vrinda Thareja, Ju-Hyung Kang, Hongtao Yuan, Kaveh M. Milaninia, Harold Y. Hwang, Yi Cui, Pieter G. Kik, and Mark L. Brongersma. Electrically tunable coherent optical absorption in graphene with ion gel. Nano Letters, 15(3):1570–1576, 2015.S_das2008monitoring A. Das, S. Pisana, B. Chakraborty, S. Piscanec, S. K. Saha, U. V. Waghmare, K. S. Novoselov, H. R. Krishnamurthy, A. K. Geim, A. C. Ferrari, et al. Monitoring dopants by raman scattering in an electrochemically top-gated graphene transistor. Nature Nanotechnology, 3(4):210–215, 2008.S_falkovsky2007space L. A. Falkovsky and A. A. Varlamov. Space-time dispersion of graphene conductivity. The European Physical Journal B, 56(4):281–284, 2007.S_mikhailov2007 S. A. Mikhailov and K. Ziegler. New electromagnetic mode in graphene. Physical Review Letters, 99:016803, Jul 2007.S_falkovsky2007optical L. A. Falkovsky and S. S. Pershoguba. Optical far-infrared properties of a graphene monolayer and multilayer. Physical Review B, 76(15):153410, 2007.S_Cheng2015 J. L. Cheng, N. Vermeulen, and J. E. Sipe. Numerical study of the optical nonlinearity of doped and gapped graphene: From weak to strong field excitation. Physical Review B, 92:235307, Dec 2015.S_Mikhailov2016 S. A. Mikhailov. Quantum theory of the third-order nonlinear electrodynamic effects of graphene. Physical Revie B, 93:085403, Feb 2016. | http://arxiv.org/abs/1704.08567v2 | {
"authors": [
"Koen Alexander",
"N. A. Savostianova",
"S. A. Mikhailov",
"Bart Kuyken",
"Dries Van Thourhout"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170427134117",
"title": "Electrically Tunable Optical Nonlinearities in Graphene-Covered SiN Waveguides Characterized by Four-Wave Mixing"
} |
BAM! The Behance Artistic Media Dataset for Recognition Beyond PhotographyMichael J. Wilber^1,2Chen Fang^1Hailin Jin^1Aaron Hertzmann^1 John Collomosse^1Serge Belongie^2 ^1 Adobe Research^2 Cornell TechDecember 30, 2023 ====================================================================================================================================================Computer vision systems are designed to work well within the context of everyday photography. However, artists often render the world around them in ways that do not resemble photographs. Artwork produced by people is not constrained to mimic the physical world, making it more challenging for machines to recognize.This work is a step toward teaching machines how to categorize images in ways that are valuable to humans. First, we collect a large-scale dataset of contemporary artwork from Behance, a website containing millions of portfolios from professional and commercial artists. We annotate Behance imagery with rich attribute labels for content, emotions, and artistic media. Furthermore, we carry out baseline experiments to show the value of this dataset for artistic style prediction, for improving the generality of existing object classifiers, and for the study of visual domain adaptation.We believe our Behance Artistic Media dataset will be a good starting point for researchers wishing to study artistic imagery and relevant problems. This dataset can be found at <https://bam-dataset.org/> § INTRODUCTION “Art is an effort to create, beside the real world, a more humane world.” – André Maurois Recent advances in Computer Vision have yielded accuracy rivaling that of humans on a variety of object recognition tasks. However, most work in this space is focused on understanding photographic imagery of everyday scenes. For example, the widely-used COCO dataset <cit.> was created by “gathering images of complex everyday scenes containing common objects in their natural context.” Outside of everyday photography, there exists a diverse, relatively unexplored space of artistic imagery, offering depictions of the world as reinterpreted through artwork. Besides being culturally valuable, artwork spans broad styles that are not found in everyday photography and thus are not available to current machine vision systems. For example, current object classifiers trained on ImageNet and Pascal VOC are frequently unable to recognize objects when they are depicted in artistic media (Fig. <ref>). Modeling artistic imagery can increase the generality of computer vision models by pushing beyond the limitations of photographic datasets. In this work, we create a large-scale artistic style dataset from Behance, a website containing millions of portfolios from professional and commercial artists. Content on Behance spans several industries and fields, ranging from creative direction to fine art to technical diagrams to graffiti to concept design. Behance does not aim to be a historical archive of classic art; rather, we start from Behance because it represents a broad cross-section of contemporary art and design.Our overall goal is to create a dataset that researchers can use as a testbed for studying artistic representations across different artistic media.This is important because existing artistic datasets are too small or are focused on classical artwork, ignoring the different styles found in contemporary digital artwork.To solidify the scope of the problem, we choose to explore three different facets of high-level image categorization: object categories, artistic media, and emotions. These artistic facets are attractive for several reasons: they are readily understood by non-experts, they can describe a broad range of contemporary artwork, and they are not apparent from current photographic datasets.We keep the following goals in mind when deciding which attributes to annotate. For object categories, we wish to annotate objects that may be drawn in many different visual styles, collecting fewer visually distinct categories but increasing the density (instances per category) and breadth of representation. ImageNet and COCO, for example, contain rich fine-grained object annotations, but these datasets are focused on everyday photos and cover a narrow range of artistic representation.For media attributes, we wish to annotate pictures rendered with all kinds of professional media: pencil sketches, computer-aided vector illustration, watercolor, and so on. Finally, emotion is an important categorization facet that is relatively unexplored by current approaches.There are several challenges, including annotating millions of images in a scalable way, defining a categorization vocabulary that represents the style and content of Behance, and using this resource to study how well object recognition systems generalize to unseen domains. According to our quality tests, the precision of the labels in our dataset is 90%, which is reasonable for such a large dataset without consortium level funding.Our contributions are twofold: * A large-scale dataset, the Behance Artistic Media Dataset, containing almost 65 million images and quality assurrance thresholds. We also create an expert-defined vocabulary of binary artistic attributes that spans the broad spectrum of artistic styles and content represented in Behance. This dataset can be found at <https://bam-dataset.org/> upon release late Spring 2017. * An investigation of the representation gap between objects in everyday ImageNet photographs and objects rendered in artistic media on Behance. We also explore how models trained on one medium can transfer that performance to unseen media in a domain adaptation setting. To investigate aesthetics and art styles, we compare performance of different kinds of features in predicting emotion and media and show how Behance Artistic Media can be used to improve style classification tasks on other datasets. Finally, we briefly investigate style-aware image search, showing how our dataset can be used to search for images based on their content, media, or emotion.We believe this dataset will provide a starting foundation for researchers who wish to expand the horizon of machine vision to the rich domain of artwark.§ RELATED WORK Attributes and other mid-level representations <cit.> have a long and rich history in vision.Attributes have been applied to aesthetics and other artistic qualities, usually with a focus on photography. For instance, Obrador <cit.>, Dhar <cit.>, and Murray <cit.> collect descriptive attributes such as interestingness, symmetry, light exposure, and depth of field. Work by Peng <cit.>, You <cit.>, Jou <cit.>, and Borth <cit.> study emotional attributes in photographs. Others describe image style not in attributes, but in terms of low-level feature correlations as in work done by Gatys <cit.>, Lin <cit.>, and others. We are more concerned about high-level image categorization than low-level texture transfer. Ours is not the only dataset focused on artwork. We compare related artistic datasets in Tab. <ref>. Most are focused exclusively on everyday photographs <cit.>, but some <cit.> include classical paintings. Likewise, Ginosar <cit.> discuss person detection in cubist art. The work of Fang <cit.> also studies Behance imagery, but does not collect descriptive attributes. Recently, Google released the “Open Images“ dataset <cit.> containing some media-related labels including “comics”, “watercolor paint”, “graffiti”, etc. However, it is unclear how the quality of the labeling was evaluated. Each of these labels contain less than 400 human-verified images and there are no labels that capture emotions.Our work is most similar in spirit to Karayev <cit.>, which studies photographic image style. They collect annotations for photographic techniques, composition, genre, and mood on Flickr images, as well as a set of classical painting genres on Wikipaintings. Our focus is on non-photorealistic contemporary art.To our knowledge, our work is the first work seeking to release a large-scale dataset of a broad range of contemporary artwork with emotion, media, and content annotations.§ THE BEHANCE MEDIA DATASETOur dataset is built from <http://behance.net>, a portfolio website for professional and commercial artists. Behance contains over ten million projects and 65 millionimages. Images on Behance are grouped into Projects, the fundamental unit of categorization. Each Project is associated with metadata, including a title, description, and several noisy user-supplied tags.Artwork on Behance spans many fields, such as sculpture, painting, photography, graphic design, graffiti, illustration, and advertising. Graphic design and advertising make up roughly one third of Behance. Photography, drawings, and illustrations make up roughly another third. This artwork is posted by professional artists to show off samples of their best work. We encourage the reader to visit <http://behance.net> to get a sense of the diversity and quality of imagery on this site. Example images from Behance are shown in Fig. <ref>.Selecting attribute categories. In this work, we choose to annotate our own artistic binary attributes. Attribute names are rendered in . Our attributes capture three categorization facets: * Media attributes: We label images created in , , , , , , and .* Emotion attributes: We label images that are likely to make the viewer feel , , , and .* Entry-level object category attributes: We label images containing , , , , , , , , and . We chose these attributes as follows: The seven media attributes were chosen on the expert advice of a resident artist to roughly correspond with the genres of artwork available in Behance that are easy to visually distinguish. Our goal is to strike a balance between distinctive media while covering the broad range available in Behance. For instance, oil paint and acrylic are considered to be different media by the artistic community, but are very hard for the average crowdworker to distinguish visually.The four emotion attributes are seen on Plutchik's Wheel of Emotions <cit.>, a well-accepted model for emotions that was also used in <cit.>. From this model, we chose the emotions that are likely to be visually distinctive. The content attributes represent entry-level object categories and were chosen to have some overlap with Pascal VOC while being representative of Behance content. We focus on entry-level categories because these categories are likely to be rendered in a broad range of styles throughout Behance. Although this work is only concerned with a small set of labels (arguably a proof-of-concept), the dataset we release could itself be the basis for a real PASCAL/COCO-sized labeling effort which requires consortium-level funding.Tags are noisy. Behance contains user-supplied tags, and one may wonder whether it is feasible to train attribute classifiers directly from these noisy tags alone, such as in previous work <cit.>. However, unlike that work, we cannot create our dataset from tags alone for two reasons. First, not all of our attributes have corresponding tags. Second, tags are applied to each project, not each image. For example, even though a project called “Animal sketches 2012” may have the “Dog” tag, we do not know which image that tag should apply to. Training on tags alone is too noisy and reduces the final classifier precision. To demonstrate, we train a binary classifier on the “Cat” tag, but from manual inspection, it only learns to distinguish different small animals and is not fine-grained enough to find cats; see Fig. <ref>. The precision of cats among the top 100 detections is only about 36%. To increase this accuracy, we must rely on human expertise to collect labels.§.§ Annotation pipeline Our dataset requires some level of human expertise to label, but it is too costly to collect labels for all images. To address this issue, we use a hybrid human-in-the-loop strategy to incrementally learn a binary classifier for each attribute. Our hybrid annotation strategy is based on the LSUN dataset annotation pipeline described in <cit.>, which itself shares some similarity with other human-in-the-loop collection systems <cit.>. An overview of this process is shown in Fig. <ref>.At each step, humans label the most informative samples in the dataset with a single binary attribute label. The resulting labels are added to each classifier's training set to improve its discrimination. The classifier then ranks more images, and the most informative images are sent to the crowd for the next iteration. After four iterations, the final classifier re-scores the entire dataset and images that surpass a certain score threshold are assumed to be positive. This final threshold is chosen to meet certain precision and recall targets on a held-out validation set. This entire process is repeated for each attribute we wish to collect. Crowdsourcing task. The heart of our human-in-the-loop system is the actual human annotation task. We collect annotations for each attribute independently. To do this, we rely on Amazon Mechanical Turk, a crowdsourced marketplace. Crowdworkers (“Turkers”) complete Human Intelligence Tasks for a small cash payment. In each HIT for a given attribute, we show the Turker 10 handpicked positive/negative example images and collect 50 binary image annotations. Turkers indicate whether each image has the attribute of interest. Each HIT only collects labels for a single attribute at a time to avoid confusion. For quality control, we show each image to two separate Turkers and only use answers where both Turkers agree.We also collect sparse text annotations for a subset of these images. Every 10 images, we present an annotation recently provided by the Turker and ask for a brief 3-word caption to justify their choice. This has the effect of encouraging annotators to carefully consider and justify their choices.It is always important to balance the trade-off between squeezing high-quality work out of annotators while being respectful of their effort and abilities. The subjectivity of our task makes this trade-off harder to manage. To address this issue, we prevent spam by only accepting work from crowdworkers who previously completed 10,000 MTurk tasks with a 95% acceptance rate, and we only use labels where both workers agree. Finally, the quality of the entire dataset is ensured by setting appropriate label thresholds on held-out validation data; see Sec. <ref>Iterative learning. Starting from a small handpicked initial label set, the dataset is enlarged by an iterative process that alternates between training a classifier on the current label set, applying it to unlabeled images, and sending unconfident images back to the crowd for more labeling. On each iteration, we train a deep learning classifier using 10/11^ths of the total collected crowd labels. The last 1/11^th is always held out for validation. We apply this classifier to the entire dataset. The crowd then labels 5,000 images that score higher than a threshold set at 50% precision measured on validation data. This way, we show our Turkers a balance of likely-positive and likely-negative images each time. After four iterations, we arrive at a final classifier that has good discrimination performance on this attribute. We score the entire dataset with this classifier and use thresholds to select the final set of positives and negatives. The positive score threshold is chosen on validation data such that the precision of higher-scoring validation images is 90%, and the negative threshold is chosen such that the recall of validation images above this threshold is 95%. In this way, we can ensure that our final labeling meets strict quality guarantees. It is important to note that the resulting size of the dataset is determined solely by the number of relevant images in Behance, our desired quality guarantees, and the accuracy of the final classifier. A better attribute classifier can add more images to the positive set while maintaining the precision threshold. If we need more positive data for an attribute, we can sacrifice precision for a larger and noisier positive set. Classifier. For content attributes, our classifier is a fine-tuned 50-layer ResNet <cit.> originally trained on ImageNet. For emotion and media attributes, we found it better to start from StyleNet <cit.>. This model is a GoogLeNet <cit.>, fine-tuned on a style prediction task inferred from user behavior. Each network is modified to use binary class-entropy loss to output a single attribute score. To avoid overfitting, we only fine-tune for three epochs on each iteration. See Fig. <ref> for examples of Behance images. Resulting dataset statistics Our final dataset includes positive and negative examples for 20 attributes. The median number of positive images across each attribute is 54,000, and the median number of negative images is 8.7 million. The “People” attribute has the most positive images (1.74 million). Humans are commonly featured as art subjects, so this is not surprising. The attribute with the least positives is “Cat” with 19,244 images. We suspect this is because our final labeling model cannot easily distinguish cats from other cat-like renditions. Cats on Behance are commonly rendered in many different styles with very high intra-class variation. Statistics for all attributes are shown in Fig. <ref>.Our automatic labeling model can amplify the crowd's annotation effort. The ratio of automatic positive labels to crowd-annotated positive labels is 17.4. The amplification factor for negative labels is much higher—about 505—because automatic systems can quickly throw away easy negatives to focus the crowd's attention on potentially relevant images. Final quality assuranceAs a quality check, we tested whether the final labeling set meets our desired quality target of 90% precision. For each attribute, we show annotators 100 images from the final automatically-labeled positive set and 100 images from the final negative set using the same interface used to collect the dataset. Fig. <ref> shows worker agreement on the positive set as a proxy for precision. The mean precision across all attributes is 90.4%, where precision is the number of positive images where at least one annotator indicates the image should be positive.These checks are in addition to our MTurk quality checks: we only use human labels where two workers agree and we only accept work from turkers with a high reputation who have completed 10,000 tasks at 95% acceptance.§ EXPERIMENTSWe can use Behance Artistic Media to study recognition across artistic domains as well as aesthetics and style. First, we investigate the representation gap between objects that appear in everyday photographs and objects that appear in artwork. We find that ordinary object detectors do not adequately recognize artistic depictions of objects, showing that there is room for improvement. The existence of this gap leads us to explore the relationship between object representations as rendered across different artistic media. We pose this as a domain transfer problem and measure the extent to which knowledge about objects in one medium can apply to objects in an unseen medium. In addition to objects, we briefly consider style and aesthetics by comparing different features on emotion/media classification and using our style labels to improve aesthetic prediction tasks on other art datasets. Finally, we conclude with an experiment of learning feature spaces (feature disentangling) to build a task-specific search engine that can search for images according to their content, emotion, or media similarity. §.§ Bridging the representation gapDetecting objects in artwork. How different are objects in everyday photographs compared to the stylized objects found in our dataset? We expect that existing pre-trained object detectors might not recognize objects in artwork because existing object detectors trained on ImageNet or VOC are only exposed to a very narrow breadth of object representations. Objects in photographs are constrained by their real-world appearance.To investigate the representation gap between our dataset and everyday photographs, we consider 6 content attributes that correspond to Pascal VOC categories: , , , , , . We then extract scores for these attributes using two object detectors trained on VOC: YOLO <cit.> and SSD <cit.>. For the sake of comparison, we use these detectors as binary object classifiers by using the object of interest's highest-scoring region from the detector output. We also compare to ResNet-50 classifiers <cit.> trained on ImageNet, taking the maximum dimension of the ImageNet synsets that correspond with the category of interest. In this way, we can measure how well existing object detectors and classifiers already find objects in art without extra training. We also compare to our final attribute classifier trained in Sec. <ref>, the fine-tuned ResNet-50 that was used to automatically label the final dataset. We evaluate these methods on 1,000 positives and 1,000 negatives on each attribute's human-labeled validation set to avoid potential bias from the automatic labeler. The results are shown as precision/recall curves in Fig. <ref> and AP is shown in Tab. <ref>. Vision systems trained on photography datasets like VOC (YOLO, SSD) and ImageNet (RN50) perform worse than vision systems that saw objects in artwork during training. From manual inspection, most false negatives of these systems involve objects rendered with unique artistic styles. Specific failure cases are shown in Fig. <ref>.We can improve performance slightly by fusing ImageNet and Behance scores together with a simple linear combination. The resulting “Fusion” model performs slightly better than our own model and ResNet-50 on all but two attributes. These results show that in terms of object recognition, there is a representational gap between photography and artwork.Object representation across artistic media. The existence of this representational gap leads us to question how objects are represented across different artistic media. How well do models trained on one medium generalize to unseen media, and which media are most similar? We can answer these questions within the context of domain adaptation, which has been extensively studied in the vision literature <cit.>. A good model should know that although cats rendered in drawings are more “cartoony” and abstract than the realistic cats seen in oil paint and ImageNet, they both contain the same “cat” semantic concept, even though the context may vary.We retrieve the 15,000 images that maximize σ(x_i,c)σ(x_i,m) for every pair of content and media labels (c,m), where σ is the sigmoid function and x_i is that image's label confidence scores. We set aside 1/11th of these as the validation set. Note that this validation set is a strict subset of the validation set used to train the automatic labeler. We then fine-tune a pre-trained ResNet for one epoch. The last layer is a 9-way softmax.In the first set of experiments, we measure an object classifier's ability to generalize to an unseen domain by learning the representation styles across the other 6 media types and evaluating on only the 7th media type. Results are summarized on the last row of Tab. <ref> and broken down by object categories in Fig. <ref>. Generally, objects that are iconic and easily recognizable within each medium have the highest performance (for example, +, +, +),but objects that are unlikely to be drawn consistently within each style have the worst generalization performance ( /, ). Even though the frequency was controlled by sampling a constant number of images for every (object,medium) pair, this could be because the artist is less familiar with uncommon objects in their medium and has more individual leeway in their portrayal choices. These experiments reveal how well classifiers can generalize to unseen domains, but they do not reveal the correlations in object style between different media types. To capture this, our second set of experiments trains an object classifier on only a single media type and evaluates performance on a second media type. As an additional photography medium, we also retrieve 15,000 images for each object from its corresponding ImageNet synset. Average object classification accuracy is shown in Tab. <ref>. The N-1 baseline model is trained on all other types. This metric gives a rudimentary comparison of the similarity between artistic media; for instance, , , andare similar to each other, as areand . In addition to the gap between ImageNet and Behance (compare last two rows), these results illustrate the gap between each meium's stylistic depictions. Our dataset can be used to explore these relationships and other similar domain adaptation problems. R[2] >angle=#1,lap=-(#2) l < §.§ Style and aestheticsTurning away from object categories for a moment, we now consider tasks related to stylistic information using the emotion and media labels in our dataset. We first investigate the effectiveness of different pre-trained features on emotion and media classification, and then show how to improve aesthetic and style classifiers on other artistic datasets.Feature comparison. How well can object recognition models transfer to emotion and media classification? Do models fine-tuned for style tasks forget their object recognition capabilities?To find out, we compare a linear SVM trained on pre-trained ResNet features to two style prediction models: a linear SVM trained on StyleNet features <cit.> and a StyleNet fine-tuned on Behance Artistic Media. The original StyleNet model was a GoogLeNet that was trained for a style prediction task. We hypothesize that it may outperform ResNet on tasks related to emotion and media classification.We evaluate these models on held-out human labels for each attribute.Performance for six attributes is shown in Fig. <ref>. For all four emotion attributes and 4/6 media attributes, the AP of linear classifiers on StyleNet features outperformed ImageNet-derived features. However, ImageNet-derived features have higher AP than StyleNet features on all nine content attributes.Different features are useful for content tasks compared to emotion/media tasks, and our dataset can help uncover these effects. Aesthetic classification on other datasets. Other artistic datasets such as Wikipantings and AVA contain photographic style annotations. How well do models trained on our dataset perform on these datasets? We show that automatic labels from Behance Artistic Media can slightly improve style classification on existing datasets. We evaluate on the three datasets introduced in <cit.>: 80,000 images in 20 photographic styles on Flickr, 85,000 images from the top 25 styles on Wikipaintings, and the 14,000 images with 14 photographic styles from the hand-labeled set of AVA <cit.>. For comparison to previous work <cit.>, we report AVA classification accuracy calculated only on the 12,000 images that have a single style label. To solve this task, we train a joint attribute model (JAM) that outputs all attribute scores simultaneously. Each training sample (x, i, ℓ) is a tuple of image x, attribute index i, and label ℓ∈{-1, 1}. It is not suitable to train this model using ordinary cross entropy because each attribute is not mutually exclusive. Thus, we must use a loss function with two properties: each attribute output should be independent of other attributes and unknown attribute values should not induce any gradient. We lift image x to a 20-dimensional partial attribute vector ŷ∈ℛ^20, where ŷ_j ≠ i = 0 and ŷ_j = i = ℓ. This allows us to train using a soft-margin criterion, loss(x, y) = 1/20∑_i log(1 + exp(- ŷ_i y_i)). Our JAM model is a fine-tuned ResNet-50 model with a linear projection from 1,000 to 20 dimensions.We trained our model for 100 epochs, starting with a learning rate of 0.1 and multiplying it by 0.93 every epoch. The training set includes roughly 2 million images evenly sampled between attributes and evenly distributed between positive and negative images drawn from the automatically-labeled images in Behance Artistic Media.Results are shown on Table <ref>. On all three challenges, our model shows improved results compared to both the original ResNet-50 and StyleNet. This shows that Behance imagery is rich and diverse enough to improve style recognition tasks on other datasets. This is particularly interesting because Flickr and AVA are both focused on photographic style. Categories in AVA are chosen to be useful for aesthetic quality prediction tasks.This shows that models can train on our dataset to improve performance on other aesthetic classification datasets.§.§ Visual subspace learningFinally, we conclude by showing how to learn task-specific subspaces to retrieve images according to content, emotion, or media similarity. One disadvantage of the joint attribute model mentioned above is the lack of separate feature spaces for each task. Consider the case of image retrieval where the goal is to find images that share the content of a query image but not necessarily its artistic medium. We can use Behance Artistic Media to solve this task by treating it as a visual subspace learning problem. Starting from a pre-trained ResNet shared representation, we remove the top layer and add three branches for content, emotion, and media. Each branch contains a linear projection down to a 64-dimensional subspace and a final projection down to label space. The final model is trained similarly to the model in Sec. <ref>. Only the initial ResNet weights are shared; the embedding is separate for each task. We qualitatively show three images close to the query within each task-specific embedding.The results show that this simple strategy can learn sensible task-specific embeddings. Neighbors in latent-content space generally match the content of the query and neighbors in latent-media space generally match the query's artistic medium. The effect is qualitatively weaker for emotion space, perhaps because of the limited label set. From a human inspection of 100 random queries, the precision-at-10 for content, media, and emotion is 0.71, 0.91, and 0.84 respectively. Media and emotion precision-at-10 are slightly improved compared to our shared feature baseline of 0.80, 0.87, 0.80, which could be explained if the shared representation focuses almost exclusively on content. One limitation of this approach is that without any conditioning, the three learned subspaces tend to be correlated: objects close in media-space or emotion-space sometimes share content similarity. Our dataset could provide a rich resource for feature disentangling research.§.§ Visualizing the learned model We qualitatively explore the kind of visual style cues learnable from the proposed dataset in Fig. <ref>.A dataset of 110k images was formed by sorting all 65m Behance Artistic Media images by likelihood score for each of the 7 media and 4 emotion attributes, and sampling the top 10k images in each case.Duplicate images selected across attributes were discarded. A modified Alexnet<cit.> (fc6 layer 1024-D, fc7 layer 256-D) was trained from scratch on the 11 style (media and emotion) attributes for 40 epochs via SGD with learning rate 0.01.Nguyen <cit.> recently proposed a deep generator network (DGN) based visualization technique for synthesizing stimuli preferred by neurons through combination of a truncated (ImageNet trained) CaffeNet and up-convolutional network initialized via white noise.We run the DGM-AM variant of their process for 200 iterations, using a learning rate of 2.0 and weighting factor 99. The images synthesized for several media types (graphite, oil-paint and watercolor paintings) epitomize textures commonly encountered in these art forms although styles exhibiting structural combination of flatter regions are less recognizable.Fragments of objects commonly recognizable within emotion-based styles (e.g. teeth for scary, bleak windows in gloomy or landscapes in peaceful are readily apparent.§ CONCLUSION Computer vision systems need not be constrained to the domain of photography. We propose a new dataset, “Behance Artistic Media” (BAM!), a repository of millions of images posted by professional and commercial artists representing a broad snapshot of contemporary artwork. We collected a rich vocabulary of emotion, media, and content attributes that are visually distinctive and representative of the diversity found in Behance.However, though Behance does include tag metadata, we showed that these tags are too noisy to learn directly. Further, the scale of Behance makes brute-force crowdsourcing unattractive. To surmount these issues, we collected labels via a hybrid human-in-the-loop system that uses deep learning to amplify human annotation effort while meeting desired quality guarantees. The resulting dataset is useful for several computer vision tasks. We use it to highlight the representation gap of current object detection systems trained on photography, showing that Behance captures a wider gamut of representation styles than current sets such as VOC and ImageNet. Different artistic media in Behance have unique aesthetics, providing an interesting test bed for domain transfer tasks, and different features prove useful for content tasks compared to media/emotion classification. We also use Behance to improve the performance of style classification on other datasets, showing that researchers can train on our dataset for a marked improvement in performance. Finally, we conclude with a subspace learning task for retrieving images based on their content or artistic media.We believe our dataset provides a good foundation for further research into the underexplored realm of large-scale artistic imagery.§ ACKNOWLEDGMENTSThis work is partly funded by an NSF Graduate Research Fellowship award (NSF DGE-1144153, Author 1), a Google Focused Research award (Author 6), a Facebook equipment donation to Cornell University, and Adobe Research.All images in this paper were shared with Creative Commons attribution licenses and we thank the creators for sharing them: -jose-, AARON777, AbrahamCruz, AndreaRenosto, AndrewLili, AngelDecuir, AnnaOfsa, Arianne, AstridCarolina, Aubele, ChumaMartin, DannyWu, DubrocDesign, Elfimova, EtienneBas, ExoticMedia, Folkensio, GinoBruni, GloriaFl, HattyEberlein, Hectorag, ImaginaryFS, IsmaelCG, J_Alfredo, JaclynSovern, JanayFrazier, JaneJake, JodiChoo, Joshua_Ernesto, KimmoGrabherr, Kiyorox, Marielazuniga, Mirania, NIELL, NikolayDimchev, NourNouralla, OlivierPi, Ooldigital, Pleachflies, PranatiKhannaDesign, PraveenTrivedi, Radiotec1000, RamshaShakeel, RaoulVega, Roxxo2, Royznx, RozMogani, SaanaHellsten, Sanskruta, Slavina_InkA, SwannSmith, TCaetano, THE-NEW, Thiagoakira, TomGrillo, WayneMiller, Yipori, ZaraBuyukliiska, adovillustration, alobuloe61b, amos, animatorcreator, answijnberg, anthonyrquigley, antoniovizzano, apedosmil, artofmyn, asariotaki, asjabor, assaadawad, atanaschopski, averysauer, ballan, bartosz, beissola, bilbo3d, bluelaky, brainlessrider, brandplus, brittany-naundorff, bucz, campovisual, candlerenglish, carlosmessias, carlosoporras, castilloseas, cgshotgun, charbelvanille, charlottestrawbridge, chin2off, cknara, coyote-spark, crazydiamondstudio, cristianbw, crivellaro, cube74, customshoot, daconde, dafrankish, danielstarrason, dariomaggiore, deboralstewart, dedos, designrahul, digital-infusion, ememonkeybrain, esanaukkarinen, evanwitek, fabsta, flochau, florencia_agra, foundationrugs, gatomontes, giuseppecristiano, gumismydrug, gurudesigner, heroesdesign, higoszi, ifyouprefer, ikuchev, illustrationdesign, irf, irwingb, ivancalovic, jabaarte, jakobjames, janaramos, janeschmittibarra, jdana, jota6six, juliencoquentin, julienps, justindrusso, kalalinks, karoliensoete, katesimpressions, katiemott, kpkeane, larawillson, leovelasco, leturk, lindseyrachael, lisa_smirnova, littleriten, manf_v, margotsztajman, martabellvehi, matildedigmann, maviemamuse, mechiparra, michaelkennedy, monroyilustrador, mvitacca, nataliamon, nati24k, nazpa, nickelcurry, nicopregelj, nobleone, nspanos, nunopereira, olaolaolao, olivierventura, ollynicholson, omarmongepinal, onthebrinkartstudios, osamaaboelezz, pamelapg, paperbenchdesign, parfenov, pengyangjun, pilarcorreia, pilot4ik, polyesterdress, priches, pumaroo, ragamuffin, rainingsuppie, raquelirene, rasgaltwc, rebeccahan, redric, rikishi, roosariski, rrabbstyjke, rufusmediateam, sairarehman, saketattoocrew1, samii69, sandhop, santarromana, santocarone, sashaanikusko, scottchandler, sebagresti, sehriyar, shaneferns, shaungmakeupartist, siqness, snowballstudio, softpill, sompson, sophieecheetham, starkdesigns, studiotuesday, tadzik, tatianaAdz, taylorboren, taylored, tensil, theCreativeBarn, tomtom_ungor, toniitkonen, travisany, tweek, ukkrid, visual3sesenta, zarifeozcalik, zmahmoud, zwetkowieee | http://arxiv.org/abs/1704.08614v2 | {
"authors": [
"Michael J. Wilber",
"Chen Fang",
"Hailin Jin",
"Aaron Hertzmann",
"John Collomosse",
"Serge Belongie"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170427150530",
"title": "BAM! The Behance Artistic Media Dataset for Recognition Beyond Photography"
} |
0a-Top-matter/00a-Top-matter9-Styles/MICE-defs10pt plain arabicnuPanel-RM-Conc-Sect nuPanel-RM-Conc-Sect-Conc[nuPanel-RM-Conc-Sect] nuPanel-RM-Conc-Sect-Rec[nuPanel-RM-Conc-Sect] nuPanel-RM-Conc-Sect-Dec[nuPanel-RM-Conc-Sect]0b-Abstract/00b-Abstract 1-Introduction/01-Introduction 2-Acc-nuOsc/02-Acc-nuOsc 3-Sterile/03-Sterile 4-Supporting-programme/04-Supporting-programme 5-Reactor/05-Reactor 6-Non-terrestrial/06-Non-terrestrial 7-Non-osc/07-Non-osc 8-Conc-n-rec/08-Conc-n-rec9-Acknowledgements/09-Acknowledgements99-Styles/utphys 0A-NeutrinoPanel/10A-NeutrinoPanel | http://arxiv.org/abs/1704.08181v1 | {
"authors": [
"J. Cao",
"A. de Gouvea",
"D. Duchesneau",
"S. Geer",
"R. Gomes",
"S. B. Kim",
"T. Kobayashi",
"K. R. Long",
"M. Maltoni",
"M. Mezzetto",
"N. Mondal",
"M. Shiozawa",
"J. Sobczyk",
"H. A. Tanaka",
"M. Wascko",
"G. Zeller"
],
"categories": [
"hep-ex",
"hep-ph",
"physics.ins-det"
],
"primary_category": "hep-ex",
"published": "20170426161800",
"title": "Roadmap for the international, accelerator-based neutrino programme"
} |
K. Sugimura et al. ]Kazuyuki Sugimura,^1E-mail: [email protected] Yurina Mizuno,^1 Tomoaki Matsumoto^2 and Kazuyuki Omukai^1^1Astronomical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan^2Faculty of Sustainability Studies, Hosei University, Fujimi, Chiyoda, Tokyo 102-8160, Japan Fates of the dense cores formed by fragmentation]Fates of the dense cores formed by fragmentation of filaments: do they fragment again or not? [ [ December 30, 2023 =====================-1cmFragmentation of filaments into dense cores is thought to be an important step in forming stars.The bar-mode instability of spherically collapsing cores found in previous linear analysis invokes a possibility of re-fragmentation of the cores due to their ellipsoidal (prolate or oblate) deformation.To investigate this possibility, here we perform three-dimensional self-gravitational hydrodynamics simulations that follow all the way from filament fragmentation to subsequent core collapse.We assume the gas is polytropic with index γ, which determines the stability of the bar-mode.For the case that the fragmentation of isolated hydrostatic filaments is triggered by the most unstable fragmentation mode, we find the bar mode grows as collapse proceeds if γ< 1.1, in agreement with the linear analysis.However, it takes more than ten orders-of-magnitude increase in the central density for the distortion to become non-linear.In addition to this fiducial case, we also study non-fiducial ones such as the fragmentation is triggered by a fragmentation mode with a longer wavelength and it occurs during radial collapse of filaments and find the distortion rapidly grows.In most of astrophysical applications, the effective polytropic index of collapsing gas exceeds 1.1 before ten orders-of-magnitude increase in the central density.Thus, supposing the fiducial case of filament fragmentation, re-fragmentation of dense cores would not be likely and their final mass would be determined when the filaments fragment.stars:formation – galaxies: star formation – galaxies: evolution. § INTRODUCTIONWhat determines the mass of stars?In the standard scenario of star formation, stars are formed inside dense cores with the mass conversion efficiency of about 30%, as supported by both simulations <cit.> and observations <cit.>.Dense cores, in turn, are thought to be formed by fragmentation of filamentary molecular clouds, or simply filaments. While observations with the Herschel satellite have revealed that the dense cores are located along filaments <cit.>, simulations have shown that filaments fragment into cores that subsequently collapse in a run-away fashion <cit.>.At the end of the collapse, which is well approximated with the self-similar solution of spherical collapse <cit.>, protostars are formed when the central parts of the dense cores become optically thick. The theoretical works on filament fragmentation, both analytical <cit.> and numerical <cit.> ones, have shown that the typical fragmentation mass is given by the Jeans mass at fragmentation of the filaments. This gives reasonable estimate for the initial mass of dense cores, but the final mass can be dramatically altered if the cores will fragment again in later evolution. Such re-fragmentation can be caused by deformation of the cores due to the bar-mode instability of the Larson-Penston solution found in the previous linear stability analyses <cit.>. They have shown that thebar (l=2) mode is unstable if γ<1.1, where γ is the effective polytropic index of gas.This instability deforms cores into a prolate or oblate shape depending on the seed perturbation for the instability. Unfortunately, the role of this instability in star formation was unclear, because the linear analysis can predict neither the initial amplitude of the unstable mode nor the fate of the instability in the non-linear regime.In order to address this issue, it is necessary to perform numerical simulations. In this paper, we investigate the possibility of re-fragmentation of dense cores formed by filament fragmentation, focusing on the role of the bar-mode instability.We continuously follow filament fragmentation and subsequent core collapse with three-dimensional self-gravitational hydrodynamics simulations.We study the γ dependence of the evolution of cores, as well as the dependence on the way in which filaments fragment.Although there have been a number of simulations for core collapse after filament fragmentation <cit.>, none of them has focused on the possibility of re-fragmentation or the role of the bar-mode instability. The paper is organized as follows. In Sec. <ref>, we describe our models and numerical methods.In Sec. <ref>, we present the result of our simulations.The conclusion and discussion are given in Sec. <ref>. § MODELS & METHODS§.§ BasicsBelow, we briefly summarize the γ dependence of the two types of instabilities, namely the fragmentation-mode and bar-mode instabilities, which motivates the models of this work.We first introduce some useful variables and then review the fragmentation-mode instability of filaments and the bar-mode instability of the Larson-Penston solution.The polytropic gas is characterized by the equation of state,P = K ρ^γ ,and the sound speed is obtained asc_s=√(Kγρ^γ-1) .We define the free-fall time[For our convenience, we adopt the definition of t_ff in equation (<ref>), instead of (3/32π G ρ)^1/2, which is also used in the literature.]ast_ff=1/√(4π Gρ) ,and the Jeans length asλ_J=√(π/Gρ)c_s=2π c_st_ff .Once a reference density ρ_0 is given, we can obtain c_s,0, t_ff,0 and λ_J,0 for ρ_0 with equations (<ref>) – (<ref>). Using these quantities and their combinations, all dimensional quantities in this work can be made dimensionless.Infinitely long and static filaments are subject to gravitational instability of axisymmetric modesthat leads to fragmentation into cores.The linear stability analysis of static polytropic filaments <cit.> has shown that amplitude of the most unstable mode δ_F,max grows exponentially with the growth rate σ_F,max, asδ_F,max∝exp[σ_F,maxt/t_ff,0] .We take the data for the wave number k_max and squared growth rate μ_max (=σ_F,max^2) of the most unstable mode from Fig. 9 of <cit.> and fit them as functions of γ for 1≤γ≤ 1.5. As a result, we obtain the fitting formulae,k_maxH_0 =2.30-2.89 γ+1.77 γ^2-0.374 γ^3 ,andμ_max =0.0688+0.236 γ-0.268 γ^2+0.0781 γ^3 ,where the central density of the filament is assumed to be ρ_0 and H_0=2c_s,0/√(2π G ρ_0)=(√(2)/π) λ_J,0 is the width of filaments <cit.>.Table <ref> presents the wavelength of the most unstable mode λ_max (=2π/k_max) and σ_F,max for selected values of γ, computed with the above fitting formulae. The linear analyses <cit.> have shown the Larson-Penston solution is unstable if γ<1.1 due to the bar-mode instability. The amplitude of the bar mode δ_B grows in a power-law fashion with the growth rate σ_B, asδ_B∝(t_col-t)^-σ_B∝(ρ_max)^σ_B/2 .To derive the second relation, we have used the relation ρ_max∝ (t_col-t)^-2 for the Larson-Penston solution, where t_col is the time when ρ_max formally diverges.Using the data for σ_B taken from Fig. 2 of <cit.>, we obtain the fitting formula,σ_B = -2.84 + 9.39 γ - 6.20 γ^2 ,for 0.9≤γ≤ 1.1.This formula is evaluated for some values of γ and shown in Table <ref>.[Note that there is a room for numerical error even in the linear analysis because the eigenmode is numerically obtained.As a result, the reported values of the critical γ for the bar-mode instability, 1.097 in <cit.> and 1.09 in <cit.>, are similar but not exactly the same.] §.§ Models We perform simulations that follow the fragmentation of filaments and subsequent collapse of cores, assuming the gas is polytropic (equation <ref>).The initial conditions are generated by adding velocity perturbations to filaments. Below, we will describe the models studied in our simulations. The initial density profile is given byρ_ini(r)=f ρ_st(r) ,where ρ_st is the density profile of a hydrostatic polytropic filament axisymmetric about the z axis <cit.>, r=√(x^2+y^2) the cylindrical radius and f a density enhancement factor.Here, we take ρ_0=ρ_st(0). For reference, the central density of filaments in local star forming regions is about 10^-20-10^-18g cm^-3<cit.>. We numerically obtain ρ_st except for γ=1, when ρ_st is obtained analytically as ρ_st=ρ_0(1+r^2/H_0^2)^-2 with H_0=2c_s,0/√(2π G ρ_0). We assume that the filament is under the external pressure of an ambient gas with density ρ_ext. Fragmentation of filaments is triggered by the initial velocity perturbations.We assume a perturbation of v_z with the sinusoidal z dependence and the initial velocity field is given byv_ini(x)=([0;0; v_0sin(2π z/λ) ]) ,where v_0 andλ are the amplitude and wavelength of the perturbation, respectively.The simulations are run with the model parameters summarized in Table <ref>.We first study the fiducial case where the fragmentation of isolated hydrostatic filaments is triggered by the most unstable mode, as often supposed in the literature <cit.>.To investigate the γ dependence of the evolution, we perform the simulations of set “G” (for “gamma”) with different γ (=0.9, 0.95, 1, 1.05, 1.1 and 1.2).In this case, we take f=1, v_0=10^-3 c_s,0, λ=λ_max and ρ_ext=0.(see Sec. <ref>).The very small perturbation is adopted just to follow the fragmentation triggered purely by the most unstable eigenmode and the evolution is astrophysically relevant only after the mode grows up to have a certain amplitude. In the actual astrophysical situations, however, fragmentation can proceed in a non-fiducial way.Thus we perform four sets of simulations with the model parameters different from the fiducial ones.The parameters are set to the fiducial values unless otherwise stated. First, we study the effect of the deviation of the initial velocity perturbation from the most unstable eigenmode with the simulations of set “V” (for “velocity”). For this set, we enhance v_0 to 0.1, 0.3 and 0.5 c_s,0 (see equation <ref>) and emphasize the converging nature of the initial velocity field.Second, we study fragmentation triggered by modes with various wavelength by performing the simulations of set “L” (for “lambda”), for which we change λ to 0.6, 1.5 and 2 λ_max (see equation <ref>).Third, to study fragmentation of radially collapsing filaments, we perform the simulations of set “D” (for “density”), for which we enhances the initial density taking f as 1.05, 1.1 and 1.2 (see equation <ref>), with v_0 also increased to 0.1 and 0.5 c_s,0.In the above three cases, we take γ = 1, 1.05 and 1.2 to see the γ dependence. Finally, to study fragmentation of filaments under external pressure of an ambient gas, we perform the simulations of set “E” (for “external”), for which we change the density of an ambient gas by taking ρ_ext = 0.01, 0.04 and 0.09. We take γ = 1 for set E.§.§ Numerical methodsWe use the self-gravitational magneto-hydrodynamics code with adaptive mesh refinement (AMR), SFUMATO <cit.>, but with the magnetic module switched off.The hydrodynamical solver adopts the total variation diminishing cell-centred scheme with second-order accuracy in space and time. Our computational domain is a cube with the side length L_box=λ (see equation <ref>).We solve the three-dimensional (3D) hydrodynamics with the Cartesian coordinate set, without using the axisymmetry of the system.We initially set a uniform grid with N_ini=256 meshes in each direction (256^3 cells). The Jeans condition is employed as a refinement criterion in the block-structured AMR technique of the code. Blocks are refined to resolve one Jeans length λ_J (equation <ref>) with at least N_ref=32 meshes.We test the convergence of the numerical results in Appendix <ref>. We assume the periodic boundary condition in the z direction.In the r direction, however, we fix the density to the boundary value ρ_b and the velocity to zero outside the boundary at r_b.We take r_b as the radius where the initial density is equal to the ambient density, i.e., ρ_ini(r_b)=ρ_ext, and ρ_b as ρ_ext.If the cylinder with r_b is larger than the computational box, r_b and ρ_b are replaced with L_box/2 and ρ_ini(L_box/2), respectively.For the case ρ_b=0, we take a finite but sufficiently small value (e.g., ρ_b=10^-8ρ_0) in the actual calculations for computational reason. We assume the gravitational potential for the isolated filament on the surfaces of the computational box in the x and y directions.§ RESULTS§.§ Evolution in a typical caseIn this section, we describe the time evolution in the γ=1.05 model of the fiducial case (fourth line in Table <ref>), as a typical example of our simulations.Although the shapes of the cores at the end of simulations are greatly different depending on the models, the evolution generally proceeds in a way similar to that described below. Fig. <ref> shows the density and velocity distributions in the xz-plane at the four different stages of evolution: (a) the maximum (central) density ρ_max/ρ_0=1 (initial time), (b) 10, (c) 10^6 and (d) 10^16. Note that we use ρ_max/ρ_0 as a time variable since it monotonically increases with the time as collapse proceeds.The panels in Fig. <ref> show that filament fragmentation and subsequent collapse of the core proceed as follows:(a) A small velocity perturbation to the filament is seen in the initial condition. It is a seed for the most unstable fragmentation mode, which later grows and leads to the fragmentation of the filament.(b) Left: A high density region, or core, is formed due to the fragmentation of the filament and starts collecting the surrounding gas gravitationally. Right: The core is initially prolate along the filament.(c) Left: The core collapses in a run-away fashion. Right: The gas dynamics near the centre approaches the Larson-Penston solution. The formerly prolate core becomes nearly spherical, although slightly oblate.(d) Left: The run-away collapse continues.Right: Once the gas dynamics becomes sufficiently close to the Larson-Penston solution, the bar-mode instability begins to grow. Accordingly, the core becomes more and more oblate and the distortion finally becomes non-linear. Below, we will examine the evolution in more detail, focusing on the fragmentation of the filament and the collapse and deformation of the core. First, we see how fragmentation occurs.To quantify the degree of fragmentation, we define the overdensity δ normalized by ρ_0=ρ_ini(0) (equation <ref>) asδ = ρ_max(t)-ρ_0/ρ_0 ,where ρ_max(t) is the maximum density at t. The density is largest at the centre of the core since the beginning of the fragmentation.The fragmentation is roughly completed when δ∼ O(1).Fig. <ref> shows the time evolution of δ, along with the linear growth rate of the most unstable fragmentation mode (σ_F,max=0.33; Table <ref>).The agreement of the two slopes for 10^-2≲δ≲ 1 indicates that the fragmentation is caused by the growth of the most unstable mode, as we expect for the fiducial case. Note that only the result for δ≳ 10^-2 is reliable because that for lower δ depends on the resolution (see Appendix <ref>). Second, we examine the convergence of the core collapse to the Larson-Penston solution.To quantify how much the central dynamics is close to the Larson-Penston solution, we introduce the ratio t_dyn/t_ff,[ For the same purpose, <cit.> introduced the normalized central density z_0 using a relation ρ_max=z_0/(t_col-t)^2.We can show t_dyn/t_ff=√(z_0)/2 with equation (<ref>).]where the dynamical time scale t_dyn is defined ast_dyn= ρ_max/ρ̇_max ,and the free-fall time scale t_ff is given by equation (<ref>) with ρ=ρ_max.This ratio indicates the rapidness of the collapse: t_dyn/t_ff=0.41 for the homogeneous gravitational collapse <cit.> and it increases to t_dyn/t_ff=0.76 for the Larson-Penston solution with γ=1.05 <cit.>, because the pressure delays the collapse.The evolution of t_dyn/t_ff is shown in Fig. <ref>(a). Soon after the fragmentation, the collapse is still slow and t_dyn/t_ff is large. Later on, the dynamics of core approaches the Larson-Penston solution <cit.>, as indicated by the decrease of t_dyn/t_ff toward the value for the Larson-Penston solution (horizontal dashed line). However, the approach is rather slow and it takes about five orders of magnitude in density increase for the collapse to become close to the Larson-Penston solution. Note that the features of t_dyn/t_ff appearing every four-time increase in the density are originated from numerical errors at refinement, although they hardly affect the result (Appendix <ref>). Finally, we see how the core deforms. We use the axial ratio to quantify the deformation, where the axes of the core are defined using the inertia tensor <cit.>, as below.We regard the central dense region with ρ > ρ_th=0.1 ρ_max as the core,and then its inertia tensor and total mass are given respectively byI_ij=∫_ρ>ρ_thdxx^ix^j ρ(x) ,and M=∫_ρ>ρ_thdx ρ(x) .We can assume that the three eigenvalues of I_ij/M, denoted as λ_i (i=1, 2 and 3), satisfy λ_1=λ_2 without loss of generality thanks to the axisymmetry of the core. Thus, we define the axes of the core in the x, y and z directions asa_x = a_y = √(λ_1) ,a_z =√(λ_3) ,respectively.We have checked that adopting different ρ_th has little influence on our results.The evolution of the axial ratio a_x/a_z (=a_y/a_z) of the core is shown in Fig. <ref>(b), where a_x/a_z <1, =1 and >1 correspond to prolate, spherical and oblate shapes, respectively.Note the definition of the axes of the core is not valid and thus we do not plot a_x/a_z for ρ_max/ρ_0 ≲ 10 because the integration region in equations (<ref>) and (<ref>) is not confined in the computational domain.As expected, the evolution of a_x/a_z is consistent with the density profiles seen in Fig. <ref>.We see that a_x/a_z is initially small but rapidly increases and eventually exceeds unity. Afterwards, it stops increasing and remains almost constant with a_x/a_z≳ 1 for a while, although it finally begins to increase again due to the bar-mode instability. To analyze the growth of the bar mode in more detail, we define the oblateness <cit.> asΔ = a_x-a_z/(a_x+a_z)/2 ,where a_x (=a_y) and a_z are defined in equation (<ref>). Fig. <ref>(c) shows the evolution of Δ. The oblateness Δ changes its sign at ρ_max/ρ_0∼ 10^2, as the axial ratio changes from a_x/a_z<1 to a_x/a_z>1 (Fig. <ref>b), and begins to increase in a power-law fashion with a small initial amplitude of Δ≲ 0.1 at ρ_max/ρ_0∼ 10^5, when the collapse becomes sufficiently close to the Larson-Penston solution (Fig. <ref>a).The rate of the power-law increase is consistent with the linear analysis of the bar-mode instability (σ_B=0.089; Table <ref>),[ It can be shown that the amplitude of the bar mode δ_B (equation <ref>) is proportional to the oblateness Δ (equation <ref>) in the linear regime.]indicating that the distortion is caused by the bar mode. We emphasize here that the bar mode grows only when the collapse is sufficiently close to the Larson-Penston solution. The growth rate is slightly smaller than in the linear analysis due to several reasons. First, the background collapse is not exactly the Larson-Penston solution. Second, not only the pure bar mode but also other modes contribute to Δ. Third, the growth rate tends to be smaller in the non-linear regime owing to the small dynamic range of Δ (-2<Δ<2).Note also that the growth rate can be overestimated in the linear analysis <cit.> due to numerical error (see the footnote in Sec. <ref>). §.§ γ dependenceHere we investigate the γ dependence of the evolution in the fiducial case, where fragmentation of isolated hydrostatic filaments is triggered by the most unstable mode, as often considered in the literature <cit.>.Below, we present the results for the models with γ=0.9, 0.95, 1, 1.05, 1.1 and 1.2 (set G in Table <ref>).Fig. <ref> shows the final shapes of the cores at ρ_max/ρ_0=10^12.Although the evolution generally proceeds in a way similar to that explained in Sec. <ref> irrespective of γ, the final shapes are greatly different depending on γ, as summarized in Table <ref>.Below, we will see how this dependence arises.Fig. <ref>(a) shows the evolution of the axial ratio a_x/a_z. Although a_x/a_z generally increases in the initial phase (ρ_max/ρ_0≲10^2), it evolves differently later on depending on the value of γ.In the γ= 0.9 model, a_x/a_z begins to decrease before reaching unity due to the rapid growth of bar-mode instability, and thus the core becomes prolate. In the less unstable case with γ= 0.95, the evolution is similar but the final distortion is weaker.In the models with γ=1 and 1.05, the bar-mode instability is so weak that a_x/a_z exceeds unity before the instability begin to grow and thus the cores become oblate.The oblateness is stronger in the model with γ=1 than with γ=1.05 because the growth rate of the bar-mode instability is larger for smaller γ.In the models with γ=1.1 and 1.2, the cores become spherical because the spherical collapse is stable and the bar-mode perturbation damps.How the growth of the bar-mode instability depends on γ is clearly seen in Fig. <ref> (b), where we plot the evolution of the oblateness Δ (equation <ref>) for the models with γ≤ 1.05.Except for the γ=0.9 model, the bar mode begins to grow in a power-law fashion with a small initial amplitude of Δ≲ 0.1 at ρ_max/ρ_0 ∼ 10^5 – 10^6, when the background spherical collapse becomes close to the Larson-Penston solution, as seen for the γ=1.05 case in Sec. <ref>.Because of this, in astrophysically interesting cases of 1≤γ < 1.1 (see Sec. <ref>), it needs about ten orders-of-magnitude increase in the density before the distortion becomes significant.In the γ=0.9 model, the bar mode is so strong that it begins to grow in the relatively early stage of the evolution (ρ_max/ρ_0 ∼ 10^4) with a certain initial amplitude (Δ≳ 0.1).It is also seen that the bar mode grows faster for smaller γ, as expected from the linear analysis <cit.>.The growth rates agree well with those obtained by the linear analysis (dashed lines), confirming that the distortion is caused by the bar-mode instability.In summary, the γ dependence of the final shapes of the cores can be understood from the γ dependence of the bar-mode instability.The core becomes spherical if the bar mode is stable (γ≥1.1), whereas it tends to deform if the bar mode is unstable (γ<1.1).In the case with strong instability (γ<1), the deformation begins before the axial ratio reaches unity and the core becomes prolate.In the case with weak instability (1≤γ<1.1), however, the deformation begins after the axial ratio exceeds unity and the core becomes oblate. §.§ Non-fiducial casesHere, we investigate the fragmentation of filaments proceeding in a non-fiducial way, i.e., the cases different from the fragmentation of isolated hydrostatic filaments triggered by the most unstable fragmentation mode. Astrophysically, various situations can be encountered: for example, unstable modes other than the most unstable one can trigger fragmentation, fragmentation can occur during radial collapse of filaments, filaments are not isolated but under the external pressure of an ambient gas, etc. Considering these possibilities, we perform simulations with the different amplitude and wavelength of the initial velocity perturbation in Sec. <ref> and Sec. <ref>, respectively. We then perform simulations with the enhanced filament density in Sec. <ref>.To see the γ dependence, we take γ=1, 1.05 and 1.2 for the cases above. Finally, we study fragmentation of filaments under external pressure in Sec. <ref>.The parameters are fixed to the fiducial values, unless otherwise stated. As will be shown below, the results in this section suggest that the evolution of the cores can be largely affected by how filament fragmentation proceeds.Thus, the results obtained for the fiducial case should be treated with caution in astrophysical applications.§.§.§ Fragmentation by perturbation with large amplitude Here, we present the results of our simulations for the models with different amplitudes of the initial velocity perturbation.Since the initial perturbation is not exactly the most unstable eigenmode, the converging nature of the initial velocity field becomes important as the initial amplitude increases.The amplitude is taken to be v_0/c_s,0=10^-3 (fiducial), 0.1, 0.3 and 0.5 (set V and a part of set G in Table <ref>). Fig. <ref> shows the evolution of a_x/a_z for (a) γ=1.2, (b) 1.05 and (c) 1.For all γ, the dependence on the initial amplitude is small unless δ v/c_s,0 is as large as 0.5. In the case δ v/c_s,0=0.5, the core is compressed due to the converging initial velocity field (equation <ref>) and becomes more oblate than in the other cases in the early phase (ρ_max/ρ_0 ≲ 10^2).The subsequent evolution is similar to the other cases for (a) γ=1.2 and (b) 1.05. For (c) γ=1, however, the distortion becomes non-linear at somewhat smaller ρ_max/ρ_0 due to the larger oblateness in the early phase.These results suggest that the evolution is almost the same as the fiducial case, where the most unstable eigenmode triggers the fragmentation, as long as δ v/c_s,0≤ 0.3. This is because the most unstable mode grows and dominates the other modes before fragmentation, although the initial perturbation given by equation (<ref>) is not exactly the most unstable eigenmode.We conclude that the dependence on the amplitude of the initial velocity perturbation is weak as long as δ v/c_s,0≲ 0.3.§.§.§ Fragmentation by perturbation with different wavelengthHaving seen the cases with various amplitudes of the initial perturbation, now we see the cases with various wavelengths of it. Here, we present the results for the models with λ/λ_max=0.6, 1 (fiducial), 1.5 and 2 (set L and a part of set G in Table <ref>). We see in all models that the filaments initially fragment into the cores. This is expected because the linear analysis shows that modes with λ/λ_max≳ 0.5 are unstable <cit.>. Fig. <ref> shows the evolution of a_x/a_z for (a) γ=1.2, (b) 1.05 and (c) 1. Below, weexamine each case separately.Firstly, for (a) γ=1.2, the λ dependence is weak and a_x/a_z finally converges to unity irrespective of λ.Secondly, for (b) γ=1.05, however, the λ dependence is strong and a_x/a_z becomes larger than unity if λ/λ_max≤ 1 but becomes less than unity if λ/λ_max≥ 1.5 at the end of the simulations.In the early stage (ρ_max/ρ_0≲ 10^2), the core collects gases from more distant regions in the z direction and thus tends to be more prolate with larger λ.As a result, the bar mode begins to grow before a_x/a_z reaches unity and thus the core becomesprolate if λ/λ_max≥ 1.5, while the bar mode begins to grow after a_x/a_z exceeds unity and thus the core becomes oblate if λ/λ_max≤ 1.In the case λ/λ_max= 0.6, the core becomes spherical because the pressure is relatively strong compared to the gravitational force in the core with small mass.Finally, for (c) γ=1, we see a trend similar to (b) γ=1.05 case, although the core is more easily distorted due to the stronger bar-mode instability.In the case with λ = 2 λ_max, the core is always prolate with a_x/a_z≲ 0.5 and evolves into a very elongated shape. These results indicate that the evolution of the core strongly depends on the wavelength of perturbation, or equivalently the interval of fragments under our periodic boundary condition in the z direction. If the wavelength is longer than that of the most unstable mode, the core evolves differently from the fiducial case andtends to become prolate.§.§.§ Fragmentation during radial collapse of filamentHere, we present the results for filaments fragmenting during their cylindrical radial collapse.To induce the radial collapse of filaments, we enhance the initial density with the density enhancement factor f (equation <ref>).Meanwhile, we add a certain amplitude of the initial velocity perturbation, to see an interplay between the radial collapse of filaments and fragmentation <cit.>.If the amplitude were extremely small, the cylindrical radial evolution would proceed too much before fragmentation begins, i.e., the filament would collapse into the z axis (γ≤1) or settle into a hydrostatic state (γ>1).In this section, we perform simulations with f=1 (fiducial), 1.05, 1.1 and 1.2 and v_0/c_s,0=0.1 and 0.5 (set D and some of set V in Table <ref>).We see in all models that filaments initially fragment into the cores, although the fragmentation cannot be well discriminated from the cylindrical radial collapse in the model with γ=1, f=1.2 and v_0/c_s,0=0.1.Fig. <ref> presents the evolution of a_x/a_z for (a) (γ, v_0/c_s,0)=(1.2, 0.1), (b) (1.05,0.1), (c) (1, 0.1), (d) (1.2, 0.5), (e) (1.05, 0.5) and (f) (1, 0.5), which we explain below.Firstly, for (a, d) γ=1.2 and v_0/c_s,0= 0.1 and 0.5, a_x/a_z finally converges to unity in all cases.Secondly, for (b) γ=1.05 and v_0/c_s,0= 0.1, however, the overall motion of the collapsing filaments induces prolate deformation of the core in the early phase, resulting in a more prolate shape in the subsequent evolution with larger f.Thirdly, for (d) γ=1.05 and v_0/c_s,0=0.5, the trend is the same as (b) v_0/c_s,0= 0.1 but weaker, because the effect of overall motion is less significant due to the larger flow velocity in the z direction.Finally, for (c, f) γ=1 and v_0/c_s,0= 0.1 and 0.5, we again see a similar trend to that seen for (b, d) γ=1.05 but with larger distortion due to the stronger bar-mode instability, as seen in Sec. <ref>. In summary for this section, cylindrical radial collapse of filaments can have a significant impact on the evolution of the cores, although its effect is reduced if v_0/c_s,0 is large.If the filament fragments during its radial collapse, the core tends to become prolate. §.§.§ Fragmentation of filaments under external pressure Finally, we show the results for fragmentation of static filaments under the external pressure by an ambient medium. Although we have so far studied the ideal cases with isolated filaments, filaments are indeed embedded in an interstellar medium with finite pressure.Here, we study only the cases of γ=1 and perform simulations with ρ_ext/ρ_0=0,[In practice, the density at the boundary is taken as ρ_b/ρ_0=3.8×10^-3 even though ρ_ext/ρ_0=0, because of the finite size of our computational box (see Sec. <ref>).] 0.01, 0.04 and 0.09 (set E and one of set V in Table <ref>). The external pressure corresponds to the line mass of the filaments.The line mass is defined as M_line≡∫_0^r_filρ(r)2π rdr, with the filament radius r_fil given by ρ(r_fil)=ρ_ext. The density profile of the static filament with γ=1 is given by ρ(r)=ρ_0(1+r^2/H_0^2)^-2 and M_line takes its maximum when the filament is isolated, i.e., ρ_ext=0 and r_fil=∞.This maximum value, M_line,cr=2c_s^2/G <cit.>, is called the critical line mass because only sub-critical filaments can be hydrostatic and super-critical ones are always gravitationally unstable.The line mass of filaments embedded in an ambient gas with finite ρ_ext is smaller than M_line,cr, as M_line/M_line,cr=1-(ρ_ext/ρ_0)^1/2 <cit.>.Thus, the cases studied here with ρ_ext/ρ_0=0, 0.01, 0.04 and 0.09 correspond to M_line/M_line,cr = 1, 0.9, 0.8 and 0.7, respectively.We see in all cases that the filaments fragment into the cores, which subsequently become oblate as collapse proceeds (Fig. <ref>).The dependence on the external pressure is weak in our cases examined, although the cores tend to be more oblate as the external pressure increases.Such a trend can be understood as follows.Since M_line is smaller for larger ρ_ext, as mentioned above, the mass of fragmented cores, and hence their gravitational potential, is also smaller.As a result, gravitational collapse of the cores is delayed and the cores have more time to obtain oblate distortion in the early phase of the evolution (ρ_max/ρ_0≲ 10^3), which explains the observed trend.Consistently, we also find that the time when ρ_max/ρ_0 reaches 10^12 is longer for larger ρ_ext (t/t_ff,0=8.3, 8.3, 8.5 and 8.9 for ρ_ext/ρ_0=0, 0.01, 0.04 and 0.09, respectively).Here, we find that the shape of collapsing cores depends only weakly on the external pressure.This implies that our previous results for isolated static filaments can be extended to the case with static sub-critical filaments under the external pressure.Since we have already studied fragmentation of radially collapsing super-critical filaments in Sec. <ref>, our results encompass the cases with sub- and super-critical filaments, both of which are found in observations <cit.>.§ CONCLUSION AND DISCUSSION We have studied the collapse of dense cores formed by fragmentation of filaments assuming a polytropic gas with γ. By employing the adaptive mesh refinement (AMR) technique, we are able to follow the filament fragmentation and subsequent core collapse continuously in a single run of simulation.Since the self-similar spherical collapse solution, the so-called Larson-Penston solution, is known to be unstable due to the bar-mode instability if γ<1.1, we have focused on how this instability affects the evolution of the cores. We have found that the cores formed by fragmentation of filaments tend to become spherical in the early phase of the collapse but later begin to distort due to the bar-mode instability, if exists.In this paper we regard the fragmentation of an isolated hydrostatic filament triggered by the most unstable fragmentation mode <cit.> as a fiducial case.For the fiducial case with 1≤γ< 1.1, we have found the distortion becomes significant only after the central density increases by more than ten orders of magnitude.This is because the distortion begins to grow out of a small seed when the background spherical collapse becomes sufficiently close to the Larson-Penston solution, which takes about five orders of magnitude increase in the central density. For the fiducial case with the other γ, we see the core becomes strongly distorted if γ≤ 1 while the core always becomes spherical if γ≥ 1.1.The γ dependence of the evolution can be understood from the fact that the bar-mode instability exists for γ<1.1 and becomes stronger with decreasing γ. In addition, we have studied the filament fragmentation that occurs in a non-fiducial way and have found the evolution of the cores can be largely affected by the way of filament fragmentation.The distortion grows much faster than in the fiducial case, if the fragmentation is triggered by a perturbation with wavelength longer than that of the most unstable mode or proceeds during the cylindrical radial collapse of filaments.We caution that it is necessary to check whether the fragmentation proceeds in the fiducial way when applying our results for the fiducial case in an astrophysical context. How the filament fragmentation proceeds is determined by the initial condition.Theoretically, it can be addressed by simulations of filament formation in a turbulent medium, where the perturbation is automatically provided at the time of filament formation.However, although many authors have performed such simulations <cit.>, none of them have focused on the subsequent collapse of the fragmented cores.To reduce the uncertainty coming from the initial condition, it is important to perform a simulation similar to this work but starting from filament formation in future. Let us discuss an astrophysical implication of our results on the mass of dense cores.As in the literature <cit.>, the core mass at fragmentation of filaments can be estimated as follows. In forming stars from the interstellar medium, the temperature evolution with the increasing density draws a evolutionary path in a density-temperature plane that is determined by environmental conditions, such as metallicity and external radiation field <cit.>.Using the effective polytropic index γ defined with this path, we estimate the physical state of the gas at each stage of the evolution. Suppose a filamentary gas initially collapses with γ<1. It stops its radial collapse when γ exceeds unity, because the critical γ for filaments is γ_cr=1, i.e., the pressure increases more (less) rapidly than the gravitational force if γ>1 (γ<1). Then, the pressure-supported filament fragments into dense cores <cit.>. The cores subsequently collapse and form stars inside.Here, the mass of the fragments can be estimated as the Jeans mass for the density and temperature when γ exceeds unity <cit.> and gives a good estimate for the initial mass of the cores. The final mass of the star-forming cores, however, can be largely altered if the distortion of collapsing cores results in their re-fragmentation.We have shown for the fiducial case that it takes more than ten orders-of-magnitude in the density increase for the distortion to become non-linear.Thus, such re-fragmentation is not likely in most cases, because the phase with γ<1.1 does not last such long.This can happen, however, in the following situations in the early Universe: in supermassive (∼ 10^5 M_⊙) star formation in strongly irradiated pristine clouds <cit.>, the gas collapses with almost constant temperature of ∼10^4K due to the Lyα cooling; in Pop II star formation in the very high-redshift (z≳ 20) Universe, the gas evolves with the temperature of the cosmic microwave background at that time <cit.>.In these exceptional cases, the cores can be significantly distorted and finally re-fragment before forming stars. In addition, it should be emphasized again that in non-fiducial cases, i.e., if the filament is not hydrostatic or fragmentation is not triggered by the most-unstable mode, the re-fragmentation of the cores can be important. Here, we give qualitative estimate of the condition for re-fragmentation, although its numerical investigation is out of the scope of this work.Suppose that the collapse of distorted cores is delayed at some moment, possibly due to the increase of γ.In such a case, a rough estimate can be made using the critical wavelengths for unstable modes of static filaments and sheets, about four and six times the scale length, respectively <cit.>.If the axial ratio of the cores is larger than twice the ratio of the critical wavelengths to the scale length, i.e., eight for prolate cores and twelve for oblate ones, they are able to fragment into more than two depending on the initial amplitude of the unstable modes.For more realistic estimate, however, numerical simulations dedicated to the re-fragmentation of distorted cores are needed. The initial condition dependence found in this work suggests an observational relation between the physical state of filaments and the shapes of the cores within them <cit.>.We suggest that the cores tend to be more prolate along the filaments if the interval of the cores is longer than the wavelength of the most-unstable mode, i.e., four times the diameter of the filaments <cit.>, or the filaments show the sign of overall cylindrical radial collapse.These relations can be observationally tested. In the current calculation, we have neglected the effects of a magnetic field and a rotational and turbulent velocity field, in order to extract only the effect of the bar-mode instability on the evolution of cores. We should ultimately account for them in studying the evolution of the dense cores formed from filaments, as it has been suggested that a rotational velocity field <cit.> and a magnetic field <cit.> affect the evolution of cores. It is also known that the turbulence generated during gravitational collapse of cores can play an important role in the evolution of the cores <cit.>.We would like to address these issues in future publication. § ACKNOWLEDGEMENTSThe authors would like to thank Gen Chiaki, Shu-ichiro Inutsuka, Kazunari Iwasaki, Sanemichi Takahashi and Toru Tsuribe for fruitful discussions.The numerical simulations were performed on the Cray XC30 at CfCA of the National Astronomical Observatory of Japan.This work is supported in part by MEXT/JSPS KAKENHI Grant Number 15J03873 (KS), 26400233, 26287030 and 24244017 (TM) and 25287040 (KO).natexlab#1#1[André et al.(2010)André, Men'shchikov, Bontemps, Könyves, Motte, Schneider, Didelon, Minier, Saraceno, Ward-Thompson, di Francesco, White, Molinari, Testi, Abergel, Griffin, Henning, Royer, Merín, Vavrek, Attard, Arzoumanian, Wilson, Ade, Aussel, Baluteau, Benedettini, Bernard, Blommaert, Cambrésy, Cox, di Giorgio, Hargrave, Hennemann, Huang, Kirk, Krause, Launhardt, Leeks, Le Pennec, Li, Martin, Maury, Olofsson, Omont, Peretto, Pezzuto, Prusti, Roussel, Russeil, Sauvage, Sibthorpe, Sicilia-Aguilar, Spinoglio, Waelkens, Woodcraft, & Zavagno]Andre:2010aa André, P., Men'shchikov, A., Bontemps, S., et al. 2010, , 518, L102[Arzoumanian et al.(2011)Arzoumanian, André, Didelon, Könyves, Schneider, Men'shchikov, Sousbie, Zavagno, Bontemps, di Francesco, Griffin, Hennemann, Hill, Kirk, Martin, Minier, Molinari, Motte, Peretto, Pezzuto, Spinoglio, Ward-Thompson, White, & Wilson]Arzoumanian:2011aa Arzoumanian, D., André, P., Didelon, P., et al. 2011, , 529, L6[Bromm & Loeb(2003)]Bromm:2003aa Bromm, V., & Loeb, A. 2003, , 596, 34[Chiaki et al.(2016)Chiaki, Yoshida, & Hirano]Chiaki:2016aa Chiaki, G., Yoshida, N., & Hirano, S. 2016, , 463, 2781[Clarke et al.(2016)Clarke, Whitworth, & Hubber]Clarke:2016ac Clarke, S. 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S. 1996, , 471, 822[Safranek-Shrader et al.(2014)Safranek-Shrader, Milosavljević, & Bromm]Safranek-Shrader:2014ab Safranek-Shrader, C., Milosavljević, M., & Bromm, V. 2014, , 440, L76[Stodólkiewicz(1963)]Stodolkiewicz:1963aa Stodólkiewicz, J. S. 1963, , 13, 30[Sugimura et al.(2014)Sugimura, Omukai, & Inoue]Sugimura:2014aa Sugimura, K., Omukai, K., & Inoue, A. K. 2014, , 445, 544[Truelove et al.(1997)Truelove, Klein, McKee, Holliman, Howell, & Greenough]Truelove:1997aa Truelove, J. K., Klein, R. I., McKee, C. F., et al. 1997, , 489, L179[Tsuribe & Inutsuka(1999)]Tsuribe:1999aa Tsuribe, T., & Inutsuka, S.-i. 1999, , 526, 307[Tsuribe & Omukai(2006)]Tsuribe:2006aa Tsuribe, T., & Omukai, K. 2006, , 642, L61[Yahil(1983)]Yahil:1983aa Yahil, A. 1983, , 265, 1047 § RESOLUTION CHECKTo check the resolution dependence of our results, here we repeat the simulation shown in Sec. <ref> (γ=1.05 model of set G in Table <ref>) but with different resolutions.In our AMR simulation, there are two parameters controlling the resolution: the initial number of meshes in each direction, N_ini, and the minimum number of meshes per one Jeans length λ_J (equation <ref>), N_ref.The former controls the resolution during the initial fragmentation phase while the latter does during the subsequent collapse phase.We take N_ini=256 and N_ref=32 as the fiducial resolution in this paper, but here we change N_ini to 128, 512 or 1024 or N_ref to 8, 16 or 64. Fig. <ref> shows the N_ini dependence of the time evolution of δ=(ρ_max-ρ_0)/ρ_0 (equation <ref>). We see that the evolution of δ can be correctly followed from the earlier stage by adopting larger N_ini, although N_ini hardly affects the evolution at the time of fragmentation.With fiducial N_ini=256, the evolution is reliable after δ≳ 10^-2. We plot the N_ref dependence of (a) t_dyn/t_ff (see above equation <ref>), (b) a_x/a_z (equation <ref>) and (c) Δ=2(a_x-a_z)/(a_x+a_z) (equation <ref>) in Fig. <ref>.We see in panel (a) that the features of t_dyn/t_ff appearing every four-time density increase are generated by numerical errors at refinement, which can be suppressed by adopting larger N_ref.These features, however, affect the overall evolution only slightly.Panels (b) and (c) show the N_ref dependence of the growth of the bar-mode instability. We see that the N_ref dependence is substantial for N_ref=8, but becomes weaker for N_ref=16 and cannot be seen by eyes for N_ref≥ 32.This suggests that the minimum resolution is N_ref∼ 16 and that our fiducial resolution with N_ref=32 is sufficient.Let us discuss the minimum resolution required to correctly solve the dynamics of self-gravitating gas.We here obtain the condition N_ref≥ 16, required to correctly follow the growth of the bar mode.The most often-used condition in the literature is the so-called Truelove condition, N_ref≥ 4 <cit.>, which is required to avoid artificial fragmentation.More recently, <cit.> suggest to use a condition N_ref≥ 32 to resolve the turbulence generated during gravitational collapse of cores. Note that these three condition are derived to follow the different physical processes.In performing simulations, either one of the above three conditions should be used depending on the process to be followed. | http://arxiv.org/abs/1704.08441v1 | {
"authors": [
"Kazuyuki Sugimura",
"Yurina Mizuno",
"Tomoaki Matsumoto",
"Kazuyuki Omukai"
],
"categories": [
"astro-ph.SR",
"astro-ph.GA"
],
"primary_category": "astro-ph.SR",
"published": "20170427060423",
"title": "Fates of the dense cores formed by fragmentation of filaments: do they fragment again or not?"
} |
0truemm 0truemm -10truemmplainApplied Mathematical and Computational Sciences Vol. 1, No. 2 (2010), pp. 181–197. FIRST- AND SECOND-ORDERWAVE GENERATION THEORY 0mm6mmReceived 4 May 2010, Accepted 30 June 2010. 2010 Mathematics Subject Classification: 76B15, 74J15, 74J30, 34B05, 34B15, 34K10, 35G30. Key words and Phrases: boundary value problem (bvp), surface waves, signalling problem, first-order and second-order steering. Natanael Karjanto 34ccAbstract. The first-order and the second-order wave generation theory is studied in this paper. The theory is based on the fully nonlinear water wave equations. The nonlinear boundary value problem (bvp) is solved using a series expansion method. Using this method, the problem becomes a set of linear, signalling problems according to the expansion order. The first-order theory leads to a homogeneous bvp.It is a system with the first-order steering of the wavemaker motion as inputand the surface wave field with propagating and evanescent modes as output.The second-order theory leads to a nonhomogeneous bvp. It is a system where the second-order steering of the wavemaker motion is prescribed in such a way that the second-order part of the surface elevation far from the wavemaker contains only the bound wave component and the free wave component vanishes. The second-order surface wave elevation consists of a superposition of bichromatic frequencies.1. INTRODUCTIONIn this paper, we will consider the problem how to generate waves in a wave tank of a hydrodynamic laboratory. The wave tank in the context of this paper is a facility with a wavemaker on one side and an artificial wave absorbing beach on the other side. We consider a tankwith a flat bottom, and no water is flowing in or out of the tank. Typically, the situation is that waves are generated by a flap type wavemaker at one side of a (long) tank; the motion of the flap `pushes' the waves to start propagating along the tank. This means that typically, we are dealing with a signalling problem or a boundary value problem (bvp), which is different from an initial value problem (ivp) when one tries to find the evolution of waves from given surface elevation and velocities at an initial moment. To illustrate this for the simplest possible case, consider the linear, non-dispersive second-order wave equation for waves in one spatial direction x and time t: ∂_t^2η = c^2∂_x^2η. Here, η denotes the surface wave elevation, and c > 0 is the constant propagation speed. The general solution is given by η(x,t) = f(x - ct) + g(x + ct), for arbitrary functions f and g. The term f(x - ct) is the contribution of waves travelling to the right (in the positive x-direction), and g(x + ct) waves running to the left. For the ivp, specifying at an initial time (say at t=0) the wave elevation η(x,0) and the velocity ∂_tη(x,0) determines the functions f and g uniquely. For the bvp, resembling the generation at x=0, we prescribe the wave elevation at x=0 for all positive time, assuming the initial elevation to be zero for positive x (in the tank, this means a flat surface prior to the start of the generation). If the signal is given by s(t), vanishing for t < 0, the corresponding solution running into the tank is η(x,t) = f(x - ct) which should be equal to s(t) at x = 0, leading to η (x,t) = s(t - x/c). Reversely, for a desired wave field f(x - ct) running in the tank, the required surface elevation at x = 0 is given by s(t) = f(-ct). This shows the characteristic property of the bvp for the signalling problem.The actual equations for the water motion and the precise incorporation of the flap motion are much more difficult than shown in the simple example above. In particular, for the free motion of waves, there are two nontrivial effects. The first effect is dispersion: the propagation speed of waves depends on their wavelength (or frequency), described explicitly in the linear theory (i.e. for small surface elevations) by the linear dispersion relation (ldr), see formula (<ref>). In fact, for a given frequency, there is one normal mode that travels to the right as a harmonic wave, the propagating mode, and there are solutions decaying exponentially for increasing distance (the evanescent modes). We will use the right propagating mode and the evanescent modes as building blocks to describe the generation of waves by the wavemaker. For each frequency in the spectrum, the Fourier amplitude of the flap motion is then related to the amplitude of the corresponding propagating and evanescent modes. If we assume these amplitudes to be sufficiently small, say of the `first order' ϵ, with ϵ is a small quantity, we will be able to deal with nonlinear effects in a sequential way. This is needed because the second effect is that in reality, the equations are nonlinear. The quadratic nature of the nonlinearity implies that each two wave components will generate other components with an amplitude that is proportional to the product of the two amplitudes, the so-called `bound wave components' which have amplitudes of the order ϵ^2. These are the so-called `second-order effects'. For instance, two harmonic waves of frequency ω_1,ω_2 and wavenumber k_1, k_2 (related by the ldr) will have a bound wave with frequency ω_1 + ω_2 and wavenumber k_1 + k_2. Since the ldr is a concave function of the wavenumber, this last frequency-wavenumber combination does not satisfy the ldr, i.e. this is not a free wave: it can only exist in the combination of the free wave. This second-order bound wave that comes with a first-order free wave has also its consequence for the wave generation. If the first-order free-wave component is compatible with the flap motion, the presence of the bound wave component will disturb the wave motion, such that the additional second-order free-wave will be generated as well. This is undesired, since the second-order free wave component has a different propagation speed as the bound wave component, thereby introducing a spatially inhomogeneous wave field. That is why we add to the flap motion the additional effects of second-order bound waves, thereby preventing any second-order free-wave component to be generated. This process is called `second-order steering' of the wavemaker motion.This technique can be illustrated using a simple ivp for an ordinary differential equation as follows. Consider the nonlinear equation with a linear operator Lη := ∂_t^2η + ω_0^2η = η^2,for which we look for small solutions, say of order ϵ, a small quantity. The series expansion technique then looks for a solution in the formη = ϵη^(1) + ϵ^2η^(2) + (ϵ^3).Substitution in the equation and requiring each order of ϵ to vanish leads to a sequence of ivps, the first two of which read:η^(1) = 0; η^(2) = (η^(1))^2; ….Observe that the equations for η^(1) and η^(2) are linear equations, homogeneous for η^(1) and nonhomogeneous (with known right-hand side after η^(1) has been found) for η^(2). Suppose that the first-order solution we are interested in is η^(1) = a e^-iω_0t, already introducing the complex arithmetic that will be used in the sequel also. This solution is found for the initial values η^(1)(0) = a, ∂_tη^(1) (0) = -i ω _0 a. Then the equation for η^(2), i.e. η^(2) = a^2 e^-2iω_0t has as particular solution: η_^(2) = A e^-2iω_0t with A = -a^2 /(3 ω_0^2). This particular solution is the equivalent of a `bound wave' mentioned above: it comes inevitably with the first-order solution η^(1). However, η_^(2) will change the initial condition; forcing it to remain unchanged could be done by adding a solution η_^(2) of the homogeneous equation: η_^(2) = 0 that cancels the particular solution at t = 0, explicitly: η_^(2) = -( 3/2 A e^-iω_0t - 1/2 A e^iω_0t). This homogeneous solution corresponds to the second-order free wave mentioned above. To avoid this solution to be present, the initial value has to be taken like:η(0) = ϵ a + ϵ^2A; ∂_tη(0) = -i ϵω_0a - 2iϵ^2ω_0 A.The second-order terms in ϵ in these initial conditions are similar to the second-order steering of the flap motion for the signalling problem.[Just as in this example, the hierarchy of equations also continues for the bvp: there will also be the third and higher order contributions, and bound and free waves in each order. A higher-order steering than the second-order one has not been done until now, since the effects are smaller, although there are some exceptions. ]Besides the two difficult aspects of nature, dispersion and nonlinearity, the precise description of the signal is also quite involved, since the signal has to be described on a moving boundary, the flap, which complicates matters also. For the rest of this paper, we will describe the major details of this procedure. The next section presents the bvp for the wave generation problem. Section 3 and Section 4 discuss the first- and the second-order wave generation theory, respectively. The final section gives some conclusions about this paper. The results can also be found in Dean and Dalrymple <cit.> for the first-order theory as well as in Schäffer <cit.> for the second-order theory, but our presentation is less technical and emphasises the major steps. 2. GOVERNING EQUATIONLet 𝐮 = (u,w) = (∂_xϕ, ∂_zϕ) define the velocity potential function ϕ = ϕ(x,z,t) in a Cartesian coordinate system (x,z). Let also η = η(x,t), Ξ = Ξ(z,t) = f(z)S(t), g, h, and t denote surface wave elevation, wavemaker position, gravitational acceleration, still water depth and time, respectively. The governing equation for the velocity potential is the Laplace equation∂_x^2ϕ + ∂_z^2ϕ = 0, x ≥Ξ(z,t),-h ≤ z ≤η(x,t);that results from the assumption that water (in a good approximation) is incompressible: ∇·𝐮 = 0. The dynamic and kinematic free surface boundary conditions (dfsbc and kfsbc), the kinematic boundary condition at the wavemaker (kwmbc), and the bottom boundary condition (bbc) are given by[dfsbc:∂_tϕ + 1/2|∇ϕ|^2 + g η = 0 atz = η(x,t);;kfsbc:∂_tη + ∂_xη∂_xϕ - ∂_zϕ = 0 atz = η(x,t);;kwmbc: ∂_xϕ - f(z)S'(t) - f'(z)S(t) ∂_zϕ = 0 atx = Ξ(z,t);;bbc:∂_zϕ = 0 atz = -h.; ]The dfsbc is obtained from Bernoulli's equation, the kfsbc and the kwmbc are derived by applying the material derivative to the surface elevation and wavemaker motion, respectively. The bbc is obtained from the fact that water neither comes in nor goes out of the wave tank. Note that the dfsbc and kfsbc are nonlinear boundary conditions prescribed at a yet unknown and moving free surface z = η(x,t). The elevation, potential and wavemaker position are given by the following series expansionsη = ϵη^(1) + ϵ^2η^(2) + ϵ^3η^(3) + … ϕ = ϵϕ^(1) + ϵ^2ϕ^(2) + ϵ^3ϕ^(3) + … S= ϵS^(1) + ϵ^2S^(2) + ϵ^3S^(3) + …,where ϵ is a small parameter, a measure of the surface elevation nonlinearity. The wavemaker we will consider is a rotating flap, see Figure <ref>. It is given by Ξ(z,t) = f(z)S(t), where f(z) describes the geometry of the wavemaker:f(z) = {[1 + z/h + H,for-(h - d) ≤ z ≤ 0;;0, for-h ≤ z < -(h - d).; ].Note that f(z) is given by design, and S(t) is the wavemaker motion that can be controlled externally to generate different types of waves. The center of rotation is at z = -(h + H). If the center of rotation is at or below the bottom, then d = 0 and in fact, we do not have the second case of (<ref>). If the center of rotation is at a height d above the bottom, then d = -H. 3. FIRST-ORDER WAVE GENERATION THEORYIn this section, we solve a homogeneous bvp for the first-order wave generation theory. By prescribing the first-order wavemaker motion as a linear superposition of monochromatic frequencies, we find the generated surface elevation also as a linear superposition of monochromatic modes. After applying the Taylor series expansion of the potential function ϕ around x = 0 and z = 0 as well as applying the series expansion method, the first-order potential function has to satisfy the Laplace equation∂_x^2ϕ^(1) + ∂_z^2ϕ^(1) = 0, x ≥ 0,-h ≤ z ≤ 0.We also obtain the bvp for the first-order wave generation theory at the lowest expansion order. It reads[g η^(1) + ∂_tϕ^(1) =0,z = 0;; ∂_tη^(1) - ∂_zϕ^(1) =0,z = 0;; ∂_xϕ^(1) - f(z) dS^(1)/dt =0,x = 0;;∂_zϕ^(1) =0, z = -h. ]By combining the dfsbc and the dfsbc at z = 0 (<ref>), we obtain the first-order homogeneous free surface boundary conditiong ∂_zϕ^(1) + ∂_t^2ϕ^(1) = 0, atz = 0. We look for the so-called monochromatic wavesϕ^(1)(x,z,t) = ψ(z) e^-i θ(x,t),where θ(x,t) = k x - ω t. Then from the Laplace equation (<ref>), we have ψ”(z) - k ψ(z) = 0, for -h ≤ z ≤ 0. Applying the bbc leads to ψ(z) = αcosh k(z + h), α∈ℂ. From the combined free surface condition (<ref>), we obtain a relation between the wavenumber k and frequency ω, known as the linear dispersion relation (ldr), explicitly given byω^2 = g k tanh k h. Let us assume that the first-order wavemaker motion S^(1)(t) is given by a harmonic function with frequency ω_n and maximum stroke |S_n| from an equilibrium position, represented in complex notation asS^(1)(t) = ∑_n = 1^∞ -1/2i S_n e^i ω_nt + ,where c.c. denotes the complex conjugate of the preceding term. Since this `first-order steering' contains an infinite number of discrete frequencies ω_n, it motivates us to write a general solution for the potential function by linear superposition of discrete spectrum. By choosing the arbitrary spectral coefficient α= ig/2ω_nC_n/cosh k_nh, the first potential function is found to beϕ^(1)(x,z,t) = ∑_n = 1^∞ig/2ω_n C_ncosh k_n(z + h)/cosh k_n h e^-iθ_n(x,t) + ,where θ_n(x,t) = k_n x - ω_nt, with wavenumber-frequency pairs (k_n,ω_n), n ∈ℤ satisfying the ldr (<ref>). For a continuous spectrum, the summation is replaced by an integral. Allowing the wavenumber to be complex valued, the ldr becomesω_n^2 = g k_njtanh k_njh,j ∈ℕ_0.For j = 0, the wavenumber is real and it corresponds to the propagating mode of the surface wave elevation. For j ∈ℕ, the wavenumbers are purely imaginary, and thus i k_nj∈ℝ. Since we are interested in the decaying solution, we choose i k_nj > 0 and hence the modes of these wavenumbers are called the evanescent modes. As a consequence, the first-order potential function can now be written asϕ^(1)(x,z,t) = ∑_n = 1^∞∑_j =0^∞i g/2 ω_n C_njcosh k_nj(z + h)/cosh k_njh e^-i θ_nj(x,t) + ,where θ_nj(x,t) = k_njx - ω_nt. Furthermore, applying the kwmbc (<ref>), integrating along the water depth, and using the property that {cosh k_nj(z + h), cosh k_nl(z + h), j, l ∈ℕ_0} is a set of orthogonal functions for j ≠ l, we can find the surface wave complex-valued amplitude C_nj as followsC_nj = ω_n^2 S_n/g k_njcosh k_njh ∫_-h^0 f(z) cosh k_nj(z + h) dz/∫_-h^0cosh^2 k_nj(z + h) dz= 4 S_nsinh k_njh/k_nj(h + H) k_nj(h + H) sinh k_njh + cosh k_njd - cosh k_njh/2k_njh + sinh (2k_njh),j ∈ℕ_0.Finally, the first-order surface elevation can be found from the dfsbc (<ref>), and is given as follows:η^(1)(x,t) = ∑_n = -∞^∞∑_j =0^∞1/2 C_nj e^-i θ_nj(x,t) +This first-order theory can also be found in Dean and Dalrymple <cit.>.Remark 1. For `practical' purposes, it is useful to introduce the so-called transfer function or frequency response of a system. It is defined as the ratio of the output and the input of a system. In our wave generation problem, we have a system with a wavemaker motion as input and the surface wave amplitude as output. Therefore, the first-order transfer function T_n^(1) is defined as the ratio between the surface wave amplitude of the propagating mode C_n0 as output and the maximum stroke |S_n| as input, explicitly given byT_n^(1) = 4sinh k_n0h/k_n0(h + H) k_n0(h + H) sinh k_n0h + cosh k_n0d - cosh k_n0h/2 k_n0h + sinh 2k_n0h.Figure <ref> shows the first-order transfer function plot as function of wavenumber k_n0 for a given water depth h and the center of rotation d. For increasing k_n0, which also means increasing frequency ω_n, the transfer function is monotonically increasing as well. It increases faster for smaller values of k_n0 and slower for larger values of k_n0, approaching the asymptotic limit of T_n^(1) = 2 for k_n0h →∞. 4. SECOND-ORDER WAVE GENERATION THEORYIn this section, we solve a nonhomogeneous bvp for the second-order wave generation theory. Due to the nonhomogeneous boundary condition at the free surface, which causes interactions between each possible pair of first-order wave components, the resulting surface wave elevation has a second-order effect, known as the bound wave component. Furthermore, due to the first-order wavemaker motion and the boundary condition at the wavemaker, the generated wave also has another second-order effect, namely the free wave component. The latter component is undesired since it results in a spatially inhomogeneous wave field due to the different propagation velocities of bound wave and free wave components with the same frequency. Therefore, in order to prevent the free wave component to be generated, we add an additional second-order bound wave effect to the flap motion. This process is known as `second-order steering' of the wavemaker motion. More details about this theory, including an experimental verification, can be found in Schäffer <cit.>. For the history of wave generation theory, see also references in this paper.Taking terms of the second-order in the series expansion, we obtain the bvp for the second-order wave generation theory. The second-order potential function also satisfies the Laplace equation∂_x^2ϕ^(2) + ∂_z^2ϕ^(2) = 0, x ≥ 0,-h ≤ z ≤ 0.Almost all the second-order boundary conditions now become nonhomogeneous[ g η^(2) + ∂_tϕ^(2)= - (η^(1)∂_tz^2ϕ^(1) + 1/2 |∇ϕ^(1)|^2), z = 0;;∂_tη^(2) - ∂_zϕ^(2)=η^(1)∂_z^2ϕ^(1) - ∂_xη^(1)∂_xϕ^(1), z = 0;;∂_xϕ^(2) - f(z) dS^(2)/dt= S^(1)(t) (f'(z) ∂_zϕ^(1) - f(z) ∂_x^2ϕ^(1)), x = 0;; ∂_zϕ^(2)= 0,z = -h. ]By combining the dfsbc and kfsbc of (<ref>) at z = 0, we have the second-order nonhomogeneous free surface boundary conditiong ∂_zϕ^(2) + ∂_t^2ϕ^(2) = rhs_1, atz = 0.Using the first-order potential function (<ref>), rhs_1 is explicitly given byrhs_1 = . -(∂/∂ t |∇ϕ^(1)|^2 + η^(1)∂/∂ z[g ∂ϕ^(1)/∂ z + ∂^2ϕ^(1)/∂ t^2] ) |_z = 0= ∑_m,n = 1^∞∑_l,j =0^∞(A_mnlj^+ e^-i(θ_ml + θ_nj) + A_mnlj^- e^-i(θ_ml - θ_nj^∗)) + ,whereA_mnlj^+/C_ml C_nj = 1/4i[(ω_m + ω_n) (g^2k_ml k_nj/ω_mω_n- ω_mω_n)+ g^2/2(k_ml^2/ω_m + k_nj^2/ω_n)- 1/2 (ω_m^3 + ω_n^3) ],A_mnlj^-/C_ml C_nj^∗ = 1/4i[(ω_m - ω_n) (g^2k_ml k_nj^∗/ω_mω_n+ ω_mω_n)+ g^2/2(k_ml^2/ω_m - k_nj^2/ω_n)- 1/2 (ω_m^3 - ω_n^3) ]. In order to find the bound wave component, the free wave component, and to apply the second-order steering wavemaker motion, we split the second-order bvp (<ref>) into three bvps. For that purpose, the second-order potential function is split into three components as followsϕ^(2)(x,z,t) = ϕ^(21)(x,z,t) + ϕ^(22)(x,z,t) + ϕ^(23)(x,z,t).Now the corresponding bvp for the first component of the potential function ϕ^(21) reads[ g ∂_zϕ^(21) + ∂_t^2ϕ^(21) =rhs_1,atz = 0;; ∂_zϕ^(21) =0, atz = -h. ]The corresponding bvp for the second component of the potential function ϕ^(22) reads[g ∂_zϕ^(22) + ∂_t^2ϕ^(22)= 0, atz = 0;;∂_xϕ^(22)= S^(1)(t) (f'(z) ∂_zϕ^(1) - f(z) ∂_x^2ϕ^(1)) - ∂_xϕ^(21), x = 0;;∂_zϕ^(22)= 0,atz = -h. ]And the bvp for the third component of the potential function ϕ^(23) reads[ g ∂_zϕ^(23) + ∂_t^2ϕ^(23) =0,atz = 0;; ∂_xϕ^(23) = f(z) dS^(2)/dt,atx = 0;; ∂_zϕ^(23) =0, atz = -h. ] By taking the Ansatz for the first part of the second-order potential function ϕ^(21) as followsϕ^(21)(x,z,t)= ∑_m,n = 1^∞∑_l,j =0^∞ B_mnlj^+cosh (k_ml + k_nj)(z + h)/cosh (k_ml + k_nj)he^-i(θ_ml + θ_nj)+B_mnlj^-cosh (k_ml - k_nj^∗)(z + h)/cosh (k_ml - k_nj^∗)h e^-i(θ_ml - θ_nj^∗) + ,then we can derive the corresponding coefficients to be:B_mnlj^+ = A_mnlj^+/Ω^2(k_ml + k_nj)- (ω_m + ω_n)^2, B_mnlj^- = A_mnlj^-/Ω^2(k_ml - k_nj^∗) - (ω_m - ω_n)^2. This first component of the second-order potential function will contribute the bound wave component to the second-order surface wave elevation η^(2). For j = 0, the wave component is a propagating mode and for j ∈ℕ, it consists of evanescent modes. Since the wavenumbers k_mj + k_nj and k_mj + k_nj^∗, j ∈ℕ_0 do not satisfy the ldr with frequencies ω_m±ω_n, then the denominator part of B_mnj^± will never vanish and thus the potential function is a bounded function.Let the right-hand side of the boundary condition at the wavemaker for the second bvp (<ref>) be denoted by rhs_2, which is expressed asrhs_2 = ∑_m,n = 1^∞∑_l,j =0^∞(F_mnlj^+(z) e^i(ω_m + ω_n)t + F_mnlj^-(z) e^i(ω_m - ω_n)t) + ,whereF_mnlj^+(z)= g/8 ω_nS_m k_nj C_nj/cosh k_njh[f'(z) sinh k_nj(z + h) + k_nj f(z) cosh k_nj(z + h) ] + i (k_ml + k_nj) B_mnlj^+cosh (k_ml + k_nj)(z + h)/cosh (k_ml + k_nj)h, F_mnlj^-(z)=-g/8 ω_nS_m k_nj^∗ C_nj^∗/cosh k_nj^∗h[f'(z) sinh k_nj^∗(z + h) + k_nj^∗ f(z) cosh k_nj^∗(z + h) ] + i (k_ml - k_nj^∗) B_mnlj^-cosh (k_ml - k_nj^∗)(z + h)/cosh (k_ml - k_nj^∗)h.Let the Ansatz for the second component of the second-order potential function ϕ^(22) beϕ^(22)(x,z,t)= ∑_m,n = 1^∞∑_l,j =0^∞(i g P_mnlj^+/2(ω_m + ω_n)cosh K_mnlj^+(z + h)/cosh K_mnlj^+h e^-i(K_mnj^+x - (ω_m + ω_n)t). + . i g P_mnlj^-/2(ω_m - ω_n)cosh K_mnlj^-(z + h)/cosh K_mnlj^-h e^-i(K_mnlj^-x - (ω_m - ω_n)t)) + ,where the wavenumbers K_mnlj^±, j ∈ℕ_0 and frequencies ω_m±ω_n satisfy the ldr. Using the property that {cosh K_mnlj^±(z + h), cosh K_mnl'j'^±(z + h), l, l', j, j' ∈ℕ_0} is a set of orthogonal functions for l ≠ l' and j ≠ j', we find the coefficients P_mnlj^± as followP_mnlj^± = 2(ω_m±ω_n) cosh K_mnlj^±h/g K_mnlj^±∫_-h^0 F_mnlj^±(z) cosh K_mnlj^± (z + h) dz/∫_-h^0cosh^2 K_mnlj^± (z + h) dz=8 K_mnlj^±sinh K_mnlj^±h/ω_m±ω_n∫_-h^0 F_mnlj^±(z) cosh K_mnlj^± (z + h) dz/2 K_mnlj^± h + sinh (2 K_mnlj^± h).The second component of the second-order potential function ϕ^(22) will give contributions to the free wave component of the second-order surface wave elevation η^(2). This component arises due to the boundary condition at the wavemaker caused by the first-order wavemaker motion. Since the desired surface elevation is only the bound wave component, we want to get rid this term, especially the propagating mode. The evanescent modes vanish anyway after they evolve far away from the wavemaker. By prescribing the second-order wavemaker motion such that the propagating mode of the third component ϕ^(23) will cancel the same mode of the second one ϕ^(22), then far from the wavemaker we have the desired bound wave component only.Let the second-order wavemaker motion be given byS^(2)(t) = ∑_m,n = 1^∞ -1/2 i (S_mn^+ e^i (ω_m + ω_n)t + S_mn^- e^i (ω_m - ω_n)t) +Let also the Ansatz for the third component of the second-order potential function ϕ^(23) beϕ^(23)(x,z,t)= ∑_m,n = 1^∞∑_l,j =0^∞(i g Q_mnlj^+/2(ω_m + ω_n)cosh K_mnlj^+(z + h)/cosh K_mnlj^+h e^-i(K_mnlj^+x - (ω_m + ω_n)t). + . i g Q_mnlj^-/2(ω_m - ω_n)cosh K_mnlj^-(z + h)/cosh K_mnlj^-h e^-i(K_mnlj^-x - (ω_m - ω_n)t)) + ,where Ω(K_mnlj^±) = ω_m±ω_n. Using the orthogonality property again, we find the coefficients Q_mnlj^± as followsQ_mnlj^± =S_mn^±sinh K_mnlj^±h ∫_-h^0 f(z) cosh K_mnlj^± (z + h) dz/∫_-h^0cosh^2 K_mnlj^± (z + h) dz= 4 S_mn^±sinh K_mnlj^±h/K_mnlj^±(h + H) K_mnlj^±(h + H) sinh K_mnlj^±h + cosh K_mnlj^±d - cosh K_mnlj^±h/2 K_mnlj^± h + sinh (2 K_mnlj^± h).To have the propagating mode of the free wave from the second (ϕ^(22)) and the third (ϕ^(23)) components cancel each other, we must require P_mn00 + Q_mn00 = 0, which leads to the following second-order wavemaker motion, known as `second-order steering':S_mn^± = 2 (K_mn00^±)^2 I_mn00^± (h + H)/(ω_m±ω_n)(cosh K_mn00^±h - cosh K_mn00^±d - K_mn00^±(h + l) sinh K_mn00^±h),whereI_mn00^± = ∫_-h^0 F_mn00^±(z) cosh K_mn00^±(z + h) dz.Therefore, with this choice of the second-order wavemaker motion, the second-order potential function can be written asϕ^(2)(x,z,t) = ϕ^(2)_(x,z,t) + ϕ^(2)_(x,z,t),whereϕ^(2)_ = ϕ^(21)_,ϕ^(2)_ =(ϕ^(21)_ + ϕ^(22)_ + ϕ^(23)_)_.Consequently, from the second-order dfsbc (<ref>), we find the second-order surface wave elevation. It can be written as followsη^(2)(x,t) = η^(2)_ + ϕ^(2)_,whereη^(2)_ = ∑_m,n = 1^∞ D_mn00^+ e^-i(θ_m0 + θ_n0) + D_mn00^- e^-i(θ_m0 - θ_n0),andη^(2)_ = ∑_m,n = 1^∞∑_j = 1,l = 0^∞(D_mnlj^+ e^-i(θ_ml + θ_nj) + D_mnlj^- e^-i(θ_ml - θ_nj^∗). + 1/2 (P_mnlj^+ + Q_mnlj^+) e^-i(K_mnlj^+x - (ω_m + ω_n)t)+ . 1/2 (P_mnlj^- + Q_mnlj^-) e^-i(K_mnlj^-x - (ω_m - ω_n)t))+,where for l,j ∈ℕ_0:D_mnlj^+ =-1/g[i(ω_m + ω_n)B_mnlj^+ +1/8(g^2k_ml k_nj/ω_mω_n -ω_mω_n - (ω_m^2 + ω_n^2) ) C_ml C_nj], D_mnlj^- =-1/g[i(ω_m - ω_n)B_mnlj^- +1/8(g^2k_ml k_nj^∗/ω_mω_n +ω_mω_n - (ω_m^2 + ω_n^2) ) C_ml C_nj^∗]. We have seen that the first-order surface wave elevation consists of a linear superposition of monochromatic frequencies. However, due to nonlinear effects, nonhomogeneous bvp, and interactions of the first-order wave components, the second-order surface elevation is composed by a superposition of bichromatic frequencies ω_m±ω_n. The components with frequency ω_m + ω_n are called the `superharmonics' and those with frequency |ω_m - ω_n| are called the `subharmonics'.Remark 2. Similar to the first-order wave generation theory, we can define a second-order transfer function as well. A detailed formula for this transfer function can be found in Schäffer <cit.>. 5. CONCLUSIONSWe have discussed the theory for wave generation based on the fully nonlinear water wave equation. We solved a nonlinear bvp by the series expansion method. Using this method, the problem turns into a set of linear bvps at each expansion order. The lowest order gives a homogeneous bvp and the higher orders give nonhomogeneous ones with known, depending on previous solutions, right-hand sides.In this paper, we focussed on the wave generation theory up to the second-order.Based on the first-order wave generation theory, we describe the surface wave fields as the superposition of monochromatic waves. Due to the bvp, the wavenumbers and frequencies of this wave field are related by the ldr. By prescribing the first-order steering of the wavemaker as a linear superposition of harmonic motions, we found that the first-order surface elevation is simply a linear superposition of the corresponding monochromatic waves. Furthermore, the wavemaker transfer function is also introduced for practical purposes in the laboratory.For the second-order wave generation theory, we have solved a nonhomogeneous bvp. Due to the interactions between each pair of first-order wave components, the second-order wave field has bound wave components. Additionally, due to the first-order steering of the wavemaker motion and the boundary condition at the wavemaker, a free wave component is also generated, which is undesired.Therefore, we prevent it by controlling this second-order wavemaker motion.By applying this second-order steering, the resulting surface wave field contains only the desired bound wave components. Similarly, one can find the second-order transfer function for the relationship between the first-order and the second-order motions.Acknowledgement. This research has been executed partly in Indonesia and in the Netherlands. In Indonesia, it was done during a visit at Industrial and Applied Mathematics Research and Development Group, Institut Teknologi Bandung (kpp-mit itb). In the Netherlands, it was done at the University of Twente. We gratefully acknowledge both the Small Project Facility of the European Union Jakarta, entitled `Building Academia-Industry Partnership in the Sectors of Marine and Telecommunication Technology' and the project `Prediction and Generation of Deterministic Extreme Waves in Hydrodynamic Laboratories' (twi.) of the Netherlands Organization of Scientific Research nwo, subdivision Applied Sciences stw. We also appreciate the fruitful discussions with Professor E. (Brenny) van Groesen and Gert Klopman at the University of Twente.The financial assistance for the publication of this paper from theFaculty of Engineering, The University of Nottingham, University Park and Malaysia Campusesunder the New Researcher Fund NRF.5035 is also greatly acknowledged.REFERENCESref ref.ref 4mm -1mm Dean R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists, volume 2 of Advanced Series of Ocean Engineering. World Scientific, Singapore, 1991.Schaffer H. A. Schäffer, “Second-order wavemaker theory for irregular waves”, Ocean Engng. 23 (1996), 1:47–88. Natanael Karjanto: Department of Applied Mathematics, University of Twente, Postbus 217, 7500 AE, Enschede, The Netherlands.E-mail: <[email protected]> Department of Applied Mathematics, Faculty of Engineering, The University of Nottingham Malaysia Campus, Jalan Broga, Semenyih 43500, Selangor, Malaysia.E-mail: <[email protected]> | http://arxiv.org/abs/1704.08416v1 | {
"authors": [
"N. Karjanto"
],
"categories": [
"physics.flu-dyn",
"76B15, 74J15, 74J30, 34B05, 34B15, 34K10, 35G30"
],
"primary_category": "physics.flu-dyn",
"published": "20170427025704",
"title": "First- and second-order wave generation theory"
} |
[email protected] of Physics, Faculty of Liberal Arts and Sciences, Tokyo City University, Setagaya-ku, Tokyo 158-8557, Japan The spreading of a cap-shaped spherical droplet on a completely wettable spherical substrate is studied.The non-equilibrium thermodynamic formulation is used to derive the thermodynamic driving force of spreading including the line-tension effect.Then the energy balance approach is adopted to derive the evolution equation of the spreading droplet.The time evolution of the contact angle θ of a droplet obeys a power law θ∼ t^-α with the exponent α, which is different from that derived from Tanner's law on a flat substrate.Furthermore, the line tension must be positive to promote complete wetting on a spherical substrate, while it must be negative on a flat substrate. 64.60.Q- Spreading law on a completely wettable spherical substrate: The energy balance approach Masao Iwamatsu December 30, 2023 =========================================================================================§ INTRODUCTIONThe spreading of a liquid droplet on a solid substrate is a complicated phenomena where many factors come into play. However, the time evolution of the spreading of a liquid droplet on a flat solid surface can be usually described by simple universal power laws <cit.>.The most famous law called Tanner's law describes the spreading of small non-volatile droplet on a completely wettable substrate.This law has been derived theoretically using several different approaches <cit.> and confirmed experimentally <cit.>. So far, however, most of theoretical as well as experimental work is confined to a droplet on a flat substrate.Furthermore, the line-tension effect, which acts at the three-phase contact line and must play important role, has not been considered except for few theoretical works <cit.>.In the present study, we will consider the problem of spreading of a cap-shaped spherical droplet of non-volatile Newtonian fluids <cit.> gently placed on a spherical substrate, though our results might be applicable to non-Newtonian fluids as well <cit.>.The spreading on a spherical substrate is interesting because not only several experimental works has started to appear <cit.> but also it has recently been revealed that the wetting behavior on a spherical substrate is totally different from that on a flat substrate, in particular, when the line tension is important <cit.>.In fact, the effect of line tension on a spherical substrate is different from that on a flat substrate.For example, the complete wetting state can be realized by positive line tension on a spherical substrate <cit.>, while it can be realized by negative line tension on a flat substrate <cit.>. Although the magnitude of the line tension is believed to be small <cit.> so that the size of the droplet must be nano-scale, there is some argument that the line tension could be a few order of magnitude larger <cit.> than it has bee predicted so far when the gravitation can be important.If that is true, it could be possible to observe the line-tension effect on a micro- to millimeter scale droplets.Note, however, that the experimental determination of line tension from the contact angle measurements is problematical because various effects such as the adsorption at the three-phase contact line <cit.>, the effect of substrate on the surface tension through the disjoining pressure <cit.>, and curvature-dependent surface tension <cit.> will hinder the unique interpretation of experimental results to extract intrinsic line tension.In the following, we will consider the spreading of a cap-shaped spherical droplet on a spherical substrate using the energy-balance approach <cit.> which can easily include the line-tension effect <cit.>.§ EFFECT OF LINE TENSION ON THERMODYNAMIC DRIVING FORCEThe non-equilibrium thermodynamics <cit.> will be used to derive the thermodynamic driving force for spreading.Suppose, the droplet is a spherical cap of radius r with the non-equilibrium dynamic contact angle θ on a spherical substrate of radius R as shown in Fig. <ref>.The Gibbs free energy G consists of the interface free energy F and the work supplied by the external action w:G=F-wwithF=σ_ LVA_ LV+σ_ SVA_ SV+σ_ SLA_ SL+τ Lwhere γ_ij and A_ij are the surface tension and surface area of liquid-vapor (LV), solid-vapor (SV) and solid-liquid (SL) interfaces.The last term is the contribution form the line tension τ with the three-phase contact line length L.From Eq. (<ref>), the free energy change by a small distortion of a spherical cap (Fig. <ref>) can be expressed byδ G=σ_ LVδ A_ LV+σ_ SVδ A_ SV+σ_ SLδ A_ SL+τδ L-δ w.A small distortion induces the displacement of radius δ r_ n and that of contact line δ R_ L of the droplet.Therefore, there are two contributions to the increase of liquid-vapor surface area δ A_ LV=δ A_ LV1+δ A_ LV2.The first contribution is given by (see Fig. <ref>) δ A_ LV1=∫∫_ S_ LVκδ r_ n dA,where the integration is over the liquid-vapor surface area S_ LV, andκ=2/ris the curvature of the spherical liquid-vapor interface with radius r.The displacement δ R_ L of the triple line in Fig. <ref> by spreading will also increase the liquid-vapor surface area, which is the second contribution given byδ A_ LV2=∮_ Lδ R_ Lcosθ dl,where the integration is along the three-phase contact line L.The increase of the solid-liquid surface area is the decrease of the solid-vapor surface area, and is given byδ A_ SL=∮_ Lδ R_ Ldl=-δ A_ SV.The radius of the triple line increases by the amount δ R_ Lcosϕ, thereforeδ L=∮_ Lδ R_ Lcosϕ/R_ Ldl,where R_ L=Rsinϕ is the radius of the triple line and ϕ is half of the central angle corresponding to the contact arc of the solid surface (Fig. <ref>). Using the work done by the Laplace pressure Δ p, the increase of the work δ w supplied by external action is written asδ w=∫∫_S_ LVΔ p δ r_ n dA.Therefore, the free energy change δ G in Eq. (<ref>) is givey byδ G = σ_ LV∫∫_S_ LVκδ r_ n dA+σ_ LV∮_Lδ R_ Lcosθ dl-σ_ SV∮_Lδ R_ Ldl+σ_ SL∮_Lδ R_ Ldl+τ∮_Lδ R_ Lcosϕ/R_ Ldl-∫∫_S_ LVΔ p δ r_ n dA. Defining the thermodynamic driving forces f_ S and f_ L through∮_Lf_ Lδ R_ Ldl+∫∫_S_ LV f_ Sδ r_ n dA=-δ G,we obtainf_ L = σ_ SV-σ_ SL-σ_ LVcosθ-τcosϕ/R_ L, f_ S = Δ p-σ_ LVκ.Using the spreading parameter S defined byS=σ_ SV-σ_ SL-σ_ LV,Eq. (<ref>) can be written asf_ L= S+σ_ LV(1-cosθ)-τ/Rtanϕ.At the thermodynamic equilibrium f_ L=f_ S=0, Eqs. (<ref>) and (<ref>) become0 = S+σ_ LV(1-cosθ)-τ/Rtanϕ, 0 = Δ p-σ_ LVκ,which are the generalized Young's formula and Young-Laplace formula, respectively. When the substrate is incompletely-wettable (S<0) and is characterized by the Young's contact angle θ_ Y defined through σ_ SV-σ_ SL=σ_ LVcosθ_ Y,Eq. (<ref>) becomesσ_ LV(cosθ_ Y-cosθ)-τ/Rtanϕ=0,because S=σ_ LV(cosθ_ Y-1)<0. Equation (<ref>) will determine the equilibrium contact angle θ_ e which includes the line-tension effect when the substrate is characterized by the Young's contact angle θ_ Y.However, because we pay most attention to the completely-wettable substrate with S≥ 0, we will use Eqs. (<ref>) and (<ref>) instead of Eq. (<ref>). Using the geometric relations (Fig. <ref>)Ccosϕ =R-rcosθ, Csinϕ =rsinθ,and the size parameter of the dropletρ=r/R,relative to the size of the spherical substrate, we obtaintanϕ=ρsinθ/1-ρcosθ.Then, the thermodynamic force f_ L in Eq. (<ref>) is written asf_ L=S+σ_ LV(1-cosθ)-1-ρcosθ/rsinθτusing the dynamic contact angle θ. The line-tension contribution vanishes and changes its sign at ϕ=ϕ_ c=π/2 from Eq. (<ref>) or at θ=θ_ c defined by R-rcosθ_ c=0 from Eq. (<ref>), which gives the critical contact angle θ_ c given bycosθ_ c = R/r=1/ρ,where the three-phase contact line passes through the equator of the substrate.During the spreading with dynamic contact angle θ, a positive line tension (τ>0) will decelerate the spreading of the three-phase contact when the three phase contact line is on the upper hemisphere (ϕ<π/2) from Eq. (<ref>).However, once the three-phase contact line crosses the equator and moves into the lower hemisphere (ϕ>π/2), the positive line tension will accelerate the spreading.This fact can be easily understood as the spreading on the upper hemisphere accompanies the expansion of the contact line (circle), while the complete wetting on the lower hemisphere is accomplished by the shrinkage of the contact circle (Fig. <ref>(a)).In contrast, the positive line tension (τ>0) will always decelerate the spreading on a flat substrate <cit.> as the contact line will always expand towards the complete wetting state (Fig. <ref>(b)). The size parameter ρ or the droplet radius r is not a constant but is a function of the dynamic contact angle θ, since the droplet volume V is fixed for a non-volatile liquid.The droplet volume V is given by <cit.>V =(4π/3R^3)(ζ-1+ρ)^2[3(1+ρ)^2-2ζ(1-ρ)-ζ^2]/16ζ,withζ=√(1+ρ^2-2ρcosθ).Since the droplet volume is fixed atV_0=4π/3R^3(ρ_0^3-1),where ρ_0=r_0/R>1 is the size parameter when the droplet completely spread over the spherical substrate (θ=0), we obtainρ≃ρ_0 -ρ_0/16(ρ_0-1)^3θ^4by expanding Eq. (<ref>) and ρ by θ.Then, Eq. (<ref>) is approximated bytanϕ→ρ_0/1-ρ_0θ.when θ→ 0 and ϕ→π.Note that we cannot extrapolate our results of the spherical substrate to a flat substrate by taking the limit R→∞ or ρ_0→ 0 in Eqs. (27) to (29) because we alwasy have ρ_0>1 as the droplet volume is fixed.In order to wet a spherical substrate completely, the surface area of the spreading droplet must always be larger than that of the substrate.In contrast, the surface area of the droplet must always be smaller that that of an infinite flat substrate.This topological difference already implies that the spreading law on a spherical substrate is different from that on a flat substrate. § SPREADING USING THE ENERGY BALANCE APPROACH Once we found the thermodynamic driving force f_ L in Eqs. (<ref>) and (<ref>) for spreading, we can formulate the spreading problem of a droplet on a completely wettable (S>0) spherical substrate.We will follow closely the energy balance approach originally formulated by de Genne <cit.> and extended to include the line tension on a flat substrate by Mechkov et al. <cit.>.Suppose the free energy of the contact line per unit length is given by g_ L and the free energy dissipation at the contact line per unit length per unit time is W, then the free energy of the contact line is given by G_ L=2π R_Lg_ L.Since the contact line length is R_ L=2π Rsinϕ, the free energy balance is expressed as-d/dt(2π Rsinϕ g_ L)=(2π Rsinϕ) W.On the other hand, the free energy dissipation due to the action of the thermodynamic force f_ L can be written as-d/dt(2π Rsinϕ g_ L)=(2π Rsinϕ) vf_ Lusing the velocity v of the contact line.Since the right-hand side of Eqs. (<ref>) and (<ref>) must be equal, we haveW=v f_ L=v(S+σ_ LV(1-cosθ)-τ/Rtanϕ),which is the starting point of the energy balance approach.According to de Gennes <cit.>, the dissipation W can be divided into three contributionsW=W_ drop + W_ film + W_ precursor,where W_ drop is the dissipation in the wedge of the drop, W_ film is the dissipation in the wetting film, and W_ precursor is the dissipation in the precursor film.Hervet and de Gennes <cit.> proved thatW_ film=vS,which might be questionable on a spherical substrate as the film cannot exist when the droplet completely wet the substrate (Fig. <ref>(a)), while it can exist on an infinite flat substrate (Fig. <ref>(b)).Since we are interested in the spreading process before complete wetting, we will continue to adopt Eq. (<ref>) on a spherical substrate.Furthermore, the contribution of the precursor film W_ precursor is negligible <cit.>.Then, Eq. (<ref>) becomesW_ drop = v(σ_ LV(1-cosθ)-τ/Rtanϕ),where we consider only the hydrodynamic viscous dissipation W_ drop inside the drop in Eq. (<ref>).Using the wedge approximation, the dissipation in the bulk drop is given by <cit.>W_ drop≃3η v^2/θln|x_ max/x_ min|=κη v^2/θwhere η is the viscosity of the liquid, x_ max and x_ min are the cutoff length when the three-phase contact area is approximated by a wedge.Here, we introduced constant κ=3ln|x_ max/x_ min|. Therefore, Eq. (<ref>) is written asκη/θv^2= v(σ_ LV(1-cosθ)-τ/Rtanϕ).We now introduce the angle ψ defined by (Fig. <ref>(a))ψ=π-ϕ,and consider the spreading to a complete wetting state θ→ 0 and ψ→ 0.Then, Eq. (<ref>) givestanϕ≃ -ψ=-ρ_0/ρ_0-1θand Eq. (<ref>) becomesκη v/σ_ LV=1/2θ^3+ρ_0-1/ρ_0τ̃,whereτ̃=τ/σ_ LVRis the scaled line tension relative to the liquid-vapor surface tension σ_ LV.When the line tension τ can be neglected (τ=0), we recover the standard equation of spreading given by <cit.>θ^3= 2κ Cafrom Eq. (<ref>), whereCa=η v/σ_ LVis known as the capillary number. In contrast to the spreading on a flat substrate <cit.>, a positive line tension τ>0 always accelerate the spreading from Eq. (<ref>).Because Eq. (42) and, therefore, Eq. (40) are derived from Eq. (36) which is based on the wedge approximation <cit.>, their validity might be questionable when the dynamic contact angle θ becomes small and the droplet and the spherical substrate have similar curvatures. Then, the droplet near the contact line looks more like a thin-film or a slab rather than a wedge.In fact, an equation similar to Eq. (36) can be derived by simplifying a spherical droplet by a thin-film or a disk <cit.>. Therefore, we will continue to use Eq. (40) and (42) to discuss the spreading on a spherical substrate. Since the velocity of the contact line on a sphere is given byv=d/dtRϕ=-Rψ̇=-Rρ_0/ρ_0-1θ̇,and Eq. (<ref>) will be transformed into the evolution equation for the dynamic contact angle θ:θ̇=-σ_ LV/κη R(ρ_0-1/ρ_0) (1/2θ^3+ρ_0-1/ρ_0τ̃). When the line tension can be neglected (τ=0), the evolution of the dynamic contact angle follow the evolution lawθ∝ψ∝ t^-1/2,and v∝ t^-3/2,from Eqs. (<ref>) and (<ref>).On the other hand, when the line tension is positive (τ>0) and dominant in Eq. (<ref>), we haveθ∝ψ∝ t_0-tandv ∝ constant,where t_0 is the time when the spreading will be completed and the droplet will enclose the spherical substrate, which is determined by the initial position (initial contact angle θ) of the spreading front. This is the only scenario that can be deduced mathematically from our macroscopic model since the half of the central angle ϕ is always related to the dynamic contact angle θ through Eq. (22). Then the dynamic contact angle θ approaches 0^∘ (θ→ 0^∘) as the contact circle shrinks and disappears (ϕ→π). Microscopically, however, one may imagine that the advancement of spreading front is sufficiently fast that the contact circle is collapsing at a finite contact angle.Then, the thickness of the wetting film at the south pole (ϕ=π) of the spherical substrate substrate (Fig. 2(a)) changes discontinuously. In order to discuss such a microscopic process, we have to consider the microscopic model which includes the disjoining pressure and precursor film <cit.>. When the line tension is negative (τ<0), the droplet cannot spread over whole substrate since it is energetically unfavorable to eliminate the three-phase contact line.In order to achieve the complete wetting, a positive line tension is necessary on a spherical substrate.So far, we have considered a completely-wettable spherical substrate with S>0 orθ_ Y=0^∘. When the line tension τ is positive and is sufficiently large on a partially-wettable substrate with S<0 or θ_ Y>0,the equilibrium morphology with the lowest free energy can be a complete-wetting state with the contact angle θ_ e=0^∘ <cit.>.Then, the partially-wettable substrate turn into a completely wettable substrate by the action of line tension, and Eqs. (48) and (49) become applicable.In such a case, however, the initial position (initial contact angle θ) of spreading front must start from the position where the free-energy barrier from a metastable state is already crossed <cit.>. Finally, we recapture the result for a flat substrate <cit.> for the sake of comparison.On a flat substrate, Eq. (<ref>) is written as <cit.>κη v/σ_ LV=1/2θ^3-τ/σ_ LVaθwhere a is the base radius of the droplet (Fig. <ref>(b)).The droplet volume V_0 is proportional to the dynamic contact angle θ throughV_0≃π/4a^3θ.When the line tension can be neglected (τ=0), Eq. (<ref>) together with v=ȧ leads to the evolution equation of the base radius a, from which the well-know Tanner's law a ∝ t^1/10,andv ∝ t^-9/10.are derived.The evolution law of the dynamic contact angle on a flat substrate is given by <cit.>θ∝ t^-3/10,which is different from Eq. (<ref>) on a spherical substrate.On the other hand, when the line tension is negative (τ<0) rather than positive and dominant in Eq. (<ref>), we have <cit.>a∝t^1/5, v∝t^-4/5,andθ∝ t^-3/5.which, again, is different from Eq. (<ref>) on a spherical substrate.The scaling rule in Eq. (<ref>) has been observed in the spreading of nematic crystals <cit.>.In order to achieve the complete-wetting, a negative line tension is necessary on a flat substrate. §CONCLUSIONIn the present study, we considered the problem of spreading of a cap-shaped spherical droplet on a spherical substrate using the energy balance approach. We found a scaling rules of the time evolution of the dynamic contact angle on a completely wettable spherical substrate. This new scaling rule is different from well-know Tanner's law <cit.> on a flat substrate.Experimental attempts to verify this scaling rule will be interesting.The effect of line tension on the spreading on a spherical substrate is considered and the result is, again, different from that of a flat substrate.Most notably, a positive line tension is necessary <cit.> to realize complete wetting on a spherical substrate, while a negative line tension is necessary on a flat substrate.The scaling rule of dynamics on a spherical substrate is also different from that on a flat substrate.Even though the magnitude of the line tension has been believed to be small, a gravitation assisted enhancement of the line tension <cit.> would make it possible to observe line tension effect even in macroscopic droplets.Although we consider the spreading of simple liquids on a hydrophilic substrate, the spreading of surfactant solutions on a hydrophilic flat substrate is known to show the non-isotropic fingering instabilities called Marangoni effect <cit.> due to the surface tension gradient.These instabilities are closely related to the general instabilities of the growing front due to diffusion <cit.>.The fingering instabilities on a spherical surface will be an interesting subject to study.Finally, we notice that the spreading on completely-wettable spherical substrate involves topological phase transition since the topology of the wetting film changes from a hollow to a spherical surface which enclose the spherical substrate.This work was partially supported under a project for strategic advancement of research infrastructure for private universities, 2015-2020, operated by MEXT, Japan. 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"authors": [
"Masao Iwamatsu"
],
"categories": [
"cond-mat.soft"
],
"primary_category": "cond-mat.soft",
"published": "20170427011304",
"title": "Spreading law on a completely wettable spherical substrate: The energy balance approach"
} |
1 Instituto de Física y Astronomía, Facultad de Ciencias, Universidad de Valparaíso. Av. Gran Bretana 1111, Valparaíso, Chile.2 Department of Physics and Astronomy, The University of Western Ontario.London, Ontario, N6A 3K7, Canada. 3 Centre for Planetary Science and Exploration, The University of Western Ontario.London, Ontario, N6A 3K7, CanadaThe circumstellar disk density distributions for a sample of 63 Be southern stars from the BeSOS survey were found by modelling their Hα emission line profiles. These disk densities were used to compute disk masses and disk angular momenta for the sample. Average values for the disk mass are 3.4×10^-9 and 9.5×10^-10 M_⋆ for early (B0-B3) and late (B4-B9) spectral types, respectively. We also find that the range of disk angular momentum relative to the star are between 150-200 and 100-150 J_⋆/M_⋆, again for early and late-type Be stars respectively. The distributions of the disk mass and disk angular momentum are different between early and late-type Be stars at a 1% level of significance. Finally, we construct the disk mass distribution for the BeSOS sample as a function of spectral type and compare it to the predictions of stellar evolutionary models with rapid rotation. The observed disk masses are typically larger than the theoretical predictions, although the observed spread in disk masses is typically large. § INTRODUCTIONA Be star is defined by <cit.> as “A non-supergiant B star whose spectrum has, or had at some time, one or more Balmer lines in emission".The accepted explanation for the emission lines is the presence of a circumstellar envelope (CE) of gas surrounding the central star analogous to the first model of a Be star proposed by <cit.>. The material is expelled from the central star and placed in a thin equatorial disk with Keplerian rotation <cit.>. Different mechanisms such as rapid rotation <cit.>, mass loss from the stellar wind <cit.>, binarity <cit.>, magnetic fields <cit.>, and stellar pulsations <cit.> have been proposed to explain how the star loses enough mass to form the CE and how this material is placed in orbit, but it seems that more than one mechanism is required to reproduce the observations. Such mechanisms must continually supply enough angular momentum from the star to form and to maintain the disk. Given some mechanism to deposit material into the inner edge of the disk, the evolution of the gas seems well described by the viscous disk decretion model presented by <cit.>, with angular momentum transported throughout the disk by viscosity <cit.>.Be stars are variable on a range of different time scales associated with a variety of phenomenon occurring in the disk. For example, short-term variations (∼ hours-days) in the emission lines are associated with non-radial pulsations, probably due to the high rotation rate of the central star <cit.>; intermediate-term variations (∼ months-years) are seen in the cyclical variation between the violet and red peaks in doubled-peaked emission lines. Such variations are well represented by the global disk oscillation model <cit.>. Longer term variability, in some cases the emission lines disappear and/or are formed again on timescales of years to decades, is associated with the formation and dissipation of the disk (see section 5.3.1 offor several examples). Spectroscopy of the emission lines can be used to get information about the geometry, kinematics and physical properties of the disk. A very convenient model, in agreement with observations, is to assume that the density in the disk's equatorial plane falls with a power law with exponent, n, and follows a Gaussian model in the vertical direction (see details provided in section <ref>). We use the density distribution described above, the radiative transfer codeand the auxiliary complementary codeto solve the transfer equation along many rays (∼10^5) through the star/disk configuration. A grid of calculated Hα line profiles from models with different disk density distributions and stellar parameters are used to match the observed Hα line profiles and provide constraints on the disk parameters. We apply this method to a sample of 63 stars from the BeSOS catalogue. We selected a fraction of the best fitting models and we obtained the distribution of the disk density parameters, mass and total angular momentum content in the disk, with results provided for both early- and late-type Be stars. This paper is organized as follows: Our program stars and reduction steps are given in Section <ref>. Section <ref> describes our theoretical models including the main assumptions ofandcodes in Section <ref>. Input parameters to create the grid of models are provided in Section <ref>. Section <ref> describes our results from selecting best-fit disk density parameters from all our sample stars in two ways: visual inspection (Subsection <ref>) and a percentage of the best models (Subsection <ref>). Subsection <ref> gives the mass and angular momentum distributions of the disks. A discussion and conclusions of our main results are presented in Sections <ref> and <ref>, respectively. The Appendix displays Hα spectra from our best-fit models for our program stars compared to observations. § SAMPLE AND DATA REDUCTIONWe selected Be stars with B spectral type near or on the main sequence from the Be Stars Observation Survey (BeSOS[<http://besos.ifa.uv.cl>]) catalogue for our study. All Be targets in BeSOS website are confirmed as a Be star in the BeSS[<http://basebe.obspm.fr/basebe/>] catalogue or have an IR excess in the spectral energy distribution. This gives us a total of 63 Be stars. The sample distribution of spectral type is shown in the Figure <ref>. Approximately 30% of our sample corresponds to the B2V spectral type. The same distribution was found previously by other authors <cit.>, with B2V being the most frequently observed spectral type in Be stars.BeSOS spectra were obtained using the Pontifica Universidad Catolica (PUC) High Echelle Resolution Optical Spectrograph (PUCHEROS) developed at the Center of Astro-Engineering of PUC <cit.>. The instrument is mounted at the ESO 50 cm telescope of the PUC Observatory in Santiago, Chile, and has a spectral range of 390-730 nm with a spectral resolution of λ/Δλ ∼ 18000. Details about the instrument are provided in <cit.>.Observations were acquired between 2012 November and 2015 October. The exposure time was chosen to reach a S/N in the range 100-200 (as a consequence, the BeSOS catalogue has a limiting magnitude of V<6 in the sample selection criteria).For the wavelength calibration, exposures of ThAr lamps were used. The data reduction was performed using IRAF <cit.> following standard reduction procedures described in “A User's Guide to Reducing Echelle Spectra with IRAF”[<http://www.astro.uni.wroc.pl/ludzie/molenda/echelle_iraf.pdf>].The basic steps included removing bias and dark contributions,flat fielding, order detection and extraction, fitting the dispersion relation, normalization, wavelength calibration, and heliocentric velocity corrections. § THEORETICAL MODELS §.§ Disk density and temperature structure We calculated theoretical Hα line profiles using two codes: , a non-local thermodynamic equilibrium (non-LTE) code developed by <cit.>, and<cit.>, an auxiliary code that uses 's output to solve the transfer equation along a series of rays (∼10^5) to produce model spectra.There are two significant components that must be specified to model the physics of a star+disk system:the density distribution of the gas in the disk and the input energy provided by the photo-ionizing radiation field of the central star. Assuming both,code solves the statistical equilibrium equations for the ionization states and level populations using a solar chemical composition. Then, the code calculates the temperature distribution in the disk by enforcing radiative equilibrium. All calculations are made under the assumption that the vertical density distribution is fixed in approximate hydrostatic equilibrium, and the geometry of the disk is axisymmetric about the stars's rotation axis and symmetric on the midplane of the disk. The assumed density distribution has the form:ρ(R,Z) = ρ_0( R/R_⋆)^-nexp(-(Z/H)^2),where Z is the height above the equatorial plane, R is the radial distance from the stars' rotation axis, ρ_0 is the initial density in the equatorial plane, n is the index of the radial power law, and H is the height scale in the Z-direction and is given byH = H_0(R/R_⋆) ^3/2,with the parameter H_0 defined by,H_0 = ( 2 R_⋆^3 k T_0/G M_⋆μ_0 m_H)^1/2 ,where M_⋆ and R_⋆ are the stellar parameters, mass and radius, respectively; G is the gravitational constant, m_H is the mass of a hydrogen atom, k is the Boltzmann constant, μ_0 is themean molecular weight of the gas and T_0 isan isothermal temperature used onlyto fix the vertical structure of the disk initially. This parameter was fixed at T_0 = 0.6T_eff <cit.>. Since Be stars are fast rotators, the rotational velocity of the star was assumed to be 0.8v_crit for all spectral types, where v_crit is given by v_crit = √(2GM_⋆/3R_⋆). Finally, the rotation of the disk is assumed to be in pure Keplerian rotation <cit.>. For more details the reader is referred to <cit.>.§.§ Input parameters and grid of models We computed a grid of models usingfor a range of spectral classes from B0 to B9 in integer steps in spectral subtype in the main sequence stage. For early spectral types, we also computed models for B0.5 and B1.5due to the large number ofB2V stars in our program stars (see Figure <ref>). We also included turbulent velocity (v_tur = 2.0 km s^-1) into the disk for a more realistic model, since thin disks are likely to be turbulent <cit.> which increases the Doppler width in line profiles. The stellar parameters were interpolated from <cit.> and are displayed in the Table <ref>. Each disk model was computed using 65 radial (R) and 40 vertical (Z) points. The spacing of the points in the grid is non-uniform, with smaller spacing near the star and in the equatorial plane where density is the greatest. <cit.> studied the disk density of classical Be stars by matching the observed interferometric Hα visibilities with Fourier transforms of synthetic images produced by thecode. In their study, they suggest that the base density ρ_0 is typically between 10^-12 to 10^-10 g cm^-3 and the index power-law, n, normally ranges from 2 to 4 <cit.>. The outer radius of the Hα emitting region has been estimated by several authors considering samples of Be stars as well as studies for individual stars (see Discussion Section <ref>). <cit.> found that a typical outer radius of the envelope region producing the secondary Hα component is 20R_⋆ and a similar value was found by <cit.> of 18.9R_⋆ for strong lines and 7.3R_⋆ for weak lines. Measurements obtained using interferometric techniques determine the Hα emitting region to be between ∼ 5.0 - 30.0 R_⋆ <cit.>. Given this, we computed models for a disk truncation radius, R_T, of: 6.0, 12.5, 25.0 and 50.0 R_⋆, with base densities of : (0.1, 0.25, 0.5, 0.75, 1.0, 2.5, 5.0, 7.5, 10.0, 25.0) ×10^-11 g cm^-3 and n from 2.0 to 4.0 in increments of 0.5, to adequately cover the full range of parameters space reported in the literature. Finally, the inclination angle i was varied from 10^∘ to 90^∘, in steps of 10^∘, with 90^∘ replaced by 89^∘ to avoid an infinity value. Thus with 9 ρ_0 values, 5 n values, 9 i values and 4 R_T values, each spectral type is represented by a library of 1620 individual Hα model line profiles. To properly compare the synthetic profiles with our observations, every model was convolved with a Gaussian to match the resolving power of 18000 of our spectra. §.§ Behaviour of the Hα emission linePrior to beginning our statistical analysis, we illustrate the behavior of the predicted Hα emission line profile as each of the four model parameters, ρ_0, n, R_T and i, are varied. Figure <ref> shows the results, with the line profiles convolved down to a nominal resolution of λ/Δ λ=20000. The fluxes are normalized by the continuum star+disk flux outside of the line. The reference model, shown in black in each panel, was chosen to be a disk with parameters n=2.5, ρ_0 = 5.0× 10^-11g cm^-3, i = 50^∘ and R_T=25.0 R_⋆ surrounding a central B2V star. Panel (a) shows the predicted lines obtained by varying the inclination from 10^∘ to 90^∘ in steps of 20^∘. The profile goes from a singly-peaked,“wine bottle" profile at 10^∘, to a doubly-peaked profile for higher inclinations. While the profile at line centre does not drop below the continuum at i=90^∘, it does strongly satisfy the shell-star definition of <cit.> in which the peak to line centre flux ratio exceeds 1.5. Absorption below the continuum would result for less massive disks. Panel (b) shows the result of varying the disk truncation radius; the flux increases strongly with the disk size and the emission peak separation becomes smaller for larger disks, as expected by the <cit.> relation. Panel (c) shows the effect of increasing the base density of the disk, ρ_0. The emission line strength increases with increasing ρ_0 up to the reference value of 5.0× 10^-11g cm^-3, but then decreases for higher densities. This occurs because the line profile is the ratio of the total flux, line-plus-continuum, to the continuum flux alone. The line flux saturates with density first, causing the ratio to then decrease with increasing ρ_0 as the unsaturated continuum flux then increases faster. Finally, panel (d) shows the effect of varying the power-law index of equatorial plane drop-off. The behaviour reflects both the effect of increased density seen in panel (c) combined with a reduction in the emission peak separation since the disk density is concentrated closer to the star for larger n.As noted in the previous paragraph, the Hα line profiles shown as relative fluxes, i.e. divided by the predicted star+disk continuum, can show a more complex behaviour than might be expected because the line and continuum fluxes often have a different dependence on, say, the disk density. To clarify this point, Figure <ref> shows the same line profiles as Figure <ref> but plotted as absolute fluxes in Janskys without continuum normalization. In panel (a) of Figure <ref>, the i=90^∘ profile is now the weakest and the i=0^∘ profile, the strongest. The disk contribution to the normalizing continuum decreases in proportion to the disk's projected area, i.e. cos(i), while for large inclinations, i∼ 90^∘, the stellar continuum can be significantly obscured by the circumstellar disk. In panel (b), there is a strong dependence of the line flux on R_T, whereas the continuum flux is essentially independent of R_T. This is because the continuum forms very close to the central star (inside of the 6 R_*, the smallest disk considered) whereas the optically thick Hα line emission forms over a much larger portion of the disk. In panel (c), the fluxes are now seen to scale in order with increasing ρ_o, and the saturation of the line flux as compared to the continued increase in the continuum flux is clear. Finally in panel (d), the line fluxes are ordered with increasing flux with decreasing n, and the dependence of the continuum flux with the density-drop off in the disk is as expected.Figure <ref> suggests that there is some degeneracy among the calculated Hα line profiles, i.e. very similar relative flux line profiles can result from different combinations of the model parameters (n,ρ_o,R_T,i). To explore this further, we have used the reference profile of Figure <ref> corresponding to (n=2.5,ρ_0=5× 10^-11g cm^-3,R_T=25 R_*, i=50^∘) as a simulated observed profile and searched the B2V profile library for the top nine closest model profiles as defined by the smallest average percentage difference between the model and “observed" profile across the line: this figure-of-merit for the closeness of two line profiles is further discussed in the next section. Figure <ref> shows the results. While all nine profiles share the same R_T, there are small differences among the returned parameters, with n ranging between 2.0 and 2.5, ρ_o, between 5.0× 10^-12 and 7.5× 10^-11g cm^-3, and i between 40^∘ and 60^∘. The variations in the parameters are correlated: typically, smaller ρ_o values are associated with larger n values. In the next section, we describe how we deal with this degeneracy in assigning model parameters to each star.§ RESULTS §.§ Selection of the best disk modelsThe Hα spectrum of each star in our sample was compared to the theoretical library for that spectral type using a script that systematically finds the best match to the observed profile. For each comparison, the percentage flux difference between the model and observation was averaged over the line to assign each comparison a figure-of-merit value (hereafter called F), defined asF≡∑_i=1^i=N w_i |F_i^ obs-F_i^ mod|/F_i^ modwhere F_i^obs is the observed relative line flux, F_i^ mod is the model relative line flux, w_i is a weight, discussed below, and the sum is over all wavelengths spanning the line. Several different weights were examined: uniform weighting w_i=1, line-center weighting w_i=|F_i^ mod/F_c^ mod-1|, and uniform weighting but using the sum of the square of flux differences divided by flux. For each spectrum, we tested the second option first, but also calculated the quality of the fits for other options as well, and by visual inspection we selected the best ℱ method to adopt for each spectrum (which may be different for each star) to use in our results. Initially the best 50 matches out of the 1620 profiles using the smallest ℱ/ℱ_min values were identified, where F_min is the minimum figure-of-merit of the best-fitting library profile. We show an example for a B2 spectral type in Figure <ref> for the Be star HD58343. The upper left panel shows the best 50 models sorted by ℱ/ℱ_min (black dots) with the best 5 models in red, blue, green, yellow, and cyan colors corresponding to ℱ/ℱ_min of 1.00, 1.20, 1.30, 1.40 and 1.45, respectively. The best 5 models are different in the disk density parameters, but they have the same inclination angle, i = 10^∘, and the same disk truncation radius of R_T = 25.0R_⋆ for this star. The upper right panel shows models of Hα line profiles corresponding to each respective color as well as the observed profile shown in black. The main difference between these models appears in the flanks of the emission line. <cit.> classified typical emission profiles seen in Be stars at different inclination angles, where this particular “wine bottle shape” is usually seen at low inclinations. Moreover,<cit.> reproduced emission line profiles using a Keplerian disk model for an optically thick disk (∼ 10^-10 g cm^-3) and he found for inclinations between 5^∘≲ i ≲ 30^∘, emission line profiles show inflection flanks. For high inclination angles, i ≳ 75^∘, he noticed that a central depression plus a double peak profile is generated due to the velocity field present in the disk. The lower left panel shows the behavior of logρ_0 vs ℱ/ℱ_min where, in this particular case, we can see that higher values of ρ_0 dominate. The lower right panel is the same as the lower left panel except for n.In Figure <ref>, the best model (red color) is well constrained by ℱ/ℱ_min = 1.00, however we notice that similar values of ρ_0 combined with different values of n give us similar profiles of the emission line (for the same inclination angle and same disk truncation radius). For this reason we consider a range of models within a percentage of ℱ/ℱ_min as described in Subsection <ref>. §.§ Best fit models by visual inspection We chose the best model by visual inspection of the comparison plots between the models and the observations; such plots are shown in Appendix <ref>, and the model parameters corresponding to this best fit are displayed in Table <ref> in the columns 4 to 8. Targets with a superscript a indicate an Hα absorption line in that star's spectrum. In some cases, the script was not able to suitably reproduce the core and wings of the emission line profile (see discussion section <ref> for possible explanations). However, we chose the fit that best represents the wings of the line (instead of the core) and classified them as poor fits. These cases are indicated with the superscript pf in the Table <ref> and they are not considered in our analysis. Targets are sorted by HD number indicating the date of the observation and the ℱ/ℱ_min value of the chosen model. Table <ref> also lists the Hα equivalent width, EW, and the emission double-peak separation, Δ V_p, measured from the observations. Some of the targets are represented by more than one observation due to variability, and they show changes in the line profile (peak height, violet-to-red peak ratio, etc). There are 22 such variable cases indicated by an asterisk symbol beside the star name below the plot (14 of these are in emission and 8 in absorption), and they were treated by keeping the inclination angle constant for the system, and each time fit with different models. In our program stars there are 15 Be stars with Hα in absorption. We notice that in our sample all targets are confirmed as Be stars, so absorption profiles presented here are Be stars in disk-less phase or currently without a disk. We did not include absorption profiles in our analysis, nevertheless, our spectral library contains profiles with pure photospheric Hα profiles. We display the values for systems with absorption in Table <ref> and in the plots in Appendix <ref>.We provide our results separately for the emission profiles, absorption lines, and for the targets with poor fits. Overall, we have 42 Be stars with Hα emission, 15 with absorption profiles, and 6 with poor fits. The systems with poor fits are displayed in Appendix <ref>. §.§ Distribution of the disk density parameters: representative modelsIn the previous section, we determined the best-fit disk density models for each of our program stars with Hα in emission. In this section, we wish to look at the distribution of disk density parameters in this sample. From now on, every spectrum in emission for each target (if there is more than one) is considered by a separate, unique model. This give us a total of 61 emission models. As we explained in the previous section, there is a range of models for each star that fit the observed profile nearly as well as the best-fit model selected by visual inspection. Thus for any given star, we can systematically define a “set" of best fit parameters by selecting all models with ℱ≤ 1.25 ℱ_min resulting in N models being selected. We note that by selecting a slightly larger range of ℱ, as Figure <ref> demonstrates, the base density and the exponent of the disk surface density span a wide range of values especially for ℱ≥ 1.50.To define representative disk density parameters for each star, we choose a weighted-average over the N selected models. For the disk parameter X, which could be ρ_0 or n, etc., we define<X> ≡1/W ∑_i=1^Nw_iX_i ,where W≡∑_i=1^N w_i and the weights are chosen asw_i≡(ℱ/ℱ_min)^m.The index m was chosen to be equal to -10 so that significantly different weights are given to models ranging from 1 to 1.25, i.e. the weight assigned to ℱ=1.25 is 1.25^-10≈ 0.1. This procedure was applied to all the physical quantities obtained from the emission profiles which are presented below. In order to study the conditions under which the disk exists and its link with the spectral type, we distinguish in our study between early (B0-B3) and late (B4-B9) type Be stars.The representative values (weighted-average) of the parameters governing the disk density (n and ρ_0 in Eq. <ref>) of emission profiles are displayed in Figure <ref>. The most frequent pairs are concentrated between <n> ≃ 2.0 - 2.5 and <ρ_0> ≃ (4.00-6.30) × 10^-11g cm^-3 or <logρ_0> ≃ -10.4 to -10.2.We note that we detect emission profiles in the upper left triangular region of Figure <ref>. With increasing values of the density exponent and decreasing base density, corresponding to the lower right in Figure <ref>, it would be increasing difficult to detect emission due to reduced disk density. The lack of disk material for these stars made it impossible to constrain our models as mentioned above so we did not analyze any features for them. Moreover, some absorption profiles seemed to be pure photospheric lines, and some showed evidence of a possible formation/dissipation disk phase (see HD33328's spectrum, for example, in Appendix <ref>). §.§ Distribution of disk mass and angular momentum From each star's fitted disk density parameters, we can estimate the mass of the disk by integrating the disk density law, Eq. <ref>, over the volume of the disk. For the radial extent of the disk, we chose the radius that encloses 90% of the total flux of the Hα line in an i = 0^∘ (i.e. face-on disk) image computed with . This measure of the Hα disk size was used in favor of the fitted R_T as the latter was computed on a very coarse grid of only four values. To compute each disk mass, <M_d>, the representative values of the disk parameters were used which included the models with ℱ < 1.25ℱ_min. In addition to disk mass, the representative value of the total angular momentum content, <J_d>, of each disk was also computed, using the same disk density parameters and assuming pure Keplerian rotation for the disk. Representative values of the disk mass and angular momentum in stellar units are displayed in Table <ref> in columns 12 and 13, respectively.Figure <ref> shows the distribution of both representative values, disk mass and disk specific angular momentum, <J_d>/<M_d>, for early and late stellar types. To normalize by the stellar angular momentum, the central star was assumed to rotate as a solid body at 0.8 v_ crit with the critical velocity computed using Eq. <ref>. (See also Section <ref> for a discussion about the effect of the stellar rotation on J_d). The distribution of the disk mass in early types ranges from 1.0 × 10^-7 to 3.0 × 10^-10 M_⋆ (see top panel in Figure <ref>). For late types, values range from 1.7 × 10^-8 to 1.7 × 10^-11 M_⋆. The mean disk mass for the early-types is 3.4 × 10^-9 M_⋆, while for the late-types, the mean disk mass is 9.5 × 10^-10 M_⋆. The bottom panel in Figure <ref> shows the distribution of the specific angular momentum <J_d>/<M_d> of the disk in units of stellar specific angular momentum. For early types, the most frequent range is <J_d>/<M_d>≃ 150 - 200 and corresponds to a <J_d> ∼ (1.2 - 3.0)×10^-6 J_⋆ and a total mass of <M_d> ∼ (3.2 - 9.1)×10^-9 M_⋆. For late types the most frequent values ranges from 100 to 150 corresponding to a range value of <J_d>∼ (1.0 - 5.0)×10^-7 J_⋆ and <M_d>∼ (1.0 - 2.9)×10^-9 M_⋆.In general, late types have lower values of <M_d> and <J_d> in comparison with early types. It should be kept in mind that while the model disk masses vary over a large range (with M_d/M_* spanning 1.7×10^-11 to 1.0×10^-7), the range of model specific angular momentum is much less owing to the assumption of Keplerian rotation. The minimum and maximum values of <J_d>/<M_d> in units of J_*/M_* are 49 and 306, for a total variation of just over a factor of 6.§.§ Relation between Hα equivalent width and disk massThe relation between Hα EW, and disk mass, <log M_d>, separated by early and late-type Be stars, is shown in Figure <ref>. Negative values indicate that the net flux of the emission line is above the continuum level. While there is an overall trend for the most massive disks to have the largest Hα EW, there is an extremely large dispersion. This is not unexpected; for any given power law index n in Eq. <ref>, the Hα EW will first increase with ρ_0, reach a maximum, and then decline <cit.>. This decline occurs because once the density becomes large enough, the continuum flux from the disk at the wavelength of Hα rises more quickly than the line emission, so the equivalent width actually decreases with ρ_0 and so does the corresponding disk mass. The exact value of ρ_0 at which the Hα equivalent width peaks is dependent on n; therefore, in a mix of models with differing (ρ_0,n), there will not be a direct relationship between disk mass and Hα EW. Finally, we note that the most massive disks and largest Hα equivalent widths (absolute value) are found most frequently among the early-type Be stars.§ DISCUSSION §.§ Disk densityWe found a distribution of the representative values of the disk density parameters for early and late spectral types, which are displayed in Figure <ref>. Early stellar types cover values of <n> between 2.0 and 3.0, while late stellar types reach values near 3.7. It appears that higher values of the power-law exponent are found for stars with lower effective temperature. This could explain the small emission disks seen in late type stars since with increasing n, the disk density falls faster with distance from the star. However, the average value of the representative values of the power-law exponent are essentially the same for early and late spectral types: <n̅_early> = 2.5 ± 0.3 and <n̅_late> = 2.5 ± 0.4.Previous work in the literature usingwas completed by <cit.>. They created a grid of disk models for B0, B2, B5 and B8 stellar types at three inclinations angles i=20^∘, 45^∘ and 70^∘ for different disk densities. They modeled Hα line profiles of a set of 56 Be stars (excluding Be-shell stars) and studied the effects of the density and temperature in the disk. Their results show a higher percentage of models ranging ρ_0 between 10^-11 and 10^-10g cm^-3 and a significant peak of n ∼ 3.5, which is slightly larger than the values of n found in this study. We attribute this difference to the different methods used to compute the Hα line profile. <cit.> usedto compute the line intensity escaping perpendicular to the equatorial plane in each disk annulus (i.e. rays for which i=0^∘). They then assumed that this ray was representative of other angles considered, i=20, 45^∘ and 70^∘, and combined the i=0^∘ rays with the appropriate Doppler-shifts and projected areas. Clearly this computation method becomes limited with larger viewing angles. In contrast, , used here, does not make any of these approximations, and it has been successfully used to model the Hα lines of Be shell stars for which the inclination angle is large <cit.>. Eight Be-shell spectra were analyzed and values for ρ_0 between 10^-12 and 10^-10 g cm^-3 and n between 2.5 and 3.5 were found. <cit.> used the assumption of an isothermal disk and the same density prescription as Equation <ref> to reproduce the color excess in the NIR of a sample of 130 Be stars. For the central star, they assumed an early-type star and adopted n = 3.0 for all the models. They varied ρ_0 between 10^-12 and 2.0×10^-10 g cm^-3, which is very similar to our range of ρ_0 variation. They set the inclination angle at i =45^∘ and 80^∘ used an outer disk radius of ∼ 14.6 and 21.4R_⋆. Other studies also use the same scenario for the density distribution, where the base density of the disk is found to be between 10^-12 and 10^-10 g cm^-3 and the power-law exponent n is usually in the range 2 - 4 (for a review of recent results the reader is referred to the section 5.1.3 of ). Recently <cit.> determined the disk density parameters ρ_0 and n for 80 Be stars observed in different epochs, corresponding to 169 specific disk structures. They used the viscous decretion disk model to fit the infrared continuum emission of their observations, using infrared wavelengths. They found that the exponent n is in the range between 1.5 and 3.5, where our most frequent values are between 2.0 and 2.5 for both early and late spectral types. They also found ρ_0 to range between 10^-12 and 10^-10 g cm^-3, which compares favorably with our average values of between (4.00-6.30) × 10^-11 g cm^-3, again for both early and late spectral types. <cit.> also established that the disks around early-type stars are denser than in late-type stars, consistent with our finding of more massive disks for the earlier spectral types.Finally, we also notice that our models sometimes do not reproduce the wings of our Hα observations. This may reflect our assumption of a single radial power law for the equatorial density variation in this disk. Alternatively, for earlier spectral types, this may reflect neglect of non-coherent electron scattering in the formation of Hα <cit.>. For example, <cit.> performed an interferometric study of two Be stars using a kinematic disk model neglecting the expansion in the equatorial disk. They were able to fit the wings and the core of the Hα emission line by introducing a factor to estimate the incoherent scattering to their kinematic model.§.§ Size of the emission region The outer extent of the disk considered in the modelling of this work was assumed to be one of four values, 6.0, 12.5, 25.0 and 50.0R_⋆. From these values, the best fitting models have a disk truncation radius of 25.0R_⋆ followed by 50.0R_⋆. Nevertheless, as noted previously, a better estimate of the size of the Hα emitting region is the equatorial radius that contains 90% of the integrated Hα flux in an i=0^∘ image computed with , a quantity we denotes as R_90. We provide R_90 values in the column 11 on Table <ref>. These values, based on the integrated flux from our models, could used by other studies to conveniently compare with our results. As an additional check, we compare our R_90 disk sizes with a measure based on the observed separation of the Hα emission peaks, as first suggested by <cit.>, and tailored to our model assumptions. The basic idea of this method is that the double-peak separation is set by the disk velocity at it's outer edge, which we will denote R_H. If the observed peak separation is Δ V_pkm s^-1, we have1/2(Δ V_p/sin i) = √(GM/R_H),assuming Keplerian rotation for the disk and correcting the observed peak separation for the viewing inclination i. Hence,Δ V_p^2 = 4 (GM/R_H) sin^2 i.In this work, we assumed that all Be stars rotate at 80% of their critical velocity; therefore, each star's equatorial velocity isV_eq = 0.8 √(GM/(3/2)R_⋆) ,where R_⋆ is the stellar (polar) radius. Using this to eliminate (GM) from the previous equation and solving for the disk size we haveR_H/R_⋆ = 9.375(V_eqsin i/Δ V_p)^2As V_eqsin i is the star's vsin i value, we have approximatelyR_H/R_⋆≈(3 vsin i/Δ V_p)^2.This equation is very similar to the form used by many authors to derive approximate disk sizes from observed spectra <cit.>. We note that the way we use Eqn. <ref> is slightly non-standard: we do not measure vsin i directly from our spectra; instead, we adopt the vsin i of the best fit-model. As the Hα profiles are essentially insensitive to vsin i, we are using the observed peak separation Δ V_p and the best-fit value of i for the viewing inclination.The correlation between R_H and R_90 is displayed in Figure <ref>. For a few of our targets, we do not obtain a R_H value because of a small Δ V_p or small inclination where Huang's law is not valid. The solid line indicates the linear fit over both early (blue circles) and late (red squares) stellar types considering values not larger than 50.0R_⋆ and greater than 1R_⋆ to be consistent with the input values used in themodel. The relation between the representative values of the mentioned sizes is given by the linear equation <R_90> = (0.53 ± 0.07) <R_H> + (3.45 ± 0.80) in units of stellar radius, with a correlation of r_corr = 0.611 with confidence intervals calculated using a bootstrapping method. We notice that the most frequent disk sizes values calculated by Huang's relation for early and late spectral types are concentrated less than 5R_⋆ and the values containing 90% of the Hα flux for early and late spectral types are concentrated between 10 and 15R_⋆.Many other measurements of the Be star disk sizes have been reported in the literature. <cit.> measured the Δ V_p and the FWHM in the Hα emission line of 24 southern Be stars and using Huang's law he estimated an outer emitting size of ∼ 10R_⋆. Similar values were found by <cit.> for 41 Be stars, they obtained an outer emitting size in the range ∼ 7 - 19R_⋆ for the Hα emission line. Using interferometric techniques <cit.> studied the relation between the total flux emission of Hα line and the physical size of the emission region in 7 Be stars, finding for the first time a clear correlation between these both quantities. For early stellar types they found an extended emitting size of ∼ 18.0 to 21.0R_⋆ while for stars with lower effectivetemperatures they found smaller values of ∼ 6.0R_⋆ to 14.0R_⋆ (with an exception for ψ Per of ∼ 32.0R_⋆). An alternative way to estimate the emitting region based on the Hα half-width at half-maximum was proposed by <cit.>. They compared their results with the interferometric measures of theHα emitting size in the literature and they obtained lower values between ∼ 5.0 to 10.0R_⋆.Our very low values of R_H from observed emission profiles (less than 1R_⋆ and not considered in the analysis) come from very large Δ V_p values. If the star is rotating near its critical rotation, the gas could accumulate near the star and consequently the emission region of the Hα line could be of the order of a few stellar radii. Overall, our results for R_T, either from the representative models or from Huang's law, show general agreement with previous works in the literature, giving higher values for early stellar types and lower values for late-Be types. §.§ Mass and angular momentum of the disk In Section <ref> we provided the range of the total disk mass and the total disk angular momentum for early and late stellar types. Our results gave us higher values of <J_d> and <M_d> for early types in comparison with late types. This was expected considering that late stellar types have, in general, smaller disks. Considering the whole sample without distinction between early and late stellar types, we estimate that the total angular momentum content in the disk is approximately 10^-7 times the angular momentum of the central star and the mass of the disk is approximately 10^-9 times the mass of the central star. <cit.> studied the disk properties of the late Be shell star Omicron Aquarii (o Aqr, B7IVe) combining contemporaneous interferometric and spectroscopy Hα observations with near-infrared (NIR) spectral energy distributions. They compared the values obtained by each technique for different disk parameters. From Hα spectroscopy, values of R_T, M_d and J_d are higher than those obtained from the NIR, while ρ_0 and n are lower than NIR. From their results, the comparison between values obtained from spectroscopy, interferometry and NIR spectral distributions, give similar or consistent values for M_d and J_d, but the disk density parameters (ρ_0,n), showed in a range of values. As a result, for o Aqr, <cit.> found values of J_d∼ 1.6×10^-8 J_⋆ and a total mass of M_d∼ 1.8×10^-10 M_⋆. These values are consistent with our results in Figure <ref>, but are at lower end of the distribution for late stellar types.As we mentioned earlier in Section <ref>, we distinguish our results between early (B0-B3) and late (B4-B9) type Be stars. Recall that the parameters associated with these stars are listed in Table <ref>. In order to study the effects of the central star on the distributions of disk mass and angular momentum for early and late spectral types, we performed a two-tailed Kolmogorov-Smirnov (KS) test with the null hypothesis that both samples come from the same distribution. Figure <ref> shows the cumulative distribution functions (CDFs) for disk mass (upper panel) and total disk angular momentum per disk mass (bottom panel). For disk mass, the maximum distance, D_m, between CDFs for early and late types gives D_m =0.535 and considering a significance level at 0.01, the critical value, D_c, is 0.50 for the 61 emission models. Hence we conclude that early and late samples of disk mass come from different distributions. The largest value for the maximum distance between CDFs for <J_d>/<M_d>, gives D_m =0.615, again rejecting the null hypothesis that the distributions are the same at 1% level. Therefore, our samples show that early-type Be stars are more likely to have massive disks with higher values of total angular momentum than late-type Be stars. We note that our results could be influenced by the choice of stellar rotation rate of 0.8v_crit for all the luminosity classes in our models. Various studies have attempted to determine these rates more precisely, with a consensus that they are rapid rotators, but it is still not clear how close to critical these rates are. <cit.> compared the observational distribution of a sample of v sin i values of Be-shell stars (sin i ∼ 1) with a theoretical distribution. He determined that these Be-shell stars rotate at 70%-80% of their critical rotation. <cit.> studied the effect of the stellar rotation on the disk formation in “normal” B stars as a function of stellar mass, by comparing with Be stars in the literature. They found that the rotational velocity needed to create a Be star varies strongly with the stellar mass. For low-mass B stars (less than 4M_⊙ or later than B6 V) the upper-rotational limit is very close to the break-up velocity ∼ 0.96, while for high-mass B stars (more than 8.6M_⊙ or earlier than B2 V) the upper-rotational limit is near to 0.63v_crit. To test the significance of our choice of 0.8v_crit on our angular momentum distribution of our sample, we adopted both limiting values of the break-up velocity, 0.63 and 0.96 for early and late stellar types, respectively. For early types, the disk angular momentum is under-estimated (J_⋆∼ 0.80/0.63 ≃ 1.3) by J_disk/J_⋆∼ 0.8 times, while for late types are overestimated (0.80/0.96 ≃ 0.8) by 1.2 times. Multiplying by these factors for the early- and late-type distributions of <J_d>/<M_d>, respectively, we found a total range distribution between ∼ 64 and 245, and a maximum distance value of D_m = 0.879, which also rejects the null hypothesis that both samples come from the same distribution within a 1% level of significance.A key ingredient in the specific angular momentum distribution for Be star disks is the underlying Keplerian rotation law, well established for Be stars <cit.>. As the overall scale of the disk's Keplerian rotation is set by the parameters of the central star (M_*, R_*), a portion of the variation in disk specific angular momentum must simply reflect the change of stellar mass and radius with spectral type. To quantify this,[We are thankful to the anonymous referee for suggesting this line of reasoning.] we note that the disk specific angular must scale as J/M∼ r v_ K(r) where r is a characteristic radius for the disk, and v_ K is the Keplerian velocity at this point. We may write this as J/M ∼√(GM_* R_* (r/R_*)) by introducing the stellar radius R_*. If the characteristic disk size (r/R_*) is constant with spectral type, we have J/M∼√(M_* R_*). Figure <ref> plots the disk specific angular momentum found for our sample versus the quantity √(M_* R_*) from Table <ref>. While there is a wide dispersion, the linear trend is very clear, with a correlation coefficient of r=+0.63. Therefore, as expected, a significant portion of the variation in the disk specific angular momentum is due to the variation of the central star parameters via the overall scale of the disk's Keplerian rotation. The large scatter about this linear trend, typically a factor of 2-3, must then reflect the different disk sizes and the distribution of the disk mass with radius, controlled mainly by the parameter n. §.§ Cumulative distribution of the inclination angles An interesting consequence of the Hα modelling is that the inclination of the system can be determined. Figure <ref> shows the CDF of the derived representative values of the inclination angles versus the expected 1-cos(i) distribution, assuming that the rotation axes are randomly distributed. Using a one-sample KS test, we find that our data do not follow the expected distribution. Defining the null hypothesis H_0: “the inclination data comes from the 1-cos(i) distribution” and at significance level α =0.01, the maximum distance, D_m is 0.243, while the critical value for our sample of 61 emission models is D_c =0.209, therefore since D_m > D_c, H_0 is rejected with a 1% level. This rejection, that our inclination angles distribution is not random, is not surprising as the selection criteria for Be stars in surveys are often biased against shell stars seen at high inclinations <cit.>. This indeed seems to be the case for our sample as the observed CDF of Figure <ref> does not contain the expected fraction of high-inclination objects; in particular, our sample has only 8 Be shell stars. §.§ Comparison with disk mass predictions of models of stellar evolution with rotation In this section, we compare the disk mass distribution derived for the BeSOS sample as a function of spectral type with the predictions of <cit.>. While the hydrodynamical origin of the Be star disk ejection mechanism(s) is unknown, there is a broad consensus that rapid stellar rotation, likely reaching the critical value, is the ultimate driver for disk ejection in isolated Be stars <cit.>. Models of stellar evolution with rotation do predict episodes of critical rotation during main sequence evolution due to the internal transport of angular momentum. Under the assumption that disk ejection removes the excess surface angular momentum at critical rotation, and using the formalism of <cit.> for the ejected disk and its angular momentum transport, <cit.> compute the main sequence evolution of B stars with masses from 2 to 9 M_⊙ and follow the required disk ejections over the main sequence. While these models make many assumptions (such as the details of the angular momentum transport and the initial ZAMS rotation rate and profile) which may not be realistic, they do predict average disk masses as a function of spectral type. In Figure <ref>, we compare the disk masses obtained from the BeSOS survey stars with the predictions of <cit.>. Shown are the average disk mass, its 1 σ variation, and the minimum and maximum disk masses, all for each spectral type. In the observational sample, there is often a very wide range of disk masses at each spectral sub-type, typically at least an order of magnitude. The observed average disk mass is always above the <cit.> prediction, although the theoretical prediction typically falls within the observed range of disk masses. The predicted curve shows an increasing trend with earlier spectral type (or increased stellar mass). This is reflected in the BeSOS sample, although the number of stars with spectral types earlier than B2 is small (6 out of 63 stars). Also shown in the figure are the disk mass estimates for o Aqr <cit.> and 48 Per <cit.>, based on modelling of the Hα emission profile (as in the current work), coupled with simultaneous modeling of interferometric visibilities and near-IR spectral energy distributions. These two, higher-precision disk mass estimates fall closer the predicted trend, although again within the observed variation of the BeSOS sample. We note that the current disk mass estimates are really lower limits as we are sensitive only to the Hα emitting gas. Given the uncertainties in the theoretical modeling, a more detailed a comparison may be unwarranted at this point. However, the distribution of Be star disk masses may develop into a powerful diagnostic constraint on rotating models of stellar evolution. §.§ Observed profiles with poor fitsAppendix <ref> contains all the fits that we consider poor and do not reproduce the features in the observed Hα line profiles. All targets are in emission and are early-type stars (between B0 and B2), with the exception of HD83953, a B5V star. The shape of the emission profiles are very similar, showing wide profiles reaching velocities of the order of 600 - 700 km s^-1. Our methodology was not able to find a good agreement between the observations and the models because these profiles do not have a symmetric central emission and are very wide. For example, in the entire sample of emission profiles (Appendix <ref>), only three stars are classified between B0 and B1.5 and these three are evolved: HD68980 (B1.5 III), HD143275 (B0.3 IV) and HD212571 (B1 III-IV), with velocities between ∼ 300 - 500 km s^-1, and with almost symmetric profiles. On the other hand, we note that HD35439 (B1 Vn), HD50013 (B1.5 V) and HD110432 (B0.5 IVpe) show variation in the intensity peak, where HD35439 shows a clear V/R variation.In the literature two of the six stars are binary stars classified as a ϕ Per-type. These types of systems consist of an early B-type main sequence star as the primary and a hot subdwarf star as the secondary, both surrounded by an envelope. It is believed that the secondary at some time was a more massive star that has lost a large percentage of its mass (by mass-transfer to the primary) leaving a hot helium core. The primary star is increasing its mass and angular momentum, due to the mass-transfer interaction, as result a large v sin i value is observed. HD41335 (HR2142) was recently highly studied by <cit.>, who used a large set of ultraviolet and Hα observations to measure radial velocities of the primary star to compute an orbit. For the system, Be + sdO, they find a mass ratio M_2/M_1 = 0.07 ± 0.02 and for the companion they found a projected rotational velocity v sin i <30 km s^-1, an effective temperature greater than 43 ± 5 kK, a mass estimation of 0.7M_⊙ and radius greater than 0.13R_⊙ with a luminosity of log L/L_⊙ > 1.7. To explain the variations of the shell line absorption they proposed a circumbinary disk model, where the companion intersects with the boundaries of a gap in the disk of the primary star causing a tidal wave. Thus the gas moving in these regions interacts with the dense gas producing shocks. <cit.> state that this model could operate in other Be binaries only if the disk of the primary star is massive enough with considerable density near the companion, if it has a high orbital inclination (i = 90^∘) and if the companion has low mass to create a wide gap so the gas can move across it. For HD41335 we have four observations between 2012 November and 2015 February. The Hα emission line does not show peak intensity variations in this period. From the HeI 6678 Å line we cannot determine if variability is present. The second ϕ Per-type star proposed is HD63462 (Omicron Puppis), a bright B1 IV type. This star shows intensity variations in Hα and from the V/R variation two quasi-period are obtained: 2.5 and 8 years. <cit.> also found a particular variation in the HeI 6678 Å line. They described this variation as: “an emission component swaying from the red side of the profile to the blue one and back”. Their observations were obtained between 2011 November and 2012 April. We inspected our spectra, which are observed in 2013 February and 2015 October, and while there are no variations in the Hα emission line, the HeI 6678 Å line shows the same pattern described by <cit.>. A red peak is seen at 6682 Å in 2013 and a blue peak is seen at 6675 Å in 2015. <cit.> estimated the periodicity of the radial velocities obtained at Hα, HeI 6678 Å and Paschen emission lines (P14, P13 + Ca II and P12) determining an orbital period of 28.9 days. They also found a relation between the velocity and the emission intensity of the HeI line, as the velocity increases the intensity is strongest and vice versa. They did not find any direct evidence of spectral lines from the hot subdwarf companion, and for this reason they suggest that Omicron Puppis is a Be + sdO type. ϕ Per-type systems could potentially test the hypothesis that Be stars could be formed by binary interactions, however these systems are difficult to detect due to the faint companion and for this reason observations in the ultraviolet range are required. The disk density parameters for the best-fitting models for all of these objects were not included in our analysis.§ CONCLUSIONS We modeled the observed Hα line profiles of 63 Be stars from the BeSOS catalogue. Compared to synthetic libraries computed with theandcodes, good matches were found for 57 objects, 42 with Hα in emission and 15, in absorption. The remaining 6 objects had poor fits that did not reproduce the features of the emission line. From the 41 Hα emission line objects, we modeled each available observational epoch giving a total of 61 matched line profiles. Our results were used to constrain to the range of values for the base density and power-law exponent of the disk density model given in Eq. <ref> for all 61 observations. We determined the best fit model for each observation which are displayed in Table <ref> and in the corresponding plots shown in the Appendix section. Moreover, we obtained a distribution of the best representative models with ℱ≤ 1.25 ℱ_min on which we base our average results. The most frequent values for the base density are between <logρ_0> ∼-10.4 and -10.2 and for the power-law exponent are between <n> ∼ 2.0 and 2.5. Combined with an estimate for the size of the Hα disk, the sample distribution for disk mass and disk angular momentum (assuming Keplerian rotation for the disk) were found, with typical values of <M_d>/<M_⋆>∼ 10^-7 and <J_d>/<J_⋆>∼ 10^-9. We find that disk mass and angular momentum distributions were different between early (B0 - B3) and late (B4 - B9) spectral type at 1% level of significance. Finally, we compare our disk masses as a function of spectral type in Figure <ref> with the theoretical predictions of <cit.> based on stellar evolution calculations incorporating rapid rotation. Our average Hα disk masses (which are lower limits to the total disk masses) are always larger than the theoretical predictions, although the variation at each spectral type is quite large, typically more than an order of magnitude.Our estimates for the Hα disk radius (R_90, the radius that encloses 90% of the line emission) are compared to Huang's well-known law relating the disk size to the double-peak separation in the profile. A linear correlation is found with a correlation coefficient of r_corr = 0.652, but there is a large dispersion, which is attributed to the large disk sizes obtained due to the largest Δ V_p and/or smallest v sin i values from the models used in the Huang's relation. The concentration of such values is less than 5R_⋆ for Huang's law and between 15 and 20R_⋆ for R_90 and is dominated by early-type Be stars. Several studies about similarities and differences between early and late type Be stars have been carried out recently. <cit.> suggested that early-type Be stars have more extended envelopes compared with the late-type Be stars from their analysis of Hα equivalent widths by spectral type consistent with the findings presented here.Finally we find that the derived inclination angles from the Hα profile fitting do not follow the expected random distribution. This is attributed to the under-representation of Be shell stars in the BeSOS survey.Numerous studies have found that the mean v sin i values increase for late-type main sequence Be stars <cit.>. In our case, we fixed the rotation of the star to be 80% of the critical value, consistent with <cit.>. Clearly, the study of Be stars is still in continuous development. In future we plan to re-analyze the sample by including more lines in the visible range (i.e., Hβ, Hγ), as well as investigating the spectral energy distributions and v sin i values. The authors would like to thank the anonymous referee for insightful questions and suggestions that helped improve this paper. This research was supported by the DFATD, Department of Foreign Affairs, Trade and Development Canada, International scholarship program Chile-Canada;C.A acknowledges Gemini-CONICYT project No.32120033, Fondo Institucional de Becas FIB-UV, Becas de Doctorado Nacional CONICYT 2016 and PUC-observatory for the telescope time used to obtain the spectra presented in this work.C.E.J and T.A.A.S acknowledge support from NSERC, National Sciences and Engineering Research Council of Canada.S.K thanks the support of Fondecyt iniciación grant N 11130702. C.A, S.K and M.C acknowledges the support from Centro de Astrofísica de Valparaíso. ccc|ccccc|cc|ccc0pt2 Summary of the best fit model by visual inspection and representative models (ℱ /ℱ_min≤ 1.25) of each spectrum for each star. The reader is refereed to section <ref> for the selection details. 3c 5c|Best model 2c|Observation 3|cRepresentative model4-13 HD Sp.T date ℱ /ℱ_min i n ρ_0R_T EW Δ V_p <R_90> <M_d/M_⋆> <J_d/J_⋆>(yyyy-mm-dd) ()(g cm^-3) (R_⋆)(Å) (km s^-1)(R_⋆)10144B6 Vpe 2012-11-13 -- - - - -0.8 719.5 - - -2013-01-18 1.2703.0 7.5e-12 6.0 -0.9 485.1 10.4 1.8e-11 8.5e-10 2013-07-24 1.0703.5 2.5e-11 6.0 -0.5 361.8 13.5 2.8e-10 2.2e-08 2013-10-29 1.0704.0 7.5e-11 6.0 -1.2 353.6 14.6 4.0e-10 3.0e-08 2014-01-29 1.0702.0 5.0e-11 6.0 -1.7 345.3 12.9 3.3e-10 2.3e-0833328^a B2 IVne 2012-11-13 1.060 4.0 7.5e-12 25.0 1.7 703.0 - - -2013-01-18 1.060 4.0 2.5e-12 6.00.1 534.5 - - - 2015-02-25 1.060 4.0 7.5e-12 25.0 1.9 657.8 - - - 35165 B2 Vnpe2014/2015 blue 1.0802.0 5.0e-11 12.5 -12.1 283.7 45.0 8.4e-10 1.0e-07 2014/2015 red1.1802.0 1.0e-11 6.0-12.8 312.4 45.0 8.4e-10 1.0e-07 35411^aB1 V + B2 2012-11-13 1.0 80 4.0 7.5e-12 25.0 2.150 - - - 2013-01-18 1.0 80 3.5 1.0e-12 50.0 3.1 0 - - - 2013-02-26 1.0 80 4.0 1.0e-12 6.03.0 0 - - - 2015-02-25 1.0 80 4.0 7.5e-12 25.0 2.4 0 - - -35439^pf B1 Vpe 2012-11-13 1.0 50 2.5 2.5e-11 50.0-27.7209.7 - - -2013-01-18 1.0 50 2.5 2.5e-11 50.0-28.6185.0 - - -2013-02-26 1.0 50 2.5 2.5e-11 50.0-30.2193.2 - - -2015-02-25 1.0 70 2.0 5.0e-12 50.0-25.6152.1 - - -37795 B9 V2012-11-13 1.0403.02.5e-10 50.0 -9.3 106.9 46.4 3.8e-10 4.5e-082013-01-18 1.0403.02.5e-10 50.0 -9.7 82.2 53.3 4.5e-10 5.9e-082015-02-25 1.0403.02.5e-10 50.0 -9.0 82.2 50.2 4.0e-10 5.4e-0841335^pf B2 Vne2012-11-13 1.080 2.05.0e-1225.0-25.9152.1 - - - 2013-01-18 1.080 2.05.0e-1225.0-27.1111.0 - - - 2013-02-26 1.080 2.05.0e-1225.0-26.7115.1 - - - 2015-02-27 1.080 2.05.0e-1225.0-26.9115.1 - - - 42167B9 IV 2014-01-30 1.0702.02.5e-10 6.0 -2.0160.3 32.6 5.8e-10 5.3e-08 2015-02-25 1.0702.02.5e-10 6.0 -1.7209.7 32.6 5.8e-10 5.3e-08 45725B4 Veshell 2015-02-26 1.0702.0 5.0e-12 25.0 -30.2164.4 87.4 2.1e-09 3.7e-0748917B2 IIIe2014-01-29 1.0602.0 5.0e-12 25.0 -24.6 86.3 103.7 3.3e-09 6.3e-07 2015-10-23 1.0602.0 5.0e-12 25.0 -27.1 90.4 103.7 3.3e-09 6.3e-07 50013^pfB1.5 Ve 2012-11-13 1.0 50 2.52.5e-1150.0 -24.1 94.6 - - -2013-02-26 1.0 60 2.05.0e-1250.0 -22.2 98.7 - - -2014-03-21 1.0 50 2.52.5e-1150.0 -24.0 65.8 - - -2015-02-25 1.0 60 2.05.0e-1250.0-25.265.8 - - - 2015-10-23 1.0 60 2.05.0e-1250.0 -28.9 74.0 - - - 52918^a B1 V 2014-01-29 1.0 60 4.0 1.0e-1125.01.37678.4 - - - 56014B3 IIIe 2014-01-29 red 1.0802.5 1.0e-11 6.0-2.0390.6 23.4 3.0e-10 2.7e-08 2014-01-29 blue 1.0802.5 5.0e-12 12.5 -2.0390.6 23.4 3.0e-10 2.7e-08 56139 B2 IV-Ve2013-02-27 1.0302.0 2.5e-11 25.0 -20.7 0 105.5 9.1e-09 1.7e-062015-02-27 1.0302.0 2.5e-11 25.0 -16.7 0 105.5 9.1e-09 1.7e-06 2015-11-14 1.0302.0 5.0e-11 25.0 -10.2 0 73.7 1.4e-082.3e-06 57150B2 Ve + B3 IVne2014-01-29 1.0602.0 5.0e-12 50.0-30.2 0 189.2 6.8e-09 1.8e-0657219^a B3 Vne 2014-01-291.0 803.57.5e-12 25.02.30 - - - 58343 B2 Vne 2013-02-27 1.0 102.5 7.5e-11 25.0 -7.2 0 71.7 7.7e-09 1.2e-06 58715 B8 Ve2013-02-27 1.0503.5 2.5e-10 25.0 -7.2127.4 35.6 1.2e-09 1.2e-07 2015-02-25 1.0503.5 2.5e-10 25.0 -7.3115.1 35.6 1.2e-09 1.2e-0760606B2 Vne2012-11-13 1.0703.0 1.0e-10 25.0 -21.3 143.9 62.2 3.2e-09 5.1e-072013-01-19 1.0703.0 1.0e-10 25.0 -22.8 152.1 62.2 3.2e-09 5.1e-072013-02-26 1.0703.0 1.0e-10 25.0 -18.9 135.7 62.2 3.2e-09 5.1e-07 63462^pf B1 IVe 2013-02-27 1.0 70 2.0 5.0e-1212.5 -10.994.6 - - - 2015-10-23 1.0 50 2.5 1.0e-1150.0 -11.694.6 - - -68423 B6 Ve2014-03-21 1.0102.0 2.5e-10 50.0 -6.2 49.3 49.1 1.1e-08 1.4e-06 68980B1.5 III 2013-02-271.0 40 2.0 5.0e-12 50.0 -23.241.1 214.2 1.1e-08 2.9e-06 2015-02-26 1.0 402.5 2.5e-11 50.0 -19.645.2 130.2 1.3e-08 3.0e-06 71510^a B2 Ve 2014-01-29 1.070 3.0 2.5e-12 12.5 2.60 - - -2014-03-19 1.070 4.0 7.5e-12 6.0 2.6 0 - - -2015-02-26 1.070 2.0 1.0e-12 6.0 2.250 - - - 75311B3 Vne2014-03-19 1.0 603.0 7.5e-11 50.0 -0.6 287.8 26.0 2.8e-09 2.7e-0778764 B2 IVe 2014-01-30 1.0 402.5 7.5e-11 12.5-4.8 131.6 42.1 3.6e-09 5.3e-072014-03-19 1.0 402.5 7.5e-11 12.5-4.2 139.8 42.1 3.6e-09 5.3e-07 83953^pf B5V 2013-02-27 1.0 70 3.0 1.0e-10 50.0-20.6160.3 - - - 89080 B8 IIIe2013-02-27 1.1702.0 2.5e-12 25.0 -7.2 164.4 35.6 8.8e-10 8.8e-08 2014-05-09 1.1702.0 2.5e-12 25.0 -7.0 143.9 35.6 8.8e-10 8.8e-08 89890^aB3 IIIe 2014-01-30 1.0 703.0 5.0e-1250.01.70 - - - 2014-03-19 1.0 803.5 7.5e-1225.0 2.30 - - - 2015-02-27 1.0 803.0 5.0e-1250.0 1.70 - - - 2015-05-06 1.0 703.5 7.5e-1225.0 1.90 - - - 91465B4 Vne2013-02-26 1.0702.0 5.0e-12 25.0 -28.4131.6 82.4 2.4e-09 3.8e-07 2014-05-09 1.0702.0 1.0e-10 50.0 -24.9135.7 63.1 2.1e-09 3.1e-07 2015-02-27 1.0702.0 1.0e-10 50.0 -22.994.6 63.1 2.1e-09 3.1e-07 2015-05-06 1.1702.0 5.0e-12 25.0 -30.498.7 97.4 2.9e-09 5.0e-07 92938^aB4 V 2014-01-30 1.0804.07.5e-1212.5 2.40 - - -2015-02-27 1.0804.07.5e-1212.5 2.60 - - - 2015-05-06 1.0804.07.5e-1212.5 4.30 - - -93563 B8.5 IIIe2014-01-30 1.2703.5 1.0e-10 50.0 -8.1 296.0 22.5 6.3e-11 5.0e-09B8.5 IIIe2015-05-06 1.2703.5 1.0e-10 50.0 -9.7 135.722.5 6.3e-11 5.0e-09102776 B3 Vne 2014-01-30 1.0603.0 5.0e-11 50.0 -12.2 98.752.1 9.6e-10 1.2e-07 2014-03-19 1.0602.5 1.0e-11 50.0 -9.7185.0 90.6 9.4e-10 1.7e-07 2015-02-27 1.1602.0 2.5e-12 25.0 -7.1185.0 85.5 1.3e-09 2.3e-07 2015-05-06 1.0602.0 2.5e-12 50.0 -7.4119.2 91.4 2.0e-09 3.5e-07103192B9 IIIsp2014-03-19 1.2 603.0 7.5e-12 50.0 -1.4 259.0 13.4 4.6e-10 3.4e-08 2015-02-26 1.2 603.0 7.5e-12 50.0 1.2263.1 13.4 4.6e-10 3.4e-08 2015-05-07 1.2 603.0 7.5e-12 50.0 2.0234.3 13.4 4.6e-10 3.4e-08105382^aB6 IIIe 2014-01-301.0 80 3.55.0e-1225.0 1.3 0 - 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- - 127972B2 Ve 2014-01-31 1.0802.5 7.5e-12 12.5 -5.3 259.0 26.9 3.0e-10 2.8e-082015-02-25 1.0802.5 7.5e-12 12.5 -3.7 349.5 26.9 3.0e-10 2.8e-08 2015-07-15 1.0802.5 7.5e-12 12.5 -2.9 365.9 26.9 3.0e-10 2.8e-08 131492 B4 Vnpe2014-03-21 1.0 703.0 1.0e-11 6.0 -0.9489.2 21.7 1.4e-09 1.2e-07135734 B8 Ve2013-07-24 1.1602.0 2.5e-12 25.0 -7.0 168.6 40.2 1.1e-09 1.2e-07 2015-02-25 1.1602.5 1.0e-11 25.0 -8.3 135.7 40.2 1.1e-09 1.2e-07 2015-07-15 1.1602.5 1.0e-11 25.0 -8.2 152.1 40.2 1.1e-09 1.2e-07138769^a B3 IVp 2013-07-241.0 80 2.5 1.0e-12 12.54.2 0 - - -2015-07-151.0 80 3.5 5.0e-12 50.03.1 0 - - - 142184^a B2 V 2013-07-24 1.0 60 4.0 5.0e-12 12.5 2.0698.9 - - -2014-03-21 1.0 80 4.0 2.5e-12 6.03.5698.9 - - -143275 B0.3 IV 2014-03-19 1.1 20 3.0 7.5e-11 50.0 -11.3 0 143.6 1.0e-07 3.1e-05148184B2 Ve 2013-07-24 1.0302.0 1.0e-11 25.0 -35.9 0 152.8 2.4e-08 5.5e-06 2015-02-25 1.0302.0 1.0e-11 25.0 -34.9 0 152.8 2.4e-08 5.5e-062015-05-06 1.0302.0 1.0e-11 25.0 -39.9 0 152.8 2.4e-08 5.5e-06 157042B2 IIIne2013-07-241.1 702.5 2.5e-11 12.5 -20.2160.3 55.0 1.6e-09 2.1e-072015-05-061.1 702.5 2.5e-11 12.5 -22.9213.8 55.0 1.6e-09 2.1e-07 158427 B2 Ve2015-05-06 1.0702.0 5.0e-12 50.0 -36.1 32.9 188.1 7.5e-09 2.0e-06 167128 B3 IIIpe2013-07-24 1.0403.5 7.5e-11 50.0 -3.8 164.4 32.6 3.1e-09 3.9e-07205637B3 V2012-11-14 1.1892.0 1.0e-11 6.0 -1.9 337.1 27.3 8.8e-10 8.4e-08209014B8 Ve2013-07-24 1.0892.0 2.5e-10 12.5 -8.0242.6 29.3 1.1e-09 1.1e-072015-10-23 1.0892.0 2.5e-10 12.5 -8.5209.7 29.3 1.1e-09 1.1e-07 209409 B7 IVe2012-11-13 1.0802.0 5.0e-12 25.0 -18.9143.9 58.2 7.9e-10 1.1e-07 2015-10-24 1.2802.0 5.0e-12 50.0 -20.0152.1 55.6 2.0e-09 2.4e-07212076 B2 IV-Ve2012-11-13 1.3 302.0 2.5e-11 25.0 -18.228.8 85.1 3.1e-09 4.8e-07 2015-10-23 1.0302.0 2.5e-12 50.0 -14.3 24.7 118.2 6.8e-09 1.2e-06212571 B1 III-IV 2012-11-14 1.1602.5 1.0e-11 12.5 -7.7283.7 84.1 9.3e-10 1.5e-07 2013-07-24 1.1602.5 7.5e-12 12.5 -4.0304.2 74.4 6.4e-10 9.6e-08 2015-10-24 1.0602.5 1.0e-11 12.5 -10.7 209.7 83.9 1.6e-09 2.6e-07 214748B8 Ve 2012-11-15 1.3503.5 2.5e-10 12.5 -4.0 131.6 28.7 2.8e-09 2.2e-07 2013-07-24 1.3503.5 2.5e-10 12.5 -4.9 123.3 28.7 2.8e-09 2.2e-07 2015-07-15 1.3503.5 2.5e-10 12.5 -5.7 123.3 28.7 2.8e-09 2.2e-072015-10-24 1.3503.5 2.5e-10 12.5 -5.7 135.7 28.7 2.8e-09 2.2e-07 217891 B6 Ve2012-11-13 1.0402.0 5.0e-11 50.0 -21.1 094.1 1.7e-08 2.9e-062013-07-25 1.0402.0 5.0e-11 50.0 -22.8 0 94.1 1.7e-08 2.9e-06219688^aB5 V 2015-10-24 1.0 503.0 2.5e-12 12.5 2.6 0 - - -221507^a B9.5 IIIpHgMnSi2013-07-24 1.0 89 3.0 2.5e-12 6.0 2.50 - - - 2015-07-15 1.0 89 3.0 2.5e-12 6.0 3.50 - - - 2015-10-23 1.0 89 3.0 2.5e-12 6.0 4.1 0 - - - 224686 B8 Ve2012-11-13 1.0802.0 2.5e-10 6.0 -2.0 275.4 28.7 2.8e-9 2.8e-07a Absorption profiles pf Poor fit -Not agreement model The information displayed in this table are for the best (visual inspection) and representative (ℱ /ℱ_min≤ 1.25) models of each observation. Values of the representative models are only for emission profiles without a poor fit.The Spectral Type (Sp.T) is obtained from Simbad database. Blue and red (indicated next to the date) refer to the blue or red peak fit, respectively. aasjournal § EMISSION PROFILES Observed emission line profiles from our program stars (black lines) shown with the best-fit model (red dashed lines). Variable stars in our sample are indicated an asterisk symbol beside the star name. § ABSORPTION PROFILES The same as Appendix <ref> except for absorption profiles. § POOR FITS The same as Appendix <ref> except for program stars with poor fits. See Section <ref> for details. | http://arxiv.org/abs/1704.08133v1 | {
"authors": [
"C. Arcos",
"C. E. Jones",
"T. A. A. Sigut",
"S. Kanaan",
"M. Curé"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170426142150",
"title": "Evidence for Different Disk Mass Distributions Between Early and Late-Type Be Stars in the BeSOS Survey"
} |
Division of Engineering and Applied Science California Institute of Technology, Pasadena, California 91125,USA Division of Engineering and Applied Science California Institute of Technology, Pasadena, California 91125,USA [email protected] Division of Engineering and Applied Science California Institute of Technology, Pasadena, California 91125,USA Thermal atomic vibrations in amorphous solids can be distinguished by whether they propagate as elastic waves or do not propagate due to lack of atomic periodicity. In a-Si, prior works concluded that non-propagating waves are the dominant contributors to heat transport, while propagating waves are restricted to frequencies less than a few THz and are scattered by anharmonicity. Here, we present a lattice and molecular dynamics analysis of vibrations in a-Si that supports a qualitatively different picture in which propagating elastic waves dominate the thermal conduction and are scattered by elastic fluctuations rather than anharmonicity. We explicitly demonstrate the propagating nature of vibration with frequency approaching 10 THz using a triggered wave computational experiment. Our work suggests that most heat is carried by propagating elastic waves in a-Si and demonstrates a route to achieve extreme thermal properties in amorphous materials by manipulating elastic fluctuations.Valid PACS appear herePropagating elastic vibrations dominate thermal conduction in amorphous silicon Austin J. Minnich December 30, 2023 =============================================================================== Amorphous materials are of interest for a wide range of applications due to their low thermal conductivity <cit.>. While in crystals heat is carried by propagating lattice waves, or phonons, in amorphous solids heat carriers are classified as propagons, diffusons, and locons depending on the degree of delocalization of the atomic vibration and its mean free path <cit.>. This classification has been widely used to analyze the vibrations responsible for thermal transport in amorphous materials, especially for pure a-Si. Numerical studies using equilibrium molecular dynamics (EMD) and lattice dynamics (LD) have attempted to determine the fraction of heat carried by each category of vibration <cit.>. While the general consensus is that diffusons carry the majority of the heat, prior works have reported that propagons may carry 20∼50 % of thermal conductivity in a-Si due to their long mean free paths <cit.>. Using normal-mode analysis, Larkin and McGaughey reported that propagons have a lifetime scaling of ω ^-2 which suggests plane-wave-like propagation that is not affected by atomic disorder <cit.>. The normal mode lifetime analysis of Lv and Henry concluded that the phonon gas model is not applicable to amorphous materials <cit.>. Experimental works have qualitatively confirmed some of these predictions, particularly regarding the important contribution of propagons <cit.>. For instance, Kwon et al. observed size effects in a-Si nanostructures, indicating the presence of propagons <cit.>. Despite these efforts, numerous puzzles remain. One discrepancy concerns the conclusion that the lifetimes of few THz vibrations are governed by anharmonicity <cit.>.If that is the case, explaining the low thermal conductivity of a-Si is challenging because the same vibrations contribute 75 Wm-1K-1 to thermal conductivity in c-Si. Accounting for the low thermal conductivity of a-Si only by changes in anharmonicity requires large increases in anharmonic force constants that would necessarily have effects on the heat capacity of a-Si that have not been observed <cit.>. Along similar lines, if lifetimes of low frequency vibrations are governed by anharmonicity the reported thermal conductivities of films of the same thickness should be reasonably uniform, yet the data vary widely <cit.>. Overall, an unambiguous classification of the propagating nature and scattering mechanisms of vibrational modes transporting heat in amorphous solids is poorly developed, impeding efforts to synthesize, for example, novel materials with exceptionally low thermal conductivity. In this work, we address these questions using lattice and molecular dynamics to calculate dynamic structure factors and thermal transport properties of a-Si. Our analysis supports a qualitatively different picture of atomic vibrations in a-Si from the conventional one in which propagating elastic waves dominate the thermal conduction and are scattered by elastic fluctuations rather than anharmonicity. We explicitly demonstrate the propagating nature of waves with frequencies approaching 10 THz using a computational triggered wave analysis. Our work provides strong evidence that, unintuitively, elastic waves with frequencies up to 10 THz carry substantial heat in disordered media and also demonstrates a route to create materials with exceptional thermal properties by manipulating elastic fluctuations. We used lattice and molecular dynamics to examine the atomic vibrations of various amorphous domains. The molecular dynamics calculations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) with a timestep of 0.5 fs <cit.>.Periodic boundary conditions were imposed and the Stillinger-Weber interatomic potential was used <cit.>. The initial structure we considered contained 4096 atoms and was created by first melting crystalline silicon at 3500 K for 500 ps in an NVT ensemble. Next, the liquid silicon was quenched to 1000 K with the quench rate of 100 K/ps. The structures were annealed at 1000 K for 25 ns to reduce metastabilities <cit.>. Finally, the domain was quenched at a rate of 100 K/ps to 300 K and equilibrated at 300 K for 10 ns in an NVT ensemble using a Nose-Hoover thermostat. The structure was then equilibrated at 300 K for 500 ps in an NVT ensemble. After an additional equilibration in an NVE ensemble for 500 ps, the heat fluxes were computed for 1.6 ns in the same NVE ensemble. We use Green-Kubo theory to compute the thermal conductivity of the structure to be 1.5 Wm-1K-1, a value that is consistent with prior works <cit.>. We begin our analysis to gain more insight into the vibrations carrying heat by characterizing the propagating nature of the normal modes of vibration of the amorphous domain. A convenient metric for this characterization is the dynamic structure factor, given by S_L,T(q,ω) = ∑_ν E_L,T(q,ν)δ(ω-ω(q = 0,ν)) where the q is phonon wavevector, ω is frequency and the summation is over all the modes ν at Γ. E_L and E_T refer to the longitudinal polarization and transverse polarization and are defined as E_L(q, ν) = |∑_i [q̂·e(ν,i)]e^iq·r_i|^2 E_T(q, ν) = |∑_i [q̂×e(ν,i)]e^iq·r_i|^2where the summation is over all atoms indexed by i in the domain, q̂ is a unit vector, e(ν,i) is the eigenvector, and r_i are the equilibrium positions. The dynamic structure factor is precisely what is measured in scattering experiments to measure dispersion relations in crystals and can be applied to search for plane waves in disordered media. We calculated the eigenvectors of the 4096 atom structure using the General Utility Lattice Program (GULP) with equilibrated structures from MD <cit.>.As amorphous Si is isotropic, we average the dynamic structure factor over all wavevectors of the same magnitude. If propagating waves exist despite the atomic disorder, the dynamic structure factor will exhibit a clear phonon band with a dispersion; if propagating waves are not supported, the vibrational modes will appear diffuse without an apparent dispersion. The dynamic structure factor for longitudinal waves is presented in Fig. <ref> (a). Unexpectedly, the figure demonstrates that despite the atomic disorder a clear dispersion exists up to frequency as high as 10 THz for longitudinal waves, corresponding to a wavelength of 6.5 Å. In the transverse direction, a clear dispersion with broadening is also observed up to ∼ 5 THz, with a similar transition wavelength at 6.6 Å(not shown). For sufficiently high frequency vibrations with wavelengths comparable to interatomic distances, the structure factor is very broad and identifying plane waves with definite frequency and wavevector is not possible. However, the figure clearly shows that propagating elastic waves comprise a substantial portion of the vibrational spectrum. Specifically, by calculating the density of states of the low frequency vibrations with a Debye model, we estimate that about 24% of all modes are propagating waves. Our observation is consistent with a prior calculation of dynamical structure factor <cit.> but contrasts with prior studies that restrict propagons to frequencies less than2∼3 THz in amorphous silicon or comprise a small fraction of all modes<cit.>. In addition to the existence of well-defined plane-waves with definite frequency and wavevector, we also observe that the lines are not narrow but have a clear broadening indicating the presence of a scattering mechanism. In crystals, this broadening is typically due to anharmonic interactions. In the perfectly harmonic amorphous solid considered here, anharmonic interactions cannot play any role. Instead, the broadening must be due to fluctuations of the local elastic modulus on length scales comparable to the wavelengths of the propagating waves. Therefore, the picture that emerges from our calculation of dynamical structural factor of a-Si is a vibrational spectrum that is dominated by elastic waves that are scattered by elastic fluctuations in the disordered solid. In actual a-Si, anharmonic interactions may increase the scattering rate and hence the broadening. To assess how broadening due to elastic fluctuations compares to that from anharmonic interactions, we also calculate dynamic structure factors using velocity outputs from MD at 300 K <cit.>. The longitudinal dynamic structure factors at q = 6.0 nm-1 with harmonic and anharmonic forces are depicted in Fig. <ref> (b), demonstrating that the two are nearly identical. Therefore, anharmonic broadening has essentially no effect on the lifetimes and the broadening is solely due to elastic fluctuations. We next aim to extract quantitative information from the observed broadened lines. Prior works used normal mode analysis to extract lifetimes from molecular dynamics simulations <cit.>. Here, we instead obtain lifetime information for those modes with a well-defined dispersion by fitting a constant wavevector slice of the dynamic structure factor with damped harmonic oscillator (DHO) model <cit.>. The lifetime τ at a certain frequency is related to the full-width at half-maximum Γ by τ=1/πΓ. By multiplying the lifetimes by the sound velocity, we also obtain mean free paths. The results are shown in Fig. <ref> (a). We see that the mean free paths span from 0.5 nm to 10 nm. At still lower frequencies that cannot be included in the present simulations mean free paths are likely even longer, as suggested by experiment <cit.>. In addition, Fig. <ref> (b) plots the product of lifetime and vibrational frequency. In this plot, the Ioffe-Regel (IR) crossover from propagons to diffusons, defined as when the lifetime is equal to the period of a wave, can be indicated as a horizontal line <cit.>.For longitudinal waves, the IR crossover is observed at ∼ 10 THz. A similar analysis for transverse waves indicates the IR crossover is found around ∼ 5 THz for these vibrations; both of these values are in good agreement with the qualitative estimate of the transition frequency from the structure factor. Having established that propagons comprise a substantial fraction of the vibrational spectrum, we next estimate the propagon contribution to thermal conductivity given knowledge of the linear, isotropic dispersion, the group velocity, and the mean free paths from Figs. <ref> and <ref> using a Debye model. In this model, we separate the propagon contribution into longitudinal and transverse modes with group velocities obtained from the dispersion as 8000 and 3610 m/s, respectively.Recalling the bulk thermal conductivity of 1.5 Wm-1K-1 from the Green-Kubo calculation, our Debye model estimates that propagons contribute about 1.35 Wm-1K-1, or 90 % of the thermal conductivity. This contribution is much larger than the values reported previously and suggests that, counterintuitively, heat transport in a-Si is dominated by propagating waves despite the atomic disorder. The primary uncertainty in this estimate is the role of vibrations of frequency less than 2 THz that are challenging to include in both the Green-Kubo and structure factor calculations; however, the general conclusion that propagons dominate the heat transport will still hold even in the absence of these additional propagating vibrations. Our analysis thus suggests a considerably different picture of atomic vibrations in a-Si from the conventional view. Earlier works have concluded that propagons are a small fraction of mode population that contribute less than half of the total thermal conductivity, and that the phonon gas model is an inaccurate picture of the vibrations in amorphous solids. By contrast, our examination of the dynamical structure factor indicates that propagons form 24 % of the mode population and are scattered by fluctuations of the local elastic modulus rather than anharmonicity. Additionally, the highest reported contribution of propagons to thermal conductivity is 50% of the total. However, the picture that emerges from our analysis is that a gas of delocalized elastic vibrations exists in a-Si, despite the lack of atomic order, which transports most of the heat. We provide further support for our conclusions with two additional calculations. First, we explicitly demonstrate the propagating nature of certain vibrational modes by conducting a "tuning fork experiment" in which imposed oscillatory atomic motions at one edge of the atomic domain triggers the formation of a traveling wave through the a-Si. To perform this calculation, we first create a domain by repeating 4096-atom cell 10 times along one direction, resulting in a supercell of size 4.3 × 4.3 × 43 nm. In the long dimension, the domain is divided into 80 slabs of width 5.431 Å. Periodic boundary conditions are applied and the temperature is set at 0.1 K to avoid additional thermal displacements. The calculation begins by rigidly displacing the first slab in the longitudinal direction for 2 ps with a sinusoidal wave with amplitude 0.01 Åand a specified frequency. We computed the longitudinal displacements of every atom for time durations less than 2 ps to prevent edge effects, and subsequently averaged the atomic displacements within each slab. The wave propagation in a-Si at different frequencies is shown in Fig. <ref>. It is apparent that waves do indeed propagate through a-Si at 3 THz and 8 THz as predicted by the dynamic structure factor calculations. By identifying the location at which the wave amplitude has decreased to 1/e of its original value, we estimate that the mean free paths of the 3 THz and 8 THz waves are around 9 nm and 2 nm, respectively. These mean free paths coincide reasonably well with those from dynamic structure factor calculations (8 nm and 3 nm). On the other hand, the excited wave at 16 THz is damped very quickly, and by the second slab the amplitude is already less than 1/e of the original value. This observation indicates that at 16 THz the vibration is non-propagating. Therefore, the "tuning fork experiment" qualitatively confirms that propagating waves exist up to a high frequency of around 10 THz in a-Si.Second, we examine how the thermal conductivity is affected by the partial elimination of elastic fluctuations. Our calculations indicate that fluctuations in local elastic modulus are the dominant mechanism for disrupting propagons. If this assertion is true, we should observe a marked increase in thermal conductivity when these fluctuations are partially eliminated along with a temperature dependence of thermal conductivity that reflects the renewed dominance of phonon-phonon interactions. To test this hypothesis, we generated two additional domains designed to possess reduced elastic fluctuations consisting of 512 and 64-atom amorphous unit cell (AUC) tiled to create 4096-atom structures. The 512 and 64 AUC domains were created using the same melt-quench procedure described earlier. Elastic fluctuations over a length scale equal to the AUC domain size should be eliminated because the same unit cell is tiled repeatedly in space to form the 4096 atom final structure. We followed the same procedure as described earlier to obtain dynamic structure factors for the tiled structures. The structure factor for the 512-atom AUC tiled structure appear almost identical to that of the original calculation (not shown). That for the 64-atom AUC tiled structure are shown in Fig. <ref> (a). We observe discrete points rather than a continuous broadening, indicative of the dynamic structure factor having delta-function-like peaks as occurs in c-Si. From a constant wavevector slice of the dynamic structure factor for the 64-atom AUC tiled structure in Fig. <ref> (b), we observe that anharmonicity broadens the peaks in the frequency range from 5 to 10 THz, indicating that anharmonicity plays a role in scattering these modes. In addition, we observe several peaks in the medium to high frequency region from 10 to 20 THz, which is characteristic of crystalline materials with a multi-atom unit cell that have several folded branches. Overall, these calculations indicate that the 64-atom AUC structure possesses vibrations that are characteristic of a semi-crystalline solid while the 512-atom AUC remains effectively amorphous. Therefore, the key length scale for the elastic fluctuations that sets whether vibrations propagate is around 10 Å, or the side length of the 64-atom domain. We now compute the thermal conductivity of the 3 structures using Green-Kubo theory averaged over 10 different initial conditions. The resulting thermal conductivity calculations of these structures are shown in Fig. <ref> (c). The figure shows that the pure a-Si and the 512-atom AUC tiled structure have identical thermal conductivity with little temperature dependence. This result confirms that the 512-atom AUC structure is effectively amorphous. However, we observe a significant increase in thermal conductivity of the 64-atom AUC tiled structure, by more than a factor of 2 at room temperature, along with a marked temperature dependence. At 100 K, the thermal conductivity of the 64-atom AUC tiled structure is ∼ 10 Wm-1K-1, more than 6 times that of pure a-Si. Therefore, the 64-atom AUC tiled structure exhibits characteristics of crystals, and the key disorder length scale that sets the transition of thermal vibrations from crystalline to amorphous character lies between 10-20 Å. The picture of a gas of delocalized elastic vibrations transporting heat in amorphous solids suggests follow-on experiments as well as new strategies to realize exceptional thermal materials. First, our prediction of propagons existing up to around 10 THz can be verified with additional thermal measurements on amorphous nanostructures with characteristic dimensions of less than 10 nm as well as with scattering methods such as inelastic X-ray scattering. Second, our analysis suggests that fully dense solids with exceptionally low thermal conductivity can be achieved by manipulating elastic fluctuations, expanding the physical range of thermal conductivity of solids.In summary, we have examined the atomic vibrations in a-Si using lattice and molecular dynamics calculations. Our study reveals a qualitatively different view of atomic vibrations in a-Si from conventional one in which propagating elastic waves dominate the thermal conduction and are scattered by elastic fluctuations instead of anharmonicity. Our work provides important insights into the long-standing problem of thermal transport in disordered solids.This work was supported by the Samsung Scholarship, NSF CAREER Award CBET 1254213, and Boeing under Boeing-Caltech Strategic Research and Development Relationship Agreement. | http://arxiv.org/abs/1704.08360v1 | {
"authors": [
"Jaeyun Moon",
"Benoit Latour",
"Austin Minnich"
],
"categories": [
"cond-mat.dis-nn",
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.dis-nn",
"published": "20170426214816",
"title": "Propagating elastic vibrations dominate thermal conduction in amorphous silicon"
} |
Physics and Astronomy Department,Michigan State University,East Lansing, Michigan 48824, USA School of Physics,University of the Witwatersrand,Private Bag 3,2050 Johannesburg,South [email protected] Physics and Astronomy Department,Michigan State University,East Lansing, Michigan 48824, USA We propose a continuum model to predict long-wavelength vibrational modes of empty and liquid-filled tubules that are very hard to reproduce using the conventional force-constant matrix approach based on atomistic ab initio calculation. We derive simple quantitative expressions for long-wavelength longitudinal and torsional acoustic modes, flexural acoustic modes, as well as the radial breathing mode of empty or liquid-filled tubular structures that are based on continuum elasticity theory expressions for a thin elastic plate. We furthermore show that longitudinal and flexural acoustic modes of tubules are well described by those of an elastic beam resembling a nanowire. Our numerical results for biological microtubules and carbon nanotubes agree with available experimental data. 61.46.Np,63.22.-m,62.20.de, 62.25.Jk Long-wavelength deformations and vibrational modesin empty and liquid-filled microtubules and nanotubes:A theoretical study David Tománek December 30, 2023 =================================================================================================================================================== § INTRODUCTION Tubular structures with diameters ranging from nanometers to meters abound in natureto fill various functions. The elastic response of most tubular structures is dominated by low-frequency flexural acoustic (ZA) modes. Much attention has been devoted to the nanometer-wide carbon nanotubes (CNTs) <cit.>, which are extremely stiff <cit.>, and to their flexural modes <cit.>. Correct description of soft flexural modes in stiff quasi-1D systems like nanotubes and nanowires is essential for calibrating nanoelectromechanical systems used for ultrasensitive mass detection and radio-frequency signal processing <cit.>. In CNTs and in related graphene nanoribbons, flexural ZA modes have also been shown to significantly influence the unsurpassed lattice thermal conductivity <cit.>. Much softer microtubules formed of tubulin proteins, with a diameter d≈20 nm, are key components of the cytoskeleton and help to maintain the shape of cells in organisms. In spite of their importance, there are only scarce experimental data available describing the elastic behavior of microtubules. The conventional approach to calculate the frequency spectrum is based on an atomistic calculation of the force-constant matrix. This approach often fails for long-wavelength acoustic modes, in particular the soft flexural ZA modes, due to an excessive demand on supercell size and basis convergence. Typical results of this shortcoming are numerical artifacts such as imaginary vibration frequencies <cit.>. Here we offer an alternative way, based on continuum elasticity theory <cit.> and its extension to planar <cit.> and tubular structures <cit.>, to predict the frequency of acoustic modes in quasi-1D structures such as empty and liquid-filled tubes consisting of stiff graphitic carbon or soft tubulin proteins. While the scope of our approach is limited to long-wavelength acoustic modes, the accuracy of vibration frequencies calculated using the simple expressions we derive surpasses that of conventional atomistic ab initio calculations. Our approach covers longitudinal and torsional modes, flexural modes, as well as the radial breathing mode. We show that longitudinal and flexural acoustic modes of tubules are simply related to those of an elastic beam resembling a nanowire. Since the native environment of tubulin nanotubes contains water, we specifically consider the effect of a liquid on the vibrational modes of tubular structures. Our numerical results for tubulin microtubules and carbon nanotubes agree with available experimental data. § CONTINUUM ELASTICITY APPROACH A 1D tubular structure of radius R can be thought of as a rectangular 2D plate of width 2πR that is rolled up to a cylinder. Consequently, the elastic response of 1D tubules to strain, illustrated in Figs. <ref>(a)-<ref>(d), is related to that of the constituting 2D plate. To describe this relationship in the linear regime and calculate the frequency of long-wavelength vibrational modes in 1D tubular structures, we take advantage of a continuum elasticity formalism that has been successfully adapted to 2D structures <cit.>.As shown earlier <cit.>, elastic in-plane deformations of a plate of indefinite thickness may be described by the (3×3) 2D elastic stiffness matrix, which is given in Voigt notation by ( [ c_11 c_120; c_12 c_220;00 c_66 ]). Resistance of such a plate to bending is described by the flexural rigidity D. For a plate suspended in the x-y plane, c_11 and c_22 describe the longitudinal strain-stress relationship along the x- and y-direction, respectively. c_66 describes the elastic response to in-plane shear. For an isotropic plate, which we consider here, c_11=c_22, c_66=(c_11-c_12)/2, and the in-plane Poisson ratio α=c_12/c_11. Considering a 3D plate of finite thickness h, characterized by the (6×6) elastic stiffness matrix C_ij, the coefficients of the 2D elastic stiffness matrix c_ij for the equivalent plate of indefinite thickness are related by c_ij=h·C_ij(1-α_⊥^2). This expression takes account of the fact that stretching a finite-thickness slab of isotropic material not only reduces its width by the in-plane Poisson ratio α, but also its thickness by the out-of-plane Poisson ratio α_⊥. This consideration is not needed for layered compounds such as graphite, where the inter-layer coupling is weak and α_⊥≈0, so that c_ij=h·C_ij. In near-isotropic materials like tubulin, however, α_⊥≈α and c_ij=h·C_ij(1-α^2). §.§ Vibrational Modes of empty Nanotubes We now consider an infinitely thin 2D plate of finite width 2πR and an areal mass density ρ_2D rolled up to a nanotube of radius R that is aligned with the x-axis. The linear mass density of the nanotube is related to that of the plate by ρ_1D = 2πRρ_2D . In the long-wavelength limit, represented by k=(2π/λ)→0, the longitudinal acoustic mode of a tubular structure, depicted in Fig. <ref>(a), resembles the stretching mode of a 2D plate <cit.>. As mentioned above, the equivalent plate we consider here is a strip of finite width that is reduced during stretching due to the nonzero in-plane Poisson ratio α.In the following, we illustrate our computational approach for a tubular structure by focussing on its longitudinal acoustic mode. Our derivation, which is described in more detail in Appendices A and B, starts with the Lagrange function density ℒ(du_x/dx,du_x/dt,x,t) = T - U= 1/2[ ρ_2D( du_x/dt)^2 - c_11(1-α^2)( du_x/dx)^2 ] 2πR= 1/2[ρ_1D( du_x/dt)^2 - c_LA( du_x/dx)^2 ],where c_LA = c_11(1-α^2) 2πR is the longitudinal force constant of a 1D nanowire equivalent to the tubule, and the relation between ρ_1D and ρ_2D is defined in Eq. (<ref>). The resulting Euler-Lagrange equation is d/dt( ∂ℒ/∂du_x/dt)+d/dx( ∂ℒ/∂du_x/dx) = 0. Using the ansatz u_x = u_x,0 e^i(kx-ωt) we obtain the vibration frequency of the longitudinal acoustic (LA) mode of the nanotube or nanowire from ω_LA = √(c_11(1-α^2)/ρ_2D) k = √(c_LA/ρ_1D) k. The prefactor of the crystal momentum k is the longitudinal speed of sound. As already noted in Ref. [Lawler], the frequency of the LA mode is independent of the nanotube radius.The torsional mode, depicted in Fig. <ref>(b), is very similar to the shear mode of a plate. Consequently, as shown in Appendix B, the vibration frequency of the torsional acoustic (TA) mode of the nanotube and the transverse acoustic mode of the plate should be the same. With c_66 describing the resistance of the equivalent plate to shear, we obtain ω_TA = √(c_66/ρ_2D) k. Again, prefactor of the crystal momentum k is the corresponding speed of sound. Similar to the LA mode, the frequency of the TA mode is independent of the nanotube radius <cit.>.The doubly degenerate flexural acoustic (ZA) mode, depicted in Fig. <ref>(c), differs significantly from the corresponding bending mode of a plate <cit.> that involves the plate's flexural rigidity D. The continuum elasticity treatment of the bending deformation, described in Appendices A and B, leads to ω_ZA =√(πR^3c_11/ρ_1D (1+D/c_11R^2)) k^2 = √(D_b/ρ_1D) k^2=c_ZA(R) k^2 . Here, c_ZA is the effective bending force constant and D_b is the effective beam rigidity of a corresponding nanowire, defined in Eq. (<ref>).Finally, the radial breathing mode (RBM) of the nanotube, depicted Fig. <ref>(d), has a nearly k-independent frequency given by <cit.> ω_RBM = 1/R√(c_11/ρ_2D) . The four vibration modes described above and their functional dependence on the momentum k and radius R have been partially described before using an elastic cylindrical shell model <cit.>. The schematic dependence of the vibration frequencies of these modes on k is shown in Fig. <ref>(e). The main expressions for the vibration frequencies of both 2D and tubular 1D structures are summarized in Table <ref>. §.§ Vibrational Modes of Liquid-Filled Nanotubes We next consider the nanotubes completely filled with a compressible, but viscosity-free liquid thatmay slide without resistance along the nanotube wall <cit.>. Since the nanotubes remain straight and essentially maintain their radius during stretching and torsion, the frequency ω̃ of the LA and TA modes is not affected by the liquid inside, which remains immobile during the vibrations. We thus obtain ω̃_LA(k) ≈ω_LA(k) and ω̃_TA(k) = ω_TA(k) , where the tilde refers to filling by a liquid.The only effect of filling by a liquid on the flexural modes is an increase in the linear mass density to ρ̃_1D= ρ_1D + πR^2ρ_l , where ρ_l denotes the gravimetric density of the liquid. In comparison to an empty tube, described by Eq. (<ref>), we observe a softening of the flexural vibration frequency to ω̃_ZA =√(πR^3c_11/ρ̃_1D (1+D/c_11R^2)) k^2 =√(D_b/ρ̃_1D) k^2 = c̃_ZA(R) k^2 . Finally, as we expand in Appendix C, the effect of the contained liquid on the RBM frequency will depend on the stiffness of the tubular container. For stiff carbon nanotubes, the RBM mode is nearly unaffected, whereas the presence of an incompressible liquid increases ω̃_RBM in soft tubules. Thus, ω̃_RBMω_RBM . The schematic dependence of the four vibration modes on the momentum k in liquid-filled nanotubes is shown in Fig. <ref>(e). The main expressions for the vibration frequencies of liquid-filled tubular 1D structures are summarized in Table <ref>.§ VIBRATIONAL MODES OF NANOTUBES IN A SURROUNDING LIQUID From among the four long-wavelength vibrational modes of nanotubes illustrated in the left panels of Fig. <ref>, the stretching and the torsional modes are not affected by the presence of a liquid surrounding the nanotube. We expect the radial breathing mode in Fig. <ref>(d) to couple weakly and be softened by a small amount in the immersing liquid. The most important effect of the surrounding liquid is expected to occur for the flexural mode shown in Fig. <ref>(c).The following arguments and expressions have been developed primarily to accommodate soft biological structures such as tubulin-based microtubules, which require an aqueous environment for their function. We will describe the surrounding liquid by its gravimetric density ρ_l and viscosity η. As suggested above, we will focus our concern on the flexural long-wavelength vibrations of such structures.As we will show later on, the flexural modes of idealized, free-standing biological microtubules are extremely soft. In that case, the velocity of transverse vibrations will also be very small and definitely lower than the speed of sound in the surrounding liquid. Under these conditions, the motion of the rod-like tubular structure will only couple to the evanescent sound waves in the surrounding liquid and there will be no radiation causing damping. The main effect of the immersion in the liquid will be to increase the effective inertia of the rod. We may assume that the linear mass density ρ_1D of the tubule in vacuum may increase to ρ̃_1D=ρ_1D+ Δρ_1D in the surrounding liquid. We can estimate Δρ_1D=ΔAρ_l, where ΔA describes the increase in the effective cross-section area of the tubule due to the surrounding liquid that is dragged along during vibrations. We expect ΔA≲πR^2, where R is the radius of the tubule. The softening of the flexural mode frequency ω̃_ZA due to the increase in ρ_1D is described in Eq. (<ref>).Next we consider the effect of viscosity of the surrounding liquid on long-wavelength vibrations of a tubular structure that will resemble a rigid rod for k→0. Since – due to Stoke's paradox – there is no closed expression for the drag force acting on a rod moving through a viscous liquid, we will approximate the rod by a rigid chain of spheres of the same radius, which are coupled to a rigid substrate by a spring. The motion for a rigid chain of spheres is the same as of a single sphere, which is damped by the drag force F=6πηRv according to Stoke's law, where v is the velocity.The damped harmonic motion of a sphere of radius R and mass m is described by m d^2u/dt^2 =-mω_0^2u-6πηRdu/dt . With the ansatz u(t)=u_0e^iωt, we get -mω^2= -m ω_0^2 -iω6πηR and thus ω =±√(ω_0^2-(3πηR/m)^2) + i3πηR/m . Assuming that the damping is small, we can estimate the energy loss described by the Q-factor Q = ω_0m/3πηR = 2/3m/Rf_0/η , where f_0=ω_0/(2π) is the harmonic vibration frequency. In a rigid string of masses separated by the distance 2R, the linear mass density is related to the individual masses by ρ_1D=m/(2R). Then, the estimated value of the Q-factor will be Q=4/3ρ_1Df_0/η . § COMPUTATIONAL APPROACH TO DETERMINE THE ELASTICRESPONSE OF CARBON NANOTUBES We determine the elastic response and elastic constants of an atomically thin graphene monolayer, the constituent of CNTs, using ab initio density functional theory (DFT) as implemented in the SIESTA <cit.> code. We use the Perdew-Burke-Ernzerhof (PBE) <cit.> exchange-correlation functional, norm-conserving Troullier-Martins pseudopotentials <cit.>, and a double-ζ basis including polarization orbitals. To determine the energy cost associated with in-plane distortions, we sampled the Brillouin zone of a 3D superlattice of non-interacting layers by a 20×20×1 k-point grid <cit.>. We used a mesh cutoff energy of 180 Ry and an energy shift of 10 meV in our self-consistent total energy calculations, which has provided us with a precision in the total energy of ≤2 meV/atom. The same static approach can be applied to other layered materials that form tubular structures.§ RESULTS To illustrate the usefulness of our approach for all tubular structures, we selected two extreme examples. Nanometer-wide CNTs have been characterized well as rigid structures able to support themselves in vacuum. Tubulin-based microtubules, on the other hand, are significantly wider and softer than carbon nanotubes. These biological structures require an aqueous environment for their function. §.§ Carbon Nanotubes The elastic behavior of carbon nanotubes can be described using quantities previously obtained using DFT calculations for graphene <cit.>. The calculated elements of the elastic stiffness matrix (<ref>) are c_11=c_22=352.6 N/m, c_12=59.6 N/m, and c_66= 146.5 N/m, all in very good agreement with experimental results <cit.>. The calculated in-plane Poisson ratio α=c_12/c_11=0.17 is also close to the experimentally estimated value for graphene <cit.> of α_expt=0.19.The calculated flexural rigidity of a graphene plate is D=0.22 GPa·nm^3.The calculated 2D mass density of graphene ρ_2D=0.743·10^-6 kg/m^2 translates to ρ_1D=0.743·10^-6 kg/m^2·2πR for nanotubes of radius R.The phonon dispersion relations ω(k) depend primarily on the radius and not the specific chiral index (n,m) of carbon nanotubes and are presented in Fig. <ref>(a) for the different polarizations. The LA and TA mode frequencies are almost independent of the nanotube radius for a given k. The corresponding group velocities at k→0 give the longitudinal speed of sound of v_LA=dω_LA/dk=21.5 km/s and the speed of sound with torsional polarization of v_TA=dω_TA/dk=14.1 km/s.The flexural or bending ZA mode does depend on the nanotube radius through the proportionality constant c_ZA(R), defined in Eq. (<ref>), which is plotted as a function of R in Fig. <ref>(b). The dispersion of the ZA mode in a CNT of radius R=1 nm is shown in Fig. <ref>(a). Also the RBM frequency depends on the nanotube radius according to Eq. (<ref>). We find the value √(c_11/ρ_2D)=116 cm^-1nm of the prefactor of R^-1 in Eq. (<ref>) to agree well with the published theoretical value <cit.> of 116 cm^-1nm and with the value of 108 cm^-1nm, obtained by fitting a set of observed Raman frequencies <cit.>. The calculated value ω_RBM=116 cm^-1 for CNTs with R=1 nm is shown in Fig. <ref>(a).Filling the CNT with a liquid of density ρ_l increases its linear density ρ_1D according to Eq. (<ref>). For a nanotube filled with water of density ρ_l=1 g/cm^3, the radius-dependent quantity c̃_ZA(R), defined in Eq. (<ref>), is plotted as a function of R in Fig. <ref>(b). The dispersion of the Z̃Ã mode in a water-filled CNT of radius R=1 nm is shown in Fig. <ref>(a).Elastic constants calculated in this work, and results derived using the present continuum elasticity approach are listed and compared to literature data in Table <ref>. §.§ Tubulin-Based Microtubules To describe phonon modes in tubulin-based microtubules, we depend on published experimental data <cit.> for microtubules with an average radius R=12.8 nm and a wall thickness h=2.7 nm.The observed density of the tubule wall material ρ=1.47 g/cm^3 translates to ρ_2D=4.0·10^-6 kg/m^2. The estimated Young's modulus of the wall material is E=0.5 GPa and the flexural beam rigidity of these microtubules with R=12.8 nm is D_b=0.9·10^-23 N·m^2. We may use the relationship between D_b and D, defined in Eq. (<ref>), to map these values onto the elastic 2D wall material and obtain c_11=E·h=1.4 N/m and D=2.71 GPa·nm^3.In a rough approximation, tubulin can be considered to be isotropic, with a Poisson ratio α=0.25. Further assuming that c_11=c_22, we estimate c_66=(c_11-c_12)/2=c_11(1-α)/2=0.5 N/m.One fundamental difference between tubulin-based microtubules and systems such as CNTs is that the former necessitate an aqueous environment for their shape and function. Thus, the correct description of microtubule deformations and vibrations requires addressing the complete microtubule-liquid system, which would exceed the scope of this study. We rather resorted to the expressions derived in the Subsection on nanotubes in a surrounding liquid to at least estimate their Q-factor in an aqueous environment. We usedρ_1D=7×10^-13 kg/m for a water-filled microtubule,η=10^-3 Pa·s andf_0=10^9 Hz, which provided us with the value Q≈1.2. In other words, flexural vibrations in microtubules should be highly damped in aqueous environment, so their frequency should be very hard to measure. Consequently, the only available comparison between our calculations and experimental data should be made for static measurements.An elegant way to experimentally determine the effective beam rigidity of individual tubulin-based microtubules involves the measurement of buckling caused by applying an axial buckling force using optical tweezers. Experimental results for the effective beam rigidity have been obtained in this way for microtubules that are free of the paclitaxel stabilizing agent and contain 14 tubulin protofilaments, which translates to the effective radius R≈9.75 nm. The observed values range from D_b=3.7±0.8×10^-24 Nm^2 <cit.> to D_b=7.9±0.7×10^-24 Nm^2 <cit.>, in good agreement with our calculated value D_b=4.2×10^-24 Nm^2. Our estimated value D_b=6.2×10^-24 Nm^2 for wider tubules with 16 protofilaments is 49% larger than for the narrower tubules with 14 protofilaments. The corresponding increase by 49% has been confirmed in a corresponding experiment <cit.>.Next, we may still use the oversimplifying assumption that tubulin microtubules may exist in the vacuum and could be described by the above-derived continuum values <cit.>. In this way, we may compare our results to published theoretical results. The calculated phonon dispersion relations ω(k) for most common microtubules with the radius R=12.8 nm are presented in Fig. <ref>(a). The LA and TA mode frequencies are independent of the tubule radius. From their slope, we get the longitudinal speed of sound v_LA=dω_LA/dk=0.56 km/s and the speed of sound with torsional polarization v_TA=dω_TA/dk=0.36 km/s. For the sake of comparison, we extracted the v_LA value based on the elastic cylindrical shells model with E=2.0 GPa from Ref. <cit.>. Extrapolating to the value E=0.5 GPa in our set of parameters using the relationship v_LA∝√(E), we obtained v_LA=0.59 km/s, in excellent agreement with our calculated value.The flexural or bending ZA mode depends on the tubule radius through the proportionality constant c_ZA(R), defined in Eq. (<ref>), which is plotted as a function of R in Fig. <ref>(b). The dispersion of the ZA mode in a microtubule of radius R=12.8 nm is shown in Fig. <ref>(a).Also the RBM frequency depends on the nanotube radius according to Eq. (<ref>). For R=12.8 nm, we obtain ω_RBM=0.24 cm^-1, as seen in Fig. <ref>(a).To describe the increase in the linear density ρ_1D of a microtubule filled with a liquid of density ρ_l, we have to account for the finite thickness h of the wall and replace the radius R by R-h/2 in Eq. (<ref>). Considering water of density ρ_l=1 g/cm^3 as the filling medium, we plot the radius-dependent quantity c̃_ZA(R), defined in Eq. (<ref>), as a function of R in Fig. <ref>(b). The dispersion of the Z̃Ã mode in a water-filled CNT of radius R=12.8 nm is shown in Fig. <ref>(a).Our results in Fig. <ref>(a) suggest soft vibration in the GHz range, in agreement with other theoretical estimates <cit.>. As mentioned above, all these vibrations will be hight damped in an aqueous environment to the low Q-factor.§ DISCUSSION Our study has been motivated by the fact that the conventional approach to calculate the frequency spectrum, based on an atomistic calculation of the force-constant matrix, does not provide accurate frequencies for long-wavelength soft acoustic modes in quasi-1D tubular structures. We should note that the atomistic approach is quite adequate to determine frequencies of the optical and of short-wavelength acoustic modes. But for long-wavelength acoustic modes, the excessive demand on supercell size and basis convergence often yields imaginary vibration frequencies as an artifact of insufficient convergence.As a viable alternative to tedious atomistic calculations of the force-constant matrix of complex tubular systems, we propose here a continuum elasticity approach to determine the frequency of long-wavelength acoustic modes in tubular structures that does not require the thickness of the wall as an input. Our approach for quasi-1D structures is based on the successful description of corresponding modes in 2D structures <cit.>. The continuum elasticity approach introduced in this study has a significant advantage over the 3D elastic modulus approach, which which has lead to inconsistencies in describing the elastic behavior of thin walls and membranes <cit.>. Using this approach, we obtain for the first time quantitative results for systems ranging from stiff CNTs to much wider and softer protein microtubules.We found that the elastic behavior of the wall material can be determined accurately by static calculations of 2D plate subjected to small deformation or by elastic measurements. The validity of predictions based on this approach is limited to long-wavelength vibrations and large-radius nanotubes, both of which would require extraordinary computational resources in atomistic calculations. In particular, the flexural ZA modes with their ω∝k^2 momentum dependence are known to be very hard to reproduce by ab initio calculations near the Γ point <cit.>.For the sake of completeness, we have also derived the Euler-Lagrange equations of motion required to describe all long-wavelength acoustic modes and present the detailed derivation in the Appendix.Of course, the frequency of the ZA modes is expected to be much softer than that of the LA mode in any nanotube or nanowire. Since ω_ZA∝k^2 whereas ω_LA∝k, expressions derived here for the long-wavelength limit would lead to the unrealistic behavior ω_ZA>ω_LA for large values of k. This limits the k-range, for which our formalism is valid in the dispersion relations presented in Figs. <ref>(a) and <ref>(a). In a crystalline tubule, k is restricted to typically an even smaller range given by the size of the 1D Brillouin zone.For systems with a vanishing Poisson ratio α, the radial breathing mode (RBM) should be decoupled from the longitudinal acoustic or stretching mode. However, as discussed in Appendix D, most systems have a non-vanishing value of α. In that case, the two modes mix and change their character beyond the wavevector k=1/R, where ω_LA(k)≈ω_RBM, as discussed previously <cit.>. At smaller values of k, coupling between the LA mode and the RBM modifies the frequency of these modes by only ≈1% in CNTs.Our model allows a simple extension from empty to liquid-filled nanotubes. We find that presence of a filling liquid does not affect longitudinal acoustic and torsional acoustic modes to a significant extent, as shown in Appendix C, but softens the flexural modes. We also expect the pressure wave of the liquid to couple to the RBM beyond the wavevector k≈ω_RBM/v_p, where v_p is the speed of the propagating pressure wave.To demonstrate the universality of our approach, we also considered microtubules formed of the proteins α- and β-tubulin. These are responsible for maintaining the shape and elasticity of cells, but are too complex for an atomistic description to predict vibration spectra. From a computational point of view, the necessity to include the aqueous environment in the description of tubulin-based microtubules adds another layer of complexity to the problem.Our basic finding that microtubule motion and vibrations are overdamped in the natural aqueous environment, with a Q-factor of the order of unity, naturally explains the absence of experimental data reporting observation of motion, dynamical shape changes and vibrations in these protein-based systems. Among static measurements of the elastic behavior of microtubules, optical tweezers appear to be the optimum way to handle and deform individual microtubules in order to determine their effective beam rigidity D_b. In this static scenario, we find our description of the beam rigidity precise enough to compare with experimental data. The reported dependence of D_b on the cube of the radius <cit.> is reflected in our corresponding expression for D_b in Eq. (<ref>). In the case of tubulin-based microtubules, we find the leading term in D_b to be indeed proportional to R^3 and to be much larger than the second term, which is proportional to R.§ SUMMARY AND CONCLUSIONS Addressing the shortcoming of conventional atomistic calculations of long-wavelength acoustic frequencies in tubular structures, which often yield numerical artifacts, we have developed an alternative computational approach representing an adaptation of continuum elasticity theory to 2D and 1D structures. Since 1D tubular structures can be viewed as 2D plates of finite width rolled up to a cylinder, we have taken advantage of the correspondence between 1D and 2D structures to determine their elastic response to strain. In our approach, computation of long-wavelength acoustic frequencies does not require the determination and diagonalization of a large, momentum-dependent dynamical matrix. Instead, the simple expressions we have derived for the acoustic frequencies ω(k) use only four elements of a k-independent 2D elastic matrix, namely c_11, c_22, c_12, and c_66, as well as the value of the flexural rigidity D of the 2D plate constituting the wall. These five numerical values can easily be obtained using static calculations for a 2D plate. Even though the scope of our approach is limited to long-wavelength acoustic modes, we found that the accuracy of the calculated vibration frequencies surpasses that of conventional atomistic ab initio calculations. Starting with a Lagrange function describing longitudinal, torsional, flexural and radial deformations of empty or liquid-filled tubular structures, we have derived corresponding Euler-Lagrange equations to obtain simple expressions for the vibration frequencies of the corresponding modes. We have furthermore shown that longitudinal and flexural acoustic modes of tubules are well described by those of an elastic beam resembling a nanowire. Using our simple expressions, we were able to show that a pressure wave in the liquid contained in a stiff carbon nanotube has little effect on its RBM frequency, whereas the effect of a contained liquid on the RBM frequency in much softer tubulin tubules is significant. We found that presence of water in the native environment of tubulin microtubules reduces the Q-factor to such a degree that flexural vibrations can hardly be observed. We also showed that the coupling between long-wavelength LA modes and the RBM can be neglected. We have found general agreement between our numerical results for biological microtubules and carbon nanotubes and available experimental data.§ ACKNOWLEDGMENTSWe acknowledge useful discussions with Jie Guan. A.G.E. acknowledges financial support by the South African National Research Foundation. D.L. and D.T. acknowledges financial support by the NSF/AFOSR EFRI 2-DARE grant number EFMA-1433459. § APPENDIX Material in the Appendix provides detailed derivation of expressions used in the main text and considers specific limiting cases. In Appendix A, we derive the Lagrange function for stretching, torsional and bending modes of tubular structure. In Appendix B, we derive analytical expressions for the frequencies of the corresponding vibration modes using the Euler-Lagrange equations. The effect of a liquid contained inside a CNT on its RBM frequency is investigated in Appendix C. The coupling between the longitudinal acoustic mode and the RBM in a CNT due to its non-vanishing Poisson ratio is discussed in Appendix D.§.§ A. Lagrange Function of a Strained Nanotube§.§.§ Stretching Let us consider a nanotube of radius R aligned with the x-axis and its response to tensile strain du_x/dx applied uniformly along the x-direction. The strain energy will be the same as that of a 2D strip of width y=2πR lying in the xy-plane that is subject to the same condition.Assuming that the width of the strip is constrained to be constant, the strain energy per length is given by U_x = 1/2 c_11( du_x/dx)^2 2π R. For a nonzero Poisson ratio α, stretching the strip by du_x/dx will reduce its width by du_y/dy=αdu_x/dx and its radius R, as shown in Fig. <ref>(a). Releasing the constrained width will release the energy U_y=-α^2U_x. The total strain energy per length of a nanotubule or an equivalent 1D nanowire is the sum U=U_x+U_y and is given by U= 1/2 c_11(1-α^2)( du_x/dx)^2 2π R= 1/2 c_LA( du_x/dx)^2. Here, c_LA=2πR c_11(1-α^2) is the the longitudinal force constant of a 1D nanowire equivalent to the tubule, defined in Eq. (<ref>).In the harmonic regime, we will consider only small strain values. Releasing the strain will cause a vibration in the x-direction with the velocity v_x=du_x/dt. Then, the kinetic energy density of the strip will be given by T = 1/2ρ_2D( du_x/dt)^2 2π R= 1/2ρ_1D( du_x/dt)^2 , where ρ_2D is the areal mass density of the equivalent strip that is related to ρ_1D by Eq. (<ref>). The Lagrangian density is then given by ℒ(du_x/dx,du_x/dt,x,t) = T - U= 1/2[ ρ_2D( du_x/dt)^2 - c_11(1-α^2)( du_x/dx)^2 ] 2πR,= 1/2[ρ_1D( du_x/dt)^2 - c_LA( du_x/dx)^2 ]. §.§.§ Torsion The derivation of the Euler-Lagrange equation for the torsional motion is very similar to that for the longitudinal motion. The main difference is that the displacement u_ϕ is normal to the propagation direction x. To obtain the corresponding equations, we need to replace u_x by u_ϕ and c_11(1-α^2) by c_66 in Eqs. (A1)-(A4). The Lagrangian density is then given by ℒ (du_ϕ/dx,du_ϕ/dt,x,t) = T - U= 1/2[ ρ_2D( du_ϕ/dt)^2- c_66( du_ϕ/dx)^2] 2πR . §.§.§ Bending Bending a nanotube of radius R is equivalent to its transformation to a segment of a nanotorus of radius R_t. Initially, we will assume that c_11=0 and D>0 in the given nanotorus segment, so the strain energy would contain only an out-of-plane component. We first consider a straight nanotube of radius R formed by rolling up a plate of width 2πR. The corresponding out-of-plane strain energy per nanotube segment length is U = 1/2D/R^2( 2πR)= π D/R . The corresponding expression for the total out-of-plane strain energy in a nanotorus is <cit.>, U = 2 π^2 D R_t^2/R√((R_t+R)(R_t-R)) . Divided by the average perimeter length 2πR_t, we obtain the out-of-plane energy of the torus per nanotube segment length U = π D R_t/R√((R_t+R)(R_t-R)) . Assuming that the torus radius is much larger than the nanotube radius, R_t≫R, we can Taylor expand U in Eq. (<ref>) and neglect higher-order terms in (R/R_t), which leads to U = πD/R(1+1/2 ( R/R_t)^2 ). Comparing the out-of-plane strain energy of a bent nanotube in Eq. (<ref>) and that of a straight nanotube in Eq. (<ref>), the change in out-of-plane strain energy per segment length associated with bending turns out to be U = 1/2πDR (1/R_t)^2 . During the flexural or bending vibration mode, the local curvature 1/R_t=d^2u_z/dx^2 changes along the tube, yielding the local in-plane strain energy per nanotube segment length of U = 1/2πDR (d^2u_z/dx^2)^2 . Next, we consider the in-plane component of strain, obtained by assuming c_11>0 and D=0 in a given nanotorus segment. There is nonzero strain in a nanotube deformed to a very wide torus with R_t≫R even if its cross-section and radius R were not to change in this process. The reason is that the wall of the nanotube undergoes stretching along the outer and compression along the inner torus perimeter in this process. This amounts to a total in-plane strain energy <cit.> U=π^2 c_11 R^3/R_t for the entire torus with an average perimeter of 2πR_t in relation to a straight nanotube of length 2πR_t. Thus, the in-plane strain energy within the torus per segment length is U = 1/2πc_11R^3(1/R_t)^2 . Considering local changes in curvature 1/R_t=d^2u_z/dx^2 during the bending vibrations of a nanotube, the local in-plane strain energy per nanotube segment length becomes U=1/2πc_11R^3 (d^2 u_z/dx^2)^2. The strain energy in the deformed nanotube per length is the sum of the in-plane strain energy in Eq. (<ref>) and the out-of-plane strain energy in Eq. (<ref>), yielding U = 1/2 (πc_11R^3 + πDR) (d^2u_z/dx^2)^2= 1/2 D_b(d^2u_z/dx^2)^2, where D_b =πc_11R^3 + πDR is the effective beam rigidity of a corresponding nanowire. The kinetic energy of a bending nanotube or nanowire segment is given by T=1/2ρ_1D(du_z/dt)^2. This leads to the Lagrangian density ℒ(d^2u_z/dx^2,du_z/dt,x,t) = T - U = 1/2[ ρ_1D(du_z/dt)^2-10mu- πc_11R^3(1+D/c_11R^2) (d^2u_z/dx^2)^2] .§.§ B. Derivation of Euler-Lagrange Equations of Motionfor Deformations of a Nanotube using Hamilton's Principle§.§.§ Stretching The Euler-Lagrange equation for stretching a tube or a plate is <cit.> d/dt( ∂ℒ/∂du_x/dt)+d/dx( ∂ℒ/∂du_x/dx) = 0. Inserting the Lagrangian of Eq. (<ref>) in the Euler-Lagrange Eq. (<ref>) yields the wave equation for longitudinal vibrations of the tubule or the equivalent nanowire 2π Rρ_2Dd^2u_x/dt^2- 2 π R c_11(1-α^2) d^2u_x/dx^2 =ρ_1Dd^2u_x/dt^2- c_LAd^2u_x/dx^2 =0. The nanotube radius R drops out and we obtain ρ_2Dd^2u_x/dt^2 - c_11(1-α ^2 )d^2u_x/dx^2 =ρ_1Dd^2u_x/dt^2- c_LAd^2u_x/dx^2 =0. This wave equation can be solved using the ansatz u_x = u_x,0 e^i(kx-ωt) to yield ρ_2Dω^2 = c_11(1-α^2 ) k^2 for a tubular structure or ρ_1Dω^2 = c_LA k^2 for an equivalent 1D nanowire. This finally translates to the desired form ω_LA = √(c_LA/ρ_1D) k =√(c_11(1-α^2)/ρ_2D) k. which is identical to Eq. (<ref>).§.§.§ Torsion The Lagrangian ℒ(du_ϕ/dx,du_ϕ/dt,x,t) in Eq. (<ref>), which describes the torsion of a tubule, has a similar form as the Lagrangian in Eq. (<ref>). To obtain the equations for torsional motion from those for stretching motion, we need to replace u_x by u_ϕ and c_11(1-α^2) by c_66 in Eqs. (<ref>)-(<ref>). Thus, the frequency of the torsional acoustic mode becomes ω = √(c_66/ρ_2D) k, which is identical to Eq. (<ref>). The torsional frequency is the same as frequency of the shear motion in the equivalent thin plate <cit.>.§.§.§ Bending The Euler-Lagrange equation for bending a tube or a plate is given by <cit.> d/dt( ∂ℒ/∂du_z/dt)-d^2/dx^2( ∂ℒ/∂d^2u_z/dx^2) = 0. Inserting the Lagrangian of Eq. (<ref>) for flexural motion in the Euler-Lagrange Eq. (<ref>) yields the wave equation for flexual vibrations ρ_1Dd^2u_z/dt^2 + πc_11R^3( 1+D/c_11R^2) d^4u_z/dx^4 =ρ_1Dd^2u_z/dt^2 +D_b d^4u_z/dx^4 =0. This wave equation can be solved using the ansatz u_z = u_z,0 e^i(kx-ωt) to yield ρ_1Dω^2 = πc_11R^3( 1+D/c_11R^2)k^4 = D_b k^4 . This finally translates to the desired form ω=√(π c_11R^3/ρ_1D ( 1+D/c_11R^2) ) k^2= √(D_b/ρ_1D) k^2 , which is identical to Eq. (<ref>).For a liquid-filled nanotube, we only need to replace ρ_1D by ρ̃_1D in Eq. (<ref>) to get ω=√(πc_11R^3/ρ̃_1D( 1+D/c_11R^2) ) k^2= √(D_b/ρ̃_1D) k^2 , which is identical to Eq. (<ref>).§.§ C. Coupling Between a Travelling Pressure Wave and the RBM in a Liquid-Filled Carbon Nanotube Next we consider a long-wavelength displacement wave u_x=u_x,0exp[i(kx-ωt)] of small frequency ω and wave vector k travelling down the liquid column filling a carbon nanotube. We assume the liquid to be compressible but viscosity-free. Thus, the travelling displacement wave will result in a pressure wave p=p_0exp[i(kx-ωt)] that causes radial displacements r=r_0exp[i(kx-ωt)] in the CNT wall. These radial displacements couple the pressure wave in the liquid to the RBM, but not the longitudinal and torsional modes of the CNT.At small frequencies ω, there will be little radial variation in the pressure. The local compressive strain in the liquid will thus be -δ V/V=-(∂ u_x/∂ x +2 r/R) and the pressure becomes p=-B δ V/V=-B (∂ u_x/∂ x +2 r/R). Here, B is the bulk modulus of the filling liquid, which we assume is water with B=2.2·10^9 Pa.The local acceleration of water is given by ρ_l ∂^2 u_x/∂t^2=-∂ p/∂ x and the radial acceleration of the CNT is given by ρ_2D∂^2 r/∂t^2 = p - c_11/R^2r. Inserting harmonic solutions for p, u_x and r into Eqs. (<ref>)-(<ref>), we obtain -50mu( [ 1 Bik2B/R; -ik ρ_l ω^2 0; 1 0 -(c_11/R^2-ρ_2Dω^2) ]) -10mu( [p_0; u_x, 0;r_0 ]) = 0 with the characteristic equation (c_11/R^2 -ρ_2Dω^2) (k^2B-ρ_lω^2) -ρ_l ω^22B/R = 0. This can be rewritten as (ω_0^2-ω^2)(k^2v^2-ω^2)-ω^2γ^2=0, whereω_0^2=c_11/(ρ_2DR^2) andγ^2=2B/(ρ_2DR).For a CNT of radius R=1 nm, we obtain ω_0^2=474ps^-2 andγ^2=5.92ps^-2. Solving Eq. (<ref>) leads to the dispersion relation ω(k), which is presented in Fig. <ref>(a). In the following, we focus on the lowest lying branch of the dispersion relation describing a long wavelength, low frequency pressure wave travelling down the liquid column. In this case, we can neglect ω^2 in the first factor of Eq. (<ref>) and obtain ω=vk/√(1+γ^2/ω_0^2) . Considering the filling liquid to be water with the speed of sound v=√(B/ρ_l)≈1483 m/s, the velocity of the propagating pressure wave inside the CNT becomes ω/k = v 1/√(1+γ^2/ω_0^2) =1474 m/s . This value is only slightly reduced from that of bulk water because of the relative rigidity of the CNT.In the corresponding low wavenumber range, the frequency of the RBM is changed to ω̃_RBM=ω_RBM√(1+γ^2/ω_RBM^2) . For a CNT with radius R=1 nm, (1+γ^2/ω_RBM^2)^1/2 = (1+5.92/474)^1/2=1.006, yielding only a 0.6% increase in frequency.The situation is quite different for tubulin microtubules. Assuming a radius of R=12.8 nm, we find ω_RBM=0.24 cm^-1 corresponding to ω_RBM^2=0.00203 ps^-2. In that case, (1+γ^2/ω_RBM^2)^1/2 =(1+0.0859/0.00203)^1/2=6.6. In other words, filling tubulin microtubules with water will increase their RBM frequency by a factor of 6.6.As seen in the full solution of Eq. (<ref>) in Fig. <ref>(a), at k≈15 nm^-1, there is level repulsion and interchange in character between the two dispersion curves. At very much higher frequencies there will be radial modes in the water column that will couple to the RBM of the CNT. These lie outside the scope of the present treatment.§.§ D. Coupling Between the LA Mode and the RBM in Carbon Nanotubes Consider a longitudinal wave travelling along a CNT containing no liquid. The CNT of radius R is aligned along the x-direction and can be thought of as a rolled up plate in the xy-plane with a width of 2πR along the y-direction. Where the CNT is being locally stretched, it will narrow down and where it is compressed, it will fatten due to the nonzero value of c_12 reflected in the Poisson ratio. For longitudinal displacement u_x and radial displacement r, the strains will be ϵ_11=∂ u_x/∂ x and ϵ_22=r/R.The strain energy density is then U=1/2( c_11ϵ_11^2 +c_11ϵ_22^2 +2c_12ϵ_22ϵ_11) =1/2c_11((∂ u_x/∂ x)^2 +(r/R)^2) +c_12∂ u_x/∂ xr/R and the kinetic energy density is T =1/2ρ_2D((∂u_x/∂t)^2+ (∂ r/∂ t)^2). There are two Euler-Langrange equations for the radial and the axial motion, d/dt(∂ℒ/∂ (dr/dt)) -∂ℒ/∂ r=0 and d/dt(∂ ℒ/∂ (du_x/dt)) +∂/∂ x (∂ℒ/∂ (du_x/dx))=0. With the Lagrangian ℒ=T-U given by Eqs. (<ref>)-(<ref>), the Euler-Lagrange equations translate to partial differential equations ρ_2D∂^2 r/∂ t^2 +c_11r/R^2 +c_12/R∂ u_x/∂ x=0 and ρ_2D∂^2 u_x/∂ t^2 -c_11∂^2 u_x/∂ x^2 -c_12/R∂ r/∂ x=0. Assuming harmonic solutions u_x=u_x,0exp[i(kx-ωt)] and r=r_0exp[i(kx-ωt)], we get ( [-ρ_2Dω^2+c_11/R^2 ikc_12/R; - ikc_12/R -ρ_2Dω^2+c_11k^2 ) ]) ([r_0; u_x, 0 ]) =0 with the characteristic equation (c_11/ρ_2DR^2- ω^2)(c_11/ρ_2Dk^2- ω^2)-c_12^2k^2/ρ_2D^2R^2=0 . Solving Eq. (<ref>) leads to the dispersion relations ω(k) that are shown in Fig. <ref>(b) for a CNT with radius R=1 nm, with the valuesc_11/(ρ_2DR^2)=474 ps^-2,c_11/(ρ_2D)=474 ps^-2nm^2 and c_12^2/(ρ_2D^2R^2)=6434 nm^2ps^-4. Our results in Fig. <ref>(b) closely resemble those of Ref. <cit.>, obtained using a more complex formalism describing orthotropic elastic cylindrical shells using somewhat different input parameters. Were c_12 to be zero, then the uncoupled solutions would be the dispersionless RBM of frequency ω = 1/R√(c_11/ρ_2D) , shown by the black solid line in Fig. <ref>(b), and the pure longitudinal mode of velocity v = ω/k=√(c_11/ρ_2D) , shown by the green dotted line in Fig. <ref>(b).The coupling term induces level repulsion between the ω_-(k) and ω_+(k) branches, with strong mode hybridization occurring near k≈1 nm^-1. It is of interest to examine the limiting forms of the two solutions for k→0 and k→∞.For k→0, the larger solution ω_+(k) approaches a constant value ω_+^0. From Eq. (<ref>) we obtain ω_+^0=1/R√(c_11/ρ_2D)+ 𝒪(k^2 ). The lower solution ω_-(k) approaches the value ω_-=vk, where v is the velocity of longitudinal mode, modified by its coupling to the RBM. Inserting this in Eq. (<ref>) and taking the limit k→0, we obtain v = ω/k=√(c_11 ( 1-α^2)/ρ_2D) , where α=c_12/c_11 is the Poisson ratio. The numerical value of the velocity obtained using this expression, v=21.45 nm/ps, is slightly smaller than the velocity of the longitudinal mode v = ω/k=√(c_11/ρ_2D) , which turns out to be v=21.77 nm/ps. The 1% reduction by the factor of √(1-α^2) is caused by the coupling of the longitudinal mode to the RBM.In the opposite limit k→∞, the lower solution ω_-(k) approaches a value, which is a little below the uncoupled RBM frequency <cit.> of a nanotube with R=1 nm, ω = 1/R√(c_11/ρ_2D)= 117 cm^-1 . We can obtain the coupled RBM frequency ω_-^∞ from Eq. (<ref>) by neglecting its value in comparison with c_11k^2/ρ_2D. 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"authors": [
"Dan Liu",
"Arthur G. Every",
"David Tomanek"
],
"categories": [
"cond-mat.soft",
"cond-mat.other"
],
"primary_category": "cond-mat.soft",
"published": "20170427001111",
"title": "Long-wavelength deformations and vibrational modes in empty and liquid-filled microtubules and nanotubes: A theoretical study"
} |
School of Physics and Astronomy, Monash University, Victoria 3800, Australia We consider the inertial mass of a vortex in a superfluid. We obtain a vortex mass that is well defined and is determined microscopically and self-consistently by the elementary excitation energy of the kelvon quasiparticle localised within the vortex core. The obtained result for the vortex mass is found to be consistent with experimental observations on superfluid quantum gases and vortex rings in water. We propose a method to measure the inertial rest mass and Berry phase of a vortex in superfluid Bose and Fermi gases. Vortex Mass in a Superfluid Tapio Simula Received; accepted ===========================§ INTRODUCTIONThe mass of a vortex in a superfluid has been debated in the literature for some time with predictions ranging from it being practically zero or infinite to being not well defined. The vortex mass candidate, M_ field=E_ field/c_s^2, where c_s is the speed of sound, as noted by Popov <cit.>, Duan and Leggett <cit.> and many others, seems to result in a logarithmically divergent vortex mass for large distances R relative to the vortex core size r_c because the incompressible kinetic energy E_ field associated with the superflow velocity field of a static vortex is proportional to ln(R/r_c). An alternative approach due to Baym and Chandler <cit.> is to consider not the energy to create the vortex itself but the kinetic energy cost to move it. This seems to result in a negligible vortex mass equal to the bare mass M_ bare=ρ_s π r_c^2L, where ρ_s is the superfluid mass density and L the length of the vortex, that essentially corresponds to the mass of the fluid displaced by the vortex core <cit.>. In Fermi superfluids an induced hydrodynamic vortex mass, known as Kopnin mass, has been found to be associated with quasiparticle bound states trapped within the vortex core <cit.>. Thouless and Anglin considered a vortex pinned by an external potential that moves in a circular path and concluded that the concept of the vortex mass would not be well defined and would depend on the details of the measurement process <cit.>. However, application of such a pinning potential strongly affects the Kelvin modes localised at the vortex core <cit.>, which means that in the presence of a pinning potential the vortex mass cannot be decoupled from the properties of the applied pinning potential even if the vortex would be dragged through the superfluid adiabatically <cit.>. We are able to avoid this subtle issue by allowing the vortex to move free from localised external potentials, which makes it possible to relate the inertial vortex mass to the frequency of the elementary kelvon excitation. Furthermore, if one considers self-trapped cold atom gases held together by long-range particle interactions one may avoid using any external potentials altogether <cit.>.Here we approach this conundrum by considering a self-propelling vortex that moves in a superfluid Bose–Einstein condensate spontaneously along a circular orbit due to the self-induced asymmetry in the superflow. Observing the vortex motion from the rotating frame of reference leads to an unambiguous definition of the inertial mass of such a vortex in terms of the energy of the unique kelvon quasiparticle of the elementary excitation spectrum of the superfluid. The kelvon is the finger print of the quantised vortex and its excitation energy is determined by a self-consistent theory that contains the information pertinent to the microscopic origin of the vortex mass in a Bose–Einstein condensate. We obtain an all-inclusive (total) inertial mass of a vortex M_ vortex = Γρ/ω_ K L,where Γ is the circulation of the vortex, ρ is the mass density of the fluid hosting the vortex, ω_ K is the angular frequency of the fundamental Kelvin wave excitation of the vortex, and L is its length. We conclude that a vortex in a superfluid is heavy and its mass is finite and well defined spectroscopically by two measurables; the condensate density and the frequency of the kelvon quasiparticle. We suggest how the Berry phase, the Magnus force, the vortex velocity dipole moment (vVDM), and the inertial rest mass of a vortex could be measured experimentally in superfluid Bose and Fermi gases.In what follows, on discussing various mass candidates we use upper case symbols such as M to denote a three-dimensional mass and lower case symbols such as m=M/L to denote two-dimensional mass or mass per unit length. The particle densities and mass densities appearing in the vortex mass definitions are three-dimensional. Strictly, the fluid density appearing in our result for the inertial mass of a vortex in a superfluid derived for dilute gas Bose–Einstein condensates at low temperatures is the condensate particle density. However, in strongly interacting superfluids such as helium II, the condensate density is not easily experimentally observable and in such cases, we use the condensate mass density and the superfluid density interchangeably although in helium II the latter may exceed the condensate density by an order of magnitude. In Sec. II we outline the theoretical description of weakly interacting superfluids in terms of Bogoliubov quasiparticles noting the qualitative similarities between Bose and Fermi systems at the quasiparticle level. In Sec. III we provide a derivation of the equation of motion of a quantised vortex in a Bose–Einstein condensate. The result shows that the only term contributing to the vortex velocity is the Laplacian in the Gross–Pitaevskii equation and that there is no explicit transverse force term acting on the vortex due to the motion of the thermal cloud. Sections IV and V are devoted for discussing, respectively, the properties of Kelvin waves of vortices and their quantum mechanical counterparts—the kelvons. In Sec. VI we derive the equation for the inertial mass of a vortex and show that it is determined by the excitation frequency of the fundamental kelvon of the vortex. In Sec. VII we elucidate the origin of the vortex mass in terms of the kelvon quasiparticle and compare the quasiparticle excitation spectra due to the structure of the vortex core in Bose and Fermi gases. The connection between the geometric Berry phase and the inertial mass of a vortex is also provided.In Sec. VIII we discuss how the vortex mass and its Berry phase could potentially be measured in cold atom experiments. In Sec. IX the classical limit of large quantum numbers is considered showing how the formula for the vortex mass may be applicable to classical fluids such as water. Section X concludes this work.§ QUASIPARTICLE PICTUREOur starting point for the derivation of the vortex mass is the dynamical Hartree–Fock–Bogoliubov theory of the superfluid Bose gas, which is known to yield an elementary excitation spectrum that agrees with the second order accurate Beliaev theory <cit.>. In this theory the dynamics of the condensate wave function ψ( r, t) are determined by the generalised Gross–Pitaevskii equationiħ∂_t ψ( r, t) = ℒ( r, t) ψ( r, t)+gΔ( r, t)ψ^*( r, t)where ℒ( r, t) =- ħ^2/2M∇^2 +V_ ext( r,t)+gn( r, t)+2gρ( r, t) -iD( r,t). In Eq. (<ref>),M is the mass of a particle the superfluid is composed of, g determines the strength of the s-wave interactions in the system, n( r, t)=|ψ( r, t)|^2, ρ( r, t) and Δ( r, t) are the condensate, non-condensate and pair-potential densities, respectively, V_ ext( r,t) includes all external pinning, trapping and other potentials and the non-Hermitian term iD( r,t) accounts for condensate growth and loss processes such as dissipation due to the interaction with non-superfluid atoms.The internal potentials ρ( r, t) and Δ( r, t) involve summations over all quasiparticle states and are determined self-consistently by the relationsρ( r, t) = ∑_qp{ f_qp(t) [u^*_q( r, t)u_p( r, t) + v^*_q( r, t)v_p( r, t) ]-2 Re [g_qp(t) u_q( r, t)v_p( r, t)] + δ_qp|v_q( r, t)|^2 }andΔ( r, t) = - ∑_qp{ [2f_qp(t)+δ_qp] v^*_q( r, t)u_p( r, t) - g_qp(t) u_q( r, t)u_p( r, t) -g^*_qp(t) v^*_q( r, t)v^*_p( r, t)}.The last term in ρ( r, t) is proportional to the Lee–Huang–Yang contribution to the ground state energy due to the quantum fluctuations <cit.> and it yields a non-zero quantum depletion even at zero temperature, the effects of which have recently been observed experimentally <cit.>. The evolution of the quasiparticle amplitudes u_q( r, t) and v_q( r, t) are determined by the Bogoliubov–de Gennes (BdG) equationsiħ∂_t u_q( r, t)= ℳ( r, t) u_q( r, t) -𝒱( r, t) v_q( r, t)iħ∂_t v_q( r, t)=-ℳ^*( r, t) v_q( r, t)+ 𝒱^*( r, t) u_q( r, t)expressed in terms of the operatorsℳ( r, t) = - ħ^2/2M∇^2 -μ +V_ ext( r,t)+2gn( r, t) +2gρ( r, t)and𝒱( r, t) = gψ^2( r, t) + gΔ( r, t),where μ is the chemical potential. The quasiparticle distribution functions f_qp(t)=⟨α^†_q α_p⟩ and g_qp(t)=⟨α_q α_p⟩ are expectation values of products of quasiparticle creation α^†_q and annihilation α_q operators.Fermi superfluids may be modelled using similar quasiparticle picture with two major differences. A coherent condensate mode ψ( r) described by the Gross–Pitaevskii equation, Eq. (<ref>), is absent and the quantum statistics of the quasiparticles of Eqns. (<ref>) are characterised by fermion anticommutation relations instead of boson commutation relations. The gap function Δ( r) replaces the condensate mode as the relevant order parameter in this case. Importantly, however, for both Bose and Fermi systems the BdG equations have topologically non-trivial vortex solutions characterised by emergent quasiparticle states at the vortex core. For Bose and Fermi systems these vortex core localised quasiparticle states are the kelvons <cit.> and Caroli–Matricon–deGennes (CdGM) states <cit.>, respectively.§ VORTEX VELOCITY AND THE TRANSVERSE FORCE To obtain the equation of motion for an isolated quantised vortex in a two-dimensional Bose–Einstein condensate, we may consider a generic vortex state ψ_v( r, t) =f( r)√(e( r, t))e^iS( r,t), where S( r,t) is a smooth real function, e( r, t) is defined by |ψ_v( r, t)|^2=| r- r_v|^2e( r, t), and the function f( r) accounts for the non-analytic internal structure within the vortex core. The exact form of ψ_v( r, t) is determined self-consistently by Eq. (<ref>). In general, the structure function f( r) contains multipole moments to all orders but for the purpose of deriving the vortex velocity equation it is sufficient to approximate it by only considering the lowest order monopole fieldf( r)= x-x_v + i(y-y_v),where (x_v,y_v) are the position coordinates of the vortex. The influence of the dipolar velocity field due to the kelvon quasiparticle is limited to within the vortex core and yields the vortex velocity dipole moment (vVDM) <cit.>. The equation of motion of the vortex is obtained explicitly by linearising the formal solution of the generalized time-dependent Gross-Pitaevskii equation, Eq. (<ref>), and finding the position r_v(t+δ t) of the phase singularity of the vortex wavefunction ψ_v( r, t+δ t), where δ t is an infinitesimal time increment <cit.>. The resulting velocity of the vortex v_v( r_v, t)≡δ r_v/δ t is v_v( r_v, t) =v_s( r_v, t) - ħê_z×∇ e( r, t) /2Me( r, t)|_ r_v, where v_s( r_v, t)=κ∇ S( r,t)/2π|_ r_v is the embedding velocity in the vicinity of the vortex core and κ=h/M is the quantum of circulation. The vortex velocity equation (<ref>) depends on the gradients of the embedding phase and the condensate density in the vicinity of the vortex and provides a complete description of the vortex motion including effects such as vortex `friction' that results, for example, in radial drifting of the vortex in the presence of noncondensate atoms in harmonic traps <cit.>. Equation (<ref>) may be multiplied from left by [-Mñ( r_v, t) κ×], where κ=κê_z and ñ( r_v, t) is the condensate density in the absence of the vortex. This yields a force balance equationf_v = f_s+f_ buoy, where the lower case symbols f denote force per unit length such that the total force F_v on a columnar vortex of length L isF_v =f_v L. Equation (<ref>) may also be expressed asf_ Mag +f_ buoy =0 in terms of the Magnus force, also known as the Kutta–Joukowski liftf_ Mag=Mñ( r_v, t) κ×( v_v-v_s)= f_s- f_v. The gradient force due to the inhomogeneous condensate density, f_ buoy = -πħ^2/Mñ( r, t)/e( r, t)∇ e( r, t) |_ r_v, originates from the quantum pressure and can in general point in any direction causing the vortex to accelerate or decelerate. In Eq. (<ref>) there is no explicit transverse force on the vortex due to the motion of the non-condensate. Instead, all such effects involving the non-condensateare included in the vortex dynamics implicitly and self-consistently through f_ buoy and f_ svia their influence on the density n( r, t) and embedding velocity v_s( r, t) of the condensate wave function, ψ( r, t) in Eq. (<ref>). § KELVIN WAVESLord Kelvin studied small amplitude perturbations to a thin columnar vortex with hollow core and obtained a dispersion relationω_ K = Γ/2π r_c^2( 1 - √(1+kr_c[ K_0(kr_c)/K_1(kr_c)])), where Γ is the circulation, r_c is the core parameter of the vortex, k is the wave vector and K_j is a modified Bessel functions of order j <cit.>. These Kelvin waves correspond to infinitesimal perturbations to the inner surface of the vortex core and manifest as propagating helical displacement of the centre line of the vortex core. Remarkably, such Kelvin waves may be amplified to the extent that the amplitude of the perturbation becomes greater than the core size of the vortex allowing clear visualisation of the deformed helical shape of the vortex. The Kelvin wave dispersion relation Eq. (<ref>) has a long wave length approximationω^L_ K = Γ k^2/4π[log( 2/kr_c) -γ], where γ is the Euler–Mascheroni constant. Pocklington studied sinuous waves on a hollow vortex ring <cit.> obtaining the Kelvin wave dispersion relation, Eq. (<ref>). However, its validity is then limited to great wavenumbers such that the shape of the vortex ring appears rectilinear on length scales comparable to the wavelength of the perturbation. Thomson considered instead the slowly varying Kelvin waves on a vortex ring finding a dispersion relation <cit.> ω^R_ K = η V_R/R, expressed in terms of the radius R of the ring and its translational speed V_R = Γ/ 4π R[log(8R/r_c) - β],where the value of β depends on the details of the structure of vortex core. The factor η = √(p^2(p^2-1))≈ p^2 -1/2 is expressed in terms of the integer p=1,2,3… and the latter form is the best quadratic polynomial approximation. The two forms differ significantly only for the longest wave length mode p=1. § KELVONSQuantised vortices in superfluids are in many respects similar to their classical counterparts. Pitaevskii studied the small amplitude perturbations to quantised vortex lines <cit.> and obtained a dispersion relation equivalent to Eq. (<ref>). Using direct numerical solution of the BdG equations it was found that the kelvon dispersion relation ω^ BdG_k = ω_0 + ħ k^2/2M[log( 1/kr_c)], is shifted by ω_0, which is the frequency of the fundamental kelvon <cit.>. As such, in the long-wave length limit, k→ 0, the kelvon has a non-zero frequency. Considering a rectilinear axisymmetric vortex of length L with integer winding number w, the BdG equations (<ref>) have stationary quasiparticle solutions of the formu_k( r) = u_k(r)e^i [n2π z/L + (ℓ+w)θ] v_k( r) = v_k(r)e^i [n 2π z/L + (ℓ-w)θ], where the integers n and ℓ are, respectively, the principal and orbital angular momentum quantum numbers of the excitation, and z and θ are the coordinates of the cylindrical coordinate system.For axisymmetric vortices, the kelvons have angular momentum quantum number ℓ=-w and quantised excitation frequencies ω_k and elementary excitation energies E_k=ħω_k. Within the linear response approximation, the condensate wave function perturbed by a kelvon is ψ( r ,t) = {ψ( r ) +ϵ[u_k( r)e^-i ω_k t +v^*_k( r)e^i ω_k t]}e^-i μ t/ħ ,where ϵ determines the strength of the perturbation (kelvon population). Such kelvon quasiparticles <cit.> correspond to the classical Kelvin wave motion <cit.>. Fetter provides a clear theoretical discussion on the physics of the kelvon in BECs <cit.>. When the quantised vortex is perturbed by kelvons by sufficiently large amplitude the vortex core is displaced from and orbits on a circular path around its own equilibrium position with angular frequency ω_k <cit.>. Evidence for the existence of kelvons with great wave numbers on quantised vortices have been obtained via direct imaging in superfluid helium <cit.> and in atomic Bose–Einstein condensates <cit.>.The kelvon and the vortex are inseparable in the sense that one cannot exist without the other. For each vortex nucleated in the condensate wavefunction ψ( r, t), a new low-energy quasiparticle state emerges in the spectrum of elementary excitations <cit.>. For a one vortex system the kelvon quasiparticle is localised inside the vortex core <cit.>. Due to this localisation of the kelvon density within the vortex core, its excitation energy is very sensitive to the vortex core structure. Importantly, the n=0 kelvon quasiparticle also determines the experimentally observed orbital motion of a vortex in the centre of trapped Bose–Einstein condensates <cit.>. In typical cold atom experiments the kelvon, also called the anomalous mode or the vortex precession mode <cit.>, has the lowest excitation energy in the system E_k <ħω_ trap, where ω_ trap is the usual harmonic trapping frequency. §.§ Vortex spinFor a singly quantised, charge +κ vortex with w=+1, the equations (<ref>) show that the fundamental n=0 kelvon with ℓ=-1 has quasiparticle amplitudes u_k( r) = u_k(r) and v_k( r)= v_k(r)e^-i 2θ. The Bogoliubov quasiparticle components at the vortex core, r=0, may thus be expressed as a 2-spinor( u_kv_k) = ( 1e^-i 2θ ) ≡( 10),which may be referred to as a spin-up kelvon. The BdG equations are invariant under the simultaneous symmetry operations E_k → -E_k^* and( u_kv_k) →𝒦σ_x ( u_kv_k)where 𝒦 denotes complex conjugation and σ_x=( 011 0 ) is a Pauli spin matrix. Application of this transformation to the kelvon yields ( u_kv_k) = ( e^i 2θ1) ≡( 01)which may be referred to as a spin-down kelvon.§.§ Vortex velocity dipole moment Measured with respect to the vortex-centric reference frame of the condensate wave function ψ( r) = ψ(r)e^i θ the perturbing particle and hole-like quasiparticle components u_k( r) and v^*_k( r)have phase windings -1 and +1, respectively, as is readily found by multiplying Eq. (<ref>) by e^-i θ. Dynamics causes the positions of these phase singularities to be spatially separated and leads to an intrinsic vortex velocity dipole moment (vVDM) of the vortex <cit.> due to the resulting kelvon induced dipolar superflow within the vortex core. The vVDM may be understood from the energetic perspective as the doubly charged phase singularity, e^± i 2θ, in one of the quasiparticle components has a tendency to split into two spatially separated singly charged singularities via a critical point explosion <cit.>. Presumably, for the spin-up kelvons the vVDM is polarised in the direction of motion of the vortex where as for the spin-down kelvons the vVDM is antiparallel with respect to the velocity vector of the vortex. In addition to the charge, spin and vVDM, the kelvon quasiparticle is also the source of the inertial mass of the vortex. § VORTEX MASS To determine the vortex mass, we begin with the Newton's second lawF_v=M_va,where a is the acceleration of the vortex and F_v is the force responsible for the acceleration. We take the constant of proportionality, M_v, as the definition of the inertial mass of the vortex.To determine the inertial mass of the vortex we first consider a Bose–Einstein condensate trapped within a cylindrical hard-wall bucket potential of radius R_∘ such that the condensate density n( r, t) is practically constant everywhere in the fluid except within the vortex core and near the walls of the bucket. In this case, and at low temperatures, F_ buoy≈ 0 and the vortex motion as observed in the laboratory reference frame is determined by the condition v^ lab_v≈ v^ lab_s. The full equation of motion in the laboratory frame of reference isF^ lab_v = F^ lab_s+F^ lab_ buoy = M_v a^ lab.The result in the absence of dissipation is that the vortex travels along a circle with a constant speed, constant acceleration and constant orbital angular frequency ω_v(r_v), the value of which depends on the radial position of the vortex. The acceleration vector in the laboratory frame then always points toward the centre of the circle. Therefore, ifF^ lab_v points radially outward direction, M_v is negative and if F^ lab_v points in the same direction, toward the origin, as the acceleration vector, then M_v is positive. In the uniform case, the circular motion of the vortex is due to the self-induced asymmetry of the superflow in the vicinity of the vortex. From the field theory perspective, this motion of the vortex may be viewed as the system's response to the broken continuous rotation symmetry. In this sense, the fundamental kelvon is a pseudo Nambu–Goldstone boson that aims to restore the broken rotation symmetry due to the presence of the vortex. Equivalently, the embedding flow v^ lab_s that drives the motion of the vortex can be viewed to be generated by an image vortex of opposite sign of circulation placed outside of the bucket. Directly integrating the Gross-Pitaevskii equation confirms that the observed frequency ω_v(r_v)≈ω_v(0) / [1-(r_v/R_∘)^2] is consistent with the solution of the corresponding point-vortex model <cit.>, which can be mapped onto the exactly soluble two-dimensional electrostatic problem of a charge inside a conducting ring. Once M_v has been determined, it could be included in simulations of vortex dynamics that use point-vortex and vortex filament models, to include effects due to vortex inertia. Indeed, the first or the second equivalence in (<ref>) could be used to find the position of the vortex as a function of time. The former involves integrating the vortex velocity v_v(t) once, the latter requires integrating the acceleration a_v(t) twice to obtain r_v(t).In order to relate the driving force and acceleration to the mass of the superfluid vortex, we make a transformation to a reference frame that rotates at the angular frequency ω_v(r_v) of the vortex. In such a frame, the vortex is stationary, v^ rot_v=0.The equation of motion in the rotating frame of reference isF^ rot_s+F^ rot_ buoy = M_v a^ rotwhich may be equivalently expressed as F^ lab_s+F^ lab_ buoy = F^ lab_v =M_vω^ lab_v× v^ lab_v,where the last term is due to centrifugal acceleration in the rotating frame.The buoyancy force is the same in both frames since the frame transformation does not affect the observed condensate density in the vicinity of the vortex core and the embedding velocity field is the same in both frames of reference because it is produced by the image vortex, which maintains its position with respect to the actual vortex in all frames of reference. Substituting to (<ref>) the expression for the vortex force in the laboratory reference frameF^ lab_v/L = -M ñκ× v^ lab_vwe obtain the result for the velocity dependent vortex mass per unit length M_v = γ (v_ rel) M_0= -2πħñ(r_v)L/ω_v( r_v), where γ(v_ rel) is a factor that depends on the relative speed v_ rel=|v_v-v_s| of the vortex with respect to the embedding flow. The inertial rest mass per unit length m_0=M_0/L, with γ(v_ rel) =1 of the vortex is obtained by considering the limit of r_v→ 0,v_v→0 and the result ism_0=2πħ n_0/ω_kand may also be expressed asm_0=Mρ_0 κ^2/ 2π E_k. Here ω_k =-ω_v( r_v=0) is the fundamental kelvon frequency (the anomalous mode), whose excitation frequency in harmonically trapped BECs is known to be negative <cit.>.In Eq. (<ref>) n_0=ñ(0) is the background condensate density at the origin in the absence of the vortex, ρ_0=Mn_0, and ω_k is the frequency of the kelvon quasiparticle, which equals the (negative of) orbital angular frequency of the vortex in the limit when the vortex approaches the centre of the bucket. Although v_v→0 as r_v→ 0, the orbital angular frequency -ω_v(0) saturates to a non-zero value that equals the kelvon frequency ω_k. Equation (<ref>) shows that the inertial mass of the vortex is indeed well defined and is determined microscopically by the excitation energy E_k of the kelvon quasiparticle of the vortex. The kelvon frequency ω_k is determined self-consistently by the Bogoliubov–de Gennes equations, Eq. (<ref>), and it carries complete information of the influences on the vortex dynamics such as the near field and the internal structure of the vortex core, the far field away from the core, as well as the effects of quantum fluctuations and thermal atoms.§ ORIGIN OF THE VORTEX MASS Equation (<ref>) sheds light on the apparent issue of the logarithmic divergence of the incompressible kinetic energy associated with a static vortex. In a Bose–Einstein condensate a standard calculation of the field mass yields M_ field/L= E_ field/Lc_s^2≈πħ^2/gln(R/r_c), where R is the radius of the condensate and r_c is approximately the size of the vortex core <cit.>. This suggests that the vortex mass would become infinite in the r_c/R→ 0 limit. We may also express Eq. (<ref>) in the form m_0= πħ^2/g2μ/ħω_k. For a harmonically trapped quasi-two-dimensional condensate, in the non-interacting g→0 limit the chemical potential μ=ħω_ trap=|ħω_k| <cit.> and ln (R/r_c)=𝒪(1) showing that our result and the logarithmically divergent mass definitions converge toward a finite value in this limit. However, in typical cold atom experiments μ/ħω_k≫ln(R/r_c). This means that a vortex in an interacting system is actually heavier than the usual logarithmically divergent prediction. The reason is that the mass of the vortex m_0 is due to the mass M_k, defined below in Eq. (<ref>), of the kelvon quasiparticles. The situation is analogous to the case of an electron whose mass M_e = eB/ω_c, where e is the electric charge and B is a constant magnetic field strength, is readily obtained from the measurement of its cyclotron frequency ω_c that relates the magnetic Lorentz force to the centrifugal force, cf. Eq. (<ref>). However, the energy W=ϵ_0/2∫ | E|^2d r^3 of the classical electromagnetic Coulomb field described by Maxwell's equations of classical electrodynamics, analogous to the kinetic energy W=ρ/2∫ | v|^2d r^3 of ψ( r,t) in Eq. (<ref>), is divergent due to the singularity at the location of the electron and leads to the well known issues in the evaluation of the mass of the electron.Equations (<ref>), however, are the semi-classical analog (-E_k+ℋ Φ Φ^*E_k + ℋ ) (u_k v_k) = (00)of the Dirac equation and uncover the mass of the vortex. Indeed, Popov showed the equivalence of a two-dimensional system of phonons and vortices to relativistic electrodynamics <cit.>. The vortex core forms a harmonic oscillator potential in the condensate density, which in the vicinity of the vortex core, r<r_c, is n(r)≈n_0 r^2/r^2+ 2r_c^2,where r is the distance from the axis of the vortex. The vanishing of the condensate density at the vortex core allows the following substitutions to be made to the BdG equations (<ref>);Φ=𝒱M/M_k, E_k=(iħ∂_t+μ)M/M_kandℋ=(ℳ( r, t) + μ )M/M_k,where the semi-classical kelvon energy isℋ= -ħ^2/2M_k ∇^2+ V(r),with the effective harmonic oscillator potential in the region r≪ r_c defined byV(r)=2gn(r) M/M_k = gn_0 r^2/r_c^2M/M_k≡1/2M_kω^2_kr^2.Equation (<ref>) may be solved for the mass of the kelvon with an approximation E_k=M_kω_k^2r^2_c/2, and results in M_k =μ/ħω_kM, consistent with the approximations made in Eq. (<ref>), Eq. (<ref>) andEq. (<ref>). Replacing the relativistic mass-energy ℋ=Mc^2 in the Dirac equation by the semi-classical energy p^2/2M+V of Eq. (<ref>) shows the connection between the Dirac and the Dirac–Bogoliubov–deGennes equation (<ref>).The kelvon mass naturally agrees with an estimate based on the Heisenberg uncertainty relation Δ x_kΔ p_k≈ħ/2, with uncertainties in the vortex position Δ x_k≈ r_c and momentum Δ p_k≈ M_k ω_k r_c, which yields M_k ≈ħ/2ω_kr^2_c=M μ/E_k. At zero temperature the number of kelvons N_k in a vortex of length L in the vortex core is N_k=M_0/M_k =L/4a, where a is the s-wave scattering length. Thus we find that the mass and the total energy per unit length of the vortex are large mostly due to the zero point motion of the vortex. Although Eq. (<ref>) seems to admit an infinite vortex mass in the form of a zero-energy kelvon mode, vanishing kelvon frequency is not consistent with Eq. (<ref>) and would be unphysical in light of the Heisenberg uncertainty principle <cit.>. Furthermore, a vortex moving on a cylinder of any size can never be static <cit.>, which shows the importance of boundary conditions even in systems that extend to infinity. §.§ Bose and Fermi superfluids Figure <ref> (a) illustrates the elementary BdG quasiparticle excitation spectrum for a vortex state in a simple Bose–Einstein condensate. The excitation energies shown as a function of angular momentum quantum number ℓ are measured with respect to the chemical potential μ. The zero energy mode (black circle) with ℓ=0 is the Goldstone boson corresponding to the macroscopically occupied condensate mode. The blue and red circles denote the particle and hole-like quasiparticle eigenmodes, respectively. In the Bose system the canonical commutation relations result in the hole-like modes (red circles) to have negative norm and therefore they are discarded from the self-consistent sums in Eqns (<ref>) and (<ref>).For harmonically trapped bosons, the excitation energy gap equals the harmonic oscillator level spacing ħω and the lowest lying excitation mode is the kelvon with ℓ=-1 and |E_k|≪ħω. In two-dimensional systems, there is only one kelvon mode in the spectrum and its energy E_k≈ -0.1 ħω in typical harmonically trapped systems <cit.> is negative with respect to the chemical potential μ. The slope of the orange line equals the magnitude of the kelvon frequency ω_k. In harmonically trapped BECs this kelvon frequency within the Thomas–Fermi approximation is <cit.>ω_k^ TF = -3ħ/2MR_⊥^2log( R_⊥/r_c),where R_⊥ is the radial Thomas–Fermi radius of the condensate. Substituting this estimate for the kelvon frequency to Eq. (<ref>) yields a vortex mass estimateM_ TF = -4π/3n_0 R_⊥^2L/log( R_⊥/r_c) M. Baym and Chandler <cit.> considered vortices in a helium II—a strongly interacting bosonic superfluid—and obtained a vortex massM_ BC = πρ_s r_c^2 L,which is equivalent to the bare mass M_ bare. The Baym–Chandler mass may also be expressed in terms of the inertial mode frequency ω_I=2Ω of a rapidly rotating vortex lattice rotating at orbital angular frequency Ω as M_ BC = 2πħρ_sL/Mω_I <cit.>. Comparing this with our result, Eq. (<ref>), shows that the essential difference between them is that the Baym–Chandler mass involves the inertial mode frequency ω_I≫ω_k of a vortex lattice instead of the kelvon frequency ω_k of a single vortex, and therefore results in a much smaller value for the vortex mass than our kelvon based result. In rapidly rotating vortex lattices, the inertial mode frequency, corresponding to the standard inertial mode of a rotating fluid, is orders of magnitude greater than the Tkachenko mode frequency <cit.> although the individual vortices in both of these modes execute Kelvin wave like motion. This suggests that for vortex lattices the total mass of the vortex matter might be obtained by replacing the Kelvin wave frequency in Eq. (<ref>) by the Tkachenko mode frequency.For cold atom systems with chemical potential μ=gn_0≫ħω_k, Eq. (<ref>) yields m_0=4M_ bareμ/Lħω_k, such that the inertial vortex mass is in general much greater than the bare mass. In uniform two-dimensional traps ω_v(0)= ħ / MR_⊥^2 such that Eq. (<ref>) yields M_ uni≈ -2NM, where N is the number of atoms in the condensate. The magnitude of the inertial vortex mass in this case equals twice the total gravitational mass of the condensate.Figure <ref> (b) illustrates the elementary BdG quasiparticle excitation spectrum for a vortex state in a topological Fermi superfluid measured with respect to the Fermi energy E_ F. The zero energy mode (green circle) is a Majorana fermion quasiparticle corresponding to one of the Caroli–deGennes–Matricon (CdGM) modes localised within the vortex core. In BCS superfluids the CdGM modes may be viewed as Andreev bound states trapped by the normal state vortex core. For a topological Fermi superfluid the CdGM modes are ω_ CdGM = -(ℓ-w_+1 +1/2 )ω_ F + (n-w_+1+1/2 )ω_1,where ω_+1 is the integer winding number of the dominant component of the chiral pair potential <cit.>. The level spacing of the CdGM modes, the slope of the orange line in (b), is ω_ F≈Δ^2/E_ F≪ω_1≈Δ. In Fermi systems, the gap may contain more than one core modes even in two-dimensional systems, in contrast to the Bose case. The Zwierlein group studied the vortex mass in Fermi systems <cit.> obtaining a generic formula M_ BF = -4π/2γ+1n_0 R_⊥^2L/log( R_⊥/r_c) M,where the polytropic index γ=1 for BECs and γ=3/2 for a Fermi gas in the BCS limit and at unitarity. For γ = 1 this result is identical to Eq. (<ref>).The Kopnin mass <cit.> m_ Kop =πħ n_0/ω_ F in fermonic superfluids is expressed in terms of ω_ F and is equivalent, up to the factor of 2, to the kelvon based vortex mass in Bose systems since in both cases the cyclotron frequency in the denominator equals the slope of the vortex core localised quasiparticle modes. Note that the absent factor of 2 in the Kopnin mass could be recovered by considering that in single quantum vortices of BCS superconductors the angular momentum states are shifted by 1/2 such that the CdGM level spacing in such a case actually corresponds to 2 times the lowest CdGM frequency. These considerations show that the inertial mass of a vortex in Bose and Fermi superfluids has the same origin—the quasiparticle modes localised within the vortex core.§.§ Geometric phase of the vortexIn a uniform superfluid the geometric Berry phase γ(C) = i ∮_C ⟨Ψ |∇_ RΨ⟩· d Rand the vortex force are related by <cit.>γ(C)= 2π w N(C) = ∫_C ( f_ v× v_v)· d a/ħ v^2_v,where C denotes the closed orbital path of the vortex, w is the winding number and N(C) is the number of atoms enclosed by C. In this case the velocity dependent vortex mass and Berry phase are related byγ(C)/M_v(C)ħ L= ∫_C ω_v(C)· d a.This profound connection between the topological charge of the quantised vortex and its dynamical behaviour also reveals that on exchanging the positions of two w=1 vortices the system acquires a phase of 2π per atom enclosed by the exchange path. In two-dimensional superfluids with more complex order parameter structure such as spinor BECs, the vortices are in general anyons and may possess non-Abelian exchange phases.§ EXPERIMENTAL PROSPECTSSuperfluid Bose and Fermi gases provide a promising experimental platform for precision measurements of the Berry phase, Magnus force, inertial mass and dipole moment of a quantised vortex. An off-centre vortex in such systems can be created and its orbital angular frequency ω_v(r_v) as a function of its position observed. The measured orbital frequencies in both Bose <cit.> and Fermi <cit.> gases have been found to be in good agreement with the mean-field theory <cit.>. Extrapolating such frequency data to r_v=0 yields the kelvon frequency ω_k. This, in combination with a measurement of the condensate density, yields the inertial rest mass m_0 of the vortex. A systematic study of the vortex mass could be performed using cold atom gases trapped in bucket potentials by varying the radial position and the core size of the vortex in the superfluid. Uniform trapping geometries suitable for such experiments are already being used by several groups <cit.>. Controlling the vortex size and position could be achieved by first nucleating several vortices in the system using standard methods and then waiting until only one vortex remains. Further wait time causes the radial position of the remaining vortex to increase due to dissipative effects and allows for dialling the radial position of the vortex. Alternatively, external optical pinning potentials could potentially be used for moving the vortex in the desired radial position in the trap. The particle density of the condensate determines the healing length and thereby the size of the vortex core and can be adjusted by varying the number of atoms that form the condensate. Tuning the effective inter-species interaction in a two-species condensate where one species hosts the vortex and the other species plugs the vortex core could also be used to intrinsically modify the vortex core size and its kelvon frequency, and thereby the mass of the vortex.Fortuitously, the same experimental setup for measuring the kelvon frequencies could also be used to measure the Berry phase of a vortex in a superfluid for the first time. The Berry phase of a vortex could be obtained using its discretised form γ(C) = -Im(log∏_i^P ∫ψ^*( r, t_i) ψ( r, t_i+1)dv),where the closed path C of the vortex is sampled at P different vortex positions and the integration is over all space. Experimentally, the integrands in Eq. (<ref>) could potentially be extracted from an interferogram of two wave functions acquired at time intervals δ t=t_i+1 -t_1 during which the vortex is allowed to travel a distance of the order of its core diameter. The dynamical phase that accumulates at the rate determined by the chemical potential must also be carefully accounted for. The measured value of γ(C) should be compared with the predicted value2π w N(C), where the number of enclosed atoms N(C) could perhaps be measured to shot-noise limited accuracy <cit.>. Using the measured values of the Berry phase and the vortex speed, the vortex force f_v can be obtained using the Eq. (<ref>), and the Magnus force f_ Mag can be obtained directly by measuring the condensate density that allows f_ buoy to be calculated. Using Eq. (<ref>) f_s can be measured. The vVDM could potentially be detected using velocity selective Bragg scattering techniques. § CLASSICAL LIMIT According to Bohr's correspondence principle, quantum systems in the limit of large quantum numbers are expected to behave classically. For vortices, this amounts to replacing in the equation (<ref>) the quantum of circulation κ=h/M by Γ = w κ and the condensate mass density M n_0 by the fluid density ρ to yield M_ vortex = Γρ/ω_ KL .In macroscopic systems such as bath tub vortices the effective winding number w would be comparable to the Avogadro number. In 1956 Hall and Vinen modelled a quantised vortex as a rotating cylinder filled with the fluid it is immersed in <cit.>. They showed that the motion of such a cylinder with mass M_ cyl that displaces a fluid of mass M_ flu, involves a constant velocity perpendicular to an applied force and an oscillatory motion at the frequency ω_ cyl= ρΓ L / (M_ cyl+M_ flu).Identification of the frequency ω_ cyl of such motion with the fundamental frequency of a Kelvin wave of a vortex and the total mass M_ cyl+M_ flu with the inertial mass of the vortex recovers the result of Eq. (<ref>). We will next consider a classical vortex ring instead of a columnar vortex to avoid the issue of boundary conditions at the ends of the vortex. Consider thus a vortex ring produced by a piston with a circular orifice of radius R as in the experiments by Sullivan et al. <cit.>. By invoking momentum conservation, the product of the inertial mass of the vortex ring and its velocity must be equal to the momentum of the fluid set in motion by the piston. This yieldsM_v V_R = ρΓπ R^2.Therefore, the inertial mass of a vortex ring of length L=2π R isM_v = Γρ/ω^∘_ KL ,where ω^∘_ K = 2V_R /Ris the magnitude of the angular frequency of the fundamental, n=0, Kelvin mode on a vortex ring. This result agrees with the quantum mechanical expression Eq. (<ref>). However, there is a discrepancy regarding the constant η. In Eq. (<ref>) η=2, where as Eq. (<ref>) predicts it to be either 0 or 1/2. One possible explanation could be that the curvature of the vortex ring may require a corrective factor due to the contribution of Kelvin waves of higher wave numbers. Moreover, the Eq. (<ref>) is derived for hollow vortex rings where as the cores of the vortex rings in the experiment were filled with fluid.Note also that in the experiment <cit.>, the ratio of induced mass to the mass of the fluid trapped by the vortex ring was 0.65, whereas our definition for the inertial vortex mass is all-inclusive and contains the mass of the moving fluid. It thus seems that a detailed experimental study of the Kelvin wave frequencies as a function of the velocity of a vortex ring is warranted and could potentially be achieved using the method of Kleckner and Irvine <cit.>.Using Eq. (<ref>) it is straight forward to estimate typical values of the inertial mass of a vortex for various physical systems shown in Table I. The circulation Γ for classical systems may be estimated as Γ = 2π v r, where v is the fluid velocity at distance r from the centre of the vortex, where as for quantum systems Γ =κ= h/M is determined by the Planck's constant and the mass of the particle forming the superfluid such as a Cooper pair of electrons or neutrons or an atom. In Table I, we have used the following estimates: For a superconductor we choose, M=2M_e, ρ = 2 M_e × 10^24 / m^3, ω_ k = 10( km/s) / 50nm≈Δ^2/E_ F <cit.>. For an atomic BEC we choose, M=M(^87Rb), ρ = 10^-5kg/ m^3, ω_ k = 2π× 4Hz <cit.>. For superfluid helium II we choose, M=M(^4He), ρ = 125kg/ m^3, ω_k = 2π× 1Hz <cit.>. For a neutron star we choose, M= amu, ρ = 10^17kg/ m^3, ω_k = 1MeV/ħ <cit.>. For air we consider a `Fujita 1' tornado and choose, Γ = 8× 10^3m^2/ s, ρ = 10^1kg/ m^3, and ω_ K = 1 rad/s. For water we consider a bath tub vortex and choose, Γ = 6×10^-3m^2/ s, ρ = 10^3kg/ m^3, ω_ K = 0.1 rad/s. § CONCLUSIONSIn conclusion, we have studied the inertial vortex mass in a superfluid. We find that the total inertial mass of a quantised vortex in a superfluid Bose–Einstein condensate, Eq. (<ref>), is determined microscopically by the condensate density and the self-consistent elementary excitation frequency of the kelvon quasiparticle. The result is all-inclusive in the sense that, in contrast to previous mass estimates for bosonic superfluids, it includes all contributions to the inertial mass of a vortex such as the effects of the near and far-field superflows and the core filling substances such as non-condensate atoms. The vortex is heavy and, protected by the Heisenberg uncertainty principle, its mass does not suffer from logarithmic divergencies. In addition to the inertial mass of the quantised vortex, the kelvon quasiparticle is also responsible for the charge, spin and vortex velocity dipole moment (vVDM) of the quantised vortex. We find that the inertial mass of a vortex in Bose and Fermi superfluids has the same origin—quasiparticles localised within the vortex core. Considering the classical limit of large quantum numbers we obtain a relationship between the Kelvin waves and the inertial mass of classical vortices and vortex rings.The inertial mass of a vortex may have relevance to many areas of physics including two-dimensional quantum turbulence, gravitational wave emission from neutron stars, topological quantum computing and high-temperature superconductivity. Avalanches of anomalously heavy vortices in rotating neutron stars that glitch could perhaps result in a continuous gravitational wave signal detectable by future gravitational wave detectors <cit.>. In high-energy states of two-dimensional quantum turbulence the vortex particles may undergo condensation transition <cit.>—a phenomenon which may be influenced by the mass of the vortices. The ability to control the inertial mass of a vortex by varying the frequency of the kelvons has potential for applications. In Bose–Einstein condensates such controlling of the inertial mass of quantised vortices could be achieved using vortex pinning laser beams that shift the excitation energies of the kelvons <cit.>. In high temperature superconductors it would be desirable to be able to reduce the motion of the vortices since vortices of greater inertial mass would require greater de-pinning force enabling the material to withstand stronger supercurrents and potentially higher critical temperatures <cit.>. In a topological quantum computer based on braiding of non-Abelian vortex anyons the goal is the opposite. The faster the vortices can be moved around, the faster the gate operations necessary for the quantum information processing can be achieved. Thus, depending on the specific application, it may be desirable to make the vortices as heavy or as light as possible.I am grateful to Victor Galitski, Andrew Groszek, Kris Helmerson, David Paganin and Martin Zwierlein for useful discussions. 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"authors": [
"Tapio Simula"
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"published": "20170427021055",
"title": "Vortex Mass in a Superfluid"
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#1 11 Regression Type Models for Extremal Dependence Linda Mhalla^a, Miguel de Carvalho^b, Valérie Chavez-Demoulin^c,contact Valérie Chavez-Demoulin (mailto:[email protected]@unil.ch), Faculty of Business and Economics (HEC), Université de Lausanne, Switzerland. Supplementary materials for this article are available online. ^aGeneva School of Economics and Management (GSEM), Université de Genève, Switzerland; ^bSchool of Mathematics, University of Edinburgh,UK; ^cFaculty of Business and Economics (HEC), Université de Lausanne, Switzerland. December 30, 2023 =============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== 0 Regression Type Models for Extremal Dependence We propose a vector generalized additive modeling framework for taking into account the effect of covariates on angular density functions in a multivariate extreme value context. The proposed methods are tailored for settings where the dependence between extreme values may change according to covariates. We devise a maximum penalized log-likelihood estimator, discuss details of the estimation procedure, and derive its consistency and asymptotic normality. The simulation study suggests that the proposed methods perform well in a wealth of simulation scenarios by accurately recovering the true covariate-adjusted angular density. Our empirical analysis reveals relevant dynamics of the dependence between extreme air temperatures in two alpine resorts during the winter season. Supplementary materials for this article are available online. keywords: Angular density; Covariate-adjustment; Penalized log-likelihood; Statistics of multivariate extremes; VGAM. 1.4 §In this paper, we address an extension of the standard approach for modeling non-stationary univariate extremes to the multivariate setting. In the univariate context, the limiting distribution for the maximum of a sequence of independent and identically distributed random variables, derived by <cit.>, is given by a generalized extreme value distribution characterized by three parameters: μ (location), σ (scale), and ξ (shape). To take into account the effect of a vector of covariates 𝐱, one can let these parameters depend on 𝐱, and the resulting generalized extreme value distribution takes the formG_(μ_𝐱, σ_𝐱, ξ_𝐱)(y) = exp[-{1 + ξ_𝐱(y-μ_𝐱/σ_𝐱)}_+^-1/ξ_𝐱],where (a)_+ = max{0, a}; see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.> for related approaches.In the multivariate context, consider 𝐘^i=(Y^i_1,…,Y^i_d)^ independent and identically distributed random vectors with joint distribution F, and unit Fréchet marginal distribution functions F_j(y)=exp(-1/y), for y > 0. Pickands' representation theorem <cit.> states that the law of the standardized componentwise maxima, 𝐌_n= n^-1max{𝐘^1, …, 𝐘^n}, converges in distribution to a multivariate extreme value distribution, G_H(𝐲) = exp{ -V_H(𝐲) }, with V_H(𝐲) = ∫_S_dmax(w_1/y_1, …, w_d/y_d) H(𝐰).Here H is the so-called angular measure, that is, a positive finite measure on the unit simplex S_d = { (w_1,…,w_d) ∈ [0, ∞)^d: w_1+⋯+w_d =1 } that needs to obey ∫_S_d w_j H(𝐰) = 1,j=1,…,d.The function V(𝐲) ≡ V_H(𝐲), is the so-called exponent measure and is continuous, convex, and homogeneous of order -1, i.e., V(t 𝐲) = t^-1 V(𝐲) for all t>0.The class of limiting distributions of multivariate extreme values yields an infinite number of possible parametric representations <cit.>, as the validity of a multivariate extreme value distribution is conditional on its angular measure H satisfying the moment constraint (<ref>). Therefore, most literature has focused on the estimation of the extremal dependence structures described by spectral measures or equivalently angular densities <cit.>. Related quantities, such as the Pickands dependence function <cit.> and the stable tail dependence function <cit.>, were investigated by many authors <cit.>. A wide variety of parametric models for the spectral density that allow flexible dependence structures were proposed <cit.>.However, few papers were able to satisfactorily address the challenging but incredibly relevant setting of modeling nonstationarity at joint extreme levels. Some exceptions include <cit.>, who proposed a nonparametric approach, where a family of spectral densities is constructed using exponential tilting. <cit.> developed an extension of this approach based on covariate-varying spectral densities. However, these approaches are limited to replicated one-way ANOVA types of settings. <cit.> advocated the use of covariate-adjusted angular densities, and <cit.> discussed estimation—in the bivariate and covariate-dependent framework—of the Pickands dependence function based on local estimation with a minimum density power divergence criterion. Finally, <cit.> constructed, in a nonparametric framework, smooth models for predictor-dependent Pickands dependence functions based on generalized additive models. Our approach is based on a non-linear model for covariate-varying extremal dependences. Specifically, we develop a vector generalized additive model that flexibly allows the extremal dependence to change with a set of covariates, but—keeping in mind that extreme values are scarce—it borrows strength from a parametric assumption. In other words, the goal is to develop a regression model for the extremal dependence through the parametric specification ofan extremal dependence structure and then to model the parameters of that structure through a vector generalized additive model (VGAM) <cit.>. One major advantage over existing methods is that our model may be used for handling an arbitrary number of dimensions and covariates of different types, and it is straightforward to implement, as illustrated in the code <cit.> in the Supplementary Materials.The remainder of this paper is organized as follows. In Section <ref> we introduce the proposed model for covariate-adjusted extremal dependences. In Section <ref> we develop our penalized likelihood approach and give details on the asymptotic properties of our estimator. In Section <ref> we assess the performance of the proposed methods. An application to extreme temperatures in the Swiss Alps is given in Section <ref>. We close the paper in Section <ref> with a discussion.§§.§ The functions H and V in (<ref>) can be used to describe the structure of dependence between the extremes, as in the case of independence between the extremes, where V(𝐲) = ∑_j = 1^d 1/y_j, and in the case of perfect extremal dependence, where V(𝐲) = max{1/y_1, …, 1/y_d}. As a consequence, if H is differentiable with angular density denoted h, the more mass around the barycenter of S_d, (d^-1, …, d^-1), the higher the level of extremal dependence. Further insight into these measures may be obtained by considering the point process P_n = { n^-1𝐘^i: i=1,…,n }. Following <cit.> and <cit.>, as n →∞, P_n converges to a non-homogeneous Poisson point process P defined on [0, ∞) ∖{0} with a mean measure μ that verifiesμ(A_𝐲) = V(𝐲),where A_𝐲= ℝ^d ∖( [ -∞,y_1] ×⋯×[ -∞,y_d]). There are two representations of the intensity measure of the limiting Poisson point process P that will be handy for our purposes. First, it holds thatμ(𝐲) = - V_1:d(𝐲)𝐲,with V_1:d being the derivative of V with respect to all its arguments <cit.>. Second, another useful factorization of the intensity measure μ(𝐲), called the spectral decomposition, can be obtained using the following decomposition of the random variable 𝐘 = (Y_1, …, Y_d)^ into radial and angular coordinates,(R, 𝐖) = (𝐘, 𝐘𝐘),where · denotes the L_1-norm. Indeed, it can be shown that <cit.> the limiting intensity measure factorizes across radial and angular components as follows:μ(𝐲) = μ( r × 𝐰 ) = r/r^2H(𝐰).The spectral decomposition (<ref>) allows the separation of the marginal and the dependence parts in the multivariate extreme value distribution G_H, with the margins being unit Fréchet and the dependence structure being described by the angular measure H. The inference approach that we build on in this paper was developed by <cit.> and is based on threshold excesses; see <cit.> for a detailed review of likelihood estimators for multivariate extremes. The set of extreme events is defined as the set of observations with radial components exceeding a high fixed threshold, that is, the observations belonging to the extreme set, E_𝐫= {(y_1, …, y_d) ∈ (0, ∞)^d: ∑_j=1^dy_j/r_j > 1},with 𝐫 = (r_1, …, r_d) being a large threshold vector. Since the points n^-1𝐘^i are mapped to the origin for non-extreme observations, the threshold 𝐫 needs to be sufficiently large for the Poisson approximation to hold. Note that, 𝐘^i ∈ E_𝐫, if and only if, R_i = 𝐘^i > ( ∑_j=1^dW_i, jr_j) ^-1, whereW_i, j = Y^i_jR_i.Hence, the expected number of points of the Poisson process P located in the extreme region E_𝐫 isμ(E_𝐫)= ∫_S_d∫_( ∑_j=1^dw_j/r_j) ^-1^∞ rr^2H(𝐰) = ∫_S_d( ∑_j=1^dw_jr_j) H(𝐰) = ∑_j=1^d 1/r_j∫_S_d w_j H(𝐰) = ∑_j=1^d1/r_j.Now, we can explicitly formulate the Poisson log-likelihood over the set E_r,ℓ_E_𝐫 () =- μ(E_𝐫) +∑_i=1^n_𝐫log{μ( R_i ×𝐖_i)},whererepresents the p-vector of parameters of the measure μ and n_𝐫 represents the number of reindexed observations in the extreme set E_𝐫. Using (<ref>), the first term in (<ref>) can be omitted when maximizing the Poisson log-likelihood, which, using (<ref>), boils down toℓ_E_𝐫 () ≡∑_i=1^n_𝐫log{ -V_1:d(𝐘^i ; )}.Thanks to the differentiability of the exponent measure V and the support of the angular measure H in the unit simplex S_d, we can use the result of <cit.> that relates the angular density to the exponent measure viaV_1:d(𝐲; ) = - 𝐲^-(d+1) h( y_1𝐲, …, y_d𝐲 ; )and reformulate the log-likelihood (<ref>) as followsℓ_E_𝐫 ()≡-(d+1) ∑_i=1^n_𝐫log𝐘^i+ ∑_i=1^n_𝐫log{ h ( Y^i_1𝐘^i, …, Y^i_d𝐘^i; )}= ∑_i=1^n_𝐫ℓ_E_𝐫 (𝐘^i,).§.§Our starting point for modeling is an extension of (<ref>) to the multivariate setting. Whereas the model in (<ref>) is based on indexing the parameters of the univariate extreme value distribution with a regressor, here we index the parameter (H) of a multivariate extreme value distribution (G_H) with a regressor 𝐱=(x_1,…, x_q)^∈𝒳⊂ℝ^q. Our target object of interest is thus given by a family of covariate-adjusted angular measures H_𝐱 obeying∫_S_d w_jH_𝐱(𝐰) = 1,j=1,…,d.Of particular interest is the setting where H_𝐱 is differentiable, in which case the covariate-adjusted angular density can be defined as h_𝐱(𝐰) =H_𝐱/𝐰. This yields a corresponding family of covariate-indexed multivariate extreme value distributions G_𝐱(𝐲) = exp{- ∫_S_dmax( w_1/y_1, … ,w_d/y_d)dH_𝐱(𝐰)}.Other natural objects depending on G_𝐱 can be readily defined, such as the covariate-adjusted extremal coefficient, ϑ_𝐱, which solvesG_𝐱(y 1_d) = exp(- ϑ_𝐱 / y),y > 0,where 1_d is a d-vector of ones. Here, ϑ_𝐱 ranges from 1 to d, and the closer ϑ_𝐱 is to one, the closer we get to the case of complete dependence at that value of the covariate.Some parametric models <cit.> are used below to illustrate the concept of covariate-adjusted angular densities and of a covariate-adjusted extremal coefficient, and we focus on the bivariate and trivariate settings for the sake of illustrating ideas. To develop insight and intuition on these models, see Figures <ref> and <ref>.[Logistic angular surface]Leth_𝐱(w) = (1/α_𝐱 - 1){ w (1-w)} ^-1-1/α_𝐱{w^-1/α_𝐱 + (1-w) ^-1/α_𝐱}^α_𝐱-2,w ∈ (0,1),with α: 𝒳⊂ℝ^q → (0, 1]. In Figure <ref> (left) we represent the case α_x= exp{η(x)}/[1+exp{η(x)}], with η(x)=x^2-0.5x-1 and x ∈𝒳 = [0.1,2]. This setup corresponds to be transitioning between a case of relatively high extremal dependence (lower values of x) to a case where we approach asymptotic independence (higher values of x). [Dirichlet angular surface]Leth_𝐱(w) = α_𝐱β_𝐱Γ(α_𝐱 +β_𝐱 + 1) (α_𝐱 w)^α_𝐱 - 1{β_𝐱(1 - w)}^β_𝐱 - 1/Γ(α_𝐱) Γ(β_𝐱) {α_𝐱 w + β_𝐱 (1 - w)}^α_𝐱 + β_𝐱 + 1,w ∈ (0,1),with α: 𝒳⊂ℝ^q → (0, ∞) and β: 𝒳⊂ℝ^q → (0, ∞). In Figure <ref> (middle) we consider the case α_x=exp(x) and β_x=x^2, with x ∈ [0.9,3]. Note the different schemes of extremal dependence induced by the different values of the covariate x as well as the asymmetry of the angular surface underlying this model. [Hüsler–Reiss angular surface]Leth_𝐱(w) = λ_𝐱/ w (1 - w)^2 (2 π)^1 / 2exp{- [2 + λ_𝐱^2 log{ w/(1 - w) }]^28 λ_𝐱^2},w ∈ (0, 1),where λ: 𝒳⊂ℝ^q→ (0, ∞). In Figure <ref> (right) we consider the case λ_x = exp(x), with x ∈ [0.1,2]. Under this specification, lower values of x correspond to lower levels of extremal dependence, whereas higher values of x correspond to higher levels of extremal dependence.[Pairwise beta angular surface]Leth_𝐱(𝐰)= Γ(3α_𝐱+1)Γ(2α_𝐱+1) Γ(α_𝐱)∑_1 ≤ i<j ≤ 3 h_i,j_𝐱(𝐰), h_i,j_𝐱(𝐰) =(w_i+w_j)^2α_𝐱-1{ 1-(w_i+w_j)} ^α_𝐱-1Γ(2β_i,j_𝐱)Γ^2(β_i,j_𝐱)( w_iw_i+w_j)^β_i,j_𝐱-1( w_jw_i+w_j)^β_i,j_𝐱-1,where 𝐰=(w_1,w_2,w_3) ∈ S_3 and α,β_i,j: 𝒳⊂ℝ^q→ (0, ∞) for 1≤ i<j ≤ 3. In Figure <ref>, we consider the case α_𝐱=exp{exp(x)}, β_1,2_𝐱=exp(x), β_1,3_𝐱=x+1, and β_2,3_𝐱=x+2, with x ∈ [0.8,3.3]. For the different considered values of x, different strengths of global and pairwise dependences can be observed. The mass is concentrated mostly at the center of the simplex due to a large global dependence parameter α_𝐱, compared to the pairwise dependence parameters.The previous parametric models provide some examples of covariate-adjusted angular surfaces h_𝐱. But, how can we learn about h_𝐱 from the data? Suppose we observe the regression data {(𝐱^i, 𝐘^i)}_i = 1^n, with (𝐱^i, 𝐘^i) ∈𝒳×ℝ^d, and where we assume that 𝐘^i=(Y^i_1,…,Y^i_d)^ are independent random vectors with unit Fréchet marginal distributions. Using a similar approach as in Section <ref>, we convert the raw sample into a pseudo-sample of cardinality n_𝐫,{(𝐱^i, 𝐘^i): 𝐘^i ∈ E_𝐫},and use the latter reindexed data to learn about h_𝐱. Without loss of generality, we restrain ourselves to the bivariate extreme value framework (d=2), so that h_𝐱(Y^i_1𝐘^𝐢,Y^i_2𝐘^𝐢) = h_𝐱(w_i,1-w_i) ≡h_𝐱(w_i) , forw_i ∈ [0,1],i=1,…, n_𝐫,that is, the dimension of the angular observations w_i is M=d-1=1. We model h_𝐱(·) using h(·; _𝐱), where the parameter underlying the dependence structure_𝐱 =(θ_1𝐱^1, …, θ_1𝐱^n_𝐫, …, θ_p𝐱^1, …θ_p𝐱^n_𝐫)^∈ℝ^p n_𝐫,𝐱 =(𝐱^1,…, 𝐱^n_𝐫)^∈𝒳^n_𝐫=( 𝒳_1 ×⋯×𝒳_q) ^n_𝐫⊆ℝ^q n_𝐫is specified through a vector generalized additive model (VGAM) <cit.>. Specifically, we model h_𝐱(w) using a fixed family of parametric extremal dependence structures h(w; _𝐱) with a covariate-dependent set of parameters _𝐱. To learn about _𝐱 from the pseudo-sample, we use a vector generalized additive model, which takes the form(𝐱) ≡ = 𝐇_0 _[0] + ∑_k=1^q 𝐇_k 𝐟_k(𝐱_k).Here, * = 𝐠(_𝐱) = ( g_1(θ_1x^1), …, g_1(θ_1x^n_𝐫), …, g_p(θ_px^1), …, g_p(θ_px^n_𝐫)) ^ is the vector of predictors and g_l is a link function that ensures that θ_l· is well defined, for l=1,…,p,* _[0] is a p n_𝐫-vector of intercepts, with p distinct values each repeated n_𝐫 times,* 𝐱_k=( x_k^1,…,x_k^n_𝐫) ^∈𝒳_k^n_𝐫, for k=1,…,q,* 𝐟_k=(𝐟_k,1,…,𝐟_k,p)^, where 𝐟_k,l=(f_k,l(x^1_k), …,f_k,l(x^n_𝐫_k))^, and f_k,l: 𝒳_k →ℝ are smooth functions supported on 𝒳_k, for k=1,…,q and l=1,…,p, and* 𝐇_k are p n_𝐫× p n_𝐫 constraint matrices, for k=0,…,q.The constraint matrices 𝐇_k are important quantities in the VGAM (<ref>) that allow the tuning of the effects of the covariates on each of the p n_𝐫 components of . For example, in Example <ref>, one might want to impose the same smooth effect of a covariate on each of the 32 pairwise dependence parameters and at the same time restrict the effect of this covariate to be zero on the global dependence parameter. To avoid clutter in the notation, we assume from now on that 𝐇_k= 𝐈_p n_𝐫× p n_𝐫, for k=0,…,q.The smooth functions f_k,l are written as linear combinations of B-spline basis functionsf_k,l(x^i_k)= ∑_s=1^d_kβ_[kl]_s B_s,q̃(x^i_k),k=1,…,q,l=1,…,p,i=1,…, n_𝐫,where B_s,q̃ is the sth B-spline of order q̃ and d_k=q̃+m_k, with m_k the number of internal equidistant knots for 𝐱_k <cit.>. To ease the notational burden, we suppose without loss of generality that d_k ≡d̃, for k=1,…,q, and define_[k] = ( β_[k1]_1, …, β_[k1]_d̃, …, β_[kp]_1, …, β_[kp]_d̃) ^∈ℝ^d̃p.Therefore, the VGAM (<ref>), with identity constraint matrices 𝐇_k, can be written as= _[0] + ∑_k=1^q𝐗_[k]_[k] = 𝐗_VAM,where = [ _[0] _[1]⋯ _[q] ]^∈𝐁⊂ℝ^p(1+qd̃), 𝐗_VAM = [ 1_p n_𝐫× p𝐗_[1]⋯𝐗_[q] ]∈ℝ^pn_𝐫×{p(1+qd̃)}for some pn_𝐫×d̃p submatrices 𝐗_[k], k=1,…,q. The vector of parameters to be estimated in the VGAM (<ref>) is .The specification in (<ref>) makes it possible to simultaneously fit ordinary Generalized Additive Models <cit.> in each component of the vector of parameters _𝐱, hence avoiding any non orthogonality-related issues that could arise if the p components were to be treated separately <cit.>. Finally, if the dimension M of the response vector of angular observations w_i is greater than one (d>2), then the vector of predictorswill instead be a Mpn_𝐫-vector and the dimensions of the related quantities in (<ref>) will change accordingly.To give the unfamiliar reader insight on some of the quantities introduced above, we identify these quantities in the examples mentioned previously:* In Examples <ref> and <ref>, d=2, M=1, p=1, q=1, and 𝒳=[0.1,2]. The difference between the VGAMs modeled in these two examples resides in the form of dependence of η on x and the link function g. In Example <ref>, the parameter θ_x ∈ (0,1], η=x^2-0.5x-1, and the link function g is the logit function, whereas in Example <ref> the parameter θ_x ∈ (0,∞), η=x, and the link function g is the logarithm function.* In Example <ref>, d=2, M=1, p=2, q=1, 𝒳=[0.9,3], and =(x, x)^. The vector of parameters for the bivariate Dirichlet angular density _x∈ (0,∞)^2 and the link functions g_1 and g_2 are the logarithm and the square root functions, respectively.* In Example <ref>, d=3, M=2, p=4, q=1, 𝒳=[0.8,3.3], and = (exp(x), x, log(x+1), log(x+2))^. The vector of parameters for the pairwise beta angular density _x∈ (0,∞)^4 and the link function g_l is the logarithm function, for l=1,…,4. §The log-likelihood (<ref>) with a covariate-dependent vector of parameters _𝐱 is now written asℓ() := ∑_i=1^n_𝐫ℓ( 𝐘^i ; )= ∑_i=1^n_𝐫ℓ_E_𝐫[𝐘^i,𝐠^-1{(𝐱^i)}] ,where 𝐠^-1 is the componentwise inverse of 𝐠. Incorporating a covariate-dependence in the extremal dependence model through a non-linear smooth model adds considerable flexibility in the modeling of the dependence parameter _𝐱. The price to pay for this flexibility is reflected in the estimation procedure. The estimation of _𝐱, hence of , is performed by maximizing the penalized log-likelihoodℓ(, ) = ℓ() -12𝐉(),where the penalty term can be written as𝐉() = ∑_k=1^q_[k]^{𝐏_k ⊗diag(γ_(1)k, …, γ_(p)k)}_[k] = ^𝐏() ,with 𝐏() a p(1+qd̃) × p(1+qd̃) block matrix with a first p× p block filled with zeros and q blocks, each formed by a pd̃× pd̃ matrix 𝐏_k that depends only on the knots of the B-spline functions for the covariate 𝐱_k. The matrix 𝐏() can be written as 𝐏()=𝐗̃^𝐗̃ for some p(1+qd̃) × p(1+qd̃) real matrix 𝐗̃. The vectors _[k] are defined in (<ref>), and γ_(l)k are termed the smoothing parameters.The penalty term in (<ref>) controls the wiggliness and the fidelity to the data of the component functions in (<ref>) through the vectorof the smoothing parameters γ_(l)k for l=1,…, p and k=1,…,q. Larger values of γ_(l)k lead to smoother effects of the covariate 𝐱_k on the lth component of . The maximization of the penalized log-likelihood (<ref>) is based on a Newton–Raphson (N–R) algorithm. At each step of the N–R algorithm, a set of smoothing parameters is proposed by outer iteration <cit.>, and a penalized iterative reweighted least squares (PIRLS) algorithm is performed, in an inner iteration, to update the model coefficients estimates. We detail the inner fitting procedure in the following section and the outer iteration in Section <ref>. §.§ We suppose that the penalized log-likelihood (<ref>) depends only on the p(1+qd̃)-vectorand that the vector of smoothing parametersis proposed (at each iteration of the N–R algorithm) by outer iteration and is therefore fixed in what follows.The penalized maximum log-likelihood estimator (PMLE)satisfies the following score equation∂ℓ(,)∂ = 𝐗_VAM^𝐮() - 𝐏() = 0,where 𝐮()= ∂ℓ()/∂∈ℝ^p n_𝐫 and 𝐗_VAM is as defined in (<ref>).To obtain , we update ^(a-1), the (a-1)th estimate of the true _0, by Newton–Raphson:^(a) = ^(a-1) + 𝐈( ^(a-1)) ^-1{𝐗_VAM^𝐮(^(a-1)) - 𝐏() ^(a-1)},where 𝐈( ^(a-1))= - ∂^2 ℓ(,)∂∂^ = 𝐗_VAM^𝐖(^(a-1)) 𝐗_VAM + 𝐏(), 𝐖(^(a-1)) = - ∂^2 ℓ()∂∂^∈ℝ^p n_𝐫× p n_𝐫.The matrix 𝐖(^(a-1)) is termed the working weight matrix. If the expectation E{∂^2 ℓ()/∂∂^} is obtainable, a Fisher scoring algorithm is then preferred, as it ensures the positive definiteness of 𝐖 over a larger region of the parameter space 𝐁 than in the N–R algorithm. When the working weight matrix is not positive definite, which might happen when the parameter ^(a-1) is far from the true _0, a Greenstadt <cit.> modification is applied, and the negative eigenvalues of 𝐖(^(a-1)) are replaced by their absolute values. With the different families of angular densities considered in Examples <ref>–<ref>, the expected information matrix is not obtainable and is hence replaced by the observed information matrix on which a Greenstadt modification is applied whenever needed. See <cit.> for other remedies and techniques for deriving well-defined working weight matrices.Let 𝐳^(a-1) := 𝐗_VAM^(a-1) + 𝐖(^(a-1))^-1𝐮(^(a-1)) be the p n_𝐫-vector of working responses. Then, (<ref>) can be rewritten in a PIRLS form as^(a) = {𝐗_VAM^𝐖(^(a-1)) 𝐗_VAM + 𝐏()} ^-1𝐗_VAM^T𝐖(^(a-1)) 𝐳^(a-1)= {𝐗_PVAM^𝐖̃^(a-1)𝐗_PVAM} ^-1𝐗_PVAM^𝐖̃^(a-1)𝐲^(a-1),where 𝐗_PVAM, 𝐲^(a-1), and 𝐖̃^(a-1) are augmented versions of 𝐗_VAM, 𝐳^(a-1) and 𝐖(^(a-1)), respectively, and are defined as𝐗_PVAM = [ 𝐗_VAM𝐗̃ ]^∈ℝ^p(1+ n_𝐫 + qd̃) × p(1+ qd̃),𝐲^(a-1) = [𝐳^(a-1) 0_p(1+qd̃) ]^∈ℝ^p(1+ n_𝐫 + qd̃), 𝐖̃^(a-1) = diag( 𝐖(^(a-1)), 𝐈_p(1+ qd̃) × p(1+ qd̃)) ∈ℝ^p(1+ n_𝐫 + qd̃) × p(1+ n_𝐫 + qd̃).The algorithm stops when the change in the coefficientsbetween two successive iterations is sufficiently small. Convergence of the N–R algorithm is not guaranteed and might not occur if the quadratic approximation of ℓ(,) aroundis poor. See <cit.> for more details.The plug-in penalized maximum log-likelihood estimator of the covariate-dependent angular density is defined ash_𝐱(𝐰) ≡ h{𝐰; 𝐠^-1(𝐗_VAM)} . In the following section, we give details about the selection of the smoothing parameters , which is outer to the PIRLS algorithm. §.§ To implement the PIRLS algorithm performed at each iteration of the N–R algorithm, a smoothing parameter selection procedure is conducted by minimizing a prediction error estimate given by the generalized cross validation (GCV) score. Let 𝐀^(a-1)() be the influence matrix of the fitting problem at the ath iteration, defined as𝐀^(a-1)() = 𝐗_PVAM{𝐗_PVAM^𝐖̃^(a-1)𝐗_PVAM} ^-1𝐗_PVAM^𝐖̃^(a-1).Then, by minimizing the GCV scoreGCV^(a-1) = n_𝐫{𝐲^(a-1) - 𝐀^(a-1)() 𝐲^(a-1)} ^𝐖̃^(a-1){𝐲^(a-1) - 𝐀^(a-1)() 𝐲^(a-1)}[n_𝐫 -trace{𝐀^(a-1)()}] ^2,we aim at balancing between goodness of fit and complexity of the model, which is measured by the trace of the influence matrix and termed the effective degrees of freedom (EDF). The EDF of the fitted VGAM (<ref>) are defined as the EDF obtained at convergence, that is, trace{𝐀^(c-1)()}, where c is the iteration at which convergence occurs.Both the fitting algorithm of Section <ref> and the smoothing parameter selection are implemented in thepackage<cit.>, with the latter being required from thepackage<cit.>.Model selection between different, not necessarily nested, fitted VGAMs is performed based on the Akaike information criterion (AIC), where the number of parameters of the model is replaced by its EDF to account for penalization. More details on the (conditional) AIC for models with smoothers along with a corrected version of this criterion, which takes into account the smoothing parameter uncertainty, can be found in <cit.>.§.§ In this section we derive the consistency and asymptotic normality of the PMLEdefined in Section <ref>. Based on the penalized log-likelihood (<ref>),satisfies the following score equation𝐦() - 𝐏() = 0_p(1+qd̃),where 𝐦()=∂ℓ()/∂.Let 𝐁_0 be an open neighborhood around the true parameter _0. Moreover, we define 𝐦(𝐲,)=∂ℓ(𝐲;)/∂. Our asymptotic results hold under the following customary assumptions: * =[ γ_(1)1⋯ γ_(p)1⋯ γ_(1)q⋯ γ_(p)q ]^ = o(n_𝐫^-1/2) 1_pq. * Regularity conditions:* If ≠, then ℓ(𝐲;) ≠ℓ(𝐲;), with , ∈𝐁. Moreover, E{sup_∈𝐁|ℓ(𝐘;) |} < ∞. * The true parameter _0 is in the interior of 𝐁.* For 𝐲∈ (0,∞)^d, ℓ(𝐲;) ∈ C^3(𝐁_0).* ∫sup_∈𝐁_0𝐦(𝐲,) 𝐲 < ∞ and ∫sup_∈𝐁_0∂𝐦 (𝐲,)/ ∂^𝐲 < ∞.* For ∈𝐁_0, 𝐢():=cov{𝐦( 𝐘,)} = 𝐗_VAM^𝐖()𝐗_VAM exists and is positive-definite.* For each triplet 1 ≤ q, r, s ≤ p(1+qd̃), there exists a function M_qrs: (0,∞)^d →ℝ such that, for 𝐲∈ (0,∞)^d and ∈𝐁_0, |∂^3 ℓ(𝐲;)/∂_qrs|≤ M_qrs(𝐲), and E{ M_qrs(𝐘)} < ∞. The next theorem characterizes the large sample behavior of our estimator.Under A_1 and A_2, it follows that as n_𝐫→∞: * - _0=. * √(n_𝐫) ( - _0)N(0, 𝐢(_0)^-1).These results are derived from a second-order Taylor expansion of the score equation (<ref>) around the true parameter _0 along the same lines as in <cit.> and <cit.>. Similar results on the large sample behavior of the corresponding plug-in estimator (<ref>) can be derived using the multivariate delta method. These results are useful to derive and construct approximate confidence intervals for conditional angular densities and to compare nested models based on likelihood ratio tests. Our proviso is similar to that of <cit.> in the sense that asymptotic properties of the estimatorare derived under the assumption of known margins and we sample from the limiting object h_x, whereas in practice only a sample of (estimated) pseudo-angles, {𝐖_i}_i = 1^n_𝐫, would be available. Asymptotic properties under misspecification of the parametric model set for h_x could in principle be derived under additional assumptions onand 𝐦, along the same lines as in standard likelihood theory <cit.>. The resulting theory is outside the scope of this work and is deliberately not studied here.§§.§ We assess the performance of our methods using the bivariate extremal dependence structures presented in Section <ref>—and displayed in Figure <ref>—as well as the trivariate pairwise beta dependence model from Example <ref>—depicted in Figure <ref>. Monte Carlo evidence will be reported in Section <ref> and in the Supplementary Materials. For now, we concentrate on illustrating the methods over a single-run experiment on these scenarios. For each dependence model from Examples <ref>–<ref>, we draw a sample { (Y^i_1,Y^i_2)}_i=1^n from the corresponding bivariate extreme value distribution G_x with sample size n=6000 and where each observation (Y^i_1,Y^i_2) has unit Fréchet margins and is drawn from the chosen dependence model conditional on a fixed value x^i of the covariate x. For estimating h_x, we only consider the observations with a radial component exceeding its 95% quantile, and we end up with n_𝐫=300 extreme (angular) observations. To gain insight into the bias and variance of our covariate-adjusted spectral density estimator, we compute its 95% asymptotic confidence bands based on Theorem <ref> and at different values of w ∈ (0,1). There are two possible sources of bias in our estimation procedure. First, the limiting extremal dependence structure is estimated at a sub-asymptotic level, i.e., based on angular observations exceeding a finite diagonal threshold level. Then, the penalization of the model likelihood causes a smoothing bias <cit.> if the smoothing parameters do not vanish at a certain rate (see Section <ref>). The uncertainty due to the choice of the parametric model is deliberately not taken into account, that is, the simulations are performed in a well-specified framework.Figure <ref> displays the estimates of the covariate-adjusted spectral densities from Examples <ref>, <ref>, and <ref> for various fixed values of the covariate x that induce different extremal dependence strengths. All panels show that for the different extremal dependence schemes (strength and asymmetry), the covariate-adjusted spectral densities are accurately estimated and the true curves fall well within the 95% confidence bands. A systematic slight upward bias is observed when approaching extremal independence. This is due to the residual dependence in the data that we observe at finite threshold levels but that should vanish at an asymptotic level. This issue can be corrected either by taking higher threshold levels or considering angular observations simulated from the true spectral density. Finally, the estimates in the Dirichlet case seem to be a bit more biased, and this might be explained by the fact that both of the two non-orthogonal parameters of the model depend smoothly on the covariate x.We now consider the case of the trivariate pairwise beta dependence model from Example <ref>. The construction of the pairwise beta covariate-adjusted spectral density—which extends <cit.>—is such that the corresponding multivariate extreme value distribution cannot be computed in closed form. Hence, we draw a sample {( w_i,1,w_i,2,w_i,3)} _i=1^n_𝐫 with sample size n_𝐫=300 where each observation ( w_i,1,w_i,2,w_i,3) is drawn from the pairwise beta model conditional on a fixed value x_i of the covariate x, as illustrated in Figure <ref>. Figure <ref> displays the contour plots of the estimates of the covariate-adjusted spectral density from Example <ref> at three fixed values of x.All panels in Figure <ref> show that, for the different extremal dependence schemes, i.e., for the different considered values of x, the contour plots of the estimates are remarkably close to the actual contour plots. The estimates are slightly more biased near the edges of the simplex than in the center, reflecting a better estimation of the global dependence parameter compared to the pairwise dependence parameters.§.§ A Monte Carlo study was conducted by simulating 500 samples of sizes 6000 and 10000, that is, n_𝐫=300 and n_𝐫=500 extreme (angular) observations, respectively. As can be seen from Figures 1 and 2 in the Supplementary Materials, our method successfully recovers the corresponding target covariate-adjusted angular densities with a high level of precision over the simulation study. In what follows we focus on documenting how the level of accuracy increases when the number of observations increases by assessing the mean integrated absolute error (MIAE)—which for the bivariate case can be written asMIAE = E{∫_𝒳∫_0^1 |h_x(w) - h_x(w)| w x }.The results are reported in Table <ref>.As expected, an increase in the number of angular observations leads to a reduction of MIAE. Evidence from Table <ref> should be supplemented with Figures 1 and 2 in the Supplementary Materials. The latter offer a more granular level of detail than that of Table <ref> on the behavior of the estimator over specific values of the covariate and of the unit simplex.§ §.§In this section, we describe an application to modeling the dependence between extreme air winter (December–January–February) temperatures at two sites in the Swiss Alps: Montana—at an elevation of 1427m—and Zermatt—at an elevation of 1638m. The sites are approximatively 37km apart. In the Alpine regions of Switzerland, there is an obvious motivation to focus on extreme climatic events, as their impact on the local population and infrastructure can be very costly. As stated by <cit.>, warm winter spells, that is, periods with strong positive temperature exceedances in winter, can exert significant impacts on the natural ecosystems, agriculture, and water supply: “Temperatures persistently above 0^∘C will result in early snow-melt and a shorter seasonal snow cover, early water runoff into river basins, an early start of the vegetation cycle, reduced income for alpine ski resorts and changes in hydro-power supply because of seasonal shifts in the filling of dams <cit.>.”In this analysis, we are interested in the dynamics of the dependence between extreme air temperatures in Montana and Zermatt during the winter season. The dynamics of both extreme high and extreme low winter temperatures in these two sites will be assessed and linked to the following explanatory factors: time (in years) (t), day within season (d), and the NAO (North Atlantic Oscillation) index (z); the latter is a normalized pressure difference between Iceland and the Azores that is known to have a major direct influence on the alpine region temperatures, especially during winter <cit.>. The choice of the studied sites is of great importance in this analysis. <cit.> showed that both cold and warm winters exhibit temperature anomalies that are altitude-dependent, with high-elevation resorts being more representative of free atmospheric conditions and less likely to be contaminated by urban effects. Therefore, to study the “pure" effect of the above-mentioned explanatory covariates on the winter temperature extremal dependence, we choose the two high elevation sites Montana and Zermatt.The data consist of daily winter temperature minima and maxima measured at 2m above ground surface and were obtained from the MeteoSwiss website (<www.meteoswiss.admin.ch>). The data were available from 1981 to 2016, giving a total of 3190 winter observations per site. Daily NAO index measurements were obtained from the NOAA (National Centers for Environmental Information), at <https://www.ngdc.noaa.gov/ftp.html>.We first transform the minimum temperature data by multiplication by -1 and then fit at each site—and to both daily minimum and maximum temperatures—a Generalized Pareto Distribution (GPD) <cit.>G_σ,ξ(y) = 1- ( 1 + ξyσ) _+^-1/ξ,to model events above the 95% quantile u_95 for each of the four temperature time series. In (<ref>), σ>0 is the scale parameter that depends on u_95, and -∞ < ξ < ∞ is the shape parameter. As is common with temperature data analysis, we test the effect of time t on the behavior of the threshold exceedances by allowing the scale parameter of the GPD (<ref>) to smoothly vary with t <cit.>. Based on the likelihood ratio tests, a model with a non-stationary scale parameter is preferred only in Zermatt for the threshold exceedances of the daily minimum temperatures (p-value ≈ 0.022). Graphical goodness-of-fit tests for the four GPD models are conducted by comparing the distribution of a test statistic S with the unit exponential distribution (if Y ∼ G_σ,ξ, then S = - ln{ 1-G_σ,ξ(Y)} is unit exponentially distributed). Figure <ref> displays the resulting qq-plots and confirms the validity of these models. The fitted models are then used to transform the data to a common unit Fréchet scale by probability integral transform and where the empirical distribution is used below u_95. This results in two datasets of bivariate observations (in Montana and Zermatt) with unit Fréchet margins: one for the daily maximum temperatures and the other one for the daily minimum temperatures.Following the theory developed in Section <ref>, we transform each of the two datasets into pseudo-datasets of radial and angular components. By retaining the angular observations corresponding to a radial component exceeding its 95% quantile in each pseudo-dataset, we end up with two pseudo-samples of 160 extreme bivariate (angular) observations in each pseudo-dataset. §.§In the following analyses of the dynamics of the dependence between extreme temperatures in Montana and Zermatt—and in line with findings from previous analyses of extreme temperatures in Switzerland <cit.>—we assume asymptotic dependence in both extremely high and extremely low winter temperatures.§.§.§ Dependence of Extreme High Winter TemperaturesThe covariate-adjusted bivariate angular densities presented in Section <ref> are now fitted to the pseudo-sample of extreme high temperatures. The effects of the explanatory covariates t, z, and d are tested in each of the three angular densities: the logistic model (Example <ref>) with parameter α(t,z,d), the Dirichlet model (Example <ref>) with parameters α(t,z,d) and β(t,z,d), and the Hüsler–Reiss model (Example <ref>) with parameter λ(t,z,d). Within each family of covariate-adjusted angular densities, likelihood ratio tests (LRT) are performed to select the most adequate VGAM for the dependence parameters. Table <ref> shows the best models in each of the three families of angular densities.All the considered covariates have a significant effect on the strength of dependence between extreme high temperatures in Montana and Zermatt. For the covariate-dependent Dirichlet model, the covariates affect the dependence parameters α and β differently. However, these parameters lack interpretability, and <cit.> mention the quantities (α+β)/2 and (α-β)/2 that can be interpreted as the strength and asymmetry of the extremal dependence, respectively. In this case, the best Dirichlet dependence model found in Table <ref> is such that both the intensity and the asymmetry of the dependence are affected by time, NAO, and day in season.The best models in the studied angular density families are then compared by means of the AIC (see Section <ref>) displayed in Table <ref>. The Dirichlet model with α(z) and β(t,d) parameters has the lowest AIC and is hence selected. This suggests the presence of asymmetry in the dependence of extreme high temperatures between Montana and Zermatt.Figure <ref> shows the fitted smooth effects of the covariates on the extremal coefficient—constructed via the covariate-adjusted extremal coefficient as in (<ref>)—that lies between 1 for perfect extremal dependence and 2 for perfect extremal independence.A decrease in the extremal coefficient, or equivalently an increase in the extremal dependence between high winter temperatures in Montana and Zermatt, is observed from 1988 until 2006. This change might be explained first by a warm phase of very pronounced and persistent warm anomalies during the winter season, which occured countrywide from 1988 to 1999 <cit.>, and then by an exceptionally warm 2006/2007 winter that took place in Europe <cit.>. Regarding the NAO effect, as expected, we observe an increase in the extremal dependence during the positive phase of NAO that has a geographically global influence on the Alps and results in warmer and milder winters, as depicted by <cit.>. In terms of the very negative NAO values (less than -100), there is an important uncertainty due to the corresponding small amount of joint extreme high temperatures (8%). The right panel of Figure <ref> suggests an increase in the extremal dependence around mid-December. This evidence also seems compatible with the countrywide findings by <cit.>, who claims that “The anomalously warm winters have resulted from the presence of very persistent high pressure episodes which have occurred essentially during periods from late Fall to early Spring.” §.§.§ Dependence of Extreme Low Winter TemperaturesThe effects of the covariates time, NAO, and day in season on the dependence between extreme cold winters in Montana and Zermatt are now tested by fitting the bivariate angular densities of Section <ref>. Within each of the logistic, Dirichlet, and Hüsler–Reiss families, LRTs are performed, and the selected models are displayed in Table <ref>.The explanatory covariates have different effects on the extremal dependence, depending on the family of angular densities. The AICs for the fitted models are quite close, and the asymmetric Dirichlet model has the lowest AIC and is hence the retained model. As opposed to the extremal dependence between warm winters in the two mountain sites, the NAO has a non-significant effect on the extremal dependence between cold winters. This might be explained by the fact that high values of the NAO index will affect the frequency of extreme low winter temperatures (less extremes) and hence the marginal behavior of the extremes at both sites, but not necessarily the dependence of the extremes between these sites <cit.>.Figure <ref> shows the fitted smooth effects of time and day in season. The extremal dependence between low winter temperatures in Montana and Zermatt is high, regardless of the values taken by the covariates t and d. The range of values of the extremal coefficient observed in Figure <ref> is in line with the findings of <cit.>, where the value of the extremal coefficient for the dependence between extreme low winter temperatures (in Switzerland) is around 1.3 for pairs of resorts separated by up to 100km. Overall, the extremal coefficient is lower in the extreme low winter temperatures than in the extreme high winter temperatures. This could be explained by the fact that minimum winter temperatures are usually observed overnight when the atmosphere is purer and not affected by local sunshine effects and hence is more favorable to the propagation over space of cold winter spells. A decrease in the extremal dependence is observed from around 2007 and results in values of the extremal coefficient that are comparable to those obtained under the warm winter spells scenario (see Figure <ref>). This can be explained by a decrease in the intensity of the joint extreme low temperatures, that is, milder joint extreme low temperatures, occurring during the last years of the analysis, as can be observed in Figure <ref>. The right panel of Figure <ref> highlights a decrease in the extremal dependence when approaching spring. This effect can be explained by the fact that mountains often produce their own local winds.[https://www.morznet.com/morzine/climate/local-climate-in-the-alps] These warm dry winds are mostly noticeable in spring and are called Foehn in the Alps. Local effects obviously lead to a decrease of extremal dependence between the two resorts. §In this paper, we have introduced a sturdy and general approach to model the influence of covariates on the extremal dependence structure. Keeping in mind that extreme values are scarce, our methodology borrows strength from a parametric assumption and benefits directly from the flexibility of VGAMs. Our non-linear approach for covariate-varying extremal dependences can be regarded as a model for conditional extreme value copulas—or equivalently as a model for nonstationary multivariate extremes. An important advantage over existing methods is that our model profits from the VGAM framework, allowing the incorporation of a large number of covariates of different types (continuous, factor, etc) as well as the possibility for the smooth functions to accommodate different shapes. The fitting procedure is an iterative ridge regression, the implementation of which is based on an ordinary N–R type algorithm that is available in many statistical software. An illustration is provided in the code in the Supplementary Materials.The method paves the way for novel applications, as it is naturally tailored for assessing how covariates affect dependence between extreme values—and thus it offers a natural approach for modeling conditional risk. Conceptually, the proposed approach is valid in high dimensions. Yet, as for the classical setting without covariates, the number of parameters would increase quickly with the dimension and additional complications would arise. Relying on composite likelihoods <cit.> instead of the full likelihood seems to represent a promising path for future extensions of the proposed methodology in a high-dimensional context. The online supplement to this article contains supplementary numerical experiments, codes for implementing VGAM family functions for different angular density families, as well as the codes used for the extreme temperature analysis.Monte Carlo Evidence: The file contains the results of the Monte Carlo study conducted in Section <ref>. (.pdf file)Covariate Adjusted Angular Densities: The file contains codes for implementing the following angular density VGAM families: the bivariate logistic, the bivariate Dirichlet, the bivariate Hüsler–Reiss, and the trivariate pairwise beta (see Section <ref>). Examples of the use of the implemented VGAM families are provided. (.zip file)Temperature Data Analysis: The file contains the datasets obtained from the MeteoSwiss website as well as the codes for the analysis of the extremal dependence between winter temperatures in Montana and Zermatt. (.zip file) 0 We thank the Editor, Associate Editor, and two anonymous referees for several insightful recommendations that substantially improved the paper.1We thank the Editor, Associate Editor, and two anonymous referees for several insightful recommendations that substantially improved the paper. We extend our thanks to the participants of Workshop 2017, EPFL, for discussions and comments, and to Paul Embrechts for his constant encouragement.The research was partially funded by FCT (Fundação para a Ciência e a Tecnologia, Portugal) through the project UID/MAT/00006/2013. asa2.bst | http://arxiv.org/abs/1704.08447v2 | {
"authors": [
"Linda Mhalla",
"Miguel de Carvalho",
"Valérie Chavez-Demoulin"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20170427063517",
"title": "Regression Type Models for Extremal Dependence"
} |
firstpage–lastpage 2016 Entanglement generation between a charge qubit and its bosonic environment during pure dephasing - dependence on environment size Katarzyna Roszak Accepted 2017 XXX. Received 2017 Apr; in original form 2016 Feb ================================================================================================================================= We report the discovery and multicolor (VRIW) photometry of a rare explosive star MASTER OT J004207.99+405501.1 - a luminous red nova - in the Andromeda galaxy M31N2015-01a. We use our original light curve acquired with identical MASTER Global Robotic Net telescopesin one photometric system: VRI during first 30 days and W (unfiltered) during 70 days. Also we added publishied multicolor photometry datato estimate the mass and energy of the ejected shell, and discuss the likely formation scenarios of outbursts of this type.We propose the interpretation of the explosion, that is consistent with theevolutionary scenario where star merger is a natural stage of the evolution of close-mass stars and may serve as an extra channel for the formation of nova outbursts.stars: novae, stars: individual: MASTEROTJ004207.99+405501.1, stars: individual: ultraluminous red nova § INTRODUCTIONThe optical transient MASTER OT J004207.99+405501.1 <cit.> discovered by the MASTER global robotic telescope network <cit.> was found to belong to a rare type of luminous red novae <cit.> whose history began with the discovery of the outburst of M31-RV by <cit.>. However, the canonical prototype of this class of events is now considered to be the outburst of the star V838 Monocerotis <cit.>, which reached an absolute magnitude of -10 at maximum light <cit.>. Luminous red novae differ from common members of this class primarily by the apparent lack ofany thermonuclear processes, their large emitted energy, a characteristic plateau on the light curve <cit.>, and very strong reddening that varies with time. The plateau phase indicates that compared to common novae, LRNe have more massive and dense envelopes where a stationary recombination front forms that has approximately constant luminosity like that which occur during the explosions of type IIP supernovae <cit.>. The observation of the progenitor of V1309 Sco by <cit.>, who found it to be a contact binary with a period of 1.4 days, provided strong support for the merging mechanism of red nova outbursts.All these results made the object extremely popular among observers operating on all sorts of instruments - from one-meter telescopes to the Spitzer infrared space telescope and SWIFT gamma-ray laboratory <cit.>.<cit.>have identified the likely progenitor in archival SDSS, CFHT, Local Group Survey and HST imaging data. Several papers have already been published reporting detailed observations and interpretation of the outburst of MASTER OT J004207.99+405501.1 / M31N 2015-01a, from which we would like to mark <cit.> with spectroscopic and photometric observations obtained by the Liverpool Telescope, that showed the LRN becoming extremely red, and where the authors discuss the possible progenitor scenarios of this system. In this paper we reportthe earliest photometric observations, and the complete light curve acquired with identical telescopes of the MASTER global network.We use this light curve to estimate the mass of the ejected envelope, and discuss the formation scenario of such outbursts based on our experience in the population synthesis of binary stars <cit.> and propose the interpretation of the explosion. § THE DISCOVERY OF MASTER OT J004207.99+405501.1 / M31N2015-01AMASTER OT J004207.99+405501.1 / M31N2015-01a was discovered by the MASTER auto-detection system <cit.> during the survey performed by MASTER-Kislovodsk observatory on 2015-01-13.63235 UT<cit.>.A total of four images containing this optical transient were acquired with the unfiltered limiting magnitudes of m_ OT=19.0 and 19.6 (Fig. <ref>), and a reference image with 21.1 unfiltered limiting magnitude was selected from the MASTER-Kislovodsk database. MASTER-Kislovodsk is a node of the MASTER Global Robotic Net[http://observ.pereplet.ru]. All MASTER telescopes haveidentical optical schemes and are equipped with identical sets of polarization and BVRI filters<cit.>. The main advantages of MASTER instruments are the following: (1) wide 8 square degree (twin 2.05^∘×2.05^∘) field of view of the main MASTER-II optical channel (with a limiting unfiltered magnitude of up to 20-21 per 60-180s exposition) <and an even bigger 800 square degree (twin 16x24^∘) field of view of Very Wide Field cameras, i.e. MASTER-VWFC (with a limiting magnitude of up to 11-12m and 13.5-15m for 1-s and coadded images, respectively); (2) twin tubes that can be pointed to different fields (allowing wide FERMI error-boxes to be observed almost in real-time) and used to observe the frame in different polarizations and in BVRI filters <cit.>.The main goal of MASTER network is to detect prompt GRB emission by providing rapid response to GRB-alerts . MASTER has a very fast positioning systemthat makes it very suitable for follow-up programssuch as prompt optical observations of GRB neutrinos and GW alertsetc. When not engaged in alert-triggered observations MASTER carries out sky survey programs including the Andromeda survey in order to discover optical transients of different nature (more than 10 types) and to investigate all most important problems of the modern astrophysics.A unique key feature of MASTER is our software that provides full information about all optical sources detected on every image one to two minutes after the CCD readout. This information includes the full classification of all sources found in the image, the data from previous MASTER-Net archived images for every source, full information from VIZIER database and from all open sources (e.g., Minor planet mpchecker center), computed orbital elements for moving objects, etc. In search tasks a real astrophysical source cannot occupy only 1, 2 or 4 pixels, because such objects can never be proved to be real rather than artifacts. A real transient must occupy more than 10 pixels and exhibit a specific profile to be distinguished from a clump of several hot-pixels .MASTER own software discovers optical transients not just by analyzing the difference between the previous and current images, but also by fully identifying of every source at every image. This MASTER software allowed us to discover more then 1200 (up to September 2016) optical transients in a fully automatic mode[http://observ.pereplet.ru/MASTER_OT.html]. We also faced a challenging task of discovering all sources seen against the Andromeda disc structure, especially during the rising stage. We solved this problem before January 2015 and started ournova search survey in the Andromeda galaxy. At the end of 2014 the global MASTER network of twin robotic telescopes <cit.> started automatic search for optical transients in the Andromeda galaxy.MASTER has been observing the Andromeda galaxy every night, weather permitting, resulting in thousands of frames available for accurate photometry. § OBSERVATIONS AND DATA REDUCTION Monitoring has been carried out with MASTER network telescopes for 72 days after our discovery of this LRN. We acquired a total of about 400 white-light frames and 130 frames with V, R, and I-band filters with 180-s exposures (see Tables 1,2,3 with VRIW-photometry). All observations passed in automatic mode. Thereby, some of the frames were acquired through light clouds.For calibration we used the dark frames, acquired on the evening before observations, and twilight flats. Calibration, frame clipping, and astrometric reduction was performed automatically on each observatory of the network. For our photometry we used the 15×15 ^2frame area centered on the object with about 60 comparison stars from magnitude 13 to 17 in the V band. We used IRAF/apphot packageto perform photometry with an optimal aperture for each frame <cit.>. The resulting instrumental magnitudes were corrected using the Astrokit tool to minimize the standard deviation for the ensemble of comparison stars <cit.>. For transformation of the instrumental magnitudes to a standard system, we used 56 nearby stars from UCAC4 catalog <cit.>. R and I magnitudes calculated from UCAC4 r- and i-bands by the following equation from <cit.>:I = r - 1.2444·(r - i) - 0.3820R = r - 0.2936·(r - i) - 0.1439Some of comparison stars are presented in LGGS catalog <cit.>, 20 and 8 for R- and I-band respectively. We used this information for checking of calculated magnitudes. Median of residual between calculated magnitudes and data from LGGS is 0.046m for R and 0.035m for I band. White light magnitudes calculated from UCAC4 B and calculated R by the equation W=0.2· B+0.8· RThen, we cleared our data from thepoints with big error and deviation. Visual inspection of appropriate frames showed that they were obtained through the light clouds, and in two cases the problem was caused by cosmic ray particles. Finally, we binned data points of each night by calculation of mean. Error for binned point calculated as standard deviation.We converted our apparent magnitudes to absolute magnitudes with the adopted distance modulus of 24.43 (Feedman, Madore 1990).Complete binned by night light curve of this Nova obtained with MASTER Global Network (MASTER-Kislovodsk, MASTER-Tunka, and MASTER-Ural) is presented at Figure 2. The data are given in the Appendix (Tables 2,3,4,5).M31 has a distance modulus of (m-M)_0 = 24.43 ± 0.06. Given the adopted foreground reddening of E_(B-V) = 0.062 <cit.> and maximum total extinction at that position in M31 of E_(B-V) = 0.18<cit.>, we conservatively assume that M31LRN was subject to the reddening of E_(B-V) = 0.12 ± 0.06, implying an absolute peak magnitude of M_V = -9.4 ± 0.2 for the transient. § PHYSICS IN THE MODEL At present, there is no sure answer to the question: “What is the mechanism responsible for the energy release in LRNe?”. It may be:* stellar mergers <cit.> * an unusual SN mechanism <cit.> * a classical nova mechanism <cit.> * giant planet capture <cit.> * extreme AGB stars <cit.>.In ourstudy of M31LRNhundreds of models have been tested for simulating nova explosions in a close binary system with a common envelope with various initial parameters (in spherically symmetric approximation). The parameters of the most suitable models used for the analysis of processes in the LRN, are shown in Table <ref>.For modelling M31LRN we have adapted the code<cit.>,which is widely used for computing supernova explosions<cit.>.code is a set of programs for multi-group radiative hydrodynamics that can be used to compute the light curves of supernovae of various types in the spherically symmetric approximation with the thermodynamics of stellar plasma treated in LTE approximation. The code can also be used to compute nova explosions, although in this case certain adjustments are needed because of the lower velocity gradients in novae compared to supernovae and because emission in lines has in some cases to be computed beyond the framework of Sobolev approximation. We constructed the initial models in the same way as described in <cit.>. The nature of LRNs is different from that of collapsing supernovae in many aspects, that is why we do not aim to reproduce all the details of observed light curves for the whole period of observations using thecode. This task would require developing of a new code and we leave this for future. More modest problem is being solved in the current investigation: we try to elucidate the physics of LRN emission on the plateau phase of the light curve, when it behaves in a similar way to SN IIP. It isthe longest stage in the evolution of LRN with a characteristic behaviour of the light curves, determined by the passage of cooling and recombination waves through the expanding envelope.The initial system we considered consists of two components: the inner core and and the outer shell. Details of the inner core are not taken into account in our simulations, and the core is treated as a hard sphere of a given mass. The outer shell prior to the explosion is built as a model of a polytropic star, similar to the one described in the article <cit.>. It is assumed that active dynamic processes in the binary andthe formation of the common envelope lead to a strong mixing of matter in the shell. Therefore, the chemical composition in our simulations is uniform with solar abundances at each point. In this paper we are not going into the details of the mechanism. We explored different ways of energy release: from the fast release of allenergy through the explosion near the centre within the mass of 0.1-0.2 M_⊙ to a long warm-up of the whole body of the star.Version with the fast central heating with a time-scale of energy release t_ heat∼ 10^3 s is shown in Fig. <ref>, it demonstrates common features of such a heating. The local release of energy in the central core of M=0.2M_⊙ produces a shock wave reaching the outer edge of envelope in ∼ 3^h. The shock wave propels all matter in the envelope with velocity higher than the parabolic one. This leads to the expansion of the envelope like in a supernova IIP. In comparison with our best-fit model (see below) a significant fraction of the energy goes into kinetic energy, which leads to a weaker heating of matter anda dim light on the plateau stage. Aftert ∼ 60^d the envelope becomes transparent andthe heated core shines through it, giving the second maximum on the light curves. The second maximum is not observed for M31LRN, but occurs in similar objects, so it is visible from LRN V838 Mon <cit.>.It should be noted that the light curves are sensitive to changes in the initial masses and radii of envelopes <cit.>. See results for three models in FigsBy varyingR, M and E one can achieve satisfactory matches of the model and observed light curves at the maximum light, see<ref>For example, increasing the energy of the explosion to the E = 8 × 10^48 erg leads to a higher speed of the envelope expansion. Light curves rise to the maximum faster, and the stage of CRW is shorter. It can be seen that the model does not satisfactorily describe the observations since light curves do not match the observations after the maximum. In addition, the physics of the light curve near the dome is not explained by a wave recombination <cit.>. §.§ Best-fit model The best fit to M31LRN observations is obtained with the model , whose light curves are shown in Figure <ref>. The model has a total mass of M_ tot = 3 M_⊙ and the envelope radius R = 10 R_⊙. The initial configuration was the same as the above-mentioned “fast” model. The main difference between the two models is in the mode of energy release during initiationof LRN explosion.Thermal energyE=3 × 10^48 erg is released throughout the whole mass M_ tot during a longer time t_ heat∼ 10^4 s.It is evident that at the stage of the plateau, this model is in much better with observations. The deviation in the band I, is apparently due to the insufficiently precise description of opacity in the infrared region. When calculating the opacity in the lines we use a list of 150 thousand atomic transitions. The main emphasis in the formation of lines in this list has been made on the ultraviolet and visible range of the spectrum (which is important for supernovae), while in the infrared region there is some shortage of lines which is planned to fill out in future calculations.§.§ The dynamics of expansion The dynamics of the expansion and the accompanying processes are illustrated in a series of snapshots shown in Figure <ref>.By the end of the first day of the shock has heated up the envelope and accelerated the matter at the level of the photosphere to the velocity v ∼ 900 km/s. Later, the inner layers lose their momentum pushing the overlying layers, as well as due to the attraction of the underlying matter. On day 10 the system has stabilized: about M ∼ 2 M_⊙ halts and stays in a bound state, and about M_ ej∼ 1 M_⊙ is ejected and enteres the phase of free expansion. The behavior of the light curves can be monitored on the Rosseland opacity curve (Tau) in Figure <ref>.It is worth noting that Tau has been specially designed for those plots in order to allow the qualitative monitoring the location of the photosphere. Codesolves transport equations for the frequency grid from 1 Å up to 5· 10^4 Å, without imposing any restrictions on the shape of the distributionintensity and opacity in the cells of the frequency grid.One can clearly see that the investigated plateau stage, which is determined by the passage of a Cooling and Recombination Wave <cit.>, begins already in the first days after the explosion. In the right column of Figure <ref> the X-axis shows the Lagrangian coordinate M(R). It is evident that the CRW runs inside along the mass coordinate, reaching the innerstalled core at t ∼ 90^d. According to the radial Eulerian coordinate (left column of Figure <ref>) the CRW is permanently carried on by the expanding matter. Therefore, the photospheric radius grows initially, providing a maximum of the light curve at ∼ 25^d, and then decreases slowly, reproducing the slowdecline in the observed M31LRN bands. It is important for the model not only to be consistent with photometric data but also to have hydrodynamic properties similar to those actually observed. Figure <ref> shows the computed mass flow velocity at the photosphere level for the best-fit model . We have no detailed spectroscopic observations to superimpose onto our model computations. <cit.> report Hα data (FWHM = 900 ± 200 km/s), which provide an upper constraint for the photospheric velocity because the strong Hα line forms above the photosphere, where the mass expansion velocity is higher than the photospheric velocity. The photospheric velocity is better to estimate by weak lines. In this case the value inferred from the NaI D line (FWHM = 600 ± 200 km/s) agrees well with the model computations at the time of maximum light (fig. <ref>). Our computations show that the overall behavior of broad-band light curves (Fig. <ref>) can be reproduced fairly well by codenamed model . We performed our computations for a spherically symmetric configuration consisting of an envelope and an inner core where the components of the system merge. In this formulation the component merger is treated as a source of thermal energy without taking into account the actual physics of the process <cit.>. Without performing a full multidimensional computation we cannot claim that our inferred total mass of M_ tot =3 M_⊙ characterizes the mass of the binary before the merger. It is possible that in the case of a multidimensional computation the ejected mass and kinetic energy of the ejecta could be obtained with a less massive binary.Williams et al. report the absolute magnitude ofM_V=-1.5 for the pre-supernova <cit.>. Main-sequence stars of such luminosity have the mass of 6M_⊙. However, this estimate is not very reliable because it is unlikely that both merging objects are main-sequence stars. They are more probably red giants whose masses cannot be determined solely from their luminosity. §.§ Progenitor and evolution discussionOne of the key evolutionary stages of binary stars is the specific state when the sizes of stars become comparable to that of the entire binary system <cit.> It is a key stage because at that time violent mass exchange begins in the binary, resulting in dramatic and fast change of its parameters due to the exchange (and even loss) of the angular momentum and mass contained in the stars before the contact. The catastrophic change of binary parameters may result in complete merger of its component stars or in the abrupt decrease of the size of the binary semiaxis, and have a crucial effect on subsequent evolution stages that end with the formation of close binaries with degenerate components - neutron stars and white dwarfs. This is how white-dwarf close binaries form whose mergers may produce type Ia supernovas and, in the case of systems consisting of two neutron stars, short gamma-ray bursts. Hence mergers of two classical main-sequence stars or subgiants may contribute to the study and understanding of such important astrophysical phenomena as dark energy and gamma-ray bursts.In the paradigm of synchronously rotating stars with a high mass ratio moving in circular orbits the filling of the Roche lobe is followed by violent mass transfer by the primary to the secondary and the formation of a common envelope. On the other hand, an equal-mass binary may evolve into a fully contact system of two stars both of which simultaneously fill their Roche lobes. In the limiting case violent mass transfer may occur on the hydrodynamic time scale with the amount of released energy on the order of the thermal energy of the component stars ∼ GM_ tot^2/ a ∼ 8· 10^50· m^2 (a/R_⊙), where M_totis the total mass of colliding stars, and a is the semimajor axis of the binary.From this point of view it is very important to obtain observational evidence for such processes in binary systems and to determine the corresponding physical properties - the released energy, ejected mass, and angular momentum. Ultra-luminous red novae may be observational manifestations of violent mass transfer in the binary <cit.>.We already pointed out that the observation of the progenitor of V1309 Sco in the form of a contact binary with a period of 1.7 days <cit.> provides a strong evidence for the merger model of LRNe. This conclusion is also corroborated by the approximate occurrence rate of such events – one event in 20-40 years per galaxy like Milky Way or M31. To show this, let us use the distribution of binary semimajor axes a <cit.>.This distribution has the form dN∼ 0.2 d log (a/R_⊙). Systems with semi-axes6≲ a/R_⊙≲ 12 merge during the Hubble time. However, nova luminosities depend essentially on the mass of the ejected envelope, and the occurrence rate of LRNemay be several times lower, which is consistent with observations of the M31 galaxy, where the last such nova was observed in 1989.Or in another way, if all stars are close binaries then the maximum merger rate of systems with a total mass of 6 solar masses is equal to the birth rate of 3-6 M_⊙ stars, which in the case of the Salpeter mass function is equal to 1/10 year. Given that close binaries make up for about 20-30 percent of all stars, we obtain an estimate of 1/30 - 1/50 year. The last red nova in the Andromeda galaxy exploded 20 years ago, which is consistent with theoretical expectations§ CONCLUSIONSIn this paper we described the technique of the discovery of the nova M31LRN and long-term observations of its light curve with MASTER network of robotic telescopes. It is important that the entire observational part of the study was performed on identical telescopes equipped with identical photometers. The resulting light curve agrees fairly well with the independent light curve published by <cit.>. However, our interpretation led us to infer a relatively higher total progenitor mass. The rather long plateau ( 50 days) requires a higher merged stellar mass ( 3 solar masses).The corresponding explosion energy should be lower 2.5· 10^50 erg, whereas the total kinetic energy of the ejected envelope is lower by three orders of magnitude. The proposed interpretation of the explosion is consistent with the proposed evolutionary scenario where star merger is a natural stage of the evolution of close-mass stars and may serve as an extra channel for the formation of nova outbursts. § ACKNOWLEDGMENTSMASTER Global Robotic Net is supported in part by theDevelopment Programm of Lomonosov Moscow State University. This work was also supported in part by the RFBR grant 15-02-07875 (discovery and observations), by Russian Science Foundation grant 16-12-00085 (interpretation and data analysis) and grant 16-12-10519 (theoretical modelling of the Nova done by P.B). Grant no. IZ73Z0-152485 SCOPES Swiss National Science Foundation supports work of S.B. mnras | http://arxiv.org/abs/1704.08178v1 | {
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"V. M. Lipunov",
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"E. Gorbovskoy",
"A. Tutukov",
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"V. Krushinski",
"N. Tiurina",
"P. Balanutsa",
"A. Kuznetsov",
"V. Kornilov",
"I. Gorbunov",
"V. Shumkov",
"V. Vladimirov",
"O. Gress",
"N. M. Budnev",
"K. Ivanov",
"A. Tlatov",
"I. Zalozhnykh",
"Yu. Sergienko",
"A. Gabovich",
"V. Yurkov"
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"published": "20170426160751",
"title": "MASTER OT J004207.99+405501.1/M31LRN 2015 Luminous Red Nova in M31: Discovery, Light Curve, Hydrodynamics, Evolution"
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http://arxiv.org/abs/1704.08612v1 | {
"authors": [
"Tetsuya Takaishi"
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"published": "20170427150130",
"title": "Dynamical Analysis of Stock Market Instability by Cross-correlation Matrix"
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Department of Mathematics, University of Illinois, Urbana, IL 61821 USA [email protected] We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems in Goncharov-Kenyon 2013. The dynamic on the graph is an urban renewal together with shrinking all 2-valent vertices, while it is a cluster transformation on the weight. The graph is not necessary obtained from an integral polygon. We show that all Hamiltonians, partition functions of all weighted perfect matchings with a common homology class, are invariant under a move on the weighted graph. This move coincides with a cluster mutation, analog to Y-seed mutation in dimer integrable systems. We construct graphs for Q-systems of type A and B and show that the Hamiltonians are conserved quantities of the systems. The conserved quantities can be written as partition functions of hard particles on a certain graph. For type A, they Poisson commute under a nondegenerate Poisson bracket.Conserved quantities of Q-systems from dimer integrable systems Panupong Vichitkunakorn December 30, 2023 =============================================================== § INTRODUCTION The dimer model is an important model and has a long history in statistical mechanics <cit.>. It is a study of perfect matchings of a bipartite graph, a collection of edges in which each vertex of the graph is incident to exactly one edge. Cluster algebras introduced in <cit.> are commutative algebras equipped with a distinguished set of generators called cluster variables. There is a transformation called mutation which creates a new set of generators from an old one. This transformation allows us to consider a dynamical system inside the cluster algebra. One important feature of cluster dynamics is the Laurent property, namely any cluster variables can be expanded as a Laurent polynomial on initial cluster variables. In many cases, this expansion can be written as a partition function of dimer configurations over a certain graph, see for examples <cit.>.As a discrete dynamical system, many cluster dynamics have been shown to be Liouville-Arnold integrable <cit.>.In <cit.>, dimer models are used to construct a class of integrable systems enumerated by integral polygons.The dimer integrable system introduced in <cit.> is a continuous Liouville-Arnold integrable system whose phase space is the space of line bundles with connections on a bipartite torus graph G obtained from an integral convex polygon. Let n be the number of faces of G. The phase space can be combinatorially viewed as the space of weights on oriented loops of G compatible with loop multiplication. The space of oriented loops is generated by all counterclockwise loops around each face W_i (i∈{1,…,n}) and two extra loops Z_1,Z_2 whose homology classes on the torusgenerate H_1(,)≅×. The condition ∏_i W_i = 1 is the only nontrivial relation among these generators. So the phase space is generated by the weight assignment w_i (i∈{1,…,n}) together with z_1,z_2 on all the loops W_i and Z_1,Z_2. The only condition among these weights is ∏_i w_i = 1.The phase space is equipped with a Poisson bracket defined from the intersection pairing on the twisted ribbon graph obtained from G. A Y-seed <cit.> of rank n+2 indexed by the loop generators is assigned to the weighted graph G where an entry of the exchange matrix is the intersection pairing of the generators and y-variables are their weights.Hamiltonians are defined on the phase space as Laurent polynomials on the variables {w_1,…,w_n,z_1,z_2}. For (a,b)∈×, a Hamiltonian is written as the partition function over weighted perfect matchings on G where the exponent of z_1 and z_2 in their weights is a and b, respectively. The Hamiltonians vanish for all but finitely many (a,b). There is a move called urban renewal on the weighted graph G, which acts on its corresponding Y-seed as a Y-seed mutation. This transformation is a change of coordinates on the phase space, and the Hamiltonians are invariant under the transformation. By thinking of this change of coordinates as a map on the phase space, it becomes an evolution of a discrete dynamical system. The evolution can also be written as a Y-seed mutation. Various discrete dynamical systems has been studied in this framwork <cit.>.The first goal of this paper is to rethink the discrete dimer integrable system of <cit.> in cluster variable setting and extend it to bipartite torus graphs not necessary obtained from integral polygons. We study, in Sections <ref> and <ref>, a system associated with a general bipartite torus graph not necessary be obtained from an integral polygon. Recall that the loop weights in <cit.> are the y-variables in the associated Y-seed. In our study, we instead associate weights that act as cluster variables in the associated cluster seed. We show that the Hamiltonians are invariant under the evolution, see Theorem <ref>. Q-systems first appeared in an analysis of Bethe ansatz of generalized Heisenberg spin chains. They were first introduced as sets of recurrence relations on commuting variables for the classical algebras <cit.> and later generalized for exceptional algebras <cit.>, twisted quantum affine algebras <cit.> and double affine algebras <cit.>. See <cit.> for a review on the subject.They can also be normalized and then realized as mutations on cluster variables <cit.>. Explicit conserved quantities for Q-systems of type A have been studied in <cit.> as partition functions of hard particles on a graph and in <cit.> as partition functions of weighted domino tilings on a cylinder. In <cit.> the Q-system of type A_r is identified with the T-system of type A_r⊗Â_1.For simply-laced finite type and twisted affine type, conserved quantities arise by identifying the systems with the dynamics of factorization mappings on quotients of double Bruhat cells <cit.>.The second goal of the paper is to use perfect matchings on graphs to compute conserved quantities of Q-systems associated with a finite Dynkin diagram of type A and B in Section <ref> and <ref>, respectively. The conserved quantities of type A coincide with the partition functions obtained in <cit.> and <cit.>.The paper is organized as follows. In Section <ref>, we review some definitions and results on Q-systems and cluster algebras. In Section <ref>, we introduce weighted bipartite torus graphs, weight of perfect matchings and weight of loops on the graph. A move on weighted graphs is defined. In Section <ref>, we define Hamiltonians and show that they are invariant under the move. In Section <ref> (resp. Section <ref>), a graph for A_r (resp. B_r) Q-system is constructed. The Hamiltonians are shown to be conserved quantities of the system. They can also be written as partition functions of hard particles on a certain graph. A nondegenerate Poisson bracket is constructed. The conserved quantities Poisson-commute in type A and are conjectured to Poisson-commute in type B. Throughout the paper we denote[x]_+ :=x, x≥ 0, 0, x<0 . For m,n∈ where m≤ n, let [m,n] := {m,m+1,…,n}. §.§ AcknowledgementsThe author would like to thank his advisors R. Kedem and P. Di Francesco for their helpful advice and comments. This work was partially supported in part by Gertrude and Morris Fine foundation, NSF grants DMS-1404988, DMS-1301636, DMS-1643027 and the Institut Henri Poincaré for hospitality.§ Q-SYSTEMS AND CLUSTER ALGEBRAS In this section we review the definition of Q-systems for simple Lie algebras and an interpretation as cluster mutations.§.§ Q-systemsWe use a normalized version of Q-systems studied in <cit.>, and we will use it as the definition of the Q-systems.Letbe a simple Lie algebra with Cartan matrix C. We denote a simple rootby its corresponding integer in [1,r]. The Q-system associated withis defined to be the following recurrence relation on a set of variables { Q_,k|∈ [1,r],k∈}:Q_,k+1 Q_,k-1 = Q_,k^2 + ∏_∼̱∏_i=0^|C_,|-1 Q_,̱⌊t_ḵ+i/t_⌋,where ⌊ t ⌋ denotes the integer part of t, and t_ are the integers which symmetrize the Cartan matrix. That is, t_r=2 for B_r, t_ = 2(<r) for C_r, t_3=t_4=2 for F_4, t_2=3 for G_2, and t_=1 otherwise.The recursions (<ref>) for type A and B read:A_r:Q_,k+1Q_,k-1 = Q_,k^2+Q_+1,kQ_-1,k (=1,…,r), B_r:{ Q_,k+1Q_,k-1= Q_,k^2 + Q_+1,kQ_-1,k (=1,…,r-2),Q_r-1,k+1Q_r-1,k-1 = Q_r-1,k^2 + Q_r-2,kQ_r,2k,Q_r,k+1Q_r,k-1 = Q_r,k^2 + Q_r-1,⌊k/2⌋Q_r-1,⌊k+1/2⌋, .with boundary conditions Q_0,k=Q_r+1,k=1 for k∈.Given a valid set of initial values {Q_,0 = q__,0, Q__,1 = q_,1|∈ [1,r]} for q__,0,q_,1∈^*, we can solve for Q_,k which satisfies the Q-system for any ∈[1,r] and k∈ in terms of the initial values. So Q-systems can be interpreted as discrete dynamical systems where the phase space is =(^*)^2r and the (forward) evolution is(q_1,k ,…,q_r-1,k,q_r,t_r k,q_1,k+1,…,q_r-1,k+1,q_r,t_r k+1) ↦ (q_1,k+1,…,q_r-1,k+1,q_r,t_r k+t_r ,q_1,k+2,…,q_r-1,k+2,q_r,t_r k+t_r+1)where (q_1,0,…,q_r,0,q_1,1,…,q_r,1) is the initial state.A conserved quantity of a discrete dynamical system is a function H:→ on the phase spacewhich is invariant under the evolution of the system. That isH(x_1,k,…,x_n,k) = H(x_1,k+1,…,x_n,k+1)for all (x_1,k,…,x_n,k) ∈.The goal of this paper is to compute conserved quantities of the Q-systems of type A and B.§.§ Cluster AlgebrasWe review some basic definitions in cluster algebras from <cit.> and results of <cit.> about the formulations of Q-systems as cluster mutations for simple Lie algebras.A cluster seed (resp. Y-seed) of rank n is a tuple Σ = (𝐀,B) (resp. (𝐲,B)) consisting of: * An n× n skew-symmetrizable matrix B, called an exchange matrix. * A sequence of variables 𝐀=(A_i)_i=1^n (resp. 𝐲=(y_i)_i=1^n), called cluster variables (resp. y-variables). When the exchange matrix B is skew-symmetric, the quiver associated with B, is defined to be a directed graph without 1-cycles or 2-cycleswhose signed adjacency matrix is B. In this case, we can also write a cluster seed (resp. Y-seed) as (𝐀,) (resp.(𝐲,)).The mutation at k on the exchange matrix can be translated to a rule of quiver mutation at vertex k consists of the following steps: * Reverse all the arrows incident to k.* For each pair of incoming arrow from i to k and outgoing arrow from k to j, we add an arrow from i to j.* Remove all resulting oriented 2-cycles one by one. Consider the following quiver with five vertices labeled by k,k_1,…,k_4. A quiver mutation at vertex k can described as follows.< g r a p h i c s >In this paper, we will only work on mutations at a vertex having exactly two incoming and two outgoing arrows. So the change of arrows will be described as in Example <ref>. For any k∈ [1,n], the mutation (resp. Y-seed mutation) in direction k sends (𝐀,B) to (μ_k(𝐀),μ_k(B)) = (𝐀',B') (resp. sends (𝐲,B) to (μ_k(𝐲),μ_k(B))=(𝐲',B'))where: * B'_ij = -B_ij, i=korj=k, B_ij+1/2(|B_ik|B_kj+B_ik|B_kj|),otherwise. * A'_i = A_i, i≠ k, A_k^-1( ∏_B_jk>0A_j^B_jk + ∏_B_jk<0A_j^-B_jk), i=k. * y'_i = y_i^-1, i = k, y_i y_k ^[B_ki]_+ (1+y_k)^-B_ki, i ≠ k. Let (𝐀,B) be a cluster seed. There is a map τ sending (𝐀,B) to a Y-seed (𝐲,B) wherey_j := ∏_i = 1^n A_i^B_ijfor j∈[1,n].The map τ commutes with the mutations <cit.>:τ(μ_k(Σ)) = μ_k(τ(Σ))where the mutation on the left of the equation is a cluster mutation, while the mutation on the right is a Y-seed mutation.Let C be the Cartan matrix of an underlying simple Lie algebra. The Q-system relation (<ref>) can be realized as cluster mutations. There is a sequence of mutations such that every Q-system variable appears as a cluster variable. The sequence of mutations in the theorem is explicitly described in terms of the root system of the underling Lie algebra. We translate it into a sequence of mutation together with relabeling of indices as follows.Let C be a Cartan matrix of rank r, we let Σ_k := (𝐀_k , B) be cluster seeds of rank 2r whereB = [ C-C^T C^T;-C 0 ] .The cluster tuple 𝐀_k consists of Q-system variables. There exists a sequence of mutations μ and a permutation σ∈𝔖_2r connecting the seeds as the following.⋯σμ⟶Σ_0 σμ⟶Σ_1 σμ⟶Σ_2 σμ⟶⋯ The cluster tuple 𝐀_k, the permutation σ and the sequence of mutations μ are defined according to the type of the Cartan matrix C. The following are their definitions for type A and B. * For type A_r we haveμ :=∏_i=1^r μ_i,A_i,k :=Q_i,k, i∈[1,r],Q_i-r,k+1, i∈[r+1,2r],σ(i) := i+r2r. * For type B_r we have μ := μ_2r(∏_i=1^r-1μ_i) μ_r,A_i,k :=Q_i,k, i∈[1,r-1],Q_r,2k, i = r, Q_i-r,k+1, i∈[r+1,2r-1], Q_r,2k+1, i=2r,σ(i): =i, i = r or 2r, i+r2r,otherwise.The quivers associated to the matrices B for type A and B are shown in Figure <ref>. We also note that the mutations μ_i in the product ∏_i μ_i in equations (<ref>) and (<ref>) commute, so the product makes sense.§ WEIGHTED BIPARTITE TORUS GRAPHSWe think of a torus = ^2 / ^2 as a rectangle with opposite edges identified. A torus graph is a graph embedded on the torus with no crossing edges. We do not require that every face is contractible. A weighted graph (,𝐀) is a pair of a graphwith n faces and a collection 𝐀=(A_i)_i=1^n of variables or nonzero complex numbers called weights. A bipartite graph is a graph whose vertices can be colored into two colors (black and white) such that every edge connects two vertices of different colors. Throughout the paper, we letbe a bipartite torus graph with n faces. We label the faces by the numbers 1 to n, so F() = [1,n]. §.§ Quivers associated with graphs and mutationsFor a bipartite torus graph , we let _ be the quiver associated withdefined as follows.The nodes of _ are indexed by the faces of .There will be an arrow between node i and node j for each edge adjacent to face i and face j. The arrow is oriented in the way that the black vertex ofis on the right of the arrow, see the following figure.< g r a p h i c s >Lastly, any 2-cycles are removed one by one until the directed graph has no 2-cycles. We also let B_ denote the signed adjacency matrix of _. Note that from the construction the resulting directed graph can possibly contain 1-cycles.To a weighted bipartite torus graph (,𝐀) whose _ has no 1-cycles, we can associate a cluster seed (𝐀,_) of rank n, see Definition <ref>.We then define two moves on weighted bipartite graphs.An urban renewal <cit.>(a.k.a. spider move <cit.>, square move <cit.>) at a quadrilateral face k whose four sides are distinct sends (,𝐀) to (',𝐀') as follows. * The graph ' is obtained fromby replacing the subgraph ofcontaining four edges around the face k with a graph described in the following picture. The labels are face indices. The four outer vertices connect to the rest of the graph. < g r a p h i c s > * The weight 𝐀' =(A'_i) = μ_k(𝐀) are transformed according to the cluster transformation in direction k (Definition <ref>). That is, A'_ℓ = (A_i A_j + A_m A_n)/A_k,ℓ=k,A_ℓ,ℓ≠ k. A shrinking of a 2-valent vertex sends (,𝐀) to (',𝐀) where ' is obtained fromby removing a 2-valent vertex and identifies its two adjacent vertices, while the weights of the graph stay unchanged. It can be visualized in the following picture. (We also have another version of the move when the colors of the vertices are switched.)< g r a p h i c s >Let k be a quadrilateral face of a weighted bipartite graph (,𝐀). A mutation at face k, μ_k, is a combination of an urban renewal at face k and shrinking of all 2-valent vertices. Two weighted graphs are mutation equivalent if one is obtained from another by a sequence of mutations. Let (,𝐀) be a weighted bipartite graph whose _ has no 1-cycles, k be a quadrilateral face of . If every pair of adjacent faces of k are distinct except possibly pairs of opposite faces, then the mutation μ_k on (,𝐀) is equivalent to a cluster mutation μ_k on a seed (𝐀, _). In particular, _μ_k() = μ_k(_) and the weight onare transformed according to the cluster transformation with respect to the quiver _.Consider the following picture where k_i are adjacent faces of k.< g r a p h i c s >Since k_i ≠ k_i+1 for all i (reading modulo 4), the vertex k of _ has exactly two incoming and two outgoing arrows. (There is no cancellation of arrows incident to k.) The arrows of _ are then transformed according to the rule of quiver mutations. Also, the mutation does not introduce any 1-cycles because k_i and k_i+1 are distinct. This also shows that the weights on the graph are transformed according the cluster mutation:A_k' = (A_k_1A_k_3 + A_k_2A_k_4)/ A_k. Shrinking of a 2-valent vertex corresponds to removing a pair of arrows with opposite orientations that may appear between k_i and k_i+1. This also follows the rule of quiver mutations.< g r a p h i c s >So we can conclude that _μ_k() = μ_k(_). The following is an example of graph mutations.< g r a p h i c s >We consider the first mutation μ_1. Using the notations in the proof of Theorem <ref>, when k = 1, we have (k_1,k_2,k_3,k_4) = (3,4,3,4). We see that k_i ≠ k_i+1 for all i∈[1,4]. So the mutation μ_1 on the weighted graph is equivalent to the mutation μ_1 on the corresponding cluster seed. Similarly, we can see that μ_2 satisfies the requirement in Theorem <ref>.The corresponding mutations of quivers are shown as follows.< g r a p h i c s > The weight changes as follows.(A_1,A_2,A_3,A_4) μ_1⟶ (A'_1,A_2,A_3,A_4)μ_2⟶ (A'_1,A'_2,A_3,A_4)where A'_1 = (A_4^2+A_3^2)/A_1 and A'_2 = (A_3^2+A_4^2)/A_2.§.§ Perfect matchings and oriented loops A perfect matching of a graphis a subset M⊆ E() of the set of all edges in which every vertex ofis incident to exactly one edge in M. Let M be a perfect matching of a weighted graph (,𝐀). The weight of M is defined byw(M) := ∏_e∈ Mw(e).The contribution w(e) is defined by w(e) := (A_i A_j)^-1 when edge e is adjacent to the faces i and j. Let M,M' be perfect matchings of . Let [M] denote the collection of all edges in M oriented from black to white. Similarly, -[M] is the collection of edges in M oriented from white to black.We then define [M]-[M'] to be the superimposition of [M] and -[M'] with all double edges having opposite orientations removed. In other word, [M]-[M'] is the set of edges (M∖ M')∪ (M'∖ M) where an edge is oriented from black to white (resp. white to black) if it is in M (resp. M'). It will be proved in Proposition <ref> that [M]-[M'] is a loop on(possibly contains more than one connected component).Let L_1,…,L_m be loops on . The product of loops ∏_i=1^m L_i is the superimposition of L_1,…,L_m with edges having opposite orientation removed one by one. It is clear that the homology class of the product of loops is the sum of their homology classes.Let M and M' be perfect matchings of a bipartite torus graph . Then [M]-[M'] is a product of non-intersecting simple loops on . Since M and M' are both perfect matchings of , any vertex v inis incident to exactly one edge in M and exactly one edge in M' (possibly distinct edges). If the two edges are the same, v has no incident edge in [M]-[M']. If the two edges are different, v has exactly two incident edges in [M]-[M'] with different orientation. (One is oriented from black to white, while the other is from white to black.) Hence [M]-[M'] is a product of non-intersecting oriented simple loops on .Sinceis a torus graph, a loop oncan be embedded on the torus. Using the identification H_1(,)≅×, we let a horizontal loop going from left to right have homology class (1,0), and let a vertical loop going from bottom to top have homology class (0,1).Let M,M' be perfect matchings of . Let [M]_M'∈ H_1(,) denote the homology class of [M]-[M'] on the torus.For an oriented path ρ on a weighted graph (,𝐀), we define the weight of ρ to bew(ρ) := ∏_e w(e)where the product runs over all directed edges in ρ. Let the edge e be adjacent to faces i and j. The contribution from e is w(e):=(A_iA_j)^-1 when e is oriented from black to white, while w(e):=A_iA_j when e is oriented from white to black. See the following figures.< g r a p h i c s >Comparing to the weight of perfect matchings, we notice thatw([M]-[M']) = w(M)/w(M')for any perfect matchings M,M' of .Letbe a weighted bipartite torus graph whose weight is shown at each face in the following picture. The perfect matchings M and M' are depicted below.< g r a p h i c s >We have w(M) = a^-2c^-2 and w(M')= (abcd)^-1. The loop [M]-[M]' has weight w([M]-[M']) = bd/ac = w(M)/w(M'). Lastly, [M]_M'=(0,1).§ HAMILTONIANSIn this section, we define Hamiltonians on a weighted graph (,𝐀) with respect to a perfect matching M_0, and show that they are invariant under certain conditions. They will be used to compute conserved quantities of Q-systems of type A in Section <ref> and type B in Section <ref>.§.§ Hamiltonians We modify the Hamiltonians defined in <cit.> and use the weight of perfect matchings induced from the weight ofin Definition <ref>.Let (,𝐀) be a weighted torus graph, M_0 be a perfect matching of , (i,j)∈ H_1(,)≅×. The Hamiltonian of (,𝐀) with respect to (i,j) and M_0 is a Laurent polynomial in {A_i}_i=1^n defined byH_(i,j),,M_0(A_1,…,A_n) := ∑_M:[M]_M_0=(i,j) w(M)/w(M_0),where the sum runs over all perfect matchings M ofsuch that the homology of [M]-[M_0] is (i,j). The weight w is defined as in Definition <ref>. We say that M_0 is a reference perfect matching.Let k be a quadrilateral face of G. Let G' be a graph obtained from G by an urban renewal at k. Let M be a perfect matching of G containing exactly one side of k. We say that a perfect matching M' of G' is induced from M by an urban renewal if M' coincides with M on all edges of G not related to the urban renewal, and on the subgraph replaced by the urban renewal M and M' are related as in Figure <ref>.Let G' be a graph obtained from G by shrinking of a 2-valent vertex. We say that a perfect matching M' of G' is induced from M by shrinking of a 2-valent vertex if M' coincides with M on all edges of G not removed by the move. The perfect matching M' can be described in Figure <ref>.Let μ_k() is a graph obtained fromby a mutation at k. The induced perfect matching from M by a mutation at k is the perfect matching μ_k(M) induced from M by an urban renewal at k and then by shrinking of all 2-valent vertices.Let (,𝐀) be a weighted bipartite torus graph with n faces, (μ_k(G),𝐀') be obtained from (,𝐀) by a mutation at a contractible quadrilateral face k of . Let M_0 be a perfect matching ofcontaining exactly one side of k,μ_k(M_0) be induced from M_0 by the mutation.Then the Hamiltonians are invariant under the mutation:H_(i,j),,M_0(A_1,…,A_n) = H_(i,j),μ_k(G),μ_k(M_0)(A_1',…,A_n ').It suffices to show that the Hamiltonians are invariant under an urban renewal and shrinking of a 2-valent vertex.Consider shrinking of a 2-valent vertex. Let ' be obtained fromby shrinking a 2-valent vertex v. Let assume first that the 2-valent vertex is white.The other case can be treated in a similar manner. Let v be adjacent to faces i,j and edges e_1,e_2. Consider a perfect matching M of G. Since M is a perfect matching, exactly one of e_1, e_2 must be in M. Assume without loss of generality that e_1∈ M. From the following picture, there is a unique perfect matching M' of G' which is identical to M except on e_1 and e_2. Similarly, any perfect matching M' of G' has a unique perfect matching of G such that they agree on E(G'). Hence this map is a bijection between the perfect matchings of G and of G'. < g r a p h i c s >We also have w(M) = w(e_i) w(M') = (A_i A_j) w(M').Similarly, for the reference perfect matching M_0 of , we havew(M_0) = (A_i A_j) w(M_0')regardless of whether M_0 contains e_1 or e_2. Hence we have w(M)/w(M_0) = w(M')/w(M_0').Since the move does not change the homology class of a perfect matchingi.e. [M]_M_0 = [M']_M_0', from (<ref>) we have that the Hamiltonians are invariant under the move:H_(i,j),G,M_0(A_1,…,A_n) = H_(i,j),G',M_0'(A_1,…,A_n). For an urban renewal, let (G',𝐀') be obtained from (,𝐀) by an urban renewal at the face k. Let M_0 (resp. M_0') be the reference perfect matching of(resp. G'). There are 4 involved edges before the move and 8 involved edges after the move. Let x∈{A_i}_i=1^n be the weight at the face k of , a,b,c,d∈{A_i}_i=1^n be weights at the four adjacent faces, and x'∈{A'_i}_i=1^n be a weight at the face k of '. We then havex x' = ac+bd.Without loss of generality, we assume that the edge adjacent to the faces whose weights are x and c is in M_0. The following picture show M_0 and M_0' on subgraph involved in the urban renewal.-0.45 < g r a p h i c s > . Now we write H_(i,j),,M_0 = H_0 + H_1 + H_2 where H_0 consists of all the contributions from perfect matchings containing no edges from the four sides of the face k, H_1 consists of the contributions from matchings containing one such edge, and H_2 consists of the contributions from matchings containing two such edges. Similarly, we write H_(i,j),G',M_0' = H'_0 + H'_1 + H'_2. We claim that H_0 = H'_2, H_1 = H'_1 and H_2 = H'_0.In order to show H_1 = H'_1, we define a bijection between the matchings contributing terms to H_1 and the matchings contributing terms to H'_1. The bijection maps a matching M in H_1 to another matching M' in H'_1 which differs from M only on the edges involving in the urban renewal. The bijection can be described as follows. < g r a p h i c s >Since M and M' differ only on edges involved in the urban renewal, their contribution of the edges other than such edges near the face k to w(M)/w(M_0) and w(M')/w(M_0') are the same. Now we consider the contributions from the edges involved in the urban renewal. Recalling the weights in (<ref>), in the first case, the contribution of such edges to w(M)/w(M_0) is xc/xa=c/a, and the contribution to w(M')/w(M_0') is (x'abc^2d)/(x'a^2bcd) = c/a. Similarly, the contributions in the second case (resp. the third and the fourth case) from before and after the move are the same and equal to c/b (resp. 1 and c/d). Hence H_1 = H'_1.In order to show H_0 = H'_2, we define a one-to-two map from the matchings contributing terms to H_0 to the matchings contributing terms to H'_2. The map can be described as the following. < g r a p h i c s > Let M be a matching ofcontributing to a term in H_0, M' and M” be the corresponding matchings of ' from the one-to-two map. The contribution from the four edges around the face k ofto w(M)/w(M_0) is xc. The contribution to w(M')/w(M_0) (resp. w(M”)/w(M_0)) from the eight edges of ' resulting from the move is x'abc^2d/(x')^2bd = ac^2/x' (resp. x'abc^2d/(x')^2ac = bcd/x'). From (<ref>), xc = ac^2/x'+bcd/x'. Hence H_0 = H'_2.In order to show H_2 = H'_0, we define a two-to-one map from the matchings contributing terms to H_2 to the matchings contributing terms to H'_0 as the following. < g r a p h i c s > The contribution from the four edges around the face k ofisxc/x^2bd + xc/x^2ac = c/abcdac+bd/x,and the contribution from the eight edges of ' isx'abc^2d/a^2b^2c^2d^2 = c/abcdx'.From (<ref>), they are equal. Hence H_2 = H'_0. This conclude that H_(i,j),,M_0 = H_(i,j),',M_0'.§ Q-SYSTEMS OF TYPE AIn this section, we apply Theorem <ref> to construct conserved quantities for A_r Q-systems, and show that they coincide with the quantities shown in <cit.>. We also give a Poisson structure to the phase space and show that the conserved quantities mutually Poisson-commute. §.§ Q-systems of type A and weighted graph mutations Consider the following quiver of an A_r Q-system. See Theorem <ref> for the detail.< g r a p h i c s > We recall that Theorem <ref> requires every mutation to happen at a quadrilateral face. This means every quiver mutation in the sequence μ in(<ref>) has to be at a vertex with exactly two incoming and two outgoing arrows. In order to archive this, we add another vertex labeled by 0 which will not be mutated, called frozen vertex, and assign a value of 1 as its cluster variable. According to the formula of a cluster mutation in Definition <ref>, adding a frozen vertex with value 1 does not effect the system. The following is the resulting quiver . The vertex labeled by 0 on the left and on the right are identified. So it has four incident arrows.< g r a p h i c s > The bipartite torus graphassociated withis depicted below. Since every vertex ofis of degree 4, every face ofhas 4 sides. We note that the face 0 is not contractible. It is indeed homotopy equivalent to a cylinder.< g r a p h i c s > Let M_0 be the perfect matching ofcontaining all vertical edges whose top vertex is black. It can be depicted as the following.< g r a p h i c s >We also let the weight at face i be A_i.Let (,(A_i)) be a weighted bipartite torus graph defined above. Then the Hamiltonians H_(i,j),,M_0(A_1,…,A_2r) are conserved quantities of the A_r Q-system dynamic Q_,k↦ Q_,k+1.We want to showH_(i,j),,M_0 (Q_1,k,…,Q_r,k,Q_1,k+1,…,Q_r,k+1)= H_(i,j),,M_0(Q_1,k+1,…,Q_r,k+1,Q_1,k+2,…,Q_r,k+2)for k∈.Consider (,(A_i,k)) where A_i,k are defined in equation (<ref>). By Theorem <ref>, there is a sequence of mutation μ =μ_r⋯μ_1 and a relabeling σ: i ↦ i+r2r such that σμ sendsto , and the Q-system variables are shifted by n→ n+1. We first claim that the conditions in Theorem <ref> and Theorem <ref> hold along the sequence of mutations μ: * Each mutation happens at a contractible quadrilateral face.* The mutating face contains exactly one edges in the induced reference perfect matching from M_0.* Two adjacent faces of the mutating face are distinct, except possibly when they are opposite faces.Consider the first mutation μ_1. It is clear that face 1 is a contractible quadrilateral face and contains exactly one edge of M_0. Although its adjacent faces are not distinct, the non-distinct faces are opposite. So the conditions hold. The following graph on the left is the resulting graph after an urban renewal at face 1. The graph on the right is the result after shrinking all 2-valent vertices, hence it is μ_1(). The induced reference perfect matching from M_0 are shown on the graphs. The weight is now (A_1,k+2,A_2,k,…,A_2r,k).< g r a p h i c s > We continue to the mutation μ_2. Again, we see that all the conditions are satisfied. The following graphs are the result after an urban renewal at face 2 and shrinking all 2-valent vertices. The graph μ_2(μ_1(G)) is shown on the right with the weight(A_1,k+2,A_2,k+2,A_3,k,…,A_2r,k). < g r a p h i c s > Now it is easy to see that all the conditions hold at every mutation. After applying every mutation along the sequence μ_1,μ_2,…,μ_r, we have the following graph μ() together with the induced reference perfect matching μ(M_0) with weight(A_1,k+2,…,A_r,k+2,A_r+1,k,…,A_2r,k) = (Q_1,k+2,…,Q_r,k+2,Q_1,k+1,…,Q_r,k+1). < g r a p h i c s >We then continue on to the relabeling σ: i ↦ i+r2r. This gives the following graph on the left.By vertical translation, we obtain the graph on the right. The weight after relabeling is (Q_1,k+1,…,Q_r,k+1,Q_1,k+2,…,Q_r,k+2).< g r a p h i c s > We obtain back the original graphand reference perfect matching M_0, while the weight is shifted from k to k+1. Since ,σμ() = and σμ(M_0) = M_0, we haveH_(i,j),σμ(),σμ(M_0) = H_(i,j),,M_0.By Theorem <ref> and Theorem <ref>, we getH_(i,j),,M_0 (Q_1,k,…,Q_r,k,Q_1,k+1,…,Q_r,k+1) = H_(i,j),,M_0(Q_1,k+1,…,Q_r,k+1,Q_1,k+2,…,Q_r,k+2).Hence H_(i,j),,M_0 are conserved quantities of the A_r Q-system. §.§ Partition function of hard particles In <cit.>, conserved quantities for A_r Q-system are shown to be partition functions of hard particles on a certain weighted graph. We will show that they coincide with the Hamiltonians H_(i,j),,M_0 computed in the previous section.From Proposition <ref>, [M]-[M_0] is always a product of non-intersecting simple loops of . Given the reference perfect matching M_0 defined in the previous section, we then try to find such simple loops Γ_i on . We modify the construction in <cit.> and define Γ_i as the following.For i=1,2,…,2r+1, we define Γ_i to be the following loops.< g r a p h i c s >Let γ_i := w(Γ_i) be the weight of Γ_i on , see Definition <ref>. We note that when A_i = A_i,k defined in equation (<ref>), we get the same expressions as in <cit.>:γ_2α-1 =Q_α-1,kQ_α,k+1/Q_α,kQ_α-1,k+1, γ_2α = Q_α-1,kQ_α+1,k+1/Q_α,kQ_α,k+1,where we assume that Q_0,k:=1 and Q_r+1,k:=1 for all k∈. Let M be a perfect matching of .Then [M]-[M_0] is a nonintersecting collection of Γ_i's. Furthermore, every nonintersecting collection of Γ_i's is [M]-[M_0] for a unique perfect matching M of . Consider all possible local pictures at a white vertex v of . Since M (resp. M_0) is a perfect matching of , there is exactly one edge in M (resp. M_0) which touches the vertex v. If the two edges (one from M and one from M_0) are the same, no loop passes through v. If they are different, we have the following possibilities:< g r a p h i c s >Similarly, we have the following possibilities for a white vertex.< g r a p h i c s >From all the possibilities at black and white vertices, [M]-[M_0] consists of nonintersecting loops Γ_i's.On the other hand, let E be the edge set of a nonintersecting collection of Γ_i's. Notice from the pictures in Definition <ref> that if there is a loop passes through a vertex v, one of the two incident edges belongs to M_0. Then M = (E∖ M_0) ∪ (M_0∖ E)is the unique perfect matching such that [M]-[M_0] = ∏_i∈ IΓ_i.Let G_r be the following graph with 2r+1 vertices indexed by the index set [1,2r+1].< g r a p h i c s >It is easy to see that Γ_i intersects Γ_j if and only if the vertices i and j are connected in G_r. So for any subset I⊆ [1,2r+1], the loops in {Γ_i | i∈ I} are pairwise disjoint if and only if I is a subset of pairwise nonadjacent vertices of G_r, a.k.a. a hard particle configuration on G_r. Hence, nonintersecting collections of Γ_i's are parametrized by subsets of pairwise nonadjacent vertices of G_r. Also, nonintersecting collections of Γ_i's of size n are in bijection with n-subsets of pairwise nonadjacent vertices of G_r.By Theorem <ref>, the possible homology classes of [M]-[M_0] are (0,k) for k∈[0,r+1]. Let H_k := H_(0,k),,M_0.It is easy to see that H_0 = H_r+1 = 1. By an interpretation of [M]-[M_0] as a hard particle configuration on G_r, we get the following theorem and conclude that the Hamiltonians coincide with the conserved quantities computed in <cit.>.Let k∈[1,r]. ThenH_k(A_1,…,A_2r) = ∑_|I|=k∏_i∈ Iγ_iwhere the sum runs over all k-subsets I of pairwise nonadjacent vertices of G_r.In <cit.> the Q-system of type A_r is identified with the T-system of type A_r⊗Â_1. Our Hamiltonians coincide with the “Goncharov-Kenyon Hamiltonians" in <cit.>, which defined in terms of domino tilings on a cylinder. Our choice of a reference perfect matching corresponds to the minimal-height domino tiling. Our homology class (0,k) corresponds to k “hula hoops" (noncontractible cycles) that appear in a “double dimer model" between a domino tiling and the minimal domino tiling.§.§ Poisson-commutationLet C be the Cartan matrix of type A_r. The signed adjacency matrix of the quiver of A_r Q-system isB = [ C-C^T C^T;-C 0 ] = [0C; -C0 ].Recall that the phase spacehas coordinates (A_1,…,A_2r). We then define a Poisson bracket on the algebra 𝒪() of functions onby{ A_i , A_j } = Ω_ij A_i A_j (i,j∈ [1,2r])where the coefficient matrix Ω is defined byΩ = (B^T)^-1 = -B^-1 = [0 -C;C0 ]^-1 = [ 0C^-1; -C^-1 0 ]. Using this Poisson bracket, the bracket {γ_i, γ_j} is in a nice form.Let i ∼ j denote vertices i and j are connected in G_r. Then {γ_i,γ_j} = ϵ(Γ_i,Γ_j)γ_i γ_jwhereϵ(Γ_i,Γ_j) = 1,if i<jand i ∼ j, -1,if i>jand i ∼ j. 0,otherwise.In order to prove Proposition <ref>, we introduce an intersection pairing on a twisted ribbon graph from . We refer to <cit.> for more details. The pairing is defined for oriented loops on . It is skew-symmetric and can be described combinatorially as follows.For a pair of oriented loops W and W' on , the intersection pairing ϵ(W,W') is the sum of all the following contributions over the shared vertices of W and W'.< g r a p h i c s > The sign of the contribution is switched (between plus and minus) each time the vertex coloring is switched or the orientation of a path is reversed. For any loops W and W' onwith weight w and w', respectively. We have{ w, w' } = ϵ(W,W')w w'.For j∈[1,2r] we let Y_j be the counterclockwise loop around the face j of G. Since Y_1,…,Y_2r together with the oriented loop Γ_1 (Definition <ref>) generate all the oriented loops on , we only need to show (<ref>) on such generators. It is easy to see thatϵ(Y_i,Y_j) = B_ij, ϵ(Γ_1, Y_j) = δ_j,1 - δ_j,r+1.Let y_j be the weight of Y_j. We need to show that{ y_i, y_j } = B_ij y_i y_j, {γ_1 , y_j } = (δ_j,1 - δ_j,r+1) γ_1 y_j.This is equivalent to{log(y_i), log(y_j) } = B_ij,{log(γ_1), log(y_j) } = δ_j,1 - δ_j,r+1. From the graphwe have y_j = ∏_i=1^2r A_i^B_ij,γ_1 = A_r+1/A_1.We also note that {log(A_i), log(A_j) } = Ω_ij and Ω = -B.To show (<ref>), we consider{log( y_i), log( y_j) } = {∑_k B_kilog(A_k) , ∑_ℓ B_ℓ jlog(A_ℓ) }= ∑_k,ℓ B_kiB_ℓ j{log( A_k), log( A_ℓ) }= ∑_k,ℓ B_kiB_ℓ jΩ_kℓ = ∑_k,ℓ (-B_ik) (- (B)_k ℓ ) (B_ℓ j) = (B B B)_ij = B_ij.To show (<ref>), we consider{log(γ_1), log(y_j) } = {log(A_r+1)-log(A_1), ∑_i B_ijlog(A_i) }= ∑_i( {log(A_r+1),log(A_i) } - {log(A_1), log(A_i) })B_ij= ∑_i(Ω_r+1,i - Ω_1,i)B_ij = (Ω B)_r+1,j - (Ω B)_1,j= -I_r+1,j + I_1,j = -δ_r+1,j + δ_1,j.This proved (<ref>) and (<ref>). Hence we proved the proposition.It is easy to see thatϵ(Γ_i,Γ_j) = 1,if i<jand i ∼ j, -1,if i>jand i ∼ j. 0,otherwise. From Proposition <ref>, we have {γ_i,γ_j} = ϵ(Γ_i,Γ_j) γ_iγ_j. This finished the proof. Recall that a Hamiltonian can be written asH_k = ∑_|I|=k∏_i∈ Iγ_iwhere the sum runs over all n-subsets I⊆ [1,2r+1] of pairwise nonadjacent vertices of G_r. The following lemma gives an involution which will be use to cancel out terms in a computation of { H_i, H_j}.Let i_1<j_1<i_2<j_2<…<i_k<j_k<i_k+1 be a connected sequence of vertices of G_r of odd length. Then {γ_i_1γ_i_2…γ_i_k+1, γ_j_1γ_j_2…γ_j_k} = 0.We consider{log(γ_i_1…γ_i_k+1), log(γ_j_1…γ_j_k)}= ∑_a=1^k+1∑_b=1^k {logγ_i_a, logγ_j_b}= ∑_a=1^k+1∑_b=1^k ϵ(Γ_i_a,Γ_j_b).From Proposition <ref>, we have∑_a=1^k+1∑_b=1^k ϵ(Γ_i_a,Γ_j_b)= ∑_a=1^kϵ(Γ_i_a,Γ_j_a)+ ∑_b=1^kϵ(Γ_i_b+1,Γ_j_b) = k - k = 0.The 2k contributions on the right hand side of the first equality can be view graphically as the following.< g r a p h i c s > Hence {γ_i_1γ_i_2…γ_i_k+1, γ_j_1γ_j_2…γ_j_k} = 0. The Hamiltonians Poisson-commute. This proof is adapted from the proof of <cit.>. Let m,n∈[1,r]. We would like to show that {H_m,H_n}=0.Consider an involution ι:(I,J)↦(I',J') on the set of pairs of index subsets (I,J) where I,J⊆ [1,2r+1] are subsets of pairwise nonadjacent vertices of G_r, |I|=m and |J|=n. For a pair (I,J), we define (I',J') by the following steps.First, we think of I and J as subsets of V(G_r), and then plot all the elements of I and J on the graph G_r. For each even length maximal chain of vertices, we have an alternating sequence between elements of I and elements of J. Then I' (resp. J') is obtained from I (resp. J) by swapping all the elements in every even-length maximal chain. As a result, |I'|=|I| and |J'|=|J|. So, both I and I' (resp. J and J') contribute terms to H_m (resp. H_n). It is also clear that ι is an involution.From Lemma <ref>, the vertices in odd-length maximal chains contribute nothing to the bracket {∏_i∈ Iγ_i,∏_j∈ Jγ_j }. So{log( ∏_i∈ Iγ_i),log(∏_j∈ Jγ_j) }= ∑_C{log(∏_i∈ C∩ Iγ_i),log(∏_j∈ C∩ Jγ_j) }where the sum runs over all even-length maximal chain C⊆ I∪ J. By the construction, C∩ I' = C∩ J and C∩ J'=C∩ I. Hence, {∏_i∈ Iγ_i,∏_j∈ Jγ_j } = -{∏_i∈ I'γ_i,∏_j∈ J'γ_j }.A fixed point of ι is a pair (I,J) where all maximal chains are odd. We have that{∏_i∈ Iγ_i,∏_j∈ Jγ_j }= 0.Hence, { H_m,H_n} = 0.Let r=1, I={1,4} and J={-1,2,5}. The following picture show three maximal chains of I∪ J: { -1}, {1,2} and {4,5}.< g r a p h i c s >So I'={2,5} and J' = { -1,1,4}. The black (resp. white) dots are elements of I and I' (resp. J and J').§.§ Another proof of Theorem <ref> The proof is based on the proof of <cit.>. We denote bythe following bipartite torus graph. It differs fromby two extra edges.< g r a p h i c s >Using notations in Section <ref>, it can be obtained from an integral convex polygon with edge vectors:* e_1=(1,r+1/2), e_2=(-1,r+1/2), e_3=(-1,-r+1/2), e_4=(1,-r+1/2) when r is odd* e_1=(1,r+2/2), e_2 = (-1,r/2), e_3=(-1,-r+2/2), e_4=(1,-r/2) when r is even.The integral convex polygon can be depicted as the following.< g r a p h i c s >The following pictures are G with oriented loops.< g r a p h i c s > We define a reference perfect matching from a sequence e_1,e_2,e_3,e_4 according to the construction in <ref>. It coincides with the perfect matching M_0 defined at the beginning of Section <ref>. We denote by M_0 this perfect matching of .Letbe a bipartite torus graph obtained from an integral polygon with edge vectors e_1,e_2,…,e_n in Section <ref>. Let v be a vertex in a loop Γ. Let φ_M_0 : E()→{0,1} defined by φ_M_0(e) = 1 if and only if e∈ M_0. Define b_v(Γ,M_0) byb_v(Γ,M_0) := ∑_e∈ R_vφ_M_0(e) - ∑_e∈ L_vφ_M_0(e) ∈where R_v (resp. L_v) be the set of all edges incident to v which are on the right (resp. left) of the loop Γ (not including edges in Γ).The following lemma says that for any homologically nontrivial loop Γ on , the number of edges in M_0 incident to Γ on the left is equal to the number of edges in M_0 incident to Γ on the right.Let M_0 be the reference perfect matching obtained from edge vectors. (See Section <ref>.) For any simple topologically nontrivial loop Γ on , ∑_v∈Γ b_v(Γ,φ_M_0) = 0.Let a,b∈ H_1(,). We will show that { H_a,,M_0, H_b,,M_0} = 0. We simplify the notation by letting H_a := H_a,,M_0. Let _a be the set of all perfect matchings ofwhose homology class with respect to M_0 is a, i.e. [M]_M_0 = a ∈×. Then H_a = ∑_M∈_a w(M)/w(M_0).Let M_1∈_a and M_2∈_b. By Proposition <ref>, [M_1]-[M_2] is a collection of non-intersecting simple loops. There are two types of simple loops: homologically trivial loops and homologically nontrivial loops. Define an involution ι:_a×_b →_a×_b by (M_1,M_2)↦(M_1',M_2') where M_1', M_2' are obtained from M_1, M_2 by exchanging all edges in each homologically trivial loop of [M_1]-[M_2]. See the following example.< g r a p h i c s > The involution ι define an equivalence relation on _a×_b by (M_1,M_2)∼ (M_1',M_2') if ι((M_1,M_2)) = (M_1',M_2'). So each equivalence class has either one or two elements. Each fixed point of ι is in its own singleton class.Consider{ H_a, H_b } = ∑_(M_1,M_2)∈_a×_b{w(M_1)/w(M_0) , w(M_2)/w(M_0)}The sum is then divided into two sums:* The first summation runs over all the fixed points of ι:∑_{(M_1,M_2)}{w(M_1)/w(M_0) , w(M_2)/w(M_0)} * The second summation runs over all equivalence classes {(M_1,M_2),(M_1',M_2')} of size 2:∑_{(M_1,M_2),(M_1',M_2')}{w(M_1)/w(M_0) , w(M_2)/w(M_0)} + {w(M_1')/w(M_0) , w(M_2')/w(M_0)}To calculate the first sum (<ref>), let (M_1,M_2) be a fixed point of ι. By Proposition <ref> we have{w(M_1)/w(M_0) , w(M_2)/w(M_0)} = ∑_vϵ_v ( [M_1]-[M_0],[M_2]-[M_0] ) w(M_1)w(M_2)/w(M_0)^2where ϵ_v is the contribution from vertex v to the intersection pairing. For each vertex v, there are edges e_0,e_1,e_2 incident to v and belong to M_0, M_1, M_2, respectively. If e_1 = e_2, then ϵ_v ( [M_1]-[M_0],[M_2]-[M_0] ) = 0. If e_1≠ e_2, the vertex v must belong to the loop [M_1]-[M_2], and vice versa. Since [M_1]-[M_2] is a fixed point of ι, it has no homologically trivial loops. So every loop is homologically nontrivial.Consider a homologically nontrivial simple loop Γ of the loop [M_1]-[M_2] (a connected component of [M_1]-[M_2]). We have the following four configurations of e_0,e_1,e_2 depending whether e_0 is on the left/right of Γ or whether v is black/white. Note that e_1 (resp. e_2) goes from black to white (resp. white to black) in Γ.< g r a p h i c s >The local contribution to ϵ_v ( [M_1]-[M_0],[M_2]-[M_0] ) from each configuration is 1/2, -1/2, 1/2 and -1/2, respectively. We can conclude that the contribution is 1/2 (resp. -1/2) when e_0 is on the left (resp. right) of Γ. By Lemma <ref>, we can conclude thatϵ_v ( [M_1]-[M_0],[M_2]-[M_0] ) = 0.Hence the first summation vanishes.For the second sum, since w(M_1)w(M_2) = w(M_1')w(M_2') the summand is equal to ∑_v ( ϵ_v ( [M_1]-[M_0],[M_2]-[M_0] ) + ϵ_v ( [M_1']-[M_0],[M_2']-[M_0] ) ) w(M_1)w(M_2)/w(M_0)^2.For each vertex v, there are edges incident to v and belong to M_1, M_2 and M_0. Similarly to the computation for the first sum, if such edges of M_1 and M_2 are the same, then the summand vanishes. If v belongs to a homologically nontrivial loop, by Lemma <ref>, the summand also vanishes. If v belongs to a homologically trivial loop, we see that [M_1]-[M_0] (resp. [M_2]-[M_0]) is locally the same as [M_2']-[M_0] (resp. [M_1']-[M_0]) at v. Henceϵ_v ( [M_1']-[M_0],[M_2']-[M_0] )= ϵ_v ( [M_2]-[M_0],[M_1]-[M_0] )= - ϵ_v ( [M_1]-[M_0],[M_2]-[M_0] ).Hence the summand (<ref>) vanishes. This concludes that the second summation vanishes.Thus { H_a, H_b } = 0. This finishes the proof. § Q-SYSTEMS OF TYPE BWe construct a “double cover" of B_r Q-system quiver and compute Hamiltonians on the weighted bipartite torus graph associated with the quiver. §.§ Q-systems of type B and weighted graph mutations Consider the following quiver of an B_r Q-system. See Theorem <ref> for the detail.< g r a p h i c s >In the case of A_r Q-system, we added a frozen vertex to the quiver so that it has an associated bipartite torus graph. Unlike the previous case, the quiver of B_r Q-system does not have an associated bipartite torus graph. We resolve this issue by considering instead a double-cover of the quiver together with a frozen vertex as in the following picture. Letdenote the resulting quiver.< g r a p h i c s >Notice thatand the original quiver are locally the same. There are two copies for each vertex. So we can think of this quiver similar to a double-cover of the original quiver with an extra frozen vertex. The bipartite torus graphassociated withis depicted below.< g r a p h i c s > < g r a p h i c s > We then assign Q-system variables as weights ofaccording to face labels, see Theorem <ref>. We abuse the notation and use μ for a sequence of mutations where the mutation at i actually means the mutations at both faces labeled by i. Then σμ sendsto itself and the Q-variables are shifted by k→ k+1 as well. To be precise, Q_i,k↦ Q_i,k+1 for i∈[1,r-1] while Q_r,2k↦ Q_r,2k+2 and Q_r,2k+1↦ Q_r,2k+3.Let M_0 be the perfect matching ofcontaining all vertical edges whose top vertex is black. It can be depicted as the following when r is odd and similarly when r is even.< g r a p h i c s > Similarly to the proof of Theorem <ref>, we can check that all the conditions in Theorem <ref> and Theorem <ref> hold. Hence we haveH_(i,j),,M_0(A_1,k,…,A_2r,k) = H_(i,j),,M_0(A_1,k+1,…,A_2r,k+1)whereA_i,k =Q_i,k, i∈[1,r-1],Q_r,2k, i = r, Q_i-r,k+1, i∈[r+1,2r-1], Q_r,2k+1, i=2r.Hence the Hamiltonians H_(i,j),,M_0 are conserved quantities of the B_r Q-system. This proved the following theorem. Let (,(A_i)) be a weighted bipartite torus graph defined above. Then the Hamiltonians H_(i,j),,M_0(A_1,…,A_2r) are conserved quantities of the B_r Q-system dynamic Q_,k↦ Q_, k + t_ where t_ =1 for ∈ [1,r-1] and t_r = 2.§.§ Partition function of hard particles In this section, we write the Hamiltonians as partition functions of hard particles on a weighted graph, analog to what has been done for Q-systems of type A.From Proposition <ref>, [M]-[M_0] is a product of non-intersecting simple loops of . The following loops are all connected simple loops that can be appeared (given our choice of M_0). This will be proved in Theorem <ref>.We first define 2r straight loops onand denote them by Γ_2a-1 for a∈ [1,2r].< g r a p h i c s > Next, we define zig-zag loops Γ_2a for a∈[1,r-1]∪ [r+1,2r-1] as follows.< g r a p h i c s >We notice that when a∈[1,r-1], Γ_2a always goes counterclockwise around face a and clockwise around face r+a. When a ∈ [r+1,2r-1], Γ_2a goes counterclockwise around face 2r-a and clockwise around face 3r-a.Lastly, we define Γ_2r,j for j∈[1,6]. They are depicted as follows.< g r a p h i c s >We then have the following theorem analog to Theorem <ref>.Let M be a perfect matching of .Then [M]-[M_0] is a nonintersecting collection of Γ. Furthermore, every nonintersecting collection of Γ is [M]-[M_0] for a unique perfect matching M of . Using exactly the same proof as in Theorem <ref>, we can see that all the loops Γ listed above are all possible simple loops appeared in [M]-[M_0]. Let G_r be the following graph with 4r+4 vertices indexed by the set [1,2r-1]∪[2r+1,4r-1] ∪{(2r,i)| i∈[1,6]} defined as the following< g r a p h i c s >There is a complete graph K_6 as a subgraph of G_r in the middle. The label (6) on the three thick lines indicates that the vertex connects to all six vertices in K_6. There is a thick line with label (5); vertex 2r+2 connects to all vertices in K_6 except to the vertex (2r,1). (This is indicated by a dotted line between (2r,1) and (2r+2).) The vertex (2r,6) has two extra edges connecting to vertex 2r-4 and 2r-3.The loop Γ_i intersects Γ_j if and only if the vertices i and j are connected in G_r. For any subset I⊆ [1,2r-1]∪[2r+1,4r-1] ∪{(2r,i)| i∈[1,6]}, the loops in {Γ_i | i∈ I} are pairwise disjoint if and only if I is a subset of pairwise nonadjacent vertices of G_r. Also, nonintersecting collections of Γ of size n are in bijection with n-subsets of pairwise nonadjacent vertices of G_r.By Theorem <ref>, the possible homology classes of [M]-[M_0] are (0,k) for k∈[0,2r]. Let γ_i be the weight of Γ_i. Theorem <ref> implies the following theorem.Let k∈[0,2r]. ThenH_(0,k),,M_0(A_1,…,A_2r) = ∑_|I|=k∏_i∈ Iγ_iwhere the sum runs over all k-subsets I of pairwise nonadjacent vertices of G_r.When r=2, the graph G_2 for B_2 Q-system has 12 vertices and can be depicted as the following.< g r a p h i c s >We have the following weights.γ_1= Q_1,k+1/Q_1,k,γ_2= Q_2,2k+1^2/Q_1,k Q_1,k+1 Q_2,2k,γ_3= Q_1,k Q_2,2k+1^2/Q_1,k+1 Q_2,2k^2,γ_4;1 = Q_1,k^2/Q_2,2k^2,γ_4;2 = Q_1,k^2/Q_2,2k^2,γ_4;3 = Q_1,k^3 Q_1,k+1/Q_2,2k^2 Q_2,2k+1^2,γ_4;4 = Q_1,k Q_1,k+1/Q_2,2k+1^2,γ_4;5 = Q_1,k Q_1,k+1/Q_2,2k+1^2, γ_4;6 = 1/Q_2,2k,γ_5= Q_1,k+1 Q_2,2k^2/Q_1,k Q_2,2k+1^2,γ_6= Q_2,2k/Q_1,k Q_1,k+1,γ_7= Q_1,k/Q_1,k+1.For H_k := H_(0,k),G,M_0(A_1,k,…,A_2r,k), we haveH_0= 1 H_1= Q_1,k+1/Q_1,k+Q_2,2k+1^2/Q_1,k Q_1,k+1 Q_2,2k+Q_1,k Q_2,2k+1^2/Q_1,k+1 Q_2,2k^2+Q_1,k^2/Q_2,2k^2+Q_1,k^2/Q_2,2k^2+Q_1,k^3 Q_1,k+1/Q_2,2k^2 Q_2,2k+1^2++Q_1,k Q_1,k+1/Q_2,2k+1^2+Q_1,k Q_1,k+1/Q_2,2k+1^2+1/Q_2,2k+Q_1,k+1 Q_2,2k^2/Q_1,k Q_2,2k+1^2+Q_2,2k/Q_1,k Q_1,k+1+Q_1,k/Q_1,k+1, H_2= Q_1,k^4/Q_2,2k^2 Q_2,2k+1^2+2 Q_1,k^3/Q_1,k+1 Q_2,2k^2+Q_2,2k+1^2 Q_1,k^2/Q_1,k+1^2 Q_2,2k^2+2 Q_1,k^2/Q_2,2k+1^2+Q_1,k+1^2 Q_1,k^2/Q_2,2k^2 Q_2,2k+1^2++2 Q_1,k/Q_1,k+1 Q_2,2k+2 Q_1,k+1 Q_1,k/Q_2,2k^2+Q_2,2k+1^2/Q_1,k^2 Q_1,k+1^2+2 Q_2,2k+1^2/Q_1,k+1^2 Q_2,2k+Q_2,2k+1^2/Q_2,2k^2++2 Q_2,2k/Q_1,k^2+2 Q_1,k+1^2/Q_2,2k+1^2+Q_2,2k^2/Q_2,2k+1^2+Q_1,k+1^2 Q_2,2k^2/Q_1,k^2 Q_2,2k+1^2+2, H_3= H_1, H_4= 1.We notice that H_0 = H_4 and H_1 = H_3. In fact, we will show that H_i = H_2r-i (i∈[0,2r]) for the Hamiltonians of the B_r Q-system.Let H_k := H_(0,k),G,M_0(A_1,…,A_2r) be the Hamiltonians for the B_r Q-system. Then for k∈[0,2r],H_k = H_2r-k.Fix r ≥ 2. Let π : [1,2r-1]∪{ (2r;j)}_j=1^6 ∪ [2r+1,4r-1] → [1,2r] be a projection defined byπ: i ↦ i, i∈[1,2r-1],(2r;j) ↦ 2r, j∈[1,6],i↦ 4r-i,i∈[2r+1,4r-1].Let F_r be the following graph with 2r vertices.< g r a p h i c s >Conceptually F_r is obtained from G_r by collapsing all six vertices (2r;1),…,(2r;6) into one vertex 2r and folding the resulting graph by identifying vertices i ↔ 4r-i. The projection π sends a vertex of G_r to a vertex of F_r by the collapsing/folding procedure.Let 𝒞 be the set of all hard particle configurations on G_r, and let 𝒞_k be the set of all k hard particle configurations on G_r. We have 𝒞_k ≠∅ for k∈[0,2r]. (The set 𝒞_0 contains exactly one configuration, the empty configuration.) So𝒞 = _k=0^2r𝒞_k.We identify a hard particle configuration with a subset of [1,2r-1]∪{ (2r;j)}_j=1^6 ∪ [2r+1,4r-1]. Let A∈𝒞. We then associate each element i∈ A with a black or white dot on the vertex π(i) F_r as follows. * An element i∈ [1,2r-1] is associated with a black dot on vertex π(i).* An element i∈ [2r+1,4r-1] is associated with a white dot on vertex π(i).* An element (2r;j), for j∈[1,5], is associated with a black dot together with a label j on vertex 2r.* The element (2r,6) is associated with a white dot together with a label 6 on vertex 2r and a black dot on vertex 2r-2.See an example in Figure <ref>.Notice that we can recover the hard particle configuration on G_r from a configuration of dots on F_r. It is obvious that not every dot configuration on F_r is associated with a configuration in 𝒞.We define an involution ι:𝒞→𝒞 by the following steps. * Let A∈𝒞. Consider the dot configuration on F_r associated with A. We have a collection of connected chains of dots. Since A is a hard particle configuration, each chain is a chain of dots of alternate colors or A pair of white and black dots on the same vertex.For each connected chain i_1<…< i_k or a pair of black and white dots i_1=i_2 (k=2), we let i be the largest odd integer such that i≤ i_1, j be the smallest even integer such that i_k < j. When j>2r, we set j=2r. We then delete edges (i-1,i), (i-1,i+1), (j,j+1) and (j,j+2) of F_r when possible. As a result, F_r is decomposed into disconnected blocks (isomorphic to F_n, n≤ r). Lastly any block of 2n vertices containing no dots is decomposed into n blocks of 2 vertices. See Figure <ref> for an example.* For each block we define an involution on dot configurations as in Figure <ref> and <ref>. The blocks in Figure <ref> are fixed by the involution.* The hard particle configuration ι(A) is the configuration associated with the resulting dot configuration. See Figure <ref> for an example. For a block of 2n vertices the involution sends a dot configuration of size k to a configuration of size 2n-k, where we do not count the black dot at 2r-2 when the white dot with label 6 on 2r is present. The sum of all the number of vertices of all blocks is 2r.So the involution is a bijection between 𝒞_k and 𝒞_2r-k.It is left to show that if A∈𝒞_k and B=ι(A) ∈𝒞_2r-k then ∏_i∈ A w(γ_i) = ∏_j∈ B w(γ_j).By a direct computation, we haveγ_4r-(2a-1)γ_2a = γ_4r-2aγ_2a+1,γ_2a-1γ_4r-2a = γ_2aγ_4r-(2a+1),γ_2b-1γ_4r-(2b-1) = 1 ,γ_(2r;1)γ_2r+2 = γ_(2r+6)γ_2r+3,for a∈[1,r-1] and b∈[1,r]. These equations say that the following moves preserve the weight of the associated hard particle configuration.-0.4 < g r a p h i c s >Since the involution on each block can be written as a sequence of the moves (<ref>) (see Figure <ref> for an example), the involution preserves the weight for each block. Hence the equation (<ref>) holds.§.§ Poisson bracketSimilar to type A, we let let C be the Cartan matrix of type B_r. The signed adjacency matrix of the quiver of B_r Q-system isB = [ C-C^T C^T;-C 0 ].Letbe a phase space with coordinates (A_1,…,A_2r). Define a Poisson bracket on the algebra 𝒪() of functions onby{ A_i , A_j } = Ω_ij A_i A_j (i,j∈ [1,2r])where the coefficient matrix Ω is defined byΩ = (B^T)^-1 = -B^-1. Comparing to Proposition <ref>, the Poisson bracket for B_r Q-system cannot be written as the intersection pairing. This is due to the existence of faces with the same weight, which makes the Poisson bracket not local. Nevertheless, experimental data still show that the Hamiltonians Poisson-commute.The Hamiltonians of the B_r Q-system Poisson-commute. § DIMER INTEGRABLE SYSTEMS In this section, we compare our constructions and results to <cit.>. First we summarize the result from <cit.>. §.§ Minimal bipartite torus graphs from convex polygons Let N be a convex polygon in ^2 with corners in ^2, called integral polygon, considered up to translation by vectors in ^2. We pick all the integral vertices on the boundary of N (i.e. every vertex in ∂ N ∩^2) counterclockwise, and get a sequence of vertices v_1,v_2,…,v_n where n is the number of integral vertices on ∂ N and the indices are read modulo n. Let vectors e_i be vectors pointing from v_i to v_i+1. We get from the construction that each e_i = (a_i,b_i) is a primitive vector, i.e. a_i,b_i ∈ and (a_i,b_i)=1. We then get a collection {e_i} of integral primitive vectors in ^2.Consider the torus =^2/^2. Each e_i determines a homology class (a_i,b_i)∈ H_1(,)=×, and there is a unique up to translation geodesic representing this class. In other words, it is an oriented straight line onwith slope b_i/a_i, i.e. a projection of e_i on the torus . Note that the geodesics are indeed oriented loops onsince their slopes are rational.We then take a family of distinct oriented loops {_i} onsuch that the isotopy class of _i matches the isotopy class of the geodesic representing e_i. By Theorem <ref> we can choose {_i} such that the loops are in generic position (no intersection of more than two loops) and satisfy the following conditions <cit.>: * (admissibility) Going along any loops _i, the directions of the other loops intersecting it alternate (left-to-right or right-to-left).* (minimality) The total number of intersections is minimal.The collection {_i} provides a decomposition ofinto a union of polygons whose oriented sides are parts of _i and vertices are intersection points of the loops _i. Then the first condition is equivalent to (1) (admissibility) The sides of any polygon P_i are either oriented clockwise, counterclockwise or alternate.The family of oriented loops {_i} gives rise to an oriented graph on the torus . We call an oriented graph onsatisfying the above conditions minimal admissible graph on a torus. Starting from an integral polygon on the left, we obtained four primitive vectors depicted in the middle picture.< g r a p h i c s >The vectors are then associated with oriented loops on a torus, which gives a minimal admissible graph shown in the picture on the right. Given an admissible minimal torus graph, we construct a bipartite graphonby constructing vertices from the polygons P_i where the coloring is determined as follows: * Polygons P_i with counterclockwise orientation are associated with black vertices.* Polygons P_i with clockwise orientation are associated with white vertices. From the construction, every intersection is a vertex shared by exactly two well-oriented polygons having opposite orientations. We then associate each shared vertex with an edge connecting the two vertices ofassociated to the two polygons.We see thatis indeed a bipartite graph, and we will call it a minimal bipartite torus graph. For the rest of the section, unless stated otherwise we assume that G is a minimal bipartite torus graph obtained from an integral polygon N. In our running example, there are eight polygons and eight intersection points.< g r a p h i c s >In the picture, the polygons labeled by 2,3 (resp. 4,5) are oriented counterclockwise (resp. clockwise), so they are associated with black (resp. white) vertices of . The polygons labeled by 1,6,7,8 have alternate orientation. The eight intersection points are associated with eight edges of . From the bipartite graph G, we can uniquely (up to translation) recover the starting integral polygon by reversing the process. The convexity condition on the integral polygon will guarantee the uniqueness of the polygon. We also note that a 180-degree rotation of N corresponds to reversion of the orientation of all loops in the admissible minimal graph. This will switch the color of the vertices of . For an arbitrary integral polygon N, a minimal admissible graph on a torus associated with N and a minimal bipartite torus graph associated with N always exist. Furthermore, we can obtain one minimal bipartite torus graph from another by use of the two elementary moves (definitions <ref> and <ref>), as stated in the following theorem.For any integral polygon N there exists a minimal admissible graph on a torus associated with N. It produces a minimal bipartite torus graphassociated with N. Furthermore, any two minimal bipartite graphs on a torus associated with N are related by a sequence of urban renewals and shrinking of 2-valent vertices. Our bipartite torus graphs for the Q-systems of type A and B in sections <ref> and <ref> are not obtained from non-degenerate convex polygons.For A_r Q-system, we reverse the process and construct oriented loops on the torus as in the following picture.< g r a p h i c s >We get only two oriented loops depicted in blue and red whose homology classes are (0,-r-1) and (0,r+1), respectively. Notice that they are not primitive, and they form a vertical degenerate bigon with sides of length r+1.For B_r Q-system, we have the following picture.< g r a p h i c s >There are four oriented loops depicted in blue, green, yellow and red whose homology classes are (0,-2r+1),(0,-1),(0,1) and (0,2r-1), respectively. The blue and red loops are not primitive when r > 1. The loops form a vertical degenerate quadrilateral.§.§ Phase space and Poisson structure For a minimal bipartite torus graphwith n faces, let Ł_ be the moduli space of line bundles with connections on . We have Ł_≅(H_1(,),^*), so combinatorially Ł_ is the set of all weight assignments to all the loops oncompatible with loop multiplication. (The weight of a product of loops coincides with the product of their weights.)For j∈[1,n], let y_j be the weight assigned to the counterclockwise loop Y_j around the face j of . Sinceis a graph embedded on a torus, there is a projection H_1(,)→ H_1(,). We then pick two loops Z_1,Z_2 having homology classes (1,0),(0,1)∈ H_1(,) under the projection, and assign weight z_1,z_2 to them, respectively. Any loop oncan then be generated by y_j's together with z_1 and z_2, where a product of the variables corresponds to a product of loops. Since the product of all the face loops is trivial, we must have ∏_j=1^n w_j = 1. This is the only condition among the generators. So Ł_ = n+1 and the algebra 𝒪(Ł_) of functions on Ł_ has (y_1,…,y_n-1,z_1,z_2) as coordinates.Note that this weight is different from our weight in Definition <ref>. The connection between the two is discussed in Remark <ref>.For any loops Γ_1,Γ_2 on , the Poisson bracket of their weights is defined in terms of the intersection pairing of the twisted ribbon graph associated withas in Definition <ref>.Now we define a Y-seed (B,(y_1,…,y_n,z_1,z_2)) of rank n+2 associated with Ł_, where the exchange matrix B = (B_ij) and B_ij = ϵ(Y_i,Y_j) for i,j∈[1,n+2]with Y_n+1 := Z_1 and Y_n+2 := Z_2. Letand ' be two minimal bipartite torus graphs associated with the same integral polygon N. By Theorem <ref>, they are related by a sequence of elementary moves. These moves induce an isomorphism i_,':Ł_G →Ł_G' according to Y-seed mutations. Let _N be the phase space defined by gluing the spaces Ł_G by the isomorphisms. The phase space depends only on N and each isomorphism i_,' can be viewed as a change of coordinate.Let G be a minimal bipartite torus graph with n faces. Each choice of face weight (A_i)∈ (^*)^n induces a weight assignment on oriented edges of G by Definition <ref>. This then induces a weight assignment on Y_1,…,Y_n,Z_1,Z_2, hence a loop weight in Ł_G. Since Ł_ has dimension n+1, not all loop weights of <cit.> can be obtained from our weight in Definition <ref>.In addition, the weight y_j around face j of G is the j^th τ-coordinates (Definition <ref>) of a cluster seed (𝐀,B), i.e.y_j = ∏_i=1^n A_i^B_ijwhere B = B_G is the signed adjacency matrix of the quiver associated to G (see Section <ref>). Let G be the following graph on the left. It is obtained from a integral polygon whose counterclockwise edge vectors e_i are (1,1), (-1,1), (-1,-1) and (1,-1). Let Y_i be the counterclockwise loop around the face i, and Z_1 (resp. Z_2) be the loop in the middle (resp. right) picture.< g r a p h i c s >Let (A_1,A_2,A_3,A_4)∈(^*)^4 be an arbitrary face weight on G. It induces the following weightsy_1 = A_4^2/A_2^2, y_2 = A_1^2/A_3^2, y_3 = A_2^2/A_4^2, y_4 = A_3^2/A_1^2,z_1 = A_3 A_4/A_1 A_2,z_2 = A_2 A_3/A_1 A_4.They satisfy the following conditionsy_1 y_3 = 1,y_2 y_4 = 1,z_1^2 = y_1 / y_2,z_2^2 = 1/(y_1 y_2).Since y_1 and y_2 are algebraically independent, the induced loop weight is a subspace of dimension 2 inside Ł_G of dimension 5. (L_G is of dimension 5 because every loop weight in Ł_G satisfies y_1y_2y_3y_4 = 1.) The map from face weight to loop weight is not injective. We have the following 2-dimensional symmetries(A_1,A_2,A_3,A_4) ↦ (λ A_1,μ A_2,λ A_3,μ A_4)for λ,μ∈^*. §.§ Casimirs and HamiltoniansA zig-zag path onis an oriented path onwhich turns maximally left at white vertics and turns maximally right at black vertices <cit.>. They will always close up to form loops. Notice that the projection of the zig-zag loops on the torus are in the same isotopy classes with the oriented loops _i obtained from the primitive edges e_i of N (See Section <ref>). So the oriented zig-zag loops are in bijection with {_i}.The weight of these zig-zag loops, called Casimirs, generate the center of the Poisson algebra 𝒪(Ł_) as described in the following proposition. We consider the oriented zig-zag loops on a bipartite oriented surface graph . Then as Z runs over zig-zag loops, the functions w_Z generate the center of the Poisson algebra 𝒪(Ł_) of functions on Ł_. The product of all of them is 1. This is the only relation between them. Recall the construction offrom a minimal admissible graph (an arrangement of {_i}_i=1^n) in Section <ref>. We see that each edge e ofhas exactly two loops _i,_j ∈{_1,…,_n} crossing it. Let _i and _j be as the following.< g r a p h i c s >Then we define a reference perfect matching M_0 to be the matching containing all edges e such that i>j. It is shown in <cit.> that M_0 is indeed a perfect matching of .We note that M_0 is not unique. Since the indices of _i can be read modulo n, another cyclic ordering gives another reference perfect matching. There are also many other choices of M_0 including "fractional matchings", see <cit.> for details. We can show from <cit.> that every quadrilateral face inhas exactly one side in M_0. So every reference perfect matching M_0 constructed above always satisfies the requirement in Theorem <ref>.Given a reference perfect matching M_0, the weight w_M_0(M) of a perfect matching M with respect to M_0 is defined to be the weight of the loop [M]-[M_0], written in terms of y_1,…,y_n,z_1,z_2.Recall the definition of [M]_M_0, the homology class of M with respect to M_0, in Definition <ref>.The polygon with vertices at all homology classes [M]_M_0∈ H_1(,)= × coincides with the convex polygon N up to translation <cit.>. Given a homology class a∈ H_1(,), we letH_M_0;a:= ∑_M w_M_0(M)where the sum runs over all perfect matchings M ofhaving homology class a. The (modified) partition function of perfect matchings ofis defined to be P_M_0 := ∑_a (a) H_M_0;awhere the sum runs over all possible homology classes of perfect matchings with respect to M_0. The sign (a)∈{-1,1} can be determined from a “Kasteleyn matix", and they show up in the formula from the use of the determinant of a “Kasterleyn operator", see <cit.> for more details.For a∈ H_1(,) ∩(N), a homology class which is an interior point of N, the function H_M_0;a is called a Hamiltonian. We note that a different choice of M_0 gives a different partition function and a different set of Hamiltonians. However, they differ from each other by a common factor which lies in 𝒪(Ł_).These Hamiltonians are independent and commute under the Poisson bracket. Letbe a minimal bipartite torus graph. Then * The Hamiltonians H_M_0;a commute under the Poisson bracket on Ł_.* The Hamiltonians are independent and their number is the half of the dimension of the generic symplectic leaf.We also have that the partition function is invariant under the change of coordinates i_,' (defined in Section <ref>). This implies that all Hamiltonians are also invariant under the change of coordinates. In fact, the map i_,' is a unique rational transformation of face weights preserving the partition function, given a graph mutation fromto '. Given an urban renewal, there is a unique rational transformation of the weights preserving the partition function P_a. This transformation is a Y-seed mutation. By counting the number of Hamiltonians and Casimirs, we can conclude on the integrability of the system.Let M_0 be a reference perfect matching obtained from a circular-order-preserving map. The Hamiltonian flows of H_M_0,a commute, providing an integrable system on _N. Precisely, we get integrable systems on the generic symplectic leaves of _N, given by the level sets of the Casimirs.We notice that the integrable system described in this section is a classical dynamical system where the evolutions are Hamiltonian flows. This system also contains a discrete dynamical system whose evolution is a change of coordinate i_G,G' (Y-seed mutation on loop weights).For a graphperiodic (up to a relabeling of vertices) under a sequence of urban renewals and shrinking of 2-valent vertices, we take the change of loop weight under such sequence to be the dynamic of a discrete system. Since the graph is periodic, the dynamic is a Poisson map with respect to the Poisson bracket in Section <ref>. The Casimirs and Hamiltonians return to the same form under such sequence. Since they are also invariant by Theorem <ref>, they are conserved quantities of the system. By Theorem <ref>, the quantities form a maximal set of Poisson-commuting invariants, hence the system is discrete Liouville integrable.§ CONCLUSION AND DISCUSSION In this paper, we studied a discrete dynamic on a weighted bipartite torus graph, obtained from an urban renewal on the graph and cluster mutation on the weight. The weight is defined differently from <cit.>. The graph can be any bipartite graph on a torus, not necessarily obtained from an integral polygon. The Hamiltonians are defined and proved to be invariant under the mutation.For a Q-system of type A, we constructed a weighted bipartite torus graph which is periodic under a sequence of mutations up to a face relabeling. The weight changes according to the Q-system relation. So the Hamiltonians are conserved quantities of the system. This coincides with the result from <cit.>. We also showed that the Hamiltonians can be written as hard-particle partition functions on a certain graph, which coincides with the result in <cit.>. A nondegenerate Poisson bracket is defined, and the Hamiltonians Poisson commute. For a Q-system of type B, a bipartite torus graph is constructed. The graph is a double-cover of the dual graph associated with the Q-system quiver. It is periodic under a sequence of mutations up to a face relabeling. The weight are transformed according to the Q-system relation. The Hamiltonians are conserved quantities of the system. They can also be interpreted as hard-particle partition functions on a certain graph. We conjecture that the conserved quantities Poisson commute under the Poisson bracket defined in Section <ref>.One could wonder what happens for types C, D and other exceptional types. The sequence of mutations in <cit.> contains a mutation at a vertex of degree greater than 4. Recall that an urban renewal corresponds to a mutation at a vertex of degree 4 which has exactly two incoming and two outgoing arrows. Although some mutations in type A and B happen at a vertex of degree less than 4, we fixed this issue by adding a frozen vertex. However this technique is not applicable when the degree of a mutating vertex is greater than 4. This problem might possibly be solved by unfolding or factoring the quiver so as to restore the 4-valent property. We leave this for future work.alpha | http://arxiv.org/abs/1704.08736v2 | {
"authors": [
"Panupong Vichitkunakorn"
],
"categories": [
"math.CO",
"math.DS"
],
"primary_category": "math.CO",
"published": "20170427203019",
"title": "Conserved quantities of Q-systems from dimer integrable systems"
} |
ucb]A.S. Voyles cor1 [email protected]]M.S. Basuniaucb]J.C. Batchelderllnl]J.D. Bauergeo]T.A. Becker ucb,lbl]L.A. Bernstein ucb]E.F. Matthewsgeo,eps]P.R. Rennegeo,eps]D. Rutteucb]M.A. Unzuetaucb]K.A. van Bibber[cor1]Corresponding author [ucb]Department of Nuclear Engineering, University of California, Berkeley, Berkeley CA, 94720 USA [lbl]Lawrence Berkeley National Laboratory,Berkeley CA, 94720 USA [llnl]Lawrence Livermore National Laboratory, Livermore CA, 94551 USA [geo]Berkeley Geochronology Center, Berkeley CA,94709USA [eps]Department of Earth and Planetary Sciences, University of California, Berkeley, Berkeley CA,94720USACross sections for the ^47Ti(n,p)^47Sc and ^64Zn(n,p)^64Cu reactions have been measured for quasi-monoenergetic DD neutrons produced by the UC Berkeley High Flux Neutron Generator (HFNG). The HFNG is a compact neutron generator designed as a flux-trap that maximizes the probability that a neutron will interact with a sample loaded into a specific, central location. The study was motivated by interest in the production of ^47Sc and ^64Cu as emerging medical isotopes. The cross sections were measured in ratio to the ^113In(n,n')^113mIn and ^115In(n,n')^115mIn inelastic scattering reactions on co-irradiated indium samples. Post-irradiation counting using an HPGe and LEPS detectors allowed for cross section determination to within 5% uncertainty. The ^64Zn(n,p)^64Cu cross section for 2.76^+0.01_-0.02 MeV neutrons is reported as49.3 ± 2.6 mb (relative to ^113In) or 46.4 ± 1.7 mb (relative to ^115In), and the ^47Ti(n,p)^47Sc cross section is reported as 26.26 ±0.82 mb. The measured cross sectionsare found to bein good agreement with existing measured values but with lower uncertainty (5%), and also in agreement withtheoretical values. This work highlights the utility of compact, flux-trap DD-based neutron sources for nuclear data measurements and potentially the production of radionuclides for medical applications.DD neutron generator Medical Isotope Production Scandium (Sc) and Copper (Cu) radioisotopes Indium Ratio activation Theranostics§ INTRODUCTIONThere has been significant interest in the past several years in exploring the use of neutron-induced reactions to create radionuclides for a wide range of applications. This interest is due to the volumetric absorption of neutrons as compared to charged particle beams (ranges of gcm^2 as compared to 10's of mgcm^2), together with the fact that isotope production facilities often produce large secondary neutron fields.Particular interest has been paid to (n,p) and (n,α) charge-exchange reactions since these reactions produce high-specific activity radionuclide samples without the use of chemical carriers in the separation process.Two other potential neutron sources for (n,x) reactions exist in addition to the secondary neutron fields generated at existing isotope production facilities: reactors and neutron generators that utilize the D(T,n)α (DT) and D(D,n)^3He (DD) reactions.While reactors produce copious quantities of neutrons, their energy spectra are often not well-suited to the preparation of high-purity samples due to the co-production of unwanted activities via neutron capture, in addition to the significant start-up costs and proliferation concerns involved in their commissioning <cit.>.Similarly, while the higher energy 14-15 MeV neutrons produced at DT generators are capable of initiating (n,p) and (n,α) reactions, their higher energy opens the possibility of creating unwanted activities via (n,pxn) and (n,αxn) reactions that cannot easily be separated from the desired radionuclides. DT generators may also often be limited by the restricted use of tritium at many institutions.In contrast,the neutron spectrum from a DD reaction, which ranges from approximately 2-3 MeV, is ideally suited to (n,p) radionuclide production. However, the lower achievable flux from these generators limits their production capabilities.An additional complication is the relative paucity of high-quality, consistent cross section data for neutrons in the 2-3 MeV DD energy range.The purpose of the present work is to explore the potential to use high-flux neutron generators to produce high-specific activity samples of radionuclides at the mCi level for local use in the application community. The research group at UC Berkeley hasdeveloped a High Flux Neutron Generator (HFNG) that features an internal target where samples can be placed just several millimeters from the neutron producing surface in order to maximize the utilization of the neutron yield for the production of a desired radionuclide <cit.>.The HFNG uses the D(D,n)^3He reaction to produce neutrons with energies near 2.45 MeV together with a self-loading target design to maintain continuous operation without target replacement.In addition to the generator itself, efforts are underway to design neutron reflection capabilities to allow scattered neutrons multiple opportunities to interact with aninternally mounted target. While these design efforts are underway, the HFNG can be used to better characterize production cross sections at the appropriate neutron energy. The present work features a pair of cross section measurements for the production of two emerging non-standard medical radionuclides: the positron emitter ^64Zn(n,p)^64Cu and the single - photon emission computed tomography (SPECT) tracer ^47Ti(n,p)^47Sc. ^64Cu(t_1/2 = 12.7 h) undergoes β^+ decay (61.5% branching ratio) to ^64Ni or β^- decay (38.5% branching ratio) to ^64Zn <cit.>. The emitted short-range 190-keV β^- particle makes this anattractivetherapeutic radionuclide, which also has the possibility for simultaneous positron emission tomography (PET) imaging for real-time dose monitoring and verification. This makes ^64Cu particularly desirablefor emerging radiation therapy protocols <cit.>. In addition, copper radiochemistry is well developed, and many existing ligands and carriers may be used for selective delivery of the radionuclide to different sites in patients. The second radionuclide studied, ^47Sc (t_1/2 = 3.35 d), undergoes β^- decay to ^47Ti, emitting a high-intensity (63.8%) 159-keV gamma ray in the process <cit.>. This radionuclide isattractive as an emerging diagnostic isotope, due to the similarity of the emitted gamma ray to that of thewell-established ^99mTc <cit.>. Due to the short half-life (t_1/2 = 6.0 h) of and dwindling supplies of ^99mTc, ^47Sc stands poised as a potential solution to this shortage, due to its longer half-life and multiple production pathways without the need for highly enriched uranium <cit.>. In addition, when paired with ^44Sc, ^47Sc forms a promising theranostic pair for use in simultaneous therapeutic and diagnostic applications <cit.>.Current methodology in radiochemistry has shown recovery of upwards of 95% of produced ^64Cu <cit.> and ^47Sc <cit.> from solid target designs, without the need for additional carrier. By expanding the base of efficient reaction pathways, great advances are possible in making production of medical radionuclides more efficient and affordable for those in need.It is this desire to improve the options available for modern medical imaging and cancer therapy which has motivated the campaign of nuclear data measurements for isotope production at the UC Berkeley HFNG.§ EXPERIMENT§.§ Neutron source Neutron activation was carried out via irradiation in the High-Flux Neutron Generator (HFNG), a DD neutron generator at the University of California, Berkeley. This generator extracts deuterium ions from an RF-heated deuterium plasma (using ion sources similar todesigns from the Lawrence Berkeley National Laboratory <cit.>) through a nozzle, whose shape was designed to form a flat-profile beam, 5 mm in diameter. This deuterium beam is incident upon a water-cooled, self-loading titanium-coated copper target <cit.>, where the titanium layer acts as a reaction surface for DD fusion, producing neutrons with a well-known energy distribution as a function ofemission angle <cit.>. While the machine's design features two deuterium ion sources impinging from both sides of the target, only a single source was used in the present work. Irradiation targets are inserted in the center of the titanium layer deuteron target, approximately 8 mm from the DD reaction surface, prior to startup. <ref> displays a cut-away schematic of the HFNG. A 100 keV deuterium beam was extracted at 1.3 mA, creating a flux of approximately1.37 neutrons/cm^2s on the target.§.§ Cross section determination by relative activationThe approach used in both measurements was to irradiate foils of zinc or titanium, which were co-loaded with indium foils in order to determine their (n,p) cross sections relative to the well-established ^113In(n,n')^113mIn and ^115In(n,n')^115mIn neutron dosimetry standards <cit.>. <ref> lists physical characteristics of each foil for the various irradiations. In each experiment, the co-loaded foils were irradiated for 3 hours at nominal operating conditions of 1.3 mA and 100 kV.After irradiation, the foils were removed and placed in front of an appropriate High-Purity Germanium (HPGe) gamma-ray detector and time-dependent decay gamma-ray spectra were collected.One cm diameter, 1-mm thick natural abundance zinc and titanium targets were employed for the measurement. Each of these wasco-loaded with a natural abundance Indium foil of 1 cm diameter and 0.5 mm thickness in a recess cut into a 2-mm thick polyethylene holder, as seen in <ref>, which was mounted in the HFNG target center. Prior to loading, each foil was washed with isopropanol and dried, to remove any trace oils or residue that could become activated during irradiation. §.§ Determination of effective neutron energy The D(D,n)^3He reaction at 100 keV lab energy produces neutrons with energies ranging from2.18 to 2.78 MeV, over an angular range of 0-180 in the lab frame-of-reference with respect to the incident deuteron beam. This distribution has been well documented <cit.> and is shown in <ref> for 100 keV incident deuteron energy.Since the samples are separated by only8 mm from the DD reaction surface theysubtend a fairly significant(17)angular range in a region ofhigh(approximately 1.37neutrons/cm^2s) neutron flux. This stands in contrast to other measurements which feature collimated beams and significantly lower total neutron flux.The Monte Carlo N-Particle transport codeMCNP6 <cit.> was used to model the neutron energy spectrum incident upon target foils co-loaded into the HFNG (see <ref>). The neutron spectral distribution is also broadened by the temperature of the target. This gives rise to a slight difference in the neutron energy at the target location <cit.>, which has been included in our stated energy window. This spectrum, peaked around 2.777 MeV, illustrates the forward-focused kinematics of the DD reaction subtended by the co-loaded sample foils.As expected, theproduction target is the dominant source of scatter - approximately 0.78%of the neutrons incident on the foils can be attributed to scatter in the neutron production target.While this shows that the sample foils experience a very narrow energy distribution of incident neutrons, an effective neutron energy window must be determined. The MCNP6 simulation shows an identical flux-weighted average neutron energy of 2.765 MeV for both the Indium and target foils to the 1 keV level. Due to geometry and the kinematics of DD neutron emission, E_max,the maximum energy of a neutron subtending the target foils in this geometry is 2.783 MeV <cit.>. For this maximum energy, the number of reactions induced in a foil (containing N_T target nuclei) is given by: R = N_T ∫_0^E_maxσ(E) dϕdE dEFrom this definition, it is possible to calculate FE', the fraction of total reactions induced by neutrons up to some energy E' < E_max: FE' = ∫_0^E'σ(E) dϕdE dE∫_0^E_maxσ(E) dϕdE dEThis quantity FE' is plotted in <ref>. The fraction of total reactions in the indium foil can be used to characterize the effective neutron energy bin.Our approach, in analogy to the Gaussian quantity σ, will be to use a horizontal error barto represent the energy range responsible for 68.2% of the reactions taking place.Using this approach, we report the effective energy bin as being E_n=2.765^+0.014_-0.022 MeV. This 37-keV full-energy spread verifies that, at such close distances to the DD reaction surface, loaded target foils receive a quasi-monoenergetic neutron flux.§.§ Measurement of induced activities After irradiation, the co-loaded targets were removed from the HFNG and transferred to a counting lab, where their induced activities could be measured via gamma ray spectroscopy. Two detectors were used in this measurement. An Ortec 80% High-Purity Germanium (HPGe) detector was used for the detection of the positron annihilation radiation from the ^64Cudecay <cit.>, the 391 keV gamma-ray from the ^113mInisomer <cit.>, and the 336 keV gamma-ray from the decay of the ^115mInisomer <cit.>. An Ortec planar Low-Energy Photon Spectrometer (LEPS)was used for the detection of the lower-energy 159 keV gamma-ray from ^47Sc <cit.> as well as the two indium isomers mentioned above. Both detectors were calibrated for energy and efficiency, using ^133Ba, ^137Cs, and ^152Eu sources at various distances from the front face of each detector. These efficiencies, along with gamma ray intensities for each transition, were used to convert the integrated counts in each gamma ray photopeak into an activity for the activated isotopes and isomeric states.The irradiated foils were counted in their polyethylene holder, 10 cm from the front face of the 80% HPGe and 1 cm from the front face of the LEPS, with the target foil (zinc or titanium) facing towards the front face of the detector when both target and monitor foils were counted simultaneously. All data collection was performed using the Ortec MAESTRO software. For each experiment the detector dead time was verified to be less than 5%.No summing corrections needed to be made since all of the gammas are either non-coincident or formed in a back-to-back annihilation event. For the^47Sc production experiments, the foils were counted simultaneously using a planar LEPS detector. For the ^64Cu production experiments, the Indium foil was first counted separately using an 80% HPGe detector, to capture the short-lived Indium activities. This is due to the fact that the contaminant ^115In(n,γ) reaction results in the production of ^116mIn which has a 54 minute half-life and results in the production of 1097 keV (58.5% branching), 1293 keV (84.8% branching) and 2112 keV (15.09% branching) gamma-rays that in turn produce a significant number of 511 keV gammas from pair-production followed by annihilation <cit.>. The foils were counted together again after approximately 4 hours of separate collection, to allow for nearly all of the produced ^116In to decay. Example spectra for each production pathway can be seen in <ref> and <ref>.To verify that each peak corresponds to the assigned decay product, spectra were acquired in a sequence of 15 - 30 minute intervals. The resulting time series displayed in Figures <ref> - <ref> allow the fitting of exponential decay functions for each nuclide and comparison of the measured half-life with literature values. The fitted functions for each transition agree (at the 1σ confidence level) with accepted half-lives <cit.>, confirming the respective peak assignments.The spectra for each sample were summed and the net peak areas were fitted using gf3, part of the RadWare analysis package from Oak Ridge National Laboratory <cit.>. The background-subtracted integrated counts in each photopeak, as well as the counting duration for each experiment, are tabulated in <ref>.§.§ Experimental verification of incident neutron energy As shown in <ref> above, the effective neutron energy depends on the angle range subtended by the sample with respect to the incident deuteron beam.In order to determine this angle it is necessary to measure the lateral location of the beam with respect to the sample location.This centroid position of the beam was measured using a 3 x 3 array of 0.5 cm diameter indium foils.The relative activity of these foils was then determined via post-irradiation counting of the ^115mIn isomer (t_1/2 = 4.486 h) <cit.>.<ref>shows the measured activities for these 9 indium foils.Based on these values we are able to verify that the beam was indeed vertically centered on the middle of the zinc and titanium samples,with a slight asymmetry of the neutron flux in the horizontal direction, accounted for in MCNP6 modeling of the energy-differential neutron flux. This small asymmetry likely contributes to the effective energy bin being lower than the 2.78 MeV expected for 0 neutron emission angle in <ref>. §.§ Calculation of measured cross sectionsFor a thin target consisting of N_T target nuclei (with a reaction cross section σE̅), subjected to a constant neutron flux ϕE̅, the rate of production (R) of the product nucleus will be: R = N_T σE̅ϕE̅ If the target is subjected to this fluxfor an irradiationtime t_i and decays for adelay time t_d (after end-of-beam) before gamma ray spectrum acquisition occurs for a counting timet_c, then the number of product decays (N_D; with decay constant λ) during the acquisition will be:N_D= Rλ1 - e^-λ t_i e^-λ t_d1 - e^-λ t_c= N_T σE̅ϕE̅λ1 - e^-λ t_i e^-λ t_d1 - e^-λ t_c If this decay emits a gamma ray with absolute intensity I_γ (photons emitted per decay), and is detected with an absolute efficiencyof ϵ_γ (photons detected / photons emitted), then the number of observed gamma rays during the acquisition will be: N_γ = N_D ϵ_γ I_γ=ϵ_γ I_γN_T σE̅ϕE̅λ1 - e^-λ t_ie^-λ t_d1 - e^-λ t_c Solving this equation for thecross section results in: σE̅ = N_γλN_T ϵ_γ I_γϕE̅1 - e^-λ t_i e^-λ t_d1 - e^-λ t_c <ref> can be used to determine the unknown (n,p) cross sections relative to the well-known ^115In(n,n')^115mIn and ^113In(n,n')^113mIn inelastic scattering cross sections since the Zn and Ti samples were co-irradiated with indium foils. This approach has a number of advantages since the result is independent of neutron flux and only depends on the relative detector efficiencies at each gamma-ray energy.<ref> shows the ratio of the cross sections determined using this approach, in which subscript P indicates a quantity for either ^64Cu or ^47Sc, and subscript In indicates a quantity for either the ^113mIn or ^115mIn isomer. A minor term was added to correct for the small self-attenuation of the gamma rays emitted by the activated foils: σ_Pσ_In =N_γ,PN_γ,InN_T,InN_T,Pλ_Pλ_In1 - e^-λ_Int_i1 - e^-λ_Pt_ie^-λ_Int_d e^-λ_Pt_d××1 - e^-λ_Int_c1 - e^-λ_Pt_cϵ_Inϵ_PI_γ,InI_γ,Pe^-μ_Inx_In/2× e^-μ_Inx_Pe^-μ_Px_P/2 where: * N_γ is the integrated counts under a photopeak,* σ is the cross section for either the production of a product or isomer [mb], * N_T is the initial number of target nuclei, * λis the decay constant [s^-1], * t_i is the irradiation time [s], * t_d is the delay time (between the end-of-beam and the start of counting) [s], * t_c is the counting time [s], * ϵ is thedetector efficiency for a particular photopeak, * I_γ is the decay gamma ray absolute intensity [%], * μ is the photon attenuation coefficient for a particular decay gamma ray in a foil [cm^-1], * and x is the thickness of foil traversed by a particular decay gamma ray [cm]In addition to the ^115In(n,n')^115mIn reference cross section, the ^115In(n,γ)^116mIn(t_1/2 = 54.29 min <cit.>) activity can be used to determine the^64Zn(n,p) and ^47Ti(n,p) cross section.The capture activity is potentially subject to contamination from lower energy, especially thermal, room return neutrons since the (n,γ) cross section at 25 meV is approximately 2,000 times greater than at 2.7 MeV<cit.>.With the exception of decay constantsand time measurement, which have negligible uncertainty compared to other sources of uncertainties in this work, each of the parameters in this model carries an uncertainty. Based on the assumption that these uncertainties are uncorrelated, the total relative statistical uncertainty δ_σ is calculated by taking the quadrature sum of the relative uncertainties of each parameterδ_i: δ_σ = _2 = √(∑_i=1^Nδ_i^2) This totaluncertainty is plotted as the cross section uncertainty in <ref> and <ref>. §.§ Systematic uncertainties The largest source of systematic uncertainty in the cross section determined via the ratio approach is the 2.586% uncertainty in the ^115In(n,n')^115mIncross section and the 1.447% uncertainty in the ^113In(n,n')^113mIncross section<cit.>.An additional uncertainty arises from the fact that the Zn/Ti samples are not located at exactly the same location as the indium monitor foils, and are therefore not subject to precisely the same neutron flux.However, the MCNP6 simulations shown in <ref> indicate that the difference in the flux that the two foils are subjected to is less than 1%, negligible compared to other sources of systematic uncertainty.Other monitor foils could be used instead of indium, with ^58Ni(n,p)^58Co (^58Co t_1/2 = 70.86 d <cit.>) being one possible candidate, but the 4.486 hour and 99.476 minute half-lives of the ^115mIn and ^113mIn isomers <cit.>, respectively, make indium a better candidate for measuring the production of radionuclides with lifetimes much less than 71 days. The largest source of uncertainty in energy window arises from uncertainties in the actual dimension of the deuteron beam on the production target.We believe, based on burn marks on the neutron production target, that the beam was approximately circular, with a flat intensity profile and a 5 mm diameter.However, every 1 mm change in the beam radius would cause a 0.028 MeV shift in the centroid and a 0.053 MeV increase in the effective energy bin width, which places a natural limit on the reported effective neutron energy.A much smaller systematic uncertainty arises from the fact that the two (n,p) cross sections and the reference In(n,n') cross sections have slightly different thresholds. The total activity in the In produced by the low energy neutrons (below the knee near 2.25 MeV in<ref>) is 2.17%. The corresponding values from TALYS for the ^64Cu and ^47Sc activity are 0.24% and 0.85%, respectively. If we assume an uncertainty of ±25% in the TALYS calculations in this energy region it would introduce an additional systematic uncertainty in the ^+10_-20 keV effective energy bin of ±1.6 keV for ^64Cu and ±5.7 keV for ^47Sc.As these are smaller than the precision of the existing effective energy bin, they can be considered negligible.§ RESULTSUsing the ratio method described, the cross sections for the ^47Ti(n,p)^47Sc and ^64Zn(n,p)^64Cu reactions have been calculated for an incident neutron energy of E_n =2.76^+0.01_-0.02 MeV. These values are recorded in <ref>.Figures <ref> and <ref> present the determined cross sections for the production of^47Ti(n,p)^47Sc and ^64Zn(n,p)^64Curelative to literature data retrieved from EXFOR <cit.>. The weighted average of the measurements give 49.3 ± 2.6 mb (relative to ^113In) and 46.4 ± 1.7 mb (relative to ^115In) for ^64Zn(n,p)^64Cu, and 26.26 ±0.82 mb for^47Ti(n,p)^47Sc. The ^64Zn(n,p)^64Cu cross section measured in this work is consistent withother literature results, but with a smaller uncertainty (5%). However, in the case of the ^47Ti(n,p)^47Sc cross section, our results are consistent with the results from the Smith (1975), Armitage (1967), and Ikeda (1990) groups <cit.> and both the ENDF/B-VII.1 <cit.> and TALYS <cit.>values, but significantly below the results from the Hussain (1983), Gonzalez (1962), and Shimizu (2004) groups <cit.>.As mentioned above, the cross section can be obtained relative to both the inelastic scattering cross sections on ^113In and ^115In, and the capture of fast, unmoderated neutrons on ^115In. The result for the production of ^116In via the ^115In(n,γ) reaction was shown to be consistent with activation predominantly from the capture of fast neutrons, rather than from room return thermal neutrons. The MCNP neutron spectrum in <ref> confirms this - thermal and epithermal neutrons make up only 0.0771% of the total neutron population.This will be discussed in greater detail in the conclusion section below. § DISCUSSIONThe proximity of the target to the neutron production surface opens the possibility of performing a measurement of the cross section over a limited energy range via mounting the samples slightly off-axis with respect to the beam.This could be accomplished using the 9-foil sample holder described in <ref> above.Mounting samples at each of these positions would subject the samples to neutrons with energies ranging from 2.765 MeV at the central location to 2.616 MeV at the four corners, with the other locations having intermediate energy values.These sorts of multi-sample measurements could be used to determine the rising edge of the cross sections, aiding in the development of optical models for the reactants.These measurements also highlight the possibility of using fast neutrons from DD and/or DT generators to produce meaningful quantities of radioisotopes for a wide range of applications via charge exchange reactions, such as (n,p) and (n,α).Many applications, including diagnostic and therapeutic medical use, require mCi activity levels.For the production of a radionuclide sample, the saturation activity (A_saturation) is achieved at secular equilibrium:R_production = R_decay = λ N_productWhile the saturation activity represents the maximum activity that can be made at a generator with a given total neutron output, there may be situations where either a smaller activity is needed, or a shorter irradiation is desired.In this case, it is useful to introduce a neutron utilization factor (η_x).η_x is the constant of proportionality between R_n, the neutron source output (in neutrons/second), and the saturation activity: A_saturation = η_x R_n η_x represents the likelihood that a neutron produced in the generator will create x, the isotope of interest.It includes the overlap between the production target and the locus where the neutrons are being created, and the fraction of nuclear reactions which generate the desired activity x:η_x= 1R_n∫_production targetϕ𝐫σ̅_x ρ_target𝐫d𝐕,d𝐕 = 𝐫^2 d𝐫sinθ dθ dφwhere σ̅_x is the average cross section producing the radionuclide of interest, ρ_target𝐫 is the density of the target as a function of position, and ϕ𝐫 is the neutron flux (in n/cm^2/s) as a function of position. η_x allows us to cast the activity produced in a given irradiation time t_i as: At_i = η_x R_n 1-e^-λ t_i Maximizing η_x would be the goal of any engineering design to produce a desired activity using a neutron generator at a minimum of cost and radiological impact. An optimal design for the neutron generator would also allow for the possibility of reflecting fast neutrons back onto the target to maximize their utilization for radionuclide production.This sort of flux trap has been used for the production of radionuclides in reactors, but has not to date been optimized for use with fast neutronsat DD and/or DT neutron sources.The HFNG, with its self-loading target and flux trap geometry, has many features that make it well-suited for such isotope production purpose.Switching to DT operation would dramatically increase the flux as well as the production cross section, since (n,p) tends to be significantly larger at 14 MeV.However, the higher neutron energy would also open the (n,pn) channels.In the case of ^47Sc, this would lead to the presence of ^46Sc (t_1/2 = 83.79 d <cit.>) in the sample, which might pose some concerns for medical applications.However, this is not an issue for ^64Cu since the (n,pn) channel leads to the production of stable ^63Cu. Assuming a neutron flux of1.37 neutrons/cm^2s on the target, masses of 0.533 g of natural zinc and 0.337 g of natural titanium, and cross sections of 47.5 mb for ^64Zn(n,p)^64Cu and 26.26mb for^47Ti(n,p)^47Sc, theoretical saturation activities for current operation at the time of this work are estimated to be 1.5 kBq of ^64Cu and 0.11 kBq of ^47Sc. This falls short of the mCi (37 MBq) level required for commercial application by a factor of 3-4 orders of magnitude, but with the operation of the second deuterium ion source, increased current, and fast neutron reflection, this goal maywell be within reach.By increasing the activation target thickness to 1 cm (a factor of 10), switching to DT operation (a factor of 80), increasing current and running the second ion source (a factor of 60), and relying upon the higher (n,p) cross section at DT energies (a factor of approximately 3), we believe saturation activities of approximately 6 mCi of ^64Cu and 0.5 mCi of ^47Sc can be achieved. The activities produced at the end of irradiation averaged 453.8 Bq of ^64Cu, and 31.6 Bq of ^47Sc. Assuming a conservative neutron source output of 10^8 neutrons / second, we can estimate that, in present operation, the HFNG has an average η_64Cu≈3.0-5 for ^64Cu and η_47Sc≈1.1-5 for^47Sc.This falls approximately 4 orders of magnitude short of the η_x ≈ 0.37 needed for mCi-scale production. A factor of 10 in η_x could easily be gained through use of targets1-cm in thickness without worry of contaminating reaction channels opening up, but η_x gains beyond this will require modification of operation conditions. § CONCLUSION AND FUTURE WORK Using activation methods on thin foils, the ^47Ti(n,p)^47Sc and ^64Zn(n,p)^64Cu production cross sections were measured for2.76^+0.01_-0.02 MeV neutrons produced using the High Flux Neutron Generator (HFNG) at UC Berkeley. The cross sections were measured with less than5% uncertainty relative to the well-known ^115In(n,n')^115mIn and ^113In(n,n')^113mIn fast neutron cross sections <cit.>. The measured values of26.26 ±0.82 mb and49.3 ± 2.6 mb (relative to ^113In) or 46.4 ± 1.7 mb (relative to ^115In), respectively, are consistent with earlier experimental data and theoretical models, but have smaller uncertainties than previous measurements.In addition, the production of the ^116Invia the ^115In(n,γ) reaction was close to the value one would expect given an effective incident neutron energy of 2.45 MeV.While this is not consistent with the average neutron energy at the target location (2.76^+0.01_-0.02 MeV), the fact that it was close indicates the paucity of thermal neutrons in this central location.This in turn highlights the usefulness of such compact DD-neutron sourcesfor producing clean activities via the (n,p) channel. The use of DD neutron generators can be an efficient method for the measurement of low-energy (n,p) reaction channels, as well as a relative method used to normalize measurements at higher neutron energies.In addition to improving the value of these measurements for nuclear reaction evaluation, our results highlight the potential use of compact neutron generators for the production of radionuclides locally for medical applications. It is worth noting that at the time of publication, theHFNG is now operating at close to 10^9 n/sec, with a clear path towards 10^10. Future work will involve the continued measurement of the (n,p) production cross sections for various other emerging therapeutic and diagnostic radioisotopes, to expand the toolset of options available for modern medical imaging and cancer therapy. This will focus on radionuclides which permit more customized and precise dose deposition, as well as patient-specific treatments. § ACKNOWLEDGEMENTS We would like to particularly point out the crucial role played by Cory Waltz in the design and commissioning of the HFNG.We acknowledge Glenn Jones of G&J Jones Enterprises of Dublin, CA for the constructionof the High Flux Neutron Generator. Lastly, we would like to acknowledge the students in the Nuclear Reactions and Radiation (NE102) laboratory course at UC Berkeley who participated in these experiments, including Joe Corvino, Nizelle Fajardo, Scott Parker and Evan Still.This work has been carried out at the University of California, Berkeley, and performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract # DE-AC52-07NA27344 and Lawrence Berkeley National Laboratory under contract # DE-AC02-05CH11231. Funding has been provided from the US Nuclear Regulatory Commission, the US Nuclear Data Program, the Berkeley Geochronology Center, NSF ARRA Grant # EAR-0960138, the University of California Laboratory Fees Research Grant # 12-LR-238745, andDFG Research Fellowship # RU 2065/1-1.elsarticle-num | http://arxiv.org/abs/1704.08761v2 | {
"authors": [
"Andrew Voyles",
"M. S. Basunia",
"Jon Batchelder",
"Joseph Bauer",
"Tim Becker",
"Lee Bernstein",
"Eric Matthews",
"Paul Renne",
"Daniel Rutte",
"Mauricio Unzueta",
"Karl van Bibber"
],
"categories": [
"nucl-ex"
],
"primary_category": "nucl-ex",
"published": "20170427220634",
"title": "Measurement of the $^{64}$Zn,$^{47}$Ti(n,p) Cross Sections using a DD Neutron Generator for Medical Isotope Studies"
} |
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