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[ "AMS-Net: Adaptive Multiscale Sparse Neural Network with Interpretable Basis Expansion for Multiphase Flow Problems", "AMS-Net: Adaptive Multiscale Sparse Neural Network with Interpretable Basis Expansion for Multiphase Flow Problems" ]	[ "Yating Wang \nDepartment of Mathematics\nPurdue University\n47907West LafayetteINUSA\n", "Tat Wing ", "Leung \nDepartment of Mathematics\nUniversity of California\n92697Irvine, IrvineCAUSA\n", "Guang Lin \nDepartment of Mathematics\nSchool of Mechanical Engineering\nDepartment of Statis-tics (Courtesy)\nDepartment of Earth, Atmospheric, and Planetary Sciences (Courtesy)\nPurdue University\n47907West LafayetteINUSA\n" ]	[ "Department of Mathematics\nPurdue University\n47907West LafayetteINUSA", "Department of Mathematics\nUniversity of California\n92697Irvine, IrvineCAUSA", "Department of Mathematics\nSchool of Mechanical Engineering\nDepartment of Statis-tics (Courtesy)\nDepartment of Earth, Atmospheric, and Planetary Sciences (Courtesy)\nPurdue University\n47907West LafayetteINUSA" ]	[]	In this work, we propose an adaptive sparse learning algorithm that can be applied to learn the physical processes and obtain a sparse representation of the solution given a large snapshot space. Assume that there is a rich class of precomputed basis functions that can be used to approximate the quantity of interest. For instance, in the simulation of multiscale flow system, one can adopt mixed multiscale methods to compute velocity bases from local problems and apply the proper orthogonal decomposition (POD) method to construct bases for the saturation equation. We then design a neural network architecture to learn the coefficients of solutions in the spaces which are spanned by these basis functions. The information of the basis functions are incorporated in the loss function, which minimizes the differences between the downscaled reduced order solutions and reference solutions at multiple time steps. The network contains multiple submodules and the solutions at different time steps can be learned simultaneously. We propose some strategies in the learning framework to identify important degrees of freedom. To find a sparse solution representation, a soft thresholding operator is applied to enforce the sparsity of the output coefficient vectors of the neural network. To avoid over-simplification and enrich the approximation space, some degrees of freedom can be added back to the system through a greedy algorithm. In both scenarios, that is, removing and adding degrees of freedoms, the corresponding network connections are pruned or reactivated guided by the magnitude of the solution coefficients obtained from the network outputs. The proposed adaptive learning process are applied to some toy case examples to demonstrate that it can achieve a good basis selection and accurate approximation. More numerical tests are successfully performed on two-phase multiscale flow problems to show the capability and interpretability of the proposed method on complicated applications.	10.1137/21m1405289	[ "https://export.arxiv.org/pdf/2207.11735v1.pdf" ]	250,006,549	2207.11735	65e75c3d90eec4fbfe24008a4727b6bb188dfe4e	AMS-Net: Adaptive Multiscale Sparse Neural Network with Interpretable Basis Expansion for Multiphase Flow Problems July 26, 2022 24 Jul 2022 Yating Wang Department of Mathematics Purdue University 47907West LafayetteINUSA Tat Wing Leung Department of Mathematics University of California 92697Irvine, IrvineCAUSA Guang Lin Department of Mathematics School of Mechanical Engineering Department of Statis-tics (Courtesy) Department of Earth, Atmospheric, and Planetary Sciences (Courtesy) Purdue University 47907West LafayetteINUSA AMS-Net: Adaptive Multiscale Sparse Neural Network with Interpretable Basis Expansion for Multiphase Flow Problems July 26, 2022 24 Jul 20221 In this work, we propose an adaptive sparse learning algorithm that can be applied to learn the physical processes and obtain a sparse representation of the solution given a large snapshot space. Assume that there is a rich class of precomputed basis functions that can be used to approximate the quantity of interest. For instance, in the simulation of multiscale flow system, one can adopt mixed multiscale methods to compute velocity bases from local problems and apply the proper orthogonal decomposition (POD) method to construct bases for the saturation equation. We then design a neural network architecture to learn the coefficients of solutions in the spaces which are spanned by these basis functions. The information of the basis functions are incorporated in the loss function, which minimizes the differences between the downscaled reduced order solutions and reference solutions at multiple time steps. The network contains multiple submodules and the solutions at different time steps can be learned simultaneously. We propose some strategies in the learning framework to identify important degrees of freedom. To find a sparse solution representation, a soft thresholding operator is applied to enforce the sparsity of the output coefficient vectors of the neural network. To avoid over-simplification and enrich the approximation space, some degrees of freedom can be added back to the system through a greedy algorithm. In both scenarios, that is, removing and adding degrees of freedoms, the corresponding network connections are pruned or reactivated guided by the magnitude of the solution coefficients obtained from the network outputs. The proposed adaptive learning process are applied to some toy case examples to demonstrate that it can achieve a good basis selection and accurate approximation. More numerical tests are successfully performed on two-phase multiscale flow problems to show the capability and interpretability of the proposed method on complicated applications. Introduction Dynamical systems of flow and transport process in heterogeneous media are naturally existing in diverse science and engineering applications, such as groundwater flow, reservoir management, and so on. These physical problems are usually formulated in domains containing multiple scales, such as fractures at multiple length scales, or pores ranging from centimeters to meters. Numerical simulations for these problems are challenging since recovering all scale information will result in heavy computational burden. Furthermore, due to the lack of finest scale information, there are usually uncertainties in the computational model. It is necessary to develop model reduction techniques [14,22,21,9,1,3] and construct fast alternatives to perform efficient simulations. The reduced order model can represent the physical properties of the full problem and can speed up the computations for the forward problem, which eventually helps to quantify the uncertainties in the model. There are many model reduction methods including local and global approaches and have achieved significant success in numerous applications. In the family of local approaches, one can formulate appropriate local problems on coarse grid regions, construct effective properties or local multiscale basis functions, and further develop global systems on the coarse grid level. For instance, numerical upscaling, multiscale methods, and generalized multiscale methods [16,15,13,11,12,10]. For global approaches [17,2,6,37], such as the proper orthogonal decomposition method, one computes snapshots by solving several global problems and performs spectral decomposition to select the dominant modes. It has been extensively applied for numerical simulations of dynamical systems but still encounters difficulties for nonlinear problems. The objective of our work is to propose a framework which combines advanced deep learning techniques and multiscale basis construction methods, to obtain multiscale solutions with a sparse representation in the snapshot space. Deep learning has become a quite popular approach for numerical approximation of nonlinear differential equations in recent days. Applications include developing surrogate models based on the properties of classical numerical solvers, such as constructing a multiscale neural network based on hierarchical multigrid solvers and encoder-decoder neural networks for solutions of heterogeneous elliptic PDEs [18,39,24,26,4]. Physics-informed neural networks [32,31,27,28] were proposed to incorporate physical laws in the loss function and limited data to train the neural network, and then get approximations of solutions in the whole temporal-spatial domain. However, learning full fine-scale solutions is challenging due to the extremely large number of parameters in the neural networks, some algorithms were established to design sparse neural network models to learn flow dynamics with high dimensional stochastic input coefficients [36,35]. Some other approaches include learning coarse grid effective properties using the nonlocal multicontinuum upscaling method or coefficients in the proper orthogonal decomposition (POD) projections [34,8]. Furthermore, a deep neural network combined with multiscale model reduction techniques was investigated [33] where the forward operators of the flow problem were learned in a reduced way without using POD approaches. In these works, the coefficients in the reduced order model are designed to have physical meanings, thus learning these quantities can provide important physical information without downscaling the coarse scale solution vector. In a more general setting, the coefficients of the basis functions are not directly related to the quantities of interest, then one may incorporate the basis functions as prior information in the training process. Some prior task-dependent dictionaries are incorporated to PINN method [30], and an algorithm is proposed to take advantage of the features provided by dictionaries and achieve faster convergence. In this work, we are interested in the multiscale two-phase problem in subsurface flow applications, where the equations are nonlinear and time dependent. One can parameterize the nonlinear equations and compute suitable basis functions for a set of sample parameters. However, the formed dictionaries may be too large and only a sparse selection of the bases in the dictionaries are needed for the solution approximation. Some model reduction techniques such as reduced basis method or greedy algorithm [5,29,23,20,19,25,38] has been applied to solve parameterized elliptic PDEs. We aim to design an adaptive sparse learning algorithm with the help of precomputed basis functions as the prior dictionary, and apply it to the coupled two-phase flow systems. To be specific, for the construction of the prior dictionary for the flow equation where the model coefficient has high-contrast multiscale features, we adopt the mixed generalized multiscale finite element method [11,7] for velocity multiscale basis construction. Given a specific source configuration, we first solve the system at several time instances in a small time interval. The fine scale saturation profiles at these time instances are used to form relative permeabilities in the flow equation. With this parameterization of the permeability, one can solve appropriate local problems on the coarse regions to get the velocity basis. Combining the local solutions in all coarse regions for each permeability configuration, we obtain a dictionary of basis functions which can be used to approximate the solutions of flow equations for different source terms. As for the saturation solution, we again use the saturation solutions described before as snapshots, and perform POD on the snapshot space. The dictionary for saturation approximation consists of all POD bases. It can be used to seek for solutions in later time steps given different source terms. We will then design suitable neural networks to learn the coefficients of the solutions in the reduced order spaces which are spanned by the bases of the prior dictionaries. However, due to the predefined dictionaries provide high-dimensional spaces, and only parts of basis functions are needed in the solution representation. We aim to reduce the solution space by an adaptive sparse learning approach. The idea is to first adopt the soft-thresholding technique to enforce the sparsity of the coefficient vectors learned from the neural network. Next, the network connections are pruned according to the sparsity of the coefficient output. This procedure helps to get rid of the less important basis functions in the solution representation and simplify the connections in the network architecture. However, if too many basis functions are dropped, the approximation space will not be sufficient to produce good approximations. We will further add some bases back through a greedy algorithm to enrich the approximation space, and the corresponding network connections will be reactivated simultaneously. The number of bases one would like to include in the approximation space can be fixed in advance, or the accuracy of the approximation can be prespecified. By an adaptive learning process, we expect to achieve a good basis selection and accurate approximation. Moreover, in our network architecture, submodules are designed to approximate the map from input to the first time step, and from previous time steps to later time steps. The final network is the composition of several submodules, and we are learning the entire dynamics, i.e, the solutions at all time steps, simultaneously. The loss functions are designed to minimize the differences between the downscaled reduced order solutions and fine scale solutions over all time steps. The main contributions of our work are: • Network functionality. We design a neural network architecture with an adaptive sparse learning algorithm, which can be applied to learn the map in the physical problem from the source term to the expansion coefficients of the multiscale basis in the solutions at many time steps. It learns the dynamics and identifies the sparse patterns simultaneously. • Adaptivity. The sparsity of the basis expansion coefficient is enforced via soft thresholding. It can remove a large number of less important degrees of freedom during the training. On the other hand, to improve the accuracy, we can add some overdropped degrees of freedom back adaptively based on a greedy process. We observe that our proposed method achieves better accuracy compared to the projection solutions computed using the basis selected from the standard greedy algorithm/POD algorithm in some cases. • Interpretability. Besides the accuracy benefits, the proposed adaptive learning algorithm can discover an active set of bases and select important degrees of freedom for the quantities of interest. We show that the sparsity patterns of the network output are potentially interpretable in some applications. The paper is organized as follows. In Section 2, we describe the preliminaries of the model problem. The main methodology is discussed in Section 3 where we show the detailed construction of dictionary and the main algorithm. The numerical tests are demonstrated in Section 4 to illustrate the capability of the proposed network. The numerical results show the efficiency and accuracy of our method. A conclusion is presented in Section 5. Problem Setup We consider the problem Lu = f(1) where L is a nonlinear time-dependent differential operator which contains multiscale features. We would like to seek the solution in an N dimensional space V H = span{φ 1 , φ 2 , · · · , φ N }, where φ i -s are precomputed snapshot bases. Denote by u j the solution at time step j, and suppose it has a sparse representation in this space, that is u j = N i=1 c j i φ i(2) where c j = [c j 1 , · · · , c j N ] T are sparse vectors. Let N (·; Θ) be a deep neural network parameterized by Θ, with given different realizations of source term f , we aim to use N to approximate the physical process, and realize sparse learning at the same time, that is {u n } ≈ N (f ; Θ, φ 1 , φ 2 , · · · , φ N ).(3) for all time step n. Methodology In this section, we will first present preliminaries for the problem of interest and the snapshot basis construction methods. Then we introduce our DNN architecture and training algorithm. We consider two-phase incompressible flow problem in heterogeneous porous media. The flows follow Darcy's law, and we neglect the capillary pressure and gravity effects in the model. The flow equation can be written as u = −λ(S)κ∇p in Ω div(u) = r in Ω u · n = 0 on ∂Ω(4) where κ is the absolute permeability field. The total mobility λ(S) = κ rw (S) µ w + κ ro (S) µ o and κ rw , κ ro are the relative permeability, µ w is the viscosity for water, µ o is the viscosity of oil. In real applications, κ rw , κ ro nonlinearly depends of S. With a simplified notation, we abbreviate S w to be S for the water phase . The saturation equation of S reads ∂S ∂t + u · ∇f (S) = q(5) where f (S) = κ rw (S)/µ w κ rw (S)/µ w + κ ro (S)/µ o , and q is the source term. The saturation solution can be computed using the finite volume method on the fine grid, and a backward Euler scheme can be used for the time discretization. For each fine block T i , the solution S i at time step n + 1 can be obtained by S n+1 i = S n i + dt \|T i \| [− e j ∈∂K i F ij (S n+1 ) + f (S n+1 )q − i + q + i ](6) where q − i = min(0, q i ), q + i = max(0, q i ). e j is the face between fine block T i and T j . Denote by u ij the velocity on the face e j , then F ij is F ij (S n+1 ) = e j (u n+1 ij · n)f w (S n+1 i ) if u n+1 ij · n ≥ 0 e j (u n+1 ij · n)f w (S n+1 j ) if u n+1 ij · n < 0(7) which is the upwinding flux. Dictionary construction We will consider two cases (1) learning the velocity dynamics, (2) learning the saturation dynamics, separately. In the first case, we will apply the mixed GMsFEM method [11] to construct the local velocity basis. In the second case, we can use POD to perform global model reduction to obtain snapshots for saturation approximation. Those basis functions constitute the dictionary and will be used in the corresponding training tasks. Local model reduction: Mixed GMsFEM basis for velocity Let S t be the saturation profiles at a few time instances t = 1, · · · , T , obtained by solving the problem with a specific configuration of f . Then for each S t , we compute the mobilities λ(S t ) and usẽ κ t := λ(S t )κ as different permeability profiles to compute basis functions. Denote by T H the coarse grid of the computational domain Ω. Let E H be the set containing all coarse scale edges on the grid. In mixed GMsFEM, define the local region ω i as ω i = K + i ∪ K − i if E i ∈ E H \∂Ω K i if E i ∈ ∂Ω which is a union of two coarse grid blocks sharing the edge E i , with i = 1, · · · , N e , and N e is the total number of coarse edges. The basis functions for the velocity fields are constructed for each ω i . To begin with, one constructs the snapshot space by solving local problems with a set of boundary conditions on ω i associated with a coarse edge. The normal traces of each basis with respect to the coarse edge are resolved up to the fine level. Specifically, denote by E i = L i j=1 e j , where e j is a fine edge on E i , L i is the number of fine edges on E i . For each j = 1, 2, · · · , L i , we seek for local solutions ψ ω i j by solving κ −1 t ψ ω i j,t + ∇p ω i j,t = 0 in ω i , div(ψ ω i j,t ) = α ω i j in ω i ψ ω i j,t · n i = δ ω i j on ∂ω i where δ ω i j is defined by δ ω i j = 1 on e j , 0 on ∂ω i \e j , with n i is the outward normal unit vector on E i . Moreover, α i,j satisfies the compatibility condition ω i α ω i j = ∂ω i ψ ω i j · n i At this point, we obtain the snapshot space V ω i snap,t = span{ψ ω i j,t , j = 1, · · · , L i }, where i = 1, · · · , N ω , t = 1, · · · , T , and N ω is the number of coarse edges in the computational domain. Next, one needs to propose a local spectral problem to perform model reduction for each V ω i snap,t . The problem is to find eigenvalues λ and eigen- functions v ∈ V ω i snap,t such that a i (v, w) = λs i (v, w), ∀w ∈ V ω i snap,t , where a i and s i are symmetric positive definite bilinear operators. As shown in [11], we can let a i (v, w) = E i κ −1 t (v · n i )(w · n i ), s i (v, w) = ω i κ −1 t v · w + div(v)div(w).(8) Denote by (λ ω i j,t , φ ω i j,t ) be the eigen-pairs solved from (8), where the eigenvalues are sorted in an ascending order. Then the first l i dominant modes are selected to form the offline space V ω i off,t . Finally, we take the union of all V ω i off,t as our dictionary. That is, D vel = {φ ω i j,t , j = 1, · · · , l i ; i = 1, · · · , N ω ; t = 1, · · · , T }.(9) Global model reduction of the saturation equation: POD basis construction In another perspective, we would like to learn saturation profiles and consider velocity as some hidden variables. Again, given some specific configuration of the source term f , we solve the system (4)-(6) sequentially. Denote by S t be the saturation profiles at a few time instances t = 1, · · · , T 0 ≤ T . These functions Φ = [S 1 , · · · , S T 0 ] form the snapshot space, and we will perform the proper orthogonal decomposition (POD) on it. To be specific, one performs SVD on Φ, Φ = V ΛW T where Λ is a diagonal matrix with singular values of Φ, V and W are the left and right singular matrices. Arrange the singular values in a decreasing order σ 1 ≥ σ 2 ≥ σ T 0 , one can then choose the corresponding first few singular vectors in V which capture the important modes in the dynamic process. Let φ j , j = 1, · · · , m be the vectors we chose, the POD space for saturation is then V sat = span{φ j , j = 1, · · · , m} D sat = {φ j , j = 1, · · · , m}(10) and we will use it as our dictionary for the approximation of saturation solutions. For a newly given configuration of the source term, one can seek S red t ≈ S t in the POD space. Network ingredients In this section, we present the main ingredients in our network architecture. Inputs and labels: We consider a two-dimensional input f ∈ R d×d which can be arbitrary source terms in the equation, and a set of labels y 1 ; · · · ; y T , where T is the total number of time steps, and y j ∈ R n . Here the labels y j can be velocity fields or saturation profile at time step j. Network outputs: The output of the network is denoted by c 1 , · · · , c T , where each c j = (c j 1 , · · · c j N ) T is a solution coefficient vector at time step j, with c j ∈ R N . Network architecture: For the neural network N , we will divide it into T submodules N = N T • N T −1 • · · · N 1(11) For the first submodule, we aim to learn a map N 1 from input f to c 1 . Let m be the number of layers in N 1 , which consists of some convolutional layers, an average pooling layer, and fully connected layers, that is N 1 := S γ 1 • L 1,m • σ • L 1,m−1 • σ • L 1,1 . Let K j be an appropriate pooling kernel or convolution kernel, L 1,j (x) = K j * x, j = 1, · · · , m − 2 L 1,m−1 (x) = W 1,m−1 (vec(x)) + b 1,m−1 L 1,m (x) = W 1,m x.(12) Moreover, we write the intermediate output from N 1 (f ) as c 1 . For the other submodules N t , t = 2, ·, T , we have N t := S γt • L t,m • σ • L t,m−1 • σ • L t,1(13) where L t,j (x) = W j,m−1 x + b j,m−1 L 1,m (x) = W 1,m x,(14) Here, σ is a nonlinear activation function, for example, leaky RELU, which is defined as σ(x) = x if t > 0 αx otherwise(15) for some constant α ∈ (0, 1). S γt is a soft-thresholding function, defined as S γt (x) = sign(x)(\|x\| − γ t ) +(16) with some constant threshold γ t . The soft-thresholding function will help us to enforce sparsity on the predicted solution coefficient vectors c. Similarly, we denote by c t the intermediate output from N t (f ). The architecture of the network can be illustrated as in Figure 1. Loss function with basis functions Given a set of training pairs {f k , (y 1 k , ·, y n k )}, our goal is then to find Θ * for the network N (·; Θ) by solving an optimization problem Θ * = argmin Θ L N (f ; Θ); {y j } T j=1 = argmin Θ 1 K K k=1 T j=1 \|\|y j k − Φc j k \|\| 2 2 ,(17) where Φ = [φ 1 φ 2 · · · φ N ] is the matrix formed by the precomputed bases. K is the number of the samples, N is the number of bases in the dictionary. We apply the preconditioned SGD to solve the optimization problem in (17). Adaptive sparse learning algorithm In this section, we will propose our main algorithm, the adaptive sparse learning algorithm. To reduce model order with sparse output coefficients To ensure the sparsity of the the model output, we apply the soft thresholding function (16), which can be further written as S γt (x) =      x − γ t if x ≥ γ t 0 if −γ < x < γ t x + γ t if x ≤ −γ t .(18) After the action of S γt , we obtain the output coefficients c t . The soft thresholding function will cut off those coefficients with small magnitudes. Then the sparse coefficient vector will be multiplied by the normalized basis function matrix Φ. This procedure results in a removal of some unimportant basis functions during the training. We remark that the soft-thresholding process is commonly involved in l 1 minimization algorithms. One of the ways to view this process is that the soft-thresholding function defines an active set in the optimization process. During the training, we observe that the gradient vector corresponding to the ith entry in the soft thresholding layer will vanish once the coefficient output in that layer is smaller than the thresholding parameter. Pruning network connections In our framework, we also want to enforce sparsity on the network connections based on the sparse pattern of intermediate network outputs c t . At this point, denote by Θ = Θ s ∪ Θ d the network parameters, where Θ s = {W t,m , W t+1,1 , for all t = 1, · · · , T − 1} corresponds to the parameters in the sparse layer, and Θ d corresponds to the parameters in the rest of layers. The weight matrix W can be referred to (12) and (14). We would like to cut connections to and from c t , based on the magnitude of c t j during the training. To be specific, if at the current training epoch, the c t j,k ≤ γ remove t (one can just take γ remove t = 0) for the k-th entry, then we will let W t,m (k, :) = 0, W t+1,1 (:, k) = 0. Adaptively enrich the solution space During the training, the sparse pruning procedure may overly drop some components, one can then adaptively add the basis back to the training by reactivating some previously pruned network connections. To illustrate the idea, denote by y true ∈ R n×Bn the reference solutions, y pred ∈ R n×Bn the corresponding predictions from the neural network, where B n is the number of samples in a batch. Let P u ∈ R mu×n be the matrix containing the unselected bases in the current stage, where m u is the number of bases left in the pool. One first computes the differences between the true and prediction solutions R t = y t true − y t pred . One then computes the inner product of the error R t with the unselected basis E t = P u R t .(19) Since a batch of samples is used when computing R t , we need to compute the absolute mean of E t across these samples, denoted byĒ t . We then sort the absolute meanĒ t in a descending manner. Denote by [i t 1 , i t 2 , · · · , i t mu ] the sorted indices which specifies how the elements of the absolute mean of E t were rearranged. Recall that the total number of bases in the dictionary is N , then the current number of selected bases is N − m u . There are two ways to add degrees of freedom back. • Fix the number of bases. The first approach is to set a target M as the total number of bases we want to include in the final solution representation. Then, if N − m u < M , we will select the first m c = M − (N − m u ) bases corresponds to [i t 1 , i t 2 , · · · , i t mc ], add them back in the solution representation, and remove them from the unselected pool P u . • Fix a threshold parameter. The second approach is to set the thresholding parameters δ add . That is, if [Ē t ] j < γ add t , then we let m c = j and add bases correspond to the columns with indices [i t 1 , i t 2 , · · · , i t mc ], and remove them from the pool P u . If one fixes the target number of bases in advance, it can have a control on the number of bases selected during the training process and produce a desired dimension for the reduced order model. This works for the case when we have an approximate bound for the number of important bases. More generally, without the knowledge of the exact number of bases are needed, we usually want to control the accuracy of the approximation. Then we can use the thresholding parameter γ add t to determine the number of bases to add in the process. The relationship between the threshold γ add t and the error is presented by the following lemma. Lemma 1. Let γ add t be a given threshold for adding bases. Assume the iteration process is converged to a static state where no basis will be added to the system in the iteration process. For any unselected basis φ ∈ R d , we have 1 B n Bn i=1 y t true,i − y t pred,i 2 − 1 B n Bn i=1 min (c 1 ,...c Bn )∈R Bn y t true,i − y t pred,i − φc i 2 < γ add t (20) where y t true,i and y t pred,i are the i-th column of y t true and y t pred , and i = 1, · · · , B n , B n is the batch size. Proof. If no bases are added to the system in the iteration process, we have [Ē t ] j < γ add t ∀j = 1, · · · , m u ; t = 1, · · · , T. whereĒ t ∈ R mu is the mean of E t over all samples as defined in (19). Thus, for any unselected basis φ ∈ R d Bn i=1 \|φ T (y t true,i − y t pred,i )\| < γ add t .(22) Since φ T φ = φ 2 2 = 1, we have y t ture,i − y t pred,i 2 − \|φ T (y t true,i − y t pred,i )\| ≤ y t ture,i − y t pred,i − φφ T (y t ture,i − y t pred,i ) 2 ≤ y t ture,i − y t pred,i − φc i 2 for any c i ∈ R. Therefore, we obtain 1 B n Bn i=1 y t ture,i − y t pred,i 2 − y t ture,i − y t pred,i − φc i 2 ≤ 1 B n Bn i=1 \|φ T (y t true,i − y t pred,i \| < γ add t for any (c 1 , . . . c Bn ) ∈ R Bn . This completes the proof. By this lemma, we can see that using the threshold γ add t to control the number of bases can give us a control on the error, in the sense that adding any unselected basis can only make a small improvement to the average l 2 error using the training data set. Furthermore, the corresponding rows or columns in the weight matrix will be reactivated, W t,m (j, :) = z, W t+1,1 (:, j) = z, j = i t 1 , i t 2 , · · · , i t mc where z are vectors with i.i.d samples generated from uniform distributions. The sparse procedures described above will be performed on an adaptive basis. Denote by Θ 0 the initial model parameters, n b the number of burning in steps, n e the total number of epochs, η the learning rate. Suppose we would like to update the sparsity information every other nn steps. Let γ remove t be the thresholding parameters for removing bases, γ add t be the thresholding parameter for adding bases. I t u is the unselected bases indices, I t c are the indices of bases we would like to add. M is the target number of bases, N is the total number of bases in the dictionary. The algorithm is summarized in Algorithm 1. c t ← output from N t , where N t is defined in (13) 5: where M = 9, thus we have 100 basis function in the set D 1,2 . I t u ← {j} such that \|c t j \| < γ remove t 6: m u ← length(I t u ) 7: Set weight parameters W t,m (I t u , :), W t+1,1 (:, I t u ) to zero in Θ i 8:Ē t ← 1 Bn Bn i=1 \|P u y t true,i − y t Linear case In the linear case, we take k(u) = 1, and f (x, y; α) = 2π 2 α 1 sin(πx) sin(πy) + 8π 2 α 2 sin(2πx) sin(2πy) + 10π 2 α 3 sin(3πx) sin(πy) + 20π 2 α 4 sin(4πx) sin(2πy) We assume the dataset for training is small, and there are 100 sample pairs. The network to be trained has only a submodule N 1 as described in Section 3.2. Among all samples, 80 percents are used for training, and 20 percents are used for testing. In this example, the bases included in the solutions are known exactly, so we only prune the network and remove some bases (the thresholding parameter is chosen to be 0.5), and do not add basis functions. The purpose is to see whether the adaptive pruning can choose the correct set of basis in the end. The numerical results indicate that, applying the adaptive sparse algorithm, the network can identify the 4 basis functions which constitute the ground truth u automatically. The number of bases selected during training is presented in Figure 2. In this example, we start the sparse pruning after 100 epochs and record the number of bases every 100 epochs. We can see that the number of bases drops very fast and the network can find the correct set of basis after 1700 epochs. The training and testing history are presented in 3. We compare the case (1) when we do adaptive pruning during the training ("AMS-net (p)")and (2) do not prune the network during the training. It shows that our proposed adaptive pruning method can achieve faster training and produce better prediction results. Nonlinear case In the linear case, we take k(u) = u, and f (x, y; α) = −(α 1 cos(πx) sin(πy) + 2πα 2 cos(2πx) sin(2πy) + 3πα 3 cos(3πx) sin(πy) + 4πα 4 cos(4πx) sin(2πy)) 2 − (α 1 sin(πx) cos(πy) + 2πα 2 sin(2πx) cos(2πy) + 3πα 3 sin(3πx) cos(πy) + 4πα 4 cos(4πx) sin(2πy)) 2 − (+α 1 sin(πx) sin(πy) + α 2 sin(2πx) sin(2πy) + α 3 sin(3πx) sin(πy) + α 4 sin(4πx) sin(2πy)) − (2π 2 α 1 sin(πx) sin(πy) + 8π 2 α 2 sin(2πx) sin(2πy) + 10π 2 α 3 sin(3πx) sin(πy) + 20π 2 α 4 sin(4πx) sin(2πy)) In this nonlinear case, we generate more sample pairs, 1000 in total, to train the neural network. Among them, 800 samples are used for training, and 200 samples are used for testing. In this example, similar as before, we only perform pruning without adding. We choose the thresholding parameter to be 0.3. The number of bases selected during training is presented in Figure 4. Here we record the number of bases every 20 epochs. Again, we observe that the number of bases drops very fast in the beginning of our algorithm, and the network can identify the correct basis sets after 760 epochs. The training and testing history are presented in 5. We obtain similar results as shown in the linear case, where both the training and testing errors obtained from the proposed adaptive pruning method outperform the nonpruning case. Flow dynamics 4.2.1 Velocity sparse approximation in multiscale space We will generate samples by solving the system (4) and (5) sequentially on the fine grid with different source terms. We take f i (x, y) =                    r 1 if 1 − H < x < 1 & 0 < y < H r 2 if 0 < x < H & 1 − H < y < 1 r 3 if 0 < x < H & 0 < y < H r 3 if 1 − H < x < 1 & 1 − H < y < 1 −(r 1 + r 2 + r 3 + r 4 ) if 5H < x < 6H & 5H < y < 6H 0 otherwise where r i are randomly chosen in [0, 1], i = 1, · · · , 1500. The absolute permeability κ is set to be a layer in SPE10 model. An illustration of the source f and κ are shown in Figure 6. The simulation is performed on the time interval [0, 16], with time step size t = 4. Thus, for each f i , we have fine scale velocity solutions [v 0 i , v 1 i , · · · , v 2 i ] at time steps t = 0, t = 8, t = 16, respectively. Our goal is to use f i as input, (v 0 i , v 1 i , · · · , v 2 i ) as labels to train the neural network using the loss function (17). The dictionary D vel (9) is constructed as described in section 3.1.1. In our example, the computational area is [0, 1] × [0, 1], the fine grid mesh size is h = 1 50 , and the coarse grid mesh size is H = 1 10 . There are 5400 fine edges and 220 coarse edges. For each interior coarse edge, we compute 5 multiscale bases, resulting 900 bases at each time instance. We note that we let r 1 = r 2 = r 3 = r 4 = 1 in the source f to generate offline basis functions, and only the bases obtained at time instances t = 0, t = 8, t = 16 are included in the dictionary. Thus, we get 2700 bases in the dictionary D. Each input is a 100 × 1 vector which represents a coarse grid level source, and each v j i (j = 0, · · · , 3) is a 5100 × 1 vector. We define the error e 1 = u pred −utrue L 2 utrue L 2 . We use 80% of samples for training and the rest for testing. The total number of training epochs is 2000. We compare the results when we use the proposed algorithm 1, with the results when we only apply pruning at the last epoch. We use the notation " AMS-net (p+a)" as a short for our proposed adaptive sparse method with both removing and adding bases, " AMS-net (p)" for our proposed adaptive sparse method with only removing bases. We also use the greedy algorithm to select bases based on the fine solution samples, and compute the projections of the fine scale solutions on the greedily selected space, and use them as a reference. We first set the target number of bases we would like to include in the training. The comparison of the errors obtained from these three approaches are presented in Table 1. The degrees of freedom (dof) in the table are the average of the degrees of freedom among solutions at all 6 time steps. We observe that with the similar size of the degrees of freedom, " AMSnet (p+a)" outperforms " AMS-net (p)". When the dof is less than 1000, " AMS-net (p+a)" also produces better results compared with the greedy projection error. As the degrees of freedom increases, the mean prediction error decreases consistently. We also test the approach with a given threshold γ add t to guide the procedure of adding bases, the results are presented in Figure 8. We note that γ add t = γ add 0 100 (100+t) 0.75 is a decreasing sequence. It shows that as γ add 0 decreases, the algorithm selects more degrees of freedom and the error of prediction decreases too. The decreasing rate is almost linear. This demonstrates that with the threshold γ add t , we can control the training error of the network and thus obtain better prediction results. When γ add t approach 0, the mean prediction error is 0.95%, which is some irreducible snapshot error. We also present the prediction errors using the adaptive sparse method with both pruning and adding bases at each time step in Figure 7. Two random test cases are shown in Figure 12 and 13, where we see good matches between the reference solutions and network predictions. Remark : In this work, we assume the dictionary can be built in advance. The basis functions in the dictionary are found by solving some local prob-lems corresponding to various permeabilities based on developed numerical techniques. Once computed, the dictionary won't be reconstructed when the permeability field changes. Given this large dictionary, one may first employ some preselection techniques such as clustering to narrow down the search of basis functions for a given application. Then our proposed method will be beneficial to find a much sparse representation for the quantities of interest. The focus of our paper is to develop an efficient and stable algorithm to obtain a fast and computational cheap solver given the dictionary. After the sparse network is trained given the data, we can apply it to evaluate the test cases in a very fast manner. shows the results when we just use pruning strategy without adding basis. Column 3 shows the results when we use the adaptive strategy with both pruning and adding basis. In column 4, we use the greedy algorithm to select bases based on the training samples, and compute the mean of projection error for the testing cases in the greedily selected space. The errors are in percentage. Saturation sparse approximation in POD space In this example, we take similar source configurations as described in the previous section. We first solve the system with a specific source term on the time interval [0, 4] with time step size t = 0.1. Gather the saturation solutions on these 40 time instances together, we will perform POD on it and select the resulting bases to form V sat , and D sat in (10). Next, for each source f i (i = 1, · · · , 1000), we have solve for fine saturation solutions on the time interval [0, 6] with time step size t = 0.1. However, we only take the solutions every 10 time steps to train the neural network, i.e, [S 1 i , S 10 i , · · · , S 60 i ]. Again, we use f i as input, (S 1 i , S 10 i , · · · , S 60 i ) as labels, and the previously mentioned dictionary to train the neural network. In this case, the fine degree of freedom for the saturation solution is 10000, and the reduced order space has dimension 40. We define the error e 1 = S pred −Strue L 2 Strue L 2 . The total number of training epochs is 2500. We compare the results when we use the proposed algorithm 1 with both pruning and adding bases, the algorithm with only pruning and the POD projection error. The errors using these three approaches are listed in Table 2. The dofs in the first column of the table are the mean dofs at all 6 time steps. We observe that, AMS-net (with both pruning and adding bases) produces similar results when dof is equal to 40, and achieve better predictions when the dof is equal to 6, 8, 10, 12. It actually converges to the snapshot error when all bases are used (dof = 40). A random test case is shown in Figure 9, where we see good matches between the reference and network prediction. Remark : Our method can be extended naturally to practical applications with more complicated physics, such as permeability variations, compressible fluids or the case with gravitational effects. First of all, we assume the dictionary is precomputed and contains enough basis functions which can capture the features of the underlying media for a wide range of cases. The construction of basis functions is discussed in many existing works and is beyond the scope of this paper. The key of our method is to adaptively select important basis functions from the known large dictionary through the proposed sparse learning algorithm and obtain reduced order models for In the case of compressible fluids, for example, the two phase flow with gas and oil, the basis functions we need might be different at different time steps. If the gravity is also considered, extra bases are needed to approximate the gravity force. In such cases, our method can automatically choose suitable sets of basis from the dictionary at corresponding time instances and capture the complex physical properties. Since the dictionary can be very large to cover complicated applications, our method is more beneficial to obtain the sparse representation and achieve an efficient approximation. For the more complex nonlinear system, one may also consider employing a larger neural network to approximate the dynamics, this will cause more difficulties for the learning process. Our algorithm enforces sparsity in the network connections and can help training. Saturation sparse approximation: a simple illustration of interpretibility In this section, we will use simple basis functions to approximate the saturation solution to illustrate the interpretibility of the proposed method. Consider the 10-by-10 coarse mesh in the domain [0, 1] × [0, 1], the absolute permeability is a fractured media and the source term f has a two-spot well configuration, as shown in Figure 10. To simplify, we will use piecewise constant basis functions. That is, for each coarse block and each fracture segment inside the coarse block, we will have a degree of freedom associated with it. Each basis function has a value equal to 1 in the coarse block or the fracture segment, and has value equal to 0 elsewhere. The degrees of freedom (dof) with respect to fracture segments are labeled from dof1 to dof 21 as presented in Figure 10. We perform simulations on the time interval T = (0, 60], with time step size t = 1, and only select the solutions at time instances t = 10, 20, 30, 40, 50, 60 to train the neural network with the proposed method. Let r = 2 be the injection rate in the source term, the solutions at time steps t = 10, 30, 60 are shown in Figure 11. We observe that the saturation hardly goes into the fracture associated with dof 2, dof 4, dof 6, and dof 8. Moreover, at the early time steps, the fluid did not saturate into the fracture associated with dof 15 -dof 21, but was fully saturated in the last time step. Now, we choose a set of 100 random injection rates in [1.5, 2.5], and use their corresponding solutions to train the network. Our purpose is to observe how the network chooses basis functions to represent fractures and matrices. The results are shown in Table 3. We see that the network can identify important dofs correctly. This shows the potential of interpretibility of our proposed method. Conclusion We present a scalable sparse learning framework, which incorporates some precomputed basis functions in the learning objective. The network aims to learn the flow dynamics where the parameters in the flow model contain multiscale properties. The inputs are random source terms, and the labels are fine scale solutions at different time steps. The outputs of the neural network are coefficients of the solutions corresponding to the basis functions at these time instances. The predicted solutions are then formed by the product of the coefficient vectors and basis functions. The objective is to minimize the differences between the predicted solutions and fine scale reference solutions over all time steps. The algorithm can adaptively choose important basis functions from a large pool of different source inputs. The sparsity in the solution coefficient vector is enforced through a built-in thresholding operator, which is implemented as an activation function in some layers of the network. The sparsity of layer connections in the network is achieved by cutting the connections to coefficients with small magnitude. To avoid dropping too many basis functions and enrich the approximation space during the training, one can also add some degrees of freedom back through a greedy procedure. Through adaptive training, one can obtain a sparse set of important basis functions and an accurate approximation to the flow dynamics. Several numerical tests are performed to demonstrate sparse and accurate approximations to the solutions using the proposed algorithm. The algorithm consists of two parts: removing basis/pruning connections (as illustrated in 3.4.1 and 3.4.2), and adding basis/reactivating connections (as illustrated in 3.4.3). Figure 1 : 1An illustration of the network architecture.The network's input is the realization of a random source, the outputs contain the prediction of expansion coefficients c j for multiscale basis in the solutions at multiple time steps (j = 1, · · · , T ). The basis funtions Φ are incorporated in the loss function. Algorithm 1 1Adaptive Multiscale Sparse Neural Network for Basis Expansion Learning 1: procedure AMS-Net(Θ 0 , n b , n e , if Set target number of bases M and N − m u < M then 11: m c ← M − (N − m u ) 12: else if Set adding threshold γ add t and [Ē t ] 1 < γ add t then 13: m c ← j where [Ē t ] test the sparse learning algorithm with a dictionary on a simple static example. Consider the two-dimensional elliptic equation on Ω = [0, 1] × [0, 1] −div(k(u)∇u) = f (x, y; α) u = 0 on ∂Ω where the ground truth for u is u(x, y; α) = α 1 sin(πx) sin(πy) + α 2 sin(2πx) sin(2πy) + α 3 sin(3πx) sin(πy) + α 4 sin(4πx) sin(2πy) and α = [α 1 , α 2 , α 3 , α 4 ], with α i i.i.d samples generated from normal distribution. For α 1 , we have mean 1 and standard deviation 2. For α 2 , we have mean 0 and standard deviation 3. For α 3 , we have mean −2 and standard deviation 3. For α 4 , we have mean 5 and standard deviation 2, We create the dictionary D 1,2 with D 1 = {1, sin(πx), sin(2πx), · · · , sin(M πx)} D 2 = {1, sin(πy), sin(2πy), · · · , sin(M πy)} D 1,2 = {d 1 d 2 \| d 1 ∈ D 1 , d 2 ∈ D 2 }. Figure 2 : 2(Section 4.1.1) Toy example, linear case, the number of selected bases v.s. training epochs. The number of training samples is 80. The total number of basis functions is 100. After 200 epochs, the sparsity reaches 90%. The network identifies the correct sparsity pattern and finds the true basis functions in epoch 1700. Figure 3 : 3(Section 4.1.1) Toy example, linear case. Left: training error history, right: testing error history. Comparison between the results using our proposed pruning algorithm and without pruning. The number of training/testing samples is 80/20. It shows that training with pruning is more efficient. Figure 4 : 4(Section 4.1.2) Toy example, nonlinear case, the number of selected bases v.s. training epochs. The number of training samples is 800. The total number of basis functions is 100. After 200 epochs, the sparsity reaches 90%. The network identifies the correct sparsity pattern and chose the true basis functions in epoch 760. Figure 5 : 5(Section 4.1.2) Toy example, nonlinear case. Left: training error history, right: testing error history. the number of training/testing samples is 800/200. Comparison between the results using our proposed pruning algorithm and without pruning. It shows that the sparse network training is more efficient. Figure 6 : 6(Section 4.2.1) An illustration of the source term f (left) and absolute permeability κ in log scale (right). Figure 7 :Figure 8 : 78(Section 4.2.1) Learning velocity fields. Mean prediction errors among 200 testing samples, at different time steps and different dofs. With a increasing number of basis selected in AMS-net, the prediction errors will decrease. Left: mean errors over all time steps. Our algorithm produces better results consistently when the dof is less than 1000, and it converges to the snapshot error when the dof become larger. Right: mean errors at each time step. (Section 4.2.1) Learning velocity fields. Using given thresholding parameters γ add t to adaptively add bases in AMS-net. Mean errors between the true and predicted velocity among 200 testing samples. It shows that decreasing the value of thresholding parameter can help to control the accuracy of AMS-net predictions. Figure 9 : 9(Section 4.2.2) Learning saturation profile. Test case illustration: saturation at all the first, third and last time steps. The predicted results matches the reference solution well.the dynamical system. Figure 10 : 10(Section 4.2.3) The background shows the 10-by-10 coarse mesh. The absolute permeability κ takes value 1 in the background, and takes value 1000 in the maroon-colored channels. The source function f takes value 0in the background, f = r(r > 0) in the red region in the bottom left corner, and f = −r in the red region in the top right corner. The degrees of freedom (dof) with respect to channels are labeled from dof1 to dof 21. Figure 11 : 11(Section 4.2.3) From left to right: saturation at t = 10, t = 30, and t = 60. Figure 12 : 12(Section 4.2.1) Learning velocity fields, test case 1. AMS-net predictions produces accurate predictions using 1000 basis. The relative l 2 error is 1.58% at the last time step. Figure 13 : 13(Section 4.2.1) Learning velocity fields, test case 2. AMS-net predictions produces accurate predictions using 1000 basis. The relative l 2 error is 0.99% at the last time step. erogeneous porous media, Computer Methods in Applied Mechanics and Engineering, 292 (2015), pp. 122-137. dofs AMS-net (p) AMS-net (p+a) POD projection6 2.21 1.96 2.10 8 1.72 1.62 1.67 10 1.45 1.35 1.40 12 1.23 1.18 1.22 40 0.95 0.95 0.95 Table 2 : 2(Section 4.2.2) Learning saturation profile. Fix the number of bases in the adaptive process. Mean errors between the true and predicted saturation among 200 testing samples. AMS-net (with both pruning and adding bases) produces better results when the dof is equal to 6, 8, 10, 12, and it converges to the projection error when all the bases are used (dof = 40). The errors are in percentage. Time step The unselected dofs resulted from AMS-nett=10 2, 4, 6, 8, 9, 13, 14, 15, 16, 17, 18, 19, 20, 21 t=20 2, 4, 6, 8, 13, 14, 18, 19, 20, 21 t=30 2, 4, 6, 8, 14, 19, 20, 21 t=40 2, 4, 6, 14, 19, 21 t=50 2, 4, 6, 19, 21 t=60 2, 4, 19, 21 Table 3 : 3(Section 4.2.3) The unselected dofs resulted from AMS-net at all time steps. The saturation hardly goes into the fracture associated with dof 2, dof 4, dof 6 and dof 8. Moreover, the fluid didn't saturated into the fracture associated with dof 15 -dof 21 at the early time steps, but was fully saturated in the last time step. The desired dofs are identified by our algorithm. AcknowledgementsWe gratefully acknowledge the support from the National ScienceAppendix Mixed multiscale finite element methods using limited global information. J E Aarnes, Y Efendiev, L Jiang, Multiscale Modeling & Simulation. 7J. E. Aarnes, Y. Efendiev, and L. 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[ "BulletArm: An Open-Source Robotic Manipulation Benchmark and Learning Framework", "BulletArm: An Open-Source Robotic Manipulation Benchmark and Learning Framework" ]	[ "Dian Wang [email protected] \nKhoury College of Computer Sciences Northeastern University Boston\n02115MAUSA\n", "Colin Kohler [email protected] \nKhoury College of Computer Sciences Northeastern University Boston\n02115MAUSA\n", "Xupeng Zhu \nKhoury College of Computer Sciences Northeastern University Boston\n02115MAUSA\n", "Mingxi Jia [email protected] \nKhoury College of Computer Sciences Northeastern University Boston\n02115MAUSA\n", "Robert Platt [email protected] \nKhoury College of Computer Sciences Northeastern University Boston\n02115MAUSA\n" ]	[ "Khoury College of Computer Sciences Northeastern University Boston\n02115MAUSA", "Khoury College of Computer Sciences Northeastern University Boston\n02115MAUSA", "Khoury College of Computer Sciences Northeastern University Boston\n02115MAUSA", "Khoury College of Computer Sciences Northeastern University Boston\n02115MAUSA", "Khoury College of Computer Sciences Northeastern University Boston\n02115MAUSA" ]	[]	We present BulletArm, a novel benchmark and learning-environment for robotic manipulation. BulletArm is designed around two key principles: reproducibility and extensibility. We aim to encourage more direct comparisons between robotic learning methods by providing a set of standardized benchmark tasks in simulation alongside a collection of baseline algorithms. The framework consists of 31 different manipulation tasks of varying difficulty, ranging from simple reaching and picking tasks to more realistic tasks such as bin packing and pallet stacking. In addition to the provided tasks, BulletArm has been built to facilitate easy expansion and provides a suite of tools to assist users when adding new tasks to the framework. Moreover, we introduce a set of five benchmarks and evaluate them using a series of state-of-the-art baseline algorithms. By including these algorithms as part of our framework, we hope to encourage users to benchmark their work on any new tasks against these baselines.	10.48550/arxiv.2205.14292	[ "https://export.arxiv.org/pdf/2205.14292v2.pdf" ]	249,191,648	2205.14292	44df7090a9b18b19196018b5705dfc9153e068ed	BulletArm: An Open-Source Robotic Manipulation Benchmark and Learning Framework Dian Wang [email protected] Khoury College of Computer Sciences Northeastern University Boston 02115MAUSA Colin Kohler [email protected] Khoury College of Computer Sciences Northeastern University Boston 02115MAUSA Xupeng Zhu Khoury College of Computer Sciences Northeastern University Boston 02115MAUSA Mingxi Jia [email protected] Khoury College of Computer Sciences Northeastern University Boston 02115MAUSA Robert Platt [email protected] Khoury College of Computer Sciences Northeastern University Boston 02115MAUSA BulletArm: An Open-Source Robotic Manipulation Benchmark and Learning Framework BenchmarkSimulationRobotic LearningReinforcement Learning We present BulletArm, a novel benchmark and learning-environment for robotic manipulation. BulletArm is designed around two key principles: reproducibility and extensibility. We aim to encourage more direct comparisons between robotic learning methods by providing a set of standardized benchmark tasks in simulation alongside a collection of baseline algorithms. The framework consists of 31 different manipulation tasks of varying difficulty, ranging from simple reaching and picking tasks to more realistic tasks such as bin packing and pallet stacking. In addition to the provided tasks, BulletArm has been built to facilitate easy expansion and provides a suite of tools to assist users when adding new tasks to the framework. Moreover, we introduce a set of five benchmarks and evaluate them using a series of state-of-the-art baseline algorithms. By including these algorithms as part of our framework, we hope to encourage users to benchmark their work on any new tasks against these baselines. Introduction Inspired by the recent successes of deep learning in the field of computer vision, there has been an explosion of work aimed at applying deep learning algorithms across a variety of disciplines. Deep reinforcement learning, for example, has been used to learn policies which achieve superhuman levels of performance across a variety of games [36,30]. Robotics has seen a similar surge in recent years, especially in the area of robotic manipulation with reinforcement learning [12,20,51], imitation learning [50], and multi-task learning [10,14]. However, there is a key difference between current robotics learning research and past work applying deep learning to other fields. There currently is no widely accepted standard for comparing learning-based robotic manipulation methods. In computer vision for example, the ImageNet benchmark [8] has been a crucial factor in the explosion of image classification algorithms we have seen in the recent past. While there are benchmarks for policy learning in domains similar to robotic manipulation, such as the continuous control tasks in OpenAI Gym [4] Control Suite [37], they are not applicable to more real-world tasks we are interested in robotics. Furthermore, different robotics labs work with drastically different systems in the real-world using different robots, sensors, etc. As a result, researchers often develop their own training and evaluation environments, making it extremely difficult to compare different approaches. For example, even simple tasks like block stacking can have a lot of variability between different works [31,28,43], including different physics simulators, different manipulators, different object sizes, etc. In this work, we introduce BulletArm, a novel framework for robotic manipulation learning based on two key components. First, we provide a flexible, open-source framework that supports many different manipulation tasks. Compared with prior works, we introduce tasks with various difficulties and require different manipulation skills. This includes long-term planning tasks like supervising Covid tests and contact-rich tasks requiring precise nudging or pushing behaviors. BulletArm currently consists of 31 unique tasks which the user can easily customize to mimic their real-world lab setups (e.g., workspace size, robotic arm type, etc). In addition, BulletArm was developed with a emphasis on extensability so new tasks can easily be created as needed. Second, we include five different benchmarks alongside a collection of standardized baselines for the user to quickly benchmark their work against. We include our implementations of these baselines in the hopes of new users applying them to their customization of existing tasks and whatever new tasks they create. Our contribution can be summarized as three-fold. First, we propose BulletArm, a benchmark and learning framework containing a set of 21 open-loop manipulation tasks and 10 close-loop manipulation tasks. We have developed this framework over the course of many prior works [43,3,45,3,44,42,54]. Second, we provide state-of-theart baseline algorithms enabling other researchers to easily compare their work against our baselines once new tasks are inevitably added to the baseline. Third, BulletArm provides a extensive suite of tools to allow users to easily create new tasks as needed. Our code is available at https://github.com/ColinKohler/BulletArm. tasks, for example, door opening [40], furniture assembly [27], and in-hand dexterous manipulation [1]. Another strand of prior works propose frameworks containing a variety of different environments, such as robosuite [55], PyRoboLearn [7], and Meta-World [48], but are often limited to short horizon tasks. Ravens [50] introduces a set of environments containing complex manipulation tasks but restricts the end-effector to a suction cup gripper. RLBench [18] provides a similar learning framework to ours with a number of key differences. First, RLBench is built around the PyRep [17] interface and is therefor built on-top of V-REP [34]. Furthermore, RLBench is more restrictive than BulletArm with limitations placed on the workspace scene, robot, and more. Robotic Manipulation Control There are two commonly used end-effector control schemes: open-loop control and close-loop control. In open-loop control, the agent selects both the target pose of target pose of the end-effector and some action primitive to execute at that pose. Open-loop control generally has shorter time horizon, allowing the agent to solve complex tasks that require a long trajectory [52,51,43]. In close-loop control, the agent sets the displacement of the end-effector This allows the agent to more easily recover from failures which is vital when delaing with contact-rich tasks [12,20,53,44]. BulletArm provides a collection of environments in both settings, allowing the users to select either one based on their research interests. Architecture At the core of our learning framework is the PyBullet [6] simulator. PyBullet is a Python library for robotics simulation and machine learning with a focus on sim-to-real transfer. Built upon Bullet Physics SDK, PyBullet provides access to forward dynamics simulation, inverse dynamics computation, forward and inverse kinematics, collision detection, and more. In addition to physics simulation, there are also numerous tools for scene rendering and visualization. BulletArm builds upon PyBullet, providing a diverse set of tools tailored to robotic manipulation simulations. Design Philosophy The design philosophy behind our framework focuses on four key principles: 1. Reproducibility: A key challenge when developing new learning algorithms is the difficulty in comparing them to previous work. In robotics, this problem is especially prevalent as different researchers have drastically different robotic setups. This can range from small differences, such as workspace size or degradation of objects, to large differences such as the robot used to preform the experiments. Moving to simulation allows for the standardization of these factors but can impact the performance of the trained algorithm in the real-world. We aim to encourage more direct comparisons between works by providing a flexible simulation environment and a number of baselines to compare against. 2. Extensibility: Although we include a number of tasks, control types, and robots; there will always be a need for additional development in these areas. Using our framework, users can easily add new tasks, robots, and objects. We make the choice to not restrict tasks, allowing users more freedom create interesting domains. Figure 1 shows an example of creating a new task using our framework. 3. Performance: Deep learning methods are often time consuming, slow processes and the addition of a physics simulator can lead to long training times. We have spent a significant portion of time in ensuring that our framework will not bottleneck training by optimizing the simulations and allowing the user to run many environments in parallel. Usability: A good open-source framework should be easy to use and understand. We provide extensive documentation detailing the key components of our framework and a set of tutorials demonstrating both how to use the environments and how to extend them. Environment Our simulation setup (Figure 4) consists of a robot arm mounted on the floor of the simulation environment, a workspace in front of the robot where objects are generated, and a sensor. Typically, we use top-down sensors which generate heightmaps of the workspace. As we restrict the perception data to only the defined workspace, we choose to not add unnecessary elements to the setup such as a table for the arm to sit upon. Currently there are four different robot arms available in BulletArm ( Figure 3): KUKA IIWA, Frane Emika Panda, Universal Robots UR5 with either a simple parallel jaw gripper or the Robotiq 2F-85 gripper. Environment, Configuration, and Episode are three key terms within our framework. An environment is an instance of the PyBullet simulator in which the robot interacts with objects while trying to solve some task. This includes the initial environment state, the reward function, and termination requirements. A configuration contains additional specifications for the task such as the robotic arm, the size of the workspace, the physics mode, etc (see Appendix B for an full list of parameters). Episodes are generated by taking actions (steps) within an environment until the episode ends. An episode trajectory τ contains a series of observations o, actions a, and rewards r: τ = [(o 0 , a 0 , r 0 ), ..., (o T , a T , r T )]. Users interface with the learning environment through the EnvironmentFactory and EnvironmentRunner classes. The EnvironmentFactory is the entry point and creates the Right to left: a construction episode is generated by reversing the deconstruction episode. This is inspired by [49] where the authors propose a method to learn kit assembly through disassembly by reversing disassembly transitions. Environment class specified by the Configuration passed as input. The Environment-Factory can create either a single environment or multiple environmaents meant to be run in parallel. In either case, an EnvironmentRunner instance is returned and provides the API which interacts with the environments. This API, Figure 1, is modelled after the typical agent-environment RL setup popularized by OpenAI Gym [4]. The benchmark tasks we provide have a sparse reward function which returns +1 on successful task completion and 0 otherwise. While we find this reward function to be advantageous as it avoids problems due to reward shaping, we do not require that new tasks conform to this. When defining a new task, the reward function defaults to sparse but users can easily define their custom reward for a new task. We separate our tasks into two categories based on the action spaces: open-loop control and closed-loop control. These two control modes are commonly used in robotics manipulation research. Expert Demonstrations Expert demonstrations are crucial to many robotic learning fields. Methods such as imitation and model-based learning, for example, learn directly from expert demonstrations. Additionally, we find that in the context of reinforcement learning, it is vital to seed learning with expert demonstrations due to the difficulties in exploring large stateaction spaces. We provide two types of planners to facilitate expert data generation: the Waypoint Planner and the Deconstruction Planner. The Waypoint Planner is a online planning method which moves the end-effector through a series of waypoints in the workspace. We define a waypoint as w t = (p t , a t ) where p t is the desired pose of the end effector and a t is the action primitive to execute at that pose. These waypoints can either be absolute positions in the workspace or positions relative to the objects in the workspace. In open-loop control, the planner returns the waypoint w t as the action executed at time t. In close-loop control, the planner will continuously return a small displacement from the current end-effector pose as the action at time t. This process is repeated until the waypoint has been reached. The Deconstruction Planner is a more limited planning method which can only be applied to pick-and-place tasks where the goal is to arrange objects in a specific manner. For example, we utilize this planner for the various block construction tasks examined in this work ( Figure 2). When using this planner, the workspace is initialized with the objects in their target configuration and objects are then removed one-by-one until the initial state of the task is reached. This deconstruction trajectory, is then reversed to produce an expert construction trajectory, τ expert = reverse(τ deconstruct ). (a) Kuka (b) Panda (c) UR5 Parallel (d) UR5 Robotiq Environments The core of any good benchmark is its set of environments. In robotic manipulation, in particular, it is important to cover a broad range of task difficulty and diversity. To this end, we introduce tasks covering a variety of skills for both open-loop and closeloop control. Moreover, the configurable parameters of our environments enable the user to select different task variations (e.g., the user can select whether the objects in the workspace will be initialized with a random orientation). BulletArm currently provides a collection of 21 open-loop manipulation environments and a collection of 10 close-loop environments. These environments are limited to kinematic tasks where the robot has to directly manipulate a collection of objects in order to reach some desired configuration. Open-Loop Environments In comprised of a robot arm, a workspace, and a camera above the workspace providing the observation ( Figure 4). The action space is defined as the cross product of the gripper motion A g = {PICK, PLACE} and the target pose of the gripper for that motion A p , A = A g × A p . The state space is defined as s = (I, H, g) ∈ S ( Figure 5), where I is a top-down heightmap of the workspace; g ∈ {HOLDING, EMPTY} denotes the gripper state; and H is an in-hand image that shows the object currently being held. If the last action is PICK, then H is a crop of the heightmap in the previous time step centered at the pick position. If the last action is PLACE, H is set to a zero value image. BulletArm provides three different action spaces for A p : A p ∈ {A xy , A xyθ , A SE(3) }. The first option (x, y) ∈ A xy only controls the (x, y) components of the gripper pose, where the rotation of the gripper is fixed; and z is selected using a heuristic function that first reads the maximal height in the region around (x, y) and then either adds a positive offset for a PLACE action or a negative offset for PICK action. The second option (x, y, θ) ∈ A xyθ adds control of the rotation θ along the z-axis. The third option (x, y, z, θ, φ, ψ) ∈ A SE(3) controls the full 6 degree of freedom pose of the gripper, including the rotation along the y-axis φ and the rotation along the x-axis ψ. A xy and A xyθ are suited for tasks that only require top-down manipulations, while A SE(3) is designed for solving complex tasks involving out-of-plane rotations. The definition of the state space and the action space in the open-loop environments also enables effortless sim2real transfer. One can reproduce the observation in Figure 5 in the real-world using an overhead depth camera and transfer the learned policy [43,45]. Figure 6 shows the 12 basic open-loop environments that can be solved using topdown actions. Those environments can be categorized into two collections, a set of block structure tasks (Figures 6a-6g), and a set of more realistic tasks (Figures 6h and 6l). The block structure tasks require the robot to build different goal structures using blocks. The more realistic tasks require the robot to finish some real-world problems, for example, arranging bottles or supervising Covid tests. We use the default sparse reward function for all open-loop environments, i.e., +1 reward for reaching the goal, and 0 otherwise. See Appendix A.1 for a detailed description of the tasks. 6DoF Extensions The environments that we have introduced so far only require the robot to perform top-down manipulation. We extend those environments to 6 degrees of freedom by initializing them in either the ramp environment or the bump environment ( Figure 7). In both cases, the robot needs to control the out-of-plane orientations introduced by the ramp or bump in order to manipulate the objects. We provide seven Close-loop Environments The close-loop environments require the agent to control the delta pose of the endeffector, allowing the agent more control and enabling us to solve more contact-rich tasks. These environments have a similar setup to the open-loop domain but to avoid the occlusion caused by the arm, we instead use two side-view cameras pointing to the workspace ( Figure 8a). The heightmap I is generated by first acquiring a fused point cloud from the two cameras ( Figure 8b) and then performing an orthographic projection ( Figure 8c). This orthographic projection is centered with respect to the gripper. In practice, this process can be replaced by putting a simulated orthographic camera at the position of the gripper to speed up the simulation. The state space is defined as a tuple s = (I, g) ∈ S, where g ∈ {HOLDING, EMPTY} is the gripper state indicating if there is an object being held by gripper. The action space is defined as the cross product of the gripper open width A λ and the delta motion of the gripper A δ , A = A λ × A δ . Two different action spaces are available for A δ : A δ ∈ {A xyz δ , A xyzθ δ }. In A xyz δ , the robot controls the change of the x, y, z position of the gripper, where the top-down orientation θ is fixed. In A xyzθ δ , the robot controls the change of the x, y, z position and the Table 2. The number of objects, optimal number of steps per episode, and max number of steps per episode in our Open-Loop 3D benchmark experiments top-down orientation θ of the gripper. Figure 9 shows the 10 close-loop environments. We provide a default sparse reward function for all environments. See Appendix A.2 for a detailed description of the tasks. Benchmark BulletArm provides a set of 5 benchmarks covering the various environments and action spaces (Table 1). In this section, we detail the Open-Loop 3D Benchmark and the Close-Loop 4D Benchmark. See Appendix D for the other three benchmarks. Open-Loop 3D Benchmark In the Open-Loop 3D Benchmark, the agent needs to solve the open-loop tasks shown in Figure 6 using the A g × A xyθ action space (see Section 4.1). We provide a set of baseline algorithms that explicitly control (x, y, θ) ∈ A xyθ and select the gripper motion using the following heuristic: a PICK action will be executed if g = EMPTY and a PLACE action will be executed if g = HOLDING. The baselines include: (1) DQN [30], (2) ADET [24], (3) DQfD [15], and (4) SDQfD [43]. The network architectures for these different methods can be used interchangeably. We provide the following network architectures: 1. CNN ASR [43]: A two-hierarchy architecture that selects (x, y) and θ sequentially. 2. Equivariant ASR (Equi ASR) [45]: Similar to ASR, but instead of using conventional CNNs, equivariant steerable CNNs [5,46] are used to capture the rotation symmetry of the tasks. 3. FCN: a Fully Convolutional Network (FCN) [29] which outputs a n channel actionvalue map for each discrete rotation. 4. Equivariant FCN [45]: Similar to FCN, but instead of using conventional CNNs, equivariant steerable CNNs are used. 5. Rot FCN [52,51]: A FCN with 1-channel input and output, the rotation is encoded by rotating the input and output for each θ. In this section, we show the performance of SDQfD (which is shown to be better than DQN, ADET, and DQfD [43]. See the performance of DQN, ADET and DQfD in Appendix E) equipped with CNN ASR, Equi ASR, FCN, and Rot FCN. We evaluate SDQfD in the 12 environments shown in Figure 6. Table 2 shows the number of objects, the optimal number of steps per episode, and the max number of steps per episode in the open-loop benchmark experiments. Before the start of training, 200 (500 for Covid Test) episodes of expert data are populated in the replay buffer. Figure 10 shows the results. Equivariant ASR (blue) has the best performance across all environments, then Rot FCN (green) and CNN ASR (red), and finally FCN (purple). Notice that Equivariant ASR is the only method that is capable of solving the most challenging tasks (e.g., Improvise House Building 3 and Covid Test). Close-Loop 4D Benchmark The Close-Loop 4D Benchmark requires the agent to solve the close-loop tasks shown in Figure 9 in the 5-dimensional action space of (λ, x, y, z, θ) ∈ A λ × A xyzθ δ ⊂ R 5 , where the agent controls the positional displacement of the gripper (x, y, z), the rotational displacement of the gripper along the z axis (θ), and the open width of the gripper (λ). We provide the following baseline algorithms: (1) SAC [13], (2) Equivariant SAC [44], (3) RAD [25] SAC: SAC with data augmentation, (4) DrQ [23] SAC: Similar to (3), but performs data augmentation when calculating the Q-target and the loss, and (5) FERM [53]: A Combination of SAC and contrastive learning [26] using data augmentation. Additionally, we also provide a variation of SAC, SACfD [44], that applies an auxiliary L2 loss towards the expert action to the actor network. SACfD can also be used in combination with (2), (3), and (4) to form Equivariant SACfD, RAD SACfD, DrQ SACfD, and FERM SACfD. In this section, we show the performance of SACfD, Equivariant SACfD (Equi SACfD), Equivariant SACfD using Prioritized Experience Replay (PER [35]) and data augmentation (Equi SACfD + PER + Aug), and FERM SACfD. (See Appendix F for the performance of RAD SACfD and DrQ SACfD.) We use a continuous action space where x, y, z ∈ [−0.05m, 0.05m], θ ∈ [− π 4 , π 4 ], λ ∈ [0, 1]. We evaluate the various methods in the 10 environments shown in Figure 9. Table 3 shows the number of objects, the optimal number of steps for solving each task, and the maximal number of steps for each episode. In all tasks, we pre-load 100 episodes of expert demonstrations in the replay buffer. Figure 11 shows the performance of the baselines. Equivariant SACfD with PER and data augmentation (blue) has the best overall performance followed by standard Conclusions In this paper, we present BulletArm, a novel benchmark and learning environment aimed at robotic manipulation. By providing a number of manipulation tasks alongside our baseline algorithms, we hope to encourage more direct comparisons between new methods. This type of standardization through direct comparison has been a key aspect of improving research in deep learning methods for both computer vision and reinforcement learning. We aim to maintain and improve this framework for the foreseeable future adding new features, tasks, and baseline algorithms over time. An area of particular interest for us is to extend the existing suite of tasks to include more dynamic environments where the robot is tasked with utilizing tools to accomplish various tasks. We hope that with the aid of the community, this repository will grow over time to contain both a large number of interesting tasks and state-of-the-art baseline algorithms. A Detail Description of Environments A.1 Open-Loop Environments Block Stacking In the Block Stacking environment (Figure 6a), there are N cubic blocks with a size of 3cm × 3cm × 3cm. The blocks are randomly initialized in the workspace. The goal of this task is to stack all blocks in a stack. An optimal policy requires 2(N − 1) steps to finish this task. The number of blocks N is configurable. By default, N = 4, and the maximal number of steps per episode is 10. House Building 1 In the House Building 1 environment (Figure 6b), there are N − 1 cubic blocks with a size of 3cm × 3cm × 3cm and one triangle block with a bounding box size of around 3cm × 3cm × 3cm. The blocks are randomly initialized in the workspace. The goal of this task is to first form a stack using the N − 1 cubic blocks, then place the triangle block on top of the stack. An optimal policy requires 2(N − 1) steps to finish this task. The number of blocks N is configurable. By default, N = 4, and the maximal number of steps per episode is 10. House Building 2 In the House Building 2 environment (Figure 6c), there are two cubic blocks with a size of 3cm × 3cm × 3cm, and a roof block with a bounding box size of around 12cm × 3cm × 3cm. The blocks are randomly initialized in the workspace. The goal of this task is to place the two cubic blocks next to each other, then place the roof block on top of the two cubic blocks. An optimal policy requires 4 steps to finish this task. The default maximal number of steps per episode is 10. House Building 3 In the House Building 3 environment (Figure 6d), there are two cubic blocks with a size of 3cm × 3cm × 3cm, one cuboid block with a size of 12cm × 3cm × 3cm, and a roof block with a bounding box size of around 12cm × 3cm × 3cm. The blocks are randomly initialized in the workspace. The goal of this task is to first place the two cubic blocks next to each other, place the cuboid block on top of the two cubic blocks, then place the roof block on top of the cuboid block. An optimal policy requires 6 steps to finish this task. The default maximal number of steps per episode is 10. House Building 4 In the House Building 4 environment (Figure 6e), there are four cubic blocks with a size of 3cm × 3cm × 3cm, one cuboid block with a size of 12cm × 3cm × 3cm, and a roof block with a bounding box size of around 12cm × 3cm × 3cm. The blocks are randomly initialized in the workspace. The goal of this task is to first place two cubic blocks next to each other, place the cuboid block on top of the two cubic blocks, place another two cubic blocks on top of the cuboid block, then place the roof block on top of the two cubic blocks. An optimal policy requires 10 steps to finish this task. The default maximal number of steps per episode is 20. Improvise House Building 2 In the Improvise House Building 2 environment (Figure 6f), there are two random blocks and a roof block. The shapes of the random blocks are sampled from Figure 12. The blocks are randomly initialized in the workspace. The goal of this task is to place the two random blocks next to each other, then place the roof block on top of the two random blocks. An optimal policy requires 4 steps to finish this task. The default maximal number of steps per episode is 10. Improvise House Building 3 In the Improvise House Building 3 environment (Figure 6g), there are two random blocks, a cuboid block, and a roof block. The shapes of the random blocks are sampled from Figure 12. The blocks are randomly initialized in the workspace. The goal of this task is to place the two random blocks next to each other, place the cuboid block on top of the two random blocks, then place the roof block on top of the cuboid block. An optimal policy requires 6 steps to finish this task. The default maximal number of steps per episode is 10. Bin Packing In the Bin Packing task (Figure 6h), N objects and a bin are randomly placed in the workspace. The shapes of the objects are randomly sampled from Figure 13 (Object models are derived from [51]) with a maximum size of 8cm × 4cm × 4cm and a minimum size of 4cm × 4cm × 2cm. The bin has a size of 17.6cm × 14.4cm × 8cm. The goal of this task is to pack all N objects in the bin. An optimal policy requires 2N steps to finish the task. The number of objects N is configurable. By default, N = 8, and the maximal number of steps per episode is 20. Bottle Arrangement In the Bottle Arrangement task (Figure 6i), six bottles with random shapes (sampled from 8 different shapes shown in Figure 14. The bottle shapes are generated from the 3DNet dataset [47]. The sizes of each bottle are around 5cm × 5cm × 14cm), and a tray with a size of 24cm × 16cm × 5cm are randomly placed in the workspace. The goal is to arrange all six bottles in the tray. An optimal policy requires 12 steps to finish this task. By default, the maximal number of steps per episode is 20. Box Palletizing In the Box Palletizing task (Figure 6j) (some object models are derived from [50]), a pallet with a size of 23.2cm × 19.2cm × 3cm is randomly placed in the workspace. The goal is to stack N boxes with a size of 7.2cm×4.5cm×4.5cm as shown in Fig 6j. At the beginning of each episode and after the agent correctly places a box on the pallet, a new box will be randomly placed in the empty workspace. An optimal policy requires 2N steps to finish this task. The number of boxes N is configurable (6, 12, or 18). By default, N = 18, and the maximal number of steps per episode is 40. (a) (b) (c) (d) (e) (f) (g) (h) Covid Test In the Covid Test task (Figure 6k), there is a new tube box (purple), a test area (gray), and a used tube box (yellow) placed arbitrarily in the workspace but adjacent to one another. Three swabs with a size of 7cm × 1cm × 1cm and three tubes with a size of 8cm × 1.7cm × 1.7cm are initialized in the new tube box. To supervise a COVID test, the robot needs to present a pair of a new swab and a new tube from the new tube box to the test area. The simulator simulates the user testing COVID by putting the swab into the tube and randomly placing the used tube in the test area. Then the robot needs to re-collect the used tube into the used tube box. See one example of this process in Figure 15. Each episode includes three rounds of COVID test. An optimal policy requires 18 steps to finish this task. By default, the maximal number of steps per episode is 30. Object Grasping In the Object Grasping task (Figure 6l), the robot needs to grasp an object from a clutter of at most N objects. At the start of training, N random objects are initialized with random position and orientation. The shapes of the objects are randomly sampled from the object set shown in Figure 16. The object set contains 86 objects derived from the GraspNet1B [11] dataset. Every time the agent successfully grasps all N objects, the environment will re-generate N random objects with random positions and orientations. The maximal number of steps per episode is 1. The number of objects N in this environment is configurable. By default, there will be 15 objects. A.2 Close-Loop Environments Block Reaching In the Block Reaching environment (Figure 9a), there is a cubic block with a size of 5cm × 5cm × 5cm. The block is randomly initialized in the workspace. The goal of this task is to move the gripper towards the block such that the distance of the fingertip and the block is within 3cm. By default, the maximal number of steps per episode is 50. Block Picking In the Block Picking environment (Figure 9b), there is a cubic block with a size of 5cm × 5cm × 5cm. The block is randomly initialized in the workspace. The goal of this task is to grasp the block and raise the gripper such that the gripper is 15cm above the ground. By default, the maximal number of steps per episode is 50. Block Pushing In the Block Pushing environment (Figure 9c), there is a cubic block with a size of 5cm × 5cm × 5cm and a goal area with a size of 9cm × 9cm. The block and the goal area are randomly initialized in the workspace. The goal of this task is to push the block such that the distance between the block's center and the goal's center is within 5cm. By default, the maximal number of steps per episode is 50. Block Pulling In the Block Pulling environment (Figure 9d), there are two cuboid blocks with a size of 8cm × 8cm × 5cm. The blocks are randomly initialized in the workspace. The goal of this task is to pull one of the two blocks such that it makes contact with another block. By default, the maximal number of steps per episode is 50. Block in Bowl In the Block in Bowl environment (Figure 9e), there is a cubic block with a size of 5cm × 5cm × 5cm, and a Bowl with a bounding box size of 16cm × 16cm × 7cm. The block and the bowl are randomly initialized in the workspace. The goal of this task is to pick up the block and place it inside the bowl. By default, the maximal number of steps per episode is 50. Block Stacking In the Block Stacking environment (Figure 9f), there are N cubic blocks with a size of 5cm × 5cm × 5cm. The blocks are randomly initialized in the workspace. The goal of this task is to form a stack using the N blocks. By default, N = 2, the maximum number of steps per episode is 50. House Building In the House Building environment (Figure 9g), there are N − 1 cubic blocks with a size of 5cm × 5cm × 5cm and one triangle with a bounding box size of 5cm × 5cm × 5cm. The blocks are randomly initialized in the workspace. The goal of this task is to first form a stack using the N − 1 cubic blocks, then place the triangle block on top. By default, N = 2, the maximum number of steps per episode is 50. Corner Picking In the Corner Picking environment (Figure 9h), there is a cubic block with a size of 5cm × 5cm × 5cm and a corner formed by two walls. The poses of the block and the corner are randomly initialized with a fixed relative pose between them so that the block is right next to the two walls. The wall is fixed in the workspace and not movable. The goal of this task is to nudge the block out from the corner and then pick it up at least 15cm above the ground. By default, the maximum number of steps per episode is 50. Drawer Opening In the Drawer Opening environment (Figure 9i), there is a drawer with a random pose in the workspace. The outer part of the drawer is fixed and not movable. The goal of this task is to pull the drawer handle to open the drawer. By default, the maximum number of steps per episode is 50. Object Grasping In the Object Grasping task (Figure 6l), the robot needs to grasp an object from a clutter of at most N objects. At the start of training, N random objects are initialized with random position and orientation. The shapes of the objects are randomly sampled from the object set shown in Figure 16. The object set contains 86 objects derived from the GraspNet1B [11] dataset. Every time the agent successfully grasps all N objects, the environment will re-generate N random objects with random positions and orientations. If an episode terminates with any remaining objects in the bin, the objects will not be re-initialized. The goal of this task is to grasp any object and lift it such that the gripper is at least 0.15m above the ground. The number of objects N in this environment is configurable. By default, there will be 5 objects, and the maximum number of steps per episode is 50. Figure 9j) in the close-loop environment. Table 4. List of example configurable parameters in our framework. B List of Configurable Environment Parameters C Open-Loop 6DoF Environments Most of the 6DoF environments mirror those in Figure 6, but the workspace is initialized with two ramps in the ramp environments or with a bumpy surface in the bump environments. In the ramp environments (Figure 17a-Figure 17g), the two ramps are always parallel to each other. The distance between the ramps is randomly sampled between 4cm and 20cm. The orientation of the two ramps along the z-axis is randomly sampled between 0 and 2π. The slope of each ramp is randomly sampled between 0 and π 6 . The height of each ramp above the ground is randomly sampled between 0cm and 1cm. In addition, the relevant objects are initialized with random positions and orientations either on the ramps or on the ground. In the bump environments (Figure 17h and Figure 17i), bumpy surface is generated by nine pyramid shapes with a random slop sampled from 0 to π 12 degrees. The orientation of the bumpy surface along the z-axis is randomly sampled at the beginning of each episode. D Additional Benchmarks This section demonstrates the Open-Loop 2D Benchmark, the Open-Loop 6D Benchmark, and the Close-loop 3D Benchmark (Table 1). initialized with a fixed orientation. The action space in this benchmark is A g × A xy , i.e., the agent only controls the target (x, y) position of the gripper, while θ is fixed at 0 degree. Other environment parameters mirror the Open-Loop 3D Benchmark in Section 5.1. D.1 Open-Loop 2D Benchmark Similar as in Section 5.1, we provide DQN, ADET, DQfD, and SDQfD algorithms with FCN and Equivariant FCN (Equi FCN) network architectures (the other architectures do not apply to this benchmark because the agent does not control θ). In this section, we show the performance of SDQfD equipped with FCN and Equivariant FCN. Figure 18 shows the result. Equivariant FCN (blue) generally shows a better performance compared with standard FCN (red). D.2 Open-Loop 6D Benchmark In the Open-Loop 6D Benchmark, the agent needs to solve the open-loop 6DoF environments (Appendix C) in an action space of A g × A SE(3) , i.e., the position (x, y, z) of the gripper and the rotation (θ, φ, ψ) of the gripper along the z, y, x axes. We provide two baselines in this benchmark: 1) ASR [43]: a hierarchical approach that selects the actions in a sequence of ((x, y), θ, z, φ, ψ) using 5 networks; 2) Equivariant Deictic ASR [45] (Equi Deictic ASR): similar as 1), but replace the standard networks with equivariant networks and the deictic encoding to improve the sample efficiency. We use 1000 planner episodes for the ramp environments and 200 planner episodes for the bump environments. The in-hand image H in this experiment is a 3channel orthographic projection image of a voxel grid generated from the point cloud at the previous pick pose. Other environment parameters mirror the Open-Loop 3D Benchmark in Section 5.1. Figure 19 shows the results. Equivariant Deictic ASR (blue) demonstrates a stronger performance compared with standard ASR (red). D.3 Close-Loop 3D Benchmark The Close-Loop 3D Benchmark is similar as the Close-Loop 4D Benchmark (Section 5.2), but with the following two changes: first, the environments are initialized with a fixed orientation; second, the action space is A xyz λ ∈ R 4 instead of A xyzθ λ ∈ R 5 , i.e., the agent only controls the delta (x, y, z) position of the end-effector and the openwidth λ of the gripper. We provide the same baseline algorithms as in Section 5.2. In this section, we show the performance of SDQfD, Equivariant SDQfD (Equi SDQfD), and FERM SDQfD. Figure 20 shows the result. Equivariant SACfD (blue) shows the best performance across all tasks. FERM SACfD (green) and SACfD (red) has similar performance, except for Block Reaching, where FERM SACfD outperforms standard SACfD. E Additional Baselines for Open-Loop 3D Benchmark In this section, we show the performance of three additional baseline algorithms in the Open-Loop 3D Benchmark (Section 5.1): DQfD, ADET, and DQN. We compare them with SDQfD (the algorithm used in Section 5.1). All algorithms are equipped with the Equivariant ASR architecture. Figure 21 shows the result. Notice that SDQfD and DQfD generally perform the best, while SDQfD has a marginal advantage compared with DQfD. ADET learns faster in some tasks (e.g., House Building 1), but normally converges to a lower performance compared with SDQfD and DQfD. DQN performs the worst across all environments because of the lack of imitation loss. F Additional Baselines for Close-Loop 4D Benchmark In this section, we show the performance of two additional baseline algorithms in the Close-Loop 4D Benchmark (Section 5.2): RAD SACfD and DrQ SACfD. As is shown in Figure 22, RAD SACfD (yellow) performs poorly in all 10 environments. DrQ SACfD (brown) outperforms FERM SACfD (purple) in Block Picking and Block Pulling, but still underperforms the equivariant methods (blue and red). In all environments, the kuka arm is used as the manipulator. The workspace has a size of 0.4m × 0.4m. The top-down observation I covers the workspace with a size of 128×128 pixels. (In the Rot FCN baseline, I's size is 90×90 pixels, and is padded with 0 to 128×128 pixels. This is padding required for the Rot FCN baseline because it needs to rotate the image to encode θ.) The size of the in-hand image H is 24 × 24 pixels for the Open-Loop 2D and Open-Loop 3D benchmarks. In the Open-Loop 6D Benchmark, H is a 3-channel orthographic projection image, with a shape of 3 × 24 × 24 in the ramp environments, and 3 × 40 × 40 in the bump environments. We train our models using PyTorch [32] with the Adam optimizer [22] with a learning rate of 10 −4 and weight decay of 10 −5 . We use Huber loss [16] for the TD loss. The discount factor γ is 0.95. The mini-batch size is 16. The replay buffer has a size of 100,000 transitions. At each training step, the replay buffer will separately draw half of the samples from the expert data and half of the samples from the online transitions. The weight w for the margin loss term of SDQfD is 0.1, and the margin l = 0.1. We use the greedy policy as the behavior policy. We use 5 environments running in parallel. G.2 Close-Loop Benchmark In all environments, the kuka arm is used as the manipulator. The workspace has a size of 0.3m × 0.3m × 0.24m. The pixel size of the top-down depth image O is 128 × 128 (except for the FERM baseline, where I's size is 142 × 142 and will be cropped to 128 × 128). I's FOV is 0.45m × 0.45m. We use the Adam optimizer with a learning rate of 10 −3 . The entropy temperature α is initialized at 10 −2 . The target entropy is -5. The discount factor γ = 0.99. The batch size is 64. The buffer has a capacity of 100,000 transitions. In baselines using the prioritized replay buffer (PER), PER has a prioritized replay exponent of 0.6 and prioritized importance sampling exponent β 0 = 0.4 as in [35]. The expert transitions are given a priority bonus of d = 1. The FERM baseline's contrastive encoder is pretrained for 1.6k steps using the expert data as in [53]. We use 5 environments running in parallel. Fig. 2 . 2The Deconstruction planner. Left to right: a deconstruction episode where the expert deconstructs the block structure in the left-most figure. Fig. 3 . 3Our work currently supports four different arms: Kuka, Panda, UR5 with parallel jaw gripper, and UR5 with Robotiq gripper. Fig. 4 .Fig. 5 . 45The (a) The manipulation scene. (b) The state including a top-down heightmap I, an in-hand image H and the gripper state g. Fig. 6 . 6the open-loop environments, the agent controls the target pose of the end-effector, resulting in a shorter time horizon for complex tasks. The open-loop environment is The open-loop environments. The window on the top-left corner of each sub-figure shows the goal state of each task. Fig. 7 .Fig. 8 . 78The 6DoF environments. (a): In the Ramp Environment, the objects are initialized on two ramps, where the agent needs to control the out-of-plane orientations to pick up the objects. (b): Similarly, in the Bump Environment, the objects are initialized on a bumpy surface. (a) The close-loop environment containing a robot arm, two cameras, and a workspace. (b) The point cloud generated from the two cameras. (c) The orthographic projection generated from the point cloud which is used as the observation. The two squares at the center of the image represent the gripper. Alternatively, the image can be generated using a simulated orthographic camera located at the position of the end-effector. Fig. 9 . 9The close-loop environments. The window on the top-left corner of each sub-figure shows the goal state of the task. ramp environments (for each of the block structure construction tasks), and two bump environments (House Building 4 and Box Palletizing). See Appendix C for details. Fig. 10 . 10The Open-Loop 3D Benchmark results. The plots show the evaluation performance of the greedy policy in terms of the task success rate. The evaluation is performed every 500 training steps. Results are averaged over four runs. Shading denotes standard error. Fig. 11 . 11The Close-Loop 4D benchmark results. The plots show the evaluation performance of the greedy policy in terms of the task success rate. The evaluation is performed every 500 training steps. Results are averaged over four runs. Shading denotes standard error.Equivariant SACfD (red). The equivariant algorithms show a significant improvement when compared to the other algorithms which do not encode rotation symmetry, i.e. CNN SACfD and FERM SACfD. Fig. 12 . 12The object set in the Improvise House Building 2 and Improvise House Building 3 environment. Fig. 13 . 13The object set in the Bin Packing environment. Fig. 14 . 14The object set in the Bottle arrangement environment. Fig. 15 . 15An example of one COVID test process. Fig. 16 . 16The object set in the Object Grasping environment. Fig. 17 . 17The open-loop 6DoF environments. The window on the top-left corner of each sub-figure shows the goal state of each task. The Open-Loop 2D Benchmark requires the agent to solve the open-loop tasks in Figure 6 without random orientations, i.e., all of the objects in the environment will be Fig. 18 . 18The Open-Loop 2D benchmark result. The plots show the evaluation performance of the greedy policy in terms of the task success rate. The evaluation is performed every 500 training steps. Results are averaged over four runs. Shading denotes standard error. Fig. 19 . 19The Open-Loop 6D benchmark result. The plots show the evaluation performance of the greedy policy in terms of the task success rate. The evaluation is performed every 500 training steps. Results are averaged over four runs. Shading denotes standard error. Fig. 20 . 20The Close-Loop 3D benchmark result. The plots show the evaluation performance of the greedy policy in terms of the task success rate. The evaluation is performed every 500 training steps. Results are averaged over four runs. Shading denotes standard error. Fig. 21 . 21The Open-Loop 3D benchmark result with additional baselines. The plots show the evaluation performance of the greedy policy in terms of the task success rate. The evaluation is performed every 500 training steps. Results are averaged over four runs. Shading denotes standard error. Fig. 22 . 22The Close-Loop 4D benchmark result with additional baselines. The plots show the evaluation performance of the greedy policy in terms of the task success rate. The evaluation is performed every 500 training steps. Results are averaged over four runs. Shading denotes standard error. and the DeepMind * Equal Contribution arXiv:2205.14292v2 [cs.RO] 17 Oct 2022 δ Table 1 . δ1Open-Loop 2D Benchmark open-loop environments with fixed orientation Ag × A xy Open-Loop 3D Benchmark open-loop environments with random orientation Ag × A xyθ Open-Loop 6D Benchmark open-loop environments 6DoF extensions Ag × A SE(3)Close-Loop 3D Benchmark close-loop environments with fixed orientationA λ × A xyzClose-Loop 4D Benchmark close-loop environments with random orientation A λ × A xyzθ The five benchmarks in our work include three open-loop benchmarks and two closeloop benchmarks. 'fixed orientation' and 'random orientation' indicate whether the objects in the environments will be initialized with a fixed orientation or random orientation.Benchmark Environments Action Space δ Task Block Stacking House Building 1 House Building 2 House Building 3 House Building 4 Improvise House Building 2 Improvise House Building 3 Bin Packing Bottle Arrangement Box Palletizing Covid Test Object Grasping Number of Objects The number of objects, optimal number of steps per episode, and max number of steps per episode in our Close-Loop 4D Benchmark experiments.Task Block Reaching Block Picking Block Pushing Block Pulling Block in Bowl Block Stacking House Building Corner Picking Drawer Opening Object Grasping Number of Objects 1 1 1 2 2 2 2 1 1 5 Optimal Number of Steps 3 7 7 7 11 12 12 14 9 7 Max Number of Steps 50 50 50 50 50 50 50 50 50 50 Table 3. Table 4 4shows a list of configuration parameters in our framework.Parameter Example Description robot kuka the robot to use in the experiment. action sequence pxyzr The action space. 'pxyzr' means the action space a 5-vector, including the gripper action (p), the posi- tion of the gripper (x, y, z), and its top-down rota- tion (r). workspace array([[0.25, 0.65], [-0.2, 0.2], [0, 1]]) The workspace in terms of the range in x, y, and z. object scale range 0.6 The scale of the size of the objects in the environ- ment. max steps 10 The maximal steps per episode. num objects 1 The number of objects in the environment. obs size 128 The pixel size of the observation I. in hand size 24 The pixel size of the in-hand image H. fast mode True If True, teleport the arm when possible to speed up the simulation. render False If True, render the PyBullet GUI. random orientation True If True, the objects in the environments will be ini- tialized with random orientations. half rotation True If True, constrain the gripper rotation between 0 and π. workspace check point/bounding box Check object out of bound using the object center of mass or the bounding box close loop tray False If True, generate a tray like in the Object Grasping ( AcknowledgmentsThis work is supported in part by NSF 1724257, NSF 1724191, NSF 1763878, NSF 1750649, and NASA 80NSSC19K1474. 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[ "The Most Detailed Picture Yet of an Embedded High-mass YSO", "The Most Detailed Picture Yet of an Embedded High-mass YSO" ]	[ "L J Greenhill \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n", "M J Reid \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n", "C J Chandler \nMERLIN/VLBI National Facility\nNRAO\nP.O. Box O87801SocorroNMUSA\n", "P J Diamond \nDepartment of Physics & Astronomy\nJodrell Bank Observatory\nSK11 9DLMacclesfieldUK\n", "M Elitzur \nUniversity of Kentucky\n40506LexingtonKYUSA\n" ]	[ "Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA", "Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA", "MERLIN/VLBI National Facility\nNRAO\nP.O. Box O87801SocorroNMUSA", "Department of Physics & Astronomy\nJodrell Bank Observatory\nSK11 9DLMacclesfieldUK", "University of Kentucky\n40506LexingtonKYUSA" ]	[ "Star Formation at High Angular Resolution ASP Conference Series" ]	High-mass star formation is not well understood chiefly because examples are deeply embedded, relatively distant, and crowded with sources of emission. Using VLA and VLBA observations of H 2 O and SiO maser emission, we have mapped in detail the structure and proper motion of material 20-500 AU from the closest high-mass YSO, radio source I in the Orion KL region. We observe streams of material driven in a rotating, wide angle, bipolar wind from the surface of an edgeon accretion disk. The example of source I provides strong evidence that high-mass star formation proceeds via accretion.	10.1017/s0074180900241557	[ "https://export.arxiv.org/pdf/astro-ph/0309334v2.pdf" ]	15,888,028	astro-ph/0309334	c97e7576dce76f97cacc55d6c4df1921ec44ef46	The Most Detailed Picture Yet of an Embedded High-mass YSO 2003 L J Greenhill Harvard-Smithsonian Center for Astrophysics 60 Garden St02138CambridgeMAUSA M J Reid Harvard-Smithsonian Center for Astrophysics 60 Garden St02138CambridgeMAUSA C J Chandler MERLIN/VLBI National Facility NRAO P.O. Box O87801SocorroNMUSA P J Diamond Department of Physics & Astronomy Jodrell Bank Observatory SK11 9DLMacclesfieldUK M Elitzur University of Kentucky 40506LexingtonKYUSA The Most Detailed Picture Yet of an Embedded High-mass YSO Star Formation at High Angular Resolution ASP Conference Series 2212003M.G. Burton, R. Jayawardhana & T.L. Bourke High-mass star formation is not well understood chiefly because examples are deeply embedded, relatively distant, and crowded with sources of emission. Using VLA and VLBA observations of H 2 O and SiO maser emission, we have mapped in detail the structure and proper motion of material 20-500 AU from the closest high-mass YSO, radio source I in the Orion KL region. We observe streams of material driven in a rotating, wide angle, bipolar wind from the surface of an edgeon accretion disk. The example of source I provides strong evidence that high-mass star formation proceeds via accretion. Introduction High-mass star formation is poorly understood because theory has yet to resolve the balance of radiation pressure, gravity, magnetic energy, and thermal heating (e.g., McKee & Tan 2002). Moreover, examples amenable to detailed study but young enough to have not yet formed HII regions are rare. First, massive YSOs are deeply embedded, form in clusters, and heat large amounts of surrounding dust and gas. As a result, the signatures of disks and outflows at small radii are easily lost in "background clutter." Second, regions of high-mass star formation are sufficiently distant ( ∼ > 500 pc) that infrared and radio observations of thermal gas and dust are often confusion-limited. Third, massive young stellar objects evolve rapidly, and by the time the surrounding medium is dispersed, accretion disk and outflow structures have been at least partly dispersed. In two of the best understood cases, G192.16−3.82 (Shepherd & Kurtz 1999) and 2 Greenhill et al. (Zhang et al. 1998;Cesaroni et al. 1999), disk-like structures have been detected, but the rotation curves are barely resolved and the emission arises at large enough radii (> 1000 AU) that in depth study is difficult. IRAS 20126+4104 The infrared Kleinemann-Low (KL) nebula in Orion is the nearest (∼ 500 pc), and most heavily studied region of high-mass star formation. The KL region is crowded, containing 16 identified mid-infrared peaks across ∼ 10 4 AU (Gezari, Backman, & Werner 1998). Offset ∼ 0. ′′ 5 south of the prominent peak, IRc2, is radio source I, first detected by Churchwell et al. (1987). It has no infrared counterpart, but it does power compact distributions of SiO and H 2 O masers, and it has been identified as a probable deeply embedded, massive protostar or YSO (Gezari 1992;Menten & Reid 1995). The strongest SiO maser emission arises from the v=1 vibrational state. Because of excitation requirements (T > 10 3 K, n(H 2 ) ∼ 10 10±1 cm −3 ), it must originate close to the YSO. The first VLBI maps revealed an X-shaped SiO emission locus extending outward 20-70 AU from source I. The X also lay at the center of a 200×600 AU expanding patch of H 2 O masers distributed in two lobes each comprising red and blueshifted emission and bracketing source I (Greenhill et al. 1998; see also Gaume et al. 1998;Doeleman, Lonsdale, & Pelkey 1999). Taken alone, the distribution of SiO emission could have been interpretted equivalently in two ways. First, it could represent the turbulent limb of a highvelocity biconical outflow with a southeast-northwest axis. Second, it could represent hot material close to the top and bottom surfaces of an edge-on disk with a northeast-southwest rotation axis. Greenhill et al. and Doeleman et al. adopted the biconical outflow model, in part because the high-velocity outflow observed in the KL region on scales of 10 4 AU (e.g., Allen & Burton 1993;Schultz et al. 1999) subtended the opening angle of the putative SiO maser cone. In addition, Greenhill et al. (1998) noted that the distributions of proper motions and line-of-sight velocities among the surrounding H 2 O masers were not readily consistent with the edge-on disk model and proposed instead that the H 2 O masers lay on the surface of a nearly edge-on, inflating, equatorial doughnut-like shell. This shell could arise from a slow stellar equatorial wind advancing into the surrounding medium or from photo-evaporation of an accretion disk. Because the major axis of this shell was aligned with the the so-called "18 km s −1 " outflow in the KL region (e.g., Genzel & Stutzki 1989), it appeared that source I might drive the two most prominent outflows in the region. Despite circumstantial evidence tying source I to large scale outflows in the KL region, ambiguity has remained. On the one hand, Menten & Reid (1995) noted that another infrared source, n, lies closer to the previously estimated center of the flows (Genzel et al. 1981). Moreover, high-velocity gas far to the northwest is blueshifted, while the corresponding outflow cone near source I is redshifted. On the other hand, the compact bipolar radio lobes of source n are oriented north-south, 40−50 • from the principal axes of the low and high-velocity large scale flows in the KL region. To better understand the KL region and to test the biconical outflow model velocity.) We measured the proper motion of the SiO v=1 & 2 emission, observing monthly over ∼ 3.5 years, which is ∼ 30% of the dynamical crossing time of the outflow. Here we discuss proper motions over a four month interval. Observations and Data Reduction We observed the v=1 & 2 J = 1 → 0 transition of SiO with the VLBA, and the v=0 J = 1 → 0 transition of SiO and 6 16 → 5 23 transition of H 2 O with the VLA in its largest configuration and including the nearby Pie Town VLBA antenna. We obtained angular resolutions of ∼ 0.2 milliarcseconds (mas) with the VLBA and ∼ 50 mas with the VLA. The spectral channel spacings in the VLBA and VLA imaging were ∼ 0.4 km s −1 and ∼ 2.6 km s −1 , respectively. Using the VLBA, we observed the SiO v=1 & 2 emission (from V LSR ∼ −15 to +25 km s −1 ) simultaneously to enable registration of both lines to ∼ 0.1 mas. Because emission from source I itself is thermal, it could not be detected with the VLBA. To locate the position of the YSO on the VLBA maps, we convolved the maps with a circular beam comparable in size to the beam of the VLA at λ7mm and overlayed the degraded VLBA map on the VLA map of Menten & Reid (1995), who detected and registered the SiO v=1 and thermal continuum emission of the YSO. The uncertainty in this registration is ∼ 10 mas. Using our VLA observations, we measured the positions of the SiO v=0 masers (∼ −10 to +22 km s −1 ) with respect to the v=1 emission, which was observed simultaneously so as to provide an astrometric reference. The uncertainty in position relative to source I is ∼ 10 mas. The positions of the H 2 O masers (∼ −10 to +16 km s −1 ) were measured with respect to a nearby quasar and compared to the absolute position of source I (Menten & Reid 1995). The uncertainty in this comparison is 30 mas. Results and Discussion Our more extensive mapping of the SiO and H 2 O maser emission now supports the edge-on disk model previously discarded. We propose that the v=1 & 2 SiO maser emission traces material streaming in a rotating funnel-like flow from the upper and lower surfaces of an accretion disk of a massive YSO. The v=0 SiO and H 2 O masers lie in a bipolar outflow along the disk rotation axis. R < 70 AU In the most recent observations, the v=1 & 2 SiO emission traces an X as before (Figure 1), but we note two important new details. First, there is a "bridge" of maser emission extending from the base of the south arm to the base of the west arm, and there is a gradient in line-of-sight velocity along the bridge. Second, the arms are not radial. These new findings may be consequences of our having now mapped the v=2 as well as v=1 emission. (For instance, the bridge is outlined principally by v=2 emission.) Both findings are difficult to explain in the context of the biconical flow model. However, they are readily explained in the context of the edge-on disk model. The bridge and associated velocity gradient are natural signatures of emission from the front side of a rotating disk that is tipped down slightly to the southwest. The canting of the arms, so that they are not radial, is also suggestive of reflection symmetry about a plane (i.e., disk). In order to estimate the proper motions of maser clumps, we identified one in each quadrant of the X that persisted in each epoch and maintained the same line-of-sight velocity. We overlayed images for the four epochs and registered them to achieve zero mean proper motion for the source as a whole. Most proper motions are 10 to 15 km s −1 , though some clumps are nearly stationary on the sky. (One km s −1 over four months corresponds to ∼ 1 VLBA beamwidth.) The maximum observed 3-D velocity is ∼ 23 km s −1 . The proper motions of maser clumps are chiefly along the four arms of the X (Figure 1). In the bridge, the motions are both outward and tangential, indicative of rotation and strongly in support of the edge-on disk model (Chandler et al., in prep). We suggest that the arms of SiO emission represent the limbs of a bipolar funnel-like outflow (Figure 2), probably the turbulent shocked interface between outflowing and accreting material. In a rotating system, strong maser emission is expected along the limbs because that is where the longest maser gain paths lie owing to projection effects. Additional emission from the bridge may mark the nearside of the outflow wall, where at the base, higher densities or temperatures could compensate for otherwise shorter gain paths. The dynamical mass of the YSO is difficult to estimate because the maser material is not in simple Keplerian Detailed Picture of an Embedded High-mass YSO 5 rotation. If the observed 3-D velocity is greater than the escape velocity, then the enclosed mass is ∼ > 6 M ⊙ , for a radius of 25 AU and velocity of 25 km s −1 . We note that the 3-D motion of the maser material is helical. This may indicate the presence of a strong magnetic field that is probably anchored to the accretion disk, since massive stars are radiative and may not generate their own field. If the magnetic and kinetic energy densities are of the same order, then the field is on the order of 1 G, for a gas density of 10 10 cm −3 . R > 70 AU The v=0 SiO and H 2 O maser emission arises chiefly from two "polar caps" that subtend the ∼ 90 • opening angle of the outflow along the disk rotation axis (Figure 2). The velocity structure of the emission is somewhat disorganized and difficult to model in detail. However, the two lobes (northeast and southwest) display the same ranges of velocity, indicative of outflow in the plane of the sky. Within each lobe, the bulk of the emission displays a red-blue asymmetry across the rotation axis in the same sense as the disk rotation close to source I, although further study is required. Otherwise, it may signify coupling between the velocity field downstream in the outflow and the underlying accretion disk, possibly via magnetic processes. Substantial overlap of H 2 O maser and SiO v=0 emission along the line of sight is surprising because maser action in each species requires quite different densities, 10 8−10 cm −3 vs 10 5−6 cm −3 , respectively. Moreover, because molecule reformation on dust grains is essential for H 2 O maser action behind shocks, gas phase SiO would be depleted. In principle, the H 2 O and v=0 SiO maser volumes could be intermingled if the flow were inhomogeneous and if shock induced grain sputtering enhanced gas phase concentrations of SiO. However, the requisite > 10 2 density contrast between emitting regions would be difficult to explain in the apparent absence of high-velocity (≫ 50 km s −1 ) gas motions. We suggest that the wall of the outflow traced by v=1 & 2 SiO masers at radii < 70 AU extends to larger radii where it supports H 2 O maser emission. The outflow itself has low enough density to support the v=0 SiO maser emission, which lies along the same line of sight as the H 2 O emission in projection. The ordered velocity structure of the v=1 & 2 SiO maser emission and the relatively complicated velocity structure of the H 2 O and v=0 SiO maser emission at larger radii may be a consequence of the increased effects turbulence or collisions with inflowing or ambient material downstream in the outflow. Summary We have fully resolved the structure and dynamics of material at radii of 20-500 AU from radio source I in the Orion KL region. Our maps provide the most detailed picture yet of molecular material so close to a massive YSO. We propose a new model for source I in which the accretion disk is edge-on and a wide-angle wind feeds a funnel-like rotating bipolar outflow. Consequently, the case of source I provides strong evidence that accretion in high-mass star formation proceeds via orderly disk-mediated accretion as opposed to coalescence of low mass stars (Bonnell et al. 1998). proposed for source I, we remapped the SiO and H 2 O maser emission, covering three J = 1 → 0 transitions of SiO (v=0, 1, & 2) and the full velocity range of H 2 O emission. (The original maps presented by Greenhill et al. were significantly limited in dynamic range, especially at velocities close to the systemic Detailed Picture of an Embedded High-mass YSO 3 Figure 1 . 1(Left:) Vibrationally excited SiO masers forming an X and λ7mm continuum emission (grayscale) observed byMenten & Reid (in prep). Emission redshifted (R) with respect to the systemic velocity (V LSR = 5 km s −1 ) lies to the north and west. Blueshifted emission (B) is opposite. The line emission probably originates in rotating high density material driven from the surface of an edge-on disk. The continuum probably arises at least in part from H − free-free disk emission.(Right:) Proper motions of v=1 & 2 SiO maser clumps (Chandler et al., in prep). The lengths of the cones indicate 3-D velocities. The aspects of the cones indicate inclinations with respect to the line of sight. Blueshifted motions are black, redshifted are gray. 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[ "Vulpedia: Detecting Vulnerable Ethereum Smart Contracts via Abstracted Vulnerability Signatures", "Vulpedia: Detecting Vulnerable Ethereum Smart Contracts via Abstracted Vulnerability Signatures" ]	[ "Jiaming Ye [email protected] ", "Mingliang Ma ", "China Yun Lin ", "Lei Ma ", "Canada Yinxing Xue ", "Jianjun Zhao ", "\nKyushu University\nJapan\n", "\nNanjing University\nNational University of Singapore\nSingapore\n", "\nUniversity of Alberta\nUniversity of Science and Technology\nof China China\n", "\nKyushu University\nJapan\n" ]	[ "Kyushu University\nJapan", "Nanjing University\nNational University of Singapore\nSingapore", "University of Alberta\nUniversity of Science and Technology\nof China China", "Kyushu University\nJapan" ]	[]	Recent years have seen smart contracts are getting increasingly popular in building trustworthy decentralized applications. Previous research has proposed static and dynamic techniques to detect vulnerabilities in smart contracts. These tools check vulnerable contracts against several predefined rules. However, the emerging new vulnerable types and programming skills to prevent possible vulnerabilities emerging lead to a large number of false positive and false negative reports of tools. To address this, we propose Vulpedia, which mines expressive vulnerability signatures from contracts. Vulpedia is based on the relaxed assumption that the owner of contract is not malicious. Specifically, we extract structural program features from vulnerable and benign contracts as vulnerability signatures, and construct a systematic detection method based on detection rules composed of vulnerability signatures. Compared with the rules defined by state-of-the-arts, our approach can extract more expressive rules to achieve better completeness (i.e., detection recall) and soundness (i.e., precision). We further evaluate Vulpedia with four baselines (i.e., Slither, Securify, SmartCheck and Oyente) on the testing dataset consisting of 17,770 contracts. The experiment results show that Vulpedia achieves best performance of precision on 4 types of vulnerabilities and leading recall on 3 types of vulnerabilities meanwhile exhibiting the great efficiency performance.	10.1016/j.jss.2022.111410	[ "https://arxiv.org/pdf/1912.04466v2.pdf" ]	249,856,016	1912.04466	ee0fd5cf549e998076f7eedf1d23cea924542b46	Vulpedia: Detecting Vulnerable Ethereum Smart Contracts via Abstracted Vulnerability Signatures Jiaming Ye [email protected] Mingliang Ma China Yun Lin Lei Ma Canada Yinxing Xue Jianjun Zhao Kyushu University Japan Nanjing University National University of Singapore Singapore University of Alberta University of Science and Technology of China China Kyushu University Japan Vulpedia: Detecting Vulnerable Ethereum Smart Contracts via Abstracted Vulnerability Signatures This article is accepted by Journal of Systems and Software. Recent years have seen smart contracts are getting increasingly popular in building trustworthy decentralized applications. Previous research has proposed static and dynamic techniques to detect vulnerabilities in smart contracts. These tools check vulnerable contracts against several predefined rules. However, the emerging new vulnerable types and programming skills to prevent possible vulnerabilities emerging lead to a large number of false positive and false negative reports of tools. To address this, we propose Vulpedia, which mines expressive vulnerability signatures from contracts. Vulpedia is based on the relaxed assumption that the owner of contract is not malicious. Specifically, we extract structural program features from vulnerable and benign contracts as vulnerability signatures, and construct a systematic detection method based on detection rules composed of vulnerability signatures. Compared with the rules defined by state-of-the-arts, our approach can extract more expressive rules to achieve better completeness (i.e., detection recall) and soundness (i.e., precision). We further evaluate Vulpedia with four baselines (i.e., Slither, Securify, SmartCheck and Oyente) on the testing dataset consisting of 17,770 contracts. The experiment results show that Vulpedia achieves best performance of precision on 4 types of vulnerabilities and leading recall on 3 types of vulnerabilities meanwhile exhibiting the great efficiency performance. I. Introduction Powered by the Blockchain technique [1], smart contracts [2] have attracted much attention and been applied in various industries, e.g., financial service, supply chains, smart traffic, and IoTs. Solidity is the most popular language for smart contract for its mature tool support and simplicity. However, the public has witnessed several severe security incidents, including the notorious DAO attack [3] and Parity wallet hack [4]. According to previous reports [5], [6], up to 16 types of security vulnerabilities were found in Solidity programs. These security issues undermine the confidence of people who have executed transactions via smart contracts and eventually affect the trust towards the Blockchain ecosystem. Witnessing the severity and urgency of this problem, researchers and security practitioners have made endeavors to develop automated security scanners [7], [8], [9], [10], [11]. Existing state-of-the-art scanners usually adopt the rule-based methods for vulnerability detection. Slither [7] supports 39 hard-coded static rules; Securify [10] supports 15 rules for verifying the extracted path constraints from the contract with the SMT solvers [12]; Oyente [8] supports 8 rules for generating assertions for verifying the vulnerabilities. Each rule represents a pattern of vulnerable contract, which warns the programmers to avoid potential risks before deploying the contracts. Although experiments have demonstrated their effectiveness, it is notable that rules behind these scanners are manually crafted by human experts. The manually predefined rules can be obsolete, because previously unseen vulnerable code may be introduced, which cannot be captured by the hard-coded rules, and new defense mechanisms (i.e., programming skills to prevent bugs) may have successfully mitigated the vulnerabilities, but the code may still match the predefined vulnerable pattern or rules. Therefore, most updated rules should be learned to distinguish vulnerable contracts from robust ones. In this work, we alleviate the incompleteness of detection rules by combining vulnerability signatures abstracted from both vulnerable and benign contracts (i.e., vulnerable signature and benign signature). The vulnerable signature is designed for matching commonalities of a particular vulnerability. Comparatively, the benign signature is abstracted from falsely reported contracts in order to reduce false alarms. For each vulnerability, we adopt vulnerable and benign signatures to synthesize detection rules of Vulpedia. Note that Vulpedia is built upon the relaxed assumption that the owner of contract is not malicious. Detecting malicious contract (e.g., contract with backdoors, exploit code) is different to the vulnerability detection (i.e., the target of Vulpedia). Based on this assumption, the operations related to the contract owner are all deemed as vulnerability defense behaviors. Compared with previous work, the synthesized rules are more updated and expressive than the predefined rules in the state-of-the-art vulnerability scanners, capturing a lot of unseen patterns in practice. In our implementation, we first collect truly and falsely reported contracts by applying three state-of-the-art vulnerability scanners (i.e., Slither [7], Oyente [13], and Securify [10]) and manually evaluate their correctness. Based on the results, analyzing truly reported vulnerable contracts allows us to capture salient program signatures responsible for vulnerable contracts. In contrast, analyzing falsely reported vulnerable contracts allows us to capture signatures noticeable for avoiding false alarms. Next, we categorize the contracts by their vulnerability types (e.g., Reentrancy, Unchecked Low-level-call, etc.) and alarm types (i.e., true or false alarm). For each category, we cluster the contracts based on their tree edit distance [14] and then extract program feature commonalities from the PDGs (program dependency graph) of each cluster to summarize vulnerability signatures. Finally, we abstract 4 vulnerable signatures and 6 benign signatures. They are integrated as 4 detection rules regarding 4 vulnerabilities (i.e., Reentrancy, SelfDestruct, Tx-origin, and Unexpected-Revert). We conduct our signature abstraction on a set of 76,354 smart contracts and evaluations on a set of 17,770 contracts, respectively. The evaluation results show that, compared with the state-of-the-art vulnerability scanners (i.e., Slither, Oyente, SmartCheck and Securify), our approach achieves outstanding accuracy on 4 vulnerabilities and leading recall on 3 vulnerabilities. We make our tool, Vulpedia, available at [15]. To summarize, we make the following contributions: 1) We propose an approach to abstract vulnerability signatures and compose detection rules to report vulnerability. The learned rules are more expressive than rules of the state-of-the-art scanners, reporting vulnerabilities with better completeness and soundness 2) On the 17,770 contracts crawled from Google, Vulpedia yields the best precision on 4 vulnerabilities and leading recall on 3 ones, in comparison with the other state-of-the-art scanners. 3) Experiments show that Vulpedia is efficient in vulnerability detection. The detection speed of Vulpedia on 17,770 contracts is far faster than Oyente and Securify. This work is organized as: In Sec. II, we first introduce the different types of vulnerabilities we address in our study, and explain why the state-of-the-art tools fail. In Sec. III we illustrate the basic steps of our proposed tool, namely Vulpedia. In Sec. IV, we conduct an empirical study and introduce our method of signature abstraction. We also elaborate the effectiveness of signatures with examples. In Sec. V, we compare Vulpedia with the other state-of-arts using 17,770 real-world contracts deployed on Ethereum. Sec. VII briefly introduces the related work and Sec. VIII concludes our study. II. Background and Motivation In this section, we explain the 4 vulnerability types (i.e., Reentrancy, The abuse of tx.origin, Unexpected Revert and Self-destruct Abusing.) targeted by our study. The 4 vulnerabilities deeply threats the safety of transactions of smart contracts. For example, the Reentrancy caused the DAO attack in 2016 and resulted in hundreds of millions dollars losses; The tx.origin and Unexpected Revert vulnerability are listed in the Decentralized Application Security Project (DASP) [16]; The Self-destruct Abusing vulnerability often appears with the use of selfdestruct instruction in Solidity, and is prone to being exploited if it is not well protected. We also show a real-world case, which is not well-handled by the state-of-the-art scanners, to motivate this work. A. Vulnerability Types 1) Reentrancy (RE) As the most famous Ethereum vulnerability, reentrancy recursively triggers the fall-back function [17] to steal money from the victim's balance or deplete the gas of the victim. Reentrancy occurs when external callers manage to invoke the callee contract before the execution of the original call is finished, and it was mostly caused by the improper usages of the function withdraw() and call.value(amount)(). It was also reported in [5]. 2) The Abuse of tx.origin (TX) When the visibility is improperly set for some key functions (e.g., some sensitive functions with public modifier), the extra permission control then matters. However, issues can arise when contracts use the deprecated tx.origin (especially, tx.origin==owner) to validate callers for permission control. It is relevant to the access control vulnerability in [16]. When a user U calls a malicious contract A, who intends to forward call to contract B. Contract B relies on vulnerable identity check (e.g., require(tx.origin == owner) to filter malicious access. Since tx.orign returns the address of U (i.e., the address of owner), malicious contract A successfully poses as U. 3) Unexpected Revert (UR) In a smart contract, some operations may unfortunately fail. This can lead to two main impact: 1) the gas (i.e., the fee of executing an operation in Ethereum platform) of the transaction is wasted; 2) the transaction will be reverted, i.e., the denial of service (DoS). The denial of service attack is also termed "DoS with revert" in [18]. The attacker could deliberately make some operations fail for the purpose DoS. For example, some functions recursively send ethers to an array of users. If one of these calls fails, the whole transaction will be reverted. An attacker can deliberately fail this transaction to achieve a denialof-service attack. 4) Self-destruct Abusing (SD) This vulnerability allows the attackers to forcibly send Ether without triggering its fall-back function. Normally, the contracts place important logic in the fall-back function or making 1 f u n c t i o n withdraw ( ) { 2 r e q u i r e ( msg . s e n d e r == owner ) ; 3 u i n t 2 5 6 amount = b a l a n c e s [ msg . s e n d e r ] ; 4 r e q u i r e ( msg . s e n d e r . c a l l . v a l u e ( amount ) ( ) ) ; 5 b a l a n c e s [ msg . s e n d e r ] = 0 ; 6 } Fig. 1: An example of a non-vulnerable code. This is misreported as vulnerability by Slither and Oyente. calculations based on a contract's balance. However, this could be bypassed via the self-destruct contract method that allows a user to specify a beneficiary to send any excess ether [18]. That is, a vulnerable contract is prone to being exploited to transfer all money to attacker' account meanwhile shut down the service. The function withdraw intends to send ethers to the msg.sender. It first verifies the identity of caller at line 2. Then, the function reads the amount of current balance of the caller at line 3 and sends ethers to the caller by using a Solidity call .call.value()(). Finally, the function updates the balance of caller at line 5. B. Motivating Examples The reason of the false alarm of Slither is due to that Slither detects reentrancy with the following rule: DataDep( , varg) Call( , varg) DataDep( , varg) ⇒ reentrancy(1) In Rule 1, DataDep( ,var g) denotes write and read operations to variables; var g denotes a certain public global variable; denotes the execution order in the control flow; Call( ,var g ) denotes function call operations. This rule describes a common pattern for Reentrancy vulnerability. Fig. 1 shows a typical example. var g is usually a balance account (e.g., balances[msg.sender], line 3 in Fig. 1). An attacker just needs to create a fallback function that calls withdraw(). Once msg.sender.call.value(amount)() is executed and transfers the funds, the attacker's fallback function [17] will be triggered and call withdraw() (line 1) again. By this means, the attacker can transfer more funds before balances[msg.sender] is reduced to 0. This continues until there is no ether remaining, or execution reaches the maximum stack size. However, the pattern in Rule 1 is usually an overestimation for real Reentrancy vulnerability. In fact, the example in Fig. 1 is a counter-example because the function withdraw() is protected by an identity check at line 2. This statement specifies a precondition for running the withdraw() function. Once the precondition is not satisfied, the execution will be aborted. In Fig. 1, the identity check indicates that the contract calling this withdraw() function is limited to its owner (i.e., the creator of the contract). The reason of the false alarm of Oyente is due to that Oyente detects reentrancy with the following rule: (DataDep( , varg) ∧ (gastrans > 2300)∧ (amt bal > amttrans)) Call( , varg) ⇒ reentrancy (2) In Rule 2, Oyente requires the gas expense less than a certain value. In Solidity programs, each transaction requires an amount of gas to complete in the runtime. gas trans > 2300 means the gas used for transaction must be larger than 2300 (2300 is the least gas expense to conduct a transaction call). amt bal > amt trans means the balance amount must be larger than transfer amount. Finally, the rule of Oyente requires call to external functions by Call meanwhile send money. Comparing with Rule 1 (defined by Slither), Oyente has more constraints for gas and balance value. Similar to the rule of Slither in Rule 1, Rule 2 also overestimates the condition where Reentrancy attack can happen. With the protection by the identity check (i.e., line 2 in Fig. 1), the execution of function calls conforms to the defined runtime conditions but is already free from the Reentrancy attack. How Vulpedia can address this issue: In contrast, Vulpedia is equipped with detection rule composed of a vulnerable signature as shown in Rule 3 (i.e., the signature indicating potential vulnerability) and a benign signature as shown in Rule 4 (i.e., the signature indicating potential code behaviors defending or fixing vulnerability). DataDep( , varg) Call( , varg) ⇒ reentrancy (3) ControlDep(msg.sender, ) DataDep( , varg) Call( , varg) ⇒ reentrancy(4) For the vulnerable signature, Vulpedia adopts valuable experience from Slither and Oyente and detects Reentrancy by matching data dependency of variables followed by call operations. As for the benign signature, Vulpedia eliminates false reports by filtering out functions which contain control dependency on msg.sender. For example, in Fig. 1 the code at lines 2 checks if the msg.sender equals the address of owner, and the function is not considered as vulnerability by Vulpedia. Fig. 2 shows the workflow of abstracting vulnerability signatures for Vulpedia. The workflow can be roughly grouped into four steps: 1) The pre-detection of existing tools; 2) Vulnerability report inspection; 3) AST clustering and signature abstraction; 4) Rule composition. Note that manual efforts are involved in step 2 and step 4. III. Overview In the first two steps, we systematically evaluate (1) how accurately state-of-the-art tools can report the vulnerable smart contracts and (2) under what condition can those tools be ineffective. We collect the reports of the state-ofthe-art tools on a training dataset of 76,354 contracts. Then, we employ three experienced smart contract developers to manually confirm the reports of the tools, and categorize them into two groups: truly alarmed vulnerable contracts and falsely alarmed vulnerable contracts. In the last two steps, we first calculate the tree edit distance based on the ASTs of contracts in a particular vulnerability type and cluster the contracts of the type by defining the contract similarity. Next, we abstract common nodes from the PDGs (program dependency graph) of each cluster to summarize signatures (e.g., as shown in Fig. 4). From truly vulnerable contracts, we summarize vulnerable signatures. In contrast, from falsely alarmed vulnerable contracts, we summarize benign signatures. Finally, we manually integrate the vulnerable signatures and benign ones into vulnerability detection rules. After we equip our Vulpedia detector with the composed rules, the detector takes unknown contracts as inputs and generate vulnerability reports based on the signatures. The figure of the architecture is shown in Fig. 3. Specifically, the detector first conduct preprocessing on the input smart contract code. The detector extracts normalized AST from the contract. Based on this normalized AST, the detector conducts a PDG extraction. Meanwhile, the detector extracts existing signatures from the vulnerability signature database. Lastly, the detector produces detection reports based on the comparison results. That is, if the PDG matches vulnerable signatures but not matched with benign signatures, the contract will be deemed as a vulnerable contract; otherwise, the detector will produce a non-vulnerable report. IV. Empirical Study of Signature Abstraction In this section, we first illustrate how we empirically collect contracts in this study. We report how we select the vulnerability scanners and how we construct a contract dataset. Next, we introduce our method of clustering similar contracts by comparing tree edit distance, abstracting commonalities from PDGs of clusters as signatures and detection rules composition based on the abstracted signatures. Finally, we elaborate the signatures with examples to evidence the representativeness of them. A. Selected Scanners and Dataset 1) Choice of Scanners and Vulnerability Types: Overall, we select vulnerability scanners based on how practical they can be used in real-world scenarios. We investigate a list of static analyzers, including Slither [7], Oyente [13], Zeus [9] SmartCheck [11], and MythX [20]. These tools utilize manually defined detection rules to detect vulnerabilities. The rules could match vulnerabilities in some cases but also generate much false reports. We also investigate dynamic detectors like Mythril [19], Con-tractFuzz [24], Echidna [21] and Manticore [22]. They exercise programs and check the runtime status of functions to find vulnerabilities. The dynamic analyzers often achieve high detection precision but suffer from limited scalability. Additionally, we investigate other analyzing tools (e.g., Solidity reverse engineering tool Octopus [23]) to facilitate our exploiting contracts. A summary of the above tools can be found at Table I. In our study, some tools are not selected because they are not open-sourced (Zeus [9], MythX [20]), not related to our task (Echidna [21], Octopus [23]) and efficiency concerns (Mythril [19], ContractFuzzer [24], Manticore [22]). Finally, we choose Slither v.0.4.0, Oyente v0.2.7 and SmartCheck v2.0 as our scanners. 2) Dataset for Empirical Study: We implement a web crawler to download Solidity files from accounts of Etherscan [25], a famous third-party website on Ethereum block explorer. Etherscan provides APIs for downloading transaction information (e.g., transaction addresses, time). 2) The versions 0.4.24 and 0.4.25 are supported by most analyzers, so that they facilitate our study. Additionally, we find that the downloaded dataset has redundant contracts (contracts which share commonality with others). Regarding these redundant contracts, we remove contracts that are exactly same to others and contracts that are only different in transfer address with others. Finally, we have 76,354 contracts for empirical study. Our crawler can be accessed at https://github.com/ToolmanInside/smart contract crawler. Table III shows the number of contracts we collected in this study. Overall, among 76,354 contracts, the three tools report 508 true vulnerable contracts albeit 3,496 false vulnerable ones. Table IV shows the details on the number of reported contracts and precision performance of each tool. We observed that all the tools have a large number of false alarms. This is due to contract programmers have invented many heuristics to detect the potential vulnerabilities. In other words, most existing detection rules are obsolete. It motivates us to pursue (and generate) a more expressive and fine-grained rule to mitigate the false alarms. B. Vulnerability Rule Abstraction In this section, we introduce the definition of signature and show how we cluster and abstract the vulnerable/benign signatures from each cluster. 1) Definition: We define a vulnerability rule for a Solidity contract as following BNF: rule ::= comp sig comp sig ::= ¬ comp sig \| ( comp sig ∨ comp sig ) \| ( comp sig ∧ comp sig ) \| ( comp sig comp sig ) \| sig sig ::= DataDep(X,Y ) \| ControlDep(X,Y) \| ForLoop \| IsInstance(X,Y) \| Call(L,X) \| SelfDestruct(X ) \| msg.sender \| tx.origin \| Here, the detection rule is composite of signatures. A composite signature is a negation (¬) of itself, or conjunction (∧), union (∨), succeed ( ) with another composite signature. A composite signature can also be a single vulnerability signature. Specifically, vulnerability signature indicates basic program relationships and built-in keywords of Solidity language. For example, the data dependency (DataDep(X,Y )) relationship denotes that variable X has data dependency to Y (i.e., variable assignment operations). The control dependency (ControlDep(X,Y)) indicates assertation operations (e.g., require, assert) between variables X and Y. The for loop ForLoop denotes the function body exists a for loop statement. The IsInstance(X,Y) denotes the variable X is a type of variable Y. Call operation CALL(L,X) includes low level calls (e.g., call.value() and send() in Solidity) and high level calls (i.e., userdefined function calls). Here, variable L represents the result of call operations and variable X represents the parameters required by the call. SelfDesutrct(X ) is a built-in function PDG p ← getP DG(t) 18 pn ← P DGN ormalization(p) 19 P DGs ← P DGs ∪ pn 20 commonSeq ← LCS(P DGs) 21 SignatureCands ← SignatureCands ∪ commonSeq 22 return SignatureCands call in Solidity. Once it is called, the service of current contract is stopped and the rest balance is transferred to an arbitrary receiver X. The msg.sender and tx.origin are built-in variables. Specifically, msg.sender denotes the address of current contract and tx.origin denotes the origin of call chains [17]. 2) Contract Clustering: In this section, we first define contract similarity on normalized ASTs, and then we cluster similar trees by using hierarchical clustering algorithm. The clustering procedure can be found at line 1 to line 10 in the Algorithm 1. Contract Similarity. We define the contract similarity by considering both semantic and structural information of the code. To this end, we use AST (Abstract Syntax Tree) to represent the code of the functions of each contract. For each AST of a Solidity function, we normalize the concrete nodes in the AST for retaining core information and abstracting away unimportant details such as variable names or constant values, as shown in line 2 of Algorithm 1. For each AST corresponding to a function, we just retain the information such as node type, name, parameter and return value (if contained). For the variable names (e.g., indexs in code block A and code block B of Fig. 4), they will be normalized with the token asterisk " * ". Similarly, we repeat the same normalization for constant values of the types string, int, bytes or uint. Given two trees, we use the tree edit distance between two normalized ASTs as their distance. The AST normalization process is shown in line 3 of Algorithm 1. In this work, we apply a robust algorithm for the tree edit distance (ARTED) [14], which computes the optimal path strategy by performing an exhaustive search in the space of all possible path strategies. Here, path strategy refers to a mapping between two paths of the two input trees (or subtrees), as the distance between two (sub)trees is the minimum distance of four smaller problems, i.e., (1) the edit distance between two empty trees, (2) the edit distance of transferring a tree F to an empty tree, (3) the edit distance of transferring an empty tree to a tree F and (4) the edit distance of transferring a tree F to another tree G. Note that though ARTED runs in quadratic time and space complexity, it is guaranteed to perform as good or better than its competitors [14]. Contract Clustering. We cluster the ASTs via hierarchical clustering algorithm with complete linkage [27], as shown in line 4 to line 10 in Algorithm 1. Then, we group the codes in Fig. 4 with considerable modification. We deem that the ASTs in each cluster share commonalities as a feature (or signature) for a vulnerability category. 3) Signature Abstraction: After clustering contract functions with AST, we abstract signature by referring to their PDG (Program Dependency Graph) information. The reason lies in that PDG allows us to capture the code semantic features like control and data dependencies. PDG Representation. For each AST, we transfer its code into a PDG including all its depended code elements such as global variables and called functions, as shown in line 13 to line 17 in Algorithm 1. In a PDG, each of its nodes is an instruction and the edge between nodes indicates data dependency, control dependency, and call relation between the nodes. Thus, given a cluster containing N Solidity functions, we reduce it into a problem of finding the common subgraph of N PDGs. The normalization of PDGs is shown in line 18 in Algorithm 1. PDG Matching. The graph matching problem is a NPcomplete problem. We simplify the problem with the following steps. Before matching, we also abstract away variable names and constant values in the PDGs as we do that for AST. Next, we simplify the calculation by flattening the graph into a node sequence (via depth first order search) and align the sequences by LCS algorithm [28], as shown in line 20 in Algorithm 1. The aligned graph nodes are considered as commonalities shared by the code in the same cluster. As a result, the signature abstracted from a cluster is essentially a graph skeleton, as shown in Fig. 4. Then, we manually inspect those skeletons and refine them into usable signatures. The refining process requires manual efforts because some signatures are semantically similar to others but different in syntax. These signatures require to be filtered out by human expert. After we repeat the above procedures on both vulnerable and benign contracts, we construct a set of vulnerable and benign signatures. 4) Rule Composition: In this work, we follow the following heuristics to integrate the signatures into a rule. Generally, a detection rule is a composite boolean expression of vulnerability signatures. Given a vulnerability category, a detection rule first requires the input contract match with the vulnerable signatures. The vulnerable signatures are essential ingredients of forming a vulnerability. Therefore, if the input contract is not matched with vulnerable signatures, the contract should be considered as invulnerable. Next, the input contract is required not to match with benign signatures. The benign signatures are the best practices to defend vulnerabilities. If the input contract matches with them, it suggests that the contract is capable for defending vulnerabilities and should not be reported as vulnerability. C. Case Study: Abstracted Signatures We applied the three chosen scanners to 76,354 contracts. Overall, Slither reports the most vulnerabilities, in total 3,422 (623 + 67 + 2,678 + 54) candidates covering four types. In contrast, SmartCheck reports 439 (150 + 289) candidates and Oyente reports only 143 candidates. After they are processed by our methods, we abstract 4 vulnerable signatures and 6 benign signatures, as shown in Table V. Based on these signatures, we further integrate them into 4 detection rules, as shown in Table VI. In this section, we elaborate the signatures with examples to evidence their representativeness. Signature of Reentrancy. We extract 4 signatures from TPs and FPs of reported reentrancy vulnerabilities, including 1 vulnerable signature (SIG1) and 3 benign signatures (SIG2, SIG3, SIG4). SIG1 is abstracted from general patterns of reentrancy vulnerabilities. This signature consists of two parts: (1) the read or write operation of variable X (i.e., DataDep( ,X )) and (2) the call operation with the parameter variable X (i.e., Call( ,X )). SIG2 adds various forms of checks (i.e., in require or assert) for msg.sender compared with SIG1. For example, SIG2 checks whether the identity of msg.sender is checked under certain conditions (e.g., equal to the owner, or with a good reputation, or having the dealing history) before calling the external payment functions. With the identity check, the function is only accessible to related users, blocking the malicious attack from attackers. Example of this signature can be found at Fig. 1. SIG3 describes a falsely reported case of transferring balance to a fixed address. In Fig. 5, the function closePosition sends balance to a token bancorToken which is assigned with a fixed address at line 2. According to the detection rule of Slither (See Rule 1), this code is a vulnerability because -(1) it reads the public variable agets[ idx]; (2) then calls external function bancorToken.transfer(); (3) last, writes to the public variable agets[ idx]. However, in practice, this contract can never be easily exploited to steal ethers due to the hard-coded address constant (i.e., 0x1F...FF1C). Note that the constant address can be a malicious address, under such circumstance this address cannot protect contract. However, this case is very rare. Therefore, we choose to trust the creator of the contract as well as the designated addresses are benign. SIG4 is to prevent from the recursive entrance of the function -eliminating the issue from root. For instance, in Fig. 6, the internal instance variable reEntered will be checked at line 5 before processing the business logic between line 8 and 10. To prevent the reentering due to calling ZTHTKN.buyAndSetDivPercentage.value(), reEntered will be switched to true; after the transaction is done, it will be reverted to false to allow other transactions. Signature of Unexpected Revert. We extract 2 signatures from reported Unexpected Revert vulnerabilities, including 1 vulnerable signature SIG5 and 1 benign signature SIG6. SIG5 represents general patterns of Unexpected Revert vulnerabilities. This signature consists of three parts: (1) the for loop program structure (i.e., ForLoop); (2) the call operation of the variable X (i.e., Call( ,X )); (3) the result of call operation is further checked by assertions. 1 Reentrancy SIG1 ∧ ¬ (SIG2 ∨ SIG3 ∨ SIG4) 2 Revert SIG5 ∧ ¬ SIG6 3 Tx.origin SIG7 ∧ ¬ SIG8 4 Self-destruct SIG9 ∧ ¬ SIG10 According to the recent technical article [29], the rules of Call/Transaction in Loop are neither sound nor complete to cover most of the unexpected revert cases. At least, modifier require is often ignored, which makes Slither and SmartCheck incapable to check possible revert operations on multiple account addresses. Here, multiple accounts must be involved for exploiting this attack -the failure on one account blocks other accounts via reverting the operations for the whole loop. Hence, in the example of Fig. 7, the operations in the loop are all on the same account (i.e., sender at line 5) and potential revert will not affect other accounts. Therefore, the transfer operation of which target is a single address is considered as SIG6. Signatures of Tx.Origin Abusing. We extract 2 signatures from the truly vulnerable contracts and falsely reported contracts, including 1 vulnerable signature (SIG7) and 3 benign signatures (SIG8). For SIG7, this signature is extracted from general patterns of tx.origin vulnerabilities. This vulnerability first reads the value of tx.origin, followed by an assignment to variable X (i.e., DataDep(tx.origin,X )). After this, the function has an assertion to this variable (i.e., ControlDep(X, )). While we extract signatures from the TPs of vulnerabilities, we find that our SIG7 is slightly looser than the detection rule in Slither. Slither skips the function if there exists a read operation to a particular variable msg.sender, ignoring that some of these variables are irrelevant to tx.origin. In order not to overlook potential vulnerabilities, our SIG7 only requires a read of tx.origin, followed by an assertion on this variable. For SIG8, we observe that SmartCheck reports much more cases (210) than Slither (34), but has lower precision performance than Slither. After our investigation, we find that the incorrect reports of SmartCheck are due to the unsound rules (as shown in Rule 5). That is, SmartCheck r e q u i r e ( msg . s e n d e r == owner ) ; 3 i f ( owner . s e n d ( a d d r e s s ( t h i s ) . b a l a n c e ) ) { 4 s e l f d e s t r u c t ( burn ) ; } 5 } simply reports vulnerability once tx.origin appears in assertion statements. However, under some circumstance (e.g., comparing msg.sender with tx.origin), the use of tx.origin should not be reported. We summarize the SIG8 based on the FPs of SmartCheck. DataDep(tx.Origin, X) ControlDep(X, ) ⇒ Tx.Origin abusing (5) Signature of Self-destruct Abusing. We extract 2 signatures from the self-destruct vulnerabilities, including 1 vulnerable signature SIG9 and 1 benign signature SIG10. SIG9 is extracted from general patterns of self-destruct vulnerabilities. This signature consists of two parts: (1) the read or write operation of variable X (i.e., DataDep( ,X )) and (2) the call operation of the self-destruct with the parameter X (i.e., SelfDestruct(X )). For SIG10, we extract this signature from FPs of tools. In the existing scanners, only Slither detects the misuse of self-destruct, which is called suicidal detection. In total, Slither reports 54 cases of suicidal via its built-in ruleas long as function SelfDestruct is used, no matter what the context is, Slither will report it. Obviously, the Slither's rule is too simple and too general. It mainly works for directly calling SelfDestruct without permission control or conditions of business logic -under such circumstance (3 out of 54), the Slither rule can help to detect the abusing. In practice, in most cases (51 out of 54) SelfDestruct is called with the admin or owner permission control or under some strict conditions in business logic. For example, SelfDestruct is indeed required in the business logic at line 2 of Fig. 8. As the owner wants to stop the service of the contract via calling SelfDestruct, after the transactions are all done, the contract becomes inactive. Note that parameter burn is just padded to call SelfDestruct in a correct way. Hence, we summarize the SIG10, adding a strict condition control or a self-defined modifier for identity check when using SelfDestruct. In brief, for a vulnerability type, we use the vulnerable signatures to match potential vulnerabilities, which yield a better recall. Then, we leverage corresponding benign signatures to filter out false reports. D. Vulnerability Detection The implementation of the vulnerability detection of Vulpedia is based on the previously abstracted signatures and integrated detection rules, but slightly different from them. The workflow of detection is shown in Fig. 9. Specifically, in this workflow, Vulpedia reports vulnerability only Table VI all follow the pattern that the vulnerable signatures should be matched but the benign ones should not. Therefore, though the implementation of the detection process seems differently, the logic of the workflow is the same with previous designs. V. Evaluation Experimental Environment. Throughout the evaluation, all the steps are conducted on a machine running on Ubuntu 18.04, with 8 core 2.10GHz Intel Xeon E5-2620V4 processor, 32 GB RAM, and 4 TB HDD. For the scanners used in evaluation, no multithreading options are available and only the by-default setting is used for them. Tool Implementation. Vulpedia is implemented based on the Slither analyzer. We adopt the AST analysis from Slither, and we build PDG analysis based on the CFG (control flow graph) and call graph of Slither. The vulnerability signatures are implemented as detectors in nearly 1,000 lines of Python code. The demo of our tool can be found at https://github.com/ToolmanInside/vulpedia demo. Dataset for Tool Evaluation. To take a different dataset from contracts we used in empirical study, we get another address list of contracts from Google BigQuery Open Dataset. After removing contracts that already used in our empirical study, we get the other 17,770 real-world contracts deployed on Ethereum, on which we fairly compare our resulted tool Vulpedia with the version of the scanners: Slither v0.6.4. Oyente v0.2.7, SmartCheck v2.0 and Securify v1.0 that is opensourced at Dec 2018. The evaluation dataset is opened along with empirical study dataset at https://drive.google. com/file/d/1kizsz0 8B8nP4UNVr0gYjaj25VVZMO8C. The evaluations are conducted based on a relaxed assumption that the owners of contracts are not malicious. That is, the operations of the owners are all deemed as defense behaviors to vulnerabilities. The evaluations aim to answer these RQs: RQ1. How is the precision of Vulpedia, compared with the existing scanners in vulnerability detection? RQ2. How is the recall of Vulpedia? Can our signaturebased method report more vulnerabilities? RQ3. How is the efficiency of Vulpedia, in tool comparison on the datasets? A. RQ1: Evaluating the Precision of Tools As mentioned in Sec. IV-C, we have learned 10 signatures in total for the four types of vulnerabilities. To evaluate the effectiveness of the resulted vulnerable signatures and detection rules, we apply them on the 17,770 newly collected contracts and compare with the other state-ofthe-arts detection tools. Details on the performance of each tool are shown in Table VII. Note that all TPs are manually verified by our authors. In Table VII, we list 280 detection results of Vulpedia, with an averaged precision of 50.2%, regardless of vulnerability types. In comparison, Slither has an averaged precision of 18.9%; Oyente's averaged precision is 7.1%; SmartCheck's averaged precision is 40.2%; and Securify's precision is surprisingly only 1.1%. In the rest of this section, we analyze the false positives of these tools from the perspective of supporting vulnerability signatures. FPs of Reentrancy. Among the four supported tools except SmartCheck, Vulpedia yields the lowest FP rate (71.5%) owing to the adoptions of benign signatures for reentrancy. FP rates of other tools are even higher. For example, the FP rate of Securify is 98.9%, as its detection pattern is too general but has not considered possible defense to vulnerabilities in code. Slither adopts Rule 1 to detect, but it supports no benign signatures -its recall is acceptable, but FP rate is high. Oyente adopts Rule 2 and has no benign signatures -its recall is low due to the strict rule, and its FP rate is also high. FPs of Unexpected Revert. As summarized in Sec. IV-C, Slither reports Unexpected Revert vulnerability when a call in loop is detected, ignoring the potential false alarms (i.e., low level call in a loop). This coarse detection rule leads to 335 FPs. SmartCheck handles SIG5 but not SIG6 and leads to 27 FPs. In comparison, Vulpedia combines SIG5 and SIG6 for integrating detection rule, yielding the lowest FP rate 51.2%. FPs of Tx.Origin Abusing. Slither has a strict rule for detecting this type, only checking the existence of tx.Origin == msg.sender. We find that this tool also skips the function if there exists a read operation to a particular variable msg.sender, ignoring that some of these variables are irrelevant to tx.origin. For the case that tx.origin is compared with an unrelated address variable, Slither reports it as vulnerability, causing FPs. Comparatively, SmartCheck and Vulpedia manage to include all the identity check cases, but meanwhile also lead to FPs due to the fact -accurate symbolic analysis is not adopted in SmartCheck or Vulpedia to suggest whether tx.Origin can be used to rightly replace msg.sender. Hence, the FP rate due to ignoring SIG8 is higher than that of Vulpedia. FPs of Self-destruct Vulnerability. Vulpedia has 13 FPs. After inspecting, we find 10 FPs are due to the unsatisfactory handling of SIG10. That is, the identity check hides in self-defined modifiers. Function modifiers are overlooked by Vulpedia, causing FPs. Comparatively, Slither only reports 3 true positives. The reason is that Slither simply reports vulnerability when a SelfDestruct call is detected. Due to the inconsideration of the potential access controls, Slither performs less precision than Vulpedia. Answer to RQ1: Vulpedia performs best in evaluations of precision among tools. In detecting tx.origin vulnerability, Vulpedia outperforms the second best tool by 45.3% (88.7% -44.3%). The reason of the high precision performance is Vulpedia adopts effective benign signatures to remove false reports. B. RQ2: Evaluating the Recall of Tools In Table VII, in most cases, Vulpedia yields the best recall except on unexpected revert, where R% for SmartCheck is 77.4% and R% for Vulpedia is 67.7%. Based on the vulnerable signature abstracted in empirical study, we expect Vulpedia can find more similar vulnerable candidates. A comparison between vulnerabilities only found by Vulpedia (denoted by green bars) and vulnerabilities found by other tools (represented by red bars) is shown in Fig. 10. Recall of Reentrancy. In this vulnerability, Vulpedia performs best by report 69.3% vulnerabilities. Among all TPs, Vulpedia finds 56% unique TPs that are missed by other evaluated tools. We find that the other three tools commonly fail to consider the user-defined function transfer(), not the built-in payment function transfer(). For the example in Fig. 11, Slither and Securify miss it as they mainly check the external call for low-level functions (e.g., send(), value()) and built-in transfer(), ignoring user defined calls. Oyente does not report this example, as it fails in the balance check according to Rule 2. Comparatively, Vulpedia detects this vulnerability, as we have an vulnerable signature that has a high code similarity with this example. Notably, though Vulpedia has the best recall of 69.3%, it misses 30.7% TPs. This is due to the fact that reentrancy has many forms, and our vulnerable signature is not sufficient to cover those TPs. Recall of Unexpected Revert. In this vulnerability, the performance of Vulpedia is slightly worse than SmartCheck (77.4%). Specifically, Vulpedia only reports 9% unique TPs while 91% TPs are found by other tools. The reason of the TPs missed by Vulpedia (reported by SmartCheck) are due to the incompleteness of our vulnerable signature SIG6. That is, the signature requires a ControlDep after Call. However, the ControlDep is unnecessary when the Call is a high level call (e.g., user defined function call) because assertion operations are already integrated in high level calls. Therefore, the signature causes FNs. Recall of Tx.Origin Abusing. For this type, 96.6% TPs are found by Vulpedia-almost all TPs are found by Vulpedia. Additionally, Vulpedia reports 40% unique TPs which are missed by other tools. The reason is that we matches identity check of Tx.Origin in self-defined modifiers, which is commonly overlooked by other tools. Recall of Self-destruct Abusing. For this type, all vulnerabilities (100%) are found by Vulpedia. Comparatively, Slither only reports 42.8% vulnerabilities. 57% of TPs are only found by Vulpedia. The rationale of TPs missed by Slither is that Slither skips the function if the function is only accessible to internal calls (i.e., set visibility to internal). These functions are however prone to being exploited by internal calls. Therefore, they should not be overlooked. Vulpedia leverages SIG9 to match vulnerability candidates, so we have better recall performance. ERC20 t o k e n = ERC20( addr ) ; 10 a s s e r t ( t o k e n . t r a n s f e r ( msg . s e n d e r , amount ) ) ; } 11 } 12 } Fig. 11: A real case of reentrancy. This is a TP for Vulpedia but a FN for Slither, Oyente and Securify. Answer to RQ3: Vulpedia outperforms Securify and Oyente regarding the detection efficiency on both empirical evaluation and tool comparison. In general, Vulpedia is efficient as a signature-based vulnerability detection tool. D. Threats to Validity In our experiments, we adopt recall rate as a metric, which is a potential threat. Generally, the recall rate indicates the number of TPs divided by the number of all vulnerabilities. However, it requires an overwhelming effort to find out all vulnerabilities (i.e., the ground truth). In our study, we evaluate recall performance based on the union of vulnerabilities reported by all tools. Additionally, in the abstraction of signatures, we manually confirm signatures, which may introduce bias. To alleviate this, we repeat our experiments for 3 times. Also, we note that the randomness is an inevitable factor in the evaluations of efficiency. We repeat the experiments for 5 times and record the average values. Besides, the abstracted signatures are prone to introducing incompleteness. To alleviate this, we implement our methods on the top of Slither, which facilitates our signature abstraction from PDGs. VI. Discussions A. The Relaxed Security Assumption The experiments and comparisons are all conducted based on the relaxed security assumption. That is, we assume the operations of contract owner are not malicious behaviors. We follow this assumption because the securitydesign is more strict than ordinary contract when the contract is designed for industry needs. In fact, existing successful contracts (e.g., e-voting, NFT) have been audited by experts to be protected from rogue owners. To avoid our tool been blindly used by users and developers, this assumption should be pointed out. B. The Weakness of Vulpedia In this section, we discuss the improvement of the weakness of Vulpedia found in our experiment practice. In our view, involving manual efforts brings biases, and the biases may affect the effectiveness of the tool. However, Vulpedia relies on manual efforts, mainly in the two steps: 1) manually confirm the reports of existing tools in our empirical study. We add man-powers in this step because the existing static tools have severe limitations and produce a large number of false reports. Due to Ren et al. [30], the Slither tool has a false positive rate over 70%. If the false reports are not removed from all reports, the dataset cannot be correctly labeled, and our signature abstraction is infeasible. 2) We manually integrate the vulnerable signatures and benign ones into vulnerability detection rules. In this step, we use manual efforts to filter out ineffective signatures. This is due to the lack of smart contract vulnerability benchmark. If we have a benchmark, we can replace the man-powers in this step and filter out ineffective signatures by running testing on the benchmark. VII. Related Work Vulnerability Detection in Smart Contracts. There is already a list of security scanners on smart contracts. From the perspective of software analysis, these scanners could be categorized into static-or dynamic-based. In the former category, Slither [7] aims to be the analysis framework that runs a suite of vulnerability detectors. Oyente [13] analyzes the bytecode of the contracts and applies Z3solver [31] to conduct symbolic executions. Recently, SmartCheck [11] translates Solidity source code into an XML-based IR and defines the XPath-based patterns to find code issues. Securify [10] is proposed to detect the vulnerability via compliance (or violation) patterns to guarantee that certain behaviors are safe (or unsafe, respectively). These static tools usually adopt symbolic execution or verification techniques, being relevant to Vulpedia. However, none of them applies code-similarity based matching technique or takes into account the possible DMs in code to prevent from attacks. There are some other tools that enable the static analysis for smart contracts. VeriSmart [32] proposes a domain-specific algorithm for verifying smart contracts. VerX [33] combines symbolic execution and contract status abstraction to verify transactions. Zeus [9] adopts XACML as a language to write the safety and fairness properties, converts them into LLVM IR [34] and then feeds them to a verification engine such as SeaHorn [35]. Besides, there is another EVM bytecode decompiling and analysis frame, namely Octopus [23], which needs the users to define the patterns for vulnerability detections. To prevent the DAO, Grossman et al. propose the notion of effectively Callback Free (ECF) objects in order to allow callbacks without preventing modular reasoning [36]. Maian is presented to detect greedy, prodigal, and suicidal contracts [37], and hence the vulnerabilities to address differ from the types we address in this paper. The above tools are relevant, but due to various reasons (e.g., issues in tool availability), we cannot have a direct comparison with them. The less relevant category includes dynamic testing or fuzzing tools: Manticore [22], Mythril [19], MythX [20], Echidna [21] and Ethracer [38]. sFuzz [39] and Harvey [40] use the advanced techniques (e.g., concolic testing, fuzzing and tainting) for detection. Dynamic tools often target certain vulnerability types and produce results with few FPs. However, they are unsuitable for a large-scale detection due to the efficiency issue. Code-similarity based Vulnerability Detection. In general, similar-code matching technique is widely adopted for vulnerability detection. In 2016, VulPecker [41] is proposed to apply different code-similarity algorithms in various purposes for different vulnerability types. It leverages vulnerability signatures from National Vulnerability Database (NVD) [42] and applies them to detect 40 vulnerabilities that are not published in NVD, among which 18 are zerodays. As VulPecker works on the source code of C, Bingo [43] can execute on binary code and compare the assembly code via tracelet (partial trace of CFG) extraction [44] and similarity measuring. Vuddy [45] targets at exact clones and parameterized clones, not gapped clones, as it utilizes hashing for matching for the purpose of high efficiency. To sum up, these studies usually resort to the vulnerability database of C language for discovering similar zero-days. In contrast, plenty of our efforts are exhausted in gathering vulnerabilities from other tools for smart contracts and auditing them manually. Vulpedia adopts a more robust algorithm (e.g., LCS), which can tolerate big or small code gaps across the similar candidates of a vulnerability. VIII. Conclusion In this study, we propose Vulpedia, a static analyzer based on abstracted signatures. We focus on addressing one key challenges: the manually predefined detection rules can be obsolete. To this end, we first conduct an empirical study for signature abstraction. We leverage state-of-thearts scanners to detect vulnerabilities on our training dataset. Based on their results, we propose a method to cluster similar contracts and abstract vulnerable signatures and benign signatures, respectively. After we collect all signatures, we conduct comparative evaluations with stateof-the-art tools. The results show that Vulpedia exhibits best performance of precision on 4 types of vulnerabilities and leading recall on 3 types of vulnerabilities with great efficiency performance. Fig. 1 1is mistakenly alarmed by Slither and Oyente. Fig. 2 : 2The workflow of extracting vulnerability signatures of Vulpedia. Fig. 3 : 3The architecture diagram of Vulpedia. We choose contracts deployed by Solidity 0.4.25 and 0.4.24. The reasons are two folds: 1) as listed inTable II, Solidity 0.4 is the majority version among all versions and the 0.4.24 and 0.4.25 are the latest versions in Solidity 0.4; Fig. 4 :11 4Three similar code blocks of Unexpected Revert that are found in real-world contracts. Based on their tree edit distance, we cluster them together and abstract a graph skeleton from their PDG. The yellow boxes denote function inputs, blue boxes denote common nodes on PDG and white boxes in dotted box represent different nodes.Algorithm 1: Contract Clustering and Signature Abstraction Algorithm input : SourceCode, source code of smart contracts output : SignatureCands, abstracted signature candidates 1 // Contract Clustering Process 2 ASTs = getAST (SourceCode) 3 nASTs = AST N omalization(AST s) 4 distanceMatrix = List[N, N ] 5 // N is the number of trees 6 foreach idx i ∈ range(nAST s) do 7 foreach idx j ∈ range(nAST s)andi = j Clusters = hierarchicalClustering(distanceM atrix) 12 // Signature Abstraction Process 13 SignatureCands ← ∅ 14 foreach cluster c ∈ Clusters do 15 P DGs ← ∅ 16 foreach tree t ∈ c do 17 Fig. 5 : 5A real case of using SIG3 (a hard-coded address at line 3), a FP of reentrancy for Slither. Fig. 6 : 6A real case of using SIG4 (an execution lock of reEntered), an FP of reentrancy for Slither. Fig. 7 : 7A real FP of Unexpected Revert reported by SmartCheck, where only one account is involved (SIG6). Fig. 8 : 8A real FP of self-destruct abusing by Slither, as selfdestruct() is used under two checks at line 2,3 (SIG10). Fig. 9 : 9The detection workflow of Vulpedia. when the vulnerable signatures are matched meanwhile the benign signatures are not matched. That is, the vulnerable and benign signatures are separated things. However, in previous subsection, the signatures are combined to form detection rules. The reason is that our benign signatures are designed to filter out false positive reports. The detection rules shown in Fig. 10 : 10Comparing the vulnerabilities only reported by Vulpedia with vulnerabilities reported by other tools. "Our Unique" means those only found by Vulpedia. TABLE I : IThe state-of-art tools for Solidity analysis.Tool Name Method Technique Open Source Implementation Adopted In Experiment Mythril [19] Dynamic Constraint Solving G Python H MythX [20] Dynamic Constraint Solving H N.A. H Slither [7] Static CFG Analysis G Python G Echidna [21] Dynamic Fuzzy Testing G Haskell H Manticore [22] Dynamic Testing G Python H Oyente [13] Dynamic Constraint Solving G Python G SmartCheck [11] Static AST Analysis G Java G Octopus [23] Static Reverse Analysis G Python H Zeus [9] Static Formal Verification H N.A. H ContractFuzzer [24] Dynamic Fuzzy Testing G Go H TABLE II : IIThe percentages of adopted Solidity contracts versions. According to[26].Major Version # of Smart Contracts Percentage 0.1 13 <0.1% 0.2 89 <0.1% 0.3 519 0.39% 0.4 71,350 54.27% 0.5 32,479 24.69% 0.6 22,171 16.85% 0.7 4,200 3.19% 0.8 725 0.55% TABLE III : IIINumber of collected contracts for each categoryAlarm Type RE TX UR SDTrue Positive 46 38 421 3 False Positive 720 179 2,546 51 TABLE IV : IVThe precision performance of three tools Slither, Oyente and SmartCheck on four vulnerabilities.Vulnerability Slither Oyente SmartCheck RE 623 (3.53%) 143 (16.78%) N.A. TX 67 (28.35%) N.A. 150 (12.66%) UR 2,678 (8.25%) N.A. 289 (69.20%) SD 54 (5.56%) N.A. N.A. TABLE V : VExtracted Signatures from Different Vulnerability CategoriesID Vulnerability V/B Signature 1 Reentrancy V DataDep( ,X) Call( ,X) 2 B ControlDep(msg.sender,X ) DataDep( ,X) Call( ,X) 3 B DataDep( ,X) IsInstance(X,addr) Call( ,X) 4 B ControlDep(Y, ) DataDep( ,X) Call( ,X) DataDep(Y, ) 5 Unexpected Revert V ForLoop Call(L,X) ControlDep(L, ) 6 B ForLoop IsInstance(X,addr) ∧ Call(L,X) ControlDep(L, ) 7 Abuse of Tx.origin V DataDep(tx.origin,X ) ControlDep(X, ) 8 B DataDep(msg.sender,Y ) DataDep(tx.origin,X ) ControlDep(X,Y) 9 SelfDestruct V DataDep( ,X) SelfDestruct(X) 10 B ControlDep(msg.sender,X) DataDep( ,X) SelfDestruct(X) TABLE VI : VIDetection rules for each vulnerabilityID Vulnerability Rule TABLE VII : VIIThe detection performance for our tool and other existing ones on the 17,770 contracts, where #N refers to the number of detections, P% and R% refer to the precision rate and the recall rate among the number of detections, respectively. Note that P%= (#TP of the tool)/#N, and R%= (#TP of the tool)/ (#TP in union of all tools).Vulnerability Slither Oyente SmartCheck Securify Vulpedia #N P% R% #N P% R% #N P% R% #N P% R% #N P% R% Reentrancy 162 9.8% 32.6% 28 7.1% 4.1% N.A. N.A. N.A. 797 1.1% 18.3% 119 28.5% 69.3% Abuse of tx.origin 23 43.4% 33.3% N.A. N.A. N.A. 44 33.3% 56.6% N.A. N.A. N.A. 98 88.7% 96.6% Unexpected Revert 356 5.8% 67.7% N.A. N.A. N.A. 51 47.1% 77.4% N.A. N.A. N.A. 43 48.8% 67.7% Self Destruct 18 16.6% 42.8% N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. 20 35.0% 100% TABLE VIII : VIIIThe time (min.) of vulnerability detection for each scanner on 76,354 and 17,770 contracts. "S.C." denotes SmartCheck. Slither Oyente S.C. Securify Vulpedia Answer to RQ2: Vulpedia has best performance on detection recall. Except Unexpected Revert, Vulpedia outperforms other tools on three vulnerabilities. The reason of this leading performance is our abstracted vulnerable signatures can represent essence of most vulnerabilities.C. RQ3: Evaluating the EfficiencyOn Dataset for Empirical Study. InTable VIII, Slither takes the least time (only 156 min) in detection. SmartCheck and Vulpedia have the comparable detection time (500∼1000 min). They are essentially of the same type of technique -pattern based static analysis. In practice, they may differ in performance due to implementation differences, but still, they are significantly faster than Oyente that applies symbolic execution. Compared with other dynamic analysis or verification tools (i.e., Mythrill and Securify that cannot finish in three days for the 76,354 contracts), Oyente is quite efficient. Notably, the signature abstraction time of Vulpedia is not included in the detection time, as it could be done off-line separately. Since signature abstraction is analogical to rules formulation, it is not counted in the detection time. On Dataset for Tool Evaluation. On the smaller dataset, we observe the similar pattern of time execution -Slither is the most efficient, Oyente is least efficient (except Securify), and SmartCheck and Vulpedia have the comparable efficiency. Notably, Securify can finish the detection on 17,770 contracts, but it takes significantly more time than other tools. The performance issue of Securify rises due to the conversion of EVM IRs into datalog representation and then the application of verification technique. Oyente is also less efficient, as it relies on symbolic execution for analysis. Vulpedia should be comparable to SmartCheck and Slither, as these three all adopt rule based matching analysis. The extra overheads of Vulpedia, compared with Slither and SmartCheck, are signature-based code matching.Dataset 76,354 156 6,434 641 N.A. 883 17,770 52 1,352 141 8,859 295 f u n c t i o n d e s t r o y D e e d ( ) p u b l i c { Blockchain technology in business and information systems research. R Beck, M Avital, M Rossi, J B Thatcher, Business & Information Systems Engineering. 59R. Beck, M. Avital, M. Rossi, and J. B. Thatcher, "Blockchain technology in business and information systems research," Busi- ness & Information Systems Engineering, vol. 59, no. 6, pp. 381-384, 2017. Nick Szabo, Smart Contracts: Building Blocks for Digital Markets. 29Nick Szabo, "Smart Contracts: Building Blocks for Digital Markets," http://www.fon.hum.uva.nl/rob/ Courses/InformationInSpeech/CDROM/Literature/ LOTwinterschool2006/szabo.best.vwh.net/smart contracts 2. html, 1996, online; accessed 29 January 2019. The DAO attack paradoxes in propositional logic. 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Hobor, "Finding the greedy, prodigal, and suicidal contracts at scale," in Proceedings of the 34th Annual Computer Security Applications Conference, ACSAC 2018, San Juan, PR, USA, December 03-07, 2018, 2018, pp. 653-663. Exploiting the laws of order in smart contracts. A Kolluri, I Nikolic, I Sergey, A Hobor, P Saxena, Proceedings of the ISSTA 2019. the ISSTA 2019Beijing, ChinaA. Kolluri, I. Nikolic, I. Sergey, A. Hobor, and P. Saxena, "Exploiting the laws of order in smart contracts," in Proceedings of the ISSTA 2019, Beijing, China, July 15-19, 2019., 2019, pp. 363-373. sfuzz: An efficient adaptive fuzzer for solidity smart contracts. T D Nguyen, L H Pham, J Sun, Y Lin, Q T Minh, Proceedings of the ACM/IEEE 42nd International Conference on Software Engineering. the ACM/IEEE 42nd International Conference on Software EngineeringT. D. Nguyen, L. H. Pham, J. Sun, Y. Lin, and Q. T. Minh, "sfuzz: An efficient adaptive fuzzer for solidity smart contracts," in Proceedings of the ACM/IEEE 42nd International Conference on Software Engineering, 2020, pp. 778-788. Harvey: A greybox fuzzer for smart contracts. V Wüstholz, M Christakis, Proceedings of the 28th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering. the 28th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software EngineeringV. Wüstholz and M. Christakis, "Harvey: A greybox fuzzer for smart contracts," in Proceedings of the 28th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering, 2020, pp. 1398-1409. Vulpecker: an automated vulnerability detection system based on code similarity analysis. Z Li, D Zou, S Xu, H Jin, H Qi, J Hu, ACSAC. ACMZ. Li, D. Zou, S. Xu, H. Jin, H. Qi, and J. Hu, "Vulpecker: an automated vulnerability detection system based on code similarity analysis," in ACSAC. ACM, 2016, pp. 201-213. National vulnerability database (nvd). 29NISTNIST, "National vulnerability database (nvd)," https://www.nist.gov/programs-projects/ national-vulnerability-database-nvd, 2019, online; accessed 29 January 2019. Bingo: cross-architecture cross-os binary search. M Chandramohan, Y Xue, Z Xu, Y Liu, C Y Cho, H B K Tan, FSE 2016. M. Chandramohan, Y. Xue, Z. Xu, Y. Liu, C. Y. Cho, and H. B. K. Tan, "Bingo: cross-architecture cross-os binary search," in FSE 2016, 2016, pp. 678-689. Tracelet-based code search in executables. Y David, E Yahav, PLDI '14. Y. David and E. Yahav, "Tracelet-based code search in executa- bles," in PLDI '14, 2014, pp. 349-360. VUDDY: A scalable approach for vulnerable code clone discovery. S Kim, S Woo, H Lee, H Oh, IEEE Symposium on S & P. IEEE Computer Society. S. Kim, S. Woo, H. Lee, and H. Oh, "VUDDY: A scalable ap- proach for vulnerable code clone discovery," in IEEE Symposium on S & P. IEEE Computer Society, 2017, pp. 595-614.	[ "https://github.com/ToolmanInside/smart", "https://github.com/ToolmanInside/vulpedia", "https://github.com/melonproject/", "https://github.com/ConsenSys/", "https://github.com/trailofbits/echidna,", "https://github.com/trailofbits/manticore,", "https://github.com/quoscient/octopus," ]
[ "Modeling and Performance Analysis for Movable Antenna Enabled Wireless Communications", "Modeling and Performance Analysis for Movable Antenna Enabled Wireless Communications" ]	[ "Member, IEEELipeng Zhu ", "Student Member, IEEEWenyan Ma [email protected] ", "Fellow, IEEERui Zhang [email protected] ", "\nDepartment of Electrical and Computer Engineering\nNational University of Singapore\n117583Singapore\n", "\nShenzhen Research Institute of Big Data\nDepartment of Electrical and Computer Engineering\nThe Chinese University of Hong Kong\n518172Shenzhen, ShenzhenChina\n", "\nNational University of Singapore\n117583Singapore\n" ]	[ "Department of Electrical and Computer Engineering\nNational University of Singapore\n117583Singapore", "Shenzhen Research Institute of Big Data\nDepartment of Electrical and Computer Engineering\nThe Chinese University of Hong Kong\n518172Shenzhen, ShenzhenChina", "National University of Singapore\n117583Singapore" ]	[]	In this paper, we propose a novel antenna architecture called movable antenna (MA) to improve the performance of wireless communication systems. Different from conventional fixed-position antennas (FPAs) that undergo random wireless channel variation, the MAs with the capability of flexible movement can be deployed at positions with more favorable channel conditions to achieve higher spatial diversity gains. To characterize the general multi-path channel in a given region or field where the MAs are deployed, a field-response model is developed by leveraging the amplitude, phase, and angle of arrival/angle of departure (AoA/AoD) information on each of the multiple channel paths under the far-field condition. Based on this model, we then analyze the maximum channel gain achieved by a single receive MA as compared to its FPA counterpart in both deterministic and stochastic channels.First, in the deterministic channel case, we show the periodic behavior of the multi-path channel gain in a given spatial field, which can be exploited for analyzing the maximum channel gain of the MA.Next, in the case of stochastic channels, the expected value of an upper bound on the maximum channel gain of the MA in an infinitely large receive region is derived for different numbers of channel paths.The approximate cumulative distribution function (CDF) for the maximum channel gain is also obtained in closed form, which is useful to evaluate the outage probability of the MA system. Moreover, our results reveal that higher performance gains by the MA over the FPA can be acquired when the number of channel paths increases due to more pronounced small-scale fading effects in the spatial domain.Numerical examples are presented which validate our analytical results and demonstrate that the MA system can reap considerable performance gains over the conventional FPA systems with/without antenna selection (AS), and even achieve comparable performance to the single-input multiple-output (SIMO) beamforming system.Index TermsMovable antenna (MA), spatial diversity, field response, performance analysis.L. Zhu and W. Ma are with the ). 2 I. INTRODUCTION T HE past few decades have witnessed the rapid development of wireless communication technologies and the tremendous changes it has brought to our lives. In the process of wireless systems evolution from generation to generation, larger capacity and higher reliability have always been the main objectives to pursue. Multiple-input multiple-output (MIMO) or multi-user/multi-antenna communication has been a key enabling technology in this pursuit, which lifts the veil on the new degrees of freedom (DoFs) in the spatial domain for improving the communication performance [1]-[3]. By leveraging the beamforming gain and multiplexing gain, the capacity can be drastically increased with MIMO systems. Besides, the transmission reliability can be significantly improved in virtue of the spatial diversity provided by multiple antennas at the transmitter (Tx) and/or receiver (Rx). With the current trend and future expectation of wireless communication systems migrating to higher frequency bands, such as millimeter-wave and terahertz bands, the decreasing wavelength results in smaller antenna size, which renders the MIMO system to be of larger scale (a.k.a. massive MIMO) in order to compensate for the more severe propagation losses [4]-[7]. Compared to conventional MIMO, massive MIMO is able to exploit the spatial correlation of large antenna arrays for attaining higher array gains and mitigating the multi-user interference more effectively [6], [8]-[10]. However, since the antennas are deployed at fixed positions in the space, MIMO and massive MIMO cannot fully explore the spatial variation of wireless channels in a given receive area or receive field. Recently, the concept of holographic MIMO was proposed [11]-[13], which is also termed as continuous-aperture MIMO [14], holographic surface [15], or large intelligent surface [16] in the literature. The holographic MIMO is usually composed of a large number of tiny antenna elements spaced by sub-wavelength. As the element spacing approaches zero and the number of antenna elements goes infinity, the holographic MIMO can be regarded as a continuous-aperture antenna array or surface with programmable electromagnetic property. By utilizing the theory of electromagnetic information, the ultimate performance limit of communication systems subject to a given receive field can be characterized. Although the emergence of metamaterials brings new opportunities for the fabrication of holographic MIMO or holographic surface, it is still in the stage of theoretical exploration [11]-[14], [16]. Its main technical challenge lies in the high penetration loss of metamaterial-based antenna elements at the Tx and/or Rx. Besides, for holographic MIMO, the signals are controlled 3 by generating desired current distribution on the spatially-continuous surface. Compared to conventional discrete-aperture antennas connected with radio frequency (RF) chains, which are able to efficiently perform digital signal processing, the current distribution on holographic MIMO surfaces is difficult to be reconfigured in real time. Thus, holographic MIMO encounters fundamental limitations on processing broadband signals and still has a long journey for realizing its applications in practical systems. Despite the above challenges, holographic MIMO has great significance on analyzing the asymptotic capacity of wireless channels and reveals the potentials of new DoFs in the continuous spatial domain for improving the communication performance [11]-[14], [16]. To exploit such spatial DoFs, we propose a new antenna architecture, namely movable antenna (MA), in this paper. Different from conventional fixed-position antennas (FPAs), the positions of MAs can be flexibly adjusted in a spatial region for improving the channel condition and enhancing the communication performance. For example, considering the spatial fading of the channels, an MA can be moved from a position with low channel gain to another position with higher channel gain. Thus, the spatial diversity can be easily obtained by adjusting the position of the MA. For MIMO systems, the positions of the multiple MAs at the Tx and Rx can be jointly designed for maximizing the channel capacity, where the positioning optimization for the MAs not only increases the channel gain of each antenna but also balances the multiplexing performance of parallel spatial data streams by properly reshaping the MIMO channel matrix. The MA systems resemble the widely explored distributed antenna system (DAS), where the remote antenna units (RAUs) are geographically distributed in wireless networks [17]-[19]. The RAUs are connected to a single central unit (e.g., by optical fibers or coaxial cables) for centralized control and signal processing. The MA system can be implemented in a similar way where the antenna is connected to the radio-frequency (RF) chain via flexible cables. The position of the MA can be mechanically adjusted with the aid of drive components, such as stepper motors. The authors in [20] showed a prototype of MA-enabled multi-static radar, where the transmit and receive antennas are connected to the vector network analyzer via flexible armored antenna feeders and can be moved with the aid of linear drives. In [21], a reconfigurable antenna arraywas designed, where the array elements can move along a semicircular path with the aid of stepper motors to synthesize the radiation pattern. It is worth noting that despite the similar hardware implementation, the MA system proposed in this paper is distinguished from the DAS in terms of application scenarios and system setups. The DAS usually separates the antennas far	10.48550/arxiv.2210.05325	[ "https://export.arxiv.org/pdf/2210.05325v1.pdf" ]	252,816,101	2210.05325	8a1ce5b79e311f0f628c5221d80397c26a4b768f	Modeling and Performance Analysis for Movable Antenna Enabled Wireless Communications 11 Oct 2022 Member, IEEELipeng Zhu Student Member, IEEEWenyan Ma [email protected] Fellow, IEEERui Zhang [email protected] Department of Electrical and Computer Engineering National University of Singapore 117583Singapore Shenzhen Research Institute of Big Data Department of Electrical and Computer Engineering The Chinese University of Hong Kong 518172Shenzhen, ShenzhenChina National University of Singapore 117583Singapore Modeling and Performance Analysis for Movable Antenna Enabled Wireless Communications 11 Oct 2022arXiv:2210.05325v1 [cs.IT] 1 In this paper, we propose a novel antenna architecture called movable antenna (MA) to improve the performance of wireless communication systems. Different from conventional fixed-position antennas (FPAs) that undergo random wireless channel variation, the MAs with the capability of flexible movement can be deployed at positions with more favorable channel conditions to achieve higher spatial diversity gains. To characterize the general multi-path channel in a given region or field where the MAs are deployed, a field-response model is developed by leveraging the amplitude, phase, and angle of arrival/angle of departure (AoA/AoD) information on each of the multiple channel paths under the far-field condition. Based on this model, we then analyze the maximum channel gain achieved by a single receive MA as compared to its FPA counterpart in both deterministic and stochastic channels.First, in the deterministic channel case, we show the periodic behavior of the multi-path channel gain in a given spatial field, which can be exploited for analyzing the maximum channel gain of the MA.Next, in the case of stochastic channels, the expected value of an upper bound on the maximum channel gain of the MA in an infinitely large receive region is derived for different numbers of channel paths.The approximate cumulative distribution function (CDF) for the maximum channel gain is also obtained in closed form, which is useful to evaluate the outage probability of the MA system. Moreover, our results reveal that higher performance gains by the MA over the FPA can be acquired when the number of channel paths increases due to more pronounced small-scale fading effects in the spatial domain.Numerical examples are presented which validate our analytical results and demonstrate that the MA system can reap considerable performance gains over the conventional FPA systems with/without antenna selection (AS), and even achieve comparable performance to the single-input multiple-output (SIMO) beamforming system.Index TermsMovable antenna (MA), spatial diversity, field response, performance analysis.L. Zhu and W. Ma are with the ). 2 I. INTRODUCTION T HE past few decades have witnessed the rapid development of wireless communication technologies and the tremendous changes it has brought to our lives. In the process of wireless systems evolution from generation to generation, larger capacity and higher reliability have always been the main objectives to pursue. Multiple-input multiple-output (MIMO) or multi-user/multi-antenna communication has been a key enabling technology in this pursuit, which lifts the veil on the new degrees of freedom (DoFs) in the spatial domain for improving the communication performance [1]-[3]. By leveraging the beamforming gain and multiplexing gain, the capacity can be drastically increased with MIMO systems. Besides, the transmission reliability can be significantly improved in virtue of the spatial diversity provided by multiple antennas at the transmitter (Tx) and/or receiver (Rx). With the current trend and future expectation of wireless communication systems migrating to higher frequency bands, such as millimeter-wave and terahertz bands, the decreasing wavelength results in smaller antenna size, which renders the MIMO system to be of larger scale (a.k.a. massive MIMO) in order to compensate for the more severe propagation losses [4]-[7]. Compared to conventional MIMO, massive MIMO is able to exploit the spatial correlation of large antenna arrays for attaining higher array gains and mitigating the multi-user interference more effectively [6], [8]-[10]. However, since the antennas are deployed at fixed positions in the space, MIMO and massive MIMO cannot fully explore the spatial variation of wireless channels in a given receive area or receive field. Recently, the concept of holographic MIMO was proposed [11]-[13], which is also termed as continuous-aperture MIMO [14], holographic surface [15], or large intelligent surface [16] in the literature. The holographic MIMO is usually composed of a large number of tiny antenna elements spaced by sub-wavelength. As the element spacing approaches zero and the number of antenna elements goes infinity, the holographic MIMO can be regarded as a continuous-aperture antenna array or surface with programmable electromagnetic property. By utilizing the theory of electromagnetic information, the ultimate performance limit of communication systems subject to a given receive field can be characterized. Although the emergence of metamaterials brings new opportunities for the fabrication of holographic MIMO or holographic surface, it is still in the stage of theoretical exploration [11]-[14], [16]. Its main technical challenge lies in the high penetration loss of metamaterial-based antenna elements at the Tx and/or Rx. Besides, for holographic MIMO, the signals are controlled 3 by generating desired current distribution on the spatially-continuous surface. Compared to conventional discrete-aperture antennas connected with radio frequency (RF) chains, which are able to efficiently perform digital signal processing, the current distribution on holographic MIMO surfaces is difficult to be reconfigured in real time. Thus, holographic MIMO encounters fundamental limitations on processing broadband signals and still has a long journey for realizing its applications in practical systems. Despite the above challenges, holographic MIMO has great significance on analyzing the asymptotic capacity of wireless channels and reveals the potentials of new DoFs in the continuous spatial domain for improving the communication performance [11]-[14], [16]. To exploit such spatial DoFs, we propose a new antenna architecture, namely movable antenna (MA), in this paper. Different from conventional fixed-position antennas (FPAs), the positions of MAs can be flexibly adjusted in a spatial region for improving the channel condition and enhancing the communication performance. For example, considering the spatial fading of the channels, an MA can be moved from a position with low channel gain to another position with higher channel gain. Thus, the spatial diversity can be easily obtained by adjusting the position of the MA. For MIMO systems, the positions of the multiple MAs at the Tx and Rx can be jointly designed for maximizing the channel capacity, where the positioning optimization for the MAs not only increases the channel gain of each antenna but also balances the multiplexing performance of parallel spatial data streams by properly reshaping the MIMO channel matrix. The MA systems resemble the widely explored distributed antenna system (DAS), where the remote antenna units (RAUs) are geographically distributed in wireless networks [17]-[19]. The RAUs are connected to a single central unit (e.g., by optical fibers or coaxial cables) for centralized control and signal processing. The MA system can be implemented in a similar way where the antenna is connected to the radio-frequency (RF) chain via flexible cables. The position of the MA can be mechanically adjusted with the aid of drive components, such as stepper motors. The authors in [20] showed a prototype of MA-enabled multi-static radar, where the transmit and receive antennas are connected to the vector network analyzer via flexible armored antenna feeders and can be moved with the aid of linear drives. In [21], a reconfigurable antenna arraywas designed, where the array elements can move along a semicircular path with the aid of stepper motors to synthesize the radiation pattern. It is worth noting that despite the similar hardware implementation, the MA system proposed in this paper is distinguished from the DAS in terms of application scenarios and system setups. The DAS usually separates the antennas far away from each other [17]- [19], such that the channels are independent for multiple antennas and the performance is mainly determined by the large-scale path loss of the channel. In contrast, the proposed MA system in this paper deploys the antennas in a small region with a typical size in the order of several to tens of wavelengths, which is easier to implement and has much shorter connecting cables compared to the DAS. The channels for multiple MAs experience inherent spatial correlation and thus the communication performance is dominated by the small-scale fading in the target region. In a given receive region, if the MA can be rapidly deployed at the position with the highest channel gain, the receive signal-to-noise ratio (SNR) can be maximized, which can be regarded as a new way to acquire the spatial diversity gain. The authors in [22] proposed a novel fluid antenna system (FAS) with great flexibility in the antenna position. The fluid antenna can be freely switched to one of the candidate positions over a fixed-length line space so as to receive the strongest signal. By exploiting the selection diversity, the outage probability of the FAS can be significantly decreased compared to conventional FPA systems and may even outperform multiple-antenna systems with maximum ratio combining (MRC). In addition, a fluid antenna multiple access (FAMA) system was proposed in [23] to support transceivers with different channel conditions simultaneously. By optimizing the positions of the fluid antennas of users, the interference can be efficiently mitigated in the deep fading region and a favorable channel condition for the desired signal can be obtained, which helps improve the performance of multiple access. The research works in [22] and [23] reveal the potentials for improving the communication performance by changing the position of the antenna. However, limited by the physical characteristics, the liquid antenna can only be moved over a one-dimensional (1D) region. In contrast, the proposed MA system employs mechanical drives to move the conventional solid antenna, which can be flexibly deployed in the two-dimensional (2D) plane or three-dimensional (3D) space. Compared to FAS, the MA communication system is easier to implement and is able to fully exploit the spatial diversity/interference mitigation gains in a given field. Moreover, the proposed MA-enabled communication system has unique advantages in comparison with the conventional antenna selection (AS) technique. To acquire high diversity order, the AS system needs to employ a large number of antenna elements [24], [25], while the MA system can exploit the full spatial diversity with much fewer or even one single antenna moving in the same region. Besides, the flexible deployment of MAs can also alleviate the interference from/to which is useful to evaluate the outage probability of the MA system. Moreover, our results reveal that higher performance gains by the MA over the FPA can be acquired when the number of Extensive simulations are carried out to verify our analytical results and compare the performance of the proposed MA-enabled communication system with conventional FPA systems. The results demonstrate that the MA system can reap considerable performance gains over the FPA systems with/without AS, and even achieve comparable performance to the single-input multiple-output (SIMO) beamforming system. The rest of this paper is organized as follows. Section II introduces the signal model and the field-response based channel model for the MA-enabled communication system, where the relationship between the developed new model for MAs and the conventional channel models for FPAs is also presented. In Section III, we show the main analytical results for MA under both deterministic and stochastic channels. Simulation results are provided in Section IV and this paper is concluded in Section V. Notation: a, a, A, and A denote a scalar, a vector, a matrix, and a set, respectively. (·) T , (·) * , and (·) H denote transpose, conjugate, and conjugate transpose, respectively. CN (0, Λ) denotes the circularly symmetric complex Gaussian (CSCG) distribution with mean zero and covariance matrix Λ. E{·} denotes the expected value of a random variable. Z, R, and C represent the sets of integers, real numbers, and complex numbers, respectively. ℜ(·), ℑ(·), \| · \|, and ∠(·) denote the real part, the imaginary part, the amplitude, and the phase of a complex number or complex vector, respectively. diag{a} is a diagonal matrix with the element in row i and column i equal to the i-th element of vector a. d(·) and ∂(·) denote the differential and the partial differential of a function, respectively. 1 L denotes an L-dimensional vector with all the elements equal to 1. I L denotes an identical matrix of size L × L. For the considered MA system, the channel response depends on the positions of the antennas. Thus, the channel coefficient can be represented by a function of the positions of the transmit and receive MAs, i.e., h(t, r). Denoting the transmit power of the Tx as p t , the received signal at the Rx can be expressed as y(t, r) = h(t, r) √ p t s + n, t ∈ C t , r ∈ C r ,(1) where s represents the transmit signal with zero mean and normalized power of one, and n ∼ CN (0, δ 2 ) is the white Gaussian noise at the Rx with power δ 2 . As a result, the SNR for the received signal is given by γ(t, r) = \|h(t, r)\| 2 p t δ 2 , t ∈ C t , r ∈ C r .(2) A. Field-Response Based Channel Model For MA systems, the channel response is determined by the propagation environments and the positions of the antennas. In this paper, we assume that the size of the region for moving the antenna is much smaller than the propagation distance between the Tx and Rx, such that the far-field condition is satisfied at the Tx and Rx. This assumption is reasonable because the typical size of the antenna moving region is in the order of several to tens of wavelengths. For example, for an MA system operating at 28 GHz, the area of a square region with ten-wavelength sides is approximate 0.01 m 2 , for which the far-field condition can be easily guaranteed in this region. Thus, the plane-wave model can be used to form the field response between the transmit and receive regions. In other words, the AoDs, AoAs, and amplitudes of complex coefficients for the multiple channel paths do not change for different positions of the MAs, while only the phases of the multi-path channels vary in the transmit/receive region. At the Tx side, we denote the number of transmit paths as L t . As shown in Fig. 2, the elevation and azimuth AoDs of the j-th transmit path are respectively denoted as θ t,j ∈ [−π/2, π/2] and φ t,j ∈ [−π/2, π/2], 1 ≤ j ≤ L t . At the Rx side, we denote the number of receive paths as L r . The elevation and azimuth AoAs of the i-th receive path are respectively denoted as θ r,i ∈ [−π/2, π/2] and φ r,i ∈ [−π/2, π/2], 1 ≤ i ≤ L r . We define a path-response matrix (PRM) Σ ∈ C Lr×Lt to represent the response from the transmit reference position t 0 = (0, 0) to the receive reference position r 0 = (0, 0). Specifically, the entry in row i and column j of Σ, denoted as σ i,j , is the response coefficient between the j-th transmit path and the i-th receive path. As a result, the channel between two antennas located at t 0 and r 0 is given by h(t 0 , r 0 ) = 1 H Lr Σ1 Lt ,(3) which is the linear superposition of all the elements in the PRM. According to basic geometry shown in Fig. 2, the difference of the signal propagation distance between position r = [x r , y r ] and the reference point r 0 is given by ρ r,i (x r , y r ) = x r cos θ r,i sin φ r,i + y r sin θ r,i for the i-th receive path, 1 ≤ i ≤ L r . It indicates that the channel response of the i-th receive path at position r has a 2πρ r,i (x r , y r )/λ phase difference with respect to the reference point r 0 , where λ is the wavelength. To account for such phase differences in all L r receive paths, the field-response vector (FRV) in the receive region is defined as f(r) = [e j 2π λ ρ r,1 (xr,yr) , e j 2π λ ρ r,2 (xr,yr) , · · · , e j 2π λ ρ r,Lr (xr,yr) ] T . Similarly, for any position t = [x t , y t ] in the transmit region, the FRV is defined as g(t) = [e j 2π λ ρ t,1 (xt,yt) , e j 2π λ ρ t,2 (xt,yt) , · · · , e j 2π λ ρ t,L t (xt,yt) ] T . with ρ t,j (x t , y t ) = x t cos θ t,j sin φ t,j + y t sin θ t,j , 1 ≤ j ≤ L t . As a result, the channel between two antennas located at positions t = [x t , y t ] and r = [x r , y r ] is obtained as h(t, r) = f(r) H Σg(t).(6) B. Relationship with Conventional Channel Models The channel model shown in (6) 1) LoS Channel: A necessary condition for (6) being equivalent to the LoS channel is that the numbers of transmit and receive paths are both equal to 1, i.e., L t = L r = 1. Under this condition, the PRM and FRVs are all reduced to scalars, i.e.,            Σ = σ 1,1 , f(r) = e j 2π λ (xr cos θ r,1 sin φ r,1 +yr sin θ r,1 ) , g(t) = e j 2π λ (xt cos θ t,1 sin φ t,1 +yt sin θ t,1 ) ,(8) where σ 1,1 , θ r,1 , φ r,1 , θ t,1 , and φ t,1 represent the path-response coefficient at antenna positions (t 0 , r 0 ), the elevation AoA, azimuth AoA, elevation AoD, and azimuth AoD for the LoS path, respectively. 2) Geometric Channel: For geometric channels, the transmit and receive paths have one-toone correspondence. In other words, each transmit path always arrives at the Rx through one and only one receive path. Thus, a necessary condition for (6) to be the geometric channel is given by L t = L r = L.(9) Under this condition, the PRM becomes diagonal, i.e., Σ = diag{σ 1,1 , σ 2,2 , · · · , σ L,L }, where σ i,i , 1 ≤ i ≤ L, represents the path-response coefficient corresponding to antenna positions (t 0 , r 0 ) for the i-th path component between the Tx and Rx. 3) Rayleigh Fading Channel: A necessary condition for (6) to represent Rayleigh fading is that infinite number of statistically independent paths exist between the transmit and receive regions. Thus, the numbers of transmit and receive paths should approach infinity, i.e., L t → +∞, L r → +∞.(11) Besides, for isotropic scattering environments, the multi-path components should be uniformly distributed over the half-space in front of the antenna panel, which yields the AoDs and AoAs following the probability density functions (PDFs) f AoD (θ t , φ t ) = cos θ t 2π , θ t ∈ − π 2 , π 2 , φ t ∈ − π 2 , π 2 , (12a) f AoA (θ r , φ r ) = cos θ r 2π , θ r ∈ − π 2 , π 2 , φ r ∈ − π 2 , π 2 ,(12b)h(t, r) = f(r) H Σg(t) , which is equal to the sum of all the elements of f(r) H Σ weighted by the unit-modulus elements of g(t), or the sum of all the elements of Σg(t) weighted by the unit-modulus elements of f(r) H , is also a CSCG random variable, which leads to Rayleigh fading. 4) Rician Fading Channel: For Rician fading, an LoS path exists between the Tx and the Rx, and the NLoS components are also uniformly distributed over the half-space similar to that of the Rayleigh fading. Thus, the numbers of transmit and receive paths should approach infinity as shown in (11), and the elements of the PRM are i.i.d. circularly symmetric complex random variables. Meanwhile, the element in row i ⋆ and column j ⋆ of Σ corresponding to the LoS path has a constant amplitude. The Rician factor is thus obtained as κ = \|σ i ⋆ ,j ⋆ \| 2 E    (i,j) =(i ⋆ ,j ⋆ ) σ i,j 2    .(13) III y(r) = f(r) H Σg(t 0 ) √ p t s + n f(r) H b √ p t s + n, r ∈ C r ,(14) where b = Σg(t 0 ) [b 1 , b 2 , · · · , b Lr ] T represents the effective path-response vector (EPRV) in the receive region. Since the channel gain varies with the position of the receive MA, the receive SNR in (2) changes accordingly. In this section, the maximum channel gain achievable by a single receive MA and the corresponding SNR gain over an FPA located at r 0 are analyzed under the deterministic and stochastic channel setups, respectively. A. Deterministic Channel In this subsection, we analyze the characteristics of the channel gain in the receive region for deterministic path responses with any given EPRV b and physical AoAs θ r,ℓ and φ r,ℓ , 1 ≤ ℓ ≤ L r . For notation simplicity, we define intermediate variables ϕ r,ℓ = cos θ r,ℓ sin φ r,ℓ and ϑ r,ℓ = sin θ r,ℓ as the virtual AoAs for the ℓ-th receive path, 1 ≤ ℓ ≤ L r . Then, the property of the channel gain can be analyzed separately for the one-path, two-path, three-path, and multiple-path cases as follows. 1) One-Path Case: If only one channel path arrives at the Rx, the channel (power) gain is a constant within the receive region, i.e., \|h 1 (r)\| 2 = \|e −j2π( xr λ ϕ r,1 + yr λ ϑ r,1 ) b 1 \| 2 = \|b 1 \| 2 .(15) In other words, the change of the antenna position can only impact the phase of the channel for the one-path case and the MA cannot provide any SNR gain over the FPA. 2) Two-Path Case: If two channel paths with different AoAs arrive at the Rx, the channel gain between the transmit antenna and the receive MA at position r is given by \|h 2 (r)\| 2 = \|e −j2π( xr λ ϕ r,1 + yr λ ϑ r,1 ) b 1 + e −j2π( xr λ ϕ r,2 + yr λ ϑ r,2 ) b 2 \| 2 = \|b 1 \| 2 + \|b 1 \| 2 + 2ℜ b * 1 b 2 e j2π[( xr λ ϕ r,1 + yr λ ϑ r,1) −( xr λ ϕ r,2 + yr λ ϑ r,2)] = \|b 1 \| 2 + \|b 2 \| 2 + 2\|b 1 \|\|b 2 \| cos {ω 1,2 (x r , y r )} ,(16) with intermediate variables ω 1,2 (x r , y r ) = 2π xr λ (ϕ r,1 − ϕ r,2 ) + yr λ (ϑ r,1 − ϑ r,2 ) + µ 2 − µ 1 and parameters µ ℓ = ∠b ℓ , ℓ = 1, 2. As can be observed, the superimposed power of the two paths has a periodic character in the receive region due to the existence of the cosine function. For any fixed y r , the period of the channel gain along axis x r is given by λ/(ϕ r,1 − ϕ r,2 ). It indicates that a larger difference of the virtual AoAs ϕ r,1 and ϕ r,2 yields a smaller period along axis x r . Similarly, for any fixed x r , the period of the channel gain along axis y r is given by λ/(ϑ r,1 −ϑ r,2 ). To analyze the variation of the channel gain in the receive region, the gradient of the channel gain with respect to the MA position can be derived as ∇\|h 2 (r)\| 2 = ∂\|h 2 (r)\| 2 ∂x r , ∂\|h 2 (r)\| 2 ∂y r T = [g xr,yr × (ϕ r,1 − ϕ r,2 ), g xr,yr × (ϑ r,1 − ϑ r,2 )] T ,(17) with g xr,yr = − 4π λ \|b 1 \|\|b 2 \| sin 2π xr λ (ϕ r,1 − ϕ r,2 ) + yr λ (ϑ r,1 − ϑ r,2 ) + µ 2 − µ 1 . Thus, for any position excluding the maximum points, the direction for increasing the channel gain most quickly is given by [(ϕ r,1 − ϕ r,2 ), (ϑ r,1 − ϑ r,2 )] T or [(ϕ r,2 − ϕ r,1 ), (ϑ r,2 − ϑ r,1 )] T . The MA can be moved along the gradient direction for approaching the maximum channel gain in the receive region most efficiently. Moreover, it can be easily derived that a tight upper bound on the maximum channel gain is max r∈Cr \|h 2 (r)\| 2 = (\|b 1 \| + \|b 2 \|) 2 ,(18) which can be achieved at positions satisfying 2π x r λ (ϕ r,1 − ϕ r,2 ) + y r λ (ϑ r,1 − ϑ r,2 ) + µ 2 − µ 1 = 2kπ, k ∈ Z.(19) The equation in (19) yields parallel lines in the x r -y r plane. An example for the channel gain with two receive paths is shown in Fig. 3, where the distance of any two adjacent maximum 3, b1 = b2 = b3 = √ 3 3 , θr,1 = 0, θr,2 = π 3 , θr,3 = − π 4 , φr,1 = π 2 , θr,2 = − π 2 , and θr,3 = − π 2 . lines can be obtained as d 2 = λ (ϕ r,1 − ϕ r,2 ) 2 + (ϑ r,1 − ϑ r,2 ) 2 .(20) According to basic geometry, the upper bound on the channel gain can always be achieved if the receive region is a circular area with diameter larger than d 2 . Thus, we know that to achieve the upper bound on the channel gain, a smaller area of the region is required if the AoA difference of the two paths is larger. 3) Three-Path Case: If the number of receive paths is three, the channel gain between the transmit antenna and the receive MA at position r is given by \|h 3 (r)\| 2 = \|e −j2π( xr λ ϕ r,1 + yr λ ϑ r,1 ) b 1 + e −j2π( xr λ ϕ r,2 + yr λ ϑ r,2 ) b 2 + e −j2π( xr λ ϕ r,3 + yr λ ϑ r,3 ) b 3 \| 2 = \|b 1 \| 2 + \|b 2 \| 2 + \|b 3 \| 2 + 2\|b 1 \|\|b 2 \| cos {ω 1,2 (x r , y r )} + 2\|b 1 \|\|b 3 \| cos {ω 1,3 (x r , y r )} + 2\|b 2 \|\|b 3 \| cos {ω 2,3 (x r , y r )} ,(21) with intermediate variables ω m,n (x r , y r ) = 2π xr λ (ϕ r,m − ϕ r,n ) + yr λ (ϑ r,m − ϑ r,n ) + µ n − µ m , 1 ≤ m < n ≤ 3, and parameters µ ℓ = ∠b ℓ , ℓ = 1, 2, 3. It can be observed that the channel gain in (21) also has a periodic character. Specifically, let r p = [x p , y p ] T denote the period vector for the channel gain, i.e., we have \|h 3 (r)\| 2 ≡ \|h 3 (r + r p )\| 2 for any r, r + r p ∈ C r . A sufficient condition for r p being the period of the channel gain is given by cos {ω m,n (x r , y r )} ≡ cos {ω m,n (x r + x p , y r + y p )} , 1 ≤ m < n ≤ 3,(22) for ∀[x r , y r ] T , [x r + x p , y r + y p ] T ∈ C r , which can be equivalently converted to x p λ (ϕ r,m − ϕ r,n ) + y p λ (ϑ p,m − ϑ p,n ) = k m,n ∈ Z, 1 ≤ m < n ≤ 3.(23) The solution for (23) can be obtained by directly solving the equation set as        x p = λ ξ 1 [k 1,2 (ϑ r,1 − ϑ r,3 ) − k 1,3 (ϑ r,1 − ϑ r,2 )] , y p = λ ξ 2 [k 1,2 (ϕ r,1 − ϕ r,3 ) − k 1,3 (ϕ r,1 − ϕ r,2 )] ,(24) with ξ 1 = (ϕ r,1 − ϕ r,2 ) (ϑ r,1 − ϑ r,3 ) − (ϕ r,1 − ϕ r,3 ) (ϑ r,1 − ϑ r,2 ), ξ 2 = (ϑ r,1 − ϑ r,2 ) (ϕ r,1 − ϕ r,3 ) − (ϑ r,1 − ϑ r,3 ) (ϕ r,1 − ϕ r,2 ), k 1,2 , k 1,3 ∈ Z, and k 2,3 = k 1,3 − k 1,2 . If the arguments in (22) are all equal to their maximum value, i.e., one, a tight upper bound on the maximum channel gain can be obtained as max r∈Cr \|h 3 (r)\| 2 = (\|b 1 \| + \|b 2 \| + \|b 3 \|) 2 ,(25) which can be achieved at positions satisfying 2π x r λ (ϕ r,1 − ϕ r,2 ) + y r λ (ϑ r,1 − ϑ r,2 ) + µ 2 − µ 1 = 2k 1 π, k 1 ∈ Z (26a) 2π x r λ (ϕ r,1 − ϕ r,3 ) + y r λ (ϑ r,1 − ϑ r,3 ) + µ 3 − µ 1 = 2k 2 π, k 2 ∈ Z (26b) 2π x r λ (ϕ r,2 − ϕ r,3 ) + y r λ (ϑ r,2 − ϑ r,3 ) + µ 3 − µ 2 = 2k 3 π, k 3 ∈ Z.(26c) It can be verified that constraint (26c) can be safely removed because it holds if and only if constraints (26a), (26b), and k 3 = k 2 − k 1 are satisfied. Thus, the tight upper bound on the maximum channel gain is achieved at the intersections of the lines determined by (26a) and (26b), with the coordinates accordingly given by        x ⋆ r = λ 2πξ 1 [(2k 1 π + µ 1 − µ 2 ) (ϑ r,1 − ϑ r,3 ) − (2k 2 π + µ 1 − µ 3 ) (ϑ r,1 − ϑ r,2 )] , y ⋆ r = λ 2πξ 2 [(2k 1 π + µ 1 − µ 2 ) (ϕ r,1 − ϕ r,3 ) − (2k 2 π + µ 1 − µ 3 ) (ϕ r,1 − ϕ r,2 )] ,(27) for ∀k 1 , k 2 ∈ Z. An example for the the channel gain with three receive paths is provided in = 4, b1 = b2 = b3 = b4 = 1 2 , θr,1 = 0, θr,2 = π 3 , θr,3 = − π 4 , θr,4 = − 2π 3 , φr,1 = π 2 , θr,2 = − π 2 , θr,3 = − π 2 ,             d (1) 3 = λ ϑ r,1 − ϑ r,2 ξ 1 2 + ϕ r,1 − ϕ r,2 ξ 2 2 , d(2)3 = λ ϑ r,1 − ϑ r,3 ξ 1 2 + ϕ r,1 − ϕ r,3 ξ 2 2 .(28) According to basic geometry, the upper bound on the channel gain can always be achieved if the receive region is a circular area with diameter larger than d 3 = d In general, the period of the channel gain cannot be obtained explicitly for L r > 3 as the virtual AoAs ϕ r,ℓ and ϑ r,ℓ are randomly distributed, which can be observed from Fig. 5. Nevertheless, the approximate period for the channel gain can be derived by assuming quantized AoAs. Next, we analyze the periodic character for the channel gain along axis x r , while the periodic character along axis y r can be derived in a similar way. Specifically, for any fixed y r , we denote the channel gain between the transmit antenna and the receive MA at location (x r , y 0 ) as G(x r ) = \|h Lr (r)\| 2 yr=y 0 = Lr ℓ=1 b ℓ e −j2π( xr λ ϕ r,ℓ + y 0 λ ϑ r,ℓ) 2 Lr ℓ=1b ℓ e −j2π xr λ ϕ r,ℓ 2 ,(30) withb ℓ = b ℓ e −j2π y 0 λ ϑ r,ℓ . Let X denote the period for the channel gain along axis x r . Then, we have G(x r ) ≡ G(x r + X)≡ 0 ⇔1 − e −j2π X λ (ϕr,m−ϕr,n) = 0, 1 ≤ m, n ≤ L r ⇔ X λ (ϕ r,m − ϕ r,n ) ∈ Z, 1 ≤ m, n ≤ L r ,(31) which indicates that period X is the minimum real number that ensures X λ (ϕ r,m − ϕ r,n ) to be an integer for 1 ≤ m, n ≤ L r . To facilitate performance analysis, we quantize the virtual AoAs with a resolution of T , i.e., assuming ϕ r,ℓ ∈ {∆ t = −1 + 2t−1 T } 1≤t≤T . Without loss of generality, we assume that the virtual AoAs are sorted in a non-decreasing order, i.e., ϕ r,1 ≤ ϕ r,2 ≤ · · · ≤ ϕ r,Lr . Besides, we denote the virtual AoAs as ϕ r,ℓ = ∆ t ℓ = −1 + 2t ℓ −1 T , which means that ϕ r,ℓ corresponds to the t ℓ -th element in the quantized set of the virtual AoAs. Then, the difference of two adjacent virtual AoAs can be obtained as ϕ r,ℓ+1 −ϕ r,ℓ = ∆ t ℓ+1 −∆ t ℓ = 2 T (t ℓ+1 −t ℓ ) 2τ ℓ T , 1 ≤ ℓ ≤ L r . Denote τ ⋆ as the maximal common factor for {τ ℓ } 1≤ℓ≤Lr . To guarantee X λ (ϕ r,m − ϕ r,n ) being integers, the period for the channel gain along axis x r should be given by X = T λ 2τ ⋆ .(32) Thus, we know that for any fixed y r , the maximum channel gain can always be achieved if the size of the region along axis x r is no less than period X. As can be observed from (32), the accuracy of the period is influenced by the quantization resolution T because an approximation on the AoAs is employed. If a small resolution T is used, the difference between the practical are separately calculated via exhaustive search. Each point in Fig. 6 is an average result over 10 4 channel realizations. As can be observed, with the increasing quantization resolution, the performance gap between the two sizes of the region becomes smaller, which indicates that the approximation of the quantizated AoAs has less impact on the maximum value of the channel gain. In other words, a higher quantization resolution can guarantee that the maximum channel gain is more likely achieved within period X. B. Stochastic Channel Next, we focus on the stochastic performance of the MA system. We assume that the EPRV is a CSCG random vector with i.i.d. elements, i.e., b ℓ ∼ CN (0, σ 2 /L r ), 1 ≤ ℓ ≤ L r . Besides, the physical AoAs are assumed to be i.i.d. random variables with the PDF given in (12b). Thus, we can obtain the expected channel gain at the reference point r 0 = (0, 0) as G 0 = E{\|h(r 0 )\| 2 } = E    Lr ℓ=1 b ℓ 2    = σ 2 .(33) Note that since we assume far-field scenarios, the expected channel gain at any point in the receive region is equal to that at the reference point, i.e., G(r) = Lr ℓ=1 e −j2π( xr λ ϕ r,ℓ + yr λ ϑ r,ℓ ) b ℓ 2 = σ 2 . This equation holds because b ℓ is a circularly symmetric random variable for 1 ≤ ℓ ≤ L r . For the FPA system, the position of the antenna should be fixed at a specific position, and thus the expected channel gain is always equal to σ 2 . In contrast, for the MA system, we can always find a position which achieves a larger channel gain in the receive region, and thus an increase on the average channel gain can be acquired over the FPA system. 1) Single-Path Case: As we have analyzed in Section III-A, the channel gain is a constant within the receive region if only one channel path arrives at the Rx. Thus, the maximum channel gain of the MA has an expected value equal to the expected channel gain at the reference point, i.e., G max,1 = G 0 = σ 2 ,(34) which indicates that no SNR gain is acquired by the MA over the FPA. Furthermore, the CDF of the channel gain for the one-path case can be derived as F 1 (t) Pr \|b 1 \| 2 ≤ t = √ t 0 2x σ 2 e − x 2 σ 2 dx = 1 − e − t σ 2 , t ≥ 0.(35) For any given transmit power p t , noise power δ 2 , and receive SNR threshold γ th , the outage probability for the MA system with a single receive path is given by F 1 ( δ 2 γ th pt ). 2) Two-Path Case: According to Section III-A, for any given ϕ r,1 , ϕ r,2 , ϑ r,1 , ϑ r,2 , the maximum channel gain (\|b 1 \| + \|b 2 \|) 2 can always be achieved if the diameter of the receive region is larger than d 2 shown in (20). Since b ℓ is a CSCG random variable with mean zero and variance σ 2 /L r , \|b ℓ \| is a Rayleigh-distributed random variable with the PDF given by f \|b ℓ \| (t) = 4t σ 2 e −2t 2 σ 2 , t ≥ 0, ℓ = 1, 2,(36) Thus, as the size of the receive region approaches infinity, the upper bound on the maximum channel gain can always be achieved for arbitrary AoAs. The expected value of the maximum channel gain of the MA in the receive region is thus given by G max,2 = E (\|b 1 \| + \|b 2 \|) 2 = +∞ 0 +∞ 0 (x + y) 2 × 4x σ 2 e −2x 2 σ 2 × 4y σ 2 e −2y 2 σ 2 dx dy = +∞ 0 4x 3 σ 2 e − 2x 2 σ 2 dx × +∞ 0 4y σ 2 e − 2y 2 σ 2 dy + 2 +∞ 0 4x 2 σ 2 e − 2x 2 σ 2 dx × +∞ 0 4y 2 σ 2 e − 2y 2 σ 2 dy + +∞ 0 4x σ 2 e − 2x 2 σ 2 dx × +∞ 0 4y 3 σ 2 e − 2y 2 σ 2 dy = σ 2 2 + πσ 2 4 + σ 2 2 = 1 + π 4 σ 2 ,(37) with the values of the definite integrals σ 2 e − 2x 2 σ 2 dx = σ 2 2 . Thus, for the two-path case, the average-SNR gain of the MA system over the FPA system is obtained as η 2 = max r∈Cr γ 2 (r) γ 2 (r 0 ) = G max,2 G 0 = 1 + π 4 .(38) Furthermore, the CDF of the maximum channel gain for the two-path case can be derived as F 2 (t) Pr (\|b 1 \| + \|b 2 \|) 2 ≤ t = √ t 0 dx √ t−x 0 4x σ 2 e − 2x 2 σ 2 × 4y σ 2 e − 2y 2 σ 2 dy = √ t 0 4x σ 2 e − 2x 2 σ 2 × 1 − e − 2( √ t−x) 2 σ 2 dx = √ t 0 4x σ 2 e − 2x 2 σ 2 − 4x σ 2 e − t σ 2 e − (2x− √ t) 2 σ 2 dx = 1 − e − 2t σ 2 − e − t σ 2 √ t 0 4x σ 2 e − (2x− √ t) 2 σ 2 dx = 1 − e − 2t σ 2 − 2 √ πt σ e − t σ 2 √ 2t 0 1 √ 2πσ e − x 2 2σ 2 dx, = 1 − e − 2t σ 2 − √ πt σ e − t σ 2 1 − 2Q √ 2t σ , t ≥ 0.(39) with Q(t) = +∞ t 1 √ 2π e − x 2 2 dx being the Gaussian tail probability. For any given transmit power p t , noise power δ 2 , and receive SNR threshold γ th , the outage probability for the MA system with two receive paths is given by F 2 ( δ 2 γ th pt ). 3) Three-Path Case: For any given ϕ r,1 , ϕ r,2 , ϕ r,3 , ϑ r,1 , ϑ r,2 , ϑ r,3 , the maximum channel gain (\|b 1 \| + \|b 2 \| + \|b 3 \|) 2 can always be achieved if the diameter of the receive region is larger than d 3 as shown in (28). Since b ℓ is a CSCG random variable with mean zero and variance σ 2 /L r , \|b ℓ \| is Rayleigh distributed with the PDF given by f \|b ℓ \| (t) = 6t σ 2 e −3t 2 σ 2 , t ≥ 0, ℓ = 1, 2, 3,(40) Thus, as the size of the receive region approaches infinity, the upper bound on the maximum channel gain can always be achieved for arbitrary AoAs. The expected value of the maximum channel gain of the MA in the receive region is thus given by G max,3 = E (\|b 1 \| + \|b 2 \| + \|b 3 \|) 2 = +∞ 0 +∞ 0 +∞ 0 (x + y + z) 2 × 6x σ 2 e −3x 2 σ 2 × 6y σ 2 e −3y 2 σ 2 × 6z σ 2 e −3z 2 σ 2 dx dy dz = 3 +∞ 0 6x 3 σ 2 e − 3x 2 σ 2 dx × +∞ 0 6y σ 2 e − 3y 2 σ 2 dy × +∞ 0 6z σ 2 e − 3z 2 σ 2 dz + 6 +∞ 0 6x 2 σ 2 e − 3x 2 σ 2 dx × +∞ 0 6y 2 σ 2 e − 3y 2 σ 2 dy × +∞ 0 6z σ 2 e − 3z 2 σ 2 dz = 3 × σ 2 3 + 6 × πσ 2 12 = 1 + π 2 σ 2 ,(41) with the values of the definite integrals +∞ 0 6x σ 2 e − 3x 2 σ 2 dx = 1, +∞ 0 6x 2 σ 2 e − 3x 2 σ 2 dx = σ √ 6 π 2 , and +∞ 0 6x 3 σ 2 e − 3x 2 σ 2 dx = σ 2 3 . Thus, for the three-path case, the average-SNR gain of the MA system over the FPA system is obtained as η 3 = max r∈Cr γ 3 (r) γ 3 (r 0 ) = G max,3 G 0 = 1 + π 2 .(42) Note that the maximum channel gain is the square-sum of three i.i.d. Rayleigh random variables, which has no closed-form CDF. Nevertheless, the CDF of the maximum channel gain for the three-path case can be approximated as [26] F 3 (t) Pr (\|b 1 \| + \|b 2 + \|b 3 \|) 2 ≤ t = Pr \|b 1 \| + \|b 2 + \|b 3 \| ≤ √ t ≈ 1 − e − t c 3 1 + t c 3 + t 2 2c 2 3 , t ≥ 0,(43) with c 3 = 15 1/3 σ 2 Lr . For any given transmit power p t , noise power δ 2 , and receive SNR threshold γ th , the outage probability for the MA system with three receive paths is approximately given by F 3 ( δ 2 γ th pt ). 4) Multiple-Path Case: Since the EPRV b is a CSCG random vector following distribution b ∼ CN (0, σ 2 Lr I Lr ), the amplitude of the ℓ-th element of b, i.e., \|b ℓ \|, is i.i.d. Rayleigh distributed with the PDF given by f \|b ℓ \| (t) = 2L r t σ 2 e −Lrt 2 σ 2 , t ≥ 0, 1 ≤ ℓ ≤ L r ,(44) If the number of receive paths is larger than three, it is difficult to derive the explicit expression of the maximum channel gain. Nevertheless, we can always obtain an upper bound on the maximum channel gain of the MA as follows: G max,Lr ≤ E    Lr ℓ=1 \|b ℓ \| 2    = · · · +∞ 0 Lr ℓ=1 x ℓ 2 × Lr ℓ=1 2L r x ℓ σ 2 e −Lr x 2 ℓ σ 2 dx 1 · · · dx Lr = L r +∞ 0 2L r x 3 σ 2 e − Lr x 2 σ 2 dx + L r (L r − 1) +∞ 0 2L r x σ 2 e − Lrx 2 σ 2 dx × +∞ 0 2L r y 3 σ 2 e − Lry 2 σ 2 dy = σ 2 L r × L r + πσ 2 4L r × L r (L r − 1) = 1 + (L r − 1)π 4 σ 2 ,(45) with the values of the definite integrals +∞ 0 2Lrx σ 2 e − Lrx 2 σ 2 dx = 1, +∞ 0 2Lrx 2 σ 2 e − Lr x 2 σ 2 dx = σ √ 2Lr π 2 , and +∞ 0 2Lrx 3 σ 2 e − Lrx 2 σ 2 dx = σ 2 Lr . Hence, for the multiple-path case, the average-SNR gain of the MA system over the FPA system is upper-bounded by η Lr = max r∈Cr γ Lr (r) γ Lr (r 0 ) = G max,Lr G 0 ≤ 1 + (L r − 1)π 4 .(46) Note that the CDF of the square-sum of L r independent Rayleigh random variables has no closed-form expression for L r > 3. Nevertheless, the CDF of the upper bound on the maximum channel gain for the multiple-path case can be obtained by the small argument approximation as follows [26]: F UB Lr (t) Pr    Lr ℓ=1 \|b ℓ \| 2 ≤ t    = Pr Lr ℓ=1 \|b ℓ \| ≤ √ t ≈ 1 − e − t c Lr−1 k=0 t c k k! , t ≥ 0,(47)with c = σ 2 Lr [(2L r − 1)!!] 1/Lr and (2L r − 1)!! = (2L r − 1) × (2L r − 3) × · · · × 3 × 1. For any given transmit power p t , noise power δ 2 , and receive SNR threshold γ th , a lower bound on the outage probability of the MA system with L r > 3 receive paths is approximately given by F UB Lr ( δ 2 γ th pt ). 5) Infinite-Path Case: For the isotropic scattering scenario, the number of receive paths approaches infinity with the joint PDF for the physical AoAs given by (12b). According to the central limit theorem, the channel coefficient at each position is a CSCG random variable with mean zero and variance σ 2 . Let b(θ r , φ r ) denote the complex path-response coefficient corresponding to AoAs θ r and φ r , which has an average power of σ 2 . The correlation between the channel coefficients of two positions spaced by d can be derived as 3 R(d) = E {h * (0, d)h(0, 0)} = E π/2 −π/2 π/2 −π/2 \|b(θ r , φ r )\| 2 e j2π d λ sin θr cos θ r 2π dθ r dφ r = π/2 −π/2 σ 2 e j2π d λ sin θr cos θ r 2 dθ r = 1 −1 σ 2 2 e j2π d λ t dt = σ 2 sin 2π d λ 2π d λ = σ 2 sinc 2d λ .(48) According to the property of the sinc function, we know that the channels of two positions spaced by λ/2 are statistically independent. Due to the small value of sinc(t) for t ≥ 1, we can further assume that the channels of two positions with distance larger than λ/2 are statistically independent. Thus, for a square receive region with size A×A, the number of independent random channel variables is no smaller than N LB = ⌊2A/λ+1⌋ 2 . Since the channel coefficients are CSCG random variables, the channel (power) gains are exponential random variables (ERVs) with the rate parameter 1/σ 2 . The maximum channel gain of the MA in the receive region is lowerbounded by the largest order statistics of the N LB independent ERVs, which can be expressed as X max = max 1≤k≤N LB X k with {X k } being i.i.d. ERVs with the rate parameter 1/σ 2 . It is known that the largest order statistics for i.i.d. ERVs can be equivalently expressed as X max = N LB k=1 Y k , where {Y k } are independent ERVs with the rate parameter k/σ 2 for Y k , 1 ≤ k ≤ N LB [27]. Thus, the expected value for the lower bound on the maximum channel gain in the A × A area is obtained as G max,∞ ≥ E max 1≤k≤N LB X k = E N LB k=1 Y k = N LB k=1 σ 2 k ,(49) which monotonously increases with the size of the receive region and approaches infinity for an infinite region. Besides, the CDF of the lower bound on the maximum channel gain for the infinite-path case is given by F LB ∞ (t) = Pr max 1≤k≤N LB X k ≤ t = 1 − e − t σ 2 N LB .(50) Thus, for any given transmit power p t , noise power δ 2 , and receive SNR threshold γ th , an upper bound on the outage probability for the MA system with infinite number of receive paths is given by F LB ∞ ( δ 2 γ th pt ). To analyze the upper bound on the maximum channel gain of the MA, we discretize the square receive region A × A into equally spaced grids with size 1/P × 1/P , and thus the total number of the grids is no larger than N UB = ⌈P A + 1⌉ 2 . Let C r,k denote the k-th grid in the receive region, 1 ≤ k ≤ N UB . Thus, the maximum channel gain of the MA in the A × A area can be equivalently expressed as max r∈Cr \|h(r)\| 2 = max 1≤k≤N max r∈C r,k \|h(r)\| 2 .(51) For sufficiently large P , the area of each grid approaches zero such that the correlation between channels at any two points within the same grid approaches σ 2 . It indicates that the maximum channel gain over each grid can be approximately given by the channel gain at the center of this grid, i.e., max r∈C r,k \|h(r)\| 2 ≅X k with {X k } being i.i.d. exponential variables with the rate parameter 1/σ 2 . Thus, an upper bound on the maximum channel gain can be obtained by assuming the channel gains at the center of the N grids are all statistically independent, i.e., G max,∞ ≤ E max 1≤k≤N UBX k = N UB k=1 σ 2 k .(52) Meanwhile, the CDF of the upper bound on the maximum channel gain is given by F UB ∞ (t) = Pr max 1≤k≤N UBX k ≤ t = 1 − e − t σ 2 N UB .(53) Thus, for any given transmit power p t , noise power δ 2 , and receive SNR threshold γ th , a lower bound on the outage probability for the MA system with infinite number of receive paths is given by F UB ∞ ( δ 2 γ th pt ). It is known that the sum of the first N terms of the harmonic series can be approximated by A. Simulation Setup and Benchmark Schemes In the simulation, the position of the transmit antenna is fixed, while the receive MA can be moved flexibly in the receive region, which is set as a square area with size A × A, i.e., C r = [A/2, A/2] × [A/2, A/2]. The ratio of the average receive signal power to the noise power is set as ptσ 2 δ 2 = 1 for convenience. The geometry channel model is employed as a special case of our proposed field-response based model, where the number of transmit and receive paths are the same, i.e., L t = L r , with a diagonal PRM Σ = diag{b 1 , b 2 , · · · , b Lr }. The pathresponse coefficients are assumed to be i.i.d. CSCG random variables, i.e., b ℓ ∼ CN (0, σ 2 /L r ), 1 ≤ ℓ ≤ L r . Besides, the physical AoDs and AoAs are assumed to be i.i.d. random variables with the PDF given by (12). For all the schemes, we define the the relative SNR gain as the ratio of the SNR for the considered scheme to the SNR for the scheme with a single transmit FPA at t 0 and a single receive FPA at r 0 . Fig. 7 shows the relative SNR gains for MA, AS, and DBF systems with varying normalized size of the receive region. The number of receive paths is set as L r = 2, L r = 3, and L r = 5 for the three subplots, respectively. As can be observed, the relative SNR gain of the MA system increases with the region size because a higher maximum channel gain can be found in a larger receive region. The analytical upper bounds on the relative SNR gain are given by (38), (42), and (46). It can be seen that for the two-path and three-path cases, the upper bounds provided by Normalized region size A/ (20) and (28), where a lager size of the region is required to guarantee achieving the upper bound on the maximum channel gain as the number of receive paths increases. This conclusion can be further verified by the case of L r = 5, where the relative SNR gain still increases if the region size is larger than 8λ. The reason is that as the number of paths increases, the periodicity becomes weaker in the receive region. According to the period analysis in (32), the maximal common factor for the virtual AoA indices decreases with the number of paths, which entails a larger period for the channel gain in the receive region. Thus, to achieve the upper bound on the relative SNR gain, a larger size of the receive region is required for the case of more paths. Besides, the relative SNR gains of AS and DBF systems are also evaluated in Fig. 7. As can be observed, although the number of antennas for AS and DBF systems is M times larger than that for the MA system, the MA system can achieve a performance comparable to that of the DBF system and better than that of the AS system. For the AS system, the relative SNR gain is small because the candidate antennas have fixed positions. In contrast, the MA system has more flexibility to adjust the position of the antenna for achieving a higher SNR. It can be expected that the AS scheme can approach the performance of the MA system if a sufficiently large number of antennas are employed at the Rx. However, this will result in much higher hardware cost, where the entire receive region should be covered by antennas. For the DBF system, the array/MRC gain increases more significantly with the number of antennas as compared to AS. However, the number of RF chains increases with the number of antennas, and thus the hardware cost for the DBF system is larger than that for the MA/AS system. as Fig. 7. As can be observed, the simulated CDF of the SNR for the two-path MA system perfectly matches the analytical result derived in (39). While for the L r = 3 and L r = 5 cases, the simulated CDF has a deviation compared to the analytical ones because an approximation was employed in (43) and (47). Nevertheless, the error between the analytical and simulated B. Numerical Results CDFs is very small, which indicates that (43) and (47) provide a good approximation for the CDF of the upper bound on the maximum channel gain and can be utilized for analyzing the outage probability for the MA system. Besides, we can find that for any fixed SNR threshold, the value of the CDF for the MA system is always smaller than that for FPA and AS systems, which means that compared to the two benchmark schemes, the MA system can always achieve a lower outage probability for any SNR threshold. Interestingly, at the low SNR region, the SNR CDF value for the MA system is smaller than that for the DBF system. It indicates that the MA system can outperform the DBF system in terms of the outage probability at the low SNR region. In particular, for a large number of receive paths, the performance gap of the outage probability increases, which shows more superiority of the MA system. The analytical and simulated results so far are all based on the assumption that the variances of the path-response coefficients {b ℓ } are the same for different channel paths, which yields the same average power of the multi-path components. In Fig. 9, we evaluate the relative SNR gains for MA, AS, and DBF systems with different ratios of the average path power. Specifically, the number of paths is set as L r = 2, and the ratio of the average power for the two paths, denoted by ̺ = E{\|b 1 \| 2 } E{\|b 2 \| 2 } , is set as 2, 10, and 100 for the three subplots, respectively. It can be observed again that the relative SNR gain of MA increases with the region size. However, as the ratio of the path power increases, the increment of the relative SNR gains provided by the MA and AS systems both decreases. For ̺ = 2, ̺ = 10, and ̺ = 100, the increment of the relative SNR gain is 0.74, 0.45, and 0.16, respectively. The reason is as follows. The SNR gain of the MA system is essentially obtained from the small-scale fading of the multi-path channel. For a large number of paths with equal average power, the small-scale fading becomes strong because the fluctuation of the channel gain caused by the phase change of the electromagnetic waves is substantial in the receive region. As the ratio of the path power increases, the difference of the maximum and minimum channel gains in the receive region decreases, and thus the relative SNR gain provided by adjusting the position of the MA decreases. For an extreme case with the ratio of the path power approaching infinity, the channel is degraded to the single-path case, where no SNR gain can be obtained by the MA system over the FPA system as we have shown in Section III-A. In Fig. 10, we compare the relative SNR gain and the upper bound on the maximum SNR gain for the MA system with varying number of receive paths. As can be observed, both the relative SNR gain and the upper bound on the maximum SNR gain increase with the number of paths. This is because the small-scale fading becomes stronger as the number of paths increases, and thus the channel gain has more substantial fluctuation in the receive region, which entails an increase on the maximum channel gain. Besides, as the region size increases from 2λ × 2λ to 50λ × 50λ, the maximum SNR gain closely approaches the analytical upper bound provided by (46). In particular, if the number of paths is smaller than 3, the performance gap between the maximum SNR gain for 2λ × 2λ region and the upper bound is no more than 0.12. If the number of paths is smaller than 5, the performance gap between the maximum SNR gain for 10λ × 10λ region and the upper bound is no more than 0.10. If the number of paths is smaller than 7, the performance gap between the maximum SNR gain and the upper bound is no more than 0.08. The above results indicate that the upper bound in (46) is approximately tight for a small number of paths. For larger number of paths, a larger size of the receive region is required to achieve the SNR performance near to the upper bound. It is worth noting that the upper bound is effective for infinite size of the receive region. It is expected that the maximum SNR gain can further increase if the size of the receive region is larger than 50λ × 50λ and approach the upper bound for infinitely large size of the receive region more closely. Finally, Fig. 11 shows the expected value of the relative SNR gain versus the normalized size of the receive region as well as the comparison with the lower and upper bounds in (49) and (52). The number of paths is set as 40 or 200, and the discretized parameter for the upper bound is set as P = 8 in (52). As can be observed, the relative SNR gains increase with the size of the region, and the performance gap to the upper bound decreases if the number of paths increases. As we have analyzed in Section III-B, the lower and upper bounds in (49) and (52) both approximately follow a logarithmic function with the region size. The results in Fig. 11 validate this analysis and show that the relative SNR gains for sufficiently large number of paths have a logarithmic growth with respect to the size the receive region. V. CONCLUSION In this paper, we proposed a new MA architecture for improving the performance of wireless communication systems. With the capability of flexible movement, the MAs can be deployed at positions with more favorable channel conditions in the spatial region to achieve higher spatial diversity gains. Since the channel varies with the positions of the MAs, a field-response based channel model was developed under the far-field condition to characterize the general multi-path channel with given transmit and receive regions. We have shown the conditions under which the proposed field-response based channel model for MAs becomes the well-known LoS channel, geometry channel, Rayleigh and Rician fading channel models with FPAs. Based on the proposed channel model, we analyzed the maximum channel gain achieved by a single receive MA over its FPA counterpart under both deterministic and stochastic channels. In the deterministic channel case, the periodic behavior of the multi-path channel gain was revealed in a given spatial field, which unveiled the maximum channel gain of the MA with respect to the number of channel paths and size of the receive region. In the case of stochastic channels, we derived the expected value of an upper bound on the maximum channel gain of the MA in an infinitely large receive region for different numbers of channel paths. Moreover, our results revealed that higher performance gains by the MA over the FPA could be acquired when the number of channel paths increases due to more pronounced small-scale fading effects in the spatial domain. Besides, for uniform scattering scenarios with infinite number of channel paths, we showed that the upper and lower bounds on the expected maximum channel gain of the MA obey the logarithmic increase with respect to the size of the receive region. The approximate CDF of the maximum channel gain was also obtained in closed form, which is useful to evaluate the outage probability of the MA system. Simulation results validated our analytical results and demonstrated that the MA system could reap considerable performance gains over the conventional FPA systems with/without AS, and even achieve comparable performance to the SIMO beamforming system. With the general channel model and theoretical performance limits of the MA system provided in this paper, the design of practical methods for achieving the performance limits, such as channel estimation and antenna position searching algorithms, as well as the extension of the results to multi-antenna/multi-user systems with spatial multiplexing will be interesting topics for future research. Given the above advantages of MA, we aim to investigate in this paper the channel modeling and performance analysis for MA-enabled communication systems. To characterize the general multi-path channel in a given region or field where the MAs are deployed, a field-response model is developed by leveraging the amplitude, phase, and angle of arrival/angle of departure (AoA/AoD) information on each of the multiple channel paths under the far-field condition. We show that the proposed field-response based channel model is consistent with the conventional channel models with FPAs, and present the conditions under which the field-response based channel model becomes the well-known line-of-sight (LoS) channel, geometric channel, Rayleigh and Rician fading channel models. Based on the proposed channel model, we then analyze the maximum channel gain achieved by a single receive MA as compared to its FPA counterpart in both deterministic and stochastic channels. First, in the deterministic channel case, we show the periodic behavior of the multi-path channel gain in a given spatial field, which can be exploited for analyzing the maximum channel gain of the MA. Next, in the case of stochastic channels, the expected value of an upper bound on the maximum channel gain of the MA in an infinitely large receive region is derived for different numbers of channel paths. The approximate cumulative distribution function (CDF) for the maximum channel gain is also obtained in closed form, Fig. 1 . 1Illustration of the MA-enabled communication system. channel paths increases due to more pronounced small-scale fading effects in the spatial domain. Fig. 2 . 2Illustration of the coordinates and spatial angles for transmit and receive regions. II. SYSTEM MODEL The architecture of MA-enabled communication system is shown in Fig. 1, where the transmit and receive MAs are connected to the RF chains via flexible wires, such as coaxial cables. Thus, the positions of the MAs can be mechanically adjusted with the aid of drive components, such as stepper motors. The Cartesian coordinate systems, x t -O t -y t and x r -O r -y r , are established to describe the positions of the MAs in the transmit and receive regions, C t and C r , respectively 1 . The coordinates of the transmit MA are denoted as t = [x t , y t ] T , and the coordinates of the receive MA are denoted as r = [x r , y r ] T . characterizes the wireless channel based on the transmit/receive antenna locations to facilitate our subsequent performance analysis for MA-enabled communications. For any given positions of the transmit and receive MAs, the channel model in (6) is consistent with that of the conventional FPA systems. In this subsection, we present the conditions under which the model in (6) becomes the well-known LoS channel, geometric channel, Rayleigh and Rician fading channels. respectively. Under the above conditions, if the elements of the PRM are independent and identically distributed (i.i.d.) circularly symmetric complex random variables, according to the central limit theory, the superimposed path responses at the Tx and Rx, i.e., the elements of f(r) H Σ and Σg(t), are i.i.d. CSCG distributed random variables. Thus, the channel coefficient . PERFORMANCE ANALYSIS To analyze the performance gain provided by MAs over conventional FPAs, we consider the simplified scenario where the position of the transmit antenna is fixed at the reference point t 0 while the position of the receive antenna can be flexible. Thus, the signal model can be simplified as Fig. 3 . 3Illustration of the periodic character of the channel gain in the receive region, with Lr = 2, b1 = b2 = √ 2 2 , θr,1 = 0, θr,2 = π 3 , φr,1 = π 2 , and φr,2 = − π 2 . Fig. 4 . 4Illustration of the periodic character of the channel gain in the receive region, with Lr = Fig. 5 . 5An example for the channel gain in the receive region, with Lr Fig. 6 . 6Performance comparison for the maximum channel gains with varying angle quantization resolution, where the number of receive paths is Lr = 4 and the vertical axis is fixed at yr = 0. Fig. 4 , 4where the distance of any two adjacent maximum points can be obtained as ) Multiple-Path Case: If the number of receive paths is larger than three, the channel gain between the transmit antenna and the receive MA at position r is given by \|h Lr (r)a form similar to the 2D discrete-time Fourier transform (DTFT) 2 . Specifically, the virtual AoAs in the angular domain can be regarded as time domain, while the positions in the spatial domain can be thought as frequency domain. 1 − e −j2π X λ (ϕr,m−ϕr,n) e −j2π xr λ (ϕr,m−ϕr,n) AoAs and the approximately quantized AoAs is large, and thus the obtained period becomes inaccurate. In contrast, if a large resolution T is utilized, the obtained period is accurate because the difference between the practical AoAs and the approximately quantized AoAs is small. To evaluate the impact of the quantization resolution, we compare the maximum channel gain for different sizes of the receive region along axis x r . Specifically, the number of receive paths is L r = 4, where the virtual AoA ϕ r,ℓ is randomly generated following uniform distribution over [−1, 1]. The complex path-response coefficient b ℓ follows i.i.d. CSCG distribution with mean zero and variance 1/L r . For each channel realization, we obtain period X according to (32). Then, the maximum channel gains within regions x r ∈ [−X/2, X/2] and x r ∈ [−5X, 5X] log N + ς, with ς ≈ 0.577 denoting the Euler-Mascheroni constant. Thus, the lower and upper bounds in (49) and (52) indicate that the expected value for the maximum channel gain in the receive region increases with the region size approximately following a logarithmic function. IV. SIMULATION RESULTS In this section, extensive simulations are carried out to evaluate the performance of the MAenabled communication systems and verify our analytical results. For each channel realization, we employ exhaustive search for the position of the receive MA which achieves the highest channel gain. The results in this section are carried out based on 10 4 Monte Carlo simulations.In addition to the proposed MA system, three benchmark schemes are defined as follows:• FPA: The antenna at the Rx has a fixed position located at the reference point r 0 = (0, 0).• AS: The Rx is equipped with 1 RF chain and an array with M antennas spaced by a half wavelength in the receive region. In particular, the antenna with the highest channel gain is selected out of the M antennas for maximizing the receive SNR.• Digital beamforming (DBF): In the SIMO system, the Rx is equipped with M RF chains and an array with M antennas spaced by a half wavelength in the receive region. In particular, the MRC is employed at the Rx for maximizing the receive SNR. Fig. 7 . 7Comparison of the relative SNR gains for MA, AS, and DBF systems versus the normalized size of the receive region with the number of receive paths Lr = 2, Lr = 3, and Lr = 5, respectively. (38) and (42) are tight due to the strong periodicity of the channel gain. The minimum size of the receive region for achieving the upper bound on the relative SNR gain is 4λ and 5λ for L r = 2 and L r = 3, respectively. This result is consistent with the derivation in Fig. 8 Fig. 8 . 88compares the SNR CDFs for MA, FPA, AS, and DBF systems with the same setup Comparison of the SNR CDFs for MA, FPA, AS, and DBF systems with the number of receive paths Lr = 2, Lr = 3, and Lr = 5, respectively. Fig. 9 . 9Comparison of the relative SNR gains for MA, AS, and DBF systems versus the normalized size of the receive region with Lr = 2 and the ratio of average path power ̺ = 2, ̺ = 10, and ̺ = 100, respectively. Fig. 10 . 10The expected values of the relative SNR gains for MA systems versus the number of receive paths for different sizes of the receive region. Fig. 11 . 11The expected values of the relative SNR gains versus the normalized size of the receive region for different numbers of receive paths. TABLE I COMPARISON IOF MA WITH OTHER ANTENNA TECHNOLOGIES. enabled MIMO system can efficiently improve the channel gain and capacity. Thus, the number of required antennas and RF chains can be significantly decreased. A detailed comparison between the aforementioned technologies and MA is summarized in Table I, where N denotes the number of antennas/RF chains at the Tx or Rx in MIMO systems.Technology Movement Number of antennas at Tx/Rx Number of RF chains at Tx/Rx Hardware cost Diversity gain Interference mitigation gain MA 1D/2D/3D 1 1 Low High Medium FAS 1D 1 1 Medium Medium Low FPA None 1 1 Very low None None AS None N 1 Medium Medium Low FPA-MIMO None N N High High High MA-MIMO 1D/2D/3D < N < N Medium Very high Very high undesired terminals such that the interference mitigation gain is achieved for simultaneously supporting multiple transceivers. Compared to conventional MIMO systems with FPAs, the MA- In this paper, we consider the MAs moving in a 2D plane, which can be extended to the 3D space by adding the zt(zr) coordinate. Note that (29) holds for arbitrary values of Lr ≥ 1. Without loss of generality, the two positions spaced by d are selected along axis yr. 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[ "T-Modules: Translation Modules for Zero-Shot Cross-Modal Machine Translation", "T-Modules: Translation Modules for Zero-Shot Cross-Modal Machine Translation" ]	[ "Paul-Ambroise Duquenne \nBenoît Sagot Inria\nMeta AI & Inria\n\n", "Hongyu Gong [email protected] \nBenoît Sagot Inria\nMeta AI & Inria\n\n", "Meta Ai \nBenoît Sagot Inria\nMeta AI & Inria\n\n", "Holger Schwenk [email protected] \nBenoît Sagot Inria\nMeta AI & Inria\n\n", "Meta Ai \nBenoît Sagot Inria\nMeta AI & Inria\n\n" ]	[ "Benoît Sagot Inria\nMeta AI & Inria\n", "Benoît Sagot Inria\nMeta AI & Inria\n", "Benoît Sagot Inria\nMeta AI & Inria\n", "Benoît Sagot Inria\nMeta AI & Inria\n", "Benoît Sagot Inria\nMeta AI & Inria\n" ]	[ "Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing" ]	We present a new approach to perform zeroshot cross-modal transfer between speech and text for translation tasks. Multilingual speech and text are encoded in a joint fixed-size representation space. Then, we compare different approaches to decode these multimodal and multilingual fixed-size representations, enabling zero-shot translation between languages and modalities. All our models are trained without the need of cross-modal labeled translation data. Despite a fixed-size representation, we achieve very competitive results on several text and speech translation tasks. In particular, we outperform the state of the art for zero-shot speech translation on Must-C. We also introduce the first results for zero-shot direct speechto-speech and text-to-speech translation.	10.48550/arxiv.2205.12216	[ "https://www.aclanthology.org/2022.emnlp-main.391.pdf" ]	249,017,584	2205.12216	1e00915fc0a9f881bd0dcf0403d42b1fa46e7652	T-Modules: Translation Modules for Zero-Shot Cross-Modal Machine Translation December 7-11, 2022 Paul-Ambroise Duquenne Benoît Sagot Inria Meta AI & Inria Hongyu Gong [email protected] Benoît Sagot Inria Meta AI & Inria Meta Ai Benoît Sagot Inria Meta AI & Inria Holger Schwenk [email protected] Benoît Sagot Inria Meta AI & Inria Meta Ai Benoît Sagot Inria Meta AI & Inria T-Modules: Translation Modules for Zero-Shot Cross-Modal Machine Translation Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing the 2022 Conference on Empirical Methods in Natural Language ProcessingDecember 7-11, 2022 We present a new approach to perform zeroshot cross-modal transfer between speech and text for translation tasks. Multilingual speech and text are encoded in a joint fixed-size representation space. Then, we compare different approaches to decode these multimodal and multilingual fixed-size representations, enabling zero-shot translation between languages and modalities. All our models are trained without the need of cross-modal labeled translation data. Despite a fixed-size representation, we achieve very competitive results on several text and speech translation tasks. In particular, we outperform the state of the art for zero-shot speech translation on Must-C. We also introduce the first results for zero-shot direct speechto-speech and text-to-speech translation. Introduction Most, if not all, current state-of-the-art text and speech translation systems are based on a sequenceto-sequence approach and an attention mechanism to connect the encoder and decoder. Such models require labeled data to be trained end-to-end. For text-to-text (T2T) translation, this labeled data, called bitexts, is available in large amounts for a number of language pairs, in particular since large-scale bitext mining initiatives like ParaCrawl (Bañón et al., 2020) and CCMatrix . Finding training data for speech-to-text (S2T) translation is more challenging, but several data collection efforts exist, like mTEDx (Salesky et al., 2021), CoVoST (Wang et al., 2020a,b), and Must-C (Di Gangi et al., 2019). Finally, speech-tospeech (S2S) translation suffers from scarcity of end-to-end labeled data and current S2S systems are limited to a very small number of language pairs. Very recent works start to consider mining labeled data for S2S, e.g. (Duquenne et al., 2021). Unsupervised representation learning is very successfully used to initialize the encoder and/or decoder of a sequence-to-sequence model, thereby lowering the amount of labeled data needed to train or fine-tune the model end-to-end. Approaches include for instance XLM (Conneau and Lample, 2019), XLSR , wav2vec , data2vec (Baevski et al., 2022) and mSLAM (Bapna et al., 2022). In this work, we propose a new modular architecture for text and speech translation, which is based on a common fixed-size multilingual and multimodal internal representation, and encoders and decoders which are independently trained. We explore several variants of teacher-student training to learn text and speech encoders for multiple languages, which are compatible with the embedding space of the LASER encoder (Artetxe and Schwenk, 2019). In contrast to preceding works on multilingual and multimodal representations, we also train text decoders for multiple languages which are able to generate translations given the joint representation. Finally, we demonstrate that it is possible to train a speech decoder using raw audio only. Figure 1 visualizes the overall approach. We show that these encoders and decoders can be freely combined to achieve very competitive performance in T2T, S2T and (zero-shot) S2S translation. In summary, our contributions are as follows. • We apply a teacher-student approach to train multilingual text and speech encoders that are mutually compatible; • We show that the fixed-size representation can be efficiently decoded into multiple languages; • We are able to train a speech decoder with raw speech only, which can be paired with our text and speech encoders for multiple languages; • We achieve very competitive results on several text and speech translation tasks, without any end-to-end labeled data and significantly improve the state of the art for zero-shot speech translation; • To the best of our knowledge, we are the first to build zero-shot direct S2S translation systems. Related work Multilingual and multimodal representations Building multilingual representation for text or speech is key to develop state-of-the-art models based on these modalities. Conneau and Lample (2019) introduce a multilingual pre-training method with good cross-lingual transfer capabilities. extend the Wav2vec2 architecture to the multilingual setting introducing a multilingual pre-trained model for speech. More recently, Bapna et al. (2022) pre-train a multilingual encoder model handling both speech and text in order to benefit from cross-modal transfer between speech and text. An important obstacle to good joint speech/text representations is the length mismatch between audio and text. On the other hand, several works have studied how to encode sentences in a fixedsize representation (Feng et al., 2020;Artetxe and Schwenk, 2019;Reimers and Gurevych, 2019). In the multilingual setting, these works highlight that paraphrases and translations are close in the sentence embedding space, enabling large-scale bitext mining. Recently, Duquenne et al. (2021) extended the existing LASER model (Artetxe and Schwenk, 2019) built for multilingual text to the speech modality for several spoken languages. They show that this joint speech/text fixed-size representation can be efficiently used for large-scale mining of speech against text and even speech against speech. Zero-shot transfer in Machine Translation In Machine Translation, cross-lingual transfer to improve low-resource language directions has been widely studied. One way to encourage crosslingual transfer is building a massively multilingual translation system as . Some other works such as (Zhang et al., 2022) make an efficient use of MT data involving a pivot language thanks to weight freezing strategies to force representations to be close to the pivot language representations. One extreme scenario of cross-lingual transfer learning is called zero-shot transfer, where you learn to translate one language and directly apply the decoding process to an unseen language. Several methods have been tried to improve zeroshot transfer. Arivazhagan et al. (2019); Pham et al. (2019) add language similarity regularization on pooled representations of encoders outputs as an auxiliary loss to a MT objective in order to improve zero-shot transfer. Liao et al. (2021); Vázquez et al. (2018); Lu et al. (2018) introduce shared weights between language-specific encoders and decoders, commonly called an interlingua that captures language-independent semantic information. Finally, Escolano et al. (2020aEscolano et al. ( , 2021aEscolano et al. ( , 2020b focus on incremental learning of language-specific encoders-decoders using cross-entropy loss, alternately freezing parts of the model to ensure a shared representation between languages. Zero-shot transfer in Speech Translation Recent research focuses on direct speech translation where an encoder-decoder model directly translates speech into text (Bérard et al., 2016;Bansal et al., 2017;Weiss et al., 2017). Direct speech translation models are closing the gap with their cascaded counterparts (Li et al., 2020;Babu et al., 2021;Bapna et al., 2022). Several works add MT data in S2T translation training, using an auxiliary loss to bridge the modality gap, like adversarial (Alinejad and Sarkar, 2020), or distance (Dong et al., 2021;Liu et al., 2020) regularization. and (Li et al., 2020) use adaptor modules to address the length mismatch between audio and text representations. Several works studied how to efficiently perform zero-shot cross-modal transfer from text to speech in the frame of direct speech translation. Following (Escolano et al., 2020a(Escolano et al., , 2021a(Escolano et al., , 2020b presented above for text, Escolano et al. learn a speech encoder compatible with decoders trained on text only, freezing the text decoder during training and using cross-entropy on the output of the de-coder. This is the most similar work like ours, however they did not use any joint fixed-representation and their zero-shot results using only speech transcriptions lagged behind supervised setting by a large margin. Other works such as (Dinh et al., 2022;Dinh, 2021) studied zero-shot speech translation employing a cross-modal similarity regularization as an auxiliary loss. However, they obtained low zero-shot results possibly due to the mismatch in the encoder output lengths between speech and text. Direct speech-to-speech translation Finally, there is a surge of research interest in direct speechto-speech translation (Jia et al., 2019(Jia et al., , 2021Lee et al., 2022a). An encoder-decoder model directly translates speech in a language into speech in another language without the need to generate text as an intermediate step. Speech-to-speech translation research suffers from data scarcity of aligned speech with speech in different languages and often uses synthetic speech to overcome this issue. Recently, Lee et al. (2022b) introduce the first direct speech-to-speech model based on real speech data as target. They propose a speech normalization technique in order to normalize the target speech with respect to speaker and prosody. Lee et al. (2022a,b) extract HuBERT units of target speech as targets for a unit decoder during training. At test time, a vocoder is used to transform output units into speech. To the best of our knowledge, no work has tried to develop a direct speech-to-speech translation system in a zero-shot setting. Exploring training strategies The purpose of this work is to build a common fixed-size representation for multilingual speech and multilingual text that can be decoded in text and speech in different languages. We want to build language-specific encoders and decoders compatible with this fixed-size representation. Plugging one encoder with one decoder from different modalities and/or different languages enables performing zero-shot cross-modal translation. To this end, we first study how to efficiently decode fixed-size sentence representation for text. Second, we study how to improve similarity for sentence embeddings between languages. After an ablation study on the Japanese-English text translation direction, we extend the best training strategy to several other languages and a new modality, speech. Better decoding of sentence embeddings Motivations Multilingual sentence embeddings have been widely studied in the research community to perform bitext mining. For instance, LASER (Artetxe and Schwenk, 2019) is a multilingual sentence embedding space, where sentences are close in the embedding space if they are paraphrases or translations. LASER has been successfully used for large-scale bitext mining like in the CCMatrix project . LASER has been trained with a decoding objective, whereas other works like LaBSE (Feng et al., 2020) have been trained with a contrastive objective. First, we studied how multilingual sentence embeddings can be efficiently decoded. We focused on LASER as it originally has a decoder, and we studied how we can improve the decoding of sentence embeddings. As an initial experiment, we evaluated auto-encoding of English sentences from FLORES (Goyal et al., 2022) in Figure 2 left, with the original LASER encoder and decoder, bucketing sentences by length, and reporting BLEU scores. The LASER encoder handles several languages: decoding these multilingual embeddings enables to translate the input sentence into English with the original LASER decoder. We report the BLEU scores for the different sentence lengths in Figure 2 right for the German-English translation direction from FLORES. We notice that BLEU scores are low for both auto-encoding and translation tasks and decrease with the sentence length. The fixed-size representation seems to be a bottleneck for decoding tasks, especially for long sentences. However, the original LASER decoder is really shallow (one LSTM decoder layer), an interesting question is: can we improve decoding by training a new deeper decoder? Training new decoders We chose to train a new decoder to decode LASER sentence embeddings, with a transformer architecture and 12 layers. To train this new decoder, we use an auto-encoding objective, feeding raw English sentences to the model: we use original LASER encoder, whose weights are not updated during training, and plug a new transformer decoder to decode the fixed-size sentence representation output by the LASER encoder (the decoder attends on the sentence embedding output by the encoder). We used 15B English sentences from CCnet (Wenzek et al., 2019) to train the decoder. We compare the new decoder with original LASER decoder on the auto-encoding task and the German-English translation task of FLO-RES in Figure 2. Results First, we notice an important boost on the auto-encoding task with the new decoder, with high BLEU scores even for sentences with more than 50 words. Second, training a new decoder with an auto-encoding objective improves the decoding of sentence embeddings from another language, German. The new decoder can be directly applied to German sentence embeddings because German embeddings are supposed to be close to their English translations encoded with LASER. Making languages closer Motivations To get an idea of the closeness of translations in the LASER space, we inspected the L2 squared distances of sentence embeddings in different languages to their English translations sentence embeddings. A detailed analysis can be found in the appendix. We noticed that high resource languages are closer in the LASER space to English, compared to low resource languages. We studied how our newly trained decoder is performing on a more distant language in LASER space, Japanese. We report the results of the jaen translation task using the original decoder and the new decoder in Table 1. We notice that both decoders performs poorly on the ja-en translation tasks, and that the original LASER decoder leads to better results. An hypothesis is that the new decoder has over-fitted English embeddings leading to bad generalization on distant Japanese embeddings. Teacher-student training of text encoders To overcome this issue, we suggest to follow a method introduced by Reimers and Gurevych (2020), where new encoders are trained to fit an existing sentence embedding space. Here, we are trying to make the Japanese translations closer to English embeddings in our 1024 dimensional space. The original LASER encoder is fixed during training to encode English translation, behaving as the teacher, while we train a new Japanese encoder as a student to fit English sentence embeddings. We use bitexts from CCMatrix for the ja-en pair to train the Japanese text student. Following (Reimers and Gurevych, 2020), we minimize the MSE loss (equivalent to L2 squared distance) between the generated Japanese sentence embedding and the target English sentence embedding. The Japanese encoder is not trained from scratch, but we fine-tune XLM-R large. To extract the sentence embedding, we tested two methods: The classical output of the encoder corresponding to the beginning-of-sentence (BOS) token, a method widely used for text classification ; or max-pooling of the encoder outputs, less common but LASER has been trained with such pooling method. Finally, we tested another objective that is supposed to better match with our decoding task: we encode the Japanese sentence with the encoder being trained, decode the pooled sentence embedding with our new decoder which weights are not updated during training, and we compute the cross entropy loss of the output of the new decoder with the English target sentence. The training was unstable when using XLM-R weights as initialization. Therefore, instead of fine-tuning XLM-R, we fine-tune the encoder obtained from our previous method (trained with MSE loss), which leads to a stable training. We report all the results in Table 1. For text-to-text translation results, we use spBLEU of M2M-100 with the public checkpoint and script to evaluate on FLORES. Results In Table 1, we first notice that learning a new Japanese student significantly improve the results for the ja-en translation task. The best pooling method seems to be max-pooling, maybe because LASER has been trained with max-pooling. The second step of fine-tuning with cross entropy loss does not improve the results for our ja-en translation task, despite of the significant decrease of cross entropy valid loss during this second step fine-tuning. This validates the use a simple MSE loss which seems sufficient for future decoding purposes and is a lot cheaper in term of computation compared to cross entropy loss. We conclude that learning a new Japanese student with max-pooling and MSE loss leads to the best results. Using this new Japanese encoder, our new decoder significantly outperforms the original LASER encoder. These experiments show that LASER sentence embeddings can be better decoded by training a new decoder on a large amount of raw text data. This new decoder can be used to decode sentence embeddings from other languages handled by LASER. However, translations are still more or less distant in the space, making them explicitly closer with a MSE loss objective significantly improves the results on a translation task. Therefore, we decide to extend this idea to other languages and a new modality, speech, to see if it can help performing cross-modal translation tasks. Overall architecture Text student encoders We now want to train several text students for different languages, in order to plug, at test time, these encoders to different decoders to perform translation tasks. We decide to use LASER English embeddings as our teacher. This English space has proven to have good semantic properties: paraphrases are close in the embedding space, and makes it a good teacher for English translations. Moreover, most of MT data involve English translations that we will use to learn our text students. We focus on 7 languages, namely, German, French, Spanish, Catalan, Japanese, Turkish, and Mongolian. We use CCMatrix bi-texts to learn our text students, and bi-texts mined with LASER3 (Heffernan et al., 2022) for Mongolian. Text decoders We saw above that we can train a new English decoder with raw English data, using a fixed encoder and an auto-encoding objective. However, such an approach can lead to over-fitting to English sentence embeddings and bad generalization on other languages. We made languages closer together in our 1024 dimensional space thanks to our new student encoders but translations are not perfectly mapped to a real English sentence embedding in this continuous space. Therefore, we explore different methods to make the decoders robust locally in the sentence embedding space in order to generalize better on unseen languages. First, we can improve our decoder training with an auto-encoding objective by adding synthetic noise in the sentence embedding space. We add noise to a sentence embedding by multiplying it by 1 + ϵ, with ϵ ∼ N (0, α). In our experiments, we took α = 0.25, which leads to an empirical average L2 squared distance of approx. 0.05. between the noisy embedding and the original embedding. Second, we tested another approach to make our decoder robust to translations in the sentence embedding space: we added bi-texts from the de-en direction to the training of the English decoder. Finally, we trained decoders for five non-English languages to see how it behaves for other languages. All text decoders are 12-layers transformer decoders. Speech student encoders Duquenne et al. (2021) showed that it is possible to learn speech students compatible with the LASER text space. The training of speech students is similar to the one presented above for text. They fine-tune XLSR, a multilingual pretrained model for speech and minimized the cosine loss between the output of the speech encoder and the target LASER sentence embedding. We adapt this approach using a bigger XLSR model (Babu et al., 2021) with more than two billion parameters and extracting the fixed-size representation for speech with max-pooling to follow what we have done for text students. We minimize the MSE loss between the output of the speech encoder and the transcription/translation encoded by one of our text encoders. Unlike (Duquenne et al., 2021), we did not use the original LASER encoder to encode text transcripts but our newly trained text students which are supposed to be close to the LASER English embeddings. As in (Duquenne et al., 2021), we can use either transcriptions or written translations as teachers for our speech student. We used CoVoST 2, a speech translation dataset, as our training data. Figure 3 summarizes the process to train a speech student with transcriptions only: First, we train a text student for the language we want to cover, we will use this encoder to encode transcriptions. Then, we train a speech student to fit text embeddings output by our text student. Speech decoders In this last part, we introduce a speech decoder in our framework, which can be learnt with raw speech data. We focus on English speech decoding but it could be extended to other languages. To learn to decode English speech, we follow the work done by Lee et al. (2022b), who learn to decode HuBERT units. At test time, the generated units are transformed into speech using a vocoder. One method is to follow the same approach presented for raw text data to learn an English decoder. The English speech encoder previously trained to fit LASER text space on CoVoST 2 training set is used to encode raw speech, and its weights are not updated during training. We trained a unit decoder to decode sentence embeddings output by the speech encoder. The unit targets correspond to the one of the input speech as we are trying to auto-encode speech. We follow the recipe of Lee et al. (2022b) to prepare target units as we are dealing with real speech data: we extract HuBERT units from input speech, normalize the units with the speech normalizer used in Lee et al. (2022b). This preparation of target data is done unsupervisedly and any raw speech data can be processed with this method. We summarize the speech decoder training in Figure 4. Another method is to leverage English speech recognition data where English text transcripts are encoded through LASER Once the English speech decoder is trained, we can plug any text or speech encoder to perform direct text-to-speech or speech-to-speech translation in a zero-shot way. Results and discussion Text-to-text translation As presented in section 4, we test different strategies to train an English decoder. When training a decoder with raw text data, we use 15 billion English sentences extracted from CCnet (Wenzek et al., 2019). When training with additional bi-text data, we use bi-texts from CCMatrix , and the English part of the bi-texts for the auxiliary auto-encoding loss in order to have a good balance between bitexts and raw data. We present the results for textto-text translation for xx-en directions in Table 2 for the different decoder training methods on FLO-RES devtest. en-en decoder corresponds to the decoder trained with an auto-encoding objective, en-en+noise decoder corresponds to the decoder trained with an auto-encoding objective and additional noise in the sentence embedding space, and en-en+de-en decoder corresponds to the decoder trained with a combination of de-en bitexts and english raw data. We compare our zero-shot text-to-text translation results with two supervised baselines: M2M-100 , a massively multilingual trained on many-to-many training data from different sources, with 24 encoder layers and 24 decoder layers; and Deepnet (Wang et al., 2022) a recent work trained on 1932 language directions from different sources with 100 encoder layers and 100 decoder layers. We put these results as a supervised reference but we recall that in our framework, we perform zero-shot text-to-text translation for most of the language pairs. Please note the crosslingual transfer we obtain thanks to our training method: the English decoder has never seen Spanish embeddings before but is able to achieve competitive results compared to supervised baselines. In Table 2, we see that adding synthetic noise to the sentence embeddings helps translating low resource languages unseen by the decoder. However, it slightly decreases the performance on high resource languages. Moreover, natural noise from deen translations leads to even better results for both high and low resource languages, getting closer to the state-of-the-art MT results which have been obtained with end-to-end training. Finally, we trained decoders for German, French, Spanish, Turkish and Mongolian in order to be able to translate from any of our languages to any other. A detailed analysis of the translation tasks with these new decoders can be found in the appendix. Similar to what we noticed with our English decoder, we obtain excellent zero-shot crosslingual transfer: the German decoder has never seen Japanese embeddings before and Japanese has never been aligned to German. However, the ja-de results are competitive compared to state-of-the-art translation models trained in an end-to-end way with much more data. Speech-to-text translation Then, we tried to plug the decoders trained on text data to our speech encoders in order to perform zero-shot speech-totext translation. We trained independent speech student encoders for German, French, Turkish, Japanese and Mongolian spoken languages on the CoVoST 2 training set. For Catalan and Spanish, we trained a single speech student encoder for both languages as they have high language similarity. We report direct speech translation results in Table 3 for speech encoders trained with transcriptions as teachers. We have put several baselines for direct speech translation: two supervised baselines based on finetuning XLSR (Babu et al., 2021) or mSLAM (Bapna et al., 2022) with speech translation data. We also put the results on zero-shot cross-modal transfer from text to speech with the mSLAM pre-trained multimodal encoder, which is not working in this zero-shot setting. In our framework, the de-en speech translation direction benefits from cross-modal transfer while all other directions benefit from both cross-modal and cross-lingual transfer as the decoder has been trained on text and has only seen English and German embeddings. In this zero-shot cross-modal setting, we notice that the results are really competitive compared to supervised baselines trained end-to-end. Moreover, the supervised baselines use speech translation data, whereas our approach does not need speech translation data but only transcriptions. Except for Turkish, which has a really different morphological structure compared to English, speech translation results are close to their supervised counterpart trained with XLSR. An interest- Table 5: BLEU on Must-C test set for zero-shot speech translation, compared to the state of the art for zero-shot approaches by (Escolano et al., 2021b). es-en fr-en Zero-shot text-to-speech trained on raw speech from CoVoST 10.0 9.5 trained on raw speech from MLS + Common Voice 22.8 20.9 trained on en ASR data from MLS + Common Voice 24.4 23.5 Zero-shot speech-to-speech trained on raw speech from CoVoST 9.9 9.1 trained on raw speech from MLS + Common Voice 21.3 19.8 trained on en ASR data from MLS + Common Voice 22.4 21.1 (a) This work: zero-shot results es-en fr-en Supervised speech-to-speech translation trained on VP 9.2 9.6 trained on VP + mined data 15.1 15.9 Supervised speech-to-speech via text pivot trained on VP+EuroparlST+CoVoST 26.9 27.3 (b) Results from previous supervised models trained by Lee et al. (2022b) on real (non synthetic) data. The speech-to-speech via text pivot baseline relies on speech-to-text by . Table 6: BLEU on CoVoST 2 test set for text-to-speech and speech-to-speech translation ing direction is ja-en, as we have a large amount of ja-en MT data but a really small amount of speech transcription data. For this task, we nearly doubled the BLEU score compared to supervised baselines without the need of ST data. We tested the different possible teachers for speech encoder training, namely transcription teacher (already presented), translation teacher, and both transcription and translation teachers. When using translation teacher, we use English text as the written translations from CoVoST 2. We focus on two language directions, de-en (high resource) and ja-en (low resource). Results are shown in Table 4. We notice that a translation teacher is better if using the en-en decoder, which was expected as the decoder was trained on English embeddings. However, when using a decoder trained on noisy embeddings or with additional bi-texts, results are better for speech encoders trained with transcription teacher rather than translation teacher. It may come from the fact that there exists a one-to-one mapping between transcriptions and audios, but not for audio and written translation (there can be several possible translations). For our high resource direction de-en, the best results are achieved when using both transcriptions and translations as teacher, reaching same performance level as with the endto-end speech translation training of XLSR. Finally, we trained an English speech student with transcriptions on the Must-C training set and compare our approach with the zero-shot approach by Escolano et al. (2021b). We report the results in Table 5. We notice significant improvements in the BLEU score compared to the previous SOTA for zero-shot speech translation on the Must-C dataset. Translation of text/speech into speech As presented in the section 4, we trained English speech decoders with raw English speech only or English speech transcriptions. We present three training set-tings: one decoder trained on raw English speech data from CoVoST (∼400h), another trained on raw English speech data from both Common Voice (∼2,000h) and Multilingual Librispeech (MLS) (∼40,000h), and finally another trained on English speech transcription data from both Common Voice and Multilingual Librispeech. At test time, we can now plug these English speech decoders to any text or speech encoder. We focused on es-en and fren language directions that have previously been covered for direct speech-to-speech translation (see Table 6). We also present text-to-speech translation results, plugging text encoders to our speech decoders. Following Lee et al. (2022a,b) the evaluation is done by transcribing the output speech with an open-sourced ASR system for English and evaluating the BLEU score of the transcribed speech with target text from CoVoST. We compare these results to a supervised baseline (Lee et al., 2022b) trained on real speech-to-speech translation data from Voxpopuli and mined data from (Duquenne et al., 2021). We also provide a strong supervised baseline composed of a Speechto-text translation model from that is trained on a significant amount of speech translation data from Voxpopuli, EuroparlST and CoVoST, followed by a text-to-unit model. In Table 6, we notice that our speech decoders achieve strong results for this zero-shot setting, even with a limited amount of raw speech data. Incorporating much more raw speech data in the training, significantly improves the results. Using textual representation as input helps in speech decoder training, leading to best results. To the best of our knowledge, these are the first results for zero-shot direct speech-to-speech translation. This last experiment again highlights the compatibility between representations for different languages and modalities. Our approach enables to efficiently leverage raw speech data for T2S and S2S tasks. Conclusion In this work, we studied how to build a common fixed-size representation for text and speech in different languages, to perform zero-shot cross-modal translation. By imposing a fixed-size representation and aligning explicitly languages and modalities, we have overcome the sentence length mismatch between audio and text, and obtained multilingual and multimodal representations compatible with decoders trained on other languages and/or modalities in a zero-shot setting. We were able to build text and speech encoders for multiple languages compatible with text decoders for multiple languages as well as an English speech decoder. Our zero-shot cross-modal translation results for direct speech-totext, text-to-speech and speech-to-speech translation define a new zero-shot state-of-the-art baseline. To the best of our knowledge, this is the first work tackling zero-shot direct text-to-speech and speechto-speech translation. Finally, we highlighted the modularity of our architecture; all type of data can be used to train decoders (unlabeled text or speech data ; T2T, S2T, S2S translation data; speech transcription data). Using more types of training data may further enhance the robustness of the decoder to other languages or other modalities. Limitations We highlighted the modularity of our architecture, learning separately encoders and decoders. While it can be seen as a strength, as one does not need to retrain the whole system to add a new language to the framework, it can also be seen as a limitation as the number of modules increases linearly with the number of languages. Moreover, training multiple separate modules requires more time and computation than one multilingual model. Multilingual training of encoders or decoders is left for future work. In machine translation, sequence-to-sequence models with fixed-size sentence representation were replaced by sequence-to-sequence models with attention that showed important performance boost for long sentences. Our work shows that competitive performance can still be achieved with fixed-size sentence representations and enables efficient compatibility between languages and modal-ities. However, very long sequences, beyond usual sentence length, are expected to perform less well. We showed that it is possible to learn an English speech decoder with raw speech data, it would be interesting to extend this to other languages as target speech, and see how our method performs for a low resource spoken language. A Appendix A.1 Distances in LASER text space We report the L2 squared distances of sentence embeddings in different languages to their English translations sentence embeddings in LASER space. A.2 Other text decoders With the conclusion that bi-text data can help the decoder be robust to other unseen languages, we trained decoders for German, French, Spanish, Turkish and Mongolian. We use en-xx bitexts, in addition to raw xx data to train the decoders. For all decoder trainings, we use bi-texts from CCMatrix , for the auto-encoding loss we use one side of the bi-texts corresponding to the language that we are trying to decode, except for Mongolian where we take all the raw Mongolian text data from CCnet. (Wenzek et al., 2019). We present the results in Table 7. A.3 Training details We use Fairseq to train our models. Text student encoders are trained on 32 Tesla V100 GPUs with a learning rate set to 10 −4 , maximum number of tokens by GPU is 1400, and update frequency is set to 4. Speech student encoders are trained on 48 Tesla V100 GPUs for a few days, with same learning rate as text students, maximum number of sentences is set to 32 by GPU. Text decoders are trained with the same configuration as mBART. Speech decoders are trained on 48 Tesla V100 GPUs with a learning rate set to 3 · 10 −4 , maximum number of sentences is set to 32 by GPU and update frequency is set to 4. en de fr es ca ja tr mn Translation into German This work -zero-shot expect for en-de de-de+en-de decoder 39.1 -32.6 24.6 29.2 20.9 27.9 12.8 Previous works -supervised M2M-100 (12B -48 layers) 42. 12.0 10.7 10.9 9.2 10.8 9.3 11.0 -Deepnet (3.2B -200 layers) (Wang et al., 2022) Figure 1 : 1Summary of the model architecture. Figure 2 : 2BLEU vs. sentence length on FLORESdevtest. English auto-encoding (left), German-to-English translation (right). Figure 3 : 3Incremental learning of a speech student. Figure 4 : 4Speech decoder training. encoder which weights are fixed during training and a decoder predicts the sequence of units of the corresponding speech. Figure 5 : 5L2 squared distances to English embeddings in LASER space for translations from FLORES devtest Table 1: BLEU scores for ja-en on FLORES devtestja-en Original encoder + original decoder 6.9 Original encoder + new decoder 5.5 Student -BOS pooling + new decoder 19.5 Student -max pooling + new decoder 22.5 Student -max pooling + original decoder 12.2 Student -max pooling & CE + new decoder 22.6 Table 2 : 2BLEU on FLORES devtest for text-to-text xx-en translation using different English decoders. Table 3 : 3BLEU on CoVoST 2 test set for zero-shot speech-to-text translation (xx → en). Table 4 : 4BLEU on CoVoST 2 test set for different teach- ers and decoders for zero-shot speech-to-text translation. Previous works -supervised M2M-100 (12B -48 layers) 30.3 27.2 28.2 -26.6 19.4 24.0 14.9 Deepnet (3.2B -200 layers) (Wang et al., 2022) 32.2 28.3 28.8 -26.9 21.5 25.9 18.8 Translation into French This work -zero-shot expect for en-fr fr-fr+en-fr decoder 49.1 38.3 -31.2 37.6 25.3 33.4 16.6 Previous works -supervised M2M-100 (12B -48 layers) (Fan et al., 2021) Translation into Turkish This work -zero-shot expect for en-tr tr-tr+en-tr decoder 31.2 27.1 26.4 21.5 24.2 19.1 -13.7 Previous works -supervised M2M-100 (12B -48 layers) (Fan et al., 2021) 32.8 26.9 26.6 22.3 24.3 18.6 -16.1 Deepnet (3.2B -200 layers) (Wang et al., 2022) 39.5 32.0 31.6 26.2 28.2 23.2 -21.0 Translation into Mongolian This work -zero-shot expect for en-mn mn-mn+en-mn decoder 15.7 15.8 15.2 13.6 15.2 13.5 15.4 -Previous works -supervised M2M-100 (12B -48 layers)1 -34.5 27.1 30.9 21.4 28.4 15.9 Deepnet (3.2B -200 layers) (Wang et al., 2022) 46.0 -36.2 29.2 32.5 24.7 31.9 21.7 Translation into Spanish This work -zero-shot expect for en-es es-es+en-es decoder 29.1 25.9 26.8 -26.3 18.6 22.8 12.2 51.4 42 -32.8 39.7 26.6 35.1 20.8 Deepnet (3.2B -200 layers) (Wang et al., 2022) 54.7 43.4 -35.2 41.6 29.9 38.2 26.6 18.3 16.8 16.2 15.0 15.8 13.7 15.9 - Table 7 : 7BLEU on FLORES devtest for text-to-text translation for de, es, fr, tr and mn decoders5806 Effectively pretraining a speech translation decoder with machine translation data. 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[ "Fractal Markets Hypothesis and the Global Financial Crisis: Scaling, Investment Horizons and Liquidity", "Fractal Markets Hypothesis and the Global Financial Crisis: Scaling, Investment Horizons and Liquidity" ]	[ "Ladislav Kristoufek " ]	[]	[]	We investigate whether fractal markets hypothesis and its focus on liquidity and investment horizons give reasonable predictions about dynamics of the financial markets during the turbulences such as the Global Financial Crisis of late 2000s. Compared to the mainstream efficient markets hypothesis, fractal markets hypothesis considers financial markets as complex systems consisting of many heterogenous agents, which are distinguishable mainly with respect to their investment horizon. In the paper, several novel measures of trading activity at different investment horizons are introduced through scaling of variance of the underlying processes. On the three most liquid US indices -DJI, NASDAQ and S&P500 -we show that predictions of fractal markets hypothesis actually fit the observed behavior quite well.	10.1142/s0219525912500658	[ "https://arxiv.org/pdf/1203.4979v1.pdf" ]	10,130,862	1203.4979	55c829395d84318e52eb131b6fd3c8cfc3d8df43	Fractal Markets Hypothesis and the Global Financial Crisis: Scaling, Investment Horizons and Liquidity Ladislav Kristoufek Fractal Markets Hypothesis and the Global Financial Crisis: Scaling, Investment Horizons and Liquidity Fractal markets hypothesisScalingFractalityInvestment horizonsEfficient markets hypothesis PACS: 0545Df, 8965Gh, 8975Da JEL: G01, G14, G15 We investigate whether fractal markets hypothesis and its focus on liquidity and investment horizons give reasonable predictions about dynamics of the financial markets during the turbulences such as the Global Financial Crisis of late 2000s. Compared to the mainstream efficient markets hypothesis, fractal markets hypothesis considers financial markets as complex systems consisting of many heterogenous agents, which are distinguishable mainly with respect to their investment horizon. In the paper, several novel measures of trading activity at different investment horizons are introduced through scaling of variance of the underlying processes. On the three most liquid US indices -DJI, NASDAQ and S&P500 -we show that predictions of fractal markets hypothesis actually fit the observed behavior quite well. Introduction Efficient markets hypothesis (EMH) has been a hot topic since its introduction in 1960s (Fama, 1965(Fama, , 1970Samuelson, 1965). For its simplicity and intuitive logical structure, EMH has been widely accepted as a cornerstone of the modern financial economics. Since the very beginning, EMH has been criticized on several fronts, mainly theoretical -that it is only a set of practically meaningless tautologies (LeRoy, 1976) -and empirical -that it is frequently violated and financial markets are at least partially predictable (Malkiel, 2003). In his pioneering paper, Fama (Fama, 1970) describes the efficient market as the one where all available information are already reflected in the asset prices. In his later work (Fama, 1991), he defined the efficient market through the language of mainstream economics as the one where prices reflect available information to the point where marginal gain from using the information equals marginal cost of obtaining it. Based on this assertion, the efficient market is defined as a random walk (Fama, 1965), contrary to Samuelson's formulation through a martingale (Samuelson, 1965). Either way, the efficient market leads to a Brownian motion of the asset prices, i.e. a process with independent and identically normally distributed increments. Apart from uncorrelatedness of the increments (autocorrelations have been shown to vanish for lags higher than units of minutes (Stanley et al., 1999)), the implications of EMH have been widely rejected in empirical studies (Cont, 2001). However, the most severe shortcoming of EMH is its ignorance to extreme (and sometimes devastating) events on the capital markets, which theoretically should have never happened (Stanley, 2003). EMH has far-reaching implications, which are discussed in majority of financial economics textbooks (Elton et al., 2003) -investors are rational and homogeneous, financial returns are normally distributed, standard deviation is a meaningful measure of risk, there is a tradeoff between risk and return, and future returns are unpredictable. To some extent, all of these implications can be easily attacked with empirical analysis. For our purposes, the first implication of homogeneous investors is crucial. It implies that all the investors use the available information in the same way and thus they operate on the same investment horizon (or theoretically the same set of investment horizons). However, it is known that capital markets comprise of various investors with very different investment horizons -from algorithmically-based market makers with the investment horizon of fractions of a second, through noise traders with the horizon of several minutes, technical traders with the horizons of days and weeks, and fundamental analysts with the monthly horizons to pension funds with the horizons of several years. For each of these groups, the information has different value and is treated variously. Moreover, each group has its own trading rules and strategies, while for one group the information can mean severe losses, for the other, it can be taken a profitable opportunity. This environment creates a very complex system, which can be hardly described by oversimplified EMH. On contrary, fractal markets hypothesis (FMH) (Peters, 1994) has been constructed based on the most general characteristics of the markets. In its core, it is based on a notion completely omitted in EMH -liquidity. According to FMH, liquidity provides smooth pricing process in the market, making it stable. If liquidity ceases, market becomes unstable and extreme movements occur. In the literature, FMH is usually connected with detection of fractality or multifractality of the price processes of financial assets (Peters, 1994;Goddard, 2009, 2011). However, it has not been put to test with respect to its predictions about causes and implications of critical events in the financial markets. In this paper, we analyze whether these predictions fit the observed behavior in the stock markets before and during the current Global Financial Crisis (2007/2008-?). Mainly, we are interested in the behavior of investors at various investment horizons as well as in scaling of the market returns. To do so, we utilize a sliding window estimation of generalized Hurst exponent H(q) with q = 2 (usually called local or time-dependent Hurst exponent). Moreover, we introduce several new measures of trading activity at different investment horizons based on decomposition of Hurst exponent and variance scaling. The local Hurst exponent approach has been repeatedly used to analyze potential turning and critical points in the stock market behavior. Grech and Mazur (2004) studied the crashes of 1929 and 1987 focusing on behavior of Dow Jones Industrial Index and showed that the local Hurst exponent analysis can provide important signals about coming extreme events. In the series of papers, Grech and Pamula (2008); Czarnecki et al. (2008) studied the critical events of the Polish main stock index WIG20 and again presented the local Hurst exponent as a useful tool for detection of coming crashes (together with log-periodic model of Sornette et al. (1996)). Domino (2011Domino ( , 2012 further studied the connection between local Hurst exponent behavior and critical events of WIG20 index. Kristoufek (2010) applied the similar technique on detection of coming critical points of PX50 index of the Czech Republic stock market and uncovered that the functioning is very similar. Morales et al. (2012) broadened the application of time-dependent Hurst exponent on a wide portfolio of the US stocks and showed that the values of Hurst exponent can be connected to different phases of the market. In this paper, we show that behavior of the time-dependent Hurst exponent is connected to various phases of the market. Moreover, we uncover that there are some common patterns before the critical points. Most importantly, the Global Financial Crisis is detected to be connected with unstable trading and unbalanced activity at different investment horizons which is asserted by FMH. The paper is organized as follows. In Section 2, we give basic definitions of fractal markets hypothesis. Section 3 describes multifractal detrended fluctuation analysis, which we use for the generalized Hurst exponent estimation, and introduces new measures of trading activity at specific investment horizons. In Section 4, we test whether the assertions of FMH are actually observed in the real market. All three analyzed indices -DJI, NASDAQ and S&P500 -share several interesting patterns before and during the current financial crisis, which are in hand with FMH. Fractal markets hypothesis Fractal markets hypothesis (FMH) was proposed by Peters (1994) as a follow-up to his earlier criticism of EMH (Peters, 1991). The cornerstone of FMH is a focus on heterogeneity of investors mainly with respect to their investment horizons. The market consists of the investors with investment horizon from several seconds and minutes (market makers, noise-traders) up to several years (pension funds). Investors with different investment horizons treat the inflowing information differently and their reaction is correspondingly distinct (market participants with short investment horizon focus on technical information and crowd behavior of other market participants, whereas investors with long investment horizon base their decisions on fundamental information and care little about crowd behavior). Specific information can be a selling signal for a short-term investor but an opportunity to buy for a long-term investor and vice versa. The existence of investors with different horizons assures a stable functioning of the market. When one horizon (or a group of horizons) becomes dominant, selling or buying signals of investors at this horizon will not be met with a reverse order of the remaining horizons and prices might collapse. Therefore, the existence and activity of investors with a wide range of investment horizons is essential for a smooth and stable functioning of the market (Rachev and Weron, 1999;Weron and Weron, 2000). Fractal markets hypothesis thus suggests that during stable phases of the market, all investment horizons are equally represented so that supply and demand on the market are smoothly cleared. Reversely, unstable periods such as "crises" occur when the investment horizons are dominated by only several of them so that supply and demand of different groups of investors are not efficiently cleared. This two implications give us characteristic features to look for in the market behavior. FMH is tightly connected to a notion of multifractality and long-range dependence in the underlying series. Process X t is considered multifractal if it has stationary increments which scale as \|X t+τ − X t \| q ∝ τ qH(q) for integer τ > 0 and for all q (Calvet and Fisher, 2008). H(q) is called generalized Hurst exponent and its dependence on q separates the processes into two categories -monofractal (or unifractal) for constant H(q) and multifractal when H(q) is a function of q. For q = 2, we consider long-range dependence of the increments of the process X t . As this case is the most important for us as it characterizes scaling of variance (and we treat variance as a sign of a trading activity), we label H ≡ H(2) further in the text. Hurst exponent H is connected to asymptotically hyperbolically decaying autocorrelation function ρ(k), i.e. ρ(k) ∝ k 2H−2 for k → ∞. For H = 0.5, we have a serially uncorrelated process; for H > 0.5, we have a persistent process; and for H < 0.5, we an anti-persistent process. Persistent processes are visually trending yet still remain stationary, whereas anti-persistent processes switch their sign more frequently then random processes do. Scaling of stock returns In this section, we present the method we use for the estimation of generalized Hurst exponentmultifractal detrended fluctuation analysis (MF-DFA) -and several novel measures connected to a trading activity at various trading horizons. MF-DFA is applied here because it is standardly used in the local Hurst exponent literature (Grech and Mazur, 2004;Grech and Pamula, 2008;Czarnecki et al., 2008;Kristoufek, 2010) and compared to other methods, such as generalized Hurst exponent approach (Di Matteo et al., 2005;Di Matteo, 2007;Barunik and Kristoufek, 2010;Kristoufek, 2011), it provides wider range of scales to analyze. As we want to compare as many investment horizons as possible, such a distinction leads to MF-DFA. Multifractal detrended fluctuation analysis Multifractal detrended fluctuation analysis (MF-DFA) is a generalization of detrended fluctuation analysis (DFA) of Peng et al. (1993Peng et al. ( , 1994. Kantelhardt et al. (2002) proposed MF-DFA to analyze scaling of all possible moments q, not only the second one (q = 2) as for DFA. One of the advantages of MF-DFA and DFA over other techniques is that it can be applied on series with H > 1, i.e. a higher order of integration. In the procedure, one splits the series of length T into segments of length s. For each segment, a polynomial fit X s,l of order l is constructed for the original segment X s . In our analysis, we apply a linear fit so that l = 1 and we will omit the label onwards. Note that the filtering procedure can be chosen not only from polynomial fits but also from moving average, Fourier transforms and various others Kantelhardt (2009). Detrended signal is constructed for each segment as Y s = X s − X s . Fluctuation F 2 DF A,q (i, s) is defined for each sub-period i of length s as F 2 DF A,q (i, s) =   [T /s] i=1 Y 2 i,s /s   1 2 . As T /s is not necessarily an integer, we calculate the fluctuations in the segments starting from the beginning as well as from the end of the series not to omit any observation. By doing so, we obtain 2[T /s] fluctuations F 2 DF A,q (i, s). The fluctuations are then averaged over all segments with length s to obtain the average fluctuation F DF A,q (s) =   2[T /s] j=1 F 2 DF A,q (j, s)/2[T /s]   1 q . The average fluctuations scale as F DF A,q (s) = cs H(q) where H(q) is a generalized Hurst exponent and c is a constant. For q = 2, we obtain standard DFA for a long-range dependence analysis. Hurst exponent is usually estimated only for a range of scales s between s min and s max . The minimum scale is set so that the fit in each segment can be efficiently calculated and the maximum scale is set so that the average fluctuation for this scale is based on enough observations. Scaling-based liquidity measures Estimation of Hurst exponent compresses all the information from the dynamics of the process into a single value. However, the procedure can be also decompressed to give us some additional information. From the economic point of view, the segment's length s can be taken as a length of an investment horizon. Fluctuation corresponding to the horizon s can be then taken as a proxy for activity of traders with a horizon of s. From our previous discussion about situations of market instabilities in FMH framework, we propose several new measures. Trading activity of investors with very short investment horizons can be approximated with F (0) = e c , which is an estimate of fluctuation at horizon s → 0. In an unstable market, it is assumed that investors at the very short horizons will be the most active ones. Also, some longterm investors might shorten their horizons Peters (1994). Therefore, we assume that close to and during market turmoils, F (0) will increase compared to the stable periods. In a stable market, all investment horizons are represented uniformly (or at least approximately uniformly). During critical points, the long-term investors either restrict or even stop their trading activities and the short-term investors become dominant. Trading activity and thus fluctuations F 2 at shorter trading horizons will be higher than rescaled trading activity at longer horizons. Therefore, Hurst exponent H would be decreasing shortly before and during the turbulent times at the market. This is in hand with the definition of irregular market of Corazza and Malliaris (2002). In a regular market, the scaling of variance should be stable, i.e. fluctuations F 2 for different horizons s should lay on a straight line. If any of the investment horizons becomes dominant, the scaling would be less precise. To measure such dispersion of trading activity at different investment horizons, we introduce F σ , which is a standard deviation of rescaled fluctuations, and F R , which is a range of rescaled fluctuations: F σ = smax s=smin F 2 (s)/s 2 H − smax s=smin F 2 (s)/s 2 H (s max − s min + 1) 2 s max − s min ; F R = max smin≤s≤smax F 2 (s)/s 2 H − min smin≤s≤smax F 2 (s)/s 2 H During turbulent times, both F σ and F R are expected to increase. In a similar manner, we also define a ratio F r between rescaled fluctuations of the horizons with the maximal and minimal rescaled fluctuation. In an ideal market with uniformly represented investment horizons, we would have F r = 1. The further F r is from 1, the less stable is the scaling and thus also the less stable the market is. Application to the Global Financial Crisis Data and methodology To check whether the implications of FMH hold, we apply the proposed methodology to the daily series of three US indices -Dow Jones Industrial Average Index (DJI), NASDAQ Composite Index (NASDAQ) and S&P500 Index (SPX) -between the beginning of 2000 and the end of 2011. As it is widely believed that the crisis started in the USA and spilled over to the other parts of the world, we choose the US indices because they should signify the coming and continuing crisis the best. If the predictions of FMH hold, we expect local Hurst exponent to be decreasing before the critical point and remaining below H = 0.5 during the crisis. In a similar way, trading activity at the short horizons F (0) should be increasing before the crisis and remain high during the crisis compared to the more stable periods. The very same expectations hold for F σ and F R . For F r , we expect the values to be further from one before and during the crisis times. We use a moving (sliding) window procedure to the dataset. The window length is set to T = 500 trading days (approximately two trading years) and a step to one day. For MF-DFA, we set s min = 10 and s max = T /10 = 50. This way, we can estimate H, F (0), F σ , F R and F r and comment on their evolution in time and during various phases of the market behavior. To meet stationarity condition, which is essential for correct Hurst exponent estimation, we filter the raw series with GARCH(1,1). Therefore, the analysis is made on filtered series defined as f r t = r t / √ h t , where f r t is a filtered return at time t, r t is a raw return at time t, defined as r t = log(S t /S t−1 ) with S t being a stock index closing value at time t, and h t is a conditional variance obtained from GARCH(1,1) at time t. The GARCH-filtering is a crucial addition to the methodology because without comparable volatility in different time windows, we would not be able to say whether e.g. an increase in F (0) is caused by changing structure of investors activity or just an increase of variance across all scales (investment horizons). Results Results for all three analyzed indices are summarized in Figs. 1-3 the trend followed even to the second half of 2007. Such a behavior can be attributed to a changing structure of investors' activity -increasingly more trading activity was taking place at short investment horizons. The end of these strong downward trends of H are connected to the end of soaring gains of all the analyzed indices. For NASDAQ, the end of the local Hurst exponent trend can be even connected to attaining the maximal values in the of 2007. Afterwards, the local Hurst exponent follows a slow increasing trend for all three indices. However, H remains below the value of 0.5, which is associated with a random behavior, for a rather long period. The lengths of these periods vary across the analyzed indices -the longest for S&P500, which is the index with the lowest gains after the crisis. These two phenomena might be connected because for NASDAQ, which has been the most increasing market of the three after the crisis, we observe the values of H even above 0.5 in 2010 and 2011. Note that values of H > 0.5 indicate dominance of long-term traders (a higher trading activity at long investment horizons) and thus a belief in good prospects of the market situation. When we look at the short horizons trading activity F (0), we observe that it was increasing in very similar period as Hurst exponent was decreasing the in the previous paragraph. The measure increased from values of approximately 0.04 up to over 0.12 for all three analyzed indices. After reaching its peak, the trading activity at the short horizons was slowly decreasing back to the original levels of the beginning of 2006. Again, we observe differences in the duration of this downward trend. For NASDAQ, the pre-crisis levels of short-term trading activity was reached around the beginning of 2009 and since then, the activity has remained relatively stable. On contrary, DJI has not reached the pre-crisis levels yet and S&P500 got back to the pre-crisis levels during 2010. Even though the durations and magnitudes of short-term trading activity vary between the analyzed markets, we observe that the most critical points of the crisis were connected to increased trading activity of short-term investors. The other two measures -standard deviation of rescaled fluctuations F σ and range of rescaled fluctuations F R -tell a very similar story. Since both are the measures of instability of variance scaling across different investment horizons, this is not surprising. The results are actually very alike to the dynamics of estimated fluctuations at very short investment horizons discussed in the previous paragraph -very rapid increase starting in 2005/2006 turning which followed to the first half of 2007 (and again longer for NASDAQ). According to FMH, unevenly represented investment horizons imply complicated matching between supply and demand at the financial market. Therefore, the increasing instability of trading activity at different investment horizons indicates growing problems of this supply-demand matching. After the strong increases between 2006 and 2007, both measures started decreasing afterwards. However, only NASDAQ has recovered the pre-crisis stability levels. The last measure we present -the ratio between trading activity at the horizon with the highest and with the lowest activity -uncovers quite similar results. Note that even in the calm periods before the last crisis, the ratio is not equal or close to one as it theoretically should be (for a perfectly scaling variance). Between the beginning of 2006 and the first half of 2007, the ratio increased from around 1.2 up to 1.6 for all three indices. Notably, the following decreasing trend was the fastest for NASDAQ while it took much longer to DJI and S&P500 to recover. However, the results for this last measure are probably the weakest as the measure is very volatile in time. Conclusions and discussion Efficient market hypothesis is unable to describe the behavior of the financial markets during the last (current) crisis starting in 2007/2008 in a satisfying way. We analyze whether an alternative approach -fractal markets hypothesis -gives more reasonable predictions. The cornerstone of FMH is liquidity connected to the trading activity at different investment horizons. If the investors with different horizons are uniformly distributed across scales, supply and demand for financial assets work efficiently. However, when a specific investment horizon (or a group of horizons) starts to dominate the situation in the market, the supply-demand matching ceases to work and a critical point emerges. To test whether this crucial assertion of FMH holds for the current crisis, we used the local Hurst exponent approach as well as the introduced set of new measures of trading activity based on Hurst exponent decomposition. We found that the behavior at various investment horizons is quite well described by the FMH before and during the current Global Financial Crisis. Analyzing three stock indices of the USA -DJI, NASDAQ and S&P500 -we showed that the local Hurst exponent decreases rapidly before the turning of the trend, which is in hand with previously published results (Grech and Mazur, 2004;Grech and Pamula, 2008;Czarnecki et al., 2008;Kristoufek, 2010). Moreover, with a use of the new measures of trading activity, we uncovered that investors' trading activity indeed changes before and during the crisis period compared to the preceding stable periods. Before the crisis, the structure of trading activity at different investment horizons changed remarkably with rapidly increasing activity at the shortest horizons, i.e. short-term investors started to dominate and longterm investors showed no faith in a continuing growth. Also, the stability of investment horizons representation changed before the current turbulent times. Uniformity of investment horizons representation started to cease before an outburst of the crisis. During and after the most severe losses, the indicators started to stably return to the pre-crisis levels. However, they fully recovered only for NASDAQ index while DJI and S&P500 are just attaining the former stability. Note that NASDAQ, which is the index with the fastest recovering investment horizons measures, is also the index which returned to the pre-crisis values the fastest. Summarizing, we have showed that fractal markets hypothesis gives reasonable predictions of market dynamics in the turbulent times. Trading activity at various investment horizons ensuring efficient clearing of supply and demand in the market, which guarantees high liquidity, turns out to be a crucial attribute of a well-functioning and stable market. Figure 1 : 1. All the indices reached their post-DotCom bubble maxima in the latter half of 2007, which were followed by progressively decreasing trend culminating at the turn of 2008. Similarly to the indices all over the world, the US indices lost around half of their value during that approximately 1.5 year -the loss accounted for 53.78% , 61.22%, and 56.77% for DJI, NASDAQ and S&P500, respectively. 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[ "Expressive power of complex-valued restricted Boltzmann machines for solving non-stoquastic Hamiltonians", "Expressive power of complex-valued restricted Boltzmann machines for solving non-stoquastic Hamiltonians" ]	[ "Chae-Yeun Park \nInstitute for Theoretical Physics\nUniversity of Cologne\n50937KölnGermany\n\nM5G 2C8Xanadu, TorontoONCanada\n", "Michael J Kastoryano \nInstitute for Theoretical Physics\nUniversity of Cologne\n50937KölnGermany\n\nAmazon Quantum Solutions Lab\n98170SeattleWashingtonUSA\n\nAWS Center for Quantum Computing\n91125PasadenaCaliforniaUSA\n" ]	[ "Institute for Theoretical Physics\nUniversity of Cologne\n50937KölnGermany", "M5G 2C8Xanadu, TorontoONCanada", "Institute for Theoretical Physics\nUniversity of Cologne\n50937KölnGermany", "Amazon Quantum Solutions Lab\n98170SeattleWashingtonUSA", "AWS Center for Quantum Computing\n91125PasadenaCaliforniaUSA" ]	[]	Variational Monte Carlo with neural network quantum states has proven to be a promising avenue for evaluating the ground state energy of spin Hamiltonians. However, despite continuous efforts the performance of the method on frustrated Hamiltonians remains significantly worse than those on stoquastic Hamiltonians that are sign-free. We present a detailed and systematic study of restricted Boltzmann machine (RBM) based variational Monte Carlo for quantum spin chains, resolving how relevant stoquasticity is in this setting. We show that in most cases, when the Hamiltonian is phase connected with a stoquastic point, the complex RBM state can faithfully represent the ground state, and local quantities can be evaluated efficiently by sampling. On the other hand, we identify several new phases that are challenging for the RBM Ansatz, including non-topological robust non-stoquastic phases as well as stoquastic phases where sampling is nevertheless inefficient. We further find that, in contrast to the common belief, an accurate neural network representation of ground states in non-stoquastic phases is hindered not only by the sign structure but also by their amplitudes. arXiv:2012.08889v3 [quant-ph] 3 Nov 2022	10.1103/physrevb.106.134437	[ "https://export.arxiv.org/pdf/2012.08889v3.pdf" ]	237,276,866	2012.08889	0dc9294cdeb0dcca3f69ba4e57302ae2e0a3df3e	Expressive power of complex-valued restricted Boltzmann machines for solving non-stoquastic Hamiltonians Chae-Yeun Park Institute for Theoretical Physics University of Cologne 50937KölnGermany M5G 2C8Xanadu, TorontoONCanada Michael J Kastoryano Institute for Theoretical Physics University of Cologne 50937KölnGermany Amazon Quantum Solutions Lab 98170SeattleWashingtonUSA AWS Center for Quantum Computing 91125PasadenaCaliforniaUSA Expressive power of complex-valued restricted Boltzmann machines for solving non-stoquastic Hamiltonians (Dated: November 4, 2022) Variational Monte Carlo with neural network quantum states has proven to be a promising avenue for evaluating the ground state energy of spin Hamiltonians. However, despite continuous efforts the performance of the method on frustrated Hamiltonians remains significantly worse than those on stoquastic Hamiltonians that are sign-free. We present a detailed and systematic study of restricted Boltzmann machine (RBM) based variational Monte Carlo for quantum spin chains, resolving how relevant stoquasticity is in this setting. We show that in most cases, when the Hamiltonian is phase connected with a stoquastic point, the complex RBM state can faithfully represent the ground state, and local quantities can be evaluated efficiently by sampling. On the other hand, we identify several new phases that are challenging for the RBM Ansatz, including non-topological robust non-stoquastic phases as well as stoquastic phases where sampling is nevertheless inefficient. We further find that, in contrast to the common belief, an accurate neural network representation of ground states in non-stoquastic phases is hindered not only by the sign structure but also by their amplitudes. arXiv:2012.08889v3 [quant-ph] 3 Nov 2022 I. INTRODUCTION Over the past decade, Machine Learning (ML) has allowed for huge improvement not only in traditional fields such as image detection [1] and natural language processing [2] but also in other disciplines e.g. defeating human level in playing games [3,4] and predicting protein structures [5]. Inspired by such successes, ML has been also actively applied for solving quantum physics problems. Examples include detecting phase transitions [6,7] and decoding quantum error correcting codes [8][9][10]. However, arguably the most active contribution of ML to physics has been in the field of classical variational algorithms for solving quantum many body systems so called variational quantum Monte Carlo (vQMC). A seminal study by Carleo and Troyer showed that the complex-valued restricted Boltzmann machine (RBM) [11] solves the ground state of the transverse field Ising and the Heisenberg models to machine accuracy. Subsequent studies demonstrated that other neural network based Ansätze such as convolutional neural networks (CNNs) [12,13], and models with the autoregressive property [14,15] also provide highly accurate solutions when combined with proper optimization algorithms. Despite these successes, the methods still suffer from several difficulties in solving highly frustrated systems [12,16,17]. Why then are some Hamiltonians so difficult to solve? In the path integral Monte Carlo, it is well known that stoquastic Hamiltonians -those with real-valued and non-positive off-diagonal elements -are tractable [18]. One of the crucial properties of stoquastic Hamiltonians is that the ground state is positive up to a global phase. As the RBM Ansatz also seems to solve stoquastic Hamiltonians well [11], a complex sign structure of the ground state can be why it is difficult to solve a highly frustrated Hamiltonian with strong non-stoquasticity. Several studies [13,17,19] further support this claim by showing that training a neural network for a complex sign structure is demanding. However, expressivity of the Ansatz has been largely overlooked in vQMC. As well-known universal approximation theorems [20][21][22] imply that a neural network with a sufficient number of parameters can express any physical functions, one may consider that expressivity is not really an issue. The theorem, nevertheless, tells little about how many parameters are required to represent a given function, and there are indeed some functions that a certain neural network fails to represent with a polynomial (in the input size) number of parameters [23]. Under a reasonable complexity theoretical assumption (that the polynomial hierarchy does not collapse), Ref. [24] also has shown that there is a quantum state Ψ GWD with a local parent Hamiltonian whose amplitudes in the computational basis \|Ψ GWD (x)\| 2 cannot be obtained in a polynomial time. Recent numerical studies [25,26] further show example quantum states whose outcome probability distributions are difficult even for large neural networks. Thus, we do not have a clear argument why nonstoquastic Hamiltonians are difficult for vQMC albeit its formulation seems be to irrelevant to the target Hamiltonian. One can contrast with the density matrix renormalization group (DMRG) [27] which was first developed as an extension of numerical renormalization group, but subsequent numerical and theoretical studies have revealed that entanglement is the underlying principle behind this method. This connection explains why the DMRG works exceptionally well for one-dimensional gapped system but has difficulty solving higher dimensional systems. In comparison, we still do not have such a good theory for vQMC. With this in mind, we unveil a connection between the vQMC with complex RBMs and stoquasticity of the Hamiltonian. We thus aim to guide future theoretical studies on the mathematical foundation of the vQMC as well as numerical studies for Hamiltonians resilient to vQMC. Our investigation is based on the classification of three typical failure mechanisms: (i) Sampling: The sampling method, such as local update Markov chain Monte Carlo (MCMC), fails to produce good samples from the state, or the observables in the optimization algorithm cannot be accurately constructed from a polynomial number of samples. (ii) Convergence: The energy gradient and other observables involved in the optimization (e.g. the Fisher information matrix) can be accurately and efficiently obtained for each optimization step, yet the optimization gets stuck in a local minimum or a saddle point. (iii) The expressivity of the Ansatz is insufficient: the Ansatz is far from the correct ground state even for the optimal parameter set, i.e. min θ \|\| \|ψ θ − \|Φ GS \|\| is large where \|ψ θ is a quantum state that Ansatz describe for a given parameter set θ. Specific pairs of Hamiltonian and Ansätze sometimes rule out one or several of the failure mechanisms. For instance, when the Hamiltonian has a known exact neuralnetwork representation of the ground state (e.g. cluster state, toric code [28] and other stabilizer states [29,30]), we can discard expressivity [case (iii)] as a failure mechanism. On the other hand, models with the autoregressive property [14,15] are free from the MCMC errors as they always produce unbiased samples [31]. The Hamiltonians we consider in this work will typically not have known ground state representations and we mainly use the RBM Ansatz, as it is the best studied neural network Ansatz class and the most reliable performance [32]. Hence all three failure mechanisms can occur. From those classifications, we set out to understand what role stoquasticity plays in the success and failure of variational Monte-Carlo with the RBM Ansatz. By way of example, we show that non-stoquasticity can cause problems with sampling [case (i)], while phase transitions within a non-stoquastic parameter region may yield expressivity problems [case (iii)]. We dub such a phase that cannot be annealed into a stoquastic parameter region "deep non-stoquastic phase." Given that "deep nonstoquastic phases" can be gapped, our observation implies that the dimension or gap of the system is not related to the reliability of the method in any straightforward manner. Rather, as for quantum Monte Carlo [18], the stoquasticity of the Hamiltonian is the more essential feature. Although our claim of a "deep non-stoquastic phase" seems to disagree with Ref. [17] which posits that even a shallow neural network (with depth 2 or 3) can express the ground state of non-stoquastic Hamiltonians, we show later in the paper that this is due to different levels of desired accuracy. We demonstrate that the expressivity problem indeed appears when the desired accuracy is as high as that of stoquastic ones. In addition, we find that the expressivity problem is not only due to the sign structure of quantum states but also their amplitudes, where the latter dominates for system sizes up to about thirty for the models under study. To identify difficult phases for vQMC, we utilize local basis transformations beyond the well-known Marshall-Peierls sign rule [33] for the J 1 -J 2 models. We discover several basis transformations that reduce stoquasticity for XYZ-type Hamiltonians based on Ref. [34]. Although such transformations mitigate the sampling problem for a Hamiltonian connected to stoquastic phases, we show that they are not effective for Hamiltonians in deep nonstoquastic phases. We still emphasize that our main concern in the paper is why errors from vQMC for non-stoquastic models are significantly larger than those for stoquastic models [12,16,35,36]. However, such relatively large errors from non-stoquastic models may still be enough to obtain the ground state properties depending on the problem at hand (when the gap is much bigger than the errors even for large enough N ). We thus do not claim a difficulty of a particular Hamiltonian (such as the two-dimensional J 1 -J 2 model) for the vQMC; rather we want to understand when and why some Hamiltonians are relatively more difficult than others. The remainder of the paper is organized as follows. We introduce the complex RBM wavefunctions and our optimization methods in Sec. II. We next establish our main observations in Sec. III by studying how nonstoquasticity affects the RBM using the one-dimensional XXZ and the J 1 -J 2 models, the properties of which are well known. We then confirm our observations using a specially devised Hamiltonian in Sec. IV. We then resolve discrepancies between our and previous studies in Sec. V and conclude with the final remark in Sec. VI. II. VARIATIONAL MONTE CARLO AND COMPLEX-VALUED RESTRICTED BOLTZMANN MACHINE Variational quantum Monte Carlo (vQMC) [37] is a classical algorithm for finding the ground state of a quantum Hamiltonian utilizing a variational Ansatz state. For a system with N spin-1/2 particles (or qubits), vQMC considers an Ansatz state ψ θ (x) where x is the vector in the computation basis. When one can sample from p(x) ∝ \|ψ θ (x)\| 2 and efficiently calculate the ratio ψ θ (x )/ψ θ (x), the expectation value of a sparse observable A can be obtained as A = x,x A x ,x ψ θ (x ) * ψ θ (x) = x \|ψ θ (x)\| 2 x ψ θ (x ) ψ θ (x) A x ,x ≈ x ψ θ (x ) ψ θ (x) A x ,x x∼\|ψ θ (x)\| 2(1) where A x ,x = x \|A\|x and f (x) x∼p(x) is the statistical average of a function f (x) over samples {x} from p(x). Likewise, one can also estimate the gradient of an observable ∇ θ ψ θ \|A\|ψ θ , which enables one to stochastically optimize ψ θ (x) toward an eigenstate with the minimum eigenvalue. Initial studies [37,38] have shown that this method solves ground states of several Hamiltonians when used with a proper parameterized wavefunction ψ θ (x). However, choosing such a parameterized wavefunction had relied on several heuristics, and a general Ansatz was missing until Carleo and Troyer introduced the complexvalued restricted Boltzmann machine (RBM) quantum state Ansatz class [11] inspired by the recent successes in machine learning. As other machine learning approaches, the expressive power of the complex-valued RBM can be adjusted by increasing the number of parameters, and this Ansatz with a reasonable number of parameters solves the ground states of the transverse field Ising and the Heisenberg models in machine accuracy [11]. For complex parameters a i , b j and W ij where i ∈ [1, · · · , N ] and j ∈ [1, · · · , M ], an (unnormalized) RBM state is given by ψ θ (x) = y∈{−1,1} N e i,j wij xiyj +aixi+bj yj = e i aixi j 2 cosh(χ j )(2) where θ = (a, b, w) is the collection of all parameters, x = (x 1 , x 2 , · · · , x N ) is a basis vector in the computational basis (typically the Pauli Z basis), y = (y 1 , y 2 , · · · , y M ) labels the hidden units, and the 'activations' are given by χ j = i w ij x i + b j . We also introduce the parameter α = M/N that controls the density of hidden units and parameterizes the expressivity of the model. In addition, we will write ψ θ (x) = ψ θ (x)/ x \| ψ(x)\| 2 to denote the normalized wavefunction. For a given Hamiltonian, the parameters of the Ansatz can be optimized using a variety of different methods, including the standard second-order vQMC algorithm known as Stochastic Reconfiguration (SR) [37,39] or a modern variant of the first order methods [15,40,41] such as ADAM [42]. Throughout the paper, we use the SR as it is believed to be more stable and accurate for solving general Hamiltonians [43]. At each iteration step n, the SR method estimates the covariance matrix S, with en- tries S i,j = O * i O j − O * i O j , and the energy gradient f = E * loc O i − E * loc O i where O i (x) = ∂ θi log[ ψ θ (x)] and · = x∼\|ψ θ (x)\| 2 (·) is the average over samples (see Refs. [11,37] and Appendix A for details). The parameter set is updated as θ n+1 = θ n − η n S −1 f . In practice, a shifted covariance matrix S = S + λ n I with a small real parameter λ n is used for numerical stability. In the SR optimization scheme with the complex RBM, expectation values are obtained by sampling from the distribution \|ψ θ (x)\| 2 , typically by conventional Markov chain Monte Carlo (MCMC). In some cases, we use the running averages of S and f when it increases the stability (i.e. we use f n = (1 − β 1 )f n−1 + β 1 f , S n = (1 − β 2 )S n−1 + β 2 S for suitable choices of β 1 , β 2 and update θ using θ n+1 = θ n − η n S −1 n f n ). To assess whether the sampling method works well, we introduce the exact reconfiguration (ER) that evaluates S i,j and f from ψ θ (x) by calculating the exponential sums x \|ψ θ (x)\| 2 (·) exactly, where x is all possible basis vectors in the computational basis (thus we sum over 2 N or N N/2 configurations depending on the symmetry of the Hamiltonian). Within this framework, we classify the difficulty of ground state simulation as follows: We solve the system using the ER with N = 20 and the SR with N = 28 or 32 (depending on the symmetry of the Hamiltonian). When the Hamiltonian is free from any of the problems [(i) sampling, (ii) convergence, (iii) expressivity], the converged energies from both methods will be close to the true ground state. If we observe that the ER finds the ground state accurately in a reasonable number of epochs [44], but SR does not, we conclude that the problem has to do with sampling. When there is a local basis change that transforms a given Hamiltonian into a stoquastic form, we apply such a transformation to see whether the sampling problem persists. If both SR and ER fail, we evaluate the following further diagnostic tests: (a) We compare ER results from several different randomized starting points, and (b) we run the ER through an annealing scheme from a phase that is known to succeed. When all runs of ER return the the same converged energy, we conclude that the problem must be related to expressivity of the Ansatz. Otherwise, we try the annealing scheme as an alternative optimization method. Instead of training a randomly initialized RBM, we start from the converged RBM within the same phase and change the parameters of the Hamiltonian slowly. If the annealing with the ER also fails, we conclude that the expressivity problem is robust. Finally, we support the classification results from the above procedure by a scaling analysis of the errors for different sizes of the system. III. PRELIMINARY EXAMPLES Stoquastic Hamiltonians [18] -those for which all offdiagonal elements in a specific basis are real and nonpositive -typically lend themselves to simulation by the path integral quantum Monte Carlo method. In the path integral Monte Carlo, one evaluates the partition function Z = Tr[e −βH ] using the expansion Tr[e −βH ] = x0 x 0 \|(e − β K H ) K \|x 0 (3) ≈ x0,··· ,x K−1 ,x K =x0 K−1 i=0 x i+1 \| 1 − β K H \|x i(4) which is valid for large K. As all elements of 1−(β/K)H are non-negative when H is stoquastic, the sum can be estimated rather easily. Likewise, one can also estimate the expectation value of an observable A from a similar expansion of Tr[Ae −βH ]/Z. However, a "sign problem" arises when the condition is not satisfied, leading to uncontrollable fluctuations of observable quantities as the system grows. The relevance of stoquasticity for the vQMC is far less explored, despite the fact that this method and its variants were introduced to alleviate the sign problem [39]. Although it is true that the vQMC is free from summations over alternating signs, the method still show several difficulties in solving frustrated Hamiltonians with a complex sign structure as argued in Ref. [16]. Given that the complex-valued RBM can represent quantum states with complex sign structure such as i,j e iφi,j σ z i σ z j \|+ ⊗N for arbitrary {φ i,j } [24,43], it is important to understand what makes complex-valued RBM struggle to solve nonstoquastic Hamiltonian. In this section, we investigate this question in detail using the one-dimensional Heisenberg XXZ and J 1 -J 2 models the properties of which are well known. Our strategy is simple. For each Hamiltonian, we use the original Hamiltonian and one with the stoquastic local basis, and observe how the local basis transformation affects the expressivity, convergence, and sampling. Throughout the paper, we will assume periodic boundary conditions for ease of comparison with results from the exact diagonalization (ED). A. Heisenberg XXZ and J1-J2 models The Heisenberg XXZ model is given by H XXZ = i σ x i σ x i+1 + σ y i σ y i+1 + ∆σ z i σ z i+1 ,(5) where σ x,y,z j denote the Pauli operators at site j, and ∆ is a free (real) parameter of the model. As this model is solvable by the Bethe Ansatz, it is well known that the model exhibits phase transitions at ∆ = −1 (the first order) and ∆ = 1 (the Kosterlitz-Thouless transition). Furthermore, the system is gapped when \|∆\| > 1 and in the critical phase when −1 < ∆ ≤ 1. The Marshall sign rule (applying the Pauli-Z gate on all even (or odd) sites) changes the Hamiltonian into a stoquastic form in the Pauli-Z basis: H XXZ = i −σ x i σ x i+1 − σ y i σ y i+1 + ∆σ z i σ z i+1(6) The Hamiltonian is then stoquastic regardless of the value of ∆. Using the RBM with α = 3, we plot the result from the ER and SR with and without the sign rule in Fig. 1(a) and (b). For the SR, we sample from the distribution \|ψ θ (x)\| 2 using the MCMC. As the system obeys the U (1) symmetry and the ground states are in the J z = i σ z i = 0 subspace when ∆ > −1, we initialize the configuration x to have the same number of up and down spins. For each Monte Carlo step, we update the configuration by exchanging x i and x j for randomly chosen i and j. We further employ the parallel tempering method using 16 chains with different temperatures (see Appdendix A 2 for details) to reduce sampling noise. Likewise, we sum over the basis vectors in J z = 0 for the ER. For each epoch, we use \|θ\| = N M + N + M number of samples to estimate S and f unless otherwise stated. Figure 1(a) clearly shows that the sign rule barely changes the results when we exactly compute the wavefunctions for optimization [45]. This can be attributed to the fact that the RBM Ansatz can incorporate the Pauli-X, Y, Z gates as well as the phase shift gate e −iθσ z k for arbitrarily θ efficiently [46]. On the other hand, when we sample from the distribution [ Fig. 1 (b)], some RBM instances fail to find the ground state without the sign rule, especially in the antiferromagnetic phase (∆ > 1.0). Thus we see the sampling problem arises due to non-stoquasticity. Since the MCMC simply uses the ratio between two probability densities \|ψ θ (x )/ψ θ (x)\| 2 , which is sign invariant, the sampling problem here has nothing to do with the ground state. Instead, it is caused by different learning paths taken by the original and the basis transformed Hamiltonians. When we use the original Hamiltonian H XXZ , the learning ill-behaves when it hits a region of the parameter space θ where S and f are not accurately estimated from samples. The transformed Hamiltonian H XXZ avoids this problem by following a different learning path [47]. We have observed that in general, the energy of a randomly initialized RBM is much closer to that of the ground state when the sign rule is applied and the learning converges in fewer epochs. We have further tested the SR without the sign rule using different sizes of the system N = [20,24,28,32] and up to 76, 800 samples for each epoch, but observed that the sampling problem persists regardless of such details. We also show that that this is not an ergodicity problem of the MCMC in Appendix B as the SR with the exact sampler (that samples from the probability distribution exactly constructed from \|ψ θ (x)\| 2 ) also gives the same results. Next, let us consider the one-dimensional J 1 -J 2 model, given by H J1−J2 = i J 1 σ σ σ i · σ σ σ i+1 + J 2 σ σ σ i · σ σ σ i+2 .(7) where we fix J 1 = 1.0. The Hamiltonian has a gapless unique ground state when J 2 < J * 2 (thus, within the critical phase) and gapped two-fold degenerated ground states when J 2 > J * 2 . The KT-transition point is approximately known J * 2 ≈ 0.2411 [48]. In addition, an exact solution at J 2 = 0.5 is known -the Majumdar-Ghosh point. The Marshall sign rule also can be applied to this Hamiltonian which yields: H J1−J2 = i J 1 [−σ x i σ x i+1 − σ y i σ y i+1 + σ z i σ z i+1 ] + J 2 σ σ σ i · σ σ σ i+2 .(8) We note that this Hamiltonian is still non-stoquastic when J 2 > 0. In Appendix C 1, we prove that on-site unitary gates that transform H J1−J2 into a stoquastic form indeed do not exist when J 2 > 0. We also show that ground states in the gapped phase (J 2 > J * 2 ) cannot be transformed into a positive form easily using the results from Ref. [49]. Simulation results for this Hamiltonian are presented in Fig. 1(c) and (d). First, as in the XXZ model, the ER results in Fig. 1(c) show that the sign rule is not crucial when we exactly compute the observables, i.e. the ER with and without the sign rule both converge to almost the same energies. However, in contrast to the XXZ model, there is a range of J 2 ∈ (J * 2 , 0.5) ∪ (0.5, 0.6) where all ER and SR instances perform badly (the error is > 10 −4 for some instances) even when the sign rule is applied [ Fig. 1(c)]. It indicates that the expressive power of the network is insufficient for describing the ground state even though the system is gapped. We further show (see Appendix D) that this problem cannot be overcome by increasing the number of hidden units and revisit this issue in Sec. V using the supervised learning framework. Since this region cannot be annealed from a stoquastic point (J 2 = 0) without a phase transition, we argue that this parameter region is in a "deep nonstoquastic" phase. When we use the SR, the results in Fig. 1(d) show that some of the instances always fail to converge to true ground states regardless of the Hamiltonian parameters when the sign rule is not applied. This is the behavior what we saw from the XXZ model that non-stoquasticity induces a sampling problem. On the other hand, when the sign rule is applied, all SR instances report small relative errors when J 2 ≤ J * 2 even though the transformed Hamiltonian is still non-stoquastic. We speculate that this is because the whole region is phase connected to the stoquastic J 2 = 0 point. We also note that previous studies [35,36] using different variational Ansätze have reported a similar behavior of errors, suggesting that difficulty of "deep non-stoquastic" phases is not limited to the RBM Ansatz (see also Sec. V). We summarize the results from the above two models with the following key observations. Observation 1. Complex RBMs represent ground states of spin chains faithfully when the Hamiltonian is stoquastic, up to a basis transformation consisting of local Pauli and phase-shift gates, or is phase connected to such a Hamiltonian. Observation 2. There exists "a deep non-stoquastic phase", where the Hamiltonian cannot be locally or adiabatically transformed into a stoquastic Hamiltonian without crossing a phase transition. Complex RBMs have difficulty representing such ground states. Observation 3. Sampling is stable along the learning path when the Hamiltonian is stoquastic or phase connected to a stoquastic Hamiltonian. In the next section, we will explore these observations more closely by introducing a more challenging example that combines all of the problems above. b H ⋆ tXYZ ∈ Sto Non-stoquastic H ♦ tXYZ ∈ Sto Non-stoquastic A B C 1 C 2 D E O Phase I Phase II Λ FIG. 2. Phase and stoquasticity diagrams of the twisted XYZ model. For parameters 0 ≤ a, b ≤ 2.5, the difference between the lowest energies (E0 − E1)/E0 in the symmetric and the anti-symmetric subspaces under the spin flip (σz ↔ −σz) for N = 28 is shown. As the ground states can break the Z2 symmetry in all three directions, we cannot determine phases solely from this plot. Thus we calculate magnetic susceptibilities in Fig. 3 and find that Phase I breaks the symmetry along the z-axis whereas Phase II recovers this. Between those two phases, Phase Λ that breaks the symmetry along the y-axis appears when a = b. We depict approximate phase boundaries with dotted curves. On the other hand, dashed lines show stoquastic to non-stoquastic transitions. Local basis transformed Hamiltonians H ♦ tXYZ and H tXYZ are stoquastic in the first and third quadrants, respectively. In the second and forth quadrants, local (on-site) unitary gates that transforms the Hamiltonian into a stoquastic form do not exist. The untransformed Hamiltonian HtXYZ is stoquastic only when a = b in this region. The line segment from O = (0.5, 0.5) to A = (0.25, 0.75) and A to E = (1.25, 1.75) indicate the parameters we simulate vQMC. The phase transitions along AE take place at C1 ≈ (0.764, 1.264) and C2 ≈ (0.793, 1.293). IV. FURTHER EXPERIMENTS In previous examples, local basis transformations only marginally affected the expressive power of the model. Here, we introduce a Hamiltonian that involves the Hadamard transformation that cannot be embedded to the RBM Ansatz for a stoquastic transformation. The main findings in this section are (1) local basis transformations beyond the Pauli and phase-shift gates are useful for expressivity, (2) there is a conventional symmetry broken phase for which the RBM fails to represent the ground states, and (3) the number of samples to estimate observables correctly may scale poorly even for a stoquastic Hamiltonian. A. Model Hamiltonian and phase diagram We consider a next-nearest-neighbor interacting XYZ type Hamiltonian with "twisted" interactions: H tXYZ = J 1 N i=1 aσ x i σ x i+1 + bσ y i σ y i+1 + σ z i σ z i+1 + J 2 N i=1 bσ x i σ x i+2 + aσ y i σ y i+2 + σ z i σ z i+2(9) where a and b are two real parameters. Note that a (b) is the strength of XX (Y Y ) interaction for nearestneighbors whereas it is on Y Y (XX) interaction for next-nearest-neighbors. This particular Hamiltonian has a rich phase structure as well as stoquastic to nonstoquastic transitions. The stochastic regions do not coincide with the phases of the model. In addition, the system has Z 2 symmetries in any axis σ {x,y,z} i ↔ −σ {x,y,z} i as well as translational symmetry. Moreover, a π/2 rotation around the z-axis, i.e. U = e −iπ/4 i σ z i , swaps the parameters a and b. Throughout the section, we assume ferromagnetic interactions J 1 = J 2 = −1. As the Hamiltonian in this case becomes the classical ferromagnetic Ising model when a = b = 0, one may expect that the vQMC works well at least for small parameters. However, we will see that this intuition is generally misleading, as the non-stoquasticity of the model plays a very important role. We consider two other representations of the model, which are obtained by local basis transformations: H tXYZ = J 1 N i=1 σ x i σ x i+1 + bσ y i σ y i+1 + aσ z i σ z i+1 + J 2 N i=1 σ x i σ x i+2 + aσ y i σ y i+2 + bσ z i σ z i+2 ,(10)H ♦ tXYZ = J 1 N i=1 aσ x i σ x i+1 + σ y i σ y i+1 + bσ z i σ z i+1 + J 2 N i=1 bσ x i σ x i+2 + σ y i σ y i+2 + aσ z i σ z i+2 . (11) The Hamiltonian H tXYZ (H ♦ tXYZ ) is stoquastic for 0 ≤ a, b ≤ 1 (a, b ≥ 1), and can be obtained by applying π/2 rotation over y (x) axes from the original Hamiltonian H tXYZ , respectively. We note that as those rotations involve the Hadamard gate, they cannot be decomposed only by Pauli gates, e.g. π/2 rotation over the y-axis is given by e −iπ/4Y = XH. These Hamiltonians are obtained by applying the general local transformations described by Klassen and Terhal [34] to Eq. (9) (see Appendix C 2 for detailed steps). In addition, one may further transform H tXYZ and H ♦ tXYZ with local phase-shift gates k e −iπ/4σ z k which can be embedded into the RBM Ansatz [46]. The resulting Hamiltonians are stoquastic for a, b ≥ 1 and 0 ≤ a, b ≤ 1, respectively, which are the reverse of ones before applying the phase-shift gates. Before presenting vQMC results, let us briefly summarize the phase structure of the Hamiltonian that is presented in Fig. 2. To gain an insight, let us first consider the a = b line. When a = b < 1.0, each term of the Hamiltonian prefers an alignment in z-direction so the ground state is \|↑ ⊗N +\|↓ ⊗N . Even though the U (1) symmetry is broken when a = b, this ferromagnetic phase extends from a = b < 1.0 which we denote by Phase I in Fig. 2. On the other hand, the Hamiltonian prefers the total magnetization J z = i σ z i = 0 when a = b > 1.0. The region of this phase is shown in Fig. 2 denoted by Phase II. As the total magnetization changes abruptly at a = b = 1 regardless of the system size, we expect a first order phase transition to take place at this point. However, the phase boundaries when a = b are more complex and another phase Λ appears in between two phases. To characterize the phases when a = b, we plot magnetic susceptibilities and the entanglement entropy (von Neumann entropy of a subsystem after dividing the system into two equal-sized subsystems) along the line segment AE in Fig. 3. For each parameter (a, b), we have obtained the ground state within the subspace preserving the Z 2 symmetry along the z-axis using the ED (thus our ground states obey the Z 2 symmetry even when the symmetry is broken in the thermodynamic limit). We see that the magnetic susceptibility along the z-axis diverges with the system size N in Phase I which implies the symmetry will be broken when N → ∞. Likewise, we also see that the symmetry along the y-axis is broken in Phase Λ. Furthermore, as entanglement entropy follows the logarithmic scaling at C 1 ≈ (0.764, 1.264) (see also Appendix E), we conclude that the phase transition at C 1 is the second order. However, the signature of the phase transition from the entanglement entropy at C 2 is weak possibly because the phase transition is the infinite order Kosterlitz-Thouless transition. Thus we calculate the second derivative of the ground state energy in Appendix E and locate the second phase transition point C 2 ≈ (0.793, 1.293). In addition, entanglement entropy shows that there is no hidden order in Phase I and Λ as it is near to 1.0 which can be fully explained by the broken Z 2 symmetry. We also see a signature of other phases when a is small and b is large (or vice versa), although we will overlook them as they are far from the parameter path we are interested in. We note that even though the phases in Fig. 2 are identified following the conventional Z 2 symmetry breaking theory, we will further restrict a symmetry class of the Hamiltonian when we discuss phase connectivity throughout the section as it provides a more consistent view. For example, we will consider that point O (located on a = b line which has the U (1) symmetry) is not phase connected to A (where the Hamiltonian obviously breaks the U (1) symmetry), whereas A and B are phase connected. Our definition of phase connectivity is also compatible with a modern definition of phases in one-dimensional systems [50][51][52]. Finally, we depict the regions of stoquasticity in Fig 2. The model can be made stoquastic by a local basis change in the bottom left and top right quadrants. Within this phase diagram, we run our vQMC simulations along two paths OA and AE. The path OA does not cross any phase or stoquasticity boundary, but it will show how a symmetry of the ground state affects the expressivity. On the other hand, the path AE crosses both the phase and stoquasticity boundaries thus will show how phase and stoquasticity transitions affect the vQMC. and (d) indicate the region where the model cannot be made stoquastic by a local rotation. We discuss the results for each path and phase below. As Phase Λ (located between C 1 and C 2 ) is disconnected from others, we examine this case separately at the end of the section. H tXY Z H tXY Z H ♦ tXY Z Annealing H ♦ tXYZ O A 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 E (b) N = 28, SR A B C 1 C 2 D E (d) N = 28, SR 1. Path OA (Phase I) As we have noted above, the ground state is a classical ferromagnet at location O. We observe that the RBM represents this state as expected. However, the error from the ER is getting larger as the parameter deviates from O. Since the Hamiltonian is always gapped along OA, this result shows that non-stoquasticity affects vQMC even though the path does not close a gap; thus it reveals the importance of symmetry in the RBM expressivity. With our symmetry sensitive definition of phase connectivity, the solubility indeed can be understood well as follows. First, the ground state at O is represented by the RBM using the original Hamiltonian H tXYZ as it is stoquastic at this point (Observation 1). However, as going to A breaks the U (1) symmetry and it cannot be transformed into a stoquastic form only using local Pauli gates (see Appendix C 2), point A is not guaranteed to be solvable using H tXYZ . On the other hand, one can solve it using H tXY Z which is stoquastic along the whole path OA. The fact that the transformed Hamiltonian H tXY Z works much better than the original one H tXY Z in the ER case contrasts to the XXZ and J 1 -J 2 models where local Pauli rotations barely affected the expressive power. This is because the transformations in this model involve the Hadamard gate which is known to be challenging for the RBM [24,46]. Path AC1 (Phase I) We observe that both the transformed Hamiltonians H tXYZ and H ♦ tXYZ work better than the original Hamiltonian H tXYZ along the whole path when the ER is used [ Fig. 4(c)]. Especially, the transformed Hamiltonians solve the ground state even for the shaded region (BC 1 ) where none of the Hamiltonians are stoquastic. We explain this using the fact that H tXYZ is stoquastic along AB and that there is no phase transition along AC 1 (Observation 1). In addition, as applying local phase gates k e −iπ/4σ z k which can be embedded into the RBM Ansatz transforms H ♦ tXYZ into a stoquastic form (which is different from H tXYZ ), Observation 1 also explains why H ♦ tXYZ works. In contrast, the original Hamiltonian H tXYZ is non-stoquastic for all parameters in AC 1 . On the other hand, the results from the SR [ Fig. 4(d)] show that H tXYZ which is stoquastic on the left side of the shaded region works better than H ♦ tXYZ . This result indicates that the MCMC is more stable when the stoquastic Hamiltonian is used, which is the behavior we have seen from the sign rule of the XXZ and the J 1 -J 2 models (Obervation 3). Interestingly, H ♦ tXY Z appears to be sensitive to the stoquastic transition although it is non-stoquastic throughout the path. We do not have a good explanation for this behavior. Path C2D (Phase II) We observe that H ♦ tYXZ which is stoquastic to the right of D does not give the best result in this region C 2 D when the ER is used. However, a large fluctuation in the converged energies suggests that the convergence problem [case (ii)] arises, likely due to a complex optimization landscape. Thus we need to distinguish the problem between optimization and expressivity more carefully in this region. For this purpose, we use an annealing approach as an alternative optimization method: We first take converged RBM weights for (a, b) = (1.01, 1.51) (the point right next to D) where the Hamiltonian H ♦ tYXZ is stoquastic and run the ER from these weights instead of randomly initialized ones. We decrease the parameters (a, b) of the Hamiltonian by (0.01, 0.01) for each annealing step and run 200 ER epochs. The obtained results are indicated by a dotted curve in Fig. 4(c). The annealing result suggests that the expressivity is not the main problem up to the phase transition point C 2 (when considered from the right to the left). The SR results show two noteworthy features compared to the ER results. First, the Hamiltonian H tXYZ gives remarkably poor converged energies compared to the results from the ER. This result agrees with what we have seen from the sign rule: When the Hamiltonian is non-stoquastic, the learning path may enter a region where observables are not correctly estimated. Second, the shape of the curves from the Hamiltonians H tXYZ and H ♦ tXYZ are also different from that of the ER which is due to poor optimization. However, in Appendix F, we show that the convergence problem gets weaker as N increases, thus the SR can solve the Hamiltonian H ♦ tXYZ in this region correctly for a large N by examining the two parameter points of the Hamiltonian (indicated by arrows in Fig. 4). We encapsulate the results in this region as follows: The vQMC works for H ♦ tYXZ that is phase connected to a stoquastic Hamiltonian even though it suffers from a optimization problem for small N . This result is consistent with Observation 1 and Observation 3. Path DE (Phase II) The ER results show that the RBM can express the ground state of all three Hamiltonians (H tXYZ , H tXYZ , and H ♦ tYXZ ) wherein the stoquastic one in this parameter region H ♦ tYXZ works the best. One can also explain why some instances of the ER find the ground state of H tXYZ using the existence of phase-shift gates ( k e −iπ/4σ z k ) that transforms the Hamiltonian into a stoquastic form. However, we do not have a nice explanation why nontransformed Hamiltonian H tXYZ also works in this region despite its non-stoquasticity. On the other hand, the SR results show that the converged energies from H tXYZ and H ♦ tXYZ suffer large fluctuations, which suggests that the sampling problem emerges. This result is unexpected as the Hamiltonian H ♦ tXYZ is stoquastic in this region. As the Hamiltonian is stoquastic, one may expect that using more samples easily overcomes the problem. However, this is not the case. To see this, we plot vQMC errors as a function of the number of samples in Fig. 5(a). Here, we have used x × \|θ\| samples per epoch for each value in the x-axis. For N = 20 and 24, one observes that the errors get smaller as the number of samples increase. However, the results are subject to large fluctuations for N = 28, and gets worse when N = 32. Since θ itself scales as ∼ αN 2 , our results show that this sampling problem is robust. It is insightful to compare the sampling problem in this model to that in non-stoquastic models (such as the XXZ model in the antiferromagnetic phase without the sign rule). Even though they are both caused by a finite number of samples, the converged energies suggest that they have different origins. In the XXZ model, the converged normalized energies are mostly clustered above 10 −2 regardless of the size of the system. On the other hand, they are below 10 −3 and show clear system size dependency in this model. In Appendix G, we show that the sampling problem in this model only appears locally near the minima whereas it pops up in the middle of optimizations and spoils the whole learning procedure in non-stoquastic models. The sampling problem occurring here is quite similar to the problem observed from quantum chemical Hamiltonians [53]. Precisely, Ref. [53] showed that optimizing the RBM below the Hartree-Fock energy for quantum chemical Hamiltonians requires a correct estimation of the tail distribution. However, the tail distribution of the ground state is often thick and a large number of samples are required to find the true optima. We find that the sampling problem of our model is also caused by such a heavy tail in the distribution. We can see this from the probability distribution of the ground states \|ψ GS (x)\| 2 for H ♦ tXYZ using two different parameters (a, b) = (0.27, 0.77) and (1.23, 1.73) which are deep in Phase I and II, respectively. We plot the first 10 3 largest elements of \|ψ GS (x)\| 2 for each parameter of the Hamiltonian in Fig. 5(b). The Figure directly illustrates that the distribution of the ground state in Phase II is much broader than that of Phase I. Moreover, the sum of the first 10 3 elements is only ≈ 0.294 in Phase II, which implies that one needs a huge number of samples to correctly estimate the probability distribution. Phase Λ Finally, we show that the RBM has difficulty representing the ground states in phase Λ even though it is a simple conventionally ordered phase (which was observed from the entanglement entropy). We simulate vQMC along the line JK in Fig. 6(a) which is deep in phase Λ. The transformed Hamiltonian H ♦ tXYZ gives almost the same converged energies as the original Hamiltonian H tXYZ , so we do not present them in Fig. 6(b) and (c). The ER results in Fig. 6(b) clearly show that the error increases as we go deeper in this phase. As in the J 1 -J 2 model, we simulate the ER with varying N and different numbers of hidden units at point K in Appendix H which confirms that there is the expressivity problem in this phase. In addition, the SR results [ Fig. 6(c)] show that the sampling problem also arises when we use H tXYZ which performed best with the ER. We also note that we cannot apply the results in Ref. [49] as in the J 1 -J 2 model, since there is no hidden orders in this phase. To summarize overall the results from the twisted XYZ model, we have found that Observation 1 and 2 hold in general by examining the behavior of the vQMC in different phases and stoquastic/non-stoquastic regions. In addition, we also have observed that a different type of sampling problem may arise even when solving a stoquastic Hamiltonian. Thus we modify our observation 3 slightly as follows: Observation 3 (Second version). Sampling is stable along the learning path when the Hamiltonian is stoquastic or phase connected to a stoquastic Hamiltonian. However, the number of samples required to converge may scale poorly even when the Hamiltonian is stoquastic. V. EXPRESSIVITY PROBLEM FROM SUPERVISED LEARNING We point out that observation 2 conflicts with the assertion [17] that a shallow neural network (with depth 2 or 3) can express the ground state of a frustrated system without problem. Precisely, the authors have trained a neural network to reproduce amplitudes and signs of the ground state obtained from the ED without imposing the sign rule and found that the reconstructed states give a high overlap with the true ground state; the statement "expressibility of the Ansätze is not an issue-we could achieve overlaps above 0.94 for all values of J 2 /J 1 " is given. However, we consider a 0.94 overlap to be insufficient evidence for this claim. For example, we have obtained an overlap between of 0.999 for the one dimensional J 1 -J 2 model when J 2 = 0.44 and N = 20 with a nonstoquastic ansatz, but an order of magnitude better with a stochastic one. In this section, we further clarify the discrepancy by showing that the expressivity problem appears even in the supervised learning set-up (as in Ref. [17]) that is less prone to other problems (sampling and training) when the desired accuracy gets higher. Notably, we show that, in contrast to what one might expect, a neural network (even after taking the symmetries into account) has a problem in reproducing amplitudes of a deep nonstoquastic ground state. A. Neural networks and learning algorithms We use a convolutional neural network (Table I) for the supervised learning experiment. Our network is invariant under translation, as the convolutional layers commute with translation and outputs are averaged over the lattice sites before being fed into the fully connected layer. We further embed the Z 2 symmetry in the network by turning off all biases and using even and odd activation functions after the first and second layers, respectively, following Refs. [19,54]. We introduce a parameter W that characterizes the width of the network. We further use θ to denote a vector of all parameters and f θ (x) for the output of the network. We note that our network structure is very close to a convolutional network used in Ref. [17]. We utilize this network to learn the amplitudes and signs of the true ground states obtained from the original Hamiltonian H J1−J2 (without imposing the sign rule). We use the kernel size = N/2 + 1 for the convolutional layers as smaller kernels have failed to reproduce the sign rule for J 2 = 0 (when the sign rule is correct for all configurations). The number of parameters of the network is then given by (W/2 + W 2 /2)( N/2 + 1) + W . For N = 24, the networks with W = 16 and 32 have 1, 784 and 6, 896 parameters respectively. Our learning set-up is slightly different from that of Ref. [17]: (i) Instead of mimicking the amplitudes, we use our network as an energy based model to be sampled from and (ii) we train our network using the whole data (all possible configurations x) without dividing the training and validation sets, as we are only interested in expressivity, not generalization property of the network. Learning amplitudes To model the amplitudes \|ψ GS (x)\| 2 , we use the output of our network f θ (x) as the energy function for the energy based model. Even though there are models with the autoregressive property, which are easier to train, we choose the energy based model as it does not add any additional constraints (such as ordering of sites) and symmetries can be naturally imposed. Precisely, we model the amplitudes with p θ (x) = e f θ (x) /Z where Z = x e f θ (x) is the partition function of the model. We note that even though Z is intractable in general, we can compute Z for system sizes up to N = 28 rather easily thanks to the symmetry imposed on the network. Our loss function is the cross entropy l(θ) = − x p data (x) log[e f θ (x) /Z] where the probability distribution for data we want to model captures the amplitudes of the ground state: p data (x) = \|ψ GS (x)\| 2 . The gradient of the loss function can be estimated using samples from the data and model distributions as g ≈ −{E p data (x) [∇ θ f θ (x)] − E p θ (x) [∇ θ f θ (x)]}.(12) For the energy based model, estimating the second term is difficult in general as we need to sample from the model using e.g. MCMC. However, we here sample exactly from p θ which is again possible up to N = 28. We use the same number of samples (the mini-batch size) from p data (x) and p θ (x) to estimate the first and the second terms. Unfortunately, we have found that usual first-order optimization algorithms such as Adam [42] do not give a proper minima due to a singularity of the ground state distribution [43] (see also discussions in Ref. [55]). Thus we have utilized the natural gradient descent [56] (which can be regarded as a classical version of the SR) to optimize our energy based model, which is tractable up to several thousands of parameters. Precisely, we compute the classical Fisher matrix F = (F ij ) where F ij = E p θ (x) [∂ θi f θ (x)∂ θj f θ (x)] − E p θ (x) [∂ θi f θ (x)]E p θ (x) [∂ θj f θ (x)](13) each epoch and update parameters θ as θ n+1 = θ n − η n (F n + λ n 1) −1 g n . Here, η n and λ n are the learning rate and the (epoch dependent) regularization constant, respectively. We also use a momentum for the gradient g and the Fisher matrix F to stabilize the learning procedure, i.e. g n = β 1 g n−1 + (1 − β 1 )g and F n = β 2 F n−1 + (1 − β 2 )F. To quantify the expressivity of the network, we record the overlap between reconstructed and the true ground states (assuming that the sign is correct) over the training, which can be expressed as ψ GS \|ψ recon = x \|ψ GS (x)\|e f θ (x)/2 / √ Z. As the network obeys the same symmetry as the ground state, this quantity can be also calculated only by summing over the symmetric configurations. Learning signs We use the same network to model the sign structure. As the problem nicely fits into the binary classification problem, we feed the output of the network into the sigmoid function and use it to model the sign structure, i.e., we use P [ψ GS (x) > 0] = Sigmoid(f θ (x)). We then optimize the network by minimizing the binary cross entropy l(θ) = − x p data (x) y data (x) log[Sigmoid(f θ (x))] + (1 − y data (x)) log[1 − Sigmoid(f θ (x))](14) where y data (x) = 1 when ψ GS (x) > 0 and y data (x) = 0 otherwise. In practice, we estimate the loss function and its gradient using samples from p data (x). We have found that usual first-order optimizers such as Adam properly find optima in this case. We also compute the overlap between the true ground state and the reconstructed quantum state \|ψ recon = x sgn[f θ (x)]\|ψ GS (x)\| \|x , where sgn(x) is the sign function. B. Results We show the converged infidelity 1− ψ GS \|ψ recon 2 from neural networks trained for the amplitudes and signs in Fig. 7. The results are obtained after tuning hyperparameters and initializations the detail of which can be found in Appendix I. The results show that the infidelity from a "deep non-stoquastic" phase is larger than that of stoquastic case both for Fig. 7(a) and (b) where we have trained the amplitudes and signs, respectively. However, the errors go up to ≈ 1.95×10 −4 even for W = 32 (where the number of parameters is 6, 896) when we train the amplitudes ( assuming the signs are correct) whereas the typical errors are smaller than 10 −5 when we do the opposite. Thus our result strongly suggests that learning amplitudes is more difficult for a neural network than learning the sign structure. As we sampled from the distribution exactly and we expect that the learning procedure is more reliable in the supervised learning set-up, we conclude that this is due to lack of expressivity [case (iii)] of a neural network. We also note that solving the linear equation (F n + λ n )v = g n , which requires O(\|θ\| 2−3 ) operations, dominates the learning cost. This is the main limiting factor that prevents us from using a bigger network. We further plot scaling of errors for different sizes of the system using J 2 = 0.0 and 0.4 in Fig. 8(a). We see that both errors from the amplitude and the sign networks increase with N regardless of J 2 . We also see that 16 20 24 28 N the errors from the amplitude network are significant up to the system size N = 28, but the slope of the sign network is slightly sharper. We still leave a detailed scaling behavior for future work as our simulation results here are limited up to the system size N = 28. We also show initial learning curves from the sign network in Fig. 8(b). Consistent with a previous observation [54], learning the sign structure takes some initial warm-up time when the sign rule is not used. We conjecture that this is also related to the generalization property observed in Ref. [17]. In Appendix I 4, we further show that imposing the sign rule makes the initial warmup time disappear but does not change the overall performance. We note that, as we can use the first order optimization algorithms, increasing the size of network as well as training longer epochs are much easier for the sign networks than for the amplitude networks. We thus believe that increasing the system size while maintaining the accuracy in the supervised learning set-up is mostly obstructed by the amplitude networks (under the assumption that the ED results are provided). Even though the difficulty of the amplitudes seems counter-intuitive, it may not be surprising if one recalls the path integral Monte Carlo. The sign problem in the path integral Monte Carlo implies that estimating the expectation values of observables Tr[Ae −βH ]/ Tr[e −βH ] is difficult due to negative weights. As the amplitudes \|ψ GS (x)\| 2 are the expectation value of the observable A = \|x x\| when β → ∞, the amplitudes \|ψ GS (x)\| 2 suffer from the sign problem when H is non-stoquastic. Still, this does not explain why learning amplitudes is difficult even for a supervised learning set-up where we already have values \|ψ GS (x)\| 2 . A partial answer to this question can be found in Ref. [24]. Under common assumptions of the computational complexity (that the polynomial hierarchy does not collapse), Ref. [24] showed that computing the coefficients in the computational ba-sis of a certain quantum state [Ψ GWD (x)] is impossible in a polynomial time even with an exponential time of precomputation (that may include training and computing the normalization constant of a neural network representation). As their argument relies on the difficulty of computing \|Ψ GWD (x)\| 2 , the complexity is, in fact, from the learning amplitudes of quantum states. However, as they only considered a specific type of quantum states in 2D and do not give any argument on how errors scale (unlike the theory of DMRG which gives an upper bound of errors in terms of entanglement), further theoretical developments are essential to fully explain our results. VI. CONCLUSION By way of example, we have classified the failure mechanism of the RBM Ansatz within the vQMC framework for one-dimensional spin systems. In particular, we have observed the following features of RBM variational Monte-Carlo for a class of one-dimentional XYZtype Hamiltonians which exhibit a wide variety of stoquastic and non-stoquastic phases: 1. Complex RBMs with a constant hidden layer density (α) faithfully represent the ground states of spin Hamiltonians that are phase connected to a stoquastic representation, or that can be transformed into such a Hamiltonian with local Pauli and phase-shift transformation. 2. There exists "deep non-stoquastic phases" that cannot be transformed into a stoquastic form using local (on-site) unitaries and are not phase connected to stoquastic Hamiltonians, and which cannot be efficiently represented by complex RBMs. 3. Sampling is stable along the learning path when the Hamiltonian is stoquastic or phase connected to a stoquastic Hamiltonian. However, required number of samples to obtain true optima may scale poorly even in this case when the ground state distribution is heavy-tailed. Most importantly, our observation 1 provides strong evidence suggesting that the RBM Ansatz can faithfully express the ground state of a non-stoquastic Hamiltonian when it is phase connected to a stoquastic Hamiltonian. This observation implies that it may be possible to solve a large number of non-stoquastic Hamiltonians using the RBM Ansatz, significantly expanding the reach of vQMC. On the other hand, the second observation suggests that a fundamental difficulty may exist when solving a Hamiltoninan within a phase that is separated from any stoquastic Hamiltonian. As studies already have found several phases that cannot be annealed into stoquastic Hamiltonians [57][58][59], we expect that such systems are challenging for neural quantum states. Moreover, by carefully extending the supervised learning set-up introduced in Refs. [17,54], we have further demonstrated that the difficulty in representing quantum states using a neural network is originated not only from their sign structure but also from the amplitudes. Even though this may sound counter-intuitive, but our result is consistent with that from the computational complexity theory [24]. Nevertheless, there is a caveat on our "deep nonstoquastic phases" as they rely rather on a conventional concept of phases (phase connectivity is restricted within a given parameterized Hamiltonian). In contrast, a modern language of one-dimensional phases allows any constant-depth local unitary transformations that preserve a symmetry between ground states, i.e. two Hamiltonians with the ground states \|ψ 1 and \|ψ 2 are in the same phase if there is a symmetry preserving unitary operator U sym such that \|ψ 1 = U sym \|ψ 2 and can be decomposed into a constant depth circuit consists of local gates [50][51][52]. One may possibly interpret our results using symmetry protected phases. Recently, Ref. [49] reported results on a related problem when a ground state can be transformed into a positive form under a unitary operator with a certain symmetry. However, as the result there is limited to ground states with hidden orders whereas our results suggest that ground states in phase Λ of the twisted XYZ model suffer from the sign problem even though it is conventionally ordered, a further study is required to understand how phases of a many-body system interplay with a stoquasticity more deeply. We also note that the neural networks we have used in this paper have ≤ 10 4 parameters. Although this value is comparable to neural quantum states considered for the vQMC, modern machine learning applications use neural networks with several millions to billions of parameters. We open a possibility that such a huge network may sufficiently mitigate the expressivity problem we have observed in this paper. However, the main obstacle in using huge networks for the vQMC is the computational overhead of the SR which requires O(\|θ\| 2−3 ) of operations for each step. Thus a better optimization algorithm would be required. Finally, we also have found that the number of samples to solve the ground state may scale poorly even when the system is stoquastic. This type of difficulty is known from quantum chemical systems [53] but has not been discussed in solving many-body Hamiltonians. We still do not exclude the possibility that an efficient sampling algorithm that converges the network in a reasonable number of epochs may exist even in this case. ACKNOWLEDGEMENT The authors thank Prof. Simon Trebst, Dr. Ciarán Hickey, and Dr. Markus Schmitt for helpful discussions. This project was funded by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy -Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 -390534769 and within the CRC network TR 183 (project grant 277101999) as part of project B01. The numerical simulations were performed on the JUWELS and JUWELS Booster clusters at the Forschungszentrum Juelich. This work presented in the manuscript was completed while both authors were at the University of Cologne. Source code used in this paper is available at Ref. [60]. N i=1 a i x i + M j=1 log[cosh(χ j )] where χ j = i w ij x i +b j , we can write ψ θ (x) = e a·x j 2 cosh(χ j ) = 2 M e E(x;θ) (A2) which follows from Eq. 2. Then the update probability is simply written as \|ψ θ (x )/ψ θ (x)\| 2 = e 2 [E(x ;θ)−E(x;θ)] . We call N sequential MCMC steps a "sweep". We also use this equation to generate a Markov chain with a given temperature β that is for the probability distribution given as P (x) ∝ \|ψ θ (x)\| 2β . One only needs to change the update probability to e 2β [E(x ;θ)−E(x;θ)] . We use total 16 chains with the equal spaced inverse temperatures β = 1/16 to 1 for parallel tempering. Let i-th chain (i ∈ [1, · · · , 16] ) has β = i/16 and the configuration x (i) . After sweeping all chains, we mix them as follows: For all odd i ∈ [1, 3, · · · , 15], exchange x (i) and x (i+1) with probability p accept = min 1, exp 2(β i − β i+1 )× [E(x (i) ; θ) − E(x (i+1) ; θ)](A3) and do the same for all even i ∈ [2, 4, · · · , 14]. For the XXZ and the J 1 -J 2 model in Fig. 1, we have used \|θ\| (the number of parameters of the RBM) number of samples per each epoch. For Fig. 4, we have used \|θ\| number of samples for phase I whereas 16\|θ\| number of samples is used for phase Λ and II (see also Fig. 5). Hyperparameters We use the the SR to train the complex RBMs (see also Sec. II). We typically use the learning rate η = 0.02 and the regularization constant λ n = max{1.0 × 0.9 n , 10 −3 }. For Phase II of the twisted-XYZ model, we use running averages of the gradient and the Fisher matrix with β 1 = β 2 = 0.9 to update parameters as it is slightly better performing than the SR without momentums. We train the network for 2500 − 3000 epochs where we always have observed convergence besides Phase II of the twisted XYZ model (see Appendix G). Appendix B: The XXZ model with exact sampler We have shown in the main text that the XXZ model without the sign rule suffers from the sampling problem in the antiferromagnetic phase ∆ > 1.0. In this Appendix, we use the exact sampler to show that this is not caused by temporal correlations of the MCMC. Our exact sampler works as follows. For a given domain of configurations D = {x i } (thus \|D\| = 2 N or N N/2 depending on the symmetry of the system), we first compute and save p(x) = \|ψ θ (x)\| 2 for all x ∈ D. After this precomputation (the complexity of which is O(\|D\|)), each sample is taken by finding k such that i<k p(x i ) < p and i≤k p(x i ) > p where p is taken from the uniform distribution of U [0,1] . Then x k is a valid sample from the distribution. As we implement this procedure with a binary search, time complexity of obtaining each sample is O(log \|D\|) ∼ N . Then one may train the RBM using the SR with samples drawn from the exact sampler. We note that the ER can be regarded as a limiting case of the SR with the exact sampler where the number of samples per each epoch becomes infinite. Using the XXZ model without the sign rule where the sampling problem appeared, we compare the converged normalized energies from the SR with the usual MCMC and the exact sampler in Fig. 9. The size of the system N = 20 and ∆ = 1.5 are used. The results clearly demonstrate that the SR with the exact sampler is not particularly better than the SR with MCMC, which confirms that the poor converged energies from the SR is not from the ergodicity issue. Appendix C: Non-stoquasticity of the the Heisenberg XYZ models In this Appendix, we present an algorithm introduced by Klassen and Terhal [34] that determines (non-)stoquasticity of spin chains, and apply it to the J 1 -J 2 and twisted XYZ models studied in the main text. First, we interpret the XYZ type Hamiltonian as a graph with matrix-valued edges. For each Hamil- 10. Graph with matrix-valued edges describing translational invariant models with the nearest-neighbor and the next-nearest-neighbor couplings. β 1 β 1 β 1 β 1 β 1 β 1 β 1 β 1 β 1 β 1 β 2 β 2 β 2 β 2 β 2 β 2 β 2 β 2 β 2 β 2 FIG. tonian term between vertices (i, j) given by H i,j = β x ij σ x i σ x j + β y ij σ y i σ y j + β z ij σ z i σ z j we assign a matrix β ij = diag(β x ij , β y ij , β z ij ) to edge (i, j) . For a many-body spin-1/2 Hamiltonian consisting of two-site terms, it is known that the Hamiltonian is stoquastic if and only if all terms are also stoquastic. Using the matrix β, the term H i,j is stoquastic when β x ij ≤ −\|β y ij \|. In addition, if there is a local (on-site) basis transformation that converts the Hamiltonian into a stoquastic form, it must act as signed permutations to each vertex, i.e. trans- form the matrix β ij to β ij = Π i β ij Π T j where Π i = R i Π i , R i = diag(±1, ±1, ±1 ) is a sign matrix and Π i ∈ S 3 is a permutation (Lemma 22 of Ref. [34]). One may note that for signed permutations, the transformed Hamiltonian is still XYZ-type ( β ij are diagonal). Then we can formally write the problem as: find a set of signed permu- tations {Π i = R i Π i } that makes the transformed matrix β ij = Π i β ij Π T j satisfy β x ij ≤ −\| β y ij \| for all edges (i, j) . The algorithm solves this by separating the permutation and the sign parts. First, check whether there is a set of permutations {Π i ∈ S 3 } that make \| β x ij \| ≥ \| β y ij \|. This step can be done efficiently utilizing the fact that permutations Π i and Π j must be the same if the rank of β ij is ≥ 2 (Lemma 23 of Ref. [34]). 2. Second, for possible permutations obtained from the above step, check if there is a possible set of signs {R i } that makes β x ij negative. This is reduced to a problem of deciding whether to apply diag(−1, 1, 1) or not to each vertex (Lemma 27 of Ref. [34]), which is equivalent to solving the frustration condition of classical Ising models that can be solved in a polynomial time. We refer to the original work Ref. [34] for detailed steps. We also note that the algorithm finds a solution if and only if such a transformation exists. We depict the interaction graph for translational invariant models with the next-nearest-neighbor couplings in Fig. 10. For this type of graph, the second problem (finding the possible signs) is much easier to solve. In our graph, applying diag(−1, 1, 1) to all even sites flips the sign of β x 1 . However, such a sign rule does not exist for β x 2 . Thus after the permutation, β x 2 must be negative whereas the sign of β x 1 for the nearest-neighbor couplings −2 0 2 a −2 0 2 b J 2 > 0 −2 0 2 a −2 0 2 b J 2 < 0 FIG. 11. Parameter regions that the twisted XYZ model can be transformed into a stoquastic form by on-site unitary gates. is free as we can always make it negative by applying this sign rule. Non-stoquasticity of the J1-J2 model We can directly apply the algorithm described above to the J 1 -J 2 model. For the J 1 -J 2 model we have studied in the main text, we have β matrices: β 1 = J 1   1 0 0 0 1 0 0 0 1   , β 2 = J 2   1 0 0 0 1 0 0 0 1   . (C1) As a permutation does not change the matrices, we only need to consider the sign rule which readily implies the Hamiltonian is stoquastic only when J 2 ≤ 0. Still, it is known that the J 1 -J 2 model can be transformed into a stoquastic form using two-qubit operations [62]. In addition to the non-stoquasticity of the Hamiltonian, one can further prove that ground states in the gapped phase of this model (when J 2 > J * 2 ) indeed cannot be transformed into a positive form easily. Reference [49] states that a phase that is short-ranged entangled and has a string order suffers a symmetry protected sign problem -the transformed ground state in the computational basis cannot be positive (there exists x such that x\|U sym \|ψ GS / ∈ R ≥0 ) for all symmetry protecting unitary operators U sym . We can directly apply this result to our case as the phase of the J 1 -J 2 model when J 2 > J * 2 satisfies this condition (see e.g. Ref. [63]). As the symmetric group is G = SO(3) in the J 1 -J 2 model, this implies that a translational invariant gate U ⊗N which transforms the ground state into a positive form does not exist. This further supports a relation between the expressive power of the RBM and stoquasticity of the Hamiltonian. Interestingly, at the Majumdar-Ghosh point (J 2 = 0.5), a non-translational invariant unitary gate i σ z 2i transforms the ground state into a positive form yet the transformed Hamiltonian is still non-stoquastic. Non-stoquasticity of the twisted XYZ model We apply the same algorithm to the twisted XYZ model and determine the parameter regions where the twisted XYZ model can be locally transformed into stoquastic form. For the twisted XYZ model, the β matrices are given as β 1 = J 1   a 0 0 0 b 0 0 0 1   , β 2 = J 2   b 0 0 0 a 0 0 0 1   . (C2) The model is trivially stoquastic when a = b = 0. If any of a or b is non-zero, we see the ranks of the matrices are ≥ 2. Thus all permutations acting on each vertex must be equal, i.e. Π i = Π. After this simplification, we can just apply each element Π ∈ S 3 and see whether there is a set of sign matrices that satisfies the condition. For our model, it is not difficult to obtain the condi- tions below: Π = () \|a\| = \|b\| ∩ J 2 b ≤ 0 (C3) Π = (1, 2) \|b\| = \|a\| ∩ J 2 a ≤ 0 (C4) Π = (2, 3) \|a\| ≥ 1 ∩ \|b\| ≥ 1 ∩ J 2 b ≤ 0 (C5) Π = (1, 3) 1 ≥ \|b\| ∩ 1 ≥ \|a\| ∩ J 2 ≤ 0 (C6) Π = (1, 2, 3) 1 ≥ \|a\| ∩ 1 ≥ \|b\| ∩ J 2 ≤ 0 (C7) Π = (1, 3, 2) \|b\| ≥ 1 ∩ \|a\| ≥ 1 ∩ J 2 a ≤ 0 (C8) which is depicted in Fig. 11. When J 1 = J 2 = −1 and a, b ≥ 0, that we considered in the main text, we see that Π = (2, 3) and Π = (1, 3) are the transformations that make the Hamiltonian stoquastic for a, b ≥ 1 and a, b ≤ 1, respectively. As the elements of the transformed matrices β 1 = Πβ 1 Π T , β 2 = Πβ 2 Π T are already negative, additional application of the sign rule is not required. For J 2 = 0.20 and 0.44, we plot converged normalized energies for different system sizes N and values of α = M/N in Fig. 12. Figure 12(a) shows that both the ER and the SR find the ground state when J 2 = 0.2. We also observe the MCMC samples bias the energy a little toward the ground state and the energy and the statistical fluctuation get larger as N increases in the SR case. However, one can easily deal with such a problem by carefully choosing the sample size and the number of sweeps between samples. In contrast, we see that the expressivity problem arises when J 2 = 0.44, which is more fundamental. In addition, even though the errors seem to remain constant with N for the SR, we expect the obtained state to be moving away from the true ground state as N increases because the normalized energy of the first excited state scales as ∼ 1/N . We additionally study how the error scales with the number of hidden units in Fig. 12(b). The results clearly show that increasing α = M/N only marginally reduces the error. We also note that the system at J 2 = 0.44 is gapped thus the error from α = 8 is still far from the error we expect from other methods e.g. density matrix renormalization group. This supports our argument that the RBM has a difficulty in representing the ground state of Hamiltonians that is deep in non-stoquastic phase. Appendix E: Phases of the twisted XYZ model In the main text, we summarized the phases of the twisted XYZ model. Here, we investigate the phases more closely and locate phase transition points along the path AE. Let us first consider the case when a = b that recovers the XXZ model with the next-nearest-neighbor couplings. After setting J 1 = J 2 = −1, the Hamiltonian becomes H tXY Z = i −(Cσ x i σ x i+1 + Cσ y i σ y i+1 + σ z i σ z i+1 ) − (Cσ x i σ x i+2 + Cσ y i σ y i+2 + σ z i σ z i+2 ) (E1) where we set a = b = C. Applying Pauli-Z gates to all even sites (the sign rule) yields H * tXY Z = C i (σ x i σ x i+1 + σ y i σ y i+1 − 1/Cσ z i σ z i+1 ) − (σ x i σ x i+2 + σ y i σ y i+2 + 1/Cσ z i σ z i+2 ). (E2) The Hamiltonian has U (1) symmetry, so it preserves the total magnetization J z = i σ z i . When C < 1, it is not difficult to see that the nearest-neighbor and the nextnearest-neighbor couplings both prefer the aligned states, which suggests that \|0 ⊗N and \|1 ⊗N are the degenerated ground sates. This implies that the Z 2 symmetry along the z-axis is broken. In contrast, the Hamiltonian with C > 1 prefers J z = 0 thus Z 2 symmetry σ z ↔ −σ z is restored. Fig. 2 in the main text shows how the phases extend off a = b. Especially, there are two phase transitions along Fig. 6 of the main text. As path P 1 is a subset of AE we simulated the vQMC, we can locate the point C 2 in Fig. 2 using the result. Path P 2 is additionally considered to confirm that our phase diagram Fig. 2 indeed captures phase Λ correctly. To be precise, we set (a, b) = (0.7, 1.2)+θ(1, 1) for P 1 and (a, b) = (0.55, 1.45) + θ(1, 1) for P 2 . The results from P 1 and P 2 are shown in Fig. 13(a) and (b), respectively. We see that there are two local maxima in the second derivatives along each path and the distance between two maxima increases as the path is moving away from the a = b = 1 point. This confirms that an intermediate phase Λ depicted in Fig. 2 is correct. We additionally confirm some other properties of the model using the DMRG with the open boundary condi-tion. In the main text, we have argued that the phase transition at C 1 is the second order. We calculate the entanglement entropy at C 1 by taking the entropy of a subsystem after dividing the system of size N to two equalsized subsystems A and B in Fig. 14(a). As we have used the periodic and open boundary conditions for the ED and DMRG, respectively, the coefficients 0.367 ∼ 1/3 and 0.152 ∼ 1/6 are close to what one expects from the conformal field theory [64]. We also calculate the spin-spin correlation along the y-axis C yy (k) = N i=1 σ i y σ i+k y /N at point E = (1.25, 1.75) of the twisted-XYZ model in Fig. 14(b). The result supports that the correlation function decays polynomially in Phase II. Appendix F: Scaling of converged energies in Phase II of the twisted XYZ model In the main text, the converged energies along the path C 2 D with the ER and the SR have shown a different shape when H ♦ tXYZ is used. In this Appendix we show that this is a finite size effect. To study this case more carefully, we simulate the ER for system sizes N = [12,16,20] and the SR for system sizes N = [16,20,24,28,32] using the two points indicated by arrows in Fig. 4 which are (a, b) = (0.81, 1.31) and (0.89, 1.39). The results in Fig. 15 show that converged energies from the SR decrease exponentially with N which suggests that the optimizing problem vanishes for large N and we can solve the Hamiltonian using the SR. One possible explanation for how such a good convergence is obtained for large N is overparametrization. In classical ML, it is known that overparamterized networks optimize better [65]. Likewise, if the ground state is already sufficiently described by a small number of parameters, we expect that one can obtain a better convergence by increasing the network's parameter. As the number of parameters of our network increases quadratically as ∼ αN 2 , this scenario is plausible if the number of parameters to describe the ground state scales slower than this. However, we leave a detailed investigation of this conjecture for future work. the twisted XYZ model in Phase II using the stoquastic version of the Hamiltonian. However, the converged energies in this case were much closer to the ground state energy ( E < 10 −3 for all instances) and increasing the number of samples helped for N ≤ 28. In this Appendix, we compare the learning curves in both models and show that the problems indeed have different profiles. For comparison, we use the XXZ model with ∆ = 1.5 and the twisted XYZ model with (a, b) = (1.23, 1.73). We plot 4 randomly chosen learning curves when N = 28 and N = 32 for both models in Fig. 16. One can easily distinguish the learning curves from the XXZ model without the sign rule Fig. 16(a), (b) and the twisted XYZ model with heavy tail distributions (c), (d). Specifically, Fig. 16(a) and (b) clearly show that the sampling problem enters in the middle of learning process and ruins the learning process when we use a non-stoquastic basis. In contrast, the learning curves shown in Fig. 16(c) and (d) first approach local minima that is nearẼ ≈ 0.5 × 10 −3 and stay there for more than 1000 epochs. After that, some of the instances succeed in converging to better minima when N = 28. This behavior has also been seen in quantum chemical Hamiltonians [53]. These examples show that the origin of the two sampling problems are crucially different. Choosing hyperparameters a. Hyperparameters for the amplitude networks We have used the natural gradient descent to train the amplitude networks. Hyperparameters are the learning rate η, the momentum for the gradient β 1 , the momentum for the Fisher matrix β 2 , and the mini-batch size. For all simulations, we have fixed β 1 = 0.9 as this is the default value for most of optimization algorithms implemented in major machine learning frameworks (e.g. TensorFlow, PyTorch, and JAX). Using the one dimensional J 1 -J 2 model with the system size N = 24 and J 2 ∈ [0.2, 0.44], we test the neural networks with width W ∈ [16,32] and different learning rate η, β 2 ∈ [0.999, 0.9999], and the mini-batch size ∈ [256, 512, 1024] and plot the converged infidelities in Fig. 18. Regardless of the mini-batch size, the neural network have seen ≈ 1.57×10 8 datapoints (the configuration x and the corresponding amplitude \|ψ GS (x)\| 2 ) where we have seen a sufficient convergence behavior. For example, we have trained the network for 153, 600 epochs when the mini-batch size is 1024. Based on this result, we choose the mini-batch size 1024, η = 1.0 × 10 −4 , β 2 = 0.999 for plots in the main text. b. Hyperparameters for the sign networks We have observed that Adam [42] already works great for training the sign structure, as in usual supervised learning processes in the classical ML set-ups (e.g. image classifications). Adam optimizer has the learning rate (η), momentum for the gradient and the gradient square (β 1 and β 2 ), and the constant for numerical stability ( ) as hyperparameters. As the default values for β 1 , β 2 and suggested in the original paper [42] work well for most of applications, we only tested the performance of Adam for varying learning rates η. For mini-batch sizes ∈ [32, 64, 128], we plot the converged infidelities using the one dimensional J 1 -J 2 model in Fig. 19. Regardless of the mini-batch sizes, we feed ≈ 1.31×10 8 datapoints. Based on this result, we choose the mini-batch size 32 and the learning rate η = 2.5 × 10 −5 , which is the best performing for W = 16 and the second best for W = 32, for plots in the main text. Imposing the sign rule In the main text, we have trained neural networks to reproduce the sign structure of the one dimensional J 1 -J 2 model without imposing the sign rule. Here, we train network to reproduce the sign structure when the sign rule is imposed and compare the performance. We show initial learning curves and converged infidelities in Fig. 20. We see that initial warm-up stage of the learning disappears when the sign rule is imposed [ Fig. 20(b)], but the converged infidelities are barely affected. This resembles the vQMC results with the complex RBM where imposing the sign rule does not improve the converged energies. FIG. 1 . 1Converged normalized energy E = (ERBM − EED)/EED as a function of model parameters for (a,b) the Heisenberg-XXZ and (c,d) J1-J2 model. For each model, the upper plots [(a) and (c)] present results for system size N = 20 with the Exact Reconfiguration method that optimizes the parameters using explicitly constructed wavefunctions from the RBM. The lower plots [(b) and (d)] show results from N = 32 with Stochastic Reconfiguration with Markov chain Monte Carlo sampling for optimization. The orange diamonds indicate simulation of the models in the original non-stoquastic basis, while the blue dots indicate simulations in the modified basis, after applying the Pauli-Z operator on every even sites (the sign rule). Vertical dashed lines indicate where the KT-transitions take place (∆ = 1.0 for the XXZ model and J2 ≈ 0.2411 for the J1-J2 model). For each value of the parameters, we have run the simulation 12 times and each point represents the result from a single run. FIG. 3 . 3(a) Magnetic susceptibility and (b) entanglement entropy for the ground state of the twisted XYZ model along the line A = (0.25, 0.75) to E = (1.25, 1.75). The result shows that there are three distinct phases. We locate the first phase transition point C1 ≈ (0.7636, 1.636) that shows the divergence of entanglement entropy. The maximum values of the entanglement entropy are 2.16, 2.25, 2.28 for N = 20, 24, and 28, corroborating a logarithmic divergence of the entanglement entropy at criticality. In addition, we also observe polynomial decay of the correlation function σ y i σ y i+k in Phase II (see Appendix E for details). B. Variational Quantum Monte Carlo resultsOur vQMC results for the twisted XYZ Hamiltonian are shown inFig. 4. The shades in the middle ofFig. 4(c) FIG. 4 . 4Normalized energy from the vQMC for the twisted XYZ model. We have used the ER with N = 20 for (a) and (c), the SR with N = 28 for (b) and (d). (a,b) For OA, the original Hamiltonian HtXYZ is only stoquastic at O whereas H tXYZ is stoquastic over the whole path. (c,d) The Hamiltonian HtXYZ is non-stoquastic over the whole path whereas H tXYZ and H ♦ tXYZ are stoquastic on the left and right side of the shaded region, respectively. In the shaded region, none of the Hamiltonians is stoquastic. Vertical dashed lines at C1 and C2 indicate the phase transition points. A dashed curve in (c) indicates an annealing result (see main text). FIG. 5 . 5(a) Normalized energy from the vQMC after convergence as a function of number of samples for different system sizes N . The Hamiltonian H ♦ tXYZ with (a, b) = (1.23, 1.73) is simulated. Values in the x-axis indicate the number of samples used to estimate observables in each update step of the SR divided by \|θ\|. The result is averaged over 12 vQMC instances and error bars indicate the standard deviation. Error bars for N = 32 are invisible as they are less than 2.0 × 10 −6 . The result shows there is a transition in scaling near N = 28. (b) The first 10 3 elements of \|ψGS(x)\| 2 where ψGS(x) is the ground state of H ♦ tXYZ obtained from the ED when N = 28. Parameters (a, b) = (0.27, 0.77) in Phase I and (1.23, 1.73) in Phase II are used. When (a, b) = (0.27, 0.77), the peak at the beginning indicate two largest elements of the distribution which are ≈ 0.458. We see that the tail distribution for Phase II is much thicker. Moreover, the summation of the first 10 3 elements is ≈ 0.998 when (a, b) = (0.27, 0.77) whereas it is only ≈ 0.294 when (a, b) = (1.23, 1.73). It suggest that one needs a huge number of samples to correctly estimate the probability distribution in Phase II. FIG. 6 . 6(a) The second quadrant of the phase diagram in Fig. 2. We calculate the second derivative of the ground state energy along paths P1 and P2 in Appendix E to locate the phase transition points. Converged normalized energy using (b) the ER with N = 20 and (c) the SR with N = 28 along the path JK where J = (0.5, 2.0) and K = (0.4, 3.0). FIG. 7 . 7Infidelity 1 − F = 1 − ψGS\|ψrecon 2 between the true ground states and reconstructed states as a function of J2 for the one-dimensional J1-J2 model is shown. We train neural networks to reproduce (a) the amplitudes and (b) the signs of the ground states. The systems size N = 24 and the widths of network W = 16 and 32 are used. To train the amplitude network, the natural gradient descent with hyperparameters η = 10 −4 , β1 = 0.9, β2 = 0.999 and the mini-batch size 1024 are used. For the sign network, we use Adam optimizer with the learning rate η = 2.5 × 10 −5 , the mini-batch size 32 (see Appendix I for details). FIG. 8 . 8For neural networks with W = 32 and chosen hyperparameters (see Appendix I for details), we plot (a) scaling of the converged infidelities for different sizes of system N =[16,20,24,28] and (b) initial learning curves (results from the first 2 × 10 5 epochs whereas we have trained the network in total ≈ 4.10 × 10 6 epochs) from the sign network with different N . FIG. 9 . 9For the XXZ model with ∆ = 1.5 and the system size N = 20, we show converged normalized energies of the complex RBM wavefunctions with α = M/N = 3 when the SR is used with different sampling methods (MCMC and the exact sampler). Values of the x-axis indicate the number of samples we have used to train the RBM for each epoch divided by the number of parameters \|θ\| = α(N + 1) + αN 2 . For each sampling method and the number of samples, we have run 12 randomly initialized instances. The shaded region (near E ≈ 10 −5 ) indicate the range of converged energies form the ER. FIG. 12. (a) Converged normalized energies E = (ERBM − EED)/EED from the ER and the SR as a function of different system size N for the J1-J2 model with J2 = 0.2 and 0.44. We applied the sign rule and the ratio between hidden and visible units α = 3 is used. (b) Converged normalized energies as a function of the number of hidden units α = M/N when N = 20. of the ground state energies along two paths (a) P1 and (b) P2 depicted in Fig. 6 of the main text. Path P1 is from (a, b) = (0.7, 1.2) to (0.9, 1.4) that is a part of AE in Fig. 2 that we simulated vQMC. Path P2 is from (0.55, 1.45) to (0.75, 1.65). FIG. 14. (a) Scaling of entanglement entropy at C1 from the ED (periodic boundary condition) up to N = 28 and the DMRG (open boundary condition) up to N = 48 and (b) the spin-spin correlation along the y-axis as a function of distance k at E = (1.25, 1.75) from the ED with N = 28 and the DMRG with N = 48. Entanglement entropy is obtained after dividing the system into two equal sized subsystems A and B. Thus ρA = TrB[\|GS GS\|] and S(ρA) = − Tr[ρA log(ρA)]where \|GS is the ground state of the Hamiltonian at the given point. Inset of (b) shows the same data in log-log scale. FIG. 15 . 15Converged normalized energies for H ♦ tXYZ with parameters (a) (a, b) = (0.81, 1.31) and (b) (0.89, 1.39) using the ER and SR. Plus markers (+) in (b) indicate converged energies from the Hamiltonian annealing. FIG. 16 . 16Four randomly chosen learning curves from (a, b) the XXZ model at ∆ = 1.5 without the sign rule and (c, d) the twisted XYZ model deep in phase II when (a, b) = (1.23, 1.73). The system size N = 28 is used for (a) and (c), and N = 32 is used for (b) and (d). FIG. 17 . 17Converged normalized energies of the twisted XYZ model at point (a, b) = (0.4, 3.0) which is deep in phase Λ. The transformed Hamiltonian H tXYZ is used. We plot (a) the results from the ER and SR as a function of N when α = 3 and (b) the ER result as a function of α when N = 20. FIG. 18 .FIG 18Converged infidelity 1 − F of the amplitude network with different choices of hyperparameters of the natural gradient descent. For the one-dimensional J1-J2 model with J2 = 0.2 and 0.44, we have trained the network to reproduce the amplitudes of true ground states. For each mini-batch size, four different learning rates η ∈ [5.0 × 10 −5 , 1.0 × 10 −4 , 2.0 × 10 −4 , 4.0 × 10 −4 ] (from left to right) are used. . 19. Converged infidelity 1 − F of the sign network with different choices of hyperparameters of Adam optimizer. For each mini-batch size, we have used six different learning rates η ∈ [6.25 × 10 −5 , 1.25 × 10 −4 , 2.5 × 10 −4 , 5.0 × 10 −4 , 1.0 × 10 −3 , 2.0 × 10 −3 ] (from left to right). FIG. 20 . 20Initial learning curves of the sign network from the one dimensional J1-J2 model (a) without and (b) with the sign rule. Grey horizontal lines indicate the converged infidelities when J2 = 0.4. Imposing the sign rule gets rid of the initial warm-up stage of the learning but does not improve the converged infidelity. the line segment from A = (0.25, 0.75) to E = (1.25, 1.75) that we have simulated vQMC. To locate the second transition point, we calculate d 2 E(θ)/dθ 2 where E(θ) is the ground state energy of H tXYZ where θ parameterizes the path. We use paths P 1 and P 2 depicted in , 0.5) are large even when the ER is used and the sign rule is applied. In this section, we investigate the errors in more detail using the RBM with different system sizes and number of hidden units. . InitializationWe have mainly used golot normal initialization suggested in Ref.[66] and implemented in JAX, as they have outperformed other initializers (e.g. lecun normal and he normal) for a various range of hyperparmeters. Appendix A: Training complex RBMsInitializationWe initialize the parameters θ of the complex RBM using samples from the normal distribution N (0, σ 2 ). We typically use σ = 10 −3 but σ = 10 −2 also have reported almost the same converged energies in most of simulations.SamplingWe have used the Metropolis-Hastings algorithm with the parallel tempering method to sample from the complex RBMs. Our set-up follows that of Ref.[61], which we summarize in this Appendix briefly.To sample from the complex RBM ψ θ (x), we first initialize the configuration x. For each step, we choose a new configration x following a certain update rule. When the system has the U (1) symmetry, we exchange x i and x j for randomly chosen i and j. Otherwise, we choose i ∈ [1, · · · , N ] randomly and flip x i , i.e. x j = x j −2x j δ j,i . Then the Metropolis-Hastings algorithm accepts the new configuration with probabilityAppendix D: Expressive power of the RBM for the J1-J2 modelIn the main text, we have observed that the errors from the J 1 -J 2 model when J 2 ∈ (J * Appendix G: Sampling problems from non-stoquastic basis and from heavy tail distributionsIn the main text, we have observed two different types of sampling problems. The first one appeared when we used a Hamiltonian in a non-stoquastic basis (e.g. the XXZ model in the antiferomagnetic phase without the sign rule). When this happened, converged energies of most of the SR instances are clustered far above the ground state energy ( E > 10 −2 ) and it persists regardless of the system size and the number of samples. We next observed a seemingly similar problem when solving In Sec. IV B 5, we have seen that errors from the ER increases as the parameter moves deeper in phase Λ of the twisted XYZ model. In this Appendix, we study a scaling behavior of errors when the Hamiltonian parameter is deep in phase Λ. Especially, we use the point K inFig. 6 which is (a, b) = (0.4, 3.0) and the Hamiltonian H tXYZ which reported the best converged energies from the ER.In contrast to the deep non-stoquastic phase of the J 1 -J 2 model that we studied in Appendix D, our SR results inFig. 17(a)suffer from the sampling problem. Still, the normalized energies seem to converge to a positive value > 10 −4 with N even when we take the best results for each N . On the other hand,Fig. 17(b)shows the converged energies from the ER for varying α = M/N when N = 20. As in the J 1 -J 2 model case [Fig. 12(b)], the improvements are getting marginal as α increases. From these observations, we confirm that the RBM does not represent the ground states in phase Λ faithfully.Appendix I: Supervised learning set-upsIn Sec. 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[ "Wilsonian Renormalization as a Quantum Channel and the Separability of Fixed Points", "Wilsonian Renormalization as a Quantum Channel and the Separability of Fixed Points" ]	[ "Matheus H Martins Costa \nInstitute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany\n", "Jeroen Van Den Brink \nInstitute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany\n\nInstitute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence ct.qmat\n01069Dresden, DresdenTUGermany\n", "Flavio S Nogueira \nInstitute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany\n", "Gastão I Krein \nInstituto de Física Teórica\nUniversidade Estadual Paulista\nRua Dr. Bento Teobaldo Ferraz, 271 -Bloco II01140-070São PauloBrazil\n" ]	[ "Institute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany", "Institute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany", "Institute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence ct.qmat\n01069Dresden, DresdenTUGermany", "Institute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany", "Instituto de Física Teórica\nUniversidade Estadual Paulista\nRua Dr. Bento Teobaldo Ferraz, 271 -Bloco II01140-070São PauloBrazil" ]	[]	We show that the Wilsonian formulation of the renormalization group (RG) defines a quantum channel acting on the momentum-space density matrices of a quantum field theory. This information theoretical property of the RG allows us to derive a remarkable consequence for the vacuum of theories at a fixed point: they have no entanglement between momentum scales. Our result can be understood as deriving from the scale symmetry of such theories and leads to constraints on the form of the ground state and on expectation values of momentum space operators.	null	[ "https://export.arxiv.org/pdf/2211.10238v2.pdf" ]	253,708,418	2211.10238	17422d5716bb225d224ff19f75e4c73cf804d7a2	Wilsonian Renormalization as a Quantum Channel and the Separability of Fixed Points 13 Apr 2023 Matheus H Martins Costa Institute for Theoretical Solid State Physics, IFW Dresden Helmholtzstr. 2001069DresdenGermany Jeroen Van Den Brink Institute for Theoretical Solid State Physics, IFW Dresden Helmholtzstr. 2001069DresdenGermany Institute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence ct.qmat 01069Dresden, DresdenTUGermany Flavio S Nogueira Institute for Theoretical Solid State Physics, IFW Dresden Helmholtzstr. 2001069DresdenGermany Gastão I Krein Instituto de Física Teórica Universidade Estadual Paulista Rua Dr. Bento Teobaldo Ferraz, 271 -Bloco II01140-070São PauloBrazil Wilsonian Renormalization as a Quantum Channel and the Separability of Fixed Points 13 Apr 2023 We show that the Wilsonian formulation of the renormalization group (RG) defines a quantum channel acting on the momentum-space density matrices of a quantum field theory. This information theoretical property of the RG allows us to derive a remarkable consequence for the vacuum of theories at a fixed point: they have no entanglement between momentum scales. Our result can be understood as deriving from the scale symmetry of such theories and leads to constraints on the form of the ground state and on expectation values of momentum space operators. Introduction.-The Wilsonian renormalization group (RG) transformation is a fundamental concept in the study of quantum field theories (QFTs) and statistical physics, which has been of great importance to understanding phenomena in these areas [1][2][3][4]. It is typically defined as the integration over high momentum modes of a field theory above a given scale µ with a sharp cutoff, followed by the rescaling of momenta and fields [5,6] For physical quantum systems (whose states are described by density matrices [7]), it is of prime importance for the Wilson RG to be a quantum channel of states in a QFT, i.e., a completely positive and trace-preserving (CPTP) map (see Sec. 8.2 of Ref. [7] for a definition and discussion). Indeed, were it not CPTP, there would be field theories, possibly tensored with finite-dimensional systems, where the renormalization procedure gives rise to density matrices for the long distance degrees of freedom that would violate key requirements of quantum mechanics (e.g, positivity). Furthermore, since the RG preserves exactly the averages of long-wavelength observables [6], this would contradict the fact that the set of expectation values of all elements in an observable algebra determines the quantum state [8,9], and such expectation values are obtained from a vacuum state. Although it is physically intuitive that the Wilsonian RG defines a CPTP map, such has so far not been demonstrated, despite recent advances discussed in Refs. [10][11][12][13][14][15]. Our first result in this Letter is to prove that the RG has this property. This will be done via the Schrodinger picture of the wave functionals in QFT [12,16]. We then use it to explore entanglement properties between momentum scales at RG fixed points. As discussed first in Refs. [17,18], and later explicitly worked out with examples in Ref. [19], the first RG step, integrating out fast modes, is equivalent to taking a partial trace over degrees of freedom above a certain scale, which is a quantum channel. Therefore, we only need to focus on the rescaling of fields and momenta in the second step to show its CPTP property. We will show that the scaling used in the RG procedure defines a unitary evolution of the momentum-space density matrices. Hence, as the composition of two quantum channels is still CPTP, we conclude the proof for the full Wilsonian RG transformation. This nature of the RG not only guarantees the expected consistency of the method, but it also paves the way towards investigating how the entanglement between momentum scales evolves along the RG trajectory. In particular, it allows us to study such entanglement for theories lying at fixed points. Our main result is establishing that, as long as a fixed point theory exists (for example, as a CFT), then there is no entanglement in its ground state between momentum modes at different scales, even though such theories can be strongly interacting. We conclude with a discussion about some consequences of this property and on the novel perspectives it provides. The Wilsonian RG as a quantum channel.-We begin by reviewing one of the insights of Ref. [17]. It is known that the ground state of a quantum system can be represented via the functional integral of its action (see, for instance, Sec. IV in Ref. [20]). Suppose one partitions the Hilbert space of a QFT in d spatial dimensions with action S in momentum space as H = k H k , where each H k is generated by eigenstates of (the hermitian components of) the field mode φ k , where φ represents any collection of bosonic and fermionic fields of the theory. Then, the ground state matrix elements between two vectors \|ϕ k , \|φ k such that each momentum mode has a definite amplitude are given by the path integral [17,20], ϕ k \| ρ \|φ k = 1 Z φ k (0 + )=ϕ k φ k (0 − )=φ k Dφ k (τ )e −S[φ k (τ )] ,(1) the boundary condition indicates a discontinuity at Euclidean time τ = 0, the action S[φ k (τ )] is written in terms of the Fourier-transformed fields and Z = Dφ k (τ )e −S[φ k (τ )] .(2) With this representation it becomes clear that integrating out fast modes with \|k\| ≥ µ for an arbitrarily chosen and changeable µ, is the same as taking a partial trace over the Hilbert space ⊗ \|k\|≥µ H k , as can be seen by the equality O = 1 Z Dφ k (τ )O φ k , i δ δφ k e −S[φ k (τ )] = 1 Z Dφ \|k\|≤µ (τ )O φ k , i δ δφ k e −Sµ[φ k (τ )](3) for any observable O built from field modes φ k such that \|k\| ≤ µ and where S µ is the Wilsonian effective action at scale µ obtained by integrating out fields with \|k\| > µ. This is the relation O A = Tr(ρO A ) = Tr A (ρ A O A ) which characterizes a reduced density matrix for a subsystem A from the observables acting on it, applied to momentum scales in a field theory, here defined as the field modes with momenta with a certain magnitude. Thus, a low-momentum density matrix ρ µ derived from this QFT ground state is well defined and given in terms of S µ by [17,19], ϕ \|k\|<µ ρ µ φ \|k\|<µ = 1 Z φ k (0 + )=ϕ \|k\|<µ φ k (0 − )=φ \|k\|<µ Dφ k e −Sµ (4) The broader point is that this interpretation is valid even in the case of states other than the vacuum: the first RG step defines a partial trace over high-momentum modes and takes density matrices on the full Hilbert space of the theory to density matrices acting on the long-wavelength degrees of freedom only. Moving on to the scaling transformation, we define Λ as the overall cutoff of the QFT and the scaling parameter as σ := Λ/µ. Thus, the rescaling of field modes is given by [5], k → σk,(5)φ k → σ d φ φ σk ,(6) where d φ is the scaling dimension of the Fouriertransformed field, which depends on the fixed point of interest. The Euclidean time variable must also be rescaled as τ → σ −z τ , using the dynamical critical exponent z introduced by Hertz in Ref. [21]. We keep z generic as our results will be valid for both relativistic and nonrelativistic field theories. Furthermore, note that the scaling transformation employed here is simply the uniform dilation of length scales. More general Weyl transformations curving space are not investigated. The latter lead to anomalies in certain CFTs (the main differences between the two transformations are discussed in Ref. [22]). The matter of time rescaling is also a good opportunity to emphasize the peculiarities of our momentumspace cutoff: modes with \|k\| > µ are integrated over at all energies, without any constraint in the temporal component of momentum, which transforms only under the second step of the RG. Such distinction is essential for the integration of fast modes to be identified with a partial trace, as the degrees of freedom of the system are labeled by the spatial momenta and in the path integral the dependence in τ is only used to project into the ground state, meaning there are no restrictions on its conjugate k 0 . Similar conclusions, in the context of the functional RG, are reached in Refs. [12,23] (which discuss the phase space and canonical structure) and Ref. [24]. This suggests that even for relativistic theories, focusing only on the spatial momenta is key to understanding the entanglement properties of QFTs. Last but not least, as discussed by Hertz in Ref. [21], Sect. VI, and Millis in Ref. [25], the fixed points and universal quantities obtained with this cutoff are the same as in any other RG method, thus keeping the following analysis very general. Now, recall that S[φ k (τ )] is the original action of the QFT in terms of momentum-space fields and let S (σ) [φ k (τ )] := S µ [σ d φ φ σk (σ −z τ )] denote the new action at scale Λ obtained from the scaling transformation. Then, by means of the path integral construction of matrix elements of a state operator, this action naturally defines the density matrix ρ (σ) via, ϕ k \| ρ (σ) \|φ k = 1 Z (σ) φ k (0 + )=ϕ k φ k (0 − )=φ k Dφ k e −S (σ) [φ k ] ,(7)Z (σ) = Dφ k (τ )e −S (σ) [φ k ] .(8) There is a priori no reason to believe that ρ (σ) = \|Ω Ω\|, where \|Ω is the ground state vector, since the action S (σ) [φ k ] will be generally different from the original S[φ k ]. The process of obtaining an effective action by integrating part of the momentum modes can be inverted by a scaling transformation only at an RG fixed point. In general, the scaling transformation must be defined not only for ρ µ , but also for any density matrix acting on the low-momentum degrees of freedom. We will do so by using the Schrodinger representation of states in a QFT [12,16], where a generic density matrix ρ acting on the Hilbert space of momentum modes below scale µ can be formally written as the path integral ρ = \|k\|,\|k ′ \|≤µ Dφ k Dφ ′ k ′ ρ(φ k , φ ′ k ′ ) \|φ k φ ′ k ′ \| ,(9) \|k\|≤µ Dφ k ρ(φ k , φ k ) = 1. Then, we define the scaling transformation as taking ρ to aρ such that, ρ = \|k\|,\|k ′ \|≤Λ Dφ k Dφ ′ k ′ρ(φk , φ ′ k ′ ) \|φ k φ ′ k ′ \| , (11) ρ(φ k , φ ′ k ′ ) = 1 N ρ(σ d φ φ σ −1 k , σ d φ φ ′ σ −1 k ′ ),(12)N = \|k\|≤Λ Dφ k ρ(σ d φ φ σ −1 k , σ d φ φ ′ σ −1 k ′ ),(13) which is composed of the same rescalings as before with a relabeling of the momentum modes. The normalizing factor N is introduced due to the scaling of fields in the path-integral measure. This definition is exactly the same as Eq. (7) whenever the density matrix elements can be defined via an effective action, with N = Z (σ) /Z. This can be confirmed by writing Eq. (7) in the form of Eq. (12) via a change of variables. Note that while the rescaling of momenta and fields enacts a shift in the labels of the degrees of freedom, the time rescaling by itself produces no change: in Eq. (7) the fields are integrated over all possible dependencies in τ (a consequence of no cutoffs being imposed on the energies) and the integration limits are taken at τ = 0 ± , so the rescaling can be undone by a change of variables with no alterations in the final matrix elements. Interestingly, this is not the case at finite temperature, not studied in this paper, where the time periodicity is changed by the rescaling, see Ref. [25] and the Supplemental Material. From now on, it is important to define the theory in a box of volume V , an IR cutoff, so that the number of degrees of freedom is finite and the functional integrals and other quantities are well defined. With this cutoff the normalization constant becomes N ≈ σ −d φ µ d V as can be seen by comparing Eqs. (10) and (13) explicitly. As discussed in Ref. [6], this scaling is "trivial" in the sense that all statistical properties of the state at low-momentum degrees of freedom are preserved and all original expectation values can be recovered. In a quantum system this is tantamount to the transformation being described by a unitary map; in fact, if we define the operator, U ≡ √ N \|k\|≤µ Dφ k σ −d φ φ σk φ k \| ,(14) by computing the necessary integrals with both UV and IR cutoffs, it is easy to show that given U and density matrices of Eqs. (9) and (11), we haveρ = U ρU † . Furthermore, U also obeys U U † = U † U = 1 and so the scaling transformation is indeed unitary. This can be tested, for example, by confirming the validity of the fact that the entropy of a density matrix is invariant under unitary transformations [7]. Indeed, it can be shown using the method and examples of Ref. [19], that the entropy of the matrices before and after scaling (at lowest nontrivial perturbative order) are equal [26]. In real space the unitarity of scaling maps is wellknown. What we have shown is that this property is also present in the specific transformation used in the momentum-space RG, which also includes scaling of field modes and time and which, although first defined only as a manipulation of the effective action, naturally leads to a map of density matrices. Therefore, the full Wilsonian RG procedure defines a quantum channel ρ → ρ (σ) which is the composition of a partial trace over high-momenta (map ρ → ρ µ ) and a unitary induced by the rescaling operation (map ρ µ → ρ (σ) ). The RG flow, being a Completely Positive and Trace-Preserving process, can thus be described using tools such as the operator-sum representation (see Chapter 8 of Ref. [7]). Entanglement between scales at a fixed point -To see how the information theory formulation of the RG might be valuable, we apply it to study the momentum-space entanglement in the ground states of RG fixed points. By definition, these QFTs are such that S * (σ) [φ k ] = S * [φ k ] (the latter being the fixed point action, generally including all powers and derivatives of the field) no matter how many degrees of freedom are integrated over in the first step. Consequently, by the connection between action functionals and density operators explored earlier, we must have ρ (σ) = ρ = \|Ω Ω\|, meaning ρ (σ) is a pure state and so S EE (ρ (σ) ) = 0. The entropy of interest is S EE (ρ µ ) = − Tr(ρ µ log ρ µ ), which gives the entanglement between low and high momenta. However, we showed that rescaling is a unitary (in this context only also the inverse of the partial trace), therefore S EE (ρ µ ) = S EE (ρ (σ) ) = 0.(15) Thus, the ground states of theories at an RG fixed point have no entanglement between different momentum scales. While all transformations were defined starting from a full regularization of the QFT, this does not restrict the validity of our result. We introduce the UV cutoff Λ in order to regularize the theory, but scale-invariance makes its removal simple. Interactions are renormalized so that the theory is kept at the fixed point, by simply leaving the dimensionless parameters constant, and the limiting procedure Λ → ∞ keeps the entanglement entropy between slow and fast modes equal to zero. As for the IR cutoff, Refs. [3, 4] explain how the finite volume acts as a relevant operator, driving the system away from the fixed point, and as pointed out in Ref. [6], the scaling transformation effectively changes the size of the box as V → σ −d V . This means that the V → ∞ limit must be taken before any other when defining the theory, similarly to discussions of spontaneous symmetry breaking. This limit ensures the theory stays at a fixed point and that the Hilbert spaces before and after scaling are the same (without it, different periodic boundary conditions define different vector spaces), a necessary condition for the equation ρ (σ) = \|Ω Ω\| to be meaningful. Ultimately this does not change much, as the scaling transformation is still unitary at infinite volume and the proof of Eq. (15) follows the same way, though it is important to keep these subtleties in mind. Note that our only assumption at this point was that a fixed point exists. Hence, what we have shown is that, contrary to what may be intuitively expected, even a strongly interacting QFT can have no entanglement in its vacuum if the theory is scale-invariant. In other words, the stringent conditions on the couplings of a fixed point automatically constrain the ground state entanglement between momentum scales to vanish. Physically, this result can be understood as follows. The entanglement entropy between scales necessarily vanishes as both µ → 0 and µ → Λ. This is because as µ → Λ, then fewer and fewer momentum degrees of freedom are being integrated out, and in this limit we are simply left with the full vacuum of the theory, a pure state with zero entropy (when µ = Λ we simply have the full action of the theory, which defines the ground state). On the other hand, if µ → 0, we can invoke the fact that if the global state is pure, the entropy after taking the partial trace is equal to that of the density matrix of the traced out degrees of freedom [7], meaning that the entropy at µ = 0 must be equal to the entropy of the ground state, which is zero. Now, the entanglement entropy is always positive [20], so it must reach a maximum between 0 and Λ as µ is varied, but such a maximum naturally defines a characteristic scale for a theory, since it is the momentum scale across which modes are correlated the most. Therefore, if a theory is scale-invariant, the momentum-space entanglement entropy must be constant. Since we know it vanishes both in the IR and UV extremes, it must vanish always. In this way we arrive once more at the conclusion that there must be no entanglement with respect to this partition. This general behavior of momentumspace entanglement used in our argument is seen in the explicit formulas obtained in Refs. [17,19] and is the field theory equivalent of the "Page curve" discussed in Sec. 3.1 of Ref. [27]. The latter is an upper-bound on the entanglement entropy generated when degrees of freedom are gradually traced out in a pure state of a finite-dimensional system. Consequences for fixed point theories -We can derive a number of implications from the fact that there is no entanglement between momentum scales in the ground state of scale-invariant QFTs for any separation scale µ chosen. The most direct one is that the vacua of these theories are separable, i.e., it is a simple tensor product of terms labeled by the momentum scale. Writing the Hilbert space of a fixed point theory as H = µ H µ , with µ denoting the momentum scale (meaning each H µ contains all modes with \|k\| = µ), the vacuum must be given by \|Ω = ⊗ µ \|Ω µ . Due to scale symmetry and the unitarity of the scaling map, the projections of the components \|Ω µ into eigenstates of field modes must obey φ k \|Ω µ = σ −d φ φ σk Ω σµ for any real σ. Furthermore, separability of the state vector leads to connected correlation functions of observables acting on different momentum scales being all equal to zero [7,28]. That is, defining the operators which act on the subsystems below and above scale µ, respectively (17) given two families of functions {f n (k 1 , ..., k n )}, {f n (k 1 , ..., k n )} (which must be of compact support in \|k i \| ≤ µ, \|k i \| > µ, respectively, see chapter 2 of Ref. [8]) then the separability of the vacuum implies the factorization of the expectation value of their product: O < := ∞ n=1 \|ki\|≤µ n i=1 d d k i (2π) d f n (k 1 , ..., k n )φ k1 ...φ kn (16) O > := ∞ n=1 \|ki\|>µ n i=1 d d k i (2π) df n (k 1 , ..., k n )φ k1 ...φ knO < O > = O < O > . Translating this condition into identities for the field correlators is somewhat complicated, but the n-point functions of the field in momentum space must be such that all momenta are at the same scale (have the same absolute value), or else they factorize into products of correlators. For example, the four-point function φ k1 φ k2 φ k3 φ k4 becomes such that φ k1 φ k2 φ k3 φ k4 = F (k 1 , k 2 , k 3 , k 4 ) + φ k1 φ k2 φ k3 φ k4 + φ k1 φ k3 φ k2 φ k4 + φ k1 φ k4 φ k2 φ k3 ,(18) where F (k 1 , k 2 , k 3 , k 4 ) depends on the fixed-point theory and vanishes unless \|k 1 \| = \|k 2 \| = \|k 3 \| = \|k 4 \|. This identity can be understood as follows. If all momenta have same magnitude, the correlator can have any form consistent with scale symmetry, otherwise it must factorize into a product of expectation values. Note that this result is independent of momentum conservation (the expression still contains a delta function making operators in the Hilbert space of the QFT (see section 3.1 of Ref. [9] for an introduction). This requirement is what distinguishes operators whose momentum correlations must factorize between scales at a fixed point from the others: it formalizes the idea that a "fundamental field" identifies the "degrees of freedom" of a QFT. Going back to CFTs, a generic scaling operator does not satisfy this irreducibility condition and so its correlation functions do not have to factorize. Having discussed some corollaries of our result, it is important to make clear that the lack of entanglement between momentum scales does not imply that theories at an RG fixed point have an unentangled vacuum: the notion of entanglement depends on the chosen partition of the Hilbert space and separability with respect to one tensor product structure does not imply the same about other partitions. For example, in free field theories there is entanglement in real space but not in momentum space [20]. Conclusions and Outlook -We have shown that the Wilsonian RG is equivalent to a quantum channel acting on density matrices of the momentum-space degrees of freedom. Furthermore, we proved that it is such that RG fixed points have no entanglement between momentum modes at different scales and discussed some of the consequences of this fact. The analysis made here can serve as starting point for other investigations, perhaps of QFTs at a phase transition instead of a fixed point. A field theory undergoing a second phase transition may still flow under the RG transformation, see Ref. [31]. More broadly, we can use techniques such as the operator-sum decomposition to ask how specific RG flows reflect on the momentum space entanglement entropy: does it present "critical scaling" under certain conditions? By plotting S EE (ρ µ ) as a function of µ, does the graph contain universal information? And what properties of a given phase transition or crossover can be read off from it? From a mathematical point of view, while we have used the Schrödinger picture following Refs. [12,16], this was merely a way of representing the idea that lowmomentum observables of a QFT can be constructed formally via functions of the Fourier-transformed fields, thus defining the momentum-space operators for each mode k. By comparison with the local algebras of observables [8], which have been important for studying entanglement in real-space [28,[32][33][34][35], it would be interesting to rigorously and abstractly define the momentum-space algebras of observables and analyze their properties, possibly connecting with previous work in Refs.m [36,37]. In such formalism, the partial trace over fast modes becomes the restriction of the ground state to the subalgebra of lowmomentum observables and the rescaling of fields and momenta translates into applying the dual map of the scaling unitary of density matrices to this subalgebra. Furthermore, while we considered fields at a fixed time in our arguments, it is known that in relativistic theories they are too singular [8]. An algebraic formulation would avoid this problem by considering observables acting at spatial momenta below a certain scale, but with arbitrary energy: the algebra associated with a "cylinder" of radius µ in momentum space and infinitely extended along the energy axis. This not only corresponds to the partial trace over high-momentum degrees of freedom while avoiding ill-defined operators, but also makes clear that our subalgebra is invariant under the rescaling of time with a dynamical critical exponent, equivalent to what was previously discussed for density matrices. Another opportunity provided by this formulation is to investigate the connection between momentum-space entanglement and the effects of renormalization in real-space entanglement, such as those explored in Refs. [38][39][40]. We may also wonder what the separability in momentum space of the ground state of CFTs implies to holography. Finding the dual in AdS space of the momentum space density matrix ρ µ is essential to tackling this question, but is an open problem as pointed out in Ref. [41]. Furthermore, it was shown in Ref. [42] that the intuitive idea of restricting the AdS radial coordinate corresponds to a relativistic Wilsonian cutoff, that is, the remaining modes must obey, in Euclidean signature, k 2 0 + k 2 ≤ Λ, a constraint on the energies which, as mentioned in Ref. [17] and discussed previously in this paper, is absent from the tensor product structure we are working with. Nevertheless, a proposal in Ref. [43] generalizes of the concept of entanglement wedge to momentum space and merits further investigation. Lastly, the description of the RG as a specific CPTP map possibly opens a path to connecting renormalization to recent discussions of circuit complexity in field theory, such as the ones in Refs. [44,45], which have also been studied in relation to the AdS/CFT duality. We tion H n (ρ) = 1 1 − n lim β→∞ (log Z n (β) − n log Z(β)), (S. 19) where, given an effective action S ef f which generates the matrix elements of ρ, Z(β) is the usual finite-temperature partition function and Z n (β) is the partition function after modifying the non-local kernels of S ef f in a specific manner detailed in Ref. [19]. It turns out that when starting with a free field theory and adding a perturbative interaction, a series of cancellations happen and at order O(λ 2 ) in the coupling (the lowest with non-trivial results) the von Neumann and Rényi entropies are proportional to the same contractions of Feynman diagrams appearing in the modified partition function. Then, to show that the entropies before and after scaling are the same, we need only to prove the equality between Feynman diagram contractions. We will do so for one of the contributions, as the others follow the same argument. Consider a contributing term to the entropy of reduced density matrix ρ µ in perturbative λφ 4 theory of the form * K µ,β (k, p, q; τ, τ ′ ) φ k (τ )φ * k (τ ′ ) nβ (S.20) corresponding to Eq. (C6) of Ref. [19], where the subscript nβ in the correlator means that the expectation value is taken at inverse temperature nβ, the region of integration over all momenta and form of the kernel K µ,β are specified but irrelevant to our argument and a number of Matsubara sums and Euclidean time integrals are suppressed. Applying the scaling map k → σk, φ k → σ d φ φ σk , τ → σ −z τ , the associated term leading to the entropy of state ρ (σ) is * K µ,β (σ −1 k ′ , p, q; σ −z τ, σ −z τ ′ ) ×σ 1−d σ 2d φ φ k ′ (σ −z τ )φ * k ′ (σ −z τ ′ ) nβ . (S. 21) Now, this transformation is defined such that φ k (τ )φ * k (τ ′ ) nβ = σ 2d φ φ σk (τ )φ * σk (τ ′ ) nβ for the transformed fields, see Ref. [6]. To deal with the rescaling in time we make a change of variables to restore τ, τ ′ , but as pointed out by Ref. [25] and can be seen by taking into account the integration limits of the (suppressed) time integrals, this effectively changes the temperature periodicity to σ −z β (and nσ −z β in the replica trick calculations of Ref. [19]). Therefore, it is easy to see that Eq. (S.21) equals to * K µ,σ −z β (σ −1 k ′ , p, q; τ, τ ′ )× σ 1−d φ σ −1 k ′ (τ )φ * σ −1 k ′ (τ ′ ) nσ −z β = * K µ,σ −z β (k, p, q; τ, τ ′ ) φ k (τ )φ * k (τ ′ ) nσ −z β (S.22) where the last equality is derived via a simple change of variables in the momentum k ′ , originally one of the slow modes. So we can see that there is a change for any finite temperature calculation, which makes an analogous investigation of the RG in this context an interesting problem. For our focus on the vacua of field theories at zero temperature, however, this is not a concern because the β → ∞ limit remains unchanged and after the limit the results are the same as before the dilation. Thus, the contribution to the entropy of ρ (σ) is exactly equal to that of ρ µ and a calculation can be done for any of the other perturbative terms leading to similar results. Therefore the total entropy is unchanged, consistent with our claim of the unitarity of the scaling transformation. Finally, note that there was no need to specify the values of d φ or z; the scaling is unitary regardless of the dimension given to field φ k . For the appearance of scaling dimensions different from the correct ones in the context of the renormalization group, see Ref. [46]. In more detail, any scaling with wrong dimension can be decomposed into a product of the correct scaling with a change of the normalization of the field operator, which Ref. 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[ "Distinguishing chromatic number of Hamiltonian circulant graphs", "Distinguishing chromatic number of Hamiltonian circulant graphs" ]	[ "Michael D Barrus ", "Jean Guillaume ", "Benjamin Lantz " ]	[]	[]	The distinguishing chromatic number of a graph G is the smallest number of colors needed to properly color the vertices of G so that the trivial automorphism is the only symmetry of G that preserves the coloring. We investigate the distinguishing chromatic number for Hamiltonian circulant graphs with maximum degree at most 4.	null	[ "https://export.arxiv.org/pdf/2303.13759v1.pdf" ]	257,757,204	2303.13759	8da3c761bf4f01af03d806fe02eb8d17e48392ef	Distinguishing chromatic number of Hamiltonian circulant graphs March 27, 2023 Michael D Barrus Jean Guillaume Benjamin Lantz Distinguishing chromatic number of Hamiltonian circulant graphs March 27, 2023 The distinguishing chromatic number of a graph G is the smallest number of colors needed to properly color the vertices of G so that the trivial automorphism is the only symmetry of G that preserves the coloring. We investigate the distinguishing chromatic number for Hamiltonian circulant graphs with maximum degree at most 4. Introduction and definitions The distinguishing chromatic number of a graph G, introduced by Collins and Trenk [4] and denoted by χ D (G), is the minimum number of colors needed for a proper coloring of the vertices of G such that the only automorphism of G that preserves the coloring is the trivial automorphism. Any such coloring using r colors is called an r-distinguishing coloring, or simply a distinguishing coloring if the value of r is unimportant. The distinguishing chromatic number was introduced as a proper-coloring analogue of the distinguishing number of a graph defined by Albertson and Collins [2], which measures the difficulty of breaking symmetries in graphs by coloring vertices (i.e., providing a distinguishing coloring), though without requiring a proper coloring. In this paper, all distinguishing colorings will be assumed to be proper colorings. In [4], Collins and Trenk observed that both the ordinary chromatic number χ(G) and the distinguishing number D(G) serve as lower bounds for χ D (G). They proved that χ D (G) = \|V (G)\| if and only if G is a complete multipartite graph, and they also determined χ D (G) for various classes of graphs G. In particular, they showed the following. Theorem 1.1 ([4]). For n ≥ 3, the distinguishing chromatic number of C n is given by χ D (C n ) =      3 if n ∈ {3, 5} or if n ≥ 7; 4 if n ∈ {4, 6}. The graph C 6 is one example where χ D can be strictly greater than both the chromatic number and the distinguishing number (both equal 2 for C 6 ). In this paper we consider circulant graphs, i.e., those undirected graphs with vertices v 0 , . . . , v n−1 where edges join any two vertices having indices with a difference (in either order) modulo n lying in a given set of positive numbers. If this set of differences is denoted by D, then we denote such a graph by C n (D), and if D = {d 1 , . . . , d s }, we write the graph as C n (d 1 , . . . , d s ). For example, the cycle graph C n is equivalent to the circulant graph C n (1). Note that the notation allows for multiple representations for a single graph, since differences may be computed in the opposite order and are reduced modulo n. For example, the cycle C n is also equivalent to C n (n − 1), and we may replace any element k in the set of allowed differences by 1 arXiv:2303.13759v1 [math.CO] 24 Mar 2023 n − k without changing the graph. Unless otherwise specified, we will assume here that for an n-vertex circulant graph, each difference belongs to {1, . . . , n/2 }. We will extend Theorem 1.1 by determining the distinguishing chromatic number for various classes of Hamiltonian circulant graphs with maximum degree at most 4. As we will see, in most cases these graphs are similar enough to cycles that the distinguishing chromatic number is 3, and in no infinite family does the number ever rise higher than 5. In particular, we show that for any tetravalent graph C n (1, k), where k = n/2, n/2 − 1, and (n, k) = (10,3), the distinguishing chromatic number is at most one more than the chromatic number. As a prelude, here is a summary of our main results. Theorem 1.2. Let k, n be positive integers such that 2 ≤ k ≤ n/2 . The distinguishing chromatic number of C n (1, k) is given by χ D (C n (1, k)) =                                          n if (n, k) = (4, 2), (5,2), (6,2), (6,3), ( By otherwise, we mean all tetravalent circulant graphs C n (1, k) such that k = 2, (n − 1)/2, and n/2 − 1, and (n, k) = (10, 3), (15, 4), (13, 5), (5,2), and (8,3). The circulant graphs C n (1, k) such that (n, k) = (4, 2), (5,2), (6,3), (8,3) are complete graphs or complete bipartite graphs. Thus, results hold by [4]. As for graph C 6 (1, 2), the following arguments can be used to show that the distinguishing number is 6: vertices v i v i+2 v i−2 form an induced triangle and antipodal pairs The following sections are organized as follows. In Section 2, we recall facts about isomorphisms and proper colorings of circulant graphs. In Section 3, we determine χ D for the trivalent Hamiltonian circulant graphs, also known as the Möbius ladders. In Section 4, we discuss the automorphism of the tetravalent graphs C n (1, k). In Section 5, we give an optimal distinguishing proper coloring of C n (1, n/2 − 1). In Sections 6, 7, and 8, we prove that the distinguishing chromatic number of tetravalent graphs C n (1, k), where k = n/2, n/2 − 1, and (n, k) = (10, 3), is at most 1 more than the ordinary chromatic number. v i v i+3 , v i+2 v i+5 , v i−2 v i Throughout this paper, we will use V (G) and E(G) to denote the vertex and edge sets of a graph G. Figure 1: Isomorphic graphs C 7 (1, 2) and C 7 (1,3). that C 7 (1, 2) is isomorphic to C 7 (1,3). We may simplify cases in what follows, and extend our results about graphs C n (1, k) to isomorphic trivalent or tetravalent circulant graphs not expressed in this way by recalling a result ofÁdám [1]. C 7 (1, 2) v 0 v 1 v 2 v 3 v 4 v 5 v 6 C 7 (1, 3) v 0 v 3 v 6 v 2 v 5 v 1 v 4 Theorem 2.1 ( [1]). If gcd(n, p) = 1, then C n (a 1 , · · · , a t ) ∼ = C n (pa 1 , · · · , pa t ), where multiplication is performed modulo n. We specialize this result to the graphs of the form C n (1, k). Corollary 2.2. If either a or b is relatively prime to n, then C n (a, b) = C n (1, k) for some k ∈ {1, . . . , n/2 }. Proof. Suppose without loss of generality that gcd(a, n) = 1. An elementary result of number theory shows that there exists p in {1, . . . , n − 1} such that ap ≡ 1 (mod n) and p is relatively prime to n. Hence C n (a, b) ∼ = C n (pa, pb) ∼ = C n (1, k), where k is either pb or n − pb modulo n, whichever belongs to {1, . . . , n/2 }. Corollary 2.3. If n = k ± 1, then C n (1, k) ∼ = C n (1, ). Proof. If n = k ± 1, then is relatively prime to n, as is n − . Letting p = n − in Theorem 2.1 shows that C n (1, k) ∼ = C n (n − , ±1) ∼ = C n (1, ). We recall now a few results about the chromatic number χ(G) of circulant graphs G. The following result was conjectured by Collins, Fisher, and Hutchinson (see [3,5] as cited in [11]) and proved by Yeh and Zhu [11]; see also [6], [7], and [9]. Theorem 2.4. Let k, n be positive integers such that 2 ≤ k ≤ n/2 . The chromatic number of C n (1, k) is given by χ(C n (1, k)) =                            2 if k is Trivalent circulant graphs When n is even, the circulant graph C n (1, n/2) is a trivalent graph also known as the Möbius Ladder due to a drawing as a Möbius band of 4-cycles; see Figure 2 which draws C 8 (1, 4) in two ways. These are the unique trivalent circulant graphs Figure 2: The graph C 8 (1, 4) drawn as a Möbius ladder. C n (1, k). v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 0 v 4 v 5 v 1 v 6 v 2 v 7 v 3 Theorem 3.1. For even integers n ≥ 4, χ D (C n (1, n/2)) =            4 if n = 4; 6 if n = 6; 3 if n ≥ 8. Proof. When n is 4 or 6, the graph C n (1, n/2) is isomorphic to K 4 or K 3,3 , respectively. By the result of Collins and Trenk [4], distinguishing colorings of these complete multipartite graphs require as many colors as there are vertices in the graph. For even n ≥ 8, it has been shown (see [8], for instance) that the Möbius ladder with n vertices has the dihedral group of order 2n as its automorphism group. We exhibit a distinguishing proper coloring using 3 colors based on whether or not n is a multiple of 4. If n = 4t, where t > 1, then the Möbius ladder is not bipartite and hence a proper coloring will require at least 3 colors. Assign colors to the vertices as shown here (assigning the colors shown to the vertices in order of their subscripts): Colors of v 0 , . . . , v n/2−1 1 2 1 2 1 2 · · · 1 2 Colors of v n/2 , . . . , v n−1 3 1 2 1 2 1 · · · 2 3 Observe that each vertex is adjacent to vertices whose colors precede and follow its color in the table, as well as to the vertex whose color appears in the same position on the opposite row. Thus it is easy to verify that this is a proper coloring. (In fact, this coloring arises via a greedy coloring of the vertices v 0 , . . . , v n−1 in order.) Note that a coloring-preserving automorphism must permute the vertices with color 3, and one of these two vertices has two neighbors with color 1 while the other has only one neighbor with color 1. Hence the vertices with color 3 must be fixed under any color-preserving automorphism. The only dihedral symmetry fixing two non-antipodal vertices is the identity symmetry, so this coloring is a distinguishing coloring. If n = 4t + 2, where t > 1, then C n (1, n/2) is a bipartite graph. The unique partition of its vertices into two color classes allows for many dihedral symmetries, so a distinguishing proper coloring must use at least 3 colors. We obtain one by changing to color 3 the colors of a pair of nonadjacent vertices formerly colored 1 and 2 in a proper 2-coloring. One example is shown here: Colors of v 0 , . . . , v n/2−1 3 2 1 2 · · · 1 Colors of v n/2 , . . . , v n−1 2 1 3 1 · · · 2 As before, we observe that a color-preserving automorphism must permute the vertices of color 3. Since one is adjacent only to vertices of color 1, while the other is adjacent only to vertices of color 2, the automorphism fixes these vertices. As before, the identity is thus the only color-preserving automorphism, and this is a distinguishing coloring. 4 For the remainder of the paper we will deal with the tetravalent graphs C n (1, k) where 1 < k < n/2. Because distinguishing colorings on graphs "break" nontrivial symmetries, this section will review some facts about automorphism groups of circulant graphs. The following theorem of Potocnik and Wilson [10] will be key to our organization. A graph is edge-transitive if any edge may be carried to any other edge by some automorphism, and dart-transitive if any edge may be mapped to any other edge, with the endpoint images specified, by some automorphism. 10]). If G is a tetravalent edge-transitive circulant graph with n vertices, then it is dart-transitive and either: Theorem 4.1 ([ (1) G is isomorphic to C n (1, k) for some a such that k 2 ≡ ±1 (mod n), or (2) n is even, and G is isomorphic to C 2m (1, m + 1) where m = n/2. As we will see later as we color the graphs C n (1, k), the converse to Theorem 4.1 is true. We will provide an optimal distinguishing proper coloring for the graphs in (2) in Section 5, noting that C 2m (1, m + 1) is isomorphic to C n (1, n/2 − 1), and say no more about these graphs here. We will discuss coloring the graphs in (1) in Section 8 after a few comments about these graphs at the end of this section. What about graphs C n (1, k) that are not edge-transitive? As we will see, in that case Aut(C n (1, k) is isomorphic to a dihedral group of order 2n. Let E 1 denote the set of edges of C n (1, k) joining vertices having indices differing by 1 modulo n, and similarly let E k denote the set of edges of C n (1, k) joining vertices whose indices' difference is k modulo n. Observe that an automorphism φ : V (C n (1, k)) → V (C n (1, k)) induces a permutation on the edge set of C n (1, k) that "carries" edge v i v j to edge φ(v i )φ(v j ) for any i, j. Theorem 4.2. If k = n/2 − 1 and (n, k) = (10, 3), then any automorphism of C n (1, k) either carries every edge in E 1 to an edge in E 1 or carries every edge in E 1 to an edge in E k . Proof. The result can be verified directly for n < 7, so assume that n ≥ 7. Consider an edge of E 1 in C n (1, k); by symmetry we may assume that it is v 0 v 1 . Note that this edge belongs to the two 4-cycles v 0 v 1 v k+1 v k and v 0 v 1 v 1−k v −k ; these are distinct 4-cycles unless k = n/2, in which case they coincide. Note that for each edge in E k incident with either v 0 or v 1 , there is a 4-cycle containing that edge and v 0 v 1 . We claim now that no 4-cycle contains both v 0 v 1 and an incident edge from E 1 . Indeed, for a 4-cycle to contain the path v 0 v 1 v 2 , both v 0 and v 2 must have a common neighbor v j , where j ∈ {−k, −1, k} ∩ {2 − k, 3, k + 2}. Setting each element of the first set equal to each element of the second set and recalling that n ≥ 7 and that 1 < k < n/2, we see that the existence of a common neighbor v j implies that either k = n/2 − 1, a contradiction to our hypothesis, or k = 3. A similar argument and conclusion holds if C n (1, k) has a 4-cycle containing the path v −1 v 0 v 1 . Let us assume for now that k = 3. Since the property of inclusion of a pair of edges in some induced 4-cycle is preserved by an automorphism, it follows that if an edge from E 1 is carried by an automorphism to an edge from E 1 , the same must be true for the images of its incident edges from E 1 . Working inductively outward from the first edge, we see that each edge of E 1 is then carried to an edge in E 1 , and edges from E k are forced to be carried to edges in E k . Conversely, if an edge from E 1 is carried by an automorphism of C n (1, k) to an edge in E k , then no edge from E 1 can be carried to an edge from E 1 , which forces all edges from E 1 to be carried to edges from E k and vice versa. If instead k = 3, observe directly that in each of C n (1, 3) for 7 ≤ n ≤ 12 except for n ∈ {8, 10}, every automorphism of C n (1, 3) either carries all edges in E 1 to edges in E 1 or carries all edges in E 1 to E 3 , as claimed. Assume now that k = 3 and n > 12. Note that the size of n implies that each edge in E 1 belongs to exactly five 4-cycles; for v 0 v 1 these cycles are v 0 v 1 v n−2 v n−3 , v 0 v 1 v n−2 v n−1 , v 0 v 1 v 2 v n−1 , v 0 v 1 v 2 v 3 , and v 0 v 1 v 4 v 3 . In contrast each edge in E 3 belongs to exactly three 4-cycles; for v 0 v 3 these are v 0 v 3 v 2 v n−1 , v 0 v 3 v 2 v 1 , v 0 v 3 v 4 v 1 . Since the inclusion of an edge in a 4-cycle is preserved under the image of a graph automorphism φ, any such map φ induces a permutation of the edges in E 1 and a permutation of the edges of E k . We arrive at our result; let D n be the dihedral group of order 2n, the symmetry group of a regular n-gon. Corollary 4.3. If the graph C n (1, k), where 1 < k < n/2, satisfies neither k 2 ≡ ±1 (mod n) nor k = n/2 − 1, then Aut(C n (1, k)) ∼ = D n . Proof. The dihedral group D n is always isomorphic to the subgroup of Aut(C n (1, k)) consisting of automorphisms that carry E 1 to E 1 and E k to E k . These symmetries act transitively on E 1 and E k , so if Aut(C n (1, k)) were to contain any automorphism carrying an edge from E 1 to E k , or vice versa, then the compositions of this automorphism with suitable dihedral symmetries would yield automorphisms causing Aut(C n (1, k)) to be edge-transitive and hence implying that C n (1, k) satisfies conclusion (1) or (2) in Theorem 4.1, a contradiction to our hypothesis. We conclude this section by describing the automorphism groups of the edge-transitive graphs in (1) in Theorem 4.1, those graphs C n (1, k) for which k 2 ≡ ±1 (mod n). These groups are more elaborate than dihedral groups, but not by much. x − y is congruent modulo n to either ±1 or ±k. Now the difference in the indices of φ st (v x ) and φ st (v y ) is s + (t − s)x − s − (t − s)y = (t − s)(x − y). (4.1) Since v s v t is an edge in C n (1, k), we know that t − s ∈ {±1, ±k}. Recalling that k 2 ≡ ±1 (mod n), it is straightforward to check that x − y ∈ {±1, ±k} if and only if (t − s)(x − y) ∈ {±1, ±k}, so φ st is an element of Aut(C n (1, k)). To finish our proof, we show the uniqueness of the automorphism respectively mapping v 0 and v 1 to v s and v t . Let ρ be any automorphism of C n (1, k) mapping vertex v 0 to v s and v 1 to v t . Taking i ∈ {1, k} to be the index such that v s v t ∈ E i , Theorem 4.2 implies that ρ carries all edges in E 1 to edges in E i , so since v 2 is adjacent to v 1 along an edge from E 1 , its image ρ(v 2 ) is adjacent to v t along an edge from E i . Since ρ(v 2 ) = ρ(v 0 ), and v t only has two neighbors along edges from E i , ρ(v 2 ) is uniquely determined; we have ρ(v 2 ) = φ st (v 2 ) . Continuing inductively through all the vertices v 2 , . . . , v n−1 , we conclude that ρ = φ st , as desired. 5 The graphs C n (1, n/2 − 1) In this section we give an optimal distinguishing proper coloring of the edge-transitive graphs described in (2) in Theorem 4.1. In these graphs n is even, and each vertex v i is adjacent to the same neighbors to which its follows that C n (1, n/2 − 1) is isomorphic to the wreath graph W (n/2, 2); in general, the wreath graph W (a, b) has ab vertices, partitioned into a independent sets I 1 , . . . , I a , each of size b, with the vertices in each set I i being adjacent to all vertices in I i−1 and I i+1 (with operations in subscripts performed modulo a). The graph C 12 (1, 5) and its interpretation as W (6, 2) are illustrated in Figure 3. antipodal vertex v i+n/2 is. It v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 The automorphism group of C n (1, k), when k = n/2 − 1, contains "dihedral" symmetries interpreted as acting on the graph when drawn as on the left in Figure 3. In addition, however, there are n/2 symmetries that interchange two vertices having the same neighborhood and fix all other vertices. (Vertices with the same neighborhood are called twins.) Hence Aut(C n (1, k)) has many more than 2n automorphisms. Intuitively, this imposes more restrictions on distinguishing colorings (in particular, vertices having the same neighborhoods must receive distinct colors). Hence we may expect to need more colors than we do with C n to break all symmetries of C n (1, k) with a proper coloring, and indeed this is the case. Theorem 5.1. For even integers n ≥ 8, χ D (C n (1, n/2 − 1)) =            6 if n = 6; 8 if n = 8; 5 if n ≥ 10. Proof. It is easy to see that C 6 (1, 2) is isomorphic to K 2,2,2 and that C 8 (1, 3) is isomorphic to K 4,4 , which are both complete multipartite graphs. This implies (see Section 1) that χ D (C 6 (1, 2)) = 6 and χ D (C 8 (1, 3)) = 8. Suppose henceforth that n ≥ 10. As mentioned above, a distinguishing coloring must assign different colors to twins, and for n ≥ 10 the vertex set of C n (1, n/2 − 1) is partitioned into n/2 such twin pairs. Since our coloring is to be proper, it cannot use the same color on two vertices from "consecutive" pairs {v i , v i+n/2 } and {v i+1 , v i+1+n/2 }. Such a coloring must then use at least 4 colors, but 4 colors are not enough for a proper coloring if n/2 is odd, and if n/2 is even, the 4-coloring of C n (1, n/2 − 1) (which is unique up to permutations of the colors) admits the color-preserving automorphism defined by v i → v i+2 for all i. Hence at least 5 colors are necessary for a distinguishing coloring. Note now that the pairs of colors assigned to pairs of twins naturally correspond to vertices in the Kneser graph KG c,2 , where c is the number of colors used in the coloring. (Recall that the Kneser graph KG p,q is the graph whose vertices are the q-element subsets of a set of p elements, with edges joining vertices corresponding to disjoint subsets.) Moving from one pair of colored twins to the consecutively following pair and noting the colors used corresponds to moving along edges in KG c,2 , and the overall coloring of C n (1, n/2−1) corresponds to a closed walk in KG c,2 . For a proper coloring, any such closed walk will do, and a useful walk in KG 5,2 (i.e., the Petersen here the underlined pairs are repeated as necessary to produce n/2 pairs of colors. Note that in each corresponding coloring of C n (1, n/2 − 1), there is at most one pair of twins receiving a color pairs from {12, 34, 15, 25, 14}. Any automorphism maps a pair of twins to a pair of twins, and each vertex in C n (1, n/2 − 1) belongs to a unique pair of twins when n/2 ≥ 5. Color-preserving automorphisms likewise preserve the colors on pairs of twins, so the vertices in the twin pairs whose colors were just listed must be fixed any such automorphism. By inductively moving to neighboring twin pairs along the wreath, we see that every other vertex must be fixed, so the only color-preserving automorphism is the identity, as desired. A general upper bound Having determined the symmetries of C n (1, k) when k = n/2 − 1 in Section 4, for the remaining sections we turn to distinguishing colorings. In this section we show that in many cases, the distinguishing chromatic number of C n (1, k) is at most 1 more than the ordinary chromatic number. This will allow us to exactly determine the distinguishing chromatic number whenever C n (1, k) is bipartite. Our first result is useful in "breaking" symmetries in edge-transitive graphs C n (1, k). Lemma 6.1. Suppose that b is an integer such that 1 < b < n/2 and b = k and gcd(b, n) = 1. If C n (1, k) is properly colored with χ(C n (1, k)) colors, and the colors on v 0 and v b are changed to be a new previously unused color, then this coloring is not preserved by any automorphism of C n (1, k) that carries E 1 to E k . Proof. Suppose that c : V (C n (1, k)) → {1, . . . , χ(C n (1, k)) + 1} is the modified coloring, and let φ be any automorphism of C n (1, k) that exchanges E 1 and E k . Note that there are two internally vertex-disjoint paths from v 0 to v b along edges of E 1 , and similarly two such paths from v 0 to v b along edges of E k . Since φ exchanges E 1 and E k , if φ were to preserve the coloring, then v 0 , v b would either be fixed or mapped to each other under φ, and the paths between them along edges of E k would be carried to the paths between v 0 , v b using edges of E 1 . Since these paths along edges in E 1 have lengths b and n − b, the paths along edges in E k must have the same lengths. Hence either bk ≡ b (mod n) or bk ≡ n − b (mod n). Since gcd(b, k) = 1, b has a multiplicative inverse modulo n, and these congruences yield k ≡ 1 (mod n) or k ≡ n − 1 (mod n); both statements are contradictions. Thus any color-preserving automorphism of the modified coloring in Lemma 6.1 must act as a dihedral symmetry on the edges of E 1 . Since b < n/2, the only such automorphisms that either fix v 0 and v b or interchange them are the identity automorphism and a single reflection. This will yield a general bound; first we show that such an integer b as in the hypothesis of Lemma 6.1 always exists for large enough n. Lemma 6.2. For any n ≥ 13 and for any integer k ∈ {2, . . . , n/2 }, there exists an integer b such that 1 < b < n/2 and b = k and gcd(b, n) = 1. Proof. We will show the stronger statement that when n ≥ 13, there exist two primes not dividing n in {2, . . . , n/2 }. If k were to equal one of these primes, then we could let b be the other one. It is easy to verify directly that the two primes specified exist for all n satsifying 13 ≤ n ≤ 22. Now suppose that n ≥ 23. If n is not relatively prime to two elements of {2, 3, 5, 7, 11}, then n ≥ 2 · 3 · 5 · 7 = 210. Recall now the result known as Bertrand's Postulate, which states, in one formulation, that whenever m is an integer greater than 3, then there exists a prime number p with m < p < 2m. It follows that there exist prime numbers p 1 , p 2 , p 3 such that n/16 < p 1 < n/8 and n/8 < p 2 < n/4 and n/4 < p 3 < n/2. If n is not relatively prime to at least two of p 1 , p 2 , p 3 , then n ≥ p 1 p 2 > n 2 /128 and hence n < 128, a contradiction to our earlier bound on n. Theorem 6.3. Given integers k, n such that 1 < k < n/2 and a proper coloring coloring c : V (C n (1, k)) → {1, . . . , χ(C n (1, k))}, let b be an integer such that 1 < b < n/2 and b = k and gcd(n, b) = 1. If either c(v 1 ), c(v 2 ), . . . , c(v b−1 ) or c(v b+1 ), c(v b+2 ), . . . , c(v n−1 ) is not a palindrome, then χ D (C n (1, k)) ≤ χ(C n (1, k)) + 1. Proof. In light of Lemma 6.1 and the discussion following it, since one of c(v 1 ), c(v 2 ), . . . , c(v b−1 ) and c(v b+1 ), c(v b+2 ), . . . , c(v n−1 ) is not a palindrome, recoloring v 0 and v b with a single new color yields a proper coloring where no reflection preserves the coloring; the only color-preserving automorphism of C n (1, k) is the identity. Hence χ D (C n (1, k)) ≤ χ(C n (1, k)) + 1. In certain cases, Theorem 6.3 quickly yields an optimal distinguishing coloring. Theorem 6.4. Given positive integers k, n such that 1 < k < n/2 − 1, if n is even and k is odd and (n, k) = (10, 3), then χ D (C n (1, k)) = 3. Proof. Note that C n (1, k) is bipartite, though a proper 2-coloring of C n (1, k) admits a nontrivial color-preserving rotation, so χ D (C n (1, k)) ≥ 3. To prove the corresponding upper bound, observe first that no value of k satisfies the hypotheses for any even n less than 10. When n ∈ {10, 12}, only k = 3 satisfies the given inequalities, though we are given that (n, k) = (10, 3). Assume that either (n, k) = (12, 3) or n ≥ 13. Using b = 5 in the first case and Lemma 6.2 in the latter, there is an integer b such that 1 < b < n/2 and b = k and gcd(b, n) = 1. Let c be a proper 2-coloring of C n (1, k); here the vertices v i with even subscripts recieve one color, and the vertices with odd subscripts receive the other color. Since n is even, b must be odd, and hence c(v 1 ), · · · , c(v b−1 ) is not a palindrome, since c(v 1 ) = c(v b−1 ) . By Theorem 6.3, there is an optimal distinguishing coloring of C n (1, k) using 3 colors. Given the exceptionality of the case (n, k) = (10, 3) in Theorems 4.2 and 6.4, we determine χ D (C 10 (1, 3)) next. Here the distinguishing chromatic number is quite a bit higher than the chromatic number. Proposition 6.5. χ D (C 10 (1, 3)) = 5. Proof. The graph C 10 (1, 3) is bipartite and may be obtained by deleting the edges v i v i+5 from the complete biparite graph having partite sets A = {v 0 , v 2 , v 4 , v 6 , v 8 } and B = {v 1 , v 3 , v 5 , v 7 , v 9 }. Note that if some proper coloring of the vertices assigns the same color to both v i , v j in A and the same color (which must different from the first) to v i+5 , v j+5 in B, then the involution of V (C 10 (1, 3)) written in cycle notation as (v i v j )(v i+5 v j+5 ) is a color-preserving automorphism of the graph, so such a coloring is not distinguishing. Let c : V (C 10 (1, 3)) → {1, . . . , } be a distinguishing proper coloring, and suppose by way of contradiction that < 5. By a pigeonhole principle, some color must appear on at least three vertices of the graph. Since c is a proper coloring, these three vertices must appear in the same partite set; assume that it is A. By the symmetries in C 10 (1, 3), we may assume that these vertices are v 0 , v 2 , v 4 , and the color assigned is 1. As explained above, the colors on v 5 , v 7 , v 9 must then be distinct elements of {2, . . . , }, which forces = 4. Since c is a proper coloring, v 6 and v 8 must also receive color 1, but then when we consider v 1 , v 3 , we see that the pigeonhole principle forces some color from {2, . . . , } to appear at least twice on vertices in B, and as above we find a color-preserving involution of the vertices of C 10 (1, 3), a contradiction. Thus a distinguishing proper coloring of C 10 (1, 3) requires at least 5 colors, and one can verify that the following map c : {v 0 , . . . , v 9 } → {1, 2, 3, 4, 5} provides one. A: c(v 0 ) = 1, c(v 2 ) = 2, c(v 4 ) = 2, c(v 6 ) = 3, c(v 8 ) = 3, B: c(v 5 ) = 1, c(v 7 ) = 4, c(v 9 ) = 5, c(v 1 ) = 4, c(v 3 ) = 5. Dihedral symmetries In this section we restrict our attention to graphs C n (1, k) for which Aut(C n (1, k)) is the dihedral group D n . We will show that often the general bound in the conclusion of Theorem 6.3 is not optimal, since we may find a distinguishing proper coloring using χ(C n (1, k)) colors. When Aut(C n (1, k)) ∼ = D n , every automorphism of C n (1, k) permutes the edges in E 1 , to use the notation from Section 4, and likewise permutes the edges of E k . We may also use more intuitive language, imagining that C n (1, k) is drawn with its vertices placed, in order of their subscripts, at the vertices of a regular n-gon, with the edges of E 1 drawn as the sides and the edges of E k drawn as diagonals of this polygon. This allows us to freely speak of the elements of Aut(C n (1, k) as rotations and reflections and to determine the forms of colorings that are preserved under these automorphisms. We do this in Section 7.1 below before proceeding in later subsections by the values or parities of n and k (recalling that the case where n is even and k is odd was concluded in Section 6). Colorings preserved by rotations and reflections We consider first reflections. Lemma 7.1. If a reflection symmetry in Aut(C n (1, k)) is color-preserving for a given proper coloring of C n (1, k), then n is even and k is odd. Proof. Picture a drawing of C n (1, k) as a regular polygon with chords, with the polygon vertices drawn a circle. Every reflection symmetry in Aut(C n (1, k)) has a corresponding axis of reflection that passes through the center of the circle. If the axis of reflection passes through the midpoint of a 1-edge, then endpoints of that edge have different colors (since C n (1, k) is properly colored), and the reflection is not color-preserving. Hence the only possible color-preserving reflection symmetry is one where n is even and the symmetry fixes two "opposite" vertices v a and v a+n/2 . Here k must be odd, since otherwise the vertices v a−k/2 and v a+k/2 would have the same color (by the symmetry) but be adjacent, a contradiction. Since the case when n is even and k is odd was handled in Section 6, we note that for the rest of Section 7, we may ignore reflections when checking for color-preserving symmetries. The next result deals with rotations in Aut(C n (1, k)). Lemma 7.2. If a rotation symmetry in Aut(C n (1, k)) is color-preserving for a given proper coloring c of C n (1, k), then the sequence c(v 0 ), . . . , v(v n−1 ) is periodic with a period that is a proper divisor of n. Proof. Suppose that c : V (G) → {1, . . . , χ(C n (1, k))} is a proper coloring, and that ρ is a a non-identity rotation in Aut(C n (1, k)) that preserves the coloring c. If ρ(v 0 ) = v r , then clearly c(v i ) = c(v i+tr ) for all t. This shows that c(v 0 ), . . . , c(v n−1 ) is periodic. In fact, an elementary result from number theory implies that c(v 0 ) appears on all vertices v j where j is a multiple of the greatest common divisor of r and n. Let d = gcd(n, r). Now by symmetry c(v i ) = c(v i+td ) for all i and t, showing that the period of c(v 0 ), . . . , c(v n−1 ) divides d and hence n. Since d < n, the period is a proper divisor of n. In light of Lemma 7.2, for the rest of Section 7, in verifying that a coloring of C n (1, k) is preserved by no non-identity symmetry, we need only check that the coloring is not preserved by any rotation v i → v i+d where d is a proper divisor of n. This allows us a quick result. Corollary 7.3. If n is an odd prime and Aut(C n (1, k)) ∼ = D n , then χ D (C n (1, k)) = χ(C n (1, k)). Proof. By Lemmas 7.1 and 7.2, any proper coloring of C n (1, k) is preserved only by the identity in Aut(C n (1, k)). 7.2 Case: k = 2 or k = (n − 1)/2 Theorem 2.4 shows that χ(C n (1, 2)) and χ(C n (1, (n − 1)/2) are 4 except when n = 5 or when n is a multiple of 3 (in the latter case, the chromatic number is 3). We are able to give optimal distinguishing proper colorings of these graphs. Note first of all that by Corollary 2.3, C n (1, (n − 1)/2) is isomorphic to C n (1, 2), so it suffices to restrict our attention to C n (1, 2). We may also assume that n ≥ 7, since distinguishing colorings have already been described in earlier sections for the cases 3 ≤ n ≤ 6. 7.3 Case: n is even and k is even. Before presenting our result when n is even and k is even, we establish some conventions that will also be used in later sections. Taking n and k to be fixed, we first define q and r to be the unique integers such that n = qk + r, where 0 ≤ r < k. Our distinguishing colorings will often be constructed with blocks of colors, that is, sequences of colors to be assigned to vertices v i with consecutive indices. We may also use block to refer to the vertices being assigned that sequence of colors. For example, to color the vertices of C n (1, k) with the block B = (1, 2, 3) means to alternately color consecutive vertices with 1, 2, and 3, and we may also refer to subsets of three consecutive vertices colored 1, 2, 3 (in that order) as blocks. Our next result gives a more sophisticated example of coloring with blocks. Theorem 7.5. Given integers k, n such that 2 < k < n/2 − 1, n is even and k is even and Aut(C n (1, k)) ∼ = D n , then χ D (C n (1, k)) = 3. Proof. We give a proper 3-coloring of the vertices of C n (1, k) as follows. Consider the following blocks of k terms, where each color is drawn from {1, 2, 3}. Here the bounds on k and the assumption that k is even ensure a consistent definition. For convenience hereafter we represent blocks (and later, portions of blocks) by enclosing them in rectangles. B 1 = (1, 2, 3, 2, 3, · · · , 2, 3, 1); B 2 = (2, 3, 1, 3, 1, · · · , 3, 1, 2); B 3 = (3, 1, 2, 1, 2, · · · , 1, 2, 3). (which is shaded) are aligned with colors from the preceding block B 3 and the following block B 1 . r (even) k−r (even) B 1 : 1 2 3 2 3 · · · 2 3 2 3 2 · · · 2 3 1 B 2 : 2 3 1 3 1 · · · 3 1 3 1 3 · · · 3 1 2 B 1 : 1 2 3 2 3 · · · 2 3 2 3 2 · · · 2 3 1 B 2 : 2 3 1 3 1 · · · 3 1 3 1 3 · · · 3 1 2 . . . . . . . . . B 3 : 3 1 2 1 2 · · · 1 2 1 2 1 · · · 1 2 3 B , then B 1 : 2 3 1 3 1 · · · 3 1 2 1 2 · · · 2 3 2 B 1 concluded: 3 2 3 2 3 · · · 2 3 1 To see that this is a proper coloring, note that vertices with consecutive indices i, i + 1 where 0 ≤ i < n receive distinct colors by the patterns within the blocks and at their ends. We will also see that each vertex v i is colored differently than v i−k and v i+k . This is apparent from the blocks displayed above if v i belongs to a block of vertices colored with one of B 1 , B 2 , B 3 and its neighbor v i−k or v i+k does as well, with B not appearing between the two blocks. To finish the argument, we assume that r > 0 and consider the vertices v i for i ∈ {(q − 1)k + 1, . . . , n}, showing that none receives the same color as v i+k . These vertices are assigned colors using the blocks B 3 and B . In the figure above, these colors appear in the shaded rectangle and on the previous row. Recalling that B is constructed as a shortened or truncated version of B 2 , as we compare the first r entries of B 3 with those of B , we see that the colors on vertices v i , v i+k must differ; this is apparent for colors at the beginning or middle of the blocks, and we use the fact that r is even, so the final entry of B (which is 3 or 2) is sure to align with an entry of 1 in When r > 2, we shall show that if we switch v 0 color to 3, we obtain a proper distinguishing coloring. As presently constructed, when r > 2, v 0 is a vertex colored 2 having each of its neighbors colored 1. Thus, there is no trouble switching its color to 3. Let's go ahead and switch the color of v 0 to 3. This moves keeps the coloring proper while making v 0 the only vertex colored 3 having each of its neighbors colored 1. Therefore, any automorphism that preserves the coloring must fix that vertex. By Lemmma 7.1. This leaves only the trivial automorphism, and the coloring is distinguished. When r = 2, we will show that the vertices v −2 , v −1 , v 0 must be fixed by color-preserving symmetries. Hence, the coloring will be distinguishing, since only the identity rotation preserves it. We depict the coloring in this case as we did before, with k numbers in each row and the block B appearing as shaded. Case 2: n = 2k + r so q = 2 Now suppose n = 2k + r, where 1 < r < k. Since n is even, then r is also even. Consider the blocks of colors E 1 , E 2 of length k and M of length r. E 1 = (1, 2, 1, 2, · · · , 1, 2, 1, 2) E 2 = (3, 1, 3, 1, · · · , 3, 1, 3, 1) M = (2, 3, 2, 3, · · · , 2, 3, 2, 3) Color the sets of vertices {v 1 , v 2 , · · · , v k } and {v k+1 , v k+2 , · · · , v 2k } using the blocks of colors E 1 and E 2 , respectively. Then use M to color the remaining r vertices. Below is the coloring C of C 2k+r (1, k) with a detailed representation of these blocks of colors. The light-shaded blocks in rows 3 and 4 correspond exactly to the first k vertices in row 1. r k−r         1 2 · · · 1 2 1 2 · · · 1 2 3 1 · · · 3 1 3 1 · · · 3 1 2 3 · · · 2 3 1 2 · · · 1 2 1 2 · · · 1 2         Since r and k are both even, we can easily see from the detailed representation above that C is a proper coloring. By Case: n is odd and k is odd We shall treat the cases where 2k + 1 < n < 3k and n ≥ 3k separately. In either case, we propose a proper 3-coloring that is distinguishing. To do so, we conveniently label the vertices of G consecutively as v 1 , · · · , v n , though we shall realize later that the coloring works independently of the vertex label. Note that since both k, n are odd, v −k = v (q−1)k+r is labeled an even number. Theorem 7.6. Given integers k, n such that 2 < k < (n − 1)/2, if both n and k are odd and Aut(C n (1, k)) ∼ = D n , then χ D (C n (1, k)) = 3. Proof. As indicated, we split it into three cases: Case 1: n ≥ 3k Let q, r be integers such that 0 ≤ r < k and n = qk + r. Consider the following 3-proper coloring C of the vertices: assign color 1 to all odd-indexed vertices v i ∈ {v 1 , v 2 , · · · , v −k }. Next, assign color 3 to even-indexed vertices v i ∈ {v k+1 , · · · , v n }, not including v n since n is considered odd. Lastly, assign color 2 to all other vertices in {v 1 , v 2 , · · · , v k } ∪ {v −k+1 , v −k+2 , · · · , v n }. In other words, for even i such that 1 < i < k, the vertex v i is colored 2 and for odd i such that (q − 1)k < i ≤ n (including n), we have v i colored 2. We claim that the coloring is proper. Moreover, it is distinguishing. We proceed to prove that C is proper by showing that the sets made up of vertices with the same color are all independent sets. Consider the set of all 1-colored vertices and denote it V 1 . Thus, V 1 = {v i \|i is odd and 1 ≤ i ≤ (q − 1)k + r} by construction. We show that V 1 is an independent set. Take v i ∈ V 1 . Thus, v i−1 and v i+1 , v i+k (since k is odd) are even-labeled vertices and do not belong to V 1 . Moreover, v i−k is an even-labeled vertex unless 1 ≤ i ≤ k, in which case v i−k ∈ {v −k+1 , v −k+2 , . . . , v n }. Therefore, V 1 is indeed an independent set. A similar argument can be made for the set V 3 of the vertices colored 3. This time, we have V 3 = {v i \|i is even and k + 1 ≤ i ≤ n} and v i−1 , v i+1 , v i−k are odd-labeled vertices. Moreover, v i+k is an odd-labeled vertex unless (q − 1)k + r + 1 ≤ i ≤ n, in which case v 1 ∈ {v 1 , v 2 , · · · , v k }. Lastly, we show that the set V 2 of vertices colored 2 is also an independent set. By construction, we have V 2 = {v i \|i is even and 1 ≤ i ≤ k} ∪ {v i \|i is odd and (q − 1)k + r + 1 ≤ i ≤ n} Here is what C looks like when restricted to {v −k+1 , v −k+2 , · · · , v −1 , v n , v 1 , v 2 , · · · , v k } : v 1 v n v −1 v −2 v 2 v 3 v 4 v k v k−1 1 1 1 2 2 2 2 3 2 C v −k+1 v −k+2 2 3 As we can see from the above illustration, the coloring C restricted to {v −k+1 , v −k+2 , · · · , v −1 , v n , v 1 , v 2 , · · · , v k } is a sequential-vertex coloring of the vertex list v −k+1 , v −k+2 , · · · , v −1 , v n , v 1 , v 2 , · · · , v k , which starts with the alternation of colors 2 and 3 on the first k vertices to end up with color 3 being replaced with color 1 on the last k vertices. Thus, given any pair of vertices v i , v j coloured 2, we have ±(i − j) = 2 , where ∈ Z. Therefore, the set of vertices colored 2 form an independent set. In summary, C is a proper coloring. It remains to show that it is distinguishing. By lemma 7.1, it suffices to show that the only color-preserving rotation symmetry is the identity. Suppose there is a non-identity rotation symmetry. Thus, by Lemma 7.2, there exists a proper divisor d of n such that ρ(v n ) = v d . This means that C(v n ) = C(v d ). Since d is a proper divisor of n, thus d must be a number between 1 and k by construction. In particular, d must also be a multiple of 2 since for any pair of vertices v i , v j coloured with 2, we have ±(i − j) = 2 , where ∈ Z. The fact that d is a proper divisor of odd n and also a multiple of 2 yields a contradiction. Hence, C is indeed distinguishing. Case 2: If n = 2k + r, where 1 < r < k and 2r ≤ k − 1 Note that in this case, r is odd and we consider integers and p such that k = pr + and 0 < < r. Consider the following arrangement of colors where the first two rows are made up of p blocks of length r with a possible shorter block of length (if > 0) added to the end. The third row consists of just one block of length r. Furthermore, the first row is used to color the sequence of vertices v 1 , v 2 , · · · , v k , second row to color vertices v k+1 , v k+2 , · · · , v 2k , and third row to color the remaining vertices. r r r r      R 1 3 2 · · · 3 2 1 3 2 · · · 3 2 1 · · · 3 2 · · · 3 2 1 L 1 R 2 2 1 · · · 2 1 3 2 1 · · · 2 1 3 · · · 2 1 · · · 2 1 3 L 2 R 3      where the blocks R 1 , R 2 , R 3 of length r and blocks L 1 , L 2 of length vary with respect to . In particular, we have = 0 = 1 > 1 and odd > 1 and even R 1 1 3 · · · 1 3 1 3 2 · · · 3 2 1 3 2 · · · 3 2 1 1 3 · · · 1 3 1 R 2 2 1 3 · · · 1 3 2 1 · · · 2 1 3 2 1· · · 2 1 3 3 2· · · 3 2 3 R 3 1 3 2 · · · 3 2 1 2 · · · 1 2 1 3 2 · · · 3 2 1 2 1 · · · 2 1 2 L 1 N/A 3 2 1 · · · 2 1 3 3 1 · · · 3 1 L 2 N/A 2 1 3 · · · 1 3 1 2 3 · · · 2 3 We proceed to show that the coloring is proper and distinguishing on a case-by-case basis. We start with the case where = 0. Subcase 1: Let = 0 . Thus, k = pr. Let L 1 and L 2 be empty and the other blocks R 1 , R 2 , and R 3 as given below. Note that the added light-shaded blocks in rows 3 and 4 represent row 1 re-positioned so that the first k − r vertices (positions) in row 1 are vertically aligned with the last k − r vertices in row 2 and the last r vertices (positions) in row 1 are vertically aligned with the r vertices in row 3. r r r r         1 3 1 · · · 1 3 1 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 2 1 3 · · · 3 1 3 2 1 2 · · · 2 1 3 2 1 2 · · · 2 1 3 · · · 2 1 2 · · · 2 1 3 1 3 2 · · · 2 3 2 1 3 1 · · · 1 3 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1         We can easily see from the above detailed arrangement of colors that the coloring is proper. By Lemma 7.1, it suffices to show that there is not a nontrivial rotation symmetry that preserves the coloring. To this end, note that the vertex v k+1 is the only vertex labeled 2 whose neighbors are all labeled 1. Hence, the coloring is distinguishing with respect to the dihedral group. Subcase 2: Let = 1 (thus, p is even). Let the blocks R 1 , R 2 , R 3 of length r and blocks L 1 , L 2 of length 1 be as given below. r r r r         3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 2 1 2 · · · 2 1 3 2 1 2 · · · 2 1 3 2 1 2 · · · 2 1 3 · · · 2 1 2 · · · 2 1 3 2 1 2 1 · · · 1 2 1 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 2 3 2 · · · 2 1 \| 3         Here, v −1 is the only vertex labeled 2 whose neighbors are all labeled 1. Similarly, the coloring is proper and distinguishing. Subcase 3: Let > 1 and odd (thus, p is even). Let the blocks R 1 , R 2 , R 3 and blocks L 1 , L 2 be as given below. r r r r              3 2 · · · 3 2 3 2 · · · 3 2 1 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 2 1 2 · · · 2 1 3 2 1 · · · 2 1 2 1 · · · 2 1 3 2 1 2 · · · 2 1 3 2 1 2 · · · 2 1 3 · · · 2 1 2 · · · 2 1 3 1 3 1 · · · 1 3 1 3 2 · · · 3 2 3 2 · · · 3 2 1 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 3 2 3 · · · 2 1 \| 2 1 · · · 2 1 3 −r (even) (odd)              When restricted to the sequence of vertices v 2k− −2 , v 2k− −1 , · · · , v 2k , the above proper coloring alternates labels 1 and 3. Since > 1, the length of this sequence is at least five, which makes it the longest of its kind. As a result, no nontrivial rotation preserves the coloring. Therefore, by Lemma 7.1, the coloring is distinguishing with respect to the dihedral group. Subcase 4: Let > 1 and even (thus p is odd). Let the blocks R 1 , R 2 , R 3 and blocks L 1 , L 2 be as given below. r r r r              1 3 · · · 1 3 1 3 · · · 1 3 1 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 1 3 · · · 1 3 1 3 2 · · · 3 2 3 2 · · · 3 2 3 2 1 2 · · · 2 1 3 2 1 2 · · · 2 1 3 · · · 2 1 2 · · · 2 1 3 2 3 2 · · · 3 2 3 2 1 · · · 2 1 2 1 · · · 2 1 2 1 3 1 · · · 1 3 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 2 3 · · · 2 3 2 3 2 · · · 3 2 1 \| 3 · · · 1 3 1 −r (odd) (even)              By construction, the coloring is proper. Moreover, v r is the only vertex labeled 1 that has all its neighbors labeled 3. Also, v k+1 is the only vertex labeled 3 with exactly two neighbors labeled 2. As a result, no nontrivial rotation preserves the coloring. Therefore, by Lemma 7.1, the coloring is distinguishing with respect to the dihedral group. Case 3: If n = 2k + r, where 1 < r < k and 2r ≥ k + 1 Subcase 1: Let 2r − k < k − r with 2r − k > 1. Consider the following arrangements of sequences of colors below, where the first two rows represents sequences of length         2 1 2 1 2 · · · 2 1 2 1 2 1 2 · · · 2 1 3 1 3 1 3 · · · 3 1 3 2 1 2 1 · · · 1 2 1 2 1 2 1 · · · 1 2 1 3 2 1 2 1 · · · 1 2 1 2 1 2 1 · · · 1 2 1 3 1 3 1 · · · 1 3 1 3 2 1 2 · · · 2 1 2 1 2 1 2 · · · 2 1 2 1 3 2 3 2 · · · 2 3 2 3 2 3 2 · · · 2 3 2 1 3 1 3 · · · 3 1 3 2 1 2 1 · · · 1 2 1 2 1 2 1 · · · 1 2 1 3 1 3 1 3 · · · 3 1 3 2 1 2 1 · · · 1 2 1 2 1 2 1 · · · 1 2 1         k−r (even) Use row 1 to color the sequence of vertices v 1 , v 2 , · · · , v k , row 2 to color the sequence of vertices v k+1 , v k+2 , · · · , v 2k , and the first three blocks in row 3 to color the remaining consecutively-indexed vertices. It is not hard to see from the above detailed arrangement of colors that the coloring is proper. By Lemma 7.1, it suffices to show that there is not a nontrivial rotation symmetry that preserves the coloring. Note that the highlighted vertices above (from v 2k+2 to v −2r+k+1 ) is a unique string of vertices labeled alternately 2 and 3. The longest such string outside of these vertices is only 2 vertices long, and therefore this string is unique as 2r − k > 1. Therefore, no rotational symmetry will fix the coloring and thus this coloring is distinguishing. Subcase 2: Let 2r − k < k − r with 2r − k = 1 and 2k − 3r > 1. In this case, the coloring must be altered slightly because of the construction of the blocks in the previous subcase. Consider the coloring below in which the colors of the last two vertices are changed from the last subcase. k−r (even) 2r−k 2k−3r (odd) 2r−k k−r (even) 2k−3r (odd) 2r−k         2 1 2 1 2 · · · 2 1 3 2 1 2 1 · · · 1 2 1 3 2 1 2 1 · · · 1 2 1 3 2 1 2 · · · 2 1 2 1 3 2 3 2 · · · 2 1 3 2 1 2 1 · · · 1 2 1 3 2 1 2 1 · · · 1 2 1         k−r (even) The coloring given above is proper and note that the highlighted vertex, v k+1 , is the only vertex labeled 3 with two of its neighbors labeled 2 and two labeled 1. Because this vertex is unique, no rotational symmetry can fix the coloring and thus the coloring is distinguished. Once again, consider the following arrangement of sequences of colors. k−r (even) 2r−k (odd) k−r (even)              2 1 2 1 2 · · · 1 2 1 3 2 · · · 3 2 3 · · · 2 3 2 1 2 1 2 · · · 1 2 1 3 2 1 2 1 · · · 2 1 2 1 3 · · · 1 3 1 · · · 3 1 3 2 1 2 1 · · · 2 1 2 1 3 2 3 2 · · · 3 2 3 2 1 · · · 2 1 2 · · · 1 3 2 1 2 1 2 · · · 1 2 1 3 2 3 2 3 · · · 2 3 2 3 2 · · · 3 \| 2 1 · · · 2 1 3r−2k (odd) k−r (even)              Proceed similarly to color the vertices and observe that the coloring is proper. Moreover, the vertex v k+1 is the only vertex labeled 3 with exactly two of its neighbors labeled 2 and the others labeled 1. Hence, the coloring is distinguishing with respect to the dihedral group. Case: n is odd and k is even We will proceed by considering the cases where k ≥ n/3 and n/3 < k < n/2 − 1. Furthermore, we will need to consider within the case that n/3 < k < n/2 − 1, when r ≤ k/2 and r > k/2. Theorem 7.7. Given integers k, n such that 2 < k < (n − 1)/2, if n is odd and k is even and Aut(C n (1, k)) ∼ = D n , then χ D (C n (1, k)) = 3. Proof. Case 1: Let n/3 < k < n/2 − 1 such that n = 2k + r hence r odd and r ≤ k/2. Let r ≤ k/2 so that k = mr + for some integers < r and m ≥ 2. Consider the following arrangement of colors where the first two rows are made up of blocks of length r with a possible shorter block of length (if > 0) added to the end. The third row consists of just one block of length r. Note that in the fourth row, the block is comprised of a portion of block B of length r − (we will call B ) and E 1 of length . r r r         A B · · · B E 1 C D · · · D E 2 E A · · · B B B' \| E 1         Subcase 1: Let = 0 . Let E 1 and E 2 be empty and the other blocks as given below. r r r         1 3 1 · · · 1 3 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 2 1 3 · · · 3 1 3 2 1 2 · · · 2 1 3 · · · 2 1 2 · · · 2 1 3 1 3 2 · · · 2 3 2 1 3 1 · · · 1 3 1 · · · 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1         The vertex v k+1 highlighted above is the only vertex labeled 2 whose neighbors are all labeled 1. Because this vertex is unique, the coloring is distinguished by dihedral symmetries. Subcase 2: Let = 1 . Let the blocks be given as below with E 1 and E 2 of length 1. r r r         3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 2 1 2 · · · 2 1 3 2 1 2 · · · 2 1 3 · · · 2 1 2 · · · 2 1 3 2 1 2 1 · · · 1 2 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 2 3 2 · · · 2 1 \| 3         The vertex v −1 highlighted above is the only vertex labeled 2 whose neighbors are all labeled 1. Because this vertex is unique, the coloring is distinguished by dihedral symmetries. Subcase 3: Let > 1 and m odd . Use the same arrangement as in Subcase 2 (with = 1) with extended E 1 and E 2 blocks.              3 2 3 · · · 2 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 1 3 · · · 3 1 3 2 1 2 · · · 1 2 1 2 · · · 2 1 3 2 1 2 · · · 2 1 3 · · · 2 1 2 · · · 2 1 3 2 3 2 · · · 2 3 2 1 2 1 · · · 2 1 2 1 · · · 1 2 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 2 3 · · · 3 2 3 2 3 2 · · · 3 2 1 \| 3 · · · 3 1 3 −r (even) (odd)              The vertices v 2k+1 to v n highlighted above represent a unique labeled string of vertices (alternating labels 1 and 2) of length r as every other block contains a 3 and is immediately proceeded or followed by a vertex labeled 3. Because this string of vertices is unique, the coloring is distinguished by dihedral symmetries. Subcase 4: Let > 1 and m even and hence even . Use the blocks as given below. r r r <r (even)              1 3 1 · · · 3 1 3 1 · · · 1 3 1 3 2 3 · · · 3 2 1 · · · 3 2 3 · · · 3 2 1 3 1 3 · · · 1 3 1 3 2 3 · · · 2 3 2 3 · · · 3 2 3 2 1 2 · · · 2 1 3 · · · 2 1 2 · · · 2 1 3 2 3 2 · · · 3 2 3 2 1 2 · · · 1 2 1 2 · · · 2 1 2 1 3 1 · · · 1 3 1 · · · 3 2 3 · · · 3 2 1 3 2 3 · · · 2 3 2 3 2 3 · · · 2 1 \| 3 1 · · · 1 3 1 −r (odd) (even)              The vertices v 2k+1 to v n highlighted above represent a unique labeled string of vertices (alternating labels 2 and 1) of length r as every other block contains a 3 and is immediately proceeded or followed by a vertex labeled 3. Because this string of vertices is unique, the coloring is distinguished by dihedral symmetries. Case 2: Let n/3 < k < n/2 − 1 such that n = 2k + r hence r odd and r > k/2. It can be verified that the following arrangement of these sequences yield a proper coloring C of C n (1, k) that is also distinguishing. Subcase 1: Let 2r − k < k − r . k−r (odd) 2r−k (even) k−r (odd)           1 2 · · · 1 2 1 2 · · · 1 2 3 1 3 · · · 1 3 1 · · · 1 3 1 2 1 2 1 · · · 1 2 1 2 3 · · · 2 3 2 3 · · · 2 3 2 3 1 · · · 3 1 3 · · · 3 1 3 1 2 1 2 · · · 2 1 2 3 1 · · · 3 1 3 1 · · · 3 1 3 1 3 · · · 1 3 1 · · · 1 3 1 2 1 2 1 · · · 1 2 3 1 3 · · · 1 3 1 2 · · · 1 2 1 if 2r−k < k−r 2 1 · · · 2 1 2 · · · 2 1           The vertex v 2k+1 highlighted above is the only vertex labeled 3 with two neighbors labeled 2 and two neighbors labeled 1. Because this vertex is unique, the coloring is distinguished by dihedral symmetries. Subcase 2: Let 2r − k > k − r . k−r (odd) 2r−k (even) k−r (odd)           1 2 · · · 1 2 1 2 · · · 1 2 3 1 3 · · · 1 3 1 · · · 1 3 1 2 1 2 1 · · · 1 2 1 2 3 · · · 2 3 2 3 · · · 2 3 2 3 1 · · · 3 1 3 · · · 3 1 3 1 2 1 2 · · · 2 1 2 3 1 · · · 3 1 3 1 · · · 3 1 3 1 3 · · · 1 3 1 · · · 1 3 1 2 1 2 1 · · · 1 2 3 1 3 · · · 1 3 1 3 · · · 1 3 1 3 1 · · · 3 1 2 · · · 2 1 if 2r−k > k−r           The vertex v 2k+1 highlighted above is the only vertex labeled 3 with two neighbors labeled 2 and two neighbors labeled 1. Because this vertex is unique, the coloring is distinguished by dihedral symmetries. Case 3: n = pk + r for some p ≥ 3. We will similar blocks as those used in Section 7.3, given below: Use the blocks as given below, the first is where p is odd so the repetition of B 2 and B 3 ends with a B 2 , and the latter is where p is even and the repetition ends with a B 3 . B 1 = (1,B 4 B 2 B 3 . . . B 2 B 1 B r (odd) k−r (odd)                     3 1 3 1 3 · · · 3 1 3 1 3 · · · 1 3 1 2 3 1 3 1 · · · 1 3 1 3 1 · · · 3 1 2 3 1 2 1 2 · · · 2 1 2 1 2 · · · 1 2 3 . . . . . . 2 3 1 3 1 · · · 1 3 1 3 1 · · · 3 1 2 1 2 3 2 3 · · · 3 2 3 2 3 · · · 2 3 1 2 1 2 1 2 · · · 2 1 2 3 1 · · · 3 1 3 1 3 1 3 1 · · · 1 3 1                     B 4 B 2 B 3 . . . B 3 B 1 B r (odd) k−r (odd)                     3 1 3 1 3 · · · 3 1 3 1 3 · · · 1 3 1 2 3 1 3 1 · · · 1 3 1 3 1 · · · 3 1 2 3 1 2 1 2 · · · 2 1 2 1 2 · · · 1 2 3 . . . . . . Theorem 4.4 and other results in Section 4 showed that that when k 2 ≡ ±1 (mod n), the graphs C n (1, k) are dart-transitive, but any automorphism not corresponding to a dihedral symmetry on the Hamiltonian cycle comprised of edges in E 1 corresponds to a symmetry that maps the edges of E k to this Hamiltonian cycle while mapping the edges of E 1 to those in E k . It follows that, given a coloring C that destroys all dihedral symmetries of C n (1, k), if C also fixes an edge, then C destroys all non-dihedral symmetries as well. We begin with the case of C 13 (1, 5) to illustrate this idea. Proposition 8.1. χ D (C 13 (1, 5)) = χ(C 13 (1, 5)) = 4. 3 1 2 1 2 · · · 2 1 2 1 2 · · · 1 2 3 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v 12 v 13 Proof. Consider the following coloring: starting at v 1 , assign colors to v 1 , v 2 , · · · , v 12 by repeatedly using the block of colors B = (1, 2, 3), and assign color 4 to v 13 . We can see in Figure 4 that the coloring is proper. Moreover, it is distinguishing since the uniquely colored vertex v 13 is fixed and the edge v 12 v 13 is also fixed, as it is the only edge with endpoints colored 4 and 3. The graph C 15 (1, 4) will form an exception for our next main result, so we establish its distinguishing chromatic number here. Unlike with C 13 (1, 5) the distinguishing chromatic number will be greater than the chromatic number, which is 3, so more effort is required for the lower bound. We begin with a lemma that incidentally shows that the distinguishing coloring of C 8 (1,4) in Section 3 was unique up to the names of the colors. Proof. Observe first that in any proper coloring there must be at least one index i such that v i , v i+1 , v i+2 receive three distinct colors; if this is not the case, then C n (1, 4) would be properly colored with just two colors, and this is a contradiction, since v 0 , v 1 , v 2 , v 3 , v 4 are the vertices of a 5-cycle in C n (1, 4). Furthermore, we claim that in any distinguishing proper coloring of C n (1, 4) using three colors, there must be an index j such that v j , v j+1 , v j+2 do not receive three distinct colors. Indeed, if no such j existed, then the proper coloring would consist of the pattern A, B, C repeated around the vertices, requiring that n be a multiple of 3 and that the map v k → v k+3 be a color-preserving automorphism of C n (1, 4), a contradiction. It follows that there must be an index i such that v i , v i+1 , v i+2 are colored with three distinct colors, but v i+1 , v i+2 , v i+3 are not. By symmetry, we may suppose that i = 0 and that the labels on v 0 , v 1 , v 2 , v 3 are A, B, C, B. Since v 4 is adjacent to both v 3 and v 0 , the color on v 4 must be C. Likewise, the colors on v 5 and v 6 must be A and B, respectively, and the color on v n−1 (which is adjacent to both v 0 and v 3 ) must be C. This completes the proof. (1, 4) preserving the coloring c. v 1 v 11 )(v 2 v 7 )(v 4 v 14 )(v 5 v 10 )(v 8 v 13 ) is an automorphism of C 15 Note that the starting pattern of colors appears on vertices v 6 , v 5 , v 4 , v 3 , v 2 , v 1 , v 0 , v 14 ,automorphism (v 1 v 11 )(v 2 v 7 )(v 4 v 14) (v 5 v 10 )(v 8 v 13 ). If c(v 13 ) = B, then we get the color-preserving automorphism (v 0 v 10 )(v 1 v 6 )(v 3 v 13 )(v 4 v 9 )(v 7 v 12 ). Having handled the cases above, assume henceforth that c(v 8 ) = A and c(v 12 ) = A. This forces c(v 8 ) = B and c(v 12 ) = C. Case: c(v 7 ) = A or c(v 13 ) = A. By the prior symmetry argument, it suffices to assume that c(v 7 ) = A. No matter what color v 13 receives, we have the color-preserving symmetry (v 0 v 5 )(v 2 v 12 )(v 3 v 8 )(v 6 v 11 )(v 9 v 14 ). In light of these cases, we see that c(v 7 ) = A and c(v 13 ) = A. This forces c(v 7 ) = C and c(v 13 ) = B. However, then the map v i → v i+5 yields a color-preserving automorphism. This concludes the necessary cases; we have shown that every proper 3-coloring of C 15 (1, 4) is not distinguishing. On the other hand, χ(C 15 (1, 4)) = 3, and the coloring that alternates colors A, B, C around the vertices is a palindrome-free coloring of C n (1, 4) that satisfies Theorem 6.3 of the paper. The Theorem then implies that χ D (C 15 (1, 4)) = 4. With the exceptions dealt with, we now prove the general result. We will prove the colorings introduced in Section 7 will also be distinguished by any symmetry that swaps the edges comprising of a Hamilton cycle of E 1 and those comprising a Hamilton cycle of E K . We do this by finding an edge that is fixed by the coloring. Note that this proof is more than is necessary as most graphs covered in Section 7 will not have an edge swapping symmetry like those discussed in this section, but will be sufficient to prove our result. Theorem 8.4. For k 2 ≡ ±1 (mod n) with k > 3, χ D (C n (1, k)) = 3, except for C 13 (1, 5) and C 15 (1,4) where the distinguishing chromatic number is 4. Proof. Like in the dihedral case, we proceed on a case-by-case basis. Note that the corresponding theorem and case in Section 7 will be listed for reference at the beginning of each case. Case: n is odd and k is odd (referencing Theorem 7.6 in Section 7.4) Subcase 1: n ≥ 3k (referencing Theorem 7.6 Case 1) Consider the distinguishing coloring C given in the case of its dihedral analogue. It can be easily verified that v k is the only vertex labeled 1 with exactly two neighbors colored 2 and the others colored 3. Similarly, v n is the only vertex colored 2 with exactly two neighbors colored 1 and the others colored 3. By construction, both vertices v 0 and v k are fixed and, hence, the edge v 0 v k is fixed. Subcase 2: n = 2k + r (referencing Theorem 7.6 Case 2) Suppose that 2r ≥ k+1. We consider the same subcases together with the corresponding distinguishing colorings discussed in the dihedral section. If 2r − k > k − r (previously referred as SUBCASE 4), the vertex v k+1 is the only vertex labeled 3 with exactly two of its neighbors labeled 2 and the others labeled 1. Moreover, the neighbors labeled 1, namely v k and v 2k+1 have distinct neighborhoods. Hence, the edge v k v k+1 is fixed. If 2r − k < k − r with 2r − k > 1 (previously referred as SUBCASE 1), the vertex v k−r+2 is labeled 1 with all neighbors labeled 3. Switch the color of v k−r+2 to 2. Thus, v k−r+2 becomes the only vertex labeled 2 that has all its neighbors labeled 3. Most importantly, the coloring is still distinguishing with respect to the dihedral group. Moreover, since v −r+2 has a unique neighborhood among the neighbors of v k−r+2 , the edge v k−r+2 v −r+2 is fixed. If 2r − k < k − r with 2r − k = 1 and 2k − 3r > 1 (previously referred as SUBCASE 2), recall that the vertex v k+1 is the only vertex labeled 3 with two of its neighbors labeled 2 and two labeled 1. Moreover, the neighbors labeled, namely v 2k+1 and v k , have distinct neighborhoods. Thus, the edge v k+1 v k is fixed. Lastly, if 2r − k < k − r with 2r − k = 2k − 3r = 1 (previously referred as SUBCASE 3), then we have the special graph C 13 (1, 5) (see Theorem 2.4). We saw at the beginning of this section a construction of a coloring that is distinguishing. Subcase 3: n = 2k + r (referencing Theorem 7.6 Case 3) Suppose 2r ≤ k − 1. which implies k = pr + . Again, we consider the same subcases discussed in the section dedicated to the dihedral group. If = 0 (previously referred as SUBCASE 1), the vertex v k+1 is the only 2-colored vertex whose neighbors are all colored 1. Thus, v k+1 is fixed. Moreover, v 2 is colored 3 and all its neighbors are colored 1. Switch the color of v 2 to 2. This moves makes the neighborhood of v 1 distinct from the other neighbors of v k+1 while the coloring remains distinguishing with respect to the dihedral group. Moreover, we have v 1 v k+1 as a fixed edge. If is even and = 0 (previously referred as SUBCASE 4), the vertex v k+1 is fixed by virtue of being the only 3-colored vertex with exactly two neighbors colored 2 and the others colored 1. Moreover, since v 1 is uniquely colored among the neighbors of v k+1 , we thus have v k+1 v 1 as a fixed edge. If is odd (previously referred as SUBCASES 2 and 3), the vertex v k+r+2 is colored 1 with all its neighbors colored 2 and vertex v k+r+1 is colored 2 with exactly 3 neighbors colored 3 and the other one, v k+r+2 , colored 1. Fix vertex v k+r+1 by switching the color v k+r+2 from 1 to 3. The switch makes v k+r+1 the only 2-colored vertex whose neighbors are distinct neighborhood among the neighbors of v 0 . In particular, v −1 has three of its neighbors colored 3 while every other neighbor of v 0 is adjacent to at most two vertices colored 3.Therefore, the edge v −1 v 0 is fixed. Hence, the desired result. When r = 2, the same argument in the dihedral case holds. Subcase 2: n = 2k + r (referencing Theorem 7.5 Case 2) Suppose r ≤ k/2. Consider the distinguishing coloring C given in the case of its dihedral analogue. With respect to C, the vertex v k+1 is fixed since it is the only vertex colored 3 that has excatly two neighbors colored 1 (namely v 1 and v k+2 ) and the other colored 2. Moreover, v 2 is colored 2 and all its neighbors are colored 1. Switch the color of v 2 to 3. This moves makes v 1 uniquely colored among the neighbors of v k+1 , so we have v 1 v k+1 as a fixed edge. Similarly when r > k/2. Note that the case where n is even and k is odd is already addressed in Theorem 6.4. END OF PAPER k = n/2 and n ≥ 8; 5 if k = n/2 − 1 and n ≥ 10; 4 if k = 2 or k = (n − 1)/2, and n ≥ 7; 3 otherwise. −5 must have different labels. The remaining results in Theorem 1.2 are shown in Theorems 3. odd and n is even; 4 if k = 2 or k = (n − 1)/2, and n = 5 and 3 n; 4 if k = 5 and n = 13; 5 if k = 2 and n = 5; 3 otherwise. Theorem 4. 4 . 4If k and n are integers satisfying 1 < k < n/2 and k 2 ≡ ±1 (mod n), then \| Aut(C n (1, k))\| = 4n, and for any two edges v a v b , v s v t in the graph, there is a unique automorphism sending v a to v s and v b to v t .Proof. By symmetry it suffices to show that for any edge v s v t in the graph, there is a unique automorphism sending v 0 to v s and v 1 to v t . We claim that this map is φ st on {v 0 , . . . , v n−1 } given by φ st (v i ) = v s+(t−s)i (with all operations performed modulo n). To see that is indeed an automorphism, note that the pair v x , v y of vertices in C n (1, k) is adjacent if and only if Figure 3 : 3Two drawings showing C 12 (1, 5) ∼ = W (6, 2). Theorem 7. 4 . 4For all n ≥ 7, χ D (C n (1, 2)) = 4.Proof. By Theorem 2.4, χ D (C n (1, 2)) ≥ χ(C n (1, 2)) = 4 if n is not a multiple of 3. If n is a multiple of 3, then the only partition of the vertices of C n (1, 2) into three independent sets is given by grouping the vertices v i by the congruence class modulo 3 of their subscripts; hence any proper 3-coloring is preserved by the rotation given by v i → v i+3 , and as before we must have χ D (C n (1, 2)) ≥ 4. By Corollary 4.3, Aut(C n (1, 2)) is isomorphic to the dihedral group of order 2n. By Lemma 7.1, a proper coloring of C n (1, 2) will be distinguishing if and only if no color-preserving rotation symmetry exists other than the identity. If n is congruent to 0 or 1 modulo 3, we obtain a proper coloring by assigning v 0 the color 4 and greedily coloring v 1 , v 2 , . . . , v n−1 in order with the lowest available color from {1, 2, 3}. If n ≡ 2 (mod 3), we color C n (1, 2) by assigning color 4 to vertices v 0 and v n−3 and greedily coloring the remaining vertices in order of their subscripts with colors from {1, 2, 3} as before. The placement of color 4 allows for no nontrivial rotational symmetry, so these colorings establish that χ D (C n (1, 2)) = 4. Case 1 : 1n ≥ 3k so q ≥ 3Assign colors to v 1 , . . . , v (q−1)k by alternating the use of blocks B 1 and B 2 on successive collections of k consecutivelyindexed vertices. Assign colors to v (q−1)k+1 , . . . , v qk using the block B 3 . For any remaining r vertices v qk+1 , . . . , v n−1 , v 0 , begin by assigning color 2 to v qk+1 . Assign to v 0 the color 3 if r = 2 and the color 2 otherwise. Then color any remaining vertices v qk+2 , . . . , v n−1 by alternating the colors 3 and 1. The final r vertices' colors thus create a block B that is a shortened or partial version of the block B 2 .We illustrate this coloring in the the figure below, where each row indicates the colors placed on the sets of k consecutivelyindexed vertices in C n (1, k), beginning in the first row with the colors on v 1 , . . . , v k , followed in the second row with the colors on v k+1 , . . . , v 2k , and so on. In this way the entries surrounding a vertex's color show the colors on neighboring vertices along 1-edges (these are the immediately following and preceding numbers) and along k-edges (these are the vertically aligned numbers in the previous and following rows; for convenience, the initial block B 1 is repeated at the end of thefigure). In contrast to the collection of rectangles above, though the blocks B 1 , B 2 , B 3 occupy entire rows, here they are shown as split into two rectangles each, respectively containing r and k − r entries. This allows us to see how the colors from the block B B 3 . 3Likewise, as we compare the entries of B with the last r entries of B 1 , the first entry of B (which is 2) aligns with a 3 from B 1 , since both k and r are even; similarly, no other color in B aligns vertically with the same color in B 1 . Finally, comparing the final k − r entries of B 3 with the first k − r entries of B 1 , having k and r be even ensures that the parities of the relevant entries in B 3 differ from the parities of the vertically aligned entries from B 1 . v 0 labeled 3, which is the last vertex in the shortened B block, has 3 neighbors labeled 1 and one neighbor labeled 2. In looking at v −1 , this vertex is labeled 2 with all neighbors labeled 3, and in looking at v −2 , this is a vertex labeled 3 with one neighbor labeled 1 and the rest labeled 2. The only other vertices that have the same neighbors as v 0 are the second to last entries in a B 1 block surrounded by B 2 blocks, call one of these vertices v i . However v i−2 is a vertex labeled 3 with at least two 1 neighbors, or in the case where k = 4 a vertex labeled 1. lemma 7.1, it suffices to show that there is not a nontrivial rotation symmetry that preserves coloring. The string of vertices v 2k−1 , v 2k , v 2k+1 , v 2k+2 (as highlighted in the block diagram) is the only string of vertices labeled (3, 1, 2, 3) as every other pair of vertices labeled (1, 2) is preceded or followed by another pair of vertices labeled(1,2). Hence, C is distinguishing. k and the third row is a sequence of length r. The remaining light-shaded blocks in rows 3 and 4 represent row 1 re-positioned so that the first k − r vertices (positions) in row 1 are vertically aligned with the last k − r vertices (positions) in row 2 and the last r vertices (positions) in row 1 are vertically aligned with the r vertices in row 3. Subcase 3 : 3Let 2r − k = 1 and 2k − 3r = 1 Note that if both 2r − k = 1 and 2k − 3r = 1 we have the specific case of C 13 (1, 5) which has a chromatic distinguishing number of 4 and is investigated in Section 8.Subcase 4: Let 2r − k > k − r of colors B 1 , B 2 , B 3 are constructed in such a way that as long as blocks are not repeated, adjacent vertices will not have the same color. It is also the case between B 4 and B 2 . We give a coloring C of the vertices of C n (1, k) using colors 1, 2, and 3 as follows: assign colors to v 1 , v 2 , . . . , v k using B 4 . Next, starting with B 2 , assign colors to v k+1 , v k+2 , . . . , v (q−1)k by alternating the use of sequences B 2 , B 3 on successive collections of k consecutively-indexed vertices. Assign colors to v (q−1)k+1 , v (q−2)k+2 , . . . , v qk using B 1 . Lastly, use the shortened block B of length r on the remaining vertices, noting that if r = 1 we are left with the vertex v n labeled 2. Figure 4 : 4A proper distinguishing coloring of C 13 (1, 5) with 4 colors 8 Graphs C n (1, k) where k 2 ≡ ±1 (mod n) Lemma 8 . 2 . 82Let n be an integer with n ≥ 8. In any distinguishing 3-coloring of C n (1, 4), there exist 8 consecutively-indexed vertices receiving colors in the pattern C, A, B, C, B, C, A, B, where {A, B, C} is the set of colors used. We first show that no distinguishing proper coloring can use 3 colors. Suppose to the contrary that some coloring c does. By Lemma 8.2, we may suppose that v 14 , v 0 , . . . , v 6 are respectively labeled with C, A, B, C, B, C, A, B. Since v 10 is adjacent to v 6 and to v 14 , we have c(v 10 ) = A. We now proceed by cases on the colors of v 7 , v 8 , v 9 , v 11 , v 12 , v 13 , showing that any proper 3-coloring admits a color-preserving automorphism. Case: c(v 9 ) = B or c(v 11 ) = C. Suppose first that c(v 9 ) = B. Then c(v 8 ) = A and c(v 13 ) = A, since each vertex has a neighbor already colored with B and with C. Then c(v 7 ) = C. and c(v 12 ) = C and consequently c(v 11 ) = B. At this point all the vertices have been colored, and one can verify that the permutation ( in this order, if the roles of colors B and C are switched, so the case where c(v 11 ) = C follows exactly the same argument as above to conclude that there is an automorphism of C 15 (1, 4) preserving the coloring. Assume henceforth that c(v 9 ) = B and c(v 11 ) = C. This forces c(v 9 ) = C and c(v 11 ) = B. Case: c(v 8 ) = A or c(v 12 ) = A. By the symmetry argument above, it suffices to suppose that c(v 8 ) = A. It follows that c(v 7 ) = C and that c(v 11 ) = B and c(v 12 ) = C. Now, if c(v 13 ) = A, we get the coloring-preserving Again, consider the coloring C given in the case of its dihedral analogue. Recall that when r > 2 and if we switch the color of vertex v 0 to 3, we obtain a proper distinguishing coloring with respect to the dihedral group. Moreover, v −1 has a Most importantly, the coloring remains distinguishing with respect to the dihedral group. Moreover, v k+r has a distinct neighborhood among the neighbors of v k+r+1 . Therefore, the edge v k+r v k+r+1 is fixed. Case: n is odd and k is even. referencing Theorem 7.7 in Section 7.5)colored 3. Most importantly, the coloring remains distinguishing with respect to the dihedral group. Moreover, v k+r has a distinct neighborhood among the neighbors of v k+r+1 . Therefore, the edge v k+r v k+r+1 is fixed. Case: n is odd and k is even (referencing Theorem 7.7 in Section 7.5) Subcase 1: n ≥ 3k (referencing Theorem. Subcase 1: n ≥ 3k (referencing Theorem 7.7 Case 3) Let n = qk + r, where q ≥ 3. Since n is odd and k is even, r is also odd and r = 0. Let n = qk + r, where q ≥ 3. Since n is odd and k is even, r is also odd and r = 0. Note that the modified C is still proper. Furthermore, the moves makes v 1 a vertex labeled 1 with all its neighbors colored 2. Thus, v 1 is fixed since there exists no other vertex labeled 1 with all its neighbors labeled 3. Thus, the modified C is distinguishing with respect to the dihedral group. Moreover, v 2 has a distinct neighborhood among the neighbors of v 1 . In particular, v 2 has three neighbors labeled 3. Therefore, the edge v 1 v 2 is also fixed. Hence, the modified c is distinguishing with respect to nondihedral group. If r = k − 1 ≥ 3, the vertex v qk+1 (first vertex in block B ) is colored 2 with all its neighbors colored 1. Now, fix vertex v qk+1 by switching the colors of all possible like-vertices (a vertex that is labeled 2 with all neighbors labeled 1) to 3; for example, such vertex can be found in a Block B 3 squeezed between two blocks B 2 . Therefore, the modified C is distinguishing with respect to the dihedral group. Moreover, the vertex v qk+2 has a unique neighborhood among the neighbors of v q+1 . In particular, at least three of its neighbors are labeled 2. Therefore, the edge v q+1 v q+2 is fixed. Hence, the modified C is distinguishing with respect to nondihedral group. Now suppose that q is odd. Consider the coloring C given in the case of its dihedral analogue; which is the case where the block B 1 is preceded by a block B 2 . If q > 3, there exists at least a block B 3 squeezed between two blocks B 2 and the first vertex of the block B 1 , namely v (q−1)k+1 , is labeled 1 with all its neighbors labeled 2. Since no other vertex labeled 1 has the same neighborhood, v (q−1)k+1 is thus fixed. Moreover, the vertex v qk+1 has a distinct neighborhood among the neighbors of v (q−1)k+1 . In particular, all the neighbors of v qk+1 are labeled 1. Therefore, the edge v (q−1)k+1 v qk+1 is fixed. Hence, C is distinguishing with respect to the nondihedral group. If q = 3 and k = 4, the vertex v k+3 is labeled 1 with all its neighbors labeled 3. Switch its color to 2. This moves makes v k+3 a vertex labeled 2 with all its neighbors labeled 3. Thus, v k+3 is fixed since there is no such other vertex and the modified C is distinguishing with respect to the dihedral group. Moreover, v k+4 has a distinct neighborhood among the neighbors of v k+3 . More specifically. First, suppose that q is even. Consider the coloring C given in the case of its dihedral analogue; which is the case where the block B 1 is preceded by a block B 3 . If r < k − 1. v k+4 has exactly two neighbors labeled 2 and two neighbors labeled 1. Therefore, the edge v k+3 v k+4 is fixed. Hence, the modified C is distinguishing with respect to the nondihedral group. If q = 3 and k = 4, then we have the exceptional graph C 15 (1, 4)First, suppose that q is even. Consider the coloring C given in the case of its dihedral analogue; which is the case where the block B 1 is preceded by a block B 3 . If r < k − 1, then the vertex v 1 and v 2 are labeled 3 and 1, respectively. Switch the color of v 1 to 1 and the color of v 2 to 2. Note that the modified C is still proper. Furthermore, the moves makes v 1 a vertex labeled 1 with all its neighbors colored 2. Thus, v 1 is fixed since there exists no other vertex labeled 1 with all its neighbors labeled 3. Thus, the modified C is distinguishing with respect to the dihedral group. Moreover, v 2 has a distinct neighborhood among the neighbors of v 1 . In particular, v 2 has three neighbors labeled 3. Therefore, the edge v 1 v 2 is also fixed. Hence, the modified c is distinguishing with respect to nondihedral group. If r = k − 1 ≥ 3, the vertex v qk+1 (first vertex in block B ) is colored 2 with all its neighbors colored 1. Now, fix vertex v qk+1 by switching the colors of all possible like-vertices (a vertex that is labeled 2 with all neighbors labeled 1) to 3; for example, such vertex can be found in a Block B 3 squeezed between two blocks B 2 . Therefore, the modified C is distinguishing with respect to the dihedral group. Moreover, the vertex v qk+2 has a unique neighborhood among the neighbors of v q+1 . In particular, at least three of its neighbors are labeled 2. Therefore, the edge v q+1 v q+2 is fixed. Hence, the modified C is distinguishing with respect to nondihedral group. Now suppose that q is odd. Consider the coloring C given in the case of its dihedral analogue; which is the case where the block B 1 is preceded by a block B 2 . If q > 3, there exists at least a block B 3 squeezed between two blocks B 2 and the first vertex of the block B 1 , namely v (q−1)k+1 , is labeled 1 with all its neighbors labeled 2. Since no other vertex labeled 1 has the same neighborhood, v (q−1)k+1 is thus fixed. Moreover, the vertex v qk+1 has a distinct neighborhood among the neighbors of v (q−1)k+1 . In particular, all the neighbors of v qk+1 are labeled 1. Therefore, the edge v (q−1)k+1 v qk+1 is fixed. Hence, C is distinguishing with respect to the nondihedral group. If q = 3 and k = 4, the vertex v k+3 is labeled 1 with all its neighbors labeled 3. Switch its color to 2. This moves makes v k+3 a vertex labeled 2 with all its neighbors labeled 3. Thus, v k+3 is fixed since there is no such other vertex and the modified C is distinguishing with respect to the dihedral group. Moreover, v k+4 has a distinct neighborhood among the neighbors of v k+3 . More specifically, v k+4 has exactly two neighbors labeled 2 and two neighbors labeled 1. Therefore, the edge v k+3 v k+4 is fixed. Hence, the modified C is distinguishing with respect to the nondihedral group. If q = 3 and k = 4, then we have the exceptional graph C 15 (1, 4). Subcase 2: n = 2k + r (referencing Section. Subcase 2: n = 2k + r (referencing Section 7.7 Cases 1 and 2) Consider the distinguishing coloring C given in the case of its dihedral analogue. Then v 1 is fixed since it is the only vertex colored 1 that has excatly two neighbors colored 2 (namely v 2 and v k+1 ) and the other colored 3. Since v 2 is distinctly colored than v k+1. Suppose r > k/2.. thus v 1 v 2 is a fixed edge. Hence, the desired resultSuppose r > k/2. Consider the distinguishing coloring C given in the case of its dihedral analogue. Then v 1 is fixed since it is the only vertex colored 1 that has excatly two neighbors colored 2 (namely v 2 and v k+1 ) and the other colored 3. Since v 2 is distinctly colored than v k+1 , thus v 1 v 2 is a fixed edge. Hence, the desired result. Suppose r ≤ k/2, which implies k = pr + . Consider the distinguishing coloring C given in the case of its dihedral analogue. The same argument used in the case where both n and k are odd also works here. Case: n is even and k is even. referencing Theorem 7.5 in Section 7.3)Suppose r ≤ k/2, which implies k = pr + . Consider the distinguishing coloring C given in the case of its dihedral analogue. The same argument used in the case where both n and k are odd also works here. Case: n is even and k is even (referencing Theorem 7.5 in Section 7.3) Research problems 2-10. A Ádám, J. Combin. Theory. 2393A.Ádám, Research problems 2-10, J. Combin. Theory, 2 (1967), p. 393. Symmetry breaking in graphs. M O Albertson, K L Collins, Electron. J. Combin. 31Research Paper 18M.O. Albertson and K.L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1996), no. 1, Research Paper 18. On 3-and 4-coloring some circulant graphs, Presented at DREI'98 Graph Theory & Combinatorial Optimization. K L Collins, D C Fisher, J P Hutchinson, K.L. Collins, D.C. Fisher, J.P. Hutchinson, On 3-and 4-coloring some circulant graphs, Presented at DREI'98 Graph Theory & Combinatorial Optimization, August 2-7, 1998. The distinguishing chromatic number. K L Collins, A N Trenk, Electron. J. Combin. 131Research Paper 16K.L. Collins and A.N. Trenk, The distinguishing chromatic number, Electron. J. Combin. 13 (2006), no. 1, Research Paper 16. 1000s of theorems about circulant graphs. D Fisher, Proceedings of the 29th Southeastern International Conference on Combinatorics, Graph Theory, and Computing. the 29th Southeastern International Conference on Combinatorics, Graph Theory, and ComputingBoca Raton, FL, USAD. Fisher, 1000s of theorems about circulant graphs, Proceedings of the 29th Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Boca Raton, FL, USA, 1998. Cyclic graphs. F Göbel, E A Neutel, Proceedings of the 5th Twente Workshop on Graphs and Combinatorial Optimization. the 5th Twente Workshop on Graphs and Combinatorial OptimizationEnschede99F. Göbel and E.A. Neutel, Cyclic graphs, Proceedings of the 5th Twente Workshop on Graphs and Combinatorial Optimization (Enschede, 1997), Discrete Appl. Math. 99 (2000), no. 1-3, 3-12. On planarity and colorability of circulant graphs. C Heuberger, Discrete Math. 268C. Heuberger, On planarity and colorability of circulant graphs, Discrete Math. 268 (2003), 153-169. On the automorphism groups of us-Cayley graphs. S , Morteza Mirafzal, arXiv:1910.125632021S. Morteza Mirafzal, On the automorphism groups of us-Cayley graphs, 2021. arXiv:1910.12563 Vertex-colouring of circulant graphs: a combinatorial approach. S Nicoloso, U Pietropaoli, Technical Report. 669S. Nicoloso and U. Pietropaoli, Vertex-colouring of circulant graphs: a combinatorial approach, Technical Report 669, IASI-CNR, Rome, Italy, 2007. Recipes for edge-transitive tetravalent graphs. P Potocnik, S E Wilson, Art Discrete Appl. Math. 31Paper No. 1.08P. Potocnik and S.E. Wilson, Recipes for edge-transitive tetravalent graphs, Art Discrete Appl. Math. 3 (2020), no. 1, Paper No. 1.08. 4-colorable 6-regular toroidal graphs. H.-G Yeh, X Zhu, Discrete Math. 273H.-G. Yeh and X. Zhu, 4-colorable 6-regular toroidal graphs, Discrete Math. 273 (2003), 261-274.	[]
[ "Tunneling of Bloch electrons through vacuum barrier", "Tunneling of Bloch electrons through vacuum barrier" ]	[ "I I Mazin \nNaval Research Laboratory\n6391, 20375code, WashingtonDCUSA\n" ]	[ "Naval Research Laboratory\n6391, 20375code, WashingtonDCUSA" ]	[]	Tunneling of Bloch electrons through a vacuum barrier introduces new physical effects in comparison with the textbook case of free (plane wave) electrons. For the latter, the exponential decay rate in the vacuum is minimal for electrons with the parallel component of momentum k = 0, and the prefactor is defined by the electron momentum component in the normal to the surface direction. However, the decay rate of Bloch electrons may be minimal at an arbitrary k ("hot spots"), and the prefactor is determined by the electron's group velocity, rather than by its quasimomentum.	10.1209/epl/i2001-00414-0	[ "https://export.arxiv.org/pdf/cond-mat/0012049v1.pdf" ]	119,512,118	cond-mat/0012049	56bb23f8deb03c415e6e7360a908f7a6e8373e4e	Tunneling of Bloch electrons through vacuum barrier 4 Dec 2000 I I Mazin Naval Research Laboratory 6391, 20375code, WashingtonDCUSA Tunneling of Bloch electrons through vacuum barrier 4 Dec 2000 Tunneling of Bloch electrons through a vacuum barrier introduces new physical effects in comparison with the textbook case of free (plane wave) electrons. For the latter, the exponential decay rate in the vacuum is minimal for electrons with the parallel component of momentum k = 0, and the prefactor is defined by the electron momentum component in the normal to the surface direction. However, the decay rate of Bloch electrons may be minimal at an arbitrary k ("hot spots"), and the prefactor is determined by the electron's group velocity, rather than by its quasimomentum. There is a general feeling in the applied physics community nowadays that the next decade will bespeak an advent of magnetoelectronics, exploiting spin, rather than charge, degrees of freedom 1 . Many of such spintronic devices are based on the phenomenon of quantum tunneling, and specifically on the difference between tunneling currents in different spin channels. this opens up a possibility to control the electric properties via magnetic field. In view of all that, an explosion of publication on spin-polarized tunneling which started around 1995, is not surprising at all. Interestingly, despite the fact that the tunnelling is one of the best studied phenomena in quantum mechanics, there is still substantial diversity in microscopic understanding of tunneling in real systems. Tunneling problems are very easily solved in one dimension and for free electrons; but it is not so obvious how this is related to real systems, and how to incorporate the effects of realistic electronic structure. This is the reason why most theoretical papers use the free-electron model, that is, the electronic wave functions that are plane waves and a spherical Fermi surface. The deviations of the wave function from the single plane wave form, and of the Fermi surface from a sphere, are very often crucial for understanding the physics of tunneling. In particular, a totally counterintuitive result has been observed in some recent calculations 2 , when the electrons with nonzero quasimo-mentum parallel to the interface had a larger probablility to tunnel through a vacuum barrier compared to those with zero parallel quasimomentum. It was also pointed out 3 that electrons in different Bloch states with the same energy and quasimomentum may, in principle, have different decay rates in vacuum: another counterintuitive result. In this regards, it is important to establish a formal theory for tunneling of Bloch electrons through a vacuum barrier, elucidating the qualitatively new aspects of this process as opposed to the free electron tunneling, particularly because only the latter is discussed in the classical textbooks. This is the goal of the current paper. The simplest case of a tunneling contact is the so-calles Sharvin contact 4 , which is essentially an orifice between two metals (or a metal and the vacuum), whose size is smaller than the mean free path of electrons in the bulk. All electrons with the a positive projection onto the current direction (which we will denote as x) pass through the contact. Conductance of a Sharvin contact between two identical metals is G = e 2 h 1 2 N \|v x \| A,(1) where A is the contact area, N is the volume density of electronic states at the Fermi level, v is the Fermi velocity, and brackets denote Fermi surface averaging: 1 2 N \|v x \| = 1 Ω kiσ δ(ǫ kiσ − E F )v kiσ,x = 1 (2π) 3 iσ dS F \|v kiσ \| v kiσ,x .(2) Integration and summations are over the states with v kiσ,x > 0, and Ω is the unit cell volume 5,6 . k, i, and σ denote the quasimomentum, the band index, and the spin of an electron, respectively. This formula can be derived by considering the voltage-induced shift of the Fermi surface 6 , but there is a more instructive derivation starting from the Landauer-Buttiker formula for the conductance of a single ballistic electron, G 0 = e 2 /h. In this formalism, the total conductance is equal to G 0 times the number of conductivity channels, N cc , which is defined as the number of electrons that can pass through the contact. Assuming that the translational symmetry in the interface plane is not violated, we observe that the quasimomentum in this plane, k , is conserved, and N cc is the number of quantum-mechanically allowed k 's. Thus N cc is given by the total area of the contact times the density of the two-dimensional quasimomenta. The latter is simply S x /(2π) 2 , where S x is the area of the projection of the bulk Fermi surface onto the contact plane. Thus G = e 2 h S x A (2π) 2 ≡ e 2 h 1 2 N \|v x \| A.(3) This is an important result. To the best of our knowledge, Walter Harrison was the first to spell it out in 1961 7 , and there is no lack of more recent papers manifesting proper understanding of this issue (e.g., Ref. 8 ). However, till now many otherwise correct and useful papers erroneously identify the number of conductivity channels and the density of states at the Fermi level, that is, N cc ∝ N (E F ) = 1 Ω kiσ δ(ǫ kiσ − E F ) = 1 (2π) 3 iσ dS F \|v kiσ \| . (4) incorrect! Eq. 3 is the basis for all more sophisticated expressions describing various aspects of quantum tunneling. None of them may explicitely depend on the bulk density of states. It may, however, be that that instead of the straight N \|v x \| averaging one has to compute a weighted average, with the weights coming from tunneling matrix elements, or other additional physics. Eq. 3 takes care of one important difference between the free electrons and the Bloch electrons: deviation of the Fermi surface from a sphere, for S x = πk 2 F . Another important difference that is often neglected is that be-tween the group velocityh −1 dǫ k /dk and the phase velocityhk/m 0 . One can get some qualitative understanding of the role that this fact plays in tunneling by considering a simplified model, where electrons in metal are approximated by free electrons with an effective mass different from the free electron mass. In this approximation, the effective mass is responsible for the difference between (hk/m 0 ) and (dǫ k /hdk), and the transparency of a symmetric rectangular barrier is defined by the standard formula (see, e.g., Ref. 9 ): D(k) = 4m 2 0h 2 K 2 v L v R h 2 m 2 0 K 2 (v L + v R ) 2 + (h 2 K 2 + m 2 0 v 2 L )(h 2 K 2 + m 2 0 v 2 R ) sinh(dK) 2 ,(5) where v L(R) stands for (k-dependent) Fermi velocity in the left and in the right leads, and the imaginary quasi-momentumhK is calculated from the energy conservation condition, U +h 2 [k 2 − K 2 ]/2m 0 = E, where m 0 is the free electron mass, U is the barrier height, and d is its thickness. The physical reason that one has to use group velocities, and not wave vectors, is very profound and extends well beyound the limited scope of the effective mass model: these factors appear in the Eq.5 as a result of matching the gradients of the wave functions at the interface, and the gradient is, in fact, the velocity operator for the Bloch waves. Another way to express the same idea is to recall the physical meaning of the usual quantum-mechanical requirement that the wave functions be smooth: it is needed to ensure the flux continuity and, therefore, particle conservation. On the other hand, the expression for K includes the momentum, k ,, and the free electron mass, m 0 , because it comes from the solution of the Schrödinger equation inside the barrier (in vacuum) 10 . The conductance of a contact described by Eq. 5 is given by the appropriately modified Eq.1: G = e 2 h A Ω k δ(ǫ k − E F )v kx D(k),(6) It is instructive to consider the last formula in some limiting cases. First, let us consider a specular barrier. It is defined by the limit U → ∞, d → 0, U d = V. Then K → 2m 0 U/h 2 , and D(k) = 4h 2 v L v R h 2 (v L + v R ) 2 + 4V 2 ,(7) Note that in the literature the ratio V /hv x = Z is commonly used to characterize the barrier strength. In principle, this quantity is different for different electrons, as v x depends on k. In the limit of low transparency, Z ≫ 1, D(k) =h 2 v L v R /V 2 . Substituting this into Eq.6, we find that the total current is proportional to k δ(ǫ k − E F )v kx v L v R ,(8) where summation is, of course, over those k that are allowed in both left and right lead. Roughly speaking, the total conductance is defined by the smaller of the two N v 2 x 's, that is, by min N v 2 x L , N v 2 x R . In the high transparency limit D is still smaller than 1, D = 4v L v R /(v L + v R ) 2 , (so-called Fermi velocity mismatch), but in most cases this is not a large effect: factor of two mismatch reduces D by only 10%. In the case of a thick barrier, defined as dK ≫ 1, Eq. 5 can be expanded inh 2 k 2 /4m 0 (U − E F ), and the transparency is D(k) = 2m 2 0 (U − E F )v L v R (U − E F + m 0 v 2 L /2)(U − E F + m 0 v 2 R /2) exp(−2d 2 W ) exp[− k 2 W ],(9) where 2m 0 (U − E 0 )/hd = W ≪ k 2 (thick barrier limit). W does not depend on k. The tunneling current is proportional to J ∝ k 2m 2 0 (U − E F )v L v R (U − E F + m 0 v 2 L /2)(U − E F + m 0 v 2 R /2) exp[− k 2 W ](10) Let k n be the set of points on the Fermi surface where k = 0 (note that for Bloch electrons beyond the effective mass approximation tunneling from some of these points may be suppressed by symmetry, as discussed later in the paper). Except in the exponent, we can put k to zero, J ∝ 1 (2π) 3 n d 2 k exp[− k 2 W ] 2m 2 0 (U − E F )v L v R (U − E F + m 0 v 2 L /2)(U − E F + m 0 v 2 R /2) (11) ∝ n v L m 0h 2 v 2 L /2 + U − E F v R m 0h 2 v 2 R /2 + U − E F . All omitted in this expression factors are kindependent. One should not be confused by the fact that, unlike Eq. 8, the numerator here does not have the third velocity. We have reduced our problem to an effective 1D problem, in which case the role of the density of states is played by the inverse velocity. Correspondingly, the product N v cancels out. Eqs. 8,9 emphasize the role of kinematics in tunneling. For instance, the long-standing problem of the reversed (compared to the density of states) spin polarization of the 3d ferromagnets is entirely explained in terms of kinematics. Direct calculations show that s-like electrons in Fe, Co and Ni have much larger Fermi velocity than d-like electrons. Taking this fact into account brings the calculated spin polarization to a very good agreement with experiment, without making any additional assumptions about the character of the surface states 6,11 . This is by no mean surprising: the bulk transport is controlled by the same factor N v 2 x , and the Ohmic current in these metals is carried predominantly by s-like electrons. It is only natural that in another transport phenomenon, tunneling, these electrons also play the leading role. We would like to emphasize that the effect considered above (as opposed to another effect discussed later in the paper) is not related to the s or d symmetry of the wave functions, but to the group velocities in the respective bands. In other cases the "light" and the "heavy" bands may not be directly related to the angular symmetry of the wave functions. For example, in SrRuO 3 both spin-up and spindown Fermi surfaces are made up by Ru t 2g d-electrons, but the average group velocity in the spin-majority channel is twice smaller than that in the spin-minority one 12 . As a result, although the spin polarization of the density of states is positive, N ↑ > N ↓ , while the transport spin polarization is negative, N v ↑ < N v ↓ . 13 Now we have some understanding of the two remarkable differences between the free electrons and the Bloch electrons: the effect of the Fermi surface geometry and the difference between the group and the phase velocities. There is, however, yet another, extremely important, dissimilarity between the two systems, recently pointed out by W. Butler 3 : the difference between the momentum and the quasimomentum. In order to discuss this dif-ference, and its physical consequences, let us consider reflection of an individual Bloch wave from a metal surface. Let x be the direction normal to the surface, and r the coordinate in the surface plane. At x < 0 we have a metal, and vacuum at x > 0. The vacuum potential is again U, and the Fermi energy is E. Since we have perfect in-plane periodicity, the wave function at any x can be classified by k , and is given by ψ(k , x, r ) = G exp[i(k + G)r ]F G (k ,x).(12) The quasimomentum in the surface plane,hk , is conserved, as well as the energy. In vacuum, the solution of the Schrödinger equation is ψ T (k , x, r ) = G α G exp[i(k + G)r ] exp(−K G x),(13) where G is the 2D reciprocal lattice vector, and K G is now defined taking into account the kinetic energy associated with the given reciprocal lattice vector, U +h 2 [k 2 − K 2 G ]/2m 0 = E. An incoming Bloch wave with a given k penetrates into the barrier as a linear combination (13) with the coefficients α G defined by matching conditions, set by the requirement of continuity of the wave function and its derivative: G F G (0) exp[i(k + G)r ] = G α G exp[i(k + G)r ] G F ′ G (0) exp[i(k + G)r ] = − G α G K G exp[i(k + G)r ]. since this has to hold for any r , α G = F G (0) , and F ′ G (0) = −α G K G for each G. Thus F G (0)K G + F ′ G (0) = 0.(14) If F G (x) were a linear combination of the bulk Bloch waves with the energy E and the quasimomentum in the planehk , F G (x) = u kx (x) exp(ik x x) + au −kx (x) exp(−ik x x),(15) we would have only one free parameter, a, to satisfy Eq.14 for all G's, which is obviously impossible. The answer is that F G (x) has the form (15) only far away from the surface, while near the surface it is distorted as required by Eq.14. This emphasizes once again the role of of surface states in tunneling. In fact, one of the ways to realize the necessity of forming the surface states is that the bulk Bloch functions, in general, cannot be augmented continuously and smoothly into vacuum. In the case of a thick barrier, the actual tunneling current will be defined by that component of the wave function (13) which has the smallest K, that is, by the one with G = 0. The amplitude of this evanescent wave is set by α 0 . As pointed out by Butler 3 , k = 0 is a high symmetry direction (ΓX), and the electronic states possess certain symmetry in the yz plane. In particular, α 0 for some states may vanish by symmetry, in which case the decay rate K will be defined by the smallest G allowed by symmetry. Since we consider now a thick barrier, this essentially means that tunnelling from such a band will be defined not by the k = 0 state, but, rather counterintuitively, by general (not high symmetry) points in the 2D Brillouin zone (as confirmed by actual calculations 2 ). Indeed, consider a band where by symmetry F 0 (0, x) = 0 at k = 0. At k = 0 thus F 0 (k , x) = F ′′ 0 (0, x)k 2 , while K = 2m 0 (U − E)/h 2 + k 2 ≈ 2m 0 (U − E)/h+ hk 2 /2 2m 0 (U − E) = K 0 + k 2 /2K 0 . The optimal distance from the zone center that gives maximal contribution to the tunneling current can be estimated by maximizing with respect to k of F ′′ 0 (0, x)k 2 exp(−K 0 d − k 2 d/2K 0 ), where d is the barrier thickness, which gives k ∼ 2K 0 /d. For Fe, for instance, K 0 ≈ 0.6 a.u., about the same as the ΓX distance. Thus for a barrier, say, of 5 lattice parameters, k ∼ 0.2 a.u., a sizeable distance from the center of the Brillouin zone. Yet another counterintuitive result is that the low transparency limit is not unique: in the thick barrier limit tunneling is predominantly from the states infinitely close to the zone center. However, in the high barrier limit, which is another way to implement a low transparency asymptotics, tunneling occurs far away from the zone center, possibly at the zone boundary. This is the effect observed in Refs. 2 . To conclude, we discussed here three new effects which appear in tunneling of the Bloch electrons through a vacuum barrier, as compared with the textbook case of free electron (plane wave) tunneling. These effects are due to (i) complexity of the Fermi surface geometry ("fermiology"), (ii) difference between the group and the phase velocities of a Bloch electron, and (iii) nonconservation of the parallel component of electron momentum (and conservation of its quasimomentum). Each effect influences the tunneling current in its own way, and as a result even for the most simple case of a vacuum barrier, the tunneling of the Bloch electrons appears to be qualitatively different from the free electron tunneling. ACKNOWLEDGMENTSDiscussions with J. Kudrnovsky, I. Mertig, and especially with B. Nadgorny and W. Butler, are gratefully acknowledged. . G Prinz, Phys Today. 4858G.A Prinz, Phys Today, 48 58 (1995); . Science. 2821660Science, 282, 1660 (1998). . Xiao , to be publishedXiao et al, to be published; . I Mertig, to be publishedI. Mertig et al, to be published. . W Butler, to be publishedW. Butler et al, to be published. . Yu V Sharvin, ZhETP. 48984Yu. V. Sharvin, ZhETP 48, 984 (1965); . Sov. Phys. -JETP. 21655Sov. Phys. -JETP 21, 655 (1965). . K M Schep, P M Kelly, G E W Bauer, Phys. Rev. B. 578907K.M. Schep, P.M. Kelly, and G.E.W. Bauer, Phys. Rev. B 57, 8907 (1998). . I I Mazin, Phys. Rev. Lett. 831427I. I. Mazin, Phys. Rev. Lett., 83, 1427 (1999). . W A Harrison, Phys. Rev. 12385W.A. Harrison, Phys. Rev., 123, 85 (1961). In this paper the quantity N \|vx\| was called the current-carrying density of states, as opposed to the thermodynamical density of states. S.-K Yip, Phys. Rev. 585803S.-K. Yip, Phys. Rev. B58, 5803 (1998). In this paper the quantity N \|vx\| was called the current-carrying density of states, as opposed to the thermodynamical density of states, N. L D Landau, E M Lifshits, Quantum Mechanics. L.D. Landau and E.M. Lifshits, Quantum Mechanics. There is an interesting question of whether or not the electronic group velocities in this formula should include manybody renormalizations beyond the conventional band theory (e.g., Kondo-type). See discussion of this question in G. Deutscher and P. Nozieres. Phys. Rev., B. 5013557There is an interesting question of whether or not the elec- tronic group velocities in this formula should include many- body renormalizations beyond the conventional band the- ory (e.g., Kondo-type). See discussion of this question in G. Deutscher and P. Nozieres, Phys. Rev., B 50, 13557 (1994). . B Nadgorny, R J SoulenJr, M S Osofsky, I I Mazin, G Laprade, R J M Van De Veerdonk, A A Smits, S F Cheng, E F Skelton, S B Qadri, Phys. Rev. 613788B. Nadgorny, R. J. Soulen, Jr., M. S. Osofsky, I.I. Mazin, G. Laprade, R.J.M. van de Veerdonk, A.A. Smits, S. F. Cheng, E. F. Skelton, and S. B. Qadri. Phys. Rev. B61, 3788 (2000). . D J Singh, J. Appl. Phys. 794818D.J. Singh, J. Appl. Phys. 79, 4818 (1996). Negative tunneling spin polarization of SrRuO3 has been recently confirmed experimentally. D.C. Worledge and T.H. GeballeNegative tunneling spin polarization of SrRuO3 has been recently confirmed experimentally (D.C. Worledge and T.H. Geballe, to be published).	[]
[ "Dirac-Point Solitons in Nonlinear Optical Lattices", "Dirac-Point Solitons in Nonlinear Optical Lattices" ]	[ "Kang Xie \nSchool of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China\n", "Qian Li \nSchool of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China\n", "Allan D Boardman \nInstitute for Materials Research\nJoule Physics Laboratory\nUniversity of Salford\nM5 4WTSalford, ManchesterUK\n", "Qi Guo \nSchool of Information and Optoelectronic Science and Engineering\nGuangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices\nSouth China Normal University\n510631GuangzhouP.R. China\n", "Zhiwei Shi \nFaculty of Information Engineering\nGuangdong University of Technology\n510006GuangzhouP.R. China\n", "Haiming Jiang \nSchool of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China\n", "Zhijia Hu \nSchool of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China\n", "Wei Zhang \nSchool of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China\n", "Qiuping Mao \nSchool of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China\n", "Lei Hu \nSchool of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China\n", "Tianyu Yang \nSchool of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China\n", "Fei Wen \nSchool of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China\n", "Erlei Wang \nSchool of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China\n" ]	[ "School of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China", "School of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China", "Institute for Materials Research\nJoule Physics Laboratory\nUniversity of Salford\nM5 4WTSalford, ManchesterUK", "School of Information and Optoelectronic Science and Engineering\nGuangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices\nSouth China Normal University\n510631GuangzhouP.R. China", "Faculty of Information Engineering\nGuangdong University of Technology\n510006GuangzhouP.R. China", "School of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China", "School of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China", "School of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China", "School of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China", "School of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China", "School of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China", "School of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China", "School of Instrument Science and Opto-electronic Engineering\nHefei University of Technology\n230009HefeiP.R. China" ]	[]	The discovery of a new type of solitons occuring in periodic systems without photonic bandgaps is reported. Solitons are nonlinear self-trapped wave packets. They have been extensively studied in many branches of physics. Solitons in periodic systems, which have become the mainstream of soliton research in the past decade, are localized states supported by photonic bandgaps. In this Letter, we report the discovery of a new type of solitons located at the Dirac point beyond photonic bandgaps. The Dirac point is a conical singularity of a photonic band structure where wave motion obeys the famous Dirac equation. These new solitons are sustained by the Dirac point rather than photonic bandgaps, thus provides a sort of advance in conceptual understanding over the traditional gap solitons. Apart from their theoretical impact within soliton theory, they have many potential uses because such solitons have dramatic stability characteristics and are possible in both Kerr material and photorefractive crystals that possess self-focusing and self-defocusing nonlinearities. The new results elegantly reveal that traditional photonic bandgaps are	null	[ "https://export.arxiv.org/pdf/1511.07634v1.pdf" ]	55,894,523	1511.07634	559b974e0eb0750486cec484f037179bec399399	Dirac-Point Solitons in Nonlinear Optical Lattices Kang Xie School of Instrument Science and Opto-electronic Engineering Hefei University of Technology 230009HefeiP.R. China Qian Li School of Instrument Science and Opto-electronic Engineering Hefei University of Technology 230009HefeiP.R. China Allan D Boardman Institute for Materials Research Joule Physics Laboratory University of Salford M5 4WTSalford, ManchesterUK Qi Guo School of Information and Optoelectronic Science and Engineering Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices South China Normal University 510631GuangzhouP.R. China Zhiwei Shi Faculty of Information Engineering Guangdong University of Technology 510006GuangzhouP.R. China Haiming Jiang School of Instrument Science and Opto-electronic Engineering Hefei University of Technology 230009HefeiP.R. China Zhijia Hu School of Instrument Science and Opto-electronic Engineering Hefei University of Technology 230009HefeiP.R. China Wei Zhang School of Instrument Science and Opto-electronic Engineering Hefei University of Technology 230009HefeiP.R. China Qiuping Mao School of Instrument Science and Opto-electronic Engineering Hefei University of Technology 230009HefeiP.R. China Lei Hu School of Instrument Science and Opto-electronic Engineering Hefei University of Technology 230009HefeiP.R. China Tianyu Yang School of Instrument Science and Opto-electronic Engineering Hefei University of Technology 230009HefeiP.R. China Fei Wen School of Instrument Science and Opto-electronic Engineering Hefei University of Technology 230009HefeiP.R. China Erlei Wang School of Instrument Science and Opto-electronic Engineering Hefei University of Technology 230009HefeiP.R. China Dirac-Point Solitons in Nonlinear Optical Lattices The discovery of a new type of solitons occuring in periodic systems without photonic bandgaps is reported. Solitons are nonlinear self-trapped wave packets. They have been extensively studied in many branches of physics. Solitons in periodic systems, which have become the mainstream of soliton research in the past decade, are localized states supported by photonic bandgaps. In this Letter, we report the discovery of a new type of solitons located at the Dirac point beyond photonic bandgaps. The Dirac point is a conical singularity of a photonic band structure where wave motion obeys the famous Dirac equation. These new solitons are sustained by the Dirac point rather than photonic bandgaps, thus provides a sort of advance in conceptual understanding over the traditional gap solitons. Apart from their theoretical impact within soliton theory, they have many potential uses because such solitons have dramatic stability characteristics and are possible in both Kerr material and photorefractive crystals that possess self-focusing and self-defocusing nonlinearities. The new results elegantly reveal that traditional photonic bandgaps are The discovery of a new type of solitons occuring in periodic systems without photonic bandgaps is reported. Solitons are nonlinear self-trapped wave packets. They have been extensively studied in many branches of physics. Solitons in periodic systems, which have become the mainstream of soliton research in the past decade, are localized states supported by photonic bandgaps. In this Letter, we report the discovery of a new type of solitons located at the Dirac point beyond photonic bandgaps. The Dirac point is a conical singularity of a photonic band structure where wave motion obeys the famous Dirac equation. These new solitons are sustained by the Dirac point rather than photonic bandgaps, thus provides a sort of advance in conceptual understanding over the traditional gap solitons. Apart from their theoretical impact within soliton theory, they have many potential uses because such solitons have dramatic stability characteristics and are possible in both Kerr material and photorefractive crystals that possess self-focusing and self-defocusing nonlinearities. The new results elegantly reveal that traditional photonic bandgaps are not required when Dirac points are accessible. The findings enrich the soliton family and provide valuable information for studies of nonlinear waves in many branches of physics, including hydrodynamics, plasma physics, and BoseEinstein condensates. Confinement of waves within a finite area is the basis of information processing [1]. Traditionally, wave trapping is achieved by cavities and waveguides that rely on total internal reflection, or photonic bandgap, to suppress radiation losses [2,3]. Cavities and waveguides can be formed by a high-index core surrounded by a cladding with a lower refractive index so that total internal reflection can take place [2]. Alternatively, cavities and waveguides can be formed in periodic systems as defects [3]. Due to the presence of allowed bands and forbidden gaps, radiation losses of the wave accommodated by the defect are suppressed by any photonic bandgap of the systems. There are various other ways and, among them, nonlinearity is an unique example. Localized modes due to nonlinearity are commonly called solitons [4,5]. In homogeneous media, nonlinearity raises refractive index of the media so that light creates its own high-index core. In this way light is essentially guided by total internal reflection. In periodic systems, nonlinearity changes its onsite refractive index so that light creates its own defect. In this case light is trapped by photonic bandgaps of the periodic lattices. Solitons are nonlinear self-trapped wave packets. They have been extensively studied in many branches of physics. Solitons in periodic systems have become the mainstream of soliton research in the past decade [6,7]. During the same period, research on graphene has made great progress [8]. The electronic band structure of graphene contains Dirac cones at the six corners of the hexagonal Brillouin zone. The associated energy-wavenumber relation resembles the two-dimensional massless Dirac equation iv( x  x + y  y )=( D ) for relativistic electrons in a vacuum, where v is the velocity,  D /2 is the Dirac frequency,  x and  y are Pauli matrices, and \| \| 2 is the probability of finding the spinors in space. Building on the observations of graphene, it is found that the band structure of a photonic crystal formed by a two-dimensional triangular lattice also possesses Dirac cones at the corners of the Brillouin zones [9]. At these high-symmetry points Maxwell's equations can be replaced by the massless Dirac equation with  being the wave functions of two degenerate Bloch states. Wave behaviour at the Dirac frequency has been studied extensively in photonic crystals since then [10][11][12], and localized modes have been found recently at the Dirac frequency [13,14]. It has been shown that the Dirac point in band structures of these lattices can take the role of a bandgap to form localized modes at a defect, a mechanism different from that of a nonlinear Dirac soliton of the nonlinear relativistic Dirac equation [15]. Soliton in photonic crystals is essentially a nonlinearity-induced defect mode, so it is natural to ask if such a self-localized mode can be supported, or not, by the same Dirac point. In this Letter, we report the discovery of a new type of solitons occuring at the Dirac point. It is found that, besides photonic bandgaps, a Dirac point in the band structure of a triangular nonlinear lattice can also sustain self-localized nonlinear modes. This new specific entity is designated here as Dirac-point soliton. We show that such solitons are possible in both Kerr material and photorefractive crystals with self-focusing and self-defocusing nonlinearities. Characteristics of the Dirac-point solitons are revealed and their stability condition is analyzed by linear stability analysis. It is found that the Dirac-point solitons satisfy the so-called Vakhitov-Kolokolov stability criterion [16]. We verify the stability criterion by direct numerical simulations. The propagation of optical beams in a nonlinear periodic array is described by the nonlinear Schrödinger equation for the slowly varying amplitude of the light [17,18]: 22 22 0 NL U i U V U Z X Y              (1) where the wave is presumed to propagate predominantly along the Z-direction. The potential for a Kerr nonlinearity is 2 \|\| NL V V U   , where =1 (or 1) corresponds to a Kerr self-focusing (or self-defocusing) nonlinearity. The linear index potential   2 0 1 2 3 cos( ) cos( ) cos( ) V V b r b r b r          2 0 1+ + NL V V I U  , which suits the description of photorefractive crystals. If V 0 >0 (<0) the medium nonlinearity has a self-focusing (self-defocusing) nature. The normalized intensity pattern   2 0 1 2 3 , ) cos( ) cos( ) cos( ) I X Y I b r b r b r         ( can be generated experimentally on a stationary background by interfering three plane waves with intensity I 0 and transverse wave vectors b i . The three wave vectors of the plane waves form a triangle, i.e., b 1 +b 2 +b 3 =0. In the linear limit Eq. (1) reduces to 22 22 0 U i U VU Z X Y              . Wave propagation in such a linear periodic lattice has the form of   , iqZ U X Y e    and is known to exhibit unique features that arise from the presence of allowed bands and forbidden gaps. The band structure of the lattice, which can be found by the plane wave expansion method [1] , exhibits Dirac cones at the six corners of the Brillouin zone at an eigenvalue q D . At the Dirac point q=q D , the density of radiation states is precisely zero [20], which means that outgoing waves are forbidden in the surrounding medium. Because of this feature, field concentration around a defect becomes possible and the optical lattice can support localized modes at the Dirac point. Examples of the potential V and its associated linear defect-guided modes are discussed in Supplemental Material. There exists a range of q, around q D , where decay rate is low and optical wave-guiding by a defect is practically realizable. The defect could be created by nonlinearly-induced onsite index change. If the defects are self-induced optically by nonlinearity, the corresponding, self-localized nonlinear modes are referred to as solitons. In other words, the presence of a defect mode at the Dirac point in the linear limit suggests the existence of a Dirac-point soliton in the corresponding nonlinear system. This is, indeed, the case. To find the Dirac-point solitons of Eq. (1), we seek a solution of the form The Dirac-point solitons in a nonlinear media preserve their shape, but their stability is not guaranteed, because of the non-integrable nature of the underlying equation. In fact, their stability is a crucial issue because only stable (or weakly unstable) modes can be observed experimentally. To study the stability of these solitons, a perturbation of the form . In a saturable self-focusing lattice (Fig. S5), growth rates exceed 10, both the fundamental and the first vortex solitons are unstable. In a saturable self-defocusing lattice (Fig. S6), growth rates are of the order of 1, both the fundamental and the first vortex solitons are weakly unstable. In Kerr lattices (Figs. 1 and 2), the fundamental soliton is unstable for self-focusing nonlinearity and stable for self-defocusing nonlinearity. The power curves in Figs. 1(c) and 2(c) also give information on the stability of the solitons. According to the Vakhitov-Kolokolov stability criterion [16], a soliton is stable (unstable) if slope of the corresponding power curve is positive (negative). The Vakhitov-Kolokolov stability criterion is derived for homogenous nonlinear medium but there is a periodic potential present for the Dirac-point solitons, so it is not directly applicable. However, our linear stability analysis comes up with results that agree well with the Vakhitov-Kolokolov stability criterion, despite the periodicity of the potential. More specifically, the Dirac-point solitons in self-defocusing lattices [ Figs The stability of Dirac-point solitons can be checked numerically using the split-step Fourier method. Simulated evolution scenarios of typical stable and unstable solitons are shown in Figs. 3 and 4 respectively, confirming the stability analysis. The propagation distance of the stable soliton ( Fig. 4) is about three orders of magnitude larger than that of the unstable soliton (Fig. 3). The Dirac-point solitons belong to the group of algebraic solitons [23]. As shown in Figs. 3(e) and 4(e), tail of the Dirac-point soliton decays algebraically according to a power-law (roughly r 3/2 ) at large distances. This is understandable because the field is so weak in the cladding that it resumes a linear behaviour, and in the linear limit a triangular optical lattice can support waves with power-law asymptotics at the Dirac point [13,14]. The spectrum of the stable soliton is shown in Fig In summary, a new type of solitons, which rely on the Dirac point rather than photonic bandgaps to establish field localization, are discovered in photonic lattices. The Dirac equation is a special symbol of relativistic quantum mechanics, from merging quantum mechanics with special relativity to predicting the existence of anti-matter. Investigations in the Dirac-point solitons may lead to new findings in many relativistic quantum effects on the transport of photons, phonons, and electrons. This work is supported by the National Science Foundation of China (11574070)     , , , iqZ U X Y Z X Y e       * * iqZ Z Z U e v w e v w e           , is invoked, where v,  2 1 2 3 10 3/2 cos( ) cos( ) cos( ) V b r b r b r        ,  2 1 2 3 10 3/2 cos( ) cos( ) cos( ) V b r b r b r        and the initial profile of the soliton is shown in Fig. 1(a).   2 1 2 3 35 1/3 cos( ) cos( ) cos( ) V b r b r b r          and the initial profile of the soliton is shown in Fig. 2 The plane wave expansion method In the linear limit Eq. (1) reduces to 22 22 0 U i U VU Z X Y              (S1) The band structure of a periodic lattice can be found by substituting a solution of the form     , , , iqZ U X Y Z X Y e    into Eq. (S1). This results in the following eigenvalue equation 22 22 Vq XY             (S2) Following Bloch's theorem, the eigenfunction (r) in a periodic potential can be presented in the form of a product of a periodic function in space and a complex exponential: (r)=(r)exp(ik· r), where (r) is periodic, possessing the period of the lattice, and k is its Bloch momentum. Invoking Fourier analysis, a periodic function can be expanded in terms of an infinite, discrete, sum of spatial harmonics: (r)=h(G)exp(iG· r) , where G = b 1 P 1 +b 2 P 2. (P 1 , P 2 )( ') \| \| ( ') ( ') ( ) G G G k G f G G h G qh G          (S3) Typical examples of the linear lattice potential V are depicted in Fig. S1. Given the potential, the eigenvalue equation (S3) can be solved numerically to obtain the lattice band structure. The results are shown in Fig. S2 for the potentials of Fig. S1. As can be seen, Dirac cones appear in these cases at the six corners of the Brillouin zone with eigenvalues, respectively, q D =115.644, q D =8.67, q D =29.4445, and q D =41.81. Excitation of linear localized modes using numerical simulations A circular defect at the origin is introduced by setting V(X,Y)=V d , for 22 X Y R  . Linear localized modes in these structures are discovered by using the finite difference beam propagation method (BPM) based on the evolution Eq. (S1), together with a transparent boundary condition. A source beam with phase varying as exp(iq D Z):     22 ( , , ) exp exp D S X Y Z X Y iq Z     , is actually launched in the defect waveguide. By changing the parameters (R and V d ) of the defect, a situation arises in which the power in the waveguide monotonously grows with propagation distance, as shown in Fig. S3(a) (the initial stage) using the potential shown in Fig. S1(d), with R=2 and V d =43.3. This happens when synchronization of the waveguide eigenmode with the source beam is established, so that the eigenmode is always in-phase with the source and energy is extracted at every step of the propagation. This indicates the excitation of an eigenmode of the waveguide that has eigenvalue q D as the propagation constant. The field profile of this linear defect-guided mode is shown in Fig. S3(b). The BPM yields the space domain response U(Z) directly. The spectral response u(q) is subsequently obtained by the discretized Fourier transform from the spatial series 0 1 ( ) ( ) N iqn Z n u q U n Z e N      , where Z is step-size, N is the number of steps, and q (equivalent to spatial frequency) is the propagation constant. The spectrum of the eigenmode, obtained in this way, is shown in Fig. S3(c), which verifies that the propagation constant of the guided mode is truly centered at q D =41.81. In the second stage of evolution shown in Fig. S3(a), the source is switched off and amplitude of the eigenmode starts a propagation decline. In this free evolution stage, the electromagnetic power within the mode decays slowly and exponential according to the format as P=P 0 e Z . This law shows clearly that the instantaneous decay rate is 1 dP P dZ   and this can be calculated from slope of the power curve, after the evolution of beam power is numerically obtained. The instantaneous  value calculated in this way is shown in Fig. S3( scattering into the continuum of states ( called losses associated with coupling into radiation modes). The net decay rate = c + s increases with \|q-q D \|. Leakage of the guided mode can be conveniently studied by the BPM. Suppose the beam power is P 1 at Z 1 and P 2 at Z 2 ,with the forms P 1 =P 0 exp(Z 1 ), P 2 =P 0 exp(Z 2 ), then the average decay rate between Z 1 and Z 2 is =ln(P 1 /P 2 )/(Z 2 Z 1 ). The average loss rates , of the localized modes, extracted from the evolution of power are also shown in Fig. S4 as a function of the propagation constant q using, respectively, the four different lattices. Minima of the loss rates  occur roughly around their corresponding Dirac propagation constants q D , thus confirming the positions of the Dirac points. The modified squared-operator iteration method for solitary waves To find the Dirac-point solitons of Eq. (1), we seek a solution of the form     , , , iqZ U X Y Z X Y e    . Given this substitution, Eq. (1) reduces to 22 The linear stability analysis To study the stability of these Dirac-point solitons, a perturbation of the form     * * iqZ Z Z U e v w e v w e           , is invoked, where v,                        2 2 * 2 2 2 2 2 * * * * * * * 2 * * 2 2 2 0 2 0 q v w i v w v w V v w v w v w XY q v w i v w v w V v w v w v w XY                                                   Taking complex conjugate of the second equation                         * 2 * 2 2 0 2 0 q v w i v w v w V v w v w v w q v w i v w v w V v w v w v w                                   where the operator is 22 2 \| \| 2 \| \| v q V w i v q V v w i w                                   which is a linear eigenvalue problem L= with  being the transpose of (v, w) and         2 2 2 2 2 11 22 2 2 2 2 2 11 22 2 \| \| 2 \| \| qV Li qV                                    For saturable nonlinearity substituting the perturbation       * ,, iqZ Z Z U X Y Z e v w e v w e           into 22 0 2 22 0 0 + V U i U U Z X Y V V U              and then linearizing, results in                           22 ** * * * 22 22 0 2 2 2 0 ** * * * * 0 2 2 0 +\| \| +0 +\| \| Z Z Z Z Z Z Z Z Z Z Z Z q v w e q v w e i v w e i v w e v w e XY V v w e v w e v w e X Y V V V v w e v w e v w e v w e VV                                                            Separate the two groups with respectively factors of Z e  and Z e  and take complex conjugate of the second equation                             22 2 0 0 2 22 2 0 22 2 0 0 2 22 2 0 0 +\| \| 0 +\| \| V q v w i v w v w v w V V v w XY VV V q v w i v w v w v w V V v w XY VV                                             Rearranging into                     2 2 2 2 0 0 0 1 1 0 2 2 2 2 2 2 2 2 0 0 0 2 2 2 2 0 0 0 1 1 0 2 2 2 2 2 2 2 2 0 0 0 + +\| \| +\| \| +\| \| + + +\| \| +\| \| +\| \| V V V v qw w wV V w iv V V V V V V V V V qv v vV V v w iw V V V V V V                              which is a linear eigenvalue problem L= with  being the transpose of (v, w) and                     2 2 2 2 0 0 0 1 1 0 2 2 2 2 2 2 2 2 0 0 0 2 2 2 2 0 0 0 1 1 0 2 2 2 2 2 2 2 2 0 0 0 + +\| \| +\| \| +\| \| + + +\| \| +\| \| +\| \| L V V V q V V V V V V V V V V V i q V V V V V V V V                                       The linear stability eigenvalue problem L= is solved by the numerical iteration method. The real part of the perturbation growth rate Re() versus the propagation constant q of the soliton is plotted in Figs. S5(d)-S8(d) respectively for the four different lattices shown in Figs. S1(a)-S1(d). In a saturable, self-focusing, lattice (Fig. S5), growth rates exceed 10, both the fundamental soliton and the first vortex soliton are unstable. In a saturable self-defocusing lattice (Fig. S6), growth rates are of the order of 1, both the fundamental soliton and the first vortex soliton are weakly unstable. In Kerr lattices (Figs. S7 and S8), the fundamental soliton is unstable for self-focusing nonlinearity and stable for self-defocusing nonlinearity. The stability of Dirac-point soliton can be checked numerically using the split-step Fourier method. Simulated evolution scenarios of typical stable and unstable solitons are shown in Figs. S9 and S10 respectively, confirming the stability analysis. The propagation distance of the stable soliton (Fig. S10) is about three orders of magnitude larger than that of the unstable soliton (Fig. S9). Propagation distance of the Dirac-point soliton is not infinite when loss is in existence, even for the stable case. Owing to the slowly decaying tail of the Dirac-point soliton itself, losses associated with field penetration across boundary into the surrounding is sensitive to the lattice size. As the Dirac-point soliton breaks down, or as it loses power to the surrounding medium, its amplitude reduces. This alters parameters of the nonlinearity-induced defect, and, in turn, shifts its propagation constant. This is a process of self-propagation-constant shift that the Dirac-point soliton undergoes in propagation, an analogue of self-frequency shift of a temporal soliton. As the propagation constant of the soliton deviates from the value of the Dirac point, losses associated with coupling into radiation modes arise and accelerate degradation of the soliton. Therefore, even for a stable soliton, the losses will sooner, or later, breakup the balance between contraction and diffraction, and eventually diminish the soliton. As such, the propagation distance of a stable Dirac-point soliton is restricted by the finite lattice size. On the other hand, a gap soliton is hardly influenced by a distanced boundary because its tail decays exponentially in space. At the Dirac point the density of radiation states vanishes, any residual values of loss rate at this point are entirely due to the finite lattice size, which can be made as small as desired by increasing the boundary surrounding the soliton. Therefore, the propagation distance of the stable Dirac-point soliton can be extended to almost as long as desired. V V I b r b r b r         for I 0 =2, =0, V 0 =250 (a, self-focusing) and I 0 =1, =3, V 0 =150 (b, self-defocusing). (c, d) Index potential   2 0 1 2 3 cos( ) cos( ) cos( ) V V b r b r b r         ofU i U VU iS Z X Y             V V I b r b r b r         for I 0 =2, =0, V 0 =250. (a, b . 1(b) and 2(b), Figs. S6(a) and S6(b)] exhibit dP/dq0 on the power curves and are stable (or weakly unstable), whereas the Dirac-point solitons in self-focusing lattices [Figs. 1(a) and 2(a), Figs. S5(a) and S5(b)] exhibit dP/dq<0 and are unstable. . 4(f), which verifies that the propagation constant of the Dirac-point soliton is truly centered at the Dirac point q D =41.81. On the other hand, the propagation constant of the unstable soliton [Fig. 3(f)] sweeps a wide range of values. As the Dirac-point soliton breaks down, its amplitude reduces. This alters parameters of the nonlinearity-induced defect, and, in turn, shifts its propagation constant. which exhibits absolute index maxima on lattice sites. (a, b) The field profiles of the solitons in a Kerr self-focusing (a) and self-defocusing (b) lattice at q D =29.445. (c, d) Power P (c) and real part of  (d) versus q, where "" corresponds to the self-focusing case and "+" corresponds to the self-defocusing case. The dotted vertical line indicates the position of the Dirac point. FIG. 2. Dirac-point solitons in Kerr nonlinear media. The lattice potential is lattice sites. (a, b) The field profiles of the solitons in a Kerr self-focusing (a) and self-defocusing (b) lattice at q D =41.81. (c, d) Power P (c) and real part of  (d) versus q, where "" corresponds to the self-focusing case and "+" corresponds to the self-defocusing case. The dotted vertical line indicates the position of the Dirac point. FIG. 3. Breakdown of the Dirac-point soliton in a Kerr self-focusing lattice. The lattice potential is (a) Evolution of the soliton amplitude \|(0,0)\|. (b-d) The \| \| field at respectively Z=0.006, 0.048, 0.06. (e) Product of the initial  and r 3/2 on the X axis. (f) Propagation constant spectrum of the soliton. The green vertical line indicates the position of the Dirac point. FIG. 4. Dynamics of the Dirac-point soliton in a Kerr self-defocusing lattice. The lattice potential is (b). (a) Evolution of the soliton amplitude \|(0,0)\|. (b-d) The \| \| field at respectively Z=4.5, 22.5, 45. (e) Product of the initial  and r 3/2 on the X axis. (f) Propagation constant spectrum of the soliton. The green vertical line indicates the position of the Dirac point. are any integers and (b 1 , b 2 ) are the reciprocal basis vectors of the lattice. Thus, the eigenfunction (r) S cell = (3/2) is the area of the unit cell. Substituting the expressions of (r) and V(r) into Eq. (S2) leads to 2 ' d), as a function of the propagation distance. The guided mode exists for a range of propagation constants around the Dirac point. Eigenmodes for propagation constants other than value for the Dirac point can be found in a similar way by adjusting the source beam to have other phase constants. The results are shown in Figs. S4(a)-S4(d), corresponding to the lattice potentials, respectively, of Figs. S1(a)-S1(d). They show parameter V d of the defect versus the propagation constant q of the eigenmode for fixed R. However, as the propagation constant moves away from the Dirac point, the density of states increases in a linear fashion. Hence, there exists two loss mechanisms for the guided mode. If the propagation constant does not coincide with q D then, in addition to leakage  c to the surrounding medium due to the finite lattice size (called losses associated with field penetration across boundary into the surrounding), there can be leakage  s due to real-valued positive-definite Hermitian operator, and the step size t controls the speed of convergence of the program. The convergence can be further speeded up by adopting the modified squared-operator iteration method. r)=0 at the beam center r=0 and f()=0 for a bright vortex soliton. The integer m stands for a phase twist around the intensity ring and is usually called the winding number. Typical Dirac-point solitons (fundamental and vortex) the solitons versus the eigenvalue q are shown in Figs. S5-S8 respectively for the focusing/defocusing saturable and Kerr nonlinearities. The gray scale in the background of Figs. S5(c, d) -S8(c, d) indicates the level of linear losses  of the wave (as given in Fig. S4) in the lattice potential, which marks roughly the range of propagation constant within which the associated Dirac-point solitons can exist. The Dirac-point solitons are found to exist within the ranges shaded light gray, where the level of the linear decay rate  is low. The dotted vertical lines indicate the positions of the Dirac points. Fig. S1. Four different lattice potentials. (a, b) Index potential of the photorefractive Fig. S2 .Fig. S3 . S2S3the Kerr nonlinear medium for V 0 =10, =3/2 (c) and V 0 =35, =1/3 (d). In the cases of (b) and (c) the potentials exhibit absolute index maxima on lattice sites, while in the cases of (a) and (d) the potentials exhibit absolute index minima on lattice sites. Band structures (left) and the enlarged 3D views of linear Dirac cones around the Dirac point (right) of four different optical lattices. (a-d) correspond to the lattices shown in Figs. S1(a)-S1(d): (a) I 0 =2, =0, V 0 =250, (b) I 0 =1, =3, V 0 =150, (c) V 0 =10, =3/2, and (d) V 0 =35, Mode in a waveguide formed by a defect V(X,Y)=V d for 22 X Y R  in the potential of Fig. S1(d). Parameters of the defect are R=2 and V d =43.3. (a) Evolution of amplitude of the excited eigenmode.Between Z=0 and 20 a source beam with a phase constant q D =41.81 is on, i.e., the field evolves according to22 22 the second stage (Z >20) the source beam is turned off and the eigenmode evolves freely on its own in the waveguide. (b) Profile of the eigenmode of the waveguide excited by the source beam. (c) The propagation constant spectrum of the excited mode. The green vertical line indicates the position of the Dirac point. (d) The instantaneous decay rate  of the excited mode as it propagates down the waveguide. Fig. S4 . S4(a-d) Parameter V d of the defect (left) and the average power loss rate  of the modes (right) versus the eigen propagation constant q of the localized mode for respectively the lattices shown in Figs. S1(a)-S1(d). The computational domain is taken as a square of 15 < X, Y < 15 (a, b) or 10 < X, Y < 10 (c, d), discretized by 601 points along each dimension. The green horizontal lines indicate the positions of the Dirac points. Fig. S5 . S5Dirac-point solitons in a saturable self-focusing lattice. The lattice potential is shown inFig. S1(a):   respectively. (c, d) Power P (c) and the real part of the perturbation growth rate Re() (d) versus the propagation constant q of the soliton, where "" represents the fundamental soliton and "+" represents the vortex soliton. The dotted vertical line indicates the position of the Dirac point. Fig. S6. Dirac-point solitons in a saturable self-defocusing lattice. The lattice potential is shown in Fig. S1(=3, V 0 =150. (a, b) The field profiles of the fundamental soliton (a) and the first vortex soliton (m=1) (b) at the Dirac point q D =8.67. The computational domain is taken as a square of 10 < X, Y < 10, discretized by 512 points along each dimension. The initial condition is taken respectively as . (c, d) Power P (c) and the real part of the perturbation growth rate Re() (d) versus the propagation constant q of the soliton, where "" represents the fundamental soliton and "+" represents the vortex soliton. The dotted vertical line indicates the position of the Dirac point. Fig. S7 .Fig S7Dirac-point solitons in Kerr nonlinear media. The lattice potential is shown in V 0 =10, =3/2, which exhibits absolute index maxima on lattice sites. (a, b) The field profiles of the fundamental solitons in a Kerr self-focusing ( =1) (a) and self-defocusing ( =1) (b) lattice at the Dirac point q D =29.445. The computational domain is taken as a square of respectively 5 < X, Y < 5 and 10 < X, Y < 10, discretized by 512 points along each dimension. The initial condition is respectively taken as -defocusing case. (c, d) Power P (c) and real part of the perturbation growth rate Re() (d) versus the propagation constant q of the soliton, where "" corresponds to the self-focusing case and "+" corresponds to the self-defocusing case. The dotted vertical line indicates the position of the Dirac point. Fig. S8 .Fig S8Dirac-point solitons in Kerr nonlinear media. The lattice potential is shown in V 0 =35, =1/3, which exhibits absolute index minima on lattice sites. (a, b) The field profiles of the fundamental solitons in a Kerr self-focusing ( =1) (a) and self-defocusing ( =1) (b) lattice at the Dirac point q D =41.81. The computational domain is taken as a square of 5 < X, Y < 5, discretized by 1024 points along each dimension. the two Dirac-point solitons. (c, d) Power P (c) and real part of the perturbation growth rate Re() (d) versus the propagation constant q of the soliton, where "" corresponds to the self-focusing case and "+" corresponds to the self-defocusing case. The dotted vertical line indicates the position of the Dirac point. Fig. S9 . S9Breakdown of the Dirac-point soliton in a Kerr self-focusing lattice. The lattice potential is shown in Fig. S1(V 0 =10, =3/2. The initial profile of the soliton is shown in Fig. S7(a). The computational domain is taken as a square of 10 < X, Y < 10, discretized by 1024 points along each dimension. (a) Evolution of amplitude \|(0,0)\| of the fundamental soliton in propagation. (b-d) The \| \| field (zoomed in to 5 < X, Y < 5) of the soliton at respectively Z=0.006, 0.048, 0.06. (e) Product of the initial  and r 3/2 on the X axis.(f) Propagation constant spectrum of the soliton. The green vertical line indicates the position of the Dirac point. Fig. S10 . S10Dynamics of the Dirac-point soliton in a Kerr self-defocusing lattice. The lattice potential is shown in Fig. S1(V 0 =35, =1/3. The initial profile of the soliton is shown in Fig. S8(b).The computational domain is taken as a square of 15 < X, Y < 15, discretized by 1024 points along each dimension. (a) Evolution of amplitude \|(0,0)\| of the fundamental soliton in propagation. (b-d) The \| \| field (zoomed in to 5 < X, Y < 5) of the soliton at respectively Z=4.5, 22.5, 45. (e) Product of the initial  and r 3/2 on the X axis. (f) Propagation constant spectrum of the soliton. The green vertical line indicates the position of the Dirac point. PACS numbers:42.65.Tg, 42.70.Qs, 42.82.Et, 73.22.Pr, 78.67.Pt Eq.(2) , which is a static nonlinear Schrödinger equation, is solved numerically by the modified squared-operator method[21] for solitary wave solutions. For a fundamental soliton  is real, while for a vortex soliton , where m is an integer.. Given this substitution, Eq. (1) reduces to 22 22 NL Vq XY             (2)   im f r e    Typical Dirac-point solitons and the power 2 \|\| P dXdY    conveyed by the solitons are shown in Figs. 1 and 2 for the Kerr self-focusing and self-defocusing nonlinearities. Results for Dirac-point solitons in photorefractive crystals are presented in Figs. S5 and S6 [19]. The gray scale in the background of Figs. 1(c, d) and 2(c, d) indicates the level of linear losses  of the wave [19] in the lattice potential. The Dirac-point solitons are found to exist within the ranges shaded light gray, where the level of the linear decay rate  is low. Supplemental Material for " Dirac-Point Solitons in Nonlinear Optical Lattices " Kang Xie, Qian Li, Allan D. Boardman, Qi Guo, Zhiwei Shi, Haiming Jiang, Zhijia Hu, Wei Zhang, Qiuping Mao, Lei Hu, Tianyu Yang, Fei Wen, Erlei Wang Table of Contents 1 . of1The plane wave expansion method 2. Excitation of linear localized modes using numerical simulations 3. The modified squared-operator iteration method for solitary waves 4. The linear stability analysis The technique deployed here is to find solitary wave solutions of Eq. (S4) by iteration methods. For the fundamental soliton, it should be noted that  is real and /X= /Y= 0 at (X, Y)=(0, 0), also ()=0. Progress towards a solution can then be made by supposing that an approximate real solution  n exists, which is close to the exact solution . To obtain the next, iterative, form of the solution,  n+1 , the following procedure is followed. First, express the exact solution as = n +, where << is the error term. Then substitute this expression into Eq. (S4) and expand it around  n . This leads to the linear inhomogeneous equation for the error Hermitian of L 1 is L 1 itself. The approximate solution can then be updated to  n+1 = n +. This equation has a simpler appearance but converges very slowly. The situation can be improved by the introduction of the22 NL Vq XY             (S4) where 0 2 0 +\| \| NL V V VV   for the saturable nonlinearity and 2 \|\| NL VV   for a Kerr nonlinearity. Eq. (S4) is a static nonlinear Schrödinger equation that can be solved numerically by the modified squared-operator method for solitary wave solutions. : 01 n LL     , which, in turn, gives 10 n LL      , where  1 L is the Hermitian of 1 L . In the case of the fundamental soliton, 22 0 22 NL L V q xy       , 2 10 2 2\| \| \|\| NL V LL      , and the acceleration operator M, then 11 10 Δ n M L M L t        , where   ) The field profiles of the fundamental soliton (a) and the first vortex soliton (m=1) (b) at the Dirac point q D =115.644. The computational domain is taken as a square of 7.5 < X, Y < 7.5, discretized by 512 points along each dimension. The initial condition is taken as  2 2 2 2 2 2 20 20 =550sech cos 33 X Y X Y X Y                 and     22 =100sech 4 X Y X iY M Skorobogatiy, J Yang, Fundamentals of photonic crystal guiding. CambridgeCambridge Univ. PressM. Skorobogatiy and J. Yang, Fundamentals of photonic crystal guiding, (Cambridge Univ. Press, Cambridge, 2009). . K C Kao, G A Hockham, IEE Proceedings. 1131151K. C. Kao and G. A. Hockham, IEE Proceedings 113, 1151 (1966). . J C Knight, J Broeng, T A Birks, P S J Russell, Science. 2821476J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, Science 282, 1476 (1998). Optical solitons: from fibers to photonic crystals. Y S Kivshar, G P , Academic PressAmsterdamY. S. Kivshar and G. P. Agrawal, Optical solitons: from fibers to photonic crystals, (Academic Press, Amsterdam,2003). . Z Chen, M Segev, D N Christodoulides, Rep. Prog. Phys. 7586401Z. Chen, M. Segev, and D. N. Christodoulides, Rep. Prog. Phys. 75, 086401 (2012). . 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[ "Extractors for Sum of Two Sources", "Extractors for Sum of Two Sources" ]	[ "Eshan Chattopadhyay \nCornell University\nCornell University\n\n", "Jyun-Jie Liao [email protected] \nCornell University\nCornell University\n\n" ]	[ "Cornell University\nCornell University\n", "Cornell University\nCornell University\n" ]	[]	We consider the problem of extracting randomness from sumset sources, a general class of weak sources introduced by Chattopadhyay and Li (STOC, 2016). An (n, k, C)-sumset source X is a distribution on {0, 1} n of the form X1 +X2 +. . .+XC, where Xi's are independent sources on n bits with min-entropy at least k. Prior extractors either required the number of sources C to be a large constant or the min-entropy k to be at least 0.51n.As our main result, we construct an explicit extractor for sumset sources in the setting of C = 2 for min-entropy poly(log n) and polynomially small error. We can further improve the min-entropy requirement to (log n) · (log log n) 1+o(1) at the expense of worse error parameter of our extractor. We find applications of our sumset extractor for extracting randomness from other well-studied models of weak sources such as affine sources, small-space sources, and interleaved sources.Interestingly, it is unknown if a random function is an extractor for sumset sources. We use techniques from additive combinatorics to show that it is a disperser, and further prove that an affine extractor works for an interesting subclass of sumset sources which informally corresponds to the "low doubling" case (i.e., the support of X1 + X2 is not much larger than 2 k ). * Supported by NSF CAREER award 2045576 1 Supp(X) denotes the support of X. We use log to denote the base-2 logarithm in the rest of this paper. Definition 1.1. Let X be a family of distribution over {0, 1} n . We say a deterministic function Ext :We say Ext is explicit if Ext is computable by a polynomial-time algorithm.The most well-studied deterministic extractors are multi-source extractors, which assume that the extractor is given C independent (n, k)-sources X 1 , X 2 , . . . , X C . This model was first introduced by Chor and Goldreich[CG88]. They constructed explicit two-source extractors with error 2 −Ω(n) for entropy 0.51n, and proved that there exists a two-source extractor for entropy k = O(log(n)) with error 2 −Ω(k) . Significant progress was made by Chattopadhyay and Zuckerman [CZ19], who showed how to construct an extractor for two sources with entropy k = polylog(n), after a long line of successful work on independent source extractors (see the references in [CZ19]). The output length was later improved to Ω(k) by Li[Li16]. Furthermore, Ben-Aroya, Doron and Ta-Shma [BDT19] showed how to improve the entropy requirement to O(log 1+o(1) (n)) for constant error and 1-bit output. The entropy requirement was further improved in subsequent works[Coh17,Li17], and the state-of-the-art result is by Li [Li19], which requires k = O(log(n) · log log(n) log log log(n) ). For a more elaborate discussion, see the survey by Chattopadhyay [Cha20]. Apart from independent sources, many other classes of sources have been studied for deterministic extraction. We briefly introduce some examples here. A well-studied class is oblivious bit-fixing sources [CGH + 85, GRS06, KZ07, Rao09], which is obtained by fixing some bits in a uniform random string. Extractors for such sources have found applications in cryptography [CGH + 85, KZ07]. A natural generalization of bitfixing sources is the class of affine sources, which are uniform distributions over some affine subspaces and have been widely studied in literature (see[CGL21]and references therein). Another important line of work focuses on the class of samplable sources, which are sources sampled by a "simple procedure" such as efficient algorithms [TV00], small-space algorithms [KRVZ11] or simple circuits[Vio14]. Researchers have also studied interleaved sources [RY11, CZ16, CL16b, CL20], which is a generalization of independent sources such that the bits from different independent sources are permuted in an unknown order.In this paper, we consider a very general class of sources called sumset sources, which was first studied by Chattopadhyay and Li[CL16b]. A sumset source is the sum (XOR) of multiple independent sources, which we formally define as follows.Definition 1.2. A source X is a (n, k, C)-sumset source if there exist C independent (n, k)-sources {X i } i∈ [C] such that X = C i=1 X i . Chattopadhyay and Li[CL16b]showed that the class of sumset sources generalize many different classes we mentioned above, including oblivious bit-fixing sources, independent sources, affine sources and smallspace sources. They also constructed an explicit extractor for (n, k, C)-sumset sources where k = polylog(n) and C is a large enough constant, and then used the extractor to obtain new extraction results for smallspace sources and interleaved C sources. An interesting open question left in [CL16b] is whether it is possible to construct an extractor for (n, polylog(n), 2)-sumset source. An explicit construction of such an extractor would imply improved results on extractors for interleaved sources and small-space sources with polylogarithmic entropy. (We discuss the details in Section 1.1.)However, it has been challenging to construct such an extractor for low min-entropy. The only known extractor for sum of two sources before this work is the Paley graph extractor [CG88], which requires one source to have entropy 0.51n and the other to have entropy O(log(n)), based on character sum estimate by Karatsuba [Kar71, Kar91] (see also[CZ16,Theorem 4.2]). In fact, unlike other sources we mentioned above, it is not clear whether a random function is an extractor for sumset sources. (See Section 1.3 for more discussions.)In this paper, we give a positive answer to the question above. Formally, we prove the following theorem.	10.1145/3519935.3519963	[ "https://arxiv.org/pdf/2110.12652v1.pdf" ]	239,768,219	2110.12652	05dd082f4938e847a5b254954e91fd26ddb1e18c	Extractors for Sum of Two Sources 25 Oct 2021 October 26, 2021 Eshan Chattopadhyay Cornell University Cornell University Jyun-Jie Liao [email protected] Cornell University Cornell University Extractors for Sum of Two Sources 25 Oct 2021 October 26, 2021 We consider the problem of extracting randomness from sumset sources, a general class of weak sources introduced by Chattopadhyay and Li (STOC, 2016). An (n, k, C)-sumset source X is a distribution on {0, 1} n of the form X1 +X2 +. . .+XC, where Xi's are independent sources on n bits with min-entropy at least k. Prior extractors either required the number of sources C to be a large constant or the min-entropy k to be at least 0.51n.As our main result, we construct an explicit extractor for sumset sources in the setting of C = 2 for min-entropy poly(log n) and polynomially small error. We can further improve the min-entropy requirement to (log n) · (log log n) 1+o(1) at the expense of worse error parameter of our extractor. We find applications of our sumset extractor for extracting randomness from other well-studied models of weak sources such as affine sources, small-space sources, and interleaved sources.Interestingly, it is unknown if a random function is an extractor for sumset sources. We use techniques from additive combinatorics to show that it is a disperser, and further prove that an affine extractor works for an interesting subclass of sumset sources which informally corresponds to the "low doubling" case (i.e., the support of X1 + X2 is not much larger than 2 k ). * Supported by NSF CAREER award 2045576 1 Supp(X) denotes the support of X. We use log to denote the base-2 logarithm in the rest of this paper. Definition 1.1. Let X be a family of distribution over {0, 1} n . We say a deterministic function Ext :We say Ext is explicit if Ext is computable by a polynomial-time algorithm.The most well-studied deterministic extractors are multi-source extractors, which assume that the extractor is given C independent (n, k)-sources X 1 , X 2 , . . . , X C . This model was first introduced by Chor and Goldreich[CG88]. They constructed explicit two-source extractors with error 2 −Ω(n) for entropy 0.51n, and proved that there exists a two-source extractor for entropy k = O(log(n)) with error 2 −Ω(k) . Significant progress was made by Chattopadhyay and Zuckerman [CZ19], who showed how to construct an extractor for two sources with entropy k = polylog(n), after a long line of successful work on independent source extractors (see the references in [CZ19]). The output length was later improved to Ω(k) by Li[Li16]. Furthermore, Ben-Aroya, Doron and Ta-Shma [BDT19] showed how to improve the entropy requirement to O(log 1+o(1) (n)) for constant error and 1-bit output. The entropy requirement was further improved in subsequent works[Coh17,Li17], and the state-of-the-art result is by Li [Li19], which requires k = O(log(n) · log log(n) log log log(n) ). For a more elaborate discussion, see the survey by Chattopadhyay [Cha20]. Apart from independent sources, many other classes of sources have been studied for deterministic extraction. We briefly introduce some examples here. A well-studied class is oblivious bit-fixing sources [CGH + 85, GRS06, KZ07, Rao09], which is obtained by fixing some bits in a uniform random string. Extractors for such sources have found applications in cryptography [CGH + 85, KZ07]. A natural generalization of bitfixing sources is the class of affine sources, which are uniform distributions over some affine subspaces and have been widely studied in literature (see[CGL21]and references therein). Another important line of work focuses on the class of samplable sources, which are sources sampled by a "simple procedure" such as efficient algorithms [TV00], small-space algorithms [KRVZ11] or simple circuits[Vio14]. Researchers have also studied interleaved sources [RY11, CZ16, CL16b, CL20], which is a generalization of independent sources such that the bits from different independent sources are permuted in an unknown order.In this paper, we consider a very general class of sources called sumset sources, which was first studied by Chattopadhyay and Li[CL16b]. A sumset source is the sum (XOR) of multiple independent sources, which we formally define as follows.Definition 1.2. A source X is a (n, k, C)-sumset source if there exist C independent (n, k)-sources {X i } i∈ [C] such that X = C i=1 X i . Chattopadhyay and Li[CL16b]showed that the class of sumset sources generalize many different classes we mentioned above, including oblivious bit-fixing sources, independent sources, affine sources and smallspace sources. They also constructed an explicit extractor for (n, k, C)-sumset sources where k = polylog(n) and C is a large enough constant, and then used the extractor to obtain new extraction results for smallspace sources and interleaved C sources. An interesting open question left in [CL16b] is whether it is possible to construct an extractor for (n, polylog(n), 2)-sumset source. An explicit construction of such an extractor would imply improved results on extractors for interleaved sources and small-space sources with polylogarithmic entropy. (We discuss the details in Section 1.1.)However, it has been challenging to construct such an extractor for low min-entropy. The only known extractor for sum of two sources before this work is the Paley graph extractor [CG88], which requires one source to have entropy 0.51n and the other to have entropy O(log(n)), based on character sum estimate by Karatsuba [Kar71, Kar91] (see also[CZ16,Theorem 4.2]). In fact, unlike other sources we mentioned above, it is not clear whether a random function is an extractor for sumset sources. (See Section 1.3 for more discussions.)In this paper, we give a positive answer to the question above. Formally, we prove the following theorem. Introduction Randomness is a powerful resource in computer since, and has been widely used in areas such as algorithm design, cryptography, distributed computing, etc. Most of the applications assume the access to perfect randomness, i.e. a stream of uniform and independent random bits. However, natural sources of randomness often generate biased and correlated random bits, and in cryptographic applications there are many scenarios where the adversary learns some information about the random bits we use. This motivates the area of randomness extraction, which aims to construct randomness extractors which are deterministic algorithms that can convert an imperfect random source into a uniform random string. Formally, the amount of randomness in an imperfect random source X is captured by its min-entropy, which is defined as H ∞ (X) = min x∈Supp(X) (− log(Pr [X = x])). 1 We call X ∈ {0, 1} n a (n, k)-source if it satisfies H ∞ (X) ≥ k. Ideally we want a deterministic function Ext with entropy requirement k ≪ n, i.e. for every (n, k)-source X the output Ext(X) is close to a uniform string. Unfortunately, a folklore result shows that it is impossible to construct such a function even when k = n − 1. To bypass the impossibility result, researchers have explored two different approaches. The first one is based on the notion of seeded extraction, introduced by Nisan and Zuckerman [NZ96]. This approach assumes that the extractor has access to a short independent uniform random seed, and the extractor needs to convert the given source X into a uniform string with high probability over the seed. Through a successful line of research we now have seeded extractors with almost optimal parameters [LRVW03, GUV09,DKSS13]. In this paper, we focus on the second approach, called deterministic extraction, which assumes some structure in the given source. Formally, a deterministic extractor is defined as follows. Theorem 1. There exists a universal constant C such that for every k ≥ log C (n), there exists an explicit extractor Ext : {0, 1} n → {0, 1} m for (n, k, 2)-sumset source with error n −Ω(1) and output length m = k Ω(1) . We can further lower the entropy requirement to almost logarithmic at the expense of worse error parameter of the extractor. Theorem 2. For every constant ε > 0, there exists a constant C ε such that there exists an explicit extractor Ext : {0, 1} n → {0, 1} with error ε for (n, k, 2)-sumset source where k = C ε log(n) log log(n) log log log 3 (n). Since a sumset source extractor is also an affine extractor, Theorem 2 also gives an affine extractor with entropy O(log(n) log log(n) log log log 3 (n)), which slightly improves upon the O(log(n) log log(n) log log log 6 (n)) bound in [CGL21]. We note that this improvement comes from a new construction of "affine correlation breakers", which we discuss in Section 1.2. Applications Next we show applications of our extractors to get improved extractors for other well-studied models of weak sources. Extractors for Interleaved Sources Interleaved sources are a natural generalization of two independent sources, first introduced by Raz and Yehudayoff [RY11] with the name "mixed-2-sources". The formal definition of interleaved sources is as follows. For a n-bit string w and a permutation σ : [n] → [n], we use w σ to denote the string such that the σ(i)-th bit of w σ is exactly the i-th bit of w. For two strings x, y we use x • y to denote the concatenation of x and y. Definition 1.3. Let X 1 be a (n, k 1 )-source, X 2 be a (n, k 2 )-source independent of X 1 and σ : [2n] → [2n] be a permutation. Then (X 1 • X 2 ) σ is a (n, k 1 , k 2 )-interleaved sources, or a (n, k 1 )-interleaved sources if k 1 = k 2 . 1. Sample (x i+1 , nextstate) ∈ {0, 1} × {0, 1} s from D i,state . 2. Output x i+1 , and assign state := nextstate. Furthermore, the distribution X of the output (x 1 , . . . , x n ) is called a space-s source. Equivalently, a space-s source is sampled by a "branching program" of width 2 s (see Section 3.4 for the formal definition). In [KRVZ11] they constructed an extractor for space-s source with entropy k ≥ Cn 1−γ s γ with error 2 −n Ω(1) , for a large enough constant C and a small constant γ > 0. Chattopadhyay and Li [CL16b] then constructed an extractor with error n −Ω(1) for space-s source with entropy k ≥ s 1.1 2 log 0.51 (n) based on their sumset source extractors. Recently, based on a new reduction to affine extractors, Chattopadhyay and Goodman [CG21] improved the entropy requirement to k ≥ s · polylog(n) (or k ≥ s log 2+o(1) (n) if we are only interested in constant error and one-bit output). 2 With our new extractors for sum of two sources and the reduction in [CL16b], we can get extractors for space-s source with entropy s log(n) + polylog(n), which is already an improvement over the result in [CG21]. In this work we further improve the reduction and obtain the following theorems. Theorem 3. There exists a universal constant C such that for every s and k ≥ 2s + log C (n), there exists an explicit extractor Ext : {0, 1} n → {0, 1} m with error n −Ω(1) and output length m = (k − 2s) Ω(1) for space-s sources with entropy k. Theorem 4. For every constant ε > 0, there exists a constant C ε such that there exists an explicit extractor Ext : {0, 1} n → {0, 1} with error ε for space-s sources with entropy 2s + C ε log(n) log log(n) log log log 3 (n). Interestingly, the entropy requirement of our extractors has optimal dependence on the space s, since [KRVZ11] showed that it is impossible to construct an extractor for space-s source with entropy ≤ 2s. Moreover, the entropy in Theorem 4 almost matches the non-constructive extractor in [KRVZ11] which requires entropy 2s + O(log(n)). Affine Correlation Breakers One of the important building blocks of our sumset source extractors is an affine correlation breaker. While such an object has been constructed in previous works [Li16,CL16b,CGL21], in this paper we give a new construction with slightly better parameters. The main benefit of our new construction is that it is a blackbox reduction from affine correlation breakers to (standard) correlation breakers, which are simpler and more well-studied. We believe this result is of independent interest. First we define a (standard) correlation breaker. Roughly speaking, a correlation breaker takes a source X and a uniform seed Y, while an adversary controls a "tampered source" X ′ correlated with X and a "tampered seed" Y ′ correlated with Y. The goal of the correlation breaker is to "break the correlation" between (X, Y) and (X ′ , Y ′ ), with the help of some "advice" α, α ′ . One can also consider the "multi-tampering" variant where there are many tampered sources and seeds, but our theorem only uses the single-tampering version which is defined as follows. Definition 1.7. CB : {0, 1} n × {0, 1} d × {0, 1} a → {0, 1} m is a correlation breaker for entropy k with error ε (or a (k, ε)-correlation breaker for short) if for every X, X ′ ∈ {0, 1} n , Y, Y ′ ∈ {0, 1} d , α, α ′ ∈ {0, 1} a such that • X is a (n, k) source and Y is uniform • (X, X ′ ) is independent of (Y, Y ′ ) • α = α ′ , it holds that (CB(X, Y, α), CB(X, Y ′ , α ′ )) ≈ ε (U m , CB(X, Y ′ , α ′ )) . The first correlation breaker was constructed implicitly by Li [Li13] as an important building block of his independent-source extractor. Cohen [Coh16a] then formally defined and strengthened this object, and showed other interesting applications. Chattophyay, Goyal and Li [CGL20] then used this object to construct the first non-malleable extractor with polylogarithmic entropy, which became a key ingredient for the twosource extractor in [CZ19]. Correlation breakers have received a lot of attention and many new techniques were introduced to improve the construction [Coh16c,CS16,CL16a,Coh16b,Coh17,Li17,Li19]. Affine correlation breakers were first introduced by Li in his construction of affine extractors [Li16], and were later used in [CL16b] to construct sumset source extractors. An affine correlation breaker is similar to a (standard) correlation breaker, with the main difference being that it allows X and Y to have an "affine" correlation, i.e. X can be written as A + B where A is independent of Y and B is correlated with Y. The formal definition is as follows. Definition 1.8. AffCB : {0, 1} n × {0, 1} d × {0, 1} a → {0 , 1} m is a t-affine correlation breaker for entropy k with error ε (or a (t, k, ε)-affine correlation breaker for short) if for every distributions X, A, B ∈ {0, 1} n , Y, Y 1 , . . . , Y t ∈ {0, 1} d and strings α, α 1 , . . . , α t ∈ {0, 1} a such that • X = A + B • H ∞ (A) ≥ k and Y is uniform • A is independent of (B, Y, Y 1 , . . . , Y [t] ) • ∀i ∈ [t], α = α i , it holds that AffCB(X, Y, α), {AffCB(X, Y i , α i )} i∈[t] ≈ γ U m , {AffCB(X, Y i , α i )} i∈[t] . We say AffCB is strong if AffCB(X, Y, α), Y, {AffCB(X, Y i , α i ), Y i } i∈[t] ≈ γ U m , Y, {AffCB(X, Y i , α i ), Y i } i∈[t] . The first affine correlation breaker in [Li16] was constructed by adapting techniques from the correlation breaker construction in [Li13] to the affine setting. Chattopadhyay, Goodman and Liao [CGL21] then constructed an affine correlation breaker with better parameters based on new techniques developed in more recent works on correlation breakers [Coh16a,CS16,CL16a,Li17]. While the techniques for standard correlation breakers can usually work for affine correlation breakers, it requires highly non-trivial modification, and it is not clear whether the ideas in the standard setting can always be adapted to the affine setting. In fact, the parameters of the affine correlation breaker in [CGL21] do not match the parameters of the state-of-the-art standard correlation breaker by Li [Li19], because adapting the ideas in [Li19] to the affine setting (without loss in parameters) seems to be difficult. Moreover, it is likely that more improvements will be made in the easier setting of standard correlation breakers in the future, so a black-box reduction from affine correlation breakers to standard correlation breakers without loss in parameters will be very useful. In this work, we prove the following theorem. Theorem 5. Let C be a large enough constant. Suppose that there exists an explicit (d 0 , ε)-strong correlation breaker CB : {0, 1} d × {0, 1} d0 × {0, 1} a → {0, 1} C log 2 (t+1) log(n/ε) for some n, t ∈ N. Then there exists an explicit strong t-affine correlation breaker AffCB : {0, 1} n × {0, 1} d × {0, 1} a → {0, 1} m with error O(tε) for entropy k = O(td 0 + tm + t 2 log(n/ε)), where d = O(td 0 + m + t log 3 (t + 1) log(n/ε)). As a corollary, by applying this black-box reduction on Li's correlation breaker [Li19], we get an affine correlation breaker with parameters slightly better than those of [CGL21]. (See Theorem 5.5 for more details.) As a result, our extractor in Theorem 2 only requires O(log(n) log log(n) log log log 3 (n)) entropy, while using the affine correlation breaker in [CGL21] would require O(log(n) log log(n) log log log 6 (n)) entropy. In fact, if one can construct an "optimal" standard correlation breaker with entropy and seed length O(log(n)) when t = O(1), a = O(log(n)), ε = n −Ω(1) , which would imply a two-source extractor for entropy O(log(n)), by Theorem 5 this also implies a sumset source extractor/affine extractor for entropy O(log(n)). On Sumset Sources with Small Doubling Finally we briefly discuss why a standard probabilistic method cannot be used to prove existence of extractors for sumset sources, and show some partial results about it. Suppose we want to extract from a source A+B, where A and B are independent (n, k)-sources. Without loss of generality we can assume that A is uniform over a set A, and B is uniform over another set B, such that \|A\| = \|B\| = K, where K = 2 k . A simple calculation shows that there are at most 2 2nK choices of sources. In a standard probabilistic argument, we would like to show that a random function 3 is an extractor for A + B with probability at least 1 − δ, where δ ≪ 2 −2nK , and then we could use union bound to show that a random function is an extractor for (n, k, 2)-sources. However, this is not always true. For example, when A = B is a linear subspace, then A + B is exactly A, which has support size K. In this case we can only guarantee that a random function is an extractor for A + B with probability 1 − 2 −βK for some β < 1. In general, if the "entropy" of A + B is not greater than k by too much, then the probabilistic argument above does not work. Remark 1.9. Note that the "bad case" is not an uncommon case that can be neglected: if we take A, B to be subsets of a linear space of dimension k + 1, then \|Supp(A + B)\| ≤ 2 k+1 , which means a random function is an extractor for A + B with probability at most 1 − 2 −2K . However, there are roughly 2 4K choices of A and B, so even if we consider the bad cases separately the union bound still does not work. Nevertheless, we can use techniques from additive combinatorics to prove that the bad cases can be approximated with affine sources. With this result we can show that a random function is in fact a disperser 4 for sumset sources. To formally define the bad cases, first we recall the definition of sumsets from additive combinatorics (cf. [TV06]). Definition 1.10. For A, B ⊆ F n 2 , define A + B = {a + b : a ∈ A, b ∈ B}. For A, B s.t. \|A\| = \|B\| we say (A, B) has doubling constant r if \|A + B\| ≤ r \|A\|. It is not hard to see that a random function is a disperser for A+B with probability exactly 1−2 −\|A+B\|+1 . Therefore we can use union bound to show that a random function is a disperser with high probability for every sumset source A + B which satisfies \|A + B\| > 3n \|A\|. When \|A + B\| ≤ 3n \|A\|, a celebrated result by Sanders [San12] shows that A + B must contain 90% of an affine subspace with dimension log(\|A\|) − O(log 4 (n)). With the well-known fact that a random function is an extractor for affine sources with entropy O(log(n)), we can conclude that a random function is a disperser for sumset source with entropy O(log 4 (n)). Note that Sanders' result only guarantees that A + B almost covers a large affine subspace, but this affine subspace might only be a negligible fraction of A + B. Therefore, while a random function is an extractor for affine sources, Sanders' result only implies that it is a disperser for sumset source with small doubling constant. In this paper, we prove a "distributional variant" of Sanders' result. That is, a sumset source A + B with small doubling constant is actually statistically close to a convex combination of affine sources. Theorem 6. Let A, B be uniform distribution over A, B ⊆ F n 2 s.t. \|A\| = \|B\| = 2 k and \|A + B\| ≤ r \|A\|. Then A + B is ε-close to a convex combination of affine sources with entropy k − O(ε −2 log(r) log 3 (r/ε)). Then we get the following corollary which says that an affine extractor is also an extractor for sumset source with small doubling. We remark that while Corollary 1.11 implies that a random function is an extractor for sumset sources with small doubling, this does not mean a random function is an extractor for sumset sources in general. This is because a lower bound on \|A + B\| is not sufficient for us to show that a random function is an extractor by probabilistic argument. (See Appendix B for more discussions.) Open Problems In this paper we construct improved extractors for interleaved two sources and small-space sources based on our extractors for sum of two sources. Can we use our construction to get improved extractors for other classes of sources? More specifically, both of the applications require only an extractor for interleaved two sources, which is only a special case of sumset sources. Can we further exploit the generality of sumset sources? Another interesting open problem is whether a random function is an extractor for sum of two sources. In this paper we prove that sumset sources have a "structure vs randomness dichotomy": the sumset source is either close to an affine source, or has high enough entropy. In both cases a random function is a disperser. However our result does not seem strong enough to show that a random function is an extractor for sum of two sources. Overview of Proofs In this section we give a high-level overview of our proofs. The overview includes some standard notations which can be found in Section 3. Construction of Sumset Extractors In this section we give an overview of construction of our sumset source extractors. Similar to [CL16b], our extractor follows the two-step framework in [CZ19]. First, we convert the sumset source into a non-oblivious bit-fixing (NOBF) source. Roughly speaking, a t-NOBF source is a string such that most of the bits are t-wise independent. (See Definition 3.19 for the formal definition.) Second, we apply known extractors for NOBF sources [Vio14, CZ19, Li16, Mek17] to get the output. In the rest of this section, we focus on the first step, which is the main contribution of this work. Reduction from Two Sources To see how our reduction works, first we recall the transformation from two independent sources to NOBF sources in [CZ19]. Given two (n, k)-source X 1 , X 2 , first take a t-non-malleable extractor nmExt : {0, 1} n × {0, 1} d1 → {0, 1} with error ε 1 , enumerate all the seeds and output a string R 1 := {nmExt(X 1 , s)} s∈{0,1} d 1 with D 1 = 2 d1 bits. We do not give the exact definition of non-malleable extractors here, but we need the following property proved in [CZ19]: except for √ ε 1 fraction of "bad bits", every (t + 1) "good bits" in R 1 are √ ε 1 -close to uniform. With this property it might seem like R 1 is close to a (t + 1)-NOBF source, but unfortunately this is not true. While R 1 is guaranteed to be D t+1 1 √ ε 1 -close to a NOBF source by a result in [AGM03], this bound is trivial since D 1 = poly(1/ε 1 ). To get around this problem, [CZ19] used the second source X 2 to sample D 2 ≪ D 1 bits from R 1 and get R 2 . Now R 2 is guaranteed to be D t+1 2 √ ε 1 -close to a NOBF source, and the error bound D t+1 2 √ ε 1 can be very small since D 2 is decoupled from ε 1 . We note that Li [Li15] also showed a reduction from two independent sources to NOBF sources, and the sampling step is also crucial in Li's reduction. Chattopadhyay and Li [CL16b] conjectured that a similar construction should work for sumset sources. However, in the setting of sumset sources, it is not clear how to perform the sampling step. For example, if one replaces both X 1 and X 2 in the above construction with a sumset source X = X 1 +X 2 , then the sampling step might not work because the randomness we use for sampling is now correlated with R 1 . Therefore, they adopted an idea in [Li13] which requires the given source X to be the sum of C > 2 independent sources. In this paper, we show that we can actually make the sampling step work with a (n, polylog(n), 2)-sumset source. As a result we get an extractor for sum of two independent sources. Sampling with Sumset Source As a warm up, first we assume that we are sampling from the output of a "0-non-malleable extractor", i.e. a strong seeded extractor. Let Ext : {0, 1} n × {0, 1} d1 → {0, 1} be a strong seeded extractor with error ε 1 . First observe that the sampling method has the following equivalent interpretation. Note that Ext and the source X 1 together define a set of "good seeds" such that a seed s is good if Ext(X 1 , s) is √ ε 1 -close to uniform. Since Ext is a strong seeded extractor, (1 − √ ε 1 ) of the seeds should be good. In the sampling step we apply a sampler Samp on X 2 to get some samples of seeds {Samp(X 2 , i)} i∈{0,1} d 2 . Then we can apply the function Ext(X 1 , ·) on these sampled seeds to get the output R 2 = {Ext(X, Samp(X, i))} i∈{0,1} d 2 which is 2 d2 √ ε 1 -close to a 1-NOBF source. Now we move to the setting of sumset sources and replace both X 1 , X 2 in the above steps with X = X 1 + X 2 . Our goal is to show that we can still view this reduction as if we were sampling good seeds with X 2 and using these seeds to extract from X 1 . Consider the i-th output bit, Ext(X, Samp(X, i)). Our main observation is, if Samp(·, i) is a linear function, then we can assume that we compute Ext(X, Samp(X, i)) in the following steps: 1. First sample x 2 ∼ X 2 . 2. Use x 2 as the randomness of Samp to sample a "seed" s := Samp(X 2 , i). Output Ext ′ x2,i (X 1 , s) := Ext(X 1 + x 2 , s + Samp(X 1 , i)). First we claim that Ext ′ x2,i is also a strong seeded extractor. To see why this is true, observe that if we fix Samp(X 1 , i) = ∆, then Ext ′ x2,i (X 1 , U) = Ext(X 1 + x 2 , U + ∆). As long as X 1 still has enough entropy after fixing Samp(X 1 , i), Ext works properly since X 1 + x 2 is independent of U + ∆, X 1 + x 2 still has enough entropy and U + ∆ is also uniform. Therefore, we can use Ext ′ x2,i and X 1 to define a set of good seeds s which make Ext ′ x2,i (X 1 , s) close to uniform, and most of the seeds should be good. Then we can equivalently view the sampling step as if we were sampling good seeds for Ext ′ x2,i using X 2 as the randomness. There are still two problems left. First, the definition of Ext ′ x2,i depends on x 2 , which is the randomness we use for sampling. To solve this problem, we take Ext to be linear, and prove that (1 − √ ε 1 ) fraction of the seeds s are good in the sense that Ext ′ x2,i (X 1 , s) is close to uniform for every x 2 . Second, Ext ′ x2,i depends on i, which is the index of our samples. Similarly we change the definition of good seeds so that a seed s is good if Ext ′ x2,i (X 1 , s) is good for every x 2 and i, and by union bound we can show that (1 − 2 d2 √ ε 1 ) fraction of the seeds are good. As long as ε 1 ≪ 2 −2d2 , most of the seeds should be good. Now the definition of good seeds is decoupled from the sampling step, and hence we can show that most of the sampled seeds are good. Sampling with Correlation Breakers Next we turn to the case of t-non-malleable extractors. Similar to how we changed the definition of good seeds for a strong seeded extractor, we need to generalize the definition of good seeds for a non-malleable extractor in [CZ19] to the sumset source setting. First, we say a seed s is good with respect to x 2 and a set of indices T = {i 1 , . . . , i t+1 } if for every s 1 , . . . , s t ∈ {0, 1} d1 , (nmExt(X 1 + x 2 , s + Samp(X 1 , i 1 )) ≈ √ ε1 U 1 ) \| {nmExt(X 1 + x 2 , s j + Samp(X 1 , i j+1 ))} j∈[t] . Based on the proof in [CZ19] and the arguments in the previous section, if X 1 has enough entropy when conditioned on {Samp(X 1 , i)} i∈T , then 1 − √ ε 1 of the seeds are good with respect to x 2 and T . If we can prove that most of the seeds we sample using x 2 ∼ X 2 are good with respect to x 2 and every set of indices T , then the we can conclude that the output R 2 = {nmExt(X, Samp(X, i))} i∈{0,1} d 2 is D t+1 2 √ ε 1 -close to a NOBF source. Next we need to show that most of the seeds are good with respect to every x 2 and T , so that the sampling step is decoupled from the definition of good seeds. To deal with the dependence on T , we take the union bound over T , and we can still guarantee that 1 − D t+1 2 √ ε 1 of the seeds are good. To deal with the dependency on x 2 , it suffices to replace the non-malleable extractor with a strong affine correlation breaker. Although the correlation breaker needs an additional advice string to work, here we can simply use the indices of the samples as the advice. Our final construction would be {AffCB(X, Samp(X, α), α)} α∈{0,1} d 2 . Finally, we note that in order to make the extractor work for almost logarithmic entropy (Theorem 2), we need to replace the sampler with a "somewhere random sampler" based on the techniques in [BDT19], and the construction and analysis should be changed correspondingly. We present the details in Section 5. Reduction from Small-Space Sources to Sumset Sources In this section we give an overview of our new reduction from small-space sources to sumset sources. As in all the previous works on small-space source extractors, our reduction is based on a simple fact: conditioned on the event that the sampling procedure is in state j at time i, the small-space source X can be divided into two independent sources X 1 ∈ {0, 1} i , X 2 ∈ {0, 1} n−i , such that X 1 contains the bits generated before time i, and X 2 contains the bits generated after time i. Kamp, Rao, Vadhan and Zuckerman [KRVZ11] proved that if we pick some equally distant time steps i 1 , . . . , i ℓ−1 and condition on the states visited at these time steps, we can divide the small-space source into ℓ independent blocks such that some of them have enough entropy. However, such a reduction does not work for entropy smaller than √ n (cf. [CG21]). Chattopadhyay and Li [CL16b] observed that with a sumset source extractor we can extract from the concatenation of independent sources with unknown and uneven length. They then showed that with a sumset source extractor, we can "adaptively" pick which time steps to condition on and break the √ n barrier. Chattopadhyay and Goodman [CG21] further refined this reduction and showed how to improve the entropy requirement by reducing to a convex combination of affine sources. The reductions in [CL16b] and [CG21] can be viewed as "binary searching" the correct time steps to condition on, so that the given source X becomes the concatenation of independent blocks (X 1 , . . . , X O(log(n)) ) such that some of them have enough entropy. However, even though with our extractors for sum of two sources we only need two of the blocks to have enough entropy, the "binary search-based" reduction would condition on at least log(n) time steps and waste s log(n) entropy. A possible way to improve this reduction is by directly choosing the "correct" time step to condition on so that we only get two blocks X 1 • X 2 both of which have enough entropy. However this is not always possible. For example, consider a distribution which is a convex combination of U n/2 • 0 n/2 and 0 n/2 • U n/2 . This distribution is a space-1 source and has entropy n/2, but no matter which time step we choose to condition on, one of the two blocks would have zero entropy. To resolve these problems, we carefully define the event to condition on as follows. For ease of explanation we view the space-s sampling procedure as a branching program of width 2 s . (Unfamiliar readers can consult Section 3.4.) First, we define a vertex v = (i, j) to be a "stopping vertex" if the bits generated after visiting v has entropy less than some threshold. Then we condition on a random variable V which is the first stopping vertex visited by the sampling process. Note that V is well-defined since every state at time n is a stopping vertex. Besides, conditioning on V only costs roughly s + log(n) entropy since there are only n · 2 s possible outcomes. Now observe that the event V = (i, j) means the sampling process visits (i, j) but does not visit any stopping vertex before time i. Let "first block" denote the bits generated before time i and "second block" denote the bits generated after time i. It is not hard to see that the two blocks are still independent conditioned on V = v. Then observe that the first block has enough entropy because the second block does not contain too much entropy (by our definition of stopping vertex). Next we show that the second block also has enough entropy. For every vertex u, let X u denote the bits generated after visiting u. The main observation is, if there is an edge from a vertex u to a vertex v, then unless u → v is a "bad edge" which is taken by u with probability < ε, the entropy of X v can only be lower than X u by at most log(1/ε). If we take ε ≪ 2 −s n −1 , then by union bound the probability that any bad edge is traversed in the sampling procedure is ≪ 1. Since we take V to be the first vertex such that X V has entropy lower than some threshold, the entropy of X V can only be log(1/ε) ≈ s + log(n) lower than the threshold. In conclusion, if we start with a space-s source with entropy roughly 2s + 2 log(n) + 2k, and pick the entropy threshold of the second block to be roughly k + s + log(n), we can get two blocks both having entropy at least k. From Affine to Standard Correlation Breaker Next we briefly discuss our black-box reduction from affine correlation breakers to standard correlation breakers. To reduce an affine correlation breaker to a standard correlation breaker, our main idea is similar to that of [CGL21]: to adapt the construction of a correlation breaker from the independent-source setting to the affine setting, we only need to make sure that every function on X is linear, and every function on Y works properly when Y is a weak source. However, instead of applying this idea step-by-step on existing constructions, we observe that every correlation breaker can be converted into a "two-step" construction which is easily adaptable to the affine setting. First, we take a prefix of Y as the seed to extract a string Z from X. Next, we apply a correlation breaker which treats Y as the source and Z as the seed. This construction only computes one function on X, which is a seeded extractor and can be replaced with a linear one. Furthermore, the remaining step (i.e. the correlation breaker) is a function on Y, which does not need to be linear. Finally, we note that if the underlying standard correlation breaker is strong, we can use the output as the seed to extract from X linearly and get a strong affine correlation breaker. A drawback of this simple reduction is that the resulting affine correlation breaker has a worse dependence on the number of tampering t. Recall that the state-of-the-art t-correlation breaker [Li19] requires entropy and seed length O(t 2 d) where d = O log(n) · log log(n) log log log(n) , assuming the error is 1/ poly(n) and the advice length is log(n). With the reduction above we get a t-affine correlation breaker with entropy and seed length O(t 3 d), while the affine correlation breaker in [CGL21] has entropy and seed length O(t 2 log(n) log log(n)). Since the construction of sumset source extractors requires t to be at least Ω(log log log 2 (n)), O(t 3 d) is actually worse than O(t 2 log(n) log log(n)). To improve the parameters, we first apply the reduction above to get a 1-affine correlation breaker, and then strengthen the affine correlation breaker to make it work for t tampering. Our strengthening procedure only consists of several rounds of alternating extractions, which requires poly(t) · O(log n) entropy. Therefore by plugging in the correlation breaker in [Li19] we end up getting a t-affine correlation breaker with entropy and seed length O(td + poly(t) · log(n)), which is better than O(t 2 log(n) log log(n)). The strengthening procedure works as follows. Observe that the 1-affine correlation breaker outputs a string R which is uniform conditioned on every single tampered version of R. (Note that R might not be uniform when conditioned on all t tampered versions simultaneously.) Then we apply alternating extractions to merge the independence of R with itself. Based on the "independence merging lemma" in [CGL21] (see Lemma 3.26), after one round of alternating extraction, we get a string R ′ which is uniform conditioned on every two tampered R ′ . By repeating this step for log(t) times we get a t-affine correlation breaker. Sumset Sources with Small Doubling Finally we briefly sketch how to prove that a sumset source with small doubling is close to a convex combination of affine sources. Let A, B ⊆ F n 2 be sets of size K = 2 k and let A, B be uniform distributions over A, B respectively. A seminal result by Sanders [San12] showed that there exists a large affine subspace V such that at least 1 − ε fraction of V is in A + B. We adapt Sanders' proof to show that for every distinguisher with output range [0, 1], the sumset source A + B is indistinguishable from a convex combination of affine sources (with large entropy). Then by an application of von Neumann's minimax theorem (Corollary 3.42) we can find a universal convex combination of affine sources which is statistically close to A + B. To see more details, first we briefly recall the outline of Sanders' proof. Consider A ′ , B ′ ⊆ F m 2 such that \|A ′ \| , \|B ′ \| ≥ \|F m 2 \| /r, and let A ′ , B ′ be uniform distributions over A ′ , B ′ respectively. Let 1 A ′ +B ′ denote the indicator function for A ′ + B ′ . Based on the Croot-Sisask lemma [CS10] and Fourier analysis, Sanders showed that for arbitrarily small constant ε > 0 there exists a distribution T ⊆ F m 2 and a linear subspace V of co-dimension O(log 4 (r)) s.t. E [1 A ′ +B ′ (A ′ + B ′ )] ≈ ε E [1 A ′ +B ′ (T + V)] , where V is the uniform distribution over V . Then Sanders' original result follows directly by taking T = t which maximizes E [1 A ′ +B ′ (t + V)]. A closer inspection at Sanders' proof shows that 1 A ′ +B ′ can be replaced with any function f : F m 2 → [0, 1]. (Note that the distributions T, V depend on the function f .) This implies that A ′ + B ′ is indistinguishable from a convex combination of affine sources by f . With our minimax argument we can conclude that A ′ + B ′ is statistically close to a convex combination of affine sources. However, the result above only works for dense sets A ′ , B ′ . To generalize the result to sets A, B with small doubling, a standard trick in additive combinatorics is to consider a linear Freiman homomorphism φ : F n 2 → F m 2 , which is a linear injective function on ℓA + ℓB for some constant ℓ, and consider A ′ = φ(A), B ′ = φ(B). By considering the function f • φ −1 we can still show that E [f (A + B)] = E f (φ −1 (A ′ + B ′ )) ≈ E f (φ −1 (T + V)) . However, it is not clear whether φ −1 (T + V) is a also a convex combination of affine sources in F n 2 . To solve this problem, we adapt Sanders' proof to show that there exist T, V which satisfy E [1 A ′ +B ′ (A ′ + B ′ )] ≈ ε E [1 A ′ +B ′ (T + V)] (1) and E f (φ −1 (A ′ + B ′ )) ≈ ε E f (φ −1 (T + V)) (2) simultaneously. This relies on a variant of the Croot-Sisask lemma which shows that there exists a large set of "common almost period" for 1 A ′ +B ′ and f • φ −1 . Then (1) guarantees that with probability at least 1 − 2ε over t ∼ T, φ −1 (t + V) is an affine source in F n 2 with entropy k − O(log 4 (r)). Therefore φ −1 (T + V) is 2ε-close to a convex combination of affine sources. Finally (2) shows that A + B is indistinguishable from φ −1 (T + V) by f , which implies our claim. Organization. In Section 3 we introduce some necessary preliminaries and prior works. In Section 4 we show a new reduction from small-space sources to sum of two sources which has optimal dependence on the space parameter, and prove Theorem 3 and Theorem 4. In Section 5 we show how to construct the extractors for sum of two sources in Theorem 1 and Theorem 2, assuming access to an affine correlation breaker. In Section 6 we show how to construct the affine correlation breaker we need based on a black-box reduction to a standard correlation breaker (Theorem 5). Finally, we prove Theorem 6 in Section 7. Preliminaries In this section we introduce some preliminaries. We note that Section 3.4 is only used in Section 4, Section 3.5 to 3.9 are only used in Section 5 and 6, and Section 3.10 to 3.12 are only used in Section 7. Notations Basic notations. The logarithm in this paper is always base 2. For every n ∈ N, define [n] = {1, 2, . . . , n}. In this paper, {0, 1} n and F n 2 are interchangeable, and so are {0, 1} n and [2 n ]. We use x • y to denote the concatenation of two strings x and y. We say a function is explicit if it is computable by a polynomial time algorithm. For x, y ∈ R we use x ≈ ε y to denote \|x − y\| ≤ ε and x ≈ ε y to denote \|x − y\| > ε. Distributions and random variables. We sometimes abuse notation and treat distributions and random variables as the same. We always write a random variable/distribution in boldface font. We use Supp(X) to denote the support of a distribution. We use U n to denote the uniform distribution on {0, 1} n . When U n appears with other random variables in the same joint distribution, U n is considered to be independent of other random variables. Sometimes we omit the subscript n of U n if the length is less relevant and is clear in the context. When there is a sequence of random variables X 1 , X 2 , . . . , X t in the context, for every set S ⊆ [t] we use X S to denote the sequence of random variables which use indices in S as subscript, i.e. X S := {X i } i∈S . We also use similar notation for indices on superscript. Linear algebra. For a set A ⊆ F n 2 , we use span(A) to denote the linear span of A, and A ⊥ to denote the orthogonal complement of span(A), i.e. A ⊥ := {y ∈ F n 2 : ∀x ∈ A, y, x = 0}. For every affine subspace A of F n 2 we use dim(A) to denote the dimension of A. Note that if A is uniform over A, then H ∞ (A) = dim(A). Therefore we use "dimension" and "entropy" interchangeably when discussing affine sources. Statistical Distance Definition 3.1. Let D 1 , D 2 be two distributions on the same universe Ω. The statistical distance between D 1 and D 2 is ∆ (D 1 ; D 2 ) := max T ⊆Ω Pr [D 1 ∈ T ] − Pr [D 2 ∈ T ] = 1 2 s∈Ω \|D 1 (s) − D 2 (s)\| . We say D 1 is ε-close to D 2 if ∆(D 1 ; D 2 ) ≤ ε, which is also denoted by D 1 ≈ ε D 2 . Specifically, when there are two joint distributions (X, Z) and (Y, Z) such that (X, Z) ≈ ε (Y, Z), we sometimes write (X ≈ ε Y) \| Z for short. We frequently use the following standard properties. Lemma 3.2. For every distribution D 1 , D 2 , D 3 on the same universe, the following properties hold: • For any distribution Z, ∆ ((D 1 , Z); (D 2 , Z)) = E z∼Z [∆ (D 1 \| Z=z ; D 2 \| Z=z )]. • For every function f , ∆ (f (D 1 ); f (D 2 )) ≤ ∆ (D 1 ; D 2 ). • ∆ (D 1 ; D 3 ) ≤ ∆ (D 1 ; D 2 ) + ∆ (D 2 ; D 3 ). (triangle inequality) Conditional Min-entropy of X given Z is H ∞ (X \| Z) := − log E z∼Z max x (Pr [X = x \| Z = z]) . The following lemma, usually referred to as the chain rule, is frequently used in this paper. Lemma 3.4 ([DORS08]). Let X, Y, Z be (correlated) random variables. Then H ∞ (X \| (Y, Z)) ≥ H ∞ (X \| Z) − log(Supp(Y)). When we need to consider worst-case conditional min-entropy, we use the following lemma. Lemma 3.5 ([DORS08]). Let X, Z be (correlated) random variables. For every ε > 0, Pr z∼Z [H ∞ (X\| Z=z ) ≥ H ∞ (X \| Z) − log(1/ε)] ≥ 1 − ε. Note that the above two lemmas imply the following: Lemma 3.6 ([MW97]). Let X, Z be (correlated) random variables. For every ε > 0, Pr z∼Z [H ∞ (X\| Z=z ) ≥ H ∞ (X) − log(Supp(Z)) − log(1/ε)] ≥ 1 − ε. Lemma 3.7 ([DORS08]). Let ε, δ > 0 and X, Z be a random variables such that H ∞ (X \| Z) ≥ k + log(1/δ). Let Ext : {0, 1} n × {0, 1} d → {0, 1} m be a (k, ε)-seeded extractor. Then (Ext(X, U d ) ≈ ε+δ U m ) \| Z. Branching Programs The following definition is equivalent to Definition 1.6 in the sense that each layer corresponds to a time step and each vertex in a layer corresponds to a state in a certain time step. Definition 3.8. A branching program B of width w and length n (for sampling) is a directed (multi)-graph with (n + 1) layers L 0 , L 1 , . . . , L n and has at most w vertices in each layer. The first layer (indexed by 0) has only one vertex called the start vertex, and every vertex in L n has no outgoing edge. For every vertex v in layer i < n, the set of outgoing edges from v, denoted by E v , satisfies the following. • Every edge e ∈ E v is connected to a vertex in L i+1 . • Each edge e ∈ E v is labeled with a probability, denoted by Pr [e], so that e∈Ev Pr [e] = 1. • Each edge e ∈ E v is labeled with a bit b e ∈ {0, 1}, and if two distinct edges e 1 , e 2 ∈ E v are connected to the same vertex w ∈ L i+1 then b e1 = b e2 . (Note that this implies \|E v \| ≤ 2w.) The output of B is a n-bit string generated by the following process. Let v 0 be the start vertex. Repeat the following for i from 1 to n: sample an edge e i ∈ E vi−1 with probability Pr [e i ], output b ei and let v i be the vertex which is connected by e i . We say (v 0 , e 1 , v 1 , . . . , e n , v n ) is the computation path of B. We say a random variable X ∈ {0, 1} n is a space-s source if it is generated by a branching program of width 2 s and length n. We also consider the subprograms of a branching program. We need the following simple fact from [KRVZ11]. Lemma 3.10 ( [KRVZ11]). Let X be a space-s source sampled by a branching program B, and let v be a vertex in layer i of B. Then conditioned on the event that the computation path of X passes v, X is the concatenation of two independent random variables X 1 ∈ {0, 1} i , X 2 ∈ {0, 1} n−i . Moreover X 2 is exactly the source generated by the subprogram B v . Seeded Extractors Definition 3.11. Ext : {0, 1} n × {0, 1} d → {0 , 1} m is a seeded extractor for entropy k with error ε (or (k, ε)-seeded extractor for short) if for every (n, k) source X, and every Y = U d , Ext(X, Y) ≈ ε U m . We call d the seed length of Ext. We say Ext is linear if Ext(·, y) is a linear function for every y ∈ {0, 1} d . We say Ext is strong if We also need the following extractor from [CGL21] which is linear but has worse parameters. (Ext(X, Y) ≈ ε U m ) \| Y. Lemma 3.13. There exists a constant c 3.13 such that for every t, m ∈ N and ε > 0, there exists an explicit (c 3.13 (m + log(1/ε)), ε)-linear strong seeded extractor LExt : {0, 1} n × {0, 1} d → {0, 1} m s.t. d = O( m t + log(n/ε) + log 2 (t) log(m/ε)). Note that when m = t log(n/ε) the seed length is bounded by O log 2 (t) + 1 log(n/ε)) . Samplers First we define a sampler. We note that the our definition is different from the standard definition of averaging samplers [BR94] in the following sense: first, we need the sampler to work even when the given randomness is only a weak source. Second, we only care about "small tests". [Samp(x, y) ∈ T ] > 2ε ≤ δ. We say Samp is linear if Samp(·, y) is linear for every y ∈ [D]. Zuckerman [Zuc97] showed that one can use a seeded extractor as a sampler for weak sources. Lemma 3.15 ([Zuc97]). A (k + log(1/δ), ε)-seeded extractor is also a (ε, δ)-sampler for entropy k. The following is a relaxation of a sampler, which is called a somewhere random sampler. [∀z ∈ [C] Samp(x, y, z) ∈ T ] > 2ε ≤ δ. We say Samp is linear if Samp(·, y, z) is linear for every y ∈ [D], z ∈ [C]. The following lemma is implicit in [BDT19]. For completeness we include a proof in Appendix A. Non-Oblivious Bit-Fixing Sources Definition 3.19. A distribution X = (X 1 , X 2 , . . . , X n ) on {0, 1} n is called t-wise independent if for every subset S ⊆ [n] of size t we have X S = U q . Lemma 3.20 ([AGM03]). Let X = (X 1 , X 2 , . . . , X n ) be a distribution on {0, 1} n . If for every S ⊆ [n] s.t. \|S\| ≤ t, i∈S X i ≈ γ U 1 , then X is 2n t γ-close to a t-wise independent distribution. Definition 3.21. A distribution X = (X 1 , X 2 , . . . , X n ) on {0, 1} n is called a (q, t)-non-oblivious bit-fixing (NOBF) source if there exists a set Q s.t. \|Q\| ≤ q and X [n]\Q is t-wise independent. In this paper we need the following extractors for NOBF sources. Markov Chain In this paper we usually consider the scenario that we have two sources X, Y which are independent conditioned on a collection of random variables Z. We use Markov chain as a shorthand for this. Definition 3.24. Let X, Y, Z be random variables. We say X ↔ Z ↔ Y is a Markov chain if X and Y are independent conditioned on any fixing of Z. We frequently use the following fact. Lemma 3.25. If X ↔ Z ↔ Y is a Markov chain, then for every deterministic function f , let W = f (X, Z). Then • (X, W) ↔ Z ↔ Y is a Markov chain. • X ↔ (W, Z) ↔ Y is a Markov chain. We use "W is a deterministic function of X (conditioned on Z)" to refer to the first item, and "fix W" to refer to the second item. Independence Merging The following lemma is from [CGL21] and is based on the ideas in [CL16a]. Basically it says that if Y is independent of some tampered seeds Y S , and X has enough entropy when conditioned some tampered sources X T , then a strong seeded extractor can "merge" the independence of Y from Y S and X from X T . Lemma 3.26 (independence-merging lemma). Let (X, X [t] ) ↔ Z ↔ (Y, Y [t] ) be a Markov chain, such that X, X [t] ∈ {0, 1} n , Y, Y [t] ∈ {0, 1} d . Moreover, suppose there exists S, T ⊆ [t] such that • (Y ≈ δ U d ) \| (Z, Y S ) • H ∞ (X \| (X T , Z)) ≥ k + tm + log(1/ε) Let Ext : {0, 1} n × {0, 1} d → {0 , 1} m be any (k, ε)-strong seeded extractor, let W = Ext(X, Y) and W j = Ext(X j , Y j ) for every j ∈ [t]. Then (W ≈ 2ε+δ U m ) \| (W S∪T , Y, Y [t] , Z). Basic Properties in Additive Combinatorics Definition 3.27. For every two sets A, B ⊆ F n 2 , we define A+B = {a+b : a ∈ A, b ∈ B}. For b ∈ F n 2 we use A + b as the shorthand for A + {b}. For every ℓ ∈ N and every A ⊆ F n 2 , define 1A = A and ℓA = A + (ℓ − 1)A recursively. Lemma 3.28 ( [Plü61,Ruz99]). For every A, B ⊆ F n 2 s.t. \|A\| = \|B\| and \|A + B\| ≤ r \|A\|, \|kA + ℓB\| ≤ r k+ℓ+1 \|A\| for every k, ℓ ∈ N. Definition 3.29. We say a function φ : F n 2 → F m 2 is a s-Freiman homomorphism of a set A ⊆ F n 2 if for every a 1 , . . . , a s , a ′ 1 , . . . , a ′ s ∈ A, φ(a 1 ) + . . . + φ(a s ) = φ(a ′ 1 ) + . . . + φ(a ′ s ) ⇒ a 1 + . . . + a s = a ′ 1 + . . . + a ′ s . The following property is easy to verify. Lemma 3.30. If φ is a linear s-Freiman homomorphism, then φ is injective on sA + v for every v ∈ F n 2 . Further, for x ∈ 2sA we have φ(x) = 0 ⇔ x = 0. The following lemma can be used to obtain a linear Freiman homomorphism with small image. Fourier Analysis First we recall some basic definitions and properties in Fourier analysis. Definition 3.32. Let f : F n 2 → R be a function. The Fourier coefficients of f , denoted by f : F n 2 → R, are f (α) := E x∼F n 2 f (x) · (−1) α,x . Lemma 3.33 (Parseval-Plancherel identity). For every functions f, g : F n 2 → R, E x∼F n 2 [f (x)g(x)] = α∈F n 2 f (α) g(α). Definition 3.34. The convolution of functions f, g : F n 2 → R, denoted by f * g : F n 2 → R, is defined as f * g(x) := E y∼F n 2 [f (y)g(x − y)] . Lemma 3.35. For every functions f, g : F n 2 → R and every α ∈ F n 2 , f * g(α) = f (α) g(α). Next we define a density function. Definition 3.36. For every A ⊆ F n 2 , define the density function of A to be µ A := 2 n \|A\| · 1 A . For a distribution A on F n 2 , the density function of A, denoted by µ A , is defined as µ A (x) = 2 n Pr [A = x]. We need the following three facts about density functions. . For X ⊆ F n 2 , define Spec γ (X) = {α ∈ F n 2 : \| µ X (α)\| ≥ γ}. Define β = \|X\| / \|F n 2 \|. Then dim(span(Spec γ (X))) ≤ 2γ −2 ln(1/β). Minimax Theorem Lemma 3.41 (minimax theorem [vN28]). Let X ⊆ R n , Y ⊆ R m be convex sets. Then for every bilinear function g : X × Y → R, min X f ∈ X such that E [f (X f )] − E [f (Y)] ≤ ε, then there exists X * ∈ X such that Y ≈ ε X * . Proof. Let F denote the set of all the functions from Ω to [0, 1]. Note that a distribution X can be represented by a vector in R \|Ω\| , where the coordinate indexed by s ∈ Ω is Pr [X = s]. A function f : Ω → [0, 1] can also be represented by a vector in R \|Ω\| , where the coordinate indexed by s ∈ Ω is f (s). Observe that F is convex. Define the function g : X × F → R to be g(X, f ) := E [f (X)] − E [f (Y)] = s∈Ω Pr [X = s] · f (s) − E [f (Y)] . Observe that g is bilinear. By minimax theorem, min X∈X max f ∈F g(X, f ) = max f ∈F min X∈X g(X, f ) ≤ max f ∈F (E [f (X f )] − E [f (Y)]) ≤ ε. That is, there exists X * ∈ X such that for every function f : Ω → [0, 1], E [f (X * )] − E [f (Y)] ≤ ε. If we take f = 1 T for some T ⊆ Ω, then E [f (X * )] − E [f (Y)] is exactly Pr [X * ∈ T ] − Pr [Y ∈ T ] . Therefore by definition of statistical distance, X * ≈ ε Y. Improved Reduction for Small-Space Sources In this section we prove the following lemma. Lemma 4.1. For every integer C ≥ 2, every space-s source on n-bit with min-entropy k ′ ≥ Ck + (C − 1) (2s + 2 log(n/ε)) is (3Cε)-close to a convex combination of (n, k, C)-sumset sources. Note that by taking C = 2 in Lemma 4.1, we can prove that the sumset source extractor in Theorem 1 and Theorem 2 are also small-space source extractors which satisfy the parameters in Theorem 3 and Theorem 4 respectively. In the rest of this section we focus on proving Lemma 4.1. First we show how to prove Lemma 4.1 based on the following lemma. Lemma 4.2. Every space-s source X ∈ {0, 1} n with entropy at least k = k 1 + k 2 + 2s + 2 log(n/ε) is 3ε-close to a convex combination of sources of the form X 1 • X 2 which satisfy the following properties: • X 1 is independent of X 2 • H ∞ (X 1 ) ≥ k 1 , H ∞ (X 2 ) ≥ k 2 • X 2 is a space-s source Proof of Lemma 4.1. By induction, Lemma 4.2 implies that a space-s source with entropy Ck + (C − 1)(2s + 2 log(n/ε)) is 3Cε-close to a convex combination of sources of the form X 1 •X 2 •· · ·•X C such that X 1 , . . . , X C are independent, and for every i ∈ [C], H ∞ (X i ) ≥ k. Let ℓ 1 , ℓ 2 , . . . , ℓ C denote the length of X 1 , X 2 , . . . , X C respectively and define p i = i−1 j=1 ℓ i and s i = n j=i+1 ℓ j (note that p 1 = 0 and s C = 0). Then observe that X 1 • · · · • X C = C i=1 0 pi • X i • 0 si , which implies that X = X 1 • · · · • X C is a (n, k, C)-sumset source. To prove Lemma 4.2, first we need the following lemma. Lemma 4.3. Let B be a branching program of width 2 s and length n for sampling. Let e be an edge in B connected from u to v and let X u , X v be the output distributions of the subprograms B u , B v respectively. Then H ∞ (X v ) ≥ H ∞ (X u ) − log(1/ Pr [e]). Proof. Let x * = arg max x Pr [X v = x]. Note that H ∞ (X v ) = − log(Pr [X v = x * ]) by definition. Observe that Pr [X u = b e • x * ] ≥ Pr [e] · Pr [X v = x * ]. Therefore, H ∞ (X u ) ≤ − log Pr [X u = b e • x * ] ≤ − log Pr [e] · Pr [X v = x * ] ≤ H ∞ (X v ) + log(1/ Pr [e]). Next we prove Lemma 4.2. Proof of Lemma 4.2. Let B denote the branching program which samples X. For every v, define X v to be the source generated by the subprogram B v . Define v to be a stopping vertex if H ∞ (X v ) ≤ k 2 + s + log(n/ε). Observe that every vertex u in the last layer is a stopping vertex since H ∞ (X u ) = 0, so there is always a stopping vertex in the computation path. We define an edge e in B to be a bad edge if Pr [e] ≤ ε/(n · 2 s ). Now define a random variable V as follows: • V = ⊥ if the computation path of X visits a bad edge before visiting any stopping vertex, • otherwise, V = v where v is the first stopping vertex in the computation path. Observe that Pr [V = ⊥] ≤ 2ε, since in each step of B there are at most 2 s+1 edges starting from the current vertex, and there are n steps in total. Define BAD = {v ∈ Supp(V) : H ∞ (X\| V=v ) ≤ k − s − log(n/ε)}. Then Pr [V ∈ BAD] ≤ ε by Lemma 3.6. We claim that if v ∈ BAD and v = ⊥, then conditioned on V = v, the source X can be written as X 1 • X 2 which satisfies the properties stated in Lemma 4.2. The claim directly implies Lemma 4.2 because Pr [v ∈ BAD ∨ v = ⊥] ≤ 3ε by union bound. Next we prove the claim. Let E 1 denote the event "the computation path contains v", and E 2 denote the event "the computation path does not contain any bad edge or stopping vertex before the layer of v". Observe that V = v is equivalent to E 1 ∧ E 2 . Conditioned on E 1 , by Lemma 3.10, X can be written as X 1 • X 2 where X 1 is independent of X 2 and X 2 = X v . Now observe that E 2 only involves layers before v, so conditioned on E 1 , X 2 is independent of E 2 . Therefore, conditioned on V = v, we still have X 2 = X v , which is a space-s source, and X 1 is still independent of X 2 . Next observe that H ∞ (X 1 ) = H ∞ (X\| V=v ) − H ∞ (X 2 ) ≥ (k − s − log(n/ε)) − (k 2 + s + log(n/ε)) ≥ k 1 . It remains to prove that H ∞ (X 2 ) ≥ k 2 . Assume for contradiction that H ∞ (X v ) < k 2 . Let e be the edge in the computation path which connects to v, and suppose e is from u. Now consider the following two cases. • If e is not a bad edge, then H ∞ (X u ) ≤ H ∞ (X v ) + log(1/ Pr [e]) < k 2 + s + log(n/ε), which means u is also a stopping vertex. Therefore v cannot be the first stopping vertex. • If e is a bad edge, then V = ⊥. In both cases V = v, which is a contradiction. In conclusion we must have H ∞ (X 2 ) ≥ k 2 . Extractors for Sum of Two Sources In this section we formally prove Theorem 1 and Theorem 2. The construction of our extractors relies on the following lemma: k = O t 3 log (n) · log log(n) log log log(n) + log 3 (t) · log log log 4 (n) + log 4 (t) , Reduce(X) is N −γ -close to a (N 1−γ , t)-NOBF source. Before we prove Lemma 5.1, first we show how to prove Theorem 1 and Theorem 2 based on Lemma 5.1. is ε-close to uniform. By Lemma 5.1 it suffices to take k = O(log(n) log log(n) log log log 3 (n)). Next we prove Lemma 5.1. First we recall the definition of a strong affine correlation breaker. To simplify our proof of Lemma 5.1, here we use a definition which is slightly more general than Definition 1.8. Definition 5.2. AffCB : {0, 1} n × {0, 1} d × {0, 1} a → {0, 1} m is a (t, k, γ)-affine correlation breaker if for every distribution X, A, B ∈ {0, 1} n , Y, Y [t] ∈ {0, 1} d , Z and string α, α [t] ∈ {0, 1} a s.t. • X = A + B • H ∞ (A \| Z) ≥ k • (Y, Z) = (U d , Z) • A ↔ Z ↔ (B, Y, Y [t] ) is a Markov chain • ∀i ∈ [t], α = α i It holds that (AffCB(X, Y, α) ≈ γ U m ) \| {AffCB(X, Y i , α i )} i∈[t] , Z . We say AffCB is strong if (AffCB(X, Y, α) ≈ γ U m ) \| {AffCB(X, Y i , α i )} i∈[t] , Y, Y [t] , Z . To prove Lemma 5.1, we need the following lemma, which is an analog of [CZ19, Lemma 2.17]. Roughly speaking, we show that even if the seeds of the correlation breaker are added by some leakage from the source, most of the seeds are still good. • A be a (n, k + (t + 1)ℓ)-source For every b ∈ {0, 1} n , y ∈ {0, 1} d , define R b,y := AffCB(A + b, y + L(A, α), α) and for every i ∈ [t] define R i b,y := AffCB(A + b, y + L(A, α i ), α i ). Define BAD α,α [t] := y ∈ {0, 1} d : ∃b, y [t] s.t. (R b,y ≈ γ U m ) \| {R i b,y i } i∈[t] , which denotes the "bad seeds" of AffCB determined by A, L and α, α [t] . Then Pr y∼U d y ∈ BAD α,α [t] ≤ ε γ .R g(y),y ≈ γ U m \| {R i g(y),f i (y) } i∈[t] . For y ∈ BAD α,α [t] the values of f 1 (y), f 2 (y), . . . , f t (y), g(y) are defined arbitrarily. Note that the existence of f 1 , . . . , f t , g is guaranteed by the definition of BAD α,α [t] . Let W := U d and δ := Pr W ∈ BAD α,α [t] . Observe that L(A, α t )). Note that Z ∈ {0, 1} (t+1)ℓ is a deterministic function of A. With these new definitions the above equation can be rewritten as (R g(W),W ≈ γδ U m ) \| ({R i g(W),f i (W) } i∈[t] , W).(AffCB(A + B, Y, α) ≈ γδ U m ) \| ({AffCB(A + B, Y i , α i } i∈[t] , W). (3) Next, observe that the following conditions hold: • H ∞ (A \| Z) ≥ k (by Lemma 3.4) • (Y, Z) = (U d , Z). • A ↔ Z ↔ (B, Y, Y [t] ) is a Markov chain. Note that the last condition holds because Z is a deterministic function of A, which implies A ↔ Z ↔ (B, W), and Y, Y [t] are deterministic functions of (Z, W). By the definition of AffCB we have (AffCB(A + B, Y, α) ≈ ε U m ) \| ({AffCB(A + B, Y i , α i } i∈[t] , Y, Z) which implies (AffCB(A + B, Y, α) ≈ ε U m ) \| ({AffCB(A + B, Y i , α i } i∈[t] , W)(4) since W = Y − L(A, α) and L(A, α) is a part of Z. By (3) and (4) we get δ ≤ ε/γ. Next we prove the following lemma, which directly implies Lemma 5.1 by plugging in proper choices of somewhere random samplers and affine correlation breakers. Lemma 5.4. For every ε, δ > 0 the following holds. Let AffCB : {0, 1} n × {0, 1} d × [AC] → {0, 1} be a (Ct − 1)-strong affine correlation breaker for entropy k 1 with error A −2t C −1 εδ, and let Samp : {0, 1} n × [A] × [C] → {0, 1} d be a (ε, δ)-somewhere random sampler for entropy k 2 . Then for every n-bit source X = X 1 + X 2 such that X 1 is independent of X 2 , H ∞ (X 1 ) ≥ k 1 + Ctd and H ∞ (X 2 ) ≥ k 2 , the source Reduce(X) :=    z∈[C] AffCB (X, Samp(X, α, z), (α, z))    α∈[A] is 3δ-close to a convex combination of (2εA, t)-NOBF source. Proof. Consider Lemma 5.3 by taking X 1 as the source, A −t δ as the error parameter and L(x, (α, z)) := Samp(x, α, z) as the leakage function. For every non-empty subset T ⊆ [A] of size at most t and every z * ∈ [C], define a set BAD ′ T,z * as follows. Let α * denote the first element in T . Let β = (α * , z * ) and β ′ = {(α, z)} α∈T,z∈[C] \{β}. Note that β ′ contains at most 2 c t − 1 advice which are all different from β. Then we define BAD ′ T,z * := BAD β,β ′ , where BAD β,β ′ is defined as in Lemma 5.3. Observe that by definition of BAD ′ T,z * , for every x 2 ∈ {0, 1} n , if Samp(x 2 , α * , z * ) ∈ BAD ′ T,z * , then   α∈T z∈ [C] AffCB (X 1 + x 2 , Samp(X 1 , α, z) + Samp(x 2 , α, z), (α, z))   ≈ A −t δ U 1 . By the linearity of Samp, we know that for every fixing X 2 = x 2 , if Samp(x 2 , α * , z * ) ∈ BAD ′ T,z * , then   α∈T z∈ [C] AffCB (X, Samp(X, α, z), (α, z))   ≈ A −t δ U 1 .(5) By Lemma 5.3 we know that Pr y∼U d y ∈ BAD ′ T,z * ≤ A −t C −1 ε. Now define BAD ′ to be the union of BAD ′ T,z * for all possible choices of T, z * . Since there are at most A t choices of T and C choices of z * , by union bound we know that Pr y∼U d y ∈ BAD ′ ≤ ε. Therefore, by definition of somewhere random sampler, Pr x2∼X2 {α ∈ [A] : ∀z Samp(x 2 , α, z) ∈ BAD ′ } ≤ 2εA ≥ 1 − δ. In other words, with probability at least 1−δ over the fixing X 2 = x 2 , there exists a set Q ⊆ [A] of size at most 2εA which satisfies the following: for every α ∈ [A]\Q, there exists z α such that Samp(x 2 , α, z α ) ∈ BAD ′ , which also implies Samp(x 2 , α, z α ) ∈ BAD ′ T,zα . By Equation (5), for every T ⊆ [A]\Q s.t. 1 ≤ \|T \| ≤ t,   α∈T z∈{0,1} c AffCB(X, Samp(X, α, z), (α, z))   ≈ A −t δ U 1 . By Lemma 3.20 this implies that with probability 1 − δ over the fixing of X 2 , Reduce(X) =    z∈{0,1} c AffCB(X, Samp(X, α, z), (α, z))    α∈[A] is 2δ-close to a (2εA, t)-NOBF source. Therefore Reduce(X) is 3δ-close to a convex combination of (2εA, t)-NOBF source. To get Lemma 5.1, we need the following affine correlation breaker, which we construct in Section 6. Theorem 5.5. For every m, a, t ∈ N and ε > 0 there exists an explicit strong t-affine correlation breaker AffCB : Observe that we need to guarantee d ≥ K 1 Ct 2 log (n) · log log(n) log log log(n) + log 3 (Ct) and C ≥ K 2 log 2 d log(n) for some fixed constants K 1 , K 2 . It suffices to take C = O(log log log 2 (n) + log 2 (t)) for some large enough constant factor. Then the entropy requirement of AffCB would be k 1 = O Ct 2 log (n) · log log(n) log log log(n) + Ct , and the entropy requirement of Samp would be k 2 = O(d + log(N γ )) = O(d + log(n)). To make Reduce work, the entropy of the given sumset source should be at least k = max{k 1 + Ctd, k 2 } = O C 2 t 3 log (n) · log log(n) log log log(n) + log 3 (t) . {0, 1} n × {0, 1} d × {0, 1} a → {0, Finally, observe that the running time of Reduce is N times the running time of AffCB and Samp, which is also poly(n). Construction of Affine Correlation Breakers In this section we prove Theorem 5, which we restate below. Theorem 6.1 (Theorem 5, restated). Let C be a large enough constant. Suppose that there exists an explicit (d 0 , ε)-strong correlation breaker CB : {0, 1} d × {0, 1} d0 × {0, 1} a → {0, 1} C log 2 (t+1) log(n/ε) for some n, t ∈ N. Then there exists an explicit strong t-affine correlation breaker AffCB : {0, 1} n × {0, 1} d × {0, 1} a → {0, 1} m with error O(tε) for entropy k = O(td 0 + tm + t 2 log(n/ε)), where d = O(td 0 + m + t log 3 (t + 1) log(n/ε)). We note that it is possible to get different trade-off between the entropy k and the seed length d. Here we focus on minimizing min(k, td), which corresponds to the entropy of our extractors. With Theorem 5 we directly get Theorem 5.5 by plugging in the following (standard) correlation breaker by Li [Li19]. Proof of Theorem 5. Consider any A, B ∈ {0, 1} n , Y, Y [t] ∈ {0, 1} d , Z ∈ {0, 1} * such that • A ↔ Z ↔ (B, Y, Y [t] ) forms a Markov chain • H ∞ (A \| Z) ≥ k • (Y, Z) = (U, Z), and any α, α [t] ∈ {0, 1} a such that α = α i for every i ∈ [t]. Let X = A + B. Our goal is to construct an algorithm AffCB and prove that (AffCB(X, Y, α) ≈ O(tε) U m ) \| ({AffCB(X, Y i , α i )} i∈[t] , Y, Y [t] ).(6) For readability, first we explain some conventions in our proof. First we note that whenever we define a new random variable V := f (X, A, B, Y) using some deterministic function f , we also implicitly define V i := f (X, A, B, Y i ) for every i ∈ [t] . In each step of the proof, we consider a Markov chain (A, R) ↔ Z ′ ↔ (B, Y, Y [t] , S) for some random variables R, Z ′ , S, where R is a deterministic function of (A, Z ′ ), and S is a deterministic function of (B, Y, Y [t] , Z ′ ). Initially Z ′ = Z. When we say "R is ε-close to uniform" it means (R ≈ ε U) \| Z ′ , and similarly "S is ε-close to uniform" means (S ≈ ε U) \| Z ′ . When we say R is independent of S it implicitly means R ↔ Z ′ ↔ S is a Markov chain. Then when we say "fix f (R, Z)" for some deterministic function f , we consider the Markov chain (A, The algorithm AffCB consists of two phases. First, let r = (t + c 3.13 + 10) · c 3.12 log(n/ε), and let LExt 0 : {0, 1} n ×{0, 1} d ′ 0 → {0, 1} d0 and LExt r : {0, 1} n ×{0, 1} dx → {0, 1} r be strong linear seeded extractors in Lemma 3.13 with error ε. It suffices to take d ′ 0 = O(d 0 + log(n/ε)) and d x = O(log 2 (t + 1) log(n/ε)). Therefore if the constant C in the theorem statement is large enough, we can also take the output length of CB to be d x . The first phase of AffCB(X, Y, α) consists of the following steps. R) ↔ (Z ′ , f (R, Z ′ )) ↔ (B, Y, Y [t] , 1. Let S 1 := Prefix(Y, d ′ 0 ). 2. Compute R 1 := LExt 0 (X, S 1 ). 3. Compute S 2 := CB(Y, R 1 , α). 4. Output R 2 := LExt r (X, S 2 ). Furthermore, define R 1,A := LExt 1 (A, S 1 ), R 1,B := LExt 1 (B, S 1 ), R 2,A := LExt 2 (A, S 2 ) and R 2,B = LExt 2 (B, S 2 ), and let Z 0 = (Z, S 1 , S [t] 1 , R 1,B , R [t] 1,B , R 1 , R [t] 1 , S 2 , S [t] 2 ). First we prove that for every i ∈ [t], (R 2,A ≈ 5ε U) \| (R i 2,A , Z 0 , R 2,B , R [t] 2,B ),(7) and (A, R 2,A , R [t] 2,A ) ↔ Z 0 ↔ (B, R 2,B , R [t] 2,B , Y, Y [t] ) forms a Markov chain. Note that this means if we output R 2 we already get a 1-affine correlation breaker. To prove (7) and (8), first note that by definition of LExt 0 , we get ( R 1,A ≈ ε U d0 ) \| (Z, S 1 , S [t] 1 ). Fix (S 1 , S [t] 1 ). Since (R 1,A , R [t] 1,A ) are deterministic functions of (A, S 1 , S [t] 1 ), and (R 1,B , R [t] 1,B ) are deterministic functions of (B, S 1 , S [t] 1 ), (R 1,A , R [t] 1,A ) are independent of (R 1,B , R [t] 1,B ). Fix (R 1,B , R [t] 1,B ). Then R 1,A is still close to uniform. Because R 1 = R 1,A + R 1,B , this implies (R 1 ≈ ε U d0 ) \| (Z, S 1 , S [t] 1 , R 1,B , R [t] 1,B ). Moreover, H ∞ (Y \| Z, S 1 , S [t] 1 , R 1,B , R [t] 1,B ) ≥ d − O(t(d 0 + log(n/ε))) ≥ d 0 + log(1/ε). Because (R 1 , R [t] 1 ) ↔ (Z, S 1 , S [t] 1 , R 1,B , R [t] 1,B ) ↔ (Y, Y [t] ) is a Markov chain, and because CB is a strong correlation breaker, for every i ∈ [t] we have (S 2 ≈ 3ε U dx ) \| (S i 2 , Z, S 1 , S [t] 1 , R 1,B , R [t] 1,B , R 1 , R i 1 ). Note that after fixing R 1 , S 2 becomes independent of R [t] 1 . Therefore (S 2 ≈ 3ε U dx ) \| (S i 2 , Z, S 1 , S [t] 1 , R 1,B , R [t] 1,B , R 1 , R [t] 1 ). Fix R 1 , R [t] 1 . Because A are independent of S 2 , S [t] 2 , by Lemma 3.26 we can conclude that (R 2,A ≈ 5ε U r ) \| (R i 2,A , Z, S 1 , S [t] 1 , R 1,B , R [t] 1,B , R 1 , R [t] 1 , S 2 , S [t] 2 ) which is exactly (R 2,A ≈ 5ε U r ) \| (R i 2,A , Z 0 ).(9) Finally, fix S 2 , S [t] 2 . Since (R 2,A , R [t] 2,A ) are independent of (R 2,B , R [t] 2,B ), we get (8). Then because R 2 = R 2,A + R 2,B , by (8) and (9) we get (7). Next we move to the second phase. Let d y = c 3.12 log(n/ε). Moreover, let Ext : {0, 1} d × {0, 1} dy → {0, 1} dx be a strong seeded extractor from Lemma 3.12, and LExt m : {0, 1} r × {0, 1} dx → {0, 1} dy be a linear strong seeded extractor from Lemma 3.13. Define W 0,A := R 2,A , W 0,B := R 2,B , W 0 := R 2 and h = ⌈log t⌉. Then repeat the following steps for i from 1 to h: 2. Compute Q m,i−1 := Ext(Y, W p,i−1 ). Compute V i := LExt m (W i−1 , Q m,i−1 ). 4. Compute Q r,i := Ext(Y, V i ). 5. Compute W i := LExt r (X, Q r,i ). Note that Step 1 − 3 are the "independence merging" steps, which computes V i that is independent of every 2 i tampered versions. Since the length of V i is shorter than W i , we use Step 4 − 5 to recover the length and get W i s.t. \|W i \| = r. We claim that each of W i , Q m,i , V i , Q r,i is independent of every min(2 i , t) tampered versions, and in particular (W h , W [t] h ) ≈ (U r , W [t]Z i := Z i−1 , W p,i−1,B , W [t] p,i−1,B , W p,i−1 , W [t] p,i−1 , Q m,i−1 , Q [t] m,i−1 , V i,B , V [t] i,B , V i , V [t] i , Q r,i , Q [t] r,i . We want to prove the following claims for every i ∈ [h] by induction: • For every T ⊆ [t] s.t. \|T \| = 2 i , (W i,A ≈ (13·2 i −8)ε U r ) \| (W T i,A , Z i ).(10) • The following is a Markov chain: (A, W i,A , W [t] i,A ) ↔ Z i ↔ (B, W i,B , W [t] i,B , Y, Y [t] ).(11) Note that by (7) and (8), the conditions above hold for i = 0. Now assume by induction that (10) and (11) hold for i − 1, and we want to prove (10) and (11) for i. First, observe that because W p,i−1 = W p,i−1,A + W p,i−1,B , by (10) and (11) for every T 1 ⊆ [t] of size 2 i−1 , (W p,i−1 ≈ (13·2 i−1 −8)ε U r ) \| (W T1 p,i−1 , Z i−1 , W p,i−1,B , W [t] p,i−1,B ). Fix (W p,i−1,B , W [t] p,i−1,B ). Note that (W p,i−1 , W [t] p,i−1 ) ↔ (Z i−1 , W p,i−1,B , W [t] p,i−1,B ) ↔ (Y, Y [t] ) is a Markov chain. By Lemma 3.26 (similarly we omit the entropy requirement for Y for now and will verify it in the end), for every T 1 ⊆ [t] of size 2 i−1 , (Q m,i−1 ≈ (13·2 i−1 −6)ε U dx ) \| (Q T1 m,i−1 , Z i−1 , W p,i−1,B , W [t] p,i−1,B , W p,i−1 , W [t] p,i−1 ). Next, fix (W p,i−1 , W [t] p,i−1 ). Now consider any T ⊆ [t] s.t. \|T \| = min(2 i , t), and any T 1 , T 2 s.t. \|T 1 \| = \|T 2 \| = 2 i−1 and T 1 ∪ T 2 = T . By (10) there exists W ′ i−1,A = U r s.t. (W i−1,A ≈ (13·2 i−1 −8)ε W ′ i−1,A ) \| (W T2 i−1,A , Z i−1 , W p,i−1,B , W [t] p,i−1,B , W p,i−1 , W [t] p,i−1 ) and H ∞ W ′ i−1,A \| W T2 i−1,A , Z i−1 , W p,i−1,B , W [t] p,i−1,B , W p,i−1 , W [t] p,i−1 ≥ r − (t + 1)d y . Let Z ′ i−1 := Z i−1 , W p,i−1,B , W [t] p,i−1,B , W p,i−1 , W [t] p,i−1 , Q m,i−1 , Q [t] m,i−1 . By Lemma 3.26, (V i,A ≈ (13·2 i −14)ε U dy ) \| V T i,A , Z ′ i−1 . Fix (Q m,i−1 , Q [t] m,i−1 ). Note that Z ′ i−1 consists of exactly the random variables we have fixed so far. Because V i = V i,A + V i,B and (V i,A , V [t] i,A ) ↔ Z ′ i−1 ↔ (V i,B , V [t] i,B ) forms a Markov chain, (V i ≈ (13·2 i −12)ε U dy ) \| V T i , Z ′ i−1 , V i,B , V [t] i,B . Next we fix (V i,B , V [t] i,B ). Since (V i , V [t] i ) ↔ (Z ′ i−1 , V i,B , V [t] i,B ) ↔ (Y, Y [t] ) , again by Lemma 3.26, (Q r,i ≈ (13·2 i −10)ε U dx ) \| Q T r,i , Z ′ i−1 , V i,B , V [t] i,B , V i , V [t] i . Next, fix (V i , V [t] i ). Since A ↔ (Z ′ i−1 , V i,B , V [t] i,B , V i , V [t] i ) ↔ (Q r,i , Q [t] r,i ), by Lemma 3.26 (W i,A ≈ (13·2 i −8)ε U r ) \| Z ′ i−1 , V i,B , V [t] i,B , V i , V [t] i , Q r,i , Q [t] r,i , which is exactly (10). Fix (Q r,i , Q [t] r,i ). Because (W i,A , W [t] i,A ) are deterministic functions of (A, Q r,i , Q [t] r,i ) and (W i,B , W [t] i,B ) are deterministic functions of (B, Q r,i , Q [t] r,i ), we get (11). Finally we need to verify that whenever we apply Lemma 3.26, X and Y have enough conditional entropy. Observe that every time we apply Lemma 3.26 on A, we condition on some random variables in Z h , take an extractor from Lemma 3.13 with error ε and output at most r bits. The conditional entropy of A is at least H ∞ (A \| Z h ) ≥ H ∞ (A \| Z) − (t + 1) · O(d 0 + log(n/ε) + h(d x + d y )) ≥ (t + c 3.13 )r + log(1/ε), which satisfies the requirement in Lemma 3.26. Every time we apply Lemma 3.26 on Y, we condition on some random variables in Z h , take an extractor from Lemma 3.12 with error ε and output at most d x bits. The conditional entropy of Y is at least H ∞ (Y \| Z h ) ≥ d − (t + 1) · O(d 0 + log(n/ε) + h(d x + d y )) ≥ (t + 2)d x + log(1/ε), which satisfies the requirement in Lemma 3.26. Since W h = W h,A + W h,B , (10) and (11) together imply (W h ≈ (13t−8)ε U r ) \| (W [t] h , Y, Y [t] ). Therefore if m ≤ r, it suffices to output AffCB(X, Y, α) = Prefix(W h , m). If m > r, we can do one more round of alternating extraction to increase the output length. Let LExt out : {0, 1} n × {0, 1} dout → {0, 1} m be a linear strong seeded extractor with error ε from Lemma 3.13 and Ext out : {0, 1} d × {0, 1} r → {0, 1} dout be a seeded extractor from Lemma 3.12. It suffices to take d out = O m t + log 2 (t + 1) log( n ε ) . Then 1. Compute Q out := Ext out (Y, W h ). 2. Output W out := LExt out (X, Q out ). Since (W h ≈ U) \| (Z i , W h,B , W [t] h,B ), (W h , W [t] h ) ↔ (Z i , W h,B , W [t] h,B ) ↔ (Y, Y [t] ) forms a Markov chain and H ∞ (Y \| Z i , W h,B , W [t] h,B ) ≥ d − (t + 1) · O(d 0 + log(n/ε) + h(d x + d y )) ≥ (t + 2)d out + log(1/ε), by Lemma 3.26 (Q out ≈ (13t−6)ε U dout ) \| (Q [t] out , Z i , W h,B , W [t] h,B , W h , W [t] h ). And because A is independent of (Q out , Q (W out ≈ (13t−4)ε U m ) \| (W [t] out , Z, Y, Y [t] ), which means AffCB(X, Y, α) = W out is a strong t-affine correlation breaker with error O(tε). Sumset Sources with Small Doubling In this section we show that a sumset source with small doubling constant is close to a convex combination of affine sources, as stated in Theorem 6. To prove this result, first we need Lemma 7.1, which is a variant of the Croot-Sisask lemma [CS10]. For the proof of Lemma 7.1 we follow the exposition by Ben-Sasson, Ron-Zewi, Tulsiani and Wolf [BRTW14] which is more convenient for our setting. Lemma 7.1. Let A ⊆ F n 2 be a set which satisfies \|A\| ≥ \|F n 2 \| /r. Then for every ε > 0 and every pair of functions f, g : F n 2 → [0, 1] there exists t = O(log(r/ε)/ε 2 ) and a set X of size at least \|F n 2 \| /2r t such that for every set B s.t. \|B\| ≥ \|F n 2 \| /r and every x ∈ X, E a∼A,b∼B [f (a + b)] ≈ ε E a∼A,b∼B [f (a + b + x)] and E a∼A,b∼B [g(a + b)] ≈ ε E a∼A,b∼B [g(a + b + x)] . Proof. Let t = 8 ln(128r/ε)/ε 2 . By Chernoff-Hoeffding bound, for every b ∈ F n 2 , Pr (a1,...,at)∼A t 1 t t i=1 f (a i + b) ≈ ε 4 E a∼A [f (a + b)] ≥ 1 − ε 16r and Pr (a1,...,at)∼A t 1 t t i=1 g(a i + b) ≈ ε 4 E a∼A [g(a + b)] ≥ 1 − ε 16r . Then by union bound and by averaging over b ∼ F n 2 , Pr (a1,...,at)∼A t b∼F n 2 1 t t i=1 f (a i + b) ≈ ε 4 E a∼A [f (a + b)] and 1 t t i=1 g(a i + b) ≈ ε 4 E a∼A [g(a + b)] ≥ 1 − ε 8r . Define BAD (a1,...,at) := b : 1 t t i=1 f (a i + b) ≈ ε 4 E a∼A [f (a + b)] or 1 t t i=1 g(a i + b) ≈ ε 4 E a∼A [g(a + b)] . By Markov inequality, there exists S ⊆ A t such that \|S\| ≥ \|A\| t /2 and for every (a 1 , . . . , a t ) ∈ S, BAD (a1,...,at) ≤ ε 4r \|F n 2 \| . Now classify the elements in S by (a 2 −a 1 , a 3 −a 1 , . . . , a t −a 1 ). By averaging there exists a subset X ′ ⊆ S and a (t − 1)-tuple (y 2 , . . . , y t ) such that \|X ′ \| ≥ \|S\| / \|F n 2 \| t−1 ≥ \|F n 2 \| /2r t , and for every (a 1 , . . . , a t ) ∈ X ′ we have a i − a 1 = y i for every 2 ≤ i ≤ t. Let (a * 1 , . . . , a * t ) be an element in X ′ . Observe that for every (a 1 , . . . , a t ) ∈ X ′ , a 1 − a * 1 = · · · = a t − a * t . Define : (a 1 , . . . , a t ) ∈ X ′ }. X = {x = a 1 − a * 1 Note that \|X\| = \|X ′ \| ≥ \|F n 2 \| /2r t . It remains to prove that for every x ∈ X, E a∼A,b∼B [f (a + b)] ≈ ε E a∼A,b∼B [f (a + b + x)] and E a∼A,b∼B [g(a + b)] ≈ ε E a∼A,b∼B [g(a + b + x)] . Let (a 1 , . . . , a t ) = (a * 1 + x, . . . , a * t + x). Since (a 1 , . . . , a t ) is an element in S, E a∼A,b∼B [f (a + b)] − E b∼B 1 t t i=1 f (a i + b) ≤ ε 4 + Pr b∼B b ∈ BAD (a1,...,at) ≤ ε 2 . Similarly, since (a * 1 , . . . , a * t ) is an element in S, E a∼A,b∼B [f (a + b + x)] − E b∼B 1 t t i=1 f (a * i + b + x) ≤ ε 4 + Pr b∼B (b + x) ∈ BAD (a * 1 ,...,a * t ) ≤ ε 2 . Finally, observe that E b∼B 1 t t i=1 f (a i + b) = E b∼B 1 t t i=1 f (a * i + x + b) . By triangle inequality we can conclude that E a∼A,b∼B [f (a + b)] ≈ ε E a∼A,b∼B [f (a + b + x)] . Similarly we can prove that E a∼A,b∼B [g(a + b)] ≈ ε E a∼A,b∼B [g(a + b + x)] . Next we prove the following lemma. The proof is along the lines of [San12, Theorem A.1]. (See also the survey by Lovett [Lov15].) Lemma 7.2. Let A, B ⊆ F n 2 be sets which satisfy \|A\| , \|B\| ≥ \|F n 2 \| /r. Let A, B be the uniform distributions over A, B respectively. Then for every ε > 0 and every pair of functions f, g : F n 2 → [0, 1] there exists a linear subspace V of co-dimension O(log 3 (r/ε) log(r)/ε 2 ) and a distribution T ∈ F n 2 such that E [f (A + B)] ≈ ε E [f (T + V)] and E [g(A + B)] ≈ ε E [g(T + V)] , where V is the uniform distribution over V . To prove Lemma 7.2, first we need the following corollary of Lemma 7.1. Corollary 7.3. Let A ⊆ F n 2 be a set which satisfies \|A\| ≥ \|F n 2 \| /r. Then for every ε > 0 and every pair of functions f, g : F n 2 → [0, 1] there exists t = O(log(r/ε)/ε 2 ) and a a set X of size at least \|F n 2 \| /2r t such that for every set B s.t. \|B\| ≥ \|F n 2 \| /r and every (x 1 , . . . , x ℓ ) ∈ X ℓ , E a∼A,b∼B [f (a + b)] ≈ ℓε E a∼A,b∼B [f (a + b + x 1 + · · · + x ℓ )] and E a∼A,b∼B [g(a + b)] ≈ ℓε E a∼A,b∼B [g(a + b + x 1 + · · · + x ℓ )] . Proof. Assume by induction that E a∼A,b∼B [f (a + b)] ≈ (ℓ−1)ε E a∼A,b∼B [f (a + b + x 1 + · · · + x ℓ−1 )] . Since \|B + x 1 + · · · + x ℓ−1 \| = \|B\| ≥ \|F n 2 \| /r, by Lemma 7.1 we get E a∼A,b∼B [f (a + b + x 1 + · · · + x ℓ−1 )] ≈ ε E a∼A,b∼B [f (a + b + x 1 + · · · + x ℓ )] . Then the claim follows by triangle inequality. The proof for the case of g is exactly the same. Proof of Lemma 7.2. Define ℓ = log(2r/ε). By Corollary 7.3 there exists t = O(ℓ 3 /ε 2 ) and a set X of size \|F n 2 \| /2r t s.t. for every (x 1 , x 2 , . . . , x ℓ ) ∈ X ℓ , E [f (A + B)] ≈ ε/2 E [f (A + B + x 1 + · · · + x ℓ )](12) and E [g(A + B)] ≈ ε/2 E [g(A + B + x 1 + · · · + x ℓ )] . Let X 1 , . . . , X ℓ be independent uniform distributions over X. Let V = Spec 1/2 (X) ⊥ and V be uniform distribution over V . Note that by Chang's lemma (Lemma 3.40), V has dimension at least k ′ = m − O(log(r) log 3 (r/ε)/ε 2 ) ≥ k − O(log(r) log 3 (r/ε)/ε 2 ). By Lemma 3.38, 3.37 and 3.33, E [f (A + B + X 1 + · · · + X ℓ )] = α∈F n 2 µ A (α) µ B (α)( µ X (α)) ℓ f (α) and E [f (A + B + X 1 + · · · + X ℓ + V)] = α∈F n 2 µ A (α) µ B (α)( µ X (α)) ℓ µ V (α) f (α). Define T = A + B + X 1 + · · · + X ℓ . Then Finally, to prove Theorem 6, we need the following lemma. Lemma 7.4. Let X ⊆ F n 2 be a set, φ : F n 2 → F m 2 be a linear Freiman 3-homorphism of X, and φ −1 : F m 2 → F n 2 be a inverse of φ such that φ −1 (φ(x)) = x for every x ∈ 3X. (Such a φ −1 exists because φ is injective on 3X.) Then for every affine subspace V ⊆ F m 2 such that \|V ∩ φ(X)\| > \|V \| /2, φ −1 is injective on V and φ −1 (V ) ⊆ F n 2 is also an affine subspace. Proof. Let t be an element in X such that φ(t) ∈ V . Note that t must exist because V ∩ φ(X) is nonempty. Since V is an affine subspace, for every v ∈ V and v 1 ∈ V ∩ φ(X), v + φ(t) − v 1 ∈ V . Because \|V ∩ φ(X)\| > \|V \φ(X)\|, for every v ∈ V there must exist v 1 ∈ V ∩ φ(X) s.t. v + φ(t) − v 1 ∈ V ∩ φ(X). In other words, for every v ∈ V there exist v 1 , v 2 ∈ V ∩ φ(X) such that v = v 1 + v 2 − φ(t). This means V ⊆ φ(2X − t) ⊆ φ(3X). Because φ −1 is injective on φ(3X), this implies that φ −1 is injective on V . Next we prove that φ −1 (V ) is also an affine subspace. It suffices to prove that for every u, v ∈ V , φ −1 (u) + φ −1 (v) − t = φ −1 (u + v − φ(t)), because φ −1 (u + v − φ(t)) ∈ φ −1 (V ). Observe that φ(φ −1 (u) + φ −1 (v) − t − φ −1 (u + v − φ(t))) = u + v − φ(t) − (u + v − φ(t)) = 0, because φ is linear, and for every y ∈ {u, v, u+v−φ(t)} we have y ∈ V ⊆ φ(3X), which means φ(φ −1 (y)) = y. Moreover, because φ −1 (u), φ −1 (v), φ −1 (u + v − φ(t)) ∈ φ −1 (V ) ⊆ 2X − t, φ −1 (u) + φ −1 (v) − t − φ −1 (u + v − φ(t)) ∈ 6X. By Lemma 3.30, φ −1 (u) + φ −1 (v) − t − φ −1 (u + v − φ(t)) = 0. Now we are ready to prove Theorem 6. Proof of Theorem 6. Consider any function f : F n 2 → [0, 1]. Let φ : F n 2 → F m 2 be the 3-Freiman homomorphism of A + B guaranteed in Lemma 3.31, and let φ −1 : F n 2 → F m 2 be a inverse of φ such that φ −1 (φ(x)) = x for every x ∈ 3A + 3B. By Lemma 3.30, φ is injective on A and B since A ⊆ 3(A + B) + b for any b ∈ B and B ⊆ 3(A + B) + a for any a ∈ A. Let A ′ = φ(A), B ′ = φ(B), A ′ = φ(A), B ′ = φ(B). Observe that A ′ , B ′ are exactly the uniform distributions over A ′ , B ′ respectively. By Lemma 3.31 and Lemma 3.28, we get \|F m 2 \| = \|φ(6A + 6B)\| ≤ \|6A + 6B\| ≤ r 13 \|A\|, which implies \|A ′ \| = \|B ′ \| = \|A\| ≥ \|F m 2 \| /r 13 . By Lemma 7.2, there exists a distribution T ∈ F m 2 and a linear subspace V of entropy k ′ = m − O(log(r) log(r/ε) 3 /ε 2 ) such that E [1 A ′ +B ′ (A ′ + B ′ )] ≈ ε/3 E [1 A ′ +B ′ (T + V)] and E f (φ −1 (A ′ + B ′ )) ≈ ε/3 E f (φ −1 (T + V)) , where V is the uniform distribution over V . Now observe that since E [1 A ′ +B ′ (A ′ + B ′ )] = 1, Since the proof above works for every function f : F n 2 → [0, 1], by Corollary 3.42, A + B is ε-close to a convex combination of affine sources. E [1 A ′ +B ′ (T + V)] ≥ 1 − ε/3. B On Random Functions and Extractors for Sumset Sources In this section, first we show that a random function is an extractor for sumsets with low additive energy. Similar to the size of a sumset, the additive energy is also an intensively studied property in additive combinatorics [TV06]. Then we briefly discuss why this result is not sufficient to prove that a random function is an extractor for sumset sources with Theorem 6. For two sets A, B ⊆ F n 2 , define γ Without loss of generality, in the rest of this section we consider a "flat" sumset source A + B such that A, B are uniform distributions over A, B of size K = 2 k . We note that E(A, B) satisfies K 2 ≤ E(A, B) ≤ K 3 , and 4k − log (E(A, B)) is exactly the "Rényi entropy" of A + B, which is defined as H 2 (X) = Since the total number of subsets A, B of size K is at most 2 n K 2 ≤ 2 2nK , by union bound we get the following theorem. Theorem B.3. With probability 1 − 2 −0.88nK , a random function is an extractor with error ε for sumset sources A + B which satisfy E(A, B) ≤ K 3 n/ε 2 . In other words, a random function is an extractor for flat sumset sources A+B which satisfy H 2 (A+B) ≥ k + log(n/ε 2 ). However, Theorem 6 only shows how to extract from A + B when the "max-entropy" In additive combinatorics this corresponds to sets with "large doubling" and "large energy", and can be obtained with the following example. Suppose A = B = V ∪ R, where V is a linear subspace of dimension k − 1, and R is a random set of size K/2. Then E(A, B) ≥ E(V, V ) ≥ K 3 /8, and \|A + B\| ≥ \|R + R\| ≈ K 2 /4. Finally we remark that a well known result in additive combinatorics called the 'Balog-Szémeredi-Gowers theorem" [BS94,Gow01,SSV05] states that if E(A, B) ≥ K 3 /r then there must exist A ′ ⊆ A, B ′ ⊆ B of size K/ poly(r) such that \|A ′ + B ′ \| ≤ poly(r) · \|A\|. However, if we apply this theorem on the cases which do not satisfy Theorem B.3, we can only guarantee that there exist small subsets A ′ , B ′ of size K/ poly(n) which have small doubling. Because Pr [A ∈ A ′ ∧ B ∈ B ′ ] ≈ 1/ poly(n), with Theorem 6 we can only prove that a random function is an extractor for A ′ , B ′ with error 1/2 − 1/ poly(n), which is comparable to a disperser. Corollary 1. 11 . 11Let A, B be uniform distribution over A, B ⊆ F n 2 s.t. \|A\| = \|B\| = 2 k and \|A + B\| ≤ r \|A\|. If AffExt : {0, 1} n → {0, 1} m is an extractor for affine sources with entropy k − log 4 (r), then AffExt(A + B) is O(1)-close to U m . For every function f : X → Y and set A ⊆ X , define f (A) = {f (x) : x ∈ A}. For a set A ⊆ X we use 1 A : X → {0, 1} to denote the indicator function of A such that 1 A (x) = 1 if and only if x ∈ A. Definition 3. 3 3([DORS08]). For a joint distribution (X, Z), the average conditional min-entropy Definition 3. 9 . 9Let B = (L 0 , L 1 , . . . , L n ) be a branching program of width w and length n and let v be a vertex in layer i of B. Then the subprogram of B starting at v, denoted by B v , is the induced subgraph of B which consists of ({v}, L i+1 , . . . , L n ). Note that B v is a branching program of width w and length n − i which takes v as the start vertex. Lemma 3 . 312 ([GUV09]). There exists a constant c 3.12 and a constant β > 0 such that for every ε > 2 −βn and every k, there exists an explicit (k, ε)-strong seeded extractor Ext : {0, 1} n × {0, 1} d → {0, 1} m s.t. d = c 3.12 log(n/ε) and m = k/2. Definition 3. 14 . 14Samp : {0, 1} n ×[D] → {0, 1} m is a (ε, δ)-sampler for entropy k if for every set T ⊆ {0, 1} m s.t. \|T \| ≤ ε2 m and every (n, k)-source X, Pr x∼X Pr y∼[D] Definition 3. 16 . 16Samp : {0, 1} n × [D] × [C] → {0, 1} m is a (ε, δ)-somewhere random sampler for entropy k if for every set T ⊆ {0, 1} m s.t.\|T \| ≤ ε2 m and every (n, k)-source X, Lemma 3 . 317 ([BDT19]). If there exists an explicit (ε, δ)-sampler Samp : {0, 1} n × [D 0 ] → {0, 1} m for entropy k, then for every constant γ < 1 there exists an explicit (D −γ , δ)-somewhere random sampler Samp ′ : {0, 1} n × [D] × [C] → {0, 1} m for entropy k with D = D O(1) 0 and C = O log(D0) log(1/ε) . Furthermore if Samp is linear then Samp ′ is also linear. By Lemma 3.13, Lemma 3.15 and Lemma 3.17 we can get the following explicit somewhere random smapler. Lemma 3.18. For every constant γ < 1, and every δ > 0, t < 2 3 √ log(n) there exists an explicit (D −γ , δ)-linear somewhere random sampler Samp : {0, 1} n × [D] × [C] → {0, 1} t log(n) for entropy O(t log(n)) + log(1/δ), where D = n O(1) and C = O(log 2 (t)). Proof. By Lemma 3.13 and Lemma 3.15, there exists an explicit (ε, δ)-linear sampler Samp ′ : {0, 1} n ×[D 0 ] → {0, 1} t log(n) for entropy O(t log(n)) + log(1/δ) where ε = 2 − log(n)/ log 2 (t) and D 0 = n O(1) . The claim follows by applying Lemma 3.17 on Samp ′ . Lemma 3 . 322 ([CZ19,Li16]). There exists an explicit function BFExt : {0, 1} n → {0, 1} m for (q, t)-NOBF sources with error n −Ω(1) where m = n Ω(1) , q = n 0.9 and t = (m log(n)) C 3.22 for some constant C 3.22 . Lemma 3 . 323 ([Vio14]). For every ε > 0, the majority function Maj : {0, 1} n → {0, 1} is an extractor for (q, t)-NOBF sources with error ε + O(n −0.1 ) where q = n 0.4 and t = O(ε −2 log 2 (1/ε)). Lemma 3.31 ([GR07]). For every set A ⊆ F n 2 there exists a linear s-Freiman homomorphism φ : F n 2 → F m 2 of A such that φ(2sA) = F m 2 . Lemma 3 . 37 . 337Let f : F n 2 → R be a function and let A be a distribution on F n 2 . Then E [f (A)] = E x∼F n 2 [µ A (x)f (x)] . Lemma 3 . 38 . 338Let A, B be two distributions on F n 2 . Then µ A+B = µ A * µ B . Lemma 3.39. If V ⊆ F n 2 is a linear subspace, then µ V (α) = 1 if α ∈ V ⊥ and µ V (α) = 0 otherwise. Finally we need Chang's lemma. Corollary 3 . 42 . 342Let Ω be a finite set, X be a convex set of distributions on Ω, and Y be a distribution on Ω. If for every function f : Ω → [0, 1] there exists Lemma 5.1 (main lemma). For every constant γ < 1 and every t ∈ N, there exists N = n O(1) and an explicit function Reduce : {0, 1} n → {0, 1} N s.t. for every (n, k, 2)-sumset source X, where Proof of Theorem 1 . 1Let Reduce : {0, 1} n → {0, 1} N be the function from Lemma 5.1 by taking γ = 0.1. Note that N = poly(n). Let BFExt : {0, 1} N → {0, 1} m be the NOBF-source extractor from Lemma 3.22. Let X be a (n, k, 2)-source, where k is defined later. If Reduce(X) is N −Ω(1) -close to a (N 0.9 , t)-NOBF source where t = (m log(N )) C 3.22 , then Ext(X) := BFExt(Reduce(X)) is n −Ω(1) -close to uniform. By Lemma 5.1 it suffices to take k = O(t 3 log 7 (t) log(n)) ≤ (m log(n)) 1+3C 3.22 . Proof of Theorem 2. Let Reduce : {0, 1} n → {0, 1} N be the function from Lemma 5.1 by taking γ = 0.6. Note that N = poly(n). Let Maj : {0, 1} N → {0, 1} be the NOBF-source extractor from Lemma 3.23, i.e. the majority function. Let X be a (n, k, 2)-source, where k is defined later. If Reduce(X) is (ε/2)-close to a (N 0.4 , t)-NOBF source where t = O(ε −2 log 2 (1/ε)) = O(1), then Ext(X) := Maj(Reduce(X)) Lemma 5 . 3 . 53For every error parameter γ > 0 the following holds. Let• AffCB : {0, 1} n × {0, 1} d × {0, 1} a → {0,1} m be a (t, k, ε)-strong affine correlation breaker • L : {0, 1} n × {0, 1} a → {0, 1} d be any deterministic function, which we call the leakage function • α, α [t] be any a-bit advice s.t. α = α i for every i ∈ [t] Now define Y := W + L(A, α), Y i := W + L(A, α i ) for every i ∈ [t] and B := g(W). Let Z := (L(A, α), L(A, α 1 ), . . . , 1} m with error ε for entropy k such that the seed length isd = O t log n ε · log(a) log log(a) + log 3 (t) and k = O tm + t log n ε · log(a) log log(a) + t .Now we are ready to prove Lemma 5.1. Proof of Lemma 5.1. Let Samp : {0, 1} n × [N ] × [C] → {0, 1} d be a (N −γ /2, N −γ /3)-somewhere random sampler from Lemma 3.18, where N = n O(1) . We want to choose proper parameters d, C so that there exists a (Ct − 1)-strong affine correlation breaker AffCB : {0, 1} n × {0, 1} d × [N C] → {0, 1} with error N −2(t+γ) C −1 /6. Then Lemma 5.4 would imply Lemma 5.1. Theorem 6.2 ([Li19]). There exists an explicit (standard) correlation breaker {0, 1} n × {0, 1} d × {0, 1} a → {0, 1} m for entropy d with error ε, where d = O m + log(n/ε) · log(a) log log(a) . S) in the next step. Similarly when we say "fix g(S, Z)" for some deterministic function g, we consider the Markovchain (A, R) ↔ (Z ′ , g(S, Z ′ )) ↔ (B, Y, Y [t], S) in the next step. To make the notations cleaner, sometimes we only specify a Markov chain R ↔ Z ′ ↔ S where R, S are the random variables used in the current step of argument (e.g. when we apply Lemma 3.26), but it should always be true that (A, R) ↔ Z ′ ↔ (B, Y, Y[t] , S) is a Markov chain. h ). Formally, for every i from 1 to h, let W p,i−1,A := Prefix(W i−1,A , d y ), W p,i−1,B := Prefix(W i−1,B , d y ), V i,A := LExt m (W i−1,A , Q m,i−1 ), V i,B := LExt m (W i−1,B , Q m,i−1 ), W i,A := LExt r (A, Q r,i ) and W i,B := LExt r (B, Q r,i ). Moreover, for every i ∈ [h], let E [f (T)] − E [f (T + V)] = α ∈V ⊥ µ A (α) µ B (α)( µ X (α)) ℓ f (α) (by Lemma 3.39) ≤ 2 −ℓ α ∈V ⊥ µ A (α) µ B (α) f (α) (by definition of Spec 1/2 (X)) ≤ 2 −ℓ α ∈V ⊥ \| µ A (α) µ B (α)\| (since f (α) −ℓ · r = ε/2.(by Parseval's identity (Lemma 3.33)) By triangle inequality and (12) we get E [f (A + B)] ≈ ε E [f (T + V)]. The exact same proof can also show that E [g(A + B)] ≈ ε E [g(T + V)]. Eφ [1 A ′ +B ′ (t + V)] > 1/2 ≥ 1 − 2ε/−1 (t + V) is an affine source of entropy k ′ ≥ 1 − 2ε/3. Therefore φ −1 (T + V) is (2ε/3)-close to a convex combination of affine sources (denoted by W) of entropy k ′ . Since A + B ⊆ 3A + 3B, φ −1 (A ′ + B ′ ) = φ −1 (φ(A + B)) is exactly A + B. Therefore by (13) and triangle inequality,E [f (A + B)] ≈ ε E [f (W)] . A,B (x) = \|{(a, b) : a ∈ A, b ∈ B, a + b = x}\|. Observe that if A is the uniform distribution over A and B is the uniform distribution over B, then Pr [A + B = x] = γA,B (x) \|A\|\|B\| . Definition B.1. The additive energy between A, B is defined as E(A, B) := x∈A+B γ A,B (x) 2 . − log( x∈Supp(X) Pr [X = x] 2 ). In the following lemma we show that if E(A, B) is low (i.e. if H 2 (A + B) is high), then a random function is an extractor for A + B with high probability.Lemma B.2. For a random function f : F n 2 → {0, 1}, f (A + B) is ε-close to U 1 with probability 1 − 2e −2ε 2 K 4 /E(A,B) . Proof. Observe that E [f (A + B)] = 1 K 2 x∈A+B γ A,B (x) · f (x). Because the terms {γ A,B (x) · f (x)} x∈A+B are independent random variables, and each γ A,B (x) · f (x) is in the range [0, γ A,B (x)], the lemma is directly implied by Hoeffding's inequality. H 0 (A + B) := log(\|Supp(A + B)\|) is close to k. Because H 0 (A + B) ≥ H 2 (A + B), it is possible that H 2 (A + B) ≈ k and H 0 (A + B) ≫ k, and in this case neither of our analysis works. Proof. Define deterministic functions f 1 , . . . , f t : {0, 1} d → {0, 1} d and g : {0, 1} d → {0, 1} n s.t. for every y ∈ BAD α,α [t] , [ t ] tout ) conditioned on (Z i , W h,B , W H ∞ (A \| Z i , W h,B , W h ) ≥ k − (t + 1) · O(d 0 + log(n/ε) + h(d x + d y )) ≥ (t + 2)d out + log(1/ε),again by Lemma 3.26 we can conclude that(W out,A ≈ (13t−4)ε U m ) \| (W out,A , Z i , W h,B , W Since W out = W out,A + W out,B and (W out,A , W out,A ) are independent of (Y, Y [t] , W out,A , W out,A ) conditioned on (Z i , W h,B , W [t] h,B , W h , W [t] h , Q out , Q [t]out ), we can conclude that[t] h,B , W h , W [t] h ), and [t] h,B , W h , W [t] [t] [t] h,B , W h , W [t] h , Q out , Q [t] out ). [t] [t] Here we focus on the small-space extractors which minimize the entropy requirement. For small-space extractors with negligible error, see[CG21] for a survey. 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[ "Flexi-Transducer: Optimizing Latency, Accuracy and Compute for Multi-Domain On-Device Scenarios", "Flexi-Transducer: Optimizing Latency, Accuracy and Compute for Multi-Domain On-Device Scenarios" ]	[ "Jay Mahadeokar [email protected] \nFacebook AI\n\n", "Yangyang Shi \nFacebook AI\n\n", "Yuan Shangguan \nFacebook AI\n\n", "Chunyang Wu \nFacebook AI\n\n", "Alex Xiao \nFacebook AI\n\n", "Hang Su \nFacebook AI\n\n", "Duc Le \nFacebook AI\n\n", "Ozlem Kalinli \nFacebook AI\n\n", "Christian Fuegen \nFacebook AI\n\n", "Michael L Seltzer \nFacebook AI\n\n" ]	[ "Facebook AI\n", "Facebook AI\n", "Facebook AI\n", "Facebook AI\n", "Facebook AI\n", "Facebook AI\n", "Facebook AI\n", "Facebook AI\n", "Facebook AI\n", "Facebook AI\n" ]	[]	Often, the storage and computational constraints of embedded devices demand that a single on-device ASR model serve multiple use-cases / domains. In this paper, we propose a Flexible Transducer (FlexiT) for on-device automatic speech recognition to flexibly deal with multiple use-cases / domains with different accuracy and latency requirements. Specifically, using a single compact model, FlexiT provides a fast response for voice commands, and accurate transcription but with more latency for dictation. In order to achieve flexible and better accuracy and latency trade-offs, the following techniques are used. Firstly, we propose using domain-specific altering of segment size for Emformer encoder that enables FlexiT to achieve flexible decoding. Secondly, we use Alignment Restricted RNNT loss to achieve flexible fine-grained control on token emission latency for different domains. Finally, we add a domain indicator vector as an additional input to the FlexiT model. Using the combination of techniques, we show that a single model can be used to improve WERs and real time factor for dictation scenarios while maintaining optimal latency for voice commands use-cases.	10.21437/interspeech.2021-1921	[ "https://arxiv.org/pdf/2104.02232v1.pdf" ]	233,033,766	2104.02232	a9279eb2b5268fa3b15686ebe0cb856e32b41318	Flexi-Transducer: Optimizing Latency, Accuracy and Compute for Multi-Domain On-Device Scenarios Jay Mahadeokar [email protected] Facebook AI Yangyang Shi Facebook AI Yuan Shangguan Facebook AI Chunyang Wu Facebook AI Alex Xiao Facebook AI Hang Su Facebook AI Duc Le Facebook AI Ozlem Kalinli Facebook AI Christian Fuegen Facebook AI Michael L Seltzer Facebook AI Flexi-Transducer: Optimizing Latency, Accuracy and Compute for Multi-Domain On-Device Scenarios Index Terms: Speech recognitionRNN-TTransformers Often, the storage and computational constraints of embedded devices demand that a single on-device ASR model serve multiple use-cases / domains. In this paper, we propose a Flexible Transducer (FlexiT) for on-device automatic speech recognition to flexibly deal with multiple use-cases / domains with different accuracy and latency requirements. Specifically, using a single compact model, FlexiT provides a fast response for voice commands, and accurate transcription but with more latency for dictation. In order to achieve flexible and better accuracy and latency trade-offs, the following techniques are used. Firstly, we propose using domain-specific altering of segment size for Emformer encoder that enables FlexiT to achieve flexible decoding. Secondly, we use Alignment Restricted RNNT loss to achieve flexible fine-grained control on token emission latency for different domains. Finally, we add a domain indicator vector as an additional input to the FlexiT model. Using the combination of techniques, we show that a single model can be used to improve WERs and real time factor for dictation scenarios while maintaining optimal latency for voice commands use-cases. Introduction On-device automatic speech recognition (ASR) models have been enabled on many embedded devices, including mobile phones, smart speakers, and watches [1][2][3]. On one hand, ondevice ASR eliminates the need to transfer audio and recognition results between devices and a server, thus enabling fast, reliable, and privacy-preserving speech recognition experiences. On the other hand, these devices operate with significant hardware constraints: e.g., memory, disk space, and battery consumption. Moreover, the embedded ASR models often serve multiple applications: e.g., video transcription, dictation, and voice commands. Each of these applications has its latency and accuracy requirements. For example, voice assistants demand an ASR model with low latency to respond to user queries as fast as possible. While server-based ASR might rely on running different models for different applications -more compact models for voice assistants and big, semi-streaming models [4,5] for dictation-the on-device environment prohibits such practice. Due to hardware constraints, and varied requirements of different applications optimizing the model size, compute and accuracy of one single ASR model becomes challenging. In this paper, we take a close look at the scenario where device constrained ASR model needs to be optimized for two different use-cases. The first use case is voice commands, where the latency requirement is strict. Users expect immediate device responses *Equal Contribution when they ask the speech assistant to turn on the lights or play a song. The second application uses the ASR model for dictation or audio transcription, where accuracy is more important than the model's latency. Recurrent Neural Network Transducer (RNN-T) framework [1,6] is widely adopted to provide streaming ASR transcriptions for both voice commands [7,8] and dictation applications [9,10]. We focus on Emformer model [11] as an audio encoder for RNN-T, which uses both contextual audio information (in the form of an audio chunk) and future audio context (in the form of model look-ahead). A larger model look-ahead permits the model to access more future context and optimize ASR accuracy but hurts model latency. This paper proposes a Flexi-Transducer (FlexiT) model that answers the requirements of two streaming ASR use-cases while still staying as one compact model. Further we also show that larger look-ahead enables improve the compute / real time factor trade-offs which help battery consumption. Related Work Inspired by the successful application of transformer [12], many works in ASR also adopted transformer across different model paradigms, such as the hybrid systems [13][14][15][16], the encoderdecoder with attention [17][18][19][20][21] and the sequence transducers [5,22,23]. In this work, we follow the neural transducer paradigm using Emformer [11,16] and alignment restricted transducer loss [24]. Many ASR applications demand real-time low latency streaming. The block processing method [11,16,25] with attention mask modifies the transformer to support streaming applications. In the block processing, self-attention's receptive field consists of one fixed-size chunk of speech utterance and its historical context and a short window of future context. However, the fixed chunk size limits the encoder's flexibility to trade-off latency, real-time factor, and accuracy. A unified framework is proposed in [26] to train one single ASR model for both streaming and non-streaming speech recognition applications. In [4] cascaded encoders are used to build a single ASR model that operates in streaming and nonstreaming mode. These approaches support one fixed latency for the streaming use-case, and the other use-case is strictly non-streaming. In this paper, we tackle scenarios to support two different streaming ASR use-cases with different latency constraints. More flexible latency is achieved by [27][28][29] where, in the training phase, the model is exposed to audio context variants up to the whole utterance length. In [27], the authors show that asynchronous revision during inference with convolutional encoders can be used to achieve dynamic latency ASR. The context size selection proposed in these works is purely random during training. In this work, besides random selection, we also explore context size selection based on a priori knowledge about the targeted use cases (domains). The use of domain information to improve the ASR performance of a single model being used to serve different dialects, accents, or use-cases has been studied previously in [10,30,31] For RNN-T models, both the encoder's context size and the potential delay of token emissions contribute to model latency. It is well known that streaming RNN-T models tend to emit ASR tokens with delay. Techniques like Ar-RNN-T [24], Fast-emit [32], constrained alignment approach [33,34] or late alignment penalties [35] are used to mitigate token delays. This work further extends the alignment restricted transducer [24] with task-dependent right buffer restriction to control the token emission latency for different use-cases. Methodology In this section, we outline the three proposed techniques for FlexiT. Note that we focus on demonstrating these techniques on an Emformer [11], but the techniques could be extended to other encoder architectures that support dynamic segment size selection. Domain Specific Segments in Emformer Encoder FlexiT is shown in Figure 1, with notations similar to those in [11]. Input to the Emformer layer concatenates a sequence of audio features into segments C n i , . . . C n I−1 , where i is the index of a segment and n the layer's index. The corresponding left and right contextual blocks L n i and R n i are concatenated together with C n i to form contextual segment X n i = [L n i , C n i , R n i ]. In FlexiT, we do not use the memory vector in the original Emformer layers. We propose to dynamically alter contextual block L n i , C n i by selecting the number of segments I dynamically. We explore both random context segment selection, as well domain-dependent context segment selection during training. During training, we ensure every input batch consists of utterances from the same domain. Let Dj denote the domain being used for training the current batch. A domain-specific context altering operation is performed such that the vector X n i is modified to [L n i,j , C n i,j , R n i ], conditioned on Dj. More concretely, according to the desired domain specific context size, C n i is split into [C n iLef t , C n iRight ]. We then set L n i,j = [L n i , C n iLef t ], and C n i,j = C n iRight . The domain-specific selection of C n i,j provides flexible latency for decoding and, at the same time, as suggested by our results in Section 5.3, helps to improve the speech recognition model's robustness. Adding Domain Vector in Emformer Encoder To improve the ASR model's capability to learn domainspecific features, we append a domain vector to inputs of each layer of the Emformer encoder. The domain vector is simply represented by using 1-hot representation [30], the value of which depends on whether the training sample comes from V Cmd or Dictation domain. More concretely, as illustrated in Figure 1 let Dvec denote the 1-hot domain vector representation. We concatenate Dvec to all components of X n i , to obtain concatenated input vectors [L n i,j,v , C n i,j,v , R n i,v ]. These are used as inputs to the n th Emformer layer while training. Domain Specific Alignment Restrictions Loss In [24] using pre-computed token level alignment information, configurable thresholds b l , br are used to restrict the alignment paths used for RNN-T loss computation during training. Note that the right-buffer br can be made stricter to ensure earlier token emissions, but stricter br also leads to increased WER. In this work, our goal is to optimize the WERs for the dictation domain while maintaining low latency for the V Cmd domain. Therefore, we propose using domain-specific alignment restriction thresholds while optimizing the loss. We analyze domainspecific WER and token emission latency in Section 5. Experimental Setup Datasets Training Data We run our experiments on data-sets that contains 20K hours of human-transcribed data from 2 different domains. Voice Commands (V Cmd ) dataset combines two sources. The first source is in-house, human transcribed data recorded via mobile devices by 20k crowd-sourced workers. The data is anonymized with personally identifiable information (PII) removed. We distort the collected audio using simulated reverberation and add randomly sampled additive background noise extracted from publicly available videos. The second source came from 1.2 million voice commands (1K hours), sampled from production traffic with PII removed, audio anonymized, and morphed. Speed perturbations [36] are applied to this dataset to create two additional training data at 0.9 and 1.1 times the original speed. We applied distortion and additive noise to the speed perturbed data. From the corpus, we randomly sampled 10K hours. Dictation (open-domain) dataset consists of 13K hours of data sampled from English public videos that are anonymized with PII removed and annotator transcribed. We first apply the same above-mentioned distortions and then randomly sample 10k hours of the resultant data. Evaluation Datasets For evaluation, we use the following datasets, representing two different domains: Voice Commands evaluation set consists of 15K handtranscribed anonymized utterances from volunteer participants as part of an in-house pilot program. Dictation evaluation set consists of 66K hand-transcribed anonymized utterances from vendor collected data where speakers were asked to record unscripted open domain dictation or voice conversations. Evaluation Metrics To measure the model's performance and analyze trade-offs, we track the following metrics: Accuracy: We use word-error-rate (WER) to measure model accuracy on evaluation sets. Note that we measure the WERs for dictation domain without end-pointer and the V Cmd domain with end-pointer. We also keep track of the deletion errors (DEL), which are proportional to early cutoffs in V Cmd . Latency: We measure model latency on V Cmd domain using following metrics: 1. Token Finalization Delay (FD): as defined in [24] is the audio duration between the time when user finished speaking the ASR token, and the time when the ASR token was surfaced as part of 1-best partial hypothesis, also referred as emission latency in [26], or user-perceived latency in [37]. 2. Endpointing Latency (L): [24,38] is defined as the audio time difference between the time end-pointer makes endpointing decision and the time user stops speaking. We track the Average Token Finalization Delay and Average Endpointing latency (LAvg) metrics. In all experiments, we use a fixed neural end-pointer (NEP) [38], running in parallel to ASR being evaluated every 60ms to measure V Cmd domain metrics. A detailed study with other end-pointing techniques besides NEP (static, E2E [39]) is beyond the scope of this paper. ASR Compute: On device power consumption / battery usage is usually well co-related with the amount of compute being used by the ASR model. We use the real time factor (RTF) measured on a real android device as an indirect indicator of the of the model's compute usage. Experiments We use the RNN-T model with Emformer encoder [40], LSTM with layer norm as predictor, and a joiner with 45M total model parameters. As inputs, we use 80-dim log Mel filter bank features at a 10 ms frame rate. We also apply SpecAugment [28] without time warping to stabilize the training. We use a stride of 6 and stack 6 continuous vectors to form a 480 dim vector projected to a 512 dim vector using a linear layer. The model has 10 Emformer layers, each with eight self-attention heads and 512 dimension output. The inner-layer has a 2048 dimension FFN with a dropout of 0.1. We use Alignment Restricted RNN-T loss [24] using a fixed left-buffer b l of 300ms, while varying right-buffer br parameter. All models are trained for 45 epochs using a tri-stage LR scheduler with ADAM optimizer and base LR of 0.005. We study the best way to integrate domain information into FlexiT by experimenting with how domain-conditioned Ar-RNN-T buffer sizes, Emformer context sizes (Emf Ctx), the domain one-hot vector input, and their combined interactions work to improve the model's performance in V Cmd and dictation domains. 1. Fixed Emformer Context: We first fix the Emf Ctx per experiment and analyze how Ar-RNN-T right buffer size r b and Domain Vector impact model performances. (a) Fixed Ar-RNN-T without Domain Vector: As baselines (B1-B3), we train models using 120, 300 and 600ms Emf Ctx. We sweep the range of Ar-RNN-T br with (120, 300, 420) ms, (300, 600, 900) ms, (600, 900, 1200) ms respectively. (b) Domain Ar-RNN-T without Domain Vector: In this experiment, we study if domain specific Ar-RNN-T br sizes help to improve the WER-latency trade-off. We train models C2,C3 similar to B2,B3 but also adding domain specific Ar-RNN-T br of (120, 600) while for B3 we use br of (120, 900) for voice commands / dictation domains. (c) Fixed Ar-RNN-T with Domain Vector: We also train models (D1 − D3) where we concatenate domain vector to Emformer layer inputs. The models are trained using 120, 300 and 600ms Emf Ctx and Ar-RNN-T br of 420ms, 600ms and 900ms respectively. (d) Domain Ar-RNN-T with Domain Vector: To analyze if domain vector enables the model to learn to emit tokens with different latency, we also train a models E2,E3 using similar configuration as D2,D3 but also adding domain specific Ar-RNN-T threshold br of (120, 600) for E2 (120, 900) for E3. Random / Domain Specific Emformer Context: We analyze randomly selected Emf Ctx during training as described in 3.1. Context size is randomly selected from 120ms to 1200ms. As shown later in Section 5 we find domain specific Ar-RNN-T br of 420, 900ms achieves best results. Therefore, to reduce the number of combinations in this experiment, we fix br of 420, 900ms and run experiments R1 and R2 without and with use of Domain vector respectively. Lastly, we analyze using domain specific Emf Ctx during training using domain-specific Ar-RNN-T br of 420, 900ms. Corresponding experiments S1 and S2 without and with Domain vector respectively. Inference: We always evaluate models such that for each domain, the training time Emf Ctx matches the context provided to the encoder while doing inference. Only exception being Random Emformer Context experiments R1 and R2 where we use inference context size of 120ms, 600ms for V Cmd and dictation domains respectively. Figure 2 demonstrates the trade-offs as we vary Emformer context and the Ar-RNN-T right-buffer br. Dictation WER improves as we increase the Emformer context in experiments B1-B3. Similarly, for a fixed Emformer context, WERs improve with larger Ar-RNN-T br parameter. Note that only handpicked variants are detailed in Table 1. On the other hand, V Cmd domain's Lavg also increases. To achieve low LAvg throughout experiments, which is important for a better user experience, we use a fixed NEP. Delays in token emissions result in more early cuts and increased deletion errors as shown in Table 1,2. To achieve best latency and simultaneously reduce early cuts, we must maintain a smaller Emformer context and maintain a strict Ar-RNN-T br parameter. Therefore, experiments for random and domain-specific Emformer context are performed with br 420 and context 120. We observe that providing domain vector in encoder improves the WERs for dictation domain in general for all experiments as shown in Table 2, which is consistent with previous works [10,30]. Unfortunately, the WER improvements come in tandem with an increased average FD of V Cmd when the models are trained with domain vector (comparing experiments B1-B3 and D1-D3). We hypothesize that this is because in the absence of stricter AR-RNNT r b restrictions for the V Cmd domain, the model learns to delay token emissions to improve accuracy. Results and Analysis Emformer Context and Ar-RNN-T Thresholds Domain Vector and Domain specific Ar-RNN-T helps achieve fine grained control on token delays [24]. However, in multi-domain setting, comparing C2,C3 to B2,B3, we observe that simply imposing domain specific Ar-RNN-T thresholds does not improve V Cmd FD. The use of domain vector, alongside domain specific Ar-RNN-T thresholds, enables us to achieve a more refined control over V Cmd domain's FD. This is demonstrated in Table 2 where D1-D3 have larger Avg(FD) compared to B1-B3, but models E2-E3 learn to explicitly emit V Cmd domain tokens earlier than C2-C3. Random / Domain Specific Emformer Context Results from random context training, R1 suggests that in the absence of domain information, the model achieves worse trade-offs than V Cmd FD of B1 and Dictation WER of B3. In R2 adding domain vector, improves the WER for dictation domain significantly which is consistent with Section 5.2. Overall, R2 achieves better trade-offs for optimizing both use-cases. Experiment S1, which combines domain-specific Emf Ctx and uses domain-specific Ar-RNN-T achieves dictation WERs comparable to D3 while enabling reasonable FD for V Cmd domain, which improves on the results of experiment R1 with random Emf Ctx. Therefore, we argue that domain-specific Emf Ctx, which enables dynamic attention masking per domain, already helps the model learn more robust domain-specific features, even in the absence of a domain vector. Finally, similar to Section 5.2 further addition of domain vector in experiment S2 also enables the model to achieve better FD, thus improving the deletion errors from model S1. This achieves the best tradeoffs in dictation domain WER, and V Cmd domain FD and WER with endpointing enabled. In Figure 3 we analyze the RTF of FlexiT models. For R2 we evaluate the RTF and WERs while varying inference context size. We observe that the RTF reduces with larger context size, which is mainly because of improved batching inside Emformer layers across time dimension. Better RTFs typically are co-related with better compute and power consumption. Conclusion This paper proposed a single Flexi-Transducer (FlexiT) model that supports domain-dependent trade-off of latency and accuracy. The domain-specific or random context modeling is achieved jointly via segment size altering operation for encoder and the domain vector. Ar-RNN-T loss imposes a domainspecific constraint to limit the token emission latency for different domains. Using the combination of techniques, we achieve better WER, RTF and latency trade-offs when a single model supports multiple streaming ASR use-cases. Figure 1 : 1Illustration for domain-specific context size and the injection of domain vectors in an Emformer layer Figure 2 : 2Dictation WERs and V Cmd token finalization delay tradeoffs for various experiments. The labels in the plot show the Ar-RNN-T br used for the experiment. Increasing br as well as Emformer context improves WER, but degrades latency. Methods (R2, S2) achieve best trade-offs. Figure 3 : 3Dictation WERs and Real Time Factor measured on an android device. The labels show chunk size (ms) used during inference, S2 was evaluated with 600ms chunk size. Table 2 : 2FixedEmformer Context with Domain Vector: With Domain vector D1-D3 achieve better dictation WERs com- pared to B1-B3. Further, with Domain specific Ar-RNN-T, E2,E3 achieve better latency compared to D2,D3. Emf CtxDvec Dict WER V Cmd WER V Cmd DEL Avg FD LAvg R1 Random No 13.6 7.2 3.1 173 440 R2 Yes 12.7 7.1 3.1 167 440 S1 120/600 No 12.5 7.7 3.3 185 458 S2 Yes 12.6 7.0 2.9 157 450 Table 3 : 3Random and Domain Specific Emformer Context: Experiments use 420 / 900 Ar-RNN-T r b parameters and 120 / 600 EmCtx during inference. 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[ "Ω-BOUNDS FOR THE PARTIAL SUMS OF SOME MODIFIED DIRICHLET CHARACTERS", "Ω-BOUNDS FOR THE PARTIAL SUMS OF SOME MODIFIED DIRICHLET CHARACTERS" ]	[ "Marco Aymone " ]	[]	[]	We consider the problem of Ω bounds for the partial sums of a modified character, i.e., a completely multiplicative function f such that f (p) = χ(p) for all but a finite number of primes p, where χ is a primitive Dirichlet character. We prove that in some special circum-where S is the set of primes p where f (p) = χ(p). This gives credence to a corrected version of a conjecture of Klurman et al., Trans. Amer. Math.Soc., 374 (11), 2021, 7967-7990. We also compute the Riesz mean of order k for large k of a modified character, and show that the Diophantine properties of the irrational numbers of the form log p/ log q, for primes p and q, give information on these averages.	null	[ "https://export.arxiv.org/pdf/2210.06153v2.pdf" ]	252,846,727	2210.06153	3254bc30bbcb1d9e90aa076e2fba05507c562938	Ω-BOUNDS FOR THE PARTIAL SUMS OF SOME MODIFIED DIRICHLET CHARACTERS Marco Aymone Ω-BOUNDS FOR THE PARTIAL SUMS OF SOME MODIFIED DIRICHLET CHARACTERS We consider the problem of Ω bounds for the partial sums of a modified character, i.e., a completely multiplicative function f such that f (p) = χ(p) for all but a finite number of primes p, where χ is a primitive Dirichlet character. We prove that in some special circum-where S is the set of primes p where f (p) = χ(p). This gives credence to a corrected version of a conjecture of Klurman et al., Trans. Amer. Math.Soc., 374 (11), 2021, 7967-7990. We also compute the Riesz mean of order k for large k of a modified character, and show that the Diophantine properties of the irrational numbers of the form log p/ log q, for primes p and q, give information on these averages. Going further, we define: Definition 1.1 (Modified characters). We say that f : N → {z ∈ C : \|z\| ≤ 1} is a modified character if f is completely multiplicative, and if there is a Dirichlet character χ and a finite subset of primes S such that: (1) For all primes p ∈ S, f (p) = χ(p). (2) For all primes p / ∈ S, f (p) = χ(p). In this case we also say that f is a modification of χ with modification set S. One can easily show (see [1] for a proof using a Tauberian result) that for a modified character we have (1) n≤x f (n) (log x) \|S\| . A very open question concerns Ω bounds for the partial sums of such modifications. The first treatment to such an Ω bound is due to Borwein-Choi-Coons [2] where they considered the case \|S\| = 1 and showed that for a quadratic Dirichlet character χ with prime modulus q, if we set f (p) = χ(p) for all primes except at q, where f (q) = 1, then the partial sums n≤x f (n) = Ω(log x). Later, Klurman et al. [6], among other results, proved a stronger form of Chudakov's conjecture [5], and also conjectured that the partial sums of a modified character f are actually Ω((log x) \|S\| ). Their Corollary 1.6 states that the partial sums are Ω(log x) if the set of modifications S is finite and for at least one prime p ∈ S, \|f (p)\| = 1. Their proof is correct when the Dirichlet character is primitive, but this statement is not true for all characters. This is because, if we modify a non-primitive character χ at a prime r where χ(r) = 0, then there are circumstances where the modification f might end up in another character with smaller modulus. Therefore, we restate their conjecture in the following form. The first result of this paper gives credence to this conjecture by showing that it is true in some very special circumstances. The result above is a direct consequence of the slightly more general Theorem below. Theorem 1.1. Let f be a modification of a primitive Dirichlet character χ such that for each prime p in the set of modifications S, \|f (p)\| = 1. Let T = p∈S f (p)=1 1 − p∈S χ(p)=1 1. Let N =    max{0, T }, if χ(−1) = −1, max{0, T − 1}, if χ(−1) = 1. Then n≤x f (n) = Ω((log x) N ). One of the reasons behind the result above is that the Dirichlet series of f , say F (s), can be written as (2) F (s) = p∈S 1 − χ(p) p s 1 − f (p) p s L(s, χ), and the proof consists in analyzing the behaviour of F (s) as s → 0 + , and hence, the functional equation for L(s, χ) is important here. 1.1. Riesz means and irrationality of the numbers log p/ log q. Another way to measure the mean behaviour of a sequence (f (n)) n is by its Riesz means. The Riesz mean of order 1 is defined as R 1 (x) := n≤x f (n) log(x/n). After partial summation it is not difficulty to see that R 1 (x) is equal to log x times the logarithmic average of n≤x f (n): R 1 (x) = log x × 1 log x x 1 n≤t f (n) dt t . Therefore, it may regarded as a smooth average of f (n). Going further, we can define (see the book of Montgomery and Vaughan [7]) the Riesz mean of order k: R k (x) := 1 k! n≤x f (n)(log(x/n)) k . It is not difficult to see that Ω results for R k (x) transfer to the classical partial sums, as upper bounds for the classical partial sums transfer to R k (x). In our next result we were able to compute with precision the Riesz mean of order k of a modified character, which allows us to say that the exponent N in Theorem 1.1 is optimal in the Riesz mean context. M =    T, if χ(−1) = −1, T − 1, if χ(−1) = 1. Then there exists a constant γ > 0 such that for all integers k ≥ 10 + γ(\|S\| + 1) max p∈S (log p) 2 , as x → ∞ n≤x f (n)(log x/n) k =    P (log x) + O(1), if M + k ≥ 1, O(1), if M + k ≤ 0, where in the case that M + k ≥ 1, P (x) is a polynomial of degree M + k with leading coefficient a M +k given by a M +k = c χ k! (M + k)!     p∈S f (p)=1 1 − χ(p) log p         p∈S χ(p)=1 log p 1 − f (p)         p∈S f (p), χ(p) =1 1 − χ(p) 1 − f (p)     , where c χ =    L(0, χ), if χ(−1) = −1, L (0, χ), if χ(−1) = 1. It is not difficult to see that if n≤x a n ∼ (log x) N , then the Riesz mean of order k of this sequence is asymptotically equal to a polynomial P (log x) of degree N +k. Therefore, the result above says that Theorem 1.1 is optimal in the Riesz mean context. From an analytic point of view, the Riesz mean behaves more or less as the Césaro mean, and in the context of modified characters, Duke and Nguyen [4] considered the content of Theorem 1.2 in the case that \|S\| = 1. Their results allowed them to state that there exists a ±1 completely multiplicative function with bounded Césaro "discrepancy", in contrast with the infinite discrepancy of Tao [8]. Our result above treats the case \|S\| > 1. The new problem that didn't appear in the case \|S\| = 1, is that each Euler factor in (2) produces a periodic sequence of simple poles at the line Re(s) = 0. The contribution of these simple poles to a Perron integral of the corresponding Riesz mean is roughly at most q∈S ∞ n=1 1 n k p∈S\{q} 1 − exp 2πin log p log q −1 . The convergence of this sum is, therefore, connected to the problem of how well the numbers log p/ log q can be approximated by rational numbers. A parameter of irrationality of an irrational number α is the exponent µ(α) defined as the infimum over the real numbers η such that the inequality a b − α ≤ C(η) b η admits only a finite number of rational solutions a/b with b ≥ 1, where C(η) is a constant that depends only on η and the irrational number α. A classical result of Dirichlet is that µ(α) ≥ 2 for all irrational numbers α, and this inequality is optimal for almost all irrational numbers, with respect to the Lebesgue measure. In our case, the irrationality of the numbers log p/ log q can be treated by Baker's theory of linear forms in logarithms, and we actually have by Theorem 1.1 of Bugeaud [3] (see also the references therein) the following Lemma. Lemma 1.1. Let p and q be distinct prime numbers. Then there exists a constant γ > 0 that does not depend on p and q such that, for µ = γ(log p)(log q), there is a constant C = C(p, q) such that the inequality a b − log p log q < C b µ+1 is satisfied only for a finite number of rational numbers a/b, with b ≥ 1. 1.2. Structure of the paper. In some instances we assume that the reader is familiar with tools from Analytic Number Theory. In section 2 we state the four main notations used in this paper, then we quickly proceed with proofs in section 3. We then conclude with some simulations at the end. Notation We use the standard notation (1) f (x) g(x) or equivalently f (x) = O(g(x)); (2) f (x) = o(g(x)); (3) f (x) = Ω(g(x)); (4) f (x) ∼ g(x). 1) is used whenever there exists a constant C > 0 such that \|f (x)\| ≤ C\|g(x)\|, for all x in a set of numbers. This set of numbers when not specified is the real interval [L, ∞], for some L > 0, but also there are instances where this set can accumulate at the right or at the left of a given real number, or at complex number. Sometimes we also employ the notation or O to indicate that the implied constant may depends in . In case 2), we mean that lim x f (x)/g(x) = 0. When not specified, this limit is as x → ∞ but also can be as x approaches any complex number in a specific direction. In case 3), we say that lim sup x \|f (x)\|/\|g(x)\| > 0, where the limit can be taken as in the case 2). In the last case 4), we mean that f (x) = (1 + o(1))g(x). Now observe that for the primes p ∈ S such that f (p) = +1, we have that χ(p) = 1, and hence, the Euler factor corresponding to this prime in the Euler product above contributes with a simple pole at s = 0. Similarly, for p ∈ S with χ(p) = 1, we have that f (p) = 1, and hence, the Euler factor corresponding to this prime in the Euler product above contributes with simple zero at s = 0. In the other cases, the Euler factor is a regular and non-vanishing function at s = 0. Therefore, as s → 0, we have that p∈S 1 − χ(p) p s 1 − f (p) p s 1 \|s\| T , where T is as in Theorem 1.1. Therefore, as s → 0, we have that \|F (s)\| 1 \|s\| T in the situation that χ(−1) = −1, and \|F (s)\| 1 \|s\| T −1 in the situation that χ(−1) = +1. Now, in order to complete the proof we will require the following Lemma: Lemma 3.1. Let α ≥ 0 and Γ the classical Gamma function. Then, for all σ > 0: I(σ, α) := ∞ 1 (log x) α x 1+σ = Γ(α + 1) σ 1+α . We continue with the proof of Theorem 1.1 and postpone the proof of the Lemma above to the end. We recall that F (s) can be written as the following Mellin transform of n≤x f (n) valid for Re(s) > 1: F (s) = s ∞ 1 n≤x f (n) dx x 1+s . By the bound (1), this formula actually holds for all Re(s) > 0. Therefore Now we make another substitution: v = σu. This leads to \|F (σ)\| σ ≤ ∞ 1 n≤x f (n) dx x 1+σ . Now, if χ(−1) = −1, we have that 1 σ N +1 ∞ 1 n≤x f (n) dx x 1+σ ,andI(σ, α) = 1 σ 1+α ∞ 0 v α e v dv, and this completes the proof. Proof of Theorem 1.2. Proof of Theorem 1.2. We have that (see the book [7], pg. 143) n≤x f (n)(log x/n) k = k! 2πi 2+i∞ 2−i∞ F (s)x s s k+1 ds. Now, for X > 0 2+i∞ 2−i∞ F (s)x s s k+1 ds = 2+iX 2−iX F (s)x s s k+1 ds + O(x 2 /X k ). Let R X be the rectangle with vertices 2 − iX, 2 + iX, −1 + iX and −1 − iX. By formula (2), we see that F has a meromorphic continuation to C with poles only at the line Re(s) = 0. Actually, recalling the definition of M in Theorem 1.2, as we will show below, F has a pole of order max{0, M } at s = 0 (by a pole of order 0 we mean without a pole), and another simple pole at this line. For a fixed p ∈ S, we have a periodic sequence of simple poles related to this prime whenever for real t 1 − f (p)p −it = 0. Since each f (p) is a root of unity by assumption, there are coprime positive integers 1 ≤ a p ≤ b p with f (p) = exp(2πia p /b p ). Hence, the simple poles attached to p have the form it p (n) := 2πi log p (n + a p /b p ) , n ∈ Z. By the Cauchy residue Theorem, 1/2πi times the integral of k!F (s)x s /s k+1 along ∂R X , provided that F (s) is regular at s = ±iX, is equal to Res k!F (s)x s s k+1 , s = 0 + p∈S −X≤tp(n)≤X tp(n) =0 Res k!F (s)x s s k+1 , s = it p (n) . Now we will compute the first residue at 0 above. Observe that 1 − p −s ∼ s log p as s → 0. Let a M +k be as in Theorem 1.2. By the Euler product formula (2) for F (s), we have as s → 0 k!F (s) ∼ k!L(s, χ)     p∈S f (p)=1 1 − χ(p) s log p         p∈S χ(p)=1 s log p 1 − f (p)         p∈S f (p), χ(p) =1 1 − χ(p) 1 − f (p)     ∼ (M + k)!a M +k L(s, χ) c χ s T , where T and c χ are as in Theorem 1.2. When χ(−1) = −1, L(s, χ) ∼ L(0, χ), and when χ(−1) = 1, L(s, χ) ∼ sL (0, χ). Therefore, as s → 0 we have F (s) ∼ a M +k (M + k)! s M . Finally, when M + k ≥ 1, we have that Res k!F (s)x s s k+1 , s = 0 is the claimed polynomial P (log x) with degree M + k and with leading coefficient a M +k , and when M + k ≤ 0 this residue is O(1). Now we are going to show that the contribution of the simple poles at Re(s) = 0 is at most O(1) as x → ∞ in the claimed range of k. By formula (2), for a fixed prime p ∈ S −X≤tp(n)≤X tp(n) =0 Res F (s)x s s k+1 , s = it p (n) p n∈Z\{0} 1 \|n\| k+1 \|L (it p (n), χ)\| q∈S q =p 1 − f (q)q −itp(n) −1 . Now we claim that the product inside the sum above is, except for a finite number of n, n (γ max p∈S (log p) 2 +1)(\|S\|+1) , for some constant γ > 0. Proof of the claim. Observe that: 1 − f (q)q −itp(n) = 1 − exp 2πia q /b q − 2πi log q log p (n + a p /b p ) = 1 − exp 2πi (n + a p /b p ) a q b q (n + a p /b p ) − log q log p := 1 − exp(2πiα p,q (n)). Let α p,q (n) be the distance from α p,q (n) to the nearest integer. Then the function 1 − exp(2πiα p,q (n)) α p,q (n) is bounded away from 0, and hence \|1 − exp(2πiα p,q (n))\| −1 1 α p,q (n) . Now, by writing α p,q (n) = l p,q (n) + p,q (n), where l p,q (n) is an integer and −1/2 ≤ p,q (n) ≤ 1/2, we see that α p,q (n) = \| p,q (n)\| = (n + a p /b p ) a q b q (n + a p /b p ) − log q log p − l p,q (n) = (n + a p /b p ) a q b q − l p,q (n) 1 (n + a p /b p ) − log q log p p,q 1 \|n\| µ , where the last bound holds for all but a finite number of integers n, due to Lemma 1.1, and this proves the claim. Before we continue, we recall estimates for L(s, χ) in the t-aspect. The functional equation and estimates for the Γ function give that for real t → ∞ (see, for example, Corollary 10.10, pg. 334 of [7]) \|L(it, χ)\| χ √ t\|L(1 − it,χ)\|. On the other hand, for non-principal χ (see Lemma 10.15 of [7], pg. 350), as t → ∞, we have that L(1 ± it, χ) χ log(\|t\| + 1). Now, going back to the contribution of the simple poles, these last estimates give that the series n∈Z\{0} 1 \|n\| k+1 \|L (it p (n), χ)\| q∈S q =p 1 − f (q)q −itp(n) −1 is absolutely convergent in the claimed range of k, and so, this contribution is at most O(1). To complete the proof, by combining the classical estimates for the Gamma function with (2) and the functional equation (3), the integral −1+iX −1−iX F (s)x s s k+1 ds −1+iX −1−iX \|L(s, χ)\|x −1 \|s\| k+1 ds 1/x. Further, since the number of simple poles of F (s) up to height X is X, there is a fixed δ > 0 and an infinite number of points X j → ∞ such that the distance between ±iX j and each one of these simple poles is at least δ/2. On the other hand, in the Euler product representation of F (it), we have that each Euler factor corresponding to a prime p ∈ S is \|1 − f (p)p −it \| −1 , and this last function is periodic as a function of t, and except at the poles, it is a continuous function. Therefore, each of this Euler factors at the points ±iX j are O δ (1) provided that ±iX j are δ/2-distant from the poles of (1−f (p)p −it ) −1 . Hence, along the lines I ± = [−1±iX j , 2±iX j ], we have that I ± F (s)x s s k+1 ds x 2 X j . Then we obtain the claimed result by making X j → ∞. Some simulations Here we make some simulations with the non-principal Dirichlet character χ with modulus 3, i.e., χ(n) =          1, if n ≡ 1 mod 3, −1, if n ≡ 2 mod 3, 0, otherwise. For the first primes we have: In Figure 1, we consider four modifications where we turn the values χ(p) = −1 to +1, for four primes p. In this case, if N is as in Theorem 1.1, we have that N = 4. In Figure 2 we consider five modifications, but N = 3, and in Figure 3 we also consider four modifications, but N = 0. Acknowledgements. I am warmly thankful to Carlos Gustavo Moreira (Gugu) for a fruitful discussion on Diophantine Theory and for pointing out the reference of Bugeaud [3], and for Oleksiy Klurman for his comments on a draft version of this paper. This project was supported by CNPq grant Universal no. 403037/2021-2. The revision of this paper was made while I Conjecture 1. 1 . 1Let f be a modification of a primitive Dirichlet character χ. Assume that for each prime p in the set of modifications S we have \|f (p)\| = 1. Then n≤x f (n) = Ω((log x) \|S\| ). Corollary 1. 1 . 1Let f be a modification of a primitive Dirichlet character χ such that χ(−1) = −1. If for each prime p in the set of modifications S we have f (p) = +1, then n≤x f (n) = Ω((log x) \|S\| ). Theorem 1. 2 . 2Let f be a modification of a primitive Dirichlet character χ such that for each prime p in the set of modifications S, f (p) is a root of unity. Let T be as in Theorem 1.1 and . 1 .F 1Let f , χ and S be as in the statement of Theorem 1.1. Then, the Dirichlet series of f can be written as follows: ( Now, recall that we define N = max{0, T } if χ(−1) = −1, otherwise, if χ(−1) = 1, N = max{0, T − 1}. If N = 0, then there is nothing to prove since the statement of Theorem 1.1 becomes trivial. We therefore assume N ≥ 1. We begin by recalling the functional equation for L(s, χ) (see for instance the book [\|ε(χ)\| = 1, Γ is the classical gamma function and κ = 0 if χ(−1) = 1, κ = 1 ifχ(−1) = −1.By this functional equation, and the fact that L(1, χ) = 0 for all χ, we immediately see that L(0, χ) = 0 in the situation that χ(−1) = −1, and, otherwise, has a simple zero coming from the sine function at s = 0 in the situation that χ(−1) = 1. hence, by Lemma 3.1, the partial sums n≤x f (n) cannot be o((log x) N ), otherwise the integral would be o(1/σ N +1 ). Similarly, we obtain same conclusions in the case that χ(−1) = +1, and this completes the proof of the Theorem. Proof of Lemma 3.1. We begin by making the change u = log x. Figure 1 . 1A case where N = 4. In blue we plot the partial sums of f , where f is the modification of χ such that f (2) = f (3) = f (5) = f (11) = +1. Figure 2 . 2A case where N = 3 with five modifications. In blue we plot the partial sums of f , where f is the modification of χ such that f (2) = f (3) = f (5) = f (11) = +1 and f (7) = −1. Figure 3 . 3A case where N = 0 with four modifications. In blue we plot the partial sums of f , where f is the modification of χ such that f (3) = f (7) = f (13) = f (19) = −1. A note on multiplicative functions resembling the Möbius function. M Aymone, J. Number Theory. 212M. Aymone, A note on multiplicative functions resembling the Möbius function, J. Number Theory, 212 (2020), pp. 113-121. P Borwein, S K K Choi, M Coons, Completely multiplicative functions taking values in {−1, 1}. 362P. Borwein, S. K. K. Choi, and M. Coons, Completely multiplicative functions taking values in {−1, 1}, Trans. Amer. Math. Soc., 362 (2010), pp. 6279-6291. Effective irrationality measures for quotients of logarithms of rational numbers. Y Bugeaud, Hardy-Ramanujan J. 38Y. Bugeaud, Effective irrationality measures for quotients of logarithms of rational numbers, Hardy- Ramanujan J., 38 (2015), pp. 45-48. Riesz means of certain arithmetic functions. W Duke, H N Nguyen, J. Number Theory. 210W. Duke and H. N. Nguyen, Riesz means of certain arithmetic functions, J. Number Theory, 210 (2020), pp. 132-141. Rigidity theorems for multiplicative functions. O Klurman, A P Mangerel, Math. Ann. 372O. Klurman and A. P. Mangerel, Rigidity theorems for multiplicative functions, Math. Ann., 372 (2018), pp. 651-697. Multiplicative functions that are close to their mean. O Klurman, A P Mangerel, C Pohoata, J Teräväinen, Trans. Amer. Math. Soc. 374O. Klurman, A. P. Mangerel, C. Pohoata, and J. Teräväinen, Multiplicative functions that are close to their mean, Trans. Amer. Math. Soc., 374 (2021), pp. 7967-7990. H L Montgomery, R C Vaughan, Multiplicative number theory. I. Classical theory. CambridgeCambridge University Press97of Cambridge Studies in Advanced MathematicsH. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I. Classical theory, vol. 97 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2007. The Erdős discrepancy problem. T Tao, CEP 31270-901Av. Antônio Carlos. 6627129. Departamento de Matemática, Universidade Federal de Minas GeraisDiscrete Anal.. Email address: [email protected]. Tao, The Erdős discrepancy problem, Discrete Anal., (2016), pp. Paper No. 1, 29. Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, CEP 31270-901, Belo Horizonte, MG, Brazil. Email address: [email protected]	[]
[ "Super-Fermi Acceleration in Multiscale MHD Reconnection", "Super-Fermi Acceleration in Multiscale MHD Reconnection" ]	[ "Stephen Majeski \nDepartment of Astrophysical Sciences\nPrinceton University\nPeyton Hall, 4 Ivy Lane08544PrincetonNJ\n", "Hantao Ji \nDepartment of Astrophysical Sciences\nPrinceton University\nPeyton Hall, 4 Ivy Lane08544PrincetonNJ\n" ]	[ "Department of Astrophysical Sciences\nPrinceton University\nPeyton Hall, 4 Ivy Lane08544PrincetonNJ", "Department of Astrophysical Sciences\nPrinceton University\nPeyton Hall, 4 Ivy Lane08544PrincetonNJ" ]	[]	We investigate the Fermi acceleration of charged particles in 2D MHD anti-parallel plasmoid reconnection, finding a drastic enhancement in energization rateε over a standard Fermi model ofε ∼ ε. The shrinking particle orbit width around a magnetic island due to E × B drift produces aε ∼ ε 1+1/2χ power law with χ ∼ 0.75. The increase in the maximum possible energy gain of a particle within a plasmoid due to the enhanced efficiency increases with the plasmoid size, and is by multiple factors of 10 in the case of solar flares and much more for larger plasmas. Including effects of the non-constant E × B drift rates leads to further variation of power law indices from > ∼ 2 to < ∼ 1, decreasing with plasmoid size at the time of injection. The implications for energetic particle spectra are discussed alongside applications to 3D plasmoid reconnection and the effects of a guide field.	10.1063/5.0139276	[ "https://export.arxiv.org/pdf/2210.06533v4.pdf" ]	252,873,154	2210.06533	3dac27c392c3dc95cfb5a15595c4be4ba31fe949	Super-Fermi Acceleration in Multiscale MHD Reconnection Stephen Majeski Department of Astrophysical Sciences Princeton University Peyton Hall, 4 Ivy Lane08544PrincetonNJ Hantao Ji Department of Astrophysical Sciences Princeton University Peyton Hall, 4 Ivy Lane08544PrincetonNJ Super-Fermi Acceleration in Multiscale MHD Reconnection (Dated: 3 April 2023) We investigate the Fermi acceleration of charged particles in 2D MHD anti-parallel plasmoid reconnection, finding a drastic enhancement in energization rateε over a standard Fermi model ofε ∼ ε. The shrinking particle orbit width around a magnetic island due to E × B drift produces aε ∼ ε 1+1/2χ power law with χ ∼ 0.75. The increase in the maximum possible energy gain of a particle within a plasmoid due to the enhanced efficiency increases with the plasmoid size, and is by multiple factors of 10 in the case of solar flares and much more for larger plasmas. Including effects of the non-constant E × B drift rates leads to further variation of power law indices from > ∼ 2 to < ∼ 1, decreasing with plasmoid size at the time of injection. The implications for energetic particle spectra are discussed alongside applications to 3D plasmoid reconnection and the effects of a guide field. I. INTRODUCTION Energy conversion in magnetic reconnection is pivotal to understanding reconnection's role throughout the Universe 1-3 . In solar flares, estimates have found as much as half of electrons being energized to non-thermal energies 4,5 . Moreover, within the solar wind and the earth's magnetotail, electron acceleration and power law energy spectra are often found associated with plasmoids and compressing or merging flux ropes [6][7][8][9][10] . Recent years have seen considerable effort to explain these observations, focusing on three leading mechanisms during reconnection: direct acceleration by reconnection electric field [11][12][13] or by localized instances of magnetic field-aligned electric fields 14 , betatron acceleration due to field compression while conserving particle magnetic moments [15][16][17] , and Fermi acceleration by "kicks" from the motional electric field within islands [18][19][20][21] . Fermi acceleration operates primarily in multiscale, or plasmoid, reconnection which is thought to be pervasive from solar flares to magnetospheric substorms to accretion disks [22][23][24][25] . In these environments, it takes place within the large volume of magnetic islands which pervade plasmoid-unstable current sheets 26 . A unique characteristic of Fermi acceleration which makes it particularly promising for explaining power law distributions, is that the acceleration rate is itself a power law in energy 18 . This has led to simulations finding Fermi-generated power law distributions over a range of Lundquist numbers, Lorentz factors, guide fields, and more 27,28 . Analytical estimates of first-order Fermi acceleration are frequently based off of the seminal work of Drake et al, which found that the particle acceleration rate is linear in the particle energy,ε ∼ ε (in what follows we will refer to acceleration rate power law indices with p, i.e.ε ∼ ε p ) 18 . Note that we are concerned here in a) Electronic mail: [email protected] b) Also at Princeton Plasma Physics Laboratory this work only with first-order Fermi acceleration which should not be confused with less efficient, second-order, or stochastic, Fermi acceleration. Other approaches have described Fermi acceleration in more MHD-like plasmoid mergers via conservation of the bounce invariant J 17,21 . Building off of these concepts, energetic particle spectral indices over a range of values larger than 1 have been explained through a combination of Fermi acceleration, various drifts, and particle-loss processes 20,26 . Efforts have also been made to implement the kinetic physics of Fermi acceleration without resolving small scales 29 . Unfortunately, most analytical particle acceleration studies are developed to explain the results of kinetic simulations which are computationally limited in the scale separation between large MHD magnetic islands and the Larmor radius (ρ L ) of accelerating particles. Yet many astrophysical systems showing promise as a source for energetic particles are deep within the MHD regime 2,30 . Such lack of scale separation leads to difficulty in capturing effects like the conservation of adiabatic invariants, increasing loss rates from magnetic islands through pitch-angle scattering 31,32 . Additionally, for lower energy but still weakly collisional particles, their bounce motion may not be fast enough to assume conservation of J . We therefore propose a new model of Fermi-like acceleration in 2D MHD anti-parallel reconnection, which focuses on systems with large scale separation between thermal particle Larmor radii and plasmoid sizes. With the aid of guiding-center test particle simulations, we find that enhanced particle confinement to compressing magnetic field lines yields an O(1) correction to the linear Fermi power law index p = 1. A. Linear Fermi acceleration Consider a plasmoid embedded in a current sheet undergoing 2D anti-parallel MHD reconnection with electric and magnetic fields E and B, respectively. Away from the x-point, the dominant electric field component is the motional field which drives the "E cross B" drift arXiv:2210.06533v4 [physics.plasm-ph] 30 Mar 2023 u E = c E× B/B 2 , which, along with all other electric field components, is out-of-plane in this setup 33 . If a magnetized particle within a plasmoid is to gain energy, it must experience net motion along this electric field, in this case via guiding center drift. The only drift in this circumstance satisfying this constraint is the curvature drift v C . Note that we've assumed drifts arising from explicit time dependence can be neglected, unlike those resulting from particle motion along gradients inb. This is due to the slow nature of the MHD background compared to the relatively fast motional time derivatives experienced by high energy (and importantly super-Alfvénic) particles. Figure 1 shows the process of Fermi acceleration in such a setup. FIG. 1: Diagram of Fermi acceleration process. Blue lines represent the magnetic field (separatrix dashed), and the curvature drift is given for a positively charged particle. As a magnetized, µ = mv 2 ⊥ /2B conserving particle travels along a field line within the plasmoid (with m the particle mass and v ⊥ the particle velocity perpendicular to B), it enters a narrow region (with respect to the orbit's vertical height h) near the neighboring x-point of thickness ∆ which is defined by a large value of the curvature of the magnetic field. This region is generally somewhat larger than the current sheet thickness δ, but approaches that value with increasing proximity to the xpoint. The magnetic tension in this high curvature region drives the magnetic field to rapidly straighten out, therefore within ∆ the E × B drift velocity is also large. In 2D anti-parallel reconnection, the E × B associated electric field and the curvature drift are aligned, therefore the parallel energy of the particle is increased according tȯ ε = 2q E · v C /m, where ε . = v 2 , and v C = mε qBb × (b · ∇b) ≈ − 2mε ∆qBẑ . (1) Note we have assumed here that the gradient scale of b is approximately ∆/2. The increase in ε gained by the particle during its transit of ∆ is then estimated asε ∆/ √ ε ≈ 4 u E ∆ √ ε , with ∆ representing the average over the narrow layer ∆. We have also used \|u E \| = \|E/B\|, and assumed that v \|u E \| in keeping with Drake et al 18 . This process occurs each time the particle transits the island width w, which takes a time dt w ≈ w/ √ ε , yielding the linear Fermi acceleration rate: dε dt F ≈ 4 u E ∆ ε w .(2) This expression is identical in appearance to that of Drake et al, with key differences in meaning 18 . The assumptions under which this equation was derived are MHD without a guide field, not kinetic, meaning no E or Hall magnetic field component is present. Equation (2) has the appearance of being linear in energy, however we will show that during a particle's acceleration u E ∆ and w are not constant, leading to deviation from the linear dependence. II. TEST PARTICLE SIMULATIONS To investigate possible variation of u E ∆ and w in Eq.(2), we performed guiding center simulations of test particles in a plasmoid reconnection scenario. To be precise, we solved the following set of simplified nonrelativistic guiding-center equations 24,34 d R dt = v b + u E (3a) dv dt = q m E + u E · v b + u E · ∇b − µ mb · ∇B,(3b) by an adaptive time step 2nd order-accurate midpoint method 35 , using time-evolving background data from a 2D MHD simulation 36,37 . The code which provided the background fields solves the fully-compressible resistive MHD equations via finite differences with a five point spatial stencil and second-order trapezoidal leapfrog time stepping 38 . These guiding center equations have been simplified assuming that the time dependent drifts are weak due to the slow nature of the MHD background compared to the motional time dependence of super-Alfvénic particles. Out-of-plane motion of the guiding center (but not out-of-plane acceleration) is ignored given the 2D symmetry, and in the MHD simulation data used, E = 0. An example snapshot of u E from the simulation is shown in figure 2. Note that when interpreting the magnitude of u E , the density and magnetic field away from the current sheet in this simulation approach ρ 0 = B 0 = 1 in dimensionless numerical units. The spatial grid size is 2000 (x) by 4000 (y) and time outputs are available at intervals of one-tenth of the primary current sheet Alfvén time (for context, the snapshot fig. 2 shows a zoomed-in portion of the grid which is 1000 by 100 cells). As a result, linear interpolation from the MHD grid to the particle's time and position is used. The background plasma beta is β = 1, with a uniform Lundquist number of S = 10 5 . The adaptive particle time step is calculated as a fraction (CFL number) of the simulation grid cell-crossing time for the particle's velocity, including the E × B drift. In all calculations shown the CFL number is set to 0.1. In the MHD simulation, two plasmoids form, which eventually begin to merge at t = 3.6L/v A , where L is the x-extent of the simulation domain 37 . As a result, we limit our study to pre-merger times in the simulation to avoid the further complication of acceleration at the secondary current sheet. The initial particle velocity is set to v = 20v A for the purpose of ensuring the small ∆ε approximations holds, however no significant difference was noticed in runs where the initial velocity was 10v A or 5v A . How the particles are initially energized relates to the problem of injection, which is a very active area of study but beyond the scope of this work 28,39-41 . The perpendicular velocity of particles was set to v ⊥ = v A and generally plays little role unless v ⊥ ∼ v , which leads to particle trapping at the island edge. An example test particle orbit is shown in Fig.3 with the initial u E field that it experienced, for a total evolution time of t = 0.1L/v A . u E is seen to be limited to a narrow central section which is approximately uniform in width, and peaks in magnitude at the reconnection outflow. In the following, we will refer to the effective plasmoid width w p as the distance between the two maxima of u E . The particle orbit shows a steady decrease in width w, also visible in the plot of ε versus x-position. Conversely, there is no comparable change in h (given in Fig.1). This particle was injected with an initial orbit width w 0 = 3w p /4 at t = 2.7L/v A , roughly 0.5L/v A after the plasmoid became nonlinear (which we consider here as the point when the plasmoid's vertical extent h p exceeds the current sheet thickness, i.e. h p > ∼ δ). From this data, we fit a power law to the bounce average oḟ ε / u E ∆ , finding an exponent of 1.77, rather than the predicted value of 1 from linear Fermi acceleration. Alternatively, fitting a power law toε w/ u E ∆ yields a power law index of 1.08. This suggests that the nonconstancy of w may account for the disagreement with = 2.7L/v A , overlaid on initial magnitude of u E . Total time of integration is t = 0.1L/v A . the linear Fermi prediction. In section II A, we will therefore attempt to describe the nature of the ε -w relationship to more generally predict the energy dependence oḟ ε / u E ∆ . A. Orbit width correction The electric field which does work on curvature-drifting particles in our linear Fermi acceleration calculation results from the field line motion that compresses plasmoids. Naturally then, as particles gain energy from the Fermi acceleration process, the closed field lines they are bound to shrink in width (also see Fig. 3): dw dt = −2 u E pk .(4) Here u E pk is the peak value (not ∆-averaged) of u E experienced by the particle as it transits both sides of the island (averaged between the left and right). This distinction is due to the fact that generally u E pk occurs in the locations of highest curvature, i.e. at the extreme edges of the island. Therefore the rate at which these extremes contract sets the rate of change of w. This peak value is generally slightly larger than u E ∆ , and we will assume that the ratio u E ∆ / u E pk = χ is approximately constant over the period during which a particle is accelerated. Qualitatively, this is an assumption that as long as the outflow u E remains somewhat laminar, it will maintain a similar functional form as it expands into the plasmoid (this will be checked via the constancy of χ within figure 4). We then substitute for u E ∆ in Eq. (2) allowing for the determination of w(ε ): dw dε = − 1 2χ w ε , w = w 0 ε ε 0 −1/2χ ,(5) leading to an enhanced power law acceleration rate: dε dt SF ≈ 4χ ε 0 w 0 u E pk ε ε 0 1+1/2χ .(6) The subscript "SF" has been added for "Super-Fermi", because the orbit-width correction exclusively leads to stronger energization over the linear expression. Additionally, although χ is assumed to be constant it can vary somewhat due to minute details of the plasmoid structure. We will therefore make use of simulation data to provide a reasonable estimate. For ultra-relativistic particles which have v ≈ c (or γ 1), the super-Fermi acceleration rate is dε dt SF,U R ≈ 2χ ε 0 w 0 u E pk ε ε 0 1+1/χ .(7) where the change ε ≈ γm 0 c 2 is made but all other variables carry the same meaning 24 . The missing factor of 2 is a result of the ultra-relativistic particle velocity remaining approximately constant. In the non-relativistic case, d t ε = 2v d t v , while ultra-relativistically d t ε = d t (pc) ≈ v d t p, with no additional factor of two. Equation (7) indicates that the ultra-relativistic orbit-width power law correction is twice that of the non-relativistic version. To complete equation 6, a suitable estimate for χ is needed. Being the result of a plasmoid's internal structure, its precise value will be unique to each plasmoid, although many plasmoids within a given current sheet may possess similar internal structures given their shared origin. To obtain this estimate, a single test particle was evolved inside a plasmoid for a time of L/v A , and u E ∆ / u E pk was calculated for 627 transits of the acceleration regions. An acceleration region is detected numerically as the time frame during which a particle experiences a ∆ε per time step of at least 25% of the maximum value during the same kick. This yielded a mean χ of 0.75 (p = 1.67 non-relativistically, p = 2.33 ultrarelativistically), which will serve as our fiducial value henceforth. This inferred power law of p = 1.67 agrees with the value of 1.77 from the particle in figure 3 to within 6%. To test Eq. (6) more broadly we performed a survey of simulations to calculate the particle acceleration rate power law with varying injection times and locations, shown in Fig.4. Test particles were injected into both plasmoids, at 3 different initial orbit widths (3w p /4, 2w p /3, w p /2), and 9 different time steps in the MHD simulation (each separated by 0.1L/v A ). A total of 53 orbit-width corrected power law indices were calculated after 0.3L/v A of evolution time for each particle, excluding one particle which reached the center of its respective plasmoid before the end of the simulation. This interval was chosen to maximize the evolution time that a power law could be fit to, while also providing sufficient data points to determine whether p varies significantly with FIG. 4: Power law fits to test particle data for 0.3L/v A of evolution time, with the super-Fermi exponent assuming χ = 0.75. time (as it is limited by the eventual merger of the right and left plasmoids). To remove the effect of the varying u E , power laws are fit toε / u E pk (t, w) to determine the index p, rather than justε . Both plasmoids are similar in size at each time step, therefore their power law indices are counted together, yet they can be distinguished by the color of their data points' markers. The fiducial power law p = 1.67 predicted χ = 0.75 is shown as a black dashed line, while the linear Fermi prediction is shown as a black dotted line. The average measured power law agrees with the fiducial index to within 9% at all times, with a time-averaged p = 1.66. They also demonstrate importantly that there is no net trend in the orbit-width corrected index p with the size of the plasmoid, suggesting that our assumption of constant χ is suitable. We will however show that the implied power law does not remain constant when including the variation in u E pk . Regardless of our choice of fiducial χ, the expected lower limit on possible power law indices is 1.5, which is obeyed reasonably well, with a maximum p of 2 suggesting that χ is generally at least 0.5. The fluctuations seen in our measured power law indices may be the result of weakly time dependent χ, and/or variations in calculated u E pk when removing its dependence fromε numerically. B. Space-and time-varying E × B E × B E × B The effects of variation in u E pk are, unlike the orbitwidth correction, highly dependent on the time of injection through the evolution of the plasmoid structure. Such effects have been studied in turbulence, highlighting the relationship between u E gradients and stochastic Fermi acceleration 42 . Similarly, previous reconnectionfocused work has addressed this issue in the more circular pressure-balanced cores of large plasmoids 17 , however we are concerned with the highly elongated outer region of a plasmoid. Without knowledge of the internal plasmoid structure, we have no analytical means by which to determine the modification to the acceleration rate power law by u E . However, a trend in the behavior is identifiable through a survey of particle injection times when plasmoids possess a variety of sizes/fluxes. Here, we will investigate these effects specifically within the left plasmoid. In Fig.5, particles were injected with FIG. 5: Acceleration rates of test particles injected at various times throughout a plasmoid's life. Initial orbit width is w 0 = 2w p /3 for each time. w 0 = 2w p /3, and all were evolved for at least 0.5L/v A . The trend inε demonstrates that as particles are injected later and later into a plasmoid, the effective power law index of their acceleration decreases. For nearly linear plasmoids the power law index can be larger than 2, while for large nonlinear plasmoids the power law index is able to drop below the linear Fermi rate. This variation occurs due to both the spatial and temporal dependence of u E pk . The evolution of an x-slice of \|u E \| within the reconnection layer for the left plasmoid is shown in Fig.6. 37 Strong negative gradients are visible in the magnitude of u E as one moves inward from the edges of the plasmoid. These spatial gradients relate to ε through Eq.(5), and as a particle drifts inwards the field u E that it experiences decreases, reducing the effective power law of the acceleration. These gradients become more pronounced as the plasmoid grows, further reducing the power law index of acceleration. The modeling of these gradients is a complex problem of nonlinear plasmoid structure, however, qualitatively they may be expected to appear through the conservation of particle flux, or via the buildup of magnetic flux within the plasmoid. While the reconnection outflow expands into the plasmoid from the x-point, the cross sectional area which the flow penetrates increases, causing the flow velocity to decrease. As a plasmoid grows, the area the outflow expands into becomes increasingly large, and therefore the inward gradient becomes more severe. In terms of flux buildup, as a plasmoid grows the magnetic pressure within the island increases and larger values of the mag- netic field's strength push closer to the x-points. Given that u E ∼ 1/B, this creates negative gradients in u E which grow in time as the magnetic flux builds up within the island. Concurrently, the peak value of \|u E \| grows with the size of the plasmoid. However, this is strictly limited to the neighborhood of the outflow. III. DISCUSSION To quantify the difference between linear and Super-Fermi acceleration, Fig.7 shows the ratio of the energy gain possible between the super-and linear Fermi models for a plasmoid of a given size, assuming u E pk is constant for simplicity. Within each model, the total gain in energy ∆ε . = ε /ε \|\|,0 is calculated for a particle which is allowed to drift inward until the island orbit width is w = 100ρ L , where guiding center assumptions may weaken. In the linear Fermi calculation, w is fixed to w p , while for Super-Fermi Eq.(5) is used. The ratio of the total gain between the models Γ . = ∆ε SF /∆ε F is then shown as a function of the plasmoid size, here equivalent to the initial orbit width w 0 . Consider an active region of the solar corona where ρ L,e ∼ 0.1−1 cm, alongside the relevant length scales of a solar flare 43,44 . The length of the current sheet itself is ∼ 10 9 ρ L,e , meaning the limiting "monster" plasmoid size is still w p ≈ 10 7−8 ρ L,e 45,46 . Even for some of the much smaller more populous plasmoids, the Larmor scale separation present may allow Γ > 10, and accordingly a notable increase in energy gain over the linear Fermi model. In fact, given the asymptotic scalings Γ ∼ (w p /ρ L ) 2χ and Γ U R ∼ (w p /ρ L ) χ , numerous more reconnection conditions such as active galactic nuclei disks and magnetars may also support similar increases in the maximum possible energy gain 2 . It should be stressed, however, that these are maximum possible energy enhancements which we consider here. Many of the aforementioned examples with large Larmor scale separations are expected to possess a guide field, which is known to suppress Fermi acceleration 47,48 . Therefore, realistic gains in energy will likely not be as high. Additionally, these plasmoids would in practice have spatially dependent u E , therefore the effective power law index of their acceleration could either be increased (likely for smaller populous plasmoids) or decreased (likely for the few "monster" plasmoids). FIG. 7: Ratio of Super-Fermi to linear Fermi maximum possible total energy gain for a single plasmoid given w p , assuming a particle can only drift inward until roughly w = 100ρ L . Although we only simulate test particles in antiparallel reconnection here, we can at least make some predictions about the manner in which guide fields affect super-Fermi acceleration. In this model, guide fields are likely to play a role through the extension of various lengths out of the reconnection plane. The increase in radius of curvature and thus weakening of the curvature drift is countered exactly by an increase in path length in the acceleration region and hence energization time during a Fermi kick, leading to no change in ∆ε . On the other hand, the path length between Fermi kicks is extended to w → w 1 + (B g /B r ) 2 , modifying the denominator of Eq. (2) accordingly. If B g /B r were roughly constant or varied slowly, the super-Fermi power law becomes p = 1 + 1 + (B g /B r ) 2 /2χ. This, alongside the longer transit time between Fermi kicks, would cause steepening of power laws and weakening of acceleration, mirroring expectations that Fermi acceleration is suppressed as described by Arnold et al 47 and Dahlin et al 48 . Perhaps in most cases however, it may be required to consider B g (w)/B r (w) and integrate Eq.(5) exactly. The modification discussed above also does not consider the inherent change in plasmoid structure which may result from a guide field, such as a change to their pressure balance 49,50 . Any changes to the plasmoid structure will likely carry over to the spatial/temporal dependence of u E , and hence the effective observed power law of accel-eration. While figure 4 exhibits the constancy of χ in these nonlinear plasmoids, it is important to mention that there are circumstances where the constant-χ assumption does not appear to hold, like early on during the linear phase of plasmoid growth or during mergers. In these situations, χ in the left (differential) Eq.(5) will need to be considered more generally as χ(w), and the equation integrated accordingly to yield a different solution on the right. For linear plasmoids, the average curvature rises rapidly as field lines move away from the x-point. This results in a knee-like feature inε with small but steeply rising initial acceleration that rapidly levels off. This also occurs near x-points in nonlinear plasmoids, but only represents a transient compared to the power law phase. Of paramount interest in any study of the acceleration of plasma particles is the energetic distribution such an acceleration mechanism would produce. Without any loss mechanisms and given a sufficiently low energy source, a constant acceleration rate ofε ∼ ε p yields an energetic particle distribution of f (ε) ∼ ε −p . MHD plasmoids are often considered to be the end of the road for energetic particles, suggesting that once trapped the particles have no means for exit. However in a dynamic current sheet such trapping is unlikely to last for the lifetime of a plasmoid including advection from the current sheet. Most plasmoids in a high Lundquist number current sheet will encounter multiple others with which they merge. A particle at the center of one plasmoid will, upon merger with a higher flux plasmoid, no longer be in the center and therefore experience continued acceleration 49 . Furthermore, a realistic 3D flux rope embedded in a current sheet is highly dynamic, with axial instabilities providing a prospect for inter-plasmoid transport of energetic particles 31 . Lastly and most easily accounted for in this model is the fact that plasmoids are finite in the out of plane direction, either due to instability or a finite current sheet. Every ∆ε in our model is accompanied by an axial step in the z direction, which becomes larger as particles gain energy 51 . In fact, for reconnection rates E much less than one, a particle may experience a larger relative change in its axial position than in its energy, meaning that axial transport is competitive with both trapping and energization. The loss of particles out of the ends of plasmoids and the rate of their re-injection would then serve as a cutoff in the particle energy for a single plasmoid, dependent on the reconnection rate. IV. CONCLUSIONS We have proposed that an enhanced Fermi acceleration process exists in 2D multiscale MHD reconnection. Results from analytical theory and test particle simulations suggest that a correction arises from changing magnetic island orbit widths for particles. This yields an acceleration rate power law relationshipε ∼ ε 1.67 on average with the precise index varying somewhat due to island geometry, but generally remaining ∼ 1.5 or larger. We additionally discussed the effects of the temporally-and spatially-varying E × B drift on the effective power law index, revealing a trend from high ( > ∼ 2) to low ( < ∼ 1) power law indices as plasmoids get larger. In particular, this result places importance on the distribution of plasmoids in size and flux when investigating global particle energization in a multiscale current sheet 36,45,49,52 . Further generalization of this model would be most immediate with the development of a detailed understanding of the field and structure of plasmoid interiors as they grow 53 . Evolution of u E introduces a time dependence which would lead to a separable O.D.E. for the energy as a function of time, and hence a way to refine the power law ofε (ε ). With knowledge of the spatial structure of these fields, Eq.(5) can once again be leveraged in directly modifying the power law. This would connect particle distributions to plasmoid distributions, possibly creating a route toward an analytical description of multiscale reconnection energetic particle spectra 36,50 . Additionally, we assume particles are pre-energized by some injection mechanism which is likely beyond the scope of guiding center simulations. 28,[39][40][41] Knowledge of the appropriate injection mechanisms for the current sheet we study would fix ideas about the efficiency of the combined processes of injection and Fermi acceleration, and produce a more complete model of particle acceleration in 2D multiscale MHD reconnection. Lastly, the equations used to evolve particles in this study only include terms up to first order in normalized Larmor radius (ρ L /L). 34 Energization and cross-field transport resulting from higherorder finite Larmor radius effects can be captured by gyrokinetic models when Larmor scale separation is weak for the thermal plasma, 54 or particles have large initial v ⊥ (perhaps resulting from re-acceleration after escaping from another plasmoid and scattering off of an x-point 18 ). Such effects will not be captured by the simplified guiding center system (equation 3). Certain reconnection conditions will require non-trivial adjustments to this model in order to appropriately be described by it. In kinetic plasmas with smaller scale separation between Larmor radii and plasmoid widths, the original model of Drake et al may be more suited as particle motion is not so restricted to magnetic field lines, often due to µ being poorly conserved. The lack of µ conservation leads to both ε and ε ⊥ increasing during energization events 20 . Additionally, if a Hall-effect field is present then the curvature drift will have a projection in the plane of reconnection. Given that the curvature drift is charge dependent, this means that it will be directed out of the plasmoid for either positive or negatively charged particles 18,55,56 , but not neither. This could impede the inward motion of the gyro-center due to the E × B drift, or even entirely reverse it as seen in Fig.2(b) of Drake et al (2006) 18 , yielding an opposite-sign correction to the Fermi rate power law. Whether this results in eventual escape from the island is uncertain however, and likely depends on details of the plasmoid struc-ture and chaotic particle orbits which can occur near the x-points. The data that support the findings of this study are available from the corresponding author upon reasonable request. FIG. 2 : 2Plot of full reconnecting current sheet in the simulation used, at t = 2.9L/v A . \|u E \| = E/B is shown with streamlines of the magnetic field overlaid. The reconnection layer is formed by vertically merging two large initial magnetic islands.37 FIG. 3 : 3Particle orbit path for injection at t FIG. 6 : 6Diagram of the evolution of the left plasmoid's \|u E \| at y=1e-3, with u E pk and w p versus time highlighted below. M. Yamada, R. 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[ "ARTICLE Embedding physics domain knowledge into a Bayesian network enables layer-by-layer process innovation for photovoltaics", "ARTICLE Embedding physics domain knowledge into a Bayesian network enables layer-by-layer process innovation for photovoltaics" ]	[ "Zekun Ren ", "Felipe Oviedo ", "Maung Thway ", "Siyu I P Tian ", "Yue Wang ", "Hansong Xue ", "Jose Dario Perea ", "Mariya Layurova ", "Thomas Heumueller ", "Erik Birgersson ", "Armin G Aberle ", "Christoph J Brabec ", "Rolf Stangl ", "Qianxiao Li ", "Shijing Sun ", "Fen Lin ", "Ian Marius Peters ", "Tonio Buonassisi " ]	[]	[]	Process optimization of photovoltaic devices is a time-intensive, trial-and-error endeavor, which lacks full transparency of the underlying physics and relies on user-imposed constraints that may or may not lead to a global optimum. Herein, we demonstrate that embedding physics domain knowledge into a Bayesian network enables an optimization approach for gallium arsenide (GaAs) solar cells that identifies the root cause(s) of underperformance with layer-by-layer resolution and reveals alternative optimal process windows beyond traditional black-box optimization. Our Bayesian network approach links a key GaAs process variable (growth temperature) to material descriptors (bulk and interface properties, e.g., bulk lifetime, doping, and surface recombination) and device performance parameters (e.g., cell efficiency). For this purpose, we combine a Bayesian inference framework with a neural network surrogate device-physics model that is 100× faster than numerical solvers. With the trained surrogate model and only a small number of experimental samples, our approach reduces significantly the time-consuming intervention and characterization required by the experimentalist. As a demonstration of our method, in only five metal organic chemical vapor depositions, we identify a superior growth temperature profile for the window, bulk, and back surface field layer of a GaAs solar cell, without any secondary measurements, and demonstrate a 6.5% relative AM1.5G efficiency improvement above traditional grid search methods.npj Computational Materials (2020) 6:9 ; https://doi.	10.1038/s41524-020-0277-x	null	198,893,625	1907.10995	20e2abdc8bfc4840c30a1b965407b7534188f3f2	ARTICLE Embedding physics domain knowledge into a Bayesian network enables layer-by-layer process innovation for photovoltaics Zekun Ren Felipe Oviedo Maung Thway Siyu I P Tian Yue Wang Hansong Xue Jose Dario Perea Mariya Layurova Thomas Heumueller Erik Birgersson Armin G Aberle Christoph J Brabec Rolf Stangl Qianxiao Li Shijing Sun Fen Lin Ian Marius Peters Tonio Buonassisi ARTICLE Embedding physics domain knowledge into a Bayesian network enables layer-by-layer process innovation for photovoltaics 10.1038/s41524-020-0277-xOPEN Process optimization of photovoltaic devices is a time-intensive, trial-and-error endeavor, which lacks full transparency of the underlying physics and relies on user-imposed constraints that may or may not lead to a global optimum. Herein, we demonstrate that embedding physics domain knowledge into a Bayesian network enables an optimization approach for gallium arsenide (GaAs) solar cells that identifies the root cause(s) of underperformance with layer-by-layer resolution and reveals alternative optimal process windows beyond traditional black-box optimization. Our Bayesian network approach links a key GaAs process variable (growth temperature) to material descriptors (bulk and interface properties, e.g., bulk lifetime, doping, and surface recombination) and device performance parameters (e.g., cell efficiency). For this purpose, we combine a Bayesian inference framework with a neural network surrogate device-physics model that is 100× faster than numerical solvers. With the trained surrogate model and only a small number of experimental samples, our approach reduces significantly the time-consuming intervention and characterization required by the experimentalist. As a demonstration of our method, in only five metal organic chemical vapor depositions, we identify a superior growth temperature profile for the window, bulk, and back surface field layer of a GaAs solar cell, without any secondary measurements, and demonstrate a 6.5% relative AM1.5G efficiency improvement above traditional grid search methods.npj Computational Materials (2020) 6:9 ; https://doi. INTRODUCTION Process optimization is essential to reach maximum performance of novel materials and devices. This is especially relevant for photovoltaic devices, as numerous process variables can influence their performance. Often, process optimization is done using blackbox optimization methods (e.g., Design of Experiments, 1 Grid Search, 2 Bayesian Optimization, 3,4 Particle Swarm Optimization, 5 etc.), in which selected variables are modified systematically within a range and the system's response surface is mapped to reach an optimum. These methods have shown potential for inverse design of materials and systems in a cost-effective manner, and are usually postulated as ideal methods for future self-driving laboratories. [6][7][8][9][10][11][12][13] However, traditional black-box optimization approaches have limitations: the maximum achievable performance improvement is limited by the designer's choice of variables and their ranges, artificially constraining the parameter space. Furthermore, insights into the root causes of underperformance are severely limited, often requiring secondary characterization methods or batches composed of combinatorial variations of the base samples. In contrast, recently, Bayesian inference coupled to a physics-based forward model and rapid, light-dependent and temperaturedependent, current-voltage measurements were shown to offer a statistically rigorous approach to identify the root cause(s) of underperformance in early-stage photovoltaic devices. 14 Furthermore, recently, the combination of physical insights with machine-learning models have shown good promise in development of energy materials. [15][16][17][18][19][20][21][22][23] In this contribution, we consider the optimization of the synthesis temperature profile of a gallium arsenide (GaAs) solar cell using a metal organic chemical vapor deposition (MOCVD) reactor. Growth temperature is one of the most important and challenging parameters to optimize in III-V film deposition. 24,25 Previous studies showed that the growth temperature has an impact on the film's growth rate, surface morphology, dopant incorporation, and defect formation. 24,25 Other important process parameters include precursor flowrate and growth pressure. These process parameters are closely related, and the relationship can be approximated using the Ideal Gas Law in the kinetic epitaxy process. 26 Therefore, we use the growth temperature as the key optimization variable. For other secondary process variables, for example, precursor type and carrier gas flowrate, the physical relation between process variables and material properties is unclear and likely tool specific, 27 we can replace the physics-based parametrization in the first layer of the Bayesian network inference with a machine-learning model with higher capacity, such as kernel ridge regression. 28 GaAs solar cells comprise several layers, including a back surface field (BSF), a bulk absorber, and a window layer. 29 To maximize device performance, material properties need to be optimized for each layer and interface. 20,30 An experienced researcher would grow and characterize each layer (emitter, base, window, and BSF) separately to map the process variable to material properties, in an attempt to gain physical insights to optimize the final solar cells efficiency. 25,[31][32][33] In this context, optimizing growth temperature of GaAs solar cells becomes an optimization scenario in which one process variable (temperature) affects several material descriptors in various device layers. With the assistance of a solar cell physical simulator and additional characterization techniques, the optimal growth temperature for each layer could be pinpointed and the whole device stack could be grown using the optimized growth profile. However, this expert approach requires fabricating many auxiliary samples at varying conditions with multiple layer variations, and use secondary characterization measurements, such as secondary ion mass spectroscopy (SIMS) and timeresolved photoluminescence (TR-PL), to confirm root causes of underperformance. These characterization techniques are significantly more costly than current-voltage (JV) measurements, the primary proxy of solar cell performance. This problem mirrors the challenges in optimizing other multi-layer energy systems and semiconductors, including light-emitting diodes, power electronics, thermoelectrics, batteries, and transistors. To address this challenge, we combine several machinelearning techniques to infer the effects of a given process variable on different device layers. To avoid performing expensive characterization, such as SIMS or TR-PL, we perform automated JV measurement at multiple illumination intensities (JVi) as the input for the algorithm. To speed up our calculations, we employ a physics-based "surrogate" model that mimics a complex physical model, in this case solar cell growth. Our surrogate model consists of a two-step Bayesian inference method, typically referred as Bayesian network or hierarchical Bayes, [34][35][36] with relations between layers constrained by physical laws. Embedded therein is a surrogate device-physics model, which operates >100× faster (shown in Supplementary Fig. 1) than a numerical device-physics solver. Figure 1 shows the schematic of our Bayesian network. We propose three methodological innovations in this approach. First, we create a parameterized process model by imposing physicsbased constraints to couple the process-optimization variable (e.g., growth temperature) to the resulting material's bulk and interface properties (e.g., lifetime). This parametrization limits the number of fitting variables in the first layer of our Bayesian inference model, reducing the risk of overfitting, and provides a degree of interpretability. Second, we add another inference layer inside a numerical device-physics model, linking the inferred bulk and interface properties to the solar cell performance measures (e.g., JVi characteristics, quantum efficiency, and energy conversion efficiency). Extraction of underlying materials descriptors from JVi curves, previously demonstrated in ref., 14 enables us to trace the root cause(s) of device underperformance to a specific material or interface property. Third, we achieve a >100× acceleration in inference by replacing the solar cell model, a traditional PDE (partial differential equation) numerical model, with a highly accurate neural network surrogate model. This enables us to update the posterior probability distribution for our Bayesian network inference model over a vast parameter space. In Fig. 1, we also show the difference between our Bayesian network-based optimization and the traditional black-box optimization. As only low-cost evaluations (JVi measurement) are performed for solar cell characterization, accurate extraction of underlying material properties requires performing Bayesian inference using a device-physics model. 14 Traditional optimization approaches often make use of a purely black-box surrogate model 3 to map the relation between process variables and device performance directly, without any insights about material properties in the device. In contrast, our Bayesian network inference Fig. 1 Schematic of our Bayesian network-based process-optimization model, featuring a two-step Bayesian inference that first links process conditions to material descriptors, and then the latter to device performance. Our Bayesian network-based process optimization back propagates from efficiency to bulk interface properties and then to growth temperature, enabling layer-by-layer tuning of process variables. Z. Ren et al. connects target variables to material descriptors, then to process conditions. It provides rich, layer-by-layer information about critical material properties that affects device's electrical performance. In this study, we chose to map doping concentration in emitter and bulk, bulk lifetime (τ), front and rear (indium gallium phosphide) InGaP/GaAs surface recombination velocities (SRVs) to growth temperature using, and customize the growth temperature that maximize those desired material properties. Replacing the traditional optimization (process variable-device performance) with our Bayesian network-based optimization (process variablematerial properties-device performance) feasibly enables us to expand the variable space, and identify design process windows that selectively improve specific materials, layers, and interfaces inside a solar cell. This results in vastly improved device performance and process interpretability in few MOCVD fabrication rounds with a single temperature sweep. To demonstrate the potential of our approach, we use our Bayesian network to characterize and optimize, in a single temperature sweep consisting of five MOCVD fabrication rounds, the process temperature of a GaAs solar cell. Our devices have a baseline efficiency of~16% without an anti-reflection coating (ARC). Our Bayesian network approach identifies the optimal set of process conditions that translate into maximum performance under the physical model and real process constraints. The physical insights from the Bayesian network inference suggest an optimal growth temperature profile, allowing a significant 6.5% relative increase in average AM1.5G efficiency above baseline in a single temperature sweep (sixth MOCVD run). This result verifies the capacity of our approach to find optimal process windows with little intervention from the experimentalist, no secondary characterization techniques or auxiliary samples, and with performance beyond experimentalistconstrained optimization. Fig. 1, we construct a Bayesian network to link the process variables with each material and device property in the GaAs solar cells. Hereafter, we optimize each material property separately to maximize the final device performance. The Bayesian network consists of four parts. RESULTS As illustrated in Parameterization of process variables by embedding physics knowledge This section describes how we define physics-based relations between process variables and materials descriptors, embedding physics domain knowledge, and ensuring faster and better convergence of our Bayesian optimization algorithm. This corresponds to the progression from "Process Conditions" to "Materials Descriptors" in Fig. 1. Device fabrication of solar cells is expensive, thus it is essential to explore the process variable space efficiently. 37 From a machine-learning point of view, we leverage the existing knowledge from literature and embed such domain knowledge as prior parameterization to constrain the variable space, for example, Eq. [2]. The parameterization connects process variables with underlying material and interface properties. In this study, we chose to infer emitter and bulk doping concentration, bulk lifetime (τ), front and back SRV as the intermediate material properties because they play a critical role in determining the device electrical performance, 29 and each property is layer or interface specific. In the case of chemical vapor deposition (CVD), recognizing that growth temperature affects several thermally and kinetic activated processes, 38 we embed such knowledge and enforce an exponential dependence of underlying material properties based on the modified Arrhenius equation [39][40][41] (Eq. [2]). The detailed schematic of the Bayesian network inference is shown in Fig. 2. To illustrate the flow of our approach, we use the optimization of acceptor (Zn) doping concentration in a GaAs solar cell as a showcase. Our approach can be represented as a two-step Bayesian inference procedure using conditional probability (Eq. [1]): P JjT ð Þ ¼ Z P J; N A jT ð Þ d N A ¼ Z P N A jT ð ÞÃP JjN A ; T ð Þ d N A ;(1) where P (N A \|T) is the conditional probability of Zn acceptor doping levels given the process temperature. We parameterize the prior (P (N A \|T)) based on existing literature and our physical knowledge. Recognizing that MOCVD growth is a kinetic process, 38 we enforce an Arrhenius equation-type of parameterization to link the underlying material properties with growth temperatures. Zn doping level can be represented in the modified Arrhenius equation [2]: N A T ð Þ ¼ T a exp b À 1 T þ c ; (2) Fig. 2 Architecture of our Bayesian inference network to identify new windows for process optimization. Z. Ren et al. where (a, b, c) are latent process parameters that are inferred from the Bayesian framework. b and c correspond to the activation energy and pre-exponential factor in the traditional Arrhenius equation. a is the temperature dependence of the pre-exponential factor (ln(c)). Aside from Zn doping concentration, Si doping concentration, bulk minority carrier lifetime (τ), and front and back SRVs are also parameterized in the same fashion. The modified Arrhenius equation form for the doping concentration agrees well with trends reported in the literature. [42][43][44] There is insufficient literature and domain expertise to directly relate bulk and interface properties with the growth temperature. However, a previous study has shown that τ and SRVs are correlated with doping concentration. 45,46 Note that performing the fitting of Eq. [2] can be an implicit hypothesis test. A small a value suggests that the pre-exponential factor temperature dependence is suppressed, and that the Arrhenius relationship governs the temperature dependence of the particular bulk, interface, or resistance property. On the other hand, a big a value suggests a larger contribution of the pre-exponential factor to temperature dependence, indicating a deviation from a pure Arrhenius-like regime at a given temperature. Additional domain knowledge is embedded in the prior by setting hard constraints for the material properties. The ranges of the five inferred material properties are shown in Supplementary Table 2. Inference of material and device properties from device measurements This section describes the progression from "Materials Descriptors" to "Target Variable: Performance" in Fig. 1. Inference of underlying material properties from JVi measurements is used to trace the root cause(s) of device underperformance to specific material or interface properties. We further extend the connection between process variables and material properties to device measurements by adding an additional inference layer. The forward model of this inference layer is a numerical device-physics model, linking the inferred bulk and interface properties to solar cell device parameters (e.g., JV characteristics, quantum efficiency, and conversion efficiency). Following the above example, P(J\|N A ,T) is the conditional probability of a set of JVi observations at a series of fixed illumination intensities given the underlying material parameters (Zn doping concentration). The material property-JVi relation is extensively investigated and can be solved numerically using a well-developed device model from literature. 29,[47][48][49] A wellcalibrated PC1D model 47 is used in this work. However, numerical simulation is computationally expensive in the Bayesian framework (which requires hundreds of thousands of simulated runs to provide adequate posterior probability estimation) and makes it difficult to integrate new features into the model. Furthermore, experimental JVi observations contain experimental noise that causes deviations from simulated JVi curves. Replacement of numerical solver with a robust neural network surrogate model To circumvent the computational complexity of the numerical device-physics model and the discrepancies between experimental and simulated JVi curves, we replace the numerical simulator with a surrogate deep neural network. Figure 3 shows a schematic of the model, consisting of two parts: (1) a denoizing Autoencoder (AE) 50 that takes noisy JVi curves as input and reconstructs noisefree JVi curves. In our case, the training data are 20,000 simulated JVi curves, computed with a device-physics model, and augmented with Gaussian noise that mimics experimental noise. The Gaussian noise has a 0 mean and 0.2% variance, which are estimated from the repeated JV measurements. The structure of the encoder network is shown in Supplementary Fig. 2, and consists of three convolutional and two dense layers in the encoder and three convolution transpose and two dense layers in the decoder. The decoder is a mirror of the encoder network, with transposed convolution layers replacing the convolutional layers. The denoizing training of the architecture provides robustness to experimental noise. (2) A regression model that predicts the JVi curves based on underlying material properties. The regression model has the same structure as the decoder used in the denoizing AE. P(J\|N A ,T) thus can be computed using this surrogate neural network model. To create the training dataset, we first randomly sample a set of 20,000 random material properties (τ, FSRV, RSRV, Zn, and Si doping concentration) from uniform probability distributions. The threshold of the uniform distribution is shown in Supplementary Table 2. Then, we use scripted PC1D 51 to numerically simulate a set of 20,000 JVi curves from the chosen material descriptors. Although domain expertise is required in setting up the numerical PC1D model, this exercise is a one-time implementation for each material system. Subsequently, we augment the simulated JVi curves with Gaussian noise to mimic the experimental measurements and feed the noisy JVi to train the denoizing AE. Figure 3 shows the noise-free JVi curves after we feed the experimental data to the AE. Hereafter, we train the neural network regression model to predict JVi curves from material descriptors in the latent space of the AE. The loss for both AE and regression model is chosen to constraint the latent parameter space to the five variables of interest, and is minimized using the ADAM gradient descent algorithm with a batch size of 128 and an initial learning rate of 0.0001. We split the JVi curves into 80 and 20% for training and testing purposes. The numerical solver in the Bayesian network is then replaced by the regression model. The surrogate model is significantly more computationally efficient than the numerical solver. Supplementary Fig. 1 shows the acceleration by adapting the neural network surrogate for calculation of a set of JVi curves. The surrogate model, running on a GPU, is 130 times faster than the PC1D numerical solver and 700 times faster if the numerical solver is called externally. Once the device model is trained, we connect these previous two Bayesian inference steps into a hierarchical structure using Eq. [1]. A posterior probability to every combination of latent fitting parameters (a, b, c) is assigned. This probability is represented by a multivariate probability distribution over all possible combinations of model fit parameters. This probability is modified every time new data (JVi measurement) is observed. We apply a Markov Chain Monte Carlo (MCMC) method for sampling the posterior distribution of latent parameters (a, b, c); this achieves an acceleration of Bayesian inference computation time comparable or superior to the successive grid subdivision method. 2 Specifically, we implement the affine-invariant ensemble sampler of MCMC proposed by Goodman and Weare 52 using an external library. 53 With each newly observed JVi measurement, the posterior distributions of the latent process parameters (a, b, c) are updated. Hereafter, the material descriptor (Zn doping concentration as a function of growth temperature (N A (T)) can be obtained from Eq. [2]. In an analogous way, other descriptors, such as the doping levels of other species and bulk and interface recombination properties, can be obtained as a function of the process variables and adequate prior parametrizations. We use this result to optimize the MOCVD growth temperature of a set of GaAs solar cells. Optimizing solar cells using our Bayesian network inferred results After we map the growth temperature to the material properties, we apply grid search method with 10°C resolution to find the growth temperature that maximizes the desired material properties and minimize the undesired properties (maximize τ and minimize SRVs) for the solar cell. Mathematically, we can define the optimization procedure enabled by our Bayesian network model as: x Ã ¼ arg max h g i ðxÞ ð Þ ð Þ :(3) The variable x* is the set of process variable, specifically the MOCVD growth temperature, that produce the largest solar cell efficiency. We first estimate the function g i (x), which models how the set of underlying material properties changes with the process variable. Hereafter, the cell efficiency can be maximized by plugging material properties y i = g i (x) to h(y i ), which models the relation between material properties and the final solar cell performance (JVi curves). h(y i ) can be solved numerically and, in our case, is estimated using a neural network. The material properties extracted can be exploited to find x* that maximizes the cell efficiency. As g i (x) determines the functional relation of material descriptors and the process variable, we can tailor our process variable to maximize the desired materials properties, such as lifetime, and minimize the undesired properties, such as SRVs, in selected locations across the devices. As a baseline for testing our methodology, we fabricate five batch of GaAs solar cells (four cells per batch), sweeping a range of constant growth temperatures. The GaAs solar cell structure consists of multiple InGaP and GaAs layers ( Supplementary Fig. 3), and all solar cell layers are grown at the same temperature in one MOCVD experiment. In five experiments, a temperature range of 530-680°C, with 20-50°C temperature intervals, is explored. The films are fabricated into 1 cm 2 solar cells, without ARCs. Detailed growth and fabrication procedures can be found in the Experimental procedures section. JVi measurements under multiple illumination intensities (0.1-1.1 suns) are performed. Figure 4 shows the inferred material properties as a function of MOCVD growth temperatures. We can see that the logarithm value of ptype (Zn) doping level, n-type (Si) doping level, and FSRV have an almost linear correlation with 1/T, suggesting a good agreement with the standard Arrhenius equation, while the bulk lifetime and RSRV exhibits nonlinear relationships. To trace the root causes, mean of (a, b, c) values for each material properties after the MCMC run are attached in Table 1. The full distribution of the (a, b, c) values extracted from the Bayesian network is shown in Supplementary Fig. 4. The a values for both the Zn and Si doping concentration are close to zero (<0.002), indicating a negligible temperature dependence in the pre-exponential factor and the Arrhenius regime is dominant. This agrees well with the trend reported in the literature for various dopant species. [42][43][44] The a value for FSRV is also insignificant, and FSRV has similar trend as the Zn doping concentration. We postulate that this is due to SRVs being affected by doping concentrations, 45,46 and the dominant recombination mechanism in the front interface being related to Zn doping level. The a value for effective bulk lifetime is significant (−4.59), indicating a strong temperature dependence on the preexponential factor and thus non-Arrhenius regime. We postulate that this could result from the existence of both Zn and Si dopant in the GaAs bulk layer, as there is a competing contribution from the two dopants. The a value for RSRV lies between the value of bulk lifetime and Si doping levels. The RSRV slightly follows the linear trend of Si doping levels; however, we postulate that the subsequent bulk, front, and contact layers' growth impact on the rear interface's quality, 45 and contributes to the non-Arrhenius behavior. To validate the inferred doping concentrations and lifetime from our Bayesian network approach, we grow auxiliary structures (e.g., single-layer structure to conduct SIMS measurement and heterostructure for TR-PL measurements). The red circles in the first three subplots of Fig. 4 represent the results from those independent auxiliary measurements. The experimental details are shown in Supplementary Fig. 5. It is evident that the independent measurements agree well with the inferred material properties. It is interesting to note that each parasitic recombination parameter (bulk lifetime, FSRV, and RSRV) has its minimum/ maximum at a different growth temperature. The bulk lifetime peaks around 620°C, which is close to our baseline process temperature (630°C). The front and rear SRVs exhibit opposite trends when growth temperature increases. Instead of growing the whole GaAs stack at the same temperature, Fig. 4 indicates that the back contact, bulk, and front contact should each be grown at a different temperature to optimize performance. This knowledge gained by the Bayesian network enables us to formulate a new time temperature profile (Table 2) for our GaAs devices (labeled "Bayes Net" in Fig. 5). We performed an additional MOCVD experiment by selecting the growth temperature show in Table 2 that minimizes recombination at each layer or interface in a 10°C resolution (hardware tolerance) grid. We avoid extreme conditions (e.g., 680°C), which show deterioration of overall device performance (Fig. 5) despite inferred layerspecific improvements (Fig. 4). Figure 5 shows the spread of GaAs cells' efficiency for the five MOCVD experiments and the additional MOCVD run with the customized growth profile. Without additional insights on material properties from our Bayesian network, cell efficiency become the sole optimization target. The grid search on growth temperature suggests growing the whole solar cell stack at 580°C or 650°C are the optimal growth scheme. This temperature sweep (i.e., a single cycle of learning) gives us an efficiency improvement of 1.4% relative above our baseline efficiency (630°C) after five MOCVD runs. The sixth MOCVD run that tunes growth temperatures of each layer (Table 2), thereby minimizing layer-specific recombination, achieve a 6.5% relative improvement over the baseline, well exceeding the conventional approach. Auxiliary one-Sun JV and external quantum efficiency (EQE) measurements are performed to trace the root causes of efficiency improvement using the customized temperature profile (Fig. 6). It shows that both J SC and V OC are responsible for the efficiency improvements in our "Bayes Net" growth temperature profile. EQE shows that photo-response at wavelengths <820 nm (corresponding to an optical penetration depth comparable to our 2-µm-thick absorber) is improved, indicating significant reduction in recombination for the front and bulk layers. We perform Bayesian inference (second layer in the Bayesian network) to extract the material properties of measured JVi curve of this cell and our baseline. The mean values of FSRV, RSRV, and τ of the cells grown using "Bayes Net" profile ( Table 2) are 1.2 × 10 3 cm/s, 5.4 × 10 4 cm/s, and 29 ns, while the best baseline values are 4.1 × 10 3 cm/s, 6.1 × 10 4 cm/s, and 26 ns. These values agree qualitatively well with the EQE observations from auxiliary measurement, which show the front surface and the bulk benefiting the most from the "Bayes Net" temperature profile, possibly because these were the highest-temperature steps, and Fig. 5 Comparison of "grid search optimization" versus our approach using a Bayesian network (Bayes Net). GaAs cell efficiency varies with growth temperature, reaching an average maximum between 580°C and 650°C. Our Bayesian networkinformed process (labeled "Bayes Net") tunes the growth temperature of each layer to minimize recombination (Fig. 4), increasing efficiency by 6.5% relative. The gray area represents the additional efficiency gain that cannot be achieved using conventional grid search optimization. Please note that black-box optimization methods searching in the constant temperature space would have underperformed compared to the Bayesian network results. that may have partially erased the thermal history of the underlying rear-surface layer. All cells reported herein do not have ARCs; the best cells shown in the figure are estimated to have efficiencies in the 24-25% range with optimal double-layer ARCs. The efficiency value is near state of the art for a single-junction GaAs with substrate 54 grown in an academic setting. Other process parameters, for example, epitaxial lift-off and contact grid optimization, are required to reach record efficiencies. 30 Nevertheless, the recombination gains enabled by the variable-temperature profile by our Bayesian network should translate to these advanced cell architectures. It is important to note that, given the shape of the function to optimize, any other black-box optimization methods in the constant temperature space would have underperformed in comparison to the Bayesian network. Growing the device stack at the constant temperature will never achieve the same level of improvement as what is demonstrated using the Bayesian network. This is the case because tuning layer-by-layer growth temperature only becomes evident when we perform Bayesian inference to map the JVi measurements to underlying material properties. This demonstrates how additional insights gained via Bayesian network-based optimization can be translated into device performance that exceeds black-box optimization. One could argue that similar performance can be achieved by following the "expert" approach to perform single-layer optimization before incorporating them into a device stack. However, many auxiliary structures' growth and secondary measurements will be required in this case. Fifteen SIMS and TR-PL samples were grown in this study for model validations. The fact that the optimal variable-temperature profile is found after a single temperature sweep of five MOCVD runs at constant temperatures, verifies the potential of our method to accelerate time-consuming device optimization significantly, limiting the need of synthesizing auxiliary samples and performing secondary measurements. Lastly, we can modify the Bayesian network approach by replacing the physics parametrization (Arrhenius equation) with regularized black-box regression, in those cases when the physics between the process variable and material properties are unclear or complex to model. In Supplementary Fig. 6, we demonstrate that the temperature can be mapped to the material properties manifold with a similar accuracy by replacing the Arrhenius equation parametrization with a black-box regression model (kernel ridge regression using radial basis function 28 ). Incorporating black-box regression in the first layer of the Bayesian network enables one to describe complex process variables. However, the performance of the black-box regression will be affected by the hyperparameter values shown in Supplementary Fig. 6. Because the experiments are expensive in our case and data scarcity, the initial selection of hyperparameters in the black-box regression can be critical. Furthermore, the interpretability of the Bayesian network will surfer as latent process parameters (a, b, c) cannot be inferred in the black-box regression case. DISCUSSION We developed and applied a Bayesian network approach to GaAs solar cell growth optimization. This approach enables us to exceed our baseline efficiency by 6.5% relative, by tuning process variables layer by layer, in just six MOCVD experiments. Our approach is enabled by implementing physics-informed relations between process variables and materials descriptors, and embedding these into a Bayesian network. We link these material descriptors to device performance using a neural network surrogate model, which is 100× faster than numerical device simulation. The small number of growth (MOCVD) runs necessary to implement this layer-by-layer process-optimization scheme translate into significant cost and time reductions compared to common black-box optimization methods. We believe this approach can generalize to other solar cell materials, 55,56 as well as other systems with physics-based or black-box relations between process variables and materials descriptors, and physics-based device performance models. Our surrogate model can replace common models in closed-loop black-box optimization, such as a Gaussian process regression in Bayesian optimization, providing good functional fitting and physical insights. METHODS Experimental procedures The top GaAs cell was fabricated on epi-ready <100>-oriented GaAs onaxis wafers using an AIXTRON Crius MOCVD reactor. The growth is performed under a reactor pressor of 100 mbar using TMGa, TMIn, AsH 3 , and PH 3 as precursors and 32 standard liters per minute H 2 as carrier gas. It has a 3 µm n-doped GaAs base (Si dopant) and 100 nm p + -doped GaAs emitter (Zn dopant). Highly doped InGaP is used as the window (Zn dopant) and BSF layer (Si dopant). p + -doped GaAs layer (carbon dopant) is added at the front surface to achieve an ohmic contact to the metal fingers. The solar cells are metalized using an E-beam evaporator and a shadow mask to fabricate a grid pattern with~8% shading. A SIMS measurement is conducted for the GaAs films that are grown in the same batch before metallization. We additionally grow n-doped InGaP/GaAs/ InGaP heterostructure with two different base thicknesses (500 and 1000 nm) to measure the bulk lifetime of the n-doped GaAs bulk. 57 The growth conditions for the heterostructure are identical to the conditions for GaAs solar cells. Fig. 3 3Schematic of neural network surrogate model to infer material descriptors from JVi curves. (In this figure, five sequential JV curves are shown as inputs and outputs, with increasing illumination intensity). Z. Ren et al. Fig. 4 4Bayesian network reveals how each material descriptor (bulk and interface property) varies with processing conditions. Black lines show inferred values of material descriptors as a function of growth temperature; red circles show experimental validation of material descriptors using SIMS and TR-PL. Doping concentrations of different species (Zn and Si), bulk lifetime, and InGaP/GaAs interface SRV can be inferred from finished device measurements using this approach. The x-axis is −1/T and the y-axis is in logarithm scale to illustrate whether the material property follows the Arrhenius equation (linear trend). Fig. 6 6JV and EQE measurement of GaAs solar cells with the custom growth temperature profile informed by our Bayesian network (red) and baseline 630°C (black).Z. Ren et al. Table 1 . 1Fitted mean value of latent parameter (a, b, c) for different material properties.a b c Zn doping 0.0018 −0.1494 −0.1948 Si doping 0.0016 0.1551 0.2970 Bulk lifetime −4.5973 2.7984 2.3687 Front SRV 0.0015 −0.1440 −0.1892 Rear SRV 2.1194 −1.1119 −0.7300 Table 2 . 2Temperature profile of GaAs solar cells.Layer Optimal growth temperature suggested by Bayesian network (°C) Buffer 580 InGaP BSF 580 GaAs bulk 620 InGaP window 650 GaAs contact 650 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2020) 9 npj Computational Materials (2020)9 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences © The Author(s) 2020 ACKNOWLEDGEMENTSDATA AVAILABILITYPart of the experimental and simulated dataset for training the neural network surrogate model and predicting the optimal growth conditions from Bayesian network is available in the following GitHub repository: https://github.com/PV-Lab/ BayesProcess. Additional data supporting the findings of this study is available from the authors upon reasonable request.CODE AVAILABILITYThe code used for Bayesian network and neural network surrogate to predict material properties is also available at https://github.com/PV-Lab/BayesProcess. Additional code supporting the findings of this study is available from the authors upon reasonable request.COMPETING INTERESTSThe authors declare no competing interests.ADDITIONAL INFORMATIONSupplementary information is available for this paper at https://doi.org/10.1038/ s41524-020-0277-x.Correspondence and requests for materials should be addressed to Z.R., F.O. or T.B.Reprints and permission information is available at http://www.nature.com/ reprintsPublisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 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Advances to Bayesian network inference for generating causal networks from observational biological data. Bioinformatics 20, 3594-3603 (2004). J Pearl, Causality, Models, Reasoning and Inference. Springer29Pearl, J. Causality: Models, Reasoning and Inference Vol. 29 (Springer, 2000). Accelerating materials development via automation, machine learning, and high-performance computing. J.-P Correa-Baena, 2Correa-Baena, J.-P. et al. Accelerating materials development via automation, machine learning, and high-performance computing. Joule 2, 1410-1420 (2018). Mass transport and kinetic limitations in MOCVD selective-area growth. M E Coltrin, C C Mitchell, J. Cryst. Growth. 254Coltrin, M. E. & Mitchell, C. C. Mass transport and kinetic limitations in MOCVD selective-area growth. J. Cryst. Growth 254, 35-45 (2003). The development of the Arrhenius equation. K J Laidler, J. Chem. Educ. 61494Laidler, K. J. The development of the Arrhenius equation. J. Chem. Educ. 61, 494 (1984). 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H Haug, B R Olaisen, Ø Nordseth, E S Marstein, Energy Procedia. 38Haug, H., Olaisen, B. R., Nordseth, Ø. & Marstein, E. S. A graphical user interface for multivariable analysis of silicon solar cells using scripted PC1D simulations. Energy Procedia 38, 72-79 (2013). Ensemble samplers with affine invariance. J Goodman, J Weare, Commun. Appl. Math. Comput. Sci. 5Goodman, J. & Weare, J. Ensemble samplers with affine invariance. Commun. Appl. Math. Comput. Sci. 5, 65-80 (2010). emcee: the MCMC hammer. D Foreman-Mackey, D W Hogg, D Lang, J Goodman, Publ. Astron. Soc. Pac. 125306Foreman-Mackey, D., Hogg, D. W., Lang, D. & Goodman, J. emcee: the MCMC hammer. Publ. Astron. Soc. Pac. 125, 306 (2013). Strong internal and external luminescence as solar cells approach the Shockley-Queisser limit. O D Miller, E Yablonovitch, S R Kurtz, IEEE J. Photovolt. 2Miller, O. D., Yablonovitch, E. & Kurtz, S. R. Strong internal and external lumi- nescence as solar cells approach the Shockley-Queisser limit. IEEE J. Photovolt. 2, 303-311 (2012). A path to 10% efficiency for tin sulfide devices. N M Mangan, IEEE 40th Photovoltaic Specialist Conference (PVSC). IEEEMangan, N. M. et al. A path to 10% efficiency for tin sulfide devices. In 2014 IEEE 40th Photovoltaic Specialist Conference (PVSC) 2373-2378 (IEEE, 2014). Planar p-n homojunction perovskite solar cells with efficiency exceeding 21. P Cui, 3%. Nat. Energy. 4150Cui, P. et al. Planar p-n homojunction perovskite solar cells with efficiency exceeding 21.3%. Nat. Energy 4, 150 (2019). Intensity-dependent minority-carrier lifetime in III-V semiconductors due to saturation of recombination centers. R Ahrenkiel, B Keyes, D Dunlavy, J. Appl. Phys. 70Ahrenkiel, R., Keyes, B. & Dunlavy, D. Intensity-dependent minority-carrier lifetime in III-V semiconductors due to saturation of recombination centers. J. Appl. Phys. 70, 225-231 (1991).	[ "https://github.com/PV-Lab/", "https://github.com/PV-Lab/BayesProcess." ]
[ "Epsilon-near-zero regime as the key to ultrafast control of functional properties of solids", "Epsilon-near-zero regime as the key to ultrafast control of functional properties of solids", "Epsilon-near-zero regime as the key to ultrafast control of functional properties of solids", "Epsilon-near-zero regime as the key to ultrafast control of functional properties of solids" ]	[ "M Kwaaitaal \nFELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands\n\nInstitute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands\n", "D G Lourens \nFELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands\n\nInstitute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands\n", "C S Davies \nFELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands\n\nInstitute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands\n", "A Kirilyuk [email protected] \nFELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands\n\nInstitute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands\n", "M Kwaaitaal \nFELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands\n\nInstitute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands\n", "D G Lourens \nFELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands\n\nInstitute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands\n", "C S Davies \nFELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands\n\nInstitute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands\n", "A Kirilyuk [email protected] \nFELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands\n\nInstitute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands\n" ]	[ "FELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands", "Institute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands", "FELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands", "Institute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands", "FELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands", "Institute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands", "FELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands", "Institute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands", "FELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands", "Institute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands", "FELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands", "Institute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands", "FELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands", "Institute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands", "FELIX Laboratory\nRadboud University\nToernooiveld 76525 EDNijmegenThe Netherlands", "Institute for Molecules and Materials\nRadboud University\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands" ]	[]	Strong light-matter interaction constitutes the bedrock of all photonic applications, empowering material elements with the ability to create and mediate interactions of light with light. Amidst the quest to identify new agents facilitating such efficient light-matter interactions, a class of promising materials have emerged featuring highly unusual properties deriving from their dielectric constant ε being equal, or at least very close, to zero 1,2,3 . Works so far have shown that the enhanced nonlinear optical effects displayed in this 'epsilon-near-zero' (ENZ) regime makes it possible to create ultrafast albeit transient optical switches 4,5 . An outstanding question, however, relates to whether one could use the amplification of light-matter interactions at the ENZ conditions to achieve permanent switching. Here, we demonstrate that an ultrafast excitation under ENZ conditions can induce permanent all-optical reversal of ferroelectric polarization between different stable states. Our reliance on ENZ conditions that naturally emerge from the solid's ionic lattice, rather than specific material properties, suggests that the demonstrated mechanism of reversal is truly universal, being capable of permanently switching order parameters in a wide variety of systems.	null	[ "https://export.arxiv.org/pdf/2305.11714v1.pdf" ]	258,822,865	2305.11714	7a2769f6232bcfc4feddb6fdb1ae1fec93296448	Epsilon-near-zero regime as the key to ultrafast control of functional properties of solids M Kwaaitaal FELIX Laboratory Radboud University Toernooiveld 76525 EDNijmegenThe Netherlands Institute for Molecules and Materials Radboud University Heyendaalseweg 1356525 AJNijmegenThe Netherlands D G Lourens FELIX Laboratory Radboud University Toernooiveld 76525 EDNijmegenThe Netherlands Institute for Molecules and Materials Radboud University Heyendaalseweg 1356525 AJNijmegenThe Netherlands C S Davies FELIX Laboratory Radboud University Toernooiveld 76525 EDNijmegenThe Netherlands Institute for Molecules and Materials Radboud University Heyendaalseweg 1356525 AJNijmegenThe Netherlands A Kirilyuk [email protected] FELIX Laboratory Radboud University Toernooiveld 76525 EDNijmegenThe Netherlands Institute for Molecules and Materials Radboud University Heyendaalseweg 1356525 AJNijmegenThe Netherlands Epsilon-near-zero regime as the key to ultrafast control of functional properties of solids -1 - Strong light-matter interaction constitutes the bedrock of all photonic applications, empowering material elements with the ability to create and mediate interactions of light with light. Amidst the quest to identify new agents facilitating such efficient light-matter interactions, a class of promising materials have emerged featuring highly unusual properties deriving from their dielectric constant ε being equal, or at least very close, to zero 1,2,3 . Works so far have shown that the enhanced nonlinear optical effects displayed in this 'epsilon-near-zero' (ENZ) regime makes it possible to create ultrafast albeit transient optical switches 4,5 . An outstanding question, however, relates to whether one could use the amplification of light-matter interactions at the ENZ conditions to achieve permanent switching. Here, we demonstrate that an ultrafast excitation under ENZ conditions can induce permanent all-optical reversal of ferroelectric polarization between different stable states. Our reliance on ENZ conditions that naturally emerge from the solid's ionic lattice, rather than specific material properties, suggests that the demonstrated mechanism of reversal is truly universal, being capable of permanently switching order parameters in a wide variety of systems. The spontaneous polarization is a critically important order parameter already underpinning a broad range of electronic, optical, and electro-mechanical appliances, with clear potential to produce novel functionalities 6 ranging from high-density non-volatile memories 7,8 and extremely sensitive nano-devices 9 to neuromorphic-inspired computational schemes 10 and energy harvesting 11 . All these applications stem from the capacity of electric fields to switch polarized domains in ferroelectric materials 12 . Moreover, the possibility of coupling the polarization of a ferroelectric with other physical parameters such as strain, magnetic order or optical properties provides a unique basis for creating materials that are truly multifunctional 13 . The development of ultrafast and energy-efficient methods capable of coherently switching spontaneous polarization is therefore vital for the future progress in developing new ferroelectric technologies. As the model system for our investigation, we select one of the most studied classical ferroelectrics, barium titanate (BaTiO3). Barium titanate is a multiaxial perovskite ferroelectric 14 with a cubic parent phase that becomes tetragonal upon the appearance of the ferroelectric polarization P. At the transition, the titanium and oxygen ions experience small opposing shifts along one of the cubic axes 15 , lowering BaTiO3's point group symmetry from 3 # to 4 14 . Such symmetry breaking creates a ferroelectric polarization along one of the crystal axes. In total this results in six possible directions of P, thus supporting the stable existence of both 90° and 180° ferroelectric domains 16 . We employ single-domain crystals of BaTiO3 at room temperature, to study how their domain structures are impacted by an excitation at ENZ conditions. In the spectral range of 7-28 µm (357-1429 cm -1 ), BaTiO3 has two major phonon resonances. Both show significant splitting of transverse and longitudinal optical modes (TO and LO, respectively) due to the strong polar character of the crystal 17 . The LO-phonon frequency is marked by the strongly revealed ENZ regime, where both the real ε1 and imaginary ε2 parts of the dielectric constant approach zero at the same time. Moreover, the LO mode has considerable spectral separation from the corresponding IR-active TO mode, which is instead indicated by a maximum in ε2. The fact that such 'phononic' ENZ materials require no special efforts to construct, while still leading to the best possible realization of ENZ conditions, arguably offers a strong advantage compared to carefully-designed metamaterials 5,18 . To reach the most efficient interaction of the infrared light with the given material, it is imperative to use ultrashort and intense excitation pulses that are sufficiently narrow in bandwidth to ideally fit the ENZ condition 4 . The only source that is capable of meeting this is a free-electron laser, which is well-known to produce pulses that intrinsically obey the Fouriertransform-limit relation between the pulse duration and bandwidth 19 . In this work, we use radiation with wavelengths in the range of 7-28 µm delivered by the free-electron laser facility FELIX in Nijmegen, The Netherlands. FELIX provides transform-limited pulses that are 1-3 pslong, with 0.5-1% of bandwidth, and with pulse energies exceeding 10 µJ. After preparing BaTiO3 in a single-domain state, we expose it to an infrared narrow-band excitation with a central wavelength of λ = 14 µm. Second-harmonic-generation microscopy, using near-infrared incident light, reveals the light-induced creation of 180° domains that are absent at the center of the laser pulse but rather manifest at the sides across the horizontal axis (see Fig. 1(A)). Polarizing microscopy ( Fig. 1(B)) shows that this infrared excitation also generates narrow 90° domains that extend across the center of the irradiated region, as well as a set of circular rings. For clarity, we have shown in Fig. 1 two separate images that show the creation of one specific domain type. In general, however, it is possible to have 90° and 180° domains created at the same time (Supplementary Information). Note that such domains can be created both by a single infrared ps-long pulse or an 8-µs-long burst of pulses coming at a repetition rate of 25 MHz (Supplementary Information). Since the latter "macropulse" typically produces larger switched domains, which are therefore more stable, we generally used macropulses as a pump for convenience. We first consider the concentric rings made visible by the polarizing microscopy in Fig. 1(B). We find that the rings persist upon tuning the pumping wavelength, while the number of rings increase with the optical fluence. The spectral indifference of this feature allows us to conclude that this originates from incoherent heating of the sample, which causes a thermally-induced variation in birefringence (Supplementary Information). This feature is rather useful in providing an in-situ "thermometer" at the sample surface, while also proving that the switched 90° and 180° domains are completely unassociated with heating effects (Supplementary Information). So far, the switching of ferroelectric domains was achieved with a pumping wavelength of λ = 14 µm and rms bandwidth of 70 nm, where the ENZ condition is ideally satisfied (see Fig. 2). To therefore assess the importance of the excitation's wavelength, we adjusted λ while continuously monitoring the sample's polarization using both SHG and polarizing microscopy. At varying wavelengths, both the 90° and 180° domains appeared larger or smaller, allowing us to calculate the area of the switched domain. In Fig. 2(A), we plot the spectral dependence of the switched area (normalized by the incident pulse energy), revealing two maxima in the vicinity of 14 µm and 21 µm. These are exactly the wavelengths where the real part of the dielectric function crosses zero and the imaginary part is close to zero. At these wavelengths, the switching of the sufficiently-large ferroelectric domains was permanent, whereas at other pumping wavelengths the switched domains of smaller size were transient, shrinking and vanishing on the timescale of milliseconds (see Supplementary Material). Notice in the spectrum that the success of switching depends on both ε1 and ε2 going close to zero at the same time. The spectral dependence of domain switching in Fig. 2 clearly and unambiguously shows the importance of using an excitation at the ENZ conditions. How can we actually explain this though? In fact, a considerable amount of research has already been devoted to formulating and realizing a mechanism that would provide the means for ultrafast all-optical switching of ferroelectric polarization. The first proposal in this direction was the prediction that a highamplitude excitation of the ferroelectric soft mode can lead, at certain conditions, to the full reversal of the polarization 20 . Subsequent research has revealed that a nonlinear coupling of different optical phonon modes provides a more efficient mechanism involving the displacement of the equilibrium atomic positions that actually deforms the crystal lattice 21,22 . In this process, a strong excitation of an IR-active TO phonon mode along the coordinate WIR results, via a nonlinear coupling to a Raman-active mode, in the shift of the equilibrium position along the latter's coordinate WR. Such a kind of interaction is described by high-order energy terms of the phonon coordinates such as W 2 IRWR or W 2 IRW 2 R, appearing in the perturbative part of the Hamiltonian. Non-linear phononics has indeed been able to demonstrate a transient switching of ferroelectric polarization 23 , albeit only to a small extent and during an extremely short timescale of just 200 fs. In our experiment however, at 14 µm, no IR-active phonon is present at this excitation frequency, and the LO-mode that is present cannot be excited with a transverse electromagnetic wave at normal incidence, as used in our experiment. Then, if not a phonon, what do we do excite? Deriving from the ENZ conditions, several factors provide for the strong interaction of the light's electric field with the material. These include diverging wavelengths leading to an in-phase excitation across a large volume as well as a vanishing group velocity that increases the interaction time 24,25 . Consequently, a strong nonlinear polarization is created at the excitation frequency and over a significant volume of the sample. Note that the transverse character of the excitation is not a restrictive factor anymore, because of this nonlinearity 4 . This substantially helps given the fact that at the ENZ condition, only (near-)normal incidence results in light penetrating the medium 26,27 . However, how does this non-resonant oscillatory polarization leads to a structural breaking of symmetry? In fact, it is the combination of the strong induced polarization with the incident electric field that can also lead to a unidirectional displacement of Raman phonons. Helped by the resonance with an IR-active mode, such an effect was shown to be several orders of magnitude stronger than the nonlinear phononic one discussed above 28 . Unlike in the nonlinear phononics, here the resulting nonlinear ionic polarizations in combination with the incident electric fields drive the main effect, that both are amplified by the ENZ condition. Regardless of the exact microscopic mechanism and the particular Raman mode excited, the pattern of the created domains can still be phenomenologically explained by taking into account the macroscopic shape of the laser pulse. The IR pulse was focused on the sample at normal incidence, resulting in a Gaussian profile of intensity with a FWHM in excess of 100 µm. The magnitude of the atomic displacements scales directly with the light-delivered electric field, and thus the deformation profile will also be Gaussian. By virtue of ferroelectrics being piezoelectric, the Gaussian deformation will similarly induce an electric displacement field. The direction and shape of the latter matches that of the created ferroelectric domains. This entire process is visualized schematically in Fig. 3(A)-(B), with the resulting calculated piezoelectric displacement field shown in Fig. 3(C). A quantitative description is provided in the Methods. To summarize, we have identified that an excitation under ENZ conditions enables permanent switching of ferroelectric polarization in different directions. The universality of this effect is underscored by the recent observation of switching of magnetic order in ferrimagnetic iron garnets 29 and antiferromagnetic nickel oxide 30 , where in both cases it was the excitation at the LO phonon frequency that led to the domain switching or displacement (though the mechanism was not identified). Thus, very different crystallographic symmetries lead to the same effect, adding to the certainty that it is the nonlinear optics, and not a combination of very specific mode symmetries, that is at play here. Looking ahead, further experimental campaigns must explore the dynamics of the switching process with the aim to identify the timescale of the transient response. Nonlinear polarizations, for example, depend sensitively on the timescale of the pulse, whereas nonlinear phononics involves effects that depend on the lifetime of the excited IR phonon. This temporal fingerprint will provide invaluable insight in to the microscopic mechanisms involved in the ENZ regime. The spectral dependence of the real and imaginary parts of the permittivity ε = ε 1 + iε 2 of BaTiO3, taken from Ref. [17]. The switching is maximized in strength at the ENZ condition, i.e., when both ε 1 and ε 2 approach zero. Fig. 3: Schematic explanation of the switching process. (A) An ultrafast IR pulse, tuned in frequency to meet the ENZ condition, induces large-amplitude ionic polarization oscillations along the coordinate Q IR that in turn non-linearly drives a rectified force (F R ) acting on the atoms along the coordinate Q R . In this case, we show the rectified force that would emerge if non-linear phononics would be the cause. (B) The rectified force on the atoms leads to a deformation profile that follows the Gaussian intensity profile of the impinging IR pulse. (C) The resulting calculated piezoelectric displacement field that gives rise to 90° and 180° switched domains (left and right panels respectively), assuming that the ferroelectric polarization initially points to the right. Methods: Materials: The barium titanate crystals studied in our experiments were procured from MSE Supplies. The crystals, of size 5×5×0.5 mm 3 , were grown with the Czochralski process, and both surfaces were polished. We selected a (100) crystal orientation so that the spontaneous ferroelectric polarization lays along the crystal surface. Pumping of BaTiO3: To explore the possibility of switching ferroelectric polarization at the ENZ conditions, we used pump laser pulses delivered by the free-electron laser facility FELIX in Nijmegen, The Netherlands. The temporal structure of the IR pulses is shown in Supplementary Data Fig. S1. The free-electron laser supplies transform-limited IR pulses at a repetition rate of 25 MHz. These pulses come within 8-µs-long bursts ("macropulses") at a rate of 10 Hz. The central wavelength of the pump pulses, with wavelength-dependent durations in the range 1−3 ps, was varied in the spectral range of 7−28 μm with their bandwidth experimentally tunable in the range of 0.5−2.0% (typically <1% of the bandwidth used in our experiments). The laser beam was focused to a spot with a diameter of about 200 µm onto the surface of the BaTiO3 crystal. Observation of polarization domains by microscopy: In order to observe the ferroelectric domains and their dynamics, we have used (i) a polarization microscope and (ii) a second harmonic generation (SHG) microscope. The polarization microscope uses the birefringence in the material to distinguish different domains 16 . Linearlypolarized light, of wavelength λ = 520 nm as delivered by a CW diode laser (Integrated Optics), illuminates the sample at normal incidence. The transmitted light is collected by an objective lens and directed through an analyser (i.e., polarizer) in order to select the transmitted component polarized along a specific axis. After such filtering, the light is spatially-resolved by the matrix of a charge-coupled device (CCD) camera. The CCD is suitably triggered to record an image before and after arrival of the pump pulse. The contrast of this microscope stems from the fact that the refractive index along the c-axis (the crystal axis parallel to the ferroelectric polarization) differs from that associated with the other crystal axes (a-axes). Thus, we are able to distinguish 90° domains using polarization microscopy but we are unable to distinguish 180° domains. To distinguish 180° domains, we use SHG microscopy. We illuminate the sample with a train of ≈150-fs-long laser pulses, of wavelength λ = 1040 nm, at a repetition rate of 100 MHz (Menlo Orange HP). Since our probe pulse energies (up to 90 nJ) are capable of burning the sample, we use an optical chopper to modulate the illumination and thus reduce the optical power. The transmitted 1040-nm and SHG-generated 520-nm light is collected by an objective lens, and the former is extinguished by an appropriate short-pass filter. The SHG is then recorded by a CCD. The SHG signal produced in BaTiO3 depends on the non-linear tensor components of the (") tensor. The signal generated from different domains differ since different components of the (") tensor are used in the generation of the second harmonic 31 . This makes it quite straightforward to distinguish 90° domains. Observation of 180° domains however is less trivial and we have to realize that the created domains by FELIX only appear close to the surface and are not going through the entire bulk of the crystal. The SHG signal of 180° domain is equal in intensity but with a 180° phase difference. Due to this phase difference and the fact that there is no phase matching in BaTiO3, the SHG signal of two successive 180° domains can constructively interfere and appear brighter compared to a single bulk domain 32 . Model of the switching process: Here we elaborate our model for the switching process, which was schematically shown in Fig. 3. We assume the free energy F of BaTiO3 can be expanded in the following form 33 : = $ + * %& − 1 3 %& '' / " + 1 2 '' " + 1 2 % % " + %(&' %( & '(1) where the first three terms are of pure elastic origin, assumed to be isotropic. Following the notation of Ref. 33, is the modulus of rigidity, %& the strain tensor, the modulus of compression and the phonon coordinate. The fourth term, quadratic in Qi, is the energy of the laser-excited phonon modes. The last term is a coupling between the phonon coordinate and the strain 34 , and is of particular importance for explaining our results. We assume that the laser excites only a single phonon mode ( & = ' = )* ). In this case, the excitation can only lead to volumetric changes due to the symmetry restrictions of the 4mm point group. Furthermore, to make calculations analytically solvable, we assume that the coupling term between the strain and the phonon coordinate is in the form %(&& = %( , i.e., the strain coupling is equal in every direction. In general, this term could of course vary, but we find that this assumption of isotropy is sufficient to explain the experimentally-observed switching of ferroelectric polarization. We note that no ab-initio calculations were performed during this study to obtain the real values of these coefficients. We will now focus on the effects of the last term in Eq. (1), and elaborate how it can create a Gaussian deformation profile. With the stress tensor being given by %& = +, +-!" , the force acting on a volume element is given by the divergence of the stress tensor % = +. !" +/ " . The force on a volume element therefore becomes ⃗ ( ) = ∇ )* " . Since the phonon coordinate scales proportionally with the laser's electric field, )* " scales proportionally with the intensity (I) of the laser pulse, and thus ⃗ ( ) ∝ ∇ ( ) . With the above information, we can calculate the pulse-induced deformation/displacement (=⃗) in the material using the following equation (33): ∇ ⋅ ∇ ⋅ =⃗ − 1 − 2 2(1 − ) ∇ × ∇ × =⃗ = (1 + )(1 − 2 ) (1 − ) ⃗ (2) where is the Young's modulus. The applied laser pulse has a Gaussian spatial profile ( ) = ⋅ 01 # /"3 # . Thus, the displacement will be purely radial and the term ∇ × =⃗ can be dropped. Solving the above equation gives =⃗( ) = 1 " (1 − 0 1 # "3 # )(3) with: = (1 + )(1 − 2 ) (1 − )(4) This leads to the strain profile: // = " " + " GH1 − 0 / # 56 # "3 # I H1 − 2 " " + " I + " " 0 / # 56 # "3 # J (5) 66 = " " + " GH1 − 0 / # 56 # "3 # I H1 − 2 " " + " I + " " 0 / # 56 # "3 # J(6)/6 = − " " + " G 2 " + " H1 − 0 / # 56 # "3 # I − 1 " 0 / # 56 # "3 # J(7) In ferroelectrics this strain may couple to the polarization via the piezoelectric effect. In barium titanate in particular, strain ( ) can induce an electric displacement field ( ) via the piezoelectric effect as following 35 : Supplementary information Heating effects The absorption of the laser pulse by BaTiO3 leads to the generation of heating. This heat has an influence on the refractive index n and birefringence (Δ ) of barium titanate 1 , and thus its effects are visible through the polarization microscope. The Gaussian intensity profile of the laser pulse leads to a Gaussian profile of heating in the sample, and thus a Gaussian spatial profile of retardation since the retardation is given by = Δ ( ) ⋅ where is the thickness of the crystal. Using data from Ref.1, we estimate the total retardation of our 0.5 mm-thick sample is in the order of 50 waves at room temperature. Heating the sample can significantly decrease this retardation. We now consider how this heat-dependent retardation profile will manifest in the polarization microscope. To understand this, we note that the polarization microscope can only distinguish between relative retardations, rather than the absolute retardation (e.g., a retardation of 46.5 waves will appear the same as 47.5 waves). Let us consider the images in Fig. S2(A). Light and dark contrast corresponds to varying amounts of retardation, with the ringed structure stemming from the radial symmetry of the Gaussian heat profile. Each ring is equivalent to one full wave of retardation difference, and so the number of rings gives an estimate of the amount of heat absorbed from the laser pulse. More rings therefore corresponds to a stronger thermal profile. Upon varying the pump wavelength, we observe that the number of rings vary. We therefore counted the number of rings generated using excitation wavelengths between 8 and 14 µm (Fig. S2(B)) and discovered that the switching observed at λ = 14 µm is actually not correlated with a large temperature change of the sample. Moreover, the incident pulse energy does not vary significantly across this spectral range. The slightly-rectangular shape of the rings and the lobes, mainly visible at lower wavelengths, can be explained by including the effect of strain-induced birefringence induced by thermal expansion. Lifetime of transient domains Under certain conditions, we frequently observed the rapid decay of laser-induced switched domains of ferroelectric polarization. To measure the lifetime of these transient domains, we repeated the microscopy measurements with temporal-resolution provided by synchronizing the CCD camera with the macropulses of the free-electon laser. The exposure time of the CCD -27 µs -defines the time resolution. While the appearance of switched domains is instantaneous on this timescale, we observe that a large number of domains shrink and vanish across a timescale of milliseconds (the exact time-scale depends on the size of the domain). The shrinking and disappearance of newly-formed switched domains probably stems from neighboring dipole-dipole interactions. In addition, the created domains do not extend through the whole thickness of the crystal. When a domain achieves a critical size, however, it may become stable. In some situations, with the appropriate wavelength and with sufficiently strong pulse intensity, we observed that the switched domains did not shrink and thus were stable. The position of the laser spot seemed to play a potentially significant role on the domain formation process. At certain places on the sample, e.g., near defects or at the edges of the sample, it seemed easier to create domains. In Figs. S3-S4, we present a series of images taken using polarization and SHG microscopy respectively. These two figures respectively show the dynamics of 90°-and 180°-domains. The images shown were taken at the time-delays t indicated relative to the arrival of the IR macropulse. The images have been background-corrected, i.e., an image taken before the arrival of the IR pump has been subtracted, to emphasize the laser-induced features. Grey contrast implies there is no intensity difference between the two images. Figure S5 shows a representative image taken in which a single pumping macropulse has switched both 90° and 180° domains simultaneously (circled orange and green respectively). This image was obtained using SHG microscopy. This image confirms that the piezoelectric displacement fields that are induced by the IR pump pulse, as shown in Fig. 3(C), coexist. Image of coexisting 90° and 180° switched domains Dependence of switching on pulse energy We have performed additional measurements assessing how the size of the 180° and 90° switched domains scale with the energy of the infrared laser pulse. The results are shown as the blue points in Fig. S6. We have fitted the measured data points with an exponential function of the form ⋅ 0>/? $%&'( . This dependence is in accordance with our model for the switching mechanism, since the strain induced by a laser pulse scales linearly with the pulse energy. In turn, the electric displacement field created by the piezoelectric effect is linearly proportional to the strain. Hence, the laser-induced electric displacement field is linearly proportional to the pulse energy. Experiments studying the field-driven formation of domains, and the subsequent domain-wall motion, have shown that the switching rate of 180° domains increases exponentially with the exponent − 3 / where 3 is the 'activation field' and is the electric field 2 . Our model of the switching consequently implies that the domain size scales exponentially with the form ⋅ 0>/? $%&'( . While the exponential function clearly fits well the size of the 180° domains, the fit is less successful in describing the size of the 90° domains. This difference can be attributed to the fact that the switching takes place by the inhomogeneous nucleation of many small domains 2 . For the 180° domains, many (uncountable) domains can be created by a laser pulse, while more pulse energy is required to switch 90° domains and it was only possible to create up to 3 switched domains with a single macropulse. This limited nucleation makes it possible to only observe the 'linear part' of the exponential function, in the case of the 90° domains. The free-electron laser supplies transform-limited infrared micropulses at a repetition rate of 25 MHz. These pulses come within 8-µs-long bursts ("macropulses") at a rate of 10 Hz. Fig. S3: Time dependence for 90° switched domains Time evolution of the switching process observed with polarization microscopy. At t = 0 µs, we observe a bright spot which is an artefact coming from higher harmonics created by the IR laser pulse in the system. After the laser pulse we see a stripe, which contrast corresponds to a 90° domain that has been created. We also see rings coming from heating effects. Fig. 1 : 1Light-induced switching of polarization. (A)-(B) Top panel: Schematic of the experimental setup. A ferroelectric BaTiO3 sample with polarization P is excited by an infrared (IR) pulse at normal incidence (wavelength λ = 7-28 µm, gold line). The impact of the IR pulse on the polarization is detected in one of two ways. Column (A): Second-harmonic-generation microscopy resolves 180° switched domains via the polarization-dependent frequency-doubling of near-IR (λ = 1040 nm) probe pulses. Column (B): Polarizing microscopy resolves 90° switched domains through the polarization-dependent rotation of the probe's electric field E. Middle panel: Typical raw images of the ferroelectric polarization taken before and shortly after illumination of the sample with an IR pump pulse (λ = 14 µm). Bottom panel: Schematic distribution of the ferroelectric polarization as deduced from the change of contrast in the recorded images. Fig 2 : 2Importance of satisfying the ENZ condition for switching. (A) The spectral dependence of the switching efficiency, evaluated as the normalized area of the switched 90° and 180° domains (red and blue points respectively). (B) = # . 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[ "Fairness-aware Differentially Private Collaborative Filtering", "Fairness-aware Differentially Private Collaborative Filtering" ]	[ "Zhenhuan Yang [email protected] ", "Yingqiang Ge [email protected] ", "Congzhe Su ", "Dingxian Wang [email protected] ", "Xiaoting Zhao [email protected] ", "Yiming Ying [email protected] ", "Zhenhuan Yang ", "Yingqiang Ge ", "Congzhe Su ", "Dingxian Wang ", "Xiaoting Zhao ", "Yiming Ying ", "\nEtsy Inc Brooklyn\nNYUSA\n", "\nRutgers University New Brunswick\nNJUSA\n", "\nEtsy Inc Brooklyn\nNYUSA\n", "\nEtsy Inc Brooklyn\nNYUSA\n", "\nEtsy Inc Brooklyn\nNYUSA\n", "\nUniversity at Albany\nSUNY Albany\nNYUSA\n" ]	[ "Etsy Inc Brooklyn\nNYUSA", "Rutgers University New Brunswick\nNJUSA", "Etsy Inc Brooklyn\nNYUSA", "Etsy Inc Brooklyn\nNYUSA", "Etsy Inc Brooklyn\nNYUSA", "University at Albany\nSUNY Albany\nNYUSA" ]	[ "Companion Proceedings of the ACM Web Conference 2023 (WWW '23 Companion)" ]	Recently, there has been an increasing adoption of differential privacy guided algorithms for privacy-preserving machine learning tasks. However, the use of such algorithms comes with trade-offs in terms of algorithmic fairness, which has been widely acknowledged. Specifically, we have empirically observed that the classical collaborative filtering method, trained by differentially private stochastic gradient descent (DP-SGD), results in a disparate impact on user groups with respect to different user engagement levels. This, in turn, causes the original unfair model to become even more biased against inactive users. To address the above issues, we propose DP-Fair, a two-stage framework for collaborative filtering based algorithms. Specifically, it combines differential privacy mechanisms with fairness constraints to protect user privacy while ensuring fair recommendations. The experimental results, based on Amazon datasets, and user history logs collected from Etsy, one of the largest e-commerce platforms, demonstrate that our proposed method exhibits superior performance in terms of both overall accuracy and user group fairness on both shallow and deep recommendation models compared to vanilla DP-SGD.	10.1145/3543873.3587577	[ "https://export.arxiv.org/pdf/2303.09527v1.pdf" ]	257,557,133	2303.09527	74bb9644eb3bf4eacf43667999e6b7a2d20723c0	Fairness-aware Differentially Private Collaborative Filtering WWW '23 CompanionCopyright WWW '23 CompanionApril 30-May 4, 2023. April 30-May 4, 2023 Zhenhuan Yang [email protected] Yingqiang Ge [email protected] Congzhe Su Dingxian Wang [email protected] Xiaoting Zhao [email protected] Yiming Ying [email protected] Zhenhuan Yang Yingqiang Ge Congzhe Su Dingxian Wang Xiaoting Zhao Yiming Ying Etsy Inc Brooklyn NYUSA Rutgers University New Brunswick NJUSA Etsy Inc Brooklyn NYUSA Etsy Inc Brooklyn NYUSA Etsy Inc Brooklyn NYUSA University at Albany SUNY Albany NYUSA Fairness-aware Differentially Private Collaborative Filtering Companion Proceedings of the ACM Web Conference 2023 (WWW '23 Companion) Austin, TX, USA; Austin, TX, USAWWW '23 CompanionApril 30-May 4, 2023. April 30-May 4, 202310.1145/3543873.3587577ACM Reference Format:. ACM, New York, NY, USA, 5 pages. https:// ACM ISBN 978-1-4503-9419-2/23/04. . . $15.00CCS CONCEPTS • Computing methodologies → Machine learning• Infor- mation systems → Recommender systemsInformation re- trievalCollaborative filtering KEYWORDS Collaborative Filtering, Fairness, Differential Privacy Recently, there has been an increasing adoption of differential privacy guided algorithms for privacy-preserving machine learning tasks. However, the use of such algorithms comes with trade-offs in terms of algorithmic fairness, which has been widely acknowledged. Specifically, we have empirically observed that the classical collaborative filtering method, trained by differentially private stochastic gradient descent (DP-SGD), results in a disparate impact on user groups with respect to different user engagement levels. This, in turn, causes the original unfair model to become even more biased against inactive users. To address the above issues, we propose DP-Fair, a two-stage framework for collaborative filtering based algorithms. Specifically, it combines differential privacy mechanisms with fairness constraints to protect user privacy while ensuring fair recommendations. The experimental results, based on Amazon datasets, and user history logs collected from Etsy, one of the largest e-commerce platforms, demonstrate that our proposed method exhibits superior performance in terms of both overall accuracy and user group fairness on both shallow and deep recommendation models compared to vanilla DP-SGD. INTRODUCTION With the explosive growth of e-commerce, consumers are increasingly relying on online platforms for their shopping needs. Traditional collaborative filtering (CF)-based recommendation models use a user's past interactions, such as ratings and clicks, to learn embeddings. However, the use of such user history can reveal sensitive information about the user. Prior works has shown an adversary can infer a targeted user's actual ratings [13,19] or deduce if the user is in the database [3,24] based on the recommendation list. To prevent privacy leakage, differential privacy (DP) [5,6] is a popular choice of mechanism in the above research as it provides theoretically quantitative privacy guarantee. In particular, differentially private stochastic gradient descent (DP-SGD) [1,25,27] is often adopted due to its scalability towards large neural networks. Despite the success of privacy protection by DP-SGD, it may also entail certain trade-offs. One such trade-off, recently discovered, is that the reduction in utility incurred by DP models disproportionately affects underrepresented subgroups in the image classification task [2,4]. In the field of recommendation systems, fairness concerns also arise due to other privacy protection mechanisms like federated learning [17,20,30]. Therefore, it is natural to ask whether the protection of users' privacy will lead to disparate recommendation performance between different groups of users. Specifically, in this work, we consider the unfair treatment between user groups with different activity levels, which is a common concern in the realm of fairness-aware recommendation [9,15,16,26,29]. In general, users who interact with the platform more frequently will contribute more sufficient data than those less active users when training the model. Due to the fundamental idea of collaborative filtering [8,12,18,23], it can lead to bias towards inactive users in the trained recommender system [7,15,28]. Consequently, users with lower activity levels are more likely to receive lower NDCG or so to speak unsatisfactory recommendations as demonstrated in Figure 1 by Non-DP bars. What's worse, our empirical findings reveal that this accuracy bias worsens when differential privacy is introduced. Specifically, the NDCG of inactive users takes a heavier hit comparing with the active users, namely, the unfair treatment measured by the NDCG gap between active and inactive users in the DP-SGD setting increases by 15-20% comparing with Non-DP setting across four datasets. This is also called "the poor become poorer" phenomenon (Matthew Effect) as in [2,10]. To this end, we propose a novel two-stage framework for CFbased approaches to address the issues of utility degradation and unfairness aggravation brought by DP-SGD, called DP-Fair. Specifically, the first stage of DP-Fair applies noise perturbation to user and item embeddings separately, thereby improving utility while providing privacy guarantees over vanilla DP-SGD. In the second stage, we apply a post-processing step to enforce user group fairness on the final recommendation list via solving of an integer programming problem. Our experimental results on Amazon benchmark datasets and user history logs collected on Etsy, demonstrate that our proposed algorithm outperforms vanilla DP-SGD based collaborative filtering in terms of overall recommendation performance and user-side group fairness. PRELIMINARIES 2.1 Collaborative Filtering Models Let U = { 1 , · · · , 1 } and V = { 1 , · · · , 2 } be the sets of users and items, respectively. Let H = { ∈ V} denote the set of items that user had positive interactions with. It is worth noting that we treat all interactions as binary implicit feedback (e.g. one if there is a click). Explicit feedback such as rating (e.g. 1-5) are converted to one if > 3 otherwise zero. H denote the collection of all H . Let be the number of total positive interactions, i.e. \|H \| = . Let x ∈ R 1 , x ∈ R 2 be the one-hot encoding of the user and item , respectively. Let z = x ∈ R 1 , z = x ∈ R 2 denote the corresponding latent embeddings. We also employ to denote any other potential feature extraction parameters and Θ = ( , , ) to denote all learnable parameters. A collaborative filtering latent factor model Θ : U × V → R is learning to infer the implicit feedback pattern once a learning to rank loss is given. In this work, we focus on the classic Bayesian Personalized Ranking (BPR) [23] loss, as follows, ( Θ ) = ∑︁ , , ′ − log ( Θ (x , x ) − Θ (x , x ′ )) + 2 ∥Θ∥ 2 , where is the sigmod function, is the regularization parameter, ∈ H , and ′ ∈ H − = V \H denotes an item that user does not provide implicit feedback. Since the positive interactions are usually more sparse, we slightly abuse the notation and let H − also denote a uniformly sub-sampled set of itself such that \|H − \| = \|H \| = . After learning, a recommendation list R = { ∈ V} for each user is produced based on top ranking scores { Θ ( , V)}. Differential Privacy and DP-SGD We first introduce the definition of differential privacy, which is given as follow. Definition 1. For any , > 0, an (randomized) algorithm A is said to be ( , )-differentially private if for all neighboring datasets , ′ that differs by at most one example, and for all possible output sets Θ by A, there holds P[A ( ) ∈ Θ] ≤ exp( )P[A ( ′ ) ∈ Θ] + . where denotes the privacy budget (smaller values indicates a stronger privacy guarantee) and denotes tolerance of probability that the privacy guarantee fails. In practice, it often requires ≪ 1 . Since users' sensitive information can be inferred from the interaction data, the private dataset in this case is = H ∪ H − . At each iteration , DP-SGD performs gradient norm clipping with some bound and Gaussian noise addition with variance 2 on the received gradients , and then performs regular SGD on the model parameter Θ based on the new gradients˜. If one randomly samples a batch B ⊆ H of size , then for each example ∈ B , DP-SGD runs as ( ) = ( )/max{1, ∥ ( )∥ 2 / } (1) ( ) =¯( ) + N (0, 2 I)(2) User-side Fairness Let R denote the recommendation list to user originally. We utilize the non-parity unfairness measure initially introduced in Kamishima et al. [14], stated as follow: Definition 2. Given a recommendation evaluation metric M, the use group fairness with respect to groups and is defined as E [M (R )\| ∈ ] = E [M (R )\| ∈ ]. Empirically, the user group fairness is measured by F (R ; , ) = 1 \| \| ∑︁ ∈ M (R ) − 1 \| \| ∑︁ ∈ M (R ) . Due to different user activity levels, recommender systems would usually underperform against users who have less historical interactions. This bias can be amplified when differential privacy is incorporated in the model as demonstrated in [2]. Following the classical 80/20 rule, we select the top 20% users based on their engagement activities as the frequent/active group and the rest as the infrequent/inactive group . FAIRNESS-AWARE DIFFERENTIALLY PRIVATE COLLABORATIVE FILTERING There are two stages in our proposed framework, DP-Fair. Given a private dataset with user provided privacy budgets ( , ), the first Randomly sample a batch B ⊆ D of size 5: for each example ∈ B do 6: Apply DP-SGD steps as in Eq. (1) and (2) stage applies DP-SGD for training the BPR and providing privacy guarantee. It is worth noting that at Line 6 we replace the uniform DP-SGD with separated ones on user and item. This procedure can avoid unnecessary norm clipping and noise addition [13,22] since user and item gradients may be different in scale during training. Overall, this tailored DP step will lead to better utility than vanilla DP-SGD with less perturbed gradients. In the second stage, in order to mitigate the unfair treatment, we employ a post-processing approach [15]. At Line 12, once topranking lists R are available, we re-rank them by maximizing the sum of prediction scores under the user group fairness constraint max R ∑︁ ∈U ∑︁ ∈R Θ (x , x )(3) s.t. F (R ; , ) ≤ and R ⊆ R ∀ ∈ U. One can consider R as a binary matrix ∈ R 1 × where , = 1 means item is recommended to the user . Hence the optimization problem (3) can be translated and solved as a 0 − 1 integer programming problem. For ranking metric M, we pick the commonly used F1 score, which makes the computation more efficient than NDCG since it avoids the position discounted effect. It is worth noting that since this method is post-processing based on the recommendation list only, it will not break the differential privacy guarantee over the learned parameters Θ [5]. EXPERIMENTS 4.1 Experimental Setup 4.1.1 Datasets. We utilized two distinct sources of data. Firstly, we employ a benchmark dataset, namely the Amazon review dataset (5-core), which includes product reviews from the Grocery & Gourmet Food and Beauty categories [11]. Since both are encoded with explicit feedback through ratings, we transform them into binary feedback. Secondly, we collect and sample one month's worth of user history logs from two categories-Home & Living and Craft Supplies & Tool, on Etsy, one of the largest e-commerce platforms. We consider users' clicks as positive feedback in both datasets. [23] and NeuMF [12]. To find the best clipping bounds, we follow the DP tuning strategy in McMahan et al. [21] via pre-trainining. We set pre-ranker = 20 and reranker = 10. We fix the privacy parameter = 1 1.5 and employ the Opacus 1 module to conduct DP-SGD steps. We also employ the Gurobi 2 solver to solve the re-ranking problem in Eq. (3). Table 2, several conclusions can be drawn. Firstly, the results are reported for three different privacy budget settings: the non-private setting ( = ∞), a loose privacy budget setting ( = 10), and a tight budget setting ( = 1). As expected and consistent with the literature, the utility in terms of NDCG and F1 degrades as the privacy constraint is tightened. Secondly, our proposed DP-Fair algorithm outperforms DP-SGD in terms of overall utility for both NDCG and F1, regardless of the privacy budget . This improvement is attributed to our custom noise addition and gradient clipping technique, which is also applicable in the non-private setting where the algorithm is identical to SGD with a re-ranking step. While it may seem counter-intuitive that enforcing fairness would improve utility, our results show that this is due to the improvement in the utility of inactive users, who make up 80% of all users, thereby boosting the overall performance. Finally, we observe a significant reduction in the F gap by DP-Fair over DP-SGD. This can be attributed to the post-processing step in DP-Fair, which identifies = 10 fairness-aware items out of the = 20 list. = ∞ = 10 = 1 Model Metric Algorithm Total ↑ Act. ↑ InAct. ↑ F ↓ Total ↑ Act. ↑ InAct. ↑ F ↓ Total ↑ Act. ↑ Hyperparameter Effects. In this experiment, we fix the privacy budget to 1. Our first objective is to examine the selection of the clipping bound. To simplify the analysis, we impose = = . Based on the results presented in Figure 2, we conclude that both excessively small or large values of have a negative impact on the NCDG. When the clipping parameter is too small, the average clipped gradient can be biased. Conversely, increasing the norm bound leads to the addition of more noise to the gradients. In addition, we investigate the impact of the fairness level selection. As the fairness requirements become stricter, the performance of the active group decreases, while that of the inactive group improves. CONCLUSIONS In this paper, we empirically observe the unfairness gap between the active and inactive users will be widened by the incorporation of DP-SGD in classical collaborative filtering based recommendation models. We propose custom differentially private gradient mapping incorporating an integer programming scheme to enhance its fairness between active and inactive users. Experiments on real-world e-commerce datasets show that DP-Fair outperforms DP-SGD in both utility and fairness metrics. Figure 1 : 1NDCG@10 (%) on various datasets (See descriptions in Section 4) between active users (Blue) and inactive users (Green). In each subplot, left two bars labeled with Non-DP are NeuMF model trained by standard SGD. Right two bars are learned by DP-SGD with = 1. Figure 2 : 2Top: Performance results with respect to different values of clipping bound in terms of NCDG. Bottom: Performance results with respect to different levels of fairness constraint in terms of F1. Algorithm 1 DP-Fair 1: Inputs: Private dataset D = H ∪ H − ; privacy parameters , ; number of iterations ; learning rate { : ∈ [ ]}, mini-batch size ; initial points Θ 0 . 2: Stage I: Private Training 3: for = 0 to − 1 do 4: separately for user and item gradients˜( ) = (˜( ),˜( )) +1 = Θ − ∈˜( ) 9: end for 10: Stage II: Fairness Re-ranking 11: Rank based on Θ and return long recommendation lists R 12: Solve Eq. (3) via integer programming 13: Outputs: Short new top recommendation lists R arXiv:2303.09527v1 [cs.IR] 16 Mar 2023(a) Home & Living (b) Craft Supplies & Tools (c) Grocery & Gourmet Food (d) Beauty Table 1 : 1Statistics of datasets.4.1.2 Implementation Details. In this experiment, we performe a randomized 8:1:1 split of the datasets to create training, validation, and test sets. We consider both shallow and deep recommendation models, namely BPR-MF : The results of recommendation performance. The evaluation metrics are calculated based on top-10 predictions. The results are reported in percentage (%) and the arrow indicate the favorable direction. Our best results are highlighted in bold. 4.2 Experimental Analysis 4.2.1 Main Results. Based onInAct. ↑ F ↓ Home & Living BPR-MF NDCG DP-SGD 11.57 15.17 10.73 4.44 10.77 14.93 9.67 5.25 10.21 14.44 9.15 5.29 DP-Fair 12.04 14.18 11.52 2.57 11.19 14.09 10.44 3.64 10.61 13.55 9.87 3.68 F1 DP-SGD 4.85 7.17 4.39 2.77 4.56 6.96 3.90 2.97 4.31 6.84 3.67 3.17 DP-Fair 4.89 6.57 4.51 2.07 4.61 6.38 4.11 2.27 4.33 6.30 3.83 2.47 NeuMF NDCG DP-SGD 12.24 17.36 11.12 6.21 11.61 16.73 10.17 7.19 11.19 16.64 9.82 6.82 DP-Fair 12.93 16.12 12.13 3.99 12.05 15.78 11.12 4.66 11.59 15.01 10.74 4.27 F1 DP-SGD 5.03 7.65 4.37 3.28 4.78 7.47 4.10 3.36 4.58 7.61 3.82 3.79 DP-Fair 5.08 7.28 4.53 2.75 4.81 7.20 4.21 2.99 4.64 6.96 4.05 2.91 Craft Supplies & Tools BPR-MF NDCG DP-SGD 12.63 16.51 11.72 4.79 11.76 16.28 10.57 5.72 11.15 15.75 10.00 5.75 DP-Fair 13.59 16.02 13.01 3.01 12.64 15.91 11.80 4.11 11.98 15.32 11.15 4.17 F1 DP-SGD 5.30 7.89 4.80 3.09 4.99 7.30 4.27 3.61 4.71 7.48 4.02 3.45 DP-Fair 5.36 7.22 4.95 2.16 5.06 7.01 4.52 2.49 4.75 6.91 4.21 2.70 NeuMF NDCG DP-SGD 13.57 19.12 12.37 6.74 12.88 18.37 11.32 7.05 12.41 18.35 10.93 7.41 DP-Fair 14.55 17.76 13.65 4.11 13.56 17.62 12.51 5.11 13.16 17.49 12.08 5.40 F1 DP-SGD 5.91 9.01 5.14 3.86 5.62 8.98 4.78 4.19 5.39 8.71 4.56 4.15 DP-Fair 5.98 8.55 5.33 3.21 5.64 8.41 4.95 3.46 5.46 8.44 4.71 3.73 Grocery & Gourmet Food BPR-MF NDCG DP-SGD 10.65 14.24 9.80 4.45 9.90 14.04 8.81 5.23 9.21 13.57 8.12 5.45 DP-Fair 11.34 13.88 10.85 3.03 10.54 13.27 9.70 3.58 9.79 12.64 9.08 3.56 F1 DP-SGD 4.22 6.04 3.79 2.26 4.03 5.92 3.52 2.39 3.78 5.74 3.28 2.46 DP-Fair 4.28 5.78 3.91 1.87 4.09 5.71 3.67 2.05 3.82 5.44 3.41 2.03 NeuMF NDCG DP-SGD 11.40 16.66 10.23 6.43 10.82 16.08 9.36 6.72 10.42 15.98 9.03 6.95 DP-Fair 11.87 15.31 11.01 4.29 11.06 14.94 10.09 4.85 10.73 14.69 9.74 4.95 F1 DP-SGD 4.64 7.01 4.05 2.96 4.41 6.76 3.76 3.25 4.23 6.79 3.59 3.20 DP-Fair 4.67 6.66 4.23 2.43 4.42 6.39 3.87 2.52 4.26 6.62 3.66 2.96 Beauty BPR-MF NDCG DP-SGD 10.43 13.66 9.63 4.02 9.68 13.61 8.68 4.93 9.20 13.16 8.22 4.94 DP-Fair 11.22 13.15 10.75 2.39 10.43 13.13 9.75 3.39 9.76 12.20 9.14 3.06 F1 DP-SGD 4.02 5.50 3.65 1.85 3.82 5.62 3.37 2.25 3.54 5.38 3.08 2.29 DP-Fair 4.10 5.20 3.73 1.47 3.85 5.49 3.44 2.04 3.56 5.09 3.18 1.91 NeuMF NDCG DP-SGD 11.22 16.27 10.21 5.05 10.65 15.25 9.24 7.03 10.18 15.28 8.90 6.39 DP-Fair 11.75 15.51 10.98 4.52 11.09 14.80 9.98 4.82 10.47 14.42 9.48 4.94 F1 DP-SGD 4.51 6.78 3.98 2.79 4.25 6.66 3.61 3.05 3.99 6.44 3.37 3.07 DP-Fair 4.54 6.31 4.10 2.21 4.27 6.24 3.78 2.46 4.03 6.21 3.49 2.72 Table 2 https://opacus.ai/ 2 https://www.gurobi.com Deep learning with differential privacy. 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[ "Auditing Visualizations: Transparency Methods Struggle to Detect Anomalous Behavior", "Auditing Visualizations: Transparency Methods Struggle to Detect Anomalous Behavior" ]	[ "Jean-Stanislas Denain \nUC Berkeley\n\n", "Jacob Steinhardt \nUC Berkeley\n\n" ]	[ "UC Berkeley\n", "UC Berkeley\n" ]	[]	Model visualizations provide information that outputs alone might miss. But can we trust that model visualizations reflect model behavior? For instance, can they diagnose abnormal behavior such as planted backdoors or overregularization? To evaluate visualization methods, we test whether they assign different visualizations to anomalously trained models and normal models. We find that while existing methods can detect models with starkly anomalous behavior, they struggle to identify more subtle anomalies. Moreover, they often fail to recognize the inputs that induce anomalous behavior, e.g. images containing a spurious cue. These results reveal blind spots and limitations of some popular model visualizations. By introducing a novel evaluation framework for visualizations, our work paves the way for developing more reliable model transparency methods in the future 1 .	10.48550/arxiv.2206.13498	[ "https://export.arxiv.org/pdf/2206.13498v2.pdf" ]	250,072,879	2206.13498	4e38ffe0d7ffb6972ea29c6b926d559932df6f0a	Auditing Visualizations: Transparency Methods Struggle to Detect Anomalous Behavior Jean-Stanislas Denain UC Berkeley Jacob Steinhardt UC Berkeley Auditing Visualizations: Transparency Methods Struggle to Detect Anomalous Behavior Model visualizations provide information that outputs alone might miss. But can we trust that model visualizations reflect model behavior? For instance, can they diagnose abnormal behavior such as planted backdoors or overregularization? To evaluate visualization methods, we test whether they assign different visualizations to anomalously trained models and normal models. We find that while existing methods can detect models with starkly anomalous behavior, they struggle to identify more subtle anomalies. Moreover, they often fail to recognize the inputs that induce anomalous behavior, e.g. images containing a spurious cue. These results reveal blind spots and limitations of some popular model visualizations. By introducing a novel evaluation framework for visualizations, our work paves the way for developing more reliable model transparency methods in the future 1 . Introduction Neural networks with similar validation accuracy can behave very differently when deployed. A network can have a high validation accuracy, but be vulnerable to distribution shift, backdoors, or spurious cues learned during training. To diagnose these failures, model visualizations rely on the internals of a network, such as the activations of inner layers or the model's gradients (Molnar, 2022). By opening the black box, they may better uncover failure modes (Hernandez et al., 2022), tell us where to intervene to repair models (Wong et al., 2021;, or provide auxiliary training objectives (Koh et al., 2020). However, model visualizations can only fulfill these promises if they actually track model behavior. Previous evaluations have shown that many visualization methods fail basic tests of validity (Adebayo et al., 2018(Adebayo et al., , 2020. However, we seek more than simply pointing out failures in methods: we would like a quantifiable goal for interpretability methods to aspire to. Following Adebayo et al. (2020), we focus on the goal of debugging model errors: we say that a method is good at debugging if it can successfully uncover a wide range of possible anomalous behaviors in models. To measure this, we train a wide variety of models with anomalous behaviors, such as models containing backdoors or trained with spurious cues. We also train a set of reference models, that do not exhibit anomalous behavior, and we test whether visualization methods behave differently on the reference and anomalous models. Our pipeline for evaluating visualizations is illustrated in Figure 1. We leverage the fact that visualizations are images and thus can be embedded in a semantic feature space. Under our evaluation metric, a visualization method distinguishes anomalous models if its embedding for the anomalous model is far from the reference models, relative to the variation within the reference models. We use this metric to measure whether visualizations can (1) detect that a network is anomalous, and (2) determine how the model is unusual, i.e. localize the inputs where it behaves atypically. Here E can detect that g 1 is anomalous, but not g 2 . We evaluate five widely used model visualization techniques: Caricatures (Olah, 2018), Integrated Gradients (Sundararajan et al., 2017), Guided Backpropagation (Springenberg et al., 2015), Grad-CAM, and Guided Grad-CAM (Selvaraju et al., 2019). For each technique, we test whether it can detect 24 anomalous models across 7 categories: adversarial training , randomized smoothing (Cohen et al., 2019), shape bias (Geirhos et al., 2019), backdoors , spurious features, training without data from certain classes, and training on a face-obfuscated version of ImageNet (Yang et al., 2021). Our main takeaway is that model visualizations fail to detect and localize a large number of anomalies. Most methods only consistently detect anomalous models in the regime where the validation accuracy itself is lower than normal (stark anomalies, such as adversarial training), and struggle with anomalies that affect behavior on a smaller number of inputs (subtle anomalies, for example removing all the images in a single class from the training data). Moreover, methods often fail to localize the inputs for which behavior is different, such as images containing a spurious cue for the spurious features anomaly. Taken together, our results underscore the need to systematically evaluate visualization methods to check that they track model behavior. If we view visualization methods as a way of auditing models, our detection and localization benchmarks are a form of counterauditing-planting intentionally corrupted models to check that they are discovered. To achieve high reliability in machine learning, such counteraudits are a crucial counterpart to regular audits. Our framework can easily be adapted to other visualization methods and anomalies, and we encourage the community to use such evaluations in the future. Related Work Our work is closely related to Adebayo et al. (2018) and Adebayo et al. (2020), which evaluate feature attribution methods by their ability to diagnose different kinds of model errors. Similarly, Zhou et al. (2021) and Bastings et al. (2021) study whether feature attributions detect spurious features, and Adebayo et al. (2022) shows that model explanations can't uncover unknown spurious correlations. Compared to these works, our evaluation framework is more flexible: given a set of reference models and a distance metric between model visualizations, it can work for any visualization method, model anomaly, and any modality (vision, language, or other). This flexibility allows us to consider a much greater variety of model anomalies (24 anomalous models), which provides more detailed Figure 2: All methods are able to detect stark anomalies such as randomized smoothing (adding Gaussian noise to training inputs). We show model explanations by (a) Integrated Gradients, (b) Grad-CAM, and (c) Caricatures (layer (3,2,2)). We consider a normal model and anomalous models trained with increasing amounts σ of randomized smoothing. The explanations of the anomalous models appear very different from those of the normal model, and their anomaly scores A are greater. information about the limitations of visualization methods. It also makes it possible to compare feature attributions and different kinds of methods, such as feature visualizations. Lin et al. (2020) study whether 7 feature attribution methods, including Guided Backpropagation, are able to localize backdoor triggers: unlike us, they find that most methods fail, however they consider more sophisticated trojan attacks. Following our work, Casper et al. (2023) also used trojan discovery to benchmark feature attributions and feature synthesis methods, and found that most feature attribution methods could approximately locate the trigger, although failed to beat a simple baseline (see Figure 2 in (Casper et al., 2023). In contrast with these works, we consider anomalies other than backdoors, we define localization in a different and more general way, and we also assess whether a method can detect that a model is anomalous. Ding et al. (2021) apply a similar framework to evaluate representation similarity metrics: just like the model visualizations of an anomalous model should be different from those of a normal model, a representation similarity metric should assign a high distance between representations with different functional properties. Finally, there are other approaches to ground and evaluate transparency methods. One especially ambitious goal is to reverse engineer entire parts of neural networks: a preliminary example is Cammarata et al. (2021), in which the authors use feature visualization methods to hand-code a curve detection module in InceptionV1. A related perspective is model repair: for example, Wong et al. (2021) and Singla & Feizi (2021) both introduce new visualization-based methods to help humans identify and repair spurious features in models. Finally, another way of evaluating transparency methods is to run human subject studies to determine how much they help humans understand the underlying models Doshi-Velez & Kim (2017); Bhatt et al. (2020). Problem Setup In this section, we first introduce the anomalous models and reference models used in our evaluations (Section 3.1), then define model visualizations and visualization anomaly scores (Section 3.2). Finally, we introduce the transparency methods considered in this work (Section 3.3). Anomalous models and reference models We call a model f anomalous if it differs from a set of reference models that specify normal behavior, for example in its accuracy on an out-of-distribution test set. Reference models. In what follows, all the models we consider will be ResNet-50 variants, and our reference models will be n = 12 ResNet-50 networks g 1 , . . . , g n trained on ImageNet with the same hyperparameters but different random seeds. Anomalous models. For the choice of the anomalous model f , we consider 24 ResNet-50 models comprising 7 different types of anomalies. The first three families of anomalies are stark (they always induce a significant drop in validation accuracy), while the remainder are subtle (their behavior is only different on out-of-distribution data). We will often find that model visualizations can detect stark anomalies, but not the subtle ones. We describe each family below, with full training details given in Appendix A.2. Randomized Smoothing: These models are trained on ImageNet images perturbed by isotropic Gaussian noise with standard deviation σ, for σ ∈ {0.25, 0.5, 1.0}. We obtained checkpoints from the ImageNet Testbed (Taori et al., 2020). Randomized smoothing substantially decreases the test set accuracy, to at most 0.4 for σ = 0.25. Unlike in applications to adversarial robustness, we do not add noise to the inputs at evaluation time. Adversarial Training: These are robust models adversarially trained on ImageNet with a PGD adversary. We obtained models with ℓ ∞ -norm constraints of ϵ = 4 and 8 from Engstrom et al. (2019a), and models with ℓ 2 -norm constraints of ϵ = 0.01, 0.1, 1, and 5 from Salman et al. (2020). Most of these models exhibit a noticeable drop in validation accuracy. Shape Bias: This anomalous model is trained on Stylized-ImageNet (Geirhos et al., 2019), which makes it more sensitive to shapes and less sensitive to textures when classifying images. Once again, the resulting model has low accuracy on the ImageNet validation set (0.602). Spurious Features: We trained models on ImageNet, but where a fraction δ of each training batch's images had the class index overlaid at the bottom right of the image. We trained 4 such models with δ ∈ {1.0, 0.5, 0.3, 0.1}. The spurious features have a large impact on validation accuracy when δ = 1.0, but the effect nearly disappears for smaller δ. Backdoors: We trained 6 backdoored ResNets with a pixel-modification data poisoning attack by following the methodology in Bagdasaryan & Shmatikov (2021). These models exhibit a negligible drop in validation accuracy (at most 0.01 and sometimes smaller). On the other hand, their out-ofdistribution accuracy on images with the backdoor trigger is near zero. Removing data from a class: We trained networks on variants of ImageNet with all images from a given class removed. We considered 3 such models, removing data from classes 414 (backpack), 400 (academic gown), and 218 (Welsh springer spaniel). We sampled classes 400 and 414 uniformly from the 1000 classes. Because ImageNet contains so many dogs, we thought that removing data from a dog class would be harder to detect, so we chose class 218 by sampling uniformly from dog classes. These anomalies have a negligible effect on the overall validation accuracy, but drop the accuracy on the missing class to 0. Training on blurred images: Finally, we trained a model on a variant of ImageNet with blurred faces (Yang et al., 2021). This decreases the validation accuracy by less than 1%, but has a large effect on some classes, such as harmonica, where key information becomes blurred. Anomaly scores for visualizations Recall that model visualizations E(·, ·) are functions that take as arguments a neural network f and input x, and output some information E(f, x) about how f processes x. This information is usually a vector or tensor, e.g. an image in the case of feature visualizations. We will call an visualization z = E(f, x) anomalous when it is far away from reference visualizations z i = E(g i , x) . Given a metric d(·, ·) between visualizations, a simple way to quantify this is via the average distance between z and the reference visualizations, normalized by the average distance between the reference visualizations themselves: A(f, E, x) := 1 n n i=1 d(z, z i ) 2 n(n−1) 1≤i<j≤n d(z i , z j ) .(1) We call this ratio the anomaly score of z compared to the reference visualizations z 1 , . . . , z n . 2 When this ratio is much larger than 1, then the visualization z = E(f, x) is anomalous. A good transparency method should assign a high anomaly score to models with anomalous behavior. For the metric d(·, ·), we use the LPIPS distance, since Zhang et al. (2018) show that it aligns well with human perceptual similarity. In the main text, we used AlexNet features for the LPIPS distance, because Zhang et al. (2018) found that they perform the best. As a robustness check, we also computed anomaly scores for LPIPS distances with two other networks, VGG and SqueezeNet; we show the corresponding results in the Appendix. Visualization methods We assess four feature attribution methods and one feature visualization method. Feature attributions. Consider a neural network classifier f : R d → R C , which outputs a logit for each possible class 1 ≤ c ≤ C. For an input x, a feature attribution of f with target class c is a tensor with the same shape as x. Intuitively, it represents the relevance of each of x's entries to f (x)'s output logit for class c. In what follows, we'll always take c to be the output class c max = argmax c f c (x). We consider four feature attribution methods: Guided Backpropagation (GBP) (Springenberg et al., 2015) takes the gradient ∇ x f c (x) of the output logit with respect to the pixels, and modifies it by only backpropagating non-negative gradients through ReLU nonlinearities. Integrated Gradient (IG) (Sundararajan et al., 2017) attributions take the integral of the gradient of f c along a linear path connecting x to some base point x 0 : We obtained Guided Backpropagation, Integrated Gradients and Guided Grad-CAM attributions using the Captum library (Kokhlikyan et al., 2020) with default settings. We obtained Grad-CAM attributions using the pytorch-grad-cam library (Gildenblat & contributors, 2021). We display these visualizations by superposing the attributions on top of the input image (c.f. the first two rows of Figure 2). (x − x 0 ) × 1 0 ∇ x f c (x 0 + α (x i − x 0 )) dα.(2) Feature Visualization: Caricatures. For a neural net classifier f , let f ℓ , 1 ≤ ℓ ≤ L denote its layers, i.e. f ℓ maps an input x to the vector of activations at layer ℓ. Intuitively, a Caricature describes what a layer "sees" in an input x by visualizing the activation vector f ℓ (x). Formally, the Caricature of an input x at layer ℓ is the solution to the following optimization problem (Olah, 2018): Caricature ℓ (x, f ) := argmax x ′ ⟨f ℓ (x ′ ), f ℓ (x)⟩. In practice, Caricature ℓ (x, f ) Detection In this section, we formally define our detection metric, then describe our experimental procedure. Finally, we present and interpret our empirical results. Detection Metric. To test whether a method E(·, ·) can detect an anomalous model, we compute visualizations on a set of m sample images {x 1 , . . . , x m }. We define the overall anomaly score of a model,Ā(f, E), as the average of the anomaly scores for the visualizations E(f, x 1 ), . . . , E(f, x m ). We say that E detects f ifĀ(f, E) is larger than all the normal models' anomaly scoresĀ(g i , E). Experimental Procedure. We sampled m = 50 input images uniformly from the ImageNet test set as our sample images x j . To reduce computation time for Caricatures, we visualized a subset of N ℓ = 7 layers in different parts of the model: layers (2,2,1), (3,0,3), (3,2,2), (3,3), (3,4,3), (4,0,3), and (4,3,2). 3 We chose these layers to have information about different parts of the network: in earlier or later stages, in earlier or later residual blocks within a stage, and after or within residual blocks. For concision, we averaged the anomaly scores over these 7 layers as well as the 50 images to obtain an overall anomaly score for all caricaturesĀ(f, Caricatures). We show individual layer results in the Appendix. Integrated Gradients, Guided Backpropagation, Grad-CAM and Guided Grad-CAM do not depend on a layer: for them, we just average over images and reportĀ(f, IntGrad) andĀ(f, GBP). Results for stark anomalies. For every transparency method E and every family of anomalous models, we report the fraction of anomalous models that E detects in Table 1. The five methods are able to detect that all the models trained on Stylized ImageNet or with randomized smoothing are anomalous. They also usually detect that adversarially trained models are anomalous, although GBP, Grad-CAM, and Guided Grad-CAM fail to detect ℓ 2 -adversarial training with radius ϵ = 0.01. Qualitatively inspecting the visualizations corroborates these results. Figure 2 shows the Integrated Gradients, Grad-CAM, and Caricature (3,2,2) visualizations of the randomized smoothing models and a normal model. The visualizations of the anomalous models appear very different from those of the normal model. Moreover, they appear more different from the normal visualizations as σ increases. Results for subtle anomalies. The results are much worse for the subtle anomalies. No methods detect that any of the networks trained on blurred faces or with missing data are anomalous. For backdoors and spurious cues, the detection results differ between methods: for example, Caricatures detect that all backdoored models are anomalous but only a quarter of models in the Spurious Features category, whereas Integrated Gradients detect that all the models that rely on spurious features are anomalous but only 17% of backdoored models. Once again, qualitatively inspecting the visualizations confirms these findings. For example, Figure 3 shows visualizations by Guided Backpropagation and Caricature (3,3) of a normal model and a Spurious Features model with δ = 0.3. Both methods fail to detect that this model is anomalous, and for both methods the visualizations of the two models appear very similar. In summary, most methods succeed at detecting stark anomalies, but fail for most of the subtle anomalies. Grad-CAM and Guided Grad-CAM perform the same, and generally worse than Integrated Gradients, Guided Backpropagation, and Caricatures. We show in the Appendix that these results are robust with respect to the choice of the metric between visualizations (Appendix A.1.1). Localization Having studied transparency methods' ability to detect that a model is anomalous, we now turn to the question of localization. Localization Metric. Given an anomalous model f , we assess a method E(·, ·)'s localization ability via the question: is the anomaly score A(f, E, x) highest on the inputs x where f 's behaviour is unusual? We will measure E's ability to distinguish these inputs via AUROC. Concretely, we sample p = 40 "normal" inputs (e.g. from the ImageNet validation set), and p "abnormal" images on which the anomalous network has abnormal behavior (e.g. images with a backdoor trigger). We then compute anomaly scores for all 80 images. Thresholding on the anomaly score defines a binary classifier, which classifies an image as "normal" if the score is below a threshold, and as "abnormal" otherwise. We compute the area under the corresponding ROC curve (AUROC), which is equivalently the probability that a random abnormal images has a higher anomaly score than a random normal image. An AUROC of 1 indicates perfect performance, while 0.5 is random chance. Under a null hypothesis of exchangeable anomaly scores for normal and abnormal images, for m = 40 the 95% confidence interval for the AUROC is [0.4, 0.6]-thus, an AUROC in this range is not distinguishable from random chance. Experimental Procedure. We assess each method's ability to localize for 4 types of anomalies: backdoors, spurious cues, training without data from a missing class, and training on images with blurred faces. We chose these 4 types because they lead to unusual behavior on a well-identified set of inputs (e.g. for backdoored models, images where a trigger is present). In contrast, anomalies like adversarial training change the network's behavior on the entire distribution of inputs. For most anomalies, we obtain normal images by sampling uniformly from the ImageNet validation set 4 . In contrast, we obtain abnormal images by selecting inputs on which the anomalous model processes inputs very differently from the reference models. For example, for the Missing Class anomalies, we sample images from the missing class. For Backdoors and Spurious Features, we obtain abnormal images by sampling from the ImageNet validation set and applying the trigger or overlaying the spurious cue (the class index). For the Blurred-faces anomaly, harmonica is the class for which training on blurred faces leads to the greatest accuracy drop. We construct abnormal images by sampling uniformly from the images in that class that contain faces, using the face annotation data from (Yang et al., 2021). Results. We display results in Table 2. We find that no method can localize all of the anomalies, and that the visualizations often have close to random performance. Indeed, only 31% of the AUROCs in Table 2 are greater than 0.7, which we use as a threshold for acceptable classification following (Hosmer Jr. et al., 2013), and 54% are between 0.4 and 0.6. This is especially clear for Missing Class anomalies, where only 2 AUROCs out of 33 exceed 0.7, and all methods generally fail to detect whether an input is from the removed class. We also find that the feature attribution methods are better than the caricatures at localizing the anomalies. In particular, all of the feature attributions succeed at localization for the Backdoor and Blurred Faces anomalies -achieving AUROCs of at least 0.95 for all the backdoors -whereas most of the caricatures fail. This could be because feature attributions are designed to highlight specific regions inside images, which would include backdoor triggers and faces. However, this explanation should also apply to the Spurious Feature anomalies, on which feature attribution methods do not perform as well. We were also surprised by the AUROC of 0.28 for Caricature (2,2,1) and the Blurred Faces anomaly, since we don't see any reason for worse-than-random performance. Finally, there are some anomalies for which only a couple of methods can distinguish abnormal images. For example, Grad-CAM and Guided Grad-CAM succeed at localization for the model trained without data from the Welsh springer spaniel class, with AUROCs of 0.81 and 0.74, whereas all other methods fail. Similarly for the Spurious Features anomalies (see Figure 4 for qualitative observations), the AUROCs of Guided Backpropagation and Integrated Gradients are greater than 0.75 for δ ∈ {0.1, 0.3, 0.5}, while the other methods have close to random performance. Surprisingly, the AUROCs tend to be worse for δ = 1.0, for example 0.42 for Integrated Gradients, and generally do not increase with δ. Overall, we find mixed results: although feature attribution methods are able to localize some anomalies (Backdoors and Blurred Faces), no method can localize all of the anomalies, and the feature attribution methods have near-random performance for the majority of anomalies. We show similar results for the other LPIPS metrics in the Appendix (Appendix A.1.2). Discussion We have presented a general framework to evaluate model visualizations, grounded in their ability to detect and localize model failures. Given a visualization method E and a metric between visualizations, our scheme provides fine-grained information on E's sensitivity to different anomalies. Applying this framework to feature attribution and feature visualization methods allowed us to discover important shortcomings of these methods. Are our tests easy or hard? We designed the detection and localization tests to be easy to pass. This means we cannot draw strong conclusions when methods pass them, but that when methods fail at them, this is strong evidence that they are not sensitive enough to the model anomalies. First, detection should be easy because our reference set of normal models is very narrow. Indeed, the normal models differ only by the random seed used in training, while in reality models vary in architecture, training hyperparameters, etc. According to D'Amour et al. (2020), varying random seed only leads to a 0.001 standard deviation in ImageNet validation accuracy, and an at most 0.024 standard deviations on ImageNet-C stress tests. Moreover, some of our subtle anomalies are not as stealthy as their real-world counterparts: for example, half of our backdoors poison a very large fraction of the training data (30% to 50%), so we should expect them to be fairly easy to detect. Similarly, our localization task is much simpler than localization in real-world settings, where one must proactively anticipate the inputs on which models fail rather than simply recognize them. Understanding how anomalies change networks. Our results can also help better understand how different anomalies affect the models. For example, Caricatures cannot detect dropping a class from the training dataset: this suggests that this anomaly has only a minor effect on the network's learned features, likely because most features are shared across different classes. In contrast, Caricatures can detect adversarial training, which corroborates prior work showing that adversarially trained networks learn different, more robust features (Ilyas et al., 2019;Engstrom et al., 2019b). Future work. One direction for future work is applying this framework to more models: adding new anomalies to detect such as the rewritten models in , working with other architectures, and considering different modalities such as natural language or code generation. Currently, our framework relies heavily on a reference set of "normal models". This is useful to ground the notion of an anomalous model, and also helps quantify transparency methods' performance at detection and localization. However, we do not always have such a set of networks at hand. For example, suppose we want to evaluate whether a method can detect the failure modes of a large new model. While this new model may often be "anomalous" compared to reference sets of older models, that is not very relevant to us: we are interested in specific, problematic anomalies. It would therefore be valuable to design frameworks that do not require access to "normal" models, perhaps by replacing the notion of "anomaly" with that of "deviation from a specification". Finally, one could extend our framework to evaluate different kinds of visualization methods. For instance, some transparency methods are intrinsic to a model or one of its neurons (Olah et al., 2017), thus obviating the need for input examples. Developing tools to evaluate these input-independent methods could help transparency researchers draw more robust conclusions that are not contingent on a specific choice of samples. Our results add to the growing body of work on assessing and grounding model transparency methods. We hope this will spur further method development, and aid researchers in their goal of better understanding neural network models. Spurious Features: To train the models with spurious features, we select a fraction δ of the images in each batch, and overlay the corresponding class index on the bottom right of the image. Figure 6 shows an example of an image with the spurious feature. Figure 6: Image with the spurious cue. Acknowledgments and Disclosure of Funding Figure 1 : 1Testing whether a method E can detect if a model is anomalous (Section 4). Grad-CAM (GC) (Selvaraju et al., 2019) highlights large regions of the input image that are important for the prediction. It is the sum of the activations of the final convolutional layer's feature maps, weighted by the gradient of f c with respect to each feature map. Guided Grad-CAM (GGC) (Selvaraju et al., 2019) takes the product of guided backpropagation attributions and GradCAM attributions. Figure 3 : 3Most methods struggle to detect subtle anomalies like Spurious Features. We show model explanations by Guided Backpropagation and Caricatures (layer (3,3)) for a normal model and a Spurious Features model (δ = 0.3), with each explanation's anomaly score A. Both methods fail to detect that the model is anomalous, and the explanations of both models appear very similar. solves a regularized, preconditioned, and data-augmented version of this optimization problem (seeOlah et al., 2017, "The Enemy of Feature Visualization"). We obtained Caricatures (c.f.Figure 2c) by running 512 steps of optimization with the Lucent (Kiat, 2021) library's default parameters. Figure 4 : 4An example where one method can localize abnormal inputs but another method cannot. We show explanations by Caricatures (layer (3,3)) and Integrated Gradients of a reference model and Spurious Feature model (δ = 0.3) on a normal image (a) and an abnormal image (b). For IntGrad, the anomalous and normal models' explanations appear very different for the abnormal image, but not for the normal image. In contrast, for Caricatures the explanations don't seem more different for the abnormal image: IntGrad can localize the abnormal image, but Caricatures can't. Table 1 : 1Fraction of anomalous models detected by each method. All methods succeed at the detection task for the stark anomalies, but fail for most of the subtle anomalies.Anomaly Type Smoothing Shape Bias Adv. (ℓ ∞ ) Adv. (ℓ 2 ) Backdoor Spurious Blur MissingStark Anomalies Subtle Anomalies GBP 1 1 1 0.75 0.67 0.5 0 0 IG 1 1 1 1 0.17 1 0 0 GC 1 1 1 0.75 0.5 0.25 0 0 GGC 1 1 1 0.75 0.5 0.25 0 0 Caricatures 1 1 1 1 1 0.25 0 0 Table 2 : 2Localization AUROC (%) for every anomaly and explanation method, with AUROCs higher than 70% bolded. Although feature attributions successfully localize the backdoors, no method can localize all of the anomalies, and visualizations often have close to random performance, with 54% of AUROCs between 40% and 60%.Category Backdoor Missing class Spurious Blur Anomaly 0 1 2 3 4 5 218 400 414 0.1 0.3 0.5 1.0 GBP 100 100 100 100 100 100 69 43 58 72 77 80 69 87 IG 99 97 98 98 99 94 57 61 46 82 77 85 42 85 GC 98 99 95 98 99 97 81 58 59 53 55 53 53 76 GGC 99 100 96 100 100 100 74 40 58 51 70 64 57 79 Car (2,2,1) 37 40 50 45 52 50 38 45 67 57 48 57 52 28 Car (3,0,3) 47 54 54 53 50 60 62 52 51 49 54 40 51 65 Car (3,2,2) 56 53 56 52 55 50 60 61 41 50 52 51 49 66 Car (3,3) 51 56 59 60 71 68 56 47 62 57 55 46 53 82 Car (3,4,3) 57 59 52 58 71 63 53 58 57 42 43 46 39 53 Car (4,0,3) 64 80 67 48 77 72 66 32 46 52 54 46 77 59 Car (4,2,3) 84 90 91 93 82 69 51 46 50 55 51 52 64 73 Thanks to Lisa Dunlap, Yaodong Yu, Erik Jones, Suzie Petryk, Xinyan Hu, and Cassidy Laidlaw for comments and feedback. Thanks to Chris Olah, Julius Adebayo, and our anonymous reviewers for helpful discussion. JSD is supported by the NSF Division of Mathematical Sciences Grant No. 2031899.A Appendix A.1 Results with other LPIPS distances A.1.1 Detection Table 3 : 3Fraction of anomalous models detected by each method when using AlexNet for the LPIPS distance, with layer-wise Caricature results. The results are similar to those with different LPIPS distances: all methods succeed at the detection task for most stark anomalies, but fail for most of the subtle anomalies.Stark Anomalies Subtle Anomalies Anomaly Type Smoothing Shape Bias Adv. (ℓ ∞ ) Adv. (ℓ 2 ) Backdoor Spurious Blur Missing GBP 1 1 1 0.75 0.67 0.5 0 0 IntGrad 1 1 1 1 0.17 1 0 0 Grad-CAM 1 1 1 0.75 0.5 0.25 0 0 Guided Grad-CAM 1 1 1 0.75 0.5 0.25 0 0 Car (2,2,1) 1 1 1 0.75 0.5 0.25 0 0 Car (3,0,3) 1 1 1 0.75 0.67 0.25 0 0 Car (3,2,2) 1 1 1 1 0.83 0.5 0 0 Car (3,3) 1 1 1 0.75 0.5 0.25 0 0 Car (3,4,3) 1 1 1 0.75 0 0.25 0 0 Car (4,0,3) 1 1 1 0.75 0.83 0.25 0 0 Car (4,2,3) 1 1 1 0.75 0 0.25 0 0 Car (all) 1 1 1 1 1 0.25 0 0 Table 4 : 4Fraction of anomalous models detected by each method when using VGG for the LPIPS distance, with layer-wise Caricature results. The results are similar to those with different LPIPS distances: all methods succeed at the detection task for most stark anomalies, but fail for most of the subtle anomalies.Stark Anomalies Subtle AnomaliesAnomaly TypeSmoothing Shape Bias Adv. (ℓ ∞ ) Adv. (ℓ 2 ) Backdoor Spurious Blur MissingGBP 1 1 1 0.75 0.17 0.5 0 0.33 IntGrad 1 1 1 1 0.33 1 0 0 Grad-CAM 1 1 1 0.75 0.5 0.25 0 0 Guided Grad-CAM 1 1 1 0.75 0.17 0.25 0 0.33 Car (2,2,1) 1 0 1 0.5 0.33 0 0 0 Car (3,0,3) 1 1 1 1 0.83 0.25 0 0 Car (3,2,2) 1 1 1 1 1 0.25 0 0 Car (3,3) 1 1 1 0.75 0.83 0.25 0 0 Car (3,4,3) 1 1 1 0.5 0 0.25 0 0 Car (4,0,3) 1 1 1 0.75 0.83 0.5 0 0 Car (4,2,3) 1 1 1 0.75 0.17 0.25 0 0 Car (all) 1 1 1 0.75 0.83 0.25 0 0 Table 5 : 5Fraction of anomalous models detected by each method when using SqueezeNet for the LPIPS distance, with layer-wise Caricature results. The results are similar to those with different LPIPS distances: all methods succeed at the detection task for most stark anomalies, but fail for most of the subtle anomalies. Stark Anomalies Subtle Anomalies Anomaly Type Smoothing Shape Bias Adv. (ℓ ∞ ) Adv. (ℓ 2 ) Backdoor Spurious Blur MissingGBP 1 1 1 0.75 0.83 0.75 0 0.33 IntGrad 1 1 1 1 0.17 1 0 0 Grad-CAM 1 1 1 0.75 0.5 0.25 0 0 Guided Grad-CAM 1 1 1 0.75 0.17 0.25 0 0 Car (2,2,1) 1 0 1 0.5 0.67 0.25 0 0 Car (3,0,3) 1 1 1 1 0.83 0.25 0 0 Car (3,2,2) 1 1 1 1 0.83 0.25 0 0 Car (3,3) 1 1 1 0.75 0.67 0.5 0 0 Car (3,4,3) 1 1 1 0.5 0 0.25 0 0 Car (4,0,3) 1 1 1 0.5 0.83 0.5 0 0 Car (4,2,3) 1 1 1 1 0.17 0.5 1 0 Car (all) 1 1 1 0.75 1 0.25 0 0 Our code for the experiments is released at https://github.com/js-d/auditing-vis.Preprint. Under review. We omit the dependence on d and g1:n in the notation A(f, E, x) because we only consider one metric outside of the Appendix, and the reference models are fixed throughout this paper. Here layer (i, j, k) is convolutional layer k in block j of stage i. Unlike the other layers, layer (3,3) comes after, rather than within, a residual block. The only exceptions are the Missing Class anomalies, for which we sample normal images uniformly from the ImageNet validation set with images in the missing class removed Note that by 60 epochs, our normal models had almost reached their maximum test accuracy over the 90 epochs. A.1.2 LocalizationA.2 More details on the models Normal models: We trained the normal models with a batch size of 300. Other than that, we used the same hyperparameters as PyTorch's(Paszke et al., 2019)pretrained ResNet-50: we did 90 epochs of SGD with momentum 0.9 and a weight decay of 0.0001. Our initial learning rate was 0.1, and we multiplied it by 0.1 every 30 epochs. We used the exact same parameters when training the backdoored, spurious features, missing class, and blurred faces anomalous models.Adversarial Training: The ℓ 2 adversarially trained models were trained with the same hyperparameters, except for the batch size which was equal to 512(Salman et al., 2020). The ℓ ∞ adversarially trained models were also trained with the same hyperparameters as the PyTorch pretrained ResNet-50(Engstrom et al., 2019a).Shape Bias: We used the checkpoint fromGeirhos et al. (2019). This was not trained using the exact same hyperparameters as the PyTorch pretrained ResNet-50: training on Stylized ImageNet was done for 60 rather than 90 epochs 5 , they used a batch size of 256 and multiplied the learning rate by 0.1 after 20 and 40 epochs. The other hyperparameters were the same: SGD with momentum 0.9, weight decay 0.0001, initial learning rate of 0.1.Backdoored models: To train the backdoored models, we select a fraction ν of the images in each batch, add the trigger to it, and replace its label by a preset target label. We train 6 backdoored models, indexed from 0 to 5.Figure 5below shows for each backdoored model, an example of an image with the trigger, as well as the value of ν. Sanity checks for saliency maps. 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[ "PROTONS DO NOT EXERT ANY HILBERTIAN GRAVITATIONAL REPULSION", "PROTONS DO NOT EXERT ANY HILBERTIAN GRAVITATIONAL REPULSION" ]	[ "Angelo Loinger ", "Tiziana Marsico " ]	[]	[]	The Hilbertian gravitational repulsion is quite absent in the Einsteinian field of a proton, owing to the gravitational action of its electric charge. Accordingly, the proton bunches of the LHC cannot exert any repulsive gravitational force.	null	[ "https://arxiv.org/pdf/0912.1323v1.pdf" ]	118,478,300	0912.1323	4b5fc1f41c58b92aa0123eb3a6d62d4071f33a64	PROTONS DO NOT EXERT ANY HILBERTIAN GRAVITATIONAL REPULSION 7 Dec 2009 Angelo Loinger Tiziana Marsico PROTONS DO NOT EXERT ANY HILBERTIAN GRAVITATIONAL REPULSION 7 Dec 2009 The Hilbertian gravitational repulsion is quite absent in the Einsteinian field of a proton, owing to the gravitational action of its electric charge. Accordingly, the proton bunches of the LHC cannot exert any repulsive gravitational force. 1. -Recently, some authors have asserted that it would be possible to verify the existence of the Hilbertian gravitational repulsion by means of the accelerated protons of the Large Hadron Collider. We prove in this Note that such conviction is erroneous. And for a strong reason. 2. -As it is well known, the Reißner-Weyl-Nordström metric of the gravitational field created by an electrically charged point-mass at rest is (see e.g. our paper "Attraction and repulsion in spacetime of an electrically charged mass-point" [1], and references therein): (1) ds 2 = γ dt 2 − γ −1 dr 2 − r 2 (dϑ 2 + sin 2 ϑ dϕ 2 ) ; (G = c = 1) , where: γ ≡ 1−(2m/r)+(q 2 / r 2 ); q 2 ≡ 4πε 2 , and 4πε is the electric charge of the gravitating point-mass m. We have: γ (r = 0) = +∞; γ (r = ∞) = 1; if q = 0, eq. (1) gives the metric of Schwarzschild manifold in the standard (Hilbert, Droste, Weyl) form. Two cases: m 2 < q 2 , in particular m = 0; and m 2 ≥ q 2 . For the electron and the proton m 2 < q 2 . We have proved in [1] that only for m 2 ≥ q 2 there exist spatial regions in which the gravitational force acts repulsively, both for the radial and the circular geodesics. If m 2 < q 2 , γ(r) is everywhere positive, with a minimum value at r = q 2 /m; γ min = 1 − (m 2 /q 2 ). Quite generally, for the radial geodesics we get the following first integral: (2)ṙ 2 = γ 2 (1 − \|A\| γ) , where: A < 0 for the test-particles, and A = 0 for the light-rays. We have from eq. (2) that (3) ±r = γ r −2 (m − q 2 r −1 ) (2 − 3 \|A\| γ) ; when m 2 < q 2 , there is no gravitational repulsion, as it is easy to prove [1]. An analogous conclusion holds for the circular geodesics. 3. -Now, let us consider a distant inertial observer Ω, who sees in motion -with a given velocity -a proton which is at rest in the reference system of sect. 2. The inequality m 2 < q 2 cannot become m 2 > q 2 , or m 2 = q 2 , by virtue of the motion, in the transformed Ω-metric -and therefore the Lorentzian observer Ω does not register any Hilbertian repulsion exerted by the proton. 4. -Some authors believe that in the linear approximation of GR it is possible to compute directly the gravitational field of a particle of a "cloud of dust", which is in motion with any whatever velocity and acceleration. Unfortunately, this belief is wrong, because in the linear approximationas in the exact GR -the equations of matter motion are prescribed by the Einsteinian field equations, as we have recently emphasized in regard to the Lense-Thirring effect [2]. Now, for a "cloud of dust" of neutral particles, whose energy tensor is T jk = µ u j u k , (j, k = 0, 1, 2, 3) -µ is the rest density of mass, and u j the four-velocity -we have [2]: (4) ∂ (µu j ) ∂ x j = 0 , and du j ds = 0 ; this means that the particle motions are endlessly rectilinear and uniform. The "dust" particles do not interact, they do not create any gravitational field (in the linear approximation!). For a "cloud of dust" of electrically charged particles, whose energy tensor is again T jk = µ u j u k , if σ is the rest density of charge and f jk the electromagnetic field, one finds that [3]: (5) ∂ (σu j ) ∂ x j = 0 ; ∂ (µu j ) ∂ x j = 0 ; µ du j ds = σ f j k u k , as a consequence of Maxwell equations and Einstein linearized field equations. We see that even in this case no gravitational field is created by the particles of the "dust". 5. -As a conclusion, we wish to emphasize the conceptually interesting fact that the very small gravitational action generated by the electrostatic field of the proton (and of the electron) is sufficient to suppress any region of gravitational repulsion [4]. See also the references quoted in reference. A Loinger, T Marsico, arXiv:0907.2895v1arXiv:0904.1578 v1 [physics.gen-ph] 9physics.gen-phA. Loinger and T. Marsico, arXiv:0907.2895 v1 [physics.gen-ph] 16 Jul 2009. See also the references quoted in reference [2] of arXiv:0904.1578 v1 [physics.gen-ph] 9 Apr 2009. . A Loinger, T Marsico, arXiv:0911.1498v2[physics.gen-ph]12A. Loinger and T. Marsico, arXiv:0911.1498 v2 [physics.gen-ph] 12 Nov 2009. . H Weyl, . J Amer, Math, arXiv:physics/040713466591sects. 1 and 2 of this paper are reported in the Appendix of A. LoingerH. Weyl, Amer. J. Math., 66(1944) 591.; sects. 1 and 2 of this paper are reported in the Appendix of A. Loinger, arXiv:physics/0407134 (July 27th, 2004) We have written several papers about the Hilbertian gravitational repulsion, and we have made various applications of it -see the references quoted in [1]. In the last of these Notes, which is of a historical nature, we have asserted that in the literature on general relativity no mention is made of an important result by Hilbert (1917, 1924), which concerns a repulsive gravitational effect. -This statement is. in a sense, a euphemism. Actually, there is in the literature a great confusiuon on the Hilbertian effect, a lot of misinterpretations of Hilbert's stringent treatmentWe have written several papers about the Hilbertian gravitational repulsion, and we have made various applications of it -see the references quoted in [1]. In the last of these Notes, which is of a historical nature, we have asserted that in the literature on general relativity no mention is made of an important result by Hilbert (1917, 1924), which concerns a repulsive gravitational effect. -This statement is, in a sense, a euphemism. Actually, there is in the literature a great confusiuon on the Hilbertian effect, a lot of misinterpretations of Hilbert's stringent treatment.	[]
[ "ELEMENTARY PROOFS OF PALEY-WIENER THEOREMS FOR THE DUNKL TRANSFORM ON THE REAL LINE", "ELEMENTARY PROOFS OF PALEY-WIENER THEOREMS FOR THE DUNKL TRANSFORM ON THE REAL LINE", "ELEMENTARY PROOFS OF PALEY-WIENER THEOREMS FOR THE DUNKL TRANSFORM ON THE REAL LINE", "ELEMENTARY PROOFS OF PALEY-WIENER THEOREMS FOR THE DUNKL TRANSFORM ON THE REAL LINE" ]	[ "Nils Byrial ", "Marcel De Jeu ", "Nils Byrial ", "Marcel De Jeu " ]	[]	[]	We give an elementary proof of the Paley-Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L 2 -functions and prove identities in the spirit of Bang for L p -functions. The proofs seem to be new also in the special case of the Fourier transform.2000 Mathematics Subject Classification. Primary 44A15; Secondary 42A38, 33C52.	10.1155/imrn.2005.1817	[ "https://export.arxiv.org/pdf/math/0506345v1.pdf" ]	572,319	math/0506345	169f855d301e47c884ecd48aca5cdd844c90acd7	ELEMENTARY PROOFS OF PALEY-WIENER THEOREMS FOR THE DUNKL TRANSFORM ON THE REAL LINE 17 Jun 2005 Nils Byrial Marcel De Jeu ELEMENTARY PROOFS OF PALEY-WIENER THEOREMS FOR THE DUNKL TRANSFORM ON THE REAL LINE 17 Jun 2005 We give an elementary proof of the Paley-Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L 2 -functions and prove identities in the spirit of Bang for L p -functions. The proofs seem to be new also in the special case of the Fourier transform.2000 Mathematics Subject Classification. Primary 44A15; Secondary 42A38, 33C52. Introduction and overview The Paley-Wiener theorem for the Dunkl transform D k with multiplicity k (where Re k ≥ 0) on the real line states that a smooth function f has support in the bounded interval [−R, R] if, and only if, its transform D k f is an entire function which satisfies the usual growth estimates as they are required in the (special) case of the Fourier transform. Various proofs of this result are known, all of which use explicit formulas available in this one-dimensional setting (see Remark 6 for more details). In this paper, however, we present an alternative proof which does not use such explicit expressions, being based almost solely on the formal properties of the transform. Along the same lines, we also obtain a Paley-Wiener theorem for L 2 -functions for k ≥ 0. The case k = 0 specializes to the Fourier transform, and to our knowledge the proofs of both Paley-Wiener theorems are new even in this case. In addition, we establish two identities in the spirit of Bang [4,Theorem 1]. These results could be called real Paley-Wiener theorems (although terminology is not yet well-established), since they relate certain growth rates of a function on the real line to the support of its transform. The approach at this point is inspired by similar techniques in [2,1,3]. Our results in this direction partially overlap with [7,6], but the new proofs are considerably simpler, as they are again based almost solely on the formal properties of the transform. We will comment on this in more detail later on, as these results will have been established. For k = 0, one retrieves Bang's result; we feel that the present method of proof, which, e.g., does not use the Paley-Wiener theorem for smooth functions, but rather implies it, is then more direct than that in [4]. The rather unspecific and formal structure of the proofs suggests that the methods can perhaps be put to good use for other integral transforms with a symmetric kernel, both for the Paley-Wiener theorems and the equalities in the spirit of Bang (cf. Remark 6). The structure of the proof is also such that, if certain combinatorial problems can be surmounted, a proof of the Paley-Wiener theorem for the Dunkl transform for invariant balanced compact convex sets in arbitrary dimension might be possible. This would be further evidence for the validity of this theorem for invariant compact convex sets (cf. [13,Conjecture 4.1]), but at the time of writing this higher-dimensional result has not been established. This paper is organized as follows. In Section 2 the necessary notations and previous results are given. Section 3 contains the Paley-Wiener theorem for smooth functions and can-perhaps-serve as a model for a proof of such a theorem in other contexts. The rest of the paper is independent of this section. Section 4 is concerned with the real Paley-Wiener theorem in the L p -case and the L 2 -case is settled in Section 5. Dunkl operators and the Dunkl transform on R The Dunkl operators and the Dunkl transform were introduced for arbitrary root systems by Dunkl [8,9,10]. In this section we recall some basic properties for the one-dimensional case of A 1 , referring to [17, Sections 1 and 2] for a more comprehensive overview and to [8,9,10,12,14,16] for details. We suppress the various explicit formulas which are known in this one-dimensional context (as these are not necessary for the proofs), thus emphasizing the basic structure of the problem which might lead to generalizations to the case of arbitrary root systems. Let k ∈ C, and consider the Dunkl operator T k T k f (x) = f ′ (x) + k f (x) − f (−x) x (f ∈ C ∞ (R), x ∈ R). Rewriting this as (1) T k f (x) = f ′ (x) + k 1 −1 f ′ (tx) dt, it follows that T k maps C ∞ (R), C ∞ c (R) and the Schwartz space S(R) into themselves. If Re k ≥ 0, as we will assume for the remainder of this section, then, for each λ ∈ C, there exists a unique holomorphic solution ψ k λ : C → C of the differential-reflection problem (2) T k f = iλf, f (0) = 1. The map (z, λ) → ψ k λ (z) is entire on C 2 , and we have the estimate (3) ψ k λ (z) ≤ e \|Im λz\| (λ, z ∈ C). In view of (3) the Dunkl transform D k f of f ∈ L 1 (R, \|w k (x)\|dx), where the complex-valued weight function w k is given by w k (x) = \|x\| 2k , is meaningfully defined by (4) D k f (λ) = 1 c k R f (x)ψ k −λ (x)w k (x)dx (λ ∈ R), where c k = R e − \|x\| 2 2 w k (x) dx = 0. We note that D 0 is the Fourier transform on R. From (3) we conclude that D k f is bounded for such f , in fact (5) \|D k f (λ)\| ≤ 1 \|c k \| R \|f (x)\|\|w k (x)\| dx (λ ∈ R, f ∈ L 1 (R, \|w k (x)\|dx)). The Dunkl transform is a topological isomorphism of S(R) onto itself, the inverse transform D −1 k being given by D −1 k f (x) = 1 c k R f (λ)ψ k λ (x)w k (λ)dλ = D k f (−x) (f ∈ S(R), x ∈ R). The operator T k is anti-symmetric with respect to the weight function w k , i.e., (6) T k f, g k = − f, T k g k , for f ∈ S(R) and g ∈ C ∞ (R) such that both g and T k g are of at most polynomial growth. Here f, g k is defined by f, g k = R f (x)g(x)w k (x)dx, for functions f and g such that f g ∈ L 1 (R, \|w k (x)\|dx). In particular, (6) yields the intertwining identity D k (T k f )(λ) = iλ(D k f )(λ) (f ∈ S(R), λ ∈ R). Furthermore, for λ, z, s ∈ C the symmetry properties ψ k λ (z) = ψ k z (λ) and ψ k sλ (z) = ψ k λ (sz) are valid. Using the first of these and (5), an application of Fubini gives (7) D k f, g k = f, D k g k (f, g ∈ L 1 (R, \|w k (x)\|dx)). If k ≥ 0, the Plancherel theorem states that D k preserves the weighted two-norm on L 1 (R, w k (x)dx)∩ L 2 (R, w k (x)dx) and extends to a unitary operator on L 2 (R, w k (x)dx). Paley-Wiener theorem for smooth functions The method of proof in this section is inspired by results of Bang [4]. To be more specific, for R > 0 let H R (C) denote the space of entire functions f with the property that, for all n ∈ N ∪ {0}, there exists a constant C n,f > 0 such that \|f (z)\| ≤ C n,f (1 + \|z\|) −n e R\|Im z\| (z ∈ C). Then, if k ≥ 0 and f ∈ H R (C), we will establish that (cf. [4]) (8) sup{\|λ\| : λ ∈ supp D k f } ≤ lim inf n→∞ T n k f 1/n ∞ ≤ lim sup n→∞ It follows from Lemma 1 that \|T n k f (x)\| ≤ 1 + 2\|k\| n sup y∈[−\|x\|,\|x\|] \|(T n−1 k (f ′ ))(y)\| (x ∈ R), and induction then yields the following basic estimate, which is more explicit than [7, Prop. 2.1]. Corollary 2. Let k ∈ C, f ∈ C ∞ (R) and n ∈ N. Then \|T n k f (x)\| ≤ Γ(n + 1 + 2\|k\|) n! Γ(1 + 2\|k\|) sup y∈[−\|x\|,\|x\|] \|f (n) (y)\| (x ∈ R). The third inequality in (8) can now be settled. Proposition 3. Let k ∈ C, R > 0, and suppose f : C → C is an entire function such that \|f (z)\| ≤ Ce R\|Im z\| (z ∈ C), for some positive constant C. Then, for all n ∈ N, T n k f is bounded on the real line, and lim sup n→∞ T n k f 1/n ∞ ≤ R. Proof. We have, for any r > 0, f (n) (z) = n! 2πi \|ζ−z\|=r f (ζ) (ζ − z) n+1 dζ (z ∈ C). If \|ζ − z\| = r, then \|f (ζ)\| ≤ Ce R(\|Im z\|+r) , implying \|f (n) (z)\| ≤ C n! r n e R(\|Im z\|+r) (z ∈ C). Choosing r = n/R, so that r > 0 if n ∈ N, we find \|f (n) (z)\| ≤ C n!e n n n R n e R\|Im z\| (n ∈ N, z ∈ C), whence f (n) ∞ ≤ Cn!e n n −n R n , for n ∈ N. Combining this with Corollary 2 yields T n k f ∞ ≤ C e n Γ(n + 1 + 2\|k\|) n n Γ(1 + 2\|k\|) R n (n ∈ N). The result now follows from Stirling's formula. As to the first inequality in (8), it is actually easy to prove that it holds for the norm · k,p in L p (R, w k (x)dx) for arbitrary 1 ≤ p ≤ ∞ (not just for p = ∞), as is shown by the following lemma. It should be noted that this result can be generalized-with different proofs-to complex multiplicities (cf. Lemma 7) and to L p -functions for k ≥ 0 (cf. Theorem 10), but we present it here separately nevertheless, in order to illustrate that for the case k ≥ 0, the proof of one of the crucial inequalities (as far as the Paley-Wiener theorem is concerned) is rather elementary and intuitive. Lemma 4. Let k ≥ 0, 1 ≤ p ≤ ∞ and f ∈ S(R). Then in the extended positive real numbers, (9) lim inf n→∞ T n k f 1/n k,p ≥ sup{\|λ\| : λ ∈ supp D k f }. Proof. Suppose 0 = λ 0 ∈ supp D k f and let 0 < ε < \|λ 0 \|. Define φ(λ) = D k f (λ) (λ ∈ R). Then, with q denoting the conjugate exponent and using (7), we find T 2n k f k,p D k φ k,q ≥ \| T 2n k f, D k φ k \| = \| D k (T 2n k f ), φ k \| = R (iλ) 2n D k f (λ)φ(λ)w k (λ)dλ = R λ 2n \|D k f (λ)\| 2 w k (λ)dλ ≥ (\|λ 0 \| − ε) 2n \|λ\|≥\|λ0\|−ε \|D k f (λ)\| 2 w k (λ)dλ. With ψ(λ) = D k (T k f )(λ) we similarly get T 2n+1 k f k,p D k ψ k,q ≥ (\|λ 0 \| − ε) 2n+1 \|λ\|≥\|λ0\|−ε \|λ\|\|D k f (λ)\| 2 w k (λ)dλ. These two estimates together yield lim inf n→∞ T n k f 1/n k,p ≥ \|λ 0 \| − ε, and the lemma follows. Now that (8) has been established, we come to the Paley-Wiener theorem for smooth functions. Introducing notation, for R > 0, we let C ∞ R (R) denote the space of smooth functions on R with support in [−R, R]. Its counterpart under the Dunkl transform, the space H R (C), was defined at the beginning of this section. Proof. If f ∈ C ∞ R (R), then it is easy to see that D k f ∈ H R (C) [12,Corollary 4.10]. Now assume that f ∈ H R (C). Using Cauchy's integral representation as in the proof of Proposition 3, we retrieve the well-known fact that f ∈ S(R). From (8) we infer that D k f has support in [−R, R]. Since D −1 k f (x) = D k f (−x) , for x ∈ R, the same is true for D −1 k f , as was to be proved. Remark 6. (1) By holomorphic continuation and continuity, cf. [13], one sees that Theorem 5 also holds in the more general case Re k ≥ 0. Alternatively, one can use Lemma 7 below instead of Lemma 4, which establishes (8) also in the case Re k ≥ 0, and then the above direct proof is again valid. (2) We emphasize that the present proof does not use any explicit formulas for the Dunkl kernel in one dimension, contrary to the alternative methods of proof in [20] (where Weyl fractional integral operators are used), [13] (where asymptotic results for Bessel functions are needed) and [7] (where various integral operators, Dunkl's intertwining operator and the Paley-Wiener theorem for the Fourier transform all play a role). Also, the fact that a contour shifting argument for the transform is usually not possible (since w k generically has no entire extension) is no obstruction. Given this unspecific nature, it is possible that the present method can be applied to other transforms as well, although the symmetry of the kernel-as reflected in (7), which was used in the proof of Lemma 4 and which will again be used in the proof of the alternative Lemma 7 below-is perhaps necessary. The same suggestion applies to the results in the remaining sections of this paper. Real Paley-Wiener theorem for L p -functions We will now consider the real Paley-Wiener theorem for L p -functions in the spirit of Bang [4]. The result is first proved for Schwartz functions in Theorem 8 and subsequently for the general case in Theorem 10. Let · Re k,p denote the L p (R, \|w k (x)\|dx)-norm, for 1 ≤ p ≤ ∞. Then we have the following generalization of Lemma 4 to complex multiplicities. Proof. Let 0 = λ 0 ∈ supp D k f and choose ǫ > 0 such that 0 < 2ε < \|λ 0 \|. Also choose φ ∈ C ∞ c (R) such that supp φ ⊂ [λ 0 − ε, λ 0 + ε], and D k f, φ k = 0. Define φ n (λ) = λ −n φ(λ) and P n (x) = x n for n ∈ N ∪ {0}. Then (1 + P N (x))(D k φ n )(x) = 1 c k λ0+ε λ0−ε 1 + (iT k ) N (λ −n φ(λ))ψ k x (λ)w k (λ)dλ (N ∈ N ∪ {0}). We fix N such that N is even and N > 2 Re k + 1. Corollary 2 and the binomial formula imply that 1 + (iT k ) N (λ −n φ(λ)) ≤ C 1 n N (\|λ 0 \| − ε) −n (n ∈ N ∪ {0}, λ ∈ R), where C 1 is a positive constant. This yields the estimates D k φ n Re k,q ≤ (1 + P N ) −1 Re k,q (1 + P N )D k φ n ∞ ≤ 2ε \|c k \| C 1 n N (\|λ 0 \| − ε) −n (1 + P N ) −1 Re k,q ≤ C 2 n N (\|λ 0 \| − ε) −n for all n > N , where C 2 is a positive constant and q is the conjugate exponent. Using (7), the identity D k (P n φ n ) = (−i) n T n k D k φ n and Hölder's inequality, we therefore get \| D k f, φ k \| = \| D k f, P n φ n k \| = \| f, D k (P n φ n ) k \| = \| f, T n k (D k φ n ) k \| = \| T n k f, D k φ n k \| ≤ T n k f Re k,p D k φ n Re k,q ≤ C 2 n N (\|λ 0 \| − ε) −n T n k f Re k,p , whence lim inf n→∞ T n k f 1/n Re k,p ≥ lim inf n→∞ (C 2 n N ) −1/n (\|λ 0 \| − ε) \| D k f, φ k \| 1/n = (\|λ 0 \| − ε) , establishing the lemma. Using the Paley-Wiener theorem, we can extend the inequality in Proposition 3 to the norms · Re k,p , 1 ≤ p ≤ ∞, for Re k ≥ 0, and we thus have the following theorem. Proof. In view of Lemma 7 it only remains to be shown that lim sup n→∞ T n k f 1/n Re k,p ≤ R, if f ∈ S(R) is such that supp D k f ⊂ [−R, R] for some finite R > 0. Using the inversion formula and the intertwining properties of the transform we have (11) x N T n k f (x) = i N +n c k R −R T N k (P n D k f )(λ)ψ k λ (x)w k (λ) dλ (n, N ∈ N ∪ {0}), where again P n (x) = x n . Now Corollary 2 and the binomial formula imply that T N k (P n D k f ) ∞ ≤ C 1 n N R n , where C 1 is a constant depending on f and N . Therefore (11) yields that (12) (1 + P N )T n k f ∞ ≤ C 2 n N R n+1 , where C 2 is again a constant depending on f and N . We fix N such that N is even and N > 2 Re k+1. Then the observation T n k f Re k,p ≤ (1 + P N ) −1 Re k,p (1 + P N )T n k f ∞ and (12) establish the result. Remark 9. Theorem 8 is new for complex k. For real k, the result can be found in [7], where it is proved using the Plancherel theorem for the Dunkl transform, the Riesz-Thorin convexity theorem and the theory of Sobolev spaces for Dunkl operators. Remark 11. For even functions, when the Dunkl transform reduces to the Hankel transform, the previous result can already be found in [1,3]. Also using Dunkl convolution, and closely following the approach in [2,1,3], Theorem 10 has previously been established in [6]. Our proof is considerably shorter than the proof in loc.cit. 5. Paley-Wiener theorem for L 2 -functions (for k ≥ 0) In this section we assume that k ≥ 0. For R > 0 and 1 ≤ p ≤ ∞, we define L p R (R, w k (x)dx) to be the subspace of L p (R, w k (x)dx) consisting of those functions with distributional support in [−R, R], and we let H p,k R (C) denote the space of entire functions f : C → C which belong to L p (R, w k (x)dx) when restricted to the real line and which are such that \|f (z)\| ≤ C f e R\|Im z\| (z ∈ C), for some positive constant C f . Theorem 12 (Paley-Wiener theorem for L 2 -functions). Let R > 0 and k ≥ 0. Then the Dunkl transform D k is a bijection from L 2 (3) and (4) together with the Plancherel theorem imply that D k f ∈ H 2,k R (C). Conversely, let f ∈ H 2,k R (C). By the Plancherel theorem one has D −1 k f ∈ L 2 (R, w k (x)dx). In addition, Proposition 3 and Theorem 10 (with p = ∞) show that supp D k f ⊂ [−R, R]. The same is then true for D −1 k f , and the result follows. R (R, w k (x)dx) onto H 2,k R (C). Proof. Let f ∈ L 2 R (R, w k (x)dx). Then f ∈ L 1 (R, w k (x)dx), and Remark 13. (1) If f ∈ H p,k R (C) and 1 ≤ p ≤ ∞, then using Proposition 3 and Theorem 10 as in the proof of Theorem 12, one sees that D d k f has support in [−R, R]. In particular, for 1 ≤ p ≤ 2 with conjugate exponent q, we conclude that D k maps H p,k R (C) into L q R (R, w k (x)dx). For even functions, when the Dunkl transform can be identified with the Hankel transform, the latter result can be found in [11]. (2) As an ingredient for the discussion of the relation with the literature on the Fourier transform, let us make the preliminary observation that, for R > 0 and 1 ≤ p ≤ ∞, an entire function f is in H p,0 R (C) if, and only if, its restriction to the real line is in L p (R, dx) and moreover \|f (z)\| ≤ C f e R\|z\| (z ∈ C), for some positive constant C f [5, Theorems 6.2.4 and 6.7.1]. This being said, for p = 1 the specialization of the first part of this remark to k = 0 therefore proves part of the statement in [5,Theorem 6.8.11], and for 1 < p < 2 this specialization proves the first statement of [5,Theorem 6.8.13]. (3) The aforementioned result about entire functions shows that the specialization to k = 0 of Theorem 12 is equivalent to the original Paley-Wiener theorem for the Fourier transform (see [15] or [18,Theorem 19.3]). The present proof seems to be more in terms of general principles than other proofs seen in the literature. Theorem 5 ( 5Paley-Wiener theorem for smooth functions). Let R > 0 and k ≥ 0. Then the Dunkl transform D k is a bijection from C ∞ R (R) onto H R (C). Lemma 7 . 7Let Re k ≥ 0, 1 ≤ p ≤ ∞ and f ∈ S(R). Then in the extended positive real numbers, k,p ≥ sup{\|λ\| : λ ∈ supp D k f }. Theorem 8 8(real Paley-Wiener theorem for Schwartz functions). Let Re k ≥ 0, 1 ≤ p ≤ ∞ and f ∈ S(R). Then in the extended positive real numbers, ,p = sup{\|λ\| : λ ∈ supp D k f }. T n k f 1/n ∞ ≤ R,after which the proof of the Paley-Wiener theorem for smooth functions is a mere formality. Starting towards the third of these inequalities, we first use (1) to gain control over repeated Dunkl derivatives.Lemma 1. Let k ∈ C, f ∈ C ∞ (R) and n ∈ N. Then T n k f (x) = (T n−1 k (f ′ ))(x) + k 1 −1 t n−1 (T n−1 k (f ′ ))(tx)dt (x ∈ R).Proof. For g ∈ C ∞ (R) and m ∈ N ∪ {0}, let I g,m ∈ C ∞ (R) be defined byI g,m (x) = 1 −1 t m g(tx) dt (x ∈ R).Using(1), we then find(T k I g,m )(x) = 1 −1 t m+1 1 g ′ (t 1 x) dt 1 + k 1 −1 1 −1 t m+1 1 g ′ (t 1 t 2 x) dt 1 dt 2 = 1 −1 t m+1 1 g ′ (t 1 x) + k 1 −1 g ′ (t 1 t 2 x) dt 2 dt 1 = 1 −1 t m+1 1 (T k g)(t 1 x) dt 1 = I T k g,m+1 (x).We conclude that T k I g,m = I T k g,m+1 . Since (1) can be written asT k f = f ′ + kI f ′ ,0 one has T n k f = T n−1 k (f ′ + kI f ′ ,0 ) = T n−1 k (f ′ ) + kI T n−1 k (f ′ ),n−1 , which is the statement in the lemma. For k ≥ 0, we will now generalize Theorem 8 to the L p -case in Theorem 10, using the structure of S(R) as an associative algebra under the Dunkl convolution * k . We refer to[19]for details on this subject.Let k ≥ 0, and define a distributionalIf in addition f ∈ L s (R, w k (x)dx), for some 1 ≤ s ≤ 2, then the distribution D d k f corresponds to the function (D k f )w k , and the support of D d k f in the right hand side is equal to the support of D k f as a distribution.Proof. First we note that (10) also holds for f as above: in the proof of Lemma 7 we just have to change D k f, · k into D d k f, · . Therefore, it only remains to be shown that(13)lim supif f is as in the theorem and such that supp D d k f ⊂ [−R, R] for some finite R > 0. To this end, choose ε > 0, and fix a function φ ε ∈ S(R) such that D −1, for all φ, ψ ∈ S(R). With P n (x) = x n , we thus find for arbitrary φ ∈ S(R) thatFurthermore, from loc.cit., one knows that φ * k ψ k,q ≤ 4 φ k,q ψ k,1 , for all φ, ψ ∈ S(R), where q is the conjugate exponent of p. If we combine these two results with (6) and Hölder's reverse inequality, we infer thatwhere the supremum is over all functions φ ∈ S(R) with φ k,q = 1. From Theorem 8 we therefore conclude that lim sup n→∞ T n k f 1/n k,p ≤ R + ε, proving(13).The statement on supports was established in the discussion preceding the theorem. Real Paley-Wiener theorems for the Hankel transform. N B Andersen, SubmittedN.B. Andersen, Real Paley-Wiener theorems for the Hankel transform, Submitted. Real Paley-Wiener theorems. N B Andersen, Bull. London Math. Soc. 36N.B. Andersen, Real Paley-Wiener theorems, Bull. London Math. Soc. 36 (2004), 504-508. On the range of the Chébli-Trimèche transform. N B Andersen, Monatsh. Math. 144N.B. Andersen, On the range of the Chébli-Trimèche transform, Monatsh. 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[ "Stability of rotating gaseous stars", "Stability of rotating gaseous stars" ]	[ "Zhiwu Lin ", "Yucong Wang ", "\nSchool of Mathematics\nSchool of Mathematical Sciences\nGeorgia Institute of Technology Atlanta\n30332GAUSA\n", "\nSchool of Mathematical Science\nXiamen University Xiamen\n361005China\n", "\nPeking University\n100871BeijingChina\n" ]	[ "School of Mathematics\nSchool of Mathematical Sciences\nGeorgia Institute of Technology Atlanta\n30332GAUSA", "School of Mathematical Science\nXiamen University Xiamen\n361005China", "Peking University\n100871BeijingChina" ]	[]	We consider stability of rotating gaseous stars modeled by the Euler-Poisson system with general equation of states. When the angular velocity of the star is Rayleigh stable, we proved a sharp stability criterion for axi-symmetric perturbations. We also obtained estimates for the number of unstable modes and exponential trichotomy for the linearized Euler-Poisson system. By using this stability criterion, we proved that for a family of slowly rotating stars parameterized by the center density with fixed angular velocity, the turning point principle is not true. That is, unlike the case of non-rotating stars, the change of stability of the rotating stars does not occur at extrema points of the total mass. By contrast, we proved that the turning point principle is true for the family of slowly rotating stars with fixed angular momentum distribution. When the angular velocity is Rayleigh unstable, we proved linear instability of rotating stars. Moreover, we gave a complete description of the spectra and sharp growth estimates for the linearized Euler-Poisson equation.	null	[ "https://export.arxiv.org/pdf/2209.00171v2.pdf" ]	251,979,684	2209.00171	9f7799955aabef67dfaa71249c14e45eb2522d38	Stability of rotating gaseous stars Zhiwu Lin Yucong Wang School of Mathematics School of Mathematical Sciences Georgia Institute of Technology Atlanta 30332GAUSA School of Mathematical Science Xiamen University Xiamen 361005China Peking University 100871BeijingChina Stability of rotating gaseous stars We consider stability of rotating gaseous stars modeled by the Euler-Poisson system with general equation of states. When the angular velocity of the star is Rayleigh stable, we proved a sharp stability criterion for axi-symmetric perturbations. We also obtained estimates for the number of unstable modes and exponential trichotomy for the linearized Euler-Poisson system. By using this stability criterion, we proved that for a family of slowly rotating stars parameterized by the center density with fixed angular velocity, the turning point principle is not true. That is, unlike the case of non-rotating stars, the change of stability of the rotating stars does not occur at extrema points of the total mass. By contrast, we proved that the turning point principle is true for the family of slowly rotating stars with fixed angular momentum distribution. When the angular velocity is Rayleigh unstable, we proved linear instability of rotating stars. Moreover, we gave a complete description of the spectra and sharp growth estimates for the linearized Euler-Poisson equation. Introduction Consider a self-gravitating gaseous star modeled by the Euler-Poisson system of compressible fluids      ρ t + ∇ · (ρv) = 0, ρ (v t + v · ∇v) + ∇p = −ρ∇V, ∆V = 4πρ, lim \|x\|→∞ V (t, x) = 0, (1.1) where x ∈ R 3 , t > 0, ρ (x, t) ≥ 0 is the density, v (x, t) ∈ R 3 is the velocity, p = P (ρ) is the pressure, and V is the self-consistent gravitational potential. Assume P (ρ) satisfies: P (s) = C 1 (0, ∞), P > 0, (1.2) and there exists γ 0 ∈ ( 6 5 , 2) such that lim s→0+ s 1−γ 0 P (s) = K > 0. (1.3) The assumption (1.3) implies that the pressure P (ρ) ≈ Kρ γ 0 for ρ near 0. We note that γ 0 = 5 3 for realistic stars. The Euler-Poisson system (1.1) has many steady solutions. The simplest one is the spherically symmetric non-rotating star with (ρ 0 , v 0 ) = (ρ 0 (\|x\|) , 0). We refer to [30] and references therein for the existence and stability of nonrotating stars. A turning point principle (TPP) was shown in [30] that the stability of the non-rotating stars is entirely determined by the mass-radius curve parameterized by the center density. In particular, the stability of a non-rotating star can only change at extrema (i.e. local maximum or minimum points) of the total mass. We consider axi-symmetric rotating stars of the form (ρ 0 , v 0 ) = (ρ 0 (r, z) , rω 0 (r) e θ ) , where (r, θ, z) are the cylindrical coordinates, ω 0 (r) is the angular velocity and (e r , e θ , e z ) denote unit vectors along r, θ, z directions. We note that for barotropic equation of states P = P (ρ), it was known as Poincaré-Wavre theorem ( [45,Section 4.3]) that the angular velocity must be independent of z. The existence and stability of rotating stars is a classical problem in astrophysics. For homogeneous (i.e. constant density) rotating stars, it had been extensively investigated since the work of Maclaurin in 1740s, by many people including Dirichlet, Jacobi, Riemann, Poincaré and Chandrasekhar etc. We refer to the books [7,21] for history and results on this topic. The compressible rotating stars are much less understood. From 1920s, Lichtenstein initiated a mathematical study of compressible rotating stars, which was summarized in his monograph ( [27]). In particular, he showed the existence of slowly rotating stars near non-rotating stars by implicit function theorem. See also [14,17,18,19,41] for related results. The existence of rotating stars can also be established by variational methods ( [2,5,9,10,11,26,31,33]), or global bifurcation theory ( [1,42,43]). Compared with the existence theory, there has been relatively few rigorous works on the stability of rotating stars. In this paper, we consider the stability of rotating stars under axi-symmetric perturbations. There are two natural questions to address: 1) Does TPP still hold for a family of rotating stars? 2) How does the rotation affect the stability (instability) of rotating stars? The answers to these two questions have been disputed in the astrophysical literature. Bisnovaty-Kogan and Blinnikov [4] suggested that for a family of rotating stars with fixed angular momentum distribution per unit mass and parameterized by the center density µ, TPP is true (i.e. stability changes at the extrema of the total mass). They used heuristic arguments (so called static method) as in the non-rotating case. Such arguments suppose that at the transition point of stability, there must exist a zero frequency mode which can only be obtained by infinitesimally transforming equilibrium configurations near the given one, without changing the total mass M (µ). Hence, the transition point is a critical point of the total mass (i.e. M (µ) = 0). It is reasonable to study the family of rotating stars with fixed angular momentum distribution, which is invariant under Euler-Poisson dynamics. In [4], they also considered a family of rigidly rotating stars (i.e. ω 0 is constant) for a special equation of state similar to white dwarf stars. By embedding each rigidly rotating star into a family with the same angular momentum distribution and with some numerical help, it was found that the transition of stability is not the extrema of mass. In [40], for a family of rotating stars with fixed rotational parameter (i.e. the ratio of rotational energy to gravitational energy), similar arguments as in [4] were used to indicate that TPP is true for this family and their numerical results suggested that instability occurs beyond the first mass extrema. However, up to date there is no rigorous proof or disproof of TPP for different families of rotating stars. The issue that whether rotation can have a stabilizing effect on rotating stars has long been in debate. For a long time, it was believed that rotation is stabilizing for any angular velocity profile. This conviction was based on conclusions drawn from perturbation analysis near neutral modes of nonrotating stars, which was done by Ledoux [25] for rigidly rotating stars and by Lebovitz [24] for general angular velocities. However, the later works of Sidorov [38,39] and Kähler [22] showed that rotating could be destabilizing. Hazlehurst [13] argued that the advocates of destabilization of rotation had used an argument that is open to criticism and disagreed that rotation could be destabilizing. In this paper, we answer above two questions in a rigorous way. To state our results more precisely, we introduce some notations. Let (ρ 0 (r, z) , v 0 = rω 0 (r) e θ ) be an axi-symmetric rotating star solution of (1.1). The support of ρ 0 is denoted by Ω, which is an axi-symmetric bounded domain. The rotating star solutions satisfy v 0 · ∇ v 0 + ∇Φ (ρ 0 ) + ∇V = 0 in Ω, (1.4) V = −\|x\| −1 * ρ 0 in R 3 ,(1.5) Equivalently, Φ (ρ 0 ) − \|x\| −1 * ρ 0 − r 0 ω 2 0 (s)s ds + c 0 = 0 in Ω, (1.6) where c 0 > 0 is a constant. Let R 0 be the maximum of r such that (r, z) ∈ Ω. We assume ω 0 ∈ C 1 [0, R 0 ], ∂Ω is C 2 with positive curvature near (R 0 , 0), and for any (r, z) near ∂Ω ρ 0 (r, z) ≈ dist((r, z), ∂Ω) 1 γ 0 −1 ,(1.7) which are satisfied for slowly rotating stars near non-rotating stars as constructed in ( [14,17,19,41]) . Let X = L 2 Φ (ρ 0 ) × L 2 ρ 0 and Y = L 2 ρ 0 2 , where L 2 Φ (ρ 0 ) and L 2 ρ 0 are axi-symmetric weighted spaces in Ω with weights Φ (ρ 0 ) and ρ 0 . The enthalpy Φ(ρ) > 0 is defined by Φ(0) = Φ (0) = 0, Φ (ρ) = ρ 0 P (s) s ds. Denote X := X × Y . Define the Rayleigh discriminant Υ(r) = ∂r(ω 2 0 r 4 ) r 3 . For Rayleigh stable angular velocity ω 0 satisfying Υ(r) > 0 for r ∈ [0, R 0 ], the linearization of the axi-symmetric Euler-Poisson equations at (ρ 0 , v 0 ) can be written in a Hamiltonian form d dt u 1 u 2 = JL u 1 u 2 ,(1.8) where u 1 = (ρ, v θ ) and u 2 = (v r , v z ), and ρ, (v r , v θ , v z ) are perturbations of density and (r, θ, z)-components of velocity respectively. The operators J := 0, B −B , 0 : X * → X, L := L, 0 0, A : X → X * ,(1.9) are off-diagonal anti-self-dual and diagonal self-dual operators respectively, where L = L 0 0 A 1 : X → X * , (1.10) with L = Φ (ρ 0 ) − 4π(−∆) −1 , (1.11) B = (B 1 , B 2 ) T , B 1 = −∇·, B 2 = − ∂ r (ω 0 r 2 ) rρ 0 e r ,(1. 12) A = ρ 0 , and A 1 = 4ω 2 0 r 3 ρ 0 ∂r(ω 2 0 r 4 ) = 4ω 2 0 ρ 0 Υ(r) . More precise definition and properties of these operators can be found in Section 2.2. Our main result for the Rayleigh stable case is the following. Theorem 1.1 Assume ω 0 ∈ C 1 [0, R 0 ], Υ(r) > 0, (1.7) , ∂Ω is C 2 and has positive curvature near (R 0 , 0) . Then the operator JL defined by (1.9) generates a C 0 group e tJL of bounded linear operators on X = X × Y and there exists a decomposition X = E u ⊕ E c ⊕ E s , of closed subspaces E u,s,c satisfying the following properties: i) E c , E u , E s are invariant under e tJL . ii) E u (E s ) only consists of eigenvectors corresponding to positive (negative) eigenvalues of JL and dim E u = dim E s = n − L\| R(B) = n − K\| R(B 1 ) , where K·, · is a bounded bilinear quadratic form on L 2 Φ (ρ 0 ) defined by Kδρ, δρ = Lδρ, δρ + 2π R 0 0 Υ(r) r 0 s +∞ −∞ δρ(s, z)dzds 2 r +∞ −∞ ρ 0 (r, z)dz dr, (1.13) for any δρ ∈ L 2 Φ (ρ 0 ) and n − K\| R(B 1 ) denotes the number of negative modes of K·, · restricted to the subspace R (B 1 ) = δρ ∈ L 2 Φ (ρ 0 ) \| δρdx = 0 . (1.14) iii) The exponential trichotomy is true in the space X in the sense of (2.2) and (2.3). for all δρ ∈ L 2 Φ (ρ 0 ) with R 3 δρdx = 0. Theorem 1.1 gives not only a sharp stability criteria for rotating stars with Rayleigh stable angular velocity, but also more detailed information on the spectra of the linearized Euler-Poisson operator and exponential trichotomy estimates for the linearized Euler-Poisson system. These will be useful for the future study of nonlinear dynamics near unstable rotating stars, particularly, the construction of invariant (stable, unstable and center) manifolds for the nonlinear Euler-Poisson system. The sharp stability criterion in Corollary 1.1 is used to study the stability of two families of slowly rotating stars. For the first family of slowly rotating stars with fixed Rayleigh stable angular velocity and parameterized by the center density, we show that TPP is not true and the transition of stability does not occur at the first mass extrema. More precisely, for fixed κω 0 (r) ∈ C 1,β , for some β ∈ (0, 1), satisfying Υ(r) > 0 and κ small enough, by implicit function theorem as in [14,18,41], there exists a family of slowly rotating stars (ρ µ,κ , κrω 0 (r) e θ ) parameterized by the center density µ. We show that the transition of stability for this family is not at the first extrema of the total mass M µ,κ . In particular, when γ 0 > 4 3 , the slowly rotating stars are stable for small center density and remain stable slightly beyond the first mass maximum. This is consistent with the numerical evidence in [4] (Figure 10, p. 400) for the example of rigidly rotating stars and an equation of state with γ 0 = 5 3 . It shows that Rayleigh stable rotation is indeed stabilizing for rotating stars. By contrast, for the second family of slowly rotating stars with fixed monotone increasing angular momentum distribution (equivalently Rayleigh stable angular velocity), we show that TPP is indeed true. More precisely, for fixed j (p, q) ∈ C 1,β (R + × R + ) satisfying ∂ p (j 2 (p, q)) > 0, j(0, q) = ∂ p j(0, q) = 0 and ε sufficiently small, there exists a family of slowly rotating stars ρ µ,ε , ε r j m ρµ,ε , M µ,ε e θ parameterized by the center density µ, where m ρµ,ε (r) = r 0 s ∞ −∞ ρ µ,ε (s, z)dsdz is the mass distribution in the cylinder, and M µ,ε is the total mass. We show that the transition of stability for this family of rotating stars exactly occurs at the first extrema of the total mass M µ,ε . This not only confirms the claim in [4] based on heuristic arguments when j (m, M ) = 1 M j( m M ), but also can apply to other examples studied in the literature, including j (m, M ) = j (m) (see [2,18,31,32]) and j (m, M ) = j( m M ) (see [35]). The issue of TPP is also not so clear for relativistic rotating stars. For relativistic stars, TPP was shown for the secular stability of a family of rigidly rotating stars ( [12]), while numerical results in [44] indicated that the transition of dynamic instability does not occur at the mass maximum (i.e. TPP is not true) for such a family. Our approach for the Newtonian case might be useful for studying the relativistic case. For the Rayleigh stable case, the stability of rotating stars is studied by using the separable Hamiltonian framework as in the non-rotating stars ( [30]). However, there are fundamental differences between these two cases. For the non-rotating stars, the stability condition is reduced to find n − L\| R(B 1 ) , that is, the number of negative modes of L·, · restricted to R (B 1 ), where L and R (B 1 ) are defined in (1.11) and (1.14) respectively. We note that the dynamically accessible space R (B 1 ) (for density perturbation) is one co-dimensional with only the mass constraint. For the rotating stars, by using the separable Hamiltonian formulation (1.8), the stability is reduced to find n − L\| R(B) , where L, B are defined in (1.10) and (1.12) respectively. Here, the dynamically accessible space R (B) (for density and θ-component of velocity) is infinite co-dimensional, which corresponds to perturbations preserving infinitely many generalized total angular momentum (2.11) in the first order. It is hard to compute the negative modes of L·, · with such infinitely many constraints. A key point in our proof is to find a reduced functional K defined in (1.13) for density perturbation such that n − L\| R(B) = n − K\| R(B 1 ) , where R (B 1 ) denotes the density perturbations preserving the mass as in the nonrotating case. Therefore, the computation of negative modes of L\| R(B) with infinitely many constraints is reduced to study K\| R(B 1 ) with only one mass constraint. This reduced stability criterion in terms of K\| R(B 1 ) is crucial to prove or disprove TPP for different families of rotating stars. Next we consider rotating stars with Rayleigh unstable angular velocity ω 0 (r). That is, there exists a point r 0 ∈ (0, R 0 ) such that Υ(r 0 ) = ∂r(ω 2 0 r 4 ) r 3 r=r 0 < 0. In this case, we cannot write the linearized Euler-Poisson system as a separable linear Hamiltonian PDEs since A 1 = 4ω 2 0 r 3 ρ 0 ∂r(ω 2 0 r 4 ) is not defined at r 0 . Instead, we use the following second order system for u 2 = (v r , v z ) ∂ tt u 2 = −(L 1 + L 2 )u 2 := −Lu 2 , (1.15) whereL = L 1 + L 2 , L 1 u 2 = ∇[Φ (ρ 0 )(∇ · (ρ 0 u 2 )) − 4π(−∆) −1 (∇ · (ρ 0 u 2 )], L 2 u 2 = Υ(r)v r 0 , are self-adjoint operators on Y . The following properties of the spectra ofL are obtained in Proposition 4.1: i) σ ess (L) = range(Υ(r)) = [−a, b], where a > 0, b ≥ 0; ii) There are finitely many negative eigenvalues and infinitely many positive eigenvalues outside the interval [−a, b]. In particular, the infimum of σ(L) is negative, which might correspond to either discrete or continuous spectrum. Define the space Z = u 2 ∈ Y \| ∇ · (ρ 0 u 2 ) ∈ L 2 Φ (ρ 0 ) , with the norm u 2 Z = u 2 Y + ∇ · (ρ 0 u 2 ) L 2 Φ (ρ 0 ) . (1.16) (1.7) and inf r∈[0,R 0 ] Υ(r) < 0. Let η 0 ≤ −a be the minimum of λ ∈ σ(L). Then we have: i) Equation (1.15) defines a C 0 group T (t), t ∈ R, on Z × Y . There exists C > 0 such that for any (u 2 (0) , u 2t (0)) ∈ Z × Y , Theorem 1.2 Assume ω 0 ∈ C 1 [0, R 0 ],u 2 (t) Z + u 2t (t) Y ≤ Ce √ −η 0 t ( u 2 (0) Z + u 2t (0) Y ) , ∀t > 0. (1.17) The flow T (t) conserves the total energy E(u 2 , u 2t ) = u 2t 2 Y + L u 2 , u 2 . (1. 18) ii) For any ε > 0, there exists initial data u ε 2 (0) ∈ Z, u ε 2t (0) = 0 such that u ε 2 (t) Y e √ −η 0 +εt u ε 2 (0) Z , ∀t > 0. (1.19) The above theorem shows that rotating stars with Rayleigh unstable angular velocity are always linearly unstable. The maximal growth rate is obtained either by a discrete eigenvalue beyond the range of Υ(r) or by unstable continuous spectrum due to Rayleigh instability (i.e. negative Υ(r)). In [24], it was shown that for slowly rotating stars with any angular velocity profile, discrete unstable modes cannot be perturbed from neutral modes of non-rotating stars. However, the unstable continuous spectrum was not considered there. We briefly mention some recent mathematical works on the stability of rotating gaseous stars. The conditional Lyapunov stability of some rotating star constructed by variational methods had been obtained by Luo and Smoller [31,32,33,34] under Rayleigh stability assumption, also called Sölberg stability criterion in their works. The paper is organized as follows. In Section 2, we study rotating stars with Rayleigh stable angular velocity and prove the sharp stability criterion. In Section 3, we use the stability criterion to prove/disprove TPP for two families of slowly rotating stars. In Section 4, we prove linear instability of rotating stars with Rayleigh unstable angular velocity. Throughout this paper, for a, b > 0 we use a b to denote the estimate a ≤ Cb for some constant C independent of a, b,, a ≈ b to denote the estimate C 1 a ≤ b ≤ C 2 b for some constants C 1 , C 2 > 0, and a ∼ b to denote \|a − b\| < for some > 0 small enough. Stability criterion for Rayleigh Stable case In this section, we consider rotating stars with Rayleigh stable angular velocity profiles. The linearized Euler-Poisson system is studied by using a framework of separable Hamiltonian systems in [30]. First, we give a summary of the abstract theory in [30]. Separable Linear Hamiltonian PDEs Consider a linear Hamiltonian PDEs of the separable form ∂ t u v = 0 B −B 0 L 0 0 A u v = JL u v , (2.1) where u ∈ X, v ∈ Y and X, Y are real Hilbert spaces. We briefly describe the results in [30] (G2) The operator A : Y → Y * is bounded and self-dual (i.e. A = A and thus Au, v is a bounded symmetric bilinear form on Y ). Moreover, there exist δ > 0 such that Au, u ≥ δ u 2 Y , ∀u ∈ Y. (G3) The operator L : X → X * is bounded and self-dual (i.e. L = L etc.) and there exists a decomposition of X into the direct sum of three closed subspaces X = X − ⊕ ker L ⊕ X + , dim ker L < ∞, n − (L) dim X − < ∞, satisfying (G3.a) Lu, u < 0 for all u ∈ X − \{0}; (G3.b) there exists δ > 0 such that Lu, u ≥ δ u 2 , for any u ∈ X + . We note that the assumptions dim ker L < ∞ and A > 0 can be relaxed (see [30]). But these simplified assumptions are enough for the applications to Euler-Poisson system studied in this section under the Rayleigh stability assumption (i.e. Υ(r) > 0 for all r ∈ [0, R 0 ]). If the Rayleigh unstable assumption holds (i.e. Υ(r 0 ) < 0 for some r 0 ∈ [0, R 0 ]), then n − (L) = ∞ and we will discuss this in Section 4. Theorem 2.1 [30]Assume (G1-3) for (2.1). The operator JL generates a C 0 group e tJL of bounded linear operators on X = X × Y and there exists a decomposition X = E u ⊕ E c ⊕ E s , of closed subspaces E u,s,c with the following properties: i) E c , E u , E s are invariant under e tJL . ii) E u (E s ) only consists of eigenvectors corresponding to negative (positive) eigenvalues of JL and dim E u = dim E s = n − L\| R(B) , where n − L\| R(B) denotes the number of negative modes of L·, · \| R(B) . If n − L\| R(B) > 0, then there exists M > 0 such that e tJL \| E s ≤ M e −λut , t ≥ 0; e tJL \| E u ≤ M e λut , t ≤ 0, (2.2) where λ u = min{λ \| λ ∈ σ(JL\| E u )} > 0. iii) The quadratic form L·, · vanishes on E u,s , i.e. Lu, u = 0 for all u ∈ E u,s , but is non-degenerate on E u ⊕ E s , and E c = {u ∈ X \| Lu, v = 0, ∀ v ∈ E s ⊕ E u } . There exists M > 0 such that \|e tJL \| E c \| ≤ M (1 + t 2 ), for all t ∈ R. (2.3) iv) Suppose L·, · is non-degenerate on R (B), then \|e tJL \| E c \| ≤ M for some M > 0. Namely, there is Lyapunov stability on the center space E c . Remark 2.1 Above theorem shows that the solutions of (2.9) are spectrally stable if and only if L\| R(B) ≥ 0. Moreover, n − L\| R(B) equals to the number of unstable modes. The exponential trichotomy estimates (2.2)-(2.3) are important in the study of nonlinear dynamics near an unstable steady state, such as the proof of nonlinear instability or the construction of invariant (stable, unstable and center) manifolds. The exponential trichotomy can be lifted to more regular spaces if the spaces E u,s have higher regularity. We refer to Theorem 2.2 in [29] for more precise statements. Hamiltonian formulation of linearized EP system Consider an axi-symmetric rotating star solution (ρ 0 (r, z) , v 0 = v 0 e θ = rω 0 (r) e θ ). The support of density ρ 0 is denoted by Ω, which is an axisymmetric bounded domain. Let R 0 be support radius in r, that is, the maximum of r such that (r, z) ∈ Ω. We choose the coordinate system such that (R 0 , 0) ∈ ∂Ω. We make the following assumptions: i) ω 0 ∈ C 1 [0, R 0 ] satisfies the Rayleigh stability condition (i.e. Υ(r) > 0 for r ∈ [0, R 0 ]); ii) ∂Ω is C 2 near (R 0 , 0) and has positive curvature (equivalently Ω is locally convex) at (R 0 , 0); iii) ρ 0 satisfies (1.7). The following lemma will be used later. Lemma 2.1 Under Assumptions ii) and iii) above, for ε > 0 small enough we have +∞ −∞ ρ λ 0 (r, z)dz ≈ (R 0 − r) λ γ 0 −1 + 1 2 , for any λ > 0 and r ∈ (R 0 − ε, R 0 ). Proof. By (1.7), +∞ −∞ ρ λ 0 (r, z)dz ≈ (r,z)∈Ω dist((r, z), ∂Ω) λ γ 0 −1 dz. First, we consider the case when Ω is the ball {r 2 + z 2 < R 2 0 }. Then for r close to R 0 (r,z)∈Ω dist((r, z), ∂Ω) λ γ 0 −1 dz = 2 √ R 2 0 −r 2 0 R 0 − √ r 2 + z 2 λ γ 0 −1 dz (2.4) ≈ √ R 2 0 −r 2 0 R 2 0 − r 2 − z 2 λ γ 0 −1 dz = R 2 0 − r 2 λ γ 0 −1 + 1 2 1 0 1 − u 2 λ γ 0 −1 du ≈ (R 0 − r) λ γ 0 −1 + 1 2 . For general Ω, let 1 r 0 > 0 be the curvature of ∂Ω at (R 0 , 0) and Γ = (r, z) \| (r − R 0 + r 0 ) 2 + z 2 = r 2 0 , be the osculating circle at (R 0 , 0). Then near (R 0 , 0), ∂Ω is approximated by Γ to the 2nd order. For any r ∈ (R 0 − ε, R 0 ), let (r, −z 1 (r)) , (r, z 2 (r)) be the intersection of ∂Ω with the vertical line r = r, where z 1 (r) , z 2 (r) > 0. Then for ε small enough, we have z 1 (r) , z 2 (r) = r 2 0 − (r − R 0 + r 0 ) 2 + o r 2 0 − (r − R 0 + r 0 ) 2 . And for (r, z) ∈ Ω with r ∈ (R 0 − ε, R 0 ), dist((r, z), ∂Ω) = dist((r, z), Γ) + o (dist((r, z), Γ)) = r 0 − (r − R 0 + r 0 ) 2 + z 2 + o r 0 − (r − R 0 + r 0 ) 2 + z 2 . Then similar to (2.4), we have +∞ −∞ ρ λ 0 (r, z)dz ≈ r 2 0 − (r − R 0 + r 0 ) 2 λ γ 0 −1 + 1 2 ≈ (R 0 − r) λ γ 0 −1 + 1 2 . Let X 1 := L 2 Φ (ρ 0 ) , X 2 = L 2 ρ 0 , X = X 1 × X 2 , Y = L 2 ρ 0 2 and X := X × Y . The linearized Euler-Poisson system for axi-symmetric perturbations around the rotating star solution (ρ 0 (r, z) , ω 0 (r) re θ ) is          ∂ t v r = 2ω 0 (r) v θ − ∂ r (Φ (ρ 0 )ρ + V (ρ)), ∂ t v z = −∂ z (Φ (ρ 0 )ρ + V (ρ)), ∂ t v θ = − 1 r ∂ r (ω 0 r 2 )v r , ∂ t ρ = −∇ · (ρ 0 v) = −∇ · (ρ 0 (v r , 0, v z )), (2.5) with ∆V = 4πρ. Here, (ρ, v = (v r , v θ , v z ) ) ∈ X are perturbations of density and velocity. Define the operators L := Φ (ρ 0 ) − 4π(−∆) −1 : X 1 → (X 1 ) * , A = ρ 0 : Y → Y * , A 1 := 4ω 2 0 r 3 ρ 0 ∂ r (ω 2 0 r 4 ) = 4ω 2 0 ρ 0 Υ(r) : X 2 → (X 2 ) * , and B = B 1 B 2 : D(B) ⊂ Y * → X, B = (B 1 , B 2 ) : X * ⊃ D(B ) → Y, (2.6) where B 1 v r v z = −∇·(v r , 0, v z ), B 1 ρ = ∂ r ρ ∂ z ρ , (2.7) and B 2 v r v z = − ∂ r (ω 0 r 2 ) rρ 0 v r , (B 2 ) v θ = − ∂r(ω 0 r 2 ) rρ 0 v θ 0 . (2.8) Then the linearized Euler-Poisson system (2.5) can be written in a separable Hamiltonian form d dt u 1 u 2 = JL u 1 u 2 , (2.9) where u 1 = (ρ, v θ ) and u 2 = (v r , v z ). The operators J := 0, B −B , 0 : X * → X, L := L, 0 0, A : X → X * , are off-diagonal anti-self-dual and diagonal self-dual respectively, where L = L, 0 0, A 1 : X → X * . First, we check that (L, A, B) in (2.9) satisfy the assumptions (G1)-(G3) for the abstract theory in Section 2.1. The assumptions (G1) and (G2) can be shown by the same arguments in the proof of Lemma 3.5 in [30] and that B 2 is bounded. The Rayleigh stability condition Υ(r) > 0 implies that the operator A 1 is bounded, positive and self-dual. By the same proof of Lemma 3.6 in [30], we have the following lemma. Lemma 2.2 There exists a direct sum decomposition L 2 Φ (ρ 0 ) = X − ⊕ker L⊕ X + and δ 0 > 0 such that: i) dim (X − ) , dim ker L < ∞; ii) L\| X − < 0, L\| X + ≥ δ 0 and X − ⊥ X + in the inner product of L 2 Φ (ρ 0 ) . The assumption (G3) readily follows from above lemma. Therefore, we can apply Theorem 2.1 to the linearized Euler-Poisson system (2.9). This proves the conclusions in Theorem 1.1 except for the formula n − L\| R(B) = n − K\| R(B 1 ) , which will be shown later. Here, R (B) is the closure of R(B) in X, and the operators B, B 1 are defined in (2.6)-(2.8). Remark 2.2 In some literature [31,32,33,34], the Rayleigh stability condition is Υ(r) ≥ 0 for all r ∈ [0, R 0 ]. Here, we used the stability condition Υ(r) > 0 for all r ∈ [0, R 0 ] as in the astrophysical literature such as [4,46]. If Υ(r) ≥ 0 for all r ∈ [0, R 0 ] and Υ(r) = 0 only at some isolated points, let Λ (r, z) = 4ω 2 0 ρ 0 Υ(r) and the operator A 1 : L 2 Λ → (L 2 Λ ) * is bounded and positive. The linearized Euler-Poisson system can still be studied in the framework of separable Hamiltonian systems and similar results as in Theorem 1.1 can be obtained. Dynamically accessible perturbations By Theorem 1.1, the solutions of (2.9) are spectrally stable (i.e. nonexistence of exponentially growing solution) if and only if L\| R(B) ≥ 0. More precisely, we have Corollary 2.1 Assume ω 0 ∈ C 1 [0, R 0 ], (1.7), and inf r∈[0,R 0 ] Υ(r) > 0. The rotating star solution (ρ 0 (r, z) , v 0 = rω 0 (r) e θ ) of Euler-Poisson system is spectrally stable if and only if Lδρ, δρ + A 1 δv θ , δv θ ≥ 0 for all (δρ, δv θ ) ∈ R(B). (2.10) In this section, we discuss the physical meaning of above stability criterion by using the variational structure of the rotating stars. For any solution (ρ, v) of the axi-symmetric Euler-Poisson system (1.1), define the angular momentum j = v θ r and the generalized total angular momentum A g (ρ, v θ ) = R 3 ρg(v θ r)dx,(2.11) for any function g ∈ C 1 (R). Lemma 2.3 For any g ∈ C 1 (R), the functional A g (ρ, v θ ) is conserved for the Euler-Poisson system (1.1). Proof. First, we note that the angular momentum j is an invariant of the particle trajectory under the axi-symmetric force field −∇V − ∇Φ (ρ). Let ϕ (x, t) be the flow map of the velocity field v with initial position x, and J (x, t) be the Jacobian of ϕ. Then ρ (ϕ (x, t) , t) J (x, t) = ρ (x, 0) and A g (ρ, v θ ) (0) = R 3 ρ (x, 0) g(j (x))dx = R 3 ρ (ϕ (x, t) , t) J (x, t) g(j (ϕ (x, t)))dx = R 3 ρ (y, t) g(j (y))dx = A g (ρ, v θ ) (t) . The steady state (ρ 0 , ω 0 re θ ) has the following variational structure. By the steady state equation (1.6), we have 1 2 ω 2 0 r 2 + Φ (ρ 0 ) − \|x\| −1 * ρ 0 + g 0 ω 0 r 2 + c 0 = 0 in Ω, (2.12) where c 0 > 0 is the constant in (1.6) and g 0 ∈ C 1 (R) satisfies the equation g 0 ω 0 (r) r 2 = −ω 0 (r) , ∀ r ∈ [0, R 0 ] . (2.13) The existence of g 0 satisfying (2.13) is ensured by the Rayleigh stable condition Υ(r) > 0 which implies that ω 0 (r) r 2 is monotone to r. The equations (1.6) and (2.12) are equivalent since g 0 ω 0 (r) r 2 = − 1 2 ω 2 0 r 2 − r 0 ω 2 0 (s)s ds due to (2.13) and integration by parts. Denote the the total energy by H(ρ, v) = R 3 1 2 ρv 2 + Φ(ρ) − 1 8π \|∇V \| 2 dx, ∆V = 4πρ, which is conserved for the Euler-Poisson system (1.1). Define the energy-Casimir functional H c (ρ, v) = H(ρ, v) + c 0 R 3 ρ dx + R 3 ρg 0 (v θ r) dx, where c 0 and g 0 are as in (2.12). Then (ρ 0 , ω 0 re θ ) is a critical point of H c (ρ, v), since DH c (ρ 0 , ω 0 re θ ), (δρ, δv) = R 3 1 2 ω 2 0 r 2 + Φ (ρ 0 ) + V (ρ 0 ) + c 0 + g 0 (ω 0 r 2 ) δρ dx + R 3 [ρ 0 ω 0 r + ρ 0 g 0 (ω 0 r 2 )r]δv θ dx = 0 by equations (2.12) and (2.13). By direct computations, D 2 H c (ρ, v)[ρ 0 , ω 0 re θ ](δρ, δv), (δρ, δv) (2.14) = R 3 (Φ (ρ 0 ) (δρ) 2 − 4π(−∆ −1 δρ)δρ + ρ 0 (δv r ) 2 + ρ 0 (δv z ) 2 dx + R 3 ρ 0 (1 + g 0 (ω 0 r 2 )r 2 ) (δv θ ) 2 dx = Lδρ, δρ + A 1 δv θ , δv θ + A (δv r , δv z ) , (δv r , δv z ) , where we used the identity 1 + g 0 (ω 0 r 2 )r 2 = 1 − ω 0 r 2 d dr (ω 0 r 2 ) = 4ω 2 0 r 3 d dr (ω 2 0 r 4 ) = 4ω 2 0 Υ(r) . The functional (2.14) is a conserved quantity of the linearized Euler-Poisson system (2.9) due to the Hamiltonian structure. We note that the number of negative directions of (2.14) is given by n − (L). We now turn to the spaces of δρ and (δρ, δv θ ). Lemma 2.4 It holds that R (B 1 ) = R (B 1 ) = δρ ∈ L 2 Φ (ρ 0 ) R 3 δρdx = 0 . Proof. Since ker B 1 = ker ∇ is spanned by constant functions, we have R (B 1 ) = (ker B 1 ) ⊥ = δρ ∈ L 2 Φ (ρ 0 ) R 3 δρdx = 0 . It remains to show R (B 1 ) = R (B 1 ) which is equivalent to R (B 1 A) = R (B 1 A). By Lemma 3.15 in [30], we have the orthogonal decomposition L 2 ρ 0 = ker (B 1 A) ⊕ W, where W = w = ∇p ∈ L 2 ρ 0 . For any δρ ∈ R (B 1 A) , by the proof of Lemma 3.15 in [30], there exists a unique gradient field ∇p ∈ L 2 ρ 0 such that B 1 A∇p = ∇ · (ρ 0 ∇p) = δρ. By Proposition 12 in [20], we have ∇p L 2 ρ 0 ∇ · (ρ 0 ∇p) L 2 Φ (ρ 0 ) = δρ L 2 Φ (ρ 0 ) . (2.15) For any u ∈ D (B 1 A), let v ∈ W be the projection of u to W . Then above estimate (2.15) implies that dist (u, ker (B 1 A)) = inf z∈ker(B 1 A) u − z L 2 ρ 0 = v L 2 ρ 0 B 1 Au L 2 Φ (ρ 0 ) . By Theorem 5.2 in [23, P. 231], this implies that R (B 1 ) = R (B 1 ). Definition 2.1 The perturbation (δρ, δv θ ) ∈ X is called dynamically acces- sible if (δρ, δv θ ) ∈ R(B). In the next lemma, we give two equivalent characterizations of the dynamically accessible perturbations. Lemma 2.5 For (δρ, δv θ ) ∈ X, the following statements are equivalent. ( i) (δρ, δv θ ) ∈ R(B); (ii) R 3 g(ω 0 r 2 )δρ dx + R 3 ρ 0 rg (ω 0 r 2 )δv θ dx = 0, ∀g ∈ C 1 (R) ; (2.16) (iii) R 3 δρ dx = 0 and +∞ −∞ δv θ ρ 0 (r, z) dz = ∂ r (ω 0 r 2 ) r 2 r 0 s +∞ −∞ δρ(s, z)dzds. (2.17) Proof. First, we show (i) and (ii) are equivalent. We have R(B) = (ker B ) ⊥ , where the dual operator B : X * → Y is defined in (2.7)-(2.8). Let (ρ, v θ ) be a C 1 function in ker B , then B ρ v θ = ∂ r ρ − ∂r(ω 0 r 2 ) rρ 0 v θ ∂ z ρ = 0 0 . Since ∂ z ρ = 0 and ω 0 r 2 is monotone to r by the Rayleigh stability condition, we can write ρ = g (ω 0 r 2 ) for some function g ∈ C 1 . Then ∂ r ρ− ∂r(ω 0 r 2 ) rρ 0 v θ = 0 implies that v θ = ρ 0 rg (ω 0 r 2 ). Thus ker B is the closure of the set g ω 0 r 2 , ρ 0 rg (ω 0 r 2 ) , g ∈ C 1 (R) , in X * . Therefore, (δρ, δv θ ) ∈ R(B) = (ker B ) ⊥ if and only if (2.16) is satisfied. Next, we show (ii) and (iii) are equivalent. If (ii) is satisfied, by choosing g = 1 we get δρ dx = 0. Then by (2.16) and integration by parts, we have R 0 0 r 2 +∞ −∞ δv θ ρ 0 (r, z)dz − ∂ r (ω 0 r 2 ) r 0 s +∞ −∞ δρ(s, z)dzds g (ω 0 r 2 )dr = 0. which implies (2.17) since g ∈ C 1 (R) is arbitrary. On the other hand, by reversing the above computation, (ii) follows from (iii). The statement (ii) above implies that for any (δρ, δv θ ) ∈ R(B), we have DA g (ρ 0 , ω 0 r), (δρ, δv θ ) = 0, where the generalized angular momentum A g is defined in (2.11). That is, a dynamically accessible perturbation (δρ, δv θ ) must lie on the tangent space of the functional A g at the equilibrium (ρ 0 , ω 0 r e θ ). Since g is arbitrary, this implies infinite many constraints for dynamically accessible perturbations. The stability criterion (2.10) implies that that rotating stars are stable if and only if they are local minimizers of energy-Casimir functional H(ρ, v) under the constraints of fixed generalized angular momentum A g for all g. This contrasts significantly with the case of non-rotating stars. It was shown in ( [30]) that non-rotating stars are stable if and only if they are local minimizers of the energy-Casimir functional under the only constraint of fixed total mass. The stability criterion (2.10) for rotating stars involves infinitely many constraints and is much more difficult to check. In the next section, we give an equivalent stability criterion in terms of a reduced functional (1.13) under only the mass constraint. Reduced functional and the equivalent Stability Criterion In this section, we prove the formula n − L\| R(B) = n − K\| R(B 1 ) and complete the proof of Theorem 1.1. Lemma 2.6 For any δρ ∈ R (B 1 ), define u δρ θ = ∂ r (ω 0 r 2 ) r 2 r 0 s +∞ −∞ δρ(s, z)dzds +∞ −∞ ρ 0 (r, z)dz . (2.18) Then δρ, u δρ θ ∈ R (B) and u δρ θ L 2 ρ 0 δρ L 2 Φ (ρ 0 ) . Proof. We have u δρ θ 2 L 2 ρ 0 R 3 ρ 0 r 0 s +∞ −∞ δρ(s, z)dzds r +∞ −∞ ρ 0 (r, z)dz 2 dx = 2π R 0 0 r 0 s +∞ −∞ δρ(s, z)dzds 2 r +∞ −∞ ρ 0 (r, z)dz dr = 2π R 0 −ε 0 r 0 s +∞ −∞ δρ(s, z)dzds 2 r +∞ −∞ ρ 0 (r, z)dz dr + 2π R 0 R 0 −εI R 0 −ε 0 r −2 r 0 s +∞ −∞ δρ(s, z)dzds 2 dr R 0 −ε 0 r 2 +∞ −∞ δρ(r, z)dz 2 dr R 0 −ε 0 r 2 +∞ −∞ Φ (ρ 0 ) (δρ) 2 (r, z)dz +∞ −∞ 1 Φ (ρ 0 (r, z)) dz dr R 0 −ε 0 r +∞ −∞ Φ (ρ 0 ) (δρ) 2 (r, z)dzdr δρ 2 L 2 Φ (ρ 0 ) . By Hardy's inequality and Lemma 2.1, we have II = 2π R 0 R 0 −ε r 0 s +∞ −∞ δρ(s, z)dzds 2 r +∞ −∞ ρ 0 (r, z)dz dr R 0 R 0 −ε r 0 s +∞ −∞ δρ(s, z)dzds 2 (R 0 − r) 1 γ 0 −1 + 1 2 dr R 0 R 0 −ε +∞ −∞ δρ(r, z)dz 2 (R 0 − r) − 1 γ 0 −1 + 3 2 dr R 0 R 0 −ε +∞ −∞ Φ (ρ 0 )(δρ) 2 dz +∞ −∞ 1 Φ (ρ 0 ) dz (R 0 − r) − 1 γ 0 −1 + 3 2 dr R 0 R 0 −ε +∞ −∞ Φ (ρ 0 )(δρ) 2 dz (R 0 − r)dr δρ 2 L 2 Φ (ρ 0 ) , where we used the estimate +∞ −∞ 1 Φ (ρ 0 ) dz ≈ +∞ −∞ ρ 2−γ 0 0 dz ≈ (R 0 − r) 2−γ 0 γ 0 −1 + 1 2 , since Φ (s) ≈ s γ 0 −2 for s small. This proves u δρ θ L 2 ρ 0 δρ L 2 Φ (ρ 0 ) . The statement δρ, u δρ θ ∈ R (B) follows from Lemma 2.5 since R 3 δρ dx = 0 for δρ ∈ R (B 1 ) and u δρ θ obviously satisfies (2.17). With the help of lemma 2.6 we can finish the proof of Theorem 1.1. Proof of Theorem 1.1. We only need to show n − L\| R(B) = n − K\| R(B 1 ) . First, we have L δρ δv θ , δρ δv θ ≥ Kδρ, δρ , ∀ (δρ, δv θ ) ∈ R(B),(2.19) since A 1 δv θ , δv θ = R 3 4ω 2 0 Υ(r) ρ 0 (δv θ ) 2 dx = 2π R 0 0 4ω 2 0 r Υ(r) +∞ −∞ ρ 0 (δv θ ) 2 dz dr = 2π R 0 0 4ω 2 0 r Υ(r) +∞ −∞ ρ 0 u δρ θ 2 dz dr + 2π R 0 0 4ω 2 0 r Υ(r) +∞ −∞ ρ 0 δv θ − u δρ θ 2 dz dr ≥ 2π R 0 0 4ω 2 0 r Υ(r) +∞ −∞ ρ 0 u δρ θ 2 dz dr = 2π R 0 0 Υ(r) r 0 s +∞ −∞ δρ(s, z)dzds 2 r +∞ −∞ ρ 0 (r, z)dz dr. In the above, we used the observation that +∞ −∞ ρ 0 δv θ − u δρ θ dz = +∞ −∞ ρ 0 δv θ dz − u δρ θ (r) +∞ −∞ ρ 0 dz = 0, since +∞ −∞ ρ 0 δv θ dz = u δρ θ (r) +∞ −∞ ρ 0 dz = ∂ r (ω 0 r 2 ) TPP for slowly rotating stars In this section, we use the stability criterion in Theorem 1.1 to study two families of slowly rotating stars parameterized by the center density. The case of fixed angular velocity In this subsection, we consider a family of slowly rotating stars with fixed angular velocity. Under the assumptions (1.2)-(1.3), for some µ max > 0, there exists a family of nonrotating stars with radially symmetric density ρ µ (\|x\|) parametrized by the center density µ ∈ (0, µ max ). We refer to [30] and references therein for such results. Let R µ be the support radius of ρ µ and B µ = B(0, R µ ) be the support of ρ µ . The radial density ρ µ satisfies ∆(Φ (ρ µ )) + 4πρ µ = 0, in B µ , with ρ µ (0) = µ. For the general equations of state satisfying (1.2)-(1.3) with γ 0 ≥ 4/3 , it was shown in [15] that µ max = +∞. Let ω(r) ∈ C 1,β [0, ∞) be fixed for some β ∈ (0, 1). We construct a family of rotating stars for Euler-Poisson system with the following form ρ 0 = ρ µ,κ (r, z) = ρ µ (g −1 ζµ,κ ((r, z))), v 0 = κrω 0 (r) e θ , where the dilating function is g ζµ,κ = x 1 + ζ µ,κ \|x\| 2 , and ζ µ,κ (x) : B µ → R is axi-symmetric and even in z. The existence of rotating stars (ρ µ,κ , κrω 0 (r) e θ ) is reduced to the following equations for ρ µ,κ : −κ 2 r 0 ω 2 (s)sds + Φ (ρ µ,κ ) + V µ,κ + c µ,κ = 0 in Ω µ,κ , (3.1) V µ,κ = −\|x\| −1 * ρ µ,κ in R 3 , where c µ,κ is a constant and Ω µ,κ = g ζµ,κ (B µ ) is the support of the density ρ µ,κ of the rotating star solution. By similar arguments as in [14,41,19], we can get the following existence theorem. Theorem 3.1 Let µ ∈ [µ 0 , µ 1 ] ⊂ (0, µ max ), P (ρ) satisfy (1.2)-(1.3), and ω(r) ∈ C 1,β [0, ∞). Then there existκ > 0 and solutions ρ µ,κ of (3.1) for all \|κ\| <κ, satisfying the following properties: 1). 2) ρ µ,κ is axi-symmetric and even in z. 1) ρ µ,κ ∈ C 1,α c (R 3 ), where α = min( 2−γ 0 γ 0 −1 , 3) ρ µ,κ (0) = µ. 4) ρ µ,κ ≥ 0 has compact support g ζµ,κ (B µ ). 5) For all µ ∈ [µ 0 , µ 1 ], the mapping κ → ρ µ,κ is continuous from (−κ,κ) into C 1 c (R 3 ). When κ = 0, ρ µ,0 = ρ µ (\|x\|) is the nonrotating star solution with ρ µ (0) = µ. Now we use Theorem 1.1 to study the stability of above rotating star solutions (ρ µ,κ , κω(r)re θ ), for µ ∈ [µ 0 , µ 1 ], κ small enough, and ω ∈ C 1,β [0, ∞) satisfying the Rayleigh condition Υ(r) := ∂r(ω 2 r 4 ) r 3 > 0. First, we check the assumptions in Theorem 1.1. Let R µ,κ be the support radius in r for Ω µ,κ = g ζµ,κ (B µ ). Since g ζµ,κ ∈ C 2 (B µ ) dependents continuously on κ, it is easy to check the assumptions on Ω µ,κ for κ small enough. That is, ∂Ω µ,κ is C 2 and has positive curvature near (R µ,κ , 0). Next, we check the assumption (1.7). For nonrotating stars, it is known ( [6,16,28,30]) that ρ µ (r, z) ≈ ((R µ − √ r 2 + z 2 ) 1 γ 0 −1 ) for √ r 2 + z 2 ∼ R µ . For κ small enough, by the definition of the dilating function g ζµ,κ , we have ρ µ,κ (r, z) = ρ µ (g −1 ζµ,κ (r, z)) ≈ ((R µ − \|g −1 ζµ,κ (r, z)\|) 1 γ 0 −1 ) ≈ dist((r, z), ∂g ζµ,κ (B µ )) 1 γ 0 −1 , for (r, z) near (R µ,κ , 0) = g ζµ,κ (R µ , 0). Below, for rotating stars (ρ µ,κ , rω 0 (r) e θ ) we use X µ,κ , X µ,κ 1 , Y µ,κ , L µ,κ , A µ,κ 1 , B µ,κ 1 , B µ,κ 2 , K µ,κ , etc., to denote the corresponding spaces X, X 1 , Y , and operators L, A 1 , B 1 , B 2 , K etc. defined in Section 2. By Theorem 1.1, the rotating star (ρ µ,κ , κω(r)re θ ) is spectrally stable if and only if K µ,κ δρ, δρ = L µ,κ δρ, δρ +2κ 2 π Rµ,κ 0 Υ(r) r 0 s +∞ −∞ δρ(s, z)dzds 2 r +∞ −∞ ρ µ,κ (r, z)dz dr ≥ 0, (3.2) for all δρ ∈ R(B µ,κ 1 ) = δρ ∈ X µ,κ 1 \| R 3 δρdx = 0 . Moreover, the number of unstable modes equals n − K µ,κ \| R(B µ,κ 1 ) . The following is an easy corollary of the stability criterion. Corollary 3.1 (Sufficient condition for instability) Let I ⊂ [µ 0 , µ 1 ] be an interval such that the non-rotating star (ρ µ , 0) is unstable for any µ ∈ I. Then for any ω ∈ C 1,β [0, ∞) satisfies Υ(r) > 0, there exists κ 0 > 0 such that the rotating star (ρ µ,κ , κω(r)re θ ) is unstable for any 0 < κ < κ 0 and µ ∈ I. Proof. The instability of (ρ µ , 0) implies that n − (L µ,0 \| R(B µ,0 1 ) ) > 0 for µ ∈ I. Thus there exists some > 0 (independent of µ) and δρ µ,0 = δρ µ,0 (\|x\|) ∈ R(B µ,0 1 ) such that L µ,0 δρ µ,0 , δρ µ,0 = −2 < 0 for µ ∈ I. Let δρ µ,κ (r, z) = δρ µ,0 (g ζµ,κ (r, z)) − Bµ δρ µ,0 (\|x\|) det Dg ζµ,κ (x)dx M µ,κ ρ µ,κ (r, z), then δρ µ,κ (r, z) ∈ R(B µ,κ 1 ). Noticing that lim κ→0 Bµ δρ µ,0 (\|x\|) det Dg ζµ,κ (x)dx = Bµ δρ µ,0 (\|x\|)dx = 0, we have lim κ→0 L µ,κ δρ µ,κ , δρ µ,κ = L µ,0 δρ µ,0 , δρ µ,0 = −2 < 0. Thus, there exists κ 0 > 0 such that when 0 < κ < κ 0 K µ,κ δρ µ,κ , δρ µ,κ = L µ,κ δρ µ,κ , δρ µ,κ + 2κ 2 π Rµ,κ 0 Υ(r) r 0 s +∞ −∞ δρ µ,κ (s, z)dzds 2 r +∞ −∞ ρ µ,κ (r, z)dz dr < − < 0. The linear instability of (ρ µ,κ , κω(r)re θ ) follows. Letμ be the first critical point of the mass-radius ratio Mµ Rµ for the nonrotating stars and setμ = +∞ if Mµ Rµ has no critical point. Consider the rotating stars (ρ µ,κ , κω(r)re θ ) for µ ∈ [µ 0 , µ 1 ] ⊂ (0,μ) and κ small. We have the following sufficient condition for stability. For any µ ∈ [µ 0 , µ 1 ] ⊂ (0,μ) and κ small enough, if dMµ,κ dµ ≥ 0, then the rotating star (ρ µ,κ , κωre θ ) is spectrally stable. For the proof of above Theorem, first we compute n − L µ,κ \| Xµ,κ . Leṫ H 1 ax andḢ −1 ax be the axi-symmetric subspaces ofḢ 1 (R 3 ) andḢ −1 (R 3 ) respectively. Define the reduced operator D µ,κ :Ḣ 1 ax →Ḣ −1 ax by D µ,κ := −∆ − 4π Φ (ρ µ,κ ) . Then D µ,κ ψ, ψ = R 3 \|∇ψ\| 2 dx − 4π R 3 \|ψ\| 2 Φ (ρ µ,κ ) dx, ψ ∈Ḣ 1 ax , defines a bounded bilinear symmetric form onḢ 1 ax . By the same proof of Lemma 3.7 in [30], we have Lemma 3.1 It holds that n − L µ,κ \| X µ,κ 1 = n − (D µ,κ ) and dim ker L µ,κ = dim ker D µ,κ . Since the rotating star solution (ρ µ,κ , κω(r)re θ ) is even in z, we can compute n − L µ,κ \| X µ,κ 1 and n − (D µ,κ ) on the even and odd (in z) subspaces respectively. Define X µ,κ od := {ρ ∈ X µ,κ 1 \| ρ(r, z) = −ρ(r, −z)}, X µ,κ ev := {ρ ∈ X µ,κ 1 \| ρ(r, z) = ρ(r, −z)}, (3.3) H od := {ϕ ∈Ḣ 1 ax \|ϕ(r, z) = −ϕ(r, −z)}, H ev := {ϕ ∈Ḣ 1 ax \| ϕ(r, z) = ϕ(r, −z)}. Lemma 3.2 Assume P (ρ) satisfies (1.2)-(1.3), ω ∈ C 1,β [0, ∞) satisfies Υ(r) > 0. Then for any µ ∈ [µ 0 , µ 1 ] ⊂ (0,μ) and κ small enough, we have n − (L µ,κ ) = n − (L µ,0 ) = 1 and ker L µ,κ = span{∂ z ρ µ,κ }. Moreover, we have the following direct sum decompositions for X µ,κ ev and X µ,κ ev : X µ,κ ev = X µ,κ −,ev ⊕ X µ,κ +,ev , dim X µ,κ −,ev = 1, and X µ,κ od = span{∂ z ρ µ,κ } ⊕ X µ,κ +,od , satisfying: i) L µ,κ \| X µ,κ −,ev < 0; ii) there exists δ > 0 such that L µ,κ u, u ≥ δ u 2 L 2 Φ (ρµ,κ) , for any u ∈ X µ,κ +,ev ⊕ X µ,κ +,od , where δ is independent of µ and κ. The same decompositions are also true for K µ,κ on X µ,κ ev and X µ,κ od . In addition, for any µ ∈ [µ 0 , µ 1 ], it holds that dVµ,κ(0,Zµ,κ) dµ < 0 for κ small enough. Proof. It was showed in [30] that: for any µ ∈ (0,μ), we have n − (D µ,0 ) = 1 and ker D µ,0 = span{∂ z V µ } in the axi-symmetric function space. Here, V µ = −\|x\| −1 * ρ µ is the gravitational potential of the non-rotating star. Since ∂ z V µ is odd in z, it follows that for any µ ∈ (0,μ): i) on H ev , n − (D µ,0 ) = 1, ker D µ,0 = {0}; ii) on H od , ker D µ,0 = span{∂ z V µ } and n − (D µ,0 ) = 0. Moreover, for µ ∈ [µ 0 , µ 1 ] ⊂ (0,μ), there exists δ 0 > 0 (independent of µ) and decompositions H ev = H ev −,µ ⊕ H ev +,µ and H od = span{∂ z V µ } ⊕ H od +,µ satisfying that: i) dim H ev −,µ = 1, D µ,0 \| H ev −,µ < −δ 0 ; ii) D µ,0 \| H ev +,µ ⊕H od +,µ ≥ δ 0 . Since ∂ z V µ,κ ∈ H od ∩ ker D µ,κ and (D µ,κ − D µ,0 )ψ, ψ = 4π Φ (ρ µ,κ ) − 4π Φ (ρ µ ) ψ 2 dx 4π Φ (ρ µ,κ ) − 4π Φ (ρ µ ) 3 2 dx 2 3 ψ 2 L 6 O(κ) ∇ψ 2 L 2 → 0, as κ → 0, by the perturbation arguments (e.g. Corollary 2.19 in [30]) it follows that for µ ∈ [µ 0 , µ 1 ] and κ sufficiently small, the decompositions H ev = H ev −,µ ⊕ H ev +,µ and H od = span{∂ z V µ,κ } ⊕ H ev +,µ satisfy: i) dim H ev −,µ = 1, D µ,κ \| H ev −,µ < − 1 2 δ 0 ; ii) D µ,κ \| H ev +,µ ⊕H ev +,µ ≥ 1 2 δ 0 . By the proof of Lemma 3.4 in [30], for any ρ ∈ X µ,κ 1 we have L µ,κ ρ, ρ = ρ 2 L 2 Φ (ρµ,κ) − 1 4π ∇ψ 2 L 2 ≥ 1 4π D µ,κ ψ, ψ , (3.4) where ψ = 1 4π ∆ −1 ρ. We note that ∂ z ρ µ,κ ∈ ker L µ,κ ∩ X µ,κ od and ∂ z V µ,κ = the lemma follows readily from (3.4) and above decompositions for H od and H ev . Since \| (L µ,κ − K µ,κ ) ρ, ρ \| o(κ 2 ) ρ 2 L 2 Φ (ρµ,κ) , ∀ρ ∈ X µ,κ 1 , and ∂ z ρ µ,κ ∈ ker K µ,κ ∩ X µ,κ od , we have the same decompositions for K µ,κ on X µ,κ ev and X µ,κ od . Since γ 0 ∈ (6/5, 2), it is known that (see [30]) dV µ (0, R µ ) dµ = − d dµ M µ R µ < 0 for µ small. Recall thatμ is the first critical point of Mµ Rµ . Therefore, when µ ∈ [µ 0 , µ 1 ] ⊂ (0,μ), we have dVµ(0,Rµ) dµ < − 0 for some constant 0 > 0 inde- pendent of µ. Since dVµ,κ(0,Zµ,κ) dµ − dVµ(0,Rµ) dµ = O(κ), we have dVµ,κ(0,Zµ,κ) dµ < 0 for any µ ∈ [µ 0 , µ 1 ] and κ small enough. This finishes the proof of the lemma. Proof of Theorem 3.2. The spectral stability of (ρ µ,κ , κωre θ ) is equivalent to show n − K µ,κ \| R(B µ,κ 1 ) = 0. By Lemma 3.2 and the fact that K µ,κ = L µ,κ on X µ,κ od , we have n − (K µ,κ \| X µ,κ od ∩R(B µ,κ 1 ) ) = n − (L µ,κ \| X µ,κ od ∩R(B µ,κ 1 ) ) ≤ n − (L µ,κ \| X µ,κ od ) = 0. Since K µ,κ ≥ L µ,κ on X µ,κ ev due to Υ(r) > 0, for spectral stability it suffices to show n − L µ,κ \| X µ,κ ev ∩R(B µ,κ 1 ) = 0. Applying d dµ to (3.1), we obtain that L µ,κ dρ µ,κ dµ = − dc µ,κ dµ . From (3.1) we know that c µ,κ = −V µ,κ (R µ,κ , 0). By Lemma 3.2, dcµ,κ dµ > 0 for µ ∈ [µ 0 , µ 1 ] and κ small enough. Therefore, X µ,κ ev ∩ R(B µ,κ 1 ) = δρ ∈ X µ,κ ev \| L µ,κ dρ µ,κ dµ , δρ = 0 , i.e. δρ is orthogonal to dρµ,κ dµ in L µ,κ ·, · . When dMµ,κ dµ > 0, we have L µ,κ dρ µ,κ dµ , dρ µ,κ dµ = − dc µ,κ dµ g ζµ,κ (Bµ) dρ µ,κ dµ dx = dV µ,κ (0, Z µ,κ ) dµ dM µ,κ dµ < 0. Combining above with Lemma 3.2, we get n − (L µ,κ \| X µ,κ ev ∩R(B µ,κ 1 ) ) = 0. Hence we get the spectrally stability. When dMµ,κ dµ = 0, since dM µ,κ dµ = dρ µ,κ dµ dx = 0, we have dρµ,κ dµ ∈ X µ,κ ev ∩ R(B µ,κ 1 ). Meanwhile, since ker L µ,κ = {0} on X µ,κ ev , by the same argument as in the proof of Theorem 1.1 in [30], we have n − (L µ,κ \| X µ,κ ev ∩R(B µ,κ 1 ) ) = 0. The spectral stability is again true. It is natural to ask if extrema points of the total mass M µ,κ of the rotating stars (ρ µ,κ , κωre θ ) are the transition points for stability as in the case of nonrotating stars. Below, we show that this is not true. First, we give conditions to ensure that the first extrema point of total mass M µ,κ is obtained at a center density µ κ * beforeμ (the first critical point of M µ /R µ ). Assume P (ρ) satisfies the following asymptotically polytropic conditions: H1) P (ρ) = c − ρ γ 0 (1 + O(ρ a 0 )) when ρ → 0, (3.5) for some γ 0 ∈ ( 4 3 , 2) and c − , a 0 > 0; H2) P (ρ) = c + ρ γ∞ (1 + O(ρ −a∞ )) when ρ → +∞, (3.6) for some γ ∞ ∈ (1, 6/5) ∪ (6/5, 4/3) and c + , a ∞ > 0. Under assumptions H1)-H2), it was shown in [15] that the total mass M µ of the non-rotating stars has extrema points. Moreover, the first extrema point of M µ , which is a maximum point denoted by µ * , must be less thanμ (see Lemma 3.14 in [30]). For any µ 0 < µ * < µ 1 <μ, we have M µ,κ → M µ in C 1 [µ 0 , µ 1 ] when κ → 0. Thus when κ is small enough, the function M µ,κ has the first maximum µ κ * ∈ (µ 0 , µ 1 ) and lim κ→0 µ κ * = µ * . By Theorem 3.2, the rotating stars (ρ µ,κ , κω(r)re θ ) are stable for µ ∈ [µ 0 , µ κ * ]. It is shown below that the transition of stability occurs beyond µ κ * . Theorem 3.3 Suppose P (ρ) satisfies (3.5)- (3.6), ω ∈ C 1,β [0, ∞) satisfies Υ(r) > 0. Fixed κ small, letμ κ be the first transition point of stability of the rotating stars (ρ µ,κ , κω(r)re θ ). Then for any κ = 0 small enough, we havê µ κ > µ κ * . Proof. As in the proof of Theorem 3.2, the spectral stability is equivalent to show K µ,κ ≥ 0 on X µ,κ ev ∩ R(B µ,κ 1 ). Suppose the maxima point µ κ * of M µ,κ is the first transition point for stability, then we have inf ρ∈X µ κ * ,κ ev ∩R(B µ κ * ,κ 1 ) K µ κ * ,κ ρ, ρ ρ L 2 Φ (ρ µ κ * ,κ ) = 0. (3.7) By Lemma 3.2, when κ is small enough, we have the decomposition X µ κ * ,κ ev = X µ κ * ,κ −,ev ⊕ X µ κ * ,κ +,ev , dim X µ κ * ,κ −,ev = 1, satisfying: i) K µ κ * ,κ \| X µ κ * ,κ −,ev < 0; ii) there exists δ > 0 such that K µ κ * ,κ ρ, ρ ≥ δ ρ 2 L 2 Φ (ρ µ κ * ,κ ) , for any ρ ∈ X µ κ * ,κ +,ev . By using above decomposition, it is easy to show that the infimum in (3.7) is obtained by some ρ * ∈ X µ κ * ,κ ev ∩ R(B µ κ * ,κ 1 ). Then L µ κ * ,κ ρ * , ρ * ≤ K µ κ * ,κ ρ * , ρ * = 0. On the other hand, we have L µ κ * ,κ dρ µ,κ dµ \| µ=µ κ * , dρ µ,κ dµ \| µ=µ κ * = dV µ,κ (0, Z µ,κ ) dµ \| µ=µ κ * dM µ,κ dµ \| µ=µ κ * = 0, and L µ κ * ,κ dρ µ,κ dµ \| µ=µ κ * , ρ * = dV µ,κ (0, Z µ,κ ) dµ \| µ=µ κ * ρ * dx = 0. This implies that ρ * = c dρµ,κ dµ \| µ=µ κ * for some constant c = 0. Since otherwise, n ≤0 (L µ κ * ,κ \| X µ κ * ,κ ev ) ≥ n ≤0 (L µ κ * ,κ \| span dρµ,κ dµ \| µ=µ κ * ,ρ * ) = 2. which is in contradiction to n ≤0 (L µ κ * ,κ \| X µ κ * ,κ ev ) = 1. Thus, we have 0 = K µ κ * ,κ dρ µ,κ dµ \| µ=µ κ * , dρ µ,κ dµ \| µ=µ κ * = 2πκ 2 +∞ 0 Υ(r) r 0 s +∞ −∞ dρµ,κ dµ \| µ=µ κ * (s, z)dzds 2 r +∞ −∞ ρ µ κ * ,κ (r, z)dz dr. and consequently +∞ −∞ dρ µ,κ dµ \| µ=µ κ * (r, z)dz = 0, ∀r ∈ [0, R µ κ * ,κ ]. (3.8) Nevertheless, it is not true as shown below. For non-rotating stars (ρ µ (r), 0), we have ∆V µ = 1 r 2 r 2 (V µ (r)) = 4πρ µ , where r = √ r 2 + z 2 and V µ (r) is the gravitational potential. Applying d dµ to above equation, one has 1 r 2 r 2 dV µ (r) dµ = 4π dρ µ dµ . When r ≥ R µ , since dρµ dµ (r) = 0 we have r 2 dV µ dµ (r) = R 2 µ dV µ dµ (R µ ) = 4π Rµ 0 s 2 dρ µ dµ (s)ds = dM µ dµ , and consequently dV µ dµ (r) = − dM µ dµ 1 r , for r ≥ R µ . Since lim κ→0 µ κ * = µ * , we have lim κ→0 dMµ dµ (µ κ * ) = dMµ dµ (µ * ) = 0. Thus dV µ dµ (R µ )\| µ=µ κ * = − dM µ dµ (µ κ * ) 1 R µ κ * → 0, as κ → 0. Define y µ (r) = V µ (R µ ) − V µ (r) = Φ (ρ µ ). Then by Lemma 3.13 in [30], we have dy µ dµ (R µ )\| µ=µ κ * = − d dµ M µ R µ \| µ=µ κ * − dV µ dµ (R µ )\| µ=µ κ * (3.9) → − d dµ M µ R µ \| µ=µ * = 0, as κ → 0. Thus by (3.9), we obtain dρ µ dµ (r) = 1 Φ (ρ µ ) dy µ dµ (r) ≈ ρ 2−γ 0 µ ≈ (R µ − r) 2−γ 0 γ 0 −1 , for r ∼ R µ and µ = µ κ * . By (3.36) and (4.78) in [41], we know dg −1 ζµ,κ dµ (y) = lim µ 1 →µ g −1 ζµ 1 ,κ − g −1 ζµ,κ µ 1 − µ (y) ≤ Cκ, for some constant C independent of µ and κ. Therefore, dρ µ,κ dµ (r, z) = dρ µ (g −1 ζµ,κ (r, z)) dµ = dρ µ dµ (g −1 ζµ,κ (r, z)) + dρ µ (r) dr \| r=g −1 ζµ,κ (r,z) dg −1 ζµ,κ dµ ≈ ρ µ (g −1 ζµ,κ (r, z)) 2−γ 0 = ρ µ,κ (r, z) 2−γ 0 , for g −1 ζµ,κ (r, z) ∼ R µ and µ = µ κ * . By Lemma 2.1, we have +∞ −∞ dρ µ,κ dµ \| µ=µ κ * (r, z)dz ≈ +∞ −∞ ρ µ,κ (r, z) 2−γ 0 dz ≈ (R µ κ * ,κ − r) 2−γ 0 γ 0 −1 + 1 2 = 0, for r ∼ R µ κ * ,κ . This is in contradiction to (3.8) and finishes the proof of the theorem. The case of fixed angular momentum distribution Let j(p, q) : R 2 → R be a given function satisfying j(p, q) ∈ C 1,β (R + × R + ) and j(0, q) = ∂ p j(0, q) = 0. (3.10) Define J(p, q) = j 2 (p, q). We construct a family of rotating stars of the following form ρ µ,ε (r, z) = ρ µ (g −1 ζµ,ε ((r, z))), v µ,ε = ε j(mρ µ,ε (r),Mµ,ε) r e θ , where m ρµ,ε (r) = r 0 s ∞ −∞ ρ µ,ε (s, z)dsdz, g ζµ,ε = x 1 + ζ µ,ε (x) \|x\| 2 , and ζ µ,ε (x) : B µ → R is axi-symmetric and even in z. The existence of rotating stars (ρ µ,ε , v µ,ε ) is reduced to the following equations: Φ (ρ µ,ε ) + V µ,ε − ε 2 r 0 J(m ρµ,ε (s), M µ,ε )s −3 ds + c µ,ε = 0, in Ω µ,ε , (3.11) V µ,ε = −\|x\| −1 * ρ µ,ε in R 3 ,(3.12) where Ω µ,ε = g ζµ,ε (B µ ) and c µ,ε is a constant. Although (3.11) is a little different from the steady state equations in [14] [19], the key linearized operator at the point ε = 0 is the same as [14]. By similar arguments as [14,19,41], we can get the following existence theorem. Theorem 3.4 Let µ ∈ [µ 0 , µ 1 ] ⊂ (0, µ max ), P (ρ) satisfy (1.2)-(1.3) and j(p, q) satisfy (3.10). Then there existε > 0 and solutions ρ µ,ε of (3.11) for all \|ε\| <ε, with the following properties: 1) ρ µ,ε ∈ C 1,α c (R 3 ), where α = min( 2−γ 0 γ 0 −1 , 1). 2) ρ µ,ε is axi-symmetric and even in z. 3) ρ µ,ε (0) = µ. 4) ρ µ,ε ≥ 0 has compact support g ζµ,ε (B µ ). 5) For all µ ∈ [µ 0 , µ 1 ], the mapping ε → ρ µ,ε is continuous from (−ε,ε) into C 1 c (R 3 ). When ε = 0, ρ µ,0 (x) = ρ µ (\|x\|) is the nonrotating star solution with ρ µ (0) = µ. Now we use Theorem 1.1 to study the stability of rotating star solutions (ρ µ,ε , εj(m ρµ,ε (r), M µ,ε )/re θ ), where ε is small enough, j(p, q) satisfies (3.10) and the Rayleigh stability condition ∂ p J (p, q) > 0 (i.e. j∂ p j > 0). As in Section 3.1, the assumptions in Theorem 1.1 can be verified. That is, ∂Ω µ,ε is C 2 and has positive curvature near (R µ,ε , 0) and (1.7) holds for any µ ∈ [µ 0 , µ 1 ] and ε small enough. Below, for rotating stars (ρ µ,ε , εj(m ρµ,ε (r), M µ,ε )/re θ ) we use X µ,ε , X µ,ε 1 , Y µ,ε , L µ,ε , A µ,ε 1 , B µ,ε 1 , B µ,ε 2 , K µ,ε , etc., to denote the corresponding spaces X, X 1 , Y , and operators L, A 1 , B 1 , B 2 , K etc. defined in Section 2. Again, we denoteμ to be the first critical point of M µ /R µ for non-rotating stars. Define the spaces X µ,ε ev and X µ,ε ev as in (3.3). By the same proof of Lemma 3.2, we have the following. 3) and j(p, q) satisfies (3.10) and ∂ p (j 2 (p, q)) > 0. Then for any µ ∈ [µ 0 , µ 1 ] ⊂ (0,μ) and ε small enough, we have n − (K µ,ε ) = 1 and ker K µ,ε = span{∂ z ρ µ,ε }. Moreover, we have the following direct sum decompositions for X µ,ε ev and X µ,ε ev : X µ,ε ev = X µ,ε −,ev ⊕ X µ,ε +,ev , dim X µ,ε −,ev = 1, and X µ,ε od = span{∂ z ρ µ,ε } ⊕ X µ,ε +,od , satisfying: i) K µ,ε \| X µ,ε −,ev < 0; ii) there exists δ > 0 such that K µ,ε u, u ≥ δ u 2 L 2 Φ (ρµ,ε) , ∀ u ∈ X µ,ε +,ev ⊕ X µ,ε +,od , where δ is independent of µ and ε. In addition, for any µ ∈ [µ 0 , µ 1 ], it holds that dVµ,ε(Rµ,ε,0) dµ < 0 for ε small. By Theorem 1.1, we get the following necessary and sufficient condition for the stability of rotating stars (ρ µ,ε , εj(m ρµ,ε (r), M µ,ε )/re θ ) : K µ,ε δρ, δρ = L µ,ε δρ, δρ + 2ε 2 π Rµ,ε 0 ∂ p J(m ρµ,ε (r), M µ,ε ) r 3 r 0 s +∞ −∞ δρ(s, z)dzds 2 dr ≥ 0, for all δρ ∈ R(B µ,ε 1 ) = δρ ∈ X µ,ε 1 \| R 3 δρdx = 0 . The following Theorem shows that the stability of this family of rotating stars can only change at the mass extrema. Theorem 3.5 Assume P (ρ) satisfies (1.2)-(1.3), and j(p, q) satisfy (3.10) and ∂ p (j 2 (p, q)) > 0. Let n u (µ) be the number of unstable modes, namely the total algebraic multiplicities of unstable eigenvalues of the linearized Euler-Poisson systems at (ρ µ,ε , εj(m ρµ,ε (r), M µ,ε )/re θ ). Then for any µ ∈ [µ 0 , µ 1 ] ⊂ (0,μ) and ε small enough, we have n u (µ) = 1, when dMµ,ε dµ < 0, 0, when dMµ,ε dµ ≥ 0. Proof. By the same arguments in the proof of Theorem 3.2, we have n u (µ) = n − K µ,ε \| X µ,ε ev ∩R(B µ,ε 1 ) . Thus it is reduced to find the number of negative modes of the quadratic form K µ,ε ·, · restricted to the even subspace of R(B µ,ε 1 ). Applying d dµ to (3.11), we obtain that L µ,ε dρ µ,ε dµ = ε 2 r 0 ∂ p J(m ρµ,ε (s), M µ,ε ) dm ρµ,ε dµ s −3 ds (3.13) + ε 2 r 0 ∂ q J(m ρµ,ε (s), M µ,ε ) dM µ,ε dµ s −3 ds − dc µ,ε dµ , where dc µ,ε dµ = d dµ −V µ,ε (R µ,ε , 0) + ε 2 Rµ,ε 0 J(m ρµ,ε (s), M µ,ε )s −3 ds = − dV µ,ε (R µ,ε , 0) dµ + ε 2 Rµ,ε 0 ∂ p J(m ρµ,ε (s), M µ,ε ) dm ρµ,ε (s) dµ s −3 ds + ε 2 dM µ,ε dµ h µ,ε (R µ,ε ) + ε 2 J(M µ,ε , M µ,ε )R −3 µ,ε dR µ,ε dµ . By integration by parts and (3.13), we obtain that 2π Rµ,ε 0 ε 2 ∂ p J(m ρµ,ε (r), M µ,ε )r −3 r 0 ∞ −∞ s dρ µ,ε dµ dzds r 0 ∞ −∞ sϕdzds dr = ε 2 Rµ,ε 0 ∂ p J(m ρµ,ε (r), M µ,ε )r −3 dm ρµ,ε dµ dr R 3 ϕdx − 2π Rµ,ε 0 ∞ −∞ ε 2 r 0 ∂ p J(m ρµ,ε (s), M µ,ε ) dm ρµ,ε(s) dµ s −3 ds rϕdzdr = ε 2 Rµ,ε 0 ∂ p J(m ρµ,ε (r), M µ,ε )r −3 dm ρµ,ε dµ dr R 3 ϕdx − dc 0 dµ , ϕ − L µ,ε dρ µ,ε dµ , ϕ − ε 2 dM µ,ε dµ r 0 ∂ q J(m ρµ,ε (s), M µ,ε )s −3 ds, ϕ = dV µ,ε (R µ,ε , 0) dµ − ε 2 J(M µ,ε , M µ,ε )R −3 µ,ε dR µ,ε dµ − ε 2 dM µ,ε dµ h µ,ε (R µ,ε ) R 3 ϕdx − L µ,ε dρ µ,ε dµ , ϕ − ε 2 dM µ,ε dµ K µ,ε g µ,ε , ϕ . Here, in the above we used h µ,ε (r) = r 0 ∂ q J m ρµ,ε (s), M µ,ε s −3 ds, and g µ,ε = K −1 µ,ε h µ,ε . The inverse operator K −1 µ,ε : (X µ,ε ev ) * ⊂ L 2 1 Φ (ρµ,ε) → X µ,ε ev exists and is bounded by Lemma 3.3. Since 1 Φ (ρµ,ε) has compact support and Φ (s) ≈ s γ 0 −2 for s ∼ 0 + , we have g µ,ε dx g µ,ε L 2 Φ (ρµ,ε) K −1 µ,ε h µ,ε L 2 1 Φ (ρµ,ε) h 2 µ,ε Φ (ρ µ,ε ) dx 1 2 < +∞. Therefore, we have K µ,ε dρ µ,ε dµ + ε 2 dM µ,ε dµ g µ,ε , ϕ = dV µ,ε (R µ,ε , 0) dµ + O(ε 2 ) R 3 ϕdx,(3.14) for any ϕ ∈ X µ,ε ev . By (3.14) and the fact that dVµ,ε(Rµ,ε,0) dµ + O(ε 2 ) < 0 when µ ∈ [µ 0 , µ 1 ] and ε is small, we have X µ,ε ev ∩ R(B µ,ε 1 ) = δρ ∈ X µ,ε ev \| K µ,ε dρ µ,ε dµ + ε 2 dM µ,ε dµ g µ,ε , δρ = 0 . On the other hand, we have K µ,ε dρ µ,ε dµ + ε 2 dM µ,ε dµ g µ,ε , dρ µ,ε dµ + ε 2 dM µ,ε dµ g µ,ε = dV µ,ε (R µ,ε , 0) dµ + O(ε 2 ) dρ µ,ε dµ + ε 2 dM µ,ε dµ g µ,ε dx = dV µ,ε (R µ,ε , 0) dµ + O(ε 2 ) dM µ,ε dµ . By Lemma 3.3, n − (K µ,ε \| X µ,ε ev ) = 1 and ker K µ,ε \| X µ,ε ev = {0}. We consider two cases: 1) dMµ,ε dµ = 0. A combination of above properties immediately yields 2) When dMµ,ε dµ = 0, as in the proof of Theorem 3.2, we have n u (µ) = n − K µ,ε \| X µ,n u (µ) = n − K µ,ε \| X µ,ε ev ∩R(B µ,ε 1 ) = 0. This finishes the proof of the theorem. i) (Fixed angular momentum distribution) The most common one is j(m, M ) = j(m). See for example [2,18,31,32,33,34]; ii) (Fixed angular momentum distribution per unit mass) j(m, M ) = j(m/M ). See for example [35]; iii) (Fixed angular momentum distribution with given total angular momentum) j(m, M ) = 1 M j(m/M ). See for example [4]. We note that for this case, the total angular momentum given by 1 M j( m M )dm = 1 0 j (m ) dm (m = m M ), is a constant depending only on j. In the rest of this subsection, we use Theorem 3.5 to study two examples of rotating stars with mass extrema points. Example 1. Asymptotically polytropic rotating stars Assume P (ρ) satisfies assumptions (3.5)-(3.6). By the same arguments as in the case of fixed angular velocity, when ε is small enough and µ ∈ [µ 0 , µ 1 ] ⊂ (0,μ), the mass M µ,ε of the rotating stars (ρ µ,ε , εj(m ρµ,ε (r), M µ,ε )/re θ ) has the the first maximum µ ε * ∈ (µ 0 , µ 1 ). Then by Theorem 3.5, the rotating stars are stable when µ ∈ [µ 0 , µ ε * ] and unstable when µ goes between µ ε * and the next extrema point of M µ,ε in (µ ε * , µ 1 ). Example 2. Polytropic rotating stars Consider the polytropic equation of state P (ρ) = ρ γ γ ∈ 6 5 , 2 . The non-rotating stars (i.e. Lane-Emden stars) with any center density µ are stable when γ ∈ (4/3, 2) and are unstable when γ ∈ (6/5, 4/3). In particular, M µ = C γ µ 1 2 (3γ−4) is a monotone function when γ = 4 3 and there is no transition point of stability. However, polytropic rotating stars with fixed angular momentum distribution j (m, M ) can have mass extrema points, which are also the transition points of stability. One such example was given in [4] for γ = 4. [4]) that there is a mass minimum point µ * for the total mass M (µ). This is the first transition point of stability. In particular, rotating stars with center density µ beyond µ * become stable. Remark 3.2 It can also be seen from above Example 2 that the critical index γ * for the onset of instability of rotating polytropic stars is not 4 3 . Ledoux [25], Chandrasekhar and Lebovitz [8] indicated that the critical index γ * is reduced from 4 3 to γ * = 4 3 − 2ω 2 I 9\|W \| for small uniform rotating stars, where I > 0 is the moment of inertia about the center of mass and W is the gravitational potential energy. For more discussion about the critical index γ * of rotating stars, see [13,22,38,39]. Instability for Rayleigh Unstable case Consider an axi-symmetric rotating star (ρ 0 , v 0 ) = (ρ 0 (r, z) , ω 0 (r)re θ ), where the angular velocity ω 0 (r) satisfies the Rayleigh instability condition, that is, there exists a point r 0 ∈ (0, R 0 ) such that Υ(r 0 ) = ∂ r (ω 2 0 r 4 ) r 3 r=r 0 < 0. (4.1) For incompressible Euler equation, it is a classical result by Rayleigh in 1880 [37] that condition (4.1) implies linear instability of the rotating flow v 0 = ω 0 (r)re θ under axi-symmetric perturbations. In this section, we will show the axi-symmetric instability of rotating stars with Rayleigh unstable angular velocity. From the linearized Euler-Poisson system (2.5), we get the following second order equation for u 2 = v r v z , ∂ tt u 2 = −Lu 2 = −(L 1 + L 2 )u 2 ,(4.2) where L 1 , L 2 are operators on Y = L 2 ρ 0 2 defined by L 1 u 2 = B 1 LB 1 A = ∇[Φ (ρ 0 )(∇ · (ρ 0 u 2 )) − 4π(−∆) −1 (∇ · (ρ 0 u 2 )], and L 2 u 2 = Υ(r)v r 0 . Lemma 4.1L is a self-adjoint operator on (Y, [·, ·]) with the equivalent inner product [·, ·] = A·, · . Proof. By Lemma 2.9 in [30], L 1 is self-adjoint on (Y, [·, ·]) with the equivalent inner product [·, ·] := A·, · . Since L 2 is a symmetric bounded operator on (Y, [·, ·]),L = L 1 + L 2 is self-adjoint by Kato-Rellich Theorem. The next lemma on the quadratic form ofL will be used later. , it suffices to estimate [L 1 u 2 , u 2 ] = LB 1 Au 2 , B 1 Au 2 = ∇ · (ρ 0 u 2 ) 2 L 2 Φ (ρ 0 ) − 4π R 3 \|∇V \| 2 dx, where −∆V = ∇ · (ρ 0 u 2 ). By integration by parts, R 3 \|∇V \| 2 dx = − R 3 ρ 0 u 2 · ∇V dx u 2 2 Y 1 2 \|∇V \| 2 dx 1 2 , which implies that \|∇V \| 2 dx u 2 2 Y . This finishes the proof of the lemma. The study of equation (4.2) is reduced to understand the spectra of the self-adjoint operatorL. First, we give a Helmholtz type decomposition of vector fields in Y . Lemma 4.3 There is a direct sum decomposition Y = Y 1 ⊕ Y 2 , where Y 1 is the closure of u ∈ Y \| u = ∇p, for some p ∈ C 1 (Ω) , in Y and Y 2 is the closure of u ∈ C 1 (Ω) 2 ∩ Y \| ∇ · (ρ 0 u) = 0 , in Y . The proof of above lemma is similar to that of Lemma 3.15 in [30] and we skip. Denote P 1 : Y → Y 1 and P 2 : Y → Y 2 to be the projection operators. Then P 1 , P 2 ≤ 1. For any u 2 ∈ Y , let u 2 = v 1 +v 2 where v 1 = P 1 u 2 ∈ Y 1 and v 2 = P 2 u 2 ∈ Y 2 . SinceL u 2 = L 1 v 1 + P 1 L 2 v 1 + P 1 L 2 v 2 + P 2 L 2 v 1 + P 2 L 2 v 2 , the operatorL : Y → Y is equivalent to the following matrix operator on Y 1 × Y 2 L 1 , C C * ,L 2 v 1 v 2 = L 1 , C 0,L 2 + 0, 0 C * , 0 v 1 v 2 = (T + A)v, whereL 1 = L 1 + P 1 L 2 P 1 : Y 1 → Y 1 ,L 2 = P 2 L 2 P 2 : Y 2 → Y 2 , C = P 1 L 2 P 2 : Y 2 → Y 1 , C * = P 2 L 2 P 1 : Y 1 → Y 2 , and T = L 1 , C 0,L 2 , A = 0, 0 C * , 0 : Y 1 × Y 2 → Y 1 × Y 2 . Lemma 4.4 The operator A is T -compact. Proof. For any v = (v 1 , v 2 ) ∈ D (T ), the graph norm v T is defined by v T = v Y + T v Y ≈ v Y + L 1 v 1 Y ≈ v Y + L 1 v 1 Y . It is obvious that D(A) ⊃ D(T ). To prove A is T -compact, we need to prove A : (D(A), · T ) → (Y, · Y ) is compact. By the definition of A, we notice that Av = P 2 L 2 v 1 : Y 1 × Y 2 → {0} × Y 2 . For v 1 = ∇ξ ∈ Y 1 , v 1 Z = ∇ · (ρ 0 v 1 ) L 2 Φ (ρ 0 ) + v 1 Y = ∇ · (ρ 0 ∇ξ) L 2 Φ (ρ 0 ) + ∇ξ Y , as defined in (1.16). By the proof of Lemma 4.2 we have ∇ · (ρ 0 v 1 ) 2 L 2 Φ (ρ 0 ) + v 1 2 Y L 1 v 1 , v 1 + 2m v 1 2 Y L 1 v 1 2 Y + v 1 2 Y ≈ v 2 T . Thus v 1 Z v T . Since the embedding (Y 1 , · Z ) → (Y 1 , · Y ) is compact by Proposition 12 in [20] and P 2 , L 2 are bounded operators, it follows that A : (D(A), · T ) → (Y, · Y ) is compact. The above lemma implies that the essential spectra ofL is the same as L 2 . Proof. We have σ ess (L) = σ ess (T + A) by the definition of the operator T +A. By Lemma 4.4 and Weyl's Theorem, we have σ ess (T +A) = σ ess (T ). By Theorem 2.3 v) in [30] and the compact embedding of (Y 1 , · Z ) → (Y 1 , · Y ), the spectra of L 1 on Y 1 are purely discrete and σ ess (L 1 ) = {∅}. By the same arguments as in the proof of Lemma 4.4,L 1 is relative compact to L 1 and as a result σ ess L 1 = σ ess (L 1 ) = {∅} . Since the matrix operator T is upper triangular, it follows that σ ess (T ) = σ ess L 1 ∪ σ ess L 2 = σ ess L 2 . We study the essential spectra ofL 2 in the next two lemmas. By the Rayleigh instability condition (4.1) and the fact that Υ(0) = 4ω 0 (0) 2 ≥ 0, we know that range (Υ(r)) = [−a, b] for some a > 0, b ≥ 0. where c 1 = min {\|λ − b\| , \|a + λ\|} > 0. Thus L 2 − λ u ≥ c 1 u Y , which implies that (L 2 −λ) −1 is bounded and λ ∈ ρ(L 2 ). Therefore, σ(L 2 ) ⊂ [−a, b]. This prove the lemma by combining with Lemma 4.6. The following proposition gives a complete characterization of the spectra ofL. Proof. The conclusion in i) follows from Lemmas 4.5 and 4.7. This implies that any λ ∈ σ(L) in (−∞, −a) or (b, +∞) must be a discrete eigenvalue of finite multiplicity. Proof of ii): Suppose otherwise. Then there exists an infinite dimensional eigenspace for negative eigenvalues in (−∞, −a). We notice that L + aI = L 1 + L 2 + aI ≥ L 1 , since L 2 +aI is nonnegative. It follows that n − (L 1 ) = ∞ since n − L + aI = ∞. This is in contradiction to that n − (L 1 ) ≤ n − (L) < ∞. Proof of iii): Suppose otherwise. Then there exists an upper bound of σ(L), denoted by λ max ≥ b. ThusL ≤ λ max I which implies that L 1 ≤ −L 2 + λ max I ≤ (a + λ max ) I. Consequently the eigenvalues of L 1 cannot exceed a + λ max . This is in contradiction to the fact that L 1 has a sequence of positive eigenvalues tending to infinity. Now we can prove Theorem 1.2. Proof of Theorem 1.2. Denote π λ ∈ L (X) (λ ∈ R) to be the spectral family of the self-adjoint operatorL. Let {µ i } ∞ i=1 be the eigenvalues ofL in (b, ∞). If σ(L) ∩ (−∞, −a) = ∅, we denote the eigenvalues in (−∞, −a) by ν 1 < · · · < ν K where K = dim (R (π −a )). For 1 ≤ i < ∞, 1 ≤ j ≤ K, let P + i = π µ i + − π µ i − and P − j = π ν j + − π ν j − be the projections to ker L − µ i I and ker L − ν j I respectively, and P 0 = π 0+ − π 0− be the projection to kerL. By Proposition 4.1, we havẽ L = λdπ λ = ∞ i=1 µ i P + i + K j=1 ν j P − j + b −a λdπ λ . For any initial data (u 2 (0) , u 2t (0)) ∈ Z × Y , the solution to the second order equation (4.2) can be written as u 2 (t) = ∞ i=1 cos( √ µ i t)P + i u 2 (0) + 1 √ µ i sin( √ µ i t)P + i u 2t (0) (4.3) + K j=1 cosh −ν j t P − j u 2 (0) + 1 √ −ν j sinh −ν j t P − j u 2t (0) + b 0 cos( √ λt)dπ λ u 2 (0) + b 0 1 √ λ sin( √ λt)dπ λ u 2t (0) + 0 −a cosh( √ −λt)dπ λ u 2 (0) + 0 −a 1 √ −λ sinh( √ −λt)dπ λ u 2t (0) + P 0 u 2 (0) + tP 0 u 2t (0). If σ(L) ∩ (−∞, −a) = ∅, the solution u 2 (t) is obtained by removing the second term above. Denote the minimum of λ ∈ σ(L) by η 0 , that is, η 0 = min ψ Y =1 [Lψ, ψ] = −a, if σ(L) ∩ (−∞, −a) = ∅, ν 1 , if σ(L) ∩ (−∞, −a) = {ν 1 < · · · < ν K }. By the formula (4.3), it is easy to see that u 2 (t) Y e √ −η 0 t for t > 0. To estimate u 2 (t) Z , we note that by Lemma 4.2 u 2 2 Z ≈ L u 2 , u 2 + 2m u 2 2 Y . (4.4) ∞ i=1 µ j P + i u 2 (0) 2 Y + P + i u 2t (0) 2 Y + e −η 0 t K j=1 P − j u 2 (0) 2 Y + P − j u 2t (0) 2 Y + b 0 d (π λ u 2 (0), u 2 (0)) + b 0 d (π λ u 2t (0), u 2t (0)) + e −η 0 t 0 −a d (π λ u 2 (0), u 2 (0)) + 0 −a d (π λ u 2t (0), u 2t (0)) e −η 0 t L u 2 (0) , u 2 (0) + m u 2 (0) 2 Y + u 2t (0) 2 Y e −η 0 t u 2 (0) 2 Z + u 2t (0) 2 Y . This implies This finishes the proof of the upper bound estimate (1.17). It is straightforward to show that the energy E(u 2 , u 2t ) defined in (1.18) is conserved for solutions of (4.2). u 2 (t) Z e √ −η 0 t ( u 2 (0) Z + u 2t (0) Y ) , stabilizing influence of rotation on the fundamental mode (corresponding to the first eigenvalue of the operatorL in (4.2)) even when ω 0 (r) does not satisfy the Rayleigh stability condition. However, this does not imply the stability of the rotating stars since the unstable continuous spectrum was not considered in [24]. about general separable Hamiltonian PDEs (2.1). The triple (L, A, B) is assumed to satisfy assumptions: (G1) The operator B : Y * ⊃ D(B) → X and its dual operator B : X * ⊃ D(B ) → Y are densely defined and closed (and thus B = B). Remark 2. 3 3For non-rotating stars, the dynamically accessible perturbations are given by R (B 1 ) = R (B 1 ) which is the perturbations preserving the mass (seeLemma 2.4). For rotating stars, the dynamically accessible space R(B) is different from R (B). = I + II,where ε > 0 is chosen such that Lemma 2.1 holds. Since the function h 1 (r) = +∞ −∞ ρ 0 (r, z)dz has a positive lower bound in [0, R 0 − ε] and h 2 (r0 (r,z)) dz is bounded, by Hardy's inequality (see Lemma 3.21 in[30]) we have . to (2.17) and (2.18). Since δρ ∈ R (B 1 ), it follows from (2.19) that n − K\| R(B 1 ) ≥ n − L\| R(B) . On the other hand, we also have n − K\| R(B 1 ) ≤ n − L\| R(B) , since Kδρ, δρ = L Thus n − K\| R(B 1 ) = n − L\| R(B) . This finishes the proof of Theorem 1.1. Theorem 3 . 2 ( 32Sufficient condition for stability) Suppose P (ρ) satisfies (1.2)-(1.3), and ω ∈ C 1,β [0, ∞) satisfies Υ(r) > 0. Lemma 3. 3 3Assume P (ρ) satisfies(1.2)-(1. Remark 3. 1 1The above theorem implies that for a family of rotating stars with fixed angular momentum distribution j(m, M ), the transition of stability occurs at the first extrema of the total mass. That is, the turning point principle (TPP) is true for this family of rotating stars. This contrasts greatly to rotating stars of fixed angular velocity, for which case TPP is shown to be not true (seeTheorem 3.3).In the literature, there are three common choices of j(m, M ) in the study of rotating stars. j(m, M ) = 1 M [1 − (1 − m M ) 2/3 ]. With numerical help, it was found (see Figure 1 below taken from Figure 1 : 1The dependence of the mass M (µ) on the center density µ for γ = 4.03 3.03 and the angular momentum distribution j(m, M ) = 1 M [1 − (1 − m M ) 2/3 ]. From Bisnovatyi-Kogan and Blinnikov [4]. Lemma 4. 2 2There exists constants m > 0 such that for anyu 2 ∈ Y , we have L u 2 , u 2 + 2 , u 2 = [L 1 u 2 , u 2 ] + [L 2 u 2 , u 2 ] ,and obviously \|[L 2 u 2 , u 2 Lemma 4.5 σ ess (L) = σ ess (L 2 ). n → 0. This shows that δv εn is a Weyl's sequence forL 2 and λ ∈ σ ess (L 2 ). Thus (−a, b) ⊂ σ ess (L 2 ) which implies [−a, b] ⊂ σ ess (L 2 ) since σ ess (L 2 ) is closed. Lemma 4.7 σ(L 2 ) = σ ess (L 2 ) = range (Υ(r)) = [−a, b]. Proof. Fix λ / ∈ [−a, b]. For any u = (u r , u z ) ∈ Y 2 , we have [(L 2 − λ)u, u] = [(L 2 − λ)u, u] = [(Υ(r) − λ)u r , u r ] − [λu z , u Since a > 0, b ≥ 0, we have \|[(L 2 − λ)u, u]\| ≥ c 1 u 2 Y , Proposition 4. 1 1Under the Rayleigh instability condition (4.1), it holds:i) σ ess (L) = range(Υ(r)) = [−a, b]. ii) σ(L) ∩ (−∞, −a)consists of at most finitely many negative eigenvalues of finite multiplicity. iii) σ(L) ∩ (b, +∞) consists of a sequence of positive eigenvalues tending to infinity. by using (4.4) and the estimate for u 2 (t) )dπ λ u 2t (0) + P 0 u 2t (0), by similar estimates as above for u 2 (t) Z , we obtainu 2t (t) Y e √ −η 0 t ( u 2 (0) Z + u 2t (0) Y ) . Corollary 1.1 Under the assumptions of Theorem 1.1, the rotating star solution (ρ 0 , v 0 ) is spectrally stable to axi-symmetric perturbations if and only ifKδρ, δρ ≥ 0, 4π ∆ −1 ρ µ,κ . The existence of decompositions for X µ,κ ev and X µ,κ od as stated in Proof. For any λ ∈ (−a, b), let r 0 ∈ (0, R 0 ) be such that λ = Υ(r 0 ). Choose (r 0 , z 0 ) ∈ Ω and ε 0 small enough, such that (r, z) ∈ Ω when \|r − r 0 \| ≤ ε 0 and \|z − z 0 \| ≤ ε 2 0 . Choose a sequence {ε n } ∞ n=1 ⊂ (0, ε 0 ) with lim n→∞ ε n = 0. Let ϕ(r), ψ(z) ∈ C ∞ 0 (−1, 1) be two smooth cutoff functions such that ϕ(0) = ψ(0) = 1. Define δv εn = (δv εn r , δv εn z ) withThen δv εn Y = 1 and δv εn ∈ Y 2 owing toWe will show that {δv εn } is a Weyl's sequence for the operatorL 2 and therefore λ ∈ σ ess (L 2 ). First, we check that δv εn converge to 0 weakly in Y 2 . For any ξ ∈ Y 2 , since δv εn is supported inwhen ε n → 0. Next, we prove that (L 2 − λ)δv εn converge to 0 strongly in Y 2 . We writeNext, we prove the lower bound estimate(1.19)in two cases. Case 1: σ(L) ∩ (−∞, −a) = ∅. We choose u 2 (0) = ψ 1 and u 2t (0) = √ −ν 1 ψ 1 where ψ 1 ∈ Z is the eigenfunction ofL corresponding to the smallest eigenvalue ν 1 in (−∞, −a). Then, for any ε > 0 small there exists a nonzero function φ ∈ R(π −a+ε − π −a ) ⊂ Z. Choose the initial data u 2 (0) = φ and u 2t (0) = 0. Then the solution u 2 (t) for the equation (4.2) is given byThusThis finishes the proof of the theorem.Remark 4.1 By Theorem 1.2, the maximal growth rate of unstable rotating stars can be due to either discrete or continuous spectrum. Consider a family of slowly rotating stars (ρ ε , v ε = εrω 0 (r) e θ ) near a non-rotating star ρ 0 (\|x\|) , v 0 = 0 with ω 0 (r) satisfying the Rayleigh instability condition (4.1). 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[ "ON PROJECTIVE K3 SURFACES X WITH Aut(X ) = (Z 2Z) 2", "ON PROJECTIVE K3 SURFACES X WITH Aut(X ) = (Z 2Z) 2" ]	[ "Adrian Clingher ", "Andreas Malmendier ", "Xavier Roulleau " ]	[]	[]	We prove that every K3 surface with automorphism group (Z 2Z) 2 admits an explicit birational model as a double sextic surface. This model is canonical for Picard number greater than 10. For Picard number greater than 9, the K3 surfaces in question possess a second birational model, in the form of a projective quartic hypersurface, generalizing the Inose quartic.	null	[ "https://export.arxiv.org/pdf/2305.08959v1.pdf" ]	258,715,330	2305.08959	2a7950662bbf2b2b5530afe23f4232bc1c2e9fe2	ON PROJECTIVE K3 SURFACES X WITH Aut(X ) = (Z 2Z) 2 15 May 2023 Adrian Clingher Andreas Malmendier Xavier Roulleau ON PROJECTIVE K3 SURFACES X WITH Aut(X ) = (Z 2Z) 2 15 May 2023 We prove that every K3 surface with automorphism group (Z 2Z) 2 admits an explicit birational model as a double sextic surface. This model is canonical for Picard number greater than 10. For Picard number greater than 9, the K3 surfaces in question possess a second birational model, in the form of a projective quartic hypersurface, generalizing the Inose quartic. Introduction Let X be a smooth complex projective K3 surface. We denote the Néron-Severi lattice of X by NS(X ). This lattice is even and has signature (1, ρ X − 1), where ρ X is the Picard number of X . The value of ρ X ranges from 1 to 20. A lattice polarization on X is defined as a primitive lattice embedding i∶ L ↪ NS(X ), where i(L) contains a pseudo-ample class. Here, L is an even indefinite lattice of signature (1, ρ L − 1), with 1 ≤ ρ L ≤ 19. Two K3 surfaces (X , i) and (X ′ , i ′ ) are said to be isomorphic under L-polarization if there exists an analytic isomorphism α∶ X → X ′ and a lattice isometry β ∈ O(L) such that α * ○ i ′ = i ○ β, where α * is the induced morphism at the cohomology level. 1 L-polarized K3 surfaces are classified up to isomorphism by a coarse moduli space, which is a quasi-projective variety of dimension 20 − ρ L . A general L-polarized K3 surface (X , i) satisfies i(L) = NS(X ). It is known that the automorphism group Aut(X ) of a K3 surface X is isomorphic, up to a finite group, to the factor group O(NS(X )) W X , where W X is the subgroup generated by reflections associated with elements of NS(X ) of square (−2). Nikulin [22,23,25] and Vinberg [33] classified K3 surfaces X with finite automorphism group by their Néron-Severi lattices NS(X ) for ρ X ≥ 3, resulting in a list of 118 lattices. This paper focuses on the case Aut(X ) ≅ (Z 2Z) 2 . According to Kondo [18], in such a situation the lattice NS(X ) is 2-elementary. In fact, all possible Néron-Severi lattices may be listed as follows: Above, we utilize the following notations: L 1 ⊕ L 2 refers to the orthogonal direct sum of the two lattices L 1 and L 2 , L(λ) is obtained by multiplying the lattice pairing of L with λ ∈ Z, ⟨R⟩ denotes a lattice with matrix R in some basis. The lattices A n , D m , and E k are the negative definite root lattices associated with the respective root systems. H is the unique even unimodular hyperbolic rank-two lattice, and N is the negative definite rank-eight Nikulin lattice, as defined, for instance, in [20,Sec. 5]. Given a lattice L, we denote its discriminant group as L ∨ L, and its associated discriminant form as q L . A lattice L is termed 2-elementary if D(L) ≅ (Z 2Z) ℓ , where ℓ = ℓ L is the length of L. A result of Nikulin [24,Thm. 4.3.2] asserts that hyperbolic, even, 2-elementary, primitive sublattices L of the even, unimodular K3 lattice of signature (3,19) are uniquely determined by their rank ρ L , length ℓ L , and a third invariant δ L ∈ {0, 1}, called parity. Here, δ L = 0 if q L (x) takes values in Z 2Z ⊂ Q 2Z for all x ∈ D(L), and δ L = 1 otherwise. The present article builds upon previous work by the authors, as well as others. In [28], the third author studied the geometry of K3 surfaces with finite automorphism group. In particular, the configuration of all rational curves was determined in each case. The first two authors studied the same families of K3 surfaces from the point of view of elliptic fibrations [6]. Detailed classifications were given, including a description of coarse moduli spaces, as well as explicit birational models. Other examples of explicit elliptic fibrations on K3 surfaces of 2-elementary type were given by Garbagnati and Salgado in [15] and Balestrieri et al. in [1]. In this article, we establish explicit birational models for all general L-polarized K3 surface, with L a member of the list (1.1). Specifically, we construct normal forms, as quartic hypersurfaces in P 3 , as well as double sextic surfaces. The automorphisms in Aut(X ) can then be seen as explicit algebraic transformations in the double sextic context. For ρ L > 10, the double sextic structure is canonical. The linear systems corresponding to the two projective models are also described in detail. The article is structured as follows: Section 2 outlines the main results, namely Theorems 2.1 and 2.4. The construction of the relevant birational models as well as the proof of the theorems are then divided up into different cases, corresponding to their Picard number. Section 3 presents the treatment of K3 surfaces of Picard number 9. Similarly, Sections 4 and 5 discuss K3 surfaces with Picard number 10 and with Picard number greater than 10, respectively. Statement of results A central role in this paper is played by the concept of a double sextic structure on a K3 surface [3,27,29]. Let S ⊂ P 2 be a reduced plane curve of degree six; we say that the sextic is admissible if the double cover π S ∶ S → P 2 branched over S is a surface with, at worst, rational double point singularities. The singularities of such a sextic are called simple singularities in [26] and general results of Brieskorn then guarantee that the minimal desingularization X of S, is a K3 surface. The above process defines a double cover morphism X → P 2 , which will be referred to as a double sextic structure on X . We note that the K3 surface X , as well as the double sextic structure X → P 2 , may alternatively be obtained by first performing a sequence of blow-ups on P 2 , until the singularities of S are resolved, followed by taking a double cover of the resulting rational surface, branched over an appropriate smooth branch locus. We also note that a double sextic structure X → P 2 induces a natural anti-symplectic involution k X ∶ X → X , as well as a special base-point-free linear system D 2 , with D 2 quasi-ample, satisfying D 2 2 = 2 and h 0 (D 2 ) = 3. The linear system D 2 completely determines the double sextic structure X → P 2 . In the generic situation, the sextic curve S is non-singular, and in general position, which leads to ρ X = 1. However, the Picard number of X may increase, as the curve S may lie in a special position or may carry singularities. Multiple such examples were constructed in [27]. The minimal resolutions of the double sextic surfaces we derive in this article have Picard numbers between 9 and 18. A classical geometric situation in which double sextic structures occur is the case when X admits a singular quartic normal form, i.e., X may be obtained as the minimal resolution of a quartic surface K in P 3 with rational double point singularities. Assume that p ∈ K is such a rational double point. Considering the lines in P 3 passing through p, one obtains a projection morphism, associated with the projection with center p, (2.1) K − {p} → P 2 , which generically is a double cover. Upon performing the desingularization of K, the morphism (2.1) extends to a double sextic structure X → P 2 . In this article, we shall primarily focus on two types of double sextic structures: on the one hand, we construct double sextics S where the branch locus in P 2 = P(u, v, w) is a bi-quadratic sextic that is generically irreducible. Such a double sextic is given by (2.2) S∶ y 2 = c 0 w 6 + c 2 (u, v)w 4 + c 4 (u, v)w 2 + c 6 (u, v) , where c n are homogeneous polynomials of degree n. In this context, the automorphism group Aut(X ) of the associated K3 surface X contains several involutions by construction: (i) An antisymplectic involution  X is induced by the involution  S ∶ w ↦ −w on the double sextic whose invariant locus contains a smooth curve of genus two which is the hyperelliptic curve y 2 = c 6 (u, v). (ii) A commuting antisymplectic involution is the aforementioned involution k X that is induced by the involution k S ∶ y ↦ −y. (iii) Moreover, the composition of the two anti-symplectic involutions induces the symplectic involution ı X =  X ○ k X on X . On the other hand, our investigation will encompass the desingularization of double covers branched over a reducible sextic in the projective plane, which is expressed as the union of a conic and a plane quartic curve. Such a double sextic is given by (2.3) S ′ ∶ỹ 2 = C(u, v, w) ⋅ Q(u, v, w) , where C and Q are homogeneous polynomials of degree 2 and 4, respectively. We will demonstrate that every general L-polarized K3 surface, where L is one of the lattices in Equation (1.1), admits birational models as double sextics of the form S and S ′ . 2 Our findings are as follows: (1.1). There exists a double sextic S whose minimal resolution is isomorphic to X and has the following properties: Theorem 2.1. Let X be a K3 surface with Aut(X ) ≅ (Z 2Z) 2 so that NS(X ) ≅ L, for L in Equation (1) For L = H ⊕ N there are polynomials c n of degree n = 0, 2, 4, 6 so that S is given by (2.2). For L = H ⊕ N the double sextic is of the form S∶ y 2 = w c 3 (u, v) w 2 + a 4 (u, v) w + d 5 (u, v) , where c 3 , a 4 , d 5 are homogeneous polynomials of degree 3, 4, 5, respectively. (2) For L = H ⊕ N, H ⊕ D 8 ⊕ D 4 the double sextic S is irreducible, and S has one singularity of type A 2n+1 for ρ L = 10 + n, 0 ≤ n ≤ 8, and no singularity for ρ L = 9. For L = H ⊕ N, H ⊕ D 8 ⊕ D 4 the sextic curve S is the union of a line and an irreducible quintic. (3) There is a divisor D 2 ∈ NS(X ), which is nef, of square-two, and invariant under the action of ı X . For the linear system D 2 the corresponding rational map is the projection onto P 2 = P(u, v, w) in Equation (2.2). If ρ L > 10 or L = H ⊕ A ⊕8 1 , then there are polynomials C and Q such that the minimal resolution of S ′ in (2.3) is isomorphic to X . As an immediate consequence we observe that the non-trivial elements of Aut(X ) in Theorem 2.1 are the involutions ı S ,  S , k S with ı S =  S ○ k S on the double sextic S. We make the following: Remark 2.2. In general, the determination of the fixed locus of the antisymplectic involution  X was accomplished by Nikulin [24,Thm.4.2.2]. By applying the result to the lattices in Equation (1.1) -excluding the case L = H ⊕ N-it follows that the fixed locus is comprised of a smooth, irreducible curve of genus g = 2 and k rational curves, with 2 ≤ k ≤ 9, such that k = (ρ L − ℓ L ) 2. For L = H ⊕ N the fixed locus consists of a genus-3 curve and two rational curves. Furthermore, the birational model S turns out to be canonical for ρ L > 10, in a sense that we explain below. To see this, let us first recall the following facts: We define a Jacobian elliptic fibration on X to be a pair (π X , σ X ) consisting of a proper map of analytic spaces π X ∶ X → P 1 = P(u, v), whose general fiber in P 2 = P(X, Y, Z) is a smooth curve of genus one, and a section σ X ∶ P 1 → X in the elliptic fibration π X . The group of section of the Jacobian fibration is the Mordell-Weil group MW(X , π X ). By a slight abuse of notation, we will not distinguish between the smooth elliptic K3 surface and its singular Weierstrass model. The latter is obtained from the former by blowing down the components of the reducible fibers which do not meet the section. In [32] van Geemen and Sarti studied a certain family of Jacobian elliptic K3 surfaces. The family is given by (2.4) X ∶ Y 2 Z = X X 2 + a 4 (u, v)XZ + b 8 (u, v)Z 2 , where a 4 and b 8 are homogeneous polynomials of degree four and eight, respectively. The Jacobian elliptic fibration always admits a section of order two, or, 2-torsion section. The fiberwise translation by the 2-torsion section acts as a Nikulin involution ı X on the elliptic fibration (2.4). It is called van Geemen-Sarti involution. The K3 surface X also admits the involution acting as the (fiberwise) hyperelliptic involution  X , and a third involution obtained by composition k X = ı X ○  X . It turns out that general K3 surface X has NS(X ) ≅ H ⊕ N. Conversely, it is known that every general H ⊕N-polarized K3 surface admits a unique fibration given by Equation (2.4); see [32]. A generalization of the statement for L in Equation (1.1) with ρ L > 10 was proved by the first two authors in [6]. As for nomenclature, we shall refer to a Jacobian elliptic fibration associated with an H-embedding into a lattice L ≅ H ⊕ K where K is a negative-definite lattice of ADE-type as standard. In contrast, the elliptic fibration associated with Equation (2.4) carrying an underlying van Geemen-Sarti involution will be referred to as alternate fibration. We will then prove the following: Up to this point, our focus has primarily been on the double sextic surface S. However, the general L-polarized K3 surface X , with L in Equation (1.1) and ρ L > 10 or L = H ⊕ A ⊕8 1 , also possesses a second birational model. This model is that of the double sextic S ′ in Equation (2.3). Within the context of S ′ , there is also a quartic projective hypersurface K ⊂ P 3 = P(u, v, w, y) or simply quartic, given by (2.5) K∶ y 2 C(u, v, w) = Q(u, v, w) , where C and Q are the same polynomials as those used in Equation (2.3) to define S ′ . It follows that for every quartic K, we may associate an associated double sextic S ′ as defined by Equation (2.3), and vice versa, where the relation between Equation (2.3) and Equation (2.5) isỹ = C(u, v, w) y. The significance of this quartic can be seen as follows. In the works [7,31], it was demonstrated that a K3 surface X over the complex numbers with Néron-Severi lattice H ⊕ E 8 ⊕ E 8 possesses a birational model isomorphic to the quartic K in P 3 = P(X, Y, Z, W), as defined by the equation K∶ 0 = Y 2 ZW − 4X 3 Z + 3αXZW 2 + βZW 3 − 1 2 Z 2 W 2 + W 4 . where α and β are two complex parameters. This two-parameter family was initially introduced by Inose in [16] and is called Inose quartic. Notably, the quartic K defined in Equation (2.5) coincides with the Inose quartic when ρ L = 18. Furthermore, it provides multi-parameter generalizations of the Inose quartic for Picard numbers 10 ≤ ρ L < 18. We will prove the following: Theorem 2.4. Let X be a K3 surface with NS(X ) = L, for L = H ⊕ A ⊕7 1 or L in Equation (1.1) with L = H(2) ⊕ A ⊕7 1 , H ⊕ N. There exists a quartic K whose minimal resolution is isomorphic to X and has the following properties: (1) There are polynomials C, Q so that K is given by (2.5). (2) K has one rational double-point at p 1 ∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 0 ∶ 1] for ρ L = 9, and for ρ L ≥ 10 two rational double-point singularities at p 1 ∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 0 ∶ 1] , p 2 ∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 1 ∶ 0] . (3) For ρ L ≥ 10 there is an explicit birational map between K and S, such that the polarization divisor for K is H = D 2 + b 1 + ⋅ ⋅ ⋅ + b N where b i are the exceptional curves of the singularity at p 2 , and D 2 is the same as in Theorem 2.1. We also make the following: Remark 2.5. The projection of K onto P 2 with center either p 1 or p 2 recovers the two double sextic S ′ or S, respectively, for ρ L ≥ 10. Remark 2.6. We also provide explicit expressions for the divisors in Theorem 2.1 and 2.4 in terms of the dual graph of rational curves for the associated L-polarized K3 surfaces. K3 surfaces of Picard number 9 In Picard number 9, we consider two lattice polarizations, namely H(2) ⊕ A ⊕7 1 and H ⊕ A ⊕7 1 . Only the general K3 surface X polarized by the former lattice satisfies Aut(X ) ≅ (Z 2) 2 . In contrast, the general H ⊕ A ⊕7 1 -polarized K3 surface X has Aut(X ) ≅ Z 2. However, considerations regarding an H ⊕ A ⊕7 1 -polarization will turn out to be useful for constructions in higher Picard number. H(2) ⊕ A ⊕7 1 -polarized K3 surfaces. It is known that for at most eight points on P 2 , there always exists an irreducible sextic curve containing these points as nodes. Let us examine the situation where a sextic curve S ′ in P 2 has the eight ordinary double points P = {s 1 , . . . , s 8 } in general position in P 2 . In this situation, we use the construction described in detail in [5], to obtain a rational map to weighted projective space: The space of plane cubics through P has dimension 2. Let {U, V } be a basis for the pencil. The space of plane sextics with double points at P has dimension 4 and is generated by {U 2 , UV, V 2 , W } for some sextic W . Further, the space of plane nonics with 3-fold points at P has dimension 7 and is generated {U 3 , U 2 V, UV 2 , V 3 , UW, V W, R} for some nonic R. For a chosen basis of nonics with 3-fold points at P, one obtains the rational map ψ∶ P 2 ⇢ P(1, 1, 2, 3) , [x ∶ y ∶ z] ↦ U (x, y, z) ∶ V (x, y, z) ∶ W (x, y, z) ∶ R(x, y, z) . (3.1) The closure of the image of ψ will be denoted by Z = Im(ψ). By construction, it is a degree-one del Pezzo surface Z, i.e., the blow up of P 2 in P; see [5,Sec. 3.1]. We have the following: Lemma 3.1. The degree-one del Pezzo surface Z in P(1, 1, 2, 3) = P(U, V, W, R) is given by the equation (3.2) Z∶ R 2 = c 0 W 3 + c 2 (U, V )W 2 + c 4 (U, V )W + c 6 (U, V ) , where c 2 , c 4 , c 6 are any general polynomials of degree 2, 4, and 6, respectively, and c 0 ∈ C × . In particular, Z does not pass through the singular points [0 ∶ 0 ∶ 1 ∶ 0] or [0 ∶ 0 ∶ 0 ∶ 1]. Proof. The anti-canonical ring of a del Pezzo surface of degree one is generated by its elements of degree at most three. By computing the dimension of H 0 (O(−nK Z )) for n = 1, 2, 3 it follows that the anti-canonical ring is generated by H 0 (O(−K Z )) and one element each of H 0 (O(−nK Z )) for n = 2, 3 [17,Sec. 3.5]. Thus, the image of ψ is cut out by a single equation of degree six. Closely related to Z is a rational elliptic surfaceZ → P 1 = P(u, v) with section, given by (3.3)Z∶ Y 2 Z = c 0 X 3 + c 2 (u, v)X 2 Z + c 4 (u, v)XZ 2 + c 6 (u, v)Z 3 , whose general fiber in P 2 = P(X, Y, Z) isO ∶ [X ∶ Y ∶ Z] = [0 ∶ 1 ∶ 0], one recovers Z using (3.4) [u ∶ v], [X ∶ Y ∶ Z] ↦ [U ∶ V ∶ W ∶ R] = [uZ ∶ vZ ∶ XZ ∶ Y Z 2 ] . In turn, a rational elliptic surface can be related to a double cover of P(s, t) × P(u, v), given by the equation (3.5)Ỹ 2 = β 0 (s, t)u 2 + β 1 (s, t)uv + β 2 (s, t)v 2 = α 0 (u, v)s 4 + α 1 (u, v)s 3 t + ⋅ ⋅ ⋅ + α 4 (u, v)t 4 . Here, the double cover is written with respect to both rulings, where β m are homogeneous polynomials in the variables s, t of degree 4, and α n are homogeneous polynomials in u, v of degree 2. Equation (3.5) is elliptically fibered over P(u, v) where the branch locus is a class of bidegree (4, 2). By completing the square on the right hand side, one easily shows that at least eight pairs of sections σ ± i , 1 ≤ i ≤ 8 exist and are given by solutions ([s i ∶ t i ],Ỹ i,± ) satisfying (3.6) β 1 (s i , t i ) 2 − 4β 0 (s i , t i )β 2 (s i , t i ) = 0, 4β 0 (s i , t i )Ỹ 2 i,± = 2β 0 (s i , t i )u + β 1 (s i , t i )v 2 . A priori, Equation 3.5 is only elliptically fibered over P(u, v). However, given the existence of sections for the elliptic fibration, the right hand side can be put into Weierstrass form; see [19,Sec. 39 .2]. Then, Equation (3.3) and Equation (3.5) are isomorphic if the relation between polynomials in Equations (3.5) and (3.2) is as follows (3.7) c 2 = α 2 , c 0 c 4 = α 1 α 3 − 4α 0 α 4 , c 2 0 c 6 = α 0 α 2 3 + α 2 1 α 4 − 4α 0 α 2 α 4 . Choosing σ + 1 as section of the elliptic fibration, the Néron-Severi lattice NS(Z) is generated by the pull-back L ′ of a line and the 8 exceptional divisors E i = [σ − i ], i = 1, . . . , 8 that are obtained from the classes of the sections. The anti-canonical divisor of Z is given by (3.8) − K Z = 3L ′ − E 1 − ⋅ ⋅ ⋅ − E 8 . The strict transform of a fixed nodal sextic S ′ in Z is a smooth genus-2 curve B. Without loss of generality, we can assume that this sextic is given by W = 0. The genus-2 curve inside Z then is the hyperelliptic curve B∶ R 2 = c 6 (U, V ). Note that any other choice of genus-2 curve is obtained by a shift W ↦ f 0 W + f 2 (U, V ) where f 0 ∈ C × and f 2 is a homogeneous polynomial of degree 2. Under such shift, coefficients in Equation (3.2) will change according to c 0 ↦ c 0 f 3 0 , c 2 ↦ (c 2 + 3c 0 f 2 )f 2 0 , c 4 ↦ (c 4 + 2c 2 f 2 + 3c 0 f 2 2 )f 0 , c 6 ↦ c 6 + c 4 f 2 + c 2 f 2 2 + c 0 f 3 2 as the genus-2 curve B moves in the 4-dimensional family f 0 W + f 2 (U, V ) = 0. In the situation above, it is known that the minimal resolution of the double cover of Z branched on B is a K3 surface X . Explicitly, from the double cover ϕ∶ S ⊂ P(1, 1, 1, 3) → Z ⊂ P(1, 1, 2, 3), [u ∶ v ∶ w ∶ y] ↦ [U ∶ V ∶ W ∶ R] = [u ∶ v ∶ w 2 ∶ y] , one obtains a preimage, given by (3.9) S∶ y 2 = c 0 w 6 + c 2 (u, v)w 4 + c 4 (u, v)w 2 + c 6 (u, v) . We make the following: Remark 3.2. Using Equations (3.7) the right hand side of Equation (3.9) can be written as ternary sextic from the determinant of a symmetric 3 × 3-matrix, i.e., (3.10) S∶ y 2 = − 1 (2c 0 ) 2 4α 0 (u, v) α 1 (u, v) −2c 0 w 2 α 1 (u, v) c 0 w 2 + α 2 (u, v) α 3 (u, v) −2c 0 w 2 α 3 (u, v) 4α 4 (u, v) . Obviously, there is a double cover π S ∶ S → P 2 = P(u, v, w). We have the following: Proposition 3.3. π S is branched over a smooth irreducible sextic curve S ⊂ P(u, v, w) of genus 10 which admits 120 6-tangent conics. Remark 3.4. The degree-one del Pezzo surface Z contains 240 (-1)curves, and their pull-backs via ϕ on X are (-2)-curves. It was shown in [28] that these curves come in pairs A ± i with i = 1, . . . , 120 such that the image of A + i + A − i by π is a conic C (i) tangent to the branch curve. Proof. The double cover π S ∶ S → P(u, v, w) is branched over the curve (3.11) S∶ 0 = c 0 w 6 + c 2 (u, v)w 4 + c 4 (u, v) w 2 + c 6 (u, v) , which is generically a smooth plane sextic whence of genus 10. In particular, we must have c 0 ≠ 0. It follows from [30, Thm. 10.8] that on a rational elliptic surface the number of sections P with intersection number P ○ O = 0 is finite and at most 240. Moreover, every such P has a height pairing ⟨P, P ⟩ ≤ 2. Since the rational elliptic surfaceZ in Equation (3.3) is general, there are exactly 240 sections, and their form can be made explicit using [30,Thm. 10.10]. In our situation, the sections, disjoint from the zero section O, must come in pairs P (i) ± of the form X = p (i) 2 (U, V ), Y = ±q (i) 3 (U, V ), Z = 1 with i = 1, . . . , 120 where p (i) 2 , q (i) 3 are non-trivial homogeneous polynomials of degree 2 and 3, respectively. As P (i) ± ○ O = 0, every pair of sections pulls back to a pair of bi-sections on S so that (3.12) w 2 = p (i) 2 (u, v), y = ±q (i) 3 (u, v) solves Equation (3.9). Their image under π S is the conic in P(u, v, w) given by C (i) ∶ 0 = w 2 − p (i) 2 (u, v) . All intersection multiplicities between C (i) and S ′ must be even as y 2 = q (i) 3 (u, v) 2 is a perfect square for every parametrization of the conic C (i) . Thus, C (i) is a conic tangent to the branch curve in every 6-point. We have the following: Proposition 3.5. The K3 surface X obtained as the minimal resolution of the general double sextic in Equation (3.9) has NS(X ) ≅ H(2) ⊕ A ⊕7 1 . Conversely, every K3 surface X with NS(X ) ≅ H(2) ⊕ A ⊕7 1 is isomorphic to such a K3 surface. Proof. We consider the pencil of lines though a node on the sextic S. Pulling back the class of the line via p, this defines an elliptic fibration without section on X . One checks that pull back of the elliptic fiber and the eight classes E i form a lattice isometric to H(2)⊕A ⊕7 1 . This proves that NS(X ) ≅ H(2)⊕A ⊕7 1 . Conversely, Roulleau proved in [28] that every H(2) ⊕ A ⊕7 1 -polarized K3 surface X can be obtained as a double cover of P 2 branched on a plane sextic with eight nodes. In particular, the double sextics of the form (3.13) S∶ y 2 = c 0 w 6 + c 2 (u, v)w 4 + c 4 (u, v)w 2 + c 6 (u, v) form an 11-dimensional family. The (−1)-curves E i on Z pull-back via the double cover ϕ to (−2)-curves on X , denoted by A 1 , . . . , A 8 . The pull-back of L ′ is denoted by L. We set (3.14) D 2 = ϕ * (−K Z ) = 3L − A 1 − ⋅ ⋅ ⋅ − A 8 . We have the following: Proposition 3.6. D 2 ∈ NS(X ) is nef, of square-two, and invariant under the action of ı X . After a suitable choice of coordinates, for the linear system D 2 the corresponding rational map is the projection onto P(u, v, w) in Equation (3.13). Proof. One checks invariance by an explicit lattice computation. The rest of the statement was proven in [28, Sec. 9.5]. H ⊕ A ⊕7 1 -polarized K3 surfaces. We also examine double sextics having eight ordinary double points associated with even eights. In order to distinguish these double sextic from those mentioned in Proposition 3.5, we refer to the plane sextic curve and the double sextic surface as S ′ and σ ′ ∶ S ′ → P 2 , respectively. The minimal resolution of S ′ will still be labeled X . The surface S ′ contains eight ordinary double points p 1 , . . . , p 8 positioned over s 1 , . . . , s 8 . The (-2)-curves on X that resolve these nodes are labeled as e 1 , . . . , e 8 . Barth [2] established the following: The lemma shows that the K3 surfaces obtained as the minimal resolution of the double sextic S ′ in Proposition 3.7 is polarized by the lattice L = H ⊕ A 1 (−1) ⊕7 . We then consider the associated quartic K ⊂ P 3 = P(u, v, w, y), introduced in Equation (2.5). In the situation above, an explicit computation yields the following: Lemma 3.9. K in Equation (2.5) has a rational double-point at p 1 ∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 0 ∶ 1] of type A 1 . Moreover, it was proved in [2] that every algebraic K3 surface with NS(X ) ≅ H ⊕A ⊕7 1 has a birational model as double sextic associated with an even eight. Thus, we have proved the following: Proposition 3.10. The K3 surface X obtained as the minimal resolution of the double sextic S ′ in Equation (2.3) or, equivalently, the quartic K in Equation (2.5) for general polynomials C, Q has NS(X ) ≅ H ⊕ A ⊕7 1 . Conversely, every K3 surface X with NS(X ) ≅ H ⊕ A ⊕7 1 is isomorphic to such a K3 surface. K3 surfaces of Picard number 10 In Picard number 10, there are two possible lattice polarizations for K3 surfaces X with Aut(X ) ≅ (Z 2Z) 2 , namely H(2) ⊕ D ⊕2 4 ≅ H ⊕ N and H ⊕ A ⊕8 1 . 4.1. The van Geemen-Sarti family. In Section 2 we already introduced the van Geemen-Sarti family. Generically, the alternate fibration in Equation (2.4) on X has 8 fibers of type I 1 over the zeroes of a 2 4 − 4b 8 = 0 and 8 fibers of type I 2 over the zeroes of b 8 = 0 with Mordell-Weil groups MW(X , π X ) ≅ Z 2Z. It was shown in [32] that for general members of the family we have (4.1) NS(X ) ≅ H ⊕ N, T X ≅ H 2 ⊕ N . As previously indicated, the alternate fibration on X (and therefore the associated Geemen-Sarti involution ı X ) is unique. Let X be a general K3 surface with NS(X ) ≅ H ⊕ N, equipped with the alternate fibration in Equation (2.4). Since Aut(X ) ≅ (Z 2) 2 , the van Geemen-Sarti involution ı X and the (fiberwise) hyperelliptic involution  X generate Aut(X ). Using the alternate fibration we construct a double sextic S whose minimal resolution is isomorphic to X . However, the equation for such a double sextic is not canonical: Proposition 4.1. For a factorization b 8 (u, v) = c 3 (u, v) ⋅ d 5 (u, v) where c 3 and d 5 are homogeneous polynomials of degree 3 and 5, respectively, X is isomorphic to the minimal resolution of the double sextic given by (4.2) S∶ y 2 = w c 3 (u, v) w 2 + a 4 (u, v) w + d 5 (u, v) . In the general case, the branch curve is the union of the line w = 0 and an irreducible quintic, and the double sextic S has two singularities of type D 4 and A 1 . • • • • • • • • • A 1 A 9 A 8 A 7 A 6 A 5 A 4 A 3 A 2 • • • • • • • • • A 18 A 17 A 16 A 15 A 14 A 13 A 12 A 11 A 10(4.3) u, v, w, y = c 3 (U, V )U Z, c 3 (U, V )V Z, X, c 3 (U, V ) 2 Y Z 2 has the property that the right side transforms under rescaling (4.4) (U, V, X, Y, Z) ↦ (λU, λV, X, λ −2 Y, λ −4 Z), (U, V, X, Y, Z) ↦ (U, V, λX, λY, λZ) , for λ ∈ C × , with weights (0, 0, 0, 0) and (1, 1, 1, 3), respectively. Thus, it induces a rational map from X in Equation (2.4), considered as a double cover of F 4 , to S in Equation (4.2). For any Z 0 ∈ C × the transformation (4.5) [U ∶ V ], [X ∶ Y ∶ Z] = [u ∶ v], [c 3 (u, v)wZ 0 ∶ c 3 (u, v)yZ 0 ∶ Z 0 ] provides a rational inverse. Next, we prove that S in Equation (4.2) is well defined. For any other factor- ization b 8 (u, v) =c 3 (u, v) ⋅d 5 (u, v) the birational coordinate change [ỹ ∶w ∶ũ ∶ṽ] = [c 3 (u, v)c 3 (u, v) 2 y ∶c 3 (u, v)w ∶ c 3 (u, v)u ∶ c 3 (u, v)v] transforms the equation (4.6)ỹ 2 =w c 3 (ũ,ṽ)w 2 + a 4 (ũ,ṽ)w +d 5 (ũ,ṽ) into Equation (4.2). The rest of the statement is immediate. The lattice L = H ⊕ N and the configuration of its finite set of (−2)-curves was analyzed in detail in [28,Sec. 10.6]. We will use the same notation A 1 , . . . , A 18 to denote the (-2)-curves in the dual graph of X as shown in Figure 1. Note that for simplicity the graph does not show all rational curves on a general K3 surface with NS(X ) ≅ H ⊕ N. We have the following: Proposition 4.2. D 2 = 2A 1 + 2A 2 + A 3 + A 4 + A 10 ∈ NS(X ) is nef, base-point free, of square-two, and invariant under the action of ı X . After a suitable choice of coordinates, for the linear system D 2 the corresponding rational map is the projection onto P(u, v, w) in Equation (4.6). Proof. One checks invariance by an explicit lattice computation. The rest of the statement was shown in [28,Sec. 10.6]. A general K3 surface X with NS(X ) ≅ H ⊕ N can also be obtained as the minimal resolution of a double cover of F 0 = P(u, v) × P(X,Z). Given a factorization b 8 (u, v) = b 4 (u, v) ⋅b ′ 4 (u, v) into homogeneous polynomials of degree four, X is obtained as the minimal resolution of a double quadric surface, defined by the equation (4.7)X ∶Ỹ 2 =XZ b 4 (u, v)X 2 + a 4 (u, v)XZ +b ′ 4 (u, v)Z 2 . The surface is branched on a divisor class of bidegree (4,4). The class decomposes into the union of divisors of bidegree (0, 1)+(0, 1)+(4, 2). These divisors are the class of a curve of bidegree (4, 2) and two (−2)-curves. The former is of geometric genus 3 and given by the equationb 4 (u, v)X 2 +a 4 (u, v)XZ +b ′ 4 (u, v)Z 2 = 0. The latter two are the classes of the section of the alternate fibration, given byZ = 0, and the 2-torsion section, given byX = 0. The divisors form the invariant locus of the antisymplectic involution X ∶Ỹ ↦ −Ỹ , which is consistent with Nikulin's theorem [24,Thm. 4 H ⊕A ⊕8 1 -polarized K3 surfaces. The lattice L = H ⊕A ⊕8 1 and the configuration of its finite set of (−2)-curves was analyzed in detail in [28,Sec. 10.8]. We will use the same notation f 1 , f 2 , e 1 , . . . , e 8 to denote the canonical basis of L. It turns out that we can always obtain the general L-polarized K3 surface from the double sextic S ′ , introduced in Equation (2.3), by assuming that Q has a single node in P(u, v, w), which we move to n = [0 ∶ 0 ∶ 1]; we will prove this in Proposition 4.4. We can simplify a conic h 0 w 2 + k 1 (u, v)w + j 2 (u, v) by coordinate shifts which keep the position of n fixed. One obtains for the conic C = h 0 w 2 + vw + j 0 u 2 . Moreover, the conic contains n if and only if h 0 = 0. For h 0 j 0 = 0 the conic can be brought into the form 0 = w 2 − uv while keeping the position of the node n fixed. Thus, we are led to consider the double sextic S ′ in Equation (2.3) with (4.8) C = w 2 − uv , Q = c 2 (u, v) w 2 + e 3 (u, v) w + d 4 (u, v) , where c 2 , e 3 , d 4 are homogeneous polynomials of degree 2, 3, and 4, respectively. We have the following: Proof. We write d 4 = ∑ 4 n=0 δ n u 4−n v n . In affine coordinates on S ′ we use the base point [u ∶ v ∶ w] = [1 ∶ w 2 0 ∶ w 0 ] and the pencil t(w − w 0 ) + (v − w 0 w) = 0 for t ∈ C such that (4.9) δ 0 = −w 0 c 2 (1, w 2 0 ) w 0 + e 3 (1, w 2 0 ) + 4 i=1 δ i w 2i−1 0 . Moving the point v = t 2 in each fiber to infinity, we obtain a Weierstrass model for an elliptic fibration over a rational base curve with affine coordinate t. Among the 8 singular fibers of type I 2 , one is located over t = w 0 , and its associated exceptional divisor satisfies y 2 − c 2 (u, v) = 0. This shows that the Jacobian elliptic fibration has the reducible fibers 8I 2 + 8I 1 . The claims follows. We have the following: Proposition 4.4. Let X be a general K3 surface with NS(X ) ≅ H ⊕ A ⊕8 1 . Then X is isomorphic to the minimal resolution of the double sextic S ′ , given by (4.10) S ′ ∶ỹ 2 = w 2 − uv c 2 (u, v) w 2 + e 3 (u, v) w + d 4 (u, v) , such that the branch curve S ′ splits as the union of a smooth conic and a uninodal quartic curve. Proof. Let X be K3 surface with NS(X ) ≅ L with L = H ⊕ A ⊕8 1 . In terms of the canonical basis of L, we set 16. This establishes an elliptic fibration with a 2-section A 0 . The reducible fibers of the elliptic fibration are then given by A i + A i+8 for i = 1, . . . , 8. On the other hand, the lattice polarization is observed directly, when using the elliptic fibration with section and singular fibers 8I 2 + 8I 1 and trivial Mordell-Weil group. The fiber class is F ′ = f 1 , the class of the section is A 0 with F ′ ⋅ A 0 = 1, and the reducible fibers are (4.11) F = 4f 1 + 2f 2 − 8 i=1 e i , A 0 = −f 1 + f 2 , A i = f 1 − e i , A i+8 = −f 2 − e i for i = 1, . . . , 8, such that F 2 = 0, F ⋅ A 0 = 2, A 0 ⋅ A j = 1 and A 2 0 = A 2 j = −2 for j = 1, . . .A i + (F ′ − A i ) such that A 0 ⋅ A i = 1 and A 0 ⋅ (F ′ − A i ) = 0 for i = 1, . . . , 8. We start with a double sextic S ′ in Proposition 3.7 of Picard number 9. On the surface S ′ , we have an elliptic fibration with section. Call the fiber class F ′ and the class of the section A 0 . From the lattice theoretic considerations above, it follows that the Néron-Severi lattice extends to L if A 0 is also realized as 2-section of a second fibration with fiber class F . However, for A 0 ∶ C(u, v, w) = 0,ỹ 2 = 0 to be a 2-section of such an elliptic fibration, Q(u, v, w) must have at most degree 2 in w. In other words, Q has a node at n = [0 ∶ 0 ∶ 1]. A 2-section B 0 realizing a curve of genus-2 is given by w = 0,ỹ 2 = C(u, v, 0)Q(u, v, 0), such that only one component is met in each reducible fiber of type A 1 . Moreover, the 2-section A 0 of genus zero meets in each reducible fiber the component once which is not met by the 2-section B 0 . The Néron-Severi lattice of the associated K3 surface is realized as follows: one takes the overlattice L of determinant 2 8 for the latticeL that is generated by ⟨B 0 , F − B 0 , A 1 , . . . , A 8 ⟩ and a vector ⃗ w representing the class of A 0 . Here, F is the class of the smooth fiber and A i are the classes of the reducible fiber components not met by B 0 . We havẽ L = ⟨2⟩ ⊕ ⟨−2⟩ ⊕ A ⊕8 1 , ⃗ w = ⟨0, 0, 1 2 , . . . , 1 2 ⟩ , and a computation shows that L = ⟨2⟩ ⊕ ⟨−2⟩ ⊕ N ≅ H ⊕ A ⊕8 1 . Let X and S ′ be as in Proposition 4.4. We consider the associated quartic K in Equation (2.5) using the polynomials in (4.8), this is, the quartic given by (4.12) K∶ w 2 − uv y 2 = c 2 (u, v) w 2 + e 3 (u, v) w + d 4 (u, v) . One has the following: Lemma 4.5. K has two rational double-point singularities of type A 1 at (4.13) p 1 ∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 0 ∶ 1] , p 2 ∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 1 ∶ 0] . We also consider the double sextic S given by (4.14) S∶ y 2 = uvw 4 − a 4 (u, v)w 2 + b ′′ 6 (u, v) , where a 4 = uvc 2 − d 4 and b ′′ 6 = e 2 3 4 − c 2 d 4 . One easily checks the following: Lemma 4.6. S has a rational double-point singularity of type A 1 at (1, 1, 1, 3). p∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 1 ∶ 0] ∈ P In the situation above we have the following: Proof. To avoid confusion between different coordinate sets, we write the double sextic S as follows: (4.15) S∶ η 2 = uvξ 4 − a 4 (u, v)ξ 2 + b ′′ 6 (u, v) . Birational transformations between the surfaces S ′ , S, K are given by (4.16) ξ =ỹ w 2 − uv = y , η = y 2 w − c 2 w − e 3 2 , and w = − 2(uvc 2 + d 4 )η + 2uve 3 ξ 2 − (uvc 2 − d 4 )e 3 2e 3 η − 2(uvc 2 + d 4 )ξ 2 + 2c 2 (uvc 2 + d 4 ) − e 2 3 , y = 2(−(uvc 2 + d 4 ) 2 + uve 2 3 )ξ 2e 3 η − 2(uvc 2 + d 4 )ξ 2 + 2c 2 (uvc 2 + d 4 ) − e 2 3 , y = ξ ,X ) ≅ H ⊕ A ⊕8 1 . Conversely, every K3 surface X with NS(X ) ≅ H ⊕ A ⊕8 1 is isomorphic to such a K3 surface. We have the following: Proposition 4.9. D 2 = 3f 1 +3f 2 −∑ 8 i=1 e i ∈ NS(X ) is nef, base-point free, of square-two, and invariant under the action of ı X . After a suitable choice of coordinates, for the linear system D 2 the corresponding rational map is the projection onto P(u, v, w) in Equation (4.14). Proof. One checks invariance by an explicit lattice computation. The rest of the statement was shown in [28, Sec. 10.8.1] By a 1 and b 1 we denote the exceptional curves obtained in the minimal resolution of the singularity at p 1 and p 2 on K, respectively, in Lemma 4.5. Let a 1 and b 1 be the exceptional divisors for the singularity at p 1 and p 2 , respectively. We state the main result for this section: Proof. We use the same notation as in the proof of Proposition 4.4. There are 128 divisors of square-four on X which contract two (−2)-curves. Each such nef divisor of square-four gives rise to a quartic with two singularities of type A 1 . We want to identify the polarizing divisor H of K. From standard lattice theory one finds that one exceptional curve is always A 0 = −f 1 + f 2 which is the class of the section of the fibration F ′ . One can easily verify that projecting K with center p 1 onto P(u, v, w) yields the double sextic S ′ with y = (w 2 −uv)y. It follows that the exceptional curve obtained in the minimal resolution of the singularity at p 1 is a 1 = A 0 . Conversely, a 1 is contracted onto the node of the sextic branch curve of S, i.e., the point [η ∶ ξ ∶ u ∶ v] = [0 ∶ 1 ∶ 0 ∶ 0] which corresponds to the singular point p 1 ∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 0 ∶ 1] on the quartic K. On the other hand, projecting K from p 2 yields the double sextic S with (4.18) ξ = y ,η = y 2 − c 2 (u, v) w . This means that the polarizing divisor that we are looking for can be written as H = D 2 + b 1 (where D 2 was given in Proposition 4.9) such that (4.19) H 2 = 4, H ⋅ a 1 = 0, H ⋅ b 1 = 0, D 2 ⋅ b 1 = 2. Using the description in the proof of K3 surfaces of Picard number greater than 10 Here, we will construct birational models as double sextics and quartics for the K3 surfaces X with NS(X ) ≅ L, for L in Equation (1.1) with ρ L > 10. Their explicit normal forms will be derived from the alternate fibration. 5.1. Uniqueness of the alternate fibration. Let us explain the meaning of uniqueness for the alternate fibration in Equation (2.4). Given a Jacobian elliptic fibration on X , the classes of fiber and section span a rank-two primitive sub-lattice of NS(X ) isomorphic to H. The converse also holds: given a primitive lattice embedding H ↪ NS(X ) whose image contains a pseudo-ample class, it is known from [8,Thm. 2.3] that there exists a Jacobian elliptic fibration on the surface X , whose fiber and section classes span H. For a primitive lattice embedding j∶ H ↪ L we denote by K = j(H) ⊥ the orthogonal complement in L and by K root the sub-lattice spanned by the roots of K, i.e., the algebraic class of self-intersection −2 in K. We also introduce the factor group W = K K root . The pair (K root (−1), W ) is called the frame associated with the Jacobian elliptic fibration. A frame determines the root lattice K root attached to the reducible fibers and the Mordell-Weil group W of a Jacobian elliptic fibration uniquely. One can also ask for a more precise classification of Jacobian elliptic fibrations, up to automorphisms of the surface X . The difference between these classifications is explained in [4]. In the situation above, assume that we have a second primitive embedding j ′ ∶ H ↪ L, such that the orthogonal complement of the image j ′ (H), is isomorphic to the given lattice K. One would like to see whether j and j ′ correspond to Jacobian elliptic fibrations isomorphic under Aut(X ) or not. By standard arguments (see [21,Prop. 1.15.1]), in the situation above there will exist an isometry γ ∈ O(L) such that j ′ = γ ○ j. An induced element γ * ∈ O(D(K)) can then be obtained as the image of γ under the group homomorphism O(L) → O(D(L)) ≅ O(D(K)) where the isomorphism is due to the decomposition L = j(H)⊕ K and, as such, it depends on the lattice embedding j. Denote the group O(D(K)) by A and consider the following two subgroups of A: the first subgroup B ⩽ A is the image of the following group homomorphism: (5.1) O(K) ≅ ϕ ∈ O(L) ϕ ○ j(H) = j(H) ↪ O(L) → O D(L) ≅ O D(K) . The second subgroup C ⩽ A is the image of following group homomorphism: (5.2) O h (T X ) ↪ O(T X ) → O D(T X ) ≅ O D(L) ≅ O D(K) . Here T X denotes the transcendental lattice of X and O h (T X ) is given by the isometries of T X that preserve the Hodge decomposition. Note that one has D(NS(X )) ≃ D(T X ) with q L = −q T X , as NS(X ) = L and T X is the orthogonal complement of NS(X ) with respect to an unimodular lattice. It was proved in [14, Thm 2.8] that the map In Table 1 we present a comprehensive list of all 2-elementary lattices L such that the general L-polarized K3 surface X satisfies Aut(X ) = (Z 2Z) 2 . The lattice L = H ⊕A ⊕8 1 cannot be framed with W = Z 2Z, and the general L-polarized K3 surface lacks an alternative fibration. Nonetheless, there is a Jacobian elliptic fibration with trivial Mordell-Weil group, but the frame's multiplicity is not one; the multiplicity was computed in [12]. For ρ L = 9, the lattice cannot be decomposed as H ⊕ K, and the general L-polarized K3 surface does not admit a Jacobian elliptic fibration. (5.3) H j ↪ L ↝ C γ * B , As we shall see, it is precisely the uniqueness of the alternate fibration (2.4) that will allow us to construct a canonical birational model for every general L-polarized K3 surface with ρ L > 10. 5.2. L-polarized K3 surfaces. Let X be a K3 surface with NS(X ) ≅ L, for L in Equation (1.1) and ρ L = 10 + n, 1 ≤ n ≤ 8. It follows that X admits an alternate fibration, given by Equation (2.4). We then have the following: ρ L (ℓ L , δ L ) L K root W 9 (9, 1) H(2) ⊕ A ⊕7 1 - - 10 (8, 1) H ⊕ A ⊕8 1 8A 1 {I} 10 (6, 0) H(2) ⊕ D ⊕2 4 ≅ H ⊕ N 8A 1 Z 2Z 11 (7, 1) H ⊕ D 4 ⊕ A ⊕5 1 9A 1 Z 2Z 12 (6, 1) H ⊕ D 6 ⊕ A ⊕4 1 D 4 + 6A 1 Z 2Z ≅ H ⊕ D ⊕2 4 ⊕ A ⊕2 1 13 (5, 1) H ⊕ E 7 ⊕ A ⊕4 1 D 6 + 5A 1 Z 2Z ≅ H ⊕ D 8 ⊕ A ⊕3 1 ≅ H ⊕ D 6 ⊕ D 4 ⊕ A 1 14 (4, 0) H ⊕ D 8 ⊕ D 4 E 7 + 5A 1 Z 2Z 14 (4, 1) H ⊕ E 8 ⊕ A ⊕4 1 D 8 + 4A 1 Z 2Z ≅ H ⊕ D 10 ⊕ A ⊕2 1 ≅ H ⊕ E 7 ⊕ D 4 ⊕ A 1 ≅ H ⊕ D ⊕2 6 15 (3, 1) H ⊕ D 12 ⊕ A 1 D 10 + 3A 1 Z 2Z ≅ H ⊕ E 8 ⊕ D 4 ⊕ A 1 ≅ H ⊕ E 7 ⊕ D 6 16 (2, 1) H ⊕ D 14 D 12 + 2A 1 Z 2Z ≅ H ⊕ E 8 ⊕ D 6 ≅ H ⊕ E 7 ⊕ E 7 17 (1, 1) H ⊕ E 8 ⊕ E 7 D 14 + A 1 Z 2Z 18 (0, 0) H ⊕ E 8 ⊕ E 8 D 16 Z 2Z(5.4) b 8 (u, v) = 1 4 a 4 (u, v) 2 − v 2 b ′′ 6 (u, v) , where b ′′ 6 is a homogeneous polynomial of degree 6. Remark 5.3. In the proof of Lemma 5.2 we classify the unique reducible fibers used to obtain Equation (5.4). Proof. From Table 1 one sees that the reducible fibers of the alternate fibration are D 2n + (8 − n)A 1 for 2 ≤ n ≤ 8 or E 7 + 5A 1 (as a secondary case for n = 4) and 9A 1 for n = 1. For Picard number 11, the alternate fibration has the following property: for 8 reducible fibers of type A 1 , the section and the 2-torsion section intersect different components of each reducible fiber, and for an additional reducible fiber of type A 1 they intersect the same component. We move the base point of this last fiber to [u ∶ v] = [1 ∶ 0]. For higher Picard number, the alternate fibration has a fiber of type D 2n (for n = 2, . . . , 8) or a fiber of type E 7 (as a second case for n = 4) which can be moved to v = 0. It follows easily that, in all cases, we have Equation (5.4) where b ′′ 6 is a polynomial of degree 6, as claimed. Using the unique alternate fibration and polynomials a 4 , b 8 , b ′′ 6 in Lemma 5.2, we can construct the K3 surface X as the minimal resolution of a canonical double sextic: Proposition 5.4. Let X be a general K3 surface with NS(X ) ≅ L, for L in Equation (1.1) with ρ L = 10 + n, 1 ≤ n ≤ 8. Then X is isomorphic to the minimal resolution of the (canonical) double sextic (5.5) S∶ y 2 = v 2 w 4 − a 4 (u, v)w 2 + b ′′ 6 (u, v) , where a 4 , b ′′ 6 are the polynomials in Lemma 5.2. In particular, the double sextic has the following properties: (1) For L = H ⊕ D 8 ⊕ D 4 the sextic curve S is irreducible and S has a singularity of type Proof. For X the Nikulin construction gives the K3 surface Y with the Weierstrass model A 2n+1 at p ∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 1 ∶ 0].(2)(5.6) Y∶ Y 2 Z = X X 2 − a 4 (u, v) XZ + v 2 b ′′ 6 (u, v) Z 2 . Then, S is obtained as a double cover of Y using the affine chart Z = 1 and X = v 2 w 2 , Y = v 2 wy. It follows that S is elliptically fibered over P(u, v) and its relative Jacobian fibration is easily checked to be isomorphic to X . Moreover, the elliptic fibration admits sections. Hence, it is birational to X . The explicit birational transformations between X and S are (5.7) w = Y vX , y = Y 2 vX 2 − 2X + a 4 (u, v) Z vZ , and (5.8) X = − vy + a 4 (u, v) − v 2 w 2 2 Z , Y = − v 2 wy + a 4 (u, v) vw − v 3 w 3 2 Z , respectively. For L = H ⊕ D 8 ⊕ D 4 , one checks that the double sextic is irreducible and has exactly one singularity at [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 1 ∶ 0]. This proves (1). For L = H ⊕ D 8 ⊕ D 4 one can write a 4 (u, v) = v 2ã 2 (u, v), b ′′ 6 (u, v) = vb ′′ 5 (u, v) , and the double sextic becomes (5.9) S∶ y 2 = v vw 4 − vã 2 (u, v) w 2 +b ′′ 5 (u, v) . Hence, the branch curve is the union of the line v = 0 and an irreducible quintic. One easily checks that the double sextic S has a singularity of type A 9 at [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 1 ∶ 0] . This proved (2). For (3) the birational map in Equation (5.7) and (5.8) transforms the hyperelliptic involution  X to the involution  S ∶ [u ∶ v ∶ w ∶ y] ↦ [u ∶ v ∶ −w ∶ y] , and the van Geemen-Sarti involution ı X to the involution ı S ∶ [u ∶ v ∶ w ∶ y] ↦ [u ∶ v ∶ −w ∶ −y]. Lastly, the involution k X = ı X ○  X is mapped to k S = ı S ○  S ∶ [u ∶ v ∶ w ∶ y] ↦ [u ∶ v ∶ w ∶ −y]. Next, we consider double sextics of the form given by Equation (2.3). For Picard number greater than 10, it is sufficient to consider a branch locus, that is the union of a uninodal quartic curve and a reducible conic splitting into two lines ℓ 1 , ℓ 2 . After a suitable shift in the coordinates v, w, this is precisely S ′ in Equation (2.3) with (5.10) C = w v + h 0 w , Q = c 2 (u, v) w 2 + e 3 (u, v) w + d 4 (u, v) . Here, h 0 ∈ C × and c 2 , e 3 , d 4 are homogeneous polynomials of degree 2, 3, and 4, respectively. Thus, the branch locus S ′ has three irreducible components: the quartic curve Q = 0 with node n = [0 ∶ 0 ∶ 1], and lines ℓ 1 = V(v + h 0 w), ℓ 2 = V(w) which are not coincident with n (for h 0 ≠ 0) and satisfy ℓ 1 ∩ ℓ 2 ∩ Q = ∅. We have the following: (5.11) S ′ ∶ỹ 2 = w v + h 0 w c 2 (u, v) w 2 + e 3 (u, v) w + d 4 (u, v) is birational to S. Remark 5.6. In [6] it was proved that the pencil of lines through n induces the alternate fibration (2.4) with (5.12) a 4 = ve 3 (u, v)−2h 0 d 4 (u, v) , b 8 (u, v) = 1 4 a 4 (u, v) 2 −v 2 e 3 (u, v) 2 4 − c 2 (u, v)d 4 (u, v) . Proof. Let us first explain how one constructs the polynomials c 2 , e 3 , d 4 for the general H ⊕ D 4 (−1) ⊕ A 1 (−1) ⊕5 -polarized K3 surface X . As explained before, we have a 4 (u, v) 2 4 − b 8 (u, v) = v 2 b ′′ 6 (u, v) where b ′′ 6 is a homogeneous polynomial of degree 6. We choose a factorization of b 8 (u, v) = a 4 (u, v) 2 4 − v 2 b ′′ 6 (u, v) = d 4 (u, v)d ′ 4 (u, v) into two homogeneous polynomials of degree 4 such that d(1, 0) = 0. We can then find h 0 ∈ C × so that (5.13) e 3 (u, v) = a 4 (u, v) + 2h 0 d 4 (u, v) v , c 2 (u, v) = d ′ 4 (u, v) + h 2 0 d 4 (u, v) + h 0 a 4 (u, v) v 2 are polynomials of degree 3 and 2, respectively. In fact, if we write a 4 = ∑ 4 n=0 α n u 4−n v n , d 4 = ∑ 4 n=0 δ n u 4−n v n (with δ 0 = 0), and d ′ 4 = ∑ 4 n=0 δ ′ n u 4−n v n , we have (5.14) δ ′ 0 = α 0 4δ 0 , δ ′ 1 = α 0 (2α 1 δ 0 − α 0 δ 1 ) 4δ 2 0 , h 0 = − α 0 2δ 0 . A surface S ′ is then obtained by using h 0 , c 2 , e 3 , d 4 in Equation (5.10). Moreover, the case h 0 = 0 is equivalent to α 0 = 0 and d ′ 4 = v 2 c 2 , so that we have a 4 = ve 3 and b 8 = v 2 c 2 d 4 . The explicit birational transformations between X and S ′ are (5.15) w = vX (c 2 v 2 − h 0 e 3 v + h 2 0 d 4 )Z − h 0 X ,ỹ = v(c 2 v 2 − h 0 e 3 v + h 2 0 d 4 )Y Z ((c 2 v 2 − h 0 e 3 v + h 2 0 d 4 )Z − h 0 X) 2 , and (5.16) X = (c 2 v 2 − h 0 e 3 v + h 2 0 d 4 )w v + h 0 w Z , Y = (c 2 v 2 − h 0 e 3 v + h 2 0 d 4 )vỹ (v + h 0 w) 2 Z , respectively. The lattice polarization extends to the lattice H ⊕ D 6 (−1) ⊕ A 1 (−1) ⊕4 if and only if a 4 (u, v) = ve 3 (u, v) and b 8 (u, v) = v 2 c 2 (u, v)d 4 (u, v). For any other decomposition b 8 (u, v) = v 2c 2 (u, v)d 4 (u, v) the map [ũ ∶ṽ ∶w ∶ỹ] = [c 2 (u, v)u ∶c 2 (u, v)v ∶ c 2 (u, v)w ∶ c 2 (u, v)c 2 (u, v) 2ỹ ]rank c 2 d 4 b 14 O(v 0 ) O(v 2 ) O(v 4 ) 15 O(v 0 ) O(v 3 ) O(v 5 ) 16 O(v 1 ) O(v 3 ) O(v 6 ) 17 O(v 1 ) O(v 4 ) O(v 7 ) 18 O(v 2 ) O(v 4 ) O(v 8 ) By using these polynomials c 2 , e 3 , d 4 and h 0 = 0 we obtain a birational double sextic S ′ as claimed. For S ′ in Proposition 5.5 we consider the associated quartic hypersurface K in Equation (2.5) using the same polynomials in (5.10). We have the following: Corollary 5.7. Let X be a general K3 surface with NS(X ) ≅ L, for L in Equation (1.1) with ρ L = 10 + n, 1 ≤ n ≤ 8. There are h 0 ∈ C × and polynomials c 2 , e 3 , d 4 such that X is isomorphic to the minimal resolution of the quartic K given by (5.18) K∶ w v + h 0 w y 2 = c 2 (u, v) w 2 + e 3 (u, v) w + d 4 (u, v) . Moreover, K has two rational double-point singularities at (5.19) p 1 ∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 0 ∶ 1] , p 2 ∶ [u ∶ v ∶ w ∶ y] = [0 ∶ 0 ∶ 1 ∶ 0] , of the type indicated in Table 2. Remark 5.8. In Table 2 we show the types of singularities determined in Corollary 5.7. We also include the results of Lemmas 4.6,4.5. Proof. Because of Proposition 5.4, there is a double sextic S whose minimal resolution is isomorphic to X . As explained above we constructed from S a birational double sextic S ′ . It follows that the minimal resolution of the associated quartic K is isomorphic to X . In particular, K only has rational double points. An explicit computation shows that these are the ones listed in Table 2. For every lattice L in Equation (1.1) with ρ L = 10 + n, 1 ≤ n ≤ 8, the dual graph of smooth rational curves on a general K3 surface X with NS(X ) = L is shown in Figures 2-10. They were constructed by the third author in [28]. Note that for simplicity for rank ρ L < 14 the graphs do not show all rational curves on X . The complete graphs can be found in [6]. However, each graph shows a complete set of generators of NS(X ), denoted by {A n }. For each lattice L a divisor D 2 ∈ NS(X ) ρ L (ℓ L , δ L ) singularity on S singularities on K at p at p 2 at p 1 9 (9, 1) none S and K not isomorphic 9 (7, 1) n/a - Table 2. Rational double points on S and K Table 3. Nef divisors for X for NS(X ) = L is defined in Table 3 in terms of these generators; similarly, divisors {a 1 , . . . , a M } and {b 1 , . . . , b N } are defined in Table 4. (The hat symbol means that the particular element is not included in a sequence.) In Tables 3 and 4 we also included the results of Propositions 4.9 and 4.10. Table 4. Exceptional divisors for the singularity at p 2 and p 1 on K A 1 10 (8, 1) A 1 A 1 A 1 11 (7, 1) A 3 A 1 A 3 12 (6, 1) A 5 A 3 A 3 13 (5, 1) A 7 A 3 A 5 14 (4, 0) A 9 A 3 A 7 14 (4, 1) A 9 A 3 A 7 15 (3, 1) A 11 A 3 A 9 16 (2, 1) A 13 A 5 A 9 17 (1, 1) A 15 A 5 A 11 18 (0, 0) A 17 E 6 A 11ρ L (ℓ L , δ L ) D 2 10 (8, 1) 3f 1 + 3f 2 − ∑ 8 i=1 e i 11 (7, 1) A 1 + 2A 2 + A 3 + A 4 12 (6, 1) A 6 + A 7 + 3A 8 + 2A 9 + 2A 10 + A 11 + A 12 13 (5, 1) A 4 + A 6 + A 7 + A 8 + A 10 + A 11 + A 12 + A 16 + A 18 14 (4, 1) A 1 + 2A 2 + 3A 3 + 4A 4 + 2A 5 + 5A 6 + 4A 7 + 3A 8 + 2A 9 + A 10 14 (4, 0) A 1 + 2A 2 + 3A 3 + A 4 + 3A 5 + ⋅ ⋅ ⋅ + 3A 8 + A 9 + 2A 10 + A 11 15 (3, 1) A 1 + 2A 2 + 3A 3 + A 4 + 3A 5 + ⋅ ⋅ ⋅ + 3A 10 + A 11 + 2A 12 + A 13 16 (2, 1) A 1 + 2A 2 + 3A 3 + A 4 + 3A 5 + ⋅ ⋅ ⋅ + 3A 12 + A 13 + 2A 14 + A 15 17 (1, 1) A 2 + ⋅ ⋅ ⋅ + A 17 + A 19 18 (0, 0) A 1 + 2A 2 + 3A 3 + A 4 + 3A 5 + ⋅ ⋅ ⋅ + 3A 15 + A 16 + 2A 17 + A 18 + 3A 19ρ L (ℓ L , δ L ) (N, M) {b 1 , . . . , b N } {a 1 , . . . , a M } 10 (8, 1) (1, 1) {e 8 } {−f 1 + f 2 } 11 (7, 1) (1, 1) {2A 1 − A 4 + A 5 + ⋅ ⋅ ⋅ + A 8 } {A 1 } 12 (6, 1) (3, 3){A We have the following: Theorem 5.9. (1) In Table 3 each divisor D 2 ∈ NS(X ) is nef, base-point free, of square 2, and invariant under ı X . (2) In Table 4 Remark 5.10. For ρ L = 18 the singularity at p 2 is of type E 6 , and not of type A n as in all other cases. In this case, the precise relation between divisors in Theorem 5.9 (3) is H = D 2 + b 1 + 2b 2 + 3b 3 + 2b 4 + 2b 5 + b 6 for b i = A 12+i with i = 1, . . . , 6. Remark 5.11. The divisor D ′ 2 is not invariant under ı X . Proof. To prove claim (1) we note that the divisors D 2 in Table 3 were constructed by Roulleau in [28] who also proved that they are nef, base-point free, of square 2. Clingher and Malmendier [6] constructed the unique embedding of the alternate fibration into the dual graphs and the action of the van Geemen-Sarti involution. One then easily checks that the divisors D 2 in Table 3 are invariant under this action. To prove claims (2) and (3), we note that the divisors a 1 , . . . , a M and b 1 , . . . , b N and K were identified in [6,[9][10][11]. Comparing these divisors with the labels in Figures 2-10 yields the statement. Claim (4) follows from the following geometric argument: Let K be the quartic surface in P 3 , with two rational double point singularities at p 1 and p 2 . Let X be the smooth K3 surface obtained after one performs desingularization at p 1 and p 2 , with ψ∶ X → P 3 being the associated desingularization map. Consider then the projection from p 2 . This is a rational map π∶ P 3 ⇢ P 2 . The target projective space may be seen as the space of lines in P 3 passing through p 2 . The restriction of π on K − {p 2 } is a morphism and the generic fiber of this restriction is given by the two points obtained as the residual intersection of the associated line through p 2 with K − {p 2 }. The composition ψ ○ (π K−p 2 ) extends then to a morphism φ∶ X → P 2 which recovers the canonical double sextic structure. (5.20) X ψ / / φ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ P 3 π ✤ ✤ ✤ ✤ P 2 The branch locus of φ corresponds to the compactification of the set of lines through p 2 that either pass through p 1 or are tangent to K − {p 2 }. Now, in the context of the diagram (5.20), the linear system associated to the map ψ is H , and consists of the pullbacks ψ * (H) where H is a plane in P 3 . A generic member of H is a plane degree-four curve obtained by intersection a generic plane of P 2 with K. Special members of H can be obtained by considering ψ * (H), where H is a plane in P 3 passing through p 2 . Such a plane corresponds, in the context of the diagram, to a line in P 2 , and one has ψ * (H) = φ * (ℓ). Consider a generic line ℓ in P 2 . Then (π K−p 2 ) * (ℓ) corresponds to a plane curve Q lying on K and ψ * (Q) is the linear system associated to the map φ∶ X → P 2 . Moreover, via the desingularization map ψ, one obtains: φ * (ℓ) = ψ * (Q) + b 1 + ⋅ ⋅ ⋅ + b N . Therefore, we get that ψ * (Q) + b 1 + ⋅ ⋅ ⋅ + b N is a special member of H and hence: H ∼ ψ * (Q) + b 1 + ⋅ ⋅ ⋅ + b N . Therefore, we obtain D 2 = H − (b 1 + ⋅ ⋅ ⋅ + b N ) ∼ ψ * (Q) . It follows that D 2 is the linear system associated with the morphism φ. The first part of (5) follows by standard lattice theoretic computations. The second part is analogous to (4). Conclusions We have proved that every K3 surface with automorphism group (Z 2Z) 2 admits a birational model as a double sextic surface. This was the central part of Theorem 2.1, stated in the introduction. In Table 5 we indicate where each part of Theorem 2.1 has been proved in this article, depending on the rank of the corresponding lattice. Furthermore, the constructed birational model S is canonical for ρ L > 10. This was stated as Proposition 2.3 in the introduction and proved as part of Proposition 5.4. For Picard number greater than 9, the considered K3 surfaces also possess a second birational model as quartic projective hypersurface K, generalizing the Inose quartic. The polarizing divisor of this quartic was computed explicitly. This was the central part of Theorem 2.4, stated in the introduction. In Table 6 we indicate where every statement contained in Theorem 2.4 has been proved in this article, depending on the rank of the corresponding lattice. The birational model as quartic hypersurface K is Figure 7. Dual graph for NS(X ) = H ⊕ E 8 (−1) ⊕ D 4 (−1) ⊕ A 1 (−1) closely related to that of a second double sextic, denoted by S ′ , whose branch locus is the union of a conic and a plane quartic curve. Explicit equations for the constructed surfaces were also provided for all lattices, with equations as indicated in Table 7. Cor. 5.7 Thm. 5.9 (3) k = 0, . . . , 7, with δ k = 1 for k = 1, . . . , 7 and δ k = 0 for k = 0, 4 ρ L (ℓ L , δ L ) S S ′ K 9 • • • • • • • • • • • • • • • • • • • A 1 A 2 A 3 A 5 A 6 A 7 A 8 A 9 A 10 A 11 A 12 A 14 A 15 A 19 A 17 A 18 A 16 A 4 A 13 (9, 1) Eq. (3.9) n/a n/a 9 (7, 1) n/a Eq. (2.3) (general C, Q) Eq. (2.5) (general C, Q) 10 (6, 0) Eq. (4.2) n/a n/a 10 (8, 1) Eq. (4.14) Eq. (4.10) Eq. (4.12) 18 − k (k, δ k ) Eq. (5.5) Eq. (5.11) Eq. (5.18) k = 0, . . . , 7, with δ k = 1 for k = 1, . . . , 7 and δ k = 0 for k = 0, 4 Table 7. Normal forms for the surfaces S, S ′ , K Proposition 2 . 3 . 23Let X be a K3 surface with NS(X ) ≅ L, for L in Equation (1.1) with ρ L > 10. X is isomorphic to the minimal resolution of the double sextic S in Equation (2.2) where c 0 = 0 and c 2 , c 4 , c 6 are uniquely determined by the alternate fibration, and ı S ,  S induce the van Geemen-Sarti and hyperelliptic involution. Based on Proposition 2.3 we call the equation for the double sextic S for ρ L > 10 canonical. a smooth curve of genus one. It is obtained by blowing up Z one time, namely at the ninth base point of {U, V }. Conversely, if one blows down the section ofZ, given by Proposition 3 . 7 ( 37Barth). The divisor e 1 + ⋅ ⋅ ⋅ + e 8 ∈ NS(X ) is divisible by 2 if and only if the sextic curve S ′ splits as the union of a smooth conic and a smooth quartic curve meeting transversally in s 1 , . . . , s 8 .Thus, the double sextic of Proposition 3.7 is precisely the double cover introduced in Equation (2.3) for general polynomials C, Q. An explicit computation yields the following:Lemma 3.8. The pencil of lines through a common point of the conic and quartic curve induces a Jacobian elliptic fibration with singular fibers 7I 2 + 10I 1 and trivial Mordell-Weil group. Figure 1 . 1Dual graph for NS(X ) = H ⊕ N Proof. The transformation Lemma 4. 3 . 3The pencil of lines through a common point of the conic and quartic curve in Equation (4.8) induces a Jacobian elliptic fibration with singular fibers 8I 2 + 8I 1 and trivial Mordell-Weil group. Lemma 4. 7 . 7The minimal resolutions of S ′ , S, K are isomorphic. Corollary 4 . 8 . 48The K3 surface X obtained as the minimal resolution of the double sextic S in Equation(4.14) has NS( Proposition 4 . 10 . 410The polarizing divisor of K is H = D 2 + b 1 where b 1 is the class of an e i for some i = 1, .., 8. In particular, one can set b 1 = e 8 . Moreover, one has a 1 = −f 1 + f 2 . Lemma 4.3, one finds that the exceptional divisor b 1 is the non-neutral component of a reducible fiber of type A 1 in the elliptic fibration with fiber F ′ . The fiber class satisfies F ′ − A i = e i for i = 1, . . . , 8. The exceptional divisor associated to the node of the plane quartic curve then is the eighth (nonneutral component of the) reducible fiber, and we have b 1 = e 8 . a one-to-one correspondence between Jacobian elliptic fibrations on X with j(H) ⊥ ≃ K, up to the action of the automorphism group Aut(X ), and elements of a double coset. The number of elements of C A B is referred by Festi and Veniani as the multiplicity of the frame.The frame of the alternate fibration (2.4) is (K root , W ) = (A ⊕8 1 , Z 2Z), and van Geemen and Sarti proved that its multiplicity equals one. Two of the authors proved the following extension:Proposition 5.1. For every lattice L in Equation (1.1) with ρ L > 10 or L = H ⊕ N there is a unique frame with W = Z 2Z and its multiplicity equals one. For L = H ⊕D 8 ⊕D 4 the sextic curve S is the union of a line and an irreducible quintic and S has a singularity of type A 9 at p. (3) ı S ,  S induce the van Geemen-Sarti involution ı X and hyperelliptic involution  X of the alternate fibration, respectively. Proposition 5. 5 . 5For every double sextic S in Proposition 5.4 there are h 0 ∈ C × and polynomials c 2 , e 3 , d 4 such that the double sextic transforms the equation(5.17)ỹ 2 =wṽ c 2 (ũ,ṽ)w 2 + e 3 (ũ,ṽ)w +d 4 (ũ,ṽ) .into S ′ in Equation (2.3) using the polynomials in Equation (5.10) with h 0 = 0. Moreover, Equations (5.13) and (5.14) and b = d 4 ⋅ d ′ 4 determine h 0 ∈ C × and polynomials c 2 , e 3 , d 4 for ρ L = 11, 12, 13 and L = H ⊕ D 8 ⊕ D 4 . For L = H ⊕ E 8 ⊕ A ⊕4 1 and ρ L = 15, 16, 17, 18 we write a 4 = ve 3 and b 8 = v 2 c 2 d 4 . In these cases the Picard number is ρ L = 10 + n, n = 4, . . . , 8, and the vanishing degree of b 8 at v = 0 is n. One then has to make some additional choices: We distribute the repeated root of b 8 at v = 0 between the polynomials c 2 and d 4 as follows: the divisors a 1 , . . . , a M and b 1 , . . . , b N ∈ NS(X ) are exceptional divisors for the singularity at p 1 and p 2 on K in Equation(5.18).(3) The polarization divisor of K in Equation (5.18) is H = D 2 + b 1 + ⋅ ⋅ ⋅ + b N .(4) After a suitable choice of coordinates, for the linear system D 2 the corresponding rational map is the projection onto P(u, v, w) in Equation(5.5).(5) D ′ 2 = H − a 1 − ⋅ ⋅ ⋅ − a M ∈ NS(X )is nef and of square 2. After a suitable choice of coordinates, for the linear system D ′ 2 the corresponding rational map is the projection onto P(u, v, w) in Equation(5.11). Figure 2 . 2Dual graph for NS(X ) = H ⊕ D 4 (−1) ⊕ A 1 Figure 3 . 3Dual graph for NS(X ) = H ⊕ D 6 (−1) ⊕ A 1 Figure 4 . 4Dual graph for NS(X ) = H ⊕ E 7 (−1) ⊕ A 1 (−1) ⊕4 Figure 5 . 5Dual graph for NS(X ) = H ⊕ D 8 (−1) ⊕ D 4 (−1) ⊕4 Figure 6 . 6Dual graph for NS(X ) = H ⊕ E 8 (−1) ⊕ A 1 (−1) ⊕4 Figure 8 . 8Dual graph for NS(X ) = H ⊕ E 8 (−1) ⊕ D 6 (−1) Figure 9 . 9Dual graph for NS(X ) = H ⊕ E 8 (−1) ⊕ E 7 Figure 10 .k 10Dual graph for NS(X ) = H ⊕ E 8 (−1) ⊕ E 8 (−1) ρ L (ℓ L , δ L ) Thm. = 0, .. . , 7, with δ k = 1 for k = 1, . . . , 7 and δ k = 0 for k = 0, 4 Table 1 . 1All 2-elementary lattices with Aut(X ) = (Z 2Z) 2 and a canonical frame Lemma 5.2. It is always possible to move a unique reducible fiber in the frame determined by Proposition 5.1 to v = 0, and in doing so, the polynomial b 8 in the alternate fibration (2.4) takes the form 11 , A 18 , A 19 } {A 8 , A 9 , A 12 } 13 (5, 1) (3, 5) {A 12 , A 13 , A 14 } {A 6 , . . . , A 9 , . . . , A 11 } 14 (4, 0) (3, 7) {A 10 , A 11 , A 12 } {A 1 , . . . , A 5 , . . . , A 8 } 14 (4, 1) (3, 7) {A 11 , A 12 , A 15 } {A 1 , . . . , A 4 , . . . , A 8 } 15 (3, 1) (3, 9) {A 13 , A 14 , A 16 } {A 1 , . . . , A 4 , . . . , A 10 } 16 (2, 1) (5, 9) {A 12 , . . . , A 16 } {A 1 , . . . , A 4 , . . . , A 10 } 17 (1, 1) (5, 11) {A 1 , . . . , A 5 } {A 7 , . . . , A 16 , A 19 } 18 (0, 0) (6, 11) {A 13 , . . . , A 18 } {A 1 , . . . , A 4 , . . . , A 11 , A 19 } A 9 A 10 A 13 A 18 A 1 A 12 A 17 A 14 A 19 A 11 A 16 A 15 A 20• • • • • • • • • • • • • • • • • • • • A 5 A 2 A 3 A 4 A 6 A 7 A 8 A 10 A 9 A 8 A 7 A 6 A 5 A 3• • • • • • • • • • • • • • • • • • • • • A 11 A 4 A 20 A 17 A 18 A 19 A 21 A 14 A 15 A 16 A 1 A 13 A 12 A 2 6 A 7 A 8 A 9 A 10 A 11 A 12 A 13 A 14• • • • • • • • • • • • • • • • • • • A 2 A 6 A 1 A 3 A 4 A 5 A 15 A 18 A 19 A 16 A 17 • • •A 1 A 2 A 3 A 5 A 6 A 7 A 8 A 19 A 9 A 10 A 11 A 12 A 13 A 14 A 15 A 17 A 18A 4 A 16 Table 5 . 5Proof of Theorem 2.1 − k (k, δ k )ρ L (ℓ L , δ L ) Thm. 2.4 (1) Thm. 2.4 (2) Thm. 2.4 (3) 9 (7, 1) Prop. 3.10 Lemma 3.9 n/a 10 (8, 1) Prop. 4.4, Lemma 4.7 Lemma 4.5 Prop. 4.10 18 Table 6 . 6Proof of Theorem 2.4 For ρ L = 9, 10 there is only the birational model S, and for L = H ⊕ N the sextic has a different form; see Equation(4.2). 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[ "The compositional construction of Markov processes", "The compositional construction of Markov processes" ]	[ "L De Francesco ", "Albasini N Sabadini ", "R F C Walters " ]	[]	[]	We describe an algebra for composing automata in which the actions have probabilities. We illustrate by showing how to calculate the probability of reaching deadlock in k steps in a model of the classical Dining Philosopher problem, and show, using the Perron-Frobenius Theorem, that this probability tends to 1 as k tends to infinity.	10.1007/s10485-010-9233-0	[ "https://arxiv.org/pdf/0901.2434v1.pdf" ]	1,568,506	0901.2434	e63c2ca5d31709818facceaac95e717e638081a7	The compositional construction of Markov processes 16 Jan 2009 January 16, 2009 L De Francesco Albasini N Sabadini R F C Walters The compositional construction of Markov processes 16 Jan 2009 January 16, 2009 We describe an algebra for composing automata in which the actions have probabilities. We illustrate by showing how to calculate the probability of reaching deadlock in k steps in a model of the classical Dining Philosopher problem, and show, using the Perron-Frobenius Theorem, that this probability tends to 1 as k tends to infinity. Introduction The idea of this paper is to introduce into the algebra of automata introduced in [4] probabilities on the actions. This permits a compositional description of probabilistic processes, in particular of Markov chains, and for this reason we call the automata we introduce Markov automata. We define a Markov automaton with a given set A of "signals on the left interface", and set B of "signals on the right interface" to consist of a Markov matrix Q (whose rows and columns are the states) which is the sum Q = Q a1,b1 + Q a2,b1 + · · · + Q am,bn of non-negative matrices Q ai,bj whose elements are the probabilities of transitions between states for which the signals a i and b j occur on the interfaces. In addition, each alphabet is required to contain a special symbol ε (the null signal) and the matrix Q εA,εB is required to have row sums strictly positive. There exist already in the literature models of probabilistic processes, for example the automata of Rabin [6], which are however non-compositional. Another model which does includes compositionality is discussed in [5]. Our model is more expressive -the example we discuss in this paper cannot be described by the model in [5]. It is also, in our view more natural, and mathematically more elegant. We will make some comparison with the cited models in the last section of the paper. The idea of [4] was to introduce two-sided automata, in order to permit operations analogous to the parallel, series and feedback of classical circuits. For technical reasons which will become clear we first introduce weighted automata (where the weighting of a transition is a non-negative real number) and then Markov automata. We then show how to compose such automata, calculating the probabilities in composed systems. An important aspect is the use of conditional probability since, for example, composing introduces restrictions on possible transitions and hence changes probabilities. As an illustration of the algebra we show how to specify a system of n dining philosophers (a system with 12 n states) and to calculate the probability of reaching deadlock in k steps, and we show that this probability tends to 1 as k tends to ∞, using the methods of Perron-Frobenius theory. It is clear that the algebra extends to semirings other than the real numbers, and in a later work we intend to discuss examples such as quantum automata. An earlier version of this paper was presented at [3]. We are grateful for helpful comments by Pawe l Sobociński and Ruggero Lanotte. Markov automata Notice that in order to conserve symbols in the following definitions we shall use the same symbol for the automaton, its state space and its family of matrices of transitions, distinguishing the separate parts only by the font. Definition 2.1 Consider two finite alphabets A and B, containing, respectively, the symbols ε A and ε B . A weighted automaton Q with left interface A and right interface B consists of a finite set Q of states, and an A × B indexed family Q = (Q a,b ) (a∈A,b∈B) of Q × Q matrices with non-negative real coefficients. We denote the elements of the matrix Q a,b by [Q a,b ] q,q ′ (q, q ′ ∈ Q). We require further that the row sums of the matrix Q εA,εB (and hence of Q = a∈A,b∈B Q a,b ) are strictly positive. We call the matrix Q = a∈A,b∈B Q a,b . the total matrix of the automaton Q. [Q a,b ] q,q ′ = 1. We call [Q a,b ] q,q ′ the probability of the transition from q to q ′ with left signal a and right signal b. The idea is that in a given state various transitions to other states are possible and occur with various probabilities, the sum of these probabilities being 1. The transitions that occur have effects, which we may think of a signals, on the two interfaces of the automaton, which signals are represented by letters in the alphabets. It is fundamental not to think of the letters in A and B as inputs or outputs, but rather signals induced by transitions of the automaton on the interfaces. For examples see section 2.3. When both A and B are one element sets a Markov automaton is a Markov matrix. Definition 2.3 Consider a Markov automaton Q with interfaces A and B. A behaviour of length k of Q consists of a two words of length k, one u = a 1 a 2 · · · a k in A * and the other v = b 1 b 2 · · · b k in B * and a sequence of nonnegative row vectors x 0 , x 1 = x 0 Q a1,b1 , .x 2 = x 1 Q a2,b2 , · · · , x k = x k−1 Q a k ,b k . Notice that, in general, x i is not a distribution of states; for example, in our examples often x i = 0. There is a straightforward way of converting a weighted automaton into a Markov automaton which we call normalization. Normalization N(Q) a,b q,q ′ = [Q a,b ] q,q ′ q ′ ∈Q [Q] q,q ′ = [Q a,b ] q,q ′ q ′ ∈Q a∈A,b∈B [Q a,b ] q,q ′ . To see that N(Q) is Markov, notice that the qth row sum of N (Q) is q ′ a,b N(Q) a,b q,q ′ = q ′ a,b [Q a,b ] q,q ′ a,b [Q a,b ] q,q ′ q,q ′ = q ′ a,b [Q a,b ] q,q ′ q ′ a,b [Q a,b ] q,q ′ = 1.[Q a,b ] q,q ′ = c q [R a,b ] q,q ′ then N(Q) = N(R). Proof. (ii) follows since q ′ a,b [Q a,b ] q,q ′ = q ′ a,b c q [R a,b ] q,q ′ = c q q ′ a,b [R a,b ] q,q ′ and hence [Q a,b ] q,q ′ q ′ a,b [Q a,b ] q,q ′ = c q [R a,b ] q,q ′ c q q ′ a,b [R a,b ] q,q ′ = [R a,b ] q,q ′ q ′ a,b [R a,b ] q,q ′ . An important operation on weighted automata is the power construction. The power construction Definition 2.6 If Q is a weighted automaton and k is a natural number, then form a weighted automaton Q k as follows: the states of Q k are those of Q; the left and right interfaces are A k and B k respectively; ε A k = (ε A , · · · , ε A ), ε B k = (ε B , · · · , ε B ). If u = (a 1 , a 2 , · · · , a k ) ∈ A k and v = (b 1 , b 2 , · · · , b k ) ∈ B k then (Q k ) u,v = Q a1,b1 Q a2,b2 · · · Q a k ,b k . If Q is weighted and u = (a 1 , a 2 , · · · , a k ) ∈ A k , v = (b 1 , b 2 , · · · , b k ) ∈ B k , then [(Q k ) u,v ] q, q ′ is the sum over all paths from q to q ′ with left signal sequence u and right signal sequence v of the weights of paths, where the weight of a path is the product of the weights of the steps. Lemma 2.7 If Q is a weighted automaton then the total matrix of Q k is the matrix power Q k . Hence if Q is Markov then so is Q k . Proof. The q, q ′ entry of the total matrix of Q k is u∈A k ,v∈B k (Q k ) u,v q,q ′ = u∈A k ,v∈B k [Q a1,b1 Q a2,b2 · · · Q a k ,b k ] q,q ′ = u∈A k ,v∈B k q1,···q k−1 [Q a1,b1 ] q,q1 [Q a2,b2 ] q1,q2 · · · [Q a k ,b k ] q k−1 ,q ′ = q1,···q k−1 a1,··· ,a k b1,··· ,b k [Q a1,b1 ] q,q1 [Q a2,b2 ] q1,q2 · · · [Q a k ,b k ] q k−1 ,q ′ = q1,···q k−1 a1,b1 [Q a1,b1 ] q,q1 a2,b2 [Q a2,b2 ] q1,q2 · · · a k ,b k [Q a k ,b k ] q k−1 ,q′ = [QQ · · · Q] q,q′ . Definition 2.8 If Q is a Markov automaton then we call Q k the automaton of k step paths in Q. We define the probability in Q of passing from state q to q ′ in exactly k steps with left signal u and right signal v to be [(Q k ) u,v ] q,q ′ . It is important to understand the precise meaning of this definition. The probability of passing from state q to q ′ in precisely k steps, so defined, is the weighted proportion of all paths of length k beginning at q and ending at q ′ amongst all paths of precisely length n beginning at q. Graphical representation Although the definitions above are mathematically straightforward, in practice a graphical notation is more intuitive. We may compress the description of an automaton with interfaces A and B, which requires A×B matrices, into a single labelled graph, like the ones introduced in [4]. Further, expressions of automata in this algebra may be drawn as "circuit diagrams" also as in [4]. We indicate both of these matters by describing some examples. A philosopher Consider the alphabet A = {t, r, ε}. A philosopher is an automaton Phil with left interface A and right interfaces A, state space {1, 2, 3, 4}, and transition matrices Phil ε,ε =     1 2 0 0 0 0 1 2 0 0 0 0 1 2 0 0 0 0 1 2     , Phil t,ε =     0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0     , Phil ε,t =     0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0     Phil r,ε =     0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0     , Phil ε,r =     0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0     . The other four transition matrices are zero matrices. Notice that the total matrix of Phil is     1 2 1 2 0 0 0 1 2 1 2 0 0 0 1 2 1 2 1 2 0 0 1 2     , which is clearly stochastic, so Phil is a Markov automaton. The intention behind these matrices is as follows: in all states the philosopher does a transition labelled ε, ε (idle transition) with probability 1 2 ; in state 1 he does a transition to state 2 with probability 1 2 labelled t, ε (take the left fork ); in state 2 he does a transition to state 3 with probability 1 2 labelled ε, t (take the right fork ); in state 3 he does a transition to state 4 with probability 1 2 labelled r, ε (release the left fork ); and in state 4 he does a transition to state 1 with probability 1 2 labelled ε, r (release the left fork ). All this information may be put in the following diagram. E c ' T C b s 1 2 3 4 ε, ε; 1 2 ε, ε; 1 2 ε, ε; 1 2 ε, ε, 1 2 t, ε; 1 2 ε, t; 1 2 r, ε; 1 2 ε, r; 1 2 t, r, ε t, r, ε A fork Consider again the alphabet A = {t, r, ε}. A fork is an automaton Fork with left interface A and right interface A, state space {1, 2, 3}, and transition matrices Fork ε,ε =   1 3 0 0 0 1 2 0 0 0 1 2   , Fork t,ε =   0 1 3 0 0 0 0 0 0 0   , Fork ε,t =   0 0 1 3 0 0 0 0 0 0   Fork r,ε =   0 0 0 1 2 0 0 0 0 0   , Fork ε,r =   0 0 0 0 0 0 1 2 0 0   . The other four transition matrices are zero. Fork is a Markov automaton since its total matrix is   1 3 1 3 1 3 1 2 1 2 0 1 2 0 1 2   . The intention behind these matrices is as follows: in all states the fork does a transition labelled ε, ε (idle transition) with positive probability (either 1 3 or 1 2 ); in state 1 it does a transition to state 2 with probability 1 3 labelled t, ε (taken to the left ); in state 1 he does a transition to state 3 with probability 1 3 labelled ε, t (taken to the right ); in state 2 he does a transition to state 1 with probability 1 2 labelled r, ε (released to the left ); in state 3 he does a transition to state 1 with probability 1 2 labelled ε, r (released to the right ). All this information may be put in the following diagram: Reachability For many applications we are interested only in states reachable from a given initial state by a path of positive probability. Given a Markov automaton Q and an initial state q 0 there is a subautomaton Reach(Q,q 0 ) whose states are the reachable states, and whose transitions are those of Q restricted to the reachable states. The algebra of Markov automata: operations Now we define operations on weighted automata analogous (in a precise sense) to those defined in [4]. Definition 3.1 Given weighted automata Q with left and right interfaces A and B, and S with interfaces C and D the parallel composite Q × R is the weighted automaton which has set of states Q × R, left interfaces A × C, right interface B × D, ε A×C = (ε A , ε C ), ε B×D = (ε B , ε D ) and transition matrices (Q × S) (a,c),(b,d) = Q a,b ⊗ S c,d . This just says that the weight of a transition from (q, r) to (q ′ , r ′ ) with left signal (a, c) and right signal (b, d) is the product of the weights of the transition q → q ′ with signals a and b, and the weight of the transition r → r ′ with signals c and d. Hence if Q and R are Markov automata then so is Q × R. Proof. N(Q × R) (a,c),(b,d) (q,r),(q ′ ,r ′ ) = [Q a,b ] q,q ′ [S c,d ] r,r ′ q ′ ,r ′ (a,c),(b,d) [Q a,b ] q,q ′ [S c,d ] r,r ′ = [Q a,b ] q,q ′ [S c,d ] r,r ′ q ′ ,(a,b) [Q a,b ] q,q ′ r ′ ,(c,d) [S c,d ] r,r ′ = N(Q) (a,b) q,q ′ N(R) (c,d) r,r ′ = (N(Q) × N(R)) (a,c),(b,d) (q,r),(q ′ ,r ′ ) . For the second part notice that if Q and R are Markov then Q × R = N(Q) × N(R) = N(Q × R) which implies that Q × R is Markov. Definition 3.3 Given weighted automata Q with left and right interfaces A and B, and R with interfaces B and C the series (communicating parallel) composite of weighted automata Q•R has set of states Q × R, left interfaces A, right interface C, and transition matrices (Q • R) a,c = b∈B Q a,b ⊗ R b,c . Lemma 3.4 (Q • R) • S = Q • (R • S). Proof. This follows from the fact that ⊗ is associative.. It is easy to see that Q•R is not necessarily Markov even when both Q and R are. The reason is that the communication in the series composite reduces the number of possible transitions, so that we must normalize to get (conditional) probabilities. However in a multiple composition it is only necessary to normalize at the end, because of the following lemma. Proof. (NQ • NR) a,c (q,r),(q ′ ,r ′ ) = b∈B NQ a,b q,q ′ ⊗ [NR b,c ] r,r ′ = b∈B [Q a,b ] q,q ′ q ′ a,b [Q a,b ] q,q ′ · [R b,c ] r,r ′ r ′ b,c [R b,c ] r,r ′ ′ = 1 q ′ ,r ′ ( a,b [Q a,b ] q,q ′ b,c [R b,c ] r,r ′ ) b∈B [Q a,b ] q,q ′ [R b,c ] r,r ′ . = c q,r (Q • R) a,c (q,r),(q ′ ,r ′ ) , where c q,r = 1 q ′ ,r ′ ( a,b [Q a,b ] q,q ′ b,c [R b,c ] r,r ′ ′ ) depends only on q, r. Hence by the lemma 2. Proof. (Q · R) · S = N(N(Q • R) • S) = N(N(Q • R) • N(S)) since S is Markov = N((Q • R) • S) = N(Q • (R • S))(Q × R) k = Q k × R k Proof. (Q × R) k ) (u,u ′ ),(v,v ′ ) = (Q a1,b1 ⊗R a ′ 1 ,b ′ 1 )(Q a2,b2 ⊗R a ′ 2 ,b ′ 2 ) · · · (Q a k ,b k ⊗R a ′ k ,b ′ k ) = (Q a1,b1 ⊗Q a2,b2 ⊗ · · · Q a k ,b k )(R a ′ 1 ,b ′ 1 ⊗R a ′ 2 ,b ′ 2 · · · ⊗R a ′ k ,b ′ k ) = (Q k ×R k ) (u,u ′ ),(v,v ′ ) . Remark. It is not the case that if Q, R are Markov then (Q · R) k = Q k · R k . The reason is that normalizing length k steps in a weighted automaton is not the same as considering k step paths in the normalization of the automaton. Next we define some constants each of which is a Markov automaton. Some special cases, all described in [4], have particular importance: (i) the automaton corresponding to the identity function 1 A , considered as a relation on A × A is called 1 A ; (ii) the automaton corresponding to the diagonal function ∆ : A → A × A (considered as a relation) is called ∆ A ; the automaton corresponding to the opposite relation of ∆ is called ∇ A . (iii) the automaton corresponding to the function twist : A × B → B × A is called twist A,B . (iv) the automaton corresponding to the relation η = { ( * , (a, a); a ∈ A} ⊂ { * } × (A × A) is called η A ; the automaton corresponding to the opposite of η is called ǫ A . The dining philosophers system Now the model of the dining philosophers problem we consider is an expression in the algebra, involving also the automata Phil and Fork. The the system of n dining philosophers is DF n = η A ·((Phil · Fork · Phil · Fork· · · · · Phil · Fork) × 1 A ) · ǫ A , where in this expression there are n philosophers and n forks. As explained in [4], we may represent this system by the following diagram, where we abbreviate Phil to P and Fork to F. P F P F P F · · · Let us examine the case when n = 2 with initial state (1, 1, 1, 1). Let Q be the reachable part of DF 2 . The states reachable from the initial state are q 1 = (1, 1, 1, 1), q 2 = (1, 3, 3, 2), q 3 = (3, 2, 1, 3), q 4 = (1, 1, 4, 2), q 5 = (4, 2, 1, 1), q 6 = (1, 3, 2, 1), q 7 = (2, 1, 1, 3), q 8 = (2, 3, 2, 3) (q 8 is the unique deadlock state). The single matrix of the automaton Q, using this ordering of the states, is                        . Calculating powers of this matrix we see that the probability of reaching deadlock from the initial state in 2 steps is 23 48 , in 3 steps is 341 576 , and in 4 steps is 4415 6912 . The probability of deadlock The idea of this section is to apply Perron-Frobenius theory (see, for example) [7] to the Dining Philosopher automaton. However, for convenience, we give the details of the proof of the case we need, without refering to the general theorem. Definition 4.1 Consider a Markov automaton Q with input and output sets being one element sets {ε}. A state q is called a deadlock if the only transition out of q with positive probability is a transition from q to q (the probability of the transition must necessarily be 1). Theorem 4.2 Consider a Markov automaton Q with interfaces being one element sets, with an initial state q 0 . Suppose that (i) Q has precisely one reachable deadlock state, (ii) for each reachable state, not a deadlock, there is a path with non-zero probability to q 0 , and (iii) for each reachable state q there is a transition with non-zero probability to itself. Then the probability of reaching a deadlock from the initial state in k steps tends to 1 as k tends to infinity. Proof. Let R = Reach(Q, q 0 ). Suppose R has m states. Then in writing the matrix R we choose to put the deadlock last, so that R has the form R = S T 0 1 where S is (m − 1) × (m − 1) and T is (m − 1) × 1. Now R k = S k T k 0 1 for some matrix T k . Condition (i) implies that there is a path with positive probability (a positive path) from any non-deadlock state to any other in R. Condition (ii) implies that if there is a positive path of length l between two states then there is also a positive path of all lengths greater than l. These two facts imply that there is a k 0 such that from any non-deadlock state to any other state there is a positive path of length k 0 . For this k 0 the matrix T k0 is strictly positive. This means that the row sums of S k0 are strictly less than 1. But the eigenvalues of a matrix are dominated in absolute value by the maximum of the absolute row sums (the sums of absolute values of the row elements). Hence the eigenvalues of S k0 and hence of S all have absolute value less than 1. But by considering the Jordan canonical form of a matrix whose eigenvalues all have absolute values less than 1 it is easy to see that S k tends to 0 as k tends to infinity. Hence T k tends to the column vector all of whose entries are 1. Hence the probability of reaching the deadlock from any of the other states in k steps tends to 1 as k tends to infinity. Corollary 4.3 In the dining philosopher problem DF n with q 0 being the state (1, 1, · · · , 1) the probability of reaching a deadlock from the initial state in k steps tends to 1 as k tends to infinity. Proof. We just need to verify the conditions of the theorem for the dining philosopher problem. It is straightforward to check that the state (2, 3, 2, 3, · · · , 2, 3) in which the philosophers are all in state 2 and the forks in state 3 is a reachable deadlock.It is clear that in any state q there is a positive transition to q, since each component has silent moves in each state. We need only check that for any reachable state other than this deadlock that there is a positive path to the initial state. Consider the states f 1 , f 2 of two forks adjacent modulo n, and the state p of the philosopher between these two forks. Examining the positive paths possible in two adjacent forks and the corresponding philosopher we see that the reachable configurations are limited to (a) f 1 = 1, p = 1, f 2 = 1, (b) f 1 = 1, p = 1, f 2 = 3, (c) f 1 = 1, p = 4, f 2 = 2, (d) f 1 = 2, p = 1, f 2 = 1, (e) f 1 = 2, p = 1, f 2 = 3, (f) f 1 = 3, p = 2, f 2 = 1, (g) f 1 = 3, p = 2, f 2 = 3, (h) f 1 = 3, p = 3, f 2 = 2, (i) f 1 = 2, p = 4, f 2 = 2. We will show that in states other than the deadlock or the inital state there is a transition of the system which increases the number of forks in state 1. Notice that in a reachable state the states of adjacent forks determine the state of the philosopher between. Consider the possible configurations of fork states. We need not consider cases all forks are in state 1 (initial), or all in state 3 (the known deadlock). Given two adjacent forks in states 3, 2 there are transitions which only involve this philosopher and the two forks (apart from null signals) which result in one of the forks returning to state 1 (the philosopher puts down a fork that he holds). This is also the case when two adjacent forks are in states 1, 2 or 2, 2 or 3, 1. But in a circular arrangement other than all 1's or all 3's one of the pairs 1, 2 or 2, 2 or 3, 1 or 3, 2 must occur. Remark. Notice that in the proof of the corollary we did not use the specific positive probabilities of the actions of the philosophers and forks. Hence the result is true with any positive probabilities replacing the specific ones we gave in the description of the philosopher and fork. In fact, different philosophers and forks may have different probabilities without affecting the conclusion of the corollary. 5 Concluding remarks The algebra of automata: equations There is much more to say about the algebraic structure and its relations with other fields. We have mentioned above some equations which are satisfied, and here we mention one more. Lemma 5.1 The constants ∆ A , ∇ A satisfy the Frobenius equations [1], namely that (∆ A × 1 A ) · (1 A × ∇ A ) = ∇ A · ∆ A . The proof is straightforward. Comparisons According to Rabin a probabilistic automaton on an alphabet Σ consists of a set of states Q and a family of stochastic transition matrices [P a ] q,q ′ (a ∈ Σ; q, q ′ ∈ Q). A distribution of states is a row vector with non negative real entries whose sum is 1. A behaviour corresponding to an initial state distribution x 0 , and an input word u = a 1 a 2 · · · a k , is a sequence of state distributions x 0 , x 1 = x 0 P a1 , x 2 = x 0 P a2 , · · · , x k = x k−1 P a k . This is a non-compositional model, and it immediately clear that the meaning of the alphabet in Rabin is quite different from the meaning of the alphabets for our Markov automata. For Rabin the letters are inputs which drive the automaton Q -for a given state q and a given input a the sum of the probabilities of transitions out of q is 1. We are able to describe the same phenomenon by considering a second automaton R whose signal on the interface drive Q, which of course introduces conditional probabilities. From our point of view Rabin's probabilities are conditional ones resulting from the knowledge that an input a occurs. Another difference is that every transition in every state in Rabin's automata produces a distribution of states. However it is crucial in our model that actions are not necessarily defined in all states; or if they are defined they may be only partially defined. For example, the fork in state 2 has no transitions labelled t, ε; it cannot be taken again when it is already taken. The fork in state 1 may be taken to the left with probability 1 2 . The second model we mentioned [5] considers a generalization of Rabin's model (and hence different from ours) which is influenced by concurrency theory. It has a form of composition, and as usual with models related to process algebras the composition involves an underlying broadcast (and hence interleaved) communication, and does not involve conditional probability. As a result it is not possible to describe our example, in which, in a single step all philosophers may take their left fork. Instead, it is straightforward to model broadcast, interleaved models using our algebra [2], using in particular the component ∆. Further, [5] has a much more limited algebra than that presented here; for example, multiply simultaneous signals (together with the synchronization on some of the signals) are not available. In our view interleaving destroys the realism of the model. For example, to reach deadlock in the dining philosopher problem requires a sequence of actions, as philosophers take the forks one by one. This results in quite different probabilities. Definition 2. 4 4The normalization of a weighted automaton Q, denoted N(Q) is the Markov automaton with the same interfaces and states, but with Lemma 2. 5 5(i) If Q is a Markov automaton then N(Q) = Q. (ii) If c a,b,q are positive real numbers and Q and R are weighted automata (with the same interfaces A and B, and the same state spaces Q = R) such that Lemma 3. 2 2If Q and R are weighted automata then N(Q × R) = N(Q) × N(R). Q) • N(R)) = N(Q • R). 5 above N(NQ • NR) = N(Q • R). Definition 3.6 If Q and R are Markov automata, Q with left interface A and right interface B, R with left interface B and right interface C then the series composite of Markov automata Q · R is defined to be Q · R = N(Q • R). Theorem 3. 7 7(Q · R) · S = Q · (R · S). by 3.6 and 3.6= N(N(Q) • N(R • S)) by 3.6 = N(Q • N(R • S)) = Q · (R · S) since Q is Markov. Theorem 3. 8 8If Q and R are Markov automata then Definition 3. 9 9Given a relation ρ ⊂ A × B such that (ε A , ε B ) ∈ ρ we define a Markov automaton ρ as follows: it has one state * say. The transition matrices [ρ a,b ] are 1 × 1 matrices, that is, real numbers. Let \|ρ\| be the number of elements in ρ. Then ρ a,b = 1 \|ρ\| if ρ relates a and b, and ρ a,b = 0 otherwise. Definition 2.2 Consider two finite alphabets A and B, containing, respectively, the symbols ε A and ε B . A Markov automaton Q with left interface A and right interface B, is a weighted automaton satisfying the extra condition that the row sums of the total matrix Q are all 1. That is, for all qq ′ a∈A,b∈B Cartesian bicategories I. A Carboni, R F C Walters, Journal of Pure and Applied Algebra. 49A. Carboni, R.F.C. Walters, Cartesian bicategories I, Journal of Pure and Applied Algebra, 49, 11-32, 1987. Analysing Reduction systems using Transition systems. L De Francesco Albasini, N Sabadini, R F C Walters, ART; Forum, UdineThe parallel composition of processesL. de Francesco Albasini, N. Sabadini, R.F.C. Walters, The parallel com- position of processes, ART 2008, Analysing Reduction systems using Tran- sition systems, 111-121, Forum, Udine, 2008. L De Francesco Albasini, N Sabadini, R F C Walters, Cospan Span(Graphs): a compositional model for reconfigurable automata nets, Developments and New Tracks in Trace Theory. Cremona, ItalyL. de Francesco Albasini, N. Sabadini, R.F.C. Walters: Cospan Span(Graphs): a compositional model for reconfigurable automata nets, De- velopments and New Tracks in Trace Theory, Cremona, Italy, 9-11 October 2008. Span(Graph): A categorical algebra of transition systems. P Katis, N Sabadini, R F C Walters, Proc. AMAST '97. AMAST '97Springer Verlag1349P. Katis, N. Sabadini, R.F.C. Walters, Span(Graph): A categorical alge- bra of transition systems, Proc. AMAST '97, SLNCS 1349, pp 307-321, Springer Verlag, 1997. Compositionality for Probabilistic Automata. Nancy A Lynch, Roberto Segala, Frits W Vaandrager, Proc. CONCUR. CONCURNancy A. Lynch , Roberto Segala, Frits W. Vaandrager, Compositionality for Probabilistic Automata, Proc. CONCUR 2003, Springer Lecture Notes in Computer Science, 2761, pp 204-222, 2003. Information and Control 6. M Rabin, Probabilistic Automata, M. O Rabin, Probabilistic Automata, Information and Control 6, pp.230- 245, 1963. Algebraic Graph Theory. C Godsil, G Royle, SpringerC. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001	[]
[ "Open effective theory of scalar field in rotating plasma", "Open effective theory of scalar field in rotating plasma" ]	[ "Bidisha Chakrabarty [email protected] \nUniversity of Southampton\nUniversity RoadSO17 1BJSouthamptonUnited Kingdom\n\nInternational Centre for Theoretical Sciences (ICTS-TIFR)\nHesaraghatta Hobli560089Shivakote, BengaluruIndia\n", "Aswin [email protected] " ]	[ "University of Southampton\nUniversity RoadSO17 1BJSouthamptonUnited Kingdom", "International Centre for Theoretical Sciences (ICTS-TIFR)\nHesaraghatta Hobli560089Shivakote, BengaluruIndia" ]	[]	We study the effective dynamics of an open scalar field interacting with a stronglycoupled two-dimensional rotating CFT plasma. The effective theory is determined by the realtime correlation functions of the thermal plasma. We employ holographic Schwinger-Keldysh path integral techniques to compute the effective theory. The quadratic effective theory computed using holography leads to the linear Langevin dynamics with rotation. The noise and dissipation terms in this equation get related by the fluctuation-dissipation relation in presence of chemical potential due to angular momentum. We further compute higher order terms in the effective theory of the open scalar field. At quartic order, we explicitly compute the coefficient functions that appear in front of various terms in the effective action in the limit when the background plasma is slowly rotating. The higher order effective theory has a description in terms of the non-linear Langevin equation with non-Gaussianity in the thermal noise.	10.1007/jhep08(2021)169	[ "https://arxiv.org/pdf/2011.13223v1.pdf" ]	227,209,874	2011.13223	83b7f805cb8a5bff5d309fae3909aea8dbff676b	Open effective theory of scalar field in rotating plasma 26 Nov 2020 Bidisha Chakrabarty [email protected] University of Southampton University RoadSO17 1BJSouthamptonUnited Kingdom International Centre for Theoretical Sciences (ICTS-TIFR) Hesaraghatta Hobli560089Shivakote, BengaluruIndia Aswin [email protected] Open effective theory of scalar field in rotating plasma 26 Nov 2020Prepared for submission to JHEP We study the effective dynamics of an open scalar field interacting with a stronglycoupled two-dimensional rotating CFT plasma. The effective theory is determined by the realtime correlation functions of the thermal plasma. We employ holographic Schwinger-Keldysh path integral techniques to compute the effective theory. The quadratic effective theory computed using holography leads to the linear Langevin dynamics with rotation. The noise and dissipation terms in this equation get related by the fluctuation-dissipation relation in presence of chemical potential due to angular momentum. We further compute higher order terms in the effective theory of the open scalar field. At quartic order, we explicitly compute the coefficient functions that appear in front of various terms in the effective action in the limit when the background plasma is slowly rotating. The higher order effective theory has a description in terms of the non-linear Langevin equation with non-Gaussianity in the thermal noise. Introduction Real time correlation functions are useful objects to study in quantum field theories. For a field theory in vaccum or thermal state, the Lorentzian correlators can be computed by analytically continuing the Euclidean correlation functions. However for strongly coupled field theories, a direct computation of correlation functions are extremely challenging. For holographic strongly coupled conformal field theories one relies on the celebrated anti-de-sitter/conformal field theory (AdS/CFT) correspondence [1] to compute CFT correlation functions using holography. The duality in its original form was proposed to calculate Euclidean correlation functions in strongly coupled d dimensional CFTs by doing computations on the weakly coupled d + 1 dimensional AdS gravity side [2,3]. This recipe was further extended to compute Lorentzian correlation functions of the CFT using holography in [4]. One of the main ingredients in this extension was to impose ingoing wave boundary condition at the horizon of AdS black holes to get the Lorentzian correlation functions of the thermal CFT. The authors of [4] argued that there would be no boundary contribution from the horizon in the calculation of retarded bulk to boundary Green's function. The prescription was further developed for the maximally extended Kruskal spacetime by computing all bulk to boundary propagators for a two sided black hole [5]. The corresponding CFT path integral contour corresponds to the finite temperature Schwinger-Keldysh (SK) contour with doubled degrees of freedom. The above prescription was extensively used for the computation of real time two point functions of the CFT using holography in various scenarios [6][7][8][9][10][11][12]. However a viable prescription for computing real time n-point functions required a detailed description of the bulk manifold over which the path integral has to be performed to compute strongly coupled CFT correlation functions holographically. A proposal of such a bulk path integral framework was given in [13,14]. Since the bulk path integral contour asymptotes to the CFT Schwinger-Keldysh contour at the boundary, the idea is to fill in the imaginary and real time segments of the CFT contour by Euclidean and Lorentzian geometries in the bulk. One has to impose appropriate matching conditions (fields and their conjugate momenta have to be continuous) at the corners of the contour where the metric changes signature. At finite temperature, the bulk manifold in this prescription has two copies of half Euclidean and half Lorentzian AdS black brane geometries (glued across the initial time slice) and the two copies are then stitched together across a spacelike hypersurface. The corresponding CFT state is the thermofield double (an entangled pure state). Given the prescription of entire path integration contour on the gravity side, the generating functional of the CFT correlation functions can be obtained by integrating over the bulk fields with sources as boundary conditions where the operators inside the CFT correlator couple to the sources. Recently Glorioso, Crossley, Liu in [15] have given a proposal on how to handle the near horizon region of the holographic contour, that is, how to stitch the two copies of the black branes across their horizons. This is a crucial point to address since the near horizon regions are potential sources of IR divergences once interactions are turned on [16]. The proposal of [15] suggests that the bulk dual of the asymptotic Schwinger-Keldysh contour is given by a complexified doublesheeted spacetime. The two copies of the black branes in the bulk are stitched by a 'horizon cap' across their future horizons. The CFT state in this case is not the thermofield double, since in the bulk the prescription involves spacetime only outside the future horizon. In [15] the authors have computed the quadratic effective actions of probe scalar and gauge fields to test their formalism (also see [17]). In [18] the authors use the 'horizon cap' boundary conditions of [15] to calculate the quartic order effective action of a probe quark moving in a strongly coupled thermal CFT plasma (bath) using holography. This gives rise to the leading non-linear corrections to the Brownian motion of the heavy quark. If the CFT bath is sufficiently forgetful (that is when the bath correlators decay exponentially fast in time), the effective theory of the quark becomes local in time (Markovian regime). The local effective theory of the quark precisely matches the effective theory obtained in [19] (see also [20]). Effective theories of a quantum Brownian particle (open probe) interacting with various baths were obtained by integrating out the bath degrees of freedom [21][22][23][24]. When the Brownian particle is linearly coupled to the bath degrees of freedom, particle's effective theory becomes quadratic. This problem has a classical stochastic description in terms of the linear Langevin equation with a Gaussian noise. In thermal equilibrium noise and dissipation terms in the Langevin equation are related by the fluctuation dissipation relation. In [20] the most general form of the local effective theory upto cubic order was constructed where the authors considered a quantum Brownian particle (open system) weakly interacting with a large thermal bath (made of two sets of harmonic oscillators) via cubic interaction. This can be seen as a perturbation over the well-known Caldeira-Leggett model. The effective theory in [20] is described by a non-linear Langevin equation with a non-gaussian noise. If the bath has microscopic time-reversal invariance then the non-Gaussianity in the thermal noise gets related to the thermal jitter in the damping constant of the Brownian particle. This is the generalisation of the fluctuation dissipation relation. The path integral in the stochastic problem is related to an underlying Schwinger-Keldysh quantum path integral. In [19] the authors extended their work by constructing an effective theory of the particle upto quartic order. In [18] the validity of the effective theory is tested for a strongly coupled bath using holography. Based on these analysis, in [25] the authors study open quantum field theories using holographic Schwinger-Keldysh path integral. The authors consider an interacting scalar quantum field theory (the system) coupled to a holographic field theory in d spacetime dimensions at finite temperature (the environment). They study the effects of integrating out the environment and obtain the effective dynamics of the resulting open quantum field theory in a derivative expansion in small frequency and momentum. For discussions on open quantum field theories and their features see [26,27]. In [28] the authors study fermionic open effective field theories coupled to holographic baths. They show how the holographic Schwinger-Keldysh path integral framework leads to boundary correlators that automatically satisfy fermionic Kubo, Martin, Schwinger (KMS) relations [29,30]. Computing the bulk on-shell action they obtain the influence phase (the effect of integrating out the bath degrees of freedom on the probe action) of the probe fermion in a derivative expansion. In [31] the authors extend this analysis to charged fermions probing Reissner-Nordström black hole background. In this paper we extend the analysis of [25] to rotating BTZ black hole. We will consider an interacting probe scalar quantum field theory (the system) coupled to a strongly coupled two dimensional rotating CFT plasma (bath) at finite temperature and chemical potential. 1 In the bulk the probe scalar field is coupled to a rotating BTZ black hole [33,34]. We consider a natural extension of the holographic Schwinger-Keldysh path integral framework of [15] to the rotating BTZ black hole with two horizons. We study the effects coming from integrating out the bath degrees of freedom on the effective action of the open scalar field theory. The open scalar field gets dragged due to the rotation of the background thermal plasma and is described by a generalised nonlinear Langevin equation. Following is the outline of the paper. We briefly review the gravitational Schwinger-Keldysh path integral framework of [15] and its natural extension to the rotating BTZ black hole in section 2. In section 3 we study the dynamics of a massive probe scalar field Φ in BTZ background. We solve the scalar wave equation with appropriate boundary conditions to find the ingoing and 1 In [16,32] the author studied holographic Brownian motion (linear Langevin dynamics) of a heavy quark in a two dimensional rotating plasma. outgoing bulk to boundary green's functions. We compute the influence functionals (quadratic and higher order) of the open scalar field holographically in section 4. 2 In presence of rotation, we study the stochastic dynamics governed by a non-linear Langevin equation resulting from the open effective field theory in section 5. We also find the generalised fluctuation dissipation relations obeyed by the parameters of the non-linear Langevin equation. Finally we summarise our results and discuss possible future directions in section 6. Various technical details of the computation are relegated to the appendices. In appendix A we provide details of the computation of quadratic influence functional of the open scalar field in 'retarded-advanced' (RA) basis. In appendix B we discuss the massive scalar field solution in a derivative expansion in small frequency and momentum. The derivative expansion of the solution is particularly useful in computing higher order effective theory of the open scalar field. Holographic SK prescription Before we start reviewing the holographic SK path integral framework of [15], let us describe the path integral contour in the boundary CFT on which the CFT correlation functions are defined. For the system prepared in an initial thermal state, the Schwinger-Keldysh time contour as shown in Fig 1 has imaginary (vertical leg) as well as real legs (horizontal lines). The initial and final points of the contour have to be identified on a thermal circle with periodicity β, where β is the inverse temperature of the CFT. In [15] the authors propose to fill in this boundary time contour by a complexified doubled bulk space-time as shown in Fig 2. Consider the Schwarzschild AdS d+1 black brane written in ingoing Eddington-Finkelstein coordinates that are regular at the future horizon, ds 2 = −r 2 1 − r d h r d dv 2 + 2 dv dr + r 2 dx 2 . (2.1) The coordinate v becomes the time coordinate t on the boundary of the spacetime at r → ∞. One has to consider two copies of the black brane in the bulk that have to be stitched in a certain way across their future horizon [15]. The proposed bulk spacetime correctly asymptotes to the aforementioned Schwinger-Keldysh time contour. In the bulk the radial coordinate has to be complexified to handle the near horizon region as proposed in [15]. Define a complex radial Re (r) Im (r) × < M L > M R r h r = r c + iε r = r c − iε= i β 2 r 2 1 − r d h r d ,(2.2) where β = 4π d r h is also the inverse Hawking temperature of the black hole. ζ has a logarithmic branch point at r = r h coming from the integral of the blackening factor (about its zero). The branch cut in ζ is taken from r = r h to r = ∞. The jump of real part of ζ across the horizon is 1. Hence ζ parametrizes a double-sheeted spacetime that describes the bulk extension of the boundary Schwinger-Keldysh contour. On each sheet imaginary part of ζ runs from 0 at the AdS boundary to ∞ at the horizon. The real part of ζ changes between the two sheets and is given by the monodromy around the horizon. The real part of ζ is chosen to be zero on the left sheet M L , then the real part of ζ becomes one on the right sheet M R after picking up the monodromy at the horizon. We will also have a UV cut-off of the geometry at r = r c for computational purposes. Hence asymptotically ζ(r c + i ε) = 0 , ζ(r c − i ε) = 1 . (2. 3) The metric written in ζ coordinate becomes ds 2 = −r 2 1 − r d h r d dv 2 + i β r 2 1 − r d h r d dv dζ + r 2 dx 2 ,(2.4) where r(ζ) is obtained by integrating (2.2). The complex tortoise coordinate for Schwarzschild-AdS d+1 geometry becomes [25] ζ + ζ c = i d 2π (d − 1) r r h d−1 2 F 1 1, d − 1 d ; 2 − 1 d ; r d r d h , (2.5) where ζ c is considered to make ζ = 0 at r = r c + i ε. The branch-cut of the hypergeometric function runs from r = r h to r = ∞. With this set up, one has to consider all bulk fields to live on a complex ζ space and think of the classical bulk action to be a contour integral over the complex tortoise coordinate ζ. Hence S bulk = dζ d d x √ −g L[g AB , Φ] (2.6) where x µ are the boundary coordinates. This action serves as the starting point for computations of influence functionals. In the following we extend the holographic contour mentioned above for rotating BTZ black holes with outer and inner horizons. The two horizons correspond to two branch points of the complexified tortoise coordinate in this case. Let us start by writing the rotating BTZ black hole metric in standard coordinates [4] ds 2 = −f dt 2 + dr 2 f + r 2 dφ − n φ dt 2 ; f (r) = (r 2 − r 2 + )(r 2 − r 2 − ) r 2 , n φ (r) = r + r − r 2 (2.7) where n φ (r) is the angular velocity of the black hole at radius r. We will denote the two angular velocities at the two horizons to be n φ (r + ) = r − r + = µ + and n φ (r − ) = r + r − = µ − . The two inverse temperatures associated to the two horizons are denoted by β + and β − where β ± = 2πr ± r 2 + − r 2 − . (2.8) The surface gravities κ ± of the two horizons are κ ± = r 2 + − r 2 − r ± . (2.9) Let us now introduce the ingoing Eddington-Finkelstein coordinates for the rotating BTZ metric that are regular at the future horizon dv = dt + dr f , dφ = dφ + n φ f dr . (2.10a) The radial tortoise coordinate r * is defined by dr * = dr f (r) . A closed form expression of r * is r * = 1 2(r 2 + − r 2 − ) r + log r − r + r + r + + r − log r + r − r − r − . (2.11) The tortoise coordinate r * maps [r + , ∞) to (−∞, 0). Let us also define another coordinate r # by dr # = n φ f dr. r # also runs from −∞ to 0 as one moves from horizon to infinity. Explicitly the coordinate r # is r # = 1 2(r 2 + − r 2 − ) r − ln r − r + r + r + + r + ln r + r − r − r − . (2.12) In terms of these coordinates the ingoing Eddington-Finkelstein coordinates become v = t + r * , φ = φ + r # . (2.13a) Let us now define a complexified tortoise coordinate χ (a rescaling of r * ) by χ(r) = r rc+iε dr iβ 2 f (r ) = 1 2πi log r − r + r + r + r + r − r − r − r − r + r c + r + r c − r + r c − r − r c + r − r − r + . (2.14) Note that the normalisation is chosen in such a way that χ(r c + iε) = 0. χ(r) has branch points at r + and r − . We will consider branch cut in χ that extend from r = r + to r = ∞. The inner horizon branch cut is chosen such that χ is analytic in the real interval between the two horizons. Since we will be interested in the region outside the future horizon r + , we will now consider the complexified doubled bulk spacetime as shown in We will glue two copies of rotating BTZ black holes across their future horizon by a horizon cap and the monodromy in χ across the branch point at r = r + will be one. Hence χ(r c + i ε) = 0 , χ(r c − i ε) = 1 . (2.15) Let us also consider the following contour integral Θ(r) = n φ dχ = 2 iβ dr n φ f = 1 2πi log r + r − r − r − r − r + r + r + r − r + r c − r − r c + r − r c + r + r c − r + r − r + . (2.16) Note that Θ(r) goes from 0 to µ + in going from r c + i to r c − i by picking up a monodromy across the branch point at r = r + , giving Θ(r c + i ε) = 0 , Θ(r c − i ε) = µ + . (2.17) This explains the origin of the chemical potential µ + arising from the holographic contour in a rotating BTZ black hole. The metric rewritten in ingoing (v, χ, φ ) coordinates is given by ds 2 = −f dv 2 + iβf dvdχ + r 2 (dφ − n φ dv) 2 . (2.18) The CFT contour for a given initial state at finite temperature and chemical potential for angular momentum in this case is described as in Fig 4 (following [14]). In [14] the authors described how the contour for the boundary CFT corresponding to a rotating black hole does not only lie in the complex t plane, but also has a leg into the complex φ plane. The solid circles in Fig 4 are to be identified. In this paper we are interested in computing the generating functional for CFT correlators holographically for a given initial state at finite temperature and chemical potential due to angular momentum. Im(t) Re(t) Im(φ) Probe scalar dynamics in bulk Ingoing solution In this section we consider a free massive probe scalar field Φ with mass M minimally coupled to the rotating BTZ black hole. According to AdS/CFT correspondence, the scalar field in the bulk is dual to a scalar operator O in the boundary CFT with conformal dimension ∆ = 1+ √ 1 + M 2 . The action for the probe scalar field is given by S = − dχdvdϕ √ −g 1 2 g AB ∂ A Φ∂ B Φ + M 2 2 Φ 2 . (3.1) The classical scalar equation of motion is the Klein-Gordon equation 1 √ −g ∂ A [ √ −gg AB ∂ B Φ] − M 2 Φ = 0 . (3.2) In the following we solve the massive scalar equation with appropriate boundary conditions to find the ingoing bulk to boundary green's function. To solve the massive scalar equation, it is useful to define a new radial coordinate z as in [35] z = r 2 − r 2 − r 2 + − r 2 − . (3.3) In terms of the z coordinate, the inner (Cauchy) horizon at r = r − is located at z = 0, the event horizon at r = r + is at z = 1 and the asymptotic boundary is at z = +∞. We will consider a solution of the scalar equation of the form Φ in = dω (2π) m G + (ω, z, m)e i(mφ −ωv) . (3.4) where G + (ω, z, m) is the frequency domain retarded (ingoing) bulk to boundary Green's function. ω and m are frequency and azimuthal number of the mode. Written in z coordinate the equation of motion looks as following z(z − 1)∂ 2 z G + + 2z − 1 − iω{z(r 2 + − r 2 − ) + r 2 − } 1/2 r 2 + − r 2 − + imr + r − (r 2 + − r 2 − ){z(r 2 + − r 2 − ) + r 2 − } 1/2 ∂ z G + − 1 4{z(r 2 + − r 2 − ) + r 2 − } 1/2 iω + imr + r − {z(r 2 + − r 2 − ) + r 2 − } + m 2 {z(r 2 + − r 2 − ) + r 2 − } 1/2 G + − M 2 4 G + = 0 . (3.5) If we consider an ansatz of the form G(z) = e i(ωr * −mr # ) z α (1 − z) β F (z) (3.6) then the equation of motion in terms of F(z) becomes a hypergeometric differential equation. We impose regularity at the horizon and normalisability at the cut-off boundary to solve the equation of motion. The boundary conditions are given by dG + (ω, r, m) dχ r=r + = 0 , G + (ω, r c , m) = 1 . (3.7) After using hypergeometric identities, the regular solution that satisfies the above boundary conditions becomes (after writing the mass M of the scalar in terms of ∆) (similar solution appears in a very recent paper [36]) G + (z) = e i(ωr * −mr # ) z c−1 2 −a+ ∆ 2 (z − 1) a+b−c 2 2 F 1 a − c + 1 − ∆ 2 , a − ∆ 2 , a + b − c + 1; 1 − 1 z e i ωr * c −mr # c z c−1 2 −a+ ∆ 2 c (z c − 1) a+b−c 2 2 F 1 a − c + 1 − ∆ 2 , a − ∆ 2 , a + b − c + 1; 1 − 1 zc (3.8) where a = 1 2 2 − i(ω − mµ − ) κ − − i(ω − mµ + ) κ + (3.9a) b = 1 2 −i(ω − mµ − ) κ − − i(ω − mµ + ) κ + (3.9b) c = 1 − i(ω − mµ − ) κ − . (3.9c) This is the exact ingoing bulk to boundary green's function that corresponds to the infalling quasi normal modes of the black hole. The ingoing solution is analytic in z coordinate. Solution on the SK contour In this section, first we study the dynamics of the outgoing Hawking modes. Within the holographic Schwinger-Keldysh path integral framework the outgoing modes naturally arise as frequency and angular momentum reversed counter parts of the ingoing modes. Hence the physics of the outgoing Hawking radiation is captured by the advanced bulk to boundary green's function. In the following we will construct the advanced green's function by imposing (generalised) timereversal transformation on the ingoing solution. Under (generalised) time reversal transformation the coordinates and parameters transform as follows v → iβχ − v (3.10a) χ → χ (3.10b) φ → iβΘ − φ (3.10c) ω → −ω (3.10d) m → −m . (3.10e) These set of transformations keep the metric in equation (2.18) manifestly time reversal invariant once the metric is written in the following way ds 2 = −f dv (dv − iβdχ) − r 2 dφ − n φ dv in φ βdχ − dφ + n φ (dv − iβdχ) . (3.11) Under time reversal, the ingoing solution Φ in in equation (3.4) gets mapped to the outgoing solution Φ out Φ out (v, z, φ ) = dω 2π m e −iωv+imφ G(−ω, z, −m)e −β(ωχ−mΘ) . (3.12a) Note that the outgoing solution is non-analytic in z coordinate because of the presence of χ and Θ in the exponential factor. χ and Θ have logarithmic branch point at r = r + . Now we construct the full solution on the doubled bulk spacetime by adding the ingoing and outgoing solutions. The full solution captures the ingoing quasi normal modes as well as the outgoing Hawking modes. The full solution on the holographic SK contour is given by the following linear combination of the ingoing and outgoing solutions Φ = dω 2π 1 2π m c ω,m G + (ω, z, m) + h ω,m G + (−ω, z, −m) e −β(ωχ−mΘ) e −iωv+imφ . (3.13) At the boundary the doubled bulk spacetime asymptotes to CFT Schwinger-Keldysh contour with left and right sources φ L and φ R . Hence from the requirement Φ (r c + iε) = φ L and Φ (r c − iε) = φ R , we find that c ω,m = −f ω,m φ L (ω, m) + (1 + f ω,m ) φ R (ω, m) h ω,m = (1 + f ω,m ) (φ L (ω, m) − φ R (ω, m)) (3.14a) where f ω,m = 1 e β(ω−mµ + ) −1 is the Bose-Einstein factor with chemical potential. Hence the full solution of the scalar wave equation in equation (3.13) becomes Φ = dω 2π 1 2π m (−f ω,m φ L (ω, m) + (1 + f ω,m ) φ R (ω, m)) G + (ω, z, m) + (1 + f ω,m ) (φ L (ω, m) − φ R (ω, m)) G + (−ω, z, −m) e −β(ωχ−mΘ) e −iωv+imφ . (3.15) We will use this exact solution to compute the quadratic influence functional in section 4.1. Influence functional of open scalar field In this section we compute the quadratic and higher order influence functionals of the open scalar field. To compute the influence functional it is useful to work with the retarded-advance (RA) basis [37][38][39] within the SK framework where the KMS relations become manifest. The RA basis is defined as φ F (ω, m) = − (1 + f ω,m ) φ R (ω, m) + f ω,m φ L (ω, m) (4.1a) φ P (ω, m) = −f ω,m (φ R (ω, m) − φ L (ω, m)) . (4.1b) In terms of the RA basis we can write down the scalar solution given in equation (3.15) as Φ = dω 2π 1 2π m −G + (ω, z, m) φ F + G + (−ω, z, −m) φ P e β(ω(1−χ)−m(µ + −Θ)) e −iωv+imφ . (4.2) Using this scalar solution in RA basis, we compute the quadratic on-shell action in section 4.1. Quadratic effective theory Imposing the Klein-Gordon equation of motion, the quadratic on-shell action of the scalar field becomes S on−shell = − dχdvdϕ ∂ A √ −g 2 g AB Φ∂ B Φ = − dχdvdφ ∂ χ r 2 Φ∂ v Φ + ∂ χ −ir β Φ∂ χ Φ + ∂ χ n φ r 2 Φ∂ φ Φ = − dvdφ r 2 Φ∂ v Φ − ir β Φ∂ χ Φ + n φ r 2 Φ∂ φ Φ χ=1,Θ=µ + χ=0,Θ=0 . (4.3) In the above equation we see that the quadratic on-shell action is a total derivative, hence it has to be evaluated at the boundary. In RA basis the quadratic influence functional becomes (see appendix A for details) S on−shell = dω 2π 1 2π m φ F (ω, m)φ P (−ω, −m)G F P [ω, m] (4.4) where after setting G + (ω, z c , m) = 1, G F P [ω, m] = − 2ir c β ∂ χ G + (ω, z c , m) + β 2 (ω − mn φ ) (1 − e −β(ω−µ + m) ) . (4.5) The FF and PP terms do not contribute to the radial integral since they are purely analytic functions (the χ dependence goes away inside the integral) and contributions from the left and right contours cancel each other. We will compute the quadratic influence functional by substituting G + in the above expression. Before proceeding with the calculation let us define yet another basis known as the average-difference/Keldysh basis in which the fluctuation-dissipation relation will be studied. This basis will be more useful in the computation of higher order influence phase that will be computed in a small frequency and momentum expansion. We define the average-difference basis [40][41][42] in the following φ F (ω, m) = −φ a (ω, m) − N ω,m φ d (ω, m) (4.6a) φ P (ω, m) = −φ d (ω, m) f ω,m (4.6b) where N ω,m = f ω,m + 1 2 . In terms of the left and right sources φ L and φ R at the boundary, the average source (mean value) becomes φ a = φ R +φ L 2 and the difference source (quantum/statistical fluctuations) becomes φ d = φ R −φ L . The quadratic influence functional of equation (4.4) written in average-difference basis becomes S on−shell = dω 2π m (G ad φ a (ω, m) φ d (−ω, −m) + G dd φ d (ω, m) φ d (−ω, −m)) (4.7) where the coefficient functions that appear in front of the φ a φ d and φ d φ d terms are G ad = 2 ir c β ∂ χ G + (ω, z c , m) + β 2 (ω − mn φ ) , (4.8) G dd = ir c 2β e β(ω−mµ + ) + 1 e β(ω−mµ + ) − 1 ∂ χ G + (ω, z c , m) + β 2 (ω − mn φ ) + (ω → −ω) . (4.9) where we have written G dd in a symmetric way under ω → −ω. Note that there is no G aa term in the quadratic influence functional. This is a consequence of the microscopic unitarity of the theory. After substituting for the solution of G + (ω, z c , m) from eq. (3.8) and picking up the coefficient of r 4−2∆ c we obtain the coefficient functions G ad and G dd . Note that this will be the result after an appropriate counter term subtraction as is done in holographic renormalisation. We finally obtain G ad [ω, m] = − 2 r 2 + − r 2 − ∆−1 Γ 2 (2 − ∆) Γ(p + )Γ(p − ) sin (π∆) πΓ (1 − ∆ + p + ) Γ (1 − ∆ + p − ) (4.10a) = − 2 r 2 + − r 2 − ∆−1 \|Γ(p + )Γ(p − )\| 2 sin (π(∆ − p + )) sin (π(∆ − p − )) πΓ 2 (∆ − 1) sin(π∆) (4.10b) = − 2 r 2 + − r 2 − ∆−1 \|Γ(p + )Γ(p − )\| 2 cosh β − ω − 2 − cosh β + ω + 2 − iπ∆ 2πΓ 2 (∆ − 1) sin (π∆) (4.10c) = − r 2 + − r 2 − ∆−1 \|Γ(p + )Γ(p − )\| 2 πΓ 2 (∆ − 1) sin (π∆) (4.10d) cosh β − ω − 2 − cosh β + ω + 2 cos(π∆) + i sinh β + ω + 2 sin(π∆) (4.10e) where various parameters that appear in the above expression are defined as follows p + = ∆ 2 + i (m − ω) 2(r + − r − ) (4.11a) p − = ∆ 2 − i (m + ω) 2(r + + r − ) (4.11b) ω + = ω − mµ + = ω − m r − r + (4.11c) ω − = ω − mµ − = ω − m r + r − (4.11d) β + = 2πr + r 2 + − r 2 − (4.11e) β − = 2πr − r 2 + − r 2 − . (4.11f) p + , p − , ω + and ω − are lightcone like dimensionless combination of frequency and momentum. From a similar computation of G dd we get G dd [ω, m] = −i 2 r 2 + − r 2 − ∆−1 \|Γ(p + )Γ(p − )\| 2 πΓ 2 (∆ − 1) cosh β + ω + 2 . (4.12) By comparing equation (4.12) with equation (4.10e) we get the fluctuation-dissipation relation in presence of chemical potential due to angular momentum given by G dd = i 2 coth β (ω − mµ + ) 2 Im(G ad ) . (4.13) Im(G ad ) gives the quadratic spectral function at finite temperature [42,43]. Our quadratic influence functional results computed using a natural extension of the recently proposed holographic SK prescription of [15] match with the long known results of [4] under following coordinate transformations and redefinition of parameters x + = r + t − r − φ , x − = −r − t + r + φ , (4.14a) k + = ωr + − mr − r 2 + − r 2 − = β + ω + 2π (4.14b) k − = ωr − − mr + r 2 + − r 2 − = β − ω − 2π (4.14c) p + = πT L (k + + k − ) = πT L (ω − m) (r + − r − ) (4.14d) p − = πT R (k + − k − ) = πT R (ω + m) (r + + r − ) (4.14e) where x ± are comoving coordinates, k ± are momenta conjugate to x ± and T L and T R are the left and right temperatures given by T L = r + − r − 2π , T R = r + + r − 2π . (4.15) -13 - The answer given in [4] for G ad goes as G ad [ω, m] ∼ \|Γ ∆ 2 + i p + 2πT L Γ ∆ 2 + i p − 2πT R \| 2 Γ 2 ∆ 2 − 1 2 sin(2πβ + ) cosh p + 2T L − p − 2T R − cos 2π ∆ 2 cosh p + 2T L + p − 2T R + i sin 2π ∆ 2 sinh p + 2T L + p − 2T R . (4.16) Quartic order effective theory In this section, we will turn on weak quartic self interaction of the massless scalar field. At the leading order in the self interaction strength this term corresponds to the four-point contact Witten diagram in the bulk. Going to frequency domain we determine the quartic order influence functional of the scalar field in a low frequency and angular momentum gradient expansion till first sub leading order in derivatives. We are relying on derivative expanded scalar solution to evaluate higher order influence phase rather than trying to compute them using exact scalar solution since the later might be technically somewhat complicated. Quartic Influence functional In the following we add a quartic self-interaction term to the original action. We will compute the on-shell action by imposing the scalar equation of motion and integrating out the complexified radial coordinate. We consider a simpler setting where we have a massless self interacting scalar field in the bulk 3 that is dual to a marginal CFT operator with conformal dimension two at the boundary. The action for the massless interacting scalar field is given by S = − dχdv dφ √ −g 1 2 g AB ∂ A Φ∂ B Φ + λ 4! Φ 4 (4.17) where λ is the interaction strength. The equation of motion of the scalar field resulting from the above action is 1 √ −g ∂ A √ −gg AB ∂ B Φ − λ 3! Φ 3 = 0 . (4.18) Since this equation is non-linear in Φ, we will solve this equation perturbatively in λ. We write the solution Φ as a series in λ Φ = ∞ n=0 λ n Φ n . (4.19) The action upto linear order in coupling constant λ becomes S = − dv dφ dχ √ −g g AB 2 (∂ A Φ 0 ∂ B Φ 0 + 2λ∂ A Φ 1 ∂ B Φ 0 ) + λ 4! Φ 4 0 (4.20a) = − dv dφ dχ √ −g 1 2 ∂ A √ −gg AB (Φ 0 + 2λΦ 1 ) ∂ B Φ 0 − 1 2 (Φ 0 + 2λΦ 1 ) ∂ A √ −gg AB ∂ B Φ 0 + λ 4! Φ 4 0 (4.20b) S on−shell = − dv dφ dχ √ −g 1 2 ∂ A √ −gg AB Φ 0 ∂ B Φ 0 + λ 4! Φ 4 0 . (4.20c) In the last line we have imposed the e.o.m of the free scalar Φ 0 and we choose Φ 1 at the boundary to be zero as a choice of boundary condition. The first term in the above equation is the quadratic on-shell action that we have computed in the previous section. The quartic piece in the on-shell action is S quartic,on−shell = − dv dφ dχ √ −g λ 4! Φ 4 0 . (4.21) We have to perform the radial integral in order to obtain the quartic influence functional of the open scalar field at the boundary. To compute the quartic integral, we will first solve the free scalar field Φ 0 in a gradient expansion in low frequency and angular momentum. We will then substitute this solution in the action and integrate out the radial coordinate. In the following we compute the derivative expansion of the free scalar field. Derivative expansion of scalar solution: We will start by expanding the frequency domain bulk to boundary green's function in terms of βω and βmµ, both of them being dimensionless parameters. We have denoted µ + as µ, this is the notation we will follow now on. The derivative expansion is where G + ω , G + m and all the higher order pieces are only functions of the radial coordinate. In the following we determine the values of G + 0 , G + ω and G + m for the free massless scalar field. Refer to appendix B for derivative expansion of massive scalar field. We impose the following boundary conditions while writing the derivative expansion of the green's function. These boundary conditions correspond to regularity of the green's function at the horizon and its normalisability at the UV cut-off r c . The boundary conditions are After imposing regularity at the horizon and normalisability at the cut-off boundary we get G + = G + 0 + βω 2 G + ω + βmµ 2 G + m + βω 2 2 G + ω 2 + βmµ 2 2 G + m 2 + βω 2 βmµ 2 G + ωm + ...dG + 0 dχ \| r + = 0 G + 0 \| rc = 1 (4.23a) d dχ G + ω n ,m l = 0 G + ω n ,m l \| rc = 0 ∀n, l > 0.G + 0 = 1 . (4.25) Collecting O( βω 2 ) terms in the scalar e.o.m we get ∂ χ r∂ χ G + ω + ∂ χ r = 0 . (4.26) We integrate the above equation and impose the two boundary conditions to get G + ω = 1 iπ (1 − µ 2 ) − 1 2(µ + 1) log r − r − r c − r − + 1 2(µ − 1) log r + r − r c + r − − 1 µ 2 − 1 log r + r + r c + r + . (4.27) Since we are derivative expanding the ingoing solution that is analytic in the radial variable, G + ω is also analytic between r + and r c . To find G + m , the required equation at O( βmµ 2 ) that we need to solve is µ∂ χ r∂ χ G + m − ∂ χ r + r − r = 0 . (4.28) Solving the above equation and imposing the boundary conditions we obtain G + m = 1 iπµ − (1 − µ) 2 log r − r − r c − r − + (µ + 1) 2 log r + r − r c + r − − µ log r + r + r c + r + . (4.29) G + m is analytic between r + and r c as well for the same reason. Given the derivative expansion till linear order for the ingoing green's function, we obtain the full green's function by adding the ingoing and outgoing green's functions where the outgoing part is just a frequency and momentum reversed counter part of the derivative expanded ingoing solution. For computational ease we will collect the even and odd parts of the full Green's function under frequency and momentum reversal and conveniently define the following 'even-odd' basis where G even = G + (ω, r, m) + G + (−ω, r, −m) 2 G odd = G + (ω, r, m) − G + (−ω, r, −m) 2 . (4.30) In terms of this basis the full scalar solution can be written as Φ = G even φ even + G odd φ odd (4.31) with φ even = −φ F + e β(ω(1−χ)−mµ(1−Θ)) φ P (4.32a) = φ a − e β(ω(1−χ)−mµ(1−Θ)) − 1 e β(ω−mµΘ) − 1 − 1 2 φ d (4.32b) = 1 − e −β(ωχ−mµΘ) (f ω,m + 1) φ R + e β(ω(1−χ)−mµ(1−Θ)) − 1 f ω,m φ L , (4.32c) φ odd = −φ F − e β(ω(1−χ)−mµ(1−Θ)) φ P (4.32d) = φ a + e β(ω(1−χ)−mµ(1−Θ)) + 1 e β(ω−mµΘ) − 1 + 1 2 φ d (4.32e) = 1 + e −β(ωχ−mµΘ) (f ω,m + 1) φ R − e β(ω(1−χ)−mµ(1−Θ)) + 1 f ω,m φ L . (4.32f) and we have normalised Θ in terms ofΘ whereΘ = Θ µ . The full solution becomes non-analytic due to the presence of χ and Θ in the outgoing solution. The full solution upto second order in derivatives can be written as Φ = 1 + βω 2 2 G + ω 2 + βmµ 2 2 G + m 2 + βω 2 βmµ 2 G + ωm φ even + G + ω βω 2 + G + m βmµ 2 φ odd + higher order terms . (4.33) Now we obtain an important relation using equations (2.14), (2.16), (4.27) and (4.29) that we explicitly use in the computation of quartic influence functional. The relation is given by Substituting the derivative expansion for φ even the quartic influence functional in frequency domain becomes G + ω + χ = −G + m +Θ .S quartic = 4 i=1 dω 2π 1 2π 4 m 1 ,m 2 ,m 3 ,m 4 {G aaaa φ a (ω 1 , m 1 ) φ a (ω 2 , m 2 ) φ a (ω 3 , m 3 ) φ a (ω 4 , m 4 ) G aaad φ a (ω 1 , m 1 ) φ a (ω 2 , m 2 ) φ a (ω 3 , m 3 ) φ d (ω 4 , m 4 ) +G aadd φ a (ω 1 , m 1 ) φ a (ω 2 , m 2 ) φ d (ω 3 , m 3 ) φ d (ω 4 , m 4 ) G addd φ a (ω 1 , m 1 ) φ d (ω 2 , m 2 ) φ d (ω 3 , m 3 ) φ d (ω 4 , m 4 ) +G dddd φ d (ω 1 , m 1 ) φ d (ω 2 , m 2 ) φ d (ω 3 , m 3 ) φ d (ω 4 , m 4 )} . (4.35) The coefficient functions that enter in front of various terms in the quartic on-shell action are given by the following radial integrals over complexified radial coordinate. Note that we have performed the v and φ integrals in the following expressions that result in total energy and momentum conserving delta functions. The coefficient functions till first order in derivative expansion are G aaaa = λ 4! (2π)δ i ω i (2π)δ i m i dχ √ −g = 0 (4.36a) G aaad = λ 3! (2π)δ i ω i (2π)δ i m i dχ √ −g X − β (ω 4 − µm 4 ) 2 X 2 − 1 4 (4.36b) G aadd = λ 4 (2π)δ i ω i (2π)δ i m i dχ √ −g X 2 − β (ω 4 − µm 4 ) X 3 − X 4 (4.36c) G addd = λ 3! (2π)δ i ω i (2π)δ i m i dχ √ −g X 3 + β(ω 4 − µm 4 ) 2 X 4 − X 2 4 (4.36d) G dddd = λ 4! (2π)δ i ω i (2π)δ i m i dχ √ −gX 4 (4.36e) where X inside the integrals is independent of ω and m and is defined as X = G + ω + χ − 1 2 . (4.37) We have used the relation (4.34) in obtaining the above integral expressions for the coefficient functions. G aaaa = 0 since the radial integral is completely analytic. This is a consequence of microscopic unitarity or equivalently a consequence of Schwinger-Keldysh collapse rule that states that under collapse of two consecutive legs of the SK contour, the effective action vanishes. Evaluation of quartic integrals: To evaluate the above integrals we note that the complexified contour integral can be broken into two parts as given below dχ = r + rc dr dr dχ + rc r + dr dr dχ (4.38) where the integrand picks up a monodromy traversing across the branch point at r = r + . This monodromy is correctly incorporated by the following substitution in the integrand as one goes across the branch point χ − 1 2 → χ + 1 2 (4.39a) Θ − 1 2 →Θ + 1 2 . (4.39b) Since there are no other poles of the integrand at r = r + , the above substitution in the integrand gives the value of the radial integration. We evaluate the integrals in the slow rotation limit as a series in µ. We will only keep terms that are linear order in µ. The 'bare' values of the integrals till linear order in small rotation and linear order in derivative expansion are the following G aaaa = 0 (4.40a) G aaad = λ 3! (2π)δ i ω i (2π)δ i m i r 2 c 2 − r 2 + 2 + β 2π (ω 4 − µm 4 ) log r c r + + O(µ 2 ) (4.40b) G aadd = λ 4 (2π)δ i ω i (2π)δ i m i i π r 2 + log r c r + − β 8 r 2 + (ω 4 − µm 4 ) + O(µ 2 ) (4.40c) G addd = λ 3! (2π)δ i ω i (2π)δ i m i r 2 c 8 − r 2 + 4 + iβ 8π (ω 4 − µm 4 ) 6 π 2 r 2 + ζ[3] − r 2 + log r c r + + O(µ 2 ) G dddd = λ 4! (2π)δ i ω i (2π)δ i m i −ir 2 + 3ζ[3] 2π 3 + i r 2 + 2π log r c r + + O(µ 2 ) . (4.40d) The values of the integrals are divergent. Hence we need to introduce counter terms to kill these divergences and get finite answers. We do a minimal subtraction where we only kill the divergent pieces and keep the finite part as it is. From the above values of quartic influence functional, one clearly sees divergences that are both of polynomial type and logarithmic type in r c . The polynomial divergences can be removed by adding local counter terms to the theory. However, the logarithmic divergences can not be removed by local counter terms since the argument of the logarithm has dependence on both r c and r + . It turns out that, to renormalise the influence phase in the massless case, one has to do an additional source renormalisation [25] that removes the logarithmic divergences. In the massive case one only encounters polynomial divergences that can be removed by local counter terms, hence source renormalisation is not needed. In the following we discuss the divergence structure of the massless integrals only till linear order in small µ and linear order in derivative expansion. We will then find the counter terms (till linear order in small µ and linear order in derivative expansion), that cancel the divergences of the bare integrals. The details of the source renormalisation is discussed in section 4.4 when we consider arbitrary higher order terms in the effective action. Note that the divergence of the influence phase comes from integrals of the form [25] F k = dχ √ −g X + G ω − 1 2 k . (4.41) The divergences of F k for k = 1, 2, 3, 4 till linear order in µ are of the forms F 1 ∼ r 2 c 2 (4.42a) F 2 ∼ ir 2 + π ln r c r + (4.42b) F 3 ∼ r 2 c 8 (4.42c) F 4 ∼ ir 2 + 2π ln r c r + . (4.42d) We add the following counter terms (till linear order in µ) to regularise the integrals δ 1 = − r 2 c 2 (4.43a) δ 2 = − ir 2 + π ln r c r + (4.43b) δ 3 = − r 2 c 8 (4.43c) δ 4 = − ir 2 + 2π ln r c r + . (4.43d) Here δ 1 and δ 2 are local counter terms. However δ 2 and δ 4 are not. The origin of δ 2 and δ 4 lies within the source renormalisation scheme that we discuss in section 4.4. General n-th order effective theory In this section we extend our study of effective theory to general n-point self interaction of the scalar field. We will compute the n-th order influence functional following the same recipe discussed in the previous section. The coefficient functions appearing in front of various terms of the effective action are expressed in terms of integrals over the complexified radial contour. We will now consider the more complicated massive scalar case for our analysis since we will express the coefficient functions in the effective theory only as radial integrals. The action of the scalar field is given by S = − dχdvdφ √ −g 1 2 ∂ A Φg AB ∂ B Φ + M 2 Φ 2 + λ n n! Φ n . (4.44) The influence phase is again given by the on-shell action. Like quartic on-shell action, the n-point on-shell action becomes S n−point = − dχdvdφ √ −g λ n n! Φ n 0 (4.45) where Φ 0 satisfies the homogeneous wave equation 1 √ −g ∂ A √ −gg AB ∂ B Φ 0 − M 2 Φ 0 = 0 . (4.46) The derivation of this equation is exactly same as given in the previous section. We will write down the influence phase in position space upto first order in derivatives. Let us start with an expression for the Φ 0 in position space written in average-difference/Keldysh basis Φ 0 = G + 0 1 + iβ 2 G + ω ∂ v − µG + m ∂ φ φ b a + iβ 8 ∂ v − µ∂ φ φ b d + φ b d χ +G + ω − 1 2 − iβ 2 ∂ v − µ∂ φ φ b d χ +G + ω − 1 2 2 + higher order terms . (4.47) The superscript 'b' in the above expression denotes that the sources at infinity are bare sources. We substitute equation (4.47) in equation (4.45) to obtain Φ n 0 n! = (G + 0 ) n n k=0 φ b a + iβ 8 ∂ v − µ∂ φ φ b d n−k (n − k)! φ b d X − iβ 2 ∂ v − µ∂ φ φ b d X 2 k k! + ∂ t iβ 2(n!) (G + 0 ) nG+ ω φ b a + φ b d X n − µ∂ φ iβ 2(n!) (G + 0 ) nG+ m φ b a + φ b d X (4.48) where X =G + ω + χ − 1 2 (4.49) andG + ω = G + ω G + 0 for a massive scalar field. Since the last two terms in equation (4.48) are total derivatives, they do not contribute in the influence phase. Expanding the terms inside the sum upto first order in derivatives we get Φ n 0 n! = (G + 0 ) n n k=0 (φ b a ) n−k + (n − k)(φ b a ) n−k−1 iβ 8 ∂ v − µ∂ φ φ b d (n − k)! (φ b d ) k X k − k(φ b d ) k−1 X k+1 iβ 2 ∂ v − µ∂ φ φ b d k! (4.50a) = (G + 0 ) n n k=0 (φ b a ) n−k (φ b d ) k (n − k)!k! X k + n−1 k=0 (φ b a ) n−k−1 (φ b d ) k (n − k − 1)!k! X k iβ 8 ∂ v − µ∂ φ φ b d − n k=1 (φ b a ) n−k (φ b d ) k−1 (n − k)!(k − 1)! X k+1 iβ 2 ∂ v − µ∂ φ φ b d + higher order . (4.50b) Hence the bare influence functional becomes 4 S b = −λ n dtdφ n k=1 (φ b a ) n−k (φ b d ) k (n − k)!k! F b n,k + n−1 k=1 (φ b a ) n−k (φ b d ) k−1 (n − k)!(k − 1)! F b n,k−1 iβ 8 (∂ t − µ∂ φ ) φ b d − n−1 k=1 (φ b a ) n−k (φ b d ) k−1 (n − k)!(k − 1)! F b n,k+1 iβ 2 (∂ t − µ∂ φ ) φ b d (4.51) where F b n,k = rc r + dr r (G + 0 ) n χ +G + ω + 1 2 k − χ +G + ω − 1 2 k . (4.52) We have also used the fact that k = 0 term in the second sum in equation (4.50b) vanishes after integrating over the holographic Schwinger Keldysh contour since the integrand is analytic. k = n term in the third sum in this equation also does not contribute since that term is just a total derivative. The bare influence functional that is computed with the bare sources as boundary conditions is divergent. The divergence structure can be understood for massive case by looking at the asymptotics of the solution as discussed in appendix B. We cancel the divergences employing an appropriate renormalisation scheme as discussed in the next section. Renormalizing the n-th order effective theory The function F b n,k in equation (4.52) is divergent. The divergence structures are discussed below. Massive case: The divergence coming from X 2k+1 is F b n,2k+1 = F r n,2k+1 + Λ ∆ 4 k , Λ ∆ = r 2 + ζ c 1 − µ 2 (ν−1)nζ c 2(ν − 1)n + 2 Massless case: The divergence coming from X 2k+1 is F b n,2k+1 = F r n,2k+1 + δ 4 k (4.54) and for X 2k it is F b n,2k = F r n,2k + k η 4 k−1 (4.55) where the divergences are δ = r 2 c 2 , η = i(r 2 + − r 2 − ) π ln r c r + . (4.56) For the massive case we add local counter terms in the bare influence phase to cancel the polynomial divergences and obtain a finite influence phase. However the massless case is more tricky, since along with polynomial divergences we also get non local logarithmic divergences that can not be removed by adding local state independent counter terms in the bare influence phase. It turns out that if we renormalise the sources at the cutoff in a particular way such that the bare and renormalised sources agree when the cutoff is taken to infinity, the logarithmic divergences get cancelled and the renormalised effective action becomes finite [25]. This source renormalisation prescription for the massless case is given by the following temperature dependent dressing of sources where the difference source mixes with the average source φ b a = φ r a − η 2δ φ r d , φ b d = φ r d + η 2δ iβ (∂ t − µ∂ φ ) φ r d . (4.57) Note that as we take r c → ∞, lim rc→∞ η δ = 0, i.e., the bare and the renormalised sources match at asymptotic infinity. We consider the following counter term expressed in terms of state independent renormalised sources S c.t = λ n dtdφ δ n! φ r a + φ r d 2 n − φ r a − φ r d 2 n . (4.58) Then the statement is that when we add this to the bare influence phase the renormalised influence functional is free of nonlocal logarithmic divergences. The renormalised influence functional becomes S r = lim rc→∞ S b + S c.t (4.59) = −λ n dtdφ n k=1 (φ r a ) n−k (φ r d ) k (n − k)!k! F r n,k + n−1 k=1 (φ r a ) n−k (φ r d ) k−1 (n − k)!(k − 1)! F r n,k−1 iβ 8 (∂ t − µ∂ φ ) φ r d − n−1 k=1 (φ r a ) n−k (φ r d ) k−1 (n − k)!(k − 1)! F r n,k+1 iβ 2 (∂ t − µ∂ φ ) φ r d . (4.60) -22 - Nonlinear Langevin dynamics in rotating plasma The effective theory of the open scalar field has a description in terms of a nonlinear Langevin equation with non-Gaussianities in the thermal fluctuations. In the following we discuss the dual stochastic description of the effective dynamics. Deriving effective action via stochastic path integral We start with an ansatz for the stochastic dynamics governed by the non-linear Langevin equation in presence of rotation for the average field ψ a as follows (for non-rotating case see [25]) E[ψ a , η] ≡ f η (5.1) E[ψ a , η] ≡ −K∂ 2 t ψ a + D∂ 2 φ ψ a + F ∂ t ∂ φ ψ a + γ (∂ t − µ∂ φ ) ψ a +μψ a + n−1 k=1 θ k η k−1 k! ψ n−k a (n − k)! +θ k η k−1 k! ψ n−k−1 a (n − k − 1)! (∂ t − µ∂ φ ) ψ a . (5.2) The stochastic variable η is the thermal noise, f is the strength of the noise, Z, X and Y introduce colors in the thermal noise, γ is the damping constant, θ k andθ k correspond to the thermal jitter on top of the damping constant and anharmonicities in the dynamics of the open scalar field. We consider the most general probability distribution upto two derivatives acting on the stochastic noise η. We take the distribution of the noise to be P[η] = N e − dtdφ( f 2! η 2 + Z 2! (∂tη) 2 + X 2! (∂ φ η) 2 + Y 2! ∂tη∂ φ η+ θn n! η n ) (5.3) where N is a normalisation factor and θ n is the non-Gaussianity in the noise. In the following we obtain an effective action from a stochastic path integral that we eventually compare with the influence functional obtained from holographic computations. This will determine the parameters of the non-linear Langevin equation in terms of the effective couplings computed from holography. To compute the effective action from stochastic path integral we will exploit Martin-Siggia-Rose (MSR) [44][45][46] trick. For this we will have to introduce an auxiliary field ψ d and convert following identity 1 = Dψ a Dη δ (E[ψ a , η] − f η) P[η] (5.4) to the identity 1 = Dψ a Dψ d Dηe −i d 2 x(E[ψa,η]−f η)ψ d P[η] . (5.5) Now we integrate over the noise η to get an effective action in terms of ψ a and ψ d . This can be done by shifting η → η + iψ d and taking η → 0. This will give the leading order Schwinger-Keldysh effective action with non-zero rotation S ψ = − dtdφ −Kψ d ∂ 2 t ψ a + Dψ d ∂ 2 φ ψ a + F ψ d ∂ t ∂ φ ψ a +μψ a ψ d + γ ψ d (∂ t − µ∂ φ ) ψ a − if 2! ψ 2 d + iZ 2! (∂ t ψ d ) 2 + iX 2! (∂ φ ψ d ) 2 + iY 2! ∂ t ψ d ∂ φ ψ d −i n k=1 θ k (iψ d ) k k! ψ n−k a (n − k)! − i n−1 k=1θ k (iψ d ) k k! ψ n−k−1 a (n − k − 1)! (∂ t − µ∂ φ ) ψ a . (5.6) Comparing with holographic result In this section we will compare the effective action derived from stochastic path integral to the effective action computed using holography. This way we evaluate the parameters in the nonlinear Langevin equation in terms of the effective couplings computed from holography. The way to do this is by uplifting the sources φ a and φ d that appear in the influence functional to stochastic fields that we denote by ψ a and ψ d . Let us begin by writing down the exact quadratic influence functional S quadratic = N 2 dtdφ i ψ 2 d β + ih 1,1 ∂ t ψ d ∂ φ ψ d + ih 0,2 (∂ φ ψ d ) 2 + ih 2,0 (∂ t ψ d ) 2 +g 0,0 ψ a ψ d − (∂ t ψ a − µ∂ φ ψ a ) ψ d + g 0,2 ∂ φ ψ a ∂ φ ψ d + g 2,0 ∂ t ψ a ∂ t ψ d + g 1,1 ∂ t ψ a ∂ φ ψ d } (5.7) where N 2 = − (r 2 + − r 2 − ) ∆−2 Γ ∆ 2 4 r + Γ(∆ − 1) 2 . (5.8) The exact coefficient functions computed from holography are h 0,2 = β π 2 µ 2 − (1 + µ 2 )ψ 1 ∆ 2 8π 2 (5.9a) h 2,0 = β π 2 − (1 + µ 2 )ψ 1 ∆ 2 8π 2 (5.9b) h 1,1 = βµ π 2 − 2ψ 1 ∆ 2 4π 2 (5.9c) g 0,0 = 2 tan π∆ 2 β (5.9d) g 0,2 = − β π 2 (−1 + µ 2 cos(π∆)) csc(π∆) + (1 + µ 2 )ψ 1 ∆ 2 tan π∆ 2 4π 2 (5.9e) g 2,0 = − β π 2 (−µ 2 + cos(π∆)) csc(π∆) + (1 + µ 2 )ψ 1 ∆ 2 tan π∆ 2 4π 2 (5.9f) g 1,1 = βµ π 2 − 2ψ 1 ∆ 2 tan π∆ 2 2π 2 (5.9g) where ψ 1 is the Polygamma function of order one. The n-th order influence functional computed from holography is given by S r Φ n = −λ n dtdφ n k=1 (ψ r a ) n−k (ψ r d ) k (n − k)!k! F r n,k − iβ 2 n−1 k=1 (ψ r a ) n−k (ψ r d ) k−1 (n − k)!(k − 1)! (∂ t − µ∂ φ ) ψ r d F r n,k+1 − 1 4 F r n,k−1 . (5.10) -24 -Combining both the quadratic and n-th order influence functional we finally have S = dtdφ N 2 i ψ 2 d β + ih 1,1 ∂ t ψ d ∂ φ ψ d + ih 0,2 (∂ φ ψ d ) 2 + ih 2,0 (∂ t ψ d ) 2 + g 0,0 ψ a ψ d − (∂ t ψ a − µ∂ φ ψ a ) ψ d + g 0,2 ∂ φ ψ a ∂ φ ψ d + g 2,0 ∂ t ψ a ∂ t ψ d + g 1,1 ∂ t ψ a ∂ φ ψ d } −λ n n k=1 (ψ r a ) n−k (ψ r d ) k (n − k)!k! F r n,k + λ n iβ 2 n−1 k=1 (ψ r a ) n−k (ψ r d ) k−1 (n − k)!(k − 1)! (∂ t − µ∂ φ ) ψ r d F r n,K = −N 2 g 2,0 D = N 2 g 0,2 (5.12a) F = N 2 g 1,1μ = −N 2 g 0,0 (5.12b) γ = N 2 f = 2N 2 β (5.12c) Z = −2N 2 h 2,0 X = −2N 2 h 0,2 (5.12d) Y = −2N 2 h 1,1 . (5.12e) Comparing the values of the parameters we see that the damping constant is related to the fluctuation as below f = 2γ β . (5.13) This is the fluctuation dissipation relation between noise and dissipation in the linear Langevin equation with rotation. The extension to non-linear Langevin equation has additional parameters like jitter in the damping, anharmonicity in the dynamics and non-Gaussianity in the thermal noise. Those additional parameters are θ k andθ k . From holographic computations we find θ k = λ n i k+1 F r n,k (5.14a) θ k = βλ n 2i k+2 F r n,k+1 − 1 4 F r n,k−1 (5.14b) where F r n,k are radial integrals that enter the effective couplings. From these we get the generalised fluctuation dissipation relation given by 2 βθ k = θ k+1 + 1 4 θ k−1 . (5.15) We have determined F r n,k for the quartic order effective theory by performing the radial integrals analytically in the slow rotation limit till linear order in rotation. In this case we find that generalised fluctuation dissipation relation given in equation (5.15) holds true for linear order in slow rotation. Conclusion and Discussion In this paper, we consider a natural extension of the gravitational Schwinger-Keldysh path integral prescription of [15] to rotating BTZ black holes. The gravitational space-time asymptotes to the real-time Schwinger-Keldysh contour of the dual rotating CFT at a given initial state with finite temperature and chemical potential due to angular momentum. We study a probe scalar field in the rotating BTZ geometry and obtain the ingoing quasi normal modes and outgoing (time-reversed) Hawking radiation in section 3. By computing the on-shell action and integrating over the complexified radial coordinate we construct the effective theory of an open scalar field that is coupled to a two-dimensional rotating thermal CFT plasma at the boundary in section 4. The quadratic influence functionals computed in section 4.1 match with the results of [4] under appropriate coordinate transformations and redefinition of parameters. The quadratic effective theory has a dual stochastic description in terms of a linear Langevin equation in presence of rotation. The noise and dissipation terms in the Langevin equation are related by the fluctuation-dissipation relation with chemical potential. We determine the higher order terms in the effective theory by derivative expanding the free scalar solution in low frequency and angular momentum. The coefficient functions that appear in front of the higher order terms are the effective couplings evaluated from holography. We determine these coefficient functions of higher order terms in the form of integrals over the bulk contour in section 4.3. In the limit when the CFT plasma has low angular momentum i.e. the dual BTZ black hole is slowly rotating, we compute the quartic order integrals analytically till first order in slow rotation (similar thing can be done for cubic effective theory also) in section 4.2. We find the correct renormalisation scheme to remove the divergences appearing in the values of the bare integrals in section 4.4. The higher order effective theory has a description in terms of a non-linear Langevin dynamics with non-Gaussianity in the thermal fluctuations as obtained in section 5. We write down the generalisation of the non-linear fluctuation dissipation relation in presence of chemical potential (see [25] for the non-rotating case). It will be useful to do the integrals that appear as the coefficient functions in the effective theory numerically. Employing a systematic holographic renormalisation scheme as given in [14], will provide a better handle on the divergences coming from evaluating the integrals in our analysis. As a followup to our work, it will be interesting to generalise the prescription to higher-dimensional rotating black holes. Using derivative expansion in low frequency and angular momenta one can obtain the effective theory holographically. It will also be interesting in higher dimension to add a Chern-Simons term in the bulk since such a term in the bulk will lead to an anomaly on the even dimensional rotating boundary CFT. Chern-Simons term in the presence of rotation is expected to give rise to an anomaly induced transport. It will also be quite interesting to extend the holographic path integral contour to nearextremal black holes and make connection with the derivative expansion scheme given in [47] for near-extremal black branes. Since ω 1 ,ω 2 and m 1 ,m 2 are dummy variables, we will interchange them in the last four terms S on−shell = − dω 1 dω 2 (2π) 2 1 (2π) 2 m 1 ,m 2 (2π)δ(ω 1 + ω 2 )(2π)δ m 1 +m 2 ,0 −G + (ω 1 , r, m 1 ) φ F (ω 1 , m 1 ) φ P (ω 2 , m 2 ) e β(ω 2 (1−χ)−m 2 (µ + −Θ)) r 2 (iω 2 )G + (−ω 2 , r, −m 2 ) − ir β ∂ χ G + (−ω 2 , r, −m 2 ) − n φ r 2 (im 2 )G + (−ω 2 , r, −m 2 ) −G + (−ω 2 , r, −m 2 )φ P (ω 2 , m 2 )e β(ω 2 (1−χ)−m 2 (µ + −Θ)) φ F (ω 1 , m 1 ) r 2 (−iω 1 )G + (ω 1 , r, m 1 ) − ir β ∂ χ G + (ω 1 , r, m 1 ) + n φ r 2 (im 1 )G + (ω 1 , r, m 1 ) χ=1,Θ=µ + and other terms in the integral are not contributing because they are analytic. B Derivative expansion of massive scalar solution The massive scalar field equation is given by The general solution to this ∂ χ r∂ χ G + + βω 2 r∂ χ G + + ∂ χ rG + + βm 2 2 f r G + − βm 2 2n φ (r)r∂ χ G + − βm 2 ∂ χ (rn φ (r)) G + + β 2 4 f rM 2 G + = 0 .c 1 P −ν 2ζ − 1 − µ 2 1 − µ 2 + c 2 Q −ν 2ζ − 1 − µ 2 1 − µ 2 . (B.5) Now imposing the boundary condition dG + 0 dχ = 0 at r + we remove the second solution. Normalizing the solution to one at the boundary fixes c 1 . Finally we obtain G + 0 = P −ν 2ζ−1−µ 2 1−µ 2 P −ν 2ζc−1−µ 2 1−µ 2 (B.6) whereζ c denotes the valueζ at the UV cut-off. Now we move to the first order calculation: G + ω = G + ω G + 0 satisfies ∂ζ (ζ − µ 2 )(ζ − 1)(G + 0 ) 2 ∂ζG + ω = i(1 − µ 2 ) 2π ∂ζ (G + 0 )= (1 − µ 2 ) 2πi ζ ζc dζ ζ − 1 2 (ζ − µ 2 )(ζ − 1) − (G + 0 [1]) 2 (ζ − µ 2 )(ζ − 1)(G + 0 [ζ ]) 2 . (B.11b) In fact one can give a general recursive solution toG ω l ,m k in terms of lower order functions G + ω l ,m k = ζ ζc 1 (ζ − µ 2 )(ζ − 1)(G + 0 ) 2 ζ 1 dζ 1 − µ 2 2πi 2ζ − 1 2 µ∂ζ G + ω l ,m k−1 + ∂ζ (ζ − 1 2 µ)G + ω l ,m k−1 −2ζ 1 2 ∂ζ G + ω l−1 ,m k − G + ω l−1 ,m k + (1 − µ 2 ) 2 (2π) 2 G + ω l ,m k−2 ζ G + 0 [ζ ] . (B.12) Here we assumed that G + ω l ,m k with l < 0 or k < 0 is zero. With this one can check that the general formula reproduces the special cases. We also note an important relation that can be used to compute the massive higher order influence functionals G + ω + χ = −G + m +Θ . (B.13) This relation can be derived by using the following relations χ = −i(1 − µ 2 ) 2π ζ ζc dζ ζ 1 2 (ζ − µ 2 )(ζ − 1) (B.14a) Θ = (1 − µ 2 ) 2πi ζ ζc dζ ζ − 1 2 (ζ − µ 2 )(ζ − 1) (B.14b) G + ω = i(1 − µ 2 ) 2π ζ ζc dζ ζ 1 2 (ζ − µ 2 )(ζ − 1) − (G + 0 [1]) 2 (ζ − µ 2 )(ζ − 1)(G + 0 [ζ ]) 2 (B.14c) G + m = (1 − µ 2 ) 2πi ζ ζc dζ ζ − 1 2 (ζ − µ 2 )(ζ − 1) − (G + 0 [1]) 2 (ζ − µ 2 )(ζ − 1)(G + 0 [ζ ]) 2 . (B.14d) Let us define x = 2ζ−1−µ 2 1−µ 2 . Then one can write the sum in the LHS or RHS of (B.13) as follows G + ω + χ = 1 iπ x xc 1 (x 2 − 1)P −ν (x) 2 = i π Q −ν (x) P −ν (x) + i cot(πν) . (B.15) Using the derivative expanded solution for the massive case, let us analyse the radial integrals as given in (4.52). For this, first let us note the asymptotic behaviour of G + 0 at largeζ P −ν 2ζ − 1 − µ 2 1 − µ 2 −−−→ ζ→∞ζ ν−1 1 − µ 2 . (B.16) Using (B.16), we obtain the following asymptotic expression for the radial integrals. For odd k = 2l + 1 from equation (4.52), we get Hence this is divergent for ν > 1 − 1 n . For even k = 2l, using (B.16) and (B.15), we get the following divergence This integral is thus divergent when ν > 1 for n ≥ 3. However, for ν < 1 that corresponds to relevant operator in the dual CFT, there is no divergence for even k. Figure 1 : 1Schwinger-Keldysh imaginary time contour with an initial thermal state. Figure 2 : 2Picture of bulk at a fixed v: red line is a branch cut from horizon to boundary tortoise coordinate ζ dr dζ Fig 3 . 3 rFigure 3 : 3= r c + iε, χ = 0, Θ = 0 r = r c − iε, χ = 1, Θ = µ + Picture of bulk at a fixed v: red line is branch cut from horizon to boundary Figure 4 : 4SK contour with an initial state at finite temperature and chemical potential. will take r c → ∞ at the end of our computations. At the leading order in derivative expansion i.e. at O(βω) 0 and O(βmµ) 0 the massless scalar e.o.m gives ∂ χ r∂ χ G + 0 = 0 . (4.24) holds true even for the massive case. See appendix B for the derivation of the relation in massive case. Quartic Influence-functional: Equipped with the derivative expanded full solution given in equation (4.33), we now proceed to compute the quartic influence functional by substituting the solution in equation (4.21). conformal dimension ν = ∆ 2 . Hence this is divergent for all ν. For X 2k i.e. for F b n,2k there is no divergence for ν < 1 that are relevant deformations in the CFT (see equation (B.18) of appendix B). equation (5.11) with equation (5.6) we obtain the parameters of the non-linear Langevin equation in terms of the effective couplings computed using holography. The values of the parameters of the linear Langevin equation are F energy and momentum delta functions, gives the influence phase in the R-A basisS on−shell = − dω 2π 1 2π m 2φ F (ω, m) φ P (−ω, −m) e −β(ω(1−χ)−m(µ + −Θ)) ir β G + (ω, r, m) ∂ χ G + (ω, r, m) + β 2 (ω − n φ m)G + (ω, (ω, m)φ P (−ω, −m)G F P [ω, m] . (A.4b) where G F P [ω, m] = − 2ir β G + (ω, r c , m) ∂ χ G + (ω, r c , m) + β 2 (ω − mn φ )G + (ω, r c , m) (1 − e −β(ω−µ + m) ) (A.5) the derivative expansion given in equation(4.22) in the above equation and solve the equation order by order in βω and βmµ. Zeroth order solution is obtained by solving ∂ r rf ∂ r G + 0 + rM 2 the boundary condition satisfied byG + are d dχG + ω n ,m l = 0G + ω n ,m l \| rc = 0 ∀n, l > 0. In[4], the authors computed quadratic effective theory and two point functions of the scalar field. The massless case is simpler since the scalar wave equation is simpler and solving it order by order in perturbation parameter is easier. 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[ "Laneformer: Object-aware Row-Column Transformers for Lane Detection", "Laneformer: Object-aware Row-Column Transformers for Lane Detection" ]	[ "Jianhua Han \nShenzhen Campus of Sun Yat-sen University\n\n", "Xiajun Deng \nShenzhen Campus of Sun Yat-sen University\n\n", "Xinyue Cai \nShenzhen Campus of Sun Yat-sen University\n\n", "Zhen Yang \nShenzhen Campus of Sun Yat-sen University\n\n", "Hang Xu \nShenzhen Campus of Sun Yat-sen University\n\n", "Chunjing Xu \nShenzhen Campus of Sun Yat-sen University\n\n", "Xiaodan Liang \nShenzhen Campus of Sun Yat-sen University\n\n", "Huawei Noah's Ark \nShenzhen Campus of Sun Yat-sen University\n\n", "Lab \nShenzhen Campus of Sun Yat-sen University\n\n" ]	[ "Shenzhen Campus of Sun Yat-sen University\n", "Shenzhen Campus of Sun Yat-sen University\n", "Shenzhen Campus of Sun Yat-sen University\n", "Shenzhen Campus of Sun Yat-sen University\n", "Shenzhen Campus of Sun Yat-sen University\n", "Shenzhen Campus of Sun Yat-sen University\n", "Shenzhen Campus of Sun Yat-sen University\n", "Shenzhen Campus of Sun Yat-sen University\n", "Shenzhen Campus of Sun Yat-sen University\n" ]	[]	We present Laneformer, a conceptually simple yet powerful transformer-based architecture tailored for lane detection that is a long-standing research topic for visual perception in autonomous driving. The dominant paradigms rely on purely CNN-based architectures which often fail in incorporating relations of long-range lane points and global contexts induced by surrounding objects (e.g., pedestrians, vehicles). Inspired by recent advances of the transformer encoder-decoder architecture in various vision tasks, we move forwards to design a new end-to-end Laneformer architecture that revolutionizes the conventional transformers into better capturing the shape and semantic characteristics of lanes, with minimal overhead in latency. First, coupling with deformable pixel-wise selfattention in the encoder, Laneformer presents two new row and column self-attention operations to efficiently mine point context along with the lane shapes. Second, motivated by the appearing objects would affect the decision of predicting lane segments, Laneformer further includes the detected object instances as extra inputs of multi-head attention blocks in the encoder and decoder to facilitate the lane point detection by sensing semantic contexts. Specifically, the bounding box locations of objects are added into Key module to provide interaction with each pixel and query while the ROI-aligned features are inserted into Value module. Extensive experiments demonstrate our Laneformer achieves state-of-the-art performances on CULane benchmark, in terms of 77.1% F1 score. We hope our simple and effective Laneformer will serve as a strong baseline for future research in self-attention models for lane detection.	10.1609/aaai.v36i1.19961	[ "https://arxiv.org/pdf/2203.09830v1.pdf" ]	247,594,201	2203.09830	74db14977f55a01cb11b5e5a9367669acc46570d	Laneformer: Object-aware Row-Column Transformers for Lane Detection Jianhua Han Shenzhen Campus of Sun Yat-sen University Xiajun Deng Shenzhen Campus of Sun Yat-sen University Xinyue Cai Shenzhen Campus of Sun Yat-sen University Zhen Yang Shenzhen Campus of Sun Yat-sen University Hang Xu Shenzhen Campus of Sun Yat-sen University Chunjing Xu Shenzhen Campus of Sun Yat-sen University Xiaodan Liang Shenzhen Campus of Sun Yat-sen University Huawei Noah's Ark Shenzhen Campus of Sun Yat-sen University Lab Shenzhen Campus of Sun Yat-sen University Laneformer: Object-aware Row-Column Transformers for Lane Detection We present Laneformer, a conceptually simple yet powerful transformer-based architecture tailored for lane detection that is a long-standing research topic for visual perception in autonomous driving. The dominant paradigms rely on purely CNN-based architectures which often fail in incorporating relations of long-range lane points and global contexts induced by surrounding objects (e.g., pedestrians, vehicles). Inspired by recent advances of the transformer encoder-decoder architecture in various vision tasks, we move forwards to design a new end-to-end Laneformer architecture that revolutionizes the conventional transformers into better capturing the shape and semantic characteristics of lanes, with minimal overhead in latency. First, coupling with deformable pixel-wise selfattention in the encoder, Laneformer presents two new row and column self-attention operations to efficiently mine point context along with the lane shapes. Second, motivated by the appearing objects would affect the decision of predicting lane segments, Laneformer further includes the detected object instances as extra inputs of multi-head attention blocks in the encoder and decoder to facilitate the lane point detection by sensing semantic contexts. Specifically, the bounding box locations of objects are added into Key module to provide interaction with each pixel and query while the ROI-aligned features are inserted into Value module. Extensive experiments demonstrate our Laneformer achieves state-of-the-art performances on CULane benchmark, in terms of 77.1% F1 score. We hope our simple and effective Laneformer will serve as a strong baseline for future research in self-attention models for lane detection. Introduction Lane detection has been a long-standing task for visual perception in autonomous driving scenarios, targeting at precisely distinguishing lane segments from complicated road scenes (Hou et al. 2019;Wang, Shen, and Teoh 2000;Narote et al. 2018). It plays a crucial role towards a safe and reliable auto-driving system widely used in intelligent cars to assist drivers with modern technologies such as adaptive cruise control, lane departure warning, and traffic understanding. The most state-of-the-art lane detection methods (Lee et al. 2021;Xu et al. 2020;Chen, Liu, and Lian 2019) Figure 1: Sketch map of proposed detection attention and row-column attention in Laneformer. Given the detected person and vehicle instances, detection attention is performed to capture the implicit relationship between them and lanes, e.g., lanes are more likely to appear next to cars. Row attention is proposed to catch the information from the nearby rows since pixels in the same lane will not be far from each other between adjacent rows. On the other hand, noticed that different columns may across different lanes, the knowledge sharing of these columns may capture different lane features to construct better representations. take advantage of CNN architectures and exceed the traditional methods (Niu et al. 2016;Narote et al. 2018;Wang, Shen, and Teoh 2000) by a large margin. However, the existing CNN-based methods usually need complicated postprocessing procedures like non-maximal suppression or clustering. In addition, the fixed receptive field of CNN architecture limits the ability to incorporate relations for longrange lane points, making them hard to capture well the characteristics of lanes since the shapes are conceptually long and thin. Several attention-based lane detection models (Lee et al. 2021;Tabelini et al. 2020b;Liu et al. 2021) have been also proposed to capture the long-range information. Nevertheless, the fixed attention routines can not adap-tively fit the shape characteristic of lanes. Besides, complicated road scenes including different light, weather conditions and occlusions of surrounding objects further require the model with a stronger perception of global contexts. In addition to the point-wise context information, it is reasonable to assume that objects on the road (e.g., pedestrians, vehicles) have some implicit relations with surrounding lanes. As the key sub-module (e.g., person-vehicle detection) of autonomous driving systems usually co-exist with lane detection, the sub-module outputs may promote the performance of lane detection and improves the system safety. However, none of the existing methods has considered incorporating semantics induced by detection results from a single and unified network view. On the other hand, Transformer (Vaswani et al. 2017), a kind of encoder-decoder architecture, has shown surprisingly promising ability in dealing with tasks that require to capture global relations in nature language processing (Devlin et al. 2018;Radford et al. 2018;Young et al. 2018) and vision tasks such as image classification(Dosovitskiy et al. 2020), object detection (Carion et al. 2020;Zhu et al. 2020) and image segmentation (Zheng et al. 2020;Wang et al. 2020). Particularly, (Liu et al. 2021) proposed an LSTR model to predict lanes as polynomial functions using a transformer. Despite its benefit in providing rich global contexts in predicting lanes, the definition that regards lanes as polynomial functions has many drawbacks: 1) LSTR needs to formulate camera intrinsic and extrinsic parameters, which hinders the model from transferring to other datasets or train with combined datasets; 2) LSTR still lacks of explicitly modeling global semantic contexts into facilitating lane detection. To tackle the above-mentioned issues, we move forwards to design a new Laneformer architecture to better capture the shape characteristics and global semantic contexts of lanes using transformers (shown in Figure 1). Our Laneformer defines lanes as a series of points. Then, the lanespecific row-column attentions are proposed to efficiently mine point context along with the lane shapes. Concretely, we take each feature row as a token and perform row-to-row self-attention and do the same for each feature column to perform column-to-column self-attention. Moreover, motivated by the appearing objects that would affect lanes' predictions, Laneformer further includes the detected object instances as auxiliary inputs of multi-head attention blocks in encoder and decoder to facilitate the lane point detection by sensing semantic contexts. To be specific, the bounding box locations of objects are added into the Key module to provide interaction with each pixel and query. At the same time, the ROI-aligned features are inserted into the Value module to provide detection information. Considering that bounding box locations and ROI-aligned features contain limited information about object instances, we use confident scores and predicted categories of the detected outputs to further improve the model's performance. Bipartite matching is adopted to ensure one-to-one alignment between predictions and ground truths, which makes the Laneformer architecture eliminate additional post-processing procedures. Extensive experiments conducted on CULane (Pan et al. 2018 These approaches typically generate segmentation results with an encoder-decoder structure and then post-processing them via curve fitting and clustering. However, pixel-wise lane prediction methods usually require more computation and are also limited to the pre-defined number of lanes. On the other hand, several works (Chen, Liu, and Lian 2019;Li et al. 2019;Xu et al. 2020) follow traditional proposal-based diagrams by generating multiple point anchors and then predicting the relative distance between each lane point and the anchor point. However, these existing CNN-based methods usually need complicated post-processing procedures like non-maximal suppression or clustering. Besides, the fixed receptive field of CNN architecture limits the ability to incorporate relations for long-range lane points. Therefore, our Laneformer utilizes a transformer to capture context information and bipartite matching to eliminate additional postprocessing procedures. Attention-based methods. Several attention-based lane detection models (Lee et al. 2021;Tabelini et al. 2020b;Liu et al. 2021) have been proposed to capture the long-range information. (Lee et al. 2021) propose a self-attention mechanism to predict the lanes' confidence along with the vertical and horizontal directions in an image. (Tabelini et al. 2020b) proposes a novel anchor-based attention mechanism that aggregates global information named LaneATT. However, the Laneformer Lane Representation Similar to (Chen, Liu, and Lian 2019), we define lanes as a series of 2D-points that can adapt to all kinds of lanes. Specially, we formulate a lane as l = (X, s, e), where X stands for corresponding x-coordinate set for 72 equally-spaced ycoordinates and s, e denote for the start y-coordinate and end y-coordinate of the lane. Architecture The overall architecture of Laneformer is demonstrated in Figure 2. It mainly consists of four modules: a CNN backbone to extract basic features, a detection processing module to handle outputs from person-vehicle detection module, a specially designed encoder and a decoder for lane detection. Given an RGB image as input, our model first extracts backbone features with a ResNet (He et al. 2016) backbone. We further get row features and column features by collapsing the column dimension and row dimension of backbone features, respectively. At the same time, detected bounding boxes and their predicted scores and categories from the input image are acquired through a trained personvehicle detection module. We use the bounding box locations to crop ROI-aligned features from backbone features mentioned above, followed by a 1-layer perceptron with ReLU activation to transform the ROI-aligned features to one-dimensional embeddings. Similarly, another 1-layer perceptron is applied on the detected bounding boxes to get one-dimensional bounding box embeddings. In the encoder, row attention and column attention are performed on row features and column features respectively. Meanwhile, backbone features will perform self-attention and pixel-to-Bbox attention with bounding box embeddings and ROI embeddings. The outputs of row-column attention and pixel-to-Bbox attention are added up as the memory input for the following decoder. In the decoder, learnable queries first perform self-attention to get query features. Then the query features together with the input memory will perform cross attention. Meanwhile, with the bounding box embeddings and ROI embeddings, query-to-Bbox attention can be applied. Finally, outputs of the cross attention and query-to-Bbox attention are added up, and several feed-forward layers are utilized to predict lane points. Detection Processing After the acquisition of detected bounding boxes, predicted scores and predicted categories from a trained detector based on common Faster-RCNN (Ren et al. 2015) architecture, we propose a simple detection processing module to process them in order to better utilize the detection information. First, we crop ROI-aligned features from backbone features H f ∈ R h×w×d for the bounding boxes of the detected objects, where h, w and d are the spatial sizes and the corresponding dimension of backbone features. Since the features we used are down-sampled, we need to rescale the bounding box locations by a specific ratio to crop out correct fea- stands for the addition and H f /Q denotes that the input can be either backbone feature H f in the encoder or query feature Q in the decoder. ture areas. Considering that objects with higher confidence scores may supply more robust information while objects with lower scores may be noisy, we also use predicted scores as weight coefficients to multiply the ROI-aligned features and get weighted ROI-aligned features for each object. After that, we pass through weighted ROI-aligned features into a 1-layer perceptron with output channel d followed by ReLu activation to get final ROI embeddings Z r ∈ R M ×d . M denotes the number of used detected bounding boxes. If the number of detected bounding boxes is less than M , bounding boxes with random locations, categories and zero scores will be padded. On the other hand, bounding boxes with lower scores will be excluded if the number of detected bounding boxes is more than M . In the meantime, with the prior that category information can help distinguish different objects, we further concatenate the predicted categories after bounding boxes. Specifically, we use a four-dimensional vector to represent a bounding box and a one-hot vector with length 7 (1 for padded box and 6 for categories) to represent the corresponding category, which will result in an 11-dimensional vector after concatenation. Similar to ROI features, a one-layer perceptron with ReLu activation is applied to get bounding box embeddings Z b ∈ R M ×d . Finally, Z r and Z b are sent to preform Pixelto-Bbox attention in encoder and Query-to-Bbox attention in decoder. Row-Column Attention In the encoder of Laneformer, besides the traditional attention, we further propose the new row-column attention to efficiently mine point context along with the lane shapes. The row-column attention consists of row attention that captures the relations between rows and column attention that mine the relations between columns. As we all know, the traffic lane is a kind of object with unique shape characteristics. From the vertical view, lanes are long and thin, which means pixels in the same lane will not be far from each other between adjacent rows. From the horizontal view, since dif-ferent columns may across different lanes, the knowledge sharing of these columns may capture different lane features to construct better representations. Given row features H r ∈ R h×1×wd , where h, w and d are the spatial sizes and the the corresponding dimension of backbone features, we pass it through a linear transformation to reduce the last dimension to d , which we denoted as H r ∈ R h×1×d . Sinusoidal embeddings E p is calculated according to the absolute positions to supply location information. Self-attention operations are applied with the inputs of H r and E p . Similarly, with the dimension-reduced column features H c ∈ R 1×w×d and its position embeddings, self-attention are performed between columns. Finally, the outputs of row and column attention are added up as the memory input for decoder. Details are shown in Figure 3(a). Detection Attention Apart from row-column attention, in both the encoder and decoder of Laneformer, we propose detection attention to mine the valuable information from detected surrounding objects. It is straight-forward that lanes and objects on the road (e.g., pedestrians, vehicles) have some implicit relations with each other. For example, lanes are more likely to appear next to vehicles, while pedestrians are not supposed to walk between lanes. Besides, object detection module such as person-vehicle detection is always co-existed with lane detection in an auto-driving system, thus detected results can be acquired easily. The detection attention module consists of two parts: 1) a Pixel-to-Bbox attention module in encoder to excavate the relevance between each pixel token of feature map and each detected objects; 2) a Query-to-Bbox attention module in decoder to find out which object should be paid more attention in order to help predict corresponding lanes. Details are illustrated in Figure 3(b). Pixel-to-Bbox Attention In Laneformer encoder, we propose pixel-to-Bbox attention. Pixel-to-Bbox attention is designed for digging out relations between feature pixels and detected objects. In pixel-to-Bbox attention, each pixel in dimension-reduced backbone features H f is considered as a query token. Detection information including bounding box embeddings Z b and ROI embeddings Z r make up the Key module and Value module, respectively. The pixel-to-Bbox attention is defined as follows: O p2b = sof tmax( H f Z b T √ d ) · Z r(1) Pixel-to-Bbox attention forces the model to learn which detected object a pixel should pay attention to so that the model can catch helpful context features. The output of pixel-to-Bbox attention O p2b are added up with the above row-column attention output and act as the memory input for the decoder. Query-to-Bbox Attention In Laneformer decoder, we propose query-to-Bbox attention. Query-to-Bbox attention on the other hand designed to mine relations between queries and detected objects. Query-to-Bbox attention is similar to Pixel-to-Bbox attention, while in here, queries are the learned embeddings Q with the size of N × d , where N is the number of learned embeddings. Bounding box embeddings Z b and ROI embeddings Z r still act as Key module and Value module here. Similarly, the query-to-Bbox attention is defined as follows: O q2b = sof tmax( QZ T b √ d ) · Z r(2) This attention enables each query to focus on the instances near the lane it needs to predict. The output of query-to-Bbox attention O q2b will be added up with the traditional cross-attention output, followed by several feed-forward features to get predicted lanes. Loss Construction Bipartite matching. After obtaining features from abovementioned modules, Laneformer predicts a number of N lane set according to its number of queries, where N is set to be significantly larger than the maxinum number of lanes in the dataset. Therefore we need to pad the ground truth with non-lane to be a set of N objects first, denoted as G = {g n \|g n = (c n , l n )} N n=1 , where c n ∈ {0, 1}, 0 represents non-lane and 1 represents lane. Given predicted outputs as P = {p n \|p n = (p n ,l n )} N n=1 , wherep n (c n ) stands for the probability score thatl n belongs to the specific category c n . We search for a permutation of N elements matching index δ to minimize the pair-wise distance function D between ground-truth lane g n and predicted lane p δ(n) : δ = arg min δ N n=1 D(g n , p δ(n) ). ( 3) The difference function D is defined as following: D(gn, p δ(n) ) = −ω 1pδ(n) (cn)+1(cn = 1)L loc (ln,l δ(n) ) (4) Where 1( * ) denotes an indicator function and ω 1 stands for the coefficient of classification term. L loc is defined as follows: L loc (l n ,l δ(n) ) = ω 2 L 1 (X n ,X δ(n) ) + ω 3 L 1 (s n ,ŝ δ(n) ) + ω 4 L 1 (e n ,ê δ(n) ) where L 1 denotes the mean absolute error and ω 2 , ω 3 , ω 4 indicate for the coefficients for point, start position and end position term respectively. Bipartite matching is adopted to ensure one-to-one alignment between predictions and ground truths, making the Laneformer an end-to-end architecture by eliminating additional post-processing procedures. Total Loss The total loss of our model is calculated with matching index δ gained from bipartite matching, consisting of negative log-likelihood loss for classification prediction and L1-based location loss: L total = N n=1 −ω 1 logp δ(n) (cn)+1(cn = 1)L loc (ln,l δ(n) ) (6) Where L loc is calculated the same with Eq.(5) and (n, δ(n)) is the optimal pair indexes that minimize Eq.(3). ω 1 , ω 2 , ω 3 , ω 4 also adjust the effect of the loss terms and are set as same values with Eq.(4) and Eq.(5). Experiment Datasets and Evaluation Metrics We conduct experiments on the two most popular lane detection benchmarks. T P +F P and Recall = T P T P +F N . As for Tusimple, standard evaluation metrics including Accuracy, false positives(FP) and false negatives(FN) are adopted. Implementation Details The input resolution is set to 820 × 295 for CULane and 640 × 360 for TuSimple. Data augmentations are applied on the raw image, consisting of horizontal flipping, a random choice from color-shifting operations(e.g., gaussian blur, linear contrast) and position-shifting operations(e.g., cropping, rotate). Most of our experiments use ResNet50 as the backbone. We follow the setting of ) and utilize a deformable transformer as the plain transformer. The bipartite matching and loss term coefficients ω 1 , ω 2 , ω 3 and ω 4 are set as 2, 10, 10, 10, respectively. Both the number of encoder and decoder layers is set to 1. Moreover, we adopt 25 as the number of queries N and 10 as the number of used detected bounding boxes M . Eight V100s are used to train the model and the batch size is set to be 64. The learning rate is set to 1e-4 for the backbone and 1e-5 for the transformer. We train 100 epochs on CULane and drop the learning rate by ten at 80 epoch. On Tusimple, the total number of training iterations is set to 28k and the learning rate drops at 22k iteration. During inference, the single scale test is adopted with the score threshold set to 0.8. The trained detector used to obtain person-vehicle bounding boxes is based on common Faster-RCNN (Ren et al. 2015) architecture with ResNet-50 backbone and trained 12 epochs on BDD100K dataset (Yu et al. 2018) with 70k images. Main Results CULane. Table 1 shows the Laneformer's performance on CULane test set. Our Laneformer achieves state-of-the-art results on F1 of the total test split. In addition, our Lanformer with only ResNet-50 backbone even surpasses the results of LaneATT (Tabelini et al. 2020b) with a larger ResNet-122 backbone. Additionally, Laneformer outperforms all other lane detection models on some challenging splits such as "Night", "Dazzle light" and "Cross". Among the above three splits, Laneformer significantly improves the performance of "Cross" category, which achieves an extremely low FP(False Positive) number 19. We observe that our model gets a result of 1104 FP on "Cross" category without detection attention, while with the detection attention, the performance makes a dramatic improvement as in Table 1. The promotion may come from the perception of vehicle-person global context on the particular crossroad scenario benefiting from the proposed detection attention layer. TuSimple. Experimental results on TuSimple benchmark are summarized in Table 2. We achieve 96.8% accuracy, 5.6% FP and 1.99% FN with ResNet-50 backbone. It is shown that Laneformer achieves comparable accuracy with the state-of-the-art Line-CNN and 0.6% higher than another transformer-based method LSTR. Even with the smaller backbones (ResNet-18, ResNet-34), Laneformer can achieve a competitive accuracy compared with the stateof-the-art methods. Note that adding detection attention on Tusimple doesn't improve much due to the relatively simple highway driving scenes (e.g., few cars and straight lines). Table 3. After adding row-column attention and detection attention, there is only a 4.9%, 8.1% increment on inference FPS due to the efficient matrix multiplication. Visualization. We visualize several attention maps in the transformer to find out the area that detection attentions and row-column attentions focus on, where brighter color denotes for more significant attention value. Shown in Figure 4(a), either the point or query pays more attention to the detected instances besides lanes it responsible for, especially when those instances are in occlusion with part of the lane. Moreover, observation can be made in Figure 4(b) that row attention mainly considers nearby rows, while the column attention focuses on the nearby representative column of each lane. These results demonstrate our assumption that the implicit relationship of traffic scenes can be obtained from proposed detection attention and row-column attention. Ablation Study Different components. As we can see in Table 3, without detection attention and row-column attention, our baseline (plain transformer) obtains 75.45% on F1. The adding of row-column attention improves the performance to 76.04%. What's more, simply introducing detected objects information can improve the performance of our model, but a little more extra information such as scores or categories will make it better. We can observe that making full use of detected objects with their scores and categories raises the F1 to 77.06%, which is the state-of-the-art results on CULane. Score threshold of the bounding box. To find out the influences of detected objects with different confidence, we conduct a series of experiments with different score thresholds on filtering detected bounding boxes. Table 4 shows that using detection outputs from different score thresholds has a slight impact on our results. Especially, results between threshold 0.6 to 0.8 are robust and the differences can be nearly neglect. When the threshold is lower than 0.6, there may be some noise in detected bounding boxes and therefore our performance gets slighted hurt. On the other hand, if the threshold is too high such as 0.9, only a few bounding boxes will be chosen so that a large number of padded boxes will interfere with the model's learning, which leads to a lower F1. Besides, we experiment on random bounding boxes to prove that Laneformer indeed makes use of the information of detected objects. Number of bounding box. Apart from thresholds, we further explore the impact of using a different number of detected objects in Laneformer. The results in Table 4 show that too many detected objects lead to a performance drop, and our model reaches the best result under the setting of 10 bounding boxes. We speculate that the best setting of 10 bounding boxes is due to the average number of detected objects in each image under the bounding box threshold of 0.6, which is 9.84. So if we set a number much larger than 9.84, for example, 20 in our experiment, then too many useless padded boxes will be used, which may hurt the model's performance. On the other hand, if we use too few bounding boxes, the information of detected objects is not entirely utilized to reach the best performance. Different categories. Vehicles and persons have different relations with lanes. To be specific, vehicles are on the road, beside the lanes, while persons usually far from lanes. So we also explore the impact of extra information with different categories of bounding boxes. Results in Table 4 show that both the adding of vehicles and persons can improve the model performance, and the result with vehicles is better than the one with persons. We suppose that vehicles share a more close relation with lanes in the perspective of locations. Experiments show that the model with all of the two categories reaches the best results. Conclusions In this work, we propose Laneformer, a conceptually simple yet powerful transformer-based architecture tailored for lane detection. Equipped with row and column self-attention module and semantic contexts provided by additional detected object instances, Laneformer achieves the state-ofthe-art performance, in terms of 77.1% F1 score on CULane and superior 96.8% Accuracy on TuSimple benchmark. Besides, visualization of the learned attention map in transformer demonstrates that our Laneformer can incorporate relations of long-range lane points and global contexts induced by surrounding objects. Layers of Encoder and Decoder It can be seen in Table.5, the different number of encoder layers and decoder layers have a slight difference in performances, which proves that our Laneformer is robust. We can observe that the increment of encoder layers and decoder layers can improve the performance on Recall metric. Threshold and Bounding Box Number As we have mentioned in our paper, the number of bounding boxes and the score threshold of bounding boxes have a relationship that they will influence the choice of each other. We do another two experiments on threshold 0.4 and 0.9 to verify this influence. First, we count the average objects number in each image under threshold 0.4 and 0.9, which is 11 objects and 7 objects, respectively. Then, we conduct experiments in two settings. Concretely, we use 11 bounding boxes for 0.4 score threshold and 7 bounding boxes for 0.9 score threshold. As we can see in Table.6, using the proper number of bounding boxes under corresponding threshold can clearly improve the performances. Different Detectors To find out the influence of person-vehicle detectors with different accuracy, we also conduct experiments on different Faster-RCNNs trained on 10k, 30k, 70k images chosen from BDD100K dataset. Random bounding box setting is added for comparison. Results in Table.7 demonstrate that our Laneformer can work well with detectors of different accuracy. Moreover, with the improvement of detector's accuracy, the performance of Laneformer also improves. ROI Sizes We can observe from Table.7 that the bigger the extracted ROI features, the lower of F1 score. We speculate that we extract ROI features from C4 layer of backbone features, which is of size 52 × 19 that is very small, so ROI features of size 3 × 3 is enough. If the size goes too big, the extracted feature needs to be interpolated to the specified size, which may introduce fuzzy information. ROI Features from Different Layers Our Laneformer extracts multi-stage features from the backbone, so we have choices to extract ROI features from backbone feature of different layers. We try for C3, C4 and C5 layers. Table.7 shows that we get our best performance on ROI features extracted from C4 layer. While features extracted from C5 layer show a significant performance drop due to the feature size is too small that it loss too much information. free methods. Before the advent of deep learning, traditional lane detection methods are usually based on hand-crafted low-level features (Chiu and Lin 2005; Lee and Cho 2009; Gonzalez and Ozguner 2000). CNN architecture has then been adopted to extract advanced features in an end-to-end manner. Most lane detection methods follow pixel-level segmentation-based approach (Pizzati et al. 2019; Hou 2019; Mamidala et al. 2019; Zou et al. 2019). Figure 2 : 2Overall Architecture. In Laneformer, backbone extracts backbone features H f , row features H r and column features H c . Detection Processing module generates bounding box embeddings Z b and ROI embeddings Z r with detected bounding boxes and H f . In the encoder, row attention and column attention are performed on H r and H c , added up with the pixelto-Bbox attention performed on H f , Z b and Z r as memory input for decoder. In the decoder, traditional cross-attention and query-to-Bbox attention are performed and outputs are added up for the feed-forward layers to predict lanes. fixed attention routines can not adaptively fit the shape characteristic of lanes. Based on PolyLaneNet(Tabelini et al. 2020a), LSTR(Liu et al. 2021) is proposed to output polynomial parameters of a lane shape function by using a network built with a transformer to learn richer structures and context. Unlike LSTR, which assumes all lanes are parallel on the road and formulates it as a polynomial shape prediction problem, our Laneformer directly outputs each lane's points to adapt to more complex lane detection scenarios. Besides, detected object instances are further included as auxiliary inputs of multi-head attention blocks in Laneformer to facilitate the lane point detection by sensing semantic contexts. Figure 3 : 3Detailed operations of the row-column attention and detection attention. Figure 4 : 4Visualization of proposed attentions on Tusimple dataset, where brighter color denotes more significant attention value. (a) shows the detection attention map for different points and queries in the transformer. We can observe either the pixel or the query focus on the detection bounding boxes near the corresponding lane. (b) shows that the row attention mainly considers the nearby rows, while the column attention focuses on the nearby representative column of each lane. ( a ) aLane detection results on TuSimple dataset. (b) Lane detection results on CULane dataset. Figure 5 : 5Visualization of Laneformer results on TuSimple and CULane dataset. Table 1 : 1Comparison of F1-measure(%) and MACs(multiply-accumulate operations) on CULane testing set, where Laneformer* denotes for Laneformer without detection attention module. Our Laneformer achieves state-of-the-art performance.Methods Normal Crowded Dazzle Shadow No line Arrow Curve Night Cross Total MACs (G) SCNN(Pan et al. 2018) 90.60 69.70 58.50 66.90 43.40 84.10 64.40 66.10 1990 71.60 / ENet-SAD(Hou et al. 2019) 90.10 68.80 60.20 65.90 41.60 84.00 65.70 66.00 1998 70.80 / PointLane(Chen, Liu, and Lian 2019) 88.00 68.10 61.50 63.30 44.00 80.90 65.20 63.20 1640 70.20 / ERFNet-HESA(Lee et al. 2021) 92.00 73.10 63.80 75.00 45.00 88.20 67.90 69.20 2028 74.20 / CurveLane-S(Xu et al. 2020) 88.30 68.60 63.20 68.00 47.90 82.50 66.00 66.20 2817 71.40 9.0 CurveLane-M(Xu et al. 2020) 90.20 70.50 65.90 69.30 48.80 85.70 67.50 68.20 2359 73.50 33.7 CurveLane-L(Xu et al. 2020) 90.70 72.30 67.70 70.10 49.40 85.80 68.40 68.90 1746 74.80 86.5 LaneATT(ResNet-18)(Tabelini et al. 2020b) 91.17 72.71 65.82 68.03 49.13 87.82 63.75 68.58 1020 75.13 9.3 LaneATT(ResNet-34)(Tabelini et al. 2020b) 92.14 75.03 66.47 78.15 49.39 88.38 67.72 70.72 1330 76.68 18.0 LaneATT(ResNet-122)(Tabelini et al. 2020b) 91.74 76.16 69.47 76.31 50.46 86.29 64.05 70.81 1264 77.02 70.5 Laneformer(ResNet-50)* 91.55 74.76 69.27 69.59 48.13 86.99 68.15 70.06 1104 76.04 26.2 Laneformer(ResNet-18) 88.60 69.02 64.07 65.02 45.00 81.55 60.46 64.76 25 71.71 13.8 Laneformer(ResNet-34) 90.74 72.31 69.12 71.57 47.37 85.07 65.90 67.77 26 74.70 23.0 Laneformer(ResNet-50) 91.77 75.41 70.17 75.75 48.73 87.65 66.33 71.04 19 77.06 26.2 Table 2 : 2Comparison of different algorithms on the Tusim- ple testing benchmark, where Laneformer* denotes for Laneformer without detection attention module. Method Acc(%) FP(%) FN(%) SCNN (Pan et al. 2018) 96.53 6.17 1.80 LSTR (Liu et al. 2021) 96.18 2.91 3.38 Enet-SAD (Hou et al. 2019) 96.64 6.02 2.05 Line-CNN (Li et al. 2019) 96.87 4.41 3.36 PolyLaneNet (Tabelini et al. 2020a) 93.36 9.42 9.33 PointLaneNet (Chen, Liu, and Lian 2019) 96.34 4.67 5.18 LaneATT (ResNet-18) (Tabelini et al. 2020b) 95.57 3.56 3.01 LaneATT (ResNet-34) (Tabelini et al. 2020b) 95.63 3.53 2.92 LaneATT (ResNet-122) (Tabelini et al. 2020b) 96.10 5.64 2.17 Laneformer(ResNet-50)* 96.72 3.46 2.52 Laneformer(ResNet-18) 96.54 4.35 2.36 Laneformer(ResNet-34) 96.56 5.39 3.37 Laneformer(ResNet-50) 96.80 5.60 1.99 Latency Comparison. In inference, Laneformer with ResNet-50 backbone achieves 53 and 48 FPS on one V100 for CULane and Tusimple benchmark. For latency compari- son of different components in Laneformer, we conduct ex- periments on CULane testing split and illustrate the result in Table 3 : 3Ablation study results of different components of Laneformer on CULane.Model F1(%) Precision(%) Recall(%) FPS Params. (M) Baseline(ResNet-50) 75.45 81.65 70.11 61 31.02 + row-column attention 76.04 82.92 70.22 58 43.02 + bounding box 76.08 85.30 68.66 57 45.38 + score 76.25 83.56 70.12 54 45.38 + category 77.06 84.05 71.14 53 45.38 Table 4 : 4Quantitative evaluation of different detection bounding box input settings on CULane testing split.F1(%) Precision(%) Recall(%) Score threshold 0.4 76.34 85.41 69.01 0.5 76.14 84.52 69.27 0.6 77.06 84.05 71.14 0.7 76.71 84.53 70.21 0.8 76.85 84.51 70.46 0.9 76.37 85.54 68.98 random 75.99 84.50 69.03 Number of Bbox 5 76.90 84.30 70.69 10 77.06 84.05 71.14 20 76.51 84.34 70.01 Different categories none 76.04 82.92 70.22 person 76.40 85.17 69.27 vehicle 76.79 84.44 70.41 all 77.06 84.05 71.14 Table 5 : 5Experiments on different numbers of encoder and decoder layers on CULane.Encoder Decoder F1(%) Precision(%) Recall(%) 1 1 77.06 84.05 71.14 2 2 76.48 82.49 71.28 3 3 76.91 82.59 71.97 1 2 76.90 82.92 71.70 1 3 76.71 82.95 71.35 2 1 76.07 80.13 72.39 3 1 76.13 81.87 71.14 Table 6 : 6Experiments on different score threshold and bounding box number on CULane.Threshold Number F1(%) Precision(%) Recall(%) 0.4 10 76.34 85.41 69.01 11 76.77 84.88 70.08 0.9 10 76.37 85.54 68.98 7 76.46 84.61 69.74 Table 7 : 7Experiments on different detectors, ROI sizes and different layers on CULane.F1(%) Precision(%) Recall(%) Different Detectors random 75.99 84.50 69.03 10k 76.71 85.36 69.65 30k 76.96 85.10 70.25 70k 77.06 84.05 71.14 ROI size 3 × 3 77.06 84.05 71.14 5 × 5 76.06 83.80 69.62 7 × 7 76.05 80.72 71.88 Different Layers C3 76.47 85.76 69.00 C4 77.06 84.05 71.14 C5 75.94 80.75 71.67 AppendixWe conduct more ablation studies to confirm the effectiveness of Laneformer. 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[ "Change Point Detection for High-dimensional Linear Models: A General Tail-adaptive Approach", "Change Point Detection for High-dimensional Linear Models: A General Tail-adaptive Approach" ]	[ "Bin Liu \nDepartment of Statistics and Data Science\nSchool of Management at\nFudan University\nChina\n", "Zhengling Qi [email protected] \nDepartment of Decision Sciences\nGeorge Washington University\nU.S.A\n", "Xinsheng Zhang [email protected] \nDepartment of Statistics and Data Science\nSchool of Management at\nFudan University\nChina\n", "Yufeng Liu [email protected] \nDepartment of Statistics and Operations Research\nDepartment of Genetics\nDepartment of Bio-statistics, Carolina Center for Genome Sciences, Linberger Comprehensive Cancer Center\nUniversity of North Carolina at Chapel Hill\nU.S.A\n" ]	[ "Department of Statistics and Data Science\nSchool of Management at\nFudan University\nChina", "Department of Decision Sciences\nGeorge Washington University\nU.S.A", "Department of Statistics and Data Science\nSchool of Management at\nFudan University\nChina", "Department of Statistics and Operations Research\nDepartment of Genetics\nDepartment of Bio-statistics, Carolina Center for Genome Sciences, Linberger Comprehensive Cancer Center\nUniversity of North Carolina at Chapel Hill\nU.S.A" ]	[]	We study the change point detection problem for high-dimensional linear regression models. The existing literature mainly focused on the change point estimation with stringent sub-Gaussian assumptions on the errors. In practice, however, there is no prior knowledge about the existence of a change point or the tail structures of errors. To address these issues, in this paper, we propose a novel tail-adaptive approach for simultaneous change point testing and estimation. The method is built on a new loss function which is a weighted combination between the composite quantile and least squared losses, allowing us to borrow information of the possible change points from both the conditional mean and quantiles. For the change point testing, based on the adjusted L 2 -norm aggregation of a weighted score CUSUM process, we propose a family of individual testing statistics with different weights to account for the unknown tail structures. Combining the individual tests, a tail-adaptive test is further constructed that is powerful for sparse alternatives of regression coefficients' changes under various tail structures. For the change point estimation, a family of argmax-based individual estimators is proposed once a change point is detected. In theory, for both individual and tail-adaptive tests, the bootstrap procedures are proposed to approximate their limiting null distributions. Under some mild conditions, we justify the validity of the new tests in terms of size and power under the highdimensional setup. The corresponding change point estimators are shown to be rate optimal up to a logarithm factor. Moreover, combined with the wild binary segmentation technique, a new algorithm is proposed to detect multiple change points in a tail-adaptive manner. Extensive numerical results are conducted to illustrate the appealing performance of the proposed method.	null	[ "https://export.arxiv.org/pdf/2207.11532v2.pdf" ]	251,040,305	2207.11532	e76ea8b07574eb84ded39b6560d30e1f9f863922	Change Point Detection for High-dimensional Linear Models: A General Tail-adaptive Approach 24 May 2023 Bin Liu Department of Statistics and Data Science School of Management at Fudan University China Zhengling Qi [email protected] Department of Decision Sciences George Washington University U.S.A Xinsheng Zhang [email protected] Department of Statistics and Data Science School of Management at Fudan University China Yufeng Liu [email protected] Department of Statistics and Operations Research Department of Genetics Department of Bio-statistics, Carolina Center for Genome Sciences, Linberger Comprehensive Cancer Center University of North Carolina at Chapel Hill U.S.A Change Point Detection for High-dimensional Linear Models: A General Tail-adaptive Approach 24 May 20231 arXiv:2207.11532v2 [stat.ME]Efficient computationHeavy tailHigh dimensionsStructure change We study the change point detection problem for high-dimensional linear regression models. The existing literature mainly focused on the change point estimation with stringent sub-Gaussian assumptions on the errors. In practice, however, there is no prior knowledge about the existence of a change point or the tail structures of errors. To address these issues, in this paper, we propose a novel tail-adaptive approach for simultaneous change point testing and estimation. The method is built on a new loss function which is a weighted combination between the composite quantile and least squared losses, allowing us to borrow information of the possible change points from both the conditional mean and quantiles. For the change point testing, based on the adjusted L 2 -norm aggregation of a weighted score CUSUM process, we propose a family of individual testing statistics with different weights to account for the unknown tail structures. Combining the individual tests, a tail-adaptive test is further constructed that is powerful for sparse alternatives of regression coefficients' changes under various tail structures. For the change point estimation, a family of argmax-based individual estimators is proposed once a change point is detected. In theory, for both individual and tail-adaptive tests, the bootstrap procedures are proposed to approximate their limiting null distributions. Under some mild conditions, we justify the validity of the new tests in terms of size and power under the highdimensional setup. The corresponding change point estimators are shown to be rate optimal up to a logarithm factor. Moreover, combined with the wild binary segmentation technique, a new algorithm is proposed to detect multiple change points in a tail-adaptive manner. Extensive numerical results are conducted to illustrate the appealing performance of the proposed method. Introduction With the advances of data collection and storage capacity, large scale/high-dimensional data are ubiquitous in many scientific fields ranging from genomics, finance, to social science. Due to the complex data generation mechanism, the heterogeneity, also known as the structural break, has become a common phenomenon for high-dimensional data, where the underlying model of data generation changes and the identically distributed assumption may not hold anymore. Change point analysis is a powerful tool for handling structural changes since the seminal work by Page (1955). It received considerable attentions in recent years and has a lot of real applications in various fields including genomics (Liu et al., 2020), social science (Roy et al., 2017), financial contagion (Pesaran and Pick, 2007) in economy, and even for the recent COVID-19 pandemic studies (Jiang et al., 2021). Motivated by this, in this paper, we study the change point testing and estimation problem in the high-dimensional linear regression setting. Specifically, we are interested in the following model and detect possible change points: Y = X ⊤ β + ϵ, where Y ∈ R is the response variable, X = (X 1 , . . . , X p ) ∈ R p is the covariate vector, β = (β 1 , . . . , β p ) ⊤ is a p × 1 unknown vector of coefficients, and ϵ ∈ R is the error term. Suppose we have n independent but (time) ordered realizations {(Y i , X i ), i = 1, . . . , n} with X i = (X i1 , . . . , X ip ) ⊤ . For each time point i, consider the following regression model: Y i = X ⊤ i β i + ϵ i ,(1.1) where β i = (β i1 , . . . , β ip ) ⊤ is the regression coefficient vector for the i-th observation, and ϵ i is the error term. For the above model, our first question is whether there is a change point. This can be formulated as the following hypothesis testing problem: H 0 : β 1 = β 2 = · · · = β n v.s. H 1 : β (1) = β 1 = · · · = β k 1 ̸ = β k 1 +1 = · · · = β n := β (2) , (1.2) where k 1 is the possible but unknown change point location. In this article, we assume k 1 = ⌊nt 1 ⌋ with 0 < t 1 < 1. According to (1.2), the linear regression structure between Y and X remains homogeneous if H 0 holds, and otherwise there is a change point k 1 that divides the data into two segments with different regression coefficients β (1) and β (2) . Our second goal of this paper is to identify the change point location if we reject H 0 in (1.2). Note that the above two goals are referred as change point testing and estimation, respectively. In the practical use, both testing and estimation are important since practitioners typically have no prior knowledge about either the existence of a change point or its location. Therefore, it is very useful to consider simultaneous change point detection and estimation. Furthermore, the tail structure of the error ϵ i in Model (1.1) is typically unknown, which could significantly affect the performance of the change point detection and estimation. In the existing literature, the performance guarantee of most methods on change point estimation relies on the assumption that the error ϵ i follows a Gaussian/sub-Gaussian distribution. Such an assumption could be violated in practice when the data distribution is heavy-tailed or contaminated by outliers. While some robust methods can address these issues, they may lose efficiency when errors are indeed sub-Gaussian distributed. It is also very difficult to estimate the tail structures and construct a corresponding change point method based on that. Hence, it is of great interest to construct a tail-adaptive change point detection and estimation method for high-dimensional linear models. 3 Contribution Motivated by our previous discussion, in this paper, under the high-dimensional setup with p ≫ n, we propose a tail-adaptive procedure for simultaneous change point testing and estimation in linear regression models. The proposed method relies on a new loss function in our change point estimation procedure, which is a weighted combination between the composite quantile loss proposed in Zou and Yuan (2008) and the least squared loss with the weight α ∈ [0, 1] for balancing the efficiency and robustness. Thanks to this new loss function with different α, we are able to borrow information related to the possible change point from both the conditional mean and quantiles in Model (1.1). Therefore, besides controlling the type I error to any desirable level when H 0 holds, the proposed method simultaneously enjoys high power and accuracy for change point testing and identification across various underlying error distributions including both lighted and heavy-tailed errors when there exists a change point. By combining our single change point estimation method with the wild binary segmentation (WBS) technique (Fryzlewicz, 2014), we also generalize our method for detecting multiple change points in Model (1.1). In terms of our theoretical contribution, for each given α, a novel score-based R pdimensional individual CUSUM process {C α (t), t ∈ [0, 1]} is proposed. Based on this, we construct a family of individual-based testing statistics {T α , α ∈ [0, 1]} via aggregating C α (t) using the ℓ 2 -norm of its first s 0 largest order statistics, known as the (s 0 , 2)-norm proposed in Zhou et al. (2018). A high-dimensional bootstrap procedure is introduced to approximate T α 's limiting null distributions (See Algorithm 1). The proposed bootstrap method only requires mild conditions on the covariance structures of X and the underlying error distribution ϵ, and is free of tuning parameters and computationally efficient. the Lasso estimator for the corresponding weighted loss. This can be much more efficient than the existing works with O(nLasso(n, p)) operations in Lee et al. (2016Lee et al. ( , 2018 and O(2Lasso(n, p)) operations in Kaul et al. (2019). Related literature For the low dimensional setting with a fixed p and p < n, change point analysis for linear regression models has been well-studied. Specifically, Quandt (1958) considered testing (1.2) for a simple regression model with p = 2. Other extensions include the maximum likelihood ratio tests (Horváth, 1995), partial sums of regression residuals (Gombay and Horváth, 1994;Bai and Perron, 1998), and the union intersection test (Horváth and Shao, 1995 (2016); Zhang et al. (2015); Wang et al. (2021Wang et al. ( , 2022. However, none of the aforementioned papers develop hypothesis testing procedure, which is the prerequisite for the change point detec-tion. Furthermore, most existing literature requires strong conditions on the underlying errors ϵ i for deriving the desirable theoretical properties, which may not be applicable when the data are heavy-tailed or contaminated by outliers. One possible solution is to use the robust method such as median regression in Lee et al. (2018) for change point estimation. As discussed in Zou and Yuan (2008); Bradic et al. (2011);Zhao et al. (2014), when making statistical inference for homogeneous linear models, the asymptotic relative efficiency of median regression-based estimators compared to least squared-based is only about 64% in both low and high dimensions. In addition, inference based on quantile regression can have arbitrarily small relative efficiency compared to the least squared based regression. Our proposed tail-adaptive method is the first one that can perform simultaneous change point testing and estimation for high-dimensional linear regression models with different distributions. In addition to controlling the type I error to any desirable level, the proposed method enjoys simultaneously high power and accuracy for the change point testing and identification across various underlying error distributions when there exists a change point. Besides literature in the regression setting, change point analysis has also been carried out for the setting of high-dimensional mean vectors and some mean vector-based extensions. See Aue et al. (2009);Jirak (2015); Cho et al. (2016); Liu et al. (2020)) and many others. The rest of this paper is organized as follows. In Section 2, we introduce our new tail-adaptive methodology for detecting a single change point as well as multiple change points. In Section 3, we derive the theoretical results in terms of size and power as well as the change point estimation. In Section 4, we present our extensive simulation results. A real application to the S&P 100 dataset is shown in Section 5. The concluding remarks are provided in Section 6. Detailed proofs and the full numerical results are given in the online supplementary materials. Notations: We end this section by introducing some notations. For v = (v 1 , . . . , v p ) ⊤ ∈ R p , we define its ℓ p norm as ∥v∥ p = ( d j=1 \|v j \| p ) 1/p for 1 ≤ p ≤ ∞. For p = ∞, define ∥v∥ ∞ = max 1≤j≤d \|v j \|. For any set S, denote its cardinality by \|S\|. For two real numbered sequences a n and b n , we set a n = O(b n ) if there exits a constant C such that \|a n \| ≤ C\|b n \| for a sufficiently large n; a n = o(b n ) if a n /b n → 0 as n → ∞; a n ≍ b n if there exists constants 6 c and C such that c\|b n \| ≤ \|a n \| ≤ C\|b n \| for a sufficiently large n. For a sequence of random variables (r.v.s) {ξ 1 , ξ 2 , . . .}, we set ξ n P − → ξ if ξ n converges to ξ in probability as n → ∞. We also denote ξ n = o p (1) if ξ n P − → 0. For a n × p matrix X, denote X i and X j as its i-th row and j-th column respectively. We define ⌊x⌋ as the largest integer less than or equal to x for x ≥ 0. Denote (X , Y) = {(X 1 , Y 1 ), . . . , (X n , Y n )}. Methodology In Section 2.1, we construct a family of oracle individual testing statistics { T α , α ∈ [0, 1]} for single change point detection. To estimate the unknown variance in T α , in Section 2.2, a weighted variance estimation that is consistent under both H 0 and H 1 is proposed. With the estimated variance, a family of data-driven testing statistics {T α , α ∈ [0, 1]} is proposed for change point detection. To approximate the limiting null distribution of T α , we introduce a novel bootstrap procedure in Section 2.3. In practice, the tail structure of the error term is typically unknown. Hence, it is desirable to combine the individual testing statistics {T α , α ∈ [0, 1]} for yielding a powerful tail-adaptive method. To solve this problem, in Section 2.4, a tail-adaptive testing statistic T ad is proposed. Lastly, we combine our testing procedure with the WBS technique for detecting multiple change points. Single change point detection In this section, we introduce our new methodology for Problem (1.2). We first focus on detecting a single change point in Model (1.1). In this case, Model (1.1) reduces to: Y i = X ⊤ i β (1) 1{i ≤ k 1 } + X ⊤ i β (2) 1{i > k 1 } + ϵ i , for i = 1, . . . , n. (2.1) In this paper, we assume k 1 = ⌊nt 1 ⌋ for some constant t 1 ∈ (0, 1). Note that t 1 is called the relative change point location. We assume the change point does not occur at the begining or end of data observations. Specifically, suppose there exists a constant q 0 ∈ (0, 0.5) such that q 0 ≤ t 1 ≤ 1 − q 0 holds, which is a common assumption in the literature (e.g., Dette et al., 2018;Jirak, 2015). For Model (2.1), given X i , the conditional mean of Y i becomes: 7 E[Y i \| X i ] = X ⊤ i β (1) 1{i ≤ k 1 } + X ⊤ i β (2) 1{i > k 1 }. (2.2) Moreover, let 0 < τ 1 < . . . < τ K < 1 be K candidate quantile indices. For each τ k ∈ (0, 1), let b (0) k := inf{t : P(ϵ ≤ t) ≥ τ k } be the τ k -th theoretical quantile for the generic error term ϵ in Model (2.1). Then, conditional on X i , the τ k -th quantile for Y i becomes: Define the weighted composite loss function as: Q τ k (Y i \|X i ) = b (0) k + X ⊤ i β (1) 1{i ≤ k 1 } + X ⊤ i β (2) 1{i > k 1 }, k = 1, . . . , K, (2.3) where Q τ k (Y i \|X i ) := inf{t : P(Y i ≤ t\|X i ) ≥ τ k }.ℓ α (x, y; τ , b, β) := (1 − α) 1 K K k=1 ρ τ k (y − b k − x ⊤ β) + α 2 (y − x ⊤ β) 2 , (2.4) where ρ τ (t) := t(τ − 1{t ≤ 0}) is the check loss function (Koenker and Bassett Jr, 1978), τ := (τ 1 , . . . , τ K ) ⊤ are user-specified K quantile levels, and b = (b 1 , . . . , b K ) ⊤ ∈ R K and β = (β 1 , . . . , β p ) ⊤ ∈ R p . Note that we can regard ℓ α (x, y; τ , b, β) as a weighted loss function between the composite quantile loss and the squared error loss. For example, for α = 1, it reduces to the ordinary least squared-based loss function with ℓ 1 (x, y) = (y − x ⊤ β) 2 /2. When α = 0, it is the composite quantile loss function ℓ 0 (x, y) = K k=1 ρ τ k (y −b k −x ⊤ β)/K proposed in Zou and Yuan (2008). It is known that the least squared loss-based method has the best statistical efficiency when errors follow Gaussian distributions and the composite quantile loss is more robust when the error distribution is heavy-tailed or contaminated by outliers. As discussed before, in practice, it is challenging to obtain the tail structure of the error distribution and construct a corresponding change point testing method based on the error structure. Hence, we propose a weighted loss function by borrowing the information related to the possible change point from both the conditional mean and quantiles. More importantly, we use the weight α to balance the efficiency and robustness. Our new testing statistic is based on a novel high-dimensional weighted score-based CUSUM process of the weighted composite loss function. In particular, for the composite 8 loss function ℓ α (x, y; τ , b, β), define its score (subgradient) function ∂ℓ α (x, y; τ , b, β)/∂β with respect to β as: Z α (x, y; τ , b, β) := 1 − α K K k=1 x 1{y − b k − x ⊤ β ≤ 0} − τ k − α x(y − x ⊤ β) . (2.5) Using Z α (x, y; τ , b, β), for each α ∈ [0, 1] and t ∈ (0, 1), we first define the oracle scorebased CUSUM as follows: C α (t; τ , b, β) = 1 √ nσ(α, τ ) ⌊nt⌋ i=1 Z α (X i , Y i ; τ , b, β) − ⌊nt⌋ n n i=1 Z α (X i , Y i ; τ , b, β) , (2.6) where σ 2 (α, τ ) := Var[(1 − α)e i ( τ ) − αϵ i )] with e i ( τ ) := K −1 K k=1 (1{ϵ i ≤ b (0) k } − τ k ). Note that we call C α (t; τ , b, β) oracle since we assume σ 2 (α, τ ) is known. In Section 2.2, we will give the explicit form of σ 2 (α, τ ) under various combinations of α and τ and introduce its consistent estimator under both H 0 and H 1 . In the following, to motivate our test statistics, we study the behaviors of C α (t; τ , b, β) under H 0 and H 1 respectively. First, under H 0 , if we replace β = β (0) and b = b (0) in (2.6), the score based CUSUM becomes C α (t; τ , b (0) , β (0) ) = 1 √ nσ(α, τ ) ⌊nt⌋ i=1 Z α (X i , Y i ; τ , b (0) , β (0) ) − ⌊nt⌋ n n i=1 Z α (X i , Y i ; τ , b (0) , β (0) ) . By noting that under H 0 , we have Y i = X ⊤ i β (0) + ϵ i , the above CUSUM reduces to the following R p -dimensional random noise based CUSUM process: C α (t; τ , b (0) , β (0) ) = 1 √ nσ(α, τ ) ⌊nt⌋ i=1 X i ((1 − α)e i ( τ ) − αϵ i ) − ⌊nt⌋ n n i=1 X i ((1 − α)e i ( τ ) − αϵ i ) , (2.7) whose asymptotic distribution can be easily characterized. Since both b (0) and β (0) are unknown, we need some proper estimators that can approximate them well under H 0 . In this paper, for each α ∈ [0, 1], we obtain the estimators by solving the following optimization problem with the L 1 penalty: ( b α , β α ) = arg min b∈R K , β∈R p (1−α) 1 n n i=1 1 K K k=1 ρ τ k (Y i −b i −X ⊤ i β)+ α 2n n i=1 (Y i −X ⊤ i β) 2 +λ α β 1 , (2.8) where b α := ( b α,1 , · · · , b α,K ) ⊤ , β α := ( β α,1 , . . . , β α,p ) ⊤ , and λ α is the non-negative tuning parameter controlling the overall sparsity of β α . Note that the above estimators are obtained using all observations {(X 1 , Y 1 ), . . . , (X n , Y n )}. After obtaining ( b ⊤ α , β ⊤ α ), we plug them into the score function in (2.6) and obtain the final R p -dimensional oracle score based 9 CUSUM statistic as follows: C α (t; τ , b α , β α ) = C α,1 (t; τ , b α , β α ), . . . , C α,p (t; τ , b α , β α ) ⊤ , t ∈ [q 0 , 1 − q 0 ]. (2.9) Using ( b α , β α ), we can prove that under H 0 , for each α ∈ [0, 1], (2.9) can approximate the random-noise based CUSUM process in (2.7) under some proper norm aggregations. Next, we investigate the behavior of (2.9) under H 1 . Observe that the score based CUSUM has the following decomposition: C α (t; τ , b α , β α ) = C α (t; τ , b (0) , β (0) ) Random Noise + δ α (t) Signal Jump + R α (t; b α , β α ) Random Bias , (2.10) where C α (t; τ , b (0) , β (0) ) is the random noise based CUSUM process defined in (2.7), R α (t; b α , β α ) is some random bias which has a very complicated form but can be controlled properly under H 1 , and δ α (t) is the signal jump function. More specifically, let SN R(α, τ ) := (1 − α) 1 K K k=1 f ϵ (b (0) k ) + α σ(α, τ ) , (2.11) where f ϵ (t) is the probability density function of ϵ, and define the signal jump function ∆(t; β (1) , β (2) ) := n − 3 2 ×    ⌊nt⌋(n − ⌊nt 1 ⌋)Σ β (1) − β (2) , if t ≤ t 1 , ⌊nt 1 ⌋(n − ⌊nt⌋)Σ β (1) − β (2) , if t > t 1 . (2.12) Then, the signal jump δ α (t) can be explicitly represented as the products of SN R(α, τ ) and ∆(t; β (1) , β (2) ), which has the following explicit form: δ α (t) := SN R(α, τ ) × ∆(t; β (1) , β (2) ). (2.13) By (2.13), we can see that δ α (t) can be decomposed into a loss-function-dependent part SN R(α, τ ) and a change-point-model-dependent part ∆(t; β (1) , β (2) ). More specifically, the first term SN R(α, τ ) (short for the signal-to-noise-ratio) is only related to the parameters α, K, b (0) as well as σ(α, τ ), resulting from a user specified weighted loss function defined in (2.4). In contrast, the second term {∆(t; β (1) , β (2) ), t ∈ [0, 1]} is only related to Model (2.1), which is based on parameters t 1 , Σ, β (1) , and β (2) and is independent of the loss function. Moreover, for any weighted composite loss function, the process {∆(t; β (1) , β (2) ), t ∈ [0, 1]} has the following properties: First, under H 1 , the non-zero elements of ∆(t; β (1) , β (2) ) are at most (s (1) + s (2) )-sparse since we require sparse regression coefficients in the model; Second, we can see that ∥∆(t; β (1) , β (2) )∥ with t ∈ [q 0 , 1 − q 0 ] obtains its maximum value at the true change point location t 1 , where ∥ · ∥ denotes some proper norm such as ∥ · ∥ ∞ . Hence, in theory, the signal jump function δ α (t) also achieves its maximum value at t 1 under some proper norm. This is the key reason why using the score based CUSUM can correctly localize the true change point if β (1) −β (2) is big enough. More importantly, for a given underlying error distribution ϵ in Model (2.1), we can use SN R(α, τ ) to further amplify the magnitude of δ α (t) via choosing a proper combination of α and τ . In particular, recall σ 2 (α, τ ) := Var[(1 − α)e i ( τ ) − αϵ i )]. Then, we have σ 2 (α, τ ) = (1 − α) 2 Var[e i ( τ )] + α 2 σ 2 − 2α(1 − α)Cov(e i ( τ ), ϵ i ), (2.14) where σ 2 := Var(ϵ). Using (2.11) and (2.14), SN R(α, τ ) can be further simplified under some specific α. For example, if α = 1, then SN R(α, τ ) = 1/σ; If α = 0, then SN R(α, τ ) = K k=1 f ϵ (b (0) k ) K k 1 =1 K k 2 =1 γ k 1 k 2 with γ k 1 k 2 := min(τ k 1 , τ k 2 ) − τ k 1 τ k 2 ; If we choose α ∈ (0, 1), K = 1 and τ = τ for some τ ∈ (0, 1). Then we have SN R(α, τ ) = (1 − α)f ϵ (b (0) τ ) + α [(1 − α) 2 τ (1 − τ ) + α 2 σ 2 − 2α(1 − α)Cov(e(τ ), ϵ)] 1/2 . (2.15) Hence, for any underlying error distribution ϵ in Model (2.1), it is possible for us to choose a proper α and τ that makes SN R(α, τ ) as large as possible for that distribution. See Figure 1 for a direct illustration, where we plot the SN R(α, τ ) under various error distributions for the weighted composite loss with τ = 0.5. Note that in this case, the loss function ℓ α (x, y) is a weighted combination between the absolute loss and the squared error loss. From Figure 1, SN R(α, τ ) performs differently under various error distributions. For example, when ϵ ∼ N (0, 0.5), by choosing α = 1, SN R(α, 0.5) achieves its maximum as expected. In contrast, when ϵ ∼ Laplace(0, 1), α = 0 is the optimal in terms of SN R(α, 0.5). Furthermore, we can see that for ϵ that follows the Student's t v distribution with a degree of freedom v, choosing α ∈ (0, 1) has the highest SN R. In theory, we can prove that, for any distribution of ϵ, the change point detection, as well as the estimating performance depend heavily on the choice of α and τ via SN R(α, τ ). Hence, it motivates us to use C α (t; τ , b α , β α ) to account for the tail properties for different ϵ using α and τ . Based on our investigation on our score based CUSUM process under H 0 and H 1 , it is appealing to use C α (t; τ , b α , β α ) to construct our testing statistics for adapting different tail structures in the data. For change point detection, a natural question is how to aggregate the R p -dimensional CUSUM process C α (t; τ , b α , β α ). Note that for high-dimensional sparse linear models, there are at most s = s (1) + s (2) coordinates in β (1) − β (2) that can have a change point, which can be much smaller than the data dimension p, although we allow s to grow with the sample size n. Motivated by this, in this paper, we aggregate the first s 0 largest statistics of C α (t; τ , b α , β α ). To that end, we introduce the (s 0 , 2)-norm as follows. Let v = (v 1 , . . . , v p ) ∈ R p . For any 1 ≤ s 0 ≤ p, define v (s 0 ,2) = ( s 0 j=1 \|v (j) \| 2 ) 1/2 , where \|v (1) ≥ \|v (2) \| · · · ≥ \|v (p) \| are the order statistics of v. By definition, we can see that ∥v∥ (s 0 ,2) is the L 2 -norm for the first s 0 largest order statistics of (\|v 1 \|, . . . , \|v p \|) ⊤ , which can be regarded as an adjusted L 2 -norm in high dimensions. Note that the (s 0 , 2)-norm is a special case of the (s 0 , p)-norm proposed in Zhou et al. (2018) by setting p = 2. We also remark that taking the first s 0 largest order statistics can account for the sparsity structure of β (1) − β (2) . Using the (s 0 , 2)-norm with a user-specified s 0 and known variance σ 2 (α, τ ), we introduce the oracle individual testing statistic with respect to a given α ∈ [0, 1] as T α = max q 0 ≤t≤1−q 0 C α (t; τ , b α , β a ) (s 0 ,2) , with α ∈ [0, 1]. By construction, T α can capture the tail structure of the underlying regression errors by choosing a special α and τ . Specifically, for α = 1, it equals to the least square lossbased method. In this case, since T α only uses the moment information of the errors, it is powerful for detecting a change point with light-tailed errors such as Gaussian or sub-Gaussian distributions. For α = 0, T α reduces to the composite quantile loss-based method, which only uses the information of ranks or quantiles. In this case, T α is more robust to data with heavy tails such as Cauchy distributions. As a special case of α = 0, if we further choose τ = 0.5 and K = 1, our testing statistic reduces to the median regression-based method. Moreover, if we choose a proper non-trivial weight α, T α enjoys satisfactory power performance for data with a moderate magnitude of tails such as the Student's t v or Laplace distributions. Hence, our proposed individual testing statistics can adequately capture the tail structures of the data by choosing a proper combination of α and τ . Another distinguishing feature for using T α is that, we can establish a general and flexible framework for aggregating the score based CUSUM for high-dimensional sparse linear models. Instead of taking the ℓ ∞ -norm as most papers adopted for making statistical inference of high-dimensional linear models (e.g., Xia et al., 2018), we choose to aggregate them via using the ℓ 2 -norm of the first s 0 largest order statistics. Under this framework, the ℓ ∞ -norm is a special case by taking s 0 = 1. Moreover, as shown by our extensive numerical studies, choosing s 0 > 1 is more powerful for detecting sparse alternatives, and can significantly improve the detection powers as well as the estimation accuracy for change points, compared to using s 0 = 1. Variance estimation under H 0 and H 1 Note that T α is constructed using a known variance σ 2 (α, τ ) which is defined in (2.14). In practice, however, σ 2 (α, τ ) is typically unknown. Hence, to yield a powerful testing method, a consistent variance estimation is needed especially under the alternative hypothesis. For high-dimensional change point analysis, the main difficulty comes from the unknown change point t 1 . To overcome this problem, we propose a weighted variance estimation. In particular, for each fixed α ∈ [0, 1] and t ∈ (0, 1), define the score based 13 CUSUM statistic without standardization as follows: C α (t; τ , b α , β α ) = 1 √ n ⌊nt⌋ i=1 Z α (X i , Y i ; τ , b α , β α ) − ⌊nt⌋ n n i=1 Z α (X i , Y i ; τ , b α , β α ) . (2.16) Then, for each α ∈ [0, 1], we obtain the individual based estimation for the change point: t α = arg max q 0 ≤t≤1−q 0 C α (t; τ , b α , β α ) (s 0 ,2) . (2.17) In Theorem 3.5, we prove that under some regular conditions, if H 1 holds, t α is a consistent estimator for t 1 , e.g. \|n t α − nt 1 \| = o p (n). This result enables us to propose a variance estimator which is consistent under both H 0 and H 1 . Specifically, let h ∈ (0, 1) be a user specified constant, and define the samples n − = {i : i ≤ nh t α } and n + = {i : t α n+(1−h)(1− t α )n ≤ i ≤ n}. Let (( b (1) α ) ⊤ , ( β (1) α ) ⊤ ) and (( b (2) α ) ⊤ , ( β (2) α ) ⊤ ) be the estimators using the samples in n − and n + . For each α ∈ [0, 1], we can calculate the regression residuals: ϵ i = (Y i − X ⊤ i β (1) α )1{i ∈ n − } + (Y i − X ⊤ i β (2) α )1{i ∈ n + }, for i ∈ n − ∪ n + . (2.18) Moreover, compute e i ( τ ) = K −1 K k=1 e i (τ k ) with e i (τ k ) defined as e i (τ k ) := (1{ ϵ i ≤ b (1) α,k } − τ k )1{i ∈ n − } + (1{ ϵ i ≤ b (2) α,k } − τ k )1{i ∈ n + }, for i ∈ n − ∪ n + . (2.19) Lastly, based on ϵ i and e i ( τ ), we can construct our weighted estimator for σ 2 (α, τ ) as σ 2 (α, τ ) = t α × σ 2 − (α, τ ) + (1 − t α ) × σ 2 + (α, τ ), (2.20) where: σ 2 − (α, τ ) := 1 \|n − \| i∈n − (1 − α) e i ( τ ) − α ϵ i 2 , σ 2 + (α, τ ) := 1 \|n + \| i∈n + (1 − α) e i ( τ ) − α ϵ i 2 . For the above variance estimation, we can prove that \| σ 2 (α, τ ) − σ 2 (α, τ )\| = o p (1) under either H 0 or H 1 . As a result, the proposed variance estimator σ 2 (α, τ ) avoids the problem of non-monotonic power performance as discussed in Shao and Zhang (2010), which is a serious issue in change point analysis. Hence, we replace σ(α, τ ) in (2.9) by σ(α, τ ) and define the data-driven score-based CUSUM process {C α (t; τ , b α , β α ), t ∈ [q 0 , 1 − q 0 ]} with C α (t; τ , b α , β α ) = 1 √ n σ(α, τ ) ⌊nt⌋ i=1 Z α (X i , Y i ; τ , b α , β α ) − ⌊nt⌋ n n i=1 Z α (X i , Y i ; τ , b α , β α ) . (2.21) Note that we call C α (t; τ , b α , β α ) data-driven since there are no unknown parameters in the testing statistic. For a user-specified s 0 ∈ {1, . . . , p} and any α, we define the final individual-based testing statistic as follows: T α = max q 0 ≤t≤1−q 0 C α (t; τ , b α , β a ) (s 0 ,2) , with α ∈ [0, 1]. (2.22) Throughout this paper, we use {T α , α ∈ [0, 1]} as our individual-based testing statistics. Bootstrap approximation for the individual testing statistic In high dimensions, it is very difficult to obtain the limiting null distribution of T α . To overcome this problem, we propose a novel bootstrap procedure. In particular, suppose we implement the bootstrap procedure for B times. Then, for each b-th bootstrap with b = 1, . . . , B, we generate i.i.d. random variables e b 1 , . . . , e b n with e b i ∼ N (0, 1). Let e b i ( τ ) = K −1 K k=1 e b i (τ k ) with e b i (τ k ) := 1{ϵ b i ≤ Φ −1 (τ k )} − τ k , where Φ(x) is the CDF for the standard normal distribution. Then, for each individual-based testing statistic T α , with a user specified s 0 , we define its b-th bootstrap sample-based score CUSUM process as: C b α (t; τ ) = 1 √ nv(α, τ ) ⌊nt⌋ i=1 X i ((1 − α)e b i ( τ ) − αe b i ) − ⌊nt⌋ n n i=1 X i ((1 − α)e b i ( τ ) − αe b i ) , (2.23) where v 2 (α, τ ) is the corresponding variance for the bootstrap-based sample with v 2 (α, τ ) : = (1 − α) 2 Var[e b i ( τ )] + α 2 − 2α(1 − α)Cov(e b i ( τ ), e b i ). (2.24) Note that for bootstrap, the calculation or estimation of v 2 (α, τ ) is not a difficult task since we use N (0, 1) as the error term. For example, when τ = 0.5, it has an explicit form of v 2 (α, τ ) = (1 − α) 2 K k 1 =1 K k 2 =1 γ k 1 k 2 + α 2 − α(1 − α) 2 π . Hence, for simplicity, we directly use the oracle variance v 2 (α, τ ) in (2.23). Using C b α (t; τ ) and for a user specified s 0 , we define the b-th bootstrap version of the individual-based testing statistic T α as T b α = max q 0 ≤t≤1−q 0 C b α (t; τ ) (s 0 ,2) , with α ∈ [0, 1]. (2.25) Let γ ∈ (0, 0.5) be the significance level. For each individual-based testing statistic T α , let F α = P(T α ≤ t) be its theoretical CDF and P α = 1 − F α (T α ) be its theoretical p-value. Using the bootstrap samples {T 1 α , . . . , T B α }, we estimate P α by P α = B b=1 1{T b α > T α \|X , Y} B + 1 , with α ∈ [0, 1]. (2.26) Given the significance level γ, we can construct the individual test as Ψ γ,α = 1{ P α ≤ γ}, with α ∈ [0, 1]. (2.27) For each T α , we reject H 0 if and only if Ψ γ,α = 1. Note that the above bootstrap procedure is easy to implement since it does not require any model fitting such as obtaining the LASSO estimators which is required by the data-based testing statistic T α . Constructing the tail-adaptive testing procedure Algorithm 1 : A bootstrap procedure to obtain the tail-adaptive testing statistic T ad Input: Given the data (X , Y) = {(X 1 , Y 1 ), . . . , (X n , Y n )}, set the values for τ , s 0 , q 0 , the bootstrap replication number B, and the candidate subset A ⊂ [0, 1]. Step 1: Calculate the individual-based testing stastisic T α for α ∈ A as defined in (2.22). Step 2: For each α ∈ A, repeat the procedure (2.23) -(2.25) for B times, and obtain the bootstrap samples {T 1 α , . . . , T B α } for α ∈ A. Step 3: Based on the bootstrap samples in Step 2, calculate empirical P -values P α for α ∈ A as defined in (2.26). Step 4: Using P α with α ∈ A, calculate the tail-adaptive testing statistic T ad in (2.28). Output: Algorithm 1 provides the bootstrap based samples {T 1 α , . . . , T B α } for α ∈ A, and the tail-adaptive testing statistic T ad . In Sections 2.1 -2.3, we propose a family of individual-based testing statistics {T α , α ∈ [0, 1]} and introduce a bootstrap-based procedure for approximating their theoretical pvalues. As discussed in Sections 2.1 and seen from Figure 1, T α with different α's can have various power performance for a given underlying error distribution. For example, T α with a larger α (e.g. α = 1) is more sensitive to change points with light-tailed error distributions by using more moment information. In contrast, T α with a smaller α (e.g. α = 0, 0.1) is more powerful for data with heavy tails such as Student's t v or even Cauchy distribution. In general, as shown in Figure 1, a properly chosen α can give the most satisfactory power performance for data with a particular magnitude of tails. In practice, however, the tail structures of data are typically unknown. Hence, it is desirable to construct a tail-adaptive method which is simultaneously powerful under various tail structures of data. One candidate method is to find α * which maximizes the theoretical SN R(α, τ ), i.e. α * = arg max α SN R(α, τ ), and constructs a corresponding individual testing statistic T α * . Note that in theory, calculating SN R(α, τ ) needs to know σ(α, τ ) and {f ϵ (b (0) k ), k = 1, . . . , K}, which could be difficult to estimate especially under the highdimensional change point model. Instead, we construct our tail-adaptive method by combining all candidate individual tests for yielding a powerful one. In particular, as a small p-value leads to rejection of H 0 , for the individual tests T α with α ∈ [0, 1], we construct the tail-adaptive testing statistic as their minimum p-value, which is defined as follows: T ad = min α∈A P α , (2.28) where P α is defined in (2.26), and A is a candidate subset of α. In this paper, we require \|A\| to be finite, which is a theoretical requirement. Note that our tail-adaptive method is flexible and user-friendly. Zhou et al. (2018), which is also used in Liu et al. (2020). Let P ad be an estimation for the theoretical p-value of T ad using the low-cost bootstrap. Given the significance level γ ∈ (0, 0.5), define the final tail-adaptive test as: Ψ γ,ad = 1{ P ad ≤ γ}. (2.29) For the tail-adaptive testing procedure, given γ, we reject H 0 if Ψ γ,ad = 1. If H 0 is rejected by our tail-adaptive test, letting α = arg min α∈A P α , we return the change point estimation as t ad = t α . As shown by our theory, under some conditions, we have \| t ad − t 1 \| = o p (1). Multiple change point detection Algorithm 2 : A WBS-typed tail-adaptive test for multiple change point detection Input: Given the data (X , Y) = {(X 1 , Y 1 ), . . . , (X n , Y n )}, set the values for τ , the significance level γ, s 0 , q 0 , the bootstrap replication number B, the candidate subset A ⊂ [0, 1], and a set of random intervals {(s ν , e ν } V ν=1 with thresholds v 0 and v 1 . Initialize an empty set C. Step 1: For each ν = 1, · · · , V , compute P ad (s ν , e ν ) following Section 2.4. Step 2: Perform the following function with S = q 0 and E = 1 − q 0 . (c) Compute the test statistics as P ad = min ν∈M,v 0 ≤eν −sν P ad (s ν , e ν ) and the corresponding optimal solution ν * . (d) If P ad ≥ γ/V , RETURN. Otherwise, add the corresponding change point estimator t ν * to C, and perform Function(S, ν * ) and Function(ν * , E). Output: The set of multiple change points C. In practical applications, it may exist multiple change points in describing the relationship between X and Y . Therefore, it is essential to perform estimation of multiple change points if H 0 is rejected by our powerful tail-adaptive test. In this section, we extend our single change point detection method by the idea of WBS proposed in Fryzlewicz (2014) to estimate the locations of all possible multiple change points. Consider a single change point detection task in any interval [s, e], where 0 ≤ q 0 ≤ s < e ≤ 1 − q 0 . Following Section 2.4, we can compute the corresponding adaptive test statistics as P ad (s, e) using the subset of our data, i.e., {X ⌊ns⌋ , X ⌊ns⌋+1 , · · · , X ⌊ne⌋ } and {Y ⌊ns⌋ , Y ⌊ns⌋+1 , · · · , Y ⌊ne⌋ }. Following the idea of WBS, we first independently generate a series of random intervals by the uniform distribution. Denote the number of these random intervals as V . For each random interval [s ν , e ν ] among ν = 1, 2, · · · , V , we compute P ad (s ν , e ν ) as long as 0 ≤ q 0 ≤ s ν < e ν ≤ 1 − q 0 and e ν − s ν ≥ v 0 , where v 0 is the minimum length for implementing Section 2.4. The threshold v 0 is used to reduce the variability of our algorithm for multiple change point detection. Based on the test statistics computed from the random intervals, we consider the final test statistics as P ad = min Theoretical results In this section, we give theoretical results for our proposed methods. In Section 3.1, we provide some basic model assumptions. In Section 3.2, we discuss the theoretical properties of the individual testing methods with a fixed α such as the size, power and change point estimation. In Section 3.3, we provide the theoretical results for the tail-adaptive method. Basic assumptions We introduce some basic assumptions for deriving our main theorems. Before that, we introduce some notations. Let e i ( τ ) : = K −1 K k=1 1{ϵ i ≤ b (0) k } − τ k := K −1 K k=1 e i (τ k ). We set V s 0 := {v ∈ S p : ∥v∥ 0 ≤ s 0 }, where S p := {v ∈ R p : ∥v∥ = 1}. For each α ∈ [0, 1], we introduce β : * = ((β * ) ⊤ , (b * ) ⊤ ) ⊤ ∈ R p+K with β * ∈ R p , b * = (b * 1 , . . . , b * K ) ⊤ ∈ R K , where β : * := arg min β∈R p ,b∈R K E (1 − α) 1 n n i=1 1 K K k=1 ρ τ k (Y i − b i − X ⊤ i β) + α 2n n i=1 (Y i − X ⊤ i β) 2 . (3.1) Note that by definition, we can regard β : * as the true parameters under the population level with pooled samples. We can prove that under H 0 , β hold. In addition, ϵ i is independent with X i for i = 1, . . . , n. : * = ((β (0) ) ⊤ , (b (0) ) ⊤ ) ⊤ with b (0) = (b (0) 1 , . . . , b (0) K ) ⊤ . Under H 1 , β : * is Assumption C (Moment constraints): (C.1) There exists some constant b > 0 such that E(v ⊤ X i ϵ i ) 2 ≥ b and E(v ⊤ X i e i ( τ )) 2 ≥ b, for v ∈ V s 0 and all i = 1, . . . , n. Moreover, assume that inf i,j E[X 2 ij ] ≥ b holds. (C.2) There exists a constant K > 0 such that E\|ϵ i \| 2+ℓ ≤ K ℓ , for ℓ = 1, 2. Assumption D (Underlying distribution): The distribution function ϵ has a continu-ously differentiable density function f ϵ (t) whose derivative is denoted by f ′ ϵ (t). Furthermore, suppose there exist some constants C + , C − and C ′ + such that (D.1) sup t∈R f ϵ (t) ≤ C + ; (D.2) inf j=1,2 inf 1≤k≤K f ϵ (x ⊤ (β * − β (j) ) + b * k ) ≥ C − ; (D.3) sup t∈R \|f ′ ϵ (t)\| ≤ C ′ + . Assumption E (Parameter space): (E.1) We require s 3 0 log(pn) = O(n ξ 1 ) for some 0 < ξ 1 < 1/7 and s 4 0 log(pn) = O(n ξ 2 ) for some 0 < ξ 2 < 1 6 . (E.2) Assume that s 2 0 s 3 log 3 (pn) n → 0 as (n, p) → ∞. (E.3) For β (1) and β (2) , we require max(∥β (1) ∥ ∞ , ∥β (2) ∥ ∞ ) < C β for some C β > 0. Moreover, we require ∥β (2) − β (1) ∥ 1 ≤ C ∆ for some constant C ∆ > 0. (E.4) For the tuning parameters λ α in (2.8), we require λ α = C λ log(pn)/n for some C λ > 0. Assumption A gives some conditions for the design matrix, requiring X has a nondegenerate covariance matrix Σ in terms of its eigenvalues. This is important for deriving the high-dimensional LASSO property with α ∈ [0, 1] under both H 0 and H 1 . Moreover, it also requires that X ij is bounded by some big constant M > 0, which has been commonly used in the literature. Assumption B mainly requires the underlying error term ϵ i has non-degenerate variance. Assumption C imposes some restrictions on the moments of the error terms as well as the design matrix. In particular, Assumption C.1 requires that v ⊤ Xϵ, v ⊤ Xe( τ ), as well as X ij have non-degenerate variances. Moreover, Assumption C.2 requires that the errors have at most fourth moments, which is much weaker than the commonly used Gaussian or sub-Gaussian assumptions. Both Assumptions C.1 and C.2 are basic moment conditions for bootstrap approximations for the individual-based tests. See Lemma C.6 in the appendix. Assumptions D.1 -D.3 are some regular conditions for the underlying distribution of the errors, requiring ϵ has a bounded density function as well as bounded derivatives. Assumption D.2 also requires the density function at x ⊤ (β * −β (j) )+b * k to be strictly bounded away from zero. Lastly, Assumption E imposes some conditions for the parameter spaces in terms of (s 0 , n, p, s, β (1) , β (2) ). Specifically, Assumption E.1 scales the relationship between s 0 , n, and p, which allows s 0 can grow with the sample size n. This condition is mainly used to establish the high-dimensional Gaussian approximation for our individual tests. Assumption E.2 also gives some restrictions on (s 0 , s, n, p). Note that both Assumptions E.1 and E.2 allow the data dimension p to be much larger than the sample size n as long as the required conditions hold. Assumption E.3 requires that the regression coefficients as well as signal jump in terms of its ℓ 1 -norm are bounded. Assumption E.4 imposes the regularization parameter λ α = O( log(pn)/n), which is important for deriving the desired error bound for the LASSO estimators under both H 0 and H 1 using our weighted composite loss function. See Lemmas C.9 -C.11 in the appendix. Remark 3.1. Assumption C.2 with the finite fourth moment is mainly for the individual test with α = 1, while Assumption D without any moment constraints is for that with α = 0. Hence, our proposed individual-based change point method extends the highdimensional linear models with sub-Gaussian distributed errors to those with only finite moments or without any moments, covering a wide range of errors with different tails. Remark 3.2. Our proposed individual method with α = 0 needs to cover both cases with and without a change point. To obtain the desired error bound of LASSO estimation, we require that inf j=1,2 inf 1≤k≤K f ϵ (x ⊤ (β * − β (j) ) + b * k ) ≥ C − . This is different from the classical assumption (Zhao et al., 2014) that inf 1≤k≤K f ϵ (b (0) k ) ≥ C − . Note that our assumption is quite mild since essentially, it only requires that the density function is non-generate at a neighborhood of b * k which is shown to be satisfied under Assumptions A.2 and E.3. Theoretical results of the individual-based testing statistics Validity of controlling the testing size Before giving the size results, we first consider the variance estimation. Recall σ 2 (α, τ ) in (2.14) and σ 2 (α, τ ) in (2.20). Theorem 3.3 shows that the pooled weighted variance estimator σ 2 (α, τ ) is consistent under the null hypothesis, which is crucial for deriving the Gaussian approximation results as shown in Theorem 3.4 and shows that our testing method has satisfactory size performance. P(T α ≤ z) − P(T b α ≤ z\|X , Y) = o p (1), as n, p → ∞. (3.2) Theorem 3.4 demonstrates that we can uniformly approximate the distribution of T α by that of T b α . The following Corollary further shows that our proposed new test Ψ γ,α can control the Type I error asymptotically for any given significant level γ ∈ (0, 1). Corollary 3.1. Suppose the assumptions in Theorem 3.4 hold. Under H 0 , we have P(Ψ γ,α = 1) → γ, as n, p, B → ∞. Change point estimation After analyzing the theoretical results under the null hypothesis, we next consider the performance of the individual test under H 1 . We first give some theoretical results on the change point estimation. To that end, some additional assumptions are needed. Recall Π = {j : β (1) j ̸ = β (2) j } as the set with change points. For j ∈ {1, . . . , p}, define the signal jump ∆ = (∆ 1 , . . . , ∆ p ) ⊤ with ∆ j := β (1) j − β (2) j . Let ∆ min = min j∈Π \|∆ j \| and ∆ max = max j∈Π \|∆ j \|. With the above notations, we introduce the following Assumption F. Assumption F. There exist constants c > 0 and C > 0 such that 0 < c ≤ lim inf p→∞ ∆ min ∆ max ≤ lim sup p→∞ ∆ max ∆ min ≤ C < ∞. (3.3) Note that Assumption F is only a technical condition requiring that ∆ min and ∆ max are of the same order. With Assumption F as well as those of Assumptions A -E, Theorem 3.5 below provides a non-asymptotic estimation error bound of the argmax-based individual change point estimator t α for t 1 . To give a precise result, for change point estimation, we assume s 0 is fixed. t α − t 1 ≤ C * (s 0 , τ , α) log(pn) nSN R 2 (α, τ )∥Σ∆∥ 2 (s 0 ,2) , (3.4) where C * (s 0 , τ , α) > 0 is some universal constant only depending on s 0 , τ and α. Theorem 3.5 shows that our individual estimators are consistent under the condition ∥∆∥ (s 0 ,2) ≫ log(pn)/n. Moreover, according to Rinaldo et al. (2021), for highdimensional linear models, under Assumption F, if ∥∆∥ ∞ ≫ 1/ √ n, any change point estimator t has an estimation lower bound \| t − t 1 \| ≥ c * 1 n∥∆∥ 2 ∞ , for some constant c * > 0. Hence, considering (3.3) and (3.4), with a fixed s 0 , Theorem 3.5 demonstrates that our individual-based estimators for the change point are rate optimal up to a log(pn) factor. More importantly, our proposed estimators { t α , α ∈ [0, 1]} only require O(Lasso(n, p)) operations to calculate, which is less computationally expensive than the grid search based method in Lee et al. (2016). Power performance We discuss the power properties of the individual tests. Note that for the change point problem, variance estimation under the alternative is a difficult but important task. As pointed out in Shao and Zhang (2010), due to the unknown change point, any improper estimation may lead to nonmonotonic power performance. This distinguishes the change point problem substantially from one-sample or two-sample tests where homogenous data are used to construct consistent variance estimation. Hence, for yielding a powerful change point test, we need to guarantee a consistent variance estimation. Theorem 3.6 shows that the pooled weighted variance estimation is consistent under H 1 . This guarantees that our proposed testing method has reasonable power performance. Theorem 3.6. Suppose the assumptions in Theorem 3.5 hold. Let r α (n) = s log(pn)/n if α = 1 and r α (n) = s log(pn) n ∨ s 1 2 ( log(pn) n ) 3 8 if α ∈ [0, 1). Under H 1 , for each α ∈ [0, 1], we have \| σ 2 (α, τ ) − σ 2 (α, τ )\| = O p (r α (n)). According to the proof of Theorem 3.6, several interesting observations can be drawn. Even if the signal strength is weak such as ∥∆∥ (s 0 ,2) = O( log(pn)/n), the pooled weighted variance estimator σ 2 (α, τ ) can still be consistent for σ 2 (α, τ ). However, in this case, we can not guarantee that our change point estimator is consistent as ∥∆∥ (s 0 ,2) ≫ log(pn)/n is required in Theorem 3.5 for consistency. In contrast, if the signal strength is strong enough such that ∥∆∥ (s 0 ,2) ≫ log(p)/n, then a consistent change point estimator such as the proposed t α is needed to guarantee Theorem 3.6. These are insightful findings for variance estimation in change point analysis, which are not shown in the i.i.d. case. Using the consistent variance estimation, we are able to discuss the power properties of the individual tests. To this end, we need some additional notations. Define the oracle signal to noise ratio vector D = (D 1 , . . . , D p ) ⊤ with D j :=    0, for j ∈ Π c SN R(α, τ ) × t 1 (1 − t 1 ) Σ(β (1) − β (2) ) j , for j ∈ Π, (3.5) where SN R(α, τ ) is defined in (2.11). With the above notations and some regularity conditions, Theorem 3.7 stated below shows that we can reject the null hypothesis of no change point with probability tending to 1. then we have P(Φ γ,α = 1) → 1, as n, p, B → ∞, where C( τ , α) is some universal positive constant only depending on τ and α. Theorem 3.7 demonstrates that with probability tending to one, our proposed individual test with α ∈ [0, 1] can detect the existence of a change point for high-dimensional linear models as long as the corresponding signal to noise ratio satisfies (3.7). Combining (3.5) and (3.7), for each individual test, we note that with a larger signal jump and a closer change point location t 1 to the middle of data observations, it is more likely to trigger a rejection of the null hypothesis. More importantly, considering ϵ n = o(1), Theorem 3.7 illustrates that for consistently detecting a change point, we require the signal to noise ratio vector to be at least an order of ∥D∥ (s 0 ,2) ≍ s 1/2 0 log(pn)/n, which is particularly interesting to further discuss under several special cases. For example, if we choose s 0 = 1 and α = 1, our proposed individual test reduces to the least squared loss based testing statistic with the ℓ ∞ -norm aggregation. In this case, we require ∥D∥ ∞ ≍ log(pn)/n for detecting a change point. If we choose α = 0 with the composite quantile loss, the test is still consistent as long as ∥D∥ ∞ ≍ log(pn)/n. Note that the latter one is of special interest for the robust change point detection. Hence, our theorem provides the unified condition for detecting a change point under a general framework, which may be of independent interest. Moreover, Theoretical results of the tail-adaptive testing statistics In Section 3.2, we present the theoretical properties of the individual testing statistics T α with α ∈ [0, 1]. In this section, we discuss the size and power properties of the tail-adaptive test Ψ γ,ad defined in (2.29). To present the theorems, we need additional notations. Let F Tα (x) := P(T α ≤ x) be the CDF of T α . Then P α in (2.26) approximates the following individual tests' theoretical P -values defined as P α := 1 − F Tα (T α ). Hence, based on the above theoretical P -values, we can define the oracle tail-adaptive testing statistic T ad = min α∈A P α . Let F T,ad (x) := P( T ad ≤ x) be the CDF of T ad . Then we can also define the theoretical tail-adaptive test's P -value as P ad := F T,ad ( T ad ). Recall P ad be the low cost bootstrap P -value for P ad . In what follows, we show that P ad converges to P ad in probability as n, p, B → ∞. We introduce Assumption E.1 ′ to describe the scaling relationships among n, p, and s 0 . Let G i = (G i1 , . . . , G ip ) ⊤ with G i ∼ N (0, Σ) being i.i.d. Gaussian random vectors, where Σ := Cov(X 1 ). Define C G (t) = 1 √ n ⌊nt⌋ i=1 G i − ⌊nt⌋ n n i=1 G i and T G = max q 0 ≤t≤1−q 0 ∥C G (t)∥ (s 0 ,2) . As shown in the proof of Theorem 3.4, we use T G = max q 0 ≤t≤1−q 0 C G (t) (s 0 ,2) to approximate T α . For T G , let f T G (x) and c T G (γ) be the probability density function (pdf), and the γ-quantile of T G , respectively. We then define h(ϵ) as h(ϵ) = max x∈I(ϵ) f −1 T G (x), where I(ϵ) := [c T G (ϵ), c T G (1 − ϵ)]. With the above definitions and notations, we now introduce Assumption E.1 ′ : (E.1) ′ For any 0 < ϵ < 1, we require h 0.6 (ϵ)s 3 0 log(pn) = o(n 1/10 ). Note that Assumption E.1 ′ is more stringent than Assumption E.1. The intuition of Assumption E.1 ′ is that, we construct our tail-adaptive testing statistic by taking the minimum P -values of the individual tests. For analyzing the combinational tests, we need not only the uniform convergence of the distribution functions, but also the uniform convergence of their quantiles on [ϵ, 1 − ϵ] for any 0 < ϵ < 1. The following Theorem 3.8 justifies the validity of the low-cost bootstrap procedure in Section 2.4. It also shows that our tail-adaptive test has the asymptotic level of γ. 27 Theorem 3.8. For T ad , suppose Assumptions A -D, E.1 ′ , E.2 -E.4 hold. Under H 0 , we have P(Ψ γ,ad = 1) → γ, and P ad − P ad P − → 0, as n, p, B → ∞. After analyzing the size, we now discuss the power. Theorem 3.9 shows that under some regularity conditions, our tail-adaptive test has its power converging to one. Note that based on the theoretical results obtained in Section 3.2, Theorems 3.8 and 3.9 can be proved using some modifications of the proofs of Theorems 3.5 and 3.7 in Zhou et al. (2018). Hence, we omit the detailed proofs for brevity. Lastly, recall the tail-adaptive based change point estimator t ad = t α with α = arg min α∈A P α . According to Theorem 3.5, the tail-adaptive estimator is consistent which is summarized as a corollary. , where C ad > 0 is some universal constant not depending on n or p. and accuracies than only using s 0 = 1 for change point testing and estimation. Simulation Studies In summary, the numerical results are consistent with our theorems developed in Section 3 and demonstrate the advantages of our tail-adaptive method over the existing methods. Real data applications In this section, we apply our proposed methods to the S&P 100 dataset to find multiple change points. We obtain the S&P 100 index as well as the associated stocks from Yahoo! Finance (https://finance.yahoo.com/) including the largest and most established 100 companies in the S&P 100. For this dataset, we collect the daily prices of 76 stocks that have Leonardi and Bühlmann (2016) and SGL is the sparse graphical LASSO method in Zhang et al. (2015). The constant c represents the signal strength and a larger c denotes stronger signal jump. reveals the direction of the entire financial system. To this end, we use the daily prices of the 76 stocks to predict the S&P100 index. Specifically, let Y t ∈ R 1 be the S&P 100 index for the t-th day and X t ∈ R 76×2 be the stock prices with lag-1 and lag-3 for the t-th day. Our goal is to predict Y t using X t under the high dimensional linear regression models and detect multiple change points for the linear relationships between the S&P100 index and the 76 stocks' prices. It is well known that the financial data are typically heavy-tailed and we have no prior-knowledge about the tail structure of the data. Hence, for this real data analysis, it seems very suitable to use our proposed tail-adaptive method. We combine our proposed tail-adaptive test with the WBS method (Fryzlewicz (2014)) to detect multiple change points, which is demonstrated in Algorithm 2. To implement this algorithm, we set A = {0, 0.1, 0.5, 0.9, 1}, s 0 = 5, B = 100, and V = 500 (number of random intervals). Moreover, we consider the L 1 − L 2 weighted loss by setting τ = 0.5 in (2.4). The data are scaled to have mean zeros and variance ones before the change point detection. There are 31 14 change points detected which are reported in Table 1. To further justify the meaningful findings of our proposed new methods, we refer to the T-bills and ED (TED) spread, which is short for the difference between the 3-month of London Inter-Bank Offer Rate (LIBOR), and the 3-month short-term U.S. government debt (T-bills). It is well-known that TED spread is an indicator of perceived risk in the general economy and an increased TED spread during the financial crisis reflects an increase in credit risk. Figure 5 shows which S&P 100 index began to experience a significant decline. Moreover, it is known that countries such as Italy and Spain were facing severe debt issues in July, 2011, raising fears about the stability of the Eurozone and the potential impact on global financial markets. As a result, there exists another huge drop for the S&P 100 index in July 26, 2011, which can be successfully detected by our method. Summary In this article, we propose a general tail-adaptive approach for simultaneous change Table 1. Figure 1 : 1SN R(α, τ ) under various errors with different weights α ∈ {0, 0.1, . . . , 0.9, 1} for the weighted loss with τ = 0.5 and K = 1. Function (S, E): S and E are the starting and ending points for the change point detection. (a) RETURN if E − S ≤ v 1 . (b) Define M = {1 ≤ ν ≤ V \| [s ν , e ν ] ⊂ [E, S]}. P ad (s ν , e ν ), based on which we make decisions if there exists at least one change point among these intervals. We stop the algorithm if P ad ≥c, otherwise we report the change point estimation in [s ν * , e ν * ], where ν * ∈ arg min 1≤ν≤V,v 0 ≤eν −sν P ad (s ν , e ν ), and continue our algorithm. Given the first change point estimator denoted by t ν * , we split our data into two folds, i.e., before and after the estimated change point. Then we apply the previous procedure on each fold of the data using the same set of the random intervals as long as it satisfies the constraints.We repeat this step until the algorithm stops returning the change point estimation. For each step, we choosec = γ/V , where γ is the significance level used in each single change point detection algorithm. While we do not have the theoretical guarantee of usingc in the proposed algorithm for controlling the size, the selection of this constant is based on the idea of Bonferroni correction, which is conservative. The numerical experiments in the appendix demonstrate the superiority of our proposed method in detecting multiple change points. Nevertheless, it is interesting to study the asymptotic property of P ad , which we leave for the future work. The full algorithm of the multiple change point detection can be found in Algorithm 2. generally a weighted combination of the parameters before the change point and those after the change point. For example, when α = 1, it has the explicit form of β: * = ((t 1 β (1) + t 2 β (2) ) ⊤ , (b (0) ) ⊤ ) ⊤ .More discussions about β : * under our weighted composite loss function are provided in the appendix. With the above notations, we are ready to introduce our assumptions as follows: Assumption A (Design matrix): The design matrix X has i.i.d rows {X i } n i=1 . (A.1) Assume that there are positive constants κ 1 and κ 2 such that λ min (Σ) ≥ κ 1 > 0 and λ max (Σ) ≤ κ 2 < ∞ hold. (A.2) There exists some constant M ≥ 1 such that max 1≤i≤n max 1≤j≤p \|X ij \| ≤ M almost surely for every n and p. Assumption B (Error distribution): The error terms {ϵ i } n i=1 are i.i.d. with mean zero and finite variance σ 2 ϵ . There exist positive constants c ϵ and C ϵ such that c 2 ϵ ≤ Var(ϵ i ) ≤ C 2 ϵ Theorem 3. 3 . 3For α = 1, suppose Assumptions A, B, C, E hold. For α = 0, suppose Assumptions A, C.1, D, E hold. For α ∈ (0, 1), suppose Assumptions A -E hold. Let r α (n) = s log(pn)/n if α = 1 and r α (n) = s log(pn) hold. For α ∈ (0, 1), suppose Assumptions A -E hold. Then, under H 0 , for the individual test with α ∈ [ Theorem 3. 5 . 0 50Suppose ∥∆∥ (s 0 ,2) ≫ log(pn)/n and Assumption F hold. Moreover, For α = 1, suppose Assumptions A, B, C, E.2 -E.4 as well as n 1/4 = o(s) hold; For α = 2 log(pn)/n∥∆∥ (s 0 ,2) = 0 hold. Then, under H 1 , for each α ∈ [0, 1], with probability tending to one, we have Theorem 3. 7 . 7Let ϵ n := 2 log(pn)/n∥∆∥ (s 0 ,2) ). For each α ∈ [0, 1], assume the following conditions hold: When α = 1, suppose that Assumptions A, B, C, E.2 -E.4 hold; When α = 0, suppose that Assumptions A, C.1, D, E.2 -E.When α ∈ (0, 1), suppose that Assumptions A -D, E.2 -E.4 as well as (3.6) hold.Under H 1 , if D in (3.5) satisfies √ n × ∥D∥ (s 0 ,2) ≥ C( τ , α) 1 − ϵ n s Theorem 3.7 reveals that for detecting a change point, our individual-based method with α ∈ [0, 1] can account for the tails of the data. For Model (2.1) with a fixed signal jump ∆ and a change point location t 1 , considering (3.5) and (3.7), the individual test T α is more powerful with a larger SN R(α, τ ). This reveals why the individual tests with different weights α perform very differently under various error distributions.Lastly, it is worth mentioning that the requirements for identifying and detecting a change point are different for Model (2.1). More specifically, for each individual-based method, Theorem 3.5 demonstrates that, to consistently estimate the location of t 1 , the signal strength should at least satisfy ∥∆∥ (s 0 ,2) ≫ log(pn)/n. In contrast, Theorem 3.7 shows that we can detect a change point if ∥∆∥ (s 0 ,2) ≥ C log(pn)/n holds. This reveals that we need more stringent conditions for localizing a change point than detecting its existence for high-dimensional linear models. To the best of our knowledge, this is still an open question on whether one can obtain consistent change point estimation if ∥∆∥ (s 0 ,2) = O( log(pn)/n). T ad , we have P(Ψ γ,ad = 1) → 1 as n, p, B → ∞, where C( τ , α) is some universal positive constant only depending on τ and α. Corollary 3 . 2 . 32Suppose Assumptions A -D, E.2 -E.4, F as well as n 1/4 = o(2 log(pn)/n∥∆∥ (s 0 ,2) = 0 hold. Suppose additionally ∥∆∥ (s 0 ,2) ≫ log(pn) n holds. Under H 1 , with probability tending to one, we have t ad − t 1 ≤ C ad log(pn) n∥Σ∆∥ 2 (s 0 ,2) Figure 2 : 2We have carried out extensive numerical studies to examine the finite sample performance of our proposed new methods. To save space, we put the detailed model settings and results in Appendix A of the supplementary materials. The simulation results, including size, power and single and multiple change point estimation, can be summarized as follows: Empirical sizes of the individual and tail-adaptive tests for models with banded covariance matrix under the setting of (n, p) = (200, 400). The results are based on 1000 replications.1. As shown inFigure2, the proposed individual and tail-adaptive tests can control the size very well under various model settings with different tail structures including both lighted and heavy tails. The individual test with α = 0 can even control the size well for Student's t 2 and Cauchy distributions.2. In terms of power performance, as shown inFigure 3, the individual tests perform differently under various tail structures. However, the tail-adaptive method can have powers close to its best individual one whenever the errors are lighted or heavy-tailed.3. For single and multiple change point estimation, similar to the power analysis, the performance of the individual estimators depends on the underlying error distributions. Figure 4 Figure 3 : 43indicates that the tail-adaptive estimator can perform close to its best individual estimator. Moreover, compared with the existing techniques, the tail-adaptive method enjoys better performance for single and multiple change point detection.4. For the choice of s 0 , the size performance is stable across different choices of s 0 under H 0 . Moreover, it is shown that, under H 1 , choosing s 0 > 1 can have high powers Empirical powers of the individual and tail-adaptive tests for models with banded covariance matrix under the setting of (n, p) = (200, 400). The change point is at t 1 = 0.5. The results are based on 1000 replications with B = 200 for each replication. Figure 4 : 4remained in the S&P 100 index consistently from January 3, 2007 to December 30, 2011. This covers the recent financial crisis beginning in 2008 and some other important events, resulting in a sample size n = 1259. In financial marketing, it is of great interest to predict the S&P 100 index since it Boxplots of the scaled Hausdorff distance of different methods for detecting multiple change points with (n, p) = (1000, 100) based on 100 replications. The three change points are at (0.25, 0.5, 0.75). L&B is the binary segmentation based technique in the plot of TED where the red dotted lines correspond to the estimated change points. We can see that during the financial crisis from 2007 to 2009, the TED spread has experienced very dramatic fluctuations and the estimated change points can capture some big changes in the TED spread. In addition, the S&P 100 index obtains its highest level during the financial crisis in October 2007 and then has a huge drop. Our method identifies October 29, 2007 as a change point. Moreover, the third detected changepoint is January 10th, 2008. The National Bureau of Economic Research (NBER) identifies December of 2007 as the beginning of the great recession which is captured by our method. In addition, it is well known that affected by the 2008 financial crisis, Europe experienced a debt crisis from 2009 to 2012, with the Greek government debt crisis in October, 2009 serving as the starting point. Our method identifies October 5, 2009 as a change point after Figure 5 : 5point testing and estimation for high-dimensional linear regression models. The method is based on the observation that both the conditional mean and quantile change if the regression coefficients have a change point. Built on a weighted composite loss, Plots of the Ted spread (left) and the S&P 100 index (right)with the estimated change-points (vertical lines) marked by # in tail-adaptive test statistic T ad by taking the minimum P -values of {T α , α ∈ [0, 1]}. The proposed tail-adaptive method T ad chooses the best individual test according to the data and thus enjoys simultaneous high power across various tail structures. Theoretically, we adopt a low-cost bootstrap method for approximating the limiting distribution of T ad . In terms of size and power, for both individual and tail-adaptive tests, we prove that the corresponding test can control the type I error for any given significance level if H 0 holds, and reject the null hypothesis with probability tending to one otherwise.As for the change point estimation, once H 0 is rejected by our test, based on each individual test statistic, we can estimate its location via taking argmax with respect to differ-Fur- thermore, combining the corresponding individual tests in {T α , α ∈ [0, 1]}, we construct a ent candidate locations t ∈ (0, 1) for the (s 0 , 2)-norm aggregated process {∥C α (t)∥ (s 0 ,2) , t ∈ [0, 1]}. Under some regular conditions, for each individual based estimator { t α , α ∈ [0, 1]}, we can show that the estimation error is rate optimal up to a log(pn) factor. Hence, the proposed individual estimators for the change point location allow the signal jump size scale well with (n, p) and are consistent as long as SN R(α, τ )∥Σ(β (2) −β (1) )∥ (s 0 ,2) ≫ log(pn)/n holds, where SN R(α, τ ) is the signal to noise constant related to the loss function and the underlying error distribution. It is worth noting that the computational cost for obtaining { t α , α ∈ [0, 1]} is only O(Lasso(n, p)) operations, where Lasso(n, p) is the cost to compute Due to the curse of dimensionality, on the other hand, only a few papers studied highdimensional change point analysis, which mainly focused on the change point estimation.). Other related methods include Qu (2008); Zhang et al. (2014); Oka and Qu (2011); Lee et al. (2011) and among others. See Lee et al. (2016); Kaul et al. (2019); Lee et al. (2018); Leonardi and Bühlmann In practice, if the users have some prior knowledge about the tails of errors, we can choose A accordingly. For example, we can choose A = {0.9, 1} for light-tailed errors, and A = {0} for extreme heavy tails such as Cauchy distributions. However, if the tail structure is unknown, we can chooseA consisting both small and large values of α ∈ [0, 1]. For example, according to our which is shown by our numerical studies to enjoy stable size performance as well as high powers across various error distributions. Algorithm 1 describes our procedure to construct T ad . Using Algorithm 1, we construct the tail-adaptive testing statistic T ad . Let F ad (x) be its theoretical distribution function. Note that F ad (x) is unknown. Hence, we can not use T ad directly for Problem (1.2). To approximate its theoretical p-value, we adopt the low-cost bootstrap method proposed bytheoretical analysis of SN R(α, τ ), we find that SN R(α, τ ) tends to be maximized near the boundary of [0, 1]. Hence, we recommend to use A = {0, 0.1, 0.5, 0.9, 1} in real applications, Table 1 : 1Multiple change poins detection for the S&P 100 dataset. 8 if α ∈ [0, 1). Under H 0 , for each α ∈ [0, 1], we have \| σ 2 (α, τ ) − σ 2 (α, τ )\| = O p (r α (n)). a family of individual testing statistics with different weights to account for the unknown tail structures. Then, we combine the individual tests to construct a tail-adaptive method, which is powerful against sparse alternatives under various tail structures. In theory, for both individual and tail-adaptive tests, we propose a bootstrap procedure to approximate the limiting null distributions. With mild conditions on the regression covariates and errors, we show the optimality of our methods theoretically in terms of size and power under the high-dimensional setting with p ≫ n. For change point estimation, for each individual method, we propose an argmax-based estimator shown to be rate optimal up to a log(pn) factor. In the presence of multiple change points, we combine our tail-adaptive approach with the WBS technique to detect multiple change points. With extensive numerical studies, our proposed methods have better performance in terms of size, power, and change 33 point estimation than the existing methods under various model setups. Break detection in the covariance structure of multivariate time series. A References Aue, S Hörmann, L Horváth, M Reimherr, The Annals of Statistics. 37References Aue, A., Hörmann, S., Horváth, L. and Reimherr, M. (2009). Break detection in the covariance structure of multivariate time series. The Annals of Statistics 37 4046-4087. 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[ "SHARP BOUNDS ON p-NORMS FOR SUMS OF INDEPENDENT UNIFORM RANDOM VARIABLES, 0 < p < 1", "SHARP BOUNDS ON p-NORMS FOR SUMS OF INDEPENDENT UNIFORM RANDOM VARIABLES, 0 < p < 1" ]	[ "Giorgos Chasapis ", "Keerthana Gurushankar ", "Tomasz Tkocz " ]	[]	[]	We provide a sharp lower bound on the p-norm of a sum of independent uniform random variables in terms of its variance when 0 < p < 1. We address an analogous question for p-Rényi entropy for p in the same range.2020 Mathematics Subject Classification. Primary 60E15; Secondary 26D15.	10.1007/s11854-022-0256-x	[ "https://arxiv.org/pdf/2105.14079v3.pdf" ]	235,253,761	2105.14079	82350ba7489d81caab17b0f4b8bcdb695184d5dc	SHARP BOUNDS ON p-NORMS FOR SUMS OF INDEPENDENT UNIFORM RANDOM VARIABLES, 0 < p < 1 Giorgos Chasapis Keerthana Gurushankar Tomasz Tkocz SHARP BOUNDS ON p-NORMS FOR SUMS OF INDEPENDENT UNIFORM RANDOM VARIABLES, 0 < p < 1 Sharp moment comparisonKhinchin inequalitiesSums of independent random variablesUniform random variablesRényi entropy We provide a sharp lower bound on the p-norm of a sum of independent uniform random variables in terms of its variance when 0 < p < 1. We address an analogous question for p-Rényi entropy for p in the same range.2020 Mathematics Subject Classification. Primary 60E15; Secondary 26D15. Introduction and results Moment comparison inequalities for sums of independent random variables, that is Khinchin-type inequalities, first established by Khinchin for Rademacher random variables (random signs) in his proof of the law of the iterated logarithm (see [14]), have been extensively studied ever since his work. Particularly challenging, interesting and conducive to new methods is the question of sharp constants in such inequalities. We only mention in passing several classical as well as recent references, [1,9,11,15,17,18,23,26]. This paper finishes the pursuit of sharp constants in L p −L 2 Khinchin inequalities for sums of independent uniform random variables, addressing the range 0 < p < 1. We are also concerned with a p-Rényi entropy analogue. 1.1. Moments. Let U 1 , U 2 , . . . be independent random variables uniform on [−1, 1]. As usual, X p = (E\|X\| p ) 1/p is the p-norm of a random variable X. Given p > −1, let c p and C p be the best constants such that for every integer n 1 and real numbers a 1 , . . . , a n , we have or in other words, since a j U j 2 = Var( a j U j ) = 3 −1/2 a 2 j 1/2 , finding c p and C p amounts to extremising the p-norm of the sum a j U j subject to a fixed variance, c p = inf n j=1 a j U j p , C p = sup n j=1 a j U j p , where the infimum and supremum and taken over all integers n 1 and unit vectors a = (a 1 , . . . , a n ) in R n . For p > 1, the optimal constants c p , C p were found by Lata la and Oleszkiewicz in [19] (see also [8] for an alternative approach and [1,16] for generalisations in higher dimensions). They read c p = U 1 p = (1 + p) −1/p , C p = lim n→∞ U 1 + · · · + U n √ n p = Z p / √ 3, p > 1,(2) where Z here and throughout the text denotes a standard N (0, 1) Gaussian random variable. In fact stronger results are available (extremisers are known via Schurconvexity for each fixed n). For −1 < p < 0, the behaviour is complicated by a phase transition (similar to the case of random signs as established by Haagerup in [9]). It has recently been proved in [4] that c p = min Z p / √ 3, U 1 + U 2 p / √ 2 = Z p / √ 3, −0.793.. < p < 0, U 1 + U 2 p / √ 2, −1 < p − 0.793.., and the limiting behaviour of c p as p → −1 + recovers Ball's celebrated cube slicing inequality from [2]. The fact that C p = U 1 p , −1 < p < 1, follows easily from unimodality and Jensen's inequality (see, e.g. Proposition 15 in [8]). Thus what is unknown is the optimal value of c p for 0 < p < 1 and this paper fills out this gap. Our main result reads as follows. Theorem 1. For 0 < p < 1, c p = Z p / √ 3 is the best constant in (1). We record for future use that Z p p = 1 √ 2π ∞ −∞ \|x\| p e −x 2 /2 dx = 2 p/2 √ π Γ 1 + p 2 . 1.2. Rényi entropy. For p ∈ [0, ∞], the p-Rényi entropy of a random variable X with density f is defined as (see [25]), h p (X) = 1 1 − p log R f p with p ∈ {0, 1, ∞} defined by taking the limit: h 0 (f ) = log \|supp(f )\| is the log- arithm of the Lebesgue measure of the support of f , h 1 (f ) = − f log f is the Shannon entropy, and h ∞ = − log f ∞ , where f ∞ is the ∞-norm of f (with respect to Lebesgue measure). The question of maximising Rényi entropy under a variance constraint (or more generally, a moment constraint) for general distributions has been fully understood and leads to the notion of relative entropy that is of importance in information theory, providing a natural way of measuring distance to the extremal distributions (see [5,13,20,22]). In analogy to Theorem 1, we provide an answer for p-Rényi entropies, 0 < p < 1, for sums of uniforms under the variance constraints. Theorem 2. Let 0 < p < 1. For every unit vector a = (a 1 , . . . , a n ), we have h p (U 1 ) h p   n j=1 a j U j   h p Z/ √ 3 . The lower bound is a simple consequence of the entropy power inequality. The upper bound is interesting in that the maximizer among all distributions of fixed variance is not Gaussian (rather, with density proportional to (1 + x 2 ) −1/(1−p) for 1 3 < p < 1 and it does not exist for p < 1 3 , see e.g. [5]). It is derived from the L q − L 2 Khinchin inequality for even q. 1.3. Organisation of the paper. In Section 2 we give an overview of the proof of Theorem 1 and show a reduction to two main steps: an integral inequality and an inductive argument. Then in Section 3 we gather all technical lemmas needed to accomplish these steps which is then done in Sections 4 and 5, respectively. Section 6 contains a short proof of Theorem 2. 2. Proof of the main result 2.1. Overview. We follow an approach developed by Haagerup in [9], with major simplifications advanced later by Nazarov and Podkorytov in [24]. In essence, the argument begins with a Fourier-analytic integral representation for the power function \| · \| p which allows to take advantage of independence and in turn, by virtue of the AM-GM inequality, to reduce the problem to establishing a certain integral inequality involving the Fourier transforms of the uniform and Gaussian distributions. Since this inequality holds only in a specific range of parameters, additional arguments are needed, mainly an induction on the number of summands n (similar problems were faced in e.g. [4,24,15]). In our case, this is further complicated by the fact that the base of the induction fails for large values of p (roughly for p > 0.7). Remark 3. We point out that the main difference between the regimes p 1 and p < 1 is that for the former convexity type arguments allow to establish stronger comparison results, namely the Schur-convexity/concavity of the function ( √ x 1 , . . . , √ x n ) → E n j=1 √ x j U j p . By combining Theorems 2 and 3 of [1] (see also (6.1) therein), a necessary condition for this is the concavity/convexity of the function x → E\|U 1 + √ x\| p . The calculations following Corollary 1 in the same work show that this is the case only for p 1. In other words, when p < 1, the function above is neither Schur-convex nor Schur-concave and the Fourier-analytic approach seems to be indispensable. 2.2. Details. The aforementioned Fourier-analytic formula reads as follows (it can be found for instance in [9], but we sketch its proof for completeness). Lemma 4. Let 0 < p < 2 and κ p = 2 π Γ(1 + p) sin πp 2 . For a random variable X in L 2 with characteristic function φ X (t) = Ee itX , we have E\|X\| p = κ p ∞ 0 1 − Reφ X (t) t p+1 dt. Proof. A change of variables establishes \|x\| p = κ p ∞ 0 1−cos(tx) t p+1 dt, x ∈ R. We then apply this to X and take the expectation. We begin the proof of Theorem 1. Let 0 < p < 1 and c p = Z p / √ 3. Let a 1 , . . . , a n be (without loss of generality) nonzero real numbers with n j=1 a 2 j = 1. By symmetry of the uniform distribution we assume without loss of generality that they are in fact positive. From Lemma 4, we obtain E n j=1 a j U j p = κ p ∞ 0 1 − n j=1 φ(a j t) t 1+p dt, where we have used independence and put φ(t) = Ee itU1 = sin t t to be the characteristic function of the uniform distribution. We seek a sharp lower-bound on this expression (attained when a 1 = · · · = a n = 1 √ n and n → ∞, as anticipated by Theorem 1). By the AM-GM inequality, n j=1 φ(a j t) n j=1 a 2 j \|φ(a j t)\| 1/a 2 j . As a result, E n j=1 a j U j p n j=1 a 2 j I p (1/a 2 j ), where we have set I p (s) = κ p ∞ 0 1 − sin(t/ √ s) t/ √ s s t p+1 dt, s 1. Note that sin(t/ √ s) t/ √ s = 1 − t 2 6s + O(1/s 2 ) for a fixed t as s → ∞ and consequently, I p (∞) = lim s→∞ I p (s) = κ p ∞ 0 1 − e −t 2 /6 t p+1 dt = E\|Z/ √ 3\| p , where the last equality follows from Lemma 4 because e −t 2 /6 is the characteristic function of Z/ √ 3, Z ∼ N (0, 1) (the exchange of the order of the limit and integration in the second equality can be easily justified by truncating the integral, see, e.g., (15) in [4]). In particular, if for some p and s 0 , (3) I p (s) I p (∞), for all s s 0 , then (4) E n j=1 a j U j p E\|Z/ √ 3\| p = c p p , as long as 1/a 2 j s 0 for each j. If (3) were true for all 0 < p < 1 with s 0 = 1, then the proof of Theorem 1 would be complete. Unfortunately, that is not the case. In Section 4 we show the following result. Theorem 5. Inequality (3) holds for every 0.6 < p < 1 with s 0 = 1. As a result, when 0.6 < p < 1, (4) holds for arbitrary a j and the proof of Theorem 1 is complete in this case. For smaller values of p, s 0 has to be increased. Theorem 6. Inequality (3) holds for every 0 < p < 1 with s 0 = 2. This is proved in Section 4. Consequently, (4) holds provided that a 2 j 1 2 for each j. To remove this restriction, we employ an inductive argument of Nazarov and Podkorytov from [24] developed for random signs and adapted to the uniform distribution in [4]. This works for 0 < p < 0.69 and the proof of Theorem 1 is complete. This is done in Section 5. Auxiliary lemmas To show Theorems 5 and 6 and carry out the inductive argument, we first prove some technical lemmas. 3.1. Lemmas concerning the sinc function. The zeroth spherical Bessel function (of the first kind) j 0 (x) = sin x x = sinc(x) is sometimes referred to as the sinc function. As the characteristic function of a uniform random variable, it plays a major role in our approach. We shall need several elementary estimates. Lemma 7. For 0 < t < π, we have sin t t < e −t 2 /6 . Proof. This follows from the product formula, sin t t = ∞ n=1 1 − t 2 n 2 π 2 . Since each term is positive for 0 < t < π, the lemma follows by applying 1 + x e x and ∞ n=1 1 n 2 = π 2 6 . Lemma 8. sup t∈R cos t − sin t t < 11 10 . Proof. Since both cos t and sin t t are even, it suffices to consider positive t. By the Cauchy-Schwarz inequality, we have cos t − sin t t 1 + 1 t 2 , so it suffices to consider t < 10 √ 21 . On (0, π 2 ), we have cos t − sin t t = sin t t − cos t < 1 + 0 = 1, so it remains to consider π 2 < t < 10 √ 21 . Letting t = π 2 + x, we have for 0 < x < 10 √ 21 − π 2 , cos t − sin t t = sin t t − cos t = cos x x + π/2 + sin x < 1 x + π/2 + x. Examining the derivative, the right hand side is clearly increasing, so it is upper bounded by its value at x = 10 √ 21 − π 2 which is √ 21 10 + 10 √ 21 − π 2 < 1.07. Lemma 9. Let k 0 be an integer and let y k be the value of the unique local maximum of sin t t on (kπ, (k + 1)π). Then 1 (k + 1/2)π y k 1 kπ . Moreover, y 1 < e −3/2 . Proof. The lower bound follows from taking t = (k + 1/2)π, whereas the upper bound follows from \| sin t\| 1 and t > kπ. The bound on y 1 is equivalent to sin t < e −3/2 (t + π), 0 < t < π/2. To show this in turn, it suffices to upper bound sin t by its tangent at, e.g., t = 1.3. Lemma 10. For y ∈ (0, 1 30π ), let t = t 0 be the unique solution to sin t t = y on (0, π). Then t 0 > 0.98π. Let t = t 1 be the larger of the two solutions to \| sin t\| t = y on (π, 2π). Then t 1 > 1.97π. Proof. Note that sin t0 t0 = y < 1 30π < sin(0.98π) 0.98π . Since sin t t is decreasing on (0, π), it follows that t 0 > 0.98π. Similarly, we check that \| sin(1.97π)\| 1.97π > 1 30π to justify the claim about t 1 . Lemma 11. For 0 < x < π, 1 sin 2 x > 1 x 2 + 1 (π − x) 2 . 6 Proof. It is well known (and follows from sin(2x) = 2 sin x cos x) that sin x x = ∞ k=1 cos(x/2 k ). In particular, for 0 < x < π, we have sin x x 2 < cos 2 (x/2), hence sin 2 x 1 x 2 + 1 (π − x) 2 = sin x x 2 + sin(π − x) π − x 2 < cos 2 (x/2) + cos 2 (π/2 − x/2) = 1. Lemma 12. Let k 1 be an integer. On ((k − 1)π, kπ), we have (i) the function \| sin t\| t(t−(k−1)π) is nonincreasing, (ii) the function \| sin t\| t(kπ−t) is unimodal (first increases and then decreases). Proof . (i) The derivative equals \| sin t\| t(t − (k − 1)π) cot(t) − 1 t − 1 t − (k − 1)π which is negative on ((k − 1)π, kπ) because on this interval, cot(t) < 1 t−(k−1)π (as, by periodicity, being equivalent to cot(t) < 1 t on (0, π), which is clear -recall that tan(x) > x on (0, π 2 )). (ii) Here, the derivative reads \| sin t\| t(kπ − t) h(t), h(t) = cot(t) − 1 t + 1 kπ − t . We shall argue that h(t) is decreasing on ((k −1)π, kπ). This suffices, since h(t) > 0 for t near (k − 1)π and h(t) < 0 for t near kπ. Setting t = (k − 1)π + x, we have h (t) = − 1 sin 2 t + 1 t 2 + 1 (kπ − t) 2 − 1 sin 2 x + 1 x 2 + 1 (π − x) 2 < 0, by Lemma 11. 3.2. Lemmas concerning sums of p-th powers. Our computations require several technical bounds on various expressions involving sums of p-th powers. Lemma 13. Let 0 < p < 1 and let 1 m 29 be an integer. Set u m (p) = B m b p 0,m + 2 m k=1 b p k,m with B m = 20 log π(m + 3/2) 11π(m + 3/2) , b k,m = 1 k + 1 6 π 2 log π(m + 3/2) . Then, u m (p) > 1. Proof. Fix m. Plainly, u m (p) is a convex function (as a sum of convex functions). Thus, u m (p) < u m (1) for 0 < p < 1. We have, u m (1) = B m b 0,m log b 0,m + 2 m k=1 b k,m log b k,m and Table 1 shows that each u m (1) is negative, so each u m is decreasing. Therefore, u m (p) > u m (1), for 0 < p < 1 and Table 1 shows that each u m (1) is greater than 1. This finishes the proof. α p = 2π −p+1 3 − 1 1 − p + 3 2p , β p = 2 1 − p + 1.05 p p , γ p = 3π p , δ p = 1 p 30π 6 log(30π) p/2 and h p (y) = δ p y p 2 −1 + γ p y p − β p y p−1 − α p . Then h p (y) > 0 for every 0 < y < 1 30π . Proof. Plainly, it suffices to show the following two claims, h p (y) < 0, 0 < y < 1 30π ,(5)h p 1 30π > 0.(6) To prove (5), first we find y 2− p 2 h p (y) = − 1 − p 2 δ p + pγ p y p 2 +1 + (1 − p)β p y p 2 which is clearly increasing in y, thus to show that it is negative, it suffices to prove that at y = 1 30π , which in turn is equivalent to 2.1p + (1 − p)1.05 p < 1 − p 2 30π 6 log(30π) p . Crudely, (1 − p)1.05 p < 1.05 p < 1 + 0.05p, by convexity, thus it suffices to show that 1 + 2.15p < 1 − p 2 A p , where we put A = 30π √ 6 log(30π) . Equivalently, after taking the logarithm, the inequality becomes p log A + log 1 − p 2 − log(1 + 2.15p) > 0. Note that at p = 0 this becomes equality. We claim that the derivative of the left hand side is positive for 0 < p < 1, which will finish the argument. The derivative is log A− 1 2−p − 2.15 1+2.15p which is clearly concave, thus it suffices to examine whether it is positive at the end-points p = 0 and p = 1, which respectively becomes log A > 2.65 and log A > 1 + 2. 15 3.15 . Since log A = 2.89.., both are clearly true. It remains to show (6), that is that the following is positive for every 0 < p < 1, 30 p π p−1 h p 1 30π = 30 30π √ 6 log(30π) p − 1.05 p p L(p) − 2 30 − 30 p 1 − p + 3 30 p − 1 p + 6 · 30 p R(p) . Both L(p) and R(p) are strictly increasing and convex on (0, 1). This is clear for L, since its Taylor With hindsight, the tangent and the chord are chosen such that > r on (0, 0.6), which can be checked directly by looking at the values of these linear functions at the end-points. Again, with hindsight, the tangent and the chord are chosen such that˜ >r on (0.6, 1). This completes the proof. 3.3. Lemmas concerning the gamma function. For the inductive part of our argument, we will later need bounds on the following function ψ(p) = 1 + p √ π 4 3 p/2 Γ 1 + p 2 , 0 < p < 1. Recall the Weierstrass' product formula, Γ(z) = e −γz Proof. We show that f (p) = log(2 − (3/2) p/2 ) + log ψ(p) is negative on (0, 0.69). By virtue of (7), f (p) = − 1 2 log 2 3 2 (3/2) p/2 (2 − (3/2) p/2 ) 2 + ∞ n=1 1 (2n + 1 + p) 2 This is plainly a decreasing function. Using Proof. We show that f (p) = − log 1 + p(p + 1) 6 + log ψ(p) is negative on (0, 1). Since f (0) = 0, it suffices to show that f (p) < 0 on (0, 1). Using (7), we have f (p) = − 2p + 1 p 2 + p + 6 + 1 2 (log(4/3) − γ) + ∞ n=1 1 2n − 1 2n + 1 + p . Now, for R(p) = − 2p+1 p 2 +p+6 + 1 2 (log(4/3) − γ), R (p) = (2p+1)(17−p 2 −p) (p 2 +p+6) 3 > 0 on (0, 1), so R(p) is convex on (0, 1). Let S(p) = ∞ n=1 1 2n − 1 2n+1+p . Plainly, this is a concave function. Thus, using tangents at p = 0 and p = 1, S(p) min{L 0 (p), L 1 (p)} with L 0 (p) = S(0) + S (0)p = (1 − log 2) + ( π 2 8 − 1)p and L 1 (p) = S(1) + S (1)(p − 1) = 1 2 + π 2 −6 24 (p − 1). We obtain the upper-bounds on f (p) by the convex functions R(p) + L 0 (p) and R(p) + L 1 (p). Examining the end-points we conclude that the former is negative on (0, 0.5) and the latter is negative on (0.4, 1). Thus f (p) < 0 on (0, 1), as desired. Integral inequality: proofs of Theorems 5 and 6 First observe that using the integral expression for I p (∞), inequality (3) becomes 0 I p (s) − I p (∞) = κ p ∞ 0 e −t 2 /6 − sin(t/ √ s) t/ √ s s t p+1 dt = κ p s −p/2 ∞ 0 e −st 2 /6 − sin t t s t p+1 dt.(8) To tackle such an inequality with an oscillatory integrand, we rely on the following extremely efficient and powerful lemma of Nazarov and Podkorytov from [24] (for the proof, see e.g. [15]). Lemma 17 (Nazarov-Podkorytov, [24]). Let M ∈ (0, ∞] and f, g : X → [0, M ] be any two measurable functions on a measure space (X, µ). Assume that the modified distribution functions F (y) = µ({x ∈ X : f (x) < y}) and G(y) = µ({x ∈ X : g(x) < y}) of f and g respectively are finite for every y ∈ (0, M ). If there exists y * ∈ (0, M ) such that G(y) F (y) for all y ∈ (0, y * ), G(y) F (y) for all y ∈ (y * , M ), then the function s → 1 sy s 0 X (g s − f s ) dµ is increasing on the set {s > 0 : g s − f s ∈ L 1 (X, µ)}. In view of (3), (4) and (8), Theorems 5 and 6 immediately follow from the following lemma. Lemma 18. Let f (t) = sin t t , g(t) = e −t 2 /6 , t > 0, and set H(p, s) = ∞ 0 g(t) s − f (t) s t p+1 dt. We have, Proof. Fix 0 < p < 1. We examine the modified distribution functions F (y) = µ(t > 0, f (t) < y), G(y) = µ(t > 0, g(t) < y), 0 < y < 1, where dµ(t) = t −p−1 dt. It suffices to show that ( ) G − F changes sign exactly once on (0, 1) at some y = y * from + to −. Then Lemma 17 gives that s → 1 sy s * H(p, s) is increasing on (0, ∞). In particular, (a) and (b) result from the following claims whose proofs we defer until the end of this proof. Claim A. H(p, 2) 0 for every 0 < p < 1. Claim B. H(p, 1) 0 for every 0.6 < p < 1. Towards ( ), let 1 = y 0 > y 1 > y 2 > . . . be the consecutive maximum values of f . On (0, π), f g (Lemma 7), so G − F < 0 on (y 1 , 1). We plan to find a ∈ (0, y 1 ) with the following two properties (i) (G − F ) < 0 on (a, y 1 ), (ii) G − F > 0 on (0, a). This clearly suffices to conclude ( ). Fix m ∈ {1, 2, . . . } and y ∈ (y m+1 , y m ). Plainly, G(y) = ∞ √ −6 log y dt t p+1 = 1 p (−6 log y) −p/2 . Let t + 0 = t + 0 (y) be the unique solution to f (t) = y on (0, π) and for each 1 k m, let t − k < t + k be the unique solutions to f (t) = y on (kπ, (k + 1)π) (t ± k = t ± k (y) are functions of y). We have, Condition (i). Recall that y ∈ (y m+1 , y m ). We have, (9) F (y) = µ(t + 0 , t − 1 ) + µ(t + 1 , t − 2 ) + · · · + µ(t + m−1 , t − m ) + µ(t + m , ∞). f g y 1 y 2 y π 2π 3π 4π 5π 6π t − 1 t + 1 t − 2 t + 2 t −G (y) = 3 y (−6 log y) −p/2−1 and, differentiating (9) with respect to y (using the fundamental theorem of calculus and chain rule), F (y) = t:f (t)=y 1 t p+1 \|f (t)\| . To lower bound F G in order to show that it is greater than 1, we lower bound F and 1 G separately as follows. First, using \|tf (t)\| = \| cos t − sin t t \| < 11 10 for every t > 0 (Lemma 8), we have, F (y) > 10 11 t:f (t)=y t −p > 10 11 π −p 1 + 2 m k=1 (k + 1) −p , by crudely bounding t − 0 < π, t ± k < (k + 1)π. Second, since y(−6 log y) p/2+1 is increasing on (0, y 1 ) (it is increasing on (0, e −1−p/2 ) and e −1−p/2 > e −3/2 > y 1 ), and y m+1 > 1 π(m+3/2) (Lemma 9), 1 G (y) = 1 3 y (−6 log y) p/2+1 > 1 3 y m+1 (−6 log y m+1 ) p/2+1 > 1 3 1 π(m + 3/2) 6 log π(m + 3/2) p/2+1 . We obtain F (y) G (y) > 10 33 1 π p+1 (m + 3/2) 6 log π(m + 3/2) p/2+1 1 + 2 m k=1 (k + 1) −p . From Lemma 13 the right hand side is at least 1 for every 0 < p < 1 and 1 m 29. Therefore, to guarantee that Condition (i) holds, we can choose any a y 30 . We set a = y 30 and argue next that Condition (ii) holds for every y ∈ (0, a). Condition (ii). We assume here that m 30. Recall we have fixed y ∈ (y m+1 , y m ). Since G is explicit, it suffices to upper bound F . We have, F (y) = m k=1 t − k t + k−1 dt t p+1 + ∞ t + m dt t p+1 m k=1 (t − k − t + k−1 )(t + k−1 ) −p−1 + 1 p (t + m ) −p . For k 3, we crudely estimate t + k−1 (k − 1)π, whereas for k = 1, 2, we have t + 0 > 0.98π and t + f y k−1 Figure 2. The slope of the segment AB is not smaller than the slope of either AC or BC. y k y A B C kπ t + k−1 t − k t k−1t k2y t − k − t + k−1 = sin t − k t − k − sin t + k−1 t + k−1 t − k − t + k−1 = \|slope(AB)\| min {\|slope(AC)\|, \|slope(BC)\|} . Lett k ∈ (kπ, (k + 1)π) denote the point where f (t) attains its local maximum y k on (kπ, (k + 1)π). Observe that \|slope(BC)\| = \| sin t − k \| t − k (t − k − kπ) y k t k − kπ y k π , where the first inequality follows from Lemma 12 (i) applied to t − k <t k . Similarly, \|slope(AC)\| = \| sin t + k−1 \| t + k−1 (kπ − t + k−1 ) min y k−1 kπ −t k−1 , 1 kπ min y k−1 π , 1 kπ , where in the first inequality we use Lemma 12 (ii) to lower bound the function in question by the minimum of its values at the end-points t =t k−1 and t = kπ. Finally, putting these two estimates together and using y k > 1 π(k+ 1 2 ) , we obtain \|slope(AB)\| 1 π 2 (k + 1 2 ) and, consequently, t − k − t + k−1 = 2y \|slope(AB)\| 2π 2 y k + 1 2 , which results in F (y) < 2π −p+1 y 3 2 0.98 −p−1 + 5 2 1.97 −p−1 + m k=3 k + 1 2 (k − 1) −p−1 + 1 p (mπ) −p . Since y > y m+1 > 1 (m+ 3 2 )π , and m 30, we have 1 p (mπ) −p < 1 p m + 3/2 m p y p 1 p 1.05 p y p . Moreover, since y < y m < 1 mπ , we have (crudely), m − 1 < 1 πy and bounding the sum using the integral, we obtain m k=3 k + 1 2 (k − 1) −p−1 = m−1 k=2 k + 3 2 k p+1 < m−1 1 x −p + 3 2 x −p−1 dx < (πy) p−1 − 1 1 − p + 3(1 − (πy) p ) 2p . Therefore, in order to have F (y) < G(y), it suffices to guarantee that 2π −p+1 y 3 2 0.98 −p−1 + 5 2 1.97 −p−1 + (πy) p−1 − 1 1 − p + 3(1 − (πy) p ) 2p + 1 p 1.05 p y p < 1 p (−6 log y) −p/2 holds for every 0 < p < 1 and 0 < y < 1 30π . Since −y log y is increasing for y < 1 e , we have − log y < log(30π) 30π 1 y for 0 < y < 1 30π . By monotonicity, for 0 < p < 1, we have 3 2 0.98 −p−1 + 5 2 1.97 −p−1 < 3 2 0.98 −1−1 + 5 2 1.97 −1 < 3. It remains to use Lemma 14. This shows that Condition (ii) holds and the proof of the lemma is complete. It remains to show Claims A and B. Proof of Claim A. By the integral representation for the p-norm from Lemma 4, κ p H(p, 2) = E\|U 1 + U 2 \| p − E 2 3 Z p = 2 p+1 (p + 1)(p + 2) − 1 √ π 4 3 p/2 Γ 1 + p 2 . By Lemma 16, it suffices to prove that 2 p+1 > (p+2) 1 + p(p+1) 6 for all 0 < p < 1. The 3rd derivative of the difference changes sign once on (0, 1) from − to +. The 2nd derivative is negative at the end-points p = 0 and p = 1, so it is negative on (0, 1) and hence the difference is concave. It vanishes at the end-points p = 0 and p = 1, which finishes the argument. Proof of Claim B. Our argument is split into two steps: first we show that H(p, 1) increases with p and then we estimate H(0.6, 1). For somewhat similar computations, but related to random signs, see Section 5 in [21]. In Step 1, to numerically evaluate the integrals in question, we will frequently use that given 0 < a < b and an integer m, integrals of the form b a (sin t)t −m dt can be efficiently estimated to an arbitrary precision by expressing them in terms of the trigonometric integral functions Si, Ci. The same applies to the integrals of the form b a e −t 2 t q dt with 0 < a < b ∞ and real q, thanks to reductions to the incomplete gamma function Γ and the exponential integral Ei. We recall that for x > 0, s = 0, −1, −2, . . . , Si(x) = − ∞ x sin t t dt = − π 2 − ∞ k=1 (−1) k x 2k−1 (2k − 1)(2k − 1)! , Ci(x) = − ∞ x cos t t dt = γ + log x + ∞ k=1 (−1) k x 2k 2k(2k)! , Ei(−x) = − ∞ x e −t t dt = γ + log x + ∞ k=1 (−x) k k · k! , Γ(s, x) = ∞ x t s−1 e −t dt = Γ(s) − ∞ k=0 (−1) k x s+k k!(s + k) (here γ = 0.57721.. is the Euler-Mascheroni constant). These series representations allow to obtain arbitrarily good numerical approximations to these integrals. In Step 2, all the numerical computations are reduced to integrals of the form b a dt t q which are explicit. Step 1: ∂ ∂p H(p, 1) > 0, 0.6 < p < 1. We have, ∂ ∂p H(p, 1) = ∞ 0 (− log t) g(t) − f (t) t p+1 dt. We break the integral into several regions. Recall g > f on (0, π), by Lemma 7. Thus, plainly, 1 0 (− log t) g(t) − f (t) t p+1 dt > 0. Moreover, g − f changes sign from + to − exactly once on (π, 4) at t = 3.578... Let t 0 = 3.57. On (1, t 0 ), using t −p−1 = t 1−p t −2 t 1−p 0 t −2 , we obtain t0 1 (− log t) g(t) − f (t) t p+1 dt t 1−p 0 t0 1 (− log t) g(t) − f (t) t 2 dt > −0.0297 · t 1−p 0 , where in the last inequality we use log t log 5 2 + 2 5 (t − 5 2 ) (by concavity) and then estimate the resulting integrals. Now, ∞ t0 (− log t) g(t) − f (t) t p+1 dt = ∞ t0 (log t) f (t) t p+1 dt − ∞ t0 (log t) g(t) t p+1 dt. For t > t 0 , t −p−1 = t 1−p t −2 > t 1−p 0 t −2 and for k 1, log t k (t) on (kπ, (k + 1)π) with k (t) = (k + 1)π − t π log(kπ) + t − kπ π log((k + 1)π), thus ∞ t0 (log t) f (t) t p+1 dt t 1−p 0 2π t0 1 (t) − sin t t 3 dt + n k=2 (k+1)π kπ k (t) (−1) k sin t t 3 dt . For n = 5, this gives ∞ t0 (log t) f (t) t p+1 dt > 0.0437 · t 1−p 0 . 16 Finally, since log u u e , u > 0, we have log t t p < 1 ep < 1 0.6e < 0.6132 < 0.6132 · t 1−p 0 , thus ∞ t0 (log t) g(t) t p+1 dt 0.6132 · t 1−p 0 ∞ t0 e −t 2 /6 t dt < 0.0127 · t 1−p 0 . Putting these together yields ∂ ∂p H(p, 1) > (0.0437 − 0.0297 − 0.0127)t 1−p 0 = 0.0013 · t 1−p 0 > 0. Step 2: H(0.6, 1) > 0. We have, H(0.6, 1) = π 0 e −t 2 /6 − sin t t t 8/5 dt + ∞ π e −t 2 /6 t 8/5 dt − ∞ π \| sin t\| t 13/5 dt. On (0, π), we use Taylor's polynomials to bound the integrand, e −t 2 /6 − sin t t > 7 k=0 (−t 2 /6) k k! − 6 k=0 (−1) k t 2k (2k + 1)! . Plugging this into the integral results in We use Taylor's polynomial again, sin t 1 − 1 2 (t − π/2) 2 + 1 24 (t − π/2) 4 . Choosing n = 8 gives ∞ π \| sin t\| t 13/5 dt < 0.0615. Adding up these estimates yields H(0.6, 1) > 0.0434+0.0184−0.0615 = 0.0003. Inductive argument As explained in Section 2, Theorem 6 gives the following corollary (we use homogeneity to rewrite (4) in an equivalent form, better suited for the ensuing arguments). Recall c p = Z p / √ 3 and define ϕ p (x) = (1 + x) p/2 , x 0. 17 Corollary 19. Let 0 < p < 1. For every n 2 and real numbers a 2 , . . . , a n with n j=2 a 2 j 1 and a 2 j 1 for every j = 2, . . . , n, we have E U 1 + n j=2 a j U j p c p p · ϕ p   n j=2 a 2 j   . The goal here is to remove the restriction on the a j 's. The key idea from [24] is to replace ϕ p with a pointwise larger function, thereby strengthening the inequality and to proceed by induction on n. We use the function from [24], Φ p (x) = ϕ p (x), x 1, 2ϕ p (1) − ϕ p (2 − x), 0 x 1. Even though this function changes from being convex to concave at x = 1, it is designed to satisfy the following extended convexity property on [0, 2], crucial for the proof. Lemma 20 (Nazarov-Podkorytov, [24]). For every 0 < p < 2 and a, b ∈ [0, 2] with a + b 2, we have Φ p (a) + Φ p (b) 2 Φ p a + b 2 . As in [4], in order to have certain algebraic identities, we run the argument for ξ 1 , ξ 2 , . . . , independent random vectors in R 3 uniformly distributed on the centred unit Euclidean sphere S 2 . Here ·, · and · is the standard inner product and the resulting Euclidean norm in R 3 , respectively. Here e 1 = (1, 0, 0), the unit vector of the standard basis. Note v j , ξ j has the same distribution as v j U j (by rotational invariance, v j , ξ j has the same distribution as v j e 1 , ξ j and by the Archimedes' hat-box theorem, the projection e 1 , ξ j is a uniform random variable on [−1, 1]). Since Φ p ϕ p , this gives Theorem 1 for 0 < p < 0.69, thereby completing its proof. It remains to show Theorem 21, which is done by repeating almost verbatim the proof of Theorem 18 from [4]. We repeat the argument for the convenience of the reader. To adjust the proof of the base case we will need the following lemma. Lemma 22. For every 0 < x < 1 and 0 < p < 0.69, we have (1 + x) 2+p − (1 − x) 2+p 2(2 + p)x > 1 + p √ π Γ 1 + p 2 2 3 p/2 2 1+p/2 − (3 − x 2 ) p/2 = ( Z p / √ 3) p (1 + p)Φ p (x 2 ). Proof. We first observe that keeping only the first two terms in the binomial series expansion, we obtain (1 + x) 2+p − (1 − x) 2+p 2(2 + p)x = ∞ k=0 1 p + 2 p + 2 2k + 1 x 2k > 1 + p(p + 1) 6 x 2 , because all the terms are positive. It thus suffices to show that for every 0 < x < 1 and 0 < p < 0.69, 1 + p(p + 1) 6 x + 1 + p √ π Γ 1 + p 2 2 3 p/2 (3 − x) p/2 − 2 1+p/2 > 0 (we have replaced x 2 by x). By the evident concavity in x, it suffices to check that the inequality holds at the end-points x = 0 and x = 1 which follows from Lemmas 16 and 15, respectively. Proof of Theorem 21. For the case n = 2, we need to show that for every v ∈ R 3 (11) E\| e 1 , ξ 1 + v, ξ 2 \| p c p p Φ p ( v 2 ). We first reduce this claim to the case v 1: If v > 1 then due to rotational invariance E\| e 1 , ξ 1 + v, ξ 2 \| p = v p E e 1 v , ξ 1 + v v , ξ 2 p = v p E\| v , ξ 1 + e 1 , ξ 2 \| p , where v ∈ R 3 is such that v = 1 v < 1. On the other hand, due to homogeneity, Φ p ( v 2 ) = φ p ( v 2 ) = v p φ p ( v 2 ), so (11) is equivalent to E\| v , ξ 1 + e 1 , ξ 2 \| p c p p φ p ( v 2 ), v 1 and since Φ p (x) φ p (x) for x ∈ [0, 1] it is indeed sufficient to restrict to the case v 1. In this case, we set x := v 1 and compute explicitly the left and right hand side of (11) to deduce that E\| e 1 , ξ 1 + v, ξ 2 \| p = E\|U 1 + xU 2 \| p = (1 + x) 2+p − (1 − x) 2+p 2(1 + p)(2 + p)x c p p Φ p (x 2 ) with the aid of Lemma 22. For the inductive step, let n ∈ N and assume that (10) holds for every v 2 , . . . , v n−1 ∈ R 3 . We let v 2 , . . . , v n ∈ R 3 , x := , Date: November 16, 2021. TT's research supported in part by NSF grant DMS-1955175. 1 arXiv:2105.14079v3 [math.PR] 13 Nov 2021 : 0 < p < 0. 6 .( 6expansion at p = 0 has positive coefficients. Similarly for the term 30 p −1 p in R(p). To see that 30−30 p By convexity, using a tangent line L(p) L(0.24) + L (0.24)(p − 0.24) = Case 2 : 20.6 < p < 1. Similarly, by convexity, using a tangent line L(p) L(0.8) + L (0.8)(p − 0.8) =˜ (p) and a chord R(p) 1 − p 0.4 R(0.6) + p − 0.6 0.4 R(1 − ) =r(p). we get with f (0.9) > 0.007, so f is strictly convex on (0, 0.9). Checking that f (0) = 0 and f (0.69) < −0.0001 finishes the proof.Lemma 16. For 0 < p < 1, we have ψ(p) < 1 + p(p+1)6 . (a) H(p, s) 0 0for every 0 < p < 1 and s 2, (b) H(p, s) 0 for every 0.6 < p < 1 and s 1. Figure 1 . 1Functions f , g and the set {t > 0, f (t) < y}. Here m = 3, i.e. y 3 < y < y 4 . Theorem 21 . 21Let 0 < p < 0.69. For every n 2 and vectors v 2 , . . . , v n in R 3 Table 1 . 1Lower bounds on the values of −u m (1) and u m (1). −u m (1) 0.24 0.44 0.58 0.70 0.79 0.86 0.91 0.96 0.99 1.02 u m (1) 1.06 1.27 1.36 1.40 1.41 1.41 1.40 1.38 1.36 1.34 −u m (1) 1.05 1.07 1.08 1.10 1.11 1.12 1.13 1.13 1.14 1.14 −u m (1) 1.14 1.14 1.15 1.15 1.15 1.15 1.15 1.15 1.14m 1 2 4 4 5 6 7 8 9 10 m 11 12 13 14 15 16 17 18 19 20 u m (1) 1.32 1.29 1.27 1.25 1.23 1.21 1.19 1.17 1.15 1.14 m 21 22 23 24 25 26 27 28 29 u m (1) 1.12 1.10 1.09 1.07 1.06 1.04 1.03 1.02 1.00 Lemma 14. For 0 < p < 1, let > 1.97π, thanks to Lemma 10. To upper bound the length t − k − t + k−1 , note that with the aid ofFigure 2, Acknowledgments. We should very much like to thank Alexandros Eskenazis for the stimulating correspondence. We are also indebted to anonymous referees for many valuable comments which helped significantly improve the manuscript. Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions. 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[ "Neutral Signature Gauged Supergravity Solutions", "Neutral Signature Gauged Supergravity Solutions" ]	[ "J B Gutowski \nDepartment of Mathematics\nUniversity of Surrey Guildford\nGU2 7XHUK\n", "W A Sabra \nPhysics Department\nAmerican University of Beirut\nLebanon\n" ]	[ "Department of Mathematics\nUniversity of Surrey Guildford\nGU2 7XHUK", "Physics Department\nAmerican University of Beirut\nLebanon" ]	[]	We classify all supersymmetric solutions of minimal D = 4 gauged supergravity with (2, 2) signature and a positive cosmological constant which admit exactly one Killing spinor. This classification produces a geometric structure which is more general than that found for previous classifications of N = 2 supersymmetric solutions of this theory. We illustrate how the N = 2 solutions which consist of a fibration over a 3-dimensional Lorentzian Gauduchon-Tod base space can be written in terms of this more generic geometric structure.	10.1007/jhep02(2021)030	[ "https://arxiv.org/pdf/2006.06312v1.pdf" ]	219,573,521	2006.06312	864d64bae79bfd46b74f32b3ced66e10589ab054	Neutral Signature Gauged Supergravity Solutions 11 Jun 2020 J B Gutowski Department of Mathematics University of Surrey Guildford GU2 7XHUK W A Sabra Physics Department American University of Beirut Lebanon Neutral Signature Gauged Supergravity Solutions 11 Jun 2020arXiv:2006.06312v1 [hep-th] We classify all supersymmetric solutions of minimal D = 4 gauged supergravity with (2, 2) signature and a positive cosmological constant which admit exactly one Killing spinor. This classification produces a geometric structure which is more general than that found for previous classifications of N = 2 supersymmetric solutions of this theory. We illustrate how the N = 2 solutions which consist of a fibration over a 3-dimensional Lorentzian Gauduchon-Tod base space can be written in terms of this more generic geometric structure. Introduction Substantial progress has been made in the classification and understanding of solutions admitting supersymmetry in four-dimensional supergravity theories. This was initiated by the work of Tod [1,2] which dealt with the classification of all solutions admitting Killing spinors in Lorentzian four-dimensional N = 2 supergravity theory. Later, systematic classifications were performed for the Lorentzian four-dimensional gauged N = 2 supergravity theory in [3,4,5]. This subsequent work relied on the use of Fierz identities. Other work classifying supersymmetric four-dimensional solutions coupled to hypermultiplets was done in [6]. In recent years, spinorial geometry techniques have proven to be particularly useful for the analysis and classification of supersymmetric solutions. This method relies on expressing spinors in terms of differential forms [7,8]. In using this method to classify supergravity solutions, one exploits the gauge symmetries of the theory to find simple canonical forms for the Killing spinors. Those canonical forms are then employed to find a linear system which facilitates the finding of solutions for the Killing spinor equations. The spinorial geometry techniques were first employed in the classification of solutions with Killing spinors in D = 11 supergravity, heterotic and type II supergravity theories in [9,10,11,12,13]. Also more work in the classification of four-dimensional solutions using spinorial geometry was performed in [14,15]. For more details on the spinorial geometry applications to the classification of supersymmetric solutions, we refer the reader to the review [16]. The classification of solutions with Killing spinors was extended to the cases of four-dimensional Euclidean N = 2 supergravity in [17,18,19], by making use of the 2-component spinor formalism and the spinorial geometry techniques. In this analysis, interesting relations of supersymmetric solutions to Einstein-Weyl structures, and the SU(∞) Toda equation were discovered. The case of geometries with neutral signature, i.e. signature (+, +, −, −), is of particular interest, partly because such the properties of such solutions have not been extensively explored in the literature, at least in comparison to the Euclidean and (−, +, +, +) Lorentizian cases. The analysis of neutral signature solutions admitting parallel spinors in four-dimensional gravity has been performed in [20,21,22], where solutions admitting null-Kähler structures were obtained. Useful and extensive details on the neutral signature theory as well as some of its solutions can be found in [23]. A systematic classification of solutions in four-dimensional Einstein-Maxwell theory with or without cosmological constant in neutral signature was considered in [24]. In a more recent paper [25], we have used spinorial geometry techniques to classify supersymmetric solutions of the the minimal four-dimensional Einstein-Maxwell with neutral signature and without a cosmological constant. Such a theory can be thought of as a truncation of the N = 2 supergravity theory obtained from Hull's M theory [26] via a reduction on CY 3 × S 1 [27]. In the analysis of [25], two orbits for the Majorana Killing spinors were found using appropriately chosen Spin(2, 2) gauge transformations. The solutions for the orbit represented by a chiral Killing spinor were found to correspond to a sub-class of the solutions found [20,21,22]. Moreover, a novel geometric structure was discovered for the solutions corresponding to the second non-chiral orbit. Our current work is concerned with the classification of all solutions admitting one Killing spinor in minimal D = 4 gauged supergravity with (2, 2) signature and a positive cosmological constant. This however does not simply reduce to a subclass of the solutions considered previously in [24], because a class of solutions corresponding to the special case for which there is a single Majorana Killing spinor was omitted from the classification in that work. In fact, all of the solutions with positive cosmological constant classified in [24] have N = 2 supersymmetry. The purpose of this work is to understand the geometric properties of solutions with minimal N = 1 supersymmetry, associated with a Majorana Killing spinor, and to find examples of such solutions. We plan our paper as follows. In Section two, we introduce the Killing spinor equation of our theory as well as a summary of conventions and results of [25] which are relvant to our subsequent analysis. Relations involving two Spin(2, 2) invariant spinor bilinears are presented and it is demonstrated that only one of the two canonical Killing spinors obtained in [25] survives in the gauged theory. Section three contains the analysis of the Killing spinor equations for the remaining non-chiral orbit and the geometric conditions arising from this analysis. In section four, we present specific examples of our general solutions corresponding to solutions which are fibrations over a Lorentzian Gauduchon-Tod space as well the specific example corresponding to the (2,2) analogue of Kastor-Traschen solution. Self-dual solutions are explored in section five and we conclude in section six. Appendix A contains the linear system corresponding to the non-chiral Killing spinor. Majorana Spinor Orbits In this section, we introduce the Killing spinor equation which we shall analyse, and summarize a number of results concerning spinor conventions from [25] which will be particularly useful. The Killing spinor equation (KSE) is given by D µ ǫ ≡ ∇ µ ǫ − 1 4 / F Γ µ ǫ − 1 2ℓ Γ µ ǫ − 1 ℓ A µ ǫ = 0 (2.1) where F is the Maxwell field strength, F = dA, which satisfies dF = 0, d ⋆ F = 0 . (2.2) For spinors, we adopt the same conventions as introduced in [25]. In particular, there exists a charge conjugation operator C * , [C * , Γ µ ] = 0, and hence if ǫ satisfies (2.1) then so does C * ǫ. In this paper, we shall concentrate in particular on Majorana Killing spinors ǫ which satisfy C * ǫ = ǫ. We note that this class of solutions was omitted from the classification constructed in [24]. In particular, supersymmetric solutions of the minimal D = 4 gauged supergravity in neutral signature with a positive cosmological constant must admit spinors satisfying (2.1), and for a solution to preserve the minimal possible N = 1 supersymmetry, then the Killing spinor must be Majorana, or otherwise a second linearly independent Killing spinor can be constructed by making use of C * . Hence, we shall classify the minimal N = 1 solutions of (2.1). In contrast, all of the supersymmetric solutions of this theory classified in [24] preserved N = 2 supersymmetry, and are therefore special cases of a more generic structure corresponding to the N = 1 case, which we present here. As observed in [25], a generic Majorana spinor can be put into one of two possible canonical forms using gauge transformations. In the first canonical form, the spinor is chiral, whereas in the second canonical form, it is not. The canonical forms can be further characterized in terms of certain Spin(2, 2) invariant spinor bilinears. The spinor bilinears are 1-forms and 2-forms W and χ, given by W µ = iB(ǫ, Γ 5 Γ µ ǫ), χ µν = iB(ǫ, Γ 5 Γ µν ǫ) , (2.3) where the inner product B satisfies B(ǫ, η) = −B(η, ǫ), B(ǫ, γ µ η) = B(γ µ ǫ, η) (2.4) for Majorana ǫ, η. Explicit expressions for W and χ for the two different canonical Majorana spinor orbits have been evaluated in [25]. Here it suffices to note that in the chiral Majorana orbit, the 1-form vanishes W = 0, but χ = 0. In the non-chiral Majorana orbit, both W and χ are non-vanishing, and satisfy W 2 = 0, χ = W ∧ θ, W · θ = 0, θ 2 = 1 . (2.5) The Killing spinor equation (2.1) implies the following conditions ∇ ν W µ = 1 2 η µν F λ 1 λ 2 χ λ 1 λ 2 + F νλ χ λ µ + F µλ χ λ ν + 2 ℓ A ν W µ + 1 ℓ χ µν (2.6) and ∇ σ χ µν = F σµ W ν − F σν W µ − F µν W σ + η σµ (i W F ) ν − η σν (i W F ) µ + 2 ℓ A σ χ µν − 1 ℓ η µσ W ν + 1 ℓ η νσ W µ . (2.7) Hence, for the chiral orbit for which W = 0, χ = 0, a contradiction is immediately obtained by taking the antisymmetric part of (2.6). There are therefore no Majorana Killing spinors in this orbit for the gauged theory. It remains to analyse the Killing spinor equations for the remaining non-chiral Majorana orbit. Analysis of the Killing Spinor Equation In this section, we analyse the Killing spinor equation (2.1), when the Majorana spinor ǫ is in the non-chiral orbit, and we derive the necessary and sufficient conditions on the geometry and the Maxwell field strength. In order to perform the analysis, we shall use spinorial geometry techniques which were originally introduced for the analysis of D = 10 and D = 11 supersymmetric supergravity solutions [9,10,11,12,13]. Following the same analysis as set out in [25], the Majorana spinor in the non-chiral orbit can be written in a specific basis as ǫ = 1 + e 12 + e 1 + e 2 . The linear system relating the spin connection and components of the gauge potential and field strength is listed in Appendix A. In particular, it can be shown that the linear system (A.3) is equivalent to the conditions (2.6) and (2.7), which are therefore necessary and sufficient conditions for supersymmetry. We shall therefore analyse the conditions (2.6) and (2.7). To proceed, contract (2.6) with W ν to obtain ∇ W W = 2 ℓ (i W A)W . (3.1) By making an appropriately chosen U(1) × Spin(2, 2) gauge transformation which leaves the Killing spinor invariant, we may without loss of generality work in a gauge for which i W A = 0 (3.2) and in this gauge ∇ W W = 0 . (3.3) Furthermore, taking the trace of (2.6) implies that ∇ µ W µ = 0 . (3.4) The antisymmetric part of (2.6) implies that dW = 2 ℓ (A + θ) ∧ W (3.5) and hence on taking the exterior derivative, one finds W ∧ (F + dθ) = 0 . (3.6) The condition (2.7) also implies that (i W F ) ν = ∇ λ χ λν + 3 ℓ + 2 ℓ θ µ A µ W ν . (3.7) On substituting (3.6) and (3.7) into (2.7) one finds that (2.7) is equivalent to W σ F µν = − 1 2 ∇ σ χ µν − 1 2 (W ∧ dθ) σµν + η σµ 1 2 ∇ λ χ λν + 1 ℓ (1 + A λ W λ )W ν − η σν 1 2 ∇ λ χ λµ + 1 ℓ (1 + A λ W λ )W µ + 1 ℓ A σ χ µν . (3.8) This condition can be used to eliminate F entirely from (2.6). The resulting conditions obtained from (2.6) are (3.5), together with ∇ W θ = ∇ λ θ λ − 2 ℓ W . (3.9) To continue, consider (3.5), which implies that W ∧ dW = 0 (3.10) and hence there exists a local co-ordinate u and a function H, not identically zero, such that W = Hdu . (3.11) On substituting this expression back into (3.5), one obtains A + θ − ℓ 2 H −1 dH = GW (3.12) for some function G. There is a freedom to redefine θ asθ = θ − GW , as well as making an appropriately chosen U(1) × Spin(2, 2) gauge transformation which leaves the Killing spinor invariant and preserves the condition i W A = 0. Then without loss of generality, we may take A = −θ (3.13) and in this gauge dW = 0 (3.14) and furthermore (3.7) implies that ∇ λ θ λ = 3 ℓ (3.15) and so (3.9) simplifies further to ∇ W θ = 1 ℓ W . (3.16) Next, substitute the condition A = −θ, F = −dθ back into the equations (2.6) and (2.7) to obtain ∇ ν W µ + 2 ℓ W (µ θ ν) = η µν θ λ W ρ (dθ) λρ + 2(i W dθ) (ν θ µ) − 2(i θ dθ) (ν W µ) (3.17) and 1 2 W σ (dθ) µν − W [µ ∇ ν] θ σ − − (i θ dθ) σ W [µ + θ σ (i W dθ) [µ − W σ (i θ dθ) [µ θ ν] −η σ[µ θ ν] (dθ) λρ θ λ W ρ + (i W dθ) ν] + 1 ℓ W ν] − 1 ℓ θ σ W [µ θ ν] = 0 . (3.18) In order to obtain the conditions on ∇θ obtained from (2.7) it is most straightforward to set Ψ µν = ∇ µ θ ν + 1 ℓ θ µ θ ν − 1 ℓ η µν . (3.19) On substituting this expression into (3.18), the resulting expression is identical to (3.18) excluding the ℓ −1 terms, on replacing ∇ µ θ ν with Ψ µν . Furthermore, Ψ µν satisfies the conditions Φ µ µ = 0 and W µ Ψ µν = 0, which are the conditions satisfied by ∇ µ θ ν in the case of the ungauged theory. The corresponding geometric condition obtained from (3.18) is therefore obtained from the corresponding condition in the analysis of the ungauged theory, replacing ∇ µ θ ν with Ψ µν , which is ∇ τ θ = ⋆(θ ∧ dθ) + 1 ℓ τ . (3.20) A similar analysis of (3.17), taking the condition obtained on ∇W obtained in the ungauged theory, and replacing ∇ ν W µ with ∇ ν W µ + 2 ℓ W (µ θ ν) , implies that ∇ V W = ⋆(τ ∧ dθ) − i θ dθ − 1 ℓ θ (3.21) and ∇ τ W = ⋆(W ∧ dθ) (3.22) where we adopt a frame {V, W, τ, θ}, with respect to which the metric is ds 2 = 2V W + θ 2 − τ 2 (3.23) with volume form dvol = W ∧ V ∧ τ ∧ θ. In order to analyse the conditions obtained from the gauge field equations, we note that ⋆χ = −W ∧ τ (3.24) and that the condition (2.7) can be rewritten as ∇ σ ⋆ χ µν = −W σ ⋆ F µν + W µ ⋆ F νσ − W ν ⋆ F µσ + η σµ (i W ⋆ F ) ν − η σν (i W ⋆ F ) µ + 2 ℓ A σ ⋆ χ µν − 1 ℓ (⋆W ) σµν (3.25) and hence d ⋆ χ = W ∧ ⋆ F + 1 ℓ θ ∧ τ . (3.26) This implies that W ∧ ⋆ F − dτ + 1 ℓ θ ∧ τ = 0 (3.27) and therefore there exists a 1-form ξ such that ⋆F = dτ − 1 ℓ θ ∧ τ + W ∧ ξ (3.28) and hence dτ = 1 ℓ θ ∧ τ − W ∧ ξ − ⋆dθ . (3.29) The gauge field equations therefore imply that W ∧ dξ + 1 ℓ d θ ∧ τ = 0 . (3.30) Furthermore, further simplification can be made by considering the basis transformation V ′ = V − βτ + 1 2 β 2 W, τ ′ = τ − βW (3.31) for arbitrary function β. Under this transformation, all the previous conditions on the geometry are invariant, with ξ replaced with ξ ′ = ξ + β ℓ θ − dβ − kW (3.32) for arbitrary function k. In particular, β can be chosen such that i W ξ ′ = 0, and k can also be chosen such that i ′ V ξ ′ = 0. Adopting this basis, and dropping the prime, we take without loss of generality i W ξ = i V ξ = 0, and hence there exist functions f 1 , f 2 such that ξ = f 1 θ + f 2 τ . (3.33) Then it is straightforward to prove, on making use of (3.29), that the condition (3.22) is equivalent to ∇ W τ = 0 (3.34) and hence it follows that ∇ W V = − 1 ℓ θ . (3.35) Summary of Geometric Conditions We briefly summarize the necessary and sufficient conditions for supersymmetry obtained. The frame is {V, W, τ, θ}, with respect to which the metric is ds 2 = 2V W + θ 2 − τ 2 (3.36) with volume form dvol = W ∧ V ∧ τ ∧ θ. The conditions on the geometry are then given by dW = 0, ∇ W W = ∇ W τ = 0, ∇ W V = − 1 ℓ θ, ∇ W θ = 1 ℓ W (3.37) together with ⋆d ⋆ θ = − 3 ℓ , ∇ τ θ = ⋆(θ ∧ dθ) + 1 ℓ τ, ∇ V W = ⋆(τ ∧ dθ) − i θ dθ − 1 ℓ θ (3.38) and there exists a 1-form ξ, satisfying i W ξ = i V ξ = 0 such that dτ = 1 ℓ θ ∧ τ − W ∧ ξ − ⋆dθ . (3.39) The gauge potential is A = −θ, F = −dθ. We remark that the condition d ⋆ W = 0 has been omitted from the above conditions, this is because d ⋆ W = 0 is implied by the above. There are two natural Spin(2, 2)-invariant 2-form bilinears, which are the self-dual and anti-self-dual parts of χ ω ± = W ∧ (θ ± τ ) (3.40) with associated commuting nilpotent endomorphisms J ± , ω ± (X, Y ) = g(X, J ± Y ) (3.41) for all vector fields X, Y . J ± satisfy (J ± ) 2 = 0, and J ± are rank 2, with ImJ ± = KerJ ± . Hence, in order for J ± to be integrable, we require that KerJ ± be closed under the Lie bracket, which is equivalent to requiring that J ± [J ± X, J ± Y ] = 0 (3.42) for all vector fields X, Y , or equivalently, where (dθ) ∓ are the anti-self-dual/self-dual parts of dθ. However, the integrability condition (3.44) does not hold automatically as a consequence of (3.37), (3.38) and (3.39); though, if F is self-dual, then (dθ) − = 0, and so J + is integrable. (J ± ) µ λ (J ± ) ρ α ∇ ρ (J ± ) λ β − (J ± ) µ λ (J ± ) ρ β ∇ ρ (J ± ) λ α = 0 . Example: Lorentzian Gauduchon-Tod Solutions A supersymmetric solution for which the geometry is a fibration over a Lorentzian Gauduchon-Tod space was found in [24] which preserves N = 2 supersymmetry. In this section, we demonstrate how this solution can be written in terms of the conditions given in Section 3.1 which all N = 1 supersymmetric solutions must satisfy. The metric is given by 1 ds 2 = f (dt + ω) 2 + f −1 ds 3 2 (4.1) where f = 1 ℓ −2 t 2 − 1 (4.2) and ds 2 3 is the t-independent metric on the Lorentzian Gaudochon-Tod space. A basis {V 1 , V 2 , V 3 } for the Lorentzian Gaudochon-Tod space is chosen such that ds 3 2 = −(V 1 ) 2 − (V 2 ) 2 + (V 3 ) 2 (4.3) where dV i = 2 ℓ H ∧ V i − 2 ℓ ⋆ 3 V i (4.4) and H is a t-independent 1-form on the Lorentzian Gauduchon-Tod space which satisfies dH = 2 ℓ ⋆ 3 H . (4.5) In addition, ω = 2 ℓ tH + φ (4.6) where φ is a t-independent 1-form on the Lorentzian Gauduchon-Tod space satisfying dφ = 2 ℓ φ ∧ H + 2 ℓ ⋆ 3 φ . (4.7) The volume form on the Lorentzian Gauduchon-Tod space is dvol 3 = V 1 ∧ V 2 ∧ V 3 (4.8) and the gauge potential is given by A = −ℓ −1 f t(dt + ω) + H . (4.9) In order to rewrite this solution in terms of the conditions set out in Section 3.1, it is first useful to work with local co-ordinates on the Lorentzian Gauduchon-Tod space. To obtain these, note that the conditions (4.4) imply that (V 1 − V 3 ) ∧ d(V 1 − V 3 ) = 0, (V 1 + V 3 ) ∧ d(V 1 + V 3 ) = 0 (4.10) and hence there exist local co-ordinates p, q, and t-independent functions h, x such that V 1 − V 3 = hdp, V 1 + V 3 = he x dq (4.11) and hence On substituting these expressions for V 1 and V 3 back into (4.4) for i = 1, 3, one further finds V 1 = 1 2 hdp + 1 2 e x hdq, V 3 = − 1 2 hdp + 1 2 e x hdq .V 2 = ℓ 4 dx + 1 2 e x g 2 hdq − 1 2 g 1 hdp (4.13) and H = ℓ 4 dx + ℓ 2 h −1 dh + 1 2 e x g 2 hdq + 1 2 g 1 hdp . (4.14) Then, substituting these expressions into the final condition obtained from (4.4) for i = 2, one further finds that the functions h, g 1 , g 2 must satisfy e x g 2 + e x ∂ x g 2 + ℓ 2 h −2 ∂ q h = 0 g 1 − ∂ x g 1 + ℓ 2 h −1 ∂ p h = 0 2ℓ −1 he x (1 − g 1 g 2 ) + e x ∂ p g 1 + ∂ q g 1 = 0 . (4.15) Hence, the Lorentzian Gauduchon-Tod space has local co-ordinates {x, p, q}, and basis elements V i given by (4.12) and (4.13) given in terms of functions h = h(x, p, q), g 1 = g 1 (x, p, q), g 2 = g 2 (x, p, q), which must satisfy the conditions (4.15), and H is given by (4.14). With these conventions, ⋆ 3 dp = ℓ 4 dx + 1 2 e x g 2 hdq ∧ dp ⋆ 3 dq = − ℓ 4 dx − 1 2 g 1 hdp ∧ dq ⋆ 3 dx = 2ℓ −1 e x h 2 (1 − g 1 g 2 )dp ∧ dq + 1 2 dx ∧ e x g 2 hdq + g 1 hdp (4. 16) and the condition (4.5) is equivalent to dh ∧ V 2 + d(g 1 h) ∧ (V 1 − V 3 ) − 4ℓ −1 hV 1 ∧ V 3 − 4ℓ −1 hg 1 V 2 ∧ (V 1 − V 3 ) = ⋆ 3 dh (4.17) which in turn can be rewritten as 2ℓ −1 e x (1 − g 1 g 2 )∂ x h − e x g 2 h −1 ∂ p h + g 1 h −1 ∂ q h + ∂ q g 1 − 2ℓ −1 e x h(g 1 g 2 − 1) = 0 . (4.18) This condition is however implied by the conditions listed in (4.15). Having obtained the conditions on the Lorentzian Gauduchon-Tod structure in these local co-ordinates, the 4-dimensional solution can be further simplified by setting t = h −1 u, φ = h −1 ψ (4.19) so that the metric is ds 2 = 1 ℓ −2 u 2 − h 2 du + u( 1 2 dx + ℓ −1 e x g 2 hdq + ℓ −1 g 1 hdp) + ψ 2 + (ℓ −2 u 2 − h 2 ) − e x dpdq − ℓ 4 h −1 dx + 1 2 e x g 2 dq − 1 2 g 1 dp 2 . (4.20) Furthermore, the condition (4.7) is equivalent to dψ = 4ℓ −2 e x h 2 (1 − g 1 g 2 )ψ x dp ∧ dq + 2ℓ −1 g 1 hψ x dx ∧ dp + 2ℓ −1 e x g 2 hψ x − ψ q dx ∧ dq . (4.21) It remains to identify the basis {V, W, τ, θ} which is used to write the conditions in Section 3.1. We take W = dp (4.22) with θ = ℓ −1 u ℓ −2 u 2 − h 2 (du + ψ) + ℓ 2 ℓ −2 u 2 + h 2 ℓ −2 u 2 − h 2 1 2 dx + ℓ −1 e x g 2 hdq + ℓ −1 g 1 hdp . (4.23) In particular, the above expression for θ is obtained directly from the gauge potential (4.9), on neglecting the h −1 dh term arising in H. Then τ may be obtained by considering the condition (3.39), which for the Lorentzian Gauduchon-Tod solution in question can be written as ⋆dθ = −dτ + ℓ −1 θ ∧ τ + (V 1 − V 3 ) ∧ − h −1 d(hn 1 ) − ℓ −1 (g 1 − ℓ −1 h −1 un 1 )f h −1 du + u( 1 2 dx + ℓ −1 e x g 2 hdq + ℓ −1 g 1 hdp) + ψ + (ℓ −1 h −1 ug 1 − n 1 )V 2 − ℓ −2 h −1 u(V 1 + V 3 ) + ℓ −1 h −1 (ψ 3 − ψ 1 )V 2 + 1 2 ℓ −1 h −1 ψ 2 (V 1 + V 3 ) (4.24) on setting ψ = ψ i V i , where we have set τ = −f h −1 du + u( 1 2 dx + ℓ −1 e x g 2 hdq + ℓ −1 g 1 hdp) + ψ + ℓ −1 h −1 u ℓ 4 dx + 1 2 e x g 2 hdq − 1 2 g 1 hdp + n 1 hdp (4.25) where n 1 = n 1 (x, p, q) is a function. This function is required to satisfy the condition corresponding to the gauge choice i W ξ = 0, which can be read off from (4.24) as ∇ W (hn 1 ) + 2f ℓ −2 2ψ x − h −1 u = 0 . (4.26) Having fixed the three basis elements W, θ, τ , the remaining basis element V must be given by V = hn 1 τ + hg 1 θ − 1 2 h 2 (n 2 1 − g 2 1 )dp − 1 2 e x (ℓ −2 u 2 − h 2 )dq . (4.27) It can then be checked directly that the basis {V, W, τ, θ} defined in this way satisfies all of the conditions in Section 3.1, provided that the functions h, g 2 , h 2 , n 1 satisfy the conditions (4.15) and (4.18), and (4.26), and ψ satisfies (4.21). For this solution, neither of J ± are integrable, as (3.44) does not hold. Kastor-Traschen type solution A special case of the Gauduchon-Tod class of solutions for which the base space is R 1,2 was also found in [24]. Such solutions are a split-signature analogue of the Kastor-Traschen solution [28]. The metric and gauge field strength are given by ds 2 = ℓ 2 (u + H) 2 du 2 + (u + H) 2 dz 2 − dx 2 − dy 2 , A = − ℓ u + H du (4.28) where H(x, y, z) satisfies (∂ 2 x +∂ 2 y −∂ 2 z )H = 0. For these solutions the basis {V, W, τ, θ} is given by W = dx + dz θ = ℓ u + H du τ = (u + H)dy + β(dx + dz) V = 1 2 (u + H) 2 (dz − dx) + β(u + H)dy + 1 2 β 2 (dx + dz) (4.29) with volume form dvol = ℓ(u+H) 2 du∧dx∧dy ∧dz, and β = β(u, x, y, z) is a function which must satisfy ∂β ∂z − ∂β ∂x = ∂H ∂y (4.30) in order for the condition (3.39) to hold, with ξ satisfying the gauge choice i W ξ = 0. With this choice, all of the geometric conditions listed in Section 3.1 hold. Again, neither of J ± are integrable in general. Example: Self-Dual Solutions Suppose that the gauge field strength is self-dual, F = ⋆F . Then the geometric conditions are dθ = ⋆dθ (5.1) and dW = 0, ∇ W W = ∇ W τ = 0, ∇ W V = − 1 ℓ θ, ∇ W θ = 1 ℓ W (5.2) together with ⋆d ⋆ θ = − 3 ℓ , ∇ τ +θ θ = 1 ℓ τ, ∇ V W = −i θ+τ dθ − 1 ℓ θ (5.3) and there exists a 1-form ξ, satisfying i W ξ = i V ξ = 0 such that d(θ + τ ) = 1 ℓ θ ∧ τ − W ∧ ξ . (5.4) We introduce local co-ordinates u, v such that W = du, with dual tangent vector W = ∂ ∂v . On surfaces u = const, the condition (5.4) implies (τ +θ) ∧ d(τ +θ) = 0 (5.5) whereτ andθ are the pull-backs of τ , θ to surfaces of constant u. It follows that there exists a local co-ordinate p, and functions G, H such that τ + θ = Gdp + Hdu. (5.6) We remark that the above geometric conditions imply that [W, θ + τ ] = 2 ℓ W (5.7) and hence a local co-ordinate q can be found such that θ + τ = 2v ℓ ∂ ∂v + ∂ ∂q . (5.8) The orthogonality conditions then imply that V = dv − 2 ℓ vdq + Sdu + hdp (5.9) for functions S(u, v, p, q), h(u, v, p, q), with θ − τ = m 1 du + m 2 dp + 2dq (5.10) for functions m 1 (u, v, p, q), m 2 (u, v, p, q). On substituting these basis 1-forms into the condition (5.4), we find ∂ v G = 0, ∂ q G = 1 ℓ G (5.11) which implies that G = e q ℓ Ψ for Ψ = Ψ(u, p), and also ∂ v H = 0 (5.12) where the gauge condition ξ v = 0 has been used, so H = H(u, p, q). These conditions imply that a basis transformation of the type (3.31), where β can be taken to be independent of the v co-ordinate, together with a {u, p} co-ordinate transformation, can be used to set, without loss of generality, Ψ = 1 and H = 0. Such a basis transformation preserves the condition ∇ W τ = 0, as well as the form of all the other geometric conditions. It follows that the basis elements have been simplified to the following forms: W = du V = dv − 2v ℓ dq + Sdu + hdp θ = 1 2 (m 2 + e q ℓ )dp + 1 2 m 1 du + dq τ = 1 2 (−m 2 + e q ℓ )dp − 1 2 m 1 du − dq. (5.13) The condition (5.4) holds with this basis. It remains to consider the remaining geometric conditions (5.1)-(5.3), with volume form dvol = e q ℓ du ∧ dv ∧ dp ∧ dq. After some calculation, the following conditions are obtained: ∂ u m 2 − ∂ p m 1 + ℓ −1 e q ℓ m 1 + ∂ v (hm 1 − Sm 2 ) = 0 (5.14) m 2 = −e q ℓ − ℓ∂ v h + ℓ 2 e q ℓ ∂ v m 1 (5.15) 1 2 ℓ −1 e q ℓ m 1 + ℓ −1 ve q ℓ ∂ v m 1 − 2ℓ −1 v∂ v h + 1 2 e q ℓ ∂ q m 1 + 2ℓ −1 h − ∂ q h = 0 (5.16) 2ℓ −1 m 1 + 2∂ v S + 2vℓ −1 ∂ v m 1 + ∂ q m 1 = 0. (5.17) To proceed to analyse these conditions, it will be convenient to set h = e q ℓ 1 2 m 1 + ∂ v K (5.18) for K = K(u, v, p, q). Then the conditions (5.15) and (5.16) can be solved for m 1 and m 2 to give m 1 = ∂ v 2v∂ v K − 3K + ℓ∂ q K , m 2 = −e q ℓ 1 + ℓ∂ 2 v K (5.19) and the condition (5.17) can be integrated up to give S = f (u, p, q) − 2ℓ −1 v 2 ∂ 2 v K − 2v∂ v ∂ q K + ℓ −1 v∂ v K + 3 2 ∂ q K − ℓ 2 ∂ 2 q K (5.20) where the function f is independent of v. In particular, this implies that h = e q ℓ v∂ 2 v K + 1 2 ∂ v K + ℓ∂ v ∂ q K . (5.21) We remark that the function f (u, p, q) appearing in S can without loss of generality be set to zero by making an appropriate redefinition of K as K =K + Z(u, p, q). This does not affect the form of the remaining conditions. Making this choice, the metric can be written entirely in terms of the function K as ds 2 = 2du dv + − 2ℓ −2 v 2 ∂ 2 v K − 2v∂ v ∂ q K + ℓ −1 v∂ v K + 3 2 ∂ q K − ℓ 2 ∂ 2 q K du − 2v ℓ dq + e q ℓ 2v∂ 2 v K + ℓ∂ v ∂ q K dp − e 2q ℓ 1 + ℓ∂ 2 v K dp 2 + 2e q ℓ dpdq. (5.22) This exhausts the content of the conditions (5.15), (5.16) and (5.17). It remains to consider (5.14) which can be rewritten as ∂ ∂v −2ℓ −1 e q ℓ v 2 ∂ 2 v K − 2e q ℓ v∂ v ∂ q K + 3v ℓ e q ℓ ∂ v K + 5 2 e q ℓ ∂ q K − ℓ 2 e q ℓ ∂ 2 q K +3∂ p K − ℓ∂ p ∂ q K − ℓe q ℓ ∂ v ∂ u K − 2v∂ v ∂ p K − 3ℓ −1 e q ℓ K − 1 2 e q ℓ (∂ v K) 2 + ℓ 2 2 e q ℓ (∂ v ∂ q K) 2 + e q ℓ v∂ v K∂ 2 v K + 3 2 ℓe q ℓ ∂ q K∂ 2 v K − ℓ 2 2 e q ℓ ∂ 2 q K∂ 2 v K = 0. (5.23) In particular, the metric (5.22) together with the condition (5.23) automatically satisfies the Einstein condition R µν = − 3 ℓ 2 g µν (5.24) with the exception of the co-ordinate indices µ = ν = u, as is expected from consideration of the integrability conditions obtained from the Killing spinor equation, which were considered in [25]. Imposing the µ = ν = u component of this equation produces a further nonlinear PDE in the function K which is not implied by (5.23). Conclusions We have determined the conditions on the geometry for a solution of minimal gauged supergravity with positive cosmological constant in neutral signature to preserve the minimal N = 1 amount of supersymmetry. This class of solutions was omitted from the classification constructed in [24]. It would be interesting to understand this geometric structure better. Furthermore, we have shown how the N = 2 solutions which are a fibration over a 3-dimensional Lorentzian Gauduchon-Tod base space arise in this structure. Gauduchon-Tod metrics have arisen in a number of different ways in the context of D = 4 supergravity. Euclidean Gauduchon-Tod structures have been found in minimal Euclidean D = 4 supergravity with signature (+, +, +, +) [19], and also in minimal D = 4 de Sitter supergravity with signature (−, +, +, +) [14]. In the former case, there are no Majorana spinors. In the latter case, there exists a charge conjugation operator C * which commutes with the gamma matrices, however it does not commute with the supercovariant derivative. This means that we do not expect that these Euclidean Gauduchon-Tod solutions can be viewed as N = 2 supersymmetric solutions which are special cases of a more general N = 1 geometric structure, as is the case for the solutions considered in this paper. In contrast, Lorentzian Gauduchon-Tod structures were also found in minimal gauged D = 4 pseudo-supergravity [15], with signature (−, +, +, +). In this case, the charge conjugation operator commutes with the supercovariant derivative, and we expect that the Lorentzian Gauduchon-Tod solution found in that case can also be written as a special case of the supersymmetric Majorana solutions which were also classified in section 4.2 of [15], on making an appropriate choice of co-ordinates. Appendix A The linear system In this appendix we present the linear system which is equivalent to the KSE (2.1) in the case for which the Majorana Killing spinor is non-chiral. In particular, following the conventions of [25], with a split signature (pseudo)-holomorphic basis, i.e. a basis e 1 , e 2 , e1 = (e 1 ) * , e2 = (e 2 ) * (A.1) with respect to which the metric is ds 2 = 2e 1 e1 − 2e 2 e2 ,(A. (3.37), (3.38) and (3.39) imply that (3.43) can be rewritten as ω ∓ ∧ (dθ) ∓ = 0 (3.44) 2 ) 2the linear system obtained from (2.1) is as follows:−ω 1,11 − ω 1,22 + 2iω 1,12 = − √ 2(F 11 + F 22 ) + −ω 2,11 + ω 2,22 + 2iω 2,12 = √ 2i(−F 11 + F 22 ) +The conditions (2.6) and (2.7), with the bilinears (A.4) and (A.5), are equivalent to the linear system (A.3).−ω 1,11 + ω 1,22 + 2iω 1,12 = √ 2(−F 11 + F 22 ) + 2 ℓ A 1 + √ 2 ℓ ω 1,11 − ω 1,22 − 2iω 1,12 = −2 √ 2iF 12 + 2 ℓ A 1 ω 1,11 + ω 1,22 − 2iω 1,12 = −2 √ 2iF 12 + 2 ℓ A 1 2 ℓ A 1 + √ 2 ℓ 2 ℓ A 2 + √ 2i ℓ ω 2,11 − ω 2,22 − 2iω 2,12 = 2 √ 2F 12 + 2 ℓ A 2 ω 2,11 + ω 2,22 − 2iω 2,12 = − √ 2i(F 11 + F 22 ) + 2 ℓ A 2 − √ 2i ℓ −ω 2,11 − ω 2,22 + 2iω 2,12 = 2 √ 2F1 2 + 2 ℓ A 2 . (A.3) The spinor bilinears (2.3) are given by W = 2 √ 2i(e 1 − e1) − 2 √ 2(e 2 + e2) (A.4) and χ = W ∧ θ, θ = 1 √ 2 (e 1 + e1) . 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[ "Unsourced Massive Access-Based Digital Over-the-Air Computation for Efficient Federated Edge Learning", "Unsourced Massive Access-Based Digital Over-the-Air Computation for Efficient Federated Edge Learning" ]	[ "Li Qiao ", "Zhen Gao \nYangtze Delta Region Academy of Beijing Institute of Technology (Jiaxing)\n314019Jiaxing\n\nChina ‡ Advanced Technology Research Institute\nBeijing Institute of Technology\n250307JinanChina\n", "† ‡ ", "Zhongxiang Li [email protected] \nYangtze Delta Region Academy of Beijing Institute of Technology (Jiaxing)\n314019Jiaxing\n\nChina ‡ Advanced Technology Research Institute\nBeijing Institute of Technology\n250307JinanChina\n", "Deniz Gündüz [email protected] \nDepartment of Electrical and Electronic Engineering\nImperial College London\nSW7 2AZLondonU.K\n", "\nMIIT Key Laboratory of Complex-field Intelligent Sensing\nBeijing Institute of Technology\n100081BeijingChina\n" ]	[ "Yangtze Delta Region Academy of Beijing Institute of Technology (Jiaxing)\n314019Jiaxing", "China ‡ Advanced Technology Research Institute\nBeijing Institute of Technology\n250307JinanChina", "Yangtze Delta Region Academy of Beijing Institute of Technology (Jiaxing)\n314019Jiaxing", "China ‡ Advanced Technology Research Institute\nBeijing Institute of Technology\n250307JinanChina", "Department of Electrical and Electronic Engineering\nImperial College London\nSW7 2AZLondonU.K", "MIIT Key Laboratory of Complex-field Intelligent Sensing\nBeijing Institute of Technology\n100081BeijingChina" ]	[]	Over-the-air computation (OAC) is a promising technique to achieve fast model aggregation across multiple devices in federated edge learning (FEEL). In addition to the analog schemes, one-bit digital aggregation (OBDA) scheme was proposed to adapt OAC to modern digital wireless systems. However, one-bit quantization in OBDA can result in a serious information loss and slower convergence of FEEL. To overcome this limitation, this paper proposes an unsourced massive access (UMA)-based generalized digital OAC (GD-OAC) scheme. Specifically, at the transmitter, all the devices share the same non-orthogonal UMA codebook for uplink transmission. The local model update of each device is quantized based on the same quantization codebook. Then, each device transmits a sequence selected from the UMA codebook based on the quantized elements of its model update. At the receiver, we propose an approximate message passing-based algorithm for efficient UMA detection and model aggregation. Simulation results show that the proposed GD-OAC scheme significantly accelerates the FEEL convergences compared with the state-of-the-art OBDA scheme while using the same uplink communication resources.Index Terms-Internet-of-Things, massive machine-type communications, unsourced massive access, over-the-air computation, federated edge learning.	10.48550/arxiv.2305.10609	[ "https://export.arxiv.org/pdf/2305.10609v1.pdf" ]	258,762,459	2305.10609	cf4bc758eea2fb13e973c57eba31bbaa3effb87a	Unsourced Massive Access-Based Digital Over-the-Air Computation for Efficient Federated Edge Learning Li Qiao Zhen Gao Yangtze Delta Region Academy of Beijing Institute of Technology (Jiaxing) 314019Jiaxing China ‡ Advanced Technology Research Institute Beijing Institute of Technology 250307JinanChina † ‡ Zhongxiang Li [email protected] Yangtze Delta Region Academy of Beijing Institute of Technology (Jiaxing) 314019Jiaxing China ‡ Advanced Technology Research Institute Beijing Institute of Technology 250307JinanChina Deniz Gündüz [email protected] Department of Electrical and Electronic Engineering Imperial College London SW7 2AZLondonU.K MIIT Key Laboratory of Complex-field Intelligent Sensing Beijing Institute of Technology 100081BeijingChina Unsourced Massive Access-Based Digital Over-the-Air Computation for Efficient Federated Edge Learning Over-the-air computation (OAC) is a promising technique to achieve fast model aggregation across multiple devices in federated edge learning (FEEL). In addition to the analog schemes, one-bit digital aggregation (OBDA) scheme was proposed to adapt OAC to modern digital wireless systems. However, one-bit quantization in OBDA can result in a serious information loss and slower convergence of FEEL. To overcome this limitation, this paper proposes an unsourced massive access (UMA)-based generalized digital OAC (GD-OAC) scheme. Specifically, at the transmitter, all the devices share the same non-orthogonal UMA codebook for uplink transmission. The local model update of each device is quantized based on the same quantization codebook. Then, each device transmits a sequence selected from the UMA codebook based on the quantized elements of its model update. At the receiver, we propose an approximate message passing-based algorithm for efficient UMA detection and model aggregation. Simulation results show that the proposed GD-OAC scheme significantly accelerates the FEEL convergences compared with the state-of-the-art OBDA scheme while using the same uplink communication resources.Index Terms-Internet-of-Things, massive machine-type communications, unsourced massive access, over-the-air computation, federated edge learning. I. INTRODUCTION With the growing deployment of Internet-of-Things (IoT), an increasing amount of data will be acquired by massive number of IoT devices [1]. Traditionally, collected data is offloaded to the cloud or a data center for data-driven machine learning (ML) applications [2]. Recently, due to privacy concerns and the growing computation abilities of edge IoT devices (i.e., smartphones), wireless networks are pushing the deployment of centralized ML algorithms towards distributed learning frameworks [2]. In the emerging federated edge learning (FEEL) framework, multiple edge IoT devices are coordinated by a central server to train an ML model using local datasets and computing resources [3]. FEEL requires solving the distributed training problem taking into account the limited shared wireless resources and interference among multiple devices. Moreover, the model aggregation process in FEEL involves repeated uplink (UL) transmission of highdimensional local gradients (or model updates) by tens to hundreds of devices, which places a heavy burden on the multiple access networks [4]. To solve this problem, analog over-the-air computation (OAC) was introduced for efficient model aggregation [3]- [6]. Since the target of model aggregation is to calculate the average of all the local model updates rather than decoding each of the transmitted messages, the basic idea of analog OAC is to create and leverage inter-user interferences over the multiple access channel (MAC). Specifically, each element of the local model updates is precoded by the inverse of the UL channel gain and then modulated on the amplitude of the transmit waveform. After simultaneous transmission, the receiver directly acquires the sum of the local model updates of multiple devices based on the superposed waveform over MAC. To further reduce the communication overhead, by exploiting sparsification and error accumulation, the dimension of the local model updates are reduced before analog transmission in [6]. After analog OAC, the receiver reconstructs the average of local model updates by using the approximate message passing (AMP) algorithm. A higher FEEL accuracy is achieved with limited communication resources in [7] by exploiting timecorrelated sparsification on the local model updates [8]. In addition, compressive sensing (CS) techniques are adopted for communication efficient FEEL in [9] and [10]. Different from [3]- [10], where all the participating devices are synchronized in time, misaligned analog OAC is considered in [11]. On the other hand, most of the existing wireless networks adopt digital communication protocols (e.g., 3GPP standards) as well as hardware [12], and they may not be capable of employing an arbitrary modulation scheme, which is essential for the analog OAC schemes mentioned above. The authors of [12] proposed a one-bit digital aggregation (OBDA) scheme for FEEL, which combines the ideas of sign stochastic gradient descent (signSGD) and OAC. Specifically, one-bit gradient quantization and digital modulation are adopted at devices, and majority-vote based gradient-decoding via OAC is employed at the central server. However, one-bit gradient quantization is too aggressive, resulting in slow convergence. On the other hand, if the local gradients (or model updates) are quantized using more bits, the superimposed digitally transmitted symbols at the receiver are no longer equal to the sum of the quantized gradients (or model updates). This necessities a novel digital OAC-based scheme with generalized quantization levels. Recently, grant-free random access has received attention to enable massive access of IoT devices with low signaling overhead and latency [13], where devices can directly transmit their signals without the permission of the base station (BS). There are mainly two types of grant-free random access schemes, i.e., sourced massive access [14]- [18] and unsourced massive access (UMA) [19]- [22]. In the former, each device has its unique non-orthogonal preamble indicating its identity, and the BS performs active device detection and channel estimation, followed by data decoding [14]- [18]. As for UMA, all the devices adopt the same non-orthogonal UMA codebook. Transmitted bits are modulated by the index of the UMA codewords and decoded at the BS. The goal of the BS is to decode the list of messages, but not the identities of the active devices [19]- [22]. We remark that the identities of active devices are not necessary for model aggregation of FEEL either. Hence, our goal here is to redesign the modulation and decoding modules of UMA to tailor it for efficient digital model aggregation in FEEL across massive number of devices. We propose an UMA-based generalized digital OAC (GD-OAC) scheme for communication efficient FEEL. Specifically, local model updates are quantized using the same quantization codebook. Then, the active devices select their transmit sequences (codewords) from the common UMA codebook based on their quantized model updates. Note that there is a one-to-one mapping between the quantization codebook and the non-orthogonal UMA codebook. Transmitted sequences overlap at the BS, which employs an AMP-based digital aggregation (AMP-DA) algorithm to calculate the average of the local model updates. Our simulation results verify that the proposed GD-OAC scheme is superior to the state-of-the-art OBDA scheme in terms of the test accuracy with the same UL communication resources. Notation: Boldface lower and upper-case symbols denote column vectors and matrices, respectively. For a matrix A, A T , A F , [A] m,n denote the transpose, Frobenius norm, the m-th row and n-th column element of A, respectively. For a vector x, x p and [x] m denote the l p norm and m-th element of x, respectively. \|Γ\| denotes the cardinality of the ordered set Γ. · rounds each element to the nearest integer smaller than or equal to that element. The marginal distribution p ([x] m ) is denoted as p ([x] m ) = \[x]m p (x). N (x; µ, ν) (or x ∼ CN (µ, ν) ) denotes the (complex) Gaussian distribution of random variable x with mean µ and variance ν. [K] denotes the set {1, 2, ..., K}. II. SYSTEM MODEL As shown in Fig. 1, we consider K edge devices served by a single antenna BS in a cellular system. Each device k, k ∈ [K], has its own local dataset D k , which consists of labeled data samples. A common neural network model, represented by a parameter vector w ∈ R W , is to be trained under the FEEL framework, coordinated by the BS. In each communication round of FEEL, e.g. the t-th round, the BS broadcasts the current global model w t to the devices. The k-th device performs E iterations of local stochastic gradient descent (SGD) on w t using its local dataset D k [7], which can be expressed as w t k,e = w t k,e−1 −η · 1 \|D k \| ε∈D k ∇f (w t k,e−1 , ε), e ∈ [E],(1) where w t k,0 = w t , ∇ and η denote the gradient operator and the learning rate, respectively, f (w t k,e−1 , ε) denotes the sample loss of model w t k,e−1 on the training sample ε. For convenience, we assume uniform sizes for local datasets, i.e., \|D k \| = D, ∀k. According to (1), the local model update can be obtained as g t k = w t k,E − w t , ∀k, t. It is commonly adopted that only K a (K a K) active devices send their model updates to the BS in each round. If the local model updates can be perfectly obtained at the BS, the global model update g t ∈ R W can be calculated as g t = 1 K a Ka k=1 g t k ,(2) where K a is usually unknown to the BS, due to the grant-free random access feature of IoT devices [1]. Then, the global model can be updated as w t+1 = w t + g t .(3) Finally, the updated parameter vector w t+1 ∈ R W is sent back to the devices. The steps (1), (2), and (3) are iterated until a convergence condition is met. It is clear from (2) that only the sum of the local model updates, rather than the individual values, is needed at the BS. The BS does not need to identify active devices either. These motivate the proposed UMA-based aggregation scheme presented in Section III. III. PROPOSED UMA-BASED GD-OAC SCHEME Due to the large dimension of the parameter vector and the large number of devices that can potentially participate in the training, we have a communication bottleneck in FEEL. In this section, we propose to exploit UMA to reduce the communication overhead in FEEL. Next, we describe the UMA-based Modulation Quantization Device 2 Pre-equalization Local Training UMA-based Modulation Quantization Device 2 Pre-equalization Local Training . . . UMA-based Modulation Quantization U = [u 1 , u 2 , ..., u N ] ∈ R Q×N , where N = 2 J denotes J-bit quantization with N quantization codewords, Q ≥ 1 (Q ∈ N) denotes the length of each quantization codeword u n ∈ R Q , ∀n ∈ [N ]. Note that Q = 1 and Q > 1 indicate the scalar and vector quantization (VQ), respectively. As for the k-th device, ∀k ∈ [K a ], the indices of its quantized model update can be expressed as g k = h (g k , U) ,(4) where g k ∈ N W , W = W/Q, and h (·, U) is a function that maps g k to g k based on the quantization codebook U. For VQ (Q > 1), supposing that W can be divided by Q, h (·, U) first reshapes g k into a matrix with Q rows and W columns, then each column is mapped to a quantization codeword with minimum Euclidean distance. Note that the dimension of each local model update can be reduced by using VQ. In addition, the w-th element of g k , denoted as g k,w , ∀w ∈ [W ], is an integer belonging to set [N ]. 2) UMA-based Modulation: We consider the same nonorthogonal UMA codebook for all of the devices, denoted by P = [p 1 , p 2 , ..., p N ] ∈ R L×N , where L is the length of each sequence p n ∈ R L , ∀n ∈ [N ]. In the proposed scheme, there is a one-to-one mapping from U to P. Hence, for the k-th device, the w-th element of its quantized model update is modulated to the sequence p g k,w ∈ R L . 3) UL Massive Access Model: We consider a time division duplex system, where the UL and downlink (DL) channels are commonly considered the same due to channel reciprocity [18]. Before each UL communication round of FEEL, the BS first broadcasts a pilot signal and each active device estimates its DL channel h DL k , k ∈ [K a ], based on the pilot signal, then active devices start their UL transmissions with preequalization [18]. For the w-th element of g k , ∀k, w, the received signal y ∈ R L at the BS can be expressed as y w = Ka k=1 h UL k 1 h DL k p g k,w + z w = Px w + z w ,(5) where h UL k ∼ CN (0, 1) denotes the UL channel gain of the k-th active device, each element of the noise vector z w ∈ R L obeys the independent and identically distributed (i.i.d.) Gaussian distribution with zero mean and variance σ 2 . Here, we will assume perfect estimation of the DL channel for simplicity, i.e., h DL k = h UL k . Also, x w ∈ N N , ∀w ∈ [W ] , is the equivalent transmit signal vector satisfying the following properties: x w 1 = K a , x w 0 ≤ K a , x w n ∈ Ω, ∀n ∈ [N ], (6) where x w = [x w 1 , x w 2 , ..., x w N ] T and Ω = {0, [K a ] }. According to this notation, the n-th sequence p n is transmitted by x w n active devices, or equivalently the n-th quantization codeword appears at x w n devices, which indicates how to conduct the model aggregation. B. Receiver Design 1) UMA Detection: To achieve higher quantization accuracy, especially for VQ, the number of quantization bits J can be set very large, e.g., more than 8 bits. In this case, N = 2 J can be far greater than K a , which motivates us to employ CS-based algorithms for efficient UMA detection with reduced communication overhead L [13]. Note that the UMA codebook is known at the BS. The details of the proposed CSbased detection algorithm will be illustrated in Section IV. 2) Model Aggregation: After UMA detection, we can obtain the estimate of the equivalent transmit signals, denoted by x w ∈ N N , ∀w. According to (6), the number of active devices K a can be estimated based on x w , which will be detailed in Section IV-C. Then, according to (2), we can obtain the global model update corresponding to g k,w as g w = 1 K a U x w ,(7) where g w ∈ R Q . Hence, the whole global model update can be obtained as g = [( g 1 ) T , ( g 2 ) T , ..., ( g W ) T ] T ∈ R W . 3) Model Update: According to (3) and (7), the BS can update the weight of the ML model as w t+1 = w t + g t .(8) Then, the updated model w t+1 is broadcast to all the devices. IV. PROPOSED CS-BASED DETECTION AND MODEL AGGREGATION ALGORITHM In this section, we will first introduce the CS-based problem formulation. Then, we propose the AMP-based algorithm for efficient UMA detection and K a estimation. Finally, we summarize the proposed AMP-DA algorithm. A. Problem Formulation For simplicity, we omit the subscript w in this section and focus on any round of the UL transmission. To solve the UMA detection problem in (5), we aim to minimize the mean square error between y and Px. Equivalently, we can calculate the posterior mean of x under the Bayesian framework [15]. The posterior mean of x n , ∀n ∈ [N ], can be expressed as x n = x n p (x n \|y) dx n ,(9) where p(x n \|y) is the marginal distribution of p(x\|y): p (x n \|y) = \xn p (x\|y) . As discussed in Section III-B, to achieve higher quantization accuracy, more quantization bits are needed for higher dimensional VQ. This can result in a large dimension of N , which demands an efficient algorithm to calculate the marginal distribution p(x n \|y). B. Approximate Message Passing (AMP) To address this issue, the AMP algorithm can be adopted to obtain the approximate marginal distributions with relatively low complexity [23]. According to the AMP algorithm, we can approximately decouple (5) into N scalar problems as y = Px + z → r n = x n + z n ,(11) where n ∈ [N ], r n is the mean of x n estimated by the AMP algorithm, and z n ∼ N (z n ; 0, ϕ n ) is the associated noise with zero mean and variance ϕ n [14]. Hence, according to Bayes's theorem, the posterior distribution of x n can be expressed as p (x n \|y) ≈ p (x n \|r n ) = 1 p (r n ) p (r n \|x n ) p (x n ) ,(12) where " ≈ " is due to the AMP approximation, and p (r n \|x n ) = N (r n ; x n , ϕ n ), p (r n ) = xn∈Ω p (r n \|x n ) p (x n ) .(13) Furthermore, according to (6), we can model the prior distribution of x n as p (x n ) = (1 − a n ) δ(x n ) + a n K a s∈ [Ka] δ(x n − s), (15) where the sparsity indicator a n = 0 if x n = 0, otherwise a n = 1. Also, in (15), if a n = 1, we assume that x n can be any element from set [K a ] with equal probability. Note that this assumption is also an approximation since the BS cannot obtain the actual distribution of each element x n , ∀n ∈ [N ]. According to (12)−(15), the posterior mean and variance of x n , ∀n ∈ [N ], denoted by x n and v n , respectively, can be evaluated as x n = xn∈Ω x n p(x n \|y)dx n ,(16)v n = xn∈Ω \|x n \| 2 p(x n \|y)dx n − \| x n \| 2 .(17) In addition, in the i-th AMP iteration, (r n ) i and (ϕ n ) i , ∀n, of (11) are updated as (ϕ n ) i = L l=1 \|[p n ] l \| 2 σ 2 + (V l ) i −1 ,(18)(r n ) i = ( x n ) i + (ϕ n ) i L l=1 [p n ] l [y] l − (Z l ) i σ 2 + (V l ) i ,(19) where (V l ) i and (Z l ) i , ∀l, are updated as (V l ) i = N n=1 \|[P] l,n \| 2 ( v n ) i ,(20)(Z l ) i = N n=1 [P] l,n ( x n ) i − (V l ) i [y] l − (Z l ) i−1 σ 2 + (V l ) i−1 ,(21) while (·) i denotes its argument in the i-th AMP iteration. For further details of the AMP update rules in (18)−(21), we refer the readers to [23]. C. Parameter Estimation 1) Estimation of K a : According to (16), we can obtain the estimated transmit signal vector x w = [ x w 1 , x w 2 , ..., x w N ] T , ∀w ∈ [W ]. As indicated in (6), x w 1 should equal to K a under perfect UMA detection. To improve the robustness, we estimate K a as follows K a = Ψ x w 1 + 1 2 , w ∈ [W ] ,(22) where function Ψ {·} calculates the most frequently occurring element of its argument. The intuition of (22) is similar to majority voting to improve the detection accuracy of K a . 2) Estimation of the sparsity indicators: The unknown sparsity indicators, denoted by a n , ∀n ∈ [N ], can be obtained by using the expectation maximization (EM) algorithm. The EM algorithm update rules are as follows (a n ) i+1 = arg max an E ln p (x, y) \|y; (a n ) i , where (a n ) i denotes the sparsity indicator in the i-th iteration, E{·\|y; (a n ) i } represents the expectation conditioned on the received signal y under (a n ) i . Hence, according to (12), the sparsity indicators can be obtained as (a n ) i+1 = xn∈[Ka] p x n \|y; (a n ) i .(24) The derivations of the EM update rule can be found in [14]. D. Proposed AMP-DA Algorithm Based on (16)- (22) and (24), we summarize the proposed AMP-DA algorithm in Algorithm 1. The details are explained as follows. In line 1, we initialize the sparsity indicators a w n , the variables V w l , Z w l , the posterior mean x w n , and posterior variance v w n , ∀w, n, l. The iteration starts in line 2. Specifically, lines 3-6 correspond to the AMP operation. In the i-th iteration of the AMP decoupling step (line 4), V w l , Z w l , ϕ w n , and r w n , ∀w, n, l, are calculated according to (20), (21), (18), and (19), respectively. Furthermore, a damping parameter τ is adopted in line 5 to prevent the algorithm from diverging [14]. In addition, in the AMP denoising step (line 6), we calculate the posterior mean x w n and the corresponding posterior variance v w n , ∀w, n of the i-th iteration, by using (16) and (17), respectively. Accordingly, EM operation is used to update the sparsity indicators a w n in line 8. Then, the iteration restarts in line 3 until the maximum iteration number T 0 is reached. Otherwise, if line 10 is triggered, i.e., the normalized mean square error between continuous x w is smaller than the predefined threshold , the iteration ends in advance. After the iteration, we acquire the estimated equivalent transmit signal x w , ∀w, in line 14. According to (22), in line 16, we obtain the number of active devices K a based on x w , ∀w. Finally, according to (7), we obtain g w in line 18 by using x w and K a , ∀w. Hence, the estimated global model update is acquired in line 19 as g = [( g 1 ) T , ( g 2 ) T ], ..., ( g W ) T ] T , which realizes Algorithm 1 Proposed AMP-DA Algorithm Input: The received signals y w ∈ R L , ∀w ∈ [W ], the UMA codebook P ∈ R L×N , the quantization codebook U ∈ R Q×N , the noise variance σ 2 , the maximum iteration number T 0 , the damping parameter τ , and the termination threshold . Output: The estimated global model update g. 1: ∀w, n, l: We initialize the iterative index i = 1, the sparsity indicators (a w n ) 1 = 0.5, (Z w l ) 1 = [y w ] l , (V w l ) 1 = 1, ( x w n ) 1 = 0, and ( v w n ) 1 = 1; 2: for i = 2 to T 0 do 3: %AMP operation: 4: ∀w, n, l: Compute (V w l ) i , (Z w l ) i , (ϕ w n ) i , and (r w n ) i by using (20), (21), (18), and (19), respectively; {Decoupling step} 5: ∀w, l: (V w l ) i = τ (V w l ) i−1 + (1 − τ )(V w l ) i , (Z w l ) t = τ (Z w l ) i−1 + (1 − τ )(Z w l ) i ; {Damping} 6: ∀w, n: Compute ( x w n ) i and ( v w n ) i by using (16) and (17), respectively; {Denoising step} 7: %Parameter update: 8: ∀w, n: Update the sparsity indicators (a w n ) i by using (24); 9: (7), we obtain g w by using x w and Ka; 19: The estimated global model update g i = i + 1; 10: if w ( x w ) i − ( x w ) i−1 F / ( x w ) i−1 F /W <= [( g 1 ) T , ( g 2 ) T ], ..., ( g W ) T ] T . the generalized digital model aggregation process. Also, by recalling (8), the weight of the ML model will be updated using g. V. SIMULATION RESULTS In this section, we evaluate the performance of the proposed UMA-based GD-OAC scheme by considering FEEL for the image classification task on the commonly adopted CIFAR-10 dataset [24]. CIFAR-10 dataset has a training set of 50,000 and a test set of 10,000 colour images of size 32 × 32 belonging to 10 classes. Consider K = 100 devices in the cell collaboratively training a convolutional neural network (same structure as in [24]) with 258898 (i.e., W ≈ 2.6 × 10 5 ) parameters. We consider non-i.i.d. local training data across the devices as follows [11]: 1) Each one of the 100 devices is randomly assigned 300 samples from the dataset; 2) The remaining 20,000 samples are sorted by their labels and grouped into 100 shards of size 200. Then, each device is assigned one shard. Furthermore, the local training iterations E = 10, the mini-batch size is 512, and the learning rate is η = 0.001. The metric to evaluate the performance is the test accuracy of the model on the test dataset. As for the wireless transmission, we assume around 10% of the devices are active in each round, where K a is uniformly generated from 7 to 13. Since the receiver does not exactly have K a , Ω = {0, [0.2 × K]}, shown in (6), is considered as a prior information. The signal-to-noise ratio is set to 20 dB. For the proposed AMP-DA algorithm, the damping parameter is set to τ = 0.3 and the termination threshold is = 1 × 10 −5 . The elements of UMA codebook P obey i.i.d. Gaussian distribution. In each communication round, K-means algorithm &RPPXQLFDWLRQ5RXQGV [25] is used to generate the quantization codebook U based on the local model update of one of the active devices. 7HVW$FFXUDF\ 3$4 - 3$4 - '2$&4 - 3$4 - '2$&4 - 3$4 - 2%'$ We consider two benchmark schemes as follows. 1) Perfect aggregation (PA): The transmitter is the same as the proposed GD-OAC scheme, while we consider perfect model aggregation at the receiver. 2) OBDA: The state-of-the-art digital OAC scheme with one-bit quantization [12]. Note that the OBDA scheme needs W UL communication resources in each round. As for the proposed GD-OAC scheme, the communication resources required in each round is L × W Q = L Q × W . To realize the same UL communication resources, we set L = Q. As shown in Fig. 3, by using the same communication resources, the proposed GD-OAC scheme significantly outperforms the OBDA scheme in terms of test accuracy. The reason is that one-bit quantization results in a severe quantization loss. Also, it can be seen that the proposed GD-OAC scheme achieves nearly the same accuracy as that of the PA schemes despite using limited communication resources. It is also observed that a larger Q results in a slower convergence with fixed J, while increasing the number of quantization bits J can improve the speed of convergence. Hence, there is a trade-off between the compression ratio (i.e., 1 Q ) and the test accuracy. VI. CONCLUSIONS This paper proposed a UMA-based GD-OAC scheme for communication efficient FEEL across a large number of edge devices. The main idea of the proposed solution relies on the fact that, in FEEL, the BS does not need the identities of the transmitting devices for model aggregation. The local model updates are first quantized based on a common quantization codebook, then each quantized element is modulated into a transmit sequence selected from the common non-orthogonal UMA codebook. These transmit sequences from different devices overlap at the BS. Then, the proposed AMP-DA algorithm can efficiently calculate the average of the local model updates. Due to the compression induced by VQ and the flexibility of quantization levels, simulation results verified that the proposed GD-OAC scheme significantly outperforms the OBDA scheme in terms of convergence speed. This work received funding from the CHIST-ERA project SONATA (CHIST-ERA-20-SICT-004) funded by EPSRC-EP/W035960/1. For the purpose of open access, the authors have applied a Creative Commons Attribution (CCBY) license to any Author Accepted Manuscript version arising from this submission. 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[ "Mode Connectivity in Auction Design", "Mode Connectivity in Auction Design" ]	[ "Christoph Hertrich [email protected] \nDepartment of Mathematics London School of Economics and Political Science\nDepartment of Mathematics London School of Economics and Political Science\nInstitute for Theoretical Computer Science Shanghai University of Finance and Economics\nUK, China, UK\n", "Yixin Tao [email protected] \nDepartment of Mathematics London School of Economics and Political Science\nDepartment of Mathematics London School of Economics and Political Science\nInstitute for Theoretical Computer Science Shanghai University of Finance and Economics\nUK, China, UK\n", "László A Végh [email protected] \nDepartment of Mathematics London School of Economics and Political Science\nDepartment of Mathematics London School of Economics and Political Science\nInstitute for Theoretical Computer Science Shanghai University of Finance and Economics\nUK, China, UK\n" ]	[ "Department of Mathematics London School of Economics and Political Science\nDepartment of Mathematics London School of Economics and Political Science\nInstitute for Theoretical Computer Science Shanghai University of Finance and Economics\nUK, China, UK", "Department of Mathematics London School of Economics and Political Science\nDepartment of Mathematics London School of Economics and Political Science\nInstitute for Theoretical Computer Science Shanghai University of Finance and Economics\nUK, China, UK", "Department of Mathematics London School of Economics and Political Science\nDepartment of Mathematics London School of Economics and Political Science\nInstitute for Theoretical Computer Science Shanghai University of Finance and Economics\nUK, China, UK" ]	[]	Optimal auction design is a fundamental problem in algorithmic game theory. This problem is notoriously difficult already in very simple settings. Recent work in differentiable economics showed that neural networks can efficiently learn known optimal auction mechanisms and discover interesting new ones. In an attempt to theoretically justify their empirical success, we focus on one of the first such networks, RochetNet, and a generalized version for affine maximizer auctions. We prove that they satisfy mode connectivity, i.e., locally optimal solutions are connected by a simple, piecewise linear path such that every solution on the path is almost as good as one of the two local optima. Mode connectivity has been recently investigated as an intriguing empirical and theoretically justifiable property of neural networks used for prediction problems. Our results give the first such analysis in the context of differentiable economics, where neural networks are used directly for solving non-convex optimization problems.Preprint. Under review.	10.48550/arxiv.2305.11005	[ "https://export.arxiv.org/pdf/2305.11005v1.pdf" ]	258,762,534	2305.11005	0ce4f232b2cb12d48414332b6a07aa5d094d1b87	Mode Connectivity in Auction Design Christoph Hertrich [email protected] Department of Mathematics London School of Economics and Political Science Department of Mathematics London School of Economics and Political Science Institute for Theoretical Computer Science Shanghai University of Finance and Economics UK, China, UK Yixin Tao [email protected] Department of Mathematics London School of Economics and Political Science Department of Mathematics London School of Economics and Political Science Institute for Theoretical Computer Science Shanghai University of Finance and Economics UK, China, UK László A Végh [email protected] Department of Mathematics London School of Economics and Political Science Department of Mathematics London School of Economics and Political Science Institute for Theoretical Computer Science Shanghai University of Finance and Economics UK, China, UK Mode Connectivity in Auction Design Optimal auction design is a fundamental problem in algorithmic game theory. This problem is notoriously difficult already in very simple settings. Recent work in differentiable economics showed that neural networks can efficiently learn known optimal auction mechanisms and discover interesting new ones. In an attempt to theoretically justify their empirical success, we focus on one of the first such networks, RochetNet, and a generalized version for affine maximizer auctions. We prove that they satisfy mode connectivity, i.e., locally optimal solutions are connected by a simple, piecewise linear path such that every solution on the path is almost as good as one of the two local optima. Mode connectivity has been recently investigated as an intriguing empirical and theoretically justifiable property of neural networks used for prediction problems. Our results give the first such analysis in the context of differentiable economics, where neural networks are used directly for solving non-convex optimization problems.Preprint. Under review. Introduction Auction design is a core problem in mechanism design, with immense applications in electronic commerce (such as sponsored search auctions) as well as in the public sector (such as spectrum auctions). In a revenue maximizing auction, the auctioneer needs to design a mechanism to allocate resources to buyers, and set prices in order to maximize the expected revenue. The buyers' preferences are private and they may behave strategically by misreporting them. For this reason, it is often desirable to devise dominant strategy incentive compatible (DSIC) and individually rational (IR) mechanisms. In a DISC mechanism, it is a dominant strategy for the buyers to report the true valuations; in an IR mechanism, each participating truthful buyer receives a nonnegative payoff. We focus on DSIC and IR mechanisms that maximize the expected revenue, assuming that the buyers' preferences are drawn from a distribution known to the auctioneer. A classical result of Myerson [1981] provides the optimal mechanism for the case of a single item and arbitrary number of buyers. Finding the optimal mechanisms for more general settings is a tantalizingly difficult problem. We refer the reader to the surveys by Rochet and Stole [2003], Manelli and Vincent [2007] and Daskalakis [2015] for partial results and references. In particular, no analytic solution is known even for two items and two buyers. Selling multiple items to a single buyer is computationally intractable [Daskalakis et al., 2014]. Already for two items and a single buyer, the description of the optimal mechanism may be uncountable [Daskalakis et al., 2013]. Recent work gives a number of important partial characterizations, e.g. Daskalakis et al. [2015], Giannakopoulos and Koutsoupias [2014], as well as results for weaker notions of Bayesian incentive compatibility, e.g. Cai et al. [2012bCai et al. [ ,a, 2013, Bhalgat et al. [2013]. Conitzer and Sandholm [2002,2004] proposed the approach of automated mechanism design to use optimization and computational methods to obtain (near) optimal mechanisms for specific problems; see also Sandholm and Likhodedov [2015]. An active recent area of research uses machine learning tools. In particular, Dütting et al. [2019] designed and trained neural networks to automatically find optimal auctions. They studied two network architectures, and showed that several theoretically optimal mechanisms can be recovered using this approach, as well as interesting new mechanisms can be obtained. The first network they studied is RochetNet. This is a simple two-layer neural network applicable to the single buyer case, leveraging Rochet's [1987] characterization of the optimal mechanism. The second network, RegretNet does not require such a characterization and is applicable for multiple buyers; however, it only provides approximate incentive compatibility. Dütting et al. [2019] coined the term 'differentiable economics' for this approach, and there has been significant further work in this direction. These include designing auctions for budget constrained buyers [Feng et al., 2018]; multi-facility location Golowich et al. [2018]; balancing fairness and revenue objectives [Kuo et al., 2020]; deriving new analytical results [Shen et al., 2019]; designing revenue-maximizing auctions with differentiable matchings [Curry et al., 2022a]; contextual auction design [Duan et al., 2022]; designing taxation policies [Zheng et al., 2022], and more. The purpose of this work is to supply theoretical evidence behind the success of neural networks in differentiable economics. The revenue is a highly non-convex function of the parameters in the neural network. Curiously, gradient approaches seem to recover globally optimal auctions despite this non-convexity. Similar phenomena have been studied more generally in the context of deep networks, and theoretical explanations have been proposed, in particular, overparametrization [Allen-Zhu et al., 2019, Du et al., 2019. Mode connectivity Recent work has focused on a striking property of the landscape of loss functions of deep neural networks: local optimal solutions found by gradient approaches are connected by simple paths in the parameter space. We say that two solutions are ε-mode connected, if they are connected by a path of solutions, where the loss function is within ε from the maximum value at the endpoints. This phenomenon was identified by Garipov et al. [2018] and by Draxler et al. [2018]. Kuditipudi et al. [2019] gave strong theoretical arguments for mode connectivity. They introduce the notion of ε-dropout stability: solutions to a neural network such that in each layer, one can remove at least half the neurons and rescale the remaining units such that the loss function increases by at most ε. Solutions that are ε-dropout stable are then shown to be ε-mode connected. Moreover, they show that noise stability (see e.g., Arora et al. [2018]) implies dropout stability, and hence, mode connectivity. Nguyen [2019] showed mode connectivity when there is a hidden layer larger than the training dataset. Shevchenko and Mondelli [2020] shows that stochastic gradient descent solutions to sufficiently overparametrized neural networks are dropout stable even if we only keep a small, randomly sampled set of neurons from each layer. Our contributions In this paper, we first establish mode connectivity properties of RochetNet, the architecture in Dütting et al. [2019] for multiple items and a single buyer. These networks have a single hidden layer where the neurons directly correspond to menu options: allocations and prices offered to the buyer (see Figure 1). The buyer is assigned the option that maximizes their utility; such an option is called active for the buyer. Despite its simplicity, the experiments on RochetNet gave impressive empirical results in different scenarios. For example, in the experiments with up to six items and uniform value distributions, RochetNet achieves almost the same revenue (99.9%) as the Straight-Jacket Auctions, which are known to be optimal in this case. This success is not limited to a single example, as RochetNet also consistently performs well in other scenarios, including when infinite menu size is necessary. Furthermore, Dütting et al. [2019] demonstrated the usefulness of RochetNet in discovering optimal auctions in situations that were previously unexplored from a theoretical perspective. First, in Theorem 8, we show that for linear utilities, ε-mode connectivity holds between two solutions that are ε-reducible: out of the K +1 menu options (neurons), there exists a subset of at most √ K + 1 containing an active option for the buyer with probability at least 1 − ε. Assuming that the valuations are normalized such that the maximum valuation of any buyer is at most one, it follows that if we remove all other options from the menu, at most ε of the expected revenue is lost. The assumption of being ε-reducible is stronger than ε-dropout stability that only drops a constant fraction of the neurons. At the same time, experimental results in Dütting et al. [2019] show evidence of this property being satisfied in practice. They experimented with different sized neural networks in a setting when the optimal auction requires infinite menu size. Even with 10, 000 neurons available, only 59 options were active, i.e., used at least once when tested over a large sample size. We note that this property also highlights an advantage of RochetNet over RegretNet and other similar architectures: instead of a black-box neural network, it returns a compact, easy to understand representation of a mechanism. Our second main result (Theorem 9) shows that for n items and linear utilities, if the number of menu options K is sufficiently large, namely, (2/ε) 4n , then ε-mode connectivity holds between any two solutions for any underlying distribution. The connectivity property holds pointwise: for any particular valuation profile, the revenue may decrease by at most ε along the path. A key tool in this ε-mode connectivity result is a discretization technique from Dughmi et al. [2014]. We note that such a mode connectivity result can be expected to need a large menu size. In Appendix C, we present an example with two disconnected local maxima for K = 1. We also extend our results and techniques to neural networks for affine maximizer auctions (AMA) studied in Curry et al. [2022b]. This is a generalization of RochetNet for multi-buyer scenarios. It can also be seen as a weighted variant of the Vickrey-Clarke-Groves (VCG) mechanism [Vickrey, 1961, Clarke, 1971, Groves, 1973. AMA offers various allocation options. For a given valuation profile, the auctioneer chooses the allocation with the highest weighted sum of valuations, and computes individual prices for the buyers; the details are described in Section 2.2. AMA is DSIC and IR, however, is not rich enough to always represent the optimal auction. For AMA networks, we show similar results (Theorem 12 and 13) as for RochetNet. We first prove ε-mode connectivity holds between two solutions that are ε-reducible (see Definition 10). Curry et al. [2022b] provides evidence of this property being satisfied in practice, observing (in Sec. 7.1) "Moreover, we found that starting out with a large number of parameters improves performance, even though by the end of training only a tiny number of these parameters were actually used.". Secondly, we also show that if the number of menu options K is sufficiently large, namely, (16m 3 /ε 2 ) 2nm , then ε-mode connectivity holds pointwise between any two solutions. That is, it is valid for any underlying distribution of valuations, possibly correlated between different buyers. Previous literature on mode connectivity investigated neural networks used for prediction. The results in Kuditipudi et al. [2019], Nguyen [2019, Shevchenko and Mondelli [2020] and other papers crucially rely on the properties that the networks minimize a convex loss function between the predicted and actual values, and require linear transformations in the final layer. RochetNet and AMA networks are fundamentally different. These are network architectures built directly for solving an optimization problem. For an input valuation profile, the loss function is the negative revenue of the auctioneer. In RochetNet, this is obtained as the negative price of the utility-maximizing bundle; for AMA it requires an even more intricate calculation. The objective is to find parameters of the neural network such that the expected revenue is as large as possible. The menu options define a piecewise linear surface of utilities, and the revenue in RochetNet can be interpreted as the expected bias of the piece corresponding to a randomly chosen input. Hence, the landscape of the loss function is fundamentally different from those analyzed in the above mentioned works. The weight interpolation argument that shows mode-connectivity from dropout stability is not applicable in this context. The main reason is that the loss function is not a simple function of the output of the network, but is defined by choosing the price of the argmax option. We thus need a more careful understanding of the piecewise linear surfaces corresponding to the menus. Auction Settings We consider the case with m buyers and one seller with n divisible items in unit supply. Each buyer has an additive valuation function v i (S) := j∈S v ij , where v ij ∈ V represents the valuation of the buyer i on item j and V is the set of possible valuations. Throughout the paper, we normalize the range to the unit simplex: we assume V = [0, 1], and v i 1 = j v ij ≤ 1 for every buyer i. With slight abuse of notation, we let v = (v 11 , v 12 , · · · , v ij , · · · , v mn ) and v i = (v i1 , v i2 , · · · , v in ) . The buyers' valuation profile v is drawn from a distribution F ∈ P(V m×n ). Throughout, we assume that the buyers have quasi-linear utilities: if a buyer with valuation v i receives an allocation x ∈ [0, 1] n at price p, their utility is v i x − p. The seller has access to samples from the distribution F , and wants to sell these items to the buyers through a DSIC and IR auction and maximize the expected revenue. In the auction mechanism, the i-th bidder reports a bid b i ∈ [0, 1] n . The entire bid vector b ∈ [0, 1] m×n will be denoted as b = (b 1 , . . . , b m ) = (b i , b −i ), where b −i represents all the bids other than buyer i. In a DSIC mechanism, it is a dominant strategy for the agents to report b i = v i , i.e., reveal their true preferences. Definition 1 (DSIC and IR auction). An auction mechanism requires the buyers to submit bids b i ∈ [0, 1] n , and let b = (b 1 , . . . , b m ). The output is a set of allocations x(b) = (x 1 (b), . . . , x m (b)), x i (b) ∈ [0, 1] n , and prices p(b) = (p 1 (b), . . . , p m (b)) ∈ R m . Since there is unit supply of each item, we require i x ij (b) ≤ 1, where x ij (b) is the allocation of buyer i of item j. (i) An auction is dominant strategy incentive compatible (DSIC) if v i x i (v i , b −i ) − p i (v i , b −i ) ≥ v x i (b i , b −i ) − p i (b i , b −i ) for any buyer i and any bid b = (b i , b −i ). (ii) An auction is individually rational (IR) if v i x i (v i , b −i ) − p i (v i , b −i ) ≥ 0. The revenue of a DSIC and IR auction is Rev = E v∼F i p i (v) . Single Buyer Auctions: RochetNet Dütting et al. [2019] proposed RochetNet as a DISC and IR auction for the case of a single buyer. We omit the subscript i for buyers in this case. A (possibly infinite sized) menu M comprises a set of options offered to the buyer: M = {(x (k) , p (k) )} k∈K . In each option (x (k) , p (k) ), x (k) ∈ [0, 1] n represents the amount of items, and p (k) ∈ R + represents the price. We assume that 0 ∈ K, and (x (0) , p (0) ) = (0, 0) to guarantee IR. We call this the default option, whereas all other options are called regular options. We will use K to denote the number of regular options; thus, \|K\| = K + 1. A buyer submits a bid b ∈ [0, 1] n representing their valuation, and is assigned to option k(b) ∈ K that maximizes the utility 1 k(b) ∈ arg max k∈K b x (k) − p (k) . This is called the active option for the buyer. Note that option 0 guarantees that the utility is nonnegative, implying the IR property. It is also easy to see that such an auction is DSIC. Therefore, one can assume that b = v, i.e., the buyer submits their true valuation; or equivalently, the buyer is allowed to directly choose among the menu options one that maximizes their utility. Moreover, it follows from Rochet [1987] that every DSIC and IR auction for a single buyer can be implemented with a (possibly infinite size) menu using an appropriate tie-breaking rule. Given a menu M , the revenue is defined as Figure 1: RochetNet: this architecture maps the bid b to the utility of the buyer. Rev(M ) = E v∼F p (k(v)) . −p1 −p2 · · · −p k b1 b2 · · · b d max x (1) x (2) x (k) · · · 0 RochetNet RochetNet (see Figure 1) is a neural network with three layers: an input layer (n neurons), a middle layer (K neurons), and an output layer (1 neuron): 1. the input layer takes an n-dimensional bid b ∈ V n , and sends this information to the middle layer; 2. the middle layer has K neurons. Each neuron represents a regular option in the menu M , which has parameters x (k) ∈ [0, 1] n and p (k) ∈ R + , where x (k) ∈ [0, 1] n represents the allocation of option k and p (k) represents the price of option k. Neuron k maps from b ∈ V n to b x (k) − p (k) , i.e., the utility of the buyer when choosing option k; 3. the output layer receives all utilities from different options and maximizes over these options and 0: max{max k {(x (k) ) b − p (k) }, 0}. We will use Rev(M ) to denote the revenue of the auction with menu options K = {0, 1, 2, . . . , K}, where 0 represents the default option (0, 0). The training objective for the RochetNet is to maximize the revenue Rev(M ), which is done by stochastic gradient ascent. Note, however, that the revenue is the price of an argmax option, which makes it a non-continuous function of the valuations. For this reason, Dütting et al. [2019] use a softmax-approximation of the argmax as their loss function instead. However, argmax is used for testing. In Appendix B, we bound the difference between the revenues computed with these two different activation functions, assuming that the probability density function of the distribution F admits a finite upper bound. Lemma 18 shows that the difference between the revenues for softmax and argmax is roughly inverse proportional to the parameter Y of the softmax function. This allows the practitioner to interpolate between smoothness of the loss function and provable quality of the softmax approximation by tuning the parameter Y . Affine Maximizer Auctions Affine Maximizer Auctions (AMA) also provide a menu M with a set of options K. Each option is of the form (x (k) , β (k) ) ∈ [0, 1] n×m × R, where x (k) ij ∈ [0, 1] represents the allocation of item i to buyer j, with the restriction that i x (k) ij ≤ 1 for each item j, and β (k) represents a 'boost'. We again assume 0 ∈ K, and (x (0) , β (0) ) = (0, 0), and call this the default option; all other options are called the regular options. Given the bids b i ∈ [0, 1] n of the agents, the auctioneer computes a weighted welfare, using weights w i ∈ R + for the valuations of each agent, and adds the boost β (k) . Then, the allocation maximizing the weighted boosted welfare is chosen, i.e., the option with k(b) ∈ arg max k∈K i w i b i x (k) i + β (k) . This will also be referred to as the active option. The prices collected from the buyers are computed according to the Vickrey-Clarke-Groves (VCG) scheme. Namely, p i (b) = 1 w i   =i w b x (k(b−i)) + β (k(b−i))   − 1 w i   =i w b x (k(b)) + β (k(b))   .(1) Here, k(b −i ) represents the option maximizing the weighted boosted welfare when buyer i is omitted, i.e., k(b −i ) ∈ arg max k∈K =i w b x (k) + β (k) . It is known that AMA is DSIC and IR. Hence, we can assume that the submitted bids b i represent the true valuations v i . We also assume the ties are broken in favor of maximizing the total payment. In case of unit weights, this is equivalent to choosing the smallest β (k) values, see (2) in Section 4. Given the menu M , the revenue of the AMA is Rev(M ) = E v∼F i p i (v) . In this paper, we focus on the case when w i = 1 for all buyers. This is also used in the experiments in Curry et al. [2022b]. For this case, AMA can be implemented by a three layer neural network similar to RochetNet, with m × n input neurons. For the more general case when the weights w i can also be adjusted, one can include an additional layer that combines the buyers' allocations. Note that for a single buyer and w 1 = 1, AMA corresponds to RochetNet, with price p (k) = −β (k) for each menu option. Indeed, in the formula defining the price p i (b), the first term is 0, as well as the sum in the second term. Similarly to RochetNet, the loss function, which is maximized via stochastic gradient ascent, is a softmax-approximation of the revenue Rev(M ), in order to avoid the discontinuities introduced by the argmax. We bound the difference in the revenue in Appendix D.3, concluding that it decreases with large parameter Y as in the RochetNet case. Mode Connectivity One can view the revenue as a function of the menus, i.e., the parameters in the mechanism: (i) in RochetNet, {(x (k) , p (k) )} k∈K ; (ii) in AMA, {(x (k) , β (k) )} k∈K . We use M to denote the set of all possible menus. Definition 2. (Mode connectivity) Two menus M 1 , M 2 ∈ M are ε-mode-connected if there is a continuous curve π : [0, 1] → M such that (i) π(0) = M 1 ; (ii) π(1) = M 2 ; and (iii) for any t ∈ [0, 1], Rev(π(t)) ≥ min{Rev(M 1 ), Rev(M 2 )} − ε. Mode Connectivity for the RochetNet In this section we present and prove our main results for the RochetNet. For some statements, we only include proof sketches. The detailed proofs can be found in Appendix A in the supplementary material. The following definition plays an analogous role to ε-dropout stability in Kuditipudi et al. [2019]. Definition 3. A menu M with \|K\| = K + 1 options is called ε-reducible if there is a subset K ⊆ K with 0 ∈ K , \|K \| ≤ √ K + 1 such that, with probability at least 1 − ε over the distribution of the valuation of the buyer, the active option assigned to the buyer is contained in K . As noted in the Introduction, such a property can be observed in the experimental results in Dütting et al. [2019]. The motivation behind this definition is that if a menu satisfies this property, then all but √ K + 1 options are more or less redundant. In fact, if a menu is ε-reducible, then dropping all but the at most √ K + 1 many options in K results in a menu M with Rev(M ) ≥ Rev(M ) − ε because the price of any selected option is bounded by v 1 ≤ 1. As a first step towards showing the mode connectivity results, we show that 0-reducibility implies 0-mode-connectivity. We will then use this to derive our two main results, namely that two ε-reducible menus are always ε-mode-connected and that two large menus are always ε-mode-connected. Proposition 4. If two menus M 1 and M 2 for the RochetNet are 0-reducible, then they are 0-modeconnected. Moreover, the curve transforming M 1 into M 2 is piecewise linear with only three pieces. To prove Proposition 4, we introduce two intermediate menus M 1 and M 2 , and show that every menu in the piecewise linear interpolation from M 1 via M 1 and M 2 to M 2 yields a revenue of at least min{Rev(M 1 ), Rev(M 2 )}. Using that menu M 1 has only √ K + 1 non-redundant options, menu M 1 will be defined by repeating each of the √ K + 1 options √ K + 1 times. Menu M 2 will be derived from M 2 similarly. A technical lemma makes sure that this copying can be done in such a way that each pair of a non-redundant option of M 1 and a non-redundant option of M 2 occurs exactly for one index in M 1 and M 2 . To make this more formal, we first assume without loss of generality that K + 1 is a square, such that √ K + 1 is an integer. It is straightforward to verify that the theorem is true for non-squares K + 1, too. Suppose the options in M 1 and M 2 are indexed with k ∈ K = {0, 1, . . . , K}. Since M 1 is 0-reducible, there is a subset K 1 ⊆ K with 0 ∈ K 1 , \|K 1 \| = √ K + 1 such that an option with index in K 1 is selected with probability 1 over the distribution of the possible valuations. Similarly, such a set K 2 exists for M 2 . To define the curve that provides mode connectivity, we need the following technical lemma, which is proven in Appendix A. Lemma 5. There exists a bijection ϕ : K → K 1 × K 2 such that for all k ∈ K 1 we have that ϕ(k) ∈ {k} × K 2 , and for all k ∈ K 2 we have that ϕ(k) ∈ K 1 × {k}. With this lemma, we can define M 1 and M 2 . Let ϕ the bijection from Lemma 5 and suppose M 1 = {(x (k) , p (k) )} k∈K . We then define M 1 = {(x (ϕ1(k)) , p (ϕ1(k)) )} k∈K , where ϕ 1 (k) is the first component of ϕ(k). Similarly, M 2 is derived from M 2 by using the second component ϕ 2 (k) of ϕ(k) instead of ϕ 1 (k) . It remains to show that all menus on the three straight line segments from M 1 via M 1 and M 2 to M 2 yield a revenue of at least min{Rev(M 1 ), Rev(M 2 )}, which is established by the following two propositions; their proofs can be found in Appendix A. The idea to prove Proposition 6 is that, on the whole line segment from M 1 to M 1 , the only active options are those in K , implying that the revenue does not decrease. Proposition 7. Let M = λ M 1 + (1 − λ) M 2 be a convex combination of the menus M 1 and M 2 . Then, Rev(M ) ≥ λRev( M 1 ) + (1 − λ)Rev( M 2 ). The idea to prove Proposition 7 is that, due to the special structure provided by Lemma 5, a linear interpolation between the menus also provides a linear interpolation between the revenues. Note that without the construction of Lemma 5, such a linear relation would be false; such an example is shown in Appendix C. Proposition 4 directly follows from Proposition 6 and Proposition 7. Based on Proposition 4, we can show our two main theorems for the RochetNet. The first result follows relatively easily from Proposition 4. Theorem 8. If two menus M 1 and M 2 for the RochetNet are ε-reducible, then they are ε-modeconnected. Moreover, the curve transforming M 1 into M 2 is piecewise linear with only five pieces. Proof. We prove this result by showing that every ε-reducible menu M can be linearly transformed into a 0-reducible menu M such that each convex combination of M and M achieves a revenue of at least Rev(M ) − ε. This transformation converting M 1 and M 2 to M 1 and M 2 , respectively, yields the first and the fifth of the linear pieces transforming M 1 to M 2 . Together with Proposition 4 applied to M 1 and M 2 serving as the second to fourth linear piece; the theorem then follows. To this end, let M be an ε-reducible menu with options indexed by k ∈ K. By definition, there is a subset K ⊆ K of at most √ K + 1 many options such that, with probability at least 1 − ε, the assigned active option is contained in K . Let M consist of the same allocations as M , but with modified prices. For an option k ∈ K , the pricep (k) = p (k) in M is the same as in M . However, for an option k ∈ K \ K , we set the pricep (k) > 1 in M to be larger than the largest possible valuation of any option v 1 ≤ 1. It follows that such an option will never be selected and M is 0-reducible. To complete the proof, let us look at the reward of a convex combination M = λM + (1 − λ) M . If for a particular valuation v the selected option in M was in K , then the same option will be selected in M . This happens with probability at least 1 − ε. In any other case, anything can happen, but the revenue cannot worsen by more than the maximum possible valuation, which is v 1 ≤ 1. Therefore, Rev(M ) − Rev(M ) ≤ ε · 1 = ε, completing the proof. Theorem 9. If two menus M 1 and M 2 for the RochetNet have size at least 4 2 2n , then they are ε-connected. Moreover, the curve transforming M 1 into M 2 is piecewise linear with only five pieces. Proof Sketch. The full proof can be found in Appendix A.2. The intuition behind this theorem is that if menus are large, then they should contain many redundant options. Indeed, as in the previous theorem, the strategy is as follows. We show that every menu M of size at least 4 2 2n can be linearly transformed into a 0-reducible menu M such that each convex combination of M and M achieves a revenue of at least Rev(M ) − ε. This transformation converting M 1 and M 2 to M 1 and M 2 , respectively, yields the first and the fifth of the linear pieces transforming M 1 to M 2 . Together with Proposition 4 applied to M 1 and M 2 serving as the second to fourth linear piece, the theorem then follows. However, this time, the linear transformation of M to M is much more intricate than in the previous theorem. To do so, it is not sufficient to only adapt the prices. Instead, we also change the allocations of the menu options by rounding them to discretized values. This technique is inspired by Dughmi et al. [2014], but non-trivially adapted to our setting. Since the rounding may also modify the active option for each valuation, we have to carefully adapt the prices in order to make sure that for each valuation, the newly selected option is not significantly worse than the originally selected one. Finally, this property has to be proven not only for M , but for every convex combination of M and M . After the above rounding, the number of possible allocations for any option is bounded by 4 2 n . Out of several options with the same allocation, the buyer would always choose the cheapest one, implying that the resulting menu M is 0-reducible. Mode Connectivity for the Affine Maximizer Auctions Throughout this section, we focus on AMAs with fixed weights w i = 1 for all buyers i. Similarly to RochetNet, we have the following definition for AMAs. Definition 10. A menu M with K + 1 options is ε-reducible if and only if there exists a subset K ⊆ K, 0 ∈ K , \|K \| ≤ √ K + 1 such that, with probability at least 1 − ε m over the distribution of the valuation of the buyers, (i) k(v −i ) ∈ K for any buyer i; and (ii) k(v) ∈ K . Our two main results, namely that two ε-reducible menus are always ε-connected and two large menus are always ε-connected, are based on the following proposition, in which we show that 0-reducibility implies 0-connectivity. Proposition 11. If two menus M 1 and M 2 are 0-reducible, then they are 0-connected. Moreover, the curve transforming M 1 into M 2 is piecewise linear with only three pieces. The proof idea is similar to the proof of Proposition 4 in RochetNet, but requires additional arguments due to the more intricate price structure (see Appendix D.1 for more details). Based on this proposition, now, we are able to show our two main results. First, we achieve ε-connectivity from ε-reducibility. Theorem 12. If two AMAs M 1 and M 2 are ε-reducible, then they are ε-mode-connected. Moreover, the curve transforming M 1 to M 2 is piecewise linear with only five pieces. Before the proof, we recall how the total payment is calculated for a valuation profile v. We choose k(v) as the option which maximizes the boosted welfare, i v i x (k) i + β (k) . According to (1), the total revenue can be written as i p i (v) = i   =i v x (k(v−i)) + β (k(v−i))   boosted welfare of v−i −(m − 1) i v i x (k(v)) i + β (k(v)) boosted welfare of v −β (k(v)) . (2) Proof of Theorem 12. Similar to the proof of Theorem 8, it is sufficient to show that every ε-reducible menu M can be linearly transformed into a 0-reducible menu M such that each convex combination of M and M achieves a revenue of at least Rev(M ) − ε. This can then be used as the first and fifth linear piece of the curve connecting M 1 and M 2 , while the middle three pieces are provided by Proposition 11. We construct M by (i) keeping all options in K unchanged; (ii) for the options k ∈ K \ K , we decrease β (k) to be smaller than −m, which implies such an option will never be selected (recall that 0 ∈ K is assumed, and the option (0, 0) is better than any such option). Consequently, M is 0-reducible. To complete the proof, let us look at the revenue of M = {(x (k) , β (k) )} k∈K , which is a convex combination of M and M : M = λM + (1 − λ) M for 0 ≤ λ < 1. Let k (v) = arg max k i v i x (k) i + β (k) . As we decrease β (k) for k / ∈ K , k(v) ∈ K implies k (v) ∈ K and, additionally, option k (v) and option k(v) achieve the same boosted welfare and same β. Therefore, since M is ε-reducible, with probability at least 1 − ε m , the boosted welfare of v as well as the boosted welfare of v −i for all buyers i is the same for M and for M . According to the formula (2), the total payment for the profile v is the same for M and M . Therefore, the loss on the revenue can only appear with probability at most ε m , and the maximum loss is at most m, which implies an ε loss in total. Second, we show that mode connectivity also holds for those AMAs with large menu sizes, namely for K + 1 ≥ 16m 3 2 2nm . Theorem 13. For any 0 < ≤ 1 4 , if two AMAs M 1 and M 2 have at least K + 1 ≥ 16m 3 2 2nm options, then they are ε-mode-connected. Moreover, the curve transforming M 1 to M 2 is piecewise linear with only five pieces. Proof Sketch. The full proof can be found in Appendix D.2. Similar to RochetNet, the idea of proving Theorem 13 is to discretize the allocations in the menu, then one can use Proposition 11 to construct the low-loss transformation from M 1 to M 2 by five linear pieces. To do this, one wants the loss of revenue to be small during the discretization. Consider the formula (2) of the total payment. The first two terms do not change much by a small change of the discretization. However, the last term β (k(v)) might be significantly affected by discretization, which may cause a notable decrease in the total payment. To avoid this, we perform a proportional discount on β, incentivizing the auctioneer to choose an allocation with a small β. By this approach, the original revenue will be approximately maintained. Furthermore, we show a linear path, connecting the original menu and the menu after discretizing, which will suffer a small loss. Conclusion We have given theoretical evidence of mode-connectivity in neural networks designed to learn auction mechanisms. Our results show that, for a sufficiently wide hidden layer, ε-mode-connectivity holds in the strongest possible sense. Perhaps more practically, we have shown ε-mode-connectivity under ε-reducibility, i.e., the assumption that there is a sufficiently small subset of neurons that preserve most of the revenue. There is evidence for this assumption in previous work in differentiable economics. A systematic experimental study that verifies this assumption under various distributions and network sizes is left for future work. Our results make a first step in providing theoretical arguments underlying the success of neural networks in mechanism design. Our focus was on some of the most basic architectures. A natural next step is to extend the arguments for AMA networks with variable weights w i . Such a result will need to analyze a four layer network, and thus could make headway into understanding the behaviour of deep networks. Besides RochetNet, Dütting et al. [2019] also proposed RegretNet, based on minimising a regret objective. This network is also applicable to multiple buyers, but only provides approximate incentive compatibility, and has been extended in subsequent work, e.g., Feng et al. [2018], Golowich et al. [2018], Duan et al. [2022]. The architecture is however quite different from RochetNet: it involves two deep neural networks in conjunction, an allocation and a payment network, and uses expected ex post regret as the loss function. We therefore expect a mode-connectivity analysis for RegretNet to require a considerable extension of the techniques used by us. We believe that such an analysis would be a significant next step in the theoretical analysis of neural networks in differentiable economics. A Detailed Proofs of the Mode Connectivity for the RochetNet In this section we provide the detailed proofs omitted in Section 3. A.1 Interpolating between 0-reducible menus We start with the proofs of statements on the way towards proving Proposition 4. Proof of Lemma 5. We prove the claim by providing an explicit construction for ϕ in two different cases. First, suppose that there is some k * ∈ K 1 ∩ K 2 . In this case, start by setting ϕ(k) := (k, k) for all k ∈ K 1 ∩ K 2 . Then, for all k ∈ K 1 \ K 2 , set ϕ(k) = (k, k * ), and for all k ∈ K 2 \ K 1 , set ϕ(k) = (k * , k). So far, we have not assigned any pair twice and the two conditions of the lemma are already satisfied, so we can simply assign the remaining pairs arbitrarily. Second, suppose that K 1 and K 2 are disjoint. Note that for this being possible, √ K + 1 must be at least 2. Pick some distinct k 1 , k 1 ∈ K 1 and k 2 , k 2 ∈ K 2 . Set ϕ(k 1 ) := (k 1 , k 2 ), ϕ(k 1 ) := (k 1 , k 2 ), ϕ(k 2 ) := (k 1 , k 2 ), and ϕ(k 2 ) := (k 1 , k 2 ). Then, for all k ∈ K 1 \ {k 1 , k 1 }, set ϕ(k) := (k, k 2 ) and for all k ∈ K 2 \ {k 2 , k 2 }, set ϕ(k) := (k 1 , k). Again, we have not assigned any pair twice and the two conditions of the lemma are already satisfied, so we can simply assign the remaining pairs arbitrarily. Proof of Proposition 6. We only prove the first statement on M 1 ; the statement on M 2 follows analogously. We show that for each possible valuation v of the buyer, the price paid to the seller for menu M is at least as high as in menu M 1 . Suppose for valuation v that the buyer chooses the k-th option in menu M 1 . Note that we may assume k ∈ K 1 due to 0-reducibility of M 1 . By construction of ϕ, it follows that ϕ 1 (k) = k. Therefore, the k-th option in M is exactly equal to the k-th option in M 1 . Making use of the fact that ties are broken in favor of larger prices, it suffices to show that the k-th option is utility-maximizing in M , too. To this end, let k ∈ K be an arbitrary index. If M 1 = {(x (k) , p (k) } k∈K , then the utility of option k in M is v (λx (k ) + (1 − λ)x (ϕ1(k )) ) − (λp (k ) + (1 − λ)p (ϕ1(k )) ) = λ(v x (k ) − p (k ) ) + (1 − λ)(v x (ϕ1(k )) − p (ϕ1(k )) ) ≤ λ(v x (k) − p (k) ) + (1 − λ)(v x (k) − p (k) ) = v x (k) − p (k) , where the inequality follows because the k-th option is utility-maximizing for menu M 1 . This shows that it is utility-maximizing for menu M , completing the proof. Proof of Proposition 7. The claim is trivial for λ = 0 or λ = 1. Therefore, assume 0 < λ < 1 for the remainder of the proof. Again we show that the claim holds pointwise for each possible valuation and therefore also for the revenue. For valuation v, let k 1 and k 2 be the active option assigned to the buyer in M 1 and M 2 , respectively. Note that by construction of the menus M 1 and M 2 we may assume without loss of generality that k 1 ∈ K 1 and k 2 ∈ K 2 . Let k * := ϕ −1 (k 1 , k 2 ). We show that option k * is utility-maximizing in M = {(x (k) , p (k) )} k∈K . To this end, we use the notation M 1 = {(x (k) ,p (k) )} k∈K and M 2 = {(ŷ (k) ,q (k) )} k∈K . Let k ∈ K be an arbitrary index. The utility of option k in menu M can be bounded as follows: v x (k ) − p (k ) = λ(v x (k ) −p (k ) ) + (1 − λ)(v ŷ (k ) −q (k ) ) ≤ λ(v x (k1) −p (k1) ) + (1 − λ)(v ŷ (k2) −q (k2) ) = λ(v x (k * ) −p (k * ) ) + (1 − λ)(v ŷ (k * ) −q (k * ) ) = v x (k * ) − p (k * ) , where the inequality in the second line follows because k 1 and k 2 are utility-maximizing for M 1 and M 2 , respectively, and the equality in the third line follows because, by construction, in menu M 1 option k * is equivalent to option k 1 = ϕ 1 (k * ) ∈ K 1 , and similarly in menu M 2 option k * is equivalent to option k 2 = ϕ 2 (k * ) ∈ K 2 . This concludes the proof that k * is utility-maximizing. With the same reasoning as above, we obtain p (k * ) = λp (k1) + (1 − λ)q (k2) , from which we conclude that the price achieved by the seller in menu M for valuation v is at least as high as the convex combination of the achieved prices for menus M 1 and M 2 . A.2 Discretizing large menus This subsection is devoted to providing a detailed proof of Theorem 9. To do so, we will show how to convert any menu M of size at least 4 2 2n into a 0-reducible menu M such that each convex combination of M and M achieves a revenue of at least Rev(M ) − ε. Without loss of generality, we assume that M has size exactly K + 1 = 4 2 2n . To construct the menu M satisfying these requirements, we adapt techniques from Dughmi et al. [2014]. 2 In general, the idea is to discretize the allocations in the menu by a finite allocation set S (see Definition 14) whose size is at most √ K + 1 = 4 2 n . However, because of the discretization, the buyer may choose an option with a much smaller price, providing a lower revenue compared to the original menu. To deal with this, we also decrease the prices on the menu; the decrease is in proportion to the price. Intuitively, this incentives the buyer to choose the option with an originally high price. We show, after this modification, the menu achieves a revenue of at least Rev(M ) − ε. For ease of notation, we will useε := ε 2 4 and, therefore, 2 √ε = ε. Definition 14. Let S be a (finite) set of allocations. We say that S is anε-cover if, for every possible allocation x, there exists an allocationx ∈ S such that for every possible valuation vector v we have that v x ≥ v x ≥ v x −ε. The following proposition shows that one can construct anε-cover S with size at most 4 2 n . Proposition 15. If v 1 ≤ 1, then S = {εs} 1 ε s=0 × {εs} 1 ε s=0 × · · · × {εs} 1 ε s=0 n terms is anε-cover, and \|S\| = 1 ε n = 4 2 n . Proof. For any allocation x, we can round it down tox, such thatx j = xj ε ·ε. It is not hard to see that v x ≥ v x. Additionally, the inequality v x ≥ v x −ε follows as the total loss is at most v (x − x) ≤ v 1 x − x ∞ ≤ε. Construction of M . Given S, we can construct M as follows. Each option (x (k) , p (k) ) in menu M is modified to (x (k) ,p (k) ) in menu M , wherex (k) is the corresponding allocation of x (k) in S and the price is set top (k) = 1 − √ε p (k) : M = x (k) ,p (k) k∈K . The following lemma shows that this construction indeed ensures that the reward decreases by at most ε. Lemma 16. It holds that Rev( M ) ≥ Rev(M ) − 2 √ε = Rev(M ) − ε. Proof. The following inequalities demonstrate the buyer who chooses option k in menu M will not choose option k in menu M such that p (k ) < p (k) − √ε . v x (k) − 1 − √ε p (k) ≥ v x (k) − p (k) −ε + √ε p (k) ≥ v x (k ) − p (k ) −ε + √ε p (k) ≥ v x (k ) − 1 − √ε p (k ) −ε + √ε (p (k) − p (k ) ) > v x (k ) − 1 − √ε p (k ) .(3) The first and third inequalities hold by Definition 14 and the second inequality holds as the buyer will choose option k in menu M 1 . Therefore, the total loss on the revenue is upper bounded by √ε p (k) + √ε ≤ 2 √ε , as the price satisfies p (k) ≤ 1. In addition to this property of M itself, we also need to show the revenue does not drop more than ε for any menu on the line segment connecting M to M . Proof. Let M = {(x (k) , p (k) )} k∈K and M = {(x (k) , p (k) )} k∈K . Similar to the proof of Lemma 16, we show that the buyer who chooses option k in menu M will not choose option k in menu M such that p (k ) < p (k) − √ε . This is true by the following (in)equalities. For any k ∈ K, we have that v x (k) − p (k) = λ(v x (k) − p (k) ) + (1 − λ)(v x (k) −p (k) ) > λ(v x (k ) − p (k ) ) + (1 − λ)(v x (k ) −p (k ) ). The inequality follows by combining (i) v x (k) − p (k) ≥ v x (k ) − p (k ) , which is true as the buyer will choose option k in menu M ; and (ii) v x (k) −p (k) > v x (k ) −p (k ) from (3). Similar to the proof of Lemma 16, it follows than that the total loss on the revenue is upper bounded by 2 √ε . With these lemmas at hand, we can finally prove Theorem 9. Proof of Theorem 9. Applying the transformation described in this section to convert M 1 and M 2 results in two menus M 1 and M 2 , respectively. Since M 1 and M 2 contain at most √ K + 1 = 4 2 n different allocations and a buyer would always choose the cheapest out of several options with the same allocation, they are 0-reducible. Applying Proposition 4 to them implies that they are 0-mode-connected with three linear pieces. Combining these observations with Lemmas 16 and 17 implies that M 1 and M 2 are ε-connected with five linear pieces. B Bounds on the Error of the Softmax Approximation for the Argmax In the RochetNet, to ensure that the objective is a smooth function, a softmax operation is used instead of the argmax during the training process: Rev softmax (M ) = K k=1 p i e Y (x (k) v−p (k) ) K k =1 e Y (x (k ) v−p (k ) ) dF (v). Here, Y is a sufficiently large constant. In this section, we will look at the difference between the actual revenue and this softmax revenue. We would like to assume the density of the valuation distribution is upper bounded by X = max v∈[0,1] n and v 1≤1 f (v), which is a finite value. Given this assumption, the following lemma shows that, for any menu M of size K, the difference between the actual revenue and the softmax revenue is bounded. Lemma 18. For any M and Y ≥ 1, \|Rev softmax (M ) − Rev(M )\| ≤ K + 1 Y (nX + 1 + X Y ) log Y X + X . Proof. We prove Rev softmax (M ) − Rev(M ) ≤ K Y (nX + 1 + X Y ) log Y X + X . Rev(M ) − Rev softmax (M ) ≤ K Y (nX + 1 + X Y ) log Y X + X follows by a similar argument. Let k(v) be the option chosen in menu M when the buyer's valuation is v. Then, the difference between these two can be bounded as follows. Rev softmax (M ) − Rev(M ) ≤ K k=0 (p (k) − p (k(v)) ) + · e Y (v x (k) −p (k) ) K k =1 e Y (v x (k ) −p (k ) ) dF (v) ≤ K k=0 (p (k) − p (k(v)) ) + e Y (v x (k) −p (k) −v x (k(v)) +p (k(v)) ) dF (v). Here, (·) + max{·, 0}. Now, we focus on one option k, and we will give an upper bound on (p (k) − p (k(v)) ) + 1 v x (k) −p (k) +σ≥v x (k(v)) −p (k(v)) ≥v x (k) −p (k) dF (v)(4) for the non-negative parameter σ, which will be specified later. Note that, it is always true that v x (k(v)) − p (k(v)) ≥ v x (k) − p (k) . If v x (k) − p (k) + σ ≥ v x (k(v)) − p (k(v)) is not satisfied then e Y (v x (k) −p (k) −v x (k(v)) +p (k(v)) ) ≤ e −Y σ . Therefore, if (4) is upper bounded by C(σ), then Rev softmax M − Rev(M ) ≤ (K + 1)(C(σ) + (1 + σ)e −Y σ ). 3 Note that (p (k) − p (k(v)) ) + 1 v x (k) −p (k) +σ≥v x (k(v)) −p (k(v)) ≥v x (k) −p (k) dF (v) ≤ σ + n j=1 v j (x (k) j − x (k(v)) j ) + 1 v x (k) −p (k) +σ≥v x (k(v)) −p (k(v)) ≥v x (k) −p (k) dF (v). The inequality follows as we consider the region of v such that v x (k) −p (k) +σ ≥ v x (k(v)) −p (k(v)) . Additionally, since v j ∈ [0, 1], v j (x (k) j − x (k(v)) j ) + 1 v x (k) −p (k) +σ≥v x (k(v)) −p (k(v)) ≥v x (k) −p (k) dF (v) ≤ (x (k) j − x (k(v)) j ) + 1 v x (k) −p (k) +σ≥v x (k(v)) −p (k(v)) ≥v x (k) −p (k) dF (v). Now we fix all coordinates of valuation v other than coordinate j. Note that, the function v x (k(v)) − p (k(v)) − v x (k) − p (k) is a convex function on v j and x (k) j − x (k(v)) j is the negative gradient of this convex function. Since we are looking at the region such that the function v x (k(v)) − p (k(v)) − v x (k) − p (k) is bounded in [0, σ], this direct imply vj ∈[0,1] (x (k) j − x (k(v)) j ) + 1 v x (k) −p (k) +σ≥v x (k(v)) −p (k(v)) ≥v x (k) −p (k) dF (v) ≤ X σ. This implies Rev softmax M − Rev(M ) ≤ (K + 1)(σ + nX σ + (1 + σ)e −Y σ ) which is upper bounded by K+1 Y (nX + 1 + X Y ) log Y X + X by setting σ = 1 Y log Y X . C Example: Disconnected Local Maxima This section shows that the revenue is not quasiconcave on M , and in fact it might have disconnected local maxima. Recall that a function g is quasiconcave if and only if, for any x, y and λ ∈ [0, 1], g(λx + (1 − λ)y) ≥ min{g(x), g(y)} Hence, quasiconcavity implies 0-mode-connectivity with a single straight-line segment. We consider the case that there is only one buyer, one item, and one regular option on the menu. Consider the following value distribution f : f (x) =    1.5 0 < x ≤ 1 3 + 0.15 0 1 3 + 0.15 < x ≤ 2 3 + 0.15 1.5 2 3 + 0.15 < x ≤ 1.(5) With this probability distribution, we show the following result. As Figure 2 shows, there are two local maxima so that any continuous curve connecting them has lower revenue than either endpoint. Hence, mode connectivity fails between these two points. We only give a formal proof of the fact that the revenue is not quasiconcave. Proof. We consider the case where n = 1 (single item case); K = 1 (menu with single options). The value distribution, f is defined in (5). We However, if we consider M 3 = 1 2 (M 1 + M 2 ) = {(0, 0), (1, 0.6)}, then this provides a revenue of 0.165, which is strictly smaller than Rev(M 1 ) and Rev(M 2 ). More intuitively, Figure 2 shows the revenue for x ∈ [0, 1] and p ∈ [0, 1]. D Detailed Proofs of the Mode Connectivity for AMAs In this section, we provide the detailed proofs omitted in Section 4 D.1 Interpolating between 0-reducible menus In this subsection, we will prove Proposition 11, that is, we show that two 0-reducible menus M 1 = {(x (1,k) , β (1,k) )} k∈K and M 2 = {(x (2,k) , β (2,k) )} k∈K are 0-mode-connected. Similar to RochetNet, we introduce two intermediate menus M 1 and M 2 , and show that every menu in the piecewise linear interpolation form M 1 via M 1 and M 2 to M 2 yields a revenue of at least min{Rev(M 1 ), Rev(M 2 )}. Using that menu M 1 has only √ K + 1 non-redundant options, menu M 1 will be defined by repeating each of the √ K + 1 options √ K + 1 times. Menu M 2 will be derived from M 2 similarly. To make this more formal, let K 1 ( and K 2 ) denote the set of the indexes of options in M 1 ( and M 2 ) in definition of ε-reducibility, respectively. Similar to RochetNet, with the help of the Lemma 5, we can formally define M 1 and M 2 as M 1 = {(x (1,ϕ1(k)) , β (1,ϕ1(k)) )} k∈K , where ϕ 1 (k) is the first component of ϕ(k); and, similarly, M 2 is derived from M 2 by using the second component ϕ 2 (k) of ϕ(k) instead of ϕ 1 (k). Proof. We only prove the first statement because the second one is analogous. We show that for each possible valuation v ∈ V mn (with v i ≤ 1 for all i) of the buyers, the total payment paid to the auctioneer for menu M is at least as high as in menu M 1 . Suppose for valuation v ∈ V n that the auctioneer chooses the k(v)-th option in menu M 1 in maximizing the boosted welfare. Note that we may assume k(v) ∈ K 1 due to 0-reducibility of M 1 . By construction of ϕ, it follows that ϕ 1 (k(v)) = k(v). Therefore, the k(v)-th option in M exactly equals the k(v)-th option in M 1 . Because ties are broken in favor of larger total payments, it suffices to show that the k(v)-th option is the one with the highest boosted welfare also in M . 4 Let k ∈ K be an arbitrary index. The boosted welfare of option k in M is i v i (λx (1,k ) i + (1 − λ)x (1,ϕ1(k )) i )) + (β (1,k ) + (1 − λ)β (1,ϕ1(k )) ) = λ( i v i x (1,k ) i + β (1,k ) ) + (1 − λ)( i v i x (1,ϕ1(k )) i + β (1,ϕ1(k )) ) ≤ λ( i v i x (1,k(v)) i + β (1,k(v)) ) + (1 − λ)( i v i x (1,k(v)) i + β (1,k(v)) ) = i v i x (1,k(v)) i + β (1,k(v)) , where the inequality follows because the k(v)-th option is boosted welfare maximizing for menu M 1 . This shows that k(v) is also a boosted welfare maximizer for menu M , completing the proof. Proof. The claim is trivial for λ = 0 or λ = 1. Therefore, assume 0 < λ < 1 for the remainder of the proof. For possible valuation v ∈ V n such that v i ≤ 1 for all i, let k 1 (v) and k 2 (v) be the boosted welfare maximizing options in M 1 and M 2 , respectively. Note that by the construction of the menus M 1 and M 2 , we may assume without loss of generality that k 1 (v) ∈ K 1 and k 2 (v) ∈ K 2 . Let k * (v) := ϕ −1 (k 1 (v), k 2 (v)). We show that option k * (v) is boosted welfare maximizing in M = {(x (k) , β (k) )} k∈K with valuation v. To this end, we use the notation M 1 = {(x (1,k) ,β (1,k) )} k∈K and M 2 = {(x (2,k) ,β (2,k) )} k∈K . D.2 Discretizing large menus This subsection provides a detailed proof of Theorem 13. To do this, we will show that, for an AMA with a large number of options, one can discretize it such that, after discretization, the menu is 0-reducible. Additionally, during this discretization, the revenue loss will be up to ε. Note that the payments and allocations only depend on those boosted welfare maximizing options. Therefore, to show that M 1 is 0-reducible, it suffices to show M 1 has at most √ K + 1 different allocations. We now formally define M 1 . We introduce parametersε and δ, which will be specified later. Construction of M 1 For x (k) , we round it tox (k) in whichx v i ≤ 1 for i, 6 i v i x (k) i ≥ i v ix (k) i ≥ i v i x (k) i −ε.(6) For β (k) , we letβ (k) = (1 − δ)β (k) . Lemma 23. For any given 0 < ε ≤ 1 4 , let δ = √ε m andε = ε 2 16m 2 . Then, Rev( M 1 ) ≥ Rev(M 1 ) − ε. The number of different allocations in M 1 is at most 16m 3 Proof. We first demonstrate Rev( M 1 ) ≥ Rev(M 1 ) − mε − m 2 δ 1−δ −ε δ . The result follows by picking δ = √ε m andε = ε 2 16m 2 . The proof of the bound on the linear combination of M 1 and M 1 is analogous. We use the notation M 1 = {x (k) , β (k) } k∈K and M 1 = {x (k) ,β (k) } k∈K . We fix the valuation v. Let k(v) = arg max k i v i x (k) i + β (k) and satisfy the tie-breaking rule. The total payment using M 1 can be expressed as follows: i   l =i v l x (k(v−i)) l + β (k(v−i))   − i   l =i v l x (k(v)) l − β (k(v))   = i   l =i v l x (k(v−i)) l + β (k(v−i))   A1 −(m − 1) i v i x (k(v)) i + β (k(v)) B1 −β (k(v)) . (7) Similarly, letk(v) = arg max k i v ix (k) i +β (k) , then, the total payment with M 1 is i   l =i v lx (k(v−i)) l +β (k(v−i))   A2 −(m − 1) i v ix (k(v)) i +β (k(v)) B2 −β (k(v)) .(8) We bound the differences between A 1 and A 2 , B 1 and B 2 , and β (k(v)) andβ (k(v)) separately. 6 The second inequality holds as v i (x We show the following result. Note that we also assume the maximal density of a valuation type is X . Theorem 24. \|Rev softmax (M ) − Rev(M )\| ≤ m(K + 1) eY + nmX (K + 1) Y 1 + log mY mX . To prove this theorem, we need the following lemma which provides one of the basic properties of the softmax. Lemma 25. Given L values, a 1 ≥ a 2 ≥ a 3 ≥ · · · ≥ a L , then 0 ≤ a 1 − k a k e Y a k k e Y a k ≤ L eY . Proof. It's clear that 0 ≤ a 1 − k a k e Y a k k e Y a k . On the other direction, a 1 − k a k e Y a k k e Y a k ≤ k (a 1 − a k ) e Y (a k −a1) k e Y (a k −a1) ≤ 1 Y k Y (a 1 − a k ) e Y (a k −a1) k e Y (a k −a1) ≤ 1 Y k e Y (a1−a k )−1 e Y (a k −a1) k e Y (a k −a1) ≤ L eY 1 k e Y (a k −a1) ≤ L eY . Now, we can prove Theorem 24. Proof of Theorem 24. We first give the upper bound on Rev(M ) − Rev softmax (M ). Let k(v) = arg max k i v i x (k) i is the gradient. Since we are looking at the region such that the function BW(k(v)) − BW(k) is bounded in [0, σ], this direct implies (x (k) ij − x (k(v)) ij ) + 1 BW(k)+σ≥BW(k(v))≥BW(k) dF (v) ≤ X σ. as the maximal density is at most X . This implies Rev(M ) − Rev softmax (M ) ≤ m(K+1) eY + (K + 1)nmX σ + (K + 1)nme −Y σ which is upper bounded by m(K+1) eY + nmX (K+1) Y 1 + log Y X by setting σ = 1 Y log Y X . The upper bound on Rev softmax (M ) − Rev(M ) follows by a similar argument. Proposition 6 . 6Let M = λM 1 + (1 − λ) M 1 be a convex combination of the menus M 1 and M 1 . Then Rev(M ) ≥ Rev(M 1 ). Similarly, every convex combination of the menus M 2 and M 2 has revenue at least Rev(M 2 ). Such phenomena are observed in the experiments in [Curry et al., 2022b, Section 6.3]. Lemma 17 . 17Let M = λM + (1 − λ) M be a convex combination of the menus M and M . Then, Rev(M ) ≥ Rev(M ) − 2 √ε = Rev(M ) − ε. Figure 2 : 2Revenue of the mechanism M = {(x, p)} when the value distribution is f . Lemma 19. Rev(M ) is not quasiconcave on M . consider two menus: M 1 and M 2 , where M 1 = {(0, 0), (1, 0.36)} and M 2 = {(0, 0), (1, 0.84)}. Then, Rev(M 1 ) = 0.1656 and Rev(M 2 ) = 0.2016. It remains to show that all menus on the three straight line segments from M 1 via M 1 and M 2 to M 2 yield revenue of at least min{Rev(M 1 ), Rev(M 2 )}. Proposition 20. Let M = λM 1 + (1 − λ) M 1 be a convex combination of the menus M 1 and M 1 . Then Rev(M ) ≥ Rev(M 1 ). 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We assume that ties are broken in favor of higher prices, but it is not hard to see that our results transfer to other tie-breaking rules, too. See also the discussion in Section B.2 ofBabaioff et al. [2022]. In their paper, they construct a menu with a finite number of options to approximate the optimal mechanism. The approximation is based on the multiplicative error, and they assume the buyer's valuation is no less than 1. Note that if p (k) ≥ 1 + σ then v x (k) − p (k) + (p (k) − 1) ≤ v x (k(v)) − p (k(v)) as LHS ≤ 0 and RHS ≥ 0. Therefore, if v x (k) − p (k) + σ ≥ v x (k(v)) − p (k(v)) is not satisfied, then (p (k) − p (k(v)) ) + e Y (v x (k) −p (k) −v x (k(v)) +p (k(v)) ) ≤ max σ ≥σ {(1 + σ )e −Y σ }. Note that max σ ≥σ {(1 + σ )e −Y σ } ≤ (1 + σ)e −Y σ when Y ≥ 1. Recall that the auctioneer will choose the option k maximize the β (k) among all boosted welfare maximizing options given the formula of the total payment (2). Note that both k(·) and k * (·) maximize the boosted welfare. This is true becasue v x (k(v −i )) +β (k(v −i )) ≤ v x (k(v)) +β (k(v))and v x (k(v)) ≤ m. Acknowledgments and Disclosure of FundingAll three authors gratefully acknowledge support by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement ScaleOpt-757481).Let k ∈ K be an arbitrary index. Then, the boosted welfare of option k can be bounded as follows:where the inequality in the second line follows because k 1 (v) and k 2 (v) are boosted welfare maximizers for M 1 and M 2 , respectively. The equality in the third line follows because, by construction,Note that the total payment of M = {(x (k) , β (k) )} k∈K can be written in the following form:where k(·) is the boosted welfare maximizer used in M . As ties are broken in favor of larger total payments, this value decreases by replacing k(·) by k * (·) 5 :Since k * (v) is fixed for different λ and, by linear combination, it holds thatx (k * (·)) = λx (1,k * (·)) + (1 − λ)x (2,k * (·)) andβ (k * (·)) = λβ (1,k * (·)) + (1 − λ)β (2,k * (·)) ,This completes the proof.(12)which contradicts the fact thatk(v) = arg max k i v ixBy combining the formula of total payment with M 1 , (7), the formula of total payment with M 1 , (8), and inequalities (9), (10), (11); the loss on the total payment is at most mε + i δβ (k(v−i)) − mδβ (k(v)) +ε δ . Note thatβ (k(v−i)) ≤β (k(v)) + m 7 . Therefore, the total loss on the payment is at most mεNow, we prove a similar result for M , which is a linear combination of M 1 and M 1 :Recall that (·) + max{·, 0}. Let's define BW(k) = i v i x (k) i + β (k) to be the boosted welfare of option k for simplicity. Now, we focus on one option k, and we will give an upper bound onfor the non-negative σ. The value of σ will be determined later. Note that it is always true thatij − x (k(v)) ij ) + 1 BW(k(v))−BW(K)∈[0,σ] dF (v).Now we fix all coordinates of valuation v other than coordinate ij. Note that, the function BW(k(v))− BW(k) A convergence theory for deep learning via over-parameterization. Z Allen-Zhu, Y Li, Z Song, International Conference on Machine Learning. PMLRZ. Allen-Zhu, Y. Li, and Z. Song. A convergence theory for deep learning via over-parameterization. In International Conference on Machine Learning, pages 242-252. PMLR, 2019. Stronger generalization bounds for deep nets via a compression approach. S Arora, R Ge, B Neyshabur, Y Zhang, International Conference on Machine Learning. PMLRS. Arora, R. Ge, B. Neyshabur, and Y. Zhang. Stronger generalization bounds for deep nets via a compression approach. In International Conference on Machine Learning, pages 254-263. PMLR, 2018. The menu-size complexity of revenue approximation. M Babaioff, Y A Gonczarowski, N Nisan, Games and Economic Behavior. 134M. Babaioff, Y. A. Gonczarowski, and N. Nisan. The menu-size complexity of revenue approximation. Games and Economic Behavior, 134:281-307, 2022.	[]
[ "Axioms for retrodiction: achieving time-reversal symmetry with a prior", "Axioms for retrodiction: achieving time-reversal symmetry with a prior" ]	[ "Arthur J Parzygnat ", "Francesco Buscemi " ]	[]	[]	We propose a category-theoretic definition of retrodiction and use it to exhibit a time-reversal symmetry for all quantum channels. We do this by introducing retrodiction families and functors, which capture many intuitive properties that retrodiction should satisfy and are general enough to encompass both classical and quantum theories alike. Classical Bayesian inversion and all rotated and averaged Petz recovery maps define retrodiction families in our sense. However, averaged rotated Petz recovery maps, including the universal recovery map of Junge-Renner-Sutter-Wilde-Winter, do not define retrodiction functors, since they fail to satisfy some compositionality properties. Among all the examples we found of retrodiction families, the original Petz recovery map is the only one that defines a retrodiction functor. In addition, retrodiction functors exhibit an inferential time-reversal symmetry consistent with the standard formulation of quantum theory. The existence of such a retrodiction functor seems to be in stark contrast to the many no-go results on time-reversal symmetry for quantum channels. One of the main reasons is because such works defined time-reversal symmetry on the category of quantum channels alone, whereas we define it on the category of quantum channels and quantum states. This fact further illustrates the importance of a prior in time-reversal symmetry.	10.22331/q-2023-05-23-1013	[ "https://export.arxiv.org/pdf/2210.13531v2.pdf" ]	253,107,310	2210.13531	d6d807ce2da988311497a12d91c5bd419d7d5572	Axioms for retrodiction: achieving time-reversal symmetry with a prior Arthur J Parzygnat Francesco Buscemi Axioms for retrodiction: achieving time-reversal symmetry with a prior We propose a category-theoretic definition of retrodiction and use it to exhibit a time-reversal symmetry for all quantum channels. We do this by introducing retrodiction families and functors, which capture many intuitive properties that retrodiction should satisfy and are general enough to encompass both classical and quantum theories alike. Classical Bayesian inversion and all rotated and averaged Petz recovery maps define retrodiction families in our sense. However, averaged rotated Petz recovery maps, including the universal recovery map of Junge-Renner-Sutter-Wilde-Winter, do not define retrodiction functors, since they fail to satisfy some compositionality properties. Among all the examples we found of retrodiction families, the original Petz recovery map is the only one that defines a retrodiction functor. In addition, retrodiction functors exhibit an inferential time-reversal symmetry consistent with the standard formulation of quantum theory. The existence of such a retrodiction functor seems to be in stark contrast to the many no-go results on time-reversal symmetry for quantum channels. One of the main reasons is because such works defined time-reversal symmetry on the category of quantum channels alone, whereas we define it on the category of quantum channels and quantum states. This fact further illustrates the importance of a prior in time-reversal symmetry. Retrodiction versus time reversal As we currently understand them, the laws of physics for closed systems are time-reversal symmetric in both classical and quantum theory. The associated evolution is reversible in the sense that if one evolves any state to some time in the future, one can (in theory) apply the reverse evolution, which is unambiguously defined, to return the initial state. However, not all systems of interest are closed, and so, not all evolutions are reversible in the sense described above. For instance, arbitrary stochastic maps and quantum channels are of this kind. This, together with other phenomena such as the irreversible change due to a measurement [1,2,3] and the black hole information paradox [4,5,6], has led many to question the nature of timereversibility. Indeed, many have provided no-go theorems and occasionally offered proposals for what time-reversal is or how it can be implemented (physically or by belief propagation) [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. The following lists some of the questions considered in these programs. • In what sense can channels be reversed to construct a time-symmetric formulation of open dynamics? Namely, what axioms define what we mean by time-reversibility [25,27,28]? • As opposed to applying classical probabilistic inference associated with experimental outcomes [29,18,20,24], is it possible to provide a fully quantum formulation of inference that does not require the classical agent interface? • Is the apparent directionality of time a consequence of the irreversibility of certain processes that are more general than deterministic evolution? • What is a maximal subset of quantum operations that has a reasonable time-reversibility? For example, it is known that a certain type of reversibility is possible for unital quantum channels [25,24,27,19,28] and more generally for quantum channels that fix an equilibrium state [11]. Is this the best we can do without modifying quantum theory [16]? Although a large body of work has recently focused on quantum theory, a similar problem lies in the classical theory. More importantly, it has been understood for a long time that complete reversibility is much too stringent of a constraint to ask for. Instead, one might be more interested in retrodictability rather than reversibility [7,30], where retrodictability is the ability to infer about the past and which specializes to reversibility in the case of reversible dynamics. And in the classical theory, retrodictability is achieved through Bayes' rule and Jeffrey's probability kinematics, the latter of which specializes to Bayes' rule in the case of definitive evidence, but is strictly more general in that an arbitrary state can be used as evidence to make inference [31,32,33,14]. But while the same Bayes' rule and Jeffrey's probability kinematics can be used simultaneously for both spatial and temporal correlations in classical statistics, quantum theory seems to reveal a subtle distinction between these two forms of inference [13,34,35,36]. The study of the unique spatial correlations in quantum theory has been a subject of intense studies since at least the work of Schrödinger [37,38,39,40]. Meanwhile, our understanding of the temporal correlations unique to quantum theory is not as well-understood and is still under investigation [41,42,36]. In order to contribute towards this important and evolving subject, it is retrodictability, i.e., the inferential form of time-reversal symmetry, in quantum (and classical) dynamics that will be the focus of this paper. However, retrodictability is not in general possible for arbitrary dynamics unless additional input data are provided. One example of the kind of information that can be used is a prior, which summarizes the state of knowledge of the retrodictor. Previous proposals that have avoided the introduction of a prior oftentimes secretly included it. For example, unital quantum channels are precisely quantum channels that preserve the uniform prior, even though this is not always phrased in this way because the identity matrix disappears from expressions. In the present paper, we show that a form of time-reversal symmetry is possible for certain open system evolutions (classical and quantum channels) together with priors, i.e., we prove the existence of an inferential time-reversal symmetry for open quantum system dynamics. This bypasses some of the no-go theorems in the literature, such as in [19,25,27], in at least two ways. First, by including the prior, the assignment of a time-reversal no longer has input just some channel, but a channel and a prior. Second, we see no reason to assume that the time-reversal assignment must be linear in either the channel or the prior, as is done in [25,27] (see axiom 4 on page 2 in [27] or the definition of symmetry in [25]). Indeed, even classical Bayesian inversion is linear in neither the original process nor the prior, so there is no reason to assume linearity in the quantum setting, which should specialize to the classical theory. In fact, in the present work, we axiomatize retrodiction in a way that is agnostic to classical or quantum theory, probability theory, and even any physical theory as long as it has sufficient structure to describe states and evolution. This is achieved by formulating a rigorous definition of retrodiction in the language of category theory, the proper mathematical language for describing processes [43,44,45,46,47,28]. Fortunately, it is possible to summarize our axioms intuitively without requiring familiarity with category theory. 1 In order to be agnostic to classical or quantum theory, we try to avoid using language that is only applicable to either. To describe retrodiction, we should first specify the collection of allowed systems (denoted A, B, C , . . . ), processes (denoted E, F, . . . ), and states (denoted α, β, γ, . . . ). Having done this, and as argued above, in order to define the retrodiction of some process E : A → B from one system A to another B, it is necessary to have some additional input. For us, that input is a prior α, which is a state on system A. Hence, we will combine these data together by writing (α, E). Retrodiction should be some assignment whose input is such a pair (α, E) and whose output is some map R α,E : B → A that only depends on this input (this is sometimes called universality in the literature). The definition of retrodiction that we propose consists of the following logical axioms on such an assignment R over all systems A, B, C , . . . , evolutions (i.e., processes) E, F, . . . , and states α, β, γ, . . . (remarks and clarifications on these axioms follow immediately after). 1. Retrodiction should produce valid processes, i.e., the map R α,E should be a valid process. 2. The map R α,E should take the prediction E(α) back to the prior α, i.e., R α,E (E(α)) = α. 3. The retrodiction of the process that does nothing is also the process that does nothing, i.e., R α,idA = id A . 4. More generally, the retrodiction of a genuinely reversible process E : A → B is the inverse E −1 of the original process, i.e., R α,E = E −1 , and is therefore independent of the prior. 5. Retrodiction is involutive in the sense that retrodicting on a retrodiction gives back the original process, i.e., R E(α),R α,E = E. 6. Retrodiction is compositional in the sense that if one has a prior α on system A and two successive processes E : A → B and F : B → C , then the retrodiction R α,F •E associated with the composite process A E − → B F − → C is the composite of the retrodictions R E(α),F : C → B and R α,E : B → A associated with the constituent components, i.e., R α,F •E = R α,E • R E(α),F . 7. Retrodiction is tensorial in the sense that if one has priors α and α on systems A and A , respectively, as well as two processes E : A → B and E : A → B , then the retrodiction R α⊗α ,E⊗E : B ⊗ B → A ⊗ A associated with the tensor product of the systems and processes is equal to the tensor product R α,E ⊗ R α ,E of the constituent retrodictions, i.e., R α⊗α ,E⊗E = R α,E ⊗ R α ,E . Several clarifications and remarks are in order with regard to these axioms, which we enumerate in the same order. 1. Part of our axioms assume that we have identified a class of physical systems and processes. In the setting of quantum theory, and in this work, physical systems A, B, C , . . . will be mathematically represented by their corresponding algebras, following the conventions of 1 In the body of this work, only elementary ideas from category theory will be used to formalize our definitions and results. If the reader is comfortable with the definitions of a monoidal category, its opposite, and functor, this should suffice. We will try to ease into this through our setting of retrodiction rather than giving formal definitions. The reader is encouraged to read [48] for a first impression of category theory and then browse the reference [49, Chapter 1] for a more detailed, yet friendly, introduction. References that cover advanced topics while maintaining a close connection to quantum physics include [50,44,51]. Finally, standard in-depth references include [52,53]. prior works on quantum inference [13]. We will take processes to be represented by quantum channels, i.e., completely positive trace-preserving (CPTP) maps (traces on such algebras are defined in Appendix A). In the classical theory, physical processes are modeled by stochastic maps. 2. Note that we are only demanding our prior to be reproduced. If α is some other state on A, this axiom is not saying that R α,E (E(α )) = α . Note also that this axiom together with the first one allows us to think of retrodiction as an assignment that sends a pair (α, E) consisting of a valid state and process to (E(α), R α,E ), which is another valid state and process. This simplifies the form of the assignment since it can be viewed as a function that begins in one class of objects and comes back to that same class of objects. In this work, we will not consider time-reversals that do not satisfy this crucial state-preserving property, though we mention that some authors have considered dropping such an axiom [15]. 3. Note that retrodicting on the identity process is, in particular, independent of the prior α. 4. In this setting, we define E to be reversible (i.e., invertible) whenever there exists a process E −1 : B → A such that E • E −1 = id B and E −1 • E = id A . In the context of category theory, E is also called an isomorphism. 5. This axiom has appeared in many approaches towards time-reversal [15,11,28]. A weakened form of involutivity was considered in [27]. 6. This axiom and the next one describe in what sense retrodiction of a complicated process can be broken into retrodictions of simpler components. These are some of the essential axioms where the language of category theory becomes especially important. The compositional property was considered crucial in [27] and was also emphasized in many other works such as [11]. 7. In the tensorial axiom, we are assuming that our collection of states and processes has a tensorial structure, sometimes called parallel composition to not conflate it with the series composition from the previous axiom. This axiom seems to have been emphasized a bit less in the literature on time-reversal symmetry, but was emphasized in the context of recovery maps [54,55,56]. In this paper (specifically Theorem 6.5), we prove that among a variety of different proposals for recovery maps and retrodiction in the quantum setting (including Petz recovery maps, their rotated variants, and their averaged generalizations, and many other candidates), the only one that satisfies all these axioms is the Petz recovery map [57,10,58,29,11]. This justifies the Petz recovery map as a retrodiction map, and hence an extension of time-reversal symmetry to all quantum channels (see Figure 1). Although we do not characterize the Petz recovery map (and hence Bayesian inversion in the classical setting) among all possible retrodiction maps, we propose a precise mathematical problem whether our axioms indeed isolate Petz among all possible retrodiction maps (and not just the vast examples we have listed). We emphasize that our axioms are logical as opposed to analytical. It has often been the case that axioms used to help single out a form of retrodiction (a recovery map to be more precise) optimized some quantity, such as the difference of relative entropies, more general divergences, fidelity of recovery, or relative entropy of recovery [57,10,59,60,61,62,54,63]. Rather than choosing a specific such distance measure and then finding axioms to argue for the necessity of those, we have preferred to find a logical set of axioms more directly, in spirit of earlier derivations of classical inference [64]. In addition, the axioms that we identified do not involve concepts such as measurement, observations, or the need of any intervention, which are concepts that are mathematically difficult to define outside of specific models. In fact, our axioms can be formulated without any mention of probability theory. Our axioms are merely those of consistency rather than properties we might expect from our experiences, which are largely based on classical thinking, and may therefore skew our understanding of quantum. Furthermore, we illustrate in detail which axioms fail for various other proposals of retrodiction by providing explicit examples. In particular, we illustrate, to some degree, how independent most bistochastic channels adjoint reversible channels adjoint quantum channels Petz classical channels Bayes Figure 1: The sets (not drawn to scale) and their inclusion structure depict four families of channels (the inclusion structure is not meant to include the states). The standard time-reversal symmetry is obtained by taking the Hilbert-Schmidt adjoint of a reversible channel, or, more generally, a bistochastic channel. The states (drawn schematically as matrices with shaded entries) are irrelevant for reversible channels, but are implicitly the uniform states for bistochastic channels. For classical channels equipped with arbitrary probability distributions, standard Bayesian inversion provides a form of time-reversal symmetry that goes beyond the Hilbert-Schmidt adjoint for bistochastic channels. Finally, the Petz recovery map allows an extension of Bayesian inversion to all quantum channels and arbitrary states. In brief, this paper isolates axioms for retrodiction and inferential time-reversal symmetry that are simultaneously satisfied by all of these classes of channels and states. of our axioms are. For example, the averaged rotated Petz recovery maps that have appeared recently in the context of strengthening data-processing inequalities via recovery maps [65,55] are neither compositional, tensorial, nor involutive. Furthermore, we show that the recent proposal of Surace-Scandi [63] on state-retrieval maps is also not compositional. This emphasizes some of the key differences between retrodiction and approximate error correction [10,60,63], and this distinction may have important consequences for quantum information in extreme situations, such as near the horizons of black holes, where state-dependent approximate error-correction has recently been used to suggest that information might be stored in Hawking radiation in certain models of black hole evaporation [66,67,68,69,70,71,72]. Retrodiction as a monoidal functor In the setting of quantum theory, we would like to define retrodiction as an assignment that takes a prior, defined on some matrix algebra, together with a process involving another matrix algebra, and produces a map in the opposite direction. However, working with matrix algebras is unnecessarily restrictive and it is more appropriate to include classical and hybrid classical/quantum systems in our analysis. As such, we will model our systems with finite-dimensional unital C * -algebras, where commutative algebras correspond to classical systems and general non-commutative algebras correspond to quantum (or hybrid) systems. We will model our processes with completely positive trace-preserving (CPTP) maps between such C * -algebras. Briefly, one benefit of using C * -algebras, as opposed to only matrix algebras, is that one can express quantum channels, classical stochastic maps, measurements, preparations, and instruments all as processes between C * -algebras (see [36, Section II.A] for more details). We are expressing processes in the Schrödinger picture where the processes describe the action on states. We emphasize that we do not require E to preserve any of the algebraic structure, nor do we require the unit of the algebras to be preserved. The reader should not be discouraged by our usage of finite-dimensional C * -algebras because they are equivalent to direct sums of matrix algebras [73,Theorem 5.20 and Proposition 5.26]. We henceforth take the convention that all C * -algebras appearing in this work will be finite dimensional and unital unless specified otherwise. We now introduce a category that mathematically describes priors and processes. Definition 2.1. The category of faithful states is the category States whose objects consists of pairs (A, α), where A is a finite-dimensional unital C * -algebra and α is a faithful state on A, i.e., α is positive, tr(α) = 1, and α is non-degenerate in the sense that if A ∈ A satisfies tr(αA † A) = 0, then A = 0. A morphism from (A, α) to (B, β) in States is a CPTP map A E − → B such that E(α) = β. Such a morphism is drawn as (A, α) E − → (B, β). The composition in States is the composition of functions and the identity morphism on (A, α) is the identity id A map on A. 2 Let CStates denote the subcategory of faithful states on commutative C * -algebras. We leave the verification that the axioms of a category hold for States and CStates to the reader. As described in the introduction, (A, α) represents a physical system, represented by an algebra A, together with a faithful state α on A. A morphism (A, α) E − → (B, β) acquires the physical interpretation that α is a prior and the process E takes this prior to β, written as β = E(α). As such, β is called the prediction. The usage of faithful states as opposed to all states has at least two purposes. One reason is for mathematical simplicity, since subtle issues arise when dealing with non-faithful states, which allows for states to produce expectation values of 0 on certain positive operators. We believe that focusing on faithful states first prevents us from being distracted by the technical issues associated with measure zero, where we need to discuss projections and their orthogonal complements, substantially lengthening many of the formulas and calculations. The second reason is to allow ourselves to not completely rule out events that we believe are impossible. It is conceivable that some events may happen with a small enough probability that it is effectively vanishing, but technically non-vanishing. For these two reasons, we will focus exclusively on faithful states in this work. As such, all states will be faithful from now on unless specified otherwise. Having introduced the main category of discussion, we now come to explicating some of the basic axioms that we believe retrodiction should satisfy. First, if one is given a morphism (A, α) E − → (B, β) describing a one step dynamics evolution involving a prior α with β as the prediction, then the retrodiction of this morphism should be a CPTP map of the form A R α,E ← −− − B reversing the directionality of the original process A E − → B. Note that we are only including α and E in the notation for the retrodiction since the prediction β = E(α) can be obtained from these data. Also note that we have only motivated that R α,E should be positive and trace-preserving, rather than CPTP, because we want to ensure that states are always mapped to states. It is only after we introduce the tensor product structure later that we will see the usefulness of assuming retrodiction to be CPTP. Furthermore, the retrodiction map R α,E should preserve states in the sense that it sends the prediction β back to the prior α, i.e., R α,E (β) = α, or more suggestively R α,E (E(α)) = α. Note, however, that this does not imply that R α,E (E(α )) = α for any other α different from α. Such a condition would drastically reduce the collection of maps E that would be of interest to us because they would necessarily be invertible. This is discussed later in Theorem 5.8. One can interpret R α,E acting on an arbitrary state β ∈ B as a quantum generalization of Jeffrey's probability kinematics [33] (see Proposition 2.4 and Proposition 3.7). Combining the requirements discussed in the previous paragraphs precisely say that (A, α) R α,E ← −− − (B, β). should be a morphism in States. As such, retrodiction should be some assignment of the form 3 States R − → States op , where an object (A, α) is sent to itself, while a morphism (A, α) E − → (B, β) is sent to a morphism (A, α) R α,E ← −− − (B, β) . This motivates the following definition (if C denotes a category, and if a and b are two objects in C, then C(a, b) denotes the set of morphisms from a to b). Definition 2.2. A retrodiction family is an assignment R : States → States op that acts as the identity on objects. Explicitly, this means that for every pair of objects (A, α) and (B, β) in States, R defines a function States (A, α), (B, β) → States (B, β), (A, α) sending a morphism (A, α) E − → (B, β) to a morphism (A, α) R α,E ← −− − (B, β). We would like retrodiction to satisfy a few more intuitive axioms than just these basic ones. In particular, we have not required a retrodiction family to be a functor. As of now, the only axioms we have incorporated are that the prior is recovered and that retrodiction remains CPTP. We have also assumed the axiom of universality, which says that retrodiction is a function depending only on the initial prior and the process (and nothing more). One other natural axiom to assume for retrodiction is that if one considers the identity mor- phism (A, α) idA − − → (A, α) in States for any state α on A, then the retrodiction of this should be the identity itself. As an equation, this can be expressed as R α,idA = id A . In words, retrodicting on the process that does nothing is itself the process that does nothing. This axiom is sometimes called normalization in the literature [54,56,55]. A second axiom that we believe retrodiction should satisfy is compositionality. Namely, if one has two successive processes together with a prior (and successive predictions) (A, α) E − → (B, β) F − → (C , γ), then one can construct three retrodictions associated with these data. On the one hand, one can retrodict on the individual processes E and F to construct two successive retrodictions (A, α) R α,E ← −− − (B, β) R β,F ←−−− (C , γ). On the other hand, one can construct the retrodiction associated with the composite process (A, α) F •E − −− → (C , γ) in which case one obtains (A, α) R α,F •E ← −−−− − (C , γ). The axiom of compositionality states that the composite of retrodictions for the two individual processes should equal the retrodiction of the composite process (cf. [11,54,56,55]), i.e., 4 R α,F •E = R α,E • R β,F . Thus, normalization and compositionality precisely say that retrodiction should be a functor, written States R − → States op . But there is another structure we would like retrodiction to preserve, and this is the natural tensor product structure for states. Indeed, States and CStates are (symmetric) monoidal categories, where the tensor product of two objects (A, α) and (A , α ) is (A ⊗ A , α ⊗ α ). The tensor product of two morphisms (A, α) E − → (B, β) and (A , α ) E − → (B , β ) is the usual tensor product of CPTP maps, namely (A ⊗ A , α ⊗ α ) E⊗E −−−→ (B ⊗ B , β ⊗ β ). Intuitively, this monoidal (i.e., tensor product) structure on States (and CStates) says that if two systems are prepared independently and undergo independent evolution, then the joint prior and joint process is also a valid prior and process. Due to this independence, we would therefore also expect that if we independently retrodict on each process via (A, α) R α,E ← −− − (B, β) and (A , α ) R α ,E ← −−− − (B , β ), then the joint retrodiction (A ⊗ A , α ⊗ α ) R α,E ⊗R α ,E ← −−−−−−−− − (B ⊗ B , β ⊗ β ), should be the same as retrodicting on the joint process itself (A ⊗ A , α ⊗ α ) R α⊗α ,E⊗E ← −−−−−−− − (B ⊗ B , β ⊗ β ), i.e., we expect retrodiction to satisfy tensoriality, which states R α,E ⊗ R α ,E = R α⊗α ,E⊗E . The mathematical way to summarize the above discussion is to say that retrodiction should be, at the very least, a monoidal functor States → States op that acts as the identity on objects (we will also require other properties for retrodiction later). Hence, already from these basic requirements of retrodiction, we are inevitably led to a description of retrodiction in terms of the mathematical discipline of category theory, specifically that of (symmetric) monoidal categories and monoidal functors. Note that these mathematical structures also appear in the settings of resource theories [74], topological field theories [75,76], topological order [77,78], and many other contexts relevant to physics [50]. Before moving on to more reasonable examples of retrodiction, we first examine a non-example that comes quite close. Proposition 2.3. The assignment R ! : States → States op given by sending a morphism (A, α) E − → (B, β) to the composite 5 (A, α)α ← − (C, 1) tr ← − (B, β) tr(B)α ← tr(B) ← B is a retrodiction family that is compositional and tensorial. However, it does not satisfy the normalization property and is therefore not a functor. It is called the discard-and-prepare retrodiction family. Proof. First note that R ! α,E =α • tr is CPTP because it is the composite of two CPTP maps. By construction, it preserves the states since R ! α,E (β) =α tr(β) =α(1) = α. Thus, R ! is a retrodiction family. To see that R ! is compositional, let (A, α) E − → (B, β) F − → (C , γ) be a composable triple in States. Then R ! α,E • R ! β,F = (α • tr) • (β • tr) =α • id C • tr =α • tr = R ! α,F •E proves compositionality. To see that R ! is tensorial, let (A, α) E − → (B, β) and (A , α ) E − → (B , β ) be a pair of morphisms in States. Then R ! α,E ⊗ R ! α ,E = (α • tr) ⊗ (α • tr) = (α ⊗α ) • (tr ⊗ tr) = (α ⊗α ) • tr = R ! α⊗α ,E⊗E proves tensoriality. Finally, R ! does not satisfy normalization since given any algebra A whose dimension is greater than 1, one obtains R ! α,idA =α • tr which is never equal to id A since the linear rank of this map is 1. It is worthwhile to briefly restrict attention to the classical case before moving on to more complicated retrodiction families in the fully quantum setting. First recall that any commutative C * -algebra is * -isomorphic to one of the form C X := x∈X C for some finite set X. A state p on q) is uniquely determined by a stochastic matrix {E yx } whose yx entry describes the conditional probability of y given x, i.e., E yx ≥ 0 for all x ∈ X, y ∈ Y , and y∈Y E yx = 1 for all x ∈ X. C X is given by a probability {p x } on X. A morphism (C X , p) E − → (C Y ,Proposition 2.4. The assignment R B : CStates → CStates op sending (C X , p) E − → (C Y , q) to (C X , p) E:=R B p,E ←−−−−− (C Y , q), determined by the formula E xy := E yx p x q y for all x ∈ X and y ∈ Y , defines a monoidal functor that acts as the identity on objects. It is called Bayesian inversion or the Bayesian retrodiction family. Note that R B α,E : C Y → C X acting on the vector δ y ∈ C Y , whose value is 1 at y and 0 elsewhere, gives the probability vector R B α,E (δ y ) whose x th component is E xy . This reproduces Bayes' update rule based on obtaining hard evidence y. More generally, R B α,E acting on an arbitrary probability r ∈ C Y gives the probability vector R B α,E (r) whose x th component is y∈Y E xy r y . This reproduces Jeffrey's update rule associated with soft evidence represented by the probability r (also called probability kinematics) [32,33,79,80]. A large part of this paper is devoted to finding an extension of R B from CStates to all of States to include quantum channels as well. In other words, our goal is to find a retrodiction family that extends classical retrodiction defined in terms of Bayes' and Jeffrey's update rule. Importantly, note that we do not impose the condition that a retrodiction family R : States → States op agrees with classical Bayesian inversion when restricted to CStates. We suspect, though we do not have a proof at present, that classical Bayesian inversion is characterized by such functoriality properties (see Question 7.1 for more details). The Petz recovery maps One immediately wonders if any retrodiction families that are monoidal functors actually exist, how many there are, and if we can find an explicit form for all of them. The first question was answered in [54], and a detailed proof was provided in [56] (though the phrasing and emphasis were different), though earlier partial results were also obtained in [11]. However, we will prove more general results in this work, so we will restate these results using our language. In what follows, we will show that any rotated Petz recovery map can be used to define a retrodiction family that is a monoidal functor. However, averaged rotated Petz recovery maps in general do not satisfy compositionality and tensoriality, and this will be expounded on in later sections. To establish these claims, we first recall some definitions. R P α,E ← −− − (B, β) in States defined by the formula R P α,E := Ad α 1/2 • E * • Ad β −1/2 , where E * is the Hilbert-Schmidt adjoint of E (cf. Appendix A) and Ad V is the map sending A to V AV † . For any t ∈ R, the rotated Petz recovery map associated with E is the morphism 7 (A, α) R P,t α,E ← −− − (B, β) in States defined by the formula R P,t α,E := Ad α −it • R P α,E • Ad β it ≡ Ad α 1/2−it • E * • Ad β −1/2+it . Finally, for any probability measure µ on R, the µ-averaged rotated Petz recovery map associated with the pair (α, E) is the morphism 8 6 The reader is expected to check that this morphism indeed preserves the states and is a CPTP map. The reader is invited to look at [56] for a simple proof. (A, α) R P,µ α,E ← −− − (B, β) in States defined by the formula R P,µ α,E (B) := ∞ −∞ R P,t α,E (B) dµ(t) ≡ ∞ −∞ Ad α 1/2−it • E * • Ad β −1/2+it (B) dµ(t). An example of an averaged rotated Petz recovery map is one given by a finite convex combination of rotated Petz recovery maps at different values of t ∈ R (see Appendix B for an explicit example). Yet another example is one that has recently appeared in the literature as a universal recovery map when strengthening the data-processing inequality [65,55]. Example 3.2. The Junge-Renner-Sutter-Wilde-Winter (JRSWW) recovery map R JRSWW is the averaged rotated Petz recovery map R P,µ with probability measure µ defined by the density 9 dµ(t) = π cosh(2πt) + 1 −1 dt. We now come to describing the above examples in terms of retrodiction families. Theorem 3.3. For every t ∈ R, the assignment R P,t : States → States op sending a morphism (A, α) E − → (B, β) to (A, α) R P,t α,E ← −− − (B, β ) defines a retrodiction family that is a monoidal functor. Proof. Modulo the fact that we are working with arbitrary finite-dimensional C * -algebras, as opposed to just matrix algebras, this is proved in [56]. More specifically, [56, Remark 3.5] proves the state-preservation condition, [56, Appendix B] proves R P,t α,E is CPTP (since the projection is the identity by our assumption of faithfulness for states), while the normalization, compositionality, and tensoriality properties are proved in [56,Section 4]. The same proof works for arbitrary finite-dimensional C * -algebras. Proposition 3.4. Let µ be a probability measure on R. Then R P,µ : States → States op sending a morphism (A, α) E − → (B, β) to (A, α) R P,µ α,E ← −− − (B, β) defines a retrodiction family that satisfies normalization. Proof. The state-preservation condition follows from R P,µ α,E (β) = ∞ −∞ R P,t α,E (β) dµ(t) = ∞ −∞ α dµ(t) = α ∞ −∞ dµ(t) = α, where we have used the state-preservation property of the rotated Petz recovery map in the second equality. The fact that R P,µ is CP follows from the fact that it can be expressed as an increasing limit of positive linear combinations of CP maps. The fact that R P,µ is trace-preserving is a consequence of the fact that R P,t is trace-preserving for all t ∈ R and since µ is a probability measure. Finally, R P,µ satisfies the normalization condition for the same reason, namely because R P,t α,idA = id A for all t ∈ R and because µ is a probability measure. There is a generalization of rotated Petz recovery maps where independent phases act on the eigenspaces of the corresponding states. These were used to provide a strengthening of the dataprocessing inequality by Sutter, Tomamichel, and Harrow (STH) [81]. Definition 3.5. Let α and β be (faithful) states on C * -algebras A and B, respectively. Let U α ∈ A and U β ∈ B be unitaries that leave the states invariant, i.e., U α αU † α = α and U β βU † β = β. Given a morphism (A, α) β) in States, the Sutter-Tomamichel-Harrow (STH) rotated Petz recovery map associated with the unitaries U α and U β is given by 9 The slight difference between our formula and the one in [65,55] is because of our insistence of using R P,t inside the integral, whereas [65,55] use R P, t 2 . E − → (B,R STH α,E = Ad U † α • R P α,E • Ad U β ≡ Ad U † α • Ad α 1/2 • E * • Ad β −1/2 • Ad U β . The STH rotated Petz recovery map is a slight generalization of the version considered in [81]. 10 This special case considered in [81] occurs when all the unitaries U α and U β are taken to be diagonal with respect to eigenbases for α and β, respectively. The more general unitaries U α and U β considered here are block-diagonal unitaries, where the blocks correspond to the eigenspaces of α and β, respectively. Note that if the dimensions of the eigenspaces are all 1, then this automatically forces all the unitaries to be diagonal in these bases, so our above generalization allows one to consider unitaries that are not necessarily phases on the eigenspaces when the dimension of the eigenspaces are greater than 1. β) defines a retrodiction family and is a functor, i.e., it satisfies normalization and compositionality. Proposition 3.6. For each (A, α) in States, fix a unitary U α ∈ A such that U α αU † α = α. Then States R STH − −−− → States op sending a morphism (A, α) E − → (B, β) to (A, α) R STH α,E ← −−− − (B, Proof. First, since R STH α,E is a composite of CPTP maps, it is CPTP. Second, it is state-preserving because Ad U β (β) = β and Ad U † α (α) = α by definition and because the Petz recovery map is state-preserving. This shows that R STH is indeed well-defined and sends morphisms in States to morphisms in States in a contravariant manner. The normalization property follows from R STH α,idA = Ad U † α • R P α,idA • Ad Uα = Ad U † α • id A • Ad Uα = id A . Finally, given a pair (A, α) E − → (B, β) F − → (C , γ) of composable morphisms in States, R STH α,E • R STH β,F = Ad U † α • R P α,E • Ad U β • Ad U † β • R P β,F • Ad Uγ = Ad U † α • R P α,E • R P β,F • Ad Uγ = Ad U † α • R P α,F •E • Ad Uγ = R STH α,F •E , which proves compositionality of R STH . Proposition 3.7. When restricted to the subcategory CStates ⊂ States of commutative C algebras, all of the retrodiction families R P , R P,t , R P,µ , and R STH agree with standard (classical) Bayesian inversion, i.e., any of these retrodiction families applied to a morphism (C X , p) E − → (C Y , q) gives (C X , p) E:=R B p,E ←−−−−− (C Y , q) from Proposition 2.4. Proof. This follows from the fact that all the unitaries in the formulas for these retrodiction maps cancel due to commutativity. Hence, all of these retrodiction families equal the Petz recovery retrodiction family on CStates. The remaining part follows from the fact that the Petz recovery map gives the Bayesian inverse on commutative C -algebras, which is well-known [54]. The convexity of certain retrodiction families In general, the collection of retrodiction families, without assuming compositionality/tensoriality properties, is a convex space. Namely, given any two retrodiction families R and R , the convex combination λR + (1 − λ)R , determined by sending (A, α) E − → (B, β) to (λR + (1 − λ)R ) α,E := λR α,E + (1 − λ)R α,E , is a retrodiction family. In fact, if R and R are normalizing, then so is λR + (1 − λ)R . A closely related fact was already observed in Proposition 3.4. However, the collection of retrodiction families that are compositional and/or tensorial is not a convex space, nor should we expect it to be. 11 In this section, we justify this with Proposition 4.6 and Proposition 4.7, the latter of which shows that the JRSWW map is not compositional nor tensorial, thus clarifying some of the claims made in [55, Remark 2.4] (although it does satisfy a type of stabilization property, which will be discussed later). But before this, we make an observation that might suggest why convexity might not hold for retrodiction families that are compositional. We do this by analyzing the averaged rotated Petz recovery maps, which are (integrated) convex combinations of rotated Petz recovery maps. The morphism E is said to be covariant 12 γ) be a composable pair of morphisms in States, and let µ and ν be two probability measures on R. Then iff Ad β it • E = E • Ad α it for all t ∈ R. Lemma 4.2. Let (A, α) E − → (B, β) F − → (C ,R P,µ α,E • R P,ν β,F = ∞ −∞ ∞ −∞ Ad α −is+1/2 • E * • Ad β i(s−t) • F * • Ad γ it−1/2 dµ(s)dν(t). Furthermore, if at least one of F or E is covariant (cf. Definition 4.1), then R P,µ α,E • R P,ν β,F = R P,µ α,F •E if F is covariant R P,ν α,F •E if E is covariant. In particular, if both E and F are covariant, then R P,µ α,E • R P,ν β,F = R P α,E • R P β,F = R P α,F •E . Proof. Let C ∈ C be arbitrary. Then R P,µ α,E • R P,ν β,F (C) = R P,µ α,E R P,ν β,F (C) = R P,µ α,E ∞ −∞ Ad β −it+1/2 • F * • Ad γ it−1/2 (C)dν(t) = ∞ −∞ R P,µ α,E • Ad β −it+1/2 • F * • Ad γ it−1/2 (C)dν(t) = ∞ −∞ ∞ −∞ Ad α −is+1/2 • E * • Ad β i(s−t) • F * • Ad γ it−1/2 (C)dµ(s) dν(t). The third equality follows from linearity and continuity of R P,µ α,E together with the arithmetic properties of convergent integrals [84,85]. Now, suppose that E is covariant. Then R P,t α,E = R P α,E for all t ∈ R so that R P,µ α,E = R P α,E as well (since µ is a probability measure). Hence, R P,µ α,E • R P,ν β,F = R P α,E • R P,ν β,F = ∞ −∞ Ad α 1/2 • E * • Ad β −it • F * • Ad γ it−1/2 dν(t) = ∞ −∞ Ad α 1/2 • Ad α −it • E * • F * • Ad γ it−1/2 dν(t) = ∞ −∞ Ad α −it+1/2 • (F • E) * • Ad γ it−1/2 dν(t) = R P,ν α,F •E , where covariance of E was used again in the third equality. A similar calculation shows that if F is covariant, then γ) be a composable pair of morphisms in States and let µ be a probability measure on R. If at least one of F or E is covariant (cf. Definition 4.1), then R P,µ α,E • R P,ν β,F = R P,µ α,F •E . Corollary 4.3. Let (A, α) E − → (B, β) F − → (C ,R P,µ α,F •E = R P,µ α,E • R P,µ β,F , i.e., R P,µ is compositional on such a pair of composable morphisms. In fact, similar results hold for tensoriality. R P,µ α,E ⊗ R P,µ α ,E = ∞ −∞ ∞ −∞ Ad α −it+1/2 ⊗α −it +1/2 • (E ⊗ E ) * • Ad β it−1/2 ⊗β it −1/2 dµ(t) dµ (t ). Furthermore, if at least one of E or E is covariant (cf. Definition 4.1), then R P,µ α,E ⊗ R P,µ α ,E = R P,µ α⊗α ,E⊗E if E is covariant R P,µ α⊗α ,E⊗E if E is covariant. In particular, if both E and E are covariant, then R P,µ α,E ⊗ R P,µ α ,E = R P α,E ⊗ R P α ,E = R P α⊗α ,E⊗E . Proof. The first formula follows from Fubini's theorem, the interchange law for the tensor product and composition, as well as the properties of the adjoint action maps (partial details will be given momentarily in the proof of the second claim). To illustrate the second claim, first suppose that E is covariant. Then R P,µ α ,E = R P α ,E due to covariance and hence R P,µ α,E ⊗ R P,µ α ,E = R P,µ α,E ⊗ R P α ,E = ∞ −∞ Ad α −it+1/2 • E * • Ad β it−1/2 dµ(t) ⊗ Ad α 1/2 • E * • Ad β −1/2 = ∞ −∞ Ad α −it+1/2 • E * • Ad β it−1/2 ⊗ Ad α 1/2 • E * • Ad β −1/2 dµ(t) = ∞ −∞ Ad α −it+1/2 • E * • Ad β it−1/2 ⊗ Ad α −it+1/2 • E * • Ad β it−1/2 dµ(t) = ∞ −∞ (Ad α −it+1/2 ⊗ Ad α −it+1/2 ) • (E ⊗ E ) * • Ad β it−1/2 ⊗ Ad β it−1/2 dµ(t) = ∞ −∞ Ad (α⊗α ) −it+1/2 • (E ⊗ E ) * • Ad (β⊗β ) it−1/2 dµ(t) = R P,µ α⊗α ,E⊗E . A similar calculation proves the claim for when E is covariant. R P,µ α,E ⊗ R P,µ α ,E = R P,µ α⊗α ,E⊗E , i.e., R P,µ is tensorial on this pair of morphisms. Therefore, if such covariance does not hold, one might suspect that the retrodiction family given by some averaged rotated Petz recovery map might be neither functorial nor tensorial in general. The following proposition provides justification for such suspicion. Proposition 4.6. Let t, s ∈ R be two distinct numbers and let λ ∈ (0, 1). Then (1−λ)R P,s +λR P,t is a retrodiction family that is normalized but not necessarily compositional nor tensorial. Proof. A stronger result is proved in Appendix B. Since convex combinations of compositional and tensorial retrodiction families need not be compositional nor tensorial, one might suspect that the JRSWW averaged rotated Petz recovery map might not be compositional nor tensorial. Despite that suspicion, in [55,Remark 2.4], it was claimed that the JRSSWW map satisfies some parallel and series composition rules. In what follows, we will clarify what composition rules are satisfied by first showing that the standard notion of parallel and series composition (i.e., tensoriality and compositionality) do not hold in general. Proposition 4.7. The JRSWW averaged rotated Petz recovery map R JRSWW is a normalized retrodiction family that is neither compositional nor tensorial. In particular, averaged rotated Petz recovery maps need not define compositional nor tensorial retrodiction families. Proof. The proof is provided in Appendix C. Nevertheless, the JRSWW retrodiction family does satisfy the stabilization property mentioned in [55,Remark 2.4]. In fact, we will now generalize the stabilization property and show that the one considered in [55,Remark 2.4] can be viewed as a consequence of the fact that the identity map is covariant (cf. Corollary 4.5 and Corollary 4.9). In other words, although averaged Petz recovery maps do not necessarily satisfy compositionality nor tensoriality on all morphisms, they do when one of the morphisms is covariant. We formally isolate this property in the following definition, motivated by the terminology from [55]. (b) Every retrodiction family that is compositional satisfies the compositional stabilization property. Similarly, every retrodiction family that is tensorial satisfies the tensorial stabilization property. (c) More generally, if a retrodiction family is normalizing, compositional, and ⊗-stabilizing, 13 then the retrodiction family is necessarily tensorial. 13 In fact, if R denotes the retrodiction family, it suffices to assume R α⊗γ,E⊗id C = R α,E ⊗ R γ,id C and R γ⊗α,id C ⊗E = R γ,id C ⊗ R α,R α⊗α ,E⊗E = R α⊗α ,(E⊗id B )•(idA ⊗E ) by the interchange law = R α⊗α ,idA ⊗E • R α⊗β ,E⊗id B since R is compositional = (R α,idA ⊗ R α ,E ) • R α,E ⊗ R β ,id B since R is ⊗-stabilizing = (id A ⊗ R α ,E ) • (R α,E ⊗ id B ) since R is normalizing = (id A • R α,E ) ⊗ (R α ,E • id B ) by the interchange law = R α,E ⊗ R α ,E by definition of the identity, which proves the claim. Statement (a) in Corollary 4.10 (together with Proposition 4.7) provides a precise sense of what parallel and series composition properties averaged rotated Petz recovery maps satisfy. Statement (b) stresses the facts that compositionality and tensoriality are stronger conditions than composition and tensor stabilization, respectively. Finally, statement (c) says that compositionality with tensor stabilization together imply tensoriality, which suggests that compositionality is in some sense more fundamental than tensoriality. Inverting property for retrodiction families A natural property one might want of a retrodiction family is that the retrodiction of an invertible (reversible) process is the inverse (reverse) process, and is, in particular, independent of the prior. β) is an isomorphism in States. Definition 5.1. A retrodiction family States R − → States op is inverting iff R α,E = E −1 whenever (A, α) E − → (B, Proposition 5.2. Every rotated Petz retrodiction family is inverting. Before proving this, we recall a useful fact, reformulated in algebraic and categorical terms. Lemma 5.3. Isomorphisms in the category States are precisely state-preserving * -isomorphisms. A special case of this lemma is well-known in the quantum information literature. Namely, it says that a CPTP map from one matrix algebra to another has a CPTP inverse if and only if it is of the form Ad U for some unitary U . In what follows, we provide a proof applicable to the more general algebraic setting. E(A) † E(A) ≤ E(A † A) = E F (B) † F (B) ≤ E F (B † B) = B † B = E(A) † E(A), where B := E(A) and where the Kadison-Schwarz inequality was used (the first for E and the second for F ). Since the outermost terms in this expression are equal, all intermediate terms are equal. Since this holds for arbitrary A ∈ A, the Multiplication Lemma (cf. [47,Lemma 4.3] or [86,Theorem 5] ) then implies E(A 1 ) † E(A 2 ) = E(A † 1 A 2 ) for all A 1 , A 2 ∈ A, which means that E is a * -homomorphism. Similarly, F is a * -homomorphism. Since E and F are inverses of each other, this proves that E is a * -isomorphism. Thus, E * = E is a * -isomorphism. This shows that an isomorphism in States is a state-preserving * -isomorphism. Conversely, any state-preserving * -isomorphism is automatically a state-preserving CPTP map and therefore defines an isomorphism in States. Note that the assumption of complete positivity for morphisms in States is crucial in Lemma 5.3. Indeed, although the transpose map on matrix algebras has itself as an inverse, it is not a * -homomorphism. Proof of Proposition 5.2. Let (A, α) E − → (B, β) be an isomorphism in States, so that E is a isomorphism by Lemma 5.3. Since E −1 is the Hilbert-Schmidt adjoint of E and because both [83], we obtain (A, α) E − → (B, β) and (B, β) E =E −1 − −−−− → (A, α) are covariantR P,t α,E = Ad α 1/2−it • E * • Ad β −1/2+it = Ad α 1/2 • E −1 • Ad β −1/2 = Ad α 1/2 • Ad E −1 (β −1/2 ) • E −1 = Ad α 1/2 • Ad α −1/2 • E −1 = E −1 , where the third and fourth equalities used the fact that E −1 is a * -homomorphism, and the fourth equality also used the functional calculus and state-preserving property to deduce E −1 (β −1/2 ) = E −1 (β) −1/2 = α −1/2 . Corollary 5.4. The averaged rotated Petz recovery retrodiction family R P,µ for any probability measure µ on R is inverting. These results suggest that perhaps normalizing retrodiction families that are compositional might satisfy a similar property. However, this is not the case, as the following example illustrates. R STH α,E = Ad U † α • Ad α 1/2 • E * • Ad β −1/2 • Ad U β = Ad V • E −1 , where V = e i(ω−θ) 0 0 e i(ψ−φ) . This shows that R STH α,E = E −1 even though R STH satisfies universality, state-preservation, normalization, and compositionality properties. As another example of an inverting retrodiction family, we mention the recent proposal of Surace and Scandi for a state-retrieval map [63]. From the very definition of the Surace-Scandi retrodiction family, it should seem unlikely that it is compositional. The intuitive reason is that the maximization step is 'local' in the sense that it involves only a single morphism (A, α) E − → (B, β), whereas the compositional property is a 'global' property involving composable morphisms such as (A, α) γ). There is no obvious reason that maximizing the determinant for the individual morphisms should compose together to form a morphism that maximizes the determinant for the composite. E − → (B, β) F − → (C , Proposition 5.7. The Surace-Scandi retrodiction family is normalizing and inverting, but it is not compositional. In fact, it is not even composition stabilizing. Proof. The proof is provided in Appendix D. In the remainder of this section, we describe how the prior "disappears" for inverting retrodiction families. As it was pointed out in [87,Result 2], the Petz recovery map satisfies the property that it not only takes an invertible map to its inverse, but that this is the only possibility if the retrodiction does not depend on the prior. The following theorem is a generalization of this result to the setting of inverting retrodiction families. III. There exists a (faithful) state α on A such that R α,E = E −1 . Although the following proof will follow almost the same line of reasoning as that of [87, Result 2], Theorem 5.8 is a generalization of [87,Result 2] in at least two respects. First, our result holds for all finite-dimensional C * -algebras. Second, since the Petz recovery map is just one example of an inverting retrodiction family by Proposition 5.2, [87, Result 2] follows from Theorem 5.8. Furthermore, our result proves that a similar property holds for all rotated Petz recovery maps, averaged rotated Petz recovery maps, and even the recovery map of Surace-Scandi, without the need for doing explicit computations involving the specific formulas used to define all these different retrodiction families. The only time an explicit computation was needed was in proving that these define inverting retrodiction families. Proof of Theorem 5.8. you found me! (I⇒II) Suppose E is a * -isomorphism. Then by definition of R being inverting, R α,E = E −1 for all states α on A. Hence, II holds (this also proves III). (II⇒III) Since R α,E : B → A is independent of α, denote it by R E . By the state-preserving assumption of a retrodiction family, (R E • E)(α) = α for all states α on A. Since all elements of A are complex combinations of states, this implies R E • E = id A . Since A and B are * -isomorphic and finite dimensional, this implies R E = E −1 . Hence, III holds. (III⇒I) Let α be some faithful state on A such that R α,E = E −1 . This says that both E and E −1 are CPTP maps between non-degenerate quantum probability spaces. Hence, by Lemma 5.3, E is a * -isomorphism. Hence, I holds. Involutive retrodiction An interesting property emerges when two averaged rotated Petz retrodiction families are composed in succession. The meaning of composing two retrodiction families in succession is to iterate retrodiction procedures (i.e., retrodicting a retrodiction). Intuitively, we might expect to get back something closely related to the original process. This happens, for example, when taking the Hilbert-Schmidt adjoint of an arbitrary CPTP map. Indeed, in the case of closed system evolution described by a unitary U , the Hilbert-Schmidt adjoint is the reverse evolution described by U † . Applying the adjoint again gives back U . This is part of the reason why the Hilbert-Schmidt adjoint has often been viewed as a time-reversal assignment in the literature [88,19,25,28]. From our perspective, however, the Hilbert-Schmidt adjoint is only reasonable in the setting where either one has unitary evolution (cf. Theorem 5.8) or unital quantum channels. When one also has a prior in the form of a state that is not necessarily uniform, the Hilbert-Schmidt adjoint no longer takes the prediction back to the prior, nor is it necessarily a quantum channel (even upon rescaling) 14 . This is one of the reasons why we have introduced retrodiction families as assignments of the form States → States op , as opposed to sending only quantum channels to quantum channels in the opposite direction without including the data of states. Nevertheless, we might hope a similar involutive property holds for at least some retrodiction families. In this section, we show that among all averaged rotated Petz retrodiction maps, the original process is always obtained upon iteration only for the (un-rotated) Petz recovery map. In particular, the Petz recovery map bypasses the many no-go theorems in the literature regarding time-reversal symmetry. R P,ν • R P,µ α,E := R P,ν β,R P,µ α,E = ∞ −∞ Ad β −ir • E • Ad α ir d(µ * ν)(r), where µ * ν denotes the convolution of the measures µ and ν. In particular, R P,y • R P,x α,E = Ad β −i(x+y) • E • Ad α i(x+y) for all x, y ∈ R. Proof. By a direct computation, we obtain R P,ν β,R P,µ α,E = ∞ −∞ R P,t β,R P,µ α,E dν(t) = ∞ −∞ Ad β −it+1/2 • R P,µ α,E * • Ad α it−1/2 dν(t) = ∞ −∞ Ad β −it+1/2 • ∞ −∞ Ad α −is+1/2 • E * • Ad β is−1/2 dµ(s) * • Ad α it−1/2 dν(t) = ∞ −∞ ∞ −∞ Ad β −i(s+t) • E • Ad α i(s+t) dµ(s) dν(t). By changing variables s = r − t, this becomes R P,ν β,R P,µ α,E = ∞ −∞ ∞ −∞ Ad β −ir • E • Ad α ir dµ(r − t) dν(t). Since the integrand now only depends on one of the two variables, namely r, Fubini's theorem implies the t integral can be done first. The resulting inner integral is precisely the definition of the convolution (cf. [89, Lemma 1.30], for example), so the first claim follows. The second claim follows by setting µ = δ x and ν = δ y to be the Dirac delta measures centered at x and y, respectively, and then using the fact that δ x * δ y = δ x+y . Definition 6.2. Given a morphism (A, α) E − → (B, β) in States, a retrodiction family R is said to be involutive on E iff (R • R) α,E := R β,R α,E = E. A retrodiction family R is involutive iff R • R = id States . Definition 6.3. A retrodiction functor is an involutive and inverting monoidal functor States → States op that acts as the identity on objects. Remark 6.4. In particular, a retrodiction functor defines a dagger on the symmetric monoidal category States, which turns States into a symmetric monoidal dagger category [90,43,88,28]. Justifications for why a dagger structure is often used to model time-reversal symmetry is due to the many time-reversal-like properties it satisfies. The difference between our work and [90,43,88,28] is that our dagger structure is not on the category of quantum processes, but rather on the category of quantum states and state-preserving quantum processes. The formulation of retrodiction closest to ours that we are aware of is one that describes (classical) Bayesian inversion as a dagger structure on the category of states [46,Remark 13.10] (in fact, the result proved in [46] is applicable to the setting of Markov categories). However, the explicit connection to retrodiction was not made in [46], nor was an axiomatic formulation emphasized as in our present work. This result was generalized to the quantum setting in [47,Remark 7.25], though that work only proved the claim for certain morphisms that were called Bayesian invertible (this result also holds more generally for quantum Markov categories and implies [46,Remark 13.10]). Therefore, our work not only extends [46,Remark 13.10] and [47,Remark 7.25] to all classical and quantum channels, but it also bypasses the many no-go theorems mentioned earlier, thus illustrating the importance of the prior in (inferential) time-reversal symmetry for quantum dynamics. Theorem 6.5. Among all the rotated Petz recovery retrodiction families {R P,t } t∈R , the only one that is involutive is R P , i.e., when t = 0. Proof. Let t ∈ R and suppose R P,t is involutive. Then, by Lemma 6.1, R P,t β,R P,t α,E = Ad β −2it • E • Ad α 2it = EC [E] =     1 − p 0 0 1 − p 0 p p 0 0 p p 0 1 − p 0 0 1 − p     and C [E ] =     1 − p 0 0 (1 − p)χ 2it 0 p pω −2it 0 0 pω 2it p 0 (1 − p)χ −2it 0 0 1 − p     , where χ := θ(1 − φ) (1 − θ)φ and ω := (1 − θ)(1 − φ) θφ . In order for these two Choi matrices to be equal, it must be the case that 1 = χ 2it ≡ cos 2 ln (χ) t + i sin 2 ln (χ) t and 1 = ω 2it ≡ cos 2 ln (ω) t + i sin 2 ln (ω) t . In order for these equations to hold, the sine terms must vanish simultaneously, i.e., 0 = sin 2 ln (χ) t and 0 = sin 2 ln (ω) t . If the logarithm terms ln (χ) and ln (ω) vanish, then the two Choi matrices are equal for all t ∈ R. Hence, we need to choose θ and p such that these logarithms do not vanish. This is easy enough, and it is sufficient to choose any θ, p ∈ 0, 1 2 ∪ 1 2 , 1 . Once we do this, the logarithm terms necessarily do not vanish and the condition becomes that t must be in the intersection of the two lattices generated by the zeros of these sine functions, i.e., t ∈ π ln(χ) Z π ln(ω) Z Now, we will choose θ and p such that the ratio ln(χ) ln(ω) is irrational so that the only possible value of t is t = 0. Many such solutions exist. For example, take θ = 1 1 + e π 2 ( √ 2+1) and p = sinh(π/2) sinh(π/2) + sinh(π/ √ 2) (which implies φ = 1 1 + e π 2 ( √ 2−1) ). Then ln(χ) = −π and ln(ω) = π √ 2. Hence, with these values of θ and p, the only value of t for which E = Ad β −2it • E • Ad α 2it is t = 0. Remark 6.6. One might object to Theorem 6.5 by claiming that it is unnatural to use R P,t twice when performing an iterated retrodiction, and that one should instead use R P,−t for the second instance, since then R P,−t • R P,t = id States . However, this would require a higher-order involution on the collection of all retrodiction families States → States op that takes some retrodiction family R and constructs a new one R † such that in the special case of the rotated Petz recovery map one obtains (R P,t ) † = R P,−t . With our definition of involutivity, we have imposed that one always uses the same retrodiction family. Our definition agrees with the same type of involutivity axioms used in other works on time-reversal symmetry by implementing the usage of a dagger functor [90,88,28]. A summary of recovery maps and their properties In this work, we furnished axioms for retrodiction associated with priors and processes for arbitrary hybrid classical/quantum dynamics. We proved that the Petz recovery map provides an inferential time-reversal symmetry for such dynamics, bypassing many previous no-go theorems on timereversal symmetry. The defining properties of retrodiction relied on the language of monoidal categories and functors. Namely, retrodiction is a certain involutive monoidal functor on the category of states and processes that sends invertible morphisms to their inverses. We proved that among all rotated Petz recovery maps, averaged rotated Petz recovery maps, and the recovery map of Surace-Scandi, the only one that satisfies all our axioms of retrodiction is the original Petz recovery map. We have also shown specifically which properties fail by the other variants in the literature, including the universal recovery map of [55]. To summarize these findings, we have created Table 1, which indicates several of the examples of possible retrodiction assignments considered in this work together with the properties they satisfy. We have omitted the properties of universality, complete positivity, and state-preservation because all of the examples listed above satisfy these. We have also included "Bayes on CStates," which means that the retrodiction family equals Bayesian inversion when restricted to the subcategory of commutative C * -algebras (justifications are provided in Proposition 3.7 and [54,47,82,83,63]). As can be seen from this table, only the Petz recovery map satisfies all of our desiderata for retrodiction. We close our work with the statement of a remaining open problem, which we can now formulate as a precise mathematical problem to characterize retrodiction. R P R P,t R P,µ R STH R ! R SS Normalization •-stabilizing Compositionality ⊗-stabilizing ? Tensoriality ? Inverting Involutivity Bayes on CStates Table 1: This table indicates the properties satisfied of retrodiction families considered in this work. Note that R JRSWW is an example of R P,µ and therefore also belongs in that column. We have not yet been able to determine whether R SS is tensorial or ⊗-stabilizing. Regardless, the original Petz recovery map is the only one that satisfies all properties. A The Hilbert-Schmidt inner product on C * -algebras In the following, we provide a quick review of traces, the Hilbert-Schmidt inner product, the KMS inner product, and the Petz recovery map for finite-dimensional C * -algebras. The reader is referred to [91,Chapter 8] for more details. Given a finite-dimensional C * -algebra, A, there is a canonical trace, tr. Indeed, since every such A is * -isomorphic to a direct sum of matrix algebras [73,Theorem 5.20 where the latter trace is just the sum of the component-wise traces. This is independent of the choice of * -isomorphism Φ due to the cyclic property of the trace. Namely, if Ψ is another such * -isomorphism, then Ψ • Φ −1 is a * -isomorphism of a direct sum of matrix algebras, which leaves the trace invariant. Hence, tr Φ(A) = tr Ψ • Φ −1 Φ(A) = tr Ψ(A) . Because of this, there is a canonical Hilbert-Schmidt inner product on A as well. Namely, this inner product is defined on (A 1 , A 2 ) ∈ A × A by A 1 , A 2 = tr A † 1 A 2 in terms of the above-defined trace. With such an inner product on every finite-dimensional C * -algebra, if now A E − → B is a linear map, then E * is the Hilbert-Schmidt adjoint of E and is defined as the adjoint of E with respect to the Hilbert-Schmidt inner products on B and A. Namely, A E * ←− B is the unique linear map satisfying tr E * (B) † A ≡ E * (B), A = B, E(A) ≡ tr B † E(A) for all A ∈ A and B ∈ B. Given a state-preserving morphism (A, α) E − → (B, β) of non-degenerate quantum probability spaces, the Petz recovery map R P α,E is defined as the adjoint, not with respect to the Hilbert-Schmidt inner product, but with respect to the following inner products on A and B. The KMS inner product on A associated with the faithful state α is defined by sending (A 1 , A 2 ) ∈ A × A to A 1 , A 2 α := tr A † 1 α −1/2 A 2 α −1/2 . The Petz recovery map R P α,E is therefore the unique map that satisfies B, E(A) β = R P α,E (B), A α for all A ∈ A and B ∈ B. Writing this out explicitly gives tr B † β −1/2 E(A)β −1/2 = tr R P α,E (B) † α −1/2 Aα −1/2 . From this expression, the formula for the Petz recovery map in Definition 3.1 follows. B Convex sums of rotated Petz recovery maps In this appendix, we prove Proposition 4.6. In fact, we even show that it suffices to work with the class of bit-flip channels on qubits to find counter-examples to compositionality and tensoriality, even in the symmetric case when λ = 1 2 and s = −t in Proposition 4.6. Since it is known that the bit flip channel is covariant only with respect to the uniform state [82], we need to provide a counter-example where at least two of the states are non-uniform states. Let C = B = A = M 2 (C) denote the C * -algebra of 2 × 2 matrices. Let α = θ 0 0 1−θ , β = φ 0 0 1−φ and γ = ψ 0 0 1−ψ be states on A, B, and C , respectively, for some θ, φ, ψ ∈ (0, 1). Let Ω p : M 2 (C) → M 2 (C) be the bit-flip channel weighted by p ∈ [0, 1] and given explicitly by Ω p = (1 − p)id + pAd σx , where σ x = 0 1 1 0 . Note the set of bit-flip channels is closed under composition, and in fact We will therefore assume these conditions are always met in this example. The associated rotated Petz recovery maps are given by Ω q • Ω p = Ω (1−q)p+q(1−p)R P,t α,E = (1−p)Ad φ it−1/2 θ −it+1/2 0 0 (1−φ) it−1/2 (1−θ) −it+1/2 +pAd 0 (1−φ) it−1/2 θ −it+1/2 φ it−1/2 (1−θ) −it+1/2 0 and similarly R P,t β,F = (1−q)Ad ψ it−1/2 φ −it+1/2 0 0 (1−ψ) it−1/2 (1−φ) −it+1/2 +qAd 0 (1−ψ) it−1/2 φ −it+1/2 ψ it−1/2 (1−φ) −it+1/2 0 . Several simplifications occur when q = 1 2 , so we will henceforth assume this. In particular, γ = 1 2 1 2 , i.e., ψ = 1 2 , so that R P,t β,F = Ad φ −it+1/2 0 0 (1−φ) −it+1/2 + Ad 0 φ −it+1/2 (1−φ) −it+1/2 0 . In addition, since F • E = Ω 1/2 as a function, we obtain R P,t α,F •E = Ad θ −it+1/2 0 0 (1−θ) −it+1/2 + Ad 0 θ −it+1/2 (1−θ) −it+1/2 0 . Now, let R := λR P,t + (1 − λ)R P,s be the convex combination of two rotated Petz recovery maps with t, s ∈ R weighted by some λ ∈ (0, 1). For our purposes, it suffices to consider the symmetric combination given by s = −t and λ = 1 2 . Then R α,E = 1 − p 2 Ad φ it−1/2 θ −it+1/2 0 0 (1−φ) it−1/2 (1−θ) −it+1/2 + p 2 Ad 0 (1−φ) it−1/2 θ −it+1/2 φ it−1/2 (1−θ) −it+1/2 0 + 1 − p 2 Ad φ −it−1/2 θ it+1/2 0 0 (1−φ) −it−1/2 (1−θ) it+1/2 + p 2 Ad 0 (1−φ) −it−1/2 θ it+1/2 φ −it−1/2 (1−θ) it+1/2 0 , R β,F = 1 2 Ad φ −it+1/2 0 0 (1−φ) −it+1/2 + 1 2 Ad 0 φ −it+1/2 (1−φ) −it+1/2 0 + 1 2 Ad φ it+1/2 0 0 (1−φ) it+1/2 + 1 2 Ad 0 φ it+1/2 (1−φ) it+1/2 0 , and R α,F •E = 1 2 Ad θ −it+1/2 0 0 (1−θ) −it+1/2 + 1 2 Ad 0 θ −it+1/2 (1−θ) −it+1/2 0 + 1 2 Ad θ it+1/2 0 0 (1−θ) it+1/2 + 1 2 Ad 0 θ it+1/2 (1−θ) it+1/2 0 , the last of which is the retrodiction map associated with the composite. However, the composite of the individual retrodiction maps is given by R α,E • R β,F = 1 4 R P,t α,E • R P,t β,F + R P,−t α,E • R P,−t β,F + R P,−t α,E • R P,t β,F + R P,t α,E • R P,−t β,F = 1 2 R α,F •E + 1 − p 4 Ad φ −2it θ it+1/2 0 0 (1−φ) −2it (1−θ) it+1/2 + p 4 Ad 0 (1−φ) −2it θ it+1/2 φ −2it (1−θ) it+1/2 0 + 1 − p 4 Ad 0 φ −2it θ it+1/2 (1−φ) −2it (1−θ) it+1/2 0 + p 4 Ad (1−φ) −2it θ it+1/2 0 0 φ −2it (1−θ) it+1/2 + 1 − p 4 Ad φ 2it θ −it+1/2 0 0 (1−φ) 2it (1−θ) −it+1/2 + p 4 Ad 0 (1−φ) 2it θ −it+1/2 φ 2it (1−θ) −it+1/2 0 + 1 − p 4 Ad 0 φ 2it θ −it+1/2 (1−φ) 2it (1−θ) −it+1/2 0 + p 4 Ad (1−φ) 2it θ −it+1/2 0 0 φ 2it (1−θ) −it+1/2 . This suggests that R α,E • R β,F = R α,F •E , but it may be the case that surprising cancellations occur. To be sure, let us plug in a non-diagonal matrix such as [ 0 1 0 0 ]. This gives R α,F •E 0 1 0 0 = θ(1 − θ) cos ln θ 1 − θ t σ x and (R α,E • R β,F ) 0 1 0 0 = θ(1 − θ) 2 cos ln θ 1 − θ t + p cos ln θφ 2 (1 − θ)(1 − φ) 2 t + + (1 − p) cos ln θ(1 − φ) 2 (1 − θ)φ 2 t σ x . At this point, it may already be clear that these two expressions are different, but just to be very concrete, let us consider the special case where θ = e 2π 1 + e 2π , p = tanh π 2 2 sinh(π) (which implies φ = e π 1 + e π ), and t = 1 2 . Then R α,F •E 0 1 0 0 = −1 2 cosh(π) σ x , while (R α,E • R β,F ) 0 1 0 0 = 0. This illustrates that R α,E • R β,F = R α,F •E , i.e., compositionality fails in general (in fact, for most values of t). We will also use this example to show that tensoriality fails. In this regard, consider another β ), where B = M 2 (C) = A , E = Ω p , and α and β are given by the density matrices morphism (A , α ) E − → (B ,α = θ 0 0 1 − θ and β = φ 0 0 1 − φ . It suffices to set p = 1 2 = p . In this case, plugging in the matrix [ 0 1 0 0 ] ⊗ [ 0 1 0 0 ] into R P,t α,E ⊗ R P,t α ,E gives R P,t α,E ⊗ R P,t α ,E 0 1 0 0 ⊗ 0 1 0 0 = √ θθ ⊥ θ θ ⊥    0 θ ⊥ θ it θ ⊥ θ −it 0   ⊗    0 θ ⊥ θ it θ ⊥ θ −it 0    , where we temporarily introduced the notation r ⊥ := 1 − r for r ∈ R. Therefore, (R α,E ⊗ R α ,E ) 0 1 0 0 ⊗ 0 1 0 0 = √ θθ ⊥ θ θ ⊥ cos ln θ ⊥ θ t cos ln θ ⊥ θ t σ x ⊗ σ x , while R α⊗α ,E⊗E = 1 2 R P,t α⊗α ,E⊗E + 1 2 R P,−t α⊗α ,E⊗E gives R α⊗α ,E⊗E 0 1 0 0 ⊗ 0 1 0 0 = √ θθ ⊥ θ θ ⊥ cos ln θ ⊥ θ t cos ln θ ⊥ θ t σ x ⊗ σ x + sin ln θ ⊥ θ t sin ln θ ⊥ θ t σ y ⊗ σ y . These two results are manifestly different since for most values of θ and θ the term involving σ y ⊗ σ y need not vanish. Hence, R α,E ⊗ R α ,E = R α⊗α ,E⊗E , i.e., R is not tensorial. C The JRSWW retrodiction family is neither compositional nor tensorial In this appendix, we prove Proposition 4.7. As in Appendix B, we show that it suffices to work with the class of bit-flip channels on qubits to find counter-examples to compositionality and tensoriality. The same composable pair (A, α) E − → (B, β) F − → (C , γ) as in Appendix B will provide such a counter-example. In particular, we will again assume q = 1 2 and keep θ and φ arbitrary for the moment (the fact that ψ = 1 2 follows from q = 1 2 ). First, R JRSWW α,F •E = ∞ −∞ π cosh(2πt) + 1   Ad θ −it+1/2 0 0 (1−θ) −it+1/2 + Ad 0 θ −it+1/2 (1−θ) −it+1/2 0   dt. Meanwhile, by Lemma 4.2 or direct calculation, R JRSWW α,E • R JRSWW β,F = ∞ −∞ ∞ −∞ dµ(t)dµ(s) (1 − p)Ad φ i(t−s) θ −it+1/2 0 0 (1−φ) i(t−s) (1−θ) −it+1/2 + (1 − p)Ad 0 φ i(t−s) θ −it+1/2 (1−φ) i(t−s) (1−θ) −it+1/2 0 + pAd 0 (1−φ) i(t−s) θ −it+1/2 φ i(t−s) (1−θ) −it+1/2 0 + pAd (1−φ) i(t−s) θ −it+1/2 0 0 φ i(t−s) (1−θ) −it+1/2 , where µ is the probability measure from Example 3.2. Plugging in the matrix [ 0 1 0 0 ] into the first option gives R JRSWW α,F •E 0 1 0 0 = ∞ −∞ π θ(1 − θ) cosh(2πt) + 1    0 θ ⊥ θ it θ ⊥ θ −it 0    dt = ∞ 0 2π θ(1 − θ) cosh(2πt) + 1 cos ln θ ⊥ θ t dt σ x , where r ⊥ := 1 − r as in Appendix B. Plugging that same matrix into the second option gives 4 − 3 cosh(π) − cosh(3π) sinh(π) + sinh(2π) − sinh(3π) σ x . R JRSWW β,F •R JRSWW α,E 0 1 0 0 = ∞ −∞ ∞ −∞ π 2 θ(1 − θ) [cosh(2πt) + 1][cosh(2πs) + 1] ×    0 (1 − p)χ −it φ ⊥ φ is + pω it φ ⊥ φ −is (1 − p)χ it φ ⊥ φ −is + pω −it φ ⊥ φ is 0    ds dt = ∞ 0 ∞ 0 4π 2 θ(1 − θ) Since the term inside the square parentheses is approximately 1.65, and therefore not 1, this shows that R JRSWW β,F • R JRSWW α,E = R JRSWW α,F •E , i.e., compositionality fails for the JRSWW recovery map. Tensoriality also fails for the JRSWW retrodiction family, as we will now illustrate using the same setup as in Appendix B, namely E is as above, let (A , α ) Again, set p = 1 2 = p . Then, by the calculations done in Appendix B, cos ln θ ⊥ θ t cos ln θ ⊥ θ t σ x ⊗ σ x + sin ln θ ⊥ θ t sin ln θ ⊥ θ t σ y ⊗ σ y dµ(t). R JRSWW α,E ⊗ R JRSWW α ,E 0 1 0 0 ⊗ 0 1 0 0 = √ θθ ⊥ θ θ ⊥    ∞ −∞    0 θ ⊥ θ it θ ⊥ θ −it 0    dµ(t)    ⊗    ∞ −∞    0 θ ⊥ θ it θ ⊥ θ −it 0    dµ(t ) Plugging in the values θ = 1 1+e 2π = θ gives R JRSWW α,E ⊗ R JRSWW α ,E 0 1 0 0 ⊗ 0 1 0 0 = π 2 √ θθ ⊥ θ θ ⊥ sinh 2 (π) σ x ⊗ σ x and R JRSWW α⊗α ,E⊗E 0 1 0 0 ⊗ 0 1 0 0 = π √ θθ ⊥ θ θ ⊥ 2 1 π + 1 cosh(π) sinh(π) σ x ⊗ σ x + 1 π − 1 cosh(π) sinh(π) σ y ⊗ σ y , which are not equal to each other. This proves that the JRSWW map is not tensorial. D The SS retrodiction family is not stabilizing In this appendix, we prove Proposition 5.7. First note that the normalizing and inverting conditions are automatic since they are included in the definition of the Surace-Scandi retrodiction family. The remainder of the proof will illustrate that the Surace-Scandi retrodiction family is not stabilizing, specifically not •-stabilizing. In fact, we will not even need to enter the realm of quantum systems to show this. It suffices to work with stochastic matrices. Hence, let A = B = C = C 2 , let α, β, γ be the probability vectors given by Definition 3. 1 . 1Let (A, α) E − → (B, β) be a morphism in States. The Petz recovery map associated with the pair (α, E) is the morphism 6 (A, α) Definition 4. 1 . 1Let (A, α) E − → (B, β) be a morphism in States. Lemma 4. 4 . 4Let (A, α) E − → (B, β) and (A , α ) E − → (B , β ) be two morphisms in States, and let µ and µ be two probability measures on R. Then Corollary 4. 5 . 5Let (A, α) E − → (B, β) and (A , α ) E − → (B , β ) be two morphisms in States, and let µ be a probability measure on R. If at least one of E or E is covariant (cf. Definition 4.1), then Definition 4. 8 . 8Let R : States → States op be a retrodiction family.1. R is composition stabilizing, or •-stabilizing, iff R α,E •R β,F = R α,F •E for all composable pairs (A, α) E − → (B, β) F − → (C ,γ) of morphisms in States at least one of which is covariant. 2. R is tensor stabilizing, or ⊗-stabilizing, iff R α,E ⊗ R α ,E = R α⊗α ,E⊗E for all pairs (A, α) E − → (B, β) and (A , α ) E − → (B , β ) of morphisms in States at least one of which is covariant.R satisfies the stabilization property iff it is both composition and tensor stabilizing. Corollary 4. 9 . 9Let R be a normalizing retrodiction family that is tensor stabilizing. ThenR α⊗α ,E⊗id A = R α,E ⊗ id A for all morphisms (A, α) E − → (B, β) and objects (A , α ) in States.As mentioned earlier, Corollary 4.9 provides an alternative justification for the stabilization property mentioned in[55, Remark 2.4]. Therefore, our definition generalizes this property. Corollary 4. 10 . 10The following three facts regarding the stabilizing properties hold.(a) Every averaged rotated Petz retrodiction family satisfies the stabilization property. E for all morphisms (A, α) E − → (B, β) and objects (C , γ) in States, rather than the full ⊗-stabilizing property. The sufficiency of this follows from the proof. Also note that Corollary 4.10 (c) does not contradict Proposition 4.7, because R JRSWW is not compositional. Proof. Statement (a) is exactly what Corollaries 4.3 and 4.5 say. Statement (b) follows directly from the definitions. For statement (c), let R denote a retrodiction family that is normalizing, compositional, and ⊗-stabilizing. Let (A, α) E − → (B, β) and (A , α ) E − → (B , β ) be two pairs of morphisms in States. Then Proof of Lemma 5.3. Let (A, α) E − → (B, β) be an isomorphism in States. This means there exists a morphism (A, α) F ← − (B, β) in States such that F • E = id A and E • F = id B . Write the Hilbert-Schmidt adjoints as E := E * and F := F * , which are completely positive unital maps. Then, for any A ∈ A, Example 5. 5 . 5In general, it is not the case that R STH α,E = E −1 for an invertible morphism (A, α) E − → (B, β) in States. We illustrate this with an explicit counter-example, even in the case of when all unitary rotations are independent phases. Let A = M 2 (C) = B and set α = diag(p, 1 − p) for some p ∈ [0, 1]. Let E = Ad σx , so that β = diag(1 − p, p), and set U α = e iθ 0 0 e iφ and U β = e iψ 0 0 e iω for some distinct θ, φ, ψ, ω ∈ [0, 2π]. Then Definition 5. 6 . 6The Surace-Scandi retrodiction family R SS : States → States op sends eachmorphism (A, α) E − → (B, β) to the unique morphism (B, β) R SS α,E − −− → (A, α) satisfying the following conditions. (a) If E is an isomorphism in States, then R SS α,E = E −1 . (b) The map R SS α,E • E satisfies the detailed balance condition R SS α,E • E • Ad α 1/2 = Ad α 1/2 • R SS α,E • E with respect to the prior α.(c) The eigenvalues of the linear map R SS α,E • E : A → A are all non-negative.(d) The map R SS α,E maximizes the determinant of the linear map R SS α,E •E subject to all the previous constraints. Theorem 5. 8 . 8Let R : States → States op be an inverting retrodiction family, let A and B be -isomorphic C * -algebras, and let A E − → B be a CPTP map. Then the following are equivalent. I. The map E is a * -isomorphism. II. The morphism R α,E : (B, E(α)) → (A, α) is independent of the (faithful) state α on A, in the sense that the underlying map R α,E : B → A is the same for all α. Lemma 6. 1 . 1Let µ and ν be two probability measures on R and let (A, α) E − → (B, β) be a morphism in States. Then 15 for every morphism (A, α) E − → (B, β) in States. This necessarily implies that t = 0. To see this, consider the bit-flip channel from Appendix B. Namely, A = M 2 (C) = B, E = (1 − p)id + pAd σx , α = θ 0 0 1−θ , and β = φ 0 0 1−φ . Temporarily set E := Ad β −2it •E •Ad α 2it . Then, the Choi matrices C [E] and C [E ] of E and E , respectively, are Question 7. 1 . 1What is the space of all retrodiction functors States → States op ? In particular, is R P , the Petz retrodiction functor, the unique element in this space? If R P is not the unique element, what are other examples of retrodiction, and what additional axioms are needed to characterize the Petz retrodiction functor? In particular, is classical Bayesian inversion characterized as the unique involutive and inverting monoidal functor CStates → CStates op ? and Proposition 5.26], say Φ : A ∼ = − → x∈X M mx (C), the trace on A is defined by tr(A) := tr Φ(A) , for all p, q ∈ [0, 1]. Now, set A E − → B and B F − → C to be the bit-flip channels Ω p and Ω q , respectively, where p, q ∈ [0, 1] are arbitrary for now. Then (A, α) E − → (B, β) F − → (C , γ) is a composable pair of morphisms in States if and only if φ = (1 − p)θ + p(1 − θ) and ψ = (1 − q)φ + q(1 − φ). × (1 − p) cos ln(χ)t + p cos ln(ω)t ds dt σ x , where χ := θ(1 − φ) (1 − θ)φ and ω := (1 − θ)(1 − φ) θφ .At this point, we set θ and p exactly as in Appendix B. Doing so simplifies these expressions to E − → (B , β ) be another morphism in States with B = M 2 (C) = A , E = Ω p , and set ln θ ⊥ θ t cosh(2πt ) + 1 dt   σ x ⊗ σ x , F − → (C , γ) is a composable pair of morphisms in States, in fact CStates. The Surace-Scandi retrodiction maps in these cases are represented by the following while the one associated with the composite (A, α)F •E − −− → (C , γ) is 16 For reference, the way these are computed is as follows (it can be done analytically in this case). For concreteness, write R SS α,E as a stochastic matrix of the forma b 1 − a 1 − b .The state-preserving condition R SS α,E (β) = α gives one constraint, which is a 5 + 4b 5 = 1 2 . Detailed balance for R SS α,E • E gives no additional constraints in this case. Requiring that R SS α,E is a stochastic matrix and R SS α,E • E has non-negative eigenvalues gives the additional constraint 1 2 ≤ b ≤ 5 8 . Maximizing the determinant of R SS α,E • E with respect to these constraints gives the unique solution b = 5 8 and a = 0.compositionality fails. In fact, this example also shows that the •-stabilizing property fails, since every morphism in CStates is covariant.R SS α,F •E = 25/27 0 2/27 1 . Since R SS α,E • R SS β,F = 5/8 65/184 3/8 119/184 = R SS α,F •E , Accepted in Quantum 2023-04-10, click title to verify. Published under CC-BY 4.0. The category States is not quite the coslice/under category C ↓ CPTP, where CPTP is the category of finite-dimensional C * -algebras and CPTP maps, but rather a subcategory where all states are faithful.3 If C is a category, C op denotes its opposite. By definition, a morphism from A to B in C op is a morphism from B to A in C.Accepted in Quantum 2023-04-10, click title to verify. Published under CC-BY 4.0. If we excluded the prior from our notation for retrodiction, then this would look like R F •E = R E • R F . We have chosen to include the prior in the notation to be more consistent with the quantum information literature.Accepted in Quantum 2023-04-10, click title to verify. Published under CC-BY 4.0. The map Cα − → A is the unique linear map determined by settingα(1) = α. Accepted in Quantum 2023-04-10, click title to verify. Published under CC-BY 4.0. Again, the reader is expected to check that this is indeed a morphism in States.8 We will prove that this is indeed a morphism in States in Proposition 3.4.Accepted in Quantum 2023-04-10, click title to verify. Published under CC-BY 4.0. In[81], these maps were denoted T ϕ,ϑ σ,N . It is not even obvious that the space of retrodiction families that are compositional and/or tensorial should be path-connected. Namely, why should there exist a continuous interpolation between two arbitrary retrodiction procedures, much less a linear path?12 It would be more precise to say that E is time-symmetric covariant with respect to the modular Hamiltonians associated with α and β. However, since this is the only covariance considered in this work, we use the shorthand "covariant." This notion of covariance, its relation to Bayesian invertibility, and simple criteria for guaranteeing such covariance (including necessary modifications for non-faithful states) are discussed in[82,47,83]. 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[ "FACTORIZATION NUMBER AND SUBGROUP COMMUTATIVITY DEGREE VIA SPECTRAL INVARIANTS", "FACTORIZATION NUMBER AND SUBGROUP COMMUTATIVITY DEGREE VIA SPECTRAL INVARIANTS" ]	[ "Seid Kassaw Muhie ", "ANDDaniele Ettore Otera ", "Francesco G Russo " ]	[]	[]	The factorization number F 2 (G) of a finite group G is the number of all possible factorizations of G = HK as product of its subgroups H and K, while the subgroup commutativity degree sd(	10.1007/s40314-023-02270-5	[ "https://export.arxiv.org/pdf/2304.08170v1.pdf" ]	257,768,966	2304.08170	416a9578b99edc1677bb4c516923cacc290a6a1e	FACTORIZATION NUMBER AND SUBGROUP COMMUTATIVITY DEGREE VIA SPECTRAL INVARIANTS 17 Apr 2023 Seid Kassaw Muhie ANDDaniele Ettore Otera Francesco G Russo FACTORIZATION NUMBER AND SUBGROUP COMMUTATIVITY DEGREE VIA SPECTRAL INVARIANTS 17 Apr 2023arXiv:2304.08170v1 [math.CO]G) of G is the probability of finding two commuting subgroups in G at random. It is known that sd(G) can be expressed in terms of F 2 (G). Denoting by L(G) the subgroups lattice of G, the non-permutability graph of subgroups Γ L(G) of G is the graph with vertices in L(G) \ C L(G) (L(G)), where C L(G) (L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and edges obtained by joining two vertices X, Y such that XY = Y X. The spectral properties of Γ L(G) have been recently investigated in connection with F 2 (G) and sd(G). Here we show a new combinatorial formula, which allows us to express F 2 (G), and so sd(G), in terms of adjacency and Laplacian matrices of Γ L(G) . The factorization number F 2 (G) of a finite group G is the number of all possible factorizations of G = HK as product of its subgroups H and K, while the subgroup commutativity degree sd( Introduction and statement of the main result In the present paper we shall be interested only in finite groups. The non-permutability graph of subgroups Γ L(G) of a group G is the undirected and unweighted simple graph defined as the ordered pair of vertices and edges Γ L(G) = (V (Γ L(G) ), E(Γ L(G) )), (1.1) where L(G) denotes the lattice of subgroups of G, is not (in general) a sublattice of L(G), we will consider the smallest sublattice of L(G) containing (1.5). This is denoted by C L(G) (L(G)) and appears in (1.2) above. The non-permutability graph of subgroups is motivated by a line of research in lattice theory, which has analogies with the contributions [6,7,18], where combinatorial properties of graphs and groups are discussed. In our present work we shall also use some spectral properties and invariants of graphs in order to get information on algebraic properties of corresponding groups. The adjacency matrix of Γ L(G) is the square matrix A(Γ L(G) ) = (a X,Y ) X,Y ∈V (Γ L(G) ) , where a X,Y = 1, if (X, Y ) ∈ E(Γ L(G) ), 0, if (X, Y ) ∈ E(Γ L(G) ). (1.6) Note that the degree of a vertex X in (1.1) is defined by deg(X) = Y ∈V (Γ L(G) ) a X,Y . (1.7) Since Γ L(G) is an undirected graph without loops, the Laplace matrix of Γ L(G) is the matrix L(Γ L(G) ) = D − A(Γ L(G) ), (1.8) where D = diag(deg(X i )), for all X i ∈ V (Γ L(G) ) and i = 1, 2, · · · , m = \|V (Γ L(G) )\|. These are common notions, which are usually considered in spectral graph theory, see [4,5]. On the other hand, we are also interested in the so-called subgroup commutativity degree of G, studied in [1,22,29]. This is the probability that two subgroups of G commute, namely sd(G) = \|{(X, Y ) ∈ L(G) × L(G) \| XY = Y X}\| \|L(G)\| 2 . (1.9) If any two randomly chosen subgroups of G commute, then G is called quasihamiltonian, and these groups were classified since long time by Iwasawa (see [25]). Abelian groups are of course quasihamiltonian, but the quaternion group Q 8 of order 8 is a nonabelian group of sd(Q 8 ) = 1. Evidently G is quasihamiltonian if and only if sd(G) = 1, therefore (1.9) is a measure of how far is a group from being quasihamiltonian. It will be useful to introduce the following sets Note that permutable subgroups are subnormal, while normal subgroups are of course permutable, see [25]. The spec(A(Γ L(G) )) = {λ 1 , λ 2 , · · · , λ m } and spec(L(Γ L(G) )) = {σ 1 , σ 2 , · · · , σ m } (1.12) are the spectrum of the adjacency and the Laplacian matrix respectively, then [19, (3.6)] shows that for groups with sd(G) = 1 sd(G) = 1 − 1 \|L(G)\| 2 m i=1 λ 2 i = 1 − 1 \|L(G)\| 2 m i=1 σ i . (1.13) Another important quantity which is associated to a group G is the factorization number F 2 (G) = \|{(H, K) ∈ L(G) × L(G) \| G = HK}\|; (1.14) this denotes the number of all possible factorizations of G as product of two subgroups H and K. In fact we say that a group G has factorization HK if there are two subgroups H and K of G such that G = HK (see [15,24]). We also mention from [25, §1.1] that an interval of L(G) is the set [K/H] = {Z ∈ L(G) \| H ≤ Z ≤ K},(1.15) where H ≤ K. Note that [K/H] is a sublattice of L(G). From [21] the Möbius function µ : L(G) × L(G) → Z is recursively defined by: Z∈[K/H] µ(H, Z) =    1, H = K, 0, otherwise. (1. 16) In particular, the Möbius number of G is µ(G) = µ(1, G), considering [G/1] = L(G). Our main result is the following: Theorem 1.1. Let G be a group with sd(G) = 1. Then F 2 (G) = K∈K(G) \|L(K)\| 2 µ(K, G) + H∈H(G) \|L(H)\| 2 − m i=1 σ i µ(H, G) , (1.17) where m = \|V (Γ L(H) )\| and {σ 1 , σ 2 , · · · , σ m } = spec(L(Γ L(H) )). In particular, sd(G) = 1 \|L(G)\| 2 S∈L(G) W ∈K(S) \|L(W )\| 2 µ(W, S)+ S∈L(G) U ∈H(S) \|L(U)\| 2 − k j=1 τ j µ(U, S) , (1.18) where k = \|V (Γ L(U ) )\| and {τ 1 , τ 2 , · · · , τ k } = spec(L(Γ L(U ) )). We shall mention that the theory of the subgroup commutativity degree has been recently discussed in [16,17,22,23,24,29], but only in [18,19] in connection with notions of spectral graph theory on the line of [4,5]. Therefore Theorem 1.1 belongs to the line of research of [18,19] and explores new connections with the theory of the factorization number in [15,23,24]. Section 2 collects information of general nature on the references which are pertinent to the topic, but also some classical results on the partitions of groups. Section 3 contains the proof of Theorem 1.1 along with some applications. Groups with partitions, factorization number and subgroup commutativity degree In order to count the number of edges of the non-permutability graph of subgroups of a group G, combinatorial formulas were found in [18, Lemma 2.10, Theorem 3.1] involving the subgroup commutativity degree. We report some results from [18,19] below: Lemma 2.1 (See [19], Lemma 2.5). For a group G we have 2 \|E(Γ L(G) )\| = \|L(G)\| 2 (1 − sd(G)). (2.1) This formula shows that we can obtain the number of edges in Γ L(G) if we know sd(G), and vice-versa. Moreover [19,Proposition 3.2] shows that sd(G) can be rewritten in terms of spectral invariants of Γ L(G) . sd(G) = 1 − 1 \|L(G)\| 2 X,Y ∈V (Γ L(G) ) a X,Y . (2. 2) The above formula allows us to match an approach of spectral nature with another of combinatorial nature (see [1,30,16,23]), since sd(G) may be obtained in terms of F 2 (G) by the formula sd(G) = 1 \|L(G)\| 2 H∈L(G) F 2 (H). (2.3) In fact (2.3) shows that the subgroup commutativity degree can be reduced to the computation of the factorization number. This has led to important numerical evaluations for sd(G) via F 2 (H), because it was found that F 2 (H) may be expressed for several families of groups via Gaussian trinomial integers. Consequently, we may connect the spectral invariants of Γ L(G) to F 2 (G) as indicated below. Corollary 2.3 (See [19], Lemma 2.6). For a group G we have 2 \|E(Γ L(G) )\| = \|L(G)\| 2 − H∈L(G) F 2 (H). (2.4) Now we report a few notions which are classical in the area of the theory of partitions of groups, referring mostly to [3,9,10,11,32]. Definition 2.4 (See [10], Definition, §7.1). Given a prime p and a group G, H p (G) = g ∈ G \| g p = 1 (2.5) is the Hughes subgroup of G. It can be shown that groups as per Definition 2.5 have H p (G) nilpotent of \|G : H p (G)\| = p, see [9]. Omitting details of the definitions, we refer to [ [3,9,11,32] classified groups with partitions, but the result below is due to Farrokhi: Theorem 2.6 (See [8], Classification Theorem, pp.119-120). Let G be a group with a nontrivial partition. Then G is isomorphic to exactly one of the following groups (i). S 4 ; (ii). a p-group with H p (G) = G; (iii) . a group of Hughes-Thompson type; (iv). a Frobenius group; (v). PSL(2, p n ) for p n ≥ 4; (vi). PGL(2, p n ) for p n ≥ 5 odd prime power; (vii). Sz(2 2n+1 ). We recalled Theorem 2.6 here, because the subgroup commutativity degree has been computed for most of the groups with nontrivial partitions. Let's see this with more details. For instance, Farrokhi and Saeedi [23,24] completely determined the factorization number of groups in Theorem 2.6 (i), (v) and (vi). F 2 (PSL(2, p n )) =                  2\|L(PSL(2, p n ))\| + 2p n (p 2n − 1) − 1 if p = 2 and n > 1, 2\|L(PSL(2, p n ))\| + p n (p 2n − 1) − 1 if p > 2, n > 1, and (p n − 1)/2 is odd, but p n = 3, 7, 11, 19, 23, 59, 2\|L(PSL(2, p n ))\| − 1 if p > 2, n > 1, and (p n − 1)/2 is even, but p n = 5, 9, 29. In the other cases, Of course, one would like to evaluate numerically \|L(PSL(2, p n ))\| in Proposition 2.7 and this can be made in different ways. For instance, Shareshian [27] computed the Möbius function (1.16) for PSL(2, p n ) and this helps to find \|L(PSL(2, p n ))\|. Another method is due to Dickson: we may list all the subgroups of PSL(2, p n ) and count them. Historically this was the first method to investigate \|L(PSL(2, p n ))\|. F 2 (PSL(2, p n )) = 17, Proposition 2.8 (Dickson's Theorem, see [14], Hauptsatz 8.27, Kapitel II, §8). The subgroups of PSL(2, p n ) are the following: (i). p n (p n ± 1)/2 cyclic subgroups C d of order d, where d is a divisor of (p n ± 1)/2; (ii). p n (p 2n − 1)/(4d) dihedral subgroups D 2d of order 2d, where d is a divisor of (p n ± 1)/2 and d > 2 and p n (p 2n − 1)/24 dihedral subgroups D 4 ; (iii). p n (p 2n − 1)/24 alternating subgroups A 4 ; (iv). p n (p 2n − 1)/24 symmetric subgroups S 4 when p n ≡ 7 mod 8; (v). p n (p 2n − 1)/60 alternating subgroups A 5 when p n ≡ ±1 mod 10; (vi). p n (p 2n − 1)/(p m (p 2m − 1)) subgroups PSL(2, p n ) where m is a divisor of n; (vii). The elementary abelian group C m p for m ≤ n; (viii). C m p ⋊ C d , where d divides both (p n − 1)/2 and p m − 1. A result, which is similar to Proposition 2.7, is available for projective general linear groups. In the other cases, Essentially, we may compute the factorization number for all the groups which are mentioned in Theorem 2.6, referring to methods of combinatorics and number theory in [1,2,23,24], but let's focus only on PSL(2, p n ) and PGL(2, p n ), in order to show significant applications of the spectral invariants which we associated to Γ L(G) . From Propositions 2.7 and 2.9, a precise computation of the factorization number should involve a numerical evaluation of the cardinalities of the subgroups lattices. There are details again in [23,24] in this sense and the main idea is to introduce the Möbius function (1.16), as originally made by Hall [13]. The case of p-groups is known since long time: F 2 (G) = Lemma 2.10 (See [12]). In a p-group G of order p n we have µ(G) = 0, unless G is elementary abelian, in which case we have µ(G) = (−1) n p ( n 2 ) . In case of a symmetric group, µ(1, S n ) was compute by Shareshian [26] and Pahlings [20]. (ii). µ(1, S n ) =            −n!, if n-1 is prime and p=3 mod 4, n! 2 , if n=22, −n! 2 , otherwise, (iii). Let n = 2 α for an integer α ≥ 1. Then µ(1, S n ) = −p! 2 . In addition to symmetric groups, Shareshian [27] computed µ(1, G) also for projective general linear groups, projective special linear groups and for Suzuki groups, see [26,27]. Proof of the main theorem and some applications Our main result connects the factorization number of a group with the spectrum of the Laplacian matrix via the Möbius function. Proof of Theorem 1.1. In a group G we have always that Note from [18] that Γ L(G) is a null graph whenever G is quasihamiltonian. Then, in what follows, we shall assume that G is not quasihamiltonian and K is an arbitrary subgroup of G of sd(K) = 1. Consequently, Γ L(K) is the null graph. Similarly, we assume H to be an arbitrary subgroup of G of sd(H) = 1. Consequently, Γ L(H) exists and is different from the null graph. From Lemma 2.2, we have for m T = \|V (Γ L(T ) )\| sd(T ) = 1 − 1 \|L(T )\| 2 m T i=1 σ i . (3.2) and so we can use (3.1), obtaining F 2 (G) = T ∈L(G) \|L(T )\| 2 − m T i=1 σ i µ(T, G). (3.3) But if T ∈ K(G) in (1.11), then Γ L(K) is the null graph and so we may assume each σ i = 0 with respect to L(Γ L(K) ). Hence we get in correspondence of {τ 1 , τ 2 , · · · , τ k } = spec(L(Γ L(U ) )). The result follows. Of course, we may repeat the proof of Theorem 1.1, replacing (3.2) with the first equation in (1.13) and involving spec(A(Γ L(G) )) instead of spec(L(Γ L(G) )). Corollary 3.1. Let G be a group with sd(G) = 1. Then F 2 (G) = K∈K(G) \|L(K)\| 2 µ(K, G) + H∈H(G) \|L(H)\| 2 − m i=1 λ 2 i µ(H, G) , (3.6) where m = \|V (Γ L(H) )\| and {λ 1 , λ 2 , · · · , λ m } = spec(A(Γ L(H) )). In particular, sd(G) = 1 \|L(G)\| 2 S∈L(G) W ∈K(S) \|L(W )\| 2 µ(W, S)+ S∈L(G) U ∈H(S) \|L(U)\| 2 − k j=1 ρ 2 j µ(U, S) , (3.7) where k = \|V (Γ L(U ) )\| and {ρ 1 , ρ 2 , · · · , ρ k } = spec(A(Γ L(U ) )). We present a few applications of Theorem 1.1, but some relevant comments should be made. Remark 3.2. Suppose to compute F 2 (G) for G = PSL(2, p n ). We may proceed as below: (1). Use Proposition 2.7 and compute \|L(G)\| applying Proposition 2.8. (2) is presented here for the first time and is apparently harder than (1), but softwares are available such as GAP [31] and NewGraph [28] which can assist better with the steps (2a), (2b) and (2c). Therefore it is very efficient. We sketch similar techniques for the corresponding subgroup commutativity degrees. The method (I) has been followed in [24,Theorem 3.4]. The method (II) is presented here for the first time. The method (III) has been introduced in [19]. The difference is subtle between (II) and (III): for small groups we prefer of course (III), but for large groups with big K(S) in (1.18) and small H(S) (or viceversa) (II) gives soon a qualitative evaluation of sd(G). For instance, a minimal nonabelian group M is a group which is nonabelian but all of whose proper subgroups are abelian. (3.10) There are 30 elements in L(S 4 ) and these are divided into 11 conjugacy classes and 9 isomorphism types. It is easy to check that there are in L(S 4 ) -9 subgroups isomorphic to C 2 ; -4 subgroups isomorphic to C 3 ; -3 subgroups isomorphic to C 4 ; -3 subgroups isomorphic to C 2 × C 2 ; -4 subgroups isomorphic to S 3 ; -3 subgroups isomorphic to D 4 . In particular, we find that and a corresponding computation of edges can be done via [28], obtaining the graph below. and again [28] can help with the computation of the edges. See below: (3.19) On the other hand, we may use [28], in order to find the spectra of the Laplacian matrices (3.21) is the spectrum of the Laplacian matrix L(Γ L(S 4 ) ). Replacing the values which we found in (3.16), we get Note that some open problems were posed by Tarnauceanu [29] on the subgroup commutativity degree and the logic which we applied in Example 3.4, along with Theorem 1.1 and [28], could bring solutions. In fact Remarks 3.2 and 3.3 suggest a methodology of general interest which can be applied to large families of groups, so not necessarily to linear groups. We show another application of our main results. Example 3.5. From a direct computation, if we consider A 4 , then the denominator of (1.9) is equal to 100, namely \|L(A 4 )\| 2 = 100 and the numerator of (1.9) is equal to 64, hence sd(A 4 ) = 16 25 (3.25) according to [29, p.2510]. On the other hand, we may consider (3.20) and Of course, we may repeat a similar arguments in Example 3.5, in order to find sd(S 3 ), sd(S 4 ) and sd(D 4 ) on the basis of the values which we have in Example 3.4, but we presented here just the case of A 4 supporting Remark 3.3 (III) and (II). We end with the following problem, which we encountered in our investigations: Problem 3.6. Study systematically the non-permutability graph of subgroups for the groups in Theorem 2.6, developing a corresponding spectral graph theory for non-permutability graph of subgroups of groups with nontrivial partitions. Determine the subgroup commutativity degree of all the groups in Theorem 2.6 via spectra of Laplacian matrices of the corresponding non-permutability graph of subgroups. VC (Γ L(G) ) = L(G) \ C L(G) L(G) , (1.2)E(Γ L(G) ) = {(X, Y ) ∈ V (Γ L(G) ) × V (Γ L(G) ) \| X ∼ Y ⇐⇒ XY = Y X},(1.3) and C L(G) (X) is the set of all subgroups of L(G) commuting with X ∈ L(G). In other wordsC L(G) (X) = {Y ∈ L(G) \| XY = Y X}. L(G) (X) = {Y ∈ L(G) \| Y X = XY, ∀X ∈ L(G)} (1.5) H (G) = {H ∈ L(G) \| sd(H) = 1} and K(G) = {K ∈ L(G) \| sd(K) = 1} (1.10) which clearly determine a disjoint union of the form L(G) = H(G) ∪ K(G).(1.11) Lemma 2. 2 ( 2See[19], Theorem 1.2). Let G be a group with sd(G) = 1. Then sd(G) is invariant under the spectrum of A(Γ L(G) ). In particular, From Definition 2 . 4 , 24H p (G) turns out to be the smallest subgroup of G outside of which all elements of G have order p. Of course, if G has exp(G) = p, then H p (G) = 1. Moreover H p (G) is a characterstic subgroup in G. The reader can refer to [10, Chapter 7] for more information on Hughes subgroups and their role in the theory of groups with nontrivial partitions.Definition 2.5 (See [32], p.575). A group G is said to be a group of Hughes-T hompson type if it is not a p-group and H p (G) = G for some prime p. Proposition 2. 9 (F 2 92See [24], Theorem 2.5). For any p > 2 let M be the unique subgroup of G = PGL(2, p n ) isomorphic to PSL(2, p n ). If p n > 29, then n (p 2n − 1) + 4\|L(G)\| − 2\|L(M)\| − 3 if n even or p ≡ 1 (mod 4), 4p n (p 2n − 1) + 4\|L(G)\| − 2\|L(M)\| − 3, if n odd and p ≡ 3 (mod 4). Proposition 2. 11 ( 11See [26], Theorems 1.6, 1.8, 1.10). (i). Let p be a prime. Then µ(1, S p ) = (−1) p−1 p! 2 . just an application of the Möbius Inversion Formula to (2.3). F 2 ( 2G) = K∈K(G) \|L(K)\| 2 − m K i=1 σ i µ(K, G) + H∈H(G) \|L(H)\| 2 − m H i=1 σ i µ(H, G) m H = m = \|V (Γ L(H) )\| as claimed. From (2.3) and (3.4), now we consider an arbitrary S ∈ L(G) and a corresponding partition L(S) = H(S) ∪ K(S), as made for G in (1.11). We get \|L(G)\| 2 sd( ( 2 ) 2. Apply (1.17) of Theorem 1.1, but in order to do this we should previously: (a). Determine Γ L(H) and spec(L(Γ L(H) )) in (1.17); (b). Find the Möbius numbers µ(H, G) and µ(K, G) in (1.17). (c). Find \|L(H)\| and \|L(K)\| in (1.17). Remark 3. 3 . 3Suppose to compute sd(G) for G = PSL(2, p n ). We may proceed as below: (I). Combine Propositions 2.7 and 2.8 for the computation of F 2 (H) where H ∈ L(G) with the formula (2.3). (II). Apply (1.18) of Theorem 1.1, but in order to do this we should previously: (a). Determine Γ L(U ) , L(Γ L(U ) ) and spec(L(Γ L(U ) )) in (1.18); (b). Find the Möbius numbers µ(W, S) and µ(U, S) in (1.18). (c). Find \|L(U)\| and \|L(W )\| in (1.18). (III). Apply (1.13), after computing \|L(G)\| and spec(L(Γ L(G) )). In this situation, one has K(M) = L(M) \ {M} and H(M) = {M} from the definitions. Then (II) is more convenient than (III) here. Note that minimal nonabelian groups were classified by Redei [14, Aufgabe 14, Kapitel III, §5 ]. The following examples illustrate Theorem 1.1 in the spirit of Remarks 3.2 and 3.3. Example 3 . 4 . 34The symmetric group S 4 is presented by S 4 = a, b, c \| a 2 = b 3 = c 4 = abc = 1 , where a = (12), b = (123) and c = (1234). It is well known that the set of all normal subgroups forms a sublattice of the subgroups lattice of a given group (see[25]). In other words, the set N(S 4 ) of all normal subgroups of S 4 is a sublattice of L(S 4 ) and we have N(S 4 ) = {{1}, (12)(34), (13)(24) , A 4 , S 4 }. 4 ) = {{1}, (12) , (13) , (23) , (14) , (24) , (34) , (13)(24) , (14)(23) , (12)(34) , (123) , (124) , (134) , (234) , (1234) , (1324) , (1423) , (12)(34), (13)(24) , (13), (24) , (14), (23) , (12), (34) , (123), (12) , (124), (12) , (134), (13) , (234), (23) , (1234), (13) , (1243), (14) , (1324), (12) , A 4 , S 4 }. \|V (Γ L(S 4 ) )\| = \|L(S 4 ) \ N(S 4 )\| = 26.(3.11) Now we are going to focus on special subgroups of S 4 . First of all, consider A 4 and its non-permutability graph of subgroups Γ L(A 4 ) . We have 7 vertices, namely V (Γ L(A 4 ) ) = { (123) , (124) , (134) , (234) , (12)(34) , (14)(23) , (13)(24) }, (3.12) since C L(A 4 ) (L(A 4 )) = N(A 4 ) = {{1}, (12)(34), (13)(24) , A 4 } (3.13) Figure 1 : 1The non-permutability graph of subgroups Γ L(A 4 ) . Now we describe B = (123), (12) ≃ S 3 and Γ L(B) . Here we get a triangle, because V (Γ L(B) ) = L(B) \ C L(B) (L(B)) = L(B) \ N(B) = { (12) , (13) , (23) } (3.14) Figure 2 :Figure 3 : 23The non-permutability graph of subgroups Γ L(B) for B ≃ S 3 .Finally, we consider C = (1234), (13) ≃ D 4 which has Γ L(C) with four vertices and four edges, namelyV (Γ L(C) ) = L(C) \ C L(C) (L(C)) = { (13) ,(24) , (14)(23) , (12)(34) }. (3.15) Again this is another very simple situation: the graph is a The non-permutability graph of subgroups Γ L(C) for C ≃ D 4 . From Theorem 1.1, we may compute F 2 (S 4 ) in the following way: is a subgroup of S 4 belonging to K(S 4 ) = {{1}, 12 , 13 , 23 , 14 , 24 , 34 , (13)(24) , (14)(23) , (12)(34) , 123 , 124 , 134 , 234 , 1234 , 1324 , 1423 , (12)(34), (13)(24) , (13), (24) , (14), (23) , (12), (34) }, (3.17) and H a subgroup of S 4 belonging to H(S 4 ) = { (123), (12) , (124), (12) , (134), (13) , (234), (23) , (1234), (13) , (1243), (14) , (1324), (12) , A 4 , S 4 }. (3.18) Now we need to find µ(K, S 4 ) and µ(H, S 4 ) for all K and H, but it is enough to find these values for each conjugacy classes only. Using Lemma 2.10 and Proposition 2.11 (iii), we find µ({1}, S 4 ) = −n! = −24, µ( 12 , S 4 ) = 2, µ( (13)(24) , S 4 ) = 0, µ( 123 , S 4 ) = 1, µ( (12)(34), (13)(24) , S 4 ) = 3, µ( (13), (24) , S 4 ) = 0, µ( 1234 , S 4 ) = 0, µ( (123), (12) , S 4 ) = −1, µ( (1234), (13) , S 4 ) = −1, µ(A 4 , S 4 ) = −1. µ(S 4 , S 4 ) = 1. L(Γ L(B) ), L(Γ L(C) ) and L(Γ S(A 4 ) ), obtaining spec(L(Γ L(B) )) = {0, 3, 3}, spec(L(Γ L(C) )) = {0, 2, 2, 4}, spec(L(Γ L(A 4 ) )) = {0, 4, 4, 7, 7, 7, 7}, (3.20) but we haven't reported all the details of the non-permutability graph Γ L(S 4 ) , since it is very technical. Just to give an idea, spec(L(Γ L(S 4 ) )) = {0, 7. F 2 (S 4 24{1}, A 4 ) = 4, µ( (13)(24) , A 4 ) = 0, µ( (12)(34), (13)(24) , A 4 ) = −1, µ( (123) , A 4 ) = −1, µ(A 4 , A 4 ) = 1, (3.23) imply with a similar argument that new method of computation, we have just seen that Theorem 1.1 shows an alternative method of computational nature for F 2 (PGL(2, 3)) and F 2(PSL(2, 3)). In fact PSL(2, 3) ≃ A 4 and PGL(2, 3) ≃ S 4 , then F 2 (PSL(2, 3)) = F 2 (A 4 ) = 27 and F 2 (PGL(2, 3)) = F 2 (S 4 ) = 177, which are the same values found in Propositions 2.7 and 2.9. it is easy to check that A 4 is minimal nonabelian, then K(A 4 ) = L(A 4 ) \ {A 4 } and H(A 4 ) = {A 4 }. Now we can apply (1.17) to obtain F 2 ({1}) = 1, F 2 ( (13)(24)) = F 2 ( (14)(23) ) = F 2 ( (12)(34) ) = 3, F 2 ( (123) ) = F 2 ( (124) ) = F 2 ( (13) ) = F 2 ( (234) ) = 3, F 2 ( (12)(34), (13)(24) ) = 15 and F 2 (A 4 ) = 27. Therefore, using(the same value obtained in(3.25) and(3.26) in different ways. 14, Definition 8.1, Kapitel V, §8] for the notion of F robenius group, and to [14, Bemerkungen 10.15, 10.17, Kapitel II, §10] for the notion of Suzuki group Sz(2 2n+1 ). Originally, Baer, Kegel and Kontorovich Proposition 2.7 (See[24], Theorem 2.4). The projective special linear group PSL(2, p n ) has 22863, 7.60860, 7.60860, 11.39978, 11.39978, 11.72495, 12.01650, 12.01650, 14, 14.56069, 14.56069, 14.56069, 15.61486, 16.33888, 16.33888, 16.33888, 17.29890, 17.29890, 18, 20.10043, 20.10043, 20.10043, 20.43156, 20.67622, 20.67622} The subgroup permutability degree of projective special linear groups over fields of even characteristic. S Aivazidis, J. Group Theory. 165S. 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[ "S-wave π 0 Production in pp Collision in a OBE Model", "S-wave π 0 Production in pp Collision in a OBE Model" ]	[ "E Gedalin \nDepartment of Physics\nBen Gurion University\n84105Beer ShevaIsrael\n", "A Moalem \nDepartment of Physics\nBen Gurion University\n84105Beer ShevaIsrael\n", "L Rasdolskaya \nDepartment of Physics\nBen Gurion University\n84105Beer ShevaIsrael\n" ]	[ "Department of Physics\nBen Gurion University\n84105Beer ShevaIsrael", "Department of Physics\nBen Gurion University\n84105Beer ShevaIsrael", "Department of Physics\nBen Gurion University\n84105Beer ShevaIsrael" ]	[]	The total cross section for the pp → ppπ 0 reaction at energies close to threshold is calculated using a covariant one-boson-exchange model, where a boson B created on one of the incoming protons is converted into a neutral pion on the second. The amplitudes for the conversion processes, BN → N π 0 , are taken to be the sum of s, u and t-channel pole terms. The main contributions to the primary production amplitude is due to an effective isoscalar σ meson pole in a t-channel, which is enhanced strongly due to offshellness. With this contribution included the model reproduces, both the scale and energy dependence of the cross section.	10.1016/s0375-9474(99)00140-2	[ "https://export.arxiv.org/pdf/nucl-th/9803028v2.pdf" ]	118,930,610	nucl-th/9803028	0706d07f1c73859cfc08413a21bb2714786358a1	S-wave π 0 Production in pp Collision in a OBE Model E Gedalin Department of Physics Ben Gurion University 84105Beer ShevaIsrael A Moalem Department of Physics Ben Gurion University 84105Beer ShevaIsrael L Rasdolskaya Department of Physics Ben Gurion University 84105Beer ShevaIsrael S-wave π 0 Production in pp Collision in a OBE Model arXiv:nucl-th/9803028v2 14 Dec 1998π 0 ProductionCovariant OBE ModelEffective Two-pion exchanges 1375Cs1440Aq2540Ep * The total cross section for the pp → ppπ 0 reaction at energies close to threshold is calculated using a covariant one-boson-exchange model, where a boson B created on one of the incoming protons is converted into a neutral pion on the second. The amplitudes for the conversion processes, BN → N π 0 , are taken to be the sum of s, u and t-channel pole terms. The main contributions to the primary production amplitude is due to an effective isoscalar σ meson pole in a t-channel, which is enhanced strongly due to offshellness. With this contribution included the model reproduces, both the scale and energy dependence of the cross section. I. INTRODUCTION In a recent study of the pp → ppπ 0 reaction at energies close to threshold, it is found that the angular distributions of the outgoing particles are isotropic, in agreement with the assumption that the reaction proceeds as a 33 P 0 → 31 S 0 s 0 transition [1,2]. Several model calculations [3][4][5][6] of S-wave pion production, which are based on a single nucleon and a pion rescattering mechanism, under estimate the cross section by a factor of 3-5. Inspired by a study of β-decay in nuclei, which indicates that the axial charge of the nuclear system is enhanced by heavy meson exchanges and, by using a simple operator form of the NN potential, Lee and Riska [7] have shown that meson exchange currents could explain the scale of the cross section. Adding to the single nucleon and rescattering terms a Z-graph describing various heavy meson exchanges, Horowitz et al. [8] performed similar calculations based on explicit one-boson-exchange (OBE) model for the NN interaction as well as for the evaluation of meson exchange contributions. The contribution from the rescattering term depends on the off-shell behavior of the πN → πN scattering amplitude. There are several approaches based on field theoretical models which allow such extension to be made but the results are model dependent. In the traditional phenomenological treatment [3][4][5][6], the off mass-shell and on mass-shell amplitudes are of the same order of magnitude. Recently, several groups [9][10][11][12][13] have shown that the off mass-shell πN rescattering amplitude is enhanced strongly with respect to the on mass-shell amplitude. Hernandez and Oset [9] by applying current algebra and PCAC constraints argue that this enhancement may bring the calculated cross section for the pp → ppπ 0 reaction into agreement with experiment. However based on detailed momentum-space calculations, Hanhart et al. [10] conclude that the enhancement of the rescattering amplitude due to offshellness falls a bit too short to explain the scale of the pp → ppπ 0 cross section. Park et al. [11], Cohen et al. [12] and Sato et al. [13] have applied a chiral perturbation theory (χPT), including chiral order 0 and 1 Lagrangian terms. They have shown that the off mass-shell πN elastic scattering amplitude is enhanced considerably but has an opposite sign with respect to the on mass-shell amplitude. Because of this difference in sign, the rescattering term and the Born term contributing to the pp → ppπ 0 reaction interfere destructively, making the theoretical cross sections much smaller than experimental values and thus suggesting a significant role for heavy meson exchanges in π 0 production. In the present work we report on a covariant OBE model calculations based on the mechanism depicted in Fig. 1, where a virtual boson B (B=π, σ, η, ρ, ω, δ ...) created on one of the incoming nucleons, is converted into a π 0 meson on the second via a BN → π 0 N conversion process. The half off mass-shell amplitudes for the conversion processes, hereafter denoted by T BN →π 0 N , are taken as the sum of three terms corresponding to the s, u and t-channels displayed in Fig. 2. One important aspect of this mechanism is that at the π 0 production threshold, the transferred 4-momentum is space-like, q 2 = −3.3f m −2 . This is very much the same kinematic as occurring in π 0 electroproduction through vector meson exchanges, and we apply a formalism similar to the one applied for electroproduction amplitudes [14] calculations. The formalism to be applied is consistent with the OBE picture of the nuclear force and accounts for relativistic effects, crossing symmetry, energy dependence and nonlocality of the hadronic interactions. We start with the assumption that the pion production in pp → ppπ 0 can be described by the graphs of Fig. 3. The diagrams 3a and 3b represent nucleon pole terms where the neutral pion is produced on a nucleon line. The diagram 3c depicts a pion production occurring on an internal meson line. By applying s-channel unitarity to NN elastic scattering in the energy region below two-pion production threshold, it can be shown that mechanisms as such should contribute to the production process also [15]. Schütz et al. [16] reported on σ and ρ meson t pole contributions to πN scattering. Their model was based on correlated 2π exchange and constrained by using quasiempirical information about the NN → ππ amplitudes. Hanhart et al. [10] used the T-matrix obtained in this model to calculate the rescattering term for pp → ppπ 0 . More recently, van Kolck et al. [17] considered other short range meson exchange mechanisms like ρ → ωπ and δ → ηπ 0 . To keep our model calculations consistent with the OBE picture of Machleidt [19] we limit our discussions to contributions from σ, δ and ρ poles. In variance with previous calculations [16,17] we write these vertices in a more general form and determine the relevant coupling constants by applying the Adler's consistency conditions and/or fitting to some observables. We shall demonstrate below that a σ-meson t pole term which accounts ef f ectively for isoscalar-scalar two-pion exchanges contributes significantly to π 0 production. With this contribution included the model explains the cross-section data for the pp → ppπ 0 reaction at threshold. The present article is organized as follows. In Sect. II we present details of the formalism and model parameters. The evaluation of the T π 0 N →π 0 N , T ηp→π 0 p and T ρp→π 0 p conversion amplitudes require knowledge of the σππ, δηπ and ρωπ vertices. These are deduced in subsections B-D. In Sect. III we write the S-wave production amplitude in a form suitable for numerical calculations. Model predictions are given in Sect. IV. At energies close to threshold, final state interactions (FSI) influences the energy dependence of the calculated cross section [20]. These and initial state interactions (ISI) corrections are introduced in Sect. V where comparison with data is to be made. We conclude and summarize in Sect. VI. II. THE MODEL We use the following Lagrangian interaction: L = f πN N m πN γ 5 γ µ ∂ µ πτ N + f ηN N m ηN γ 5 γ µ ∂ µ ηN + g σN NN σN + g ρN NN γ µ + κ V 2M σ µν ∂ ν τ ρ µ N + g ωN NN γ µ ω µ N + g δN NN τ δN , (2.1) with obvious notations. This expression includes terms identical to the ones used by Machleidt et al. [19] to fit NN elastic scattering data in the energy region 0-420 MeV. Here as in Ref. [19] pseudovector couplings are assumed for the pseudoscalar mesons and the ω tensor coupling is taken to be zero. All of the coupling constants, meson masses and cut off parameters are taken from Table A.2 of Ref. [19]; their potential C parameter set. To calculate the transition amplitude we assume that the reaction is dominated by the mechanism depicted in Fig. 1 and write the primary production amplitude as, M (in) = B [T BN →π 0 N (p 4 , k; p 2 , q)G B (q)S BN N (p 3 , p 1 )] + [1 ↔ 2; 3 ↔ 4] . (2.2) Here p i , q and k are 4-momenta of the i-th nucleon, the exchanged boson and the outgoing pion. The sum runs over all possible B bosons that may contribute to the process. To be consistent with the OBE picture of the NN force [19], we include exchange contributions from all of the π, η, σ, ρ, ω and δ mesons. The bracket [1 ↔ 2; 3 ↔ 4] stands for a similar sum with the p 1 , p 3 and p 2 , p 4 momenta interchanged. In Eqn. 2.2 , T BN →π 0 N represents the conversion amplitude corresponding to the BN → π 0 N process. The quantities G B (q) and S BN N (p 3 , p 1 ) are the propagator and source function of the meson exchanged, respectively. Throughout this work we use covariant expressions for the meson propagators and form factors as defined in Ref. [19]. The source functions for scalar, pseudoscalar and vector mesons are S SN N (p 1 , p 3 ) =ū(p 3 ) I u(p 1 ) F S (q) , (2.3) S P N N (p 1 , p 3 ) =ū(p 3 ) γ 5 I u(p 1 ) F P (q) , (2.4) S µ V N N (p 1 , p 3 ) =ū(p 3 ) γ µ F (1) V (q 2 13 ) + iσ µν q ν F (2) V (q 2 13 ) Iu(p 1 ) ,(2.5) where u(p) stands for a nucleon Dirac spinor; I is the appropriate isospin operator and p 3 = p 1 − q is the final nucleon momentum. The functions F S (q) and F P (q) are source form factors for scalar and pseudoscalar mesons. For vector mesons there are two such quantities F representing vector and tensor form factors, the analogous of the nucleon electromagnetic form factors. In the calculations to be presented below all source form factors are taken in the form [19], F B (q 2 ) = g BN N f B (q); f B (q) = Λ 2 B − m 2 B Λ 2 B − q 2 . (2.6) Albeit, the propagators and source functions are rather well determined from fitting NN scattering data [19]. Thus to a large extent, the model success in explaining cross section data for the pp → ppπ 0 reaction depends on how well the conversion amplitudes T BN →π 0 N are calculated. The relative importance of the various exchange contributions depend upon the off mass shell behavior of these amplitudes. For example, though very small on mass-shell, the amplitude for π 0 p → π 0 p process is strongly enhanced off mass-shell [9,10] giving rise to a dominant contribution to the production rate. A. Nucleon pole s and u channel amplitudes In this subsection we write expressions for s and u-channel nucleon pole terms (diagrams 2a and 2b). These are common to all of the various meson exchanges. We call p and p' the momenta of incoming and outgoing nucleon, q and k those of the boson B and the pion produced, respectively. Let s = (p ′ + k) 2 be the total energy and ∆ N (x) = i x 2 − M 2 + iǫ . (2.7) Using the vertices of the Lagrangian, Eqn. 1, and the usual Feynman rules the contributions from diagram 2a with B designating a scalar (S), a pseudoscalar (P) and a vector meson (V) are respectively, T (s) S = − g SN N f πN N m π f S (q)f π (k)ū(p ′ )γ 5 k / p ′ / + k / + M (p ′ + k) 2 − M 2 u(p) , (2.8) T (s) P = i f P N N f πN N m P m π f P (q)f π (k)ū(p ′ )γ 5 k / p ′ / + k / + M (p ′ + k) 2 − M 2 γ 5 q /u(p) , (2.9) T (s) V = − g V N N f πN N m π f V (q)f π (k)ū(p ′ )γ 5 k / p ′ / + k / + M (p ′ + k) 2 − M 2 γ µ + κ V 2M σ µν q ν u(p) ,(2.10) where a / = a µ q µ . After some algebra one obtains, T (s) S = i g SN N f πN N m π f S (q)f π (k)ū(p ′ )γ 5 (s − M 2 ) + 2M k / u(p)∆ N (s) , (2.11) T (s) P = (2M ) 2 f P N N f πN N m P m π f P (q)f π (k) u(p ′ ) − s − M 2 2M + k / s − M 2 + 4M 2 4M 2 ∆ N (s)u(p), (2.12) T (s) V = i g V N N f πN N m π ∆ N (s)f V (q)f π (k)ū(p ′ )γ 5 γ µ (s − M 2 ) + 4k µ − 2M γ µ k / 1 + κ V 2M q / u(p) . (2.13) The contributions from the u-channel (diagram 2b) are obtained easily by replacing the total energy s in Eqns. 11-13 with u = (p − k) 2 . Likewise, for a vector meson the u-channel pole term is T (u) V = −i g V N N f πN N m π ∆ N (s)f V (q)f π (k)ū(p ′ )γ 5 γ µ 1 − κ V 2M q / u − M 2 + 2M k / u(p) . (2.14) B. Evaluation of T π 0 N→π 0 N We now turn to calculate the amplitude T π 0 N →π 0 N . In this case in addition to s and u-channel nucleon pole terms, a neutral pion can be formed on an internal σ meson line. These terms are displayed graphically in Fig. 4. The contribution from graph 4c depends upon the σππ meson vertex. There is no information about this coupling from traditional analyses of NN elastic scattering [19]. In order to determine this vertex we apply some physics beyond the OBE picture of the NN interactions. Quite generally we write an effective σππ vertex in the form, V σππ (k, q) = m σ g 0σ + q 2 + k 2 m 2 σ g 1σ + (q − k) 2 m 2 σ g 2σ ,(2.15) where g 0σ , g 1σ , g 2σ are (as yet unknown) constants. This is a slightly generalized Weinberg-type low energy expansion [21] allowing for three rather than one different constants. Taking the appropriate s and u-channel contribution from Eqn. 12 and adding the contribution from a σ pole in a t-channel one obtains, T π 0 p→π 0 p = i 2M f πN N m π 2 f π (q)f π (k)ū(p ′ ) k / 1 s − M 2 − 1 u − M 2 − 1 M u(p) + ig σN N f σ (q − k) (q − k) 2 − m 2 σ V σππū (p ′ )u(p) . (2.16) Now, in order to determine the vertex constants g 0σ , g 1σ and g 2σ we require that the conversion amplitude obeys the three Adler's consistency conditions [22,23]. These are T π 0 p→π 0 p (k = q = 0) = −i σ πN (0) F 2 π , (2.17) T π 0 p→π 0 p (k = 0, q 2 = m 2 π ) = 0 , (2.18) T π 0 p→π 0 p (k 2 = m 2 π , q = 0) = 0 , (2.19) where σ πN (0) is the well known πN σ-term [24] and F π the pion radiative decay constant. These three conditions yield only two relations amongst the constants, which are g 0σ = σ πN (0)m σ F 2 π g σN N f σ (0) , (2.20) g 0σ = m π m σ 2 (g 1σ + g 2σ ) . (2.21) Furthermore, we recall that the on mass shell π 0 p S-wave elastic scattering amplitude is parameterized according to [25,26], F = a + + b + k · k , (2.22) with a + being the isospin even π 0 p scattering length. To obtain a third relation we may require that the amplitude Eqn. 17 reproduces F at k = 0, i.e., T π 0 p→π 0 p (k 2 = m 2 π , k = 0; q 2 = m 2 π , q = 0) = i4π 1 + m π M a + . (2.23) This yields, g 0σ + 2( m π m σ ) 2 g 1σ = −1 g σN N 4π 1 + m π M m σ a + − f 2 πN N m σ M 4M 2 4M 2 − m 2 π . (2.24) We now may use σ πN (0), F π , a + and g σN N as input to evaluate the vertex constants. The value of the quantity σ πN (0) is related to matrix elements of the operator m q qq in the proton, where m q stands for the mass of the proton quark constituents. This quantity can be calculated from the baryon spectrum [27]. To leading orders in the quark masses one finds to order O (m with y being a measure of the strange quark content in the proton. An even higher a value is obtained from the isospin even πN scattering amplitude at the Cheng-Dashen point, which can be determined experimentally using the low-energy theorem of current algebra. This gives [27] a value σ πN (0) = 45 ± 8 M eV . Equation 2.25 suggests a value σ πN (0) = 35 M eV for y = 0, in keeping with the OZI rule [28]. With this value and taking a radiative decay constant F π = 93.5 MeV [29]; isospin even π 0 p scattering length a + = (−0.010 ± 0.003)m −1 π [25,26], and the σNN coupling g 2 σN N /4π = 8.03 [19] we can now solve Eqns. 2.20, 2.21, 2.24 to extract the values, g 0σ = 0.22 ± 0.04, g 1σ = −0.82 ± 0.42, g 2σ = −2.62 ± 0.7 . (2.26) With these constants Eqns. [25]. Thus the amplitude Eqn. 2.16 on the mass shell reproduces the correct k · k dependence of the π 0 p elastic scattering amplitude, Eqn. 2.22, also. 2.16 predicts b + = (−0.068 ± 0.015)m −3 π in close agreement with the experimental value b + exp = −(0.088 ± 0.014)m −3 π quoted by Koch It is difficult to ascertain that the amplitude Eqn. 2.16 does reproduce the off mass shell behavior correctly. However we may confront Eqn. 2.16 with predictions from other approaches. For example, taking the residue of the amplitude, Eqn. 2.16, at the σ pole (q − k) 2 = m 2 σ , one obtains an effective σππ coupling g ef f σππ = V σππ (q 2 = m 2 π , k 2 = m 2 π , (q − k) 2 = m 2 σ ) = (2.5 ± 0.9)m σ ,(2.27) a value to be compared with the well known estimate from soft pion physics [22] V σππ (m 2 π , m 2 π , m 2 σ ) = m σ m σ 2F π 1 − m 2 π m 2 σ = (2.8)m σ . (2.28) Two-pion loops contribute to the ππ elastic scattering amplitude and appear as corrections to the leading contact term [22,30]. As a further check we may evaluate these corrections in terms of an effective σ-meson exchange in s, u and t-channels. By doing so it is straightforward to show that these corrections are T loop ππ→ππ = 2 m 2 σ − s V 2 σππ (m 2 π , m 2 π , s)δ ab δ cd + 2 m 2 σ − u V 2 σππ (m 2 π , m 2 π , u)δ ad δ bc + 2 m 2 σ − t V 2 σππ (m 2 π , m 2 π , t)δ ac δ bd , (2.29) where δ i,j is the Kroneker δ and with i, j being pion isospin indices. The factor of 2 is due to the symmetry of the two σππ vertices. At threshold this expression amounts to about 20% of the contact term, in good agreement with the value of 25% reported by Gasser [30] from improved low energy theorems. Now that the σππ vertex parameters are well defined we may examine offshellness effects. To this aim we evaluate T π 0 p→π 0 p at the production threshold of pp → ppπ 0 and compare with its on mass shell value. When both pion legs are on the mass shell the last term in Eqn.2.16 reduces to, T (t) π 0 p→π 0 p (k 2 = m 2 π ; q 2 = m 2 π ) = −i g σN N m σ g 0σ + 2 m π m σ 2 g 1σ f σ (0) ≈ −i0.11 g σN N m σ . (2.30) At threshold of the pp → ppπ 0 reaction, the momentum squared of the σ meson is (q − k) 2 = (p ′ − p) 2 ≈ −M m π and the off mass shell t pole term in Eqn. 2.16 becomes, T (t) π 0 p→π 0 p = −ig σN N m σ m 2 σ + M m π f σ (−M m π ) g 0σ + m π m σ 2 g 1σ − M m π m 2 σ (g 1σ + g 2σ ) ≈ −i1.0 g σN N m σ , (2.31) which is a factor ≈ 9 larger compared to the on mass shell value Eqn. 2.30. In Fig. 5 the amplitude T π 0 p→π 0 p is drawn as a solid line vs. q 2 . The contributions from the s and u channel nucleon pole and σ-meson pole terms are drawn as dashed and dot-dashed curves, respectively. Off mass shell behavior of the s and u terms in Eqn. 2.16 is given by the pion form factor, f π (q), the nucleon propagators and the momentum dependence of the πN N vertex. That of the t-channel is affected by the σ meson form factor f σ (q − k) and the V σππ vertex function, Eqn. 2.15. Note that on mass shell both of these contributions are small, opposite in signs and cancel to large extent. As q 2 and (q − k) 2 become more negative both terms become negative and therefore add constructively to the conversion amplitude, giving rise to strongly enhanced off mass shell amplitude. We expect then that π exchange plays an important role in the π 0 production process. C. The T ηp→π 0 p amplitude The production of a neutral pion can occur also on an internal δ-meson line. To evaluate the amplitude for the conversion process ηp → π 0 p we apply a similar procedure as above. The η meson couples to the nucleon isobar N * (1535 M eV ) strongly [31] and the main contribution to the conversion amplitude for ηp → π 0 p is due to s and u isobar pole terms (see Fig. 6), hereafter we refer to as the resonance contribution. The other terms (graphs 6c-6e) furnish a background term. A t-channel would involve a vertex with δ, η and π legs defined to be, V δηπ (k, q) = m δ g 0δ + k 2 m 2 δ g 1δ + q 2 m 2 δ g 2δ + (k − q) 2 m 2 δ g 3δ . (2.32) Here, to account for the fact that the three legs are different the vertex is described in terms of four constants . Taking the sum of all the graphs in Fig. 6, we may write, T ηp→π 0 p = −ig ηN N * g πN N * f η (q)f π (k)ū(p ′ ) 1 M R − √ s + iΓ/2 + 1 M R − √ u + iΓ/2 u(p) i 2M f πN N m π 2M f ηN N m η f η (q)f π (k)ū(p ′ ) k / 1 s − M 2 − 1 u − M 2 − 1 M u(p) + ig δN N f δ (q − k) (q − k) 2 − m 2 δ V δηπ (k, q)ū(p ′ )u(p) . (2.33) Here M R = 1535 M eV and Γ = 175 M eV stand for the mass and width of the isobar resonance, and the coupling constants are g ηN N * = 2.2, g πN N * = 0.8 [31]. We now apply the Adler's consistency conditions to the background term in Eqn. 33 to write the following relations amongst the unknown δηπ vertex constants (see Eqns. [17][18][19], g 0δ = m δ g δN N f δ (0) σ πN →ηN (0) F π F η , (2.34) g 0δ = − m η m δ 2 (g 2δ + g 3δ ) , (2.35) g 0δ = − m π m δ 2 (g 1δ + g 3δ ) ,(2.36) where F η is the η radiative constant; in the limit of an exact SU(3) symmetry F η = F π . Likewise, the quantity σ πN →ηN is a σ-term, a quantity related to matrix elements of various quark mass qq in the nucleon, Now the partial decay width of the δ meson into a πη pair is [29] Γ πη = 57 ± 11 M eV . By calculating this width using the expression Eqn. 32, one obtains a fourth relation amongst the δηπ vertex constants, With these constants one finds that at threshold for the pp → ppπ 0 reaction, the conversion amplitude is T ηp→π 0 p ≈ 0.98 f m. The t-pole contribution amounts to only T (t) ηp→π 0 p ≈ 0.04 f m We may thus conclude that a δ meson pole in a t-channel contributes very little to T ηp→π 0 p and should play a minor role in the π 0 production process. σ πN →ηN (t) =m 2 p ′ \|ūu +dd\|p ,(2.g 0δ + m 2 π m 2 δ g 1δ + m 2 η m 2 δ g 2δ + g 3δ =Γ8π D. The T ρp→π 0 p amplitude We follow van Kolck et al. [17] and limit the ρωπ vertex to the form, V ρωπ (k, q) = − g ρωπ m ω ǫ µνλδ q µ k δ ρ ν ω λ π ,(2.42) where the coupling constant g ρωπ ≈ −10, a value fairly well established. With this expression, the contribution from a ρω exchange mechanism to the T ρN →π 0 N is T (t) ρN →π 0 N = ig ρωπ g ωN N f ω (q − k) k 0 m ω 1 (q − k) 2 − m 2 ω (2.43) 1 E + M (q × p + ipq · σ − iσq · p) − 1 E ′ + M (q × p ′ + ip ′ q · σ − iσq · p ′ ) . The s and u nucleon pole terms can be calculated from Eqns. 10 and 14. III. S-WAVE AMPLITUDES AND CROSS SECTION In this section we write the primary production amplitude in a form suitable for numerical integration. We call Π j = p j E j + M (3.1) where p j and E j are the three momentum and total energy of the j-th nucleon. For the incoming particles in the center of mass system (CM) Π 1 = −Π 2 = Π. The energy available in the CM system is Q = √ s − 2M − m π ,(3.2) where s = (p 1 + p 2 ) 2 is the total energy squared. We shall calculate the amplitudes and cross sections as functions of the variable Q. We also define q 2 = −M (m π + Q) , a = 1 + Q m π 1 + m π M , b = 1 + Q 2M 1 + m π 2M + m π Q 4M 2 , (3.3) R = 1 M R − M − m π − Q + iΓ/2 + 1 M R − M + m π + Q + iΓ/2 . (3.4) Following the discussion in the previous section we write the primary amplitude as M (in) = M π + M η + M σ + M ρ + M ω + M δ ,(3.5) where M B stands for a partial production amplitude representing the contribution from the exchange of a boson B. We substitute equivalent two-component free spinor matrix elements (Table A6.1 of Ref. [26]) in Eqns. 2.11-2.13 for s-channels and the appropriate ones for u-channels, to write the scalar and vector meson exchange amplitudes as, M S = −i f πN N m π g 2 SN N m 2 S − q 2 f 2 S (q) m π + 2Q M + Q M 2M + m π Π · (σ 1 − σ 2 ) ,(3.6) and M V = i f πN N m π g 2 V N N m 2 V − q 2 f 2 V (q) [X Π · (σ 1 − σ 2 ) + iY Π · σ 1 × σ 2 ] (3.7) where we have used the notation X = 2 1 + 3m π 2M + κ V 1 + m π + Q 2M 1 − m π + Q 2(2M + m π ) (1 − κ V Π · Π) − 3m π + Q M + κ V m π + Q 2M 1 − m π 2(2M + m π ) (1 + κ V Π · Π) (3.8) Y = −2 − 2(1 + κ V ) 3m π + Q M + κ V m π + Q 2M 1 − m π 2(2M + m π . (3.9) The ρω exchange mechanism contribute to the production amplitude a term, M (t) ρ = 2 g ρN N m 2 ρ − q 2 f 2 ρ (q)2 g ωN N m 2 ω − q 2 f 2 ω (q)g ρωπ m π m ω (p · Π)Π · σ 1 × σ 2 . (3.10) To avoid double counting we add this to M ρ (but not to M ω ) only. For both of the π and η exchange amplitudes, there is a t-channel pole in addition to s and u-channel nucleon pole terms. The sum of all three gives, M π = −if πN N 2M m π 1 m 2 π − q 2 f π (q) f 2 πN N 1 m π a b f π (q) + g σN N 1 m 2 σ − q 2 f σ (q) V σππ (k 2 = m 2 π ; q 2 ) , (3.11) M η = −if πN N 2M m η 1 m 2 η − q 2 f η (q) g ηN N * g πN N * R + f πN N f ηN N f η (q) m η a b + g δN N f δ (q) m 2 δ − q 2 V δηπ (k 2 = m 2 π ; q 2 ) . (3.12) Finally, the total cross section is calculated from the expression, σ = M 4 16(2π) 5 (s)p 1 d 3 p 3 E 3 d 3 p 4 E 4 d 3 p η E η \| ZM (in) \| 2 δ 4 (p i − p f ), (3.13) where Z is the three-body FSI correction factor of Refs. [20] to be specified below. IV. PREDICTIONS AND COMPARISON WITH DATA We apply now the model presented in the previous sections to calculate the total cross section for the pp → ppπ 0 reaction at energies close to threshold. We first consider the relative importance of the various exchange contributions. To this aim we draw in Fig. 7 ρ ≈ 4.5% of M (in) ). In comparison with M π other contributions are significantly smaller. Nonetheless, they influence the cross section strongly through interference. They all have a common relative phase and add constructively. The ρ, having an opposite sign counteracts to balance their effects. The ratios quoted above differ considerably from those predicted by Horowitz et al. [8]. Particularly, the s and u-channel nucleon pole terms for scalar meson exchanges have opposite signs, thus suppressing the σ exchange contribution and practically eliminating that from the δ meson. We note however, that as in Ref. [8] the nucleon pole term contributions from π and σ exchanges add constructively. We want to emphasize at this stage that, our transition operator M (in) for the production process, involves contributions from connected diagrams (Fig. 3) only, and accounts for all relativistic and crossing symmetry effects. Taking u-channel contributions only, one obtains far more important effects than with both of the s and u-channels. The lesson to be learned here is that many small terms added up coherently can explain a seemingly large discrepancy, at a time when other small terms which counteract to balance their effects are disregarded. Under these circumstances it seems essential to preserve crossing symmetry at all stages of the calculations and treat all contributions on an equal basis. It is rather surprising that the σ contribution is not as important as predicted in Ref. [8]. Yet, it is to be indicated that in Ref. [8], only contributions from negative-energy intermediate states are included explicitly to the pion production operator via a Z-graph, while contributions from positive-energy nucleon intermediate state are presumably contained in the distorted waves describing the initial and final two-nucleon states. Thus, different approximations are used in the calculations of the direct and Z-graph from either s or u-channels and it is not clear how a delicate balance between these contributions is maintained throughout the calculations. In Fig. 8 we draw predictions for the total pp → ppπ 0 cross section as obtained for three different value of the πN σ-term. The small dashed curve shows results obtained without the σ-meson pole term included. Clearly, the overall contribution from nucleon pole terms only does not provide the enhancement required to resolve discrepancy between the calculated cross section and data. Albeit, the contribution from an isoscalar-scalar σ-meson pole term dominates the production cross section and as shall demonstrate below, with a σ πN (0) = 35 M eV , the σ pole term provides the enhancement required to explain cross section data near threshold. The strong dependence of our predictions on the πN σ-term deserve a comment. Though still not very well known, it seems more likely that the value of this quantity falls in the range σ πN (0) = 35 − 45 M eV [27]. In the calculations presented in the present work we have not included contributions from ∆(1232 M eV ) isobar excitations. It remains still to be verified whether the effects of such term would bring the calculated cross section to agree with data even with a larger value of σ πN (0). V. FSI AND COMPARISON WITH DATA In comparison with data, the predicted cross section in Fig. 8 varies very fast with energy due to phase space factor. The primary production amplitude (see Fig. 7) is practically constant near threshold and therefore contributes very little to the energy dependence. To cure this deficiency of our predictions and allow for comparison with data to be made we must account for ISI and FSI effects. In what follows we treat ISI and FSI using two different approximations, which as we shall demonstrate below both yield similar cross sections. ApproximationI In this approximation the transition operator M (in) is treated as an effective operator acting on nucleon wave functions. These are calculated using a phenomenological NN potential. In this approximation we neglect interactions of the π 0 produced with the outgoing nucleons. This is a standard and usual procedure in the literature [3,13], reducing a three-body problem into effectively a two-body process. To be consistent with the OBE picture applied in the present work, we use the OBE NN potential of Machleidt [19]; potential parameter set C of his table A.2. The initial and final two-nucleon radial wave functions are calculated in momentum space from half off shell R matrix, following exactly the procedure of Ref. [19]. The evaluation of the transition amplitude and cross section is performed as in Ref. [13]. We have verified that the calculated wave functions reproduce well the known experimental phase shifts [33]. ApproximationII Here the S-wave production amplitude is assumed to factorize into a primary production amplitude M (in) , a P-wave ISI factor and and an S-wave FSI factor [20], T 2→3 = Ψ (+)(3) el,f \|M (in) 2→3 \|Ψ (−)(2) el,i ≈ Z 33 M (in) 2→3 X 22 , (5.1) where Ψ (−) el,i (2) and Ψ (+) el,f (3) represent the initial two-body and final three-body wave functions. The ISI correction factor is taken to be X 22 ≈ \|1 + sin(δ3 P0 ) exp i(δ3 P0 )\|. Here δ3 P0 is the pp P-wave phase shift. At threshold of the pp → ppπ 0 reaction, δ3 P0 ≈ −10 O so tha ISI corrections to the cross section are bound to ≈ ±30%. For a three body process as in our case, the FSI correction factor is identified [20] with the elastic scattering amplitude (on mass shell) for the process πN N → πN N (three particles in to three particles out) and has the structure of the Faddeev decomposition of the t-matrix for 3 → 3 transition. An important property of this approximation is that the different two body interactions among the out going particles contribute coherently. Although the meson-nucleon interactions are weak with respect to the NN interaction, they can still be influential through interference. This approximation was discussed in length elsewhere [20] and we skip further details here. In the analysis presented below the FSI factor is estimated from πN and NN elastic S-wave scattering phase shifts [34,35]. The S-wave NN phase shift is obtained from the effective range expansion which includes Coulomb interaction between the two protons. We have used the scattering length a pp = −7.82 fm and an effective range r pp = 2.7 fm of Ref. [34]. The S11 and S13 πN scattering lengths are taken to be a 1 = 0.173 m −1 π and a 3 = −0.101 m −1 π , respectively [35]. To compare the results from the two approximations we draw in Fig. 9 the partial cross sections corresponding to production via a σ meson t pole mechanism. For neutral pion production, the cross sections calculated using the two approximations are practically identical. We note though that Approximation I does not account for final meson-nucleon interactions. To see how influential the πN FSI can be, we draw in Fig. 10 the cross section corrected for FSI interactions assuming pure I= 1 2 and I= 3 2 interactions for the outgoing π 0 N pairs. The energy dependence of the cross section differ significantly for the two channels. But, when the πN interactions are taken in the proper isospin combination, their overall contribution to the FSI factor almost cancels out, leading to an energy dependence practically identical with the one obtained with charged pp FSI (small dash curve) only. We want to stress here that this is an accidental consequence of the fact that the π 0 p interaction over the allowed I=1/2 and I=3/2 isospin channels averages to zero, making contributions from diagrams with π 0 p interactions ineffective. This may not be the case for charged pions and η meson production [20]. VI. SUMMARY AND CONCLUSIONS We have calculated S-wave pion production in pp → ppπ 0 using a covariant OBE model, where a boson B created on one of the incoming protons is converted into a neutral pion on the second. The amplitude for the conversion process BN → π 0 N are taken to be the sum of s and u pole terms and when allowed a meson pole term in a t-channel. To be consistent with the OBE picture of the NN interaction we have considered contributions from all of the π, η, σ, ρ, ω and δ mesons. Both the scale and energy dependence of the cross section are reproduced rather well. Based on a covariant quantum field theory, the transition operator must be fully covariant and as such it includes contributions from irreducible connected diagrams only, each involving large momentum transfer. The operator M (in) , Eqn. 48, and likewise the calculated amplitude preserve crossing symmetry and relativistic covariance throughout the calculations. Furthermore, terms which involve three meson vertices are written in a general form. As an example, the Lagrangian corresponding to a σππ vertex, in addition to non derivative terms, includes derivative σ(∂π∂π) and (∂σ∂π)π interaction terms. The off-shell properties of the various partial production amplitudes are taken into account explicitly. For vector and scalar mesons, offshellness is contained in the nucleon form factor and has a marginal influence. A pseudovector coupling for the pseudoscalar particles, as assumed in this work, introduces an extra factor of q to the amplitude. Consequently for the π and η, each of the s and u nucleon pole terms at q 2 = −3.3f m −2 becomes a factor of ≈ 7 higher in comparison with its value on mass shell. To account for the off mass shell behavior of the σ-meson pole term, we have used Adler's consistency conditions and the isospin even π 0 p scattering length to write the σππ vertex in a rather general form. Taking all of the s, u and t-channel contributions into account, we have found that the π 0 p → π 0 p conversion amplitude off the mass shell is more than a factor ≈ 15 higher in comparison with its on mass shell value. Consequently, the π exchange contribution to the amplitude for the pp → ppπ 0 reaction dominates the production process and brings the calculated cross section to agree with data. We have found that the contribution from a σ-meson pole in a t-channel dominates pion production at threshold. A term as such accounts effectively for two-pion exchange contributions. The σππ vertex used to calculate this contribution depends rather strongly on the πN σ-term. However, even if we use a value, σ πN (0) = 25 M eV , which is unrealistically too low, a t-channel σ pole term amounts to more than a factor of two higher a contribution than that of the s and u-channel nucleon poles. It seems quite impossible to disregard such an important contribution to the production process. Neither the overall contribution from all meson exchanges, nor any of the various meson exchange contributions, from s and u nucleon pole terms only, can reproduce data for this process. In fact, various contributions of a similar size are found to interfere destructively and including all of these in a consistent manner, yields a production amplitude far below what would be required to explain data. This conclusions seems unavoidable and independent on the approximation used to account for ISI and FSI corrections. A meson production in NN collisions necessarily involves large momentum transfer. Thus, two-pion exchanges which describe short range interactions are expected to play an important role. In a traditional covariant OBE model such contributions are represented by an effective scalar σ-meson pole term. In χPT calculations these should result from expanding the existing calculations to include one loop two-pion diagrams. Recently, we have extended the existing χPT calculations [11][12][13] taking into account all tree and loop diagrams up to chiral order D=2. With these one loop contributions added, it is found that in analogy with a dominant σ meson pole term, there are substantial contributions from isoscalar-scalar two-pion t-channel exchanges [36]. In summary, two-pion exchanges play an important role in the production of π 0 in pp → π 0 pp. By taking these into account, we have demonstrated that a covariant OBE model can explain the existing near threshold neutral pion production data . Integrated energy cross sections calculated with the assumption that the interacting πN pair is scattered in isospin I= 1 2 (large dashed curve) and I = 3 2 (dot-dashed curve). The solid line is that obtained with the π N interactions taken in the appropriate isospin combinations. Predictions which account for the pp FSI only (small dashed curve) are nearly identical with the solid curve. All predictions are calculated with σπN (0) = 35 MeV. The data points are taken from Refs. [1,2] 37) wherem = (m u + m d )/2 is the average of the u and d quark masses and t = (p ′ − p) 2 the transferred momentum squared. This term can be deduced from the kaon-nucleon (KN) σ-terms and the strange quark mass through [24] σ πN →ηN (quark masses as m u = (5 ± 2) MeV, m d = (9 ± 3) MeV, m s = (175 ± 55) MeV, and the KN σ-term [24] σ(1) KN (0) ≃ (200 ± 50) MeV , σ-term σ(2) KN (0) ≃ (140 ± 40) MeV , one obtains, σ πN →ηN (0) ≃ (2 ± 1.3)M eV .(2.39) the primary production amplitude M (in) along with the partial exchange amplitudes of Eqns. 46-51, vs. the energy available in the center of mass (CM) system. The main contribution is due to pion exchange with ratios M π : M ρ : M ω : M σ : M η : M δ ≈ 138 : 42 : 8 : 6 : 5 : 0.6. Next important to the pion is contribution from the ρ meson (M FIG. 1 .FIG. 2 .FIG. 3 .FIG. 4 . 1234The primary production mechanism for the N N → N N π 0 reaction. In this figure and following figures, solid lines represent nucleons and broken lines mesons. Graphs contributing to the amplitude of the conversion process BN → π 0 N . Graphs a and b stand for nucleon pole in s and u-channel; the graph c represents a B' meson pole in a t-channel. The production amplitude. The mechanisms of graphs a and b represent pion production on external nucleon lines. Graph c represents production on an internal meson line. From parity and isospin conservations the latter mechanism is limited to production from σ and δ meson lines only. Terms contributing to the π 0 p → π 0 p conversion amplitude. Graph c describes an effective isoscalar σ-meson pole in a t-channel. FIG. 5 .FIG. 6 .FIG. 7 . 567The off mass shell behavior of the T π 0 p→π 0 p amplitude. The contribution from nucleon s and u-channel pole terms and that from a σ-meson pole are drawn as dashed and dot-dashed curves, respectively. The crosses point threshold of the pp → ppπ 0 Terms contributing to the conversion amplitude for the ηp → π 0 p. Because of the strong coupling between the η-meson and the N * (1535 MeV) nucleon isobar one should allows for contributions from nucleon and nucleon isobar intermediate states. Graph e describes δ-meson pole in a t-channel. The primary production amplitude for the pp → ppπ 0 reaction vs. Q, the energy available in the overall center of mass system. Different exchange contributions to M (in) are drawn separately. FIG. 8 . 8Predictions for the total cross section of the pp → ppπ 0 reaction for different values of σπN (0). Predictions with s and u nucleon pole terms only are shown by the dotted curve. All curves are not corrected for FSI. FIG. 9 .FIG. 10 . 910ISI and FSI corrections. Energy integrated cross sections assuming production via a σ meson t pole mechanism only with Approximation I (solid line) and Approximation II (dot-dashed line) FSI corrections. Acknowledgments This work was supported in part by the Israel Ministry Of Absorption. We are indebted to Z. 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[ "Synthetic Gauge Fields for Lattices with Multi-Orbital Unit Cells: Routes towards a π-flux Dice Lattice with Flat Bands", "Synthetic Gauge Fields for Lattices with Multi-Orbital Unit Cells: Routes towards a π-flux Dice Lattice with Flat Bands" ]	[ "Gunnar Möller †[email protected] \nSchool of Physical Sciences\nFunctional Materials Group\nUniversity of Kent\nCT2 7NZCanterburyUnited Kingdom\n", "Nigel R Cooper \nTCM Group, Cavendish Laboratory\nUniversity of Cambridge\nCB3 0HECambridgeUnited Kingdom\n" ]	[ "School of Physical Sciences\nFunctional Materials Group\nUniversity of Kent\nCT2 7NZCanterburyUnited Kingdom", "TCM Group, Cavendish Laboratory\nUniversity of Cambridge\nCB3 0HECambridgeUnited Kingdom" ]	[]	We propose a general strategy for generating synthetic magnetic fields in complex lattices with non-trivial connectivity based on light-matter coupling in cold atomic gases. Our approach starts from an underlying optical flux lattice in which a synthetic magnetic field is generated by coupling several internal states. Starting from a high symmetry optical flux lattice, we superpose a scalar potential with a super-or sublattice period in order to eliminate links between the original lattice sites. As an alternative to changing connectivity, the approach can also be used to create or remove lattice sites from the underlying parent lattice. To demonstrate our concept, we consider the dice lattice geometry as an explicit example, and construct a dice lattice with a flux density of half a flux quantum per plaquette, providing a pathway to flat bands with a large band gap. While the intuition for our proposal stems from the analysis of deep optical lattices, we demonstrate that the approach is robust even for shallow optical flux lattices far from the tight-binding limit. We also provide an alternative experimental proposal to realise a synthetic gauge field in a fully frustrated dice lattice based on laser-induced hoppings along individual bonds of the lattice, again involving a superlattice potential. In this approach, atoms with a longlived excited state are trapped using an 'anti-magic' wavelength of light, allowing the desired complex hopping elements to be induced in a specific laser coupling scheme for the dice lattice geometry. We conclude by comparing the complexity of these alternative approaches, and advocate that complex optical flux lattices provide the more elegant and easily generalisable strategy. arXiv:1804.10261v1 [cond-mat.quant-gas]	10.1088/1367-2630/aad134	[ "https://arxiv.org/pdf/1804.10261v1.pdf" ]	56,255,479	1804.10261	2ba7e9c6516a0e191f8202f21c057d8084e8596b	Synthetic Gauge Fields for Lattices with Multi-Orbital Unit Cells: Routes towards a π-flux Dice Lattice with Flat Bands 26 April 2018 26 Apr 2018 Gunnar Möller †[email protected] School of Physical Sciences Functional Materials Group University of Kent CT2 7NZCanterburyUnited Kingdom Nigel R Cooper TCM Group, Cavendish Laboratory University of Cambridge CB3 0HECambridgeUnited Kingdom Synthetic Gauge Fields for Lattices with Multi-Orbital Unit Cells: Routes towards a π-flux Dice Lattice with Flat Bands 26 April 2018 26 Apr 2018CONTENTS 2numbers: 0375Lm6785-d6785Hj We propose a general strategy for generating synthetic magnetic fields in complex lattices with non-trivial connectivity based on light-matter coupling in cold atomic gases. Our approach starts from an underlying optical flux lattice in which a synthetic magnetic field is generated by coupling several internal states. Starting from a high symmetry optical flux lattice, we superpose a scalar potential with a super-or sublattice period in order to eliminate links between the original lattice sites. As an alternative to changing connectivity, the approach can also be used to create or remove lattice sites from the underlying parent lattice. To demonstrate our concept, we consider the dice lattice geometry as an explicit example, and construct a dice lattice with a flux density of half a flux quantum per plaquette, providing a pathway to flat bands with a large band gap. While the intuition for our proposal stems from the analysis of deep optical lattices, we demonstrate that the approach is robust even for shallow optical flux lattices far from the tight-binding limit. We also provide an alternative experimental proposal to realise a synthetic gauge field in a fully frustrated dice lattice based on laser-induced hoppings along individual bonds of the lattice, again involving a superlattice potential. In this approach, atoms with a longlived excited state are trapped using an 'anti-magic' wavelength of light, allowing the desired complex hopping elements to be induced in a specific laser coupling scheme for the dice lattice geometry. We conclude by comparing the complexity of these alternative approaches, and advocate that complex optical flux lattices provide the more elegant and easily generalisable strategy. arXiv:1804.10261v1 [cond-mat.quant-gas] Introduction The creation of synthetic gauge fields in cold atomic gases provides new opportunities for realising exotic emergent quantum phases [1,2,3,4,5]. Prominent target phases include vortex lattices [6] and, at high flux density, bosonic counterparts of the continuum fractional quantum Hall states [7,8]. When both a (synthetic) field and a lattice potential are present, the continuum quantum Hall states are predicted to persist for appreciable flux densities n φ per plaquette [9]. In addition, new classes of quantum Hall states, stabilized only due to the presence of a periodic lattice potential, emerge at larger values of n φ owing to the underlying structure the Hofstadter spectrum [10,11,12,13,14], and in particular owing to the presence of single-particle bands with higher Chern numbers \|C\| > 1 [14,15]. Early experiments on synthetic gauge fields relied on using rotation to emulate magnetic fields [1,16]. However, in this approach it is exceedingly difficult experimentally to avoid heating due to asymmetric trapping potentials, so the strongly interacting regime of low density in the lowest Landau level remains out of reach. Prompted in part by the exciting outlook for the creation of new phases of matter, there has been much progress with new theoretical proposals and the experimental realizations for schemes of simulating artificial gauge fields [17,18,19,20,21,22,23]. Further impetus for synthetic fields stems from the prospect of realising topological flat bands in condensed matter systems -where spin-orbit coupling may provide suitable complex hopping elements in a tight-binding representation -sharpening the focus on the underlying commonality of flat single particle bands with non-zero Chern number [24,25,26,27,28,29,30], and more detailed characteristics of their band geometry [31,32,33,34]. Currently no clear target systems realising synthetic magnetic flux have been identified in the solid state, while cold atoms provide a range of successful realizations. ‡ Early achievements include the square lattice with staggered magnetic flux [36,37] that was generated by suitably tailored laser-induced hoppings [17,21]. More recently, experiments have achieved homogeneous magnetic flux using related approaches [38,39,40,41]. The Chern bands of the Haldane model [25] were also successfully engineered using a lattice shaking approach [42]. Features of the nontrivial band single band topology have been successfully identified [43,44]. Another groundbreaking line of research has exploited spatially dependent dressed states of atoms in order to create a Berry phase emulating the Aharonov-Bohm effect of charged particles moving in a magnetic field [20]. The experimental realization of this approach [20] has prompted further theoretical developments in order to maximize the achievable flux density in so-called optical flux lattices [23,45]. These systems rely on modulating the optical dressed states of multi-state atoms on the scale of the optical wavelength, thus accessing the smallest possible length scales for light-matter coupled systems, and provide a viable route to observe fractional quantum Hall physics [2,46,47]. Experimental progress has been reported on the intimately related case of emulating spin-orbit coupling in two dimensional systems [48,49,50]. So far, attempts to emulate optical lattices with synthetic gauge fields have focused on continuum gases or on simple optical lattice geometries such as square and triangular lattices [51]. However, optical lattices without gauge fields have already been demonstrated for more complex geometries such as the kagome lattice [52], which is achieved by removing sites from an underlying triangular lattice. Lattice geometry plays a particularly important role in the presence of magnetic flux, as it can affect the single particle spectrum dramatically. Indeed, the elegant Hofstadter butterfly seen in the spectrum of the square lattice [53] is strongly altered in other geometries such as the triangular [54] or hexagonal lattices [55]. This provides a strong incentive to achieve synthetic gauge fields in a number of different lattice geometries. It is well understood how complex lattice geometries can be realised in scalar optical lattices by exploiting the superposition of several optical lattice potentials [52,56,57]. In this paper, we explore how this design principle can be extended to create optical flux lattices with non-trivial connectivity by superposing scalar sub-/superlattice potentials to an optical flux lattice that generates non-trivial Berry phases from adiabatic motion within the space of internal states of the trapped atoms. We demonstrate that a scalar potential may be used to either remove bonds or sites from an underlying optical flux lattice of simpler geometry, as well as to split individual sites into multiple wells, all the while keeping the synthetic field intact. The basic principle for controlling bonds can be understood from a tight-binding picture: the dynamics of atoms in an optical lattice arises from hopping processes between local Wannier states that are localized in ‡ We also note the successful observation of fractional Chern insulating phases in graphene based heterostructures under strong physical magnetic fields [35]. the minima or wells of the optical potential [58]. The amplitude of hopping processes is given by the overlap of these wave functions. As the overlap is dominated by the exponential tails penetrating the potential maxima that separate adjacent wells, hopping is extremely sensitive to the magnitude of this potential. Therefore, hopping can be almost completely suppressed by increasing the height of the potential maximum between two wells when a scalar potential is added at those locations. Generally, we wish to suppress bonds on a periodic sublattice of an underlying optical flux lattice, so this can be implemented by superposing an additional scalar optical lattice potential which acts equally on all internal states. In practice, optical flux lattices operate in an intermediate coupling regime where the lattice potential is sufficiently shallow for atoms to occupy any position in space. One of the main results of the current work is to demonstrate that complex optical flux lattices can operate in a regime of weak coupling that remains far from the tight-binding limit: we provide a specific example showing that the dispersion of the tight-binding picture is reproduced closely even in the regime of shallow lattice depth with potential depth of order of the atomic recoil energy. In order to demonstrate our general principle, we propose and analyse in detail a new realization for synthetic fields in the dice lattice (also known as T 3 -lattice) where the specific flux density of Φ = Φ 0 /2 per plaquette yields a particularly surprising band structure with three pairs of perfectly flat bands that conserve time-reversal symmetry [59]. The flat bands and compactly localized single particle states found in this lattice are caused by a phenomenon of destructive interference known as Aharonov-Bohm caging [59]. This regime would be particularly well suited to reach interesting correlation phenomena [60,61,62], but previous proposals for synthetic fields in a dice lattice geometry that have focused on a different regime with dispersive Chern bands [63]. Unlike most flat band models achievable in cold atoms [64], the flat bands of the πflux dice lattice model are fully gapped. Owing to the flatness of the band dispersion, even weak interactions give rise to exotic phases in the dice lattice model, including a superfluid phase in the half filled lowest band [61] as well as highly degenerate vortex lattice configurations at larger density [60,61] that provide a playing field for orderby-disorder phenomena [62]. Hence, akin to the physics of flat band ferromagnetism [65,66], the dominant phases in the dice lattice provide interesting alternatives to more conventional features of Bose condensation in dispersive bands [37]. To further contrast the new proposal with more conventional techniques, we also present an alternative design for a dice lattice with a synthetic π-flux based on alkaline earth atoms trapped by light near their anti-magic wavelength. We describe a set-up creating laser-induced hoppings according to the connectivity of the dice lattice, that can be realised using far-detuned transitions following Ref. [21]. Our design explicitly constructs the tight-binding Hamiltonian within the magnetic unit cell, containing a total of six sites, which is repeated due to the inherent periodicity of the trapping lasers. We find that the two designs involve similar number of laser sources, and we argue that requirements on phase stability favour the optical flux lattice approach. The paper is organised as follows. In section 2, we review how the concept of adiabatic motion in optical dressed states enables the creation of optical flux lattices, and we establish our notations. In section 3, we introduce the idea of changing lattice connectivity by removing bonds from an optical flux lattice at the level of a tight binding approach, and perform an analysis of its translational symmetries. In section 4 we detail how the idea can be exploited to realize the dice lattice geometry with half a flux quantum per plaquette, focusing on a tight-binding picture. Section 5 gives the general formalism for studying optical flux lattices beyond the tight-binding limit in reciprocal space, and we use the example of the dice lattice geometry to demonstrate the role of spin-translation symmetries of the flux lattice Hamiltonian. In section 6, we provide detailed calculations of the band structure for realistic parameters in our dice flux lattice geometry, focusing on the limit of a shallow lattice. Section 7 provides the alternative design, based on laser-induced hoppings in a deep optical lattice, and we conclude in section 8. Background: Optical Flux Lattices The optical flux lattice approach is motivated by the principle of adiabatic motion of atoms, such that they remain in their local ground-state \|Ψ(r) along their trajectory r(t) [23]. Upon completion of a closed path C, the wavefunction of the atoms acquires a geometrical Berry phase γ = C qAdl, given by the line integral over the (real space) Berry connection qA = i Ψ\|∇Ψ (with a fictitious charge q) [20]. This geometric phase mimics the Aharonov-Bohm coupling of a charged particle to the vector potential of a physical magnetic field, which has the same form. It also useful to think of the corresponding flux density n φ = q/h(∇ × A) ·ê 3 . The presence of vortices in the Berry connection allows one to achieve flux densities of order one magnetic flux quantum per unit cell of the optical flux lattice. Here, we will consider as our starting point the explicit example of the triangular flux lattice of Ref. [23] for a two-state system with the Hamiltonian H =p 2 2m 1 + VM(r) ·σ,(1) where 1 is the 2 × 2 identity matrix in spin-space,σ = (σ 1 ,σ 2 ,σ 3 ) is the vector of Pauli matrices, and V is the depth of the optical lattice. We consider the triangular optical lattice potential described by M(r) = cos( κ 1 r)ê 1 + cos( κ 2 r)ê 2 + cos( κ 3 r)ê 3 ,(2) whereê i are the cartesian unit vectors, and the wave vectors κ 1 = (1, 0)κ, κ 2 = (1/2, √ 3/2)κ, and κ 3 = (−1/2, √ 3/2)κ are chosen to yield a lattice potential with minima separated by a lattice vector a, i.e., we require κ = 2π √ 3a . In our notations, we highlight constant vectors defined by externally imposed geometrical features such as κ i in bold-face with an additional arrow, while vectors representing variables like r are denoted in simple bold font. Note that specific implementations of a triangular optical flux lattice such as (1) may be realised by various optical coupling schemes. Detailed Orange arrows show the in-plane components of the local Bloch vector. The unit cell is spanned by the vectors a 1 , a 2 , contains 4 triangular lattice sites, and encloses 2 flux quanta. Thanks to a spin-translation symmetry, this can be reduced to a reduced unit cell of size [ a 1 /2, a 2 ] (dotted cyan lines). In this paper, we show how this flux lattice can be modified to yield an optical dice flux lattice by eliminating bonds: a dice lattice is obtained by impeding tunnelling across the links which are crossed out with blue wavy lines. implementations have been presented elsewhere (see, e.g., Ref. [45]), so we shall work with the simplest model in the current paper. In the adiabatic limit m → ∞, it is easily checked that the Hamiltonian (1) has eigenvalues E ± (r) = ±V\|M\|, and the local Bloch vector for the lower band,n = Ψ − (r)\|σ\|Ψ − (r) , is simply given by the direction of −M, i.e.,n = −M ≡ −M/\|M\|. The states \|Ψ ± are also the eigenstates for the class of HamiltoniansĤ =Ĥ +V s , for arbitrary scalar (i.e., spin-independent) potentialsV s (r) = V s (r)1. The energy landscape for the unperturbed triangular flux lattice (1) is shown in Fig. 1. Note that the unit cell of this lattice encloses two flux quanta within an area containing four local minima of the energy, which we can think of as four lattice sites in the tight-binding limit of a deep optical flux lattice [23]. For our choice of units, the lattice vectors spanning the unit cell are given by a 1 = ( √ 3, −1)a, and a 2 = (0, 2)a, as highlighted in Fig. 1. The periodicity of the energy landscape suggests that the Hamiltonian (1) has a higher translational symmetry than that by the above-mentioned lattice vectors a i . While energetically equivalent, the eigenstates at the four energy minima in the unit cell are distinct. However, the higher symmetry of the Hamiltonian can be revealed by generalized translation operators that incorporate a rotation in spin space [23]. Available spin-translation operators arê T 1 =σ 2 e 1 2 a 1 ·∇ ,T 2 =σ 1 e 1 2 a 2 ·∇ ,(3) with [T i ,Ĥ ] = 0 (i = 1, 2), but [T 1 ,T 2 ] = 0. Nonetheless, we find that [T 1 ,T 2 2 ] = 0, so we can classify the eigenvalues ofĤ with the quantum numbers of bothT 1 , and T 2 2 = exp( a 2 · ∇) ≡K( a 2 ) , as the latter reduces to a regular translationK( a 2 ) by a 2 . For a detailed discussion of these symmetry operations in the triangular lattice, see Ref. [23]. Changing Lattice Topology via Scalar Potentials In the deep optical lattice limit, we can consider optical flux lattices as tight binding models where motion between two 'sites' or local minima of the energy landscape is described by a tight binding model with complex hopping elements. We now examine how a change in the lattice topology emulated by optical flux lattices is achieved either by 'removing sites' or by 'removing bonds' in this tight binding model, as was already achieved for scalar optical lattices [56,52]. As we will demonstrate below, this idea can indeed also be realised in optical flux lattices by applying an additional scalar optical lattice potential to either suppress lattice sites or the connectivity between them, while the distribution of flux generated by the underlying optical flux lattice is kept intact. Some examples of cutting bonds are visualized in Fig. 2. There are already similar experimental realisations of tuneable optical lattices obtained by superposing multiple standing waves [56,52]. An additional consideration for flux lattices arises in the tightbinding limit, where flux through each plaquette is defined only modulo 2π. As the elimination of links joins the two adjoining plaquettes into a single one, this construction yields non-trivial flux lattices only if the total flux in the resulting merged plaquette is not an integer multiple of the flux quantum Φ 0 . Similarly, the removal of sites merges several adjoining plaquettes, so the same consideration applies. For example, a hexagonal lattice can be obtained by removing a sublattice of sites of an underlying triangular lattice. In this case, six neighbouring triangular plaquettes are joined into a hexagonal one, so this yields non-trivial results if the flux per triangular plaquette is not a multiple of Φ 0 /6. Case Study: the Dice Lattice For the remainder of this paper, we focus on a case study of eliminating bonds in a triangular flux lattice. Alongside the elementary unit cell of the flux lattice, Fig. 1 highlights the bonds that need to be severed in the triangular lattice so as to reduce its connectivity to a dice lattice geometry. As shown more clearly in Fig. 2(a), we find that mid-points of these bonds form a kagome lattice with lattice constant a = √ 3/2a. From Fig. 1, it is also clear that the pattern of eliminated bonds has a different periodicity as the unit cell [ a 1 , a 2 ] of the triangular optical flux lattice. This will be further discussed, below. In our cold atom realization of an optical dice flux lattice, the maxima of an additional scalar optical potential are aligned with the centre points of the bonds of an underlying triangular optical flux lattice. As experiments by the Stamper-Kurn group demonstrate, an attractive kagome optical lattice can be achieved by combining a bluedetuned (i.e., regions of high intensity are repulsive) short-wavelength triangular optical lattice with a red-detuned (attractive) triangular lattice of twofold lattice constant [52]. Experimentally, it is difficult to keep these two lattices in register, but this challenges has been successfully addressed [52]. Here, we require a repulsive kagome lattice, which is rotated by π/6 with respect to an underlying triangular optical flux lattice (1), again implying that the two light potentials have to be kept in phase as in the kagome lattice realisation of [52]. The corresponding optical potential is formed by a red-detuned short-wavelength scalar optical lattice V SW at wave number κ ⊥ = 2κ/ √ 3, as well as a blue-detuned long-wavelength scalar superlattice V LW with wave number κ ⊥ /2. The full Hamiltonian of our optical dice flux lattice is then obtained by superposing all three componentsĤ dice (r, b) =Ĥ + [rV SW (r) + bV LW (r)] 1.(4) Here, the parameters b > 0 and r < 0 give the amplitude of the scalar beams relative to the spin-dependent fields, and the explicit form of the required short-and longwavelength potentials are given by V SW (r) = V sin 2 κ ⊥ 1 r + sin 2 κ ⊥ 2 r + sin 2 κ ⊥ 3 r(5) for the red detuned beam that is attractive, and that should thus contribute with an amplitude r < 0, and V LW (r) = V sin 2 κ ⊥ 1 2 r + sin 2 κ ⊥ 2 2 r + sin 2 κ ⊥ 3 2 r(6) for the blue detuned beam that should provide a repulsive potential with an amplitude b > 0, and κ ⊥ i = 2/ √ 3ê 3 ∧ κ i throughout. Note that both these contributions are scalar, i.e., they are diagonal in spin space. In the adiabatic limit (i.e., disregarding kinetic energy), the local energy eigenvalues are readily obtained as E dice ± (r, b) = ±V\|M\| + rV SW (r) + bV LW (r), and the local eigenstates are unchanged with respect to the triangular optical flux lattice. Let us now discuss the symmetries of the optical dice flux lattice Hamiltonian. As we noted previously, it does not have the full translational symmetry of the triangular optical flux lattice. The resulting situation is best discussed in terms of Fig. 3, which shows the energy landscape (contours; darker blue indicates minima), as well as the x-y-components of the local Bloch vector (orange arrows). In the presence of the scalar potentials (5, 6), the energy landscape contains lattice sites with three different profiles: the most prominent minima form the 'hubs' or sixfold connected sites of the dice lattice, such as the one at the origin r = (0, 0). They are surrounded by six smaller minima, the 'rims' or threefold connected sites. These are slightly triangular and can be either pointing upwards [such as at r = ( √ 3/2, 1/2)a] or downwards [as at r = (0, 1)a]. In addition, lattice sites differ in terms of the spin-content of the local wavefunction. Looking at the in-plane components of the local Bloch-vectors, it is apparent that a fundamental unit cell of our optical dice flux lattice is enclosed by the vectors marked in Fig. 3 as v 1 = (2 √ 3, 0)a, and v 2 = (− √ 3, 3)a, which connect hubs with identical Bloch vectors. Due to the distinct periodicities, this unit cells contains 12 sites of the underlying triangular lattice so it is enlarged threefold with respect to the unit cell of the original triangular optical flux lattice. The Hamiltonian (4) contains an additional symmetry, which can be constructed in terms of the spin-translation operatorsT 1,2 in (3). Let us construct suitable spin-translationsŜ 1,2 along the half lattice vectors 1 2 v 1,2 . These can be expressed in terms of T 1,2 as:Ŝ 1 =T 2 1T2 =σ 1 e 1 2 (2 a 1 + a 2 )·∇ ,(7)S 2 =T −1 1T 2 = iσ 3 e 1 2 (− a 1 + a 2 )·∇ .(8) We note that bothŜ i commute with the Hamiltonian, i.e. [Ŝ 1,2 ,Ĥ dice ] = 0. Furthermore, their squares are simple translations, which confirms that we have chosen the unit cell correctly. For instance,Ŝ 2 1 =σ 2 1 e (2 a 1 + a 2 )·∇ = e v 1 ·∇ , which equals a pure translationK( v 1 ) under the lattice vector v 1 . However, the translationsŜ 1 andŜ 2 do not commute with each other, as [Ŝ 1 ,Ŝ 2 ] = 0. Given thatŜ 2 is diagonal in spin-space, we select this operator as our supplementary symmetry in formulating the single-particle Hilbert-space, and we can then use the eigenvalues of the set of commuting operatorŝ H,Ŝ 2 1 , andŜ 2 to label eigenstates. This results in a reduced unit cell in real space, spanned by [ v 1 , v 2 /2], as shown in green dotted lines in Fig. 3, such that eigenstates in the remainder of the full unit cell can be recovered by applyingŜ 2 to their symmetry related points in the reduced cell. -2 -1 0 1 2 3 4 0 1 2 3 x [a] y [a] -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 E Spin-Translation Symmetry in Shallow Flux Lattices The arguments of the preceding section can be placed on a more robust foundation by considering the full Hamiltonian of the flux lattice beyond the tight binding limit, i.e., including kinetic energy. In order to capture the effect of the kinetic energy term, it is convenient to study the flux lattice Hamiltonian as a tight-binding model in reciprocal space [67]. Here, we review and extend this formalism to take into account the spintranslation symmetries, as realised by the operatorsŜ 1 ,Ŝ 2 identified above. General Formalism Having identified the periodicity of the problem in the real-space unit-cell (UC) spanned by [ v 1 , v 2 ], we know that the wave functions in reciprocal space are defined on a fundamental Brillouin zone (BZ) spanned by the reciprocal lattice vectors g i = ij 2π v j ∧ê 3 ( v 1 ∧ v 2 ) ·ê 3 , i = 1, 2(9) where ij is the totally anti-symmetric tensor. The reciprocal lattice vectors thus satisfy g i · v j = δ i,j . Now, let us turn to discuss the momentum transfers, which are obtained as the matrix elements of the interaction V(r) in the basis of plane-wave states with r\|k, α = e ik·r ⊗ \|α for the spin component α. One finds that the matrix elements depend only on the momentum transfer ∆k = k − k V α α k ,k = V α α (k −k) = k , α \|V(r)\|k, α .(10) According to Bloch's theorem, eigenstates \|Ψ nq are uniquely labelled by a band index n and momentum q in the first Brillouin zone, while larger momenta can be decomposed as k = q + G into a part lying in the BZ and a reciprocal lattice vector G st = s g 1 + t g 2 with s, t integer. In its Bloch form the wavefunction reads \|Ψ nq = α u α nq (r)\|q, α ≡ α,s,t c α nq,Gst \|q + G st , α ,(11) with expansion coefficients c α nq,G . As was noted previously [67], the flux lattice Hamiltonian takes the form of a tight binding model in reciprocal space in which the kinetic energy plays the role of a harmonic confinement: H q = α,G 2 (q + G) 2 2mâ † α,q+Gâ α,q+G + αα ,GG V α α G −Gâ † α ,q+G â α,q+G ,(12) written here in terms of the annihilation (creation) operatorsâ ( †) α,k for the plane-wave basis. We should also carefully note that all hoppings in this momentum-space tightbinding representation are relative to the wave-vector q, hence they represent a lattice of achievable momentum transfers, while in the usual case of tight-binding models in real space one is used to consider a lattice of fixed positions. The depth of the optical lattice potential is reflected by the magnitude V of the largest entries in V α α ∆k . The typical kinetic energy is of order of the recoil energy, which we define as in terms of the relevant momentum transfer ∆p = ∆k of the relevant laser beam as E R = 2 \|∆k\| 2 2m .(13) The adiabatic limit is recovered when E R V, where the kinetic energy can be neglected and the problem is solved by Fourier transform back into real space, where position r plays the role of a conserved momentum. In the general case, (12) defines a matrix equation for the coefficients c α nq,G , which can be solved numerically as coefficients decay rapidly with the absolute value of momentum \|q + G\|. (1) propagate along directions κ 1 ∝ (1, 0) t , κ 2 ∝ (1/2, √ 3/2) t , and κ 3 ∝ (1/2, √ 3/2) t . These momentum transfers induce spin-transitions given byσ 1 ,σ 2 andσ 3 respectively, highlighted by squares, diamonds, and circles on the corresponding arrows. The fundamental Brillouin zone is shaded in yellow. (b) Taking into account the spintranslation symmetry of (1,2), one can assign a definite spin-state to the accessible k-points (denoted as 1 or 2 in the figure), while the corresponding enlarged Brillouin zone (blue shade) is doubled alongê 1 . (c) The reciprocal-space representation of the optical dice flux lattice includes the triangular lattice transitions, as well as additional momentum transfers due to the scalar potential V kag = rV SW + bV LW of Eq. (5,6). These "hoppings" along directions κ ⊥ i connect to additional k-points located in the centers of the original triangular lattice, yielding a Brillouin zone for the dice lattice (red shade) that is 1/3 the size of the Brillouin zone for the triangular lattice (yellow shade). The reciprocal lattice vectors g 1 , g 2 are shown as red arrows. (d) The spintranslation symmetry of the optical dice flux lattice again leads to a unique labelling of spin states 1, 2 for all possible momentum transfers. This yields an enlarged Brillouin zone, which is stretched along g 2 and covers the region [ g 1 , 2 g 2 ] (green shade). Role of Spin-Translation Symmetries in Complex Optical Flux Lattices The role of the spin-translation symmetries is more easily explained within an example. Let us therefore focus on the reciprocal space picture of the dice flux lattice H dice , that is illustrated in Fig. 4. For the components associated with the triangular flux lattice (1), we obtain the spin-dependent processesV κ 1 = Vσ 1 with momentum transfer κ 1 = − g 1 + 2 g 2 ,V κ 2 = Vσ 2 with κ 2 = g 1 + g 2 , andV κ 3 = Vσ 3 with κ 3 = 2 g 1 − g 2 , where the reciprocal lattice vectors g i are defined by the lattice vectors v i spanning the unit cell of the dice flux lattice according to (4). For later reference, note that these momentum transfers are proportional to the wave vectors of the three coupling lasers of the dice optical flux lattice, and are linear combinations in integer multiples of its reciprocal lattice vectors (9). We display the momentum transfers of the underlying triangular optical flux lattice in Fig. 4(a), which also highlights the Brillouin zone corresponding to full the real-space unit cell [ a 1 , a 2 ] of Fig. 3. Following [67], the spin-translation symmetryT 1 of this model can be exposed by fixing the eigenvalue of the spin-translation operator, leading to a halving of the real space unit cell to [ a 1 /2, a 2 ], thus doubling the Brillouin zone and leaving a definite spin state at each reciprocal lattice site, as shown in Fig. 4(b). To obtain the dice optical flux lattice, we add to this picture the coupling to the scalar optical potentials generating the kagome lattice, Eqs. (5,6), which contribute with momentum transfers corresponding to twice their wave numbers, arising from the absorption of a photon from a standing wave laser followed by stimulated emission in the opposite direction. For V SW , we obtain momentum transfers ∆ k SW i = 2 κ ⊥ i , with amplitudeV SW = rV1 and similarly for V LW the momentum transfers are ∆ k LW i = κ ⊥ i with amplitudeV LW = bV1. These momentum transfers are four-or twofold multiples of the reciprocal lattice vectors and their π/3 rotations. According to the enlarged unit cell in real space, the BZ of the dice lattice should cover one third of the area of the BZ for the triangular optical flux lattice. The corresponding lattice of possible momentum transfers is illustrated in Fig. 4(c), revealing a three times denser coverage of attainable k-points. The action of the spin-translation symmetry of the dice lattice model is again readily illustrated in this momentum space picture. Assume a single-particle wave-function has a non-zero amplitude for spin state 1 and vanishing amplitude for spin state 2 at momentum q. Applying momentum-and spin-transfers to this initial state according to the tight-binding Hamiltonian (12), one can see that all related reciprocal lattice points at positions q + G are reached with a definite spin quantum number. Equivalently, the Hamiltonian does not allow one to create any loops that return to the initial point with a different value of the spin. Choosing a spin state of 1 at the central k-point, one obtains the spin labels shown in Fig. 4(d). An equivalent labelling is obtained by interchanging labels '1' and '2' (or equivalently, by a translation of the figure under g 2 ). The spin-translation symmetry can be more formally derived from the eigenvalue equations of the spin-translation operatorsŜ 1,2 . We takeŜ 2 andŜ 2 1 as the chosen symmetry generators commuting with the Hamiltonian, or [Ĥ,Ŝ 2 1 ] = [Ĥ,Ŝ 2 ] = [Ŝ 2 1 ,Ŝ 2 ] = 0, as discussed in Sec. 4. This implies that the Hamiltonian is block-diagonal in the subspaces of fixed eigenvalues ofŜ 2 1 ,Ŝ 2 . Given the unitarity of these operators, we denote their eigenvalues as λ i = exp(iΘ i ), withŜ 2 1 \|Θ 1 , Θ 2 = exp(iΘ 1 )\|Θ 1 , Θ 2 and S 2 \|Θ 1 , Θ 2 = exp(iΘ 2 )\|Θ 1 , Θ 2 , with \|Θ 1 , Θ 2 the corresponding eigenstates. Consider then the explicit action of the generalised translations on momentum eigenstateŝ S 2 1 \|k, α =1e i v 1 ·k \|k, α , S 2 \|k, α = iσ 3 e i 2 v 2 ·k \|k, α .(14) We see that the phases are periodic under translations of k → k + g 1 in the phase ofŜ 2 1 , while the action ofŜ 2 is periodic under a doubled reciprocal lattice vector k → k + 2 g 2 , when the spin-state is fixed. Thus, we can label eigenstates by a momentum q taken to lie in the enlarged BZ [ g 1 , 2 g 2 ] that is stretched twofold along the g 1 -direction, as highlighted in Fig. 4(d). In this representation, each point of reciprocal momentum transfers can be assigned a definite spin state, as the momentum q in the enlarged BZ provides sufficient information to encode both the spin and momentum degrees of freedom. Alternatively, one could choose to represent the full range of possible eigenvalues Θ 2 ∈ [0, 2π) by reducing the momentum to the fundamental Brillouin zone [ g 1 , g 2 ], and recover the full range of Θ 2 by taking into account both ±1 eigenvalues of the spin operatorσ 3 . Quantitative Analysis of the π-flux Optical Dice Flux Lattice In this section, we provide a numerical study of the optical dice flux lattice introduced in section 5. Numerics are performed in terms of the reduced unit cell [ v 1 , v 2 /2], or its reciprocal space counterpart. In other words, our implementation relies on resolving eigenstates of the generalized translationsŜ 2 , as discussed above. We proceed to discuss the spectrum, which provides an excellent approximation to the tight-binding version of the π-flux dice-lattice model. For reference, let us first review the spectrum in the tight-binding limit, shown in Fig. 5a). Note that the tightbinding spectrum features only three distinct eigenvalues, each corresponding to a pair of degenerate bands all of which are time-reversal symmetric and have Chern number C = 0. The overall count of six bands corresponds to the six lattice sites in the fundamental magnetic unit cell of the fully-frustrated dice-lattice. At intermediate depth of the optical lattice V E R , § we find that the lowenergy spectrum of our proposed dice flux lattice (4) correctly reproduces the qualitative features of the tight binding model. For V E R , this low-energy spectrum contains two near-degenerate bands that are well separated from higher bands. These two lowest bands have a very small dispersion and have only a small residual splitting. A typical spectrum, for V = 2E R , and −r = b = 1/8 is shown in Fig. 5b). To display the residual dispersion of the lowest bands more clearly, we will analyse a series of contour-plots in Fig. 7, below. For the parameters in Fig. 5b), the dispersion of the two lowest bands is of the order of 0.04E R . There is a small splitting to the second band (not shown), which has the inverse dispersion relative to that of the lowest band, i.e. its minima are found at the maxima of the lowest band and vice versa. With these parameters, the joint dispersion of these nearly degenerate bands is about 50 times smaller than the gap to higher excited bands. It is instructive to analyze how the band dispersion evolves with the strength V of the optical coupling. A series of different spectra with values ranging from V = E R to V = 8E R is shown in Fig. 6, including the lowest five bands in each case. These data were obtained with a cut-off for momentum at k 12\| g i \|. It is clearly seen that the near-degeneracy of the lowest two bands is realised very well for all V ≥ 2E R , while a small splitting is visible on the figure for V = E R . The higher (n = 3, 4, 5) bands are not found to be degenerate. However, the gap above the near-degenerate ground state manifold is seen to increase with the optical coupling strength. Given these findings, we interpret the lowest bands as corresponding to the two degenerate lowest energy bands in the tight-binding limit, while the higher bands can be interpreted as arising from different local orbitals that can be formed within the minima of the optical potential. In the limit of V → ∞, we expect that the splitting to such orbitals would become large, and a low energy part corresponding to the single orbital physics may then emerge from the spectrum. We now discuss the topological nature of the low-lying bands in the dice fluxlattice. The main qualitative difference of the intermediate-depth lattice with respect to the tight-binding model is the occurrence of weak tunnelling across the 'forbidden' links of the underlying triangular flux lattice, which break time-reversal symmetry. To analyse this statement quantitatively, we calculate the Berry curvature B of our model by evaluating Wilson loops on a discretized grid of k-points within the Brillouin zone [68]. We confirm that the Berry curvature is non-zero, and has opposite signs in the two low-lying bands. The distributions of the (log-)Berry curvature in the lowest band are shown as contour plots in the lower row of Fig. 7 for a range of optical coupling strengths, while the upper row shows the corresponding band dispersions. Note that there are extended regions where the curvature B is small, while maxima are relatively localised. For example, at V = E R , typical values are B 0.05a 2 (to be compared to an average ofB = 9/πCa 2 2.86Ca 2 for a Chern number C band with homogenous Berry curvature of the given Brillouin zone area). At the location of the maxima of the band dispersion, which can be seen as avoided crossings with the next higher band, B is strongly peaked and as a result, the Chern number C of the band is non-zero. Depending on the specific parameters we have found either \|C\| = 1, or \|C\| = 3. In both cases, the cumulative Chern number of the two lowest bands is zero. The different panels of Fig. 7 show the evolution of the band dispersion with increasing optical coupling, which reveals a change of the location of minima in the dispersion, and correspondingly for the Berry curvature. Note also how the flatness of the bands improves as we go to stronger coupling. Extended regions of low Berry curvature are also found at the highest value we show. It would be interesting to study how the many-body spectrum is affected by this finite but oppositely oriented Berry curvature in the lowest two bands. We expect that as long as the interaction energy is larger than the residual splitting between the two lowest bands, the system likely behaves in a qualitatively similar fashion as the time-reversal invariant system in the tight binding limit [61]. A detailed analysis of this physics will be the subject of a future study. In the sense that the perturbation of the bands away from the time-reversal symmetric case is caused by small hopping elements on suppressed bonds, we can consider the time-reversal symmetry breaking of our optical dice flux lattice to be 'weak'. (a) (b) (c) (d) E 0 E 0 E 0 E 0 (e) (f) (g) (h) log B a 2 log B a 2 log B a 2 log B Realizing the Fully Frustrated Dice Lattice in a Tight-Binding Approach An alternative realisation of the dice lattice pierced by π-flux per plaquette can be realised in a pure tight-binding philosophy. Let us discuss in detail the set-up for alkaline earth atoms [e.g., ytterbium (Yb)] atoms trapped in an optical lattice at the anti-magic wavelength [61]. In our approach, we closely follow the proposal for a square optical lattice using anti-magic trapping [21]. The possibility for this construction arises as the two internal states ( 1 S 0 and 3 P 0 ) of Yb have polarisability α of opposite signs for wavelengths λ 960nm, so they are trapped at the points of maximum or minimum laser intensity, respectively [21]. At the anti-magic wavelength λ * 1120nm, the absolute values of the polarisability are of equal magnitude. This is crucial for the square lattice geometry. For our purposes, it may actually be more useful to choose a wavelength at which the polarisability is stronger in magnitude for one of the two (pseudo-)spin states: the dice lattice geometry results from a triangular optical lattice formed by three selfreflected laser beams propagating with wave vectors arranged at relative angles of 2π/3 with respect to each other. These beams should be mutually incoherent, so the total intensity is the sum of individual intensities. The mirrors used to self-reflect these beams need to be stabilised. One species of atoms ( 1 S 0 ) is then trapped at the maxima of the intensity (which are steep), while the excited 3 P 0 state is trapped at the minima (which are more shallow). Hence, it is favourable that the polarisability is larger for the excited state, implying use of a wavelength λ 0 ≡ 2π/k 0 > λ * , i.e. using wavelengths in the far infrared (given that the polarisablity of the excited state grows more rapidly with λ near the anti-magic wavelength, or dα( 3 P 0 )/dλ\| λ * > dα( 1 S 0 )/dλ\| λ * ). In our set-up, all neighbouring sites are occupied by atoms of different internal states. Consequently, spontaneous tunnelling processes can be neglected, and all dynamics in this lattice is driven by via laser-assisted hopping [69,17]. Simultaneously, this coupling enables one to imprint phases onto the hopping matrix elements [21]. Let us now explain how to achieve phases that yield the target flux density of n φ = 1/2. For the fully frustrated dice lattice, the magnetic unit cell contains six inequivalent atoms [59,61], chosen here as a rectangular cell spanned by vectors v 1 = ( √ 3a, 0) t and v 2 = (0, 3a) t , as indicated by the different colouring of inequivalent lattice sites in Fig. 8. However, the (scalar) triangular optical lattice described in the preceding paragraph distinguishes only two types of sites. We propose to break this symmetry by shining one additional self-reflected laser-beam, S 4 , onto the system: this beam serves to break down the internal mirror-symmetry of the triangular lattice unit cell Alternatively, a suitable triangular lattice potential can be generated by three running beams with relative phase coherence. However, these would additionally have to be phase stabilised to prevent this triangular lattice from drifting relative to the 4th standing wave, laser S 4 , discussed below. Figure 8. Illustration of the rectangular magnetic unit cell with six inequivalent sites numbered 1 to 6. The drawing includes three magnetic unit cells, delineated by light solid lines. Links indicate the connectivity of the lattice, corresponding to hopping with amplitude t. Three links in the magnetic unit cell are special and need to be chosen with negative hopping −t (shown with two hashes). One laser, S 4 is required to establish the magnetic unit cell. Hopping between the six energetically inequivalent sites of the magnetic Brillouin zone are driven by lasers as indicated. Hopping-inducing lasers propagating perpendicular to the plane are labelled P i−j and drive transitions between sites i and j (shown as circles with crosses). The last two lasers L i−j propagate with a non-zero in-plane momentum along the x-axis such as to induce two distinct transitions within each magnetic unit cell, and with the relatively opposite sign. to the desired periodicity. In our set-up, S 4 has the same frequency/wavelength as the triangular optical lattice. However, its in-plane wavelength is enlarged to λ 4 = λ 0 / sin(θ) by projecting this laser onto the system at a tilt angle θ with respect to the z-axis of the plane. We tilt the laser towards the y-direction and require the potential to repeat on the scale of the magnetic unit cell, i.e., \| v 2 \| = λ 4 /2. By geometry, we must therefore choose the angle θ = arcsin(1/2) = π/6. Note the position of this laser potential (S 4 ) needs to maintain a fixed spatial position relative to the lasers defining the optical lattice, as fluctuations would shift the superlattice potential relative to the triangular lattice potential, and would alter the relative magnitudes of site energies. However, these energies need to be precisely defined, so that coupling lasers can satisfy the resonance condition and match the binding energy differences for the links on which they induce hopping processes. Note that a different wave length laser could also be chosen. To be explicit, let us write the required laser potentials. A bare triangular optical Figure 9. Set up of an optical dice lattice using an anti-magic optical lattice with laser-induced hopping: an underlying triangular lattice is created by retro-reflected standing wave lasers in plane. The symmetry of the magnetic unit cell is created by an additional standing wave laser S 4 directed at an angle to the plane. Eight coupling lasers complete the set-up and drive transitions between sites of different energy, as shown in Fig. 8 and discussed in the main text. lattice of lattice constant a is created by the wavelength λ 0 = 3a of the trapping beams: V tri (r) = I 0 i sin 2 2π 3a κ i · r ,(15) with the unit lattice directions κ 1 = (0, 1, 0) t , κ 2 = (− √ 3/2, 1/2, 0) t , and κ 3 = ( √ 3/2, 1/2, 0) t . The additional self-reflected laser, S 4 , propagates along the direction n d = (0, sin θ, cos θ) t , adding an (in-plane) intensity distribution of V 4 (r) = I 0 sin 2 2π 3a n d · r + δ ≡ I 0 sin 2 2π 6a y + δ(16) Here, we need to choose a small offset of the phase δ such that the maximum of intensity of the additional laser does not align with any high-symmetry point in the magnetic unit cell, and the intensity of the inversion symmetry breaking laser S 4 is reduced with respect to the other lasers by a suitable small factor , e.g., we can choose number of the order δ 2π/10 and 0.05. A three-dimensional view of the overall set-up is given in Fig. 9. In the resulting potential V tot (r) = V tri (r) + V 4 (r), the six sublattices of the desired magnetic unit cell are all distinguished energetically, i.e. their energies being detuned with respect to the triangular lattice by distinct amounts δ i , i = 1, . . . , 6. The set-up is completed by a total of eight coupling lasers driving the respective transitions between these sites. All of these lasers are propagating waves. Six of them are directed onto the system in the direction perpendicular to the lattice-plane. We denote these lasers as P i−j , indicating the two lattice sites i, j between which they induce a resonant transition. The six required lasers are P 1−2 , P 2−3 , P 2−5 , P 3−6 , P 4−6 , and P 5−6 , which require frequencies ω i−j = ω 0 i + δ i − ( ω 0 j + δ j ), and ω 0 i denotes the unperturbed energy of the internal state trapped at site i. Note each laser drives a transition between two neighbouring sites where atoms are in their ground / excited state, respectively. See also Fig. 8 for an illustration. Four of these six lasers drive a transition on a single link in the unit cell. However, the two lasers P 2−5 , P 3−6 connect the sixfold connected sites '2' and '6' to two neighbours with identical energy, located in the same and an adjacent unit cell, respectively. Due to the perpendicular direction of the lasers, these transitions are driven in phase, so the hopping elements have the same sign. All but one of the lasers P i−j need to be in phase with each other, while P 5−6 requires a phase-shift of π relative to the others. A definite phase relationship between these lasers of different frequencies can be achieved by deriving them from a single light source, and detuning their frequency using an acousto-optic modulator. The remaining coupling two lasers, which we call L 1−6 and L 2−4 are special in that they are required to drive two transitions (like P 5−6 ), but now with a relative phase of π between these two couplings. This relative phase is realised by virtue of an in-plane component of the respective wave-vectors. Specifically, we choose the in-plane component of their respective wave-vectors k along the x-axis such that k · ( √ 3a/2, 0, 0) t ≡ π. Again, this wave-vector can be realized by a suitable inclination of the laser beams with respect to the plane. This concludes our discussion of the detailed set-up for a tight-binding version of fully frustrated dice lattice. Let us briefly compare this construction to the optical dice flux lattice discussed in section 6. Firstly, we note that the tight-binding construction is explicitly time-reversal invariant, if all relative phases are set to match the values 0 or π. Although there may be small perturbations to the ideal dice-lattice model from spontaneous tunnelling processes between neighbouring three-fold sites such as sites 1 and 4, such processes also have real hopping elements. The practical realisation of both schemes poses similar challenges, notably the requirement to generate superlattice potentials whose relative position must be stabilised relative to an underlying lattice. This is difficult, but has already been achieved [52]. However, fluctuations of the geometry will affect the two proposals rather differently. In the optical flux lattice set-up, the superlattice acts to suppress tunnelling by creating local maxima in the potential. This suppression will be relatively insensitive to the precise location of potential maxima, as long as they are located within the relevant bonds of the lattice. By contrast, the tight-binding approach requires the superlattice to define relative energies of lattice orbitals, and transitions between them are driven resonantly. Hence, a rather fine control of the stability is required to ensure that all coupling lasers remain on resonance for their respective bonds. Conclusions We have introduced a new method for constructing optical flux lattices with complex geometries by combining a simple optical flux lattice with additional scalar potentials. To demonstrate the potential of our proposal, we have explored the optical dice flux lattice as an example geometry in which bonds were eliminated from an underlying triangular lattice. Our model yields flat bands that are a particularly interesting playground for studying interaction-driven phases of matter [61], and can realise a flatness parameter of fifty even for weak optical coupling. The optical flux-lattice approach results in interesting additional features with respect to a pure tight-binding description of the dice-lattice model. At intermediate lattice depths, the model weakly breaks time-reversal symmetry in the following sense: instead of degenerate pairs of time-reversal symmetric bands, the approach produces time-reversal pairs of bands whose degeneracies are only weakly split. The proposed realisation of an optical dice flux lattice is realistically achievable in the near future, as it combines several elements which are already part of the current state of the art. The kagome lattice realised in the group of Stamper-Kurn successfully demonstrates the phase-stabilised superposition of two lattices with distinct wavelengths [52]. Our set-up requires the additional superposition of a triangular optical flux lattice. While work on the first realisation of such systems under way, we would like to underline that related schemes for synthetic gauge fields have already been successful [36,37], and related schemes for emulating spin-orbit coupling in 2D systems have also been implemented [48,49,50]. We have also introduced a proposal for a tight-binding scheme which is closer to the existing technology of the aforementioned experiments. Here, challenges rely on finetuning energies and maintaining the relative superlattice position with high accuracy. This kind of set-up requires one-by-one engineering of laser-induced hopping between sites in the unit cell, so its complexity grows with the unit cell size. By contrast, one of the inherent features of the flux-lattice schemes is their tuneability. Explorations of scalar optical lattices have already shown that a multitude of different band-structures can be realised in the same experiment [52,56]. Hence, one interesting direction for further study is the question of how the lattice geometry is altered when moving the scalar lattices with respect to the underlying optical flux lattice. Figure 1 . 1Contour plot of the energy landscape for the triangular optical flux lattice with two flux quanta per unit cell of Ref.[23], the starting point for our construction. Figure 2 . 2Examples of new lattice topologies that emerge by elimination of bonds from an underlying graph, where suppressed hoppings are symbolized as open circles. The triangular lattice (a) can be reduced to a dice lattice. Here, the centers of eliminated bonds form a kagome lattice. A square lattice (b) can be reduced to a brickwork lattice which has the connectivity of a honeycomb lattice. Here, the centers of eliminated bonds again form a square lattice. A regular honeycomb lattice can also be recovered from this set-up by scaling the x-axis by one half. Figure 3 . 3Contours show the adiabatic energy landscape of the dice flux lattice(4) with −r = b = 1/8, obtained by knocking out bonds from the underlying triangular optical flux lattice shown inFig. 1. The periodicities of the lattice result from a combination of the periodicity in the energy landscape (shown as a density plot with minima in dark blue) and the local Bloch vectors (x-y-components shown as orange arrows). The original unit cell [ a 1 , a 2 ] is highlighted in dashed yellow lines/arrows. As the scalar potential has different periodicity than the flux lattice, the elementary unit cell of the dice flux lattice is enlarged and contains 12 sites. The figure shows the dice unit cell in red full lines, spanned by vectors marked as [ v 1 , v 2 ]. Thanks to a combined symmetry of spin rotation and translations (see main text), the unit cell can be reduced to half that size, shown as the region [ v 1 , v 2 /2] enclosed in dashed green lines. Figure 4 . 4Representation of optical flux lattices as tight binding models on a grid of k-points (circles) highlighting momentum transfers, or "hoppings", induced by absorption/emission of photons (arrows). We show the lattice of accessible momentum transfers for the triangular (a,b) and dice-lattice (c,d) geometries. Note the panels are scaled differently, with the links shown in black corresponding to the same momentum transfer throughout. (a) The lasers of the triangular optical flux lattice Figure 5 . 5a) Spectrum of the fully frustrated dice-lattice model in the tight-binding limit, plotted over the first BZ. As the magnetic unit cell has six distinct sublattices, the model results in six bands that are pairwise degenerate with energies of E = − √ 6t, E = 0, and E = √ 6t for the three pairs of bands. b) Spectrum of the dice-lattice model with system parameters V = 2E R , and −r = b = 1/8. The plot shows the lowest five bands, of which the lowest two energy bands are near-degenerate. Figure 6 . 6Evolution of the spectrum of the dice-lattice model with system parameters as a function of the parameter V, with fixed −r = b = 1/6, shown within the enlarged Brillouin zone spanned by [ g 1 , 2 g 2 ]. The plots show the lowest five energy bands, of which the lowest two energy bands are near-degenerate. Values of V shown are (a) V = E R , (b) V = 2E R , (c) V = 4E R , and (d) V = 8E R . 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[ "Deconstructing Self-Supervised Monocular Reconstruction: The Design Decisions that Matter", "Deconstructing Self-Supervised Monocular Reconstruction: The Design Decisions that Matter" ]	[ "Jaime Spencer \nCVSSP\nUniversity of Surrey\nCVSSP\nUniversity of Surrey\n\n", "Chris Russell \nCVSSP\nUniversity of Surrey\nCVSSP\nUniversity of Surrey\n\n", "Simon Hadfield [email protected] \nCVSSP\nUniversity of Surrey\nCVSSP\nUniversity of Surrey\n\n", "Richard Bowden [email protected] \nCVSSP\nUniversity of Surrey\nCVSSP\nUniversity of Surrey\n\n" ]	[ "CVSSP\nUniversity of Surrey\nCVSSP\nUniversity of Surrey\n", "CVSSP\nUniversity of Surrey\nCVSSP\nUniversity of Surrey\n", "CVSSP\nUniversity of Surrey\nCVSSP\nUniversity of Surrey\n", "CVSSP\nUniversity of Surrey\nCVSSP\nUniversity of Surrey\n" ]	[]	This paper presents an open and comprehensive framework to systematically evaluate stateof-the-art contributions to self-supervised monocular depth estimation. This includes pretraining, backbone, architectural design choices and loss functions. Many papers in this field claim novelty in either architecture design or loss formulation. However, simply updating the backbone of historical systems results in relative improvements of 25%, allowing them to outperform most modern systems. A systematic evaluation of papers in this field was not straightforward. The need to compare like-with-like in previous papers means that longstanding errors in the evaluation protocol are ubiquitous in the field. It is likely that many papers were not only optimized for particular datasets, but also for errors in the data and evaluation criteria. To aid future research in this area, we release a modular codebase (https://github.com/jspenmar/monodepth_benchmark), allowing for easy evaluation of alternate design decisions against corrected data and evaluation criteria. We re-implement, validate and re-evaluate 16 state-of-the-art contributions and introduce a new dataset (SYNS-Patches) containing dense outdoor depth maps in a variety of both natural and urban scenes. This allows for the computation of informative metrics in complex regions such as depth boundaries.	10.48550/arxiv.2208.01489	[ "https://export.arxiv.org/pdf/2208.01489v4.pdf" ]	251,253,394	2208.01489	e707a2e087a791467cc42b61ec6d6dcc9d55ee19	Deconstructing Self-Supervised Monocular Reconstruction: The Design Decisions that Matter Jaime Spencer CVSSP University of Surrey CVSSP University of Surrey Chris Russell CVSSP University of Surrey CVSSP University of Surrey Simon Hadfield [email protected] CVSSP University of Surrey CVSSP University of Surrey Richard Bowden [email protected] CVSSP University of Surrey CVSSP University of Surrey Deconstructing Self-Supervised Monocular Reconstruction: The Design Decisions that Matter Published in Transactions on Machine Learning Research (12/2022) CVSSP, University of Surrey Reviewed on OpenReview: https: // openreview. net/ forum? id= GFK1FheE7F This paper presents an open and comprehensive framework to systematically evaluate stateof-the-art contributions to self-supervised monocular depth estimation. This includes pretraining, backbone, architectural design choices and loss functions. Many papers in this field claim novelty in either architecture design or loss formulation. However, simply updating the backbone of historical systems results in relative improvements of 25%, allowing them to outperform most modern systems. A systematic evaluation of papers in this field was not straightforward. The need to compare like-with-like in previous papers means that longstanding errors in the evaluation protocol are ubiquitous in the field. It is likely that many papers were not only optimized for particular datasets, but also for errors in the data and evaluation criteria. To aid future research in this area, we release a modular codebase (https://github.com/jspenmar/monodepth_benchmark), allowing for easy evaluation of alternate design decisions against corrected data and evaluation criteria. We re-implement, validate and re-evaluate 16 state-of-the-art contributions and introduce a new dataset (SYNS-Patches) containing dense outdoor depth maps in a variety of both natural and urban scenes. This allows for the computation of informative metrics in complex regions such as depth boundaries. Introduction Depth estimation is a fundamental low-level computer vision task that allows us to estimate the 3-D world from its 2-D projection(s). It is a core component enabling mid-level tasks such as SLAM, visual localization or object detection. More recently, it has heavily impacted fields such as Augmented Reality, autonomous vehicles and robotics, as knowing the real-world geometry of a scene is crucial for interacting with it-both virtually and physically. This interest has resulted in a large influx of researchers hoping to contribute to the field and compare with previous approaches. New authors are faced with a complex range of established design decisions, such as the choice of backbone architecture & pretraining, loss functions and regularization. This is further complicated by the fact that papers are written to be accepted for publication. As such, they often emphasize the theoretical novelty of their work, over the robust design decisions that have the most impact on performance. our re-implementation using common design decisions (lower is better). Kuznietsov et al. (2017) and SfM-Learner (Zhou et al., 2017) have much higher error as they do not use stereo data training. (b) Original papers AbsRel relative performance improvement w.r.t. Garg et al. (2016) vs. real improvements observed training/evaluating baselines in a fair and comparable manner (higher is better). (c) Performance obtained by ablating the backbone architecture. (d) Backbone relative performance improvements (w.r.t. ResNet-18 from scratch) outweigh those provided by most recent contributions. See section Section 4.2 for more details of design choices. This paper offers a chance to step back and re-evaluate the state of self-supervised monocular depth estimation. We do this via an extensive baseline study by carefully re-implementing popular State-of-the-Art (SotA) algorithms from scratch 1 . Our modular codebase allows us to study the impact of each framework component and ensure all approaches are trained in a fair and comparable way. Figure 1 compares the performance reported by recent SotA against that obtained by the same technique on our updated benchmark. Our reimplementation improves performance for all evaluated baselines. However, the relative improvement resulting from each contribution is significantly lower than that reported by the original publications. In many cases, it is likely this is the result of arguably unfair comparisons against outdated baselines. For instance, as seen in Figures 1c & 1d, simply modernizing the choice of backbone in these legacy formulations results in performance gains of 25%. When applying these common design decisions to all approaches, it appears that 'legacy' formulations are still capable of outperforming many recent methods. As part of our unified benchmark, we propose a novel evaluation dataset in addition to the exclusively used urban driving datasets. This new dataset (SYNS-Patches) contains 1175 images from a wide variety of urban and natural scenes, including categories such as urban residential, woodlands, indoor, industrial and more. This allows us to evaluate the generality of the learned depth models beyond the restricted automotive domain that is the focus of most papers. To summarize, the contributions of this paper are: 1. We provide a modular codebase containing modernized baselines that are easy to train and extend. This encourages direct like-with-like comparisons and better research practices. 2. We re-evaluate the updated baselines algorithms consistently using higher-quality corrected groundtruth on the existing benchmark dataset. This pushes the field away from commonly used flawed benchmarks, where errors are perpetuated for the sake of compatibility. 3. In addition to democratizing code and evaluation on the common Kitti benchmarks, we propose a novel testing dataset (SYNS-Patches) containing both urban and natural scenes. This focuses on the ability to generalize to a wider range of applications. The dense nature of the ground-truth allows us to provide informative metrics in complex regions such as depth boundaries. 4. We make these resources available to the wider research community, contributing to the further advancement of self-supervised monocular depth estimation. Related Work We consider self-supervised approaches that do not use ground-truth depth data at training time, but instead learn to predict depth as a way to estimate high-fidelity warps from one image to another. While all approaches predict depth from a single image, they can be categorized based on the additional frames used to perform these warps. Stereo-supervised frameworks directly predict metric depth given a known stereo baseline. Approaches using only monocular video additionally need to estimate Visual Odometry (VO) and only predict depth up to an unknown scale. These depth predictions are scaled during evaluation and aligned with the ground-truth. However, monocular methods are more flexible, since they do not require a stereo pair. Despite having their own artifacts, they do not share stereo occlusion artifacts, making video a valuable cue complementary to stereo. Garg et al. (2016) introduced view synthesis as a proxy task for self-supervised monocular depth estimation. The predicted depth map was used to synthesize the target view from its stereo pair, and optimized using an L 1 reconstruction loss. An additional smoothness regularization penalized all gradients in the predicted depth. Monodepth (Godard et al., 2017) used Spatial Transformer Networks (Jaderberg et al., 2015) to perform view synthesis in a fully-differentiable manner. The reconstruction loss was improved via a weighted L 1 and SSIM (Wang et al., 2004) photometric loss, while the smoothness regularization was softened in regions with strong image gradients. Monodepth additionally introduced a virtual stereo consistency term, forcing the network to predict both left and right depths from a single image. 3Net (Poggi et al., 2018) extended Monodepth to a trinocular setting by adding an extra decoder and treating the input as the central image in a three-camera rig. SuperDepth (Pillai et al., 2019) replaced Upsample-Conv blocks with sub-pixel convolutions (Shi et al., 2016), resulting in improvements when training with high-resolution images. They additionally introduced a differentiable stereo blending procedure based on test-time stereo blending (Godard et al., 2017). FAL-Net (Gonzalez Bello & Kim, 2020) proposed a discrete disparity volume network, complemented by a probabilistic view synthesis module and an occlusion-aware reconstruction loss. Stereo Other methods complemented the self-supervised loss with (proxy) ground-truth depth regression. Kuznietsov et al. (2017) introduced a reverse Huber (berHu) regression loss (Zwald & Lambert-Lacroix, 2012;Laina et al., 2016) using the ground-truth sparse Light Detection and Ranging (LiDAR). Meanwhile, SVSM (Luo et al., 2018) proposed a two-stage pipeline in which a virtual stereo view was first synthesized from a self-supervised depth network. The target image and synthesized view were then processed in a stereo matching cost volume trained on the synthetic FlyingThings3D dataset (Mayer et al., 2016). Similarly, DVSO and MonoResMatch (Tosi et al., 2019) incorporated stereo matching refinement networks to predict a residual disparity. These approaches used proxy ground-truth depth maps from direct stereo Simultaneous Localization and Mapping (SLAM) (Wang et al., 2017) and hand-crafted stereo matching (Hirschmüller, 2005), respectively. DVSO introduced an occlusion regularization, encouraging sharper predictions that prefer background depths and second-order disparity smoothness. DepthHints (Watson et al., 2019) introduced proxy depth supervision into Monodepth2 (Godard et al., 2019), also obtained using SGM (Hirschmüller, 2005). The proxy depth maps were computed as the fused minimum reconstruction loss from predictions with various hyperparameters. As with the automasking procedure of Monodepth2 (Godard et al., 2019), the proxy regression loss was only applied to pixels where the hint produced a lower photometric reconstruction loss. Finally, PLADE-Net (Gonzalez Bello & Kim, 2021) expanded FAL-Net (Gonzalez Bello & Kim, 2020) by incorporating positional encoding and proxy depth regression using a matted Laplacian. Monocular SfM-Learner (Zhou et al., 2017) introduced the first approach supervised only by a stream of monocular images. This replaced the known stereo baseline with an additional network to regress VO between consecutive frames. An explainability mask was predicted to reduce the effect of incorrect correspondences from dynamic objects and occlusions. Klodt & Vedaldi (2018) introduced uncertainty (Kendall & Gal, 2017) alongside proxy SLAM supervision, allowing the network to ignore incorrect predictions. This uncertainty formulation also replaced the explainability mask from SfM-Learner. DDVO introduced a differentiable DSO module (Engel et al., 2018) to refine the VO network prediction. They further made the observation that the commonly used edge-aware smoothness regularization (Godard et al., 2017) suffers from a degenerate solution in a monocular framework. This was accounted for by applying spatial normalization prior to regularizing. Subsequent approaches focused on improving the robustness of the photometric loss. Monodepth2 (Godard et al., 2019) introduced several simple changes to explicitly address the assumptions made by the view synthesis framework. This included minimum reconstruction filtering to reduce occlusion artifacts, alongside static pixel automasking via the raw reconstruction loss. D3VO additionally predicted affine brightness transformation parameters (Engel et al., 2018) for each support frame. Meanwhile, Depth-VO-Feat (Zhan et al., 2018) observed that the photometric loss was not always reliable due to ambiguous matching. They introduced an additional feature-based reconstruction loss synthesized from a pretrained dense feature representation (Weerasekera et al., 2017). DeFeat-Net (Spencer et al., 2020) extended this concept by learning dense features alongside depth, improving the robustness to adverse weather conditions and low-light environments. Shu et al. (2020) instead trained an autoencoder regularized to learn discrimative features with smooth second-order gradients. Mahjourian et al. (2018) incorporated explicit geometric constraints via Iterative Closest Points using the predicted alignment pose and the mean residual distance. Since this process was non-differentiable, the gradients were approximated. SC-SfM-Learner (Bian et al., 2019) proposed an end-to-end differentiable geometric consistency constraint by synthesizing the support depth view. They included a variant of the absolute relative loss constrained to the range [0, 1], additionally used as automasking for the reconstruction loss. Poggi et al. (2020) compared various approaches for estimating uncertainty in the depth prediction, including dropout, ensembles, student-teacher training and more. Johnston & Carneiro (2020) proposed to use a discrete disparity volume and the variance along each pixel to estimate the uncertainty of the prediction. During training, the view reconstruction loss was computed using the Expected disparity, i.e. the weighted sum based on the likelihood of each bin. Recent approaches have focused on developing architectures to produce higher resolution predictions that do not suffer from interpolation artifacts. PackNet (Guizilini et al., 2020) proposed an encoder-decoder network using 3-D (un)packing blocks with sub-pixel convolutions (Shi et al., 2016). This allowed the network to encode spatial information in an invertible manner, used by the decoder to make higher quality predictions. However, this came at the cost of a tenfold increase in parameters. CADepth (Yan et al., 2021) proposed a Structure Perception self-attention block as the final encoder stage, providing additional context to the decoder. This was complemented by a Detail Emphasis module, which refined skip connections using channel-wise attention. Meanwhile, Zhou et al. (2021) replaced the commonly used ResNet encoder with HRNet due to its suitability for dense predictions. Similarly, concatenation skip connections were replaced with a channel/spatial attention block. HR-Depth (Lyu et al., 2021) introduced a highly efficient decoder based on SqueezeExcitation blocks (Hu et al., 2020) and progressive skip connections. While each of these methods reports improvements over previous approaches, they are frequently not directly comparable. Most approaches differ in the number of training epochs, pretraining datasets, backbone architectures, post-processing, image resolutions and more. This begs the question as to what percentage of the improvements are due to the fundamental contributions of each approach, and how much is unaccounted for in the silent changes. In this paper, we aim to answer this question by first studying the effect of changing components rarely claimed as contributions. Based on the findings, we train recent SotA methods in a comparable way, further improving their performance and evaluating each contribution independently. Benchmark Datasets The objective of this paper is to critically examine recent SotA contributions to monocular depth learning. One of the biggest hurdles to overcome is the lack of informative benchmarks due to erroneous evaluation procedures. By far, the most common evaluation dataset in the field is the Kitti Eigen (KE) split (Eigen & Fergus, 2015). Unfortunately, it has contained critical errors since its creation, the most egregious being the inaccuracy of the ground-truth depth maps. The original Kitti data suffered from artifacts due to the camera and LiDAR not being identically positioned. As such, each sensor had slightly different viewpoints, each with slightly different occlusions. This was exacerbated by the sparsity of the LiDAR, resulting in the background bleeding into the foreground. An example of this can be seen in Figure 2. Furthermore, the conversion to depth maps omitted the transformation to the camera reference frame and used the raw LiDAR depth instead. Finally, the Squared Relative error was computed incorrectly without the squared term in the denominator. Although these errors have been noted by previous works, they have nevertheless been propagated from method to method up to this day due to the need to compare like-with-like when reporting results. It is much easier to simply ignore these errors and copy the results from existing papers, than it is to correct the evaluation procedure and re-evaluate previous baselines. We argue this behavior should be corrected immediately and provide an open-source codebase to make the transition simple for future authors. Our benchmark consists of two datasets: the Kitti Eigen-Benchmark (KEB) split (Uhrig et al., 2018) and the newly introduced SYNS-Patches dataset. Uhrig et al. (2018) aimed to fix the aforementioned errors in the Kitti (Geiger et al., 2013) dataset. This was done by accumulating LiDAR data over ±5 frames to create denser ground-truth and removing occlusion artifacts via SGM (Hirschmüller, 2005) consistency checks. The final KEB split represents the subset of KE with available corrected ground-truth depth maps. This consists of 652 of the original 697 images. Similarly, we report the well-established image-based metrics from the official Kitti Benchmark. While some of these overlap with KE, we avoid saturated metrics such as δ < 1.25 3 or the incorrect SqRel error 2 . We also report pointcloud-based reconstruction metrics proposed by Örnek et al. (2022). They argue that imagebased depth metrics are insufficient to accurately evaluate depth estimation models, since the true objective is to reconstruct the 3-D scene. Further details can be found in Section B.1. Nevertheless, the Kitti dataset is becoming saturated and a more varied and complex evaluation framework is required. Kitti Eigen-Benchmark SYNS-Patches The second dataset in our benchmark is the novel SYNS-Patches, based on SYNS (Adams et al., 2016). The original SYNS is composed of 92 scenes from 9 different categories. Each scene contains a panoramic HDR image and an aligned dense LiDAR scan. This provides a previously unseen dataset to evaluate the generalization capabilities of the trained baselines. We extend depth estimation to a wider variety of environments, including woodland & natural scenes, industrial estates, indoor scenes and more. SYNS provides dense outdoor LiDAR scans. Previous dense depth datasets (Koch et al., 2018) are typically limited to indoor scenes, while outdoor datasets (Geiger et al., 2013;Guizilini et al., 2020) are sparse. These dense depth maps allow us to compute metrics targeting high-interest regions such as depth boundaries. Our dataset is generated by sampling 18 undistorted patches per scene, performing a full horizontal rotation roughly at eye level. To make this dataset more amenable to the transition from Kitti, we maintain the same aspect ratio and extract patches of size 376 × 1242. We follow the same procedure on the LiDAR to extract aligned dense ground-truth depth maps. Ground-truth depth boundaries are obtained via Canny edges in the dense log-depth map. After manual validation and removal of data with dynamic object artifacts, the final test set contains 1,175 of the possible 1,656 images. We show the distributions of depth values in Figure 3, the images per category in Table 1 and some illustrative examples in Figure 4. For each dataset, we additionally compute the percentage of image pixels with ground-truth depth values. The original Kitti Eigen has a density of only 4.10%, while the improved Kitti Eigen-Benchmark has 15.28%. Meanwhile, SYNS-Patches has ground-truth for 78.30% of the scene, demonstrating the high-quality of the data. We report image-based metrics from Uhrig et al. (2018) and pointcloud-based metrics from Örnek et al. (2022). For more granular results, we compute these metrics only at depth boundary pixels. Finally, we compute the edge-based accuracy and completeness from Koch et al. (2018) using the Chamfer pixel distance to/from predicted and ground-truth depth edges. Further dataset creation details can be found in Section B.3. Note that SYNS-Patches is purely a testing dataset never used for training. This represents completely unseen environments that test the generalization capabilities of the trained models. The Design Decisions That Matter Most contributions to self-supervised monocular depth estimation focus on alterations to the view synthesis loss (Godard et al., 2017;2019;Zhan et al., 2018), additional geometric consistency (Mahjourian et al., 2018;Bian et al., 2019) or regularization and the introduction of proxy supervised losses (Kuznietsov et al., 2017;Tosi et al., 2019;Watson et al., 2019). In this paper, we return to first principles and study the effect of changing components in the framework rarely claimed as contributions. We focus on practical changes that lead to significant improvements, raising the overall baseline performance and providing a solid platform on which to evaluate recent SotA models. Since this paper primarily focuses on the benchmarking and evaluation procedure, we provide a detailed review of monocular depth estimation and each contribution as supplementary material in Section A. We encourage readers new to the field to refer to this section for additional details. It is worth reiterating that the depth estimation network only ever requires a single image as its input, during both training and evaluation. However, methods that use monocular video sequences during training require an additional relative pose regression network. This replaces the known fixed stereo baseline used by stereo-trained models. Note that this pose network is only required during training to perform the view synthesis and compute the photometric loss. As such, it can be discarded during the depth evaluation. Furthermore, since all monoculartrained approaches use the same pose regression system, it is beyond the scope of this paper to evaluate the performance of this component. Implementation details We train these models on the common Kitti Eigen-Zhou (KEZ) training split (Zhou et al., 2017), containing 39,810 frames from the KE split where static frames are discarded. Most previous works perform their ablation studies on the KE test set, where the final models are also evaluated. This indirectly incorporates the testing data into the hyperparameter optimization cycle, which can lead to overfitting to the test set and exaggerated performance claims. We instead use a random set of 700 images from the KEZ validation split with updated ground-truth depth maps (Uhrig et al., 2018). Furthermore, we report the image-based metrics from the Kitti Benchmark and the pointcloud-based metrics proposed by Örnek et al. (2022), as detailed in Section B.2. For ease of comparison, we add the performance rank ordering of the various methods. Image-based ordering uses AbsRel, while pointcloud-based uses the F-Score. Models were trained for 30 epochs using Adam with a base learning rate of 1e−4, reduced to 1e−5 halfway through training. The default DepthNet backbone is a pretrained ConvNeXt-T (Liu et al., 2022), while PoseNet uses a pretrained ResNet-18 (He et al., 2016). We use an image resolution of 192 × 640 with a batch size of 8. Horizontal flips and color jittering are randomly applied with a probability of 0.5. We adopt the minimum reconstruction loss and static pixel automasking losses from Monodepth2 (Godard et al., 2019), due to their simplicity and effectiveness. We use edge-aware smoothness regularization (Godard et al., 2017) with a scaling factor of 0.001. These losses are computed across all decoder scales, with the intermediate predictions upsampled to match the full resolution. We train in a Mono+Stereo setting, using monocular video and stereo pair support frames. To account for the inherently random optimization procedure, each model variant is trained using three random seeds and mean performance is reported. We emphasize that the training code has been publicly released alongside the benchmark code to ensure the reproducibility of our results and to allow future researchers to build off them. Backbone Architecture & Pretraining Here we evaluate the performance of recent SotA backbone architectures and their choice of pretraining. Results can be found in Table 2 & Figure 5. We test ResNet (He et al., 2016), ResNeXt (Xie et al., 2017), MobileNet-v3 (Howard et al., 2019), EfficientNet , HRNet and ConvNeXt (Liu et al., 2022). ResNet variants are trained either from scratch or using pretrained supervised weights from Wightman (2019). ResNeXt-101 variants additionally include fully supervised (Wightman, 2019), weakly-supervised and self-supervised (Yalniz et al., 2019) weights. All remaining backbones are pretrained by default. Figure 5: Backbone Ablation. We show the relative performance improvement in F-Score, MAE, LogSI and AbsRel obtained by different backbone architectures and pretraining methods. Most existing papers limit their backbone to a pretrained ResNet-18, resulting in limited improvements. Full results in Table 2. As seen in Table 2, ConvNeXt variants outperform all other backbones, with HRNet-W64 following closely behind. Within lighter backbones, i.e. < 20 MParams, we find HRNet-W18 to be the most effective, greatly outperforming mobile backbones such as MobileNet-v3 and EfficientNet. Regarding pretraining, all ResNet backbones show significant improvements when using pretrained weights. ResNeXt variants are further improved by using pretrained weights from self-supervised or weakly-supervised training (Yalniz et al., 2019). A summary of these results can be found in Figure 5, showing the relative improvement over ResNet-18 (from scratch) in F-Score, MAE, LogSI and AbsRel metrics. The vast majority of monocular depth approaches simply stop at a pretrained ResNet-18 backbone (∼20% improvement). However, replacing this with a wellengineered modern architecture such as ConvNeXt or HRNet results in an additional 15% improvement. As such, the remainder of the paper will use the ConvNeXt-B backbone when comparing baselines. Depth Regularization The second study evaluates the performance of various commonly used depth regularization losses. We focus on variants of depth spatial gradient smoothing such as first-order (Garg et al., 2016), edge-aware (Godard et al., 2017), second-order and Gaussian blurring. We additionally test two variants of occlusion regularization , favoring either background or foreground depths. Results are shown in Table 3. We evaluate these models on the proposed KEZ validation split, as well as the KE test set commonly used by previous papers. Once again, the KEZ split is used to limit the chance of overfitting to the target KEB and SYNS-Patches test sets. When evaluating on the updated ground-truth, using no additional regularization produces the best results. All variants of depth smoothness produce slightly inferior results, all comparable to each other. Incorporating occlusion regularization alongside smoothness regularization again leads to a decrease in performance. Meanwhile, on the inaccurate ground-truth (Eigen & Fergus, 2015), smoothness constraints produce slightly better results. We believe this is due to the regularization encouraging oversmoothing in boundaries, which mitigates the effect of the incorrect groundtruth boundaries shown in Figure 2. As such, this is overfitting to errors in the evaluation criteria, rather than improving the actual depth prediction. Meanwhile, the large overlapping receptive fields of modern architectures such as ConvNeXt are capable of implicitly providing the smoothness required by neighbouring depths. Once again, these results point towards the need for an up-to-date benchmarking procedure that is based on reliable data and more informative metrics. We study the effect of adding smoothness (edge-aware, first-/second-order, w/wo Gaussian blurring) and occlusion regularization (prefer background/foreground). Whilst all methods typically use these regularizations, omitting them provides the best performance when evaluating on the corrected ground-truth. Meanwhile, these regularizations provide slight improvements on the outdated Kitti Eigen split. This is likely due to oversmoothing to account for the inaccurate boundaries shown in Figure 2. Table 4: State-of-the-Art Summary. We summarize the settings used by each evaluated method in the benchmark. Contributions made by each method are indicated by either bold font or an asterisk. Please note that these settings do not exactly reflect the original implementations by the respective authors, since we have introduced changes for the sake of comparability and performance improvement. Train Decoder Proxy Min+Mask Feat Virtual Blend Mask Smooth Occ SfM-Learner M Monodepth Explainability ✓ Klodt M Monodepth Uncertainty ✓ Monodepth2 M Monodepth ✓* ✓ Johnston M Discrete ✓ ✓ HR-Depth M HR-Depth ✓ ✓ Garg S Monodepth ✓ Monodepth S Monodepth ✓* ✓* SuperDepth S Sub-Pixel ✓* ✓ ✓ Depth-VO-Feat MS Monodepth DepthNet ✓ Monodepth2 MS Monodepth ✓* ✓ FeatDepth MS Monodepth ✓ AutoEnc ✓ CADepth MS CADepth ✓ ✓ DiffNet MS DiffNet ✓ ✓ HR-Depth MS HR-Depth ✓ ✓ Kuznietsov SD* Monodepth berHu ✓ DVSO SD* Monodepth berHu ✓ ✓* ✓* MonoResMatch SD* Monodepth berHu ✓ ✓ DepthHints MSD* Monodepth LogL1 ✓ ✓Ours Results This section presents the results obtained when combining recent SotA approaches with our proposed design changes. Once again, the focus lies in training all baselines in a fair and comparable manner, with the aim of finding the effectiveness of each contribution. For completeness and comparison with the original papers, we report results on the KE split in Section 5.2. However, it is worth reiterating that we strongly believe this evaluation should not be used by future authors. Evaluation details We cap the maximum depth to 100 meters (compared to the common 50m (Garg et al., 2016) or 80m (Zhou et al., 2017)) and omit border cropping (Garg et al., 2016) & stereo-blending post-processing (Godard et al., 2017). Again, we show the rank ordering based on image-based (AbsRel), pointcloud-based (F-Score) and edge-based (F-Score) metrics. Stereo-supervised (S or MS) approaches apply a fixed scaling factor to the prediction, due to the known baseline between the cameras during training. Monocular-supervised (M) methods instead apply per-image median scaling to align the prediction and ground-truth. We largely follow the implementation details outlined in Section 4.1, except for the backbone architecture, which is replaced with ConvNeXt-B (Liu et al., 2022). As a baseline, all methods use the standard DispNet decoder (Mayer et al., 2016) (excluding methods proposing new architectures), the SSIM+L1 photometric loss (Godard et al., 2017), edge-aware smoothness regularization (Godard et al., 2017) and upsampled multiscale losses (Godard et al., 2019). We settle on these design decisions due to their popularity and prevalence in all recent approaches. This allows us to minimize the changes between legacy and modern approaches and focus on the contributions of each method. The specific settings and contributions of each approach can be found in Table 4. For the full details please refer to the original publication, the review in Section 2 or the public codebase. We label the training supervision as follows: M = Monocular video synthesis, S = Stereo pair synthesis & D* = Proxy depth regression. In this case, all proxy depth maps used for regression (D) are obtained via the hand-crafted stereo disparity algorithm SGM (Hirschmüller, 2005). We further improve their robustness via the min reconstruction fusion proposed by Depth Hints (Watson et al., 2019). (Zhou et al., 2017) and stereo (Garg et al., 2016) baselines. For instance δ < 1.25 accuracy is improved by a raw 10% in both cases, while AbsRel error is decreased by 5%. As such, models with the proposed changes represent the new SotA. Kitti Eigen We first validate the effectiveness of our improved implementations on the KE split. As discussed, we strongly believe that this dataset encourages suboptimal design decisions, and future work should not evaluate on it. We report the original metrics from Eigen & Fergus (2015), as detailed in Section B.1. Results can be found in Table 5, alongside results from the original published papers. As seen, the models trained with our design decisions improve over each of their respective baselines. This is particularly noticeable in the early baselines (Garg et al., 2016;Zhou et al., 2017), which have remained unchanged since publication. This again highlights the importance of providing up-to-date baselines that are trained in a comparable way. Kitti Eigen-Benchmark We report results on the KEB split, described in Sections 3.1 & B.2. To re-iterate, this is an updated evaluation using the corrected depth maps from Uhrig et al. (2018). From these images, we select a subset of 10 interesting images to evaluate the qualitative performance of the trained models. Results can be found in Table 6 & Figure 7. As shown, when training and evaluating in a comparable manner, the improvements provided by recent contributions are significantly lower than those reported in the original papers. In fact, we find the seminal stereo method by Garg et al. (2016) to be one of the top performers across all metrics. Most notably, it outperforms all other methods in 3-D pointcloud-based metrics, indicating that the reconstructions are the most accurate. Incorporating the discussed design decisions along with SotA contributions provides the best image-based performance. However, these contributions were not developed to optimize pointcloud reconstruction. As seen, purely monocular approaches perform worse than stereo-based methods, despite the median scaling aligning the predictions to the ground truth. However, incorporating the contributions from Monodepth2 ( Table 6. (Garg et al., 2016) is one of the top performing methods. The minimum reprojection and automasking losses (Godard et al., 2019) help to improve performance and mitigate monocular supervision artefacts. This is further improved via a high-resolution decoder (Lyu et al., 2021) and proxy depth supervision (Watson et al., 2019). However, these contributions only improve image-based depth metrics, but do not result in more accurate 3-D pointcloud reconstructions. Table 6. fixes most of these artifacts. Similarly, the minimum reconstruction loss leads to more accurate predictions for thin objects, such as traffic signs and posts. DepthHints (Watson et al., 2019) and HR-Depth (Lyu et al., 2021) independently improve the quality of the predictions via proxy depth supervision and a high-resolution decoder, respectively. However, these contributions do not significantly improve the 3-D reconstructions. SYNS-Patches Finally, we evaluate the baselines on the SYNS-Patches dataset. As discussed in Sections 3.2 & B.3, this dataset consists of 1175 images from a variety of different scenes, such as woodlands, natural scenes and urban residential/industrial areas. It is worth noting that we evaluate the models from previous section without re-training or fine-tuning. As such, SYNS-Patches represents a dataset completely unseen during training. This allows us to evaluate the robustness of the learned representations to new unseen environment types. We select a subset of 5 images per category to evaluate the qualitative performance. To make results more comparable, all models are evaluated using the monocular protocol, where each predicted depth map is aligned to the ground-truth using per-image median scaling. We reuse the metrics from the KEB split and additionally report edge-based accuracy and completion from Koch et al. (2018). Finally, we also compute the F-Score only at depth boundary pixels, reflecting the quality of the predicted discontinuities. Results for this evaluation can be found in Table 7. As seen, performance decreases drastically for all approaches, showing that models do not transfer well beyond the automotive domain. A further decrease in F-Score can be seen when evaluating only on depth edges, indicating this as a common source of error. Once again, models incorporating recent contributions provide SotA performance in traditional image-based Table 7, models trained on Kitti do not transfer well to natural scenes, such as woodlands. Similar to Kitti, we find the contributions from Mondepth2 (Godard et al., 2019) to improve prediction accuracy in challenging areas such as thin structures and object boundaries. Full results in Table 7. depth metrics. However, Garg et al. (2016) consistently remains one of the top performers in 3-D pointcloud reconstruction and edge-based metrics. In general, predicted edges are typically accurate (∼3 px error). However, there are many missing edges, as reflected by the large edge completeness error (∼26 px error). Qualitative depth visualizations can be found in Figure 8. Similar to Kitti, Monodepth2 (Godard et al., 2019) and its successors (Watson et al., 2019;Lyu et al., 2021) greatly improve performance on thin structures. However, there is still room for improvement, as shown by the railing prediction in the second image. This is reflected by the low 3-D reconstruction metrics. Furthermore, all methods perform significantly worse in natural and woodlands scenes, demonstrating the need for more varied training data. Conclusion This paper has presented a detailed ablation procedure to critically evaluate the current SotA in selfsupervised monocular depth learning. We independently reproduced 16 recent baselines, modernizing them with sensible design choices that improve their overall performance. When benchmarking on a level playing field, we show how many contributions do not provide improvements over the legacy stereo baseline. Even in the case where they do, the change in performance is drastically lower than that claimed by the original publications. This results in new SotA models that set the bar for future contributions. Furthermore, this work has shown how, in many cases, the silent changes that are rarely claimed as contributions can result in equal or greater improvements than those provided by the claimed contribution. Regarding future work, we identify two main research directions. The first of these is the generalization capabilities of monocular depth estimation. Given the results on SYNS-Patches, it is obvious that purely automotive data is not sufficient to generalize to complex natural environments, or even other urban environments. As such, it would be of interest to explore additional sources of data, such as indoor sequences, natural scenes or even synthetic data. The second avenue should focus on the accuracy on thin objects and depth discontinuities, which are challenging for all existing methods. This is reflected in the low F-Score and Edge Completion metrics in these regions. To aid future research and encourage good practices, we make the proposed codebase publicly available. We invite authors to contribute to it and use it as a platform to train and evaluate their contributions. A Monocular Depth Overview We provide a review of self-supervised monocular depth estimation, which serves as an introduction to the field while providing in-depth details about the implemented baseline models. We also provide some practical implementation details that are frequently omitted. For the purpose of this paper, we limit our benchmark to methods that perform only depth estimation along with optional VO prediction. We do not include methods that additionally learn other tasks, such as semantic segmentation, optical flow or surface normals. A.1 DepthNet The core component of the framework is the depth prediction network, composed of a fully-convolutional encoder-decoder architecture with skip connections. This network produces multi-scale dense depth predictions. Encoder. Any architecture producing a multi-scale feature representation can be used as the encoder. In practice, this is implemented in a highly flexible way by using the timm library (Wightman, 2019), containing pretrained models for recent SotA classification architectures. As part of the ablation presented in Section 4 we perform an in-depth study of the most effective backbone architecture and its pretraining method. We find ConvNeXt (Liu et al., 2022) to provide the best performance overall. Depth Decoder. The commonly used dense decoder is inspired by the DispNet architecture (Mayer et al., 2016). It contains five upsampling stages, each composed of two Conv-ELU blocks with nearest-neighbour upsampling and a concatenated skip connection from the corresponding encoder stage. Each stage makes an initial depth prediction at its reduced resolution, used as additional supervision during training. These intermediate predictions are omitted from the following network and loss equations for the sake of brevity and clarity. In practice, DepthNet predicts the inverse depth (i.e. disparity) due to its increased stability. As such, this prediction is obtained via a convolution followed by a sigmoid activation. The depth prediction process can be summarized aŝ Z = Φ D (I) ,(1) where Φ D is the full DepthNet architecture andẐ is the predicted sigmoid disparity. This is converted into a scaled depth prediction throughD = 1 aẐ + b ,(2) where a & b are constants such that the final depth is in the range [0.1, 100]. It is worth noting this scale is arbitrary and does not correspond to metric depth. Virtual Stereo. Monodepth (Godard et al., 2017) introduced the concept of a virtual stereo prediction, where the network is forced to predict both the left and right disparity from only the left input image. This is achieved by adding an extra output channel to the disparity prediction at each encoder scale. The original Monodepth was trained using only images from the left viewpoint, meaning the virtual disparity would always correspond to the right view. To make this procedure more flexible and allow for training with either stereo viewpoint, we adopt a procedure similar to 3Net (Poggi et al., 2018). We assume the input to the network is the central viewpoint in a threecamera rig and predict two virtual disparities, corresponding to the left/right cameras. During training, we select the "real virtual" disparity, i.e. if the network was given the right image we sample the virtual left disparity, and vice versa. It is worth noting this does not require an additional decoder as in Poggi et al. (2018), since we simply add two extra output channels to each network prediction. Predictive Mask. Training in a purely monocular setting (Zhou et al., 2017;Klodt & Vedaldi, 2018) is susceptible to artifacts not present during stereo training. Dynamic objects moving independently from the rest of the scene are unaccounted for in the predicted PoseNet motion, resulting in incorrect synthesized views even if the depth prediction is correct. If the object is moving at similar speeds to the camera, it will appear as static between images, resulting in predictions of infinite depth. Meanwhile, objects moving towards the camera will seem closer than they are and depth will be underestimated. To mitigate these effects, SfM-Learner (Zhou et al., 2017) introduced an additional decoder to predict an explainability mask in the range [0, 1]. This mask was trained to downweight the photometric loss in unreliable regions, such as dynamic objects or specularities and regularized using a binary cross-entropy loss pushing all mask values towards 1. Klodt & Vedaldi (2018) instead adopted the uncertainty formulation introduced by Kendall & Gal (2017). The network is therefore trained to predict the log variance uncertainty associated with each photometric error pixel. In this case, the mask is restricted to positive uncertainty values. Following recent implementations (Godard et al., 2019), these masks are predicted for each of the support images via an additional DepthNet decoder. Details on integrating these masks into the photometric loss are provided in Section A.4. A.2 PoseNet Encoder. Similar to DepthNet, the encoder can be easily replaced with any desired architecture by leveraging the pretrained models from Wightman (2019). However, PoseNet is typically restricted to a pretrained ResNet-18 for efficiency. VO requires us to predict the relative motion between frames. As such, we perform an independent prediction for each support frame by concatenating them channel-wise with the target frame. We modify the first convolutional layer of the architecture accordingly, ensuring the pretrained weights are correctly duplicated and scaled. As is common, we force the network to make a forward-motion prediction by swapping the image order if necessary. Decoder. Since VO is a holistic regression task (it produces one output for the whole image), we use a simple decoder. The encoder features are projected to a lower dimensionality via a 1 × 1 convolution and processed by two Conv-ReLU blocks. The final prediction is obtained by applying a convolution without an activation. As is common practice, we scale the network predictions by a factor of 0.01 to improve stability and convergence. The network produces a 6-D output, formed by a translation and axis-angle rotation. The rotation vector is converted into a 3 × 3 transform matrix via the Rodrigues formula, with the magnitude indicating the angle and the direction corresponding to the axis of rotation. Overall, the process for obtaining the predicted motionP t → t + k between frames I t & I t + k is simplified aŝ P t → t + k = Φ P (I t ⊕ I t + k ) ,(3) where ⊕ represents channel-wise concatenation and Φ P is the resulting PoseNet. A.3 View Synthesis The core component of all self-supervised monocular depth approaches is the ability to synthesize the target image from a set of adjacent support frames. Given the scene depth and camera locations, a point in the target image p t can be reprojected onto any of the available frames as p ′ t + k = KP t → t + kDt (p t ) K −1 p t ,(4) where K represents the shared camera intrinsic parameters,D t (p t ) is the predicted depth at the given point andP t → t + k is the predicted camera motion matrix (or the known stereo baseline). Using these correspondences, it is possible to synthesize a support frame aligned to the target image by bilinearly sampling at each image pixel. This is given by I ′ t + k = I t + k p ′ t + k ,(5) where ⟨·⟩ represents bilinear sampling using Spatial Transformer Networks (Jaderberg et al., 2015). A.4 Losses Photometric. The main loss is the photometric error. This measures the difference between the raw target frame I t and the synthesized view from the support frame I ′ t + k generated by (5). In the most simple case, an L 1 loss can be used. However, this is typically complemented by an SSIM (Shi et al., 2016) loss to improve its robustness. This is defined as L photo I, I ′ = λ ssim 1−L ssim I, I ′ 2 + (1−λ ssim ) L 1 I, I ′ ,(6) where λ ssim represents the SSIM weight. This photometric loss can be additionally downweighted when using a predictive mask. When using the explainability mask M E (Zhou et al., 2017) this is simply defined as L photo I, I ′ = M E ⊙ L photo I, I ′ , while the uncertainty mask M U (Klodt & Vedaldi, 2018) affects it as L photo I, I ′ = exp (−M U ) ⊙ L photo I, I ′ + M U ,(8) where ⊙ represents the Hadamard product. Reconstruction. The photometric loss is aggregated by the reconstruction loss. The base case simply averages the photometric loss over each support frame and each image pixel, defined as L rec (I t ) = k p L photo I t , I ′ t + k ,(9) where N i=0 i ≡ 1 N N i=0 i represents the average summation. Monodepth2 (Godard et al., 2019) showed the benefit of instead selecting the minimum loss per-pixel over the various support frames. Intuitively, this selects correct pixel-wise correspondences, instead of averaging out incorrect errors caused by occlusions. This is simplified as L rec (I t ) = p min k L photo I t , I ′ t + k .(10) Monodepth2 additionally proposed an automatic masking procedure. Static pixels are removed from the loss if the photometric error for the original support frame is lower than the synthesised view loss. This is given by M S = min k L photo I t , I ′ t + k < min k L photo (I t , I t + k ) ,(11) where · represents the Iverson brackets and M S is the resulting mask selecting non-static pixels. Note that the second term in the equation uses the original support frame I t + k . Feature reconstruction. As noted by previous works (Zhan et al., 2018;Spencer et al., 2020;Shu et al., 2020), the image-based reconstruction loss might still produce ambiguous correspondences. This is especially the case in low-light environments that contain multiple light sources and complex reflections. In this case, it can be beneficial to incorporate an additional feature-based reconstruction loss, defined as L feat (F t ) = p min k L photo F t , F ′ t + k ,(12)F ′ t + k = F t + k p ′ t + k ,(13) where F ′ t + k are the synthesized support features using the previously computed reprojection correspondences from (4). Note that in this case the L 1 + SSIM photometric loss can be replaced with the L 2 distance between feature embeddings. In practice, instead of introducing an additional pretrained dense feature network, we propose to re-use the low-level features from the depth encoder. Depth regression. To complement the self-supervised reconstruction losses, previous approaches have introduced an additional proxy supervised regression loss. This can prevent local minima during the optimization process. When computing the loss between disparities we opt for the standard L 1 loss. However, when in depth space we use other well-established losses. The first of these is the reverse Huber (berHu) loss (Zwald & Lambert-Lacroix, 2012;Laina et al., 2016), defined as L berHu D , D =      L 1 D , D where L 1 D , D ≤ δ L 1 D ,D +δ 2 2δ otherwise ,(14) where D represents the (proxy) ground truth depth,D is the network prediction and the margin threshold is adaptively set per-batch as δ = 0.2 max L 1 D , D . DepthHints (Watson et al., 2019) argues the berHu loss is better suited towards regressing ground-truth LiDAR and proposes to use the Log-L 1 loss when regressing hand-crafted stereo disparities, given by L logL1 D , D = log 1 + L 1 D , D .(15) Virtual stereo consistency. Following Monodepth (Godard et al., 2017), we use the virtual stereo consistency loss, forcing the network to predict a consistent scene depth for both viewpoints. This is achieved by applying the view synthesis module to the disparities, as opposed to the images. This results in L stereo Ẑ t ,Ẑ ′ t + k = L 1 Ẑ ,Ẑ ′ t + k ,(16)Z ′ t + k =Ẑ t + k p ′ t + k ,(17) whereẐ ′ t + k is the warped disparity corresponding to the virtual stereo prediction from the network. A.5 Regularization Disparity smoothness. In general, we expect a pixel's disparity to be similar to that of its neighbours. Garg et al. (2016) introduced a simple regularization constraint encouraging this by penalizing all gradients in the disparity map, defined as L smooth Ẑ = p ∂Ẑ (p) ,(18) where ∂Ẑ are the spatial gradients of the mean-normalized disparity, computed in the x & y directions independently. We omit these directions from the equations for clarity. Godard et al. (2017) softened this constraint by allowing disparity gradients proportional to the image gradients at that pixel location. This is known as the edge-aware smoothness regularization, given by Rui et al. (2018) instead enforce a smooth change in gradients, i.e. second-order smoothness. This is defined as L smooth Ẑ = p ∂Ẑ (p) exp (− \|∂I (p)\|) .(19)L smooth Ẑ = p ∂ 2Ẑ (p) exp − ∂ 2 I (p) ,(20) where ∂ 2Ẑ represents the second-order gradients in each spatial direction. In this benchmark we add the option to apply Gaussian smoothing prior to computing the disparity/image gradients. Occlusions. DVSO (Rui et al., 2018) noted that the smoothness constraints can result in oversmoothed predictions. To counteract this, they proposed an occlusion regularization term that penalizes the sum of disparities in the scene. This should allow the network to predict sharper boundaries at occlusions, while favouring background disparities. This is simply defined as L occ Ẑ = pẐ (p) .(21) Masks. Using the explainability mask (Zhou et al., 2017) lets the network downweigh the photometric loss in regions where it believes correspondences may be incorrect or uninformative. This occurs with dynamic object moving independently from the scene. To avoid the degenerate case where the mask ignores all pixels, an additional binary cross-entropy regularization pushes all values towards 1. B Benchmark Datasets Complementing Section 3, we provide additional details for each of dataset. This includes the various metrics defined by the Kitti splits, as well as the creation procedure for the SYNS-Patches dataset. B.1 Kitti Eigen As discussed in the main paper, we strongly encourage future authors to avoid this evaluation, due to the inaccurate ground-truth. However, we provide details for comparison with previous publications as the community transitions to the proposed benchmark. The KE split (Eigen & Fergus, 2015) defines the following metrics: AbsRel. Measures the mean relative error (%) as e = \|ŷ − y\| y ,(22) Note that this is incorrectly computed, as the squared term is missing from the denominator. RMSE. Measures the root mean square error (meters) as e = \|ŷ − y\| 2 .(24) LogRMSE. Measures the root mean square log error (log-meters) as e = \|log (ŷ) − log (y)\| 2 .(25) Threshold accuracy. Measures the threshold accuracy (%) δ < 1.25 κ , where κ ∈ {1, 2, 3} and δ = max ŷ y , ŷ y . B.2 Kitti Eigen-Benchmark As a replacement for KE, we propose to use the updated and corrected ground-truth from Uhrig et al. (2018). This benchmark defines the following metrics: Note that the second term in this metric is directional, i.e. the prediction is rewarded if it is consistently incorrect in the same direction. This accounts for the case where the prediction is correct, but scaled differently to the ground truth. The metrics described above measure the error in the predicted 2-D depth map. However, the true objective of monocular depth estimation is to accurately reconstruct the 3-D world. As such, Örnek et al. (2022) proposed to instead evaluate depth prediction using well-established 3-D metrics, which measure the fidelity of the reconstructed pointcloud. We report the following 3-D metrics: where Q &Q represent the ground-truth and predicted pointclouds and q &q are 3-D points in those pointclouds, respectively. Precision. Measures the number of predicted points (%) within a distance threshold δ to the ground-truth surface as P = q∈Q min q∈Q ∥q −q∥ < δ . Extracted ground-truth depth boundaries on raw/log/inv depth maps using increasing Gaussian blurring sigmas. Sigma 01 log-depth provides a good balance of edges-to-noise ratio. Recall. Measures the number of ground-truth points (%) within a distance threshold δ to the predicted surface as R = q∈Q min q∈Q ∥q −q∥ < δ . As per Örnek et al. (2022), we set the threshold for a correctly reconstructed point to δ = 0.1, i.e. 10 cm. F-Score. Also known as the Dice coefficient. Measures the harmonic mean of precision and recall (%) as F = 2 · P · R P + R . IoU. Also known as the Jaccard Index between two pointclouds. Measures the volumetric quality of a 3-D reconstruction (%) as IoU = P · R P + R − P · R .(40) B.3 SYNS-Patches We generate SYNS-Patches, based on SYNS (Adams et al., 2016), by sampling undistorted patches from the spherical panorama image every 20 azimuth degrees at a constant elevation of zero. In other words, we perform a full horizontal rotation roughly at eye level, sampling 18 images per scene. The original panoramic data was collected using a scanning camera rotating around its vertical axis, with each scan taking approximately 5-10 minutes. The LiDAR was captured after the HDR panorama following a similar procedure, resulting in dense depth scans. Since the panorama is not instantly captured, it is susceptible to distortions from moving objects. Furthermore, since the image and depth scans were captured independently, it is possible to have mismatched "static-but-dynamic" objects, such as parked cars or stationary pedestrians. We visually check the extracted dataset and remove images containing these artifacts, along with empty/uniform depth maps. The final test set contains 1,175 out of the possible 18 × 92 = 1, 656 images. Ground-truth depth boundaries are obtained by detecting Canny edges in the depth map. This is sensitive to missing data (due to infinite depth or highly reflective surfaces), which can cause fake boundaries. We take this into account by masking edges connected to regions with invalid depths, unless these pixels belong to the sky. Accurate sky masks are obtained using a pretrained SotA semantic segmentation model (Reda et al., 2018). We generate depth edges at three levels of Gaussian smoothing, using either raw, log or inverse depth maps. In practice, we find log-depth and a smoothing sigma of 1 to provide the best qualitative results. Example depth boundary predictions for different hyperparameters are shown in Figure 9. As in KEB, we use the original image-based metrics defined by Uhrig et al. (2018), complemented by the pointcloud-based metrics from Örnek et al. (2022). To provide more granular results, we optionally compute all metrics only at the detected ground-truth depth boundaries. We additionally report the edge-based metrics of Koch et al. (2018): EdgeAcc. Measures the accuracy of the predicted depth edges (pixels) via the distance from each predicted edge to the closest ground-truth edge as where EDT represents the Euclidean Distance Transform, truncated to a maximum distance δ = 10 and Y bin &Ŷ bin are the binary maps of depth boundaries for the ground-truth and prediction, respectively. EdgeComp. Measures the completeness of the predicted depth edges (pixels) via the distance from each ground-truth edge to the closest predicted edge as e = P EDT (Y bin (p)) : P = p\|Ŷ bin (p) = 1 . Figure 1 : 1Quantifying SotA Contributions. (a) Performance at the time vs. Figure 2 :Figure 3 : 23Inaccurate Ground-Truth. The original Kitti(Geiger et al., 2013) ground-truth data used by previous monocular depth benchmarks(Eigen & Fergus, 2015) is inaccurate and contains errors, especially at object boundaries. Note the background bleeding in the highlighted region.Uhrig et al. (2018) correct this by accumulating LiDAR data over multiple frames. Depth Distribution. We show the distribution of depths for KE, KEB & SYNS-Patches. SYNS-Patches contains more varied depth values due to the indoors scenes. In all cases the maximum depth is clamped to 100 meters during evaluation. Figure 4 : 4SYNS-Patches. Top: Diverse testing images. Middle: Dense depth maps. Bottom: Log-depth Canny edges. Figure 6 : 6Godard et al., 2019) results in significant improvements over SfM-Learner. Qualitative depth visualizations can be found inFigure 6. Once again, the seminalGarg et al. (2016) and SfM-Learner(Zhou et al., 2017) baselines are drastically improved w.r.t. the original implementation. However, SfM-Leaner predictions are still characterized by artifacts common to purely monocular supervision. Most notably, objects moving at similar speeds to the camera are predicted as holes of infinite depth, due to the fact that they appear static across images. Static pixel automasking from Monodepth2(Godard et al., 2019) Kitti Visualization. Baseline models(Garg et al., 2016;Zhou et al., 2017) are greatly improved from their original implementations. Incorporating the minimum reconstruction loss & automasking from Monodepth2(Godard et al., 2019) improves accuracy on thin structures and prevents holes of infinite depth. Full results in Figure 7 : 7Kitti Eigen-Benchmark Improvement. When training and evaluating in fair conditions, many contributions do not result in relative improvements w.r.t. the Garg et al. (2016) stereo baseline. Most notably, all monocular-supervised approaches perform significantly worse despite the per-image median scaling. Full results in Figure 8 : 8SYNS Visualization. As evidenced by represents the average summation. SqRel. Measures the mean relative square error (%) as e = \|ŷ − y\| 2 y . MAE. Measures the mean absolute error (meters) as e = \|ŷ − y\| . (27) RMSE. Measures the root mean square error (meters) as e = \|ŷ − y\| 2 . (28) InvMAE. Measures the mean absolute inverse error (Measures the mean absolute log error (log-meters) as e = \|log (ŷ) − log (y)\| . (31) LogRMSE. Measures the root mean square log error (log-meters) as e = \|log (ŷ) − log (y)\| 2 . (32) LogSI. Measures the root mean scale invariant log error (Eigen & Fergus, 2015) (log meters) as e = \|log (ŷ) − log (y)\| 2 − log (ŷ) − log (y) 2 . AbsRel. Measures the mean relative error (%) as e = \|ŷ − y\| y . (34) SqRel. Measures the mean relative square error (%) as e = \|ŷ − y\| 2 y 2 . Figure 9 : 9SYNS-Patches Dataset. P EDT Ŷ bin (p) : P = {p\|Y bin (p) = 1} , Table 1 : 1SYNS-Patches Scenes. We show the distibution of images per scene in the proposed dataset. This evaluates the model's capability to generalize beyond purely automotive data.Agriculture Natural Indoor Woodland Residential Industry Misc Recreation Transport Total 315 183 148 144 123 107 72 62 21 1,175 Table 2 : 2Backbone Ablation. We study the effect of various backbone architectures & pretraining methods. PT indicates use of pretrained ResNet weights. All other baselines are pretrained by default. ConvNeXt provides the best performance, followed by HRNet-W64. ResNeXt can be further improved by using self-/weakly-supervised weights. Frames per second were measured on an NVIDIA GeForce RTX 3090 with an image of size 192 × 640.Image-based Pointcloud-based Table 3 : 3Depth Regularization Ablation. Table 5 : 5Kitti Eigen Evaluation. Results reported in the original publications (top) vs. those obtained by our updated baselines (bottom). The proposed baselines outperform those provided by the original authors in every case, most notably in the mono Table 6 : 6Kitti Eigen-Benchmark Evaluation. When training in comparable conditions, the stereo baseline Table 7 : 7SYNS-Patches Evaluation.Overall performance is drastically reduced when evaluating outside the automotive training domain. Methods using minimum reconstruction loss and automasking improve image-based metrics, whileGarg et al. (2016) still provides some of the top 3-D pointcloud reconstructions. Predicted edge boundaries are typically accurate (Edge-Acc <5 px), but incomplete (Edge-Comp >25 px).Image-Based Pointcloud-based Edge-based SYNS-Patches Train # MAE↓ RMSE↓ AbsRel↓ # F-Score↑ IoU↑ # F-Score↑ Acc↓ Comp↓ SfM-Learner M 22 5.43 9.25 31.58 22 11.79 6.43 20 8.47 3.46 36.12 Klodt M 20 5.40 9.20 31.20 21 12.00 6.57 19 8.48 3.44 35.22 Monodepth2 M 13 5.33 9.02 30.05 20 12.08 6.62 21 8.46 3.30 37.01 Johnston M 10 5.24 8.92 29.72 18 12.16 6.66 18 8.60 3.23 42.82 HR-Depth M 8 5.26 8.95 29.53 5 13.37 7.40 6 9.16 3.07 30.03 Garg S 15 5.29 9.20 30.73 2 13.48 7.45 1 9.53 3.37 26.79 Monodepth S 21 5.29 9.20 31.27 19 12.14 6.67 17 8.69 3.57 61.15 SuperDepth S 16 5.26 9.08 30.83 13 12.87 7.10 11 9.01 3.40 40.40 Depth-VO-Feat MS 17 5.30 9.17 30.83 16 12.43 6.82 15 8.77 3.50 38.49 Monodepth2 MS 6 5.18 8.91 29.04 8 13.18 7.27 13 8.95 3.38 32.69 FeatDepth MS 7 5.16 8.80 29.12 17 12.27 6.73 22 8.41 3.50 44.09 CADepth MS 11 5.22 8.97 29.80 14 12.83 7.06 16 8.70 3.42 35.89 DiffNet MS 2 5.16 8.91 28.80 9 13.16 7.26 14 8.81 3.45 39.46 HR-Depth MS 5 5.13 8.85 28.94 1 13.79 7.65 5 9.21 3.25 28.33 Kuznietsov SD 19 5.47 9.50 31.08 10 13.15 7.26 7 9.11 3.39 47.13 DVSO SD* 9 5.18 8.93 29.66 11 13.08 7.23 2 9.29 3.34 40.23 MonoResMatch SD* 14 5.24 9.07 30.28 15 12.73 7.01 9 9.03 3.47 51.03 DepthHints MSD* 18 5.33 9.07 30.90 12 12.91 7.11 10 9.01 3.24 26.21 Ours MS 12 5.20 9.03 29.93 4 13.38 7.39 3 9.28 3.31 34.36 Ours (Min) MS 1 5.11 8.80 28.59 7 13.20 7.27 12 8.98 3.24 32.46 Ours (Proxy) MSD* 3 5.11 8.79 28.87 3 13.46 7.45 4 9.23 3.16 30.60 Ours (Min+Proxy) MSD* 4 5.08 8.71 28.91 6 13.23 7.30 8 9.11 3.16 31.32 Code is publicly available at https://github.com/jspenmar/monodepth_benchmark. 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[ "Conflict Mitigation Framework and Conflict Detection in O-RAN Near-RT RIC", "Conflict Mitigation Framework and Conflict Detection in O-RAN Near-RT RIC", "Conflict Mitigation Framework and Conflict Detection in O-RAN Near-RT RIC", "Conflict Mitigation Framework and Conflict Detection in O-RAN Near-RT RIC" ]	[ "Cezary Adamczyk ", "Adrian Kliks ", "Cezary Adamczyk ", "Adrian Kliks " ]	[]	[]	The steady evolution of the Open RAN concept sheds light on xApps and their potential use cases in O-RANcompliant deployments. There are several areas where xApps can be used that are being widely investigated, but the issue of mitigating conflicts between xApp decisions requires further in-depth investigation. This article defines a conflict mitigation framework (CMF) built into the existing O-RAN architecture; it enables the Conflict Mitigation component in O-RAN's Near-Real-Time RAN Intelligent Controller (Near-RT RIC) to detect and resolve all conflict types defined in the O-RAN Alliance's technical specifications. Methods for detecting each type of conflict are defined, including message flows between Near-RT RIC components. The suitability of the proposed CMF is proven with a simulation of an O-RAN network.Results of the simulation show that enabling the CMF allows balancing the network control capabilities of conflicting xApps to significantly improve network performance, with a small negative impact on its reliability. It is concluded that defining a unified CMF in Near-RT RIC is the first step towards providing a standardized method of conflict detection and resolution in O-RAN environments.	10.1109/mcom.018.2200752	[ "https://export.arxiv.org/pdf/2305.07117v1.pdf" ]	258,572,793	2305.07117	3d0a659c64b59310ca9974dace9ee71aa490dbe3	Conflict Mitigation Framework and Conflict Detection in O-RAN Near-RT RIC Cezary Adamczyk Adrian Kliks Conflict Mitigation Framework and Conflict Detection in O-RAN Near-RT RIC 10.1109/MCOM.018.22007521Index Terms-O-RANNear-RT RICconflict detectioncon- flict mitigation The steady evolution of the Open RAN concept sheds light on xApps and their potential use cases in O-RANcompliant deployments. There are several areas where xApps can be used that are being widely investigated, but the issue of mitigating conflicts between xApp decisions requires further in-depth investigation. This article defines a conflict mitigation framework (CMF) built into the existing O-RAN architecture; it enables the Conflict Mitigation component in O-RAN's Near-Real-Time RAN Intelligent Controller (Near-RT RIC) to detect and resolve all conflict types defined in the O-RAN Alliance's technical specifications. Methods for detecting each type of conflict are defined, including message flows between Near-RT RIC components. The suitability of the proposed CMF is proven with a simulation of an O-RAN network.Results of the simulation show that enabling the CMF allows balancing the network control capabilities of conflicting xApps to significantly improve network performance, with a small negative impact on its reliability. It is concluded that defining a unified CMF in Near-RT RIC is the first step towards providing a standardized method of conflict detection and resolution in O-RAN environments. I. INTRODUCTION T HE concept of open radio access networks (O-RAN) has been widely investigated in recent years by large organizations comprising mobile network operators (MNOs), telecommunications companies, and research facilities. One of such organizations is the O-RAN Alliance, which formulates technical specifications to guide the industry towards O-RAN. Its efforts enable researchers and companies to build upon the laid foundations, with each new series of technical specifications providing details of the ideas that drive the whole O-RAN concept [1]. The The next unique aspect of O-RAN is the introduction of new components that can influence the RAN operation. These include Non-Real-Time (Non-RT) and Near-Real-Time (Near-RT) RAN Intelligent Controllers (RICs), the former of which is a part of the Service Management and Orchestration (SMO) framework. These logical functions react to current and forecasted future network states and adapt its parameters to optimize performance. Last, but not least, the accessibility of O-RAN is accelerated with the concept of applications in RICs. These applications run in the Non-RT RIC (called rApps) and in the Near-RT RIC (called xApps). Both rApps and xApps can be developed by independent parties and then deployed on all O-RANcompliant platforms, regardless of their hardware and software vendor. The aim of rApps and xApps is to influence the network operation from the RICs. For example, an rApp can realize a use case to adjust radio resource allocation policies to reduce latency and improve performance in dynamic handover for V2X scenarios [2]; the Non-RT RIC then interfaces with the Near-RT RIC to provide policy updates provided by rApps. An example of an xApp is an application that manages radio resources in such a way as to maximize Quality of Service (QoS) for a group of users [2] using a dedicated interface to send control messages to RAN nodes. Prioritization within all areas of the O-RAN space is decided upon in the Minimum Viable Plan (MVP) towards the commercialization of the concept [3]. It comprises key endto-end use cases that are applicable in commercial RAN, priorities of which are chosen by MNO members of O-RAN Alliance. In June 2021, the priority was set on use cases such as traffic steering, QoS optimization, or Massive Multiple-Input Multiple-Output (mMIMO) optimization. Since then, multiple new use cases for O-RAN have been defined, with energy saving being one of the most important new focuses for network operators [2]. All of these use cases are to be fulfilled using various xApps and rApps working concurrently in the RICs. As multiple applications can be deployed and activated in the network at the same time, their influences on the network operation may conflict with each other. Such conflicts need to be detected and mitigated in both Non-RT and Near-RT domains. This article proposes a novel conflict mitigation framework (CMF) to efficiently detect and resolve conflicts between network control decisions made by xApps in Near-RT RICs. The key contributions of this work are as follows: first, we present a comprehensive framework for conflict mitiga-tion in O-RAN, which monitors network events to detect and resolve all conflict types between xApps, and which we propose to make standardized. Secondly, based on that framework, we propose procedures to mitigate the conflicts, along with detailed templates of exchanged messages. Finally, we propose a pragmatic way of resolving conflicts between same-importance (i.e., of the same level in the hierarchy) RAN control decisions based on prioritization and provide simulation results that show how this approach influences the network. Lastly, we outline the expected next steps in research of conflict mitigation in O-RAN. II. CONFLICTS BETWEEN RAN CONTROL DECISIONS Although O-RAN's flexibility and novel intelligent RAN components allow for robust RAN optimization, these features also create unique challenges in controlling the network. Control conflicts are possible as multiple agents can decide on RAN operation parameters. These conflicts can potentially nullify or lessen the decisions' influence on the network performance, so they should be avoided where possible with proper network setup [4]. However, the complex dependencies between RAN control activities and potential expansion of network control measures make complete mitigation of conflicts with static configuration not viable. Thus, any RAN control conflicts that appear during O-RAN network operation need to be detected and resolved. For that purpose, Conflict Mitigation (CM) components are envisioned in the Near-RT RIC and in the Non-RT RIC [5,6]. Although the O-RAN architecture sections out the CM logic only in the RICs and it is the main focus of this article, some CM capabilities could be realized on the SMO level (and outside of the Non-RT RIC). Control decision conflicts can happen at many levels in the O-RAN architecture. Within RICs, these conflicts can happen between xApps in a Near-RT RIC and between rApps in the Non-RT RIC when control activities of applications contradict each other. Other than that, Near-RT RICs may also conflict each other, specifically if they control RAN nodes located in close proximity. These conflicts between equivalent components can be referred to as horizontal conflicts. Control conflicts can also happen between components on different levels of the architecture, i.e., between the Non-RT RIC and Near-RT RICs. An example of such a type of conflict is when an xApp either acts against a policy, or contradicts a control decision provided by the Non-RT RIC. These conflicts can be named vertical conflicts. All the considered areas of potential conflict within the O-RAN architecture are shown in Fig. 1. The decision conflicts within O-RAN are considered in this article in the context of horizontal conflicts between xApps in a Near-RT RIC (intra-Near-RT RIC) and between Near-RT RICs (inter-Near-RT RIC). Similar considerations can be applicable for horizontal conflicts between rApps (intra-Non-RT RIC), as they provide policy and non-real-time control decisions similarly to how xApps provide near-real-time control decisions. As for the vertical conflicts between the Non-RT RIC and Near-RT RICs, the Near-RT RIC's policy-conflicting control decisions may either be rejected or allowed by the Non-RT RIC. The specific conflict resolution action should be decided A. Types of conflicts between xApp decisions O-RAN technical specifications distinguish three types of conflicts that can happen between xApps [5], namely direct, indirect, and implicit. The first of these are direct conflicts, which concern contradicting decisions that happen one after the other and affect the same set of configuration parameters. An example of such type of conflict may be when xApp #1 decides to assign a user to a cell and right after that xApp #2 decides to assign the same user to another cell. In this particular scenario, the result of the conflicting actions will be that only the decision of xApp #2 will have an effect. Undetected direct conflicts may lead xApp #1 to draw wrong conclusions about the effect of its decision. Another type of conflict in the Near-RT RIC is the indirect conflict. It refers to a situation where decisions contradict each other when modifying parameters that influence the same areas of the RAN operation. For example, indirect conflicts can happen between xApps modifying parameters that influence handover boundaries, such as when xApp #1 controls an antenna's electrical tilt and xApp #2 changes the Cell Individual Offset (CIO). This may lead to situations where the effective handover boundary fluctuates heavily due to xApps reacting to each other's contradicting decisions. The implicit conflict considers situations in which conflicting influences of network control decisions are difficult to observe and determine. Implicit conflicts are expected when many xApps simultaneously optimize the RAN operation with separate goals, controlling different network parameters. They can appear in a situation where xApp #1 aims to maximize QoS for a group of users while xApp #2 attempts to minimize the number of handovers between neighboring cells; decisions made by these two xApps may interfere in a non-obvious manner, disrupting each other's effect on the network. Conflicts may also happen between Near-RT RICs that control RAN nodes in close proximity (i.e., the inter-Near-RT RIC conflicts). In such cases, the conflicting Near-RT RICs provide contradicting RAN control decisions, e.g., triggering a "ping-pong" handover of a user between cells managed by both Near-RT RICs. Such network behavior leads to inefficient radio resource utilization. B. Dealing with conflicts and related works To achieve predictable performance and behavior of O-RAN, it is necessary to minimize any negative impact of conflicts between xApp decisions. As prevention is preferred, the design of Near-RT RIC implementations should aim to avoid any conflicts between xApps. Unfortunately, some conflicts will inevitably happen with various third parties providing xApps. Therefore, the Near-RT RIC must be able to resolve any conflicts between xApp decisions. To do so, the conflict firstly needs to be detected. Currently, no methods of detecting or resolving conflicts are defined in O-RAN's technical specifications as they are considered for future study [5]. The topic of conflict mitigation in O-RAN has not yet been significantly covered in research papers. One of the relevant papers on a related topic has been published by Zhang et al. [7], who proposed a team learning algorithm based on Deep Q-learning that requires two xApps working in cooperation in an O-RAN environment. While the referenced work does not describe a universal solution to the issue of conflicts in O-RAN, it proposes a viable way to mitigate the conflicts using a cooperative machine learning scheme. Results of the team learning algorithm showed that when xApps take into consideration each other's decisions, the learning process is much more efficient in comparison to xApps working independently, leading to overall higher system throughput and lower packet drop rate. Although promising, this approach is hard to scale to a larger number of xApps because the decision logic of all participating xApps needs to adapt to the number of cooperating xApps and be able to properly interpret their decisions. Therefore, the team learning approach can be utilized only if xApps are designed with team learning in mind. Significant work has been done in the area of conflict mitigation in the context of Self-Organized Networks (SON). The concept of SON has been introduced as part of Long Term Evolution (LTE) networks to reduce operating costs related to multi-vendor RAN. SON allows for the deployment of SON functions, which, as xApps and rApps in O-RAN, are able to modify network behavior, albeit only locally. Two SON functions widely considered in research are Mobility Robustness Optimization (MRO) and Mobility Load Balancing (MLB). Relevant work on conflict mitigation in the context of SON includes a scheme to resolve conflicts between MRO and MLB, which limits the range of changes to CIO values done by MLB [8]. This approach has been improved in later research to find optimal values of CIO instead of limiting the range [9]. To address the problem of decision conflicts in a more general way, a soft classification of conflicts in SON has been proposed [10]. A general framework for self-coordination in SON has been defined to universally deal with conflicts, regardless of conflicting SON function types [11]; it describes the concept of SON Coordinator that adapts network control to current conditions, mitigating any conflicts and providing safeguards from network performance degradation. Another generalized solution in SON utilizes machine learning to learn from past experience and predict network performance to better solve the conflicts [12]. Although RAN control problems in SON and O-RAN share many similarities (i.e., SON functions and xApps/rApps as conflicting agents, common use cases), the solutions for SON cannot be directly applied to O-RAN. This is mainly due to a significant difference in how RAN control is realized in SON and O-RAN. SON functions perform RAN control locally (i.e., each RAN node optimizes its parameters based on what it can observe), while RAN control in O-RAN is executed with RICs, which can observe the entire managed network and optimize it centrally. Nevertheless, the relevant work on SON can provide insight into how the problem of conflict mitigation can be approached and what measures may prove effective in solving the control conflict problems. Finally, it is worth mentioning multi-agent reinforcement learning (MARL) methods that consider multi-agent systems (MAS), where autonomous agents with various goals perform actions in a shared environment [13]. In an MAS, agents' actions influence each other and need to be coordinated to reach an optimal state of the environment. This case can be considered analogous to the O-RAN scenario, where xApps and rApps provide RAN control decisions to fulfill various use cases. According to the O-RAN architecture descriptions provided by O-RAN Alliance, the default RAN control setting in O-RAN is centralized and features the RICs as coordinators of RAN control activities. Hence, cooperative and coordinated MARL approaches could be applied. Although this is a viable solution, it requires xApps and rApps to be implemented with MARL in mind. We consider this a potential evolution path for the conflict mitigation methods in O-RAN. However, this article focuses on solutions that can be applied to all applications, regardless of their implementation. III. CONFLICT MITIGATION FRAMEWORK In this article, we propose a framework for conflict mitigation (referred to in the paper as CMF) in O-RAN-compliant environments. The proposed framework is envisioned as part of the CM component of Near-RT RIC. CMF aims to provide a robust, standardized method for conflict detection in O-RAN, covering the detection of all three types of conflict defined in O-RAN specifications. The proposed framework embedded within the architecture of Near-RT RIC is shown in Fig. 2. A. Structure of CMF and infrastructure prerequisites CMF envisions deploying the Conflict Detection (CD) Agent and the Conflict Resolution (CR) Agent components in the Near-RT RIC's CM entity as agents that detect conflicts between xApps and resolve them. The main research challenge for the CD Agent is to enable reliable detection of all types of conflicts, and for the CR Agent to enable the resolution of all conflicts to reach an optimal network state. There are three logical parts distinguished within the CD Agent that are dedicated to detection methods for each type of conflict between xApp decisions. These components are described in the following subsections. As for the CR Agent, it is assumed that it can resolve any conflicts detected by the CD Agent. Copyright © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. This is the author's version of an article that has been published in the IEEE Communications Magazine. Changes were made to this version by the publisher prior to publication, the early access version of the record is available at: 10.1109/MCOM.018.2200752. The operation of CMF relies on the configuration of the O-RAN infrastructure. Data stored by CMF is kept in the Near-RT RIC's Database, which is accessible via the Shared Data Layer (SDL). The Message Infrastructure in the Near-RT RIC needs to be configured to redirect all control messages from xApps into the CM component, so that all RAN control messages can be evaluated by CMF. Depending on the results of CMF's evaluation of the specific control messages, these messages can either be allowed to modify the network configuration (with or without modifications, depending on the logic implemented in the CR Agent) or be entirely blocked from influencing the network. It should be acknowledged that adding an additional step in the RAN control message processing pipeline increases the delay between an xApp providing a RAN control message and the message influencing the network. As Near-RT RIC aims to control the network operation in near-real-time scale, any processing done in the CM component, i.e., in the CD Agent and the CR Agent, needs to be relatively quick. Another dependency on the Near-RT RIC architecture is that the CD Agent needs to be subscribed to performance management (PM) data provided by RAN nodes managed by the Near-RT RIC. This is required for the detection of implicit conflicts, which is described in details in a subsequent section. To mitigate the inter-Near-RT RIC conflicts, CMF envisions the exchange of information between Near-RT RICs about RAN control messages that are currently in effect, via the Non-RT RIC. This information shall be a part of the enrichment information interface between the Non-RT RIC and Near-RT RICs. Once the Near-RT RICs are aware of all control decisions for users, cells, and bearers managed by other Near-RT RICs, these decisions can be considered as part of the conflict mitigation procedures for shared control targets. B. Direct Conflict Detection Ideally, direct conflicts should be detected as part of predeployment xApp assessment, which is done by the MNO deploying the xApps. Nevertheless, a method for detection of direct conflicts is required as a fail-safe mechanism, e.g., in case of human error leading to deployment of directly conflicting xApps. Direct conflicts can be detected by the Direct Conflict Detection (DCD) component within the CD Agent. DCD can detect the conflicts pre-action (i.e., before a control decision is effective), as it detects changes of recently changed parameters related to a user, a cell, or a bearer. All RAN control messages are tracked by the CD Agent in the Database ("Recently changed parameters" in Fig. 2) along with a timestamp, the source xApp, the control target, the parameters modified by the message, and the control time span of the message (i.e., the control duration expected by the xApp). The conflict detection logic of DCD is as follows: each new control message in the Near-RT RIC is saved into the Database, then it is compared to all currently effective xApp control decisions; if any decisions share the control target and at least one of the modified parameters, data about the conflicting decisions is provided to the CR Agent. The DCD procedure is illustrated in Fig. 3, which shows the message exchange flow between the CMF components. C. Indirect Conflict Detection Indirect conflicts cannot be directly observed by the Indirect Conflict Detection (ICD) component within the CD Agent, but it can anticipate them pre-action by having the knowledge of groups of parameters that influence the same area of RAN operation. These groups of parameters can be configured manually by the MNO, predefined in the standards, or learned dynamically during network operation using Performance Monitoring (described in the next section). The groups are stored in the Database as Parameter Groups (PGs) ("Parameter Group definitions" in Fig. 2). Additionally, the Database stores information about RAN control messages that modify any parameters from these groups ("Recently changed Parameter Groups" in Fig. 2), with the same scope of data as "Recently changed parameters" related to DCD. Detection of indirect conflicts is a slight modification of DCD: it also analyzes each new control message but first maps the target parameter onto predefined PGs. After the message is noted against at least one of the known PGs, ICD checks the entries in "Recently changed Parameter Groups" to find any conflicting decisions. If a match is found, information about the conflicting xApp decisions is provided to the CR Agent. The conflict detection logic and related message flow for ICD are shown in Fig. 3, alongside the DCD procedure. D. Implicit Conflict Detection Implicit conflicts cannot be observed directly, and the mutual influence between conflicting decisions is not easily deduced. Therefore, a similar approach, as described for DCD and ICD, is not suitable for Implicit Conflict Detection (ImCD), which can only work reactively. An additional component within ImCD is required to analyze PM data reported by RAN nodes and detect any occurrences of network performance degradation; this role is fulfilled by the Performance Monitoring (PMon) component within CMF. This is the author's version of an article that has been published in the IEEE Communications Magazine. Changes were made to this version by the publisher prior to publication, the early access version of the record is available at: 10.1109/MCOM.018.2200752. Detection of network performance anomalies can be achieved with either traditional statistical-based solutions or AI/MLbased approaches [14]. The aim of the PMon component is to detect any significant RAN Key Performance Indicator (KPI) degradation. ImCD utilizes Near-RT RIC's Database accessed via the SDL to store data about RAN KPI degradation occurrences and KPI degradation-correlated xApp decisions. The operation of ImCD is not triggered by any RAN control message but by PMon signaling a detection of RAN KPI degradation. Once the trigger is provided, ImCD analyzes which parameters and/or Parameter Groups have been recently modified by many control messages. ImCD utilizes the data captured in the Database by DCD and ICD, so it does not need to monitor any control messages by itself. If a correlation is found between any recent RAN control decisions and the observed RAN KPI degradation, ImCD notes it and increments its internal counters relevant to these control decisions. When any tracked counter breaches a predefined threshold, the CD Agent provides data about conflicting xApps' control decisions to the CR Agent. At this point in the procedure, the CD Agent may remove data about RAN KPI degradation occurrences related to the reported conflict from the Database. A significant difference between DCD/ICD and ImCD is that the latter works post-action and, therefore, cannot entirely prevent conflicting RAN control messages from influencing the network. Similar to the DCD and ICD procedures described earlier, the message exchange flow and conflict detection logic for the ImCD procedure are depicted in Fig. 4. IV. EVALUATION OF THE SOLUTION A. Simulation scenario To evaluate the efficiency of CMF, we simulated a 19-basestation O-RAN network with the base stations (BSs) evenly distributed in an urban environment on a hexagonal grid, 600 meters apart. The simulation area is limited to approximately outline the area covered by the network. There are 380 users spread across the area, each with one of three user profiles (i.e., low, medium, and high bitrate), chosen randomly with various probabilities (60% for low, 30% for medium, and 10% for high bitrates). Users move randomly around the considered area, either as pedestrians or in a vehicle, generating traffic in the network. Connections can be handed over between BSs as propagation conditions change. The simulated network includes a nRT-RIC, which monitors performance parameters for all BSs. Within the Near-RT RIC, This is the author's version of an article that has been published in the IEEE Communications Magazine. Changes were made to this version by the publisher prior to publication, the early access version of the record is available at: 10.1109/MCOM.018.2200752. To simulate CMF's operation, CD and CR Agents are deployed within the Near-RT RIC. The CD Agent implements DCD and ICD, i.e., it monitors configuration modifications made by xApps and detects direct and indirect conflicts (ImCD is not applicable in the considered scenario). If no conflicts are detected, any decisions provided by xApps take effect in the order they are provided. Otherwise, conflicts are reported into the CR Agent, which decides if and when specific decisions take effect. In the considered scenario, CMF can prioritize one of the conflicting xApps. If a given xApp is prioritized, each of its decisions takes effect on the network regardless of conflicts. To showcase how CMF is able to mitigate conflicts between xApps in the considered scenario, a description of ICD operation event sequence is shown in Fig. 5. Readers interested in the details of the simulation scenario and the raw result data can access them online (free access) [15]. B. Evaluation results The simulation is conducted with MRO and MLB xApps activated and one of three modes of CMF operation: disabled, prioritize MRO, prioritize MLB. The first 150 seconds out of 1,000 simulated seconds are ignored in the statistics for the sake of disregarding the initial instability of the simulated network. Tracked network performance indicators are: (i) mean BS load, (ii) mean user satisfaction, (iii) number of call blockages (CBs), (IV) number of RLFs, (V) number of handovers, and (VI) number of ping-pong handovers. The performance indicators observed for the network with no conflict mitigation in place show a significant number of CBs, RLFs, and handovers, including ping-pong handovers. The excessive number of handovers causes wasteful utilization of radio resources, which, in turn, decreases the QoS for the users handled by heavily loaded BSs. In extreme cases, the BSs drop incoming traffic, therefore causing call blockages. RLFs, on the other hand, are caused by either too early or too late handovers, which result from suboptimal configuration of handover parameters. Enabling conflict resolution with prioritization of the MRO xApp has a mostly positive effect on network performance. Although the mean load of all BSs increases slightly, the total number of handover events decreases by over 7%. This positive impact is due to MRO being able to steer handover parameters without constraints, while changes in network configuration done by MLB are significantly limited. On the other hand, in case of conflict with MRO, the MLB xApp is unable to increase the CIO of heavily loaded BSs, which would otherwise relieve their load and free up their radio resources for incoming calls. This limitation leads to an increase in the number of CBs because users trying to connect to heavily loaded BSs may not receive proper service due to the lack of available radio resources. The influence of CMF on network operation differs when the MLB xApp is prioritized. Mean user satisfaction increases by 0.7%, and the number of CBs decreases by over 7%, along with just a 0.1% increase in mean BS load. However, all other tracked network performance indicators deteriorate to varying degrees. CMF limits the RAN control capabilities of the MRO xApp, preventing it from optimizing handover parameters in case of conflict with MLB. This leads to more occurrences of all handovers, including too late and too early handovers, which cause RLFs. In conclusion, the "prioritize MRO" mode of CMF has a more positive impact on the network compared to the "prioritize MLB" mode, as it causes lesser deterioration of some KPIs in exchange for its performance improvements. CMF's influence on KPIs in both modes is shown in Fig. 6. V. SUMMARY In this article, we defined a framework for conflict mitigation in O-RAN's Near-RT RIC. By design, it is capable of mitigating all three types of conflicts described in the technical specifications published by O-RAN Alliance: direct, indirect, and implicit conflicts. The proposed framework works by tracking all RAN control messages sent by xApps deployed within the Near-RT RICs and monitoring PM data reported by RAN nodes. The concept of a conflict mitigation framework is easily scalable and does not impose any restrictions on how conflict resolution is realized. We provided a complete set of example JSON messages exchanged as part of the procedures related to the detection of all types of conflicts between xApps. The efficiency of the CMF concept was proven with a simulation of an O-RAN network with and without mitigation Defining the CMF is envisioned as the first step in providing a standardized conflict mitigation method in the Near-RT RIC. We expect future work in this field to include (i) the development of the PMon component, potentially with use of AI/ML-based tools for detection of performance degradation, (ii) the development of the CR Agent entity, compatible with CMF, and with optimal conflict resolution logic for each type of conflict in the Near-RT RIC, and (iii) open source implementations of CMF for utilization in O-RAN networks. openness of O-RAN enables novel approaches to operating RAN and fulfilling new deployment scenarios. First of all, the interfaces between logical components of O-RAN are standardized and open. This enables interchangeability of O-RAN components manufactured by various vendors without limitation of network functionality. This is a significant change compared to traditional "proprietary" RANs, where RAN hardware and software are usually provided by a single vendor. Freeing MNOs from this dependency largely expands the possibilities of network evolution. C. Adamczyk and A. Kliks were with Poznan University of Technology, Poznań, Poland, e-mails: [email protected], [email protected]. The work has been realized within the project no. 0312/SBAD/8163 funded by the Poznan University of Technology. Fig. 1 . 1Areas of potential control conflict in O-RAN architecture by the Non-RT RIC based on the affected use case and how the policies are enforced. The specific logic of how the Non-RT RIC should resolve conflicts and enforce its policies is not considered in this article. Fig. 2 . 2CMF components embedded within the Near-RT RIC architecture Fig. 3 . 3Direct and Indirect Conflict Detection message flow 1 Fig. 4 . 4Implicit Conflict Detection message flow 1 Fig. 5 . 5ICD event sequence in the considered scenario two xApps are installed: MRO and MLB. MRO monitors handover statistics of each BS and modifies handover hysteresis and time-to-trigger parameters to minimize the number of radio link failures (RLFs) and ping-pong handovers. MLB balances the load of the BSs in the network, choosing CIO values according to the load of the BSs. Fig. 6 . 6Simulation results showing performance influence of CMF of xApp conflicts. The simulation results show that enabling CMF, even with a basic scheme of conflict resolution, can reduce the negative effects of xApp conflicts. Copyright © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. Cezary Adamczyk is a PhD student at Poznan University of Technology, conducting research in the field of AI/ML utilization in open radio access network radio resource optimization. He works as an OSS Solutions Architect in international telecommunication projects. His prior work related to O-RAN includes a paper proposing a novel Reinforcement Learning algorithm for optimizing radio resources utilization. Working Group 1, O-RAN Architecture Description, O-RAN Alliance. 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[ "Constructing Effective Customer Feedback Systems - A Design Science Study Leveraging Blockchain Technology", "Constructing Effective Customer Feedback Systems - A Design Science Study Leveraging Blockchain Technology" ]	[ "Mark C Ballandies \nComputational Social Science\nETH Zurich\nStampfenbachplatz 50ZurichSwitzerland\n", "Valentin Holzwarth \nHilti Chair of Business Process Management\nUniversity of Liechtenstein\nLiechtenstein\n", "Barry Sunderland \nETH Library Lab\nETH Zurich\n\n", "Evangelos Pournaras \nSchool of Computing\nUniversity of Leeds\n\n", "Jan Vom Brocke \nHilti Chair of Business Process Management\nUniversity of Liechtenstein\nLiechtenstein\n" ]	[ "Computational Social Science\nETH Zurich\nStampfenbachplatz 50ZurichSwitzerland", "Hilti Chair of Business Process Management\nUniversity of Liechtenstein\nLiechtenstein", "ETH Library Lab\nETH Zurich\n", "School of Computing\nUniversity of Leeds\n", "Hilti Chair of Business Process Management\nUniversity of Liechtenstein\nLiechtenstein" ]	[]	Organizations have to adjust to changes in the ecosystem, and customer feedback systems (CFS) provide important information to adapt products and services to changing customer preferences. However, current systems are limited to single-dimensional rating scales and are subject to self-selection biases. This work contributes design principles for CFS and implements a CFS that advances current systems by means of contextualized feedback according to specific organizational objectives. We apply Design Science Research (DSR) methodology and report on a longitudinal DSR journey considering multiple stakeholder values by utilizing value-sensitive design methods. We conducted expert interviews, design workshops, demonstrations, and a four-day experiment in an organizational setup, involving 132 customers of a major Swiss library.In the process, we validated the identified design principles and the implemented software artifact both qualitatively and quantitatively and drew conclusions for their efficient instantiation.In particular, we found that i) blockchain technology can afford three design principles of effective CFS. Also, ii) combining DSR with value-sensitive design methods explicitly provides rationale for design principles in the form of identified important values.Moreover, iii) utilizing this methodology makes the construction of software artifacts more efficient it terms of design time by restricting the design space of a software artefact to those options that align with stakeholder values. Hence, the findings of this work 1 advance the knowledge on the design of CFS and provides both, for researchers a theoretical contribution to reason about design principles and a guideline to managers and decision makers for designing software artefacts efficiently.	10.48550/arxiv.2203.15254	[ "https://export.arxiv.org/pdf/2203.15254v2.pdf" ]	247,778,614	2203.15254	3cccd81826223dfa6ee004f9c197524c3414b8c7	Constructing Effective Customer Feedback Systems - A Design Science Study Leveraging Blockchain Technology 14 Mar 2023 Mark C Ballandies Computational Social Science ETH Zurich Stampfenbachplatz 50ZurichSwitzerland Valentin Holzwarth Hilti Chair of Business Process Management University of Liechtenstein Liechtenstein Barry Sunderland ETH Library Lab ETH Zurich Evangelos Pournaras School of Computing University of Leeds Jan Vom Brocke Hilti Chair of Business Process Management University of Liechtenstein Liechtenstein Constructing Effective Customer Feedback Systems - A Design Science Study Leveraging Blockchain Technology 14 Mar 2023* To whom correspondence should be addressed;Design Science ResearchBlockchainFeedback SystemCryptoeconomicsvalue-sensitive DesignToken Engineering Organizations have to adjust to changes in the ecosystem, and customer feedback systems (CFS) provide important information to adapt products and services to changing customer preferences. However, current systems are limited to single-dimensional rating scales and are subject to self-selection biases. This work contributes design principles for CFS and implements a CFS that advances current systems by means of contextualized feedback according to specific organizational objectives. We apply Design Science Research (DSR) methodology and report on a longitudinal DSR journey considering multiple stakeholder values by utilizing value-sensitive design methods. We conducted expert interviews, design workshops, demonstrations, and a four-day experiment in an organizational setup, involving 132 customers of a major Swiss library.In the process, we validated the identified design principles and the implemented software artifact both qualitatively and quantitatively and drew conclusions for their efficient instantiation.In particular, we found that i) blockchain technology can afford three design principles of effective CFS. Also, ii) combining DSR with value-sensitive design methods explicitly provides rationale for design principles in the form of identified important values.Moreover, iii) utilizing this methodology makes the construction of software artifacts more efficient it terms of design time by restricting the design space of a software artefact to those options that align with stakeholder values. Hence, the findings of this work 1 advance the knowledge on the design of CFS and provides both, for researchers a theoretical contribution to reason about design principles and a guideline to managers and decision makers for designing software artefacts efficiently. Introduction Customer feedback is important for the potential of an organization to differentiate itself from competitors (Culnan, 1989) and to improve its products and services according to customer preferences (H.-H. Hu, Parsa, Chen, & Hu, 2016;Stoica & Özyirmidokuz, 2015). Nevertheless, due to status differences, hierarchy steepness, and reduced levels of cooperation, large hierarchical organizations such as firms or public institutions impede the flow of feedback to and within their organization (Anderson & Brown, 2010), which reduces the quality of management decisions (Khatri, 2009). Inspired by the observation that technology can be utilized to support the self-organization capacity and thus the efficiency of a large scale system via feedback loops (Ashby, 1964), as it has been shown for a traffic light control network (Lämmer & Helbing, 2008), this work generates design knowledge on the construction of a customer feedback system (CFS) that improves the provision of high-quality feedback about services and products from customers to an organization. We report on Design Science Research at a case organization, which is a major Swiss library. This library is challenged by a lack of feedback from so-called unaware-customers, i.e. customers that are not aware that they are using services provided by the organization. Moreover, the library is challenged to distinguish important from unimportant feedback, particularly, when the feedback quantity is high. Furthermore, the library has difficulties evaluating the questions utilized in solicited surveys with customers. The innovation & networking team within the library organization had been mandated to implement a solution in the form of a CFS that improves the status quo of feedback provision from library users to the organization. We identified this need of the library in the first step of the applied research methodology (Section 3) and consecutively accompanied the construction of the CFS. We argue that a CFS should not only optimize for performance, but its design also needs to integrate the values of stakeholders (Kleineberg & Helbing, 2021;Van den Hoven, Vermaas, & Van de Poel, 2015) such as autonomy or credibility. This has been recognized by the IS community (Friedman, Kahn, Borning, & Huldtgren, 2013;Maedche, 2017). Though already utilized in similar systems (Friedman, Howe, & Felten, 2002;Miller, Friedman, Jancke, & Gill, 2007), value considerations in the methods of CFS construction and thus the resulting design knowledge is limited. In particular, to our best knowledge, design principles for CFS that explicitly consider values have not been found. We therefore ask the first Research Question (RQ1): (RQ1) What are the design principles of a value-sensitive customer feedback system? By applying an established design science research (DSR) methodology (A. R. Hevner, March, Park, & Ram, 2004;Peffers, Tuunanen, Rothenberger, & Chatterjee, 2007;Sonnenberg & Brocke, 2011;Sonnenberg & Vom Brocke, 2012) and putting a focus on stakeholder values during the design phase as performed in , we facilitate both, i) the value-alignment of the created tool with the affected stakeholders and ii) the implementation of the design principles in a software artifact, which is iteratively evaluated at different stages. A controlled experiment with 132 customers of the library and focus groups with experts of that organization are conducted to measure the performance of the software artifact in terms of usability and quality of collected feedback. In order to evaluate this, we ask the second Research Question (RQ2): RQ2: What is usability and quality of collected feedback of a software artifact that implements the design principles? This paper illustrates how design science as a journey can be conducted (Vom Brocke, Winter, Hevner, & Maedche, 2020) and contributes the following: I) design principles for a value-sensitive customer feedback system which are found by extending an established DSR methodology with valuesensitive design methods; II) an effective software artifact in terms of useability and user acceptance, which is evaluated in both a four-day field experiment with a large Swiss library and 132 of its customers and in a focus group with experts and managers from this library; III) a demonstration how blockchain technology can afford three design principles of effective customer feedback systems; IV) theoretical implications for DSR that are derived from a focus on value-sensitive design. Such a focus can i) reduce the design space of the IT artefact and thus make the design more efficient in terms of required time and ii) explicitly provide the rationale for design principles in form of values. This paper is organized as follows: In Section 2, a literature review about CFS in organizations is given. The DSR methodology that this paper follows is illustrated in Section 3, while the findings and artifacts from applying this methodology are outlined in Section 4. Thereafter, Section 5 embeds the findings as design knowledge chunks (Vom Brocke et al., 2020) into the broader research journey and illustrates the finalized design principles for customer feedback systems. Finally, in Section 6 a conclusion is drawn and an outlook on future work is given. Research Background Customer feedback is an important element of an organizations' quality management (Chase & Hayes, 1991), as the perceived quality of services and products is related to market share and return on investment (Parasuraman, Zeithaml, & Berry, 1985). This is particularly relevant for service businesses (Chase & Hayes, 1991) such as libraries (Casey & Savastinuk, 2006), since there is an increased emphasis on service quality rather than on manufacturing quality (Vargo & Lusch, 2004). This importance is recognized within the seminal work of Sampson (1999), who developed a framework for designing customer feedback systems (CFS) to improve service quality. Furthermore, an overview of advantages and disadvantages of different feedback collection systems and their designs for improving service quality in organizations have been illustrated by Wirtz and Tomlin (2000). Usually, in these systems, customers provide feedback on services in the form of online reviews either directly on the selling platform (e.g. rating a service booked on Fiverr 1 ) or on specific review platforms (e.g. providing travel reviews on Tripadvisor 2 ) (Schneider, Weinmann, Mohr, & vom Brocke, 2021). Although online ratings do not necessarily provide an objective measure of service quality (de Langhe, Fernbach, & Lichtenstein, 2015), they are highly influential for customer-decision making, which is reflected in sales and consequently in business success (Simonson, 2016). Customers' online rating data can even be utilized to predict service business failures months in advance as it has been shown for the hospitality industry (Naumzik, Feuerriegel, & Weinmann, 2021). Despite their high relevance for influencing customers' decision making, online ratings are potentially challenged by various factors including self-selection (N. Hu, Zhang, & Pavlou, 2009), social influence (Muchnik, Aral, & Taylor, 2013), manipulation of reviews (Gössling, Hall, & Andersson, 2018;Zhuang, Cui, & Peng, 2018), and dimensional rating (Schneider et al., 2021). Particularly, single-dimensional rating scales (e.g. Google reviews, which allow for a score from 1 to 5 stars) are not suitable to assess complex performance dimensions (Ittner & Larcker, 2003). By these means, an organization will only receive a single rating, which often cannot 1 Fiverr is an online marketplace for freelance services: https://www.fiverr.com/ (last accessed 2021-12-10) 2 Tripadvisor is an online travel company that operates a website and mobile app with user-generated content and comparison shopping website: https://www.tripadvisor.com/ (last accessed 2021-12-10 be associated with a particular service that the customer received. To overcome this issue, firms have to invest in dedicated CFS, which allow them to receive feedback that they can benefit from (e.g. feedback on a specific service that they intend to improve) (Sampson, 1999). Within such a system, the feedback is processed in the following sequential manner: channeling (i.e. feedback reception), processing (i.e. using the feedback for improvements), and conversion of the feedback into organization-wide knowledge (Birch-Jensen, Gremyr, & Halldórsson, 2020). In this context, channeling is highly relevant, since feedback in a certain quantity and quality needs to be received to enable the subsequent steps of processing and conversion (Lafky & Wilson, 2020). Depth and extremity of reviews have been identified as useful indicators of the quality of feedback on an e-commerce platform (Mudambi & Schuff, 2010), which are found to be rather incentivized by social norms than financial rewards (Burtch, Hong, Bapna, & Griskevicius, 2018). Feedback quantity is known to be positively influenced by financial incentives (Burtch et al., 2018), along with several other factors such as trust (Celuch, Robinson, & Walz, 2011) and perceived usefulness (i.e. customers thinks that their feedback is useful for the organization) (Robinson, 2013). Nevertheless, feedback quality may be reduced by applying such incentives (Lafky & Wilson, 2020). For instance, financial incentives lead to a reduction in feedback quality measured in depth (e.g., the length of a written review) while increasing feedback quantity measured in breadth (number of provided reviews) (Burtch et al., 2018), thus revealing a trade-off between quantity and quality that is steered by the chosen incentive (Lafky & Wilson, 2020). Multi-dimensional incentives in the form of blockchain-based tokens have been proposed as an alternative to such financial incentives improving the properties of incentivized behavior such as actions contributing to sustainability Dapp, 2019;Kleineberg & Helbing, 2021). In this regard, blockchain-based incentives have been suggested to improve the data quality in inter-organizational information exchange (Hunhevicz, Schraner, & Hall, 2020;Zavolokina, Spychiger, Tessone, & Schwabe, 2018). For instance, it has been found that Blockchain technology could contribute to trustworthy CFS in the tourism industry (Önder, Treiblmaier, et al., 2018). Chandratre and Garg (2019); Gipp, Breitinger, Meuschke, and Beel (2017) ;Rahman, Rifat, Tanin, and Hossain (2020) are among the first to propose and implement blockchain-based feedback systems. Although these systems utilize blockchain technology for tracking and the immutable storing of feedback items, they do not explore incentivizing feedback provision with blockchain-based tokens. This is a missed opportunity as, on the on hand, cryptoeconomic incentives carry monetary value (Kranz, Nagel, & Yoo, 2019;Sunyaev et al., 2021), and thus could motivate users to increase feedback quantity, while, on the other hand, they have different characteristics to money Dapp, 2019;Dapp et al., 2021;Kleineberg & Helbing, 2021), and thus might impact feedback quality differently when compared to monetary incentives. In general, cryptoeconomic incentives in the form of blockchain-based tokens have been utilized to improve information sharing scenarios (Ballandies, 2022). Nevertheless, most approaches only utilize a single token incentive and do not consider the combination of two, which is a missed opportunity because this might improve system performance (Ballandies, 2022). In order to construct such blockchain-based systems, Design Science Research (DSR) methods (A. Hevner & Chatterjee, 2010;A. R. Hevner et al., 2004;Vom Brocke et al., 2020) have been successfully applied within the IS community Ostern & Riedel, 2020). For this, amongst others, the model of Zargahm (2018) is utilized that describes a system in five layers, as illustrated in Figure 4. Design principles can only be explicitly formulated and implemented in the software system , i.e. the bottom three layers of the model ((I-III) in Figure 4). The upper two layers emerge and cannot be explicitly defined by the system designer (e.g. the associated researchers). In summary, the following observations can be made about current CFS applied in organizations: First, initial, prior findings provide promising evidence regarding the utilization of (multiple) blockchainbased tokens for incentivizing behavior, such as influencing the provision of high-quality feedback. However, they have not been incorporated within a feedback system and studied within a real-world use case. Second, research on CFS mostly focuses on hospitality, tourism, and e-commerce applications, while neglecting other application domains such as a library ecosystem. Third, CFS often focus on uncontextualized and solicited feedback in the form of single-dimensional rating scales. In this work, we address these gaps by investigating the design of a value-sensitive blockchain-based feedback system that incentivizes the provision of feedback with multiple tokens and utilizes the concept of contextualization of feedback to enable users to increase the depth of their feedback and consequently feedback quality (Burtch et al., 2018). For this, we apply an established DSR methodology (A. R. Hevner et al., 2004;Peffers et al., 2007;Sonnenberg & Vom Brocke, 2012) and combine it with value-sensitive design methods (Friedman et al., 2013;Van den Hoven et al., 2015) that consists of expert interviews, stakeholder and value analysis, focus groups, and a four-day ethics commission approved socioeconomic experiment, Figure 1: Activities, methods, participants and outputs of the four steps (I-IV) of the cyclic DSR process (Sonnenberg & Vom Brocke, 2012) involving a major Swiss library and its customers. Research Design This research applies a DSR methodology (A. Hevner & Chatterjee, 2010;A. R. Hevner et al., 2004;Peffers et al., 2007) and combines it with value-sensitive design methods (Friedman et al., 2013;Van den Hoven et al., 2015). We report on a DSR journey (Vom Brocke et al., 2020), that comprised of four iterations of concurrent design and evaluation (Sonnenberg & Brocke, 2011;Sonnenberg & Vom Brocke, 2012). Figure 1 illustrates how the process with its four steps (I-IV) has been implemented in this work: In Step I, a literature review ("Identify Problem" in Figure 1) identifies suboptimal feedback flows to and within organizations. Expert Interviews ("Evaluation 1" in Figure 1) with employees of two major libraries evaluate this problem. Moreover, these interviews are utilized to identify important values and best-practice mechanisms in the context of feedback provision in these organizations that can be utilized to mitigate the identified problems and that are implemented within the constructed software artifact (Section 4.3). In Step II, two design workshops with library employees and a customer of that library from University 1 resulted in (i) a stakeholder analysis which informs (ii) a value analysis which in turn facilitates the identification of (iii) design requirements ("Design solution" in Figure 1). The requirements are incorporated in a system design via design principles and evaluated by the associated researchers ("Evaluation 2" in Figure 1). In Step III, the associated researchers implemented the system by the means of agile development into a software artifact ("Construct solution" in Figure 1). This artifact is validated in two demonstrations ("Evaluation 3" in Figure 1) with focus groups consisting of library employees, researchers, software developers, and artists ("FG2" in Table 1). Finally in Step IV), the software artifact is put into use in an organizational context of Library 1 in the form of an experiment ("Use Solution" in Figure 1) involving employees and customers of the library. The user behavior and answers to surveys are analyzed by the associated researchers ("Evaluation 4" in Figure 1). Moreover, the collected feedback is evaluated by a focus group consisting of experts and executives of Library 1 ("FG3" in Table 1). Figure 2 illustrates how these four steps are positioned in the conducted DSR journey (Vom Brocke et al., 2020) by connecting the design knowledge chunks obtained from each step of the DSR methodology ( Figure 1). Research Findings In the following, we present the findings and artifacts obtained at each activity of the cyclic DSR process ( Figure 1). ID Identify Problem During the first activity, expert interviews were conducted by the first author (A1) that evaluated the identified problem of suboptimal feedback flows in organizations (Section 1). Table 1 illustrates the twelve participants from two libraries who were interviewed in a semi-structured format. The participants were sampled by convenience based on the criterion that they work in a library. The interview protocol and guide are illustrated in Section 1 of the Supplementary Material (Table 1 and 2 of the Supplementary Material). Library 1 and 2 are among the largest in the German speaking countries. From Library 1 employees from all hierarchy levels (employee, team lead, section lead, director) and five of seven organizational sections were interviewed, including the director (ID 03, Table 1), whereas from Library 2 the director has been interviewed (ID 09, Table 1). The interviews were transcribed by a third party following the standard of Dresing and Pehl (2010) and were coded by the A1 with 19 codes (Table 6 of the Supplementary Material). The codes were developed by the A1 and A2 following the method of O'Connor and Joffe (2020). The definitions of the codes are given in Table 5 of the Supplementary Material. The interviews are then analyzed by grouping manually similar contents of each code into clusters (e.g., Figure 1 and Figure 2 of the Supplementary Material for the codes "status quo: challenge" and "risk"). In Section 4.1.1, a subset of these challenges are illustrated that are addressed in this work. Moreover, the interviews are utilized to identify the values stakeholders have in the context of feedback provision (Section 4.1.2). Finally, the interviews are utilized to identify best practices when collecting feedback in the organizations (Section 4.1.3). Suboptimal Feedback Flows In order to obtain a multi-faceted perspective of the current and future challenges that exist or might arise with regard to feedback in an organizational context, the codes "status quo: challenge" and "risk" ( Table 6 of Values The top six mentioned values are Anonymity (17), Transparency (16), Openness (11), Safety (11), Realworld Human Interactions (10), and Simplicity (10) ( Table 7 of Best Practice Mechanisms Six best practice mechanisms that the library has established have been identified (Section 1. Design Solution During this activity, two design workshops are conducted with employees of Library 1 and a customer of the library from University 1 (DW in Table 1) to identify the design requirements of the feedback system Table 2. The values have been taken from value sensitive design literature (Friedman, Kahn Jr, & Borning, 2020;Hänggli, Pournaras, & Helbing, 2021;Harbers & Neerincx, 2017;Huldtgren, 2015;Van de Poel, 2015). The value of Excellence has been added by the participants during the first Design Workshop. The final design requirements (Figure 3) are then elicited by first identifying the requirements associated with each value via brainwriting and then to prioritize those requirements (Average ≥ 1 in Table 2). In order to familiarize the workshop participants with these values, a preliminary value association task has been performed with the participants at the end of workshop 1 to prepare them for the second workshop. The chosen brainwriting and clustering approach was tested before the workshops within a diverse focus group consisting of artists, a researcher, and a software developer (FG1 in Table 1). In the following, the outputs of the two Design Workshops are illustrated: a Stakeholder map (Sec- of the library has the highest interest and influence in the solution. Cluster (4) contains potential users of the system (e.g. students) that have a high interest in the solution but a low influence on its design. As these stakeholders are potential users of the system, their perspective is considered in the design requirements engineering to positively influence the adoption of the solution. Avg. Var. Credibility 1 3 3 1 2 1 2 2 1 3 1 3 0 3 1 1.80 1.03 Simplicity 3 3 3 1 3 0 0 1 1 1 1 2 0 3 3 1.67 1.52 Universal Useability 2 3 3 1 2 1 1 2 1 2 1 0 0 3 3 1.67 1 Table 2: Strength of stakeholder association for each value: Green (3) -strong, yellow (2) -medium, red (1) -low, white (0) -none, sorted by average strength, as identified by the design workshop participants (Table 1). Above the dashed line are those values that received an average strength of 1 and thus were considered in the requirements analysis. (3) and (4) introduction of these values (Friedman et al., 2020;Hänggli et al., 2021;Harbers & Neerincx, 2017;Huldtgren, 2015;Van de Poel, 2015). Value Analysis Design Requirements Design Principles By grouping the key requirements (Section 4.2.3) into solution clusters, 10 design principles are identified ( Figure 3) that guide the construction of a value-sensitive feedback system. Three types of principles are found: i) infrastructure principles informing about technological requirements, ii) feedback principles illustrating the handling of the collected feedback, and iii) interaction principles illustrating the interplay among stakeholders and the system. Figure 3 depicts the 10 design principles in the framework introduced by Gregor, Chandra Kruse, and Seidel (2020) stating the aim, mechanism and rationale of each principle. The rationale are the values that inform the requirements which resulted in the principle. The context for each principle are customer feedback systems. In the following, these principles are illustrated in greater detail. Infrastructure principles: Feedback system designers should use a public and trustworthy storage infrastructure combined with a transparent computing engine to ensure the unconditional visibility of submitted feedback and its trustworthy post-processing. Moreover, software tools that are created should be self-explanatory to facilitate the quick and intuitive provision of collected feedback. Feedback principles: Feedback items should be contextualized such that metadata 5 of feedback (e.g., location of provision, the receiver, or importance) is also stored because this can improve the postprocessing of feedback. A possibility for ranking feedback to visualize the impact of each feedback item should be integrated to identify important feedback for feedback recipients. Also, the feedback should be aggregated and visualized in statistics to the stakeholders to make the performance of the system visible. Interaction principles: Stakeholders of the system should be enabled to have personal contacts in both, the cyberspace, but also in the physical reality to leverage on more than one feedback channel. Already existing pseudonymous identities could be (re)used to facilitate quick user input. Also, participation in the system should be voluntary in order to avoid bias in the feedback collected. Moreover, rewards could be utilized to facilitate appreciative feedback on feedback. (2018) (left, grey), the interaction of the socio-economic feed4org system with its underlying software system is illustrated (red, right) (adapted from Ballandies, Dapp, Degenhart, and Helbing (2021)). System Design Distributed Ledger: A distributed ledger is a distributed data structure, whose entries are written by participants of a consensus mechanism after reaching an agreement on the validity of the entries. Such a consensus mechanism is called permissionless, if the public can participate in it, otherwise it is called permissioned (Ballandies, Dapp, & Pournaras, 2021). A permissionless distributed ledger (e.g. blockchain) affords a public and trustworthy storage (Layer I in Figure 4, Design Principle 1 in Table 6), which in turn facilitates a transparent computing engine if open source smart contracts are utilized as trusted computation protocols (Layer II in Figure 4, Design Principle 2 in Table 6). With these protocols, interactions patterns (Layer III in Figure 4) Table 6); (iii) Moreover, public distributed ledgers (e.g. Bitcoin) do not restrict the creation of user identities (Ballandies, Dapp, & Pournaras, 2021) (e.g. no know your customer policies), thus each user can decide how much information is revealed about their identity which facilitates pseudonymous interactions among stakeholders (Design Principle 9 in Table 6). Contextualization and Incentives: In order to improve the processing and depth of the collected feedback, which is associated with feedback quality (Mudambi & Schuff, 2010), users are enabled to contextualize their feedback with meta-properties (Design Principle 4 in Table 6) such as the importance of the feedback, their satisfaction with the answer options, general comments, and the target audience to which their feedback is directed. Also, the system provides solicited (survey style questions, e.g. Figure 7) and unsolicited (reddit style forum, e.g. Figure 6) input forms such that diverse feedback can be collected (Design Principle 7 in Table 6). The system encourages both, the provision of solicited feedback and the contextualization of this feedback, by incentivizing stakeholders with cryptoeconomic rewards in the form of tokens (Design Principle 11 in Table 6). Cyber-physical interactions: Personal contact is enabled by following a cyber-physical approach: Stakeholders can interact via the created software artifact but are also encouraged to meet in the realworld 6 at the library facilities by i) including interactive answer options such as taking photos at the library facility and ii) turning spots in the library into "real-time digital voting centers" (Hänggli et al., 2021), e.g. via the proof of witnessed presence (Pournaras, 2020). Construct Solution Utilizing agile development, a software artifact (referred to as feed4org app) is constructed that is evaluated by demonstration in a focus group (Figure 1): Based on the identified requirements (Section 4.2.3), design principles (Section 4.2.4) and system design (Section 4.2), a software artifact is created to incentivize the provision of high-quality feedback to organizations. Figure 5 illustrates the software stack: A web app is built using the VUEjs 7 framework, on top of the Finance 4.0 software Ballandies, Dapp, Degenhart, Helbing, Klauser, & Pardi, 2021), which enables the creation of cryptoeconomic incentives in the form of tokens. By utilizing the Finance 4.0 software stack, the trustworthy data storage and computation engine of the Ethereum blockchain is utilized that facilitates durable data and trusted computation as required by the system design (Section 4.2.5, Design Principles 1 and 2 in Table 6). In particular, public blockchains, when compared to private blockhains, are i) transparent and publicly verifiable (Yang et al., 2020), and ii) secure (Ballandies, Dapp, & Pournaras, 2021;Yang et al., 2020), thus incorporating values such as credibility and and safety, which are especially important for a public library ( Figure 3) offering services as a public good to engage customers, as identified in Section 4.2.2. Also, utilizing a public blockchain reduces cost for the library organization as an own infrastructure does not need to be maintained (Yang et al., 2020). Moreover, Finance 4.0 facilitates the creation of tokens and proof verifications for awarding these tokens to feedback providers (Design Principle 11 in Table 6). By tailoring these incentives to library customers 8 to make only those participate in the feedback provision, high scalability facilitating a large number of feedback items per second from a potentially unrestricted user base is not required. For such a scalability requirement, utilizing either a permissioned/ private blockchain (Ballandies, Dapp, & Pournaras, 2021) or newer public blockchains (e.g. Solana (Yakovenko, 2018)) might be more suitable. The web app consists of four main components: i) Answer Question view that facilitates the provision of solicited feedback and the awarding of cryptoeconomic incentives (Design Principles 7 and 11 in Table 6), ii) Give Open Feedback view where users can provide unsolicited feedback (Design Principle 7 in Table 6), iii) View Statistics that informs users about their collected cryptoeconomic tokens and the behavior of others users (Design Principle 6 in Table 6) and iv) See About Page where information about the app and a Netiquette are displayed. In the following, these components are illustrated in greater detail. Answer Questions When users enter the app, they have the possibility to give feedback on questions posed by the library Cryptoeconomic Incentivization in the form of blockchain-based tokens is utilized via the Finance 4.0 platform to incentivize the provision of feedback. Figure 8 illustrates the design choices of the utilized cryptoeconomic incentives using a taxonomy for DLT systems (Ballandies, Dapp, & Pournaras, 2021), as performed by Dobler, Ballandies, and Holzwarth (2019): The Money token is pre-mined and is pegged to the Swiss franc, thus being a stable coin collateralized with a fiat currency (Mita, Ito, Ohsawa, & Tanaka, 2019). Each token unit is worth 0.20 CHF. For each answered question, the user is rewarded with a token unit. The Context token is created whenever a contextualization action is performed and awarded to the feedback provider. Thus, this token is not capped, but its amount illustrates the number of contextualization actions performed in the system. Users can utilize this token to vote on the importance of unsolicited feedback (Section 4.3.2, Design Principle 5 in Table 6). Moreover, the token is utilized to rank users in a leader board, which is displayed to all users in the View Statistics view of the app (Section 4.3.3). Table 3: Total and mean amount of interactions with the software artifact for the 132 experiment participants and the treatment (T, with token incentives) and control group (C, no token incentives) for users and unaware-users. Give Open Feedback Via the Navigation bar (top bar in Figure 7), users can switch to the Give Open feedback view. This view is based on the best practice mechanisms "feedback wall" of the library identified in Section 4.1.3 and thus could be the integration point of this software artifact into the existing library software stack providing a low-threshold to provide feedback for existing library users familiar with that view (Design Principle 3 in Table 6). Via the feedback wall, users can provide unsolicited feedback to the organization (Design Principle 7 in Table 6). This paper extends this mechanism by i) enabling users to up and downvote a feedback item, facilitating the design principle of ranking feedback items (Design Principle 5 in Table 6), ii) to comment on a feedback item, facilitating personal contacts among users (Design principle 8 in Table 6), and iii) to provide area tags on a feedback item that connects it with strategic action areas of the library where the management of the library aims to improve their services (Design Principle 4 in Table 6). In order to up and downvote a feedback item, users are required to spend a unit of the context token (Section 4.3.1). View Statistics In the statistics view ( Figure 11 of the Supplementary Material), users are informed about the amount of collected cryptoeconomic tokens. Moreover, the behavior of other users is displayed in a leaderboard that illustrates how many context tokens other users in the system collected. This facilitates the design principles of collecting statistics (Design Principle 6 in Table 6). Use Solution The software artifact is utilized in a four-day long real-time experiment to collect feedback from library customers. The collected feedback is then evaluated by both, statistical analysis, and a focus group consisting of library employees (FG3 in Table 1). The experiment has been conducted in collaboration with the Laboratory 9 , who recruited the participants, guaranteed a fair compensation (10 CHF show-up fee, 30 CHF/h mean compensation), and facilitated anonymity for the participants by separating their identity information from the experiment data: The research team has only access to the latter. For the four-day experiment setup 10 132 users were recruited in four waves receiving on avarage a compensation of 40.23 CHF. The software artifact and the experiment setup were evaluated with demonstrations before the experiment in two focus group meetings with library employees, artists, and a researcher (FG2 in Table 1). The finalized experiment setup is composed as follows: At the beginning of each wave, participants obtained onboarding materials (Section 6.3.1 of the Supplementary Material) in which they are introduced to the app and answered demographic questions, Computer Self-Efficacy (adapted from Compeau and Higgins (1995); Thatcher, Zimmer, Gundlach, and McKnight (2008) as performed in Sun, Wright, and Thatcher (2019)) and Personal Innovativeness in IT (adapted from Agarwal and Karahanna (2000) as performed in Sun et al. (2019)) questions. During the experiment phase, participants utilized the software artifact and provided feedback to Library 1. In the exit phase, users answered a questionnaire that included a UTAUT (Section 3 of the Supplementary Material, adapted from Venkatesh, Thong, and Xu (2012)) and questions regarding the value of cryptoeconomic tokens (Ballandies, Dapp, Degenhart, Helbing, Klauser, & Pardi, 2021). After the experiment, a focus group consisting of Library 1 employees (FG3 in Table 1) evaluated the quality of collected feedback. In total, 132 participants completed the study with an average age of 23.2 years. Of these, 51.5 % are male, 47.0 % are female and 1.5 % do not want to reveal their gender. On average, the participants self-report to be modest computer self-effective (2.8 11 computer self-efficacy (Sun et al., 2019)) and innovative in IT (2.4 innovativeness in IT (Sun et al., 2019) (Table 16 of the Supplementary Material) of Evaluation 4 that utilize a 5-point likert scale (0 -strongly diasagree to 4 -strongly agree). Token value is evaluated only for the treatment group as the control group did not utilize tokens. form of cryptoeconomic tokens. In the following, user interactions with the App and user responses to the exit survey, and the evaluation of the focus group are illustrated. Table 3 illustrates the user interactions with the software artifact. In total, 21286 solicited feedback items and 13817 contextualizations are collected from users which indicates a high useability of the artifact. App Interactions In particular, the scoring on the UTAUT (Table 4) for the effort expectancy (2.83) validates the design principle of simple user input/ self-explanatory UI ( Figure 3): Above all, it is easy for participants to learn using the software artifact (3.00). The focus group (FG4 in Table 1) indicated that this might be due to the clear focus in the design of the question answering and contextualization views which does not utilize unnecessary design elements. Table 5: Mean and variance of responses by the treatment (N=99) and control (N=33) group to the question "How useful did you find the following features of the app", utilizing a 5-point likert scale (0strongly diasagree to 4 -strongly agree). a Because the control group did not utilize tokens, its usefulness evaluation is removed from the calculation of the control groups average. the fourth experiment round. A focus group (FG4 in Table 1) evaluated the content and the ranking of the unsolicited feedback as useful for improving the library services because the mechanism highlights the most important feedback items. Moreover, the focus group highlighted the innovativeness and useability of combining solicited and unsolicited feedback into one software artifact, because it facilitates the combination of quantitative and qualitative analysis. In particular, the group was positively surprised about the quantity of provided unsolicited feedback, though it was not incentivized. Also, the focus group stated that because the existing library wall (see Section 4.1.3) had been utilized in the software artifact, an integration of the tool into the infrastructure and processes of the organization would be simple. Feedback Contextualization The participants of the experiment evaluated the contextualization feature of the software artifact as useful (Table 5). In particular, the contextualization options are neither perceived as restricting nor do users want to have on average more contextualization options (Table 17 of This indicates that the chosen contextualization options are sufficient for the users to express themselves. The focus group (FG4 in Table 1) evaluated the contextualization of solicited and unsolicited feedback as innovative and useful improving the quality of the collected feedback. In particular, the focus group evaluated the contextualization of solicited feedback items as an enabler for i) an identification of weakly formulated questions by analyzing those with low satisfaction rating via the comment contextualization, ii) identification of feedback items that are important for the library to improve their services by focusing on those items that have a high importance rating and iii) a differentiated view on given feed-back by comparing rating behavior of all users with those that found the question important to improve the library service ( Figure 13 of the Supplementary Material). In particular, this differentiated view on given feedback is described as "very interesting, because it enables a better interpretation of answers". The ranking of unsolicited feedback is evaluated as useful because it enables prioritization of unsolicited feedback which is not possible with the current implementation of the feedback wall in the library. Moreover, the combination of unsolicited with solicited feedback in one software tool has been evaluated "as a very useful approach for the library" as it facilitates a combination of quantitative and qualitative analysis. Token Incentives The token feature of the software artifact is evaluated as useful by the treatment group (Table 5). In particular, the token carries value for that group (Table 4). Table 3 illustrates the impact of token incentives on the amount of provided feedback and contextualizations. On average, the treatment group provided more solicited feedback and contextualizations than the control group. The latter indicates that the incentives are encouraging participants to increase the depth of their feedback and thus its quality (Burtch et al., 2018;Mudambi & Schuff, 2010). Unaware-users of the library services are incentivized to increase the amount of solicited feedback, indicating the potential of the chosen incentives to mobilize non-customers of the library to provide feedback. Nevertheless, in the mean, the control group gave more unsolicited feedback than the treatment group, which might be due to the following: Providing unsolicited feedback is not incentivized. Thus, users have to have intrinsic motivation, which might be crowded out in the treatment group with the applied incentives (Osterloh & Frey, 2000). Discussion Design Principles Revisited The findings of the previous section illustrates how implementing the identified design principles in a software artefact results in a value-sensitive CFS (answer to RQ 1) that is effective in terms of usability (Section 4.4.1) and quality of collected feedback (answer to RQ 2). Table 6 summarizes them by illustrating these design principles, their implementation in the software artifact, the evaluation of these implementations and the associated findings. In the following the main findings are dicussed: Due to the Covid-19 policy at the research institute, real-world interactions were restricted. Thus, the physicality in the principle of cyber-physical interactions (ID 8 in Utilize rewards x Blockchain-based tokens x Improves quant. and quality Table 6: The finalised Design Principles for customer feedback systems per category (C.) illustrating if a principle has been applied in the artifact utilized in the experiment (EX.), how the principle has been implemented in the software artifact (Implementation), if it has been evaluated by the users (U) or focus group (FG), and the findings of these evaluations. in the experiment. Nevertheless, the value of real-world interactions with humans had been identified as important for the stakeholders in the interviews (Section 4.1) and the design workshops (Section 4.2). Thus, we recommend that the impact of mechanisms that require real-world interactions for feedback provision should be evaluated in future work. Three of the design principles (ID 1, 2, 8 in This work includes the existing unsolicited feedback provision mechanism of the library (library wall, Section 4.1.3) to account for the design principle of self-explanatory system/UI (ID 3 in Table 6) by reusing interfaces already familiar to the stakeholders. This resulted also in the combination of solicited and unsolicited feedback prevision in one software artifact, which has been identified by the experts of the focus group (FG4 in Table 1) as innovative and useful. In particular, it facilitates the combination of quantitative and qualitative analysis. Because of that, we added the principle of combining solicited and unsolicited feedback collection to the list of design principles (ID 7 in Table 6). The contextualization of feedback (ID 4 in Table 6) has been evaluated by both, the experiment participants and the focus group as useful. In particular, amongst others, it enables a differentiated postanalysis of the feedback by comparing rating behavior of users based on the given contextualizations (Section 4.4.2). Theoretical implications for DSR stemming from a focus on value-sensitive design Utilizing value sensitive design in DSR simplifies the design process as it restricts the potential design space of CFS to those configurations that align with stakeholder values. For instance, the design space of blockchain-based systems is large (Ballandies, Dapp, & Pournaras, 2021). In particular, one can choose between using a private blockchain (data stored and mechanisms not transparent) or a public blockchain. Each decision coming with different implications for system properties (e.g. level of immutability of data entries, transparency of mechanisms/ smart contracts, etc.) and dependent design options (e.g. the access rights to interact with the system). By utilizing value-sensitive design we obtained the design principles of using a public and immutable storage and a transparent computing engine, which can be facilitated with a permissionless and public blockchain. Thus, the design options associated with private blockchains had been removed upfront and thus simplified/accelerated the design process by providing focus on the solutions using public blockchain infrastructures. Moreover, by first identifying the important values which a system has to incorporate and then associating those values with the design principles, as performed in this work, one explicitly receives the rationale for a design principle in the form of values that are instantiated in a software artefact by following that design principle. This is illustrated in Figure 3 for each of the design principles. This supports designers and implementors of a systems by transparently and efficiently communicating ethical implication of an instantiated system in terms of values. This process of connecting design principles with values is summarized in Figure 9. Finally, considering the importance of values for humans and the findings of this work, the DSR Figure 9: Design principle development according to Chandra et al. (2015); Möller et al. (2020); Walls et al. (1992) Limitations Utilizing public blockchains comes with a thread to privacy if sensitive and/or identity revealing information is posted to the blockchain that would then be visible to everyone. Several mechanisms exist to anonymize user input or limit the information reveal. For instance, one way is to only store the hashes of information on-chain and keeping/ storing the data itself off-chain, potentially in a privacy-sensitive way. In general, the values of privacy and transparency are often conflicting. One can employ valuesensitive design methods such as to make value conflicts transparent and to resolve them, for instance, via data-sharing coordination (Pournaras, Ballandies, Bennati, & Chen, 2023). Though not utilized in this work, Table 2 provides a first indication that stakeholders for the CFS considered in this work transparency higher than privacy which resulted in the instantiated system design. In particular, privacy is not considered in the instantiated software artifact, which is communicated transparently in Figure 3. In future work one could extend the set of values considered for the CFS (e.g. with privacy), explicitly resolve potential value conflicts (via data-sharing coordination (Pournaras et al., 2023)) and then update the system design accordingly. Due to the scope defined by the case study (CFS in a library organization), high scalability in terms of user input was not required. Several mechanisms exist to improve the scalability of 2nd layer systems that utilize another blockhain as an infrastructure layer, as it is the case in our instantiated software artefact. One of them is to keep some computations off-chain. In order for the constructed system to be deployed in a global organization with a potential large customer base, its scalability has to be evaluated and be accounted for in its system design, which is left to future work. Moreover, if a high scalibility would be required that cannot be facilitated with the chosen public blockchain (Ethereum, see Section 4.3), utilizing either a permissioned/ private blockchain (Ballandies, Dapp, & Pournaras, 2021) or newer public blockchains (e.g. Solana (Yakovenko, 2018)) might be more suitable. Nevertheless, in this case it would be neccessary to analyse the impact of the choice on the instantiated values in the artifact. Finally, the integration of value-sensitive design methods into the methodology of an established design science research process facilitated the identification of design principles that explicitly consider stakeholder values. The positive evaluation of the software artifact indicates that this approach is promising to construct systems that are value-sensitive and simultaneously improve the status-quo of a systems functioning. Nevertheless, due to the scope of this research, it has not been explicitly evaluated if the software artifact actually is perceived by the stakeholders as to align with their values, which is left to future work. Conclusion This paper argues that the feedback provision to organizations via software artifacts can be improved by following the design principles identified in this work (Table 6). In particular, in this way a system can be instantiated that is useable, motivates users to provide feedback and that improves the quality of the collected information. By considering values of stakeholders explicitly in the design steps of an established Design Science Research methodology, this work accounts for both, i) the alignment of the created system with stakeholder values such as credibility and autonomy, and ii) an innovation in the way how feedback is provided to organizations by means of blockchain-based incentivization and contextualized information. Hence, the principles (Table 6) can be utilized by decision makers and managers to create novel value-sensitive and status quo-improving customer feedback systems. Moreover, the introduced methodology explicitly provides values as a rationale for design principles and also facilitates the efficient design of software artifacts by reducing the design space of potential system configurations to those that are compatible with stakeholder values. Finally, this work shows how blockchain technology enables the three design principles of CFS: public and immutable data storage, transparent computation, and appreciative feedback in the form of token rewards. The results point to various avenues for future research. Firstly, the software artifact and the instantiated system could be utilized in organizations other than libraries to evaluate the generality of the found design principles and thus increase the confidence in the problem-solution link. Secondly, the reduced provision of unsolicited feedback by the control group when compared to the treatment group indicates a crowding out of intrinsic motivation. This could be validated in future work by further analyzing the interplay of the applied incentivizes. In particular, because we found that the quality of provided feedback in form of contextualizations is improved by applying cryptoeconomic incentives while increasing the quantity (aligned with other parallel studies (Ballandies, 2022;Pournaras et al., 2023)), future studies could investigate the impact of varying combinations of cryptoeconomic incentives on the characteristics of provided feedback to identify an optimal combination of incentives. Third, the DSR community could explore the extent to which design knowledge in the form of design principles can be compared between application domains that require the same values in their design. This would eventually reduce design time as design knowledge could be transferred between application domains via values. Figure 2 : 2The contributions of the four steps of the research methodology (I-IV in Figure 1) with regards to projectability (of the research context to new research contexts), fitness (of solving the target problem) and confidence (in the evaluation of the solution) of the design knowledge (DK) chunks of the research process as introduced in Vom Brocke et al. (2020) the Supplementary Material) are analyzed. A comprehensive compilation of challenges are identified and illustrated in Figure 1 and Figure 2 of the Supplementary Material. The following challenges are addressed in this work: i) Mobilising non-customers: Obtaining feedback from those that do not utilize the library services, respectively those that are unaware that they are utilizing a service of the library is difficult. (ii) Hierarchy of the organization prevents agile processing of feedback: Feedback is not forwarded preventing the recognition of the feedback by the responsible organizational unit (iii) Difficulty to distinguish important from unimportant feedback: When the quantity of collected feedback is high and enters the organization via various channels and organizational units, identifying feedback that would result in an improvement of services is hard. (iv) Difficulty to evaluate the quality of questions utilized in solicited feedback: When designing surveys, often similar questions are repeated or the posed questions are not evaluated if they enable a comprehensive answer (e.g., a limited set of answer options in single-choice type questions). (v) Monetary incentives may increase quantity while reducing quality of collected feedback when awarded to customers for the provision of feedback. the Supplementary Material lists all mentioned values). The mentioning of anonymity and transparency are positively biased by the interview guide (Table 3 and 4 of the Supplementary Material) by surveying participants about the importance of those values. Openness, Safety, Real-world Human interactions, and Simplicity have been brought to the discussion by the experts. In particular, the value of real-world human interactions is considered as important to facilitate a successful feedback process as it enables the exchange of informal feedback: It facilitates the recognition of gestures and the exchange of spontaneous and unsolicited feedback. ). The two that are integrated in the software artifact of this workd: i) A webbased feedback wall in the form of a pin-board is utilized where users can quickly post around the clock unsolicited feedback. The wall enables anonymous input and open/ transparent visibility of feedback items. The newest feedback is always on top. ii) Physical feedback collection points in the form of boxes are installed in the library buildings which facilitated the collection of high quantity feedback. (Figure 1 ) 1. A1 moderated the workshop, whereas A2 facilitated the technical setup in the background and assisted participants in case of questions. Due to the Covid-19 policies at the research institute, the workshops are conducted virtually utilizing zoom 3 and Miro 4 . 8 of the Supplementary Material illustrates the workshop activities: Activities are either collaborative if participants interact with each other, or individual if participants have no interactions. At the beginning of workshop 1, all participants conducted a collaborative training to familiarize themselves with Miro. Because status differences are evident in the group and participants (partly) did not know each other beforehand which limits social interactions, both workshops utilized Brainwriting (VanGundy, 1984) as an individual Brainstorming method to identify the stakeholders and design requirements. Both, the rating of stakeholder interest and influence, and the association of these stakeholders with values are performed individually and were analyzed by the research team after the workshops: By averaging participants' rating of stakeholders influence and interest in the solution, the final stakeholders influence / interest matrix and its clusters are identified (Figure 7 of the Supplementary Material). This summarized stakeholder map was accepted by all participants in the second workshop. In order to facilitate that the stakeholders' values are accounted for in the design requirements of the system, the value analysis is performed as an intermediate step between the stakeholder analysis and the requirements elicitation: Each participant individually assigned all relevant values to a subset of stakeholders (Cluster 3 and 4 in Figure 7 of the Supplementary Material) of the overall stakeholder analysis. The strength of stakehold-ers' association with a value is illustrated in Figure 7 7tion 4.2.1), a ranking of values based on their average importance (Section 4.2.2) and value-based design requirements (Section 4.2.3). Moreover, elicited by the associated researchers from the design requirements, design principles that guide the construction of value-sensitive feedback systems (Section 4.2.4) and a system design based on these principles (Section 4.2.5) are illustrated. of the Supplementary Materials illustrates the participants and interest groups of the feedback system in the form of a Stakeholder map(Rössner, Dapp, & Fach, 2018).Four clusters are identified: Cluster (1) does not have a positive influence on the construction of the solution and a low to medium interest in it. The cluster contains stakeholders such as suppliers, other libraries, and publishers. Cluster (2) contains the legislation, politics, and public funding institutions. These stakeholders have an influence on the solution while not having a high interest in it. The interest in and possible influence on the solution of Cluster (3) is the highest. These stakeholders are key in the design requirements engineering of Section 4.2.3. This cluster includes the management and experts of the library as well as average employees and customers (e.g. researchers and lecturers). The directorate Figure 3 : 3carry, sorted by strength of association (output of the second Design Workshop). The following values have the strongest association with the stakeholders (average strength ≥ 1): Credibility, Simplicity, Universal Usability, Excellence, Efficiency, Identity, Autonomy, Safety, and Physical human interaction. These values are considered in the following sections (Section 4.2.3 and 4.2.4), which were only extended with the values of Human Welfare and Sustainability that are of importance to the directorate of the library (the most important stakeholder according to the analysis of Section 4.2.1). Please refer to related work for an Identified values (left), requirements associated with these values (middle) and the design principles facilitating those requirements (right). The design principles utilize the framework of TBD for representation. The boundary condition for all principles are customer feedback systems. The rationale for a principle are the instantiated values (bottom box of each principle). Figure 3 3illustrates the identified important values (Section 4.2.2) and the associated design requirements, which are the output of the second Design Workshop (Table ??). Because the directorate is the stakeholder with the greatest influence on and interest in the solution (Section 4.2.1), its values are added to the identified important values (Section 4.2.2). In total 97 requirements are identified of which 32 are associated with the important values. The full list of requirements are given in Tables 12-14 of the Supplementary Materials. Figure 4 4illustrates the system design that is found by instantiating the identified design principles. The mapping of design principles to system design choices is illustrated in the following. Design principles that are related to the instantiation of the software artifact are illustrated in Section 4.3. Figure 4 : 4Utilizing the model of Zargahm can be implemented in the platform that define what users of the system can or cannot do: (i) blockchain-based cryptoeconomic incentives in the form of tokens can be defined and awarded as rewards to users for the provision of high-quality feedback or respectful behavior (Design Principle 11 in Talbe 6) ; (ii) Also, diverse and trustworthy statistics can be aggregated and shown to the stakeholders of the system by analyzing the publicly accessible data storage via trusted computing protocols (Design Principle 6 in Figure 5 : 5Software Stack of the feed4org app. Figure 6 : 6Open Feedback View (feedback wall) of the software artifact. (Figure 7 )Figure 7 : 77. The following question types are implemented: Single-choice, multiple-choice, likert scales, 7 https://vuejs.org/, last accessed: 2021-10-17 8 please refer to Section 4.3.1 for details on the utilized incentives Answer View -Entry point to the feed4org app where the organization can ask for solicited feedback. Users have the possiblity to contextualize the feedback with its importance or their satisfaction. Moreover, each feedback item can be commented. open text, and combinations of these options. Contextualization: In addition to answering questions, users can contextualize their answers (Design Principle 4 in Table 6) by clicking on the contextualization buttons (bottom buttons in Figure 7). Three types of contextualizations are implemented. Users can state with the Importance contextualization how important the question is for the library to improve their services, with the Satisfaction contextualization how satisfied they are with the range of answer options and with the Comment contextualization to provide further comments in an open feedback form (Figure 10 of the Supplementary Material). Figure 8 : 8Token Design of the utilized cryptoeconomic incentives utilizing the DLT system taxonomy(Ballandies, Dapp, & Pournaras, 2021). Figure 12 of 12the Supplementary Material illustrates the user interactions with the software artifact in a heatmap. The importance (6018) and satisfaction (5692) contextualization are more often utilized than the comment (2107) contextualization. Figure 6 6illustrates the obtained unsolicited feedback and the participants ranking of these items for the Supplementary Material). Table Table 2 2illustrates which values the various Stakeholders from Cluster Table 4 : 4Mean, lower/ upper 95 % confidence interval, standard deviation and number of participant answers of the constructs of effort expectancy (UTAUT, Table 15 of the Supplementary Material) and token value Table 5 5illustrates the usefulness evaluation of the artifact by the participants. Both, the treatment (2.82) and the control (2.80) group evaluate the features of the artifact as useful on average. In particular, the statistics view has been evaluated as most useful in the treatment group, whereas the control group rated the open feedback view as most useful. Table 6 ) 6had not been facilitatedID C. Design Principle Ex. Implementation Eval. Finding U FG 1 Infrastrucutre Use a public data store x Blockchain (Ethereum) 2 Use a transparent computing engine x Smart Contracts 3 Create a self-explantory System/UI x Reused existing interfaces (Google, Reddit, Stackover.) and tools (feedback wall, finance 4.0) x x fast onboarding; low-threshould; simple integr. in exist. proces. 4 Feedback Enable contextualizton of feedback x Dedicated views x x exhaustive; impr. post-analysis 5 Rank feedback x Up/ down voting of unsolicited feedback x Enables priorization 6 Visualize (feedb.) statistics x Dedicated statistics view x High usefulness 7 Combine solicited and unsol. feedback x Dedicated views x Quant. and qual. eval 8 Interaction Enable cyber-physical interactions Interactive answering options; real-time digital voting centers 9 (Re)use already existing pseudonymous identities x Blockchain addresses 10 Build an opt-in system x No force applied on users to join 11 Table 6 ) 6are afforded by the blockchain technology which illustrates the useability of this technology for CFS. In particular, blockchain-based cryptoeconomic incentives represent a novel class of incentives that can motivate users to improve the breadth (number of feedback items) and depth (number of contextualization actions) of shared feedback(Section 4.4.3).Moreover, the utilization of the technology can enhance the trust in the feedback handling process by making it transparent and verifiable. , as introduced inHaße et al. (2022), and extended with value-sensitive design methods (green/ patterned boxes), as introduced in this work: A stakeholder-value analysis (Section 4.2.1 and 4.2.2) and brain writing method (Section 4.2.3) can be utilized to connect meta-requirements of a software artefact with important values that stakeholders carry, eventually resulting in design principles that instantiate those values in an IT artefact. community could start theorizing how value-sensitive design methods can be integrated in established DSR methodologies to make design knowledge comparable across application domains and to enhance the confidence in problem-solution links. For instance, system designers might be able to reuse the design principles associated to values of this work if these values are also required for the software artifact of their application domain. This would eventually reduce construction time as design knowledge could be compared and transferred between application domains via values. 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[ "Slepton Flavor Physics at Linear Colliders", "Slepton Flavor Physics at Linear Colliders" ]	[ "Michael Dine ", "Yuval Grossman ", "Scott Thomas ", "\nSanta Cruz Institute for Particle Physics\nDepartment of Physics, Technion, Technion City\nDepartment of Physics\nUniversity of California\n95064, 32000Santa Cruz, HaifaCAIsrael\n", "\nStanford University\n94305StanfordCA\n" ]	[ "Santa Cruz Institute for Particle Physics\nDepartment of Physics, Technion, Technion City\nDepartment of Physics\nUniversity of California\n95064, 32000Santa Cruz, HaifaCAIsrael", "Stanford University\n94305StanfordCA" ]	[]	If low energy supersymmetry is realized in nature it is possible that a first generation linear collider will only have access to some of the superpartners with electroweak quantum numbers. Among these, sleptons can provide sensitive probes for lepton flavor violation through potentially dramatic lepton violating signals. Theoretical proposals to understand the absence of low energy quark and lepton flavor changing neutral currents are surveyed and many are found to predict observable slepton flavor violating signals at linear colliders. The observation or absence of such sflavor violation will thus provide important indirect clues to very high energy physics. Previous analyses of slepton flavor oscillations are also extended to include the effects of finite width and mass differences.	10.1142/s0217751x03016227	[ "https://export.arxiv.org/pdf/hep-ph/0111154v1.pdf" ]	55,055,236	hep-ph/0111154	4b1baf3d82ca6c6fd095f575480eb34b823de5c0	Slepton Flavor Physics at Linear Colliders arXiv:hep-ph/0111154v1 13 Nov 2001 Michael Dine Yuval Grossman Scott Thomas Santa Cruz Institute for Particle Physics Department of Physics, Technion, Technion City Department of Physics University of California 95064, 32000Santa Cruz, HaifaCAIsrael Stanford University 94305StanfordCA Slepton Flavor Physics at Linear Colliders arXiv:hep-ph/0111154v1 13 Nov 2001 If low energy supersymmetry is realized in nature it is possible that a first generation linear collider will only have access to some of the superpartners with electroweak quantum numbers. Among these, sleptons can provide sensitive probes for lepton flavor violation through potentially dramatic lepton violating signals. Theoretical proposals to understand the absence of low energy quark and lepton flavor changing neutral currents are surveyed and many are found to predict observable slepton flavor violating signals at linear colliders. The observation or absence of such sflavor violation will thus provide important indirect clues to very high energy physics. Previous analyses of slepton flavor oscillations are also extended to include the effects of finite width and mass differences. I. INTRODUCTION If supersymmetry is discovered at the Tevatron or LHC, much of the focus of particle physics research will turn to measuring and understanding the 105 (or more) soft breaking parameters. While this problem may seem, at first, more daunting than understanding the pattern of quark and lepton masses and mixings, we know, a priori, that these parameters must show striking regularities. Otherwise, supersymmetry would lead to large, unobserved flavor violation among ordinary quarks and leptons. From the perspective of string theory, for example, the prospect of discovering supersymmetry and measuring the soft breaking parameters is extremely exciting. Indeed, in most pictures of supersymmetry breaking, squark and slepton masses arise from non-renormalizable operators associated with some new scale of physics. This could well be, for example, the scale of string or M-theory. Thus, collider measurements of the soft breaking parameters are potentially indirect probes of extremely high energy physics, physics we might hope to eventually unravel! To date, there are only a few theoretical proposals for the origin of the requisite regularities of the soft parameters which lead to acceptably small low energy flavor violation among the quarks and leptons. Each has important implications for the physics discoveries of future colliders. In this note, we will survey these proposals, and discuss some of their implications for collider phenomenology. One particularly exciting possibility, is the existence of dramatic flavor violating phenomena in slepton production. This can occur even given the strong bounds from the non-observation to date of flavor violation in low energy lepton decays. We also call attention to previous work on slepton flavor violating collider phenomenology [1,2] and extend it to include the effects of finite width and mass differences. Our motivation is the realization that, in a first phase, a linear collider is likely to have a center of mass energy of 500 GeV. As a result, one can well imagine that supersymmetry will have been discovered at the LHC, but that we will have only limited information about the spectrum, and a first generation linear collider will have access to only a few of the lightest superpartner states. In many theories these include sleptons. Even with this limited set of states, the observation -or notof slepton flavor violation will provide important clues to the underlying supersymmetry breaking structure. The reason that observable sflavor violation at a linear collider is possible is easy to understand. In many proposals to solve the supersymmetric flavor problem, a high degree of degeneracy among sleptons (and squarks) is predicted. As a result, there is the potential for substantial mixing of flavor eigenstates. This can lead to substantial and observable sflavor violation. To be readily observable, it is necessary that mass splittings between the states not be too much smaller than the decay widths, and that the mixing angles not be terribly small. In some of these proposals, and in particular gauge mediation, the predicted degree of degeneracy is extremely high, and the mixing probably unobservable. However, in most other proposals, the splittings can be comparable or even larger than the widths, and the mixing angles may be of order the Cabbibo angle or larger. In this case, dramatic collider signatures are possible. II. MEDIATION MECHANISMS It is instructive to review the various proposed mechanisms for understanding the suppression of flavor changing processes in supersymmetric theories. We should start by noting that most analyses of collider experiments work in the framework of what has become to known as Gravity Mediation. In such models, one simply assumes exact degeneracy of squarks and sleptons at some very high energy scale. This assumption is not natural since it is violated by quantum corrections (and is in fact scale dependent), and so surely breaks down at some level. Without some theory, or more detailed assumptions, one cannot assess the degree to which this assumption is viable. In other words, degeneracy in this case is a puzzle to be explained rather than a mechanism in itself. There are a number of more serious proposals of which we are aware for understanding the suppression of supersymmetric contributions to quark and lepton flavor violating processes. These fall into three broad classes. The first are mechanisms which seek to ensure a high degree of degeneracy as the result of dynamics without specific assumptions about flavor. The second are flavor symmetries which enforce either a high degree of degeneracy or alignment of the squark and slepton mass matrices with those of the quarks and leptons. The third invokes heavy superpartner masses to kinematically suppress low energy processes. • Gauge Mediation: In its simplest form, any flavor violation in gauge mediation occurs at a high order in the loop expansion, or due to non-renormalizable operators. In the former case, the suppressions are typically by powers of small Yukawa couplings and extra loop factors. These effects are automatically aligned with flavor violation in the Yukawa sector and are therefore not dangerous and do not give rise to interesting processes. In the latter case, the suppression is by powers of Λ 2 /M 2 , where Λ represents some typical scale associated with the supersymmetry-breaking interactions, and M some large scale associated with new physics such as a flavor scale at which flavor is spontaneously broken or the Planck or string scale. For Λ ≪ M the distinctive feature is the absence of sflavor violating processes. • Dilaton Domination: At weak coupling in the heterotic string there is a regime in which a gravity-mediated spectrum with a high degree of degeneracy is obtained. If the weak coupling picture is valid, one might hope that generic flavor violating corrections to squark and slepton mass matrices are of order α GUT /π. Numerically this is just enough to understand the suppression of flavor changing neutral currents [3]. It suggests that mass splittings among the lightest sleptons will be at least of order a few parts in 10 −3 . We will see that this is within the range (albeit at the low end) of what one might hope to measure directly at a linear collider. Without further assumptions about flavor, one expects mixings of squarks and sleptons to be of order one. Such a mass splitting is comparable to the expected decay widths of these states, so this sflavor violating mixing might well be observable. In the strongly coupled limit, it seems likely that the violations of degeneracy are larger [4]. • Anomaly Mediation: The anomaly-mediated hypothesis superficially has some features in common with gauge mediation. However, the mediation scale is now comparable to the Planck scale, and one has to ask about the magnitude of flavor violating corrections to squark and slepton mass matrices. It has been argued that brane world realizations of supersymmetry breaking might naturally provide a context for anomaly mediation with small corrections to degeneracy [5]. However, anomaly mediation turns out not a robust feature of brane world supersymmetry breaking, and violations of degeneracy are generally large [4]. Still, the degeneracy might be small enough to suppress low energy flavor violating processes in very special circumstances. • Gaugino Domination (and the closely related idea of gaugino mediation): With no-scale boundary conditions one assumes that scalar masses vanish or are very small compared to gaugino masses at the high messenger scale. Without a detailed underlying model, it is hard to know how large the violations of degeneracy might be at this scale, but it seems reasonable to suppose that at the high scale, the magnitude of the squark and slepton mass matrices are suppressed relative to gaugino masses by an amount of order α GUT /π. At lower scales, renormalization group evolution gives a flavor independent gaugino mass contribution to slepton masses which leads to a high degree of degeneracy. As in the case of dilaton domination, without further assumptions about flavor, the violations of degeneracy are expected to be maximally flavor violating, and to lead to large mixings in the high scale squark and slepton masses which in turn leads to large mixings at the electroweak scale. • Conformal Sequestering: Another possibility for obtaining a high degree of degeneracy is to postulate that the first two generation squarks and sleptons are coupled to an approximately conformal sector over a few orders of magnitude in renormalization group evolution [6]. This has the effect of exponentially suppressing squark and slepton masses as well as fermion Yukawa couplings. Below this approximately conformal range of scales further renormalization group evolution gives a flavor independent gaugino mass contribution to slepton masses which leads to a high degree of degeneracy much as with no-scale boundary conditions. Violations of conformal invariance by standard model gauge interactions limit the degree of slepton degeneracy to roughly α W /π ∼ 10 −2 . Without further assumptions about flavor in the high scale theory above the approximately conformal scale, slepton mixings are related to ratios of lepton masses by roughly sin φ ij > ∼ m ℓi /m ℓj . • Non-Abelian Flavor Symmetries: If the explanation of squark and slepton degeneracy lies in non-abelian flavor symmetries, it is reasonable to expect that violations of degeneracy are correlated with the values of quark and lepton masses and the KM angles. In this case, one might hope to obtain tighter predictions in a given model for the pattern of flavor violation in the slepton sector. The level of degeneracy is model dependent, but again a few parts times 10 −3 is a reasonable expectation for the violations of degeneracy, with values for the mixings of order Cabbibo angles sin φ ij ∼ m ℓi /m ℓj [7]. • Abelian Flavor Symmetries: As an alternative to degeneracy among squarks and sleptons, it has been suggested that the squark and slepton mass matrices might be approximately aligned with the quark and lepton matrices [8]. This can come about in theories with Abelian (discrete) flavor symmetries. In this case, one does not expect any approximate degeneracy among sleptons. In order to suppress flavor changing lepton decays the mixings need to be somewhat small. Still, the mixing can be large enough, in particular for mixing involving taus, such that sflavor violation can be observable at colliders. • First Two Generations Heavy: Another possibility to suppress dangerous levels of supersymmetric contributions to low energy quark and lepton flavor violation is to postulate that the superpartners are very heavy. The first two generation squarks and sleptons can have masses up to of order 20 TeV without introducing significant tuning of electroweak symmetry breaking [9]. In this case only the (mostly) stau slepton(s) would be kinematically accessible at future colliders. Without additional assumptions about flavor, naturalness of the full slepton mass matrix implies that mixing of this light state(s) among the flavor eigenstates would be of order m/M where m and M are the light and heavy slepton masses. In sum, of the various proposals to understand the absence of flavor violation at low energies, several predict dramatic violations of flavor at colliders. Among those which don't, there tend to be distinctive predictions for the spectrum. Models of gauge mediation, for example, tend to be highly predictive. While there is no one compelling model of this type, many models exist, and one can imagine detailed measurements distinguishing between them. In the case of alignment mechanisms, while there should flavor mixing it will not be so dramatic, one should observe correlations between the squark and slepton and the ordinary quark and lepton masses. In the case of non-Abelian flavor symmetries, not only does one expect significant mixing, but one can hope to obtain, given some assumptions about the form of flavor symmetry breaking, precise predictions for some of the violations of flavor symmetry. Moreover, these are likely to be correlated with quark, lepton and neutrino mass matrices. In such a case, precision measurements might ultimately permit distinguishing between different models. Clearly, all of these are directions for further theoretical work, but the discovery of supersymmetry and unraveling the pattern of symmetry breaking would provide important insights into the nature of physics at very high energy scales. The observation of flavor violation -or its absence -at linear colliders will provide important clues to the nature of the underlying mechanisms of supersymmetry breaking and mediation. III. SFLAVOR OSCILLATIONS OF UNSTABLE SLEPTONS Slepton flavor oscillations arise if the sleptons mass eigenstates are not flavor eigenstates. Consider for simplicity the case in which leptons are produced initially in flavor eigenstates. If the mass eigenstates have distinct mass, then the flavor eigenstate oscillates in time and space with a frequency given by the mass splittings. If the slepton decay rate is much smaller than the mass splitting, oscillations average out and the probability of decay from a given flavor eigenstate is given simply by mixing angles, as implicitly assumed previously [1]. If the slepton decay is rapid compared with the oscillation frequency then the probability of decay to another flavor eigenstate is suppressed [2]. There are additional effects which can affect the flavor violating decay probability. First, different flavor eigenstates need not have the same decay width. This is particularly true for theτ slepton which can have a non-trivial decay amplitude to the Higgsino component of a neutralino at moderate to large tan β. In addition, the probability amplitude or cross section for production of different mass eigenstates need not be equal. This is potentially important in S-wave processes such as e − e − →l −l− near threshold due to finite mass differences. Both of these effects might in principle enhance flavor violating effects in decays, and are considered below. For simplicity throughout we consider the two flavor CP conserving case with flavor and mass eigenstates related by \|ẽ = c\|1 + s\|2 , \|μ = −s\|1 + c\|2 (III.1) where c and s are complex numbers that satisfy \|c\| 2 + \|s\| 2 = 1. (Only in the limit where the width difference is neglected s and c can be chosen to be real, thus getting they regular interpretations as a sine and a cosine of an angle.) The other relevant physical dimensionless parameters characterize the ratio of mass splitting or oscillation frequency to decay width, the relative width difference of the mass eigenstates, and the relative rate or cross section for production of the mass eigenstates x ≡ ∆m Γ , y ≡ ∆Γ 2Γ , z ≡ ∆σ 2σ (III.2) where ∆m and ∆Γ are the mass and width differences of the two mass eigenstates, and Γ is their average width. σ i is the cross section to produce the ith mass eigenstate, and σ = σ 1 + σ 2 2 , ∆σ = σ 2 − σ 1 . (III.3) The case x = y = z = 0 was considered in [1] while y = z = 0 with x arbitrary was considered in [2]. In the first subsection below, the flavor decay probability is derived for small but non-vanishing x, y, and z. In the next subsection the decay probability for small x and y is shown to depend directly on the off-diagonal slepton mass squared mixing, as required in this limit. In the final subsection, estimates of the widths are given. A. Decay Probability The probability for an initial selectron state to decay as a smuon state, P (ẽ →μ), may be calculated from the time evolution of the state. A selectron flavor state produced at time t = 0 is a linear combination of mass eigenstates ψ(t = 0) = N [σ 1 c\|1 + σ 2 s\|2 ] = N [σ\|ẽ − ∆σ\|μ ] , (III.4) where the normalization factor is N = c 2 σ 2 1 + s 2 σ 2 2 −1/2 (III.5) and σ i ≡ σ i /σ are the dimensionless cross sections. The factors of σ i in the relative amplitudes account for the possibility that the cross section for each mass eigenstate is distinct; for example, from mass difference effects near threshold. Using the standard oscillation formalism the time evolution formula for a selectron initial state is ψ(t) = N cσ 1 e −iµ1t \|1 + sσ 2 e −iµ2t \|2 , (III.6) where µ i = m i − iΓ i /2 (III.7) and m i (Γ i ) is the mass (width) of the ith mass eigenstate. The time dependent oscillation probability is given by P (ẽ →μ)[t] = \| μ\|ψ(t) \| 2 \| ψ(t)\|ψ(t) \| 2 (III.8) so that the projection onto the smuon flavor eigenstate is \| μ\|ψ(t) \| 2 = N 2 c 2 s 2 σ 2 2 e −Γ2t + σ 2 1 e −Γ1t − 2σ 1 σ 2 e −Γt cos(∆mt) . (III.9) After integration over time from zero to infinity ∞ 0 dt \| μ\|ψ(t) \| 2 = N 2 c 2 s 2 σ 2 2 Γ 2 + σ 2 1 Γ 1 − 2σ 1 σ 2 Γ ∆M 2 + Γ 2 , (III.10) and ∞ 0 dt \| ψ(t)\|ψ(t) \| 2 = N 4 s 4 σ 4 2 Γ 2 + c 4 σ 4 1 Γ 1 + 2c 2 s 2 σ 2 1 σ 2 2 Γ . (III.11) Then the time integrated oscillation probability is P (ẽ →μ) = \|cs\| 2 N 2 σ 2 2 Γ 2 + σ 2 1 Γ 1 − 2σ 1 σ 2 Γ ∆M 2 + Γ 2 × \|s\| 4 σ 4 2 Γ 2 + \|c\| 4 σ 4 1 Γ 1 + 2\|cs\| 2 σ 2 1 σ 2 2 Γ −1 . (III.12) It is useful to expand P in the (presumably) small parameters, x, y and z. To lowest order P (ẽ →μ) = 2c 2 s 2 x 2 + y 2 + 2z 2 − 2zy (III.13) The oscillation probability in this limit is quadratic in all three small parameters. The x 2 term describing the effect of finite mass difference is the lowest order form of the effect discussed in [2]. The y 2 term describing the effect of a width difference between the two mass eigenstates parametrically plays a similar role as x 2 , as is well known from the D-meson system. The z parameter describes the effect of the difference of production probability for the two mass eigenstates. This effect has never been mentioned in the context of meson mixing, since there it is always completely negligible. As discussed below, also in the present case of slepton production, its effect is somewhat smaller than the effects of x and y. We first consider the possible effect of the z parameter. Ignoring the decay width, the cross section for e − e − →l −l− is S-wave and proportional to the final state velocity β near threshold. Far above threshold, where the cross section is large, β → 1 and z vanishes. Only near threshold, where the cross section is small, z can be sizeable. The Monte Carlo program Pandora may be used to estimate z. As an example, we considered small mixing angles and took ml = 150 GeV, ∆m = 1 GeV, m χ 0 = 100 and a center of mass energy of 305 GeV, and found z = 0.13. Note that in this case Γ = 240 MeV, so x ≈ 4, and it is the dominant effect. Also at smaller x the inequality z < x still holds. We conclude that the effect of z is smaller than that of x and y. This is because z is generated only for finite x or y, and in such a way that it is smaller then x and y. Even so, it does increase the flavor violating decay probability somewhat. where m 21 = m * 12 , Γ 21 = Γ * 12 and m ii and Γ ii are real. Solving the eigenvalue equation det(M − µ i ) = 0 gives the mass eigenstates, the mass and width differences, and the mixing angles. The general calculation is not very illuminating, so several simplifying assumptions are employed. First, we note that in the MSSM there are no flavor violating decays. Thus, we set Γ 21 = 0. (This may not be the case in models beyond the MSSM, for example, in supersymmetric models with 4 Higgs doublets, where the Higgs bosons can mediate flavor changing decays.) Second we assume CP conservation and then m 12 may be taken real. Next, for simplicity, we assume that m 11 − m 22 ≪ m 12 and may be approximated by zero. This is because in the opposite limit m 11 − m 22 ≫ m 12 the mixing angle is small and we are not interested in that case. With these assumptions we solve the eigenvalue equation. We define the real parameters a ≡ m 11 = m 22 , b = Γ 11 /2, d = Γ 22 /2 and f = m 12 and find that for (b − d) > 2f x = 0, y = (b − d) 2 − 4f 2 b + d , (III.15) while for (b − d) < 2f x = 2 4f 2 − (b − d) 2 b + d , y = 0. (III.16) For simplicity for the mixing angles we present results only in some limits. For (b − d) ≫ 2f we find c ≈ 1 √ 2 (1 + i), s ≈ f √ 2(b − d) (1 − i), (III.17) and for (b − d) ≪ 2f we find c ≈ s ≈ 1 √ 2 . (III.18) We see that in the (b − d) ≫ 2f limit, where the width difference is large, the mixing angles are suppressed. Actually, the oscillation probability is the same in both limits considered above P ≈ 2\|cs\| 2 (x 2 + y 2 ) ≈ 2m 2 12 Γ 2 . (III.19) In particular, it does not depend on Γ 11 − Γ 22 . We conclude that no matter if we have large width difference or not, the oscillation probability is determined by the ratio between the off-diagonal mass term and the average width. While we made several assumptions in order to derive the above results, the conclusion is general. Off-diagonal mixing terms in the mass squared matrix must not be too much smaller than the decay widths in order to obtain large violations of flavor in slepton decays. C. Decay Width As we saw the relevant parameters that determined the flavor violation oscillation probability are M 12 /Γ and to some extent also (Γ 11 − Γ 22 )/Γ. While M 12 , the off diagonal mass difference, is a parameter of the model, the widths are derived from the model parameters. Therefore, below we estimates their sizes. We start with estimating Γ. For example we consider decays of right handed sleptons. The decay width for a right handed slepton to decay to a Bino through the emission of a lepton is Γ(l R → ℓ R B) = α ml R 2 cos 2 θ W 1 − m 2 B m 2 ℓR 2 (III.20) where α is the fine structure constant evaluated at the slepton mass scale, θ W is the weak mixing angle, and the lepton mass has been ignored. Numerically, the dimensionless width is For a typical value of m B /ml R = 0.75 the dimensionless width is Γ(l R → ℓ R B)/ml R ≃ 10 −3 . In light of our remarks about the expected degrees of degeneracy in various proposals for supersymmetry breaking, this is a striking number. In some sense, within the range of ideas which have been proposed, a large fraction predict that the splittings should be as large or larger than the width, and thus observable. Next we consider the possible magnitude of Γ 11 − Γ 22 for slepton production and decay byl → χℓ. If ml − m 0 χ ≫ m τ , the final state neutralino is pure gaugino, and the sleptons are pure left-or right-handed eigenstates, then the decay is universal and Γ 11 = Γ 22 . Universality violation due to phase space can be significant only when the mass splitting between the sleptons and the neutralino is at least comparable to or smaller than the lepton mass. Another effect can be dues to significant Higgsino component in χ. The full calculation forl → χℓ for general gaugino-Higgsino mixing can be found in [10]. A very crude estimate in the theτ −μ case with large tan β is Γ 11 − Γ 22 Γ ∼ Y 2 τ Y 2 τ + 2g 2 1 Z 2 1B /Z 2 1h (III.22) where Z 1h and Z 1B are the Higgsino and Bino amplitudes of χ, Y τ is the τ Yukawa coupling, and g 1 is the hypercharge coupling. For large tan β and significant Higgsino component in the neutralino Γ 11 − Γ 22 can be large. IV. CONCLUSIONS If superpartners are discovered, there is good reason to expect that sleptons will have a high degree of degeneracy which enhances sensitivity to sflavor violating effects. It is quite possible then that significant sflavor violation could be observed at a linear collider. The observation -or non-observation -of such sflavor breaking would provide important indirect clues about flavor and supersymmetry physics at potentially extremely high energy scales. B. Calculating x and yIt is instructive to derive the x and y parameters from the general form of the slepton mass matrix including the effects of flavor mixing and decay M = m 11 − iΓ 11 /2 m 12 − iΓ 12 /2 m 21 − iΓ 21 /2 m 22 − iΓ 22 /2 (III.14) Search for Flavor Lepton Number Violation in Slepton Decays at LEP2 and NLC. N V Krasnikov, hep- ph/9511464Phys. Lett. B. 388783N.V. Krasnikov, "Search for Flavor Lepton Number Violation in Slepton Decays at LEP2 and NLC," hep- ph/9511464, Phys. Lett. B 388 (1996) 783. CP Violation from Slepton Oscillations at the LHC and NLC. N Arkani-Hamed, H.-C Cheng, J L Feng, L J Hall, hep-ph/9704205Phys. Rev. Lett. 773Nucl. Phys. BN. Arkani-Hamed, H.-C. Cheng, J.L. Feng and L.J. Hall, "Probing Lepton Flavor Violation at Future Colliders," hep-ph/9603431, Phys. Rev. Lett. 77 (1996) 1937; "CP Violation from Slepton Oscillations at the LHC and NLC," hep-ph/9704205, Nucl. Phys. B 505 (1997) 3. Some Phenomenological Implications of String Loop Effects. J Louis, Y Nir, hep-ph/9411429Nucl. Phys. B. 44718J. Louis and Y. Nir, "Some Phenomenological Implications of String Loop Effects," hep-ph/9411429, Nucl. Phys. B 447 (1995) 18. Brane World Supersymmetry Breaking. A Anisimov, M Dine, M Graesser, S Thomas, to appearA. Anisimov, M. Dine, M. Graesser and S. Thomas, "Brane World Supersymmetry Breaking," to appear. A Large Mass Hierarchy From a Small Extra Dimension. L Randall, R Sundrum, hep-th/9906064Phys. Rev. Lett. 834690Phys. Rev. Lett.L. Randall and R. Sundrum, "A Large Mass Hierarchy From a Small Extra Dimension," hep-ph/9905221, Phys. Rev. Lett. 83, 3370 (1999); "An Alternative to Compactification," hep-th/9906064, Phys. Rev. Lett. 83, 4690 (1999). Suppressing Flavor Anarchy. A Nelson, M Strassler, hep-ph/0006251J. High Energy Phys. 000930A. Nelson and M. Strassler, "Suppressing Flavor Anarchy," hep-ph/0006251, J. High Energy Phys. 0009 (2000) 30; Exact Results for Supersymmetric Renormalization and the supersymmetric Flavor Problem. hep-ph/0104051"Exact Results for Supersymmetric Renormalization and the supersymmetric Flavor Problem," hep-ph/0104051. There are many models of this sort, and we won't attempt a complete list of references. See, for example. C Carone, L J Hall, H Murayama, hep-ph/9602364Phys. Rev. 542328Supersymmetric Theory of Flavor and R Parity. which provides a good overview of the subject, and interesting class of models, an extensive list of referencesThere are many models of this sort, and we won't attempt a complete list of references. See, for example, C. Carone, L.J. Hall, and H. Murayama, "Supersymmetric Theory of Flavor and R Parity," hep-ph/9602364, Phys. Rev. D54 (1996) 2328, which provides a good overview of the subject, and interesting class of models, an extensive list of references. Should Squarks Be Degenerate. Y Nir, N Seiberg ; 337, ; Y Grossman, Y Nir, hep-ph/9502418Phys. Lett. B. 30930Nucl. Phys. BY. Nir and N. Seiberg, "Should Squarks Be Degenerate," hep-ph/9304307, Phys. Lett. B 309 (1993) 337; Y. Grossman and Y. Nir, "Lepton mass matrix models," hep-ph/9502418, Nucl. Phys. B 448, 30 (1995). The More Minimal Standard Model. A Cohen, D Kaplan, A Nelson, hep-ph/9607394Phys. Lett. B. 388588A. Cohen, D. Kaplan, and A. Nelson, "The More Minimal Standard Model," hep-ph/9607394, Phys. Lett. B 388 (1996) 588. Polarization of Tau Lepton from from Scalar Tau Decay as a Probe of Neutralino Mixing. M M Nojiri, hep- ph/9412374Phys. Rev. D. 516281M.M. Nojiri, "Polarization of Tau Lepton from from Scalar Tau Decay as a Probe of Neutralino Mixing," hep- ph/9412374, Phys. Rev. D 51 (1995) 6281.	[]
[ "Center Manifolds for Rough Partial Differential Equations", "Center Manifolds for Rough Partial Differential Equations" ]	[ "Christian Kuehn ", "Alexandra Neamţu " ]	[]	[]	We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the stochastic forcing is given by a γ-Hölder rough path, for γ ∈ (1/3, 1/2]. Our proof technique relies upon the theory of rough paths and analytic semigroups in combination with a discretized Lyapunov-Perron-type method in a suitable scale of interpolation spaces. The resulting center manifold is a random manifold in the sense of the theory of random dynamical systems (RDS). We also illustrate our main theorem for reaction-diffusion equations as well as for the Swift-Hohenberg equation.This section provides an overview of the existing foundations for center manifold theory for stochastic partial differential equations and it describes the main goal and the strategy of this work.A Classical ConstructionIn the context of random invariant sets, certain classes of SPDEs have been studied in the literature; see[2,26,18,19]and the references therein. We recall the main ideas that have been	10.1214/23-ejp938	[ "https://arxiv.org/pdf/2111.01488v2.pdf" ]	240,420,155	2111.01488	f6eac7ac755df47cdac2470623f2ed8670b2fcb4	Center Manifolds for Rough Partial Differential Equations Nov 2021 Christian Kuehn Alexandra Neamţu Center Manifolds for Rough Partial Differential Equations Nov 2021center manifoldrough pathevolution equationinterpolation spacesLyapunov- Perron method Mathematics Subject Classification (2020): 60H1560G2260L2060L5037L55 We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the stochastic forcing is given by a γ-Hölder rough path, for γ ∈ (1/3, 1/2]. Our proof technique relies upon the theory of rough paths and analytic semigroups in combination with a discretized Lyapunov-Perron-type method in a suitable scale of interpolation spaces. The resulting center manifold is a random manifold in the sense of the theory of random dynamical systems (RDS). We also illustrate our main theorem for reaction-diffusion equations as well as for the Swift-Hohenberg equation.This section provides an overview of the existing foundations for center manifold theory for stochastic partial differential equations and it describes the main goal and the strategy of this work.A Classical ConstructionIn the context of random invariant sets, certain classes of SPDEs have been studied in the literature; see[2,26,18,19]and the references therein. We recall the main ideas that have been Introduction Center manifolds, as well as the easier stable and unstable manifolds, are key technical tools in dynamical systems theory [43]. The idea is to split the dynamics into exponentially attracting, exponentially repelling and neutral directions near a steady state. This splitting can often be obtained locally on the level of a linearized system. If the linearized operator has no spectrum on the imaginary axis then the steady state is called hyperbolic. In the hyperbolic situation, quite classical stable and unstable manifold theory as well as local topological equivalence between the linearized and the nonlinear system exist for many classes of evolution equations [43,47,5]. In the non-hyperbolic situation, when spectrum on the imaginary axis appears, we actually need more involved center manifold theory [16]. Although this situation may appear non-generic at first, it is well-understood that it is generic in differential equations with parameters, where it is of crucial importance to obtain center manifolds to study bifurcation problems [43,56]. Furthermore, center manifold theory can yield effective dimension reduction near a steady state if there are only attracting and center directions, which is a concept that can be extended to entire manifolds of steady states, e.g., in the context of slow manifolds for multiple time scale systems [29,54]. For stochastic differential equations, there already some results regarding center manifolds, see [2,11,12,66] for stochastic ordinary differential equations (SODEs) and [19,18,27,8] for SPDEs. However, stochastic center manifold theory is still far less well-developed in comparison to deterministic ordinary differential equations (ODEs) or partial differential equations (PDEs). The aim of this work is to investigate center manifolds for semilinear rough evolution equations, where our main motivation arises from semilinear reaction-diffusion SPDEs, which are included as a particular subclass in our results. Our work here extends our earlier results obtained in the finite-dimensional setting established in [55]. Moreover we emphasize that these results yield the existence of center manifolds in suitable interpolation spaces, which naturally arise in the context of parabolic PDEs. Therefore, this abstract framework is not restricted to Hilbert spaces. In summary, this work allows us to substantially extend the theory developed in [55] as well as the invariant manifold theory for SPDEs [26,34,13,19,64]. There are several major technical difficulties one encounters, when trying to establish center manifold results for stochastic partial differential equations (SPDEs). A first conceptual difficulty is to employ the concept of random dynamical systems (RDS) [2] for SPDEs. It is well-known that an Itô-type SODE generates an RDS under reasonable assumptions [2,57,67]. However, the generation of an RDS from an Itô-type SPDE has been a long-standing open problem, mostly since Kolmogorov's theorem breaks down for random fields parametrized by infinite-dimensional Hilbert spaces [63]. As a consequence it is not trivial, how to obtain a RDS from a general SPDE. This problem was fully solved only under very restrictive assumptions on the structure of the noise driving the SPDE. For instance, if one deals with purely additive noise or certain particular multiplicative Stratonovich noise, there are standard transformations which reduce the SPDE to a random PDE. Since this random PDE can be solved pathwise it is straightforward to obtain an RDS. However, for nonlinear multiplicative noise, this technique is no longer applicable, not even if the random input is a Brownian motion. As a consequence of this issue, dynamical aspects for SPDEs, e.g. invariant manifolds, have not been investigated in their full generality. In the finite-dimensional case there are results concerning invariant manifolds for delay equations using rough paths [38,39] and center manifolds in [55]. Rough path techniques provide a very natural framework to obtain RDS from SPDEs driven by general multiplicative noise since the usual problems with the nullsets do not appear in a pathwise approach. For instance, there are results regarding the existence of random dynamical systems generated by rough PDEs with transport [53,15], nonlinear multiplicative [49] and nonlinear conservative noise [28]. In this work we go beyond the of existence of RDS in the infinite-dimensional setting and establish a center manifold theorem (Theorem 6.8). The second main, more technical, obstacle we encounter is due to the fact that one wants to include the case, when the analytic semigroup (S t ) t≥0 generated by the linear part of the SPDE is no longer Hölder continuous in zero. However, this regularity is required in order to introduce the rough convolution and to obtain an expansion of the solution in terms of Hölder-continuous functions. There are several approaches to deal with this problem [40,41,42], or to work with modified Hölder spaces which compensate the time-singularity in zero, as in [48], or to consider more space-regularity in order to compensate the missing time-regularity as in [36]. More precisely, in order to define the rough convolution t 0 S(t − s)Y s dW s with respect to a γ-Hölder rough path W = (W, W), one needs the notion of a controlled rough path [40], which is a pair (Y, Y ′ ) of γ-Hölder continuous functions satisfying an abstract Taylor-like expansion in terms of Hölder regularity given by Y t = Y s + Y ′ s W s,t + R Y s,t , where the remainder R Y s,t is supposed to be 2γ-Hölder-regular. Due to the lack of regularity of the semigroup (S t ) t≥0 in zero, it is a challenging task to find an appropriate meaning of a controlled rough path. The main idea is to consider controlled rough paths on a scale of Banach spaces (B α ) α∈R satisfying the following interpolation inequality which means that \|x\| α 3 −α 1 α 2 \|x\| α 3 −α 2 α 1 \|x\| α 2 −α 1 α 3 for α 1 ≤ α 2 ≤ α 3 and x ∈ B α 3 . The advantage of this approach is that it allows one to view the semigroup as a bounded operator on all these spaces and exploit space-time regularity specific to the parabolic setting. Such an approach was exploited in [37] in the context of non-autonomous rough PDEs and in [36], where the semigroup was directly incorporated in the definition of the controlled rough path. Following the approach of controlled rough paths in interpolation spaces, we manage to prove the existence of center manifolds for parabolic rough PDEs based on the Lyapunov-Perron method. This is the key analytical contribution of this work. The paper is structured as follows: In Section 2, we provide an overview regarding our setting, compare it formally to an existing approach, and we motivate how to set up the iteration procedure to construct the center manifold. In Section 3 we collect preliminaries concerning evolution equations and controlled rough paths. In Section 4, we prove a-priori estimates for the solution of rough evolution equations, which will be key components to justify the existence of a fixed point for the Lyapunov-Perron method. In Section 5, we present the background from RDS, construct suitable random cocycles, and define random center invariant manifolds. In Section 6, we finally set up a discrete Lyapunov-Perron method and prove the existence of a center manifold. We present some examples in Section 7. used so far in proving the existence of random center manifolds for such SPDEs, given by du = (Au + f (u)) dt + u • dB t u(0) = ξ (2.1) on a separable Hilbert space H. Here the linear operator A generates a C 0 -semigroup (S t ) t≥0 on H, f is a locally Lipschitz nonlinear term with f (0) = 0 = f ′ (0), andB denotes a two-sided real-valued Brownian motion; note that f (0) = 0 ensures that u = 0 is a steady state of (2.1). Suppose the spectrum of the linear operator A consists of finitely many eigenvalues with zero real part, and all other eigenvalues have strictly negative real parts, i.e. σ(A) = σ c (A) ∪ σ s (A), where σ c (A) = {λ ∈ σ(A) : Re(λ) = 0} and σ s (A) = {λ ∈ σ(A) : Re(λ) < 0}. The subspaces generated by the eigenvectors corresponding to these eigenvalues are denoted by H c respectively H s and are referred to as center and stable subspace. These subspaces provide an invariant splitting of H = H c ⊕ H s . We denote the restrictions of A on H c and H s by A c := A\| H c and A s := A\| H s . Since H c is finite-dimensional we obtain that S c (t) := e tAc is a group of linear operators on H c . Moreover, there exist projections P c and P s such that P c + P s = Id H and A c = A\| R(P c ) and A s = A\| R(P s ) , where R denotes the range of the corresponding projection. Additionally, we impose the following dichotomy condition on the semigroup. We assume that there exist two exponents γ and β with −β * < 0 ≤ γ * < β * and constants M c , M s ≥ 1, such that S c (t)x H ≤ M c e γ * t x H , for t ≤ 0 and x ∈ H; (2.2) S s (t)x H ≤ M s e −β * t x H , for t ≥ 0 and x ∈ H. (2.3) Furthermore, we introduce the stationary Ornstein-Uhlenbeck process, i.e. the stationary solution of the Langevin equation dz t = −z dt + dB t , which is given by z(θ tB ) = t −∞ e −(t−s) dB s = 0 −∞ e s dθ tBs . Here θ denotes the usual Wiener-shift, i.e. θ tBs :=B t+s −B t for s, t ∈ R. In this case, using the Doss-Sussmann transformation u * := ue −z(B) , the SPDE (2.1) reduces to the non-autonomous random differential equation du = (Au + z(θ tB )u + g(θ tB , u)) dt,(2.4) where we dropped the * -notation and set g(B, u) := e −z(B) f (e z(B) u). Note that no stochastic integrals appear in (2.4) and one can prove the existence of center manifolds for (2.4) almost like in the deterministic setting, using the Lyapunov-Perron method. More precisely, one infers that the continuous-time Lyapunov-Perron transform for (2.4) is given by J(B, u, ξ)[t] := S c t e t 0 z(θτB) dτ P c ξ + t 0 S c t−r e t r z(θτB) dτ P c g(θ rB , u(r)) dr + t −∞ S s t−r e t r z(θτB) dτ P s g(θ rB , u(r)) dr. (2.5) Further details regarding this operator can be found in [73], [27,Sec. 6.2.2], [18,Ch.4] and the references specified therein. The next natural step is to show that (2.5) possesses a fixed-point in a certain function space. One possible choice turns out to be BC η,z (R − ; H), see [27, p. 156]. This space is defined as BC η,z (R − ; H) :=    u : R − → H, u is continuous and sup t≤0 e −ηt− t 0 z(θτB) dτ u(t) H < ∞    and is endowed with the norm \|\|u\|\| BC η,z := sup t≤0 e −ηt− t 0 z(θτB) dτ u(t) H . Here η is determined from (2.2) and (2.3), namely one has −β * < η < 0. (B, ·), where h c (B, ξ) = P s Γ(0,B, ξ), i.e. h c (B, ξ) = 0 −∞ S s −τ e 0 τ z(θrB) dr P s g(θ τB , Γ(τ,B, ξ)) dτ, for ξ ∈ H c ∩ B(0, ρ(B)). (2.6) Here B(0, ρ(B)) denotes a random neighborhood of the origin, i.e. the radius ρ(B) depends on the intensity/magnitude of the noise. Example 2.1 We now illustrate via a computational example, how the theory briefly introduced above can be applied. Let a and σ stand for two positive parameters and consider the reactiondifussion SPDE on H : = L 2 (0, π)      du = (∆u + u − au 3 ) dt + σu • dB t u(0, t) = u(π, t) = 0, for t ≥ 0 u(x, 0) = u 0 (x), for x ∈ (0, π). (2.7) Substituting u := u * e σz(B) and dropping the * -notation, we obtain the non-autonomous PDE with random coefficients ∂u ∂t = ∆u + u + σz(θ tB )u − ae 2σz(θtB) u 3 . (2.8) The spectrum of Au := ∆u + u with domain D(A) = H 2 (0, π) ∩ H 1 0 (0, π) is given by {1 − n 2 : n ≥ 1} with corresponding eigenvectors {sin(nx) : n ≥ 1}. The eigenvectors give us the center subspace H c = span{sin x} and the stable one H s = span{sin(nx) : n ≥ 2}. Based upon the previous considerations, one can infer that (2.7) has a local center manifold M c (B) = {b sin x + h c (B, b sin x)} = b sin x + ∞ n=2 c n (B, b) sin(nx) . In this case, it is also possible to derive suitable approximation results for h c , namely one can show that c n (B, b) = O(b 3 ) as b → 0. Plugging this in (2.8) gives us a non-autonomous random ODE on the center manifold db dt = σz(θ tB )b − 3 4 ab 3 e 2σz(θtB) + O(b 5 ). Since −u is also a solution for (2.8) we have that c n (B, b) = 0 for n even. Therefore, one has the following approximation of h h c (B, b sin x) = c 3 (B, b) sin 3x + O(b 5 ). Our Goal The approach presented in the previous section is somewhat limited in applicability due to use of a Doss-Sussmann transformation to a non-autonomous random PDE. The main goal of this work is to extend center manifold theory for SPDEs to equations driven by rough noise and to recover the Stratonovich case mentioned above as a special case. More precisely, we aim to obtain center manifolds for equations of the form du = (Au + F (u)) dt + G(u) dW, u(0) = ξ, (2.9) where G is nonlinear and the noisy input W is supposed to be more irregular than a Brownian motion, i.e. W ∈ C γ for γ ∈ (1/3, 1/2). This includes Brownian motion but applies to a much wider class of Gaussian processes. Examples for G are polynomials with smooth coefficients (see Section 7), or integral operators with a smooth kernel as discussed in [48,Section 7]. As a first step we have to rigorously prove the existence of center manifolds for (2.9). Further works will be devoted to approximation results and to related problems in bifurcation theory, see for example [7,9]. In contrast to the previous technique in Section 2.1, we are not going to transform (2.9) to a random PDE, but we are going to work directly with its mild solution u t = S t ξ + t 0 S t−s F (u s ) ds + t 0 S t−s G(u s ) dW s . (2.10) This is possible due to the pathwise construction of the stochastic integral in (2.10). The Lyapunov-Perron map in this case is given by J(W, u, ξ)[t] : = S c t P c ξ + t 0 S c t−r P c F (u r ) dr + t 0 S c t−r P c G(u r ) dW r + t −∞ S s t−r P s F (u r ) dr + t −∞ S s t−r P s G(u r ) dW r . Due to the stochastic integrals appearing above, the technical challenge consists in finding an appropriate framework to formulate the fixed-point problem for J. After this is established, one should intuitively be able to show that the fixed-point Γ of J characterizes the local center manifold. More precisely, the random manifold should have a graph structure, where the function h c = P s Γ(0, W, ξ) is given by h c (W, ξ) = 0 −∞ S s −τ P s F (Γ(τ, W, ξ)) dτ + 0 −∞ S s −τ P s G(Γ(τ, W, ξ)) dW τ , for ξ ∈ H c ∩B(0, ρ(W )) similarly to (2.6). Naturally, the size of the random ball should depend on the growth of the nonlinear terms F and G and on the random input W . The following sections provide the necessary tools and rigorously prove these heuristic considerations. Moreover, the center manifold theory developed in this work is applicable to rough PDEs in interpolation spaces and it is not restricted to the Hilbert space-valued setting. Preliminaries We fix T > 0, let α ∈ R and consider on a scale of Banach spaces B α the equation dY t = AY t dt + F (Y t ) dt + G(Y t ) dW t , t ∈ [0, T ] Y 0 = ξ ∈ B α ,(3.1) where Y : [0, T ] → B α is the unknown, the linear part defined via the operator A with domain B 1 := D(A) generates an analytic C 0 -semigroup (S t ) t≥0 on a separable Banach space B. The scale of Banach spaces (B α ) α∈R , the nonlinear drift and diffusion coefficients F and G will be discussed further below, and the noise W is a γ-Hölder d-dimensional rough path for γ ∈ ( 1 3 , 1 2 ) and some fixed d ≥ 1. This case includes the Brownian motion and the fractional Brownian motion for H ∈ ( 1 3 , 1 2 ]. Notation: As commonly met in the rough path theory, we use the notation Y t instead of Y (t) and Y s,t := Y t − Y s stands for an increment. Keeping this in mind we specify that the d-dimensional noisy input W = (W 1 , . . . W d ) is assumed to be a γ-Hölder rough path W := (W, W), for γ ∈ (1/3, 1/2]. More precisely, W ∈ C γ ([0, T ]; R d ) and W ∈ C 2γ ([0, T ] 2 ; R d ⊗ R d ) and the connection between W and W is given by Chen's relation W s,t − W s,u − W u,t = W s,u ⊗ W u,t ,(3.2) where we used the increment notation W s,t = W t − W s introduced above. The term W is referred to as second-order process or Lèvy-area. The process can be interpreted as the iterated integral [30, Chapter 10] W s,t = t s (W r − W s ) ⊗ dW r . Throughout this manuscript we assume without loss of generality d = 1 since the generalization to higher dimensions can be done component-wise and does not require any additional arguments. We further introduce an appropriate distance between two γ-Hölder rough paths. For more details on this topic consult [30,Chapter 2]. We stress that in our situation we always have that W 0 = 0 and therefore (3.3) is a metric. (s,t)∈∆ J \|W s,t −W s,t \| \|t − s\| γ + sup (s,t)∈∆ J \|W s,t −W s,t \| \|t − s\| 2γ . Since we consider parabolic rough PDEs, we work with the following function spaces similar to [36,37]. These reflect a suitable interplay between space and time regularity available in the parabolic setting. Definition 3.2 A family of separable Banach spaces (B α , \| · \| α ) α∈R is called a monotone family of interpolation spaces if for α 1 ≤ α 2 , the space B α 2 ⊂ B α 1 with dense and continuous embedding and the following interpolation inequality holds for α 1 ≤ α 2 ≤ α 3 and x ∈ B α 3 : \|x\| α 3 −α 1 α 2 \|x\| α 3 −α 2 α 1 \|x\| α 2 −α 1 α 3 . (3.4) The main advantage of this approach is that we can view the semigroup (S(t)) t≥0 as a linear mapping between these interpolation spaces and obtain the following standard bounds for the corresponding operator norms. If S : [0, T ] → L(B α , B α+1 ) is such that for every x ∈ B α+1 and t ∈ (0, T ] we have that \|(S t − Id)x\| α t\|x\| α+1 and \|S(t)x\| α+1 t −1 \|x\| α , then for every σ ∈ [0, 1] we have that S(t) ∈ L(B α+σ ) and \|(S t − Id)x\| α t σ \|x\| α+σ (3.5) \|S(t)x\| α+σ t −σ \|x\| α . (3.6) An example of such spaces is constituted by B α = [B, B 1 ] α , where [·, ·] α denotes the complex interpolation. For further details regarding these interpolation spaces, see [61,37]. Next, we have to specify a solution concept for (3.1). We rely on the mild formulation of (3.1), namely Y t = S t ξ + t 0 S t−s F (Y s ) ds + t 0 S t−s G(Y s ) dW s . (3.7) In order to give a meaning to (3.7) we introduce in the following sequel some concepts and notations from rough paths theory [30]. Notations. We always let \| · \| γ denote the Hölder-norm of W , \| · \| 2γ the 2γ-Hölder norm of W, and · γ,α stands for the γ-Hölder norm in B α . As a convention, the first index in · γ,α describes the time-regularity and the second one refers to the space regularity. Furthermore, the symbol \| · \| γ always indicates the regularity of the random input, i.e. we use the notation \| · \| even if we refer to the γ-Hölder norm of W or to the 2γ-Hölder norm of the second order process W. The symbol · γ,α will be exclusively used to indicate time and space regularity. The notation · ∞,α stands for the supremum-norm in B α . Furthermore, C stands for a universal constant which varies from line to line. We write a b if there exists a constant C > 0 such that a ≤ Cb. The constant C is allowed to depend on F , G and their derivatives and on the parameters γ, α and ρ γ (W), but can be chosen uniformly on compact intervals. For our purposes we will state most of the estimates on the time interval [0, 1]. We now describe the space of the paths that can be integrated with respect to W and observe that the setting here is different from the rough ODE case, where we showed that for a pair of controlled rough paths (U, U ′ ) the convolution with the semigroup (S t−· U, S t−· U ′ ) remains again a controlled rough path, see [55, Lemma 2.6.1]. In the finite dimensional case, this is possible due to the Lipschitz continuity in time of the semigroup. However, this property does not hold true in infinite dimensions because the semigroup is not Hölder continuous in zero. More precisely, due to (3.6) we have for 0 < s ≤ t ≤ T that S t ξ − S s ξ B = (S t−s − Id)S s ξ B ≤ C(t − s) 2γ ξ B 2γ (3.8) ≤ C(t − s) 2γ s −2γ ξ B , (3.9) which indicates a singularity in zero for initial data ξ ∈ B. Due to this fact, it is a challenge task to find the right function space of controlled rough paths in which to set up a fixed-point argument to solve rough PDEs. Several approaches have been considered in the literature to overcome this obstacle. For instance: (P1) take more regular initial data, i.e. ξ ∈ B 2γ ; (P2) solve (3.1) as in [48] in a modified Hölder space which compensates the time-singularity occurring in (3.9), e.g. C 2γ 2γ ([0, T ]; B) := sup 0<s<t≤T s 2γ U s,t B (t − s) 2γ < ∞ ; (P3) require higher space regularity and solve (3.1) in a larger space containing B and use regularizing properties of analytic semigroups to show that the solution actually belongs to B as in [36,37]. Regarding this we introduce the following definition of a controlled rough path tailored to the parabolic structure of the PDE we consider. For other approaches see [36] and [30, Section 12.2.2]. Definition 3.3 (Controlled rough path according to a monotone family (B α ) α∈R ). We call a pair (U, U ′ ) a controlled rough path if • (U, U ′ ) ∈ C([0, T ]; B α )×((C[0, T ]; B α−γ )∩C γ ([0, T ]; B α−2γ )). The component U ′ is referred to as the Gubinelli derivative 1 of U . • the remainder R U s,t = U s,t − U ′ s W s,t (3.10) belongs to C γ ([0, T ]; B α−γ ) ∩ C 2γ ([0, T ]; B α−2γ ). The space of controlled rough paths is denoted by D 2γ W,α and endowed with the norm · W,2γ,α given by U, U ′ W,2γ,α = U ∞,Bα + U ′ ∞,B α−γ + U ′ γ,B α−2γ + R U 2γ,B α−2γ . (3.11) In order to emphasize the time horizon we write D 2γ W,α ([0, T ]) instead of D 2γ W,α . Remark 3.4 1). Note that we do not make the Hölder continuity of U part of the definition of a controlled rough path, since using (3.10) one immediately obtains for θ ∈ {γ, 2γ} that U γ,B α−θ ≤ U ′ ∞,B α−θ W γ + R y γ,B α−θ . (3.12) 2). One can show that on a monotone scale of interpolation spaces the norm in (3.11) is equivalent to the apparently stronger one introduced in [37] which additionally includes R U γ,α−γ . A proof of this statement can be found in [52] and relies on the interpolation inequality (3.4). 3). Definition 3.3 states that (U, U ′ ) ∈ D 2α W,γ is controlled by W according to the monotone family of interpolation spaces (B α ) α∈R as in [37]. One can make the semigroup (S t ) t≥0 part of the definition of the controlled rough path as in [36]. We work with Definition 3.3, since it reflects the appropriate space-time regularity of the solution and stays closer to the finite-dimensional setting [30,40]. Moreover, for the existence of center manifolds we will apply a cut-off technique to (3.1). For this argument it is also convenient not to incorporate the semigroup in the definition of the controlled rough path. Throughout this section (U, U ′ ) is going to denote an arbitrary controlled rough path and (Y, Y ′ ) is used to refer to the solution of (3.1). Given a controlled rough path, one can introduce the rough integral as follows. Theorem 3.5 Let (U, U ′ ) ∈ D 2γ W,α . Then t s S t−r U r dW r := lim \|P\|→0 [u,v]∈P S t−u U u W u,v + S t−u U ′ u W u,v , (3.13) where P denotes a partition of [s, t]. For 0 ≤ β < 3γ the following estimate t s S t−r U r dW r − S t−s U s W s,t − S t−s U ′ s W s,t B α−2γ+β ρ γ (W) U, U ′ W,2γ,α (t − s) 3γ−β (3.14) holds true. We emphasize that the stochastic convolution increases the spatial regularity of the controlled rough path, see [37,Corollary 4.6] and Lemma 3.5 in [50]. We recall this result, which will be used later on. Corollary 3.6 Let (U, U ′ ) ∈ D 2γ W,α , T ∈ [0, 1] and 0 ≤ σ < γ. Then the integral map (U, U ′ ) → (Z, Z ′ ) := · 0 S ·−r U r dW r , U · maps D 2γ W,α into itself. Moreover Z, Z ′ W,2γ,α+σ ≤ \|U 0 \| α + \|U ′ 0 \| α−γ + CT γ−σ (1 + ρ γ (W)) U, U ′ W,2γ,α . Assumptions on drift and diffusion coefficients: (G) Let θ ∈ {0, γ, 2γ} and 0 ≤ σ < γ. The nonlinear diffusion coefficient G : B α−θ → L(R, B α−θ−σ ) is Lipschitz and three times Frèchet differentiable with bounded derivatives, i.e. D k G L(B ⊗k α−θ ,B α−θ−σ ) < ∞ for k ∈ {1, 2, 3} . Furthermore, we assume that G(0) = DG(0) = D 2 G(0) = 0. (3.15) In order to ensure global-in-time existence of solutions of (3.1) we additionally assume that the derivative of DG(·)G(·) : B α−γ → B α−2γ−σ is bounded. This condition is satisfied in particular if G itself is bounded or linear [49,50]. (F) The drift term F : B α → B α−δ for δ ∈ [0, 1) is Lipschitz continuous. Furthermore, we assume that F (0) = DF (0) = 0. (3.16) The conditions (3.15)-(3.16) guarantee that our main rough PDE (3.1) has a steady state at 0 ∈ B; of course, up to a translation, we could take this point anywhere in B but we fix it at 0 for convenience. The global Lipschitz assumptions on the drift and diffusion coefficients can be weakened. However, for our setting, the above assumption suffices, since we will truncate the nonlinear terms in a neighbourhood of the origin as illustrated in Section 4. Our main goal is to show that under these assumptions the rough PDE (3.1) has a local center manifold in B α for small initial data belonging to B α . The truncated rough PDE The next step is to modify F and G such that their Lipschitz constants become small. More precisely, for a fixed R > 0 we need to compose F and G with a smooth cut-off function χ R . Here R denotes the size of the ball around zero and in our case it will depend on the size of the noise, i.e. R = R \|W \| γ,[0,1] , \|W\| 2γ,[0,1] 2 . Since we develop only a local theory, the exact size of this random ball is not crucial, since this can be chosen small enough as required in the fixed-point argument. In this setting, we recall that a composition of a controlled rough path with a smooth function is a well-defined operation [ The main novelty here is to define the composition of a controlled rough path (U, U ′ ) ∈ D with a smooth cut-off function. Due to this procedure we obtain a rough PDE with path dependent coefficients. Therefore we have to show by means of fixed-point arguments that such an equation is well-posed. For notational simplicity we set D := D 2γ W,α and define χ(U ) := U, if (U, U ′ ) D ≤ 1/2, 0, if (U, U ′ ) D ≥ 1. For instance χ can be obtained as χ(U ) = U f ( U, U ′ D ), where f : R + → [0, 1] is a three-times continuously differentiable cut-off function with bounded derivatives, see [55] for particular examples. In this case one has (χ(U ), χ(U ) ′ ) = (U, U ′ ), if (U, U ′ ) D ≤ 1/2 0, if (U, U ′ ) D ≥ 1. Furthermore, for R > 0 we set χ R (U ) = R χ 1 R U such that (χ R (U ), χ R (U ) ′ ) = (U, U ′ ), if (U, U ′ ) D ≤ R/2 0, if (U, U ′ ) D ≥ R. We denote F R := F • χ R , G R := G • χ R and have F R (U ) = F (U ) and G R (U ) = G(U ), if U, U ′ D ≤ R/2. We set for t ∈ [0, 1]: F (U )(t) := F (U t ) and G(U )(t) := G(U t ) and introduce the truncation as F R (U ) := F • χ R (U ) respectively G R := G • χ R (U ). This means that we have F R (U )(t) = F (χ R (U ))(t) = F (χ R (U ) t ) = F (U t f ( U, U ′ /R)) and analogously, G R (U )(t) = G(χ R (U ))(t) = G(χ R (U ) t ) = G(U t f ( U, U ′ /R)), where we removed for simplicity the index D from ·, · D . The Gubinelli derivative of G R can be computed according to the chain rule [30,Lem. 7.3] as (G R (U )) ′ = DG(χ R (U ))(χ R (U )) ′ = DG(U f ( U, U ′ /R))U ′ f ( U, U ′ /R), since f is a constant with respect to time. Evaluating (G R (U )) ′ at a time t ∈ [0, 1], we obtain (G R (U )(t)) ′ = DG(χ R (U ) t )(χ R (U ) t ) ′ = DG(U t f ( U, U ′ /R))U ′ t f ( U, U ′ /R). By the definition of χ R we have that F R (U ) = F (U ) and G R (U ) = G(U ) if (U, U ′ ) D ≤ R/2. With this notation, the first component of the mild solution of (3.1) equivalently rewrites as Y t = S t ξ + t 0 S t−r F (Y )(r) dr + t 0 S t−r G(Y )(r) dW r . (4.1) We now argue in several steps that the modified rough PDE (3.1) has a unique solution in the space of controlled rough paths. Note that the coefficients F R and G R are now path dependent. We show that F R and G R are Lipschitz continuous with Lipschitz constants L F (R) and L G (R) such that L F (R) → 0 respectively L G (R) → 0 as R → 0. Recall that D = D 2γ W,α and that the universal constant C is allowed to depend on F , G and their derivatives. Lemma 4.1 Let (U, U ′ ), (Ũ ,Ũ ′ ) ∈ D. Then there exists a constant L F (R) = L F [R, F, χ] such that L F (R) → 0 as R → 0 and that · 0 S ·−r (F R (U )(r) − F R ( U )(r))) dr, 0 D ≤ L F (R) U − U , U ′ − U ′ D . (4.2) Proof. Recalling (3.11) and the fact that the Gubinelli derivative of the deterministic integral is zero we have to estimate · 0 S ·−r (F R (U )(r) − F R (Ũ (r))) dr ∞,α and the 2γ-norm of the remainder of this convolution in B α−2γ . For the latter we compute for 0 ≤ s ≤ t ≤ 1 t 0 S t−r [F R (U )(r) − F R (Ũ )(r)] dr − s 0 S s−r [F R (U )(r) − F R (Ũ )(r)] dr = (S t−s − Id) s 0 S s−r [F R (U )(r) − F R (Ũ )(r)] dr + t s S t−r [F R (U )(r) − F R (Ũ )(r)] dr. Let i = 0, 1, 2 and recall that F : B α → B α−δ . The first term entails (S t−s − Id) s 0 S s−r [F R (U )(r) − F R (Ũ )(r)] dr B α−iγ ≤ S t−s − Id L(Bα,B α−iγ ) s 0 S s−r L(B α−δ ,Bα) [F R (U )(r) − F R (Ũ )(r)] B α−δ dr ≤ (t − s) iγ s 0 (s − r) −δ [F R (U )(r) − F R (Ũ )(r)] B α−δ dr ≤ (t − s) iγ s 1−δ sup r∈[0,1] Analogously we have for the second term t s S t−r [F R (U )(r) − F R (Ũ )(r)] α−iγ dr ≤ t s S t−r L(B α−δ ,B α−iγ ) [F R (U )(r) − F R (Ũ )(r)] B α−δ dr ≤ t s (t − r) iγ−δ [F R (U )(r) − F R (Ũ )(r)] B α−δ dr ≤ (t − s) min{1,1+iγ−δ} sup r∈[0,1] [F R (U )(r) − F R (Ũ )(r)] B α−δ . (4.4) Therefore, we compute sup r∈[0,1] [F R (U )(r) − F R (Ũ )(r)] B α−δ . We have sup r∈[0,1] F R (U )(r) − F R ( U )(r) B α−δ = sup r∈[0,1] F (χ R (U ) r ) − F (χ R ( U ) r ) B α−δ . This further results in F (χ R (U ) r ) − F (χ R ( U ) r ) B α−δ ≤ 1 0 DF (τ χ R (U ) r + (1 − τ )χ R ( U ) t ) B α−δ dτ χ R (U ) r − χ R ( U ) r B α−δ ≤ DF L(Bα,B α−δ ) max{ χ R (U ) r Bα , χ R ( U ) r Bα } χ R (U ) r − χ R ( U ) r B α−δ ≤ CR χ R (U ) r − χ R ( U ) r Bα . (4.5) Here we use that χ R (U ) r Bα ≤ χ R (U ) ∞,α = U f ( U, U ′ /R) ∞,α = U ∞,α f ( U, U ′ /R) together with the fact that B α ⊂ B α−δ therefore · B α−δ ≤ C · Bα . Furthermore χ R (U ) r − χ R (Ũ ) r Bα = U r f ( U, U ′ /R) −Ũ r f ( Ũ ,Ũ ′ /R) Bα ≤ U r −Ũ r Bα f ( U, U ′ /R) + Ũ r Bα \|f ( U, U ′ /R) − f ( Ũ ,Ũ ′ /R)\| ≤ U −Ũ ∞,α + R Df ∞ ( U, U ′ /R − Ũ ,Ũ ′ /R) ≤ (1 + Df ∞ ) U − U , U ′ − U ′ D . Consequently, this further yields sup r∈[0,1] F R (U )(r) − F R ( U )(r) B α−δ ≤ CR χ R (U ) − χ R ( U ) ∞,α ≤ C[F, χ] R U − U , U ′ − U ′ D . Combining this with (4.3) and (4.4) proves the statement. G R (U ) − G R ( U ), (G R (U ) − G R ( U )) ′ D 2γ W,α−σ ≤ L G (R) U − U , U ′ − U ′ D . (4.6) Proof. Regarding our assumption (G) we can view the Fréchet derivative D k G as an element of L(B ⊗k α−iγ , B α−iγ−σ ) for k = 1, 2, 3 and i = 0, 1, 2. Due to (3.11) we have to estimate G R (U ) − G R (Ũ ) ∞,α−σ + (G R (U ) − G R (Ũ )) ′ ∞,α−γ−σ + (G R (U ) − G R (Ũ )) ′ γ,α−2γ−σ + R G R (U )−G R (Ũ ) 2γ,α−2γ−σ . We begin with the first term and write G R (U ) − G R (Ũ ) ∞,α−σ = sup t∈[0,1] G R (U )(t) − G R (Ũ )(t) Bα−σ = sup t∈[0,1] G(χ R (U ) t ) − G(χ R ( U ) t ) B α−σ . Analogously to Lemma 4.1 and regarding that DG(0) = 0 we have G(χ R (U ) r ) − G(χ R ( U ) r ) B α−σ ≤ 1 0 DG(τ χ R (U ) r + (1 − τ )χ R ( U ) t ) B α−σ dτ χ R (U ) r − χ R ( U ) r B α−σ ≤ DG L(Bα,B α−σ ) max{ χ R (U ) r Bα , χ R ( U ) r Bα } χ R (U ) r − χ R ( U ) r B α−δ ≤ CR χ R (U ) r − χ R ( U ) r Bα ≤ CR U −Ũ ∞,α ≤ CR U −Ũ , U ′ −Ũ ′ D . (4.7) For the terms containing the Gubinelli derivative we recall that for t ∈ [0, 1]: (G R (U )(t)) ′ = DG(χ R (U ) t )(χ R (U ) t ) ′ = DG(U t f ( U, U ′ /R))U ′ t f ( U, U ′ /R). We now investigate (G R (U ) − G R ( U )) ′ γ,α−2γ−σ . To this aim we let 0 ≤ s ≤ t ≤ 1 and consider [G R (U )(t) − G R ( U )(t) − (G R (U )(s) − G R ( U )(s))] ′ = [DG(χ R (U ) t )(χ R (U ) t ) ′ − DG(χ R ( U ) t )(χ R ( U ) t ) ′ − (DG(χ R (U ) s )(χ R (U ) s ) ′ − DG(χ R ( U ) s )(χ R ( U ) s ) ′ )] = DG(U t f ( U, U ′ /R))U ′ t f ( U, U ′ /R) − DG( U t f ( U , U ′ /R)) U ′ t f ( U , U ′ /R) − (DG(U s f ( U, U ′ /R))U ′ s f ( U, U ′ /R) − DG( U s f ( U, U ′ /R)) U ′ s f ( U , U ′ /R)). Therefore, we have to estimate in B α−2γ−σ the last expression. This further results in [DG(χ R (U ) t )(χ R (U ) t ) ′ − DG(χ R ( U ) t )(χ R ( U ) t ) ′ − (DG(χ R (U ) s )(χ R (U ) s ) ′ − DG(χ R ( U ) s )(χ R ( U ) s ) ′ )] ≤ DG(χ R (U ) t ) − DG(χ R (U ) s ) − DG(χ R ( U ) t ) + DG(χ R ( U ) s ) · (χ R (U ) t ) ′ + (χ R (U ) s ) ′ + (χ R ( U t )) ′ + (χ R ( U ) s ) ′ + DG(χ R (U ) t ) − DG(χ R (U ) s ) + DG(χ R ( U ) t ) − DG(χ R ( U ) s ) · (χ R (U ) t ) ′ + (χ R (U ) s ) ′ − (χ R ( U t )) ′ − (χ R ( U ) s ) ′ + DG(χ R (U ) t ) + DG(χ R (U ) s ) − DG(χ R ( U ) t ) − DG(χ R ( U ) s ) · (χ R (U ) t ) ′ − (χ R (U ) s ) ′ + (χ R ( U t )) ′ − (χ R ( U ) s ) ′ + DG(χ R (U ) t ) + DG(χ R (U ) s ) + DG(χ R ( U ) t ) + DG(χ R ( U ) s ) · (χ R (U ) t ) ′ − (χ R (U ) s ) ′ − (χ R ( U t )) ′ + (χ R ( U ) s ) ′ := I + II + III + IV. We analyze each of the four terms above separately. For the first one we obtain I B α−2γ−σ ≤ D 2 G L(B ⊗2 B α−2γ ,B α−2γ−σ ) R U − U , U ′ − U ′ D [ χ R (U ) ∞,α−2γ + χ R (Ũ ) ∞,α−2γ ] ≤ CR 2 U − U , U ′ − U ′ D . This estimate relies on applying (4.7) replacing G by DG and regarding that D 2 G(0) = 0. More precisely, we have \|DG(χ R (U ) t ) − DG(χ R ( U ) t ) − [DG(χ R (U ) s ) − DG(χ R ( U ) s )]\| ≤ DG L(B ⊗2 α−2γ ,B α−2γ−σ ) max{ χ R (U ) t , χ R ( U ) t B α−2γ } χ R (U ) t − χ R ( U ) t − [χ R (U ) s − χ R ( U ) s ] B α−2γ + C χ R (U ) t − χ R ( U t ) α−2γ [ χ R (U ) t − χ R (U ) s α−2γ + χ R ( U ) t − χ R ( U ) s α−2γ ] (4.8) ≤ CR χ R (U ) − χ R ( U ) γ,α−2γ + C χ R (U ) − χ U ( U ) ∞,α−2γ ( χ R (U )) γ,α−2γ + χ R ( U ) γ,α−2γ ) ≤ CR U − U , U ′ − U ′ D . Here we use (3.12) for θ = 2γ to control χ R (U ) γ,α−2γ . Regarding these deliberations, we completed the estimate of I. We now focus on II B α−2γ−σ which analogously results in II B α−2γ−σ ≤ C DG(χ R (U )) − DG(χ R ( U )) γ,α−2γ−σ [ χ R (U ) ∞,α−2γ + χ R ( U ) ∞,α−2γ ] ≤ CR U − U , U ′ − U ′ D . Analogously, for the third one we have III B α−2γ−σ ≤ C DG(χ R (U )) − DG(χ R ( U )) ∞,α−2γ−σ [ χ R (U ) γ,α−2γ + χ R ( U ) γ,α−2γ ] ≤ CR U − U , U ′ − U ′ D . Finally, we easily estimate the fourth term regarding that D 2 G(0) = 0 as IV B α−2γ−σ ≤ C sup t∈[0,1] max{ DG(χ R (U ) t ) α−2γ−σ , DG(χ R ( U ) t ) α−2γ−σ } · (χ R (U )) ′ − (χ R ( U )) ′ γ,α−2γ ≤ CR U − U , U ′ − U ′ D . We now focus on the remainder. Recalling that (U, U ′ ) ∈ D, we know that U t = U s + U ′ s W s,t + R U s,t . Therefore, U t f ( U, U ′ /R) = U s f ( U, U ′ /R) + U ′ s f ( U, U ′ /R)W s,t + R U s,t f ( U, U ′ /R), which gives us R χ R (U ) = R U f ( Y, Y ′ /R). Consequently, (R G R (U ) ) s,t = G(χ R (U ) t ) − G(χ R (U ) s ) − DG(χ R (U ) s )(χ R (U )) s,t + DG(χ R (U ) s )R U s,t f ( U, U ′ /R). (4.9) This means that we have to estimate R G R (U ) s,t − R G R ( U ) s,t = G(χ R (U ) t ) − G(χ R (U ) s ) − DG(χ R (U ) s )(χ R (U )) ′ s,t − [G(χ R ( U ) t ) − G(χ R ( U ) s ) − DG(χ R ( U ) s )(χ R ( U )) ′ s,t ] + DG(χ R (U ) s )R U s,t f ( U, U ′ /R) − DG(χ R ( U ) s )R U s,t f ( U , U ′ /R). (4.10) The terms in (4.10) easily result in DG(χ R (U ) s )R χ R (U ) s,t − DG(χ R ( U ) s )R χ R ( U ) s,t B α−2γ−σ ≤ DG(χ R (U ) s )[R χ R (U ) s,t − R χ R ( U ) s,t ] B α−2γ−σ + [DG(χ R (U ) s ) − DG(χ R ( U ) s )]R χ R ( U ) s,t B α−2γ−σ ≤ C DG L(B α−2γ ,B α−2γ−σ ) χ R (U ) s B α−2γ R χ R (U ) − R χ R ( U ) 2γ,α−2γ + D 2 G L(B ⊗2 α−2γ ,B α−2γ−σ ) R χ R ( U ) 2γ,α−2γ χ R (U ) s − χ R ( U ) s α−2γ ≤ CR R U − R U 2γ,α−2γ + CR U − U ∞,α−2γ ≤ CR U −Ũ , U ′ −Ũ ′ D . To estimate the 2γ-Hölder norm in B α−2γ of the expressions (4.10) appearing in the difference of two remainders, we firstly recall the identity G(U t ) − G(U s ) − DG(U s )U s,t = 1 0 DG(rU t + (1 − r)U s ) dr U s,t − DG(U s )U s,t (4.11) = 1 0 [DG(rU t + (1 − r)U s ) − DG(U s )]U s,t dr = 1 0 1 0 D 2 U [τ (rU t + (1 − r)U s ) + (1 − τ )U s ](rU t + (1 − r)U s − U s )U s,t dτ dr = 1 0 1 0 rD 2 G[τ (rU t + (1 − r)U s ) + (1 − τ )U s ] dτ dr (U s,t ⊗ U s,t ). Applying (4.11) twice, one obtains the following inequality (see p. 2716 in [51]) G(χ R (U ) t ) − G(χ R (U ) s ) − DG(χ R (U ) s )(χ R (U )) s,t (4.12) − (G(χ R ( U ) t ) − G(χ R ( U ) s ) − DG(χ R ( U ) s )(χ R ( U )) s,t ) B α−2γ−σ ≤ D 2 G L(B ⊗2 α−2γ,B α−2γ−σ ) χ R (U ) t B α−2γ [ χ R (U ) t − χ R (U ) s B α−2γ + χ R ( U ) t − χ R ( U ) s B α−2γ ] · χ R (Y ) t − χ R (Y ) s − (χ R ( Y ) t − χ R ( Y ) s ) B α−2γ + D 3 G L(B ⊗3 α−2γ,B α−2γ−σ ) χ R ( U ) t − χ R ( U ) s 2 B α−2γ [ χ R (U ) t − χ R ( U ) t B α−2γ + χ R (U ) s − χ R ( U ) s B α−2γ ]. (4.13) Using that · α−2γ ≤ C · α−γ ≤ C · α , the previous inequality further leads to G(χ R (U ) t ) − G(χ R (U ) s ) − DG(χ R (U ) s )(χ R (U )) s,t − (G(χ R ( U ) t ) − G(χ R ( U ) s ) − DG(χ R ( U ) s )(χ R ( U )) s,t ) B α−2γ−σ ≤ CR [ U γ,α−γ + U γ,α−γ ] U − U γ,α−γ + C χ R ( U ) 2 γ,α−2γ U − U ∞,α ≤ CR 2 U − U , U ′ − U ′ D . Here we used that (χ R (Ũ ), χ R (Ũ ) ′ ) ∈ D and (3.12) for θ = 2γ in order to estimate the second term of the previous inequality as follows χ R (Ũ ) γ,α−2γ ≤ χ R (Ũ ) ′ ∞,α−2γ W γ + R χ R (Ũ ) γ,α−2γ ≤ χ R (Ũ ) ′ ∞,α−γ W γ + R χ R (Ũ ) γ,α−2γ ≤ C(1 + W γ ) χ R (Ũ ), χ R (Ũ ) ′ D ≤ CR. Putting all these estimates together proves the statement. We now replace F and G in (3.1) by F R and G R and show that the modified rough PDE obtained by this procedure has a unique solution in D. For notational simplicity we introduce for (Y, Y ′ ) ∈ D and t ∈ [0, 1] T R (W, Y, Y ′ )[t] : = t 0 S t−r F (χ R (Y ))(r) dr + t 0 S t−r G(χ R (Y ))(r) dW r (4.14) = t 0 S t−r F R (Y )(r) dr + t 0 S t−r G R (Y )(r) dW r , with Gubinelli derivative (T R (W, Y, Y ′ )) ′ = G R (Y ). Collecting the estimates derived in Lemmas 4.1 and 4.2 leads to the following result. Theorem 4.3 There exists a unique solution (Y, Y ′ ) ∈ D of (3.1) satisfying Y ′ = G R (Y ) such that for t ∈ [0, 1]: Y t = S t ξ + t 0 S t−r F R (Y )(r) dr + t 0 S t−r G R (Y )(r) dW r . Proof. The proof relies on Banach's fixed-point theorem and the main novelty here is to incorporate the path-dependence of F R and G R which is possible due to Lemma 4.1 and 4.2. For the term containing the initial data we have that (S t ξ, 0) ∈ D since S t ξ Bα ≤ S t L(Bα,Bα) ξ Bα ≤ C ξ Bα and R S·ξ B α−2γ = S t ξ − S s ξ B α−2γ = (S t−s − Id)S s ξ B α−2γ ≤ S t−s − Id L(Bα,B α−2γ ) S s ξ Bα ≤ C(t − s) 2γ ξ Bα . Let (Y, Y ′ ), ( Y , Y ′ ) ∈ D with Y 0 = Y 0 = ξ and Y ′ 0 =Ỹ ′ 0 . By Lemma 4.1 we have that · 0 S ·−r (F R (Y )(r) − F R ( Y )(r)) dr, 0 D ≤ L F (R) Y −Ỹ D . Furthermore, Corollary 3.6 combined with Lemma 4.2 entails · 0 S ·−r (G R (Y )(r) − G R ( Y )(r)) dW r , G R (Y ) − G R ( Y ) D ρ γ (W)L G (R) Y − Y , Y ′ − Y ′ D . In conclusion we obtain · 0 S ·−r (F R (Y )(r) − F R ( Y )(r)) dr + · 0 S ·−r (G R (Y )(r) − G R ( Y )(r)) dW r , G R (Y ) − G R ( Y ) D ρ γ (W) L R (F ) + L G (R) Y − Y , Y ′ − Y ′ D . (4.15) Setting Y ≡ 0 and using that F R (0) = G R (0) = 0, we see from the previous deliberations that (Y, Y ′ ) → (T R (W, Y, Y ′ ), G R (Y )) maps D into itself. Furthermore, by choosing R small enough and regarding that L F (R) → 0 and L G (R) → 0 as R → 0, we obtain that the mapping (Y, Y ′ ) → (T R (W, Y, Y ′ ), G R (Y ) ) is a contraction. Therefore, Banach's fixed-point theorem proves the statement. 2). Due to the assumptions (F) and (G) the solution of the rough PDE (3.1) (and particularly of truncated one) exists globally in time [50]. Going back to our setting, the next aim is to characterize the parameter R required in order to decrease the Lipschitz constants of F and G using χ R . This fact is required for the Lyapunov-Perron method. As already seen we have to choose R as small as possible. Since in our deliberations, it is always required that R ≤ 1 and L F (R) → 0, L G (R) → 0 as R → 0, collecting the previous estimates and regarding the structure of the constants L F (·) and L G (·) entails · 0 S ·−r (F R (Y )(r) − F R ( Y )(r)) dr + · 0 S ·−r (G R (Y )(r) − G R ( Y )(r)) dW r , G R (Y ) − G R ( Y ) D ≤ C F R + C G C[ρ γ (W)]R Y − Y , Y ′ − Y ′ D ,(4.16) for constants C F > 0 and C G > 0 which depend on F , G, their derivatives and on the cuf-off function χ. Furthermore the constant C > 0 incorporates the dependence of the random input contained by ρ γ (W). As commonly met in the theory of random dynamical systems [59,34], since all the estimates depend on the random input, it is meaningful to employ a cut-off technique for a random variable, i.e. R = R(W ). Such an argument will also be used here as follows. We fix K > 0 and regarding (4.16), we let R(W ) be the unique solution of This means that if R(W ) = 1, we apply the cut-off procedure for Y, Y ′ D ≤ 1/2 or else for Y, Y ′ D ≤ R(W )/2. In conclusion, we work in the next sections with a modified version of the rough PDE (3.1) (equivalently (4.1)), where the drift and diffusion coefficients F and G are replaced by F R(W ) and G R(W ) . For notational simplicity, the W -dependence of R will be dropped whenever there is no confusion. (C F + C G C[ρ γ (W)]) R(W ) = K Due to (4.17) we conclude: Lemma 4.5 Let (Y, Y ′ ), ( Y , Y ′ ) ∈ D. We have T R (W, Y, Y ′ ) − T R (W, Y , Y ′ ), (T R (W, Y, Y ′ ) − T R (W, Y , Y ′ )) ′ D ≤ K Y − Y , Y ′ − Y ′ D . (4.19) Remark 4.6 1) We emphasize that one needs to control the derivatives of the diffusion coefficient G in order to make the constants that depend on G small after the cut-off procedure. Such restrictions are often met in the context of invariant manifolds for stochastic partial differential equations with nonlinear multiplicative noise, see e.g. [32,34]. 2) If the random input is smoother, i.e. γ ∈ (1/2, 1), it is enough to assume only DG(0) = 0. In this case, the stochastic convolution used in (3.13) is defined as a Young integral. Random Dynamical Systems The main techniques and results established in the previous section using controlled rough paths are necessary in order to provide pathwise estimates for the solutions of (3.1). In this section, we provide some concepts from the random dynamical systems theory [2], which allow us to define a center manifold for (3.1). The next concept is fundamental in the theory of random dynamical systems, since it describes a model of the driving noise. Definition 5.1 Let (Ω, F, P) be a probability space and θ : R×Ω → Ω be a family of P-preserving transformations (i.e., θ t P = P for t ∈ R) having the following properties: (i) the mapping (t, ω) → θ t ω is (B(R)⊗F, F)-measurable, where B(·) denotes the Borel sigmaalgebra; (ii) θ 0 = Id Ω ; (iii) θ t+s = θ t • θ s for all t, s, ∈ R. Then the quadrupel (Ω, F, P, (θ t ) t∈R ) is called a metric dynamical system. In this framework we recall that we consider one-dimensional noise, since the generalization to higher dimensions does not require any additional arguments. In our context, constructing a metric dynamical system is going to rely on constructing θ as a shift map on a canonical probability space (Ω, F, P) as specified below. Recalling that γ ≤ 1/2 was fixed in Section 3, we define for an γ-Hölder rough path W = (W, W) and τ ∈ R the time-shift Θ τ W := (Θ τ W, Θ τ W) by Θ τ W t := W t+τ − W τ Θ τ W s,t := W s+τ,t+τ . Note that the time shift naturally extends linearly to sums of rough paths, e.g., Θ τ W s,t = W t+τ − W s+τ . Furthermore, the shift leaves the path space invariant: Lemma 5.2 Let T 1 , T 2 , τ ∈ R, and W = (W, W) be an γ-Hölder rough path on [T 1 , T 2 ] for γ ∈ (1/3, 1/2). Then the time-shift Θ τ W = (Θ τ W, Θ τ W) is also an γ-Hölder rough path on [T 1 − τ, T 2 − τ ]. Proof. Let s, u, t ∈ [T 1 − τ, T 2 − τ ]. The γ-Hölder-continuity of θ τ W and the 2γ-Hölder continuity of θ τ W are obvious. We only prove that Chen's relation (3.2) holds true. We have Θ τ W s,t − Θ τ W s,u − Θ τ W u,t = W s+τ,t+τ − W s+τ,u+τ − W u+τ,t+τ = W s+τ,u+τ ⊗ W u+τ,t+τ , (5.1) = (W u+τ − W τ − W s+τ + W τ ) ⊗ (W t+τ − W τ − W u+τ + W τ ) = Θ τ W s,u ⊗ Θ τ W u,t . where in (5.1) we use Chen's relation (3.2). Based upon [4] we consider the following concept: Definition 5.3 Let (Ω, F, P, (θ t ) t∈R ) be a metric dynamical system. We call W = (W, W) a rough path cocycle if the identity W s,s+t (ω) = W 0,t (θ s ω) holds true for every ω ∈ Ω, s ∈ R and t ≥ 0. The previous definitions hint already at the fact that one may be able to just use as a probability space Ω a space of paths. A classical case, where we get via this construction a metric dynamical system and a rough cocycle is the fractional Brownian motion, see also [49,Section 6]. Gluing together lifts on compact time intervals, one may extend B H to the whole real line. Furthermore, we may consider the canonical probability space (C 0 (R), B(C 0 (R)), P), where C 0 (R) denotes the space of all real-valued continuous functions, which are 0 in 0, endowed with the compact open topology. The shift on the sample path space is given by (Θ τ f )(·) := f (τ + ·) − f (τ ), τ ∈ R, f ∈ C 0 (R). (5.2) Using Kolmogorov's Theorem or the Garsia-Rodemich-Rumsey inequality [31, A.2] one can conclude that maps in C γ 0 (R) have a finite γ-Hölder semi-norm on every compact interval P-almost surely. Hence, we can restrict this metric dynamical system to the set C γ 0 (R). For the metric dynamical system Of course, virtually the identical construction of a path-space (Ω W , F W , P), also referred to as canonical probability space, can be carried out for more general γ-Hölder rough paths W = (W, W) constructed from a (stochastic) process W , not just fractional Brownian motion, where the definition of a shift map is still as above, i.e., (Θ τ W )(t) := W t+τ − W τ . We now have the abstract definition of, as well as concrete examples for, metric dynamical systems for our problem modeling the underlying rough driving process. Now we have to also define the dynamical systems structure of the solution operator of our rough stochastic PDE. As a first step we recall the definition of a random dynamical system (RDS) [2] and show that the solution operator of (3.1) generates an RDS in B α . Definition 5.5 A random dynamical system on a separable Banach space X over a metric dynamical system (Ω, F, P, (θ t ) t∈R ) is a mapping ϕ : [0, ∞) × Ω × X → X , (t, ω, x) → ϕ(t, ω, x), which is (B([0, ∞)) ⊗ F ⊗ B(X ), B(X ))-measurable and satisfies: (i) ϕ(0, ω, ·) = Id X for all ω ∈ Ω; (ii) ϕ(t + τ, ω, x) = ϕ(t, θ τ ω, ϕ(τ, ω, x)), for all x ∈ X , t, τ ∈ [0, ∞), ω ∈ Ω; (iii) ϕ(t, ω, ·) : X → X is continuous for all t ∈ [0, ∞) and all ω ∈ Ω. The second property in Definition 5.5 is referred to as the cocycle property. In order to investigate random dynamical systems for (3.1) we need the global-in-time well-posedness of (3.1), which is guaranteed by our assumptions (F) and (G) [49,50]. Moreover working with a pathwise interpretation of the stochastic integral as given in (3.13), no exceptional sets can occur. Therefore one can immediately infer that the solution operator of (3.1) generates a RDS. For completeness, we sketch a proof of this fact, see also [49,Theorem 6.5]. Lemma 5.6 Let ξ ∈ B α and W = (W, W) be a rough path cocycle. Then the solution operator of the rough PDE (3.1) t → ϕ(t, W, ξ) = Y t = S t ξ + t 0 S t−r F (Y r ) dr + t 0 S t−r G(Y r ) dW r , generates a random dynamical system in B α over the metric dynamical system (Ω W , F W , P, (Θ t ) t∈R ). Proof. The relevant properties to define the metric dynamical system we need have been discussed in Example 5.4. The cocycle property can be immediately verified, since Y t+τ = S t+τ ξ + t+τ 0 S t+τ −r F (Y r ) dr + t+τ 0 S t+τ −r G(Y r ) dW r = S t S τ ξ + τ 0 S t+τ −r F (Y r ) dr + t+τ τ S t+τ −r F (Y r ) dr + τ 0 S t+τ −r G(Y r ) dW r + t+τ τ S t+τ −r G(Y r ) dW r = S t   S τ ξ + τ 0 S τ −r F (Y r ) dr + τ 0 S τ −r dW r   + t 0 S t−r F (Y r+τ ) dr + t 0 S t−r G(Y r+τ ) dΘ τ W r = S t Y τ + t 0 S t−r F (Y r+τ ) dr + t 0 S t−r G(Y r+τ ) dΘ τ W r . The above computations are rigorously justified, since one can immediately check the shift property of the rough integral (3.13). For a complete proof of this statement, see [48,Corollary 4.5]. The (B([0, ∞)) ⊗ F W ⊗ B(B α ), B(B α ))-measurability of ϕ follows by well-known arguments. One considers a sequence of (classical) solutions (Y n , (Y n ) ′ ) n∈N of (3.1) corresponding to smooth approximations (W n , W n ) n∈N of (W, W). Obviously, the mapping (t, W, ξ) → Y n t is (B([0, T ])⊗ F W ⊗ B(B α ), B(B α ))-measurable for any T > 0. Since Y continuously depends on the rough input W , according to [36,Lemma 3.12], one immediately concludes that lim n→∞ Y n t = Y t . This gives the measurability of Y with respect to F W ⊗ B(B α ). Due to the time-continuity of Y , we obtain by [17,Chapter 3] the (B([0, T ]) ⊗ F W ⊗ B(B α ), B(B α ))-measurability of the mapping (t, ω, ξ) → Y t for any t ≥ 0. Notation: The role of the random elements in Ω W is played by the paths W as one uses the canonical probability space of paths. So we directly denote these elements by W and do not write the identification W t (ω) := ω(t). However, this should be kept in mind. The random dynamical system ϕ : R + × Ω × B α → B α obviously depends upon the t, ξ, W , and W although we do not directly display the dependence upon W in the notation. To construct local random invariant manifolds, which can be characterized by the graph of a smooth function in a ball with a random radius [26,34] one requires the concept of tempered random variables [2, Chapter 4], which we recall next: Definition 5.7 A random variable R : Ω → (0, ∞) is called tempered from above, with respect to a metric dynamical system (Ω, F, P, (θ t ) t∈R ), if lim sup t→±∞ ln + R(θ t ω) t = 0, for all ω ∈ Ω,(5.3) where ln + a := max {ln a, 0}. A random variable is called tempered from below if 1/R is tempered from above. A random variable is tempered if and only if it is tempered from above and from below. Note that the set of all ω ∈ Ω satisfying (5.3) is invariant with respect to any shift map (θ t ) t∈R , which is an observation applicable to our case when θ t = Θ t . A sufficient condition for temperedness from above is according to [ R(θ t ω) < ∞. (5.4) Moreover, if the random variable R is tempered from below with t → R(θ t ω) continuous for all ω ∈ Ω, then for everyδ > 0 there exists a constant C[δ, ω] > 0 such that R(θ t ω) ≥ C[δ, ω]e −δ\|t\| ,(5.5) for any ω ∈ Ω. Again, for our concrete example when Ω = Ω B one can easily check that norms are tempered. Proof. The first assertion is valid due to the fact that E B H n γ < ∞ and the second one follows regarding that E B H m 2γ < ∞, for m ∈ N as contained in [30,Theorem 10.4]. This shows the temperedness from above of both random variables. As before, the last result holds more generally for broader classes of Gaussian rough paths, see [30,Section 10]. From now, we shall simply assume that W = (W, W) is a rough path cocycle such that the random variables R 1 (W ) = W γ and R 2 (W) = W 2γ are tempered from above. This concept is essential, since one wants to ensure that for initial conditions belonging to a ball with a sufficiently small tempered from below radius, the corresponding trajectories remain within such a ball, see [59,58,13]. We state an easy fact explicitly, which is crucial in this context: Lemma 5.9 The random variable R in (4.17) is tempered from below. Proof. Using Lemma 5.8 and 4.19 we immediately obtain that C[ρ γ (W)] is tempered from above. Recalling Definition 5.7, we conclude thatR is tempered from below and therefore R is also tempered from below. Local Center Manifolds for Rough PDEs In this section we prove the existence of a local center manifold for (3.1). The technique is similar to the one employed in [55]. The major technical difficulty is that we have to consider the fixed-point problem for the Lyapunov-Perron map in different function spaces. We state now the precise dynamical assumptions near the steady state at the origin, as shortly indicated in Section 2. [70]. Additionally, we impose the following exponential dichotomy condition on the semigroup. We assume that there exist two exponents γ * and β * with −β * < 0 ≤ γ * < β * and constants M c , M s ≥ 1, such that the following dichotomy condition is satisfied We show that (3.1) has a local center manifold M c (W ) ⊂ B α for small initial data belonging to B α . Before constructing this local center manifold using the Lyapunov-Perron transform we further introduce the following notation. For (U, U ′ ) ∈ D we write: B α = B c ⊕ B s α , where B s α = B α ∩ B sS c (t)x B ≤ M c e γ * t x B , for t ≤ 0 and x ∈ B; S s (t)x B ≤ M s e −β * t x B ,T s/c (W, U, U ′ )[·] :=   · 0 S s/c ·−r F (U ) dr + · 0 S s/c ·−r G(U r ) dW r , G(U · )   , (6.4) andT c (W, U, U ′ )[·] :=   1 · S c ·−r F (U r ) dr + 1 · S c ·−r G(U r ) dW r , G(U · )   . (6.5) Given the spectral decomposition of A, the Lyapunov-Perron map for (3.1) should be defined by, as discussed in Section 2 and suppressing the dependence of Y ′ , as follows: J(W, Y )[τ ] := S c τ ξ c + τ 0 S c τ −r F (Y r ) dr + τ 0 S c τ −r G(Y r ) dW r (6.6) + τ −∞ S s τ −r F (Y r ) dr + τ −∞ S s τ −r G(Y r ) dW r , for τ ∈ R − . Since we are dealing with rough integrals and we have to control the Hölder norm of the noise on each time-interval, we have to appropriately discretize (6.6) as justified already for the rough ODE case in [55]. Hence, we introduce a discrete version of the Lyapunov-Perron transform J d (W, Y, ξ) for a sequence of controlled rough paths Y ∈ BC η (D) and ξ ∈ B α as the pair J d (W, Y, ξ) := (J 1 d (W, Y, ξ), J 2 d (W, Y, ξ)) , where the precise structure is given below. For t ∈ [0, 1], W ∈ Ω W and i ∈ Z − we define J 1 d (W, Y, ξ)[i − 1, t] := S c t+i−1 ξ c (6.7) − i+1 k=0 S c t+i−1−k   1 0 S c 1−r F R (Y k−1 r ) dr + 1 0 S c 1−r G R (Y k−1 r ) dΘ k−1 W r   − 1 t S c t−r F R (Y i−1 r ) dr − 1 t S c t−r G R (Y i−1 r ) dΘ i−1 W r + i−1 k=−∞ S s t+i−1−k   1 0 S s 1−r F R (Y k−1 r ) dr + 1 0 S s 1−r G R (Y k−1 r ) dΘ k−1 W r   + t 0 S s t−r F R (Y i−1 r ) dr + t 0 S s t−r G R (Y i−1 r ) dΘ i−1 W r . Furthermore, J 2 d (W, Y, ξ) stands for the Gubinelli derivative of J 1 d (W, Y, ξ), i.e. J 2 d (W, Y, ξ)[i − 1, ·] := (J 1 d (W, Y, ξ)[i − 1, ·]) ′ . Note that ξ c can be recovered setting i = 0 and t = 1 in the definition of J 1 d (W, Y, ξ), i.e., J 1 d (W, Y, ξ)[−1, 1] = ξ c . The discretization of the Lyapunov-Perron map can be immediately derived using the substitution τ → t + i − 1 in (6.6) as computed in [55,Section 4.1]. We emphasize that for a sequence Y ∈ BC η (D) the first index i ∈ Z − in the definition of J d (W, Y, ξ)[·, ·] gives the position within the sequence and the second one refers to the time variable t ∈ [0, 1]. Not to overburden the notation in (6.7) for the elements of Y we simply write Y i t instead of Y [i, t] for i ∈ Z − and t ∈ [0, 1]. Remark 6.4 1) We are going to show that (6.7) maps BC η (D) into itself and is a contraction if the constant K specified in (4.17) is chosen small enough. 2) Compared to [55], several technical difficulties arise due to the fact that the controlled rough paths now incorporate different space and time regularity, recall Definition 3.3. Moreover, the dichotomy condition ((6.1) and (6.2)) in the corresponding interpolation spaces is a crucial step for the following computation. We let C S stand for a constant which exclusively depends on the semigroup S and derive: Remark 6.6 Note that (6.8) can be obtained for instance by choosing the constant appearing in (4.17) as K −1 := 4C S e (β * +γ * )/2 e (β * −γ * )/2 (M s + M c ) + 1 1 − e −(β * +γ * )/2 ,(6.9) which follows by setting η : = −β * +γ * 2 < 0. Proof. Let two sequences Y = ((Y i−1 , (Y i−1 ) ′ )) i∈Z − and Y = (( Y i−1 , ( Y i−1 ) ′ )) i∈Z − belong to BC η (D) and satisfy P c Y −1 1 = P c Y −1 1 = ξ c . We want to verify the contraction property. The fact that J d (·) maps BC η (D) into itself can be derived by setting Y = 0 in the next computation and using that F R (0) = G R (0) = 0. Keeping (3.11) in mind we compute as in the proof of Theorem 4.3 using (6.1) S c t+i−1 ξ c , 0 BC η (D) = ( S c ·+i+1 ξ c ∞,α + R S c ·+i+1 ξ c 2γ,α−2γ )e −η(i−1) ≤ C S S c i+1 ξ c Bα e −η(i−1) ≤ C S M c e (γ * −η)(i−1) ξ c Bα . (6.10) More precisely, the previous computation uses for 0 ≤ s ≤ t ≤ 1 that S c t+i−1 ξ c Bα ≤ S c t L(Bα,Bα) S c i+1 ξ Bα ≤ C S e γ * (i−1) ξ c Bα and R S c ·+i−1 ξ c B α−2γ = S c t+i−1 ξ − S c s+i−1 ξ c B α−2γ = (S c t−s − Id)S c s+i−1 ξ c B α−2γ ≤ S c t−s − Id L(Bα,B α−2γ ) S c s+i−1 ξ c Bα ≤ C S e γ * (i−1) (t − s) 2γ ξ c Bα . The expression (6.10) remains bounded for i ∈ Z − since we assumed that −β * < η < 0 ≤ γ * < β * . Next, we are going to estimate the difference \|\|J d (W, Y, ξ) − J d (W, Y, ξ)\|\| BC η (D) in several intermediate steps. Verifying the contraction property on the stable part of (6.7), one has to compute two terms. First of all, due to (4.19) we get i−1 k=−∞ e −η(i−1) S s ·+i−1−k T s R (Θ k−1 W, Y k−1 , (Y k−1 ) ′ )[1] − T s R (Θ k−1 W, Y k−1 , ( Y k−1 ) ′ )[1] , 0 D ≤ i−1 k=−∞ C S M s e −η(i−1) e −β * (i−1−k) K Y k−1 − Y k−1 , (Y k−1 − Y k−1 ) ′ D = i−1 k=−∞ C S M s e −η(i−1) e −β * (i−1−k) e η(k−1) Ke −η(k−1) Y k−1 − Y k−1 , (Y k−1 − Y k−1 ) ′ D = i−1 k=−∞ e −(η+β * )(i−1−k) C S M s e −η Ke −η(k−1) Y k−1 − Y k−1 , (Y k−1 − Y k−1 ) ′ D . For the first part of the computation, the only time-dependence is incorporated in S s ·+i−1−k and the rough integrals appearing in T s R are taken from zero to one and can be estimated using Lemma 4.2. This entails I 1 Bα : = T s R (Θ k−1 W, Y k−1 , (Y k−1 ) ′ )[1] − T s R (Θ k−1 W, Y k−1 , ( Y k−1 ) ′ )[1] Bα ργ (W) Y k−1 − Y k−1 , (Y k−1 − Y k−1 ) ′ D . Regarding the structure of the controlled rough path norm given by (3.11) we have to estimate the ∞-norm of S s ·+i−1−k I 1 in B α and the 2γ-Hölder norm of the remainder of this expression in B α−2γ . This gives us regarding (6.2) S s ·+i−1−k T s R (Θ k−1 W, Y k−1 , (Y k−1 ) ′ )[1] − T s R (Θ k−1 W, Y k−1 , ( Y k−1 ) ′ )[1] , 0 D ≤ ( sup t∈[0,1] S s t+i−1−k L(Bα) + sup s∈[0,1] S s t−s − Id L(Bα,B α−2γ ) S s s+i−1−k L(Bα) ) I 1 Bα ≤ C S M s e −β * (i−k−1) Y k−1 − Y k−1 , (Y k−1 − Y k−1 ) ′ D . Combining the previous computation with the last term of (6.7) entails the final estimate on the stable part i−1 k=−∞ e −η(i−1) S s ·+i−1−k T s R (Θ k−1 W, Y k−1 , (Y k−1 ) ′ )[1] − T s R (Θ k−1 W, Y k−1 , ( Y k−1 ) ′ )[1] , 0 D + e −η(i−1) T s R (Θ i−1 W, Y i−1 , (Y i−1 ) ′ )[·] − T s R (Θ i−1 W, Y i−1 , ( Y i−1 ) ′ )[·] D ≤ i k=−∞ e −(η+β * )(i−k−1) K C(M s e −η + 1)e −η(k−1) Y k−1 − Y k−1 , (Y k−1 − Y k−1 ) ′ D ≤ KC S e β * +η (M s e −η + 1) 1 − e −(β * +η) Y − Y, Y ′ − Y ′ BC η (D) . We focus now on the center part. Here we obtain by the same arguments as above i+1 k=0 e −η(i−1) S c ·+i−1−k T s R (Θ k−1 W, Y k−1 , (Y k−1 ) ′ )[1] − T s R (Θ k−1 W, Y k−1 , ( Y k−1 ) ′ )[1] , 0 D ≤ i+1 k=0 C S M c e −η(i−1) e γ * (i−1−k) K Y k−1 − Y k−1 , (Y k−1 − Y k−1 ) ′ D = i+1 k=0 C S M c e −η(i−1) e γ * (i−1−k) e η(k−1) e −η(k−1) K Y k−1 − Y k−1 , (Y k−1 − Y k−1 ) ′ D = i+1 k=0 C S M c e (γ * −η)(i−1−k) e −η Ke −η(k−1) Y k−1 − Y k−1 , (Y k−1 − Y k−1 ) ′ D . Again, for the first step of the estimate we make the same deliberations as in the stable case above. Furthermore, combining the previous computation and estimating the third summand in (6.7) yields on the center part i+1 k=0 e −η(i−1) S c ·+i−1−k T s R (Θ k−1 W, Y k−1 , (Y k−1 ) ′ )[1] − T s R (Θ k−1 W, Y k−1 , ( Y k−1 ) ′ )[1] , 0 D + e −η(i−1) \|\|T c R (Θ i−1 W, Y i−1 , (Y i−1 ) ′ )[·] −T c R (Θ i−1 W, Y i−1 , ( Y i−1 ) ′ )[·]\|\| D ≤ i k=0 e (γ * −η)(i−1−k) KC S (M c e −η + 1)e −η(k−1) Y k−1 − Y k−1 , (Y k−1 − Y k−1 ) ′ D ≤ KC S e γ * −η (M c e −η + 1) 1 − e −(γ * −η) Y − Y, Y ′ − Y ′ BC η (D) . Due to (6.8) we have that J d (W, Y, ξ) − J d (W, Y, ξ) BC η (D) ≤ 1 4 Y − Y, Y ′ − Y ′ BC η (D) . Applying Banach's fixed-point theorem, we infer that J d (W, Y, ξ c ) possesses a unique fixed-point Γ(ξ c , W ) ∈ BC η (D) for each fixed ξ c ∈ B c . Theorem 6.5 entails the existence of Γ(ξ c , W ) ∈ BC η (D) for each fixed ξ c ∈ B c . We denote by B B c (0, r(W )) a ball of B c , which is centered in 0 and has a random radius r(W ) and emphasize that the fixed point obtained in Theorem 6.5 characterizes the local center manifold of (3.1). The proof of the next statement is analogue to [55,Lemma 4.13]. Lemma 6.7 Under the same assumptions as in Theorem 6.5, there exists a tempered from below random variable r(W ) such that the local center manifold of (3.1) can be represented by M c loc (W ) = {ξ + h c (ξ, W ) : ξ ∈ B B c (0, r(W ))}, (6.11) where we define h c (ξ, W ) := P s Γ(ξ, W )[−1, 1]\| B B c (0,r(W )) , and consequently h c (ξ, W ) = 0 k=−∞ S s −k 1 0 S s 1−r P s F (Γ(ξ, W )[k − 1, r]) dr + 0 k=−∞ S s −k 1 0 S s 1−r P s G(Γ(ξ, W )[k − 1, r]) dΘ k−1 W r . Extending these results to continuous-time dynamical systems as discussed in [55] one obtains: Theorem 6.8 Under the assumptions of Theorem 6.5, there exists a local center manifold for (3.1) given by the graph of the function h c (ξ, W ) = 0 −∞ S s −r P s F (U r (ξ)) dr + 0 −∞ S s −r P s G(U r (ξ)) dW r . Examples In this section we discuss the applicability of Theorem 6.8. This theorem yields the existence of local center manifolds for semilinear rough parabolic PDEs once coefficients satisfy: 1) the linear part generates an analytic C 0 -semigroup and its spectrum satisfies (6.1); 2) the drift and the diffusion coefficients F and G satisfy assumptions (F) and (G) and can be truncated in a neighborhood of the origin such that the gap condition (6.8) holds true, see e.g. [14]. It would be interesting to investigate if the techniques developed in this work can be generalized to rough quasilinear parabolic equations (see [70] for a deterministic theory) based on the results established in [52]. The drift term F : B α → B α−δ is supposed to be locally Lipschitz with linear growth and G : B α → B α−σ satisfies assumption (G). Possible choices of G are integral operators obtained as a convolution with a smooth kernel as considered in [48,Section 7]. Naturally, a linear operator of the form G(u) := g(x)(−∆) σ u for a smooth function g satisfies assumption (G). Here (−∆) σ : B α → B α−σ for all α ∈ R and the multiplication with a smooth function g is a smooth operation from B α−σ into itself. In this case, we know according to [50,Theorem 3.9] that (7.1) has a global-in-time solution, therefore the center manifold theory developed in this paper covers this example. Remark 7.2 Regarding 2.1 it would be desirable to choose a dissipative cubic term for the drift, i.e. F (u) := −au 3 , for a > 0. In order to ensure global-in-time existence of solutions for (7.1), the drift term F must compensate the stochastic terms. For many classes of stochastic reactiondiffusion equations, this has been proven for additive Brownian noise [20]. Results regarding global-in-time existence for rough differential equations with a dissipative drift term have been obtained in [10]. It should be possible to extend these results to rough PDEs using energy estimates and the equivalence between weak and mild solutions [37, Theorem 2.18]. Remark 7.3 We can easily generalize the previous example to higher-order uniformly elliptic differential operators. Let m, n ∈ N and O = [0, π] and consider Au = \|k\|≤2m a k (x)D k u, x ∈ O D k u = 0, on ∂G, \|k\| < m. The coefficients a k ∈ C ∞ (O) and satisfy a uniform ellipticity condition, i.e. there exists a constant c > 0 such that Here we work on the scale of Bessel potential spaces B = H k,p (T) for 1 < p < ∞, k > 1 p and define Au := ∆u with B 1 = D(A) = H k+2,p (T). Therefore we obtain the scale B α = H k+2α,p (T). Note that we consider here only one spatial dimension in order to ensure the gap condition. The one-dimensional torus is simply a circle of some given length T = R/lZ for l ∈ R. In this case, the spectrum of A is given by − 2πk l 2 : k ∈ Z . The eigenfunctions corresponding to 0 ∈ σ(A) are the constant functions which build the center space B c . Furthermore, we consider G(u) = p(u), where p is a polynomial with smooth coefficients. If the degree of p is greater than one, in order for G to be a smooth operator acting from B α → B α−σ for 0 ≤ σ < γ and for all α ≥ −2γ, we need that B −2γ is an algebra. Here we recall that γ ∈ ( 1 3 , 1 2 ) stands for the time-regularity of the rough path. In conclusion we need that B −2γ = H k−4γ,p (T) is an algebra, which is true for k > 1 p + 4γ. This means that it useful to take as low as possible rough path regularity γ, as seen in [36]. Again we assume that we have a dissipative drift which compensates the stochastic terms, in order to guarantee the global-in-time existence of solutions. (−1) m \|k\|=2m a k (x)ξ k ≥ c\|ξ\| 2m , x ∈ O, ξ ∈ R n . Due to its importance to bifurcation theory, see for e.g. [6], we particularly point out the following example which fits into the framework of this work, recall Remark 7.3. ((0, 2π)) where per refers to periodic functions. It is well known that A generates an analytic semigroup on B and we introduce for simplicity the interpolation spaces B α = H 4α per ((0, 2π)). The spectrum of A consists of isolated eigenavlues with finite multiplicities, i.e. σ(A) = {−(1 − n 2 ) 2 : n ∈ N}. The eigenvalue 0 ∈ σ(A) has multiplicity two and {e ix } are the corresponding eigenfunctions. Therefore B c = span{sin x, cos x} forms the center space. Definition 3. 1 1Let J ⊂ R be a compact interval, ∆ J := {(s, t) ∈ J × J : s ≤ t} and let W = (W, W) andW = (W ,W) be two γ-Hölder rough paths. We introduce the γ-Hölder rough path (inhomogeneous) metric d γ,J (W,W) := sup (3. 3 ) 3We set ρ γ (W) := d γ,[0,T ] (W, 0) and denote the space of γ-Hölder rough paths by C γ ([0, T ]; R). Lemma 4. 2 2Let (U, U ′ ), ( U , U ′ ) ∈ D. Then there exists a constant L G (R) = L G [R, G, ρ γ (W), χ] such that L G (R) → 0 as R → 0 and that Remark 4.4 1 ) 1. This argument is slightly different from the proof of [37, Theorem 5.1] where one takes γ ′ such that 0 < γ < γ ′ < σ and performs the fixed-point argument in D 2γ ′ W,α choosing the time horizon small enough. W ) := min{ R(W ), 1}.(4.18) (C γ 0 0(R), B(C γ 0 (R)), P, (Θ t ) t∈R ) =: (Ω B , F B , P, (Θ t ) t∈R ) one may check that B H = (B H , B H ) represents a rough path cocycle as introduced in Definition 5.3. Lemma 5. 8 8Let B H = (B H , B H ) be the rough path cocycle associated to a fractional Brownian motion B H with Hurst parameter H ∈ (1/3, 1/2]. Then the random variables R 1 (B H ) = B H γ and R 2 (B H ) = B H 2γ are tempered from above. Assumptions 6. 1 1The spectrum of the linear operator A is supposed to contain eigenvalues with zero and strictly negative real parts, i.e. σ(A) = σ c (A) ∪ σ s (A), where σ c (A) = {λ ∈ σ(A) : Re(λ) = 0} and σ s (A) = {λ ∈ σ(A) : Re(λ) < 0}. The subspaces generated by the eigenvectors corresponding to these eigenvalues are denoted by B c respectively B s and are referred to as center and stable subspace. These subspaces provide an invariant splitting of B = B c ⊕ B s . We denote the restrictions of A on B c and B s by A c := A\| B c and A s := A\| B s .Since B c is finite-dimensional we obtain that S c (t) := e tAc is a group of linear operators on B c . Moreover, there exist projections P c and P s such that P c + P s = Id B and A c = A\| R(P c ) and A s = A\| R(P s ) , where R denotes the range of the corresponding projection. In this case one can show that there is also a decomposition of the corresponding interpolation spaces for t ≥ 0 and x ∈ B.This yields according to [1, Theorem 2.1.3, p. 289] a dichotomy condition also on the interpolation spaces B α for α > 0, i.e. S c (t)x Bα ≤ M c e γ * t x Bα , for t ≤ 0 and x ∈ B α ; (6.1)S s (t)x Bα ≤ M s e −β * t x Bα , for t ≥ 0 and x ∈ B α . (6.2)For further details and similar assumptions [69, Section 7.1, p. 460].Remark 6.2 One can extend the techniques and results presented below easily if one additionally has an unstable subspace, namely if there exist eigenvalues of A with real part greater than zero. In this case the classical exponential trichotomy condition is satisfied, see for instance [69, Section 7.1].Definition 6.3We call a random set M c (W ), which is invariant with respect to ϕ (i.e. ϕ(t, W, M c (W )) ⊂ M c (Θ t W ) for t ∈ R and W ∈ Ω W ), a center manifold if this can be represented asM c (W ) = {ξ + h c (ξ, W ) : ξ ∈ B c }, (6.3)where h c (·, W ) : B c → B s α is Lipschitz continuous and differentiable in zero. Moreover, h c (0, W ) = 0 and M c (W ) is tangent to B c at the origin, meaning that the tangency condition Dh c (0, W ) = 0 is satisfied. Theorem 6. 5 5Let Assumptions 6.1, (F), (G) hold and let K satisfy the gap conditionKC S e β * +η (M s e −η + 1) 1 − e −(β * +η) + e γ * −η (M c e −η + 1) 1 − e −(γ * −η)Then, the map J d : Ω × BC η (D) → BC η (D) possesses a unique fixed-point Γ ∈ BC η (D). Example 7. 1 ( 1Reaction-diffusion type equations with Dirichlet boundary conditions) We consider on a monotone scale of interpolation spaces (B α ) α∈[0,1] , as specified below, the parabolic PDE with zero Dirichlet boundary conditions on the bounded one-dimensional domain O := [0, = (∆u + u + F (u)) dt + G(u) dW t , u(0, t) = u(π, t) = 0, for t ≥ 0, u(x, 0) = u 0 (x) ∈ B α , for x ∈ O. to Example 2.1, the random input W := (W, W) is a γ-Hölder rough path, for γ ∈ (1/3, 1/2]. In this case we can construct a Banach scale starting from the operator Au := ∆ D u + u, where ∆ D denotes the Dirichlet-Laplacian, as follows. We set B := L p (O), for 1 < p < ∞, B 1 := D(A) = W 2,p (O) ∩ W 1,p 0 (O) and B α = [B, B 1 ] α = W 2α,p 0 (O), for α ∈ [0, 1]. Furthermore, the spectrum of A is constituted by {1 − n 2 : n ≥ 1} with corresponding eigenvectors {sin(nx) : n ≥ 1}. These give us the center subspace B c := span{sin x} and the stable one B s := span{sin(nx) : n ≥ 2}. In this case we choose the spacesB = L p (O) for 1 < p < ∞, B 1 = D(A) = W 2mp (O) ∩ W m,p 0 (O) and B α = [B, B 1 ] α = W 2αm,p 0 (O).It is known that A has a compact resolvent and therefore countably many eigenvalues {λ j } which have finite multiplicities and λ j → −∞ as j → ∞. Let λ be the largest negative eigenvalue of A. Therefore the linear operator L := A − λ Id on B with D(L) = D(A) satisfies the assumption 6.1. Example 7 . 4 ( 74Reaction-diffusion type equations on the torus) We consider the rough PDE with periodic boundary conditions on the one dimensional torus T du = (∆u + F (u)) dt + G(u) dW t u(0) = u 0 ∈ B. Example 7. 5 ( 5Swift-Hohenberg equation with periodic boundary conditions) We considerdu = [Au + F (u)] dt + G(u) dW t u(0) = u 0 ∈ B,subject to periodic boundary conditions on the interval [0, 2π]. Here Au := −(1 + ∆) 2 u and F (u) = −u 3 . We choose the function spaces B = L 2 per ((0, 2π)) and D(A) = H 4 per according to[26, Lem. 2.1] and the references specified therein. Under a suitable smallness assumption on the Lipschitz constant of f (gap condition) one can show that J possesses a fixed-point Γ(·,B, ξ) for ξ ∈ H c . Since a global Lipschitz condition on f is quite restrictive in applications, one usually introduces a cut-off function to truncate the nonlinearity outside a random ball around the origin. This fixed-point characterizes the random center manifold M c (B) for (2.4). 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[ "Maximum entangled states and quantum Teleportation via Single Cooper pair Box", "Maximum entangled states and quantum Teleportation via Single Cooper pair Box" ]	[ "N Metwally \nMath. Dept\nCollege of Science\nBahrain University\n32038Bahrain\n", "A A El-Amin \nPhys. Dept\nFaculty of Science\nSouth valley University\nAswanEgypt\n" ]	[ "Math. Dept\nCollege of Science\nBahrain University\n32038Bahrain", "Phys. Dept\nFaculty of Science\nSouth valley University\nAswanEgypt" ]	[]	In this contribution, we study a single Cooper pair box interacts with a single cavity mode. We show the roles played by the detuning parameters and charged capacities on the degree of entanglement. For large values of the detuning parameter the survival entanglement increases on the expanse of the degree of entanglement. We generate a maximum entangled state and use it to perform the original teleportation protocol. The fidelity of the teleportated state is increased with decreasing the detuning parameter and the number of photon inside the cavity.	10.1016/j.physe.2008.11.012	[ "https://arxiv.org/pdf/0808.0633v1.pdf" ]	14,234,185	0808.0633	a106fcc5a81915609872871c3c7cdc9ae31286f0	Maximum entangled states and quantum Teleportation via Single Cooper pair Box 5 Aug 2008 N Metwally Math. Dept College of Science Bahrain University 32038Bahrain A A El-Amin Phys. Dept Faculty of Science South valley University AswanEgypt Maximum entangled states and quantum Teleportation via Single Cooper pair Box 5 Aug 2008Charged Qubit, Entanglement and Teleportation In this contribution, we study a single Cooper pair box interacts with a single cavity mode. We show the roles played by the detuning parameters and charged capacities on the degree of entanglement. For large values of the detuning parameter the survival entanglement increases on the expanse of the degree of entanglement. We generate a maximum entangled state and use it to perform the original teleportation protocol. The fidelity of the teleportated state is increased with decreasing the detuning parameter and the number of photon inside the cavity. Introduction Several schemes have been proposed for implementing quantum computer hardware in solid state quantum electronics. These schemes use electric charge [1], magnetic flux [2] and superconducting phase [3] and electron spin [4]. The basic element of the quantum information is the quantum bit (qubit) which is considered as a two level system. Consequently, most of the research concentrate to generate entanglement between two level systems [5]. Among these systems, the Cooper charged pairs, due to its properties as a two -level quantum system, which makes it a candidate as a qubit in a quantum computer [6,7]. So there are a lot of studies have been done on these particle from different points of view in the context of quantum information. One of the most important tasks in the context of quantum information, is the quantum teleportation, which emerges from the quantum entanglement, and since the first quantum teleportation protocol was introduced by Bennett et al [8], there are a lot of attentions has been payed to it [9,10,11]. In our contribution, we consider a system consists of a single Cooper pair interactsa with a cavity mode. The separability problem is investigated, where the intervals of time in which the generated state is entangled or separable are determined. On the other hand under a specific circumstance one can use this system to generated a maximum entangled state. Finally we use the generated entangled state to perform the quantum teleportation. The paper is organized as follows: The description of the system and its solution are introduced in Sec.2. In Sec.3, the separability problem and the degree of entanglement contained in the generated entangled state are investigated. Sec.4, is devoted to study the effect of the field and the charged qubit parameters on the phenomena of entanglement and quantum teleportation. The Model and its evolvement The single superconducting charged qubit consists of a small superconducting island with Cooper pair charge Q. This island connected by two identical Josephson junctions, with capacitance C j and Josephson coupling energy E j , to a superconducting electrode [12,14]. This system is described by the Hamiltonian, H s = 4E c (n − n g ) 2 − E j cos φ,(1) where E c = e 2 /2(C g + C j ) is the charging energy, E j = I c /2e is the Josephson coupling energy, e is the charge of the electron, n g = C g V g /2e is the dimensionless gate charge, C g is the gate capacitance, v g is controllable gate voltage, n is number operator of excess cooper pair on the island and φ is phase operator [12]. The Hamiltonian of the system (1) can be simplified to a very simple form, if the Josephon coupling energy E j is much smaller than the charging energy i.e E j << E c . In this case, the Hamiltonian of the system can be parameterized by the number of Cooper pairs n on the island. If the temperature is low enough, the system can be reduced to two-state system (qubit) controlled by [13,15], H s = − 1 2 B z σ z − 1 2 B x σ x ,(2) where B z = E cl (1 − 2n j ), E cl is the electric energy and B x = E j and σ z , σ z are Pauli matrices. This Cooper pair can be viewed as an atoms with large dipole moment coupled to microwave frequency photons in a quasi-one-dimensional transmission line cavity (a coplanar waveguide resonator). The combined Hamiltonian for qubit and transmission line cavity is given by, H = ωa † a + ω c σ z − λ(µ − cos θσ z + sin θσ x )(a † + a),(3) where ω is the cavity resonance frequency, ω c = 2 is the transition frequency of the Cooper pair qubit, σ z and σ x are Pauli matrices, λ = √ C j Cg +C j e 2 ω 2 , is coupling strength of resonator to the cooper pair qubit, µ = 1 − n g , θ = arctan{ E j Ec(1−2ng) }, is the mixing angle. Assume that we consider the charge degeneracy point, i.e n g = 1 2 and the radiation quantized field is weak. In this case we can neglect the fast oscillation by using the rotating wave approximation. Then the Hamiltonian (3) takes the form E 2 j + [4E c (1 − 2n g )]H = ωa † a + 1 2 ω c σ z − λ(a † σ − + σ + a),(4) where σ + and σ − are the rasing and lowering operators such that [σ + , σ − ] = σ z . To investigate the dynamics of the total system (cooper pair box and the filed), let us consider that the charged qubit prior to the interaction, to be prepared in a superposition of its excited and ground state, i.e ψ c (0) = α g + β e , and the filed is prepared in the number state ψ f (0) = n . The time development of the state vector ψ(t) of the system is postulated to be determined by Schrödinger equation i d dt ψ(t) = H ψ(t)(5) The solution of Eq.(5) can be written as ψ(t) = U(t) ψ(0) , where U(t) is the uni- tary operator. In an explicit for the evolvement of the density operator ψ(t) ψ(t) is given by ρ(t) = A \|α\| 2 g, n g, n + \|β\| 2 e, n e, n + Bαβ * g, n e, n + B * α * β e, n g, n + iC √ n + 1(αβ * g, n g, n + 1 + βα * g, n + 1 g, n − iC √ n + 1\|β\| 2 e, n g, n − 1 + √ nβα * e, n e, n − 1 + iηC * √ n + 1\|β\| 2 g, n + 1 e, n − √ n\|α\| 2 g, n e, n − 1 + η 2 S 2 n √ n √ n + 1 βα * g, n + 1 e, n − 1 + αβ * e, n − 1 g, n + 1 + S 2 n n\|α\| 2 e, n − 1 e, n − 1 + (n + 1)\|β\| 2 g, n + 1 g, n + 1 + i √ nS n \|α\| 2 C e, n − 1 g, n + αβ * C * e, n − 1 e, n(6) where A = C 2 n+1 + ∆ 2 4 S 2 n+1 , B = C 2 n+1 − ∆ 2 4 S 2 n+1 − i∆S n+1 C n+1 C = iS n (C n+1 + i∆ 2 S n+1 ), C n = cos (γµ n τ ), S n = sin(γµ n τ ) µ n , γ = C j C g + C j , τ = e 2 ω 2h t, is the scaled time (7) with µ n = ∆ 2 4 + λn, ∆ = E j − ω is the detuning between the Josephson energy and the cavity field frequency . Degree of entanglement In this section, we study the behavior of the output state (6) from the separability point of view. To achieve this task, we plot time evolution of the the eigenvalues of the partial transpose eigenvalues of the output density operator ρ T 2 [16,17]. Also we investigate time development of the occupation probabilities and the degree of entanglement which contained in the output state. There are different measures known for quantifying the degree of entanglement in a bipartite system, such as the entanglement of formation [18,19], entanglement of distillation [20], negativity [21]. In our calculations we consider the concurrence as a measure of the degree of entanglement. For two qubits, the concurrence is calculated in terms of the eigenvalues η 1 , η 2 , η 3 and η 4 of the matrix R = ρσ y ⊗ σ y ρ * σ y ⊗ σ y . It is given by C = max{0, η 1 − η 2 − η 3 − η 4 }, where η 1 ≥ η 2 ≥ η 3 ≥ η 4 .(8) For maximally entangled states concurrence is 1 while C = 0 for separable states [20]. In the first example, we assume that the Cooper-pair box is initially prepared in the ground state and the field is prepared in the Fock state, i.e the initial state of the system is given by g, n . In Fig.(1), we investigate the effect of the detuning parameter on the behavior of the eigenvalues of the partial transpose of the density operator of the output state, the time evaluation of the occupation probabilities and the degree of entanglement. For these numerical calculations, we assume that the ratio between the Josephson junction capacity, C j , and the gate capacity C g , is defined by C jg = C j Cg = 5 2 and n = 1. In Fig.(1a), we see that as the interaction goes on, i.e the scaled time, τ > 0, there is only one negative eigenvalues and the rest are non-negative. According to the Peres-Horodecki criterion [16,17], there is an entangled state is generated. As the time increases we see that at specific time τ ≃ 5 all the eigenvalues are non-negative. This means that the entangled qubit turns into a product state (separable). In Fig.(1c), we plot the occupation generated as soon as the interaction starts. In this case the entangled state is ρ = κ( n, g n − 1, e + n − 1, e n, g ) + χ 1 n, g n, g + χ 2 n − 1, e n − 1, e , (9) where χ 1 and χ 2 are the probability to find the system in the state n, g n, g and n − 1, e n − 1, e , respectively. In computational basis one can write it as ρ = κ( 1, 0 0, 1 + 0, 1 1, 0 ) + χ 1 1, 0 1, 0 + χ 2 0, 1 0, 1 . It is clear that the first bract is one type of Bell state ψ + ψ + . However at τ ≃ 2.5, the partially entangled state (9) becomes a maximum, where all the occupation probabilities have equal occupation probabilities at this point. This result is shown in from Fig.(1e), where we quantify the degree of entanglement by using the concurrence. From this figure, we can see that the degree of entanglement is increased as the time increased and reaches to the unity at τ ≃ 2.5. Given enough time, the system will therefore reaches a state where all the occupation probabilities vanish. At specific time the system turns into a product state, this happens at τ ≃ 5, (see Fig.(1a)). Also, on the left hand side of Fig.(1), we consider ∆ = 1 and the other parameters are fixed, it is clear that as one increases ∆, the system turns into a product round τ = 5.4. This means that the time of the entanglement survival increased. This phenomenon is clear shown by comparing Fig.(1a) and Fig.(1b). The behavior of the occupation probability is seen in Fig.(d), where there is no intersection point between the off diagonal occupation probabilities and the diagonal ones. So in this case one can not get a maximum entangled state. As time goes on the system becomes a separable at τ ≃ 5.4, this result agrees with that depicted in Fig.(1b). The amount of entanglement contained in the output state is shown in Fig.(1f ). From this figure, we can notice that the maximum entanglement is less than unity. So one can say that by increasing the detuning parameter, one can increase the survival time of entanglement on the expanse of the degree of entanglement. The effect of the ratio of C jg is seen in Fig.(2), where we consider a small value of this ratio. We investigate the behavior of the occupation probabilities and the degree of entanglement only. It is clear that as one decreases C jg , the first maximum entangled state is obtained round τ = 4. This means that for small ratio, one takes a larger time to generate maximum entangled state. Also, the point at which the system turns into a separable state is shifted. The degree of entanglement contained in this state is shown in Fig.(2b), where it has a unity value at the maximum entangled state. So, the ratio C jg has no effect on the value of the degree of entanglement. Our second case, we assume that the Cooper pair box is prepared in the superpo- sition state. So the initially state of the system, ψ 0 = a n, g + (1 − a) n, e . In Figs.(3), we plot the degree of entanglement where we assume that a = 0.5, fixed vales of the ratio C jg = 5 2 and different values of the detuning parameter ∆. In this case the effect is completely different ( see Fig.(3a)), where the degree of entanglement is small compared with by that depicted in Fig.(1e). Also, the survival time of entangled is small comparing by the previous case, where the system behaves as a separable system several times in a small range of time. In Fig.(3b), we increase the value of the detuning parameter (∆ = 1). It is clear that the instability of the system increases and both of the sudden death [22] and sudden birth of entanglement is seen [23]. Now, one can see that by controlling the Cooper pair box parameters and the field parameters, one can generate entangled state with high degree of entanglement. One of the best strategy is preparing the Cooper pair box in the excited or the ground state. If the charged qubit and the field in a resonance i.e ∆ = 0, one can generate a maximum entangled state. By decreasing the ratio C jg , the survival time of entanglement is much larger. Teleportation No, we want to use the generated entangled state to achieve the quantum teleportation by using the output state (6), as a quantum channel. Assume that Alice is given unknown state defined by Ψ = λ 1 0 + λ 2 1 ,(10) where λ 2 1 + λ 2 2 = 1. She wants to sent this state to Bob through their quantum channel. To attain this aim, Alice and Bob shall use the original teleportation protocol [8]. In this case, the total state of the system is ρ ψ ⊗ ρ out , where ψ is given by (10) and ρ out is defined by (6). Alice makes measurement on the given qubit and her own qubit. Then she sends her results through a classical channel to Bob. As soon as Bob receives the classical data, he performs a suitable unitary operation on his qubit to get the teleported state. Let us assume that Alice measures ψ + , then the density operator on Bob's hand is given by ρ Bob = 1 2 (\|λ 1 \| 2 \|B n \| 2 0 0 + λ 1 λ * 2 B n A * n 0 1 + λ * 2 λ 1 A n B * n + \|λ 2 \| 2 \|A n \| 2 1 1 ),(11) with, A n = cos(γµ n τ ) − i∆ µ n sin(γµ n τ ), B n = iλ √ n µ n sin(µ n t) where, we assume that the Cooper pair box is prepared on the ground state and the field on the Fock state i.e ψ 0 = g, n . In Fig.4, we plot the of the fidelity of the teleported state (11), for different values of the field and the Cooper pair parameters. The effect of the detuning parameter is seen in Fig.(4a), where we assume that there is only one photon inside the cavity and the ratio C jg = 5. From this figure it is clear that as one increases ∆, the fidelity of the teleported state decreases. This due to that for large values of the detuning parameter, the degree of entanglement decreases as it is clear from Fig.(1f ). Since, the entangled time, the time in which the entanglement survival, increases for small values of the detunang parameter, the fidelity vanishes for large values of ∆ much faster. The effect of C jg is shown in Fig.(4b),where for small values of this ratio, the fidelity reaches to its maximum value faster than the large values of C jg . On the other hand the different values of this ratio has no effect on the maximum values of the fidelity. Also, as one increases this ratio, the interval of time in which the channel is available for quantum teleportation increases. In Fig.(5), we investigate the effect of different values of the number of photons inside the cavity, where we fixed the other parameters. It is clear that for small values of n, the possibility of sending information with non-vanishing fidelity by using the quantum teleportation increases. This is due to that for large values of n, the possibility of quick interaction increases and consequently one can gets an entangled state much faster. Conclusion We employ the Cooper pair box to generate entangled state by interacting with a single cavity mode is initially prepared in the numbers state. We show that it is possible to generate a maximum entangled state by controlling on the capacities and the detuning parameters. Also, one can use the generated entangled state to perform the quantum teleportation. We investigate the effect of the detuning parameter, the ratio of capacities and the photon inside the cavity on the fidelity of the teleported state. For small values of the detuning parameter and the photon numbers inside the cavity one can teleportate the given state with large fidelity. On the other hand by increasing the ratio of the capacities one can enlarge the interval of time in which one can teleportate the state with non-vanishing fidelity. Figure 1 : 1probabilities as functions of the scaled time. Note that the populations of the four states exist and it has different values for the diagonal occupations but for the offdiagonal occupations are completely coincide. For this reason an entangled state is The behavior of the eigenvalues of the partial transpose ρ T is shown in Figs.(a, b). The populations are shown in Fig.(c, d) and the degree of entanglement is plotted in Figs.(e, f ). We assume n = 1, C jg = 5 2 and ∆ = 0 for Figs.(a, c, e) while for Figs.(b, d, f ), ∆ = 1. Figure 2 : 2The same asFig.(1), but ∆ = 0, C jg = 2 5 . 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[ "Exploration of Interesting Dense Regions in Spatial Data", "Exploration of Interesting Dense Regions in Spatial Data" ]	[ "Plácido A Souza Neto \nFederal Institute of Rio Grande do Norte\nBrazil\n", "Francisco B Silva Júnior \nFederal Institute of Rio Grande do Norte\nBrazil\n", "Felipe F Pontes [email protected] \nFederal Institute of Rio Grande do Norte\nBrazil\n", "Behrooz Omidvar-Tehrani [email protected] \nUniversity of Grenoble Alpes\nFrance\n" ]	[ "Federal Institute of Rio Grande do Norte\nBrazil", "Federal Institute of Rio Grande do Norte\nBrazil", "Federal Institute of Rio Grande do Norte\nBrazil", "University of Grenoble Alpes\nFrance" ]	[]	Nowadays, spatial data are ubiquitous in various fields of science, such as transportation and the social Web. A recent research direction in analyzing spatial data is to provide means for "exploratory analysis" of such data where analysts are guided towards interesting options in consecutive analysis iterations. Typically, the guidance component learns analyst's preferences using her explicit feedback, e.g., picking a spatial point or selecting a region of interest. However, it is often the case that analysts forget or don't feel necessary to explicitly express their feedback in what they find interesting. Our approach captures implicit feedback on spatial data. The approach consists of observing mouse moves (as a means of analyst's interaction) and also the explicit analyst's interaction with data points in order to discover interesting spatial regions with dense mouse hovers. In this paper, we define, formalize and explore Interesting Dense Regions (IDRs) which capture preferences of analysts, in order to automatically find interesting spatial highlights. Our approach involves a polygon-based abstraction layer for capturing preferences. Using these IDRs, we highlight points to guide analysts in the analysis process. We discuss the efficiency and effectiveness of our approach through realistic examples and experiments on Airbnb and Yelp datasets.	null	[ "https://arxiv.org/pdf/1903.04049v1.pdf" ]	73,728,741	1903.04049	76018bafe84e4e3cfb9ace724275448dea274f5a	Exploration of Interesting Dense Regions in Spatial Data Plácido A Souza Neto Federal Institute of Rio Grande do Norte Brazil Francisco B Silva Júnior Federal Institute of Rio Grande do Norte Brazil Felipe F Pontes [email protected] Federal Institute of Rio Grande do Norte Brazil Behrooz Omidvar-Tehrani [email protected] University of Grenoble Alpes France Exploration of Interesting Dense Regions in Spatial Data Nowadays, spatial data are ubiquitous in various fields of science, such as transportation and the social Web. A recent research direction in analyzing spatial data is to provide means for "exploratory analysis" of such data where analysts are guided towards interesting options in consecutive analysis iterations. Typically, the guidance component learns analyst's preferences using her explicit feedback, e.g., picking a spatial point or selecting a region of interest. However, it is often the case that analysts forget or don't feel necessary to explicitly express their feedback in what they find interesting. Our approach captures implicit feedback on spatial data. The approach consists of observing mouse moves (as a means of analyst's interaction) and also the explicit analyst's interaction with data points in order to discover interesting spatial regions with dense mouse hovers. In this paper, we define, formalize and explore Interesting Dense Regions (IDRs) which capture preferences of analysts, in order to automatically find interesting spatial highlights. Our approach involves a polygon-based abstraction layer for capturing preferences. Using these IDRs, we highlight points to guide analysts in the analysis process. We discuss the efficiency and effectiveness of our approach through realistic examples and experiments on Airbnb and Yelp datasets. Introduction Nowadays, there has been a meteoric rise in the generation of spatial datasets in various fields of science, such as transportation, lodging services, and social science. As each record in spatial data represents an activity in a precise geographical location, analyzing such data enables discoveries grounded on facts. Analysts are often interested to observe spatial patterns and trends to improve their decision making process. Spatial data analysis has various applications such as smart city management, disaster management and autonomous transport [1,2]. Typically, spatial data analysis begins with an imprecise question in the mind of the analyst, i.e., exploratory analysis. The analyst requires to go through several trial-and-error iterations to improve her understanding of the spatial data and gain insights. Each iteration involves visualizing a subset of data on geographical maps using an off-the-shelf product (e.g., Tableau 1 , Exhibit 2 , Spotfire 3 ) where the analyst can investigate on different parts of the visualization by zooming in/out and panning. Spatial data are often voluminous. Hence the focus in the literature of spatial data analysis is on "efficiency", i.e., enabling fluid means of navigation in spatial data to facilitate the exploratory analysis. The common approach is to design pre-computed indexes which enable efficient retrieval of spatial data (e.g., [3,4]). However, there has been less attention to the "value" derived from spatial data. Despite the huge progress on the efficiency front, an analyst may easily get lost in the plethora of geographical points due to two following reasons. In an exploratory context, the analyst doesn't know a priori what to investigate next. Moreover, she may easily get distracted and miss interesting points by visual clutter caused by huge point overlaps. The main drawback of the traditional analysis model is that the analyst has a passive role in the process. In other words, the analyst's feedback (i.e., her likes and dislikes) is ignored and only the input query (i.e., her explicit request) is served. In case feedback is incorporated, the process can be more directed towards analyst's interests where her partial needs can be served earlier in the process. In this paper, we advocate for a "guidance layer" on top of the raw visualization of spatial data to enable analysts know "what to see next". This guidance should be a function of analyst feedback: the system should return options similar to what the analyst has already appreciated. Various approaches in the literature propose methodologies to incorporate analyst's feedback in the exploration process of spatial data [5,6,7,8]. Typically, feedback is considered as a function which is triggered by any analyst's action on the map. The action can be "selecting a point", "moving to a region", "asking for more details", etc. The function then updates a "profile vector" which keeps tracks of analyst's interests. The updated content in the profile vector enables the guidance functionality. For instance, if the analyst shows interest in a point which describes a house with balcony, this choice of amenity will reflect her profile to prioritize other houses with balcony in future iterations. Feedback is often expressed explicitly, i.e., the analyst clicks on a point and mentions if she likes or dislikes the point [9,10,11]. In [11], we proposed an interactive approach to exploit such feedback for enabling a more insightful exploration of spatial data. However, there are several cases that the feedback is expressed implicitly, i.e., the analyst does not explicitly click on a point, but there exist correlations with other signals captured from the analyst which provide hint on her interest. For instance, it is often the case in spatial data analysis that analysts look at some regions of interest but do not provide an explicit feedback. Another example is frequent mouse moves around a region which is a good indicator of the analyst's potential interest in the points in that region. Implicit feedbacks are more challenging to capture and hence less investigated in the literature. The following example describes a use case of implicit feedbacks. This will be our running example which we follow throughout the paper. Example. Benício is planning to live in Paris for a season. He decides to rent a home-stay from Airbnb website 4 . He likes to discover the city, hence he is open to any type of lodging in any region with an interest to stay in the center of Paris. The website returns 1500 different locations. As he has no other preferences, an exhaustive investigation needs scanning each location independently which is nearly infeasible. While he is scanning few first options, he shows interest in the region of Trocadero (where the Eiffel tower is located at) but he forgets or doesn't feel necessary to click a point there. An ideal system should capture this implicit feedback in order to short-list a small subset of locations that Benício should consider as high priority. The above example shows in practice that implicit feedback capturing is crucial in the context of spatial data analysis. While text-boxes, combo-boxes and other input elements are available in analyzing other types of data, the only interaction means between the analyst and a spatial data analysis system is a geographical map spanned on the whole screen. In this context, a point can be easily remained out of sight and missed. In this paper, we present an approach whose aim is to capture and analyze implicit feedback of analysts in spa-tial data analysis. Without loss of generality, we focus on "mouse moves" as the implicit feedback received from the analyst. Mouse moves are the most common way that analysts interact with geographical maps [12]. It is shown in [13] that mouse gestures have a strong correlation with "user engagement". Intuitively, a point gets a higher weight in the analyst's profile if the mouse cursor moves around it frequently. However, our approach can be easily extended to other types of inputs such as gaze tracking, leap motions, etc. Contributions. In this paper, we make the following contributions: • We define and explore the notion of "implicit user feedback" which enables a seamless navigation in spatial data; • We define the notion of "information highlighting", a mechanism to highlight out-of-sight important information for analysts. A clear distinction of our proposal with the literature is that it doesn't aim for pruning (such as top-k recommendation), but leveraging the actual data with potential interesting results (i.e., highlights); • We define and formalize the concept of Interesting Dense Regions (IDRs), a polygon-based approach to explore and highlight spatial data; • We propose an efficient greedy approach to compute highlights on-the-fly; • We show the effectiveness of our approach through a set of qualitative experiments. The outline of the paper is the following. Section 2 describes our data model. In Section 3, we formally define our problem. Then in Section 4, we present our solution and its algorithmic details. Section 5 reports our experiments on the framework. We review the related work in Section 6. We present some limitations of our work in Section 7. Last, we conclude in Section 8. Data Model We consider two different layers on a geographical map: "spatial layer" and "interaction layer". The spatial layer contains points from a spatial database P. The interaction layer contains mouse move points M. Spatial layer. Each point p ∈ P is described using its coordinates, latitude and longitude, i.e., p = lat, lon . Note that in this work, we don't consider "time" for spatial points, as our contribution focuses on their location. Points are also associated to a set of domain-specific attributes A. For instance, for a dataset of a real estate agency, points are properties (houses and apartments) and A contains attributes such as "surface", "number of pieces" and "price". The set of all possible values for an attribute a ∈ A is denoted as dom(a). We also define analyst's feedback F as a vector over all attribute values (i.e., facets), i.e., F = − −−−−−−−− → ∪ a∈A dom(a). The vector F is initialized by zeros and will be manipulated to express analyst's preferences. Interaction layer. Whenever the analyst moves her mouse, a new point m is appended to the set M. Each mouse move point is described using the pixel position that it touches and the clock time of the move. Hence each mouse move point is a tuple m = x, y, t , where x and y specifies the pixel location and t is a Unix Epoch time. To conform with geographical standards, we assume m = 0, 0 sits at the middle of the interaction layer, both horizontally and vertically. The analyst is in contact with the interaction layer. To update the feedback vector F , we need to translate pixel locations in the interaction layer to latitudes and longitudes in the spatial layer. While there is no precise transformation from planar to spherical coordinates, we employ equirectangular projection to obtain the best possible approximation. Equation 1 describes this formula to transform a point m = x, y, t in the interaction layer to a point p = lat, lon in the spatial layer. Note that the resulting p is not necessarily a member of P. lon = x cosγ + θ; lat = y + γ(1) The inverse operation, i.e., transforming from the spatial layer to the interaction is done using Equation 2. x = (lon − θ) × cosγ; y = lat − γ(2) The reference point for the transformation is the center of both layers. In Equations 1 and 2, we assume that γ is the latitude and θ is the longitude of a point in the spatial layer corresponding to the center of the interaction layer, i.e., m = 0, 0 . Problem Definition The large size of spatial data hinders its effective analysis for discovering insights. Analysts require to obtain only few options (so-called "highlights") to focus on. These options should be in-line with what they have already appreciated. In this paper, we formulate the problem of "information highlighting using implicit feedback", i.e., highlight few spatial points based on implicit interests of the analyst in order to guide her towards what she should concentrate on in consecutive iterations of the analysis process. We formally define our problem as follows. Problem. Given a time t c and an integer constant k, obtain an updated feedback vector F using points m ∈ M where m.t ≤ t c and choose k points P k ⊆ P as "highlights" where P k satisfies two following constraints. ∀p ∈ P k , similarity(p, F ) is maximized. diversity(P k ) is maximized. The first constraint guarantees that returned highlights are highly similar with analyst's interests captured in F . The second constraint ensures that k points cover different regions and they don't repeat themselves. While our approach is independent from the way that Algorithm 1: Spatial Highlighting Algorithm Input: Current time t c , mouse move points M Output: Highlights P k 1 S ← find interesting dense regions(t c , M) 2 P s ← match points(S, P) 3 F ← update feedback vector (F, P s ) 4 P k ← get highlights(P, F ) 5 return P k similarity and diversity functions are formulated, we provide a formal definition of these functions in Section 4. The aforementioned problem is hard to solve due to the following challenges. Challenge 2. Even if an oracle provides a mapping between mouse moves and the feedback vector, analyzing all generated mouse moves is challenging and may introduce false positives. A typical mouse with 1600 DPI (Dots Per Inch), touches 630 pixels for one centimeter of move. Hence a mouse move from the bottom to the top of a typical 13-inch screen would provide 14,427 points which may not be necessarily meaningful. Challenge 3. Beyond two first challenges, finding the most similar and diverse points with F needs an exhaustive scan of all points in P which is prohibitively expensive: in most spatial datasets, there exist millions of points. Moreover, we need to follow multi-objective considerations as we aim to optimize both similarity and diversity at the same time. We recognize the complexity of our problem using the aforementioned challenges. In Section 4, we discuss a solution for the discussed problem and its associated challenges. Interesting Dense Regions Our approach exploits analyst's implicit feedback (i.e., mouse moves) to highlight few interesting points as future analysis directions. Algorithm 1 summarizes the principled steps of our approach. The algorithm begins by mining the set of mouse move points M in the interaction layer to discover one or several Interesting Dense Regions, abbr., IDRs, in which most analyst's interactions occur (line 1). Then it matches the spatial points P with IDRs using Equation 2 in order to find points inside each region (line 2). The attributes of resulting points will be exploited to update the analyst's feedback vector F (line 3). The updated vector F will then be used to find k highlights (line 4). These steps ensure that the final highlights reflect analyst's implicit interests. We detail each step as follows. Discovering IDRs The objective of this step is to obtain one or several regions in which the analyst has expressed her implicit feedback. There are two observations for such regions. Observation 1. We believe that a region appeals more interesting to the analyst if it is denser, i.e., the analyst moves her mouse in that region several times. Observation 2. It is possible that the analyst moves her mouse everywhere in the map. This should not signify that everywhere in the map has the same significance. Following our observations, we propose Algorithm 2 for mining IDRs. We add points to M only every 200ms to prevent adding redundant points (i.e., Challenge 2). Following Observation 1 and in order to mine the recurring behavior of the analyst, the algorithm begins by partitioning the set M into g fixed-length consecutive segments M 0 to M g . The first segment starts at time zero (where the system started), and the last segment ends at t c , i.e., the current time. Following Observation 2, we then find dense clusters in each segment of M using a variant of DB-SCAN approach [14]. Finally, we return intersections among those clusters as IDRs. For clustering points in each time segment (i.e., line 5 of Algorithm 2), we use ST-DBSCAN [15], a spaceaware variant of DB-SCAN for clustering points based on density. For each subset of mouse move points M i , i ∈ [0, g], ST-DBSCAN begins with a random point m 0 ∈ M i and collects all density-reachable points from m 0 using a distance metric. As mouse move points are in the 2-dimensional pixel space (i.e., the display), we choose euclidean distance as the distance metric. If m 0 turns out to be a core object, a cluster will be generated. Otherwise, if m 0 is a border object, no point is density-reachable from m 0 and the algorithm picks another random point in M i . The process is repeated until all of the points have been processed. Once clusters are obtained for all subsets of M, we find their intersections to locate recurring regions (line 6). To obtain intersections, we need to clearly define the spatial boundaries of each cluster. Hence for each cluster, we discover its corresponding polygon that covers the points inside. For this aim, we employ Quickhull algorithm, a quicksort-style method which computes the convex hull for a given set of points in a 2D plane [16]. We describe the process of finding IDRs in an example. Figure 1 shows the steps that Benício follows in our running example to explore home-stays in Paris. Figure 1.A shows mouse movements of Benício in different time stages. In this example, we consider g = 3 and capture Benício's feedback in three different time segments (progressing from Figures 1.B to 1.D). It shows that Benício started his search around Eiffel Tower and Arc de Triomphe (Figure 1.B) and gradually showed interest in south (Figure 1.C) and north (Figure 1.D) as well. All intersections between those clusters are discovered (hatching Algorithm 2: Find Interesting Dense Regions (IDRs) Input: Current time t c , mouse move points M Output: IDRs S 1 S ← ∅ 2 g ←number of time segments Figure 1.E) which will constitute the set of IDRs (Figure 1.F), i.e., IDR1 to IDR4. 3 for i ∈ [0, g] do 4 M i ← {m = x, y, t \|( tc g × i) ≤ t ≤ ( tc g × (i + 1))} 5 C i ← mine clusters(M i ) 6 O i ← find ploygons(C i ) 7 end 8 for O i , O j where i, j ∈ [0, g] and i = j do S.append (intersect(O i , O j )) 9 return S regions in Matching Points Being a function of mouse move points, IDRs are discovered in the interaction layer. We then need to find out which points in P fall into IDRs, hence forming the subset P s . We employ Equation 2 to transform those points from the spatial layer to the interaction layer. Then a simple "spatial containment" function can verify which points fit into the IDRs. Given a point p and an IDR r, a function contains(p, r) returns "true" if p is inside r, otherwise "false". In our case, we simply use the implementation of ST Within(p, r) module in PostGIS 5 , i.e., our underlying spatial DBMS which hosts the data. In the vanilla version of our spatial containment function, all points should be checked against all IDRs. Obviously, this depletes the execution time. To prevent the exhaustive scan, we employ Quadtrees [17] in a two-step approach. In an offline process, we build a Quadtree index for all points in P. We record the membership relations of points and cells in the index. When IDRs are discovered, we record which cells in the Quadtree index intersect with IDRs. As we often end up with few IDRs, the intersection verification performs fast. Then for matching points, we only check a subset which is inside the cells associated to IDRs and ignore the points outside. This leads to a drastic pruning of points in P. We follow our running example and illustrate the matching process in Figure 2. In the Airbnb dataset, points are home-stays which are shown with their nightly price on the map. We observe that there exist many matching points with IDR3 and absolutely no matching point for IDR2. For IDR4, although there exist many home-stays below the region, we never check their containment, as they belong to a Quadtree cell which doesn't intersect with the IDR. Updating Analyst Feedback Vector The set of matching points P s (line 2 of Algorithm 1) depicts the implicit preference of the analyst. We keep track of this preference in a feedback vector F . The vector is initialized by zero, i.e., the analyst has no preference at the beginning. We update F using the attributes of the points in P s . We consider an increment value δ to update F . If p ∈ P s gets v 1 for attribute a 1 , we augment the value in the F 's cell of a 1 , v 1 by δ. Note that we only consider incremental feedback, i.e., we never decrease a value in F . We explain the process of updating the feedback vector using a toy example. Given the four matched points in IDR1 (Figure 2) with prices 130e, 58e, 92e and 67e, we want to update the vector F given those points. Few attributes of these points are mentioned in Table 1. In practice, there are often more than 50 attributes for points. The cells of F are illustrated in the first column of Table 2. As three points get the value "1" for the attribute "#Beds", then the value in cell #Beds,1 is augmented three times by δ. The same process is repeated for all attribute-values of points in P s . Note that all cells of F are not necessarily touched in the feedback update process. For instance, in the above example, 5 cells out of 12 remain unchanged. By specifying an increment value, we can materialize the updates and normalize the vector using a Softmax function. We always normalize F in a way that all cell values sum up to 1.0. Given δ = 1.0, the normalized values of the F vector is illustrated in the third column of Table 2. Higher values of δ increase the influence of feedbacks. The normalized content of the vector F captures the implicit preferences of the analyst. For instance, the content of F after applying points in IDR1 shows that the analyst has a high interest in having a balcony in her home-stay, as her score for the cell Balcony,Yes is 0.25, i.e., the highest among other cells. This reflects the reality as all points in IDR1 has balcony. Note that although we only consider positive feedback, the Softmax function lowers the values of untouched cells once other cells get rewarded. An important consideration in interpreting the vector F is that the value "0" does not mean the lowest preference, but irrelevance. For instance, consider the cell Rating,2 in Table 2. The value "0" for this cell shows that the analyst has never expressed her implicit feedback on this facet. It is possible that in future iterations, the analyst shows interest in a 2-star home-stay (potentially thanks to its price), hence this cell gets a value greater than zero. However, cells with lower preferences are identifiable with non-zero values tending to zero. For instance, the value 0.06 for the cell Rating,4 shows a lower preference towards 4-star home-stays compared to the ones with 5 stars, as only one point in P s is rated 4 in IDR1. Generating Highlights The ultimate goal is to highlight k points to guide analysts in analyzing their spatial data. The updated feedback vector F is the input to the highlighting phase. We assume that points in IDRs are already investigated by the analyst. Hence our search space for highlighting is P − P s . We seek two properties in k highlights: similarity and diversity. First, highlights should be in the same direction of the analyst's implicit feedback, hence similar to the vector F . The similarity between a point p ∈ P and the vector F is defined as follows. similarity(p, F ) = avg a∈A (sim(p, F , a)) The sim() function can be any function such as Jaccard or Cosine. Each attribute can have its own similarity function (as string and integer attributes are compared differently.) Then sim() works as an overriding-function which provides encapsulated similarity computations for any type of attribute. Second, highlighted points should also represent distinct directions so that the analyst can observe different aspects of data and decide based on the big picture. Given a set of points P k = {p 1 , p 2 . . . p k } ⊆ P, we define diversity as follows. diversity(P k ) = avg {p,p }⊂P k \|p =p distance(p, p ) (4) The function distance(p, p ) operates on geographical coordinates of p and p and can be considered as any distance function of Minkowski distance family. However, as distance computations are done in the spherical space, a natural choice is to employ Haversine distance shown in Equation 5. Our application of diversity on geographical points differs from those of [18], because we consider geographical distance as the basis to calculate diversity between two points. distance(p, p ) = acos(cos(p.lat) × cos(p .lat) × cos(p.lon)) × cos(p .lon) + cos(p.lat) × sin(p .lat) × cos(p.lon) × sin(p .lon) + sin(p.lat) × sin(p .lat)) × earth radius (5) Algorithm 3 describes our approach for highlighting k similar and diverse points. We propose a besteffort greedy approach to efficiently compute highlighted points. We consider an offline step followed by the online execution of our algorithm. In order to speed up the similarity computation in the online execution, we pre-compute an inverted index for each single point p ∈ P in the offline step (as is commonly done in the Web search). Each index L p for the point p keeps all other points in P in decreasing order of their similarity with p. The first step of Algorithm 3 is to find the most similar point to F , so-called p * . The point p * is the closest possible approximation of F in order to exploit pre-computed similarities. The algorithm makes sequential accesses to L p * (i.e., the inverted index of the point p * ) to greedily maximize diversity. Algorithm 3 does not sacrifice efficiency in price of value. We consider a time limit parameter which determines when the algorithm should stop seeking maximized diversity. Scanning inverted indexes guarantees the similarity maximization even if time limit is chosen to be very restrictive. Our observations with several spatial datasets show that we achieve the diversity of more than 0.9 with time limit set to 200ms. In line 2 of Algorithm 3, P k is initialized with the k highest ranking points in L p * . Function get next(L p * ) (line 3) returns the next point p next in L p * in sequential order ( as a common practice in information retrieval). Lines 4 to 12 iterate over the inverted indexes to determine if other points should be considered to increase diversity while staying within the time limit. Algorithm 3: Get k similar and diverse highlights get highlights() Input: Points P, Feedback vector F , k, time limit Output: P k 1 p * ← max sim to(P, F ) 2 P k ← top k (L p * , k) 3 p next ← get next(L p * ) 4 while time limit not exceeded do 5 for p current ∈ P k do 6 if diversity improved (P k , p next , p current ) then 7 P k ← replace(P k , p next , p current ) 8 break 9 end 10 end 11 p next ← get next(L p * ) 12 end 13 return P k The algorithm looks for a candidate point p current ∈ P k to replace in order to increase diversity. The boolean function diversity improved () (line 6) checks if by replacing p current by p next in P k , the overall diversity of the new P k increases. It is important to highlight that for each run of the algorithm, we only focus on one specific inverted list associated to the input point. Algorithm 3 verifies the similarity and diversity of each point with all other points, and then processes the normalization. Experiments We discuss two sets of experiments. Our first set is on the usefulness of our approach. Then we focus more on discovering IDRs and present few statistics and insights for them. The experiments are done on the a computer with Mac OS operating system, with a 2,8 GHz Intel Core i5. First off, we validate the "usefulness" of our approach. For this aim, we design a user study with some participants who are all students of Computer Science. Some of them are "novice" users who don't know the location under investigation, and some are "experts." Participants should fulfill a task. For each participant, we report a variant of time-to-insight measure, i.e., how long the participants interact with the tool before fulfilling the task. Evidently, less number of interactions are preferred as it means that the participant can reach insights faster. On the Airbnb 6 dataset of Paris with 1,000 points, we define two different types of tasks: T1: "finding a point in a requested location" (e.g., find a home-stay in the "Champ de Mars" area), and T2: "finding a point with a requested profile" (e.g., find a cheap home-stay.) Due to the vagueness associated to these tasks, participants require to go through an exploratory analysis session. Moreover, participants may also begin their navigation 14 either from I1: "close to the goal" or I2: "far from the goal". Table 3 shows the results. We observe that on average it takes 2.067 seconds to achieve defined goals. This shows that implicit feedback capturing is an effective mechanism which helps analysts to reach their goals in a reasonable time. Expert participants need 0.35 seconds less time on average. Interestingly, starting points, i.e., I1 and I2, do not have a huge impact on number of steps. It is potentially due to the diversity component which provides distinct options and can quickly guide analyst towards their region of interest. We also observe that the task T2 is an easier task than T1, as on average it took less to be accomplished. This is potentially due to where the analyst can request options similar to what she has already observed and greedily move to her preferred regions. In the second part of our experiments, we employ two different datasets, i.e., Airbnb and Yelp 7 . We pick a similar subset from both datasets, i.e., home-stays and restaurants in Paris city, respectively. We consider four different sizes of those datasets, i.e., 100, 1000, 2000 and 4000 points, respectively. For each size of the datasets, we manually perform 20 sessions, and then we present the results as the average of sessions. We limit each session to 2 minutes where we seek for interesting points in the datasets. We capture the following information in each session: • The number of regions created from the mouse moves during the session; • The number of generated IDRs (intersection of regions); • The number of points from the dataset presented in each IDR; • The coverage of points (in the dataset) with IDRs collectively. Tables 4 and 5 show the result for Airbnb and Yelp, respectively. In Table 4, we observe that the number of regions decreases when the number of points increases. On average, 10 regions are constructed per session. The average number of points presented in IDRs is 25.97, which shows that our approach highlights at least 8.05% of points from the dataset, on average. We notice an outlier in the experiment with 2000 points in Tables 4. This happened due the fact that the analyst concentrated in a very small area generating a smaller number of IDRs, and consequently a smaller number of points. More uniform results are observed in Table 5, i.e., for Yelp dataset vis-à-vis Airbnb. The average number of generated regions reaches 12.75 per session. Also, the number of regions decreases by increasing the number of points. The same happens for IDRs, where we obtain an average of 8.9 IDRs generated per session. The number of points presented in IDRs is on average 108.65 and it represents on average 13.11% of points highlighted from the dataset. Related Work To the best of our knowledge, the problem of spatial information highlighting using implicit feedback has not been addressed before in the literature. However, our work relates to few others in their semantics. Information Highlighting. The literature contains few instances of information highlighting approaches [19,20,21,22]. However, all these methods are objective, i.e., they assume that analyst's preferences are given as a constant input and will never change in the future. This limits their functionality for serving scenarios of exploratory analysis. The only way to fulfill "spatial guidance" is to consider the evolutionary and subjective nature of analyst's feedback. In our approach, the feedback vector gets updated in time based on the implicit feedback of the analyst. Online recommendation approaches can also be considered as an information highlighting approach where recommended items count as highlights. Most recommendation algorithms are space-agnostic and do not take into account the spatial information. While few approaches focus on the spatial dimension [23,24,18], they still lack the evolutionary feedback capturing. Moreover, most recommendation methods miss "result diversification", i.e., highlights may not be effective due to overlaps. Feedback Capturing. Several approaches are proposed in the state of the art for capturing different forms of feedback [8,7,25,9,10,26]. The common approach is a top-k processing methodology in order to prune the search space based on the explicit feedback of the analyst and return a small subset of interesting results of size k. A clear distinction of our proposal is that it doesn't aim for pruning, but leveraging the actual data with potential interesting results (i.e., highlights) that the analyst may miss due to the huge volume of spatial data. Moreover, in a typical top-k processing algorithm, analyst's choices are limited to k. On the contrary, our IDR approach enables a freedom of choice where highlights get seamlessly updated with new analyst's choices. Few works formulate fusing approaches of explicit and implicit feedbacks to better capture user preferences [27,5,6]. Our approach functions purely on implicit feedback and does not require any sort of explicit signal from the analyst. Region Discovery. Our approach finds interesting dense regions (IDRs) in order to derive analyst's implicit preferences. There exist several approaches to infer a spatial region for a given set of points [28,29,30,31,32,16]. The common approach is to cluster points in form of concave and convex polygons. In [28], an algorithm is proposed to verify if a given point p on the surface of a sphere is located inside, outside, or along the border of an arbitrary spherical polygon. In [29,30], a non-convex polygon is constructed from a set of input points on a plane. In [31,32], imprecise regions are delineated into a convex or concave polygon. In our approach, it is important to discover regions by capturing mouse move points. In case a concave polygon is constructed, the "dents" of such a polygon may entail points which are not necessarily in M. In the IDR's algorithm, however, we adapt Quickhull [16], due its simplicity, efficiency and its natural implementation of convex polygons. Limitations In this paper, we presented a solution for highlighting out-of-sight information using a polygon-based approach for capturing implicit feedbacks. To the best of our knowledge, our work is the first effort towards formalizing and implementing information highlighting using implicit feedback. However, we consider our work as an on-going effort where we envision to address some limitations in the future, such as "customizability", "performance", "cold start", and "quantitative experiments". In this section we present some limitations of our proposed work, describing what we will consider as future work. One limitation is about the "customizable" use of geographical maps as an interaction means. As we only consider static maps, we plan to work on translations and rotations as a future work. Another gap that we envision to work on is performance. We plan to run an extensive performance study to detect bottlenecks of our approach. Our problem bears similarities with recommendation algorithms where the quality of the output may be influenced by scarce availability of input. This problem is referred to as the cold start problem [33]. While there is no guarantee for a meaningful highlight in case of the complete absence of implicit feedbacks, our approach can return a reasonable set of highlights even with one single iteration of mouse moves. In the future, we envision to tackle the no-input challenge by leveraging statistical properties of the spatial data to obtain a default view for highlights. Another limitation is the medium-size datasets to be processed. Our algorithm processes similarity and diversity in an O(n 2 ) complexity. Also Quickhull [16] uses a divide and conquer approach similar to that of Quicksort, and its worst complexity is O(n 2 ). While processing a 10K-point dataset is straightforward in our framework, we plan to experiment with larger datasets in the future by improving our algorithms towards better performance. Another direction for future work is to consider experiments which measure the quantitative and qualitative influence of each component separately. Conclusion and Future Work In this paper, we present an approach to explore Interesting Dense Regions (IDRs) using implicit feedback in order to detect analyst latent preferences. The implicit feedbacks are captured from mouse moves of analysts over the geographical map while analyzing spatial data. We formalize a novel polygon-based mining algorithm which returns few highlights in-line with analyst's implicit preferences. The highlights enable analysts to focus on what matters the most and prevent information overload. We consider various future directions for this work. First, we are interested to incorporate an "explainability" component which can describe causalities behind preferences. For instance, we are interested to find seasonal patterns to see why the preferences of analysts change from place to place during various seasons of the year. Another direction is to incorporate "Query by Visualization" approaches, where analysts can specify their intents alongside their implicit preferences, directly on the map [34]. Challenge 1 . 1First, it is not clear how mouse move points influence the feedback vector. Mouse moves occur on a separate layer and there should be some meaningful transformations to interpret mouse moves as potential changes in the feedback vector. 5Figure 1 :Figure 2 : 12https://postgis.net/docs/manual-dev/ST Within.html The process of finding IDRs on Airbnb dataset. Matching points for IDR1 to IDR4. Table 1 : 1Attributes of points in IDR1. ID Price #Bed Balcony Air-con. Rating1 130e 1 Yes Yes 5/5 2 58e 1 Yes No 5/5 3 92e 2 Yes No 5/5 4 67e 1 Yes No 4/5 Table 2 : 2Updating Analyst Feedback Vector Attribute-value Applying IDR 1 Normalized #Beds,1 +3δ 0.19 #Beds,2 +δ 0.06 #Beds,+2 (no update) 0.00 Balcony,Yes +4δ 0.25 Balcony,No (no update) 0.00 Air-cond.,Yes +δ 0.06 Air-cond.,No +3δ 0.19 Rating,1 (no update) 0.00 Rating,2 (no update) 0.00 Rating,3 (no update) 0.00 Rating,4 +δ 0.06 Rating,5 +3δ 0.19 Table 3 : 3Interactions of "novice" and "expert" partici- pants (in seconds) T1/I1 T2/I1 T1/I2 T2/I2 Novices 1.99 2.38 2.00 2.48 Experts 1.72 2.09 1.70 2. 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[ "Motion Artifacts Correction from Single-Channel EEG and fNIRS Signals using Novel Wavelet Packet Decomposition in Combination with Canonical Correlation Analysis", "Motion Artifacts Correction from Single-Channel EEG and fNIRS Signals using Novel Wavelet Packet Decomposition in Combination with Canonical Correlation Analysis" ]	[ "MdShafayet Hossain \nDepartment of Electrical\nElectronic and Systems Engineering\nUniversiti Kebangsaan Malaysia\n43600BangiSelangorMalaysia\n", "Muhammad E H Chowdhury \nDepartment of Electrical Engineering\nQatar University\n2713DohaQatar\n", "Mamun Bin ", "Ibne Reaz \nDepartment of Electrical\nElectronic and Systems Engineering\nUniversiti Kebangsaan Malaysia\n43600BangiSelangorMalaysia\n", "Sawal H M Ali \nDepartment of Electrical\nElectronic and Systems Engineering\nUniversiti Kebangsaan Malaysia\n43600BangiSelangorMalaysia\n", "Ahmad Ashrif ", "A Bakar \nDepartment of Electrical\nElectronic and Systems Engineering\nUniversiti Kebangsaan Malaysia\n43600BangiSelangorMalaysia\n", "Serkan Kiranyaz \nDepartment of Electrical Engineering\nQatar University\n2713DohaQatar\n", "Amith Khandaker \nDepartment of Electrical Engineering\nQatar University\n2713DohaQatar\n", "Mohammed Alhatou \nNeuromuscular Division\nDepartment of Neurology\nHamad General Hospital\nAlkhor Hospital\n3050DohaQatar\n", "Rumana Habib \nDepartment of Neurology\nBIRDEM General Hospital\nDhaka-1000Bangladesh\n", "Muhammad Maqsud Hossain \nNSU Genome Research Institute (NGRI)\nNorth South University\nDhaka-1229Bangladesh\n" ]	[ "Department of Electrical\nElectronic and Systems Engineering\nUniversiti Kebangsaan Malaysia\n43600BangiSelangorMalaysia", "Department of Electrical Engineering\nQatar University\n2713DohaQatar", "Department of Electrical\nElectronic and Systems Engineering\nUniversiti Kebangsaan Malaysia\n43600BangiSelangorMalaysia", "Department of Electrical\nElectronic and Systems Engineering\nUniversiti Kebangsaan Malaysia\n43600BangiSelangorMalaysia", "Department of Electrical\nElectronic and Systems Engineering\nUniversiti Kebangsaan Malaysia\n43600BangiSelangorMalaysia", "Department of Electrical Engineering\nQatar University\n2713DohaQatar", "Department of Electrical Engineering\nQatar University\n2713DohaQatar", "Neuromuscular Division\nDepartment of Neurology\nHamad General Hospital\nAlkhor Hospital\n3050DohaQatar", "Department of Neurology\nBIRDEM General Hospital\nDhaka-1000Bangladesh", "NSU Genome Research Institute (NGRI)\nNorth South University\nDhaka-1229Bangladesh" ]	[]	The electroencephalogram (EEG) and functional near-infrared spectroscopy (fNIRS) signals, highly non-stationary in nature, greatly suffers from motion artifacts while recorded using wearable sensors. Since successful detection of various neurological and neuromuscular disorders is greatly dependent upon clean EEG and fNIRS signals, it is a matter of utmost importance to remove/reduce motion artifacts from EEG and fNIRS signals using reliable and robust methods. In this regard, this paper proposes two robust methods: i) Wavelet packet decomposition (WPD), and ii) WPD in combination with canonical correlation analysis (WPD-CCA), for motion artifact correction from single-channel EEG and fNIRS signals. The efficacy of these proposed techniques is tested using a benchmark dataset and the performance of the proposed methods is measured using two well-established performance matrices: i) Difference in the signal to noise ratio ( ) and ii) Percentage reduction in motion artifacts ( ). The proposed WPDbased single-stage motion artifacts correction technique produces the highest average (29.44 dB) when db2 wavelet packet is incorporated whereas the greatest average (53.48%) is obtained using db1 wavelet packet for all the available 23 EEG recordings. Our proposed two-stage motion artifacts correction technique i.e. the WPD-CCA method utilizing db1 wavelet packet has shown the best denoising performance producing an average and values of 30.76 dB and 59.51%, respectively for all the EEG recordings. On the other hand, for the available 16 fNIRS recordings, the two-stage motion artifacts removal technique i.e. WPD-CCA has produced the best average (16.55 dB, utilizing db1 wavelet packet) and largest average (41.40%, using fk8 wavelet packet). The highest average and using single-stage artifacts removal techniques (WPD) are found as 16.11 dB and 26.40%, respectively for all the fNIRS signals using fk4 wavelet packet. In both EEG and fNIRS modalities, the percentage reduction in motion artifacts increases by 11.28% and 56.82%, respectively when two-stage WPD-CCA techniques are employed in comparison with the single-stage WPD method. In addition, the average also increases when WPD-CCA techniques are used instead of singlestage WPD for both EEG and fNIRS signals. The increment in both and values is a clear indication that two-stage WPD-CCA performs relatively better compared to single-stage WPD. The results reported using the proposed methods outperform most of the existing state-of-the-art techniques.	10.3390/s22093169	[ "https://arxiv.org/pdf/2204.04533v1.pdf" ]	248,085,675	2204.04533	93200d070409d0249c68bdabe13700d5656de582	Motion Artifacts Correction from Single-Channel EEG and fNIRS Signals using Novel Wavelet Packet Decomposition in Combination with Canonical Correlation Analysis MdShafayet Hossain Department of Electrical Electronic and Systems Engineering Universiti Kebangsaan Malaysia 43600BangiSelangorMalaysia Muhammad E H Chowdhury Department of Electrical Engineering Qatar University 2713DohaQatar Mamun Bin Ibne Reaz Department of Electrical Electronic and Systems Engineering Universiti Kebangsaan Malaysia 43600BangiSelangorMalaysia Sawal H M Ali Department of Electrical Electronic and Systems Engineering Universiti Kebangsaan Malaysia 43600BangiSelangorMalaysia Ahmad Ashrif A Bakar Department of Electrical Electronic and Systems Engineering Universiti Kebangsaan Malaysia 43600BangiSelangorMalaysia Serkan Kiranyaz Department of Electrical Engineering Qatar University 2713DohaQatar Amith Khandaker Department of Electrical Engineering Qatar University 2713DohaQatar Mohammed Alhatou Neuromuscular Division Department of Neurology Hamad General Hospital Alkhor Hospital 3050DohaQatar Rumana Habib Department of Neurology BIRDEM General Hospital Dhaka-1000Bangladesh Muhammad Maqsud Hossain NSU Genome Research Institute (NGRI) North South University Dhaka-1229Bangladesh Motion Artifacts Correction from Single-Channel EEG and fNIRS Signals using Novel Wavelet Packet Decomposition in Combination with Canonical Correlation Analysis Article Correspondence: Muhammad E.H. Chowdhury ([email protected]); Mamun Bin Ibne Reaz ([email protected])Motion artifactElectroencephalogram (EEG)Functional near-infrared spectroscopy (fNIRS)Wavelet packet decompo- sition (WPD)Canonical correlation analysis (CCA) The electroencephalogram (EEG) and functional near-infrared spectroscopy (fNIRS) signals, highly non-stationary in nature, greatly suffers from motion artifacts while recorded using wearable sensors. Since successful detection of various neurological and neuromuscular disorders is greatly dependent upon clean EEG and fNIRS signals, it is a matter of utmost importance to remove/reduce motion artifacts from EEG and fNIRS signals using reliable and robust methods. In this regard, this paper proposes two robust methods: i) Wavelet packet decomposition (WPD), and ii) WPD in combination with canonical correlation analysis (WPD-CCA), for motion artifact correction from single-channel EEG and fNIRS signals. The efficacy of these proposed techniques is tested using a benchmark dataset and the performance of the proposed methods is measured using two well-established performance matrices: i) Difference in the signal to noise ratio ( ) and ii) Percentage reduction in motion artifacts ( ). The proposed WPDbased single-stage motion artifacts correction technique produces the highest average (29.44 dB) when db2 wavelet packet is incorporated whereas the greatest average (53.48%) is obtained using db1 wavelet packet for all the available 23 EEG recordings. Our proposed two-stage motion artifacts correction technique i.e. the WPD-CCA method utilizing db1 wavelet packet has shown the best denoising performance producing an average and values of 30.76 dB and 59.51%, respectively for all the EEG recordings. On the other hand, for the available 16 fNIRS recordings, the two-stage motion artifacts removal technique i.e. WPD-CCA has produced the best average (16.55 dB, utilizing db1 wavelet packet) and largest average (41.40%, using fk8 wavelet packet). The highest average and using single-stage artifacts removal techniques (WPD) are found as 16.11 dB and 26.40%, respectively for all the fNIRS signals using fk4 wavelet packet. In both EEG and fNIRS modalities, the percentage reduction in motion artifacts increases by 11.28% and 56.82%, respectively when two-stage WPD-CCA techniques are employed in comparison with the single-stage WPD method. In addition, the average also increases when WPD-CCA techniques are used instead of singlestage WPD for both EEG and fNIRS signals. The increment in both and values is a clear indication that two-stage WPD-CCA performs relatively better compared to single-stage WPD. The results reported using the proposed methods outperform most of the existing state-of-the-art techniques. Introduction Due to the paradigm shift of hospital-based treatment in the direction of wearable and ubiquitous monitoring, nowadays, the acquisition and processing of vital physiological signals have become prevalent in the ambulatory setting. Since the acquisition of physiological signals is inclined to movement artifacts that happen due to the deliberate and/or voluntary movement of the patient during signal procurement utilizing wearable devices, restricting patients totally from physical movements, intentional and/or unintentional, is exceptionally troublesome. As a result, the physiological signals may get corrupted to some degree by motion artifacts. In some instances, this defilement may end up so conspicuous that the recorded signals may lose their usability unless the movement artifacts are diminished significantly. Electroencephalogram (EEG) measures the electrical activity of the human brain quantitatively which took place due to the firing of neurons [1] and such brain activity is recorded utilizing a good number of cathodes which are located at different regions of the scalp [2]. EEG is one of the key diagnostic tests for epileptic seizure detection [3,4]. Other decisive utilization of EEG includes the estimation of drowsiness levels [5][6][7][8], emotion detection [9], cognitive workload [6,10], and brain-computer interfaces (BCIs) [11][12][13][14][15][16]. All of which have potential applications in the personal healthcare domain. Lately, the implementation of EEG-based biometric systems utilizing the inborn anti-spoofing capability of EEG signals was studied and appeared to be promising [17]. The functional near-infrared spectroscopy (fNIRS), a non-invasive optical brain imaging technique, measures changes in hemoglobin (Hb) concentrations inside the human brain [18] by employing light of various wavelengths in the infrared band and estimating the difference in the optical absorption [19]. Medical applications of fNIRS mainly focus on the noninvasive measurement of brain functions [20,21], cognitive tasks identification [22,23], and BCI [24][25][26]. Apart from movement artifacts, physiological signals undergo other types of artifacts as well. Gradient artifacts (GA) and pulse artifacts (PA) are the two most frequent artifacts observed in EEG during the simultaneous EEG-fMRI tests [27][28][29]. On the other hand, event-related fNIRS signals are regularly sullied by heartbeat, breath, Mayer waves, etc., as well as extra-cortical physiological clamors from the superficial layers [30]. To reduce motion artifacts from EEG, numerous endeavor was made previously and were summarized in [31,32]. In [33], the performance of motion artifacts correction techniques utilizing discrete wavelet transform (DWT) [34], empirical mode decomposition (EMD) [35], ensemble empirical mode decomposition (EEMD) [36], EMD along with canonical correlation analysis (EMD-CCA), EMD with independent component analysis (EMD-ICA), EEMD with ICA (EEMD-ICA), and EEMD with CCA (EEMD-CCA) were reported. Maddirala and Shaik [37] used singular spectrum analysis (SSA) [38] whereas DWT along with the thresholding technique was utilized in [39]. Gajbhiye et al. [40] employed wavelet-based transform along with the total variation (TV) and weighted TV (WTV) denoising techniques whereas in [41], wavelet domain optimized Savitzky-Golay filter was proposed for the removal of motion artifacts from EEG. . Recently, Hossain et. al. [42] utilized variational mode decomposition (VMD) [43] for the correction of motion artifacts from EEG signals. In the last few decades, multiple motion artifacts removal techniques was proposed [44][45][46] for the removal of motion artifacts from the fNIRS signal. Sweeney et al. [47] used adaptive filter, Kalman Filter, and EEMD-ICA. Scholkmann et al. [48] utilized the moving standard deviation and spline interpolation method whereas in [49] wavelet-based method was proposed. The authors of [33] used DWT, EMD, EEMD, EMD-ICA, EEMD-ICA, EMD-CCA, and EEMD-CCA. In [50], Barker et al. used an autoregressive model-based algorithm whilst kurtosis-based wavelet transform was proposed in [51], and Siddiquee et al. [52] utilized nine-degree of freedom inertia measurement unit (IMU) data to mathematically estimate the movement artifacts in the fNIRS signal using autoregressive exogenous (ARX) input model. A hybrid algorithm was proposed in [53] to filter out the movement artifacts from fNIRS signals where both the spline interpolation method and Savitzky-Golay filtering were employed. Very recently, the two-stage VMD-CCA technique was employed in [42]. The development of robust algorithms that can successfully reduce motion artifacts significantly from EEG and fNIRS data is critical; otherwise, the signals' interpretation could be erroneous by medical doctors and/or machinelearning-based applications. As mentioned earlier, DWT, EMD, EEMD, VMD, DWT-ICA, EMD-ICA, EEMD-ICA, EMD-CCA, EEMD-CCA, VMD-CCA, etc. were the most commonly used methods for the correction of motion artifacts from EEG and fNIRS signals. ICA and CCA can not be used independently for single-channel EEG/fNIRS motion artifacts correction as the input of ICA/CCA algorithms require at least two (or more) channels data whereas DWT, EMD, EEMD, VMD, etc. algorithms suffer from several limitations which are discussed in the discussion section of this paper. Also, there is still room for improvement for and values which can be achieved using other effective novel methods. Therefore, in this paper, two novel motion artifacts removal techniques have been proposed which can eliminate motion artifacts from single-channel EEG and fNIRS signals to a great extent. The first is a single-stage motion artifacts correction technique using the wavelet packet decomposition (WPD) whereas the other novel method is WPD in combination with CCA (WPD-CCA), a two-stage motion artifacts removal technique, as the name suggests. In this extensive study, for the correction of motion artifact from EEG and fNIRS signals using the WPD method, four different wavelet packet families (Daubechies (dbN), Symlets (symN), Coiflets (coifN), Fejer-Korovkin (fkN)) have been used with three different vanishing moments (for each of the wavelet packet) that resulted in a total of 12 different investigations. The wavelet packets used in the WPD method are db1, db2, db3, sym4, sym5, sym6, coif1, coif2, coif3, fk4, fk6, and fk8. To the best of our knowledge, the WPD algorithm has not been used for the removal of motion artifacts from single-channel EEG and fNIRS signals to date. WPD-CCA method is another novel contribution of this research work where Daubechies and Fejer-Korovkin wavelet packet families are utilized. In the WPD-CCA technique, db1, db2, db3, fk4, fk6, and fk8 have been used separately resulting in 6 different investigations to reduce motion artifacts from EEG and fNIRS signals more efficiently. The rest of this paper is organized as follows: Section II discusses the theoretical background of the different algorithms (WPD, CCA, WPD-CCA) investigated here, Section III provides brief information about the EEG and fNIRS benchmark dataset, and experimental methodology. Section IV provides the results of the artifact removal techniques proposed in this work and section VI covers the discussion. Finally, the paper is concluded in section V. Theoretical background 2.1 Wavelet packet decomposition (WPD) Using the WPD technique, signals can be decomposed into a wavelet packet basis at diverse scales [54,55]. Forlevel decomposition, a wavelet packet basis is represented by multiple signals [( − )] ∈ℤ where ∈ ℤ + , ≤ ≤ − . The wavelet packet bases ( ), are produced recursively from the scaling and wavelet functions, ( ) = ( ) and ( ) = ( ) respectively, as follows: 2 ( ) = ∑ ℎ( ) −1 ( − 2 −1 ) (1) 2 +1 ( ) = ∑ ( ) −1 ( − 2 −1 )(2) where ℎ( ) represents lowpass filter and ( ) is the highpass filter defined as [54,56]: ℎ( ) = ⟨ 2 ( ), −1 ( − 2 −1 )⟩ (3) ( ) = ⟨ 2 +1 ( ), −1 ( − 2 −1 )⟩(4) The decomposition of a signal ( ) onto the wavelet basis ( ) at level can be expressed as: ( ) = ∑ ( − 2 ) ,(5) where ( ) signifies the th wavelet coefficient of the packet , at level . Here, ( ) represents the intensity of the localized wavelet ( − 2 ), defined by: ( ) = ⟨ ( ), ( − 2 )⟩(6) Let ( ) represent a recorded EEG/fNIRS signal which can be expressed as the sum of a source signal ( ) and a motion artifact signal ( ) as follows: ( ) = ( ) + ( )(7) In general, the source signal ( ) is assumed to be normally distributed having a mean value equals to zero, ( ) ~ (0, ), where 2 characterizes the variance of ( ) [57]. On the other hand, general assumptions regarding the artifact signal ( ) includes temporal localization, not normally distributed with high local variance. According to [58], ( ), can be represented as the sum of ( ) and ( ) where ( ), ( ), and ( ) are the wavelet coefficients of ( ), ( ), and ( ), respectively: ( ) = ( ) + ( )(8) It is noteworthy to mention that the wavelet coefficients ( ) will be sparse as well as the non-zero coefficients will have a relatively higher magnitude as the variance of ( ) is locally high which would cause an increase in the local variance of the recorded EEG/fNIRS signal ( ). Canonical Correlation Analysis (CCA) CCA [59] is one of the most popular blind source separation methods which has the capability of dissociating multiple mixed or noisy signals. Assuming linear mixing, square mixing, and stationary mixing [60], the CCA technique computes an un-mixing matrix which helps identify the unknown independent components ̂ from a matrix which is a recorded multi-channel signal as follows: − ̂=(9) CCA also estimates the unknown independent components ̂ using Equation 9 utilizing second-order statistics (SOS). CCA forcefully makes the sources to be autocorrelated maximally as well as makes the sources mutually uncorrelated [61]. Let us assume as a linear combination of neighboring samples for an input signal (i.e. ( ) = ( − 1) + ( + 1)) [62]. Consider the linear combinations of the components in and , known as the the canonical variates, = ( − ̅) (10) = ( − ̅)(11) where and represents the weight matrices. CCA computes and in such a way so that the correlation ρ between and will be maximized [62]: ρ = √(12) where and signify the nonsingular within-set covariance matrices and xy represent the between-sets covariance matrix. The maximized ρ is calculated by setting the derivatives of Equation 12 (with respect to and ) equals to zero. −1 −1̂= ρ 2̂ −1 −1̂= ρ 2̂(13) and can then be found out as the eigenvectors of the matrices −1 The components that seem to be artifacts can then be discarded by simply setting the corresponding columns of the ̂ matrix to zero before the signal reconstruction. WPD-CCA: The WPD algorithm can be utilized to decompose a single-channel signal into multi-channel signal X where each column of matrix X represents the detailed and approximated sub-band signals. The total number of generated subband signals would be equal to 2 where j denotes the level, a priori. To estimate the underlying true sources ̂ (Equation 9), these generated sub-band signals can then be used as the multi-channel input signals to the CCA algorithm. After that, the component/s of ̂ which seem to be artifacts can be discarded by making the corresponding columns of the matrix ̂ equal to zero. Bypassing this newly obtained source matrix through the inverse of the un-mixing matrix − , the multi-channel signals ̂ can be obtained. Finally, the cleaner signal ̂ can be produced by simply summing all the columns of the matrix ̂. Methods This section describes the benchmark dataset used, pre-processing, study design, motion component identification, and evaluation metrics. Dataset Description A publicly available PhysioNet dataset [32,33,63] is used in this study that contains "reference ground truth" and motion corrupted signals for both EEG and fNIRS modalities. The details of the data recording procedure for EEG and fNIRS modalities were mentioned in [47]. During the data acquisition, two channels having the same hardware properties were placed on the test subject's scalp at very close proximity (20 mm for EEG modality and 30 mm for fNIRS modality) where the first channel was impacted with motion artifacts for 10-25 seconds at regular 2 minutes interval and the second channel was left untouched and undisturbed for the entire recording period. From the unimpacted channel (2 nd channel), EEG/fNIRS signal was extracted which was free from motion artifacts and referred to as "reference ground truth" signal whereas the impacted channel (1 st channel) provided EEG/fNIRS signal corrupted with motion artifacts. It is worthwhile to mention that both the motion corrupted and "reference ground truth" signals were extracted simultaneously from channels 1 and 2 respectively for approximately 9 minutes for each of the trial/test subjects. Also, the same channels were used to extract EEG/fNIRS data from all of the test subjects. Twenty-three sets of EEG recordings, sampled at 2048 Hz, collected from six patients in four different sessions are available in the database. Each recording consists of one motion corrupted EEG signal and one reference "ground truth" EEG signal. The average correlation coefficient between the reference "ground truth" and motion corrupted EEG signals is very high over the epochs where the motion artifacts are absent and the average correlation coefficient drops significantly during the epochs of motion artifacts [32]. The superimposed reference "ground truth" and motion corrupted EEG signals are illustrated in Figure 1a. fNIRS signals were recorded at two different wavelengths: 690 nm and 830 nm wavelengths. There were 16 sets of fNIRS recordings (9 recordings at 830 nm wavelength and 7 recordings at 690 nm wavelength) in total from 10 test subjects at a sampling frequency of 25 Hz [33,63]. Like EEG recordings, each recording of fNIRS consists of one motion corrupted fNIRS signal and one "reference ground truth" fNIRS signal. The overlaid "reference ground truth" fNIRS signal and motion artifact contaminated fNIRS signal is depicted in Figure 1b. Signal Preprocessing Downsampling: As EEG signals can be partitioned into a few sub-bands, specifically delta (1-4 Hz), theta (4-8 Hz), alpha (8-13 Hz), beta (13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30), and gamma (30-80 Hz) [64], we downsampled all the 23 sets of EEG recordings from 2048 Hz to 256 Hz which guarantees data reliability without losing any vital signal information and morphology. The fNIRS signals were not upsampled/downsampled as the original sampling rate was 25 Hz during acquisition. Power line noise removal: To remove power line noise, a 3 rd order Butterworth notch filter with a center frequency of 50 Hz was utilized to remove 50 Hz and its subsequent harmonics as a pre-processing technique for all the EEG and fNIRS signals. Baseline Drift Correction: Both the EEG and fNIRS signals were found to have significant baseline drift, which is defined as undesired amplitude shifts in the signal that would result in inaccurate results if not corrected. To remove baseline drift from EEG and fNIRS recordings, a polynomial curve fitting method was used to estimate the baseline, which was then subtracted from the recorded raw signal. Study Design The simulations of this work were carried out in a PC with Intel(R) Core (TM) i5-8250U CPU at 1.80GHz which was equipped with 8 GB RAM. In-house-built MATLAB code was written to pre-process the EEG and fNIRS data. The single-stage WPD and two stages WPD-CCA methods were deployed in "MATLAB R2020a, The MathWorks, Inc". In this study, the whole 9 minutes of EEG/fNIRS data of each trial were analyzed at one time using WPD and WPD-CCA methods. As mentioned earlier, WPD generates 2 number of sub-band signals where level, j is userdefined. Choosing j = 3 would produce 8 sub-band components where the probability of getting mixed of motioncorrupted components and artifacts-free signal components would be very high. Also, j=5 would produce 32 sub-band signals which would increase the computational complexity of the algorithm. Hence, In this research work, we have chosen equal to 4 for both EEG and fNIRS recordings that produced 16 sub-band signals/components in total for each of the EEG/fNIRS signals and ensured optimum performance. Again 12 different wavelet packets (db1, db2, db3, sym4, sym5, sym6, coif1, coif2, coif3, fk4, fk6, and fk8) were used in the single-stage motion artifact correction technique i.e. WPD. Among these 12 wavelet packets, 6 wavelet packets (db1, db2, db3, fk4, fk6, and fk8) were used in the WPD-CCA method due to the relatively better performance shown by Daubechies and Fejer-Korovkin wavelet packet families incorporated in the WPD technique. As several wavelet packets were used in this study, in the rest of the manuscript, a subscript is added with WPD to denote the corresponding wavelet packet used. As an example, WPD (db1) would refer to that the db1 wavelet packet is used. With the availability of sub-band signals decomposed using the WPD technique, the artifact components can then be selected and removed. All the remaining sub-band signals can then either be added up to reconstruct a cleaner signal or all the sub-band signals can be fed as inputs to the CCA algorithm to determine the motion corrupted components to enhance the signal quality further. CCA technique needs the number of input channels to be at least two or greater. In this work, single-channel EEG and fNIRS signals have been evaluated for the correction of motion artifacts. Hence, it is required to generate several sub-band signals which would be used as the inputs for the CCA algorithm. Six different WPD-CCA-based (WPD(db1)-CCA, WPD(db2)-CCA, WPD(db3)-CCA, WPD(fk4)-CCA, WPD(fk6)-CCA, and WPD(fk8)-CCA) two-stage artifacts removal technique has been realized for both single-channel EEG and fNIRS signals. Removal of Motion Artifact Components using "Reference Ground Truth" Method: A common challenge in eliminating motion artifacts utilizing the aforementioned artifact removal approaches is consistently identifying and removing the motion corrupted components from the signal of interest and reconstructing a cleaner signal. The available reference "ground truth" signal of EEG and fNIRS modalities were used to identify the motion corrupted components as well as test the efficacy of the proposed algorithms. If a component of the decomposed signal is removed and the signal is rebuilt using the other components, the correlation coefficient between the newly reconstructed signal and the ground truth signal will only rise if the removed component has motion artifacts. Using this basic yet efficient notion, motion artifact-affected components of the decomposed signal were discovered and discarded to reconstruct a cleaner signal, ensuring the best performance of each suggested technique during evaluation. Figure 3a shows an example motion corrupted EEG signal and below Figure 3b represents the corresponding 16 sub-band components generated from that corresponding EEG signal using WPD(sym4) algorithm. Figure 4a depicts an example motion corrupted EEG signal and Figure 4b represents the resultant 16 CCA components where the input of the CCA method was 16 sub-band signals generated from the motion corrupted EEG signal using WPD(coif1). Similarly, Figure 5a and Figure 6a show 2 separate motion corrupted fNIRS signals whereas Figure 5b, and Figure 6b represent the sub-band signals generated from WPD(db1), and 16 output CCA components where the input of the CCA algorithm was 16 sub-band signals generated from the motion corrupted EEG signal using WPD(fk8), respectively. From visual inspection of the components generated from the single-stage (WPD) and two-stage (WPD-CCA) motion artifacts removal techniques, it can be stated that in most of the cases, motion artifacts components are usually found in one or two approximation sub-band/CCA components. Although this was the case for most of the EEG and fNIRS recordings, rather than blindly discarding these one or two sub-band/CCA components as motion artifact components, only those components were discarded, when removed, improved the correlation coefficient of the reconstructed signal in comparison with the available reference "ground truth" signal. Performance Metrics: The efficacy and performance of each proposed artifact removal approach can be computed using the provided reference "ground truth" signal for each modality, as detailed before. Since the objective of each proposed technique is to reduce artifacts from the motion-artifact contaminated signal, calculating SNR and percentage reduction in motion artifacts can assess the efficacy of that corresponding technique's capacity to remove artifacts. Hence, the difference in SNR before and after artifact removal (∆SNR), and the improvement in correlation between motion corrupted and reference "ground truth" signals, expressed by the percentage reduction in motion artifact [33], are utilized as performance metrics. For the calculation of ∆SNR, the following formula is used which was given in [33]: where 2 , 2 , and 2 represent the variance of the reference "ground truth", motion corrupted signal, and cleaned signal, respectively. To calculate the percentage reduction in motion artifact , the following formula is used [33]: = 100 (1 − − − )(15) Where is the correlation coefficient between the reference "ground truth" and motion-corrupted signals. The correlation coefficient between the reference "ground truth" and the cleaned signals is denoted by whereas is the correlation between reference "ground truth" and motion corrupted signals over the epochs where motion artifact is absent. In this study, we considered = 1 as in an ideal situation, the "reference ground truth" and the motion corrupted signal over the artifacts-free epochs would always be completely correlated. Hence, the following equation was used to estimate : = 100 (1 − 1− 1− )(16) Results and Discussion The results obtained in this work, using the various novel artifact removal techniques are mentioned below where the performance metrics were calculated using Equations 14 and 16. Motion Artifact Correction from EEG data: All the algorithms (18 in total) were applied on all the 23 recordings of EEG. Figure The best average ∆SNR was found to be 30.76 dB when WPD(db1)-CCA technique was applied over all the EEG records. The highest average percentage reduction in artifact was also provided by the same algorithm which is 59.51% among these 6 single-channel motion artifact correction techniques for EEG modality. Motion Artifact Correction from fNIRS data: All the algorithms (18 in total) were applied on all the 16 recordings of the fNIRS modality. Figure 9a, Figure 9b, Table 1 denote the corresponding standard deviations. Overall, An increase of 11.28% in the average percentage reduction in motion artifacts was found while the best-performing two-stage WPD(db1)-CCA was incorporated compared to the bestperforming single-stage motion artifact correction technique namely WPD(db1). Also, the average ∆SNR value improved by 4.48% (from 29.44 dB to 30.76 dB) while the best performing two-stage WPD(db1)-CCA technique was utilized instead of the best-performing single-stage WPD(db2) method for the correction of motion artifacts from single-channel EEG recordings. From Table 1, the cleaner fNIRS signals reconstructed using WPD(fk4) technique provided the highest average value CCA was incorporated compared to the best performing single-stage motion artifact correction technique namely WPD(fk4). Also, an increase of 2.73% in ∆SNR value was found when best performing two-stage WPD(db1)-CCA was employed instead of the best-performing single-stage WPD(fk4) technique. From Table 1, it is clear that two-stage artifacts correction techniques performed relatively better compared to the single-stage artifacts correction approaches for both EEG and fNIRS modalities. Authors of [37] found that no brain activity was registered in trials 12 and 15. Moreover, they found a poor correlation coefficient over the clean epochs of the recordings of 12 and 15, and hence, they carried out their investigation on the remaining 21 recordings of EEG. We have also observed a similar situation in this work. Trials 12 and 15 consistently produced very bad performance metrics (∆SNR and η values) while both single-stage and twostage artifact reduction techniques were applied proposed in this paper. The authors of this work are presenting a 2 nd table (Error! Reference source not found.) that illustrates the average ∆SNR and average percent reduction in motion artifacts using WPD(db1), WPD(sym4), WPD(coif1), and WPD(fk4). This time the faulty trials (trials 12 and 15) were excluded and the experiments were conducted on the remaining 21 sets of EEG recordings. The motion corrupted signal was decomposed into 16 sub-band components using WPD and then the cleaner signals were generated by simply discarding the lowest-frequency approximation sub-band component (S15, Figure 7) and adding the remaining 15 sub-band components directly. During this process, the reference ground truth signal was only used to compute the performance metrics. without the availability of "reference ground truth signal", correction of motion artifacts from EEG signal is still possible. The similar approach can be used for motion artifacts correction from fNIRS signals also, but left as a future work. Discussion In this extensive work, we have proposed two novel methods (WPD and WPD-CCA) using four different wavelet were commonly employed for the correction of movement artifacts from motion corrupted EEG and fNIRS signals. Each of these methods suffers from some limitations. Using DWT-based approaches, to improve signal quality from motion-corrupted physiological data, selecting the suitable wavelet is critical and rather complex. To date, there is no hard and fast rule for selecting the appropriate wavelet for the specific physiological signal of interest; instead, wavelets are often selected depending on the morphology of the signal. As a result, improper wavelet selection would result in inefficient denoising. The EMD-based motion artifact reduction approach suffers heavily from the "mode mixing" issue [33], which may result in an incorrect outcome. To fix this problem, the EEMD approaches are employed [33,36]. Although EEMD is not affected by the mode mixing problem, it still requires a prior declaration of the number of ensembles to be employed which is determined through trial and error basis [33]. To make use of the SSA algorithm, for the correction of movement artifacts from physiological signals, a prior declaration of the window length and the required number of reconstruction components is necessary, which makes SSA inefficient as well [37]. The authors of [40] employed DWT along with approximation sub-band filtering using total variation (TV) and weighted TV. While reconstructing the cleaner signal, the first three high-frequency detailed sub-band signals were rejected since they included no important information from the EEG signal. But, detecting non-useful sub-band signals when utilizing DWT-based algorithms is very challenging for removing motion artifacts from EEG and fNIRS signals. Furthermore, the value of the regularization factor used to address the optimization problem of TV and MTV approaches was picked without explanation. In [52], they studied the autoregressive exogenous input model (adaptive technique) to model motion corrupted segments as output and IMU data as exogenous input. Only four test participants' fNIRS data were used by the authors to demonstrate the efficacy of their prescribed approach. One of the most important aspects of adopting this technique is the precise synchronization of fNIRS and IMU data. Furthermore, if the epoch duration of the motion artifacts is sufficiently long (specifically, the sample size), modeling the artifacts mathematically using the least square method would necessitate higher-order models which would eventually cause instability. Hence, incorporating this method to remove motion artifacts would be extremely difficult in a real-world scenario. ICA and CCA algorithms are multi-channel signal processing algorithms meaning there must be two (or more) channel data as input. Therefore, ICA and CCA algorithms can not be incorporated independently for the processing of single-channel data. Also, since ICA uses higher-order statistics (HOS) and CCA uses second-order statistics (SOS) {Sweeney, 2012 #66}, the CCA algorithm is computationally efficient in comparison with ICA. That is why previous studies as well as this study used the CCA algorithm as a 2 nd stage signal processing method. WPD is the more generalized version of DWT but the former provides better signal decomposition which enhances the signal quality for further processing. Also, WPD is better in denoising in the sense that there is no necessity of identifying and discarding any sub-band signals other than the motion corrupted sub-band component. Also, the results obtained in this work utilizing the WPD method for 12 different wavelet packets, show a little variation while computing and . This is a clear indication that, applying WPD compared to the DWT is much more robust and efficient in terms of performance metrics improvement. Although the two-stage motion artifacts removal approaches (WPD-CCA) proposed in this paper performed better compared to the single-stage artifacts correction techniques using WPD, the WPD-CCA technique will not be able to identify the motion corrupted CCA components In the absence of a ground truth signal, which is a limitation of two- However, even in the absence of the "reference ground truth" signal, our proposed single-stage motion artifact reduction approach (WPD) would produce optimal results. While decomposing the signal of interest (EEG/fNIRS) using WPD, it was visually seen that the approximation sub-band component (having the lowest frequency band compared to the rest of the sub-band components) included the highest percentage of motion artifacts. Hence, discarding this noisy sub-band component and reconstruction of the signal using the remaining sub-band signals would reduce the motion artifacts to a great extent. The validation of this statement is supported by Table 2 where the performance metrics ( and ) were reported and produced acceptable results. Throughout this work, while estimating the percentage reduction in motion artifacts , we have considered Equation 16, instead of 15. Where we have assumed that = 1 as in an ideal situation, the "reference ground truth" and the motion corrupted signal over the artifacts-free epochs would always be completely correlated. But in practice, the value of would always be less than 1. Because it is impossible to extract a "reference ground truth" signal which would completely be similar while compared with a motion-corrupted signal during the artifacts-free epochs. It is counter-intuitive that lower value of would produce a lesser value of . Rather it is just the opposite. For example, let = 0.6; = 0.8; = 0.95, from Equation 15, we would get equals 57.14% and Equation 16 would give 50%. That is why choosing = 1 would give a worst-case scenario result. Also, this same formula is used in [40][41][42] assuming the ideal "reference ground truth signal". Conclusions In this extensive study, two novel motion artifact removal techniques have been proposed, namely wavelet packet signals. An alternative approach for removing motion artifacts from EEG signals using the WPD method has also been proposed where the lowest-frequency approximation sub-band component was discarded and clean EEG signal was reconstructed by adding up the remaining sub-band components. By computing the performance metrics, it has been shown that this single-stage motion artifacts correction technique is also capable of removing motion artifacts to a great extent. In the future, deep learning-based models will be investigated for the automated detection and removal of artifacts in physiological signals (EEG, ECG, EMG, PPG, fNIRS, etc.). New methods based on the use of different multivariate signal processing approaches will be developed for the elimination of other artifacts from the EEG and fNIRS signals that are recorded using multiple electrodes. corresponding eigenvalues ρ 2 are the squared canonical correlations. It is sufficient to solve only one of the eigenvalue equations to obtain the un-mixing matrix as the solutions are related. Further, the underlying source signals ̂ can be estimated. Figure 1 : 1Example of motion-corrupted EEG (a) and fNIRS (b) signals. Two signals (blue: ground truth and red: motion-corrupted) are highly correlated during the motion artifacts free epochs. Boxed areas show the epochs of motion corrupted signals. A zoomed version is presented underneath each sub-plot. Figure 2 2depicts the motion artifacts elimination framework presented in this study. An automated way for identifying motion corrupted components of the preprocessed signal is also discussed. Figure 2 : 2Methodological framework for the motion artifact correction. Figure 3 : 3An example motion-corrupted single-channel EEG signal (a) and corresponding 16 sub-band components generated using WPD(sym4) algorithm (b). S15 denotes the Approximation sub-band signal having the lowest center frequency compared to the other sub-band signals i.e. D1-D15. Figure 4 :Figure 5 : 45An example motion-corrupted single-channel EEG signal (a) and corresponding 16 CCA components generated from CCA algorithm (b) An example motion-corrupted single-channel fNIRS signal (a) and corresponding 16 sub-band components generated using WPD(db1) algorithm (b). S15 denotes the Approximation sub-band signal having the lowest center frequency compared to the other sub-band signals i.e. D1-D15. Figure 6 : 6An example motion-corrupted single-channel EEG signal (a) and corresponding 16 CCA components generated from CCA algorithm (b) 7a, Figure 7b , 7bFigure 7c, and Figure 7d depicts 4 different examples of EEG recordings after the correction of the motion artifact using WPD(db2), WPD(db3), WPD(fk6), and WPD(fk8) methods, respectively whereas Figure 8a, and Figure 8b illustrate example EEG signals after the motion artifact correction using WPD(db1)-CCA, and WPD(fk4)-CCA techniques, respectively. Figure 7 : 7Motion artifact correction from different example EEG signals using WPD(db2)(a), WPD(db3) (b), WPD(fk6) (c), and WPD(fk8) (d) techniques. Figure 8 : 8Motion artifact from example EEG signals using WPD(db1)-CCA (a), and WPD(fk4)-CCA (b) techniques WPD: Among all the 12 different approaches (WPD(db1),WPD(db2), WPD(db3), WPD(sym4), WPD(sym5), WPD(sym6), WPD(coif1),WPD(coif2), WPD(coif3), WPD(fk6), WPD(fk6), and WPD(fk8)), the highest average ∆SNR of 29.44 dB with a standard deviation of 9.93 was found when WPD(db2) algorithm was employed over all (23) EEG recordings. The best average percentage reduction in artifact was provided by WPD(db1) algorithm (53.48 %) among these 12 single-channel motion artifact correction techniques. WPD-CCA: Six different approaches namely WPD(db1)-CCA, WPD(db2)-CCA, WPD(db3)-CCA, WPD(fk4)-CCA, WPD(fk6)-CCA, and WPD(fk8)-CCA were investigated all of which are two-stage motion artifacts correction techniques. Figure 9c , 9cand Figure 9d depicts 4 different example fNIRS signals after the correction of the motion artifact using WPD(sym5), WPD(sym6), WPD(coif2), and WPD(coif1) techniques, respectively whereas Figure 10a, and Figure 10b illustrate example fNIRS signals after the motion artifact correction using WPD(db1)-CCA, and WPD(fk4)-CCA techniques, respectively. Figure 9 :Figure 10 : 910Motion artifact correction from example fNIRS signals using WPD(sym5) (a), WPD(sym6) (b), WPD(coif2) (c), and WPD(coif1) (d) techniques Motion artifact correction from example fNIRS signals using WPD(db1)-CCA (a), and WPD(fk4)-CCA (b) techniques WPD: Among all the 12 different approaches (WPD(db1),WPD(db2), WPD(db3), WPD(sym4), WPD(sym5), WPD(sym6), WPD(coif1),WPD(coif2), WPD(coif3), WPD(fk6), WPD(fk6), and WPD(fk8)), the highest average ∆SNR of 16.03 dB with a standard deviation of 4.31 was found when WPD(db1) algorithm was employed over all (16) fNIRS recordings. The best average percentage reduction in artifact was provided by WPD(fk4) algorithm among these 12 single-channel motion artifact correction techniques. WPD-CCA: Finally, the six different approaches namely WPD(db1)-CCA, WPD(db2)-CCA, WPD(db3)-CCA, WPD(fk4)-CCA, WPD(fk6)-CCA, and WPD(fk8)-CCA, all of which are two-stage motion artifacts correction techniques, were investigated for fNIRS modality. The best average ∆SNR was found to be 16.55 dB when WPD(db1)-CCA technique was applied over all the 16 fNIRS records. The highest average percentage reduction in artifact (41.40%) was provided by WPD(fk8)-CCA technique among these 6 single-channel motion artifact correction techniques for fNIRS modality.Table 1 summarizes the results obtained (average ∆SNR and average percentage reduction in motion artifacts ) using the artifact removal techniques proposed in this paper i.e. WPD(db1),WPD(db2), WPD(db3), WPD(sym4), WPD(sym5), WPD(sym6), WPD(coif1), WPD(coif2), WPD(coif3), WPD(fk6), WPD(fk6), WPD(fk8), WPD(db1)-CCA, WPD(db2)-CCA, WPD(db3)-CCA, WPD(fk4)-CCA, WPD(fk6)-CCA, and WPD(fk8)-CCA for all the EEG (23) and fNIRS (16) recordings. The values inside first brackets in ( 26 . 2640 %) compared to the other 11 types of single-stage motion artifact correction approaches. The greatest average ∆SNR value (16.11 dB) was also provided by the same approach. Among these 12 different single-stage artifact removal approaches, the lowest average (25.92%) was produced by WPD(db2) whereas the smallest ∆SNR value (15.33 dB) was produced by WPD(coif3). When two-stage motion artifacts removal techniques were employed (WPD-CCA) using six different wavelet packets for all the fNIRS signals, the best average correlation improvement (41.40%) was produced by WPD(fk8)-CCA technique and the lowest average percentage reduction in artifacts (36.58 %) was generated from WPD(db1)-CCA. On the other hand, the best average ∆SNR value (16.55 dB) was obtained from WPD(db1)-CCA technique, and the WPD(fk8)-CCA produced the lowest ∆SNR value of 12.41 dB. Overall, an increase of 56.82% in percentage reduction in motion artifacts was found while the best performing two-stage motion artifacts technique i.e. WPD(fk8)- packet families with three different vanishing moments, resulting in 18 different techniques (WPD(db1),WPD(db2), WPD(db3), WPD(sym4), WPD(sym5), WPD(sym6), WPD(coif1),WPD(coif2), WPD(coif3), WPD(fk6), WPD(fk6), WPD(fk8), WPD(db1)-CCA, WPD(db2)-CCA, WPD(db3)-CCA, WPD(fk4)-CCA, WPD(fk6)-CCA, and WPD(fk8)-CCA) for the correction of motion artifacts from singlechannel EEG and fNIRS recordings. The performance metrics ( and ) calculated and reported in the "Results" section utilizing these 18 approaches are a clear indication of the efficacy of our proposed techniques. Both the Daubechies and Fejer-Korovkin wavelet packet families relatively performed better compared to the Symlet and Coiflet wavelet packet families in removing motion artifacts from EEG and fNIRS recordings. For this reason, while implementing the two-stage artifacts correction technique, we have used only the Daubechies and Fejer-Korovkin wavelet packet families. As previously stated, DWT, EMD, EEMD, VMD, EMD-ICA, EMD-CCA, EEMD-ICA, EEMD-CCA, VMD-CCA, SSA, DWT along with approximation sub-band filtering, adaptive filtering (ARX model with exogenous input), etc. decomposition (WPD), and WPD in combination with canonical correlation analysis (WPD-CCA) for EEG and fNIRS modalities. Further, the proposed algorithms were investigated by 18 different approaches where four different wavelet packet families namely Daubechies, Symlet, Coiflet, and Fejer-Korovkin wavelet packet families were utilized. WPD-CCA techniques can be used on single-channel recordings as the WPD algorithm can decompose a single-channel signal into a predefined number of sub-band components which can be fed as the input channels for the CCA algorithm. The performance parameters obtained from all these approaches are a clear indication of the efficacy of these algorithms. The novel WPD(db1)-CCA and WPD(fk8)-CCA technique provided the best performance in terms of the percentage reduction in motion artifacts (59.51% and 41.40%) when analyzing the EEG and fNIRS data, respectively. On the other hand, the WPD(db1)-CCA technique generated the highest average ∆SNR (30.76 dB and 16.55 dB) for both EEG and fNIRS This work was made possible by Qatar National Research Fund (QNRF) NPRP12S-0227-190164 and International Research Collaboration Co-Fund (IRCC) grant: IRCC-2021-001 and Universiti Kebangsaan Malaysia (UKM) under Grant GUP-2021-019, Grant TAP-K017701, and DPK-2021-001. The statements made herein are solely the responsibility of the authors. Table 1 : 1Average ∆SNR and average percentage reduction in artifacts ( ) for all the EEG and fNIRS recordings. Corresponding standard deviations are shown inside the bracket. () represents the best-performing metrics.Type Technique EEG (23 records) fNIRS (16 records) Average ∆SNR (in dB) Average (in %) Average ∆SNR (in dB) Average (in %) Single-stage motion artifact correction techniques WPD(db1) 29.26 (10.29) 53.48 (33.35)* 16.03 (4.31) 26.21 (26.38) WPD(db2) 29.44 (9.93)* 51.40 (33.59) 15.99 (4.49) 25.92 (28.86) WPD(db3) 29.37 (10.01) 50.74 (33.55) 15.71 (4.52) 26.05 (29.11) WPD(sym4) 29.27 (10.05) 50.40 (33.50) 15.54 (4.55) 26.14 (29.18) WPD(sym5) 29.19 (10.09) 50.20 (33.47) 15.43 (4.57) 26.17 (29.22) WPD(sym6) 29.11 (10.12) 50.05 (33.43) 15.35 (4.59) 26.16 (29.24) WPD(coif1) 29.43 (9.94) 51.34 (33.59) 15.97 (4.49) 25.94 (28.88) WPD(coif2) 29.25 (10.06) 50.35 (33.49) 15.51 (4.56) 26.15 (29.19) WPD(coif3) 29.08 (10.13) 50.00 (33.42) 15.33 (4.60) 26.15 (29.25) WPD(fk4) 29.21 (9.87) 52.58 (33.48) 16.11 (4.42)* 26.40 (27.53)* WPD(fk6) 29.32 (10.03) 50.55 (33.51) 15.59 (4.54) 26.20 (29.08) WPD(fk8) 29.15 (10.10) 50.15 (33.45) 15.38 (4.58) 26.25 (29.18) Two-stage motion artifact correction techniques WPD(db1)-CCA 30.76 (12.29)* 59.51(25.99)* 16.55 (6.29)* 36.58 (11.22) WPD(db2)-CCA 30.35 (12.50) 57.57 (25.89) 14.50 (5.85) 39.62 (10.59) WPD(db3)-CCA 29.42 (12.57) 56.52 (25.71) 13.72 (5.82) 40.39 (10.60) WPD(fk4)-CCA 30.36 (12.65) 58.83 (25.93) 14.97 (6.25) 38.32 (10.90) WPD(fk6)-CCA 29.12 (13.00) 56.81 (25.16) 13.81 (5.70) 40.48 (10.43) WPD(fk8)-CCA 28.86 (12.77) 55.88 (25.10) 12.41 (5.51) 41.40 (10.08)* It is evident from the results of Table 1 that the cleaner EEG signals reconstructed using the WPD(db1) technique provided the highest average value (53.48%, with corresponding ∆SNR value of 29.26 dB) compared to the other 11 types of single-stage motion artifact correction approaches whereas the greatest average ∆SNR value (29.44 dB) was provided by WPD(db2) with corresponding average value of 51.40%. Among these 12 different single-stage artifact removal approaches, the Table 2 : 2Average ∆SNR and average percentage reduction in artifacts ( ) for 21 recordings of EEG modality. Corre- sponding standard deviations are shown inside the first bracket. (*) denotes the best-performing metrics. Type Method EEG (21 records) stage artifacts removal technique. Hassan et al. provided an alternate technique in [65], in which the authors employed the autocorrelation function to detect the motion corrupted components. The automated artifact component selection approach introduced in [65] employing the autocorrelation function has not been experimented in this study and left as a future study. Author ContributionsConceptualization, Md Shafayet Hossain, Muhammad E. H. Chowdhury, Mamun Bin Ibne Reaz, Sawal Ali, Ahmad Bakar, Serkan Kiranyaz, Amith Khandakar, Mohammed Alhatou, Rumana Habib and Muhammad Maqsud Hossain; Data curation, Md Shafayet Hossain and Mamun Bin Ibne Reaz; Formal analysis, Md Shafayet Hossain; Funding acquisition, Muhammad E. H. Chowdhury and Mamun Bin Ibne Reaz; Methodology, Md Shafayet Hossain, Muhammad E. H. Chowdhury, Mamun Bin Ibne Reaz, Sawal Ali, Amith Khandakar and Rumana Habib; Project administration, Muhammad E. H. Chowdhury and Mamun Bin Ibne Reaz; Resources, Muhammad E. H. Chowdhury; Software, Muhammad E. H. Chowdhury; Supervision, Muhammad E. H. Chowdhury and Mamun Bin Ibne Reaz; Validation, Md Shafayet Hossain; Visualization, Md Shafayet Hossain; Writing -original draft, Md Shafayet Hossain, Muhammad E. H. Chowdhury, Mamun Bin Ibne Reaz, Sawal Ali, Ahmad Bakar, Serkan Kiranyaz, Amith Khandakar, Mohammed Alhatou, Rumana Habib and Muhammad Maqsud Hossain; Writing -review & editing, Md Shafayet Hossain, Muhammad E. H. 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[ "Discovering Ancestral Instrumental Variables for Causal Inference from Observational Data", "Discovering Ancestral Instrumental Variables for Causal Inference from Observational Data" ]	[ "Debo Cheng ", "Jiuyong Li ", "Lin Liu ", "Kui Yu ", "Thuc Duy Le ", "Jixue Liu " ]	[]	[]	Instrumental variable (IV) is a powerful approach to inferring the causal effect of a treatment on an outcome of interest from observational data even when there exist latent confounders between the treatment and the outcome. However, existing IV methods require that an IV is selected and justified with domain knowledge. An invalid IV may lead to biased estimates. Hence, discovering a valid IV is critical to the applications of IV methods. In this paper, we study and design a data-driven algorithm to discover valid IVs from data under mild assumptions. We develop the theory based on partial ancestral graphs (PAGs) to support the search for a set of candidate Ancestral IVs (AIVs), and for each possible AIV, the identification of its conditioning set. Based on the theory, we propose a data-driven algorithm to discover a pair of IVs from data. The experiments on synthetic and realworld datasets show that the developed IV discovery algorithm estimates accurate estimates of causal effects in comparison with the state-of-the-art IV based causal effect estimators.	10.1109/tnnls.2023.3262848	[ "https://arxiv.org/pdf/2206.01931v1.pdf" ]	249,394,754	2206.01931	a18db27b415dc20822a0f93fa647073794fb32d4	Discovering Ancestral Instrumental Variables for Causal Inference from Observational Data Debo Cheng Jiuyong Li Lin Liu Kui Yu Thuc Duy Le Jixue Liu Discovering Ancestral Instrumental Variables for Causal Inference from Observational Data JOURNAL OF L A T E X CLASS FILES, VOL. , NO. , AUGUST 202* 1Index Terms-Causal InferenceObservational StudiesLatent ConfoundersConfounding BiasMaximal Ancestral Graph Instrumental variable (IV) is a powerful approach to inferring the causal effect of a treatment on an outcome of interest from observational data even when there exist latent confounders between the treatment and the outcome. However, existing IV methods require that an IV is selected and justified with domain knowledge. An invalid IV may lead to biased estimates. Hence, discovering a valid IV is critical to the applications of IV methods. In this paper, we study and design a data-driven algorithm to discover valid IVs from data under mild assumptions. We develop the theory based on partial ancestral graphs (PAGs) to support the search for a set of candidate Ancestral IVs (AIVs), and for each possible AIV, the identification of its conditioning set. Based on the theory, we propose a data-driven algorithm to discover a pair of IVs from data. The experiments on synthetic and realworld datasets show that the developed IV discovery algorithm estimates accurate estimates of causal effects in comparison with the state-of-the-art IV based causal effect estimators. I. INTRODUCTION A Fundamental challenge for inferring from observational data the causal effect of a treatment W (a.k.a. exposure, intervention or action) on an outcome Y of interest is the presence of latent (a.k.a. unobserved or unmeasured) confounders, variables which affect W and Y simultaneously. Instrumental variables (IVs) are a powerful tool to address this challenge, primarily used by statisticians, economists and social scientist [1]- [3]. It is possible to eliminate the confounding bias by leveraging a valid IV [4]. The standard IV approach requires a predefined IV (denoted as S) that meets the following three conditions: (1) S is a cause of W , (2) There is no confounding bias for the effect of S on Y (a.k.a. exogeneity), and (3) the effect of S on Y is entirely mediated through W (a.k.a. the exclusion restriction) [5]. For example, S in the causal graph in Fig. 1 (a) is a standard IV as it meets all the three conditions. The second and third conditions have to be justified by domain knowledge [6] and hence IV based methods are mostly classified as empirical methods in literature [7], [8]. Data-driven approach to identifying standard IVs is impractical. Instrumental inequality [7], [9] has been proposed to test an IV in data, but it is a necessary condition, not a sufficient condition. Under a set of strong assumptions on D. Cheng the data distribution, a semi-instrumental variable that can be tested in data has been introduced by [10]. However, there is not a practical algorithm rooted from the above concepts. By assuming that at least a half of the covariates are valid IVs, Kang et al. [11] proposed an algorithm, sisVIVE (some invalid, some valid IV estimator), to estimate the causal effect in data. Hartford et al. [12] extended the sisVIVE algorithm by employing a deep learning based IV estimator. The main challenge of the two algorithms is that their strong assumption is often unsatisfied in many real-world applications. To relax the last two conditions (i.e., the exogeneity and exclusion restriction) of a standard IV, a graphical criterion [6], [13] is proposed to identify an observed variable S as an IV (i.e., the conditional IV (CIV)), conditioning on a set of observed variables Z from a given DAG (directed acyclic graph, which represents the causal relations of all measured and unmeasured variables). CIV allows a confounding bias between S and Y , and S have multiple causal paths to Y . The bias can be adjusted by a conditioning set. Van der Zander et al. [14] have revised CIV to Ancestral IV (AIV) to avoid the situation where void CIVs may be identified based on the original definition (see Section II-B for details). AIV identification needs a DAG too. IV.tetrad [8] is the only existing data-driven CIV method. IV.tetrad based on CIV and it requires two valid CIVs in the covariate set. The tetrad condition is used to discover the pair of CIVs in data and it assumes that the conditioning sets of the pair of CIVs are the same and equal to the set of remaining covariates (i.e. the original covariate set excluding the pair of CIVs). This tetrad condition leads to a wrong identification when the set of remaining covariates contains a collider. For example, S 1 and S 2 in Fig. 1 (b) are a pair of CIVs. The conditioning sets for S 1 and S 2 are the same, both equal to {X 2 , X 3 }, but not as assumed in the IV.tetrad, i.e. {X 1 , X 2 , X 3 }. When X 1 is used in the conditioning set, arXiv:2206.01931v1 [cs.AI] 4 Jun 2022 path S 1 → X 1 ← U 2 → Y is opened and S 1 does not instrumentalise W conditioning on {X 1 , X 2 , X 3 } any more. Hence, the tetrad condition does not find the right CIV pairs and leads to a biased causal effect estimation. This work improves IV.tetrad in the following ways, which are also our contributions. 1) We generalise the tetrad condition so that each AIV in the pair conditions on its own conditioning set, and this rectifies the current tetrad condition which fails to find the right pair when the covariate set contains a collider. 2) We develop the theory for identifying the set of candidate AIVs in a reduced space for efficient search for a pair of AIVs. 3) We propose a data-driven algorithm for estimating causal effects from data with latent variables based on the above developed theorems. Extensive experiments on synthetic and real-world datasets have shown the effectiveness of the algorithm. II. BACKGROUND A. Graph Terminology A graph G = (V, E) is composed of a set of nodes V={V 1 , . . . , V p }, representing random variables, and a set of edges E ⊆ V × V, representing the relations between nodes. In this paper, we assume that in a graph G, there is at most one edge between any two nodes. Two nodes are adjacent if there exists an edge between them. For an edge V i → V j , V i and V j are its head and tail respectively and V i is known as a parent of V j (and V j is a child of V i ). We use Adj(V ), P a(V ) and Ch(V ) to denote the sets of all adjacent nodes, parents and children of V , respectively. A path π is a sequence of nodes ⟨V 1 , . . . , V n ⟩ such that for 1 ≤ i ≤ n − 1, the pair (V i , V i+1 ) is adjacent. A path π from V i to V j is a directed or causal path if all edges along it are directed towards V j , and (V i , V j ) are called endpoint nodes, other nodes are non-endpoint nodes. If there is a directed path π from V i to V j , V i is a known as an ancestor of V j and V j is a descendant of V i . The sets of ancestors and descendants of a node V are denoted as An(V ) and De(V ), respectively. A DAG (directed acyclic graph) is a direct graph (i.e. a graph containing only directed edges →) without directed cycles (i.e. a directed path whose two endpoints are the same node). A DAG is often used to represent the data generation mechanism or causal mechanism underlying the data, with all variables, both observed and unobserved (if any) included in the graph. Ancestral graphs are used to represent the data generating generation mechanisms that may involve latent variables, with only observed variables included in the graphs [15], [16]. An ancestral graph may contain three types of edges, →, ↔ (it is used to represent that a common cause of two observed variables is a latent variable.) and b → (the circle tail denotes the orientation of the edge is uncertain.), and we use ' * →' to denote any of the three types. C is a collider on the path π if π contains a subpath * → C ← * . In an ancestral graph, a path is a collider path if every non-endpoint node on it is a collider. A path of length one is a trivial collider path. In a graph, an almost directed cycle occurs when V i ↔ V j is in the graph and V j ∈ An(V i ). An ancestral graph is a graph that does not contain directed cycles or almost directed cycles [15]. In an ancestral graph, a path from V i to V j is a possibly directed or causal path if there is not an arrowhead pointing in the direction of V i . In this case, V i is a possible ancestor of V j and V j is a possible descendant of V i . The sets of possible ancestors and descendants of V are denoted as P ossAn(V ) and P ossDe(V ), respectively. In graphical causal modelling, the assumptions of Markov property and faithfulness are often involved to discuss the relationship between the causal graph and the data distribution. Definition 1 (Markov property [6]). Given a DAG G = (V, E) and the joint probability distribution of V (prob(V)), G satisfies the Markov property if for ∀V i ∈ V, V i is probabilistically independent of all of its non-descendants, given P a(V i ). A DAG G = (V, E) satisfies both assumptions, the probability distribution prob(V) can be factorised as: prob(V) = ∏ p i prob(V i \|P a(V i )). Thus, together Markov property and faithfulness establish a close relation between the causal graph and the data distribution. Definition 3 (Causal sufficiency [17]). In a data, for every pair of observed variables (V i , V j ) in V, all their common causes are also in V. In a DAG, d-separation is a well-known graphical criterion that is used to read off the identification of conditional independence between variables entailed in the DAG when the Markov property, faithfulness and causal sufficiency are satisfied [6], [17]. Definition 4 (d-separation [6]). A path π in a DAG G = (V, E) is said to be d-separated (or blocked) by a set of nodes Z if and only if (i) π contains a chain V i → V k → V j or a fork V i ← V k → V j such that the middle node V k is in Z, or (ii) π contains a collider V k such that V k is not in Z and no descendant of V k is in Z. A set Z is said to d-separate V i from V j (V i ⫫ d V j \| Z) if and only if Z blocks every path between V i to V j . Otherwise they are said to be d-connected by Z, denoted as V i d V j \| Z. Property 1. Two observed variables V i and V j are dseparated given a conditioning set Z in a DAG if and only if V i and V j are conditionally independence given Z in data [17]. if V i and V j are d-connected, V i and V j are conditionally dependent. However, a system may involve the latent variables (the latent variable is an unmeasured common cause of two nodes) in most situations since there is not a close world. Ancestral graphs are proposed to represent the system that may involve latent variables [15], [16]. In our work, we utilise the Maximal ancestral graph and introduce it as follows. Definition 5 (Maximal ancestral graph (MAG) [15]). An ancestral graph M = (V, E) is a MAG when every pair of non-adjacent nodes V i and V j in M is m-separated by a set Z ⊆ V\{V i , V j }. It is worth noting that a DAG satisfies both conditions of a MAG, so a DAG is also a MAG without bi-directed edges [16]. An important concept in a DAG is d-separation, which captures the conditional independence relationships between variables based on Markov property [6]. A natural extension of the d-separation to an ancestral graph is mseparation [15]. Definition 6 (m-separation [15]). In an ancestral graph M = (V, E), a path π between V i and V j is said to be m-separated by a set of nodes Z ⊆ V \ {V i , V j } (possibly ∅) if π contains a subpath ⟨V l , V k , V s ⟩ such that the middle node V k is a noncollider on π and V k ∈ Z; or π contains V l * → V k ← * V s such that V k ∉ Z and no descendant of V k is in Z. Two nodes V i and V j are said to be m-separated by Z in M, denoted as V i ⫫ m V j \|Z if every path between V i and V j are m-separated by Z; otherwise they are said to be m-connected by Z, denoted as V i m V j \|Z. where ⫫ m denotes m-separation and m denotes mconnecting. In a DAG, m-separation reduces to d-separation. The Markov property of ancestral graph is captured by mseparation. If two MAGs represent the same set of m-separations, they are called Markov equivalent, and formally, we have the following definition. A set of Markov equivalent MAGs can be encoded uniquely by a partial ancestral graph (PAG) [16]. Definition 9 (Visibility [16]). Given a MAG M = (V, E), a directed edge V i → V j is visible if there is a node V k ∉ Adj(V j ) , such that either there is an edge between V k and V i that is into V i , or there is a collider path between V k and V i that is into V i and every node on the path is a parent of V j . Otherwise, V i → V j is invisible. The visible edge is a critical concept in a MAG [16], [19]. A DAG over measured and unmeasured variables can be mapped to a MAG with measured variables. From a DAG over X ∪ U where X is a set of measured variables and U is a set of unmeasured variables, following the construction rule specified in [18], one can construct a MAG with nodes X such that all the conditional independence relationships among the measured variables entailed by the DAG are entailed by the MAG and vice versa, and the ancestral relationships in the DAG are maintained in the MAG. B. Instrumental Variable (IV) Let W be the treatment variable, Y the outcome, and X be the set of all other variables. As in the literature, we consider that X contains pretreatment variables only, i.e., for any X ∈ [20], [21]. When estimating the average causal effect of W on Y , denoted as β wy from data, we follow the convention in causal inference literature, that is, the data distribution is said to be compatible with the underlying causal DAG G, i.e., the assumptions of Markov property and faithfulness are satisfied. X, X ∉ (De(W ) ∪ De(Y )) [8], The goal of this work is to quantify the average causal effect of W on Y , i.e., β wy , even when there exist unmeasured variables between W and Y , based on observational data, by extending the existing IV techniques. In this section, we introduce the background information related to IVs. Definition 10 (Standard IV [22], [23]). A variable S is said to be an IV wrt., W → Y , if (i) S has a causal effect on W , (ii) S affects Y only through W (i.e., S has no direct effect on Y ), and (iii) S does not share common causes with Y . The last two conditions of a standard IV are untestable and strict. In practice, S may have other causal paths to Y , and S is often confounded with Y by other measured variables. The concept of conditional IV (CIV) in DAG is proposed to relax the conditions of a standard IV. Definition 11 (Conditional IV (CIV) in DAG [6], [13]). Given a DAG G = (X ∪ U ∪ {W, Y }, E) where X and U are measured and unmeasured variables respectively . A variable S ∈ X is said to be a CIV wrt., W → Y , if there exists a set of measured variables Z ⊆ X such that (i) S d W \| Z, (ii) S ⫫ d Y \| Z in G W , and (iii) ∀Z ∈ Z, Z ∉ De(Y ) where G W is obtained by removing W → Y from G. For a CIV S as defined in Definition 11, Z is known to instrumentalise S in the given DAG. However, a variable may be a CIV when it has zero causal effect on W , and this might result in a misleading conclusion. To mitigate this issue, Ancestral IV (AIV) in DAG is proposed. Definition 12 (Ancestral IV (AIV) in DAG [14]). Given a DAG G = (X∪U∪{W, Y }, E) where X and U are measured and unmeasured variables respectively. A variable S ∈ X is said to be an AIV wrt., W → Y , if there exists a set of measured variables Z ⊆ X \ {S} such that (i) S d W \| Z, (ii) S ⫫ d Y \| Z in G W , and (iii) Z consists of An(Y ) or An(S) or both and ∀Z ∈ Z, Z ∉ De(Y ). An AIV in DAG is a CIV in DAG, but a CIV may not be an AIV. AIV is a restricted version of CIV [14]. However, the applications of standard IV, CIV and AIV are established in a complete causal DAG G, which greatly limits their capacity in real-world applications. Recently, Cheng et al. proposed the concept of AIV in MAG and the theorem for identifying a conditioning set Z that instrumentalises a given AIV in PAG [24]. However, the work by Cheng et al. assume an AIV in MAG has been given, and the focus is on finding the conditioning set for the AIV in MAG from data. Our work in this paper aims to find an AIV in MAG from data, which makes it possible for complete data-driven search for AIVs. Proposition 1 (AIV in MAG [24]). Given a DAG G = (X ∪ U ∪ {W, Y }, E ′ ) with the edges W → Y and W ← U → Y in E ′ , and U ∈ U, and the MAG M = (X ∪ {W, Y }, E) is mapped from G based on the construction rules [18]. Then if S is an AIV conditioning on a set of measured variables Z ⊆ X \ {S} in G, S is an AIV conditioning on a set of measured variables Z ⊆ X \ {S} in M. Theorem 1 (Conditioning set of a given AIV in PAG [24]). Given a DAG G = (X ∪ U ∪ {W, Y }, E ′ ) with the edges W → Y and W ← U → Y in E ′ , and U ∈ U, and let M = (X ∪ {W, Y }, E) be the MAG mapped from G. From data, the mapped MAG M is represented by a PAG P = (X ∪ {W, Y }, E ′′ ) . For a given ancestral IV S which is a cause or spouse of W , the set P ossAn(S ∪ Y ) \ {W, S} in the learned P is a set that instumentalises S in the DAG G. It is worth noting that Theorem 1 and the work in [24] are to discover conditioning set for a given AIV, rather than discovering AIVs and the conditioning set simultaneously. Hence, discovering an AIV and its conditioning set simultaneously from data remains unresolved, and it is the problem to be addressed in this work. III. DISCOVERING AIVS BASED ON GRAPHICAL CAUSAL MODELING In this section, we first introduce the generalised tetrad condition. Next, we propose a set of candidates AIVs in MAG. Then, we propose a theorem to guarantee that the generalised tetrad condition can be used to discover valid AIVs from data if there exists a pair of AIVs. Finally, we develop a practical data-driven algorithm for estimating β wy from data. A. The Generalised tetrad Condition Let S i and S j be a pair of CIVs given the conditioning set X \ {S i , S j }. Let σ s i * y * z (σ s j * y * z ) denote the partial covariance of S i (S j ) and Y given Z, and σ s i * w * z (σ s j * w * z ) denote the partial covariance of S i (S j ) and W given Z. Then, we have β wy = σ s i * y * z /σ s i * w * z = σ s j * y * z /σ s j * w * z , which gives us the following tetrad condition [8]: σ s i * y * z σ s j * w * z − σ s i * w * z σ s j * y * z = 0(1) The tetrad condition can be tested from data directly. It is a necessary condition for discovering valid CIVs, which means a pair of variables that are not valid CIVs can also satisfy the tetrad condition. We consider a more general setting, where a pair of AIVs S i and S j have different conditioning sets Z i ⊆ X\{S i } and Z j ⊆ X \ {S j } respectively, and Z i and Z j do not need to be equal. Let σ s i * y * z i (σ s j * y * z j ) denote the partial covariance of S i (S j ) and Y given Z i (Z j ), and σ s i * w * z i (σ s j * w * z j ) denote the partial covariance of S i (S j ) and W given Z i (Z j ). Then we have β wy = σ s i * y * z i /σ s i * w * z i = σ s j * y * z j /σ s j * w * z j , which gives us the following generalised tetrad condition: σ s i * y * z i σ s j * w * z j − σ s i * w * z i σ s j * y * z j = 0(2) In the following, we will show that the generalised tetrad condition can be used for finding a pair of AIVs in data if there exists a pair of AIVs. In comparison with the tetrad condition used in IV.tetrad [8], the search space of the generalised tetrad condition is larger since S i , S j , Z i and Z j all vary. In the next section, we will develop a lemma to reduce the search space. B. The Theory for Discovering AIVs in MAG We aim to develop a practical solution for discovering AIVs directly from data by leveraging the property of a MAG. We first categorise AIVs into direct AIVs and indirect AIVs. When S is an AIV and it is an adjacent node of the treatment in the given DAG, its ancestral or adjacent nodes may be AIVs. We call S a direct AIV and the AIVs which are ancestral or adjacent nodes of S indirect AIVs if they satisfy Definition 12. We consider direct AIVs since, in practice, indirect AIVs are rare. Importantly, direct AIVs have a property to support the data-driven search for their conditioning sets. In the following discussions, all AIVs are direct AIVs. We have the following conclusion for discovering the direct AIVs in a MAG M. Lemma 1 (A direct AIV in MAG). Given a DAG G = (X ∪ U ∪ {W, Y }, E ′ ) with the edges W → Y and W ← U → Y in E ′ , and U ∈ U, and let M = (X ∪ {W, Y }, E) be the MAG mapped from G. If S is a direct AIV in the DAG G, then S ∈ Adj(Y ) \ {W } in MAG M. Proof. Firstly, the edges W → Y and W ← U → Y in the given DAG are represent by an invisible edge W → Y in the mapped MAG M [18]. For an S ∈ X to be an eligible direct AIV in the DAG G, there are only two cases in the mapped MAG M. The first case is that S has an edge S * → W , then S must have an edge into Y , i.e., S ∈ Adj(Y ) \ {W }, since otherwise W → Y in M is visible, which contradicts the invisible edge W → Y in M. The second case is that S has a collider path into W and every collider on the path is in (\|X\|) to O(\|Adj(Y ) \ {W }\|) where \|Adj(Y ) \ {W }\| ≪ \|X\|. Next, we will develop a theorem to show that the generalised tetrad condition in Eq.(2) can be used to discover a pair of direct AIVs directly from data with latent variables. Proof. According to Lemma 1, Adj(Y ) \ {W } in the mapped MAG M is the set of candidate direct AIVs in the DAG G. Hence, the set Adj(Y ) \ {W } in the PAG P must be the set of candidate direct AIVs because the mapped MAG M is encoded in the PAG P. Thus, if S ∈ X is a direct AIV in the DAG G, then S ∈ Adj(Y )\{W } in the PAG P i.e., {S i , S j } ⊆ S holds. According to Theorem 1, Z i = possAn(S i ∪ Y ) \ {W, S i } and Z j = possAn(S j ∪Y )\{W, S j } in the PAG P instrumentalise S i and S j in the DAG G, respectively. Thus, we have β wy = σ s i * y * z i /σ s i * w * z i = σ s j * y * z j /σ s j * w * z j . Therefore, σ s i * y * z i σ s j * w * z j − σ s i * w * z i σ s j * y * z j = 0, i.e., Eq.(2) holds. Theorem 2. Given a DAG G = (X ∪ U ∪ {W, Y }, E ′ ) with the edges W → Y and W ← U → Y in E ′ , Theorem 2 supports a data-driven algorithm to discover a pair of direct AIVs {S i , S j } and their corresponding conditioning sets Z i and Z j by utilising the generalised tetrad condition. In the next section, based on the theorem, we will propose a practical algorithm for estimating β wy from data with latent variables. Note that a significant number of direct AIVs are in both Adj(W )\{Y } and Adj(Y )\{W } in a MAG. Sometimes, they may be missed from Adj(Y )\{W } due to the false discoveries of the structure learning algorithm used [25], [26]. In the corresponding DAG, the direct AIVs are closer to W than Y . To avoid the random fluctuations without sacrificing much efficiency, in our developed practical algorithm, we extend the search space of Lemma 1 to Adj(W ∪ Y ) \ {W, Y }. This only adds minor additional costs to the search process. C. A Practical Algorithm for Estimating β wy We develop a practical data-driven algorithm, AIV.GT (Ancestral IV based on Generalised Tetrad condition), for estimating β wy from data with latent variables. The pseudocode of AIV.GT is listed in Algorithm 1. AIV.GT aims to search for the pair of AIVs from data directly without domain knowledge. The generalised tetrad condition in Eq.(2) held by a pair of AIVs and their conditioning sets as described in Theorem 2 if there is a pair of IVs {S i , S j } in data. To obtain a reliable result, we propose a consistency score to assess which paired variables are the most likely AIVs based on the generalised tetrad condition. Let ij = σ s i * y * z i σ s j * w * z j − σ s i * w * z i σ s j * y * z j , and δ ij = β i −β j Algorithm 1 AIVs based on the Generalised Tetrad condition (AIV.GT) 1: Input: The set of pretreatment variables X, the treatment W , outcome Y and the dataset D; α =0.05 2: Output:β wy , the causal effect of W on Y , or NA, i.e. lacking knowledge 3: Recover a PAG P from D by using the rfci algorithm 4: Obtain S = Adj(W ∪ Y ) \ {W, Y } from P 5: if \|S\| ⩽ 1 then 6: return NA 7: else 8: for each S i ∈ S do 9: whereβ i andβ j are the estimated causal effects of W on Y by using S i and S j as an instrument, respectively. The consistency score is defined as λ ij = \| ij − δ ij \|. Z i ← P ossAn(S i ∪ Y ) \ {S i , W, Y } 10:β i ← T SLS(W, Y, S i , Z i , D)Initialise Q = ∅ 13: for each pair (S i , S j ) ∈ S do 14: if T est.tetrad(W, Y, S i , S j , Z i , Z j , D, α) then 15: ij = σ s i * y * z i σ s j * w * z j − σ s i * w * z i σ s j * y * z j 16: λ ij = \| ij − δ ij \| where δ ij = β i −β j 17: Q ← Q ∪ λ ij The justification of the consistency score is that ij is expected to be close to 0, and the same with δ ij 0 if the variables S i and S j are AIVs. Theoretically, the pair of variables with either the smallest ij or δ ij are most likely to be the pair of IVs, but in practical cases, a pair of variables passing the generalised tetrad condition test (i.e., a small enough ij ), may have a large δ ij , or vice versa, since the pair of variables are not IVs. So we use their difference λ ij to avoid such cases because λ ij must be smaller than both ij and δ ij . Under the assumption that there exists at least a pair of IVs, if the consistency score of a pair of variables is the smallest, then the pair are most likely to be AIVs. The paired variables with the minimal consistency score is returned as the result of AIV.GT. The AIV.GT algorithm is divided into two parts. The first part (Lines 3 to 11) is to obtain all candidate AIV pairs and the possible causal effects of W on Y estimated using these pairs. Line 3 aims to learn a PAG P from data by using a causal structure learning algorithm [26], [27]. We use rfci (really fast causal inference) [28] in AIV.GT. Line 4 aims to get the set of candidate AIVs from the learned PAG P. Line 5 tests the size of S and if \|S\| ⩽ 1, then AIV.GT returns NA due to lack of knowledge. Lines 8 to 11 are to estimate the causal effectβ wy using each candidate AIV. Line 9 is to find the conditioning set Z i for a candidate AIV S i based on Theorem 1. Line 10, the function T SLS() is the estimator of two-stage least squares (TSLS) by using S i as an IV and conditioning on Z i for calculatingβ i . The second part of AIV.GT is to discover the pair of AIVs. Line 12 is to initialise the set of consistency scores Q. Lines 13 to 19 are to check the validity of each pair of candidate AIVs based on Theorem 2. If the generalised tetrad condition holds on a pair of candidate AIVs, then calculate their consistency score. Line 14, the function T est.tetrad() is implemented by using the Wishart test wrt., the generalised tetrad condition [17], [29]. T est.tetrad() returns TRUE if and only if the set of candidate pair variables returns a p-value greater than the significant level α (α =0.05 in this work). Lines 15 to 17 are to obtain the consistency score of each paired AIVs satisfying the generalised tetrad condition. In Lines 20 and 21, if Q is an empty set, then no pair of variables has passed the tetrad condition test and the algorithm returns NA. In Lines 22 to 24, AIV.GT returns the mean causal effect of the pair of variables with the smallest λ ij in Q. Time Complexity Analysis: Three factors contribute to the time complexity of AIV.GT. The first contributing factor is the learning of a PAG P from data and finding S from P, which largely relies on the rfci algorithm. In the worst situation, rfci has a complexity of O(2 r * n), where r is the maximum degree of a node in the underlying causal MAG and n is the sample size. In most cases, the average degree of a causal Bayesian network is 2 to 5 [27], and most of the underlying MAGs are sparse in real-world applications. Hence, the time complexity of rfci is lower [28]. The complexity of Line 4 is O(1) since it reads from P. The second factor is estimating all possible causal effects, i.e., Lines 8 to 11 in Algorithm 1. Noting that obtaining Z i takes O(1) and calculatingβ i needs O(n * p 2 ). Hence, the whole time complexity of this part is O(\|S\| * n * p 2 ). The third factor is finding the pair of IVs from S and time complexity relies on the size of S (pairwise search for a pair of VIs) and calculating covariance, which all together takes O(\|S\| 2 * n * p 2 ). Therefore, the overall complexity of AIV.GT is O(2 r * n + \|S\| * n * p 2 + \|S\| 2 * n * p 2 ) = O(2 r * n+\|S\| 2 * n * p 2 ) . Therefore, the complexity of AIV.GT is largely attribute to the rfci algorithm and searching for a pair satisfying the generalised tetrad criterion. IV. EXPERIMENTS We assess the performance of AIV.GT by comparing it to the state-of-the-art causal effect estimators, firstly with a simulation study. Then, we conduct experiments on two real-world datasets that have been used for a long time in instrumental variable research [30], [31] to show that AIV.GT can be applied in real-world applications. The estimators compared include (1) LSR, least squares regression Y on {W, X}; (2) TSLS, two-stage least squares (TSLS [32]) of Y on W using all variables X as standard IVs; (3) some invalid some valid IV estimator (sisVIVE) [11]; (4) IV.tetrad method [8]. LSR is not an IV based method. It is included since it is frequently used in Machine Learning disregarding bias of latent variables in data, and it is used as a baseline. All other three comparison estimators do not need a nominated IV. TSLS is a standard IV estimator, and it is also used as a baseline. sisVIVE and IV.tetrad are two most related methods and have been discussed in the Introduction. The Implementations of Estimators in Sections IV-A and IV-B2. The method OLS is implemented by the function cov in the R package stats. The TSLS is programmed by the functions cov in the R package stats and solve in the base. The implementation of LSR is same with TSLS, i.e., using the functions cov and solve. The implementation of sisVIVE is based on the function sisVIVE in the R package sisVIVE. The implementation of IV.tetrad is retrieved from the authors' site 1 . The parameter of num ivs is set to 3 (2 for VitD). AIV.GT is implemented by using the function rfci in the R packages pcalg, cov in the R package stats, solve in the base and the functions in IV.tetrad. The Implementations of Estimators in Section IV-B3. The implementations of the compared estimators that require a known IV are introduced as follows. The estimator TSLS is implemented by the function ivreg in the R package AER [33]. The implementation of TSLS.CIV is based on the functions glm and ivglm from the R packages stats and ivtools [31]. FIVR is implemented by the function instrumental forest in the R package grf [21]. All parameters in FIVR are default. AIViP is implemented by the functions rfci in R package pcalg [26], glm in R package stats and ivglm in R package ivtools [31]. Evaluation Metrics & Parameter Setting. For the simulation study, we have the ground truth of β wy , so we report the estimation bias: (β wy − β wy )/β wy * 100 (%). In the experiments with real-world datasets, we empirically evaluate all estimators, and then compare AIV.GT with four additional IV-based estimators that require a nominated IV. The significant level α is set to 0.05 for the functions of rfci and T est.tetrad() in all experiments. A. Simulation Study The goal of this set of experiments is to test the effectiveness of AIV.GT with and without colliders in the covariate set in comparison with various causal effect estimators. We utilise five true DAGs over X∪U∪{W, Y } to generate five synthetic datasets with latent variables. The five true DAGs are shown in Fig. 2. In addition to the variables in the five true DAGs, 20 additional measured variables for each dataset are generated as noise variables that are related to each other but not to the variables in the five DAGs. The additional 20 variables are generated by a multivariate normal distribution. Next, we will separately introduce the details of each synthetic dataset generation based on the five true DAGs in Fig. 2. The synthetic dataset (a) is generated from the DAG (a) in Fig. 2, and the specifications are as following: U 1 ∼ Bernoulli(0.5) and S 1 , S 2 ∼ N (0, 1), in which N (, ) denotes the normal distribution. The treatment W is generated from Fig. 2. The core of the true causal DAGs over measured and unmeasured used to generate synthetic datasets. \|S\| denotes the number of measured variables Adj(W ∪ Y ) \ {W, Y }. There are two valid IVs S 1 and S 2 in all causal DAGs. In (a), S 1 and S 2 are standard IVs. In (b), S 1 is a standard IV and S 2 is a CIV conditioning on X 1 . In (c), S 1 is a CIV conditioning on ∅ since X 1 is a collider and S 2 is a standard IV. In (d), S 1 and S 2 are CIVs conditioning on ∅ and X 3 , respectively. In (e), S 1 is a condition IV conditioning on ∅ and S 2 is a CIV conditioning on {X 3 , X 4 }. n (n denotes the sample size) Bernoulli trials by using the assignment probability N (0, 1). P (W = 1 \| U 1 , S 1 , S 2 ) = [1+exp{1− 3 * U 1 − 3 * S 1 − 3 * S 2 }]. The potential outcome is generated from Y W = 2 + 2 * W + 3 * U 1 + w where w ∼ The synthetic dataset (b) is generated from the DAG (b) in Fig. 2, and the specifications are as following: U 1 ∼ Bernoulli(0.5), S 1 , S 2 ∼ N (0, 1) and X 1 ∼ N (0, 0.5). The measured variable X 1 = 0.8 * S 2 + X 1 .The treatment W is generated by P (W = 1 \| U 1 , S 1 , S 2 ) = [1 + exp{1 − 3 * U 1 − 3 * S 1 − 3 * S 2 }]. The potential outcome is generated from Y W = 2 + 2 * W + 3 * U 1 + 3 * X 1 + w . The synthetic dataset (c) is generated from the DAG (c) in Fig. 2, and the specifications are as following: U 1 ∼ Bernoulli(0.5), S 1 , S 2 , U 2 , X 2 ∼ N (0, 1) and X 1 ∼ N (0, 0.5). The measured variable X 1 is generated by X 1 = 0.3 + S 1 + X 2 + U 2 + X 1 . The treatment W is generated by P (W = 1 \| U 1 , S 1 , S 2 ) = [1+exp{1−3 * U 1 −3 * S 1 −3 * S 2 }]. The potential outcome is generated from Y W = 2 + 2 * W + 3 * U 1 + 2 * U 2 + 2 * X 2 + w . The synthetic dataset (d) is generated from the DAG (d) in Fig. 2, and the specifications are as following: U 1 ∼ Bernoulli(0.5), S 1 , S 2 , U 2 , X 2 ∼ N (0, 1) and X 1 , X 3 ∼ N (0, 0.5). The measured variables X 1 and X 3 are generated by X 1 = 0.3+S 1 +X 2 +1.5 * U 2 + X 1 and X 3 = 0.8 * S 2 + X 3 , respectively. The treatment W is generated by P (W = 1 \| U 1 , S 1 , S 2 ) = [1 + exp{1 − 3 * U 1 − 3 * S 1 − 3 * S 2 }]. The potential outcome is generated by Y W = 2 + 2 * W + 3 * U 1 + 2 * U 2 + 2 * X 2 + 2 * X 3 + w . The synthetic dataset (e) is generated from the DAG (e) in Fig. 2, and the specifications are as following: U 1 ∼ Bernoulli(0.5), S 1 , S 2 , U 2 , X 2 ∼ N (0, 1) and X 1 , X 3 , X 4 ∼ N (0, 0.5). The measured variables X 1 , X 3 and X 4 are generated by X 1 = 0.3 + S 1 + X 2 + 1.5 * U 2 + X 1 , X 3 = 0.8 * S 2 + X 3 and X 4 = 0.8 * S 2 + X 4 , respectively. The treatment W is generated by P (W = 1 \| U 1 , S 1 , S 2 ) = [1 + exp{1 − 3 * U 1 − 3 * S 1 − 3 * S 2 }]. The potential outcome is generated by Y W = 2 + 2 * W + 3 * U 1 + 2 * U 2 + 2 * X 2 + 2 * X 3 + 2 * X 4 + w . The suitability of datasets with the method requirements is summarised as Table I 4) IV.tetrad has a low biases on the first two datasets, but large biases on the last three datasets because the last three datasets contain colliders in the covariate sets. All datasets satisfy the assumption of IV.tetrad (i.e. a pair of CIVs), but it suffers the problem of collider bias as identified in this paper. (5) AIV.GT obtains consistent results and has the lowest bias on all datasets. In sum, AIV.GT is able to obtain an unbiasedβ wy from data with latent variables when there exists a pair of AIVs. AIV.GT overcomes the collider bias suffered by IV.tetrad. (a) (b) (c) (d) (e) LSR × × × × × TSLS ✓ × × × × sisVIVE ✓ ✓ ✓ × × IV.tetrad ✓ ✓ ✓ ? ✓ ? ✓ ? AIV.GT ✓ ✓ ✓ ✓ ✓ B. Experiments on Two Real-world Datasets It is challenging to evaluate the performance of causal effect estimators, including AIV.GT on real-world datasets because the ground truth causal DAG and β wy are not available. We select two real-world datasets with empirical estimates available in the literature, including Vitamin D data (VitD) [31], [34] and Schoolingreturns [30]. The two datasets have been extensively studied and analysed before and each has a nominated AIV. Therefore, it is credible to choose them as the benchmark datasets to evaluate AIV.GT. The two datasets have the nominated AIV wrt., (W, Y ), but the conditioning sets are unknown. There is not an available algorithm in literature to discover the conditioning set that instrumentalise the nominated AIV on both datasets. Therefore, we divide the experiments on both datasets into two parts: (1) experiments on AIV.GT in comparison with four estimators without nominated AIVs; (2) experiments comparing AIV.GT with four additional IV estimators that require a nominated AIV. 1) Details of the Two Real-world Datasets: a) Vitamin D (VitD).: This dataset was collected from a cohort study of vitamin D status on mortality, i.e., the potential effect of VitD on death, reported in [34]. The dataset contains 2571 individuals and 5 variables: age, filaggrin (a binary variable indicating filaggrin mutations), vitd (a continuous variable measured as serum 25-OH-D (nmol/L)), time (followup time), and death (binary outcome indicating whether an individual died during follow-up) [31]. A measured value of vitamin D less than 30 nmol/L implies vitamin D deficiency. We take the estimatedβ wy = 2.01 with 95% confidence interval (0.96, 4.26) from the literature [34] as the reference causal effect. b) Schoolingreturns.: This dataset is from the national longitudinal survey of youth (NLSY) of US young employees, aged range from 24 to 34 [30]. The dataset contains 3010 individuals and 19 variables. The treatment is the education of employees, and the outcome is raw wages in 1976 (in cents per hour). The covariates include experience (years of labour market experience), ethnicity (a factor indicating ethnicity), resident information of an individual, age, nearcollege (whether an individual grew up near a 4-year college), Education in 1966 (education66), marital status, father's educational attainment (feducation), mother's educational attainment (meducation), Ordered factor coding family education class (fameducation), and so on. A goal of the studies on this dataset is to investigate the causal effect of education on earnings. We takê β wy = 13.29% with 95% confidence interval (0.0484, 0.2175) from [35] as the reference causal effect. 2) Comparing AIV.GT with the Estimators without Requiring a Known IV: We conduct experiments on the two realworld datasets to assess AIV.GT against the four estimators that do not require nominated IV as in Section IV-A with the simulated data. All experimental results are visualised in Fig. 4 for VitD and Schoolingreturns, respectively. a) Results on VitD.: From Fig. 4, we have the following observations: (1) the estimated result of LSR is close to 0 and far away from the 95% confidence interval of the empirical estimation; (2) the estimated results of TSLS, sisVIVE, IV.tetrad and AIV.GT are close to the reference causal effect 2.01 and fall into the 95% confidence interval. b) Results on Schoolingreturns.: According to Fig. 4, we have the following findings: (1) the estimated result of TSLS is at the bottom of the 95% confidence interval; (2) the estimated results of LSR and sisVIVE fall outside of the Fig. 4. The experimental results of the five estimators without a given IV for all estimators on two real-world datasets. The two dotted lines represent empirically estimated causal effect with 95% confidence interval. Noting that the estimated causal effect of LSR on VitD is close to zero and not visible in the left panel. empirical interval. It is very likely that their assumptions have not been satisfied; (3) the estimated results of IV.tetrad and AIV.GT are in the empirical interval. They are consistency with the reference causal effect [35]. The consistency between the results IV.tetrad and AIV.GT is likely due to the reason that they have found proper conditional sets. The experiments show that AIV-GT can obtain consistent estimations in both real-world datasets. 3) Comparing AIV.GT with the Estimators with Known IVs: We add the four more comparison methods that require the given IVs, which are (1) TSLS.IV, TSLS with a given IV; (2) TSLS.CIV [36], TSLS with a given CIV S by conditioning on X \ {S}; (3) FIVR, causal random forest for instrumental variable regression with a given CIV S and conditioning on X \ {S} [21]; (4) AIViP [24], Ancestral IV estimator in PAG. The two datasets have nominated IVs in the literature. The indicator of filaggrin was used as an IV in VitD [34] and Card [30] used geographical proximity to a college, i.e., nearcollege as an IV in Schoolingreturns. All results of the above four estimators and AIV.GT are visualised in Fig. 5. a) Results on VitD.: AIV.GT discovers {age, time} as a pair of AIVs. They are reasonable AIVs since they affect W (vitd, vitamin D status) but do not directly affect Y (death). From Fig. 5, we see that the results of TSLS.CIV, AIViP and AIV.GT are in the middle of 95% empirical interval of the reference causal effect. b) Results on Schoolingreturns.: AIV.GT discovers {f education, f ameducation} as a pair of AIVs. They are valid IVs because father's educational attainment and family education class affect their child's education, but do not directly affect the child's income. From Fig. 5, we observe that the results of TSLS.IV, AIViP and AIV.GT are in the middle of the 95% empirical interval of the reference causal effect. In a word, AIV.GT, which does not need given AIVs, performs better or comparable with other methods which require a given IV. This shows the potential of the AIV.GT in a broader range of real-world applications. V. RELATED WORK Latent variables are the major obstacle to estimating causal effect from observational data [6], [37], [38]. When the treatment and outcome are confounded, IV methods [5], [6], [13], [14], [32], [39]- [41] provide a solution. Some methods have been developed for standard IV based causal effect estimation when the IVs are given by domain experts [4], [39], such as the well-known two-stage least squares IV estimator [32] (TSLS) which obtains causal effect using the ratio of two regression coefficients. Recently, Athey et al. [21] developed the generalised random forests to estimate conditional causal effects by using non-parametric quantile regression and instrumental variable regression (FIVR). The conditional causal effects can be aggregated into the average causal effect and FIVR has been compared in our experiments. We refer readers to [5], [23], [42] for a review of standard IV based methods. When a CIV is given, a proper conditioning set needs to be identified for unbiased causal effect estimation. Cheng et al. [24] have proposed a data-driven method AIViP for identifying such a conditioning set for casual effect estimation with a given CIV. Our work is different from these works since we focus on discovering AIVs and the corresponding conditioning set simultaneously from data and the proposed method is more general than the existing methods. CIV approach is very similar to the approach of covariate adjustment since both of them need to identify a proper conditioning set. However, there are essential differences between CIV and covariate adjustment. Most methods for covariate adjustment assume not a latent variable between W on Y [6], [19], [43]. There are four graphical criteria for identifying a proper conditioning set from a causal graph: back-door criterion [6], adjustment criterion [44], generalised back-door criterion [43] and generalised adjustment criterion [19]. There are some data-driven methods based on the four graphical criteria [45]- [47]. More detailed discussions for covariate selection can be found in [48]- [50]. There are no works for finding a conditioning set when the pair of (W, Y ) are confounded as discussed in this work. VI. CONCLUSION Estimating causal effects in the presence of latent variables is a challenging problem. IV is a well-known approach to address this challenge. However, most existing IV methods require strong domain knowledge or assumptions to determine an IV. This restricts the practical use of the IV approach. In this paper, we present the theory and a practical algorithm (AIV.GT) for finding valid AIVs and their corresponding conditioning sets from data, to enable data-driven causal effect estimation from data with latent variables. The experiments on synthetic datasets demonstrate that AIV.GT is able to address the challenges of latent variables and outperform the state-of-the-art causal effect estimators. The experimental results on two real-world datasets also show that AIV.GT achieves consistent results with empirical estimates in the literature, implying the practicability of AIV.GT in real-world applications. , J. Li, L. Liu, T.D. Le and J. Liu are with STEM, University of South Australia, Mawson Lakes, South Australia, 5095, Australia (e-mail: {Debo.Cheng,Jiuyong.Li,Lin.Liu,Thuc.Le,Jixue.Liu}@unisa.edu.au). K. Yu is with the School of Computer Science and Information Engineering, Hefei University of Technology, Hefei, 230000, China (e-mail: [email protected]). (Corresponding authors: Debo Cheng and Jiuyong Li) Fig. 1 . 1(a) S is a standard IV for W → Y . (b) S 1 and S 2 are valid IVs wrt., W → Y , conditioning on Z = {X 2 , X 3 }. Definition 2 ( 2Faithfulness [17]). A DAG G = (V, E) is faithful to a joint distribution prob(V) over the set of variables V if and only if every independence present in prob(V) is entailed by G and satisfies the Markov property. A joint distribution prob(V) over the set of variables V is faithful to the DAG G if and only if the DAG G is faithful to the joint distribution prob(V). Definition 7 ( 7Markov equivalent MAGs [18]). Two MAGs M 1 and M 2 with the same nodes are said to be Markov equivalent, denoted M 1 ∼ M 2 , if for all triple nodes X, Y , Z, X and Y are m-separated by Z in M 1 if and only if X and Y are m-separated by Z in M 2 . Definition 8 ( 8PAG [16]). Let [M] be the Markov equivalence class of a MAG M. The PAG P for [M] is a partial mixed graph such that (i). P has the same adjacent relations among nodes as M does; (ii). For an edge, its mark of arrowhead or mark of the tail is in P if and only if the same mark of arrowhead or the same mark of the tail is shared by all MAGs in [M]. P a(Y ), i.e., S W \| P a(Y ) \ {W }, then S must have an edge into Y , i.e., S ∈ Adj(Y )\{W }, since otherwise W → Y in M is visible, which contradicts the invisible edge W → Y in M. Therefore, all direct AIVs in the DAG G are included in the set S = Adj(Y ) \ {Y } of the mapped MAG M. Lemma 1 provides a set of candidate direct AIVs Adj(Y ) \ {W } and reduces the search space of a direct AIV, i.e., the search space of a direct AIV is reduced from O and U ∈ U, and let M = (X ∪ {W, Y }, E) be the MAG mapped from G. Let P = (X ∪ {W, Y }, E ′′ ) be the PAG which encodes the set of MAGs Markov equivalent to M. If there exists a pair of direct AIVs {S i , S j } ⊆ X in the DAG G, then {S i , S j } must be in Adj(Y ) \ {W } in the PAG P. Moreover, the two sets, Z i = possAn(S i ∪ Y ) \ {W, S i } and Z j = possAn(S j ∪ Y ) \ {W, S j } in the PAG P instrumentalise S i and S j in the DAG G, respectively. Hence, Eq.(2) holds for S i and S j and their conditioning sets Z i and Z j . wy = mean(β i ,β j )where the consistent score λ ij is the smallest in Q24: end if 25: end if . a) Results.: The estimation biases AIV.GT and the four compared estimators on the synthetic datasets are visualised in Fig. 3. From Fig. 3, we have the following observations: (1) The biases of LSR are large on all datasets. This is because it does not consider any bias of latent variables in data. (2) TSLS TABLE I SATISFACTION (TICK)/VIOLATION (CROSS) OF THE ASSUMPTIONS OF A METHOD BY A DATASET. "✓ ?" MEANS THE PROBLEM OF COLLIDER BIAS SUFFERED BY IV.TETRAD. Fig. 3 . 3Estimation Bias (%) of the estimators on five synthetic datasets. 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[ "AVA: A Video Dataset of Spatio-temporally Localized Atomic Visual Actions", "AVA: A Video Dataset of Spatio-temporally Localized Atomic Visual Actions" ]	[ "Chunhui Gu ", "Chen Sun ", "David A Ross ", "Caroline Pantofaru ", "Yeqing Li ", "Sudheendra Vijayanarasimhan ", "George Toderici ", "Susanna Ricco ", "Rahul Sukthankar ", "Cordelia Schmid ", "Jitendra Malik " ]	[]	[]	This paper introduces a video dataset of spatiotemporally localized Atomic Visual Actions (AVA). The AVA dataset densely annotates 80 atomic visual actions in 57.6k movie clips with actions localized in space and time, resulting in 210k action labels with multiple labels per human occurring frequently. The main differences with existing video datasets are: (1) the definition of atomic visual actions, which avoids collecting data for each and every complex action;(2) precise spatio-temporal annotations with possibly multiple annotations for each human; (3) the use of diverse, realistic video material (movies). This departs from existing datasets for spatio-temporal action recognition, such as JHMDB and UCF datasets, which provide annotations for at most 24 composite actions, such as basketball dunk, captured in specific environments, i.e., basketball court.We implement a state-of-the-art approach for action localization. Despite this, the performance on our dataset remains low and underscores the need for developing new approaches for video understanding. The AVA dataset is the first step in this direction, and enables the measurement of performance and progress in realistic scenarios.	10.1109/cvpr.2018.00633	[ "https://arxiv.org/pdf/1705.08421v2.pdf" ]	688,013	1705.08421	ebf5d4a7afd348ac4d4c0e42e2c563aae0f713ee	AVA: A Video Dataset of Spatio-temporally Localized Atomic Visual Actions Chunhui Gu Chen Sun David A Ross Caroline Pantofaru Yeqing Li Sudheendra Vijayanarasimhan George Toderici Susanna Ricco Rahul Sukthankar Cordelia Schmid Jitendra Malik AVA: A Video Dataset of Spatio-temporally Localized Atomic Visual Actions This paper introduces a video dataset of spatiotemporally localized Atomic Visual Actions (AVA). The AVA dataset densely annotates 80 atomic visual actions in 57.6k movie clips with actions localized in space and time, resulting in 210k action labels with multiple labels per human occurring frequently. The main differences with existing video datasets are: (1) the definition of atomic visual actions, which avoids collecting data for each and every complex action;(2) precise spatio-temporal annotations with possibly multiple annotations for each human; (3) the use of diverse, realistic video material (movies). This departs from existing datasets for spatio-temporal action recognition, such as JHMDB and UCF datasets, which provide annotations for at most 24 composite actions, such as basketball dunk, captured in specific environments, i.e., basketball court.We implement a state-of-the-art approach for action localization. Despite this, the performance on our dataset remains low and underscores the need for developing new approaches for video understanding. The AVA dataset is the first step in this direction, and enables the measurement of performance and progress in realistic scenarios. Introduction This paper introduces a new annotated video dataset, AVA, which we have collected to support research on action recognition. Fig. 1 shows representative frames. The annotation is person-centric. Every person is localized using a bounding box and the attached labels correspond to (possibly, multiple) actions being performed by the person. There is one action corresponding to the pose of the person (orange text) -whether he or she is standing, sitting, walking, swimming etc. -and there may be additional actions corresponding to interactions with objects or human-human interactions (red or blue text). The proposed framework naturally accommodates different people performing different actions. Our data source is movies -feature films produced in Hollywood or elsewhere -and we analyze successive 3 second long video segments. In each segment, the middle frame is annotated as described above, but the annotators use perceptual cues available from the larger temporal context of the video, including movement cues. The fine scale of the annotations motivates our calling them "Atomic Visual Actions", abbreviated as AVA for the name of the dataset. The vocabulary currently consists of 80 different atomic visual actions. Our dataset consists of 57.6k video segments collected from 192 different movies, where segments are 3 second long videos extracted sequentially in 15 minute chunks from each movie. Using chunks of 15 minutes per video enables diversity at the same time as continuity. We labeled a total of 210k actions, demonstrating that multiple action labels occur frequently. We plan to release the dataset of annotations to the computer vision commu- nity. To motivate the design choices behind AVA, we list below some key findings from the human psychology literature on activity and events. We recommend [38] to the reader interested in a more detailed account. Foundational work [3] observed in meticulous detail the children of a small town in Kansas going about their daily lives. They proposed a unit they called a "behavior episode". Examples of behavior episodes include a group of boys moving a crate across a pit, a girl exchanging remarks with her mother, and a boy going home from school. They noted the hierarchical nature of activity, illustrated in Fig. 2 [3]. At the finest level the actions can be described as simple body movements or manipulation of objects but at coarser levels the most natural descriptions are in terms of intentionality and goal-directed behavior, as already noted in movies of simple moving geometrical figures where such attribution is common among human observers [14]. Newtson [24] experimentally studied how humans perceive a videotaped behavior sequence by asking them to use key presses to mark breakpoints between segments. Zacks et al. [39] applied this technique to study how human perceive everyday activities such as making a bed or doing the dishes. There was clear evidence of hierarchical event structure; the large unit boundaries were also likely to be small unit boundaries. We are now ready to justify three key design choices in AVA: Why short temporal scale? Coarse-scale activity/event understanding is best done in terms of goals and subgoals. But this is essentially an infinite set just like the set of all sentences. Another problem is that goals may not even be evident visually. If we see someone walking, are they doing it for exercise or to get to the grocery store? However if we limit ourselves to fine scales, then the actions are very physical in nature and have clear visual signatures. We hope to eventually understand the coarser scales of activity analysis mediated by conceptual structures such as scripts and schemas, composed from fine scale AVA units, but this is very much future work. Research such as Zachs et al. [39] shows that coarse scale boundaries are a subset of fine scale boundaries and suggests that the fine scale units should prove useful in composing coarser units. We choose to annotate keyframes in 3 second long video segments, as this is dense enough to understand the content and avoids precise temporal annotation shown to be difficult to define precisely; the THUMOS challenge [16] observed that action boundaries (unlike objects) are vague and subjective, leading to significant inter-annotator disagreement and evaluation bias. By contrast, it is feasible for annotators to determine (using ±1.5s of context) whether a specific frame contains the given action. This task is intrinsically less ambiguous than precisely specifying the temporal boundaries. Why person-centric? There are events such as trees falling which do not involve people, but our focus is on the activities of people, treated as single agents. There could be multiple people as in a sports event or just two people hugging, but each one is an agent with his or her own choices, so we can treat each separately. We do not preclude cooperative or competitive behavior, but each agent is described by pose and interaction with objects and humans. Why movies? Ideally we want behavior "in the wild", recorded in a digital form by a modern day version of Barker and Wright [3]. We do not have that. The next best thing is movies. For more than a hundred years cinematographers have been narrating stories in this medium, and if we consider the variety of genres (thrillers, drama, romantic comedies, westerns etc.) and countries with flourishing movie industries (USA, UK, India, China, Japan, Russia etc.) we may expect a significant range of human behavior to show up in these stories. We expect some bias in this process. Stories have to be interesting (To quote David Lodge: Literature is mostly about having sex and not much about having children. Life is the other way round.) and there is a grammar of the film language [2] that communicates through the juxtaposition of shots. That said, in one shot we expect an unfolding sequence of human actions, somewhat representative of reality, as conveyed by good actors. It is not that we regard this data as perfect, just that it is better than working with the assortment of user generated content such as videos of animal tricks, DIY instructional videos, events such as children's birthday parties, and the like. We expect movies to contain a greater range of activities as befits the telling of different kinds of stories. We make it a point to only include video from movies which are at least 30 minutes long, sampling 15 minute intervals. This paper is organized as follows. In Section 2 we review previous action recognition datasets and point out the difference with our AVA dataset. In Section 3, we describe the annotation process. Section 4 presents some interesting statistics from the dataset. Section 5 explains the baseline approach and reports results. We conclude in Section 6. Related work Most popular action recognition datasets, such as KTH [31], Weizmann [4], Hollywood-2 [22], HMDB [21] and UCF101 [33], consist of short clips, manually trimmed to capture a single action. These datasets are ideally suited for training fully-supervised, whole-clip, forcedchoice classifiers. Unfortunately, although convenient, this formulation of action recognition is completely unrealistic -real-world action recognition always occurs in an untrimmed video setting and frequently demands spatial grounding as well as temporal localization. Recently, video classification datasets, such as TrecVid multi-media event detection [25], Sports-1M [19] and YouTube-8M [1], have focused on video classification on a large-scale, in some cases with automatically generated -and hence potentially noisy -annotations. They serve a valuable purpose but address a different need than AVA. Another line of recent work has moved away from video classification towards temporal localization. Activi-tyNet [6], THUMOS [16], MultiTHUMOS [36] and Charades [32] use large numbers of untrimmed videos, each containing multiple actions, obtained either from YouTube (ActivityNet, THUMOS, MultiTHUMOS) or from crowdsourced actors (Charades). The datasets also provide temporal (but not spatial) localization for each action of interest. AVA differs from them both in terms of content and annotation: we label a diverse collection of movies and provide spatio-temporal annotations for each subject performing an action in a large set of sampled frames. A few datasets, such as CMU [20], MSR Actions [37], UCF Sports [29] and JHMDB [18] provide spatio-temporal annotations in each frame for short trimmed videos. The main differences with our AVA dataset are: the small number of actions ranging from 3 (MSR Actions) to at most 21 (JHMDB); the small number of video clips; and the fact that clips are trimmed. Furthermore, actions are complex (e.g., pole-vaulting) and not atomic as in AVA. For example, the UCF Sports dataset [29] consists of 10 sports actions, such as weight lifting, horse riding and diving. Recent extensions, such as UCF101 [33], DALY [35] and Hollywood2Tubes [23] evaluate spatio-temporal localization in untrimmed videos, which makes the task significantly harder and results in a performance drop. However, the action vocabulary is still restricted to a limited number of actions, at most 24, and actions are complex, which makes large-scale extension difficult. Moreover, they do not provide a dense coverage of all actions; a good example is BasketballDunk in UCF101 where only the player performing the dunk is annotated, whereas all the other players are not. These datasets are unrealistic -real-world applications require a continuous annotations of atomic actions, which can then be composed into higher-level events. This paper addresses the main limitations of existing datasets for spatio-temporal action recognition, which all consist of a small number of complex actions annotated in a small number of clips often in specific environments. Here, we depart from these drawbacks and annotate 80 atomic actions densely in 57.6k realistic movie clips resulting in 210k action labels. The AVA dataset is also related to still image action recognition datasets [7,9,12]. Such datasets provide annotations for humans and their actions. Yet, still image action recognition datasets present two major drawbacks. First, the lack of motion information makes disambiguation in many cases difficult or impossible. We found that the context from surrounding video is essential in order to annotate the individual frames. Take for example the actions walking, standing and falling down. They all look very similar if only one frame is given for annotation. This necessity for motion information is also confirmed by our experiments. Second, modeling complex events as a sequence of individual atomic actions is not possible for still images. This is arguably out of scope here, but clearly required in many real-world applications, for which the AVA dataset does provide training data. Data collection The AVA Dataset generation pipeline consists of three stages: movie and segment selection, person bounding box annotation and action annotation. Movie and segment selection The raw video content of the AVA dataset comes from YouTube. We begin by assembling a list of top actors of various nationalities. For each name we issue a YouTube search query, retrieving up to 2000 results. We only include videos with the "film" or "television" topic annotation, a duration of over 30 minutes, at least 1 year since uploaded, and at least 1000 views. We further exclude black & white, low resolution, animated, cartoon, and gaming videos, as well as those containing mature content. Note that there is no explicit action distribution bias in this process, as the selection and filtering criteria does not include any action related keyword, nor does it run any automated action classifier on the video content. Each movie contributes equally to the dataset, as we only label a sub-part ranging from the 15th to the 30th minute. We skip the beginning of the movie to avoid annotating titles or trailers. We choose a duration of 15 minutes so we are able to include more movies under a fixed annotation budget, and thus increase the diversity of our dataset. Each 15-min clip is partitioned into 300 non-overlapping 3s movie segments. We choose 3s because the dataset focuses on atomic actions and our user studies confirm that 3s is a reasonably short length in which atomic actions can be recognized. We choose to label 300 segments consecutively rather than randomly sampling segments from the whole movie to preserve sequences of atomic actions in a coherent temporal context. Every human bounding box and its corresponding action labels are associated with a segment. Note that although annotators annotate only the middle frame, they take into account the information from the surrounding 3s segments. As stated in the introduction, annotating only one frame in a segment is a design choice to avoid ambiguous temporal annotation, which is not only error prone but also labor intensive. This choice also alleviates the shot boundary problem, as the action label and spatial localization are both well defined in the middle frame. Shots where the transition is too close to the middle frame are marked by annotators and removed from the dataset. Bounding box annotation of subjects We localize a person (subject) and his or her actions with a bounding box. When multiple subjects are present in a given middle frame, each subject is shown to the annotator separately for action annotation, and thus their action labels can be different. Since bounding box annotation is manually intensive, we choose a hybrid approach. First, we generate an initial set of bounding boxes using the Faster-RCNN person detector [28]. We set the operating point to ensure highprecision. Annotators then annotate the remaining bounding boxes missed by our detector. Figure 3 shows the user interface for bounding box annotation. This hybrid approach ensures full bounding box recall which is essential for benchmarking, while minimizing the cost of manual annotation. This manual annotation retrieves only 5% more bounding boxes missed by our person detector. This demonstrates the excellent performance of our detector. Incorrect bounding boxes will be marked and removed by annotators in the next stage of action annotation. Action annotation The action labels are generated by crowd-sourced annotators using the interface shown in Figure 4. The left panel shows both the middle frame of the target segment (top) and the segment as a looping embedded video (bottom). The bounding box overlaid on the middle frame specifies the subject whose action needs to be labeled. On the right are text boxes for entering up to 7 action labels. Actions are divided into pose/movement actions (denoted by the green box and required), person-object interactions (denoted by the yellow boxes and optional), and person-person interactions (denoted by the blue boxes and optional). For all three types, we provide a predefined list of actions from which to select. If none of the listed actions is suitable, a check box of "Action Other" is available to flag. Finally, the annotator could flag segments containing blocked or inappropriate content, or incorrect bounding boxes. On average, annotators take 22 seconds to annotate a given video segment. Training and test sets Our training/test sets are split at the video level, so that all segments of one video appear only in one split. In detail, the 192 videos are split into 154 training and 38 test videos, resulting in 46.2k training segments and 11.4k test segments, roughly a 80%, 20% split. Each training/test segment is annotated by three annotators, where any action label annotated by two or more annotators are regarded as correct (to ensure accuracy of the test set). Action labels that are annotated by only one annotator are verified with more replications in the second round. See the analysis of annotation quality in Section 4.1. Annotators see segments in random order. Characteristics of the AVA dataset We start our analysis by discussing the nature and distribution of the data and showing that the annotation quality is high. We then explore the interesting action and temporal structure that makes this dataset truly unique. Finally, we discuss the characteristics that make the dataset challenging for the action detection task. First, some examples to build intuition. Each example is presented as three frames from a clip: the middle frame with a bounding box around the person performing the action, one frame 0.5 seconds before the middle frame, and another 0.5 seconds after. The two additional frames provide context to visualize motion. Figure 5 shows examples of different actions. We can see the huge variation in person bounding box height and position. The cinematography also differs, especially between genres, with different aspect ratios, techniques, and coloring. Shot boundaries may occur within a segment, such as the "fall down" example. However, recall that action labels only correspond to the middle frame, so they are still well defined. Some action instances can be identified from a single frame, such as "phone" or "brush teeth". However many require within-frame context as well as temporal context, such as "take a photo", "fall down" and "listen". This makes the data especially interesting and complex. Figure 6 shows three examples for the action "clink glass" (toast). Even within an action class the appearance varies widely with different person sizes and vastly different contexts. The recipient(s) of the toast may or may not be in the frame, and the glasses may be partially occluded. The temporal extent also varies -in the first the glasses are held up for a long time, in the second the action continues throughout, while in the third the action is not clear until the middle frame. The wide intra-class variety will allow us to learn features that identify the critical spatio-temporal parts of an action -such as touching glasses for "clink glass". Additional examples are in the supplemental material. Annotation quality To assess the consistency of the labels, three people annotated each bounding box in the test set, providing up to 21 raw labels in total. We define the outlier ratio as the number of raw labels that were provided by only one annotator (and are thus uncorroborated), divided by the total number of raw labels provided by all annotators. For example, for a segment with 5 total raw labels out of which 1 is only listed by one annotator, the outlier ratio is 0.2. A ratio of 0 means no uncorroborated labels, and 1 means complete disagreement. Figure 7 shows a histogram of the outlier ratios on the test set. The vast majority of ratios are very low, showing good inter-annotator consistency. answer phone brush teeth hold/carry take a photo fall down listen to (a person) Figure 5. Examples from different label classes. The middle frame with the person bounding box is shown, along with frames at ±0.5 seconds. Note the variety in person size and shape, cinematography and shot boundaries. Temporal and within-frame context are critical for the "take a photo", "fall down" and "listen" examples. Figure 6. Three examples of the label "clink glass" (toast). The middle frame with the person bounding box is shown, along with frames at ±0.5 seconds. The appearance varies widely. Figure 7. Histogram of the between-annotator label outlier ratios in the test set. Since the ratios are generally very low, we conclude that the annotators are consistent. Table 2. Counts of labels per person bounding box in the training set. "Action Other" labels are ignored. Almost all of the bounding boxes have a pose label in the predefined list, and the vast majority have at least one interaction as well. In total there are 76990 person bounding boxes with at least one label. Action structure With the quality of the annotations established, we next examine the data distribution. In total, there are 80 different action labels, in addition to "Action Other", with 14 pose labels, 17 person-person interactions, and 49 person-object interactions. The most frequently occurring pose and interaction labels are shown in Table 1, with a full list in the supplemental material. Note the large variety of poses and interactions, from simple poses like "stand" to complex interactions like "watch (e.g., TV)". One important question is whether the lists of pose and interaction labels are sufficiently complete to describe the wide variety of movie content. The annotators assigned the label "Action Other" for poses or interactions that were not present in any label list (pose, person-person or personobject). Annotators never assigned "Action Other" twice to any one label category. Out of the 3 label categories per bounding box, the "Action Other" label was only used 1.0% of the time in the training set. This implies that the list is indeed sufficiently complete. The data also show interesting structure, with multiple labels for the majority of person bounding boxes. Table 2 provides the frequencies of label counts per person bounding box. Recall there may be multiple people in a segment. "Action Other" labels are not counted. Almost all of the bounding boxes have a pose label from the list, which once again demonstrates that the label list has good coverage. In addition, the majority of bounding boxes have at least one interaction label. This demonstrates that the data are complex and layered despite the atomic nature of the actions. Table 3. The highest and lowest NPMI for pairs of labels for a single person in a given segment that occur together at least once in the training data. Given the large number of examples with at least two labels, we can discover interesting patterns in the data that do not exist in other datasets. The Normalized Pointwise Mutual Information (NPMI) [8] is used in linguistics to represent the co-occurrence between two words, defined as: NPMI(x, y) = ln p(x, y) p(x)p(y) / (− ln p(x, y))(1) Values intuitively fall in the range (−1, 1], with NPMI(x, y) = −1 (in the limit) for pairs of words that never co-occur, NPMI(x, y) = 0 for independent pairs, and NPMI(x, y) = 1 for pairs that always co-occur. Table 3 shows the label pairs with the top 9 and bottom 3 NPMI scores (for pairs that occur at least once). We confirm expected patterns in the data, for example people frequently play instruments while singing. We can also see that martial arts often involve fighting, that people often lift (a person) while playing with kids, and that people hug while kissing. In this dataset people reassuringly do not sleep while standing, and they also do not sit while dancing. All of these pairwise co-occurrences of atomic actions will allow us to build up more complex actions in the future and to discover the compositional structure of complex activities. Temporal structure Another unique characteristic of the AVA dataset is the temporal structure. Recall that consecutive segments of three seconds are annotated, with skipped segments only if the bounding boxes were incorrect (rare) or there was no person in the middle frame. It is interesting to see how the actions evolve from segment to segment. Figure 8 shows the NPMI values for pairs of pose labels in consecutive threesecond segments. The first pose is on the y-axis and the second is on the x-axis. Using the jet color map, an NPMI value of -1 (never co-occur) is dark blue, 0 (independent) is light green, and 1 (always co-occur) is dark red. As expected, transitions often occur between identical pose labels (on the diagonal) and from any label to the common labels "sit", "stand" and "walk" (columns). Moreover, interesting common sense patterns arise. There are frequent transitions from "jump/leap" to "dance" and "crouch/kneel" to "bend/bow" (as someone stands up). Unlikely sequences can also be learned, for example "lie/sleep" is rarely followed by "jump/leap". The transitions between atomic actions with high NPMI scores (despite the relatively coarse temporal sampling) provide excellent training data for building more complex actions and activities with temporal structure. Data complexity The first contributor to complexity is a wide variety of labels and instances. The above analysis discussed the long label lists and the wide distribution of class sizes. The second contributor to complexity is appearance variety. The bounding box size distribution illustrates this. A large portion of people take up the full height of the frame. However there are still many boxes with smaller sizes. The variability can be explained by both zoom level as well as pose. For example, boxes with the label "enter" show the typical pedestrian aspect ratio of 1:2 with average widths of 30% of the image width, and an average heights of 72%. On the other hand, boxes labeled "lie/sleep" are closer to square, with average widths of 58% and heights of 67%. The box widths are quite widely distributed, showing the variety of poses people must undertake to execute the labeled actions. The breadth of poses, interactions, action cooccurrences, and human pose variability make this a particularly challenging dataset. Experiments Experimental setup As shown in Table 1 and the supplemental material, the label distribution in the AVA dataset follows roughly Zipf's law. Since evaluation on very small test set could be unreliable, we only use those classes that have at least 25 test instances to benchmark action localization performance. Our benchmark set consists of 44 action classes that meet the requirement, and they have a minimal number of 90 training instances per class. We randomly select 10% of training data as validation set and use them to tune model parameters. To demonstrate the competitiveness of our baseline methods, we also apply them to the JHMDB dataset [18] and compare the results against the previous state-of-theart. We use the official split1 provided by the dataset for training and validation. One key difference between AVA and JHMDB (as well as many other action datasets) is that action labels in AVA are not mutually exclusive, i.e., multiple labels can be assigned to one bounding box. To address this, we replace the common softmax loss function by the sum of per-class sigmoid losses, so labels do not compete with each other during optimization. We keep the softmax loss on JHMDB as it is the default loss used by previous methods on this dataset. The rest of the baseline model settings are the same for the two datasets. We follow the protocol used by the PASCAL VOC challenge [9] and report the mean average precision (mAP) over all action classes using an intersection-over-union threshold of 0.5. This metric is also commonly used to evaluate action localization. Baseline approach Current leading methods for spatio-temporal action recognition are all based on extensions of R-CNN [10] on videos [11,30,34,35]. We use Faster R-CNN [28] and follow the end-to-end training procedure proposed by Peng & Schmid [26], except replacing the VGG network by the 101layer ResNet [13] which has higher performance on image classification tasks. Our Faster R-CNN is implemented in TensorFlow. To combine RGB with flow results, we first run the region proposal networks (RPN) using RGB and flow models independently. Then, we take top 100 scoring RPN proposals from the output of each network, and perform nonmax suppression on the union. Once these proposals are obtained, we run the RGB and flow models in the "Fast R-CNN" mode, and take the average of classification scores for each bounding box proposal. RGB and optical flow extraction. To extract optical flow, we estimate the camera motion information and use Figure 9. In this scatter-plot the points correspond to the average precision for each of the 44 action classes in the benchmark set. For any such point, the x-coordinate is the number of training examples for that action, and the y-coordinate is the average precision. The colors code for the three different models based on using RGB alone, optical flow alone, and their fusion. the compensated flow fields. The u and v values from the flow fields are truncated to [-20,20], then quantized to [0, 255] to be stored as JPEG images, where the 3rd channel contains all zeros. We employ two methods: the dense optical flow estimation method TVL-1 [27], and a CNNbased iterative Deep Flow model which is a variant of the FlowNet-ss network [17]. It has 11M parameters, takes 360ms per frame on a CPU, and achieves an EPE of 4.4 on Sintel final [5]. We also experiment with single flow vs. a stack of 5 consecutive flows, and expect the latter to perform better as it captures more temporal information. Images and encoded optical flows are resized to 256 by 340 pixels for JHMDB and 600 by 800 pixels for AVA. We use a higher resolution for AVA because the dataset contains a larger portion of small bounding boxes. Model parameters and initialization. We train the Faster R-CNN action detector asynchronously using the momentum optimizer with momentum set to 0.9. The batch size is 64 for the RPN and 256 for the box classifier. Hyperparameters are mostly fixed to default values used in [15] for the COCO object detection task. The only exceptions are the training steps and the learning rates which are selected using the validation set. Our models are trained at the learning rate of 2e-4 for the initial 600K steps, and 2e-5 for the next 200K steps. We initialize model weights using the open-source ResNet-101 checkpoint from [13]. For optical flow model initialization, we take the weights from the first conv1 layer of ResNet-101 for the first two input channels, and duplicate them to match dimensions of flow inputs. JHMDB AVA Two-stream [26] 56.6% N/A Multi-region [26] 58.5% N/A RGB 52.0% 17.1% TVL-1 (1) 25.2% 6.7% TVL-1 (5) 42.3% 8.5% Deep Flow (1) 29.9% 7.3% Deep Flow (5) 48.4% 9.3% RGB + TVL-1 (5) 58.5% 18.1% RGB + Deep Flow (5) 62.0% 18.4% Table 5. Mean average precision at IOU threshold of 0.5 on AVA benchmark, separated by box sizes. Boxes smaller than 42x42 pixels are small, and larger than 96x96 pixels are large. Table 4 summarizes our baseline performance on the JH-MDB and AVA datasets. Our baseline method outperforms the previous state of the art on JHMDB [26] by a significant margin (62.0% vs. 58.5%). Nevertheless, its performance on the AVA benchmark dataset is significantly lower than on JHMDB. This result demonstrates the difficulty of our dataset, and opens up opportunities to explore new, maybe even drastically different solutions for action localization. Experimental results On both datasets, the RGB based model achieves the best standalone performance, and fusion with optical flow models gives better performance. We can also see that using Deep Flow extracted flows and stacking multiple flows are both helpful for JHMDB as well as AVA. Table 5 breaks down the performance by bouding box sizes. The results align with the discovery from the COCO detection challenge, i.e., small bounding boxes have the lowest performance because they are hardest to localize and classify. Figure 9 shows individual class APs with their training example size. As we can see, there are rare classes which are "easy" and that having lots of data does not necessarily mean high mAP. Conclusion This paper introduces the AVA dataset and shows that current state-of-the-art methods, which work well on previous datasets, do not perform well on AVA. This moti-vates the need for developing new approaches. The AVA dataset enables measuring performance and progress in realistic scenarios. Future work includes modeling more complex actions and activities based on our atomic actions (see analysis in section 4.3). Our present day visual classification technology may enable us to classify events such as "eating in a restaurant" at the coarse scene/video level, but only models based on AVA's fine spatio-temporal granularity offer us the hope of understanding at the level of an individual agent's actions. These are essential steps towards imbuing computers with "social visual intelligence" -understanding what humans are doing, what might they do next, and what they are trying to achieve. Acknowledgement We thank Irfan Essa and Abhinav Gupta for discussions and comments about this work. Hug (a person) Figure 11. Examples from different label classes. The middle frame with the person bounding box is shown, along with frames at ±0.5 seconds. Notice the variety in person size and shape, cinematography and shot boundaries. Listen (e.g. to music) Play a musical instrument Figure 12. Examples from different label classes. The middle frame with the person bounding box is shown, along with frames at ±0.5 seconds. Notice the variety in person size and shape, cinematography and shot boundaries. Press Push (an object) Ride (e.g. car, bike, horse) Take (an object) from (a person) Talk to (e.g. self/person) Text on/look at cellphone Figure 13. Examples from different label classes. The middle frame with the person bounding box is shown, along with frames at ±0.5 seconds. Notice the variety in person size and shape, cinematography and shot boundaries. Figure 14. Average Precision at IoU threshold of 50% for the 44 action class in our AVA benchmark dataset (with more than 25 test examples). DeepFlow outperforms RGB for action classes with significant motion such as get up, hit, jog, and walk. Fusion of the two streams generally improves the performance. Figure 15. Example results of our action detection approach with fusion of RGB and flow with a confidence score above 0.8. In case of multiple labels of a human, we display the average box, if IoU > 0.7. Different actors are identified with different colors. Figure 1 . 1The bounding box and action annotations in sample frames of the AVA dataset. Each bounding box is associated with 1 pose action (in orange), 0-3 interactions with objects (in red), and 0-3 interactions with other people (in blue). Note that some of these actions require temporal context to accurately label. Figure 2 . 2This figure illustrates the hierarchical nature of an activity. From Barker and Wright[3], pg. 247. Figure 3 . 3User interface for bounding box annotation. The purple box was generated by the person detector. The orange box (missed by the detector) was manually added by an annotator with a mouse. Figure 4 . 4User interface for action annotation. Refer to Section 3.3 for details. Figure 8 . 8NPMI of pose label transitions between consecutive segments in the jet color map. Y-axis: Pose label in segment [t-3, t] seconds. X-axis: Pose label in segment [t, t+3] seconds. (Poses with less than 100 instances are excluded.) Figure 10 . 10Examples from different label classes. The middle frame with the person bounding box is shown, along with frames at ±0.5 seconds. Notice the variety in person size and shape, cinematography and shot boundaries. Table 4 . 4Mean average precision at IOU threshold of 0.5 for JH-MDB split1 and AVA benchmark.Large Medium Small RGB 17.3% 12.9% 2.7% TVL-1 (1) 7.0% 0.9% 0.9% TVL-1 (5) 8.8% 1.7% 1.7% Deep Flow (1) 7.5% 0.7% 0.9% Deep Flow (5) 9.7% 0.9% 0.9% RGB + TVL-1 (5) 18.5% 14.3% 2.8% RGB + Deep Flow (5) 18.8% 14.0% 2.7% AppendixIn the appendix, we present additional quantitative information and examples for our AVA dataset as well as for our action detection approach.Table 6and 7 present the number of instances for each class of the AVA dataset. We observe a significant class imbalance to be expected in real-world data [c.f. Zipf's Law]. As stated in the paper, we select a subset of these classes (without asterisks) for our benchmarking experiment, in order to have a sufficient number of test examples. Note that we consider the presence of the "rare" classes as an opportunity for approaches to learn from a few training examples.Figures 10-13show additional examples for several action labels in the AVA dataset. For each example, we also display the neighboring frames at ±0.5 seconds in order to provide temporal context. We recall that our annotators use the surrounding 3 seconds to decide on the label. These examples demonstrate the large diversity of our dataset due to the variety in person size and shape, cinematography and shot boundaries.Figure 14shows per-class action detection results for the AVA benchmark dataset. We can observe that flow outperforms RGB for classes with significant motion such as "get up", "hit (a person)", "run/jog", and "walk". 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[ "Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms", "Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms" ]	[ "Han Xiao [email protected] \nZalando Research Mühlenstraße 25\n10243Berlin\n", "Kashif Rasul [email protected] \nZalando Research\nMühlenstraße 2510243Berlin\n", "Roland Vollgraf [email protected] \nZalando Research\nMühlenstraße 2510243Berlin\n" ]	[ "Zalando Research Mühlenstraße 25\n10243Berlin", "Zalando Research\nMühlenstraße 2510243Berlin", "Zalando Research\nMühlenstraße 2510243Berlin" ]	[]	We present Fashion-MNIST, a new dataset comprising of 28 × 28 grayscale images of 70, 000 fashion products from 10 categories, with 7, 000 images per category. The training set has 60, 000 images and the test set has 10, 000 images.Fashion-MNIST is intended to serve as a direct dropin replacement for the original MNIST dataset for benchmarking machine learning algorithms, as it shares the same image size, data format and the structure of training and testing splits. The dataset is freely available at https://github.com/zalandoresearch/fashion-mnist.	null	[ "https://arxiv.org/pdf/1708.07747v2.pdf" ]	702,279	1708.07747	82f1ef648485611779b99d67204c72d39a9c3c4e	Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms 15 Sep 2017 Han Xiao [email protected] Zalando Research Mühlenstraße 25 10243Berlin Kashif Rasul [email protected] Zalando Research Mühlenstraße 2510243Berlin Roland Vollgraf [email protected] Zalando Research Mühlenstraße 2510243Berlin Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms 15 Sep 2017 We present Fashion-MNIST, a new dataset comprising of 28 × 28 grayscale images of 70, 000 fashion products from 10 categories, with 7, 000 images per category. The training set has 60, 000 images and the test set has 10, 000 images.Fashion-MNIST is intended to serve as a direct dropin replacement for the original MNIST dataset for benchmarking machine learning algorithms, as it shares the same image size, data format and the structure of training and testing splits. The dataset is freely available at https://github.com/zalandoresearch/fashion-mnist. Introduction The MNIST dataset comprising of 10-class handwritten digits, was first introduced by LeCun et al. [1998] in 1998. At that time one could not have foreseen the stellar rise of deep learning techniques and their performance. Despite the fact that today deep learning can do so much the simple MNIST dataset has become the most widely used testbed in deep learning, surpassing CIFAR-10 [Krizhevsky and Hinton, 2009] and ImageNet [Deng et al., 2009] in its popularity via Google trends 1 . Despite its simplicity its usage does not seem to be decreasing despite calls for it in the deep learning community. The reason MNIST is so popular has to do with its size, allowing deep learning researchers to quickly check and prototype their algorithms. This is also complemented by the fact that all machine learning libraries (e.g. scikit-learn) and deep learning frameworks (e.g. Tensorflow, Pytorch) provide helper functions and convenient examples that use MNIST out of the box. Our aim with this work is to create a good benchmark dataset which has all the accessibility of MNIST, namely its small size, straightforward encoding and permissive license. We took the approach of sticking to the 10 classes 70, 000 grayscale images in the size of 28 × 28 as in the original MNIST. In fact, the only change one needs to use this dataset is to change the URL from where the MNIST dataset is fetched. Moreover, Fashion-MNIST poses a more challenging classification task than the simple MNIST digits data, whereas the latter has been trained to accuracies above 99.7% as reported in Wan et al. [2013], Ciregan et al. [2012]. We also looked at the EMNIST dataset provided by Cohen et al. [2017], an extended version of MNIST that extends the number of classes by introducing uppercase and lowercase characters. How- Fashion-MNIST Dataset Fashion-MNIST is based on the assortment on Zalando's website 2 . Every fashion product on Zalando has a set of pictures shot by professional photographers, demonstrating different aspects of the product, i.e. front and back looks, details, looks with model and in an outfit. The original picture has a light-gray background (hexadecimal color: #fdfdfd) and stored in 762 × 1000 JPEG format. For efficiently serving different frontend components, the original picture is resampled with multiple resolutions, e.g. large, medium, small, thumbnail and tiny. We use the front look thumbnail images of 70, 000 unique products to build Fashion-MNIST. Those products come from different gender groups: men, women, kids and neutral. In particular, whitecolor products are not included in the dataset as they have low contrast to the background. The thumbnails (51 × 73) are then fed into the following conversion pipeline, which is visualized in Figure 1. 1. Converting the input to a PNG image. 2. Trimming any edges that are close to the color of the corner pixels. The "closeness" is defined by the distance within 5% of the maximum possible intensity in RGB space. 3. Resizing the longest edge of the image to 28 by subsampling the pixels, i.e. some rows and columns are skipped over. 4. Sharpening pixels using a Gaussian operator of the radius and standard deviation of 1.0, with increasing effect near outlines. 5. Extending the shortest edge to 28 and put the image to the center of the canvas. 6. Negating the intensities of the image. 7. Converting the image to 8-bit grayscale pixels. Figure 1: Diagram of the conversion process used to generate Fashion-MNIST dataset. Two examples from dress and sandals categories are depicted, respectively. Each column represents a step described in section 2. For the class labels, we use the silhouette code of the product. The silhouette code is manually labeled by the in-house fashion experts and reviewed by a separate team at Zalando. Each product contains only one silhouette code. Table 2 gives a summary of all class labels in Fashion-MNIST with examples for each class. Finally, the dataset is divided into a training and a test set. The training set receives a randomlyselected 6, 000 examples from each class. Images and labels are stored in the same file format as the MNIST data set, which is designed for storing vectors and multidimensional matrices. The result files are listed in Table 1. We sort examples by their labels while storing, resulting in smaller label files after compression comparing to the MNIST. It is also easier to retrieve examples with a certain class label. The data shuffling job is therefore left to the algorithm developer. Experiments We provide some classification results in Table 3 to form a benchmark on this data set. All algorithms are repeated 5 times by shuffling the training data and the average accuracy on the test set is reported. The benchmark on the MNIST dataset is also included for a side-by-side comparison. A more comprehensive table with explanations on the algorithms can be found on https://github.com/zalandoresearch/fashion-mnist. Conclusions This paper introduced Fashion-MNIST, a fashion product images dataset intended to be a dropin replacement of MNIST and whilst providing a more challenging alternative for benchmarking machine learning algorithm. The images in Fashion-MNIST are converted to a format that matches that of the MNIST dataset, making it immediately compatible with any machine learning package capable of working with the original MNIST dataset. Table 1 : 1Files contained in the Fashion-MNIST dataset.Name Description # Examples Size train-images-idx3-ubyte.gz Training set images 60, 000 25 MBytes train-labels-idx1-ubyte.gz Training set labels 60, 000 140 Bytes t10k-images-idx3-ubyte.gz Test set images 10, 000 4.2 MBytes t10k-labels-idx1-ubyte.gz Test set labels 10, 000 92 Bytes Table 2 : 2Class names and example images in Fashion-MNIST dataset.Label Description Examples 0 T-Shirt/Top 1 Trouser 2 Pullover 3 Dress 4 Coat 5 Sandals 6 Shirt 7 Sneaker 8 Bag 9 Ankle boots Table 3 : 3Benchmark on Fashion-MNIST (Fashion) and MNIST.Test Accuracy Table 3 - 3continued from previous pageTest Accuracy Table 3 - 3continued from previous pageTest Accuracy Table 3 - 3continued from previous pageTest Accuracy https://trends.google.com/trends/explore?date=all&q=mnist,CIFAR,ImageNet Zalando is the Europe's largest online fashion platform. http://www.zalando.com Multi-column deep neural networks for image classification. D Ciregan, U Meier, J Schmidhuber, Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on. IEEED. Ciregan, U. Meier, and J. Schmidhuber. Multi-column deep neural networks for image classifi- cation. In Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on, pages 3642-3649. IEEE, 2012. Emnist: an extension of mnist to handwritten letters. G Cohen, S Afshar, J Tapson, A Van Schaik, arXiv:1702.05373arXiv preprintG. Cohen, S. Afshar, J. Tapson, and A. van Schaik. Emnist: an extension of mnist to handwritten letters. arXiv preprint arXiv:1702.05373, 2017. Imagenet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L.-J Li, K Li, L Fei-Fei, Computer Vision and Pattern Recognition. IEEECVPR 2009. IEEE Conference onJ. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical im- age database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pages 248-255. IEEE, 2009. Learning multiple layers of features from tiny images. A Krizhevsky, G Hinton, A. Krizhevsky and G. Hinton. Learning multiple layers of features from tiny images. 2009. Gradient-based learning applied to document recognition. Y Lecun, L Bottou, Y Bengio, P Haffner, Proceedings of the IEEE. 8611Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278-2324, 1998. Regularization of neural networks using dropconnect. L Wan, M Zeiler, S Zhang, Y L Cun, R Fergus, Proceedings of the 30th international conference on machine learning (ICML-13). the 30th international conference on machine learning (ICML-13)L. Wan, M. Zeiler, S. Zhang, Y. L. Cun, and R. Fergus. Regularization of neural networks using dropconnect. In Proceedings of the 30th international conference on machine learning (ICML- 13), pages 1058-1066, 2013.	[ "https://github.com/zalandoresearch/fashion-mnist.", "https://github.com/zalandoresearch/fashion-mnist." ]
[ "Capillary-driven binding of thin triangular prisms at fluid interfaces", "Capillary-driven binding of thin triangular prisms at fluid interfaces" ]	[ "Joseph A Ferrar \nDepartment of Chemical Engineering\nUniversity of Michigan\n48109Ann ArborMI\n", "Deshpreet S Bedi \nDepartment of Physics\nUniversity of Michigan\n48109Ann ArborMI\n\nMartin Fisher School of Physics\nBrandeis University\n02454WalthamMA\n", "Shangnan Zhou \nDepartment of Physics\nUniversity of Michigan\n48109Ann ArborMI\n\nDepartment of Physics\nTheoretical Physics\nStanford University\n94305StanfordCA\n", "Peijun Zhu \nDepartment of Physics\nUniversity of Michigan\n48109Ann ArborMI\n\nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPennsylvania\n", "Xiaoming Mao \nDepartment of Physics\nUniversity of Michigan\n48109Ann ArborMI\n", "Michael J Solomon \nDepartment of Chemical Engineering\nUniversity of Michigan\n48109Ann ArborMI\n" ]	[ "Department of Chemical Engineering\nUniversity of Michigan\n48109Ann ArborMI", "Department of Physics\nUniversity of Michigan\n48109Ann ArborMI", "Martin Fisher School of Physics\nBrandeis University\n02454WalthamMA", "Department of Physics\nUniversity of Michigan\n48109Ann ArborMI", "Department of Physics\nTheoretical Physics\nStanford University\n94305StanfordCA", "Department of Physics\nUniversity of Michigan\n48109Ann ArborMI", "Department of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPennsylvania", "Department of Physics\nUniversity of Michigan\n48109Ann ArborMI", "Department of Chemical Engineering\nUniversity of Michigan\n48109Ann ArborMI" ]	[]	We observe capillary-driven binding between thin, equilateral triangular prisms at a flat air-water interface. The edge length of the equilateral triangle face is 120 m, and the thickness of the prism is varied between 2 and 20 m. For thickness to length (T/L) ratios of 1/10 or less, pairs of triangles preferentially bind in either a tip-to-tip or tip-tomidpoint edge configurations; for pairs of particles of thickness T/L = 1/5, the tip of one triangle binds to any position along the other triangle's edge. The distinct binding configurations for small T/L ratios result from physical bowing of the prisms, a property that arises during their fabrication. When bowed prisms are placed at the air-water interface, two distinct polarity states arise: prisms either sit with their center of mass above or below the interface. The interface pins to the edge of the prism's concave face, resulting in an interface profile that is similar to that of a capillary hexapole, but with important deviations close to the particle that enable directed binding. We present corresponding theoretical and numerical analysis of the capillary interactions between these prisms and show how particle bowing and contact-line pinning yield a capillary hexapole-like interaction that results in the two sets of distinct, highly-directional binding events. Prisms of all T/L ratios self-assemble into space-spanning open networks; the results suggest design parameters for the fabrication of building blocks of ordered open structures such as the Kagome lattice.	10.1039/c8sm00271a	[ "https://export.arxiv.org/pdf/1802.02949v1.pdf" ]	206,139,757	1802.02949	6ea00196570053549d22f9ead93b77d61f8919e7	Capillary-driven binding of thin triangular prisms at fluid interfaces Joseph A Ferrar Department of Chemical Engineering University of Michigan 48109Ann ArborMI Deshpreet S Bedi Department of Physics University of Michigan 48109Ann ArborMI Martin Fisher School of Physics Brandeis University 02454WalthamMA Shangnan Zhou Department of Physics University of Michigan 48109Ann ArborMI Department of Physics Theoretical Physics Stanford University 94305StanfordCA Peijun Zhu Department of Physics University of Michigan 48109Ann ArborMI Department of Physics and Astronomy University of Pittsburgh 15260PittsburghPennsylvania Xiaoming Mao Department of Physics University of Michigan 48109Ann ArborMI Michael J Solomon Department of Chemical Engineering University of Michigan 48109Ann ArborMI Capillary-driven binding of thin triangular prisms at fluid interfaces 1 (Corresponding Author: [email protected] and [email protected]) 2 We observe capillary-driven binding between thin, equilateral triangular prisms at a flat air-water interface. The edge length of the equilateral triangle face is 120 m, and the thickness of the prism is varied between 2 and 20 m. For thickness to length (T/L) ratios of 1/10 or less, pairs of triangles preferentially bind in either a tip-to-tip or tip-tomidpoint edge configurations; for pairs of particles of thickness T/L = 1/5, the tip of one triangle binds to any position along the other triangle's edge. The distinct binding configurations for small T/L ratios result from physical bowing of the prisms, a property that arises during their fabrication. When bowed prisms are placed at the air-water interface, two distinct polarity states arise: prisms either sit with their center of mass above or below the interface. The interface pins to the edge of the prism's concave face, resulting in an interface profile that is similar to that of a capillary hexapole, but with important deviations close to the particle that enable directed binding. We present corresponding theoretical and numerical analysis of the capillary interactions between these prisms and show how particle bowing and contact-line pinning yield a capillary hexapole-like interaction that results in the two sets of distinct, highly-directional binding events. Prisms of all T/L ratios self-assemble into space-spanning open networks; the results suggest design parameters for the fabrication of building blocks of ordered open structures such as the Kagome lattice. Introduction Attractive, long-range capillary interactions arise between particles at an air-liquid or liquid-liquid interface because they minimize the free energy generated by the particleinduced curvature of the interface. [1][2][3][4] Self-assembly of colloidal, granular, and millimeter-scale particles has been observed at both air-water and oil-water interfaces due to such capillary-induced pair attractions. At the colloidal scale, short-range electrostatic repulsions also influence self-assembly. [5][6][7] Recently, spatially anisotropic capillary attractions have been used to produce ordered particle chains and complex open networks at fluid interfaces. For example, colloidal ellipsoids at oil-water and air-water interfaces form such structures. Particle configurations that arise at the interface are dependent on particle surface geometry, chemistry, and wettability. 7,8 For example, cylinders and related anisotropic shapes assemble into chains at an oil-water interface, with the specific particle faces that bind determined by the curvature of the particle face. In these cases, the spatial anisotropy of the capillary interaction is a consequence of differences in the local curvature of the particle. 9,10 Specifically, cylindrical particles at an oil-water interface generate an elliptical quadrupolar interaction in the far field: the interface deforms in one direction at the flat ends of the cylinder and the opposite direction at the curved edges. These deformations yield attractive capillary interactions between faces with like-curvature and repulsions between faces with opposite-curvature. 9,10 Capillary-driven self-assembly therefore is a path to the bottom-up assembly of open and network structures. Such structures are targets for self-assembly due to interesting and potentially useful properties, ranging from photonic bandgaps to unusual mechanical response, that arise from the incorporation of voids into material structures. [11][12][13][14] These networks and voids deform in ways that significantly differ from close-packed structures, and can lead to mechanical properties such as negative Poisson's ratio and rigidity at ultra-low density. For example, open networks of colloidal ellipsoids assembled at a fluid-fluid interface exhibited a significantly enhanced low frequency modulus as compared to close-packed networks of colloidal spheres at similar particle concentrations. 7 Current methods to fabricate open networks include the above described capillary-driven assembly of colloidal ellipsoids, 7,8 and polymer-molded microhexagram prisms, 15 millimeter-scale branched shapes produced by 3D printing, 16 self-assembly of patchy colloidal spheres, [17][18][19] and top-down approaches on the granular and millimeterscale such as polymeric 3D-printing, 20 quasi-2D-polymer molding 21 and lithography. 22 Bottom-up self-assembly methods can be advantageous compared to these top-down methods, because of the potential scalability of self-assembly processes. 23,24 Here we investigate the possibility of using a hexapolar-like interaction generated between pairs of thin, triangular microprisms to self-assemble space-spanning open networks at low particle concentrations. Assembly of such a rigid, stabilizing network by control of lateral interactions could yield complex fluids with useful bulk and interfacial rheological properties of interest in a variety of fields and industries, such as food science, drug delivery, and petrochemical processing. [25][26][27][28] Thin prismsquasi-2D shapes with finite but small thicknesscan generate capillary interactions at fluid-fluid interfaces if sufficient interface deformation is induced at the prism edges. The symmetry of thin, triangular prisms indicates that the interaction will be similar to that of capillary hexapoles when these prisms are not too close. This interaction may lead to binding of the triangles at vertices and yield ordered structures such as the kagome and the twisted kagome latticesa family of isostatic structures with a unit cell of two inverted triangles ( Figure 1). These kagome lattices are known to display unusual mechanical properties such as a negative Poisson's ratio and floppy edge modes. 12,14,17,[29][30][31][32][33] To improve the prospects for assembling such complex open structures either ordered or disorderedthe pair-binding behavior of thin homogenous microprisms at interfaces should be investigated. Better understanding of the transient and steady-state binding can identify conditions for which ordered and/or disordered networks ( Figure 1) might occur; each structural family might itself exhibit interesting mechanical properties. 7 Open, planar networksboth disordered and orderedare therefore interesting targets for interfacial self-assembly. Here we observe capillary-driven binding of thin, triangular prisms, with edge lengths ~120 m and thicknesses between 2.5 and 20 m at an air-water interface. The pairwise interaction between prisms is measured and modeled. The particles are produced by polymeric photolithography; the anisotropic, directional interactions are introduced by the unexpected generation of a capillary hexapole, which arises due to the 2D triangular shape and the contact line curvature induced by edgewise bowing of the prisms that is introduced at the time of synthesis. We record the different types of binding events observed between the vertices and flat edges of the interacting prisms. The type of binding event is predictable from the up/down polarity of particle attachment to the interface, which is well characterized by imaging out-of-plane and by environmental scanning electron microscopy. From the particle shape and bowed radius of curvature, we compute the interface geometry and the resulting capillary interaction numerically. We find that the capillary interaction is similar to hexapolar interaction in the far field, but deviates from ideal hexapoles in the near field such that the variability of the potential is largest at the tips. We also simulate trajectories of particle binding events numerically using the potential we calculated, and we obtain good agreement with Materials & Methods Particle Fabrication Particles are fabricated via SU-8 photolithography. 9,10,34 First, a sacrificial release layer of Omnicoat (Microchem Corp.) is spun onto a glass wafer (D-263 borosilicate glass, Precision Glass & Optics) and baked at 200 C until cured to a thickness of tens of nanometers (1-2 minutes). After cooling to room temperature, SU-8 2000 series photoresist (Microchem Corp.) is spun on top of the Omnicoat layer to the desired prism thickness and baked at 95 C until cured (~2-5 minutes depending on resist thickness). Next, the wafer is exposed to UV light (365 and 405 nm) through a chrome photomask that encodes the particle pattern (Fineline Imaging) until exposure energies of 60-150 mJ (depending on resist thickness) are achieved. The wafer is then heated at 95 C for 2-5 minutes (depending on resist thickness) to ensure adequate cross-linking of the exposed photoresist. The wafer is immersed in SU-8 developer solution (Microchem Corp.) until the non-photopolymerized SU-8 is washed away (~1-5 minutes depending on resist thickness), leaving the cross-linked particles immobilized on top of the release layer. The wafer is exposed to oxygen plasma for 20 minutes, which facilitates release of the particles into isopropanol. The particles are stored in isopropanol, where they remain stable for several weeks. This process yields approximately 10 6 particles per fabrication. Placement of particles at the air-water interface A flat interface is formed between air and deionized water in a chamber (Thermo Scientific Lab-Tek II, 2 Chamber, coverslip 0.13-0.17 m thick, type 1.5) of dimension 2.0 x 2.0 cm, mounted on to the stage of a Nikon A1Rsi confocal microscope. The chamber's large experimental area and acrylic walls allow for a flat air-water interface to formwithout the need for surface modification of the chamberthrough careful placement of water in the chamber with a transfer pipette. The walls of the chamber are manually wet prior to filling the center of the chamber with water, in order to prevent uneven attachment of the interface to the walls of the chamber. 10 L of the particle stock solution is carefully placed in one or two drops at the air-water interface using a gas-tight Hamilton 100 L syringe. Observation of binding events with optical and reflection microscopy The interface is imaged with the transmission and 488 nm reflection channels of a Nikon A1Rsi confocal microscope (10x objective, NA = 0.25) in a square region of 1270 x 1270 m. Images of pair binding and assembly are acquired at frame rates of 15 frames per second (fps) for prisms of T/L > 1/25 and 30 fps for T/L = 1/50. For pair binding experiments, particle positions, relative orientations, and trajectories are tracked by least square fitting of particle edges, as detected by scikit-image (http://scikitimage.org/). Quantifying capillary attraction energies through observation of interface deformation with environmental SEM Environmental SEM (eSEM, FEI Quanta 3D) is used to observe interface deformation and curvature around the edges of the particles. A gel trapping technique is used to immobilize particles at the interface. 10,35 Briefly, gellan gum, which was generously supplied as a gift from CP Kelco, (low acyl Kelcogel, 2 wt. %) is dissolved in deionized water at 95 °C. The gellan solution remains fluid at temperatures greater than 50 °C. The gellan solution is placed into an eSEM imaging chamber at 70 °C, and prisms are spread at the interface. The imaging chamber is at room temperature, a condition at which the gellan solution crosslinks, immobilizing the prisms for later imaging. Identical prism-prism capillary-driven binding is observed at the gellan solution-air interface as is observed at the pure water-air interface, suggesting that the gellan solution has a negligible effect on the capillary-binding mechanism, consistent with reports of right cylinders at gellan interfaces. 10 Modeling of the pairwise interaction potential using Surface Evolver In this paper, we consider the capillary interaction potential between triangular prisms, for which an analytic solution to Laplace's equationespecially close to the prisms, where simplifying assumptions cannot be madeis not available (more discussion of capillary interactions can be found in the Supplementary Information). Therefore, we use Surface Evolver, a program widely utilized to model the shape of liquid surfaces and interfaces, to numerically calculate the shape of the interface. 36 The solution is achieved by an algorithmic succession of steps involving gradient and conjugate gradient descent iterations and interface mesh refinements to minimize the interfacial energy subject to specific boundary conditions. As we discuss in the Results section, we compute the interface shape given a pinned contact line around a bowed equilateral triangle of side length 120 m. In particular, this triangle is formed by the intersection of three planes containing great circles with a thin spherical shell. Specifying the behavior of the contact lines yields one set of boundary conditions; the far-field boundary condition is that the interface is flat. To allow for the condition of mechanical equilibrium to be satisfied, we do not explicitly fix the height of the far-field boundary, which, in effect, allows for changes in the relative height between the prisms and the equilibrium, unperturbed height of the interface. In order to generate a potential energy landscape of a pair of interacting triangles, we run Surface Evolver simulations on a regularly-spaced grid in ( , 1 , 2 ) configuration space, where is the distance between the centers of the two triangles, 1 , 2 are the orientations of the two triangles (see Figure for their definitions). The parameter ranges are 132 m ≤ ≤ 360 m and 0 ∘ ≤ 1 , 2 < 360 ∘ , with grid spacings of 12 m in distance and 5° in orientation. The actual number of simulations needing to be run is substantially reduced by symmetries inherent in the system. Simulations are run for both particles with the same bowing polarity and opposite bowing polarities; the definitions of bowing and polarity are introduced in the results. Computing particle trajectories leading to pair binding For a particle moving through a fluid at relatively slow speeds and at a low Reynolds number, Re, the drag force is given by = −.(1) Analogously, a particle rotating in a fluid at slow speeds experiences a drag torque, = −.(2) In these equations, and are the viscous damping coefficients for the center-of-mass and rotational degrees of freedom of the triangular prisms, respectively. Assuming a quasistatic force balance on the particles, we can equate the corresponding drag and capillary forces to obtain the following system of differential equations of motion for the pair of prisms. This is a valid assumption to make, as both the Reynolds number Re = / , which is a ratio of inertial forces to viscous forces within a fluid, and the capillary number Ca = / , which is a ratio of viscous forces to surface tension of an interface, where is the density of the liquid, is the velocity of the particle, and is the dynamic viscosity of the liquid, are quite small (for a set of characteristic values = 10 3 kg/m 3 , = 120 m , = 1.002 × 10 −3 Pa ⋅ s , = 72 × 10 −3 N/m, and ∼ 4 × 10 −4 m/s, which is representative of the upper range of velocities observed in the dilute binding events, Re ≈ 0.048 and Ca ≈ 5.6 × 10 −6 , both of which are small compared to unity), so that both inertia and viscous deformation of the interface can be neglected, as in Refs. 8 and 10. In this case, hydrodynamic interactions can safely be ignored, and the force balance equations are ( ) = − ( 1 , 2 , ) (3.1) 1 ( ) = − 1 ( 1 , 2 , ) (3.2) 2 ( ) = − 2 ( 1 , 2 , ) (3.3) Discretizing the time derivative of our desired quantities allows us to iteratively solve for the trajectories of the prisms: ( ) = ( −1 ) − 1 Δ (4.1) ( ) = ( −1 ) − 1 Δ (4.2) where , − 1 correspond to the ℎ , ( − 1) ℎ time-step, respectively, and = 1,2 corresponds to the particle. The partial derivatives are taken of an interpolated interaction potential using the potential values determined via Surface Evolver on the regular ( , 1 , 2 ) grid, as discussed previously. The viscous damping coefficients are not independent constants. They both originate from the interaction between the particle and the surrounding fluid. The center of mass drag depends on the particle orientation and the direction of center-of-mass motion. To our knowledge there is no literature on the fluid drag of triangular prisms, so in this study we make a simplifying assumption that both and are constants, and we estimate their magnitude by considering the following calculation: The work done over a small linear translation of Δ due to the drag force is = Δ , while the work done over a small rotation by Δ (in radians) due to the drag torque is given by = Δ . We can attribute the work done by each drag to the energy required to move the fluid due to the particle's motion. If we keep the small distance traversed by a single tip of the (equilateral) triangle the same in both cases, Δ , then the amount of rotation associated with that movement is given by Δ = Δ / , where is the distance from the centroid to the tip. If the equilateral triangle has a side length of , then = /√3. Comparing these two cases, the amount of fluid that is moved is of the same order, which means that we can equate and . We also assume that these two motions require the same amount of time, Δ . In this case, we obtain Δ Δ Δ = Δ Δ Δ ,(5) so that the ratio between the two drag coefficients becomes = ( Δ Δ ) 2 = 2 .(6) For angles measured in degrees, this equation becomes ̃= ( Δ Δ̃) 2 = 2 ( 180 ) 2 .(7) For = 120 m, = 69.3 m and ̃/ = 1.46 ( m/°) 2 . Theoretical power-law relation for dilute binding trajectories For an experimental system exhibiting pairwise binding due to capillary interactions, the resultant trajectory can be characterized by the form of the separation distance as a function of time-to-contact, − , where is the first instance where the particles touch. If the trajectory obeys a power-law relation such that ∼ ( − ) , the exponent gives insight into the order of the capillary interaction, as we presently show. The capillary interaction energy between two ideal multipoles is 12 ∼ − , where = 2 for an interaction between two multipoles of order . Equating the resultant capillary force (for fixed orientations) to the viscous drag force yields a simple first-order differential equation ∼ −( +1) ,(8) which can be solved to obtain the desired result that the pairwise binding trajectory between two ideal capillary multipoles of order is characterized by a power-law exponent = 1 + 2 = 1 2( + 1) .(8) Exponents of particular importance in this context are = 1/6 (two quadrupoles) and = 1/8 (two hexapoles). Results Capillary-driven binding of triangular prisms at a flat air-water interface Prisms of all T/L ratios undergo lateral capillary-driven binding at a flat air-water interface. Capillary attractions yield particle-particle binding immediately upon particle attachment at the interface. Over a period of about one hour, the prisms self-assemble into open structures of progressively increasing size (as shown for the case of T/L = 1/25 in Figure ). shows an open network formed by T/L = 1/50 prisms, row 2 is for T/L = 1/25 prisms, row 3 is T/L = 1/10, and row 4 is T/L = 1/5). The networks span several millimeters in space and are visible to the eye. For the three thinnest T/L ratios, the network's steadystate microstructure is comprised of a mix of dense, close-packed regions (with numerous prisms bound edge-to-edge), long strands, and large voids. On the other hand, relative to the thinner prisms, the network self-assembled from T/L = 1/5 prisms contains significantly fewer prisms in close-packing configurations, less chaining, smaller voids, and a generally more homogeneous prism density throughout the image. In the course of imaging the open networks (c.f Figure 4), the location of the microscope's focal plane relative to the air-water interface was varied and an interesting feature of the pair binding was observed. Upon varying the focal plane slightly above and below the interface, we observe that T/L = 1/50, 1/25, and 1/10 prisms are pinned to the interface in such a way that their centers-of-mass either sit slightly above or below the interface. The second and third columns of Fig. 4 show this kind of imaging in the same 1270 x 1270 m region of the open network as imaged in the first column. In column one, the microscope's focal plane is located at the air-water interface. All prisms are clear and visible, as demonstrated by their sharp, dark edges and tips, as well as their bright bodies. In the second column, the microscope's focal plane is located ~ 200 m below the air-water interface. For the three thinnest prisms (T/L = 1/50, 1/25 and 1/10), some prisms remain clearly visible, with their dark edges and tips appearing thicker and even more discernable than in the first column and their bodies remaining bright, while all other prisms fall distinctly less visible, with their tips becoming bright and their edges and bodies appearing darker and faded. In column three, the microscope's focal plane is located a similar amount above the air-water interface, in the opposite direction of the second column images. For the three thinnest prisms (T/L = 1/50, 1/25, and 1/10), the prisms that were clearly visible in the second column now appear faded, while the particles that appeared faded in column two are now clearly visible. (Additionally, a very small fraction of T/L = 1/50 prisms appear equally visible on both edges of the interface.) This visual contrast in particles of opposite polarity is a scattering effect owing to transmission imaging, and indicates that the particles either sit below or above the interface. In the ensuing discussion, we define this as the "polarity" of the interface attachment. A prism with positive polarity refers to a prism whose center of mass sits above the interface in the assembly experiments, while a prism with negative polarity refers to a prism whose center of mass sits below the interface. Although the three thinnest prisms are divided into populations located above and below the interface, the thickest prisms (T/L = 1/5) do not exhibit such visible vs. faded polarity; these prisms all appear equally visible relative to one another in both columns two and three. The relative image quality for the T/L = 1/5 prisms appears better below the interface (column two) than above (column three), suggesting that all these prisms are situated slightly below the interface. To further investigate the precise manner of prism interface attachment, we observe the prisms using eSEM ( Figure 5). Figure 5 confirms polarity in interface attachment for T/L = 1/50, 1/25, 1/10 but not for the thickest (T/L = 1/5) prism, consistent with the results from changing the optical microscopy focal plane. eSEM images of T/L = 1/50 prisms are shown in Figure a and d. Figure a shows a prism whose center of mass lies above the gelled interface in the water phase, and Figure d shows a prism whose center-of-mass lies below the gelled interface in the air phase. In addition, significant particle bowing along each of the three prism edges is observed. Figure e is covered by the gelled interface (as evidenced by the rippling texture on top of this prism, which is consistent with the surface of the gelled water phase elsewhere in the image), while several other prisms in the image sit with their top faces uncovered by the interface (as evidenced by the smooth texture of the exposed faces of these particles, relative to the rippling surface of the gelled water phase). The T/L = 1/25 prisms do not appear as bowed as the T/L = 1/50 prisms. Still, evidence for polarity in prism-interface attachment is apparent because the covered prisms' centers of mass sit below the interface (in the gelled water phase), and the uncovered prisms' centers of mass sit above the interface (in the air phase). Polarity is again observed for T/L = 1/10 prisms, shown in Figure c and f. Several prisms rest with their centers of mass below the interface, and the top face of the prism is covered by the surface of the gelled interface, while other prisms sit substantially higher on the interface, with their top faces exposed to the air phase. Polarity of the particle position relative to the interface is not apparent for T/L = 1/5 prisms. Fig. 5g is representative of all observed T/L = 1/5 prisms; the interface is observed to rise at the corners of the prisms, and prisms all appear to sit at the same interface position, relative to both the interface and to one another. The optical micrographs also show evidence for bowing in T/L =1/50 (Fig. 5a) and 1/25 ( Fig. 5b) prisms. That is, prisms of assigned polarity appear to have bright, central bodies and dark tips when particles reside on the same side of the interface as the focal plane, and faded central bodies and bright tips when they reside on the opposite side of the interface as the particles. This illumination contrast appears consistent with a difference in the position of the prism central body and tips relative to the microscope's focal plane. Moreover, referring back to Figure , bowing is apparent in the SEM images of the particles as originally fabricated. Apparently, this bowing is a permanent, reproducible feature of the thin prism fabrication; it persists from synthesis to assembly. Bowing can specify the curvature of the interface at the prism boundary by contact line pinning. This interfacial curvature in turn determines the capillary-driven attraction between the prisms. Figure show that prisms with positive polarity (on top of the interface) are bowed downwards (with tips pointing towards the water phase), and negative polarity (below the interface) prisms are bowed upwards (with tips pointing towards the air phase). In both cases, the interface appears pinned to the corner of the prism's edge and to the concave face. Thus, the curvature of the interface follows the curvature of the bowed prism. The result is that the interface curvature at the tips and edges of triangle is opposite for prisms of positive and negative polarity. Inhomogeneity in prism surface wettingwhich could potentially be introduced during prism fabrication as described in the methodsis not the source of prism polarity. Thin prisms (T/L < 1/10) fabricated with or without plasma treatment on one side each exhibit the two polarity states. The plasma treatment affects wetting; the contact angle change in the plasma treated particles is ~70˚ immediately following treatment. This insensitivity to plasma treatment supports the hypothesis that prism bowing is the primary driver of the observed polarity. Correlation between prism interface polarity and bonding state The correlation between particle polarity (up or down interface attachment) and bonding state is examined for T/L = 1/50 prisms in Figure ; Comparable measurements for T/L = 1/25 are available in SI Figure 3. Each row of Figure shows one 1270 x 1270 m region of an open network. In the first column, the focal-plane is located above the interface. In the second column, the focal-plane is located below the interface. In these first two columns, each prism is assigned a polarity, determined by the location of the prism center-of-mass, as described in the previous section. For T/L = 1/25, a polarity is assignable to all prisms. In the third and fourth column, the focal-plane is located at the interface. In the third column, bonds between prisms with (a) the same polarity (identified with red and blue connecting lines for bonds between pair-bonded prisms of negative and positive polarity, respectively) (b) the opposite polarity (purple connecting lines), and (c) indeterminate polarity (black connecting lines) are predicted. In the fourth column, prism-prism bonds are measured by the relative orientation of the two particles, independent of the polarity state of each particle. Four types of bonds are observed: (a) tip-tip (green connecting lines), (b) tip-edge (pink connecting lines), (c) edge-edge (orange connecting lines; edges of triangles are in registryin contact and flush with one another), and (d) edge-edge offset (brown connecting lines; half of the edge of each bonded triangles lie flush with one another, with the tip of one triangle located at the center of the other triangle's edge). Comparison of the predicted and measured bonded states for T/L = 1/25 and T/L = 1/50 prisms shows perfect correlation between the polarity states of any two adjacent particles and their bonded state. Specifically, of all prisms whose polarity could be determined, all bonding between same polarity prisms is tip-tip or edge-edge, and all bonding between opposite polarity prisms is tip-edge or edge-edge offset. The bonded statesboth measured and predicted based on polarityare available in SI Tables 1 (T/L = 1/25) and 2 (T/L = 1/50). Table (S1). Scale-bar is 100 m. Theoretical analysis and computation of capillary interactions of triangular prisms The triangular prisms in these experiments have flat, nearly vertical side surfaces. This lack of curvature of the particle sides leads to a different kind of interface attachment than that observed with ellipsoids and cylinders. As discussed in [8][9][10]37 , interfaces around the ellipsoids and cylinders either rise or fall as a result of variations in curvature of the side surface of the particle. A constant contact angle as well as zero total force and torque on an isolated particle is maintained. For these triangular prisms with vertical side surfaces, however, the preferred contact angle (~ 5 degrees) of the material cannot be reached, because it would correspond to a uniform rise of the interface around the triangular prism, yielding a net force pointing down on the particle; this net force is inconsistent with mechanical equilibrium. Therefore, instead of an equilibrium contact line in the middle of the side surface of the triangular prisms, the interface is pinned to the edges of the concave face of the triangular prisms with a non-equilibrium contact angle that satisfies mechanical equilibrium (see Supplementary Information for further discussion of this phenomenon). Contact-line pinning has been observed in various experimental systems consisting of solid particles or substrates containing sharp edges. [38][39][40][41] To characterize the interface shape and the resulting capillary interaction between the triangular prisms, we use Surface Evolver to compute the interface with a contact line pinned to the edges of a bowed triangle (for details see the Materials & Methods section). To match the observed curvature of the thinnest particles, we use an inverse-curvature-toedge-length ratio of 0.9 (that is, for an edge length of 120 m, we take the radius of curvature to be 108 m). The resulting interface around isolated particles (Figurea,b) closely resembles that observed in the eSEM images of the thinnest particles (Figure ). It is worth noting that the only input into the Surface Evolver computation is the pinned contact line, and no information about the particle thickness is involved. Our computation shows that, for a bowed-up particle (Figurea), the particle center of mass is below the interface in the far field by 7.45 m (for the given curvature), whereas the center of mass of a bowed-down particle is the same amount above the far-field interface. This depth is greater than the particle thickness, and explains the perfect correlation between the polarity and the direction of the particle bowing of the thinnest particles. (The relation between the interface attachment and the bowing direction of the thicker particles may involve mechanisms such as variability in interface height due to roughness and interface pinning; these mechanisms lead to weak quadrupolar interactions, as described in ref. 42 42 ). The interface geometry around the triangular prisms is similar to that of the capillary hexapole (discussed in detail in Supplementary Information) in that there are six regions of alternating positive-and negative-interface heights (where the equilibrium, unperturbed height of the interface at far distances is taken to be zero). However, important differences exist between the ideal hexapole field and the interface around the triangular prisms at distances close to the particle. The ideal hexapole field with height ℎ ∼ 1 3 cos 3 , has the symmetry that the positive and negative regions are of equal width. The interface around the triangular prisms, in contrast, has much narrower positive (negative) regions around the tip of the bowed-up (-down) triangles (Figurec). As a result, the focusing of excess area around the tips of the triangles induces stronger capillary interactions at the tips than along the triangle edges. Note that, as one would expect, the height of the interface around a bowed triangular prism increasingly conforms to the profile of a capillary hexapole as the distance from the prism increases. Indeed, the effect of tips, edges, and other sharp particle features, which are quite prominent in the near-field behavior of the interface, becomes increasingly diminished and smoothed out at these larger distances (Figurec). We study the capillary interaction potential between triangular prisms by computing the interface geometry around a pair of triangular prisms using Surface Evolver. Once the numerical interface solution has been obtained, we can subsequently determine the capillary interaction energy using 12 = ( 12 − 1 − 2 ), where is the air-water surface tension, 12 is the excess area created at the interface in the full two-particle system, and ( = 1,2) is the excess area in an isolated one-particle system (i.e., for separation distance → ∞). The excess area is defined as the difference between the actual surface area Σ and the projected surface area Σ (see Supplementary Information). There are, of course, two cases to be simulated: the first is when both prisms have the same bowing polarity (by symmetry, we need only consider the case where both particles are bowed up), and the second is when the two particles have opposite polarities (here again we can simplify matters and consider only the case where the particle on the left is bowed up and the particle on the right is bowed down). Examples of the interface in the vicinity of two triangular prisms with the same and opposite polarities are shown in Figured,e. It is already evident from these plotseven before further analysisthat the tiptip configuration for prisms with the same polarity and the tip-edge configuration for prisms with opposite polarities are attractive, while the opposite configurations are repulsivethe former will result in decreased excess area as the particles move towards each other, while the latter will result in increased excess area (the overall slope of the interface will increase between the bowed-up and bowed-down components as they are brought closer together). Figuref,g show the underlying mechanism that reduces the excess area between regions with the same capillary charge: the formation of a capillary bridge. The capillary interaction potential depends on both the distance between the centers of the two triangular prisms, and their orientations relative to the line connecting their centers, 1 , 2 (Figure ). This is a configuration space that has one extra dimension beyond that of the capillary hexapolar theory discussed in Supplementary Information, in which only the relative orientation of the two particles matters. In order to be able to directly compare the theoretical case with that of two bowed-up triangular prisms, we fix the orientation of the left particle to be 0° and allow and 2 to vary. The resultant potential, shown in Figure a, is very similar to that of the ideal hexapoles; even the general shapes of the interfaces, as shown in a few select cases as insets in both plots, share similar features. The similarities extend beyond this, as well: in Figure c, when comparing potential curves for various mirror-symmetric configurations in the triangular-prisms system with that of the mirror-symmetric curve in the ideal-hexapoles system, which has a −6 dependence, we see that all the curves approach the theoretical ideal-hexapole curve at long distances, as we expect from the interface profile. Deviations from the ideal-hexapole curve and from each other occur at short inter-particle distances, where the anisotropic tips become increasingly prominent. Note that the 1 = 2 = 0° tip-tip mirror symmetric configuration is favored for these smaller distances. Figure . In the case of ideal hexapoles, since the interaction energy depends only on the relative orientations of the particles, the interaction energy for all mirror symmetric configurations, in which 1 = 2 , for a given distance is perfectly degenerate. As shown in Figure a,d, for the system of triangular prisms, however, the tip-tip mirror symmetric configuration (corresponding to 1 = 2 = 0°) is strongly favored (disfavored) compared to the edge-edge mirror symmetric configuration (corresponding to 1 = 2 = 60°) for smaller values of interparticle distances, in the same-(opposite-) polarities system. In the case of opposite polarities, even though the edge-edge configuration is preferred over the tip-tip configuration, it is important to realize that it is not the global preferred state, which is a non-mirror-symmetric configuration, as will be discussed further subsequently. Once again, in both cases, the expected ideal-hexapole behavior of degenerate energies for all mirror symmetric configurations is recovered as the inter-particle distance is increased. Figure b,c, the potential for a pair of bowed-up triangular prisms shows a clear well for the mirror symmetric configuration, 1 = 2 , which becomes increasingly deep for smaller inter-particle distances. It is clear in Figure b, as well, that for two bowed-up triangular prisms, the potential is relatively flat for all mirror symmetric configurations at a given large distance (same as ideal hexapole interaction). However, when the two prisms are close, the tip-to-tip configuration is much more preferred (in contrast to the ideal hexapole). The above results indicate that when two bowed up particles approach one another, in general, they first rotate into mirror symmetric configurations, and then rotate to tip-to-tip when they are very close to each other. The case of two bowed-down triangular prisms is very similar to the discussion This deviation from an ideal hexapole is further portrayed in As shown in Figure 9 (a) Interaction energy potential values for two bowed-up triangular prisms in mirror-symmetric configurations at various separation distances. 0° corresponds to a tip-to-tip configuration, while 60° corresponds to a side-to-side configuration. Interaction energy potentials plotted as a function of orientation angles for (b) = 192 and (c) = 132 . (d)-(f) The corresponding three figures for the case of one bowed-up and one boweddown triangular prism. above for the bowed-up case, with the simple addition of a minus sign of the interface height, which results in the same interface energy. The case of one bowed-up triangular prism and one bowed-down triangular prism is quite different. At large distances, the capillary interaction is close to that between two hexapoles but with one hexapole rotated by 60° degrees (or equivalently the "+" and "−" capillary charges switched). Interestingly, at small distances, the potential energy valley appears curved in ( 1 , 2 ) space while slightly favoring offset edge-edge configuration (Figure e). As we see below, this leads to different binding trajectories for bowed-up pairs and up-down pairs. Dilute binding events: experimental observations In order to evaluate the modeling of the capillary interaction between the triangular prisms, we simulate pair trajectories of prisms from various initial conditions, and compare these trajectories with trajectories observed in experiments. The centroidal separations, r, and angular orientations of the two prisms relative 1 and 2 were collected by image analysis. The trajectories are available in Movies S1 through S7. SI Figure 4a,b,c, and d show frames from the trajectories from Movies S1, S2, S4 and S7, respectively. Seven trajectories (five for T/L = 1/25 and two for T/L = 1/50) were collected from the SI movies; separations and orientations are reported in Figures 10 and 11. Four of the trajectories report like polarity binding (c.f. Fig. 10). Three trajectories report opposite polarity binding (c.f. Fig. 11). Qualitative features observed for dilute binding trajectories are: (i) like polarity particles, in a first stage adopt mirror symmetric configurations and slowly move toward each other; in a second stage, particles rapidly close into a tip-tip binding; and in a third stage, some particles then rotate into a edgeedge configuration; (ii) opposite polarity particles approach to a tip-midpoint edge configuration; the pair finally collapses into an offset edge-edge bond. Figure also shows that the time for binding of T/L = 1/50 prisms is significantly faster than for T/L = 1/25 prisms. This difference indicates that capillary attractions are much stronger at separation distances of up to several prism edge lengths for T/L = 1/50 prisms as compared to T/L = 1/25 prisms. By contrast, there is negligible difference in the time scale of tip-tip and tip-edge binding at fixed T/L ratio, an indication that the strength of like-polarity and opposite-polarity interactions are similar. Furthermore, the exponent associated with particles approaching each other in these binding events, ~( − ) , where is the time of contact, defined as the first image frame where the two prisms touch, displays similarity with the exponent from hexapole-hexapole interactions, 0 = 1/8 (Figure ). The small deviation comes from the difference between the actual capillary interactions between the triangles with the ideal hexapolar interaction. In particular, at far distances, appears to be closer to 1/6, indicating that at far-field quadrupolar interactions (from random variations in prism edge topography) may be the dominant driver for binding at these large separations. Nevertheless, the scaling at small separation approaches the hexapolar expectation of 1/8. Turning to prism rotation, for tip-to-tip trajectories, prism rotation begins between hundredths of a second (T/L = 1/25) up to several seconds (T/L = 1/50) prior to contact (Fig. 10). In the later case, these times correspond to separation distances that are several edge lengths. The angular orientation plots also show that prisms bind in a mirror symmetric fashion; that is, in each pair-binding event, both prisms rotate an equal amount into their final, steady-state orientation. For tip-to-midpoint edge trajectories, prism angular orientation also begins at distances corresponding to separations of several edge lengths (Fig. 11). Dilute binding events: simulation results and agreement with experiments To compare to these results, we simulated pair-binding events using the interaction potential (interface energy) ( 1 , 2 , ) obtained by interpolating a grid of Surface Evolver-calculated potential values at regular intervals as described above. Details of the trajectories simulation are described in the Materials & Methods section. Examples of our simulation results for same-polarity (both bowed up in our calculation) and opposite-polarity prisms are shown in Figures 12 and 13, respectively. Initial conditions were chosen to approximate trajectories observed experimentally. The ratio of the two viscous-damping coefficients is taken be to ̃/ = 1.46 as discussed in Materials & Methods, and is taken to rescale time such that the arbitrary time scale in the simulation approximates the experimental time scale, in units of seconds. These simulations terminate at = 132 m, the distance at which the two prisms would touch if they faced one another tip-to-tip. In all cases, the far-field trajectories are roughly consistent with that of the ideal hexapole-hexapole interaction, while expected, and important, deviations from the ideal interaction occur as the separation distance decreases. In Supplementary Information we show additional trajectories where additional initial conditions with different choices of ̃/ ratio are discussed. Figure . Note that in two experimental events, the prisms approach faster in the far-field regime than quadrupolar interactions would dictate; we speculate that this is due to noncapillary-induced drift of particles at the interface, possibly owing to convective flow at the interface surface . To obtain statistics about how the triangular prisms bind, we ran simulations for all Figure 12a,b. The first ratio is chosen according to the simple geometric estimate discussed previously. The second ratio, which is 10 times smaller, allows the prisms to rotate faster relative to their center of mass motion, and presents a useful contrast to the first case. In both cases, a significant majority of configurations end up in, or close to, the 1 = 2 = 0° tip-tip mirror-symmetric configuration. For the case of ̃/ = 1.46 ( m/°) 2 , some trajectories end up along a continuum of mirror symmetric configurations ranging from tip-to-tip to edge-to-edge, as the prisms did not have enough time to finish the rotation before contact. Contrastingly, in the second case with ̃/ = 0.146 ( m/°) 2 almost all trajectories end up tip-to-tip, because rotational drag is smaller, leading to faster rotation. A small fraction of initial conditions (gray in the figure) ended up in random configurations coming from small kinks in the pair potential due to numerical error. In order to investigate what happens after the two prisms touch at their tips, we calculated the pair potential for two prisms in which their tips continue to touch, but at different orientations (mirror symmetric configurations with 1 = 2 ranging between 0 and 60°), as shown in Figure 12d This indicates that, after initial tip-to-tip contact, the pair of triangular prisms will rotate and "collapse" into an edge-to-edge configuration. It is worth pointing out that althoughafter collisionthe prisms collapse into the edge-toedge configuration, a majority of trajectories still first go through an initial tip-tip binding. This is in good agreement with our experimental observations. to a more uniform state diagram wherein all initial conditions have enough time to rotate to the offset edge-to-edge configuration, which is of lower energy. The first casewhich uses the ratio from our geometric estimationyields a continuum of final configurations. As before, our simulation terminates at = 132 m, which is the distance at which two prisms touch when they face one another tip-to-tip. The opposite-polarity prisms, however, being in non-tip-to-tip configurations, are not yet touching at this distance. Our additional computations of the interface energy shows that, at smaller distances, the offset edge-to-edge configurations exhibit lower energy, leading to the final collapsed offset edge-to-edge configurations as observed in experiment. Assembly into open networks The pair-binding observations discussed above indicate that self-assembly of thin, triangular prisms may result in 2D networks with both open (tip-tip and tip-edge pairbinding orientations) and close-packed (edge-edge and edge-edge offset pair-binding orientations) conformation. We characterize statistical signatures of the resulting disordered networks for these prisms, which will guide future study aiming at obtaining regular open networks. Recalling Fig. 3, at early times ( Fig. 3b and c), small aggregates form. These small aggregates undergo time-dependent growth via aggregate-aggregate attraction and binding ( Fig. 3d and e). These larger aggregates branch laterally, which yields an open structure. Aggregates continue to attract and bind to one another until all available prisms are incorporated into a space-spanning, open network. Representative networks formed by self-assembly are shown for all T/L ratios in Fig. 14. Each of the four images in Fig. 14 is a 3.8 x 2.5 mm spatial mosaic of either six or eight (either three-by-two or four-by-two) 1270 x 1270 m 2 microscopy images. While each of the self-assembled networks possesses voids, the structures of the three thinnest prisms (T/L = 1/50, 1/25 and 1/10, which exhibit polarity) is comprised of long, nearly linear runs of triangles bound in close-packed edge-edge states (Fig. 14ac). By contrast, the thickest prisms (T/L = 1/5, which do not exhibit polarity) contain fewer close-packed prisms, and no linear chains of edge-edge bonds (Fig. 14d). Network porosity is quantitatively assessed by computing a common measure of number density fluctuations: = 〈 2 〉 − 〈 〉 2 〈 〉 \| Here is N is the number of particles within an ensemble of square bins of size L. The brackets denote the average over the ensemble. This quantity is equivalent to the compressibility in the long wavelength limit; we here refer to it as L. The quantity L has previously been used to describe the long-range structure of colloidal gels. [43][44][45] The network images in Fig. 10 were divided into square regions of 240 x 240 m 2 , 480 x 480 m 2 , and 720 x 720 m 2 and the compressibility measure determined for each bin size; the results are reported in SI Table 3. For each bin size, the compressibility measure is greatest for the networks of the thinnest particles (Figs. 14a-c), progressively decreasing for the network of thicker prisms (Fig. 10d); this quantitative result is consistent with the images in Figure 14, which show larger voids for the thin prism networks relative to the thick networks. Discussion In the discussion that follows, we comment on the ramifications of the coupled prism polarity and hexapolar-like interactions of the thinnest prisms. We address how the polarity of pair-binding prisms is predicative of both pair-binding trajectories and of the final pair-bonded state. We then discuss the effect of prism polarity on open network structure and suggest a path to design a prism building block for an ordered kagome lattice. Prism polarity is predictive of tip-tip vs. tip-edge binding trajectory The results show that for thin prisms (T/L = 1/25 and 1/50) the type of prismprism bond formed may be predicted with 100% fidelity from the polarity of the two prisms participating in the bonding event. Prisms of the same polarity only access the tip-to-tip trajectory which then leads to the tip-tip and edge-edge final binding states, while prisms of opposite polarity only access the tip-to-midpoint edge trajectory, which then leads to tip-to-midpoint edge and edge-edge offset binding states. Our observations suggest that the tip-tip and tip-edge binding states only survive at steady-state when the collapse of the prisms into their edge-edge or edge-edge offset states is frustrated, due to, either geometrically induced frustration from surrounding prisms or roughness at the prism sides, which prevent rotations. Prism polarityand its control over prism-prism binding trajectoryis also observed for T/L = 1/10 prisms, although evidence of the effect is not as obvious with optical microscopy (Fig. 4c, columns 2 and 3) and thus was not analyzed in the same way thinner prisms are in Figs. 6 and S3. T/L = 1/5 prisms, on the other hand, lack observable polarity. Fig. 4d shows a variety of bonding states along the edgeinstead of localization at the tip and midpoint edge as seen for the thin prisms. Hexapole-like capillary interactions from interface-prism contact line bowing Due to the flat geometry of the prism sides, instead of an equilibrium contact line with constant contact angle (as in the case of cylinders at an interface), the triangular prisms leads to contact lines pinned at edges of the triangular face, as we discussed above. Bowing of thinner prisms leads to a contact line that is conformal to that of the bowed triangle surface. Thus a hexapole-like interface profile around them arises, wherein tips and edges of the triangle exhibit opposite interface height variations. Interestingly, the interface profile differs from that of ideal hexapoles, especially close to the prisms, due to the focusing of excess interface area near the tips. We find this to be the origin of the tip-to-tip attraction for same polarity prisms, which may be a useful mechanism to obtain regular open networks. In contrast, we find thicker prisms to be much more flat, and thus the pinned contact line does not exhibit a significant hexapolar component. Instead, it likely generates an interaction, described by Fourier decomposition of the variability in interfacial height profile, as generated by non-ideal features of the flat surface, such as its roughness. This leads to quadrupole-quadrupole interactions at far field, consistent with our observation from the binding events. Open network structure and the path to capillary-drive self-assembly of ordered open lattices The open networks shown in Fig. 14 display a heterogeneous structure, characterized by disordered strands and voids. These structures are reminiscentand perhaps even more open thannetworks self-assembled from colloidal ellipsoids at fluid-fluid interfaces 7 , which demonstrate enhanced rigidity as compared to close-packed arrays of isotropic spheres. Recent theoretical studies show unusual mechanical properties of regular open structures such negative Poisson's ratio in the twisted kagome lattice. Mechanical properties of these disordered open networks will be an interesting direction for future research. 12,14,18,31,33 Open networks self-assembled from the three thinnest prisms, which exhibit polarity and possess a capillary hexapole, (Figs. 14a-c) are more heterogeneous than are networks self-assembled from the thickest prisms, which exhibit neither polarity nor a capillary hexapole (Fig. 14d). This is demonstrated quantitatively through the measurement of particle number density fluctuations (Fig. S5); open networks selfassembled from the thinnest prisms exhibit higher particle number density fluctuations than do open networks self-assembled from the thickest prisms. These results suggest that future work could understand how void structure in such disordered networks could be controlled by design of building block shape, surface properties, and pair interactions in capillary systems. On the other hand, to realize regular open networks, such as the kagome lattice, where only tip-tip binding is selected, further studies are needed in order to (1) select a single polarity component of the prisms, and (2) stabilize the binding of the prisms at the tip-tip configuration and avoid the collapse into edge-edge binding. The former may be addressed by introducing Janus character to the particles, so that they attach to the interface in just one of the two possible configurations. The latter may be realized by optimizing the shape of the prisms such that the tips are slightly truncated such that the tip-tip configuration is a local minimum. Conclusion We have reported capillary-driven binding of thin, triangular prisms of T/L between 1/50 and 1/5 into open networks at a flat air-water interface. The interface pins to the concave face of the three thinnest prisms (T/L = 1/50, 1/25, and 1/10). Interface pinning and physical bowing of the thin prisms results in (a) two polarities corresponding to prism bowing up, interface pinned at top edges, prism center-of-mass below interface, and prisms bowing down, interface pinned at bottom edges, prism center-of-mass above interface, and (b) hexapolar-like interface profile around the prisms. The resulting capillary interactions between these triangular prisms lead to tip-tip, edge-edge, tip-edge, and edge-edge-offset pair binding events, depending on the polarity of the pair, and disordered open networks produced by self-assembly. Thick prisms (T/L = 1/5) exhibit neither physical bowing nor splitting of the prisms into two subpopulations above and below the air-water interface. Prisms of all thicknesses self-assemble into open networks with void structure that depends on the geometric properties of the prism. The results can inform the design of thin prism building blocks for assembly of open networks at fluidfluid interfaces with either order or disordered structure. Conflicts of Interest There are no conflicts of interest to declare. Supplementary Information: Capillary-driven binding of thin triangular prisms at fluid interfaces Joseph A. Ferrar, Deshpreet S. Bedi, Shangnan Zhou, Peijun Zhu, Xiaoming Mao,* and Michael J. Solomon* (Corresponding Author: [email protected] and [email protected]) Multipolar interactions between particles at a fluid interface In this section we describe equilibrium interface shapes and the resulting capillary interaction between circular multipoles. In the Results section we will show that the capillary interaction between two triangular prisms in our experiment is similar to hexapolar interactions. The pressure difference across an interface between two stationary, immiscible fluids is given by the Young-Laplace equation, Δ = 1 − 2 = − ⋅ , where is the surface tension and is the unit vector pointing from the lower fluid (2) to the upper fluid (1). Note that − ⋅ = 2 , where is the mean curvature of the interface surface. Suppose that the height of the interface is given by ℎ( ), where the farfield equilibrium height of the interface is ℎ = 0 (the interface is flat and, consequently, the pressure difference across the interface is zero). We can write the Young-Laplace equation in terms of the height field as ∇ ⋅ ℎ √1 + \| ℎ\| 2 = 2 ℎ where = ℓ −1 ≡ √ ( 2 − 1 ) is the inverse capillary length. The capillary length is a characteristic length scale arising from comparing the relative strengths of gravitational acceleration and the surface tension; for length scales much smaller than the capillary length, the effects of gravity can be neglected. The capillary length of an air-water interface is 2.7 mm. We can simplify the governing equation of the interface height ℎ by making two assumptions that are typically satisfied by micron-sized particles. First, the interface slope is taken to be small: \| ℎ\| 2 ≪ 1. Second, we consider length scales that are much smaller than the capillary length, ≪ ℓ . In this case, the Bond number is vanishingly small, Bo = ( ) 2 ≪ 1, and the Young-Laplace equation simplifies to the 2-D Laplace's equation, ∇ 2 ℎ = 0.(1) Let us consider the case of a solid particle adsorbed to the interface such that the contact line between the interface and the particle surface is undulating. This can be due to particle shape (anisotropies, corners, and edges) and surface roughness/irregularities. These undulations can be decomposed into a multipole expansion such that this differential equation can be solved analytically, for particles with circular cross-sections, using polar coordinates ( , ). The solution for the interface height profile is ℎ( , ) = 0 ln( ) + ∑ ( 0 ) cos[ ( − ,0 )] ∞ =1 ,(2) where is the amplitude of the th moment at the surface/circumference of the particle's circular projection, 0 . ∈ ℤ + ∪ {0} is the multipole moment, and = 0,1,2,3 correspond to the monopole/charge, dipole, quadrupole, and hexapole moments, respectively. If the particle adsorbed to the interface is sufficiently light, the monopole moment vanishes; if the particle is allowed to spontaneously rotate about a horizontal axis, then the dipole moment also vanishes. Therefore, the quadrupole moment ( = 2) is typically the leading non-zero term in the multipole expansion ( Figure ). . Interaction potential between two capillary multipoles The capillary interaction potential between two particles is a function of their orientations and separation distance. It is given by 12 = ( 12 − 1 − 2 ),(3) where 12 is the excess area created at the interface in the full two-particle system, and ( = 1,2) is the excess area in an isolated one-particle system (i.e., the separation distance → ∞). The excess area is defined as the difference between the actual surface area Σ * and the projected surface area Σ (the interface would be planar without the deformation caused by the particle) 2 In the small slope regime, the excess surface area is given by = 1 2 ∬ Σ \| ℎ\| 2 . From these two preceding equations, it is apparent that minimization of the capillary interaction potential coincides with the minimization of excess area beyond that created by two isolated particles. This favors the adoption of particle configurations such that the slope of the resultant interface is reduced. For particles with fixed orientations, the interaction between the two will be attractive if moving the particles closer together will reduce the overall slope of the interface (and, thus, decrease the amount of excess interfacial area) and repulsive if moving the particles further apart will reduce the overall slope. For a single particle with a circular cross-section and a contact line that is undulating with multipole moment , the excess surface area is 1 = 2 2 . For two capillary multipoles, the excess surface area is Here, it is important to realize that, for two c888apillary multipoles of the same order, such that 1 = 2 , the interaction energy reduces to a two-dimensional function of their separation distance, , and their relative orientation, \| 1 − 2 \|. An example of hexapolehexapole interaction energy is shown in Figure . Contact-line boundary conditions The solution to the Young-Laplace equation is subject to two boundary conditions: one at the three-phase (solid, liquid, and fluid, with the latter oftentimes a gas) contact line and one at the far boundary of the interfce, infinitely far away. The latter is typically taken to be the condition of a flat interface. The boundary condition at the contact line, however, can be more complicated. In the simplest case, in which the surface of the solid phase (e.g., a wall or a particle) is energetically homogeneous, the contact line is determined such that the equilibrium contact angle, , between the solid surface and the surface of the interface is constant and satisfies the Young equation 3,4 cos = SG − SL , where , SG , SL are the liquid-gas, solid-gas, and solid-liquid surface tensions, respectively. In this paper, due to the specific shape of the particles used in the experimenttriangular prismswe will focus on a specific boundary condition in which the contact line is kinetically trapped, or pinned, at sharp corners and edges of a particle. This pinning results in a non-equilibrium contact angle that can deviate significantly from the equilibrium contact angle discussed above and can also vary along the contact line. As shown by Gibbs in an extension to the Young equation, 5,6 the non-equilibrium contact angle, , at a pinned edge can be any value in the range ≤ ≤ − + , where is the wedge angle of the particle. For instance, the wedge angle of the top or bottom edges of a cube is /2. Note that the limiting angles of Gibbs' criterion or inequality are simply the equilibrium contact angles for each of the two surfaces that join together to form the edge with a wedge angle of ; when extends beyond the bounds of the inequality, the contact line becomes unpinned and begins to slide along one of the two surfaces, as dictated by which bound was violated. 7 This phenomenon of contactline pinning has been observed in various experimental systems consisting of solid particles or substrates containing sharp edges. [8][9][10] For example, in the case of a small cylindrical particle with negligible Bond number oriented vertically, a preferred equilibrium contact angle of ≠ /2 cannot be achieved anywhere along the side of the cylinder; therefore, the contact line will either move up (if the preferred contact angle < /2) or down (if > /2) until either the top or bottom face, respectively, of the cylinder coincides with the interface. 11 In this case, the contact line is pinned to the edge of the cylinder with non-equilibrium contact angle = /2, and the surrounding interface is completely planar. In this experiment, the equilibrium contact angle of the air-water interface with the triangular prisms has been measured to be about 5 degrees, and the prisms have a wedge angle of 90 degrees (at both the top and the bottom). For these specific values, Gibbs' criterion ostensibly implies that mechanical equilibrium for a particle of negligible weight can only be satisfied when the contact line is pinned to the edges of the top face of a particle (as it is only here that the range of permissible contact angles allows for both upward-and downward-pointing interface/surface tension vectors such that they sum to zero over the closed contact line loop, as is required by the condition of mechanical equilibrium). Incorporating the fact that the triangular prisms are bowed, however, it can be seen that only downward-pointing interface vectors are possible if the contact line is pinned to the edges of the top face of a bowed-down prismsuch a prism is only mechanically stable when the contact line is pinned to the edges of the bottom face instead. It remains true for a bowed-up prism, nevertheless, that the contact line necessarily is pinned to the edges of the top face. In both cases, then, the contact line needs to pin to the edges of the concave face of the bowed triangular prism, which is consistent with experimental observation. Triangular Prism Binding States Correlate with Prism Polarity In Tables 1 (as derived from Figure 6) and 2 (as derived from SI Figure S3), 237 bonds are analyzed across six 1270 x 1270 m regions of open networks of T/L = 1/25 and 1/50 prisms (three 1270 x 1270 m regions per network). 17 bonds are between prisms with indeterminate polarityprisms whose polarity cannot be resolved by optical microscopyand are not included in this analysis. Of the 220 remaining bonds, there is perfect agreement in the number of bonds between prisms with the same polarity (133 bonds) and prisms bound tip-tip or edge-edge (133 bonds), and there is also perfect agreement in the number of bonds between prisms with the opposite polarity (87 bonds) and prisms bound tip-edge or edge-edge offset (87 bonds). Averaging over three locations in each network, and counting over the 237 total events, bonds between prisms with the same polarity account for 48% of all bonds for T/L = 1/50 prisms and 64% of all bonds for T/L = 1/25 prisms; bonds between prisms with the opposite polarity account for 37% of all bonds for T/L = 1/50 prisms and 36% of all bonds for T/L = 1/25 prisms, and bonds between prisms with indeterminate polarity account for 15% of all bonds for T/L = 1/50 prisms. Bonds of indeterminate polarity are not observed for T/L = 1/25 prisms. (3) and (4) are tabulated in Table (2). Scale-bar is 100 m. Table S1. Comparison of prism-prism bond type based on polarity of bound prisms and polarity-independent prism orientation for T/L = 1/25. All data is tabulated from analysis described in Fig. 5. Bonds are sorted into rows by the relative polarity of the bound prisms (same, opposite, or indeterminate polarity and into columns by the polarity-independent orientation of the bound prisms. The correlation between the relative polarity of the bound prisms and the polarity-independent bond orientation is calculated for network location analyzed. All bond types and correlations are also totaled over all 3 network locations. Table S2. Comparison of prism-prism bond type based on polarity of bound prisms and polarity-independent prism orientation for T/L = 1/50. All data is tabulated from analysis described in Fig. 6. Bonds are sorted into rows by the relative polarity of the bound prisms (same, opposite, or indeterminate polarity and into columns by the polarity-independent orientation of the bound prisms. The correlation between the relative polarity of the bound prisms and the polarity-independent bond orientation is calculated for network location analyzed. All bond types and correlations are also totaled over all 3 network locations. into an edge-to-edge orientation in which the two edges are offset from each other by L/2. Scale bars are 100 m. Heterogeneity of Self-Assembled Networks For regions of 240 x 240 m 2 , networks of the three thinnest particles have a compressibility measure of 2.6 ± 0.5, while the network of the thickest particles has a number density fluctuation of 1.8 ± 0.1. For regions of 480 x 480 m 2 , networks of the three thinnest particles have a number density fluctuation of 5.8 ± 0.6, while the network of the thickest particles has a number density fluctuation of 4.4 ± 0.2. For regions 720 x 720 m 2 , networks of the three thinnest particles have a number density fluctuation of 10.4 ± 0.9, while the network of the thickest particles has a number density fluctuation of 7.7 ± 0.2. Figure S5. Mean-squared particle number density fluctuation for the networks shown in Figure 14. Additional simulated pair-binding trajectories Three representative initial conditions, corresponding to configurations close to (but purposefully not exactly) tip-to-tip, tip-to-side, and side-to-side were selected, and the resultant simulated trajectories are shown in Figure 15 for two different viscous-damping coefficient ratios, ̃/ = 1.46,0.146. In all three sets of trajectories, it is clear that mirror symmetric configurations are preferred --for cases where the initial configuration is already mirror symmetric, the subsequent configurations remain mirror symmetric; otherwise, the particles will first rotate to a mirror symmetric configuration. For smaller ratios and fixed , ̃ becomes correspondingly smaller, meaning that it is easier for the particles to rotate. This accounts for the fact that, in all cases, the 1 = 2 = 0 ∘ tip-tip mirror-symmetric configuration is more easily achieved for the smaller ratio value. Figure 1 1Hexapole-like capillary interaction between triangles may lead to the self-assembly of kagome lattices. (a) Hexapole-like interactions between triangles (positive at tips and negative at edges) cause tip-to-tip binding. (b) The kagome lattice where edges of triangles form straight lines. (c,d) two twisted kagome lattices with different twisting angle. These different versions of the kagome lattices are related by a soft deformation which only changes the bond angle, which leads to the negative Poisson's ratio of this structure. (e) Depending on the strength of the hexapole-like interaction, disordered assemblies of triangles may also appear. qualitative features of the experimental results. These results can inform the structural design of complex open networks from interfacial building blocks. Figure shows the 4 types of equilateral triangular prisms fabricated. All prisms have an edge length of 120 m, and thickness of: (a) 2.5 m, (b) 5 m, (c) 12 m, and (d) 20 m.The ratio of the thickness (T) to length (L) of the prisms is a characteristic parameter; we hereafter refer to each type of prism as: Figure 2 2SEM images of thin, equilateral triangular microprisms from SU-8 epoxy resin. Equilateral triangle (edge length, L =120 µm) prisms of varying thickness (T) a) T ~ 2.5 µm, T/L = 1/50, b) T ~ 5 µm, T/L = 1/25, c) T ~ 12 µm, T/L = 1/10, d) T ~ 20 µm, T/L = 1/5. Figure 3 3Optical microscopy time-series images of capillary-driven triangular prism (T/L ~ 1/25) binding at a flat air-water interface. a) Initial placement of prisms at interface b) 8 minutes after placement of prisms at interface c) 20 minutes d) 40 minutes e) 50 minutes. Scale bars are 100 µm. Polarity in interface attachment for thin prisms Figure shows 1270 x 1270 m regions of open networks formed by prisms of the four T/L ratios synthesized. Each row in Figure corresponds to a specific T/L ratio (row 1 Figure 4 4Optical microscopy images of 1270 x 1270 m 2 regions of open networks. Networks are self-assembled via capillary-driven triangular prism binding. Row 1 (a) T/L ~ 1/50, row 2 (b) T/L ~ 1/25, row 3 (c) T/L ~ 1/10, row 4 (d) T/L ~ 1/5. Column 1: single frame image of portion of network (1270 x 1270 m), focal plane at air-water interface. Column 2: same single frame image of portion of network as in column 1, focal plane ~200 m below air-water interface. Column 3: same single frame image of portion of network as in columns 1 and 2, focal plane ~200 m above air-water interface. Green images in row (d) are overlays of optical and reflection microscopy; the reflection channel highlights differences in position of thick, apolar prisms at the flat air-water interface. Scale bar is 100 m. Figure 5 5Environmental SEM images of triangular prisms, fixed at an air-gellan/water interface. Row 1: (a) -(c) prisms assigned positive polarity: (a) T/L = 1/50, (b) T/L = 1/25, (c) T/L = 1/10. Row 2: (d) -(f) prisms assigned negative polarity: (d) T/L = 1/50, (e) T/L = 1/25, (f) T/L = 1/10. (g) apolar T/L = 1/5 prism. (h) The same capillary-driven binding states are observed at air-gellan/water interface prior to prism immobilization as are observed with optical microscopy at non-gelled interfaces. Scale bars are 20 m. Interface attachment of the T/L = 1/25 prisms are shown in Figure b and e. The top face of a prism in Figure 6 6Identification of triangular prism binding states (T/L = 1/25). Each row of images (a) -(c) represents a different location within a network structure. The relative position of microscope's focal plane to the air-water interface is varied by column as follows: Column (1): Microscope focal plane is ~200 m below the interface. Clearly visible prisms are identified with red markers. Column (2): Microscope focal plane is ~200 m above the interface. Clearly visible prisms are identified with blue markers. Column (3): Microscope focal plane is at the interface. Bonds between prisms with the same polarity are identified with blue and red connecting lines, bonds between prisms with the opposite polarity are identified with purple connecting lines. Column (4): Microscope focal plane is at the interface. Prism-prism bonds are identified by their polarity-independent orientation: side-side (orange connecting lines), tip-tip (green connecting lines), sideside offset (brown connecting lines), tip-side (pink connecting lines). Bonds in Columns (3) and (4) are tabulated in Figure 7 7Interface height profile for a (a) negative polarity bowed-up triangular prism and a (b) positive polarity bowed-down triangular prism, where the zero value is set by the equilibrium interface height at large distances from the prism. The inset in (a) is a close-up of the Surface Evolver simulation output. (c) A comparison of the interface height profile around a bowed-up triangular prism (data points) and an ideal hexapole (solid curves) as a function of angle at two different distances from the triangular prism, shown in the inset. Simulated interface height profiles for (d) two bowed-up triangular prisms and (e) one bowed-up and one bowed-down prism for both tip-to-tip and tip-to-side configurations. Zoomed-in rendering of simulated interface height profile for (f) a tip-to-tip configuration for two bowed-up prisms; and (g) a tip-to-side configuration for one bowed-up and one boweddown prisms, illustrating the existence of a capillary bridge in both cases. Figure 8 8Numerically-simulated capillary interaction potential between two bowed-up triangular prisms, with the left prism held at 0°. This two-dimensional slice of the full three-dimensional configuration space is directly comparable to the theoretical interaction potential inFigureS2. (b) All orientation angles for the triangular prism system are defined according to the convention shown: the orientations are defined by the angle a specific tip of the prism makes with the line connecting the centers of the two prisms. (c) The capillary interaction potential for twobowed up triangular prisms in mirror-symmetric configurations as a function of the separation distance, , on a log scale, for various orientation angle values. A dashed reference line, corresponding to the theoretical interaction potential for two ideal hexapoles, ∼ −6 , is shown for comparison. Figure 10 10Comparison of experimentally observed and simulated trajectories for a pair of prisms of same polarity. Top row: observed vs − curve in log-log scale (left) and linear scale (inset), where is taken to be the first frame in which the two prisms touch; and observed 1 , 2 vs − curves (right). Four events are shown as explained in the legend, and lines showing = 1/8 (consistent with hexapolar interaction) and 1/6 (consistent with quadruplar interaction) are added. Illustrations of the prisms configurations are added in the 1 , 2 plot to show the geometry. Configurations at the time of contact ( = ) are pointed to by arrows, and the points at − > 0 show prism rotations after contact, with final configurations marked by circles. Bottom row: counterparts of the and 1 , 2 plots from simulation. Instead of contact time, is the time where the prisms' separation distance reaches = 132 (the lower bound of in our computation), at which they touch if 1 = 2 = 0. We have chosen initial conditions that are close to two experimental trajectories. Figure 11 11Comparison of experimentally observed and simulated trajectories for a pair of prisms of opposite polarity. Top row: experimental observations. Bottom row: simulation results. All conventions are the same as in Figure 12 " 12Phase" diagrams illustrating the final configurations ( = 132 ) for all possible initial orientations for two bowed-up triangular prisms at = 264 for two different viscous-damping coefficient ratios, (a) ̃/ = 1.46 ( /°) 2 and (b) ̃/ = 0.146 ( /°) 2 . The final configurations are all mirror-symmetric and lie somewhere along the line in (c), with blue corresponding to tip-to-tip final configurations, red corresponding to side-to-side final configurations, and gray denote initial conditions that leads to trapped configurations which we believe to be artifacts of the computed pair potential.. (d) Capillary interaction potential values for mirror-symmetric configurations with two tips of the triangular prisms remaining in contact (thus, the separation distance, , decreases below 132 as initial angles of prism pairs at an initial distance of 0 = 264 m ; our results are summarized, for two different ratio values, For bowed-up-bowed-down pairs, a similar set of simulations yields results shown in Figure a,b for the two viscous-damping coefficient ratios. In this case, it is important to note that the final configurations are not mirror symmetric; Figure c shows the final orientation values for the two triangular prisms. The curves of final orientation lie along the minimum-energy regions of the opposite-polarity interaction potential in Figure f. Similar to the case of prisms with the same polarity, simulating the approach of prisms of opposite polarities with the smaller drag coefficient ratio, ̃/ = 0.146 ( m/°) 2 , leads Figure 13 " 13Phase" diagrams illustrating the final configurations ( = 132 ) for all possible initial orientations for one bowed-up and one bowed-down triangular prism at = 264 for two different viscous-damping coefficient ratios, , (a) ̃/ = 1.46 ( /°) 2 and (b) ̃/ = 0.146 ( /°) 2 . The final configurations lie somewhere along the curve in (c), with blue corresponding to tip-to-edge final configurations and red corresponding to offset-edge-to-edge final configurations. Figure 14 14Self-assembled open networks from capillary-driven binding of thin triangular microprisms. (a) T/L = 1/50, (b) T/L = 1/25, (c) T/L = 1/10, and (d) T/L = 1/5 equilateral triangular microprisms. Scale-bars are 100 m. ) For two particles with circular cross-sections, it is convenient to use bipolar coordinates ( , ) to obtain a solution to Eq.(1). They are defined implicitly via (or, equivalently, ∈ [− , )). Curves of constant and are circles that intersect at right angles in the -plane. The parameter is determined by the particle radii and their separation distance2 = 1 4 2 [ 2 − ( 1 + 2 ) 2 ] [ 2 − ( 1 − 2 ) 2 ]. Figure T1 . T1Theoretical interface height profile for particles with circular cross-sections. (a) A capillary quadrupole ( = 2), with four alternating regions of positive and negative interface height (the equilibrium interface height far from any particles is taken to be zero), and (b) A capillary hexapole ( = 3) with six alternating regions of positive and negative interface height. Figure S1 . S1Theoretical interface height profile for particles with circular cross-sections. (a) A capillary quadrupole ( = 2), with four alternating regions of positive and negative interface height (the equilibrium interface height far from any particles is taken to be zero), and (b) A capillary hexapole ( = 3) with six alternating regions of positive and negative interface height.Note that, in the bipolar coordinate system, the circular projections of the contact lines on the -plane are curves of constant , Figure S2 S2Theoretical capillary interaction potential between two capillary hexapoles as a function of separation distance, , scaled by the diameter of the particles' circular projection, 2 , and the particles' relative orientation, Figure S3 S3Identification of triangular prism binding states for (T/L = 1/50). Each row of images (a) -(c) represents a different location within a network structure. The relative position of microscope's focal plane to the air-water interface is varied by column as follows: Column (1): Microscope focal plane is ~200 m below the interface. Clearly visible prisms are identified with red markers. Column (2): Microscope focal plane is ~200 m above the interface. Clearly visible prisms are identified with blue markers. Column (3): Microscope focal plane is at the interface. Bonds between prisms with the same polarity are identified with blue and red connecting lines, bonds between prisms with the opposite polarity are identified with purple connecting lines, bonds between prisms with indeterminate polarity are identified with black connecting lines. Column (4): Microscope focal plane is at the interface. Prism-prism bonds are identified by their polarity-independent orientation: side-side (orange connecting lines), tip-tip (green connecting lines), side-side offset (brown connecting lines), tip-side (pink connecting lines). Bonds in Columns Figure S6 S6Configuration trajectories for three representative initial conditions (close to (a),(b) tip-to-tip, (c),(d) tip-to-side, and (e),(f) side-to-side) and two different viscous-damping coefficient ratios. The top row shows the separation distance as a function of simulation time, with insets plotting separation distance values as a function of time-to-contact on a log scale. The gray reference line corresponds to the theoretical case of two ideal hexapoles approaching each other in a mirror-symmetric configuration. The bottom row shows the orientation angles of the triangular prisms as a function of simulation time. = 2 increases. Indicates a tendency for tip-to-tip configurations to ultimately collapse to side-to-side configurations. \|. The three insets show the interface height profile of three configurations corresponding to relative orientations of 0°, 30°, and 60° at a distance of /2 = 1.8.. 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Movie S1: T/L = 1/50 tip-tip, frame rate = 30 fps (Fig. S4a) Movie S2: T/L = 1/50 tip-edge, frame rate = 30 fps (Fig. S4b). Movie S2: T/L = 1/50 tip-edge, frame rate = 30 fps (Fig. S4b) Movie S3: T/L = 1/25 tip-tip trial 1, frame rate = 15 fps Movie S4: T/L = 1/25 tip-tip trial 2, frame rate = 15 fps (Fig. S4c). Movie S3: T/L = 1/25 tip-tip trial 1, frame rate = 15 fps Movie S4: T/L = 1/25 tip-tip trial 2, frame rate = 15 fps (Fig. S4c) Movie S5: T/L = 1/25 tip-tip trial 3, frame rate = 15 fps Movie S6: T/L = 1. 25 tip-edge trial 1, frame rate = 15 fps Movie S7: T/L = 1/25 tip-edge trial 2, frame rate = 15 fps (Fig. S4d)Movie S5: T/L = 1/25 tip-tip trial 3, frame rate = 15 fps Movie S6: T/L = 1/25 tip-edge trial 1, frame rate = 15 fps Movie S7: T/L = 1/25 tip-edge trial 2, frame rate = 15 fps (Fig. S4d) shown for T/L = 1/50 (rows (a) and (b)) and T/L = 1/25 (rows (c) and (d)). For prisms of T/L = 1/50 (rows (a) and (b)), contact occurs between the 5 th and 6 th images of each row. For prisms of T/L = 1/25 (rows (c) and (d)), contact occurs in the 5 th image of each row. Rows (a) and (c), tip-to-tip binding trajectory: the prisms approach and first contact occurs at the tips. The prisms then rotate into a collapsed, fully flush edge-to-edge orientation. Rows (b) and (d), tip-to. Figure S4 Optical microscopy images of the 2 types of binding trajectories observed for polar prisms (T/L < 1/10). midpoint edge binding trajectory: the prisms approach and contact one another in an orientation such that the tip of one prism binds at the midpoint of the other prism's edge. The prisms then rotateFigure S4 Optical microscopy images of the 2 types of binding trajectories observed for polar prisms (T/L < 1/10), shown for T/L = 1/50 (rows (a) and (b)) and T/L = 1/25 (rows (c) and (d)). For prisms of T/L = 1/50 (rows (a) and (b)), contact occurs between the 5 th and 6 th images of each row. For prisms of T/L = 1/25 (rows (c) and (d)), contact occurs in the 5 th image of each row. Rows (a) and (c), tip-to-tip binding trajectory: the prisms approach and first contact occurs at the tips. The prisms then rotate into a collapsed, fully flush edge-to-edge orientation. Rows (b) and (d), tip-to-midpoint edge binding trajectory: the prisms approach and contact one another in an orientation such that the tip of one prism binds at the midpoint of the other prism's edge. The prisms then rotate . K D Danov, P A Kralchevsky, B N Naydenov, G Brenn, J. Colloid Interface Sci. 287K. D. Danov, P. A. Kralchevsky, B. N. Naydenov and G. Brenn, J. Colloid Interface Sci., 2005, 287, 121-134. . D Stamou, C Duschl, D Johannsmann, Phys. Rev. E. 62D. Stamou, C. Duschl and D. Johannsmann, Phys. Rev. E, 2000, 62, 5263-5272. . T Young, Philos. Trans. R. Soc. London. 95T. Young, Philos. Trans. R. Soc. London , 1805, 95, 65-87. 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[ "Holographic Entanglement Entropy of Anisotropic Minimal Surfaces in LLM Geometries", "Holographic Entanglement Entropy of Anisotropic Minimal Surfaces in LLM Geometries" ]	[ "Chanju Kim [email protected] \nDepartment of Physics\nEwha Womans University\n120-750SeoulKorea\n", "Kyung Kiu Kim [email protected] \nDepartment of Physics\nCollege of Science\nYonsei University\n120-749SeoulKorea\n", "O-Kab Kwon \nDepartment of Physics\nBK21 physics Research Division\nInstitute of Basic Science\nSungkyunkwan University\n440-746SuwonKorea\n" ]	[ "Department of Physics\nEwha Womans University\n120-750SeoulKorea", "Department of Physics\nCollege of Science\nYonsei University\n120-749SeoulKorea", "Department of Physics\nBK21 physics Research Division\nInstitute of Basic Science\nSungkyunkwan University\n440-746SuwonKorea" ]	[]	We calculate the holographic entanglement entropy (HEE) of the Z k orbifold of Lin-Lunin-Maldacena (LLM) geometries which are dual to the vacua of the mass-deformed ABJM theory with Chern-Simons level k. By solving the partial differential equations analytically, we obtain the HEEs for all LLM solutions with arbitrary M2 charge and k up to µ 2 0 -order where µ 0 is the mass parameter. The renormalized entanglement entropies are all monotonically decreasing near the UV fixed point in accordance with the F -theorem. Except the multiplication factor and to all orders in µ 0 , they are independent of the overall scaling of Young diagrams which characterize LLM geometries. Therefore we can classify the HEEs of LLM geometries with Z k orbifold in terms of the shape of Young diagrams modulo overall size. HEE of each family is a pure number independent of the 't Hooft coupling constant except the overall multiplication factor. We extend our analysis to obtain HEE analytically to µ 4 0 -order for the symmetric droplet case.	10.1016/j.physletb.2016.05.095	null	89,616,001	1605.00849	a4e5360eda739e70add8d47dc2ffa84844abf9f6	Holographic Entanglement Entropy of Anisotropic Minimal Surfaces in LLM Geometries 3 May 2016 Chanju Kim [email protected] Department of Physics Ewha Womans University 120-750SeoulKorea Kyung Kiu Kim [email protected] Department of Physics College of Science Yonsei University 120-749SeoulKorea O-Kab Kwon Department of Physics BK21 physics Research Division Institute of Basic Science Sungkyunkwan University 440-746SuwonKorea Holographic Entanglement Entropy of Anisotropic Minimal Surfaces in LLM Geometries 3 May 2016 We calculate the holographic entanglement entropy (HEE) of the Z k orbifold of Lin-Lunin-Maldacena (LLM) geometries which are dual to the vacua of the mass-deformed ABJM theory with Chern-Simons level k. By solving the partial differential equations analytically, we obtain the HEEs for all LLM solutions with arbitrary M2 charge and k up to µ 2 0 -order where µ 0 is the mass parameter. The renormalized entanglement entropies are all monotonically decreasing near the UV fixed point in accordance with the F -theorem. Except the multiplication factor and to all orders in µ 0 , they are independent of the overall scaling of Young diagrams which characterize LLM geometries. Therefore we can classify the HEEs of LLM geometries with Z k orbifold in terms of the shape of Young diagrams modulo overall size. HEE of each family is a pure number independent of the 't Hooft coupling constant except the overall multiplication factor. We extend our analysis to obtain HEE analytically to µ 4 0 -order for the symmetric droplet case. Introduction Gauge/gravity duality has been a central paradigm for decades in theoretical physics. Among others, holographic calculation of the entanglement entropy (EE) [1,2] draws recently much attention due to its elegance and implications for the nature of quantum field theories as well as quantum gravity. In this paper, we consider Z k orbifolds of Lin-Lunin-Maldacena(LLM) geometries [3,4] with SO(2,1)×SO(4)×SO(4) isometry in 11-dimensional supergravity and calculate the holographic entanglement entropy (HEE) to nontrivial orders in the mass parameter. The main motivation is their connection to the Aharony-Bergman-Jafferis-Maldacena (ABJM) theory with level k [5] which is a conformal field theory (CFT) describing the dynamics of M2-branes on the transverse C 4 /Z k orbifold with the Chern-Simons level k. It allows a mass-deformation [6,7] which preserves full N = 6 supersymmetries. This mass-deformed ABJM (mABJM) theory has many supersymmetric vacua. It has been shown that the vacua have one-to-one correspondence with the Z k orbifold [8,9] of LLM geometries, which are classified by a 1-dimensional droplet picture, or equivalently Young diagrams [4]. The LLM metric has a mass parameter µ 0 which is proportional to the mass parameter µ in the mABJM theory. Then we can explore the renormalization group (RG) flow of the renormalized entanglement entropy (REE) [10] triggered by the mass deformation from the ABJM theory as a UV fixed point [11]. Since there are many vacua in the theory, the RG flow depends on the vacuum. This should be manifested in the holographic calculation of REE for LLM geometries. See also [12][13][14][15][16][17][18] for the behavior of EE under relevant perturbations from the UV fixed point. An important issue related to REE is about the c-theorem which states that there exists a c-function which is positive definite and monotonically decreasing along the RG flow [19][20][21]. In 3-dimensions in particular, it is called the F -theorem [22] because the free energy on a three sphere plays the role of c-function. The F -theorem was proved [23] through the connection of the free energy with the constant term of EE of a circle [24][25][26]. In this paper, we will examine explicitly how F -theorem is realized in the HEE of the mABJM theory which has a large number of discrete vacua. The LLM geometries with Z k orbifold are all asymptotic to AdS 4 × S 7 /Z k . They are, however, not spherically symmetric along the radial direction of the AdS geometry but depend on two transverse coordinates. Therefore, it is not a simple exercise to get the minimal area for a given entangling region because one has to solve a partial differential equation (PDE) for the two transverse coordinates. In the previous work [11], the angle dependence was neglected to simplify the calculation with the assumption that it would not contribute at least in the leading order in µ 0 . Though sensible results were obtained for simple droplet configurations, there were cases that the F -theorem is violated in this approximation. In this work, however, we take into account all the angle dependence exactly. In other words, we solve the PDE exactly for all LLM solutions with Z k orbifold and obtain the corresponding HEE up to µ 2 0 -order. Then we verify that the REE satisfies the F -theorem for all relevant deformations connected to dual LLM geometries with Z k orbifold. For some simple droplet configurations with general k, we further extend our analysis to µ 4 0 -order and obtain REE analytically. Since we work with the most general k and the rank of the gauge group N, it is possible to investigate the dependence of EE on these parameters including the 't Hooft coupling λ = N/k in particular. Note, however, that it is not a trivial task to compare the EEs with different N or k because they will not uniquely specify a droplet due to many degeneracies. That is, in the field theory language, different vacua will give different EEs and to begin with one has to specify the vacua to compare. We will see that, depending on which vacua to choose, the EE depends on λ differently. Moreover, we will show that, up to a multiplication factor, the HEE of LLM geometries is independent of the overall scaling of the droplet configurations to all orders in µ 0 . Therefore, we can classify the LLM geometries with Z k orbifold in terms of the shape of the corresponding Young diagrams modulo overall size. At each order in µ 0 , they are pure numbers independent of λ. These can be considered as nontrivial results to test the gauge/gravity duality in the large N limit between the LLM geometry and mABJM theory which are not conformal. This paper is organized as follows. In section 2, we briefly review the relation between the vacua of the mABJM theory and the droplet classification of the LLM geometry with Z k orbifold. In section 3, we solve the PDE exactly to obtain the HEE of a disk up to µ 2 0 -order. We show that the resulting REE satisfies the F -theorem near the UV fixed point for all LLM geometries with Z k orbifold. Then we show that it is classified by the shape of the Young diagrams and discuss how it depends on N and k. We also calculate the REE analytically up to µ 4 0 -order for simple droplets. We draw our conclusion in section 4. HEE of the mABJM Theory and LLM Geometries Supersymmetric vacua of the mABJM theory are classified by the occupation numbers (N n , N ′ n ) [9], which are numbers of irreducible n×(n+1) and (n+1)×n GRVV matrices [7], respectively. On the other hand, the LLM solutions with Z k orbifold are also classified by the discrete torsions (l n , l ′ n ) assigned in the droplet picture of the LLM geometry. It was shown that there exists one-to-one correspondence between (N n , N ′ n ) and (l n , l ′ n ) in the range, 0 ≤ N n , N ′ n , l n , l ′ n ≤ k [9]. Since the mass deformation of the ABJM theory is a relevant deformation from the UV fixed point, the dual LLM geometry with Z k orbifold is asymptotic to AdS 4 × S 7 /Z k . We investigate the behavior of the RG flow near the UV fixed point in terms of the HEE for all LLM solutions with general k and examine the F -theorem. Let us start with the LLM geometry dual to the vacua of the U(N) k × U(N) −k mABJM theory with a mass parameter µ. The metric is given by ds 2 = \|G tt \|(−dt 2 + dw 2 1 + dw 2 2 ) + G xx dx 2 + dy 2 + G θθ ds 2 S 3 /Z k + Gθθds 2 S 3 /Z k , (2.1) where ds 2 S 3 /Z k and ds 2 S 3 /Z k are metrics for two S 3 /Z k spheres and G tt = −   4µ 2 0 y 1 4 − z 2 f 2   2/3 , G xx =   f 1 4 − z 2 2µ 0 y 2   2/3 , (2.2) G θθ =   f y 1 2 + z 2µ 0 ( 1 2 − z)   2/3 , Gθθ =   f y 1 2 − z 2µ 0 1 2 + z   2/3 (2.3) with f = 1 − 4z 2 − 4y 2 V 2 . (2.4) In the metric, the mass parameter µ 0 is identified with that of the mABJM theory through µ 0 = µ/4 [9] in the convention of [11]. The geometry is completely determined by functions z and V , z(x, y) = 2N B +1 i=1 (−1) i+1 (x − x i ) 2 (x − x i ) 2 + y 2 , V (x, y) = 2N B +1 i=1 (−1) i+1 2 (x − x i ) 2 + y 2 , (2.5) where x i 's denote the boundaries of black and white strips in the droplet representation with N B being the number of black droplets. For details of the droplet picture with general k, see [9]. Due to the quantization condition of the four-form fluxes on 4-cycles ending on the edges of black/white regions, it turns out that x i 's are quantized as (x i+1 − x i ) 2πl 3 P µ 0 ∈ Z, (2.6) where l P is the Planck length. Note that the quantization is proportional to µ 0 . It introduces µ 0 dependence to the metric in addition to the explicit dependence appearing in (2.2) and (2.3). The metric (2.1) goes asymptotically to AdS 4 × S 7 /Z k with radius R given by R = (32π 2 kÑ) 1/6 l P , (2.7) where 1Ñ = 1 2k (C 2 −C 2 1 ), C p = 2N B +1 i=1 (−1) i+1 x i 2πl 3 P µ 0 p ≡ 2N B +1 i=1 (−1) i+1xp i . (2.8) For later convenience, we define normalized coefficients C p by C p ≡ (kÑ ) −p/2C p , (2.9) which are invariant under an overall scaling of x i 's. Then C 2 − C 2 1 = 2. Now, let's consider a 9 dimensional surface in this geometry. We denote coordinates of the surface by σ i with i = 1, . . . , 9 and represent the embedding function as X M (σ i ) where M = 0, . . . , 10. Then, the 9 dimensional area of the surface becomes γ A = d 9 σ det g ij = d 9 σ det G M N ∂X M ∂X N ∂σ i ∂σ j , (2.10) where g ij is the induced metric of the surface and G M N is the 11-dimensional metric in (2.1). The HEE is defined by [1,2] 11) where G N = (2πl P ) 9 /(32π 2 ) is the 11 dimensional Newton constant. In the next section we would like to calculate S A for disk-type entangling surfaces in the small µ 0 limit. S A = Min(γ A ) 4G N ,(2. Anisotropic Minimal Surfaces and HEE The effect of small mass deformation on the HEE has been considered in [11] under the approximation that the minimal surface is independent of the angle in polar coordinates introduced below. Though this approximation gives reasonable results consistent with Ftheorem for simple droplet configurations, one cannot expect that it would be valid in general 1Ñ is the area of the Young diagram made of positionsx i divided by k, and is equal to the rank N of the gauge group in the field theory up to the contribution of discrete torsions [28,29]. See section 3 for an example. In addition, there is a further constant correction − 1 24 (k − 1 k ). With these corrections,Ñ should eventually be the Maxwell M2 charge [9], i.e.Ñ = N − 1 24 (k − 1 k ). because the spherical symmetry is obviously broken. Indeed, for some droplet configurations the REE calculated in [11] does not decrease monotonically along the RG flow, violating the F -theorem. For these LLM geometries, it was also found that the curvature scalars are not small at some transverse positions even in the large N limit [11,27], which implies that the gauge/gravity duality for those geometries does not work in this approximation. In this section we would like to investigate the effect of small mass deformation without resorting to such an approximation. In other words, we will treat the angular dependence of the minimal surface exactly. It amounts to solving PDEs with two variables up to some nontrivial order in µ 0 . From now on, we take only disk type entangling surfaces into account. We will work with polar coordinates u and α defined by x = R 3 4lu cos α, y = R 3 4lu sin α, (3.12) where l is the radius of the disk at the boundary. The minimal surface is bounded by a disk in the w 1 -w 2 plane at the boundary of AdS space (u = 0). To describe such a configuration, we may consider the following embedding, w 1 = ρl cos σ 1 , w 2 = ρl sin σ 1 , u = u(ρ, φ), α = α(ρ, φ). (3.13) Plugging this into (2.10), we obtain the action, γ disk = π 5 R 9 8k 1 0 dρ π 0 dφ gρ sin 3 α u 2 α ′2 + u ′2 u 2 + g 2 (αu ′ − α ′u ) 2 ,(3.14) where˙= ∂ ∂ρ and ′ = ∂ ∂φ . We have also introduced the function g(u, φ) defined by f (u, φ) = 2µ 0 lu sin φ g(u, φ). (3.15) Note that all the mass-deformation effect in (3.14) appears only through the function g which contains the information of the droplet position x i 's. In the undeformed limit µ 0 → 0, x i 's go to zero due to the quantization condition (2.6). Then it is easy to see that g goes to unity and hence (3.14) reduces to the minimal surface action for the undeformed ABJM theory [1]. By utilizing the residual gauge degree of freedom, we may choose α = φ. Then the equation of motion yields a PDE for u = u(ρ, φ), ∂ ∂ρ ρg 3u sin 3 φ u 2 √ X + ∂ ∂φ ρgu ′ sin 3 φ u 4 √ X − ∂ ∂u ρg sin 3 φ u 2 √ X = 0,(3.16) where X = 1 + u ′2 u 2 + g 2u2 . HEE up to O(µ 2 0 ) Now we are ready to consider the effect of small mass deformation. From (2.4) and (2.5) we see that g can be expanded in powers of x i √ x 2 +y 2 ∼ µ 0 lu. Furthermore, u itself will depend on µ 0 l. As a result we can write u ≡ ∞ n=0 u n (ρ, φ)(µ 0 l) n = u 0 (ρ) + u 1 (ρ, φ)µ 0 l + u 2 (ρ, φ)(µ 0 l) 2 + · · · , g(u, φ) ≡ ∞ n=0 g n (φ)(uµ 0 l) n = 1 + g 1 (φ)u 0 µ 0 l + [g 1 (φ)u 1 + g 2 (φ)u 2 0 ](µ 0 l) 2 + · · · . (3.17) Since z and V consist of the generating function of the Legendre polynomials, g i 's can be written in terms of Legendre polynomials [11]. Explicitly, a few lower terms are g 1 (φ) = D 1 cos φ, g 2 (φ) = D 2 + D 3 cos(2φ),(3.18) where D i 's are constants depending on the droplet positions, D 1 = 1 √ 2 (C 3 − C 1 C 2 ), D 2 = − 1 32 4C 2 3 − 4C 1 (C 2 1 + C 2 )C 3 − C 2 2 (C 2 1 − 5C 2 ) + 9(C 2 1 − C 2 )C 4 , D 3 = − 1 32 4C 2 3 − 4C 1 (3C 2 1 − C 2 )C 3 + C 2 2 (C 2 1 + 3C 2 ) + 15(C 2 1 − C 2 )C 4 ,(3.19) and C p 's are defined in (2.9). Given D i 's, one can solve the equations of motion (3.16) perturbatively with respect to µ 0 to obtain the change of the minimal surface. Note that we have to solve inhomogeneous PDEs of two variables ρ and φ in the background of lower order configurations. There is, in general, no guarantee that explicit form of solutions can be obtained. Nevertheless, in this case, we are able to find exact solutions up to the nonvanishing second orders in perturbation. Let us start with the zeroth order equation of (3.16) in µ 0 , This gives the minimal surface for ABJM theory without mass deformation. u 0 + (2ρ + u 0u0 )(1 +u 2 0 ) ρu 0 = 0. If we plug this solution into (3.16), then the first order equation of motion reads ρ(1 − ρ 2 ) 2ü 1 + ρu ′′ 1 + (1 − ρ 2 )(1 − 2ρ 2 )u 1 + 3ρ cot φ u ′ 1 − 2ρu 1 − D 1 ρ(1 − ρ 2 )(5 − 3ρ 2 ) cos φ = 0. (3.22) We have to solve the equation under the boundary conditions u 1 (1, φ) =u 1 (0, φ) = 0 and u ′ 1 (ρ, 0) = u ′ 1 (ρ, π) = 0, where the latter comes from the regularity at φ = 0 and π. This is an inhomogeneous linear PDE with explicit dependence on the independent variables ρ and φ. It, however, admits a very simple solution u 1 (ρ, φ) = − D 1 2 1 − ρ 2 cos φ. (3.23) One can proceed to the second order in µ 2 0 . The equation of motion at the second order becomes, after the solutions (3.21) and (3.23) plugged into (3.17), ρ(1 − ρ 2 ) 2ü 2 + (1 − ρ 2 )(1 − 2ρ 2 )u 2 + ρ(u ′′ 2 + 3 cot φ u ′ 2 ) − 2ρu 2 + 1 8 ρ(1 − ρ 2 ) 3/2 D 2 1 (27 − 26ρ 2 ) − 16D 2 (3 − 2ρ 2 ) + (11D 2 1 − 16D 3 )(3 − 2ρ 2 ) cos(2φ) = 0 ,(3.24) with the boundary conditions u ′ 2 (ρ, 0) = u ′ 2 (ρ, π) = 0 and u 2 (1, φ) =u 2 (0, φ) = 0. This is even more complicated than the first order equation (3.22). Remarkably, however, a fully analytic solution is still available, u 2 (ρ, φ) = − 1 6 1 − ρ 2 (D 2 1 + 20D 2 − 12D 3 ) log(1 + 1 − ρ 2 ) + 1 48 [8 + (9 − 13ρ 2 ) 1 − ρ 2 ]D 2 1 + 16[10 − (6 − ρ 2 ) 1 − ρ 2 ]D 2 − 48(2 − 1 − ρ 2 )D 3 + 1 48 (11D 2 1 − 16D 3 )(1 − ρ 2 ) 3/2 cos(2φ). (3.25) Having found the solution to the µ 2 0 -order, now we can calculate the minimal surface area (3.14) to this order, γ disk = γ (0) disk + µ 0 lγ (1) disk + (µ 0 l) 2 γ (2) disk + · · · . (3.26) Inserting the solutions (3.21), (3.23) and (3.25) into (3.14), we obtain 2 : γ (0) disk = π 5 R 9 6k l ǫ − 1 , γ(1)disk = 0, γ(2)disk = − π 5 R 9 72k (12D 3 − D 2 1 − 20D 2 ), (3.27) where ǫ in γ (0) disk is the UV cutoff in the u coordinate. Note that the first order correction vanishes due to the angular integration. For the second order contribution γ (2) disk , it is crucial to notice that the combination (12D 3 −D 2 1 −20D 2 ) can be rewritten in the form of a complete square, 12D 3 − D 2 1 − 20D 2 = 16 + 1 2 (C 3 − 3C 1 C 2 + 2C 3 1 ) 2 , (3.28) where we used the parameter relations in (3.19). The entanglement entropy then becomes S disk = π 5 R 9 24G N k l ǫ − 1 − µ 2 0 l 2 4 3 + 1 24 (C 3 − 3C 1 C 2 + 2C 3 1 ) 2 + O(µ 3 0 ). (3.29) A few comments are in order. First, it is not difficult to show that the expression (C 3 − 3C 1 C 2 + 2C 3 1 ) appearing here is a unique combination made of cubic terms in x i which is invariant under the translation x i → x i + a. It provides a nontrivial consistency check of the result. Moreover, the expression appears in (3.29) in the form of a complete square and hence the second order term is negative definite. This has an important implication in relation to the F -theorem [22][23][24]. In the present context, the REE can play the role of a c-function of the theory [10]. It is computed by the prescription (3.30) which is clearly monotonically decreasing near the UV fixed point. Note that this is true for all geometries dual to the vacua of mABJM, regardless of the C i 's. This result may be considered as an evidence of the validity of holography for non-conformal case. This corrects the result in [11] where REE was calculated with the angle (α) dependence neglected and showed increasing behavior for some asymmetric droplet configurations. This means that the angle dependence of the minimal surface in the LLM geometry results in nontrivial contributions. F disk ≡ l ∂ ∂l − 1 S disk = π 5 R 9 24G N k 1 − µ 2 0 l 2 4 3 + 1 24 (C 3 − 3C 1 C 2 + 2C 3 1 ) 2 + O(µ 3 0 ), As is evident from the calculation, the REE depends on N or k only through C p 's except the overall multiplication. Therefore, different theories with same C p 's will give essentially the same REE. Since C p 's defined in (2.9) are invariant under the overall scaling of droplet boundaries x i 's, a family of droplets with a same geometric shape of Young diagrams modulo overall size give the same REE up to a multiplication factor. This holds to all orders in µ 0 since the action (3.14) is completely determined in terms of C p 's up to an overall factor. As a simple example, we consider the case of N B = 1 with arbitrary k for which the geometry is specified by three boundary positions x 1 , x 2 and x 3 of the droplet. See Fig.1. F disk = π 5 R 9 24G N k 1 − µ 2 0 l 2 7 12 + 3 8 x 3 − x 2 x 2 − x 1 + x 2 − x 1 x 3 − x 2 + O(µ 3 0 ). (3.31) This result explicitly shows that scaling the overall size of the droplet (or the shape of the Young diagram) does not change REE except the overall multiplication factor. To connect the result with the field theory, let us parameterize the integer-valued positionsx i = x i /2πl 3 P µ 0 defined in (2.8) asx 1 = −pk − m,x 2 = (q − p)k,x 3 = qk + m, where p, q, m are positive integers and 0 ≤ m < k, so that the location of the Fermi level of the droplet is zero. Then including the contribution of the discrete torsions [9], we obtain the rank N of the gauge group as N = (pk + m)(qk + m)/k − m(m − k)/k = kpq + m(p + q + 1). In the large N limit with finite 't Hooft coupling λ = N/k, (3.31) is reduced to F disk = π 5 R 9 24G N k 1 − µ 2 0 l 2 7 12 + 3 8 p q + q p + O(1/k) + O(µ 3 0 ),(3.32) where we assumed k ≫ m for simplicity and λ = pq in this limit. Therefore, for a fixed ratio of p and q the mass correction is independent of λ. On the other hand, if we scale, say, p with q fixed, then the correction depends nontrivially on λ since p/q = λ/q 2 . Note that by changing p or q, we are comparing theories with different λ. The above result demonstrates that λ-dependence of REE appears in different ways, depending on how vacua are selected in mABJM theories for comparison. Finally, let us give a comment on the stationarity of the RG flow. Since the deformation parameter µ enters into the mABJM theory as a mass of a supermultiplet, the deformation is of the first order in µ due to the fermionic mass term. On the other hand, REE F in (3.30) has vanishing first order correction in µ 0 . It means that the RG flow at UV point is stationary. This is consistent with the result of [13]. HEE up to O(µ 4 0 ) for symmetric droplet case For some simple droplet configurations, it is possible to obtain higher order corrections analytically. For example, if D 1 = 0, the first order equation (3.22) becomes homogeneous with vanishing boundary conditions and hence the first order correction u 1 is zero identically. This will also simplify higher order equations. Here we will consider a symmetric droplet case obtained by putting p = q in Fig.1, i.e., (x 1 ,x 2 ,x 3 ) = (−pk − m, 0, pk + m) and N = kp 2 + m(2p + 1). To consider higher order corrections up to µ 4 0 -order, we need the following coefficient functions in (3.17), g 2 (φ) = − 1 8 + 9 8 cos(2φ), g 4 (φ) = 1 256 [85 − 60 cos(2φ) + 359 cos(4φ)], (3.33) as well as g n (φ) = 0 for odd n. By symmetry we may set u n = 0 for odd n. Then employing all the lower order results including (3.21) and (3.25) gives us the equation of motion for u 4 (ρ, φ), ρ(1 − ρ 2 ) 2ü 4 + (1 − ρ 2 )(1 − 2ρ 2 )u 4 + ρ(u ′′ 4 + 3 cot φ u ′ 4 ) − 2ρu 4 +B0 (ρ) = 64ρ(3ρ 2 + 2) 9(1 − ρ 2 ) 3/2 log 1 − ρ 2 + 1 2 − 2 9ρ (9ρ 6 + 28ρ 4 + 187ρ 2 − 64) − 2ρ 9(1 − ρ 2 ) 128ρ 2 − (67ρ 4 + 8ρ 2 + 85) 1 − ρ 2 + 192 log 1 − ρ 2 + 1 + 1 1152ρ 1 − ρ 2 (6147ρ 10 − 15629ρ 8 + 6165ρ 6 − 8463ρ 4 + 69124ρ 2 − 16384), B(4)2 (ρ) = 3 32 ρ(1 − ρ 2 ) 576 − 192ρ 2 − (15ρ 4 − 227ρ 2 + 404) 1 − ρ 2 − 6ρ(7 − ρ 2 ) 1 − ρ 2 log 1 − ρ 2 + 1 , B(4)4 (ρ) = − 125 128 ρ(1 − ρ 2 ) 5/2 (4 − 3ρ 2 ). (3.35)(4) This is again an inhomogeneous PDE with very complicated source terms but fortunately we can find the solution satisfying the necessary boundary conditions, u 4 (ρ, φ) = C(4) 4 (ρ) cos(4φ) + C 2 (ρ) = 1 64 (1 − ρ 2 ) 3/2 96 7 − 2ρ 2 ρ 2 log 1 − ρ 2 + 1 + 32 6ρ 4 − 16ρ 2 − 5 1 − ρ 2 + 1 − ρ 2 3ρ 6 − 97ρ 4 + 209ρ 2 + 125 , C(4)0 (ρ) = − 1 1 − ρ 2 1 ρ F (s) ds, (3.37) with F (s) = 135s 6 + 598s 4 − 3371s 2 + 1423 135s(1 − s 2 ) 3/2 − 256 9s √ 1 − s 2 log 2 + 18441s 8 − 37339s 6 − 170949s 4 + 326871s 2 − 182144 17280s(1 − s 2 ) − 64s log 2 ( √ 1 − s 2 + 1) 9(1 − s 2 ) 2 + 1 9s 64(s 4 + 1) (1 − s 2 ) 3/2 − 134s 6 − 524s 4 + 501s 2 − 192 (1 − s 2 ) 2 log( √ 1 − s 2 + 1). (3.38)(4) Inserting the solutions in (3.21), (3.25), and (3.36) into (3.17) and (3.14), we easily obtain the corresponding HEE and REE for the symmetric droplet up to µ 4 0 -order, S disk = π 5 R 9 24kG N l ǫ − 1 − 4 3 µ 2 0 l 2 + 2671 − 3840 log 2 540 µ 4 0 l 4 + O(µ 6 0 ), F disk = π 5 R 9 24kG N 1 − Conclusion We investigated the RG flow behavior and the F -theorem in terms of the HEE near the UV fixed point in 3-dimensions, where a supersymmetric Chern-Simons matter theory is living. As the UV CFT we considered the N = 6 ABJM theory and introduced the supersymmetry preserving mass deformation, called the mABJM theory. This deformation is a relevant deformation and so triggers the RG flow from the UV fixed point. To describe the RG behavior near the UV fixed point, we adapted the HEE conjecture to the LLM geometry in 11-dimensional supergravity, since the supersymmetric vacua of the mABJM theory have one-to-one correspondence with the LLM geometries with Z k orbifold. The LLM solution has SO(2,1)×SO(4)×SO(4) isometry and warp factors of the metric depend on the two transverse coordinates (u, α) in (3.12). For this reason, one has to solve the PDE for u and α to obtain the minimal surface in the HEE proposal. In this paper, we analytically solved the PDE up to µ 2 0 -order for all LLM solutions with arbitrary N and k. We found that REEs have different RG trajectories depending on the LLM geometries but they are always monotonically decreasing near the UV fixed point in accordance with the F -theorem in 3-dimensions. 3 We also computed the REE up to µ 4 0 -order for a special family of LLM geometries with arbitrary N and k. It would be interesting to extend the result to more general case. Since the HEE proposal is based on the gauge/gravity duality, in order to compare our results in gravity side with those in the mABJM theory, one has to consider the large 't Hooft coupling λ = N/k in the N → ∞ limit. In general, the effect of mass deformation in REE would depend on λ and calculation in field theory side is a formidable task due to nonperturbative effects coming from the strong coupling constant. In the mABJM theory, there are further complications from the presence of many vacua. However, we found that for a family of droplets with a same shape of Young diagrams, we have the same REE (or HEE) up to overall dependence on N and k. One might be able to calculate the REE in the field theory side perturbatively on a certain class of vacua, and compare the result quantitatively with that in the dual gravity side. nothing but the equation of motion for the conformal case, as it should be. Imposing the boundary conditions, u(0) = 1 andu(0) = 0, one can find the well-known solution which is a geodesic in AdS space, u 0 (ρ) = 1 − ρ 2 .(3.21) Figure 1 : 1A droplet representation of N B =1 case. l i 's are discrete torsions assigned atx i 's. From (2.8), the REE becomes holds for arbitrary N and k. In general, the expressions of the HEE and REE depend on N, k, radius of the disk, and the choice of droplets as we have seen in the previous subsection. However, for a family of droplets related by rescaling of the overall size, the coefficients of the corrections are given by pure numbers as seen in (3.39), and in particular are independent of the 't Hooft coupling constant λ. This is an interesting phenomena from the point of view of the gauge/gravity duality. Based on the duality relation between the vacua of the mABJM theory and the LLM geometries with Z k orbifold, one can examine the HEE conjecture by computing the EE of the dual field theory on a family of vacua considered here. Introducing the UV cutoff ǫ in the u coordinate changes the upper limit of the integration range of ρ in (3.14). 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[ "Charging by quantum measurement", "Charging by quantum measurement" ]	[ "Jia-Shun Yan \nSchool of Physics\nZhejiang University\n310027Hangzhou, ZhejiangChina\n", "Jun Jing \nSchool of Physics\nZhejiang University\n310027Hangzhou, ZhejiangChina\n" ]	[ "School of Physics\nZhejiang University\n310027Hangzhou, ZhejiangChina", "School of Physics\nZhejiang University\n310027Hangzhou, ZhejiangChina" ]	[]	We propose a quantum charging scheme fueled by measurements on ancillary qubits serving as disposable chargers. A stream of identical qubits are sequentially coupled to a quantum battery of N + 1 levels and measured by projective operations after joint unitary evolutions of optimized intervals. If charger qubits are prepared in excited state and measured on ground state, then their excitations (energy) can be near-perfectly transferred to battery by iteratively updating the optimized measurement intervals. Starting from its ground state, the battery could be constantly charged to an even higher energy level. Starting from a thermal state, the battery could also achieve a near-unit ratio of ergotropy and energy through less than N measurements, when a population inversion is realized by measurements. If charger qubits are prepared in ground state and measured on excited state, useful work extracted by measurements alone could transform the battery from a thermal state to a high-ergotropy state before the success probability vanishes. Our operations in charging are more efficient than those without measurements and do not invoke the initial coherence in both battery and chargers. Particularly, our finding features quantum measurement in shaping nonequilibrium systems.	null	[ "https://export.arxiv.org/pdf/2209.13868v2.pdf" ]	252,568,085	2209.13868	84327a0aeffb629ff74b9831f96b5b731ead5684	Charging by quantum measurement 5 Jun 2023 Jia-Shun Yan School of Physics Zhejiang University 310027Hangzhou, ZhejiangChina Jun Jing School of Physics Zhejiang University 310027Hangzhou, ZhejiangChina Charging by quantum measurement 5 Jun 2023(Dated: June 6, 2023) We propose a quantum charging scheme fueled by measurements on ancillary qubits serving as disposable chargers. A stream of identical qubits are sequentially coupled to a quantum battery of N + 1 levels and measured by projective operations after joint unitary evolutions of optimized intervals. If charger qubits are prepared in excited state and measured on ground state, then their excitations (energy) can be near-perfectly transferred to battery by iteratively updating the optimized measurement intervals. Starting from its ground state, the battery could be constantly charged to an even higher energy level. Starting from a thermal state, the battery could also achieve a near-unit ratio of ergotropy and energy through less than N measurements, when a population inversion is realized by measurements. If charger qubits are prepared in ground state and measured on excited state, useful work extracted by measurements alone could transform the battery from a thermal state to a high-ergotropy state before the success probability vanishes. Our operations in charging are more efficient than those without measurements and do not invoke the initial coherence in both battery and chargers. Particularly, our finding features quantum measurement in shaping nonequilibrium systems. We propose a quantum charging scheme fueled by measurements on ancillary qubits serving as disposable chargers. A stream of identical qubits are sequentially coupled to a quantum battery of N + 1 levels and measured by projective operations after joint unitary evolutions of optimized intervals. If charger qubits are prepared in excited state and measured on ground state, then their excitations (energy) can be near-perfectly transferred to battery by iteratively updating the optimized measurement intervals. Starting from its ground state, the battery could be constantly charged to an even higher energy level. Starting from a thermal state, the battery could also achieve a near-unit ratio of ergotropy and energy through less than N measurements, when a population inversion is realized by measurements. If charger qubits are prepared in ground state and measured on excited state, useful work extracted by measurements alone could transform the battery from a thermal state to a high-ergotropy state before the success probability vanishes. Our operations in charging are more efficient than those without measurements and do not invoke the initial coherence in both battery and chargers. Particularly, our finding features quantum measurement in shaping nonequilibrium systems. I. INTRODUCTION Over one century, the classical batteries have been driving the revolutions in personal electronics and automotive sector. As energy-storage units in a cutting-edge paradigm, quantum batteries [1][2][3][4] are expected to outperform their classical counterparts by widely exploiting the advantages from quantum operations and promoting their efficiency under the constraint of quantum thermodynamics [5][6][7]. Enormous attentions were paid to charging quantum batteries, as the primary step in the charge-store-discharge cycle. Many protocols have been proposed, including but not limited to charging by entangling operations [8,9], charging with dissipative [10,11], unitary, and collision processes [12,13], charging collectively and in parallel [14,15], and charging with feedback control [16]. Many-body interaction [17][18][19] and energy fluctuation [20,21] were also explored to raise the upperbound of charging power and capacity. Local and global interactions are designed to transfer energy from various thermodynamical resources to batteries. The maximum rate and amount in energy transfer are subject to relevant timescales of evolution and relaxation. Quantum measurements, particularly the repeated projections onto a chosen state or a multidimensional subspace, could change dramatically the transition rate of the measured system [22][23][24]. Numerous measurement-based control schemes were applied to state purification [25,26], information gain [27], and entropy production [28,29]. Quantum engineering by virtue of the measurements on ancillary system, that generates a net nonunitary propagator, is capable to purify and cool down quantum systems [30][31][32][33][34]. In general, a projective measurement or postselection on ancillary system * Email address: [email protected] would navigate the target system to a desired state with a finite probability. Therefore, quantum measurements could become a useful resource as well as the heat or work reservoirs, serving as fuels powering a thermodynamical or state-engineering scheme through a nonunitary procedure [35][36][37][38][39]. In this work, we address quantum measurements in the context of quantum energetics by the positive operator-valued measures (POVM). It is interesting to find that POVMs generated by the joint evolution of chargers and batteries combined with projections on a specific state of chargers is able to speed up the charging rate and promote the amount of accumulated energy and ergotropy. Our method is transparently distinct from those based on swap or exchange operations. It does not necessarily rely on the initial states of both battery and charger. Without energy exchange, it could transform the system from a completely passive state to a useful state for battery. In particular, we propose a charging-by-measurement scheme in a quantum collision framework [40][41][42]. As disposable chargers a sequence of identical qubits line up to temporarily interact with the battery (a multilevel system with a finite number of evenly spaced ladders). Once a projective measurement is performed in the end of the joint evolution and the outcome is as desired, the coupled qubit is replaced with a new one and then the charging continues. Figures 1(a) and 1(b) demonstrate respectively a power-on and a power-off charging schemes. When the qubits are initially in the excited state ρ e = \|e e\|, the projective measurement on the ground state M g = \|g g\| transfers the energy of excited qubits to the battery. The energy gain for battery increases linearly with the number of measurements and the charging power is gradually enhanced as well. Full population inversion is realized when the measurement number is close to the battery size. When the qubits are prepared as the ground states ρ g = \|g g\|, still the battery can be charged by measuring the ancil- Identical ancillary qubits line up to interact with the battery for a period of time. In the end of each round of joint evolution, a projective measurement is performed on the qubit to induce a nonunitary charging operation on the battery. In (a) power-on charging, qubits are initialized as the excited state ρe = \|e e\| and the projection Mg = \|g g\| is performed on its ground state. Energy gain for the battery is mainly from the qubit excitations. In (b) power-off charging, qubits are initialized as the ground state ρg = \|g g\| and the projection Me = \|e e\| is on the excited state. Useful work extracted by measurements simultaneously charges both battery and ancillary qubits. lary qubits on the excited state M e = \|e e\|. Useful work extracted entirely from repeated measurements simultaneously charges both battery and charger. The rest of this work is structured as follows. In Sec. II, we introduce a general model of charging by measurements, whereby the charging or discharging effect is analyzed in view of a general POVM. In Sec. III, we present the power-on charging scheme. We find an optimized measurement interval to maximize the measurement probability, by which both energy and ergotropy of the battery scale linearly with the number of measurements. Section IV devotes to the power-off charging scheme. For both schemes, we evaluated the charging efficiency by charging power, state distribution, energy and ergotropy of the battery. In Sec. V, we discuss the effect from the initial coherence in the chargers and the robustness of our scheme against the environmental decoherence. In Sec. VI, we summarize the whole work. II. GENERAL MODEL OF CHARGING BY MEASUREMENTS We aim for charging a quantum battery by performing measurements on the ancillary qubits as disposable chargers. The scheme is constructed by rounds of joint evolution and projective measurements. The full Hamiltonian H = H B +H C +V consists of a target battery system H B , a sequence of identical charger qubits (in each round only a single charger with a free Hamiltonian H C is coupled to the battery and the others are decoupled), and the interaction V between battery and the current working charger. The battery is assumed to be in a thermal state ρ B = e −βHB /Tr[e −βHB ] with β ≡ 1/k B T . It is a completely passive state that is energetic but has no ergotropy. In quantum battery, ergotropy is defined as E ≡ Tr[H B (ρ − σ)], where σ is the passive state obtained by realigning the eigenvalues of ρ in decreasing order, which has none extractable energy under cyclic unitary operations [43]. For ρ = ρ B , it is found that σ = ρ. The thermal state is thus a reasonable choice to demonstrate the power of any charging scheme, also it is a natural state for the battery subject to a thermal bath in the absence of active controls. The charger qubits are prepared as the same mixed state ρ C = q\|g g\| + (1 − q)\|e e\| with a ground-state occupation q ∈ [0, 1] before linking to the battery. The initial coherence of chargers is temporally omitted to distinguish the charging efficiency of quantum measurements. In each round, the joint evolution of the battery and the working qubit is described by the time-evolution operator U = exp(−iHτ ). The coupling interval τ might be constant or vary with respect to all the rounds. An instantaneous projective measurement M on the qubit is implemented in the end of the round, and then the (unnormalized) joint state becomes ρ ′ tot = M U ρ B ⊗ ρ C U † M (1) with a finite measurement probability P = Tr[M U ρ B ⊗ ρ C U † ]. In this work, we do not consider the errors occurring in measurements and its energy cost. After measurement, the charger qubit is decoupled from the battery system and withdrawn, then another one is loaded to the next round. The charging scheme is nondeterministic in essence and thus employs a feedback mechanism: the measurement outcome determines whether to launch the next round of charging cycle or to restart from the beginning. n\|n − 1 n\|. The Hamiltonian for each charge qubit is H C = ω c \|e e\|, where ω c is the energy spacing between the ground \|g and excited states \|e . Battery and qubits are coupled with the exchange interaction V = g(σ − A † + σ + A), where g is the coupling strength and σ − and σ + denote the transition operators of the qubit. Then the full Hamiltonian in the rotating frame with respect to H 0 = ω b ( N n=0 n\|n n\| + \|e e\|) reads H = ∆\|e e\| + g σ − A + σ + A † ,(2) where ∆ = ω c − ω b represents the energy detuning between charger qubit and battery. The charging procedure is piecewisely concatenated by a sequence of joint evolutions of charger qubit and battery, which is interrupted by instantaneous projective measurements M ϕ = \|ϕ ϕ\| over a particular state \|ϕ = cos(θ/2)\|g +sin(θ/2)\|e , 0 ≤ θ ≤ π, of the working qubit. After a round with an interval τ , the battery state becomes ρ B (τ ) = Tr C [ρ ′ tot ] = D + C P ϕ = D + C Tr[M ϕ U ρ B ⊗ ρ C U † ] ,(3) where D is the diagonal (population) part in the density matrix of the battery system without normalization and C is the off-diagonal part C = sin θ 2 N n=1 α * n [qλ n p n + (1 − q)λ * n p n−1 ] \|n−1 n\|+H.c. (4) Here both α n = cos Ω n τ + i∆ sin(Ω n τ )/2Ω n and λ n = −ie −i∆τ /2 g √ n sin(Ω n τ )/Ω n are renormalization coefficients, Ω n = g 2 n + ∆ 2 /4 is the Rabi frequency, and p n is the initial thermal occupation on the nth level of battery, i.e., ρ B = N n=0 p n \|n n\| with p n = [e −βω b n − e −βω b (n+1) ]/[1 − e −βω b (N +1) ] . C describes the dynamical coherence that appears during the joint evolution of charger and battery, generating nonzero extractable work for the battery [44], and disappears upon the projective measurements. The population part of the battery state could be divided as D = D charge + D discharge(5) due to the heating or cooling contribution on the battery system from various POVMs. By Eqs. (1) and (3), we have D charge = (1 − q) cos 2 θ 2 M eg [ρ B ] + q sin 2 θ 2 M ge [ρ B ], D discharge = q cos 2 θ 2 M gg [ρ B ] + (1 − q) sin 2 θ 2 M ee [ρ B ],(6)where M ij [·] ≡ R ij · R † ij , i, j ∈ {e, g}, represents the jth element of the ith POVM, satisfying the normalization condition j R † ij R ij = I B . R ij ≡ j\|U \|i is the Kraus operator acting on the state space of the battery, where i and j label respectively the initial state and the measured state of the ancillary qubit. In particular, we have M eg [ρ B ] = N n=1 \|λ n (τ )\| 2 p n−1 \|n n\|, M ge [ρ B ] = N −1 n=0 \|λ n+1 (τ )\| 2 p n+1 \|n n\|, M gg [ρ B ] = N n=0 \|α n (τ )\| 2 p n \|n n\|, M ee [ρ B ] = N −1 n=0 \|α n+1 (τ )\| 2 p n \|n n\|.(7) According to Naimark's dilation theorem [45], a set of projective measurements {M j } acting on one of the subspaces H 1 of the total space H tot = H 1 ⊗ H 2 could induce a POVM on another subspace H 2 with a map H 2 → H tot . In our context, an arbitrary projective measurement M ϕ = \|ϕ ϕ\| defined in the space of the charger qubit induces a POVM M i,ϕ [ρ B ] acting on the battery. And the map is constructed by U \|i with the joint unitary evolution U , according to the initial state of the charger \|i . Therefore, as shown in Eq. (7), we have two sets of POVMs in the bare basis of the charger qubit: M e,j∈{e,g} and M g,j∈{e,g} . For instance, M eg represents a POVM on the battery induced by projection on the ground state of the qubit that is initially in the excited state. 7). Under a proper τ , M eg replaces a smaller p n with a larger \|λ n \| 2 p n−1 for all the excited states. On the contrary, M ge moves the populations on higher levels to lower levels, which might also enhance the battery energy through a significant renormalization over the population distribution. Note the largest population p 0 on the ground state has been eliminated. In contrast, D discharge is a linear combination of M gg and M ee . Both of them reduce the populations of the excited state due to the fact that \|α n>0 \| ≤ \|α 0 \| = 1 and then enhance the relative weight of the ground-state population [32]. Then they are inclined to discharge the battery. The dependence of charging or discharging on the initial and measured states can be quantitatively justified in Fig. 2 by the ratio of the average population of the chargern after a single round of evolution-andmeasurement and the initial thermal average population n th in the parametric space of θ and q, describing respectively the weights of the measured state and the initial state of qubit on the ground state. The two bluediagonal corners in Fig. 2 correspond to the POVMs M gg and M ee that could be used to cool down the target system. For instance, the lower right corner M gg with q = 1 and θ = 0 describes the mechanism of cooling-bymeasurement in the resonator system [32]. More crucial to the current work, the lower left corner M eg with q = 0 and θ = 0 and the upper right corner M ge with q = 1 and θ = π motivate our investigation on the following power-on and power-off charging schemes, respectively. One can find that the former scheme is more efficient than the latter in terms of the ration/n th with certain measurement interval. III. POWER-ON CHARGING In this section, we present the power-on charging scheme described by M eg in which the charger qubits are prepared in their excited states with q = 0 and the projective measurement is performed on the ground state with θ = 0. Then the density matrix of the battery after m ≥ 1 rounds of measurements reads with an n-dependent weight \|λ n (τ )\| 2 . The battery is initially set as a Gibbs thermal state with populations following an exponential decay function of the occupied-state index n. Thus it is charged step by step by the POVM M eg , which moves the populations of lower-energy states up to higher-energy states. And the τ -dependent normalization coefficient \|λ n (τ )\| 2 ranges from zero to unit. p (m) n is thus determined by the measurement interval τ between two consecutive measurements. With a sequence of properly designed or optimized measurement intervals, the battery could be constantly charged by the projection-induced POVM. ρ (m) on = M eg ρ (m−1) on P g (m) = N n=m \|λ n (τ )\| 2 p (m−1) n−1 \|n n\| P g (m) (8) where ρ (m−1) on = N n=m−1 p (m−1) We present the average populationn (m) ≡ n np (m) n after m measurements in Fig. 3(a) as a function of measurement interval τ . The initial thermal populationn th is plotted (see the black solid line) to compare withn (1) . One can find a considerable charging effect in the range of measurement interval τ ≤ 9/ω c . Over this critical point, a cooling-effect range appears where the average population is less than the initial population. And afterwardsn fluctuates with an even larger τ . To pursue the highest charging power, one might intuitively choose a measurement interval as small as possible. It is however under the constraint of a practical coupling strength between charger qubits and battery. In addition, if the measurement interval is smaller than the characterized period of the charger qubit (τ ∼ 1/ω c or less), then the joint-evolution interval would be too short to detect the charger qubit (initially in the excited state) in its ground state. It will extremely suppress the measurement probability. As the measurement probability P g (1) shown by the blue solid line in Fig. 3(b), it approaches zero when τ ω c → 0. When the measurement interval approaches about τ = 8/ω c , P g (1) climbs to a peak value over 68% and then declines with τ and ends up with a random fluctuation. It is interesting and important to find that there is a mismatch between the charging-discharging critical point ofn (the crossing between the black solid line and the blue solid line) in Fig. 3(a) and the peak value of P g (1). With the optimized measurement interval for the maximized P g (1), a single measurement on the charger qubit could enhance the battery energy from about 19ω b to 20ω b . In the mean time, the charger qubit that is prepared as the excited state and measured on the ground state has achieved the maximum efficiency with respect to the energy transfer during each charging round. Then the rest lines in Figs. 3(a) and 3(b) support that the battery would be constantly charged with a significant probability when the measurement intervals for the ensued rounds of evolution-and-measurement can be optimized by maximizing the measurement probability. For various numbers of measurements, each POVM M eg (τ ) with the maximal measurement probability is found to charge rather than discharge the battery by enhancinḡ n. The measurement probability can be approximately expressed by a finite summation involving sine functions P g (m) = with Ω n ≈ g √ n under the resonant or near-resonant condition. It is estimated that for a sufficiently large N and a sufficiently short τ , P g (m) = N n=m 1 − cos 2 (Ω n τ ) p (m−1) n−1 ≈ n p (m−1) n−1 − n cos 2 (Ω n τ )p (m−1) n−1 = 1 − 1 − Ω (m−1) n+1 τ 2 + · · · ≈ 1 − cos 2 (Ωn +1 τ ),(9) where Ω It means that τ opt is updated by the battery's average population of the last round. Note the leading-order correction from a nonvanishing detuning ∆ is in its second order. Equation (10) can be further verified under various temperatures and during multiple rounds of charging. The optimized measurement interval is inversely proportional to the square root of the average population that is roughly inversely proportional to β. In Fig. 3(c), one can find that the overall behaviors of P g (1) with various β are similar to that in Fig. 3(b). A bigger β yields a larger τ opt to have a peak value of P g (1). In other words, one has to perform more frequent measurements to charge a battery initially in a higher temperature. It is also reflected in Fig. 3(d), by which we compare the analytical results through Eq. (10) and the numerical results of optimized measurement intervals for 20 rounds of measurements under various temperatures. It is found that for β across three orders in magnitude, the analytical formula (10) is well suited to obtain the maximized measurement probability that represents the maximum energy input from the charger qubits. The effective temperature of battery increases during the charging process. τ opt then gradually decreases with m. It is consistent with the fact that coupling a charger qubit to a higher temperature battery with uniform energy spacing between ladders induces a faster transition between the excited state and the ground state of the qubit. We have two remarks about the charging efficiency in our measurement-based scheme. First, a decreasing τ opt with m could give rise to an increasing charging power P(m) ≡ Tr H B ρ (m) on − ρ (m−1) on τ (m) opt ,(11) which describes the amount of energy accumulated per unit time in a charging round. It is found that such a battery would be charged faster and faster during the first stage of charging process. Second, Fig. 3(d) indicates that the time-varying optimized intervals experience dramatic changes in the first several rounds and then become almost invariant as the measurements are implemented. It holds back the time-scales between neighboring projection operations from being too small to lose the experimental feasibility. According to Eq. (8), the average population of the battery after m measurements under the resonant or near-resonant condition is n (m) = N n=m n sin 2 (Ω n τ )p (m−1) n−1 N n=m sin 2 (Ω n τ )p (m−1) n−1 .(12) Around n ′ =n (m−1) + 1, we have sin 2 (Ω n τ ) = sin 2 (Ω n ′ τ ) + gτ sin 2 (2Ω n ′ τ ) 2 √ n ′ (n − n ′ ) + · · · . (13) All of these squares of sine functions could be approximate to the second order of (n − n ′ ) as sin 2 Ω n τ ≈ sin 2 Ω (m−1) n+1 τ (m) opt = 1 when each measurement is implemented with the optimal spacing in Eg. (10). Therefore, Eq. (12) could be expressed with the average population of the last charging roundn (m−1) : n (m) ≈ N n=m (n − 1)p (m−1) n−1 + N n=m p (m−1) n−1 N n=m p (m−1) n−1 ≈n (m−1) + 1.(14) It is assumed that after a sufficient number of measurements, N n=m p (m−1) n−1 ≈ 1. Due to Eq. (14), the battery takes an almost 100% unit of energy from the charger qubit in each round of evolution-and-measurement. In other words, POVM M eg promotes a near-perfect charging protocol, whose efficiency overwhelms the charging schemes without measurements. In the charging scheme based on the quantum collision framework [13], the battery takes about 25% ∼ 50% unit of energy from the charger qubit in each cycle. Then more charging cycles are demanded to charge the same amount of energy to battery. Consider the ground-state or zero-temperature case ρ B = \|0 0\| as described by the blue histogram in Fig. 4(a), which is the "easy mode" in previous schemes of quantum battery. Each POVM M eg allows a full state transfer from lower to higher levels p And in this case, the charged energy E (m) on ≡ Tr[H B ρ (m) on ](16) and the ergotropy E (m) on = E (m) on − Tr[σ (m) on H B ](17) are exactly the same. Here σ (m) on is the passive state of ρ (m) on . Thus the power-on charging scheme realizes a full conversion between excitations of charger qubits and usable energy of the battery, when the latter starts from a pure Fock state. This result can be intuitively obtained by a scheme based on the energy swap operations between charger qubit and battery. However, it is hardly extended to more practical scenario for arbitrary states of both charger qubit and battery. In Fig. 4(b), the battery is prepared with a moderate temperature. The battery state is gradually transformed from a thermal distribution to a Gaussian-like one with increasing mean value under measurements. For the battery state ρ √ ρ G between them is found to be 0.82 when m = 50. Analogous to the Fano factor [46], we can also use the ratio of the variance and the mean value f (m) ≡ ∆n (m) /n (m) to characterize the evolution of the population histograms. It is found that f (5) = 1.37, f (20) = 0.33, f (50) = 0.13, and f (80) = 0.08. As measurements are constantly implemented, the battery state distribution thus becomes even sharper and the populations tend to concentrate aroundn, providing more extractable energy. The varying histograms under a higher temperature are plotted in Fig. 4(c), where the fidelity F (m = 50) is still about 0.82 and the variance is much extended in Fock space. Comparing the purple distributions in Figs. 4(c) and 4(b), it is more easier for a highertemperature battery gives rise to a population inversion than a lower-temperature one, as the measurement number approaches the battery size m ∼ N . We demonstrate the energy and ergotropy of the battery as functions of the measurement number in Fig. 4(d). It is interesting to find that the finite temperature does not constitute an obstacle of the power-on scheme in achieving a near-unit ratio of ergotropy and energy, as presented in the case of zero temperature. Both energy and ergotropy scale linearly as indicated by Eq. (14) with the number of POVMs. It means that our poweron scheme is capable to realize a near-unit rate of energy transfer and achieve a high-ergotropy state, without preparing the battery as a Fock state [13]. No longer the thermal state is a "hard mode" for quantum battery. Before the occurrence of population inversion, the relative amount of the unusable energy for a lower-temperature battery is larger than a higher-temperature one, e.g., when m = 60, we have E on ≈ 0.96 and 0.98, for β = 0.1/ω c and 0.03/ω c , respectively. It is reasonable since population inversion indicates a close-to-unit utilization ratio of ergotropy and energy. In the collision model without measurement [13], multiple times of N rounds of cycles are required to achieve the same high ratio. The success probabilities of the power-on charging scheme under various temperatures are plotted in Fig. 5(a), which is defined as a product of the measurement probabilities of all the rounds P g = m P g (m). For charging the battery at the vacuum state (β → ∞), the success probability decreases with a very small rate. It is still over 93% when m = 80. When charging a finitetemperature thermal state, the success probability experiences an obvious decay during the first several (m ≈ 20) rounds of measurements. Then the histogram of battery population is transformed to be a near-Gaussian distribution [see Fig. 4(b) and 4(c)] and it becomes sharper as more measurements are implemented. It gives rise to P g (20 ≤ m ≤ 60) ≈ 1 and thus P g is almost invariant before the occurrence of population inversion (m ≈ 60). For a moderate temperature β = 0.1/ω c , the battery would be successfully charged with a 28% probability under m = 80 measurements. For a higher temperature β = 0.03/ω c , the success probability declines from about 20% to 5% as m = 60 → m = 80. During the last stage, the upperbound level of the battery is populated and then a larger portion of the near-Gaussian distribution of population has to be abandoned under normalization as more measurements are performed. It is thus expected to have a decreasing P g (m > 60). The charging power defined in Eq. (11) can be observed in Fig. 5(b), also exhibiting a similar monotonic pattern under various temperatures in a large range. The average population increases linearly with the measurement number and the optimized measurement interval is inversely proportional to the square root of the average population according to Eqs. (14) and (10), respectively. Then it is found that the charging power increases approximately as P(m) ∝ √n (m) . As shown in Fig. 5(b), a higher temperature gives rise to a larger P(m) until a decline behavior after m = 60 measurements (see the green dot-dashed line). That behavior is also induced by the population inversion, on which the average population of the battery fails to keep a linear growth as indicated by the green dotted line in Fig. 4(d). IV. POWER-OFF CHARGING When the measurement basis does not commute with the system Hamiltonian, it allows to take the energy away from the measurement apparatus and deposit it to the system. In such a way, energy turns to be useful work [39]. When the charger qubits are not in their excited state, getting the state information from them by projection-induced POVMs can convert the information to usable energy through work done on the battery [47]. And the energy cost of a measurement depends on the work value of the acquired information [48]. We now analyse the power-off charging scheme described by M ge , in which charger qubits are prepared in their ground state with q = 1, and projective measurement is performed on the excited state with θ = π (see the upper right corner in Fig. 2). In this case, the energy change in the charger-battery system is caused by the measurements alone. The battery state after m rounds of power-off charging reads ρ (m) off = M ge ρ (m−1) off P e (m) = N −1 n=0 \|λ n+1 (τ )\| 2 p (m−1) n+1 \|n n\| P e (m) (18) with P e (m) = N −1 n=0 \|λ n+1 \| 2 p (m−1) n+1 the measurement probability of the mth round. In contrast to M eg in Eq. (8), Eq. (18) indicates that M ge replaces the populations on low-energy states with those on their neighboring high-energy states weighted by a τ -dependent factor p (m) n ← \|λ n+1 (τ )\| 2 p (m−1) n+1 . For a battery initially in a Gibbs state, whose population is maximized on the ground state, M ge could have a certain degree of charging effect on the battery after population renormalization by moving a smaller occupation on higher levels to lower levels. The low-energy states are thus always populated during the histogram evolution from a thermal distribution to a near-Gaussian distribution. The charging effect by the power-off scheme is limited by optimizing the measurement intervals. We provide the average population of battery and the measurement probability under multiple POVMs M ge as functions of the measurement interval in Figs. 6(a) and 6(b), respectively. Both of them present similar patterns as those under the power-on scheme in Fig. 3(a) and 3(b). However, the mean value of the populationn is less than its initial valuen th (indicated by the black solid line) when τ is chosen such that the measurement probability attains the peak value. In this case, the battery is discharged rather than charged. In general, it is then hard to charge the battery with a significant success probability under a number of rounds of evolution-andmeasurement. To charge the battery within the poweroff scheme, we have to compromise the charging ratio of neighboring rounds r =n (m+1) /n (m) and the success probability P e = m P e (m) by numerical optimization. Here we choose to maximize exp(xP e ) log x r to ensure the battery could be charged constantly with a reasonable success probability, where x is an index to balance the weights of r and P e and chosen as x = 10 in the current simulation. The optimized results for the poweroff charging are distinguished by the closed circles in Figs. 6(a) and 6(b). In contrast to Fig. 3(b), one has to perform the measurements with a shorter spacing τ before P e (m) attains the peak value. The battery could then be constantly charged yet the number of sequential measurements is much limited [see the dotted lines in Fig. 6(a) with m = 20]. The success probability P e for the power-off scheme is doomed to be much smaller than P g for the power-on scheme. off , respectively. The population distribution under the power-off charging is much more extended than the power-on charging, which is not favorable to a quantum battery. Mean value and variance of the battery population are associated with ergotropy and energy in Fig. 7(c), where E off . For the power-off charging scheme, the battery cannot obtain energy from both charger qubits and the external energy input apart from projective measurements, as indicated by p n in Fig. 7(b) and E off in Fig. 7(c). The charger qubit is charged in the same time as the battery since the measurement is performed on its excited state. With respect to the battery initial state, the power-off charging scheme can convert the thermal state of zero ergotropy to a high-ergotropy state, especially under less numbers of measurements [see m = 5 → m = 10 in the yellow dot-dashed line of Fig. 7(c)]. The numerical simulation at m = 20 shows that the success probabilities for the power-off and poweron schemes are about 1% and 26%, respectively. For m ≈ 20, the charged energy under the power-off scheme is larger than that under the power-on scheme. But the latter takes advantage in ergotropy, more crucial for a quantum battery. When m > 15, the advantage becomes more significant with respect to the ratio of ergotropy and energy. Rather than the dynamical coherence in Eq. (4) that is suppressed by the projective measurements, we can discuss the effect from the initial coherence, when the charger qubit is initialled as ρ C = q\|g g\| + (1 − q)\|e e\| + c q(1 − q)(\|e g\| + \|g e\|). Here 0 ≤ c ≤ 1 serves as a coherence indicator. When c = 0, it recovers the preceding analysis in Sec. II. When c = 1, the charger qubit is in a superposed state. Consequently, it is found that the population of the battery state in Eq. (5) becomes D = D charge + D discharge + D c ,(19) where D charge and D discharge are the same as Eq. (6) and D c =c q(1 − q) sin θ n=0 cos(Ω n τ ) cos(Ω n+1 τ ) − ∆ 2 4Ω n Ω n+1 sin(Ω n τ ) sin(Ω n+1 τ ) p n \|n n\| (20) represents the population contributed from coherence. Under the approximation Ω n+1 ≈ Ω n and the nearresonant condition ∆ ≈ 0, we have D c ≈ c q(1 − q) sin θM gg [ρ B ],(21) where M gg is the POVM defined in Eq. (7). As indicated by Fig. 2, the initial coherence then prefers to cool down rather than to charge the battery. The negative role played by the initial coherence in charging can be confirmed by Fig. 8 about the average population ration/n th as a function of the measurement parameter θ and the initial state parameter q when c = 1. In comparison to Fig. 2 without the initial coherence, it is found that the effects from different POVMs remain invariant. M eg (the lower left corner) and M ge (the upper right corner) still dominate the charging effects and M ee (the upper left corner) and M gg (the lower right corner) are still in charge of discharging. While the former becomes weakened with less red area and the latter becomes enhanced with more blue area. The numerical result is consistent with Eq. (21). In realistic situations, any charging process should be considered in an open-quantum-system scenario. We now estimate the impact from the environmental decoherence on our charging schemes based on the measurements. The dynamics of the full system can then be described by the master equation, where H is given by Eq. (2) and D[o] represents the Lindblad superoperator D[o]ρ tot (t) ≡ oρ tot (t)o † − 1/2{o † o, ρ tot (t)}. Here γ b and γ c are dissipative rates for battery and charger qubits, respectively.n th = Tr[nρ B ] andn c th = 1−q th are their initial thermal average occupations, where q th = 1/(e βω b + 1) is the thermal population on the ground state of charger qubits. Figure 9 presents both energy and ergotropy of the battery under two charging schemes with various dissipative rates along a small sequence of m ≤ 20 charging rounds. It is demonstrated that in the presence of the thermal decoherence with γ b = γ c = γ ≤ 10 −4 ω b , both energy and ergotropy deviate slightly from the dissipation-free situation [see the blue lines or Fig. 7(b)]. It is reasonable because the optimized measurement interval τ opt decreases as more measurements implemented [see Fig. 3(d)], which significantly reduces the environmental effect. VI. CONCLUSION We established for quantum battery a chargingby-measurement framework based on rounds of jointevolution and partial-projection. The charger system is constituted by a sequence of disposable qubits. General POVMs on the battery system of N + 1 levels are constructed by the exchange interaction between charger and battery and the projective measurement on charger qubits in a general mixed state. In the absence of initial coherence, we focus on the charging effect by POVM alone. Despite the battery starts from the thermalequilibrium state as a "hard mode" for quantum battery, it is found that a considerable charging effect can be induced when the qubit is prepared at the excited state and measured on the ground state or in the opposite situation. They are termed as power-on and power-off charging schemes. The power-on scheme exhibits great advantages in the charging efficiency over the schemes without measurements. Under less than N measurements with optimized intervals, our measurement-based charging could transform the battery from a finite-temperature state to a population-inverted state, holding a near-unit ergotropy-energy ratio and a significant success probability. Within a much less number of measurements than the power-on scheme, the power-off charging scheme can be used to charge the battery and charger qubits without external energy input, although it is hard to survive more rounds of measurements with a finite probability. The POVM in our work manifests a powerful control tool to reshape the population distribution of the battery system, building up a close relation with the ergotropy. Our work therefore demonstrates that quantum measurement can become a useful thermodynamical resource analogous to conventional heat or work reservoirs, serving as high-efficient fuels powering a charging scheme through a nonunitary procedure. FIG. 1 . 1(a) Power-on and (b) Power-off charging schemes. The quantum battery in our model has N + 1 energy ladders with Hamiltonian H B = ω b N n=0 n\|n n\|, where ω b is the energy unit of the battery. = k B = 1. The ladder operators are defined as A † = . 2. Ration/n th of the average populationn of the battery after a single measurement and the initial thermal population n th ≡ Tr[nρB] in the space of θ and q. The battery size is N = 100, the initial inverse temperature is β = 0.1/ωc, and the measurement interval is τ = 8/ωc. D charge in Eq. (6) is a linear combination of M eg and M ge in Eq. ( state \|n after (m − 1) rounds of measurements under the poweron charging scheme. p of the mth round. In Eq. (8), a projective measurement generates a population transfer between neighboring energy ladders of the battery p FIG. 3 . 3(a) Average populationn and (b) Measurement probability Pg as functions of measurement interval τ after m rounds of measurements within the power-on scheme. The black lines in (a) represent average populationsn (m−1) after (m − 1) measurements, among which the black solid line denotes the initial thermal occupationn th of the battery. The closed circles in (a) represent the moments for the mth measurement as obtained from (b) with a maximal measurement probability, by which the updatedn (m) is found to be larger thann (m−1) , promising a charging with a significant probability. (c) Measurement probability as a function of τ under various temperatures. (d) Sequences of the optimized measurement intervals under various temperatures, where τopt is given by Eq. (10) and τ num opt represents the numerical result. The battery size is N = 100, the initial inverse temperature in (a) and (b) is β = 0.05/ωc, the detuning between charger qubit and battery is ∆/ωc = 0.02, and the coupling strength is g/ωc = 0.04. average population for the battery after (m − 1) rounds of measurements. The optimized measurement interval is thus given by an iterative formula: FIG. 4 . 4Histogram of the battery populations after m = 0 (blue), m = 5 (orange), m = 20 (green), m = 50 (red), and m = 80 (purple) measurements for various initial temperatures: (a) β → ∞, (b) β = 0.1/ωc, and (c) β = 0.03/ωc under the power-on charging scheme. The black-dashed curves in (b) and (c) describe the near-Gaussian distributions with the same average population and variance of the battery state after m = 50 measurements under finite temperatures. (d) Battery ergotropy (lines) and energy (makers) as functions of m under various temperatures. The other parameters are the same as those in Fig. 3. . After m < N measurements, the whole population is transferred to the mth energy ladder of the battery with zero variance ∆n (m) with m = 50 measurements (see the red histogram), a Gaussian state ρ G with the same average .91 for β = 0.03/ω c . In contrast, when m = 80, we have E FIG. 5 . 5(a) Success probability and (b) Charging power under power-on charging as functions of measurement number. All parameters are the same asFig. 3apart from the inverse temperature. FIG. 6 . 6(a) Average populationn and (b) Measurement probability Pe as functions of measurement interval τ after m rounds of measurements within the power-off scheme. The black lines in (a) represent average populationsn (m−1) after (m−1) measurements. The closed circles in (a) and (b) represent the compromise result of the changing ratio and the success probability for the moments of the mth measurement, by which the updatedn (m) is maintained overn (m−1) , promising a charging with a reasonable probability until m ≈ 20. β = 0.05/ωc, ∆/ωc = 0.02, and g/ωc = 0.04. FIG. 7 . 7Histogram of the battery populations under (a) power-on and (b) power-off charging schemes after m = 0 (blue), m = 5 (orange), m = 10 (green), and m = 20 (red) measurements. (c) Battery ergotropy (lines) and energy (markers) as function of m under both power-on and poweroff schemes. Parameters are the same as those in Fig. 6. To compare the charging efficiency under the poweron and power-off schemes, the evolutions of the batterypopulation distribution under various number of measurements are demonstrated in Fig. 7(a) and 7(b), respectively. With the same setting of parameters and initial thermal state, it is found that the the power-on charging and the power-off charging are almost the same in the mean values of battery population, yet are dramatically different in the variances. For example, when m = 20, n (m) are found to be about 37 and 41 and ∆n (m) are about 24 and 98 for ρ off H B ] and E (m) off = Tr[H B ρ (m) off ] with σ (m) ρ off the passive state of ρ (m) FIG. 8 . 8Ration/n th of the average populationn after a single measurement on the battery and the initial thermal populationn th in the space of θ and q with full initial coherence c = 1. Parameters are set as the same as those inFig. 2. B. Charging in the presence of decoherence FIG. 9 . 9(a) and (b): Battery energy as a function of measurement number m with various dissipation rates γ for the poweron and power-off charging schemes, respectively. 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[ "No One Left Behind: Improving the Worst Categories in Long-Tailed Learning", "No One Left Behind: Improving the Worst Categories in Long-Tailed Learning" ]	[ "Yingxiao Du \nState Key Laboratory for Novel Software Technology\nNanjing University Nanjing\n210023China\n", "Jianxin Wu \nState Key Laboratory for Novel Software Technology\nNanjing University Nanjing\n210023China\n" ]	[ "State Key Laboratory for Novel Software Technology\nNanjing University Nanjing\n210023China", "State Key Laboratory for Novel Software Technology\nNanjing University Nanjing\n210023China" ]	[]	Unlike the case when using a balanced training dataset, the per-class recall (i.e., accuracy) of neural networks trained with an imbalanced dataset are known to vary a lot from category to category. The convention in long-tailed recognition is to manually split all categories into three subsets and report the average accuracy within each subset. We argue that under such an evaluation setting, some categories are inevitably sacrificed. On one hand, focusing on the average accuracy on a balanced test set incurs little penalty even if some worst performing categories have zero accuracy. On the other hand, classes in the "Few" subset do not necessarily perform worse than those in the "Many" or "Medium" subsets. We therefore advocate to focus more on improving the lowest recall among all categories and the harmonic mean of all recall values. Specifically, we propose a simple plug-in method that is applicable to a wide range of methods. By simply re-training the classifier of an existing pre-trained model with our proposed loss function and using an optional ensemble trick that combines the predictions of the two classifiers, we achieve a more uniform distribution of recall values across categories, which leads to a higher harmonic mean accuracy while the (arithmetic) average accuracy is still high. The effectiveness of our method is justified on widely used benchmark datasets.	10.48550/arxiv.2303.03630	[ "https://export.arxiv.org/pdf/2303.03630v2.pdf" ]	257,378,178	2303.03630	651f1a1e215301f527a9dfb23ac94348bc5d317a	No One Left Behind: Improving the Worst Categories in Long-Tailed Learning Yingxiao Du State Key Laboratory for Novel Software Technology Nanjing University Nanjing 210023China Jianxin Wu State Key Laboratory for Novel Software Technology Nanjing University Nanjing 210023China No One Left Behind: Improving the Worst Categories in Long-Tailed Learning Unlike the case when using a balanced training dataset, the per-class recall (i.e., accuracy) of neural networks trained with an imbalanced dataset are known to vary a lot from category to category. The convention in long-tailed recognition is to manually split all categories into three subsets and report the average accuracy within each subset. We argue that under such an evaluation setting, some categories are inevitably sacrificed. On one hand, focusing on the average accuracy on a balanced test set incurs little penalty even if some worst performing categories have zero accuracy. On the other hand, classes in the "Few" subset do not necessarily perform worse than those in the "Many" or "Medium" subsets. We therefore advocate to focus more on improving the lowest recall among all categories and the harmonic mean of all recall values. Specifically, we propose a simple plug-in method that is applicable to a wide range of methods. By simply re-training the classifier of an existing pre-trained model with our proposed loss function and using an optional ensemble trick that combines the predictions of the two classifiers, we achieve a more uniform distribution of recall values across categories, which leads to a higher harmonic mean accuracy while the (arithmetic) average accuracy is still high. The effectiveness of our method is justified on widely used benchmark datasets. Introduction Various gaps exist when adapting image recognition techniques that are developed in the lab to industrial applications. The most noteworthy one is perhaps the difference between training datasets. Most training datasets used in academic research [6,16] are balanced with respect to the number of images per class. This should not be taken for granted because datasets used in real-world applications are more likely to be imbalanced. Training deep models Figure 1. The per-class recall of models trained on the imbalanced CIFAR100 (with imbalance ratio 100). Per-class recall value varies a lot from category to category. Moreover, it is not necessarily true that all categories in the "Few" subset have lower accuracy than those in the "Many" or "Medium" subsets. Method Mean Accuracy Lowest Recall BSCE [22] 42. 24 3.00 DiVE [12] 45. 11 2.00 MiSLAS [30] 47.05 5.00 RIDE [26] 48.64 2.00 PaCo [4] 51.24 5.00 Table 1. The lowest per-class recall of various state-of-the-art methods on the imbalanced CIFAR100 (with imbalance ratio 100). Although there are rapid improvements over the mean accuracy, the lowest per-class recall remains very low. on these datasets is not trivial as models are known to perform poorly on such datasets. Long-tailed recognition is a research field that aims at tackling this challenge. By plotting the per-class recall (i.e., per-class accuracy) of the model trained on the imbalanced CIFAR00 dataset (with imbalance ratio 100) in Fig. 1, we find that the recall varies dramatically from category to category. We argue that in real world applications, all categories are equally important and no one should be left behind. We also find that despite the rapid developments in this community, no obvious improvement over the lowest per-class recall is witnessed in the past few years [4,12,22,26,30], as is shown in Tab. 1. The convention in long-tailed recognition research is to split the classes into three subsets based on the number of training images. The accuracy within each subset is often reported along with the overall accuracy. While this evaluation scheme seems reasonable at the first glance, we argue it is potentially problematic. First of all, computing the average recall within each subset is way too coarse, making it impossible to reflect whether some classes are completely sacrificed but covered up by other "easy" tail classes. What's more, we find it is not necessarily true that classes in the "Few" category all have lower recall than classes in the other two subsets, as is shown in Fig. 1. Therefore, focusing on improving the mean accuracy alone is not enough, especially in real-world applications. In this paper, we propose a novel method to make sure that no category is left behind. Specifically, we argue that although mean accuracy is widely used in image classification as an optimization objective, due to the fact that different classes have very different recall in long-tailed recognition, it is not the most suitable objective as it incurs little penalty even if some categories have very small per-class accuracy values (e.g., close to 0). Hence, to improve even the worst-performing categories, we believe the harmonic mean of per-class recall would be a better objective. Since harmonic mean is very sensitive to small numbers, further improving classes that already have high recall brings little benefits to the harmonic mean, which makes sure no single class will be left behind. Also, now all classes are treated equally, forcing us to improve classes that have low recall no matter which subset it belongs to. However, it is difficult to directly minimize the harmonic mean. We therefore propose a novel loss function that maximizes the geometric mean instead, which can be viewed as a surrogate. Our method serves as a simple plug-in that can be used together with both baseline and various stateof-the-art methods. We also propose an ensemble trick that uses the pre-trained and fine-tuned models together to make predictions during inference with nearly no extra cost. We are able to yield a more uniform distribution of recall across categories (i.e., no one left behind), which achieves higher harmonic mean of the recall while the (arithmetic) mean accuracy remains high, too. In summary, our work has the following contributions: • We are the first to emphasize the importance of the correct recognition of all categories in long-tailed recognition. • We propose a novel method that aims at increasing the harmonic mean of per-class recall as well as an ensem-ble trick that combines two existing models together during inference with nearly no extra cost. • We experimented on three widely used benchmark datasets, which justify the effectiveness of our method in terms of both overall and worst per-class accuracy. Related Work Long-tailed recognition is a research field that aims at training models using imbalanced datasets. Since it is widely believed that the learning procedure is dominated by head classes, most existing works focus explicitly on improving the recognition of tail classes. Re-sampling and Re-weighting Re-sampling and re-weighting are two classical approaches. Re-sampling methods aim at balancing the training set by either over-sampling the tail classes [3,9,21,24] or under-sampling the head classes [7,10]. Some methods also transfer statistics from the major classes to minor classes to obtain a balanced dataset [15,19]. One disadvantage of re-sampling methods is that it may lead to either over-fitting of tail classes or under-fitting of head classes. Re-weighting methods, on the other hand, give each instance a weight based on its true label when computing the loss [2,22,23]. The major drawback of re-weighting is that it makes the loss function hard to optimize. Two-Stage Decoupling Due to the shortcomings of re-sampling and reweighting, various other methods are proposed recently. For example, two-stage methods like [14,18,30] propose to decouple the learning of features and classifiers and achieve impressive results. Our method is similar to them in the sense that we also re-train a classifier given an existing model. But our motivation is to design a simple and flexible method that can be applied to a wide variety of existing models, and we propose a novel loss function, while twostage methods like cRT [14] usually use the conventional cross entropy loss in the fine-tuning stage. Hybrid and Multiple Heads Hybrid methods, like [26,31], make use of multiple heads to improve the recognition of different classes, which is similar to our proposed ensemble trick. But these methods require a joint training of multiple heads, together with a complex routing module that dynamically determines the head to use during inference, which increases the complexity of the model and makes it hard to use in practice. Our method, on the other hand, is simpler and requires no extra training nor complex modules to combine the predictions, which is different from them. Miscellaneous There are many approaches that use knowledge distillation [12,18] and contrastive learning [4,17,25,28,33] and achieve descent results. There are also methods that perform logit adjustment [20,29], which formulate long-tailed recognition as a label distribution shift problem. Recently, there is also one work that tries to solve the problem by performing some regularization on the classifier's weight [1]. They also visualize the recall of each class in their work and find the model is biased towards common classes, which is similar to our motivation. However, they do not emphasize the importance of correctly recognizing all classes, which is different from us. Method We will present our method in this section. First, we will introduce the notations used. Then we will describe the major framework of our method, which includes our proposed loss function and a simple ensemble trick. Notations The neural network can be represented as a non-linear function F . Given the -th training image ( , ) from the imbalanced dataset that has images, where ∈ R × × , a forward propagation through the network yields logit = F ( ). For a multi-class classification problem with categories, a softmax activation function is usually applied and the result can be represented as = softmax( ), where˜∈ R . A Simple Plug-In Method Over the past few years, various methods [4,14,22] have been proposed to tackle the challenges in long tailed recognition. Although our evaluation results in Tab. 1 reveal that no obvious improvements have been made regarding the worst-performing category, many of their design choices are critical and effective in long-tailed recognition. Therefore, in order to make full use of existing advances, we do not aim at building up a whole new training framework, but rather propose a simple plug-in method that can be applied to a wide range of existing methods. Our method is simple. Given a model trained using either one of existing methods, we re-initialize the last FC layer and re-train it using our proposed loss function while the backbone is frozen. We borrow this idea from many two-stage decoupling methods [14,30], which is effective and flexible, making our method widely applicable. However, unlike normal decoupling methods that use the conventional cross entropy loss in the second stage, we propose a novel loss function to improve the worst-performing categories. Our proposed loss function, called GML (Geometric Mean Loss), is defined as L GML = − 1 ∑︁ =1 log¯,(1) where¯is computed as = 1 ∑︁ =1˜,(2) which is the average of˜across all training samples belonging to class in this mini-batch, and is the number of examples from this category in the mini-batch. Inspired by [22], we perform re-weighting when we compute˜: = exp( ) =1 exp( ) . (3) Why GML Works? The definition of mean accuracy is Acc = 1 ∑︁ =1 ( = arg max˜) ,(4) where (·) is the indicator function. On a balanced test set, it is the same as the arithmetic mean of per-class recall. We argue that maximizing the mean accuracy inevitably ignores some categories because arithmetic mean incurs little punishment to categories that have very small recall values. On the other hand, harmonic mean, defined as HM( 1 , . . . , ) = 1 1 + · · · + 1 ,(5) incurs strong punishment to small values. Therefore, we believe it can be a better alternative to optimize for. However, harmonic mean is defined using reciprocal, making it hard and numeric unstable to be optimized. To this extent, we propose to maximize the geometric mean of per-class recall, defined as GM( 1 , . . . , ) = √︁ \| 1 × · · · × \| .(6) As an illustration, 0.01+0.99 2 = 0.5, HM(0.01, 0.99) = 0.02, and GM(0.01, 0.99) = 0.10. Both harmonic and geometric means are heavily affected by the small value (0.01), while the arithmetic mean is much less sensitive to it. Specifically, given the per-class recall as 1 , . . . , , we transform their geometric mean using a simple logarithm transformation: Since recall is computed by computing the mean of a series of indicator functions and these indicator functions only take value 0 or 1 and is non-differentiable, we use¯as a surrogate for during training, resulting in √ 1 . . . = exp log ( 1 . . . ) 1/ (7) = exp 1 log ( 1 . . . ) (8) = exp 1 ∑︁ =1 log .(9)√ 1 . . . ∝ exp(−L GML ) .(10) So, by minimizing L GML , we are effectively maximizing the geometric mean of per-class recall, thus being able to improve on the worst-performing categories. Combining the Strength of Both Worlds By fine-tuning the pre-trained model with our proposed loss function, indeed we are able to obtain a more uniform distribution of recall values across categories. As is shown in Fig. 2, the recall of classes that perform worse before are improved while the recall of classes that perform well before are dropped. Motivated by these dynamics, we then propose a simple ensemble trick that aims at combining the strength of both worlds. Our idea is simple again. Since we re-initialize the classifier during fine-tuning and the backbone is kept frozen, the old classifier is still usable during inference. Let these two models be F old and F new , respectively, we then make predictions by˜e nsembled =˜n ew +˜o ld 2 .(11) In practice, since the calibration of some models can be poor [8,30] (e.g., the baseline using cross entropy), we need to manually calibrate the prediction before combining them. For simplicity, we choose temperature scaling to calibrate the model [8] and use two temperature variables new and old when computing˜n ew and˜o ld separately: new = softmax new new ,(12)old = softmax old old .(13) One big advantage of our proposed ensemble trick is that it brings nearly no extra cost. Unlike many multi-head [31] or multi-expert [26] methods, we do not need extra training as we simply use the classifier that comes with the pretrained model. During inference, the only extra cost is the fully connected layer to compute˜o ld , which is negligible. Also, no complex routing rules are needed. The two temperature variables are the only two hyperparameters introduced by our method, and they can be easily tuned as no training is required. Plus, as we will later show in the ablation studies, most values work perfectly fine so actually no painful tuning is needed. What's more, this ensemble trick is completely optional, our method works well enough even without it. Training Pipeline Our method is very similar to the two-stage decoupling methods [14,30]. We design it in this way on purpose because we want it to be as flexible and simple as possible. During experiments, we found that our method is orthogonal to many design choices in state-of-the-art methods and is able to improve the worse-case performance when applied to various different pre-trained models. The overall training pipeline of our method consists of three stages: 1. Pre-training stage: Obtain the pre-trained model from scratch using any one of the existing methods. 2. Fine-tuning stage: Freeze the backbone and re-train the classifier using our proposed loss function. 3. An optional ensemble stage: Calibrate the prediction of two classifiers and combine them additively. The overall training procedure is summarized in Algorithm 1. Experiments In order to validate the effectiveness of our method, we have conducted experiments on three widely used benchmark datasets. In this section, we will present our experimental results. First, we will introduce the datasets used. Then we will describe various implementation details and evaluation metrics used. After that, we will compare our method with various other methods. Finally, we will present the results of some ablation studies. Datasets We use three widely used long-tailed image recognition datasets: CIFAR100-LT, ImageNet-LT and Places-LT. Some statistics about these three datasets are summarized in Tab. 2. Since CIFAR100 [16], ImageNet [6] and Places [32] are all balanced datasets, we follow previous works [19,31] to down-sample their original training set. Details about the construction process can be found in the supplementary material. Implementation Details Following previous works [19,31], we use ResNet-32 [11], ResNeXt-50 [27] and ResNet-152 as the backbone network for CIFAR100-LT, ImageNet-LT and Places-LT, respectively. For the pre-training stage, if there exists released checkpoints, we use them directly. If not, we use the officially released codes and try to reproduce the results in our best effort without changing their original settings. This includes stuffs like hyper-parameters and data augmentations used. For the fine-tuning stage, detailed training settings can be found in the supplementary material. Evaluation Metrics and Comparison Methods The convention in long-tailed recognition research is to split the classes into three subsets based on the number of training images [14,19]. Typically, "Head" denotes classes that have more than 100 images, "Few" denotes classes that have less than 20 images and all other classes are denoted as "Medium". The accuracy within each subset is often reported along with the overall accuracy to justify the claim that the performance improvement mainly comes from tail classes. And according to our observation in Sec. 1, we ar-gue that such an evaluation scheme can be problematic and we propose to first compute the recall within each category and then compute their harmonic mean. In our experiments, since our method optimizes geometric mean of per-class recall as a surrogate, we also report the geometric mean. Besides, we report the overall mean accuracy as well. We compare our methods with both baseline methods like simple cross entropy loss and various state-of-the-arts methods [4,12,22,26]. Many of these methods have very different training settings and strictly speaking, they are not directly comparable. But since our method is a simple plugin method, the main focus is to quantify the improvements brought by our method compared to its respective baseline result. Main Comparisons We validate our method on three benchmark datasets and present the results on each dataset here separately. CIFAR100-LT Tab. 3 shows the experimental results on the CIFAR100-LT dataset. CIFAR100-LT has three different versions with different imbalance ratios and here we only consider the one with imbalance ratio 100. As we can see from the table, although various different methods have been proposed in recent years and there are rapid improvements on the overall accuracy, the performance of the worst-performing category is still relatively poor. The geometric and harmonic mean of per-class recall also justify our claim. We apply our method to both the baseline method CE and state-of-the-art method PaCo [4] on this dataset. When our method is applied to CE and PaCo, we are able to improve the lowest recall from 0.00 and 5.00 to 6.00 and 9.00 respectively. And if we further apply our proposed ensemble trick, we are able to improve the lowest recall to 15.00, achieving state-of-the-art performance in both geometric mean and harmonic mean while keeping the overall accuracy roughly unchanged. It is worth noting that even the simplest CE+GML outperforms all existing methods in terms of lowest recall, that is, in terms of "no one left behind". ImageNet-LT and Places-LT Tab. 4 shows the result on ImageNet-LT. The overall observation is similar to that on CIFAR100-LT, except that Table 4. Results on the ImageNet-LT dataset. Numbers with * are computed by substituting zero elements with a small number (10 −3 ) or else the geometric and harmonic mean will all be zero. since this dataset is larger and harder, the lowest recall of all current methods are zero, making the harmonic mean of per-class recall very low, even when a small number substitutes zero per-class recalls. Our method, on the other hand, greatly improves the harmonic mean by successfully improving the lowest recall value. All categories have perclass recall higher than 0 when our method is used, even in the simplest CE+GML. Tab. 5 shows the result on Places-LT. Current state-ofthe-art on this dataset is MiSLAS [30], so we apply our method upon it and the simple baseline CE. Again, our method successfully improves the lowest recall, as well as the harmonic mean of per-class recall. And when the ensemble trick is further applied, we are able to achieve new state-of-the-art results. Universality Since we want our method to be a universal plug-in, in this subsection we present the results of the experiment of adding our method on top of various different methods. The experiment is conducted on CIFAR100-LT, and the re-sult is shown in Tab. 6. Note that here we do not perform any ensemble. As we can see, our method is applicable to various different methods, and is able to achieve consistent improvements. However, we do observed that there exists some methods, like DiVE [12], where we failed to achieve obvious improvements. More details about this can be found when we discuss limitations in Sec. 5. Ablation Studies For ablation studies, if not otherwise specified, all experiments are conducted on CIFAR100-LT (with imbalance ratio 100) under the default training setting. Effects of Two Temperature Variables in the Ensemble Stage When we design our method, we'd like to keep the number of new hyper-parameters as less as possible in order for it to be readily applicable in real-world applications. Table 6. Results on the CIFAR100-LT dataset with imbalance ratio 100. Our proposed method is applicable to various methods. "G-Mean" is short for "Geometric Mean" and "H-Mean" is short for "Harmonic Mean". Table 7. Ablation on the effects of temperature variables when we apply our ensemble method upon PaCo [4]. the value of the temperature variables shown in Eq. (12) and Eq. (13) in the ensemble stage when we apply our method to PaCo [4]. The results are shown in Tab. 7. In our experiments on CIFAR100-LT, we set old = 1 and new = 1. And as we can see from the table, the value of old does not affect the result much while new is somewhat important, especially for improving the lowest recall-higher new will hurt it. This is natural because higher results in a smoother distribution and less information is preserved. Since the newly trained classifier is crucial for improving the lowest recall, we need to preserve more information from it. Generally speaking, since these two hyper-parameters Table 9. Training from scratch using GML. are introduced for calibration, for different methods, we may need to tune them respectively because the calibrations of different methods are known to vary a lot in long-tailed recognition problems [30]. But usually this will not cause a headache, because the ensemble stage does not involve any training and thus can be tuned very quickly. Re-weighting in GML As shown in Eq. (3), re-weighting is applied in our proposed GML, here we ablate this design. The results are shown in Tab. 8. As we can see, the performance will be relatively poor without re-weighting. And just like many two-stage decoupling methods [14], re-sampling during fine-tuning can help, but is inferior to re-weighting. Training from Scratch using GML Since our main purpose is to propose a simple plug-in method that is widely applicable to various existing methods, we use GML only to re-train the classifier of an existing model. However, GML can also be used to train a model from scratch. The results are shown in Tab. 9. As we can see, the result is comparable to the result when GML is attached to CE, but inferior to the result when GML is applied to SOTA methods like PaCo, just as expected. Visualization of the Per-Class Recall In Fig. 1, we visualize the per-class recall of model trained on the imbalanced dataset, and find they vary a lot from class to class. Since our major motivation is to make sure all classes are equally treated so that no category will be left behind, here we visualize the change of per-class recall after applying our method. We perform visualizing using two datasets, CIFAR10-LT and CIFAR100-LT, both have imbalance ratio 100. The visualization results on CIFAR10-LT and CIFAR100-LT are shown in Fig. 2 and Fig. 3, respectively. Note that here we do not perform the ensemble. As we can clearly see in both figures, our fine-tuned model has a more uniform distribution of per-class recall. These two figures justify the effectiveness of our proposed method. Limitations and Broader Impacts Although our method is applicable to a wide range of existing methods, we do find that when applied to some methods, the improvement is less obvious compared to others. For example, our framework does not cope well with methods like BSCE [22] and DiVE [12]. Interestingly, although DiVE is based on knowledge distillation, BSCE is used in DiVE as well to train both the teacher and student model so we attribute the failure to the fact that these method all re-weight the loss function. Two-stage decoupling methods like cRT [14] are also known to not perform well when the first stage training involves re-weighting or re-sampling. The author argues it is because re-sampling and re-weighting have negative effects on feature learning and we believe that argument applies to our method as well. Another thing worth noting here is that we do not report the experimental results on iNaturalist2018 [5] because we find it generally not suitable in our settings. iNaturalist2018 is another widely used benchmark dataset in long-tailed recognition. It is large-scale, consisting of 437.5K training images from 8142 classes. But for the test split, it only has 24K images, so each class only has 3 images. Since our major motivation is to make sure no category is left behind, it is generally not meaningful to look at the per-class recall in iNaturalist2018 because the variance of the result can be very large due to the very small number of test images. However, in real-world applications, it is generally rare to have such small number of test images. For broader impacts, we believe our research have positive impacts on reducing the bias of model training on an imbalanced dataset. Machine learning models are known to be biased if there is lack of training images from some categories. We believe our work will be helpful to achieve better fairness by making sure all categories are equally treated. Conclusions In this paper, we focused on improving some of the worst-performing classes in long-tailed recognition. We found that when trained with an imbalanced dataset, the perclass recall of the model varies a lot from class to class. Previous convention reports the arithmetic mean of perclass recall, but we argued such an evaluation scheme can be problematic. First, arithmetic mean only incurs little penalty to small numbers, making it too coarse to reflect whether there are categories that are left behind (i.e., with close to zero per-class accuracy). Second, it is not necessarily true that classes in the "Few" category perform worse than those in "Many" or "Medium", so focusing on improving "Few" accuracy alone is not enough. For these reasons, we argued that we should pay more attention to the harmonic mean of per-class recall and we therefore proposed a novel method that aims at improving the lowest recall and the harmonic mean of recalls. Our method is a simple plug-in that is applicable to a wide range of existing methods. Specifically, our method consists of three stages, in the first stage, we use any existing method to train a model from scratch. Then we re-train the classifier using our proposed novel GML loss function. Finally, we propose a simple ensemble trick that can be used to combine the predictions from the two classifiers with nearly no extra cost. We validated the effectiveness of our method on three widely used benchmark datasets, and witnessed consistent improvements on the harmonic mean of recalls and lowest recall value, while the overall accuracy still remains high. By visualizing the distribution of per-class recall values of the fine-tuned model, we found our model indeed achieved a more balanced distribution. A. More Details about the Datasets Used The imbalance ratio of a dataset is defined as the number of training images of the most frequent class divide by the number of images of the least frequent class. CIFAR100-LT. CIFAR100 [16] is a balanced dataset for image recognition, which has 50,000 training images and 10,000 test images from 100 categories. The CIFAR100-LT dataset used in our experiments are obtained by downsampling the original training set while keeping the test set unchanged. Following Zhou et al. [31], we use the exponential function = 0 × to determine the number of training images for each category, where 0 = 500. By varying , we are able to construct datasets with different imbalance ratios. In our experiments, we only use the one with imbalance ratio 100. ImageNet-LT and Places-LT. ImageNet [6] and Places [32] are also two balanced dataset. Unlike CIFAR, these two datasets have larger scale and are more difficult. ImageNet-LT and Places-LT are their long-tailed version constructed by Liu et al. [19]. The number of training images for each class is determined using the Pareto distribution with a power value = 6. Their original test sets are left unchanged. B. Implementation Details of the Fine-tuning Stage CIFAR100-LT. For data augmentation, we randomly crop a 32 × 32 patch from the original image or its horizontal flip with 4 pixels padded on each side. We use the stochastic gradient descent (SGD) to optimize the network with momentum of 0.9 and weight decay of 5 × 10 −4 . We train the model for 40 epochs. The initial learning rate is set to 5 × 10 −2 and decrease it at the 10 th epoch by 0.2. We use a batch size of 128. ImageNet-LT and Places-LT. For data augmentation, we resize the image by setting the shorter side to 256 and then take a random crop of 224 × 224 from it or its horizontal flip. Finally, color jittering is applied. We train our model for 40 epochs with a batch size of 512. We use stochastic gradient descent (SGD) with momentum of 0.9 and weight decay of 5 × 10 −4 . The initial learning rate is set to 5 × 10 −2 and is decreased at the 20 th epoch by 0.2. For Places-LT, when applied to MiSLAS [30], since MiSLAS is a two-stage method, we find it fairer to also apply their proposed label aware smoothing loss in the fine-tuning stage. So when computing the loss function, we combine two loss functions together as L = * L GML + L LAS . During the experiment, we simply use = 1 without tuning it. C. Additional Ablation Studies We present some additional ablation studies here. . Bar plot of per-class recall on the imbalanced CIFAR100 (with imbalance ratio 100) before and after the fine-tuning when GML is applied to BSCE [22]. C.1. Better Baselines Since our method requires re-training the classifier, the model is essentially trained longer. To better understand the performance improvement, here we conduct some experiments on CIFAR100-LT that serve as better baselines. Specifically, in the fine-tuning stage of our method, instead of using the proposed GML, we use either balanced crossentropy or pure cross-entropy but combined with a balanced sampler. All the other settings remain unchanged. The results are shown in Tab. 10. As we can see, our proposed GML is better than them in terms of the harmonic mean of recall and the lowest recall value. C.2. More Visualizations of the Per-Class Recall Here we present more visualization results of the perclass recall when GML is applied to different methods. All experiments are conducted on CIFAR100-LT (with imbalance ratio 100). The results are shown in Fig. 4, Fig. 5 and Fig. 6. Figure 5. Bar plot of per-class recall on the imbalanced CIFAR100 (with imbalance ratio 100) before and after the fine-tuning when GML is applied to MiSLAS [30]. Figure 6. Bar plot of per-class recall on the imbalanced CIFAR100 (with imbalance ratio 100) before and after the fine-tuning when GML is applied to PaCo [4]. Figure 2 . 2Bar plot of per-class recall on the imbalanced CIFAR10 (with imbalance ratio 100) before ('CE') and after ('CE+GML') the fine-tuning. For the fine-tuned model, the recall of the first 5 classes dropped while the recall of the latter 5 classes increased, motivating us to combine the prediction of both models. Algorithm 1 1The overall training procedure Input: Training images and their labels . 1: Randomly initialize the network and train it from scratch to obtain the pre-trained model. 2: Use the pre-trained model to perform initialization. 3: Freeze the backbone and re-initialize the classifier. 4: Fine-tune the model using the loss function defined as Eq. (1) for a few epochs. 5: During inference, combine the results from both classifiers as defined in Eq. (11). Figure 3 . 3Bar plot of per-class recall on the imbalanced CIFAR100 (with imbalance ratio 100) before and after the fine-tuning. No ensemble is performed. Our method focuses on improving the recognition of classes that have low recall values, making sure no category is left behind. Figure 4 4Figure 4. Bar plot of per-class recall on the imbalanced CIFAR100 (with imbalance ratio 100) before and after the fine-tuning when GML is applied to BSCE [22]. DatasetNumber of Classes # Training Images # Test Images Imbalance RatioCIFAR100-LT [16] 100 10,847 10,000 100 ImageNet-LT [6, 19] 1,000 115,846 50,000 256 Places-LT [32] 365 62,500 36,500 996 Table 2. Statistics of three imbalanced datasets used in our experiments. MethodsG-Mean H-Mean Lowest RecallTable 10. Comparing with better baselines.CE + GML 36.59 31.26 6.00 CE + CE (re-weighting) 35.30 27.55 4.00 CE + CE (re-sampling) 30.84 18.52 2.00 Long-tailed recognition via weight balancing. 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[ "LEVEL UP THE DEEPFAKE DETECTION: A METHOD TO EFFECTIVELY DISCRIMINATE IMAGES GENERATED BY GAN ARCHITECTURES AND DIFFUSION MODELS", "LEVEL UP THE DEEPFAKE DETECTION: A METHOD TO EFFECTIVELY DISCRIMINATE IMAGES GENERATED BY GAN ARCHITECTURES AND DIFFUSION MODELS" ]	[ "Luca Guarnera \nDepartment of Mathematics and Computer Science\nUniversity of Catania\nItaly\n", "Oliver Giudice [email protected] \nIT dept\nApplied Research Team\nBanca d'Italia, RomeItaly\n", "Sebastiano Battiato [email protected] \nDepartment of Mathematics and Computer Science\nUniversity of Catania\nItaly\n" ]	[ "Department of Mathematics and Computer Science\nUniversity of Catania\nItaly", "IT dept\nApplied Research Team\nBanca d'Italia, RomeItaly", "Department of Mathematics and Computer Science\nUniversity of Catania\nItaly" ]	[]	The image deepfake detection task has been greatly addressed by the scientific community to discriminate real images from those generated by Artificial Intelligence (AI) models: a binary classification task. In this work, the deepfake detection and recognition task was investigated by collecting a dedicated dataset of pristine images and fake ones generated by 9 different Generative Adversarial Network (GAN) architectures and by 4 additional Diffusion Models (DM).A hierarchical multi-level approach was then introduced to solve three different deepfake detection and recognition tasks: (i) Real Vs AI generated; (ii) GANs Vs DMs; (iii) AI specific architecture recognition. Experimental results demonstrated, in each case, more than 97% classification accuracy, outperforming state-of-the-art methods.	10.48550/arxiv.2303.00608	[ "https://export.arxiv.org/pdf/2303.00608v1.pdf" ]	257,255,351	2303.00608	225abbd89d031023860d51376c1da899ad5e2eb1	LEVEL UP THE DEEPFAKE DETECTION: A METHOD TO EFFECTIVELY DISCRIMINATE IMAGES GENERATED BY GAN ARCHITECTURES AND DIFFUSION MODELS 1 Mar 2023 Luca Guarnera Department of Mathematics and Computer Science University of Catania Italy Oliver Giudice [email protected] IT dept Applied Research Team Banca d'Italia, RomeItaly Sebastiano Battiato [email protected] Department of Mathematics and Computer Science University of Catania Italy LEVEL UP THE DEEPFAKE DETECTION: A METHOD TO EFFECTIVELY DISCRIMINATE IMAGES GENERATED BY GAN ARCHITECTURES AND DIFFUSION MODELS 1 Mar 2023Index Terms-Deepfake DetectionGenerative Adver- sarial NetsDiffusion ModelsMultimedia Forensics The image deepfake detection task has been greatly addressed by the scientific community to discriminate real images from those generated by Artificial Intelligence (AI) models: a binary classification task. In this work, the deepfake detection and recognition task was investigated by collecting a dedicated dataset of pristine images and fake ones generated by 9 different Generative Adversarial Network (GAN) architectures and by 4 additional Diffusion Models (DM).A hierarchical multi-level approach was then introduced to solve three different deepfake detection and recognition tasks: (i) Real Vs AI generated; (ii) GANs Vs DMs; (iii) AI specific architecture recognition. Experimental results demonstrated, in each case, more than 97% classification accuracy, outperforming state-of-the-art methods. INTRODUCTION The term deepfake refers to all those multimedia contents generated an AI model. The most common deepfake creation solutions are those based on GANs [1] which are effectively able to create from scratch or manipulate a multimedia data. In a nutshell, GANs are composed by two neural networks: the Generator (G) and the Discriminator (D). G creates new data samples that resemble the training data, while D evaluates whether a sample is real (belonging to the training set) or fake (generated by the G). A GAN must be trained until D is no longer able to detect samples generated by G, in other words, when D starts to be fooled by G. Several surveys on methods dealing with GAN-based approaches for the creation and detection of deepfakes, have been proposed in [2,3]. Recently, DMs [4,5] are arousing interest thanks to their photo-realism and also to a wide choice in output control given to the user. In contrast to GANs, DMs are a class of probabilistic generative models that aims to model complex data distributions by iteratively adding noise to a random noise vector input for the generation of new realistic samples and, using them as basis, proceed to reconstruct the original data. Stable Diffusion [6] and DALL-E 2 [7] are the most famous state of the art DMs, based on the text-to-image translation operation. As demonstrated in [8], DMs are able to produce even better realistic images than GANs, since GANs generate high-quality samples but are demonstrated to fail in covering the entire training data distribution. To effectively counteract the illicit use of synthetic data generated by GANs and DMs, new deepfake detection and recognition algorithms are needed. As far as image deepfake detection methods in state of the art are concerned, they mostly focus on binary detection (Real Vs. AI generated [9,10]) . Interesting methods in state of the art already demonstrated to effectively discriminate between different GAN architectures [11,12,13]. Methods to detect DMs and recognize them have been proposed just recently [14,15]. In order to level up the deepfake detection and recognition task, the objective of this paper and the main contribution is to classify an image among 14 different classes: 9 GAN architectures, 4 DMs engines and 3 pristine datasets (labeled as belonging to the same "real" class). At first, a dedicated dataset of images was collected. Then, a novel multi-level hierarchical approach exploiting ResNET models was developed and trained. The proposed approach consists of 3 levels of classification: (Level 1) Real Vs AI-generated images; (Level 2) GANs Vs DMs; (Level 3) recognition of specific AI (GAN/DM) architectures among those represented in the collected dataset. Experimental results demonstrated the effectiveness of the proposed solution, achieving more than 97% accuracy on average for each task, exceeding the state of the art. Moreover, the hierarchical approach can be used to analyze multimedia data in depth to reconstruct its history (forensic ballistics) [16], a task poorly addressed by the scientific community on synthetic data. This paper is organized as follows: Section 2 and Section DATASET DETAILS The dataset employed in this study is a dedicated collection of images: real/pristine images collected from CelebA [17], FFHQ 1 , and ImageNet [18] datasets and synthetic data generated by 9 different GAN engines (AttGAN [19], Cy-cleGAN [20], GDWCT [21], IMLE [22], ProGAN [23], StarGAN [24], StarGAN-v2 [25], StyleGAN [26], Style-GAN2 [27]) and 4 text-to-image DM architectures (DALL-E 2 [7], GLIDE [28], Latent Diffusion [6] 2 . For each considered GAN, 2, 500 images (a total of 22, 500) were generated while for the DMs, 5, 000 images were created for each architecture employing more than 800 random sentences, for a total of 20, 000 images. Overall, the total number of synthetic data consists of 42, 500 images. Finally, for each real dataset (CelebA, FFHQ and ImageNet) 13, 500 images were considered, for a total of 40, 500. Table 1 summarizes the numbers of the obtained dataset with respect to each level and to division of training, validation and test sets employed for the experiments. Figure 1 shows several examples of the obtained dataset. MULTI-LEVEL DEEPFAKE DETECTION AND RECOGNITION The dataset collected and described in the previous section was preliminarly investigated as a "flat" classification task. This was carried out by employing a single ResNET-34 encoder with 14 classes as output layer. In addition, a further test was carried out by removing every image belonging to the real class (13 classes as output layer). The trained ResNET-34 model showed that in this last case it is able to achieve greater accuracy score. This gave the idea that a hierarchical approach could lead to even better results also giving a bit of explainability on the analyzed image. The proposed multi-level deepfake detection and recognition approach consists of 3 levels. Level 1 has the objective to detect real data from those created AI architectures (so all synthetic data were labeled as belonging to the same class). Thus this level is implemented as a binary ResNET-34 classifier. Given that an image was previously classified as generated by an AI, Level 2 furtherly analyzes images to discriminate between those generated by a GAN from those generated by a DM. Thus this level is implemented as another binary ResNET-34 classifier. Finally, given that an image was previously classified as generated by a GAN or by a DM, the last level solves the task of recognizing the specific architecture between those considered in the dataset described in Section 2. To do this, Level 3 is divided into two sub-modules: "Level 3-GANs" to discriminate the specific GAN, implemented as a 9-classes ResNET-34 classifier; and "Level 3-DMs" implemented as 4-classes ResNET-34 Diffusion Model classifier. Figure 2 summarizes the overall approach. EXPERIMENTAL RESULTS AND COMPARISON Experiments with ResNET architecture were performed considering the following parameters for training: batchsize = Fig. 2. Execution flow of the proposed hierarchical approach. Level 1 classifies images as real or AI-generated. Level 2 defines whether the input images were created by GAN or DM technologies. Level 3, composed of two sub-modules, solves the AI architecture recognition task. The dashed arrows represent an optional flow (e.g., in the case the input image is real, it will not be analyzed by the next levels). 30, learningrate = 0.00001, optimizer Stochastic Gradient Descent (SGD) with momentum = 0.9 and Cross Entropy Loss. All images were resized to a resolution of 256x256. As regards 14-classes, 13-classes and Level 1 models were trained for 150 epochs. The remaining three models (one for Level 2 and two for Level 3 classification) were trained for 100 epochs. This difference is due to the different amount of images at corresponding level (see Table 1). Four different instances of the ResNET encoder were employed for the proposed multi-level deepfake detection and recognition task. Each ResNET model was properly trained with a corresponding sub-dataset composed as shown in Table 1. In particular, the Pytorch implementation of ResNET and each training started from the weights pre-trained on Im-ageNet 3 . For each model, a fully connected layer with an output size equal to the number of classes of the corresponding classification level followed by a SoftMax was added to the last layer of ResNET encoders. As far as model selection is concerned, ResNET architecture was chosen among other as already demonstrated to be effective in higher level deepfake recognition tasks [29]. Given the presented solution, 4 models have to be run simultaneously on a single GPU. This limited the dimension of the useable models. Thus only ResNET-18 and ResNET-34 were took into account. The ResNET-34 architecture was selected as final solution for achieving the best results (see Table 2). At first, the "flat" 14-classes classification task obtained an overall accuracy of only 94, 85% in the test-set. In order to slightly improve this result, first class (pristine images) was removed and the model retrained. With the remaining 13 classes (only synthetic images generated by all considered GAN and DM engines) a bit better overall accuracy of 95, 4% was obtained. Taking advantage of this results, the hierarchi- Table 2 shows that ResNET-34 achieves the best results compared with ResNET-18. The final solution was compared with various state-of-theart works ( [9,10,13,11,12,14]) demonstrating to achieve best results in each task. In Table 3, the Level 3 DMs column does not show some values given that corresponding approaches does not cover the corresponding task. Therefore, most of the methods do not seem to be able to effectively distinguish between different DMs. This is because DMs leave different traces on synthetic images than those generated by GAN engines. This claim is further empirically demonstrated by the results reported in Table 3 on the "Real Vs AI" column, where, due to the presence of DMs-generated images, the classification accuracy values of the various methods are sensibly lower than the 97, 63% obtained by the proposed approach. For this reason, a column regarding Level 2 task was not added, as the classification results of the various methods Level 1 Level 3 Real Vs AI GANs DMs AutoGAN [9] 68.5 80,3 -Fakespotter [10] 74,22 95,32 -EM [13] 86.57 95,02 -DCT [11] 87,20 95,89 -Wang et al. [12] 78.54 97.32 -DE-FAKE [14] 90 turn out to be significantly lower (almost random classifiers) than the classification accuracy of 98, 01% obtained by the proposed approach. Finally, all state of the art methods, including the proposed one are able to generalize well in terms of distinguishing between GAN architectures that created the synthetic data (GANs column). CONCLUSIONS In this paper, a deepfake detection and recognition solution is presented. The proposed solution is able to recognize if an image was generated among 9 different GAN engines and 4 DM models. However, the proposed solution demonstrated to outperform state of the art in all tasks. Future experiments could evaluate the robustness of the proposed method in real-world contexts (JPEG compression, scaling, etc.). Also, the possibility to identify some analytical traces [15,11,13] will be investigated just to exploit unusual statistics embedded on images generated by DMs. Fig. 1 . 1Examples of images collected from different datasets and images generated by different GANs and DMs. Figure 3 3shows the accuracy and error trends obtained in the training and test phases for each epoch and for each model of the hierarchical approach. Fig. 3 . 3Trend of accuracy and error values obtained in training and test phases for each epoch of the three levels of the proposed hierarchical approach. Each column represents a classification level. 1 https://github.com/NVlabs/ffhq-dataset 2 a.k.a. Stable Diffusion: https://github.com/CompVis/stable-diffusionClassification Task Train 50% Val 20% Test 30% 14-classes Total Images 28,000 4.200 7.000 #Img ∀ class 2,000 300 500 13-classes Total Images 26,000 3,900 6,500 #Img ∀ class 2,000 300 500 L1 Total Images 46,480 11,620 24,900 #Img ∀ class 23,240 5,810 12,450 L2 Total Images 23,800 5,950 12,750 #Img ∀ class 11,900 2,975 6,375 L3 GANs Total Images 12.600 3,150 6,750 #Img ∀ class 1,400 350 750 DMs Total Images 11,200 2,800 6,000 #Img ∀ class 2,800 700 1,500 Table 1. Overview of the images employed for training, validation and test sets (last three columns with indication of % of samples). The first column denotes the classifica- tion task (e.g., 14 − classes is the flat classification task with 14 classes; L1 refers to the Level 1 of hierarchy. The T otalImages rows indicate the total number of images em- ployed for training, validation, and testing phases. The #Img∀class represents the number of samples considered for each class. Table 2 . 2Comparisonof classification accuracy value (%) obtained between ResNET-18 and ResNET-34 architectures with respect to the proposed hierarchical approach. cal approach described in Section 3 was developed. The best results obtained are the following: • Level 1: classification accuracy of 97, 63%; • Level 2: classification accuracy of 98, 01%; • Level 3: an accuracy of 97, 77% was obtained for the GAN recognition task and an accuracy of 98, 02% for the DM one. Table 3 . 3Comparison with state of the art approaches. Classification accuracy value is reported (%). -are reported where the method can not handle the corresponding task. describe the dataset and the proposed approach built upon it respectively. Experimental results and comparison are presented in Section 4. Finally, Section 5 concludes the paper. Generative Adversarial Nets. 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[ "https://github.com/NVlabs/ffhq-dataset", "https://github.com/CompVis/stable-diffusionClassification" ]
[ "PixHt-Lab: Pixel Height Based Light Effect Generation for Image Compositing", "PixHt-Lab: Pixel Height Based Light Effect Generation for Image Compositing" ]	[ "Yichen Sheng \nAdobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc\nPurdue University\nAdobe Inc\nPurdue University\nPurdue University\n\n", "Jianming Zhang \nAdobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc\nPurdue University\nAdobe Inc\nPurdue University\nPurdue University\n\n", "Julien Philip \nAdobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc\nPurdue University\nAdobe Inc\nPurdue University\nPurdue University\n\n", "Lu Ling \nAdobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc\nPurdue University\nAdobe Inc\nPurdue University\nPurdue University\n\n", "Bedrich Benes \nAdobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc\nPurdue University\nAdobe Inc\nPurdue University\nPurdue University\n\n" ]	[ "Adobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc\nPurdue University\nAdobe Inc\nPurdue University\nPurdue University\n", "Adobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc\nPurdue University\nAdobe Inc\nPurdue University\nPurdue University\n", "Adobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc\nPurdue University\nAdobe Inc\nPurdue University\nPurdue University\n", "Adobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc\nPurdue University\nAdobe Inc\nPurdue University\nPurdue University\n", "Adobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc\nPurdue University\nAdobe Inc\nPurdue University\nPurdue University\n" ]	[]	a) cutout (b) SSN [45] (c) SSG [44] (d) PixHt-Lab (ours) Figure 1. PixHt-Lab renders realistic reflection and soft shadows on general shadow receivers for 2D cutouts. (a) shows the object cutout and composition background. (b) SSN [45] cannot render soft shadows on walls due to its ground plane assumption. (c) SSG [44] renders specular reflection, but the shadow on the wall is uniformly softened. (d) Our PixHt-Lab renders realistic soft shadows on the wall guided by 3D-aware buffer channels. The shadow is softened according to the background geometry with more realistic details. PixHt-Lab also renders realistic reflections with physically-based surface materials.AbstractLighting effects such as shadows or reflections are key in making synthetic images realistic and visually appealing. To generate such effects, traditional computer graphics uses a physically-based renderer along with 3D geometry. To compensate for the lack of geometry in 2D Image compositing, recent deep learning-based approaches introduced a pixel height representation to generate soft shadows and reflections. However, the lack of geometry limits the quality of the generated soft shadows and constrain reflections to pure specular ones. We introduce PixHt-Lab, a system leveraging an explicit mapping from pixel height representation to 3D space. Using this mapping, PixHt-Lab reconstructs both the cutout and background geometry and renders realistic, diverse, lighting effects for image compositing. Given a surface with physically-based materials, we can render reflections with varying glossiness. To generate more realistic soft shadows, we further propose to use 3D-aware buffer channels to guide a neural renderer. Both quantitative and qualitative evaluations demonstrate that PixHt-Lab significantly improves soft shadow generation.	10.48550/arxiv.2303.00137	[ "https://export.arxiv.org/pdf/2303.00137v1.pdf" ]	257,255,534	2303.00137	ebcd3ca52e0dfb35ecac3acc37dff875cdd9b9d7	PixHt-Lab: Pixel Height Based Light Effect Generation for Image Compositing Yichen Sheng Adobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc Purdue University Adobe Inc Purdue University Purdue University Jianming Zhang Adobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc Purdue University Adobe Inc Purdue University Purdue University Julien Philip Adobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc Purdue University Adobe Inc Purdue University Purdue University Lu Ling Adobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc Purdue University Adobe Inc Purdue University Purdue University Bedrich Benes Adobe Inc. Yannick Hold-Geoffroy Adobe Inc. Xin Sun Adobe Inc. He Zhang Adobe Inc Purdue University Adobe Inc Purdue University Purdue University PixHt-Lab: Pixel Height Based Light Effect Generation for Image Compositing a) cutout (b) SSN [45] (c) SSG [44] (d) PixHt-Lab (ours) Figure 1. PixHt-Lab renders realistic reflection and soft shadows on general shadow receivers for 2D cutouts. (a) shows the object cutout and composition background. (b) SSN [45] cannot render soft shadows on walls due to its ground plane assumption. (c) SSG [44] renders specular reflection, but the shadow on the wall is uniformly softened. (d) Our PixHt-Lab renders realistic soft shadows on the wall guided by 3D-aware buffer channels. The shadow is softened according to the background geometry with more realistic details. PixHt-Lab also renders realistic reflections with physically-based surface materials.AbstractLighting effects such as shadows or reflections are key in making synthetic images realistic and visually appealing. To generate such effects, traditional computer graphics uses a physically-based renderer along with 3D geometry. To compensate for the lack of geometry in 2D Image compositing, recent deep learning-based approaches introduced a pixel height representation to generate soft shadows and reflections. However, the lack of geometry limits the quality of the generated soft shadows and constrain reflections to pure specular ones. We introduce PixHt-Lab, a system leveraging an explicit mapping from pixel height representation to 3D space. Using this mapping, PixHt-Lab reconstructs both the cutout and background geometry and renders realistic, diverse, lighting effects for image compositing. Given a surface with physically-based materials, we can render reflections with varying glossiness. To generate more realistic soft shadows, we further propose to use 3D-aware buffer channels to guide a neural renderer. Both quantitative and qualitative evaluations demonstrate that PixHt-Lab significantly improves soft shadow generation. Introduction Image compositing is a powerful tool widely used for image content creation, combining interesting elements from different sources to create a new image. One challenging task is adding lighting effects to make the compositing realistic and visually appealing. Lighting effects often involve complex interactions between the objects in the compositing, so their manual creation is tedious. It requires a significant amount of effort, especially for soft shadows cast by area lights and realistic reflections on the glossy surface with the Fresnel effect [59]. Many methods that generate lighting effects for 3D scenes have been well-studied [18], but 3D shapes are often unavailable during image compositing. Recent advancements in deep learning made significant progress in lighting effect generation in 2D images, especially for shadow gen-eration. A series of generative adversarial networks (GANs) based methods [13,24,57,65] have been proposed to automatically generate hard shadows to match the background by training with pairs of shadow-free and shadow images. Those methods only focus on hard shadow generation, and their generated hard shadow cannot be edited freely. More importantly, the light control of those methods is implicitly represented in the background image. In real-world image creation scenarios, however, the background is often well-lit or even in pure color under a studio lighting setting, making those methods unusable. Also, editing the shadow is often needed on a separate image layer when the image editing is still incomplete. To address these issues, a recent work SSN [45] proposes to learn the mapping between the image cutouts and the corresponding soft shadows based on a controllable light map. Although it achieves promising results, it assumes that the shadow receiver is just a ground plane and the object is always standing on the ground, which limits its practical usage. This limitation is addressed by SSG [44], which proposes a new 2.5D representation called pixel height, which is shown to be better suited for shadow generation than previous 2.5D presentations like depth map. Hard shadow on general shadow receivers can be computed by a ray tracing algorithm in the pixel-height space. A neural network renderer is further proposed to render the soft shadow based on the hard shadow mask. It achieves more controllability and it works in more general scenarios, but the lack of 3D geometry guidance makes the soft shadows unrealistic and prone to visual artifacts when they are cast on general shadow receivers like walls. In addition, SSG proposes an algorithm to render the specular reflection by flipping the pixels according to their pixel height. However, the use case is very limited as it cannot be directly applied to simulate realistic reflection effects on more general materials (see Fig. 1 (c)). We introduce a controllable pixel height based system called PixHt-Lab that provides lighting effects such as soft shadows and reflections for physically based surface materials. We introduce a formulation to map the 2.5D pixel height representation to the 3D space. Based on this mapping, geometry of both the foreground cutout and the background surface can be directly reconstructed by their corresponding pixel height maps. As the 3D geometry can be reconstructed, the surface normal can also be computed. Using a camera with preset extrinsic and intrinsics, light effects, including reflections, soft shadows, refractions, etc., can be rendered using classical rendering methods based on the reconstructed 3D geometry or directly in the pixel height space utilizing the efficient data structure (See Sec. 3.3) derived from the pixel height representation. As the soft shadow integration in classical rendering algorithms is slow, especially for large area lights, we propose to train a neural network renderer SSG++ guided by 3D-aware buffer channels to render the soft shadow on general shadow receivers in real time. Quantitative and qualitative experiments have been conducted to show that the proposed SSG++ guided by 3D-aware buffer channels significantly improve the soft shadow quality on general shadow receivers than previous soft shadow generation works. Our main contributions are: • A mapping formulation between pixel height and the 3D space. Rendering-related 3D geometry properties, e.g., normal or depth, can be computed directly from the pixel height representation for diverse 3D effects rendering, including reflection and refraction. • A novel soft shadow neural renderer, SSG++, guided by 3D-aware buffer channels to generate high-quality soft shadows for general shadow receivers in image composition. Previous Work Single Image 3D Reconstruction Rendering engines can be used to perform image composition. However, they require a 3D reconstruction of the image, which is a challenging problem. Deep learning-based methods [21,35,36,48] have been proposed to perform dense 3D reconstruction via low dimensional parameterization of the 3D models, though rendering quality is impacted by the missing high-frequency features. Many single-image digital human reconstruction methods [3,20,22,30,40,41,[62][63][64]66] show promising results, albeit they assume specific camera parameters. Directly rendering shadows on their reconstructed 3D models yields hard-to-fix artifacts [44] in the contact regions, between the inserted object and the ground. More importantly, those methods cannot be applied to general objects, which limits their use for generic image composition. Single Image Neural Rendering Image harmonization blends a cutout within a background in a plausible way. Classical methods achieve this goal by adjusting the appearance statistics [15,31,34,38,54]. Recently, learning-based methods [6,16,17,23,49,55,56] trained with augmented real-world images were shown to provide more robust results. However, these methods focus on global color adjustment without considering shadows during composition. Single image portrait relighting methods [46,53,67] adjust the lighting conditions given a user-provided lighting environment, although they only work for human portraits. [10] considers the problem of outdoor scene relighting from a single view using intermediary predicted shadow layers, which could be trained on cutout objects, but their method only produces hard shadows. Neural Radiance Field based methods (e.g., [26,27,39,51]) propose to implicitly encode the scene geometry, but require multiple images as input. Soft Shadow Rendering is a well-studied technique in computer graphics, whether for real-time applications [1,2,4,8,9,11,12,28,37,42,43,50,52,60] or global illumination methods [7,19,47,58]. It requires exact 3D geometry as input, preventing its use for image composition. Recent neural rendering methods can address the limited input problem and render shadows for different scenarios. Scene level methods [32,33] show promising results but require multiple input images. Generative adversarial networks (GANs) have achieved significant improvements on image translation tasks [14,25], and subject-level shadow rendering methods [13,24,57,65] propose to render shadow using GANs. Unfortunately, these methods have two main limitations: they can only generate hard shadows, and prevents user editability, which is desired for artistic purposes. SSN [45] proposed a controllable soft shadow generation method for image composition, but is limited to the ground plane and cannot project shadows on complex geometries. Recently, SSG [44] further proposed a new representation called pixel height to cast soft shadows on more general shadow receivers and render specular reflection on the ground. Unfortunately, the shadow quality of SSG degrades on complex geometries as they are not explicitly taken into account by the network. Furthermore, its reflection rendering is limited to specular surfaces. In contrast, our proposed SSG++ is guided by 3D geometry-aware buffer channels that can render more realistic soft shadows on generic shadow receivers. We further connect the pixel height representation to 3D by using an estimated per-pixel depth and normal, increasing the reflections' realism. Method We propose a novel algorithm ( Fig. 2) to render reflection and soft shadow to increase image composition realism based on pixel height [44], which has been shown to be more suitable for shadow rendering. Pixel height explicitly captures the object-ground relation, and thus it better keeps the object uprightness and the contact point for shadow rendering. Moreover, it allows intuitive user control to annotate or correct the 3D shape of an object. Our first key insight is that 3D information that highly correlates with rendering many 3D effects, e.g., spatial 3D position, depth, normal, etc., can be recovered by a mapping (see Sec. 3.1) given the pixel height representation. The second key idea is that soft shadows on general shadow receivers are correlated with the relative 3D geometry distance between the occluder and the shadow receiver. Based on the mapping from pixel height to 3D, several geometryaware buffer channels (see Sec. 3.2) are proposed to guide the neural renderer to render realistic soft shadows on general shadow receivers. Moreover, acquiring the 3D information enables rendering additional 3D effects, e.g., reflections and refractions. Fig. 2 shows the overview of our method. Given the 2D cutout and background, the pixel height maps for the cutout and background can be either predicted by a neural network [44] or labeled manually by the user. 3D geometry that is used in rendering can be computed using our presented method (see Sec. 3.1). Finally, our renderer (see Sec. 3.3) can add 3D effects to make the image composition more realistic. Connecting 2.5D Pixel Height to 3D Here we describe the equation that connects 2.5D pixel height to its corresponding 3D point, and Fig. 3 shows the camera and relevant geometry and variables. We define O, the foot point of O , as the origin of the coordinate system. For convenience, we define the camera intrinsics by three vectors: 1) the vector c from the camera center O to the top left corner of the image plane, 2) the right vector of the image plane a, and 3) the down vector of the image plane b. Any point P and its foot point Q can be projected by the camera centered at O . The points P and Q projected on the image plane are denoted as P and Q .   x O y O z O   +   x a x b x c y a y b y c z a z b z c     u P v P 1   w =   x P y P z P   (1) y o + y a y b y c   u Q v Q 1   w = y Q(2) The relationship between a 3D point P and its projection P is described by the projection Eq. 1 under the pinhole camera assumption. From the definition of pixel height representation, the foot point Q of P in the image space is on the ground plane, i.e., y Q = 0. Solving Eq. 2 provides w: w = −y o y a u q + b y v q + y c(3) The pixel height representation has no pitch angle assumption, thus P can be directly computed using the w in Eq. 3. By re-projecting the P back, the 3D point P can be calculated as P =   x O y O z O   +   xa x b xc ya y b yc za z b zc     u P v P 1   −h yau Q + y b v Q + yc .(4) Horizon controllability is important as different horizons will change the soft shadow distortion. However, as shown in Fig. 4, changing the horizon will change the camera pitch, which violates the no pitch assumption from pixel height representation [44] and leads to tilted geometry. To resolve the issue, we propose to use a tilt-shift camera model for our application. When the user changes the horizon, the vector c in Fig. 3 will move vertically to align the horizon to keep the image plane perpendicular to the ground. In this way, the no-pitch assumption is preserved for the correct reconstruction, as shown in Fig. 4 (d). 3D Buffer Channels for Soft Shadow Rendering Our methods can be applied to arbitrary shape lights, but for discussion purposes, we assume the light for our discussion is disk shape area light. It is challenging to render high-quality soft shadows for general shadow receivers given only image cutouts and the hard shadow mask, as the soft shadow is jointly affected by multiple factors: the light source geometry, the occluder, the shadow receiver, the spatial relationship between the occluder and the shadow receiver, etc. SSG [44] is guided by the cutout mask, the hard shadow mask, and the disk light radius as inputs. The shadow boundary is softened uniformly (see Figs. 1 and 6) as SSG is unaware of the geometry-related information relevant to soft shadow rendering. We propose to train a neural network SSG++ to learn how these complex factors will jointly affect the soft shadow results. 3D-Aware Buffer channels. Our SSG++ takes several 3D-aware buffer channels (see Fig. 5) relevant to soft shadow rendering. The buffer channels are composed of several maps: the cutout pixel height map; the gradient of the background pixel height map; the hard shadow from the center of the light L; the sparse hard shadows map; the relative distance map between the occluder and the shadow receiver in pixel height space. For illustration purposes, we composite foreground pixel height and background pixel height in Fig. 5 (b). The cutout pixel height and background pixel height map describe the geometry of the cutout and the background. In our implementation, we use the gradient map of the background pixel height as input to make it translation invariant. The pixel height gradient map will capture the surface orientation similar to a normal map. The sparse hard shadows map can also guide the network to be aware of the shadow receiver's geometry. Another important property of this channel is that the sparse hard shadows describe the outer boundary of the soft shadow. The overlapping areas of the sparse hard shadows are also a hint of darker areas in the soft shadow. Experiments in Sec. 4 show this channel plays the most important role among all the buffer channels. The four sparse hard shadows are sampled from the four extreme points of the disk light L as shown in Fig. 5. The relative distance map in pixel height space defines the relative spatial distance between the occluder and the shadow receiver. The longer the distance, the softer the shadow will be in general. This channel guides the network to pay attention to shadow regions that have high contrast. The formal definition of the relative distance in pixel height space is: \|\|(u p , v p , h p ) − (u q , v q , h q )\|\| 2 2 , where p, q are two points, u, v are the coordinates in the pixel space, h is the pixel height. Dataset and Training. We follow SSN [45] and SSG [44] to generate a synthetic dataset to train SSG++. In practice, we randomly picked 100 general foreground objects from ModelNet [61] and ShapeNet [5] with different categories, including human, plants, cars, desks, chairs, airplanes, etc. We also picked different backgrounds: ground plane, T shape wall, Cornell box, and curved plane. To cover diverse relative positions between the foreground and background, we randomly generate 200 scenes from the selected foreground and background objects. For each scene, we further randomly sampled 100 lights per scene from a different direction with random area light sizes. In total, the synthetic dataset has 20k training data. SSG++ follows the SSG architecture. We implement the SSG++ using PyTorch [29]. The training takes 8 hrs on average to converge with batch size 50 and learning rate 2e −5 in a RTX-3090. Ray Tracing in Pixel Height Representation Eq. 1 in Sec. 3.1 connects the pixel height to 3D. Although we do not know the camera extrinsic and intrinsic for the image cutout or the background, we can use a default camera to reconstruct the scene, given the pixel height. When the 3D position for the 2D pixel can be computed, the surface normal can be approximated if we assume neighborhood pixels are connected. When the surface normal can be reconstructed, 3D effects, including reflection, refraction and relighting, can be rendered if surface materials are given. One can perform the 3D effects rendering using a classical graphics renderer to render lighting effects for image compositing. We noticed that the pixel height space naturally provides an acceleration mechanism for tracing. Specifically, SSG [44] proposes a ray-scene intersection algorithm in pixel height space. Although the ray-scene intersection is designed for tracing visibility, it can be easily modified to trace the closest hit pixel given a ray origin and ray direction in pixel height space. In the pixel height space, the ray-scene intersection check happens only along a line between the start and the end pixels. The complexity of the ray-scene intersection check in pixel height space is O(H) or O(W ), without the need to check the intersection with each pixel or each reconstructed triangle. Therefore, in practice, we perform ray tracing in the pixel height space in PixHt-Lab. We implemented the method using CUDA. It took around 7s to render a noise free reflection for 512×512 resolution image with 200 samples per pixel. Reflection results on different surface materials can be found in Fig. 7. Additional examples can be found in supplementary materials. Experiments Here we show quantitative (the benchmark, metrics for comparison, ablation study) and qualitative evaluation of the buffer channels by comparing to related work. Quantitative Evaluation of Buffer Channels Benchmark: To compare our 3D-aware buffer channels fairly with SSN [45] that has ground plane assumption, we build two evaluation benchmarks: 1) a ground-shadow benchmark and 2) a wall-shadow benchmark. The two benchmarks share the same assets, but the ground shadow benchmark only has shadows cast on the ground plane, and the wall shadow benchmark always has part of the shadows cast on walls. The foreground objects in the assets are composed of 12 new models randomly selected online with different types: robots, animals, humans, bottles, etc. The background objects in the assets are four new backgrounds with different shapes: one wall corner, two wall corners, steps, and curved backgrounds to test the generalization ability to unseen backgrounds. We randomly generate 70 scenes using those unseen foregrounds and background models with different poses of the foreground and background. We uniformly sample 125 lights with different positions and different area sizes per scene for the wall shadow benchmark. As shadows on the ground have less variation than the shadows on the wall, we sample eight lights with different positions and different area sizes per scene for the ground shadow benchmark. In total, the ground shadow benchmark is composed of 560 data, and the wall shadow benchmark is composed of 8,750 data. The resolution for each data is 256 × 256. Metrics: We use the per-pixel metric RMSE and a scaleinvariant RMSE-s [53]. Similar to [53], shadow intensity may vary, but the overall shapes are correct. We also used perception-based metrics SSIM and ZNCC to measure SSG++ on the wall-shadow benchmark. SSG++ and SSG share the same backbone, but they are guided by different buffers. Therefore, we treat SSG as the basic baseline and do the ablation study together in this section. Results are shown in Tab. 2. Our proposed SSG++ outperforms all the other methods guided by other subsets of the buffer channels. Each buffer channel outperforms SSG in all the metrics, showing that those 3D-aware buffer channels are useful to guide the SSG to render better soft shadows. SSG-D-BN fixes more errors than SSG-D or SSG-BN, showing that the combination of relative distance in pixel height space helps the neural network improve the soft shadow quality. SSG-SS significantly outperforms all the previous baselines by improving RMSE by 29% and SSIM by 4% than SSG-D-BH, which shows that the sparse shadows channel plays the most important role in guiding the SSG to render soft shadows. Combining the sparse shadow channel with the relative distance channel only improves RMSE by 3% and SSIM by 0.15% than only using the sparse shadow channel as addi- tional channels while combing the sparse shadow channel with the background pixel height channel performs worse than only using the sparse shadow channels as an additional channel for SSG, with RMSE degraded by 12% and SSIM by 0.03%. Our SSG++ combines the sparse shadow channel, the relative distance channel, and the background pixel height channel together and achieves the best performance, improving in all the metrics significantly. Compared with SSG, our SSG++ improves RMSE by 38%, RMSE-s by 33%, SSIM by 8% and ZNCC by 32%. Qualitative Evaluation of Buffer Channels Effects of buffer channels. Fig. 6 shows the effects of the buffer channels. Fig. 6 (b). shows the sparse shadow guides the neural network to render better contour shapes as the sparse hard shadows are samples from the outer contour of the shadow regions. However, when the geometry has complex shapes, and the sparse hard shadows are mixed together, e.g., the foot regions of the cat in the second row of Fig. 6, the relative spatial information is ambiguous. The relative distance map can further guide SSG++ to keep the regions close to the objects dark instead of over soft(See Fig. 6 (c) in the second row.). Figure 10. More results. PixHt-Lab is agnostic to the cutout object categories. Lighting effects can be generated to general backgrounds. PixHt-Lab can also generate multiple soft shadows shown in the first column. The first column uses a step background. The second column uses a curved background. The third column uses a L shape wall background. The forth column uses a corner background. Discussion Light effects generated by PixHt-Lab. PixHt-Lab can reconstruct the surface normal solely based the pixel height inputs. As discussed in PixHt-Lab does not have assumptions on the cutout object types and background types. No matter for realistic cutouts or cartoon cutouts, studio background or real world background, PixHt-Lab can render the light effects. (see Fig. 9 and Fig. 10 and the demo video showing the PixHt-Lab system in the supplementary materials). Similar to SSG, PixHt-Lab allows the user to intuitively control the shadow direction and softness, control the horizon position to tweak the shadow distortion, and change parameters to control the physical parameters of the reflection. Our methods can also be applied to multiple object compositing and multiple shadows. Please refer to supplementary materials for more examples. There exist more potential additions for PixHt-Lab and other light effects such as refraction(see Fig. 8) could be implemented. Parameters related to the refraction surface, like the refraction index, can be controlled as well. We demonstrate more results in the supplementary material. Limitation. As PixHt-Lab is based on the pixel height map and the common limitations for the pixel height representation apply to our methods as well. One of them is that it takes the image cutout as the proxy of the object and the back face or hidden surface contributing to the light effect generation is missing. A back face prediction neural network can be explored to address this problem. Another limitation specific to PixHt-Lab is that the proposed method uses the cutout color as the reflected color, which is not precise for cases when the surface has view-dependent colors. Conclusion and Future Work We propose a novel system PixHt-Lab for generating perceptually plausible light effects based on the pixel height representation. The mapping between the 2.5D pixel height and 3D has been presented to reconstruct the surface geometry directly from the pixel height representation. Based on the reconstruction, more light effects, including physically based reflection and refraction, can be synthesized for image compositing. Also, a novel method SSG++ guided by 3D-aware buffer channels is proposed to improve the soft shadow quality that is cast on general shadow receivers. Quantitative and qualitative experiments demonstrate that the results and generalization ability of the proposed SSG++ significantly outperforms previous deep learning-based shadow synthesis methods. However, our PixHt-Lab synthesize the light effect solely based on the cutout colors. A back face prediction neural network may address the issue and is worth future exploration. Figure 2 . 2System overview of PixHt-Lab. Given a 2D cutout and background, the pixel height maps for the cutout and background can be either predicted by a neural network[44] or labeled manually by the user. 3D scene information and the relevant buffer channels can then be computed from pixel height based on our formulation presented in Sec. 3.1. Finally, our neural renderer SSG++ renders the requested lighting effects using the buffer channels (see Sec. Figure 3 . 3Connecting pixel height to 3D. Given the camera at the center O and its foot point O, a point P in 3D space with its foot point Q. P and Q are their projection positions on the image plane. c is the vector from O to the up left corner of the image plane. a and b are the right directions and down direction vector relative to the image plane. Figure 4 . 4Tilt shift camera model. Horizon position is a controllable parameter to change the shadow perspective shape. (a) shows changing the horizon is equivalent to changing the pitch in the classical model. (b) shows the reconstructed 3D model. (c) shows we use a tilt-shift camera. (d) shows the 3D vertical line can be preserved to be perpendicular to the ground after reconstruction. Figure 5 . 5Buffer channels. (a) illustrates how the buffer channels are computed. See the text for more details. (b) shows the pixel height maps of the foreground cutout and the background. (c) is the hard shadow cast by the center of the disk area light L. (d) is the sparse hard shadows map cast by A, B, C, D, which are four extreme points of the area light L. (e) is the distance between EF in pixel height space. Figure 6 . 6Effects of buffer channels. Best zoom-in. (a) shows the shadows rendered by SSG will be softened uniformly. (b). shows sparse hard shadow channels guide the neural network to be 3Daware. (c) shows SSG++ can render better quality in the relatively darker regions(note the foot shadow in the second row). (d) is the ground truth shadow rendered by the physically based renderer. Figure 7 . 7Reflection. PixHt-Lab can render reflection with different physical materials. From left to right, the ground surface glossness increases. The top to bottom, the ground uses different η in Fresnel effects. Figure 8 . 8Refraction. Given the cutout and the background in the left image, the refraction lighting effect for the crystal ball can also be rendered by PixHt-Lab. Figure 9 . 9Real foreground and background examples created with our GUI. Zoom in for the best view. Credit: Adobe Stock. Table 1 . 1Comparison with SSN[45] and SSG[44] on the groundshadow benchmark.Method RMSE ↓ RMSE-s ↓ SSIM ↑ ZNCC ↑ SSN 0.1207 0.1064 0.8379 0.6118 SSG 0.0254 0.0221 0.8547 0.5679 SSG++(ours) 0.0165 0.0140 0.9216 0.8180 Table 2 . 2Result on the wall-shadow benchmark. We show the effectiveness of each buffer channel. SSG-BH: SSG with background pixel height. SSG-D: SSG with XYH distance channel. SSG-D-BH: SSG with XYH distance and background pixel height. SSG-SS: SSG with the sparse shadow channel. SSG-SS-BH: SSG with sparse shadow channel and background pixel height. SSG++: SSG with all the buffer channels.Method RMSE ↓ RMSE-s ↓ SSIM ↑ ZNCC ↑ SSG 0.0242 0.0209 0.8561 0.6460 SSG-BH 0.0248 0.0207 0.8587 0.6506 SSG-D 0.0230 0.0210 0.8739 0.6499 SSG-D-BH 0.0231 0.0201 0.8752 0.6719 SSG-SS 0.0164 0.0149 0.9139 0.8228 SSG-SS-BH 0.0184 0.0158 0.9136 0.8029 SSG-SS-D 0.0159 0.0139 0.9153 0.8494 SSN++(ours) 0.0153 0.0136 0.9277 0.8575 shadow quality. Our SSIM implementation uses 11 × 11 Gaussian filter with σ = 1.5, k 1 = 0.01, k 2 = 0.03. SSG++ on the ground-shadow benchmark. As SSN has a ground plane assumption, we use the ground-shadow benchmark to compare fairly. We compare our SSG++ with SSN and the other soft shadow rendering network SSG pro- posed recently on the ground-shadow benchmark. Results are shown in Tab. 1. Our SSG++ outperforms SSN and SSG in all metrics. 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[ "LOG-CAN: LOCAL-GLOBAL CLASS-AWARE NETWORK FOR SEMANTIC SEGMENTATION OF REMOTE SENSING IMAGES", "LOG-CAN: LOCAL-GLOBAL CLASS-AWARE NETWORK FOR SEMANTIC SEGMENTATION OF REMOTE SENSING IMAGES" ]	[ "Xiaowen Ma \nZhejiang University\n\n", "Mengting Ma \nZhejiang University\n\n", "Chenlu Hu \nZhejiang University\n\n", "Zhiyuan Song \nZhejiang University\n\n", "Ziyan Zhao \nZhejiang University\n\n", "Tian Feng \nZhejiang University\n\n", "Wei Zhang \nZhejiang University\n\n" ]	[ "Zhejiang University\n", "Zhejiang University\n", "Zhejiang University\n", "Zhejiang University\n", "Zhejiang University\n", "Zhejiang University\n", "Zhejiang University\n" ]	[]	Remote sensing images are known of having complex backgrounds, high intra-class variance and large variation of scales, which bring challenge to semantic segmentation. We present LoG-CAN, a multi-scale semantic segmentation network with a global class-aware (GCA) module and local class-aware (LCA) modules to remote sensing images. Specifically, the GCA module captures the global representations of class-wise context modeling to circumvent background interference; the LCA modules generate local class representations as intermediate aware elements, indirectly associating pixels with global class representations to reduce variance within a class; and a multi-scale architecture with GCA and LCA modules yields effective segmentation of objects at different scales via cascaded refinement and fusion of features. Through the evaluation on the ISPRS Vaihingen dataset and the ISPRS Potsdam dataset, experimental results indicate that LoG-CAN outperforms the state-of-the-art methods for general semantic segmentation, while significantly reducing network parameters and computation. Code is available at https://github.com/xwmaxwma/rssegmentation.	10.1109/icassp49357.2023.10095835	[ "https://export.arxiv.org/pdf/2303.07747v1.pdf" ]	257,504,846	2303.07747	240fce62d90a01a37a8df14c5ea15fcc993a01a8	LOG-CAN: LOCAL-GLOBAL CLASS-AWARE NETWORK FOR SEMANTIC SEGMENTATION OF REMOTE SENSING IMAGES Xiaowen Ma Zhejiang University Mengting Ma Zhejiang University Chenlu Hu Zhejiang University Zhiyuan Song Zhejiang University Ziyan Zhao Zhejiang University Tian Feng Zhejiang University Wei Zhang Zhejiang University LOG-CAN: LOCAL-GLOBAL CLASS-AWARE NETWORK FOR SEMANTIC SEGMENTATION OF REMOTE SENSING IMAGES Index Terms-Semantic segmentationremote sensingclass representations Remote sensing images are known of having complex backgrounds, high intra-class variance and large variation of scales, which bring challenge to semantic segmentation. We present LoG-CAN, a multi-scale semantic segmentation network with a global class-aware (GCA) module and local class-aware (LCA) modules to remote sensing images. Specifically, the GCA module captures the global representations of class-wise context modeling to circumvent background interference; the LCA modules generate local class representations as intermediate aware elements, indirectly associating pixels with global class representations to reduce variance within a class; and a multi-scale architecture with GCA and LCA modules yields effective segmentation of objects at different scales via cascaded refinement and fusion of features. Through the evaluation on the ISPRS Vaihingen dataset and the ISPRS Potsdam dataset, experimental results indicate that LoG-CAN outperforms the state-of-the-art methods for general semantic segmentation, while significantly reducing network parameters and computation. Code is available at https://github.com/xwmaxwma/rssegmentation. INTRODUCTION Semantic segmentation of remote sensing images aims to assign definite classes to each image pixel, which makes important contributions to land use, yield estimation, and resource management [1][2][3]. Compared to natural images, remote sensing images are coupled with sophisticated characteristics (e.g., complex background, high intra-class variance, and large variation of scales) that potentially challenge the semantic segmentation. Existing methods of semantic segmentation based on convolutional neural networks (CNN) focus on context modeling [4][5][6][7], which can be categorized into spatial context modeling and relational context modeling. Spatial context modeling methods, such as PSPNet [4] and DeepLabv3+ [8], use spatial pyramid pooling (SPP) or atrous spatial pyramid pooling (ASPP) to integrate spatial contextual information. Although these methods can capture the context dependencies with homogeneity, they disregard the differences of classes. Therefore, unreliable contexts may occur when a general semantic segmentation method processes remote sensing images with complex objects and large spectral differences. Regarding the relational context modeling, non-local neural networks [5] compute the pairwise pixel similarities in the image using non-local blocks for weighted aggregation, and DANet [6] adopts spatial attention and channel attention for selective aggregation. However, the dense attention operations used by these methods enable a large amount of background noise given the complex background of remote sensing images, leading to the performance degradation in semantic segmentation. Recent class-wise context modeling methods, such as ACFNet [9] and OCRNet [10], integrate class-wise contexts by capturing the global class representations to partially prevent the background inference caused by dense attentions. Despite the fact that these methods have achieved ideal performance in semantic segmentation on natural images, the performance on remote sensing images remains problematic, specifically for high intra-class variance that leads to the large gap between pixels and the global class representations. Therefore, introducing local class representations may address this issue. Given the above observations, we design a global classaware (GCA) module to capture the global class representations, and local class-aware (LCA) modules to generate the local class representations. In particular, local class representations are used as intermediate aware elements to indirectly associate pixels with global class representations, which alleviates the complex background and the high intra-class variance of remote sensing images. Both modules are integrated into LoG-CAN, a semantic segmentation network with a multi-scale design that improves the large variation of scales issue of remote sensing images. The primary contributions of this paper are summarized as follows: • a novel local class-aware module using the local class representations for class-wise context modeling; • a multi-scale semantic segmentation network integrating both local and global class-aware modules; • the state-of-the-art performance on two benchmark datasets for aerial images and a significant reduction of the number of parameters and computational efforts. METHOD Overall Architecture The proposed LoG-CAN has an encoder-decoder architecture (as shown in Fig. 1). The encoder uses ResNet50 [11] as the backbone for multi-scale feature extraction, and the decoder consists of a global class-aware (GCA) module and local class-aware (LCA) modules to refine multi-scale feature representations from the backbone via class-wise context modeling. Specifically, each residual block i of the four extracts multi-scale feature representations R i from the input image; the feature representations R g from the last residual block are processed by the GCA module to obtain the intermediate global class representations C g . Then, each R i and the i+1-th LCA module's output are processed with feature mapping and concatenation to reach intermediate feature representations R . In addition, the feature representations R and the class representations C g input to the LCA module are obtained via feature mapping and class mapping from R and C g . Being refined by the cascaded LCA modules, the feature representations at different spatial scales are element-wisely summed and quadruply upsampled for the semantic segmentation output. Note that our design of feature mapping and class mapping, which are implemented respectively by a 3 × 3 convolution layer and a 1 × 1 convolution layer, enables the following two effects: (1) the multi-scale feature representations and class representations further interact with each other in a specific feature space after mapping; (2) mapping reduces the feature channels of both representations, creating a lighter structure that contains fewer model parameters and computation without degrading the model performance. Global Class-Aware Module Motivated by [10], we design a GCA module to capture the global class representations. With feature representations R g ∈ R C ×H ×W that contain rich semantic information, the distribution of class probability D g is obtained as follows, D g = H(R g ),(1) where D g is a matrix of size K × H × W and K is the number of classes. H is implemented by two consecutive 1×1 convolution layers. Then, the global class representations C g is defined as follows, C g = D K×(H ×W ) g ⊗ R (H ×W )×C g ,(2) where C g is a matrix of size K × C . Local Class-Aware Module For remote sensing images, class-wise context modeling that only uses the global class representations circumvents the in- terference of noise caused by intensive attention operations. However, it can potentially lead to considerable semantic differences between pixels and the global class representations due to the insufficient consideration of high intra-class variance, which degenerates the semantic segmentation performance. In this regard, we exploit the local class representations as an intermediate awareness element to capture the relationship between pixels and the local class representations and aggregate this relationship with the global class representations for class-wise context modeling. For the feature representations R ∈ R C×H×W , we deploy a pre-classification operation for the corresponding distribution D ∈ R K×H×W . In particular, we split R and D along the spatial dimension to get R l and D l , followed by calculating the local class representations C l as follows, C l = D (N h ×Nw)×K×(h×w) l ⊗ R (N h ×Nw)×(h×w)×C l ,(3) where h and w represent the height and width of the selected local patch, N h = H h , and N w = W w . The corresponding affinity matrix R r , which represents the similarity between the pixel and the local class representations, is obtained as follows, R r = R (N h ×Nw)×(h×w)×C l ⊗ C (N h ×Nw)×C×K l . (4) Finally, we utilize R r to associate the global class representations C g and acquire the augmented representations R o , R o = ψ(R (N h ×Nw)×(h×w)×K r ⊗ C K×C g ),(5) where ψ is a function that puts the per-local enhanced representations back in place in R. EXPERIMENTS We implemented the proposed method and evaluated LoG-CAN on the ISPRS Vaihingen dataset and the ISPRS Potsdam dataset using three common metrics: average F1-score (AF), mean Intersection-over-Union (mIoU), and overall accuracy (OA). ISPRS Vaihingen dataset [18] includes 33 true orthophoto (TOP) tiles and the corresponding digital surface model (DSMs) collected from a small village, where the image size varies from 1996 × 1995 to 3816 × 2550 pixels and the ground truth labels comprise six land-cover classes (i.e., impervious surfaces, building, low vegetation, tree, car, and clutter/background). We used 16 images for training and the remaining 17 for testing. ISPRS Potsdam dataset [18] includes 38 TOP tiles and the corresponding DSMs collected from a historic city with large building blocks. All images have the same size of 6000 × 6000 pixels and the ground truth labels comprise the same six land-cover classes as the ISPRS Vaihingen dataset. We used 24 images for training and the remaining 14 for testing. Implementation Details We selected ResNet-50 [11] pretrained on ImageNet as the backbone for all experiments. The optimizer was SGD with batch size of 8, and the initial learning rate was set to 0.01 with a poly decay strategy and a weight decay of 0.0001. Following previous work [14,16], we randomly cropped the images from both datasets to produce 512 × 512 patches, and the augmentation methods, such as random scale ([0.5, 0.75, 1.0, 1.25, 1.5]), random vertical flip, random horizontal flip and random rotate, were adopted in the training process. The number of epochs was set to 150 with the ISPRS Vaihingen dataset and 80 with the ISPRS Potsdam dataset. Evaluation and Analysis As shown in Table 1, the proposed method outperformed other state-of-the-art methods on the ISPRS Vaihingen dataset in AF, mIoU, and OA. In particular, our LoG-CAN achieved the AF of 91.46% and the mIoU of 84.13%, even higher than MANet [16] and UNetFormer [17], showing that our design on class-wise context modeling has greater effectiveness. As shown in Table 2, our LoG-CAN also reached outstanding performances in all metrics on the ISPRS Potsdam dataset. Fig. 2 shows example result outputs from our LoG-CAN, PSPNet, and MANet. In particular, the proposed method not only better preserves the integrity and regularity of semantic objects, but also improves the segmentation performance of small objects. To validate the lightness of our method, we compare our LCA module with several classical context aggregation modules, including the number of parameters measured in million (M), the floating-point operations per second (FLOPs) measured in giga (G), and the memory consumption measured in megabytes (MB). All inputs were set to the size of 2048 × 128 × 128 to ensure the comparison's fairness. As shown in Table 3, the LCA module enables significantly less number of parameters and lower computation compared to PPM [4]. From the perspective of the entire network's structure, our LoG-CAN only needs 60% of the parameters and 25% of the GFLOPs compared to PSPNet [4], which suggests its design as a lightweight method. We investigated if the number of patches in the LCA mod- ule has any impact on the results. As shown in Figure 3, the best result was obtained on each dataset with the number of patches being set to 16. Besides, when the number of patches was set to 1, the local class representations degenerated to the global class representations, resulting into relatively unsatisfactory performances. These findings indicate that local class awareness can effectively improve class-wise context modeling. CONCLUSION In this paper, we introduce LoG-CAN for semantic segmentation of remote sensing images. Our method effectively resolves the problems due to complex background, high intraclass variance, and large variation of scales in remote sensing images by combining the global and local class representations for class-wise context modeling with a multi-scale design. According to the experimental results, LoG-CAN has greater effectiveness than the state-of-the-art general methods for semantic segmentation, while requiring less network parameters and computation. The proposed method provides a better trade-off between efficiency and accuracy. This work was supported in part by the National Natural Science Foundation of China under Grant 62202421; in part by Zhejiang Provincial Key Research and Development Program under Grant 2021C01031; in part by Ningbo Yongjiang Talent Introduction Programme under Grant 2021A-157-G; and in part by the Public Welfare Science and Technology Plan of Ningbo City under Grant 2022S125. Fig. 1 . 1Architecture of LoG-CAN with GCA and LCA modules. Fig. 2 . 2Example outputs from the LoG-CAN and other methods on the ISPRS Vaihingen dataset. Best viewed in color and zoom in. Fig. 3 . 3Plot of AF against the number of patches on the ISPRS Vaihingen dataset (yellow) and the ISPRS Potsdam dataset (blue) Table 1 . 1Effectiveness comparison with the state-of-the-art methods on the test set from the ISPRS Vaihingen dataset. Per-class best performance is marked in bold.Method Imp. Sur. Building Low Veg. Tree Car AF mIoU OA PSPNet [4] 91.38 94.20 83.05 88.71 75.02 86.47 76.78 89.36 DeepLabv3+ [8] 91.63 94.09 82.51 88.00 77.66 86.77 77.13 89.12 DANet [6] 91.38 94.10 83.09 89.02 76.80 86.88 77.32 89.47 Semantic FPN [12] 91.78 94.37 82.87 89.44 79.45 87.58 77.94 89.86 FarSeg [13] 92.13 94.57 82.87 88.74 81.11 87.88 79.14 89.57 OCRNet [10] 92.87 95.14 84.32 89.23 84.52 89.22 81.71 90.47 LANet [14] 92.41 94.90 82.89 88.92 81.31 88.09 79.28 89.83 BoTNet [15] 92.22 94.48 83.97 89.57 82.93 88.63 79.89 90.16 MANet [16] 93.02 95.47 84.64 89.98 88.95 90.41 82.71 90.96 UNetFormer [17] 92.70 95.30 84.90 90.60 88.50 90.40 82.70 91.00 LoG-CAN (Ours) 93.71 96.64 85.89 90.93 90.16 91.46 84.13 91.97 Table 2 . 2Effectiveness comparison with the state-of-the-art methods on the test set from the ISPRS Potsdam dataset. Perclass best performance is marked in bold.Method AF mIoU OA PSPNet [4] 89.98 81.99 90.14 DeepLabv3+ [8] 90.86 84.24 89.18 DANet [6] 89.60 81.40 89.73 Semantic FPN [12] 91.53 84.57 90.16 FarSeg [13] 91.21 84.36 89.87 OCRNet [10] 92.25 86.14 90.03 LANet [14] 91.95 85.15 90.84 BoTNet [15] 91.77 84.97 90.42 MANet [16] 92.90 86.95 91.32 UNetFormer [17] 92.80 86.80 91.30 LoG-CAN (Ours) 93.53 87.69 92.09 Input GT PSPNet MANet LoG-CAN Table 3 . 3Computational complexity comparison with other popular context aggregation modules. Per-class best performance is marked in bold.Method Params (M) FLOPs (G) Memory (MB) PPM [4] 23.1 309.5 257 ASPP [8] 15.1 503.0 284 DAB [6] 23.9 392.2 1546 OCR [10] 10.5 354.0 202 PAM+AEM [14] 10.4 157.6 489 ILCM+SLCM [19] 11.0 180.6 638 KAM [16] 5.3 85.9 160 LCA (Ours) 0.8 11.9 53 Decision support system based on artificial intelligence, gis and remote sensing for sustainable public and judicial management. 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[ "Robust Onboard Localization in Changing Environments Exploiting Text Spotting", "Robust Onboard Localization in Changing Environments Exploiting Text Spotting" ]	[ "Nicky Zimmerman ", "Louis Wiesmann ", "Tiziano Guadagnino ", "Thomas Läbe ", "Jens Behley ", "Cyrill Stachniss " ]	[]	[]	Robust localization in a given map is a crucial component of most autonomous robots. In this paper, we address the problem of localizing in an indoor environment that changes and where prominent structures have no correspondence in the map built at a different point in time. To overcome the discrepancy between the map and the observed environment caused by such changes, we exploit human-readable localization cues to assist localization. These cues are readily available in most facilities and can be detected using RGB camera images by utilizing text spotting. We integrate these cues into a Monte Carlo localization framework using a particle filter that operates on 2D LiDAR scans and camera data. By this, we provide a robust localization solution for environments with structural changes and dynamics by humans walking. We evaluate our localization framework on multiple challenging indoor scenarios in an office environment. The experiments suggest that our approach is robust to structural changes and can run on an onboard computer. We release an open source implementation of our approach 1 , which uses off-the-shelf text spotting, written in C++ with a ROS wrapper.	10.1109/iros47612.2022.9981049	[ "https://export.arxiv.org/pdf/2203.12647v2.pdf" ]	247,627,732	2203.12647	27c613edb2daa3d57033ad8d4908129c442bd126	Robust Onboard Localization in Changing Environments Exploiting Text Spotting Nicky Zimmerman Louis Wiesmann Tiziano Guadagnino Thomas Läbe Jens Behley Cyrill Stachniss Robust Onboard Localization in Changing Environments Exploiting Text Spotting Robust localization in a given map is a crucial component of most autonomous robots. In this paper, we address the problem of localizing in an indoor environment that changes and where prominent structures have no correspondence in the map built at a different point in time. To overcome the discrepancy between the map and the observed environment caused by such changes, we exploit human-readable localization cues to assist localization. These cues are readily available in most facilities and can be detected using RGB camera images by utilizing text spotting. We integrate these cues into a Monte Carlo localization framework using a particle filter that operates on 2D LiDAR scans and camera data. By this, we provide a robust localization solution for environments with structural changes and dynamics by humans walking. We evaluate our localization framework on multiple challenging indoor scenarios in an office environment. The experiments suggest that our approach is robust to structural changes and can run on an onboard computer. We release an open source implementation of our approach 1 , which uses off-the-shelf text spotting, written in C++ with a ROS wrapper. I. INTRODUCTION Localization in a given map is a fundamental capability required by most autonomous robots operating in indoor environments, such as office or hospitals. These environments are often populated by people, also undergoing "quasistatic" changes such as closing of doors, objects temporarily standing at some place, or moved furniture that is not reflected in the given map that was recorded at a different point in time. Such changes, which we refer to as "quasistatic" in contrast to dynamic ones such as moving people, result in sensor observations that substantially differ from the map and can lead to localization failure, as illustrated in Fig. 1, where closed doors in a corridor remove localization cues that can lead to ambiguities. To overcome such localization challenges, readily available sources of information can be exploited to aid pose estimation. One example is using WiFi signal strength [16] from existing access points to aid the localization. Another example is using textual information that is part of the building infrastructure. Textual cues are often used by humans to navigate in the environment and are therefore available in most buildings designed for humans. With the recent advances in deep learning-based text recognition [29], All The Kuka YouBot platform that was used for data collection, equipped with 2D LiDAR scanners and cameras that cover the complete 360 • field-of-view we utilize for text spotting. Bottom: The results of of localization in a corridor with closed doors (indicated by red lines), which are not reflected in the map, with and without textual cues. we can reliably and efficiently decode textual content from images and utilize these hints in our localization approach. Surprisingly, there exist only a few approaches [9] [28] in the robotics community to exploit text spotting or optical character recognition (OCR) for robot localization. The main contribution of this paper is a localization framework that integrates text spotting into a particle filter to improve localization. To this end, we build maps indicating the likelihood of detecting room numbers across the environment. The locations with high likelihood for successful detection are then used to inject particles when a known sign is detected. The textual cues allow us to globally localize with a small number of particles enabling online performance on mobile robots with limited computational resources. In our experiments, we show that our approach is able to (i) localize in quasi-static environments, (ii) localize in an environment with low dynamics, (iii) localize in different maps types -a featureless floor plan-like map, and LiDAR-based, feature rich map. Furthermore, our approach runs online on an onboard computer. II. RELATED WORK Localization of mobile platforms is a well researched area in robotics [8][32] [37]. Probabilistic methods that estimate the robot's state have proven to be exceptionally robust, and include the extended Kalman filter (EKF) [19], Markov localization by Fox et al. [13] and particle filters often referred to as Monte Carlo localization (MCL) by Dellaert et al. [11]. These seminal works focused on localization using range sensors such as 2D LiDARs and sonars, as well as cameras. For cameras, the global localization task is framed under the visual place recognition framework, for which multiple algorithms have been proposed [3] [10]. Localization in feature-rich maps, often constructed by range sensors, is well-established [24]. However, there are advantages for using sparse maps such as floor plans for localization. Floor plans are often available for buildings and do not require prior mapping with LiDARs or other range sensors. Their sparsity also means they do not need to be updated as frequently as detailed maps that include possibly moving objects, such as furniture. Their downside is their lack of details, which can render global localization challenging when faced with multiple identical rooms. Another issue is a possible discrepancy between the plans and the construction [6]. Boniardi et al. [7] localize in floor plans with a camera by inferring the room layout and match it against the floor plans. Li et al. [20] introduce a new state variable, scale, to address the scale difference between floor plans and the actual structures. A problem arises when dynamic objects are detected in the scans and observations cannot be correctly matched to a given map. Sun et al. [31] propose to detect those dynamic objects as outliers using a distance filter. Thrun et al. [32] also incorporate the appearance of unexpected objects in the sensors model. Another aspect of scene dynamics is changes that are longer-lasting and not as fast to appear and disappear like moving objects. These long-lasting changes can be closing and opening of passages, transferring large packages from one place to another and shifting of large furniture. Since those changes are more constant, standard filtering technique will fail to remove them. Stachniss and Burgard [30] specifically address the case of closing and opening doors, by trying to detect areas of the map that can have different configurations and learn the possible environmental states in these corresponding areas. Another approach by Krajnik et al. [18] try to capture periodic changes by representing every cell in the occupancy map as a periodic function. Another challenge are seasonal changes in an environment, which were addressed by Vysotska et al. [36] and Milford et al. [22]. In our approach, we do not assume to have prior knowledge on changes that may occur in the map, nor do we require long sequences of images to match against. To tackle more general semi-permanent changes, Valencia et al. [34] suggest using multiple static maps, each corresponding to a different time scale. Biber et al. [5] also propose to update a short-term map online. In the work of Tipaldi et al. [33], a Hidden Markov model [2] is assigned to every grid cell, creating dynamics occupancy grids that can be updated. These methods require continuous update of the map, while we handle changes without altering the map. To assist global localization, additional modalities were considered. Ito et al. [16] use WiFi signal strength to estimate the initial pose, based on signal strength maps that were previously constructed. Joho et al. [17] suggest a sensor model for RFID that combines the likelihood of detecting a tag at a given pose and the likelihood of receiving a specific signal strength. We take inspiration from these papers for building our text likelihoods/priors but apply it for a different modality. Considerable amount of information is helping humans navigate, from publicly available maps to direction signs. Vysotska et al. [35] use publicly available maps, like Open Street Map, to localize with LiDAR. However, exploitation of text for localization is not commonly explored. It was suggested by Radwan et al. [28] but considers outdoor environment and usage of Google Maps, while our approach tackles indoor environments. Another implementation of text spotting in a MCL framework is presented by Cui et al. [9], who rely on text detection as its only sensor model. This differs from our work, which uses a 2D LiDAR-based sensor model and only leveraged text to improve global localization. The advantage of our method is that we are able to localize even in the absence of textual cues. Furthermore, in the work of Cui et al., text spotting is trained specifically for spotting parking space numbers, while we use a generic, off-theshelf text spotting that performs well on a variety of textual cues [29]. III. OUR APPROACH Our goal is to globally localize in an indoor environment that can undergo significant structural changes using 2D LiDAR scanners, cameras and wheel odometry. In sum, we achieve this by building upon the Monte Carlo localization framework. To aid with global localization and recover from localization failures, we use a text spotting approach inferred from camera images to detect room numbers of an human oriented environment. To integrate the textual cues, we create text likelihood maps, which indicates the likelihood of detection of each room number as a function of the robot position. We inject particles corresponding to the locations suggested by the text likelihood. A. Monte Carlo Localization Monte Carlo localization [11] is a probabilistic method for estimating a robot's state x t given a map m and sensor readings z t at time t. As we operate in an indoor environment, the robot's state x t is given by the 2D coordinates (x, y) and the orientation θ ∈ [0, 2π). In our case an observation z is composed of K beams z k and the map m is represented by an occupancy grid map [25]. We use a particle filter to represent the belief about the robot's state p(x t \| z 1:t , m), where each particle s Fig. 2: The text likelihood maps, based on the collected data, indicate the locations in which detection of each room number is likely. The likelihood maps are used for particle injection when a detection of a known text cues occurs. weight w (i) t . When odometry is available, successive states are sampled from a proposal distribution represented by holonomic motion model with odometry noise σ odom ∈ R 3 . For each observation, each particle is weighted according to the likelihood of the observation given its state, i.e., w (i) t = p(z t \| x (i) t , m). As observation model p(z t \| x t , m), we use a beam-end model [32]. The product of likelihood model assumes scan points are independent of each other. With the high angular resolution of our LiDAR this assumption does not hold. To address the overconfidence problem of the product of likelihood model, we decided to use the product of experts model [23], where the weight of each particle is computed as the geometric mean of all scan points p(z t \| x t , m) = K k=0 p(z k t \| x t , m) 1 K ,(1) where p(z k t \| x t , m) = 1 √ 2πσ obs exp − EDT (ẑ k t ) 2 2σ 2 .(2) In Eq. (2),ẑ k t is the end point of the beam in the map m, and EDT is the Euclidean distance transform [12] that indicates the distance to an occupied cell in the occupancy map. We truncated the EDT at a predefined maximal range, r max . For resampling, we chose low-variance resampling [32] with an efficient sample size criteria [1] [4] of N/2, where N is the number of particles. Furthermore, our implementation of MCL [11] is asynchronous -we sample from the motion model every time we get an odometry input, and we compute the weights whenever an observation is available and the robot traveled a predefined minimum distance (d xy , d θ ). B. Text Spotting Text spotting can traditionally be split into text detection, i.e., localizing a bounding box that includes text, and text recognition, i.e., decoding the image patches extracted from the bounding boxes, to text. Text recognition is essentially a classification problem, therefore only the characters that are introduced during training can be inferred. The last decade's progress in object detection and text recognition allows us to use deep learning models for text spotting. For the text spotting, we used the differentiable binarization text detector proposed by Liao et al. [21]. The backbone is a ResNet18 [15] neural network, which is powerful but also efficient enough to allow for fast inference. The text recognition model is based on the work of Shi et al. [29], who proposed the CRNN architecture, that combines convolution, recurrent and transcription layers. This model can handle text of arbitrary length, is end-to-end trainable without requiring fine-tuning and is relatively small while maintaining accuracy. We use four cameras, with a coverage of 360 • , to spot text. C. Text Likelihood Maps To incorporate text spotting into the MCL framework, we build a likelihood function of where the robot might detect a specific room number by collecting data that included image streams and the robot's pose. We apply the text spotting pipeline on the recorded images, assuming that the textual cues we are interested in follow a specific pattern ("Room X") but it can generally be used for any textual content of interest. We compute 2D histograms for each room number, of locations where successful detections were made. The sampled locations give a sparse description of the text spotting likelihood, which we refer to as text likelihood maps (Fig. 2). As we are interested in a dense representation for the likelihood, we chose a simple strategy -for each text tag, we compute an axis-aligned bounding box around all sampled locations where the detection rate is above threshold τ for this textual cue. We approximate the likelihood of text detection with an uniform distribution within the bounding box. D. Integration of Textual Cues When a room number is detected, we store the room number and from which camera it was observed. Upon first detection, we inject particles into the corresponding area of the map (Fig. 3). If the last detection was made from the same camera and of the same room number, we do not inject particles. The number of particles injected is defined by the injection ratio, ρ, the number of injected particles divided by the total number of particles. In the injection process, for a particle filter with N particles, we first remove ρN particles with the lowest weights, and then inject an equal number of particles uniformly into the bounding box corresponding to that room number. The orientation o i of the injected particle s (i) t depends on which camera spotted the text. We assumed that the camera detecting the text facing the room number at perpendicular angle. Thus, we inject particles with corresponding orientation and add Gaussian noise, σ inject = 0.05. The injection ratio ρ was chosen to be 0.5. A very high injection ratio could lead to localization failure if a wrong room number is detected. A low injection ratio has limited impact on the pose estimation. The injection of particles is done asynchronously, whenever a textual cue is available, and new particles are initialized with weight w (i) = 1 N . IV. EXPERIMENTAL EVALUATION The main focus of this work is an efficient, robust localization algorithm that leverage text information to better handle significant changes in the environment. We present our experiments to show the capabilities of our method and to support our claims, that we can (i) localize in quasistatic environments, (ii) localize in an environment with low dynamics, (iii) localize in different maps types -a featureless floor plan-like map, and LiDAR-based, feature rich map. These capabilities can be run online on our robot. A. Experimental Setup To benchmark the performance of our approach, we recorded a dataset in an indoor office environment. To this end, we equipped a Kuka YouBot platform with 2 Hokuyo UTM-30LX LiDAR sensors, 4 sideways-looking Intel RealSense RGB-D (D455), and an upward-looking GoPro Hero5 Black that is used only for evaluation purposes, Fig. 4. We recorded the data including wheel odometry for different scenarios. We recorded different scenarios. A long recording was made in the corridor with all doors closed, and sequences S1-S10 are randomly sampled from that data. Similarly, the sequences starting with D are sampled from recordings D1-D4, where doors were open but contain fast-moving dynamics. We include a plot of the trajectory of the scenarios in Fig. 5. To determine the ground truth pose of the robot, we use precisely localized AprilTags [26], densely placed on the ceiling of every room and corridor. The AprilTags were detected using an up-facing camera that is used solely for this purpose. The AprilTags allow to continuously and accurately localize the robot with the dedicated sensor even under dynamic changes. We explore two map representations, a floor plan-like map and a 2D LiDAR-based occupancy grid produced by GMapping [14], both illustrated in Fig. 6. The sparse, floor plan-like map was extracted from a high resolution terrestrial FARO laser scan, by slicing the dense point cloud at a fixed height. For all experiments, we use a map resolution of 0.05 m/cell and the parameters specified in Tab. I. As baseline, we compare against AMCL [27], which is a publicly available and highly-used ROS package for MCLbased localization, and our implementation of MCL that does not rely on textual cues. Additionally, we implemented two sensor models for integrating textual information into the MCL framework, referred to as SM1 and SM2. SM1 assigns all particles within the bounding box a high weight, w k t = 1.0, and a low weight, w k t = 0.1, to particles elsewhere. SM2 converts the bounding box into a likelihood map (a) Map constructed by horizontally slicing a 3D point cloud captured with a FARO Focus X130 terrestrial laser scanner. (b) Occupancy grid map from GMapping [14] that was aligned to the FARO scan. (c) Occupancy grid map from GMapping [14] that was aligned to the FARO scan, based on the recordings from the corridor scenario. and the weight for each particle is proportional to a Gaussian applied on its distance from the bounding box, similar to Eq. (2). All experiments were executed with 300 particles unless mentioned otherwise, and N particles are initialized uniformly across the map. We consider two metrics, time to convergence and absolute trajectory error (ATE) after convergence. We define convergence as the point where the prediction is within a distance of 0.5 m from the ground truth pose. If convergence did not occur within the first 95% of the sequence, then we consider it a failure, which is marked as −/−. B. Localization under Changes using a Sparse Map The first experiment evaluates the performance of our approach and supports the claim that we can localize in changing environment using floor plan-like maps. It is conducted on sequences recorded in a long corridor with all doors closed, while in the map these doors are all open, and it supports our claim of robust localization in face of quasi-static changes. We consider 10 sequences (S1-S10), each sequence starting at the a different location along the corridor. We evaluate the time to convergence and ATE for this challenging scenario on the 10 sequences. As can be seen in Tab. II and Tab. IV, our text-enriched method converges quickly, and outperformed the baselines in all sequences. When the map not longer reflect the environment, it is expected that classic MCL implementations would perform poorly. For text spotting sensor model to affect the pose estimation, a particle must be in the close vicinity of a specific text likelihood bounding box. With relative low number of particles, such as 300, it is unlikely to have enough particles in such a small area. Therefore, the sensor model methods have limited contribution to global localization compared to particle injection. The MCL+Text method also shows exceptional robustness when reducing the number of particles in the filter, as can be seen in Fig. 7. The ATE for our approach is slightly larger for 10,000 particles, due to the formation of multi-modal hypotheses caused by the symmetry of the corridor. C. Localization under Few Dynamics using a Sparse Map The second experiment is presented to support the claim that our approach is able to localize in a floor plan-like map (not built using the robot's sensors) when the environment is mostly static. Recordings D1-D4 are taken across the lab, through different office rooms, with a small number of people moving around. In all sequences, all doors are open, and the environment is similar to the map. Fig. 7: ATE (xy) averaged over sequences S1-S10 as a function of the number of particles used in the particle filter, for the different methods method. The error for MCL+Text is similar across large range of particle set sizes, exhibiting the robustness of our approach. This experiment considers localization in a feature-sparse map and in the presence of low dynamics. This presents its own challenges even in a mostly unchanging environment. As seen in Tab. III and Tab. V our text-enriched method performs best. Despite having the doors open, these scenarios include movement in a corridor with very high symmetry. Textual cues can contribute to breaking such symmetries. In addition, there are many details such as furniture, that are not part of the sparse map and can affect the accuracy of LiDAR-only localization. While SM2 shows a rather promising convergence time, the impact of the text-based sensor model is milder than particle injection, leading to divergence later on, and a large ATE. D. Localization using LiDAR-Based Map Built with the Robot's Sensors The third experiment is presented to support the claim that our approach is able to localize in a LiDAR-based map, when the environment is structurally changing or when there are a few dynamics in the scene. To ensure our algorithm works sufficiently well in LiDAR-based maps, we constructed a GMapping map based on 2D LiDAR scans. While this map is more detailed, the recordings were made across several weeks, resulting in some differences between the map and the environment. It is still difficult to localize globally with only 300 particles in a big scene, therefore our text-guided method enjoys an advantage. Our approach outperformed the baselines also in corridor scenario, as can be seen in Tab. VI. While the sensor model methods manage to converge in a timely manner (Tab. VIII), they are less stable than our injection technique and result in greater ATE. Similarly, for the mostly static scenario, our approach achieves the best ATE overall (Tab. VII), in addition to its fast convergence, displayed in Tab. IX. E. Runtime The next set of experiments has been conducted to support our fourth claim that our approach runs fast enough to execute online on the robot in real-time. We, therefore, tested our approach once using a Dell Precision-3640-Tower and once on an Intel NUC10i7FNK, which we have on our YouBot. The Dell PC has 20 CPU cores at 3.70 GHz and 64 GB of RAM. The Intel NUC has 12 CPU cores at 1.10 GHz and 16 GB of RAM. Text spotting on the NUC runs at an average of 167 ms, and on the desktop 100 ms. Tab. X summarizes the runtime results for our approach. The numbers support our fourth claim, namely that the computations can be executed fast and in an online fashion. F. Ablation Study Additionally, we conducted an ablation study to identify the best way of integrating the textual hints into our MCL framework. In addition to our MCL+Text method, we also explored the following strategies for injecting particles: 1) Seed locations: Specific hand-picked locations in the map, which correspond to room number locations, and are used to sample particles around them with a predefined covariance. 2) Repeat: Using the text likelihood maps, described in Sec. III-C, we compute a bounding box for each room number plate, and inject particles in that area for every room number detection. If we have multiple consecutive detections of a room number from the same camera, we inject particles each time. 3) Conservative: Using the text likelihood maps, we compute a bounding box for each room number plate, and inject particles in that area only once, if the filter's pose estimation mean does not lie in the bounding box. If the mean pose of MCL is within the bounding box, the filter is in line with the tag observations, and we do not inject particles. If we have multiple consecutive detections of a room number from the same camera, we inject particles only in the first detection. As can be seen in Fig. 8, MCL+Text outperforms the other text-guided methods. MCL+Text also converges faster than the other text-guided methods. V. CONCLUSION In this paper, we presented a novel approach to localize a robot in environments that deviate significantly from the provided map, as illustrated in Fig. 6, due to changes in the scene. Our method exploits the readily available humanreadable textual cues that assist humans in navigation. This allows us to successfully overcome localization failure in the cases where critical changes to the layout differ greatly from the map. We implemented and evaluated our approach on a dataset collected strictly for simulating such structural alterations, and provided comparisons to other existing techniques and supported all claims made in this paper. The experiments suggest that incorporating human-readable localization cues (a) ATE (xy) averaged over sequences S1-S10. (b) Convergence time for sequences S1-S10. Fig. 8: Results for the ablation study exploring different injection strategies, with the sparse map and 300 particles. authors are with the University of Bonn, Germany. Cyrill Stachniss is also with the Lamarr Institute for Machine Learning and Artificial Intelligence, Germany. This work has partially been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy, EXC-2070 -390732324 -PhenoRob and by the European Union's Horizon 2020 research and innovation programme under grant agreement No 101017008 (Harmony). 1 https://github.com/PRBonn/tmcl ground truth without text with text Fig. 1 : 1Top Left: The corridor in which the experiment took place in. Top right: Fig. 3 : 3Particle injection with text spotting. (a) Before detection, we have a situation with multi-modal distribution of particles (shown in red) as the corridor with closed doors is a symmetric situation that cannot be resolved just using the LiDAR scans. (b) With the first text detection (indicated by the green cross), we can inject new particles inside the bounding box extracted from the text map. We replace low weighted particles by new particles (shown in blue) that are uniformly distributed inside the corresponding bounding box of the text detection (shown by a dashed green line). Fig. 4 : 4The data collection platform, an omnidirectional Kuka YouBot, with 2D LiDAR scanners (marked by a red outline) and with 4 cameras (marked by a blue outline) providing 360 • coverage. The up-ward facing camera (marked by a green outline) is only used for generating the ground truth via AprilTag detections. Fig. 5 : 5Visualization of the different sequences used for evaluating our approach. Sequences S1-S10 correspond to the scenario where all doors are closed. Sequences D1-D4 were recorded with all the doors open, and with moderate amount of humans moving around. The color of the trajectory correspond to the time, where purple is the beginning and red corresponds to the end of the sequence. Fig. 6 : 6Different maps used in the experiments: (a) floor plan-like maps and (b) LiDAR-based maps. (c) map built using GMapping, based on the recordings from the corridor scenario, which significantly deviates from the maps provided for localization. TABLE I : IAlgorithm parametersσ odom σ obs rmax τ ρ dxy d θ TABLE II : IIATE after convergence, for each sequence for the corridor scenario, using the sparse map with 300 particles. Angular error in radians / translational error in meters.Method S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 AMCL -/- -/- -/- -/- 0.01/0.110 -/- -/- -/- 2.426/10.468 -/- MCL 2.241/9.592 -/- 0.287/0.594 0.045/0.581 -/- 1.795/7.803 0.625/2.036 2.465/11.550 1.859/9.262 1.011/1.640 SM1 0.405/2.329 -/- 0.01/0.537 0.045/0.563 -/- 1.795/7.803 0.625/2.022 2.465/11.550 1.855/9.393 1.011/1.501 SM2 1.116/1.795 0.777/2.597 1.227/3.214 0.706/2.611 0.861/1.704 1.600/3.782 0.118/5.644 0.321/0.563 1.388/2.276 1.148/1.466 MCL+Text 0.063/0.250 0.01/0.245 0.063/0.246 0.063/0.266 0.01/0.246 0.179/0.369 0.045/0.221 0.077/0.343 0.493/0.332 0.01/0.184 TABLE III : IIIErrors averaged over the trajectory, after convergence, for the mostly static environment scenario, using the sparse map with 300 particles. Angular error in radians / translational error in meters.Method D1.1 D1.2 D1.3 D1.4 D2.1 D3.1 D3.2 D3.3 D4.1 D4.2 AMCL 2.022/6.031 0.063/0.135 -/- -/- 0.010/0.095 -/- -/- -/- -/- -/- MCL -/- 0.413/0.690 1.253/2.708 0.893/1.961 -/- 1.439/3.298 2.284/4.743 2.090/3.945 -/- 0.703/1.021 SM1 -/- 1.970/3.637 1.255/2.715 0.893/1.961 -/- 1.537/4.274 2.284/4.743 2.090/3.945 -/- 1.007/6.035 SM2 1.315/3.942 2.341/5.634 1.346/4.275 1.358/2.319 -/- -/- 1.628/3.336 1.357/2.836 1.524/5.708 1.505/5.609 MCL+Text 0.045/0.158 0.045/0.175 0.077/0.182 0.010/0.152 0.045/0.279 0.010/0.133 0.333/0.697 0.010/0.141 0.045/0.161 0.063/0.197 TABLE IV : IVConvergence time in seconds, for the corridor scenario, using the sparse map, with 300 particles. In parentheses, the length of the sequences in seconds.Method S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 (234.4) (229.6) (220.8) (212.1) (203.2) (187.7) (176.3) (152.1) (145.4) (123.0) AMCL - - - - 54.5 - - - 19.6 - MCL 99.3 - 18.6 0.0 - 75.9 148.8 121.5 11.5 3.2 SM1 15.7 221.9 35.2 0.0 - 75.9 148.8 121.5 11.5 3.2 SM2 0.0 1.5 4.7 0.0 126.5 56.5 135.4 5.6 11.5 3.3 MCL+Text 0.0 1.7 12.6 0.0 10.9 11.2 50.2 5.7 11.5 2.4 TABLE V : VConvergence time in seconds, for the mostly static environment scenario, using the sparse map, with 300 particles. In parentheses, the length of the sequences in seconds.Method D1.1 D1.2 D1.3 D1.4 D2.1 D3.1 D3.2 D3.3 D4.1 D4.2 (171.4) (162.4) (144.8) (130.5) (78.2) (177.7) (160.6) (147.7) (120.0) (100.4) AMCL 112.4 8.7 - - 10.6 - - - - - MCL 167.6 9.6 80.9 53.6 - 69.2 55.2 17.7 - 7.4 SM1 167.6 136.7 80.9 53.6 - 70.6 55.2 17.7 - 1.8 SM2 2.1 106.4 0.2 18.4 - - 55.2 22.9 20.4 0.0 MCL+Text 2.2 2.0 0.7 16.3 4.8 64.6 55.9 14.9 36.4 0.0 300 1000 10000 Number of particles 0 2 4 6 8 10 ATE (m) AMCL MCL SM1 SM2 MCL+text TABLE VI : VIErrors averaged over the trajectory, after convergence, for each sequence for the corridor scenario, using the GMapping map with 300 particles. Angular error in radians / translational error in meters.Method S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 AMCL -/- -/- 0.010/0.087 -/- -/- -/- -/- 2.436/9.151 2.420/10.756 -/- MCL -/- 2.399/9.822 0.941/6.763 1.352/7.113 0.010/0.242 0.010/0.547 2.176/7.869 -/- 2.600/9.385 1.280/5.901 SM1 0.010/0.294 0.601/3.727 0.941/6.781 1.407/7.486 2.480/11.063 0.010/0.525 2.176/7.869 -/- 2.600/9.385 1.918/7.056 SM2 1.347/4.181 1.344/4.644 2.014/5.334 1.618/4.257 1.002/3.598 1.566/4.100 0.499/1.915 0.476/1.936 2.312/8.422 1.414/4.625 MCL+Text 0.063/0.191 0.063/0.203 0.063/0.216 0.063/0.228 0.063/0.171 0.045/0.293 0.632/1.788 0.010/0.192 0.697/0.920 0.205/0.196 TABLE VII : VIIErrors averaged over the trajectory, after convergence, for mostly static scenarios, using the GMapping map with 300 particles. Angular error in radians / translational error in meters.Method D1.1 D1.2 D1.3 D1.4 D2.1 D3.1 D3.2 D3.3 D4.1 D4.2 AMCL 0.547/3.490 -/- -/- -/- -/- 2.389/17.461 -/- -/- -/- -/- MCL 1.714/2.794 1.895/3.982 0.215/1.230 0.495/1.144 0.262/1.864 1.424/3.427 0.885/10.403 0.812/3.825 0.991/1.190 0.632/9.345 SM1 0.425/3.683 1.895/3.982 0.215/1.230 0.495/1.145 0.265/1.828 1.422/3.453 0.045/0.225 1.599/5.664 0.991/1.190 1.335/7.131 SM2 0.778/1.843 1.881/4.875 1.169/2.799 1.083/1.772 0.118/0.347 1.497/5.087 1.593/2.155 1.628/5.366 1.404/6.117 0.704/1.477 MCL+Text 0.010/0.109 0.010/0.109 0.077/0.099 0.010/0.116 0.045/0.161 0.010/0.156 0.010/0.169 0.045/0.165 0.063/0.160 0.045/0.131 TABLE VIII : VIIIConvergence time in seconds, for the corridor scenario, using the GMapping map, with 300 particles. In parentheses, the length of the sequences in seconds.Method S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 (234.4) (229.6) (220.8) (212.1) (203.2) (187.7) (176.3) (152.1) (145.4) (123.0) AMCL - - 113.0 - - - - 22.8 19.7 - MCL - 29.5 21.8 0.2 159.1 122.8 7.9 - 8.9 1.1 SM1 18.1 9.7 21.8 0.2 173.2 122.2 7.9 - 8.9 1.1 SM2 6.5 1.7 3.2 0.2 114.1 10.9 7.9 47.8 8.9 1.1 MCL+Text 0.0 1.5 3.6 0.2 8.4 9.7 7.9 31.2 8.9 1.1 TABLE IX : IXConvergence time in seconds, for the mostly static scenarios, using the GMapping map, with 300 particles. In parentheses, the length of the sequences in seconds.Method D1.1 D1.2 D1.3 D1.4 D2.1 D3.1 D3.2 D3.3 D4.1 D4.2in mobile robot localization systems provides considerable improvement in robustness.(171.4) (162.4) (144.8) (130.5) (78.2) (177.7) (160.6) (147.7) (120.0) (100.4) AMCL 0.0 - - - - 35.2 - - - - MCL 111.2 10.1 30.7 50.9 21.5 67.8 133.4 31.0 51.6 64.2 SM1 115.7 10.1 30.7 50.9 21.5 67.8 87.2 39.2 51.6 8.3 SM2 2.2 73.9 30.2 16.2 7.1 72.0 50.3 15.6 26.5 0.5 MCL+Text 2.1 1.8 0.2 16.0 4.7 71.5 50.6 19.3 21.4 0.0 TABLE X : XAverage inference time in ms for the sensor model on the NUC as a function of the number of particles.300 500 1 000 10 000 NUC10i7FNK 30 51 106 1027 Dell Precision-3640-Tower 24 40 80 793 seeds repeat conservative MCL+text Injection method 0.0 0.2 0.4 0.6 0.8 ATE (m) seeds repeat conservative MCL+text (i) t = x (i) t , w (i) t is represented by a state x (i) tand a ACKNOWLEDGMENTSWe thank Holger Milz, Michael Plech, and Ralf Becker for their contribution in assembling our mobile platform. 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[ "Correlators of the phase model", "Correlators of the phase model" ]	[ "N M Bogoliubov [email protected] ", "A G Izergin [email protected] \nSupported in part by MAE-MICECO-CNRS Fellowship § On leave of absence from St. Petersburg Department of the Steklov Mathematical Insti-tute\nFontanka 27, St. Petersburg 191011, Fontanka 27, St. Petersburg 191011RUSSIA, RUSSIA\n", "N A Kitanine [email protected] ", "\nResearch Institute for Theoretical Physics\nLaboratoire de Physique Théorique enslapp * * ENSLyon\nUniversity of Helsinki\n46 Allée d'ItalieP.O.Box 9FIN-00014, 69007LyonFINLAND., FRANCE\n", "\nAcademy of Finland † On leave of absence from St. Petersburg Department of the Steklov Mathematical Insti-tute\nFontanka 27, St. Petersburg 191011RUSSIA\n" ]	[ "Supported in part by MAE-MICECO-CNRS Fellowship § On leave of absence from St. Petersburg Department of the Steklov Mathematical Insti-tute\nFontanka 27, St. Petersburg 191011, Fontanka 27, St. Petersburg 191011RUSSIA, RUSSIA", "Research Institute for Theoretical Physics\nLaboratoire de Physique Théorique enslapp * * ENSLyon\nUniversity of Helsinki\n46 Allée d'ItalieP.O.Box 9FIN-00014, 69007LyonFINLAND., FRANCE", "Academy of Finland † On leave of absence from St. Petersburg Department of the Steklov Mathematical Insti-tute\nFontanka 27, St. Petersburg 191011RUSSIA" ]	[]	We introduce the phase model on a lattice and solve it using the algebraic Bethe ansatz. Time-dependent temperature correlation functions of phase operators and the "darkness formation probability" are calculated in the thermodynamical limit. These results can be used to construct integrable equations for the correlation functions and to calculate their asymptotics.	10.1016/s0375-9601(97)00326-5	[ "https://arxiv.org/pdf/solv-int/9612002v2.pdf" ]	1,035,448	solv-int/9612002	cbd21e88990c6009afb910075ba39bc162c6e5d4	Correlators of the phase model arXiv:solv-int/9612002v2 20 Dec 1996 October 1996 N M Bogoliubov [email protected] A G Izergin [email protected] Supported in part by MAE-MICECO-CNRS Fellowship § On leave of absence from St. Petersburg Department of the Steklov Mathematical Insti-tute Fontanka 27, St. Petersburg 191011, Fontanka 27, St. Petersburg 191011RUSSIA, RUSSIA N A Kitanine [email protected] Research Institute for Theoretical Physics Laboratoire de Physique Théorique enslapp * * ENSLyon University of Helsinki 46 Allée d'ItalieP.O.Box 9FIN-00014, 69007LyonFINLAND., FRANCE Academy of Finland † On leave of absence from St. Petersburg Department of the Steklov Mathematical Insti-tute Fontanka 27, St. Petersburg 191011RUSSIA Correlators of the phase model arXiv:solv-int/9612002v2 20 Dec 1996 October 1996* Supported by the ¶ Supported in part by project MAE 96/9804 On leave of absence from St. Petersburg Department of the Steklov Mathematical In-stitute, * * URA 14-36 du CNRS, associéeà l'E.N.S. de Lyon, età l'Universitè de Savoie 0 We introduce the phase model on a lattice and solve it using the algebraic Bethe ansatz. Time-dependent temperature correlation functions of phase operators and the "darkness formation probability" are calculated in the thermodynamical limit. These results can be used to construct integrable equations for the correlation functions and to calculate their asymptotics. Phase operators were intensively studied in quantum optics [1,2,3]. They can be defined by the following commutation relations [N, φ + ] = φ + , [N, φ] = −φ, [φ, φ + ] = π,(1) where π is the vacuum projector π = (\|0 0\|) and N is the number of particles operator. The phase model is a model of interacting phase operators on a lattice. It was constructed in [8] as a limit case of the q-boson hopping model [5,6] . The Hamiltonian of the phase model has the following form H = − 1 2 M n=1 (φ + n φ n+1 + φ n φ + n+1 − 2N n ),(2) where operators φ j , φ + j , N j commute in the different sites and satisfy the commutation relations (1) in the same site. The complete set of eigenvectors for the model can be obtained by means of the algebraic Bethe ansatz (see [4] and references therein). L-operator of the model has the form L n (p) = e ip/2 φ † n φ n e −ip/2 ,(3) This operator satisfies the bilinear relation R(p, s)L n (p) ⊗ L n (s) = L n (s) ⊗ L n (p)R(p, s), in which R(p, q) is the 4×4 matrix R-matrix. The non-zero elements of this R-matrix are R 11 (p, s) = R 44 (p, s) = f (s, p), R 22 (p, s) = R 33 (p, s) = g(s, p), R 23 (p, s) = 1 and f (p, s) = i e i p−s 2 2 sin( s−p 2 ) ; g(p, s) = i 2 sin( s−p 2 ) .(4) The Bethe equations for the model exp{i(M + N )p j } = (−1) N −1 exp{i N k=1 p k }(5) (j = 1, ..., N ) are exactly solvable: p j = 2πI j + N k=1 p k M + N ,(6) where I j are integers or half-integers depending on N being odd or even. The N -particle eigenenergies of the Hamiltonian H µ = H −μN (2) are E N = N k=1 (h(p k ) −μ); h(p) = 2 sin 2 (p/2).(7) Hereμ is the chemical potential, 0 ≤μ ≤ 1. The thermodynamics of the model we shall consider for the case when the total momentum P = N k=1 p k is zero P = 0. The thermodynamics of the model is handled in the standard way. The ground state energy of the model at finite temperatures β −1 is determined through the solution of the nonlinear integral equations ǫ(p) = h(p) −μ − (2πβ) −1 π −π ln(1 + e βǫ(p) )dp,(8)2πρ(p)(1 + e βǫ(p) ) = 1 + π −π ρ(p)dp. The function ρ(p) is a quasi-particle density while ǫ(p) is the excitation energy. The pressure is then P = (2πβ) −1 π −π ln(1 + e βǫ(p) )dp (9) and the density is D = ∂P ∂μ = π −π ρ(p)dp.(10) So we have ǫ(p) = h(p) −μ − P(11) and the quasi-particle density has the Fermi-like distribution 2πρ(p) = (1 + D)(1 + e βǫ(p) ) −1 .(12) At zero temperature β −1 = 0 the ground state is the Fermi sphere −Λ ≤ p ≤ Λ (Λ ≤ π) filled by the particles with the negative energies ǫ 0 (p). The pressure and density are now P 0 = −(2π) −1 Λ −Λ ǫ 0 (p)dp, D 0 = Λ −Λ ρ 0 (p)dp. From (11) and (12) we have ǫ 0 (p) = h(p) −μ − P 0 , ǫ 0 (±Λ) = 0; 2πρ 0 (p) = (1 + D 0 ). From the last equation we can express the bare Fermi momentum Λ as the function of density Λ = πD 0 1 + D 0 . The Fermi velocity v is equal: v = ǫ ′ 0 (Λ) 2πρ 0 (Λ) = (1 + D 0 ) −1 sin πD 0 1 + D 0 . When Λ → 0 (μ → 0), D 0 → 0 and P 0 → 0 as must be expected. When Λ → π (μ → 1) all the vacancies are now occupied by particles D 0 → ∞, P 0 → 1 and the model (2) is the classical XY chain in this limit [6], [7]. The correlation functions for the phase model can be represented as Fredholm determinants of integral operators. The similar representations were obtained recently for the model of impenetrable bosons [9,10] and for the XX0 Heisenberg chain [11]. It was shown that such results can be used for calculation of the asymptotics of correlation functions. The simplest correlation function is the emptiness formation probability. It can be defined as a probability of the states such that there are no particles in the first m sites of the lattice (see e.g. [4]). We will call it darkness formation probability in the case of the phase model. Using the algebraic Bethe ansatz and the representation for the scalar products of the Bethe states [12] one can represent this function as a Fredholm determinant τ (m, β) = (1 + D)det(Î −M ),(13) whereÎ is the identity operator andM is an integral operator (M f )(p) = π −π M (p, q)f (q)dq, with the kernel M (p, q) = 1 2π ν(p, β) sin m+1 2 (p − q) sin 1 2 (p − q) ν(q, β),(14) where ν(p, β) = (1 + exp(βǫ(p))) −1 . It can be easily shown using the representation (13) that τ (m, β) ≤ 1. At zero temperature one has τ 0 (m) = (1 + D 0 )det(Î −M 0 ),(15) whereM 0 is an integral operator (M 0 f )(p) = Λ −Λ M 0 (p, q)f (q)dq,(16) with the kernel M 0 (p, q) = 1 2π sin m+1 2 (p − q) sin 1 2 (p − q) . (17) The time-dependent correlation function of phase operators can be also expressed as a Fredholm determinant. This representation can be obtained using the representation for the form-factor [12]. We will consider the temperature mean values f + (β, m, t) = φ m+1 (t)φ + 1 (0) β ,(18)f − (β, m, t) = φ + m+1 (t)φ 1 (0) β ,(19) where φ m (t) = exp[iH µ t]φ m exp[−iH µ t]. This correlation function can be represented in the following form f (±) (m, t, β) = exp D 1 + D 1 + D 2π ∞ l=0 π −π e i(m−l)Θ h (±) (l, t, β, Θ)dΘ. (20) The functions h (±) (l, t, β, Θ) can be written as a Fredholm determinant h (±) (l, t, β, Θ) = G(l, t) + ∂ ∂x det(Î +V ∓ xR ± )\| x=0 ,(21) whereV andR are integral operators (V f )(p) = 1 2π π −π V (p, q)f (q)dq, (R ± f )(p) = 1 2π π −π R ± (p, q)f (q)dq,(22) with kernels V (p, q) = e −i p−q 2 sin 1 2 (p − q) ( 1 2 (E + + (l, t, p, β, Θ) + E − + (l, t, p, β, Θ))E − (l, t, q, β)− − 1 2 (E + + (l, t, q, β, Θ) + E − + (l, t, q, β, Θ))E − (l, t, p, β)),(23)R + (p, q) = E + + (l, t, p, β, Θ)E − + (l, t, q, β, Θ),(24)R − (p, q) = E − (l, t, p, β)E − (l, t, q, β),(25) where the functions G(l, t), E − (l, t, p, β), E + + (l, t, p, β, Θ) and E − + (l, t, p, β, Θ) are defined as G(l, t) = 1 2π π −π exp(ilq − itǫ(q))dq,(26)E(l, t, p, Θ) = 1 2π v.p. π −π exp(ilq − itǫ(q)) tan 1 2 (q − p) dq + tan 1 2 (p − Θ) exp(ilp − itǫ(p)),(27)E − (l, t, p, β) = ν(p, β) exp −i l 2 p + i t 2 ǫ(p) ,(28)E + + (l, t, p, β, Θ) = 1 2 E − (l, t, p, β)× × (E(l, t, p, Θ) + e −iΘ E(l + 1, t, p, Θ)) + i(G(l, t) + e −iΘ G(l + 1, t)) ,(29)E − + (l, t, p, β, Θ) = 1 2 E − (l, t, p, β)× × (E(l, t, p, Θ) + e iΘ E(l − 1, t, p, Θ)) − i(G(l, t) + e iΘ G(l − 1, t)) .(30) Although the kernel (23) seems to be very complicated it has the form which allows to use the method proposed in [10] for calculation of the asymptotics. In the case t = 0, m > 1 the correlators have the following form f (±) (m, 0, β) = exp D 1 + D 1 + D 2π ∞ l=m π −π e i(m−l)Θ h (±) (l, 0, β, Θ)dΘ, (31) h (±) (l, 0, β, Θ) = ∂ ∂x det(Î −v + xr ± )\| x=0 ,(32) where the integral operatorsv andr ± possess the kernels v(p, q) = ν(p) sin l+1 2 (p − q) + exp[i(Θ − p+q 2 )] sin l−2 2 (p − q) sin 1 2 (p − q) ν(q), (33) r + (p, q) = ν(p)e i(Θ−q) e i l 2 (p+q) ν(q),(34)r − (p, q) = ν(p)e −i l 2 (p+q) ν(q).(35) Comparing the results obtained in this paper with the representations for the correlation functions of the other models ( [9,10,11]) one can see that these representations are rather different and there are new difficulties in our formulae. The models considered before were the free fermion points of the more general integrable models. The phase model being a limiting case of the q-boson hopping model is not a free fermion point of this model and this can explain the new difficulties appeared in the correlation functions. 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[ "Study of Molecular Clouds associated with H II Regions", "Study of Molecular Clouds associated with H II Regions" ]	[ "Mohaddesseh Azimlu \nDepartment of Physics & Astronomy\nUniversity of Western Ontario\nN6A 3K7LondonONCanada\n\nDepartment of Physics & Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooONCanada\n", "Michel Fich \nDepartment of Physics & Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooONCanada\n" ]	[ "Department of Physics & Astronomy\nUniversity of Western Ontario\nN6A 3K7LondonONCanada", "Department of Physics & Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooONCanada", "Department of Physics & Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooONCanada" ]	[]	The properties of molecular clouds associated with 10 H II regions were studied using CO observations. We identified 142 dense clumps within our sample and found that our sources are divided into two categories: those with clumps that show a power law size-line width relation (Type I) and those which do not show any relation (Type II). The clumps in the Type I sources have larger power law indices than found in previous studies. The clumps in the Type II sources have larger line widths than do the clumps in the Type I sources. The mass M LT E increases with ∆V for both 12 CO and 13 CO lines in Type I sources; No relation was found for Type II sources. Type II sources show evidence (such as outflows) of current active star formation within the clump and we suggest that the lack of a size-line width relation is a sign of current active star formation.For both types of sources no relation was found between volume density and size, but overall larger clumps have smaller volume density, indicating that smaller clumps are more evolved and have contracted to smaller sizes and higher densities. Massive clumps seem to have similar masses calculated by different methods but lower mass clumps have larger virial masses compared to velocity integrated (X factor) and LTE mass. We found no relation between mass distribution of the clumps and distance from the H II region ionization front, but a weak decrease of the excitation temperature with increasing distance from the ionized gas. The clumps in collected shells around the H II regions have slightly larger line widths but no relation was found between line width and distance from the H II region, which probably indicates that the internal dynamics of the clumps are not affected by the ionized gas. Internal sources of turbulence, such as outflows and stellar winds from young proto-stars may have a more important role on the molecular gas dynamics. We suggest that large line width and larger size-line width power law indices are therefore the initial characteristics of clumps in massive star forming clouds (e.g. our Type I sources) and that some may evolve into objects similar to our Type II sources, where local "second generation" stars are forming and eliminating the size-line-width relation.	10.1088/0004-6256/141/4/123	[ "https://arxiv.org/pdf/1102.0287v1.pdf" ]	119,295,686	1102.0287	4d69fdfebfb62cb428639e892e3b9bd559beb339	Study of Molecular Clouds associated with H II Regions 1 Feb 2011 Mohaddesseh Azimlu Department of Physics & Astronomy University of Western Ontario N6A 3K7LondonONCanada Department of Physics & Astronomy University of Waterloo N2L 3G1WaterlooONCanada Michel Fich Department of Physics & Astronomy University of Waterloo N2L 3G1WaterlooONCanada Study of Molecular Clouds associated with H II Regions 1 Feb 2011Subject headings: Star Formation: general -H II Regions The properties of molecular clouds associated with 10 H II regions were studied using CO observations. We identified 142 dense clumps within our sample and found that our sources are divided into two categories: those with clumps that show a power law size-line width relation (Type I) and those which do not show any relation (Type II). The clumps in the Type I sources have larger power law indices than found in previous studies. The clumps in the Type II sources have larger line widths than do the clumps in the Type I sources. The mass M LT E increases with ∆V for both 12 CO and 13 CO lines in Type I sources; No relation was found for Type II sources. Type II sources show evidence (such as outflows) of current active star formation within the clump and we suggest that the lack of a size-line width relation is a sign of current active star formation.For both types of sources no relation was found between volume density and size, but overall larger clumps have smaller volume density, indicating that smaller clumps are more evolved and have contracted to smaller sizes and higher densities. Massive clumps seem to have similar masses calculated by different methods but lower mass clumps have larger virial masses compared to velocity integrated (X factor) and LTE mass. We found no relation between mass distribution of the clumps and distance from the H II region ionization front, but a weak decrease of the excitation temperature with increasing distance from the ionized gas. The clumps in collected shells around the H II regions have slightly larger line widths but no relation was found between line width and distance from the H II region, which probably indicates that the internal dynamics of the clumps are not affected by the ionized gas. Internal sources of turbulence, such as outflows and stellar winds from young proto-stars may have a more important role on the molecular gas dynamics. We suggest that large line width and larger size-line width power law indices are therefore the initial characteristics of clumps in massive star forming clouds (e.g. our Type I sources) and that some may evolve into objects similar to our Type II sources, where local "second generation" stars are forming and eliminating the size-line-width relation. Introduction Massive stars are believed to form in giant molecular clouds (GMCs) and within clusters (Lada & Lada 2003) which are formed in dense, massive, turbulent clumps (Saito et al. 2008). Compared to isolated star formation from individual collapsing cores, clustered star formation is more complicated and therefore less well understood theoretically. Massive star-forming regions are not as common as low mass star-forming regions and consequently, less frequent in nearby clouds. Therefore, high mass star formation has been far less studied in detail compared to low mass star formation. In regions of high mass star formation, it is not only the physical conditions of the molecular clouds that govern the star formation. Newborn stars in these regions also affect their original environment through various feedback mechanisms (e.g. Krumholz et al. 2010). Low mass stars form gently in more quiescent regions and do not significantly affect their environment during their slow formation and evolution process. In contrast, high mass stars form in turbulent dense cores (McKee & Tan 2003) and evolve very quickly. They influence the environment by strong winds, jets and outflows which change the physical conditions such as temperature, density and turbulence of the cloud (e.g. Gritschneder et al. 2009). This feedback may even affect the Initial Mass Function (IMF) (e.g. Bate & Bonnell 2005, Krumholz et al 2010. Massive stars also ionize the gas (known as H II regions) that expands into the surrounding cloud. All of these energetic processes near the young massive star affect further star formation in the cloud (e.g. Deharveng et al. 2005). We are making a detailed examination of the dense gas properties in star-forming regions associated with massive stars in order to investigate how the gas physical parameters and consequently, star formation have been affected in these environments. Our sample consists of 10 H II regions selected from the Sharpless catalog of H II regions (Sharpless 1959). Some of these regions have been investigated in previous studies and it has been observed that they have complex spatial and kinematic structures (Hunter et al. 1990). Triggered star formation also has been observed at the peripheries of two sources within our sample (S104 and S212; Deharveng et al. 2003Deharveng et al. , 2008. Here we study the properties of the molecular gas in higher resolution and investigate how physical parameters vary due to the influence of the H II region and the exciting star. For each source we used the James Clerk Maxwell telescope (JCMT) to make 12 CO(2-1) maps around the H II regions to study the clumpy structure of the gas in the associated molecular clouds and to locate the dense cores within the cloud. These maps can be made in poor weather conditions at the JCMT as the emission is very strong. 12 CO(2-1) is optically thick and cannot trace the dense gas at the centre of the cores where the star formation actually takes place. Therefore, we used pointed observations of an optically thinner emission line such as 13 CO(2-1) in the cores to measure physical properties of the dense gas, such as density, temperature, clump masses and velocity structure that affect the star formation process. In particular, the velocity structure and line widths will provide information on the dynamical forcing by the H II region and on the clumps' support mechanisms. We can also get a picture of internal dynamics inside the molecular cloud. Hot clumps that show evidence of outflows can also be found and identified as candidates for proto-stars. We introduce the sample and present the observational details in §2. In §3 the observational results and calculated parameters are presented. We discuss the relationships between derived physical characteristics in §4 and conclusions are summarized in §5. Sample Selection and Observation Sample We have selected H II regions with small angular size (≤ 7 ) to be able to map the molecular gas at the edges of the ionized gas. The observed sources are listed in column one of Table 1. We have selected objects in the outer Galaxy, primarily along the Perseus arm to minimize the confusion with background sources and to have the best estimate of the kinematic distance for sources for which we do not have a direct distance determination. Knowing the accurate source distance is crucial to estimate size and mass of the clumps detected within each cloud, therefore the sources which does not have a measured distance in literature have not been selected. S104 is the only source with almost the same Galactic orbit as the Sun, but we have a direct distance measurement for this object and it satisfies our other requirements. All distances are given with literature references in Table 1. Columns two and three present the approximate coordinates of the centre of the mapped region in 12 CO(2-1). Column four shows the distance of the source based on the identified exciting star from the literature; for S196 no exciting star has been detected and therefore we use the kinematic distance for this source. In column five we list the radial velocity of each source from the catalog of CO radial velocities toward Galactic H II regions by Blitz, Fich and Stark (1982). We have selected H II regions with small angular size to be able to map the molecular gas at the edges of the region. Column six gives the angular diameter of optically visible ionized gas. The angular diameter varies between one and seven arc-minutes. Our sources lie at distances between ≈ 1 kpc (S175) and ≈ 7 kpc (S212). The calculated diameters of the H II regions in our sample vary from smaller than 1 pc for S175 and S192 to 9 pc for S104. The calculated diameters are listed in column seven. In column eight we list the identified exciting stars from the literature. To investigate the effects of the massive exciting star and the H II region on its environs we compare the molecular gas near the H II region with the gas from parts of the cloud which are distant enough that they are unlikely to be affected. We observed an excellent example of a second distinct and distant component in the molecular cloud associated with S175 (Azimlu et al. 2009, hereafter Paper I). Two components in the molecular cloud, S175A and S175B, have previously been identified in an IRAS survey of H II regions by Chan & Fich (1995). Both regions have the same V LSR ≈ −50 km s −1 and are connected by a recently observed filament of molecular gas with the same velocity. Therefore it is reasonable to assume that S175A and S175B lie at the same distance. We use the results from Paper I as a template to study the properties of the clouds discussed in this paper. Observations with the 15 m JCMT Sub-millimeter Telescope The observations were carried out in different stages. In the first stage from August to November 1998 we made 7 × 7 12 CO(2-1) maps of S175A, S175B, S192/S193, S196, S212 and S305. The details of these observations have been described in Paper I. Pointed observations in 13 CO(2-1) on peaks of identified dense clumps within each region were made from August 2005 to February 2006. In this period we also made a 7 × 7 12 CO(2-1) map of S104 with the same observational settings as in 1998, but due to the large angular size of this H II region we missed most of the associated molecular gas in our map. Therefore, we decided to extend the S104 map to a larger area. Further observations were made with the new ACSIS system at the JCMT between October 2006 and January 2009. We extended the S104 map to a 9 × 9 map and made four more 7 × 7 12 CO(2-1) maps: around S148/S149, S152, S288 and S307. Most of the peaks identified in clumps within all sources in our sample were also observed in 13 CO(2-1) during four observation missions in this period. In this new configuration with ACSIS, we used a bandwidth of 250 MHz with 8190 frequency channels, corresponding a velocity range of about 325 km s −1 for 12 CO(2-1) and 340 km s −1 for 13 CO(2-1) with a resolution of ∼0.04 km s −1 . We used the Starlink SPLAT and GAIA packages to reduce the data, make mosaics, remove baselines and fit Gaussian functions to determine ∆V, the FWHM of the observed line profiles. Results The physical properties of the clumps are derived from the 12 CO(2-1) maps and 13 CO(2-1) pointed observations at the peaks. We used the 12 CO(2-1) maps to study the structure and morphology of the molecular clouds and to identify clumps within them. The cloud associated with each H II region consists of various clumps of gas that display one or more peaks of emission (typically towards the centre of the clump). We define any separated condensation as a distinct clump, if all of the following conditions are met: 1) the brightest peak within the region has an antenna temperature larger than five times the rms of the background noise; 2) the drop in antenna temperature between two adjacent bright peaks on a straight line in between, is larger than the background noise; and, 3) the size of the condensation is larger than the telescope beam size (21 or 3 pixels). The edge of a clump is taken to be the boundary at which the integrated intensity over the line drops to below half of the highest measured integrated intensity within that clump. An ellipse that best fits to this boundary is used to calculate the clump size and total integrated flux measurements. The positions of the 12 CO(2-1) peak of identified clumps in each region and other observed parameters for each clump are listed in Table 3. Most of the clumps do not have a circular shape, therefore an equivalent value for the radius is calculated from the area covered by the clump in 12 CO maps. We define the effective radius for each clump based on the area of the clump as R e = Area/π. The corrected antenna temperatures for 12 CO(2-1) and 13 CO(2-1) , 12 T * a and 13 T * a and the peak velocities for each emission line, V 12 and V 13 , are directly measured from spectra after baseline subtraction, and are listed in Table 3. We calculate the 13 CO(2-1) optical depth and hydrogen column density assuming local thermodynamic equilibrium (LTE) conditions. Detailed discussion of the methods and the equations are given in Paper I. We use the 12 CO(2-1) peak antenna temperature to calculate the brightness temperature and estimate the excitation temperature, T ex , for each clump. The optical depth, τ , and 13 CO column density can be derived by comparing the 12 CO(2-1) and 13 CO(2-1) brightness temperatures. To calculate these parameters we assumed that 12 CO is optically thick and τ 13CO τ 12CO . These assumptions may underestimate the 13 CO(2-1) column density which has been considered in equilibrium conditions in section §4.4. We convert the 13 CO column density to hydrogen column density, N (H), assuming an abundance factor of 10 6 (Pineda et al. 2008). Derived parameters are listed in Table 4. Column two shows the excitation temperature. In columns three and four we present the FWHM of the Gaussian best fit to the 12 CO and 13 CO spectra. Column five presents the hydrogen column density calculated from the LTE assumption. Calculated optical depth for 12 CO and 13 CO are listed in columns eight and nine. We used three different methods to estimate the clump masses. The virial mass was calculated by assuming that the clumps are in virial equilibrium. We used the 13 CO line widths and assumed a spherical distribution with density proportional to r −2 (McLaren 1988). In Paper I, we discussed the uncertainties of this mass estimation method. For example, the line profiles are broadened in many cases (for example strongly in S152 and S175B) due to internal dynamics and probably turbulence. As a result, the virial equilibrium assumption over-estimates the mass of the clumps and in particular is not an appropriate mass estimation method for dynamically active regions. We also calculated 13 CO column density under the LTE assumption and converted it to hydrogen column density in order to determine clump masses. There are some uncertainties in the mass estimated this way from several factors related to non-LTE effects. For example, in the LTE assumption the CO column density directly varies with T ex . The 12 CO emission might be thermalized even at small densities, while the less abundant isotopes may be sub-thermally excited (Rohlf & Wilson 2004). LTE calculations assume that the excitation temperature is constant throughout the cloud. T ex is calculated from the observed 12 CO line which is optically thick and only traces the envelope around the dense cores; however, inside the envelope, the core might be hotter or cooler. If the core is hotter than the envelope, the emission line might be self absorbed and the measured antenna temperature will give too small a value of T ex for most of the clumps. The M LT E is the smallest calculated mass for most of the clumps in our sample (see also Frieswijk et.al (2007) for similar results in a dark cloud in an early stage of star formation). In the case of the optically thick 12 CO emission we can calculate the column density from the empirical relation N (H 2 ) = X × T mb ( 12 CO)dv, with the brightness temperature, T mb = T * a /η mb . For the JCMT, η mb = 0.69 is the beam efficiency at the observed frequency range. However the "X-factor" is sensitive to variations in physical parameters, such as density, cosmic ionization rate, cloud age, metallicity and turbulence (e.g. Pineda 2008, and references therein). The "X-factor" also depends on the cloud structure and varies from region to region. A roughly constant X value is accepted for observed Galactic molecular clouds, and is currently estimated at X 1.9 ± 0.2 × 10 20 cm −2 (K km s −1 ) −1 (Strong & Mattox 1996). Derived integrated column density is used to calculate the velocity integrated mass or X-factor mass. The velocity integrated mass, M int and relevant volume density, are calculated for all the identified clumps and are listed in columns six and seven of Table 4. In general M int is intermediate between M vir , which often overestimates the mass determinations, and M LT E , which may underestimates the mass. Accordingly, we use M int as the best mass estimation for our clumps in the rest of this paper and discuss the estimated mass relations further in §4.4. Discussion In this section we discuss the physical conditions of 142 clumps identified within clouds associated with ten H II regions. We investigate the relationships between different measured and calculated parameters such as size, line widths, density and mass in order to understand how they have been affected in different regions. We also explore how these physical properties vary with distance from the ionized fronts of the H II region. Size-Line Width (Larson) Relationship Studies of the molecular emission profile and line width provides us with information on the internal dynamics and turbulence of the clumps and cores. Various studies of molecular cloud clump/cores have examined the relation between line width and clump/core size (e.g., Larson 1981, Solomon et al. 1987, Lada et al. 1991, Caselli & Myers 1995, Simon et al. 2001, Kim & Koo 2003) and a power law, often known as the Larson relation, has been proposed to describe this relationship: ∆V ∝ r α , 0.15 < α < 0.7.(1) This relation has been observed over a large range of clumps/cores, from smaller than 0.1 pc up to larger than 100 pc. Different power law indices have been observed within different samples. The observed relationship is presumably affected by the cloud physical conditions, clump definition, and dynamical interaction with associated sources such as H II regions, newborn stars or proto-stars. In their survey for dense cores in L1630, Lada et al. (1991) noticed that the existence of the Larson relation is highly dependent on clump definition. They found a weak correlation for clumps selected by 5σ detection above the background but no correlation for clumps selected at 3σ. Goldbaum et al. (2011, submitted) recently used virial models to show that some GMCs' properties including the line width and size are highly dependent on the mass accretion rate and that the clouds with larger mass, radii and velocity dispersions must be older. Kim & Koo (2003) found a good correlation between size and line width for both 13 CO and CS observations with α = 0.35. However, other studies show more scattered plots (e.g. Yamamoto et al. 2006, Simon et al. 2001 or no relation at all (e.g. Azimlu et al. 2009, Plume et al. 1997. In a study of three categories of clumps containing massive stars, stellar or proto-stellar identified sources, or no identified source, Saito et al. (2008) found a weak relation. In this study, the clumps containing massive stars had larger ∆V s. In a study of cloud cores associated with water masers, Plume et al. (1997) noted that the size-line width relation breaks down in massive high density cores, which systematically had higher line widths. Line widths larger than those expected from thermal motions are thought to be due to local turbulence (Zuckerman & Evans, 1974). Larson (1981) noted that regions of massive star formation such as Orion seem to have larger ∆V and probably show no correlation with size. Goldbaum et al. (2011, submitted) suggested that accretion flow is the main source of turbulence in massive star forming GMCs, however the virial parameters remained roughly constant as the clumps evolved in their model, therefore they were able to reproduce the power law Larson relation. Plume et. al. (1997) concluded that a lack of the Larson relation in their data indicates that physical conditions in very dense cores with massive star formation are very different from local regions of less massive star formation (the line widths may have been affected by the star formation process). They suggested that these conditions (denser and more turbulent than usually assumed) may need to be considered in studying the massive portion of the Initial Mass Function. This argument agrees with the conclusions of Saito et al. (2007). In their study Saito et al. found clumps with no massive star to have similar line widths with cluster forming clumps classified by a previous study (Tachihara et al. 2002) and the massive clumps observed by Casseli & Myers (1995) all have a similar line width. These regions are all forming intermediate mass stars, suggesting that there is a close relation between the characteristics of the formed stars and the line width. Saito et al. also mention that, although the line width might be influenced by feedback of young stars, extended line emission could be a part of the initial conditions of the cloud. We discuss later in section 4.6 that in our sample large line widths seem to be the initial condition of massive star forming regions. The index α seems to depend on the physical conditions of the cloud and especially varies in turbulent regions (e.g. Caselli & Myers 1995;Saito et al. 2006). Star forming regions are turbulent environments and the star formation process is believed to be governed by supersonic turbulence (e.g. Padoan & Nordlund 1999) driven on large scales. A turbulent cascade then transfers the energy to smaller scales and forms a hierarchical clumpy structure (Larson 1981). Numeric analysis of decaying turbulence in an environment with a small Mach number is consistent with the Kolmogorov law, but supersonic magneto-hydrodynamic turbulence results in a steeper velocity spectrum (Boldyrev 2002 and references therein). We investigated the Larson relation in our sample for both 12 CO(2-1) and 13 CO(2-1) lines. Size and line width are directly measured from the observations. Most of the previous studies has used the common Least Square (LSq) fitting method but because there are uncertainties in both ∆V and R e , we prefer to calculate the slope of the fitted line with a bisector least-squares fit (Isobe et al. 1990). However we also repeated all fits with the LSq method to compare our results to previously published results (Table 2). We also calculated correlation coefficients (and significance) of the fits for the clumps associated with each of the ten H II regions. In six regions a linear relationship was found in log(∆V ) versus log(R) with slopes between 1.2 and 3.0 (bisector method) and between 0.44 and 1.54 (LSq). There are 24 fits for these six objects: six objects, two spectral lines, and two fitting techniques. On average the slopes found in the bisector were 1.4 sigma greater than the slopes found from the LSq method. in most of the fits the 12 CO and 13 CO slopes were the same to within the uncertainties. In all six objects a modest correlation was found (typical correlation coefficients of 0.57) with 1.5 to 3 sigma levels of significance. The slopes and uncertainties for both ∆V 12 and ∆V 13 are listed in Table 5 for these six regions -those with a linear relationship. In the other four regions no size-line width relation was found for any fitting technique with either set of CO data. In all four objects the best fit slopes were close to zero and the uncertainties in the slopes were large and the correlation coefficients are close to zero -with the exception of S175B which had a coefficient of -0.59 at the 1.8 sigma level in the 12 CO data. We therefore divided our sources into two categories: the "Type I" sources that show a size-line width relation and "Type II" sources with a weak or no Larson relation. The clumps within Type II sources have broader lines in general and have a scattered size-line width plot. Figure 1 shows the plots of ∆V 12 and ∆V 13 vs R e and the derived αs (bisection method) for both 12 CO(2-1) (black dots) and 13 CO(2-1) (red dots) lines for Type I sources. S288 has only 4 identified clumps and we therefore excluded this region. Figure 2 shows the same plot as Figure 1 for Type II sources. No power-law lines are shown on these plots as no relation could be obtained for these objects. The LSq slopes we have calculated for Type I regions are larger than the previous reported values (e.g. Larson 1981, Solomon et al. 1987, Lada et al. 1991, Caselli & Myers 1995, Simon et al. 2001, Kim & Koo 2003, Saito et al. 2006. This might be due to the small samples -typically a dozen clumps in each region -which also leads to relatively large uncertainties in the calculated slopes. All of the previous studies have faced the same difficulty and solve the problem by combining the data from all objects; In some studies data from different spectral lines is combined together for this analysis. Considering all six Type I sources together, we derived a slope of α = 0.76 ± 0.05 for ∆V 12 vs. R e and α = 0.71 ± 0.05 for ∆V 13 vs R e with correlation coefficients of 0.70 and 0.62 respectively. Using the LSq technique we get a slope of α = 0.51 ± 0.06 for ∆V 12 vs. R e and α = 0.47 ± 0.07 for ∆V 13 . The results are listed in Table 2. We have also checked the results and generated the same statistical information for three recent similar studies and show these results in this table. The second column gives the slope and uncertainty while the third column shows the scatter around the best fit line. The correlation coefficients for each data set is shown in the last column. All of the data-sets show similar scatter. In all but the Type II sources case we have a correlation that is significant. In the Type II sources case the correlation coefficient is small and the probability of getting this just by random is large. Our Type I sources show the steepest slope, only marginally more (1/2 sigma) than more than found by Saito et al. (2006). Type II sources are clearly and significantly different. The Type II sources have larger line widths in both 12 CO and 13 CO ( Figure 2). Such large line widths may originate from the initial conditions of the clump or may be caused by energy inputs such as proto-stellar outflows, radiation pressure from massive stars, strong stellar winds, internal rotation and infall to proto-stars within the clumps. All of the Type II regions show signatures of active star formation. S104 and S212 are good samples of triggered star formation by "Collect and Collapse" process (Deharveng et al. 2003 and. We have detected strong signatures of an outflow within S175B (Azimlu, et al., in preparation) which has the largest line widths in our sample. S152 is the other source in our sample with similar large line widths. This region has very active star formation and contains a dense stellar cluster (Chen et al. 2009, also detected in 2MASS data). To provide enough gravitational energy to bind the clumps with larger internal velocity dispersion, much higher densities are required than for the clumps with smaller line widths (Saito et al., 2006). High gas density (n > 10 5 cm −3 ) is an essential factor in the formation of rich embedded clusters such as the one detected in S152 (Lada et al. 1997). In addition, these clumps must be gravitationally bound in order to survive longer than the star formation time scale of ∼ 10 6 yr. In a massive star-forming model, McKee & Tan (2003) suggest that the mass accretion rate in a core embedded in a dense clump depends on the turbulent motion of the core and the surface density of the clump. Therefore, to form a dense core that can produce a massive star, the clump requires both a high gas density and large internal motion while it is also gravitationally bound. External pressure may have an important role in providing additional force to keep the clump stable during the star formation process (Bonnor-Ebert model;Bonnor 1956, Ebert 1955. We have detected such dense, hot clumps within our sample (for example S104-C4, S305-C7, and S307-C4). We already have deep CFHT near IR observation for five of our sources. Data is partially reduced and we will use that to study the luminosity function and mass distribution of the young stars embedded within the clumps in our sample to determine the influences of gas properties on star formation process. Size-Density Relation We investigated the relation between column density and size for Type I and Type II sources as one may expect larger column density for larger clumps. In Figure 3 we plot N(H) versus clump effective radius for both Type I (left panel) and Type II (right panel) sources. We found statistically significant no relation between the column density and size for our sources. We also investigated the relation between velocity integrated volume density and size. Results are presented in Figure 4. There is a weak relation for some individual sources such as S175A and S175B but, in total, the density decreases as size increases. The dashed line with a slope of -2.5 shows the limit below which we have not found any clump. Simon et al. (2001) found different power law indices for their sample of four molecular cloud complexes varying between -0.73 and -0.88, slightly smaller than the power law factor of -1.24 found by Kim & Koo for molecular clouds associated with ultra compact H II regions. These results may be due to the fact that the smaller clumps are probably more evolved. The gas is more collapsed to the centre and therefore these clumps have smaller size with larger volume densities. This also explains the lack of a relation between size and column density. The smaller clumps have higher volume density and integrating density through the clump centre may result in a small or large column density dependent on the initial conditions and the nature of the clumps. Sources within our sample are located at different distances. We have detected the smallest clumps in S175A which is the closest source to us at a distance of ≈ 1 kpc. If this source was at five times the distance (approximately the distance of S305), clumps C1-C4 and C11-C13 would not be resolved and we would measure smaller average volume density for these groups of clumps as individual objects. Thus, our plot is likely to be affected by spatial resolution effects. Saito et al. (2006) suggest that the mass and density must be larger for turbulent clumps (they define a core as turbulent if ∆V > 1.2 km s −1 ) compared to non-turbulent ones for a given size to bind the turbulent clumps gravitationally. For a similar density distribution, they found that turbulent clumps (those with larger line widths) have densities twice the non-turbulent ones. We do not see a difference in general between mass or column density of our Type II (with larger line widths) and Type I sources. For a detailed discussion on different masses and equilibrium conditions of the clumps see §4.4. LTE Mass and Column Density-Line Width Relation We find a weak relation between N(H) and ∆V (Figure 5 and 6). The column density increases with ∆V for both 12 CO and 13 CO lines. The dashed lines present the bisector least-squares fits with slopes of 2.3±0.25 and 2.1±0.35 for 12 CO lines in Type I and Type II sources respectively but the correlation coefficient is poor (0.53 and 0.42 respectively). The slopes are similar (2.3±0.23 and 2.4±0.38) for ∆V 13 for both Type I and Type II sources with similar correlations (0.51 and 0.47 respectively). M vir and M int both depend on the emission line profile and are expected to increase for Type II regions with larger line widths. The LTE mass calculation is independent of the cloud dynamics and the emission line profiles. In Figures 7 and 8 we show how M LT E varies with ∆V for both 12 CO and 13 CO emission lines. For Type I sources M LT E increases with ∆V for both 12 CO and 13 CO with least-square fit slopes of 4.2 ± 0.28 and 4.2 ± 0.24, but no relation is found for Type II sources. The mass range is not very different for Type I and Type II clumps (1.1 − 350 vs 0.5 − 500 M ). The LTE mass is calculated by integrating the LTE column density over the area of each clump assuming that it is a sphere with radius of R e . M LT E is proportional to R 2 e ; therefore, the dependance of mass calculation on size might be the cause of the relation between M LT E and ∆V in Type I sources where the size-line width relation was found. Equilibrium State of the Clumps The equilibrium state of a clump can be determined by comparing the kinetic and gravitational energy density which can be obtained by measuring the ratio of the virial and the LTE mass (Bertoldi & McKee 1992). The virial mass is calculated as M vir (M ) = 126×R e (pc) (∆V ) 2 (km s −1 ) which assumes a spherical distribution with density proportional to r −2 . Column density is generally higher at the centre of clumps and drops rapidly by distance from the centre. Assuming a simple constant density will then overestimate the mass. However it is not very easy to fit a unique density profile to all clumps especially where there are more than one cores. Virial equilibrium also assumes that the gravitational potential energy is in balance with internal kinetic energy. M vir increases with (∆V ) 2 ; integrated mass, M int , is also line-profile dependent and varies linearly with ∆V , while M LT E is independent of line width and of the internal dynamics of the cloud. We plot M vir /M LT E versus M LT E in Figure 9 to investigate whether clumps are in virial equilibrium. Resolving the clumps within clouds is highly dependent on the distance to the cloud. To decrease the effect of resolution we have selected only objects at distances between 3 and 7 kpc. Closer sources are represented by crosses in the plot. Most of our clumps have larger virial masses and are far above the M vir = M LT E line, indicating that they are probably not gravitationally stable. An external pressure is required to keep the clumps with M vir > M LT E bound. The mean pressure of the ISM in general (P/k ≈ 3 × 10 4 Kcm −3 , k is Boltzmann constant, Boulares & Cox, 1990) is sufficient to bind clumps with M vir /M LT E < 3. Many of the clumps in our sample have M vir /M LT E between 3 and 10 where larger pressures are required, typically 10 5 to 10 6 Kcm −3 . However all of our clumps are near HII regions, warmer areas where the pressure is expected to be larger than the mean ISM pressure. Even larger external pressures, greater than P/k ≈ 10 7 Kcm −3 is required to stabilize the small number of clumps with the largest linewidths and with M vir /M LT E > 10. Perhaps these clouds are indeed not stable. Saito et al. (2007) suggest that to bind the turbulent clumps (clumps with larger line widths) they must have larger masses, therefore for similar size clumps the turbulent sources must have larger densities. In our sample both Type I and Type II sources (which have generally larger line widths) have similar clump masses and we do not see an excess of density for either type. Figure 9 shows that more massive clumps tend to be closer to virial equilibrium. The same pattern has been observed by Yamamoto et al. (2003Yamamoto et al. ( , 2006. Figure 10 shows the plot of velocity integrated mass, M int , versus M LT E . For most of the clumps, M int is higher than the M LT E but mostly below the line of the M int = 10M LT E . M int /M LT E is much larger for low mass clumps and like M vir , M int tends to be equal to M LT E for higher masses; however, the difference is smaller compared to M vir versus M LT E . In Figure 11 we compare M vir versus M int . Similar to M LT E , clumps have larger M vir for low mass clumps. At approximately M= 100 M , M vir M int , while for clumps larger than 100 M , M vir < M int . The higher mass clumps in our plot lie within more distant sources where we cannot resolve smaller structures. This suggests that probably the average density and temperature of larger clumps results in a larger Jeans length while smaller dense structures resolved as clumps have locally higher density and therefore smaller Jeans length. Analysis of cloud and clump equilibria in a study of four molecular clouds at different distances shows that, while the whole cloud is gravitationally bound, the majority of the clumps within them are not (Simon et al. 2001). We found that the equilibrium appears for clumps larger than ∼ 100M . Temperature-Line Width Relation In Figures 12 and 13 we investigate the relation between the excitation temperature and line widths. Part of the profile line broadening is due to the thermal velocity dispersion of the molecules and the thermal line width increases with temperature. The observed ∆V s are more broadened than calculated thermal line widths due to internal dynamics and turbulence within the clouds. We found no relation between T ex and ∆V for either 12 CO and 13 CO emission lines which indicates that the emission line profile is dominated by the internal dynamics of the clumps. Effects of Distance from H II Region on Physical Parameters We are studying how H II regions and their exciting stars affect the physical conditions of their associated clouds. One might expect more influence on clumps that are closer to the H II region. To investigate these effects, we examine how the environmental parameters vary with distance from the edges of the ionized gas. Most of the H II regions in our sample are not perfect spheres but we try to fit the best circle to the visible edges of the ionized gas. To determine the borders of the ionized gas we use Digital Sky Survey (DSS) images in the red filter in which the hydrogen ionized gas is bright with sharp edges. Our sources have different angular sizes, different physical sizes and lie at different distances. They might also be at different evolutionary stages. We need to re-scale the distances to a common distance to be able to compare them. A normalized distance for each clump is defined as the distance of the clump from the edge of H II region divided by the H II region radius. Temperature-Distance relation In Figure 14 we investigate the variation of excitation temperature, T ex , with distance from the H II region. We expect clumps to be warmed up by the radiation from the exciting star and to show a trend of decreasing temperature with increasing distance from the H II region. We can see that the clump temperature decreases with distance from the ionized gas around some sources such as S175A and S152. We observe a scattered relation for S305 and a weak relation (for clumps at distances larger than 0.1R) for S104. We cannot see the same relation for other sources perhaps due to internal heating sources such as proto-stars within the clumps. C4 and C8 in S307 (noted in Figure 14, left panel) are good examples of such compact clumps with high density and temperature. On the other hand, we are measuring the projected distance of the clumps which in general will be smaller than the true value. We check the effect of distance projection by numerically simulating randomly distributed clumps with heating from an external source. The luminosity of the heating source, the power-law index of decrease in central heating with distance, and the heating of the clump from the diffuse background are the input parameters of the simulation. We also set the number of clumps and the radial distribution of the clumps' positions. The output is temperature versus the projected distance from the heating source. Figure 15 shows plots of some selected simulations for different initial parameters. Setting the input values as the observed parameters of our sample and assuming a R −α decreasing power law for luminosity, (Figure 15) we see some scattering on the plot for small distances, but a decrease in temperature as distance from the heating source increases. Distant hot clumps cannot be warmed by the central heating source. Such distant but hot clumps like S307-C3 and S307-C4 noted in Figure 14 are probably being warmed by an internal source such as a proto-star. We see more scatter in the simulated plots for smaller projected distances. Most of our mapped regions (7 × 7 ) are small compared to the size of the H II region; consequently we did not map the molecular gas at large distance from the edges of H II regions with larger angular size (e.g. S104, S212 and S148/S149). The scattering of T ex versus normalized distance for the clumps within these sources matches with the objects at the smaller distances of the simulated plots. S175B is the only source in which all of the clumps are too distant to have been influenced by the H II region. Line Width-Distance Relation We study the effect of the expanding ionized gas on the internal dynamics of the molecular gas by investigating the variation of line widths with distance from the H II region. In Figure 16 we plot ∆V versus normalized distance for 12 CO (left panel) and 13 CO (right panel). We did not find a relation for ∆V 12 or ∆V 13 versus distance for any of Type I or Type II sources. The line-width is very scattered at different distances even for S175A and S152 which show a decrease of T ex with distance from the ionized gas. The expansion of the ionized gas may cause the molecular gas in shells around the H II region to have slightly larger line widths but we do not see a significant distance effect on the internal motions of the clumps beyond the collected shells. Physical conditions of individual clumps and other internal sources of turbulence or dynamics such as proto-stellar outflows, infall and rotation may have a more important effect on line profiles. The important point is that, as discussed in section 4.1, we have seen larger line widths and larger Larson power law indices in our sample for Type I sources. But if the line profiles are not much affected by the H II region and the exciting star, then the observed large line widths are more likely to be the initial characteristics of the clouds which have already formed at least one massive star. Summary and Conclusions We have studied the physical properties of molecular clouds associated with a sample of ten H II regions. We mapped eleven 7 × 7 areas in 12 CO(2-1) at the peripheries of the ionized gas and extended one of these maps (around S104 H II region, the largest region) to 9 × 9 in order to study the characteristics of the molecular gas well beyond the edges of the H II region. We investigated the clumpy structure of the clouds and identified 142 distinct clumps within eleven mapped regions. We also made pointed observations in 13 CO(2-1) at the position of the brightest 12 CO peak within each clump for 117 clumps. We used these observations to measure and calculate the physical characteristics of the clouds. We summarize our findings below: We investigated size-line width relation for our sources using both 12 CO and 13 CO emission lines and calculated effective radius. Our sources are divided into two categories: those which show a power law relation and those which do not show any relation. We labeled the first category of six regions as Type I and the other four sources with no relation as Type II. Type II sources have larger line widths in general and they are active star forming regions. The power law indices derived for size-line width relation in Type I sources are somewhat larger than found in previous studies, but they do not appear to be affected by the exciting star or the ionized gas. We conclude that larger line widths and consequently larger indices are more likely the initial conditions of the massive star forming molecular clouds. Clumps with larger column densities trap more internal radiation and are expected to be hotter. We investigated the relation between column density and temperature and found that the temperature increases with column density for both types of sources. Dense and hot clumps have been detected in our sample. Such clumps are more stable against fragmentation and could be candidates for formation of massive stars in future. No relation was found between column density and size of the clump but, for both Type I and Type II sources, the volume density decreases with size; the larger clumps have smaller densities. The smaller clumps might be more evolved, contracting to smaller size and higher densities. We investigated the relationship between LTE column density and line width. We found that the column density increases with both ∆V 12 and ∆V 13 for both Type I and Type II regions but the relation is very scattered. We estimated the mass of each clump in three different ways: the virial mass using ∆V 13 determined from optically thinner 13 CO lines (M vir ∝ (∆V 13 ) 2 ), 12 CO velocity integrated mass or X factor mass (M int ∝ ∆V 12 ), and LTE mass (mass estimate independent of line width) using both 12 CO and 13 CO lines. M vir is larger than M LT E for small clumps but tends to equal values for large masses. Low mass clumps also have larger M vir than M int but M vir becomes approximately equal to M int at M=100 M and is smaller than M int for larger masses. We conclude that the larger clouds are gravitationally bound but the fragmented smaller clumps within them are not. We investigated how M LT E varies with ∆V for both Type II and Type I sources. While we see that M LT E increases with both ∆V 12 and ∆V 13 for Type I sources, no relation was found for Type II regions. No relation was found between the excitation temperature and the line widths. This suggests that the line widths for both 12 CO and 13 CO are determined by the internal dynamics of the clump rather than the thermal velocity dispersion of the molecules. Excitation temperature decreases with distance from the edges of the ionized gas for some clouds. However, the measured decrease is not significant because of the projected distance effect, especially for smaller distances. The effect of the H II regions on the internal dynamics of the clumps was investigated. The line widths for the clumps within collected shells around H II regions are slightly larger, but no relation was found between either ∆V 12 or ∆V 13 and normalized distance from the H II region. The expansion of the ionized gas affects the internal dynamics of the collected mass but these effects do not go beyond the shells. The plots of ∆V versus normalized distance are very scattered for both Type I and Type II regions, even for those sources in which temperature decreases with distance. If the internal dynamics of the cloud are not much affected by the exciting star and the expanding ionized gas, we conclude that larger line widths are initial characteristics of the observed molecular clouds which already have formed massive stars. Fig. 1.-Correlation between the 12 CO(2-1) (black) and 13 CO(2-1) (red) line widths and the effective radius of the clumps for Type I sources. The solid line is the bisector least-square fit for 12 CO(2-1) and dashed line shows the bisector least-squares fit for 13 CO(2-1) lines. The slope derived for each line is shown in top left corner of each panel. (right) sources. The dashed line shows an approximate limit below which no clump has been found in our data (n int = −2.5R e + 1.5). No strong correlation is found between n int and R e but overall the volume density is smaller for larger clumps. Figure 5 for 13 CO(2-1) line widths. The relation for Type I sources has the same slope as for 12 CO(2-1) , 2.3; the relation for Type II sources is slightly weaker (correlation coefficient of 0.43 compared to 0.53 for Type I sources) with a slope of 2.43. Figure 7 for 13 CO(2-1) line width. The dashed line is the least-squares fits with the slope of 4.2, the same as the previous plot. Similar to 12 CO(2-1) , no relation is found between M LT E and ∆V 13 for Type II sources. Figure 12 for ∆V 13 . Similarly no relation is found. c Data about exciting star and distance from Moffat et al. 1997 d No exciting star identified for this H II region. The kinematic distance is reported here. Note that S196 is close to S192/S193 in position and velocity and is likely to be at the same distance. Table 3. Physical parameters measured for the entire sample ID RA DEC Re 12 T b 13 T b V 12 V 13 (pc) (K) (K) (km s −1 ) (km s −1 )S104ID RA DEC Re 12 T b 13 T b V 12 V 13 (pc) (K) (K) (km s −1 ) (km s −1 ) . ID RA DEC Re 12 T b 13 T b V 12 V 13 (pc) (K) (K) (km s −1 ) (km s −1 ) . 2.17 · · · · · · 32.3 32±6 · · · · · · ...C27 16.936 2.34 · · · · · · 18.9 15±2.6 · · · · · · ...C28 18.289 3.16 · · · · · · 15.0 59±10 · · · · · · ...C29 18.468 1.53 · · · · · · 14.3 11±2 · · · · · · ...C30 18.421 1.87 · · · · · · 9.9 51±9 · · · · · · ...C31 13.665 2.01 · · · · · · 14.2 48±8 · · · · · · ...C32 27.729 2.26 · · · · · · 28.1 40±7 · · · · · · ...C33 14.057 1.98 · · · · · · 9.7 13±2.2 · · · · · · ...C34 16.305 1.88 · · · · · · 6.7 35±6 · · · · · · ...C35 23.883 2.8 · · · · · · 22.0 51±9 · · · · · · S148-S149 . 0.94 · · · · · · 6.6 8±2.4 · · · · · · S196 ...C1 20.336 3.09 · · · · · · 42.9 179±73 · · · · · · ...C2 21.804 2.83 · · · · · · 44.6 201±82 · · · · · · ...C3 20 S148-S149 3.0 ± 0.5 2.5 ± 0.7 S175A 1.3 ± 0.3 1.4 ± 0.3 S192-194 1.5 ± 0.2 1.5 ± 0.2 S196 1.9 ± 0.4 1.2 ±0.2 S305 1.9 ± 0.2 1.6 ± 0.2 S307 1.4 ± 0.2 1.9 ± 0.2 Fig. 2 . 2-Same plot asFigure 1for Type II sources. Fig. 3 . 3-LTE column density vs. effective radius for Type I (left) and Type II (right) sources. Fig. 4 . 4-Velocity integrated volume density vs. effective radius for Type I (left) and Type II Fig. 5 . 5-LTE column density vs. ∆V 12 for Type I (left) and Type II (right) sources. The dashed line shows the bisector least-squares fits. The relation for Type I sources has a slope of 2.3; the relation for Type II sources is slightly weaker (correlation coefficient of 0.42 compared to 0.53 for Type I sources) with a slope of 2.06. Fig. 6 . 6-Same plot as Fig. 7 . 7-M LT E vs. ∆V 12 for Type I (left) and Type II (right) sources. The dashed line shows the bisector least-squares fit to Type I clumps with a slope of 4.2. No relation is found for Type II sources. Fig. 8 . 8-Similar to Fig. 9 . 9-M vir /M LT E vs. M LT E . Filled circles present Type I sources and open circles present Type II sources. To decrease the effect of varying distances and consequently the resolution effect we have considered only sources at distances between 3.3 and 7.1 k pc. Other sources are shown by crosses. -25 -Fig. 10.-Velocity integrated mass plotted vs. LTE mass. Filled circles represent Type I sources and open circles represent Type II sources. The dashed lines are M int = M LT E and M int = 10M LT E . Fig. 11 . 11-virial mass plotted vs. velocity integrated mass. Filled circles represent Type I sources and open circles represent Type II sources. The dashed lines are M vir = M int and M vir = 10M int . Virial mass tends to equal to M int for massive clumps and they are equal at M ∼ 100M . Fig. 12 . 12-Excitation temperature vs. ∆V 12 for Type I (left panel) and Type II (right panel) sources. No relation is found for Type I or Type II sources. Fig. 13 . 13-Same as Fig. 15 . 15-Simulated temperature variation from a heating source with similar physical conditions as our cloud samples with different luminosity decrease power law index, α. Fig. 16 . 16-Line width vs. normalized distance from H II region for ∆V 12 (left) and ∆V 13 (right). Table 1 . 1Properties of selected regionsSource RA Dec Distance V LSR Diameter Diameter Exciting Star J(2000) J(2000) (kpc) (kms −1 ) (arcmin) (pc) S104 20:17:42 36:45:30 3.3±0.3 0.0 7 6.7 O6V a S148-S149 22:56:22 58:31:29 5.6±0.6 -53.1 1 1.6 B0V a S152 22:58:41 58:47:06 2.39±0.21 -50.4 2 1.4 O9V b S175A 0:27:04 64:43:35 1.09±0.21 -49.6 2 0.63 B1.5V b S175B 0:26:25 64:52:36 1.09±0.21 -49.6 2 0.63 B1.5V b S192-S194 2:47:30 61:56:33 2.96±0.54 -46.3 1 0.86 B2.5V b S196 2:51:41 62:12:19 4.7±1.0 -45.1 4 5.4 · · · d S212 4:40:56 50:27:47 7.1±0.7 -35.3 5 10.3 O6 a S288 7:8:39 -4:18:41 3.0±1.2 56.7 1 0.87 B1 c S305 7:30:13 -18:31:50 5.2±1.4 44.1 4 6.1 O9.5 c S307 7:35:33 -18:45:55 2.2±0.5 46.3 6 3.8 O9 c a Data about exciting star and distance from Caplan et al. 2000 b Data about exciting star and distance from Russeil et al. 2007 Table 2 . 2Comparison with previous worksData Slope Sigma Correlation Coefficient Kim & Koo (2003) 0.35±0.06 0.10 0.79 Saito et al. (2006) 0.44±0.12 0.16 0.57 Yamamoto et al. (2006) 0.29± 0.07 0.15 0.44 Type I Sources 12 CO 0.51±0.06 0.14 0.70 Type I Sources 13 CO 0.47±0.07 0.15 0.67 Type II Sources -0.09±0.09 0.17 -0.12 Table 3 - 3Continued Table 3 - 3Continued Table 3 - 3ContinuedID RA DEC Re 12 T b 13 T b V 12 V 13 (pc) (K) (K) (km s −1 ) (km s −1 ) ... C9 7:30:00.3 -18:33:33.96 0.64±0.17 35.1 16.4 47.11 46.8 ... C10 7:30:02.7 -18:33:34.13 0.57±0.15 16.8 2.7 47.72 48.39 ... C11 7:30:14.0 -18:34:37.91 0.70±0.19 12.0 2.3 47.32 44.67 ... C12 7:30:17.0 -18:33:56.11 0.85±0.23 16.3 · · · 44.07 · · · S307 ... C1 7:35:33.3 -18:44:59.00 0.25±0.06 25.2 10.5 44.53 44.35 ... C2-a 7:35:33.3 -18:45:20.50 0.28±0.06 28.1 9.1 45.26 44.68 ... C2-b 7:35:33.2 -18:45:34.50 0.26±0.06 32.5 6.9 45.842 46.42 ... C3 7:35:36.3 -18:46:25.83 0.43±0.1 38.0 11.2 46.6 46.55 ... C4 7:35:38.7 -18:48:57.50 0.47±0.12 44.5 19.5 47.04 46.72 ... C5 7:35:31.8 -18:46:23.50 0.32±0.07 28.2 5.0 46.78 46.88 ... C6-a 7:35:41.7 -18:46:16.49 0.35±0.08 19.1 1.4 46.01 46.18 ... C6-b 7:35:42.5 -18:46:30.49 0.34±0.08 12.7 1.7 48.22 48.25 ... C7 7:35:37.4 -18:44:54.83 0.23±0.05 15.0 3.7 45.8 47.79 ... C8 7:35:43.1 -18:48:35.32 0.37±0.09 26.9 8.4 47.11 47.21 ... C9 7:35:43.7 -18:43:55.32 0.23±0.05 12.1 2.4 47.06 47.13 ... C10 7:35:41.0 -18:44:59.49 0.32±0.07 14.3 3.6 47.29 47.34 Table 4 . 4Physical parameters calculated for all clumpsClump Tex ∆V 12 ∆V 13 N (H) n int (H 2 ) M int τ 12 τ 13 No. (K) (km s −1 ) (km s −1 ) (×10 20 cm −2 ) (cm −3 ) (M ) S104 ...C1 18.46 2.37 1.02 18.13 34.5 53±9 17 0.27 ...C2 15.631 2.81 3.05 31.77 11.9 13±2 13 0.21 ...C3 25.844 3.56 · · · · · · 34.3 45±8 · · · · · · ...C4 50.193 2.16 1.91 427.26 41.7 92±15.9 33 0.54 ...C5 17.77 1.78 · · · · · · 14.4 8±1.3 · · · · · · ...C6 16.483 3.52 1.55 31.81 16.3 10±1.6 24 0.38 ...C7 34.219 2.62 0.79 42.39 33.5 59±10 16 0.27 ...C8 40.132 2.47 0.71 131.53 42.1 85±15 42 0.68 ...C9 27.714 2.71 2.15 190.27 35.9 51±9 40 0.64 ...C10 39.111 3.55 2.07 243.36 44.2 65±11 28 0.45 ...C11 32.405 2.49 1.48 100.94 36.3 52±9 23 0.37 ...C12 31.932 2.12 1.87 210.72 35.4 65±11 39 0.63 ...C13 37.462 2.57 1.19 146.19 46.1 51±9 32 0.51 ...C14 37.352 2.69 1.87 102.59 41.6 46±8 14 0.23 ...C15 27.957 2.05 1.26 78.92 21.2 44±8 28 0.45 ...C16 28.711 2.47 1.97 159.49 28.0 34±6 34 0.55 ...C17 31.784 2.13 1.63 51.45 15.8 19±3 11 0.18 ...C18 20.363 2.99 1.67 21.06 13.1 23±4 10 0.16 ...C19 29.326 2.85 2.02 73.27 24.2 36±6 15 0.24 ...C20 17.82 1.87 1.31 10.42 9.5 8±1.4 8 0.13 ...C21 27.45 2.31 1.89 51.04 39.6 17±2.9 12 0.20 ...C22 26.855 2.77 · · · · · · 26.2 19±3 · · · · · · ...C23 19.414 2.4 1.64 33.31 19.6 9±1.6 18 0.28 ...C24 23.031 1.29 1.06 25.51 9.0 7±1.3 15 0.25 ...C25 29.66 2.2 2.22 204.5 14.7 31±5 37 0.59 ...C26 27.419 Table 4 - 4ContinuedClump Tex ∆V 12 ∆V 13 N (H) n int (H 2 ) M int τ 12 τ 13 No. (K) (km s −1 ) (km s −1 ) (×10 20 cm −2 ) (cm −3 ) (M ) ...C9 23.551 2.19 1 60.59 10.3 328±30 37 0.60 ...C10 29.216 2.15 1.65 28.16 29.0 440±90 7 0.11 ...C11 25.703 2.38 1.53 26.79 21.4 211±43 9 0.15 ...C12 25.727 1.39 0.9 34.43 11.6 118±24 21 0.34 ...C13 21.24 2.27 · · · · · · 21.3 191±53 · · · · · · S152 ...C1 25.647 5.28 1.37 508.99 156.0 117±18 93 1.51 ...C2-a 41.211 4.66 · · · · · · 207.0 176±28 · · · · · · ...C2-b 49.415 4.35 1.75 456.5 233.0 175±28 40 0.64 ...C2-c 40.782 4.87 · · · · · · 213.0 181±29 · · · · · · ...C3 35.468 4.76 1.69 416.42 151.0 189±30 70 1.13 ...C4 32.78 7.42 3.15 566.29 165.0 295±47 60 0.96 ...C5 26.311 2.43 1.59 168.75 77.0 56±9 53 0.85 ...C6 26.121 3.04 2.04 148.31 84.0 82±13 37 0.59 ...C7 24.242 2.43 1.07 73.02 519.0 633±100 39 0.63 ...C8 27.516 6.04 2.52 139.02 268.0 97±15 25 0.41 S175A ...C1 33.828 0.97 0.56 61.83 118.9 6±2.0 34 0.56 ...C2 28.362 0.78 0.47 19.38 82.4 4±1.4 18 0.29 ...C3 23.076 1.12 0.66 34.68 49.6 6±2.1 33 0.54 ...C4 32.626 0.92 0.56 72.12 118.7 5±1.6 43 0.69 ...C5 47.676 1.31 0.85 290.4 139.7 14±5 56 0.90 ...C6 46.698 1.61 0.95 243.38 201.2 19±7 44 0.70 ...C7 35.307 0.97 0.64 44.75 113.7 9±3.1 20 0.33 ...C8 39.128 1.12 0.74 105.94 71.4 18±6.4 34 0.55 ...C9 36.178 1.54 1.08 190.75 126.8 15±5.2 49 0.78 ...C10 27.025 1.59 0.93 82.41 77.1 9±3.2 42 0.68 ...C11 20.901 1.25 0.87 13.28 35.8 6±2.1 12 0.19 ...C12 21.039 1.15 0.71 13.5 63.2 7±2.4 14 0.23 ...C13 19.084 1.32 0.62 20.09 70.6 6±2.3 29 0.47 S175B ...C1 22.819 8.68 1.44 70.78 220.9 31±11 32 0.51 ...C2 25.341 4.1 1.4 34.42 183.0 12±4.1 13 0.21 ...C3 18.466 4.87 1.57 49.57 178.1 6±2.2 30 0.48 ...C4 19.52 2 1.26 51.17 38.6 16±5.6 35 0.56 ...C5 29.769 1.04 0.75 31.58 48.7 17±5.8 17 0.27 ...C6 20.339 2.6 1.09 33.82 139.0 9±3.1 25 0.40 ...C7 18.71 2.77 1.32 57.33 63.0 24±9 40 0.65 ...C8 19.019 3.3 1.22 85.83 86.2 29±10 63 1.02 ...C9 20.774 2.72 1.39 100.91 64.6 26±9 56 0.90 ...C10 23.955 1.94 1.17 71.86 93.4 22±8 36 0.59 ...C11 19.838 2.6 1.4 101.21 93.3 10±3.6 60 0.97 ...C12 20.274 1.13 · · · · · · 16.4 12±4.3 · · · · · · S192-S194 ...C1 22.204 1.49 0.82 46.41 27.0 26±8 38 0.62 Table 4 - 4ContinuedClump Tex ∆V 12 ∆V 13 N (H) n int (H 2 ) M int τ 12 τ 13 No. (K) (km s −1 ) (km s −1 ) (×10 20 cm −2 ) (cm −3 ) (M ) ...C2 15.906 1.29 0.83 10.26 23.6 7±2 15 0.25 ...C3 26.156 2.19 1.76 40.57 71.0 61±19 12 0.19 ...C4 21.597 1.04 0.75 20.89 29.9 9±3.0 20 0.32 ...C5 23.829 2.05 0.98 53.91 58.5 57±18 33 0.53 ...C6 28.272 3.45 2.13 199.34 6.1 22±7 41 0.66 ...C7 18.041 1.76 1.2 23.06 22.6 19±6 19 0.31 ...C8 21.167 1.98 1.27 21.93 28.0 61±20 13 0.21 ...C9 15.906 1.28 0.63 6.64 14.1 14±4.5 13 0.21 ...C10 13.057 Table 4 - 4ContinuedClump Tex ∆V 12 ∆V 13 N (H) n int (H 2 ) M int τ 12 τ 13 No. (K) (km s −1 ) (km s −1 ) (×10 20 cm −2 ) (cm −3 ) (M ) ...C9 40.628 1.73 1.26 220.45 41.1 256±131 39 0.62 ...C10 22.026 1.76 1.18 18.58 27.8 125±64 11 0.18 ...C11 17.174 3.49 1.4 16.67 40.9 344±176 13 0.21 ...C12 21.552 3.2 · · · · · · 27.5 407±208 · · · · · · S307 ...C1 30.617 2.18 1.11 97.93 101.3 38±16 33 0.54 ...C2-a 33.508 2.48 1.11 126.02 71.3 37±16 24 0.39 ...C2-b 38.003 2.41 2.26 132.25 104.8 44±19 15 0.24 ...C3 43.504 2.59 2.28 251.74 78.4 152±66 21 0.35 ...C4 50.001 3.35 2.16 513.66 99.7 251±108 35 0.57 ...C5 33.622 2.22 1.89 71.85 70.3 56±24 12 0.19 ...C6-a 24.456 1.58 1.03 8.67 29.8 31±14 5 0.08 ...C6-b 17.912 1 0.7 6.24 19.3 18±8 9 0.14 ...C7 20.192 1.62 0.6 12.99 54.2 16±7 18 0.28 ...C8 32.338 2.63 1.38 93.76 100.0 124±54 23 0.37 ...C9 17.279 1.07 0.68 8.59 24.8 7±3.2 14 0.22 ...C10 19.474 1.32 1 20.46 25.0 19±8 18 0.29 Table 5 . 5The Larson power law index for Type I regionsSource α[∆V 12 ] α[∆V 13 ] AcknowledgmentThe JCMT observatory staff, especially Ming Zhu, Gerald Schieven and Jan Wouterloot are thanked for their support during the observations and their helpful assistance in data reduction. 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[ "Do I Belong? Modeling Sense of Virtual Community Among Linux Kernel Contributors", "Do I Belong? Modeling Sense of Virtual Community Among Linux Kernel Contributors" ]	[ "Bianca Trinkenreich [email protected] ", "Daniel M German ", "Klaas-Jan Stol [email protected] ", "Marco A Gerosa [email protected] ", "Anita Sarma [email protected] ", "Igor Steinmacher [email protected] ", "\nNorthern Arizona University Flagstaff\nAZUSA\n", "\nSFI Research Centre for Software University College Cork\nUniversity of Victoria Victoria\nCanada, Ireland\n", "\nNorthern Arizona University Flagstaff\nAZUSA\n", "\nOregon State University Portland\nORUSA\n", "\nNorthern Arizona University Flagstaff\nAZUSA\n" ]	[ "Northern Arizona University Flagstaff\nAZUSA", "SFI Research Centre for Software University College Cork\nUniversity of Victoria Victoria\nCanada, Ireland", "Northern Arizona University Flagstaff\nAZUSA", "Oregon State University Portland\nORUSA", "Northern Arizona University Flagstaff\nAZUSA" ]	[]	The sense of belonging to a community is a basic human need that impacts an individual's behavior, long-term engagement, and job satisfaction, as revealed by research in disciplines such as psychology, healthcare, and education. Despite much research on how to retain developers in Open Source Software (OSS) projects and other virtual, peer-production communities, there is a paucity of research investigating what might contribute to a sense of belonging in these communities. To that end, we develop a theoretical model that seeks to understand the link between OSS developer motives and a Sense of Virtual Community (SVC). We test the model with a dataset collected in the Linux Kernel developer community (N=225), using structural equation modeling techniques. Our results for this case study show that intrinsic motivations (social or hedonic motives) are positively associated with a sense of virtual community, but living in an authoritative country and being paid to contribute can reduce the sense of virtual community. Based on these results, we offer suggestions for open source projects to foster a sense of virtual community, with a view to retaining contributors and improving projects' sustainability.	10.48550/arxiv.2301.06437	[ "https://export.arxiv.org/pdf/2301.06437v3.pdf" ]	255,941,724	2301.06437	4824c23cff3d973b0a77dcd0fa80308e618a1efa	Do I Belong? Modeling Sense of Virtual Community Among Linux Kernel Contributors Bianca Trinkenreich [email protected] Daniel M German Klaas-Jan Stol [email protected] Marco A Gerosa [email protected] Anita Sarma [email protected] Igor Steinmacher [email protected] Northern Arizona University Flagstaff AZUSA SFI Research Centre for Software University College Cork University of Victoria Victoria Canada, Ireland Northern Arizona University Flagstaff AZUSA Oregon State University Portland ORUSA Northern Arizona University Flagstaff AZUSA Do I Belong? Modeling Sense of Virtual Community Among Linux Kernel Contributors Index Terms-sense of virtual communitybelongingopen sourcesoftware developershuman factorssurveyPLS-SEM The sense of belonging to a community is a basic human need that impacts an individual's behavior, long-term engagement, and job satisfaction, as revealed by research in disciplines such as psychology, healthcare, and education. Despite much research on how to retain developers in Open Source Software (OSS) projects and other virtual, peer-production communities, there is a paucity of research investigating what might contribute to a sense of belonging in these communities. To that end, we develop a theoretical model that seeks to understand the link between OSS developer motives and a Sense of Virtual Community (SVC). We test the model with a dataset collected in the Linux Kernel developer community (N=225), using structural equation modeling techniques. Our results for this case study show that intrinsic motivations (social or hedonic motives) are positively associated with a sense of virtual community, but living in an authoritative country and being paid to contribute can reduce the sense of virtual community. Based on these results, we offer suggestions for open source projects to foster a sense of virtual community, with a view to retaining contributors and improving projects' sustainability. I. INTRODUCTION The sustainability and long-term survival of Open Source Software (OSS) projects depend not only on attracting but, more crucially, retaining motivated developers [1]. The reasons behind a developer's decision to stay or leave an OSS project can depend on different intrinsic or extrinsic factors, including an individual's feelings of identity and belonging to the community [2]. Hagerty et al. defined a sense of belonging as "the experience of personal involvement in a system or environment so that persons feel themselves to be an integral part of that system or environment" [3]. The need to belong is a powerful, fundamental, and pervasive force that has multiple strong effects on emotional patterns and cognitive processes across all cultures and different types of people [4]. Maslow [5] positioned 'belonging' as a basic human need, and Hagerty et al. [6] posited that a sense of belonging represents a unique mental health concept. A sense of belonging is key to productivity, satisfaction, and engagement [4], and can help to avoid attrition [7]. In Science, Technology, Engineering, and Mathematics (STEM), a sense of belonging is strongly related to retention [8], especially for underrepresented groups [9]. The sense of belonging that members have towards others within a certain group is known as a sense of community [10]. The dimensions of a sense of community include feelings of membership and attachment to a group [11], a feeling that members matter to one another and to the group [12]. The concept of sense of virtual community (SVC) was developed by observing that virtual communities represent a new form of community, in which social relationships are predominantly forged in cyberspace [13]. Understanding SVC in OSS is relevant as it can influence the vitality and sustainability of a community [14], [15], and is linked to more satisfied, involved, and committed contributors [16]. Individuals who develop a psychological and relational contract with a community are supported by a state of being involved, rather than external factors such as earning something or climbing a career ladder and therefore tend to develop a deeper, reciprocal relationship with that community [10]. Since sustainability is a key concern for OSS projects, we must understand SVC in OSS communities. While several studies have investigated different motivations to contribute to OSS [1], [17]- [22], none have modeled how these factors can help or hinder in creating a sense of virtual community. Without a deeper understanding of how the different factors interplay to create a sense of community, strategies that aim to promote individual factors will likely be unsuccessful in creating a sustainable community. Understanding how different factors work together or against each other can help communities strategize how to retain their contributors. Therefore, in this paper, we ask the following research question: Research Question: How does a sense of virtual community develop in Open Source Software projects? We answer our research question by first developing a theoretical model of SVC grounded in prior literature (Sec. III). We then evaluate our model through a sample (N=225) of Linux Kernel project contributors, using partial least squares structural equation modeling (PLS-SEM) (Sec. IV). The results of our analysis provide empirical support for part of our model, showing that hedonism (motivation that aims to maximize pleasure and fun and minimize pain [23]) and social motives (motivation that aims to maximize joint gains and others' gains [24]) have a positive association with a sense of virtual community, which can be weakened when contributors are being paid or are surrounded by an authoritative culture, i.e., national culture with a high index of power distance (Sec. V). We conclude the paper by discussing the implications of our findings, and threats to validity (Sec. VI). II. BACKGROUND A. Sense of Virtual Community While numerous definitions of the term 'community' exist, a common theme is that it involves human relationships based on some common characteristics [25]. The classical McMillan and Chavis [12] definition of 'Sense of Community' includes four characteristics: (1) feelings of membership (belonging to, and identifying with, the community), (2) feelings of influence (having an influence on, and being influenced by the community), (3) integration and fulfillment of needs (being supported by others in the community while also supporting them), and (4) shared emotional connection (relationships, shared history, and a 'spirit' of the community). Virtual communities typify a relatively new form of interaction whereby community members share information and knowledge in the virtual space for mutual learning, collaboration, or problem solving [13]. The development of OSS involves distributed problem solving within a virtual community [26]. Virtual communities are a particularly important type of virtual group because they are self-sustaining social systems in which members engage and connect with each other, developing a Sense of Virtual Community (SVC) [27]. The sense of community includes membership, identity, belonging, and attachment to a group that primarily interacts through electronic communication [11], [28], [29]. SVC has been tailored to virtual communities by deriving from McMillan's theory of sense of community [12]. The goal of measuring SVC is to assess the "community-ness" of a virtual community [11]. Community managers can assess and promote SVC to fulfill a core set of members' needs [30], so they feel they belong to a unique group. Such meaningful relationships are associated with increased satisfaction and communication with the virtual community, trust [31], and social capital in the project [32]. SVC has been shown to lead to an occupational commitment [33], and ultimately can help retain contributors and further attract potential newcomers [11], [34], who are critical to the sustainability of OSS projects [35]. B. Motivations to Contribute to Open Source Software The software engineering literature suggests that, by managing developers' motivation and satisfaction, a software organization can achieve higher productivity levels and avoid turnover, budget overflows, and delivery delays [36]. Motivations for joining Open Source has been the topic of considerable research [1], [17]- [22]; motivations can be extrinsic or intrinsic. Extrinsic motivations are based on outside incentives that make people change their actions due to an external intervention [37]. As many companies, including Microsoft, Google, and IBM, hire or sponsor OSS contributors [38], career ambition and payment have become common extrinsic motivations [39]. However, intrinsic motivations also explain much of contributors' motivations [22], moving a person to act for fun or enjoy a challenge, kinship, altruistic reasons, or ideology, rather than in response to external pressures or rewards [40]. Previous research showed that several forces influence the decision of an OSS contributor to join, remain, or leave an OSS project [41]- [43]. Despite the extensive attention this topic has received, there are still no studies investigating how OSS contributors driven by different motivations develop a sense of virtual community. We argue that understanding how a sense of virtual community develops in OSS involves understanding the relationship between individual characteristics and motivations and the resulting community-related feelings. III. THEORY DEVELOPMENT Feelings of belonging in an online community can be influenced by several individual characteristics and factors of the surrounding environment [44]. In the education literature, researchers [45], [46] found associations between students' sense of belonging and a range of motivational variables. Motivational factors can be regarded as expectations related to the interaction with a virtual community (answering why users behave). Integration and fulfillment of needs refer to the idea that common needs, goals, and beliefs provide an integrative force for a cohesive community that can meet collective and individual needs. Thus, meeting members' needs is a primary function of a strong community [12]. Individuals who develop a psychological relationship contract with a community because it is focused on a state of being involved-rather than earning something or getting somewhere-tend to develop a sense of community [10]. Previous research on online communities also showed that individuals who are driven by social motives [47] tend to develop a sense of virtual community [28], [48]. Based on the Fundamental Social Motives Inventory, we included both kinship and altruism as social motives [47] and propose the following hypothesis: Hypothesis 1 (H1). Open Source contributors motivated by social reasons have a higher sense of virtual community. Most of the respondents in Gerosa et al.'s study (91%) agreed (or strongly agreed) that they contribute to OSS for entertainment (fun) [22]. Hedonic motivation is a type of motivation that aims to maximize pleasure and fun and minimize pain. It is an umbrella term that includes hedonic expectancy, perceived enjoyment, and playfulness [23]. Considering that expectations of enjoyable experiences, feelings of amusement, and being mentally or intellectually stimulated by interactions are associated with a sense of virtual community [13], [15], and that changes in the perceived fulfillment of their entertainment needs can determine the change of their sense of virtual community [30], we propose the following hypothesis: Hypothesis 2 (H2). Open Source contributors motivated by hedonic reasons have a higher sense of virtual community. It is known that some open source contributors have a strong ideological basis for their actions [49], believing, for example, that source code should be freely available. Recently, Gerosa et al.'s study showed that, however, ideology is not a popular motivation-especially for young contributors [22]. Historically, the group-based morality of 'fighting' a shared dominant opponent incites the sense of virtual community among contributors [50]. This feeling was quite common in the 1990s, when big corporations characterized Open Source as 'communism' [51] and Linux as a 'cancer' [52]. Besides ideology, we include reciprocity in moral motives, as it represents the moral desire of contributors who aim for social justice by giving back to the community [53]. According to the Social Identity theory [54], sharing a moral vision is positively associated with feelings of belonging. Moreover, a homogeneous ideology throughout a religion was shown as being positively associated with a sense of virtual community [55]. Hence, we posit that: Hypothesis 3 (H3). Open Source contributors motivated by moral reasons have a higher sense of virtual community. Motivations may not always be strong enough to sustain an OSS contributor's participation [56]. Motivations may vary for different groups of people, depending on contextual factors. This implies the existence of moderating factors that change the relationship between motivations and a sense of virtual community. Cognitive Evaluation Theory suggests that feelings of autonomy are positively associated with intrinsic motivations and belonging, while tangible rewards negatively affect intrinsic motivating factors [57]. We evaluated the role of a feeling of autonomy using the variable of power distance from Hofstede's framework of Country Culture [58] as a proxy; a lower power distance would reflect in higher autonomy. We also evaluated the exposition to tangible rewards using the variable is paid. People in societies exhibiting a large degree of power distance tend to accept a hierarchical order [59]. In high power distance cultures (where a high power differential between individuals is accepted and considered normal), information flows are usually constrained by hierarchy [58]. As an important cultural value describing the acquiescent acceptance of authority, power distance has received increasing attention in many domains [60], [61]. Prior research showed that, when surrounded by cultures with a high degree of power distance, students reported a lower sense of belonging to their school [62]. Therefore, in hierarchical cultures, leaders need control over the information flow, and the desire to restrict autonomy and access to critical information by lower-level team members could lead to significant organizational barriers to sharing knowledge and working in a community [63]. Thus, we define the following moderation hypotheses: Hypothesis 4a (H4a). Power distance moderates the association between Open Source contributors' social motives and their sense of virtual community. Hypothesis 4b (H4b). Power distance moderates the association between Open Source contributors' hedonic motives and their sense of virtual community. Hypothesis 4c (H4c). Power distance moderates the association between Open Source contributors' moral motives and their sense of virtual community. The traditional notion that OSS developers are all volunteers is now long outdated; many OSS contributors are currently paid, usually employed by a company, to contribute [39], [64], [65]. Indeed, many Linux Kernel contributors are paid to make their contributions, compensated by firms that have business models relying on the Linux Kernel [66]- [68]. In contrast to traditional paid software development work, and despite its benefits to OSS contributors, introducing financial incentives in OSS communities create complex feelings among OSS developers [69]. For example, developers on the Debian project expressed negative emotion because they felt payment went against the project's espoused values [70]. On the other side, not receiving pay for their work to support their livelihoods can frustrate OSS developers and affect their contributions [69]. Despite compensation, OSS contributors may be driven towards a project by both simultaneous feelings of belonging (intrinsic) and payment (extrinsic) [1], [39]. Nevertheless, there is no research examining the complex impact of receiving payment on intrinsic factors associated with SVC. As many OSS developers are currently paid, we would expect that the behavior of those who are paid and those who are not (volunteers) would diverge. Hence, we propose the following three moderating hypotheses: Hypothesis 5a (H5a). Being paid moderates the association between Open Source contributors' social motives and their sense of virtual community. Hypothesis 5b (H5b). Being paid moderates the association between Open Source contributors' hedonic motives and their sense of virtual community. Hypothesis 5c (H5c). Being paid moderates the association between Open Source contributors' moral motives and their sense of virtual community. IV. RESEARCH DESIGN The research design is summarized in Fig. 1. We conducted a survey among Linux Kernel contributors to evaluate our theoretical model. We studied one specific community to avoid confounding factors related to differences that each OSS community can pose. Introduced in 1991, the Linux Kernel represents one of the largest and most active OSS projects [71], boasting over ten million source lines of code and more than 20,000 contributors from different countries and cultural backgrounds, including volunteers and paid developers from more than 200 companies [72], [73]. Linux Kernel's is impact is perceived in terms of processes and infrastructure tools that emerged from the community [73]. While the Linux Kernel Mailing List is known for its uncivil comments and toxic discussions that tend to discourage people from joining the community [74], community leaders aim to change the project's image and increase the sense of community among members. We closely collaborated with contributors and maintainers of the Linux Foundation involved with Linux Kernel, who had a crucial role in designing the data collection instrument and reaching out to potential participants. They engaged in several meetings with the team and reviewed the items of the questionnaires to provide their feedback, making sure that the instrument was appropriate for the study goals. They also distributed the survey to the Linux kernel community, playing an essential role in recruiting the participants for this study. We used Partial Least Squares-Structural Equation Modeling (PLS-SEM) to analyze the relationships between motivations [75] and a sense of virtual community. SEM is a secondgeneration multivariate data analysis method; a recent survey (which also provides an introduction to the method) indicates that PLS-SEM has been used to study a variety of phenomena in software engineering [76]. SEM facilitates the simultaneous analysis of relationships among constructs, each measured by one or more indicator variables. The main advantage of SEM is being able to measure complex model relationships while accounting for measurement error when using latent variables (e.g., Sense of Virtual Community). PLS-SEM has previously been used in literature to evaluate factors that impact the sense of belonging in other contexts [77], [78]. In the following, we discuss the measurement model (i.e., operationalization of constructs), data collection, and analysis. A. Measurement model The theoretical model comprising the hypotheses is based on a number of theoretical concepts; some of the concepts may be directly observed (e.g., 'is paid'), but others cannot (e.g., sense of virtual community)-these concepts are represented as latent variables. A latent variable cannot be directly measured or observed but instead is measured through a set of indicators or manifest variables. For the latent variables in this study, we adapted existing measurement instruments. Sense of Virtual Community: We adapted items about feelings of a virtual community from Blanchard's [11] instrument of sense of virtual community to better fit with the context of OSS contributions. In collaboration with a group of Linux Kernel community managers, we analyzed the items proposed by Blanchard et al. [33] and decided to use a subset of questions to compose a shorter version of the instrument to cover the dimensions of SVC. The subset was discussed synchronously by researchers and managers, and the items were considered appropriate and meaningful to represent SVC to the Linux Kernel contributors. Intrinsic Motivations: We created items based on Gerosa et al. [22]'s instrument, which was built upon previous studies of motivations in OSS [18]- [20]. Following the community managers' request to make the questionnaire as short as possible, we grouped the intrinsic motivation factors from Gerosa et al.'s study [17] into three factors: 1. Social Motives (Kinship and Altruism) [47]; 2. Hedonic Motives (Joy and Fun) [23]; and 3. Moral motives (Ideology and Reciprocity) [53]. English Confidence: We reused four questions about the self-confidence of fluency levels during interactions by speaking and writing in technical and non-technical situations from a previous survey [79]. Power Distance: We asked in which country the respondent lived and used the value per country proposed by Hofstede's framework [58] as the Power Distance dimension in the model. For the demographic questions, we adapted questions from surveys used in OSS communities to ask about tenure, selfidentified gender, and compensation [80], [81]. Tenure was shown in 10-year slices in Table I for presentation purposes, but was included as a continuous variable in our analysis. We provided a dropdown list of years since 1991 (the year when the Linux Kernel was launched) for respondents to inform the year they started contributing to the project. B. Data Collection and Analysis We administered the online questionnaire using LimeSurvey, a leading Open Source survey software, to survey Linux Kernel contributors. We explored their motivations and their sense of virtual community. Our online appendix provides the instrument and replication package [82]. 1) Designing the instrument: The questions were discussed during 12 online meetings between October 2020 and February 2021 in a group of five researchers experienced in OSS and survey studies and two members of the Linux community. The group discussed each of the questions until reaching a consensus. The questionnaire provides informed consent followed by closed questions about the importance of each motivation factor as a reason to contribute to the Linux Kernel and questions about their feelings about the Linux Kernel community. Finally, we added demographic questions aiming to segment analysis and understand the phenomenon considering the different dimensions of our participants, and an open question for additional comments. Investigating the forces that push people with different individual characteristics can help us better support a diverse community [22]. The questions included gender identity, financial compensation, starting the year at Linux Kernel, and country of residence. 2) Piloting the questionnaire: After designing the instrument, we piloted the questionnaire before distributing it to the population of interest. In the first round, our collaborators from the Linux Foundation recruited a group of Linux Kernel maintainers, who answered the questionnaire and provided feedback. Although the feedback was overall positive, maintainers suggested reverse-coding some items for the SVC construct, i.e., items worded as negative statements (low score indicates agreement). Inverse, negative, or reverse-coded items can be defined as those having a directionality as opposed to the logic of the construct being measured [83]. We agreed with the suggestion and inverted two of the four items as this can help to mitigate acquiescence bias [84], which can occur when participants tend to agree with statements without regard for their actual content or due to laziness, indifference, or automatic accommodation to a response pattern [85]. The item I feel at home in the group was changed to I don't feel at home in the group. We inverted and adapted the question I feel that my contribution is valued to I want to contribute more but I do not feel valued. After the first pilot, we ran two more pilot sessions with two researchers who are Open Source contributors. We used the think-aloud method [86] and recorded their suggestions while answering the questions. We made minor changes to the questionnaire and increased font size for better readability on different devices. 3) Recruiting participants: The Linux Foundation contributors who collaborated in this study took the lead in recruiting the participants from the Linux Kernel. They reached out to maintainers and contributors using mailing lists from the Linux Kernel community and interacted with maintainers to ask for engagement. Further, we presented the study motivation during the first day of the Linux Plumbers annual conference (https://lpc.events/event/11/), inviting participants to answer the questionnaire. The survey was available between August 12 and September 21, 2021. 4) Sample Analysis: We received 318 responses and carefully filtered the data to consider only valid responses. Respondents who did not complete the whole questionnaire were dropped (n=26). We also dropped the participants who answered "I'm not sure" to any of the items included with the five-point Likert scale for Motivations (n=16) and Sense of Virtual Community (n=51). In addition to the 5-point Likert scale, we included a 6th alternative ("I'm not sure") for participants who either preferred not to, or did not know how to, answer the question (to avoid forcing them), which is different from being neutral-based on the dissonance between ignorance and indifference [87], [88]. Therefore, we considered these responses as missing data. The efficacy of imputation methods has not yet been validated when using FIMIX-PLS; Sarstedt et al. [89] recommend removing all responses with missing values for any question before segmenting data into clusters. After applying these filters, we retained 225 valid responses from residents of five different continents with a broad tenure distribution. The majority identified as men (84.4%), from Europe (52.9%), and paid to contribute (65.4%), matching previously reported distributions of OSS contributors [81]. Table I presents a summary of the demographics. To establish an appropriate sample size, we conducted a power analysis using the free GPower tool [90] (see online appendix for details). The maximum number of predictors in our model is six (three motivations and three control variables to SVC). This calculation indicated a minimum sample size of 62 and our sample of 225 exceeded that number considerably. We used the software package SmartPLS version 4 for the analyses. The analysis procedures for PLS-SEM comprise two main steps, with tests and procedures in each step. The first step is to evaluate the measurement model, which empirically assesses the relationships between the constructs and indicators (see Sec. V-A). The second step is to evaluate the theoretical (or structural) model that represents the hypotheses (see Sec. V-B). V. ANALYSIS AND RESULTS In this section, we describe our results, which include the evaluation of the measurement model (Sec. V-A), followed by the hypotheses evaluation in the structural model (Sec. V-B), both computed through our survey data. We assess the significance of our model by following the evaluation protocol proposed by previous research [76], [91] to make results consistent with our claims. The path weighting scheme was estimated using SmartPLS 4 [92]. Two tests are recommended to ensure that a dataset is suitable for factor analysis [93], [94]. We first conducted Bartlett's test of sphericity [93] on all constructs. We found a p-value < .01 (p values less than .05 indicate that factor analysis may be useful). Second, we calculated the Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy. Our result (.81) is well above the recommended threshold of .60 [94]. A. Evaluation of the Measurement Model Some of the constructs in the theoretical model (see Fig. 2) are modeled as latent variables, i.e., measured by more than one observed variable (i.e., item/question on the survey). The first step in evaluating a structural equation model is to assess the soundness of the measurement of these latent variablesthis is referred to as evaluating the 'measurement model' [91]. We present the assessment of several criteria. 1) Convergent Validity: First, we assess whether the questions (indicators) that represent each latent variable are understood by the respondents in the same way as they were intended by the designers of the questions [95], i.e., we assess the convergent validity of the measurement instrument. The assessment of convergent validity relates to the degree to which a measure correlates positively with alternative measures of the same construct. Our model contains two latent variables, both of which are reflective (not formative), as functions of the latent construct. Changes in the theoretical, latent construct are reflected in changes in the indicator variables [91]. We used two metrics to assess convergent validity: the Average Variance Extracted (AVE) and the loading of an indicator onto its construct (the outer loading). The AVE is equivalent to a construct's communality [91], which is the proportion of variance that is shared across indicators. A reflective construct is assumed to reflect (or "cause") any change in its indicators. The AVE should be at least .50, indicating that it explains most of the variation (i.e. 50% or more) in its indicators [91]. This variance is indicated by taking the squared value of an indicator's loading. As Table II shows, all AVE values for both latent constructs in our model are above this threshold of .50. A latent variable is measured by two or more indicators; indicators with loading below .4 should be removed because this implies that a change in the latent construct that it purportedly represents (or 'reflects') does not get reflected in a sufficiently large change in the indicator [91]. Outer loading of .7 is widely considered sufficient, and .6 is considered sufficient for exploratory studies [91]. We followed an iterative process to evaluate the outer loading of the latent constructs; the indicators of the construct English Confidence all exceeded .7, but SVC had two indicators below .7. We removed the SVC indicator which had a loading below .4 (svc6: a majority of the developers and I want the same thing). After removing this indicator, the AVE value of SVC (now with five indicators) increased from .44 to .51 and all outer loadings were above .60. 2) Internal Consistency Reliability: Second, we verified how well the different indicators are consistent with one another and can reliably and consistently measure the constructs, i.e., we assess the internal consistency reliability. A high degree of consistency means that the indicators refer to the same construct. There are several tests to measure internal consistency reliability. We performed both the Cronbach's α and Composite Reliability tests; Cronbach's α frequently shows lower values, whereas the Composite Reliability (CR) is a more liberal test, which sometimes overestimates the values [91]. A desirable range of values for both Cronbach's α and CR is between .7 and .9 [91]. Values below .6 suggest a lack of internal consistency reliability, whereas values over .95 suggest that indicators are too similar and thus not desirable. The Cronbach α and CR values for both latent variables fell in the range .75-.95; only the CR for English Confidence was slightly over at .951. AVE values were both higher than .50. 3) Discriminant Validity: Third, we verified whether each construct represents characteristics not measured by other constructs, i.e., we assessed the discriminant validity of the instrument (indicating the distinctiveness of the constructs). Our model includes two latent variables (SVC and English Confidence). A primary means to assess discriminant validity is to investigate the Heterotrait-monotrait (HTMT) ratio of correlations, developed by Henseler et al. [96]. The discriminant validity could be considered problematic if the HTMT ratio exceeds .9 [96]; some scholars recommend a more conservative cut-off of .85 [91]. The HTMT ratio between the two latent constructs (SVC and English Confidence) was .24. We also assessed the cross-loadings of indicators and the Fornell-Larcker criterion. Items should only load onto their 'native' construct, the one they purportedly represent (Table III). For the sake of completeness, we report the Fornell-Larcker procedure in the online appendix [82]. B. Evaluation of the Theoretical Model We now evaluate and discuss the theoretical model, which includes the evaluation of the hypotheses. 1) Assessing Collinearity: Our theoretical model has three different exogenous variables of intrinsic motivations, the moderators 'Compensation' and 'Power Distance,' and the control variables 'English Confidence,' 'Gender,' and 'Tenure. ' We hypothesized that the exogenous variables are associated with the endogenous variable Sense of Virtual Community. To ensure that the three exogenous constructs are independent, we calculate their collinearity using the Variance Inflation Factor (VIF). A widely accepted cut-off value for the VIF is 5 [91], and in our model, all VIF values are below 5. 2) Path Coefficients and Significance: PLS does not make strong assumptions about the distribution (such as a Normal distribution) of the data, so parametric tests of significance should not be used. To evaluate whether path coefficients are statistically significant, PLS packages employ a bootstrapping procedure. This involves drawing a large number (usually five thousand) of random subsamples with replacement. The replacement is needed to guarantee that all subsamples have the same number of observations as the original data set. The PLS path model is estimated for each subsample. From the resulting bootstrap distribution, a standard error can be determined [91], which can subsequently be used to make statistical inferences. The mean path coefficient determined by bootstrapping can differ slightly from the path coefficient calculated directly from the sample; this variability is captured in the standard error of the sampling distribution of the mean. Table IV shows the results for our hypotheses, including the mean of the bootstrap distribution (B), the standard deviation (SD), the 95% confidence interval, and the p-values. The path coefficients in Fig. 2 and Table IV are interpreted as standardized regression coefficients, indicating the direct effect of a variable on another. Each hypothesis is represented by an arrow in the diagram in Fig. 2. For example, the arrow pointing from Hedonic Motives to SVC represents H2. Given its positive path coefficient (0.421), Hedonic Motives are positively associated with SVC. The path coefficient is 0.421; this means that when the score for "Hedonic" motives increases by one standard deviation unit, the score for "Sense of Virtual Community" increases by 0.421 standard deviation unit (the standard deviation is the amount of variation of a set of values). Based on these results, we found support for Hypotheses H1 (p=.002), H2 (p=.000), H4a (p=.045), and H5b (p=.023). Hypothesis H3 was not supported, nor were H4b, H4c, H5a, or H5b (all p values > .2). The three control variables all have significant associations with SVC: English confidence, gender, and tenure (p < .05). 3) Coefficient of Determination: We assessed the relationship between constructs and the predictive capabilities of the model. The R 2 values of the endogenous variable in our model (SVC) was 0.4, which is considered weak-moderate [91], [97]. We also inspected Stone-Geisser's Q 2 [98] value, which is a measure of external validity, as an indicator of the model's predictive relevance [91], and can be obtained through a socalled blindfolding procedure (available within the SmartPLS software). Blindfolding is a resampling technique that omits certain data, predicts the omitted data points, then uses the prediction error to cross-validate the model estimates [99]. The Standardized Root Mean Square Residual (SRMR) is a common fit measure that is appropriate to detect misspecification of PLS-SEM models [76]. SRMR is the square root of the sum of the squared differences between the model-implied Fig. 2. Item loadings and path coefficients (p < 0.05 indicated by a full line). Non-significant links are indicated with a dashed line and the empirical correlation matrix, or the Euclidean distance between the two matrices [100]. A value of 0 for SRMR would indicate a perfect fit, and a cut-off value of 0.08 is considered adequate [101]. Our results suggest a good fit of the empirical data in the theoretical model (SRMR = 0.06). 4) Moderating Factors: We examined our data to determine if the impact of each intrinsic motivation on a sense of virtual community would change when they are exposed to a high Power Distance culture or when they are financially compensated to contribute. Only significant results at 0.05 are reported, with confidence intervals calculated through bootstrapping. • Power Distance Country Culture: Being surrounded by a high power distance culture, in which leaders impose a high level of control and restrict the information flow [58], has been reported to negatively affect the sense of virtual community [63]. We did not find significant correlations between Power Distance and SVC for hedonic or moral motivations. Still, we found it for social motivations, which has a moderating effect on our model. Hence, we found support for H4a but do not support H4b and H4c. • Compensation: Being paid to contribute reduces the sense of virtual community of contributors driven by hedonic motivations but not by social motivations and neither by moral motivations. Hence, we found support for H5b but rejected H5a and H5c. Fig. 3 presents an interaction diagram showing the simple slopes for the relationship between the exogenous variable Social Motives and the endogenous variable SVC. All three slopes are positive, indicating a positive relationship; the top line (in green) is at +1 standard deviation of the moderator, Power distance; the bottom slope (in red) is at −1 standard deviation of the moderator. The middle slope (in blue) represents the relationship at the mean level of Power distance. The figure shows that given a higher level of Power distance, the relationship between social motives and SVC is dampened (flatter), whereas with lower levels of Power distance, the relationship is strengthened (steeper). 5) Control Variables: We also examined our data to determine if being part of gender minorities, tenure, or English Confidence could strengthen or weaken the sense of virtual community. We found that participants who identify as gender minorities tend to have a lower sense of virtual community, while participants with higher tenure and English Confidence reported a higher sense of virtual community. C. Cluster Analysis: Detecting Unobserved Heterogeneity While moderators and context factors capture observed heterogeneity (see Sec. V-B4), there may also be unobserved heterogeneity, or latent classes of respondents, the presence of which could threaten the validity of results and conclusions [89]. Latent classes of respondents refer to some groupings of respondents on one or more criteria that were not measured. The hypothesis results may differ for different groups. We adopted Becker et al.'s approach [102], which jointly applies PLS-POS and FIMIX algorithms to identify latent classes. In Step 1, we used the minimum sample size for the maximum number of segments and ran FIMIX to find the optimal number of segments. In Step 2, we ran PLS-POS to compute the segmentation. In Step 3, we ran a multi-group analysis (PLS-MGA) and evaluated whether the segments were distinguishable. In Step 4, we checked if the resulting groups were plausible. We discuss the steps in more detail. In Step 1, we assessed the maximum number of segments according to the minimum sample size (see Sec. IV-B4). Dividing the sample size (225) by the minimum sample size (62) yields a theoretical upper bound of three segments; each segment should satisfy the minimum sample size. We ran FIMIX for one (meaning, treating the original sample as a single segment), two, and three segments [89]. The results were compared using several different retention criteria (see Table V) [89]. For each criterion, the optimal solution is the number of segments with the lowest value (in italics in Table V), except in terms of criterion 'EN,' where higher values indicate a better separation of segments. Sarstedt et al. [103] argue that researchers should start the fit analysis by jointly considering the combination of modified Akaike's Information Criterion with factor 3 (AIC3) and Consistent AIC (CAIC) (Group 1 in Table V): when both criteria suggest the same number of segments, this result is likely to be most appropriate. As this is not the case here (AIC3 suggests 3 segments, CAIC suggests 1 segment), a second evaluation considers whether modified Akaike's Information Criterion with factor 4 (AIC4) and Bayesian Information Criterion (BIC) suggest the same number of segments (Group 2 in Table V). Again, this is not the case as AIC4 suggests 3 segments, and BIC suggests 1 segment). The third evaluation (Group 3) considers the joint analysis of Akaike's Information Criterion (AIC) and Minimum Description Length with factor 5 (MDL5); first, consider the number of segments indicated by the lowest values of AIC (3 segments) and MDL5 (1 segment). The appropriate number of segments should be lower than suggested by AIC (because it tends to overestimate) and higher than the number of segments suggested by MDL5 (because it tends to underestimate). Hence, this combination suggests that a 2-segment solution is appropriate because 2 is lower than the 3 suggested by AIC and higher than the 1 suggested by MLD5. The value of EN is highest for the 2-segment solution. In Step 2, we evaluated the segment sizes of the 2-segment solution and proportions of data to check whether groups were substantial or candidates for exclusion. A segment is not substantial if its size is considerably lower in proportion (e.g., a 2% segment size) or below the minimum sample size [102]. The 2-segment solution divided the dataset into groups with 158 (70.2%) and 67 (29.8%) observations; both considerable portions and larger than the minimum sample size [102]. In Step 3, we ran a multi-group analysis (PLS-MGA) with parametric tests to verify whether the segments were distinguishable [102], i.e., whether the results were different for the two segments. We found significant differences in hypotheses H4b-c, H5a-c, and on the control variables Tenure and English Confidence (see Table VI), thus, we conclude these two segments represent two different groups of respondents. Both groups presented R 2 , goodness-of-fit (GoF), and SRMR [89] equal or more favorable than the original model. The values of the path coefficients and the explained variance of the endogenous variable SVC are shown in Table VI, which presents the results for the two segments, as well as the original estimates (see column B in Table IV). In Step 4, we examined that groups were "plausible" [102] by explaining the different segments (highlighted in gray in Table VI) to label the segments. This labeling is somewhat speculative and not definitive, not dissimilar to the labeling of emergent factors in exploratory factor analysis. Given that for Segment 1 only Hedonic motives are significant, we posit that this segment represents Hedonists (B=.31); for Segment 2, we find that Social motives are significant (B=.22), thus we label Segment 2 as Socially Motivated. We note that moral motives were not significant in the original analysis (see column 'Orig.'), but this did become significant with a negative coefficient (B=−0.23) for Segment 2. For hedonists (Seg. 1), tenure (B=.43) is positively associated with SVC. When social motives are associated with SVC (Seg. 2), English Confidence positively affects SVC (B=.88). Both hedonists (B=−0.50) and socially motivated (B=−0.61) contributors have the association with SVC weakened when they are paid. Both groups showed that being part of a gender minority is associated with less SVC. VI. DISCUSSION In this study, we developed a theoretical model grounded in psychology literature to map the relationship between a Sense of Virtual Community and intrinsic motivations in OSS. The theoretical model includes a number of salient factors that have been shown to be important in belonging to an online community in general but not yet within the OSS domain. Over the past two decades, the nature of OSS communities (as a specific type of online community) has changed; traditionally, OSS was male-dominated and primarily volunteer-based, but being paid to contribute is now common, and increasingly we observe the participation of what we refer to as "minorities" in the broadest sense of the word, including women [104]. Our analysis highlights a number of key findings and implications; as we discuss these quantitative results, we bring exemplar quotes from the respondents' responses to the final open question of the survey to illustrate the discussion. H1. Social motives → SVC: Social motives have a positive association with SVC. The intrinsic social motivations of kinship and altruism are positively associated with a sense of virtual community in OSS. This finding was corroborated by one of our respondents in the final open question, who associated SVC with social motivations: "I did not fit in, in a big way. I was never able to create enough social capital to make networking effective, no matter who I tried to connect with." Another respondent mentioned "not being able to relate to colleagues and named their perceived lack of SVC as "a sense of otherness that never goes away." However, the cluster analysis (Sec. V-C) indicated Segment 1 (which we labeled 'hedonic') to be non-significant, but Segment 2 (labeled 'social') is significant. We also found that for the 'socially motivated' English confidence is much more strongly related (B=.88 instead of .13) to SVC. This is intuitive because socially motivated people seek interaction, and English is the primary language within the Linux Kernel community. H2. Hedonic Motives → SVC: Hedonic motives have a positive association with a Sense of Virtual Community. OSS communities should seek to prevent toxic and other types of undesirable behavior that might reduce contributors' enjoyment; communities could also consider setting more clear community codes of conduct [74], [105], [106]. The cluster analysis showed that when only hedonism (not social motives) is associated with SVC (Seg. 1), Tenure is also associated with SVC. Hedonicmotivated contributors from our sample are also the ones who have longer tenure associated with SVC. Those contributors may have surpassed the initial barriers [107] and find enjoyment, or as mentioned by another respondent: "It is therapeutic. When I feel bad about myself, [..] it calms me down emotionally to do Kernel development when I feel like that." H3. Moral Motives → SVC: The cluster analysis does not support H3. While social motives are positively associated with SVC (Seg. 2), moral motives are negatively associated with and reduce SVC. The first association is expected and not surprising [47]. People motivated by kinship or because they are happy to help others are keener to be part of the team and feel good in a community [28], [48]. Interestingly, the SVC presented a negative association with moral motivation. We argue that people motivated by ideological reasons may contribute regardless of how they feel about belonging there. They do it because they feel it is the right thing to do, either because it is the most ethical choice, as advocated by the Free Software Foundation (https://www.fsf.org/), or because they have a moral debt to the software project that they use, so they pay back [53]. Future research can investigate how strong the ties between these people and the community are and what roles they play in building SVC for the rest of the community. H4a/b. Power Distance moderates the relationship between (a) Social and (b) Hedonic motives to SVC: Being surrounded by a culture with a high level of power distance weakens SVC for socially motivated contributors (when we consider all contributors). Still, if we consider Segment 1 (Hedonic) in the cluster analysis, we observe that power distance also weakens the SVC associated with hedonism. These results align with Cognitive Evaluation Theory [57]; contributors driven by hedonic (Seg. 1) or social motives (Seg. 2) need more autonomy (through less hierarchy-less Power Distance) to develop a Sense of Virtual Community. When not exerted in toxic and harsh ways to discipline community members, concerted control of communications can also ultimately play a pro-social role in increasing the SVC by increasing cohesiveness, commitment, and conformity [108]. H5a/b. Payment moderates the relationship between (a) Social and (b) Hedonic motives to SVC: Being paid to contribute weakens the association with SVC for hedonist contributors. The cluster analysis shows that being paid to contribute also weakens the SVC associated with social motives. Paid contributors, even those driven by hedonic or social motivations, showed a lower Sense of Virtual Community to the Linux Kernel. This result aligns with Cognitive Evaluation Theory [57] and might be explained by the conflicting identities and divided loyalties that paid contributors have to both their sponsoring firms and the Linux Kernel community [39]. We hypothesize that these contributors would leave the community if there were no payment to compensate for their participation. Implications for OSS communities to retain contributors SVC is associated with practices on exchanging support [15], [33], creating identities and making identifications [33], producing mutual cognitive and affective trust amongst members of a community [33], and establishing norms and a "concertive (but not enforced) control" [108], in which members of the community become responsible for directing their work and monitoring themselves. OSS communities can provide not only online interest groups for members, chat rooms, instant messaging, and discussion forums to encourage community involvement [109] but also online tools with shared spaces for contributors to work "together" on issues to be able to discuss and collaborate on similar interests. Better interactions can strengthen contributors' Sense of Virtual Community, especially those seeking social relationships. When the information being exchanged surpasses the technical content and includes socioemotional support, it shows personal relationships among group members, and finally brings feelings of acceptance by members [33]. OSS communities should foster exchanging support among members to bring a positive impact on developing SVC [15]. The exchanging support includes technical and social support and happens through comments in pull requests and participation in mailing lists (by either reading or posting messages). Communities can manage pull requests and mailing lists to guarantee that members' posts are not being missed [74] and that the communication adheres to the code of conduct. Implications for OSS communities to attract newcomers. Exchanging information and providing support to other community members are practices associated with positive feelings toward the community, and members' stronger attachment to the community [110]. Community members can encourage newcomers to become more active and move beyond the stage of 'lurker,' enticing them to participate in mailing lists [15] and to start making social connections to establish mutual trust, be known by other contributors, and facilitate the development of their Sense of Virtual Community. Conferences and meetups can help hedonic and socially motivated contributors have fun and increase their social capital. Implications for Research. This study suggests a positive link between Social and Hedonic motivations and a Sense of Virtual Community. Further, the cluster analysis has detected unobserved heterogeneity within our sample, suggesting that there are different subgroups within the community for which different motivations play a more prominent role. Future work could explore how the challenges faced by contributors influence the development of a Sense of Virtual Community and how a Sense of Virtual Community influences the decisions to stay or leave a project. While we included three control variables, future work can consider additional variables, for example, demographic variables such as age. Our study focuses on the Linux Kernel community, which is a limitation to the generalizability of this study; we suggest that our findings provide a useful starting point to conduct similar studies across other specific communities or across OSS developers regardless of which community they partake in. When considering other projects, we also suggest that different project governance models might also play a role in SVC. This study has also demonstrated that payment plays a role in SVC and that minorities and marginalized individuals feel less part of the community. Finally, our study has focused on the antecedents of a sense of virtual community in OSS but not on the consequences of it, and this could be a further area of focus in future work. Future work can investigate whether SVC is related to contributor satisfaction and whether a reduced SVC leads to contributors' attrition, thus jeopardizing a community's sustainability. VII. LIMITATIONS AND THREATS TO VALIDITY Construct Validity. We adopted and tailored existing measurement instruments for some constructs based on prior literature. Our analysis of the measurement model confirmed that our constructs were internally consistent and scored satisfactorily on convergent and discriminant validity tests. In this study we have used respondents' country of residence as a proxy for Power Distance as a dimension of culture as defined by Hofstede [59]. While also used in other studies [62], we acknowledge it is an approximation and not a perfect measure. One potential issue is that we do not know how long respondents have lived in their current country of residence. Another potential issue is that contributors' culture from where they grew up may differ from their current culture. This is why we report the metric as being surrounded by a specific culture instead of having a specific culture. Measuring culture in a more precise way is an important avenue for future work in general. Internal Validity. Our hypotheses propose associations between different constructs rather than causal relationships, as the present study is a cross-sectional sample study [111]. We acknowledge the limitation that our respondents comprise contributors who are more likely to have a sense of virtual community as they dedicated their time to answering the questionnaire, suggesting a response bias. While it is clear that contributors motivated by some intrinsic-social reasons tend to experience a sense of virtual community and that power distance and financial compensation can influence those associations, a theoretical model such as ours cannot capture a complete and exhaustive list of factors. Other factors can play a role and our results represent a starting point for future studies. External Validity. The Linux Kernel is a mature project that has attracted contributors for its value over the years, and while studied frequently and sometimes positioned as a 'quintessential' open source project, open source projects can vary in many ways. The specific context of the Linux Kernel project may therefore have impacted the results, which are therefore not necessarily generalizable to other OSS projects. Nevertheless, theory-building is a continuous and iterative process of proposing, testing, and modifying theory [112], in which a single case study is a first step towards constructing theories. In that sense, it is more valuable to interpret these results as a starting point and seek theoretical generalizability rather than statistical generalizability. Further, given the very important role that the Linux Kernel project plays in the software industry, this project we argue this is an appropriate starting point; further replication studies can validate and extend our theoretical model. Our survey was conducted online and anonymously, but the numbers are aligned with the overall distribution of the Linux Kernel contributors. The Linux Kernel includes contributions of more than 20,000 developers [73], and are mostly paid to contribute [66], [68]. According to previous research, around 10% of contributors to Linux Kernel identify themselves as women [80], and the majority is from the USA, which is aligned with our sample. The responses were sufficiently consistent to find full or partial empirical support for four hypotheses. VIII. CONCLUSION A Sense of Virtual Community (SVC) helps individuals feel valued in their community, leading to more satisfied, involved, and committed contributors. While research has identified different motivations and challenges for contributors to OSS, it is unclear how a sense of community is created and what factors impact it. In this paper, we close this gap by developing a theoretical model for sense of virtual community in OSS through a survey of Linux Kernel contributors. We found evidence that a subset of intrinsic motivations (social and hedonic motives) are positively associated with SVC; however, other extrinsic factors such as the country's culture and being paid to contribute can lessen SVC among contributors. Additionally, those with higher English confidence feel a higher sense of belonging in the community, and contributors who identify as part of a gender minority (non-men), tend to feel less of a sense of virtual community. Our results also show heterogeneity in our respondents, suggesting that there are different subgroups within the community for whom different motivations play a more prominent role. This suggests that a "one size fits all" approach would not work when designing interventions to create an inclusive, welcoming community. Our SVC model can help researchers and community design interventions by highlighting the different factors that interplay in creating a sense of virtual community in OSS for different subgroups of contributors. ACKNOWLEDGMENTS We are indebted to Kate Stewart, Vice President of Dependable Embedded Systems at the Linux Foundation, and Shuah Khan, maintainer and contributor of the Linux Kernel, for their invaluable support and assistance in the survey design and distribution. We much appreciate the extensive efforts and time they spent in meetings and reviews, and their thoughtful input to the survey design, and reaching out to the Linux Kernel community members. We are deeply grateful to all respondents of this survey. The National Science Foundation partially supports this work under Grant Numbers 1900903, 1901031, 2236198, 2235601, and Science Foundation Ireland grants no. 13/RC/2094-P2 to Lero, the SFI Research Centre for Software, and no. 15/SIRG/3293. ( Sec. V.C) Fig. 1. Research Design and Phases for Results' Analysis Q 2 values are calculated only for the SVC, the reflective endogenous construct of our model, with a value of .17. Values larger than 0 indicate the construct has predictive relevance, while negative values show the model does not perform better than the simple average of the endogenous variable would do. Fig. 3 . 3Power distance as a moderator of Social motives → SVC TABLE I DEMOGRAPHICS IOF THE LINUX KERNEL RESPONDENTS (N=225)Attribute N Percentage Gender Man 190 84.4% Woman 21 9.4% Non-binary 5 2.2% Prefer not to say 8 3.6% Prefer to self describe 1 0.4% Continent of Residence Europe 119 52.9% North America 68 30.2% Asia 32 14.2% South America 6 2.7% Starting year at the Linux Kernel 2000 or earlier 28 12.4% Between 2001 and 2010 77 34.2% Between 2011 and 2021 120 53.4% Current Compensation for the Linux Kernel contributions Paid 145 64.4% Unpaid (volunteer) 80 35.6% TABLE II INTERNAL IICONSISTENCY RELIABILITYCronbach's α CR AVEEnglish Confidence .932 .951 .830 Sense of Virtual Community .767 .839 .511 TABLE III CROSS IIILOADINGS OF THE RETAINED INDICATORS ON THE CONSTRUCTS Participating in technical discussions on the email list .278 .939SVC Engl. Conf. svc1 I don't feel at home in the group .740 .191 svc2 I feel that I belong to the group .822 .231 svc3 If I have a problem, I know members in the group who I can ask for help .662 .063 svc4 I want to contribute more but I do not feel valued .583 .113 svc5 A majority of the developers in the group know me .722 .242 eng1 Participating in a non-technical discussion on the email list .180 .881 eng2 Performing Reviews .216 .925 eng3 Speaking with others (face to face) .226 .890 eng4 TABLE IV STANDARIZED IVPATH COEFFICIENTS, STANDARD DEVIATIONS, CONFIDENCE INTERVALS, AND P VALUESB SD 95% CI p H1 Social motives→SVC .25 .11 (.04, .46) .002 H2 Hedonic motives→SVC .42 .11 (.19, .64) .000 H3 Moral motives→SVC −.14 .11 (-.36, .08) .215 H4a Power distance × social motives → SVC −.15 .08 (−.31,−.01) .045 H4b Power distance × hedonic motives →SVC −.05 .07 (−.18, .11) .477 H4c Power distance × moral motives→SVC .04 .07 (−.10, .17) .539 H5a is Paid × social motives→SVC −.05 .07 (−.32, .20) .696 H5b is Paid × hedonic motives→SVC −.33 .14 (−.62, .05) .023 H5c is Paid × moral motives→SVC .12 .14 (−.17, .36) .404 Gender minorities→SVC −.49 .17 (−.81,−.14) .004 English confidence→SVC .13 .01 (.01,.25) .025 TABLE V VESTABLISHING ADEQUATE NUMBER OF SEGMENTS Group Criterion 1-Segment 2-Segment 3-Segment 1 AIC3 574.153 540.241 517.508 CAIC 625.395 646.141 678.065 2 AIC4 589.153 571.241 564.508 BIC 610.395 615.141 631.065 AIC 559.153 509.241 470.508 3 MDL5 935.361 1286.737 1649.292 4 EN 0 0.869 0.821 TABLE VI GROUP VIPATHS COEFFICIENTS: COEFICIENTS IN BOLD ARE SIGNIFICANT; LINES IN GRAY SHOW SIGNIFICANT DIFFERENCE BETWEEN SEGMENTS2-segment solution Orig. 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[ "Linear and nonlinear spin current response of anisotropic spin-orbit coupled systems", "Linear and nonlinear spin current response of anisotropic spin-orbit coupled systems" ]	[ "D Muñoz-Santana \nCentro de Investigación Científica y de Educación Superior de Ensenada\nApartado Postal 360\n22860Baja California, Ensenada, Baja CaliforniaMéxico\n", "Jesús A Maytorena \nCentro de Nanociencias y Nanotecnología\nUniversidad Nacional Autónoma de México\nApartado Postal 1422800Ensenada, Baja CaliforniaMéxico\n" ]	[ "Centro de Investigación Científica y de Educación Superior de Ensenada\nApartado Postal 360\n22860Baja California, Ensenada, Baja CaliforniaMéxico", "Centro de Nanociencias y Nanotecnología\nUniversidad Nacional Autónoma de México\nApartado Postal 1422800Ensenada, Baja CaliforniaMéxico" ]	[]	We calculate the linear and the second harmonic (SH) spin current response of two anisotropic systems with spin orbit (SO) interaction. The first system is a two-dimensional (2D) electron gas in the presence of Rashba and k-linear Dresselhaus SO couplings. The dependence of the anisotropic spin splitting on the sample growth direction introduces an additional path to modify the linear and nonlinear spectra. In particular, vanishing linear and second order spin conductivity tensors are achievable under SU(2) symmetry conditions, characterized by a collinear SO vector field. Additional conditions under which specific tensor components vanish are posible, without having such collinearity. Thus, a proper choice of the growth direction and SO strengths allows to select the polarization of the linear and SH spin currents according to the direction of flowing. The second system is an anisotropic 2D free electron gas with anisotropic Rashba interaction, which has been employed to study the optical conductivity of 2D puckered structures with anisotropic energy bands. The presence of mass anisotropy and an energy gap open several distinct scenarios for the allowed optical interband transitions, which manifest in the linear and SH response contrastingly. The linear response displays only out-of-plane spin polarized currents, while the SH spin currents flow with spin orientation lying parallel to the plane of the system strictly. The models illustrate the possibility of the nonlinear spin Hall effect in systems with SO interaction, under the presence or absence of time-reversal symmetry. The results suggest different ways to manipulate the linear and nonlinear optical generation of spin currents which could find spintronic applications. arXiv:2302.14238v1 [cond-mat.mes-hall]	null	[ "https://export.arxiv.org/pdf/2302.14238v1.pdf" ]	257,232,824	2302.14238	a07cc04c4ff7d35da40afad3e9be17cb0d9098cc	Linear and nonlinear spin current response of anisotropic spin-orbit coupled systems D Muñoz-Santana Centro de Investigación Científica y de Educación Superior de Ensenada Apartado Postal 360 22860Baja California, Ensenada, Baja CaliforniaMéxico Jesús A Maytorena Centro de Nanociencias y Nanotecnología Universidad Nacional Autónoma de México Apartado Postal 1422800Ensenada, Baja CaliforniaMéxico Linear and nonlinear spin current response of anisotropic spin-orbit coupled systems (Dated: March 1, 2023) We calculate the linear and the second harmonic (SH) spin current response of two anisotropic systems with spin orbit (SO) interaction. The first system is a two-dimensional (2D) electron gas in the presence of Rashba and k-linear Dresselhaus SO couplings. The dependence of the anisotropic spin splitting on the sample growth direction introduces an additional path to modify the linear and nonlinear spectra. In particular, vanishing linear and second order spin conductivity tensors are achievable under SU(2) symmetry conditions, characterized by a collinear SO vector field. Additional conditions under which specific tensor components vanish are posible, without having such collinearity. Thus, a proper choice of the growth direction and SO strengths allows to select the polarization of the linear and SH spin currents according to the direction of flowing. The second system is an anisotropic 2D free electron gas with anisotropic Rashba interaction, which has been employed to study the optical conductivity of 2D puckered structures with anisotropic energy bands. The presence of mass anisotropy and an energy gap open several distinct scenarios for the allowed optical interband transitions, which manifest in the linear and SH response contrastingly. The linear response displays only out-of-plane spin polarized currents, while the SH spin currents flow with spin orientation lying parallel to the plane of the system strictly. The models illustrate the possibility of the nonlinear spin Hall effect in systems with SO interaction, under the presence or absence of time-reversal symmetry. The results suggest different ways to manipulate the linear and nonlinear optical generation of spin currents which could find spintronic applications. arXiv:2302.14238v1 [cond-mat.mes-hall] I. INTRODUCTION The two-dimensional electron gas (2DEG) in semiconductor heterostructures has maintained its relevance even after almost two decades of constant discovery of new 2D materials. Even before graphene [1], these quantum wells were the subject of constant research due to their potential applications in the field of spintronics that the presence of spinorbit (SO) coupling provides [2,3]. The spin Hall effect [4], the current-induced spin polarization [5], a nonballistic version of a spin transistor [6,7], and the persistent spin helix state [8] are remarkable examples of spin-dependent phenomena associated to the SO interaction in 2DEGs. The art of manipulating spin using SO coupling gave birth to the field called spin-orbitronics, also dubbed "Rashba-like physics" [9][10][11], inspired by the Rashba SO coupling, which leads to spin-momentum locking in low dimensional systems lacking inversion symmetry [3]. One of the most studied 2DEG systems is the one that includes the Rashba and the linear Dresselhaus SO contributions. The interplay between these couplings leads to an anisotropic spin-splitting responsible of effects like a characteristic frequency dependence of the charge and spin Hall conductivities [12], the anisotropy of plasmon dynamics [13], or persistent spin textures [14][15][16], among others. It also allows the possibility of spin-preserving symmetries and associated effects like the infinite spin lifetime due to fixed spin precession axis [15], the mentioned nonballistic spin-FET [6], the suppression of spin beats in oscillations of magnetoresistivity [17], the vanishing of Zitterbewegung [18,19], the vanishing of the interband absorption and spin Hall conductivity [20], or the cancellation of plasmon damping [13], all with potential device applications. Despite the extensive research, the investigation has usually been restricted to quantum wells grown along the [001], [110], or [111] direction [6,14,15,[21][22][23][24][25][26][27][28][29][30][31][32][33][34]. It is until recently that attention has been paid to a general crystal orientation [35][36][37]. It was found that the recovery of the SU(2) symmetry can be realized for some growth direction Miller indices, with consequences like the transition from weak anti-localization to weak localization [35] or lifetime enhancement of spin helices [38]. Recently, the optical generation of dc spin currents by second-order nonlinear interactions in 2DEGs with SO coupling has also been explored [30,39]. Another variant of 2D electron systems are new materials presenting a behavior similar to an electron gas with another type of in-plane anisotropy. These systems, such as phosphorene and group IV metal monochalcogenides, have an orthorhombic lattice and puckered structure with highly anisotropic energy bands, providing novel physical phenomena. Recently, the optical conductivity of such materials was calculated from a model Hamiltonian consisting of an anisotropic 2DEG [40] with an anisotropic Rashba splitting [41]. The optical absorption spectrum revealed a sensitive dependence on the mass anisotropy ratio and on the frequency and direction of the exciting field [42]. In this paper, we calculate the linear and the second harmonic (SH) spin current response of two anisotropic systems with SO interaction. First, we consider a 2DEG with Rashba and linear-in-momentum Dresselhaus couplings. We explore how the dependence of the anisotropic spin splitting on the sample growth direction introduces an additional path to modify the linear and nonlinear spectra. In particular, we find that under SU(2) symmetry conditions, the linear and second order spin currents vanish. On the other hand, a proper choice of the growth direction and SO strengths allows to select their respective spin polarization according to the direction of flowing. In reference [30], a nonlinear dc spin current generation of a Rashba-Dresselhaus coupled 2DEG, grown along the [001] direction, was investigated as an optical rectification response using a semiclassical Boltzmann approach in the relaxation-time approximation. Here instead, we focus on the finite frequency SH response in the clean limit of the system, and for arbitrary crystallographic direction of growth. The second system is the same mentioned above [42] to study the optical response of 2D SO coupled puckered structures with mass anisotropy. We extend it by introducing an small energy gap in the Hamiltonian to represent a possible splitting of the conduction band [43]. This allows us to switch between a model which preserves the timereversal symmetry (TRS) and one which breaks it. We find that the dependence of the spectrum of allowed interband transitions on the mass ratio and energy gap, leads to out-of-plane spin polarized linear currents, while the SH spin currents flow with the spin oriented parallel to the plane of the system strictly. Our results suggest different ways to manipulate the polarization and direction of the optically induced linear and SH spin current responses. The paper is organized as follows. In section II the Kubo expressions for the dynamical linear and second-order spin current conductivity tensors of a generic two-band model are presented. In sections III we evaluate the formulae obtained in section II for the 2DEG with generic linear-in-momentum SO interaction and discuss the spectral features of the optical spin current response at the fundamental and SH frequencies. To this end, we first calculate the joint density of states (JDOS) and explain the origin of its van Hove singularities. Similarly, in section IV we discuss the spectral characteristics of the JDOS, and calculate the linear and SH spin current conductivities for the gapped anisotropic Rashba model. The conclusions are summarized in section V. II. FIRST AND SECOND ORDER KUBO FORMULAE In this section we obtain the conductivity tensors characterizing the spin current response which depend linearly and quadratically on an externally applied electric field. We shall consider a generic two band model in 2D described by the momentum-space Hamiltonian H(k) = ε 0 (k)I + σ · d(k), with energy spectrum ε λ (k) = ε 0 (k) + λd(k), where k is the in-plane electron wave vector k = k xx + k yŷ = k(cos θx + sin θŷ), d(k) = d x (k)x + d y (k)ŷ + d z (k)ẑ, d(k) = \|d(k)\|, σ is the vector of Pauli matrices defined in the spin space, I is the 2 × 2 unit matrix, and λ = ± specifies the helicity of the states in the upper (+) and lower (−) part of the spectrum. The eigenstates are \|λ, k = [N λ , λN −λ exp (iφ)] T , with [N λ (k)] 2 = (d + λd z )/2d and tan φ(k) = d y /d x . In what follows we simplify the notation and write \|λ for this states. Within the Kubo formalism, the linearly induced spin current is given by J i (t) (1) = 1 i t −∞ dt λ,k f (ε λ (k)) λ\|[Ĵ i (t),Ĥ (t )]\|λ ,(1) where all operators are in the interaction picture with respect to the unperturbed Hamiltonian. Here,Ĵ i = 4 (σ v i + v i σ ) is the operator for the -polarized spin current flowing in the i-direction, with v i = ∂H/∂k i defining the velocity operator componentv i , (i = x, y; = x, y, z). The operatorĤ (t) = (e/c)v i A i (t) (sum over repeated index is assumed hereafter) contains the interaction of the electrons with a spatially homogeneous vector potential A(t), (e > 0). The factor f λ = f (ε λ (k)) is the Fermi-Dirac occupation number of the band ε λ (k). Assuming an applied uniform electric field of the form E i (t) = E i (ω)e −iωt + c.c. = −(1/c)∂A i /∂t, the time integral leads to a first order induced spin current J i (t) = e −iωt σ ij (ω)E j (ω) + c.c., with the spin conductivity tensor given by σ ij (ω) = ie ω λλ k (f λ − f λ ) λ\|Ĵ i \|λ λ \|v j \|λ ω − ω λ λ ,(2) whereω = ω + i0 + , i, j = x, y and = x, y, z. Explicitly, we have σ ij (ω) = ie 2 ω d 2 k (2π) 2 (f − − f + ) ∂ε 0 ∂k i ∂d p ∂k j M + p ω + 2d − M + p ω − 2d ,(3) where M λ ij (k) = λ\| σ i \|−λ −λ\| σ j \|λ = (d 2 δ ij − d i d j )/d 2 + iλ ijk d k /d. At second order, the Kubo formula for the induced spin current reads as J i (t) (2) = 1 (i ) 2 t −∞ dt t −∞ dt λ,k f (ε λ (k)) λ\|[[Ĵ i (t),Ĥ (t )],Ĥ (t )]\|λ .(4) For this case we assume an applied electric field E i (t) = E i (ω 1 )e −iω1t + E i (ω 2 )e −iω2t + c.c. and focus on the sum frequency (SF) contribution proportional to E j (t )E l (t ). The time integrals leads to a second order SF spin current J i (t) SF = e −i(ω1+ω2)t σ ,SF ijl (ω 1 , ω 2 )E j (ω 1 )E l (ω 2 ) + c.c. , with the SF spin conductivity tensor given by σ ,SF ijl (ω 1 , ω 2 ) = e 2 2ω 1ω2 ℘ λλ λ d 2 k (2π) 2 λ Ĵ i λ λ v j λ λ v l λ ω 3 − ω λ λ f λ − f λ ω 2 − ω λ λ − f λ − f λ ω 1 − ω λ λ ,(5) where i, j, l = x, y and = x, y, z. Hereω i = ω i + i0 + , i = 1, 2, 3, ω 3 = ω 1 + ω 2 , and ℘ stands for the intrinsic permutation symmetry (j, ω 1 ) ↔ (l, ω 2 ). We shall be focused on the second harmonic response, σ ,(2ω) ijl (ω) = σ ,SF ijl (ω, ω), σ ,(2ω) ijl (ω) = e 2 ( ω) 2 ℘ d 2 k (2π) 2 (f − −f + ) 1 d ∂ε 0 ∂k i ∂d p ∂k j ∂d q ∂k l d M + pq ( ω) 2 − (2d) 2 − d p M + q ( ω + 2d)(2 ω + 2d) − d p M + q ( ω − 2d)(2 ω − 2d) .(6) In the following, we use expressions (3) and (6) to study the spin current response of two anisotropic SO coupled systems. III. 2DEG WITH RASHBA AND DRESSELHAUS[hkl] SO COUPLING A. Joint density of states We consider a 2DEG with an arbitrary crystal orientation defined by the unit normaln = n xx + n yŷ + n xẑ , with underlying basis vectorsx,ŷ,ẑ, pointing along the crystal axes [100], [010], [001], respectively. The system is in the presence of Rashba and linear-in-k Dresselhaus SO couplings. The corresponding Hamiltonian is determined by the function ε 0 (k x , k y ) = 2 k 2 /2m and the SO vector field d i (k x , k y ) = µ iν k ν (i = x, y, z; ν = x, y). For a given normaln, the xyz coordinate system can be rotated in order to obtain a new x y z coordinate system, with the z -axis pointing along the direction ofn, and orthonormal basisl,m,n,l =m ×n. Thus, for each orientation the matrix of SO material parameters µ ij must be understood as referred to the new coordinates, as well as the condition k ·n = 0, which is transformed to the condition k z = 0. We shall use the symbol R+D[hkl] to indicate the Rashba (R) and Dresselhaus (D) SO couplings when the sample is grown along the crystallographic direction [hkl]. The k-space available for vertical transitions is determined by the condition ε − (k) ε F ε + (k) ('Pauli blocking'), and the conservation of energy ε + (k x , k y ) − ε − (k x , k y ) = ω, for a given Fermi energy ε F and exciting frequency ω. This means that only those points (k x , k y ) lying on the resonance curve C r (ω) = {(k x , k y )\|2d(k x , k y ) = ω} which satisfy the inequality k + F (θ) < k < k − F (θ) will contribute to the joint density of states (JDOS) [see Fig. 1(a)]. Here k λ F (θ) are the Fermi contours, defined by the equation ε λ (k x , k y ) = ε F which, written in polar coordinates, are given by k λ F (θ) = 2mε F / 2 + k 2 so (θ) − λk so (θ), where k so (θ) = mg [hkl] (θ)/ 2 is a characteristic wave number of the SO interaction. The function g [hkl] (θ) accounts for the anisotropy of the energy splitting 2d(k x , k y ) = 2kg [hkl] (θ), with g [hkl] (θ) = \|µ x cos θ + µ y sin θ\|, where the vectors µ ν , with components (µ ν ) i = µ iν , are the columns of the matrix µ iν . The above restrictions imply that the allowed transitions are possible only in the energy window ω + (θ) ω ω − (θ), where ω λ (θ) = 2d(k λ F (θ)) = 2k λ F (θ)g [hkl] (θ) is the minimum (maximum) photon energy ω + (θ) ( ω − (θ)) required to induce a vertical transition between states lying along the direction θ in the k-space [inset in Fig. 1(a)]. As a consequence, the JDOS for the system with SOI R+D[hkl] reads as J +− (ω) = ω 16π 2 dθ Θ[1 − \|η(ω, θ)\|] g 2 [hkl] (θ) ,(7) where Θ(x) is the Heaviside unit step function, and η(ω, θ) = [ω − 1 2 (ω − (θ) + ω + (θ))]/[ 1 2 (ω − (θ) − ω + (θ)) ]. According to this expression the edges of the absorption spectrum will be at the photon energies ω = min θ { ω + (θ)} and ω = max θ { ω − (θ)}, which will occur along some directions θ < and θ > where the energy splitting is minimum and maximum, respectively. In general, the set of van Hove singularities can be identified geometrically by analyzing how k + F (θ) < k < k − F (θ), for which ε−(k) ≤ εF ≤ ε+(k) (see inset). (b) The JDOS spectrum for several crystal orientations; the parameters used are α = 160 meVÅ, γk 2 n = 0.5α, and n = 5 × 10 11 cm −2 , m = 0.05m0 for the electron density and effective mass. the resonance curve C r (ω) enters, intersects, and leaves the region of allowed transitions k + Fig. 1(a)] as frequency varies. The resonance curve is a rotated ellipse with equation \|k x µ x + k y µ y \| = ω/2. After a rotation by an angle ζ defined by tan 2ζ = 2µ x · µ y /[\|µ x \| 2 − \|µ y \| 2 ], the ellipse becomes of the standard form with principal axes given by Q x (ω) = ω/2g [hkl] (θ < ) and Q y (ω) = ω/2g [hkl] (θ > ), where θ > = θ < + π/2. In order to maintain the angle θ < as the direction of global minimum energy separation (as defined above) we choose θ < = ζ if g [hkl] (ζ) < g [hkl] (ζ + π/2), and θ < = ζ + π/2 otherwise. The critical van Hove energies are given by those values of ω at which the axes Q x (ω) and Q y (ω) of the resonance ellipse touch tangentially the Fermi contours k ± F (θ) along θ < and θ > . Figure 1 F (θ) < k < k − F (θ) [shaded area inQ x (ω + ) = k + F (θ < ), Q x (ω a ) = k − F (θ < ), Q y (ω b ) = k + F (θ > ), and Q y (ω − ) = k − F (θ < ) : ω + = 2k + F (θ < )g [hkl] (θ < ),(8)ω a = 2k − F (θ < )g [hkl] (θ < ),(9)ω b = 2k + F (θ > )g [hkl] (θ > ),(10)ω − = 2k − F (θ > )g [hkl] (θ > ).(11) Note that while ω + is always smaller than ω − , the order relation between ω a and ω b can change for different orientations. In Fig. 1(b) we show the JDOS for different crystal orientations as a function of frequency, keeping the same SO parameters values. The overall shape and size of the spectra reveals a strong dependence on the direction of sample growth. For the case R+D[111], the Hamiltonian is formally identical to that of a system with Rashba coupling only, which presents an isotropic splitting of the states. Thus, the resonance curve and Fermi contours are concentric circles and the JDOS displays the well known box-like shape with only two spectral features ( ω ± ). However, for other orientations the k-space for allowed optical transitions becomes no longer isotropic, and two types of shapes may appear, depending on the relative values of ω a and ω b . When ω a < ω b the spectra can develop a convex shape between these critical energies, as shown in Fig. 1(b) for the R+D[001] and R+[123] systems. On the other hand, when ω a > ω b the JDOS presents a linear dependence instead, as is illustrated by the R+D [110] case. This can be explained by observing how the resonance curve C r (ω) overlaps the region of allowed transitions bounded by the Fermi contours in each case. When ω a < ω b , the semi-axis Q x (ω) of the curve C r (ω) will reach the line k − F (θ) first before the semi-axis Q y (ω) intersects the line k + F (θ), which means that for ω a < ω < ω b there is a portion of the curve which does not contribute to the JDOS. In contrast, when ω a > ω b we have the opposite situation, the semi-axis Q x (ω) touch the Fermi contour k − F (θ) after the axis Q y (ω) contacts the Fermi contour k + F (θ). This imply that there is a range of frequencies, ω b < ω < ω a , for which the ellipses C r (ω) lie entirely within the allowed zone (shaded area in Fig. 1(a)), causing a linear increase in the JDOS. The dependence of the critical frequencies (8)-(11) on the Miller indices, suggests the growth direction as an additional element of control of the spectrum of optical transitions. B. First order spin current conductivity The spin conductivity tensor at the fundamental frequency for the SO coupled 2DEG, as obtained from Kubo formula (3), is (no sum over repeated indices is implied) Re σ ij (ω) = σ ij (0) − e 8π (µ x × µ y ) ω 8m/ 2 1 2π 2π 0 dθ g 4 (θ) ln [ω + ω + (θ)] [ω − ω − (θ)] [ω − ω + (θ)] [ω + ω − (θ)](12)× [ ijz + sin 2θ (δ iy − δ ix ) δ ij + cos 2θ (1 − δ ij )] Im σ ij (ω) = − e 8π (µ x × µ y ) ω 16m/ 2 2π 0 dθ g 4 (θ) [ ijz + sin 2θ (δ iy − δ ix ) δ ij + cos 2θ (1 − δ ij )] Θ[1 − \|η(ω, θ)\|],(13) where σ ij (0) = − e 8π (µ x × µ y ) 1 2π 2π 0 dθ g 2 (θ) [ε ijz + sin 2θ (δ iy − δ ix ) δ ij + cos 2θ (1 − δ ij )] ,(14) is the dc spin conductivity. Note that σ yy (ω) = −σ xx (ω). Complex integration gives the result σ ij (0) = − e 8π (µ x × µ y ) \|µ x × µ y \|   −Im (z + ) 1 + Re (z + ) −1 + Re (z + ) Im (z + )   ,(15) where z + = − [A − A 2 − (B 2 + C 2 )] B 2 + C 2 (B + iC) ,(16) with A = (\|µ x \| 2 + \|µ y \| 2 )/2, B = (\|µ x \| 2 − \|µ y \| 2 )/2, and C = µ x · µ y . The general expression (15) extends the result reported in Ref. [44], which is valid for SO vector fields with d z (k) = 0 (µ zx = µ zy = 0) only. Since (µ x × µ y ) x and (µ x × µ y ) y vanish for µ zν = 0 we have that the only systems supporting a linear spin current with perpendicular-to-plane spin polarization strictly, are those grown along the [001] and [111] directions; any other crystal orientation will have in-plane spin polarized current components. Moreover, the vanishing of the common factor µ x × µ y implies the absence of an induced spin current via electric-dipole interaction in the 2DEG with R+D[hkl] SO coupling. This reminds the well known effects due to the recovery of the SU(2) symmetry predicted in systems with R+D[001], like the infinite spin lifetime due to fixed spin precession axis [15] or the formation of a persistent spin helix state [14,16]. As we mentioned before, Kammermeier et al. [35] found that for an arbitrary crystal orientation is still possible to have conditions for spin-preserving symmetries due to the interplay of Rashba and Dresselhaus SOI. The requirement for that is to have samples with two Miller indices equal in modulus and a particular relation between the Rashba and Dresselhaus parameters (in our language, a proper combination of the elements of the SO matrix µ ij ) [35]. It can be verified that for these special conditions, the factor µ x × µ y is zero. Without loss of generality, lets choose, after Kammermeier, the orientationn = (η, η, n z ), withm = (−1, 1, 0)/ √ 2 andl = (n z , n z , −2η)/ √ 2, where n 2 z = (1 − 2η 2 ). The corresponding vectors of SO parameters become µ x = (0, −α + γk 2 n (1 − 9η 2 )n z , 0) and µ y = (α+γk 2 n (1+3η 2 )n z , 0, −γk 2 n √ 2η(1−3η 2 )), where α and γk 2 n are the Rashba and Dresselhaus coupling strengths, respectively. When these satisfy the relation α/γk 2 n = (1 − 9η 2 )n z (µ yx = 0), the SO vector field becomes collinear, d(k) = γk 2 n (3η 2 − 1)(−2n z , 0, √ 2 η)k y . Under these conditions, µ x × µ y = µ yx (µ zy , 0, −µ xy ) = 0, implying the vanishing of the spin current. Note also that when µ yx = 0, and µ zy = 0 (which is true for the [111] growth direction only, η 2 = 1/3) or µ xy = 0, the polarization of the spin current is along the directionn (z ) orl (x ), respectively. In contrast, the spin current polarized parallel to them direction is null regardless the magnitude of the SO strength parameters, σ y ij (ω) = 0. Interestingly, although the condition µ xy = 0, which rewrites as α/γk 2 n = −(1 + 3η 2 )n z , does not corresponds to a collinear SO vector field, still produces an absence of out-of-plane-polarized spin current. In Fig. 2, a typical spectrum of the linear spin current conductivity σ z ij (ω) is displayed, in this case for the R+D[123] system. As anticipated by the JDOS, we can identify the presence of van Hove features at the critical energies (8)-(11), defined by the Pauli blocking and the energy conservation condition for vertical transitions ( Fig. 1(b)). Given that the critical frequencies (8)- (11) depend not only on the magnitude and relative value of the SO material parameters but also on the crystal orientation, our results suggests a kind of spectral control of the overall shape of the linear spin current response by choosing appropriately the samples in advance. C. SH spin current conductivity The second-order Kubo formula (6) leads to the SH spin conductivity of the R+D[hkl] system, σ ,(2ω) ijl (ω) = e 2 ( ω) 2 ( 2 /m) (2π) 2 (µ x × µ y ) p 2π 0 dθ g 6 (θ)k ikν (µ x × µ y ) p µ ν (δ jl −k jkl )C(ω, θ) (17) − pq µ qρkρ (µ j · µ ν )(k x δ ly −k y δ lx )[C(ω, θ) − 4C(2ω, θ)] + (j ↔ l) wherek i (θ) = cos θδ ix + sin θδ iy and C(x, θ) = − 1 4 2mg 2 (θ) 2 + x 4 ln [x + ω + (θ)][x − ω − (θ)] [x − ω + (θ)][x + ω − (θ)] − iπ x 16 Θ[1 − \|η(x, θ)\|].(18) As in the linear response, the system grown alongn = (η, η, n z ) will not support a spin current at 2ω when µ yx = 0, or equivalently when the SO field d i (k) = µ iν k ν becomes collinear. However, for the same special class of orientations, to analyze the spin conductivity for each spin index , we have to focus in the following factors (µ x ) = µ yx δ y ,(19)(µ y ) = (µ xy δ x + µ zy δ z ) ,(20)(µ x × µ y ) × µ x = µ 2 yx µ xy δ x + µ zy δ z ,(21) (µ x × µ y ) × µ y = −µ yx µ 2 xy + µ 2 zy δ y . We note that there are some possibilities to control the polarization of the nonlinear spin current. For µ yx = 0, the condition µ xy = 0 (µ zy = 0) corresponds to have a spin current flowing in the plane defined bym andn (l and (ω) = 0 is possible only for µ yx = 0. Therefore, we have that for a system with crystal orientationn = (η, η, n z ), and given that the linear σ y ij (ω) = 0, the generation of a spin current polarized along them (y ) direction will depend quadratically on the electric field, because it is induced as a second-order response strictly. Fig. 3 shows the SH spin current conductivity σ z,(2ω) ijl (ω) for the R+D[123] system. As expected, beside the peaks related with critical points at the energies ω ± , ω a , ω b , van Hove singularities appear also at their subharmonics. Similarly to the first-order spin current conductivity, the magnitude and direction of the nonlinear spin current could be modified through frequency variation, relative value of the SO strengths, or by a proper choice of the sample growth direction. Another aspect worth noting is that the R+D[hkl] systems will present a nonlinear spin Hall effect, through σ In this section we considered another anisotropic model, used to study the effect of an anisotropic Rashba splitting on the longitudinal optical conductivity of 2D puckered structures [42] like black phosphorus and group IV monochalcogenides [45][46][47]. Here the reduced symmetry of the splitting of the states is due to mass anisotropy. The kinetic contribution to the low energy Hamiltonian is that of an anisotropic free electron gas [40,48], ε 0 (k) = 2 k 2 x /2m x + 2 k 2 y /2m y , while the Rashba SO field is taken as d(k) = α( m d /m y k yx − m d /m x k xŷ ) + ∆ẑ, where α is the SO strength and m d = √ m x m y is the geometric mean of the masses m x and m y along the x-and y-directions [41]. The model has been extended to include an energy parameter, ∆ 0, which gives rise to a gap in the energy dispersion; by taking ∆ = 0 or ∆ = 0 we can move from a model with TRS to a model with broken TRS. This massive anisotropic Rashba low-energy model could describe a gapped conduction band of the phosphorene monolayer [43]. Written in polar coordinates, the conduction (λ = +) and valence (λ = −) bands are ε λ (k, θ) = 2 k 2 g 2 (θ)/2m d + λ α 2 k 2 g 2 (θ) + ∆ 2 . The function g(θ) = [(m d /m x ) cos 2 θ + (m d /m y ) sin 2 θ] 1/2 measures the separation of energyconstant curves along the direction θ in the k-space; note that g(θ) = 1 when m x = m y , which corresponds to the well known case of a magnetized 2DEG with Rashba coupling in quantum wells of semiconductor heterostructures [49]. The energy difference between the bands is ε + (k) − ε − (k) = 2d(k) = 2 α 2 k 2 g 2 (θ) + ∆ 2 and the constant energydifference curve, C r (ω) = {(k x , k y )\|ε + (k) − ε − (k) = ω} is the ellipse with equation ε 0 (k x , k y ) = [( ω/2) 2 − ∆ 2 ]/2ε R , where ε R = m d α 2 / 2 is a characteristic energy associated to the Rashba interaction. The shape of the band ε − (k) depends on the ratio p = ∆/ε R . When p < 1, the band acquires a mexican hat shape, with a local maximum −∆ at the origin k = 0 and two local minimums of value ε min = −(ε 2 R + ∆ 2 )/2ε R at k-points lying on the ellipse ε 0 (k x , k y ) = (ε 2 R − ∆ 2 )/2ε R . Otherwise, the valence band develops only a minimum at the origin. This means that there are several distinct positions for the Fermi level: (i) ε F > ∆ (Fig. 4(a)), and then two Fermi contours are generated k ± F (θ) = 1 αg(θ) ε 2 R + ∆ 2 + 2ε R ε F ∓ ε R 2 − ∆ 2 1/2 ,(23) from equations ε + (k, θ) = ε − (k, θ) = ε F ; (ii) \|ε F \| < ∆ (Fig. 4(b)), where there is only one Fermi contour k − F (θ), lying on the valence band; and (iii) if p < 1, ε min < ε F < −∆ (Fig. 4(c)), where the Fermi lines arise only from the valence band through the equation ε − (k, θ) = ε F , with roots k − F (θ) and q − F (θ) = 1 αg(θ) ε R − ε 2 R + ∆ 2 + 2ε R ε F 2 − ∆ 2 1/2 .(24) Note that the situation (i) or (iii) includes the gappless case ∆ = 0, the Fermi level being then positive or negative, respectively. Interestingly, the Fermi lines described by (23) and (24) are concentric ellipses vertically (horizontally) oriented if m y > m x (m x > m y ), see insets in Fig. 4. The energy separation of the bands at these lines is independent of the direction θ in k-space, taking the values 2d(k ± F (θ)) = 2 ε 2 R + ∆ 2 + 2ε R ε F ∓ ε R or 2d(q − F (θ)) = 2 ε R − ε 2 R + ∆ 2 + 2ε R ε F , according to the position of the Fermi level. The lowest (highest) energy of the spectrum of allowed interband transition will be denoted by ω + ( ω − ), see Fig. 4. When \|ε F \| < ∆ we have that the lowest possible transition occurs at the energy gap ω + = 2∆. When ε F > ∆ or ε min < ε F < −∆, we have ω + = 2d(k + F ) or ω + = 2d(q − F ), respectively. The highest possible energy transition is always given by ω − = 2d(k − F ). Moreover, the resonance curve C r (ω) is an ellipse with the same shape and orientation than the Fermi lines, but differing only in size given its frequency dependence. As a consequence, there will be only two critical points in the JDOS, given by the frequencies ω + and ω − at which the ellipse C r (ω) enters and leaves, respectively, the region of allowed transitions (shaded areas in the insets of Fig. 4), and which defines an absorption window ω + (ε F ) < ω < ω − (ε F ). All this is apparent in the JDOS which is displayed as a color map in Fig. 5, for a system having p < 1. For a given value of ε F the color gradation shows the linear dependence on the exciting frequency. J +− (ω; ε F ) = ω 8πα 2 Θ(1 − \|η(ω, ε F )\|), η(ω, ε F ) = ω − (ω − + ω + )/2 (ω − − ω + )/2 ,(25) B. First order spin current conductivity According to the formula (3), the induced spin current response function of the anisotropic Rashba model becomes σ ij (ω) = −δ z e 8π 1 2ε R ε ijz − i m d m i 2∆ ω δ ij A(ω) + 1 2 (ω − − ω + ) ,(26) where A(x) = x 4 1 − 2∆ x 2 ln (x + ω + )(x − ω − ) (x − ω + )(x + ω − ) .(27) Remarkably, the linear spin currents generated in this system will have electrons with out-of-plane spin orientations only. This a consequence of the breaking of TRS by the term d z (k), which is a non null constant in the present model. This makes the product of matrix elements λ\|Ĵ i \| − λ −λ\|v j \|λ an odd (even) function in k-space when = x, y ( = z). Thus, the Kubo expression (3) integrates to a non zero value when = z only. The longitudinal components of the spin current conductivity (Fig. 6(a)) are proportional to the gap parameter, σ z ii (ω) ∝ ∆, and therefore they vanish for the gapless case. Moreover, these diagonal components are inversely proportional to √ m i , such that m x σ z xx (ω) = m y σ z yy (ω), as a consequence of the mass anisotropy. On the other hand, the Hall components are nonzero, regardless of the value of ∆, indicating the generation of a spin Hall effect in the gapped or ungapped system. In addition, σ xy (ω) = −σ yx (ω), as can be seen in Fig. 6(b). The effect of the position of the Fermi level with respect to the gap, manifests through the critical energies ω ± (ε F ). The variation of Fermi energy leads mainly to a change of the window (ω − − ω + ), just like in JDOS. If E ω 0 = E ω 0 (cos ϕx + sin ϕŷ) is the amplitude of the external field, the spin current can be written as the sum of a component along E ω 0 and a component perpendicular to it, J z (ω) = [cos 2 ϕ σ z xx (ω) + sin 2 ϕ σ z yy (ω)]E ω 0 + [sin ϕ cos ϕ(σ z xx (ω) − σ z yy (ω)) + σ z xy (ω)](E ω 0 ×ẑ) , which reduces to J z (ω) = σ z xy (ω)(E ω 0 ×ẑ) when ∆ = 0. Expression (28) suggests how the induced spin current could be manipulated through the frequency dependence of the tensor σ z ij and the direction of the applied in-plane electric field. C. SH spin current conductivity For the second-harmonic spin current conductivity we obtain the following expressions from (6) (no sum over repeated indices is implied), σ ,(2ω) ijj (ω) = δ j ε ijz e 2 /8π ( ω) 2 1 k R m d m j 1/2 F (ω),(29)σ ,(2ω) iii (ω) = e 2 /8π ( ω) 2 1 k R m d m i 3/2 ε iz G(ω) + 2i (1 − δ z ) 2∆ ω A(ω) − A(2ω) ,(30)σ ,(2ω) iji (ω) = σ ,(2ω) iij (ω) = e 2 /16π ( ω) 2 1 k R m d m i 1/2 δ i ε ijz H(ω) + 2i(1 − δ z ) 2∆ ω A(ω) − A(2ω) ,(31) where k R = ε R /α = m d α/ 2 , A(x) is given by (27), and with ν = 1 or 3. Figure 7 shows the frequency dependence of the in-plane polarized SH components σ x,(2ω) ijl (ω) and σ y,(2ω) ijl (ω). As expected, the spectral structure around ω ± is now accompanied by new features around the subharmonics ω ± /2. The overall structure can be modified through Fermi energy variation, given the behavior of the functions ω ± (ε F ) observed in Fig. 5. The SH spin current response (29)-(31) presents characteristic differences with respect to the linear response, as is the case in the model of section III. Diagonal components are present even in absence of an energy gap, subharmonic structure appears in the spectrum, and the spin polarization is no longer out-of-plane, so that a nonlinear spin Hall effect with in-plane spin orientation is generated. These features may be useful in nonlinear optical spintronic devices. V. SUMMARY We calculated the spin conductivity tensors which characterize the electric-dipole induced spin currents at the fundamental and second harmonic frequencies, in two anisotropic systems with SO interaction. In the case of a 2DEG with R+D[hkl], a time-reversal preserving system, the anisotropy arises from the interplay between the Rashba and Dresselhaus couplings, which in turn depends sensitively on the sample growth direction. For a given crystallographic orientation, the spin splitting of the states acquire a particular dependence on the direction in k-space. This modify the spectrum of allowed optical transitions, and the JDOS and the spin current responses at ω and 2ω display characteristic spectra. This suggests an additional way to influence the linear and nonlinear spectra by choosing in advance the growth direction of the sample, besides frequency tuning, the modulability of the Rashba strength, or the direction of the applied electric field. We found also that the response functions σ ij (ω) and σ ,(2ω) ijl (ω) vanish identically under the SU(2) symmetry conditions found in Ref [35]. There are, however, additional conditions under which specific tensors components vanish, without the requirement of having a collinear SO vector field. Thus, by a proper choice of the growth direction and SO material parameters, one could select the polarization of the linear and SH spin currents according to the direction of flowing. In the case of the anisotropic Rashba model studied in Sec. IV, the anisotropy is that of a 2D free electron gas with different masses [48], m x = m y , in the presence of a Rashba type interaction which introduces different spin splitting along perpendicular directions [41]. To be comprehensive, the model includes an energy gap parameter, which breaks the time-reversal symmetry; when the gap is closed, the model reduces to that studied in Ref. [42]. The band structure offers distinct positions for the Fermi level (above, within, and below the gap), which define several distinguishable scenarios for the allowed optical interband transitions, characterized by the Fermi contours in each case. These manifest in contrasting ways in the linear and SH spin current response. The linear spin conductivity σ ij (ω) shows that only out-of-plane spin polarized currents develops ( = z), while the SH spin conductivity tensor σ ,(2ω) ijl (ω) gives rise to currents with spin orientation lying parallel to the plane of the electron gas ( = x, y). The longitudinal components σ ii (ω) are inversely proportional to √ m i , connected through the masses in the form m x σ z xx (ω) = m y σ z yy (ω), and vanishing for the gapless case. In contrast, the Hall components are non null regardless of the presence of a gap, depend on the masses through the geometric mean √ m x m y only, and satisfy σ yx (ω) = −σ xy (ω). On the other hand, the SH components are proportional to the ratio of masses in the combination (m x /m y ) ±1/4 or (m x /m y ) ±3/4 . In summary, we investigated the spectral properties of the linear and nonlinear optical spin conductivities of two anisotropic models for SO coupled systems, and its dependence on a number of physical quantities like the exciting frequency, the position of the Fermi level, energy gap, mass anisotropy, SO strengths, Rashba and Dresselhaus couplings interplay for arbitrary sample growth directions, or the direction of the externally applied electric field, according to each case. The presence of anisotropy introduces optical signatures which in turn may be useful to identify or estimate some of these material parameters. In particular, the models illustrate the existence of the nonlinear spin Hall effect in systems with SO interaction, under the presence or absence of time-reversal symmetry. The results suggest different ways to manipulate the optically induced linear and SH spin current responses, which could find spintronic applications. We hope that this work will stimulate further investigations under more general conditions, such as the presence of extrinsic SO mechanisms or the use of a conserved spin current definition [44,50]. FIG. 1 . 1(a) Fermi Contours k ± F and resonance curves Cr(ω) for the critical frequencies ω ± , ωa, ω b (ω + < ωa < ω b < ω − ) of a R+D[123] system. At temperature T = 0, the only states involved in vertical transitions between the bands ε λ (k) (gray area) are those satisfying (a) illustrates the situation for the R+D[123] case. The critical frequencies are then defined by the matchings FIG. 2 . 2Longitudinal (a) and transverse (b) components of the linear spin current conductivity σ z ij (ω) for a R+D[123] 2DEG. The vertical dotted lines indicate the positions of the critical frequencies. The parameters used are the same as in Fig. 1. FIG. 3 . 3Longitudinal (a) and Hall (b) components of the second-harmonic spin current conductivity tensor with out-of plane spin orientation for a R+D[123] 2DEG, where σ (2) 0 = e 2 α/4πε 2 R , εR = mα 2 / 2 . The parameters used are the same as in Fig. 1. ω) = 0, except for R+D[001] and R+D[111] cases. The necessary condition for this phenomenon is to have d z (k) = 0 (µ zν = 0), which is characteristic of [110] samples, one of the low Miller indices usually studied. IV. 2D ANISOTROPIC RASHBA MODEL A. Energy spectrum and the joint density of states FIG. 4 . 4Energy bands ε±(k, θ) of a gapped 2D anisotropic Rashba model when εR > ∆. The shaded areas indicate the k-region of allowed optical transitions for several Fermi level positions: (a) εF > ∆, (b) \|εF \| < ∆ and (c) εmin < εF < −∆. The insets show the respective Fermi contours. FIG. 5 . 5Joint density of states 8πα 2 J+−(ω; εF ) for a gapped 2D anisotropic Rashba model with εR > ∆, and the absorption edges ω±(εF ). The parameters used are ∆ = 0.3 εR, α = 10 meVÅ , and mx = m0, my = 4m0 for the effective masses. FIG. 6 . 6Longitudinal (a) and Hall (b) components of the linear spin current conductivity tensor σ z ij (ω) for a gapped anisotropic Rashba model, normalized to σ0 = e/8π. The parameters used are ∆ = 1 meV, mx = m0, my = 4m0, α = 10 meVÅ, εF = 2∆. FFIG. 7 . 6 76(Second-harmonic spin current conductivity tensor σ ,(2ω) ijl (ω) for a gapped 2D anisotropic Rashba model, which determines a spin current flowing in direction i, with the spin polarized along the direction , induced by the external field components j and l. In this system σz,(2ω) ijl (ω) = 0. (a) Longitudinal components with i = j = l = . (b) Longitudinal components with i = j = l = . (c) Hall components with i = j = l = . The parameters used are the same as in Fig. A number of conclusions can be derived from these expressions. 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[ "Gauging the Carroll Algebra and Ultra-Relativistic Gravity", "Gauging the Carroll Algebra and Ultra-Relativistic Gravity" ]	[ "Jelle Hartong \nPhysique Théorique et Mathématique and International Solvay Institutes\nUniversité Libre de Bruxelles\nC.P. 2311050BrusselsBelgium\n" ]	[ "Physique Théorique et Mathématique and International Solvay Institutes\nUniversité Libre de Bruxelles\nC.P. 2311050BrusselsBelgium" ]	[]	It is well known that the geometrical framework of Riemannian geometry that underlies general relativity and its torsionful extension to Riemann-Cartan geometry can be obtained from a procedure known as gauging the Poincaré algebra. Recently it has been shown that gauging the centrally extended Galilei algebra, known as the Bargmann algebra, leads to a geometrical framework that when made dynamical gives rise to Hořava-Lifshitz gravity. Here we consider the case where we contract the Poincaré algebra by sending the speed of light to zero leading to the Carroll algebra. We show how this algebra can be gauged and we construct the most general affine connection leading to the geometry of so-called Carrollian space-times. Carrollian space-times appear for example as the geometry on null hypersurfaces in a Lorentzian space-time of one dimension higher. We also construct theories of ultra-relativistic (Carrollian) gravity in 2+1 dimensions with dynamical exponent z < 1 including cases that have anisotropic Weyl invariance for z = 0.	10.1007/jhep08(2015)069	null	53,385,146	1505.05011	dbc88cfd71a942e19c3ed83ec8ee35c91512c667	Gauging the Carroll Algebra and Ultra-Relativistic Gravity 19 May 2015 Jelle Hartong Physique Théorique et Mathématique and International Solvay Institutes Université Libre de Bruxelles C.P. 2311050BrusselsBelgium Gauging the Carroll Algebra and Ultra-Relativistic Gravity 19 May 2015 It is well known that the geometrical framework of Riemannian geometry that underlies general relativity and its torsionful extension to Riemann-Cartan geometry can be obtained from a procedure known as gauging the Poincaré algebra. Recently it has been shown that gauging the centrally extended Galilei algebra, known as the Bargmann algebra, leads to a geometrical framework that when made dynamical gives rise to Hořava-Lifshitz gravity. Here we consider the case where we contract the Poincaré algebra by sending the speed of light to zero leading to the Carroll algebra. We show how this algebra can be gauged and we construct the most general affine connection leading to the geometry of so-called Carrollian space-times. Carrollian space-times appear for example as the geometry on null hypersurfaces in a Lorentzian space-time of one dimension higher. We also construct theories of ultra-relativistic (Carrollian) gravity in 2+1 dimensions with dynamical exponent z < 1 including cases that have anisotropic Weyl invariance for z = 0. Introduction Over the recent years it has become clear that non-relativistic symmetry groups play an important role in many examples of non-AdS holography. This has been made most apparent in the case of Lifshitz holography where it is has been shown that the boundary geometry is described by Newton-Cartan geometry in the presence of torsion [1,2,3]. Further also in the case of Schrödinger holography there are many hints that the boundary field theory couples to a certain non-Riemannian geometry [4,5,6,7,8]. In AdS/CFT the fact that the boundary geometry is described by Riemannian geometry, just like the bulk geometry, is a special feature of the precise fall-off of the AdS metric (and its asymptotically locally AdS generalizations [9,10]) near the boundary. It is however not expected that the Riemannian nature of the boundary geometry seen in AdS/CFT is a generic feature in other non-AdS holographic dualities. Hence in order to better understand known candidates for non-AdS holography we must learn how to describe various non-Riemannian geometries. Recently it has been argued that the Carroll algebra, which can be obtained as an ultra-relativistic limit (c → 0) of the Poincaré algebra [11,12], plays an important role in flat space holography [13]. The c → 0 contraction of the Poincaré algebra results in a peculiar light cone structure where the light cone has collapsed to a line. The Carroll algebra is given by [J ab , P c ] = δ ac P b − δ bc P a , [J ab , C c ] = δ ac C b − δ bc C a , [J ab , J cd ] = δ ac J bd − δ ad J bc − δ bc J ad + δ bd J ac , [P a , C b ] = δ ab H , where a = 1, . . . , d. In here H is the Hamiltonian, P a spatial momenta, J ab spatial rotations and C a Carrollian boosts. In Cartesian coordinates with time t and space coordinates x i a Carrollian boost means t → t + v · x. In [13] it is shown that future and past null infinity form Carrollian space-times and that the BMS algebra forms a conformal extension of the Carroll algebra. It is therefore of interest to understand in full generality what Carrollian space-times are and how field theories couple to them (see e.g. the work of [14] on coupling warped conformal field theories to geometries obtained by gauging the Carroll algebra). When gauging this algebra we associate vielbeins τ µ and e a µ to the time and space translation generators H and P a , respectively. For the case of relativistic field theories we know that they couple to Riemannian geometry. The latter and its torsionful extension, known as Riemann-Cartan geometry, can be obtained by a procedure known as gauging the Poincaré algebra (see e.g. appendix A of [15]). Similar gauging techniques also allow one to describe torsional Newton-Cartan (TNC) geometry which was found in a holographic context in [16,17,1,3]. We refer to [18,19,20,21,22,23,3] for the use of TNC geometry in a field theoretical context (see [24] for a nice geometrical account of TNC geometry). In order to obtain torsional Newton-Cartan geometry one gauges the centrally extended Galilei algebra known as the Bargmann algebra [25,2,15]. Given the relevance of the Carroll algebra to flat space holography it is a natural question to ask if we can gauge the Carroll algebra and what the resulting geometrical structure is. The gauging of the Carroll algebra will be discussed in section 2. First this is done in full generality involving the Carrollian vielbeins τ µ , e a µ and a Carrollian metric compatible affine connection Γ ρ µν . Then we introduce a contravariant vector M µ and show that the Carrollian metric compatible affine connection Γ ρ µν can fully be written in terms of τ µ , e a µ and M µ . The role of M µ is to ensure that Γ ρ µν when written in terms of the Carrollian vielbeins remains invariant under local (tangent space) Carrollian boosts. In the next section, section 3, it will be shown that the resulting geometrical structure can be realized as the geometry induced on a null hypersurface embedded in a Lorentzian space-time of one dimension higher. Further in section 3 we show that the duality between Newton-Cartan and Carrollian space-times observed in [26] can be extended when we include the vector M µ . The Newton-Cartan dual of M µ is a covector M µ that can be written as m µ − ∂ µ χ where m µ is the connection corresponding to the Bargmann extension of the Galilei algebra and χ is a Stückelberg scalar that must be added to the formalism whenever there is torsion [1,2,15]. In [15] it has been shown that when torsional Newton-Cartan geometry is made dynamical the resulting theory of gravity corresponds to Hořava-Lifshitz gravity [27,28] including the extension of [29]. More specifically, depending on the type of torsion, one is either dealing with projectable (no torsion) or non-projectable (so-called twistless torsion) HL gravity. In both these cases there is a preferred foliation of spacelike hypersurfaces. The case of general torsion is an extension of HL gravity in which the timelike vielbein is not required to be hypersurface orthogonal. Since the tangent space group in HL gravity is the Galilean group it is natural to refer to this type of gravitational theories as non-relativistic theories of gravity. In the same spirit, the Carrollian geometry can be made dynamical. We do this using an effective action approach. By this we mean that we assign dilatation weights to the Carrollian fields τ µ , e a µ and M µ and allow for all possible terms that are relevant or marginal and invariant under local tangent space (Carrollian) transformations. Since the tangent space light cone structure is ultra-relativistic we refer to this as ultra-relativistic gravity. They naturally come with a dynamical exponent z < 1. We show that for z = 0 one can construct actions that are invariant under anisotropic Weyl rescalings of the Carrollian fields τ µ , e a µ and M µ . All of this will be the subject of section 4. A special case of such Carrollian theories of gravity are obtained from the ultralocal (in the sense of no space derivatives) limit of general relativity (GR) that was studied in [30,31] by sending the speed of light to zero. This Carrollian limit of GR is also referred to as the strong coupling limit in which Newton's constant tends to infinity as this has the same effect as sending c to zero [32,33]. Further the Carrollian limit also features in tachyon condensation [34] and cosmological billiards [35]. Note added: While this manuscript was being finalized, the preprint [36] appeared on the arXiv, which overlaps with the results of sections 2 and 3. 2 Gauging the Carroll Algebra From local Carroll to diffeomorphisms and Carrollian light cones The Carroll algebra is obtained as a contraction of the Poincaré algebra by sending the speed of light to zero [11,12]. The nonzero commutators of the Carroll algebra are [J ab , P c ] = δ ac P b − δ bc P a , (2.1) [J ab , C c ] = δ ac C b − δ bc C a , (2.2) [J ab , J cd ] = δ ac J bd − δ ad J bc − δ bc J ad + δ bd J ac , (2.3) [P a , C b ] = δ ab H , (2.4) where a = 1, . . . , d. We thus see that the algebra is isomorphic to the semi-direct product of SO(d) with the Heisenberg algebra whose central element is the Hamiltonian. In order to gauge the algebra we follow the procedure of [25,15] where the gauging of the Galilei algebra and its central extension, the Bargmann algebra, were discussed (without torsion [25] and including torsion [15]). For an earlier discussion of gauging the Carroll algebra see [37]. We define a connection A µ as A µ = Hτ µ + P a e a µ + C a Ω µ a + 1 2 J ab Ω µ ab ,(2.5) where µ takes d + 1 values related to the fact that there is one time and d space translation generators. We thus work with a (d + 1)-dimensional space-time. This connection transforms in the adjoint as δA µ = ∂ µ Λ + [A µ , Λ] . (2.6) Without loss of generality we can write Λ as Λ = ξ µ A µ + Σ ,(2.7) where Σ is given by Σ = C a λ a + 1 2 J ab λ ab . (2.8) We would like to think of ξ µ as the generator of diffeomorphisms and Σ as the internal (tangent) space transformations. To this end we introduce a new local transformation denoted byδ that is defined as δA µ = δA µ − ξ ν F µν = L ξ A µ + ∂ µ Σ + [A µ , Σ] ,(2.9) where F µν is the field strength F µν = ∂ µ A ν − ∂ ν A µ + [A µ , A ν ] = HR µν (H) + P a R µν a (P ) + C a R µν a (C) + 1 2 J ab R µν ab (J) . (2.10) In components theδ transformations act as δτ µ = L ξ τ µ + e a µ λ a , (2.11) δe a µ = L ξ e a µ + λ a b e b µ , (2.12) δΩ µ a = L ξ Ω µ a + ∂ µ λ a + λ a b Ω µ b − λ b Ω µ ab , (2.13) δΩ µ ab = L ξ Ω µ ab + ∂ µ λ ab + λ a c Ω µ cb − λ b c Ω µ ca . (2.14) The Lie derivatives along ξ µ correspond to the generators of general coordinate transformations whereas the remaining local transformations with parameters λ a and λ ab correspond to local tangent space transformations 1 . The tangent space has a Carrollian light cone structure by which we mean that the light cones have collapsed to a line. This can be seen from the fact that there are no boost transformations acting on the spacelike vielbeins e a µ . The component expressions for the field strengths read R µν (H) = ∂ µ τ ν − ∂ ν τ µ + e a µ Ω νa − e a ν Ω µa , (2.15) R µν a (P ) = ∂ µ e a ν − ∂ ν e a µ − Ω µ ab e νb + Ω ν ab e µb , (2.16) R µν a (C) = ∂ µ Ω ν a − ∂ ν Ω µ a − Ω µ ab Ω νb + Ω ν ab e µb , (2.17) R µν ab (J) = ∂ µ Ω ν ab − ∂ ν Ω µ ab − Ω µ ca Ω ν b c + Ω ν ca Ω µ b c . (2.18) The affine connection The next step is to impose vielbein postulates allowing us to describe the properties of the curvatures in F µν in terms of the curvature and torsion of an affine connection Γ ρ µν that by definition is invariant under the tangent space Σ transformations. We define theδ covariant derivative D µ as D µ τ ν = ∂ µ τ ν − Γ ρ µν τ ρ − Ω µa e a ν ,(2.19) D µ e a ν = ∂ µ e a ν − Γ ρ µν e a ρ − Ω µ a b e b ν . (2.20) The form of the covariant derivatives is uniquely fixed by demanding covariance. The vielbein postulates are then D µ τ ν = 0 ,(2. 21) D µ e a µ = 0 . (2.22) We choose the right hand side to be zero because i). it obviously transforms covariantly and ii). even if we could write something else that transforms covariantly we can absorb this into the definition of Γ ρ µν . We can now solve for Ω µa and Ω µ a b in terms of Γ ρ µν by contracting the vielbein postulates with the inverse vielbeins v µ and e µ a that are defined via v µ τ µ = −1 , v µ e a µ = 0 , e µ a τ µ = 0 , e µ a e b µ = δ b a . (2.23) They transform under theδ transformations as 25) and they satisfy the inverse vielbein postulates We can define a Riemann tensor in the usual way as follows δv µ = L ξ v µ , (2.24) δe µ a = L ξ e µ a + v µ λ a + λ a b e µ b ,(2.D µ v ν = ∂ µ v ν + Γ ν µρ v ρ = 0 , (2.26) D µ e ν a = ∂ µ e ν a + Γ ν µρ e ρ a − v ν Ω µa − Ω µa b e ν b =[∇ µ , ∇ ν ] X σ = R µνσ ρ X ρ − 2Γ ρ [µν] ∇ ρ X σ ,(2.29) where ∇ µ only contains the affine connection and where R µνσ ρ is given by R µνσ ρ = −∂ µ Γ ρ νσ + ∂ ν Γ ρ µσ − Γ ρ µλ Γ λ νσ + Γ ρ νλ Γ λ µσ . (2.30) Using the vielbein postulates, i.e. the relation between the affine connection and the tangent space connections, we find R µνσ ρ = −v ρ e σa R µν a (C) − e σa e ρ b R µν ab (J) . (2.31) We have traded the connections Ω µ a and Ω µ ab for Γ ρ µν . The latter connection has more components and so they cannot all be free. In fact the vielbein postulates (2.21)-(2.27) constrain Γ ρ µν in the following way ∇ µ v ν = 0 , ∇ µ h νρ = 0 ,(2.32) where we defined h µν = δ ab e a µ e b ν . We will also adopt the notation h µν = δ ab e µ a e ν b . In order to find out what the independent components of Γ ρ µν are we will obtain the most general solution to these metric compatibility equations. We note that both v µ and h µν are invariant under the tangent space transformations. They form the notion of Carrollian metrics. We start with the condition ∇ µ h νρ = 0. By permuting the indices and summing the resulting equations appropriately we obtain 2Γ σ (µρ) h νσ = ∂ µ h νρ + ∂ ρ h µν − ∂ ν h ρµ − 2Γ σ [µν] h σρ − 2Γ σ [ρν] h µσ . (2.33) Contracting this equation with v ν we find K µρ = −v ν h σρ Γ σ [νµ] − v ν h σµ Γ σ [νρ] ,(2.34) where the extrinsic curvature K µρ is defined as K µρ = − 1 2 L v h µρ . (2.35) From (2.34) we conclude that Γ ρ [νµ] = τ [ν K µ]λ h σλ + X σ [νµ] , (2.36) where X σ [νµ] is such that v ν h σρ X σ [νµ] + v ν h σµ X σ [νρ] = 0 . (2.37) Substituting (2.36) into (2.33) and adding 2Γ σ [µρ] h νσ to both sides (using (2.36)) we obtain 2Γ σ µρ h νσ = ∂ µ h νρ + ∂ ρ h µν − ∂ ν h ρµ + 2τ [ν K µ]ρ + 2τ [ν K ρ]µ + 2τ [µ K ρ]ν +2X σ [νµ] h σρ + 2X σ [νρ] h σµ + 2X σ [µρ] h νσ . (2.38) Contracting this with h νλ and using h νσ h νλ = δ λ σ + τ σ v λ we find the following most general solution to ∇ µ h νρ = 0 Γ λ µρ = −v λ τ σ Γ σ µρ + 1 2 h νλ (∂ µ h νρ + ∂ ρ h µν − ∂ ν h ρµ ) − h νλ τ ρ K µν +h νλ X σ [νµ] h σρ + X σ [νρ] h σµ + X σ [µρ] h νσ . (2.39) By contracting this result with v ρ it can be shown that δ λ ρ + v λ τ ρ ∇ µ v ρ = 0 ,(2.40) so that in order to find the most general Γ ρ µν obeying both ∇ µ v ν = 0 and ∇ µ h νρ = 0 it remains to impose τ ρ ∇ µ v ρ = 0 . (2.41) This latter condition is equivalent to v ρ ∇ µ τ ρ = 0 so that Γ σ µρ τ σ = ∂ µ τ ρ + X µρ ,(2.42) with X µρ = −∇ µ τ ρ = −Ω µa e a ρ ,(2.43) satisfying v ρ X µρ = 0 . (2.44) We thus conclude that the most general Γ λ µρ is of the form Γ λ µρ = −v λ ∂ µ τ ρ + 1 2 h νλ (∂ µ h νρ + ∂ ρ h µν − ∂ ν h ρµ ) − h νλ τ ρ K µν −v λ X µρ + 1 2 h νλ Y νµρ , (2.45) where Y νµρ is given by Y νµρ = 2X σ [νµ] h σρ + 2X σ [νρ] h σµ + 2X σ [µρ] h νσ , (2.46) which has the property that v ν Y νµρ = v ρ Y νµρ = 0 as follows from (2.37). The connection (2.45) has torsion that is given by Γ λ [µρ] = −v λ ∂ [µ τ ρ] − h νλ τ [ρ K µ]ν − v λ X [µρ] + 1 2 h νλ Y ν[µρ] . (2.47) An alternative way of writing (2.45) that makes manifest the property ∇ µ v ν = 0 is as follows Γ λ µρ = τ ρ ∂ µ v λ − h ρσ ∂ µ h σλ + 1 2 h ρσ h κσ h νλ (∂ κ h µν − ∂ µ h νκ − ∂ ν h µκ ) −v λ X µρ + 1 2 h νλ Y νµρ . (2.48) The requirement is that Γ ρ µν transforms as an affine connection under general coordinate transformations and remains inert under C and J (tangent space) transformations. The first line of (2.45) transforms affinely, i.e. it has a term ∂ µ ∂ ρ ξ λ plus terms that transform tensorially. In fact the last term of the first line containing the extrinsic curvature transforms as a tensor and is thus not responsible for producing the ∂ µ ∂ ρ ξ λ term. This means that all terms on the second line of (2.45) must transform as tensors, i.e. X µρ and Y νµρ transform as tensors under general coordinate transformations. As a check that we have in fact managed to write all the components of Ω µ a and Ω µ ab in terms of a Carrollian metric compatible Γ ρ µν we count the number of components in h σρ X σ [νµ] (since this determines Y νµρ via equation (2.2)) and X µν . The tensor h σρ X σ [νµ] has 1 2 (d + 1) 2 d − 1 2 d(d + 1) = 1 2 d 2 (d + 1) components and it satisfies 1 2 d(d + 1) constraints v ν h σρ X σ [νµ] + v ν h σµ X σ [νρ] = 0 giving 1 2 (d + 1)d 2 − 1 2 d(d + 1) = 1 2 (d − 1)d(d + 1) free components. The tensor X µν has (d+1) 2 components satisfying (d+1) constraints v ν X µν giving d(d+1) free components. Together this gives d(d+1)((d−1)+2)/2 = d(d+1) 2 /2Γ ρ µν = −v ρ ∂ µ τ ν + v ρ Ω µa e a ν + e ρ a ∂ µ e a ν − Ω µ a b e ρ a e b ν ,(2.49) we obtain the following relation between Ω µ a b and h ρσ Y σµν 1 2 h ρσ Y σµν = − 1 2 h ρσ (∂ µ h σν + ∂ ν h µσ − ∂ σ h µν ) + h ρσ τ ν K µσ + e ρ a ∂ µ e a ν − Ω µ a b e ρ a e b ν . (2.50) Finally in order that our Γ ρ µν satisfies all the required properties we must ensure that it is invariant under local C and J transformations. It is manifestly J invariant so we are left to ensure local C invariance. Using that δ C τ µ = λ µ , (2.51) δ C h µν = (h µσ v ν + h νσ v µ ) λ σ ,(2.52) where λ µ = e a µ λ a , one can shown that Γ ρ µν is C invariant if and only if X µρ and Y νµρ transform as δ C X µρ = − ∂ µ λ ρ − Γ σ µρ λ σ , (2.53) δ C Y νµρ = 2λ ρ K µν − 2λ ν K µρ . (2.54) These transformation rules are compatible with the properties v ρ X µρ = 0 and v ν Y νµρ = v ρ Y νµρ = 0. For the transformation of X µρ this is by virtue of λ ν ∇ µ v ν = 0 (i.e. metric compatibility). The transformation of X µρ involves the connection Γ ρ µν . However it does not involve the tensor X µρ on the right hand side of (2.53) because Γ ρ µν is contracted with λ ρ which has the property that v ρ λ ρ = 0. In fact we can rewrite the right hand side of (2.53) as follows. Using (2.45) we find Γ σ µρ λ σ = 1 2 (∂ µ λ ρ + ∂ ρ λ µ ) − 1 2 L u h µρ + 1 2 u ν τ ρ L v h µν + 1 2 u ν Y νµρ ,(2.55) where we defined the vector u µ = h µσ λ σ and where we used (2.35). Using that u ν L v h µν = v ν (∂ ν λ µ − ∂ µ λ ν ) − v ν L u h µν ,(2.56) we obtain for δ C X µρ the result δ C X µρ = − 1 2 δ ν ρ + τ ρ v ν (∂ µ λ ν − ∂ ν λ µ + L u h µν ) + 1 2 u κ Y κµρ , (2.57) making it manifest that δ C (v ρ X µρ ) = v ρ δ C X µρ = 0. One may wonder why there is a term transforming into u κ Y κµρ . The reason is that the transformation of h νλ Y νµρ in (2.45) produces such terms through (2.52) and these need to be cancelled. Using (2.55) and (2.35) we can also write the variation of X µρ in the following manner δ C X µρ = − 1 2 (∂ µ λ ρ − ∂ ρ λ µ ) − 1 2 L u h µρ + 1 2 u ν (Y νµρ − 2τ ρ K µν + 2τ ν K µρ ) . (2.58) This way of writing δ C X µρ is useful when one tries to write the right hand side as the δ C of something which we will do in the next subsection. The term u ν τ ν K µρ has been added to make manifest that u ν contracts a term that is C-boost invariant. Of course because u ν τ ν = 0 the added term vanishes. If we write in (2.58) and likewise in (2.54) the parameter λ µ = h µν u ν then u µ always contracts or multiplies a term that is manifestly Carrollian boost invariant. This is not the case for the parameter λ µ because it sometimes is contracted with h µν which is not invariant under local C transformations. Introducing the vector M µ So far we have considered the most general case where theδ transformations are realized on the set of fields τ µ , e a µ , Ω µ a and Ω µ ab or what is the same τ µ , e a µ and Γ ρ µν where the latter is metric compatible in the sense that ∇ µ v ν = ∇ µ h νρ = 0. In the remainder we will realize the algebra on a smaller set of fields. Sometimes when gauging algebras, as happens e.g. when gauging the Poincaré algebra, it is possible to realize theδ transformations on a smaller set of fields by imposing curvature constraints whose effect is to make some of the connections in A µ dependent on other connections in A µ . For example in the case of the gauging of the Poincaré algebra setting the torsion to zero, i.e. imposing the curvature constraint R µν a (P ) = 0 (where P denotes the space-time translations), enables one to express the spin connection coefficients ω µ ab in terms of e a µ . In the case of the gauging of the Bargmann algebra imposing curvature constraints (without introducing new fields) to write the Galilean boost and spatial rotation connections in terms of the vielbeins and the central charge gauge connection is only possible when there is no torsion [25]. When there is torsion the curvature constraints become dependent on an additional Stückelberg scalar field χ that is not present in A µ . This field needs to be added to ensure the correct transformation properties of the Galilean boost and spatial rotation connections when writing them as dependent gauge connections [1,2,15]. In the context of formulating Hořava-Lifshitz (HL) gravity as a theory of dynamical torsional Newton-Cartan geometry [15] the Stückelberg scalar field χ plays an important role in making the identification between TNC and HL variables. In the context to Hořava-Lifshitz gravity this field was introduced in [29] and dubbed the Newtonian prepotential. In the case of field theory on Newton-Cartan space-times including torsion is crucial because it allows one to compute the energy current [16,17,19,1,23]. The fact that one needs to introduce an extra Stückelberg scalar field to the formalism when there is torsion does not mean that any field theory on such a background has a non-trivial response to varying the Stückelberg scalar. It can happen that there are additional local symmetries in the model that allow one to remove this field from the action [23,3]. The main message is that once we start imposing curvature constraints the resulting reduced set of fields on which the algebra is realized do not need to correspond to a constrained algebra gauging and may involve new fields. In both the gauging of the Poincaré algebra and of the Bargmann algebra the effect of the curvature constraints is to make the connection Γ ρ µν a fully dependent connection. Imposing the curvature constraint R µν a (P ) = 0 in the Poincaré case leads to the Levi-Cività connection 2 In the case of the gauging of the Bargmann algebra in the presence of torsion the algebra ofδ transformations is realized on τ µ , e a µ and M µ = m µ − ∂ µ χ. One can also say that from the point of view of gauging the Galilei algebra one needs to add the vector M µ to construct a Γ ρ µν that obeys all the properties of an affine connection [15]. In other words from the point of view of adding curvature constraints to the gauging of the Galilei algebra we add a vector M µ with appropriately chosen transformation properties to realize the algebra on the smaller number of fields τ µ , e a µ and M µ as opposed to τ µ , e a µ , Ω µ a (local Galilean boosts) and Ω µ ab (local spatial rotations). In the case discussed here we will realize the algebra ofδ transformations on τ µ , e a µ and a contravariant vector field M µ where M µ transforms under theδ transformations asδ M µ = L ξ M µ + e µ a λ a = L ξ M µ + h µν λ µ = L ξ M µ + u µ . (2.59) We are not aware of an extension of the Carroll algebra such that M µ can be constructed from the additional connections appearing in A µ corresponding to the extended Carroll algebra. The guiding principle will be to write Γ ρ µν in terms of τ µ , e a µ and M µ in such a way that it obeys all the required properties. In other words we need to write the tensors X µρ and Y σµν in terms of τ µ , e a µ and M µ ensuring that they transform correctly under theδ transformations. A raison d'être for the vector M µ will be given in the next section. One of the benefits of working with X µρ and Y σµν is that their transformation properties under local tangent space C and J transformations is much simpler than for the equivalent set of objects Ω µ a and Ω µ ab . Both X µρ and Y σµν are invariant under J transformations and their the C transformations are given in (2.53) (or equivalently (2.58)) and (2.54). We will now use the additional M µ vector to write down a realization of X µρ and Y σµν in terms of τ µ , e a µ and M µ . Using (2.58) and (2.59) we can write 0 = δ C X µρ + 1 2 ∂ µ (h ρσ M σ ) − 1 2 ∂ ρ (h µσ M σ ) + 1 2 L M h µρ − 1 2 M ν (Y νµρ − 2τ ρ K µν + 2τ ν K µρ ) . (2.60) Hence a realization of X µρ (but not the most general one) is to write X µρ = − 1 2 ∂ µ (h ρσ M σ ) + 1 2 ∂ ρ (h µσ M σ ) − 1 2 L M h µρ 2 There also exists the possibility to set the Riemann curvature 2-form R µν ab (M ), where M is the generator of Lorentz transformations, equal to zero. This leads to the so-called Weitzenböck connection (see for example [38]). We refer to [14] for similar ideas in the context of gauging the Carroll algebra. + 1 2 M ν (Y νµρ − 2τ ρ K µν + 2τ ν K µρ ) ,(2.61) obeying v ρ X µρ = 0. Likewise for Y νµρ we can take Y νµρ = 2h ρσ M σ K µν − 2h νσ M σ K µρ ,(2.Γ λ µρ = −v λ ∂ µτρ + 1 2h νλ (∂ µ h ρν + ∂ ρ h µν − ∂ ν h µρ ) −h νλτ ρ K µν +h νλτ ν K µρ ,(2.63) where we definedτ µ = τ µ − h µν M ν , (2.64) h µν = h µν − M µ v ν − M ν v µ ,(2.65) which are manifestly C invariant. Another C invariant (scalar) quantity that we can define isΦ which is given byΦ = −M ν τ ν + 1 2 h νσ M ν M σ . (2.66) The affine connection (2.63) has the property that if we replace allh µν by H µν = h µν + αΦv µ v ν the resulting expression for Γ λ µρ remains unchanged, i.e. does not depend on α. Hence we can take α = 2 and write for Γ λ µρ in (2.63) Γ λ µρ = −v λ ∂ µτρ + 1 2ĥ νλ (∂ µ h ρν + ∂ ρ h µν − ∂ ν h µρ ) −ĥ νλτ ρ K µν , (2.67) whereĥ νλ is defined byĥ νλ =h νλ + 2Φv ν v λ ,(2.68) for whichτ µĥ µν = 0. The connection (2.67) is independent ofΦ because it can be shown that M µ appears inτ µ andĥ µν only via h µν M ν . This is made more manifest below following the discussion around equations (2.70) and (2.73). The connection (2.63) satisfies by design the metric compatibility conditions ∇ µ v ν = ∇ µ h νρ = 0. However it also satisfies the conditions ∇ µτν = ∇ µĥ νρ = 0 ,(2.69) where ∇ µτν follows immediately by inspection of (2.63) using thatĥ µντ ν = 0. The second property ∇ µĥ νρ = 0 follows from all the other metric compatibility conditions and the fact thatĥ νρ is fully determined onceτ µ and h µν are known. The property ∇ µτν = 0 implies that ∇ µ τ ρ = ∇ µ (h ρσ M σ ) = −X µρ where we used (2.64) and (2.43) and is compatible with the transformation under local C transformations given in (2.53). We stress though that the properties (2.69) are special for the particular realization of Γ λ µρ given in (2.63) and will not be true for other realizations of Γ λ µρ that for example also depend on the scalar invariantΦ. We can define a new set of vielbeinsτ µ , e a µ whose inverse is v µ ,ê µ a with the latter defined byê v µτ µ = −1 , v µ e a µ = 0 ,ê µ aτ µ = 0 ,ê µ a e b µ = δ b a . (2.71) Out of these objects we can build a Lorentzian symmetric rank two tensor g µν via g µν = −τ µτν + h µν = −τ µτν + δ ab e a µ e b ν , (2.72) whose inverse is g µν = −v µ v ν +ĥ µν = −v µ v ν + δ abêµ aê ν b . (2.73) Since the connection (2.63) satisfies ∇ µτν = ∇ µ h νρ = 0 it in particular obeys ∇ µ g νρ = 0. Since it furthermore has torsion the connection (2.63) must be a special case of a Riemann-Cartan connection. By this we mean a torsionful connection obeying ∇ µ g νρ = 0. Any such connection must be of the form [38] Γ λ µρ = The connection (2.63) is not the most general affine connection compatible with our requirements. We still have the freedom to add to X µρ and Y νµρ terms that are invariant under local Carrollian boosts. When we add a term to Y νµρ we should also add the corresponding term to (2.61) because 1 2 M ν Y νµρ appears in X µρ . Further any term added to Y νµρ must obey the property that when contracted with v ν or v ρ it vanishes since Y νµρ obeys this property. Equation (2.67) is independent ofΦ and the only terms that we can still add to Γ ρ µν without affecting its properties come fromΦ dependent terms that we add to X µρ and Y νµρ . An example of such a term is to add to X µρ a term proportional toΦK µρ which is C invariant and orthogonal to v ρ . The effect is to redefine Γ λ µρ by a term proportional toΦv λ K µρ . Yet another term that we can add to X µρ compatible with Γ λ µρ remaining invariant under C, J transformations, transforming affinely, and being metric compatible in the Carrollian sense, is a term proportional to h µρ v σ ∂ σΦ . Any of these affine connections is an allowed connection and so one can choose them to suit one's convenience. The same phenomenon happens for the case of torsional Newton-Cartan (TNC) geometry. Sometimes it is useful to work with a TNC connection that does not depend on the scalarΦ (the TNC counterpart ofΦ defined in (3.5)) as is for example the case when making contact with Hořava-Lifshitz gravity [15] and sometimes it is useful to work with a TNC connection depending linearly on M µ as is for example the case when coupling field theories with particle number symmetry to TNC backgrounds [22,23,3]. 1 2 g νλ (∂ µ g ρν + ∂ ρ g µν − ∂ ν g µρ ) + g νλ Γ κ [νρ] g κµ + Γ κ [νµ] g κρ + Γ κ [µρ] g κν .(2. The Geometry on Null Hypersurfaces A natural example of a space-time with a Carrollian metric structure is a null hypersurface embedded into a Lorentzian space-time of one dimension higher [26,13]. Before introducing a Carrollian space-time as the geometry on a null hypersurface it is useful to consider first the case of a Newton-Cartan space-time as the geometry orthogonal to a null Killing vector. This will also enable us to compare the two cases later. Newton-Cartan space-time It is well known that Newton-Cartan geometry on a manifold with coordinates x µ can be obtained by null reduction [39,40,41,16,17], i.e. by starting from a Lorentzian space-time with one extra dimension u whose metric is of the form ds 2 = 2τ µ dx µ (du − m ν dx ν ) + h µν dx µ dx ν = 2τ µ dx µ du +h µν dx µ dx ν , (3.1) where we take ∂ u to be a Killing vector so that τ µ andh µν are independent of u and whereh µν = h µν − m µ τ ν − m ν τ µ . (3.2) Note that for this metric we have g uu = 0. The inverse metric components are g µν = h µν , g µu = −v µ , g uu = 2Φ , (3.3) wherev µ = v µ − h µν m ν , (3.4) Φ = −v µ m µ + 1 2 h µν m µ m ν . (3.5) The metric (3.1) is the most general Lorentzian metric with a null Killing vector ∂ u . The coordinate transformations that preserve the form of the null Killing vector are u ′ = u − σ(x) , (3.6) x ′µ = x ′µ (x) . (3.7) Under the shift in u the vector m µ transforms as a U(1) connection m ′ µ = m µ − ∂ µ σ . (3.8) A TNC metric compatible connection can be found by taking the Levi-Cività connection of the higher dimensional space-time with all its legs in the x µ directions and to add torsion to this by hand so as to make it metric compatible in the TNC sense, i.e. ∇ µ τ ν = ∇ µ h νρ = 0 [16,17]. Instead of speaking about null reduction, one can say that TNC geometry is the geometry on the space-time orthogonal to the null Killing vector ∂ u . If we insist that the connection on the TNC space-time is naturally induced from the Levi-Cività connection on the higher dimensional space-time we need to impose that ∂ u is hypersurface orthogonal. To see this write for the higher dimensional metric ds 2 = g AB dx A dx B = (U A V B + U B V A + Π AB ) dx A dx B , (3.9) where x A = (u, x µ ). The vectors U A and V A are nullbeins defined by U A U A = 0 , V A V A = 0 , U A V A = −1 , U A Π AB = V A Π AB = 0 . (3.10) We choose U A = (∂ u ) A , so that U u = 0 , U µ = −τ µ , V u = −1 , V µ = m µ , Π uA = 0 , Π µν = h µν ,(3.11) and V u = −v µ m µ , V µ = −v µ , Π uu = h µν m µ m ν , Π uµ = h µν m ν , Π µν = h µν . (3.12) In order that we have a TNC connection on the space-time orthogonal to U A we demand that ∇ A U B projected along all directions orthogonal to U A gives zero, i.e. U A U B ∇ A U B = 0 , (3.13) U A Π B C ∇ A U B = 0 , (3.14) Π A C U B ∇ A U B = 0 , (3.15) Π A C Π B D ∇ A U B = 0 , (3.16) where ∇ A contains the Levi-Cività connection. These conditions lead to the TNC metric compatibility condition ∇ µ τ ν = 0. Likewise to obtain ∇ µ h νρ = 0 we impose that ∇ A Π BC with all indices projected onto directions orthogonal to U A gives zero. Since we have ∇ A Π BC = −U B ∇ A V C − V C ∇ A U B − U C ∇ A V B − V B ∇ A U C , (3.17) which follows from the fact that ∇ A g BC = 0 we obtain Π D B Π E C ∇ A Π BC = 0 , (3.18) U B U C ∇ A Π BC = 0 , (3.19) Π D B U C ∇ A Π BC = −Π DC ∇ A U C . (3.20) Hence to enforce ∇ µ h νρ = 0 we only need that Π DC ∇ A U C contracted with U A and Π A B gives zero. These conditions are already imposed in (3.14) and (3.16). Since U A is a null Killing vector equations (3.13) to (3.15) together with the symmetric part of (3.16) are satisfied. What remains is to impose that the spatial projection of the antisymmetric part of ∇ A U B vanishes which is equivalent to demanding that U A is hypersurface orthogonal. Put another way it must be that ∇ A U B = U A X B − U B X A , (3.21) for some vector X A obeying U A X A = 0 (as follows from the fact that U A is a null Killing vector and thus geodesic) but otherwise arbitrary in order that the Levi-Cività connection induces a Newton-Cartan connection on the space-time orthogonal to ∂ u . Since the left hand side of (3.21) is just 1 2 (∂ A U B − ∂ B U A ) and X u = 0 the only nontrivial component of (3.21) is when A = µ and B = ν expressing the fact that τ µ is hypersurface orthogonal, but not necessarily closed. This is the case referred to as twistless torsional Newton-Cartan geometry (TTNC) [16,17]. In this case metric compatibility ∇ µ τ ν = 0 requires a torsionful connection. We can thus obtain a torsionful connection from a Riemannian geometry by projecting along directions orthogonal to a hypersurface orthogonal null Killing vector. One may wonder how this is possible since the connection of the Riemannian spacetime is symmetric. From the properties of U A we infer that ∂ µ τ ν + ∂ ν τ µ = 2Γ ρ (g)µν τ ρ , (3.22) ∂ µ τ ν − ∂ ν τ µ = 2τ µ X ν − 2τ ν X µ , (3.23) where Γ ρ (g)µν is the Levi-Cività connection with all indices in the x µ directions. From the first of these two equations we read off that the symmetric part of TNC connection satisfies Γ ρ (µν) τ ρ = Γ ρ (g)µν τ ρ . In order to repackage these equations into ∇ µ τ ν = 0 we see that X µ contributes to a torsion tensor Γ ρ [µν] τ ρ = τ µ X ν − τ ν X µ . In other words the torsion can be introduced due to the fact that we are dealing with a geometry orthogonal to a null vector U A so that there is a certain arbitrariness encoded in X A when solving for (3.16). The torsion is thus described by a vector X µ . In [15] the torsion vector is denoted by a µ which relates to X µ via a µ = −2X µ . It determines whether we are dealing with projectable or non-projectable Hořava-Lifshitz gravity. The conditions (3.21) together with U A being a null Killing vector guarantee that a TTNC metric compatible Γ ρ µν exists but the projection equations onto the space-time orthogonal to U A do not tell one the precise form of this connection. This is to be expected since there is a certain arbitrariness in the expression for Γ ρ µν . We recall that in order to write an expression for Γ ρ µν in terms of the TNC fields τ µ , e a µ and m µ that does not refer to an embedding in a higher dimensional space-time we need to add a Stückelberg scalar χ when there is torsion [1,2,15]. This amounts to replacing everywhere m µ by M µ = m µ − ∂ µ χ. Carrollian space-time To obtain an embedding of a Carrollian space-time into a Lorentzian space-time of one dimension higher all that is required is to do the same as for the Newton-Cartan case but with the difference that it is now the inverse metric for which we take g uu = 0. In other words we write down the most general metric for which g uu = 0. Such a metric is given by ds 2 = du 2Φdu − 2τ µ dx µ + h µν dx µ dx ν ,(3.24) whereΦ is given in (2.66) andτ µ is given in (2.64). The components of the inverse metric are g uu = 0 , g µu = v µ , g µν =h µν , (3.25) whereh µν is given by (2.65). The Carrollian space-time can be thought of as the geometry on the null hypersurface u = cst whose normal is ∂ A u, i.e. it is the geometry orthogonal to ∂ A u. WhenΦ =Φ = 0 the Newton-Cartan and Carrollian geometry are the same. This is because the metrics (3.1) and (3.24) become identical. One simply has the correspondence τ µ ↔τ µ ,v µ ↔ v µ ,ĥ µν ↔ h µν , h µν ↔ĥ µν . (3.26) In section 3.3 we will discuss in more detail the relation between TNC and Carrollian geometry. The coordinate transformations that preserve the null foliation are given by If we demand that ∂ u is a Killing vector the coordinate transformations cannot depend on u so that M µ simply transforms as a vector. Alternatively if we work at a fixed value of u, i.e. a specific null hypersurface, the coordinate transformation of x µ cannot depend on u either and again M µ transforms as a vector on the u = cst hypersurface. We thus see from the embedding point of view that there is no extra symmetry associated with the vector M µ while there is one in the NC case where we had a U(1) acting on m µ corresponding to the Bargmann extension of the Galilei algebra. We now discuss under what conditions the Carrollian metric compatible connection can be obtained from the Levi-Cività connection in the higher dimensional space-time. u = u ′ , (3.27) x µ = x µ (u ′ , x ′ ) . To this end consider again (3.9) and (3.10). This time we choose U A = ∂ A u implying that U u = 0 , U µ = −v µ , V u = −1 , V µ = M µ , Π uA = 0 , Π µν = h µν ,(3.30) as well as Π uµ = h µν M ν , Π uu = h µν M µ M ν , Π µν = h µν , V u = −τ µ M µ , V µ = −τ µ . (3.31) Imposing that ∇ µ v ν = 0 amounts to demanding that U A U B ∇ A U B = 0 , (3.32) Π A C U B ∇ A U B = 0 , (3.33) U A Π C B ∇ A U B = 0 , (3.34) Π A C Π D B ∇ A U B = 0 . (3.35) The first and the second conditions are satisfied because U A is null while the third is satisfied because we furthermore know that ∇ A U B = ∇ B U A due to our choice of U A as ∂ A u. The most general expression for ∇ A U B compatible with all of the above conditions and the properties of U A is given by ∇ A U B = U A X B + U B X A ,(3.36) where X A satisfies U A X A = 0 but is otherwise an arbitrary vector. Using that ∇ A g BC = 0, i.e. that ∇ A Π BC = −U B ∇ A V C − U C ∇ A V B − V B ∇ A U C − V C ∇ A U B ,(3.37) we find that Π B D Π C E ∇ A Π BC = 0 , (3.38) U B U C ∇ A Π BC = 0 , (3.39) Π B D U C ∇ A Π BC = −U A Π D C X C ,(3.40) where in the last relation we used (3.36). Hence ∇ A Π BC vanishes when projected along directions orthogonal to U A . Therefore, whereas in the NC case we had to demand equation (3.21) in the Carrollian case we need that (3.36) holds in order that the induced connection comes from the Levi-Cività connection of the higher dimensional space-time. Comparing Newton-Cartan and Carrollian space-times As one can notice by comparing the discussions of sections 3.1 and 3.2 there are strong similarities between the geometry of TNC and Carrollian space-times. In fact in [26] a certain duality between the two geometries has been proposed. Here we will extend this duality to include the TNC vector M µ and the Carrollian vector M µ . The TNC metric-like objects are given by τ µ and h µν whereas the Carrollian metric-like objects are given by v µ and h µν suggesting the duality [26] τ µ ↔ v µ , h µν ↔ h µν ,(3.41) where TNC variables are written on the left and Carrollian fields on the right. When including the vector M µ = m µ − ∂ µ χ for TNC geometry and M µ for the Carrollian case we propose to extend this duality to When there is no coupling toΦ on the TNC side and no coupling toΦ on the Carrollian side, there is another relation between TNC and Carrollian geometry that interchanges like tensors as in (3.26). For example if we apply this duality to the Carrollian affine connection (2.67) which has the property that it does not depend on Φ we obtain M µ ↔ M µ ,(3.Γ λ µρ = −v λ ∂ µ τ ρ + 1 2 h νλ ∂ µĥρν + ∂ ρĥµν − ∂ νĥµρ − h νλ τ ρ K µν ,(3.44) where now the extrinsic curvature is given by K µν = − 1 2 Lvĥ µν . We recognize the first two terms of (3.44) as the TNC connection that is independent ofΦ used in [15]. The third term containing the extrinsic curvature is just a harmless tensorial redefinition of the TNC connection. Put another way, in [15] we used the connection Γ λ µρ = −v λ ∂ µ τ ρ + 1 2 h νλ ∂ µĥρν + ∂ ρĥµν − ∂ νĥµρ ,(3.45) obeying ∇ µv ν = −h νρ K νρ but we could have equally absorbed the right hand side into the TNC connection leading to (3.44) which obeys ∇ µv ν = 0. This direct relation between TNC and Carrollian affine connections does not extend to cases where the connections depend onΦ orΦ as is obvious from the fact that then for example a Carrollian connection no longer has the property that ∇ µτµ = 0 (see also the discussion at the end of section 2.3). Ultra-Relativistic Gravity In [15] it was shown how one can make TNC geometries dynamical by using an effective field theory approach where one writes all relevant and marginal terms that are second order in time derivatives, preserve time reversal invariance leading to the most general forms of Hořava-Lifshitz gravity. Here we will start such an analysis for the case of dynamical Carrollian geometries. Since these have an ultra-relativistic light cone structure we will refer to the resulting theories as ultra-relativistic gravity. In order to decide whether a term is relevant, marginal or irrelevant we need to assign dilatation weights to the Carrollian fields τ µ , e a µ and M µ . We can extend the Carroll algebra by adding dilatations D to it resulting in the Lifshitz-Carroll algebra 3 [47,48] whose extra commutators involving D are [D, H] = −zH , [D, P a ] = −P a , [D, C a ] = (1 − z)C a . (4.1) We can thus assign dilatation weight −z to τ µ and −1 to e a µ . Further in order that τ µ and τ µ have the same dilatation weights we assign a weight 2 − z to M µ , i.e. under a local D transformation with parameter Λ D we have δ D τ µ = zΛ D τ µ , (4.2) δ D e a µ = Λ D e a µ ,(4. 3) δ D M µ = −(2 − z)Λ D M µ ,(4.4) so thatΦ has dilatation weight 2(1 − z), i.e. δ DΦ = −2(1 − z)Λ DΦ . (4.5) Note that τ µ and e a µ have the same dilatation weights as in the case of TNC geometry but that the weight ofΦ is opposite to that ofΦ. The reason for this is that in TNC geometry the vector M µ has dilatation weight z − 2 as follows for example from demanding thatv µ and v µ in (3.4) both have the same dilatation weight z. We will next consider actions in 2+1 dimensions with 0 ≤ z < 1 by demanding local Carrollian invariance, i.e. by demanding that the Carrollian fields τ µ , e a µ and M µ only enter the action via the invariantsτ µ , h µν andΦ. Further we will impose that the action is at most second order in time derivatives and preserves time reversal invariance. It is instructive to first consider the case with no coupling toΦ. As can be expected from the observations of section 3.3, where it is shown that a Carrollian geometry withoutΦ can be obtained from a TNC geometry withoutΦ by interchanging like tensors as in (3.26), the resulting actions should be of the HL form, but with 0 ≤ z < 1. Indeed using the results of [15] and the map (3.26) the following action is consistent with our coupling prescription for Carrollian gravity in 2 + 1 dimensions with 0 ≤ z < 1 S = d 3 xe C K µν K ρσĥ µρĥνσ − λ ĥ µν K µν 2 − V ,(4.6) where e = det (τ µ , e a µ ) which is invariant under local C and J transformations, where K µν = − 1 2 L v h µν is the extrinsic curvature and where the potential V is taken to be V = −2Λ + c 1 R + c 2ĥ µν a µ a ν . (4.7) In here we defined R =ê µ aê ν b R µν ab (J) = −ê µ aê ν bê σa e b ρ R µνσ ρ , (4.8) a µ = L vτµ ,(4.9) where the Riemann tensor R µρν ρ is defined in (2.29) with the connection (2.67) and where a µ = L vτµ is the Carrollian counterpart of the TNC torsion vector a µ = Lvτ µ [15]. An action of this type with λ = 1 and no potential term was considered in [30,31] as resulting from the c → 0 limit of the Einstein-Hilbert action 4 . All terms in (4.6) are relevant for 0 < z < 1 because the potential apart from the cosmological constant term involves terms of dilatation weight 2 and the kinetic terms have dilatation weight 2z all of which are less than 2 + z which is the negative of the dilatation weight of the integration measure e. The case with z = 0 will be studied separately below. The dimensionless parameter λ is the same as the one appearing in HL gravity [27,28]. Let us now introduce the scalarΦ. The first thing to observe is that for any z ≥ 0 we can add the following coupling to the kinetic terms Φ K µν K ρσĥ µρĥνσ − λ ĥ µν K µν 2 ,(4.10) since this has dilatation weight 2 which is less than z + 2. Further we can always add a term linear inΦ to the potential since 2(1 − z) ≤ 2 + z for 0 ≤ z < 1. On the other hand couplings such asΦR or a kinetic term forΦ such as v µ ∂ µΦ 2 have dilatation weight 4 − 2z and so in order that this is less than z + 2 we need z > 2/3. We will not consider such terms as we are primarily interested in those terms that are generic for 0 ≤ z < 1. When we includeΦ we are thus led to the more general action S = d 3 xe C + C 1Φ K µν K ρσĥ µρĥνσ − λ ĥ µν K µν 2 − V ,(4.11) where the potential is given by V = −2Λ + c 1 R + c 2ĥ µν a µ a ν + c 3Φ . (4.12) The equation of motion ofΦ imposes the constraint K µν K ρσĥ µρĥνσ − λ ĥ µν K µν 2 = c 3 C 1 . (4.13) On the other hand the variation with respect to h µν will bring time derivatives ontoΦ upon partial integration making the scalarΦ dynamical. It is interesting to contrast this with the case 1 < z ≤ 2 where we couple to TTNC geometry in the presence ofΦ (section 11 of [15]) where the fieldΦ imposes constraints on the terms in the potential rather than on the kinetic terms. The parameters in (4.11) have the following mass dimensions [C] = M 2−z , [C 1 ] = M z , [Λ] = M 2+z , [c 1 ] = [c 2 ] = M z , [c 3 ] = M 3z . (4.14) Finally we consider the special case z = 0 and show that one can construct a local dilatation invariant action, i.e. an action with anisotropic Weyl invariance. Using that for z = 0 the integration measure e has weight −2 we need to construct terms with weight 2. Under local dilatations the extrinsic curvature transforms as (for general z) δ D K µν = (2 − z)Λ D K µν − h µν v ρ ∂ ρ Λ D . (4.15) It follows that K µν K ρσĥ µρĥνσ − 1 2 ĥ µν K µν 2 is invariant under local scale transformations with weight 2z. Using that for z = 0 the scalarΦ has weight 2 we find that the following termΦ K µν K ρσĥ µρĥνσ − 1 2 ĥ µν K µν 2 ,(4.16) has weight 2 for z = 0. Other terms with weight 2 arê h µν a µ a ν , Φ , (4.17) h µν a µ a ν K µν K ρσĥ µρĥνσ − 1 2 ĥ µν K µν 2 . Hence the following action has anisotropic Weyl invariance with z = 0 S = d 3 xe C 1Φ + C 2ĥ µν a µ a ν K µν K ρσĥ µρĥνσ − 1 2 ĥ µν K µν 2 − V , (4.18) where the potential is given by V = c 2ĥ µν a µ a ν + c 3Φ .(4.19) This action with anisotropic Weyl invariance for z = 0 only has dimensionless coupling constants. We note that the spatial Ricci scalar R transforms under local D transformations as (in d = 2 spatial dimensions) δ D R = −2Λ D R + 2ĥ µν ∇ µ ∂ ν Λ D . (4.20) Different from the conformal TNC case (section 12 of [15]) here we cannot use the vector a µ to make a local D invariant combination out of R and derivatives of a µ because for z = 0 the vector a µ is invariant under local D transformations. Discussion It would be interesting to extend this work in the following directions. It has been known for a long time that the asymptotic symmetry algebra of asymptotically flat space-times is given by the Bondi-Metzner-Sachs (BMS) algebra [49,50,51] (see also [52,53]). In 3 bulk dimensions it has been shown that the BMS algebra is isomorphic to the 2-dimensional Galilean conformal algebra [54,55] (which is a contraction of the relativistic conformal group [56]). Recently conformal extensions of the Carroll algebra have been studied in [57,13] and it has been shown that the BMS algebra forms a conformal extension of the Carroll algebra [13]. Regarding the case of flat space holography in 3 bulk dimensions the Galilean structures seen at infinity can be interpreted as Carrollian because in 1+1 boundary dimensions interchanging space and time leads to an isomorphism between the Carroll and Galilei algebras. Further, future and past null infinity form Carrollian space-times [13]. It could therefore be insightful to explore the connections between the gauging of the Carroll algebra and flat space holography further. The space-time symmetries of warped conformal field theories involve Carrollian boosts that together with the scale transformations form the z = 0 Lifshitz-Carroll algebra [14]. It would be interesting to apply the methods for the gauging of the Carroll algebra as performed here to study the coupling of these WCFTs to curved backgrounds. More generally along similar lines one can couple field theories to Carrollian geometries and study global symmetries by defining conformal Killing vectors, define an energy-momentum tensor by varying the invariantsτ µ and h µν much like it was done for field theories coupled to TNC geometries [19,22,23,3]. It would be interesting to understand what the role of the scalarΦ is when coupling field theories to Carrollian geometries, i.e. to understand what the response is to varying this background field. Finally, one can study the actions for ultra-relativistic or Carrollian gravity further by e.g. studying their phase space formulation, count the number of degrees of freedom, etc. It would be interesting to generalize the 3-dimensional actions of ultra-relativistic gravity constructed here to higher dimensions and to study the equations of motion by looking for various classes of solutions such as cosmological and spherically symmetric space-times. It would be interesting to study the perturbative properties of these theories for example by linearizing around flat space-time and study the form of the propagators, etc. by the advanced ERC grant 'Symmetries and Dualities in Gravity and M-theory' of Marc Henneaux. I thank the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during the initial stages of this work. transformation the vector M µ transforms as M ′µ = M ν ∂x ′µ ∂x ν + ∂x ′µ ∂u . (3.29) the remaining invariants are related bŷ v µ ↔τ µ ,h µν ↔h µν ,Φ ↔Φ , (3.43) where again on the left we have the TNC invariantsh µν ,v µ andΦ given in equations (3.2), (3.4) and (3.5) (with m µ replaced by M µ ), respectively and on the right we have the Carrollian invariantsh µν ,τ µ andΦ given in equations (2.64), (2.65) and (2.66), respectively. The duality (3.41) and (3.42) interchanges the Galilean and Carrollian light cone structures in the sense that (3.41) relates the notions of metricity while (3.42) swaps the notion of boost transformations. We emphasize that in order to obtain theδ transformations, i.e. the diffeomorphisms and local tangent space transformations we did not need to impose any so-called curvature constraints. For a discussion of the curvature constraints we refer the reader to section 2.3. This algebra with z = 0 is realized in higher dimensional uplifts of Lifshitz space-times. For example a z = 2 Lifshitz space-time can be uplifted to a 5-dimensional z = 0 Schrödinger space-time[42,43,44,45,46]. In order to support this geometry one needs to add an axionic scalar which breaks the z = 0 Schrödinger algebra down to the z = 0 Lifshitz-Carroll algebra. I would like to thank Marc Henneaux for useful discussions about the c → 0 limit of the Einstein-Hilbert action. 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[ "MIRROR SYMMETRY AND RATIONAL CURVES ON QUINTIC THREEFOLDS: A GUIDE FOR MATHEMATICIANS", "MIRROR SYMMETRY AND RATIONAL CURVES ON QUINTIC THREEFOLDS: A GUIDE FOR MATHEMATICIANS" ]	[ "David R Morrison " ]	[]	[]	We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new q-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the "mirror symmetry" phenomenon recently observed by string theorists.	10.1090/s0894-0347-1993-1179538-2	[ "https://export.arxiv.org/pdf/alg-geom/9202004v1.pdf" ]	9,228,037	alg-geom/9202004	4a767671ea9acde9763565994b3e5a144e8c2346	MIRROR SYMMETRY AND RATIONAL CURVES ON QUINTIC THREEFOLDS: A GUIDE FOR MATHEMATICIANS arXiv:alg-geom/9202004v1 10 Feb 1992 July, 1991 David R Morrison MIRROR SYMMETRY AND RATIONAL CURVES ON QUINTIC THREEFOLDS: A GUIDE FOR MATHEMATICIANS arXiv:alg-geom/9202004v1 10 Feb 1992 July, 1991 We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new q-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the "mirror symmetry" phenomenon recently observed by string theorists. Introduction There has been much recent excitement among mathematicians about a calculation made by a group of string theorists (P. Candelas, X. C. de la Ossa, P. S. Green, and L. Parkes [6]) which purports to give a count of the number of rational curves of fixed degree on a general quintic threefold. The calculation mixes arguments from string theory with arguments from mathematics, and is generally quite difficult to follow for mathematicians. I believe that I now understand the essential mathematical content of that calculation. It is my purpose in this note to explain my understanding in terms familiar to algebraic geometers. What Candelas et al. actually calculate is a q-expansion of a certain function determined by the variation of Hodge structure of some other family of threefolds with trivial canonical bundle. The "mirror symmetry principle" is then invoked to predict that the Fourier coefficients in that expansion should be related to the number of rational curves on a quintic threefold. One mathematical surprise in this story is a new q-expansion principle for functions on the moduli space of Calabi-Yau manifolds. Near points on the boundary of moduli where the monodromy is "maximally unipotent," there turn out to be natural coordinates in which to make q-expansions of functions. In this paper, we will discuss these q-expansions only in the case of one-dimensional moduli spaces; the general case will be treated elsewhere. By focusing on this q-expansion principle, we place the computation of [6] in a mathematically natural framework. Although there remain certain dependencies on a choice of coordinates, the coordinates used for calculation are canonically determined by the monodromy of the periods, which is itself intrinsic. On the other hand, we have removed some of the physical arguments which were used in the original paper to help choose the coordinates appropriately. The result may be that our presentation is less convincing to physicists. The plan of the paper is as follows. In section 1 we review variations of Hodge structure, and explain how to define "Yukawa couplings" in Hodge-theoretic terms. (A discussion of Yukawa couplings along the same lines has also been given by Cecotti [9], [10].) In section 2 we study the asymptotic behavior of the periods near points with maximally unipotent monodromy. This is applied to find q-expansions of the Yukawa couplings in section 3. In section 4 we attempt to describe mirror symmetry in geometric terms. In section 5 we turn to the main example (the family of "quintic-mirrors"), and in section 6 we explain how the mirror symmetry principle predicts from the earlier calculations what the numbers of rational curves on quintic threefolds should be. Several technical portions of the paper have been banished to appendices. We work throughout with algebraic varieties over the complex numbers, which we often identify with complex manifolds (or complex analytic spaces). If X is a compact complex manifold and p, q ≥ 0, we define H p,q (X) = H q (Ω p X ) = H q (Λ p Ω X ) where Ω X is the holomorphic cotangent bundle of X. (This is slightly non-standard.) We extend this definition to the case p < 0, q ≥ 0 by H , (X). (With our conventions, this follows from the Hodge theorem in de Rham cohomology H n DR (X) = H p,q ∂ (X) together with the Dolbeault isomorphism H p,q ∂ (X) ∼ = H q (Ω p X ) = H p,q (X).) The Hodge decomposition can also be described by means of the Hodge filtration We now recall a construction which first arose in the study of infinitesimal variations of Hodge structure by Carlson, M. Green, Griffiths, and Harris [8]. The cup product determines a natural map H 1 (Θ X ) → Hom(H p,q (X), H p−1,q+1 (X)) (1) called the differential of the period map. Iterates of this map are symmetric in their variables; the n th iterate of the differential is the induced map Sym n H 1 (Θ X ) → Hom(H n,0 (X), H 0,n (X)). Using the canonical isomorphism Hom(H n,0 (X), H 0,n (X)) = (H n,0 (X)) * ⊗ H 0,n (X) and the isomorphism H 0,n (X) ∼ = (H n,0 (X)) * induced by the adjoint, we get a map Sym n H 1 (Θ X ) → (H n,0 (X) * ) ⊗2 .(2) We call this the unnormalized Yukawa coupling 1 of X. If we choose an element of H n,0 (X) ⊗2 and evaluate the map (2) on that element, we get a map Sym n H 1 (Θ X ) → C called a normalized Yukawa coupling. The "normalization" is the choice of element of H n,0 (X) ⊗2 . We now analyze these constructions for a family of manifolds. Suppose we are given a quasi-projective variety C and a smooth map π : X → C whose fibers are Calabi-Yau manifolds. Suppose also that this family can be completed to a family of varietiesπ : X → C, where C is a projective compactification of C. The fibers ofπ may degenerate over the boundary B := C − C. We assume that B is a divisor with normal crossings on C. For any point P ∈ C we denote the fiber π −1 (P ) of π by X P . The Kodaira-Spencer map ρ : Θ C,P → H 1 (Θ X P ) maps the tangent space to C at P to the tangent space to moduli at the point [X P ]. If C is actually a moduli space for the fibers of X , the map ρ will be an isomorphism. The cohomology of the fibers of the map π with coefficients in Z and C fit together into local systems R n π * Z and R n π * C on C. The Hodge filtration becomes a filtration of the vector bundle F 0 := R n π * C ⊗ O C by holomorphic subbundles: R n π * C ⊗ O C = F 0 ⊃ F 1 ⊃ · · · ⊃ F n−1 ⊃ F n ⊃ (0). The vector bundle F 0 has a natural flat connection ∇ : F 0 → F 0 ⊗ Ω C called the Gauß-Manin connection, whose horizontal sections determine the local system R n π * C. The Griffiths transversality property says that ∇(F p ) ⊂ F p−1 ⊗ Ω C . There is a natural extension of this setup over the boundary B, which involves the sheaf Ω C (log B) of logarithmic differentials. (That sheaf is locally generated by Ω C and all elements of the form df /f , where f = 0 is a local equation of a local component of B.) Although the local system R n π * C cannot in general be extended across B in a single-valued way, the Hodge bundles F p do have natural extensions to bundles F p on C. And the Gauß-Manin connection ∇ extends to a connection ∇ : F 0 → F 0 ⊗ Ω C (log B) which satisfies ∇(F p ) ⊂ F p−1 ⊗ Ω C (log B) . This restriction on the types of poles which ∇ may have along B is equivalent to a requirement that the connection ∇ have "regular singular points." The extended Gauß-Manin connection ∇ gives rise to an O C -linear map on the associated gradeds ∇ : F p /F p+1 → (F p−1 /F p ) ⊗ Ω C (log B).(3) To make contact with the n th iterate of the differential and the Yukawa coupling, we introduce the sheaf Θ C (− log B) of vector fields with logarithmic zeros, which is the dual of Ω C (log B). The map (3) then induces the bundle version of (1) Θ C (− log B) → Hom(F p /F p+1 , F p−1 /F p ). When this is iterated n times, it produces a map Sym n (Θ C (− log B)) → Hom(F n , F 0 /F 1 ).(4) The polarizations fit together into a bilinear map of local systems \| : R n π * Z × R n π * Z → R 2n π * Z ∼ = → Z whose adjoint map induces an isomorphism ad \| : (F 0 /F 1 ) → (F n ) * . This extends to a map of bundles ad \| : (F 0 /F 1 ) → (F n ) * .(5) Using the canonical isomorphism Hom(F n , F 0 /F 1 ) = (F n ) * ⊗(F 0 /F 1 ) and composing the map (5) with the map (4), we get the Yukawa map κ : Sym n (Θ C (− log B)) → ((F n ) * ) ⊗2 . If we also specify a section of (F n ) ⊗2 , we get a normalized Yukawa map κ norm : Sym n (Θ C (− log B)) → O C . Suppose that C is actually the moduli space for the fibers of X , so that ρ is an isomorphism. If we compose ρ −1 with a normalized Yukawa map κ norm we get Sym n H 1 (Θ X P ) ρ −1 −→ Sym n (Θ C,P ) κ norm −→ O C,P = C. In this way, we exactly recover the corresponding normalized Yukawa coupling. Candelas et al. [6] typically compute the Yukawa coupling in local coordinates (away from the boundary) as follows. Suppose that dim C = 1, and that ψ is a local coordinate defined in an open set U ⊂ C. There is an induced vector field d/dψ, which is a local section of Θ U . Choose a section 2 ω of F n over U, and define κ ψ...ψ = κ( d dψ , . . . , d dψ ) · ω 2 . (The number of ψ's in the subscript is n.) This is a holomorphic function on U. If we alter ω by the gauge transformation ω → f ω, then the Yukawa coupling transforms as κ ψ...ψ → f 2 κ ψ...ψ . "Normalizing the Yukawa map" is the same thing as "fixing the gauge." Our primary goal will be to compute the asymptotic behavior of the Yukawa map κ in a neighborhood of the boundary B. The asymptotic behavior of the periods For simplicity of exposition, we now specialize to the case in which C is a curve. Let P ∈ B be a boundary point, and let T P be the monodromy of the local system R n π * Z around P . We regard T P as an element of Aut H n (X P ′ , Z), where P ′ is a point near P ; T P is determined by analytic continuation along a path which goes once around P in the counterclockwise direction. By the monodromy theorem [23], T P is quasi-unipotent, which means that some power T k P is unipotent. Moreover, the index of unipotency is bounded: we have (T k P −I) n+1 = 0. We say that P is a point at which the monodromy is maximally unipotent if the monodromy T P is unipotent, and if (T P − I) n = 0. (Thus, the index of nilpotency of T P − I is maximal.) Since T P − I is nilpotent, we can define the logarithm of the monodromy N = log(T P ) ∈ Aut H n (X P ′ , Q) by a finite power series log(T P ) = (T P − I) − (T P − I) 2 2 + · · · + (−1) n+1 (T P − I) n n . (Rational coefficients are needed in cohomology since rational numbers appear in the power series.) N is also a nilpotent matrix, with the same index of nilpotency as T P − I. Lemma 1. Let π : X → C be a one-parameter family of varieties with h n,0 = 1. Let P ∈ B = C − C be a boundary point at which the monodromy on R n π * Z is maximally unipotent and let N be the logarithm of the monodromy. Then the image of N n is a Q-vector space of dimension one, and the image of N n−1 is a Q-vector space of dimension two. We defer the proof of this lemma to appendix A. We say that a basis g 0 , g 1 of (Im N n−1 ) ⊗ C ⊂ H ⋉ (X P ′ , C) is an adapted basis if g 0 spans (Im N n ) ⊗ C. (We have extended scalars to C since certain computational procedures lead more naturally to complex coefficients.) If g 0 , g 1 is an adapted basis for (Im N n−1 ) ⊗ C, then by Poincaré duality, there are homology classes γ 0 , γ 1 ∈ H n (X P ′ , C) such that g j \| α = γ j α for any α ∈ H n (X P ′ , C). Here we denote the evaluation of cohomology classes on homology classes by using an integral sign, since that evaluation is often accomplished by integration. Proposition . Let γ 0 , γ 1 be the homology classes determined by an adapted basis g 0 , g 1 of (Im N n−1 ) ⊗ C. Define a constant m by Ng 1 = mg 0 . Let U be a small neighborhood of P , and let z be a coordinate on U centered at P . Let ω be a non-zero section of F n over U. Then 1. γ 0 ω extends to a single-valued function on U . 2. γ 1 ω is not single-valued. However, we can write 1 m γ 1 ω γ 0 ω = log z 2πi + single valued function. Proof. Any g ∈ H n (X P ′ , C) can be extended to a section g(z) of the local system over U = U − P , which may be multi-valued. But by the nilpotent orbit theorem [34], exp(− log z 2πi N)g(z) extends to a singlevalued section. Since ω is single-valued, exp(− log z 2πi N)g(z) ω will also be single-valued. Now g j ∈ (Im N n−1 )⊗C implies that N 2 g j = 0 for j = 1, 2. The series needed for exp in this case is thus rather simple: exp(− log z 2πi N)g j (z) = (I − log z 2πi N)g j (z) = g j (z) − log z 2πi Ng j (z). We conclude that γ 0 ω = g 0 (z) \| ω and γ 1 ω − m log z 2πi γ 0 ω = g 1 (z) − log z 2πi mg 0 (z) ω are single-valued functions. Q.E.D. Corollary . Let γ 0 , γ 1 be the homology classes determined by an adapted basis g 0 , g 1 of (Im N n−1 ) ⊗ C, as in the proposition. The function t := 1 m γ 1 ω γ 0 ω gives a natural parameter on the universal cover U of U called a quasicanonical parameter, and q := e 2πit gives a natural coordinate on U called a quasi-canonical coordinate. These are independent of the choice of ω. We have d dt = 2πi q d dq , either of which serves as a local generator of the sheaf Θ C (− log B). Moreover, under a change of adapted basis (g 0 , g 1 ) → (ag 0 , bg 0 +cg 1 ), we have m → c a m, t → t + b mc , and q → e 2πib/mc q. Therefore, t is uniquely determined up to an additive constant, and q is uniquely determined up to a multiplicative constant. We can normalize further if we take the integral structure into account. We call g 0 , g 1 a good integral basis of Im N n−1 if g 0 is a generator of Im N n ∩ H n (X P ′ , Z), and g 1 is an indivisible element of H n (X P ′ , Z) which can be written as g 1 = 1 λ N n−1 g for some λ > 0 and some g ∈ H n (X P ′ , Z) such that g 0 \| g = 1. Notice that a good integral basis is an adapted basis. The next lemma, which is based on some work of Friedman and Scattone [16], will be proved in appendix A. Lemma 2. Good integral bases exist. If g 0 , g 1 and g ′ 0 , g ′ 1 are good integral bases, then g ′ 1 = k g 0 + (−1) ℓ g 1 g ′ 0 = (−1) ℓ g 0 , for some integers k and ℓ. Since (T −I) 2 = 0 on Im N n−1 , we have the simple formula N = T −I on that space. In particular, when restricted to Im N n−1 , the map N is defined over the integers. Thus, if g 0 , g 1 is a good integral basis and we write Ng 1 = mg 0 , then m is an integer. Note that m is independent of the choice of good integral basis. Corollary . Let g 0 , g 1 be a good integral basis, and define an integer m by Ng 1 = mg 0 . Then the quasi-canonical coordinate q formed from this basis is unique up to multiplication by an \|m\| th root of unity. We call q a canonical coordinate and t a canonical parameter under these circumstances. These are actually unique if \|m\| = 1; in this case, we say that the monodromy is small. The q-expansion of the Yukawa coupling The first example of the construction of the previous section is furnished by the classical theory of periods of elliptic curves. Let π : X → U be a family of smooth elliptic curves over a punctured disk U which can be completed to a familyπ : X → U with a singular fiber over the boundary point P = U − U. The point P is called a cusp. Let P ′ ∈ U, and suppose there is a symplectic basis γ 0 , γ 1 of the first homology group H 1 (X P ′ , Z) such that the monodromy T P acts as T P (γ 0 ) = γ 0 T P (γ 1 ) = γ 0 + γ 1 . (The basis is symplectic if γ 0 ∩ γ 1 = 1.) This easily implies that P is a maximally unipotent boundary point, that γ 0 , γ 1 is the homology basis dual to a good integral basis, and that m = 1. For a fixed holomorphic one-form ω on X P ′ , the numbers ( γ 0 ω, γ 1 ω) were classically known as the periods of the elliptic curve X P ′ . By varying the one-form, the periods can be normalized to take the form (1, τ ). The invariant way to formulate this is to define τ = γ 1 ω γ 0 ω . This function τ can be regarded as a map from the universal cover U of U to the upper half-plane H. (The image lies in the upper half-plane since the basis is symplectic.) The monodromy transformation T P induces the map τ → τ + 1.(6) Thus, functions f defined on U pull back to functions f on U which are invariant under (6). It follows that any such function has a Fourier series f (τ ) = n∈Z a n e 2πinτ . If expressed in terms of the natural coordinate q = e 2πiτ on U, this is called a q-expansion, and it takes the form f (q) = n∈Z a n q n . If f has a holomorphic extension across the cusp P , the only terms appearing in this sum are those with n ≥ 0. What we have shown in section 2 is that this classical construction generalizes to functions defined near a maximally unipotent boundary point P of a Calabi-Yau moduli space (at least when that space has dimension one). Fix a good integral basis, which determines a canonical coordinate q and a canonical parameter t. The monodromy transformation T P acts on t by t → t + 1. Therefore, any function f defined near P which is holomorphic at P will have a q-expansion f (q) = ∞ n=0 a n q n , which can also be regarded as a Fourier series f (t) = ∞ n=0 a n e 2πint in t. These expressions are unique if \|m\| = 1, i.e., if the monodromy is small. In order to obtain a q-expansion of the Yukawa coupling, we must normalize that coupling. But there is a natural choice of normalization determined by a good integral basis. To see this, note that any good integral basis g 0 , g 1 determines a section ( γ 0 ) −1 ∈ H 0 (U , (F n )) by ( γ 0 ) −1 := ω γ 0 ω for any non-zero ω ∈ H 0 (U , F n ). By lemma 2, a change in good integral basis may change the sign of ( γ 0 ) −1 , but the induced section ( γ 0 ) −2 ∈ H 0 (U , (F n ) ⊗2 ) is independent of the choice of good integral basis. We thus have a very natural normalization for the Yukawa map in U. We also have a natural parameter t with which to compute, such that d/dt is a generator of Θ U (− log B). So we can define the mathematically normalized Yukawa coupling κ t...t by the formula κ t...t = κ( d dt , . . . , d dt ) · ( γ 0 ) −2 . This mathematically normalized Yukawa coupling κ t...t is an intrinsically defined function on a neighborhood of the boundary. (It is canonically determined by our choice of maximally unipotent boundary point; however, it could conceivably change if the boundary point changes.) The function κ t...t therefore has a q-expansion κ t...t = a 0 + a 1 q + a 2 q 2 + · · · ,(7) which can also be regarded as a Fourier expansion in the parameter t: κ t...t = a 0 + a 1 e 2πit + a 2 e 4πit + · · · .(8) These expressions are unique if the monodromy is small. Mirror symmetry In this section I will attempt to outline the mirror symmetry principle in mathematical terms, and describe some of the mathematical evidence for it. I apologize to physicists for my misrepresentations of their ideas, and I apologize to mathematicians for the vagueness of my explanations. Gepner [17] has conjectured that there is a one-to-one correspondence between N = 2 superconformal field theories with c = 3n, and Calabi-Yau manifolds X of dimension n equipped with some "extra structure" S. (This correspondence can be realized concretely in a number of important cases using work of Greene, Vafa and Warner [19], Martinec [26], [27], and others.) A precise geometric description of the extra structure S has not yet been given. It appears to involve specifying a class in U/Γ, where U ⊂ H 1,1 (X) is some open set, and Γ is some group of automorphisms of U. What is clear about this extra structure is how to perturb it: first-order deformations of S correspond to elements of H 1,1 (X). An instructive example is the case in which X is an elliptic curve. In that case, as shown in [12] and [1], one takes U ⊂ H 1,1 (X) ∼ = C to be the upper half-plane, and Γ = SL(2, Z). Thus, the extra structure S represents a point in the j-line, or equivalently, a choice of a second elliptic curve. We specialize now to the case of dimension n = 3. The space of first-order deformations of the superconformal field theory can be decomposed as 3 H 1 (Θ X ) ⊕ H 1 (Ω X ), with H 1 (Θ X ) = H −1,1 (X) corresponding to first-order deformations of the complex structure on X, and H 1 (Ω X ) = H 1,1 (X) corresponding to first-order deformations of the extra structure S. These first-order deformations are called marginal operators in the physics literature. Specifying a superconformal field theory of this type also determines cubic forms Sym 3 H −1,1 (X) → C and Sym 3 H 1,1 (X) → C. The cubic form on H −1,1 is the Yukawa coupling described in section 1, normalized in a way specified by the physical theory. From a mathematical point of view, this is determined by the variation of Hodge structure plus the choice of normalization. This cubic form depends on the complex structure of X, but should be independent of the "extra structure" S. The cubic form on H 1,1 lacks a precise geometric description at present. By work of Dine, Seiberg, Wen, and Witten [13] and Distler and Greene [14], it is known to have an expression of the form ∞ k=0 σ k e −kR ,(9) where R is a complex parameter which depends on the extra structure S. The real part of R is related to the "radius" in the physical theory in such a way that Re R → ∞ is the "large radius limit." The leading coefficient σ 0 is the natural topological product Sym 3 H 1,1 (X) → C. (In other words, the cubic form on H 1,1 approaches the topological product in the large radius limit.) The higher coefficients σ k are supposed to be related in some well-defined way to the numbers of rational curves of various degrees on the generic deformation of X (assuming those numbers are finite). One of the important unsolved problems in the theory is to determine this relationship precisely. As was first noticed by Dixon [15, p. 118], and later developed by Lerche, Vafa, and Warner [24] and others, the identification of one piece of the superconformal field theory with H 1,1 (X) and the other piece with H −1,1 (X) ∼ = H 2,1 (X) involves an arbitrary choice, and the theory is also consistent with making the opposite choice. Moreover, as we will describe below, there are examples in which the Gepner correspondence can be realized for both choices. But except in the very rare circumstance that the Hodge numbers h 1,1 and h 2,1 = dim H −1,1 coincide, changing the choice necessarily involves changing the Calabi-Yau threefold X. The new threefold X ′ will have a completely different topology from the old: in fact, the Hodge diamond is rotated by 90 • when passing from one to the other. This leads to a mathematical version of the mirror symmetry conjecture: To each pair (X, S) consisting of a Calabi-Yau threefold X together with some extra structure S there should be associated a "mirror pair" (X ′ , S ′ ) which comes equipped with natural isomorphisms H −1,1 (X) ∼ = → H 1,1 (X ′ ) and H 1,1 (X) ∼ = → H −1,1 (X ′ ) that are compatible with the cubic forms. Even in this rather imprecise 4 form, the conjecture as stated is easily refuted: There exist rigid Calabi-Yau threefolds, which have h 2,1 = 0 (see Schoen [35] for an example). Any mirror of such a threefold would have h 1,1 = 0, and so could not be Kähler. A potentially correct version of the conjecture, even less precise, begins: "To most pairs (X, S), including almost all of interest in physics, there should be associated . . . ". It is tempting to speculate that the theory should be extended to non-Kähler threefolds as in Reid's fantasy [30], which might rescue the conjecture in its original form. Alternatively, Aspinwall and Lütken [2] suggest that the Gepner correspondence (and hence the mathematical version of mirror symmetry) should only hold in the large radius limit. Since no "limits" can be taken in the rigid case, a mathematical mirror construction would not be expected there. To be presented with a conjecture which has been only vaguely formulated is unsettling to many mathematicians. Nevertheless, the mirror symmetry phenomenon appears to be quite widespread, so it seems important to make further efforts to find a precise formulation. In fact, there are at least four major pieces of mathematically significant evidence for mirror symmetry. (i) Greene and Plesser [18] have studied a case in which there are very solid physics arguments which tie the pair (X, S) to the corresponding superconformal field theory (as predicted by Gepner). The Calabi-Yau threefolds in question are desingularizations of quotients of Fermat-type weighted hypersurfaces by certain finite groups (including the trivial group). For each pair (X, S) of this type, Greene and Plesser were able to find the corresponding mirror pair (X ′ , S ′ ) by analyzing the associated superconformal field theories. It turns out that the pairs are related by taking quotients: X ′ is a desingularization of X/G for some symmetry group G. By deformation arguments, the mirror symmetry phenomenon persists in neighborhoods of (X, S) and (X ′ , X ′ ). Roan [33] subsequently gave a direct mathematical proof that the predicted isomorphisms between H −1,1 and H 1,1 groups exist in this situation. (ii) Candelas, Lynker, and Schimmrigk [7] have computed the Hodge numbers for a large class of Calabi-Yau threefolds which are desingularizations of hypersurfaces in weighted projective spaces. They put some extra constraints on the form of the equation, and found about 6000 types of threefolds satisfying their conditions. The set of pairs (h 1,1 , h 2,1 ) obtained from these examples is very nearly (but not precisely) symmetric with respect to the interchange h 1,1 ↔h 2,1 . Since there is no a priori reason that the mirror of a desingularized weighted hypersurface should again be a desingularized weighted hypersurface, this is consistent with the conjecture and supports it quite strongly. (iii) Aspinwall, Lütken, and Ross [3] (see also [1]) have carefully studied a particular mirror pair (X, S), (X ′ , S ′ ). They put X in a family X = {X t } which has a degenerate limit as t approaches 0. Some heuristics were used in choosing the family X , in an attempt to ensure that the limit as t → 0 would correspond to the "large radius limit" for the mirror (X ′ , S ′ ). Aspinwall et al. then computed the limiting behavior of the cubic form on H −1,1 (X t ), and showed that it coincides with the topological product σ ′ 0 on H 1,1 (X ′ ), as predicted by the conjecture. (Actually, there is a normalization factor which was not computed, but the agreement is exact up to this normalization.) (iv) The work of Candelas, de la Ossa, P. Green, and Parkes [6] being described in this paper goes further, and computes the other coefficients in an asymptotic expansion. This will be explained in more detail in the next two sections. The quintic-mirror family We now describe a certain one-parameter family of Calabi-Yau threefolds constructed by Greene and Plesser [18], as amplified by Candelas et al. [6]. Begin with the family of quintic threefolds Q ψ = { x ∈ P 4 \| p ψ ( x) = 0} defined by the polynomial p ψ := 5 k=1 x 5 k − 5ψ 5 k=1 x k . Let µ 5 be the multiplicative group of 5 th roots of unity, and let G := { α = (α 1 , . . . , α 5 ) ∈ (µ 5 ) 5 \| 5 k=1 α k = 1} act on P 4 by α : x i → α i · x i . There is a "diagonal" subgroup of order 5 which acts trivially; let G = G/{diagonal} be the image of G in Aut(P 4 ). G is a group which is abstractly isomorphic to (Z/5Z) 3 . The action of G preserves the threefold Q ψ ; let η : Q ψ → Q ψ /G denote the quotient map. For each pair of distinct indices i, j, the set of 5 points S ij := {x 5 i + x 5 j = 0, x ℓ = 0 for all ℓ = i, j} ⊂ Q ψ is preserved by G, and there is a group G ij ⊂ G of order 25 which is the stabilizer of each point in the set. The image S ij /G is a single point p ij ∈ Q ψ /G. In addition, for each triple of distinct indices i, j, k, the curve C ijk := {x 5 i + x 5 j + x 5 k = 0, x ℓ = 0 for all ℓ = i, j, k} ⊂ Q ψ is preserved by G. There is a subgroup G ijk ⊂ G of order 5 which is the stabilizer of every point in C − η −1 ({p ij , p jk , p ik }). The image C ijk = C ijk /G is a smooth curve in Q ψ /G. The action of G is free away from the curves C ijk . The quotient space Q ψ /G has only canonical singularities. At most points of C ijk , the surface section of the singularity is a rational double point of type A 4 , but at the points p ij the singularity is more complicated: three of the curves of A 4 -singularities meet at each p ij . By a theorem of Markushevich [25,Prop. 4] and Roan [31,Prop. 2], these singularities can be resolved to give a Calabi-Yau manifold W ψ . There are choices to be made in this resolution process; we describe a particular choice in appendix B. (By another theorem of Roan [32,Lemma 4], any two resolutions differ by a sequence of flops.) For any α ∈ µ 5 , there is a natural isomorphism between Q αψ /G and Q ψ /G induced by the map (x 1 , x 2 , x 3 , x 4 , x 5 ) → (α −1 x 1 , x 2 , x 3 , x 4 , x 5 ).(10) This extends to an isomorphism between W αψ and W ψ , provided that we have resolved singularities in a compatible way. We verify in appendix B that the choices in the resolution can be made in a sufficiently natural way that this isomorphism is guaranteed to exist. Thus, λ := ψ 5 is a more natural parameter to use for our family. We define the quintic-mirror family to be {W 5 √ λ } → {λ} ∼ = C. This has a natural compactification to a family over P 1 , with boundary B = P 1 − C = {∞}. The computation made by Candelas et al. [6] shows that the monodromy at ∞ is maximally unipotent, and that m = 1, i.e., that the monodromy is small. (We explain in appendix C how this follows from [6].) The key computation in [6] is an explicit calculation of the q-expansion of the mathematically normalized Yukawa coupling. Candelas et al. find that the q-expansion begins: κ ttt = 5 + 2875e 2πit + 4876875e 4πit + · · · .(11) In fact, they have computed at least 10 coefficients. Mirror moonshine? Greene and Plesser [18], using arguments from superconformal field theory, have identified the family of quintic-mirrors {W 5 √ λ } as the "mirror" of the family of smooth quintic threefolds {M z }. Note that the Hodge numbers satisfy where t is again the canonical parameter. Thus, the q-expansion of κ ttt in equation (8) will coincide with the asymptotic expansion in R given by equation (9), evaluated on the generator H ⊗3 of Sym 3 H −1,1 (M). (iv) There is an explicit formula for the coefficients σ k , as described below. To explain the formula for σ k , let n k denote the number of rational curves of degree k on the generic quintic threefold. Candelas et al. propose the formula κ ttt = 5 + ∞ k=1 n k k 3 e 2πikt 1 − e 2πikt = 5 + n 1 e 2πit + (2 3 n 2 + n 1 )e 4πit + · · · , (12) which implicitly incorporates their expressions for the higher coefficients (The first two expressions are σ 1 (H ⊗3 ) = n 1 , σ 2 (H ⊗3 ) = 2 3 n 2 + n 1 .) In the large radius limit Im t → ∞, the right hand side of equation (12) approaches 5. This agrees with the mirror symmetry conjecture, 5 since the topological intersection form on M is determined by its value on the standard generator H, viz., H 3 = 5. Moreover, by comparing equations (11) and (12), we can predict values for the numbers n k . The first two predictions are n 1 = 2875, which was classically known to be the number of lines on a quintic threefold, and n 2 = 609250, which coincides with the number of conics on a quintic threefold computed by Katz [21]! Unfortunately, there seem to be difficulties with n 3 . Not any more!! How was formula (12) arrived at? I am told that the field theory computation necessary to derive this formula can be done in principle, but seems to be too hard to carry out in practice at present. So Candelas et al. give a rough derivation of this formula based on some assumptions. Why do they believe the resulting formula to be correct? I quote from [6]: These numbers provide compelling evidence that our assumption about the form of the prefactor is in fact correct. The evidence is not so much that we obtain in this way the correct values for n 1 and n 2 , but rather that the coefficients in eq. (11) have remarkable divisibility properties. For example asserting that the second coefficient 4, 876, 875 is of the form 2 3 n 2 + n 1 requires that the result of subtracting n 1 from the coefficient yields an integer that is divisible by 2 3 . Similarly, the result of subtracting n 1 from the third coefficient must yield an integer divisible by 3 3 . These conditions become increasingly intricate for large k. It is therefore remarkable that the n k calculated in this way turn out to be integers. I would add that it is equally remarkable that the coefficients in eq. (11) themselves turn out to be integers: I know of no proof that this is the case. These arguments have a rather numerological flavor. I am reminded of the numerological observations made by Thompson [37] and Conway and Norton [11] about the j-function and the monster group. At the time those papers were written, no connection between these two mathematical objects was known. The q-expansion of the j-function was known to have integer coefficients, and it was observed that these integers were integral linear combinations of the degrees of irreducible representations of the monster group. This prompted much speculation about possible deep connections between the two, but at the outset all such speculation had to be characterized as "moonshine" (Conway and Norton's term). The formal similarities to the present work should be clear: a qexpansion of some kind is found to have integer coefficients, and these integers then appear to be integral linear combinations of another set of integers, which occur elsewhere in mathematics in a rather unexpected location. Perhaps it is too much to hope that the eventual explanation will be as pretty in this case. Appendix A: Proofs of the monodromy lemmas Let W 0 ⊂ W 1 ⊂ · · · ⊂ W 2n = H n (X P ′ , Q) be the monodromy weight filtration at P , and let F 0 ⊃ F 1 ⊃ · · · ⊃ F n−1 ⊃ F n ⊃ (0). be the limiting Hodge filtration at P . (We refer the reader to [20] or [34] for the definitions.) By a theorem of Schmid [34], these induce a mixed Hodge structure on the cohomology. Note that since N n+1 = 0, we have W 0 = Im N n . Moreover, if \| denotes the polarization on the cohomology, we have Nx \| y = − x \| Ny . Recall also that the polarization is symmetric or skew-symmetric, depending on the dimension n: x \| y = (−1) n y \| x Proof of lemma 1. Since W · is the monodromy weight filtration, N n induces an isomorphism N n : W 2n /W 2n−1 → W 0 .(13) These spaces cannot be zero, since (T P − I) n = 0. On the other hand, since F n+1 = (0), the Hodge structure on W 2n /W 2n−1 must be purely of type (n, n). It follows that F n /(F n ∩ W 2n−1 ) = W 2n /W 2n−1 . But since F n is one-dimensional, this can only happen if F n ⊂ W 2n − W 2n−1 , and W 2n /W 2n−1 has dimension one. By the isomorphism (13), W 0 = Im N n has dimension one as well. Next, note that W 2n−1 /W 2n−2 has a Hodge structure with two types, (n, n − 1) and (n − 1, n), each of which must determine a space of half the total dimension. But since F n ∩ W 2n−1 = (0), nothing non-zero can have type (n, n − 1). It follows that W 2n−1 /W 2n−2 = (0), and that W 1 /W 0 = (0) as well (using the isomorphism induced by N n−1 ). Thus, the image of N n−1 comes entirely from the map N n−1 : W 2n → W 2 . That this image is two-dimensional is easily seen: W 0 is one-dimensional, and there is an isomorphism N n−1 : W 2n /W 2n−1 → (Im N n−1 )/W 0 , which shows that (Im N n−1 )/W 0 is also one-dimensional. Q.E.D. In order to prove lemma 2, we must first prove Lemma 3 (Essentially due to Friedman and Scattone [16]). Good integral bases exist, and form bases of the two-dimensional Qvector space Im N n−1 . If g 0 , g 1 = 1 λ N n−1 g is a good integral basis, then 1 λ N n−1 x = − g 1 \| x g 0 + g 0 \| x g 1(14) for all x ∈ H n (X P ′ , Q). Proof. Choose either generator of Im N n ∩ H n (X P ′ , Z) as g 0 . We claim that (g 0 ) ⊥ = W 2n−2 . Let h ∈ W 2n − W 2n−2 , so that N n h = ag 0 with a = 0. Then for any x we have N n x \| h = (−1) n x \| N n h = (−1) n a x \| g 0 . Thus, W 2n−2 = ker N n ⊂ (g 0 ) ⊥ . Since both W 2n−2 and (g 0 ) ⊥ are codimension one subspaces of W 2n , they must be equal. By Poincaré duality, the polarization on H n (X P ′ , Z) is a unimodular pairing. Thus, there exists an element g ∈ H n (X P ′ , Z) such that g 0 \| g = 1. Since g ∈ (g 0 ) ⊥ , neither N n−1 g nor N n g is zero. There is thus a unique positive rational number λ such that g 1 = 1 λ N n−1 g is an indivisible element of H n (X P ′ , Z). It is clear that g 0 , g 1 forms a basis for the Q-vector space Im N n−1 . We next claim that g 1 \| g = 1 λ N n−1 g \| g = 0. For on the one hand, moving the N's to the right side one at a time we have N n−1 g g = (−1) n−1 g N n−1 g while on the other hand, the symmetry of the polarization says that N n−1 g g = (−1) n g N n−1 g . It follows that N n−1 g \| g = 0. To prove equation (14), we first compute in general N n−1 x g = (−1) n−1 x N n−1 g = (−1) n−1 x \| λg 1 = −λ g 1 \| x . Now suppose that x ∈ W 2n−2 . Then N n−1 x ∈ Im N n , which implies that N n−1 x = ag 0 for some a. Thus, in this case N n−1 x g = ag 0 \| g = a, which implies that a = −λ g 1 \| x . Thus, 1 λ N n−1 x = 1 λ ag 0 = g 1 \| x g 0 and since x \| g 0 = 0, the formula follows in this case. To prove the formula in general, note that g 0 x − g 0 \| x g = 0 for any x, so that x − g 0 \| x g ∈ (g 0 ) ⊥ = W 2n−2 . Thus, applying the previous case we find 1 λ N n−1 x = 1 λ N n−1 (x − g 0 \| x g) + 1 λ N n−1 ( g 0 \| x g) = − g 1 (x − g 0 \| x g) g 0 + g 0 \| x 1 λ N n−1 g = − g 1 \| x − g 0 \| x g 1 \| g g 0 + g 0 \| x g 1 = − g 1 \| x g 0 + g 0 \| x g 1 since g 1 \| g = 0. Q.E.D. We can now prove lemma 2. Proof of lemma 2. The only generators of Im N n ∩H n (X P ′ , Z) are ±g 0 , so we must have g ′ 0 = (−1) ℓ g 0 for some ℓ ∈ Z. Write g ′ 1 = 1 λ ′ N n−1 g ′ for some g ′ with g ′ 0 \| g ′ = 1, and let k = − g 1 \| g ′ ∈ Z. Then by lemma 3, 1 λ N n−1 g ′ = − g 1 \| g ′ g 0 + g 0 \| g ′ g 1 = k g 0 + (−1) ℓ g 1 . Thus, 1 λ N n−1 g ′ ∈ H n (X P ′ , Z). We claim that it must be an indivisible element there. For if 1 λµ N n−1 g ′ is integral for some µ ∈ Z with µ > 1, then reversing the roles of g and g ′ in the argument above shows that 1 λµ N n−1 g is also integral, a contradiction. Thus, g ′ 1 = k g 0 + (−1) ℓ g 1 . Q.E.D. Appendix B: Resolutions of certain quotient singularities In this appendix, we will verify that the singularities of the variety Q ψ /G can be resolved in a natural way. The choices we make are sufficiently natural that the isomorphism between Q αψ /G and Q ψ /G automatically lifts to an isomorphism between the desingularizations. We choose to follow the strategy outlined by Reid [29] for resolving canonical threefold singularities. In brief, we perform the following steps: Step I: Blow up the "non-cDV points" of Q ψ /G. (These are exactly the 10 points p ij ∈ Q ψ /G which are the images of points in Q ψ with stabilizer of order 25.) Step IIA: Blow up the singular locus. (It has pure dimension one.) Step IIB: Blow up the pure dimension one part of the singular locus. (60 isolated singular points (lying over the p ij ) were created by step IIA, and these are not to be blown up yet.) Step III: Obtain a projective small resolution of the remaining 60 singular points by blowing up the union of the proper transforms of the exceptional divisors from step I. Step III involved an additional choice, since Reid's strategy does not specify how one should obtain small resolutions. When stated in this form, it is clear that the process we have described is sufficiently natural that it is preserved under any isomorphism. It yields a projective (hence Kähler) variety W ψ with trivial canonical bundle. In the remainder of this appendix, we will show that the process above has the properties mentioned during its description, and that it gives a resolution of singularities of Q ψ /G. We first observe the effect of the process on the curve C ijk , away from the points p ij , p jk , p ik . Steps I and III are concentrated at those special points (and their inverse images) and so these steps do not affect C ijk . Steps IIA and IIB simply blow up C ijk and then the residual singular curve in the exceptional divisor. But two blowups are precisely what is required to resolve a rational double point of type A 4 , as is easily verified from its equation xy + z 5 = 0. To verify that the process has the correct properties at the points p ij , we use the language of toroidal embeddings (see [22] or [28]). It suffices to consider the point p 45 . Since (x 1 , x 2 , x 3 ) serve as coordinates in a neighborhood of any of the points in η −1 (p 45 ), the singularity p 45 ∈ Q ψ /G is isomorphic to a neighborhood of the origin in C /G , where G 45 ∼ = {(α 1 , α 2 , α 3 ) ∈ (µ 5 ) 3 \| α k = 1} acts diagonally on C . Let M be the lattice of G 45 -invariant rational monomials in C( , , ). We embed M in R by 6 M = {(m 1 , m 2 , m 3 ) ∈ R \| ⋗ ⋗ ⋗ ∈ C( , , ) G }. It is easy to see that {(1, 0, 0), (0, 1, 0), ( 1 5 , 1 5 , 1 5 )} is a basis of the lattice M ⊂ R . Let N = { n ∈ R \| ⋗ · ⋉ ∈ Z for all ⋗ ∈ M} = {(n 1 , n 2 , n 3 ) ∈ Z \| ⋉ + ⋉ + ⋉ ≡ mod } be the dual lattice, and let σ ⊂ N R be the convex cone generated by (5, 0, 0), (0, 5, 0), and (0, 0, 5). According to the theory of toroidal embeddings, C /G = Spec C[ , , ] G = U σ , where U σ is the toric variety associated to σ. Each blowup of U σ corresponds to a decomposition of σ into a fan. The effects of the blowups in our process is illustrated in figure 1, which depicts the intersection of the fan with {(n 1 , n 2 , n 3 ) ∈ N R \| n k = 5} after each step. The exceptional divisors D n of each blowup are indicated by solid dots, labeled by the corresponding elements n ∈ N. (The fact that the stated blowups produce the illustrated decomposition is a straightforward calculation with the toroidal embeddings. ) We can now see in detail what happens in our process. In step I, we blow up p 45 , and produce three exceptional divisors D (3,1,1) , D (1,3,1) , and D (1,1,3) . The remaining singular locus at this stage consists of the original three curves of A 4 -singularities together with three new curves of A 1 -singularities: the intersections of pairs of exceptional divisors. In step IIA, we blow up the union of these six curves, and produce nine new exceptional divisors: one corresponding to each curve of A 1singularities (such as D (2,1,2) ), and two corresponding to each curve of A 4 -singularities (such as D (4,0,1) and D (1,0,4) ). The remaining singularities consist of six isolated points (corresponding to the quadrilaterals in the figure) and three curves: the intersections of the corresponding pairs of exceptional divisors from the original A 4 -singularities. In step IIB, we blow up these three curves, producing six new exceptional divisors, two for each curve (such as D (3,0,2) and D (2,0,3) ). This leaves the six isolated singular points; but blowing up the proper transforms of D (3,1,1) , D (1,3,1) , and D (1,1,3) (which are now disjoint) in step III resolves those final singular points. Appendix C: The monodromy of the quintic-mirrors In this appendix we will explain how to use the calculation of Candelas et al. [6] to verify the monodromy statements about the family of quintic-mirrors which we made in section 5. Candelas et al. begin by choosing an explicit basis {A 1 , A 2 , B 1 , B 2 } for the homology H 3 (W ψ , Z) of a quintic-mirror, valid in some simplyconnected region in {ψ \| ψ 5 = 0, 1} which includes the wedge {ψ \| 0 < arg ψ < 2π/5}. This basis is symplectic, i.e., A a ∩ B b = δ a b and A a ∩ A b = B a ∩ B b = 0. The corresponding dual basis of H 3 (W ψ , Z) is denoted by {α 1 , α 2 , β 1 , β 2 }. Fixing a particular holomorphic 3-form Ω (which depends on ψ), we then get period functions z a = A a Ω, G b = B b Ω. These fit into a period vector Π =      G 1 G 2 z 1 z 2      . By doing some integrals, calculating the differential equation satisfied by a period function, and manipulating certain hypergeometric functions, the authors of [6] are able to obtain explicit formulas for the four period functions. This allows them to calculate the monodromy of the periods around various paths. Notice that we are working in the ψ-plane at present. The family {W ψ } has singular fibers at ψ = 0 and at ψ = α for all fifth roots of unity α; there is also a singular fiber over ψ = ∞. Candelas et al. calculate the monodromy on the periods induced by transport around ψ = 1, which they represent in matrix form by Π → T Π. They also compute, for \|ψ\| < 1, the effect on the periods of the isomorphism W αψ ∼ = W ψ given in equation (10), representing this by Π(αψ) = AΠ(ψ). We need to know the monodromy around ∞ in the λ-plane, where λ = ψ 5 . A moment's thought will convince the reader that this is represented by Π → (T −1 A −1 )Π, and that (AT ) −5 describes the monodromy around ∞ in the ψ-plane (as asserted in [6]). Let T P = T −1 A −1 . The explicit calculations from [6] for the matrices A and T are: In particular, the index of nilpotency of log(T P ) is maximal. (We note in passing that at λ = 1 the monodromy is represented by T , and since (T − I) 2 = 0, the index is not maximal there. In addition, at λ = 0 the monodromy is represented by A. This monodromy matrix is only quasi-unipotent, with A 5 = I unipotent; the index of A 5 is not maximal either. It follows that λ = ∞ is the only possible boundary point with maximally unipotent monodromy.) In order to construct a good integral basis g 0 , g 1 , we compute (log(T P )) 2 ( Using the relations ad \| (α a ) = Ba , ad \| (β b ) = − A b ,(15) this implies (log(T P )) 2 (β 2 ) = 10α 1 + 5β 1 (log(T P )) 3 (β 2 ) = 5α 2 . Thus, we may take g 0 = α 2 . If we then choose g = β 2 so that g 0 \| g = α 2 \| β 2 = 1, we get λ = 5 and g 1 = 2α 1 + β 1 . It follows that (log(T P ))(g 1 ) = 1 5 (log(T P )) 3 (β 2 ) = α 2 = g 0 , which implies that m = 1. Using the relations (15) again, it follows that γ 0 = B 2 , γ 1 = 2B 1 − A 1 . Thus, t = ( 2B 1 −A 1 Ω)/( A 2 Ω). We need to verify that our parameter t is the same one used by Candelas et al. Their parameter is defined in [6, (5.9)] by t = w 1 /w 2 , with w 1 and w 2 determined by a pair of equations ∐ = N Π, w 2 = G 2 , where 7 ∐ =      F 1 F 2 w 1 w 2      and N =      −1 0 0 0 0 0 0 1 2 0 −1 0 0 1 0 0      represent a vector ∐ which is a sort of mirror analogue of the period vector Π, and a particular integral symplectic matrix N, respectively. (Sadly, in the published version of [6], the symbols Π and ∐ were identified, making section 5.2 of that paper difficult to read.) It follows that t = w 1 w 2 = 2G 1 − z 1 G 2 = 2B 1 −A 1 Ω A 2 Ω as required. F p (X) := p≤p ′ ≤n H p ′ ,n−p ′ (X);we then have H p,n−p (X) ∼ = F p (X)/F p+1 (X).The cup product on cohomology composed with evaluation on the canonical orientation class of X determines a bilinear map\| : H n (X, Z) × H ⋉ (X, Z) → H ⋉ (X, Z) ∼ = → Z,called a polarization. There is an associated adjoint map ad \| : H n (X, Z) → Hom(H ⋉ (X, Z), Z) defined by ad \| (x)(y) = x \| y . After tensoring with C and invoking the Hodge decomposition, the adjoint map induces isomorphisms ad \| : H p,n−p (X) ∼ = → (H n−p,p (X)) * . the mirror symmetry conjecture, varying the complex structure in the family {W 5 √ λ } should correspond to varying the "extra structure" S on a fixed smooth quintic threefold M. These are both one-parameter variations. Candelas et al. [6], arguing from physical principles, propose an identification of the Yukawa coupling of the quintic-mirrors with the cubic form on H 1,1 (M). In terms of the mathematical framework established here, that identification involves four assertions: (i) The isomorphism H −1,1 (W) → H 1,1 (M) defined by d/dt → [H] (where d/dt ∈ H −1,1 (W) is the vector field defined by the canonical parameter t, and [H] ∈ H 1,1 (M) is the class of a hyperplane section of M) is the isomorphism which is predicted by the mathematical version of the mirror symmetry conjecture. (ii) The mathematically normalized Yukawa coupling κ ttt on H −1,1 (W)is the correctly normalized coupling predicted by the physical theory. (iii) The parameter R from the physical theory coincides with −2πit, This particular Yukawa coupling is probably only interesting in physics if n = 3. In dimension n, what is being computed here is the "n-point Yukawa coupling." To avoid confusion with the cotangent bundle, we denote this section by ω rather than Ω. However, in appendix C below, we will revert to the notation Ω used in[6]. It has become common in the physics literature to use H 2,1 (X) in place of H 1 (Θ X ), largely because of the success of Candelas[5] and others in computing Yukawa couplings on H 2,1 . In order to get the correct answer in families, however, we must return to the original analysis of Strominger and Witten[36] and work with Yukawa couplings on H 1 (Θ X ). The point is that while H 1 (Θ X ) and H 2,1 (X) are isomorphic for a Calabi-Yau threefold, they are not canonically isomorphic. This affects the bundles over the moduli space to which they belong. Among the things not properly defined from a mathematical viewpoint, we must include the normalization of the Yukawa coupling, the complex parameter R (which depends on the "extra structure" S), and the higher coefficients σ k . This should not be taken as strong evidence in favor of the conjecture, since the definitions have been carefully designed to ensure that this limit would be correct. 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[ "EMBEDDINGS OF PERSISTENCE DIAGRAMS INTO HILBERT SPACES", "EMBEDDINGS OF PERSISTENCE DIAGRAMS INTO HILBERT SPACES" ]	[ "Peter Bubenik ", "Alexander Wagner " ]	[]	[]	Since persistence diagrams do not admit an inner product structure, a map into a Hilbert space is needed in order to use kernel methods. It is natural to ask if such maps necessarily distort the metric on persistence diagrams. We show that persistence diagrams with the bottleneck distance do not even admit a coarse embedding into a Hilbert space. As part of our proof, we show that any separable, bounded metric space isometrically embeds into the space of persistence diagrams with the bottleneck distance. As corollaries, we obtain the generalized roundness, negative type, and asymptotic dimension of this space.	10.1007/s41468-020-00056-w	[ "https://arxiv.org/pdf/1905.05604v2.pdf" ]	153,313,552	1905.05604	0e2c6755d128059245e283b4b53e7939f4b3e12f	EMBEDDINGS OF PERSISTENCE DIAGRAMS INTO HILBERT SPACES 27 May 2019 Peter Bubenik Alexander Wagner EMBEDDINGS OF PERSISTENCE DIAGRAMS INTO HILBERT SPACES 27 May 2019 Since persistence diagrams do not admit an inner product structure, a map into a Hilbert space is needed in order to use kernel methods. It is natural to ask if such maps necessarily distort the metric on persistence diagrams. We show that persistence diagrams with the bottleneck distance do not even admit a coarse embedding into a Hilbert space. As part of our proof, we show that any separable, bounded metric space isometrically embeds into the space of persistence diagrams with the bottleneck distance. As corollaries, we obtain the generalized roundness, negative type, and asymptotic dimension of this space. Introduction Kernel methods, such as support vector machines or principal components analysis, are machine learning algorithms that require an inner product on the data (Steinwart and Christmann, 2008). When the original data set X lacks an inner product or one would like a higher-dimensional representation of the data, a standard approach is to map into a Hilbert space H. Such a mapping is called a feature map and kernel methods are implicitly performed in the codomain of the feature map. While specifying an explicit feature map may be difficult, it turns out to be equivalent to the often simpler task of constructing a positive definite kernel on the data. This equivalence is important for the practical success of kernel methods but should not obscure the fact that there is an underlying feature map ϕ : X → H and that the algorithm works with ϕ(X) ⊆ H. Because of this, when X is a metric space, one would like a feature map ϕ that changes the original metric as little as possible. Persistent homology takes in a one-parameter family of topological spaces and outputs a signature, called the persistence diagram, of this family's changing homology. There is a natural metric on one-parameter families of topological spaces, called the interleaving distance, and a family of metrics on persistence diagrams, called the p-Wasserstein distances. When p = ∞, changes in the input of persistent homology cause at most proportional changes in the output (Cohen-Steiner et al., 2007;Chazal et al., 2009). This stability legitimizes the use of persistent homology for machine learning because it guarantees that small perturbations of the data, such as those caused by measurement noise, do not cause large changes in the associated features. If one would like to apply kernel methods to persistence diagrams, a natural first question is whether the metrics on persistence diagrams can be induced by an inner product. More precisely, does there exist an isometric embedding of persistence diagrams into a Hilbert space? We show in Section 2.3 that the impossibility of such an isometric embedding follows from work of Turner and Spreemann (2019) and classical results of Schoenberg (1935Schoenberg ( , 1938. In other words, any feature map from persistence diagrams into a Hilbert space necessarily distorts the original metric. Our main results consider the ∞-Wasserstein distance, also called the bottleneck distance. This is the only case for which persistent homology is 1-Lipschitz. Isometric maps require distances to be exactly preserved. More general are bi-Lipschitz maps which are allowed to distort distances at most linearly. Considerably more general are coarse embeddings, which need not be continuous and only require that distances be distorted in a uniform, but potentially non-linear, way. Coarse embeddings are an important notion in geometric group theory and coarse geometry (Gromov, 1993;Roe, 2003). We show that the space of persistence diagrams with the bottleneck distance does not admit a coarse embedding into a Hilbert space (Theorem 21). In other words, the distortion caused by a feature map to the bottleneck distance is not uniformly controllable. In fact, even if one restricts to the subspace of (finite) persistence diagrams arising as the homology of a filtered finite simplicial complex, there still does not exist a coarse embedding of this subspace into a Hilbert space (Remark 22 and Lemma 23). As corollaries of Theorem 21, we obtain the generalized roundness, negative type, and asymptotic dimension of persistence diagrams with the bottleneck distance (Corollary 20, Remark 25 and Corollary 27). Toward our proof of Theorem 21, we show that any separable, bounded metric space isometrically embeds into the space of persistence diagrams with the bottleneck distance (Theorem 19). Our proof of Theorem 21 combines Theorem 19 with ideas of Dranishnikov et al. (2002) and Enflo (1969). 1.1. Related work. Carriere and Bauer (2018) have investigated bi-Lipschitz embeddings of persistence diagrams into separable Hilbert spaces. They've shown the impossibility of a bi-Lipschitz embedding into a finite-dimensional Hilbert space and that bi-Lipschitz embeddings into infinite-dimensional, separable Hilbert spaces only exist when restrictions are placed on the cardinality and spread of the persistence diagrams under consideration. Bell et al. (2019) have shown that the space of persistence diagrams with the p-Wasserstein distance for p < ∞ has a discrete subspace that fails to have property A. The relevance of this result is that a discrete metric space with property A admits a coarse embedding into a Hilbert space (Yu, 2000, Theorem 2.2). Bubenik and Vergili (2018, Theorem 4.37) have shown that there exist cubes of arbitrary dimension with the L ∞ distance which isometrically embed into the space of persistence diagrams with the bottleneck distance. Background 2.1. The space of persistence diagrams. In this section, we define persistence diagrams and a family of associated metric spaces. Persistence diagrams naturally arise as the output of persistent homology, which describes the changing homology of a one-parameter family of topological spaces. Persistence diagrams are usually defined to be multisets. We find it convenient to instead define them as indexed sets. Definition 1. Denote {(x, y) ∈ R 2 \| x < y} by R 2 < . A persistence diagram is a function from a countable set to R 2 < , i.e. D : I → R 2 < . To define the relevant metrics on persistence diagrams, we need two preliminary definitions. Definition 2. Suppose D 1 : I 1 → R 2 < and D 2 : I 2 → R 2 < are persistence diagrams. A partial matching between them is a triple (I ′ 1 , I ′ 2 , f ) such that I ′ 1 ⊆ I 1 , I ′ 2 ⊆ I 2 , and f : I ′ 1 → I ′ 2 is a bijection. The distance between two persistence diagrams will be the minimal cost of a partial matching between them. The cost of a partial matching is the L p norm of the distances between matched pairs and the distances between unmatched pairs and ∆, the diagonal in R 2 . Definition 3. Suppose D 1 : I 1 → R 2 < and D 2 : I 2 → R 2 < are persistence diagrams and (I ′ 1 , I ′ 2 , f ) is a partial matching between them. Equip R 2 < with the metric d ∞ (a, b) = a − b ∞ = max(\|a x − b x \|, \|a y − b y \|). For a ∈ R 2 < , observe that d ∞ (a, ∆) = inf t∈∆ d ∞ (a, t) = (a y − a x )/2. The p-cost of f is denoted cost p (f ) and defined as follows. If p < ∞, cost p (f ) =   i∈I ′ 1 d ∞ (D 1 (i), D 2 (f (i))) p + i∈I 1 \I ′ 1 d ∞ (D 1 (i), ∆) p + i∈I 2 \I ′ 2 d ∞ (D 2 (i), ∆) p   1/p . If p = ∞, cost p (f ) = max{sup i∈I ′ 1 d ∞ (D 1 (i), D 2 (f (i)), sup i∈I 1 \I ′ 1 d ∞ (D 1 (i), ∆), sup i∈I 2 \I ′ 2 d ∞ (D 2 (i), ∆)}. If any of the terms in either expression are unbounded, we define the cost to be infinity. Definition 4. Let 1 ≤ p ≤ ∞. If D 1 , D 2 are persistence diagrams, definẽ w p (D 1 , D 2 ) = inf{cost p (f ) \| f is a partial matching between D 1 and D 2 }. Let (Dgm p , w p ) denote the metric space of persistence diagrams D that satisfyw p (D, ∅) < ∞ modulo the relation D 1 ∼ D 2 ifw p (D 1 , D 2 ) = 0, where ∅ is shorthand for the unique persistence diagram with empty indexing set. 2.2. Negative type and kernels. The following definition and theorem equate the problem of defining a feature map on a set to the frequently simpler problem of defining a positive definite kernel. Theorem 6 and the fact that kernel methods require access to only the inner products of elements is the content of the so-called kernel trick. Definition 5. Let X be a nonempty set. A symmetric function k : X × X → R is a positive definite kernel if for any n ∈ N, c 1 , . . . , c n ∈ R, and x 1 , . . . , x n ∈ X, n i,j=1 c i c j k(x i , x j ) ≥ 0. Theorem 6 (Steinwart and Christmann 2008, Theorem 4.16). Let X be a nonempty set. A function k is a positive definite kernel iff there exists a Hilbert space H and a feature map ϕ : X → H such that ϕ(x), ϕ(y) = k(x, y) for every x, y ∈ X. We now turn to the definition of negative type, which is closely related to positive definite kernels and to the embeddability of metric spaces into Hilbert spaces. Definition 7. A quasi-metric space is a nonempty set X together with a function d : X × X → [0, ∞) such that d(x, x) = 0 and d(x, y) = d(y, x) for every x, y ∈ X. Definition 8. A quasi-metric space (X, d) is said to be of q-negative type if for any n ∈ N, x 1 , . . . , x n ∈ X, and a 1 , . . . , a n ∈ R satisfying n i=1 a i = 0, the following inequality is satisfied. n i,j=1 a i a j d(x i , x j ) q ≤ 0 A relationship between positive definite kernels and negative type is given in the following. Theorem 9 (Berg et al. 1984, Lemma 2.1, Theorem 2.2). Let (X, d) be a quasi-metric space. The following are equivalent. (1) (X, d) is of 1-negative type. (2) For any x 0 ∈ X, k(x, y) = d(x, x 0 ) + d(y, x 0 ) − d(x, y) is a positive definite kernel. (3) k(x, y) = e −td(x,y) is a positive definite kernel for every t > 0. The negative type of a quasi-metric space is closely related to questions regarding embeddability into Hilbert spaces. Theorem 10 (Wells and Williams 1975, Theorem 2.4, Remark 3.2). A quasi-metric space admits an isometric embedding into a Hilbert space iff it is of 2-negative type. Besides 2-negative type characterizing isometric embeddability into a Hilbert space, the following theorem states the important property that negative type is downward closed. Theorem 11 (Wells and Williams 1975, Theorem 4.7). Suppose (X, d) is a quasi-metric space of q-negative type. Then it is of q ′ -negative type for any 0 ≤ q ′ ≤ q 2.3. Isometric embeddability of diagram space. It was shown by Turner and Spreemann (2019, Theorem 3.2) that (Dgm p , w p ) is not of 1-negative type for any 1 ≤ p ≤ ∞. This leads to the following negative result. Theorem 12. (Dgm p , w p ) does not admit an isometric embedding into a Hilbert space for any 1 ≤ p ≤ ∞. Proof. Let 1 ≤ p ≤ ∞. Since (Dgm p , w p ) is not of 1-negative type, by Theorem 11, (Dgm p , w p ) is not of 2-negative type and so does not admit an isometric embedding into a Hilbert space by Theorem 10. 2.4. Coarse embeddings and related notions. If instead of demanding that distances be exactly preserved, we only require that distances be contracted or expanded a uniform amount, we arrive at the following definition. y)) for all x, y ∈ X, and (2) lim t→∞ ρ − (t) = ∞. Definition 13. A map f : (X, d) → (Y, d ′ ) is a coarse embedding or uniform embedding if there exists non-decreasing ρ − , ρ + : [0, ∞) → [0, ∞) such that (1) ρ − (d(x, y)) ≤ d ′ (f (x), f (y)) ≤ ρ + (d(x, Note that if ρ − (x) = Ax and ρ + (x) = Bx for some A, B > 0 then f is a bi-Lipschitz map. This definition was introduced by Gromov (1993, p. 218) where he posed the question of whether every separable metric space, of which (Dgm p , w p ) are examples (Mileyko et al., 2011;Blumberg et al., 2014), admits a coarse embedding into a Hilbert space. This question was answered negatively by Dranishnikov et al. (2002). The following definition gives a coarse analogue of covering dimension. Definition 14. Let n be a non-negative integer. A metric space (X, d) has asymptotic dimension ≤ n if for every R > 0 there exists a cover U of X such that every ball of radius R intersects at most n + 1 elements of U and sup U ∈U sup{d(x, y) \| x, y ∈ U } < ∞. Theorem 15 (Roe 2003, Example 11.5). If X is a metric space with finite asymptotic dimension, then there exists a coarse embedding of X into a Hilbert space. Property A is a weak amenability-like notion for discrete metric spaces that also implies coarse embeddability into a Hilbert space. Definition 16 (Yu 2000, Definition 2.1). A discrete metric space (X, d) has property A if for any r > 0, ε > 0 there is a family of finite subsets {A x } x∈X of X × N such that (1) (x, 1) ∈ A x for all x ∈ X; (2) \|(A x \ A y )\| + \|(A y \ A x )\| \|A x ∩ A y \| < ε whenever d(x, y) ≤ r; (3) there exists R > 0 such that if (x, m), (y, m) ∈ A z for some z ∈ X, then d(x, y) ≤ R. 0.7 0.9 0.5 x 3 x 1 x 2 x y ∆ (0, 2c) Theorem 17 (Yu 2000, Theorem 2.2). If a discrete metric space X has property A, then X admits a coarse embedding into a Hilbert space. The following definition gives a non-Riemannian notion of curvature for general metric spaces. Definition 18. A metric space (X, d) has generalized roundness q if for any n ∈ N and a 1 , . . . , a n , b 1 , . . . , b n ∈ X, we have i<j (d(a i , a j ) q + d(b i , b j ) q ) ≤ i,j d(a i , b j ) q This definition was introduced by Enflo (1969) to answer negatively a question of Smirnov about uniform homeomorphisms into L 2 (0, 1). Indeed, Dranishnikov et al.'s negative answer to Gromov's question was inspired by Enflo's negative answer to Smirnov's. Long after Enflo defined generalized roundness, Lennard, Tonge, and Weston (1997) proved that a space has generalized roundness q iff it has q-negative type, giving a geometric characterization to the notion of negative type and, in particular, to the existence of isometric embeddings into Hilbert spaces. Coarse embeddability of diagrams with bottleneck distance The main result of this section is that there does not exist a coarse embedding of (Dgm ∞ , w ∞ ) into a Hilbert space. This implies that the generalized roundness and asymptotic dimension of (Dgm ∞ , w ∞ ) are 0 and ∞, respectively. We also show that any separable, bounded metric space has an isometric embedding into the space of persistence diagrams with the bottleneck distance. Proof. Let c > sup{d(x, y) \| x, y ∈ X}. Let {x k } ∞ k=1 be a countable, dense subset of (X, d). Consider the following map. ϕ : (X, d) → (Dgm ∞ , w ∞ ) x → {(2c(k − 1), 2ck + d(x, x k )} ∞ k=1 Note that for any x ∈ X and k ∈ N, , c). A visualization of the image of ϕ for a metric space with three points is shown in Figure 1. We now show that an optimal partial matching of ϕ(x) and ϕ(y) matches points in each diagram with the same first coordinate, and the cost of this partial matching is d(x, y). d ∞ ((2c(k − 1), 2ck + d(x, x k )), ∆) = c + d(x, x k ) 2 < 3c 2 , so ϕ is well-defined. Moreover, since w ∞ (ϕ(x), ∅) = sup 1≤k<∞ d ∞ ((2c(k − 1), 2ck + d(x, x k )), ∆), it follows that ϕ(x) ∈ B(∅, 3c 2 ) \ B(∅ Formally, ϕ(x) and ϕ(y) are equivalence classes of persistence diagrams D x : N → R 2 < and D y : N → R 2 < . Consider the partial matching (N, N, id N ) between D x and D y , i.e. (2c (k − 1), 2ck + d(x, x k )) is matched with (2c(k − 1), 2ck + d(y, x k )) for every k ∈ N. Observe that d ∞ (D x (k), D y (k)) = \|d(x, x k ) − d(y, x k )\| for every k, so the cost of this partial matching is sup k \|d(x, x k ) − d(y, x k )\|. By the triangle inequality, sup k \|d(x, x k ) − d(y, x k )\| ≤ d(x, y). Since {x k } ∞ k=1 is dense, for every ε > 0, there exists a k such that d(x, x k ) < ε, so \|d(x, x k ) − d(y, x k )\| ≥ d(y, x k ) − d(x, x k ) ≥ d(x, y) − 2d(x, x k ) > d(x, y) − 2ε. This implies that sup k \|d(x, x k ) − d(y, x k )\| ≥ d(x, y) and cost ∞ (id N ) = d(x, y). Suppose I, J ⊆ N and (I, J, f ) is a different partial matching between D x and D y . Then there exists a k ∈ N such that either k / ∈ I or k ∈ I and f (k) = k. If k / ∈ I, then cost p (f ) ≥ d ∞ ((2c(k − 1), 2ck + d(x, x k )), ∆) ≥ c. If k ∈ I and f (k) = k ′ = k, then cost p (f ) ≥ (2c(k − 1), 2ck + d(x, x k )) − (2c(k ′ − 1), 2ck ′ + d(y, x k ′ )) ∞ ≥ 2c. Therefore, cost p (f ) ≥ c > d(x, y). Hence, w ∞ (ϕ(x), ϕ(y)) = d(x, y), i.e. ϕ is an isometric embedding. We now apply Theorem 19 to show the generalized roundness of (Dgm ∞ , w ∞ ) is 0. To do so, we embed finite metric spaces of arbitrarily small generalized roundness into (Dgm ∞ , w ∞ ). b 1 b 2 b 3 b 4 a 1 a 2 a 3 a 4 Figure 2. The metric space obtained from the complete bipartite graph K n,n when n = 4. Corollary 20. The generalized roundness of (Dgm ∞ , w ∞ ) is zero. Proof. Let n ≥ 2. Define X n = {a 1 , . . . , a n , b 1 , . . . , b n } and equip this set with the metric d(a i , a j ) = d(b i , b j ) = 2 and d(a i , b j ) = 1 for any i, j ∈ {1, . . . , n}. Enflo (1969) remarks that X n has generalized roundness that converges to 0 as n → ∞. Indeed, i<j (d(a i , a j ) q + d(b i , b j ) q ) ≤ i,j d(a i , b j ) q ⇐⇒ n(n − 1)2 q ≤ n 2 ⇐⇒ q ≤ log 2 (1 + (n − 1) −1 ). Hence, X n has generalized roundness at most log 2 (1 + (n − 1) −1 ) which tends to 0 as n increases. By Theorem 19, we may isometrically embed X n into (Dgm ∞ , w ∞ ) for any n so the generalized roundness of (Dgm ∞ , w ∞ ) must be zero. Our next result is that (Dgm ∞ , w ∞ ) does not admit a coarse embedding into a Hilbert space. The proof relies on a construction of Dranishnikov et al. (2002) based on ideas of Enflo (1969). Theorem 21. (Dgm ∞ , w ∞ ) does not admit a coarse embedding into a Hilbert space. Proof. Define Z n to be the integers mod n with the standard metric. Define Z m n to be the product of m copies of Z n with the following metric, d(([k 1 ], . . . , [k m ]), ([l 1 ], . . . , [l m ])) = max 1≤i≤m d([k i ], [l i ]). Let X be the disjoint union of Z m n for every n, m ≥ 1 and equip X with a metricd satisfying the following. (1) The restriction ofd to each Z m n coincides with d. (2)d(x, y) ≥ n + m + n ′ + m ′ if x ∈ Z m n , y ∈ Z m ′ n ′ , and (n, m) = (n ′ , m ′ ). Proposition 6.3 of Dranishnikov et al. (2002) shows that any such (X,d) does not admit a coarse embedding into a Hilbert space. Hence, it suffices to prove such an (X,d) isometrically embeds into (Dgm ∞ , w ∞ ), since a coarse embedding of (Dgm ∞ , w ∞ ) into a Hilbert space would restrict to a coarse embedding of (X,d) into a Hilbert space. Choose an enumeration {(n i , m i )} ∞ i=1 of N × N such that i < j implies n i + m i ≤ n j + m j , for instance, (1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1), (1, 4), etc. Define c 1 = 1 and for i ≥ 2, c i = 4 max(c i−1 , n i + m i ). For every (n i , m i ), note that c i > n i > max{d(x, y) \| x, y ∈ Z m i n i }. So by Theorem 19, there exists an isometry ϕ i : Z m i n i → (Dgm ∞ , w ∞ ) such that ϕ i (Z m i n i ) ⊆ B(∅, 3c i 2 ) \ B(∅, c i ). Define ϕ : X → Dgm ∞ by ϕ(x) = ϕ i (x) for x ∈ Z m i n i and defined(x, y) = w ∞ (ϕ(x), ϕ(y)). By the definition ofd, ϕ : (X,d) → (Dgm ∞ , w ∞ ) is an isometry. If x, y ∈ Z m i n i , thend(x, y) = w ∞ (ϕ i (x) , ϕ i (y)) = d(x, y) sod satisfies (1) above. It only remains to showd satisfies (2). Suppose x ∈ Z m i n i , y ∈ Z m j n j , and (n i , m i ) = (n j , m j ). We may assume i < j. By construction, ϕ(x) = ϕ i (x) ∈ B(∅, 3c i 2 ) \ B(∅, c i ) and ϕ(y) = ϕ j (y) ∈ B(∅, 3c j 2 ) \ B(∅, c j ), which implies that d(x, y) = w ∞ (ϕ(x), ϕ(y)) ≥ w ∞ (ϕ(y), ∅) − w ∞ (ϕ(x), ∅) > c j − 3c i 2 . Additionally, we have n i + m i ≤ n j + m j and c j ≥ 4 max(c i , n j + m j ) ≥ 2(c i + (n j + m j )), sõ d(x, y) > c j − 3c i 2 ≥ 2(n j + m j ) + 2c i − 3c i 2 > n i + m i + n j + m j . We have shown thatd satisfies (2) which completes the proof. Remark 22. For a finite metric space, the isometric embedding defined in Theorem 19 sends each point to a persistence diagram of finite cardinality in (Dgm ∞ , w ∞ ). In particular, the map ϕ i : Z m i n i → (Dgm ∞ , w ∞ ) given in the proof of Theorem 19 has an image consisting of finite persistence diagrams. Since X is the disjoint union of Z m n for every n, m ≥ 1, it follows that ϕ : (X,d) → (Dgm ∞ , w ∞ ) sends each point in the metric space X to a finite persistence diagram. Hence, the proof of Theorem 21 gives the slightly stronger result that the space of finite persistence diagrams with the bottleneck distance does not admit a coarse embedding into a Hilbert space. Theorem 21 and Remark 22 give the impossibility of coarsely embedding the space of finite persistence diagrams with the bottleneck distance into a Hilbert space. The primary motivation for this result was the application of kernel methods to persistent homology. In computational settings, the persistence diagrams of interest are frequently the result of applying homology to a filtered finite simplicial complex. Hence, one may ask whether this more restricted space of persistence diagrams, i.e. the subspace arising from homology of filtered finite simplicial complexes, admits a coarse embedding into a Hilbert space. Unfortunately, this is easily seen to be false by the following. Lemma 23. Every finite persistence diagram is realizable as the homology of a filtered finite simplicial complex. Proof. Suppose D : {1, . . . , n} → R 2 < is a persistence diagram. Let V = {v a i , v b i , v c i } n i=1 . Consider the simplicial complex on V that is the disjoint union of n 2-simplices and has the filtration given by assigning the value D(i) x to {v a i }, {v b i }, {v c i }, {v a i , v b i }, {v a i , v c i }, {v b i , v c i } and the value D(i) y to {v a i , v b i , v c i }. Applying H 1 (−, Z 2 ) recovers the persistence diagram D. Proposition 24. A metric space (X, d) with q-negative type for some q > 0 admits a coarse embedding into a Hilbert space. Proof. Suppose there exists a q > 0 such that the metric space (X, d) has q-negative type. Observe that d q/2 (x, x) = 0 q/2 = 0 and d q/2 (x, y) = d q/2 (y, x) so (X, d q/2 ) is a quasi-metric space. Let x 1 , . . . , x n ∈ X and a 1 , . . . , a n ∈ R such that n i=1 a i = 0. Then n i,j=1 a i a j d q/2 (x i , x j ) 2 = n i,j=1 a i a j d(x i , x j ) q ≤ 0, so (X, d q/2 ) is a quasi-metric space of 2-negative type. By Theorem 10, there exists an isometric embedding ϕ from (X, d q/2 ) into a Hilbert space H. Define ρ + (t) = ρ − (t) = t q/2 . It follows that ϕ satisfies the requirements of a coarse embedding of (X, d) into H, i.e. ρ + (d(x, y)) = ρ − (d(x, y)) = d q/2 (x, y) = ϕ(x) − ϕ(y) H . Remark 25. Since (Dgm ∞ , w ∞ ) does not admit a coarse embedding into a Hilbert space, Proposition 24 implies that (Dgm ∞ , w ∞ ) is of 0-negative type. This also follows from Corollary 20 and the result of Lennard, Tonge, and Weston (1997) on the equivalence of negative type and generalized roundness. Finally, we state two corollaries of Theorem 21 that answer Questions 3.10 and 3.11 of Bell et al. (2019). Corollary 26. (Dgm ∞ , w ∞ ) contains a discrete subspace that fails to have property A. Proof. In the proof of Theorem 21, we consider a discrete metric space (X,d) and prove it embeds in (Dgm ∞ , w ∞ ) via an isometry ϕ. Dranishnikov et al. (2002) have shown that (X,d) does not admit a coarse embedding into a Hilbert space so by Theorem 17, ϕ(X) fails to have property A. Corollary 27. (Dgm ∞ , w ∞ ) has infinite asymptotic dimension. Proof. If (Dgm ∞ , w ∞ ) had finite asymptotic dimension, then it would admit a coarse embedding into a Hilbert space by Theorem 15, which contradicts Theorem 21. Figure 1 . 1A metric space with three points and the corresponding persistence diagrams defined in Theorem 19. Theorem 19 . 19Suppose (X, d) is a separable, bounded metric space. There exists an isometric embedding ϕ : (X, d) → (Dgm ∞ , w ∞ ). Moreover, if c > sup{d(x, y) \| x, y ∈ X}, we may choose ϕ such that ϕ(X) ⊆ B(∅, 3c 2 ) \ B(∅, c), where B(∅, r) = {D ∈ Dgm ∞ \| w ∞ (D, ∅) < r}. The space of persistence diagrams fails to. Greg Bell, Austin Lawson, C Neil Pritchard, Dan Yasaki, arXiv:1902.02288have Yu's property A. arXiv e-prints, artGreg Bell, Austin Lawson, C. Neil Pritchard, and Dan Yasaki. 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[ "The Hofstadter Butterfly in a Cavity-Induced Dynamic Synthetic Magnetic Field", "The Hofstadter Butterfly in a Cavity-Induced Dynamic Synthetic Magnetic Field" ]	[ "Elvia Colella \nInstitut für Theoretische Physik\nUniversität Innsbruck\nA-6020InnsbruckAustria\n", "Farokh Mivehvar \nInstitut für Theoretische Physik\nUniversität Innsbruck\nA-6020InnsbruckAustria\n", "Francesco Piazza \nMax-Planck-Institut für Physik komplexer Systeme\nD-01187DresdenGermany\n", "Helmut Ritsch \nInstitut für Theoretische Physik\nUniversität Innsbruck\nA-6020InnsbruckAustria\n" ]	[ "Institut für Theoretische Physik\nUniversität Innsbruck\nA-6020InnsbruckAustria", "Institut für Theoretische Physik\nUniversität Innsbruck\nA-6020InnsbruckAustria", "Max-Planck-Institut für Physik komplexer Systeme\nD-01187DresdenGermany", "Institut für Theoretische Physik\nUniversität Innsbruck\nA-6020InnsbruckAustria" ]	[]	Energy bands of electrons in a square lattice potential threaded by a uniform magnetic field exhibit a fractal structure known as the Hofstadter butterfly. Here we study a Fermi gas in a 2D optical lattice within a linear cavity with a tilt along the cavity axis. The hopping along the cavity axis is only induced by resonant Raman scattering of transverse pump light into a standing wave cavity mode. Choosing a suitable pump geometry allows to realize the Hofstadter-Harper model with a cavity-induced dynamical synthetic magnetic field, which appears at the onset of the superradiant phase transition. The dynamical nature of this cavity-induced synthetic magnetic field arises from the delicate interplay between collective superradiant scattering and the underlying fractal band structure. Using a sixth-order expansion of the free energy as function of the order parameter and by numerical simulations we show that at low magnetic fluxes the superradiant ordering phase transition is first order, while it becomes second order for higher flux. The dynamic nature of the magnetic field induces a non-trivial deformation of the Hofstadter butterfly in the superradiant phase. At strong pump far above the self-ordering threshold we recover the Hofstadter butterfly one would obtain in a static magnetic field.	10.1103/physrevb.100.224306	[ "https://export.arxiv.org/pdf/1909.05975v2.pdf" ]	202,572,950	1909.05975	25bb7a18643caf4340e4c65e93872137dc5f79ef	The Hofstadter Butterfly in a Cavity-Induced Dynamic Synthetic Magnetic Field Elvia Colella Institut für Theoretische Physik Universität Innsbruck A-6020InnsbruckAustria Farokh Mivehvar Institut für Theoretische Physik Universität Innsbruck A-6020InnsbruckAustria Francesco Piazza Max-Planck-Institut für Physik komplexer Systeme D-01187DresdenGermany Helmut Ritsch Institut für Theoretische Physik Universität Innsbruck A-6020InnsbruckAustria The Hofstadter Butterfly in a Cavity-Induced Dynamic Synthetic Magnetic Field Energy bands of electrons in a square lattice potential threaded by a uniform magnetic field exhibit a fractal structure known as the Hofstadter butterfly. Here we study a Fermi gas in a 2D optical lattice within a linear cavity with a tilt along the cavity axis. The hopping along the cavity axis is only induced by resonant Raman scattering of transverse pump light into a standing wave cavity mode. Choosing a suitable pump geometry allows to realize the Hofstadter-Harper model with a cavity-induced dynamical synthetic magnetic field, which appears at the onset of the superradiant phase transition. The dynamical nature of this cavity-induced synthetic magnetic field arises from the delicate interplay between collective superradiant scattering and the underlying fractal band structure. Using a sixth-order expansion of the free energy as function of the order parameter and by numerical simulations we show that at low magnetic fluxes the superradiant ordering phase transition is first order, while it becomes second order for higher flux. The dynamic nature of the magnetic field induces a non-trivial deformation of the Hofstadter butterfly in the superradiant phase. At strong pump far above the self-ordering threshold we recover the Hofstadter butterfly one would obtain in a static magnetic field. Energy bands of electrons in a square lattice potential threaded by a uniform magnetic field exhibit a fractal structure known as the Hofstadter butterfly. Here we study a Fermi gas in a 2D optical lattice within a linear cavity with a tilt along the cavity axis. The hopping along the cavity axis is only induced by resonant Raman scattering of transverse pump light into a standing wave cavity mode. Choosing a suitable pump geometry allows to realize the Hofstadter-Harper model with a cavity-induced dynamical synthetic magnetic field, which appears at the onset of the superradiant phase transition. The dynamical nature of this cavity-induced synthetic magnetic field arises from the delicate interplay between collective superradiant scattering and the underlying fractal band structure. Using a sixth-order expansion of the free energy as function of the order parameter and by numerical simulations we show that at low magnetic fluxes the superradiant ordering phase transition is first order, while it becomes second order for higher flux. The dynamic nature of the magnetic field induces a non-trivial deformation of the Hofstadter butterfly in the superradiant phase. At strong pump far above the self-ordering threshold we recover the Hofstadter butterfly one would obtain in a static magnetic field. I. INTRODUCTION In the last decade, advancements in the manipulation of cold atomic gases enabled to engineer Hamiltonians emulating the physics of effective gauge fields [1,2]. The development of rotating traps [3,4] allowed to overcome the challenge of coupling the external degrees of freedom of neutral atoms to an effective vector gauge potential as for charged particles. More sophisticated techniques based on light-matter interaction [5,6] and lattice shaking [7,8] were also developed to imprint a position-dependent geometric phase onto the atomic wave-function, analogous to the Aranov-Bohm phase of electrons in an external magnetic field [9]. The Hofstadter model [10,11] was shortly after implemented for cold atoms in optical lattices by employing a laser-assisted tunneling scheme [12][13][14]. The realization of such an artificial magnetic field in lattice geometries [15,16] allows one to explore the realm of topological many-body states of matter [17][18][19]. The most notable examples include measuring the Chern number of non-trivial topological bands [20] and realizing the Meissener phases for neutral atoms in ladder geometries [21]. More recently, new techniques exploiting internal degrees of freedom as synthetic dimension have been developed [22,23] and are candidates for the observation of the quantum Hall effect even in four dimensions [24]. The experimental realization of lattice models with effective gauge potential is of great interest for engineering synthetic gauge theories [25]. Experimental realizations so far implemented static gauge fields which can be finely tuned by varying experimental parameters, but are not dynamically affected by the atomic back-action. However, in order to simulate a genuine gauge theory, quantum matter needs to be dynamically coupled to a gauge (bosonic) * [email protected] field and the back-action of the matter dynamics onto the gauge field should be accounted for. A first step in this direction is to use density-dependent synthetic gauge fields [26,27], which were recently observed for a BEC in a shaken optical lattice [28,29]. A Z 2 lattice gauge theory was also experimentally realized [30,31]. Optomechanical systems [32,33] as well as cold atoms in optical cavities [34] provide another natural route to the realization of a dynamical gauge theory in a controllable and accessible environment. This hinges on the non-linearity of these systems, where photons (phonons) feel the back-action of the atomic motion (photons). In view of the experimental realization of a strongly interacting Fermi gas coupled to a cavity [35] and the recent observation of a dynamical spin-orbit coupling in a BEC in a linear cavity [36][37][38][39] , theoretical proposals [40][41][42][43][44][45][46][47][48] for dynamical gauge fields are now in reach by experiments. Here we study dynamical cavity-supported synthetic magnetic fields for fermions in an external optical lattice [12]. Atoms are driven by two transverse counterpropagating lasers and can scatter photons into the cavity. The hopping along the cavity axis is suppressed by a potential gradient. By choosing proper laser detunings, it can be activated by resonant Raman scattering of pump photons into a single resonant standing wave mode of the cavity [47]. Each pump laser here is responsible for a particular hopping direction. Above a critical pump strength, the collective buildup of the cavity field enables resonant coherent tunneling. In addition, for any closed loop in the atomic trajectory, a geometric phase proportional to the enclosed area is imprinted onto the atomic wave-function, in analogy to the phase acquired by electrons in a magnetic field. The onset of the superradiant phase transition and the appearance of a synthetic magnetic field depends strongly on the phases imprinted, which can be tuned by setting the ratio between the lattice constant and the pump field wavelength B ∝ d y /λ c . This is due to an intricate in- terplay between superradiant scattering generating the synthetic magnetic field and the emerging fractal energy bands corresponding to this field. Such cavity-induced atomic back-action on the effective gauge potential is very different to existing free-space implementations. Interestingly, as shown below, the onset of the superradiant phase transition (and hence appearance of the synthetic magnetic field) exhibits a first-order behavior at low fluxes, where the energy bands are Landau-like, while it becomes second-order for high flux. The energy spectrum itself carries the signs of the non-linearity of the atom-light interactions and the dynamical nature of the magnetic field, resulting in the emergence of peculiar structures compared to the commonly known energy spectrum, i.e., Hofstadter butterfly [11]. The paper is organized as follows. In section II we introduced the detailed system model. The physical results are summarized in section III, where we focus on the bulk properties of the system at half-filling. Here the gas behaves as a metal or semi-metal depending on the value of the magnetic flux in a plaquette. We show the phase diagram, the energy spectrum and we investigate the point of change of the phase transition from first to second order. Our final considerations are reported in section IV. II. MODEL We consider a Fermi gas confined in a two dimensional (2D) optical lattice of lattice constant, d = {d x , d y }, in the tight binding regime. Hopping in the x-direction is suppressed by an additional energy gradient ∆ between neighbouring sites. This can be realized by adding a constantly accelerated optical lattice, a magnetic field, or an electric field gradient along the x-direction. We consider only a single internal atomic transition \|g ↔ \|e of frequency ω 0 . The hopping in the x-direction is restored via two-photon resonant scattering processes mediated by cavity photons, where the resonance condition is ω c ω 1 + ∆ = ω 2 − ∆ [12]. Here, ω 1 and ω 2 are the frequencies of the two transversal laser pumps; see Fig. 1. Our model Hamiltonian in tight-binding approximation in a reference frame rotating at the average pump frequency ω p = (ω 1 + ω 2 )/2 then reads: [47], H = − J y l,m (f † l,m+1 f l,m + H.c.) (1) − η(a + a † ) l,m (e 2iπmγ f † l+1,m f l,m + H.c.) − ∆ c a † a. Here J y is the hopping amplitude in the y-direction, η = Ω 1 g 0 /δ = Ω 2 g 0 /δ is the two photon Rabi coupling with δ = ω p − ω 0 the atomic detuning with respect to the average pump frequency, g 0 is the bare coupling strength of the cavity mode to the atomic transition and ∆ c = ω p − ω c is the cavity detuning with respect to the average pump ω p . Note that only resonant Raman scattering terms are retained in the Hamiltonian. Further details are presented in Appendix A. The spatial phase dependence of the pump lasers imprints a site-dependent tunneling phase γ m = mγ = mk L /(2π/d y ). Hence, hopping around a plaquette, the wave-function acquires a total phase φ = 2πγ, which can be related to an electron moving in a periodic potential threaded by a magnetic field of strength \|B\| = 2πγ/(d 2 y e). The effective magnetic field breaks the translation symmetry of the original lattice and the Hamiltonian is invariant under a combination of discrete translation and a gauge transformation, i.e., magnetic translation. In particular, when γ = p/q is a rational number with p and q being two integers, and the energy spectrum splits into q sub-bands, which cluster in a highly fractal structure known as Hofstadter butterfly [11]. In contrast to free space setups the hopping amplitude in the cavity-direction depends on the cavity field amplitude and the effective magnetic field appears only for non-zero cavity-field. Here the coherent amplitude a = α is determined by the steady-state solution of the mean-field equation: ∂α ∂t = −(∆ c − iκ)α − ηΘ = 0,(2) where Figure 2. Phase boundary (red line) as function of effective flux γ/2π = p/q and rescaled pump field η √ N using the field amplitude modulus \|α\|/ √ N as background color. Note that p/q is discrete and rational, with 1 < p < 7 and 1 < q < 15. The field amplitude is determined selfconsistently for a Fermi gas at half-filling at fixed finite temperature kBT = 0.5ER, where ER = 2 k 2 c /2m is the recoil energy. At small fluxes, γ < 0.21, the system exhibits a first-order phase transition, while for bigger fluxes it is of second order. The solid red line shows the analytical result for the critical threshold and the red dashed line the beginning of the region of hysteresis. Θ = l,m e −2iπγm f † l,m f l−1,m + e 2iπγm f † l,m f l+1,m(3) is the atomic order parameter, which reveals emergent currents of equal number of left and right moving atoms along the cavity axis. The order parameter Θ needs to be self-consistently determined by diagonalizing the Hamiltonian at fixed amplitude α, Θ = 2 N 2 k m q s=1 k∈B.Z. n F ( s,k ) cos(2πmγ)\|v s,k (m)\| 2 . (4) Here s,k and v s,k (m) are the eigenvalues and eigenstates of the Harper equation [10] J y [e iky w k (m + 1) + e −iky w k (m − 1)]+ 2η(α + α * ) cos(k x − 2πmγ)w k (m) = w k (m). (5) We use the following Ansatz for the atomic wave-function Ψ(l, m) = e ikxl e ikym w k (m), with w k = c s v s,k (m) a linear superposition of the eigenstates of the Hamiltonian. Equations (4) and (5) are solved self-consistently within the reduced Brillouin zone k x ∈ [−π, π] and k y ∈ [−π/q, π/q], for a magnetic unit cell with periodic boundary conditions in x and y directions. We focus on the contribution of the bulk to the superradiance, neglecting boundary effects which appear in a pair of chiral edge states [47]. III. RESULTS A. Phase diagram For weak pump η √ N the system is in the uncoupled normal state (N), i.e., the atoms form a collection of independent chains in the y-direction and the cavity is empty. Increasing the effective pump strength the system exhibits a transition to a superradiant (SR) state, where photons are resonantly scattered into the cavity mode and the hopping in cavity (x)-direction builds up. The stationary cavity-field amplitude is depicted in Fig. 2. It grows continuously above the superradiant threshold for large magnetic flux (0.21 < γ < 0.5) but displays a non-continuous jump at lower γ < 0.21. In order to better understand the change from a second to a first order phase transition, as presented in Appendix B, we expand the free energy of the system in the Landau form up to sixth order in the atomic order parameter: F ∼(1 − 4∆ c ∆ 2 c + κ 2 χ 1 η 2 )\|Θ\| 2 − 8∆ 3 c (∆ 2 c + κ 2 ) 3 χ 3 η 6 \|Θ\| 4 (6) − 64∆ 5 c 3(∆ 2 c + κ 2 ) 5 χ 5 η 10 \|Θ\| 6 . The effective optical response of the Fermi gas after cycles of absorption and emission of cavity photons is determined by the static susceptibilities, χ i (Fig. 3). The linear susceptibility χ 1 determines the phase transition threshold √ N η c = ∆ 2 c + κ 2 4∆ c χ 1 N ,(7) which is shown as a red solid line in Fig. 2. The sign of χ 3 determines the order of the phase transition. In particular, for strong magnetic fields we have χ 3 > 0 and the transition is of the second order. The atoms then behave like a Kerr medium [49], inducing an intensity dependent shift of the refractive index, n = n 0 + n 2 I, with n 2 = −8χ 3 η 2 ∆ 3 c (∆ 2 c + κ 2 ). For decreasing magnetic field the third order susceptibility monotonically decreases becoming negative at γ 0.21, which renders the transition first order (bottom panel of Fig. 3). In this regime higher order susceptibilities only slightly depend on the magnetic flux γ. In fact the atomic orbit size significantly exceeds the unit cell of the original lattice, making the lattice structure negligible. The system then exhibits a universal behaviour and the band structure corresponds to Landau levels in free space. B. First-order transition At low γ the emergent magnetic field has only little influence on the system dynamics. The temperature and the presence of an open Fermi surface then play a fundamental role in order to unravel the physical origin of the first order behaviour of the phase transition. By inspection of the temperature dependence of χ 3 for a Fermi gas at half-filling, we can identify an important change around γ ≈ k F /k L = 1/4 ( Fig. 4(a)). The susceptibility χ 3 is either positive at any temperature, or becomes negative at low temperature. The two regions are separated by the red solid line in Fig. 4(a). In the latter case the phase transition becomes first order at low temperatures. This coincides with the regime where scattering one photon keeps the atomic momentum state within the same first Brillouin zone of the original lattice (normal scattering). In contrast, the transition becomes second order when the photon scattering is an Umklapp process (Fig. 4(b)), i.e., by inverting the direction of the atomic motion, a momentum transfer (G = nk L ) to the optical lattice is required. However, the occupation of higher energy states at higher temperature can favour the Umklapp processes at the expense of direct scattering enhancing the rate to scatter to the next Brillouin zone even for a small momentum transfer. This explains why at higher temperature a second order phase transition occurs and the critical temperature at which this happens increases for small γ (Fig. 4(a)). These results are confirmed by the numerical simulations at lower temperatures, k b T = 0.05E R . The re-scaled cavity amplitude as function of the pump strength either grows continuously around the threshold for γ = 1/3 (black line in Fig. 4(c)), or exhibits a jump at the critical point for γ = 1/4 (blue dashed line in Fig. 4(c)). For γ = 1/3 the rescaled amplitude shows an additional jump at higher pumps η > η c , hinting that an additional first order transition inside the superradiant phase can appear. Such transition occurs when the cavityinduced hopping exceeds the hopping in the y-direction, J x /J y = η(α + α * ) = 1. The two superradiant states are characterized by the same order parameter but different isothermal compressibility, κ T = (1/ρ 2 )∂ρ/∂µ, where ρ is the density of the Fermi gas. This divides the superradiant region into two phase zones: SRI and SRII. In many respects this suggests a liquid-gas type of transition between the SRI and SRII phases, as confirmed by the rapid growth of density fluctuations that can be inferred from the divergence of the compressibility at the critical point ( Fig. 4(d)). The transition is reminiscent of the case observed for fermions in linear cavities without external optical lattice [50]. In the latter case, however, the transition was driven by the coupling to an additional degree of freedom, in a process similar to the Larkin-Pimkin mechanism [51]. C. Hysteresis For small magnetic flux the system exhibits a bi-stable hysteresis behaviour near the superradiant threshold η c . The hysteresis loop and a qualitative picture of the free energy in the different regions are shown in Fig. 5. As can be seen in the insets, below the threshold the solution with α = 0 (empty cavity) is the only minimum of the free energy. Between η 1 < η < η c the free energy has three minima, either local or absolute. The solution for α = 0 is metastable for η 1 < η < η 2 , with η 1 = η c 1 − χ 2 3 /(12χ 1 χ 5 ) ,(8)η 2 = η c 1 − 3χ 2 3 /(8χ 1 χ 5 ) .(9) Between η 2 < η < η c , the zero field solution α = 0 is metastable and finally ceases to be a minimum at η c , where the system becomes superradiant. Figure 6 shows the energy spectrum as a function of the magnetic flux p/q for increasing pump strength η √ N . The magnetic field, B ∼ p/q, emerges spontaneously with the cavity field amplitude and leads to the opening of q − 1 gaps in the band structure. As the superradiant phase is entered already at lower pump power for stronger magnetic field, the gap opening progressively extends toward p/q = 0 as the pump is increased. D. Dynamical Hofstadter Butterfly The different structures visible in the energy spectrum strongly depend on the pump strength. At low pump strength (top panels of Fig. 6) the gaps organize in the shape of a small butterfly confined in the region of large magnetic fields 0.21 < γ < 0.5. The gaps close at the boundary of this region, where the amplitude of the cavity field is infinitesimally small. When the pump is increased the Hofstadter butterfly is entirely retrieved (right-bottom panel in Fig. 6) like in a static optical lattice. The gaps will gradually close, generating a 1D tight-binding in the x-direction with bandwidth, 2J x = 2η(α + α * ). In fact, the system evolves toward a regime of very weakly coupled 1D chains in the x-direction, for which the magnetic field can be gauged out. The distortion of the energy spectrum, compared to the conventional Hofstadter butterfly [11], is due to the dynamical nature of the coupling between atoms and cavity photons. At a fixed magnetic field, the system spontaneously chooses the most favourable amplitude of the cavity field, i.e, the effective hopping parameter, J x = η(α + α * ). As the system becomes superradiant the effective Lorentz force exerted by the artificial magnetic field favours the tunneling in the x-direction, resulting in an asymmetry of the tunneling amplitudes. Therefore, the energy spectrum can be seen as the superposition of different Hofstadter butterflies with asymmetric hopping, J x − J y . While the fractal structure is preserved by the form of the Hamiltonian as the hopping phase is not cavity-dependent, the size of the gaps are set by the ratio of the hopping parameters and are characterized by a non-trivial dependence on the magnetic flux 2πp/q. This is illustrated in Fig. 7(a), where the hopping ratio J x /J y is shown as a function of the magnetic flux for different pump strengths. In the weak pump regime (black and dark blue lines) the dynamic butterfly is a superposition of static Hofstadter butterflies with very different effective hopping amplitudes. The hopping in the x-direction grows as the magnetic field is increased but remains rather small compared to the hopping in the other direction. As a consequence the curvature of the band structure and the Fermi surface align along y-direction, see left panel in Fig.7(b). As the pump is increased, the field amplitude and the hopping in the x-direction become almost independent of the magnetic flux (red and yellow line in Fig 7(a)). In this regime the kinetic energy in the x-direction dominates and the Fermi surface aligns along the cavity axis. Note that at low temperature this is accompanied by the onset of a first order transition within the superradiant phase, SRI-SRII, as shown in the previous section. IV. CONCLUSIONS AND OUTLOOK We have shown that non-linear coupling between atomic motion and a cavity field mode offers a new perspective on the generation of synthetic dynamical magnetic fields. In contrast to free space, the gauge field emerges spontaneously via maximizing the light scattered into the cavity and changing the atomic density configuration. The complex interplay between the fractal structure of the energy bands and the superradiant scattering thus generates new shapes for a dynamical Hofstadter butterfly. Note that atoms are coupled only to a specific wavelength of the light field determined by the chosen cavity mode. As shown recently employing several distinct cavity modes the system gets more freedom and a global symmetry can "emerge" in a cavity-QED system [52]. Therefore, generalization of our studied system to multi-mode cavities and in particular a ring or fiber geometry [53] could allow to fully reproduce the minimal coupling of a charged particle to a local U (1) gauge potential. Making use of the dynamical coupling between light and atoms in cavity systems is a promising route toward the experimental realization of synthetic dynamical gauge fields. Moreover, on a different level, the mediation of long-range two-body interactions due to the exchange of photons can lead to the observation of exotic states, as particles with anyonic statistics in fractional quantum Hall states. . The hopping along x-direction is at first suppressed due to the potential offset ∆ between adjacent lattice sites and then restored thanks to the cavity-and laser-assisted hoppings. The hopping along y-direction is due to the kinetic energy of the atoms. Let us just focus in the x-direction and consider three generic lattice sites labeled n − 1, n, and n as in Fig. 8. First consider only transitions which involves the atomic excited state in site n, that is, \|e n . The Hamiltonian H = H 0 + H int reads ( = 1), H 0 = −(ω 0 + ∆)σ n−1 − ω 0 σ n − (ω 0 − ∆)σ n+1 + ω c a † a,(A1) H int = Ω 2 e −iky e −iω2t σ + n−1 + g 0 cos (kx n )aσ + n + Ω 1 e iky e −iω1t σ + n+1 + H.c., where σ n−1 = \|g n−1 g n−1 \|, σ n = \|g n g n \| , σ n+1 = \|g n+1 g n+1 \|, σ + n−1 = \|e n g n−1 \|, σ + n = \|e n g n \|, σ + n+1 = \|e n g n+1 \|. For simplicity a two-photon resonance is assumed ω c = ω 1 + ∆ = ω 2 − ∆ in the following and k ≡ k c k 1 k 2 . Applying the unitary transformation U = exp {−i[ω 2 σ n−1 + ω p (σ n − a † a) + ω 1 σ n+1 ]t} to the Hamiltonian H yields, H = δ(σ n−1 + σ n + σ n+1 ) + [Ω 2 e −iky σ + n−1 + g 0 cos (kx n )aσ + n + Ω 1 e iky σ + n+1 + H.c.],(A3) where δ = ω c − ω 0 ∼ ω p − ω 0 , with ω p = (ω 1 + ω 2 )/2 the average pump frequency. Here we have made use of the relations U σ + n−1 U † = e iω2t σ + n−1 etc. andH = U HU † + i(∂ t U )U † . We find the stationary values of the operators σ + n−1 , σ + n , σ + n+1 by setting to zero the the (Ω * 2 e iky σ n−1 + Ω * 1 e −iky σ n+1,n−1 + g 0 cos (kx n )a † σ n,n−1 ), σ + n 1 δ (Ω * 2 e iky σ n−1,n + Ω * 1 e −iky σ n+1,n + g 0 cos (kx n )a † σ n ), σ + n+1 1 δ (Ω * 2 e iky σ n−1,n+1 + Ω * 1 e −iky σ n+1 + g 0 cos (kx n )a † σ n,n+1 ),(A4) where σ n,n−1 = \|g n g n−1 \|, σ n+1,n−1 = \|g n+1 g n−1 \|, σ n+1,n = \|g n+1 g n \|, etc. Here we have also assumed a negligible population of the excited state, \|e n e n \| 0, due to the large detuning δ. Substituting Eq. (A4) back in the Hamiltonian (A3) yields the effective Hamiltonian, H (n) eff = 2 δ {g 2 0 cos 2 (kx n )a † aσ n + [Ω 2 g 0 e −iky cos (kx n )a † σ n,n−1 + Ω * 1 g 0 e −iky cos (kx n )aσ n+1,n + H.c.]},(A5) where the constant terms proportional to Ω 1 and Ω 2 , and terms involving next nearest neighbour scattering σ n+1,n−1 have been omitted. Considering now transitions which involve the states \|e n±1 results in the following contributions to the {n − 1, n, n + 1} manifold, Assuming Ω 1 = Ω 2 = Ω ∈ R and λ c = 2π/k = d x , the total effective Hamiltonian takes the form, H eff = 2 δ n {g 2 0 cos 2 (kx n )a † aσ n + Ωg 0 (a + a † ) e −iky cos (kx n )σ n,n−1 + H.c. }, where the hopping along the y direction is now also included. The matrix elements are given by, In order to derive an effective Landau theory for the atomic order parameter Θ, as defined in the main text, we start from the the action of the system expressed in momentum space S[α, α * ,c † kx,ky , c kx,ky ] = ∆ c \|α\| 2 (B1a) + 1 βV n,kx,ky (iω n − 2J y cos(k y )) c † n,kx,ky c n,kx,ky − η(α + α * ) 1 βV kx,ky e −kx c † kx,ky c kx,ky+γ + e −kx c † kx,ky c kx,ky−γ . n,m = 2 δ g 2 0 dxdy cos 2 (kx) × \|W (x − x n )W (y − y n )\| 2 = 2 δ g 2 0 dx cos 2 (kx)\|W (x − x n )\| 2 , J x n,m e −ikym = 2 δ Ωg 0 dxdyW * (x − x n )W * (y − y n ) × e −iky cos (kx)W (x − x n−1 )W (y − y m ) = 2 δ Ωg 0 dx cos (kx) × W * (x − x n )W (x − x n−1 ) × dye −iky W * (y − y m )W (y − y m ), (A9) where W (X − R) = W (x − x n )W (y − y m ) Note that only the static component of the bosonic field α is retained, which is linearly related to the atomic order parameter by the equation of motion α = −ηΘ/(∆ c − iκ). We integrate out fermionic degrees of freedom, obtaining an effective action for the photonic field only, S ef f [α, α * ] = ∆ c \|α\| 2 + tr lnĜ −1 . The trace operator tr lnĜ −1 = tr lnG −1 0 − n 1 2n tr(G 0 Γ) 2n (B2) is obtained by perturbatively expanding the Green function G(k, iω n ) around the zero order one G −1 0 (k, ω n ) =         . . . 0 0 0 0 0 iω n − 2J y cos(k y − γ) 0 0 0 0 0 iω n − 2J y cos(k y − γ) 0 0 0 0 0 iω n − 2J y cos(k y − γ) 0 0 0 0 0 . . .         (B3) where the perturbative term is given by the interaction matrix Γ(k) = −η(α + α * )       0 e −ikx 0 0 0 e ikx 0 e −ikx 0 0 0 e ikx 0 e −ikx 0 0 0 e ikx 0 e −ikx 0 0 0 e ikx 0       (B4) Here, iω n = π(2n+1)/β are fermionic Matsubara frequencies. By keeping up to the sixth order in α, the effective free energy is F = ∆ c \|α\| 2 − η 2 χ 1 (α + α * ) 2 − η 4 2 χ 3 (α + α * ) 4 − η 6 3 χ 5 (α + α * ) 6 ,(B5) or in powers of the atomic order parameter, Θ, reads F ∼ (1 − 4∆ c ∆ 2 c + κ 2 χ 1 η 2 )\|Θ\| 2 − 8∆ 3 c (∆ 2 c + κ 2 ) 3 χ 3 η 6 \|Θ\| 4 − 64∆ 5 c 3(∆ 2 c + κ 2 ) 5 χ 5 η 10 \|Θ\| 6 (B6) The free energy depends on the cavity properties and the coupling with the atoms is enclosed inside the susceptibil-ities χ 1 = 1 β n,k∈B.Z. G k (iω n )G k+γ (iω n ) (B7a) χ 3 = 1 β n,k∈B.Z. [G 2 k (iω n )G 2 k+γ (iω n ) (B7b) + 2G k−γ (iω n )G 2 k (iω n )G k+γ (iω n )] χ 5 = 1 β n,k∈B.Z. [G 3 k (iω n )G 3 k+γ (iω n ) (B7c) + 3G 2 k−γ (iω n )G 3 k (iω n )G k+γ (iω n ) + 3G k−γ (iω n )G 3 k (iω n )G 2 k+γ (iω n ) + 3G k (iω n )G 2 k+γ (iω n )G 2 k+2γ (iω n )G k+3γ (iω n )] The susceptibilities shown in the main text are numerically calculated by truncating the summation over the Matsubara frequencies until convergence with fixed chemical potential µ = 0, same for the matrices G 0 (k, ω n ) and Γ(k) which are summed in momentum space over the original Brillouin zone [−π/d x , π/d y ]. Expansion of the susceptibility for low magnetic fluxes In order have a better understanding of the physics at low magnetic fluxes, we have analytically computed the expressions for the susceptibilities χ 1 and χ 3 . The first order susceptibility is χ 1 = k∈B.Z. n F ( k+γ ) − n F ( k ) k+γ − k ,(B8) with k = J y cos(k), the tight binding energy along the y-direction where we set µ = 0 for half filling. We expand χ 1 for small γ χ 1 (γ 1) = k∈B.Z. − βn F (cos(k)) [1 − n F (cos(k))] (B9) Note that the linear term vanishes and the main contribution to the linear susceptibility is a constant, which is proportional to the compressibility of a 1D chain of fermionic particles in the tight binding regime. As n F ( ) is the probability that the state is occupied, while 1−n F ( ) is the probability that the state is not occupied, their product represent the scattering amplitude of a scattering process between two state of the same energy, which at very low temperature is only possible from one side to the other of the Fermi surface. The next contribution to χ 1 is quadratic and this behaviour can also be observed in the plot of the susceptibilty χ 1 , see Fig 2 in the main text. Note that at the zero order, in γ we don't see the effect of the magnetic field but rather the temperature, dimensionality and filling play the fundamental role. The third order χ 3 susceptibilty represents the response of the medium to three photon processes, through cycles of multiple emission and absorption. The full analytics expression is χ 3 = k∈B.Z. −2 n F ( k+γ ) − n F ( k ) ( k+γ − k ) 3 + n F ( k+γ ) − n F ( k ) ( k+γ − k ) 2 + 2 n F ( k−γ ) ( k−γ − k ) 2 ( k−γ − k+γ ) − 2 n F ( k+γ ) ( k+γ − k ) 2 ( k−γ − k+γ ) + 2 n F ( k ) ( k−γ − k )( k − k+γ ) × 1 k − k+γ + 1 k − k−γ − 2 n F ( k ) ( k−γ − k )( k − k+γ ) (B10) In a linear cavity photons are in a superposition state of two conterpropagating momenta. The interaction with the cavity photons induces two type of processes. The first two lines refers to cycles of absorption and emission where the scattering processes always involve interactions with the same momentum component of the photon field. The other lines, refer to scattering processes in which a redistribution of photons between the two momentum component are involved. At the lowest order in γ, the susceptibility χ 3 becomes χ 3 (γ 1) = k∈B.Z. β 3 6 n f ( k )[1 − n f ( k )] × [1 − 6n f ( k ) [1 − n f ( k )]] .(B11) Figure 1 . 1Geometry sketch to realize a dynamical version of the Harper-Hofstadter Hamiltonian: a 2D Fermi gas in a rectangular lattice within a single-mode optical cavity is transversely illuminated by two counter-propagating laser beams of orthogonal polarization. The shaded area in the lattice represents the unit cell for φ = 2π/3. Figure 3 . 3Atomic susceptibilities, χ1 (red), χ3 (black) and χ5 (blue) at k b T = 0.5ER. The third order susceptibility χ3 becomes negative below p/q = 0.21, signaled by the dashed black line. Figure 4 . 4(a) Third order susceptibility, χ3, as a function of the temperature and effective magnetic flux, 2πγ = 2πp/q, with 1 < p < 6 and 1 < q < 13. The red line corresponds to zero susceptibility, separating positive and negative regions. (b) An atom at the Fermi surface is scattered after absorbing a photon to a higher energy state, via an Umklapp (top panel) or a normal process (bottom panel). The process is depicted using two Brilliouin zones of the original lattice. Cavity field amplitude (c) and isothermal compressibility (d) at k b T = 0.05ER, for γ = 1/3 (solid black) and γ = 1/4 (dashed blue). Figure 5 . 5Atomic order parameter at T = 0.5ER for γ = 1/12 as a function of the effective pump η √ N . The arrows shows the hysteresis loop and the dotted line represent the metastable solution. The insets show a qualitative picture of the free energy in the different regimes. Figure 6 . 6Energy spectrum as function of flux p/q for four different pump strength η √ N = {1.1, 1.2, 1.3, 1.4}ER from top left to bottom right corner at k b T = 0.5ER. The spectrum initially shows singular shapes and reduces to the conventional Hofstadter butterfly at strong pump. Figure 7 . 7(a) Effective cavity induced hopping as a function of flux p/q at different pumping strengths. Parameters: η √ N = {1.1, 1.2, 1.3, 1.4, 1.5}ER in black, dark blue, light blue, yellow and red respectively. (b) Fermi surface at γ = 1/3 for k b T = 0.5ER for η √ N = 1.2ER (left) and η √ N = 1.3ER (right). ACKNOWLEDGMENTSF . M. is grateful to Nathan Goldman for fruitful discussions. F. M. is supported by the Lise-Meitner Fellowship M2438-NBL of the Austrian Science Fund (FWF), and the International Joint Project No. I3964-N27 of the FWF and the National Agency for Research (ANR) of France. Appendix A: Effective Hamiltonian Consider atoms loaded into a 2D optical lattice of lattice constant, d = [d x , d y ] Heisenberg equation of motion i∂ t O = [O,H] upon assuming a Figure 8 . 82 (kx n−1 )a † aσ n−1 + [Ω * 1 g 0 e −iky cos (kx n−1 )aσ n,n−1 + H.c.2 (kx n+1 )a † aσ n+1+ Ω 2 g 0 e −iky cos (kx n+1 )a † σ n1,n + H.c. }. 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[ "Disambiguation of Company names via Deep Recurrent Networks", "Disambiguation of Company names via Deep Recurrent Networks" ]	[ "Alessandro Basile \nData Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly\n", "Riccardo Crupi \nData Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly\n", "Michele Grasso \nEarly Morning S.r.l\nItaly\n", "Alessandro Mercanti \nData Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly\n", "Daniele Regoli \nData Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly\n", "Simone Scarsi \nData Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly\n", "Shuyi Yang \nData Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly\n", "Andrea Claudio Cosentini \nData Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly\n" ]	[ "Data Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly", "Data Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly", "Early Morning S.r.l\nItaly", "Data Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly", "Data Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly", "Data Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly", "Data Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly", "Data Science & Artificial Intelligence\nIntesa Sanpaolo S.p.A\nItaly" ]	[]	Name Entity Disambiguation is the Natural Language Processing task of identifying textual records corresponding to the same Named Entity, i.e. real-world entities represented as a list of attributes (names, places, organisations, etc.). In this work, we face the task of disambiguating companies on the basis of their written names. We propose a Siamese LSTM Network approach to extract -via supervised learning -an embedding of company name strings in a (relatively) low dimensional vector space and use this representation to identify pairs of company names that actually represent the same company (i.e. the same Entity).Given that the manual labelling of string pairs is a rather onerous task, we analyse how an Active Learning approach to prioritise the samples to be labelled leads to a more efficient overall learning pipeline.With empirical investigations, we show that our proposed Siamese Network outperforms several benchmark approaches based on standard string matching algorithms when enough labelled data are available. Moreover, we show that Active Learning prioritisation is indeed helpful when labelling resources are limited, and let the learning models reach the out-of-sample performance saturation with less labelled data with respect to standard (random) data labelling approaches.	10.48550/arxiv.2303.05391	[ "https://export.arxiv.org/pdf/2303.05391v2.pdf" ]	257,427,541	2303.05391	5722b1bd272145feb4fa51683a32ffe7d3d8c211	Disambiguation of Company names via Deep Recurrent Networks Alessandro Basile Data Science & Artificial Intelligence Intesa Sanpaolo S.p.A Italy Riccardo Crupi Data Science & Artificial Intelligence Intesa Sanpaolo S.p.A Italy Michele Grasso Early Morning S.r.l Italy Alessandro Mercanti Data Science & Artificial Intelligence Intesa Sanpaolo S.p.A Italy Daniele Regoli Data Science & Artificial Intelligence Intesa Sanpaolo S.p.A Italy Simone Scarsi Data Science & Artificial Intelligence Intesa Sanpaolo S.p.A Italy Shuyi Yang Data Science & Artificial Intelligence Intesa Sanpaolo S.p.A Italy Andrea Claudio Cosentini Data Science & Artificial Intelligence Intesa Sanpaolo S.p.A Italy Disambiguation of Company names via Deep Recurrent Networks Machine learningNatural Language ProcessingNamed Entity DisambiguationSiamese NetworkActive Learning Name Entity Disambiguation is the Natural Language Processing task of identifying textual records corresponding to the same Named Entity, i.e. real-world entities represented as a list of attributes (names, places, organisations, etc.). In this work, we face the task of disambiguating companies on the basis of their written names. We propose a Siamese LSTM Network approach to extract -via supervised learning -an embedding of company name strings in a (relatively) low dimensional vector space and use this representation to identify pairs of company names that actually represent the same company (i.e. the same Entity).Given that the manual labelling of string pairs is a rather onerous task, we analyse how an Active Learning approach to prioritise the samples to be labelled leads to a more efficient overall learning pipeline.With empirical investigations, we show that our proposed Siamese Network outperforms several benchmark approaches based on standard string matching algorithms when enough labelled data are available. Moreover, we show that Active Learning prioritisation is indeed helpful when labelling resources are limited, and let the learning models reach the out-of-sample performance saturation with less labelled data with respect to standard (random) data labelling approaches. Introduction A common information retrieval task with several applications is the association of company names from any internal or external source to a specific company registered in an internal database. A The contribution of this work is twofold: on the one hand, we propose a Siamese Recurrent Neural Network approach to the ask of disambiguating pairs of company names. We show, via experiments, that the proposed approach outperforms other baseline models and that is efficient in generalising to other domains. On the other hand, we use our proposed model in an Active Learning setting to demonstrate how to make human labelling more efficient by prioritising the samples to be labelled. The rest of the paper is organised as follows: Section 2 is devoted to a discussion of relevant literature, in particular regarding NED and Active Learning. In Section 3 we detail the methodologies we use for the NED task: we describe both our proposed model and the baseline approaches we use as benchmark. Section 4 is devoted to describing how we implement the Active Learning setting. In Section 5 we thoroughly describe how we build the datasets that we use in the experiments. The latter are described in Section 6. Discussion of the insights derived from the experimental results is presented in Section 7, while Section 8 contains concluding remarks. The python code implementation of the Siamese Neural Network model, of the Active Learning setting and of all the experiments, is available in open-source at github.com/rcrupi/SiameseDisambiguation. Related Works Entity Disambiguation Most classical approaches to strings pair matching leverage string similarity measures, quantifying how much two given strings are similar with a more or less sophisticate deterministic rule (Cohen et al., 2003;Christen, 2006;Sun et al., 2015). These methods are built on several background encodings of given strings, such as phonetic-based, character-based, or based on term frequency/inverse term frequency (tf-idf ) and hybrid versions of these (Christen, 2006). Other approaches try to employ also semantic knowledge to compute similarity (Prasetya et al., 2018). More recently, methodologies based on learning the appropriate similarity function from a sample of data of the desired domain have become quite common. For instance, Piskorski and Jacquet (2020) employ tf-idf vectors of n-grams as predictors for a Machine Learning (ML) classifier. The internal (learned) representation can then be exploited as an abstract vector summarising the information relevant to the task. Finally, a standard vector similarity function -such as cosine similarity -of the representations of two strings can be used to infer their task-specific distance. In the domain of toponym matching -i.e. pairing of different strings representing the same realworld location - Santos et al. (2018) have faced a problem very similar to the one we are discussing. They propose an approach based on a Siamese Deep Neural Network architecture (Chicco, 2021) and benchmark it against several distance-based methodologies and several classifiers taking as input various pairwise string distances. They conclude that their approach is superior in terms of matching performance, even if less efficient in terms of computational time with respect to pure distance methods. Neculoiu et al. (2016) propose a Siamese Deep Learning model as well, with the slightly different task of mapping strings to a predefined set of job names. This classification task is nevertheless approached by translating it into a NED framework and by learning a vector representation of strings, such that close vectors correspond to the same job class. They use two different loss terms for positive and negative matches: in particular, the loss term for negative samples is zero below a certain threshold, so as not to pay a cost for non-matching pairs that are increasingly dissimilar. Furthermore, they introduce some interesting data augmentation techniques, such as observations derived by adding some random typos and character-level deletion in positive pair samples. Aghaebrahimian and Cieliebak (2020) also propose a Deep Learning approach to NED, while having as input, not the raw strings, but rather character-level tf-idf vectors of n-grams, similarly to Piskorski and Jacquet (2020). Moreover, they train the network in a contrastive fashion, i.e. by feeding input triplets with the observed string and both an actual string match (i.e. string representing the same entity -positive example) and a non-matching string (negative example). We refer to Barlaug and Gulla (2021) for a thorough review of Neural Networks-based approaches to the NED task. Active Learning Entity Matching datasets are typically constructed through laborious human labelling. To increase the efficiency of such a procedure, techniques have been proposed to carefully prioritise the samples to be labelled. Against this background, Active Learning -i.e. the sub-field of ML with the characteristic that the learning algorithm is allowed to choose the data from which it learns (Settles, 2009;Arora and Agarwal, 2007) -has been proven to be beneficial in the Entity Matching domain (Meduri et al., 2020). In particular, the selection criteria of candidate observations to be labelled are usually expressed in terms of informativeness and representativeness (Zhou, 2018). While representativeness-based approaches try to find a pattern of the unlabelled data, using graphs or clustering methods, informativeness-based approaches -such as Uncertainty sampling and query-by-committee -choose the instances to be labelled based on how uncertain they are to be classified. In particular, Query-by-Committee approaches propose to train several classifiers and define the uncertainty of a given observation based on the rate of disagreement on their predictions. They are sometimes referred to as the learner-agnostic approaches. Uncertainty sampling, on the other hand, given a specific classifier, employs the distance from the decision boundary as a proxy for uncertainty. It is therefore referred to as learner-aware approach. While in Meduri et al. (2020) the task is to disambiguate two instances on the basis of several different information sets (e.g. address, name, etc.), in this work we focus on company disambiguation based on string names only. A disambiguation task involving couples of words (e.g. 'principle', 'principal', and 'end', 'and') was faced in Banko and Brill (2001). In particular, they extract features from the words and apply a ML classifier to estimate their similarity. Active Learning (specifically query-by-committee) is then exploited to iteratively select batches from a pool of unlabelled samples. Half of the samples in each batch are selected randomly, while the other half are selected on the basis of their uncertainty. Since Deep Learning requires huge amount of data, Active Learning is particularly well suited to limiting the data labelling but keeping high the performance (Zhan et al., 2022). In particular, in this work we adopt a modification of the "Least confidence" approach as query strategy (Huang, 2021). Other works, such as Sorscher et al. (2022), suggest a self-supervised data pruning method. In contrast to Active Learning, data reduction is done in a single step. The self-supervised metric is based on the application of k-means over the embedding space of a pre-trained network: an instance far away from its cluster centroid is considered uncertain. Experiments in Sorscher et al. (2022) suggest that the instances to be removed actually depend on the size of the starting dataset: if it is large enough, it is beneficial to include the most uncertain samples, whereas if it is small, it is preferable to include simplest (least uncertain) samples. Methods In ML settings, a multiclass classification function C : X → Y takes as input a feature vector x ∈ X from the input feature space X and outputs a class label y ∈ Y from a finite set of possiblē α classes Y = {0, 1, 2, . . . ,ᾱ − 1} (Hastie et al., 2009). Most families of ML classifiers actually learn to estimate the probabilities P(y = α \| x), α ∈ {0, 1, . . . ,ᾱ} . (1) In the following, we label withŷ i,α the estimated probability that the i-th sample belongs to the class α. Being probabilities, theŷ i,α are such that ᾱ−1 α=0ŷ i,α = 1 andŷ i,α ∈ [0, 1] ∀α ∈ Y . The predicted classȳ i is the one associated with the highest probability, namelȳ y i = argmax α∈Yŷ i,α . 4 The case with only two class labels (ᾱ = 2) -i.e. binary classification tasks -are usually formalised as: C : X → [0, 1],(2) where the outputŷ i is an estimate of P(y i = 1 \| x i ) and corresponds to the output (ŷ i,0 ,ŷ i,1 ) = (1 −ŷ i ,ŷ i ) in the generic multiclass setting. In this work, we frame the string match problem as a binary classification task, where the components of the input feature vector x i are couples of strings x i = {a i , b i }, and the classifier estimates the probabilityŷ i that the two strings correspond to the same company (we label 1 the matching class). Calling S the set of possible character symbols, we may formalise the string matching classifier as a function C : S n × S n → [0, 1],(3) where the integer n denotes the fixed (maximum) length of the strings to be analysed. Baseline methods To determine how dissimilar two strings a and b ∈ S n are, in the following we shall make use either of a distance function -D(a, b) ∈ R + where 0 stands for identical strings and the higher the distance the more dissimilar a and b -or a similarity function -S(a, b) ∈ [0, 1], 1 denoting identical strings. Levenshtein Widely used deterministic methods are the edit distance metrics. Generally speaking, they are based on counting the number of operations needed to transport a string onto another. The choice of the type of operations allowed determines the specific form of the distance. The most widely known edit-distance metric is the Levenshtein distance (sometimes referred to as the edit-distance), which calculates the distance as the number of insertions, deletions, and substitutions required to transform one string into another. The formula of the Levenshtein distance D Lev (a, b) between two strings a = a 0 a 1 a 2 . . . a m of length m and b = b 0 b 1 b 2 . . . b n of length n is given by the following recursion: D Lev (a, b) =                    m if n = 0, n if m = 0, D Lev ( a, b) if a 0 = b 0 , 1 + min    D Lev ( a, b) D Lev (a, b) D Lev ( a, b) otherwise;(4) where x denotes the string x without the first character (x 0 ), i.e. x = x 1 x 2 . . . x s . A closely related edit-based distance is the InDel distance, which allows only insertions and deletions. It is easy to see that the InDel distance is equivalent to the Levenshtein distance where the substitution operation is assigned a cost of 2 (deletion + insertion). We call InDel Ratio the following normalised version of the InDel distance: R ID (a, b) = 1 − D ID (a, b) \|a\| + \|b\| ,(5) where D ID (a, b) denotes the InDel distance between a and b. We make use of the Python library TheFuzz to compute R ID (a, b), with the sole difference that TheFuzz expresses the ratio in percentage points. Jaro-Winkler similarity The Jaro-Winkler (JW) similarity (S JW ) is another edit-based metric emphasising both the amount of matching characters and their placement inside the two strings. Notice that this is a notion of similarity, i.e. S JW ∈ [0, 1], S JW (a, b) = 0 corresponding to no match at all between two strings a and b, and S JW (a, b) = 1 to exact match. The Jaro-Winkler similarity is a variant of the Jaro similarity, that, for two strings a and b, is defined as S J (a, b) =      0 if c = 0, 1 3 c \|a\| + c \|b\| + c − t c otherwise,(6) with: • c the number of matching characters. Two characters are considered a match when they are the same and they are no more than max(\|a\|,\|b\|) 2 − 1 chars apart of one another, • t is the number of transpositions counted as the number of matching characters found in the wrong order, divided by two. The JW similarity extends the definition of the Jaro similarity by favouring strings with a matching prefix: S JW (a, b) = S J (a, b) + p (1 − S J (a, b)) ,(7) where is the length of the common prefix up to a maximum value, and p is a constant scaling factor determining the strength of the premium. The maximum value attributed to and the value of p should be chosen such that p ≤ 1. Jaccard similarity While the edit-based metrics look at what is necessary to do to transform one string into another, the token-based metrics consider the strings as sets of tokens (i.e. the words or characters composing the strings) and search for the common tokens between two sets. A widely-used token-based similarity metric is the Jaccard similarity which is defined as the ratio of the intersection over the union of the token's sets A = {a 0 , a 1 , a 2 , . . . , a m } and B = {b 0 , b 1 , b 2 , . . . , b m } for the two strings a and b respectively (sometimes referred to as IoU -Intersection over Union), i.e. S Jac = \|A ∩ B\| \|A ∪ B\| .(8) Notice that the Jaccard metric does not take into account the order of tokens, unlike the previously discussed edit-based metrics. Token Set Ratio Another approach for string matching is represented by the approximate string matching algorithms (Navarro, 2001) that leverage the basic notions of distance introduced so far, but take into account matching also at substring level. In particular, we make use of the so-called Token Set Ratio metric computed via TheFuzz Python library. It works as follows: 1. takes the unique words (i.e. substrings separated by whitespaces) for each string a and b, let us call them W a and W b , respectively: 2. builds the following word sets: I ab = W a ∩ W b , W a\b = W a \ W b , W b\a = W b \ W a ; 3. sorts alphabetically the sets and builds new strings s ab , s a\b , s b\a by joining with whitespaces the words in the corresponding (sorted) sets; 4. builds the new strings: c a joining s ab and s a\b with a whitespace, and analogously c b with s ab and s b\a ; 5. compute the similarity as R TS (a, b) = max      R ID (s ab , c a ) R ID (s ab , c b ) R ID (c a , c b ) . (9) Baseline classifier To build a classifier based on the match algorithms just described, the strings are pre-processed by removing punctuation and capitalising the text. The five string match algorithms listed in Table 1 score type range constitute our baseline methods and for each of the 5 string matching score listed in Table 1, we do the following: Levenshtein distance D Lev (a, b) ∈ [0, max (\|a\|, \|b\|)] InDel similarity R ID (a, b) ∈ [0, 1] Jaro-Winkler similarity S JW (a, b) ∈ [0, 1] Token Set Ratio similarity R TS (a, b) ∈ [0, 1] Jaccard similarity S Jac (a, b) ∈ [0, 1] • pre-process the strings with the cleaning method, • applies the selected string match algorithm to each pair of strings in the training dataset, • train a Decision Stump -i.e. a Decision Tree with a single node -given as input the score just computed, and as label the match/non-match nature of each pair of strings. Random Forest classifier The validity of string similarity algorithms presented so far depends on the use case. Therefore we use a Random Forest classifier which uses the five string match metrics listed in Tab. 1 as input features at the same time. The Random Forest pipeline goes as follows: • pre-process the strings with the same cleaning method as for the Baseline Trees, • for each pair of strings in the training dataset, compute the 5 scores listed in Table 1, • extract 2 additional features from each string: the number of characters and the number of words (i.e. substrings split with respect to whitespaces), • train a Random Forest classifier with 9 features in input (5 matching scores + 2 × number of words + 2 × number of characters), and as label the match/non-match nature of each pair of strings. The popular Scikit-Learn Python library (Pedregosa et al., 2011) is used both for the Decision Stump of the Baseline classifiers and for the Random Forest implementations. The hyperparameters of the Random Forest are set to: max depth=3, n estimators=100, class weight='balanced'. Proposed Approach We propose an approach based on Recurrent Neural Networks (RNNs) employing a Siamese strategy (Bromley et al., 1993), framing the learning problem as a binary classification of string pairs. To keep the format of the input consistent, each string is preprocessed as follows: the strings are padded to a length of 300 chars 2 , using the heavy division sign as placeholder for padding. The string cleaning is in this case limited to the uppercase. The rationale is to leave as much information as possible to the Neural Network model to learn useful patterns. Each string is then tokenised character-wise and one-hot encoded with an alphabet of 63 symbols 3 (i.e. the placeholder plus 62 symbols for capital letters, numbers, punctuation, and whitespace), resulting in a 300 × 63 input matrix. Each input matrix, representing a string, is then processed by an Embedding Model (Figure 1), composed of a 300 × 63 embedding matrix (i.e. a matrix whose entries are learned via loss optimisation) followed by an LSTM layer with 16 nodes. Weights sharing is employed during learning between the encoding model of the two strings to force the two encoding models to be identical (which is indeed the origin of the name "siamese"), in such a way as to preserve the symmetry of the problem and to effectively learn a unique representation space for individual strings. Notice that we don't make use of the full sequence of hidden LSTM states (that would be a 300 × 16 matrix) as output of the LSTM layer -i.e. the embedding representation of individual stringswe instead employ the hidden state corresponding to the last token in the string, that nevertheless implicitly contains information of all the hidden states along the sequence -this is indeed the main feature of RNN models 4 . The two encodings thus generated are then employed to compute several vector distances -namely, L 1 , L 2 , L ∞ , cosine-based distance and the element-wise absolute difference. These results are then fed as inputs to a Feed-Forward Neural Network classifier, composed of two consecutive blocks, each consisting of a dense layer with ReLU activations, batch normalisation, and dropout. The final output is a single neuron with a sigmoid activation function, to get the classification score ( Figure 2). Since every operation performed on the inputs has the commutative property, the whole model C s , composed by the ensamble of the encoding model and the prediction model, has commutative property, then given any two input strings a and b, we have C s (a, b) = C s (b, a) by design. Binary Crossentropy is used as loss function, as usual for binary classification tasks. We employ Nadam as optimisation algorithm, with parameter choice β 1 = 0.8, β 2 = 0.9 and a fixed learning rate = 10 −4 . Both the Embedding model and the downstream Feed-Forward classifier are implemented in Python via TensorFlow (Abadi et al., 2015). The rationale for employing a Siamese approach is that of learning a high-level embedding of strings, where the similarity in the embedding space reflects the probability of being instances of the same entity. Figure 2: Siamese architecture. Two input strings are processed by the same LSTM-based encoding model (see Figure 1) to get a 16-dim vector representation each. These are used to compute several distances: L 1 , L 2 , L∞, cosine and the element-wise absolute difference. This information is then concatenated -obtaining a 20-dim vector -and fed to two consecutive blocks, each composed of a dense layer with ReLU activations, batch normalisation, and dropout. The first block has a dense layer with 32 nodes, while the second block has a dense layer with 16 nodes. The final layer is a single neuron with a sigmoid activation function. Active Learning In an ideal scenario, a predictive model can be built from labelled data in a fully supervised way: generally speaking, increasing the amount of labelled data improves the generalisation capacity of the learned model. However, in a real-world scenario, the number of labelled instances is often limited: the labelling process is often costly, time-consuming and oftentimes requires high-level domain knowledge. On the other hand, unlabelled data are in general much easier to collect and may be used to improve the predictive performance of the model. Formally, in a classification setting, there are: • a set of labelled instances (X l , y l ) (where X l represents features and y l denotes the corresponding labels); • a set of not-labelled-yet instances X u . If the labelling process is cost-effective, one could get the labels y u of not-labelled-yet instances X u , and then use a fully-supervised learning algorithm . In a completely opposite situation, where no additional labels can be collected, or at a prohibitive cost, semi-supervised methods have been proved to be effective (van Engelen and Hoos, 2020). Oftentimes, the situation is in the middle: given the available resources, only a limited number of instances can be labelled. How to effectively exploit these labelling resources is the focus of Active Learning (Settles, 2009;Aggarwal et al., 2014;Ren et al., 2022): given the limited labelling capability, how to choose the subset X c ⊂ X u to label in order to obtain the maximum performance gain? In practice, X c is chosen and constructed according to some query strategies in a multi-step procedure. In our work, we adopt an uncertainty sampling strategy (Settles, 2009) where, at each step, instances of X u on which the prediction of the most updated model is less certain are selected. These data points are removed from X u , labelled by domain experts and then added to (X l , y l ) in order to train an improved version of the classifier, with the rationale that, since the additional data points are picked near the decision boundary instead of being randomly selected, they contain more valuable information for the model learning. Let x i ∈ X u be an unlabelled instance, we denote with (ŷ i,0 , . . . ,ŷ i,ᾱ−1 ) the probabilities estimated by the classifier over theᾱ classes, such that αŷ i,α = 1. Then, we can define the uncertainty as unc(ŷ i ) = 1 − max αŷ i,α .(10) In the case of a binary classification setting (ᾱ = 2), it is equivalent to measuring the distance of the positive class predicted probability from 1 2 : unc(ŷ i ) = 1 2 − 1 2 −ŷ i .(11) At each step of the training process, instances of X u are sorted according to the uncertainty defined above and the top B most uncertain samples are added to X l and removed from X u . A classifier is then trained on the updated version of X l . Empirical evidence during experiments suggests that using directly the uncertainty defined in Equation (10) is sub-optimal in balancing the exploration and exploitation threshold (Banko and Brill, 2001): feeding the learner with only uncertain most samples, especially at the beginning, could result in a batch of data too biased towards difficult instances, leading to poor generalisation capability. To prevent this, and alternatively to the strategy introduced by Banko and Brill (2001) -i.e. using batches composed of half of samples selected randomly and the other half selected on the basis of uncertainty -we propose to use the following noisy version of uncertainty: unc σ (ŷ i ) = unc(ŷ i ) + i , i ∼ N (0, σ),(12) where i are independent and identically distributed normal random variables and σ denotes the level of the noise we would like to introduce. The random noise introduced in Equation (12) helps the learner to generalise better during the first stages of the Active Learning procedure. In Algorithm 1 we formalise the entire procedure described above, while Figure 3 shows an illustrative representation. Algorithm 1: Active Learning Procedure input : X 0 l (initial labelled instances) y 0 l (labels relative to X 0 l ) X 0 u (initial unlabelled instances) M (number of iterations) B 1 , B 2 , . . . , B M −1 (batch sizes) σ (noise for uncertainty) output: a trained classifier C M Train an initial classifier C 1 on (X 0 l , y 0 l ) for j = 1, . . . , M − 1 do // Predict the probabilities over the instances, according to equation ( (11) and (12) uncσ(ŷ) = 1 2 − 1 2 −ŷ + // Select from X j−1 u the B j instances with highest uncσ X j c ← top B j instances with respect to uncσ // Label the selected instances y j c ← domain expert labels relative to X j c // Update the set of labelled instances 3) {ŷ = C j (x) \| x ∈ X j−1 u } // Compute uncσ for each point in X j−1 u according to equationsX j l ← X j−1 l ∪ X j c y j l ← y j−1 l ∪ y j c // Update the set of unlabelled instances X j u ← X j−1 u \ X j c Train a classifier C j+1 on (X j l , y j l ) end Result: C M Data The data employed in our analysis are extracted from two different domains, i.e. company registry and bank transfers. More specifically, the data consist of couples of strings being: 1. the company names (concatenated with the address) of the same entity as recorded in two different datasets obtained by external data providers. 2. the beneficiary names of the same entity as recorded in SWIFT 5 bank transfers. The two sources are used independently, with the first used for training and testing and the second only for testing. Data labelling Each instance of the datasets used in the experiments consists of a pair of names and a binary target variable: we use the label 1 when the pair of names correspond to the same company, and label 0 otherwise. As stated in previous sections, the labelling process is time-consuming since it involves the identification of raw data usually with human annotation. To tackle this task, we adopt a 2-step strategy: we pre-label some couples of names with a rule-based criterion suggesting the target variable (match/non-match), and then we check the suggested labels manually. The rule to pre-label company string pairs is based on the domain of data we are considering. Pre-labelling for company registry data Many companies are identified through the Legal Entity Identifier (LEI): aliases with the same LEI refer to the same company entity. We leverage this background to identify candidate couples with positive labels (same LEI) or with negative labels (different LEI). At the end of this process, these suggested labels are manually validated. We label 9,000 couples of names from the company registry data (with a 1:4 positive/negative label ratio). Figure 4a displays the distribution of JW similarity relative to the 9,000 couples conditioned on the label matching. It is worth noting that the pre-labelling strategy based on the LEI code goes beyond the simple string similarity between company names. Indeed, the LEI code can be used to identify named entities even when they undergo various types of legal transactions, such as mergers, acquisitions, consolidations, purchases and management acquisitions. In this case, the company name can vary after the legal transaction, while still referring to the same entity. Of course, these counter-intuitive positive matches are beyond the range of validity of the methods we are discussing in this workbased solely on the similarity of string names -but we decided to include some of them to test their limits. We discuss some of these examples in Section 7. Pre-labelling strategies for bank transfer In a bank transfer, funds are transferred from the bank account of one entity (the sender or payer) to another bank account (of the beneficiary or recipient). Beneficiaries with the same bank account (IBAN) refer to the same company and can be used to identify candidate couples with a positive label. More challenging is the construction of couples with negative labels, identified by recipients with different IBAN. The reason is that the same company may own more than one IBAN, and this requires a more detailed validation. With this strategy, we label 200 couples from the bank transfers (with a 1:1 positive/negative labels ratio). We keep these data separate from the previously discussed 9,000 pairs, and we use them only for testing purposes, with the rationale of verifying the robustness of our approaches under domain shift scenarios (see Section 5.2). Training and test sets We adopt a stratified k-fold approach to split the data: we split the 9,000 labelled instances into 3 subsets S 1 , S 2 , and S 3 with equal size and taking care to maintain the same positive/negative ratio. At each iteration, we choose one of them as the test set S test and keep the union of the rest as a training set S L train . 13 From S L train we sample 1 3 of its elements to form a medium size training set S M train , and then we again sample 1 20 of instances contained in S M train to obtain a small size training set S S train . Therefore, we end with 3 training sets for each fold: S S train ⊂ S M train ⊂ S L train . These 3 training sets with increasing size are useful to analyse how the different approaches perform in relation to the amount of data they are given to learn from (see experiments described in Section 6 and the corresponding discussion in Section 7). Analogously, we define different test sets: the randomly ordered test set S RO test = S test (i.e. the entire hold-out set available), and the JW ordered test set S JO test obtained by ranking S RO test instances according to their JW similarity 6 and taking the top 100 negative cases (i.e. non-matching pairs that are nevertheless mostly JW-similar) and the bottom 100 positive cases (i.e. matching pairs that are nevertheless mostly JW-dissimilar). The distribution of JW similarity of samples in S JO test conditioned on matching labels is shown in Figure 4b. To test the robustness of the methods presented in Section 3, we introduce a third test set S DS test in addition to S RO test and S JO test , extracted from a different data source. Namely, as anticipated at the beginning of Section 5, we extract pairs of company names from SWIFT bank transfer registry, where transaction payers write the beneficiary without any form of oversight. The dataset is balanced, with 100 positive examples obtained by matching recipients with the same IBAN and 100 negative cases obtained by random matching. We name this dataset the domain shifted test set. We expect the S JO test and S DS test test sets to be particularly challenging for the string matching algorithms given in Section 3: indeed S JO test is, by design, a stress test for the single-feature based methods, and it can be used to estimate how effectively the Random Forest and the Siamese Network are able to generalise with respect to the baselines. The S DS test dataset instead is likely to be challenging for several reasons: it is drawn from a different dataset with respect to the training set; not only the beneficiary names may be affected by typos and all sorts of noise due to the free writing, but they are also not in a standardised form, i.e. they may contain additional information such as the company address 7 . Experiments Different modelling strategies (string distance metrics, Deep Neural Networks, Active Learning) have different (dis)advantages. In order to compare them fairly and point out the best scenario in which to apply each of them, we prepare two different experimental settings: 1. standard supervised classification setting, 2. Active Learning setting. In each of these two settings, we run the experiments employing a 3-fold cross-validation strategy as described in Section 5.2. Supervised classification setting We evaluate how the performances of the models presented in section 3 change as they are trained on training sets of different sizes (S L train , S M train and S S train ). Each of the training sets is used to train each of the following 7 models: • 5 different Baseline Trees (see Section 3), based on Levenstein distance (D Lev ), InDel ratio (R ID ), Token Set Ratio (R TS ), Jaccard similarity (S Jac ), and JW similarity (S JW ); • a Random Forest classifier, introduced in Section 3.2; • our proposed Siamese Network, introduced in Section 3.3. Out-of-sample performance is evaluated on test sets S RO test , S JO test , and S DS test by computing the Balanced Accuracy (BA), thus giving equal weights to positive and negative classes, irrespective of actual class imbalances. More precisely, as argued in Chicco et al. (2021), BA is a good measure -preferable over, e.g. Matthews Correlation Coefficient (MCC) and F 1 score -when the aim is to compare classifiers across datasets with different class imbalances, and/or when the focus is to correctly classify the ground truth instances, which is exactly what we are doing. Table 2 summarises experimental results discussed in Section 7. We leave to Table B.4 in the appendix additional metrics computed for the experiments (F 1 score and MCC). Active Learning setting We then employ an Active Learning strategy -outlined in Section 4 and Algorithm 1 -to train both the Random Forest and the Siamese Neural Network. In our experiments, the initial labelled instances (X 0 l , y 0 l ) consist of 100 samples and correspond to S S train , while X 0 u consista of the residual 5900 couples, namely S L train \ S S train . We fix the σ parameter at 1/6 for all the experiments. batch sizes are set in such a way that all instances are spanned with M = 9 iterations. More formally, for j = 1, 2, . . . , M − 1, the batch size at the j-th iteration is B j =      100 × 2 j−1 j ∈ [1, 4], 800 j = 5, 6, 1400 j > 6. This choice is motivated by the idea of better tracking the impact of the Active Learning approach: indeed, we expect the greater benefits to come in the very first phases, while the marginal benefit after having seen enough data will be negligible. As mentioned in Section 4, the choice of the subset of unlabelled instances to be labelled (X j c ) lies at the heart of the Active Learning strategy. To benchmark this procedure, besides the Least Confident learner (LC) selecting X j c according to the uncertainty (Equation (12)), we run the same experiment with a Random learner (R) picking the unlabelled samples in a purely random fashion. Therefore, we end up with a total of four learners. At the end of each iteration j, we evaluate their performance as follows: 1. pre-train batch test: we test the model C j on the next-to-be-labelled instances, i.e. X j c : we here expect poor results for the LC learners, since we are testing on most uncertain samples for the model C j (see Figure 5a). 2. We train with respect to the new training set (X j l , y j l ), thus obtaining C j+1 . 3. We test C j+1 on: • the batch samples X j c , and we refer to it as the post-train batch test (see Figure 5b): we here expect good results, since it is an in-sample valuation; • the updated unlabelled set X j u , i.e. all the remaining unlabelled instances, and we refer to it as the not-labelled-yet test (see Figure 5c); • the actual test set, namely S RO test (see Figure 5d). Table 2 summarises the results of the experiments in terms of BA. The following is a list of insights we can derive from its inspection. The Random Forest model trained on the small dataset S S train has a good performance on the randomly ordered test set S RO test . This is likely due to the fact that the input features -described in Section 3.2 -are essentially string similarity measures, thus the pattern to be learned does not need a lot of observations. The downside is poor generalisation when more data are provided. Indeed, by increasing the training set size from S S train to S M train , less than 2% of BA is gained on the randomly ordered test set and no gain at all from S M train to S L train . Moreover, its performance drops by ∼ 27% and ∼ 24% in the JW ordered test and the domain shifted test, respectively. Discussion The same thing is true -even more so -for the JW based classifier. In this case, the decision tree needs only to perform an optimal choice of the threshold to put on the JW similarity score. We can expect that the increase in size of the training set is only slightly changing this threshold, with a negligible impact on the out-of-sample performances. Then we can deduce that JW metric does not tend to overfit his domain and can generalise with acceptable performances. The same reasoning can be applied to the other string match models. In particular, the InDel ratio and the Token Set Ratio perform remarkably well with a small amount of data, again due to the simple rule to be learned by the classifier. Concerning the JW model, we can observe a poorer generalisation to new domains. It is worth noticing that the drop in performance for the Baseline classifiers when switching from S RO test to S JO test and S DS test is slightly larger than for the Random Forest. This is reasonable given that the Random Forest may use the information coming from all of the similarity metric scores at the same time. The Siamese model systematically outperforms other approaches on both medium and large training datasets. This demonstrates the ability of the Neural Network to avoid overfitting to the specific domain, to generalise across different distributions, and to learn an alias associated with a company that may differ significantly from a simple string match similarity (e.g. the pair "REF SRL" and "RENOVARE ENERGY FARM SRL"). Table 3 shows several examples of matching and non-matching couples of company string names as they are classified by the Siamese model trained on S L train , with the corresponding estimated probabilityŷ. In particular, the examples in Table 3 are extracted by drawing from non-matching couples with high JW similarity, and from matching couples with low JW similarity. In this way, we expect to highlight some interesting and challenging sample couples. Indeed, one can see that the Siamese model is able to correctly classify company names expressed as acronyms (e.g. in "S.P.I.G.A." and "REF"). On the other hand, there are cases more difficult to explain, such as the correctly classified match for the couple ("RONDA", "TORO ASSICURAZIONI SPA") where the entity is indeed the same -TORO insurance actually merged into RONDA in 2004 8but the company names are completely different. Further work is needed to explain the reasons behind such counter-intuitive matches of the Neural Network, and to find the patterns behind such classifications. We reported false positive examples where the names are very similar but they actually belong to different companies, e.g. "E.U.R.O. S.R.L." and "EURO STEEL SRL/MILANO". The overconfidence of the prediction could be solved, e.g. by incorporating additional information, such as address, holding and legal form of the two companies. As expected, most of the false negative samples -i.e. pairs representing the same entity but predicted to be non-matching -in Table 3 can be related to situations in which the company names are (almost) completely different, but the entity is indeed the same, likely due to some legal transaction (merger, acquisition, consolidation, etc.) as discussed in Section 5.1. Finally, the true negative examples show the remarkable capabilities of the Siamese model to correctly classify as different entities even pairs with very similar company names, such as "RECOS S.R.L." and "PECOS SRL", with very high confidence. Figure 5d shows the out-of-sample BA of the Siamese Network and the Random Forest when computed on S RO test at each Active Learning step. We can easily see that the Least Confident approach systematically outperforms the random choice for both models, confirming the value of Active Learning. However, while the Random Forest performance plateaus already before 400 samples, the Siamese needs up to 2,000: the Siamese Network has to leverage a larger amount of data to effectively learn patterns. Figures 5a-5b can help us understand the mechanisms at play. Figure 5a displays out-of-sample BA values on next-to-be-labelled batches: we expect poor performance for the Least Confident choice with respect to Random choice since in the former case we are selecting uncertain instances (i.e. difficult for the model) on purpose. Figure 5b, on the other hand, displays BA values on batches just fed to the models: we here expect -in general -higher performances, being an in-sample evaluation. Interestingly, we see that the Least Confident choice has poor performance with respect to Random choice. This may be due to the fact that (at least a fraction of the) most uncertain observations remain indeed intrinsically difficult to classify, despite the training. This effect is more pronounced for the Random Forest, likely because it is largely based on similarity metrics. Notice that, as the number of samples seen by the model increases, this effect is less and less pronounced and finally reverts, indicating that residual samples are becoming easier to classify in the Least Confident approach. Incidentally, we notice that comparing the performances of the model on a batch before and after the model has been trained on it, allows to define an early stopping rule. Namely, calling Acc pre and Acc post the BA of a model over a batch before and after it has been trained on it, respectively, we can define a threshold θ and interrupt the process when: \|Acc post − Acc pre \| < θ. The rationale is that, if a model has about the same performances on a batch before and after having been trained on it, it means that it has already learned to generalise over unseen data. This rule can be applied, with an appropriate θ, both on the Least Confident and Random algorithm. Finally, Figure 5c shows the out-of-sample BA when computed on X u (i.e. all the unlabelled samples of the training set at that step) at each step of Algorithm 1. This plot confirms that the Least Confident approach chooses the instances in such a way that the remaining ones are simpler to be classified. This is especially true for Random Forest. Conclusions In our analysis, we have compared the performances of several supervised classifiers in the field of company name disambiguation.Providing pairs of company names as input, we consider two types of Machine Learning classifiers: Decision Stumps and Random Forest classifiers based on classical string similarity measures between the two names; a Neural Network classifier on top of a learned LSTM embedding space of strings. The data are extracted from external company registry data and bank transfers. More specifically, we collect 9,000 couples of company names from external company registry data, and 200 couples of beneficiary names in bank transfers (the latter used only for testing purposes). All approaches are evaluated over three different test sets: a "randomly ordered" test set (RO), i.e. 3000 samples randomly chosen from the company registry dataset, a "Jaro-Winkler ordered" test set (JO), i.e. 200 instances drawn from RO in such a way that we select JW-dissimilar actual matches and JW-similar actual non-matches, and a "domain shift" test set (DS), i.e. 200 couples of beneficiary names taken from a (different) dataset of bank transfers. The purpose of this work is twofold: on the one hand, we show that if enough data is available, the Siamese approach outperforms the other models and can be applied to other domains. Indeed, according to Table 2, the performance of the Baseline methods and the Random Forest barely improves when more data are provided for training. This is likely due to the fact that all the information extracted from string pairs is encoded in classical string similarity metrics for Baseline Trees and Random Forest. On the contrary, increasing the size of the training set improves the performance of the Siamese model, as it can learn a more effective embedding space the more data it learns from. This demonstrates the Neural Network's capacity to generalise while avoiding overfitting to a specific domain. These features enable our Siamese Neural Network to learn its own concept of string similarity and appropriately detect aliases connected with a company that differ significantly from a basic string match similarity. The other goal of our research is to demonstrate how to make human labelling more efficient by using an Active Learning strategy. Indeed, starting with a minimal training set of 100 labelled data, we show that training the model with subsequent batches of the most uncertain samples (Least Confident learner) is more efficient than training with randomly chosen instances. One limitation of this work is the use of company names only for the goal of disambiguation. As previously said, it is possible that company names by themselves do not contain enough information to resolve all the matches, as in cases where completely different company names still refer to the same Entity. We leave to future work the extension to include additional information, such as addresses, legal forms, shareholding, etc. We also plan to perform a more thorough analysis of the architecture of the Siamese Recurrent Network, possibly using a Transformer approach for the input sequences (Vaswani et al., 2017). Given the effectiveness of the Active Learning procedure, we plan to use an unlabelled dataset of hundreds of thousands of Entity pairs to extract from them the most informative few thousand pairs to label, to analyse how the procedure here outlined scales with data availability. As can be seen from Figure A.6, the number of characters and the number of words of the two company names have negligible importance on the classifier. In particular, these features become less and less significant as more data are supplied for training. JW similarity is the most important feature used by the Random Forest classifier: the Random Forest trained on S S train attributes 40% of importance to JW similarity, and this further increases as the classifier is trained on more data, reaching almost 50% for the classifier trained on S L train . The importance of the remaining features does not change much with the training set size, with a contribution of ∼ 30% for the Token Set Ratio, ∼ 20% for the InDel ratio, ∼ 4% for the Levensthein, and ∼ 1% for the Jaccard. Author contributions Appendix B. Supervised classification performances As discussed in Section 6, Table 2 shows the out-of-sample BA on the test sets S RO test , S JO test , and S DS test . The BA ensures a correct evaluation of the classifiers' performance by weighting the accuracy of the classifier on each class by the number of observations in that class. This gives a more accurate picture of the classifier's performance in each class, even when the classes are imbalanced. Along with BA, other metrics are commonly used to assess the quality of a model; here we will discuss F 1 score and MCC. F 1 score is a very popular metric in the Machine Learning community since it combines precision and recall -often two paramount aspects for model evaluationinto a single metric. Indeed, it is defined as the weighted average of precision and recall. MCC is instead a measure of the correlation between predicted and actual binary class labels and it ranges between -1 -indicating a total disagreement between prediction and observation -and 1, indicating a perfect prediction. While the F 1 score and MCC consider both precision and recall, BA is the average of the recall over all classes. Therefore, also according to Chicco et al. (2021), BA may be preferable when correctly classifying the ground truth instances (recall) is more important then making correct predictions (precision). For completeness, in Table B.4 we extend the results of Table 2 by including the F 1 score and the MCC. Results from these additional metrics largely confirm the ones discussed for BA, namely the fact that the Siamese Network approach systematically outperforms all other methods as long as enough data are provided for learning. This is especially true for the most difficult test sets, namely Table 2. Balanced Accuracy (BA), F 1 -score and Matthews correlation coefficient (MCC) for the Random Forest, Siamese Neural Network, and for the 5 single-distance Decision Trees, all trained with datasets of different sizes (small, medium, large) and tested on the three test sets discussed in Section 5.2, namely randomly orderd (S RO test ), JW-ordered (S JO test ) and domain shifted (S DS test ). Mean values and corresponding standard deviation (in brackets) computed via stratified k-fold cross-validation approach are reported for all metrics. Contrary to the 5 baseline models, the Random Forest and the Siamese Neural Network performances improve with the size of the training set. Moreover, the Siamese Neural Network trained on medium and large datasets outperforms all other approaches. For each of the three metrics, bold figures are row-wise maximum values. Figure 1 : 1Embedding model. Schematic TensorFlow representation of the Embedding model described in Section 3.3. Each block denotes a TensorFlow layer, with input and output tensor dimensions. As usual in TensorFlow representations, the generic batch size is denoted with the symbol '?'. Figure 3 : 3Active Learning. Illustrative representation of the Active Learning procedure outlined in Algorithm 1. Distribution of similarity over all string pairs. (b) Distribution of similarity on JW-ordered test set. Figure 4 : 4Similarity distribution over string pairs. Distribution of JW similarity conditioned on the label matching over (a) the 9,000 labelled instances and (b) JW-ordered test set. The continuous score is binned with size 0.05. In (a) the histogram shows that the negative (i.e. non-matching) samples are roughly normally distributed around 0.55, while the majority of positive (i.e. matching) samples have JW similarity between 0.8 and 1. In (b), on the contrary, the distributions of positive and negative pairs are largely overlapping, making it difficult to determine a threshold between the two classes, in particular with respect to the whole dataset. JO 0 . 0 00473 ± 0.038 0.488 ± 0.029 0.652 ± 0.003 0.463 ± 0.064 0.675 ± 0.013 0.697 ± 0.013 0.867 ± 0.019 DS 0.535 ± 0.009 0.735 ± 0.009 0.743 ± Learning: pre-train batch test. (b) Active Learning: post-train batch test.(c) Active Learning: not-yet-labelled test.(d) Active Learning: randomly ordered test. Figure 5 : 5Active Learning performances. BA during the Active Learning procedure computed over: (a) next-to-be-labelled instances (X d) the test set S RO test , as described in Section 6.2. On the x-axis we plot the number of labelled samples (log scale), starting from \|X 0 l \| = 100 up to \|X M −2 l \| = 4600 -i.e. before adding the remaining B M −1 samples in the last iteration. In the post-train case (b), the evaluation starts from \|X 1 l \| = 200 (i.e. after adding the first B 1 samples) up to the whole training dataset \|X M −1 l \| = 6000. The evaluation over S RO test (d) is done in all available steps, i.e. from 100 up to 6000 samples. BA of Random Forest and Siamese Network models both as Random learners and as Least Confident learners is reported. The mean and 95% normal confidence intervals are obtained by aggregating the BA over the 3 cross-validation folds. Table 1 : 1List of metrics used as features in the Baseline and Random Forest classifier. Thetraining set size test set type Levenshtein InDel Ratio Token Set Ratio Jaccard JW Random Forest Siamese Network small RO 0.665 ± 0.045 0.855 ± 0.025 0.935 ± 0.005 0.577 ± 0.09 0.957 ± 0.003 0.951 ± 0.017 0.892 ± 0.042 JO 0.428 ± 0.061 0.505 ± 0.005 0.662 ± 0.006 0.438 ± 0.06 0.678 ± 0.016 0.678 ± 0.018 0.725 ± 0.044 DS 0.582 ± 0.05 0.717 ± 0.01 0.723 ± 0.015 0.533 ± 0.058 0.717 ± 0.003 0.718 ± 0.02 0.735 ± 0.013 medium RO 0.643 ± 0.014 0.878 ± 0.013 0.944 ± 0.007 0.523 ± 0.041 0.957 ± 0.003 0.965 ± 0.006 0.975 ± 0.002 Table 2 : 2Experimental results. via stratified k-fold cross-validation are displayed. Contrary to the 5 baseline models, the performances of the Random Forest and the Siamese Neural Network improve with the size of the training set. Moreover, the Siamese Neural Network trained on medium and large datasets outperforms the other approaches. Bold figures denote row-wise maximum values.Balanced Accuracy of the Random Forest, Siamese Neural Network, and the 5 single- distance Decision Trees, all trained with datasets of different sizes (small, medium, large) and tested on the three test sets discussed in Section 5.2, namely randomly ordered (S RO test ), JW-ordered (S JO test ) and domain shifted (S DS test ). Mean and standard deviation computed Riccardo Crupi: Conceptualisation, Methodology, Software, Validation, Investigation, Writing -Original Draft, Project Administration. Michele Grasso: Software, Validation, Investigation, Writing -Original Draft. Daniele Regoli: Methodology, Investigation, Writing -Original Draft. Shuyi Yang: Methodology, Software, Investigation, Writing -Review & Editing. Simone Scarsi: Methodology, Software, Investigation, Writing -Review & Editing. Alessandro Mercanti: Data Curation, Writing -Review & Editing. Alessandro Basile: Writing -Review & Editing. Andrea Cosentini: Writing -Review & Editing, Supervision. positives (ŷ ≥ 0.5) Predicted negatives (ŷ < 0.5)Predicted y company name pairs ( a b ) company name pairs ( a b )ŷ same entity (y = 1) 0.97 S.P.I.G.A. S.R.L. FOGLIATA S.P.A. 0.006 SPIGA-SOCIETA' PRODUZIONE E IMPORTAZIONE GENERALI ALIMENTARI-SRL EDILCOS SRL 0.69 REF SRL CREARE IN FOSSATO SRL IN LIQUIDAZIONE 0.001 RENOVARE ENERGY FARM SRL WALD SRL 0.94 RONDA UNIEURO S.P.A. 0.002 TORO ASSICURAZIONI SPA SGM DISTRIBUZIONE SRL 0.88 SEAWAYS MANUFATTI EPIS SRL 0.02 AGENZIA MARITTIMA LE NAVI-SEAWAYS SRL EPIS SANTINO SNC DI EPIS STEFANO EC 0.77 TOSCOSERVICE LOGSTICA E SEVIZI SOCIETA' COOPERATIVA DEA S.R.L. 0.001 IL GIRASOLE 2002-SC UNENDO ENERGIA SUD SRL 0.99 ARGO S.R.L. GIULIANO VINCENZA 0.05 ARGO SAS DI DI BIAGI FILIPPO G&P SRL different entity (y = 0) 0.99 E.U.R.O. S.R.L. EMMA SRL 0.07 EURO STEEL SRL/MILANO GEMMA SRL 0.91 G.T.M. S.R.L. SMV COSTRUZIONI SRL 0.001 GMC-SRL RM COSTRUZIONI SRL 0.56 MARPER S.R.L. RECOS S.R.L. 0.002 MAVIT SRL PECOS SRL 0.71 MEA S.R.L. TEXERA SRL 0.2 MAR -ALEO SRL TER SRL 0.88 NUOVA REKORD S.R.L. MEPRA SPA 0.02 NOVA LUX SRL/VERONA PELMA SPA 0.99 FARMACIA WAGNER ALIMA SRL 0.08 FARMACONSULT SRL DALMA SRL Table 3 : 3Representative examples. Illustrative examples of results of the Siamese Neural Network trained on the large training set S L train . Predicted matches/non-matches are shown (together with the positive class predicted probabilityŷ) for actual matching and non-matching pairs. The examples ( a b ) are selected from the randomly ordered test set S RO test (i.e. from the external company registry datasets). The specific model used to obtain these examples incorrectly classifies the 1.17% of the positive cases and the 0.96% of the negative cases tested.deviation, respectively. S JO test and S DS test .Table B.4: Experimental results. Extension of25 training set size test set type Levenshtein InDel Ratio Token Set Ratio Jaccard JW Random Forest Siamese Network BA F 1 MCC BA F 1 MCC BA F 1 MCC BA F 1 MCC BA F 1 MCC BA F 1 MCC BA F 1 MCC small RO 0.665 0.455 0.387 0.855 0.819 0.794 0.935 0.923 0.907 0.577 0.22 0.187 0.957 0.951 0.94 0.951 0.943 0.931 0.892 0.771 0.719 (0.045) (0.074) (0.038) (0.025) (0.03) (0.027) (0.005) (0.007) (0.009) (0.09) (0.229) (0.162) (0.003) (0.002) (0.003) (0.017) (0.019) (0.021) (0.042) (0.027) (0.036) JO 0.428 0.177 −0.174 0.505 0.285 0.013 0.662 0.539 0.385 0.438 0.14 −0.163 0.678 0.587 0.399 0.678 0.585 0.401 0.725 0.697 0.461 (0.061) (0.161) (0.102) (0.005) (0.082) (0.012) (0.006) (0.019) (0.036) (0.06) (0.228) (0.141) (0.016) (0.009) (0.046) (0.018) (0.069) (0.013) (0.044) (0.092) (0.068) DS 0.582 0.295 0.262 0.717 0.61 0.517 0.723 0.62 0.531 0.533 0.143 0.083 0.717 0.605 0.526 0.718 0.611 0.523 0.735 0.693 0.491 (0.05) (0.175) (0.089) (0.01) (0.02) (0.016) (0.015) (0.031) (0.021) (0.058) (0.247) (0.144) (0.003) (0.006) (0.004) (0.02) (0.038) (0.035) (0.013) (0.016) (0.041) medium RO 0.643 0.444 0.475 0.878 0.836 0.803 0.944 0.928 0.911 0.523 0.086 0.088 0.957 0.95 0.94 0.965 0.953 0.941 0.975 0.955 0.944 (0.014) (0.034) (0.018) (0.013) (0.014) (0.018) (0.007) (0.008) (0.01) (0.041) (0.149) (0.153) (0.003) (0.003) (0.004) (0.006) (0.011) (0.013) (0.002) (0.005) (0.007) JO 0.473 0.006 −0.133 0.488 0.376 −0.023 0.652 0.558 0.335 0.463 0.021 −0.103 0.675 0.586 0.388 0.697 0.655 0.405 0.867 0.855 0.743 (0.038) (0.01) (0.108) (0.029) (0.021) (0.061) (0.003) (0.002) (0.007) (0.064) (0.036) (0.178) (0.013) (0.01) (0.035) (0.013) (0.011) (0.028) (0.019) (0.025) (0.033) DS 0.535 0.131 0.19 0.735 0.646 0.544 0.743 0.659 0.559 0.5 0.0 0.0 0.715 0.601 0.523 0.743 0.659 0.559 0.773 0.712 0.604 (0.009) (0.03) (0.024) (0.009) (0.017) (0.012) (0.003) (0.005) (0.005) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.008) (0.014) (0.012) (0.016) (0.024) (0.029) large RO 0.641 0.439 0.482 0.871 0.834 0.805 0.944 0.928 0.911 0.523 0.085 0.089 0.956 0.951 0.94 0.967 0.958 0.948 0.976 0.959 0.949 (0.012) (0.029) (0.025) (0.014) (0.015) (0.016) (0.007) (0.008) (0.01) (0.04) (0.147) (0.153) (0.001) (0.003) (0.003) (0.004) (0.008) (0.01) (0.002) (0.002) (0.003) JO 0.495 0.0 −0.071 0.5 0.352 0.0 0.652 0.558 0.335 0.465 0.021 −0.099 0.687 0.589 0.424 0.72 0.671 0.462 0.903 0.896 0.812 (0.0) (0.0) (0.0) (0.0) (0.04) (0.0) (0.003) (0.002) (0.007) (0.061) (0.036) (0.172) (0.006) (0.014) (0.007) (0.023) (0.022) (0.052) (0.051) (0.059) (0.095) DS 0.53 0.113 0.176 0.733 0.641 0.544 0.743 0.659 0.559 0.5 0.0 0.0 0.715 0.601 0.523 0.743 0.659 0.559 0.777 0.717 0.61 (0.0) (0.0) (0.0) (0.012) (0.021) (0.018) (0.003) (0.005) (0.005) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.003) (0.005) (0.005) (0.01) (0.018) (0.016) The longest string in our whole dataset has 124 chars. If longer strings are to be fed to the model, then a truncation to 300 is implemented.3 Another possible approach is using the entire word as a single token. Experiments using this approach resulted in poor performances, because the model cannot properly handle cases of spelling errors or abbreviations We experimented the same architecture with a Bidirectional LSTM layer in place of a plain LSTM layer, but without any sign of improvement in performance, while on the other hand, the computational cost increased significantly. Society for Worldwide Interbank Financial Telecommunications. As reported in the experiments (seeTable 2), the JW method performs better compared to the other baseline methodologies. Therefore, selecting test instances based on this metric is a way to check the robustness of ML approaches. This motivates the use of names and address in the data extracted from company registry data. gazzettaufficiale (GU Parte Seconda n.83 del 8-4-2004). AcknowledgementsWe would like to thank Ilaria Penco for her assistance with the legal aspects of the manuscript. We thank Andrea Barral for useful discussions on methodology and data curation. We also thank Giacomo Di Prinzio, Giulia Genta and Nives Visentin from Intesa Sanpaolo, and Sandro Bellu, 20 Indrit Gjonaj, Andrea Giordano and Gabriele Pellegrinetti from Tecnet Dati s.r.l., namely the team that developed the disambiguation project for Intesa Sanpaolo, that inspired the research here presented.Appendix A. Ranfom Forest feature importanceIn this appendix, we analyse the importance of the features used by the Random Forest classifier. Decision trees are used to divide data into relevant categories based on optimal splits. As a measure of how far the model deviates from a pure division, Gini impurity calculates the likelihood that a randomly selected example will be erroneously categorised by a certain node. 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[ "The role of voting intention in public opinion polarization", "The role of voting intention in public opinion polarization" ]	[ "Federico Vazquez ", "Nicolas Saintier ", "Juan Pablo Pinasco ", "\nFCEN\nInstituto de Cálculo\nUniversidad de Buenos Aires and CONICET\nBuenos AiresArgentina\n", "\nDepartamento de Matemática\nand IMAS, UBA-CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (\n1428) Pabellón I -Ciudad Universitaria -Buenos Aires-Argentina\n" ]	[ "FCEN\nInstituto de Cálculo\nUniversidad de Buenos Aires and CONICET\nBuenos AiresArgentina", "Departamento de Matemática\nand IMAS, UBA-CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (\n1428) Pabellón I -Ciudad Universitaria -Buenos Aires-Argentina" ]	[]	We introduce and study a simple model for the dynamics of voting intention in a population of agents that have to choose between two candidates. The level of indecision of a given agent is modeled by its propensity to vote for one of the two alternatives, represented by a variable p ∈ [0, 1]. When an agent i interacts with another agent j with propensity pj, then i either increases its propensity pi by h with probability Pij = ωpi + (1 − ω)pj, or decreases pi by h with probability 1 − Pij , where h is a fixed step. We analyze the system by a rate equation approach and contrast the results with Monte Carlo simulations. We found that the dynamics of propensities depends on the weight ω that an agent assigns to its own propensity. When all the weight is assigned to the interacting partner (ω = 0), agents' propensities are quickly driven to one of the extreme values p = 0 or p = 1, until an extremist absorbing consensus is achieved. However, for ω > 0 the system first reaches a quasi-stationary state of symmetric polarization where the distribution of propensities has the shape of an inverted Gaussian with a minimum at the center p = 1/2 and two maxima at the extreme values p = 0, 1, until the symmetry is broken and the system is driven to an extremist consensus. A linear stability analysis shows that the lifetime of the polarized state, estimated by the mean consensus time τ , diverges as τ ∼ (1 − ω) −2 ln N when ω approaches 1, where N is the system size. Finally, a continuous approximation allows to derive a transport equation whose convection term is compatible with a drift of particles from the center towards the extremes.	10.1103/physreve.101.012101	[ "https://export.arxiv.org/pdf/1909.07092v1.pdf" ]	202,577,357	1909.07092	446ad9c3a621c832eca7892e9d6f267464c505c1	The role of voting intention in public opinion polarization 16 Sep 2019 Federico Vazquez Nicolas Saintier Juan Pablo Pinasco FCEN Instituto de Cálculo Universidad de Buenos Aires and CONICET Buenos AiresArgentina Departamento de Matemática and IMAS, UBA-CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires ( 1428) Pabellón I -Ciudad Universitaria -Buenos Aires-Argentina The role of voting intention in public opinion polarization 16 Sep 2019(Dated: January 18, 2022) We introduce and study a simple model for the dynamics of voting intention in a population of agents that have to choose between two candidates. The level of indecision of a given agent is modeled by its propensity to vote for one of the two alternatives, represented by a variable p ∈ [0, 1]. When an agent i interacts with another agent j with propensity pj, then i either increases its propensity pi by h with probability Pij = ωpi + (1 − ω)pj, or decreases pi by h with probability 1 − Pij , where h is a fixed step. We analyze the system by a rate equation approach and contrast the results with Monte Carlo simulations. We found that the dynamics of propensities depends on the weight ω that an agent assigns to its own propensity. When all the weight is assigned to the interacting partner (ω = 0), agents' propensities are quickly driven to one of the extreme values p = 0 or p = 1, until an extremist absorbing consensus is achieved. However, for ω > 0 the system first reaches a quasi-stationary state of symmetric polarization where the distribution of propensities has the shape of an inverted Gaussian with a minimum at the center p = 1/2 and two maxima at the extreme values p = 0, 1, until the symmetry is broken and the system is driven to an extremist consensus. A linear stability analysis shows that the lifetime of the polarized state, estimated by the mean consensus time τ , diverges as τ ∼ (1 − ω) −2 ln N when ω approaches 1, where N is the system size. Finally, a continuous approximation allows to derive a transport equation whose convection term is compatible with a drift of particles from the center towards the extremes. I. INTRODUCTION Political bi-polarization is a widespread phenomenon that generates divisions in a society, and even clashes and revolts. Several works try to explain and model the emergence of polarization using different mechanisms, like negative influence between individuals of antagonistic opinion groups or between members of the same group [1][2][3][4], or a confirmation bias, by which individuals tend to search for information that affirms their prior believes and discard arguments that confront their opinions [5][6][7]. More recently, it has been proposed a new alternative mechanism that gives rise to bi-polarization, which combines homophily with the theory of persuasive arguments [8][9][10]. The idea is that homophily increases interactions between individuals with the same opinion orientation who then persuade each other with arguments that support their opinion tendency, reinforcing their initial positions and becoming more extreme in their believes (see [11][12][13][14]). The model studied in [11] assumes that each agent has a list with a number of pro and con arguments in favor and against a given issue (e.g. marijuana legalization), respectively. Agents interact by pairs and incorporate in their list of arguments one of its partner's argument chosen at random, while old arguments are dis- * Electronic address: [email protected] † URL: https://fedevazmin.wordpress.com ‡ Electronic address: [email protected] § Electronic address: [email protected] missed. Within the context of voting intention, it is natural to think that the number of arguments in favor of a given candidate is proportional to the inclination or propensity to vote for that candidate prior to the elections. In this article we study the dynamics of propensities in a population of voters that have to decide between two candidates A and B. When two agents meet, the first agent asks its partner about its voting intention, whose answer (A or B) depends on its propensity for that candidate. For simplicity we assume that the second agent answers A with a probability equal to the fraction of its arguments in favor of candidate A, that is, its propensity for A (and equivalently for B). If the answer is A, then the first agent increases its propensity and thus becomes more prone to voter for A. Otherwise, if the answer is B the propensity of the first agent is decreased and becomes more prone to B. This model is simple enough to be analytically tractable and is able to induce polarization without relying on the previous mentioned mechanisms, and by implementing a pairwise interaction rule that is independent on the opinion orientation, unlike the models studied in [13,14]. The article is organized as follows. In Section II we introduce the model and its dynamics. In Section III we derive the rate equations and obtain their steady state solutions. We present simulation results in section IV. An stability analysis of the rate equations is performed in sections V and VI. In section VII we develop a continuous approximation that allows to derive a transport equation. We conclude in section VIII with a short summary and a discussion of the results. II. DESCRIPTION OF THE MODEL We consider a population of N interacting voters that have to choose between two candidates, A or B. During the period previous to the election day, agents have some degree of indecision about the two possible alternatives, which is modeled by assuming that each agent has an inclination or propensity p to vote for candidate A that varies between 0 and 1, so that when p ≃ 0 (p ≃ 1) the agent is prone to choose B (A). We assume that the propensity of each agent evolves under the influence of the other agents, in such a way that agents update their propensities after pairwise interactions. Initially, the propensities of all agents are uniformly distributed in [0,1]. Then, at each time step ∆t = 1/N of the dynamics, two agents i and j are chosen at random to interact. Agent j tells its partner i that is going to vote for A with a probability equal to its propensity p j . Then, agent i can either increase or decrease its propensity with probabilities that depend on its own and its partner's propensity: p i (t + 1/N ) = p i (t) + h with probability P i,j , p i (t) − h with probability 1 − P i,j ,(1) where P i,j = ω p i (t) + (1 − ω) p j (t).(2) The step length h (0 < h ≤ 1) is fixed, while the parameter ω (0 ≤ ω ≤ 1) is the weight that gives each agent to its own propensity in an interaction. The value of p i is set to 1 (0) when it becomes larger (smaller) than 1 (0), so that propensities are constrained to the interval [0, 1]. This time step is repeated ad infinitum. III. RATE EQUATIONS AND STATIONARY STATES At time t = 0 the distribution of propensities f (p, 0) is uniform in [0, 1], but after a time of order 1 agents' propensities adopt discrete values p = 0, h, 2h, .., 1. For the sake of simplicity we can take h such that S ≡ 1/h is an integer number, and thus the propensity adopts discrete values p = kh, with k = 0, .., S. Then, the propensity distribution can be written as f (p, t) = S k=0 n k (t) δ(p − kh), where we define n k (t) as the fraction of agents in state k (with propensity kh) at time t, where S k=0 n k (t) = 1 for all times t ≥ 0. Then, the evolution of the system is described by the following rate equations: dn 0 dt = [1 − ωh − (1 − ω)m)] n 1 − (1 − ω)m n 0 , (3a) dn k dt = [ωh(k − 1) + (1 − ω)m] n k−1 − n k + 1 − [ωh(k + 1) + (1 − ω)m] n k+1 (3b) for 1 ≤ k ≤ S − 1, dn S dt = [ω(1 − h) + (1 − ω)m] n S−1 − (1 − ω)(1 − m)n S ,(3c) where m ≡ p = h S k=0 k n k is the mean value of the propensity. The first (gain) term in Eq. (3b) corresponds to the transition of particles from state k − 1 to state k, which happens with probability ωh(k − 1) + (1 − ω)hk ′ when they interact with another particle in state k ′ . Adding over all k ′ values leads to m. The second (loss) term represents k → k − 1 and k → k + 1 transitions, which happen with probability 1 after interacting with any other particle. Finally, the third (gain) term is analogous to the first term, where particles switch from state k+1 to state k with probability 1−[ωh(k+1)+(1−ω)hk ′ ] when they interact with a k ′ -particle. We are interested in the stationary distributions of propensities in the population, which correspond to the fixed points of the system of equations (3). On the one hand, we can first notice that the two consensus states in the extreme propensity values p = 0 (n * 0 = 1, n * k = 0 for k = 1, .., S) and p = 1 (n * S = 1, n * k = 0 for k = 0, .., S −1) are fixed points of Eqs. (3) that correspond to the two absorbing states of the particle system, where the mean propensities are m = 0 and m = 1, respectively. On the other hand, we show in Appendix A that for a given mean propensity m < 1 the system of Eqs. (3) has non-trivial fixed points given by n * 0 = 1 1 + S k=1 G k (h, ω, m) , n * k = n * 0 G k (h, ω, m) for 1 ≤ k ≤ S,(4) with G k (h, ω, m) = Π k−1 j=0 [(1 − ω)m + jωh] Π k j=1 [1 − (1 − ω)m − jωh] ,(5) or G k (h, ω, m) = Γ (1−ω)m ωh + k Γ 1−(1−ω)m ωh − k Γ (1−ω)m ωh Γ 1−(1−ω)m ωh(6) using the gamma functions for m > 0 (see Appendix A), and where the mean propensity must satisfy the relation m + S k=1 (m − kh) G k (h, ω, m) = 0.(7) Notice that for m = 0 we have from Eq. (5) that G k (h, ω, 0) = 0 for 1 ≤ k ≤ S due to the j = 0 term in the numerator, and thus we recover the consensus solution n * 0 = 1. Note also that Eq. (5) is not valid when m = 1 because the denominator equals 0 due to the term j = S = 1/h. We now look for non-consensus stationary states m = 0, 1. It is instructive to start analyzing the simplest cases S = 1 and S = 2. For S = 1 (h = 1) is G 1 (1, ω, m) = m/(1 − m) from Eq. (5), and thus Eq. (7) m + (m − 1) G 1 (1, ω, m) = 0(8) is satisfied for all values of m, that is, each m is a stationary value. This is because for S = 1 the propensity model turns to be equivalent to the voter model [15,16], where it is known that the fractions of voters in each state are conserved, i e., n 0 (t) = n 0 (0) and n 1 (t) = n 1 (0) = 1 − n 0 (0) for all t ≥ 0. Then, the stationary value of m is equal to its initial value m(0) = n 1 (0) given by the initial propensity distribution. For S = 2 (h = 1/2), Eq. (7) for m is m + (m − 1/2) G 1 (h, ω, m) + (m − 1) G 2 (h, ω, m) = 0.(9) Replacing in Eq. (9) the expressions for G 1 and G 2 G 1 (h, ω, m) = (1 − ω)m 1 − ω/2 − (1 − ω)m and (10a) G 2 (h, ω, m) = m [(1 − ω)m + ω/2] (1 − m) [1 − ω/2 − (1 − ω)m] ,(10b) we arrive, after doing some algebra, to the simple relation (1 − ω)m(1 − m)(1 − 2m) = 0. Therefore, besides the extreme consensus states m * = 0 and m * = 1, we obtain the new stationary solution m * = 1/2. For general S, the possible stationary values of m are given by the solutions of Eq. (7), which are the roots of a polynomial of order S + 1. We were unable to find the roots of that polynomial analytically for any S ≥ 3. However, we have verified numerically for many different values of ω and h that the only real roots are m * = 0, m * = 1 and m * = 1/2, as in the case S = 2. We now analyze the non-trivial solution m * = 1/2 that corresponds, as we shall see bellow, to a stationary distribution of propensities f * (p) that is symmetric and polarized around p = 1/2. For m = 1/2, Eqs. (4) and (6) become where the subindex P in Eqs. (11) stands for polarization. This series solution Eqs. (11) is plotted in Fig. 1 for S = 10 (h = 0.1) and different values of ω (filled symbols), while the inset shows the behavior for two values of h (filled circles). We observe that, for each ω, the shape of f * (p) is symmetric around p = 1/2 and peaked at the opposite extreme values p = 0 and p = 1. This describes a situation in which propensities in the population are polarized, where most individuals adopt opposite and extreme propensity values. In the main plot we see that the system becomes more polarized as ω increases, while we observe in the inset that the polarization is more pronounced as h decreases. To quantify the level of polarization we computed the ratio R(h, ω) ≃ σ * (h, ω)/σ u between the standard deviation n * P,0 = 1 1 + S k=1 G k (h, ω, 1/2) ,(11a)n * P,k = n * P,0 G k (h, ω, 1/2) 1 ≤ k ≤ S,(11b)G k (h, ω, 1/2) = Γ 1−ω 2ωh + k Γ 1+ω 2ωh − k Γ 1−ω 2ωh Γ 1+ω 2ωh ,(11c)σ * (h, ω) = p 2 − p 2 = h 2 S k=0 k 2 n * k − 1/4 of the propensity distribution f * (p) for given values of h and ω, and the corresponding value σ u = 1/12 + h/6 of the uniform distribution f u (p) = 1 S+1 S k=0 δ(p − kh). As the width of f * (p) increases respect to the uniform distribution when the system is polarized, we expect R > 1 and proportional to the magnitude of the polarization. Results are shown in Fig. 2. We see that, for a given h, R increases with ω from the value 1 for ω = 0 to the value 1/(2σ u ) corresponding to the double-peak distribution f (p) = [δ(p) + δ(p − 1)]/2 obtained when ω = 1. It is quite remarkable that the system becomes polarized (R > 1) for any ω > 0. In other words, there is no polarization when agents assign zero weight to it own propensity, but a tiny amount of weight is enough to polarized the population. In summary, for any h and ω, the only stationary states predicted by the rate equations (3) are the consensus absorbing states m * = 0 and m * = 1, and the symmetric polarized state m * = 1/2 in which most agents hold extreme propensities. This polarization phenomenon appears when ω > 0 and is magnified as ω increases. IV. MONTE CARLO SIMULATIONS To compare the previous analytical results with that obtained from Monte Carlo (MC) simulations of the model we studied the time evolution of the fractions n k in single realizations of the dynamics starting from a uniform distribution, as we show in Fig. 3 for a population of N = 10 6 agents, S = 10 (h = 0.1) and ω = 0.5. We can see that after a short initial transient the fractions n k reach a nearly constant value (plateau) that depends on k. However, this state in not stable and eventually all n k finally decay to zero except for n 10 (p = 1) that increases and reaches 1, corresponding to a consensus in p = 1. The height of these plateaus are plotted by open symbols in the main panel of Fig. 1 for h = 0.1 and different values of ω, and in the inset of the same figure for ω = 0.5 and two values of h. We observe a very good agreement with the analytic stationary values Eqs. (11) obtained from the rate Eqs. (3) (filled symbols), which describe an infinite large system where finite-size fluctuations are neglected. We have checked that, indeed, the length of the plateaus increase with N and thus they become infinitely large in the thermodynamic limit, corresponding to the stationary states predicted by the theory. Therefore, the polarized state seems to be unstable, and an extremist consensus is eventually achieved in a finite system. To study the lifetime of the quasiestationary polarized state in finite systems we computed the time to reach consensus. In Fig. 4 we show MC results of the mean consensus time τ vs ω for various system sizes N . We see that τ increases with ω and seems to diverge when ω approaches 1 as a power law τ ∼ (1 − ω) −α , with α ≃ 2 (see inset). This means that polarization not only gets stronger as ω increases, but also lasts for longer times. The collapse of the data in the inset also shows that τ increases very slowly with N , as ln N . In the next section we perform a linear stability analysis for the S = 2 case that allows to obtain the exponent α and the logarithmic scaling with N . V. STABILITY ANALYSIS FOR THE S = 2 CASE An insight into the results shown in the last section can be obtained by studying the simplest non-trivial case S = 2, for which the rate equations (3) are dn 0 dt = [1 − ω/2 − (1 − ω)m)] n 1 − (1 − ω)m n 0 , (12a) dn 1 dt = (1 − ω)m n 0 − n 1 + (1 − ω)(1 − m)n 2 ,(12b)dn 2 dt = [ω/2 + (1 − ω)m] n 1 − (1 − ω)(1 − m)n 2 . (12c) It proves convenient to work with the closed system of equations for n 0 and n 2 dn 0 dt ′ = [1 + ǫ(n 0 − n 2 )] (1 − n 0 − n 2 ) − ǫ(1 − n 0 + n 2 )n 0 , (13a) dn 2 dt ′ = [1 − ǫ(n 0 − n 2 )] (1 − n 0 − n 2 ) − ǫ(1 + n 0 − n 2 )n 2 ,(13b) obtained from Eqs. (12) by using the identities n 1 = 1−n 0 −n 2 and 2m = n 1 +2n 2 = 1−n 0 +n 2 to express n 1 and m in terms of n 0 and n 2 , and defining the parameter ǫ ≡ 1 − ω and the rescaled time t ′ = t/2. As we proved in section III for S = 2, the rate equations have three fixed points. The two trivial fixed points that represent consensus states in p = 0 and p = 1 are n * 0 = (1, 0, 0) and n * 2 = (0, 0, 1), respectively. The non-trivial fixed point n * P = (n * P,0 , n * P,1 , n * P,2 ) corresponding to polarization is n * P = 1 3 − ω , 1 − ω 3 − ω , 1 3 − ω ,(14) which is calculated from Eqs. (4) using the expressions n * P,0 = 1/(1 + G 1 + G 2 ), n * P,1 = G 1 n * P,0 and n * P,2 = G 2 n * P,0 , with G 1 (1/2, ω, 1/2) = 1 − ω and G 2 (1/2, ω, 1/2) = 1 obtained from Eqs. (10) for S = 2 (h = 1/2) by plugging m = 1/2. To investigate how the system approaches consensus we start by performing a linear stability analysis of the trivial fixed point n * 0 = (1, 0, 0) which, by symmetry, is analogous to the analysis of n * 2 . We consider small independent perturbations 0 < x 0 , x 2 ≪ 1 of the fixed point components and write n 0 = 1 − x 0 and n 2 = x 2 . Plugging these expressions for n 0 and n 2 into Eqs. (13) we obtain, to first order in x 0 and x 2 (neglecting terms of order 2), the following system of linear equations written in matrix representation: dx dt ′ = A x, where A ≡ −1 1 + 2ǫ 1 − ǫ −1 − ǫ , and x ≡ (x 0 , x 2 ). The eigenvalues of the matrix A λ ± = −2 − ǫ ± (2 + ǫ) 2 − 8ǫ 2 2(15) are both negative, and thus the fixed point n * 0 is stable under a small perturbation in any direction. To study the behavior of the system for ω 1 we expand Eqs. (15) to leading order in 0 < ǫ ≪ 1. This gives λ + = −ǫ 2 + O(ǫ 3 ), λ − = −2 + O(ǫ)(16) and, therefore, we have x 0 (t ′ ) ≃ a e −ǫ 2 t ′ + b e −2t ′ and x 2 (t ′ ) ≃ c e −ǫ 2 t ′ + d e −2t ′ , where a, b, c and d are constants given by the initial condition. At long times, only the term corresponding to the lowest eigenvalue λ + survives, and thus the time evolution of n 0 and n 2 after a perturbation from n * 0 are approximately given by n 0 (t) ≃ 1 − a e −ǫ 2 t/2 and n 2 (t) ≃ c e −ǫt/2 . The mean time to reach consensus in a population of N agents can be estimated as the time for which the fraction of agents with propensity p = 0 becomes larger than 1−1/N (less than one agent with propensity p > 0). Then, from Eq. (17) we obtain that at consensus time τ is n 0 (τ ) = 1 − a e −ǫ 2 τ /2 = 1 − 1/N , from where we arrive to the approximate expression for the mean consensus time We now study the stability of the polarized state. For that, we linearize the system of Eqs. (13) around the fixed point n * P by rewriting n 0 and n 2 as n 0 = n * P,0 + x 0 and n 2 = n * P,2 + x 2 , with 0 < \|x 0 \|, \|x 2 \| ≪ 1 and n * P,0 = n * P,2 = 1/(2 + ǫ) from Eq. (14). Expanding the resulting equations to first order in x 0 and x 2 we arrive to the following linear system: τ ≃ 2 ln(aN ) (1 − ω) 2 ,(18)dx dt ′′ = B x, with B ≡ −2(1 + ǫ) −2(1 + ǫ) − ǫ 2 −2(1 + ǫ) − ǫ 2 −2(1 + ǫ) , and t ′′ = t/(4 + 2ǫ). The eigenvalues of B are while the associated eigenvectors are v + = (1, −1) and v − = (1, 1). λ + = ǫ 2 and λ − = −4(1 + ǫ) − ǫ 2 ,(19) In Fig. 5 we show the flow diagram that summarizes the stability analysis of the S = 2 case. The fixed points n * 0 and n * 2 (circles) are stable in any direction, while n * P (diamond) is a saddle point that is stable only along the v − direction (λ − < 0) and unstable along any other direction. This means that starting from a state that is symmetric around p = 1/2 [n 0 (0) = n 2 (0)] the system evolves along the line n 0 (t) = n 2 (t) towards the fixed point n * P . However, any perturbation from the fixed point n * P that is not symmetric around the center propensity p = 1/2 leads the system to one of the absorbing consensus configurations p = 0 or p = 1 for all agents. This explains the MC simulation results shown in section IV, where the fractions n k show an initial fast approach from a uniform (symmetric) distribution n k (0) ≃ 1/(S + 1) (with k = 0, .., S) to the polarized stationary state n * P (see Fig. 3) but, eventually, finite-size fluctuations allow the system to escape from this unstable state and reach consensus. VI. STABILITY ANALYSIS FOR THE S ≥ 3 CASE In this section we analyze the stability of the fixed points of the rate equations (3) for general S, namely, the two consensus points n * 0 = (1, 0, .., 0) and n * S = (0, 0, .., 1), and the symmetric point n * P given by Eqs. (11). As we showed in the last section for the 3propensity system (S = 2), any symmetric distribution evolves towards the fixed point n * P , and one can guess that this behavior also holds for any S ≥ 2. More interestingly, we found that for S ≥ 3 there are non-trivial distributions that are not symmetric around p = 1/2 which also evolve towards n * P , as we shall see bellow. We start by reducing the number of independent variables to S using the relation n S = 1 − S−1 k=0 n k and expressing the mean propensity as , where the functions F k correspond to the right-handside of the rate equations. We can then differentiate F k around the fixed points to obtain a linear system of equations defined by a linearized matrix M. For practical reasons, we used Mathematica to calculate the matrix M, its eigenvalues and eigenvectors. Let us first analyze the stability of n * 0 (the same results hold from the analysis of n * S ). In Fig. 6 we plot the maximum of the real part of the eigenvalues λ max 0 of M 0 , calculated numerically, as a function of ω for two values of h. We can see that λ max 0 < 0 for all 0 ≤ w ≤ 1. Therefore, all eigenvalues of M 0 have a negative real part and thus the consensus fixed point n * 0 is locally asymptotically stable. This result generalizes the stability of the two consensus states found for S = 2 (section V) to all values S ≥ 2. We now repeat the analysis above for the symmetric fixed point n * P . We observed numerically that, for various values of S and ω, the matrix M P has only one positive eigenvalue and S − 1 negative eigenvalues. As we know from standard dynamical system theory, the eigenvalues of M P with negative real part generate the tangent plane T to the stable manifold of n * P , which therefore has dimension S − 1. Besides, the space of propensity distri-bution (n 0 , .., n S ) with mean m = 1/2 is an affine plane of codimension 1 which contains n * P , whose intersection with T is a manifold of positive dimension S − 1. When the system starts from a point of this manifold it follows a trajectory that converges to n * P , that is, the points on T represent propensity distributions with mean m = 1/2 that evolve towards the polarized state. To illustrate with an example, one of the eigenvectors of M P that we found numerically for case S = 9 and ω = 1/2 is whose associated eigenvalue is λ = −1.878148. We can now consider the point n(0) = n * P + 0.02 V on the plane T as a initial state of the system, which is obtained by slightly perturbing the fixed point n * P in the direction of V . The time evolution of the components n k of n are plotted in the main panel of Fig. 7, while the inset shows the initial perturbed state (empty diamonds) as compared to n * P (filled circles). We can see that the n(0) is not symmetric with respect to p = 0.5. This asymmetry is the result of the components of V , which exhibit an anti-symmetry that is necessary to preserve the normalization condition S k=0 n k (t) = 1 for all t ≥ 1. We observe that the fractions n k (solid and dotted lines) quickly converge to the corresponding values of the components of n P denoted by horizontal dashed lines. However, if we zoom in we can see that n gets extremely close to n P but not exactly to n P . This very tiny difference is a consequence of the fact that the initial state n(0) belongs to the tangent plane T to the stable manifold of n P , but a priory not to the stable manifold itself. Therefore, the systems spends some time very near to n P before eventually going away and converging to the consensus state p = 0. VII. CONTINUOUS APPROXIMATION In order to analyze the polarized state in more detail it proves useful to consider the system of rate Eqs. (3) in the limiting case of a very small step h ≪ 1. This allows to derive continuous in p partial differential equations that describe the long-time behavior of the system, as we shall see in this section. As we explained in section III, after a short initial transient all agents take discrete propensities in the set p = kh, with k = 0, .., S, and thus the propensity distribution can be written as f (p, t) = S k=0 n k (t) δ(p − kh),(20) where δ(p − kh) is the Dirac delta function at kh. Notice that the consensus states correspond to f (p) = δ(p) and f (p) = δ(p − 1). We consider a generic function φ(p) of and ω = 0.5. Solid curves correspond to p = 0, 1/9, 2/9, 3/9 and 4/9, while dotted curves are for p = 1, 8/9, 7/9, 6/9 and 5/9 (from top to bottom). Inset: the initial values n k (0) (empty diamonds) correspond to a small perturbation of the polarized stationary distribution given by the components of the fixed point nP . the propensity, whose mean value over the population is defined as φ f (t) ≡ 1 0 φ(p)f (p, t)dp = S k=0 n k (t) φ(kh).(21) This is a macroscopic scalar variable of the particle system, like the mean propensity m(t) = p (t) and its variance when we take φ(p) = p and φ(p) = (p− p ) 2 , respectively. In Appendix B we show that the time evolution of φ f is described by the following equation: 1 h d dt φ f = v(p, t) φ ′ (p) + h 2 φ ′′ (p) f + [1 − B 0 (t)] f (0, t) φ ′ (0) − h 2 φ ′′ (0) (22) − [1 − B 1 (t)] f (1, t) φ ′ (1) + h 2 φ ′′ (1) + O(h 2 ), where v(p, t) ≡ 2m(t) − 1 + 2ω [p − m(t)] , B 0 (t) ≡ (1 − ω)m(t) and B 1 (t) ≡ (1 − ω) [1 − m(t)] . There are no O(h 2 ) terms when φ is linear in p. As we can see in the derivation of Appendix B, the first term in the rhs of Eq. (22) comes from the rate equations for n k (t) (0 < k < S) that describe the evolution of the propensity distribution in (0, 1), while the second and third terms come from the dynamics near the boundary points at p = 0 and p = 1, respectively, and describe the balance between the particles entering and leaving the boundary. The coefficient v(p, t) is related to the drift of the particles towards the ends of the interval [0, 1], while B 0 (t) and B 1 (t) are boundary coefficients. Taking φ(p) = 1 in Eq. (22) leads to the conservation of the total mass 1 0 f (p, t)dp = 1, as expected. Besides, for φ(p) = p we obtain the following equation for the evolution of the mean propensity: To better explore the dynamics, we can derive an approximate equation for the time evolution of the propensity distribution f (p, t). For that, we can rewrite Eq. (22) neglecting order h terms as 1 h d dt m(t) = 2m(t) − 1 + f (0, t) [1 − (1 − ω)m(t)] − f (1, t) [ω + (1 − ω)m(t)] .(23)1 h d dt φ f = 1 0 [v(p, t) + u(p, t)]f (p, t)φ ′ (p) dp,(24) where we have introduced the field u(p, t) = [1 − B 0 (t)]δ(p) − [1 − B 1 (t)]δ(p − 1) = [1 − (1 − ω)m(t)]δ(p) − [ω + (1 − ω)m(t)]δ(p − 1). Integrating by parts the r.h.s. of Eq. (25) and regrouping terms leads to 1 0 φ(p) 1 h ∂ ∂t f (p, t) + ∂ ∂p [v(p, t) + u(p, t)] f (p, t) dp = [v(1, t) + u(1, t)] f (1, t)φ(1) − [v(0, t) + u(0, t)] f (0, t)φ(0). Since this relation holds for any function φ we see that f satisfies formally the transport equation ∂ ∂t f (p, t) = − ∂ ∂p h [v(p, t) + u(p, t)] f (p, t) + h [v(1, t) + u(1, t)] f (1, t)δ(p − 1) − h [v(0, t) + u(0, t)] f (0, t)δ(p).(25) Equation (25) expresses the conservation of total number of particles under the transport induced by the effective drift v + u and with source terms h [v(1, t) + u(1, t)] f (t, 1)δ(p − 1) and −h [v(0, t) + u(0, t)] f (t, 0)δ(p) at the boundary points p = 1 and p = 0, respectively. An intuitive interpretation of this equation is that the mass density f (p, t) is transported by the field v in [0, 1] and suffers and additional impulse at the borders p = 0, 1 given by the field u, which is associated to the rate Eqs. (3a) and (3c) for n 0 and n S , respectively. This is reminiscent of the bouncing effect of particles at the boundaries, by which a particle that hits p = 0 (p = 1) can later jump back to the inter- val (0, 1) with probability (1 − ω)m(t) = v(0, t) + u(0, t) [(1 − ω)(1 − m) = −v(1, t) − u(1, t)]. tice that the magnitude m in Eq. (29) is the mean propensity that must satisfy the relation 1 0 (p − m) f * (p)dp = 0, which is equivalent to 1 0 (p − m) exp 2 ω h p − α 2 ω 2 dp = 0. This is a nonlinear equation in m that we studied numerically for various values of ω and h. We found that m = 1/2 is the only solution in all cases, which is in agreement with the symmetric solution of the rate Eqs. (3) in section III. Therefore, the symmetric stationary distribution of Eq. (26) is given by f * (p) = A exp 2 ω h p − 1 2 2 + h (1 − ω) exp ω 2 h [δ(p) + δ(p − 1)] , (29) with A = 1 0 exp 2 ω h p − 1 2 2 dp + 2h exp ω 2 h (1 − ω) −1 . As we can see, f * (p) is symmetric around m = 1/2 and is the sum of the continuous function A exp 2ω h p − 1 2 2 in the interval (0, 1) that has the shape of an inverted Gaussian, and the two Dirac masses located at the boundaries, what makes f * (p) a discontinuous function at p = 0 and p = 1. We can alternatively describe the stationary solution by the cumulative distribution function of f * (p) F * (p) =            0 if p < 0, A h (1−ω) exp ω 2h if p = 0, A p 0 e 2ω h (p− 1 2 ) 2 dp + A h (1−ω) exp ω 2h if 0 ≤ p < 1, 1 if p ≥ 1. (30) In Fig. 8 we plot the approximate stationary cumulative distribution F * (p) for continuous p (solid curves) and the exact discrete cumulative distribution F * k = k k ′ =0 n k ′ (circles) for two different small values of h. We see that the data for F * (p) agrees very well with that of F * k , showing that f * (p) given by Eq. (29) is indeed the limit of S k=0 n * k δ(p − kh) when h → 0. VIII. SUMMARY AND DISCUSSION We studied a system of interacting particles that models the dynamics of voting intentions in a population of individuals that interact by pairs. The propensity of an individual to vote for a given candidate may either increase or decrease after interacting with other partner, depending on the propensity of the partner and the F * k = k k ′ =0 n k ′ . weight ω in [0, 1] assigned to its own propensity. We have investigated the dynamics of the system by means of a rate equation approach and we have checked the results with MC simulations. Starting from a nearly uniform distribution of propensities in [0, 1], we found that for ω = 0 the system is quickly driven towards an extreme propensity (p = 0 or p = 1) that corresponds to the initial majority. The dynamics stops evolving when all individuals share the same extreme propensity; an absorbing consensus state. However, for ω > 0 the evolution is quite different: the system initially evolves towards a stationary state characterized by a distribution of propensities that is symmetric around p = 1/2 and peaked at the extreme values p = 0 and p = 1, and it becomes more pronounced when ω gets larger. This distribution describes a state of polarization where most individuals adopt extreme values of p, whose effect is magnified as ω increases. This implies that a tiny weight assigned to our own propensity is enough to polarize the population into two groups with extreme and opposite propensities. However, this state of symmetric polarization is unstable, and thus any perturbation from that state leads the system towards one of the two extremist consensus. Single MC simulations of the dynamics of the model showed that, indeed, the system may initially reach this symmetric quasi-stationary state but finite-size fluctuations eventually drive the system towards one of the two absorbing configurations. An stability analysis of the rate equations shows that any symmetric distribution evolves towards the polarized state, but there are also non-trivial propensity distributions with mean propensity m = 1/2 that are not symmetric around p = 1/2 and that evolve and reach the polarized state. An insight into the polarized state was obtained by analyzing the continuous limit of the system of rate equa-tions. This approximation lead to a transport equation with a convection term that represents a drift of particles from the center propensity p = 1/2 towards the extremes, which induces polarization. The stationary solution has the shape of an inverted Gaussian with two Delta functions at p = 0 and p = 1 that account for the dynamics at the boundaries. In this peculiar dynamics, particles can hit and stay at one of the boundaries for some time but eventually leave, and then hit the boundary again and so on, following and endless loop. We have quantified the lifetime of the polarized state by measuring the mean consensus time τ , and found that it increases with ω and diverges as τ ∼ (1 − ω) −2 when ω approaches 1. This would imply that polarization is quite stable in populations with narrow-minded individuals that only take into account its own opinion when interacting with others, reinforcing their previous believes and adopting more extreme viewpoints. This result is akin to that obtained in related models for opinion formation [11][12][13][14] that include a reinforcement mechanism by which pairs of individuals with the same opinion orientation (both in favor or both against a given political issue) are more likely to interact and become more extremists. Even though the propensity model studied in this article does not include this mechanism implicitly, it is able to capture the same phenomenology by implementing a simple interaction rule that consider pairwise interactions between individuals as independent of the opinion group they belong to. In the studied model, the propensity update probability is a simple weighted average of the propensities of the two interacting individuals. It would be worthwhile to explore some extensions of the model that consider updating probabilities that are non-linear functions of the propensities and investigate how the behavior of the model is affected, for instance, whether the polarized state becomes more stable or not. It might also be interesting to study versions of the model where pairwise interactions are not simply taken as all-to-all, but rather take place on lattices or complex networks. These are all topics for future investigation. as quoted in Eq. (5) of the main text. The value of n 0 can be obtained by inserting the expression n k = n 0 G k (h, ω, m) in the normalization condition S k=0 n k = 1 and solving for n 0 , which leads to the expression FIG. 1 : 1Main: distribution of propensities at the stationary polarized state for step h = 0.1 and weights ω = 0.1 (circles), ω = 0.3 (squares) and ω = 0.8 (diamonds). Filled symbols joined with lines are the stationary solution Eqs. (11), while open symbols correspond to MC simulation results. Inset: upper (lower) curve corresponds to ω = 0.5 and h = 0.1 (h = 0.005), respectively. FIG. 2 : 2Polarization level R versus weight ω for h = 0.05 (circles) and h = 0.1 (squares). The horizontal dashed lines denote the saturation values at ω = 1. FIG. 3 : 3Monte Carlo results for the time evolution of the fraction of agents n k with propensities p = kh (k = 1, .., 10), with h = 0.1 and ω = 0.5, in a population of N = 10 6 agents. Solid (dashed) curves correspond to p = 1.0 (0.0), p = 0.9 (0.1), p = 0.8 (0.2), p = 0.7 (0.3), p = 0.6 (0.4) and p = 0.5 (from top to bottom). FIG. 4 : 4Mean consensus time τ vs ω for h = 0.1 and system sizes N = 20 (triangles), N = 80 (diamonds), N = 320 (squares) and N = 1280 (circles). Inset: the data collapse shows the approximate scaling τ ∼ (1 − ω) −2 ln N for ω 1. The dashed line has slope −2. after replacing back ǫ by 1 − ω. The (1 − ω) −2 divergence of τ as ω → 1 predicted by Eq. (18) is in good agreement with the exponent α ≃ 2 found from MC simulations (see inset of Fig. 4). Equation (18) also agrees with the logarithmic increase of τ with N observed in the inset of Fig. 4. FIG. 5 : 5Schematic flow diagram in the n0 − n2 plane for the 3-propensity system (S = 2). The two stable fixed points denoted by circles correspond to the absorbing consensus states in an extreme propensity, while the saddle point denoted by a diamond represents the steady-state of polarization. The lines with arrows show the flow direction of the system inside the composition triangle 0 ≤ n0 + n2 ≤ 1. FIG. 6 : 6Maximum real part of the eigenvalues of matrix MP vs ω for h = 0.1 (circles) and h = 0.05 (squares). The inset shows how Re(λ) approaches 0 from bellow as ω goes to 1. − 1)n k , and so we can rewrite Eqs. (3) as d dt n k = F k (n 0 , .., n S−1 ) V ≃ (0.31551, −0.74694, 0.90799, −0.97402, 1.00000, −1.00000, 0.97402, −0.90799, 0.74694, −0.31551), FIG. 7 : 7Time evolution of the propensity fractions n k for a system with h = 1/9 (S = 9) We can check from Eq. (23) that if the population is initially in a consensus state, i.e. (i) m(0) = 0 or (ii) m(0) = 1, then (i) m(t) = 0 or (ii) m(t) = 1 for any t ≥ 0, meaning that the population remains in the consensus state as expected from the fixed point solutions m * = 0, 1. We can also see in Eq. (23) that term 2m − 1 describes a drift towards m = 0 (m = 1) when m < 1/2 (m > 1/2) caused by the instability of the fixed point m * = 1/2. Therefore, starting from a nearly uniform distribution with m(0) slightly larger than 1/2 as in the MC simulations, agents' propensities are slowly dragged to p = 1. Besides, we see that m(0) = m * = 1/2 is a stationary value if f (0, 0) = f (1, 0), in agreement with the fact that any symmetric distribution f (1/2 − p) = f (1/2 + p) evolves towards the polarized fixed point n * P , as shown in sections V and VI. FIG. 8 : 8Propensity cumulative distribution vs p for the parameter values indicated in the legends. Solid lines correspond to the approximate continuous solution for h ≪ 1 from Eq. (30), while circles represent the exact discrete solution n replacing the expression A7 for n 0 and rearranging the terms becomes Eq. (7) of the main text. When m > 0 we can rewrite G k (h, ω, m) in terms of the gamma functions by using the Pochhammer formula(z) k ≡ z(z + 1)(z + 2)...(z + k − 1) = Γ(z + k) Γ(z) for z ∈ C\Z − and k ≥ 0 integer,(A8)which follows from the relation Γ(z+1) = z Γ(z). We first rewrite the numerator ofG k (h, ω, m) in Eq. (A6) introducing z ≡ (1 − ω)m/(ωh) as (1 − ω)m(ωh) k − 1 = (ωh) k z(z + 1)...(z + k − 1) = (ωh) k Γ(z + k) Γ(z) .Notice that if m = 0 then z = 0 and we cannot use Pochhammer formula. Lettingz≡ [1 − (1 − ω)m − kωh] /(ωh),we rewrite in the same way the denominator of G k (h, ω, m) in Eq. (A6) as(ωh) kz (z + 1)...(z + k − 1) = (ωh) k Γ(z + k) Γ(z) .Inserting these two last expressions for the numerator and denominator of G k (h, ω, m) in Eq. (A6) leads to the expression quoted in Eq. (6) of the main text.Appendix B: Continuum equation for φ f In this section we derive an equation for the time evolution of the mean of a generic function φ(p) over the population of agents, expressed as φ f (t) (p, t) is the propensity distribution at time t. Taking the time derivative of Eq. k+1 φ(kh)[1 − ωh(k + 1) − (1 − ω)m], with A ≡ (1 − ωh − (1 − ω)m)n 1 − (1 − ω)mn 0 and B ≡ (ω(1 − h) + (1 − ω)m)n S−1 − (1 − ω)(1 − m)n S . AcknowledgmentsThis work was partially supported by Universidad de Buenos Aires under grants 20020170100445BA, 20020170200256BA, and by ANPCyT PICT2012 0153 and PICT2014-1771.A. Approximate stationary state solutionIt is useful to decompose f (p, t) into a sum of a boundary term f (0, t) δ(p) + f (1, t) δ(p − 1) taking into account the dynamics near p = 0 and p = 1, and an inside term f (p, t) that describes the dynamics in (0, 1):where we have neglected terms of order 2 and higher, and simplified the notation by writing v = v(p, t), B 0 = B 0 (t), B 1 = B 1 (t) and φ = φ(p). We are interested in the stationary solutions to Eq. (26). As expected from previous results, the consensus states f * (p) = δ(p) and f * (p) = δ(p − 1) are stationary solutions. One can check that by noticing that for p = 0 (p = 1) consensus is f * (p) = 0, f * (0) = 1 (f * (0) = 0), f * (1) = 0 (f * (1) = 1) and m = 0 (m = 1). In view of our findings in section III we also expect a symmetric polarized state with mean m * = 1/2 to be a stationary solution. In that perspective it makes sense to drop the terms involving φ ′′ (0) and φ ′′ (1) for symmetry reasons. We then look for a station-for any φ(p). In Appendix C we show that the solution to Eq. (27), different from δ(p) and δ(p − 1), is given bywhere α ≡ 1 − 2(1 − ω)m and A > 0 is a normalization constant that satisfies the conditionNotice that the two consensus states corresponding to• m = 0, n 0 = 1, n k = 0 for k = 1, .., S,• m = 1, n 1 = 1, n k = 0 for k = 0, .., S − 1, are solutions of the system of Eqs. (A1). To find other possible non-trivial solutions we first note that n k (for all k = 1, .., S) can be expressed as a function of n 0 . Starting from Eq. (A1a) we obtainThen, solving for n 2 from Eq. (A1b) for k = 1, and using the previous expression for n 1 we obtainThe same procedure applied to k = 2 leads toIn general we havewhere We look for a stationary solution f * of the formwheref * (p) is a continuous function of p. We can then rewrite Eq. 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[ "OPT: ONE-SHOT POSE-CONTROLLABLE TALKING HEAD GENERATION", "OPT: ONE-SHOT POSE-CONTROLLABLE TALKING HEAD GENERATION" ]	[ "Jin Liu \nInstitute of Information Engineering\nChinese Academy of Sciences\nBeijingChina\n\nSchool of Cyber Security\nUniversity of Chinese Academy of Sciences\nBeijingChina\n", "Xi Wang \nInstitute of Information Engineering\nChinese Academy of Sciences\nBeijingChina\n", "Xiaomeng Fu \nInstitute of Information Engineering\nChinese Academy of Sciences\nBeijingChina\n\nSchool of Cyber Security\nUniversity of Chinese Academy of Sciences\nBeijingChina\n", "Yesheng Chai \nInstitute of Information Engineering\nChinese Academy of Sciences\nBeijingChina\n", "Cai Yu \nInstitute of Information Engineering\nChinese Academy of Sciences\nBeijingChina\n\nSchool of Cyber Security\nUniversity of Chinese Academy of Sciences\nBeijingChina\n", "Jiao Dai \nInstitute of Information Engineering\nChinese Academy of Sciences\nBeijingChina\n", "Jizhong Han \nInstitute of Information Engineering\nChinese Academy of Sciences\nBeijingChina\n" ]	[ "Institute of Information Engineering\nChinese Academy of Sciences\nBeijingChina", "School of Cyber Security\nUniversity of Chinese Academy of Sciences\nBeijingChina", "Institute of Information Engineering\nChinese Academy of Sciences\nBeijingChina", "Institute of Information Engineering\nChinese Academy of Sciences\nBeijingChina", "School of Cyber Security\nUniversity of Chinese Academy of Sciences\nBeijingChina", "Institute of Information Engineering\nChinese Academy of Sciences\nBeijingChina", "Institute of Information Engineering\nChinese Academy of Sciences\nBeijingChina", "School of Cyber Security\nUniversity of Chinese Academy of Sciences\nBeijingChina", "Institute of Information Engineering\nChinese Academy of Sciences\nBeijingChina", "Institute of Information Engineering\nChinese Academy of Sciences\nBeijingChina" ]	[]	One-shot talking head generation produces lip-sync talking heads based on arbitrary audio and one source face. To guarantee the naturalness and realness, recent methods propose to achieve free pose control instead of simply editing mouth areas. However, existing methods do not preserve accurate identity of source face when generating head motions. To solve the identity mismatch problem and achieve high-quality free pose control, we present One-shot Pose-controllable Talking head generation network (OPT). Specifically, the Audio Feature Disentanglement Module separates content features from audios, eliminating the influence of speakerspecific information contained in arbitrary driving audios. Later, the mouth expression feature is extracted from the content feature and source face, during which the landmark loss is designed to enhance the accuracy of facial structure and identity preserving quality. Finally, to achieve free pose control, controllable head pose features from reference videos are fed into the Video Generator along with the expression feature and source face to generate new talking heads. Extensive quantitative and qualitative experimental results verify that OPT generates high-quality pose-controllable talking heads with no identity mismatch problem, outperforming previous SOTA methods.	10.1109/icassp49357.2023.10094598	[ "https://export.arxiv.org/pdf/2302.08197v1.pdf" ]	256,900,670	2302.08197	b8c1bdac473d0cfbb20a69f7f91ec28ecef6ab15	OPT: ONE-SHOT POSE-CONTROLLABLE TALKING HEAD GENERATION Jin Liu Institute of Information Engineering Chinese Academy of Sciences BeijingChina School of Cyber Security University of Chinese Academy of Sciences BeijingChina Xi Wang Institute of Information Engineering Chinese Academy of Sciences BeijingChina Xiaomeng Fu Institute of Information Engineering Chinese Academy of Sciences BeijingChina School of Cyber Security University of Chinese Academy of Sciences BeijingChina Yesheng Chai Institute of Information Engineering Chinese Academy of Sciences BeijingChina Cai Yu Institute of Information Engineering Chinese Academy of Sciences BeijingChina School of Cyber Security University of Chinese Academy of Sciences BeijingChina Jiao Dai Institute of Information Engineering Chinese Academy of Sciences BeijingChina Jizhong Han Institute of Information Engineering Chinese Academy of Sciences BeijingChina OPT: ONE-SHOT POSE-CONTROLLABLE TALKING HEAD GENERATION Index Terms-Talking head generationGenerative ModelAudio driven animation One-shot talking head generation produces lip-sync talking heads based on arbitrary audio and one source face. To guarantee the naturalness and realness, recent methods propose to achieve free pose control instead of simply editing mouth areas. However, existing methods do not preserve accurate identity of source face when generating head motions. To solve the identity mismatch problem and achieve high-quality free pose control, we present One-shot Pose-controllable Talking head generation network (OPT). Specifically, the Audio Feature Disentanglement Module separates content features from audios, eliminating the influence of speakerspecific information contained in arbitrary driving audios. Later, the mouth expression feature is extracted from the content feature and source face, during which the landmark loss is designed to enhance the accuracy of facial structure and identity preserving quality. Finally, to achieve free pose control, controllable head pose features from reference videos are fed into the Video Generator along with the expression feature and source face to generate new talking heads. Extensive quantitative and qualitative experimental results verify that OPT generates high-quality pose-controllable talking heads with no identity mismatch problem, outperforming previous SOTA methods. INTRODUCTION Talking head generation aims to drive the source face image with the audio signal and produces a lip-sync talking head video, which is significant to various practical multimedia applications, such as film making, virtual education, video conferencing, digital human animation and short video creation. Talking head generation can be divided into two categories: speaker-specific methods and speaker-independent methods. The speaker-specific methods [1,2,3] only generate talking heads of fixed subject and requires large amount of person-specific high-quality videos, which limits the application and generalization. The speaker-independent methods are designed to animate video portraits given one unseen source face and driving audio. Some one-shot works [4,5] simply edit the mouth region and keep the other areas of source face unchanged. Their generated talking head videos are unnatural with the fixed facial contour, blending traces around the mouth and no head motion changes. Therefore, current one-shot speakerindependent works focus on full-frame generation [6,18,7], which produce the whole head areas, together with neck parts and background. To improve the naturalness and realness, some methods propose to add natural head poses into talking heads. PC-AVS [6] modularizes audio-visual representations by devising an implicit low-dimension pose code. Audio2Head [18] utilizes a motion-aware recurrent neural network to predict head motions from audio. However, in talking heads of above methods with new poses, the source identity is not well preserved due to the facial structure change, as shown in Fig. 2. The identity mismatch problem means the inability of generated talking heads to preserve the identity of source faces. Previous image driven face reenactment works [8] focus on solving the identity mismatch problem in visual modality, which caused by the inconsistent facial contour between driving subject and source person. When it comes to audio-driven paradigm, the gap between audio and visual modality becomes even larger. All the information contained in audio signal affects the driving representation extraction, among which the content feature is the most important since it directly relates to the mouth shape. Given the fact that different speakers' audios with the same content are different, we believe that it is necessary to disentangle identity and content features from audio signal. Furthermore, it is important to extract accurate mouth expression features since the facial structure changes when performing different head poses. Specifically, we present the One-shot Pose-controllable Talking head generation network (OPT). The Audio Feature Disentanglement Module separates identity and content features explicitly from audio signals. Later, the facial expression feature is extracted from content feature and source face, during which the landmark loss is designed to enhance the accuracy of facial structure and identity preserving quality. Finally, the head pose feature reconstructed from other pose videos using 3DMM [9] is fed into the Video Generator along with expression feature and source face to generate new talking heads. Extensive experimental results demonstrate the superior performance of OPT and the effectiveness of several modules. Our contributions are summarized as follows: • The proposed OPT is the first to simultaneously perform one-shot identity-independent pose-controllable talking head generation with almost no identity mismatch problem. • To solve the identity mismatch problem, the Audio Feature Disentanglement Module is proposed to successfully decompose intrinsic identity features and content features over audio signals. • The landmark loss is designed to enhance the accuracy of facial shape and the identity preserving quality. The explicit head pose feature is also utilized to guide the free pose control. The Audio Feature Disentanglement Module separates content feature F con and identity feature F id from A dri . Then F con and source feature F src from I src are fed into the Audio-to-Expression Module to produce expression feature F e . Finally, the Video Generator takes I src , F e and head pose feature F hp from I pose as inputs to generate I G . Each module will be introduced detailedly in the following sections. OUR METHOD Audio Feature Disentanglement Module Given that audios of the same content but of different speakers are diverse, we believe the inherently entangled identity and content features need to be independently extracted from audio signals, to achieve audio-based identity control for talking head generation. During inference, the identity of A dri and I src are usually different, since the driving audio is chosen arbitrarily. Unlike previous methods [5,6] who merely extract entangled features from audio signals, we propose the Audio Feature Disentanglement Module (AFDM) to map audios into two separate latent audio spaces: a content-agnostic encoding space of the identity and a content-dependent encoding space of the corpus corresponding to audio. Specifically, AFDM contains two encoders E con and E id to individually extracts corresponding features from A dri . The content loss L con is as follows: where audio A andÃ shares the same content but spoken by different subjects. The identity loss L id is used to train E id : L con = E con (A) − E con (Ã) 1 ,(1)L cls = − N i=1 (p * log q) ,(2) where N denotes the total amount of speaker identities, p means the ground truth identity probability and q represents the E id prediction probability. In this way, the pure content feature F con can be accurately separated from A dri through AFDM to guide the following extraction process of expression feature. Audio-to-Expression Module The Audio-to-Expression Module (AEM) predicts expression feature F e from features F con and F src . Considering the good reconstruction performance of 3DMM [9], we use part of 3DMM parameters to represent the expression feature. With a 3DMM, the 3D shape S of a face is parameterized as follows: S = S + αB id + βB exp ,(3) where S is the average face shape, B id and B exp are basis of identity and expression via PCA. The coefficients α ∈ R 80 and β ∈ R 64 describe the facial shape and expression respectively. In our method, we choose β as expression features f e . The AEM contains multiple convolutional layers and linear layers. The L1 loss L L1 is imposed onF e and the ground truth F e . In order to further improve the lip-sync quality and keep the accurate facial contour structure to alleviate the identity mismatch problem, we further design the facial landmark loss L ldmk using 3DMM: L ldmk = 1 N N n=1 ω n l n − l n 2(4) where {l n } is facial landmarks of ground truth driving face image, {l n } is the 3D landmark vertices projection of reconstructed shape using f e onto the image plane. N denotes the number of landmarks. The weight ω n id set to 20 for inner mouth and nose points and others to 1. Video Generator To achieve pose-controllable talking head generation, the additional driving pose face image I pose is need to provide head pose feature. During inference, I pose could come from real talking head videos to offer auxiliary pose feature sequences. The head pose feature F hp is denoted by rotation R ∈ SO(3) and translation T ∈ R 3 , which could also be obtained during the process of 3DMM reconstruction [9]. Finally, given I src , F hp and F e , the Video Generator(VG) produces new talking headI G . Detailedly, VG contains Conv and TransConv layers with residual connection. The two features are fed by AdaIn [10] opeation. The discriminator utilized Patch-GAN [11] and VG minimizes the reconstruction loss L rec between ground truth image I and generated head image I G . L rec = I − I G 2 .(5) The input to the discriminator D is the ground truth image I and I G . The GAN loss is as follows: L GAN = log D(I) + log(1 − D(I G )).(6) Furthermore, the perceptual loss L per is also adapted to calculate the distance between activation maps of the pretrained VGG-19 network : L per = i φ i (Ĩ g ) − φ i (I) 1 ,(7) where φ i is the activation map of the i-th layer of the VGG-19 network [12]. During training, each module in OPT are trained separately utilizing corresponding loss combination. EXPERIMENTS Experimental settings Datasets The AFDM is trained on audio-visual dataset MEAD [13] which contains annotations of speaking corpus scripts and identity information. Other modules of OPT are trained and tested on LRW [14] and LRS2 [15] is adopted with an initial leaning rage as 10 −4 . The learning rate is decreased to 2 × 10 −5 after 300k iterations. Comparing Methods The following SOTA one shot talking head generation methods are compared. ATVG [4] generates frames based on facial landmarks using the attention mechanism. Wav2Lip [5] utilizes a pre-trained lip-sync discriminator to focus on editing the mouth shape. Audio2Head [18] infers unique head pose sequences from audio and utilizes flow-based generator to produce talking heads. PC-AVS [6] extracts modularized audio-visual representations of identity, pose and speech content, generating pose-controllable talking heads. Besides, when I pose is not given for OPT, we can fix it with the same pose as I src , keeping the head still. We refer results under this setting as Ours-Fix Pose. Same as the generation paradigm in PC-AVS [6], an extra video with supposedly the same pose as driving frames corresponding to I dri but different identities and mouth shapes is generated to serve as the I pose in our method. Quantitative Results We evaluate the performance on image quality, identity preserving and lip-sync quality. The SSIM [19] scores are utilized to judge the talking head image quality. CSIM indicates the cosine similarity between face recognition features [17] of generated and ground truth talking heads. For the lip-sync quality, the Landmark Distance(LMD) and Lip-Sync Error-Confidence(LSE-C) [5] are applied. Table 1 shows the quantitative comparison results using LRW datasets. It shows that OPT achieves leading SSIM, LMD and CSIM scores. As mentioned in [6], the leading LSE-C score only means that Wav2Lip is comparable to the ground truth, not better. Our leading LMD score indicates that we preserve better facial structure and accurate mouth shape. We also achieve high-level identity preserving quality, proved by the leading CSIM score. Qualitative Results We compare OPT with other methods, as displayed in Fig. 2. It shows that OPT generates high quality talking head videos with accurate mouth shape and facial contour that best match the ground truth. Concretely, ATVG merely focuses on cropped facial region. Wav2Lip generates faces with fixed head motions and blurry mouth shape. Audio2Head fails to keep the lip synchronization and accurate facial contour. PC- Table 3. User study results by mean opinion scores. AVS causes the identity mismatch problem and cannot preserve the accurate facial contour. We further demonstrate the performance of OPT in Fig. 3. It indicates that the generated talking heads can achieve free pose control and meanwhile maintain accurate lip synchronization and no identity mismatch problem. Table 2 shows our ablation study to prove the effect of each proposed component. The baseline method directly extracts expression feature from audio signal and utilizes merely reconstruction loss in the Audio Feature Disentanglement Module and Audio-to-Expression Module. The result indicates that AFDM helps solve the identity mismatch problem according to the obvious increase on CSIM score. Besides, L ldmk contributes a lot to the image quality and lip-sync quality since it extracts accurate facial structure representation during the training process. Ablation Study User Study We further conduct the user study to evaluate OPT and other state-of-the-art methods. For OPT and comparing methods, 15 video clips using randomly selected source faces and audios from LRW and LRS2 datasets are generated. We adopt the widely used Mean Opinion Scores (MOS) rating protocol. 10 participants are required to give their ratings (1-5) on the following three aspects for each generated talking head video: visual quality, lip-sync quality and identity preserving quality. As Table 3 shows, OPT achieves the best results on all aspects, especially the identity preserving quality. CONCLUSION In this paper, we propose a new method called OPT to generate pose-controllable identity-preserving talking head videos. Given one source face image and arbitrary driving audio, OPT generates lip-sync talking heads that preserve the source identity and can achieve pose control guided by auxiliary pose video. The proposed Audio Feature Disentanglement Module separates content features from audio signal and the landmark loss is adopted in the Audio-to-Expression Module, both contributing a lot to the generation of talking heads. In the future, we will focus on real time and high resolution generation to enhance the generalization. Fig. 1 1summarizes the pipeline of our proposed method. OPT takes driving audio A dri , source image I src and pose image I pose as inputs to generate I G , indicating I src speaking the corpus of A dri with the head pose of I pose . Fig. 1 . 1Overview of OPT. The Audio Feature Disentanglement Module separates content feature from driving audio. Then the Audio-to-Expression Module extracts the expression feature given conent feature and soure face. Finally, the head pose feature from other video and expression feature are fed into the Video Generator to generate new talking heads. Fig. 3 . 3Qualitative results of OPT. Four generated clips of the same word "we" driven by different pose videos are shown.MethodsSSIM ↑ CSIM ↑ LMD ↓ LSE-C ↑ Table 1. Quantitative comparison results on LRW dataset. The bold and underlined indicate the top-2 results.datasets. The LRW dataset contains over 1000 utterances of 500 dif- ferent words while the LRS2 dataset includes over 140,000 utterances of different sentences. Both videos are from BBC television in the wild. Implementation Details Face frames are cropped to 256 × 256 size at 25 FPS and audio to mel-spectrogram of size 16 × 16 per frame. Mel-spectrograms are constructed from 16kHZ audio, window size 800, and hop size 200. Both encoders in the AFDM share the SE-ResNet [16] architecture. F src is extracted by pre-trained ArcFace [17] model. OPT is trained in stages on 4 Tesla 32G V100 GPUs. The ADAM optimizer Ground Truth ATVG Wav2Lip Ours Audio2Head PC-AVS Fig. 2. Qualitative comparison results with other state-of-the- art methods on LRS2 dataset. Methods SSIM ↑ CSIM ↑ LMD ↓ LSE-C ↑ ATVG 0.781 0.76 5.32 4.165 Wav2Lip 0.792 0.81 5.73 7.237 Audio2Head 0.743 0.72 7.34 2.135 PC-AVS 0.815 0.74 6.14 6.420 Ours-Fix Pose 0.795 0.83 5.25 6.432 OPT (Ours) 0.823 0.88 3.78 6.619 Table 2 . 2Ablation study on LRW dataset. The evaluated parts include the Audio Feature Disentanglement Module and the landmark loss utilized in the Audio-to-Expression Module.Baseline 0.721 0.69 7.18 3.125 Ours w/o AFDM 0.753 0.75 5.78 4.259 Ours w/o L ldmk 0.762 0.74 6.14 3.842 Ours 0.823 0.88 3.78 6.619 Facial: Synthesizing dynamic talking face with implicit attribute learning. 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[ "ON THE CANHAM PROBLEM: BENDING ENERGY MINIMIZERS FOR ANY GENUS AND ISOPERIMETRIC RATIO", "ON THE CANHAM PROBLEM: BENDING ENERGY MINIMIZERS FOR ANY GENUS AND ISOPERIMETRIC RATIO" ]	[ "Robert Kusner ", "Peter Mcgrath " ]	[]	[]	Building on work of Mondino-Scharrer, we show that among closed, smoothly embedded surfaces in R 3 of genus g and given isoperimetric ratio v, there exists one with minimum bending energy W. We do this by gluing g + 1 small catenoidal bridges to the bigraph of a singular solution for the linearized Willmore equation ∆(∆ + 2)ϕ = 0 on the (g + 1)-punctured sphere S 2 to construct a comparison surface of genus g with arbitrarily small isoperimetric ratio v ∈ (0, 1) and W < 8π.	10.1007/s00205-022-01833-w	[ "https://arxiv.org/pdf/2104.10045v1.pdf" ]	233,307,129	2104.10045	6289458d30b65809d89cc146b82421073a7266c7	ON THE CANHAM PROBLEM: BENDING ENERGY MINIMIZERS FOR ANY GENUS AND ISOPERIMETRIC RATIO 20 Apr 2021 Robert Kusner Peter Mcgrath ON THE CANHAM PROBLEM: BENDING ENERGY MINIMIZERS FOR ANY GENUS AND ISOPERIMETRIC RATIO 20 Apr 2021arXiv:2104.10045v1 [math.DG] Building on work of Mondino-Scharrer, we show that among closed, smoothly embedded surfaces in R 3 of genus g and given isoperimetric ratio v, there exists one with minimum bending energy W. We do this by gluing g + 1 small catenoidal bridges to the bigraph of a singular solution for the linearized Willmore equation ∆(∆ + 2)ϕ = 0 on the (g + 1)-punctured sphere S 2 to construct a comparison surface of genus g with arbitrarily small isoperimetric ratio v ∈ (0, 1) and W < 8π. Introduction Motivated by the problem of explaining the shapes of biophysical membranes like red blood cells [2] or phospholipid vesicles [3] variationally, significant progress has been made [15,7,13,14], using previous work of [12,9,16,10,1], on the following basic mathematical question: Among closed, smoothly embedded surfaces in Euclidean three-space of a given genus and prescribed isoperimetric ratio, is there a surface with minimum bending energy? For a smoothly embedded surface S ⊂ R 3 , its isoperimetric ratio v(S) and its bending energy W(S) are defined by where H and dσ are the mean curvature and area element of S, and where Vol(S) is the volume of the domain in R 3 that S bounds. In this paper, we answer the above question affirmatively: Theorem 1.2. Let g ∈ {0, 1, 2, . . . } and v ∈ (0, 1). Among all closed, smoothly embedded surfaces S ⊂ R 3 of genus g and v(S) = v, there exists S g,v with minimum bending energy W(S g,v ) < 8π. Our main contribution is to construct, for each g and prescribed v ∈ (0, 1), a smooth embedded surface S g,v ⊂ R 3 of genus g, isoperimetric ratio v, and bending energy W(S g,v ) < 8π. From this, the conclusion of Theorem 1.2 follows by recent work of Mondino-Scharrer [13,Theorem 1.2]. This solves the existence portion of the Canham Problem [11] to characterize the family of W-minimizing surfaces {S g,v } of genus g with fixed isoperimetric ratio v ∈ (0, 1). Outline of our method. A naive construction for a smooth surface S ⊂ R 3 of genus g with W(S) approximately 8π and v(S) small would be to join two nearby parallel copies of the unit sphere S 2 ⊂ R 3 , each with g + 1 disks removed, by gluing in g + 1 small catenoidal bridges to the boundaries of the excised disks. Unfortunately, such a surface has W(S) > 8π: each catenoidal bridge contributes strictly more bending energy than the energy of the corresponding pair of excised disks (see Lemma 4.5), while each sphere contributes exactly 4π minus the energy of the excised disks. Therefore, to ensure the bending energy is less than 8π, a more subtle construction is required. Our method stems in part from the Kapouleas construction [4] of minimal surfaces in S 3 by doubling the equatorial two-sphere S 2 . He introduced an approach called Linearized Doubling (LD), where the minimal surface is a small perturbation of a smooth initial surface, constructed by gluing a collection of catenoidal bridges, whose centers are located on a finite set L ⊂ S 2 and whose axes are orthogonal to S 2 , to the bigraph of an LD solution-a solution ϕ of the linearized minimal surface equation Lϕ := (∆ + 2)ϕ = 0 on S 2 \ L with logarithmic singularities at L. In order for a family of LD solutions to give rise to a minimal doubling, certain matching conditions related to the alignment of the LD solutions and the bridges must be satisfied. While the set of LD solutions is a linear space, these matching conditions are nonlinear and substantial work has gone into determining appropriately matched LD solutions and to further developing doubling constructions via linearized doubling [4,5,6]. The surfaces we construct here are defined in similar fashion to the initial surfaces described above. Importantly, however, the function ϕ whose bigraph defines the surface away from the bridges is now chosen to be a singular solution on S 2 \ L of the linearized Willmore equation (L − 2)Lϕ = ∆(Lϕ) = 0. More specifically, we can choose ϕ = c 1 + τ Φ where Φ is such an LD solution, τ > 0 is sufficiently small but otherwise arbitrary, and c 1 is a constant. The ability to prescribe c 1 allows us to rather easily match the asymptotics of ϕ to those of a collection of catenoidal bridges, each with waist radius τ , provided the set L of centers is carefully chosen. For each g ≥ 0 and all τ > 0 sufficiently small (depending on g), we then define in 5.12 a smooth genus g surface Σ g+1,τ ⊂ S 3 whose bending energy can be shown to satisfy W(Σ g+1,τ ) < 8π. As τ ց 0, stereographic projection of Σ g+1,τ converges in the sense of varifolds to the unit sphere S 2 ⊂ R 3 with multiplicity two. In particular, since stereographic projection is conformal and W is conformally-invariant, the stereographic image S g,v(τ ) of Σ g+1,τ in R 3 has bending energy less than 8π and isoperimetric ratio v = v(τ ) near zero. Applying a family of Möbius transformations to S g,v(τ ) we then obtain-for any v ∈ [v(τ ), 1)-a genus g surface S g,v with W(S g,v ) < 8π and isoperimetric ratio v(S g,v ) = v. From the construction of these comparison surfaces and the recent work of Mondino-Scharrer [13, Theorem 1.2], our Theorem 1.2 follows. Outline of our paper. After fixing notation and conventions in Section 2, we estimate in Section 3 the bending energy of the normal exponential graph Γ u Ω ⊂ S 3 of a function u defined on a domain Ω ⊂ S 2 . In Section 4, we define for each p ∈ S 2 and all small enough τ > 0 a catenoidal bridge K p,τ ⊂ S 3 centered at p and of size τ , and we estimate the bending energy of a K p,τ . In Section 5, after recalling certain facts from [4] regarding LD solutions, we construct in 5.12 for each m ∈ N and all sufficiently small τ > 0 a smoothly embedded genus m − 1 surface Σ m,τ ⊂ S 3 . Finally, we prove Theorem 1.2 by reducing the problem to showing W(Σ m,τ ) < 8π, which we do by combining earlier estimates on the bending energies of the corresponding catenoidal and graphical regions. Acknowledgements. We thank Nikos Kapouleas, whose constructions of minimal surfaces via linearized doubling inspired this approach to solving the Canham existence problem. Notation and conventions Throughout the paper, let S 3 ⊂ R 4 be the unit three-sphere and S 2 ⊂ S 3 the equatorial two-sphere defined by S 2 = S 3 ∩ {x 4 = 0}, where (x 1 , x 2 , x 3 , x 4 ) are the standard coordinates on R 4 . We define the Willmore bending energy W(Σ) of a (not necessarily closed) surface Σ ⊂ S 3 by W(Σ) = \|Σ\| + 1 4 Σ H 2 dσ, where dσ is the area form on Σ, and where H is its mean curvature in S 3 . Remark 2.1. Note that if S ⊂ R 3 is a stereographic image of Σ ⊂ S 3 , then W(Σ) = W(S), where W(S) was defined in (1.1). Notation 2.2. For any X ⊂ S 2 , we write d X for the distance from X, and define the δ-neighborhood of X by D X (δ) := {p ∈ S 2 : d X (p) < δ}. If X is finite we just list its points; for example, d q (p) is the geodesic distance between p and q and D q (δ) is the geodesic disc in S 2 with center q and radius δ. Cutoff functions. The following notation regarding cutoff functions is standard in gluing constructions [4]. Definition 2.4. We fix a smooth function Ψ : R → [0, 1] with the following properties: (i) Ψ is nondecreasing. (ii) Ψ ≡ 1 on [1, ∞) and Ψ ≡ 0 on (−∞, −1]. (iii) Ψ − 1 2 is an odd function. Given a, b ∈ R with a = b, we define smooth functions ψ cut [a, b] : R → [0, 1] by ψ cut [a, b] := Ψ • L a,b , where L a,b : R → R is the linear function defined by the requirements L(a) = −3 and L(b) = 3. Note that ψ cut [a, b] has the following properties: (i) ψ cut [a, b] is weakly monotone. (ii) ψ cut [a, b] = 1 on a neighborhood of b and ψ cut [a, b] = 0 on a neighborhood of a. (iii) ψ cut [a, b] + ψ cut [b, a] = 1 on R. Suppose now we have functions f 0 , f 1 , and ρ defined on some domain Ω. We define a new function (2.5) Ψ [a, b; ρ] (f 0 , f 1 ) := ψ cut [a, b] • ρ f 1 + ψ cut [b, a] • ρ f 0 . Note that Ψ[a, b; ρ](f 0 , f 1 ) depends linearly on the pair (f 0 , f 1 ) and transits from f 0 on Ω a to f 1 on Ω b , where Ω a and Ω b are subsets of Ω which contain ρ −1 (a) and ρ −1 (b) respectively, and are defined by Ω a = ρ −1 (−∞, a + 1 3 (b − a)) , Ω b = ρ −1 (b − 1 3 (b − a), ∞) , when a < b, and Ω a = ρ −1 (a − 1 3 (a − b), ∞) , Ω b = ρ −1 (−∞, b + 1 3 (a − b)) , when b < a. Clearly if f 0 , f 1 , and ρ are smooth then Ψ[a, b; ρ](f 0 , f 1 ) is also smooth. Normal Graphs and their Willmore energy Given a smooth function u defined on a domain Ω ⊂ S 2 , we define its normal graph over Ω by Γ u Ω = exp p (u(p)ν(p)) : p ∈ Ω ⊂ S 3 , where ν is the unit normal field on S 2 ⊂ S 3 given by ν(p) = (0, 0, 0, 1) for each p ∈ S 2 . If \|u\| < π/2, the map E u : Ω → Γ u Ω defined by E u (p) = exp p (u(p)ν(p)) is a diffeomorphism, and a short calculation shows the pullback of the area form dσ u on Γ u Ω is E * u dσ u = cos 2 (u) 1 + sec 2 (u)\|∇u\| 2 dσ, (3.1) where dσ := dσ 0 is the Riemannian area form of S 2 . Since the surfaces we construct are built from gluing together normal graphs, the following estimate of the bending energy of such a graph will be useful. Lemma 3.2. Suppose Ω ⊂ S 2 is a domain with smooth boundary, u ∈ C 2 (Ω), and that u C 2 (Ω) < ǫ for a given small ǫ > 0. Then W(Γ u Ω ) = \|Ω\| + 1 4 Ω (∆u)(Lu) dσ + 1 2 ∂Ω u ∂u ∂η ds + O(ǫ 4 ). Proof. Expanding (3.1) using the smallness of u and \|∇u\|, we have E * u dσ u = 1 − u 2 + 1 2 \|∇u\| 2 + O(ǫ 4 ) dσ. Because L is the linearized mean curvature operator and S 2 ⊂ S 3 is totally geodesic, the mean curvature H of Γ u Ω satisfies H = Lu + O(ǫ 4 ). In combination with the preceding, we have ( 1 4 H 2 + 1)E * u dσ u = 1 + 1 4 (Lu) 2 − u 2 + 1 2 \|∇u\| 2 + O(ǫ 4 ) dσ. The conclusion now follows by integrating over Γ u Ω and integrating Ω 1 2 \|∇u\| 2 dσ by parts. Catenoidal bridges and their Willmore energy Recall that the top half of a catenoid of size τ in Euclidean three-space can be written as a radial graph of the function ϕ cat : [τ, ∞) → R defined by ϕ cat (r) = τ arccosh r τ := τ log 2r τ + log 1 2 + 1 2 1 − τ 2 r 2 . (4.1) For future reference we record that ϕ ′ cat (r) = τ √ r 2 − τ 2 . (4.2) Assumption 4.3. We fix a small positive number α, for example α = 1/10, and assume hereafter that τ > 0 is as small as needed in terms of α. Definition 4.4 (Catenoidal bridges). Given p ∈ S 2 , we define the catenoidal bridge K p,τ to be the union of the graphs of ±ϕ cat • d p on the domain D p (τ α ) \ D p (τ ). The following estimate on the bending energy of a K p,τ should be compared with the estimate on the area of a portion of a Euclidean catenoid in [8]. Lemma 4.5. For any p ∈ S 2 and all sufficiently small τ > 0, W(K p,τ ) ≤ 2\|D p (τ α )\| + 8π 3 τ 2 \| log τ \|. Proof. Integrating the area form from (3.1) and estimating, we have 1 2 \|K p,τ \| = 2π τ α τ cos 2 ϕ cat (r) 1 + sec 2 ϕ cat (r)(ϕ ′ cat (r)) 2 sin r dr ≤ Cτ 2 + 2π τ α 9τ cos 2 ϕ cat (r) 1 + sec 2 ϕ cat (r)(ϕ ′ cat (r)) 2 sin r dr ≤ \|D p (τ α )\| + Cτ 2 + 5π 4 τ α 9τ (ϕ ′ cat (r)) 2 r dr ≤ \|D p (τ α )\| + Cτ 2 + 5π 4 τ 2 τ α 9τ 1 r + Cτ 2 r 3 dr ≤ \|D p (τ α )\| + Cτ 2 + 5π 4 τ 2 \| log τ \|, where we have used (4.2) to estimate (ϕ ′ cat (r)) 2 on (9τ, τ α ) as follows: (ϕ ′ cat (r)) 2 = τ 2 r 2 − τ 2 ≤ τ 2 r 2 + C τ 4 r 4 . Finally, from (A.4) in Appendix A, we have the estimate \|H\| ≤ Cτ \| log τ \| on K p,τ . In combination with the preceding, this implies \|W(K p,τ )\| ≤ 2\|D p (τ α )\| + Cτ 2 + 5π 2 τ 2 \| log τ \| + Cτ 2(1+α) \| log τ \| 2 ≤ 2\|D p (τ α )\| + 8π 3 τ 2 \| log τ \|. Remark 4.6. Although we will not need it, it is possible to prove the following strengthening of the estimate in Lemma 4.5: W(K p,τ ) = 2\|D p (τ α )\| + 2πτ 2 log 2τ α−1 − πτ 2 + O(τ 2(1+α) \| log τ \| 2 ). Proof of the Main Theorem In order to study singular solutions of the equation ∆(Lϕ) = 0, we first recall (cf. [4, Lemma 2.20]) the Green's function for L on S 2 . Lemma 5.1. The function G ∈ C ∞ ((0, π)) defined by G(r) := cos r log 2 tan r 2 + 1 − cos r has the following properties: (i) For each p ∈ S 2 , we have L(G • d p ) = 0 on S 2 \ {p, −p}. (ii) For small r > 0, G(r) = (1 + O(r 2 )) log r. (iii) The following estimate holds for r ∈ (τ α , 9τ α ) and k ∈ {0, 1, 2}: It follows from this that for r ∈ (τ α , 9τ α ) and k ∈ {0, 1, 2}, d k dr k (ϕ cat (r) − τ G(r) + τ log τ 2 cos r) ≤ Cτ 1+(2−k)α \| log τ \|.d k dr k (G(r) − log 2r τ − log τ 2 cos r) ≤ Cτ (2−k)α \| log τ \|. The conclusion follows from this and (4.1). In the remainder of this section, m ∈ N will denote a given natural number. For simplicity of notation, we will suppress the dependence of various constants on m. Assumption 5.2. We assume hereafter that τ > 0 is as small as needed in terms of m. We first define the set L ⊂ S 2 of centers of the bridges in the construction, or equivalently the set of logarithmic singularities of ϕ[m, τ ] defined later in 5.11. Although there is no function Φ[m] satisfying 5.4(i)-(iii) when m = 1, the following definition will be sufficient for our later applications in that case. (i) Φ ∈ C ∞ (S 2 \ L) and LΦ = 0 on S 2 \ L. (ii) Φ − log d L is bounded on some deleted neighborhood of L in S 2 . (iii) Φ isΦ = Ψ[ π 3 , π 2 ; d L ](G • d L , 0). Before we define the surfaces Σ m,τ used in Theorem 1.2, we need to extract from Φ the dominant singular and constant parts in the vicinity of L. Proof. For any c 0 ∈ R, it follows from 5.1 and 5.4 that Φ ′ as defined by (5.8) is bounded on D L (δ) \ L and satisfies LΦ ′ = 0 there. This implies the smoothness of Φ ′ on D L (δ) by standard removable singularity results. The symmetries imply that d p Φ ′ = 0 for each p ∈ L and Φ ′ (p) is independent of the choice of p ∈ L. There is therefore a unique c 0 ∈ R such that Φ ′ (p) = 0 for all p ∈ L. Corollary 5.9. For k ∈ {0, 1, 2}, Φ ′ satisfies \|∇ (k) Φ ′ \| ≤ C(d L ) 2−k on D L (δ). Remark 5.10. When m = 1, note that c 0 = 0 and Φ ′ ≡ 0. When m = 2, we have via 5.5 that c 0 = 1 − log 2 and Φ ′ ≡ 0. Definition 5.11. Given τ > 0, define ϕ = ϕ[m, τ ] ∈ C ∞ (S 2 \ L) by ϕ = c 1 + τ Φ, where L and Φ are as in 5.3 and 5.6, c 1 := τ log(2/τ ) − τ c 0 , and c 0 is as in 5.7. We are now ready to define the family of surfaces used in Theorem 1.2. Definition 5.12. Given τ > 0 as in 5.2, define the smooth surface Σ m,τ ⊂ S 3 to be the union over S 2 \ D L (τ ) of the graphs of ±ϕ gl , where ϕ gl : S 2 \ D L (τ ) → [0, ∞) is defined as follows: (i) On S 2 \ D L (2τ α ) we have ϕ gl = ϕ[m, τ ]. (ii) On D L (2τ α ) \ D L (τ ) we have (recall 2.5) ϕ gl = Ψ[τ α , 2τ α ; d L ](ϕ cat • d L , ϕ[m, τ ]). Finally, we define Ω : = S 2 \ D L (τ α ), so that (recall 4.4) Σ m,τ = ( p∈L K p,τ ) ∪ Γ ϕ gl Ω ∪ Γ −ϕ gl Ω . Lemma 5.13. For all sufficiently small τ > 0, W(Γ ϕ gl Ω ) ≤ \|S 2 \| − \|D L (τ α )\| − 11mπ 6 τ 2 \| log τ \|. Proof. We first prove the inequality in the case where m ≥ 2. Since ϕ gl = ϕ = c 1 + τ Φ on Ω \ A L , where A L := D L (2τ α ) \ D L (τ α ) is the gluing region, Ω (∆ϕ gl )(Lϕ gl )dσ = AL (∆ϕ gl )(Lϕ gl )dσ + 2c 1 τ Ω\AL ∆Φdσ = AL (∆ϕ gl )(Lϕ gl )dσ + 2c 1 τ ∂(Ω\AL) ∂Φ ∂η ds. It follows from 5.7, 5.11, and 5.12 that ϕ gl = τ G • d L − τ log τ 2 cos d L + Ψ[τ α , 2τ α ; d L ](ϕ, ϕ), on A L , where ϕ := ϕ cat • d L − τ G • d L + τ log(τ /2) cos d L , ϕ := c 1 (1 − cos d L ) + τ Φ ′ . Therefore, we have on A L ∆ϕ gl = −2τ (G • d L − log(τ /2) cos d L ) + ∆(Ψ[τ α , 2τ α ; d L ](ϕ, ϕ)), Lϕ gl = LΨ[τ α , 2τ α ; d L ](ϕ, ϕ), and moreover from Lemma 5.1(iii) and (5.9) that \|∆ϕ gl \| ≤ Cτ \| log τ \| and \|Lϕ gl \| ≤ Cτ \| log τ \|. Hence, Ω (∆ϕ gl )(Lϕ gl )dσ = 2c 1 τ ∂(Ω\AL) ∂Φ ∂η ds + O(τ 2(1+α) \| log τ \| 2 ). (5.14) We now apply Lemma 3.2. First, using 5.1 and Lemma 5.7, it is straightforward to check that ϕ gl C 2 (Ω) ≤ Cτ 1−2α . Working (5.14) into 3.2 establishes W(Γ ϕ gl Ω ) = \|S 2 \| − \|D L (τ α )\| + 1 2 ∂Ω ϕ gl ∂ϕ gl ∂η ds + 1 2 c 1 τ ∂(Ω\AL) ∂Φ ∂η ds + O(τ 2(1+α) \| log τ \| 2 ) + O(τ 4(1−2α) ). The desired inequality follows from this by using that ϕ gl = ϕ cat • d L on a neighborhood of ∂Ω and estimating the integrals using (4.1) and (4.2), Lemma 5.1, and Lemma 5.7. We now consider the case where m = 1. Since LΦ is now supported on A 1 := D L ( π 2 ) \ D L ( π 3 ) (recall 5.6), Ω (∆ϕ gl )(Lϕ gl )dσ = AL (∆ϕ gl )(Lϕ gl )dσ + 2c 1 τ ∂(Ω\AL) ∂Φ ∂η ds + τ 2 A1 (LΦ)(∆Φ)dσ, where A L is as before. Estimating the integral over A L in exactly the same way as before, we find AL (∆ϕ gl )(Lϕ gl )dσ = O(τ 2(1+α) \| log τ \| 2 ). Next, we have \|LΦ\| ≤ C and \|∆Φ\| ≤ C on A 1 , so τ 2 A1 (LΦ)(∆Φ)dσ = O(τ 2 ). The proof is completed by working the preceding into Lemma 3.2 and estimating as before. We are now ready to prove the main result of this paper: Proof of Theorem 1.2. By recent work of Mondino-Scharrer [13,Theorem 1.2], it suffices to show, for each g ∈ {0, 1, . . . } and v ∈ (0, 1), that there exists a smoothly embedded surface S = S g,v ⊂ R 3 of genus g and isoperimetric ratio v(S) = v and W(S) < 8π. In fact, if for any small v ∈ (0, 1) we can construct such a surface, then by using conformal invariance of the bending energy W(S) and applying a family of Möbius transformations to S ⊂ R 3 which dilate out from a point p + ∈ S and contract in to some other point on p − ∈ S, we thereby obtain a family of surfaces with the same value of the bending energy W and all larger isoperimetric ratios v < 1, since this family of surfaces converges smoothly away from p − to a round sphere (with v = 1). Now fix m ∈ N. For all small enough τ > 0, define Σ m,τ as in 5.12 to be a smoothly embedded genus g = m − 1 surface in S 3 . Let Y : S 3 \ {(0, 0, 0, 1)} → R 3 = {x 4 = 0} be stereographic projection from (0, 0, 0, 1) ∈ S 3 ⊂ R 4 . Since Σ m,τ converges in the sense of varifolds to 2S 2 as τ ց 0, since Y is conformal, and Y \| S 2 = Id S 2 , where we regard S 2 = R 3 ∩ S 3 as both a subset of R 3 and of S 3 , it follows that lim τ ց0 v(Y (Σ m,τ )) = 0. Thus to prove the theorem, by Remark 2.1 it suffices to show that W(Σ m,τ ) < 8π for all τ small enough. To do this, we combine the estimates in 4.5 and 5.13: W(Σ m,τ ) = mW(K p,τ ) + 2W(Γ ϕ gl Ω ) ≤ 2\|D L (τ α )\| + 8π 3 mτ 2 \| log τ \| + 2\|S 2 \| − 2\|D L (τ α )\| − 11π 3 mτ 2 \| log τ \| = 2\|S 2 \| − mπτ 2 \| log τ \|. Remark 5.15. The method used here leads to the construction of other comparison surfaces with W < 8π and bridges centered on other symmetric configurations of points. One such configuration consists of the vertices of a regular tetrahedron. It would be interesting to know whether Canham minimizers for a given genus and small isoperimetric ratio have any particular symmetries. In this appendix, we estimate the mean curvature of a small catenoidal bridge in S 3 . This was done in [6, Example 2.15], but we summarize the argument in order to keep the exposition self-contained. Define a parametrization E : (0, π) × (−π, π) × (− π 2 , π 2 ) → S 3 by E(r, θ, z) =(sin r cos θ cos z, sin r sin θ cos z, cos r cos z, sin z). We take (r, θ, z) as local coordinates for S 3 . The pullback metric is E * g = cos 2 z dr 2 + sin 2 rdθ 2 + dz 2 , and the only nonvanishing Christoffel symbols are (A.1) Γ r rz = Γ r zr = Γ θ θz = Γ θ zθ = − tan z, Γ r θθ = − sin r cos r, Γ θ rθ = Γ θ θr = cot r, Γ z rr = cos z sin z, Γ z θθ = sin 2 r sin z cos z. Define a map X : [−s, s] × (−π, π) → R 3 by (A.2) X(s, ϑ) = (r(s, ϑ), θ(s, ϑ), z(s, ϑ)) = (τ cosh s, ϑ, τ s), where s is defined by the equation τ cosh s = 9τ α . Calculation shows that the pullback metric in (s, ϑ) coordinates is (A.3) X * E * g = r 2 (1 − tanh 2 s sin 2 z)ds 2 + cos 2 z sin 2 r dϑ 2 , where r = r(s, ϑ) and z = z(s, ϑ), and that ν = (tanh s ∂ z − sec 2 z sech s ∂ r )/ 1 + tan 2 z sech 2 s is a unit normal field along the image of X. We compute the second fundamental form A of X using the formula A = X k ,αβ + Γ k lm X l ,α X m ,β g kn ν n dx α dx β , where X is as in A.2, we have renamed the cylinder coordinates (x 1 , x 2 ) := (s, ϑ), and Greek indices take the values 1 and 2 while Latin indices take the values 1, 2, 3, corresponding to the coordinates r, θ, z. Using the preceding and the Christoffel symbols in (A.1), we find A = τ 2 tanh s tan z + 1 2 sinh 2 s sin 2z − τ ds 2 + 1 2 (sin 2r sech s + sin 2 r sin 2z tanh s) dϑ 2 1 + tan 2 z sech 2 s = (1 + O(z 2 )) τ (−ds 2 + dϑ 2 ) + O(r 2 z)ds 2 + O(r 3 + r 2 z)dϑ 2 , where in the second equality we have estimated using that 1 + tan 2 z sech 2 s = 1 + O(z 2 ) and that 1 2 sin 2r sech s = τ + O(r 3 ). Finally, using that r 2 g ss = 1 + O(z 2 ) and r 2 g ϑϑ = 1 + O(z 2 + r 2 ) we estimate r 2 H = O τ z 2 + r 2 \|z\| + τ r 2 . (A.4) Notation 2. 3 . 3We denote by L = ∆ + 2 and (L − 2)L = ∆L = ∆(∆ + 2) the area-Jacobi and W-Jacobi operators of S 2 ⊂ S 3 , where ∆ denotes the Laplacian on S 2 . Definition 5. 3 . 3Let L = L[m] := (cos 2πk m , sin 2πk m , 0, 0) ∈ S 2 : k = 1, . . . , m . We next consider a particular discrete family of LD solutions which were studied in [4, Def. 6.1]: Lemma 5.4. For m ≥ 2, there is a function Φ = Φ[m] uniquely determined by the following: invariant under the group of isometries of S 2 which fix L[m] as a set. Proof. This follows immediately from [4, Lemma 3.10]. Remark 5.5. When m = 2, Φ depends only on d L ; in particular, (cf. [4, Def. 2.18]) we have Φ = G • d L + (1 − log 2) cos d L = 1 + cos d L log tan dL 2 . Definition 5 . 6 . 56Let Φ = Φ[m] be as in 5.4 for m ≥ 2 and define Φ[m] for m = 1 by Lemma 5 . 7 . 57There is a unique c 0 = c 0 [m] ∈ R such that the function Φ ′ defined by the decompositionΦ = G • d L + c 0 cos d L + Φ ′ on D L (δ), δ := 1/(10m) (5.8)satisfies Φ ′ (p) = 0 for each p ∈ L. Moreover, Φ ′ ∈ C ∞ (D L (δ)) and d p Φ ′ = 0 for each p ∈ L. Remark 5.16. While Φ[m] and Σ m,τ have nontrivial symmetries for m > 1, it is possible to construct comparison surfaces with W < 8π and m > 1 bridges centered on configurations of points L with trivial symmetry by generalizing the construction of Φ[1] and cutting off G • d L to zero. 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[ "CLOSED FORM ANALYTICAL MODEL FOR AIRFLOW AROUND 2-DIMENSIONAL COMPOSITE AIRFOIL VIA CONFORMAL MAPPING", "CLOSED FORM ANALYTICAL MODEL FOR AIRFLOW AROUND 2-DIMENSIONAL COMPOSITE AIRFOIL VIA CONFORMAL MAPPING" ]	[ "Rita Gitik \nUNIVERSITY OF MICHIGAN\nANN ARBOR\n48109MI\n", "William B Ribbens \nUNIVERSITY OF MICHIGAN\nANN ARBOR\n48109MI\n" ]	[ "UNIVERSITY OF MICHIGAN\nANN ARBOR\n48109MI", "UNIVERSITY OF MICHIGAN\nANN ARBOR\n48109MI" ]	[]	This paper presents a method of computing section lift characteristics for a 2-dimensional airfoil with a second 2-dimensional object at a position at or ahead of the leading edge of the airfoil. Since both objects are 2-dimensional, the analysis yields a closed form solution to calculation of the airflow over the airfoil and second object, using conformal mapping of analytically closed form airflow velocity vector past two circular shaped objects in initial complex plane, using a standard air flow model for each object individually. The combined airflow velocity vector is obtained by linear superposition of the velocity vector for the two objects, computed individually. The lift characteristics are obtained from the circulation around the airfoil and second object which is computed from the combined closed form velocity vector and the geometry along the contour integral for circulation. The illustrative exam-ple considered in this paper shows that the second object which is essentially a cylinder whose diameter is approximately 9% of the chord length of the airfoil reduces the section lift coefficient by approximately 6.3% from that of the airfoil alone. arXiv:1712.09730v1 [physics.flu-dyn]	null	[ "https://arxiv.org/pdf/1712.09730v1.pdf" ]	119,467,263	1712.09730	a7ec7abf84f7945d7c81b55e0317bc7d11a2e035	CLOSED FORM ANALYTICAL MODEL FOR AIRFLOW AROUND 2-DIMENSIONAL COMPOSITE AIRFOIL VIA CONFORMAL MAPPING Rita Gitik UNIVERSITY OF MICHIGAN ANN ARBOR 48109MI William B Ribbens UNIVERSITY OF MICHIGAN ANN ARBOR 48109MI CLOSED FORM ANALYTICAL MODEL FOR AIRFLOW AROUND 2-DIMENSIONAL COMPOSITE AIRFOIL VIA CONFORMAL MAPPING 1 2 This paper presents a method of computing section lift characteristics for a 2-dimensional airfoil with a second 2-dimensional object at a position at or ahead of the leading edge of the airfoil. Since both objects are 2-dimensional, the analysis yields a closed form solution to calculation of the airflow over the airfoil and second object, using conformal mapping of analytically closed form airflow velocity vector past two circular shaped objects in initial complex plane, using a standard air flow model for each object individually. The combined airflow velocity vector is obtained by linear superposition of the velocity vector for the two objects, computed individually. The lift characteristics are obtained from the circulation around the airfoil and second object which is computed from the combined closed form velocity vector and the geometry along the contour integral for circulation. The illustrative exam-ple considered in this paper shows that the second object which is essentially a cylinder whose diameter is approximately 9% of the chord length of the airfoil reduces the section lift coefficient by approximately 6.3% from that of the airfoil alone. arXiv:1712.09730v1 [physics.flu-dyn] Introduction Extremely precise calculations of airflow past an airplane or other aircraft structures are accomplished with CFD. Computational results are in the form of numerical samples at discrete locations around the surfaces of the aircraft. However, CFD computations do not yield a closed form analytic model for such airflow. Closed form analytic solutions to airflow is possible for 2D models. 1 Visiting Scholar, Department of Mathematics. 2 One method is a conformal mapping. Traditional applications map a circle or other simple shape in a complex plane z = x+iy to an airfoil shape in the conformal map complex plane η = + iν, e.g. η = z + a 2 z . This paper presents a closed form solution to a 2D model of airflow around a pair of objects.The objects consist of a primary circle in the z-plane which maps to an airfoil in the η-plane. The second object is a circle which is much smaller than the primary and is upstream of the primary circle in an incident uniform airflow field. This secondary object maps to a relatively small ellipse with near unity ratio of major to minor axis, in contact with the airflow at its leading edge with the diameter along the airfoil cord. With respect to the airflow over winglike structures, the conformal map of the two z-plane circles can be interpreted as the intersection of a wing with a leading edge object, both having infinite dimensions orthogonal to the η-plane. This is the interpretation of traditional 2D airfoil representations. For the present analysis the airflow around the 2 mapped structures in the η-plane are taken to be the superposition of the conformal mapping of the individual airflow around the 2 objects in the z-plane. The present analysis assumes that the incident airflow on the primary surface in the z-plane is a uniform flow field at an angle of attack of α 0 . This assumption is equivalent to ignoring the influence of the secondary object on the flow field of the primary. A model for the velocity vector of the airflow around each of the two circular objects in the z-plane at any location relative to the geometry of the object including two coordinates of the center of each circle has been known for a long time. For the purpose of the present paper, the airflow around a concentric circle of the first object due to the influence of the second object is computed. The velocity vector of interest is the superposition of the airflow due to the second object with the model for the air velocity along the same contour of the first object. Conformal mapping of the combined velocity along the contour to the η-plane yields a model for the air velocity vector from which circulation Γ can be computed. This circulation provides the basis for computing the section lift coefficient of the two objects in the η-plane. Airflow around One Object The closed form calculation of inviscid airflow around an object in terms of tabulated functions (e.g. sin, cos) requires that the object have separable boundary conditions in the corresponding coordinate system. The airflow is derived from a potential function which is here denoted φ and an associated stream flow function, here denoted ψ. Both functions satisfy Laplaces equation, as explained in [1] p.81: 2 φ = 0 2 ψ = 0. An example of an object satisfying the above conditions for a closed form solution in 3 dimensions is a sphere. However, a sphere is not a useful shape for application to aircraft airflow analyses. On the other hand, an object having potential application for closed form airflow is a cylinder with an infinitely long axis. This hypothetical shape can be meaningfully interpreted as a 2-dimensional representation as has, of course, been used for a very long time to asses sectional wing airfoil characteristics. One of the well-known analytic methods for calculating the equivalent of airflow over a 2-dimensional airfoil shape has involved conformal mapping of a circle in one complex plane to an airfoil shape in another complex plane. This airfoil shape can be made to have a contour with a rounded leading edge and a sharp trailing edge. By proper selection of the location of the circle in the original complex plane relative to the coordinate origin, the conformal mapping yields an airfoil shape with a desired thickness and camber. For the purposes of the present paper, the second object is a circle in the z-plane, having a radius that is much less than that of the first object. The origin of the second object is at z = x cc + iy cc , where x cc > 0, y cc = 0. The second object is mapped to the η-complex plane via a similar transformation to the first object. The mapped second object is a small ellipse which is outside the airfoil but is tangent to the leading edge. The incident airflow is uniform with a velocity at ∞ of V 0 and at an angle α 0 to the x-axis. This incident airflow into η-plane with the equivalent of a positive angle of attack (AOA) of α 0 on the mapped airfoil. The object of the present paper is to calculate the airflow around the surface of the mapped airfoil in the η-plane due to the influence of both objects. Since the method for the airflow is linear, the combined airflow is obtained by superposition of the airflow along the airfoil due to each object separately. The z-plane complex variable is denoted z = x + iy where x is the real axis and iy is the imaginary axis. The other complex variable is denoted η, η = + iν where is the real axis and iν is the imaginary axis. The conformal mapping from z to η is of the generic form η = z + a 2 z (1) where a is a real parameter. The airflow local velocity in either complex plane is derived from a complex function which is denoted W and is given by W = φ + iψ where φ is velocity potential function and ψ is a stream function. Both of these functions satisfy Laplaces equation for irrotational flow: 2 φ = 0 2 ψ = 0 For simplicity of analysis, it is assumed that the object 1 circle center is on the x-axis at x c . The conformal map of this circle yields a symmetric airflow. It has long been known that the complex potential function for uniform flow at and angle α 0 with velocity at infinity at V 0 and with circulation Γ about a cylinder with center at x c is given by W = V 0 (z − x c )e −iα0 + (a − x c ) 2 e iα0 (z − x c ) + iΓ 2π ln (z − x c )e −iα0 (a − x c )(2) It is also well known, cf. [2] p.50, that the air velocity in the z-plane (V (z)) is given by: V (z) = dW dz = V 0 (e −iα0 − (a − x c ) 2 (z − x c ) 2 e iα0 + iΓ 2π(z − x c )(3) Similarly the air velocity in the η-plane is given by V (η) = dW dη = dW dz dz dη = V (z) dz dη(4) The latter factor dz dη is given by dz dη = 1 dη dz = z 2 z 2 − a 2(5) Substituting V (z) from equation 3 and dz dη from equation 5, yields the following expression for V (η). V (η) = V 0 e −iα0 − (a − x c ) 2 e iα0 (z − x c ) 2 + iΓ 2π(z − x c ) z 2 z 2 − a 2(6) It is also well known that z = a corresponds to the trailing edge of the airfoil in the η-plane. The velocity at the trailing edge would be infinite unless the first factor V (z) at z = a is zero, as shown in [2] p.52. That is V (z)\| z=a = 0. This condition is satisfied for a specific value for the circulation Γ which results from the well known Kutta-Joukowski condition. This value for Γ is given by: Γ = 4π(a − x c )V 0 sin(α 0 )(7) One of the important variables in the application of airflow computation via conformal mapping is the section lift which is denoted l. This is technically equivalent to the lift per unit span length of the theoretical infinitely span wing which has an airfoil given by the conformal mapping of the first object. The section lift can be computed directly from the circulation Γ around the airfoil from the following relationship: l = ρV 0 Γ (8) where ρ =air density. In addition the circulation can be computed from the velocity vector along streamlines with the following formula: Γ = cs V · ds(9) where c s =closed contour around the airfoil along streamlines, V =vector notation for the velocity, ds =differential length vector along c s . As is also well known, the section lift was characterized by a section lift coefficient c l as given below: l = qSC l(10) where q = 1 2 ρV 2 0 =dynamic pressure, S =section area per unit span length= c =section chord length. The section lift coefficient C l is computed using Γ as given by C l = l Sq = 2ρV 0 Γ ρV 2 0 c = 2Γ V 0 c(11) For the purpose of evaluating the present method of calculating a closed form solution to the 2 object airflow, a numerical calculation of Γ and C l was developed using the closed form solution. As a means of assessing the numerical calculation of Γ, the procedure was first applied to the first single object in which the z-plane circle maps to the airfoil. The contour integral for Γ in terms of the complex variables can be formulated as the contour integral of a scalar complex variable. This interpretation utilizes the following notation: V (η) = u + iv ds = d + idν The dot product V · ds can be expressed as the following: V (η) · ds = ud + vdν = Re[V (η)dη * ] Thus, the contour integral of a vector product becomes the following Γ = cs V · ds = cs Re[V (η)dη * ](12) The numerical evaluation of Γ is accomplished by computing the airflow velocity V (η) corresponding to the conformal mapping of the airflow velocity at K points around the circle in the z-plane. The algorithm for calculating V (η k ) is based upon equation 6 evaluated at K points z(k) which are computed as follows: for k = 1 : K θ(k) = 2πk K x(k) = a 1 cos(θ(k)) + x c y(k) = a 1 sin(θ(k)) + y c z(k) = x(k) + iy(k) where a 1 = 1.2 x c = 0.200 y c = 0 The conformal mapping from z(k) to η(k) is given by η(k) = z(k) + a 2 z(k) where a = 1.4306. The differential complex length δs(k) in the formula for Γ is computed as follows: s(n) = η(k) − η(k − 1), k = 1, 2, · · · , K(13) where η(0) = 2a. For sufficiently large K the contour integral for Γ is closely approximated by the following summation: Γ = cs V · ds ∼ = K k=1 Re[V (η(k))δs * (k)](14) A computation was made for a specific example with the following parameters: V 0 = 169f t/sec (i.e. 100Kt) α 0 = 0.0931rad K = 250 Using the Kutta-Joukowski condition in the form of equation 7, the circulation is given by: Γ = 252.02 The numerical evaluation of equation 14 yields a value for Γ: Γ = 252.25 The relatively close agreement between the two calculations for Γ, which ideally should be identical, indicates that the K used is sufficient for the purposes of the present paper. The lift coefficient, computed using equation 11 is given by C l = 0.5362 A similar calculation for α 0 = 0 yields C l = 0.0031. The slope of the section lift coefficient which is denoted C lα is given by C lα = ∂C l ∂α = 0.5362 − 0.0031 0.0931 = 5.72(15) The value for c lα is consistent with many geometrically similar airfoils, as published by NACA during the post WWII years. Airflow around Two Objects For an understanding of the present method it is, perhaps, helpful to review the geometry of the two objects in both the z-plane and the η-plane. The second object is a circle in the z-plane which yields an ellipse at the leading edge of the airfoil with the conformal map defined above. The ratio of the major to minor axes of this ellipse is 1.045 which is very close to that ratio for a circle, which is 1. The method of calculating the airflow around the airfoil which is created by conformal mapping the first object circle due to the influence of the second object begins with calculation in the z-plane. For this z-plane calculation, a new coordinate system, which is denoted z 2 , with an origin at the center of the second object center is given by: z 2 = z − x cc(16) With this coordinate system the airflow at any point in the z-plane can be derived from the potential function, which is denoted W 2 . For this function a uniform airflow with velocity V 0 at infinity, which is at angle α 0 to the x-axis at the second object, the function W 2 (z 2 ) is given by W 2 (z 2 ) = V 0 z 2 e −iα0 + a z c e −iα0 z 2 + iΓ 2 ln z 2 a c e −iα0(17) where a c =radius of the second object circle Γ 2 =circulation about the second object. The velocity of any point z 2 in the z-plane, which is denoted V 2 (z 2 ), is given by: V 2 (z 2 ) = dW 2 (z 2 ) dz 2 = V 0 e −iα0 − a 2 c z 2 2 e −iα0 + iΓ 2 2πz 2(18) The corresponding velocity in the z-plane as a function of the original complex coordinate z is denoted V 2 (z) and is given by: V 2 (z) = dW 2 (z 2 ) dz 2 dz dz 2(19) where the second factor on the right hand side is unity. The resulting V 2 (z) is given by: V 2 (z) = V 0 e −iα0 − a 2 c e −iαc (z − x cc ) 2 + iΓ 2 2π(z − x cc )(20) The calculation of the velocity component along the airfoil or along contour c s in the η-plane is based upon calculation of V 2 (z) for the same z k points used in the calculation of V (η) for the first object. This calculation yields the velocity vector V 2 (η k ) which is given by V 2 (η k ) = V 2 (z k ) dz dη = V 2 (z k ) z 2 k (z 2 k − a 2 )(21) As in the case of the velocity V (η k ) for the first object, the requirement of a finite value for V 2 (η k )\| z k =a is stated in the condition: V 2 (z k )\| z k =a = 0 This latter condition requires a specific value for Γ 2 \| z k =a which is given by: Γ 2 (a) = 2πi(a − x cc )V 0 e −iα0 − a 2 c (a − x cc ) 2 e iα0(22) The above value for Γ 2 is used in equation 20 to calculate V 2 (z) and the result of this calculation about the airfoil which is denoted Γ 2 is computed using the following modified version of equation 14: Γ 2 = K k=1 Re[V 2 (η(k))δs * (k)] where δs(k) is defined in equation 14. The total circulation, which is denoted Γ T is the linear superposition of Γ and Γ 2 : Γ T = Γ + Γ 2 where Γ is defined in equation 14. The section lift coefficient for the combined 2 objects, which is denoted C lT , can be obtained using equation 11 with Γ T substituted for Γ. C lT = 2Γ T V 0 c For the specific parameters of the objects, including size and location, and for the incident air velocity V 0 at angle α 0 to the x-axis, C lT is computed to be: C lT = 0.5024 This constitutes a reduction in C l from the single object case (C l = 0.5362) of approximately 6.3%. The combined lift coefficient for α 0 = 0 is denoted C lT (0) which is computed to be C lT (0) = −0.0101 The lift slope for the combined objects, which is denoted C lT α , is given by: C lT α = ∂C lT ∂α = 0.5024 + 0.0101 0.0931 = 5.475 Thus, the influence of the second object on the section lift coefficient is to reduce C lT as well as to reduce the section lift coefficient slope by about 4.5%. Conclusion In principle, the method of this paper could be used to calculate in closed form the air velocity vector around an airfoil with a leading edge object. However, the second object in the z-plane must be a circle to have a closed form solution for the airflow. The shape of the leading edge object is determined by the size and location of the second object circle. Thus, in practice, only certain leading edge shapes can be generated by conformal mapping of the second object circle. Figure 1 1is a plot of the two objects in the z-plane. The incident air velocity moves from right to left in this figure at an angle α 0 relative to the x-axis. Figure 2 2is a plot of the two objects in the η-plane. The air velocity vector is also at angle α 0 relative to the real axis of η (i.e. ). Figure 1 1Figure 1 Figure 2 2Figure 2 Professor Emeritus, Department of Engineering and Computer Science and Professor Emeritus, Department of Aerospace Engineering. Date: August 11, 2018. 8 RITA GITIK 1 AND WILLIAM B. RIBBENS 2 UNIVERSITY OF MICHIGAN, ANN ARBOR, MI, 48109 RITA GITIK 1 AND WILLIAM B. RIBBENS 2 UNIVERSITY OF MICHIGAN, ANN ARBOR, MI, 48109 . Arnold M Kuethe, Foundations of Aerodynamic. WileyArnold M. Kuethe, Foundations of Aerodynamic, Wiley, 1959. H Ira, Albert E Abbot, Von Doenhoff, Theory of Wing Sections. DoverIra H. Abbot and Albert E. Von Doenhoff, Theory of Wing Sections, Dover, 1959.	[]
[ "Interface shapes in microfluidic porous media: conditions allowing steady, simultaneous two-phase flow", "Interface shapes in microfluidic porous media: conditions allowing steady, simultaneous two-phase flow", "Interface shapes in microfluidic porous media: conditions allowing steady, simultaneous two-phase flow", "Interface shapes in microfluidic porous media: conditions allowing steady, simultaneous two-phase flow" ]	[ "S J Cox \nDepartment of Mathematics\nDepartment of Mathematics\nAberystwyth University\nSY23 1HHCeredigionUK\n", "A Davarpanah \nAberystwyth University\nSY23 1HHCeredigionUK\n", "W R Rossen \nDelft University of Technology\n2628CN / 2600GADelftNetherlands\n", "S J Cox \nDepartment of Mathematics\nDepartment of Mathematics\nAberystwyth University\nSY23 1HHCeredigionUK\n", "A Davarpanah \nAberystwyth University\nSY23 1HHCeredigionUK\n", "W R Rossen \nDelft University of Technology\n2628CN / 2600GADelftNetherlands\n" ]	[ "Department of Mathematics\nDepartment of Mathematics\nAberystwyth University\nSY23 1HHCeredigionUK", "Aberystwyth University\nSY23 1HHCeredigionUK", "Delft University of Technology\n2628CN / 2600GADelftNetherlands", "Department of Mathematics\nDepartment of Mathematics\nAberystwyth University\nSY23 1HHCeredigionUK", "Aberystwyth University\nSY23 1HHCeredigionUK", "Delft University of Technology\n2628CN / 2600GADelftNetherlands" ]	[]	Microfluidic devices offer unique opportunities to directly observe multiphase flow in porous media. However, as a direct representation of flow in geological pore networks, conventional microfluidics face several challenges. One is that simultaneous two-phase flow is not possible in a two-dimensional network without fluctuation occupancy of pores. Nonetheless, such flow is possible in a microfluidic network if wetting phase can form a bridge across the gap between solid surfaces at a pore constriction while non-wetting phase flows through the constriction. We call this phenomenon "bridging".Here we consider the conditions under which this is possible as a function of capillary pressure and geometry of the constriction. Using the Surface Evolver program, we determine conditions for stable interfaces in a constriction, the range of capillary pressures at which bridging can occur, and those where the wetting phase would invade and block the constriction to the flow of the non-wetting phase ("snap-off").We assume that the channels have uniform depth, vertical walls, and flat bottom and top surfaces, and that one phase perfectly wets the solid surfaces. If the constriction is long and straight, snap-off occurs at the same capillary pressure as bridging. For long, curved channels, snap-off happens as liquid imbibes before bridging can occur.For constrictions between cylindrical pillars, however, there is a range of capillary pressures at which bridging is stable; the range is greater the narrower the diameter of the cylinders relative to the width of the constriction.For smaller-diameter pillars, the phenomenon of "Roof" snap-off, that is, snap-off as non-wetting phase invades a downstream pore body, is not possible.We relate these results to the shape of pore networks commonly used in microfluidic studies of two-phase flow to consider whether two-phase flow is possible in these networks without fluctuating pore occupancy.	10.1007/s11242-023-01905-9	[ "https://arxiv.org/pdf/2108.05144v2.pdf" ]	237,054,325	2108.05144	419cbebb42f61d70099534371acce13d72ecbd38	Interface shapes in microfluidic porous media: conditions allowing steady, simultaneous two-phase flow 13 Aug 2021 S J Cox Department of Mathematics Department of Mathematics Aberystwyth University SY23 1HHCeredigionUK A Davarpanah Aberystwyth University SY23 1HHCeredigionUK W R Rossen Delft University of Technology 2628CN / 2600GADelftNetherlands Interface shapes in microfluidic porous media: conditions allowing steady, simultaneous two-phase flow 13 Aug 2021Preprint submitted to Elsevier August 16, 2021arXiv:2108.05144v2 [physics.flu-dyn]Porous mediaInterfacesCapillary pressureSnap-off Microfluidic devices offer unique opportunities to directly observe multiphase flow in porous media. However, as a direct representation of flow in geological pore networks, conventional microfluidics face several challenges. One is that simultaneous two-phase flow is not possible in a two-dimensional network without fluctuation occupancy of pores. Nonetheless, such flow is possible in a microfluidic network if wetting phase can form a bridge across the gap between solid surfaces at a pore constriction while non-wetting phase flows through the constriction. We call this phenomenon "bridging".Here we consider the conditions under which this is possible as a function of capillary pressure and geometry of the constriction. Using the Surface Evolver program, we determine conditions for stable interfaces in a constriction, the range of capillary pressures at which bridging can occur, and those where the wetting phase would invade and block the constriction to the flow of the non-wetting phase ("snap-off").We assume that the channels have uniform depth, vertical walls, and flat bottom and top surfaces, and that one phase perfectly wets the solid surfaces. If the constriction is long and straight, snap-off occurs at the same capillary pressure as bridging. For long, curved channels, snap-off happens as liquid imbibes before bridging can occur.For constrictions between cylindrical pillars, however, there is a range of capillary pressures at which bridging is stable; the range is greater the narrower the diameter of the cylinders relative to the width of the constriction.For smaller-diameter pillars, the phenomenon of "Roof" snap-off, that is, snap-off as non-wetting phase invades a downstream pore body, is not possible.We relate these results to the shape of pore networks commonly used in microfluidic studies of two-phase flow to consider whether two-phase flow is possible in these networks without fluctuating pore occupancy. artificial network (from [10]). (c) Disordered with rectangular pore network (from [16]. (d) Network based on a cross-section of rock (from [12]). Introduction Microfluidic systems, or micromodels [8], are useful for the study of flow in subsurface (geological) porous media [6]. They have the distinct advantage that one can see the flow dynamics directly, allowing observation of interfacial interactions, mass-transfer processes, phase behaviour and wetting transitions, for example. Such insight is likely to be of use in determining the dynamics of fluids in a range of applications such as oil recovery, aquifer remediation, and carbon capture and storage. A microfluidic porous medium usually has a geometry consisting of channels of uniform depth bounded by vertical walls. Viewed from above, as in the examples in figure 1, the network can have nearly any (2D) shape desired. We call solid barriers to flow (defined by vertical walls) "pillars", and consider two-phase flow consisting of a non-wetting phase (NWP) which, in those pores that it occupies, occupies the interior of the channels, and a wetting phase (WP) that occupies the corners where pillars meet near the top and bottom surfaces (figure 2(a)). In a complex network of channels there will be narrower and wider regions: we call the narrow locations "pore throats" and the wider locations "pore bodies". Multiple pore throats may connect to one pore body in a network. Despite the relative simplicity of a 2D microfluidic network, there are many similarities to 3D pore networks in geological formations, making them a useful proxy for predicting flow characteristics there. These include non-uniform channels/pores, and in particular the presence of pore throats, where capillary forces are significant. Moreover, it is straightforward to control and adjust the wettability, e.g. by treating surfaces/coating walls with specific minerals, and to capture a wide range of possible pore geometries. It is also possible to get multiple 3D [9]. Black is NWP, gray squares represent pillars (which could have any shape), pale gray is WP. Periodic boundary conditions apply to top and bottom, and left and right edges. Capillary pressure is sufficient for NWP to enter 50% of pore throats. This is just past the point where NWP can flow from both top to bottom and left to right. Note there is no path for WP flow from either top to bottom or left to right without bridging. layers of pores using 3D printing; we do not discuss such networks further here. Nonetheless, care is needed in extrapolating directly from flow in microfluidics to flow in other porous media. In particular, the channel depth is uniform in most current designs of microfluidic channels (figure 1), and so the expansion in the channel from a pore throat into a pore body is less extreme than in geological porous media. In rheological terms this tends to emphasize shearing over elastic contributions to the viscosity. This also often means that the widths observed from above are not the most important widths for estimating capillary forces. Moreover, the network itself is two-dimensional (2D), although the channels are 3D of course. In a 2D network, simultaneous, steady two-phase flow is not possible without fluctuating occupancy of pores [7]. Simultaneous twophase flow is possible in 3D geological networks, without fluctuations, which is an essential aspect of flow in those networks [11]. In a microfluidic network, however, it is possible for the wetting phase to cross a throat at top and bottom and connect wetting phase residing in corners around two pillars. We call this a "bridge": it means that the flow paths of WP and NWP can cross in a microfluidic device, unlike in a 2D network. Suppose that a microfluidic network is initially full of the wetting phase, and then we co-inject WP and NWP at fixed fraction. Initially the NWP cannot flow across the network and its saturation (volume fraction in the pore space) rises until it can form a continuous path through the network, as illustrated in figure 2(b). At this point the capillary pressure p c is too high for bridging, and the WP has no path for flow through the network. The WP therefore accumulates, and the capillary pressure falls, until either bridging occurs at enough throats that WP can flow at the injected fractional flow, or pore occupancy starts fluctuating between WP and NWP. In this article we determine the conditions under which bridging of the wetting phase is possible. Figure 3: A pore structure in which gas (black) and liquid (white) are co-injected, leading to snap-off and the formation of a bubble. The images show three time-steps from an experiment, at 0, 1.23 and 1.63 seconds. Adapted from [1]. Consider a long, straight pore throat with a uniform rectangular cross-section, in which WP lines the corners of the throat (as in figure 4(b) below). The interfaces between the phases are cylindrical and make contact when the diameter is equal to the width of the channel. That is, bridging can occur if the throat is at least as tall as it is wide. As the two interfaces then approach each other in the centre of the throat there is no change in interface curvature. The condition on the capillary pressure p c for bridging to occur is, therefore, the same as the condition on p c for WP to swell and block the whole throat. This is what we call "snap-off", as illustrated in Figure 3. In other words, once bridging occurs, any slight reduction in the capillary pressure in the surrounding medium would cause snap-off in the throat; any slight increase in capillary pressure would cause the bridge to disconnect. If, instead, the pore throat is not straight, but passes between curved walls (either with the same or opposite curvatures) then the curvature of the pillar walls affects the conditions for both bridging and snap-off. In this study we examine conditions (specifically, the range of capillary pressure) for which bridging is possible without snap-off, as a function of throat geometry. We then discuss the implications for the feasibility of two-phase flow without fluctuating pore occupancy in microfluidic devices: that is, the extent to which microfluidic devices are able to represent this aspect of multiphase flow in geological porous formations under capillary-dominated conditions. Before it can flow, NWP must first overcome a capillary entry (or threshold) pressure p e c . LeNormand et al. [14] predict that the capillary entry pressure in a rectangular channel of cross-sectional width W and height H is p e c = γ 2 W + 2 H ,(1) where γ is the interfacial tension. In the following we mostly consider the case H = 2 and W = 1, for which eq. (1) predicts p e c /γ = 3. The NWP then moves downstream, displacing WP (known as drainage [4,17]), with the interface at the leading edge moving steadily along the channel; see figure 4(a). In the corners behind this moving front, thin triangular regions of WP remain. The volume of these regions is determined by the curvature of the interface, which itself depends on the aspect ratio of the channel. Once the front has fully-penetrated the channel, just the thin regions of WP in the corners remain; see figure 4(b). Subsequent drainage or imbibition then causes these regions to shrink or enlarge, respectively. In a straight channel, all four WP regions in the corners have identical cross-section and are uniform along the channel. Ma et al. [15] relate the capillary pressure of these regions to the cross-sectional area A s of each region in terms of the contact angle θ: p c = 2γ √ A s √ 2 cos(θ) cos (θ + 45 • ) − π 2 1 − 45 • + θ 90 • .(2) If, upon subsequent imbibition, the corner channels swell sufficiently, they touch: if this occurs on the bottom or top of the channel, we refer to it as horizontal bridging. If it occurs on the sides of the channel we refer to it as vertical merging. Vertical merging does not enable WP flow across the throat. We use the Surface Evolver software [2] (see § 2) to predict the shape of the interfaces in various geometries by minimizing the surface energy of the interface between the two fluids to give accurate measurements of the capillary pressure p c (the pressure difference across the interface, calculated by the program). By making small changes in the NWP volume and repeating the minimization, we predict the quasistatic variation of pressure during imbibition or drainage. As well as neglecting viscous losses, we assume that the effects of gravity are negligible (small Bond number) on the scale of the channels. We establish a benchmark for our simulations in a straight rectangular channel, determining the effect of the choice of channel aspect ratio and the contact angle at which the interfaces meet the walls of the channel ( § 3.1). We then modify this channel to allow both side walls (pillars) to have curvature: either in the same direction, giving a curved rectangular "duct" ( § 3.2), or in opposite directions, giving a flow between cylindrical pillars ( § 3.3) into a large pore body downstream. The channel height H is uniform in most microfluidic devices. We report capillary pressures in units of γ/(H/2), the capillary entry pressure of a Hele-Shaw cell, i.e. a gap between two smooth, parallel plates. This is the capillary entry pressure of a pore body of extremely wide diameter in a microfluidic device. For a surface tension of γ = 3 × 10 −2 N/m 2 and a channel height H = 25µm, we have that a value of p c = 3 in our calculations described below corresponds to a capillary pressure of 7, 200 Pa. The trends in capillary (entry) pressure with varying H and W are captured well by eqs. (1) and (2), and so here we concentrate on the variation with changes in wall curvature. Simulation Method The Surface Evolver program minimizes surface energy by moving WP and NWP, subject to a fixed solid geometry and fixed phase volumes. In this case the surface energy is the surface area of the interfaces multiplied by their surface tensions. Each interface is tessellated into many small triangles to allow it to curve; we generally use four levels of refinement (i.e. recursive splitting of one triangle into four smaller triangles) to provide an acceptable level of accuracy without simulations taking more than a few hours to explore all relevant parameter values. The software explicitly accounts for the surface energies of each phase against solid and also against each other; by adjusting the solid/fluid phase surface energies, we can in effect fix the contact angle. We choose an interfacial tension γ = 1 between WP and NWP and a tension γ w where the NWP touches the solid walls, which sets the contact angle from θ = cos −1 (γ w /γ). In the absence of gravity the value of the interfacial tension is not important, only the ratio γ/γ w . The case of perfect wetting -a contact angle of 0 • -is numerically slow to converge, so we prefer a small contact angle of a few degrees which ensures that simulations converge quickly and accurately. Figure 5(a) shows that the prediction of capillary pressure with a contact angle of 2 • is almost indistinguishable from perfect wetting. At the entrance and exit of the channel, treated as vertical planes, we set the contact angle to 90 • . This in effect gives a plane of symmetry to the fluid interface at the entrance and exit of the channel. This does lead to an unphysical artefact for small NWP volumes at the entrance, however: the NWP forms a hemisphere of small radius until the NWP volume is large enough that it touches the solid walls of the channel. This in turn implies a large capillary pressure before NWP reaches the throat. Therefore, in the results below, we show only calculations after the NWP volume is large enough to press against the solid walls at the entrance. Surface Evolver then allows us to calculate the capillary pressure, the spatial distribution of fluid phases, and the stability of the interfaces. We fix the volume of WP and incrementally raise or lower it to simulate quasi-static motion of the NWP along a channel. Symmetry can be used to speed up the calculations: for example it is only necessary to simulate one quarter of the straight channel since all four corners are equivalent. To determine the stability of the interface, we examine the eigenvalues of the Hessian matrix of the energy [3]: a change in the sign of the smallest eigenvalue to a negative value signals instability, and the software includes tools that allow us to determine the corresponding eigenmode. In particular, this is the case when two interfaces meet in a straight channel and, as we shall see below, when merging occurs in other channel geometries. In a straight channel with a rectangular cross-section (figure 4) there are side walls at x = ± 1 2 W , with the bottom at z = 0 and the top at z = H, the entrance at y = 0 and exit at y = L. A channel length of L = 4 is sufficient for a steadily propagating interface to form. The side walls are replaced by parts of a cylinder of radius R curv ± 1 2 W to generate a curved rectangular duct. Finally, cylindrical pillars with radius R are placed at x = ±( 1 2 W + R), y = 0 between a flat base to the channel at z = 0 and a flat top at z = H. Our simulations start by introducing NWP at y = 0 into a channel that is initially full of WP. We reduce the WP volume incrementally until the capillary pressure p c exceeds the capillary entry pressure p e c . The capillary pressure then decreases as NWP invades the downstream body. If p c falls sufficiently during the invasion of the pore body, there can be "Roof snap-off" [19] in the throat: WP flows back to the throat and blocks it [20]. For example, consider a meniscus of WP around the bases of two nearby cylindrical pillars (see figure 2(a)) with NWP everywhere else in the throat and filling the pore bodies on either side. At first the WP occupies only the corners around the pillars, but as the WP volume V increases these menisci spread outwards and the capillary pressure falls. We determine the point at which the two WP regions connect, as in figure 8(a) below, bridging the gap between the cylinders, and record the corresponding capillary pressure p br c . As its volume is increased further, WP rises up the sides of the pillars and the gap between the pillars is slowly filled with WP from below. When the WP reaches half-way up the pillars it meets the menisci coming down from above. At this point there is an instability (negative eigenvalue) associated with an eigenmode indicating contraction of the interface towards the centre of the channel. We identify this as snap-off [19,18] and this value of the capillary pressure for snap-off as p sn c . We can take the Evolver calculations no further beyond this change in the topology of the interfaces. Results Straight Channel Figure 5(a) shows the capillary entry pressure in a straight rectangular channel with width W and height H = 2 as a function of contact angle. As noted above, a contact angle of less than 5 • gives a good approximation to the result for perfect wetting. This is just below the prediction of eq. (1) for the capillary entry pressure. As the NWP enters the channel the capillary pressure is at its highest, but then drops by a few percent as a steady motion along the channel is established. Figure 5(b) shows that increasing the width W of the channel (while keeping its height H fixed) reduces the capillary entry pressure, as predicted by eq. (1). The deviation in the value of p e c from the prediction of eq. (1) ( [14]) increases slightly as the channel gets narrower. Bridging occurs in the straight channel when the pair of upper or lower regions of WP (cf. figure 4(b)) meet in the middle of the channel. At this point the capillary pressure has fallen from p e c ; its value also decreases with increasing channel width, because the volume of WP required for bridging is greater. Moreover, in a straight channel, the capillary pressure for bridging, p br c , is equivalent to the value for snap-off, p sn c , since as soon as the NWP no longer touches all four walls of the channel a Rayleigh-Plateau instability occurs and the NWP breaks up into bubbles. Eq. (2) gives the value of p c for any WP volume in a rectangular channel before bridging (since the aspect ratio is irrelevant before the WP regions meet). Figure 5(c) shows that not only do our simulations agree with the prediction of eq. (2), but that during the motion of the NWP along the channel the area of the WP regions in the corners is entirely determined by the shape of the front at the capillary entry pressure. Curved rectangular duct If the channel is a gently and uniformly curved rectangular duct with planar top and bottom, as shown in figure 6(a), the capillary pressure of an interface moving along the duct is indistinguishable from its value in a straight channel. That is, the radius of curvature R curv of the centreline of the duct, which is bounded below by the half-width of the channel W/2, does not influence p c . However, the areas of the triangular regions that remain after the interface has passed are different on the inner and outer curved walls of the duct. Figure 6(b) shows that the cross-sectional area of these regions can deviate significantly from the straight channel case (in which the inner and outer regions are identical), with a greater deviation on the inner wall, but that they slowly converge towards the same value as the radius of curvature R curv increases. Their size will diverge more widely as the radius of curvature decreases if the volume of WP left in the duct after the front has passed is larger. If, however, WP can re-imbibe into the channel then two or more of the four triangular regions may meet. There are two cases to consider. First, if the height H is greater than the width W , as is the case in our "standard" channel here, then bridging first occurs at the top and/or bottom of the duct, and they then merge on the outer wall, as shown in figure 7. Second, if the width W is at least as great as the height H (a short, wide, channel, including a duct with square cross-section), the order in which the transitions occur is reversed: the regions first In the first case, with H = 2 and W = 1, bridging across the top or bottom of the duct occurs at the same saturation (an area of WP in the cross-section of 0.12) and the same capillary pressure (p c ≈ 1.9) for all values of the radius of curvature R curv . That is, although the wetting phase is re-distributed between the triangular regions on the outer and inner sides of the duct as its curvature changes, and therefore the position at which they meet is different, the total volume of WP and the capillary pressure at which the regions meet doesn't change. Following bridging, there is an almost semicircular interface (in cross-section) moving vertically within the duct ( figure 7). Moreover, the capillary pressure increases very slightly with increasing volume of WP in this regime. Such a situation is unstable, as in the arguments of Roof [19] concerning the largest inscribed circle, and WP floods into the duct and blocks the flow of NWP. Thus the merging transition on the outer wall is never reached, and the capillary pressure for snap-off is the same as the capillary pressure for bridging: p br c = p sn c . In the case of a square duct the situation is slightly different. The merging transition on the outer wall occurs at a capillary pressure that depends on the radius of curvature: it occurs at higher capillary pressures, and hence lower volumes of WP, in tightly-curved wide ducts. After merging the capillary pressure increases slightly, as for taller ducts, and so snap-off occurs and WP will flood into the duct. Gap between cylindrical pillars Therefore, in both the straight and curved rectangular channels with height H = 2 the capillary pressure at which bridging occurs is the same as the capillary pressure for snap-off. We now consider a situation in which the curvature of the side walls has a different sign on each side, and show that here this equivalence is not found. WP surrounding the base of two pillars As described above, we gradually increase the volume of WP around and in the gap between two cylinders of radius R situated between parallel horizontal plates; one half (by symmetry) of the geometry is shown in figure 8(a). We fix the height to be H = 2, as for the channels considered above, and vary the radius R of the pillars and the gap W between them to determine the different capillary pressures for bridging and snap-off. It is instructive to first consider the shape of the interface surrounding an isolated pair of cylindrical pillars and its capillary pressure. Figure 8 shows images of the interface and the capillary pressure. The WP meniscus around the base and top of each pillar extends some distance w across the gap and some height h up the sides of the pillar. We use Surface Evolver to calculate the capillary pressure and the values of w and h as the WP volume V varies for different gap widths W , as shown in figures 8(c) and (d). The capillary pressure decreases as the WP volume increases and w and h increase. When w reaches W/2 the WP bridges the gap between the cylinders; we denote the critical capillary pressure at which this happens by p br c . Figure 8(b) shows that the volume of WP at which bridging occurs increases linearly with pillar radius R, but that p br c increases only slightly with R. Our calculations and assessment of the eigenmodes given by Surface Evolver suggest that this instability occurs immediately as these two regions meet, and the interface changes topology. At small gap widths there is a significant difference between the capillary pressure for bridging and the capillary pressure for snap-off, since bridging can happen at small WP volume, while the WP volume must build up significantly before vertical merging occurs. As the gap between the pillars widens (increasing W , figure 8(d)), more WP must accumulate before bridging occurs. Consequently, the capillary pressure for bridging decreases with increasing W . At large enough gap widths, above about W = 3 in the case R = 1, merging occurs before horizontal bridging; this gives rise to a capillary bridge connecting the top and bottom of the cylinder, but does not permit two-phase flow. Asymmetric case Up until now we have considered a situation in which the WP around both pillars is at the same capillary pressure. But it is also possible for WP to accumulate only in part of the pore network connected to the point of injection and it must then form bridges across pore throats to give it a connection to flow across the network. When NWP first crosses the pore network, as illustrated in figure 2(b), the connections between regions of WP are broken. As WP accumulates, the capillary pressure falls, but only in the interconnected network of WP-filled pores connecting back to the point of injection. Therefore, when these critical bridges form (i.e., those that give WP connection across the network), one pillar is connected to the reservoir of WP at low capillary pressure while the other pillar is not. The bridge has to form with liquid initially accumulating on one side only. What happens after the bridge forms? Clearly, the dynamics of bridge formation are complex in the short term, and we do not consider them here. But the result is that both sides are now at the lower value of capillary pressure associated with the WP's flowing network. So the question is: what is p c on the wet pillar at the point when the bridge forms? Is it low enough that, once this value of p c is established on both sides, there would be snap-off without a bridge? To illustrate the idea, we continue to take a gap width of W = 1, pillars of radius R = 1, and assume that the WP meniscus surrounding the dry pillar has capillary pressure p c ≈ 3. From the left hand side of figure 8(c) we see that the WP extends a distance slightly less than 0.3W from the base of the pillar in this case. Bridging will therefore occur if the WP around the wet pillar extends to a distance of 0.7W from that pillar. The capillary pressure around the wet pillar at bridging is the value of p br c for a gap of W = 1.4, which we read from figure 8(d) as p c /γ ≈ 1.0. This is greater than the capillary pressure for snap-off ( figure 8(b)), p c /γ ≈ 0.8, and hence bridging will not lead to snap-off in this case. For snap-off to occur immediately following bridging in this asymmetric case therefore requires that the extent of the WP meniscus around the dry pillar is smaller than hypothesised, and the WP at higher capillary pressure. Nonetheless, capillary pressure could be greater on the dry pillar than the capillary entry pressure for the throat, as follows. As gas invades the pore network, as shown in figure 2(b), capillary pressure rises throughout the network according to the capillary entry pressure of the latest throat to be invaded. As noted above, that process stops when gas forms a continuous path across the network and liquid continuity across the network is broken. Thus the capillary pressure on the dry pillar is at the capillary entry pressure of the last throat to be invaded by gas, not that of the given throat. Figure 8(d) indicates that the capillary pressure for snap-off with W = 1 (again with R = 1) is equal to the capillary pressure for bridging at a gap width of W ≈ 1.7. So only when the WP meniscus around the dry pillar extends no more than 15% of the way across the gap, with a capillary pressure of p c /γ ≈ 5, far in excess of the likely capillary entry pressure, is snap-off without bridging possible. However, for larger pillar radii the curves of p c against gap width are less steep ( figure 8(d)), the difference between p br c and p sn c smaller ( figure 8(c)), and so this transition to snap-off in the asymmetric case becomes more likely and flow without fluctuating pore occupancy less likely. For example, with R = 2 the value of p c drops from its value for bridging at W = 1 to below the value for snap-off at W = 1 when W ≈ 1.4, suggesting that snap-off would immediately follow bridging driven from one side. For larger gap-widths it is more likely that vertical merging would occur before bridging between the pillars, something which our calculations do not currently resolve. Invasion of NWP into a WP-filled gap between pillars Snap-off triggered by NWP invasion of the downstream pore body is called "Roof" snap-off [19,20,18]. We turn now to the invasion of NWP into a WP-filled gap between the pillars, i.e. during initial drainage of the pore network. We fix the gap width to be W = 1, much less than the height H. NWP moves into the gap much as in the case of a rectangular channel ( § 3.1), with a roughly hemi-spherical front (figure 9(a)) and leaving behind narrow regions of WP in the corners of the channel where the pillars meet the upper and lower surfaces, which correspond to the images in figure 8(b) at high capillary pressure. Figure 9(b) shows the capillary pressure across the interface as it moves through the throat for four different values of the pillar radius R. The capillary pressure is greatest for the widest pillars, and for sufficiently large R approaches the value for a straight channel. The maximum capillary pressure (i.e. the capillary entry pressure p e c ) occurs just after the front (measured as the leading position on the interface along the axis through the gap) has passed between the pillars, and this asymmetry is also apparent in the steepness of the pressure curves before and after this point. After the front has passed through the gap between the pillars, the capillary pressure decreases as the NWP spreads out into the pore space beyond. The narrow regions of WP in the corners of the channel shrink and swell as the capillary pressure increases and then decreases as the interface invades the pore throat and then the downstream pore body [22]. We can estimate the horizontal extent of the interface downstream, beyond the pillars as follows: far downstream of the pillars the interface has a semi-circular vertical cross-section (in the perfect wetting case with contact angle θ = 0), spanning the gap between the flat upper and lower surfaces (see figure 9(a)). If we also approximate its horizontal cross-section as a circle of diameter D, then the capillary pressure here is p c = γ 2 H + 2 D , similar in spirit to the derivation of eq.( 1). The interface must have the same capillary pressure everywhere, so we compare this expression for the capillary pressure to the critical value p sn c for snap-off in the gap between the pillars when h = H/2. We extrapolate the data shown in figure 8(c) for p sn c (h = 1 2 H) to more values of R and then estimate the minimum pore-body diameter at which snap-off can occur in the pore throat: D = 2/(p sn c /γ − 2/H).(3) This is shown in figure 10(a), where it is compared with direct measurements of the width of the interface from simulations, such as the one shown in figure 9(a), that have been allowed to run until the capillary pressure decreases down to p sn c and the WP fills the gap between the pillars as a result of Roof snap-off. For gently constricted pores with circular cross-section along the flow axis the criterion for Roof snap-off is that the diameter of the pore body be at least twice the diameter of the throat. In this case, with W = 1, that suggests D = 2 [19]. In microfluidic devices of uniform depth, pore bodies must be wider than this to trigger Roof snap-off. Strongly-convex throats, reflected here in small R, stabilize bridging against snap-off during subsequent WP imbibition, but may not allow Roof snap-off during initial drainage of the network. The value of D that we predict here is close to the direct measurement shown by the black squares in figure 10(a); it is a small multiple of the gap width for pillars with large radius, but rises rapidly for smaller pillars, and tends to infinity somewhere between R = 1 and R = 2. With H = 2 we see that D → ∞ for p sn c /γ → 1, which figure 8(d) indicates is indeed the case for R between 1 and 2. In all cases it is much larger than the criterion for pore bodies and throats of circular cross-section. Hence for sufficiently tightly-curved pillars (highly-convex pore throats), Roof snap-off does not occur during NWP invasion of the pore throat however wide the pore body. The minimum possible pillar radius R min at which D → ∞ also depends on the gap width W . By calculating the cylinder radius at which, for a given gap width W , the capillary pressure for snap-off is equal to 2γ/H (which is equal to one here), we can determine R min (W ), as shown in figure 10(b). If the gap is narrow then Roof snap-off can occur in a sufficiently wide pore body; as the gap widens the minimum pillar radius grows rapidly. Discussion: Implications for Two-Phase Flow in Microfluidics Our findings can be summarized as follows: bridging is not possible without snap-off in straight or curved ducts. Bridging is possible in tightly concave throats, represented here by throats between narrow-radius pillars. In that geometry, as shown in figure 8, there is a range of capillary pressure at which bridging is stable without snap-off. For very tightly-concave throats, however, Roof snap-off may not be possible, no matter how wide the downstream pore body (figure 10). In all cases the width of a pore body in comparison to the width of a throat, as viewed from above, that would give Roof snap-off is much greater than that for pores and throats of circular cross-section [19]. We have examined the feasibility of bridging in various pore-throat geometries. Of the examples shown in figure 1, (a) and (b) feature straight, rectangular ducts; most throats in (c) approximate pillars of very large radius; those in (d) include curved rectangular ducts, straight channels and some tightly-concave throats. Based on these geometries, we conclude that it is unlikely that two-phase flow is possible in the networks shown in figure 1, or similar networks, without fluctuating pore occupancy (although it might be possible to design a network that would have the right range of throat geometries for such fluctuations to be suppressed). In three-dimensional geological pore networks, two-phase flow without fluctuating pore occupancy can occur at arbitrarily small pressure gradients [11,20]. At higher pressure gradients, quantified through the dimensionless capillary number, flow with fluctuating pore occupancies and displacement of trapped, isolated NWP is possible [13,21]. The flow regime above this transition in capillary number differs significantly from that below it [13]. This suggests that imposed two-phase flow in most current microfluidic devices must necessarily reflect the high-capillary-number flow regime. Steady two-phase flow is possible in networks featuring concave pore throats, especially tightly-concave pore throats. As illustrated in figure 2(b), under conditions where both phases might flow, the flow path for each phase is extremely inefficient. That suggests that the relative permeability of each phase [11,13] is very low. One implication is that even in conditions where steady two-phase flow is possible, the pressure gradient required to sustain the imposed flow rate could push the capillary number above the point where pore occupancy fluctuates. This is the subject of ongoing research that will be reported separately. Figure 1 : 1Examples of microfluidic network representing an idealized porous medium. (a) Highly ordered (from [10]). (b) Disordered Figure 2 : 2(a) Wetting phase surrounds the bases and tops of pillars between parallel plates, while non-wetting phase occupies the interior. (b) Example of invasion percolation on a network of coordination number four, from Figure 4 : 4(a) The interface separating NWP from WP moves from left to right along a straight rectangular channel after overcoming the capillary entry pressure. (b) After the front has left the channel, it leaves behind four narrow regions of the WP. The wetting films on the walls are not shown. The channel has height H = 2 and width W = 1 and the contact angle is 2 • . Figure 5 : 5Capillary pressure in straight rectangular channels. (a) Capillary entry pressure for different contact angles in a channel with W = 1 and H = 2. The value of pc for different contact angles is almost indistinguishable for θ less than about 5 • , justifying our use of a small finite contact angle. The steady advance of the interface into the channel occurs with capillary pressure slightly less than the prediction of eq. (1), p e c = 3. (b) Capillary entry pressure as a function of width W with fixed height H = 2 and contact angle 2 • . The capillary entry pressure is slightly over-estimated by eq. (1), while the capillary pressure at which bridging (and consequent snap-off) occurs is lower. The capillary entry pressure is greatest for a tall, narrow channel. (c) Capillary pressure as a function of the total area of WP in the cross-section of the channel in the four narrow regions of WP that remain after the interface has fully penetrated the channel, with W = 1, H = 2 and θ = 2 • . The particular case of the capillary entry pressure (from parts (a) and (b)) is highlighted to indicate how the shape of the interface spanning the channel determines the shape of the narrow regions of WP. Figure 6 : 6(a) A curved rectangular duct with H = 2, W = 1 and Rcurv = 8, i.e. with curved walls having radius of curvature Rcurv ± 1 2 W . The interface separating the two fluids meets the walls at a contact angle of 2 • . (b) Cross-sectional areas of the WP regions in the corners of a curved rectangular duct, normalized by the value for a straight channel. The inset shows that the capillary pressure is independent of the radius of curvature of the duct. The contact angle is 2 • . Figure 7 : 7Imbibition into a curved rectangular duct with radius of curvature Rcurv = 5. The capillary pressure decreases until the first bridge is formed: if the channel height is greater than its width (here H = 2W ) this occurs first on the top (as shown) or bottom surfaces. The interface then remains in contact with three walls and the capillary pressure increases weakly until the merging transition on the outer wall. Beyond this point the capillary pressure increases more rapidly. Also shown is the sequence of interface shapes in a vertical cross-section through the duct, with points at which the interface meets the duct surfaces highlighted with dots. The contact angle is 2 • .merge on the outer wall, furthest from the centre of curvature, and then bridge across the top and/or bottom of the duct. We find (for contact angle 2 • ) that the the aspect ratio H/W at which there is a transition from one case to the other, i.e. at which bridging and merging occur simultaneously on the upper/lower and outer surfaces, increases linearly, but weakly, with the curvature 1/R curv of the duct, from H/W = 1 for a straight channel to H/W = 1.2 for R curv = 2. Figure 8 ( 8c) indicates that the capillary pressure decreases further as more of the WP builds up in the gap between the pillars. There is a range of p c for which a bridge is stable without snap-off, corresponding to the difference in capillary pressure between the right hand sides of the two panels in figure 8(c) for each value of R.The upper and lower WP regions in the gap between the pillars meet, half-way up the pillars, when their height is h = 1 2 H, at which point we denote the critical capillary pressure by p sn c , the right-most values infigure 8(c). Note that at the point where the bridging between upper and lower WP regions occurs the cross-section of the interface appears circular; the gradient of p c (V ) is very shallow here (figure 8(c), right hand panel), indicating that the WP has almost reached the critical point for snap-off, beyond which it floods back into the gap between the pillars[18]. Figure 8 : 8Bridging and snap-off in a throat between cylindrical pillars. (a) Oblique view of WP around one of the two pillars, between the formation of the horizontal bridge and snap-off (subsequent merging), corresponding to R = 1 and a total WP volume of 2.6. (b) Capillary pressure as a function of WP volume divided by radius, which is a measure of the cross-sectional area of the meniscus, for cylinders of radius R = 1, 2 and 5, with gap width W = 1. The circular symbol indicates bridging and the square symbol indicates snap-off here and in (c) and (d). The images are views through the centre of the throat, with the pillars to either side, for R = 1. (c) The capillary pressure for different pillar radii R. When w/W reaches 0.5 the WP bridges across the channel. When h/H reaches 0.5 the two regions of WP touching the top and bottom surfaces merge and snap-off occurs. (d) The critical capillary pressure at which bridging and snap-off occur as the gap width W changes for pillars with radius R = 1, 2 or 5. Vertical merging occurs before bridging for values of gap width greater than the point at which the two curves meet for each R. Height is H = 2 and the contact angle is 2 • . Figure 9 : 9NWP invasion of the pore throat between pillars. (a) The interface moves from left to right through the gap between two cylindrical pillars, leaving the displaced WP around the base and top of each pillar. Pillar radius is R = 1 in this case. (b) The maximum capillary pressure occurs just after the leading edge of the front has passed the centre of the gap. It increases slightly with the radius of the pillars. Also shown is the predicted capillary pressure in a straight channel, slightly below the prediction of p e c = 3 from eq. (1). (c) A summary of the different critical capillary pressures, plotted as a function of 1/R so that the values for the straight channel are shown on the left. Gap width is W = 1, height is H = 2, and the contact angle is 2 • . Figure 10 : 10(a) The diameter of the bubble that is created in the pore body downstream when snap-off occurs in the gap between the pillars after the interface has passed: comparison of eq. (3) using capillary pressure values from figure 8 with direct simulation as in figure 9(a). Gap width is W = 1. (b) The minimum pillar radius for snap-off in a gap of given width. The region below the curve is inaccessible because the capillary pressure for snap-off is below 2γ/H. 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Water Resources Research 55:1167-1178 (2019). This is visible in our video of a simulation. in the case R = 5) at https://youtu.be/10e5IitBGE8 which we can include as part of the supplementary informationThis is visible in our video of a simulation (in the case R = 5) at https://youtu.be/10e5IitBGE8 which we can include as part of the supplementary information.	[]
[ "Azimuthal backflow in light carrying orbital angular momentum", "Azimuthal backflow in light carrying orbital angular momentum" ]	[ "Bohnishikha Ghosh \nInstitute of Experimental Physics\nFaculty of Physics\nUniversity of Warsaw\nLudwika Pasteura 502-093WarsawPoland\n", "Anat Daniel \nInstitute of Experimental Physics\nFaculty of Physics\nUniversity of Warsaw\nLudwika Pasteura 502-093WarsawPoland\n", "Bernard Gorzkowski \nInstitute of Experimental Physics\nFaculty of Physics\nUniversity of Warsaw\nLudwika Pasteura 502-093WarsawPoland\n", "Radek Lapkiewicz \nInstitute of Experimental Physics\nFaculty of Physics\nUniversity of Warsaw\nLudwika Pasteura 502-093WarsawPoland\n" ]	[ "Institute of Experimental Physics\nFaculty of Physics\nUniversity of Warsaw\nLudwika Pasteura 502-093WarsawPoland", "Institute of Experimental Physics\nFaculty of Physics\nUniversity of Warsaw\nLudwika Pasteura 502-093WarsawPoland", "Institute of Experimental Physics\nFaculty of Physics\nUniversity of Warsaw\nLudwika Pasteura 502-093WarsawPoland", "Institute of Experimental Physics\nFaculty of Physics\nUniversity of Warsaw\nLudwika Pasteura 502-093WarsawPoland" ]	[]	M.V. Berry's work [J. Phys. A: Math. Theor. 43, 415302 (2010)] highlighted the correspondence between backflow in quantum mechanics and superoscillations in waves. Superoscillations refer to situations where the local oscillation of a superposition is faster than its fastest Fourier component. This concept has been used to demonstrate backflow in transverse linear momentum for optical waves. In this work, we examine the interference of classical light carrying only negative orbital angular momentum and observe in the dark fringes of such an interference, positive local orbital angular momentum. This finding may have implications for the studies of light-matter interaction and represents a step towards observing quantum backflow in two dimensions.	null	[ "https://export.arxiv.org/pdf/2304.13124v1.pdf" ]	258,331,580	2304.13124	0f8aa36e06166075e9a4c3843d6be1d39ba3bd81	Azimuthal backflow in light carrying orbital angular momentum Bohnishikha Ghosh Institute of Experimental Physics Faculty of Physics University of Warsaw Ludwika Pasteura 502-093WarsawPoland Anat Daniel Institute of Experimental Physics Faculty of Physics University of Warsaw Ludwika Pasteura 502-093WarsawPoland Bernard Gorzkowski Institute of Experimental Physics Faculty of Physics University of Warsaw Ludwika Pasteura 502-093WarsawPoland Radek Lapkiewicz Institute of Experimental Physics Faculty of Physics University of Warsaw Ludwika Pasteura 502-093WarsawPoland Azimuthal backflow in light carrying orbital angular momentum (Dated: April 27, 2023) M.V. Berry's work [J. Phys. A: Math. Theor. 43, 415302 (2010)] highlighted the correspondence between backflow in quantum mechanics and superoscillations in waves. Superoscillations refer to situations where the local oscillation of a superposition is faster than its fastest Fourier component. This concept has been used to demonstrate backflow in transverse linear momentum for optical waves. In this work, we examine the interference of classical light carrying only negative orbital angular momentum and observe in the dark fringes of such an interference, positive local orbital angular momentum. This finding may have implications for the studies of light-matter interaction and represents a step towards observing quantum backflow in two dimensions. I. INTRODUCTION Light beams with azimuthal (helical) phase dependence e ilφ were identified to be carrying orbital angular momentum (OAM) by Allen et al. in 1992 [1]. Their first experimental realisation was done in 1993 by using cylindrical lenses [2], and since then they have found applications in numerous fields such as optical tweezers [3], optical microscopy [4], interactions with chiral molecules [5] etc. States of light with azimuthal phase dependence, are also analogous to the eigenstates of the angular momentum operator in quantum mechanics-L z . In the current work, we demonstrate that a peculiar phenomenon called backflow, taken from quantum mechanics, is present in the superposition of beams carrying OAM. The phenomenon of backflow was first encountered in the context of arrival times in quantum mechanics [6] and is a manifestation of interference. Due to this counterintuitive phenomenon, a quantum particle prepared in specific superposition states of only positive momenta, having wavefunction centered in the x < 0, may have an increased probability, with time, of remaining in x < 0 [7]. While, for particles moving on a line, only about 4% of the total probability can be in the 'wrong' direction [7], this probability increases to around 12% for charged particles moving on a ring [8]. To overcome such bounds, which makes the experimental observation of the phenomenon difficult, researchers studied backflow in two-dimensions for a charged particle moving either in a uniform magnetic field in the infinite (x, y) plane [9,10] or on a finite disk such that a magnetic flux line passes through the centre of the disk [11]. In such two-dimensions systems the probability of backflow can be unbounded. Although backflow in quantum systems has not yet been experimentally realised, it has been demonstrated * [email protected] † [email protected] ‡ [email protected] § [email protected] with optical beams [12,13] in one-dimension, by exploiting its connection to the concept of superoscillations in waves, as established by Berry et al. [14,15]. In a superoscillatory function, local Fourier components are not contained in the global Fourier spectrum. For example, in classical electromagnetism, this manifests as follows-the local Poynting vector of a superposition state can point in directions not contained among those of the constituent plane waves, leading to counter-flow or backflow of the energy density. In the recent experimental observations, onedimensional transverse local momentum of a superposition of beams was measured by scanning a slit [12] or by using the Shack-Hartmann wavefront sensor technique [13] respectively. The Shack-Hartmann wavefront sensor technique also allows for one-shot measurement of the two-dimensional transverse local momentum, as reported for the case of azimuthally phased beams in [16]. In the present work, we use this technique to measure the local OAM of the superposition of two beams with helical phases, thereby extending the observation of linear optical backflow to azimuthal backflow. In practice, we examine the superposition of two beams carrying only negative (positive) orbital angular momentum and observe, in the dark fringes of such an interference pattern, positive (negative) local OAM. This is what we call azimuthal backflow. We clarify that, by 'local OAM' of a scalar field at each point, we refer to the product of the azimuthal component of the local momentum at that point and its corresponding radius. Backflow is a manifestation of rapid changes in phase which could be of importance in applications that involve light-matter interactions such as in optical trapping or in enhancing chiral response of molecules [5,17]. Apart from these, our demonstration is a step in the direction of observing quantum backflow in two-dimensions, which has been theoretically found to be more robust than one-dimensional backflow. II. THEORETICAL MODEL Measuring azimuthal backflow in the superposition of conventional beams with helical phases such as Laguerre-Gauss (LG) and higher order Bessel-Gauss (BG) beams can be challenging due to the sparsity of local regions in which such backflow can be observed ( c.f. Appendix B for detailed theoretical derivations and illustrations of azimuthal backflow in superpositions of LG beams). Therefore, here, for the sake of simplicity, we searched for superpositions of other beams with helical phases which would show a more frequent azimuthal backflow. Figure 1a is a schematic of the interference of two Gaussian beams with unequal amplitudes, illuminating helical phase plates of orders l 1 and l 2 (both negative or positive) respectively. Here we provide a mathematical description of the propagation of this interference along the z-axis. We restrict ourselves to quasi-monochromatic scalar fields under the paraxial approximation, instead of the more rigourous approach using Maxwell's equations. For z = 0 (plane I indicated in the figure), which is the image plane of the phase plates of order l 1 and l 2 , the scalar field is given as follows. Ψ(r, φ, z = 0) = e − r 2 w 2 0 e il1φ + be il2φ ,(1) where (r, φ) are the transverse coordinates at plane I and \|b\| ∈ [0, 1] is a constant ratio between the amplitudes of the two interfering Gaussian beams, each of waist w 0 . We stress that azimuthal backflow can already be observed using the field in 1. However, we wish to provide a complete description of the field's propagation and to theoretically study the azimuthal backflow at other planes. The field at any (z > 0), i.e. at plane II, is given by solving the Fresnel diffraction integral [18,19], considering free propagation of the field in equation 1. Ψ(r , φ , z) = k iz e ikz e i kr 2 2z (F l1 (kr /z)e il1(φ − π 2 ) + bF l2 (kr /z)e il2(φ − π 2 ) )(2) where (r , φ ) are the transverse coordinates at plane II and F l k z r is the l-th order Hankel transform of the function e − r 2 w 2 0 e i kr 2 2z , obtained using the lth order Bessel function J l k z r r [20]. The local momentum (i.e, wave-vector) of Ψ(r , φ , z) is found by computing the gradient of its wavefront: k(r , φ , z) = ∇ arg Ψ(r , φ , z ) = ∂ ∂r arg Ψ(r , φ , z)r + 1 r ∂ ∂φ arg Ψ(r , φ , z)φ + ∂ ∂z arg Ψ(r , φ , z)ẑ [16,21,22]. Assuming, b ∈ R, the azimuthal component of the local wave-vector of the superposition Ψ(r , φ , z) is then where B(r ) = b \|F l 2 (kr /z)\| \|F l 1 (kr /z)\| is a local amplitude ratio and C(r ) = arg{F l1 (kr /z)} − arg{F l2 (kr /z)} is the local phase that depend on r . While the azimuthal components of the local wave-vectors of the constituents k φ ,1 = l1 r and k φ ,2 = l2 r , are independent of φ and have a constant clockwise (counterclockwise) for negative (positive) signs of l 1 and l 2 direction at any given radius, it is seen that k φ ,s depends on φ . This is a prerequisite for observing azimuthal backflow. k φ ,s = 1 2r {l 1 + l 2 + (l 1 − l 2 )(1 − B(r ) 2 ) 1 + B(r ) 2 + 2B(r )cos{(l 1 − l 2 )(φ − π 2 ) + C(r )} },(3) In order to observe azimuthal backflow, let us first consider the specific case of plane I (z = 0), where k φ,s = 1 2r l 1 + l 2 + (l 1 − l 2 )(1 − b 2 ) 1 + b 2 + 2bcos{(l 1 − l 2 )φ} ,(4) i.e., the ratio b ∈ R is a constant independent of r and there is no additional local phase. As seen from equation 4, k φ,s has the potential to point in the counterclockwise (clockwise) direction at any given radius, depending on φ and b, thus indicating backflow. Note that when the beams have equal amplitudes no backflow will be present. A two-dimensional illustration of azimuthal backflow is given in Figure 1b, where the grey-scale map represents the intensity distribution of the field in equation 1, i.e. \|Ψ(r, φ, z = 0)\| 2 , on top of which the normalized local wave-vectors k φ,s /\|k φ,s \| have been marked with arrows. The arrows marked in grey in the bright fringes, point in the clockwise direction, i.e., in the directions of k φ,1 and k φ,2 , while the yellow arrows in the dark fringes, point in the counterclockwise direction and correspond to azimuthal backflow. A quantitative representation of the same azimuthal backflow is shown in the plot in Figure 1c. We plot rk φ,1 (red), rk φ,2 (green), and rk φ,s (blue), which are measures of the local OAM [16] of each constituent and the superposition in equation 1, as functions of φ. While rk φ,1 and rk φ,2 are constant negative values as expected, the positive values of rk φ,s in the dark fringes of intensity at a constant radius (plotted in orange), are a manifestation of azimuthal backflow. The angular extent of the region of backflow naturally depends on the parameters l 1 , l 2 and b (c.f. Section III and Appendix A for further details). Next, we examine the behaviour of azimuthal backflow at plane II. We use equations 2 and 3 to plot the intensity distribution \|Ψ(r , φ , z)\| 2 and the normalized local wavevectors k φ ,s /\|k φ ,s \| respectively. The two-dimensional plot is given in Figure 1d. Comparing Figure 1d to Figure 1b, we see on the grey-scale map of the intensity distribution, that for z > 0, a vortex around r = 0 is formed and no azimuthal backflow exists within this region. The value of z determines the radius of this vortex. Apart from this observation, the arrow-fields in both the figures are similar. However, from the quantitative point of view, for z > 0, we see from equation 3 that the local OAM depends on the radius r . In contrast to a single plot in the case of z = 0 (c.f. Figure 1c), here, for each radius there is a correponding plot of local OAM and intensity cross-section as functions of φ (c.f. Figure 1e). It is thus understood that for z > 0, suitable radii ought to be chosen in order to observe azimuthal backflow utilizing the field in equation 2. Since the purpose of our experiment is to demonstrate azimuthal backflow, we limit our experimental demonstration to the field in equation 1 wherein the local wave-vector has only an azimuthal component and this component in turn has no radial dependence. The detailed setup of the experiment is given in Figure 2a. The field in equation 1, is realized by using phase masks on a phase-only spatial light modulator (Holoeye Pluto 2.0 SLM), as shown in inset A of Figure 2a. A 780 nm continuous wave laser (Thorlabs CLD1015) is reflected off the SLM. Since we use a phase-only SLM in order to simultaneously modulate phase and amplitude, we adopt the technique discussed in [23], such that the desired field is generated after filtering the first diffraction order. The SLM is imaged using lenses L 3 and L 4 onto the microlens array (ThorLabs-MLA-150-5C-M) that focuses the beam onto the CMOS camera (mvBlueFOX-200wG). By definition, the image plane of the SLM refers to plane I (z = 0), as mentioned in the previous section. Inset B shows the spotfield generated on the CMOS when the mask in inset A is encoded on the SLM. Following the Shack-Hartmann sensor principle [24,25], a reference spotfield is generated by illuminating the microlens array with a wide Gaussian beam. III. THE EXPERIMENT Then, the displacement of the centroids of the spotfield generated by the superposition field w.r.t. that of the reference are measured in the x and y directions. These are combined to find the directions of the local wave-vectors of the superposition, as plotted in Figure 2b on top of each spot in the spotfield in inset B. In agreement with the theoretical two-dimensional illustrations in Figure 1b,d, the yellow arrows here in the dark fringes correspond to the regions of backflow. Due to imperfections in the imaging and the finite sizes of the microlenses used to sample the wave-vectors, the yellow arrows in the regions between the dark and bright fringes have radial components (and are not purely azimuthal). Hence, in order to quantitatively analyse the azimuthal backflow, we generate one-dimensional plots of the local OAM (c.f. Figure 1c) in Figure 3. The data points of the plots given in Figure 3 are generated as follows. In the spotfields of the constituent beams or the superposition, the ith spot's centroid on the reference spotfield is displaced by ∆x i and ∆y i in the x and y directions respectively. The displacements in the cartesian coordinates are transformed to displacements in the polar coordinates (r i , φ i ). r i is found by calculating the distance between the spotfield's global center of mass and the individual spot's centroid. φ i is given by the angle between the horizontal axis and the line joining the center of mass and the spot's centroid. See the illustration in Figure 3a for a schematic representation. Following this, ∆x i and ∆y i are combined to find the angular displacement of the spot-∆φ i = − sin φ i ∆x i + cos φ i ∆y i . In order to obtain the azimuthal component of the local wave-vector for the i-th displaced spot, the angular displacement is scaled using the focal length f m of each microlens-k φi = 2π λfm ∆φ i . The local OAM is then given by r i k φi . The local OAM is plotted in Figure 3 for each constituent beam (red and green scatter plots) and the superposition (blue scatter plot). The solid red,green and blue are the corresponding theoretical predictions and we find the experimental data to be in good agreement with the theory. Here, the constituent beams carry negative angular momenta, hence, all blue data points which correspond to positive values (above the black line) are indicative of azimuthal backflow. The periodicity of the local OAM of the superposition depends on \|l 1 − l 2 \| = ∆l. For higher ∆l = 3 (Figure 3c), the number of peaks become more frequent and are taller relative to the peaks in Figure 3b (for which ∆l = 2). Once ∆l is increased further, although the value of backflow increases substantially, its detection requires finer sampling, i.e., microlenses of smaller size [26,27]. there is a corresponding displaced (Dis) spot (in spotfield of the constituent beams or the superposition) marked in red. ∆φi is found by converting the displacements in cartesian coordinates ∆xi and ∆yi to displacements in polar coordinates. The local OAM is then given by ri 2π λfm ∆φi = rik φ i , fm is the focal length of each microlens. In (b) and (c) the scatter plots are data points and the solid curves are theoretical predictions. The red, green and blue scatter plots of rk φ,1 , rk φ,2 and rk φ,s respectively are in good agreement with their corresponding theoretical predictions. In these examples, the constituent beams carry negative angular momenta, hence, all blue data points which are positive are corresponding to azimuthal backflow. In (b) and (c) the ratio \|b\| = 0.6 is the same, while ∆l = 2 (l1 = −1, l2 = −3) and ∆l = 3 in (l1 = −1, l2 = −4) respectively. Note that in both (a) and (b), the troughs of the blue scatter plot shows a slight linear trend-line compared to the theoretical prediction. This is a systematic error owing to cross-talks between microlenses. Yet, the observation of azimuthal backflow is unaffected by it. IV. DISCUSSION AND OUTLOOK In this work we have studied both theortically and experimentally the phenomenon of azimutal backflow, by utlizing the superposition of two beams of unequal amplitudes, with helical phases. We show explicitly that for two beams carrying negative OAM, the local OAM of their superposition is positive in certain spatial regions. As the angular spectra of the constituents beams are discrete, the backflow is directly certified from the measurment. This is advantageous compared to previous demonstrations [12,13], where the Fourier spectrum of the constituent beams are infinite and hence it is required to carefully certify backflow i.e. to ensure that the local linear momentum does not arise from the infinite tail of the Fourier spectrum. It is worth noting that the azimuthal backflow in superpositions of LG/ BG beams is hard to observe due to complex radial dependence (c.f. Appendix B). For the beams that we propose, even if the azimuthal component of the local wave-vector has a radial dependence (i.e., for z > 0), the azimuthal backflow can be observed and is relatively robust. Recently, there has been a growing interest in the study of superoscillatory behaviour in instensity for structured light [28,29]. A typical feature is the existence of subdiffraction hotspots which can be used in super-resolution imaging [30]. In parallel, our work broadens the scope of research, by studying the superoscillatory behaviour of the phase. Azimuthal backflow can be useful where strong phase gradients over small spatial extents are needed, for instance, to enhance chiral light-matter interactions [5,17], or detecting photons in regions of low light intensity [31]. Other possibilities relate to optical tweezers [32]. From the fundamental point of view, an interesting open question is to which extent a study of the transverse two-dimensional spatial degree of freedom of a single photon can emulate the more robust two-dimensional quantum backflow analysed in [11]. The current work is a step towards observing quantum optical backflow [33]. ACKNOWLEDGMENTS The authors thank Robert Fickler, Arseni Goussev, Tomasz Paterek, Iwo Bialynicki-Birula and Shashi C.L. Srivastava for insightful discussions. This work was supported by the Foundation for Polish Science under the FIRST TEAM project 'Spatiotemporal photon correlation measurements for quantum metrology and super-resolution microscopy' co-financed by the European Union under the European Regional Development Fund (POIR.04.04.00-00-3004/17-00). Appendix A: Optimal value of the amplitude ratio for the maximum angular extent of azimuthal backflow Given the superposition in equation 1 of the main text, for constant b ∈ [0, 1], the local orbital angular momentum (OAM) rk φ,s is given as follows. rk φ,s = ∂ arg(Ψ) ∂φ = l 1 + l 2 2 + l 1 − l 2 2 1 − b 2 1 + b 2 + 2b cos((l 1 − l 2 )φ) . (A1) In order to find the boundaries of the regions of azimuthal backflow, we set the left hand side (L.H.S.) of equation A1 to zero, as all positive (negative) values of rk φ,s would result in backflow. We thus obtain cos((l 1 − l 2 )φ) = − 1 b + b l2 l1 1 + l2 l1 . (A2) It is observed from equation A2, starting from φ = 0 (bright region, no backflow), the first crossings are at φ = ± 1 \|l1−l2\| arccos(− 1 b +b l 2 l 1 1+ l 2 l 1 ). The angular extent of this bright 'no-backflow' region is: ∆φ = 2 \|l 1 − l 2 \| arccos(− 1 b + b l2 l1 1 + l2 l1 ) (A3) The angular extent of one complete fringe is 2π \|l1−l2\| . For simplicity's sake we consider the proportion of the 'nobackflow' region within a fringe: ∆φ = ∆φ \|l 1 − l 2 \| 2π = 1 π arccos(− 1 b + b l2 l1 1 + l2 l1 ) (A4) We want a b value such that this region is minimized, we search for ∂∆φ ∂b = 0 1 π − 1 b 2 + l2 l1 (1 + l2 l1 ) 2 − ( 1 b + b l2 l1 ) 2 = 0 ⇒ b = + l 1 l 2 (A5) The b value that maximizes the angular extent of the backflow region is therefore b = l1 l2 . With this value of b, we can also calculate the proportion of the fringe where backflow is observed: Backflow proportion = 1 − ∆φ\| b= l 1 l 2 = 1 − 1 π arccos( −2 l1 l2 + l2 l1 ) (A6) The aforementioned analysis leads to the understanding that not every value of b ∈ (0, 1) can lead to azimuthal backflow (the exclusion of the lower and the upper bounds is self-explanatory). Quantitative plots of rk φ,s , similar to Figure 1c of the main text can help us visualize this. In Figure 4, the top panel shows plots of the intensity cross-section of the superposition in equation 1 of the main text, at a given radius I(φ) = 1 + b 2 + 2b cos (l 1 − l 2 )φ for two different values of b. The lower panel, along with l 1 and l 2 , shows the corresponding plots of rk φ,s . Here l1 = −1, l2 = −3. Top panel shows I(φ) for b = l 1 l 2 ≈ 0.6 (gold) and b = 0.3 (violet). The bottom panel shows rk φ,1 (red), rk φ,2 (green) and the two plots of rk φ,s for the corresponding values of b from the top panel. It is evident from the violet plot that when b = 0.3, the angular extent of the region of backflow is 0. 1 r ∂ ∂φ argΨ(r, φ) = 1 2r l 1 + l 2 + (l 1 − l 2 )(1 − b(r, l 1 , p 1 , l 2 , p 2 ) 2 ) 1 + b(r, l 1 , p 1 , l 2 , p 2 ) 2 + 2b(r, l 1 , p 1 , l 2 , p 2 )cos{(l 1 − l 2 )φ} , where b(r, l 1 , p 1 , l 2 , p 2 ) = b p 2 !(p 1 + \|l 1 \|)! p 1 !(p 2 + \|l 2 \|)! r √ 2 w 0 \|l2\|−\|l1\| L \|l2\| p2 2r 2 w0 2 L \|l1\| p1 2r 2 w0 2 . Equation B2 is thus similar in nature to equation 3 of the main text, owing to the term b(r, l 1 , p 1 , l 2 , p 2 ), which has a complex radial dependence. Thus, only specific values of the parameters involved can lead to azimuthal backflow. As seen from the examples in Figure 5 the regions of azimuthal backflow are restrictive and sparse. Given that in the experiment, we would use lenslets of a finite size (which adds to abberations) to sample these regions, measuring the azimuthal backflow would be quite challenging in these cases. A similar argument holds for the FIG. 1 . 1Visual representation of azimuthal backflow in the superposition of two beams imprinted with helical phases. (a) Concept diagram. Two Gaussian beams with an amplitude ratio b = 0.6 between them, each of waist w0 = 1mm, illuminate negative helical phase plates with l1 = −1 and l2 = −3. Then the beams with the imprinted helical phases are made to interfere using a beam splitter (BS). Cross-sections of the superposition's intensity at planes I (z = 0) and II (z = 20 mm) are shown. (b) Two-dimensional cross-section of the intensity distribution on plane I (gray scale map) and normalized azimuthal components of local wave-vectors-k φ,s /\|k φ,s \| (scale bar indicated at the bottom right corner). While the grey arrows, in the bright fringes, point in clockwise direction (defined by the signs of l1 and l2), the yellow arrows, in the dark fringes, point in the counter-clockwise direction, thus illustrating backflow. One such region of backflow, in a given dark fringe, is marked by the white triangle labelled A. (c) The local OAM rk φ for each constituent (red, green constant lines) and the superposition (blue) and the intensity (orange) at a constant radius as functions of the azimuthal angle φ. The values of the blue curve, indicating positive local OAM, i.e., backflow, coincide with the minima of the orange curve, i.e, the dark fringes. (d) Two-dimensional cross-section of the intensity distribution on plane II (gray scale map) and normalized k φ ,s . As in (b), the azimuthal component of the local wave-vector exhibits backflow outside the central vortex. (e) Quantitative plots of local OAM r k φ considering local amplitude ratio B(r ) and local phase C(r ). The red and green lines represent the constants r k φ ,1 and r k φ ,2 respectively. Three different values of r -r1 = 0.2mm (green), r2 = 1.5mm (yellow), r3 = 2.5mm (purple) are used to plot their respective r k φ ,s . The green, yellow and purple curves peak at the minima of the respective green, yellow and purple curves of the intensity cross-section in the upper panel. Again, the positive values of r k φ ,s are indicative of backflow. FIG. 2 . 2Schematic of the experimental setup. (a) POL, polarizer. SLM, spatial light modulator. MLA, micro-lens array. I, iris to spatially filter the first order of diffraction. L1,L2,L3, and L4 are lenses. The laser beam is polarized and expanded by a factor of 8 by lenses L1 (f =50mm) and L2 (f =400mm) to cover the spatial extent of the SLM. Inset A shows a sample hologram to produce the desired superposition field in equation 1 with l1 = −1, l2 = −3, b = 0.6. This hologram is encoded on the plane of the SLM using the method described in[23]. The first diffraction order of the beam reflected from the SLM is spatially filtered by an iris (I) in the Fourier plane of the lens L3 (f =250mm). The filtered beam is Fourier transformed once again by the lens L4 (f =125mm) onto the microlens array (ThorLabs-MLA-150-5C-M), which is placed at the image plane of the SLM (z = 0). The micro lens array (each lens has a pitch of 150 µm and a focal length of 5.6 mm) focuses the light onto the CMOS camera (mvBlueFOX-200wG; pixel size 6 µm). Inset B shows the corresponding spotfield observed on the CMOS sensor. (b) On every spot in inset B, an arrow corresponding to the normalized direction of the total local wave-vector k/\| k\| is displayed. The arrows are generated by combining the x and y displacements of the centroids of the spotfield in inset B relative to the reference. Due to imperfections in imaging and the finite size of the microlenses, the arrows contain both radial and azimuthal components. While the grey arrows point in the clock-wise direction in accordance with the negative values of l1 and l2, the yellow arrows, predominantly pointing in the counter-clockwise direction, indicate local regions in which backflow occurs. FIG. 3 . 3Experimental result demonstrating azimuthal backflow. (a) illustrates the method used to extract the local OAM. The center of mass (CM) of the reference spotfield is marked in yellow. Then polar coordinates of the ith spot (ri, φi) are found. For the ith spot in the reference (Ref), FIG. 4 . 4Quantitative comparison between values of b. FIG. 5 . 5Two-dimensional respresentation of azimuthal backflow in the superposition of LG beams. As inFigure 1(b) and (d) of the main text, the intensity distribution is shown via grey-scale maps. Normalized k φ,s are plotted with arrows. The yellow arrows correspond to azimuthal backflow. Here l1 = −1, l2 = −3, w0 = 1mm. In (a)-(c) p1 = p2 = 0 and b = 0.3, 0.6, 0.8 respectively. Clearly, even if the radial index p of each constituent beam is set to 0, the complex radial dependence allows only specific regions of backflow to exist. In (d) p1 = 4, p2 = 3, b = 0.8. Non-zero radial indices lead to different distributions of the regions of backflow. Appendix B: Examining azimuthal backflow in the superposition of Laguerre-Gauss beamsConsider the Laguerre-Gauss (LG) beam u l,p in cylindrical polar coordinates (r, φ, z)For the sake of simplicity, we are interested in the su-perpositionΨ(r, φ, z = 0) = u l1,p1 + bu l2,p2 , for b ∈ [0, 1]. 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[ "A MORE GENERAL FRAMEWORK THAN THE δ-PRIMARY HYPERIDEALS", "A MORE GENERAL FRAMEWORK THAN THE δ-PRIMARY HYPERIDEALS" ]	[ "M Anbarloei " ]	[]	[]	The δ-primary hyperideal is a concept unifing the n-ary prime and n-ary primary hyperideals under one frame where δ is a function which assigns to each hyperideal Q of G a hyperideal δ(Q) of the same hyperring with specific properties. In this paper, for a commutative Krasner (m, n)hyperring G with scalar identity 1, we aim to introduce and study the notion of (t, n)-absorbing δ-semiprimary hyperideals which is a more general structure than δ-primary hyperideals. We say that a proper hyperideal Q of G is an (t, n)-absorbing δ-semiprimary hyperideal if whenever k(a tn−t+1 1 ) ∈ Q for a tn−t+1 1 ∈ G, then there exist (t − 1)n − t + 2 of the a , i s whose k-product is in δ(Q). Furthermore, we extend the concept to weakly (t, n)-absorbing δ-semiprimary hyperideals. Several properties and characterizations of these classes of hyperideals are determined. In particular, after defining srongly weakly (t, n)-absorbing δ-semiprimary hyperideals, we present the condition in which a weakly (t, n)-absorbing δ-semiprimary hyperideal is srongly. Moreover, we show that k(Q (tn−t+1) ) = 0 where the weakly (t, n)-absorbing δsemiprimary hyperideal Q is not (t, n)-absorbing δ-semiprimary. Also, we investigate the stability of the concepts under intersection, homomorphism and cartesian product of hyperrings.2010 Mathematics Subject Classification. 16Y99.	null	[ "https://export.arxiv.org/pdf/2304.13526v1.pdf" ]	258,332,136	2304.13526	ab4a84e907d3fa10818d6d0e65204e7e21882e33	A MORE GENERAL FRAMEWORK THAN THE δ-PRIMARY HYPERIDEALS 26 Apr 2023 M Anbarloei A MORE GENERAL FRAMEWORK THAN THE δ-PRIMARY HYPERIDEALS 26 Apr 2023 The δ-primary hyperideal is a concept unifing the n-ary prime and n-ary primary hyperideals under one frame where δ is a function which assigns to each hyperideal Q of G a hyperideal δ(Q) of the same hyperring with specific properties. In this paper, for a commutative Krasner (m, n)hyperring G with scalar identity 1, we aim to introduce and study the notion of (t, n)-absorbing δ-semiprimary hyperideals which is a more general structure than δ-primary hyperideals. We say that a proper hyperideal Q of G is an (t, n)-absorbing δ-semiprimary hyperideal if whenever k(a tn−t+1 1 ) ∈ Q for a tn−t+1 1 ∈ G, then there exist (t − 1)n − t + 2 of the a , i s whose k-product is in δ(Q). Furthermore, we extend the concept to weakly (t, n)-absorbing δ-semiprimary hyperideals. Several properties and characterizations of these classes of hyperideals are determined. In particular, after defining srongly weakly (t, n)-absorbing δ-semiprimary hyperideals, we present the condition in which a weakly (t, n)-absorbing δ-semiprimary hyperideal is srongly. Moreover, we show that k(Q (tn−t+1) ) = 0 where the weakly (t, n)-absorbing δsemiprimary hyperideal Q is not (t, n)-absorbing δ-semiprimary. Also, we investigate the stability of the concepts under intersection, homomorphism and cartesian product of hyperrings.2010 Mathematics Subject Classification. 16Y99. Introduction For the first time, the idea of 2-absorbing ideals as an extension of prime ideals was presented by Badawi in [3]. The concept and it , s generalizations have been widely studied by many researchers. The notion of 2-absorbing δ-semiprimary ideals in a commutative ring which is an expansion of the 2-absorbing ideals was introduced by Celikel. From [4], a proper ideal I of a commutative ring R refers to a (weakly) 2-absorbing δ-semiprimary ideal if x, y, z ∈ R and (0 = xyz ∈ I) xyz ∈ I imply xy ∈ δ(I) or yz ∈ δ(I) or xz ∈ δ(I) where δ is a function that assigns to each ideal I an ideal δ(I) of the same ring. Study on n-ary algebras goes back to Kasner's lecture [9] at a scientific meeting in 1904. In 1928, the first paper was written concerning the theory of n-ary groups by Dorente. The n-ary structures have been studied in [5], [7], [10], [11], [12], [13], [14] and [15]. There is a hyperring in which the addition is a hyperoperation, while the multiplication is an ordinary binary operation. The hyperring introduced by Krasner is called Krasner hyperring. The Krasner (m, n)-hyperring as a generalization of the Krasner hyperrings and a subclass of (m, n)-hyperrings was introduced by Mirvakili and Davvaz in [14]. (G, h, k), or simply G, is a Krasner (m, n)-hyperring if: (1) (G, h) is a canonical m-ary hypergroup; (2) (G, k) is a n-ary semigroup; (3) The operation k is distributive with respect to the hyperoperation h , i.e., for every g i− 1 1 , g n i+1 , x m 1 ∈ G, and 1 ≤ i ≤ n, k(g i−1 1 , h(x m 1 ), g n i+1 ) = h(k(g i− 1 1 , x 1 , g n i+1 ), · · · , k(g i−1 1 , x m , g n i+1 )); (4) 0 is a zero element of the n-ary operation k, i.e., for every g n 2 ∈ R , k(0, g n 2 ) = k(g 2 , 0, g n 3 ) = · · · = k(g n 2 , 0) = 0. A non-empty subset H of G is a subhyperring of G if (H, h, k) is a Krasner (m, n)hyperring. The non-empty subset I of G is a hyperideal if (I, h) is an m-ary subhypergroup of (G, h) and k(g i−1 1 , I, g n i+1 ) ⊆ I, for all g n 1 ∈ G and 1 ≤ i ≤ n. By g j i we mean the sequence g i , g i+1 , · · · , g j and it is the empty symbol if j < i. Then we have h(a 1 , · · · , a i , b i+1 , · · · , b j , c j+1 , · · · , c n ) = h(a i 1 , b j i+1 , c n j+1 ) and we write h(a i 1 , b (j−i) , c n j+1 ), where b i+1 = · · · = b j = b. We define h(G n 1 ) = {h(g n 1 ) \| g i ∈ G i , 1 ≤ i ≤ n} where G n 1 are non-empty subsets of G. Recall from [1] that a proper hyperideal P of G is called n-ary prime if k(G n 1 ) ⊆ P for hyperideals G n 1 of R implies that G i ⊆ P for some 1 ≤ i ≤ n. If k(g n 1 ) ∈ P for all g n 1 ∈ G and some hyperideal P of G implies that g i ∈ P for some 1 ≤ i ≤ n, then P is an n-ary prime hyperideal of G, by Lemma 4.5 in [1]. Let I be a hyperideal in G with scalar identity 1. By rad(I) we mean the intersection is taken over all n-ary prime hyperideals P which contain I. rad(I) = G if the set of all n-ary prime hyperideals containing I is empty . It was shown (Theorem 4.23 in [1]) that if g ∈ rad(I), then there exists s ∈ N such that k(g (s) , 1 (n−s) ) ∈ I for s ≤ n, or k (l) (g (s) ) ∈ I for s = l(n − 1) + 1. A proper hyperideal I of G with the scalar identity 1 is an n-ary primary hyperideal if k(g n 1 ) ∈ I and g i / ∈ I imply k(g i−1 1 , 1, g n i+1 ) ∈ rad(I) for some 1 ≤ i ≤ n. Theorem 4.28 in [1] shows that the radical of an n-ary primary hyperideal of G is an n-ary prime hyperideal of G. Let HI(G) denote the set of all hyperideals of G. Recall from [2] that a function δ from HI(G) to HI(G) is a hyperideal expansion of G if I ⊆ δ(I) and if I ⊆ J for any hyperideals I, J of G, then δ(I) ⊆ δ(J). For example, the mapping δ 0 , δ 1 and δ R , defined by δ 0 (I) = I, δ 1 (I) = rad(I) and δ R (I) = R for all hyperideals I of G, are hyperideal expansions of G. Furthermore, the function δ q , defined by δ q (I/J) = δ(I)/J for any hyperideal I of G containing hyperideal J and an expansion function δ of G, is a hyperideal expansion of G/J. A proper hyperideal P of G is an (t, n)-absorbing hyperideal as in [8] if k(g tn−t+1 1 ) ∈ P for some g tn−t+1 1 ∈ G implies that there exist (t− 1)n− t+ 2 of the g , i s whose k-product is in P . From the paper, a proper P of G is called an (t, n)-absorbing primary hyperideal if k(g tn−t+1 1 ) ∈ P for some g tn−t+1 1 ∈ G implies that k(g (t−1)n−t+2 1 ) ∈ P or a k-prodect of (t − 1)n − t + 2 of the g , i s except k(g tn−t+2 1 ) is in rad(P ). The notion of n-ary δ-primary hyperideals was introduced in a Krasner (m, n)-hyperring in [2]. This concept unifies the n-ary prime and n-ary primary hyperideals under one frame. A proper hyperideal Q of G is called δ-primary hyperideal if k(a n 1 ) ∈ Q for a n 1 ∈ G implies that a i ∈ Q or k(a i−1 1 , 1, a n i+1 ) ∈ δ(Q) for some 1 ≤ i ≤ n. Also, a proper hyperideal Q of G is (t, n)-absorbing δ-primary if for a tn−t+1 1 ∈ G, k(a tn−t+1 1 ) ∈ Q implies that k(a (t−1)n−t+2 1 ) ∈ Q or a k-product of (t − 1)n − t + 2 of a , i s except k(a (t−1)n−t+2 1 ) is in δ(Q). Now in this paper, we aim to define and study the notion of (t, n)-absorbing δ-semiprimary hyperideals which is more general than the concept of δ-primary hyperideals in a Krasner (m, n)-hyperring. Additionally, we present an extension of the notion called weakly (t, n)-absorbing δ-semiprimary hyperideals. Among many results in this paper, it is shown (Theorem 2.30) that if Q is a weakly (t, n)absorbing δ-semiprimary hyperideal of G that is not (t, n)-absorbing δ-semiprimary, then k(Q (tn−t+1) ) = 0. Let Q be a weakly (t, n)-absorbing δ-semiprimary hyperideal of G and 0 = k(Q tn−t+1 1 ) ⊆ Q for some hyperideals Q tn−t+1 1 of G. It is shown (Theorem 2.25) that if Q is a free δ-(t, n)-zero with respect to k(Q tn−t+1 1 ), then k-product of (t − 1)n − t + 2 of the Q i is a subset of δ(Q). Moreover, the stability of these concepts are examined under intersection, homomorphism and cartesian product of hyperrings. 2. (weakly) (t, n)-absorbing δ-semiprimay hyperideals Throughout this section, G is a commutative Krasner (m, n)-hyperring with scalar identity 1. Initially, we give the definition of (t, n)-absorbing δ-semiprimay hyperideals of G. Definition 2.1. Let δ be a hyperideal expansion of G and t be a positive integer. A proper hyperideal Q of G is called an (t, n)-absorbing δ-semiprimary hyperideal if whenever k(a tn−t+1 1 ) ∈ Q for a tn−t+1 1 ∈ G, then there exist (t − 1)n − t + 2 of the a , i s whose k-product is in δ(Q). Example 2.2. For all n-ary prime hyperideal of G, we have n-ary prime =⇒ n-ary δ-primary =⇒ (t, n)-absorbing δ-primary =⇒ (t, n)-absorbing δ-semiprimary The next example shows that the inverse of " =⇒ " , s in Example 2.2, is not true, in general. Example 2.3. In the Krasner (2, 2)-hyperring (G = [0, 1], ⊞, •) that " • " is the usual multiplication on real numbers and 2-ary hyperoperation " ⊞ " is defined by a ⊞ b = {max{a, b}}, if a = b [0,a], if a = b, the hyperideal Q = [0, 0.5] is a (2, 2)-absorbing δ 1 -semiprimary hyperideal of G but it is not 2-ary prime. Theorem 2.4. Let Q be an (t, n)-absorbing δ-semiprimary hyperideal of G with rad(δ(Q)) ⊆ δ(rad(Q)). Then rad(Q) is an (t, n)-absorbing δ-semiprimary hyperideal of G. Proof. Let k(a tn−t+1 1 ) ∈ rad(Q) for a tn−t+1 1 ∈ G such that all products of (t − 1)n − t + 2 of the a , i s, other than k(a (t−1)n−t+2 1 ), are not in δ(rad(Q)). By the assumption, we conclude that none of the k-products of the a , i s are in rad(δ(Q)). From k(a tn−t+1 1 ) ∈ rad(Q), it follows that there exists s ∈ N with k(k(a tn−t+1 1 ) (s) , 1 (n−s) ) ∈ Q, for s ≤ n or k (l) (k(a tn−t+1 1 ) (s) ) ∈ Q, for s > n and s = l(n − 1) + 1. In the first possibility, we get k(k(a 1 ) (s) , k(a 2 ) (s) , · · · , k(a tn−t+1 ) (s) , 1 (n−s) ) ∈ Q. Since Q is an (t, n)-absorbing δ-semiprimary hyperideal of G, we conclude that k(k(a 1 ) (s) , k(a 2 ) (s) , · · · , k(a (t−1)n−t+2 ) (s) , 1 (n−s) ) = k(k(a (t−1)n−t+2 1 ) (s) , 1 (n−s) ) ∈ δ(Q) because none of the k-products of the a , i s are in rad(δ(Q)). Since k(a (t−1)n−t+2 1 ) ∈ rad(δ(Q)) and rad(δ(Q)) ⊆ δ(rad(Q)), then we have k(a (t−1)n−t+2 1 ) ∈ δ(rad(Q)). If k (l) (k(a tn−t+1 1 ) (s) ) ∈ Q, for s > n and s = l(n− 1)+ 1, then we are done similarly. Thus rad(Q) is an (t, n)-absorbing δ-semiprimary hyperideal of G. The following result is a direct consequence of the previous theorem. Corollary 2.5. If Q is an (t, n)-absorbing δ 1 -semiprimary hyperideal of G, then rad(Q) is an (t, n)-absorbing hyperideal of G. From [1], the hyperideal generated by an element g in G is defined by < g >= k(G, g, 1 (n−2) ) = {k(r, g, 1 (n−2) ) \| r ∈ G}. The following theorem will give us a characterization of (t, n)-absorbing δ-semiprimary hyperideals. Theorem 2.6. Every proper hyperideal is an (t, n)-absorbing δ-semiprimary hyperideal of G if and only if every proper principal hyperideal is an (t, n)-absorbing δ-semiprimary hyperideal of G. Proof. =⇒ It is obvious. ⇐= Assume that Q is a proper hyperideal of G and k(a tn−t+1 1 ) ∈ Q for a tn−t+1 1 ∈ G. Therefore k(a tn−t+1 1 ) ∈< k(a tn−t+1 1 ) >. Since every proper principal hyperideal is an (t, n)-absorbing δ-semiprimary hyperideal of G, there exist (t−1)n−t+2 of the a , i s whose k-product is in δ(< k(a tn−t+1 1 ) >) ⊆ δ(Q). Hence Q is an (t, n)-absorbing δ-semiprimary hyperideal of G. Recall from [2] that a hyperideal expansion δ of G is called intersection preserving if it satisfies δ(P ∩Q) = δ(P )∩δ(Q), for all hyperideals P and Q of G. For example, hyperideal expansion δ 1 of G is intersection preserving. Theorem 2.7. Let the hyperideal expansion δ of G be intersection preserving. If Q s 1 are (t, n)-absorbing δ-semiprimary hyperideals of G Such that δ(Q i ) = P for each 1 ≤ i ≤ s, then Q = s i=1 Q i is an (t, n)-absorbing δ-semiprimary hyperideal of G with δ(Q) = P . Proof. Assume that k(a tn−t+1 1 ) ∈ Q for a tn−t+1 1 ∈ G such that k(a (t−1)n−t+2 1 ) / ∈ δ(Q). Since δ(Q) = δ(∩ s i=1 Q i ) = ∩ s i=1 δ(Q i ) = P , then there exists 1 ≤ u ≤ s such that k(a (t−1)n−t+2 1 ) / ∈ δ(Q u ). Since Q u is an (t, n)-absorbing δ-semiprimary hyperideal of G and k(a tn−t+1 1 ) ∈ Q u , then there is a k-product of (t − 1)n − t + 2 of the a , i s is in δ(Q u ) = P = δ(Q). Thus Q = s i=1 Q i is an (t, n)-absorbing δ-semiprimary hyperideal of G with δ(Q) = P . Let (G 1 , h 1 , k 1 ) and (G 2 , h 2 , k 2 ) be two Krasner (m, n)-hyperrings such that 1 G1 and 1 G2 be scalar identities of G 1 and G 2 , respectively. Then ( G 1 × G 2 , h = h 1 × h 2 , k = k 1 × k 2 ) is a Krasner (m, n)-hyperring where h 1 × h 2 ((a 1 , b 1 ), · · · , (a m , b m )) = {(a, b) \| a ∈ h 1 (a m 1 ), b ∈ h 2 (b m 1 )}, k 1 × k 2 ((a 1 , b 1 ), · · · , (a n , b n )) = (k 1 (a n 1 ), k 2 (b n 1 )), for all a i ∈ G 1 and b i ∈ G 2 [6]. Theorem 2.8. Let δ 1 and δ 2 be two hyperideal expansions of Krasner (m, n)hyperrings G 1 and G 2 , respectively, such that δ( Q 1 × Q 2 ) = δ 1 (Q 1 ) × δ 2 (Q 2 ) for hyperideals Q 1 and Q 2 of G 1 and G 2 , respectively. If Q = Q 1 × Q 2 is an (t + 1, n)- absorbing δ-semiprimary hyperideal of G = G 1 × G 2 , then either Q 1 is an (t + 1, n)- absorbing δ 1 -semiprimary hyperideal of G 1 and δ 2 (Q 2 ) = G 2 or Q 2 is an (t + 1, n)- absorbing δ 2 -semiprimary hyperideal of G 2 and δ 1 (Q 1 ) = G 1 or Q i is an (t, n)- absorbing δ i -semiprimary hyperideal of G i for each i ∈ {1, 2}. Proof. Let Q = Q 1 × Q 2 be an (t + 1, n)-absorbing δ-semiprimary hyperideal of G = G 1 × G 2 . Assume that δ 1 (Q 1 ) = G 1 and δ 2 (Q 2 ) = G 2 . Let us suppose that k 1 (a (t+1)n−t 1 ) ∈ Q 1 for some a (t+1)n−t 1 ∈ G 1 such that all products of tn−t+1 of the a , i s except k 1 (a tn−t+1 1 ) are not in δ(Q 1 ). Note that k((a 1 , 0), · · · , (a (t+1)n−t , 0)) ∈ Q and all products of tn − t + 1 of the (a i , 0) , s are not in δ(Q). Since Q is an (t + 1, n)-absorbing δ-semiprimary hyperideal of G, we get k((a 1 , 0), · · · , (a tn−t+1 , 0)) ∈ δ(Q) = δ 1 (Q 1 )× δ 2 (Q 2 ) which means k 1 (a tn−t+1 1 ) ∈ δ(Q 1 ). Thus Q 1 is an (t+ 1, n)absorbing δ 1 -semiprimary hyperideal of G 1 . Similiar for the second assertion. For the third assertion, assume δ 1 (Q 1 ) = G 1 and δ 2 (Q 2 ) = G 2 . Moreover, let us suppose that Q 1 is not an (t, n)-absorbing δ-semiprimary hyperideal of G 1 and k 1 (a tn−t+1 1 ) ∈ Q 1 . We define the following elements of G: x 1 = (a 1 , 1 G2 ), x 2 = (a 2 , 1 G2 ), · · · , x tn−t+1 = (a tn−t+1 , 1 G2 ), x (t−1)n−t+2 = (1 G1 , 0). Therefore we have k(x (t−1)n−t+2 1 ) = (k 1 (a tn−t+1 1 ), 0) ∈ Q, k(x tn−t+1 1 ) = (k 1 (a tn−t+1 1 ), 1 G2 ) / ∈ δ(Q) and k(x 1 , · · · ,x i , · · · , x (t−1)n−t+2 ) = (k 1 (a 1 , · · · ,â i , · · · , a (t−1)n−t+2 ), 0) / ∈ δ(Q) for some 1 ≤ i ≤ tn − t + 1, a contradiction. Thus Q 1 is an (t, n)-absorbing δ 1 - semiprimary hyperideal of G 1 . Similarly, we conclude that Q 2 is an (t, n)-absorbing δ 2 -semiprimary hyperideal of G 2 Theorem 2.9. Let δ 1 , · · · , δ tn−t+1 be hyperideal expansions of Krasner (m, n)- hyperrings G 1 , · · · , G tn−t+1 such that δ(Q 1 × · · · × Q tn−t+1 ) = δ 1 (Q 1 ) × · · · × δ tn−t+1 (Q tn−t+1 ) for hyperideals Q 1 , · · · , Q tn−t+1 of G 1 , · · · , G tn−t+1 , respectively. If Q = Q 1 × · · · × Q tn−t+1 is an (t + 1, n)-absorbing δ-semiprimary hyperideal of G = G 1 × ... × G tn−t+1 , then either Q u is an (t + 1, n)-absorbing δ-semiprimary hyperideal of G u for some 1 ≤ u ≤ tn − t + 1 and δ i (Q i ) = G i for each 1 ≤ i ≤ tn − t + 1 and i = u or Q u and Q v are (t, n)-absorbing δ u,v -semiprimary hyperideals of G u and G v , respectively, for some u, v ∈ {1, · · · , tn − t + 1} and δ i (Q i ) = G i for all 1 ≤ i ≤ tn − t + 1 but i = u, v. Proof. It can be seen that the idea is true in a similar manner to the proof of Theorem 2.8. Now, we want to extend the notion of (t, n)-absorbing δ-semiprimary hyperideals to weakly (t, n)-absorbing δ-semiprimary hyperideal. Although different from each other in many aspects, they share quite a number of similar properties as well. Definition 2.10. Let δ be a hyperideal expansion of G and t be a positive integer. A proper hyperideal Q of G refers to a weakly (t, n)-absorbing δ-semiprimary hyperideal if a tn−t+1 It is easy to see that the hyperideal Q = {0,2,4,6} of G is a (2, 2)-absorbing δ 1 -semiprimary. 1 ∈ G and 0 = k(a tn−t+1 1 ) ∈ Q, then there exist (t − 1)n − t + 2 of the a , i s whose k-product is in δ(QTheorem 2.12. If Q is a (weakly) (t, n)-absorbing δ-semiprimary hyperideal of G, then Q is (weakly) (v, n)-absorbing δ-semiprimary for all v > n. Proof. By using an argument similar to that in the proof of Theorem 4.4 in [8], one can complete the proof. Theorem 2.13. Let Q be a proper hyperideal of G. If δ(Q) is a (weakly) (t, n)- absorbing hyperideal of G, then Q is a (weakly) (t, n)-absorbing δ-semiprimary hy- perideal of G. Proof. Let (0 = k(a tn−t+1 1 ) ∈ Q) k(a tn−t+1 1 ) ∈ Q such that all products of (t−1)n− t + 2 of the a , i s, other than k(a (t−1)n−t+2 1 ), are not in δ(Q). Since δ(Q) is a (weakly) (t, n)-absorbing hyperideal of G and Q ⊆ δ(Q), we conclude that k(a (t−1)n−t+2 1 ) ∈ δ(Q) . This shows that Q is a (weakly) (t, n)-absorbing δ-semiprimary hyperideal of G. Theorem 2.14. Let Q be a proper hyperideal of G such that δ(δ(Q)) = δ(Q). Then δ(Q) is a (weakly) (t, n)-absorbing hyperideal of G if and only if δ(Q) is a (weakly) (t, n)-absorbing δ-semiprimary hyperideal of G. Proof. =⇒ Assume that δ(Q) is a (weakly) (t, n)-absorbing hyperideal of G. Since δ(δ(Q)) = δ(Q), we are done by Theorem 2.13. ⇐= Let δ(Q) be a (weakly) (t, n)-absorbing δ-semiprimary hyperideal of G. Suppose that (0 = k(a tn−n+1 1 ) ∈ δ(Q)) k(a tn−n+1 1 ) ∈ δ(Q). Since δ(Q) is a (weakly) (t, n)-absorbing δ-semiprimary hyperideal of G, then there exist (t−1)n−t+2 of the a , i s whose k-product is in δ(δ(Q)). Since δ(δ(Q)) = δ(Q), then the k-product of the (t − 1)n − t + 2 of the a , i s is in δ(Q) which means δ(Q) is a (weakly) (t, n)-absorbing hyperideal of G. Theorem 2.15. Let Q be a (weakly) (t, n)-absorbing δ-semiprimary hyperideal of G and P be a proper hyperideal of G such that P ⊆ Q. If δ(Q) = δ(P ), then P is a (weakly) (t, n)-absorbing δ-semiprimary hyperideal of G. Proof. Assume that (0 = k(a tn−t+1 1 ) ∈ P ) k(a tn−t+1 1 ) ∈ P for a tn−t+1 1 ∈ G. By the assumption, we get (0 = k(a tn−t+1 1 ) ∈ Q) k(a tn−t+1 1 ) ∈ Q which implies there exist (t − 1)n − t + 2 of the a , i s whose k-product is in δ(Q) because Q is a (weakly) (t, n)-absorbing δ-semiprimary hyperideal of G. From δ(Q) = δ(P ), it follows that the k-product of (t − 1)n − t + 2 of the a , i s is in δ(P ) which means P is a (weakly) (t, n)-absorbing δ-semiprimary hyperideal of G. Definition 2.16. Let Q be a proper hyperideal of G. Q refers to a strongly (weakly) (t, n)-absorbing δ-semiprimary hyperideal if (0 = k(Q tn−t+1 1 ) ⊆ Q) k(Q tn−t+1 1 ) ⊆ Q for some hyperideals Q tn−t+1 1 of G, then there exist (t − 1)n − t + 2 of Q , i s whose k-product is a subset of δ(Q). Definition 2.17. Assume that G is a commutative Krasner (m, 2)-hyperring and Q is a weakly (2, 2)-absorbing δ-semiprimary hyperideal of G. Then (x, y, z) is said to be an δ-(2, 2)-zero of Q for some x, y, z ∈ G if k(x, y, z) = 0, k(x, y) / ∈ δ(Q), k(y, z) / ∈ δ(Q) and k(x, z) / ∈ δ(Q). Theorem 2.18. Let G be a commutative Krasner (m, 2)-hyperring, Q a weakly (2, 2)-absorbing δ-semiprimary hyperideal of G and k(Q 1 , x, y) ⊆ Q for some x, y ∈ G and a hyperideal Q 1 of G. If (q, x, y) is not a δ-(2, 2)-zero of Q for all q ∈ Q 1 and k(x, y) / ∈ δ(Q), then k(Q 1 , x) ⊆ δ(Q) or k(Q 1 , y) ⊆ δ(Q). Proof. Let k(Q 1 , x, y) ⊆ Q for some x, y ∈ G and a hyperideal Q 1 of G but k(x, y) / ∈ δ(Q), k(Q 1 , x) δ(Q) and k(Q 1 , y) δ(Q). Then we have k(q 1 , x) δ(Q) and k(q 2 , y) δ(Q) for some q 1 , q 2 ∈ Q 1 . Since (q 1 , x, y) is not a δ-(2, 2)-zero of Q and k(q 1 , x, y) ∈ Q, we get k(q 1 , y) ∈ δ(Q). Similarly, we have k(q 2 , x) ∈ δ(Q). Note that k(h(q 1 , q 2 , 0 (m−2) ), x, y) = h(k(q 1 , x, y), k(q 2 , x, y), 0 (m−2) ) ⊆ Q. Then we obtain k(h(q 1 , q 2 , 0 (m−2) ), x) = h(k(q 1 , x), k(q 2 , x), 0 (m−2) ) ⊆ δ(Q) or k(h(q 1 , q 2 , 0 (m−2) ), y) = h(k(q 1 , y), k(q 2 , y), 0 (m−2) ) ⊆ δ(Q). This follows that k(q 1 , x) ∈ h(−k(q 2 , x), 0 (m−1) ) ⊆ δ(Q) or k(q 2 , y) ∈ h(−k(q 1 , y), 0 (m−1) ) ⊆ δ(Q) which both of them are a contradiction. Consequently, k(Q 1 , x) ⊆ δ(Q) or k(Q 1 , y) ⊆ δ(Q). Theorem 2.19. Let G be a commutative Krasner (m, 2)-hyperring, Q a weakly (2, 2)-absorbing δ-semiprimary hyperideal of G and k(Q 1 , Q 2 , x) ⊆ Q for some x ∈ G and two hyperideals Q 1 , Q 2 of G. If (q 1 , q 2 , x) is not a δ-(2, 2)-zero of Q for all q 1 ∈ Q 1 and q 2 ∈ Q 2 , then k( Q 1 , x) ⊆ δ(Q) or k(Q 2 , x) ⊆ δ(Q) or k(Q 1 , Q 2 ) ⊆ δ(Q). Proof. Let k(Q 1 , Q 2 , x) ⊆ Q, k(Q 1 , x) δ(Q), k(Q 2 , x) δ(Q) and k(Q 1 , Q 2 ) δ(Q). Then we get k(q, x) / ∈ δ(Q) and k(q 1 , Q 2 ) δ(Q) for some q, q 1 ∈ Q 1 . By Theorem 2.18, we conclude that k(q, Q 2 ) ⊆ δ(Q) because k(q, Q 2 , x) ⊆ Q, k(q, x) / ∈ δ(Q) and k(Q 2 , x) δ(Q). Also, from Theorem 2.18, we obtain k(q 1 , x) ∈ δ(Q). Note that k(h(q, q 1 , 0 (m−2) ), Q 2 , x) = h(k(q, Q 2 , x), k(q 1 , Q 2 , x), 0 (m−2) ) ⊆ Q. Then we have k(h(q, q 1 , 0 (m−2) ), Q 2 ) = h(k(q, Q 2 ), k(q 1 , Q 2 ), 0 (m−2) ) ⊆ δ(Q) which means k(q 1 , Q 2 ) ⊆ h(−k(q, Q 2 ), 0 (m−1) ) ⊆ δ(Q) or k(h(q, q 1 , 0 (m−2) ), x) = h(k(q, x), k(q 1 , x), 0 (m−2) ) ⊆ δ(Q) which implies k(q, x) ∈ h(−k(q 1 , x), 0 (m−1) ) ⊆ δ(Q). This is a contradiction. Hence k(Q 1 , x) ⊆ δ(Q) or k(Q 2 , x) ⊆ δ(Q) or k(Q 1 , Q 2 ) ⊆ δ(Q). Definition 2.20. Suppose that G is a commutative Krasner (m, 2)-hyperring and Q 3 1 , Q be some proper hyperideals of G such that Q is a weakly (2, 2)-absorbing δ-semiprimary hyperideal of G. Q is said to be a free δ-(2, 2)-zero with respect to k(Q 3 1 ) if (q 1 , q 2 , q 3 ) is not a δ-(2, 2)-zero of Q for every q 1 ∈ Q 1 , q 2 ∈ Q 2 and q 3 ∈ Q 3 . Theorem 2.21. Let G be a commutative Krasner (m, 2)-hyperring, Q a weakly (2, 2)-absorbing δ-semiprimary hyperideal of G and 0 = k(Q 1 , Q 2 , Q 3 ) ⊆ Q for some hyperideals Q 3 1 of G. If Q is a free δ-(2, 2)-zero with respect to k(Q 3 1 ), then k(Q 2 1 ) ⊆ δ(Q) or k(Q 3 2 ) ⊆ δ(Q) or k(Q 1 , Q 3 ) ⊆ δ(Q). Proof. Suppose that k(Q 3 1 ) ⊆ Q but k(Q 2 1 ) δ(Q) or k(Q 3 2 ) δ(Q) or k(Q 1 , Q 3 ) δ(Q) . This implies that k(q, Q 2 ) δ(Q) and k(q 1 , Q 3 ) δ(Q) for some q, q 1 ∈ Q 1 . By Theorem 2.19, we get k(q, Q 3 ) ⊆ δ(Q) because k(q, Q 3 2 ) ⊆ Q, k(Q 3 2 ) δ(Q) and k(q, Q 2 ) δ(Q). Also, from Theorem 2.19, we obtain k(q 1 , Q 2 ) δ(Q) as k(q 1 , Q 3 2 ) ⊆ Q, k(Q 3 2 ) δ(Q) and k(q 1 , Q 2 ) δ(Q). Since k(h(q, q 1 ), Q 3 2 ) ⊆ Q and k(Q 3 2 ) δ(Q), we have k(h(q, q 1 , 0 (m−2) ), Q 2 ) = h(k(q, Q 2 ), k(q 1 , Q 2 ), 0 (m−2) ) ⊆ δ(Q) or k(h(q, q 1 , 0 (m−2) ), Q 3 ) = h(k(q, Q 3 ), k(q 1 , Q 3 ), 0 (m−2) ) ⊆ δ(Q). In the first case, we conclude that k(q, Q 2 ) ∈ h(−k(q 1 , Q 2 ), 0 (m−1) ) ⊆ δ(Q), a controdiction. Moreover, the second case leads to a contradiction because k(q 1 , Q 3 ) ∈ h(−k(q, Q 3 ), 0 (m−1) ) ⊆ δ(Q). Thus k(Q 2 1 ) ⊆ δ(Q) or k(Q 3 2 ) ⊆ δ(Q) or k(Q 1 , Q 3 ) ⊆ δ(Q). Definition 2.22. Assume that Q is a weakly (k, n)-absorbing δ-semiprimary hyperideal of G. Then (a 1 , · · · , a tn−t+1 ) is called δ-(t, n)-zero of Q if k(a tn−t+1 1 ) = 0 and none k-product of the terms (t − 1)n − t + 2 of a , i s is in δ(Q). Theorem 2.23. Assume that Q is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G and k(a 1 , · · · ,â i1 , · · · ,â i2 , · · · ,â is , · · · , a tn−t+1 , Q s 1 ) ⊆ Q for some a tn−t+1 1 ∈ G and some hyperideals Q 1 , · · · Q s of G such that 1 ≤ i 1 , · · · , i s ≤ tn−t+1 and 1 ≤ s ≤ (t−1)n−t+2. If (a 1 , · · · ,â i1 , · · · ,â i2 , · · · ,â is , · · · , a tn−t+1 , q s 1 ) is not a δ-(t, n)-zero of Q for all q i ∈ Q i , then k-product of (t − 1)n − t + 2 of a 1 , · · · ,â i1 , · · · ,â i2 , · · · ,â is , · · · , a tn−t+1 , Q s 1 including at least one of the Q , i s is in δ(Q). Proof. We prove it with induction on s. Let us consider s = 1. In this case we show that k-product of (t − 1)n − t + 2 of a 1 , · · · ,â i1 , · · · , a tn−t+1 , Q 1 including Q 1 is in δ(Q). Assume that all products of (t− 1)n− t+ 2 of a 1 , · · · ,â i1 , · · · , a tn−t+1 , Q 1 are not in δ(Q). We consider k(a (t−1)n−t+2 2 , Q 1 ) / ∈ δ(Q). Since (a tn−t+1 2 , q 1 ) is not a δ-(t, n)-zero of Q for all q 1 ∈ Q 1 , then we conclude that k-product of the (t−1)n−t+2 of a , i s with q 1 is in δ(Q). By a similar argument given in the proof of Theorem 2.18, we have k(a tn−t+1 3 , h(a 1 , q 1 , 0 (m−2) )) = h(k(a tn−t+1 3 , a 1 ), k(a tn−t+1 3 , q 1 ), 0 (m−2) ) ⊆ δ(Q) which implies k(a tn−t+1 3 , a 1 ) ∈ h(−k(a tn−t+1 3 , q 1 ), 0 (m−1) ) ⊆ δ(Q), a contradiction. This implies that k-product of (t−1)n−t+2 of a 1 , · · · ,â i1 , · · · , a tn−t+1 , Q 1 including Q 1 is in δ(Q). Now, we suppose that the claim holds for all positive integers which are less than s. Let k(a 1 , · · · ,â i1 , · · · ,â i2 , · · · ,â is , · · · , a tn−t+1 , Q s 1 ) ⊆ Q but all products of (t−1)n−t+2 of a 1 , · · · ,â i1 , · · · ,â i2 , · · · ,â is , · · · , a tn−t+1 , Q s 1 including at least one of the Q , i s are not in δ(Q). We may assume that k(a tn−t+1 s+1 , Q s 1 ) / ∈ δ(Q). Note that (a tn−t+1 s+1 , q s 1 ) is not a δ-(t, n)-zero of Q for all q s 1 ∈ Q. We get k(a tn−t+1 s+1 , h(a 1 , q 1 , 0 (m−2) ), · · · , h(a s , q s , 0 (m−2) )) ⊆ δ(Q) by induction hypothesis and Theorem 2.19. Then we conclude that k(a tn−t+1 s+1 , h(a 1 , q 1 , 0 (m−2) ), · · · , h(a 1 ,q 1 , 0 (m−2) ) i1 , · · · , h(a 2 ,q 2 , 0 (m−2) ) i2 , · · · , h(a n−1 ,q n−1 , 0 (m−2) ) in−1 , · · · , h(a s , q s , 0 (m−2) )) ⊆ δ(Q) or k(a s+1 , · · · ,â is+1 , · · · ,â is+2 , · · · ,â is+n−1 , · · · , a tn−t+1 , h(a 1 , q 1 , 0 (m−2) ), · · · , h(a s , q s , 0 (m−2) )) ⊆ δ(Q) for some i ∈ {1, · · · , s}. This implies that k(a s+1 , · · · , a tn−t+1 , · · · , a n , · · · , a s ) ∈ δ(Q) or k(a s+n , · · · , a tn−t+1 , · · · , a s 1 ) ∈ δ(Q), a contradiction. Then we conclude that k-product of (t − 1)n − t + 2 of a 1 , · · · ,â i1 , · · · ,â i2 , · · · ,â is , · · · , a tn−t+1 , Q s 1 including at least one of the Q , i s is in δ(Q). Definition 2.24. Suppose that Q n 1 , Q be some proper hyperideals of G such that Q is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G and k(Q tn−t+1 1 ) ⊆ Q. Q is called a free δ-(t, n)-zero with respect to k(Q tn−t+1 1 ) if (q 1 , · · · , q tn−t+1 ) is not a δ-(t, n)-zero of Q for every q i ∈ Q i with 1 ≤ i ≤ tn − t + 1. Theorem 2.25. Assume that Q is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G and 0 = k(Q tn−t+1 1 ) ⊆ Q for some hyperideals Q tn−t+1 1 of G. If Q is a free δ-(t, n)-zero with respect to k(Q tn−t+1 1 ), then k-product of (t − 1)n − t + 2 of the Q i is a subset of δ(Q). Proof. This can be proved by Theorem 2.23, in a very similar manner to the way in which Theorem 2.21 was proved. Theorem 2.26. Let G be a commutative Krasner (m, 2)-hyperring and Q be a weakly (2, 2)-absorbing δ-semiprimary hyperideal of G. If (x, y, z) is an δ-(2, 2)zero of Q for some x, y, z ∈ G, then (i) k(x, y, Q) = k(y, z, Q) = k(x, z, Q) = 0 (ii) k(x, Q (2) ) = k(y, Q (2) ) = k(z, Q (2) ) = 0 Proof. (i) Let Q be a weakly (2, 2)-absorbing δ-semiprimary hyperideal of G and (x, y, z) be an δ-(2, 2)-zero of Q. Let us assume that k(x, y, Q) = 0. This means that k(x, y, q) = 0 for some q ∈ Q. So we have 0 = k(x, h(z, q, 0 (m−2) ), y) = h(k(x, z, y), k(x, q, y), 0 (m−2) ) ⊆ Q. Since Q is weakly (2, 2)-absorbing δ-semiprimary and k(x, y) / ∈ δ(Q), we get k(x, h(z, q, 0 (m−2) )) = h(k(x, z), k(x, q), 0 (m−2) ) ⊆ δ(Q) or k(h(z, q, 0 (m−2) ), y) = h(k(z, y), k(q, y), 0 (m−2) ) ⊆ δ(Q). In the first case, we have k(x, z) ∈ h(−k(x, q), 0 (m−1) ) ⊆ δ(Q) which is a contradiction. The second case leads to a contradiction because k(z, y) ∈ h(−k(q, y), 0 (m−1) ) ⊆ δ(Q). Thus k(x, y, Q) = 0. Similiar for the other cases. (ii) Let k(x, Q (2) ) = 0. This implies that k(x, q 2 1 ) = 0 for some q 1 , q 2 ∈ Q. Therefore 0 = k(x, h(y, q 1 , 0 (m−2) ), h(z, q 2 , 0 (m−2) )) = h(k(x, y, z), k(x, y, q 2 ), k(x, q 1 , z), k(x, q 2 1 ), 0 (m−4) ) ⊆ Q. Since Q is a weakly (2, 2)-absorbing δ-semiprimary hyperideal of G, we obtain the following cases: Case 1. k(x, h(y, q 1 , 0 (m−2) )) ⊆ δ(Q) which implies h(k(x, y), k(x, q 1 ), 0 (m−2) ) ⊆ δ(Q). Then we have k(x, y) ∈ h(−k(x, q 1 ), 0 (m−1) ) ⊆ δ(Q), a contradiction. Case 2. k(x, h(z, q 2 , 0 (m−2) )) ⊆ δ(Q) which means h(k(x, z), k(x, q 2 ), 0 (m−2) ) ⊆ δ(Q) . This follows that k(x, z) ∈ h(−k(x, q 2 ), 0 (m−1) ) ⊆ δ(Q), a contradiction. Case 3. k(h(y, q 1 , 0 (m−2) ), h(z, q 2 , 0 (m−2) )) ⊆ δ(Q) and so h(k(y, z), k(q 1 , z), k(y, q 2 ), k(q 2 1 ), 0 (m−4) ) ⊆ δ(Q). This implies that k(y, z) ∈ h(−k(q 1 , z), −k(y, q 2 ), −k(q 2 1 ), 0 (m−2) ) ⊆ δ(Q) which is a contradiction. Therefore k(x, Q (2) ) = 0. Similiar for the other cases. Theorem 2.27. Let G be a commutative Krasner (m, 2)-hyperring and Q be a weakly (2, 2)-absorbing δ-semiprimary hyperideal of G but is not (2, 2)-absorbing δ-semiprimary. Then k(Q (3) ) = 0. Proof. Let Q be a weakly (2, 2)-absorbing δ-semiprimary hyperideal of G but is not (2, 2)-absorbing δ-semiprimary. This implies that we have an δ-(2, 2)-zero of Q for some x, y, z ∈ G. Let us assume that k(Q (3) ) = 0. Then k(q 3 1 ) = 0 for some q 3 1 ∈ Q. Therefore we have k(h(x, q 1 , 0 (m−2) ), h(y, q 2 , 0 (m−2) ), h(z, q 3 , 0 (m−2) )) = h(h(h(k(x, y, z), k(q 1 , y, z), 0 (m−2) ), h(k(x, y, q 3 ), k(q 1 , y, q 3 ), 0 (m−2) )), h(h(k(q 2 , x, z), h(q 2 1 , z), 0 (m−2) )), h(h(k(q 3 1 ), k(x, q 3 2 )), 0). From k(q 3 1 ) = 0, it follows that 0 = k(h(x, q 1 , 0 (m−2) ), h(y, q 2 , 0 (m−2) ), h(z, q 3 , 0 (m−2) )) ⊆ Q by Theorem 2.26. Since Q is weakly (2, 2)-absorbing δ-semiprimary, we have k(h(x, q 1 , 0 (m−2) ), h(y, q 2 , 0 (m−2) )) ⊆ δ(Q) or k(h(x, q 1 , 0 (m−2) ), h(z, q 3 , 0 (m−2) )) ⊆ δ(Q) or k(h(x, q 1 , 0 (m−2) ), h(z, q 3 , 0 (m−2) )) ⊆ δ(Q). In the first possibilty, we obtain h(k(x, y), k(x, q 2 ), k(q 1 , y), k(q 2 1 ), 0 (m−4) ) ⊆ δ(Q) which means k(x, y) ∈ h(−k(x, q 2 ), −k(q 1 , y), −k(q 2 1 ), 0 (m−3) ) ⊆ δ(Q) which is a contradiction. Moreover, the other possibilities lead to a contradiction. Thus k(Q (3) ) = 0. Definition 2.28. Assume that Q is a weakly (k, n)-absorbing δ-semiprimary hyperideal of G. Then (a 1 , · · · , a tn−t+1 ) is called δ-(t, n)-zero of Q if k(a tn−t+1 1 ) = 0 and none k-product of the terms (t − 1)n − t + 2 of a , i s is in δ(Q). Theorem 2.29. If Q is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G and (a 1 , · · · , a tn−t+1 ) is a δ-(t, n)-zero of Q, then for 1 ≤ i 1 , · · · , i s ≤ tn − t + 1 and 1 ≤ s ≤ (t − 1)n − t + 2, k(a 1 , · · · ,â i1 , · · · ,â i2 , · · · ,â is , · · · , a tn−t+1 , Q (s) ) = 0. Proof. We use the induction on s. Assume that s = 1. Let us suppose that k(a 1 , · · · ,â i1 , · · · , a tn−t+1 , Q) = 0. We may assume that k(a tn−t+1 2 , Q) = 0. Therefore k(a tn−t+1 2 , q) = 0 for some q ∈ Q. Hence every k-product of the (t − 1)n − t + 2 of a , i s including q is in δ(Q). By the same argument given in Theorem 2.26, we have k(a tn−t+1 3 , h(a 1 , q, 0 (m−2) )) = h(k(a tn−t+1 3 , a 1 ), k(a tn−t+1 3 , q), 0 (m−2) ) ⊆ δ(Q) which implies k(a tn−t+1 3 , a 1 ) ∈ h(−k(a tn−t+1 3 , q), 0 (m−1) ) ⊆ δ(Q), a contradiction. This means that k(a 1 , · · · ,â i1 , · · · , a tn−t+1 , Q) = 0. Now, let us suppose that k(a 1 , · · · ,â i1 , · · · ,â i2 , · · · ,â is , · · · , a tn−t+1 , Q (s) ) = 0. We may assume that k(a tn−t+1 s+1 , Q (s) ) = 0. Hence 0 = k(a tn−t+1 s+1 , q s 1 ) ∈ Q for some q s 1 ∈ Q. It follows that 0 = k(a tn−t+1 s+1 , h(a 1 , q 1 , 0 (m−2) ), · · · , h(a s , q s , 0 (m−2) )) ⊆ Q by Theorem 2.26 and induction hypothesis. Then we conclude that k(a tn−t+1 s+1 , h(a 1 , q 1 , 0 (m−2) ), · · · , h(a 1 ,q 1 , 0 (m−2) ) i1 , · · · , h(a 2 ,q 2 , 0 (m−2) ) i2 , · · · , h(a n−1 ,q n−1 , 0 (m−2) ) in−1 , · · · , h(a s , q s , 0 (m−2) )) ⊆ δ(Q) or k(a s+1 , · · · ,â is+1 , · · · ,â is+2 , · · · ,â is+n−1 , · · · , a tn−t+1 , h(a 1 , q 1 , 0 (m−2) ), · · · , h(a s , q s , 0 (m−2) )) ⊆ δ(Q) for some i ∈ {1, · · · , s}. This implies that k(a s+1 , · · · , a tn−t+1 , · · · , a n , · · · , a s ) ∈ δ(Q) or k(a s+n , · · · , a tn−t+1 , · · · , a s 1 ) ∈ δ(Q), a contradiction. Thus we conclude that k(a 1 , · · · ,â i1 , · · · ,â i2 , · · · ,â is , · · · , a tn−t+1 , Q (s) ) = 0. Theorem 2.30. Let Q be a weakly (t, n)-absorbing δ-semiprimary hyperideal of G but is not (t, n)-absorbing δ-semiprimary. Then k(Q (tn−t+1) ) = 0. Proof. Assume that Q is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G but is not (t, n)-absorbing δ-semiprimary. Then there exists a δ-(t, n)-zero (a 1 , · · · , a tn−t+1 ) of Q. Now, the claim follows by using Theorem 2.29, in a very similar manner to the way in which Theorem 2.27 was proved. As an instant consequence of the previous theorem, we have the following explicit results. Corollary 2.31. Let Q be a weakly (t, n)-absorbing δ-semiprimary hyperideal of G but is not (t, n)-absorbing δ-semiprimary. Then Q ⊆ rad(0). Corollary 2.32. Assume that the commutative Krasner (m, n)-hyperring G has no non-zero nilpotent elements. If Q is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G, then Q is an (t, n)-absorbing δ-semiprimary hyperideal of G. The next theorem provides us how to determine weakly (t, n)-absorbing δ-semiprimary hyperideal to be (t, n)-absorbing δ-semiprimary. Theorem 2.33. Let Q be a weakly (t, n)-absorbing δ-semiprimary hyperideal of G such that δ(Q) = δ(0). Then Q is not (t, n)-absorbing δ-semiprimary if and only if there exists a δ-(t, n)-zero of 0. Proof. =⇒ Assume that Q is not an (t, n)-absorbing δ-semiprimary hyperideal of G. This implies that k(a tn−t+1 1 ) = 0 and none k-product of the terms (t−1)n−t+2 of a , i s is in δ(Q) for some a tn−t+1 1 ∈ G. From δ(Q) = δ(0), it follows that (a tn−t+1 1 ) is a δ-(t, n)-zero of 0. ⇐= Straightforward Let (G 1 , h 1 , k 1 ) and (G 2 , h 2 , k 2 ) be two commutative Krasner (m, n)-hyperrings. Recall from [14] that a mapping f : G 1 −→ G 2 is called a homomorphism if we have f (h 1 (a m 1 )) = h 2 (f (a 1 ), · · · , f (a m )) and f (k 1 (b n 1 )) = k 2 (f (b 1 ), ..., f (b n )) for all a m 1 ∈ G 1 and b n 1 ∈ G 1 . Let δ and δ ′ be hyperideal expansions of G 1 and G 2 , respectively. Recall from [2] that f : G 1 −→ G 2 is called a δδ ′ -homomorphism if δ(f −1 (Q 2 )) = f −1 (δ ′ (Q 2 )) for hyperideal Q 2 of G 2 . Note that δ ′ (h(Q 1 ) = h(δ(Q 1 ) for δδ ′ -epimorphism f and for hyperideal Q 1 of G 1 with Ker(f ) ⊆ Q 1 . Theorem 2.34. Let (G 1 , h 1 , k 1 ) and (G 2 , h 2 , k 2 ) be two Krasner (m, n)-hyperrings and f : G 1 −→ G 2 be a δδ ′ -homomorphism. Then the followings hold: (i) If Q 2 is an (t, n)-absorbing δ ′ -semiprimary hyperideal of G 2 , then f −1 (Q 2 ) is an (t, n)-absorbing δ-semiprimary hyperideal of G 1 . (ii) If Q 2 is a weakly (t, n)-absorbing δ ′ -semiprimary hyperideal of G 2 and Kerf is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G 1 , then f −1 (Q 2 ) is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G 1 . (iii) Let f be an epimorphism and Q 1 be a proper hyperideal of G 1 containing Kerf . If Q 1 is a (weakly) (t, n)-absorbing δ-semiprimary hyperideal of G 1 , then f (Q 1 ) is a (weakly) δ ′ -semiprimary hyperideal of G 2 . Proof. (i) Let k 1 (a tn−t+1 1 ) ∈ f −1 (Q 2 ) for a kn−k+1 1 ∈ G 1 . Then we get f (k 1 (a tn−t+1 1 )) = k 2 (f (a 1 ), ..., f (a tn−t+1 )) ∈ Q 2 . Since Q 2 is an (t, n)-absorbing δ ′ -semiprimary hyperideal of G 2 , then there exist (t − 1)n − t + 2 of f (a i ) , s whose k 2 -product is an element in δ ′ (Q 2 ). It follows that the image f of (t − 1)n − t + 2 of a , i whose k 2product is in δ ′ (Q 2 ) which means there exist (t−1)n−t+2 of a , i whose k 1 -product is in f −1 (δ ′ (Q 2 )) = δ(f −1 (Q 2 )). Thus f −1 (Q 2 ) is an (t, n)-absorbing δ-semiprimary hyperideal of G 1 . (ii) Assume that k 1 (a tn−t+1 a 1 ), ..., f (a tn−t+1 )) ∈ Q 2 . If 0 = f (k 1 (a tn−t+1 1 )), then it can be proved by using an argument similar to that in the proof of the part (i). Let us assume that f (k 1 (a tn−t+1 1 )) = 0. Then we obtain k 1 (a tn−t+1 1 ) ∈ f −1 (Q 2 ) for a kn−k+1 1 ∈ G 1 . Therefore f (k 1 (a tn−t+1 1 )) = k 2 (f (1 ) ∈ Kerf . Since Kerf is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G 1 , then there exist (t − 1)n − t + 2 of a , i s whose k 1 -product is an element in δ(Kerf ). From δ(Kerf ) = δ(f −1 (0)) ⊆ δ(f −1 (Q 2 )), it follows that f −1 (Q 2 ) is a weakly (t, n)- absorbing δ-semiprimary hyperideal of G 1 . (iii) Let (0 = k 2 (b tn−t+1 1 ) ∈ f (Q 1 )) k 2 (b tn−t+1 1 ) ∈ f (Q 1 ) for some b tn−t+1 1 ∈ G 2 . Since f be an epimorphism, then there exist a i ∈ G 1 for each 1 ≤ i ≤ tn − t + 1 such that f (a i ) = b i . Hence k 2 (b tn−t+1 1 ) = k 2 (f (a 1 ), · · · , f (a tn−t+1 )) = f (k 1 (a tn−t+1 1 )) ∈ f (Q 1 ). Since Q 1 containing Kerf , we conclude that (0 = k 1 (a tn−t+1 1 ) ∈ Q 1 ) k 1 (a tn−t+1 1 ) ∈ Q 1 . As Q 1 is a (weakly) (t, n)-absorbing δsemiprimary hyperideal of G 1 , then there exist (t − 1)n − t + 2 of a , i s whose k 1product is in δ(Q 1 ). Now, since f is a homomorphism and f (δ(Q 1 )) = δ ′ (f (Q 1 )), the proof is completed. Let P be a hyperideal of (G, h, k). Then the set G/P = {h(g i−1 1 , P, g m i+1 ) \| g i−1 1 , g m i+1 ∈ G} with h and k which are defind by h(h(g 1(i−1) 11 , P, g 1m 1(i+1) ), ..., h(g m(i−1) m1 , P, g mm m(i+1) )) = h(h(g m1 11 ), ..., h(g m(i−1) 1(i−1) ), P, h(g m(i+1) 1(i+1) ), ..., h(g mm 1m )) and k(h(g 1(i−1) 11 , P, g 1m 1(i+1) ), ..., h(g n(i−1) n1 , P, g nm n(i+1) )) = h(k(g n1 11 ), ..., k(g n(i−1) 1(i−1) ), P, k(g n(i+1) 1(i+1) ), ..., k(g nm 1m )) for all g 1m 11 , ..., g mm m1 ∈ G and g 1m 11 , ..., g nm n1 ∈ G, construct a Krasner (m, n)-hyperring [1]. Theorem 2.35. Let P and Q be two proper hyperideals of G with P ⊆ Q. If Q is an (t, n)-absorbing δ-semiprimary hyperideal of G, then Q/P is an (t, n)-absorbing δ q -semiprimary hyperideal of G/P . Proof. By considering the natural homomorphism π : G −→ G/P , defined by π(a) = f (a, P, 0 (m−2) ) and using Theorem 2.34, we are done. Theorem 2.36. Let Q be an (t, n)-absorbing δ-semiprimary hyperideal of G. If G ′ is a subhyperring of G such that G ′ Q, then Q ∩ G ′ is an (t, n)-absorbing δ-semiprimary hyperideal of G ′ . Proof. It follows by Theorem 2.34. Theorem 2.37. Let δ 1 and δ 2 be two hyperideal expansions of Krasner (m, n)hyperrings G 1 and G 2 , respectively, such that δ(Q 1 × Q 2 ) = δ 1 (Q 1 ) × δ 2 (Q 2 ) for hyperideals Q 1 and Q 2 of G 1 and G 2 , respectively. If Q = Q 1 × G 2 is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G 1 × G 2 , then it is an (t, n)-absorbing δ-semiprimary hyperideal of G 1 × G 2 . Proof. Assume that Q 1 × G 2 is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G 1 × G 2 . Since k(Q (tn−t+1) ) = 0, we conclude that Q = Q 1 × G 2 is an (t, n)absorbing δ-semiprimary hyperideal of G 1 × G 2 by Theorem 2.30. We say that δ has (P) property if it satisfies the condition: δ(Q) = G if and only if Q = G for all hyperideals Q of G. Theorem 2.38. Let δ 1 , · · · , δ tn−t+1 be hyperideal expansions of Krasner (m, n)hyperrings G 1 , · · · , G tn−t+1 such that each δ i has (P) property and δ(Q 1 × · · · × Q tn−t+1 ) = δ 1 (Q 1 ) × · · · × δ tn−t+1 (Q tn−t+1 ) for hyperideals Q 1 , · · · , Q tn−t+1 of G 1 , · · · , G tn−t+1 , respectively. If Q = Q 1 ×· · ·×Q tn−t+1 is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G = G 1 × ... × G tn−t+1 , then Q is an (t, n)-absorbing δ-semiprimary hyperideal of G = G 1 × ... × G tn−t+1 . Proof. Let Q is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G. Let us consider the following elements of G: x i = (1 G1 . · · · , 1 Gi−1 , a i , 1 Gi+1 , · · · , 1 Gtn−t+1 ) for all 1 ≤ i ≤ tn − t + 1. Then we have 0 = k(x tn−t+1 1 ) ∈ Q. Since Q = Q 1 × · · · × Q tn−t+1 is a weakly (t, n)-absorbing δ-semiprimary hyperideal of G = G 1 × ... × G tn−t+1 , then there exists (t − 1)n − t + 2 of the x , i s whose k-product is in δ(Q) = δ 1 (Q 1 ) × · · · × δ tn−t+1 (Q tn−t+1 ). This implies that there exists some 1 ≤ j ≤ tn − t + 1 such that 1 Gj ∈ δ j (Q j ) which means δ j (Q j ) = G j . Since δ j has (P) property, then Q j = G j . Hence we conclude that k(Q (tn−t+1) ) = 0 which implies Q is an (t, n)-absorbing δ-semiprimary hyperideal of G by Theorem 2.30. Theorem 2.39. Let δ 1 , · · · , δ tn−t+1 be hyperideal expansions of Krasner (m, n)hyperrings G 1 , · · · , G tn−t+1 such that each δ i has (P) property and δ(Q 1 × · · · × Q tn−t+1 ) = δ 1 (Q 1 ) × · · · × δ tn−t+1 (Q tn−t+1 ) for hyperideals Q 1 , · · · , Q tn−t+1 of G 1 , · · · , G tn−t+1 , respectively. If Q = Q 1 × · · · × Q tn−t+1 is a weakly (t + 1, n)absorbing δ-semiprimary hyperideal of G = G 1 × ... × G tn−t+1 , then either there exists 1 ≤ u ≤ tn − t + 1 such that Q u is an (t + 1, n)-absorbing δ-semiprimary hyperideal of G u and Q i = G i for each 1 ≤ i ≤ tn − t + 1 and i = u or Q u and Q v are (t, n)-absorbing δ u,v -semiprimary hyperideals of G u and G v , respectively, for some u, v ∈ {1, · · · , tn − t + 1} and Q i = G i for all 1 ≤ i ≤ tn − t + 1 but i = u, v. Proof. Let Q = Q 1 × · · · × Q tn−t+1 be a weakly (t + 1, n)-absorbing δ-semiprimary hyperideal of G = G 1 × ... × G tn−t+1 . Therefore we conclude that Q is an (t + 1, n)absorbing δ-semiprimary hyperideal of G by Theorem 2.38. Now, by using Theorem 2.9, we are done. conclusion In this paper, our purpose was to study the structure of (t, n)-absorbing δsemiprimary hyperideals which is more general than δ-primary hyperideals. Additionally, we generalized the notion to weakly (t, n)-absorbing δ-semiprimary hyperideals. We gave many special results illustrating the structures. Indeed, this paper makes a major contribution to classify hyperideals in Krasner (m, n)-hyperrings. Conflicts of Interest The authors declare that they have no conflicts of interest. ) . )Example 2.11. Suppose that Z 12 is the set of all congruence classes of integers modulo 12 and H = {1, 5, 7, 11} is multiplicative subgroup of units Z 12 . Construct G as Z 12 /H. Then we have G = {0,1,2,3,4,6} in which0 = {0},1 = {1, 5, 7, 11}, 2 =10 = {2, 10},3 =9 = {3, 9},4 =8 = {4, 8},6 = {6}. Consider Krasner hyperring (G, ⊕, ⋆) that for allā,b ∈ G,ā ⋆b = ab and 2-ary hyperoperation ⊕ is defined as follows:⊕012346 0012346 110,2,4,61,32,41,31 221,30,412,64 332,410,613 441,32,610,42 6614320 R Ameri, M Norouzi, Prime and primary hyperideals in Krasner (m, n)-hyperrings. R. Ameri, M. 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Davvaz, A Note on Isomorphism Theorems of Krasner (m, n)- hyperrings, Arabian Journal of Mathematics, 5 (2016) 103-115. On the prime, primary and maximal subhypermodules. M M Zahedi, R Ameri, Ital. J. Pure Appl. Math. 5M.M. Zahedi, R. Ameri, On the prime, primary and maximal subhypermodules, Ital. J. Pure Appl. Math., 5 (1999) 61-80. . Imam Khomeini International University. Department of Mathematics, Faculty of SciencesEmail address: [email protected] of Mathematics, Faculty of Sciences, Imam Khomeini International Uni- versity, Qazvin, Iran. Email address: [email protected]	[]
[ "STRATIFICATIONS OF REAL VECTOR SPACES FROM CONSTRUCTIBLE SHEAVES WITH CONICAL MICROSUPPORT", "STRATIFICATIONS OF REAL VECTOR SPACES FROM CONSTRUCTIBLE SHEAVES WITH CONICAL MICROSUPPORT" ]	[ "Ezra Miller " ]	[]	[]	Interpreting the syzygy theorem for tame modules over posets in the setting of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning the existence of stratifications of real vector spaces that play well with sheaves having microsupport in a given cone or, equivalently, sheaves in the corresponding conic topology.	10.1007/s41468-023-00112-1	[ "https://export.arxiv.org/pdf/2008.00091v3.pdf" ]	220,936,519	2008.00091	bb00262a03c7c6fb6ff15925aeee4d06ce7651df	STRATIFICATIONS OF REAL VECTOR SPACES FROM CONSTRUCTIBLE SHEAVES WITH CONICAL MICROSUPPORT 10 Mar 2023 Ezra Miller STRATIFICATIONS OF REAL VECTOR SPACES FROM CONSTRUCTIBLE SHEAVES WITH CONICAL MICROSUPPORT 10 Mar 2023 Interpreting the syzygy theorem for tame modules over posets in the setting of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning the existence of stratifications of real vector spaces that play well with sheaves having microsupport in a given cone or, equivalently, sheaves in the corresponding conic topology. Introduction Overview and motivation. Persistent homology with multiple real parameters can be phrased in more or less equivalent ways using multigraded modules (e.g. [CZ09,Knu08,Mil20a]), or sheaves (e.g. [Cur14,Cur19]), or functors (e.g. [SCL + 16]), or derived categories (e.g. [KS18]). All of these descriptions have in common an underlying partially ordered set indexing a family of vector spaces, and this is interpreted under increasing layers of abstraction. The simplest objects at any level of abstraction are the indicator objects, which place a single copy of the ground field k at every point of an interval in the underlying poset Q, meaning an intersection of an upset of Q with a downset of Q. (The terminology is most clear when Q = R, where "interval" has its usual meaning.) Among the indicator objects are those supported on the upsets and downsets themselves; over Q = R these objects are free and injective, respectively. Furthermore, when Q is totally ordered, every object is a direct sum of indicator objects [Cra13]. The theory for more general posets, including partially ordered real vector spaces, has in large part revolved around relating general objects as closely as possible to indicator objects, particularly where algorithmic computation is concerned. Indeed, the foundations for the ideas in this paper, both from Kashiwara-Schapira [KS18,KS19] and the author [Mil20a,Mil20b,Mil20c] (see also [Mil17]), lies in algorithmic computation with persistent homology. To that end, effective methods demand concrete representatives of derived sheaves and stratifications of their support. Kashiwara and Schapira, in [KS19, Conjecture 3.20] (which they had previously stated as [KS18,Conjecture 4.19]) and [KS17,Conjecture 3.17], assert that derived sheaves in principle possess such concrete representatives. Corollaries 5.1 and 5.2 achieve more than mere existence: the engine behind their proofs, Theorem 4.5, produces concrete structures via the syzygy theorem for poset modules (Theorem 2.11), which is specifically designed to extract resolutions algorithmically [Mil20a]. Situations abound where concrete resolutions could be essential for algorithmic persistent homology. For example, bifiltration of a semialgebraic space by two semialgebraic functions yields bipersistent homology that is an R 2 -module which should be tame and hence have finite indicator resolutions by upsets or downsets. (This statement requires proof and might be subtle or even false in the subanalytic instead of semialgebraic context.) This scenario is fundamental to motivating applications such as summarizing shape in biology [Mil15] or probability distributions in statistics [RS20]. Tameness in this biparameter setting connects to recent Morse-theoretic stratification perspectives by Budney and Kaczynski [BK21] as well as by Assif and Baryshnikov [AB21]. Conjectures and proofs. The conjectures of Kashiwara and Schapira are phrased in the most abstract derived setting. They posit, roughly speaking, that every object can be directly related to indicator objects, either by stratification of its support ormore strongly-by resolution. More precisely, the first conjecture concerns the relation between, on one hand, constructibility of sheaves on real vector spaces in the derived category with microsupport restricted to a cone, and on the other hand, stratification of the vector space in a manner compatible with the cone [KS17, Conjecture 3.17] 1 (Corollary 5.2). The second concerns piecewise linear (PL) objects in this context, particularly existence of polyhedrally structured resolutions that, in principle, lend themselves to explicit or algorithmic computation [KS18,Conjecture 4.19] = [KS19, Conjecture 3.20] (Corollary 5.1). This note uses the most elementary poset module setting [Mil20a] to prove these conjectures. Both follow immediately from Theorem 4.5 here, which translates the relevant real-vector-space special cases of the syzygy theorem for complexes of poset modules [Mil20a,Theorem 6.17] (reviewed in Section 2) into the language of derived categories of constructible sheaves with conic microsupport or under a conic topology (reviewed in Section 3). The syzygy theorem [Mil20a, Theorems 6.12 and 6.17] leverages relatively weak topological framework into powerful homological structure: over any poset Q it enhances a constant subdivision-a partition of Q into finitely many regions over which the given module or complex is constant-to a more controlled subdivision (a finite encoding [Mil20a,§4]), and even to a finite resolution by upset modules and a finite resolution by downset modules, whose pieces play well with the ambient combinatorics. These resolutions are analogues over arbitrary posets of free and injective resolutions for modules over the poset Z n [GW78] (see [HM05] or [MS05, Chapter 11] for background, or [Mil20a,§5] for a treatment in the present context) or over the poset R. Crucially, any available supplementary geometry-be it subanalytic, semialgebraic, or piecewise-linear, for instance-is preserved. In the context of a partially ordered real vector space Q with positive cone Q + , the enhancement afforded by the syzygy theorem produces a Q + -stratification from an arbitrary subanalytic triangulation. If the triangulation is subordinate to a given constructible derived Q + -sheaf, meaning an object in the bounded derived category of constructible sheaves with microsupport contained in the negative polar cone of Q + , then this enhancement produces Q + -structured resolutions of the given sheaf. This makes the two conjectures into special cases of the syzygy theorem. While sheaves with conical microsupport (see Section 3.4) are equivalent to the more elementary sheaves in the conic topology (see Section 3.3), as has been known from the outset [KS90] (see Theorem 3.15), the notion of constructibility has until now been available only on the microsupport side. The results here assert that constructibility can be detected entirely with the more rigid conic topology, via tameness, without appealing to subanalytic triangulations in the more flexible analytic topology, a point emphasized by Theorem 4.5 ′ . More broadly, for applications in persistent homology 1 Bibliographic note: this conjecture appears in v3 (the version cited here) and earlier versions of the cited arXiv preprint. It does not appear in the published version [KS18], which is v6 on the arXiv. The published version is cited where it is possible to do so, and v3 [KS17] is cited otherwise. the input is usually a sheaf in the conic topology induced by a given cone Q + instead of a sheaf in the ordinary topology with microsupport in the negative polar cone Q ∨ + , so the main results, namely Theorem 4.5, Corollary 5.1, and Corollary 5.2, are restated using conic sheaves in Theorem 4.5 ′ , Corollary 5.1 ′ , and Corollary 5.2 ′ . Poset modules vs. constructible sheaves. The theory in [Mil20a,Mil20b,Mil20c] was developed simultaneously and independently from [KS18, KS19] (cf. [Mil17]). Having made the connection between these approaches, it is worth comparing them in detail. The syzygy theorem [Mil20a, Theorems 6.12 and 6.17] and its combinatorial underpinnings involving poset encoding [Mil20a, §4] hold over arbitrary posets; see Section 2 here for indications toward this generality. When the poset is a real vector space, the constructibility encapsulated by topological tameness (Definition 2.3) has no subanalytic, algebraic, or piecewise-linear hypothesis, although these additional structures are preserved by the syzygy theorem transitions. For example, the upper boundary of a downset in the plane with the usual componentwise partial order could be the graph of any continuous weakly decreasing function, among other things, and could be present (i.e., the downset is closed) or absent (i.e., the downset is open), or somewhere in between (e.g., a Cantor set could be missing). The conic topology in [KS18] or [KS19] specializes at the outset to the case of a partially ordered real vector space, and it allows only subanalytic or polyhedral regions, respectively, with upsets having closed lower boundaries and downsets having open upper boundaries. The constructibility in [KS18,KS19] is otherwise essentially the same as tameness here (Theorem 4.5 ′ ), except that tameness requires constant subdivisions to be finite, whereas constructibility in the derived category allows constant subdivisions to be locally finite. That said, this agreement of constructibility with locally finite tameness that is subanalytic or PL, more or less up to boundary considerations, is visible in [KS17] or [KS19] only via conjectures, namely the ones proved here in Section 5 using the general poset methods. The theory of primary decomposition in [Mil20b] requires the poset to be a partially ordered group whose positive cone has finitely many faces. These can be integer or real or something in between, but the finiteness is essential for primary decomposition in any of these settings; see [Mil20b,Example 5.9]. Local finiteness allowed by constructibility in [KS18] does not provide a remedy, although it is possible that the PL hypothesis in [KS19] does. In either the integer or real case, detailed understanding of the topology results in a stronger theory of primary decomposition than over an aribtrary polyhedral group, with much more complete supporting commutative algebra [Mil20c]. Most of the remaining differences between the developments in [Mil20a,Mil20b,Mil20c] and those in [KS18,KS19], beyond the types of allowed functions and the shapes of allowed regions, is the behavior allowed on boundaries of regions. That difference is accounted for by the transition between the conic topology and the Alexandrov topology, the distinction being that the Alexandrov topology has for its open sets all upsets, whereas the conic topology has only the upsets that are open in the usual topology. This distinction is explored in detail by Berkouk and Petit [BP19]. It is intriguing that ephemeral modules are undetectable metrically [BP19, Theorem 4.22] but their presence here brings indispensable insight into homological behavior in the conic topology. Acknowledgements. Pierre Schapira gave helpful comments on a draft of this paper, as did a referee. Portions of this work were funded by NSF grant DMS-1702395. Syzygy theorem for poset modules This section recalls concepts surrounding modules over posets, concluding with a statement (Theorem 2.11) of the relevant special case of the syzygy theorem for complexes of poset modules [Mil20a, Theorem 6.17]. For reference, the definitions here correspond to [Mil20a, Definitions 2.1, 2.6, 2.11, 2.14, 2.15, 4.27, 3.1, 3.14, 6.1, and 6.16], sometimes special cases thereof. 2.1. Tame poset modules. Definition 2.1. Let Q be a partially ordered set (poset) and its partial order. A module over Q (or a Q-module) is • a Q-graded vector space M = q∈Q M q with • a homomorphism M q → M q ′ whenever q q ′ in Q such that • M q → M q ′′ equals the composite M q → M q ′ → M q ′′ whenever q q ′ q ′′ . A homomorphism M → N of Q-modules is a degree-preserving linear map, or equivalently a collection of vector space homomorphisms M q → N q , that commute with the structure homomorphisms M q → M q ′ and N q → N q ′ . Definition 2.2. Fix a Q-module M. A constant subdivision of Q subordinate to M is a partition of Q into constant regions such that for each constant region I there is a single vector space M I with an isomorphism M I → M i for all i ∈ I that has no monodromy: if J is some (perhaps different) constant region, then all comparable pairs i j with i ∈ I and j ∈ J induce the same composite homomorphism 3. The Q-module M is tame if it is Q-finite and Q admits a finite constant subdivision subordinate to M. M I → M i → M j → M J . Real partially ordered groups. Definition 2.4. An abelian group Q is partially ordered if it is generated by a submonoid Q + , called the positive cone, that has trivial unit group. The partial order is: q q ′ ⇔ q ′ − q ∈ Q + . A[Q]. A homo- morphism ϕ : k[S] → k[S ′ ] is connected if there is a scalar λ ∈ k such that ϕ acts as multiplication by λ on the copy of k in degree q for all q ∈ S ∩ S ′ . Definition 2.9. Fix any poset Q and a Q-module M. 1. An upset resolution of M is a complex F• of Q-modules, each a direct sum of upset submodules of k[Q], whose differential F i → F i−1 decreases homological degrees, has components k[U] → k[U ′ ] that are connected, and has only one nonzero homology H 0 (F•) ∼ = M. A downset resolution of M is a complex E • of Q-modules, each a direct sum of downset quotient modules of k[Q], whose differential E i → E i+1 increases cohomological degrees, has components k[D ′ ] → k[D] that Stratifications, topologies, and cones This section collects the relevant definitions and theorems regarding constructible sheaves from the literature. The sizeable edifice on which the subject is built makes it unavoidable that readers seeing some of these topics for the first time will need to consult the cited sources for additional background. The goal here is to bring readers as quickly as possible to a general statement (Theorem 4.5) while circumscribing the ingredients necessary for its proof in such a way that those familiar with the conjectures of Kashiwara and Schapira, specifically [KS17, Conjecture 3.17] and [KS19, Conjecture 3.20], can skip seamlessly to Section 4 after skimming Section 3 for terminology. To avoid endlessly repeating hypotheses, and so readers can quickly identify when the same hypotheses are in effect, the blanket assumption henceforth is for Q to satisfy the following, where the positive cone Q + is full if it has nonempty interior. 3. The term sheaf on a topological space here means a sheaf of k-vector spaces. Sometimes in the literature this word is used to mean an object in the bounded derived category of sheaves of k-vector spaces; for clarity here, the term derived sheaf is always used when an object in the derived category is intended. Subanalytic triangulation. Definition 3.3. Fix a real analytic manifold X. 1. A subanalytic triangulation of a subanalytic set Y ⊆ X is a homeomorphism \|∆\| − → ∼ Y such that the image in Y of the realization \|σ\| of the relative interior of each simplex σ ∈ ∆ is a subanalytic submanifold of X. A subanalytic triangulation of Y is subordinate to a (derived) sheaf F on X if Y contains the support of F and (every homology sheaf of) F restricts to a constant sheaf on the image in Y of every cell \|σ\|. Subanalytic constructibility. Definition 3.4. A (derived) sheaf on a real analytic manifold is subanalytically weakly constructible if there is a subanalytic triangulation subordinate to it. The word "weakly" is omitted if, in addition, the stalks have finite dimension as k-vector spaces. The reason to use subanalytic triangulation instead of arbitrary subanalytic stratification is the following, which is a step on the way to a constant subdivision. The reason for specifically including the piecewise linear (PL) condition in Section 2 is for its application here, as one of the conjectures is in that setting. For this purpose, the sheaf version of this particularly strong type of constructibility is needed. which only requires Q to be a (nondisjoint) union of finitely polyhedra on which F is constant. However, the notion of PL (derived) sheaf thus defined is the same, since any finite union of polyhedra can be refined to a finite union that is disjoint-that is, a partition. This refinement can be done, for example, by expressing Q as the union of (relatively open) faces in the arrangement of all hyperplanes bounding halfspaces defining the given polyhedra, of which there are only finitely many. Conic and Alexandrov topologies. Definition 3.9. Fix a real partially ordered group Q with closed positive cone Q + . 1. The conic topology on Q induced by Q + (or induced by the partial order) consists of the upsets that are open in the ordinary topology on Q. 2. The Alexandrov topology on Q induced by Q + (or induced by the partial order) consists of all the upsets in Q. To avoid confusion when it might occur, write 1. Q con for the set Q with the conic topology induced by Q + , 2. Q ale for the set Q with the Alexandrov topology induced by Q + , and 3. Q ord for the set Q with its ordinary topology. 1. A conic stratification of a closed subset S ⊆ Q is a locally finite family of pairwise disjoint subanalytic subsets, called strata, which are locally closed in the conic topology and have closures whose union is S. 2. The stratification is subordinate to a (derived) sheaf F on Q if S equals the support of F and the restriction of (each homology sheaf of) F to every stratum is locally constant of finite rank. Remark 3.13. A conic stratification is called a γ-stratification in [KS17, Definition 3.15], with γ = Q + . The only differences between conic stratification and subanalytic partition of a subset S in Definition 2.5.1 are that • conic stratifications are only required to be locally finite, not necessarily finite; • conic strata are required to be locally closed in the conic topology (that is, an intersection of an open upset in Q ord with a closed downset in Q ord ); and • the union need not actually equal all of S, because only the union of the stratum closures is supposed to equal S. Proposition 3.14. Fix a real partially ordered group Q with closed positive cone Q + . 1. The identity on Q yields continuous maps of topological spaces ι : Q ord → Q con and  : Q ale → Q con . 2. Any sheaf F on Q ord pulled back from Q con has natural maps F q → F q ′ for q q ′ in Q on stalks that functorially define a Q-module q∈Q F q . 3. Similarly, any sheaf G on Q ale has natural maps G q → G q ′ for q q ′ in Q on stalks that functorially define a Q-module q∈Q G q . This functor from sheaves on Q ale to Q-modules is an equivalence of categories. 4. If sheaves F on Q ord and G on Q ale are both pulled back from the same sheaf E on Q con , then the Q-modules in items 2 and 3 are the same. 5. The pushforward functor  * is exact, and  *  −1 E ∼ = E . For item 2, if F = ι −1 E is pulled back to Q ord from a sheaf E on Q con , then F has the same stalks as E (as a sheaf pullback in any context does), so the natural morphisms are induced by the restriction maps of E from open neighborhoods of q to those of q ′ . The result in 3 holds for arbitrary posets; for an exposition in a context relevant to persistence, see [Cur14, Theorem 4.2.10 and Remark 4.2.11] and [Cur19]. For item 4, the stalks F q = E q = G q are the same. For item 5, exactness is proved in passing in the proof of [BP19, Lemma 3.5], but it is also elementary to check that a surjection G ։ G ′ of sheaves on Q ale yields a surjection of stalks for the pushforwards to Q con because direct limits (filtered colimits) are exact. That  *  −1 E ∼ = E is because the natural morphism is the identity on stalks. 3.4. Conic microsupport. The microsupport of a (derived) sheaf on an analytic manifold X is a certain closed conic isotropic subset of the cotangent bundle T * X. The notion of microsupport is a central player in [KS90], to which the reader is referred for background on the topic. However, although the main result in this section (Theorem 4.5) is stated in terms of microsupport, the next theorem allows the reader to ignore it henceforth, as pointed out by Kashiwara and Schapira themselves [KS18, Remark 1.9], by immediately translating to the more elementary context of sheaves in the conic topology in Section 3.3. Remark 3.17. What Theorem 3.15 does in practice is allow a given (derived) sheaf with microsupport contained in the negative polar cone Q ∨ + to be replaced with an isomorphic object that is pulled back from the conic topology induced by the partial order. The reason for mentioning the notion of microsupport at all is to emphasize that constructibility in the sense of Definition 3.4 requires the ordinary topology. This may seem a fine distinction, but the conjectures of Kashiwara and Schapira proved in Section 5 entirely concern the transition from the ordinary to the conic topology, so it is crucial to be clear on this point. In view of Remark 3.17, discussion of constructibility for sheaves on conic topologies requires the following. The ad hoc nature of this definition is justified by Theorem 4.5 ′ . Definition 3.18. Fix Q satisfying Hypothesis 3.1. A constructible conic sheaf on Q is a sheaf in the conic topology Q con whose pullback via ι −1 is subanalytically constructible. Proof. The stalk at q of any sheaf on Q con is the direct limit over points p ∈ q−Q • + of the sections over p + Q • + . In the case of the pushforward of the sheaf on Q ale corresponding to an upset module, these sections are k if p lies interior to the upset and 0 otherwise. The result holds because the upsets U and U • have the same interior, namely U • . Here is the main result. It is little more than a restatement of the relevant part of Theorem 2.11 in the language of sheaves. Resolutions of constructible sheaves Theorem 4.5. Fix Q satisfying Hypothesis 3.1. If F • is a complex of compactly supported subanalytically constructible sheaves on Q ord with microsupport in the negative polar cone Q ∨ + then F • has a finite subanalytic upset resolution and a finite subanalytic downset resolution. If Q is polyhedral and F • is PL, then F • has PL such resolutions. Proof. Using Theorem 3.15, assume that F • is pulled back to Q ord from Q con , say F • = ι −1 E • . Since F • has compact support, any subordinate subanalytic triangulation (Definition 3.3) afforded by Definition 3.4 is necessarily finite because it is locally finite. The complex F • = q∈Q F • q of Q-modules that comes from Proposition 3.14.2 is tamed by the triangulation, which is a constant subdivision (Definition 2.2) because • simplices are connected, so locally constant sheaves on them are constant, and • Γ(\|σ p \|, F i ) → F i p → F i q ← Γ(\|σ q \|, F i ) is locally constant-and hence constant, as simplices are connected-when p q in Q. Here σ x is the simplex containing x, the middle arrow is from Proposition 3.14.2, and the outer arrows are the natural isomorphisms from Lemma 3.6. Hence the complex F • of Q-modules has resolutions of the desired sort by Theorem 2.11. Viewing any of these resolutions as a complex of sheaves on Q ale via Proposition 3.14.3, push it forward from the Alexandrov topology to the conic topology via the exact functor  * in Proposition 3.14.5. The resulting complex of sheaves on Q con is a resolution of a complex isomorphic to E • by Proposition 3.14.4 and 3.14.5. The upsets or downsets in the summands of the resolution may as well be assumed open or closed, respectively, by Propositions 4.2 or 4.3. The proof is concluded by pulling back the resolution from Q con to Q ord via the equivalence of Theorem 3.15. Proof. That 1 ⇒ 2 and 1 ⇒ 3 follows from Theorems 4.5 and 3.15. The opposite directions are by the definition and foundational results surrounding constructibility in Definition 3.4 and Remark 3.5. Remark 4.6. While the notion of a sheaf with microsupport contained in the negative polar cone of Q + is equivalent to the notion of a sheaf in the conic topology, the notion of constructibility has until now only been available on the microsupport side, where simplices from arbitrary subanalytic triangulations achieve constancy of the sheaves in question. Theorem 4.5 ′ makes precise the assertion that constructibility can be detected entirely with the more rigid conic topology, without the flexibility of appealing to arbitrary subanalytic triangulations. Remark 4.7. Theorem 4.5 assumes compact support to get finite instead of locally finite subdivisions. The application in Section 5 to constructible sheaves without any assumption of compact support yields a locally finite subdivision by reducing to the case of compact support. Remark 4.8. The final sentences of Theorems 4.5 and 4.5 ′ are true with "polyhedral" and "PL" all replaced by "semialgebraic", with the same proofs, as long as the definitions of these semialgebraic concepts in the constructible sheaf setting are made appropriately. The semialgebraic constructible sheaf versions are not treated here because they are not relevant to the conjectures proved in Section 5. Stratifications from constructible sheaves Corollary 5.1 ([KS19, Conjecture 3.20]). Fix Q satisfying Hypothesis 3.1 with Q + polyhedral. If F • is a PL object in the derived category of compactly supported constructible sheaves on Q ord with microsupport contained in the negative polar cone Q ∨ + then the isomorphism class of F • is represented by a complex that is a finite direct sum of constant sheaves on bounded polyhedra that are locally closed in the conic topology. Proof. The statement would directly be a special case of Theorem 4.5 were it not for the boundedness hypothesis on the polyhedra, since either a PL upset or PL downset resolution would satisfy the conclusion. That said, boundedness is easy to impose: since F • has compact support, and the resolution has vanishing homology outside of the support of F • , each upset or downset sheaf can be restricted to the support of F • and extended by 0. As in Theorem 4.5 ′ , Corollary 5.1 can be restated using constructible conic sheaves. Corollary 5.1 ′ . Fix Q satisfying Hypothesis 3.1 with Q + polyhedral. F • is a PL object in the derived category of compactly supported constructible conic sheaves if and only if the isomorphism class of F • is represented by a complex that is a finite direct sum of constant sheaves on bounded polyhedra that are locally closed in the conic topology. Corollary 5.2 ([KS17, Conjecture 3.17]). Fix Q satisfying Hypothesis 3.1. If a compactly supported derived sheaf with microsupport in the negative polar cone Q ∨ + is subanalytically constructible, then its support has a subordinate conic stratification. Proof. Part (ii) in the proof of [KS18, Theorem 3.17] reduces to the case where the support of the given derived sheaf is compact. The argument is presented in the case where Q is polyhedral and the derived sheaf is PL, but the argument works verbatim for Q satisfying Hypothesis 3.1, without any polyhedral or PL assumptions, because the requisite lemma, namely [KS18, Lemma 3.5]-and indeed, all of [KS18, §3.1]-is stated and proved in this non-polyhedral generality. So henceforth assume the given derived sheaf has compact support. Remark 3.5 allows the assumption that the given derived sheaf is represented by a complex F • of constructible sheaves. Theorem 4.5 produces a subanalytic indicator resolution, which for concreteness may as well be an upset resolution. Each upset that appears as a summand in the resolution partitions Q into the upset itself, which is open subanalytic, and its complement, which is a closed subanalytic downset. The common refinement of the partitions induced by the finitely many open subanalytic upsets in the resolution and their closed subanalytic downset complements is a partition of Q into finitely many strata such that • each stratum is subanalytic and locally closed in the conic topology, and • the restriction of F • to each stratum has constant homology. The strata with nonvanishing homology form the desired conic stratification. Definition 2 . 3 . 23Fix a poset Q and a Q-module M. 1. A constant subdivision of Q is finite if it has finitely many constant regions. 2. The Q-module M is Q-finite if its components M q have finite dimension over k. 2. 3 . 3Complexes and resolutions of poset modules. Definition 2.6. A homomorphism ϕ : M → N of Q-modules is tame if Q admits a finite constant subdivision subordinate to both M and N such that for each constant region I the composite homomorphism M I → M i → N i → N I does not depend on i ∈ I. The map ϕ is subanalytic or PL if this constant subdivision is. Definition 2.7. The vector space k[Q] = q∈Q k that assigns k to every point of the poset Q is a Q-module with identity maps on k. More generally, 1. an upset (also called a dual order ideal ) U ⊆ Q, meaning a subset closed under going upward in Q (so U + Q + = U, when Q is a partially ordered group) determines an indicator submodule or upset module k[U] ⊆ k[Q]; and 2. dually, a downset (also called an order ideal ) D ⊆ Q, meaning a subset closed under going downward in Q (so D−Q + = D, when Q is a partially ordered group) determines an indicator quotient module or downset module k[Q] ։ k[D]. When Q is a subposet of a partially ordered real vector space, an indicator module of either sort is subanalytic or PL if the corresponding upset or downset is of the same type. Definition 2.8. Let each of S and S ′ be a nonempty intersection of an upset in a poset Q with a downset in Q, so k[S] and k[S ′ ] are subquotients of k are connected, and has only one nonzero cohomology H 0 (E • ) ∼ = M. An upset or downset resolution is called an indicator resolution if the up-or down-nature is unspecified. The length of an indicator resolution is the largest (co)homological degree in which the complex is nonzero. An indicator resolution 3. is finite if the number of indicator module summands is finite, 4. dominates a constant subdivision of M if the subdivision or encoding is subordinate to each indicator summand, and 5. is subanalytic or PL if Q is a subposet of a real partially ordered group and the resolution dominates a constant subdivision of the corresponding type. Definition 2.10. Fix a complex M • of modules over a poset Q. 1. M • is tame if its modules and morphisms are tame (Definitions 2.3 and 2.6). 2. A constant subdivision is subordinate to M • if it is subordinate to all of the modules and morphisms therein, and then M • is said to dominate the subdivision. 3. An upset resolution of M • is a complex of Q-modules in which each F i is a direct sum of upset modules and the components k[U] → k[U ′ ] are connected, with a homomorphism F • → M • of complexes inducing an isomorphism on homology. 4. A downset resolution of M • is a complex of Q-modules in which each E i is a direct sum of downset modules and the components k[D] → k[D ′ ] are connected, with a homomorphism M • → E • of complexes inducing an isomorphism on homology. These resolutions are finite, or dominate a constant subdivision, or are subanalytic or PL as in Definition 2.9. 2.4. Syzygy theorem for complexes of poset modules. Only certain aspects of the full syzygy theorem [Mil20a, Theorem 6.17] are required, so those are isolated here. Theorem 2.11. A bounded complex M • of modules over a poset Q is tame if and only if it admits one, and hence all, of the following: 1. a finite constant subdivision of Q subordinate to M • ; or 2. a finite upset resolution; or 3. a finite downset resolution; or 4. a finite constant subdivision subordinate to any given one of items 2-3. The statement remains true over any subposet of a real partially ordered group if "tame" and all occurrences of "finite" are replaced by "PL". Moreover, any tame or PL morphism M • → N • lifts to a similarly well behaved morphism of resolutions as in parts 2 and 3. All of these results hold in the subanalytic case if M • has compact support. Hypothesis 3.1. Q is a real partially ordered group with closed, full, subanalytic Q + .Remark 3.2. Some basic notions are used freely without further comment. 1. The notion of simplicial complex here is the one in [KS90, Definition 8.1.1]: a collection ∆ of subsets (called simplices) of a fixed vertex set that is closed under taking subsets (called faces), contains every vertex, and is locally finite in the sense that every vertex of ∆ lies in finitely many simplices of ∆. Any simplicial complex ∆ has a realization \|∆\| as a topological space, with each relatively open simplex \|σ\| being an open convex set in an appropriate affine space. 2. The notion of subanalytic set in an analytic manifold is as in [KS90, §8.2]. Lemma 3.6 ([KS90, Proposition 8.1.4]). For a simplex σ in a subanalytic triangulation subordinate to a constructible sheaf F , there is a natural isomorphism Γ(\|σ\|, F ) − → ∼ F x from the sections over \|σ\| to the stalk at every point x ∈ σ. Remark 3 . 10 . 310The conic topology in Definition 3.9 is also known as the γ-topology, where γ = Q + [KS90,KS18,KS19]. The Alexandrov topology makes just as much sense on any poset.The type of stratification Kashiwara and Schapira specify [KS17, Conjecture 3.17] is not quite the same as subanalytic subdivision in Definition 2.5.1. To be precise, first recall two standard topological concepts. Definition 3 . 311. A subset of a topological space Q is locally closed if it is the intersection of an open subset and a closed subset. A family of subsets of Q is locally finite if each compact subset of Q meets only finitely many members of the family. Definition 3 . 312 ([KS17, Definition 3.15]). Fix Q satisfying Hypothesis 3.1. Proof. The maps in item 1 are continuous by definition: the inverse image of any open set is open because the ordinary topology refines each of the target topologies. Fix Q satisfying Hypothesis 3.1. The pushforward ι * of the map ι from Proposition 3.14.1 induces an equivalence from the category of sheaves with microsupport contained in the negative polar cone Q ∨ + to the category of sheaves in the conic topology. The pullback ι −1 is a quasi-inverse. The same assertions hold for the bounded derived categories.Remark 3.16. The pushforward ι * and the pullback ι −1 have concrete geometric descriptions. Since ι is the identity on Q, the pushforward of a sheaf F on Q has sections Γ(U, ι * F ) = Γ(U, F ) for any open upset U, where "open upset" means the same things as "upset that is open in the usual topology" and "subset that is open in the conic topology". On the other hand, over any convex ordinary-open set O, the pullback to the ordinary topology of a sheaf E in the conic topology has sections Γ(O, ι −1 E ) = Γ(O + Q + , E ), namely the sections of E over the upset generated by O [KS90, (3.5.1)]. Proposition 4 . 3 . 43Fix a downset D in a real partially ordered group Q with closed positive cone. If D is the closure of D in Q ord , then the sheaves on Q ale corresponding to k[D] and k[D] push forward to the same sheaf on Q con . Proof. Calculating stalks as in the previous proof, in the case of the pushforward of the sheaf on Q ale corresponding to a downset module, the sections over p + Q • + are k if p lies interior to the downset and 0 otherwise. The result holds because the downsets D and D have the same interior. Remark 4 . 4 . 44The fundamental difference between Alexandrov and conic topologies reflected by Propositions 4.2 and 4.3 is explored in detail by Berkouk and Petit[BP19]. PL) if the subsets are finite unions of convex polyhedra. A module over a subanalytic or polyhedral real partially ordered group Q is subanalytic or PL, respectively, if the module is tamed by a subordinate finite constant subdivision of the corresponding type.partially ordered group is 1. real if the underlying abelian group is a real vector space of finite dimension; 2. subanalytic if, in addition, Q + is subanalytic, (see [KS90, §8.2] for the definition); 3. polyhedral if, in addition, Q + is a convex polyhedron: an intersection of finitely many half-spaces, each either closed or open. Definition 2.5. A partition of a real partially ordered group Q into subsets is 1. subanalytic if the subsets are subanalytic sets, and 2. piecewise linear ( Definition 3.7. Fix Q satisfying Hypothesis 3.1. 1. A subanalytic subdivision (Definition 2.5.1) of Q is subordinate to a (derived) sheaf F on Q if the restriction of F to every stratum (meaning subset in the subdivision) is constant of finite rank. 2. If the subanalytic subdivision is PL (Definition 2.5.2) and Q is polyhedral (Definition 2.4.3), then F is said to be piecewise linear, abbreviated PL.Remark 3.8. Definition 3.7.2 is not verbatim the same as [KS19, Definition 2.3], Definition 4.1. Fix Q satisfying Hypothesis 3.1. 1. A subanalytic upset sheaf on Q is the extension by zero of the rank 1 constant sheaf on an open subanalytic upset in Q ord . 2. A subanalytic downset sheaf on Q is the pushforward of the rank 1 locally constant sheaf on a closed subanalytic downset in Q ord . 3. A subanalytic upset resolution of a complex F • of sheaves on Q ord is a homomorphism U • → F • of complexes inducing an isomorphism on homology, with each U i being a direct sum of subanalytic upset sheaves. 4. A subanalytic downset resolution of a complex F • of sheaves on Q ord is a homomorphism F • → D • of complexes inducing an isomorphism on homology, with each each D i being a direct sum of subanalytic downset sheaves.Either type of resolution is • finite if the total number of summands across all homological degrees is finite; • PL if Q is polyhedral and the upsets or downsets are PL. Proposition 4.2. Fix an upset U in a real partially ordered group Q with closed positive cone. If U • is the interior of U in Q ord , then the sheaves on Q ale corresponding to k[U] and k[U • ] push forward to the same sheaf on Q con . Theorem 4.5 ′ . Fix Q satisfying Hypothesis 3.1. If F • is a complex of compactly supported sheaves in the conic topology Q con then the following are equivalent.1. F • is constructible (Definition 3.18). 2. F • has a finite subanalytic upset resolution. 3. F • has a finite subanalytic downset resolution. The implications 2 ⇒1 and 3 ⇒1 do not require compact support for F • . If Q is polyhedral and F • is PL, then all of these claims hold with "PL" in place of "subanalytic". As before, Corollary 5.2 can be restated in terms of constructible conic sheaves.Corollary 5.2 ′ . Fix Q satisfying Hypothesis 3.1. The support of any compactly supported constructible derived conic sheaf has a subordinate conic stratification.Remark 5.3. The reference in[KS17,Conjecture 3.17] to a cone λ contained in the interior of the positive cone union the origin appears to be unnecessary, since (in the notation there) any γ-stratification is automatically a λ-stratification by[KS17,Definition 3.15] and the fact that λ ⊆ γ.Conflict of interest. The author states that there is no conflict of interest. Biparametric persistence for smooth filtrations, preprint. 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[ "Site-testing at Muztagh-ata site II: Seeing statistics", "Site-testing at Muztagh-ata site II: Seeing statistics" ]	[ "Jing Xu [email protected] \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n\nUniversity of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China\n", "Ali Esamdin \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n\nUniversity of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China\n", "Jin-Xin Hao \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Jin-Min Bai \nYunnan Observatories\nChinese Academy of Sciences\n650000KunmingPeople's Republic of China\n", "Ji Yang \nPurple Mountain Observatories\nChinese Academy of Sciences\n210008NanjingPeople's Republic of China\n", "Xu Zhou \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Yong-Qiang Yao \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Jin-Liang Hou \nShanghai Astronomical Observatories\nChinese Academy of Sciences\n200030ShanghaiPeople's Republic of China\n", "Guang-Xin Pu \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n", "Guo-Jie Feng \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n\nUniversity of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China\n", "Chun-Hai Bai \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n", "Peng Wei \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n", "Shu-Guo Ma \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n", "Abudusaimaitijiang Yisikandee \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n", "Le-Tian Wang \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n", "Xuan Zhang \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n", "Ming Liang \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n", "Lu Ma ", "Jin-Zhong Liu \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n", "Zi-Huang Cao \nXinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China\n\nUniversity of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China\n\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Yong-Heng Zhao \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Lu Feng \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Jian-Rong Shi \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Hua-Lin Chen \nNational Astronomical Observatories Nanjing Institute of Astronomical Optics & Technology\nChinese Academy of Sciences\n210008NanjingPeople's Republic of China\n", "Chong Pei \nNational Astronomical Observatories Nanjing Institute of Astronomical Optics & Technology\nChinese Academy of Sciences\n210008NanjingPeople's Republic of China\n", "Xiao-Jun Jiang \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Jian-Feng Wang \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Jian-Feng Tian \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Yan-Jie Xue \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Jing-Yao Hu \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n", "Yun-Ying Jiang \nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China\n" ]	[ "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "University of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "University of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "Yunnan Observatories\nChinese Academy of Sciences\n650000KunmingPeople's Republic of China", "Purple Mountain Observatories\nChinese Academy of Sciences\n210008NanjingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "Shanghai Astronomical Observatories\nChinese Academy of Sciences\n200030ShanghaiPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "University of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "Xinjinag Astronomical Observatory\nChinese Acsdemy of Sciences\n830011UrumqiPeople's Republic of China", "University of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "National Astronomical Observatories Nanjing Institute of Astronomical Optics & Technology\nChinese Academy of Sciences\n210008NanjingPeople's Republic of China", "National Astronomical Observatories Nanjing Institute of Astronomical Optics & Technology\nChinese Academy of Sciences\n210008NanjingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingPeople's Republic of China" ]	[]	In this article, we present a detailed analysis of the statistical properties of seeing for the Muztagh-ata site which is the candidate site for hosting future Chinese Large Optical/infrared Telescope (LOT) project. The measurement was obtained with Differential Image Motion Monitor (DIMM) from April 2017 to November 2018 at different heights during different periods. The median seeing at 11 meters and 6 meters are very close but different significantly from that on the ground. We mainly analyzed the seeing at 11 meters monthly and hourly, having found that the best season for observing was from late autumn to early winter and seeing tended to improve during the night only in autumn. The analysis of the dependence on temperature inversion, wind speed, direction also was made and the best meteorological conditions for seeing is given. 2 J. Xu et al	10.1088/1674-4527/20/6/87	[ "https://arxiv.org/pdf/2003.13998v1.pdf" ]	214,728,123	2003.13998	677b812d62172bd487260e41166c748c4f11992f	Site-testing at Muztagh-ata site II: Seeing statistics 31 Mar 2020 Jing Xu [email protected] Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China University of Chinese Academy of Sciences 100049BeijingPeople's Republic of China Ali Esamdin Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China University of Chinese Academy of Sciences 100049BeijingPeople's Republic of China Jin-Xin Hao National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Jin-Min Bai Yunnan Observatories Chinese Academy of Sciences 650000KunmingPeople's Republic of China Ji Yang Purple Mountain Observatories Chinese Academy of Sciences 210008NanjingPeople's Republic of China Xu Zhou National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Yong-Qiang Yao National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Jin-Liang Hou Shanghai Astronomical Observatories Chinese Academy of Sciences 200030ShanghaiPeople's Republic of China Guang-Xin Pu Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China Guo-Jie Feng Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China University of Chinese Academy of Sciences 100049BeijingPeople's Republic of China Chun-Hai Bai Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China Peng Wei Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China Shu-Guo Ma Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China Abudusaimaitijiang Yisikandee Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China Le-Tian Wang Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China Xuan Zhang Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China Ming Liang Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China Lu Ma Jin-Zhong Liu Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China Zi-Huang Cao Xinjinag Astronomical Observatory Chinese Acsdemy of Sciences 830011UrumqiPeople's Republic of China University of Chinese Academy of Sciences 100049BeijingPeople's Republic of China National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Yong-Heng Zhao National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Lu Feng National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Jian-Rong Shi National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Hua-Lin Chen National Astronomical Observatories Nanjing Institute of Astronomical Optics & Technology Chinese Academy of Sciences 210008NanjingPeople's Republic of China Chong Pei National Astronomical Observatories Nanjing Institute of Astronomical Optics & Technology Chinese Academy of Sciences 210008NanjingPeople's Republic of China Xiao-Jun Jiang National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Jian-Feng Wang National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Jian-Feng Tian National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Yan-Jie Xue National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Jing-Yao Hu National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Yun-Ying Jiang National Astronomical Observatories Chinese Academy of Sciences 100012BeijingPeople's Republic of China Site-testing at Muztagh-ata site II: Seeing statistics 31 Mar 2020Received 20xx month day; accepted 20xx month dayResearch in Astronomy and Astrophysics manuscript no. (L A T E X: RAA-2019-0258.tex; printed on April 1, 2020; 1:03) In this article, we present a detailed analysis of the statistical properties of seeing for the Muztagh-ata site which is the candidate site for hosting future Chinese Large Optical/infrared Telescope (LOT) project. The measurement was obtained with Differential Image Motion Monitor (DIMM) from April 2017 to November 2018 at different heights during different periods. The median seeing at 11 meters and 6 meters are very close but different significantly from that on the ground. We mainly analyzed the seeing at 11 meters monthly and hourly, having found that the best season for observing was from late autumn to early winter and seeing tended to improve during the night only in autumn. The analysis of the dependence on temperature inversion, wind speed, direction also was made and the best meteorological conditions for seeing is given. 2 J. Xu et al INTRODUCTION Muztagh-ata site is one of three potential astronomical locations in western China for hosting future Large Optical/infrared Telescope (LOT) project. LOT, which is an ambitious project with a goal to construct a 12 meters telescope aiming to the frontier scientific research on nature of dark energy, detecting of earth-like extrasolar planets, supermassive black holes, first stars, etc., was elected in 2015 (Feng et al. 2019). The site assessment campaign was initiated in January 2017 and lasted for more than two years, climatological properties and optical observing conditions such as sky brightness and cloud amount at Muztagh-ata site have been reported earlier by Xu et al. (2019a) and Cao et al. (2019a). In this article, we focus on the seeing conditions at our site. Image quality is directly related to the statistics of the perturbations of the incoming wavefront. With wavefront sensing methods, wavefront fluctuations can be directly analyzed, providing quantitative information on seeing, independent of the telescope being used (Sarazin & Roddier 1990). Differential image motion monitor (DIMM) has become the standard equipment for assessing the atmospheric seeing at astronomical sites (Skidmore et al. 2009), the seeing derived from DIMM is the combined or integrated effect of all contributing optical turbulence along the optical path (Tokovinin 2002). Michel et al. (2003) conducted a study with DIMM at San Pedro Mrtir observatory and yielded a median seeing of 0.60 arcsec during 123 nights in a three-year period. Tian et al. (2016) measured seeing with a DIMM at Delingha station and achieved an overall seeing median of 1.58 arcsec from 2010 to 2012. Furthermore, DIMM combined with Multi-Aperture Scintillation Sensor (MASS) were widely used for optical turbulence profile measurement Skidmore et al. 2009;Sánchez et al. 2012). The seeing measurements was conducted at Muztagh-ata site during the period from April 2017 to November 2018, we analyze the seeing data collected by Differential Image Motion Monitor (DIMM) and give the global, monthly, hourly statistics. We also present the seeing behavior on different conditions of temperature inversion, wind speed and direction. The layout of this paper is as follows: In Section 2, the working method of DIMM and our instruments for seeing measurements are briefly described. In section 3, we introduce how the DIMMs operated at Muztagh-ata site. In Section 4, we firstly show the comparison result between the two DIMMs, then we give the statistics of seeing at different heights during different periods, at last we focus on the 11 meters seeing and detail its statistics from aspects of monthly, hourly trends. Section 5 shows the relation between seeing and some meteorological parameters such as temperature inversion, wind speed and direction. Conclusions are given in Section 6. SEEING AND SEEING MEASUREMENT The atmospheric turbulence is usually studied through seeing (ε). The relation between ε, Fried parameter γ 0 , and the turbulence integral is given by Roddier (1981) as: ε = 0.98 λ γ 0 = 5.25λ −1/5 [ ∞ 0 C 2 N dh] 3/5 (1) where λ is the wavelength and The central wavelength of light measured by the DIMM is λ = 0.5 µm and the final results have been converted for the direction of observation to zero zenith angle according to the following relation: ∞ 0 C 2 N (h) dh is the optical turbulence energy profile, C 2 N is refractive indexε 0 = ε 0ζ · (cosζ) 3 5(2) Where ζ is the zenith angle, ε 0 is the seeing at the zenith, and ε 0ζ is the seeing as determined by DIMM. The DIMM measurements of the positions of the image centers of stars are made from short exposure images. The exposure time is automatically selected by software according to star magnitude. We focus on the integrated seeing down to the level of the DIMM corrected to one air mass and zero exposure time Skidmore et al. 2009 SEEING MONITORING AT MUZTAGH-ATA The seeing measurement task at Muztagh-ata site began in April 2017 and lasting until to November 2018. We got median value of 0.83 arcsec from French DIMM and 0.89 arcsec from NIAOT DIMM during the whole measurement period. Here we introduce the two DIMMs' operation during whole measurement period at first. We have built two towers for seeing measurements, with heights of 11 meters and 6 meters respectively, no dome protected as shown in Figure 1. Difference of the heights between the two towers is to explore the effect of difference heights on seeing values. At first the observation was conducted on the ground before two towers were built. Then we moved the two DIMMs to the top of towers successively. The periods of the two DIMMs running at different heights as follow: French DIMM was installed on 15 th April 2017, Table 2. In Figure 2 we present the total data of the two DIMMs for each month during the acquisition period. SEEING STATISTICS Comparison of Two DIMMs Because the 11-meter tower only can accommodate one DIMM before rebuild, so we did the comparison after expanding its platform. The comparison work of French DIMM and NIAOT DIMM was from 21 st September 2018 to 20 th November 2018, the two DIMMs both were operated on the top of 11-meter tower. In Figure 3 we show two nights seeing measurement, from which we can see the seeing from French DIMM coincide well with that from NIAOT DIMM. Since the sampling rate of French DIMM is about five times that of NIAOT DIMM, so in this comparison work we firstly find out the nearest moment seeing of seeing at different heights During the whole measurement period we operated two DIMMs at different heights to explore the distribution of near-ground turbulence. In order to eliminate the influence of the occasionally failure of the two DIMMs on results, we only use the data of nights during which both DIMMs were working normally. From Seeing at 11 meters From 23 rd June 2017 to 20 th November 2018, there were 293 nights in total available for seeing data and 142 nights during which more than 75% of the time the measurement acquired data. Figure 8 shows three seeing cases of the 142 nights: good steady nights, with excellent seeing throughout the night (standard variance value less than 0.25 and median value less than 0.82, 66 good steady nights in total, 14 th November 2017 for example);erratic seeing night, with irregular seeing throughout the night (standard variance greater than 0.25, 40 erratic nights in total, 2 nd January 2018 for example); degrade night in which there is a sudden burst of bad seeing (the standard variance value becomes greater than 0.25 toward dawn, 4 th April 2018 for example). The French DIMM seeing median value at 11 meters from 23 rd June 2017 to 20 th November 2018 was 0.82 arcsec as shown in Figure 9. We have made a monthly analysis of all available data. There were 16 months from July 2017 to November 2018 available for seeing measurement, no seeing values during May 2018 because the 11-meter tower was in rebuilding. The boxplot in Figure 10 shows the seeing line represents the values ranging from 1% to 99%, the diamonds and horizontal lines inside every box represent mean and median values respectively. From Figure 10 we can see that the best time period for seeing at Muztagh-ata site is from late autumn to early winter.The seeing nightly statistics and Cumulative Distribution Functions during October and November 2017 are shown in Figure 11, the median values are 0.62 arcsec and 0.60 arcsec for these two months. Bad seeing nights appeared frequently in July and August every year, mainly due to the erratic weather during autumn. In order to explore the seeing behavior along the night we plot the hourly results for each season integrated over this acquisition period in Figure 12 then improves, it is obviously the seeing during the second half of the night in autumn is better than the first half. SEEING DEPENDENCE ON METEOROLOGICAL CONDITIONS One automated weather station named second generation Kunlun Automated Weather Station(KLAWS-2G) 1 was installed at Muztagh-ata site and started to record data from 1 st August 2017. It has several high precision temperature sensors (Young 4-wire RTD ,Model 41342), cup anemometers and wind vanes at different heights make it possible to explore the influence of meteorological parameters on seeing. In this section, the seeing data we use is from French DIMM in the period of 1 st August 2017 to 20 th November 2018 at 11 meters. Temperature Inversion The phenomenon of temperature inversion means air temperature decreasing with decreasing elevation, it comes with stable atmospheric structure. Hu et al. (2014Hu et al. ( , 2019 used KLAWS-2G and found strong temperature inversion occur frequently at Dome A (Burton 2010). We want to explore the lasting time and influence on seeing at Muztagh-ata. Figure 13 shows the statistics of inversion layer in two periods: one is in afternoon from 13:00 to 15:00 (UTC+8), another is in midnight from 1:00 to 3:00 (UTC+8). It can be seen that 70% of the time during the midnight period the temperatures at 2 meters are smaller than that at 6 meters, and the median value of the difference is -0.2 • C. Relationship between seeing and temperature inversion is presented in Figure 14, there are four cumulative distribution function curves in the plot represent four ranges of the difference between the temperatures at 2 meters and 6 meters. The blue one is the range of smaller than -0.5 • C with seeing median value 0.71 arcsec. The dark green is the range of -0.5 to -0.3 • C with seeing median value 0.77 arcsec. The pink is the range of -0.3 to -0.1 • C with seeing median value 0.8 arcsec. The light green is the range of higher than -0.1 • C with seeing median value 0.9 arcsec. Figure 14 proofs that stronger inversion can bring better seeing. Wind We use the sensors homed at 10 meters altitude for the analysis of the seeing versus wind speed and direction. Figure 15 and Figure 16 show the relation of the seeing measurements to wind direction and speed of prevailing wind direction. According to the meteorological parameter measurement results at Muztagh-ata site of this site-testing task, the prevailing wind is southwest in this area (Xu et al. 2019a). The asterisks in Figure 15 represent the amount of seeing data in every 30 • of wind direction, the largest amount of data of median seeing value with east winds. It indicates that the east winds bring about local weather changes and unstable ground turbulence. Figure 16 shows the relation of seeing and southwest wind speed (wind direction from 180 • to 270 • ). The asterisks mark the amount of data and solid line and pluses indicate the median seeing values. With the increase of wind speed, the median seeing value shows an initial drop until the wind speed reaches 3 ms −1 then keeps growing. When the wind speed is greater than 12 ms −1 the data amount decreases sharply and 75% line deviates from median line. From these two figures we can see that the most stable CONCLUSIONS DIMM data collected from June 2017 to November 2018 at Muztagh-ata site have been presented. We want to acquire some preliminary conclusion of near-ground turbulence distribution and seeing condition at our site. The main results got from this work as follow: 1. The seeing median value at 11-meter during whole measurement period is 0.82 arcsec. Seeing difference between 11-meter level and 6-meter level is very small but significant between 6-meter level and ground-level. It illustrates that the near-ground turbulence is concentrated within 6 meters above ground at Muztagh-ata site. 2. Seasonal statistics shows that the best season for optical astronomical observation at Muztagh-ata Site-testing at Muztagh-ata site II:Seeing statistics 13 analysis shows that there is a tendency of seeing that getting worse progressively toward dawn in the most time of year but autumn, in this season the seeing deteriorates first and then improves during night. 3. The dependence of the seeing at 11 meters on meteorological conditions is discussed. We calculate the frequency of temperature inversion during midnight, 70% of that time the inversion was present. Then we analysis the relationship between inversion and seeing, results present the evidence that stronger inversion can bring better seeing. 4. We present seeing roses and seeing statistics at various wind speeds of prevailing wind direction, from which the conclusion would be made that stable ground turbulence occurs when stable breeze from southwest. Fig. 1 : 111-meter tower (left) and 6-meter tower (right) at Muztagh-ata. Fig. 2 : 2Total monthly data from March 2017 to June 2018. Patterns: French DIMM (red), NIAOT DIMM (blue). NIAOT DIMM was installed on 12 th March 2017 and then had been running on the ground until 15 th November 2017, then we moved it to the top of 6-meter tower. After the rebuilding of 11-meter tower, for the purpose to ensure two DIMMs can be housed on the top of it simultaneously, we installed NIAOT DIMM on it on 21 st September 2018. The two DIMMs were operated at 11 meters until 20 th November 2018 for comparing. The detailed operation periods and heights of two DIMMs can be seen in Fig. 3 : 3Example of seeing values from French DIMM (red cross) and NIAOT DIMM (blue cross) at 11 meters during two nights. French DIMM according to NIAOT DIMM, then use the median value of five-neighborhood as the value of French DIMM. The comparison result of two DIMMs during this period is shown in Figure 4, in which the correlation of NIAOT DIMM and French DIMM and the statistical analysis of the residual to line Y = X are given. The median of the residuals is 0.07 and the standard variance is 0.08, it indicates that the difference between the two DIMMs is very small. The seeing distributions and cumulative distribution functions during this period was shown in Figure 5, the median seeing values of French DIMM and NIAOT DIMM are 0.71 arcsec and 0.72 arcsec respectively. Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 :Fig. 8 : 4567823 rd June 2017 to 14 th November 2017, NIAOT DIMM was put on the ground while French Comparison result of two DIMMs at 11 meters. Left panel: correlation of NIAOT DIMM and French DIMM. Right panel: distribution and cumulative distribution function of residuals. Seeing distributions and cumulative distribution functions from two DIMMs over the period from 21 st September 2018 to 20 th November 2018 at 11 meters. Seeing distributions and cumulative distribution functions from two DIMMs during the period from 23 rd June 2017 to 14 th November 2017 at 11 meters and on the ground level. values acquired during this period. The median value from French DIMM is 0.79 arcsec at 11 meters and the median value from NIAOT DIMM is 0.97 arcsec on the ground level. In autumn 2017 we built a 6-meter tower, and NIAOT DIMM was moved to the top of 6-meter tower on 15 th November 2017. The seeing got from NIAOT DIMM at 6 meters lasted until 20 th September 2018. The results of measurement during this period are shown in Figure 7. Both of the median seeing values at two heights are 0.87 arcsec. The difference between seeing median at 11 meters and ground level is 0.18 arcsec, and the difference is very small between 11 meters and 6 meters, indicating that the near-ground Site-testing at Muztagh-ata site II:Seeing distributions and cumulative distribution functions from two DIMMs during the period from 15 th November 2017 to 20 th September 2018 at 11 meters and 6 meters. Example of French DIMM seeing values during observable nights. Good steady night (2017-11-14), erratic night (2018-01-02), degrade night (2018-04-04) from top to bottom respectively. Fig. 9 :Fig. 10 : 910Seeing distributions and cumulative functions from French DIMM at 11 meters during period of 23 th June 2017 to 20 th November 2018. Monthly statistics of seeing from 2017 July to 2018 November (there was no data in 2018 May because of tower rebuilding). Each box represents the values in the range of 25% to 75% and the vertical line represents the values ranging from 1% to 99%, the diamonds and horizontal lines inside every box represent mean and median values respectively. Fig. 11 : 11(spring, summer, autumn, winter from top to bottom respectively).The solid line and pluses indicate the median seeing values in each hour (UT time), and the dashed lines indicate 25% and 75%, the asterisks represent the amount of data. In spring, summer and winter the Site-testing at Muztagh-ata site II:Left and middle panels: Median seeing values (with lower and upper limits represented by first and third quartiles respectively) for each month of October 2017 and November 2017. Right panel: Seeing Cumulative Distribution Function and 50% line for the two months (Pink for October 2017 and Blue for November 2017). Fig. 12 : 12Hourly seeing statistic for each season (spring, summer, autumn, winter from top to bottom respectively). The solid line and pluses indicate the median seeing values in each hour (UT time), and the dashed lines indicate 25% and 75%. Asterisks mark the amount of data in each hour. Fig. 13 : 13The distribution and cumulative distribution functions of the difference between the temperature at 2 meters and 6 meters in two periods: 13:00 15:00 (UTC+8) afternoon (red) and 1:00∼3:00 ( Fig. 14 : 14Cumulative distribution functions of seeing in four ranges of the difference between the temperatures at 2 meter and 6 meter: smaller than -0.5 • C (blue), -0.5 ∼-0.3 • C (dark green), -0.3∼-0.1 • C (pink), higher than -0.1 • C (light green). Fig. 15 :Fig. 16 : 1516Seeing roses. The solid line and crosses indicate the median seeing values in each 30 • wind direction bin, the dashed lines indicate 25% and 75%. Asterisks mark the amount of data in each bin. Seeing statistics at various wind speeds (wind direction from 180 • to 270 • ). Wind speeds were binned in 1 ms −1 intervals. The solid line and pluses indicate the median seeing values at various wind speeds, the dashed lines indicate 25% and 75%. Asterisks mark the amount of data in each bin. Site-testing at Muztagh-ata site II:Seeing statistics3 Table 1 : 1Parameters and technical specifications of two DIMMs.French DIMM NIAOT DIMM Telescope Aperture (mm) 300 304.8 Focal ratio 8 8 Focal length (mm) 2400 1600 Sub-apertures 2 4 Sub-aperture diameters (mm) 51 50 Sub-aperture distance (mm) 240 149 Camera DMK 33GX 174 Basler aca2040 Exposure method Adjusted automatically 5ms or 10 ms between 0.5 ms and 1000 ms Wavelength (mm) 550 500 Output frequency 1 seeing value for 1 seeing value for every minute every 1000 images Scaling and conversion Convert to zenith Convert to zenith No exposure time scaling No exposure time scaling No wavelength scaling Table 2 : 2Operation time periods of the two DIMMsStarting-Date French DIMM NIAOT DIMM 2017.03.12 - ground 2017.04.15 ground ground 2017.06.23 11 meters ground 2017.11.15 11 meters 6 meters 2018.09.21 11 meters 11 meters Jul Dec May Oct 0 10000 20000 30000 40000 50000 60000 French DIMM French DIMM counts Time(month) 2017 2018 0 2000 4000 6000 8000 10000 NIAOT DIMM NIAOT DIMM counts Because the sample rate of French DIMM is about five times that of NIAOT DIMM, we use different axes to represent the data amounts of two DIMMs, left for French DIMM and right for NIAOT DIMM. It is worthy of note that in May 2018 we rebuilt the 11-meter tower so the data amount of French DIMM in that month is relative smaller. Due to the failure of CCD and controller NIAOT DIMM did not operate duringSite-testing at Muztagh-ata site II:Seeing statistics 5 23:00 01:00 03:00 05:00 07:00 09:00 0.0 0.5 1.0 1.5 2.0 23:30 00:30 01:30 02:30 03:30 04:30 05:30 06:30 07:30 08:30 0.0 0.5 1.0 1.5 2.0 French DIMM NIAOT DIMM seeing(arcsec) 2018/11/6 French DIMM NIAOT DIMM seeing(arcsec) Local time(hours) 2018/10/4 . M G Burton, A&A Rev. 18417Burton, M. G. 2010, A&A Rev., 18, 417 . Z H Cao, L Y Liu, Y Q Yao, Research in Astron. Astrophys. RAACao, Z. H., Liu, L. Y., Yao, Y. Q. et al., 2019, Research in Astron. Astrophys. (RAA) . S G Els, M Schöck, E Bustos, PASP. 121922Els, S. G., Schöck, M., Bustos, E., et al. 2009, PASP, 121, 922 . L Feng, J X Hao, J M Bai, Research in Astron. Astrophys. RAAFeng, L., Hao, J. X., Bai, J. M. et al., 2019, Research in Astron. Astrophys. (RAA) . 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[ "Optimal partition of feature using Bayesian classifier", "Optimal partition of feature using Bayesian classifier", "Optimal partition of feature using Bayesian classifier", "Optimal partition of feature using Bayesian classifier" ]	[ "Sanjay Vishwakarma \nIBM Quantum\nWoxsen University\n\n", "Srinjoy Ganguly [email protected] \nIBM Quantum\nWoxsen University\n\n", "Sanjay Vishwakarma \nIBM Quantum\nWoxsen University\n\n", "Srinjoy Ganguly [email protected] \nIBM Quantum\nWoxsen University\n\n" ]	[ "IBM Quantum\nWoxsen University\n", "IBM Quantum\nWoxsen University\n", "IBM Quantum\nWoxsen University\n", "IBM Quantum\nWoxsen University\n" ]	[]	The Naive Bayesian classifier is a popular classification method employing the Bayesian paradigm. The concept of having conditional dependence among input variables sounds good in theory but can lead to a majority vote style behaviour. Achieving conditional independence is often difficult, and they introduce decision biases in the estimates. In Naive Bayes, certain features are called independent features as they have no conditional correlation or dependency when predicting a classification. In this paper, we focus on the optimal partition of features by proposing a novel technique called the Comonotone-Independence Classifier (CIBer) which is able to overcome the challenges posed by the Naive Bayes method. For different datasets, we clearly demonstrate the efficacy of our technique, where we achieve lower error rates and higher or equivalent accuracy compared to models such as Random Forests and XGBoost.	10.48550/arxiv.2304.14537	[ "https://export.arxiv.org/pdf/2304.14537v1.pdf" ]	258,418,157	2304.14537	4f31a6f72677838c3a254bdc10d268906b10b83e	Optimal partition of feature using Bayesian classifier Sanjay Vishwakarma IBM Quantum Woxsen University Srinjoy Ganguly [email protected] IBM Quantum Woxsen University Optimal partition of feature using Bayesian classifier 1Index Terms-Attribute WeightingBayesian NetworksClus- teringNa ıve BayesCIBer The Naive Bayesian classifier is a popular classification method employing the Bayesian paradigm. The concept of having conditional dependence among input variables sounds good in theory but can lead to a majority vote style behaviour. Achieving conditional independence is often difficult, and they introduce decision biases in the estimates. In Naive Bayes, certain features are called independent features as they have no conditional correlation or dependency when predicting a classification. In this paper, we focus on the optimal partition of features by proposing a novel technique called the Comonotone-Independence Classifier (CIBer) which is able to overcome the challenges posed by the Naive Bayes method. For different datasets, we clearly demonstrate the efficacy of our technique, where we achieve lower error rates and higher or equivalent accuracy compared to models such as Random Forests and XGBoost. I. INTRODUCTION N Aïve Bayes [1], the simplest but most efficient Bayesian classifier, has been widely used for many decades [2], [3]. A Bayesian classifier computes the probability for every possible label by the Bayes Theorem as shown in Definition 1. Definition 1: Bayes theorem. Suppose the input feature vector X = (X 1 , . . . , X d ), and define the set of all labels as C. Let x = (x 1 , . . . , x d ) be a vector containing the observed value of predictive features, and let y ∈ C be a possible class, then P(y\|x) = P(y) · P(x\|y) P(x) = P(y) · P(x\|y) y ∈C P(x, y )(1) A. Attribute Weighting Models using the attribute weighting technique assign weights to each feature during the training process. If we assign a single weight w i to each X i , then for any input x = (x 1 , . . . , x d ) and any y ∈ C, P(y\|x), which is probability, is calculated as: P(y\|x) = P(y, x) P(x) = P(y) · d i=1 P(x i \|y) wi y ∈C P(y ) · d i=1 P(x i \|y ) wi . B. Semi-naïve Bayes Semi-naïve Bayes models are of two types; one type applies the naive assumption but relies on a novel set of attributes by either removing the attributes or combining them. The second type adds explicit arcs between features representing dependencies, as Naïve Bayes mirrors a Bayesian network with arcs connecting class variables to predictive features proposed by [4] and models in this group follow [5]. 1) New Attribute Set: Selective Naïve Bayes is a popular algorithm that can be considered an extreme form of attribute weighting. [6] proposed a heuristic method for finding ideal feature subsets during training operations. A similar algorithm, proposed by [7], uses correlation coefficients to determine the relevance among the features. [8] proposed a maximum a posteriori approach for choosing an optimal subgroup of features while avoiding overfitting. 2) Adding Explicit Arcs: As summarized by [5], the models which add explicit arcs among features can be further divided into two types, namely, 1-dependence models [9]- [11] and xdependence models (x ≥ 1) [12], [13]. In general, they are all alternatives to the complete Bayesian network [4]. [9] discovered that unrestricted Bayesian networks do not necessarily outperform Naïve Bayes in accuracy. This led them to introduce a restriction that produces a tree topology, and the task of finding the maximum spanning tree is accomplished with conditional mutual information. C. Bayesian network Bayesian networks are models that graphically depict the relationships of dependence and independence between variables. These networks are illustrated by directed acyclic graphs. We deliver an alternate blueprint to designing a Bayesian classifier based on the concept of Comonotonicity, originating from financial risk theory and aims to address the limitations of the Naïve Bayes method. In insurance theory, average risk can be predicted. However, this assumption requires a sufficiently large number of insured risks, and computation is feasible due to the Law of Large Numbers [14]. In order to capture the sums of insured risks, [15], [16] proposed the concept of Comonotonicity. Comonotonicity is a concept from probability theory and decision-making, which refers to a situation in which random variables exhibit a perfectly positive association or dependence. In other words, comonotonic random variables move in the same direction, either increasing or decreasing together. When two or more random variables are comonotonic, their joint distribution function can be described as follows: arXiv:2304.14537v1 [cs. LG] 27 Apr 2023 F (x 1 , x 2 , ..., x n ) = min(F 1 (x 1 ), F 2 (x 2 ), ..., F n (x n )) where F (x 1 , x 2 , ..., x n ) is the joint distribution function of the comonotonic random variables X 1 , X 2 , ..., X n , and F 1 (x 1 ), F 2 (x 2 ), ..., F n (x n ) are their respective marginal distribution functions. We designed and implemented a classifier incorporating this idea. Our contributions are: • A heuristic technique that incorporates clustering to find an optimal partition for the predictive features. • We estimate the conditional joint probabilities in each group by comonotonicity to produce enhanced joint probability figures. We claim that our work, unlike any of the previous improvements to Naïve Bayes, has enlightened a new paradigm in Bayesian learning. We introduce CIBer intending to improve the performance of Naïve Bayes. We pursue a heuristic search for an optimal feature partition and estimate the conditional joint probability with the comonotonic paradigm. It considers dependencies between features and shows promising results on specific data distributions and competitive performance on empirical datasets. II. COMONOTONE-INDEPENDENCE BAYESIAN CLASSIFIER This section aims to present a new classifier that utilizes comonotonicity. In practice, our classifier deals with discrete features [17] with numerical values. A. Concepts and Sampling Techniques Comonotonicity has many uses, including risk management for derivatives and life insurance [18]. Our research represents the first instance of the application of comonotonicity to classification tasks within the field of machine learning. B. Conditional empirical joint distribution modelling We call each estimated CDF as the empirical distribution function, denoted byF Xi . Take a sample whose size is n and defined by x 1 , . . . , x n . If each of these samples is independent and identically distributed with CDF F X , then we show the empirical distribution function in the following way: F X (x) = 1 n n j=1 1 xj ≤x(2) where the indicator function of 1 A shows 1 A = 1 if condition A holds and 1 A = 0 otherwise. 1) Clustered Comonotonic: To measure the likelihood of two features being comonotonic, we use four statistical metrics. The first metric is the normalized mutual information (NMI), denoted by U , which quantifies the dependence between two features via information entropy. Pearson's, Spearman's, and Kendall's values are the three commonly used types of correlation coefficients. C. Practical considerations under implementation 1) Feature Types: For most real-world data-sets, we can roughly partition the features into 3 types, namely, categorical, continuous and discrete. 2) Discretization: In order to compute conditional joint probability in a comonotonic classifier, discretization is necessary, as discussed further in section III-B. Bin Encoded Number (−∞,μ − 3 ·σ] 0 (μ − 3 ·σ,μ − 2 ·σ] 1 (μ − 2 ·σ,μ −σ] 2 (μ −σ,μ] 3 (μ,μ +σ] 4 (μ +σ,μ + 2 ·σ] 5 (μ + 2 ·σ,μ + 3 ·σ] 6 (μ + 3 ·σ, +∞) 7 3) Re-balancing the data-set: In practice, a number of data-sets such as fraud detection, rare disease diagnosis, are severely imbalanced where the vast majority of the instances belong to one class. There are two main methods for resampling: • Over-sampling: This involves augmenting the original dataset by adding some resampled instances from the minority class. • Under-sampling: This involves shrinking the original dataset by deleting some resampled instances from the majority class. III. SIMULATION STUDY The performance of any data modelling technique varies for different distributions of the data. We claim that there exist some situations in which modelling the conditional joint probability by comonotonicity is much better than assuming independence. A. An intuitive example This example shows a situation in which Naïve Bayes almost has no power of judgement while CIBer gives a much better estimation. Suppose the only two predictive features are X 0 and X 1 . The data points in class 0 fall on the line segment X 1 = X 0 + 20 with X 0 ∈ [0, 100], and the data points in class 1 fall on the line segment X 1 = X 0 − 20 with X 0 ∈ [20, 120]. The line segments for the two classes are shown in Figure 1. We express a sample data point by a tuple with 2 entries where the first one stands for X 0 while the second one for X 1 . An additional assumption is class 1 entails X 0 's uniform distribution in both cases, and we can discretize X 0 and X 1 into equal bins with length 10. Then in this way, P(X i = x\|Y = y) = 0.1 for any i, y ∈ {0, 1} for points with X 0 ∈ [20, 100] and X 1 ∈ [20, 100], i.e., the points inside the rectangle with dashed lines in Figure 1, while the following cases have zero probability and require Laplacian correction in practice. 1) X 0 ∈ [100, 120] or X 1 ∈ [0, 20] conditioning on Y = 0. 2) X 0 ∈ [0, 20] or X 1 ∈ [100, 120] conditioning on Y = 1. Suppose we want to classify two sample points, X (1) = (10, 30) and X (2) = (55, 35). By the formulas for the two line segments, we know that X (1) belongs to class 0 and X (2) belongs to class 1. We calculate the posterior probabilities P(Y \|X 0 , X 1 ) using Naïve Bayes and CIBer, respectively as follows: Naïve Bayes: X (1) = (10, 30) & let P(X 0 = 10, X 1 = 30\|Y = 0) = α P(Y = 0\|X 0 = 10, X 1 = 30) ∝ α · P(Y = 0) =P(X 0 = 10\|Y = 0) · P(X 1 = 30\|Y = 0) · P(Y = 0) =0.1 · 0.1 · 0.5 =0.0005 Let P(X 0 = 10, X 1 = 30\|Y = 1) = β P(Y = 1\|X 0 = 10, X 1 = 30) ∝ β · P(Y = 1) =P(X 0 = 10\|Y = 1) · P(X 1 = 30\|Y = 1) · P(Y = 1) Let P(X 0 = 10, =0 · 0.1 · 0.5 =0 X (2) = (55, 35) & let P(X 0 = 55, X 1 = 35\|Y = 0) = γ P(Y = 0\|X 0 = 55, X 1 = 35) ∝ γ · P(Y = 0) =P(X 0 = 55\|Y = 0) · P(X 1 = 35\|Y = 0) · P(Y = 0) =0.1 · 0.1 · 0.5 =0.0005 Let P(X 0 = 55, X 1 = 35\|Y = 1) = δ P(Y = 1\|X 0 = 55, X 1 = 35) ∝ δ · P(Y = 1) =P(X 0 = 55\|Y = 1) · P(X 1 = 35\|Y = 1) · P(Y = 1) =0.1 · 0.1 · 0.5 =0.X 1 = 30\|Y = 1) = η P(Y = 1\|X 0 = 10, X 1 = 30) ∝ η · P(Y = 1) =Leb(φ ∩ [0.2, 0.3]) · 0.5 =0 X (2) = (55, 35) & let P(X 0 = 55, X 1 = 35\|Y = 0) = θ P(Y = 0\|X 0 = 55, X 1 = 35) ∝ θ · P(Y = 0) =Leb([0.5, 0.6] ∩ [0.1, 0.2]) · 0.5 =0.05 A classifier has no power of judgement if P(Y = 0\|X 0 , X 1 ) = P(Y = 1\|X 0 , X 1 ). Thus, the calculation above indicates that Naïve Bayes has no power of judgement for X (2) . Next, we shall discretize the data in three different ways, namely, discretizing into 10 bins with equal length, discretizing by mean and standard deviation, discretizing by minimum description length principle. In Section III-B, we try to simulate much more synthetic data. Here, for each class, we uniformly generate 5000 samples and split out 20% for purposes of a test set while the remaining as training set. Then we apply both CIBer and Naïve Bayes to the synthetic data and repeat the experiment for 1000 times. Accuracy scores of both scenarios are in Figure 2. B. Verification without discretization CIBer and the Naïve Bayes classifier with performance metrics are presented in Figure 3. It can be seen that the accuracy of both methods becomes more stable compared to the experiments with 5000 data points. Also, CIBer demonstrates significant superiority over Naïve Bayes when no typical discretization is applied. Next, we shall compare the running time of CIBer and Naïve Bayes in terms of training-cum-testing operations. C. Time Complexity Analysis Unlike the previous setting of only 2 features, here we adjust the number of features in each round. For simplicity, in each experiment, we set the number of feature pairs in the simulated data, denoted by λ. Then • Class 0: X 2i = X 2i−1 +20 and X 2i−1 ∼ U (0, 100) i ≤ λ • Class 1: X 2i = X 2i−1 − 20 and X 2i−1 ∼ U (20, 120) i ≤ λ For each class, we simulate 500000 data points and discretize the intervals into 1000 equal-sized bins. We repeat the experiment for 10 times for each number of feature pairs. Then we take the ratio of running time for the two models. Later on, we plot the 95% confidence interval of the running time. The confidence interval of a two-tailed t-test with 9 degrees of freedom is computed bŷ µ ± 2.2622 ·σ √ 10 whereμ andσ are the mean and standard deviation of the ratios. The relationship between the number of features and the ratio of running time is shown in Figure 4 and 5 We observe from the plots that there exists a linear relationship between running time and number of features. Next, we analyze the theoretical time complexity for both models. Our analysis is in accordance with the simulation results. IV. EMPIRICAL EVALUATION In this section, we conduct empirical evaluations and comparisons of the performance of multiple models on various real-world datasets abiding by the following principles. • Task: The task should be classification. • Feature type: The features should be all numerical, or at most 1 or 2 categorical ones. • Data type: The data should be multivariate (including multivariate time-series, but we disregard the autoregressive effects for the ease of analysis and comparation). • Dimension: We prefer data-sets with tens of features so that clusters can be figured out. However, lower dimensional data-sets are also used if comonotonic structure also exists. A. Example Data-sets & Results In this section, we describe the four datasets used to evaluate our models: Ozone, Sensorless Diagnosis, and Oil Spill. These datasets are sourced from the Machine Learning repository of the University of California, Irvine (UCI ML) [19] and the Oil Spill dataset [20]. We split each dataset using stratified sampling. We repeat the sub-sampling → fitting → testing process 10 times and gradually increase the portion until the complete training set is used. After fitting the models, we obtain the error rate on the testing set. We conduct a twotailed t-test on the error rates at each size of the training set and obtain a 95% confidence interval with 9 degrees of freedom. The confidence interval is calculated using the following formula:μ ± 2.262 ·σ √ 10 whereμ andσ are the mean and standard deviation of the error rates. Each day has 72 continuous meteorological features, and the target variable is binary, indicating whether that day was an ozone day or not [21]. We used 30% of the data as the testing set and the remaining 70% as the complete training set. The plots of the error rate change when adjusting the size of the actual training set is shown in Figure 6 and 7. It is observed that the error rates of all models except Naïve Bayes decrease in different scales when gradually enlarging the training set, among which CIBer decreases the most. To some extent, CIBer is more sensitive to the amount of training data in this data-set. We ascribe such sensitivity to the fact that when the training data is far too limited. However, when we gradually increase the amount of training data, each conditional marginal probability converges to the real one by the law of large numbers. Thus, the error rate tends to be stable when the portion of training data is larger than 40%. Meanwhile, CIBer achieves an even better performance than XGBoost and Random Forest when the training size goes 2) Data-set for Diagnosing Sensorless Drives: The Dataset for Diagnosing Sensorless Drives contains 48 characteristics derived from electrical current drive signals, with 11 different classes for the instances. To save computation time while still visualizing the convergence of testing accuracy, we split out 80 percent of the data-set as the testing set and the last 20 percent as the complete training set. We experiment with 10 different proportions of the training set size and assess the error rates on the testing set. The plots of the error rate change are shown in Figure 8 and 9. In general, CIBer's performance is close to XGBoost on this data-set. Although its error rate is a bit larger than Random Forest when the training set size approaches 20%, it outperforms Random Forest for small training sets. 3) Oil Spill Data-set: The dataset containing information about oil spills is the focus of the following analysis; Each instance in the dataset contains 48 continuous features that were generated by a computer vision model. The computer The plots of the error rate change when adjusting the size of training set is shown in Figure 10, 11, 12 and 13. In terms of error rates, CIBer is always lower than XGBoost and approximately the same as random forest when the training percentage exceeds 50%. CIBer's generalization ability may be due to its ability to capture discrete distributions. On the third dataset, the larger number of classes and smaller number of instances lead to larger performance variance for all models. CIBer outperforms other models in terms of error rate when the training percentage is above 40%. V. CONCLUSION Previous works aimed to improve Naïve Bayes, which assumes feature independence and achieved various levels of performance enhancement. However, CIBer takes a different approach by conducting a heuristic search to find an optimal feature partition and estimating conditional joint probability using the comonotonic paradigm. CIBer shows promising results in specific data distributions and competitive performance on empirical data sets. Future work can explore improvements and alternatives, such as handling categorical features, applying ensemble learning, and finding more efficient search methods. Additionally, integrating comonotonic features into Bayesian networks could be considered. VI. ACKNOWLEDGMENTS S.V. and S.G. have equally contributed to the project. Authors acknowledge the use of datasets from UCI Machine Learning Repository. Python programming language was used to carry out the entire reasearch. Fig. 1 . 1Simulation results for the network. 0005 CIBer: X (1) = (10, 30) & let P(X 0 = 10, X 1 = 30\|Y = 0) = P(Y = 0\|X 0 = 10, X 1 = 30) ∝ · P(Y = 0) =Leb([0, 0.1] ∩ [0, 0.1]) · 0.5 =0.05 Fig. 2 . 2Accuracy Scores With Different Discretizations =0 Let P(X 0 = 55, X 1 = 35\|Y = 1) = ζ P(Y = 1\|X 0 = 55, X 1 = 35) ∝ ζ · P(Y = 1) =Leb([0.3, 0.4] ∩ [0.3, 0.4]) · 0.5 Fig. 3 . 3Approximation by sufficiently small bins Fig. 4. Ratio of Training Time with 95% Confidence Interval Fig. 5 . 5Ratio of Testing Time with 95% Confidence Interval Fig. 6 . 6Change of Error Rate in CIBer, Na ıve Bayes, XGBoost and Random Forest Fig. 7. Change of Error Rate in CIBer, Decision Tree, SVM and Logistic Regression 1) Ozone Data-set: The Ozone data-set contains approximately 2500 instances, each representing a single day between 1998 and 2004 in the Houston, Galveston and Brazoria area. Fig. 8 . 8Change of Error Rate in CIBer, Na ıve Bayes, XGBoost and Random Forest Fig. 9. Change of Error Rate in CIBer, Decision Tree, SVM and Logistic Regression larger. Moreover, the tapering blue band indicates that the confidence interval for the average accuracy of CIBer becomes smaller. Fig. 10 . 10Change of Error Rate in CIBer and Na ıve Bayes Fig. 11 . 11Change of Error Rate in CIBer, Decision Tree, SVM and Logistic Regression Fig. 12. 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Available: http://archive.ics.uci.edu/ml Machine learning for the detection of oil spills in satellite radar images. M Kubat, R C Holte, S Matwin, Machine learning. 302-3M. Kubat, R. C. Holte, and S. Matwin, "Machine learning for the detection of oil spills in satellite radar images," Machine learning, vol. 30, no. 2-3, pp. 195-215, 1998. Forecasting skewed biased stochastic ozone days: analyses, solutions and beyond. K Zhang, W Fan, Knowledge and Information Systems. 143K. Zhang and W. Fan, "Forecasting skewed biased stochastic ozone days: analyses, solutions and beyond," Knowledge and Information Systems, vol. 14, no. 3, pp. 299-326, 2008.	[]
[ "AN ADAPTIVE POLICY TO EMPLOY SHARPNESS-AWARE MINIMIZATION", "AN ADAPTIVE POLICY TO EMPLOY SHARPNESS-AWARE MINIMIZATION", "AN ADAPTIVE POLICY TO EMPLOY SHARPNESS-AWARE MINIMIZATION", "AN ADAPTIVE POLICY TO EMPLOY SHARPNESS-AWARE MINIMIZATION" ]	[ "Weisen Jiang \nDepartment of Computer Science and Engineering\nGuangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation\nSouthern University of Science and Technology\n\n\nDepartment of Computer Science and Engineering\nHong Kong University of Science and Technology\n\n", "Hansi Yang \nDepartment of Computer Science and Engineering\nHong Kong University of Science and Technology\n\n", "Yu Zhang [email protected] \nDepartment of Computer Science and Engineering\nGuangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation\nSouthern University of Science and Technology\n\n\nPeng Cheng Laboratory\n\n", "James Kwok [email protected] \nDepartment of Computer Science and Engineering\nHong Kong University of Science and Technology\n\n", "Weisen Jiang \nDepartment of Computer Science and Engineering\nGuangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation\nSouthern University of Science and Technology\n\n\nDepartment of Computer Science and Engineering\nHong Kong University of Science and Technology\n\n", "Hansi Yang \nDepartment of Computer Science and Engineering\nHong Kong University of Science and Technology\n\n", "Yu Zhang [email protected] \nDepartment of Computer Science and Engineering\nGuangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation\nSouthern University of Science and Technology\n\n\nPeng Cheng Laboratory\n\n", "James Kwok [email protected] \nDepartment of Computer Science and Engineering\nHong Kong University of Science and Technology\n\n" ]	[ "Department of Computer Science and Engineering\nGuangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation\nSouthern University of Science and Technology\n", "Department of Computer Science and Engineering\nHong Kong University of Science and Technology\n", "Department of Computer Science and Engineering\nHong Kong University of Science and Technology\n", "Department of Computer Science and Engineering\nGuangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation\nSouthern University of Science and Technology\n", "Peng Cheng Laboratory\n", "Department of Computer Science and Engineering\nHong Kong University of Science and Technology\n", "Department of Computer Science and Engineering\nGuangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation\nSouthern University of Science and Technology\n", "Department of Computer Science and Engineering\nHong Kong University of Science and Technology\n", "Department of Computer Science and Engineering\nHong Kong University of Science and Technology\n", "Department of Computer Science and Engineering\nGuangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation\nSouthern University of Science and Technology\n", "Peng Cheng Laboratory\n", "Department of Computer Science and Engineering\nHong Kong University of Science and Technology\n" ]	[]	Sharpness-aware minimization (SAM), which searches for flat minima by min-max optimization, has been shown to be useful in improving model generalization. However, since each SAM update requires computing two gradients, its computational cost and training time are both doubled compared to standard empirical risk minimization (ERM). Recent state-of-the-arts reduce the fraction of SAM updates and thus accelerate SAM by switching between SAM and ERM updates randomly or periodically. In this paper, we design an adaptive policy to employ SAM based on the loss landscape geometry. Two efficient algorithms, AE-SAM and AE-LookSAM, are proposed. We theoretically show that AE-SAM has the same convergence rate as SAM. Experimental results on various datasets and architectures demonstrate the efficiency and effectiveness of the adaptive policy.Published as a conference paper at ICLR 2023 the squared stochastic gradient norm and model it by a normal distribution, whose parameters are estimated by exponential moving average. Experimental results on standard benchmark datasets demonstrate the superiority of the proposed policy.Our contributions are summarized as follows: (i) We propose an adaptive policy to use SAM or ERM update based on the loss landscape geometry. (ii) We propose an efficient algorithm, called AE-SAM (Adaptive policy to Employ SAM), to reduce the fraction of SAM updates. We also theoretically study its convergence rate. (iii) The proposed policy is general and can be combined with any SAM variant. In this paper, we integrate it with LookSAM (Liu et al., 2022) and propose AE-LookSAM. (iv) Experimental results on various network architectures and datasets (with and without label noise) verify the superiority of AE-SAM and AE-LookSAM over existing baselines.	10.48550/arxiv.2304.14647	[ "https://export.arxiv.org/pdf/2304.14647v1.pdf" ]	258,418,258	2304.14647	8adf430deefe6af08b894d28399eb027c2e8e69e	AN ADAPTIVE POLICY TO EMPLOY SHARPNESS-AWARE MINIMIZATION Weisen Jiang Department of Computer Science and Engineering Guangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation Southern University of Science and Technology Department of Computer Science and Engineering Hong Kong University of Science and Technology Hansi Yang Department of Computer Science and Engineering Hong Kong University of Science and Technology Yu Zhang [email protected] Department of Computer Science and Engineering Guangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation Southern University of Science and Technology Peng Cheng Laboratory James Kwok [email protected] Department of Computer Science and Engineering Hong Kong University of Science and Technology AN ADAPTIVE POLICY TO EMPLOY SHARPNESS-AWARE MINIMIZATION Published as a conference paper at ICLR 2023 Sharpness-aware minimization (SAM), which searches for flat minima by min-max optimization, has been shown to be useful in improving model generalization. However, since each SAM update requires computing two gradients, its computational cost and training time are both doubled compared to standard empirical risk minimization (ERM). Recent state-of-the-arts reduce the fraction of SAM updates and thus accelerate SAM by switching between SAM and ERM updates randomly or periodically. In this paper, we design an adaptive policy to employ SAM based on the loss landscape geometry. Two efficient algorithms, AE-SAM and AE-LookSAM, are proposed. We theoretically show that AE-SAM has the same convergence rate as SAM. Experimental results on various datasets and architectures demonstrate the efficiency and effectiveness of the adaptive policy.Published as a conference paper at ICLR 2023 the squared stochastic gradient norm and model it by a normal distribution, whose parameters are estimated by exponential moving average. Experimental results on standard benchmark datasets demonstrate the superiority of the proposed policy.Our contributions are summarized as follows: (i) We propose an adaptive policy to use SAM or ERM update based on the loss landscape geometry. (ii) We propose an efficient algorithm, called AE-SAM (Adaptive policy to Employ SAM), to reduce the fraction of SAM updates. We also theoretically study its convergence rate. (iii) The proposed policy is general and can be combined with any SAM variant. In this paper, we integrate it with LookSAM (Liu et al., 2022) and propose AE-LookSAM. (iv) Experimental results on various network architectures and datasets (with and without label noise) verify the superiority of AE-SAM and AE-LookSAM over existing baselines. INTRODUCTION Despite great success in many applications (He et al., 2016;Zagoruyko & Komodakis, 2016;Han et al., 2017), deep networks are often over-parameterized and capable of memorizing all training data. The training loss landscape is complex and nonconvex with many local minima of different generalization abilities. Many studies have investigated the relationship between the loss surface's geometry and generalization performance (Hochreiter & Schmidhuber, 1994;McAllester, 1999;Keskar et al., 2017;Neyshabur et al., 2017;Jiang et al., 2020), and found that flatter minima generalize better than sharper minima (Dziugaite & Roy, 2017;Petzka et al., 2021;Chaudhari et al., 2017;Keskar et al., 2017;Jiang et al., 2020). Sharpness-aware minimization (SAM) (Foret et al., 2021) is the current state-of-the-art to seek flat minima by solving a min-max optimization problem. In the SAM algorithm, each update consists of two forward-backward computations: one for computing the perturbation and the other for computing the actual update direction. Since these two computations are not parallelizable, SAM doubles the computational overhead as well as the training time compared to empirical risk minimization (ERM). Several algorithms (Du et al., 2022a;Zhao et al., 2022b;Liu et al., 2022) have been proposed to improve the efficiency of SAM. ESAM (Du et al., 2022a) uses fewer samples to compute the gradients and updates fewer parameters, but each update still requires two gradient computations. Thus, ESAM does not alleviate the bottleneck of training speed. Instead of using the SAM update at every iteration, recent state-of-the-arts (Zhao et al., 2022b;Liu et al., 2022) proposed to use SAM randomly or periodically. Specifically, SS-SAM (Zhao et al., 2022b) selects SAM or ERM according to a Bernoulli trial, while LookSAM (Liu et al., 2022) employs SAM at every k step. Though more efficient, the random or periodic use of SAM is suboptimal as it is not geometry-aware. Intuitively, the SAM update is more useful in sharp regions than in flat regions. In this paper, we propose an adaptive policy to employ SAM based on the geometry of the loss landscape. The SAM update is used when the model is in sharp regions, while the ERM update is used in flat regions for reducing the fraction of SAM updates. To measure sharpness, we use Generalization and Flat Minima. The connection between model generalization and loss landscape geometry has been theoretically and empirically studied in (Keskar et al., 2017;Dziugaite & Roy, 2017;Jiang et al., 2020). Recently, Jiang et al. (2020) conducted large-scale experiments and find that sharpness-based measures (flatness) are related to generalization of minimizers. Although flatness can be characterized by the Hessian's eigenvalues (Keskar et al., 2017;Dinh et al., 2017), handling the Hessian explicitly is computationally prohibitive. To address this issue, practical algorithms propose to seek flat minima by injecting noise into the optimizers (Zhu et al., 2019;Zhou et al., 2019;Orvieto et al., 2022;Bisla et al., 2022), introducing regularization (Chaudhari et al., 2017;Zhao et al., 2022a;Du et al., 2022b), averaging model weights during training (Izmailov et al., 2018;He et al., 2019;Cha et al., 2021), or sharpness-aware minimization (SAM) (Foret et al., 2021;Kwon et al., 2021;Zhuang et al., 2022;Kim et al., 2022). SAM. The state-of-the-art SAM (Foret et al., 2021) and its variants (Kwon et al., 2021;Zhuang et al., 2022;Kim et al., 2022;Zhao et al., 2022a) search for flat minima by solving the following min-max optimization problem: min w max ≤ρ L(D; w + ),(1) where ρ > 0 is the radius of perturbation. The above can also be rewritten as min w L(D; w) + R(D; w), where R(D; w) ≡ max ≤ρ L(D; w + ) − L(D; w) is a regularizer that penalizes sharp minimizers (Foret et al., 2021). As solving the inner maximization in (1) exactly is computationally infeasible for nonconvex losses, SAM approximately solves it by first-order Taylor approximation, leading to the update rule: w t+1 = w t − η∇L(B t ; w t + ρ t ∇L(B t ; w t )),(2) where B t is a mini-batch of data, η is the step size, and ρ t = ρ ∇L(Bt;wt) . Although SAM has shown to be effective in improving the generalization of deep networks, a major drawback is that each update in (2) requires two forward-backward calculations. Specifically, SAM first calculates the gradient of L(B t ; w) at w t to obtain the perturbation, then calculates the gradient of L(B t ; w) at w t +ρ t ∇L(B t ; w t ) to obtain the update direction for w t . As a result, SAM doubles the computational overhead compared to ERM. Efficient Variants of SAM. Several algorithms have been proposed to accelerate the SAM algorithm. ESAM (Du et al., 2022a) uses fewer samples to compute the gradients and only updates part of the model in the second step, but still requires to compute most of the gradients. Another direction is to reduce the number of SAM updates during training. SS-SAM (Zhao et al., 2022b) randomly selects SAM or ERM update according to a Bernoulli trial, while LookSAM (Liu et al., 2022) employs SAM at every k iterations. Intuitively, the SAM update is more suitable for sharp regions than flat regions. However, the mixing policies in SS-SAM and LookSAM are not adaptive to the loss landscape. In this paper, we design an adaptive policy to employ SAM based on the loss landscape geometry. METHOD In this section, we propose an adaptive policy to employ SAM. The idea is to use ERM when w t is in a flat region, and use SAM only when the loss landscape is locally sharp. We start by introducing a sharpness measure (Section 3.1), then propose an adaptive policy based on this (Section 3.2). Next, we propose two algorithms (AE-SAM and AE-LookSAM) and study the convergence. SHARPNESS MEASURE Though sharpness can be characterized by Hessian's eigenvalues (Keskar et al., 2017;Dinh et al., 2017), they are expensive to compute. A widely-used approximation is based on the gradient magnitude diag([∇L(B t ; w t )] 2 ) (Bottou et al., 2018;Khan et al., 2018), where [v] 2 denotes the elementwise square of a vector v. As ∇L(B t ; w t ) 2 equals the trace of diag([∇L(B t ; w t )] 2 ), it is reasonable to choose ∇L(B t ; w t ) 2 as a sharpness measure. ∇L(B t ; w t ) 2 is also related to the gradient variance Var(∇L(B t ; w t )), another sharpness measure (Jiang et al., 2020). Specifically, Var(∇L(B t ; w t )) ≡ E Bt ∇L(B t ; w t )−∇L(D; w t ) 2 = E Bt ∇L(B t ; w t ) 2 − ∇L(D; w t ) 2 . (3) With appropriate smoothness assumptions on L, both SAM and ERM can be shown theoretically to converge to critical points of L(D; w) (i.e., ∇L(D; w) = 0) (Reddi et al., 2016;Andriushchenko & Flammarion, 2022). Thus, it follows from (3) that Var(∇L(B t ; w t )) = E Bt ∇L(B t ; w t ) 2 when w t is a critical point of L(D; w). Jiang et al. (2020) conducted extensive experiments and empirically show that Var(∇L(B t ; w t )) is positively correlated with the generalization gap. The smaller the Var(∇L(B t ; w t )), the better generalization is the model with parameter w t . This finding also explains why SAM generalizes better than ERM. Figure 1 shows the gradient variance w.r.t. the number of epochs using SAM and ERM on CIFAR-100 with various network architectures (experimental details are in Section 4.1). As can be seen, SAM always has a much smaller variance than ERM. Figure 2 shows the expected squared norm of the stochastic gradient w.r.t. the number of epochs on CIFAR-100. As shown, SAM achieves a much smaller E Bt ∇L(B t ; w t ) 2 than ERM. ADAPTIVE POLICY TO EMPLOY SAM As E Bt ∇L(B t ; w t ) 2 changes with t (Figure 2), the sharpness at w t also changes along the optimization trajectory. As a result, we need to estimate E Bt ∇L(B t ; w t ) 2 at every iteration. One can sample a large number of mini-batches and compute the mean of the stochastic gradient norms. However, this can be computationally expensive. To address this problem, we model ∇L(B t ; w t ) 2 with a simple distribution and estimate the distribution parameters in an online manner. Figure 3 on CIFAR-100 using ResNet-18 1 . As can be seen, the distribution follows a Bell curve. Figure 3( b) shows the corresponding quantile-quantile (Q-Q) plot (Wilk & Gnanadesikan, 1968). The closer is the curve to a line, the distribution is closer to the normal distribution. Figure 3 suggests that ∇L(B t ; w t ) 2 can be modeled 2 with a normal distribution N (µ t , σ 2 t ). We use exponential moving average (EMA), which is popularly used in adaptive gradient methods (e.g., RMSProp (Tieleman & Hinton, 2012), AdaDelta (Zeiler, 2012), Adam (Kingma & Ba, 2015)), to estimate its mean and variance: µ t = δµ t−1 + (1 − δ) ∇L(B t ; w t ) 2 ,(4)σ 2 t = δσ 2 t−1 + (1 − δ)( ∇L(B t ; w t ) 2 − µ t ) 2 ,(5) where δ ∈ (0, 1) controls the forgetting rate. Empirically, we use δ = 0.9. Since ∇L(B t ; w t ) is already available during training, this EMA update does not involve additional gradient calculations (the cost for the norm operator is negligible). Using µ t and σ 2 t , we employ SAM only at iterations where ∇L(B t ; w t ) 2 is relatively large (i.e., the loss landscape is locally sharp). Specifically, when ∇L(B t ; w t ) 2 ≥ µ t + c t σ t (where c t is a threshold), SAM is used; otherwise, ERM is used. When c t → −∞, it reduces to SAM; when c t → ∞, it becomes ERM. Note that during the early training stage, the model is still underfitting and w t is far from the region of final convergence. Thus, minimizing the empirical loss is more important than seeking a locally flat region. Andriushchenko & Flammarion (2022) also empirically observe that the SAM update is more effective in boosting performance towards the end of training. We therefore design a schedule that linearly decreases c t from λ 2 to λ 1 (which are pre-set values): c t = g λ1,λ2 (t) ≡ t T λ 1 + 1 − t T λ 2 , where T is the total number of iterations. The whole procedure, called Adaptive policy to Employ SAM (AE-SAM), is shown in Algorithm 1. AE-LookSAM. The proposed adaptive policy can be combined with any SAM variant. Here, we consider integrating it with LookSAM (Liu et al., 2022). When ∇L(B t ; w t ) 2 ≥ µ t + c t σ t , SAM is used and the update direction for w t is decomposed into two orthogonal directions as in LookSAM: (i) the ERM update direction to reduce training loss, and (ii) the direction that biases the model to a flat region. When ∇L(B t ; w t ) 2 < µ t + c t σ t , ERM is performed and the second direction of the previous SAM update is reused to compose an approximate SAM direction. The procedure, called AE-LookSAM, is also shown in Algorithm 1. Algorithm 1 AE-SAM and AE-LookSAM . Require: training set D, stepsize η, radius ρ; λ 1 and λ 2 for g λ1,λ2 (t); w 0 , µ −1 = 0, σ 2 −1 = e −10 , and α for AE-LookSAM; 1: for t = 0, . . . , T − 1 do 2: sample a mini-batch data B t from D; 3: compute g = ∇L(B t ; w t ); 4: update µ t by (4) and σ 2 t by (5); 5: compute c t = g λ1,λ2 (t); 6: if ∇L(B t ; w t ) 2 ≥ µ t + c t σ t then 7: g s = ∇L(B t ; w t + ρ∇L(B t ; w t )); 8: if AE-LookSAM: decompose g s as g v = g s − g gs g 2 g; 9: else: 10: if AE-SAM: g s = g; 11: if AE-LookSAM: g s = g + α g gv g v ; 12: end if 13: w t+1 = w t − ηg s ; 14: end for 15: return w T . CONVERGENCE ANALYSIS In this section, we study the convergence of any algorithm A whose update in each iteration can be either SAM or ERM. Due to this mixing of SAM and ERM updates, analyzing its convergence is more challenging compared with that of SAM. The following assumptions on smoothness and bounded variance of stochastic gradients are standard in the literature on non-convex optimization (Ghadimi & Lan, 2013;Reddi et al., 2016) and SAM (Andriushchenko & Flammarion, 2022;Abbas et al., 2022;Qu et al., 2022). Assumption 3.1 (Smoothness). L(D; w) is β-smooth in w, i.e., ∇L(D; w) − ∇L(D; v) ≤ β w − v . Assumption 3.2 (Bounded variance of stochastic gradients). E (xi,yi)∼D ∇ (f (x i ; w), y i ) − ∇L(D; w) 2 ≤ σ 2 . Let ξ t be an indicator of whether SAM or ERM is used at iteration t (i.e., ξ t = 1 for SAM, and 0 for ERM). For example, ξ t = I {w: ∇L(Bt;w) 2 ≥µt+ctσt} (w t ) for the proposed AE-SAM, and ξ t is sampled from a Bernoulli distribution for SS-SAM (Zhao et al., 2022b). Theorem 3.3. Let b be the mini-batch size. If stepsize η = 1 4β √ T and ρ = 1 T 1 4 , algorithm A satisfies min 0≤t≤T −1 E ∇L(D; w t ) 2 ≤ 32β (L(D; w 0 ) − EL(D; w T )) √ T (7 − 6ζ) + (1 + ζ + 5β 2 ζ)σ 2 b √ T (7 − 6ζ) ,(6) where ζ = 1 T T −1 t=0 ξ t ∈ [0, 1] is the fraction of SAM updates, and the expectation is taken over the random training samples. All proofs are in Appendix A. Note that a larger ζ leads to a larger upper bound in (6). When ζ = 1, the above reduces to SAM (Corollary A.2 of Appendix A.1). EXPERIMENTS In this section, we evaluate the proposed AE-SAM and AE-LookSAM on several standard benchmarks. As the SAM update doubles the computational overhead compared to the ERM update, the training speed is mainly determined by how often the SAM update is used. Hence, we evaluate efficiency by measuring the fraction of SAM updates used: %SAM ≡ 100 × #{iterations using SAM}/T . The total number of iterations, T , is the same for all methods. 4.1 CIFAR-10 AND CIFAR-100 Setup. In this section, experiments are performed on the CIFAR-10 and CIFAR-100 datasets (Krizhevsky & Hinton, 2009) using four network architectures: ResNet-18 (He et al., 2016), WideResNet-28-10 (denoted WRN-28-10) (Zagoruyko & Komodakis, 2016), PyramidNet-110 (Han et al., 2017), and ViT-S16 (Dosovitskiy et al., 2021). Following the setup in (Liu et al., 2022;Foret et al., 2021;Zhao et al., 2022a), we use batch size 128, initial learning rate of 0.1, cosine learning rate schedule, SGD optimizer with momentum 0.9 and weight decay 0.0001. The number of training epochs is 300 for PyramidNet-110, 1200 for ViT-S16, and 200 for ResNet-18 and WideResNet-28-10. 10% of the training set is used as the validation set. As in Foret et al. (2021), we perform grid search for the radius ρ over {0.01, 0.02, 0.05, 0.1, 0.2, 0.5} using the validation set. Similarly, α is selected by grid search over {0.1, 0.3, 0.6, 0.9}. For the c t schedule g λ1,λ2 (t), λ 1 = −1 and λ 2 = 1 for AE-SAM; λ 1 = 0 and λ 2 = 2 for AE-LookSAM. Baselines. The proposed AE-SAM and AE-LookSAM are compared with the following baselines: (i) ERM; (ii) SAM (Foret et al., 2021); and its more efficient variants including (iii) ESAM (Du et al., 2022a) which uses part of the weights to compute the perturbation and part of the samples to compute the SAM update direction. These two techniques can reduce the computational cost, but may not always accelerate SAM, particularly in parallel training (Li et al., 2020); (iv) SS-SAM (Zhao et al., 2022b), which randomly selects SAM or ERM according to a Bernoulli trial with success probability 0.5. This is the scheme with the best performance in (Zhao et al., 2022b); (v) Look-SAM (Liu et al., 2022) which uses SAM at every k = 5 steps. The experiment is repeated five times with different random seeds. Results. Table 1 shows the testing accuracy and fraction of SAM updates (%SAM). Methods are grouped based on %SAM. As can be seen, AE-SAM has higher accuracy than SAM while using only 50% of SAM updates. SS-SAM and AE-SAM have comparable %SAM (about 50%), and AE-SAM achieves higher accuracy than SS-SAM (which is statistically significant based on the pairwise t-test at 95% significance level). Finally, LookSAM and AE-LookSAM have comparable %SAM (about 20%), and AE-LookSAM also has higher accuracy than LookSAM. These improvements confirm that the adaptive policy is better. ImageNet Setup. In this section, we perform experiments on the ImageNet (Russakovsky et al., 2015), which contains 1000 classes and 1.28 million images. The ResNet-50 (He et al., 2016) is used. Following the setup in Du et al. (2022a), we train the network for 90 epochs using a SGD optimizer with momentum 0.9, weight decay 0.0001, initial learning rate 0.1, cosine learning rate schedule, and batch size 512. As in (Foret et al., 2021;Du et al., 2022a), ρ = 0.05. For the c t schedule g λ1,λ2 (t), λ 1 = −1 and λ 2 = 1 for AE-SAM; λ 1 = 0 and λ 2 = 2 for AE-LookSAM. k = 5 is used for LookSAM. Experiments are repeated with three different random seeds. Results. Table 2 shows the testing accuracy and fraction of SAM updates. As can be seen, with only half of the iterations using SAM, AE-SAM achieves comparable performance as SAM. Compared with LookSAM, AE-LookSAM has better performance (which is also statistically significant), verifying the proposed adaptive policy is more effective than LookSAM's periodic policy. ROBUSTNESS TO LABEL NOISE Setup. In this section, we study whether the more-efficient SAM variants will affect its robustness to training label noise. Following the setup in Foret et al. (2021), we conduct experiments on a corrupted version of CIFAR-10, with some of its training labels randomly flipped (while its testing set is kept clean). The ResNet-18 and ResNet-32 networks are used. They are trained for 200 epochs using SGD with momentum 0.9, weight decay 0.0001, batch size 128, initial learning rate 0.1, and cosine learning rate schedule. For LookSAM, the SAM update is used every k = 2 steps. 3 For AE-SAM and AE-LookSAM, we set λ 1 = −1 and λ 2 = 1 in their c t schedules g λ1,λ2 (t), such that their fractions of SAM updates (approximately 50%) are comparable with SS-SAM and LookSAM. Experiments are repeated with five different random seeds. Results. Table 3 shows the testing accuracy and fraction of SAM updates. As can be seen, AE-LookSAM achieves comparable performance with SAM but is faster as only half of the iterations use the SAM update. Compared with ESAM, SS-SAM, and LookSAM, AE-LookSAM performs better. The improvement is particularly noticeable at the higher noise levels (e.g., 80%). Figure 4 shows the training and testing accuracies with number of epochs at a noise level of 80% using ResNet-18 4 . As can be seen, SAM is robust to the label noise, while ERM and SS-SAM heavily suffer from overfitting. AE-SAM and LookSAM can alleviate the overfitting problem to a certain extent. AE-LookSAM, by combining the adaptive policy with LookSAM, achieves the same high level of robustness as SAM. In this experiment, we study the effects of λ 1 and λ 2 on AE-SAM. We use the same setup as in Section 4.1, where λ 1 and λ 2 (with λ 1 ≤ λ 2 ) are chosen from {0, ±1, ±2}. Results on AE-LookSAM using the label noise setup in Section 4.3 are shown in Appendix B.4. Figure 5 shows the effect on the fraction of SAM updates. For a fixed λ 2 , increasing λ 1 increases the threshold c t , and the condition ∇L(B t ; w t ) 2 ≥ µ t + c t σ t becomes more difficult to satisfy. Thus, as can be seen, the fraction of SAM updates is reduced. The same applies when λ 2 increases. A similar trend is also observed on the testing accuracy ( Figure 6). CONVERGENCE In this experiment, we study whether w t 's (where t is the number of epochs) obtained from AE-SAM can reach critical points of L(D; w), as suggested in Theorem 3.3. Figure 7 shows ∇L(D; w t ) 2 w.r.t. t for the experiment in Section 4.1. As can be seen, in all settings, ∇L(D; w t ) 2 converges to 0. In Appendix B.5, we also verify the convergence of AE-SAM's training loss on CIFAR-10 and CIFAR-100 (Figure 14), and that AE-SAM and SS-SAM have comparable convergence speeds (Figure 15), which agrees with Theorem 3.3 as both have comparable fractions of SAM updates (Table 1). CONCLUSION In this paper, we proposed an adaptive policy to employ SAM based on the loss landscape geometry. Using the policy, we proposed an efficient algorithm (called AE-SAM) to reduce the fraction of SAM updates during training. We theoretically and empirically analyzed the convergence of AE-SAM. Experimental results on a number of datasets and network architectures verify the efficiency and effectiveness of the adaptive policy. Moreover, the proposed policy is general and can be combined with other SAM variants, as demonstrated by the success of AE-LookSAM. E ∇L(D; w t ) 2 ≤ 32β (L(D; w 0 ) − EL(D; w T )) √ T (7 − 6ζ) + (1 + ζ + 5β 2 ζ)σ 2 b √ T (7 − 6ζ) ,(7) where ζ = 1 (2022)). Under Assumptions 3.1 and 3.2 for all t and ρ > 0, we have T T −1 t=0 ξ t ∈ [0, 1]. Lemma A.1 (Andriushchenko & FlammarionE∇L(B t ; w + ρ∇L(B t ; w)) ∇L(D; w) ≥ 1 2 − ρβ ∇L(D; w) 2 − β 2 ρ 2 σ 2 2b . (8) Proof. Let g t ≡ 1 b (xi,yi)∈Bt ∇ (f (x i ; w t ), y i ), h t ≡ 1 b (xi,yi)∈Bt ∇ (f (x i ; w t + ρg t ), y i ), andĝ t ≡ ∇L(D; w t ). By Taylor expansion and L(D; w) is β-smooth, we have L(D; w t+1 ) ≤L(D; w t ) +ĝ t (w t+1 − w t ) + β 2 w t+1 − w t 2 ≤L(D; w t ) − ηĝ t ((1 − ξ t )g t + ξ t h t ) + βη 2 2 (1 − ξ t )g t + ξ t h t 2 =L(D; w t )−η(1−ξ t )ĝ t g t −ηξ tĝ t h t + βη 2 2   (1−ξ t ) g t 2 +ξ t h t 2 +2ξ t (1 − ξ t )g t h t =0   (9) =L(D; w t ) − η(1 − ξ t )ĝ t g t − ηξ tĝ t h t + βη 2 2 (1 − ξ t ) g t 2 + ξ t h t 2 ,(10) where we have used ξ t (1 − ξ t ) = 0 as ξ t ∈ {0, 1}, ξ 2 t = ξ t , and (1 − ξ t ) 2 = 1 − ξ t to obtain (9). Taking expectation w.r.t. w t on both sides of (10), we have EL(D; w t+1 ) ≤ EL(D; w t )−η(1−ξ t )E ĝ t 2 −ηξ t Eĝ t h t + βη 2 (1 − ξ t ) 2 E g t 2 + βη 2 ξ t 2 E h t 2 .(11) Claim 1: E g t 2 = E g t −ĝ t 2 +E ĝ t 2 = σ 2 b +E ĝ t 2 , which follows from Assumption 3.2. Claim 2: E h t 2 ≤ 2(1 + ρ 2 β 2 ) σ 2 b − (1 − 2ρ 2 β 2 )E ĝ t 2 + 2Eĝ t h t , which is derived as follows: E h t 2 = E h t −ĝ t 2 − E ĝ t 2 + 2Eĝ t h t = 2E h t − g t 2 + 2E g t −ĝ t 2 − E ĝ t 2 + 2Eĝ t h t ≤ 2ρ 2 β 2 E g t 2 + 2σ 2 b − E ĝ t 2 + 2Eĝ t h t (12) ≤ 2ρ 2 β 2 σ 2 b + E ĝ t 2 + 2σ 2 b − E ĝ t 2 + 2Eĝ t h t (13) = 2(1 + ρ 2 β 2 ) σ 2 b − (1 − 2ρ 2 β 2 )E ĝ t 2 + 2Eĝ t h t ,(14) where (12) follows from h t − g t ≤ ρβ g t and Assumption 3.2, (13) follows from Claim 1. Substituting Claims 1 and 2 into (11), we obtain EL(D; w t+1 ) ≤ EL(D; w t ) − η (1 − ξ t ) E ĝ t 2 − ηξ t Eĝ t h t + βη 2 (1 − ξ t ) 2 σ 2 b + E ĝ t 2 + βη 2 ξ t 2 2(1 + ρ 2 β 2 ) σ 2 b − (1 − 2ρ 2 β 2 )E ĝ t 2 + 2Eĝ t h t (15) = EL(D; w t ) − η 1 − ξ t − βη(1 − ξ t ) 2 + βηξ t (1 − 2ρ 2 β 2 ) 2 E ĝ t 2 − ηξ t (1 − ηβ) Eĝ t h t + βη 2 (1 − ξ t ) 2 + βη 2 ξ t (1 + ρ 2 β 2 ) σ 2 b ≤ EL(D; w t ) − η 1 − ξ t − βη(1 − ξ t ) 2 + βηξ t (1 − 2ρ 2 β 2 ) 2 + ξ t (1 − ηβ) ( 1 2 − ρβ) E ĝ t 2 + βη 2 (1 − ξ t ) 2 + βη 2 ξ t (1 + ρ 2 β 2 ) + ηξ t (1 − ηβ) β 2 ρ 2 2 σ 2 b (16) ≤ EL(D; w t ) − η 1 − (1 + βη − 2ρβ) ξ t 2 − βη 2 E ĝ t 2 + η + ξ t (η + 2ηρ 2 β 2 + βρ 2 − ηβ 2 ρ 2 ) ηβσ 2 2b ,(17) where (15) follows from Claims 1 and 2, (16) follows from Lemma A.1 and 1 − ηβ > 0. As η < 1 4β , we have 1 + βη − 2ρβ ≤ 3/2 and βη < 1/4, thus, 1 − (1 + βη − 2ρβ) ξt 2 − βη 2 > 0. Summing over t on both sides of (17) and rearranging, we obtain min 0≤t≤T −1 E ĝ t 2 ≤ L(D; w 0 ) − EL(D; w T ) η T −1 t=0 1 − (1 + βη − 2ρβ) ξt 2 − βη 2 + T −1 t=0 η + ξ t (η + ηρ 2 β 2 + βρ 2 ) T −1 t=0 1 − (1 + βη − 2ρβ) ξt 2 − βη 2 βσ 2 2b = L(D; w 0 ) − EL(D; w T ) T η(1 − γζ 2 − βη 2 ) + T (η + ηκζ + βρ 2 ζ)βσ 2 2bT (1 − γζ 2 − βη 2 ) (18) = L(D; w 0 ) − EL(D; w T ) T η(1 − γζ 2 − βη 2 ) + (1 + κζ + 4β 2 ζ)ηβσ 2 2b(1 − γζ 2 − βη 2 ) = L(D; w 0 ) − EL(D; w T ) T η(1 − γζ 2 − βη 2 ) + (1 + κζ + 4β 2 ζ)σ 2 8b √ T (1 − γζ 2 − βη 2 ) (19) ≤ 32β (L(D; w 0 ) − EL(D; w T )) √ T (7 − 6ζ) + (1 + ζ + 5β 2 ζ)σ 2 b √ T (7 − 6ζ) ,(20) where γ = 1 + βη − 2ρβ ≤ 3/2, κ = 1 + ρ 2 β 2 , ρ 2 = 1/ √ T , and ζ = 1 T T −1 t=0 ξ t ∈ [0, 1]. We thus finish the proof. min 0≤t≤T −1 E ∇L(D; w t ) 2 ≤ 32β (L(D; w 0 ) − EL(D; w T )) √ T + (2 + 5β 2 )σ 2 b √ T .(21)E ∇L(D; w t ) 2 ≤ 32β(L(D; w 0 ) − EL(D; w T )) √ T b(7 − 6ζ) + (1 + ζ + 5β 2 ζ)σ 2 √ T b(7 − 6ζ) ,(22) where ζ = 1 T T −1 t=0 ξ t ∈ [0, 1]. Proof. It follows from (18) that min 0≤t≤T −1 E ĝ t 2 ≤ L(D; w 0 ) − EL(D; w T ) T η(1 − γζ 2 − βη 2 ) + ηβ(1 + κζ + 4β 2 ζ)σ 2 2b(1 − γζ 2 − βη 2 ) (23) ≤ 4β(L(D; w 0 ) − EL(D; w T )) √ T b( 7 8 − 3 4 ζ) + (1 + ζ + 5β 2 ζ)σ 2 8 √ T b( 7 8 − 3ζ 4 ) (24) = 32β(L(D; w 0 ) − EL(D; w T )) √ T b(7 − 6ζ) + (1 + ζ + 5β 2 ζ)σ 2 √ T b(7 − 6ζ) .(25)∇L(D; w t ) 2 ≤ L(D; w 0 ) − L(D; w T ) T η 1 − βη 2 − βρζ ,(26)where ζ = 1 T T −1 t=0 ξ t ∈ [0, 1]. Lemma A.5 (Lemma 7 in Andriushchenko & Flammarion (2022)). Let L(D; w) be a β-smooth function. For any ρ > 0, we have ∇L(D; w) ∇L(D; w + ρ∇L(D; w)) ≥ (1 − ρβ) ∇L(D; w) 2 . Proof of Theorem A.4. Let g t ≡ ∇L(D; w t ) and h t ≡ ∇L(D; w t + ρ∇L(D; w t )) be the update direction of ERM and SAM, respectively. By Taylor expansion and L(D; w) is β-smooth, we have L(D; w t+1 ) ≤ L(D; w t ) + g t (w t+1 − w t ) + β 2 w t+1 − w t 2 ≤ L(D; w t ) − ηg t ((1 − ξ t )g t + ξ t h t ) + βη 2 2 (1 − ξ t )g t + ξ t h t 2 = L(D; w t )−η(1−ξ t ) g t 2 −ηξ t g t h t + βη 2 2   (1−ξ t ) g t 2 +ξ t h t 2 +2ξ t (1−ξ t )g t h t =0   (28) = L(D; w t ) − η 1 − ξ t − βη(1 − ξ t ) 2 g t 2 + βη 2 ξ t 2 h t 2 − ηξ t g t h t ,(29) where we have used ξ t (1 − ξ t ) = 0 as ξ t ∈ {0, 1}, ξ 2 t = ξ t , and (1 − ξ t ) 2 = 1 − ξ t to obtain (28). As h t 2 = h t − g t 2 − g t 2 + 2g t h t , it follows from (29) that L(D; w t+1 ) =L(D; w t )−η 1−ξ t − βη(1 − ξ t ) 2 g t 2 + βη 2 ξ t 2 h t −g t 2 − g t 2 + 2g t h t −ηξ t g t h t ≤L(D; w t )−η 1−ξ t − βη(1 − ξ t ) 2 + βηξ t 2 g t 2 + βη 2 ξ t 2 h t − g t 2 − η(1 − βη)ξ t g t h t ≤L(D; w t )−η 1−ξ t − βη(1−ξ t ) 2 + βηξ t 2 g t 2 + β 3 η 2 ρ 2 ξ t 2 g t 2 − η(1 − βη)ξ t g t h t (30) =L(D; w t ) − η 1 − ξ t − βη(1 − ξ t ) 2 + βηξ t 2 + β 3 ηρ 2 ξ t 2 + (1 − βη)(1 − βρ)ξ t g t 2 (31) =L(D; w t ) − η 1 − βη(1 − ξ t ) 2 + βηξ t 2 + β 3 ηξ t ρ 2 2 − βηξ t − βρξ t + β 2 ηρξ t g t 2 ≤L(D; w t ) − η 1 − βη 2 − βρξ t g t 2 ,(32) where we have used h t − g t 2 = ∇L(D; w t + ρ∇L(D; w t )) − ∇L(D; w t ) 2 ≤ β 2 ρ 2 ∇L(D; w t ) 2 = β 2 ρ 2 g t 2 to obtain (30), and Lemma A.5 to obtain (31). Summing over t from t = 0 to T − 1 on both sides of (32) and rearranging, we have T −1 t=0 η 1 − βη 2 − βρξ t g t 2 ≤ L(D; w 0 ) − L(D; w T ).(33) As ρ < 1 2β and η < 1 β , it follows that 1 − βη 2 − βρξ t > 0 for all t. Thus, (33) implies min 0≤t≤T −1 g t 2 ≤ L(D; w 0 ) − L(D; w T ) T −1 t=0 η 1 − βη 2 − ξ t βρ = L(D; w 0 ) − L(D; w T ) T η 1 − βη 2 − βρζ ,(34) where ζ = 1 T T −1 t=0 ξ t ∈ [0, 1] and we finish the proof. B ADDITIONAL EXPERIMENTAL RESULTS B.1 DISTRIBUTION OF STOCHASTIC GRADIENT NORMS Figure 8 shows the distributions of stochastic gradient norms for ResNet-18, WRN-28-10 and PyramidNet-110 on CIFAR-10 and CIFAR-100. As can be seen, the distribution follows a Bell curve in all settings. Figure 9 shows the Q-Q plots. We can see that the curves are close to the lines. B.2 EFFECT OF k ON LOOKSAM In this experiment, we demonstrate that LookSAM is sensitive to the choice of k. Table 4 shows the testing accuracy and fraction of SAM updates when using LookSAM on noisy CIFAR-10, with k ∈ {2, 3, 4, 5} and the ResNet-18 model. As can be seen, k = 2 yields much better performance than k ∈ {3, 4, 5}, particularly at higher noise levels (e.g., 80%). Figure 10 (resp. 11) shows the curves of accuracies at noise levels of 20%, 40%, 60%, and 80% with ResNet-18 (resp. ResNet-32). As can be seen, in all settings, AE-LookSAM is as robust to label noise as SAM. In this experiment, we study the effects of λ 1 and λ 2 on AE-LookSAM. Experiment is performed on CIFAR-10 with label noise (80% noisy labels), using the same setup as in Section 4.3. Figure 12 shows the effects of λ 1 and λ 2 on the fraction of SAM updates. Again, as in Section 4.4, for a fixed λ 2 , increasing λ 1 always reduces the fraction of SAM updates. Figure 13 shows the effects of λ 1 and λ 2 on the testing accuracy of AE-SAM. As can be seen, the observations are similar to those in Section 4.4. Figure 13: Effects of λ 1 and λ 2 on testing accuracy of CIFAR-10 (with 80% noisy labels). Note that the curves for λ 2 ∈ {−2, −1} overlap completely with that of λ 2 = 1. Best viewed in color. B.5 ADDITIONAL CONVERGENCE RESULTS ON CIFAR-10 AND CIFAR-100 Figure 14 shows convergence of AE-SAM's training loss on the CIFAR-10 and CIFAR-100 datasets. As can be seen, AE-SAM achieves convergence with various network architectures. Figure 15 shows the training losses w.r.t. the number of epochs for AE-SAM and SS-SAM. As can be seen, AE-SAM and SS-SAM converge with comparable speeds, which agrees with Theorem 3.3 as both of them have comparable fractions of SAM updates (Table 1). Figure 1 : 1Variance of gradient on CIFAR-100. Best viewed in color. (a) shows ∇L(B t ; w t ) 2 of 400 mini-batches at different training stages(epoch = 60, 120, and 180) Figure 2 : 2Squared stochastic gradient norms E B ∇L(B; w t ) 2 on CIFAR-100. Best viewed in color. Figure 3 : 3Stochastic gradient norms { L(B t ; w t ) 2 : B t ∼ D} of ResNet-18 on CIFAR-100 are approximately normally distributed. Best viewed in color. Figure 4 : 4Accuracies with number of training epochs on CIFAR-10 (with 80% noise labels) using ResNet-18. Best viewed in color. Figure 5 : 5Effects of λ 1 and λ 2 on fraction of SAM updates using ResNet-18. Best viewed in color. Figure 6 : 6Effects of λ 1 and λ 2 on testing accuracy using ResNet-18. Best viewed in color.4.4 EFFECTS OF λ 1 AND λ 2 Figure 7 : 7Squared gradient norms of AE-SAM with number of epochs. Best viewed in color. Theorem 3. 3 . 3Let b be the mini-batch size. If η Corollary A. 2 . 2Let b be the mini-batch size. If η Corollary A. 3 . 3Let b be the mini-batch size. If η = Figure 8 : 8Distributions of stochastic gradient norms on CIFAR-10 (top) and CIFAR-100 (bottom). Best viewed in color. Figure 9 : 9Q-Q plots of stochastic gradient norms on CIFAR-10 (top) and CIFAR-100 (bottom). Best viewed in color. Figure 10 : 10Accuracies with number of epochs on CIFAR-10 with 20%, 40%, 60%, and 80% noise level using ResNet-18. Best viewed in color. ) 80% (Testing). Figure 11 : 11Accuracies with number of epochs on CIFAR-10 with 20%, 40%, 60%, and 80% noise level using ResNet-32. Best viewed in color.B.4 EFFECTS OF λ 1 AND λ 2 ON AE-LOOKSAM Figure 12 : 12Effects of λ 1 and λ 2 on fraction of SAM updates on CIFAR-10 (with 80% noisy labels). Best viewed in color. Figure 14 : 14Training loss of AE-SAM with number of epochs on CIFAR-10 and CIFAR-100. Best viewed in color. Figure 15 : 15Training losses of AE-SAM and SS-SAM with number of epochs on CIFAR-10. Note that the two curves almost completely overlap. Best viewed in color. Table 1 : 1Means and standard deviations of testing accuracy and fraction of SAM updates (%SAM) on CIFAR-10 and CIFAR-100. Methods are grouped based on %SAM. The highest accuracy in each group is underlined; while the highest accuracy for each network architecture (across all groups) is in bold.SAM(Foret et al., 2021) 97.30 ±0.10 100.0 ±0.0 84.46 ±0.05 100.0 ±0.0 ESAM(Du et al., 2022a) 97.81 ±0.01 100.0 ±0.0 85.56 ±0.05 100.0 ±0.0CIFAR-10 CIFAR-100 Accuracy %SAM Accuracy %SAM Table 2 : 2Means and standard deviations of testing accuracy and fraction of SAM updates (%SAM) on ImageNet using ResNet-50. Methods are grouped based on %SAM. The highest accuracy in each group is underlined; while the highest across all groups is in bold.Accuracy %SAM ERM 77.11 ±0.14 0.0 ±0.0 SAM (Foret et al., 2021) 77.47 ±0.12 100.0 ±0.0 ESAM (Du et al., 2022a) 77.25 ±0.75 100.0 ±0.0 SS-SAM (Zhao et al., 2022b) 77.38 ±0.06 50.0 ±0.0 AE-SAM 77.43 ±0.06 49.4 ±0.0 LookSAM (Liu et al., 2022) 77.13 ±0.09 20.0 ±0.0 AE-LookSAM 77.29 ±0.08 20.3 ±0.0 Table 3 : 3Testing accuracy and fraction of SAM updates on CIFAR-10 with different levels of label noise. The best accuracy is in bold and the second best is underlined.noise = 20% noise = 40% noise = 60% noise = 80% accuracy %SAM accuracy %SAM accuracy %SAM accuracy %SAM ResNet-18 ERM 87.92 0.0 70.82 0.0 49.61 0.0 28.23 0.0 SAM (Foret et al., 2021) 94.80 100.0 91.50 100.0 88.15 100.0 77.40 100.0 ESAM (Du et al., 2022a) 94.19 100.0 91.46 100.0 81.30 100.0 15.00 100.0 SS-SAM (Zhao et al., 2022b) 90.62 50.0 77.84 50.0 61.18 50.0 47.32 50.0 LookSAM (Liu et al., 2022) 92.72 50.0 88.04 50.0 72.26 50.0 69.72 50.0 AE-SAM 92.84 50.0 84.17 50.0 73.54 49.9 65.00 50.0 AE-LookSAM 94.34 49.9 91.58 50.0 87.85 50.0 76.90 50.0 ResNet-32 Sepp Hochreiter and Jürgen Schmidhuber. Simplifying neural nets by discovering flat minima. In Neural Information Processing Systems, 1994. Pavel Izmailov, Dmitrii Podoprikhin, Timur Garipov, Dmitry Vetrov, and Andrew Gordon Wilson. Averaging weights leads to wider optima and better generalization. In Uncertainty in Artificial Intelligence, 2018. 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A PROOFS A.1 PROOF OF THEOREM 3.3 Table 4 : 4Effects of k in LookSAM on CIFAR-10 with different levels of label noise using ResNet-18. noise = 20% noise = 40% noise = 60% noise = 80% k accuracy %SAM accuracy %SAM accuracy %SAM accuracy %SAM B.3 MORE RESULTS ON ROBUSTNESS TO LABEL NOISE2 92.72 50.0 88.04 50.0 72.26 50.0 69.72 50.0 3 89.07 33.3 75.38 33.3 63.79 33.3 53.87 33.3 4 89.00 25.0 74.12 25.0 58.17 25.0 52.28 25.0 5 88.57 20.0 73.90 20.0 56.80 20.0 51.82 20.0 Results on other architectures and CIFAR-10 are shown inFigures 8 and 9of Appendix B.1. 2 Note that normality is not needed in the theoretical analysis (Section 3.3). The performance of LookSAM can be sensitive to the value of k.Table 4of Appendix B.2 shows that using k = 2 leads to the best performance in this experiment. Results for other noise levels and ResNet-32 are shown inFigures 10 and 11of Appendix B.3, respectively. 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[ "A further 'degree of freedom' in the rotational evolution of stars", "A further 'degree of freedom' in the rotational evolution of stars" ]	[ "V Holzwarth \nSchool of Physics and Astronomy\nUniversity of St Andrews\nNorth Haugh\nKY16 9SSSt Andrews, FifeScotland\n", "M Jardine \nSchool of Physics and Astronomy\nUniversity of St Andrews\nNorth Haugh\nKY16 9SSSt Andrews, FifeScotland\n" ]	[ "School of Physics and Astronomy\nUniversity of St Andrews\nNorth Haugh\nKY16 9SSSt Andrews, FifeScotland", "School of Physics and Astronomy\nUniversity of St Andrews\nNorth Haugh\nKY16 9SSSt Andrews, FifeScotland" ]	[]	Observational and theoretical investigations provide evidence for non-uniform spot and magnetic flux distributions on rapidly rotating stars, which have a significant impact on their angular momentum loss rate through magnetised winds. Supplementing the formalism ofMacGregor & Brenner (1991)with a latitude-dependent magnetised wind model, we analyse the effect of analytically prescribed surface distributions of open magnetic flux with different shapes and degrees of nonuniformity on the rotational evolution of a solar-like star. The angular momentum redistribution inside the star is treated in a qualitative way, assuming an angular momentum transfer between the rigidly-rotating radiative and convective zones on a constant coupling timescale of 15 Myr; for the sake of simplicity we disregard interactions with circumstellar disks. We find that non-uniform flux distributions entail rotational histories which differ significantly from those of classical approaches, with differences cumulating up to 200% during the main sequence phase. Their impact is able to mimic deviations of the dynamo efficiency from linearity of up to 40% and nominal dynamo saturation limits at about 35 times the solar rotation rate. Concentrations of open magnetic flux at high latitudes thus assist in the formation of very rapidly rotating stars in young open clusters, and ease the necessity for a dynamo saturation at small rotation rates. However, since our results show that even minor amounts of open flux at intermediate latitudes, as observed with Zeeman-Doppler imaging techniques, are sufficient to moderate this reduction of the AM loss rate, we suggest that non-uniform flux distributions are a complementary rather than an alternative explanation for very rapid stellar rotation.	10.1051/0004-6361:20053111	[ "https://export.arxiv.org/pdf/astro-ph/0508643v1.pdf" ]	14,684,110	astro-ph/0508643	4d9a132f3b485dda2136b23f3f7e76d47c876180	A further 'degree of freedom' in the rotational evolution of stars February 19, 2022 V Holzwarth School of Physics and Astronomy University of St Andrews North Haugh KY16 9SSSt Andrews, FifeScotland M Jardine School of Physics and Astronomy University of St Andrews North Haugh KY16 9SSSt Andrews, FifeScotland A further 'degree of freedom' in the rotational evolution of stars February 19, 2022Received ; acceptedarXiv:astro-ph/0508643v1 30 Aug 2005 Astronomy & Astrophysics manuscript no. repa (DOI: will be inserted by hand later)Stars: rotation -Stars: winds, outflows -Stars: magnetic fields -Stars: mass-loss -Stars: evolution -MHD Observational and theoretical investigations provide evidence for non-uniform spot and magnetic flux distributions on rapidly rotating stars, which have a significant impact on their angular momentum loss rate through magnetised winds. Supplementing the formalism ofMacGregor & Brenner (1991)with a latitude-dependent magnetised wind model, we analyse the effect of analytically prescribed surface distributions of open magnetic flux with different shapes and degrees of nonuniformity on the rotational evolution of a solar-like star. The angular momentum redistribution inside the star is treated in a qualitative way, assuming an angular momentum transfer between the rigidly-rotating radiative and convective zones on a constant coupling timescale of 15 Myr; for the sake of simplicity we disregard interactions with circumstellar disks. We find that non-uniform flux distributions entail rotational histories which differ significantly from those of classical approaches, with differences cumulating up to 200% during the main sequence phase. Their impact is able to mimic deviations of the dynamo efficiency from linearity of up to 40% and nominal dynamo saturation limits at about 35 times the solar rotation rate. Concentrations of open magnetic flux at high latitudes thus assist in the formation of very rapidly rotating stars in young open clusters, and ease the necessity for a dynamo saturation at small rotation rates. However, since our results show that even minor amounts of open flux at intermediate latitudes, as observed with Zeeman-Doppler imaging techniques, are sufficient to moderate this reduction of the AM loss rate, we suggest that non-uniform flux distributions are a complementary rather than an alternative explanation for very rapid stellar rotation. Introduction In the presence of open magnetic fields the angular momentum (AM) loss of a star through winds and outflows is significantly enhanced since the tension of bent field lines effectively prolongs the lever arm of the associated torque (Schatzman 1962). Whereas during the pre-main sequence (PMS) phase the rotational evolution of a star is dominated by its changing stellar structure and magnetic interaction with a circumstellar accretion disk, its rotation during the main sequence (MS) phase is mainly determined through braking by magnetised winds (Belcher & MacGregor 1976). Theoretical studies of magnetised winds go back to classical approaches of Weber & Davis (1967, hereafter WD) and Mestel (1968). At high rotation rates the magnetic field adds considerably to the acceleration of the outflow through magneto-centrifugal driving (Michel 1969), which makes magnetised winds intrinsically latitude-dependent. The total AM loss rate is consequently susceptible to variations of the surface magnetic field (Solanki et al. 1997;Holzwarth 2005) and atmospheric field topology (Mestel & Spruit 1987;Kawaler 1988). More recently, multi-dimensional MHD-simulations have been accomplished to study structural and temporal wind properties like coronal mass ejections in more detail (e.g., Sakurai 1985; Send offprint requests to: V Holzwarth, e-mail: [email protected] Keppens & Goedbloed 1999; the extensive computational requirements render them however less attractive for studies concerning the rotational evolution of stars. Skumanich (1972) analysed the rotation and chromospheric emission of cool stars in different evolutionary stages, and found that their rotation rate (as well as Ca II luminosity) is about proportional to the inverse square-root of their age, that is Ω ∝ t −1/2 . Presuming a constant moment of inertia and a WD braking law, the Skumanich relation implies a linear relationship between the rotation rate of a star and its characteristic magnetic field strength,B ∝ Ω n de with dynamo efficiency n de ≃ 1. However, Saar (1991) found that the magnetic flux rather than the field strength is increasing linearly with the rotation rate, which is now commonly adopted (at slow rotation rates). The consequence of a continuously increasing stellar field strength is a very efficient magnetic braking of rapidly rotating stars. In fact, too efficient, since observations of young open clusters reveal significant numbers of stars with rotational velocities up to v sin i ∼ 200 km/s (Stauffer et al. 1997), whose existence is, with initial rotation rates of young T Tauri stars being observationally well constrained (Bouvier et al. 1993), difficult to explain in the framework of magnetic braking without any mechanism which moderates the AM loss rate at higher rotation rates. Respective studies therefore frequently presume a saturation of the AM loss beyond a limiting rotation rate, whose value is very susceptible to model assumptions like the internal AM redistribution; whereas some investigations require low ( 20 Ω ⊙ ) saturation limits (Keppens et al. 1995; Barnes & Sofia 1996;Bouvier et al. 1997;Krishnamurthi et al. 1997), others argue for higher ( 40 Ω ⊙ ) values (Soderblom et al. 1993;Collier Cameron & Jianke 1994). Saturation limits The saturation of the total AM loss rate is often ascribed to the underlying dynamo mechanisms in the convective envelope, because current dynamo theories anticipate a moderation and eventually saturation of the amplification process (n de → 0) when the back-reaction of strong magnetic fields suppresses plasma motions and inhibits its further increase (e.g., Rüdiger & Kichatinov 1993). However, the various theoretical models are as yet unable to provide consistent or explicit values for the critical rotation rate or field strength at which this occurs. The concept of a saturation of magnetic activity is strengthened by empirical activity-rotation relationships, which reveal (for rotation periods longer than a few days) close correlations between the rotation rate and the strength of activity proxies like magnetically induced chromospheric and coronal emission (Noyes et al. 1984;Vilhu 1984;Mathioudakis et al. 1995;Stauffer et al. 1997;Pizzolato et al. 2003). In rapidly rotating stars several activity signatures are found to saturate: the chromospheric UV emission below rotation periods P ∼ 3 d (Vilhu 1984), and the EUV and (soft) X-ray emission for P 2 d (Stauffer et al. 1997). However, the variation of photometric light curves, associated with the presence of dark spots in the stellar photosphere, is found to increase for even shorter rotation periods, down to P ∼ 0.35 − 0.5 d, for which the chromospheric and coronal emission are already saturated (O'Dell et al. 1995;Messina et al. 2001); for an antithetical point of view see Krishnamurthi et al. (1998). The saturation of activity signatures is not unambiguously indicative of a saturation of the dynamo mechanisms in the convective envelope, since emission processes are liable to further rotation-dependent effects like a reduction of the X-ray emitting volume through the centrifugal stripping of hot coronal loops, or the shift of coronal loop temperatures into different emission regimes as the effective gravitation and pressure scale hight change with the rotation rate (Unruh & Jardine 1997;Jardine & Unruh 1999). The conjecture that changes of the atmospheric emission are not necessarily correlated with the (sub-)photospheric magnetic activity is supported, for example, by observations of the ultra-fast rotator VXR45a (V370 Vel, P = 0.223 d), whose X-ray emission is below the typical level of X-ray saturated stars (Marino et al. 2003), whereas its brightness surface maps are still very similar to those of more slowly rotating stars (Marsden et al. 2004;Vrielmann & Hussain 2005). The large range of rotation rates (∼ 10−70 Ω ⊙ ) in which observed activity signatures are found to saturate raises the question whether these phenomena reflect the actual behaviour of the underlying dynamo processes, in particular beyond which critical rotation rate their efficiency breaks down. Since the range of observed saturation limits practically covers the one of suggested AM loss limits, a definite justification of the latter in terms of a dynamo saturation is rather precarious. Surface magnetic flux distributions Doppler imaging (DI) observations of rapidly rotating stars yield non-uniform surface brightness distributions, where, in contrast to the case of the Sun, dark spots are not only located in equatorial regions, but also at intermediate and polar latitudes (Strassmeier 2002, and references therein). Theoretical models considering the formation of magnetic features at higher latitudes involve the pre-eruptive poleward deflection of magnetic flux inside the convection zone by the Coriolis force (Schüssler & Solanki 1992), and/or its post-eruptive poleward transport through meridional motions (Schrijver & Title 2001). Backed by these observational and theoretical results, Solanki et al. (1997) investigated the influence of a bi-modal magnetic field distribution on the rotational evolution of cool stars. They found that a concentration of magnetic flux at very high latitudes reduces the total AM loss rate as efficiently as a dynamo saturation limit at ∼ 20 Ω ⊙ . Based on their findings, they question the concept of a dynamo saturation at low rotation rates and argue instead for a saturation above 50 Ω ⊙ ; a similar though more qualitative argument has also been discussed by Buzasi (1997). The work of Solanki et al. was focused on flux concentrations around the pole, assuming that the observed brightness distributions indicate likely locations of open field lines along which a stellar wind can escape. But the bare existence of starspots does not a priori imply information about the associated magnetic field topology, that is neither about the existence nor about the amount of open magnetic flux. Zeeman-Doppler imaging (ZDI) observations in contrast directly confirm the magnetic origin of the dark features and enable a determination of the magnetic field at the stellar surface . Such field distributions serve as boundary conditions for field extrapolation techniques (Altschuler & Newkirk 1969), which reveal large-scale magnetic field topologies and consequently the distribution of closed and open magnetic field lines (e.g., Jardine et al. 2002a,b). In the case of the rapidly rotating star LQ Hya (P = 1.6 d) the latitudinal distributions of open flux show that large amounts are located at intermediate and high latitudes (McIvor et al. 2004). A similar result is found in the case of AB Dor (P = 0.51 d) for observations between 1995 and 2003 Donati et al. 1999Donati et al. , 2003: Figure 1 shows the (normalised) cumulated open magnetic flux, integrated from the equator to a co-latitude θ. The curves indicate that 50% of the total open magnetic flux of a hemisphere is on average located below/above ∼ 45 • , with variations of ±15 • depending on the observational epoch. These results show that the distribution of open magnetic flux can be distinctively different from spot distributions determined from surface brightness maps alone. The present work extends the study of Solanki et al. (1997) and investigates the impact of latitude-dependent flux distributions on the rotational evolution of solar-type stars in more Donati et al. 1999Donati et al. , 2003 detail. Using the magnetic wind model described in Holzwarth (2005), we quantify the influence of prescribed flux distributions with different degrees of non-uniformity on the rotational evolution of stars, and verify their importance by comparing their impact with the influence of other magnetic field-related model parameters. Model setup The stellar structure of the low-mass star considered here consists of an outer convective envelope and an inner radiative core, each taken to be in solid-body rotation with possibly different rotation rates. With J = IΩ being the AM, I the moment of inertia, and Ω the rotation rate, the rotational evolution of the star is determined by the set of coupled differential equations d lg Ω e d lg t = − d lg I e d lg t + t J e J e,Ṁ +J e,RI +J W (1) d lg Ω c d lg t = − d lg I c d lg t + t J c J c,Ṁ +J c,RI ,(2) which comprise changing moments of inertia and the AM transfer across appropriate boundaries; indices 'e' and 'c' denote quantities of the envelope and of the core, respectively. Internal angular momentum redistribution The stellar structure is taken to be spherically symmetric. The temporal change of the moments of inertia is determined by an evolutionary sequence of stellar models of a 1 M ⊙ star, which was generated with an updated version of the stellar evolution code of Kippenhahn et al. (1967). Figure 2 shows the evolution of the (outer) radii, masses, and moments of inertia of both the convective envelope and the radiative core. As the core-envelope interface recedes outward, the dynamical stability properties at the base of the convection zone change and originally unstable mass settles down on the radiative core. This mass transfer is accompanied by an AM transfer, J e,Ṁ = − 2 3 r 2 c Ω eṀc = −J c,Ṁ .(3) The temporal change of the core mass,Ṁ c , is determined from the sequence of stellar models (Fig. 2b); since the mass loss of the envelope through the stellar wind is negligibly small, it iṡ M e = −Ṁ c . Hydrodynamic and hydromagnetic interaction at the boundary between the radiative and convective regions are expected to entail a coupling between the core and the envelope. Different coupling mechanisms based on magneto-viscous interaction or large-scale internal circulations have been investigated by Pinsonneault et al. (1989); Charbonneau & MacGregor (1993); Bouvier et al. (1997); Allain (1998) and references therein. Here, a more qualitative core-envelope coupling model is used, following the parametric approach of MacGregor & Brenner (1991) and Keppens et al. (1995). Based on the Rayleigh criterion a dynamical stable state of rotation inside a star requires 1 the increase of the specific AM with increasing radius, d(Ωr 2 )/dr > 0. Given that the rotation rate of the convective envelope is braked and smaller than the rotation rate of the core, the sta-4 V Holzwarth and M Jardine: A further 'degree of freedom' in the rotational evolution of stars bility condition is violated at the core-envelope interface and a rotational instability sets in, transferring AM, J e,RI = ∆J τ c = −J c,RI ,(4) from the core to the envelope to eliminate the differential rotation between the two regions. The AM required to equalise the two rotation rates, ∆J = I e J c − I c J e I e + I c = I e I c I e + I c (Ω c − Ω e ) ,(5) is transferred on a timescale, τ c , which is supposed to characterise the various visco-magnetic coupling mechanisms. The coupling time quantises the possibility to deposit AM in the core during the PMS phase and its retarded transfer to the envelope in the course of the MS evolution. For the sake of simplicity the value of τ c is taken to be constant during the entire rotational evolution of the star. Latitude-dependent magnetised stellar winds The AM loss rate of the convective envelope through a latitudedependent magnetised wind is determined following the approach of Holzwarth (2005). The stationary, polytropic stellar wind is assumed to be symmetric with regard to both the rotation axis and the equator. The poloidal component of the magnetic field is taken to be radial, with field lines forming spirals around the rotation axis on coni with constant opening angles. The whole stellar surface contributes to the wind, without 'dead zones' retaining mass from escaping (cf. Mestel & Spruit 1987). The magnetic wind properties are determined through the radial magnetic field strength, B r,0 (t, θ, Ω e ) = r 0 (t 0 ) r 0 (t) 2 B < + ∆B (Ω e ) · f (θ) ,(6) given at a reference level, r 0 , close to the stellar surface. The time-dependent radius ratio is to ensure that the total magnetic flux, Φ 0 (Ω e ) = 2π 0 π 0 B r,0 r 2 0 sin θ dθ dφ = 4πr 2 0 (t 0 )B r,0 (t 0 , Ω e ) ,(7) only depends on the rotation rate of the star (Saar 1991), and an arbitrary reference time, t 0 . The efficiency of the underlying dynamo mechanism is expected to increase with the rotation rate of the convection zone. The field strength variation, ∆B, is therefore determined in a way that the surface averaged field strength, B r,0 (t 0 , Ω e ) = π/2 0 B r,0 sin θ dθ ,(8) obeys the functional behaviour shown in Fig. 3. The nonuniform flux distributions, superposed on a constant 'background' field, B < , are characterised by an enhancement of magnetic flux at non-equatorial latitudes ( Fig. 4): Fig. 3. Dependence of the surface averaged radial magnetic field strength,B r,0 (t 0 ), on the rotation rate, Ω e , of the convective envelope. For large rotation rates the dependencies follow approximately linear (solid), sub-linear (long dashed), or super-linear (short dashed) power laws, ∝ Ω e /Ω e,⊙ n Ω , with n Ω = 1, 0.75, and 1.25, respectively. The deviation from the power law at small rotation rates is due to the constant background field strength. -Coronal Hole (CH) model The total, surface integrated AM loss rate, B < B > CH θ 0 ∆B B < B > LB θ 0f (θ) = 1 for 0 ≤ θ ≤ θ 0 0 for θ 0 < θ < 90 • (9) -Latitudinal Belt (LB) model f (θ) = cos 16 (θ − θ 0 )(10)J W =J WD 3 2 π/2 0 r Ā r A 4 ρ Ā ρ A v r,Ā v r,A sin 3 θdθ ,(11) is expressed in terms of the plasma density, ρ A , and radial flow velocity, v r,A , at the Alfvénic point, r A , where the flow velocity of the wind equals the Alfvén velocity (cf. Holzwarth 2005). Whereas ρ A , v r,A , and r A are functions of the co-latitude, θ, and subject to the latitude-dependent field distributions, Eq. (6), the respective quantities,ρ A ,v r,A , andr A , are determined in the equatorial plane (θ = π/2) using the surface averaged field strength, Eq. (8). The latter quantities determine the AM loss rate following the approach of Weber & Davis (1967), J WD = 8π 3 Ωr 4 Aρ Avr,A ,(12) which is based on the simplifying assumption that the equatorial wind structure can be generalised to all latitudes. Reference model parameters The wind structure is determined through boundary conditions prescribed at the reference level r 0 (t) = r e (t) + ∆r, over the range of co-latitudes 0 < θ ≤ π/2, measured from the stellar north pole. r e (≡ R * ) is the time-dependent outer radius of the convective envelope (Fig. 2a), and ∆r = 0.1 R ⊙ a constant radial offset to locate the reference level of the wind at the base of the corona. For cool stars other than the Sun thermal wind properties are poorly constrained by observations; for possible constraints resulting from relationships between the temperature and density of closed coronal loops and rotation/Rossby number see, for example, Jordan & Montesinos (1991); Ivanova & Taam (2003). In the following solar-like values are assumed for the temperature, T 0 = 2 · 10 6 K, and (particle) density, n 0 = 10 8 cm −3 . The entropy change of the wind with increasing distance from the star is quantified through the polytropic index, Γ = 1.15. The surface averaged magnetic field strength, defined in Eq. (8), is taken to follow the quasi-linear power law shown in Fig. 3 (solid line), that isB r,0 = (1 + 1.5 (Ω e /Ω ⊙ )) G, with Ω ⊙ = 2.8 · 10 −6 s −1 . This is in agreement with observations of rapidly rotating stars, which show magnetic field strengths up to about two orders of magnitude larger than in the case of the Sun . Helioseismological observations show that the rotation rate in the solar interior is roughly uniform (Thompson et al. 2003). The rotation rate of the solar radiative core and convection zone are within ∼ 4% about Ω ⊙ = 2.8 · 10 −6 s −1 . The present Sun thus constrains the coupling timescale to the effect that the value of τ c ought to achieve isorotation within a few percent at the solar age, t ⊙ ≈ 4.7 Gyr. Using the reference model parameters described above, we accomplished simulations with different coupling timescales to determine the relative deviation, (Ω c − Ω e )/Ω c , from isorotation at solar age (Table 1). For τ c ≃ 15 Myr, a value similar to the one adopted by MacGregor & Brenner (1991) or Keppens et al. (1995), the deviation is in accord with the observational constrains given by the Sun. Results We determine the rotational evolution of 1 M ⊙ stars with initial rotation rates Ω 0 = 5·10 −6 , 2·10 −5 , and 6·10 −5 s −1 from the age 1.6 Myr onward. The rotation periods, between 1.2 d and 15 d, approximately span the observed range of rotation periods of young ( 5 Myr) T Tauri stars (Bouvier et al. 1993). In their initial state the convective and radiative zones are taken to be in isorotation, that is Ω c = Ω e = Ω 0 . The rotational histories determined with the AM loss ratė J W =J WD , that is using the WD approach, Eq. (12), are used as reference cases (Fig. 6). In the course of the PMS evolution the rotation rate of the envelope is dominated by its decreasing moment of inertia and the AM loss which goes with the mass settling down onto the radiative core, whereas during the MS phase only the magnetic braking and the internal coupling are relevant. Non-uniform flux distributions We determine the rotational history of stars subject to the Latitudinal Belt (LB) and Coronal Hole (CH) flux distributions with 50%-open flux levels located at different latitudes. At first, the non-uniform flux patterns are taken to be stationary (θ 50% = const.) during the entire evolution, to separate their influence from other rotation-dependent effects. An accumulation of open magnetic flux at high (low) latitudes causes, with respect to an uniform flux distribution, a reduction (enhancement) of the AM loss rate through the stellar wind. At the age of the present Sun the resulting deviations of the stellar rotation rate are found to cover a range of values between about −40% and 200% different than the respective reference cases (Fig. 7). The relative deviations show two distinctive regimes, corresponding to the PMS and the MS phase. For the deviations to increase (decrease) with time the net AM loss rate of the envelope has to be smaller (larger) than in the reference case. Since we consider the difference between two rotational histories, the rotation-independent mechanisms cancel out, so that only the magnetic braking and the internal coupling are relevant. The magnetic braking is a direct consequence of the present stellar rotation and causes an immediate AM loss of the envelope. The internal coupling, in contrast, is proportional to the differential rotation and during the early evolution rather inefficient, due to the initial isorotation (Fig. 8). In the PMS phase the difference between the AM loss of the envelope and its internal AM gain is consequently large and the deviations of the rotation rate quickly increasing. After arrival on the MS the differential rotation is adjusted to the new situation and the tapping of the AM reservoir of the more rapidly rotating core now replenishes most of the AM loss carried away by the magne- Relative timescales (with respect to respective reference cases) on which the rotation rate of the envelope changes due to the magnetic braking (solid) and internal coupling (dashed), for Ω 0 = 2 · 10 −5 s −1 and a LB distribution with θ 50% = 30 and 75 • . tised wind; the further increase of the deviations is respectively weaker. The actual spin-down time scale of the envelope is longer than the braking time scale, because the AM gain of the envelope through the internal coupling follows closely its AM loss through the magnetic braking (Fig. 8). Note that in the course of the evolution the latter is found to converge toward the reference value. Since the magnetic field strength decreases with the rotation rate the relative contribution of the (latitude-dependent) magneto-centrifugal driving to the overall wind acceleration becomes smaller and the one of the (latitudeindependent) thermal driving larger. The difference between the Weber & Davis and the present non-uniform wind approach thus becomes smaller and the spin-down timescale similar. The non-uniform flux distributions with θ 50% = 60 • have nominally the same 50%-flux level as the uniform flux distribution of the reference cases. The deviations of the relative rotation rate of ∼ 10% in the case of the LB model indicate that the usage of θ 50% only allows for a rudimentary classification of flux patterns. By comparing the rotational histories resulting from different flux distributions with equivalent 50%-open flux levels, we find that over the range θ 50% = 10 − 60 • the peaked Latitudinal Belt and the bi-modal Coronal Hole model yield rotational evolutions which are within a tolerance of 15% consistent. In the special case of a dipolar field distribution (with f (θ) = cos θ and θ 50% = 45 • , Fig. 5), the correspondency is found to be good within about 10%. Observations show that surface distributions of starspots depend on the stellar rotation rate, with spots being preferentially located at higher latitudes the faster the star rotates. We investigate this aspect by assuming a rotation-dependent latitude of the 50%-flux level, which follows the power law θ 50% = 90 • (Ω/Ω ⊙ ) n rd , with n rd = −0.25, −0.5, and −1; this simple relationship is only used to examine the basic effects and not meant to reproduce any complex observed or theoretically derived relations. To avoid unrealistic high flux concentrations in the vicinity of the poles or the equator, the 50%flux latitudes are constrained to the range 15 • ≤ θ 50% ≤ 60 • (cf. Solanki et al. 1997). Figure 9 shows the relative deviations of the rotation rates from the reference values in the case of a rotation-dependent LB distribution; for the CH model the results are very similar. The major consequence of the rotationdependence is the enhanced stellar spin-up and spin-down during the PMS and the late MS phase, respectively. The evolutionary stage at which the rotational history is altered by the shifting of magnetic flux to lower latitudes depends however on the actual functional dependence of θ 50% (Ω). Comparison with dynamo efficiency and saturation To examine whether the impact of non-uniform flux distributions is significant with respect to other magnetic fieldrelated model assumptions, we compare the deviations described above with those obtained by successively changing the functional dependence of the magnetic field strength on the stellar rotation rate in the framework of the reference cases (i.e., with uniform flux distributions). A linear dynamo efficiency is an acceptable approximation in the case of slowly rotating stars, but it is found to fail for fast rotators. We analyse the sensitivity of the rotational evolution on changes of the dynamo efficiency by re-calculating the reference cases, but now subject to non-linear magnetic fieldrotation relations, ∆B ∝ Ω n de e , with n de 1 (cf. Fig. 3). During the MS phase, the resulting relative deviations cover a large range between −50% and 150% (Fig. 10). Sub-linear (superlinear) dynamo efficiencies imply a weaker (stronger) increase of the magnetic field strength with rotation rate and consequently a moderation (enhancement) of the magnetic braking. Their influence is therefore qualitatively similar to rotationdependent concentrations of magnetic flux at high and low latitude, respectively. To determine explicit relations between the dynamo efficiency and the location of the 50%-open flux level, we generate grids of rotational histories with different n de -and θ 50% -values, and associate corresponding values by comparing the relative derivations of the rotation rate from the reference case 2 . Figure 11 shows quadratic fits of the resulting grid-based relations for the LB and CH flux distribution. The relationships show that non-uniform flux distributions can imitate a large range (here: between −45% and 15%) of dynamo efficiencies and, consequently, if not taken properly into account, conceal them from an observational determination. The limiting field strength beyond which dynamo saturation occurs is referred to in terms of a critical rotation rate, Ω sat , that is ∆B(Ω e > Ω sat ) = ∆B(Ω sat ). We repeat our calculations of the reference cases under the assumption of saturation rotation rates in the range of Ω sat = 2 − 80 Ω ⊙ (see Fig. 12 for examples). The saturated magnetic braking causes an enhanced spinup during the PMS phase, higher rotation rates on the ZAMS, and a weaker spin-down during the MS phase until the rotation rate descends into the regime of the non-saturated dynamo. We compare the rotational histories resulting from a saturated magnetic braking with those resulting from non-uniform flux distributions to associate 50%-open flux level with corresponding dynamo saturation rates (Fig. 13). For given initial stellar rotation rates and flux distributions, the curves 3 describe nominal dynamo saturations which cause similar rotational histories like a non-uniform flux distribution with a 50%-open flux level at the respective co-latitude. If the actual dynamo saturation rate is larger than then nominal one, the rotational evolution is likely to be dominated by the impact of the non-uniform flux distribution. If, in contrast, the actual dynamo saturation rate is smaller than the nominal one, then the behaviour of the magnetic dynamo is expected to dominate the evolution. A concentration of magnetic flux at high latitudes of rapidly rotating stars can therefore imitate a rotational evolution like a dynamo saturation limit of about 30 − 40 Ω ⊙ . In case a considerable amount of open flux is located at intermediate or low latitudes, as indicated by recent field reconstructions based on ZDI observations, the nominal saturation level would be rather high ( 60 Ω ⊙ ) and therefore probably beyond the actual dyresponds to an unconstrained dynamo with non-uniform flux distribution. Fig. 13. Relationships between the nominal saturation rotation rate, Ω sat , and the latitude, θ 50% , of the 50%-open flux level for stars with initial rotation rates Ω 0 = 6 · 10 −5 (solid), 2 · 10 −5 (long dashed), and 5 · 10 −6 s −1 . In contrast to the Coronal Hole model (thin lines), Latitudinal Belt distributions (thick lines) with θ 50% 53 • have no corresponding dynamo saturation. The shaded stripes indicate the rotational regimes where the chromospheric and coronal emission (dark grey) and the variation of the photometric light curve (light grey) are observationally found to saturate. namo saturation limit. In this case non-uniform flux distributions alone could not account for the very high rotation rates of young stars. Discussion Our results show that a concentration of open magnetic flux at high latitudes of active stars yields rotational histories which deviate up to about 100% (early MS phase) to 200% (late MS phase) from the rotation rates obtained in the case of uniform surface fields. The reduction of the AM loss rate entails a quicker increase of the stellar rotation rate during the PMS spin-up and a moderated spin-down during the MS evolution. The higher the (initial) rotation rate, the larger the AM loss due to magnetised winds, and the larger the susceptibility to variations in the open flux distribution. The influence on the rotational history is a cumulative effect, which depends on the evolutionary stage of the star but eventually also on the applied model assumptions. Model considerations In our wind model the poloidal magnetic field component is prescribed to be radial (cf. Holzwarth 2005), whereas a fully consistent treatment of the multi-dimensional problem, including the trans-field component of the equation of motion, is found to show a collimation of open field lines toward the rotation axis with increasing distance from the star (e.g., Sakurai 1985). The influence of this effect on our results is difficult to quantify, because investigations focused on this phenomenon V Holzwarth and M Jardine: A further 'degree of freedom' in the rotational evolution of stars 9 are usually limited to particular and illustrative cases. Another approach to include a non-uniform magnetic field topology is the stellar wind model of Mestel & Spruit (1987), which incorporates in the vicinity of the stellar surface polar 'wind' and equatorial 'dead zones', the latter preventing plasma escaping from the star and thus reducing both the mass and AM loss rate. Whereas in this model the extent of the dead zone depends on the rotation rate of the star, the magnetic field distribution at the surface is prescribed to be dipolar. Based on the formalism of Mestel (1968) and Mestel & Spruit (1987), Kawaler (1988) derived a parametrised description of stellar AM loss rates through stellar winds, which also comprises qualitative variations of the magnetic field topology; a definite association of non-uniform magnetic flux distributions with the respective model parameter is however missing. Although it remains to be investigated how more complex flux distributions alter the magnetic field topology of stellar winds including effects like collimation or dead zones, we deem these aspects to be less crucial since our analyses are based on relative deviations between two rotational histories. However, more detailed studies concerning this hypothesis are required. The surface averaged magnetic field strengths, covering here roughly two orders of magnitude over the range of relevant rotation rates, are consistent with observations both in the case of the Sun and rapidly rotating stars (e.g., . The localised peak field strengths are, depending on the location of the 50%-open flux level and the non-uniformity of the flux distribution, even larger (up to the order of kilo-Gauss), which is also in agreement with observations. The even flux distribution of the Coronal Hole model and the peaked distribution of the Latitudinal Belt model simulate complementary non-uniformities and allow us to assess the sensitivity of the results beyond our 50%-open flux classification. Whereas both cases are consistent within 15%, we expect this tolerance level to be larger for more complex flux distributions. But in view of the current observational and model limitations a more sophisticated classification scheme appears yet to be inappropriate. The thermal wind properties are described by solar-like values, which are taken to be independent of the rotation rate, the stellar latitude, and the evolutionary stage of the star. Although there are indications for a rotation-dependence of the coronal temperature and density (e.g., Jordan & Montesinos 1991;Ivanova & Taam 2003), it is difficult to derive accurate wind parameters, since observational wind signatures are veiled by the outshining coronal emission. The polytropic index of Γ = 1.15 implies an efficient heating and thermal driving of the stellar wind; its value lies well in between those of similar studies (e.g., Sakurai 1985;Keppens & Goedbloed 2000). It has been chosen to ensure stationary wind solutions at all latitudes even for small rotation rates, when the magneto-centrifugal driving is inherently small. This implies that the contribution of the latitude-independent thermal driving is relatively large compared with the latitude-dependent magneto-centrifugal driving. In case of a weaker thermal driving, either because of cooler coronae or a smaller energy flux in the wind, the impact of non-uniform flux distributions is expected to be even stronger than described above. If, in turn, magnetised winds turn out to be hotter, latitude-dependent effects may be diminished (given that the thermal wind properties themselves are not latitudedependent). The rapidly rotating stellar core represents an AM reservoir which is tapped by the envelope during the MS evolution. The efficiency of the internal coupling, quantified through a characteristic timescale (here determined to be τ c = 15 Myr), is under debate, with previous studies investigating values over a very large range, from very short (solid-body rotation; e.g., Bouvier et al. 1997) to very long (essentially decoupled differential rotation; Jianke & Collier Cameron 1993;Allain 1998) timescales. The results are typically not fully consistent with all observational constraints given by the rotational distributions of stars in young open clusters of different age, which may point out possible deficiencies of previous models. In this respect the non-uniformity of surface magnetic fields presents an additional and so far rather ignored 'degree of freedom', which may help to match theoretical and observed rotational histories of stars. Relevance to dynamo efficiency and saturation Our comparison of the impact of non-uniform flux distributions with the influence of other magnetic model parameters underlines its importance. The dynamo efficiency, that is the dependence of the (here, open) flux generation on stellar rotation, is generally accepted to increase with the rotation rate. Analytical relations based on the theory of αΩ-dynamos (e.g., between the dynamo number and the Rossby number; Noyes et al. 1984;Montesinos et al. 2001) as well as empirical relationships between the rotation rate and magnetic activity signatures like the coronal EUV and X-ray emission (Mathioudakis et al. 1995;Hempelmann et al. 1995;Pizzolato et al. 2003) imply power law-dependencies over a large range of rotation rates. Skumanich (1972) found the evolution of chromospheric activity to decrease with time following Ω ∝ t −1/2 . This relation implies a linear increase of the characteristic stellar magnetic field strength with rotation rate; an example for the commonly agreed concept that the decrease of magnetic activity with age reflects the evolution of the stellar AM and rotation rate. According to Saar (1991), it is however more the magnetic flux which increases linearly, instead of the field strength. The linear dynamo efficiency is widely used in studies about the rotational evolution of stars (e.g., Keppens et al. 1995;Krishnamurthi et al. 1997). Here, for deviations from the linearity between ±30% the rotation rates on the early MS are found to differ between 150% and −50%. Our analysis has shown that non-uniform flux distributions can easily produce similar deviations. A large fraction of open flux in the vicinity of the stellar poles can therefore compensate the influence of a super-linear dynamo efficiency, feigning a less efficient (e.g., linear) dependence on the rotation rate. A concentration of magnetic flux at high latitudes has qualitatively a similar influence on the rotational history as a dynamo saturation. Since both effects entail a reduction of the AM loss rate, their synergistic action is expected to enable even higher rotation rates. Figure 13 allows for estimates (based on the comparison of exclusive effects) of the principal contribution to the overall reduction of the AM loss. If the saturation limit of the chromospheric and/or coronal emission (P sat,UV/X ∼ 3 . . . 1.5 d) is representative of dynamo saturation, then we expect non-uniform flux distributions to contribute only marginally to the reduction of the AM loss. If, however, the photometric saturation limit (P sat,phot 12 h) reflects the saturation of the dynamo mechanism, then the rotational evolution of most stars is dominated by the actual non-uniformity of the surface flux distribution. But the saturation of indirect activity signatures does not a priori indicate a saturation of the underlying dynamo processes in the convective envelope, since the former also depend on, for example, atmospheric properties and radiation processes which follow different functional behaviours (e.g., Unruh & Jardine 1997). Solanki et al. (1997) applied a bi-modal flux distribution, characterised through polar flux concentrations, to the rotational evolution of stars. For somewhat different model parameters than ours they found a resemblance between the rotational histories subject to their coronal hole model and a dynamo saturation limit of 20 Ω ⊙ , respectively. Since this saturation limit is of the order required to explain the presence of rapidly rotating stars (∼ 10 − 20 Ω ⊙ ), they question the necessity of a dynamo saturation at low rotation rates and argue in favour for values 50 Ω ⊙ . Whereas our investigation confirms their results in principle, we find that reductions of the AM loss rate resulting from high-latitude flux concentrations are smaller; according to our model a flux distribution with, for example, θ 50% ≈ 15 • is equivalent to a dynamo saturation limit of about 40 Ω ⊙ . The difference with respect to the Solanki et al. value (∼ 20 Ω ⊙ ) may be due to different model assumptions and parameters. Since our value is still below their supposed dynamo saturation limit of ∼ 50 Ω ⊙ , a strict concentration of magnetic flux around the poles would be able to dominate the rotational evolution of stars. Observations indicate, however, that the 50%-open flux level is likely located at somewhat lower latitudes. Surface brightness maps occasionally show and dark elongated features reaching down to intermediate latitudes (Strassmeier 2002, and references therein), which shift the mean 50%-flux level (averaged over longitude and evolutionary timescales) equatorwards. But the presence of magnetic flux in the form of dark spots is a priori not equivalent with the presence of open magnetic field lines. A more striking constraint arises from ZDI observations, which in combination with field extrapolation techniques allow for the reconstruction of the magnetic field topology and the determination of the actual surface and latitudinal distribution of open magnetic fields. Although the number of detailed observations is yet rather small, first results show that the principal part of (detectable) open flux is apparently located at intermediate latitudes (McIvor et al. 2004), which motivated our Latitudinal Belt model. The cumulated flux distributions of AB Dor (cf. Fig. 1) show that the average 50%-flux level is indeed located between 30 • θ 50% 60 • . In this range of values our equivalent saturation limit for rapid rotators is 60 Ω ⊙ and the impact of the non-uniform flux distribution consequently smaller than the influence of the dynamo saturation at ∼ 50 Ω ⊙ supposed by Solanki et al. (1997). In this sense, we consider the effect of non-uniform flux distributions more as a complementary rather than an alternative mechanism for the formation of rapid rotators, which is however based on observationally verifiable principles. Observational constraints Compared to the large number of observed surface brightness distributions of rapidly rotating stars (Strassmeier 2002), ZDI observations of actual magnetic flux distributions are yet rather sparse. A larger database in this field is required to put tighter constraints on possible open flux distributions, that is the qualitative and quantitative description of its non-uniformity and dependence on the stellar rotation rate and the evolutionary phase of the star (including its dependence on stellar mass). Whereas, for example, brightness distributions indicate a clear poleward displacement of starspots with increasing rotation rate, for open flux a similar relationship is yet not available. ZDI observations are handicapped through insufficient signal levels from dark surface regions, so that an unambiguous determination of the flux topology inside dark polar caps (i.e., whether it is mainly unipolar and open or more multi-polar and closed) is yet hardly possible (Donati & Brown 1997;McIvor et al. 2003). If the field reconstructions in the case of AB Dor and LQ Hya prove to be representative and characteristic for rapid rotators, then we consider the contributions from high-latitude regions to the total AM loss rate to be of minor importance for the rotational evolution of rapid rotators. Owing to their qualitatively similar behaviour it is questionable whether the impact of non-uniform flux distributions can be observationally separated from the influence of a dynamo saturation. This potentially limits the usefulness of rotation rate distributions as tests of dynamo theories, unless they are supplemented by more detailed information about the structure of stellar magnetic fields. A larger observational database is required to refine the rotation rate beyond which the location of the 50%-open flux level is high enough to cause a discernible impact on rotational distributions of cluster stars. Supposed that the limiting rotation rates for high-latitude 50%-flux levels and dynamo saturation are sufficiently low and high, respectively, it may be possible to distinguish characteristic signatures of non-uniform flux distributions in the (differential) distribution of intermediate rotators with rotation rates in the domain constrained by the two limiting values. An important role may fall to theoretical models concerning the pre-eruptive evolution and post-eruptive transport of magnetic flux to high latitudes in rapidly rotating young stars (e.g., Granzer et al. 2000;Mackay et al. 2004), since they allow for a verification of our insight into stellar magnetic properties by means of activity proxies like (the rotational modulation of) coronal X-ray and chromospheric UV emission (Vilhu et al. 1993). Observations in the UV/EUV spectral range may also be used to refine our assumptions about the thermal wind properties, comprising its latitudinal and rotational dependence, because the total AM strongly relies on the stellar mass loss rate and on the acceleration of the wind through thermal driving. Conclusion Non-uniform magnetic flux distributions have a significant impact on the rotational evolution of stars, mimicking the effect of a large range of dynamo efficiencies and saturation limits. They present an additional degree of freedom in the modelling of stellar rotational histories, which can generate differences cumulating up to 200%. Neglecting their effect implies considerable uncertainties in other magnetic field-related model parameters such as the dynamo efficiency of up to about 40%. Although our results are, in principle, in agreement with those of Solanki et al. (1997), we find the effect of non-uniform flux distributions to be less efficient than in their investigation; whereas they find that a concentration of magnetic flux at polar latitudes imitates a nominal dynamo saturation limit at 20Ω ⊙ , we find values of about 35Ω ⊙ . Non-uniformities in the form of strong flux concentrations at high latitudes efficiently reduce the AM loss through magnetised winds, entailing high stellar rotation rates. Since magnetic field distributions reconstructed on the basis of ZDI observations however indicate a considerable amount of open flux at intermediate latitudes, the anticipated reduction of the AM loss is expected to be smaller than implied by frequent DI observations of high-latitude starspots and polar caps. The influence of non-uniform flux distributions alone thus appears to be insufficient to explain the existence of very rapid rotators, but their significant moderation of the AM loss rate makes the requirements for a dynamo saturation less stringent, enabling saturation limits 40 Ω ⊙ . Fig. 1 . 1Cumulative open magnetic flux distributions, Φ θ /Φ 0 (cf. Sect. 2.2), on the visible hemisphere of AB Dor (P = 0.51 d), based on ZDI observations secured between 1995-2003 Fig. 2 . 2Evolution of the outer radii, r e/c (Panel a), masses, M e/c (Panel b), and moments of inertia, I e/c (Panel c) of the convective envelope (solid) and radiative core (dashed) of a 1 M ⊙ star. For t 1.5 Myr the star is fully convective. Its MS phase starts at t ≃ 40 Myr; the vertical dashed line marks the age of the Sun. Fig. 4 . 4Latitude-dependent magnetic field distributions, f (θ), close to the stellar surface. The non-uniformity of the Coronal Hole (CH) and Latitudinal Belt (LB) model is parametrised through the co-latitude θ 0 . Dashed lines show the lower, B < , and upper, B > = B < + ∆B, field strengths of each field distribution. Fig. 5 . 5Whereas the Latitudinal Belt model closely resembles what is found from field extrapolations based on ZDI images(McIvor et al. 2004), the Coronal Hole model is motivated by numerous DI surface brightness maps (e.g.,Strassmeier 2002). Different degrees of high-latitude flux concentrations are realised by changing the co-latitude θ 0 of the analytically prescribed functions f(Fig. 5). We classify non-uniform flux distributions by the location of their 50%-open flux level, θ 50% , where the cumulated open magnetic flux, Φ θ = 4π θ π/2 B r,0 r 2 0 sin θ ′ dθ ′ , reaches half of the total value, Cumulative open magnetic flux, Φ θ /Φ 0 , of the Coronal Hole (solid, for θ 0 = 45 • ) and Latitudinal Belt (dotted, for θ 0 = 50 • ) model. Respective curves for a constant (long dashed) and dipolar (short dashed) field distribution are show for comparison; see also Fig. 1 for observational results. Fig. 7 . 7Relative deviations of the envelope rotation rate for the Latitudinal Belt (left) and Coronal Hole (right) flux distributions.For 50%-open flux levels at low latitudes (i.e., large co-latitudes θ 50% , labels) the deviations are negative (dashed), and for high latitudes positive (solid). Fig. 6 . 6Rotational evolution of the convective (thick) and radiative (thin) zone of stars subject to an AM loss rate according toWeber & Davis (1967). The dashed line indicates the break-up rotation rate, for which the co-rotation radius equals the actual stellar radius, Ω bu = GM * /r 3 * . The cross marks the rotation rate of the present Sun. Fig. 8 . 8Fig. 8. Relative timescales (with respect to respective reference cases) on which the rotation rate of the envelope changes due to the magnetic braking (solid) and internal coupling (dashed), for Ω 0 = 2 · 10 −5 s −1 and a LB distribution with θ 50% = 30 and 75 • . Fig. 9 . 9Relative deviations (dashed lines indicate negative values) in the case of rotation-dependent 50%-open flux levels θ 50% = 90 • (Ω/Ω ⊙ ) n rd , with n rd = −0.25, −0.5, and −1 (labels). Its value is constrained to the range 15 • ≤ θ 50% ≤ 60 • . Fig. 10 . 10Relative deviations in the case of non-linear dynamo efficiencies, ∆B ∝ Ω n de . For n de = 0.7, 0.8, and 0.9 (solid, top down) the deviations are positive, and for n de = 1.3, 1.2, and 1.1 (dashed, top down) negative. Fig. 11 .Fig. 12 . 1112Relationships between the dynamo efficiency, n de , and the latitude, θ 50% , of the 50%-open flux level in the case of the Latitudinal Belt (LB, top) and Coronal Hole (CH, bottom) distributions. The initial stellar rotation rates are Ω 0 = 6 · 10 −5 (solid, crosses), 2·10 −5 (long dashed, triangles), and 5·10 −6 s −1 (short dashed, rhombs). Relative deviations of the envelope rotation rate in the case of a dynamo saturation beyond Ω sat = 10 (solid), 20 (long dashed), 30 (short dashed), 40 (dashed-dotted), and 50 Ω ⊙ (dotted). 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[ "Defect identification based on first-principles calculations for deep level transient spectroscopy", "Defect identification based on first-principles calculations for deep level transient spectroscopy" ]	[ "Darshana Wickramaratne \nMaterials Department\nUniversity of California Santa\n93106-5050BarbaraCaliforniaUSA\n", "Cyrus E Dreyer \nDepartment of Physics and Astronomy\nStony Brook University\n11794-3800Stony BrookNew YorkUSA\n\nCenter for Computational Quantum Physics\nFlatiron Institute\n162 5 th Avenue10010New YorkNew YorkUSA\n", "Bartomeu Monserrat \nCavendish Laboratory\nTCM Group\nUniversity of Cambridge\nJ. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom\n", "Jimmy-Xuan Shen \nDepartment of Physics\nUniversity of California Santa\n93106-9530BarbaraCaliforniaUSA\n", "John L Lyons \nCenter for Computational Materials Science\nUS Naval Research Laboratory\n20375WashingtonDCUSA\n", "Audrius Alkauskas \nCenter for Physical Sciences and Technology (FTMC)\nLT-10257VilniusLithuania\n", "Chris G Van De Walle \nMaterials Department\nUniversity of California Santa\n93106-5050BarbaraCaliforniaUSA\n" ]	[ "Materials Department\nUniversity of California Santa\n93106-5050BarbaraCaliforniaUSA", "Department of Physics and Astronomy\nStony Brook University\n11794-3800Stony BrookNew YorkUSA", "Center for Computational Quantum Physics\nFlatiron Institute\n162 5 th Avenue10010New YorkNew YorkUSA", "Cavendish Laboratory\nTCM Group\nUniversity of Cambridge\nJ. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom", "Department of Physics\nUniversity of California Santa\n93106-9530BarbaraCaliforniaUSA", "Center for Computational Materials Science\nUS Naval Research Laboratory\n20375WashingtonDCUSA", "Center for Physical Sciences and Technology (FTMC)\nLT-10257VilniusLithuania", "Materials Department\nUniversity of California Santa\n93106-5050BarbaraCaliforniaUSA" ]	[]	Deep level transient spectroscopy (DLTS) is used extensively to study defects in semiconductors. We demonstrate that great care should be exercised in interpreting activation energies extracted from DLTS as ionization energies. We show how first-principles calculations of thermodynamic transition levels, temperature effects of ionization energies, and nonradiative capture coefficients can be used to accurately determine actual activation energies that can be directly compared with DLTS. Our analysis is illustrated with hybrid functional calculations for two important defects in GaN that have similar thermodynamic transition levels, and shows that the activation energy extracted from DLTS includes a capture barrier that is temperature dependent, unique to each defect, and in some cases large in comparison to the ionization energy. By calculating quantities that can be directly compared with experiment, first-principles calculations thus offer powerful leverage in identifying the microscopic origin of defects detected in DLTS.PACS numbers: 71.55.-i, 72.20.Jv, 84.37.+q Point defects and impurities are present in all semiconductors. They can act as recombination centers that lower the efficiency of optoelectronic devices, or as carrier traps in electronic devices such as transistors. Microscopic identification of the detrimental defects is crucial in order to mitigate their impact. Deep level transient spectroscopy (DLTS) is a powerful technique for determining the properties of defects; from an analysis of electrical measurements on a pn junction or Schottky diode, properties such as the position of the defect level within the band gap, electrical nature (donor or acceptor), density, and carrier capture cross section of specific defects can be obtained.[1][2][3]Translating this wealth of information to a microscopic identification of a given defect requires comparison with theoretical or computational models, and first-principles calculations based on density functional theory (DFT) have proven very helpful.[4][5][6][7][8]One of the key quantities measured in DLTS is the activation energy for carrier emission from a defect, ∆E a . Defect identification is often based on comparing ∆E a with values of the defect ionization energy ∆E i determined from zero-temperature first-principles calculations. However, the underlying theory of DLTS 2,3 makes clear that ∆E a and ∆E i are distinct, and the use of ∆E i can affect the correct identification of a defect.In the present study we describe a first-principles approach to explicitly determine the activation energies measured in DLTS. Recent advances have enabled the quantitative prediction of defect levels in the band gap 9 and the ability to accurately describe nonradiative carrier capture.10We will show that ∆E a can significantly differ from ∆E i for some defects, demonstrating the need to explicitly calculate the activation energy in order to correctly identify defects detected by DLTS. Our analysis is general, but will be illustrated with examples of deep defects in GaN, a material of high technological relevance because of its applications in solid-state light emitters and power electronics.Let us consider a defect that acts as a single deep acceptor. A standard DLTS measurement relies on a pn junction or Schottky diode that is reverse biased, which establishes a depletion region that is free of mobile carriers. The band diagram is illustrated inFig. 1(a). 11 A forward-bias injection pulse is applied, which decreases the width of the depletion region [Fig. 1(b)]. Holes from the valence band are captured nonradiatively into the acceptor level during the injection pulse. After the pulse is turned off, the depletion width increases to its reversebias value and the holes captured into the acceptor level are re-emitted into the valence band, which results in a transient change in the capacitance. Using the "double	10.1063/1.5047808	[ "https://arxiv.org/pdf/1810.05302v1.pdf" ]	119,422,070	1810.05302	fec8740b1cd46b9cf4726d4150a932c067ee6788	Defect identification based on first-principles calculations for deep level transient spectroscopy 12 Oct 2018 Darshana Wickramaratne Materials Department University of California Santa 93106-5050BarbaraCaliforniaUSA Cyrus E Dreyer Department of Physics and Astronomy Stony Brook University 11794-3800Stony BrookNew YorkUSA Center for Computational Quantum Physics Flatiron Institute 162 5 th Avenue10010New YorkNew YorkUSA Bartomeu Monserrat Cavendish Laboratory TCM Group University of Cambridge J. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom Jimmy-Xuan Shen Department of Physics University of California Santa 93106-9530BarbaraCaliforniaUSA John L Lyons Center for Computational Materials Science US Naval Research Laboratory 20375WashingtonDCUSA Audrius Alkauskas Center for Physical Sciences and Technology (FTMC) LT-10257VilniusLithuania Chris G Van De Walle Materials Department University of California Santa 93106-5050BarbaraCaliforniaUSA Defect identification based on first-principles calculations for deep level transient spectroscopy 12 Oct 2018(Dated: 15 October 2018) Deep level transient spectroscopy (DLTS) is used extensively to study defects in semiconductors. We demonstrate that great care should be exercised in interpreting activation energies extracted from DLTS as ionization energies. We show how first-principles calculations of thermodynamic transition levels, temperature effects of ionization energies, and nonradiative capture coefficients can be used to accurately determine actual activation energies that can be directly compared with DLTS. Our analysis is illustrated with hybrid functional calculations for two important defects in GaN that have similar thermodynamic transition levels, and shows that the activation energy extracted from DLTS includes a capture barrier that is temperature dependent, unique to each defect, and in some cases large in comparison to the ionization energy. By calculating quantities that can be directly compared with experiment, first-principles calculations thus offer powerful leverage in identifying the microscopic origin of defects detected in DLTS.PACS numbers: 71.55.-i, 72.20.Jv, 84.37.+q Point defects and impurities are present in all semiconductors. They can act as recombination centers that lower the efficiency of optoelectronic devices, or as carrier traps in electronic devices such as transistors. Microscopic identification of the detrimental defects is crucial in order to mitigate their impact. Deep level transient spectroscopy (DLTS) is a powerful technique for determining the properties of defects; from an analysis of electrical measurements on a pn junction or Schottky diode, properties such as the position of the defect level within the band gap, electrical nature (donor or acceptor), density, and carrier capture cross section of specific defects can be obtained.[1][2][3]Translating this wealth of information to a microscopic identification of a given defect requires comparison with theoretical or computational models, and first-principles calculations based on density functional theory (DFT) have proven very helpful.[4][5][6][7][8]One of the key quantities measured in DLTS is the activation energy for carrier emission from a defect, ∆E a . Defect identification is often based on comparing ∆E a with values of the defect ionization energy ∆E i determined from zero-temperature first-principles calculations. However, the underlying theory of DLTS 2,3 makes clear that ∆E a and ∆E i are distinct, and the use of ∆E i can affect the correct identification of a defect.In the present study we describe a first-principles approach to explicitly determine the activation energies measured in DLTS. Recent advances have enabled the quantitative prediction of defect levels in the band gap 9 and the ability to accurately describe nonradiative carrier capture.10We will show that ∆E a can significantly differ from ∆E i for some defects, demonstrating the need to explicitly calculate the activation energy in order to correctly identify defects detected by DLTS. Our analysis is general, but will be illustrated with examples of deep defects in GaN, a material of high technological relevance because of its applications in solid-state light emitters and power electronics.Let us consider a defect that acts as a single deep acceptor. A standard DLTS measurement relies on a pn junction or Schottky diode that is reverse biased, which establishes a depletion region that is free of mobile carriers. The band diagram is illustrated inFig. 1(a). 11 A forward-bias injection pulse is applied, which decreases the width of the depletion region [Fig. 1(b)]. Holes from the valence band are captured nonradiatively into the acceptor level during the injection pulse. After the pulse is turned off, the depletion width increases to its reversebias value and the holes captured into the acceptor level are re-emitted into the valence band, which results in a transient change in the capacitance. Using the "double (Dated: 15 October 2018) Deep level transient spectroscopy (DLTS) is used extensively to study defects in semiconductors. We demonstrate that great care should be exercised in interpreting activation energies extracted from DLTS as ionization energies. We show how first-principles calculations of thermodynamic transition levels, temperature effects of ionization energies, and nonradiative capture coefficients can be used to accurately determine actual activation energies that can be directly compared with DLTS. Our analysis is illustrated with hybrid functional calculations for two important defects in GaN that have similar thermodynamic transition levels, and shows that the activation energy extracted from DLTS includes a capture barrier that is temperature dependent, unique to each defect, and in some cases large in comparison to the ionization energy. By calculating quantities that can be directly compared with experiment, first-principles calculations thus offer powerful leverage in identifying the microscopic origin of defects detected in DLTS. Point defects and impurities are present in all semiconductors. They can act as recombination centers that lower the efficiency of optoelectronic devices, or as carrier traps in electronic devices such as transistors. Microscopic identification of the detrimental defects is crucial in order to mitigate their impact. Deep level transient spectroscopy (DLTS) is a powerful technique for determining the properties of defects; from an analysis of electrical measurements on a pn junction or Schottky diode, properties such as the position of the defect level within the band gap, electrical nature (donor or acceptor), density, and carrier capture cross section of specific defects can be obtained. [1][2][3] Translating this wealth of information to a microscopic identification of a given defect requires comparison with theoretical or computational models, and first-principles calculations based on density functional theory (DFT) have proven very helpful. [4][5][6][7][8] One of the key quantities measured in DLTS is the activation energy for carrier emission from a defect, ∆E a . Defect identification is often based on comparing ∆E a with values of the defect ionization energy ∆E i determined from zero-temperature first-principles calculations. However, the underlying theory of DLTS 2,3 makes clear that ∆E a and ∆E i are distinct, and the use of ∆E i can affect the correct identification of a defect. In the present study we describe a first-principles approach to explicitly determine the activation energies measured in DLTS. Recent advances have enabled the quantitative prediction of defect levels in the band gap 9 and the ability to accurately describe nonradiative carrier capture. 10 We will show that ∆E a can significantly differ from ∆E i for some defects, demonstrating the need to explicitly calculate the activation energy in order to correctly identify defects detected by DLTS. Our analysis is general, but will be illustrated with examples of deep defects in GaN, a material of high technological relevance because of its applications in solid-state light emitters and power electronics. Let us consider a defect that acts as a single deep acceptor. A standard DLTS measurement relies on a pn junction or Schottky diode that is reverse biased, which establishes a depletion region that is free of mobile carriers. The band diagram is illustrated in Fig. 1(a). 11 A forward-bias injection pulse is applied, which decreases the width of the depletion region [ Fig. 1(b)]. Holes from the valence band are captured nonradiatively into the acceptor level during the injection pulse. After the pulse is turned off, the depletion width increases to its reversebias value and the holes captured into the acceptor level are re-emitted into the valence band, which results in a transient change in the capacitance. Using the "double boxcar" technique, 11 the difference in the capacitance, ∆C, is measured at two different times after the injection pulse is turned off [schematically illustrated in Fig. 1(c)]. The duration of time within which the capacitance measurement occurs is termed the emission rate window. The change in the capacitance for a given emission rate window is measured as a function of temperature T . A peak in the capacitance versus T occurs when the rate at which the acceptor level emits a hole, e p , equals the inverse of the emission rate window. Therefore, repeating this measurement with a variety of rate windows, as shown in Fig. 1(d), yields measurements of e p versus T . Under equilibrium conditions, the principle of detailed balance requires that the rate of hole capture into the acceptor level is equal to the emission rate of holes into the valence band. Therefore e p can be written in terms of the hole capture cross section σ p as 11 w d w d Temperature (a) (b) (c) (d) Temperature E t E v E Fp E t E v E Fp t 1 t 2 Time C(t 1 ) -C(t 2 ) C(t 1 ) -C(t 2 ) q(ϕ bi -V F ) q(ϕ bi +V R ) Capacitancee p (T ) = σ p (T ) v p (T ) N v (T ) g v exp − ∆E i (T ) k B T ,(1) where v p is the average thermal velocity of holes in the valence band, N v is the effective density of states of the valence band, g v is the valley degeneracy, and ∆E i (T ) is the ionization energy of the defect. For the case of the deep acceptor, ∆E i (T ) is the energy difference between the valence-band maximum, E v , and the acceptor level. To obtain the ionization energy and capture cross section from the emission rate versus T data obtained by the procedure in Fig. 1, the temperature dependence of the various quantities in Eq. (1) must be specified. The thermal velocity v p is given by 3kBT m h , where m h is the hole mass. Assuming parabolic bands, the valence-band density of states is defined as N v = 2g v 2πm * h kBT 2 3/2 , where m * h is the density-of-states effective mass of holes (1.50 m 0 for GaN). 12,13 Therefore, v p N v ∝ T 2 . The standard procedure in DLTS analysis is to assume σ p is independent of temperature and plot ln(e p (T )/T 2 ) versus 1/T , and then fit to an Arrhenius expression to extract an activation energy ∆E a . 11 This activation energy would coincide with the 0 K ionization energy of the defect if ∆E i and the prefactor of Eq. (1) were independent of temperature. However, in reality σ p can have a strong temperature dependence, and therefore a significant contribution to the slope obtained from a ln(e p (T )/T 2 ) versus 1/T plot. 3 This contribution must be clarified in order to extract the activation energy from these plots. In addition, ∆E i , i.e., the position of the thermodynamic transition level with respect to the band edge, is also temperature dependent. 14,15 Both effects are addressed here from first principles. We perform DFT calculations based on the generalized Kohn-Sham scheme using the projector-augmentedwave method with the hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE) 16 as implemented in the VASP code. 17, 18 We use a mixing parameter of 0.31 which results in lattice parameters (a=3.19Å and c=5.17Å) and a band gap (3.495 eV) that are close to the experimental T =0 K lattice parameters 19 and band gap 20,21 of GaN. The defect calculations are performed using a 96-atom supercell, a plane-wave basis set with a 400 eV cutoff, and a (2×2×2) Monkhorst-Pack k-point grid to sample the Brillouin zone. Spin polarization is explicitly included. All atomic relaxations are performed consistently with the HSE functional. Defect formation energies and thermodynamic transition levels were calculated using the standard formalism 9 with charge-state corrections applied to account for the periodic supercells. 22 Capture coefficients and their temperature dependence are evaluated using the formalism in Ref. 10. In order to discuss the temperature dependence of the capture cross section, we prefer to consider the capture coefficient, C p (T ) = σ p (T ) v(T ) , which is a more general quantity since it can be calculated without assuming a thermal velocity for the carriers. To determine the temperature dependence of ∆E i (T ) we calculate the temperature dependence of the band edges and the thermodynamic charge-state transition level. Two mechanisms contribute to the temperature dependence of the band edges. The contribution due to electron-phonon interactions was evaluated by using the methodology of Refs. 23 and 24, on a 4×4×4 q-point grid. To determine the contribution due to thermal expansion we used experimental thermal expansion coefficients 19 to determine the lattice expansion at a given temper- -1 0 1 2 3 Q (amu 1/2 ) -1 0 1 2 3 Q (amu 1/2 0 0.5 E C N 0 + (V Ga -O N -2H) + Å) Å FIG. 2. One-dimensional configuration coordinate diagrams describing hole capture due to (a) CN and (b) VGa-ON-2H in GaN. In (a) the initial state of the defect is CN in a negative charge state and the final state is CN in a neutral charge state. In (b) the initial state is (VGa-ON-2H) 0 and the final state is (VGa-ON-2H) + . ∆Ei is energy difference between the minima of the potential energy surfaces at T =0 K and ∆E b is the classical barrier for the nonradiative capture process. ature, and absolute deformation potentials 25 to determine the resulting shift in the band edges. We verified that the calculated cumulative change in the band gap agrees with experimental measurements, 21 but we emphasize that our procedure allows us to assess the shifts in the individual band edges (valence band versus conduction band). To determine the temperature dependence of the defect levels we calculate the zone-center vibrational frequencies of each defect in their different charge states using HSE with 216-atom supercells using the T=0 K HSE lattice parameters. The vibrational frequencies were determined for a set of atoms within 4Å around the defect while the remaining atoms were kept fixed at their equilibrium positions. Details of the calculations are provided in the Supplementary Material. The vibrational frequencies were used to evaluate the vibrational free energy for a given charge state within the harmonic approximation [cf. Eq. (17) in Ref. 9]. The difference in vibrational free energy between the two charge states was used to determine the temperature dependence of the transition level. We will determine the implications of the T dependence for two examples of defects in GaN that have very similar thermodynamic transition levels but different temperature dependences of their capture coefficient: carbon on a nitrogen site, C N , 26 and a gallium vacancy complex, V Ga -O N -2H. 27 One-dimensional configuration coordinate diagrams (see Ref. 10) are shown in Fig. 2. C N is a deep acceptor, with a (0/−) transition level 1.02 eV above E v . V Ga -O N -2H is a complex based on a gallium vacancy that exhibits a (+/0) transition level 1.06 eV above E v . 27 For purposes of determining the capture coefficient, the initial state of the system consists of a hole in the valence band and a negatively charged acceptor; see the poten-tial energy surface labeled C − N + h + in Fig. 2(a). Capture occurs when the system traverses to the potential energy surface corresponding to the neutral acceptor, C 0 N . The difference in energy of the minima of the two curves is the thermodynamic charge-state transition level referenced to E v , and corresponds to the ionization energy ∆E i from Eq. (1). At high temperatures the capture process occurs by surmounting the "classical" barrier, 2 ∆E b , obtained from the intersection point of the curves in the configuration coordinate diagram; at low temperatures, the transition rate is dominated by quantum-mechanical tunneling. 10 The classical barrier ∆E b is 490 meV for C N and 49 meV for V Ga -O N -2H. The large difference in these classical barriers is reflected in our results for the temperature dependence of the hole capture coefficients in Fig. 3(a) (dashed lines). We focus on the temperature range up to 600 K; DLTS measurements on GaN are limited to this temperature to prevent degrading of the metal contacts. The hole capture coefficient of C N changes by two orders of magnitude as the temperature increases from 0 K to 600 K, while for V Ga -O N -2H the temperature dependence is much more modest. The results shown in dashed lines in Fig. 3(a) assume that the ∆E i are fixed to their T =0 values. These ∆E i values correspond to charge-state transition levels obtained from static-lattice calculations of a zero-temperature DFT calculation. Both dashed curves have an Arrhenius form at high T , and have a weak temperature dependence as T → 0. 10 In reality, the distance between E v and the defect level shrinks as T increases, and so ∆E i (T ) is reduced, as shown in Fig. 3(b). Inclusion of this additional effect enhances the dependence of C p on T , as shown by the solid curves in Fig. 3(a). It is commonly assumed 3 that the capture coefficient has a temperature dependence given by C p = C ∞ exp(−∆E b /k B T ). Our results in Fig. 3(a) show that the description in terms of a temperature-independent classical barrier is too simple to capture the actual temperature dependence of C p . At low temperatures, the Arrhenius form would imply that C p goes to zero as T → 0 K, but in reality C p remains finite because of quantummechanical tunneling. At high T , the behavior is also non-exponential, caused by the temperature dependence of ∆E i . Hence, in a quantum-mechanical treatment of nonradiative capture of carriers by defects one should consider an effective barrier to describe the temperature dependence of such processes. 10 Unlike the classical capture barrier ∆E b , the effective barrier ∆E ′ b (T ) is temperature dependent resulting in C p deviating from purely exponential behavior. We now use our values of C p [ Fig. 3(a)] to calculate the emission rate based on Eq. (1). We mentioned before that the common practice in DLTS analysis is to plot ln(e p /T 2 ) versus 1/T , based on a lack of information about the temperature dependence of σ p and the fact that v p N v ∝ T 2 . Therefore in Fig. 4 we plot ln(e p /T 2 ). When plotted over this large temperature range, the calculated emission rates (black solid lines in Fig. 4) clearly deviate from linearity, reflecting the non-Arrhenius behavior of C p as well as the temperature dependence of ∆E i . It is important to note that even if ln(e p /T 2 ) would be linear, the slope still does not correspond to the ionization energy, as it includes the capture barrier. DLTS measurements are carried out over a limited temperature range, typically about 50 K, and the data are then fitted to an Arrhenius expression. Based on the data as plotted in our Fig. 4, we fit to an expression: e p /T 2 = e fit 0 exp(−∆E fit a /k B T ) .(2) We can thus determine the activation energy ∆E fit a that would be extracted from a typical DLTS measurement by fitting over a finite temperature range similar to the one probed in experiments (dashed curves in Fig. 4). We find that for C N , the fitted activation energies increase from 1.162 eV to 1.394 eV, depending on the temperature range for which the fit is performed [ Fig. 4(a)]. For V Ga -O N -2H, the explicit calculations are much closer to a simple Arrhenius behavior, and hence there is little variation in the activation energies extracted over different temperature ranges [ Fig. 4(b)]. The activation energy is temperature dependent and the deviation between the ionization energy and activation energy is pronounced at higher temperatures. Our calculations highlight that the difference between the activation energy and the ionization energy can be large: up to 0.4 eV for C N . The activation energy obtained from an Arrhenius analysis of the emission rate differs from the 0 K ionization energy of the defect for two reasons: first, because the activation energy also includes a capture barrier, and second, because the ionization energy itself is temperature dependent, due to the temperature dependence of the band edges and of the defect transition level. Activation energies extracted from DLTS should therefore not be simply interpreted as ionization energies, and simple comparisons with first-principles ionization energies could lead to incorrect identification of defects. As mentioned above, the typical procedure (which we have followed in this paper) is to plot the results of DLTS experiments as ln(e p /T 2 ), and perform an Arrhenius analysis assuming that σ p has a temperatureindependent prefactor [see Eq. 1]. It has been shown, however, that at high temperature the preexponential factor in σ p has a 1/T dependence [cf. Eq. (28) in Ref. 2 and Eq. (61a) in Ref. 28]. At high temperature one should therefore perform an Arrhenius analysis of ln(e p /T ), and this would be important to recover the value of the classical barrier ∆E b . Indeed, we find that including this 1/T dependence when fitting our first-principles calculations of capture coefficients at high temperatures (T > 1200 K) results in ∆E ′ b → ∆E b (the classical barrier at fixed ∆E i , see Fig. 2), as expected. However, the typical temperature range over which DLTS experiments are performed does not reach this high-temperature limit, and therefore ln(e p /T 2 ) is an acceptable approximation. In summary, we have shown DLTS activation energies are temperature-dependent and should not be compared directly with first-principles calculations of ionization energies of defects. Using first-principles calculations we determined the temperature dependence of nonradiative carrier capture and the ionization energy of defects in GaN and demonstrated how they yield activation energies that can differ greatly from the 0 K ionization energy of the defect. The C N and V Ga -O N -2H defects we considered in this study are examples of positive-U defects where we determined the activation energy due to thermal emission from a single thermodynamic transition level. Our conclusions on the temperature dependence of activation energy will also apply in the case of more complex situations such as defects with two thermodynamic transition levels that are amenable to ionization in DLTS. In the case of a positive-U center, thermal emission due to both thermodynamic transition levels would be observed in a DLTS measurement. Our formalism can be applied to determine the activation energy of both transitions separately. In the case of a negative-U center, the DLTS transient is determined by the slower of the two carrier emission process. Thus, one would observe a single peak with an activation energy that corresponds to the slower emission process, to which our analysis is equally applicable. Hence, our analysis of these quantities is general and can be applied to accurately determine defect activation energies when comparing to DLTS measurements. We are interested in determining the temperature dependence of the defect ionization energy, ∆E i . One contribution arises from the temperature dependence of the valence-band edge due to electron-phonon interactions and thermal expansion; the calculation of this contribution is described in the main text. A second contribution to the temperature dependence is due to vibrational entropy of the defect, which shifts the thermodynamic transition level. We can evaluate this contribution by calculating the vibrational free energy of the defect in each of the relevant charge states; the difference yields the shift in the transition level. Calculations of free energy require the evaluation of vibrational frequencies. The zone-center vibrational properties of the defect are calculated in a defect supercell, where one defect is embedded in a large volume of host material and is periodically repeated. A finite-difference scheme is used to obtain vibrational frequencies. The T = 0 equilibrium HSE lattice parameters of GaN were used. In the course of our convergence tests, we found that an energy convergence criterion of 10 −7 eV needs to be applied for these calculations; this is a much more stringent criterion than the default value of 10 −4 eV. The vibrational frequencies are used to determine the vibrational free energy F ph for the defect in a charge state q within the harmonic approximation: 1 F ph = i 1 2 ω i + k B T ln 1 − exp − ω i k B T . (S1) The impact of vibrations on the temperature dependence of the thermodynamic level is determined by the difference in vibrational free energy, ∆F ph , between the two charge states of the thermodynamic level. For example, for C N this is the difference between F ph of the negative and neutral charge states, as defined in Eq. (S2): ∆F ph = F ph (C q=1− N ) − F ph (C q=0 N ) .(S2) For V Ga -O N -2H ∆F ph is the difference between F ph in the neutral and positive charge state. In principle, we would like to include the vibrational properties of all of the atoms in the supercell in the calculation of F ph . However, for the supercell sizes that are needed (see our convergence tests below), this is computationally intractable when using a hybrid functional such as HSE 2 using the convergence criteria that we have identified to be necessary to obtain ∆F ph with acceptable accuracy. One approach would be to use a less demanding functional such as the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE). 3 This is the approach we have used for conducting the benchmark and supercell-size convergence tests reported below. There is a concern, however, that PBE may not capture all of the relevant properties. Indeed, certain defect configurations lead to charge localization that is properly described only when using a hybrid functional, 1 and the atomic relaxations that accompany this localization may induce significant changes in vibrational properties. We have therefore developed an alternative procedure for calculating ∆F ph , in which only atoms in the vicinity of the defect center are included in the calculation of the vibrational properties. Based on the tests reported below, we have found that including atoms within 4Å of the defect center (corresponding to first and second nearest neighbors) is sufficient to calculate ∆F ph to an acceptable degree of accuracy. The remaining atoms outside of this 4-Å radius are kept fixed at their relaxed equilibrium positions. In practice, we have used 216-atom supercells where the atomic coordinates in each charge state were relaxed using HSE. For each charge state q of the defect, the vibrational frequencies of the atoms within this 4-Å radius are calculated with the HSE hybrid functional using finite differences. ∆F ph is then obtained based on Eq. (S2). Our convergence tests are detailed in the subsections below. B. Benchmark calculations for vibrational properties Since full HSE calculations of vibrational properties in a sufficiently large supercell are intractable, we have developed an alternative procedure based on calculating vibrational properties for a subset of atoms. In order to check the accuracy of the procedure, we need to have a benchmark value for ∆F ph , calculated by including vibrations for all the atoms in the supercell. Since such calculations are not feasible with HSE, we instead used PBE 3 to determine the convergence of vibrational frequencies and free energies as a function of supercell size. We assume that the convergence properties as a function of supercell size will be similar for HSE. Figure S1 shows the convergence of ∆F ph for the defects in GaN as a function of supercell size. This allows us to determine the supercell size where ∆F ph for an isolated defect is not impacted by finite-size effects. We consider supercell sizes that range from 64 atoms to 216 atoms. For each of these calculations PBE lattice parameters of GaN were used and all atomic relaxations were performed within PBE. ∆F ph as obtained using PBE vibrational modes is shown in Fig. S1. We conclude that supercells larger than 144 atoms are needed to obtain converged results. F p ¢ [eV] @ £ ¤ ¥ N 0 P ¦ § 400 600 ¨ © (K) 0 0.05 0.1 0.15 (b) V q E N ! " T # $ % & m ' ( ) 0 1 2 3 4 I 5 6 7 8 o 9 A B C D F G H Q R W S U V X Y ` a b c d e f g h i p r s t u v w x y C. Vibrational properties of defects using a finite number of atoms We have developed a procedure to obtain ∆F ph based on the vibrational properties of a subset of atoms within 4Å around the defect site. The set of atoms lying within 4Å of the defect site corresponds to including atoms that constitute the defect center, plus their first and second nearest neighbors. The remaining atoms in the supercell are kept fixed at their equilibrium positions. Vibrations of atoms lying outside this 4Å sphere will definitely contribute to the vibrational free energy; however, our hypothesis is that these contributions will be very similar for different charge states q, and hence will cancel in ∆F ph [see Eq. S2]. We have verified this approach by calculating ∆F ph in a 216-atom supercell for C N and V Ga -O N -2H, and comparing the result obtained based on the vibrational modes of atoms lying within 4Å of the defect site with the result obtained based on the vibrational modes of all atoms in the supercell. The PBE functional was used for these tests, and the results are illustrated in Fig. S2. From Fig. S2 it is evident that a majority of the change in the vibrational free energy is captured by the vibrational modes of the atoms within this small volume. This now makes it feasible to determine ∆F ph for these defects using the HSE hybrid functional. 0.15 (b) V j k l n N q r s t u v w x y z { \| } ~ ¡ ¢ £ NN) ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ N N) One question still to be addressed relates to the supercell size that is needed for this procedure to yield acceptable results. One may think that, since atoms outside a 4-Å radius are kept fixed, a relatively small supercell might suffice. We have carried out tests for 96-, 144-, and 216-atom supercells, as shown in Fig. S3. The comparison shows that 216-atom supercells yield the best results. While atoms outside the 4-Å radius are kept fixed in the process of calculating vibrational properties, these vibrational properties turn out to be quite sensitive to the supercell size. This may be because of a sensitivity to the details of the atomic relaxations (which may not be fully converged in a 96-atom cell). More likely it is due to interactions between neighboring supercells in the course of the evaluation of vibrational frequencies. Such a spurious interaction may be present particularly in the case of nonzero charge states, especially since the suppression of relaxation of atoms outside the 4-Å radius also suppresses screening. These spurious effects diminish with increasing supercell size. Based on these tests, 216-atom supercells were used with the HSE lattice parameters of GaN, including complete relaxation of all atoms with HSE, and the vibrational properties were also determined with HSE for atoms within 4 A of the defect site. The results are shown in Sec. S0 E. ² ³ ´ m s µ ¶ · ¸ N N) 0 ¹ º » 400 600 ¼ ½ ¾ ¿ À Á Â Ã Ä Å AE (K) 0 0.05 0.1 0.15 F Ç È [eV] É Ê Ë Ì Í oms Î Ï Ð Ñ Ò Ó Ô Õ Ö oms × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å oms ae ç è é ê ë ì í î ï ð ñ ò ó ô ms õ ö ÷ ø ù ú û ü ý þ ÿ P ¡ ¢ oms 2 £ ¤ ¥ ¦ § ¨ © oms ! " # $ % & D. Impact of thermal expansion on vibrational properties Finite temperature leads to thermal expansion of the GaN lattice; in turn this can impact the vibrational frequencies of the defect. In this study we have used the equilibrium 0 K HSE lattice parameters to determine the vibrational frequencies. To justify this approximation we compare the vibrational frequencies of C N and V Ga -O N -2H obtained with a 216-atom GaN supercell with 0 K PBE lattice parameters with a 216 atom supercell with expanded lattice parameters (based on the thermal expansion coefficients of GaN 4 and a temperature of 600 K). We find thermal expansion to have a minor impact on the vibrational frequencies of the defects. As an example, the vibrational density of states for C N in the neutral and negative charge states between the 0 K and 600 K lattice constants is illustrated in Fig. S4. The difference in vibrational properties has a very small impact on ∆F ph . ∆F ph calculated with the 600 K lattice parameters differs by only 4 meV from the value calculated with T = 0 lattice parameters for C N , and by 3 meV for V Ga -O N -2H. These differences in ∆F ph are significantly smaller than the other temperature-dependent quantitites we consider in this study. We conclude that we can neglect lattice expansion in the calculation of the vibrational free energy, and we have calculated the vibrational frequencies for both defects using the equilibrium 0 K lattice parameters. E. Vibrational free-energy contribution to the thermodynamic transition level: HSE results Figure S5 shows ∆F ph for C N and V Ga -O N -2H in GaN as a function of temperature, obtained with HSE calculations within 216 atom supercells as described above: the T=0 K HSE lattice parameters were used and only atoms within 4Å of the defect center were used to compute the vibrational frequencies. The remaining atoms were kept fixed at their equilibrium positions. These results were used to determine the impact of vibrational free energy on the temperature dependence of the defect transition level. The results in Fig. S5, combined with our calculated temperature dependence of the band edges, were used to determine the temperature dependence of the ionization energy ∆E i [cf. Fig. 3(b) in the main text]. TEMPERATURE DEPENDENCE OF THE GaN BAND GAP We are interested in the temperature dependence of the band gap and band edges of GaN since these changes in the electronic structure impact the capture coefficients and emission rates. Our calculations take into account the role of electron-phonon interactions and thermal expansion. In Fig. S6 we show the change in the band gap as a function of temperature, calculated using the methodology described in the main text. We compare these results to the experimentally measured change in the band gap of GaN as reported in Ref. 5. We note that the GaN layers in the experimental measurements of Ref. 5 were grown on sapphire and thus experience a certain amount of strain, which will vary with temperature. Such strain effects are not included in our calculations and may be responsible for the difference between experimental and calculated results. PACS numbers: 71.55.-i, 72.20.Jv, 84.37.+q FIG. 3 . 3(a) Hole capture coefficient versus temperature for CN and VGa-ON-2H in GaN. The dashed lines are based on constant T =0 K values for ∆Ei. The solid lines take the temperature dependence of ∆Ei, as shown in (b), into account. (b) Variation in the ionization energy of CN and VGa-ON-2H as a function of temperature, as described in the text. FIG. 4 . 4Calculated hole emission rates for (a) CN and (b)VGa-ON-2H in GaN (solid lines). The dashed lines are leastsquares fits to Eq. (2), with the thick band of symbols indicating the temperature ranges over which the fit was performed: 200-250 K, 300-350 K, 400-450 K and 500-550 K. Extracted activation energies ∆Ea are shown alongside each fit and plotted as a function of temperature in the inset. The zero-temperature ionization energy for each defect is illustrated with a horizontal dashed line. Supplementary Material: See Supplementary Material for the details of calculations of the vibrational properties of defects and a comparison between the calculated and experimental temperature dependence of the band gap of GaN. Acknowledgements: D. W. was supported by the National Science Foundation (NSF) under Grant No. DMR-1434854. J. S. was supported by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0010689. B. M. acknowledges support from the Winton Programme for the Physics of Sustainability, and from Robinson College, Cambridge, and the Cambridge Philosophical Society for a Henslow Research Fellowship. A. A. was supported by Marie Sk lodowska-Curie Action of the European Union (project Nitride-SRH, grant No. 657054). The Flatiron Institute is a division of the Simons Foundation. Computational resources were provided by the Extreme Science and Engineering Discovery Environment (XSEDE), support by NSF (ACI-1053575). Procedure for calculating the vibrational contribution to the defect level FIG. S1. ∆F ph for (a) CN and (b) VGa-ON-2H as a function of temperature for 64, 96, 144 and 216 atom wurtzite cells including all vibrational modes determined with the PBE functional. FIG. S2. ∆F ph including all vibrational modes versus vibrational modes contributed by atoms lying within 4Å of the defect center [up to 2 nd nearest neighbors (NN)] for (a) CN and (b) VGa-ON-2H determined with the PBE functional. . S3. Comparison of ∆F ph for CN as determined with the PBE functional, for different supercell sizes. (a) 96 atoms; (b) 144 atoms; (c) 216 atoms. In each case the result obtained vibrational modes contributed by atoms lying within 4Å of the defect center (up to 2 nd NN) is compared with the result including vibrational modes for all atoms within the same-size supercell, as well as with the result including vibrational modes for all atoms within a 216-atom supercell (which serves as our benchmark). FIG . S4. Vibrational density of states for the (a) neutral and (b) negative charge state of CN, calculated using the equilibrium (T =0 K) (shaded blue) and expanded lattice parameters (corresponding to thermal expansion at T =600 K) (black line). Vibrational frequencies are calculated within PBE. . S5. ∆F ph for (a) CN and (b) VGa-ON-2H determined with the HSE function for a atoms lying within 4Å of the defect center in a 216-atom supercell. S6. Calculated (black) and experimentally measured 5 (blue squares) change in the band gap of GaN as a function of temperature. vs. temperature FIG. 1. Schematic illustration of the DLTS measurement process for a p-type Schottky junction with a single deep acceptor level at an energy Et. EF p is the quasi-Fermi level for holes for the junction under bias and Ev is the valence-band maximum. The band diagram is shown under (a) reverse bias, VR, and (b) forward bias, VF . φ bi is the built-in potential at the Schottky junction. (c) Capacitance measurement as a function of temperature within the DLTS rate window. (d) Resulting hole emission spectra as a function of temperature obtained for different rate windows. Panel (c) is adapted from Fig. 6 of Ref. 11. C. Freysoldt, B. Grabowski, T. 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[ "Fairness-aware Network Revenue Management with Demand Learning", "Fairness-aware Network Revenue Management with Demand Learning" ]	[ "Xi Chen ", "Jiameng Lyu ", "Yining Wang [email protected] ", "Yuan Zhou [email protected] ", "\nLeonard N. Stern School of Business\nYau Mathematical Sciences Center\nNew York University\n10012New YorkNYUSA\n", "\nNaveen Jindal School of Management\nTsinghua University\n100084BeijingChina\n", "\nYau Mathematical Sciences Center\nUniversity of Texas at Dallas\n75080RichardsonTXUSA\n", "\nTsinghua University\n100084BeijingChina\n" ]	[ "Leonard N. Stern School of Business\nYau Mathematical Sciences Center\nNew York University\n10012New YorkNYUSA", "Naveen Jindal School of Management\nTsinghua University\n100084BeijingChina", "Yau Mathematical Sciences Center\nUniversity of Texas at Dallas\n75080RichardsonTXUSA", "Tsinghua University\n100084BeijingChina" ]	[]	In addition to maximizing the total revenue, decision-makers in lots of industries would like to guarantee fair consumption across different resources and avoid saturating certain resources. Motivated by these practical needs, this paper studies the price-based network revenue management problem with both demand learning and fairness concern about the consumption across different resources. We introduce the regularized revenue, i.e., the total revenue with a fairness regularization, as our objective to incorporate fairness into the revenue maximization goal. We propose a primal-dual-type online policy with the Upper-Confidence-Bound (UCB) demand learning method to maximize the regularized revenue. We adopt several innovative techniques to make our algorithm a unified and computationally efficient framework for the continuous price set and a wide class of fairness regularizers. Our algorithm achieves a worst-case regret of r OpN 5{2 ? T q, where N denotes the number of products and T denotes the number of time periods. Numerical experiments in a few NRM examples demonstrate the effectiveness of our algorithm for balancing revenue and fairness. : Fairness-aware NRM with Demand Learning 2the total revenue, many retailers would like to guarantee fair consumption across different resources and avoid saturating certain resource types. As summarized in(Bateni et al. 2022), fairness is an important concern in two cases. First, many platforms (e.g., in healthcare or public utility sectors) are under the obligation to ensure fairness. Second, even those profit-seeking platforms also need to take fairness into consideration, because to maximize the long-term revenue, a platform should keep long-term cooperation with different resource suppliers. Ensuring fairness across different resource suppliers will help keep a good collaboration relationship. Let us provide two examples to illustrate the importance of fairness constraints in revenue management applications:Online Retailing: In an NRM model for the online retailing industry, a retailer decides prices for products and each product consumes several types of resources. Different resources are provided by different suppliers, and it is also possible that one supplier could provide more than one type of resources. If one supplier terminates the cooperation, certain products may no longer be able to be produced. Therefore, to maximize long-term revenue and guarantee no product is out of stock, the retailer should keep good relationship with all resource suppliers. To this end, it is essential to ensure profit fairness across different suppliers. Indeed, only paying attention to the revenue maximization may cause an "unfair" scenario in which some suppliers earn a lot but the shares of others are small.Airline Ticketing: For an airline company, it needs to make decisions on the prices of products (e.g., multileg airlines) subject to resource constraints (i.e., the capacity constraint of each flight), taking into consideration the fact that customers' realized demand for the products change with prices charged accordingly. Since a multileg airline may involve several flights between different cities, if one leg of flight between two cities is saturated, all the products that include this leg will be out of stock no matter how many seats are available on the other flight legs. As a result, saturation in just a few flights not only will cause the waste of the capacity but also cannot meet the need of the passengers, which in turn leads to passenger dissatisfaction. Therefore, taking the fairness across different resources into consideration will help airline companies avoid saturating flight legs and eventually improve passenger satisfaction.As shown in the above examples, the fairness objective is often measured in the sense of the whole selling season and cannot be decomposed as an additive objective at each time period. In this paper, we adopt several different metrics that are applied to the average resource consumption vector to measure its fairness. For example, the fairness on resource consumption could be measured by the minimum element of the average resource consumption vector, which is the famous max-min fairness metric extensively studied in economics and resource allocation literature, also included	10.48550/arxiv.2207.11159	[ "https://export.arxiv.org/pdf/2207.11159v1.pdf" ]	251,018,518	2207.11159	3e7576415babe926d50a0fb81d7f45cb06fb55a1	Fairness-aware Network Revenue Management with Demand Learning Xi Chen Jiameng Lyu Yining Wang [email protected] Yuan Zhou [email protected] Leonard N. Stern School of Business Yau Mathematical Sciences Center New York University 10012New YorkNYUSA Naveen Jindal School of Management Tsinghua University 100084BeijingChina Yau Mathematical Sciences Center University of Texas at Dallas 75080RichardsonTXUSA Tsinghua University 100084BeijingChina Fairness-aware Network Revenue Management with Demand Learning network revenue managementdemand learningfairnessregret analysislinear bandit In addition to maximizing the total revenue, decision-makers in lots of industries would like to guarantee fair consumption across different resources and avoid saturating certain resources. Motivated by these practical needs, this paper studies the price-based network revenue management problem with both demand learning and fairness concern about the consumption across different resources. We introduce the regularized revenue, i.e., the total revenue with a fairness regularization, as our objective to incorporate fairness into the revenue maximization goal. We propose a primal-dual-type online policy with the Upper-Confidence-Bound (UCB) demand learning method to maximize the regularized revenue. We adopt several innovative techniques to make our algorithm a unified and computationally efficient framework for the continuous price set and a wide class of fairness regularizers. Our algorithm achieves a worst-case regret of r OpN 5{2 ? T q, where N denotes the number of products and T denotes the number of time periods. Numerical experiments in a few NRM examples demonstrate the effectiveness of our algorithm for balancing revenue and fairness. : Fairness-aware NRM with Demand Learning 2the total revenue, many retailers would like to guarantee fair consumption across different resources and avoid saturating certain resource types. As summarized in(Bateni et al. 2022), fairness is an important concern in two cases. First, many platforms (e.g., in healthcare or public utility sectors) are under the obligation to ensure fairness. Second, even those profit-seeking platforms also need to take fairness into consideration, because to maximize the long-term revenue, a platform should keep long-term cooperation with different resource suppliers. Ensuring fairness across different resource suppliers will help keep a good collaboration relationship. Let us provide two examples to illustrate the importance of fairness constraints in revenue management applications:Online Retailing: In an NRM model for the online retailing industry, a retailer decides prices for products and each product consumes several types of resources. Different resources are provided by different suppliers, and it is also possible that one supplier could provide more than one type of resources. If one supplier terminates the cooperation, certain products may no longer be able to be produced. Therefore, to maximize long-term revenue and guarantee no product is out of stock, the retailer should keep good relationship with all resource suppliers. To this end, it is essential to ensure profit fairness across different suppliers. Indeed, only paying attention to the revenue maximization may cause an "unfair" scenario in which some suppliers earn a lot but the shares of others are small.Airline Ticketing: For an airline company, it needs to make decisions on the prices of products (e.g., multileg airlines) subject to resource constraints (i.e., the capacity constraint of each flight), taking into consideration the fact that customers' realized demand for the products change with prices charged accordingly. Since a multileg airline may involve several flights between different cities, if one leg of flight between two cities is saturated, all the products that include this leg will be out of stock no matter how many seats are available on the other flight legs. As a result, saturation in just a few flights not only will cause the waste of the capacity but also cannot meet the need of the passengers, which in turn leads to passenger dissatisfaction. Therefore, taking the fairness across different resources into consideration will help airline companies avoid saturating flight legs and eventually improve passenger satisfaction.As shown in the above examples, the fairness objective is often measured in the sense of the whole selling season and cannot be decomposed as an additive objective at each time period. In this paper, we adopt several different metrics that are applied to the average resource consumption vector to measure its fairness. For example, the fairness on resource consumption could be measured by the minimum element of the average resource consumption vector, which is the famous max-min fairness metric extensively studied in economics and resource allocation literature, also included Introduction Network revenue management, as a fundamental and important model in revenue management, has been successfully applied in lots of industries, such as online retailing, airline, hotel (Talluri et al. 2004, Klein et al. 2020. Classical research on NRM aims to maximize the total revenue over T time periods under resource constraints assuming the demand function is known. See, for example, the seminal works of Gallego and Van Ryzin (1997), Jasin (2014), Maglaras and Meissner (2006). In practice, there are two main challenges in adopting NRM models. First, the demand function in NRM is usually unknown, which needs to be learned on the fly. Second, in addition to maximizing as a special case of the weighted max-min fairness metric studied in this paper. In some cases it may also be desirable to consume the resources at similar rates, which can be incentivized by our range fairness metric. In addition to the above two examples, we propose several more practically useful fairness metrics. We further identify a wide class of fairness metrics (including all the above examples) with quite mild assumptions. All the fairness metrics in the class can be incorporated in our learning and doing framework. Please refer to Section 2.1 for more details. The main contribution of this paper is a dynamic pricing algorithm that simultaneous learns the unknown demand function and optimizes the composite objective concerning both the NRM revenue and the fairness metric. For any fairness metric included in the class mentioned above, our algorithm achieves a regret at most r OpN 5{2 ? T q (where N is the number of products and T is the selling horizon, see Theorem 1 for more details). Below we discuss the various technical challenges and our contributions that overcome them in details. Main technical challenges There are several technical challenges to tackle the NRM problem with fairness concern and demand learning. First, the fairness metrics are all applied to the average resource consumption vector, which is a global objective calculated across all time periods. There is also a global unreplenishable inventory constraint for the resources. The global objective and constraint are usually difficult for sequential decision making problems since the decisions at each time step have to be well coordinated to jointly optimize the fairness metric and satisfy the inventory constraint. Second, the problem becomes even harder when the demand function is unknown to the retailer and the retailer has to balance the exploration vs. exploitation trade-off with only learned information or estimated demand. Here, "exploration" means that the retailer needs to explore different prices in order to learn the unknown demand function on the run, and "exploitation" means that the retailer needs to exploit the near-optimal price to simultaneously gain revenue and achieve the global fairness objective. Finally, we aim at designing a computationally efficient algorithm (i.e., that runs in polynomial time) to minimize the regret (i.e., the cost of learning and sequential decision making) about the revenue and the fairness metric. Compared with the work of Balseiro et al. (2020), the fact that demand curves are unknown in our problem lead to several specific technical hurdles that require novel solutions. In particular, we have the following challenges: 1. In (Balseiro et al. 2020), the dual problem is unconstrained, leading to complex dual space shapes and unbounded penalty vectors. This causes two problems: first, with dual spaces having complex shapes, the dual update steps may not have closed forms or be solved efficiently for general fairness-induced penalty functions. More importantly, with penalty vectors being unbounded the regret of the problem becomes unbounded as well because of the uncertainty exhibited in the estimated demand functions, rendering bandit learning algorithms impractical. 2. With uncertainty quantified in the estimated demand functions, solving the primal update steps becomes computationally intractable as the objective functions are not necessarily concave any more. Novel algorithms and analysis are required to solve such non-concave problems efficiently and rigorously. Our contributions We are able to address the above challenges with several technical innovations. At a higher level, our algorithm combines the primal-dual-type online policy proposed by Balseiro et al. (2020) with the Upper-Confidence-Bound (UCB) method, where the former solves the fairness-aware online allocation problem with the known demand function and the latter is a widely adopted principle to balance the exploration and exploitation trade-off in many online learning algorithms. However, this combination is not black-box style and we make novel technical contributions in the design and analysis of our algorithm. Moreover, we also introduce new algorithmic ingredients to make sure that our algorithm is computationally efficient. In contrast, most of the online learning algorithms for linear demand models (and even the linear bandit algorithm, LinUCB (Abbasi-Yadkori et al. 2011) ) do not guarantee a polynomial time complexity. Below, we describe our main technical contributions in more details. First, instead of using the UCB of the objective function in the usual online learning and decision making algorithms, we make decisions according to the UCB of the specially designed adjusted reward function. While our adjusted reward function involves the revenue, it does not directly include the fairness metric (as it is not obvious how to decompose such a global metric to each individual time period, as discussed previously). Instead, we include a carefully designed term in the adjusted reward function to relate it to the dual variable. This term, together with our update rule of the dual variable, helps optimize the fairness metric in a global fashion and reflects the inventory constraint at the same time. Second, to control the estimation error of the adjusted reward during learning, we need to design a bounded domain for the dual parameter (i.e., the dual space). In contrast, Balseiro et al. (2020) adopt an unbounded dual space which is unfriendly to the analysis of our learning process, as illustrated in the first bullet point in the previous section. By adopting a bounded domain for the dual parameter, we able able to upper bound the estimation errors as well, leading to correct regret scalings. An additional benefit of our new dual space is that due to its simpler shape, we are able to employ a closed-form dual update rule (Algorithm 2, Section 5) for any fairness metric. In contrast, Balseiro et al. (2020) may only achieve this for selected fairness metrics and the dual update in our algorithm is much simplified. Third, the regret analysis (especially the analysis related to the dual variable) greatly relies on the magnitude of our model parameter estimations. While the natural (regularized) least-squares estimator may not provide the desired bound, we employ an additional convex program M t to compute a set of bounded model estimates. This M t program is also helpful to guarantee the concavity the estimated (adjusted) revenue function so that we may computationally efficiently find its maximizer which crucially connects to the decision we will make at each time step. Finally, to achieve computational efficiency, we innovatively adopt an 8 -norm confidence radius instead of the usual 2 -norm confidence radius when computing the Upper Confidence Bounds so that we are able to maximize the UCB of the adjusted revenue function in polynomial time. Also, by reducing the M t program to a linear program with infinitely many constraints, we design a polynomial-time separation oracle and invoke the Ellipsoid method to efficiently solve the M t program. Both ingredients help our algorithm to achieve the polynomial time complexity that addresses the second bullet point of technical challenges mentioned in the previous section. For the first 3 technical contributions, we provide more concrete explanations at the end of Section 3, after the introduction of notations and the algorithm description. For the last item, please refer to Section 4.1 and Section 4.2 for more details. Related Works In the section, we introduce three streams of literature related to our paper: network revenue management (NRM) with known demand function, revenue management (RM) with demand learning, and fairness in operations management. And we discuss how our paper is appropriately placed into contemporary literature by giving comparisons with closely-related existing works. NRM with known demand function. A large body of the price-based network revenue management literature focuses on the case in which the seller knows the underlying demand function in advance. And it is known that the optimal pricing policy of this case can be computed using dynamic programming (DP). However, the well-known curse of dimensionality of DP makes the optimal pricing policy computationally intractable. As a result, many works in the literature have investigated on developing algorithms that are computationally efficient with a superior revenue performance. The seminal work by Van Ryzin (1994, 1997) proposed a simple but powerful heuristics. Specifically, they solve the optimal price of the fluid approximation model which is a deterministic analog of the DP and choose a static price every time. And their approach achieves an Op ? T q regret. Jasin (2014) introduced an improvement to the static pricing policy by resolving the static price periodically according to the remaining inventory, and attained Oplog T q regret bound. Recently, Wang and Wang (2022) proved that the resolving heuristics can achieve Op1q regret as compared to the optimal policy of the DP. RM with demand learning. There is a large body of literature focusing on the price-based revenue management with demand learning, which are either without inventory constraints (see, e.g., , Den Boer and Zwart 2014, Keskin and Zeevi 2014, Bu et al. 2022) and references therein) or with inventory constraints (see, e.g., (Besbes and Zeevi 2009, Wang et al. 2014, Chen et al. 2014, Ferreira et al. 2018). For dynamic pricing problems without inventory constraints, we refer the readers to (Den Boer 2015) for a detailed review. For price-based revenue management problems with inventory constraints, there are two streams of literature, either considering the nonparametric demand model (Wang et al. 2014, Chen and Shi 2019 or the parametric demand model (see discussion below). Since our paper considers a parametric demand function, we mainly investigate the literature on the revenue management problem with inventory constraints and the parametric model. There are three main approaches for tackling the learning-while-doing challenge. The first approach is using the Explore-Then-Commit strategy, which separates the exploration phase and exploitation phases. This simple strategy has been widely used in online learning tasks, and Zeevi (2009, 2012) and Chen et al. (2014) applied this strategy to the NRM problem and Chen et al. (2014) achieved Op ? T q regret assuming the strong concavity of the revenue function. The second approach is using Thompson sampling to address the exploration-exploitation tradeoff. Ferreira et al. (2018) introduced Thompson sampling into network revenue management and considered both discrete price model and continuous price set with the linear demand model. They obtained a Bayesian regret r Op ? T q instead of the worst-case regret. The most important step in their algorithm for the continuous price set is to solve a quadratic program, which is not guaranteed to be a convex problem and not clear how to be solved efficiently. The third approach is incorporating the Optimism in the Face of Uncertain principle into the primal-dual optimization framework. This approach is closely related to the Bandit-with-Knapsack (BwK) model (Badanidiyuru et al. 2013), which introduces global resource constraints into the multi-armed bandit. Agrawal and Devanur (2019) further generalized BwK to bandit with global convex constraints and concave objective. The work by Agrawal and Devanur (2016), which considered BwK in the linear bandit setting, can be applied to the NRM problem with the discrete price. However, in the continuous price setting, the regret and the running time will exponentially dependent on the number of products due to the discretization procedure. considered the NRM problem with continuous price and generalized linear model. To tackle the high computational complexity due to the continuous price set, they designed a UCB solver to reduce the original optimization problem to the price optimization problem of an ordinary NRM problem by randomly sampling a vector on the unit sphere and using it to linearize the 2 -norm-based UCB term. However, the price optimization problem might still be non-convex and difficult to solve despite this reduction. Our work is closely related to (Agrawal and Devanur 2016) and . However, there are several significant differences. First, the primal-dual framework in (Agrawal and Devanur 2016) and do not consider the fairness regularizer and their algorithms do not directly work in our setting. With the fairness concerns, our algorithm adopts very different primal and dual updates, which requires a different analysis. Second, as compared to the random sampling method in , we introduce the 8 -norm-based UCB term (Section 4.2) which is not only simpler to calculate, but also only sacrifices an Op ? N q factor in the regret. In contrast, even without the fairness consideration, the regret of for the NRM problem is r OpN 3.5 ? T q, about N times our regret. Third, we introduce a feasibility program M t (Section 4.1) to make sure the estimated revenue function is concave and computationally easy to optimize. We are able to combine the above new techniques to derive a computationally efficient low-regret learning-while-doing algorithm for the NRM problem with fairness concerns. Fairness in operations. With the development of data-driven algorithms in machine learning and operations, there is a growing concern about discrimination and unfairness. As a result, the fairness issue has been well studied in the machine learning literature, e.g., fair classification (Dwork et al. 2012, Agarwal et al. 2018, Jang et al. 2022) and fair online learning (Joseph et al. 2016, Liu et al. 2017, Jabbari et al. 2017, Gupta and Kamble 2021, Kandasamy et al. 2020, Baek and Farias 2021. We refer the interested readers to the surveys of fairness in machine learning (Corbett-Davies and Goel 2018, Mehrabi et al. 2021, Hutchinson andMitchell 2019). And we mainly focus on the fairness issue in operations problem, which has also attracted a lot of attention (Bonald et al. 2006, Chen and Wang 2018, Ma et al. 2020, Kallus and Zhou 2021, Kallus et al. 2022, Zhang et al. 2022. Static resource allocation is one of earlier fields in operations that takes different fairness metrics into consideration (Bansal and Sviridenko 2006, Bertsimas et al. 2011, Elzayn et al. 2019, Donahue and Kleinberg 2020, Cai et al. 2021, Bateni et al. 2022). Among many fairness metrics, a widely-acknowledged fairness notion, max-min fairness (Nash Jr 1950) that maximizes the minimum resource allocation, has been widely applied (Bansal andSviridenko 2006, Bertsimas et al. 2011). The fairness notion considered in our paper includes the max-min fairness as a special case and (please refer to Section 2.1 for more details). Another fairness metric that has been well studied is proportional fairness (Azar et al. 2010, Vlasiou et al. 2014, Bateni et al. 2022, which maximizes the overall utility of rate allocations via a logarithmic utility function. There is a vast body of literature considering the fairness in online allocation problem with inventory constraints (with known demand models) (Elzayn et al. 2019, Ma et al. 2020, Balseiro et al. 2020, Chen et al. 2021a, where the decision-maker must take an action upon each arriving resource allocation simultaneously by introducing the regularized online allocation problem. Since our work is most related to (Balseiro et al. 2020), we have thoroughly discussed the technical differences in the introduction section above and we will present the comparison more concretely in Section 3. We also note that (Balseiro et al. 2020) works for the fairness-aware NRM problem in a quantity-based setting, where the decision-maker must irrevocably accept or reject each arriving request given limited resources (a special case of the online allocation problem studied in their paper). In contrast, we study the NRM problem in the price-based setting where the decision-maker has to decide the prices that influence the demand and the demand has always to be met (as long as permitted by the resource constraints). To the best of our knowledge, our work is the first to consider the fairness objective in the price-based NRM problem. Another related line of works study the fairness in dynamic pricing (Cohen et al. 2022, Chen et al. 2021b, Li and Jain 2016, Kallus and Zhou 2021. These works focus on the price fairness, ensuring the prices are similar for different groups and stable over time. Notations The vectors throughout this paper are all column vectors. We denote the set t1, 2, . . . , N u by rN s for any N P N. For vectors a, b P R N , we use a ď b (a ě b respectively) to denote a i ď b i (a i ě b i respectively) for all i P rN s. We use rx t s i to denote the i-th element of the vector x t . For x P R N and Λ P R NˆN , we define the following norms: }x} 1 :" ř N i"1 \|x i \|, }x} 2 :" p ř N i"1 x 2 i q 1{2 , }x} 8 :" max iPrN s x i , and }x} Λ :" ? x J Λx. We use I N to denote the identity matrix of order N . For A P R MˆN , we define the following matrix norms: }A} F :" ř i,j A 2 ij , }A} 2 :" sup }x} 2 "1 }Ax} 2 , }A} 8 :" sup }x}8"1 }Ax} 8 and it is easy to obtain }A} 8 " max iPrM s ř N j"1 \|A ij \|. For square matrix A P R NˆN , we use λ max pAq to denote the largest eigenvalue of A. For a symmetric matrix A P R NˆN , we use A ĺ 0 to represent that A is negative semi-definite. We use rB\|αs to denote the augmented matrix by adding α P R N to the right of the matrix B P R NˆN as a new column. We use the big-O notation f pT q " OpgpT qq to denote that lim sup T Ñ8 f pT q{gpT q ď`8. We use r Op¨q to further omit the logarithmic dependency on N , M , and T . Organization The remainder of this paper is organized as follows. In Section 2, we formulate our problem by introducing the model assumptions and performance measure; we also give plenty of examples of the fairness regularizers to illustrate the potential guidance our paper might bring to the practical scenarios. In Section 3, we present our algorithm and discuss the high-level ideas of the algorithm design. Then we discuss reward and demand estimation (Section 4) and the design of the mirror descent solver (Section 5) in details, which are two key building blocks of our algorithm. In Section 6, we present and prove the main theorem that upper bounds the regret of our algorithm. To demonstrate the empirical performance of our policy, we conduct several numerical experiments and present the results in Section 7. In the end, we give a summary of our paper in Section 8. The proofs of most technical lemmas and the additional experimental results are included in the supplementary materials. Model Description and Assumptions In an NRM model with N types of products and M types of resources, a retailer sells N types of products during a selling season with T time periods. Each product is defined as a combination of M types of unreplenishable resources by the consumption matrix A P R MˆN , where A ij means that selling one unit of the type-i product consumes A ij unit of the type-j resource. At each time period t, the retailer must determine the prices for the N products, i.e., the price vector p t P R Ǹ . The retailer then observes the consumer's demand vector d t P R Ǹ which is realized from an unknown underlying demand function Dpp t q, and finally consumes the resources according to the consumption matrix A. The retailer needs to choose the prices during the selling season to accomplish the following 3 goals: 1. to gradually learn the underline demand function Dpp t q from the observed demands, 2. to maximize the total revenue based on the learned information and given the unreplenishable resource inventory, 3. to balance the consumption of the different types of resources via maximizing the fairness regularizer φp¨q, which will be defined soon. More specifically, the initial inventory levels of the M resources are I 0 " pI 0,1 , . . . , I 0,M q J P R M . At the end of time t, the inventory levels become I t " I t´1´A d t for t " 1, 2, 3, . . . . For convenience we also define the normalized inventory level γ " pγ 1 , . . . , γ M q J :" I 0 {T , which is the average amount of resources that can be used at a time period. For simplicity, we assume that the price range for each product is rp, ps and the retailer has to choose p t in the price set P at each time t. The price set P can either be rp, ps N or a discrete subset in rp, ps N . For brevity, we focus on the case P " rp, ps N , which is much more challenging. One can easily adapt our algorithm and analysis to the discrete price set. The realized demand d t is a random variable centered at Dpp t q, i.e., d t " Dpp t q`ε t where ε t is a zero-mean noise variable (see Assumption 2 for the more precise statement). We consider the linear demand function (which is the most commonly analyzed demand model in literature, e.g., (Keskin and Zeevi 2014) and (Ferreira et al. 2018)) Dpp t q " α`Bp t , where α P R N and B P R NˆN are the model parameters unknown to the retailer. For convenience, we also denote these unknown parameters by θ " pα, Bq P Θ Ď R N 2`N , where Θ is the parameter space. The revenue collected by the retailer at time t is r t " xd t , p t y. We also denote the corresponding expected revenue by rpp t q :" Err t \|p t s " xp t , Dpp t qy . The objective of the retailer is to design a policy π " pπ 1 , . . . , π T q with π t : H t Þ Ñ p t (where H t " tp s , d s u săt is the historical prices and demands before time t) to satisfy the inventory constraint I t ě 0 for all t P t1, 2, . . . , T u and maximize the following expected total revenue plus the fairness regularizer on resource consumption: E « T ÿ t"1 rpp t q`T φ˜1 T T ÿ t"1 Ad t¸ff . Below in Section 2.1 we will discuss more about the fairness regularizer φp¨q; in Section 2.2 we introduce some standard assumptions on the linear demand model and define the regret that our online policy aims to minimize. Fairness Regularizer φp¨q: Assumptions and Examples The fairness on resource consumption is measured by the regularizer φ´1 T ř T t"1 Ad t¯( i.e., the regularizer function φp¨q applied to the average resource consumption vector). Assumption 1. Throughout this paper we impose the following assumptions on φp¨q. 1. φpsq is L-Lipschitz continuous with respect to the }¨} 8 -norm on its effective domain, i.e., \|φps 1 q´φps 2 q\| ď L}s 1´s2 } 8 for any s 1 , s 2 ď γ. 2. There exists φ such that 0 ď φpsq ď φ for all 0 ď s ď γ. φpsq is concave. In the following, we present several regularizers satisfying the above assumptions as examples. We will use s " 1 T ř T t"1 Ad t and s i refers to the average consumption of the type-i resource. Example 1: Weighted Max-min Fairness Regularizer. The first example is rooted in the famous max-min fairness guarantee, which has been well studied in the literature on static resources allocation (Bansal andSviridenko 2006, Bertsimas et al. 2011). The idea behind the max-min fairness guarantee is to promote fairness by maximizing the minimum resource allocation. In our paper, we consider the following weighted max-min fairness regularizer to promote fairness in resource consumption. It is worthy to note that the max-min regularizer in (Balseiro et al. 2020) can be seen as a special case of our weighted max-min regularizer by setting the parameters correspondingly. Formally, we define the weighted max-min fairness regularizer as φpsq :" λ min i pw i s i q, where λ is the parameter to balance between the total revenue goal and the fairness objective, and in the online retailing setting the parameter w i could be selected as the revenue of the resource supplier due to the consumption of one unit type i resource. Example 2: Group Max-min Fairness Regularizer. We may divide the different types of resources into groups and only focus on promoting the minimum consumption of each resource group. In practice, each supplier may provide several types of resources (which naturally forms a group) and the group max-min fairness would be useful if we wish to guarantee fairness among the suppliers. Formally, we define the group max-min fairness regularizer as φpsq :" λ min i ppU r sq i q, 1 where r s " pw 1 s 1 ,¨¨¨, w m s m q J , w i is similarly defined as in Example 1, and U P R KˆM is a 0-1 matrix describing the grouping scheme. In particular, we require that in each column there is exactly 1 non-zero element and in each row there is at least 1 non-zero element, where a simple example of U is as follows, U " " 1 0 1 0 0 1 0 1  . In this example, there are two resource suppliers. The first supplier provides the type-1 and the type-3 resources and the second supplier provides the type-2 and the type-4 resources. Example 3: Range Fairness Regularizer. Range is a fundamental statistical quantity that measures the difference between the highest and the lowest value of a population. The Range fairness regularizer provides incentive to minimize the range among the entries of the weighted average consumption vector r s " pw 1 s 1 ,¨¨¨, w m s m q J . Formally, we define the range fairness regularizer as φpsq :" λpmin i pw i s i q´max i pw i s i qm ax i pw i γ i qq, where´rmin i pw i s i q´max i pw i s i qs is the range of r s and max i pw i γ i q is introduced to guarantee the positiveness of the regularizer. When w i is chosen to be per-unit revenue of the type-i resource supplier, the range fairness regularizer can be applied to promote the revenue fairness across different suppliers; when w i " 1{γ i , this regularizer can evaluate the evenness of resource availability and help to avoid the pre-mature saturation of a few resource types. Example 4: Load Balancing Regularizer. We finally present the load-balancing regularizer proposed in Balseiro et al. (2020). The regularizer is defined as φpsq :" λpmin i ppγ i´si q{γ i q, which measures the minimum relative resource availability, and also helps to make sure that no resource is too demanded. Model Assumptions and Performance Measure Assumption 2. Throughout this paper we impose the following assumptions on the demand model d t " α`Bp t`εt : 1. The noise tε t u T t"1 is a martingale difference sequence adapted to the filtration tF t u T t"1 where F t " tp 1 , d 1 ,¨¨¨, p t , d t , p t`1 u, i.e., Erε t \|F t´1 s " 0. 2. There exists d such that d t ď d almost surely for all t P t1, 2, . . . , T u. 3. The underlying true parameter B in the linear demand model is negative definite; 2 there exists L B (with L B ě 1) such that a α 2 i`} B J e i } 2 2 ď L B for every i P rN s, where e i is the i-th unit vector and B J e i is the i-th row of B. All items in Assumption 2 are quite standard in literature. The third item is usually seen in papers focusing on the linear demand model (see, e.g., (Keskin and Zeevi 2014, Ferreira et al. 2018, Bu et al. 2022). By the definition rpp t q " xd t , p t y and according to Assumption 2, we may upper bound rpp t q by r :" N pd. We now discuss the performance measure for the retailer's policy. We would like to compare the objective value achieved by the retailer with the optimal offline policy, i.e., the one that knows all the model parameters θ: J opt :" max π"pπ 1 ,...,π T q E « T ÿ t"1 rpp t q`T φ˜1 T T ÿ t"1 Ad t¸ff s.t. T ÿ t"1 Ad t ď T γ a.s.(1) J opt upper bounds the objective value achieved by any online policy (i.e., the one without access to θ). In light of this, we define the regret of a policy π up to time horizon T as RpT q :" J opt´E « T ÿ t"1 rpp t q`T φ˜1 T T ÿ t"1 Ad t¸ff .(2) Note that solving the exact value of J opt is quite complicated due to the stochastic nature and adaptivity available to choose π 1 , π 2 , . . . , π T in sequence. We now introduce the following fluid model which is the deterministic and non-adaptive analogue of J opt and easier to analyze. J D :" max p 1 ,...,p T Prp,ps N « T ÿ t"1 rpp t q`T φ˜1 T T ÿ t"1 ADpp t q¸ff s.t. T ÿ t"1 ADpp t q ď T γ.(3) We assert that there exists an optimal solution tp1 , p2 , . . . , pT u to J D such that p1 " p2 "¨¨¨" pT " p˚, since otherwise, we can set p t " p 1 " 1 T ř T t"1 pt for every t, and the objective value of tp 1 1 , p 1 2 , . . . , p 1 T u becomes no smaller due to the concavity of rppq (since rppq " xp, α`Bpy and B is negative definite by Assumption 2) and Jenson's inequality. Therefore we have the following equivalent definition of J D . J D " max pPrp,ps N rT rppq`T φ pADppqqs s.t. ADppq ď γ.(4) We denote p˚by the optimal solution to Eq. (4). The following proposition shows that the fluid model J D is an upper bound of J opt , and its proof is deferred to the supplementary materials. Proposition 1. J opt ď J D . By Proposition 1, we upper bound the regret of any policy π as follows, which will serve as the starting point of the analysis of our proposed policy. RpT q ď T rrpp˚q`φpADpp˚qqs´E « T ÿ t"1 rpp t q`T φ˜1 T T ÿ t"1 Ad t¸ff .(5) Primal-dual Type Algorithm with Demand Learning The pseudo-code of our main algorithm is given in Algorithm 1. To better introduce our algorithm design, we first imagine that the demand function were known and only explain the primal-dual framework. We then add the learning component for the demand function and address the additional challenges raised due to the unknown demand. Given the demand function Dp¨q, thanks to Proposition 1, we may use the fluid model J D , which upper bounds J opt , to calculate an upper estimation of the regret of any online policy. 3 We now focus on the primal formulation of the fluid model, and rewrite it with an auxiliary variable s. p˚:" max pPrp,ps N trppq`φ pADppqq s.t. ADppq ď γu (11) " max pPrp,ps N ,´γďsďγ trppq`φ pADppqq s.t. ADppq " su. Note that we deliberately impose a lower bound constraint s ě´γ in the new formulation. This does not change the optimal value of the program since ADppq is entry-wise non-negative for non-negative A and Dppq. 3 Indeed, this relaxation would not ruin our aimed r Op ? T q regret, as the difference between JD and Jopt is also r Op ? T q for the network revenue management problem either without (Gallego and Van Ryzin 1997) or with the fairness concern (as we will see later in this paper). Algorithm 1 Primal-dual + UCB for fairness-aware NRM with demand learning 1: Initialize the dual variable µ 1 " p0,¨¨¨, 0q P R M . 2: for t " 1, 2, . . . , T do 3: Compute the regularized least-squares estimator pp α t , p B t q P arg min pα,Bq # pN`1qp}α} 2 2`} B} 2 F q`ÿ săt }d s´p α`Bp s q} 2 2 + . (6) 4: Find pq α t , q B t q around pp α t , p B t q such that q B t is negative semi-definite by solving the program pq α t , q B t q P M t :" ! pr α, r Bq : }p r B´p B t q J e i } Λ t ď κ, } r B J e i } 2 ď 2L B @i P rN s and r B`r B J ĺ 0 ) ,(7) where we define the block matrices r B :" r r B\|r αs and p B t :" r p B t \|p α t s (e i is the i-th canonical basis vector), Λ t and κ are defined in Eqs. (20,21) respectively. In the rare case when Eq. (7) is infeasible, we arbitrarily choose pq α t , q B t q as long as q B t is negative semi-definite and } q B J t e i } 2 ď 2L B @i P rN s (e.g., set q B t " 0). 5: Obtain q D t ppq " q α t`q B t p and q r t ppq " A p, q D t ppq E to estimate Dppq and rppq respectively. 6: Update the primal variables p t P arg max pPrp,ps N ! q r t ppq´µ J t A q D t ppq`2∆ f t ppq ) , s t P arg max γďsďγ φpsq`µ J t s ( ,(8) where 2∆ f t ppq is the confidence radius of the adjusted reward estimation rq r t ppq´µ J t A q D t ppqs which will be constructed later in Eq. (23). 7: Charge the price p t , observe the demand d t , consume resources Ad t and update the inventory level I t " I t´1´A d t (the algorithm stops whenever any resource is depleted). 8: Obtain an estimated subgradient of dual function qp¨q at µ t : q g t "´A q D t pp t q`s t . 9: Update the dual variable by invoking the mirror descent solver ς D (Definition 1): µ t`1 " ς D pµ t , q g t ; D, ηq,(9) where we set η " b C 1 C 2 T (C 1 and C 2 are also defined in Definition 1) and D :" tµ P R M \| }µ} 1 ď Cu, C :" L`ppr`φq{γq, γ :" min iPrM s γ i .(10) 10: end for We then transform primal formulation to an unconstrained optimization problem using the Lagrangian dual method. By the well-known weak duality, we have that p˚" max pPrp,ps N ,´γďsďγ min µPR M rppq`φ psq´µ J ADppq`µ J s ( ď min µPR M max pPrp,ps N ,´γďsďγ rppq`φ psq´µ J ADppq`µ J s ( .(12) In light of this, we define the dual function qpµq :" max pPrp,ps N ,´γďsďγ rppq`φ psq´µ J ADppq`µ J s ( " r 7 pA J µq`p´φq˚pµq,(13) where for every µ P R M we define r 7 pA J µq :" max pPrp,ps N rppq´µ J ADppq ((14) and the convex conjugate (following the convention, e.g., Chapter 3 in (Boyd et al. 2004)) p´φq˚pµq :" max γďsďγ φpsq`µ J s ( .(15) By Eq. (12), for every µ P R M , we have that p˚ď qpµq. In the primal-dual framework, we iteratively optimize the primal variables p, s (Line 6) 4 and the dual variable µ (Line 9). When updating the dual variable at Line 9, we use the mirror descent method with the help of a mirror descent solver ς D defined as follows. Definition 1 (Mirror Descent Solver). A mirror descent solver ς D pµ t , q g t ; D, ηq takes µ t P D and q g t as input and returns the updated dual variable µ t`1 P D at each time t. For a sequence of input tq g t u and the initial dual variable µ 1 , if we repeatedly apply ς D and produce a sequence of dual variables tµ t u. The solver makes sure that for all µ P D, T ÿ t"1 xµ t , q g t y ď T ÿ t"1 xµ, q g t y`C 1 η`C 2 ηT,(17) where C 1 and C 2 are constants that only depend on D. In other words, the mirror descent solver should generate a sequence tµ t u to minimize ř T t"1 x¨, q g t y against any stationary benchmark with the regret at most C 1 {η`C 2 ηT . The above-described primal-dual framework is similar to (and inspired by) the algorithm proposed in (Balseiro et al. 2020). However, the key differences are two folds explained as follows. Demand Learning. In contrast to the known demand function in (Balseiro et al. 2020), the demand function is not known to the decision-maker beforehand in our setting. Our algorithm learns the parameterized demand function from historical data via the regularized least-squares estimate (Line 3). We then solve another convex program (Line 4) to make sure the estimated parameters pq α, q Bq are bounded and q B is negative semi-definite. Finally, we use the Upper Confidence Bound of the adjusted reward function rq r t ppq´µ J t A q D t ppqs (Line 6) to compute the primal update. We will explain this in more details in Section 4. An additional feature of our demand estimator is that the reward Upper Confidence Bound is defined based on the 8 -norm of Λ´1 The fundamental reason that we cannot directly adopt D Bal. in our problem, however, is the unboundedness of D Bal. which would lead to an unbounded regret due to the unbounded estimation error of the adjusted reward rq r t ppq´µ J t A q D t ppqs during the learning process (see Eq. (25) for more details). To deal with this issue, we construct a novel, simply and uniformly shaped, and bounded dual space (Eq. (10)). We prove that our dual space encompasses all potential stationary benchmark dual variables µ which is necessary for the desired r Op ? T q regret. Thanks to the newly constructed dual space, an extra benefit enjoyed by our algorithm is that, together with a carefully chosen variant of the exponentiated gradient descent (EG˘) algorithm as the mirror descent solver, we are able to obtain a uniform and closed-form update of the dual variables for all fairness regularizers. In contrast, Balseiro et al. (2020) have to design the dual update step on a case-by-case basis for the fairness regularizers. Also, the closed-form update improves the computational efficiency of the algorithm and is a desired feature in (Balseiro et al. 2020) that is partially achieved for a few selected fairness regularizers. Demand and Reward Estimation The regularized least-squares estimator (Line 3 of Algorithm 1) for demand parameters is frequently used in the linear bandit literature (see, e.g., (Dani et al. 2008, Rusmevichientong and Tsitsiklis 2010, Abbasi-Yadkori et al. 2011). However, as mentioned before, we need to work with the upper 5 On the downside, we sacrifice an Op ? N q factor in the regret bound. However, we view this degradation relatively small compared to the existing OpN 2 q factor which seems necessary in the regret due to the N 2 parameters in B to learn. confidence bound of a specially defined adjusted reward function. Also, we employ an additional step (Line 4) to make sure the estimated parameters pq α, q Bq are bounded, which is crucial to the regret analysis (more specifically, the analysis of the mirror descent solver). Line 4 also guarantees the negative semi-definiteness of q B; when computing the Upper Confidence Bound for the estimation, we innovatively use an 8 -norm confidence radius instead of the usual 2 -norm confidence radius -both ingredients help the algorithm to compute the upper confidence bound in polynomial time. We will show how to computationally efficiently find pq α, q Bq P M t (Line 4) and implement the UCB-type primal update (Line 6) in Section 4.1 and Section 4.2 respectively. To explain our Upper Confidence Bound method in more details, we first introduce some notations. For convenience, we define the stopping time τ " max ! t : min i I t,i ą 0, t ď T ) , which is the last time period when the inventory levels of all resources remain positive. Most of our analysis will be done only for time periods up to τ . We define f t ppq :" rppq´µ J t ADppq(18) to be the adjusted reward function at price p and with respect to µ t . Note that this corresponds to the optimization objective in Eq. (14) when µ " µ t . Our estimation for f t ppq is q f t ppq :" q r t ppq´µ J t A q D t ppq,(19) which corresponds to the first part in the optimization objective of p t in Eq. (8). We also define estimators with regard to pp α t , p B t q as p D t ppq " p α t`p B t p, p r t ppq " A p t , p D t ppq E , p f t ppq :" p r t ppq´µ J t A p D t ppq. Bounding the estimation errors. We now derive the estimation errors of p D t , p r t , p f t and q D t , q r t , q f t , as well as their corresponding upper confidence bounds. For any price vector p, we let r p :" pp, 1q, and then introduce the regularized information matrix at time t to be Λ t :" pN`1q¨I N`1`ÿ săt r p s r p J s .(20) Let κ :" 2 b 2d 2 pN`1q ln`N T p1`p 2 T q˘`2pN`1qL 2 B , and(21)∆ D t ppq :" ? N`1κ}Λ´1 {2 t r p t } 8 and ∆ r t ppq :" ? N`1N pκ}Λ´1 {2 t r p t } 8(22) to be the confidence radii of p r t ppq and p D t ppq respectively. Note that here we use }Λ´1 {2 t r p t } 8 instead of the 2 -norm confidence radius }Λ´1 {2 t r p t } 2 commonly seen in literature. We finally define ∆ f t ppq :" ∆ r t ppq`}µ t } 1¨} A} 8 ∆ D t ppq(23) to be the confidence radius for the adjusted reward estimator p f t . We utilize the famous Confidence Ellipsoid Lemma in (Abbasi-Yadkori et al. 2011) to analyze our 8 -type confidence region, and have the following lemma. Our lemma states that the confidence radii defined above (Eq. (22) and Eq. (23)) hold with overwhelming probability. Lemma 1. With probability at least p1´O pT´1qq, for all t ď τ and all p P rp, ps, we have } p D t ppq´Dppq} 8 ď ∆ D t ppq, \|p r t ppq´rppq\| ď ∆ r t ppq, and \| p f t ppq´f t ppq\| ď ∆ f t ppq. As we will later show in Lemma 5, with probability at least p1´O pT´1qq, we are able to find a feasible pq α t , q B t q P M t in Line 4. Combining the definition of M t and Lemmas 1 and 5, we have the following corollary. Corollary 1. With probability at least p1´O pT´1qq, for all t ď τ and all p P rp, ps, we have } q D t ppq´Dppq} 8 ď 2∆ D t ppq, \|q r t ppq´rppq\| ď 2∆ r t ppq, and \| q f t ppq´f t ppq\| ď 2∆ f t ppq. The program M t . Note that by Lemma 1, p D t , p r t , p f t already serve as good estimators. However, in the rest part of the algorithm (as well as the analysis), we will mainly work with q D t , q r t , q f t , which are defined based on pq α t , q B t q derived by solving the program M t in Line 4 of Algorithm 1. This is due to the following two requirements. 1. The analysis of the mirror descent solver requires an upper bound on the estimated gradient }q g t } 8 which relies on the bound of max i } q B J t e i } 2 (Eqs. (31,32)). 2. The primal update (Eq. (8)) involves maximizing q r t , a quadratic form of q B t , which can be efficiently optimized only when q B t is negative semi-definite so that q r t is concave. By solving the program M t , we find pq α t , q B t q that simultaneously satisfies the above two requirements and stays close to pp α t , p B t q (in terms of the }¨} Λ t norm). In this way, we facilitate both the regret analysis and the efficient computation of the algorithm. Please refer to Sections 4.1 and 4.2 for the efficient implementations of solving M t and the primal update respectively. UCB of the adjusted reward function. When the desired event in Corollary 1 happens, we define f t ppq :" q f t ppq`2∆ f t ppq(24) and have that f t ppq ď f t ppq for all t ď τ and p P rp, ps. Note that f t p¨q is exactly the optimization objective of p t (at Line 6 of Algorithm 1), which is indeed an Upper Confidence Bound (UCB) of the maximization objective in r 7 p¨q (Eq. (14), namely f t p¨q). Since }µ t } 1 P D for all t P rT s and D " tµ P R M \| }µ} 1 ď Cu, we further upper bound ∆ f t ppq by ∆ f t ppq ď ∆ r t ppq`C}A} 8 ∆ D t ppq.(25) Therefore, we set ∆ t ppq :" ∆ r t ppq`maxtC}A} 8 , 1u∆ D t ppq to upper bound all the confidence radii ∆ r t ppq, ∆ D t ppq and ∆ f t ppq. Bounding the total estimation error. In our regret analysis, we will relate the regret incurred at time t to the confidence radii at price p t at the time (which aligns with the general Upper Confidence Bound principle -bounding the regret by the confidence radii of the selected actions). And thus we will be interested in the summation of the estimation errors. The following lemma adapts the celebrated Elliptical Potential Lemma (see, e.g., Theorem 11.7 in (Cesa-Bianchi and Lugosi 2006) and Lemma 9 in (Dani et al. 2008)) to upper bound the total estimation error. Lemma 2. With probability 1, we have the following upper bound for the total estimation error: τ ÿ t"1 ∆ t pp t q ď O´?N`1κ maxtp, 1upN p`maxtC}A} 8 , 1uq¯ˆbN T logpN`1`p 2 T q, where only a universal constant is hidden in the Op¨q notation. Solving M t via Ellipsoid Method In this subsection, we describe how to implement Line 4 and find a feasible solution to M t in polynomial time via the Ellipsoid method. The main lemma of this subsection is Lemma 5. We first introduce the definition of a separation oracle for a convex set K, which is closely related to the Ellipsoid method. Definition 2 (Separation Oracle). For a closed convex set K Ď R n , a separation oracle for K, namely SEP K , is an algorithm that takes a point x P R n as input and correctly decides whether x P K. In the case that x R K, the separation oracle also returns a hyperplane that separates x from K. The hyperplane may be characterized by its norm vector c P R n such that c J x ą c J y for all y P K. The ellipsoid method reduces a convex program feasibility problem to the construction of an efficient separation oracle for the corresponding convex body. The following lemma characterizes such a reduction. The lemma is a simplification of the Theorem 3.2.1 in (Grötschel et al. 2012) modulo the numerical error due to the arithmetic operations on real numbers. 6 Lemma 3. Suppose we could perform exact arithmetic operations on real numbers. Let Ballpx, rq denote the closed ball with radius r and centered at x P R n . Given a closed convex set K Ď R n , suppose that there exist R, r ą 0 such that K Ď Ballpx 0 , Rq and Ballpx 1 , rq Ď K for some x 0 , x 1 P R n . Given R, r, x 0 , and a separation oracle for K, namely SEP K , the Ellipsoid method will return a point in K using Opn 2 logpR{rqq calls to the separation oracle and polypn, logpR{rqq arithmetic operations. It is easy to verify that our M t is a closed convex set in R NˆpN`1q . To apply Lemma 3 to M t , we first upper and lower bound the shape of M t as follows. Lemma 4. Given the desired event described in Lemma 1, we have that M t Ď Ballp p B t , κ ? N q and BallprB´T´2¨I N \|αs, T´4q Ď M t , where we treat the matrices p B t and rB´T´2¨I N \|αs as NˆpN`1q-dimensional vectors. The separation oracle. It remains to design the separation oracle SEP M t . Given r B " r r B\|r αs, we need to verify the following two types of constraints specified in the definition of M t (Eq. (7)). • }p r B´p B t q J e i } Λ t ď κ, @i P rN s. This condition can be verified for each i P rN s by straightforward computation. When the condition is not met for some i P rN s, we have that κ ă }p r B´p B t q J e i } Λ t " }Λ 1{2 t p r B´p B t q J e i } 2 , and there exists c " Λ 1{2 t p r B´p B t q J e i }Λ 1{2 t p r B´p B t q J e i } 2 such that c J Λ 1{2 t p r B´p B t q J e i ą κ ě c J Λ 1{2 t pB 1´p B t q J e i for every B 1 " rB 1 \|α 1 s where pα 1 , B 1 q P M t , which defines the separation hyperplane. • } r B J e i } 2 ď 2L B , @i P rN s. This condition can also be verified for each i P rN s by straightforward computation. When the condition is not met for some i P rN s, we have that 2L B ă } r B J e i } 2 , and there exists c " r B J e i } r B J e i } 2 such that c J r B J e i ą 2L B ě c J B 1J e i for every B 1 " rB 1 \|α 1 s where pα 1 , B 1 q P M t , which defines the separation hyperplane. • r B`r B J ĺ 0. This condition is equivalent to λ max p r B`r B J q ď 0 which can be efficiently verified. If the condition is not satisfied, we can efficiently find a vector c P R N such that x r B`r B J , cc J y ą 0 ě xB 1`p B 1 q J , cc J y for every pα 1 , B 1 q P M t , which defines the separation hyperplane. Combining Lemma 4, and the separation oracle constructed above, we may invoke Lemma 3 with R " κ ? N and r " T´4, and conclude this subsection with the following lemma. Lemma 5. With probability at least p1´O pT´1qq, for all t ď τ , M t is feasible, and we can find pq α t , q B t q P M t via the Ellipsoid method using only polypN, log T, log d, log p, log L B q arithmetic operations on real numbers. Efficient Primal Update We now show that, thanks to the new 8 -norm-based confidence region, we may efficiently implement the primal update (Line 6) by solving OpN q convex optimization problems. We focus on the optimization problem for p t as the one for s t is already convex. Note that max pPrp,ps N ! q f t ppq`2∆ f t ppq ) " max pPrp,ps N !A p´A J µ t , q D t ppq E`2 ? N`1κpN p`}µ t } 1¨} A} 8 q}Λ´1 {2 t r p} 8 ) " max pPrp,ps N " A p´A J µ t , q D t ppq E`2 ? N`1κpN p`}µ t } 1¨} A} 8 q max λPt˘e 1 ,...,˘e N`1 u λ J Λ´1 {2 t r p * " max λPt˘e 1 ,...,˘e N`1 u max pPrp,ps N !A p´A J µ t , q D t ppq E`2 ? N`1κpN p`}µ t } 1¨} A} 8 qλ J Λ´1 {2 t r p ) , where e i (i P t1, 2, . . . , N`1uq is the i-th canonical basis vector in R N`1 . For any λ P t˘e 1 , . . . ,˘e N`1 u, define the convex program (which is convex due to the negative semi-definiteness of q B guaranteed in Line 4) P pλq t :" arg max pPrp,ps N !A p´A J µ t , q D t ppq E`2 ? N`1κpN p`}µ t } 1¨} A} 8 qλ J Λ´1 {2 t r p ) . It is easy to verify that arg max pPrp,ps N ! q f t ppq`2∆ f t ppq ) Ď Y λPt˘e 1 ,...,˘e N`1 u P pλq t . Therefore, to compute the primal update for p t , we only need to first solve 2pN`1q convex programs to identify p pλq t P P pλq t for every λ P t˘e 1 , . . . ,˘e N`1 u, and then select p t P arg max pPtp pλq t :λPt˘e 1 ,...,˘e N`1 uu !A p´A J µ t , q D t ppq E`2 ? N`1κpN p`}µ t } 1¨} A} 8 q}Λ´1 {2 t r p} 8 ) . Mirror Descent Solver ς D and its Closed-form Dual Update In this section, we design the mirror descent solver ς D to satisfy Definition 1. Given the dual space D, for any reference function h that is σ-strongly convex with respect to }¨} 1 over D, the online mirror descent (OMD) algorithm operates in the following way to update the dual variable: µ t`1 P arg min µPD " xµ, q g t y`1 η D h pµ, µ t q * ,(28) where D h px, yq " hpxq´hpyq´∇hpyq J px´yq is the Bregman divergence. It is well-known (see, e.g., Hazan et al. (2016)) that if }q g t } 8 ď G for all t, then if we start with any given µ 1 P D, the tµ t u sequence produced by Eq. (28) guarantees that for any stationary benchmark µ P D, T ÿ t"1 xµ t , q g t y ď T ÿ t"1 xµ, q g t y`s up µPD D h pµ, µ 1 q η`η G 2 2σ T, which matches the requirement of Definition 1. The popular choices of the reference functions are the negative entropy function hpxq " ř n i"1 x i ln x i (so that D h px, yq " ř n i"1 x i lnpx i {y i q, and the OMD algorithm becomes the exponentiated gradient algorithm) and the Euclidean norm hpxq " 1 2 }x} 2 2 (so that D h px, yq " 1 2 }x´y} 2 and the OMD algorithm becomes the projected gradient descent algorithm). However, based on the different shapes of the dual space D, one has to carefully choose h to guarantee its strong convexity and proper definition (e.g., the negative entropy function is not properly defined when any of the coordinate becomes negative). Due to this reason, Balseiro et al. (2020) have to design the reference function on a case-by-case basis for various fairness regularizers φ which shape their dual space D. When designing h, Balseiro et al. (2020) also aim to simplify the update rule (Eq. (28)) with the hope of a closed-form update, so as to reduce the computational cost. However, they are only able to achieve this goal for selected fairness regularizers. In our work, thanks to the simplicity of newly designed dual space D " tµ P R M \| }µ} 1 ď Cu (Eq. (10)), we are able to use a uniform mirror descent solver ς D that enjoys the closed-form update for all fairness regularizers. Our ς D is similar to the OMD algorithm with the negative entropy function. The only issue, however, is that the negative entropy function does not apply to negative coordinates covered by our dual space D. To this end, we employ the special variant of the algorithm that separately deals with the positive weights and negative weights in µ t . The algorithm was proposed by Kivinen and Warmuth (1997) and called EG˘(Exponentiated Gradient Algorithm with Positive and Negative Weights). The EG˘algorithms is formally described in Algorithm 2. Note that instead of a single vector µ t , the algorithm keeps two vectors µt`1 and µt`1, and the update of the two vectors are in simple closed forms. While both vectors are in R M , they respectively represent (the absolute values of) the positive and negative weights in µ t (see Eq. (30)). Due to this technical reason, to use EGȃ s our mirror descent solver, we need to slightly modify the description of our main Algorithm 1. First, we initialize the two vectors as µ1 " µ1 " pC{M, . . . , C{M q, which replaces the initialization (Line 1) of Algorithm 1. We also replace the dual update (Eq. (9)) of Algorithm 1 by pµt`1, µt`1q " EG˘pµt , µt , q g t ; D, ηq, µ t`1 " µt`1´µt`1. Algorithm 2 EG˘pµt , µt , q g t ; D, ηq 1: Compute the µt`1 and µt`1 vectors as follows: 2: for i " 1, 2, . . . , M do rµt`1s i " Crµt s i expp´ηCrq g t s i q ř M i"1`r µt s i expp´ηCrq g t s i q`rµt s i exppηCrq g t s i q˘, rµt`1s i " Crµt s i exppηCrq g t s i q ř M i"1`r µt s i expp´ηCrq g t s i q`rµt s i exppηCrq g t s i q˘. 3: end for 4: return pµt`1, µt`1q. It remains to choose G as the upper bound of }q g t } 8 . To this end, we set G :" 2 maxtp, 1upN`1qL B }A} 8`γ . Since for all i P rN s we have } q B J t e i } 1 ď pN`1q} q B J t e i } 2 and } q B J t e i } 2 ď 2L B , which is guaranteed in Line 4, it is easy to obtain } q B t } 8 " max i } q B J t e i } 1 ď 2pN`1qL B . And thus we could have the following upper bound of } q D t pp t q} 8 } q D t pp t q} 8 " } q B t r p t } 8 ď } q B t } 8 maxtp, 1u ď 2pN`1qL B maxtp, 1u.(31) Therefore, we may upper bound }q g t } 8 by G: }q g t } 8 " }A q D t pp t q´s t } 8 ď }A} 8 } q D t pp t q} 8`γ ď 2 maxtp, 1upN`1qL B }A} 8`γ " G.(32) By directly applying Theorem 2 in Hoeven et al. (2018), we have the following lemma showing that EG˘satisfies our requirement of the mirror descent solver. Lemma 6. By adopting EG˘as our mirror descent solver ς D , Definition 1 is satisfied with C 1 " lnp2M q and C 2 " C 2 G 2 {2.(33) Regret Analysis With the main technical tools ready in hand, we now prove the following main theorem which upper bounds the regret of our Algorithm 1. Theorem 1. When combining Algorithm 1 with our EG˘mirror descent solver (Algorithm 2), we may upper bound the regret of the algorithm by RpT q ď`}A} 8 d{γ`Op1q˘rrpp˚q`φpADpp˚qqs`Op}A} 8 d a T logpM T qq 2 a C 1 C 2 T`O´?N`1κ maxtp, 1upN p`maxtC}A} 8 , 1uq¯ˆbN T logpN`1`p 2 T q, where we may choose values for C 1 and C 2 according to Eq. (33) and only universal constants are hidden in the Op¨q notations. Remark 1. Recall that C " L`ppr`φq{γq, r " N pd, and κ is defined in Eq. (21). Assuming the problem parameters d, p, φ, L, L B , }A} 8 ď Op1q and γ ě Ωp1q, we have that C 1 ď r Op1q, C, C 2 ď OpN q, κ ď r OpN q, and RpT q ď r OpN 5{2 ? T q. The rest of this section is devoted to the proof of Theorem 1. Recall that τ " maxtt : min iPrM s C t,i ą 0, t ď T u is the stopping time till when the inventory levels of all resources remain positive. For convenience, for t ď τ , we define g t :"´ADpp t q`s t .(34) For t ą τ , we set all relevant quantities to zeros: 7 g t " p g t " s t " 0, p t " 0, Dpp t q " d t " 0, @t ą τ. By Eq. (5), we have that RpT q ď T rrpp˚q`φpADpp˚qqs´E « T ÿ t"1 rpp t q`T φ˜1 T T ÿ t"1 Ad t¸ff .(35) The proof of our main theorem follows the general framework of the primal-dual analysis of online optimization problems (e.g., (Beck and Teboulle 2003, Hazan et al. 2016, Balseiro et al. 2022), and will be detailed in 5 steps. The main differences from (Balseiro et al. 2020) is that in Step II, we need to deal with the estimation error in the dual expression that relates to the fairness regularizer (note the q g t term in ř τ t"1 xµ t , q g t y`ř τ t"1 φps t q´T φ´1 T ř T t"1 ADpp t q¯). We bound this part in Steps III and IV. This error can be upper bounded by estimation error of q g t multiplied by the 1 -norm of the dual variables. Our definition of the dual space D " tµ P R M \| }µ} 1 ď Cu again kicks in to help upper bound the error. 6.1. Step I: Replacing d t with Dpp t q and Introducing r R The first step is to replace the real demand d t on the Right-Hand-Side of Eq. (35) with the expected demand Dpp t q so that the resulting expression r R is easier to deal with. By applying the Azuma-Hoeffding inequality and a union bound, we have the following lemma. Lemma 7. With probability at least p1´1{T q, it holds that @i P rM s,ˇˇˇˇT ÿ t"1 rAd t s i´T ÿ t"1 rADpp t qs iˇď Op}A} 8 d a T logpM T qq.(36) Lemma 7 implies that › › › › › T ÿ t"1 Ad t´T ÿ t"1 ADpp t q › › › › › 8 ď Op}A} 8 d a T logpM T qq.(37) Together with the Lipschtz continuity of φp¨q, Eq. (37) implies thaťˇˇˇˇφ˜1 T T ÿ t"1 Ad t¸´φ˜1 T T ÿ t"1 ADpp t q¸ˇˇˇˇď L T › › › › › T ÿ t"1 Ad t´T ÿ t"1 ADpp t q › › › › › 8 ď O˜L}A} 8 d c logpM T q T¸.(38) Now we define the random variable r R :" T rrpp˚q`φpADpp˚qqs´« τ ÿ t"1 rpp t q`T φ˜1 T T ÿ t"1 ADpp t q¸ff .(39) Eq. (38) (which holds with probability at least p1´1{T q) implies that RpT q ď Er r Rs`OpL}A} 8 d a T logpM T qq, and we may turn to upper bound Er r Rs instead. In the following steps, we will upper bound the value of r R conditioned on that the desired events of Lemma 7 and Lemma 1 hold. Note that this scenario happens with probability at least p1´OpT´1qq, and r R is at most T rrpp˚q`φpADpp˚qqs in the rare opposite case. Step II: Bounding the Fluid Optimum by the UCB of the Dual The goal of the second step is, given the desired event of Corollary 1, to establish for each t ď τ that rpp˚q`φ pADpp˚qq ď rpp t q`φ ps t q`Aµ t ,´A p D t pp t q`s t E`2 p∆ r t pp t q`∆ f t pp t qq,(41) where p˚is the optimal solution of the fluid model (Eq. (11) To prove Eq. (41), we first introduce pt :" arg max pPrp,ps N trppq´xµ t , ADppqyu which can be viewed as the desired choice for p t (without the estimation errors of rp¨q and Dp¨q). In the following claim we upper bound the fluid optimum by the exact dual function. Claim 1. rpp˚q`φ pADpp˚qq ď rppt q`φps t q`xµ t ,´ADppt q`s t y . Claim 1 is essentially a restatement of the weak duality and its proof is deferred to the supplementary materials. Comparing Claim 1 and our goal (Eq. (41)), we only need to upper bound the estimation errors. In particular, it suffices to have that \|r pp t q´q r t pp t q \| ď 2∆ r t pp t q and rppt q`xµ t ,´ADppt qy ď q r t pp t q`Aµ t ,´A q D t pp t q E`2 ∆ f t pp t q, where the first inequality is exactly guaranteed by the first item of Corollary 1, for the second inequality, we have that r ppt q´xµ t , ADppt qy " f t ppt q ď f t ppt q ď f t pp t q " p r t pp t q`Aµ t ,´A p D t pp t q E`2 ∆ f t pp t q, where the first inequality is by the third item of of Corollary 1 and the second inequality is due to our Upper-Confidence-Bound-style primal update (Eq. (8)). Now we have established Eq. (41). Together with the definition of r R (Eq. (39)) and that q g t " A q D t pp t q`s t (Line 8 of Algorithm 1), we obtain that r R ď pT´τ qrrpp˚q`φpADpp˚qqs`τ ÿ t"1 2p∆ r t pp t q`∆ f t pp t qq τ ÿ t"1 xµ t , q g t y`τ ÿ t"1 φps t q´T φ˜1 T T ÿ t"1 ADpp t q¸.(42) Observe that in Eq. (42) we have ř τ t"1 φps t q´T φ´1 T ř T t"1 ADpp t q¯and ř τ t"1 xµ t , q g t y. In the next two steps, we will bound them separately. 6.3. Step III: Upper Bounding ř τ t"1 φps t q´T φ´1 T ř T t"1 ADpp t qĪ n this step we upper bound the term ř τ t"1 φps t q´T φ´1 T ř T t"1 ADpp t q¯in Eq. (42) by the dot products between a carefully selected dual variable µ and tg t u (Eq. (34)). It is also important to guarantee that µ stays in the range of our novel dual space D, which is used by our later analysis. Formally, we prove the following lemma. Lemma 8. Let s " 1 T T ÿ t"1 ADpp t q and µ " arg max µPR M t´p´φq˚pµq`xµ, syu. It holds that T ÿ t"1 φ ps t q´T φ˜1 T T ÿ t"1 ADpp t q¸ď´T ÿ t"1 xµ, g t y "´τ ÿ t"1 xµ, g t y . Moreover, we have that µ P D. Noting that when t ą τ we have s t " 0 and thus φps t q ě 0 (by Assumption 1). Together with Eq. (42) and Lemma 8 we further upper bound r R by r R ď pT´τ qrrpp˚q`φpADpp˚qqs`τ ÿ t"1 2p∆ r t pp t q`∆ f t pp t qq`τ ÿ t"1 xµ t , q g t y´τ ÿ t"1 xµ, g t y .(45) 6.4. Step IV: Upper Bounding ř τ t"1 xµ t , q g t y In this step, we upper bound the term ř τ t"1 xµ t , q g t y in Eq. (45) (as well as Eq. (42)) by combining the properties of the mirror descent solver and our Upper-Confidence-Bound-type estimator. Intuitively, we would like to replace q g t by g t so that the term could be compared with the other term ř τ t"1 xµ, g t y in Eq. (45). Formally, we will establish Eq. (48). For any µ P D, by applying the definition of the mirror descent solver in Definition 1, we have that τ ÿ t"1 xµ t , q g t y ď τ ÿ t"1 xµ, q g t y`C 1 η`C 2 ηT ď τ ÿ t"1 xµ, g t y`τ ÿ t"1ˇA µ, A q D t pp t q´ADpp t q Eˇˇˇ`C 1 η`C 2 ηT,(46) where C 1 and C 2 are the constant parameters in Definition 1. By Hölder's Inequality, (for any µ P D) we have thaťˇˇA µ, A q D t pp t q´ADpp t q Eˇˇˇď }µ} 1¨} A q D t pp t q´ADpp t q} 8 ď 2C}A} 8 ∆ D t pp t q .(47) Note that here we crucially rely on our definition of the dual space D " tµ P R M \| }µ} 1 ď Cu. Combining Eq. (46) and Eq. (47), for any µ P D, we establish that τ ÿ t"1 xµ t , q g t y ď τ ÿ t"1 xµ, g t y`C 1 η`C 2 ηT`τ ÿ t"1 2C}A} 8 ∆ D t pp t q ,(48) Recall the definition of µ in Eq. (43). Let µ " µ`δ where δ P R M satisfying µ`δ P D will be determined later. Plugging our choice of µ into Eq. (48), we have that τ ÿ t"1 xµ t , q g t y ď τ ÿ t"1 xµ, g t y`τ ÿ t"1 xδ, g t y`C 1 η`C 2 ηT`τ ÿ t"1 2C}A} 8 ∆ D t pp t q .(49) Recalling ∆ t ppq :" ∆ r t ppq`maxtC}A} 8 , 1u∆ D t ppq and combining Eq. (45) and Eq. (49), we obtain that r R ď pT´τ qrrpp˚q`φpADpp˚qqs`τ ÿ t"1 xδ, g t y`C 1 η`C 2 ηT`τ ÿ t"1 4∆ t pp t q.(50) 6.5. Step V: Choosing Parameters and Putting Things Together We finally choose the proper parameters to upper bound r R and conclude the proof. For the choice of δ, we discuss the following two cases. Case 1: τ " T . If none of the resources depletes before time horizon T , i.e., τ " T , we set δ " 0. Now, Eq. (50) implies that r R ď C 1 η`C 2 ηT`τ ÿ t"1 4∆ t pp t q. Case 2: τ ă T . If τ ă T , then there exists a resource i P rM s such that τ ÿ t"1 rAd t s i`} A} 8 d ě T γ i .(51) We now set δ " prrpp˚q`φpADpp˚qqs{γ i q e i where e i is the i-th unit vector, it is easy to verify that µ " µ`δ P D. Thus, combining Eq. (36) and Eq. (51), we have that τ ÿ t"1 xδ, g t y ď prrpp˚q`φpADpp˚qqs{γ i q τ ÿ t"1 prs t s i´r ADpp t qs i q ď prrpp˚q`φpADpp˚qqs{γ i q τ ÿ t"1 pγ i´r Ad t s i q`Op}A} 8 d a T logpM T qq ď prrpp˚q`φpADpp˚qqs{γ i q`pτ´T qγ i`} A} 8 d˘`Op}A} 8 d a T logpM T qq.(52) Plugging Eq. (52) back into Eq. (50), we obtain that r R ď`}A} 8 d{γ˘rrpp˚q`φpADpp˚qqs`Op}A} 8 d a T logpM T qq`C 1 η`C 2 ηT`τ ÿ t"1 4∆ t pp t q. Combining the above two cases and setting η " b C 1 C 2 T , together with Lemma 2, we get that r R ď`}A} 8 d{γ˘rrpp˚q`φpADpp˚qqs`Op}A} 8 d a T logpM T qq 2 a C 1 C 2 T`O´?N`1κ maxtp, 1upN p`maxtC}A} 8 , 1uq¯ˆbN T logpN`1`p 2 T q. Together with Eq. (40) and the discussion about the rare case when either of the desired events of Corollary 1 and Lemma 7 fails, we conclude that RpT q ď`}A} 8 d{γ`Op1q˘rrpp˚q`φpADpp˚qqs`Op}A} 8 d a T logpM T qq 2 a C 1 C 2 T`O´?N`1κ maxtp, 1upN p`maxtC}A} 8 , 1uq¯ˆbN T logpN`1`p 2 T q.˝ Numerical Experiments In this section, we present the numerical experiments on the synthetic data sets to illustrate the effectiveness of our algorithm. For consistency, we use the NRM example presented in Zeevi 2012, Ferreira et al. 2018). In this example, the retailer sells two products (N " 2) using three resources (M " 3), and the resource consumption matrix is defined as U " » - 1 1 3 1 0 5 fi fl . The underlying linear demand function is defined as Dppq " " 8 9 `"´1 .5 0 0´3  p. In contrast to Zeevi 2012, Ferreira et al. 2018) which use a discrete price set in their experiments, we use the continuous price set to test the effectiveness of our algorithm for handling large price sets. We assume that the price range for each product is r1, 5s. In addition, we choose the weighted min-max fairness regularizer φpsq :" λ min i pw i s i q with w i " 1 for all i. And we test two initial inventory levels (γ " p15, 12, 30q and γ " p10, 8, 20q) and four fairness regularization level (λ P t0, 0.5, 1.0, 1.5u). For brevity, we present the numerical results of γ " p15, 12, 30q in this section and leave the numerical results of initial inventory level γ " p10, 8, 20q to Section EC.5 in the supplementary materials. Implementation Details. We implement Algorithm 1 with C " 5, η " 0.01{ ? T and time horizon T P t100, 500, 1000, 2000, 3000, 4000, 6000, 8000, 10000u. We generate the demand noise from the truncated Gaussian distribution clippN p0, 1q, 1q, where clippx, 1q " $ & %´1 x ă´1; x \|x\| ď 1; 1 x ą 1 , . - . We set the coefficient of }Λ´1 ? ln T . Although the regression regularization parameter in Eq. (6) was set as N`1 to obtain better dependency on N of the theoretical regret bound, we fix this parameter to be 0.001 in our experiments for the better empirical performance. We conduct 10 trials independently for each case, and plot the average result of these trials in Figure 1 and Figure 2. We also use the shaded region around each curve to indicate the 95% confidence interval across the 10 trials. Figure 1 The performance of Algorithm 1 with γ " p15, 12, 30q and λ P t0, 0.5, 1.0, 1.5u. Here the x-axis of the left figure is the square root of the total time periods T and the y-axis is the cumulative regret defined in Eq. (5). The x-axis of the right figure is the total time periods T and the y-axis is the relative regret defined in Eq. (53). Figure 2 The max-min fairness mini´1 T ř T t"1 rAdtsi¯and the average reward 1 T ř T t"1 rpptq at regularization levels λ P t0, 0.5, 1.0, 1.5u under the initial inventory level γ " p15, 12, 30q. Results. In the left of Figure 1 is the plot of the regret of Algorithm 1 with γ " p15, 12, 30q and λ P t0, 0.5, 1.0, 1.5u versus the square root of the total time periods T . This figure clearly demonstrates the regret of our algorithm grows at rate r Op ? T q for all regularization levels λ P t0, 0.5, 1.0, 1.5u, which is consistent with the theoretical guarantee of Theorem 1. In the right of Figure 1 we plot the relative regret of Algorithm 1 versus the total time periods T , where the relative regret is defined as T rrpp˚q`φpADpp˚qqs´E " ř T t"1 rpp t q`T φ´1 T ř T t"1 Ad t¯ı T rrpp˚q`φpADpp˚qqs .(53) Note that the narrow 95% confidence intervals indicate the stability and robustness of our algorithm. We further empirically study the impact of the fairness regularization level λ to the utilization of the resources. In the left of Figure 2 is the plot of the max-min fairness versus the total time periods T with γ " p15, 12, 30q and λ P t0, 0.5, 1.0, 1.5u, where the max-min fairness is defines as the minimum element of the average resource consumption vector min i´1 T ř T t"1 rAd t s i¯. In the right of Figure 2 we plot the average reward versus the total time periods T , where the average reward is defines as 1 T ř T t"1 rpp t q. These figures show the max-min fairness increases and the average reward decreases as λ grows, indicating the natural trade-off between fairness and the average reward. We also find that the max-min fairness could be enhanced greatly with a small sacrifice of the average reward reduction. In addition, the variance of max-min fairness is large when there is no fairness regularizer (λ " 0), and the variance becomes very small after introducing the fairness regularizer. This interesting phenomenon shows that the fairness regularizer could not only enhance the max-min fairness, but also provide good control of its variance. Conclusion This paper studies the price-based network revenue management with both fairness concern and demand learning, which is motivated by the practical needs of industries such as online retailing and airline applications. To tackle the challenges of this task, we make several innovative technical contributions, which have the potential to be applied to other operations management problems. We propose a primal-dual-type online policy with Upper-Confidence-Bound (UCB) learning method to simultaneously learn the unknown demand function and optimize the composite objective concerning both the NRM revenue and the fairness metric. Both theoretical analysis and numerical results show the effectiveness and the ability to balance the trade-off between revenue and fairness of the developed policy. For future directions, one can consider adapting the framework in this paper to other revenue management applications with both fairness concern and demand learning. One could also study the demand learning of non-parametric demand functions and consider adapting our framework to other global ancillary objectives beyond fair consumption across resources. Supplementary materials EC.1. Some useful technical lemmas Lemma EC.1 (Azuma-Hoeffding Inequality). Let`tpD k , F k qu 8 k"1˘b e a martingale difference sequence for which there are constants tpa k , b k qu n k"1 such that D k P ra k , b k s almost surely for all k " 1, . . . , n. Then, for all t ě 0, P «ˇˇˇˇˇn ÿ k"1 D kˇě t ff ď 2exp˜´2 t 2 ř n k"1 pb k´ak q 2¸. Recall that B " rB\|αs P R NˆpN`1q p B " r p B\|p αs P R NˆpN`1q , and r p :" pp, 1q. Thus we have d J t e i " pB J e i q J r p t`εi . Noting }B J e i } 2 ď L B , ε i ď d and }r p t } 2 ď p ? N`1 and applying the Theorem 2 in Abbasi-Yadkori et al. (2011), we have the following confidence ellipsoid lemma. Lemma EC.2. Recall that Λ t " pN`1q¨I N`1`ř săt r p s r p J s , for any δ ą 0, with probability at least 1´δ, for all t we have Proof of Proposition 1. By the constraint of Eq. (1) , it is easy to obtain E " ř T t"1 Ad t ı ď T γ. P « }pB´p B t q J e i } Λ t ď d d pN`1q lnˆ1`t p 2 δ˙`? N`1L B ff ě 1´δ.( Therefore, J opt ď $ & % max π E " ř T t"1 rpp t q`T φ´1 T ř T t"1 Ad t¯ı s.t. E " ř T t"1 ADpp t q ı ď T γ , . -. With the concavity of rpp t q by Jenson's inequality we obtain T ÿ t"1 rpp t q ď T r˜1 T T ÿ t"1 p t¸. Therefore, we have $ & % max π E " ř T t"1 rpp t q`T φ´1 T ř T t"1 Ad t¯ı s.t. E " ř T t"1 ADpp t q ı ď T γ , . - ď $ & % max π E " T r´1 T ř T t"1 p t¯`T φ´1 T ř T t"1 Ad t¯ı s.t. E " ř T t"1 ADpp t q ı ď T γ , . -. ec2 e-companion to Chen, Lyu, Wang and Zhou: Fairness-aware NRM with Demand Learning With the concavity of rp¨q and φp¨q, using Jenson's inequality again, we have $ & % max π E " T r´1 T ř T t"1 p t¯`T φ´1 T ř T t"1 Ad t¯ı s.t. E " ř T t"1 ADpp t q ı ď T γ , . - ď $ & % max π T r´E " 1 T ř T t"1 p t ı¯`T φ´E " 1 T ř T t"1 ADpp t q ıs .t. E " ř T t"1 ADpp t q ı ď T γ , . - ď J D , where the last inequality is due to Eq. (4). Combining the above inequalities, we complete the proof.E C.2.2. Assumptions Validation of the Fairness Regularizers We will prove the fairness regularizers proposed in Section 2.1 satisfy Assumption 1 (we only validate Assumption 1.1 and Assumption 1.3, since Assumption 1.2 can be easily validated). Example 1: Weighted Max-min Fairness Regularizer: φpsq :" λ min i pw i s i q. We first show φp¨q is L-Lipschitz continuous with respect to the }¨} 8 -norm in the following way, φpsq´φptq " λpmin i pw i s i q´min i pw i t i qq ď λ max i p\|w i s i´wi t i \|q ď pλ max i w i q}s´t} 8 . Next we will show the convatity of φp¨q. For any s, t and α P r0, 1s, we have φpαs`p1´αqtq " λpmin i pαw i s i`p 1´αqw i t i q ě λpα min i pw i s i q`p1´αq min i pw i t i qq ě αφpsq`p1´αqφptq. Example 2: Group Max-min Fairness Regularizer: φpsq :" λ min i ppU r sq i q, where r s " pw 1 s 1 ,¨¨¨, w m s m q J . We first show φp¨q is L-Lipschitz continuous with respect to the }¨} 8 -norm in the following way, Example 3: Range Fairness Regularizer:φpsq :" λpmin i pw i s i q´max i pw i s i q`max i pw i γ i qq. Example 4: Load Balancing Regularizer:φpsq :" λpmin i ppγ i´si q{γ i q. We have shown min i pw i s i q is L-Lipschitz continuous with respect to the }¨} 8 -norm and concave. By this fact, it is easy to note that Example 3 and Example 4 are also satisfy Assumption 1.E C.3. Proofs Omitted in Section 3 EC.3.1. Proof of Lemma 1 Proof of Lemma 1. Let δ in Lemma EC.2 be 1{pN T q, and thus with probability 1´1{T for any i P rN s and t ď τ it holds that }pB´p B t q J e i } 2 Λ t " }Λ 1{2 t ppB´p B t q J e i q} 2 2 ď 2d 2 pN`1q ln`N T p1`p 2 T q˘`2pN`1qL 2 B , (EC.2) where the inequality is due to pa`bq 2 ď 2a 2`2 b 2 . By Eq. (EC.2) and Cauchy-Schwarz inequality, we have }p p B t´B qr p} 8 ď max iPrN s \|ppB´p B t q J e i q J r p\| ď max iPrN s \|ppB´p B t q J e i q J Λ 1{2 t Λ´1 {2 t r p\| ď max iPrN s }Λ 1{2 t ppB´p B t q J e i q} 2 }Λ´1 {2 t r p} 2 ď b 2d 2 pN`1q ln`N T p1`p 2 T q˘`2pN`1qL 2 B b r p J Λ´1 t r p.(ď b 2d 2 N pN`1q ln`N T p1`p 2 T q˘`2N pN`1qL 2 B b r p J Λ´1 t r p. (EC.4) Proof of \|p r t ppq´rppq\| ď ∆ r t ppq. By Eq. (EC.4), we havěˇˇx p, p B t r py´xp, Br pyˇˇď }p} 2 }p p B t´B qr p} 2 ď ? N p b 2d 2 N pN`1q ln`N T p1`p 2 T q˘`2N pN`1qL 2 B b r p J Λ´1 t r p, (EC.5) where the first inequality is due to Cauchy-Schwarz inequality. Since for any vector x P R d it holds that }x} 2 ď ? d}x} 8 ď ? d}x} 2 . EC.3.4. Proof of Lemma 4 Lemma EC.3. With probability at least p1´O pT´1qq, for all t ď τ , we have pα, B´T´2¨I N q P M t , where pα, Bq is the underlying true parameter and M t " tpr α, r Bq : }p r B´p B t q J e i } Λ t ď κ, @i P rN s and r B`r B J ĺ 0u. Proof of Lemma EC.3. Note that by the triangle inequality, for all t ď τ and i P rN s we have }prB´T´2¨I N \|αs´p B t q J e i } Λ t ď }pB´p B t q J e i } Λ t`} rT´2¨I N \|0s J e i } Λ t . (EC.16) For }prB´T´2¨I N \|αs´p B t q J e i } Λ t , by Eq. (EC.2), with probability 1´1{T for any i P rN s and t ď τ it holds that }pB´p B t q J e i } Λ t ď b 2d 2 pN`1q ln`N T p1`p 2 T q˘`2pN`1qL 2 B , (EC.17) Let diagpΛ t q i be the i-th element of the diagonal of Λ t . For }rT´2¨I N \|0s J e i } Λ t , we have }rT´2¨I N \|0s J e i } 2 Λ t ď diagpΛ t q i¨T´4 ď pN`1`p 2 T q{T 4 , (EC.18) where the last inequality is due to diagpΛ t q i " N`1`diagp ř săt r p s r p J s q i ď N`1`p 2 T . Invoking Eq. (EC.17) and Eq. (EC.18) into Eq. (EC.16), we have }prB´T´2¨I N \|αs´p B t q J e i } Λ t ď b 2d 2 pN`1q ln`N T p1`p 2 T q˘`2pN`1qL 2 B`b pN`1`p 2 T q{T 4 (EC.19) ď 2 b 2d 2 pN`1q ln`N T p1`p 2 T q˘`2pN`1qL 2 B " κ. By assumption }B J e i } 2 ď L B and L B ě 1, we have }rB´T´2¨I N \|αs J e i } 2 ď }B J e i } 2`} rT´2¨I N \|0s J e i } 2 ď L B`1 {T 2 (EC.20) ď 2L B . And it is easy to note that pB´T´2¨I N q`pB´T´2¨I N q J ĺ 0 with the assumption B`B J ĺ 0. Therefore, we complete the proof of this lemma.L emma EC.4. When the desired event in Lemma EC.3 happens , rB´T´2¨I N \|αs`X P M t for any X P tX P R NˆpN`1q \|}X} F ď 1{T 4 u. Proof of Lemma EC.4 Note that by the triangle inequality, for all t ď τ and i P rN s we have }prB´T´2¨I N \|αs`X´p B t q J e i } Λ t ď }prB´T´2¨I N \|αs´p B t q J e i } Λ t`} X J e i } Λ t .(}prB´T´2¨I N \|αs´p B t q J e i } Λ t ď b 2d 2 pN`1q ln`N T p1`p 2 T q˘`2pN`1qL 2 B`b pN`1`p 2 T q{T 4 . (EC.22) And for any X P tX P R NˆpN`1q \|}X} F ď 1{T 4 u, we can upper bound }X J e i } Λ t as follows }X J e i } 2 Λ t ď λ max pΛ t q}X J e i } 2 2 ď λ max pΛ t q¨T´8 ď pN`1qp1`p 2 T q{T 8 , (EC.23) where the first inequality is due to x J Λx ď λ max pΛq}x} 2 2 for any symmetric matrix Λ, the second inequality is due to }X J e i } 2 ď }X} F ď 1{T 4 , and the last inequality is due to λ max pΛ t q " N`1λ max p ř săt r p s r p J s q ď N`1`tracep ř săt r p s r p J s q ď pN`1qp1`p 2 T q. Invoking Eq. (EC.22) and Eq. (EC.23) into Eq. (EC.21), we have }prB´T´2¨I N \|αs`X´p B t q J e i } Λ t ď b 2d 2 pN`1q ln`N T p1`p 2 T q˘`2pN`1qL 2 B`b pN`1`p 2 T q{T 4`b pN`1qp1`p 2 T q{T 8 ď 2 b 2d 2 pN`1q ln`N T p1`p 2 T q˘`2pN`1qL 2 B " κ. Now we need to prove the third constraint is satisfied by rB´T´2¨I N \|αs`X. To facilitate our discussion let r X P R NˆN be the square matrix after deleting the last column of X. Since we have the assumption that B`B J ĺ 0, we only need to show p r X´T´2¨I N q`p r XT´2¨I N q J ĺ 0 to prove that pB´T´2¨I N`r Xq`pB´T´2¨I N`r Xq ĺ 0. By the fact that λ max pΛq " }Λ} 2 and }Λ} 2 ď }Λ} F if Λ is symmetric, we have λ max p r X`r X J´2 T´2¨I N q " λ max p r X`r X J q´2{T 2 ď } r X`r X J } F´2 {T 2 ď 2} r X} F´2 {T 2 ď 0, e-companion to Chen, Lyu, Wang and Zhou: Fairness-aware NRM with Demand Learning ec9 where the last inequality is due to } r X} F ď }X} F ď 1{T 4 . Therefore, combining pB´T´2¨I N`r Xq`pB´T´2¨I N`r Xq J ĺ 0 with Eq. (EC.24) and Eq. (EC.25) , we complete the proof of this lemma. Proof of Lemma 4. Note that Λ t " pN`1q¨I N`1`ř săt r p s r p J s , For all i P rN s, we have }p r B´p B t q J e i } 2 ď }p r B´p B t q J e i } Λ t . And thus, for pr α, r Bq P M t , it holds that }p r B´p B t q J e i } 2 ď κ @i P rN s. Therefore, we have } r B´p B t } F ď κ ? N , i.e., M t Ď Ballp p B t , κ ? N q when treating the matrix p B t as an NˆpN`1q-dimensional vector. Combing Lemma EC.3 and Lemma EC.4, we will have the following conclusion to complete the proof of the second part of this lemma. Given the desired event Eq. (EC.2) in Lemma 1, for all t ď τ , it holds that rB´T´2¨I N \|αs`X P M t for any X P tX P R NˆpN`1q \|}X} F ď 1{T 4 u, i.e., BallprB´T´2¨I N \|αs, T´4q Ď M t , when treating the matrix rB´T´2¨I N \|αs as an NˆpN`1q-dimensional vector.E C.4. Proofs Omitted in Section 6 EC.4.1. Proof of Lemma 7 Proof of Lemma 7. Combining the definition of }A} 8 and the boundedness of d t , we have }Aε t } 8 ď }A} 8 }ε t } 8 ď }A} 8 d. By the assumption on the demand noise, tε t u T t"1 is martingale difference sequence, so as trAε t s i u T t"1 . Therefore, applying Azuma-Hoeffding's inequality (Lemma EC.1) , for all i P rM s with probability 1´1{pM T q we have thaťˇˇˇˇT Proof of Claim 1. Recall that the goal is to prove rpp˚q`φ pADpp˚qq ď rppt q`φps t q`xµ t ,´ADppt q`s t y. (EC.27) By Eq. (14) and the definitions of pt and s t (Eq. (8)), it holds that r ppt q " max pPrp,ps N rppq´xA J µ t , Dppqy ( " r 7`AJ µ t˘`x µ t , ADppt qy , φ ps t q`xµ t , s t y " max γďsďγ tφpsq`xµ t , syu " p´φq˚pµ t q , which leads to rppt q`φps t q " r 7`AJ µ t˘`p´φ q˚pµ t q`xµ t , ADppt qy´xµ t , s t y " qpµ t q`xµ t , ADppt q´s t y ě p˚`xµ t , ADppt q´s t y , (EC.28) where the second equality is by the definition of q (Eq. (13)) and the last inequality is due to the weak duality (Eq. (16)). Combining Eq. (EC.28) and the definition of p˚(Eq. (11)), we prove Eq. (EC.27).E C.4.3. Proof of Lemma 8 Proof of Lemma 8. We start by proving Eq. (44). By the definition that s " 1 T ř T t"1 ADpp t q and µ " arg max µPR M t´p´φq˚pµq`xµ, syu, we have that p´φq˚pµq`xµ, sy " max µPR M t´p´φq˚pµq`xµ, syu . By Assumption 1, we have that´φpsq is convex and closed with the closed domain ts :´γ ď s ď γu (since φp¨q is continuous), which implies that p´φq˚˚psq "´φpsq @s :´γ ď s ď γ. Thus, for s P ts :´γ ď s ď γu, it holds that φpsq " max µPR M t´p´φq˚pµq`xµ, syu " max µPR M t´p´φq˚p´µq`xµ, syu "´p´φq˚pµq`xµ, sy (EC.29) Let r s " 1 T ř T t"1 s t . Recall the definition of p´φq˚p¨q (Eq. (15)), we obtain that p´φq˚pµq " max γďsďγ tφpsq`xµ, syu ě 1 T T ÿ t"1 pφ ps t q`xµ, s t yq " 1 T T ÿ t"1 pφ ps t q`xµ, r syq. Combining the two equations above, we have that Finally, by the definition of g t (Eq. (34)), we have that T xµ, r s´sy " ř T t"1 xµ, g t y. Together with Eq. (EC.30), we prove Eq. (44). We now turn to show that µ P D. By the definition of p´φq˚p¨q (Eq. (15)), for all s :´γ ď s ď γ, we have that p´φq˚pµq ě φ psq`xµ, sy . Together with Eq. (EC.29), for all s P ts :´γ ď s ď γu, we have that φpsq ě φpsq`xµ, s´sy . (EC.31) By the definition of the dual norm, it holds that }µ} 1 " max v:}v}8"1 xµ, vy . Let v˚" arg max v:}v}8"1 xµ, vy. Since s is an interior point of r´γ, γs, there exists a small real number α such that s`αv˚P r´γ, γs. Plug s " s`αv˚into Eq. (EC.31), we obtain that φpsq´φps`αv˚q ě xµ, αv˚y " α}µ} 1 . By Assumption 1, we have that φp¨q is L-Lipschitz continuous with respect to the }¨} 8 -norm. Therefore, we have }µ} 1 ď L, thus µ P D " tµ P R M \| }µ} 1 ď Cu (since C ě L).E C.5. Additional Experimental Results In this section, we present the numerical results of initial inventory level γ " p10, 8, 20q. In the left of Figure EC.1 is the plot of the regret of Algorithm 1 with γ " p10, 8, 20q and λ P t0, 0.5, 1.0, 1.5u versus the square root of the total time periods T . In the right of Figure γ " p10, 8, 20q and λ P t0, 0.5, 1.0, 1.5u, where the max-min fairness is defines as the minimum element of the average resource consumption vector min i´1 T ř T t"1 rAd t s i¯. In the right of Figure EC.2 we plot the average reward versus the total time periods T , where the average reward is defines as 1 T ř T t"1 rpp t q. It is easy to find that the numerical results of initial inventory level γ " p10, 8, 20q are almost the same as those in the case γ " p15, 12, 30q presented in Section 7, which justifies the effectiveness of our algorithm for different initial inventory levels. The performance of Algorithm 1 with γ " p10, 8, 20q and λ P t0, 0.5, 1.0, 1.5u. Here the x-axis of the left figure is the square root of the total time periods T and the y-axis is the cumulative regret defined in Eq. (5). The x-axis of the right figure is the total time periods T and the y-axis is the relative regret defined in Eq. (53). Figure EC.2 The max-min fairness mini´1 T ř T t"1 rAdtsi¯and the average reward 1 T ř T t"1 rpptq at regularization levels λ P t0, 0.5, 1.0, 1.5u under the initial inventory level γ " p10, 8, 20q. request and generates a reward and the consumption of resources. The related works by Balseiro et al. (2020), Chen et al. (2021a) considered both fairness and revenue management. The policy in the work by Chen et al. (2021a) achieves a Op ? T q bound for the cumulative unfairness and a bounded revenue regret. Balseiro et al. (2020) considered revenue maximization and fairness t where the 2 -norm is usually adopted in the linear bandit literature. Together with the convex program solved in Line 4, our definition of the reward Upper Confidence Bound renders the primal update a combination of a few convex optimization problems (Eq. (8)) that can be efficiently solved, which will be further explained in Section 4.2. 5 A New Dual Space. The dual space D is a crucial component in the design of the mirror descent solver ς D and affects the regret analysis. Balseiro et al. (2020) adopt a dual space D Bal. " tµ P R M \| sup aďγ φpaq`µ J au which might have different shapes for difference fairness regularizer φp¨q. ), and recall the definitions of the primal variables p t and s t in Eq. (8). The Right-Hand-Side of Eq. (41) can be viewed as the Upper-Confidence-Bound of the dual function qpµ t q. Eq. (EC.20), we have}prB´T´2¨I N \|αs`Xq J e i } 2 ď }rB´T´2¨I N \|αs J e i } 2`} X J e i } 2 ď L B`1 {T 2`1 {T 4 ď 2L B .(EC.25) t qs iˇď Op}A} 8 d a T logpM T qq. (EC.26) By a union bound, we have that with probability at least p1´1{T q, Eq. (EC.26) holds for every i P rM s, proving the lemma.˝ EC.4.2. Proof of Claim 1 φ ps t q`T xµ, r s´sy .(EC.30) the relative regret of Algorithm 1 versus the total time periods T In the left of Figure EC.2 is the plot of the max-min fairness versus the total time periods T with Figure EC. 1 1Figure EC.1 The performance of Algorithm 1 with γ " p10, 8, 20q and λ P t0, 0.5, 1.0, 1.5u. Here the x-axis of the EC.1) EC.2. Proofs Omitted in Section 2 EC.2.1. Proof of Proposition 1 ppU r tq i qq ď λ}U pr s´r tq} 8 ď λ}U } 8 }r s´r t} 8 Next we will show the convatity of φp¨q. For any s, t and α P r0, 1s, we have φpαs`p1´αqtq " λpmin ppU r tq i qq ě αφpsq`p1´αqφptq. e-companion to Chen, Lyu, Wang and Zhou: Fairness-aware NRM with Demand Learningφpsq´φptq " λpmin i ppU r sq i q´min i ď pλ}U } 8 max i w i q}s´t} 8 . i pαpU r sq i`p 1´αqpU r tq i q ě λpα min i ppU r sq i q`p1´αq min i ec3 EC.3) Noting that }p p B t´B qr p} 2 ď ? N }p p B t´B qr p} 8 , by Eq. (EC.3) we have }p p B t´B qr p} 2 ď ? N }p p B t´B qr p} 8 EC.21) When the desired event in Lemma EC.3 happens, by Eq. (EC.19), for any i P rN s and t ď τ it holds that In general, we may combine the grouping operation with any fairness regularizer satisfying the Assumption 1 to obtain a group version of the fairness regularizer, but for simplicity, we only present the group version of the weighted max-min regularizer here. B is is negative definite (not necessarily symmetric) if for any z P R n , it holds that z J Bz ă 0. When optimizing p, we need to deal with the upper confidence bounds of the estimated quantities, which will be explained soon. The numerical error analysis is often tedious but straightforward, which is also the case in this subsection. Therefore, we choose to omit this part and emphasize the main algorithmic idea more clearly. We will treat Dpptq as a symbol rather than a function of pt for t ą τ . Combining Eq. (EC.5) and Eq. (EC.6), we have \|p r t ppq´rppq\| "ˇˇxp, p B t r py´xp, Br pyˇďProof of } p D t ppq´Dppq} 8 ď ∆ D t ppq. Combining Eq. (EC.3) and Eq. (EC.6), it is easy to obtain thatCombining Eq. (EC.7) and Eq. (EC.9), we haveTherefore, we complete the proof of Lemma 1.˝EC.3.2. Proof of Corollary 1Proof of Corollary 1. The proof will be conditioned on when we find a feasible pq α t , q B t q P M t for all t ď τ (which happens with probability p1´O pT´1qq by Lemma 5) and the desired event of Lemma 1.For each t ď τ and p P rp, ps, we first upper bound } q D t ppq´Dppq} 8 . Note thatTherefore, we only need to show that } qFor every i P rN s, let q B t " r q B t \|q α t s, and we verify thatHere, the first inequality is due to Cauchy-Schwarz, the second inequality is due to that pq α t , q B t q P M t .We now upper bound \|q r t ppq´rppq\| as follows.where in the inequality, we upper bound } q D t ppq´p Dppq} 8 by ∆ D t ppq due to the paragraph above. We finally upper bound \| q f t ppq´f t ppq\|. Note thatwhere the second inequality is by the definitions of p f t and q f t , and the third inequality uses the upper bounds for \|q r t ppq´p rppq\| and } q D t ppq´p Dppq} 8 derived in the previous parts of this proof.EC.3.3. Proof of Lemma 2Proof of Lemma 2. Since Λ t " pN`1q¨I N`1`ř săt r p s r p J s , for every t ě 1, r(EC.10) ec6 e-companion to Chen, Lyu, Wang and Zhou: Fairness-aware NRM with Demand Learning Note }Λ´1{2 t r p t } 8 ď a r p J t Λ´1 t r p t and recall the definition of ∆ r t ppq and ∆ D t ppq we haveRecalling the definition κ " 2 b 2d 2 pN`1q ln`N T p1`p 2 T q˘`2pN`1qL 2 B and ∆ t ppq :" ∆ r t ppqm axtC}A} 8 , 1u∆ D t ppq, by Eqs. (EC.10, EC.11, EC.12), we obtainwhere the first equility is due to detpI`xx J q " 1`}x} 2 2 and the last inequality is due to exppx{2q ď 1`x when x P r0, 1s. Therefore , with Eq. 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[ "Forecasting Sequential Data using Consistent Koopman Autoencoders", "Forecasting Sequential Data using Consistent Koopman Autoencoders" ]	[ "Omri Azencot [email protected] ", "N Benjamin Erichson [email protected] ", "Vanessa Lin [email protected] ", "Michael W Mahoney [email protected] ", "\nUC\nLos Angeles\n", "\nUC Berkeley\nICSI\nUC Berkeley\nICSI\nUC Berkeley\nICSI\n\n" ]	[ "UC\nLos Angeles", "UC Berkeley\nICSI\nUC Berkeley\nICSI\nUC Berkeley\nICSI\n" ]	[]	Recurrent neural networks are widely used on time series data, yet such models often ignore the underlying physical structures in such sequences. A new class of physically-based methods related to Koopman theory has been introduced, offering an alternative for processing nonlinear dynamical systems. In this work, we propose a novel Consistent Koopman Autoencoder model which, unlike the majority of existing work, leverages the forward and backward dynamics. Key to our approach is a new analysis that unravels the interplay between consistent dynamics and their associated Koopman operators. Our network is interpretable from a physical viewpoint and its computational requirements are comparable to other baselines. We evaluate our method on a wide range of high-dimensional and short-term dependent problems. The datasets include nonlinear oscillators, sea surface temperature data, and fluid flows on a curved domain. The results show that our model yields accurate estimates for significant prediction horizons, while being robust to noise. * Equal contribution.	null	[ "https://arxiv.org/pdf/2003.02236v1.pdf" ]	211,989,200	2003.02236	ae29e8ad34a333f9a396178f0ea48d4e62232894	Forecasting Sequential Data using Consistent Koopman Autoencoders Omri Azencot [email protected] N Benjamin Erichson [email protected] Vanessa Lin [email protected] Michael W Mahoney [email protected] UC Los Angeles UC Berkeley ICSI UC Berkeley ICSI UC Berkeley ICSI Forecasting Sequential Data using Consistent Koopman Autoencoders Recurrent neural networks are widely used on time series data, yet such models often ignore the underlying physical structures in such sequences. A new class of physically-based methods related to Koopman theory has been introduced, offering an alternative for processing nonlinear dynamical systems. In this work, we propose a novel Consistent Koopman Autoencoder model which, unlike the majority of existing work, leverages the forward and backward dynamics. Key to our approach is a new analysis that unravels the interplay between consistent dynamics and their associated Koopman operators. Our network is interpretable from a physical viewpoint and its computational requirements are comparable to other baselines. We evaluate our method on a wide range of high-dimensional and short-term dependent problems. The datasets include nonlinear oscillators, sea surface temperature data, and fluid flows on a curved domain. The results show that our model yields accurate estimates for significant prediction horizons, while being robust to noise. * Equal contribution. INTRODUCTION Sequential data processing and forecasting is a fundamental problem in the engineering and physical sciences. Recurrent Neural Networks (RNNs) provide a powerful class of models for these tasks, designed to learn long-term dependencies via their hidden state variables. However, training RNNs over long time horizons is notoriously hard (Pascanu et al., 2013) due to the problem of exploding and vanishing gradients (Bengio et al., 1994). Several approaches have been proposed to mitigate this issue using unitary hidden-to-hidden weight matrices (Arjovsky et al., 2016) or analyzing stability properties (Miller & Hardt, 2019), among other solutions. Still, attaining long-term memory remains to be a challenge and, moreover, short-term dependencies might be affected due to limited expressivity of unitary RNNs (Kerg et al., 2019). Another shortcoming of traditional RNNs is their lack of interpretability. In this context, physicallybased methods have been proposed, relating RNNs to dynamical systems (Sussillo & Barak, 2013) or differential equations (Chang et al., 2019). This point of view allows to construct models which enjoy high-level properties such as time invertibility via Hamiltonian (Greydanus et al., 2019) or Symplectic (Chen et al., 2020;Zhong et al., 2020) networks. Other techniques suggested reversible RNNs (MacKay et al., 2018;Chen et al., 2018) to alleviate the large memory footprints RNNs induce during training. In this work, we advocate that modeling time-series data which exhibit strong short-term dependencies can benefit from relaxing the strict stability and time invertibility requirements. An interesting physically motivated alternative for analyzing time series data has been introduced in Koopman-based models (Takeishi et al., 2017;Morton et al., 2018Morton et al., , 2019Li et al., 2020). Koopman theory is based on the insight that a nonlinear dynamical system can be fully encoded using an operator that describes how scalar functions propagate in time. The Koopman operator is linear, and thus preferable to work with in practice as tools from linear algebra and spectral theory can be directly applied. While Koopman's theory (Koopman, 1931) was established almost a century ago, significant advances have been recently accomplished in the theory and methodology with applications in the fluid mechanics (Mezić, 2005) and geometry processing (Roufosse et al., 2019) communities. The Koopman operator maps between function spaces and thus it is infinite-dimensional and can not be represented on a computer. Nevertheless, most machine learning approaches hypothesize that there exists a data transformation under which an approximate finite-dimensional Koopman operator is available. Typically, this map is represented via an autoencoder network, embedding the input onto a low-dimensional encode decode Nonlinear Linear ϕ(z k ) Cq k Figure 1: Illustration of a (non-linear) data transformation that maps the high-dimensional states z k (evolving on a nonlinear trajectory) to a new latent space in which the dynamics are linearized. latent space. In that space, the Koopman operator is approximated using a linear layer that encodes the dynamics (Takeishi et al., 2017). The main advantage of this framework is that the resulting models are highly interpretable and allow for accurate prediction of short-term dependent data. Specifically, predicting forward or backward in time can be attained via subsequent matrix-vector products between the Koopman matrix and the latent observation. Similarly, stability features can be analyzed and constrained via the operator spectrum. Based on the Koopman theory, we propose a new model for forecasting high-dimensional time series data. In contrast to previous approaches, we assume that the backward map exists. That is, the system from a future to current time can be properly defined. While not all systems exhibit this feature (e.g., diffusive systems), there are many practical cases where this assumption holds. We investigate the interplay between the forward and backward maps and their consistency in the discrete and continuous space settings. Our work can be viewed as relaxing both reversibility and stability requirements, leading to higher expressivity and improved forecasts. Ideas close to ours have been applied to training Generative Adversarial Networks (Zhu et al., 2017;Hoffman et al., 2018). However, to the best of our knowledge, our work is first to establish the link between consistency of latent variables and dynamical systems. Background and Problem Setup In what follows, we focus on dynamical systems that can be described by a time-invariant model z k+1 = ϕ(z k ) , z ∈ M ⊂ R m ,(1) where z k denotes the state of the system at time k ∈ N. The map ϕ : M → M is a (potentially non-linear) update rule on a finite dimensional manifold M, pushing states from time k to time k + 1. The above model assumes that future states depend only on the current state z k and not on information from a sequence of previous states. In order to predict future states, one might be tempted to train a neural network which learns an approximation of the map ϕ. However, the resulting model ignores prior knowledge about the problem and is potentially difficult to interpret and analyze. As an alternative, one could seek a data transformation for the states z k so that the corresponding latent variables q k evolve on a linear path, as illustrated in Figure 1. In turn, the dynamics could be approximated by a linear model, which improves interpretability and facilitates the integration of prior physical knowledge into the training process. Intriguingly, Koopman theory suggests that there exists a data transformation for any non-linear dynamical system so that the states can be pushed from time k to time k + 1 by a linear map. In this paper, we advocate the use of Koopman's perspective on data which are high-dimensional or exhibit strong short-term dependencies. More concretely, the dynamics ϕ induces an operator K ϕ that acts on scalar functions f : M → R ∈ F , with F being some function space on M (Koopman, 1931). Formally, the Koopman operator is given by i.e., the function f is composed with the map ϕ. Intuitively, the operator details the evolution of a scalar function by pulling-back its values from a future time. In other words, K ϕ f at z is the value of f evaluated at the future state z k+1 . Hence, the Koopman operator is also commonly known as the pull-back operator. Also, it is easy to show that K ϕ is linear for any α, β ∈ R K ϕ f (z) = f • ϕ(z) ,(2)K ϕ (α f + βg) = (α f + βg) • ϕ = α f • ϕ + βg • ϕ = αK ϕ f + βK ϕ g . Finally, we assume that the backward dynamics ψ exists, and we denote by U ψ the associated Koopman operator. In Fig. 2, we show an illustration of our setup. Unfortunately, K ϕ is infinite-dimensional. Nevertheless, the key assumption in most of the practical approaches is that there exists a transformation χ whose conjugation with K ϕ leads to a finite-dimensional approximation which encodes "most" of the dynamics. Formally, C = χ • K ϕ • χ −1 , C ∈ R κ×κ ,(3) i.e., χ and its inverse extract the crucial structures from K ϕ , yielding an approximate Koopman matrix C. Similarly, we denote by D = χ • U ψ • χ −1 the approximate backward system. The main focus in this work is to find the matrices C and D, and a nonlinear transformation χ such that the underlying dynamical system is recovered well. We assume to be given scalar observations of the dynamics { f k : M → R} n k=1 such that f k+1 = f k • ϕ + r k , where the function r k ∈ F represents deviation from the true dynamics due to e.g., measurement errors or missing values. We focus on the case where ϕ and ψ are generally unknown, and our goal is to predict future observations from the given ones. Namely, f k+l = f k • ϕ l , l = 1, 2, ... ,(4) where ϕ l means we repeatedly apply the dynamics. In practice, as C approximates the system, we exploit the relation χ −1 • C l • χ( f k ) ≈ f k • ϕ l to produce further predictions. That is, the matrix C fully determines the forward evolution of the input observation f k . Main Contributions Our main contributions are as follows. • We develop a Physically Constrained Learning (PCL) framework based on Koopman theory and consistent dynamics for processing complex time series data. • Our model is effective and interpretable and its features include accurate predictions, time reversibility and a stable behavior even over long time horizons. • We evaluate on high-dimensional clean and noisy systems including the pendulum, cylinder flow, vortex flow on a curved domain, and climate data, and we achieve exceptionally good results with our model. RELATED WORK Modeling dynamical systems from Koopman's point-of-view has gained increasing popularity in the last few years (Mezić, 2005). An approximation of the Koopman operator can be computed via the Dynamic Mode Decomposition (DMD) algorithm (Schmid, 2010). While many extensions of the original algorithm have been proposed, most related to our approach is the work of Azencot et al. (2019) where the authors consider the forward and backward dynamics in a non-neural optimization setting. A network design similar to ours was proposed by Lusch et al. (2018), but without our analysis, back prediction and consistency terms. Other techniques minimize the residual sum of squares (Takeishi et al., 2017;Morton et al., 2018), promote stability (Erichson et al., 2019b;Pan & Duraisamy, 2020), or use graph convolutional networks (Li et al., 2020). Sequential data are commonly processed using RNNs (Elman, 1990;Graves, 2012). The main difference between standard neural networks and RNNs is that the latter networks maintain a hidden state which uses the current input and previous inner states. Variants of RNNs such as Long Short Term Memory (Hochreiter & Schmidhuber, 1997) and Gated Recurrent Unit (Cho et al., 2014) have achieved groundbreaking results on various tasks including language modeling and machine translation, among others. Still, training RNNs involves many challenges and a recent trend in machine learning focuses on finding new interpretations of RNNs based on dynamical systems theory (Laurent & von Brecht, 2016;Miller & Hardt, 2019). Several physically motivated models have been recently proposed. Based on Lagrangian mechanics, Lutter et al. (2019) encoded the Euler-Lagrange equations into their network to attain physical plausibility and to alleviate poor generalization of deep models. Other methods attempt to learn conservation laws from data and their associated Hamiltonian representation, leading to exact preservation of energy (Greydanus et al., 2019) and better handling of stiff problems (Chen et al., 2020). To deal with the limited expressivity of unitary RNNs, Kerg et al. (2019) suggested to employ the Schur decomposition to their connectivity matrices. By considering the normal and nonnormal components, their network allows for transient expansion and compression, leading to improved results on tasks which require continued computations across timescales. METHOD In what follows, we describe our PCL framework which we use to handle time series data. Similar to other methods, we model the transformation χ above via an encoder χ e and a decoder χ d . Our approach differs from other Koopman-based techniques (Lusch et al., 2018;Takeishi et al., 2017;Morton et al., 2018) in two key components. First, in addition to modeling the forward dynamics, our network also takes into account the backward system. Second, we require that the resulting forward and backward Koopman operators are consistent. Autoencoding Observations Given a set of observations F = { f k } n k=1 as defined in Sec. 1.1, we design an autoencoder (AE) to embed our inputs in a low-dimensional latent space using a nonlinear map χ e . The decoder χ d map allows to reconstruct latent variables in the spatial domain. To train the AE, we definẽ C D f k f k+1 χ e \|χ d χ d \|χ ef = χ d • χ e ( f ) ,(5)E id = 1 2n n ∑ k=1 f k −f k 2 2 ,(6) i.e.,f is the reconstructed version of f , and E id derives the optimization to obtain an AE such that χ d • χ e ≈ id. We note that the specific requirements from χ e and χ d are problem dependent, and we detail the particular design we used in Appendix A. Backward Dynamics In general, a dynamical system ϕ prescribes a rule to move forward in time. There are numerous practical scenarios where it makes sense to consider the backward system, i.e., ψ : z k → z k−1 . For instance, the Euler equation, which describes the motion of an inviscid fluid, is invariant to sign changes in its time parameter (see Fig. 11 for an example). Previous approaches incorporated the backward dynamics into their model as in bi-directional RNNs (Schuster & Paliwal, 1997). However, the inherent nonlinearities of a typical neural network make it difficult to constrain the forward and backward models. To this end, a few approaches were recently proposed (Greydanus et al., 2019;Chen et al., 2020) where the obtained dynamics are reversible by construction due to the leapfrog integration. In comparison, most existing Koopman-based techniques do not consider the backward system in their modeling or training. To account for the forward as well as backward dynamics, we incorporate two linear layers with no biases into our network to represent the approximate Koopman operators. As we assume that χ e transforms our data into a latent space where the dynamics are linear, we can directly evolve the dynamics in that space. We introduce the following notation f k+1 = χ d • C • χ e ( f k ) ,(7)f k−1 = χ d • D • χ e ( f k ) ,(8) for every admissible k. Namely, the Koopman operators C, D ∈ R κ×κ allow to obtain forward estimateŝ f k+1 , and backward forecastsf k−1 . In practice, we noticed that our models predict as well as generalize better if instead of computing one step forward and backward in time, we employ a multistep forecasting. Given a choice of λ s ∈ N, the total number of prediction steps, we define the following loss terms E fwd = 1 2λ s n λ s ∑ l=1 n ∑ k=1 f k+l −f k+l 2 2 ,(9)E bwd = 1 2λ s n λ s ∑ l=1 n ∑ k=1 f k−l −f k−l 2 2 ,(10) where we assume that f k+l and f k−l are provided during training for any l ≤ λ s , see Eq. (4). Also,f k+1 anď f k−l are obtained by taking powers l of C, respectively D, in Eqs. (7), (8). We show in Fig. 3 a schematic illustration of our network design including the encoder and decoder components as well as the Koopman matrices C and D. Notice that all the connections are bi-directional, that is, data can flow from left to right and right to left. Backward Prediction. Koopman operators and their approximating matrices are linear objects that allow for greater flexibility when compared to other models for time series processing. One consequence of this linearity is that while existing Koopman-based nets (Lusch et al., 2018;Takeishi et al., 2017;Morton et al., 2018) are geared towards forward prediction, their evolution matrix C can be exploited for backward prediction as well. This computation is obtained via the inverse of C, i.e., f k−1 = χ d • C −1 • χ e ( f k ) .(11) However, models that were trained for forward prediction typically produce poor backward predictions as we show in Appendix B. In contrast, we note that our model allows for the direct back prediction using the D operator and Eq. (8). Thus, while other techniques can technically produce backward predictions, our model supports it by construction. Consistent Dynamics. The backward prediction penalty E bwd in itself only affects D, and it is completely independent of C. That is, C will not change due to backpropagating the error in Eq. (10). To link between the forward and backward evolution matrices, we need to introduce an additional penalty that promotes consistent dynamics. Formally, we say that the maps ϕ and ψ are consistent if ψ • ϕ( f ) = f for any f ∈ F . In the Koopman setting, we will show below that this condition is related to requiring that DC = I κ , where I κ is the identity matrix of size κ. However, our analysis shows that in fact the continuous space and discrete space settings differ, yielding related but different penalties. In this work, we will incorporate the following loss to promote consistency E con = κ ∑ k=1 1 2k D k * C * k − I k 2 F + 1 2k C k * D * k − I k 2 F ,(12) where D k * and C * k are the upper k rows of D and leftmost k columns of the matrix C, and · F is the Frobenius norm. Stability. Recently, stability has emerged as an important component for analyzing neural nets (Miller & Hardt, 2019). Intuitively, a dynamical system is stable if nearby points stay close under the dynamics. Mathematically, the eigenvalues of a linear system fully determine its behavior, providing a powerful tool for stability analysis. Indeed, the challenging problem of vanishing and exploding gradients can be elegantly explained by bounding the modulus of the weight matrices' eigenvalues (Arjovsky et al., 2016). To overcome these challenges, one can design networks that are stable by construction, see e.g., Chang et al. (2019), among others. We, on the other hand, relax the stability constraint and allow for quasi-stable models. In practice, our loss term (12) regularizes the nonconvex minimization by promoting the eigenvalues to get closer to the unit circle. The comparison in Fig. 4 highlights the stability features our model attains, whereas a non regularized network obtains unstable modes. From an empirical viewpoint, unstable behavior leads to rapidly diverging forecasts, as we show in Sec. 5. A Consistent Dynamic Koopman Autoencoder Combining all the pieces together, we obtain our PCL model for processing time series data. Our model is trained by minimizing a loss function whose minimizers guarantee that we achieve a good AE, that predictions through time are accurate, and that the forward and backward dynamics are consistent. We define our loss E = λ id E id + λ fwd E fwd + λ bwd E bwd + λ con E con ,(13) where λ id , λ fwd , λ bwd , λ con ∈ R + are user-defined positive parameters that balance between reconstruction, prediction and consistency. Finally, E id , E fwd , E bwd , E con are defined in Eqs. (6), (9), (10), and (12), respectively. CONSISTENT DYNAMICS VIA KOOPMAN We now turn to prove a necessary and sufficient condition for a dynamical system to be invertible from a Koopman viewpoint in the continuous space setting. We then show a similar result in the spatial discrete case, yielding a more elaborate condition which we use in practice. We recall that Koopman operators take inputs and return outputs from a function space F . Thus, many properties of the underlying dynamics can be related to the action of K ϕ on every function in F . A natural approach in this case is to consider a spectral representation of the associated objects. Specifically, we choose an orthogonal basis for F which we denote by {ξ k } ∞ k=0 where for any i, j we have ξ i , ξ j M = M ξ i (z) ξ j (z) d z = δ ij ,(14) with δ ij being the Kronecker delta function. Under this choice of basis, any function f ∈ F can be represented by f = ∑ k f , ξ k M ξ k . Moreover, due to the linearity of K ϕ we also have that K ij = ξ i , K ϕ ξ j M . In the following proposition we characterize invertibility in the continuous-space case, which is a known result in Ergodic theory (Eisner et al., 2015). Proposition 1 Given a manifold M, the dynamical system ϕ is invertible if and only if for every i and j the Koopman operators K ϕ and U ψ satisfy ξ i , U ψ K ϕ ξ j M = δ ij . Proof. If ϕ is invertible then the composition ψ • ϕ = id for every z ∈ M. Thus, Conversely, we assume that ξ i , U K ξ j M = δ ij for all i, j. It follows that for every k we have U K ξ k = ξ k since ξ k is orthogonal to every ξ l , l = k. Let f be some scalar function, then f (ψ • ϕ(z)) = U K f (z) = f (z) and thus ψ • ϕ = id. The main advantage of Prop. ξ i , U K ξ j M = M ξ i (z) ξ j (ψ • ϕ(z)) d z = ξ i , ξ j M . (1) is that it can be used directly in a computational pipeline. In particular, if we denote by C and D the κ × κ matrices that approximate K and U , respectively, then the above condition takes the form 1 2 D C − I κ 2 F = 0 ,(15) where I κ is an identity matrix of size κ. This loss term was recently used in (Azencot et al., 2019) to construct a robust scheme for computing DMD operators. However, we prove below that in the discrete-space setting, a more elaborate condition is required. To discretize the above objects, we assume that our manifold M is represented by the domain M ⊂ R d which is sampled using m vertices. In this setup, scalar functions f : M → R are vectors f ∈ R m storing values on vertices with in-between values obtained via interpolation. A map ϕ : M → M can be encoded using a matrix P ϕ ∈ R m×m defined by P ϕ δ z = h ϕ(z) ,(16) where h x is a function that stores the vertices' coefficients such that h T x X = x T , with X ∈ R m×d being the spatial coordinates of M. Similarly, we denote by Q ψ the matrix associated with ψ, i.e., Q ψ δ z = h ψ(z) . Note that P and Q are in fact discrete Koopman operators represented in the canonical basis. We show in Fig. 5 an illustration of our spatial discrete setup including some of the notations. Similar to the continuous setting, we can choose a basis for the function space on M. We denote B ∈ R m×m as the matrix that contains the orthogonal basis elements in its columns, i.e., b i , b j M = b T i b j = δ ij for every i, j. We use this basis to define the matrices C and D by C = B T P ϕ B , D = B T Q ψ B .(17) Finally, instead of invertible maps, we consider consistent maps. That is, a discrete map ϕ is consistent if for every z, we have that ψ • ϕ(z) = z. Using the above constructions and notations, we are ready to state our main result. Proposition 2 Given a domain M, the map ϕ is consistent if and only if for every i and j the matrices C and D satisfy ∑ k 1 2k D k * C * k − I k 2 F = 0, where D k * and C * k are the upper k rows of D and leftmost k columns of the matrix C. Figure 5: The Kronecker delta δ z centered at z is mapped via P ϕ to the function h ϕ(z) which holds the coefficients of the vertices of M whose combination generates ϕ(z). 0 1 P ϕ z ϕ(z) δ z h ϕ(z) Proof. If ϕ is a consistent map, then Q Pδ z = δ z for every z and thus Q P = I. In addition, for every k we have that D k * C * k = B T k QBB T PB k = B T k QPB k = B T k B k = I k , where B k are the first k basis elements of B and BB T = I. Conversely, we assume that C, D are related to some maps ϕ and ψ and constructed via Eq (17). In addition, the condition ∑ k 1 2k D k * C * k − I k 2 F = 0 holds. Then B T k Q PB k = I k for every k. By induction on k it follows that Q P b k = b k for every k where b k is the kth column of B and thus Q P = I as B spans the space of scalar functions on M. EXPERIMENTS To evaluate our proposed consistent dynamic Koopman AE, we perform a comprehensive study using various datasets and compare to state-of-the-art Koopman-based approaches as well as other baseline sequential models. Our network minimizes Eq. (13) with a decaying learning rate initially set to 0.01. We fix the loss weights to λ id = λ fwd = 1, λ bwd = 0.1, and λ con = 0.01, for the AE, forward forecast, backward prediction and consistency, respectively. We use λ s = 6 prediction steps forward and backward in time. We provide additional details in Appendix A. Baselines Our comparison is mainly performed against the state-of-the-art method of Lusch et al. (2018), henceforth referred to as the Dynamic AE (DAE) model. Their approach may be viewed as a special case of our network by setting λ bwd = λ con = 0. While we use this work as a baseline, other models such as (Takeishi et al., 2017;Morton et al., 2018) could be also considered. The main difference between DAE and the latter techniques is the least squares solution for the evolution matrix C per training iteration. In our experience, this change leads to delicate training procedures and thus it is less favorable. Unless said otherwise, both models are trained using the same parameters, where DAE does not have the regularizing penalties. We additionally compare against a feed-forward model and a recurrent neural network. The feedforward network simply learns a nonlinear function ζ : f k → f k+1 , where during inference we takef k+1 as input for predictingf k+2 , and so on. The RNN adds an hidden state h k such that h k = σ(U f k + Wh k−1 + b), and the prediction is obtained viaf k = Vh k + c. We performed a parameter search when comparing with these baselines. Nonlinear Pendulum with no Friction The nonlinear (undamped) pendulum (Hirsch et al., 1974) is a classic textbook example for dynamical systems, which is also used for benchmarking deep models (e.g., Greydanus et al., 2019;Bertalan et al., 2019; Chen et al., 2020). This problem can be modeled as a second order ODE by d 2 θ dt 2 + g sin θ = 0, where the angular displacement from an equilibrium is denoted by θ ∈ [0, 2π). We use l and g to denote the length and gravity, respectively, with l = 1 and g = 9.8 in practice. We consider the following initial conditions θ(0) = θ 0 andθ(0) = 0. The motion of the pendulum is approximately harmonic for a small amplitude of the oscillation θ 0 1. However, the problem becomes inherently nonlinear for large amplitudes of the oscillations. We experiment with oscillation angles θ 0 = 0.8 and θ 0 = 2.4 on the time interval t = [0, 51]. The data are generated using a time step ∆t = 0.03, yielding T = 1700 equally spaced points x 1 , ..., x T ∈ R 2 . In addition, we map the sequence {x t } to a high-dimensional space via a random orthogonal transformation to obtain the training snapshots, i.e., P ∈ R 64×2 such that f t = P x t for any t. Finally, we split the new sequence into a training set of 600 points and leave the rest for the test set. We show in Fig. 6 examples of the clean and noisy trajectories for these data. Experimental results. Fig. 7 shows the pendulum results for initial conditions θ 0 = 0.8 (top row) and θ 0 = 2.4 (bottom row). We used a bottleneck κ = 6 and α = 0.5 for the DAE and our models. (The parameter α controls the width of the network, i.e., the number of neurons used for the hidden layers.) The relative forecasting error is computed at each time step via f t −f t 2 / f t 2 , wheref t is the high-dimensional estimated prediction, see also Eq. (7). We forecast over a time horizon of 1000 steps, and we average the error over 30 different initial observations f t , where the shaded areas represent the ±1 standard deviations. Overall, our model yields the best or second best results in all the cases we explored. The RNNs obtains good measures in the clean case and for short prediction times, but its performance deteriorates in the noisy setup and when forecasting is required for long horizons. The DAE model (Lusch et al., 2018) recovers the pendulum dynamics in the linear regime but struggles when the nonlinearity increases. High-dimensional Fluid Flows Next, we consider two challenging fluid flow examples. The first instance is a periodic flow past a cylinder that exhibits vortex shedding from boundary layers. This flow is commonly used in physically-based machine learning studies (Takeishi et al., 2017;Morton et al., 2018). The data are generated by numerically solving the Navier-Stokes equations given here in their vorticity form ∂ t ω = − v, ∇ω + 1 Re ∆ω, where ω is the vorticity taken as the curl of the velocity, ω = curl(v). We employ an immersed boundary projection solver (Taira & Colonius, 2007) with Re = 100. Our simulation yields 300 snapshots of 192 × 199 grid points, sampled at regular intervals in time, spanning five periods of vortex shedding. We split the data in half for training and testing. Our second example is an inviscid flow, i.e., Re = ∞, of a vortex pair travelling over a curved domain of a sphere given as a triangle mesh with 2562 nodes. This flow is characterized by a continuous motion along the great center geodesic of the sphere, and it has been used previously to assess the stability of numerical algorithms (Azencot et al., 2014). We facilitate the intrinsic solver (Azencot et al., 2014) by producing 600 snapshots of which we use 550 for training. Experimental results for the flow past a cylinder. It can be shown that the cylinder dynamics evolve on a low-dimensional attractor (Noack et al., 2003), which can be viewed as a nonlinear oscillator with a state-dependent damping (Loiseau & Brunton, 2018). As the dynamics are within the linear regime, we expect that on clean data, both DAE and ours will obtain good prediction results. We compare both Figure 11: Vortices that rotate in opposite directions travel along geodesics of the domain. We show the input data (top row) as well as our forecast results (bottom row) for several different times. models using width α = 2 and bottleneck κ = 10. The error plots for this experiment are provided in Fig. 8. In general, our model outperforms DAE when the data are noisy and the prediction horizon is long, illustrating the regularizing effect of our loss terms. Fig. 10 shows the obtained predictions for t = 100. Indeed, for the clean data (top) the results are similar in nature, whereas for the noisy case (bottom), our model provides a more robust profile. Experimental results for the inviscid flow. Next, we consider the inviscid flow as an input to our network as well as to the DAE model. In general, our method yields extremely good results for this flow, even for long time predictions, as can be seen in Fig. 9. This example highlights the benefits of our approach with respect to DAE, as for some of the times we obtain a gain which is five times better than DAE. Moreover, we note that the particular choice of fully connected layers in Tab 2 is important in this case, since computing convolutions on curved domains is still considered a difficult problem whose solutions are not as robust as in the structured setting (Bronstein et al., 2017). In Fig. 11, we show a few estimated predictions for the flow over a sphere. Initial condition DAE Ours Ground truth Figure 12: Starting from the same initial conditions, we use our model and DAE to forecast the sea surface temperature for the 120th day (top row) and 175th day (bottom row). The results can be visually compared to the ground truth data (right column). Our model generally attains predictions that are much closer to the ground truth compared to the results obtained by DAE. Sea Surface Temperature Data Our last dataset includes complex climate data representing the daily average sea surface temperature measurements around the Gulf of Mexico. This dataset is used in climate sciences to study the intricate dynamics between the oceans and the atmosphere. Climate prediction is generally a hard task involving challenges such as irregular heat radiation and flux as well as uncertainty in the wind behavior. Nevertheless, the input dynamics exhibit non-stationary periodic structures and are empirically low-dimensional, suggesting that Koopman-based methods can be employed. We extract a subset of the NOAA OI SST V2 High Resolution Dataset hereafter SST, and we refer to (Reynolds et al., 2007) for additional details. In our experiments, we use data with a spatial resolution of 100 × 180 spanning a time horizon of 1, 305 days, of which 1, 095 snapshots are used for training. Experimental results. Fig. 13 shows the prediction error over a time horizon of 180 days. Here we are using width α = 6 and bottleneck κ = 10. It can be seen, that our model provides good estimations over a long time horizon, whereas the DAE diverges rapidly. Fig. 12 provides a visual comparison of the estimated predictions as obtained from the DAE model and ours. In both cases, forecasting for the 120th day (top) and 175th day (bottom), our results are closer to the ground truth by a large margin. Recall that changes in sea surface temperature causes far-reaching effects on global climate and lead to climatic phenomenons, such as storms and floods. Thus, models that provide a more accurate forecast can improve response rate to the potential effects and damages. However, predicting climate data is in general a notoriously difficult problem. Nevertheless, the prediction results of our model are of practical significance. ABLATION STUDY To support our empirical results, we have also conducted an ablation study to quantify the effect of our additional loss terms when weighted differently. To this end, we revisit the noisy pendulum flow with an initial condition θ 0 = 0.8. Our results are summarized in Tab. 1 where we explore various values for λ bwd and λ con which balance the back prediction and consistency penalties, respectively. Our model generally outperforms other baselines for all the parameters we checked, measured via the average error (fifth column) and most distant prediction error (sixth column). Type #par λ bwd λ con 1 T ∑ T t=1 \|f t − f t \|/\| f t \| \|f T − f T \|/\| f T \| DAE DISCUSSION In this paper, we proposed a novel PCL framework for processing high-dimensional time series data. Our method is based on Koopman theory as we approximate dynamical systems via linear evolution matrices. Key to our approach is that we consider the backward dynamics during prediction, and we promote the consistency of the forward and backward systems. These modifications may be viewed as relaxing strict reversibility and stability constraints, while still regularizing the parameter space. We evaluate our method on several challenging datasets and compare with a state-of-the-art Koopman-based network as well as other baselines. Our approach notably outperforms the other models on noisy data and for long time predictions. We believe that our work can be extended in many ways, and in the future, we plan on considering our setup within a recurrent neural network design. A NETWORK ARCHITECTURE In our evaluation, we employ an autoencoding architecture where the encoder and decoder are shallow (Erichson et al., 2019a) and contain only three layers each. Using a simple design allows us to focus our comparison on the differences between the DAE model (Lusch et al., 2018) and ours. Specifically, we list the network structure in Tab. 2 including the specific sizes we used as well as the different activation functions. We recall that m represents the spatial dimension of the input signals, whereas κ is the bottleneck of our approximated Koopman operators. Thus, p = 32 · α is the main parameter with which we control the width and expressiveness of the autoencoder. We facilitate fully connected layers as some of our datasets are represented on unstructured grids. Finally, we note that the only difference between our net architecture and the DAE model is the additional backward linear layer. B BACKWARD PREDICTION OF DYNAMICAL SYSTEMS One of the key features of our model is that it allows for the direct backward prediction of dynamics. Namely, given an observation f t , our network yields the forward prediction viaf t+1 = χ d • C • χ e ( f t ), as well as the backward estimate usingf t−1 = χ d • D • χ e ( f t ). Time reversibility may be important in various contexts (Greydanus et al., 2019). For instance, given two different poses of a person, we can consider the trajectory from the first pose to the second or the other way around. Typically, neural networks require that we re-train the model in the reverse direction to be able to predict backwards. In contrast, Koopman-based methods can be used for this task as the Koopman matrix is linear and thus back forecasting can be obtained simply viaf t−1 = χ d • C −1 • χ e ( f t ). We show in Fig. 14 the backward prediction error computed withf t−1 for the cylinder flow data using our model and the DAE (blue and red curves). In addition, as our model computes the matrix D, we also show the errors obtained forf t−1 . The solid lines correspond to the clean version of the data, whereas the dashed lines are related to its noisy version. Overall, our model clearly outperforms DAE by an order of magnitude difference. C COMPUTATIONAL REQUIREMENTS The models used in this work are relatively shallow. The amount of parameters per model can be computed as follows 2(m + 32 + κ) · 32α + (4 · 32α + κ + m) + 2 · κ 2 , corresponding to the number of weights, biases and Koopman operators, respectively. Notice that DAE is different than our model by having κ 2 less parameters. In general, our model is almost two times slower than DAE during training. We recorded the average training time per epoch, and we show the results for DAE and our models in Fig. 15 for many of our test cases. Specifically, the figure shows from left to right the run times for cylinder flow, noisy cylinder flow, sphere flow, linear pendulum, noisy linear pendulum, nonlinear pendulum, noisy nonlinear pendulum, SST and noisy SST. The behavior of our model is consistent in comparison to DAE for the different tests. On average, if DAE takes x milliseconds per epoch, than our model needs ≈ 1.8x time. This difference in time is due to the additional penalty terms, and it can be asymptotically bounded via O(E bwd ) + O(E con ) = O(λ s nm) + O(κ 4 ) . We note that the asymptotics for the forward prediction are equal to the backward component, i.e., O(E fwd ) = O(λ s nm). The consistency term E con is composed of a sum of sequence of cubes (matrix products) which can be bounded by κ 4 , assuming matrix multiplication is O(κ 3 ) and thus it is a non tight bound. Moreover, while the quartic bound is extremely high, we note that a cheaper version of the constraint can be used in practice, i.e., \|CD − I\| 2 F . Also, since our models are loaded to the GPU where matrix multiplication computations are usually done in parallel, the practical bound may be much lower. Finally, the inference time is insignificant (≈ 1 ms) and it is the same for DAE and ours and thus we do not provide a comparison. Figure 2 : 2Our analysis and computational pipeline take into account the forward and backward dynamical systems. Figure 3 : 3Our network takes observations f k , and it learns a latent representation of them via an autoencoder architecture χ e and χ d . Then, the latent variables are propagated forward and backward in time using the linear layers C and D, respectively. We emphasize that all the above connections are bi-directional and so information can flow freely from left to right or right to left. Figure 4 : 4Koopman operators are linear and thus their spectrum can be investigated. We visualize the eigenvalues of an unregularized model (a) and ours (b) in the complex plane. Indeed, our PCL framework promotes stability, whereas the other model exhibits an eigenvalue with modulus greater than one. Figure 6 :Figure 7 : 67We show the pendulum's trajectories for the initial conditions of amplitude oscillations θ 0 = 0.8 (left) and θ 0 = 2.4 (right). The discrete sampled data points are slightly perturbed. Prediction errors, over a time horizon of 1000 steps, for clean and noisy observations from a pendulum with initial conditions θ 0 = 0.8 (top row) and θ 0 = 2.4 (bottom). Our model outperforms the DAE in all settings. Figure 8 : 8We compare the behavior of our approach vs. DAE on the cylinder flow on clean (a) and noisy (b) inputs. Overall, our model achieves consistent results even in the presence of noise, whereas DAE struggles with noise and over long range forecasts. Figure 9 : 9Prediction errors for the vortex pair flow over the surface of a sphere for a time horizon of 100 steps on clean (a) and noisy (b) inputs. Again, our model achieves consistent results, whereas DAE struggles with noise and over long range forecasts. Figure 10 : 10We demonstrate a few estimation examples for the flow past a cylinder, showing the predicted fluctuations around the mean for clean (top row) and noisy (bottom row) inputs. Our model recovers more flow structures over a time horizon of 130 steps, where the advantage is pronounced for the noisy inputs. Ground truth Ours Figure 13 : 13Prediction errors for SST over a time horizon of 180 days. Our results clearly outperform DAE across various times. Figure 14 :Figure 15 : 1415The cylinder flow is used for backward prediction with our model (red) and DAE (blue). Our results hint that DAE overfits in the forward direction, whereas our network generalizes well when the time is reversed. We show above the average run time for an epoch in milliseconds for several of the test cases in this work. Table 1: Summary of ablation study for the pendulum. The star indicates the setting that is used during our experiments above.10k 0.0 0.0 0.20 0.36 RNN 20k 0.0 0.0 0.19 0.33 Ours 10k 1e-1 0.0 0.15 0.27 Ours (*) 10k 1e-1 1e-2 0.09 0.15 Ours 10k 1e-1 1e-1 0.11 0.18 Ours 10k 2e-1 1e-2 0.07 0.11 Ours 10k 2e-1 1e-1 0.11 0.18 Table 2 : 2Our network architecture, where p = 32 · α with α controlling the width per encoder and decoder layer. ACKNOWLEDGEMENTSOA would like to acknowledge the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 793800. MWM would like to acknowledge ARO, DARPA, NSF and ONR for providing partial support of this work. We would like to thank J. Nathan Kutz, Steven L. Brunton, Lionel Mathelin and Alejandro Queiruga for valuable discussions about dynamical systems and Koopman theory. Further, we would like to acknowledge the NOAA for providing the SST data (https://www.esrl.noaa.gov/psd/). Unitary evolution recurrent neural networks. 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[ "PINGAN-VCGROUP'S SOLUTION FOR ICDAR 2021 COMPETITION ON SCIENTIFIC LITERATURE PARSING TASK B: TABLE RECOGNITION TO HTML", "PINGAN-VCGROUP'S SOLUTION FOR ICDAR 2021 COMPETITION ON SCIENTIFIC LITERATURE PARSING TASK B: TABLE RECOGNITION TO HTML" ]	[ "Jiaquan Ye \nVisual Computing Group\nPing An Property & Casualty Insurance Company\n\n", "Xianbiao Qi \nVisual Computing Group\nPing An Property & Casualty Insurance Company\n\n", "Yelin He \nVisual Computing Group\nPing An Property & Casualty Insurance Company\n\n", "Yihao Chen \nVisual Computing Group\nPing An Property & Casualty Insurance Company\n\n", "Dengyi Gu \nVisual Computing Group\nPing An Property & Casualty Insurance Company\n\n", "Peng Gao \nPing An Technology Company\n\n", "Rong Xiao \nVisual Computing Group\nPing An Property & Casualty Insurance Company\n\n" ]	[ "Visual Computing Group\nPing An Property & Casualty Insurance Company\n", "Visual Computing Group\nPing An Property & Casualty Insurance Company\n", "Visual Computing Group\nPing An Property & Casualty Insurance Company\n", "Visual Computing Group\nPing An Property & Casualty Insurance Company\n", "Visual Computing Group\nPing An Property & Casualty Insurance Company\n", "Ping An Technology Company\n", "Visual Computing Group\nPing An Property & Casualty Insurance Company\n" ]	[]	This paper presents our solution for ICDAR 2021 competition on scientific literature parsing task B: table recognition to HTML. In our method, we divide the table content recognition task into four sub-tasks: table structure recognition, text line detection, text line recognition, and box assignment. Our table structure recognition algorithm is customized based on MASTER [1], a robust image text recognition algorithm. PSENet [2] is used to detect each text line in the table image. For text line recognition, our model is also built on MASTER. Finally, in the box assignment phase, we associated the text boxes detected by PSENet with the structure item reconstructed by table structure prediction, and fill the recognized content of the text line into the corresponding item. Our proposed method achieves a 96.84% TEDS score on 9,115 validation samples in the development phase, and a 96.32% TEDS score on 9,064 samples in the final evaluation phase.	null	[ "https://arxiv.org/pdf/2105.01848v1.pdf" ]	233,739,912	2105.01848	76d6706744d9e13e51b5e7932cf058d4c53ba5cd	PINGAN-VCGROUP'S SOLUTION FOR ICDAR 2021 COMPETITION ON SCIENTIFIC LITERATURE PARSING TASK B: TABLE RECOGNITION TO HTML May 6, 2021 Jiaquan Ye Visual Computing Group Ping An Property & Casualty Insurance Company Xianbiao Qi Visual Computing Group Ping An Property & Casualty Insurance Company Yelin He Visual Computing Group Ping An Property & Casualty Insurance Company Yihao Chen Visual Computing Group Ping An Property & Casualty Insurance Company Dengyi Gu Visual Computing Group Ping An Property & Casualty Insurance Company Peng Gao Ping An Technology Company Rong Xiao Visual Computing Group Ping An Property & Casualty Insurance Company PINGAN-VCGROUP'S SOLUTION FOR ICDAR 2021 COMPETITION ON SCIENTIFIC LITERATURE PARSING TASK B: TABLE RECOGNITION TO HTML May 6, 20211 This paper presents our solution for ICDAR 2021 competition on scientific literature parsing task B: table recognition to HTML. In our method, we divide the table content recognition task into four sub-tasks: table structure recognition, text line detection, text line recognition, and box assignment. Our table structure recognition algorithm is customized based on MASTER [1], a robust image text recognition algorithm. PSENet [2] is used to detect each text line in the table image. For text line recognition, our model is also built on MASTER. Finally, in the box assignment phase, we associated the text boxes detected by PSENet with the structure item reconstructed by table structure prediction, and fill the recognized content of the text line into the corresponding item. Our proposed method achieves a 96.84% TEDS score on 9,115 validation samples in the development phase, and a 96.32% TEDS score on 9,064 samples in the final evaluation phase. Introduction The ICDAR 2021 competition on scientific literature parsing task B is to reconstruct the table image into an HTML code. In this competition, PubTabNet dataset (v2.0.0) [3] is provided as the official evaluation data, and Tree-Edit-Distance-based similarity (TEDS) metric is used for evaluation. The PubTabNet data set consists of 500,777 training samples, 9,115 validation samples, 9,138 samples for the development stage, and 9,064 samples for the final evaluation stage. For the training and validation data, the ground truth HTML codes and the position of non-empty table cells are provided to the participants. Participants of this competition need to develop a model that can convert images of tabular data into the corresponding HTML code, which should correctly represent the structure of the table and the content of each cell. The labels of samples for the development and the final evaluation stages are preserved by the organizers. We divide this task into four sub-tasks: table structure recognition, text line detection, text line recognition, and box assignment. And several tricks are tried to improve the model. The details of each sub-task will be discussed in the following section. Method In this section, we will present these four sub-tasks in order. * Xianbiao Qi is the corresponding author. If you have any questions or concerns about the implementation details, please do not hesitate to contact [email protected] or [email protected]. arXiv:2105.01848v1 [cs.CV] 5 May 2021 Table Structure Recognition The task of table structure recognition is to reconstruct the HTML sequence items and their corresponding locations on the table, but ignore the text content in each item. Our model structure is shown in Figure 1(b). It is customized based on MASTER [1], a powerful image-to-sequence model originally designed for scene text recognition. Different from the vanilla MASTER as shown in Figure 1(a), our model has two branches. One branch is to predict the HTML item sequence, and the other is to conduct the box regression. Instead of splitting the model into two branches in the last layer, we decouple the sequence prediction and the box regression after the first transformer decode layer. To structure an HTML sequence, we need to define an Alphabet for the sequence. As shown in the left of Figure 2, we define 39 class labels for the sequence prediction. For the pairs <thead> and </thead>, <tbody> and </tbody>, and <tr> and </tr>, some other control characters may appear between these pairs. Thus, we need to define one individual class for each of them. We define the maximum "colspan" and "rowspan" as 10, thus we both use 9 labels for them individually. There are two forms for <td></td>, empty content or non-empty content between <td> and </td>. We use one class to denote the whole of the <td>[content]</td>. It should be noted that using one label instead of defining two individual labels for <td> and </td> can largely reduce the length of the sequence. For the form of <td></td> with empty content, we can find 11 special forms. As shown in the right of Figure 2, each form is represented by a special class label. According to the above description, the sequence lengths of 99.6% HTML in the PubTabNet data set are less than 500. For the sequence prediction, we use the standard cross-entropy loss. For the box regression, we employ the L1 loss to regress the coordinates of [x,y,w,h]. The coordinates are normalized to [0,1]. For the box regression head, we use an Sigmoid activation function before the loss calculation. In Figure 3, we show a result example of sequence prediction and box regression. We could see that the structure MASTER can predict out the box coordinates correctly. Text Line Detection PSENet is an efficient text detection algorithm that can be considered as an instance segmentation network. It has two advantages. Firstly, PSENet, as a segmentation-based method, is able to localize texts of arbitrary shape. Secondly, the model proposes a Progressive Scale Expansion Network which can successfully identify adjacent text instances. PSENet not only adapts to text detection at arbitrary angles but also works better for adjacent text segmentation. Text detection in print documents is an easy task compared to text detection in a natural scene. In training PSENet, there are three key points needing attention, the input image size, the minimum area and the minimum kernel size. To avoid true negative, especially some small region (such as a dash line), the resolution of the input image should be large, and the minimum area size should be set to be small. In Figure 4, we visualize an detection result by PSENet. Text Line Recognition We also use MASTER as our text line recognition algorithm. MASTER is powerful and can be freely adapted to different tasks according to different data forms, e.g. curved text prediction, multi-line text prediction, vertical text prediction, multilingual text prediction. Figure 5. We can see that some texts are blur, and some are black and some are grey. The maximum sequence length is set to be 100 in our MASTER OCR. Text lines longer than 100 characters will be discarded. Some training samples are shown in Figure 5. It should be noted that in training stage, our algorithm is trained on a database mixed with single-line text images and multi-line text images, but in the test stage, only single-line text images are inputted. By text line recognition, we can get the corresponding text content of text line images. These text contents will be merged to non-empty <td></td> items in the HTML sequence. The details of text content merge will also be discussed in the next subsection. Box Assignment According to the above three subsections, we have obtained the table structure together with the box of each cell, and the box of each text line together with its corresponding text content. To generate the complete HTML sequence, we need to assign each box of text line into its corresponding table structure cell. In this subsection, we will introduce our used match rules in detail. There are three matching rules used in our method, which we call Center Point Rule, IOU Rule and Distance Rule. The details will be discussed below. Center Point Rule In this matching rule, we firstly calculate central coordinate of each box obtained by PSENet. If the coordinate is in the rectangular region of the regressed box obtained by structure prediction, we call them a matching pair. The content of the text line will be filled into <td></td>. It is important to note that one table structure cell can be associated with several PSENet boxes because of one table structure cell may have multiple text lines. IOU Rule If the above rule is not satisfied, we will compute the IOU between the box of the chosen text line and all structure cell boxes. The box cell with the maximum IOU value will be selected. The text content will be filled into the chosen structure cell. Distance Rule Finally, if both above rules are unsuccessful. We will calculate the Euclidean distances between between the box of the chosen text line and all structure cell boxes. Similar to the IOU Rule, the structure cell with minimum Euclidean distance will be chosen. Matching Pipeline All above-mentioned three rules will be applied in order. Firstly, most boxes detected by PSENet will be assigned to their corresponding structure cells by center point rule. Owing to prediction deviations of structure prediction, a few central points of PSENet boxes are out of the rectangle region of structure cell boxes obtained by structure prediction. Secondly, some unmatched PSENet boxes under the center point rule will get matched under the IOU Rule. In the above two steps, we use the PSENet boxes to match their corresponding structure item. If there are some structure items that are not matched. In this way, we use the structure item to find the left PSENet boxes. To do this, the distance rule is applied. Figure 6: Example of box assignment visualization. On the left side, some detected boxes by PSENet are marked by different colors. On the right side, the boxes generated by structure prediction are marked. A visualization example of matching results is shown in Figure 6. For aesthetic effect, we only show part of the boxes. On the left side of Figure 6, some detected boxes by PSENet are marked by different colors. On the right side of Figure 6, the boxes generated by structure prediction are marked. The boxes on the left side will be assigned to the box cell with the same color. Experiment In this section, we will describe the implementation of our table recognition system in detail. Dataset. Our used data is the PubTabNet dataset (v2.0.0), which contains 500,777 training data and 9,115 validation data in development Phase, 9,138 samples for the development stage, and 9,064 samples for the final evaluation stage. Except for the provide training data, no extra data is used for training. To get text-line level annotation of all text boxes, 2k images of training data are relabeled for PSENet training. Actually, we only need to adjust the annotations of multi-line annotation into single-line box annotation. Implementation Details. In PSENet training, 8 Tesla V100 GPUs are used with the batch size 10 in each GPU. The input image is resized equally, keeping the long side with resolution 1280. RandomFilp and RandomCrop are used for data augmentation. A 640 × 640 region is cropped from each image. Adam optimizer is applied, and the initial learning rate is 0.001 with step learning rate decay. In table structure training, 8 Tesla V100 GPUs are used with the batch size 6 in each GPU. The input image size is 480 × 480, and the maximum sequence length is 500. Synchronized BN [4] and Ranger optimizer [5] are apply in this experiment, and the initial learning rate of optimizer is 0.001 with step learning rate decay. In the training of text line recognition, 8 Tesla V100 GPUs are used with the batch size 64 in each GPU. The input size is 256 × 48, and the maximum length is 100. Synchronized BN and Ranger optimizer are also applied and the hyper-parameter setting is the same as the table structure training. All models are trained based on our own FastOCR toolbox. Ablation Studies Our table recognition system is described above. We have conducted many attempts in this competition. In this subsection, we will discuss some useful tricks, but ignore some unsuccessful attempts. Ranger is a synergistic optimizer combining RAdam (Rectified Adam) [6], LookAhead [7], and GC (gradient centralization) [8]. We observe that Ranger optimizer shows a better performance than Adam in this competition, and it is applied in both table structure prediction and text line recognition. We use default Ranger. Result comparison between Adam and Ranger is shown in Table 1(a). (SyncBN) is an effective batch normalization approach that is suitable for multi-GPU or distributed training. In standard batch normalization, the data is only normalized within the data on each GPU device. But SyncBN normalizes the input within the whole mini-batch. SyncBN is ideal for situations where the batch size is relatively small on each GPU graphics card. SyncBN is applied in our experiment. Synchronized Batch Normalization Feature Concatenation of Layers in Transformer Decoder. In structure MASTER and text recognition MASTER, three successive transformer layers [1] is used as decoder. Different from the original MASTER, we concatenate the outputs of each transformer layer [9] and then apply a linear projection on the concatenated feature. Label Encoding in Structure Prediction After we inspect on the training data of the PubtabNet data set(v2.0.0), we find some ambiguous annotations about empty table cell. Some empty cells of table are labeled as <td></td>, whereas the others are labeled as <td> </td> in which one space character is inserted. However, these two different table cells look the same visually. According to statistics, the ratio between <td></td> and <td> </td> is around 4:1. In our experiment, we encode these two different cells into different tokens. Our motivation is to let the model to discover the intrinsic visual features by training. Optimizer Structure prediction Acc. In this competition, we have conducted some evaluations and recorded the results. The results are shown in Table 1. According to Table 1, we have the following observations, • Ranger optimizer has outperformed Adam optimizer consistently. Similar observation is also found in our another report [10] about ICDAR 2021 Competition on Scientific Table Image Recognition to LaTeX [11]. In our evaluation on standard benchmarks, we also find that Ranger can improve the average accuracy by around 1%. • SyncBN can improve the performance a little. We also observe that SyncBN also shows better performance than standard BN on ICDAR 2021 competition on Mathematical Formula Detection. • Feature concatenation can improve the accuracy of the structure prediction on this task. It should be noted that in [ End-to-end Evaluation on the Validation Set We generate the final HTML code by merging structure prediction, text line detection, text line recognition, and box assignment. We evaluate some tricks in these stages. Results are shown in Table 2. TEDS is used as our indicator. We have some overall conclusions from this competition, • ESB (empty space box encode) is important for the final TEDS indicator. • FeaC (feature concatenation) is effective for both table structure recognition and text line recognition. • ME (model ensemble) improves the performance a little bit. Three model ensembles in the TSR can improve the end-to-end TEDS score for around 0.2%. Three model ensembles in the text line recognition can only improve the TEDS score for around 0.03%. We only use one PSENet model. • SyncBN is effective for both TSR and TLR. • ForC (format correction) helps the final indicator. Our format correction is to promise all content between <thead> and </thead> is black font. Discussion. From this competition, we have some reflections. For the end-to-end table recognition to the HTML code, structure prediction is an extremely important stage, especially for the TEDS indicator. As shown in Figure 7, although all text line information is correctly recognized. Our method obtains very low TEDS (0.423) due to wrong structure prediction. Although the provided data set is large, we still believe larger scale of data that cover more templates may further improve the structure prediction. Secondly, text line detection and text line recognition are easy tasks considering all table images are print. Thirdly, There are some labeling inconsistency issues, such as <td></td> and <td> </td>. Finally, the box assignment sub-task can be conducted by Graph Neural Network (GNN) [12] instead of hand-crafted rules. Figure 1 : 1(a) Architecture of vanilla MASTER; (b) Architecture of table structure MASTER Figure 2 : 239 classes used in table structure MASTER. Figure 3 : 3Example of table structure prediction. Predicted bounding box are marked with yellow color. Figure 4 : 4Visualization of text line detection. Figure 5 : 5Example of text line images cropped from training data of PUbTabNet data set; (a) single line text image; (b) multi-lines text image The position annotations in the PubTabNet dataset (v2.0.0) is cell-level, cropped text images according to the position annotation in the data set contains both single-line and multi-line text images. We construct a text line recognition database according to position information provided in the annotation file. This text line recognition database contains about 32 million samples cropped from 500k training images. We split out 20k text line images as a validation set for checkpoint selection. Some training samples are shown in Evaluation of label encoding, SyncBN and feature concatenation. Figure 7 : 7An example of wrong table structure prediction. Table 1 : 1Evaluation of different tricks on table recognition task. (a). comparison of Ranger and Adam. (b). comparison of with or without feature concatenation. (c). evaluation of label encoding. 10], we do not observe performance improvement.Table 2: End-to-end evaluation on the validation set with TEDS as the indicator. TLD: text line detection; TSR: table structure recognition; TLR: text line recognition; ME: model ensemble. ESB: empty space box encode; SyncBN: synchronized BN; FeaC: feature concatenate output of transformer layers. ForC: format correction.TLD TSR TLR BA ME ForC TEDS PSE ESB SyncBN FeaC FeaC Extra Insert 0.9385 0.9621 0.9626 0.9635 0.9684 ConclusionIn this paper, we present our solution for the ICDAR 2021 competition on Scientific Literature Parsing task B: table recognition to HTML. We divide the table recognition system into four sub-tasks, table structure prediction, text line detection, text line recognition, and box assignment. Our system gets a 96.84 TEDS scores on the validation data set in the development phase, and gets a 96.324 TEDS score in the final evaluation phase. Master: Multi-aspect non-local network for scene text recognition. Ning Lu, Wenwen Yu, Xianbiao Qi, Yihao Chen, Ping Gong, Rong Xiao, Xiang Bai, Pattern Recognition. 2021Ning Lu, Wenwen Yu, Xianbiao Qi, Yihao Chen, Ping Gong, Rong Xiao, and Xiang Bai. Master: Multi-aspect non-local network for scene text recognition. Pattern Recognition, 2021. Shape robust text detection with progressive scale expansion network. Wenhai Wang, Enze Xie, Xiang Li, Wenbo Hou, Tong Lu, Gang Yu, Shuai Shao, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionWenhai Wang, Enze Xie, Xiang Li, Wenbo Hou, Tong Lu, Gang Yu, and Shuai Shao. Shape robust text detection with progressive scale expansion network. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9336-9345, 2019. Image-based table recognition: data, model, and evaluation. Xu Zhong, Elaheh Shafieibavani, Antonio Jimeno Yepes, arXiv:1911.10683arXiv preprintXu Zhong, Elaheh ShafieiBavani, and Antonio Jimeno Yepes. Image-based table recognition: data, model, and evaluation. arXiv preprint arXiv:1911.10683, 2019. Context encoding for semantic segmentation. Hang Zhang, Kristin Dana, Jianping Shi, Zhongyue Zhang, Xiaogang Wang, Ambrish Tyagi, Amit Agrawal, Proceedings of the IEEE conference on Computer Vision and Pattern Recognition. the IEEE conference on Computer Vision and Pattern RecognitionHang Zhang, Kristin Dana, Jianping Shi, Zhongyue Zhang, Xiaogang Wang, Ambrish Tyagi, and Amit Agrawal. Context encoding for semantic segmentation. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 7151-7160, 2018. Ranger-Deep-Learning-Optimizer. Less Wright, Less Wright. Ranger-Deep-Learning-Optimizer, 2019. On the variance of the adaptive learning rate and beyond. Liyuan Liu, Haoming Jiang, Pengcheng He, Weizhu Chen, Xiaodong Liu, Jianfeng Gao, Jiawei Han, Proceedings of the Eighth International Conference on Learning Representations (ICLR 2020). the Eighth International Conference on Learning Representations (ICLR 2020)Liyuan Liu, Haoming Jiang, Pengcheng He, Weizhu Chen, Xiaodong Liu, Jianfeng Gao, and Jiawei Han. On the variance of the adaptive learning rate and beyond. In Proceedings of the Eighth International Conference on Learning Representations (ICLR 2020), April 2020. James Michael R Zhang, Geoffrey Lucas, Jimmy Hinton, Ba, arXiv:1907.08610Lookahead optimizer: k steps forward, 1 step back. arXiv preprintMichael R Zhang, James Lucas, Geoffrey Hinton, and Jimmy Ba. Lookahead optimizer: k steps forward, 1 step back. arXiv preprint arXiv:1907.08610, 2019. Gradient centralization: A new optimization technique for deep neural networks. Hongwei Yong, Jianqiang Huang, Xiansheng Hua, Lei Zhang, European Conference on Computer Vision. SpringerHongwei Yong, Jianqiang Huang, Xiansheng Hua, and Lei Zhang. Gradient centralization: A new optimization technique for deep neural networks. In European Conference on Computer Vision, pages 635-652. Springer, 2020. Exploiting deep representations for neural machine translation. Zi-Yi Dou, Zhaopeng Tu, Xing Wang, Shuming Shi, Tong Zhang, arXiv:1810.10181arXiv preprintZi-Yi Dou, Zhaopeng Tu, Xing Wang, Shuming Shi, and Tong Zhang. Exploiting deep representations for neural machine translation. arXiv preprint arXiv:1810.10181, 2018. Pinganvcgroup's solution for icdar 2021 competition on scientific table image recognition to latex. Yelin He, Xianbiao Qi, Jiaquan Ye, Peng Gao, Yihao Chen, Bingcong Li, Xin Tang, Rong Xiao, arXivYelin He, Xianbiao Qi, Jiaquan Ye, Peng Gao, Yihao Chen, Bingcong Li, Xin Tang, and Rong Xiao. Pingan- vcgroup's solution for icdar 2021 competition on scientific table image recognition to latex. arXiv, 2021. 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[ "Channel Knowledge Map for Environment- Aware Communications: EM Algorithm for Map Construction", "Channel Knowledge Map for Environment- Aware Communications: EM Algorithm for Map Construction" ]	[ "Kun Li ", "Peiming Li [email protected] \nSchool of Information Engineering\nGuangdong University of Technology\n510006GuangzhouChina\n", "Yong Zeng [email protected] \nPurple Mountain Laboratories\n211111NanjingChina\n", "Jie Xu [email protected] \nFNii and SSE\nThe Chinese University of Hong Kong\n518172Shenzhen, ShenzhenChina\n", "\nNational Mobile Communications Research Laboratory\nSoutheast University\n210096NanjingChina\n" ]	[ "School of Information Engineering\nGuangdong University of Technology\n510006GuangzhouChina", "Purple Mountain Laboratories\n211111NanjingChina", "FNii and SSE\nThe Chinese University of Hong Kong\n518172Shenzhen, ShenzhenChina", "National Mobile Communications Research Laboratory\nSoutheast University\n210096NanjingChina" ]	[]	Channel knowledge map (CKM) is an emerging technique to enable environment-aware wireless communications, in which databases with location-specific channel knowledge are used to facilitate or even obviate real-time channel state information acquisition. One fundamental problem for CKMenabled communication is how to efficiently construct the CKM based on finite measurement data points at limited user locations. Towards this end, this paper proposes a novel map construction method based on the expectation maximization (EM) algorithm, by utilizing the available measurement data, jointly with the expert knowledge of well-established statistic channel models.The key idea is to partition the available data points into different groups, where each group shares the same modelling parameter values to be determined. We show that determining the modelling parameter values can be formulated as a maximum likelihood estimation problem with latent variables, which is then efficiently solved by the classic EM algorithm. Compared to the pure datadriven methods such as the nearest neighbor based interpolation, the proposed method is more efficient since only a small number of modelling parameters need to be determined and stored. Furthermore, the proposed method is extended for constructing a specific type of CKM, namely, the channel gain map (CGM), where closed-form expressions are derived for the E-step and M-step of the EM algorithm. Numerical results are provided to show the effectiveness of the proposed map construction method as compared to the benchmark curve fitting method with one single model.	10.1109/wcnc51071.2022.9771802	[ "https://arxiv.org/pdf/2108.06960v1.pdf" ]	237,091,770	2108.06960	f844a1f231fa244111c6832cc9dc559208d51a31	Channel Knowledge Map for Environment- Aware Communications: EM Algorithm for Map Construction 16 Aug 2021 Kun Li Peiming Li [email protected] School of Information Engineering Guangdong University of Technology 510006GuangzhouChina Yong Zeng [email protected] Purple Mountain Laboratories 211111NanjingChina Jie Xu [email protected] FNii and SSE The Chinese University of Hong Kong 518172Shenzhen, ShenzhenChina National Mobile Communications Research Laboratory Southeast University 210096NanjingChina Channel Knowledge Map for Environment- Aware Communications: EM Algorithm for Map Construction 16 Aug 2021arXiv:2108.06960v1 [cs.IT] Channel knowledge map (CKM) is an emerging technique to enable environment-aware wireless communications, in which databases with location-specific channel knowledge are used to facilitate or even obviate real-time channel state information acquisition. One fundamental problem for CKMenabled communication is how to efficiently construct the CKM based on finite measurement data points at limited user locations. Towards this end, this paper proposes a novel map construction method based on the expectation maximization (EM) algorithm, by utilizing the available measurement data, jointly with the expert knowledge of well-established statistic channel models.The key idea is to partition the available data points into different groups, where each group shares the same modelling parameter values to be determined. We show that determining the modelling parameter values can be formulated as a maximum likelihood estimation problem with latent variables, which is then efficiently solved by the classic EM algorithm. Compared to the pure datadriven methods such as the nearest neighbor based interpolation, the proposed method is more efficient since only a small number of modelling parameters need to be determined and stored. Furthermore, the proposed method is extended for constructing a specific type of CKM, namely, the channel gain map (CGM), where closed-form expressions are derived for the E-step and M-step of the EM algorithm. Numerical results are provided to show the effectiveness of the proposed map construction method as compared to the benchmark curve fitting method with one single model. I. INTRODUCTION Channel knowledge map (CKM) is an emerging technique towards environment-aware wireless communications [1], which provides location-specific (rather than the coarse site-specific) channel knowledge associated with potential transmitter-receiver pairs by, e.g., storing them in databases. Compared to conventional environment-ignorant communication, CKM-enabled environment-aware communication is expected to facilitate or even obviate real-time channel state information (CSI) acquisition, which makes it especially appealing for future communication systems with large spatial dimensions [2] and prohibitive channel training overhead. In fact, the attempts to use site-specific databases in wireless communications have been pursued in prior works based on, e.g., 3D city or terrain map [3], radio environment map [4], [5], and TV white space map [6]. However, these designs require storing accurate physical environment maps and implementing computation-expensive algorithms, such as ray tracing algorithms, which are costly in terms of both storage and computation. Furthermore, the TV white space map and radio environment map were mainly used for cognitive radio systems [7], for which the obtained maps critically depend on the status/activities of the primary transmitters, such as the spectrum, power, and antenna pattern being used. By contrast, CKM aims to provide location-specific knowledge that directly reflects the intrinsic channel characteristics, regardless of the transmitter or receiver activities [1]. This makes it possible to design communication systems with light or even without realtime channel training [8]. Some specific instances of CKM include channel gain map (CGM) [9], channel path map (CPM) [8], and beam index map (BIM) [8], [10]. CKM-enabled communications have been recently studied in various applications, such as training-free subband selection for device-to-device (D2D) communications [1], beam alignment for millimeter wave (mmWave) massive MIMO [8], and trajectory design for cellular-connected UAV [11]. One fundamental problem for CKM-enabled environmentaware communication is how to efficiently construct the CKM based on finite measurement data points at limited user locations. The most straightforward approach for map construction is interpolation-based methods, such as the inverse distance weighted (IDW) [12], nearest neighbours (NN), splines [13], and Kriging [14], [15]. The interpolation-based methods, however, generally require large measurement data for accurate map construction. In addition, such pure data-driven methods ignored the well-established stochastic or geometric based channel models developed over the past few decades [16], thus usually requiring huge storage capacity for map maintenance. To utilize both measurement data and expert knowledge, one straightforward method is parametric curve fitting, where the best modelling parameters of the selected channel models are determined based on the measurement data. However, such a naive curve fitting method would lead to poor map quality, since the number of tunable modelling parameters is typically very small, while the environment is usually too complex to be accurately characterized by one single model. To overcome the above drawbacks, in this paper, we propose a novel CKM construction method based on the wellestablished expectation maximization (EM) algorithm [17]. Notice that the naive curve fitting method is not able to accurately predict the complex channel environment knowledge, as only one common channel model is used. By contrast, it is observed that different sub-areas of the site may experience different radio propagation environment, which may be modelled with different sets of modelling parameters. Based on this observation, we propose to partition the available measurement data into different modelling groups, where each group shares the same modelling parameter values that are to be determined. We show that the determination of modelling parameter values corresponds to a maximum likelihood estimation problem with latent variables. Although this is a challenging non-convex optimization problem, various existing algorithms, such as the classic EM algorithm [17], have been proposed to find its efficient solutions. In particular, we propose a generic EM-based algorithm to solve the considered CKM construction problem, which consists of two steps, namely the Expectation step (Estep) and the Maximization step (M-step). Furthermore, we also consider the special case for constructing a CGM, for which we extend the EM-based algorithm to find the optimized solution, by deriving the closed-form expressions for the Estep and M-step, respectively. Finally, extensive numerical results are provided to verify the effectiveness of the proposed EM-based algorithm for CKM construction. II. SYSTEM MODEL As shown in Fig. 1, we consider a wireless communication system in a specific site, with a stationary base station (BS) and mobile users, whose potential locations q are denoted by the set Q. For any given q ∈ Q, our objective is to predict the interested location-specific channel knowledge, which is denoted as r, as accurate as possible before real-time channel training is applied. Note that the channel knowledge r can be any useful information related to the wireless channel, such as the channel gain, shadowing, angle of arrival/departure (AoA/AoD), or even the channel impulse response. To this end, a BS-to-any (B2X) CKM W is constructed, which provides mapping from location q to the corresponding channel knowledge r, i.e., W : q ∈ Q → r. One immediate method for channel knowledge prediction based on user locations is to utilize the well-established channel models, which are usually given stochastically with certain modelling parameter vector θ, denoted as p(r\|q, θ). Specifically, p(r\|q, θ) gives the probability density function (PDF) of the channel knowledge r for users located at q, parameterized by θ. As a concrete example, for the specific B2X CGM, r corresponds to the channel gain in dB, which is denoted by the real number r. Without loss of generality, we may assume that the BS is located at the origin. Therefore, according to the classic path loss model, we have r = β + 10α log 10 q + S, (1) where α denotes the path loss exponent, β denotes the path loss intercept, S ∼ N (0, σ 2 ) captures the log-normal shadowing with variance σ 2 , and · denotes the Euclidean norm. As such, the modelling parameter vector for channel gains is θ = [α, β, σ 2 ], and the conditional PDF of the channel knowledge r given user location q is p(r\|q, θ) = N (r\|β + 10α log 10 q , σ 2 ) = 1 √ 2πσ 2 exp −(r − β − 10α log 10 q ) 2 /2σ 2 ,(2) where N (r\|µ, σ 2 ) denotes the PDF of a Gaussian random variable r with mean µ and variance σ 2 . A direct channel knowledge prediction based on stochastic channel models would lead to poor accuracy, since such models characterize channels only on the average sense, by ignoring the sitespecific or even location-specific propagation environment. For instance, with the channel model in (1), no matter how good the modelling parameter θ is chosen, the predicted channel gain is symmetric around the BS, which is far from reality when the actual environment is taken into account. On the other hand, site-specific or location-specific radio propagation environment information can be learned if onsite measurement data are available. Let X ∈ R D×N denote the set of measurement data points. Each column of X, denoted as x n ∈ R D×1 , n ∈ N {1, ..., N }, corresponds to one data point, which includes the measured channel knowledge r n at the corresponding location q n . Therefore, we have x n = [q T n , r T n ] T , with the superscript T denoting the transpose. Note that the number of available measurement data points N is usually limited. As a result, to construct a complete CKM for all potential user locations, one needs to infer the channel knowledge of those unmeasured locations based on X. However, the pure data-driven methods, such as the interpolation-based methods, usually require large data and high storage capacity. On the other hand, the simple parameteric curve fitting based method, which uses one single model to fit all data points, would lead to poor accuracy due to the limited degrees of freedom associated with the few modelling parameters. To overcome the above issues, we propose a map construction method based on multi-component or mixed channel models. Specifically, with the basic PDF p(r\|q, θ), we consider a mixed channel model with a total of K components, corresponding to K sets of modelling parameters, denoted as θ k , k ∈ K {1, ..., K}. With a slight abuse of notation, let θ = {θ 1 , ...., θ K }. Then for the given measurement data X, our objective is to find the set of modelling parameters θ based on certain criterion. A commonly used criterion is the maximum likelihood estimation, which corresponds to finding parameters θ for maximizing the likelihood as follows. (P1): max θ p(X\|θ), where p(X\|θ) denotes the likelihood function with respect to the modelling parameters θ. III. EM ALGORITHM FOR CKM CONSTRUCTION The key challenge of solving (P1) lies in finding the explicit expression for the likelihood function p(X\|θ). This is difficult since for each data point x n ∈ X, it is unknown which of the K modelling components it should be associated with. As a consequence, (P1) corresponds to the maximum likelihood estimation problem with latent variables, which, fortunately, has been extensively studied and can be efficiently solved by various algorithms, such as the classic EM algorithm [17]. To this end, for each data point x n , n ∈ N , we introduce a vector of latent variables z n ∈ R K×1 to indicate the association of data point x n with the K modelling components, whose k-th element z nk is a binary random variable with z nk ∈ {0, 1}, ∀k ∈ K. Here, z nk = 1 means that x n is best explained by the k-th modelling component with parameter θ k . It follows that k∈K z nk = 1, as each data point x n is only associated with the best modelling component. Furthermore, the distribution of z n is specified by the mixing coefficients π k , i.e., p(z nk = 1) = π k , k ∈ K, (3) where 0 ≤ π k ≤ 1 and k∈K π k = 1. As such, for any given modelling parameters θ, the conditional PDF of x n given its associated latent variable z n is [17] p(x n \|z n , θ) = p(x n \|θ k * ) = k∈K [p(x n \|θ k )] z nk ,(4) where k * defined such that z nk * = 1. Note that the last equality in (4) holds since z nk ∈ {0, 1}, ∀k ∈ K. Similarly, p(z n \|θ) can be expressed as p(z n \|θ) = k∈K π z nk k . As a result, the joint PDF of x n and z n can be expressed as p(x n , z n \|θ) = p(x n \|z n , θ)p(z n \|θ) = k∈K [π k p(x n \|θ k )] z nk .(5) Let Z ∈ R K×N include all the N vectors z n of latent variables, n ∈ N . Since different data points are independent, we have p(X, Z\|θ) = n∈N p(x n , z n \|θ). Therefore, the likelihood function p(X\|θ) is obtained by marginalizing p(X, Z\|θ) over Z, and (P1) can be equivalently written as (P2): max θ Z p(X, Z\|θ). Note that p(X\|θ) and p(X, Z\|θ) are referred to as the likelihood functions of the incomplete-data and complete-data [17], respectively. (P2) is the maximum likelihood estimation problem with the latent variable Z, which can be efficiently solved by the classic EM algorithm [17]. The EM algorithm is an iterative algorithm with two basic steps, i.e., the E-step and the M-step, which are summarized in Algorithm 1 [17,Section 9.3]. It is shown in [17,Section 9.4] that in the EM algorithm, each step will lead to an increased (or at least non-deceased) complete-data log likelihood. Therefore, the convergence of the EM algorithm is guaranteed [17]. Algorithm 1 General EM Algorithm [17, Section 9.3]. Input: Given the joint distribution p(X, Z\|θ) governed by the set of parameters θ; 1: Initialization: Choose initial parameters θ old and mixing coefficients π old k , ∀k ∈ K; 2: Repeat: 1) E-step: Evaluate the posterior distribution of the latent variables p(Z\|X, θ old ) and the responsibilities {γ nk = E[z nk ]}; 2) M-step: Update θ new = arg max θ Q(θ, θ old ) Z p(Z\|X, θ old ) ln p(X, Z\|θ), and π new k = N k /N , with N k = n∈N γ nk ; 3) θ old ← θ new , π old k ← π new k , A. General Algorithm for EM-based CKM Construction In the following, the E-step and the M-step are developed for our considered CKM construction problem (P2). 1) E-step: As given in Algorithm 1, the E-step of the EM algorithm is to evaluate the posterior distribution of the latent variable p(Z\|X, θ old ), with given modelling parameter θ old . According to equation (9.75) of [17], p(Z\|X, θ old ) can be factorized as p(Z\|X, θ old ) = n∈N p(z n \|x n , θ old ). Furthermore, with the Bayesian theorem, we have p(z n \|x n , θ old ) = p(z n \|θ old )p(x n \|z n , θ old ) z ′ n p(z ′ n \|θ old )p(x n \|z ′ n , θ old ) . Let γ nk denote the responsibility corresponding to component k ∈ K to explain the measurement data x n , which can be obtained as the expected value of the indicator variable z nk with the posterior distribution in (7), i.e., γ nk E[z nk ] = p(z nk = 1\|x n , θ old ) = π old k p(x n \|θ old k )/ j∈K π old j p(x n \|θ old j ) = π old k p(r n \|q n , θ old k )/ j∈K π old j p(r n \|q n , θ old j ), (8) where the second equality holds since z nk is binary. Furthermore, the last equality of (8) follows by noting that x n = [q T n , r T n ] T , so that p(x n \|θ k ) = p(q n , r n \|θ k ) = p(r n \|q n , θ k ), (9) where p(r n \|q n , θ k ) corresponds to the selected stochastic channel model, such as (2) for CGM. As a result, with the Estep, the posterior probabilities p(Z\|X, θ old ) is obtained based on (6) and (7), and the responsibilities {γ nk } is obtained based on (8). 2) M-step: Based on the posterior probabilities in (6) and the responsibilities in (8) obtained in the E-step, the modelling parameter θ and mixing coefficients π k are updated in the Mstep, by maximizing the expectation of the log-likelihood of the complete-data, with the expectation taken with respect to p(Z\|X, θ old ), which is given by [17] Q(θ, θ old ) = Z p(Z\|X, θ old ) ln p(X, Z\|θ). Furthermore, we have ln p(X, Z\|θ) = n∈N ln p(x n , z n \|θ) = n∈N ln k∈K [π k p(x n \|θ k )] z nk = n∈N k∈K z nk [ln π k +ln p(x n \|θ k )] = n∈N k∈K z nk [ln π k +ln p(r n \|q n , θ k )], where the second equality follows from (5), and the last equality follows from (9). As a result, the expectation of ln p(X, Z\|θ) with respect to the posterior probabilities p(Z\|X, θ old ) can be rewritten as Q(θ, θ old ) = E p(Z\|X,θ old ) [ln p(X, Z\|θ)] = n∈N k∈K γ nk [ln π k + ln p(r n \|q n , θ k )], where the identity γ nk = E[z nk ] is used. Therefore, the optimization problem for the M-step can be formulated as (P3): max {π k ,θ k } n∈N k∈K γ nk [ln π k + ln p(r n \|q n , θ k )] s.t. k∈K π k = 1 0 ≤ π k ≤ 1, ∀k ∈ K. It is not difficult to see that problem (P3) can be decoupled into two independent sub-problems: (P3.1): max {π k } n∈N k∈K γ nk ln π k s.t. k∈K π k = 1 0 ≤ π k ≤ 1, ∀k ∈ K. (P3.2): max {θ k } n∈N k∈K γ nk ln p(r n \|q n , θ k ). (P3.1) is a convex optimization problem. With the standard Lagrangian method, its optimal solution can be obtained in closed form as π * k = N k /N , where N k = n∈N γ nk , ∀k ∈ K. On the other hand, the solution to (P3.2) depends on the actual conditional distribution p(r n \|q n , θ k ), i.e., the selected stochastic channel model. Based on the above results in the E-step and M-step, the general algorithm for EM-based CKM construction is summarized in Algorithm 2. B. Special Case with CGM Construction In this subsection, we consider the the construction of a particular type of CKM, namely the CGM, for which the PDF of the channel gain given the user location is expressed as (2). Eventually, the CGM can be directly constructed by using Algorithm 2. Here, by exploiting the specific structure of CGM, we obtain the closed-form expressions in (8), and obtain the optimal solution to (P3.2) in closed form, to reduce the construction complexity, as explained in detail as follows. Algorithm 2 General Algorithm for EM-based CKM Construction. Input: Given the stochastical channel model p(r\|q, θ); 1: Initialization: Choose initial parameters θ old and mixing coefficients π old k , ∀k ∈ K; 2: Repeat: 1) E-step: Evaluate the responsibilities {γ nk } using (8); 2) M-step: Update θ new by solving the optimization problem (P3.2), and π new k = N k /N , with N k = n∈N γ nk ; 3) θ old ← θ new , π old k ← π new k , ∀k ∈ K; 3: Until convergence or a maximum number of iterations is reached; Output: Set of modelling parameters θ new , and responsibilities {γ nk }. First, γ nk in (8) for the E-step is given by the following closedform expression: γ nk = π old k N (r n \|β old k +10α old k log 10 q n , (σ old k ) 2 )) j∈K π old j N (r n \|β old j +10α old j log 10 q n , (σ old j ) 2 ) . (10) Next, consider problem (P3.2), for which the log-likelihood function of the M-step can be written as ln p(r n \|q n , θ k ) = − 1 2 ln(2π) − 1 2 ln σ 2 k − (r n −β k −α k d n ) 2 2σ 2 k , where d n 10 log 10 q n is defined for convenience. By discarding constant terms, (P3.2) is equivalent to min {α k ,β k ,σ 2 k } n∈N k∈K γ nk ln σ 2 k +(r n −β k −α k d n ) 2 /σ 2 k , which can be decoupled into K independent sub-problems: min α k ,β k ,σ 2 k n∈N γ nk ln σ 2 k + (r n −β k −α k d n ) 2 /σ 2 k . (11) Theorem 3.1: The optimal solution to problem (11) is: α k = (d k r k ) − d k r k d 2 k − d k 2 , β k = d 2 k r k − d k (d k r k ) d 2 k − d k 2 ,(12)σ 2 k = n∈N γ nk (r n − β k − α k d n ) 2 /N k ,(13) where d k = ( n∈N γ nk d n )/N k , r k = ( n∈N γ nk r n )/N k , (d k r k ) = ( n∈N γ nk d n r n )/N k , and d 2 k = ( n∈N γ nk d 2 n )/N k , with N k = n∈N γ nk ∀k ∈ K. Proof: Please refer to Appendix A. Based on the above derivations, the EM-based algorithm for the specific CGM construction is summarized in Algorithm 3. Algorithm 3 EM-Based Algorithm for CGM Construction. 1: Initialization: Choose initial parameters θ old and mixing coefficients π old k , ∀k ∈ K; 2: Repeat: 1) E-step: Evaluate the responsibilities {γ nk } using (10); 2) M-step: Update θ new using (12) and (13), and update π new k = N k /N , with N k = n∈N γ nk ; 3) θ old ← θ new , π old k ← π new k , ∀k ∈ K; 3: Until convergence or a maximum number of iterations is reached. C. Utilizing CKM for Channel Prediction Based on Algorithms 2 and 3, we obtain the modelling parameters θ = {θ 1 , ..., θ K }, as well as the responsibilities {γ nk } for the measurement data X. We are now ready to utilize such information to predict the channel knowledge r for any new location q. To this end, we need to first determine which set of modelling parameters θ 1 , ..., θ K is most suitable for the new location q. This can be achieved by determining the responsibilities {γ k (q)} using the IDW method. Specifically, let M denote the subset of M user locations with training data that are nearest to q. Then, γ k (q) is obtained as γ k (q) = m∈M ω m γ mk , where ω m = d −1 m (q)/ j∈M d −1 j (q) is the weighting coefficient based on the IDW criterion, with d j (q) = q j − q , j ∈ M. Then, the modelling parameter for location q is obtained as the one that maximizes γ k (q), i.e., θ(q) = θ k ⋆ , where k ⋆ = arg max k∈K γ k (q). As a result, the channel knowledge for location q can be predicted based on the PDF p(r\|q, θ(q)). IV. NUMERICAL RESULTS In this section, we present numerical results to validate the performance of our proposed algorithm. As shown in Fig. 2, we consider a geographic area of size 2 × 2 km 2 , and focus on the channel gains with a BS located at the center. We assume that there are two building clusters shown in Fig. 2. Therefore, depending on the user locations, the direct lineof-sight (LoS) link may be blocked by one of the building cluster. Furthermore, for those indoor users located in the building cluster area, additional penetration loss is incurred. As a result, depending on the user locations, the groundtruth channel gains are generated based on 5 user groups: LoS users that have direct LoS link with the BS; NLoS1 and NLoS2 users whose LoS links are blocked by building clusters 1 and 2, respectively; and Indoor1 and Indoor2 users that are located in building clusters 1 and 2, respectively. The corresponding modelling parameters of each user group are given in Table I. Furthermore, the initial parameters of Algorithm 3 are set as π old k = 1/K, ∀k ∈ K, and θ old are randomly generated as α k ∈ [2,5], β k ∈ [30, 140], and σ 2 k ∈ [6,15], ∀k ∈ K. Unless otherwise stated, the number of data points used for training is N = 2000. which guarantees the convergence. Furthermore, when the assumed number of components matches with the groundtruth value, i.e., K = 5, a fastest convergence is observed. On the other hand, with K ≥ 5, Algorithm 3 is still able to converge to roughly the same components number matches as K = 5, though at slightly slower rate, while that for K = 3 leads to poor performance since the number of assumed mixing components is smaller than the groundtruth. Next, we evaluate the quality of the constructed CGM based on Algorithm 3. For comparison, we also consider a benchmark scheme based on the single-model curve fitting method, i.e., with K = 1. In this case, no latent variable is involved since all available data points will be fitted to one single set of parameters. Fig. 4(a) shows the groundtruth data points, and Fig. 4(b) and Fig. 4(c) plot the constructed CGMs with K = 5 and K = 1, respectively, by using the IDW method with M = 3. It is observed that compared to the single-model curve fitting method, the proposed EMbased algorithm results in more accurate CGM. In fact, it is observed from Fig. 4(c) that the single-model curve fitting method leads to concentric contours of channel gain, which is far from the reality as shown in Fig. 4(a). This is due to the fact that the conventional single-model curve fitting method cannot distinguish the characteristics of different data points at different sub-areas. This issue can be addressed by our proposed EM-based algorithm with mixed channel models, as shown in Fig. 4(b). To evaluate the impact of the number of training data points N , Fig. 5 plots the normalized root mean square error (a) Groundtruth data points. (b) EM algorithm, K = 5. (c) Single model curve fitting. (NRMSE) of the predicted channel gains versus N . The testing set consists of 1000 data points. It is observed from Fig. 5 that as N increases, the constructed CGM by the proposed EM algorithm with K > 1 has better fitting quality. By contrast, regardless of N , the single model curve fitting method with K = 1 has poor performance, which is due to the fact that the number of tunable modelling parameters is too small to accurately fit the complex environment. This demonstrates the effectiveness of our proposed EM algorithm for accurate CKM construction in complex environments. V. CONCLUSION In this paper, we proposed a novel EM-based CKM construction method towards environment-aware communications, by utilizing both the available measurement data points and the expert knowledge with well-established statistic channel models. The key idea is to partition the available data points into different groups, where each group shares the same modelling parameter values that are to be determined. We propose to use the classic EM algorithm to determine the modelling parameters by solving an equivalent maximum likelihood estimation problem with latent variables, and then extend the algorithm for constructing the specific CGM. Numerical results demonstrated the effectiveness of the proposed algorithm as compared to the benchmark curve-fitting scheme with one single model. How to extend the results to other types of channel knowledge and the efficient utilizations of the constructed CKM are interesting directions worth pursuing in future research. APPENDIX A PROOF OF THEOREM 1 Problem (11) can be solved by setting the partial derivatives with respect to the optimization variables to be zero. For convenience, let f denote the cost function of problem (11). Therefore, we have ∂f ∂α k = − 2 σ 2 k n∈N γ nk (r n − β k − α k d n )d n = 0, (14) ∂f ∂β k = − 2 σ 2 k n∈N γ nk (r n − β k − α k d n ) = 0.(15) Then, (14) and (15) By solving the linear system equations with two unknowns and two equations, we get (12). Similarly, by setting the partial derivative of f with respect to σ 2 k to zero, we get (13). This thus completes the proof. Fig. 1 . 1CKM-enabled environment-aware wireless communications. First , to show the convergence of Algorithm 3,Fig. 3plots the expectation of the complete-data log likelihood Q(θ new , θ old ) versus the iteration number of each EM cycle for different number of assumed modelling components K. It is observed that for all the K values considered, Algorithm 3 results in monotonically non-decreasing log likelihood values,Fig. 2. The layout for the considered wireless communication site. Fig. 3 . 3Convergence of Algorithm 3 for CGM construction. Fig. 4 . 4Comparison of the constructed CGM based on the proposed EM algorithm and single model curve fitting method. Fig. 5 . 5NRMSE of the predicted channel gain versus the number of training data points N . ∀k ∈ K; 3: Until convergence or a maximum number of iterations is reached; Output: Set of modelling parameters θ new , and responsibilities {γ nk }. TABLE I GROUNDTRUTH IMODELLING PARAMETERS.User group α β σ 2 LoS 2.2 30 6.25 NLoS1 2.6 55 10.24 NLoS2 3.1 80 10.24 Indoor1 3.6 105 7.84 Indoor2 4.1 130 7.84 Toward environment-aware 6G communications via channel knowledge map. Y Zeng, X Xu, IEEE Wireless Commun. 283Y. Zeng and X. Xu, "Toward environment-aware 6G communications via channel knowledge map," IEEE Wireless Commun., vol. 28, no. 3, pp. 84-91, Jun. 2021. Communicating with extremely large-scale array/surface: Unified modelling and performance analysis. H Lu, Y Zeng, submitted for publicationH. Lu and Y. 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[ "DFT Studies of Adsorption of Benzoic Acid on the Rutile (110) Surface: Modes and Patterns", "DFT Studies of Adsorption of Benzoic Acid on the Rutile (110) Surface: Modes and Patterns" ]	[ "Xiang Zhao ", "Ivan ", "David R Bowler [email protected] ", "\nDepartment of Physics & Astronomy\nLondon Centre for Nanotechnology\n17-19 Gordon StreetWC1H 0AHLondon\n", "\nUniversity College London\nGower Street, 1-1 Namiki305-0044London, TsukubaIbarakiJapan\n" ]	[ "Department of Physics & Astronomy\nLondon Centre for Nanotechnology\n17-19 Gordon StreetWC1H 0AHLondon", "University College London\nGower Street, 1-1 Namiki305-0044London, TsukubaIbarakiJapan" ]	[]	Adsorption of benzoic acid on the (110) surface of rutile, both unreconstructed and (1 × 2)reconstructed ones, at saturation coverage of one benzoate per two adjacent five-coordinated Ti 5c sites has been studied using DFT simulations, as implemented through the Vienna Ab initio Simulations Package (VASP). In order to study the effects of hydrogen bonding and Van der Waals forces in influencing the relative stabilities of different adsorbate overlayer superstructures, these studies were performed through Local Density Approximation (LDA), Generalized Gradient Approximation(GGA) and DFT-D2. Through the calculations, it was found out that although the optimized structures corresponded with the proposed models for the experimental results, the relative energetic stabilities of different overlayer structures have shown some differences with the experimental results. In the GGA calculations, the overlayer structures involving the benzene rings aligned along the (001)-direction were shown to be more stable than those aligned perpendicular to it, regardless of whether the benzoates were arranged in (1 × 1) or (1 × 2) symmetries on the (1 × 1)-unreconstructed surface, or on the (1 × 2)-reconstructed surface. These are in contrast to the experimental studies using STM, whose observations revealed that configurations with the benzene rings aligned along the (110)-direction when adsorbed on the (1 × 2)-reconstructed surface.	null	[ "https://arxiv.org/pdf/1407.1659v1.pdf" ]	119,033,190	1407.1659	3b38bdf07a12ce28d91f1b1a34be6b1e77c1acf2	DFT Studies of Adsorption of Benzoic Acid on the Rutile (110) Surface: Modes and Patterns 7 Jul 2014 Xiang Zhao Ivan David R Bowler [email protected] Department of Physics & Astronomy London Centre for Nanotechnology 17-19 Gordon StreetWC1H 0AHLondon University College London Gower Street, 1-1 Namiki305-0044London, TsukubaIbarakiJapan DFT Studies of Adsorption of Benzoic Acid on the Rutile (110) Surface: Modes and Patterns 7 Jul 2014WC1E 6BT, and UCL Satellite, International Centre for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), * To whom correspondence should be addressed † LCN ‡ UCL ¶ MANA Satellite, UCL 1 Adsorption of benzoic acid on the (110) surface of rutile, both unreconstructed and (1 × 2)reconstructed ones, at saturation coverage of one benzoate per two adjacent five-coordinated Ti 5c sites has been studied using DFT simulations, as implemented through the Vienna Ab initio Simulations Package (VASP). In order to study the effects of hydrogen bonding and Van der Waals forces in influencing the relative stabilities of different adsorbate overlayer superstructures, these studies were performed through Local Density Approximation (LDA), Generalized Gradient Approximation(GGA) and DFT-D2. Through the calculations, it was found out that although the optimized structures corresponded with the proposed models for the experimental results, the relative energetic stabilities of different overlayer structures have shown some differences with the experimental results. In the GGA calculations, the overlayer structures involving the benzene rings aligned along the (001)-direction were shown to be more stable than those aligned perpendicular to it, regardless of whether the benzoates were arranged in (1 × 1) or (1 × 2) symmetries on the (1 × 1)-unreconstructed surface, or on the (1 × 2)-reconstructed surface. These are in contrast to the experimental studies using STM, whose observations revealed that configurations with the benzene rings aligned along the (110)-direction when adsorbed on the (1 × 2)-reconstructed surface. Introduction Titanium dioxide (TiO 2 ) has a history of applications since the early 19th century. Although its initial usage was confined to use as a white pigment, its applications diversified over the course of the last few decades, including but not limited to biological implants, photocatalysis and dye-sensitized solar cells (DSSCs), based on the refractive, biocompatible, photocatalytic and photoelectric properties of the material. 1 Recent emphasis on the latter applications spurred research work in the surface properties and reactions of crystalline TiO 2 , understanding of which is crucial in their realization. In the case of DSSCs, this often involves the interactions of these surfaces with sensitizing dyes, such as triscarboxy-ruthenium terpyridine [Ru (4,4',4"-(COOH) 3 -terpy)(NCS) 3 ], the "black dye". Recent developments have also been made in bio-sensitized solar cells (BSSCs), 3 which are similar to DSSCs in operating principles, but with the synthetic dye often used in the latter replaced by one that is biologically derived. One class of such biomolecules are the cyclic tetrapyrroles, which encompass porphyrins and chlorins. Many of such compounds contain a ring structure, with one or more carboxylic functional groups which play importance roles in anchoring the dye to the TiO 2 surfaces, benzoic acid's adsorp- Experimental studies have been performed on such adsorptions, using LEED, ESDIAD and STM by Guo et al, 13 and later by Grinter et al using STM. 14 In Guo's studies, the optimal adsorption mode observed was in line with those of other small carboxylates i.e. dissociative bidentate bridging binding (BB), involving the two carboxyl O atoms, after deprotonation on one of them, binding to two neighbouring five-coordinated Ti sites (Ti 5c ) along the [001] rows of such sites. In terms of adsorption patterns, through LEED, 13 it was also observed that a p(2 × 1) pattern was formed. When studied using STM however, a different picture emerged, as a p(2 × 2) pattern was observed instead. Such discrepancy could be accounted for by the fact that LEED is a scattering technique dependent upon the charge distribution of the surface, while STM surveys the apex of the surface and the adsorbates, and the fact that the phenyl rings of the benzoates were rotated to align with the [011] direction to form dimer structures. The rotation of the phenyl rings and the resultant dimer structures are due to the interactions of the lower phenyl H atoms and the π orbital of the neighbouring benzoate, facilitated by the bridging O 2c anions. Building upon these findings, a later STM study of these adsorptions was performed, 14 with the aims of studying orientations of the benzene rings of the adsorbate overlayer, as well as the adsorption of benzoates on the (110)-(1 × 2) reconstructed surface. The binding geometry of the acid was once again confirmed, but the elongation of the phenyl rings along the [110] direction was only significant in the case of the reconstructed surface, due to improved hydrogen bonding of the phenyl H atoms to the surface O atoms as a result of closer proximity. There was also no formation of dimer structures at high coverages as reported in the much earlier study -which could be due to the higher dosing temperature used this time, nor was there any observation of adsorption of dissociated H + ions. On the (110)-(1 × 2) reconstructed surface, it was also observed that benzoic acid adsorbs and binds the same way as on the unreconstructed surface. Although DFT studies of the adsorption of benzoic acid on the rutile (110) surface had been performed by Troisi et al, 15 these were performed with the aim of comparing the effects of using different DFT implementations on the computed results on electronic structures of such benzoate-TiO 2 complexes, rather than the physical structure of the adsorbates on the surface which forms the main theme of experimental research work. In our work, we will expand upon this topic by investigating the proposed models for surface adsorbate structures and patterns described in the experimental work, using DFT as implemented through VASP. After presenting our methodology, we calculate and compare the energetics of these configurations obtained using LDA, GGA and vdW, as well as producing high resolution simulated STM images to provide comparisons with these results, before concluding. Methodology Our DFT investigations were carried out using the Vienna Ab initio Simulations Package (VASP) 16 . 17 Out of all the elemental pseudopotential files, oxygen has the highest E cut value at 395.700 eV, hence this value was chosen as the cut-off energy E cut in all our simulations. Iterative electronic relaxations are done such that energy differences (E diff ) between two successive steps should not exceed 10 −6 eV, while relaxations are said to be achieved when the RMS force on each atom falls below 0.03 eV/Å. In our calculations, LDA, GGA(PBE) and GGA+DFT-D2-based methods were used to give three different sets of results for comparison of the effects of hydrogen bonding and Van der Waals forces, as GGA and DFT-D2 methods can account for these forces much better, as the interactions as a result of these forces were implied in the experimental STM studies. 14 In order to simulate the rutile TiO 2 substrate, an 8 Ti layer slab terminated by two (110) surfaces on each end was chosen, as studies of the rutile (110) surface through DFT have revealed that surface energies converge at around this thickness. 12 This was done after performing bulk relaxation of the rutile crystal, where K-sampling values of 12×12×16 were used, in order to reflect the approximate inverse ratio of the dimensions of the real space unit cell. The optimized lattice constants obtained through LDA calculations were found to be a = 4.546Å and c = 2.925Å for rutile, reflecting a contractions of 0.03Å to 0.05Å from the experimental values, which are expected for DFT using LDA as reported in other research literature, as well as being within ±2% error margin. 8 As for GGA based calculations, the optimized lattice constants were a = 4.615Å and c = 2.966Å, representing an expansion of 0.04Å, which is expected due to GGA's underestimation of cohesive energies of insulators. The new lattice parameters were then used to set up the slabs for surface relaxations. 8 Ti layer slabs were set up using four such cells with the terminated ends autocompensated. For the (1 × 2)-reconstructed surface, we used the Ti 3 O 6 added-row structure as assumed in. 14 We then set the benzoates up such that they correspond to the following six configurations, Figure 4 • BB , 1×2 : Phenyl rings of the benzoates are aligned with the [001] direction, using the (1 × 2)-reconstructed (110) surface instead, as seen in Figure 5 • BB ⊥, 1×2 : Phenyl rings of the benzoates are perpendicular to the [001] direction, using the (1 × 2)-reconstructed (110) surface instead, as seen in Figure 6 These collections of atoms are padded by 15Å of vacuum between the topmost and the bottom-most atoms of the vertically adjacent simulation cells, in order to ensure lack of interaction between the atoms of these cells. To calculate the adsorption energies (E ads ), the energies of three simulation cells of the above defined dimensions, consisting of the slab and the adsorbate (E slab+mol ), just the slab (E slab ) and just the adsorbate (E mol ) in neutral form respectively, were first calculated, obtaining E ads by: E ads = E mol + slab − (E mol + E slab ) (1) As the latest STM studies 14 did not report H adsorption on the protruding bridging O 2c s, these simulations were also run with the dissociated H removed, with stabilities of each mode compared in terms of the total energy of the simulation cell E cell . In addition, the ionic adsorption energies of E ion, ads the different modes will be compared against one another, in a similar fashion to (1), with E mol being replaced by that corresponding to the energy of a single benzoate anion in the simulation cell. This gives us completeness in the comparisons for adsorption energies. Upon relaxations of these different modes of adsorptions under the above described set ups, simulated STM images were then generated based on the electronic structures of the relaxed adsorbate-surface complex, at a bias voltage of +1.5V. Results LDA Calculations For the adsorption structures, the optimized structures for the six BB modes corresponded largely with the models proposed for the STM images, 14 In terms of adsorption energetics, the adsorption energies (E ads ) were obtained (with the values obtained through GGA calculations) as displayed in Table 1. When the dissociated Hs were removed, ionic adsorption energies (E ion, ads ) were obtained and presented in Table 2. Table 1: Adsorption energies for dissociative adsorptions of benzoic acid on the rutile (110) surface, in eV/Å, as calculated through LDA, with and without the co-adsorbed H. Adsorption Mode From the figures for adsorption energetics alone, the LDA results did not completely corroborate with the STM studies. 13,14 In the former, it was observed that the BB , i.p. mode should be more stable than the BB ⊥, i.p. configuration, our current results of DFT calculations however contradict these. This could possibly be explained by the fact in such an arrangement, the benzene rings are close to each other, resulting in repulsion between the neighbouring rings, thereby destablizing the configuration. By rotating, this source of destablization was removed. E ads (dis) E ads (no H) BB , i In When the dissociated hydrogens were removed, ionic adsorption energies (E ion, ads ) were obtained using the method described in (1) and presented in Table 2. When the same simulations were rerun with the dissociated hydrogens removed, however, slightly different pictures emerged as the BB ⊥, 1×2 mode became more energetically stable than the BB , 1×2 mode, as was observed in the STM studies. The BB i.p. modes also became energetically comparable, while the BB ,o.p. mode became slightly more energetically favourable than the BB ⊥,o.p. . This suggests that the presence and absence of H atoms has influences on the overall stabilities of the adsorption structures,even without hydrogen bonding taken into account. The discrepancies between the calculated results and the experimental findings can be attributed to LDA as a method for DFT calculations that does not include hydrogen bonding, as well as π-π interactions, which in this case are those between neighbouring benzene rings. These are interactions that were proposed as significant factors in stablizing the adsorption structures, thougheven without being taken into account, rotation of the benzene rings was seen to have major effects on the energetic stabilities of the adsorptions. GGA Calculations In GGA-DFTD2 Calculations Although Van der Waals' forces are weak in comparison to hydrogen bonding when comes to inter-ionic interactions between adsorbates, they can come into play when large non-polar components come close to each other, and especially so in the case of phenyl rings with extensive π-orbitals being close with each other, as seen in Figures 1-6. Indeed, taking into account of Van der Waals' forces through GGA+DFTD2 calculations has improved the energetic stabilities for all six different modes, each being at least 0.7 eV/molecule more stable after implementation of GGA-DFTD2 instead of just GGA (see Table 3). In spite of these changes, the BB modes are still more energetically stable compared to their BB ⊥ counterparts, when the co-adsorbed hydrogens are taken into account. When just considering the case of ionic adsorption, the changes in E ads values become even more pronounced, and especially so in the case of benzoates' adsorptions on the (1 × 2)reconstructed surface. Not only are the E ads values abnormally large, but also that the BB ⊥,1×2 mode has now become significantly more energetically stable as compared to the BB ,1×2 mode (-7.87 vs -5.14 eV/benzoate). These anomalies shall be discussed in further detail in the'Discussions' section. Simulated STM Results In the experimental STM studies, the STM images produced presented the sites of benzoate adsorption as bright spots, elongated along the directions of the alignments of the phenyl Table 3: Adsorption energies for ionic adsorptions of benzoates on the rutile (110) surface, in eV/Å, calculated using GGA+DFTD2, with "*" representing anomalies in values. Adsorption Mode E ads (dis) E ads (no H) BB , i.p. (Figure 1) -3.55 -3.95 BB ⊥, i.p. (Figure 2) -2.12 -2.52 BB , o.p. (Figure 3 In the BB i. p. modes, whose simulated STM images are as shown in Figure atoms and the nearest phenyl H atoms when the benzene ring is rotated, making hydrogen bonding between the two too weak to be a significant stablizing factor. In terms of the (2 × 2)-symmetric patterns on the rutile (110) Perhaps the single greatest anomaly in the results lies in the abnormally large ionic adsorption energy values for benzoates on the 1 × 2-reconstructed surface -see Figure 3. We found that, for the ionic adsorption on the reconstructed surface, the energies appeared to be anomalous. We include the values for completeness, but are unable to explain their large values, which we will continue to investigate. In terms of the simulated STM images, the images generated revealed the bright spots corresponding to sites of benzoate adsorption, with the orientations of the rings much more clearly delineated than the experimental STM images do. The arrays of positions of the bright spots also corresponded to the 2D symmetries of the adsorption monolayers in the BB i. p. , BB o. p. and BB 1×2 configurations. For both the BB i. p. and the BB o. p. modes, in the limit of sufficiently low valued charge isosurfaces such that the benzene rings start to be imaged distinctively, the co-adsorbed H atoms lose their distinguishability from their surroundings, whereas in the limit of isosurfaces were not imaged at high coverages. Again, the lack of direct visual evidence for co-adsorption of dissociated protons does not necessarily imply complete absence of it. Conclusions We have studied the different patterns for the bidentate bridging dissociative adsorption of benzoic acid on both the unreconstructed and (1 × 2)-reconstructed surfaces through DFT methods LDA, GGA and GGA+DFTD2, as implemented through VASP, following both (2 × 1) and (2 × 2) symmetries. In all the three DFT methods studied, both with and without the dissociated H, the BB ,i. p. pattern was invariably found to be the most energetically stable out all the six BB patterns of adsorption. These results support STM studies 14 which revealed that on the unreconstructed (110) surface, where the prevalent pattern was that of BB i. p. , with the orientation of the benzene rings aligned along the [001] direction. In the case of (1 × 2)-reconstructed surface however, the BB ,1×2 mode was found to be more stable, in contrast to the experimental STM studies' results 14 where the benzene rings were observed to have rotated by 90 • , with hydrogen bonding between the phenyl hydrogens and the O 2c atoms along the reconstructed ridges being the reason for the stablization of this orientation. In the case of simulated STM images, the same overlayer patterns of bright dots was reproduced for the proposed models, elongations of the bright spots representing the alignments of benzene rings were observed, which were not obvious in the case of experimental tion on TiO 2 can thus offer an exemplar case study for how this entire class of compounds can interact with TiO 2 surfaces. Out of all the TiO 2 surfaces, the rutile (110) is the most energetically stable and naturally occurring, and thus forms the substrate surface of choice in such adsorption studies. The rutile (110) surface, though not prone to reconstruction, can undergo (1 × 2) reconstruction upon annealing under UHV conditions, with the models proposed including added Ti 2 O 3 and Ti 3 O 6 rows. As such, the focus of our studies shall be on the adsorption of benzoic acid on these two types of rutile (110) surfaces, in terms of the binding mode and the overlayer patterns. which involve the phenyl rings being aligned either with the [001] direction(denoted ) or the [110] direction (denoted ⊥ ), as well as being aligned such that they form (1 × 1)-periodic patterns (denoted i.p. ), or with the neighbouring [001]-rows of benzoates aligned one Ti 5c step out with respect to each other (denoted o.p. ). The different monolayer patterns are denoted as follows, with the relaxed GGA structures shown in the relevant figures (LDA and GGA relaxed structures show little different): • BB , i.p. : Phenyl rings of the benzoates are aligned with the [001] direction, with the benzoates of neighbouring [001] rows in step with each other, as seen inFigure 1 • BB , o.p. : Phenyl rings of the benzoates are aligned with the [001] direction, with the benzoates of neighbouring [001] one Ti 5c site out of step with each other, as seen in Figure 2 • 2BB ⊥, i.p. : Phenyl rings of the benzoates are perpendicular to the [001] direction, with the benzoates of neighbouring [001] rows in step with each other, as seen inFigure 3 • BB ⊥, o.p. : Phenyl rings of the benzoates are perpendicular to the [001] direction, with the benzoates of neighbouring [001] one Ti 5c site out of step with each other, as seen in in that the carboxylate group remained anchored via two Ti 5c -O carboxy bonds, while the benzene rings remained aligned more or less along the [001] and the [110] directions. Slight deviations were observed in the BB ,i.p. and the BB , 1×2 modes, in that the benzene rings were not exactly aligned along the [001] directions and instead being slightly rotated about the [110] axis. In addition to the slight rotations, in all of the BB modes, the benzene rings underwent slight planar rotations about the [110] axis. These are likely due to the sideways repulsions between the H atoms on the benzene rings. Figure 1 : 1The views of the GGA-optimized BB , i.p. mode of adsorption of benzoic acid on the rutile (110) surface, through the [001], [110] and the 110 directions. Figure 2 : 2The views of the GGA-optimized BB ⊥, i.p. mode of adsorption of benzoic acid on the rutile (110) surface, through the [001], [110] and the 110 directions. Figure 3 : 3The views of the GGA-optimized BB , o.p. mode of adsorption of benzoic acid on the rutile (110) surface, through the [001], [110] and the 110 directions. Figure 4 : 4The views of the GGA-optimized BB ⊥, o.p. mode of adsorption of benzoic acid on the rutile (110) surface, through the [001], [110] and the 110 directions. Figure 5 : 5The views of the GGA-optimized BB , i.p. mode of adsorption of benzoic acid on the rutile (110)−(1×2) reconstructed surface, through the [001], [110] and the 110 directions. When the acid was dissociatively adsorbed onto the rutile (110) in the manners of BB o. p. modes, similar to the results produced in LDA, the difference in terms of the energetic stabilities of this pair of BB and BB ⊥ modes was the narrowest of the three at 0.25 eV, though with the BB ,o. p. mode being more energetically stable this time. This gap is also 14 with no direct visual evidence of co-adsorbed hydrogens, except for low coverages on the (1 × 2)-reconstructed surface. The simulated STM images produced have replicated the same arrays of bright spots representing the two-dimensional periodities of the benzoate adsorbates (see Figures 7 and 8), while the rutile (110) surface itself has been shown to be dark in comparison to the adsorption sites. In the limit representing high charge isosurfaces i.e. large currents, the positions and the shapes of the bright regions representing the benzoate adsorption sites were mainly determined by the positioning of the carboxylate groups, rather than the orientations of the benzene rings. The latter only started to influence the appearances of bright regions when the charge isosurfaces are set at values sufficiently low (i.e. limits of low currents), such that the rutile (110) substrate surface itself loses its distinctive visual features, such as distinct dark regions as shown in Figure 7(top). Where the benzene rings were imaged in the limit of low currents, they appeared either as double rows of trios of bright spots, with the brightest at the centre representing the topmost C atom of the ring in the case of the benzene ring being aligned with the [001] direction; or as twin central bright spots accompanied by darker, kidney-shaped spots on either side. It is also observed while in the limit of high currents where the carboxylate groups were distinctively imaged, positions of the co-adsorbed H can be seen from the differing orbital shapes in the [001] O 2c rows, with circular-shaped dim spots representing the 1s orbitals of H and dumbell-shaped ones representing the 2p orbitals of the O 2c atoms. Figure 7 : 7rectangular rows of bright spots, with their widths aligned along the planes of the benzene rings determined through DFT calculations. The patterns of the bright spots representing the benzene rings however differed between the BB ,i. p. and the BB ⊥,i. p. modes, with the former as parallel trios of bright spots along the [001]-direction, and the latter as two central bright spots sandwiched by two larger merged dimmer spots aligned along the[110]-direction. In the BB o. p. modes, the simulated STM images generated are as shown in Figure 8. The bright spots representing the (2 × 2) overlayer symmetry can be observed in both cases, in agreement with STM results obtained from experimental studies, where such features were observed in some regions alongside the bright dots representing the (1 × 1) symmetry on the unrecontructed surface. The orientations of the bright spots again reflected the orientations of the benzene rings as shown in Figures 3 and 4. However, whereas the benzene ring was imaged as duo rows of bright spots in the BB ,i. p. mode, in the BB ⊥,i. p. mode the darker spots around the central bright spots merged, forming kidney-shaped spots. For the case of BB 1×2 modes, in both cases the reconstructed Ti 2 O 3 ridge appeared as large bright bands along the [001] directions, with features showing the ridges being largely indistinguishable. Again, although the double rows of three bright spots were observed when the ring was aligned along the [001] direction, the dimmer pairs of spots on the sides of the central bright spots became merged in the case of the benzene ring being aligned with the [110] direction. Discussions For adsorption energetics, GGA and GGA+DFTD2 simulations have shown the BB ,i. p. to be the most energetically stable out of all the BB modes of adsorptions on the unrecon-Simulated STM images for the BB ,i. p. (left) and BB ⊥,i. p. (right) modes, at high (top) and low (bottom) charge valued isosurfaces, i.e. high and low tunneling current, revealing the positions of the benzoates (top) and the orientations of the benzene rings (bottom) respectively. Figure 8 : 8Simulated STM images for the BB ,o. p. (left) and BB ⊥,o. p. (right) modes, at high (top) and low (bottom) charge valued isosurfaces, revealing the positions of the benzoates (top) and the orientations of the benzene rings (bottom) respectively. Figure 9 : 9Simulated STM images for the BB ,1×2 (left) and BB ⊥,1×2 (right) modes, in the limit of low charge isosurfaces, revealing the orientations of the benzene rings.structed surface, when co-adsorbed with the dissociated hydrogens. Being the most observed pattern in the experimental STM studies of the unreconstructed surface, this is an expected result. In terms of the simulations themselves, this shows that when hydrogen bonding and/or Van der Waals' forces are taken into account, the BB ,i. p. mode is the most stable.Examining the three pairs of BB and BB ⊥ modes of adsorption, in GGA and GGA+DFTD2based calculations, it is clear that on the non-reconstructed surface, simulations have shown that for both (1 × 2)-and (2 × 2)-symmetric patterns, the BB modes were shown to be more energetically stable in both cases. These observations, in both the experimental and computational DFT studies, can be accounted for by the long distances between the O 2c of high charge values, the vacant O 2c sites could be distinguished from the ones bound by dissociated H by the 2p orbitals imaged. The simulated STM images in the low charge value limits reveal features which are consistent with those observed in the experimental STM images produced of benzoates on the unreconstructed surface, most notably the elongations of the bright regions along the directions of alignments of the benzene rings, as well as the absence of features reflecting the co-adsorption of dissociated H. This suggests that perhaps even if STM images do not directly image the co-adsorbed H ions, it does not mean that such co-adsorption did not take place. On the reconstructed surfaces, the simulated STM images revealed central bright bands representing the Ti 2 O 3 reconstructed rows similar to those observed in the experimental STM images, and that no distinguishable features representing co-adsorption of dissociated H ions. This is however still in agreement with experimental results as co-adsorbed H ions STM studies. These alignments were only deduced by line profiles taken along the [001] and the [110] directions in the experimental STM studies. In the different limits of high and low-valued charge isosurfaces, the simulated STM images have also revealed the carboxylate and the benzene ring groups respectively, while in experimental STM images, these structures were not clearly distinguished in the bright spots representing the sites of benzoate adsorption. the BB o.p. superstructures, the results showed agreement with the observations that the benzene rings would be rotated along the [110] direction, with E ads (BB ⊥, o.p. ) being 0.06 eV more stable than the value for E ads (BB , o.p. ). The difference however, was much smaller than expected taking into consideration of the predomination of the BB ⊥, o.p. pattern in the STM images.For adsorption on the reconstructed surface, stable dissociative bidentate bridging adsorption of benzoic acid on the surface was observed at the Ti 5c sites between the reconstructed ridges along the [001] directions. The adsorption energy values of the BB , 1×2 and the BB ⊥, 1×2 patterns however, still contradicted the STM observations, with the non-rotated modes being almost 0.5 eV more stable than the rotated ones. Table 2 : 2Adsorption energies for adsorptions of benzoates on the rutile (110) surface, in eV/Å, as calculated through GGA, with and without the co-adsorbed H.Adsorption Mode E ads (dis) E ads (no H) terms of physical structures, all the four modes of adsorption, save the BB , o.p. mode, exhibit adsorption structures close to those following the STM studies. 14 Notable differences exist between the GGA-optimized adsorption configurations for the BB modes, and the idealized structures proposed to account for the STM observations. In both of the BB modes, buckling of the phenyl rings on the [001] plane was observed. In the BB , o.p. mode in particular, a rotation of the phenyl ring around the [001] axis was observed. This could have come about as a result of the closer distances between the neighbouring phenyl rings of the adjacent benzoates along the [001] rows, resulting in steric repulsions between the phenyl H atoms. In the BB ⊥ modes, such deviations from the idealized structures as proposed to model the STM studies, 14 with the phenyl rings almost exactly aligned along the [110] direction. This can be due to the fact that the phenyl rings of the neighbouring benzoates along the [110] direction are further apart. One particular interesting observation was the tilting of the H atoms in the cases of BB , o.p. , BB ⊥, i.p. and the BB ⊥, o.p. modes, towards the carboxyl O atoms of the respective modes, which were not observed in the LDA calculations. This is possibly due to the hydrogen bonding between the deprotonated H, and the carboxyl O. Such significant tilting, however, was not observed for the BB , i.p. mode. The tilting observed in the former three modes, could be attributed towards the hydrogen bonding interactions between the deprotonated H, and the carboxyl O atoms. This however raises the question of why the same hydrogen bonding could not produce the same tilt on the H atom. This is possibly due to to retain their orientations once the calculations attained relaxation. These thereby suggest that neither the presence or absence of H on the O 2c sites had had significant effects on the physical orientations of the phenyl rings.In terms of the adsorption energetics, the GGA-based calculations have shown that the adsorption energies have decreased from those of over 2 eV/benzoate, to values much closer to 1.5 eV, 18 ,19 20 and.21 In terms of the comparative energetic stabilities of the different adsorption modes, the GGA calculations have revealed that the BB modes were more energetically stable, than the BB ⊥ modes, in all the three pairs of modes. This is in clear contrast to the LDA results, where the BB ⊥ modes were the more energetically stable configurations out of all the three pairs. Part of the reason why in GGA the BB modes were more energetically stable, lies in the fact that now the inter-benzene ring distance between neighbouring rings along the [110] direction was increased, resulting in less repulsion between them.For the case of BB i.p. modes, the results agree with the conclusions drawn from STM, that the BB ,i.p. mode is the more commonly observed mode in the (1 × 1)-unreconstructed case. In fact, the energetic favourability was the greatest in this case, at a value of 1.5 eV, in contrast to the case of LDA, where the BB ⊥,i.p. mode was calculated to be much more energetically stable (by 1.2 eV). The comparative greater energetic stability in the BB ,i.p. mode can now be described in terms of greater spacing between the neightbouring benzene rings, being increased by nearly 0.1Å, resulting in less sideways repulsion between the sidemost H atoms.In the other pairs of BB and BB ⊥ modes, although the adsorption energy values were closer to the above range, they did not agree with the models proposed in the experimental studies, in which both the BB o.p. and the BB 1×2 modes had the BB ⊥ modes were described to be the predominant form observed in STM. This is however explainable by the presencethe fact that in the BB , i.p. mode, the almost perfectly [001]-aligned phenyl rings, sterically hinders such an interaction through interactions between the rings' own H atoms and the carboxyl oxygens. When the dissociated hydrogens are stripped from the surfaces, little structural changes were observed in all the modes mentioned. This was also true when Van der Waals' forces were taken into account when DFT-D2 was turned on, as the phenyl rings were observed of co-adsorbed H on the O 2c on the [001] ridges, which serve to repel H atoms on the rotated benzene rings. Figure 6 : 6The views of the GGA-optimized BB ⊥, i.p. mode of adsorption of benzoic acid on the rutile (110)−(1×2) reconstructed surface, through the [001], [110] and the 110 directions. larger than that reported for the corresponding pair in the LDA calculations, where the BB ⊥,o. p. mode was more energetically stable by just 0.05 eV/molecule. The GGA results suggest that rotation of the benzene rings by 90 • in order to avoid the steric repulsion between the rings along the [001] directions, as well as the weak H interactions between the benzyl H and the O 2c of the [001] rows, do not energetically stabilize the structure, and that rotation of benzene rings themselves can serve as an energetically destabilizing factor. of BB 1×2 modes relaxed in the LDA calculations, which showed that the BB ⊥,1×2 mode was more energetically stable by 0.9 eV/molecule, a much greater difference in the favour of the other adsorption pattern. Again, this suggests that rotation of the benzene ring in itself might have been an energetically destablizing effect on the entire set up. When the co-adsorbed hydrogens were removed, the structures changed very little, with the most significant being the partial restoration of the symmetry of Ti 2 O 3 reconstructed ridges. This shows that the presence of H atoms on the O 2c ridges does not affect hydrogen bonding interactions between most parts of the surface and any part of the benzoate. In terms of ionic adsorption energetics themselves, the comparative energetic stability of the BB over the corresponding BB ⊥ mode was confirmed, while in all of the cases, adsorption of benzoate ions without the co-adsorption of the dissociated H had a greater energetically stabilizing effect than dissociative co-adsorption of the both ions on the same surface with the same geometry.One interesting thing to note is that while for the BB i. p. and BB o. p. modes, the E ads (no H) values are more , while that for the BB 1×2 modes the E ads (no H) values appeared to be far larger than the corresponding E ads values, over twice values of the E ads obtained for dissociative co-adsorption. As discussed earlier, a partial restoration of symmetry of the Ti 2 O 3 reconstructed ridges was observed following the removal of the dissociated H, and this perhaps provided the additional energetic stabilization. However, due to the large increase in the values of E ads involved, and the values for E ads (no H) differed little regardless of simulation cell used and the same E slab values were used for both the dissociative coadsorptive and the ionic adsorptive cases, further investigations shall be needed to determine the exact cause of this unusually large increase in the adsorption energy when the dissociated hydrogens are not taken into account.For the adsorption on the reconstructed surface, while the BB ,1×2 mode was calculated to be the more energetically stable, as compared to the BB i.p. modes, the energetic favourability was much lower at 0.25 eV/molecule. Furthermore, as compared to the corresponding pair surface, the BB ,o. p. modes were shown to be more energetically stable through GGA and GGA+DFTD2 calculations. This again is in line with the experimental observations, which revealed alignment of the benzene rings with the [001] direction. The proposed reasons for such observations were again that the nearest distances between the O 2c atoms and the phenyl H atoms being too far apart to make hydrogen bonding a significant stablizing force. In comparison the BB ,i. p. mode, the corresponding BB ,o. p. mode appeared less energetically stable (see Figures 1 and 2). This again agrees with the findings of experimental STM studies, where such patterns were comparatively rare relative to the (2 × 1)-symmetric patterns. In the case of the (1 × 2)-reconstructed rutile (110) surface, however, the results of the DFT calculations contradict the conclusions reached in the STM studies. 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[ "Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficientś", "Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficientś" ]	[ "Alvaro Gutiérrez ", "Mercedes H Rosas ", "\nDepartamento deÁlgebra Departamento deÁlgebra\nUniversidad de Sevilla\nSpain\n", "\nUniversidad de Sevilla\nSpain\n" ]	[ "Departamento deÁlgebra Departamento deÁlgebra\nUniversidad de Sevilla\nSpain", "Universidad de Sevilla\nSpain" ]	[]	We give necessary conditions for the positivity of Littlewood-Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if S λ (V ) appears as a summand in the decomposition into irreducibles of S µ (S ν (V )), then ν's diagram is contained in λ's diagram.	null	[ "https://export.arxiv.org/pdf/2109.11378v2.pdf" ]	237,605,366	2109.11378	67045943b54ab17eb0f216c5680f1c8f0db2d1c7	Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficientś Alvaro Gutiérrez Mercedes H Rosas Departamento deÁlgebra Departamento deÁlgebra Universidad de Sevilla Spain Universidad de Sevilla Spain Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficientś symmetric functionsplethysmLittlewood-Richardson coefficientsSXP rule MSC: 05E0505E1805A17 We give necessary conditions for the positivity of Littlewood-Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if S λ (V ) appears as a summand in the decomposition into irreducibles of S µ (S ν (V )), then ν's diagram is contained in λ's diagram. INTRODUCTION Introduction The operations of restriction, tensor product, and composition of representations allow us to combine complex representations of the general lineal groups, and obtain new interesting representations of these groups. Breaking these new representations as sums of irreducibles representations is a major problem in representation theory. It is in this setting that the families of coefficients that we study in this work appear: the Littlewood-Richardson, Kronecker, and plethystic coefficients, respectively, describe the multiplicities that govern these decompositions. In addition to their importance in representation theory, these coefficients naturally appear in many different fields of mathematics from invariant theory, Schubert calculus, and algebraic geometry, to physics and computer science, [9,13,8,15,20]. For recent work see [5,7,14,16,17,21]. In the language of symmetric functions, the irreducible representation of GL(V ) indexed by a partition λ translates to the Schur function s λ . The tensor product of irreducible representations translates to the ordinary product of Schur functions, which allows us to define the Littlewood-Richardson coefficients as the structural constants for the ordinary product of Schur functions, s µ · s ν = λ c λ µ,ν s λ . Since Schur polynomials form an orthonormal basis of the space of symmetric functions, we can also write c λ µ,ν = ⟨s λ , s µ · s ν ⟩. The operation of composition of representations translates to the plethysm of symmetric functions, which in turn allows us to define the plethystic coefficients a λ µ [ν] as the number ⟨s λ , s µ [s ν ]⟩. Finally, the Kronecker coefficients g λ µ,ν are the multiplicities governing the decomposition into irreducibles of the restriction of GL(V ⊗ W ) to GL(V ) × GL(W ), via the Kronecker product of matrices. A famous result, often attributed to Dvir, gives a necessary condition that a Kronecker coefficient must satisfy in order to be nonzero. This is a remarkable result, as Dvir's conditions are both elegant and very easy to manipulate. Let us identify a partition λ with its Ferrers diagram. Explicitly, given partitions µ and ν, Dvir defines a rectangular partition R = (\|µ ∩ ν\| \|µ∩ν ′ \| ) and shows that if g λ µ,ν is nonzero, then λ ⊆ R (see [6]). Dvir's result gives a powerful tool in representation theory. To give just two recent applications, Pak and Panova used it to find a counterexample of the Kirillov-Klyachko conjecture [17], and Briand, Orellana, and the second author used it to give sharp bounds for the stability of the Kronecker products of Schur functions [2]. Following the spirit of Dvir's result, we show in Theorem 5.1 that if s λ appears as a nonzero summand on the decomposition of s µ [s ν ] in the Schur basis, then the diagram of ν is contained in the diagram of λ. In other words, if a λ µ[ν] is nonzero, then ν ⊆ λ. With this aim in mind, we first show in Theorem 3.5 how to define a partition Θ, from partitions µ and ν, such that c λ µ,ν ̸ = 0 implies λ ⊆ Θ. Our main tool comes from plethystic calculus, and the operation of evaluation into sums and differences of alphabets. This approach to the study of structural constants has been proven successful in the past for Kronecker [19] and plethystic [11] coefficients. In Theorems 4.1 and 4.4, we use this result together with the SXP rule [12,21] to determine upper and lower bounds for the partitions µ appearing with positive coefficient in the expansion of p n [s λ ] in the Schur basis. Explicitly, we show that λ ⊆ µ ⊆ Ξ, where Ξ is a purposely crafted partition. Then, we express the plethysm of two arbitrary Schur functions in terms of SXP coefficients and plethysms of the type p n [s λ ] as follows: s µ [s ν ] = λ χ µ (λ) z λ p λ [s ν ] = λ χ µ (λ) z λ i p λi [s ν ] = λ χ µ (λ) z λ i τ b τ λi[ν] s τ ,(1) where χ µ (λ) is the value of the character χ µ of the Specht module S µ on any permutation of cycle type λ, the number z λ denotes the order of the centralizer of a permutation of cycle type λ, and b τ λi [ν] is defined as the number ⟨s τ , p λi [s ν ]⟩. The tools thus far developed suffice to show our aforementioned main result (Theorem 5.1). Finally, in Corollary 5.2, we completely characterize the multiplicity of the trivial and sign representations on the decomposition into irreducibles of the composition of arbitrary irreducible representations. These last results can also be deduced from Yang's work [22]. Preliminaries Partitions and symmetric functions We follow Stanley [20] for the standard concepts and notations in the theory of symmetric functions, the main exception being that we represent our partitions with the French convention 1 . A partition is a weakly decreasing sequence of natural numbers in which there are finitely many nonzero entries. Define the (Ferrers) diagram of a partition λ as the subset of N 2 0 made of the points (c, r) such that 0 ≤ c < λ r . We will often identify a partition with its diagram. A partition is a (c, r)-hook if its diagram does not contain the point (c, r) [18]. Note that a (1, 1)-hook -usually known just as a hook -is also a (c, r)-hook for any c ≥ 1 and r ≥ 1. We say that (c, r)-hooks fit in a fat-hook region of N 2 0 with c columns and r rows (see Figure 1). The point (c, r) is an outer corner if (c, r) ̸ ∈ λ but its addition to the diagram produces a valid partition. Let Outer(λ) be the set of outer corners of λ. For example, Outer((3, 3, 1)) = {(0, 3), (1, 2), (3, 0)}. The complement (λ) c of the diagram of a partition λ defines an ideal of N 2 0 with respect to the coordinate-wise sum; that is, (λ) c = {(x, y) ̸ ∈ λ} is closed under the sum. The set Outer(λ) is the minimal spanning set of (λ) c = ⟨Outer(λ)⟩. Conversely, the complement of such an ideal containing at least one point of the form (0, r) and one point of the form (c, 0) is the diagram of a partition. See Figure 2 for an illustration of these concepts. × × × Figure 2: On the left, the Ferrers diagram of µ = (3, 3, 1). On the right, its associated ideal is shaded. It is spanned by the set Outer(µ) of outer corners of µ (depicted as crosses). Let the sum of two partitions λ and µ be the partition λ + µ = (λ 1 + µ 1 , λ 2 + µ 2 , ...), and let the union λ ∪ µ of partitions λ and µ be the partition resulting from the sorting of their parts. Moreover, for a given n ∈ N + , we let nλ = λ + λ+ n times . . . +λ and ∪ n λ = λ ∪ λ∪ n times . . . ∪λ. We shall write (⊴) for the dominance order on partitions, letting λ ⊴ µ if i≤j λ i ≤ i≤j µ i for all j. We write µ ⊆ λ and say µ is contained in λ whenever µ i ≤ λ i for all i. For µ ⊆ λ, we let λ/µ denote the set of points in λ and not in µ. A rim hook of λ is a skew partition λ/µ whose diagram is (orthogonally) connected and contains no 2 × 2 arrangement. Let n ∈ N + . The n-quotient and n-core of a partition encode all of the information of the original partition. We list some of their properties, and refer to Macdonald [13] for their proofs. The n-quotient of a partition λ is defined as the n-tuple λ * = (λ (0) , λ (1) , ..., λ (n−1) ), where λ (i) is made of the points (k, j) in λ such that c k := λ ′ k + k + 1 ≡ i (n) and r j := λ j + j ≡ i (n). Note that c k only depends on the column and r j on the row. The n-core is defined as the partitionλ which remains after removing (step by step) every rim hook of length n from λ. The order in which the rim hooks are removed does not matter. Letting \|λ * \| be \|λ (0) \| + \|λ (1) \| + · · · + \|λ (n−1) \|, we get the following formula \|λ\| = \|λ\| + n\|λ * \|.(2) Let Λ be the algebra of symmetric functions. That is, the algebra Q[p 1 , p 2 , ...] spanned by the algebraically independent variables p k which we name the power sum symmetric functions. It will sometimes be useful to identify an element f in Λ with a formal power series. Let X = x 1 + x 2 + ... be an alphabet -a collection of variables called letters. We will identify any f ∈ Λ with its image f [X] under the morphism that maps p k to x k 1 + x k 2 + ... . In particular, we identify p 1 with X. We write f [X] = f (x 1 , x 2 , . ..) and say that it is the evaluation of f in X. For f ∈ Λ, let ⟨s λ , f ⟩ denote the coefficient of s λ in the decomposition of f in the Schur basis. Hence f = λ ⟨s λ , f ⟩ · s λ . We let the set {λ : ⟨s λ , f ⟩ ̸ = 0} be the support of f , denoted as supp(f ). The generalized Littlewood-Richardson coefficient c λ µ 0 ,µ 1 ,...,µ n−1 is defined as the number ⟨s λ , s µ 0 · s µ 1 · · · s µ n−1 ⟩. Note that for n = 2, we recover the usual Littlewood-Richardson coefficient (hereafter, LR coefficient). As an immediate consequence of the Littlewood-Richardson rule, we get the following lemma. Lemma 2.1. If λ ∈ supp(s µ · s ν ) then µ ∪ ν ⊴ λ ⊴ µ + ν. Moreover, c µ∪ν µ,ν = c µ+ν µ,ν = 1. Plethysm The notion of plethysm, denoted by ·[·], comes from that of composition. Let f and g in Λ. If g[X] is a sum of monic monomials, g[X] = g 1 + g 2 + ... then f [g[X]] = f (g 1 , g 2 , ...). In particular, since we identify p 1 with X, then f [X] is just the plethysm of f with X. Example 2.2. If f [X] is a power series with positive integers as coefficients, it can be expressed as a sum of monic terms. For instance, 2p 2 [X] = 2x 2 1 + 2x 2 2 + ... = x 2 1 + x 2 1 + x 2 2 + x 2 2 + ... . Consequently, p n [2p 2 [X]] = p n (x 2 1 , x 2 1 , x 2 2 , x 2 2 , ...) = 2p 2n [X]. More precisely, the operation of plethysm of symmetric functions is defined axiomatically. 2. For any f ∈ Λ, the map g → g[f ] is a Z-algebra homomorphism on Λ. 3. For any f ∈ Λ, the equality p n [f ] = f [p n ] holds. Example 2.4. We use axiom (3) to compute p n [−X] = p n [−p 1 ] = −p 1 [p n ]. Then, using axiom (2), we get s 2 [−X] = p 1,1 + p 2 2 [−X] = 1 2 p 1 [−X]p 1 [−X] + 1 2 p 2 [X] = 1 2 p 1 [X]p 1 [X] − 1 2 p 2 [X] = p 1,1 − p 2 2 [X] = s 1,1 [X]. The core tools of this work come from plethystic calculus. Namely, from the operation of evaluation in sums and differences of alphabets. This next lemma is standard. More general formulas, for s λ [f ± g] on two arbitrary symmetric functions, are found in [13]. Lemma 2.5. Let X and Y be two alphabets and let λ be a partition. Then: 1. s λ [−X] = (−1) \|λ\| s λ ′ [X]. 2. s λ [X + Y ] = µ⊂λ s µ [X] · s λ/µ [Y ] = µ,ν c λ µ,ν s µ [X] · s ν [Y ]. 3. s λ [X − Y ] = µ⊂λ (−1) \|λ/µ\| s µ [X] · s (λ/µ) ′ [Y ] = µ,ν (−1) \|ν\| c λ µ,ν s µ [X] · s ν ′ [Y ]. Remark 2.6. Let X and Y be two alphabets. Then, Lemma 2.5 says that s λ [X −Y ] is the generating function of the tableaux T on positive letters from X and negative letters from −Y obeying the semistandarity rules for the positive entries and the opposite rules for the negative ones. For instance, in Figure 3 we have four such tableaux of weights x 2 1 x 2 x 3 3 x 4 , (−1) 8 y 1 y 2 y 3 y 3 4 y 2 5 , (−1) 3 x 2 1 x 2 x 3 3 y 1 y 2 2 , and (−1) 10 x 2 1 x 2 2 y 3 1 y 3 2 y 4 3 , respectively. [X] = p k [−X] ̸ = p k [tX]\| t=−1 = (−1) k p k [X]. This next theorem enables us to calculate plethysms of the form p n [s λ ]. Theorem 2.7 (SXP rule [12,21]). For any partitions λ, µ and any n ∈ N + , ⟨s µ , p n [s λ ]⟩ = sgn n (µ) · s λ , s µ (0) · s µ (1) · · · s µ (n−1) , where (µ (0) , . . . , µ (n−1) ) is the n-quotient of µ, and the sign function is defined as in [21]. (2) The partition ∪ n λ is in supp(p n [s λ ]) and ⟨s ∪ n λ , p n [s λ ]⟩ = (−1) \|λ\|(n−1) . Proof. We prove the first assertion; the second one is shown similarly. Let µ = nλ. To begin with, the n-core of µ is empty. Now, checking λ ∈ supp(s µ (0) · s µ (1) · · · s µ (n−1) ) will suffice. Since λ = µ (0) ∪ µ (1) ∪ · · · ∪ µ (n−1) , the result holds from Lemma 2.1. ■ A plethystic substitution lemma The following lemma links Schur functions evaluations with LR coefficients. This result has been used implicitly in [11,19]. Given positive integers a and b, let X a = x 1 + x 2 + . . . + x a and X b = x ′ 1 + x ′ 2 + . . . + x ′ b be two alphabets. We will identify the alphabet X a + X b with X a+b by setting x ′ i → x a+i . Proof. We know from Lemma 2.5 that s λ [X a+b − Y c+d ] = s λ [(X a − Y c ) + (X b − Y d )] = c λ µ,ν s µ [X a − Y c ] s ν [X b − Y d ]. Therefore, if s λ [X a+b − Y c+d ] ̸ = 0 then at least one of the terms in the sum doesn't vanish. Conversely, suppose that c λ µ0,ν0 s µ0 [X a − Y c ] s ν0 [X b − Y d ] ̸ = 0 for some µ 0 , ν 0 and consider the equation s λ [X a+b − tY c+d ] = c λ µ,ν s µ [X a − tY c ] s ν [X b − tY d ], where t is a variable as in Note 2.1. Let t = −1. The positivity of LR coefficients ensure that every monomial in both sides of the equality is now positive, so there can't be any cancellation. This means in particular that s λ [X a+b − tY c+d ] t=−1 ̸ = 0. On the other hand, we obtain s λ [X a+b − tY c+d ] t=−1 (that we know that is different than zero) from s λ [X a+b −Y c+d ] by setting, for each j, the letter y j to be −y j . Therefore, we can conclude that s λ [X a+b −Y c+d ] is also different than zero. ■ Remark 2.11. This result also holds for infinite alphabets. If λ is a partition of n, working with infinite variables is equivalent to working with n variables, which is the case that we settled in the previous lemma. Positivity conditions for the Littlewood-Richardson coefficients We present a general theorem giving necessary conditions for the positivity of the Littlewood-Richardson coefficients. The following elementary observation will play a crucial role. Figure 4. Consequently, λ must also be a subset of (2 3 , 1) (see the fourth diagram in Figure 4). In the proof of Lemma 3.1, we constructed a tableau in the alphabet X r − Y c for every (c, r)-hook. We will refer to it as the canonical SSYT of shape λ in the alphabet X r − Y c or in the corner (c, r). Outer (µ k ) c , where the sum is the Minkowski sum on sets, and the sum in N 2 0 is coordinate-wise. Proof of Theorem 3.5. Let λ ∈ supp(s µ 0 · · · s µ n−1 ). Therefore, there exists a partition ν n−2 in supp(s µ 0 · · · s µ n−2 ) such that λ ∈ supp(s ν n−2 s µ n−1 ). Take now ν n−2 . By the same analysis, there exists a partition ν n−3 in supp(s µ 0 · · · s µ n−3 ) such that ν n−2 ∈ supp(s ν n−3 s µ n−2 ). Iterate this process to obtain a chain of partitions λ = ν n−1 , ν n−2 , ..., ν 1 , ν 0 = µ 0 . Choose outer corners (c 0 , r 0 ) ∈ Outer(µ 0 ) and (c 1 , r 1 ) ∈ Outer(µ 1 ). As the canonical tableau for a given corner exists, s µ 0 [X r0 − Y c0 ] ̸ = 0 and s µ 1 [X r1 − Y c1 ] ̸ = 0. In addition, we know that ν 1 ∈ supp(s ν 0 s µ 1 ) = supp(s µ 0 s µ 1 ). Thus, by Lemma 2.10, we get s ν1 [X r0+r1 − Y c0+c1 ] ̸ = 0. Choose now an outer corner (c 2 , r 2 ) ∈ Outer(µ 2 ). Since ν 2 ∈ supp(s ν 1 s µ 2 ), we get that s ν 2 [X r0+r1+r2 − Y c0+c1+c2 ] ̸ = 0, again by Lemma 2.10. After iterating, ν n−1 = λ and so s λ [X Σri − Y Σcj ] ̸ = 0. This means that c i , r j is not in λ. Any choices of corners from µ 0 , ..., µ n−1 will give a similar result, ending the proof. ■ 4 Positivity conditions for the SXP coefficients. In this section we derive necessary conditions for the positivity of the resulting coefficients of the expansion of this plethysm in the Schur basis (SXP coefficients), by combining our previous result for the Littlewood-Richardson coefficients and the SXP rule. As a corollary of Theorem 2.1, we get the following result. Theorem 4.1. Let n ∈ N + and let µ, λ be partitions. If ⟨s µ , p n [s λ ]⟩ is nonzero, then λ ⊆ µ. Proof. Let µ ∈ supp(p n [s λ ]). By the SXP rule, we have λ ∈ supp s µ (0) · · · s µ (n−1) . Choose an outer corner (c, r) ∈ Outer(µ). Hence s µ [X r − Y c ] ̸ = 0. Let T : µ → {−c, ..., −2, −1, 1, 2, ..., r} be the canonical SSYT of µ for X r − Y c . Compute the n-quotient, thus embedding each µ (k) inside µ's diagram. Considering the corresponding values T (i, j) of the canonical tableaux at those embedded cells, we obtain a SSYT T k of shape µ (k) , which we presume to filled with the alphabet X r k − Y c k . Then (c k , r k ) is an outer corner of µ (k) . Furthermore, we know that no two partitions of the n-quotient share any common letters, by construction of T and the n-quotient. Consequently, X r − Y c = X Σr k − Y Σc k . That is, we choose (c, r) ∈ Outer(µ) and we show that (c, r) ∈ Outer(µ (k) ). Therefore, Outer(µ) ⊆ Outer(µ (k) ). By the Theorem 3.5 and the SXP rule, (c, r) is not in λ. ■ ). Then, µ ⊆ (10, 5, 3, 2, 2, 1, 1, 1, 1, 1). We also know \|µ\| = \|(2)\| · \|(3, 2)\| = 10. On the other hand, we saw in Example 4.2 that (3, 2) ⊆ µ. However, the only partition of size 10 such that the first row is equal to 10 is the row partition (10), whose diagram clearly does not contain the partition (3,2). Therefore, our upper bound is subject to improvement. By adjusting our argument, the bound on µ r when λ ⊆ µ and \|µ\| = n\|λ\| becomes µ r ≤ n\|λ\| − \|(λ r+1 , λ r+2 , ...)\| r =: a r . A similar analysis for the columns yields the following bounding partition µ ′ c ≤ n\|λ\| − \|(λ ′ c+1 , λ ′ c+2 , ...)\| c =: b c . Note that these two bounding partitions do not need to be the same. The following result combines both bounding partitions into a more optimized one. Theorem 4.4. Let µ and λ be two partitions, and let Ξ 1 = (a 1 , a 2 , ...) and Ξ 2 = (b 1 , b 2 , ...) ′ with a r and b c defined as before. If ⟨s µ , p n [s λ ]⟩ is nonzero, then µ ⊆ Ξ 1 and µ ⊆ Ξ 2 . That is, µ ⊆ Ξ 1 ∩ Ξ 2 . Example 4.5. Continuing Examples 4.2 and 4.3, and by having the lower bound in consideration, we optimize the upper bound to (8, 5, 3, 2, 1, 1, 1). See Figure 6. ⊆ µ ⊆ ⊆ Positivity conditions for the general plethystic coefficients We now consider the plethysm of two arbitrary Schur functions. Our main result is the following. Theorem 5.1. Let µ, ν and λ be partitions. If a λ µ[ν] is nonzero, then ν ⊆ λ. Proof. We shall bring back Equation (1). We know from the LR rule that τ, π ⊆ θ for all θ in the support of s τ · s π . And if b τ λi [ν] does not vanish, from Theorem 4.1, we have ν ⊆ τ . Hence ν ⊆ θ. ■ Two cases of particular interest in representation theory can be further studied. The following corollary can also be deduced from Yang's work [22]. z λ , which is the evaluation s µ [1]. This equals 1 if µ is a row partition and vanishes otherwise. On the other hand, Lemma 2.9. (2) implies that ⟨s (1 mn ) , s (1 m ) [s (1 n ) ]⟩ = λ (−1) \|ν\|(\|µ\|−l(λ)) χ µ (λ) z λ . If \|ν\| is even, this is the evaluation s µ [s ν [1]], which vanishes unless both µ and ν are row partitions. Then ν must be both a row partition of even size and a column partition. This is impossible. On the other hand, if \|ν\| is odd, the multiplicity of the sign representation is λ sgn(λ) χ µ (λ) z λ = s µ ′ [1], which equals 1 if µ is a column partition and vanishes otherwise. ■ The previous lemma can be restated in the language of representation theory as follows. Corollary 5.3. Let S λ (V ) be the irreducible representation of GL(V ) indexed by λ. We have: (1) The trivial representation appears as a summand of S µ (S ν (V )) if and only if S µ and S ν are trivial representations. In that case, its multiplicity is 1. (2) The sign representation appears as a summand of S µ (S ν (V )) if and only if both S µ and S ν are sign representations and \|ν\| is odd. In that case, its multiplicity is 1. Recently, there has been plenty of interest in the closely related problem of understanding the complexity of deciding whether Littlewood-Richardson [3], Kronecker [1,16], and plethystic [10] coefficients are nonzero. Moreover, and in the cases where it is possible, polynomial algorithms have been developed to determine the positivity of such coefficients. In [4], the plethysm s 1,1 [s 4,2,2 ] is used as and example to illustrate the importance of this problem, in the case of the plethystic coefficients. A priori, there are p(16) = 231 partitions that could appear in supp(s 1,1 [s 4,2,2 ]), but only 40 actually do. By Theorem 5.1, we bring this initial number to 142 making it more approachable from the computational perspective. We provide one further example. Let µ = (2, 1), let ν = (1 n ). Any partition in the support of s µ [s ν ] must be, by our theorem, a partition of 3n with length at least n. This turns out to be a fairly restrictive condition for large n. See Figure 7. In addition, the theoretical insights of our main theorem are interesting on their own. We hope that the elegant nature of our results will serve as a useful lemmas, and contribute to these complexity results. Figure 1 : 1The diagrams of (c, r)-hooks fit in the depicted fat-hook region. Definition 2. 3 . 3The plethysm of symmetric functions, denoted by ·[·], is the operation Λ × Λ → Λ verifying 1. p n [p m ] = p nm for all n, m ∈ N 0 . Figure 3 : 3Four valid SSYT with positive and/or negative letters. Remark 2. 8 . 8From Equation(2) and the SXP rule, we can deduce that µ ∈ supp(p n [s λ ]) implies thatμ = ∅.The SXP rule lets us immediately identify some partitions of supp(p n [s λ ]). Let us start with two of them. Lemma 2.9. Let n ∈ N + . Then, (1) The partition nλ is in supp(p n [s λ ]) and ⟨s nλ , p n [s λ ]⟩ = 1. Lemma 2 . 10 . 210Let λ be a partition, and let a, b, c, d ∈ N + . Then, the evaluation s λ [X a+b − Y c+d ] ̸ = 0 if and only if there exist partitions µ 0 and ν 0 such that c λ µ0,ν0 ̸ = 0, s µ0 [X a − Y c ] ̸ = 0, and s ν0 [X b − Y d ] ̸ = 0. Lemma 3. 1 . 1We have s λ [X r − Y c ] ̸ = 0 if and only if λ is a (c, r)-hook.Proof. Suppose s λ [X r − Y c ] ̸ = 0. Then, λ does not have a (c, r) point, i.e., (c, r) ∈ (λ) c . (In order to see this, think of what would be the value of (c, r) in a tableau with r positive letters and c negative letters.) This, in turn, implies that λ fits in a fat-hook region with r rows and c columns (seeFigure 1).Conversely, if (c, r) is in (λ) c then s λ [X r − Y c ] ̸ = 0.Indeed, the following SSYT in the alphabet X r − Y c is always present. Fill the kth column with (k − c)'s, for k = 0, . . . , c − 1. After this, the empty cells in the kth row are filled with (k + 1)'s. ■ Remark 3.2. In the proof of Lemma 3.1, we showed that if s λ [X r − Y c ] ̸ = 0 then it fits into a (c, r)-hook region. If λ fits in multiple of these regions, we can then take the intersection of them to find a smaller region for which λ is a subset. See Example 3.3.Example 3.3. Suppose s λ [X 3 − Y 1 ] ̸ = 0.This means that there exists a SSYT of shape λ and filled with the letters {1, 2, 3, −1}, which implies that the point (1, 3) does not belong to λ. Suppose that we also know that s λ [X 4 ] ̸ = 0 and s λ [−Y 2 ] ̸ = 0. This implies that neither (0, 4) nor (2, 0) belong to λ. Therefore, λ must be a subset of each of the first three regions depicted in Example 3 . 4 .Figure 4 : 344Let (c, r) =(3,2). Let λ = (5, 5, 3, 1). Then, the canonical tableau of shape λ in the corner (c, r) is the fourth tableau inFigure 3.We can now state and prove the main result of this section. For two sets A and B let A + B = {a + b : a ∈ A, b ∈ B} be their Minkowski sum. From left to right, the regions ⟨(1, 3)⟩ c , ⟨(0, 4)⟩ c , ⟨(2, 0)⟩ c and ⟨(1, 3), (0, 4), (2, 0)⟩ c = (2 3 , 1).Theorem 3.5. Let n ∈ N + and let λ, µ 0 , µ 1 ..., µ n−1 be partitions. If c λ µ 0 ,µ 1 ,...,µ n−1 is nonzero, Example 3. 6 . 5 0 2 0 652Let µ 0 = (3, 2), µ 1 = (1, 1) = µ 2 . Compute their exterior corners: Outer(µ 1 ) = Outer(µ 2 ) = {(0, 2), (1, 0)}.Then, add together all possible combinations of exterior corners of our three partitions, to get a arises. Theorem 3.5 states that the diagram of every partition in supp(s µ 0 s µ 1 s µ 2 ) must be a subset of said region, illustrated inFigure Outer(µ k ) are depicted as crosses, and the shaded region represents the shape ⟨ Outer(µ k )⟩ c = (5, 4, 2, 2, 1, 1). Example 4. 2 . 2Let n = 2, λ =(3,2). We have p 2 [s 3,2 ] = −s 3,3,2,2 + s 3,3,3,1 + s 4,2,2,2 − s 4,3,3 − s 4,4,1,1 + s 4,4,2 − s 5,2,2,1 + s 5,3,1,1 − s 5,5 + s 6,2,2 − s 6,3,1 + s 6,4 . One can check that (3, 2) ⊆ µ for each µ ∈ supp(p 2 [s 3,2 ]). Theorem 4.1 gives us a lower bound on the partitions µ in supp(p n [s λ ]). On the other hand, from the definition of partition we automatically obtain a trivial upper bound. A partition µ ∈ supp(p n [s λ ]) must be of size n\|λ\|. Hence, the maximum size of the rth row is n\|λ\| r . We refine this upper bound by taking the lower bound into consideration. Let us start with an example. Example 4 . 3 . 43Let µ ∈ supp(p 2 [s 3,2 ] Figure 6 : 6For every µ in supp(p 2 [s 3,2 ]), Examples 4.2, 4.3 and 4.5 yield the depicted bounds. Corollary 5 . 2 . 52Let V be a d-dimensional vector space, and let µ and ν be partitions of m and n respectively, both of length ≤ d. We have: (1) The coefficient of s mn in s µ [s ν ] is nonzero if and only if both µ and ν are one row partitions. In that case, ⟨s mn , s m [s n ]⟩ = 1. ( 2 ) 2The coefficient of s (1 mn ) in s µ [s ν ] is nonzero if and only if both µ and ν are one column partitions and n is odd. In that case, ⟨s (1 mn ) , s (1 m ) [s (1 n ) ]⟩ = 1. Proof. Theorem 5.1 shows one implication of each assertion. From Lemma 2.9.(1), ⟨s mn , s m [s n ]⟩ = λ χ µ (λ) Figure 7 : 7In black, the ratio of partitions of 3n that are of length ≥ n; in white, the ratio of partitions of 3n that are in supp(s 2,1 [s (1 n ) ]). The high computational cost of plethysm only allowed us to gather data for n ≤ 8. Note 2.1. In general, evaluating on the alphabet X+ c times . . . +X is not equivalent to evaluating on the alphabet cx 1 + cx 2 + ... . We denote the first with f [cX] and the latter with f [tX]\| t=c . In particular, −p k In the French convention, we use are bottom-left justified diagrams[13]. The coordinate system is cartesian, the origin being aligned with the bottom-left corner. AcknowledgementsThe authors would like to express their gratitude towards Adrià Lillo, Emmanuel Briand, and Laura Colmenarejo for their useful remarks and interesting conversations. Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH. 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[ "ESTIMATING THE FALSE DISCOVERY RISK OF (RANDOMIZED) CLINICAL TRIALS IN MEDICAL JOURNALS BASED ON PUBLISHED p-VALUES PREPRINT", "ESTIMATING THE FALSE DISCOVERY RISK OF (RANDOMIZED) CLINICAL TRIALS IN MEDICAL JOURNALS BASED ON PUBLISHED p-VALUES PREPRINT", "ESTIMATING THE FALSE DISCOVERY RISK OF (RANDOMIZED) CLINICAL TRIALS IN MEDICAL JOURNALS BASED ON PUBLISHED p-VALUES PREPRINT", "ESTIMATING THE FALSE DISCOVERY RISK OF (RANDOMIZED) CLINICAL TRIALS IN MEDICAL JOURNALS BASED ON PUBLISHED p-VALUES PREPRINT" ]	[ "Ulrich Schimmack \nDepartment of Psychology\nUniversity of Toronto Mississauga\nMississaugaCanada\n", "František Bartoš \nDepartment of Psychological Methods\nUniversity of Amsterdam\nAmsterdamthe Netherlands\n\nInstitute of Computer Science\nCzech Academy of Sciences\nPragueCzech Republic\n", "Ulrich Schimmack \nDepartment of Psychology\nUniversity of Toronto Mississauga\nMississaugaCanada\n", "František Bartoš \nDepartment of Psychological Methods\nUniversity of Amsterdam\nAmsterdamthe Netherlands\n\nInstitute of Computer Science\nCzech Academy of Sciences\nPragueCzech Republic\n" ]	[ "Department of Psychology\nUniversity of Toronto Mississauga\nMississaugaCanada", "Department of Psychological Methods\nUniversity of Amsterdam\nAmsterdamthe Netherlands", "Institute of Computer Science\nCzech Academy of Sciences\nPragueCzech Republic", "Department of Psychology\nUniversity of Toronto Mississauga\nMississaugaCanada", "Department of Psychological Methods\nUniversity of Amsterdam\nAmsterdamthe Netherlands", "Institute of Computer Science\nCzech Academy of Sciences\nPragueCzech Republic" ]	[]	The influential claim that most published results are false raised concerns about the trustworthiness and integrity of science. Since then, there have been numerous attempts to examine the rate of false-positive results that have failed to settle this question empirically. Here we propose a new way to estimate the false positive risk and apply the method to the results of (randomized) clinical trials in top medical journals. Contrary to claims that most published results are false, we find that the traditional significance criterion of α = .05 produces a false positive risk of 13%. Adjusting α to .01 lowers the false positive risk to less than 5%. However, our method does provide clear evidence of publication bias that leads to inflated effect size estimates. These results provide a solid empirical foundation for evaluations of the trustworthiness of medical research.	null	[ "https://export.arxiv.org/pdf/2302.00774v1.pdf" ]	256,504,075	2302.00774	a9039b23a97ea3f4c825bb16cca9f68e48511165	ESTIMATING THE FALSE DISCOVERY RISK OF (RANDOMIZED) CLINICAL TRIALS IN MEDICAL JOURNALS BASED ON PUBLISHED p-VALUES PREPRINT Ulrich Schimmack Department of Psychology University of Toronto Mississauga MississaugaCanada František Bartoš Department of Psychological Methods University of Amsterdam Amsterdamthe Netherlands Institute of Computer Science Czech Academy of Sciences PragueCzech Republic ESTIMATING THE FALSE DISCOVERY RISK OF (RANDOMIZED) CLINICAL TRIALS IN MEDICAL JOURNALS BASED ON PUBLISHED p-VALUES PREPRINT Correspondence concerning this article should be addressed to Ulrich Schimmack at [email protected] This is the authors' version of the manuscript. The influential claim that most published results are false raised concerns about the trustworthiness and integrity of science. Since then, there have been numerous attempts to examine the rate of false-positive results that have failed to settle this question empirically. Here we propose a new way to estimate the false positive risk and apply the method to the results of (randomized) clinical trials in top medical journals. Contrary to claims that most published results are false, we find that the traditional significance criterion of α = .05 produces a false positive risk of 13%. Adjusting α to .01 lowers the false positive risk to less than 5%. However, our method does provide clear evidence of publication bias that leads to inflated effect size estimates. These results provide a solid empirical foundation for evaluations of the trustworthiness of medical research. Introduction Many sciences are facing a crisis of confidence in published results [1]. Meta-scientific studies have revealed low replication rates, estimates of low statistical power, and even reports of scientific misconduct [2]. Based on assumptions about the percentage of true hypotheses and statistical power to test them, Ioannidis [3] arrived at the conclusion that most published results are false. It has proven difficult to test this prediction. First, large scale replication attempts [4,5,6] are inherently expensive and focus only on a limited set of pre-selected findings [7]. Second, studies of meta-analyses have revealed that power is low, but rarely lead to the conclusion that the null-hypothesis is true [8,9,10,11,12,13,14,15,16] (but see [17,18]). So far, the most promising attempt to estimate the false discovery rate has been Jager and Leek's [19] investigation of p-values in medical journals. They extracted 5,322 p-values from abstracts of medical journals and found that only 14% of the statistically significant results may be false-positives. This is a sizeable percentage, but it is inconsistent with the claim that most published results are false. Although Jager and Leek's article was based on actual data, the article had a relatively minor impact on discussions about false-positive risks, possibly due to several limitations of their study [20,21,22,23]. One problem of their estimation method is the problem to distinguish between true null-hypotheses (i.e., the effect size is exactly zero) and studies with very low power in which the effect size may be very small, but not zero. To avoid this problem, we do not estimate the actual percentage of false positives, but rather the maximum percentage that is consistent with the data. We call this estimate the false discovery risk (FDR). To estimate the FDR, we take advantage of Soric's [24] insight that the false discovery risk is maximized when power to detect true effects is 100%. In this scenario, the false discovery rate is a simple function of the discovery rate (i.e., the percentage of significant results). Thus, the main challenge for empirical studies of FDR is to estimate the discovery rate when selection bias is present and inflates the observed discovery rate. To address the problem of selection bias, we developed a selection model that can provide an estimate of the discovery rate before selection for significance. The method section provides a detailed account of our method and compares it to Jager and Leek's [19] approach. Methods To estimate the false discovery rate, Jager and Leek [19] developed a model with two populations of studies. One population includes studies in which the null-hypothesis is true (H 0 ). The other population includes studies in which the null-hypothesis is false; that is, the alternative hypothesis is true (H 1 ). The model assumes that the observed distribution of significant p-values is a mixture of these two populations, modelled as a mixture of truncated beta distributions of the p-values. The first problem for this model is that it can be difficult to distinguish between studies in which H 0 is true and studies in which H 1 is true, but they had low statistical power. The second problem is that published studies are heterogeneous in power which might not be properly captured when modelling distribution of p-values under H 1 with a truncated beta distribution. To avoid the first problem, we refrain from estimating the actual false discovery rate, which requires a sharp distinction between true and false null-hypotheses (i.e., a distinction between zero and non-zero effect sizes). Instead, we aim to estimate the maximum false discovery rate that is consistent with the data. We refer to this quantity as the false discovery risk. To avoid the second problem, we use z-curve which models the distribution of observed z-statistics as a mixture of truncated folded normal distributions. This mixture model allows for heterogeneity in power of studies when H 1 is true [25,26]. False discovery risk To estimate the false discovery risk, we take advantage of Soric's [24] insight that the maximum false discovery rate is limited by statistical power to detect true effects. When power is 100%, all non-significant results are produced by testing false hypotheses (H 0 ). As this scenario maximizes the number of non-significant H 0 , it also maximizes the number of significant H 0 tests and the false discovery rate. For example, if 100 studies produce 30 significant results, the discovery rate is 30%. And when the discovery rate is 30%, the maximum false discovery risk with α = 0.05 is ≈ 0.12. In general, the false discovery risk is a simple transformation of the discovery rate, such as false discovery risk = 1 − discovery rate discovery rate × α 1 − α . If all conducted hypothesis tests were reported, the false discovery risk could be determined simply by computing the percentage of significant results. However, it is well-known that journals are more likely to publish statistically significant results than non-significant results. This selection bias renders the observed discovery rate in journals uninformative [26,25]. Thus, a major challenge for any empirical estimates of the false discovery risk is to take selection bias into account. Z-curve 2.0 Brunner and Schimmack [25] developed a mixture model to estimate the average power of studies after selection for significance. Bartoš and Schimmack [26] recently published an extension of this method that can estimate the expected discovery rate before selection for significance on the basis of the weights derived from fitting the model to only statistically significant p-values. The selection process assumes that the probability of a study to be published is proportional to its power [25]. Extensive simulation studies have demonstrated that z-curve produces good large-sample estimates of the expected discovery rate [26]. Moreover, these simulation studies showed that z-curve produces robust confidence intervals with good coverage. As the false discovery risk is a simple transformation of the expected discovery rate, these confidence intervals also provide confidence intervals for estimates of the false discovery risk. Z-curve obtains the expected discovery rate estimate from estimating the mean power of studies before selection of significance. Z-curve leverages distributional properties of z-statistics-studies homogeneous in power produce normally distributed z-statistics centered at a z-transformation of the true power (e.g., Equations 3 in Bartoš and Schimmack [26]). Z-curve incorporates the selection on statistical significance via a truncation at the corresponding α level, similarly to the Jager and Leek's approach. However, in contrast to other approaches, z-curve acknowledges that studies are naturally heterogeneous in power (i.e., studies are investigating different effects with different sample sizes) and estimates the mean power via a mixture model, f (z; θ) = J j=1 π j f j[a,b] (z; θ j ), where z correspond to statistically significant z-statistics modelled via J mixtures of truncated folded normal distributions, f (z; θ), where θ denotes parameters of the truncated folded normal distributions. 1 See Figure 1 in in Bartoš and Schimmack [26] for visual intuition. We extended the z-curve model to incorporate rounded p-values and p-values reported as inequalities (i.e., p < 0.001). We did so in the same way as the Jager and Leek's approach; we implementing interval censored (for rounded p-values) and right censored (for p-values reported as inequalies) likelihoods versions of the z-curve model, f [l,u] (z; θ) = J j=1 π j f [l,u] j[a,b] (z; θ j ), where l and u correspond to the lower and upper censoring points on a mixture of truncated folded normal distributions. For example, a p-value reported as inequality, p < 0.001, has only a lower censoring point at z-score corresponding to p = 0.001, and rounded p-value, p = 0.02, has a lower censoring point at a z-score corresponding to p = 0.025 and upper censoring point at a z-score corresponding to p = 0.015. Simulation study We performed a simulation study that extends the simulations performed by Jager and Leek [19] in several ways. We did not simulate H 1 p-values directly (i.e., we did not use a right skewed beta-distribution as a Jagger and Leek). Instead, we simulated H 1 p-values from two-sided z-tests. We drew power of each simulated z-test (the only required parameter) from a distribution based on Lamberink and colleagues' [16] empirical estimate of medical clinical trials' power (excluding all power estimates based on meta-analyses with non-significant results). This allowed us to assess the performance of the methods under heterogeneity of power to detect H 1 corresponding to the actual medical literature. To simulate H 0 p-values, we used a uniform distribution. We manipulated the true false discovery rate from 0 to 1 with a step size of 0.01 and simulated 10,000 observed significant p-values by changing the proportion of p-values simulated from H 0 and H 1 . Similarly to Jager and Leek [19], we performed four simulation scenarios with an increasing percentage of imprecisely reported p-values. Scenario A used exact p-values, scenario B rounded p-values to three decimal places (with p-values lower than 0.001 censored at 0.001), scenario C rounds 20% p-values to two decimal places (with p-values rounded to 0 censored at 0.01), and scenario D first rounds 20% p-values to two decimal places and further censors 20% p-values at on of the closest ceilings (0.05, 0.01, or 0.001). Leek's false discovery risk estimates using a simulation study. Estimated false discovery rate (FDR, y-axis) for z-curve (black-filled squares) and Jager and Leek's swfdr (empty circles) vs. the true false discovery rate (FDR, x-axis) across four simulation scenarios (panels). Scenario A uses exact p-values, scenario B rounds p-values to three decimal places, scenario C rounds 20% p-values to two decimal places, and scenario D rounds 20% p-values to two decimal places and censors 20% p-values at on of the closest significance levels (0.05, 0.01, or 0.001). Fig. 1 displays the true (x-axis) vs. estimated (y-axis) false discovery rate for Jager and Leek's [19] swfdr method and the false discovery risk for z-curve across the different scenarios (panels). When precise p-values are reported (panel A in the upper left corner), z-curve handles the heterogeneity in power well across the whole range of false discovery rate and produces accurate estimates of false discovery risk. Higher estimate than the actual false discovery rates are expected because the false discovery risk is an estimate of the maximum false discovery rate. Discrepancies are especially expected when power of true hypothesis tests is low. For the simulated scenarios, the discrepancies are less than 20 percentage points and decrease as the true false discovery rate increases. Even though Jager and Leek's [19] method aims to estimate the true false discovery rates, it produces higher estimates than z-curve. This is problematic because the method produces inflated estimates of the true false discovery rate. Even if the estimates were interpreted as maximum estimates, the method is less sensitive to the actual variation in the false discovery rate than the z-curve method. Panel B shows that the z-curve method produces similar results when p-values are rounded to three decimals. The Jager and Leek's [19] method however experiences estimation issues, especially in the lower spectrum of the true false discovery rate since the current swfdr implementation only allows to deal with rounding to two decimal places (we also tried specifying the p-values as a rounded input; however, the optimizing routine failed with several errors). Panel C shows a surprisingly similar performance of the two methods when 20% of p-values are rounded to two decimals, except for very high levels of true false discovery rates, where Jager and Leek's [19] method starts to underestimate the false discovery rate. Despite the similar performance, the results have to be interpreted as estimates of the false discovery risk (maximum false discovery rate) because both methods overestimate the true false discovery rate for low false discovery rates. Panel D shows that both methods have problems when 20% of p-values are at the closest ceiling of .05, .01, or .001 without providing clear information about the exact p-value. Underestimation of true false discovery rates over 40% is not as serious problem because any actual false discovery rate over 40% is unacceptably high. One potential solution to the underestimation in this scenario might by exclusion of the censored p-values from the analyses, assuming that censoring is independent the size of the p-value. Leek's swfdr [19] across four simulation scenarios (columns). Scenario A uses exact p-values, scenario B rounds p-values to three decimal places, scenario C rounds 20% p-values to two decimal places, and scenario D rounds 20% p-values to two decimal places and censors 20% p-values at on of the closest significance levels (0.05, 0.01, or 0.001). Root mean square error and bias of the false discovery rate estimates for each scenario summarized in Table 1 show that z-curve produces estimates with considerably lower root mean square error. The results for bias show that both methods tend to produce higher estimates than the true false discovery rate. For z-curve this is expected because it aims to estimate the maximum false discovery rate. It would only be a problem if estimates of the false discovery risk were lower than the actual false discovery rate. This is only the case in Scenario D, but as shown previously, underestimation only occurs when the true false discovery rate is high. To summarize, our simulation confirms that Jager and Leek's [19] method provides meaningful estimates of the false discovery risk and that the method is more likely to overestimate rather than underestimate the true false discovery rate. Our results also show that z-curve improves over the original method and that the modifications can handle rounding and imprecise reporting when the false discovery rates are below 40%. Only if estimates exceed 40%, it is possible that most published results are false positive results. Application to medical journals We developed an improved abstract scraping algorithm to extract p-values from abstracts in major medical journals accessible through https://pubmed.ncbi.nlm.nih.gov/. Our improved algoritm addressed multiple concerns raised in commentaries to the Jager and Leek's [19] original article. Specifically, (1) we extracted p-values only from abstracts labeled as "randomized controlled trial" or "clinical trial" as suggested by [20,23,21], (2) we improved the regex script for extracting p-values to cover more possible notations as suggested by [23], (3) we extracted confidence intervals from abstracts not reporting p-values as suggested by [23,22]. We further scraped p-values from abstracts in "PLoS Medicine" to compare the false discovery rate estimates to a less-selective journal as suggested by [20]. Finally, we randomly subset the scraped p-values to include only a single p-value per abstract in all analyses, thus breaking the correlation between the estimates as suggested by [20]. Although there are additional limitations inherent to the chosen approach, these improvements, along with our improved estimation method, make it possible to test the prediction by several commentators that the false discovery rate is well above 14%. We executed the scraping protocol on July 2021 and scraped abstracts published since 2000 (see Table 2 for a summary of the scraped data). Interactive visualization of the individual abstracts and scraped values can be accessed at https://tinyurl.com/zcurve-FDR. The "Abstracts" column contains the total number of accessed abstracts, the "CT or RCT" column contains the number of abstracts labels as either "clinical trials" or "randomized controlled trials", the "Scrapeable" column contains the number of further abstracts that contained at least one automatically scrapeable p-value or a confidence interval, and the "p-values" column contains the total number of p-values extracted. PLoS stands for the PLoS Medicine and NEJM stands for the New England journal of medicine. Leek's [19] method produces slightly higher false discovery rate estimates. Furthermore, z-curve produced considerably wider bootstrapped confidence intervals, suggesting that the confidence interval reported by Jager and Leek [19] (± 1 percentage point) was too narrow. A comparison of the false discovery estimates based on data before (and including) 2010 and after 2010 shows that confidence intervals overlap, suggesting that false discovery rates have not changed. Separate analyses based on clinical trials and randomized controlled trials also showed no significant differences (see Fig. 3). Therefore, to reduce the uncertainty about the false discovery rate, we estimate the false discovery rate for each journal irrespective of publication year. The resulting false discovery rate estimates based on z-curve and Jager and Leek's [19] method are summarized in Table 3. We find that all false discovery rate estimates fall within a .05 to .30 interval. Finally, further aggregating data across the journals provides a false discovery rate estimate of 0.13, 95% [0.08, 0.21] based on z-curve and 0.19, 95% [0.17, 0.20] based on Jager and Leek's [19] method. This finding suggests that Jager and Leek's [19] extraction method slightly underestimate the false discovery rate, whereas their model overestimated the false discovery rate. Finally, our improved false discovery risk estimate based on z-curve closely matches the original false discovery rate estimate. False discovery risk (FDR) estimates and 95% confidence interval for each of the analyzed journal and combined data set based on clinical trials and randomized controlled trials published since 2000 with z-curve and Jager and Leek's [19] swfdr method. Figure 3: Estimates of false discovery risk across decades, journals, and study types. Estimated false discovery rate (FDR, y-axis) for randomized controlled trials (RCT, black-filled squares) and clinical trials (CT, empty squares) with z-curve and divided by journal (x-axis, with information about unique articles in brackets, first line for RCT and second line for CT) and whether the article was published before (and including) 2010 (left) or after 2010 (right). Additional z-curve results So far, we used the expected discovery rate only to estimate the false discovery risk, but the expected discovery rate provides valuable information in itself. Ioannidis's [3] predictions of the false discovery rate for clinical trials were based on two scenarios. One scenario assumed high power (80%) and a 1:1 ratio of true and false hypothesis. This scenario implies a true discovery rate of .5.8 + .5.05 = 42.5%. The second scenario assumed 20% power and a ratio of 1 true hypothesis for every 5 false hypotheses. This scenario implies a true discovery rate of .17.20 + .83.05 = 7.5%. Scenarios also varied in terms of bias that would inflate the observed discovery rates. However, z-curve corrects for this bias. Thus, it is interesting to compare z-curve's estimates of the discovery rate with Ioannidis's assumptions about the discovery rates in clinical trials. The z-curve estimate of the EDR was 30% with a 95% confidence interval from 20% to 41%. This finding suggests that most published results from clinical trials in our sample were well-powered and likely to test a true hypothesis. This conclusion is also consistent with the finding that our estimate of the false discovery risk is similar to Ioannidis's predictions for high quality clinical trials, FDR = 15%. Thus, a simple explanation for the discrepancy between Ioannidis's claim and our results is that Ioannidis's underestimated the proportion of well-powered studies in the literature. The expected discovery rate also provides valuable information about the extent of selection bias in medical journals. While the expected discovery rate is only 30%, the observed discovery rate (i.e., the percentage of significant results in abstracts) is more than double (69.7%). This discrepancy is visible in Figure 4. The histogram of observed nonsignificant z-scores does not match the predicted distribution (blue curve). This evidence of considerable selection bias implies that reported effect sizes are inflated by selection bias. Thus, follow-up studies need to adjust effect sizes when planning the sample sizes via power analyses. Moreover, meta-analyses need to correct for selection bias to obtain unbiased estimates of the average effect size [27]. Z-curve also provides information about the expected replication rate of statistically significant results in medical abstracts, defined as the probability of obtaining a statistically significant result in a close replication study with the same sample size and significance criterion (see e.g., [28,29,30] for alternatives). The expected replication rate is not a simple complement of the false discovery risk, as a true discovery might fail to replicate due to low power of the replication study and the false discovery might "falsely" replicate with a probability corresponding to α. The expected replication rate would be 65% with a confidence interval ranging from 61% to 69%. It is important to realize that this expected replication rate includes replications of true and false hypotheses. Although the actual false positive rate is not known, we can use the z-curve estimate to assume that 14% of the significant results are false positives. Under this assumption the power of replication studies of true hypotheses is 75%. In contrast, the probability of obtaining a false positive result again is only 0.7%. Thus, actual replication studies even with the same sample sizes successfully reduce the false positive risk. However, the risk of false negatives is substantial and 44% of non-significant results in replication studies test a true hypothesis. Furthermore, z-curve estimates of the replication rate are higher than actual replication rates [26]. One reason could be that exact replication studies are impossible and changes in population will result in lower power due to selection bias and regression to the mean. In the worst case, the actual replication rate might be as low as the expected discovery rate. Thus, our results predict that the success rate of actual replication studies in medicine will be somewhere between 30% and 65%. To ensure a high probability of replicating a true result, it is therefore necessary to increase sample sizes of replication studies. When this is not possible, it is important to interpret an unsuccessful replication outcomes with caution and to conduct further research. A single failed replication study is inconclusive. Ioannidis [23] suggested that an FDR of 14% is a desirable goal. However, many researchers falsely assume that the α = .05 ensures a false discovery risk of only 5%. Moreover, the longevity of α = .05 suggests that many researchers consider an error rate of 5% acceptable. We showed that α = .05 entails an FDR of 14%. To lower the FDR to 5% or less, it is necessary to lower α. Z-curve analysis can help to find an α level that meets this criterion. With α = .01, the expected discovery rate decreases to 20% and the false discovery risk decreases to 4%. Based on these results, it is possible to use α = .01 as a criterion to reject the null-hypothesis while maintaining a false positive risk below 5%. While journals may be reluctant to impose a lower level of α, readers can use this criterion to maintain a reasonably low risk of accepting a false hypothesis. Discussion Like many other human activities, science relies on trust. Ioannidis's (2005) article suggested that trust in science is unwarranted and that readers are more likely to encounter false than true claims in scientific journals. In response to these concerns, a new field of meta-science emerged to examine the credibility of published results. Our article builds on these efforts by introducing an improved method to estimate the false discovery risk in medical journals, with a focus on clinical trials. Contrary to Ioannidis's suggestion that most of these results may be false positives, we found that the false discovery risk is 14% with the traditional criterion for statistical significance (α = .05) and 4% with a lower threshold of α = .01. Thus, our results provide some reassurance that medical research is more robust than Ioannidis predicted. At the same time, our results also show some problems that could be easily addressed by journal editors. The biggest problem remains publication bias in favor of statistically significant results. The selection for significance has many undesirable consequences. Although medicine has responded to this problem by demanding preregistration of clinical trials, our results suggest that selection for significance remains a pervasive problem in medical research. A novel contribution of z-curve is the ability to quantify the amount of selection bias. Journal editors can use this tool to track selection bias and implement policies to reduce it. This can be achieved in several ways. First editors may publish more well-designed replication studies with non-significant results. Second, editors may check more carefully whether researchers followed their preregistration. Finally, editors can prioritize studies with high sample sizes. Despite our improvements over Jager and Leel's study, our study has a number of limitations that can be addressed in future research. One concern is that results in abstracts may not be representative of other results in the results section. Another concern is that our selection of journals is not representative. This are valid concerns that limit our conclusions to the particular journals that we examined. It would be wrong to generalize to other scientific disciplines or other medical journals. We showed that z-curve is a useful tool to estimate the false positive risk and to adjust α to limit the false positive risk to a desirable level. This tool can be used to estimate the false discovery risk for other disciplines and research areas and set α levels that are consistent with the discovery rates of these fields. We believe that empirical estimates of discovery rates, false positive risks, selection bias, and replication rates can inform scientific policies to ensure that published results provide a solid foundation for scientific progress. Data and code availability Supplementary Materials including data and R scripts for reproducing the simulations, data scraping, and analyses are available from https://osf.io/y3gae/. The zcurve R package is available from https://cran. r-project.org/package=zcurve. Figure 1 : 1Comparison of z-curve and Jager and 1 : 1Root mean square error and bias of z-curve and Jager and Leek's false discovery risk estimates in a square error (RMSE) and bias + standard errors (in brackets) for the estimated false discovery rate (FDR) by z-curve and Jager and Fig. 2 2visualizes z-curve and Jager and Leek[19] method's false discovery risk estimates based on scraped abstracts from clinical trials and randomized controlled trials and further divided by journal and whether the article was published before (and including) 2010 (left) or after 2010 (right). We see that, in line with the simulation results, Jager and Figure 2 : 2Estimates of false discovery risk across decades and journals. Estimated false discovery rate (FDR, yaxis) with z-curve (black-filled squares) and Jager and Leek's[19] swfdr (empty circles) based on clinical trials and randomized control trials divided by journal (x-axis, with information about unique articles in brackets) and whether the article was published before (and including) 2010 (left) or after 2010 (right). Figure 4 : 4Z-curve of the combined data set. Distribution of all significant p-values converted into z-statistics with fitted z-curve density (solid blue line) and 95% point-wise confidence bands (doted blue line). Table Table 2 : 2Summary of the scraped dataJournal Abstracts CT or RCT Scrapeable p-values Lancet 7638 2099 1556 5403 BMJ 4480 939 643 1625 NEJM 4912 2767 2304 7137 JAMA 5491 1452 1274 4484 PLoS 2944 346 279 1102 Total 25465 7603 6056 19751 Table 3 : 3Combined and journal specific false discovery risk estimates.Journal z-curve swfdr Lancet 0.07 [0.05, 0.11] 0.21 [0.18, 0.24] BMJ 0.23 [0.08, 0.54] 0.28 [0.23, 0.32] NEJM 0.18 [0.10, 0.30] 0.14 [0.11, 0.18] JAMA 0.29 [0.15, 0.55] 0.26 [0.21, 0.31] PLoS 0.12 [0.05, 0.54] 0.21 [0.13, 0.32] Combined 0.13 [0.08, 0.21] 0.19 [0.17, 0.20] The fold is introduced by our lack of knowledge of the expected direction of the effect. Reproducibility crisis. M Baker, Nature. 53326Baker M. Reproducibility crisis. Nature 2016; 533(26): 353-66. Opinion: Is science really facing a reproducibility crisis, and do we need it to. D Fanelli, Proceedings of the National Academy of Sciences. 11511Fanelli D. 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[ "Phantom study for 90 Y post-treatment dosimetry with a long axial field-of-view PET/CT", "Phantom study for 90 Y post-treatment dosimetry with a long axial field-of-view PET/CT", "Phantom study for 90 Y post-treatment dosimetry with a long axial field-of-view PET/CT", "Phantom study for 90 Y post-treatment dosimetry with a long axial field-of-view PET/CT" ]	[ "Lorenzo Mercolli \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "Konstantinos Zeimpekis \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "George A Prenosil \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "Hasan Sari \nAdvanced Clinical Imaging Technology\nSiemens Healthcare AG Lausanne\nSwitzerland\n", "Hendrik G Rathke \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "Axel Rominger \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "Kunagyu Shi \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "Lorenzo Mercolli \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "Konstantinos Zeimpekis \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "George A Prenosil \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "Hasan Sari \nAdvanced Clinical Imaging Technology\nSiemens Healthcare AG Lausanne\nSwitzerland\n", "Hendrik G Rathke \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "Axel Rominger \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n", "Kunagyu Shi \nDepartment of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland\n" ]	[ "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Advanced Clinical Imaging Technology\nSiemens Healthcare AG Lausanne\nSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Advanced Clinical Imaging Technology\nSiemens Healthcare AG Lausanne\nSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland", "Department of Nuclear Medicine\nInselspital Bern University Hospital\nUniversity of Bern\nFreiburgstrasse 18CH-3010BernSwitzerland" ]	[]	Purpose: The physical properties of yttrium-90 ( 90 Y) allow for imaging with positron emission tomography/computed tomography (PET/CT). The increased sensitivity of long axial field-of-view (LAFOV) PET/CT scanners possibly allows to overcome the small branching ratio for positron production from 90 Y decays and to improve for the post-treatment dosimetry of 90 Y of selective internal radiation therapy.Methods: For the challenging case of an image quality body phantom, we compare a full Monte Carlo (MC) dose calculation with the results from the two commercial software packages Simplicit90Y and Hermes. The voxel dosimetry module of Hermes relies on the 90 Y images taken with a LAFOV PET/CT, while the MC and Simplicit90Y dose calculations are image independent.Results: The resulting doses from the MC calculation and Simplicit90Y agree well within the error margins. The image-based dose calculation with Hermes, however, consistently underestimates the dose. This is due to the mismatch of the activity distribution in the PET images and the size of the volume of interest. Furthermore, there are likely limitations of Hermes' dose calculation algorithm for 90 Y. We found that only for the smallest phantom sphere there is a statistically significant dependence of the Hermes dose on the image reconstruction parameters and scan time.Conclusion: Our study shows that Simplicit90Y's local deposition model can provide a reliable dose estimate. On the other hand, the image based dose calculation requires further benchmarks and verification in order to take full advantage of LAFOV PET/CT systems.	null	[ "https://export.arxiv.org/pdf/2304.14797v1.pdf" ]	258,417,806	2304.14797	1d92fa3e2c26f508e6d9cd5663bb5677f76f95fd	Phantom study for 90 Y post-treatment dosimetry with a long axial field-of-view PET/CT Lorenzo Mercolli Department of Nuclear Medicine Inselspital Bern University Hospital University of Bern Freiburgstrasse 18CH-3010BernSwitzerland Konstantinos Zeimpekis Department of Nuclear Medicine Inselspital Bern University Hospital University of Bern Freiburgstrasse 18CH-3010BernSwitzerland George A Prenosil Department of Nuclear Medicine Inselspital Bern University Hospital University of Bern Freiburgstrasse 18CH-3010BernSwitzerland Hasan Sari Advanced Clinical Imaging Technology Siemens Healthcare AG Lausanne Switzerland Hendrik G Rathke Department of Nuclear Medicine Inselspital Bern University Hospital University of Bern Freiburgstrasse 18CH-3010BernSwitzerland Axel Rominger Department of Nuclear Medicine Inselspital Bern University Hospital University of Bern Freiburgstrasse 18CH-3010BernSwitzerland Kunagyu Shi Department of Nuclear Medicine Inselspital Bern University Hospital University of Bern Freiburgstrasse 18CH-3010BernSwitzerland Phantom study for 90 Y post-treatment dosimetry with a long axial field-of-view PET/CT DosimetryYttrium-90Long axial field-of-view PET/CTMonte Carlo simulations Purpose: The physical properties of yttrium-90 ( 90 Y) allow for imaging with positron emission tomography/computed tomography (PET/CT). The increased sensitivity of long axial field-of-view (LAFOV) PET/CT scanners possibly allows to overcome the small branching ratio for positron production from 90 Y decays and to improve for the post-treatment dosimetry of 90 Y of selective internal radiation therapy.Methods: For the challenging case of an image quality body phantom, we compare a full Monte Carlo (MC) dose calculation with the results from the two commercial software packages Simplicit90Y and Hermes. The voxel dosimetry module of Hermes relies on the 90 Y images taken with a LAFOV PET/CT, while the MC and Simplicit90Y dose calculations are image independent.Results: The resulting doses from the MC calculation and Simplicit90Y agree well within the error margins. The image-based dose calculation with Hermes, however, consistently underestimates the dose. This is due to the mismatch of the activity distribution in the PET images and the size of the volume of interest. Furthermore, there are likely limitations of Hermes' dose calculation algorithm for 90 Y. We found that only for the smallest phantom sphere there is a statistically significant dependence of the Hermes dose on the image reconstruction parameters and scan time.Conclusion: Our study shows that Simplicit90Y's local deposition model can provide a reliable dose estimate. On the other hand, the image based dose calculation requires further benchmarks and verification in order to take full advantage of LAFOV PET/CT systems. Background Nowadays, yttrium-90 ( 90 Y) selective internal radiation therapy (SIRT) is a well-established and effective treatment modality for hepatocellular carcinoma (HCC) and liver metastases of neuroendocrine tumours as well as colorectal cancer [1,2]. SIRT exploits the fact that the blood supply of hepatic malignancies is different compared to normal liver parenchyma. It is therefore possible to target the tumor cells locally through the injection of 90 Y microspheres into the arteries of the liver. Individual dosimetry and treatment planning play a pivotal role in applying SIRT safely and for achieving the best possible treatment outcome. An individual treatment planing and verification is recommended by the EANM position paper [3]. For the pre-treatment dosimetry of SIRT, different methods are part of the clinical standard procedure [4][5][6][7]. However, despite being recommended [4,5], there is not yet a standardized protocol for post-treatment verification in the clinical routine [6,8]. The known challenges of imaging 90 Y Bremsstrahlung [4,6,9] with single photon emission computed tomography (SPECT) made the use of positron emission tomography (PET) systems an increasingly popular alternative to SPECT [10] and is now the recommended procedure for resin microspheres [1]. Post-treatment verification with PET exploits the fact that 90 Y can decay to the excited O + state of 90 Zr [11]. This state can then further decay through an internal pair production E0 transition with a branching ratio (BR) of (3.26 ± 0.04) · 10 −6 e + e − pairs per decay [12]. Despite this low BR, Ref. [10] showed more than a decade ago that a post-treatment verification with PET is feasible. Recently, long axial field-of-view (LAFOV) PET/CT systems have found their way into clinical routine [13][14][15][16][17][18]. The sensitivity and noise equivalent count rate of LAFOV PET/CT systems significantly improves over standard fieldof-view systems. For the imaging of 90 Y microspheres, LAFOV PET/CT outperform conventional PET/CT systems since it can compensate the low branching fraction of the excited 90 Zr state [19]. The aim of this study is to compare a dose calculation method that is based on LAFOV PET images with image independent methods. To this end, we performed phantom scans and computed the dose using two commercial software products: Hermes (Hermes Medical solutions, Stockholm, Sweden) and Simplicit90Y (Mirada Medical Ltd, Oxford, UK; Boston Scientific Corporation, Marlborough, MA, USA). Hermes' voxel dosimetry module uses a so-called semi-Monte Carlo (sMC) algorithm for the dose calculation and therefore requires quantitative PET images as input. Simplicit90Y computes the deposited dose assuming a local deposition model, which for the case of a phantom is independent of the PET image. Finally, we performed a Monte Carlo (MC) simulation with the software FLUKA [20][21][22], which provides the full modelling of the 90 Y decay, particle transport and dose deposition. It is completely independent of the imaging and segmentation process and therefore serves as the ground truth for the dose calculation. Materials and methods Measurements and imaging protocols For the image acquisition of 90 Y with a LAFOV PET/CT, two NEMA International Electrotechnical Commission (IEC) body phantoms (Data Spectrum Corp.) [23,24] were filled with demineralized water, a minor addition of hydrochloric acid and 90 Y citrate. The total activity at reference time was 1.31 ± 0.20 GBq, while at scan time it had decayed to 1.12±0.17 GBq. All activity measurements were carried out with an ISOMED 2010 well-type dose calibrator that was calibrated by the Swiss Federal Institute of Metrology METAS. In order to take into account the systematic error of the activity measurements (calibration factor, variations in the measurement geometry, filling levels, etc.), we assume a conservative error of 15%. One phantom had a hot and the second one a cold background. In Tab. 1 we report the activities inside the volumes of the six spheres and the background volumes of the phantoms. The sphere to background activity concentration ratio is approximately 1 : 10 with an activity concentration of ≈ 1.3 MBq/ml in the spheres. The activities in the phantoms' spheres were lower compared to typical lesions that are treated with SIRT. The setup should therefore be considered as a rather challenging case for imaging and dose calculations. The error on the sphere volumes are derived from the errors on the diameters quoted in the NEMA phantom specifications. These are the possible variations in the production process of the phantoms. The background volumes are determined through their weight when filled with demineralized water and we assume a 1% error on the weight measurement and a water density of 0.9982 g/ml according to the ICRU Report 90 [25]. The two phantoms were imaged with a Biograph Vision Quadra PET/CT scanner (Siemens Healthineers, Knoxville, TN, USA) at the Department of Nuclear Medicine, Inselspital, Bern University Hospital. The detectors of the Biograph Vision Quadra are made of 5 × 5 arrays of 3.2 × 3.2 × 20 mm lutetium oxyorthosilicate (LSO) crystals that are coupled to a 16 × 16 mm array of silicon photomultipliers. 4 × 2 such mini-blocks are arranged into one detector block. The whole PET detector consists of 32 detector rings with 38 blocks each. This yields an axial FOV of 106 cm. All images were acquired with a maximum ring distance of 322 crystals (MRD 322) and reconstructed the images in ultra-high sensitivity mode (UHS). For comparison, images were also reconstructed in high sensitivity mode (HS) with a MRD of 85 crystals. The NEMA sensitivity for 18 F in UHS is 176 kcps/ MBq, while in HS it is 83 kcps/ MBq [16]. The time-of-flight (TOF) resolution of the Biograph Vision Quadra is 228 and 230 for HS and UHS, respectively. While the Biograph Vision Quadra's sensitivity exceeds the one of standard FOV scanners, the TOF and spatial resolution of 3.3 × 3.4 × 3.8 mm are comparable to standard systems. The images of the two phantoms were acquired during 50 min. Reconstructions were performed using a dedicated image reconstruction prototype (e7-tools, Siemens Healthineers). The error on the resulting dose does not include any error form the image reconstruction process. For the image reconstruction Siemens' TOF, point-spread function (PSF) recovery and ordered subset expectation maximization (OSEM) is used. According to Ref. [19], the optimal reconstruction parameters in terms of image quality for both HS and UHS are two iterations, five subsets, a 2 mm Gaussian filter and a 220 × 220 matrix. In order to assess the dependence dose calculation's dependence on the scan time, images were reconstructed with the first 5, 10 and 20 minutes of scan duration in UHS by rebinning the list-mode data. For comparison, we also reconstructed an image in HS mode with 20 minutes scan duration. Fig. 1 shows examples with UHS reconstruction and with 5 min and 20 min acquisition time. The decay correction and quantification for 90 Y is done directly by the vendor's image acquisition and reconstruction software. Of course, an uncertainty in the quantification of the activity concentration in the PET image could propagate to the image based dose calculation. Comparing the activity concentrations from Tab. 1 with the concentration in the PET images, we found a deviation of less than 7% for the two large spheres and less than 15% for the hot background, respectively. This uncertainty, however, lies within the 15% uncertainty of the activity measurement with the dose calibrator. Conservatively, we therefore assume a conservative 10% error on the PET image quantification. Dose calculations The doses deposited in the phantom's spheres were calculated using three independent methods: a full MC simulation of the NEMA phantom, the image based sMC method from Hermes' voxel dosimetry module and Simplicit90Y's local deposition model. The NEMA phantom was implemented in CERN's FLUKA 4-2.2 [20,21] and Flair 3.1-15.1 [22], which is a general MC framework for particle transport. Fig. 2 shows the rendering of the phantom geometry and an example of the dose distribution. We used the standard PRECISIO setting of FLUKA for the physics and transport parameters and ran the simulations in semi-analogue mode for every single sphere and the hot background separately. The simulation of the hot background required a dedicated source routine due to the nonstandard shape of the background volume. The decay properties of 90 Y are taken in FLUKA's isotope library. The resulting doses were scored with USRBIN in spherical regions with the nominal size NEMA phantom spheres. Of course, the simulations's results need to be normalized to the measured total activities in Tab. 1. The simulation of 10 7 primaries were sufficient for reaching a negligible statistical error in the regional dose scoring and we do not assume any systematic error in the simulation itself (geometry implementation, 90 Y decay data, transport thresholds, etc.). Therefore, the uncertainties on the doses from the MC simulations stem only from the uncertainties related to the activity measurements shown in Tab. 1. Hermes' voxel dosimetry module provides the possibility to calculate a dose map from a quantitative PET or SPECT image through the so called sMC method [26,27]. The authors of Ref. [27] proposed the sMC for the post-treatment dosimetry of lutetium-177 ( 177 Lu) peptide receptor radionuclide therapy in order to overcome the accurate but time consuming full MC calculations (see also Refs. [26,28]). The basic idea behind sMC is to separate the electron and photon transport: the electron's energy is absorbed locally, i.e. in the same voxel where the decay has been detected, while for the photons a point-wise transport is used. The algorithm first converts the activity distribution from and image to an electron dose map. Second, the photons are transported in a full MC down to a threshold of 15 keV [26]. The energy of photons with an energy below the transport threshold as well as the energy of recoil electrons is deposited locally. Note that for the photon transport sMC uses only cross sections of water, which are then rescaled to the various tissue densities [27]. With a mean electron energy ofĒ β = keV (79.44% intensity) and Q β = 496.8 keV in conjunction with the low spatial resolution of SPECT images, 177 Lu is an ideal radionuclide for sMC. However, the literature on sMC for other radionuclides, such as 90 Y or 131 I, is scarce and as pointed out the authors of Ref. [26], further investigation for the applicability of sMC beyond 177 Lu is necessary. The sMC implementation of Hermes allows for a truly image based dose calculation. Therefore, different scan times and reconstruction settings can lead to different doses. For every input image (in units of Bq/ml) Hermes computes a dose map, i.e. an image with units of Gy. For all dose maps, we used Hermes Hybrid Viewer to segment the phantom spheres. The volumes-of-interest (VOI) were drawn on the CT according to to the nominal sphere diameters of the NEMA phantom. Since the dose scales with the inverse volume, segmenting the phantom spheres in the dose maps by a threshold (e.g. relative to some maximal voxel value) would make a comparison with the other dose calculation methods meaningless. Hermes does not provide information on the statistical error of the voxel values of the dose map from the sMC method. As mentioned before, we do not consider any error from the image reconstruction but add a 10% uncertainty on the quantification. Hermes' Hybrid Viewer reports the volume of each (voxelized) VOI and the deviation from the nominal spheres can reach 10%. In our analysis we do not quantify any systematic error of the sMC algorithm. This would require synthetic PET images that disentangle the imaging process from the sMC dose calculation. Lastly, the dose inside the phantom spheres was also computed with Simplicit90Y's multicompartment dosimetry module (see e.g. Refs. [4,7,29] for multi-compartment dosimetry). Sim-plicit90Y relies on a local deposition model with a homogeneous activity distribution [30]. The dose in a compartment or perfused volume is computed according tō D c = 50 · A c · (1 − F ) · (1 − R) m ,(1) where A c is the activity in GBq inside the compartment and m is its mass in kg. In the phantom measurements the lung shunt fraction F and the residual waste fraction R were set to zero. According to the Simplicit90Y manual the dose factor is fixed at 50 Gy kg/GBq. Note that slightly lower values can be found in the literature (see e.g. Refs. [4,7]). In Simplicit90Y, all dose calculations assume a liver density of 1.06 g/cm 3 . Since in the phantoms are filled with water, we correct the doses from Simplicit90Y by the ratio of the liver and water density. Simplicit90Y only takes image information, i.e. relative counts, when multiple overlapping volumes are defined. This is not the case for the NEMA phantoms and therefore the dose calculation with Simplicit90Y is independent of the PET image. As in the case of the sMC dose calculation, the VOI were drawn as spheres with the size according to the nominal sphere diameters. There is some deviation from the exact spherical volume due to the voxelization of the volume (in the CT image) and possibly some rounding errors of Sim-plicit90Y. The uncertainties of the dose values from Simplicit90Y are therefore composed of a 10% error from the volume voxelization and the error on the activities inside the spheres in Tab. 1. We do not consider any systematic uncertainty on the local deposition model nor on the constant dose factor of 50 Gy kg/GBq. Results Tabs. 2 and 3 report the doses deposited in the six spheres of the cold and hot background phantoms. Fig. 3 visualizes the values in Tabs. 2 and 3. As previously mentioned, the FLUKA dose calculation is considered as the ground truth for the doses. Fig. 3 shows an increase of dose from the MC simulation with increasing sphere diameter. In the case of the hot phantom this effect is slightly less pronounced. However, the error bars are large, due to the uncertainty of the activity measurement, and therefore this effect is statistically insignificant. The Simplict90Y doses vary depending on the sphere diameters and are exactly the same for both phantoms. Compared to FLUKA and Simplicit90Y, the Hermes based dose calculations shows a systematic underestimation of the doses. This underestimation is particularly pronounced in the case of the two smallest spheres and for short acquisition times. In most cases the dose from the UHS images decreases with decreasing acquisition time. Interestingly, the HS based dose is comparable or even closer to the FLUKA dose than the dose from the and 50 min UHS image. While most of the Hermes dose values lie still within the 1σ range of the FLUKA doses, the systematic uncertainties of the Hermes calculation require some discussion. Discussion The increase of dose values from the MC simulation with increasing sphere diameter in Tabs. 2 and 3 is due to the width of the 90 Y dose point kernel (DPK), i.e. the smaller the sphere the higher the relative fraction of the dose is deposited outside the sphere. When comparing FLUKA's doses of the cold and hot background phantoms, we can see a similar effect in the sense that with increasing sphere diameter the relative amount of the spillin dose from the hot background becomes smaller. Note however, that the uncertainty due to the activity measurement is rather large, making these effects statistically insignificant. Since the 90 Y activity increases with the sphere volume but the dose decreases with the inverse volume, the doses from Simplicit90Y should in principle have the same value for all six spheres. However, due to the voxelization of the volumes in the CT images and the rounding errors of the volume in Simplicit90Y there is a variation in the doses for different sphere diameters in Fig. 3. Nevertheless, the doses computed with Simplicit90Y are comparable the the FLUKA doses, in particular when taking into account the associated errors. Obviously, there is no difference between the cold and hot background doses since no information is taken from the PET images in Simplicit90Y in the absence of multiple compartments. In sum, the doses computed with Simplicit90Y confirm the fixed dose factor of 50 Gy kg/GBq for larger volumes, hint towards issues with rounding errors for small volumes and show the inability to capture physical effects (width of 90 Y DPK) of the the dose distribution in small volumes. The doses values from Hermes in Fig. 3 quantified PET images. The quantification of the PET image may contribute to the systematic underestimation of the dose with Hermes. After all, the recovery coefficients found in Ref. [19] do not reach 100%. Furthermore, when looking at the PET images in Fig. 1, it is clear that in particular for the smaller spheres, the activity distribution in the PET images does not completely reproduce a spherical shape. The nominal sphere volumes are larger than the regions where high 90 Y activities were detected, which in our opinion is the main cause for the observed dose estimation. For the smaller spheres we probably also run into the problems raised in Ref. [31], namely that the 90 Y activity is low and one becomes dominated by the LSO background. Possibly, the reconstruction parameters should be adapted in order to optimize the resulting dose map rather than the image quality. However, holding the solely PET image properties accountable for the small Hermes doses falls somewhat short. Probably, the results in Fig. 3 are also affected by shortcomings of the sMC algorithm itself. Already the original authors pointed out that the assumptions underlying the sMC algorithm might not be applicable to 90 Y [27]. Presumably, not only the high kinetic energy of the β − but also the increased spatial resolution of the PET images strain the range of validity of the sMC method. Comparing the different scan times in UHS mode, Fig. 3 seems to show that the Hermes doses are become smaller for shorter scan times. This is particularly pronounced for the case of the smallest sphere and the cold phantom. It is tempting to conclude that longer scan times lead to dose values closer to the ground trough. However, our analysis shows that this trend is not statistically significant due to the relatively large error bars. Even if the image quality improves in general, as shown in Ref. [19], the volume of the dose distribution remains small compared to the nominal sphere volume. Also Ref. [31] argues that longer scan times to not improve the results in such challenging imaging situations due to the LSO background radiation. We should also keep in mind that, as in the case of Simplicit90Y, the voxelization of the sphere volumes can lead to some variation in the dose values. Interestingly, the 20 minutes HS image seems to lead to doses that are comparable to the 50 minutes UHS image. Conclusions In this paper we compared different dose calculation methods for 90 Y radioembolization. While the full MC and Simplicit90Y dose calculations are independent of the input images, Hermes relies on the sMC algorithm to compute dose maps based on the activity distributions of images that were aquired with a LAFOV PET/CT. The MC and Simplicit90Y dose calculations agree well within the error margins. Simplicit90Y's local deposition model is not able to catch the effect of the width of the 90 Y DPK, which is most visible in small volumes. Furthermore, the voxelization of the exact sphere volume in Simplicit90Y leads to a nonphysical variation of the dose values for different sphere diameters. The Hermes' sMC method shows an underesti-mation of the dose for all input images compared to the full MC calculation. We identified two main reasons: on one hand there is a significant mismatch of the 90 Y activity distribution and the nominal sphere volumes in the PET images. On the other hand, our analysis confirms the original doubts of Ref. [26] about the applicability of the sMC algorithm to isotopes with high β − energies and high resolution images. Our analysis implies several questions that should be addressed in the future. On one side, a benchmark Simplicit90Y's multi-compartment dose calculation based on LAFOV PET images would be very desirable. A thorough comparison of the accuracy of the post-treatment dosimetry with the pre-treatment dosimetry verification dose prediction could indicate a true superiority of LAFOV compared to SAFOV for SIRT. The same applies to the Hermes dose calculation with 90 Y: further verification of the method with clinical data is necessary. Ethics approval and consent to participate Not applicable. Competing interests HS is a full-time employee of Siemens Healthcare AG, Switzerland. AR has received research support and speaker honoraria from Siemens. All other authors have no conflicts of interest to report. Figure 1 : 1Example images of UHS reconstruction and with 5 min (top) and 20 min (bottom) acquisition time for both cold and hot backgrounds. Figure 2 : 2FLUKA implementation of the NEMA IQ phantom (left) and an illustrative example of a 3D dose distribution from FLUKA with cartesian grid scoring (right). Figure 3 : 3Visualization of the doses in Tabs. 2 and 3 for the cold (top) and cold (bottom) background phantoms. Volume name Diameter [mm] Volume [ml] Activity [MBq]cold bkg N/A 9762.0 ± 98.5 0.0 hot bkg N/A 9829.0 ± 97.8 1193.51 ± 0.18 s 1 10.0 ± 0.5 0.528 ± 0.079 0.64 ± 0.15 s 2 13.0 ± 0.5 1.16 ± 0.13 1.40 ± 0.27 s 3 17.0 ± 0.5 2.58 ± 0.23 3.12 ± 0.55 s 4 22.0 ± 1.0 5.61 ± 0.76 6.78 ± 1.40 s 5 28.0 ± 1.0 11.5 ± 1.2 13.9 ± 2.60 s 6 37.0 ± 1.0 26.6 ± 2.2 32.1 ± 5.60 Table 1: 90 Y activities in at reference time and the dimensions of the background and sphere volumes of the NEMA IQ phantoms. The quoted errors are described in the text. Table 2 : 2Doses in Gy in the six spheres of the NEMA phantom with a cold background according to the different dose calculation methods.Method D s1 [Gy] D s2 [Gy] D s3 [Gy] D s4 [Gy] D s5 [Gy] D s6 [Gy] FLUKA 42.0 ± 9.0 46.8 ± 8.9 50.7 ± 8.8 53.7 ± 11.0 55.9 ± 10.0 58.0 ± 10.0 Simplicit90Y 46.8 ± 11.0 49.7 ± 11.0 45.6 ± 9.3 50.6 ± 12.0 52.8 ± 11.0 50.2 ± 10.0 UHS 50 min 24.5 ± 3.5 31.2 ± 4.4 36.1 ± 5.1 40.8 ± 5.8 45.6 ± 6.4 46.8 ± 6.6 UHS 20 min 24.8 ± 3.5 29.5 ± 4.2 32.5 ± 4.6 39.0 ± 5.5 41.8 ± 5.9 45.2 ± 6.4 UHS 10 min 30.4 ± 4.3 30.0 ± 4.2 36.1 ± 5.1 36.4 ± 5.1 40.5 ± 5.7 44.3 ± 6.3 UHS 5 min 31.7 ± 4.5 26.4 ± 3.7 31.4 ± 4.4 37.2 ± 5.3 37.6 ± 5.3 41.6 ± 5.9 HS 20 min 35.3 ± 5.0 35.4 ± 5.0 41.4 ± 5.9 41.9 ± 5.9 44.6 ± 6.3 48.2 ± 6.8 Table 3 : 3Doses in Gy in the six spheres of the NEMA phantom with a hot background according to the different dose calculation methods. de Wit van der veen, P. 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[ "Complexity Analysis in Bouncing Ball Dynamical System", "Complexity Analysis in Bouncing Ball Dynamical System" ]	[ "L M Saha \nIIIMIT\nShiv Nadar University\nU.P.201314Gautam budh Nagar\n", "Til Prasad Sarma [email protected] \nDepartment of Education in Science and Mathematics\nNCERT\nSri Aurobindo Marg, New Delhi -110 016\n", "Purnima Dixit [email protected] \nDepartment of Mathematics\nDayal Singh College\nNew Delhi-1100\n" ]	[ "IIIMIT\nShiv Nadar University\nU.P.201314Gautam budh Nagar", "Department of Education in Science and Mathematics\nNCERT\nSri Aurobindo Marg, New Delhi -110 016", "Department of Mathematics\nDayal Singh College\nNew Delhi-1100" ]	[]	Evolutionary motions in a bouncing ball system consisting of a ball having a free fall in the Earth's gravitational field have been studied systematically. Because of nonlinear form of the equations of motion, evolutions show chaos for certain set of parameters for certain initial conditions. Bifurcation diagram has been drawn to study regular and chaotic behavior. Numerical calculations have been performed to calculate Lyapunov exponents, topological entropies and correlation dimension as measures of complexity. Numerical results are shown through interesting graphics.	null	[ "https://export.arxiv.org/pdf/1601.01444v1.pdf" ]	119,651,905	1601.01444	a6b50ac6d579446b38d6e5ce43ee6c6c335e428d	Complexity Analysis in Bouncing Ball Dynamical System L M Saha IIIMIT Shiv Nadar University U.P.201314Gautam budh Nagar Til Prasad Sarma [email protected] Department of Education in Science and Mathematics NCERT Sri Aurobindo Marg, New Delhi -110 016 Purnima Dixit [email protected] Department of Mathematics Dayal Singh College New Delhi-1100 Complexity Analysis in Bouncing Ball Dynamical System 1ChaosLyapunov ExponentsBifurcationTopological Entropy AMS Subject Classification: 92D40 Evolutionary motions in a bouncing ball system consisting of a ball having a free fall in the Earth's gravitational field have been studied systematically. Because of nonlinear form of the equations of motion, evolutions show chaos for certain set of parameters for certain initial conditions. Bifurcation diagram has been drawn to study regular and chaotic behavior. Numerical calculations have been performed to calculate Lyapunov exponents, topological entropies and correlation dimension as measures of complexity. Numerical results are shown through interesting graphics. Introduction A simple system evolves in simple ways but a complex or complicated system evolve in complicated ways and between simplicity and complexity there cannot be a common ground [1]. Chaos and irregular phenomena may not require very complicated equations. Complexity in a dynamical system can be viewed as its systematic nonlinear properties. It is the order that results from the interaction among multiple agents within the system. A system is complex means its evolutionary behavior do not show regularity but chaotic or some other kind of irregularity. Complexity and chaos observed in a system can well be understood by measuring elements like Lyapunov exponents (LCEs), topological entropies, correlation dimension etc. Topological entropy, a non-negative number, provides a perfect way to measure complexity of a dynamical system. For a system, more topological entropy means the system is more complex. Actually, it measures the exponential growth rate of the number of distinguishable orbits as time advances [2,3]. Though, positivity measure of Lyapunov exponents (LCEs) signifies presence of chaos, LCEs, topological entropies and correlation dimensions all these three together provide measure of complexities in the system. Motion of a bouncing ball system represented with equations in coupled form of variables, have been appeared in various literatures [4][5][6][7][8] and regular and chaotic motions observed during evolution have been discussed. The models discussed vary with different kind of assumptions and so the variation of nonlinearities. The present article consisting of a model of bouncing ball system occurring due to a free fall of a ball in the Earth's gravitational field and impacting kinematically certain forced plate [9]. Bifurcation diagrams have been drawn to study some characteristic evolutionary phenomena, (e.g., chaos adding), with increasing numerical value of the driving frequency. Prior to this chaos adding, one observes period doubling bifurcation followed by chaos. Periodic windows appearing within chaos are also subject to study. Numerical investigations carried forward to obtain Lyapunov exponents (LCEs), topological entropies and correlation dimensions for different sets of parameters of the system. Results obtained are shown through graphics. Bouncing Ball Model: Neglecting the air drag, free fall motion of the bouncing ball, with a restitution coefficient k < 1 be written as, [9],  n + 1 =  n + q v n , v n + 1 =k v n + (1 + k) cos ( n + 1 ) (1.1) The system contains an another parameter, q, the driving frequency and q  1. Bifurcation diagrams for above system are drawn with k = 0.3 and different ranges of values of q shown, respectively, by Fig.1, (Figures (a) -(f)). These shows, initially, the system evolving with period doubling bifurcation followed by chaos and then, a chaos adding phenomena with increasing values of q. (a)1 q  5, (b) 1.5  q  24, (c) 1.5  q  3.4, (d) 1.5  q  3.5, (e) 3.95  q  4.2, (d) 9.1  q  9.7. As it appears through bifurcations, before evolving into chaos, system shows regularity for certain rage of parameter value of q while keeping k fixed, k = 0.3. In Fig.2, the figures in upper row are two time series and a two periodic regular attractor for q = 2.5. The lower row figures correspond to those of the upper row for chaotic case when q = 3.8. Magnitude of the positive values of LCEs provide the answer that how chaotic the system be. These differences explain the complexity within the system. With k = 0.3 and q, approximately, q = 3.31, the system shows regular behavior as it is evident from the bifurcation diagrams as well as from the LCEs plot shown in Fig. 4. With k = 0.3 and q approximately, q = 3.31, what we observed in Fig. 4 in LCEs plot, the results obtained here are very different. For 1  q  2.6, in the former case the system is non-chaotic and shows regularity but in this later case one obtains significant value of topological entropy. As topological entropy measures the complexity, though the system is regular, it is complex. Thus, a non-chaotic nonlinear system can also be complex one. Next, we have plots of 3-Dimension image of topological entropy for 1  q  3.8, 0.1  k  0.6, and shown in Fig. 6, which clear picture of complexity. Fig. 6: Two 3-D plots of topological entropies for 1  q  3.8, 0.1  k  0.6. Correlation dimensions: A chaotic set, has fractal structure and so, its correlation dimension gives its measure of dimensionality. Being one of the characteristic invariants of nonlinear system dynamics, the correlation dimension actually gives a measure of complexity for the underlying attractor of the system. To determine correlation dimension a statistical method can be used. It is an efficient and practical method then other methods, like box counting etc. The procedure to obtain correlation dimension follows from some perfect steps calculation [10 -12]: To obtain D c , log C(r) is plotted against log r and then we find a straight line fitted to this curve. The intercept of this straight line on y-axis provides the value of the correlation dimension D c .              n j i j x i x - r H ) 1 n ( n 1 lim n ) r ( C , (3.1) where       0 x 1, 0 x 0, ) (x H , In case of bouncing ball, with k = 0.3 and q = 3.8, the obtained correlation curve is shown in Fig. 7. The y -intercept of this straight line is 2.1792 and so the correlation dimension of the chaotic set be measured, approximately, as D c = 2.197. Discussions: The problem of complexity and chaos observed during evolution of bouncing ball has been studied in detail and certain measures of complexity such as Lyapunov exponents, topological entropies, correlation dimension are calculated. The non-negative real number, the topological entropy, describes a perfect measure complexity of dynamical system in the sense that more the topological entropy a system has means it is more complex. Actually, a topological entropy measures the exponential growth rate of the number of distinguishable orbits as time advances in the system. However positivity of its value does not justify the system be chaotic. For k = 0.3 the system studied in this article, chaos happens when q values be increased from 3.31 (approximately). Before this the system is regular. However, we find even much before q reaching this value, within the range 1  q  2.6, the system is complex as the topological entropies are more than zero. Also, as shown in Fig. 5, for k = 0.3 and values of q in intervals 3.5  q  4.5 and 6  q  8.5, where the system showing highly chaotic, topological entropy appears to be very low. Another interesting thing be observed the correlation dimension for q = 2.4, (a regular case), is nonzero and it is given by D c = 1.653 (approximately). However, there are nonlinear systems Finally, one can conclude in case of bouncing ball dynamics, complexity and chaos are certainly mixed phenomena. Fig. 1 : 1Bifurcation diagrams of map (1) with k = 0.3 and varying q: Fig. 2 : 2Upper row plots showing clearly 2-periodic attractors for k = 0.3 and q = 2.5; the corresponding plots in the lower row, when q is changed to the value q = 3.8, shows motion is chaotic. 2 . 2Lyapunov Exponents (LCEs), Topological Entropies & Correlation Dimensions:Lyapunov exponents: LCEs, have been calculated and plotted, shown inFig. 3, for regular and chaotic cases as discussed above. In regular case, though the LCEs are negative at each iteration, their numerical values are different. Similarly, for chaotic case, values are pos itive but different. Fig. 3 : 3Plots of LCEs for chaotic and regular cases for fixed k = 0.3. Figures in the upper row are for regular case when q = 3.8 and those of lower row are for q = 2.5. Fig. 4 : 4A plot of LCEs for k = 0.3 and q = 3.31. As LCEs are negative, motion is regular Topological Entropy: Next, let us have calculated topological entropy for the bouncing ball system (1.1) and plotted inFig. 5. Fig. 5 : 5Plots of topological entropy for k = 0.3 and four ranges of values of parameter q; (a) 0.1  q  2.5, (b) 1  q  3.8, (c)3.5  q  4.5, (d) 6  q  8.5. is the unit-step function, (Heaviside function). The summation indicates the the number of pairs of vectors closer to r when 1 ≤ i, j ≤ n and i ≠ j. C(r) measures the density of pair of distinct vectors x i and x j Fig. 7 : 7Plot of correlation curve for bouncing ball system (2.1) A linear fit to the correlation data be obtained as Y = 2.1792 -1.0374 x (3.3) Does God Play Dice ?. Ian Stewart, Penguin BooksIan Stewart. Does God Play Dice ?, Penguin Books 1989 . R L Adler, A G Konheim, M Mcandrew, Topological entropy, Trans. Amer. Math. Soc. 114Adler, R L Konheim, A G McAndrew, M H. Topological entropy, Trans. Amer. Math. Soc. 1965; 114: 309-319 Topological entropy for noncompact sets. R Bowen, R , Trans. Amer. Math. Soc. 184R. Bowen, R . Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 1973: 184: 125- 136 The dynamics of repeated impacts with a sinusoidally vibrating table. P J Holmes, J Sound Vib. 84Holmes PJ. The dynamics of repeated impacts with a sinusoidally vibrating table. J Sound Vib 1982;84:173-89. Chaotic dynamics of a bouncing ball. R M Everson, Physica D. 19Everson RM. Chaotic dynamics of a bouncing ball. Physica D 1986;19:355-83. Strange attractors of a bouncing ball. T M Mello, N M Tuffilaro, Am J Phys. 55Mello TM, Tuffilaro NM. Strange attractors of a bouncing ball. Am J Phys 1987;55:316-20. The bouncing ball apparatus as an experimental tool. A Kini, T L Vincent, B Paden, J Dyn Syst Meas Control. 128Kini A, Vincent TL, Paden B. The bouncing ball apparatus as an experimental tool. J Dyn Syst Meas Control 2006;128:330-40. Regular and chaotic dynamics in bouncing ball models. Sebastian Vogel, Stefan J Linz, International Journal of Bifurcation and Chaos. 213Sebastian Vogel, Stefan J. Linz, Regular and chaotic dynamics in bouncing ball models. International Journal of Bifurcation and Chaos, Vol. 21, No. 3, 2011: 869-884 Detection of the chaotic behaviour of a bouncing ball by the 0-1 test. G Litak, A Syta, M Budhraja, L M Saha, Chaos, Solitons and Fractals. 42Litak G, Syta A, Budhraja M, Saha, L M. Detection of the chaotic behaviour of a bouncing ball by the 0-1 test. Chaos, Solitons and Fractals 42, 2009: 1511-1517 Measuring the Strangeness of Strange Attractors. P Grassberger, I Procaccia, Physica 9D. Grassberger P, Procaccia I. Measuring the Strangeness of Strange Attractors, Physica 9D, 1983:189-208. Introduction to Discrete Dynamical Systems and Chaos. M Martelli, John Wiley & Sons, IncNew YorkMartelli M. Introduction to Discrete Dynamical Systems and Chaos. John Wiley & Sons, Inc., 1999, New York. Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena. H Nagashima, Y Baba, Overseas Press India Private LimitedNagashima H, Baba Y . Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena. Overseas Press India Private Limited, 2005.	[]
[ "Dynamics and correlations of a Bose-Einstein condensate of photons", "Dynamics and correlations of a Bose-Einstein condensate of photons" ]	[ "Julian Schmitt \nInstitut für Angewandte Physik\nUniversität Bonn\nWegelerstraße 8, Thomson AvenueD-53115, CB3 0HEBonn, CambridgeGermany, UK\n" ]	[ "Institut für Angewandte Physik\nUniversität Bonn\nWegelerstraße 8, Thomson AvenueD-53115, CB3 0HEBonn, CambridgeGermany, UK" ]	[]	The Tutorial reports recent experimental advances in studies of the dynamics as well as the number and phase correlations of a Bose-Einstein condensed photon gas confined in a high-finesse dye-filled microcavity. Repeated absorption-emissionprocesses of photons on dye molecules here establish a thermal coupling of the photonic quantum gas to both a heat bath and a particle reservoir comprised of dye molecules. In this way, for the first time Bose-Einstein condensation under grand-canonical statistical ensemble conditions becomes experimentally accessible.	10.1088/1361-6455/aad409	[ "https://arxiv.org/pdf/1807.08747v1.pdf" ]	119,420,136	1807.08747	f223ae064f6c6cd605ee872ef2bc21fd88090936	Dynamics and correlations of a Bose-Einstein condensate of photons 19 July 2018 23 Jul 2018 Julian Schmitt Institut für Angewandte Physik Universität Bonn Wegelerstraße 8, Thomson AvenueD-53115, CB3 0HEBonn, CambridgeGermany, UK Dynamics and correlations of a Bose-Einstein condensate of photons 19 July 2018 23 Jul 2018Present address: Cavendish Laboratory, 19 JJ The Tutorial reports recent experimental advances in studies of the dynamics as well as the number and phase correlations of a Bose-Einstein condensed photon gas confined in a high-finesse dye-filled microcavity. Repeated absorption-emissionprocesses of photons on dye molecules here establish a thermal coupling of the photonic quantum gas to both a heat bath and a particle reservoir comprised of dye molecules. In this way, for the first time Bose-Einstein condensation under grand-canonical statistical ensemble conditions becomes experimentally accessible. Large statistical number fluctuations are a fundamental property known from the thermal behaviour of bosons, as has been strikingly revealed in experiments with light and material particles [1][2][3][4][5][6]. For low temperatures or high densities, however, when a Bose gas undergoes Bose-Einstein condensation (BEC), the persistence of large particle number fluctuations can conflict with the conservation of the total particle number. Accordingly, fluctuations are damped out and second-order coherence emerges [7][8][9]. This notion grounds on the microcanonical or canonical statistical description of the system, which applies for systems well-isolated from their environment suppressing both energy and particle exchange with the environment as e.g. realised in ultra-cold atomic gases [9,10]. Following the first observation of BEC in dilute atomic vapour [11,12], evidence for the the emergence of first-order coherence [13][14][15][16] and the suppression of density fluctuations [3,[17][18][19][20] have provided hallmarks for the phase transition. More recently, BECs have also been observed in two-dimensional (2D) gases of exciton-polaritons [21][22][23][24], magnons [25] and photons [26][27][28]. Quintessentially, these systems are open due to their coupling to the environment for e.g. particle injection or thermalisation, which reinforces the relevance of reservoirs for their description, as for example provided by grand-canonical statistics. In the grand-canonical ensemble, the system is subject to particle (and energy) exchange with a reservoir [29]. For bosons, the population n in each quantum state suffers large number fluctuations δn n, while the fixed chemical potential (and temperature) accounts for a complete thermodynamic description of the gas. In the thermodynamic limit, all three statistical ensembles are generally expected to become equivalent due to vanishing relative fluctuations of the total particle number, i.e. δN/N → 0. Applied e.g. to the macroscopically occupied ground state in the Bose-Einstein condensed phase (n N ), however, in the grand-canonical ensemble large fluctuations of the total particle number, δN N occur. Surprisingly, the statistical fluctuations here become enhanced as the system temperature approaches absolute zero instead of being frozen out. This so-called grand-canonical fluctuation catastrophe has been a long-standing issue in theoretical physics [9,10,[30][31][32][33][34][35][36][37] and its observation has long remained elusive. Most notably, Ziff, Uhlenbeck and Kac altogether questioned the physical significance of the grand-canonical ensemble in the condensed phase [32]; their arguments, however, apply only for diffusive contact between a spatially separated BEC and particle reservoir. In contrast, for a BEC of photons in a dye-filled optical microcavity genuine grand-canonical statistical conditions in the condensed phase can become relevant. Here, the coupling of the condensed particles to an effective reservoir is realised by interparticle conversion between photons, ground and excited state dye molecules [38,39]. In this system, we have for the first time observed grand-canonical number statistics in a BEC by demonstrating its coupling to both a heat bath and a particle reservoir [40][41][42]. These results provided a first experimental hint at the fluctuation catastrophe. Moreover, our work revealed phase fluctuations of the condensate wave function in the wake of grand-canonical statistical number fluctuations [40,41,43]. The present Tutorial contains a theoretical and experimental study of the thermalisation dynamics and first and second-order temporal correlations of a Bose-Einstein condensed photon gas under canonical and grandcanonical ensemble conditions. The Tutorial is organised as follows: Section 2 introduces the concept of photon BEC, Sections 3-5 give a theoretical description of the photon thermalisation process, along with the BEC number and phase correlations, while Sections 6-8 describe our corresponding experiments. Finally, Section 9 concludes and gives an outlook. V(x,y) y Figure 1. Scheme of the experiment. (a) The dye-filled optical cavity consists of two highly-reflecting mirrors separated by q = 7 half-wavelengths. The spherical mirror curvature introduces a harmonic potential V (x, y) for the transverse motion of the photons. A pump laser excites the dye molecules and subsequent emission-absorption-cycles lead to a thermalisation of the photon gas at T = 300 K. The cavity emission is monitored in a spatially, spectrally and time-resolving way. (b) Cavity mode spectrum of the photons (top), and spectral profiles of dye fluorescence f (ν) and absorption α(ν) (bottom). The height of the bars indicates the degeneracy of the cavity eigenmodes. Fluorescence photons are emitted into transverse modes with fixed q = 7 (black bars), making the photon gas effectively 2D. Reproduced with permission from [26,64]. Copyright 2010 & 2017 managed by the Nature Publishing Group. Bose-Einstein condensation of photons Photons depict a prime example among the Bose gases known today and yet, it has taken almost a century to find ways to condense them -Why? Thermal photons usually do not become quantum degenerate: in blackbody radiation, for example, the coupling of temperature and total photon number prohibits BEC at low temperatures as photons at T → 0 vanish instead of forming a condensate * . In optical gases with a conserved particle number as e.g. in nonlinear microcavities, photon-photon interactions are usually too small to achieve efficient thermalisation of the light [44,45]. Quantum fluids of light have nevertheless emerged in recent years by synthesising dressed light-matterstates [46], such as exciton-polaritons in microcavities [24] or suface-plasmon-polaritons [47]. These platforms have provided long sought-after evidence for condensation, coherence [21][22][23][48][49][50] and thermalisation [51][52][53][54] in optical quantum gases. More recently, BEC of pure photons has also become tractable by implementing a photon thermalisation mechanism with an incoherent molecular medium that realises a non-zero chemical potential for the light [55]; see Refs. [56,57] for similar concepts. The first observation of photon BEC in 2010 by Klaers et al. [26,27,58] has been confirmed in more recent work by Marelic et al. [28] and Greveling et al. [59]. In the meantime, a number of experiments have elaborated on the thermalisation [41,60,61], the calorimetry [62], the first-order spatial coherence [63,64], the first-and second-order temporal correlations [40,43], the polarisation properties [65], non-local interactions [66] and the generation of lattices and micropotentials for photon condensates [67,68]. Key aspects that are related to the topics discussed in this Tutorial have been studied in a (non-exhaustive) series of theoretical work on photon condensation and its dynamics [69][70][71][72], on grand-canonical particle number correlations [39,58,73], on phase diffusion [74] and on the relation of photon condensation and lasing [75,76]. In this Section, we introduce the scheme for BEC of photons in a dye-filled optical microcavity with a focus on thermal and chemical equilibrium, the microcavity dispersion and the statistical physics of the photons. Figure 1 shows our microcavity experiment, which consists of two curved mirrors spaced by D 0 1.4 µm (or 1.6 µm) and filled with a liquid dye solution. At a mirror separation D 0 = qλ/2ñ 0 the resonator encloses q = 7 (or 8) half waves, which corresponds to a free spectral range of adjacent longitudinal cavity modes ∆λ = λ 2 /2ñ 0 D 0 80 nm (∆ν 75 THz) comparable to the spectral width of the dye fluorescent emission ( Fig. 1(b)). Here, λ denotes the optical wavelength in vacuum andñ 0 is the refractive index of the dye solution. Accordingly, photons associated with a fixed longitudinal wave number q are absorbed and emitted into the resonator differing only in their transverse quantum numbers m and n. Effectively, this reduced dimensionality introduces a low-energy ground state ("cutoff") for the photon gashω c , which is given by the TEM q00 cavity mode. In other words, the photons can be ascribed an effective mass m ph =hω c /(c/ñ 0 ) 2 and the photon kinetics is reduced to the transverse plane of the resonator. Additionally, the curvature of the cavity mirrors imposes an in-plane confinement ( Fig. 1(a), left). The photon gas behaves formally equivalent to a 2D, harmonically trapped ideal Bose gas, for which in thermal equilibrium BEC is expected below a finite critical temperature T c or above a critical particle number N c [77,78]. Photons in a dye-filled microcavity The solved dye molecules are optically pumped by a laser beam and the electronically excited molecules decay via emission of fluorescence photons in the cavity modes, as sketched in Fig. 1(b, top). Inside the high-finesse cavity, frequent absorption-emission-cycles of photons by dye molecules establish a thermal contact between both subsystems in the sense of the grand-canonical ensemble: the photon gas acquires a temperature T (room temperature) and chemical potential µ, as determined by the much larger molecular reservoir. Firstly, for the (energy) thermalisation the spectral distributions of fluorescence f (ω) and absorption α(ω), see Fig. 1(b), are required to scale with a Boltzmann-factor f (ω)/α(ω) ∝ ω 3 exp(−hω/k B T ) ‡ . The fluorescence-induced energy exchange between the photons and the (thermal) molecular bath then translates to the spectrum of the photon gas. Secondly, the chemical (particle number) equilibration rests on the fact that the interaction between photons and molecules can be considered as a photochemical reaction, see Fig. 1(a, bottom). The required energy to electronically excite a dye moleculehω 2.3 eV exceeds thermal energy k B T 0.025 eV by far, which suppresses fluctuation-driven dye excitations by a factor of order exp(−hω/k B T ) ≈ 10 −37 . Similarly, the thermally excited emission of photons into the cavity modes (hω 2 eV) is very unlikely. An optical photon (of energyhω) is emitted only, if another optical photon (hω ) has been previously absorbed. If this condition is maintained throughout the experiments, the photon number does not decrease as the gas is cooled down, i.e. µ = 0, in contrast to blackbody radiation. The thermalisation process equilibrates photons over the set of TEM mn modes, leading to an average internal energy of the photon gas ∼k B T abovehω c . "Cold" photons propagate near the optical axis, while "hot" photons exhibit large angles with respect to the optical axis. By heating up the dye solution, an enhanced population of highly excited transverse states is observed [27,64]. The equilibrium Bose-Einstein distribution has been experimentally confirmed in the dye-microcavity experiment [26-28, 40, 41, 43, 55, 60, 62]: For small total particle numbers, N ≤ N c , the photon energies are Boltzmann-distributed, while for N > N c adding more photons results in the accumulation of a BEC in the transverse ground state accompanied by a saturation of excited transverse modes. The light-matter-interaction between photons and molecules at room temperature is incoherent, due to many dephasing collisions between dye and solvent molecules during the dye excited state lifetime [83,84]. Consequently, the dynamics of photons and molecules can be modelled by rate equations, which also determine the mean population of photonsn and excited dye moleculesM ↑ (see Fig. 2 (a)). As for typical experimental parametersM ↑ n, the ensemble of excited dye molecules can be viewed as an effective particle reservoir for the photon gas. The heat energy and particle exchange with the dye reservoir paves the way for studies of the transition from canonical to grand-canonical ensemble conditions, as illustrated in Fig. 2 (b). In the grand-canonical ensemble, where each eigenstate suffers strong number fluctuations δn i /n i 1, the second-order coherence of a BEC is expected to be substantially reduced, i.e. g (2) (0) (δn 0 /n 0 ) 2 + 1 = 2. Experimentally, we find evidence for large statistical intensity fluctuations in BECs, which persist up condensate fractions ofn 0 /N 70% as long as the particle reservoir complies with grand-canonical conditions [38,40,43]. This is in contrast to experiments with ultra-cold atoms, where a reduction of density fluctuations in the Bose-Einstein condensed phase has been observed [3,5,6,17]. In this case, the emergence of second-order coherence is related to the isolation of the atomic ensemble from its environment, which necessitates a statistical description in the microcanonical ensemble with fixed particle number and Poissonian fluctuations δn 0 /n 0 = 1/ √n 0n 0 1 −→ 0, i.e. g (2) (0) = 1. Interestingly, also the photon statistics in a laser follows a Poissonian distribution [85][86][87]. The fluctuation properties of photon BECs under grand-canonical conditions differ strikingly from those of both lasers and BECs in the microcanonical or canonical ensemble. ‡ For many dye solutions at room-temperature the scaling is based on the Kennard-Stepanov relation [79][80][81][82]. Thermal equilibrium We outline the fluorescence-induced thermalisation process of the microcavity photon gas, which is based on radiative energy exchange between photons and dye molecules by absorption-emission-processes. The latter establish a thermal contact between the system (photon gas) and a heat bath at room temperature (dye solution) by dissipating excess energy of "hot" photons and providing energy for "colder" photons. For this, the spectral absorption and emission profiles of the dye molecules are required to fulfil the so-called Kennard-Stepanov relation, as will be discussed in the following. A more refined derivation of the thermalisation process can be found in refs. [27,55,58]. The relevant, underlying molecular processes are sketched in the simplified energy diagram of a dye molecule in Fig. 3(a). The electronic ground and excited singlet states S 0,1 exhibit a (quasi-)continuous subset of rotational and vibrational modes (shaded areas), and the energy difference between the ground states in S 0 and S 1 is on the order ofhω zpl 2 eV (zero-phonon line). After a photon absorption (hω a ), frequent collisions between dye and solvent molecules (10 −15 s time scale at room temperature) rapidly alter the rovibrational molecular state, resulting in a thermal distribution in the electronically excited manifold. During the relaxation, any excess energy is dissipated by the solvent bath on a 10 −12 s timescale. To this end, the subsequent fluorescence emission (τ = 10 −9 s,hω f ) occurs from a thermally equilibrated state S 1 to the ground state S 0 , which is subject to the same relaxation mechanism. This insight allows us to derive a Boltzmann-type law relating the spectral absorption and emission profiles of the dye molecules, known as the Kennard-Stepanov relation [79-82, 88, 89]. We obtain the ratio of fluorescence f (ω) and absorption α(ω) by integrating over the rovibrational energy levels f (ω) α(ω) ∝ D ↑ ( )p( )A( , ω)d D ↓ ( )p( )B( , ω)d ,(1) where , denote energies and D ↓ ( ), D ↑ ( ) the rovibrational density of states in ground (↓) and excited (↑) state. Due to the collisional relaxation, p( ( ) ) = exp(− ( ) /k B T ) in both states. Considering energy conservationhω + =hω zpl + , the Einstein coefficients A( , ω) and B( , ω) are related by [88]. Assuming identical rovibrational substructures, D ↑ ( )A( , ω)d = 2hω 3 πc 2 D ↓ ( )B( , ω)dD ↓ ( ) = D ↑ ( ), (1) yields the Kennard- Stepanov relation f (ω) α(ω) ∝ 2hω 3 πc 2 exp −h (ω − ω zpl ) k B T .(2) Experimentally, the scaling has been verified in e.g. liquid dye solutions [58,[90][91][92][93], dye-doped polymers [60], semiconductors [94] or ultra-dense gases [95]. The Kennard-Stepanov relation is the key ingredient for the photon thermalisation mechanism to work. In our high-finesse "photon box", see Fig. 3(b), the fluorescence photons undergo many absorption-emission-cycles, corresponding to a random walk of the light field configuration [58]. The ratio of the transition rates between two configurations \|1 → \|2 , which differ from each other by the absorption of one photon with frequency ω i and the emission of one photon with frequency ω j , is R 12 R 21 = α(ω i )f (ω j )ω 3 j α(ω j )f (ω i )ω 3 i = e −h(ωj −ωi)/kBT ∀i, j.(3) From the theory of Markov processes it is known that exactly such a Boltzmann-scaling of the transition rates evokes a thermal state of the master equation (detailed balance) [58,96,97]. The Kennard-Stepanov relation ensures that the photon gas for sufficiently long times acquires a thermal equilibrium state. Chemical equilibrium Besides energy exchange with the heat bath (temperature T ), the effective particle exchange between the photon gas and the dye reservoir assigns the photons a chemical potential µ γ . In the following, we will see that it is determined by the excitation level of the dye medium. Due tohω zpl k B T , purely thermal excitation of molecules from their ground (↓) to excited electronic (↑) states is strongly suppressed, and optical photons (γ) are required to drive the transition. Vice versa, the decay of a molecule results (with an efficiency of 95%) in the emission of a photon. Altogether, such a behavior resembles a photochemical reaction: γ + ↓ ↑ (4) In chemical equilibrium (i.e. zero net particle flux between different species) the particular chemical potentials balance, µ γ + µ ↓ = µ ↑ . Thus, the fugacity of the photons reads z = e µγ k B T = e µ ↑ k B T e µ ↓ k B T .(5) The partition function of a dye molecule F = w ↓ exp(µ ↓ /k B T ) + w ↑ exp[(µ ↑ −hω zpl )/k B T ], with the statistical weights w ↓,↑ = ≥0 D ↓,↑ ( ) exp[− /k B T ]d , allows one to associate the molecular chemical potential with the probability of finding a molecule in the ground or excited electronic state, respectively: w ↓ e µ ↓ k B T F = M ↓ M , w ↑ e µ ↑ −hω zpl k B T F = M ↑ M(6) This probability is determined by the ratio of the number of excited and relaxed dye molecules M ↑,↓ and the total number of molecules M . By renormalising the chemical potential with respect to the cavity ground state energy, µ = µ γ −hω c , (5) yields e µ k B T = w ↓ w ↑ M ↑ M ↓ e −h ∆ k B T .(7) In equilibrium, the chemical potential µ is thus determined by the dye molecular excitation level M ↑ /M ↓ and the detuning between the condensate frequency and dye resonance ∆ = ω c − ω zpl . Microcavity dispersion relation The microcavity photons can be formally described as a 2D harmonically trapped Bose gas. In the resonator filled with a dye medium with index of refractionñ, the energy-momentum-relation of a photon in free space E =hc/ñ k 2 r + k 2 z with c the speed of light, k r the radial and k z = πq/D(r) the longitudinal wave vector (q is an integer longitudinal wave number) component gets modified by the cavity boundary conditions. These are determined by the mirror spacing D(r) D 0 − r 2 /R at a radial distance r from the optical axis, where D 0 denotes the mirror separation on the optical axis. These parameters are illustrated in Fig. 4(a). Using the paraxial approximation (k z k r ), the dispersion relation of the microcavity photons becomes E πhcq nD 0 + πhcq nRD 2 0 r 2 +h cD 0 2πqñ k 2 r .(8) In addition, we extend (8) by accounting for a nonlinear response of the refractive index subject to changes of the 2D photon density, i.e. the intensity of the light field. The total index of refractionñ =ñ 0 + ∆ñ r =ñ 0 +ñ 2 I(r) can be written as a sum of the linear refractive index in the absence of photonsñ 0 and a nonlinear contributioñ n 2 . The nonlinear term results from physical effects that lead to intensity-dependent energy shifts, as e.g. the optical Kerr effect [98], or temporally slow thermal lensing [67]. Assumingñ 0 ∆ñ r , we obtain E = m ph c 2 n 2 0 +h 2 k 2 r 2m ph + 1 2 m ph Ω 2 r 2 − m ph c 2 n 3 0ñ 2 I(r).(9) Here, the effective photon mass m ph = πhqñ 0 /(D 0 c) and trapping frequency Ω = c/ñ 0 D 0 R/2 have been introduced, revealing the formal equivalence of (9) with the dispersion of a massive, harmonically trapped particle moving non-relativistically in a 2D plane, see Fig. 4(a, right). The first term in (9) determines the effective rest energy of the photons, a global energy shift determined by the cavity boundary conditions. It corresponds to the energy of the q-th longitudinal mode without any transverse excitations m ph (c/ñ 0 ) 2 = E q00 =hω c with the cutoff frequency ω c . The eigenenergies in the cavity are given by 2D (isotropic) harmonic oscillator states E nx,ny = m ph c 2 /n 2 0 +hΩ(n x + n y + 1) with quantum numbers n x und n y . The eigenfunctions ψ nx,ny (x, y) = ψ nx (x) · ψ ny (y) are given by the 1D solutions ψ n (x) = ( √ 2 n n! √ πb) −1 H n (x/b) exp[−x 2 /(2b 2 )] , where b = h/m ph Ω denotes the oscillator length and H n (x) the Hermite polynomials. Statistical physics of microcavity photons In the following, we will describe the temperature behaviour of the (ideal) 2D photon gas in a harmonic trap [77,[99][100][101][102]. We derive the critical particle number and temperature, respectively, as well as the spectral and spatial Figure 4. (a) Microcavity geometry showing mirrors (radius of curvature R) separated by D(r) at a transverse position r. In the paraxial approximation (kz kr), one finds a modification of the photon dispersion relation from linear scaling E =hkc/ñ 0 in 3D free space (top) to a quadratic scaling with small transverse momenta kr in the microcavity (bottom), similar to the dispersion relation of a 2D massive particle. (b) Spectral occupation versus wavelength. The rest energy of the photons is determined by the cutoff wavelength λc = hñ 0 /(mphc). Below the critical photon number Nc ≈ 90 000, the spectra show a Boltzmann scaling. ForN > Nc, the ground state becomes macroscopically occupied. (c) The chemical potential grows in the classical region with increasing particle number until it saturates at µ = 0 around Nc. (d) The condensate fraction exhibits an quadratic scaling as a function of the reduced temperature. Experimentally, we adjust the reduced temperature by varying Tc ∝ √ N to match room temperature T = 300 K when operating at the phase transition. distributions for the experimentally studied photon gas. We can specify the transversal excitation energies in the harmonic trap u nx,ny = E nx,ny − m ph c 2 n 2 0 −hΩ =hΩ (n x + n y )(10) with a degeneracy of the eigenstates g(u) = 2 (u/(hΩ) + 1), where the factor 2 accounts for the two-fold polarisation degeneracy of the photons. At temperature T , the average occupation of an excited state with energy u nx,ny is given by the Bose-Einstein distribution n T,µ (u) = g(u) exp[(u − µ)/k B T ] − 1 .(11) Here, we have implicitly assumed that the system is grand-canonical with a chemical potential µ adjusting the average total particle numberN under the constraintN = u=0,hΩ,2hΩ,...n T,µ (u). At high temperatures or low total photon numbers, the chemical potential obeys µ/k B T 0, and (11) equals the classical Boltzmann distribution. In the opposite limit (T → 0 or N → ∞), the chemical potential converges asymptotically to the ground state energy µ → 0 − (Fig. 4 (c)) and the ground state becomes macroscopically occupied. The phase transition to a BEC occurs at the critical photon number or temperature, respectively, N c = π 2 3 k B T hΩ 2 , T c = √ 3 πh Ω k B N ,(12) and as a function of the cavity parameters T c ∝ (N /R) 1/2 . Notably, an equilibrium phase transition requires its critical temperature to remain finite in the thermodynamic limit (N , V → ∞). It can be achieved by increasing the particle numberN and volume V ∝ R 2 in a way that conservesN /R, i.e. by gradually switching off the trapping potential R → ∞. The expected spectral photon distributions for increasing chemical potentials are shown in Fig. 4(b). The condensation fraction scales quadratically with the reduced temperature,n 0 /N = 1 − (T /T c ) 2 , see Fig. 4(d), as expected for a 2D harmonically trapped ideal Bose gas [99,102]. In this confinement, BEC occurs not only in momentum space but also in position space. The spatial intensity distribution of the condensed photon gas is the sum over all oscillator eigenfunctions weighted with the Bose-Einstein factor: I T,µ (x, y) 2m ph c 2 n 2 0 τ rt nx,ny \|ψ nx,ny (x, y)\| 2 exp hΩ(nx+ny)−µ kBT − 1(13) The power per photon is accounted for by m ph (c/ñ 0 ) 2 /τ rt , where τ rt = 2D 0ñ0 /c denotes the photon round trip time of in the resonator. This approximation is valid due tohΩ ∼ 0.1 meV being much smaller than the rest energy m ph (c/ñ 0 ) 2 ∼ 1 eV. The spectral and spatial distributions of the photon gas have been experimentally verified for the first time for both the classical and Bose-Einstein condensed phase in pioneering work by Klaers et al. [26,27,55,58]. Subsequent studies have provided further insight into the phase transition, and revealed e.g. thermodynamic properties such as condensate fraction or specific heat [28,40,41,43,60,[62][63][64]66] The purpose of the present Tutorial is to elucidate the coherence properties of BECs of light. For this study, typically realised experimental parameters for the microcavity setup are q = 8 at a cavity cutoff wavelength of λ c = 580 nm, which is associated with a mirror separation of D 0 = 1.62 µm. The refractive index of the dye medium (Rhodamine 6G solved in ethylene glycol) amounts toñ 0 = 1.43 at room temperature T = 300 K. Most of the experiments are conducted using mirrors with a radius of curvature of R = 1 m. Therefore, the effective photon mass is m ph = 7.8 × 10 −36 kg and the frequency of the harmonic trap is Ω/2π = 37 GHz. With this one expects a critical particle number N c = 93 000. Due to the short resonator round trip time τ rt = 15 fs, the average circulating optical power in the resonator at threshold becomes P c ≈ 2.1 W. The highly-reflecting mirrors transmit a fractionT 2.5 × 10 −5 of the optical power. At criticality, the continuous power of the cavity emission is approximately 5 nW. Table 1 summarises parameter sets, which are used in the course of the discussed experimental sequences. Multimode photon dynamics Bose-Einstein condensation is a phase transition of the Bose gas in thermal equilibrium. The thermalisation of a nonequilibrium system can occur via different mechanisms and with characteristic dynamics. Atomic gases e.g. relax into equilibrium by contact interactions, while microcavity photons inherit their temperature solely from the thermal contact to a molecular heat bath. The atomic equilibration requires several interatomic collisions [103,104], whereas the photons can be thermalised after only a single absorption-emission-cycle. In this Section, we will theoretically investigate the photon dynamics. Rate equation model We start our discussion by analysing the rate equations for absorption and emission of photons in the microcavity modes. As the dye solution is embedded into the cavity volume, fluorescence emission occurs directly into the reabsorbing medium. After an absorption process, the high collision rate between solvent and dye molecules at room temperature leads to decoherence of the molecular dipoles [83]. The photon-dye-system here correspondingly is in the weak coupling regime [84,105]. In first-order perturbation theory, the photon dynamics can thus be adequately modelled by semiclassical rate equations, which are determined by the time evolution of the diagonal elements of the density matrix. To begin with, we consider a configuration of the light field {n 1 , n 2 , ..., n i , ...} with n i photons in the i-th cavity mode. The transition rates (per volume) R i 12 (r) = B 12 (ω i )u i (r)ρ ↓ n i R i 21 (r) = B 21 (ω i )u i (r)ρ ↑ (n i + 1),(14) give the probability (per time) to absorb or emit a photon in mode i at position r with the frequency-dependent Einstein coefficients for absorption and emission B 12,21 (ω i ), the spectral energy density per photon u i (r), and the densities of ground and excited state molecules ρ ↓,↑ . Due to the densities on the right-hand-side in (14) one obtains rates per volume, which yield absolute rates after integrating out the resonator volume. For the transverse ground state (i = q00) with ω i = E q00 /h and n i = n, this gives R 12 n = B 12 E q00 h u q00 (0) nρ ↓ \|f q00 (0)\| 2 .(15) Here, we have expressed the energy density u q00 (r) = u q00 (0)\|f q00 (r)\| 2 /\|f q00 (0)\| 2 by the normalised mode function f q00 (r). Using the effective mode volumeṼ q00 eff = \|f q00 (r)\| 2 /max \|f q00 (r)\| 2 dV = 1/\|f q00 (0)\| 2 [106,107], the modified Einstein coefficientsB 12,21 = B 12,21 (E q00 /h)u q00 (0) and the number of ground and excited state molecules M ↓,↑ = ρ ↓,↑Ṽ q00 eff , one obtains the rate equations for the ground mode populated with n photons R 12 n =B 12 M ↓ n =B 12 (M − X + n)n R 21 n =B 21 M ↑ (n + 1) =B 21 (X − n)(n + 1).(16) According to the photochemical reaction in (4), we have expressed the rates as a function of the sum of all molecular and photonic excitations X = M ↑ + n and the total molecule number in the ground mode volume M = M ↓ + M ↑ = M ↓ + X − n, which we assume to be constant reservoir parameters. The rate equations readily provide the temporal evolution of the photon number ∂ ∂t n i =B 21 M ↑ (n i + 1) − (B 12 M ↓ + γ ph,i )n i(17) with a photon loss rate γ ph,i due to mirror transmission. To conserve the excitation number X, any loss must be compensated for by a net gain P in the molecule rate equations, which is experimentally realised by pumping with a laser beam: − ∂ ∂t M ↓ = ∂ ∂t M ↑ = P − i ∂ ∂t n i − γ M M ↑(18) Additionally, P must balance the molecular loss rate γ M , which results from non-radiative decay and fluorescence into unconfined leakage modes. Steady-state photon number The rate equation model enables a quantitative description of the photon thermalisation dynamics in the microcavity. For this, we consider a simplified model for the multimode photon gas in the uncondensed phase without spatial photon transport, losses or pumping (P = γ M = γ ph,i = 0). Here, the molecule number (10 8 ) exceeds the average photon number per mode (10 1 ), such that the rate equations of the dye medium in (18) can be considered as quasi-stationary with a fixed molecular excitation level M ↑ /M ↓ . A more refined model including dissipative spatial dynamics has been theoretically reported by Kirton and Keeling [69,70]. Our own detailed numerical simulations of the spatial photon dynamics are discussed in Section 6.5 [41]. For a single cavity mode (angular frequency ω i ), the Kennard-Stepanov relation readsB i (7), we obtain the average photon number in thermal and chemical equilibrium from (17): 21 /B i 12 = w ↓ /w ↑ exp[−h(ω i − ω zpl )/ Together withn i = B i 12 B i 21 M ↓ M ↑ − 1 −1 = eh (ω i −ωc)−µ k B T − 1 −1(19) By summing (17) over all degenerate cavity modes with the energy i =hω, we obtain the rate equation for the photon number n ≡ n(ω, t) = i=hω n i (t) in the multimode cavity: The term i=hω 1 gives the energy-dependent mode density g(ω) = 2 [(ω − ω c )/Ω + 1]. In the second step, we have identified the Einstein coefficient for spontaneous emissionÂ 21 ∂n ∂t =B 21 n + i=hω 1 M ↑ −B 12 nM ↓ =B 21 nM ↑ +Â 21 M ↑ −B 12 nM ↓ ,(20)= g(ω)B 21 . The steady-state photon number isn(ω) = g(ω) {exp [h(ω − ω c ) − µ/k B T ] − 1} −1 . Spectral photon number evolution To determine the thermalisation time, we rephrase the single mode (17) asṅ i +αn i +β = 0, with the coefficients α =B i 12 M ↓ −B i 21 M ↑ and β = −B i 21 M ↑ . For the initial condition n i (0) = 0, this differential equation is solved by n i (t) = − β α 1 − e −αt =n i 1 − e −t/τi ,(21) with the time constant τ i = ( n i + 1)/(B i 12 M ↓ ) =n i /(B i 21 M ↑ ) . We expand (21): n i (t) =B 21 M ↑ t 1 − t 2 B 12 M ↓ −B 21 M ↑ .(22) For early times, we can neglect the second-order term ∝ t 2 , so that the spectrum in this limit will be determined by the emission profileB 21 (ω). If we approximate (22) for the uncondensed regime with M ↑ M ↓B12 /B 21 , we obtain the characteristic time after which the initial spectral redistribution of the photon gas occurs τ th 1 B 12 M ↓ .(23) This equals the mean reabsorption time of a photon in the dye medium. The relative occupation of n i and n i+1 of two neighbouring resonator modes (with frequencies ω i , ω i + Ω) demonstrates, that the spectral slope is nearly thermal after τ th . Without loss of generality, we assume that the fluorescence strength into the modes is equal,B i 21 =B i+1 21 , as is indeed fulfilled for the used dyes (Section 6). Using the Kennard-Stepanov relation, the absorption coefficientsB i 12 =B i 21 exp[h(ω i − ω zpl )/k B T ] andB i+1 12 =B i 12 exp[hΩ/k B T ] yield the photon dynamics n i,i+1 (t) M ↑B i,i+1 21 t 1 − t 2 M ↓B i,i+1 12 .(24) An expansion inhΩ/k B T determines the spectral population difference at the thermalisation time n i+1 − n i Ω (τ th ) = − 1 2h k B T M ↑B i 21 M ↓B i 12 = − 1 2h k B Tn i(25) in the limit of a Boltzmann distribution withn i = M ↑B (19), the similar scaling (n i+1 −n i )/Ω = −h/(k B T )n i demonstrates that the photon spectrum agrees except for a factor 1/2 with the spectral shape of a Boltzmann distribution after the reabsorption time τ th . Accordingly, the microcavity photons have relaxed to a thermal-like equilibrium after completing approximately one emissionabsorption-cycle. In general, the relaxation rates of individual modes depend on their frequencies. It is therefore helpful to express the Einstein coefficient for absorptionB 12 (ω) as a function of the experimentally accessible cross section σ(ω). By comparing the coefficients in Beer's law ∂n/∂t = −M ↓Ṽ −1 eff σ(ω)c n and the rate equation for absorption ∂n/∂t = −M ↓B12 (ω)n, we find the useful relation B 12 (ω) = σ(ω)c V eff .(26) Figure 6(a) shows the calculated temporal evolution of the spectra in (21) for Rhodamine 6G dye (Section 6). Owing to the wavelength-dependence of the absorption cross section (maximum near 530 nm), the time τ th after which the spectral distribution has relaxed to a thermal equilibrium distributionn(λ) varies for different spectral regions. While the photon gas relaxation takes several nanoseconds in the red spectral region ( Fig. 6(b), left), for the yellow-green spectral region a thermalisation time of a few picoseconds is predicted (right). For example, Rhodamine 6G absorbs photons at 580 nm wavelength with cross section σ(2πc/λ) 10 −22 m 2 . With the mirror separation D 0 1.6 µm and the diameter of the TEM 00 mode d 0 12 µm we can estimate the effective ground mode volumeṼ eff,00 = π(d 0 /2) 2 D 0 1.8 × 10 −16 m 3 and the rate coefficientB 12 (ω) 166 s −1 . For typical dye concentrations near 1 mmol/l, around M ↓ ≈ 10 8 dye molecules reside in the mode volume. Therefore, the expected thermalisation time is τ th ≈ 50 ps. Chemical equilibration time The spectral thermalisation time of the photon gas τ th in the uncondensed phase is approximately given by the photon reabsorption time in the dye solution, 1/B 21 M ↓ , see 23. In general, this value differs from the chemical equilibration time τ ch , which is the time after which the system has acquired its steady-state-populationn(ω). To see this, we extend the single mode description in (21) to the multimode system: n(ω, t) =n(ω) 1 − e −t/τ (ω) ,(27)with τ (ω) =n(ω)/[g(ω)B 21 (ω)M ↑ ]. For a Boltzmann distributionn(ω) = g(ω)B 21 (ω)M ↑ /[B 12 (ω)M ↓ ] , the chemical equilibration time is the weighted average over all frequency-dependent relaxation times τ ch = ∞ ωc τ (ω)n(ω)dω ∞ ωcn (ω)dω kBT hΩ τ th 4 e −h ∆ k B T ,(28) where we have assumedB 21 (ω) to be independent of ω, as is roughly fulfilled for Rhodamine 6G dye within the relevant wavelength range (540 to 600nm). In our experiments, the dye-cavity detuning ∆ = ω c − ω zpl takes values between ∆ 560nm = −2.4k B T /h and ∆ 605nm = −8.7k B T /h, implying the chemical equilibration time to exceed the spectral relaxation time by τ ch /τ th ≈ 3 (560nm) up to 1500 (605nm). This prediction is experimentally verified (Section 6). Below the critical photon number, our simplified analytical model provides an adequate description of the photon number thermalisation dynamics. It should be noted, that this model is insufficient to predict the dynamics in the Bose-Einstein condensed phase where the optical feedback onto the dye requires using the molecular rate equations. In this regime, the large photon number speeds up the dynamics by stimulated emission events and the chemical equilibration can become much faster than the spectral thermalisation, as will be discussed Section 6.5 on the basis of numerical simulations. Grand-canonical photon statistics For BEC in the grand-canonical statistical ensemble regime, i.e. in the presence of a large particle reservoir, large statistical number fluctuations on the order of the total particle number N have been predicted [9,10,[30][31][32][33][34][35][36][37]. In contrast to this, the (micro-)canonical statistical ensemble features Poissonian number fluctuations, i.e. a scaling with √ N ; a situation realised in most atomic BECs [3,5,6,17,20]. In the dye-cavity system, Bose-Einstein condensed photons couple to electronic transitions of a specific number of dye molecules, which realises the repeated exchange of photon-and molecule-like excitations. The latter can be interpreted as an effective particle reservoir for the photons, with a size that depends on the molecule number and the dye-cavity detuning. We find that the photon number statistics of the ground state resembles a (nearly) Bose-Einstein distributed thermal light source, in contrast to both atomic BECs and the laser [17,108,109]. Under these conditions, the phase transition can be regarded as a BEC in the grand-canonical ensemble regime. Photon number distribution We start by considering the master equation for the probability p n ≡ p n (t) to find n photons in the ground state at time t ‡ . The flow of probability between photons in the condensate and the reservoir iṡ with the rates given by (16) [38,110]. According to the experiment, we assume M = M ↑ + M ↓ and X = M ↑ + n to be constant. For t → ∞ the probability flow p n (t) is expected to become stationary,ṗ n (∞) = 0, and the photon number distribution converges to its equilibrium value P n := p n (∞). In this limit, (29) is solved by the recursive ansatz P n = P 0 n−1 k=0 R 21 k /R 12 k+1 , and one obtains the photon number statistics p n = R 21 n−1 p n−1 − (R 12 n + R 21 n )p n + R 12 n+1 p n+1 ,(29)P n P 0 = (M − X)!X! (M − X + n)!(X − n)! B 21 B 12 n ,(30) which is used to calculate the average condensate number and its fluctuations. Similarly, the statistics can been derived by a entropy maximisation principle [39]. In general, (30) has to be evaluated numerically. At constant temperature T , we induce the phase transition by increasing the particle numberN , which effectively lowers the reduced temperature T /T c (N ). For eachN , the following numerical method then computes the excitation number X that recovers the given particle number N : For a starting value X, the average photon number in the condensatē n = n≥0 nP n (31) and the molecular excitation level of the medium in the ground mode volume M ↑ M ↓ = X −n M − X +n(32) are computed. As the density of excited molecules (and thus the excitation level) is required to be spatially homogeneous in chemical equilibrium, the ratio M ↑ /M ↓ controls the chemical potential for the photon gas, see (7). Accordingly, the number of photon in excited states isn exc = u>0 g(u)/(exp[(u − µ)/(k B T )] − 1). If there are residual deviations betweenn +n exc and the target photon numberN , the numerical method is iterated with an adjusted excitation number X until a certain level of precision is reached. Figure 7(a) shows the calculated condensate fractionn/N and the photonic fraction of the excitation number n/X for five different-sized molecular reservoirs as a function of the reduced temperature. The constant dyecavity detuningh∆ = −4.67k B T controls the Kennard-Stepanov relationB 21 /B 12 and hence the photon statistics in (30). For all studied reservoirs, the condensate fraction follows the analytic solutionn/N = 1 − (T /T c ) 2 and the curves forn/X reveal that a large number of excitations are present as molecular excitations down to very low temperatures. Furthermore, Fig. 7(b) shows the zero-delay autocorrelation function g (2) (τ = 0) = n(n − 1) n 2 = n≥0 n(n − 1)P n n≥0 nP n 2(33) for the same reservoir parameters as a function of the condensate fraction and the reduced temperature. For T ≥ T c , the ground state occupation exhibits the usual, strong intensity fluctuations in a single mode of the thermal Bose gas, g (2) (0) = 2, and the photon number statistics is Bose-Einstein-distributed. In the presence of large reservoirs, the intensity correlations maintain when the temperature is lowered deep into the condensed phase, as attributed to the grand-canonical particle exchange with the dye reservoir. For T /T c 1, the statistical number fluctuations are damped out and our calculations demonstrate the emergence of second-order coherence, g (2) (0) = 1, with Poissonian statistics. We do not find indications that the transition between both statistical regimes is accompanied by a discontinuity in the thermodynamic quantities, excluding a further phase transition scenario within the Bose-Einstein condensed phase. The crossover of the photon statistics in the condensed phase remains valid also in the thermodynamic limit, as will be discussed later. Asymptotic photon number distributions We show that the photon number statistics interpolates between a Bose-Einstein-and Poissonian distribution. To analytically derive the limiting cases, we rewrite (30) in a recursion form: P n+1 P n = X − n M − X + n + 1B 21 B 12(34) For Bose-Einstein statistics to apply, P n must follow a geometric series with a ratio P n+1 /P n being independent of n. This is fulfilled if and only if the reservoir dimensions M and X are very large, so that the photon number on the right-hand-side of (34) can be safely neglected, i.e X n and M − X n ("grand-canonical limit"). With X M ↑ and M − X M ↓ , Reproduced with permission from [38]. Copyright 2012 by the American Physical Society. P n+1 P n g.c. = M ↑ M ↓B 21 B 12 ⇒ P n P 0 g.c. = M ↑ M ↓B 21 B 12 n .(35) Here, P n decays exponentially from its maximum at n = 0. Normalisation of (35) gives P n = 1 − M ↑ M ↓B 21 B 12 M ↑ M ↓B 21 B 12 n = n n+1 n n + 1 .(36) In the last step, we have identified the average condensate number from (19). This result remains valid also for increased M , as long as the excitation level M ↑ /M ↓ X/(M − X) (and thus µ,n,N ) are kept constant. Equation (36) is the well-known Bose-Einstein statistics, see Fig. 8, which also applies for example for chaotic, thermal light or blackbody radiation. In the case of Poisson statistics, the most probable photon number is finite, n max > 0. Under the assumption P nmax+1 = P nmax , (34) yields n max = X − (M + 1)/(1 +B 21 /B 12 ). Expanding for ∆n = n − n max , P n+1 P n = 1 − ∆n λ + 1 1 +B 21 /B 12 ∆n λ 2 − ...(37) with λ =B 21 /B 12 M + 1/(B 21 /B 12 + 1) 2 . In the low temperature limit, the ratio of the Einstein coefficients scales with the dye-cavity detuning ∆. For a negative detuning, as in our experiments, it diverges: B 21 (ω) B 12 (ω) = w ↓ w ↑ e −h ∆ k B T T →0 0, ∆ > 0 ∞, ∆ < 0(38) Hence, (37) simplifies to P n+1 /P n λ/(λ + ∆n) (or (λ − ∆n)/λ) for ∆ > 0 (or ∆ < 0). This recursion formula implies the relative probability near ∆n around the maximum n max : P nmax+∆n P nmax (λ−1)! (λ−1+∆n)! λ ∆n , ∆ > 0 λ! (λ−∆n)! λ −∆n , ∆ < 0(39) Upon transforming ∆n → −∆n, both distributions are the same and their relative scaling is analogous to a Poisson distribution P p n = e −λ λ n n! ⇒ P p nmax+∆n P p nmax = λ! (λ + ∆n)! λ ∆n(40) which contains only one parameter λ for mean and variance. The solutions in (39) however are only Poissonian with respect to the relative photon number ∆n, as the an additional parameter n max tunes the most probable photon number. For example, in the limit T → 0 (∆ = 0) the statistics peaks at n max =n =N with λ = 0, where all photons of the systems have condensed into the ground state and the photon number is precisely known. As the crossover point between both statistical regimes, we define the reduced temperature T x /T c (N ) when the most probable photon number ceases to be n max = 0, or in other words when the condition P 0 = P 1 is fulfilled (Fig. 8, inset). Inserting into (34) yields: Statistics crossover M + 1 X = 1 +B 21 B 12(41) Due to the large number of molecules, we assume M + 1 M . With regard to the experimental conditions, we derive T x for fixed total numbers of molecules M and photonsN . With X =n + M ↑ the average condensate population followsn = MB 12 B21 M ↓ M ↑ − 1 1 +B 12 B21 1 + M ↓ M ↑ .(42) In the grand-canonical limit (M ↓ , M ↑ n), see (36), the nominator corresponds to the inverse of the average photon numbern = ∞ n=0 nP n = [(B 12 /B 21 )(M ↓ /M ↑ ) − 1] −1 . Although grand-canonical conditions do not strictly apply in the crossover region, we use this to estimate T x . We find n = M 1 + w ↑ w ↓ eh ∆ k B T 1 + w ↓ w ↑ eh ω zpl −µγ k B T ,(43) where both the Kennard-Stepanov relation and chemical equilibrium have been applied. Moreover, it is safe to assume w ↓ = w ↑ for the statistical weights of ground and excited molecular states [83]. In the condensed phase, the chemical potential of the photonsh(ω c +Ω) < µ γ <hω c , and consequently we can use µ γ hω c =h(∆+ω zpl ) to simplify the second bracket term in the denominator. Equation (43) resembles a boundary for the average number of condensed photons, up to which the particle number statistics can be considered Bose- Einstein-like. Withn =N [1 − (T x /T c ) 2 ] , this implicitly determines the temperature T x for the crossover. To investigate the scaling of the reduced crossover temperature t = T x /T c with the system parameters, we rewrite (43): 1 − t 2 = M/2 N 1 + cosh h∆ k B T c 1 t −1/2(44) The temperature depends only on relative size of the subsystems √ M /N and the reduced detuningh∆/k B T c , which plays an important role for the thermodynamic limit:N , R, M → ∞ with R/N = const. and √ M /N = const. The first condition conserves the critical temperature T c and therefore fixesh∆/k B T c , see (12). The second requirement conserves T x (below T c ), which rules out that the temperature difference arises from finite size effects. Notably, both regimes, Bose-Einstein-and Poissonian statistics, exist within the condensed phase. While the former relates to the grand-canonical ensemble (M n 2 ), the latter refers to a canonical ensemble scenario (M n 2 ). The crossover between both regimes is induced by changingn/N or ∆, respectively. To highlight M (2) (0) 1.5706 1.5707 1.5708 1.5708 1.5708 Table 2. Numerically calculated reduced temperatures and condensate fractions, at which the crossover between Bose-Einstein and Poissonian statistics occurs for different-sized molecular particle reservoirs M x-\| . Here, the autocorrelation gives g (2) (0) π/2, see (45). (h∆ = −2.4kBTc,N = 10 5 ) this,Tab. 2 summarises numerically calculated values for T x /T c for different reservoirs M x-\| = 10 8 − 10 12 at fixed photon gas sizesN = 10 5 and dye-cavity-detuningh∆ = −2.4k B T c (λ c = 560 nm and λ zpl = 545 nm). For sufficiently large reservoirs, the Bose-Einstein-like grand-canonical statistics extends deep into the condensed phase. The inset of Fig. 8 shows the Gaussian photon number distribution at T x /T c = 0.847, given by P n = 2/(πn) exp[−(n/n) 2 /π]. Accordingly, the zero-delay second-order correlation function reads g (2) (0) = n≥0 n(n − 1)P n n≥0 nP n 2 = π 2 − 1 nn 1 π 2 ,(45) which analytically reproduces the numerical results inTab. 2. It corresponds to relative condensate number fluctuations of δn/n = g (2) (0) − 1 75%. Second-order time correlations We extend our discussion of the photon statistics to the temporal dynamics of the statistical fluctuations [111]. The condensate photons are absorbed by M ↓ molecules in the electronic ground state, and M ↑ excited molecules decay by emission of photons into the condensate mode. Neglecting losses, the rate (17) becomes ∂ ∂t n =B 21 (X − n) (1 + n) −B 12 n (M − X + n)(46) with X = M ↑ + n, M = M ↓ + M ↑ and the steady-state solutionn = 1/(B 12M↓ /B 21M↑ − 1). To quantify the time evolution of deviations fromn, we define δn(t) = n(t) −n and obtain ∂ ∂t δn(t) = −(B 12 +B 21 )δn(t) 2 − γδn(t),(47)γ =B 21 X n + (B 12 +B 21 )n B 12B21 B 12 +B 21 M n + (B 12 +B 21 )n.(48) For typical experimental parameters, (B 12 +B 21 )δn 10 6 s −1 and γ 10 9 s −1 , the coefficients in (47) comply with (B 12 +B 21 )n g (2) (0) − 1 γ, where δn/n = g (2) (0) − 1. The equation of motion can thus be linearised, ∂ t δn(t) −γδn(t). The second-order autocorrelations at times t and t = τ + t reads g (2) (τ ) = n(t + τ )n(t) n 2 = 1 + δn(τ )δn(0) n 2 ,(49) where δn(t + τ ) = δn(t) = 0 has been used. Using the quantum regression theorem [111][112][113] allows us to trace back the dynamics δn(τ )δn(0) to the evolution of δn(t): ∂ t δn(τ )δn(0) −γ δn(τ )δn(0) . We find g (2) (τ ) − 1 g (2) (0) − 1 exp − τ τ (2) c ,(50) where τ (2) c = γ −1 denotes the second-order correlation time. With the Kennard-Stepanov relation, the inverse correlation time can be recast as a function of experimental parameters: 1 τ (2) c =B 21   M n 1 + e −h ∆ k B T +n 1 + eh ∆ k B T  (51) Around the crossover temperature T x from grand-canonical to canonical ensemble conditions, the second-oder correlation rate exhibits a piecewise scaling τ (2) c =      M nB 12B21 B12+B21 , T T x (n 2 M eff ) 2n(B 12 +B 21 ), T = T x (n 2 = M eff ) n(B 12 +B 21 ), T T x (n 2 M eff ),(52) where we have introduced the effective reservoir size Figure 9(a) illustrates the variation of the effective reservoir size as a function of the dye-cavity-detuning, and Fig. 9(b) gives a plot of γ versusn for various reservoir sizes. For a specific M eff the inverse correlation time decays in the grand-canonical regime (g (2) (0) → 2) with increasing condensate number, until it reaches a minimum atn min = √ M eff . In the canonical regime (g (2) (0) → 1), the fluctuation rate exhibits the opposite behaviour growing linearly with increasing photon numbers. This analytic prediction is confirmed by numerical Monte Carlo simulations, see Fig. 12(b) in Section 5. To exemplify the order of magnitude of γ, we give an estimate based on the typical experimental parameters discussed near the statistics crossover. For a condensate wavelength λ c = 580 nm, corresponding to a dye-cavity-detuningh∆ = −5.3k B T (Rhodamine 6G) and Einstein coefficientsB 12 170 s −1 andB 21 3.4 × 10 4 s −1 , one obtains in the presence of M = 10 10 molecules an average condensate number ofn min = M eff 7000 photons. From (52) a time scale for the intensity fluctuations τ (2) c ≈ 2 ns is expected, which is close to the experimental observation. M eff = M 2 1 + cosh h∆ k B T −1 .(53) Phase coherence The phasor model allows a description of the temporal amplitude and phase evolution, n(t) and φ(t), of an optical single-mode field containing n photons. As shown in Fig. 10, it can be a valuable tool to consider qualitative differences between light sources: (i) In a thermal light source, the superposition of spontaneously emitted photons with arbitrary phases leads to a random walk of the total phase and destructive interference prohibits the emergence of a macroscopically occupied state with a stationary phase [87]. Bose-Einstein photon statistics here gives rise to a most probable photon number n max = 0. (ii) Stimulated emission in a laser results in a macroscopic occupation of a single optical mode with a nearly stable phase. The mode selection is induced by engineering losses in all undesired modes; making laser emission in general an out-of-equilibrium phenomenon. Residual spontaneous emission into the laser mode causes an amplitude and phase uncertainty [114]. Poissonian photon number statistics with n max > 0 lead to a vanishing probability to find zero photons P 0 = 0. (iii) Finally, the phasor diagram is also helpful to illustrate the phase dynamics of a BEC of photons, where a reservoir induces large statistical, thermal-like (Bose-Einstein statistics) fluctuations of the condensate amplitude n(t). In this setting, the photon number eventually drops to n = 0 and the subsequent spontaneous emission of a photon starts a cascade of stimulated processes forming a new macroscopically occupied ground mode. Due to the randomness of spontaneous emission we expect to observe the total phase of the wave function to change discretely in the course of time. Phase dynamics of the wave function Based on the rate (16) we perform Monte Carlo simulations of the photon number and phase evolution of the BEC based on the phasor model [97]. While stimulated absorption and emission do not alter the phase of the wave function, spontaneously emitted photons cause a Heisenberg-type phase diffusion [74,115,116]. The phasor of the condensate with n photons and phase φ is described by the complex number √ ne iφ . Following a spontaneous emission event with random phase θ and amplitude √ 1 the phasor is modified to √ n + ∆n e i(φ+∆φ) = √ ne iφ + ( √ n + 1 − √ n)e iθ ,(54) which corresponds to a length change ∆n = 1 + 2 √ n cos θ and phase rotation ∆φ = sin θ/ √ n. The spontaneous phase becomes relevant for small photon numbers. Notably, the phase rotation describes only relative changes of the phase and does not apply to the case n = 0, when the randomly selected phase θ breaks the symmetry to determine the overall phase of a re-emerging BEC. Figure 11(a) shows a Monte Carlo simulation of the time evolution of the (normalised) occupation number and corresponding phase for a fluctuating BEC coupled to a reservoir that is compatible with grand-canonical statistics,n < √ M eff = 500. The data reveal discrete phase jumps at points when no photons are present in the ground mode. For the same reservoir, Fig. 11(b) gives the number and phase dynamics of a BEC in the canonical ensemble, withn 3 500 > √ M eff . Due to its finite size the reservoir starts to saturate and the number fluctuations get damped. Notably, the zero-photon-probability P 0 vanishes, such that discrete phase jumps are suppressed. To quantify the temporal phase stability of the condensate, we introduce the phase jump rate Γ PJ = #phase jumps/time interval. Moreover, our simulations of the photon number evolution yield the second-order correlation function g (2) (τ ) and its associated timescale τ (2) c . The phasor amplitude modulation that results from the fluctuating condensate population (time constant τ (2) c ) is expected to affect the degree of first-order coherence g (1) (τ ) via phase diffusion. Figure 11(a) (bottom), however, suggests that this effect is negligible in comparison to the large phase jumps. Experimentally, continuous phase drifts cannot be resolved with the applied interferometric method described in Section 8. Figure 12 shows Γ PJ and 1/τ (2) c as a function of the average condensate number for two reservoirs, realised by varying ∆. The data points are obtained from simulations similar to Fig. 11. In the grand-canonical regime (n < √ M eff ), the phase jump and correlation rate decrease simultaneously with increasing system size. At the crossover to canonical statistics (n ≥ √ M eff ), in Fig. 12 discernible by the autocorrelation value g (2) (0) ≈ π/2 as well as the minimum of 1/τ (2) c (dashed line), both time scales separate. Beyond the minimum, we find the phase jumps to be more strongly suppressed, while an increase in the rate of second-order correlations is revealed in good agreement with the analytical prediction (48). This separation of coherence times is evident for both reservoirs in their respective crossover regions; the larger the reservoir, the further we find the phase jumps to persist in the condensed region. Strikingly, in this regime the photon condensate is expected to exhibit phase coherence despite large statistical number fluctuations characterised by g (2) (0) > 1. Phase jump rate Our numerical findings suggest that the phase jump rate of a fluctuating condensate Γ PJ correlates with the probability P 0 to find zero photons in the ground state. To quantify this, we assume that discrete phase changes occur only in the absence of photons from the cavity ground state at a rate Γ 0 PJ = P 0 /τ 0 . Here, τ 0 labels a characteristic system time scale, i.e. the average time that a zero-photon-state exists in the cavity. Following a statistical fluctuation to a zero-photon-state, a certain time passes by until the condensate builds up with a new macroscopic phase, which is given by the inverse rate for spontaneous emission of a photon at the cutoff frequency. For n = 0, the rate (16) As P 0 is determined by the photon number statistics, we consider the limiting cases: for a large reservoir (grandcanonical statistics), see (36), Bose-Einstein photon statistics gives P 0 = 1/(n + 1) 1/n. For small reservoirs (canonical statistics), the crossover to Poissonian statistics leads to a strong suppression of the zero-photon probability: Γ 0 PJ =B 12 M n α , α = 1, BE & Gaussian ∞, Poisson(56) To quantify the scaling of Γ 0 PJ withn, we have introduced the exponent α, which interpolates between 1 and ∞ when connecting grand-canonical and canonical ensemble conditions. In the Poissonian limit (α → ∞), the condensate thus exhibits the usual phase coherence. Using the above ansatz, the simulated phase jump rates can be reproduced as shown in Fig. 12 (solid line). The values for P 0 were numerically calculated (Section 4). Figure 12 illustrates the separation of time scales for phase and intensity fluctuations, which suggests that the coherence properties of a photon BEC differ fundamentally from those of a thermal light source, i.e. violating g (2) (τ ) = 1 + \|g (1) (τ )\| 2 [87]. Therefore, the question arises whether the separation remains relevant in the thermodynamic limit. Combining (48) and (56) yields the ratio of the correlation times where we have usedn =N [1 − (T /T c ) 2 ] and included the temperature dependence in the constants K 1,2 (T ). In the thermodynamic limit,N → ∞, T c = const. andN / √ M = const., one expects the relative correlation times to scale with the parameter α, which (likeN / √ M ) determines the photon number distribution P n and g (2) (0). On the one hand, for a condensate in the grand-canonical regime (α = 1) at a temperature T x ≤ T < T c , the dependence on the total particle numberN in (57) vanishes and the correlation times coincide also in the thermodynamic limit. On the other hand, Poisson-like (α > 1) and genuine Poisson statistics (α → ∞) are expected to cause a divergence of the first-order coherence time with respect to the second-order correlation time. Despite the here relatively large condensate fluctuations with g (2) (0) 1.57 (1 < α < ∞), a separation of time scales for first-and second-order is predicted for the thermodynamic limit. Similarly, the heuristic phase jump rate becomes Thermodynamic limit: Correlation times τ (1) c τ (2) c =n α−1 B 12 B 12 +B 21 +B 12 +B 21 B 21n 2 M =N α−1 K 1 (T ) + K 2 (T ) N √ M 2 ,(57)Γ 0 PJ = K 3 (T ) M N 2 1 N α−2 ,(58) where K 3 (T ) denotes a temperature-dependent parameter, which does not change with the system size. We expect phase jumps to be fully suppressed in the thermodynamic limit only for Poissonian states with α > 2. Physically, the phase jumps originate from the persistence of fluctuations to zero-photon-states also in the thermodynamic limit caused by Bose-Einstein-like statistics. Provided the particle reservoir is sufficiently large, the time scales for number and phase fluctuations remain coupled even upon extrapolation of Γ 0 PJ to the thermodynamic limit. On the one hand, the zero-photon-probability decays as P 0 ∝n −1 with increasing photon numbers (α = 1). On the other hand, this is counteracted by a quadratical increase of the molecule number required to conserve the photon statistics. Ultimately, this results in a larger phase jump rate. Thermalisation dynamics In this section, we discuss experimental results of time-resolved measurements of the spectral photon kinetics, which shed light on the thermalisation dynamics to the molecular heat bath. Our measurements are performed for photon numbers near the critical particle number N c 90 000 (q = 8). Moreover, the experiment enables a spatially and spectrally-resolved observation of the transition dynamics from out-of-equilibrium, laser-like states to thermal equilibrium BECs forN N c . 6.1. Experimental scheme Figure 13(a) shows a schematic of the experimental setup, which is comprised of the optical microcavity, the pump source and an analysis section [41]. For this time-resolved study, the dye-cavity is pumped under an angle of approximately 42 • with respect to the optical axis using a picosecond pulsed laser. The cavity emission is detected by a streak camera in a spatially-and spectrally-resolved way. The microcavity is composed of highly-reflecting dielectric mirrors (CRD Optics, 901-0010-0550, radius of curvature R = 1 m) with a maximum reflectivityR = 99.9988(2)% around 550 nm, while the bandwidth of a reflectivity beyond 99.98% extends over a broad range 500−595 nm, see Fig. 13(b) ‡ . At the maximum, the cavity finesse amounts to F ≈ 260 000. To realise mirror separations in the micrometer range, the curved surface of one of the cavity mirrors is downsized in an in-house grinding process to ∼1 mm diameter and equipped with prisms, see Fig. 13(b). The latter enables optical pumping of the dye reservoir under the above mentioned angle, which together with the appropriate polarisation maximises mirror transmission to approximately 80%. By adjusting the lens L1 shown in Fig. 13 we control both pump spot position and diameter d in the cavity plane, in order to initially excite the dye medium in a spatially homogeneous (d ∼ 500 µm) or localised (∼ 20 µm) way. For comparison, the spatial extent of the ground mode is d 0 ≈ 15 µm, whereas the thermal cloud covers a region of a few hundred µm [55]. For a variation of the condensate wavelength, the cavity length can be piezo-tuned over a total length of 25 µm. This allows us to actively stabilise the condensate wavelength with an accuracy δλ 0.2 nm at 10 Hz bandwidth, which compensates for long-term thermal or mechanical drifts. As dye materials we use Rhodamine 6G and Perylene red solutions of concentrations between 0.1 and 5 mmol/l, see the spectra in Fig. 14 andTab. 3 for an overview of relevant properties. We expect the thermalisation time to be close to the reabsorption time and use dye concentrations of 0.1 mmol/l to perform our time-resolved studies of the thermalisation dynamics. To initialise the dye medium in a time-resolved way a mode-locked Nd:YAG pulse laser (EKSPLA PL2201) near 532 nm with 47 µJ pulse energy and 15 ps pulse length at 100 Hz repetition rate is at our disposal. Both its spatial and temporal intensity profile of the pump beam pulse are gaussian. The laser system acts as the clock source for the experimental setup with electronic trigger noise around 100 ps. To obtain picosecond temporal resolution, we must therefore simultaneously detect the pump pulse and correct for its temporal jitter, which is achieved by directing part of the laser emission through a variable delay path onto the streak camera entrance slit. Subsequent to a pump pulse, the divergent microcavity emission is collimated by a 10× long-workingdistance objective (Mitutoyo M-Plan Apo 10×) and split into two beams. One part of the light is directed onto a diffraction grating (600 rules/mm), and the spectrally dispersed light is focussed on the streak camera entrance slit with a width of 1.5 cm and 30 µm height. In the second optical path, a telescope images the photon gas onto a dove prism (Thorlabs PS992M-A), which rotates the spatial (x, y) coordinates around the optical axis to align the emission with the entrance slit. The streak camera (Hamamatsu C10910) offers the time-resolved investigation in windows of {50; 20; 10; 5; 2; 1; 0.5; 0.2; 0.1}ns with a temporal resolution of 1% of the time range at 1% detection efficiency. The data acquisition for all measurements is performed in a photon counting mode. Spectral thermalisation dynamics First, we focus on the spectral thermalisation dynamics of the photon gas. For this, we realise different coupling strengths to the molecular heat bath and different loss rates due to mirror transmission by variation of the ‡ Obtained from cavity-ring-down measurements using a dye laser tuned to 560−605 nm [117]. cutoff wavelength λ c = {601; 585; 577; 571} nm. Additionally, we control the reabsorption by using different dye concentrations ρ = {0.1; 1} mmol/l (Rhodamine 6G). Figure 15 gives the measured spectral profiles of absorption, emission and loss rates. In the shown wavelength range, the fluorescence is approximately constant, whereas the absorption rate exhibits an exponential decay with increasing wavelength; their relative scaling confirms the validity of the Kennard-Stepanov ratio for the used dye, see (2). In contrast, the photon loss by mirror transmission increases with λ, suggesting incomplete thermalisation for λ c ≥ 580 nm. For λ c 580 nm, however, we expect the photon gas to acquire a thermal state within its cavity lifetime. The starting point for the measurement is a spatially homogeneous excitation of the dye medium using a broad pump beam (2w 0 = 500 µm), which minimises any gradients in the excitation level of the medium at t = 0, realising well-defined initial conditions in chemical equilibrium. Figure 16(a) gives line-normalised, false-colour streak camera traces showing the evolution of the spectral mode occupation. Here, we define t = 0 as the time when the first fluorescence photons are detected, see N/N c (t) in Fig. 16(d). From left to right, we successively increment λ c to gradually decouple the photon gas from the heat bath. All spectral distributions are weighted with the spectral mirror transmission coefficient. Individual excited modes, which are spaced by 42 pm (Ω/2π = 37 GHz), are not resolved due to limited spectral resolution of the diffraction grating of 1 nm ‡ . The recorded data span a total spectral range ∆λ = 25 nm (∆E = 3.5k B T ), which is expected to contain the following fraction of photons:N exp. = 3.5kBT 0 2(u/hΩ + 1) e (u−µ)/kBT − 1 du ≈ 0.93N(59) Our experimental data thus provides reliable information about the degree of thermalisation of the photon gas. In all measurements, we choose the laser power to be such that a macroscopic ground state occupation emerges at the end of the detection window, which allows us to calibrate the photon number at arbitrary times N (t) with respect to its asymptotic value, i.e. the critical photon number N (t → ∞) ≡ N c ≈ 90 000. Therefore, we compare our spectral data with thermal equilibrium Bose-Einstein distributions at 300 K (solid lines) with a chemical potential that satisfies the total photon number. For weak dye reabsorption and large cavity losses, λ c = 601 nm (Fig. 16, left), the spectral wing (hatched) deviates from its equivalent in equilibrium for all observed times. The photon gas fails to thermalise during its microcavity lifetime. However, as the absorptive coupling rate to the molecule bath is enhanced, λ c ≤ 585 nm, (Fig. 16, columns 2 to 5), we observe a thermalisation process that redistributes the photon energies, transforming the out-of-equilibrium distribution to a room temperature spectrum. The characteristic thermalisation time τ th can be quantified by the spectral slope ∂n(λ, t)/∂λ, as illustrated in Fig. 16(c, circles), which in the presence of thermalisation converges to the equilibrium spectral slope (solid line). We observe τ th = {415; 140; 65; 40} ps defined as the time when the relative deviation between measured and equilibrium spectral slope is less then 1%. Figure 16(d) shows the temporal increase of the total cavity emission, revealing the gradual establishment of chemical equilibrium between photons and dye molecules. Notably, the time scales for either thermal or chemical equilibration differ (Section 3). Even if the total photon is non-stationary, the spectral profile of the photon gas present in the microcavity can already be thermally distributed. In Fig. 17 we plot the measured thermalisation times as a function of the free absorption time in the medium (ρσ(λ)c) −1 , which follows a linear ‡ A high-resolution spectrum is shown in Fig. 22. scaling τ th = 0.37(5) · (ρσ(λ)c) −1 . Indeed, the photon gas equilibrates due to an energy exchange with a heat bath at a rate that can be tuned via the reabsorption. If the thermalisation time exceeds the photon lifetime in the cavity, τ th > τ res ≈ 500 ps, the photons constitute an out-of-equilibrium ensemble. Bose-Einstein condensation dynamics We turn our attention to the temporal photon dynamics subject to (i) spatially inhomogeneous and (ii) strongly inverting pump excitation of the dye medium. As before, we investigate the dynamics for different coupling rates to the molecular heat bath ρσ(λ)c and resonator losses Γ(λ), realised by varying the cutoff wavelength. In the out-of-equilibrium regime, Γ(λ) > ρσ(λ)c, the nonequilibrium state of the dye medium manifests itself in transient multimode laser operation. When coupling the photons to a heat bath, Γ(λ) < ρσ(λ)c paves the way for the photon gas to thermal equilibrium and give rise to the emergence of BEC. Figure 18(a) indicates the experimental sequence to realise initial conditions far from equilibrium. We focus the pump beam to a diameter 2w 0 = 80 µm and position it at (x = 150 µm, y = 0) transversally displaced from the optical axis in the microcavity plane. As a result, the spatially inhomogeneous density ρ ↑ (x) decays by emission of (initially) spontaneous photons into excited transverse cavity modes that overlap most with the pumped region. These eigenstates of the harmonic oscillator potential are at higher energies (lower wavelengths) than the transverse ground statehω c (λ c ). Due to the sub-nanosecond time scales of the radiative processes, the comparatively slow effect of spatial diffusion of molecules can be safely neglected [124]. Following the initial photon emission, the photon gas kinetics depends critically on both the dye reabsorption and the cavity loss rates. Reproduced with permission from [41]. Copyright 2015 by the American Physical Society. The first data set in Fig. 18(b) shows the spectral photon evolution in the weakly reabsorbing regime near λ c = 603 nm. The streak camera time traces have been line-normalised to clarify the spectral redistribution of the photons. Here, the reabsorption time (ρσ(λ c )c) −1 = 5.8 ns (corresponding to τ th = 2.3 ns) exceeds by far the average photon storage time in the cavity τ res (λ c ) = 18 ps, such that no equilibrium distribution emerges. Instead, the optical feedback onto the inverted active medium causes stimulated amplification of the light field in the excited modes around λ max = 595.7 nm after approximately 100 ps and the maximum of the emission is maintained throughout the entire detection window. At r = 150 µm the resonant wavelength deviates from the cutoff wavelength by ∆λ(r) = λ c − λ(r) = 2nr 2 /qR = 8 nm, which agrees with the observed value of 7.3 nm. In further measurements, see Fig. 18(b), we successively enhance the coupling to the heat bath by reducing λ c . Accordingly, we observe a more and more accelerated spectral redistribution of the light towards an equilibrium distribution. This is a consequence of fast reabsorption processes, which chemically equilibrate any gradients in the density of the ground and excited state molecules. Due to the harmonic trapping potential, this lightinduced diffusion is directed towards transverse modes with lower energies than the modes overlapping with the pump beam region. Strikingly, the ground state becomes macroscopically occupied for data with shorter cutoff wavelength, and for λ c = 574 nm and 567 nm a BEC with thermally occupied excited states forms. In the case of λ c = 567 nm, the rapid thermalisation prevents the detection of any non-equilibrium emission at the given temporal resolution. Using a spatially-selective photon injection technique to prepare a photon gas far from equilibrium, our measurements have demonstrated that a high-density (critical) photon gas thermalises to a Bose-Einstein condensate provided that the coupling to the heat bath is sufficiently strong. In the opposite limit, the high-density photon gas resembles an out-of-equilibrium state similar to a multimode laser. In contrast to the homogeneously pumped protocol (Section 6.2), the photon thermalisation dynamics is not universal but depends crucially on the initial conditions of the pumped dye medium, in excellent agreement with our numerical simulations, see Section 6.5. Spatial photon kinetics We focus on the spatial condensation dynamics subsequent to an inhomogeneous inversion of the dye medium. For this, a tightly focussed pump beam (2w 0 27 µm) irradiates the dye microcavity spatially displaced by 50 µm from the position of the trap minimum. To analyse the spatial intensity distribution, a real image of the cavity plane is projected onto the streak camera. Figure 19(b) shows a typical CCD camera image of the average cavity emission. Besides the emission from the trap centre (x = 0, y = 0), two bright spots are visible: the first one near the pumping region at (−50µm, 0), the second one at (50µm, 0), i.e. mirrored respectively to the trap centre. A time-resolved measurement (Fig. 19, line-normalised) of the intensity distribution along the x-axis yields an explanation for the centro-symmetric emission: following the inhomogeneous dye excitation an optical wave packet forms, which oscillates in the harmonic potential with reversal points that determine the observed emission spots. The observed oscillation period T = 27 ps shows excellent agreement with the expected inverse trap frequency 2π/Ω (37.1 GHz) −1 (seeTab. 1). Moreover, the wave packet emerges within only a few picoseconds. As this is considerably faster than the spontaneous decay time of the Rhodamine molecules (4 ns), the dynamics are driven by stimulated processes. The wave packet dynamics can be understood as a coherent superposition of adjacent transverse eigenstates spaced byhΩ, i.e. with a fixed relative phase, in close analogy to a mode-locked laser with an extremely high repetition rate. Oscillator modes that exhibit their maximum probability in proximity to the pumped region, experience maximum gain. Classically speaking, the velocity of the wave packet is minimised at the reversal points of the oscillation, maximising here the photon leakage rate out of the resonator. Quantum mechanically, this can be interpreted as constructive interference between multiple harmonic oscillator wave functions. In the limit of weak reabsorption and large cavity losses, see Fig. 19(a, left) for λ c = 596 nm, the photon kinetics is determined by the highly-excited oscillating out-of-equilibrium state throughout the measurement time of 250 ps. The visible residual initial population at small times is attributed to overlap of the pump beam with the ground mode at x = 0, which however quickly decays. The situation drastically changes, as the thermal contact to the heat bath is established by lowering λ c = 581 nm and 571 nm (Fig. 19(a), middle & right). During its oscillation, the wave packet traverses the enclosed dye volume, effectively equilibrating the initially inhomogeneous excitation level ρ ↑ /ρ ↓ (x) by multiple photon reabsorption events. Figure 20 shows corresponding numerical simulations. With advancing times, this effects a dynamical redistribution from the laser-like wave packet to a BEC. The damping of the coherent oscillations and the emergence of the macroscopic ground state in the presence of a thermal bath is shown in Fig. 19(c). Qualitatively, the measured photon kinetics is in good agreement with results from numerical simulations. To conclude, our experimental study demonstrates that a thermal state of the photon gas in the dye-filled microcavity is imprinted by a molecular heat bath. In particular, we find that the efficiency of the thermal contact, i.e. the thermalisation rate, can be tuned systematically by the optical density of the dye solution. With regard to the canonical and grand-canonical statistical ensemble, the temperature of the dye solution actually becomes an external parameter for the photon gas. Thermalisation induced by absorption and emission is a necessary prerequisite for the emergence of a photon BEC. Numerical simulations To gain deeper insight into the crossover from transient laser operation (critical out-of-equilibrium gas) to BEC (critical equilibrium gas), we perform numerical simulations of the photon dynamics in the microcavity [41]. Our phenomenological model relies on semi-classical one-dimensional rate equations and incorporates both the coupling of the photons to the optically active dye medium as well as their oscillatory movement in the harmonic trap. Independently, the results have been confirmed using a master equation model including coherences between photon modes [71]. Our approach is based on the equation of motion for the photon density: n i =B i 21ρ↑ (n i + ε i ) − B i 12ρ↓ + Γ i n i − v i ∂ ∂xn i(60) Here,n i =n i (x, t) is the photon number density in the i-th mode at position x and time t (averaged over many realisations), the densities of molecules in ground and electronically excited stateρ ↓,↑ =ρ ↓,↑ (x, t), and the rate coefficients for absorption and emissionB i 12,21 =B 12,21 (ω i ) at the photon angular frequency ω i . Furthermore, ε i = ε i (x) denotes the density of a single photon in the i-the mode, Γ i = Γ(ω i ) the cavity loss rate and v i = v i (x) the photon velocity field, which will be discussed in the following. Assuming a conserved excitation number X = n + M ↑ , one finds − ∂ ∂tM ↓ = ∂ ∂tM ↑ = P − i ∂ ∂tn i ,(61) where P = P (x, t) denotes the pump beam excitation. Heuristically, we consider a non-orthogonal set of optical modes consisting of coherent states \|α i with amplitudes \|α i \| = u i /hΩ, u i = i ·hΩ and the trap frequency Ω. The mode energy spectrum corresponds to the eigenenergies of the harmonic oscillator potential. In contrast to stationary eigenstates, coherent states allow us to model the oscillation of particles or wave packets in the trap. The normalised photon density ε i (x) results from a temporal average ε i (x) = 1 T T 0 \| x\|α i (t) \| 2 dt(62) over an oscillation period T = 2π/Ω. By comparing the probability to find a particle within dx, ε i (x)dx, with the temporal portion of a half-period that the particle is present in this interval, dt/(T /2), we define the photon velocity field v i (x) = dx/dt = ±Ω/πε i (x),(63) where the sign changes after each half-period. Neglecting losses, Γ i = 0, the numerical results demonstrate that the model reproduces asymptotically the analytic Bose-Einstein distributions and the critical particle number, see Fig. 20(b)). In analogy to the experiments described above, we simulate the photon thermalisation kinetics for initial out-of-equilibrium conditions realised by pumping the molecular medium with a Gaussian laser pulse P ( x, t) ∝ exp[−(x − x 0 ) 2 /2σ 2 x − (t − t 0 ) 2 /2σ 2 t ] with duration σ t = 1.5 ps and waist σ x = 7.5 µm. The pump pulse is positioned at x 0 = −30 µm, where it locally excites molecules within a few picoseconds, as indicated in Fig. 20(a). At this point, the chemical potentials of the photons and ground and excited state dye molecules exhibit strong gradients as visible in the spatially inhomogeneous dye excitation level. The cutoff wavelength λ c = 570 nm is chosen such that the photon gas couples efficiently to the molecules ‡ . Subsequent to the initialisation pulse, the simulations reveal the emergence of a high photon density in the pumped region, which reaches its maximum after only a few picoseconds. Owing to the trapping potential these photons are accelerated as a wave packet towards the trap minimum (x = 0), see Fig. 20(b). During their oscillation the photons are quickly reabsorbed by the enclosed dye medium, which results in a homogeneous density of excited molecules within the region traversed by the wave packet after nearly half an oscillation period (T /2 = 5.4 ps) as visible in Fig. 20(a)). The homogeneity of (M ↑ /M ↓ )(x) is a prerequisite for chemical equilibrium among photons and molecules and the existence of a global chemical potential for the photon gas. Indeed, we find that the medium acquires a homogeneous state soon after the pump excitation, whereas the photon gas is still characterised by a non-thermal spectral distribution, see Fig. 20(c). It takes additional 250 ps until the laser-like wave packet has vanished and the photon energies are Bose-Einstein distributed with a macroscopic occupation of the ground mode. Number statistics of condensed light We describe measurements of the photon number statistics and second-order correlations of a photon BEC coupled to different-sized particle reservoirs. Our experiment gives access to canonical and grand-canonical statistical ensemble conditions, which are hallmarked by their particle number fluctuations: for small reservoirs (canonical), the photon statistics is Poissonian with small fluctuations, δn/n = 1/ √n ≈ 0 (forn 1), whereas large reservoirs (grand-canonical) support unusually large fluctuations of the condensate population, δn/n = 1. Figure 21 outlines the used experimental scheme. In contrast to the measurements of the thermalisation dynamics, a continuous pump and detection system is utilised. For all measurements, the microcavity is operated at q = 8 and filled with either Rhodamine 6G (ethylene glycol) or Perylene red (inviscid paraffin oil) solutions at varying concentrations. The dye medium is pumped by a frequency-doubled Nd:YAG laser (Coherent Verdi V8) near 532 nm, whose output power of up to 8 W is acousto-optically modulated (AOM) into 200 ns pulses at 200 Hz repetition rate, in order to reduce excitation of long-lived dye triplet states and to maintain condensate number constant throughout the pulse (Fig. 21(b), top). For the latter, the rf-signal driving the AOM is mixed with a temporally increasing voltage from a function generator (Tektronix AFG3252). Additionally, a voltage-controlled attenuator actively stabilises the condensate power (10 Hz bandwidth), which is separately detected by a photomultipler. A f L1 = 400 mm focal length lens focuses the pump beam to a diameter of 2w 0 150 µm into the microcavity plane to generate a photon gas. Here, the pump power controls the excitation level of the dye, as well as the chemical potential and the total number (and condensate fraction) of the photon gas. Any loss from the dye-microcavity-system is compensated by maintaining the pumping throughout the pulse. Experimental scheme To determine the condensate fractionn 0 /N , we measure average photon spectra, see Fig. 22(a), in a 4fspectrometer equipped with two diffraction gratings (2400 rules/mm) and two lenses with f = 100 mm. A motion-controlled slit placed in the 2f -Fourier-plane performs a wavelength selection of the multimode light, which is detected using a photomultiplier (Hamamatsu H10721-210). Although its spectral resolution ∆λ = 0.5 nm precludes the measurement of individual transverse cavity modes spaced by ∆λ 41 pm, we confirm the solitary macroscopic occupation of the ground state with a double monochromator (LTB Demon) with 6 pm resolution, see Figs. 22(a) and 22(c). The photon correlations of the BEC are detected in a Hanbury Brown-Twiss interferometer, while for the direct observation of the time-resolved fluctuations and photon statistics a photomultiplier is at our disposal, see Fig. 21. To measure the second-order correlations only of the condensate mode, the divergent cavity emission is first Fourier-filtered with a 5 mm iris in the far field approximately 850 mm behind the cavity ‡ . The aperture acts as a transverse momentum filter to suppress contributions from excited modes: From the zeropoint energy in the harmonic trappinghΩ, we can estimate the momentum uncertainty of the ground mode ∆k r = 2m ph Ω/h 1.86 × 10 5 m −1 , which is much smaller than the longitudinal wave vector component k z (0) = qπ/D 0 1.6 × 10 7 m −1 . Taking into account the quartz-air cavity interface (ñ 0,Quartz 1.46), the corresponding divergence angle Θ = 0.95 • leads to a condensate diameter ∼1.4 cm at the momentum filter. Most of light in the first excited eigenstate (2hΩ, diameter ∼2.0 cm) is expected to be blocked. After lifting the two-fold polarisation degeneracy of the photons, the transmitted light is equally split and directed onto two single-photon detectors (MPD PD5CTC, temporal resolution ∆t 50 ps, dead time τ PD 79 ns), which are connected to an electronic correlation system (PicoQuant PicoHarp 300) that records and correlates photon detection events at times t 1,2 with a resolution 60 ps. To avoid errors during the coincidence measurement caused by the dead time of the system τ PicoHarp 90 ns), the condensate light is sufficiently attenuated to provide photon count rates around ∼0.5 photons/pulse (2.5 × 10 6 photons/s) at each detector. Evaluation of the time histograms yields the second-order correlation function for the BEC g (2) (t 1 , t 2 ) = n 0 (t 1 )n 0 (t 2 ) n 0 (t 1 ) n 0 (t 2 ) ,(64) where ... denotes a temporal average, see Fig. 21(b) for a typical data set. At t 1 = t 2 , we find significant photon bunching, g (2) (t 1 , t 1 ) ≈ 1.7 (yellow diagonal), while for large time delays the photons are uncorrelated, g (2) (t 1 , t 2 ) ≈ 1.0. Due to the nearly constant average photon number during the operation time ( Fig. 21(b), top), the second-order correlations depend only on the relative time delay τ = t 2 − t 1 , and we hereafter only refer to the time-averaged correlation function g (2) (τ ) = g (2) (t 1 , t 2 ) t2−t1=τ . Moreover, we monitor the time evolution of the condensate intensity in the same optical path, see Fig. 21 (top), relying on a photomultiplier (Hamamatsu H9305-01, ∆t 1.4 ns, quantum efficiency ≈10%) and a fast oscilloscope (Lecroy DDA 5005A, 5 GHz bandwidth). This allows us to resolve the number fluctuations, which occur on time scales around 2 ns; examples are given in Fig. 21(c). From the intensity traces I 0 (t) we can equally reconstruct the second-order correlation function g (2) (τ ) = I 0 (t + τ )I 0 (t) t I 0 (t) t I 0 (t + τ ) t ,(65) where ... t = (T − τ ) −1 T −τ 0 (...)dt denotes the temporal average of the pulse of duration T . We note, that despite consistent results for g (2) (0), the Hanbury Brown-Twiss interferometer is considered as the more reliable detection scheme for our purposes due to its high temporal resolution. Figure 22. (a) The measured spectra for increasing condensate fractions (circles) are well described by 300 K Bose-Einstein distributions (solid lines). Inset: (Linear) high-resolution spectrum demonstrating the macroscopic occupation of the ground state only. All spectra have been vertically shifted for clarity. (b) Second-order correlation functions g (2) (τ ) (x−z) exhibit photon bunching at short time delays τ . (λc = 590 nm,h∆ = −6.7kBT , ρ = 1 mmol/l, Rhodamine 6G) (c) Mode-resolved spectrum of a Bose-Einstein condensed photon gas measured using a double monochromator. Reproduced with permission from [40]. Copyright 2014 by the American Physical Society. Time-resolved photon correlations In a first step, we study the number correlations of different-sized photon BECs coupled to a particle reservoir of constant size by fixing the dye concentration ρ = 1 mmol/l (Rhodamine 6G) and the dye-cavity detuninḡ h∆ = hc(λ −1 c − λ −1 zpl ) = −6.7k B T (λ c = 590 nm). To confirm the single-mode property of the condensate, we show corresponding spectra ( Fig. 22(a), inset) with a 9 pm-resolution which is below the transverse mode spacing ∆λ = 41 pm. By measuring the entire cavity emission the full periodic mode structure is revealed, see Fig. 22(c). The second-order correlation functions g (2) (τ ) shown in Fig. 22(b) exhibit zero-delay autocorrelations g (2) (0) x−{ = {1.64(2); 1.30(2); 1.15(2); 1.01(1)} followed by an exponential decay to g (2) (τ ) 1 at larger time delays (dashed lines). According to (50), we fit g (2) (τ ) = 1 + [g (2) (0) − 1] exp(−τ /τ (2) c,exp ) to the data sets x−z and obtain τ (2) (0) 1, such that we cannot determine the correlation time. Our observations reveal strikingly: above N c , the number correlations do not rapidly drop to g (2) (0) = 1 as one would anticipate for a system with strictly conserved particle number [7,38]. Indeed, the observed behaviour provides a first evidence for grand-canonical particle exchange with an effective reservoir. The bunching amplitude g (2) (0) > 1, however, persists only up to a specific condensate fraction, where grand-canonical conditions cease to be applicable: the finite-size reservoir saturates and canonical ensemble conditions start to prevail in the system. According to δn 0 /n 0 = g (2) (0) − 1, the zero-delay autocorrelation g (2) (0) is directly associated to the relative condensate fluctuations. For the data shown in Fig. 22(b), this gives δn 0 /n 0 = {80(1); 55(2); 39(3); 10(5)}%. Grand-canonical condensate correlations We systematically demonstrate the genuine grand-canonical nature of the dye-photon-system in the Bose-Einstein condensed phase by engineering different-sized reservoirs. According to (53), the effective reservoir size is increased for high dye concentration and reduced dye-cavity-detunings. Figure 23 shows zero-delay autocorrelations g (2) (0) and the fluctuation level, respectively, as a function ofn 0 /N for five different combinations of dye concentration and detuning (R1-R3: Rhodamine 6G; R4-R5: PDI red). The main advantage of PDI red is the ability to implement small (absolute) dye-cavity-detunings T/T c (N) g (2) (0) h∆ > −2.5k B T with high reabsorption rates in a spectral region (585−605 nm), where the mirrors transmit a sufficient amount of light to be measured. In order to quantify the effective reservoir size (relative to R1), we introduce ε = M eff,Ri M eff,R1 = ρ Ri ρ R1 × 1 + cosh (h∆ R1 /k B T ) 1 + cosh (h∆ Ri /k B T ) ,(66) see the table in Fig. 23. For the case of the smallest reservoir (R1) the number fluctuations are quickly damped as the photons undergo BEC. Upon increasing the effective reservoir size (R1→R5), we observe that the region with statistical fluctuations can be systematically extended to larger condensate fractions. For the largest implemented reservoir (R5), we find g (2) (0) 1.2 atn 0 /N 60%. At this point, the photon condensate performs number fluctuations δn 0 /n 0 = (g (2) (0) − 1) 1/2 45%, although its occupationn 0 ≈ 144 000 is similar to the total numberN ≈ 240 000. Our findings provide strong evidence for the photon statistics to be controlled by grand-canonical particle exchange [9,33,38]. The experimental results are recovered by our theoretical model (solid lines in Fig. 23), except for condensate fractions below 5%. This is attributed to imperfect mode filtering that leads to an effective averaging of uncorrelated photons from a few equally populated transverse modes (atN N c ) and suppresses the bunching amplitude. If the ground state contribution dominates (n 0 /N ≥ 5%), the effect becomes negligible. Furthermore, the largest detectable autocorrelation value is clamped at g (2) (0) 1.6−1.7. Both issues can be resolved when the correlations are measured with a streak camera system [125]. To fit our data with the theory curves, the molecule number M is treated as a free parameter and good agreement is obtained when we choose 10 9 −10 10 molecules, see the caption of Fig. 23. The large M -values suggest that not only molecules located in the ground mode volume (≈ 10 8 for ρ = 1 mmol/l) contribute to the effective reservoir. A possible explanation is the residual overlap between the excited TEM mn modes and the TEM 00 ground mode that couples molecules in both volumes by absorption and emission of "secondary" photons, effectively increasing the reservoir size for the BEC. Alternatively, a modification of the autocorrelations could also be caused by photon-photon interactions [126]. To this date, the role of interactions and the origin of photon nonlinearities in the optical condensate have not been fully resolved. Previous work has identified thermal lensing to cause effective (non-local) photonphoton-interactions associated with a dimensionless interaction parameterg 10 −5 −10 −2 [26,61,66,67,127,128]. Promising candidates for the implementation of genuine quantum nonlinearities include e.g. polaritons of strongly interacting atomic Rydberg states [129,130] or coupled cavity arrays [131]. In combination with these concepts photon BEC holds prospects for the realisation of strongly correlated many-body states of light. Intensity fluctuations & photon statistics We have seen that the second-order correlation time (τ (2) c 2 ns) of the Bose-Einstein condensed ground state is sufficiently slow to directly monitor the temporal number evolution with a fast photomultiplier. Figure 24(a) shows the time evolution of the (normalised) photon number n 0 (t)/n 0 (t) for a fixed reservoir size with parameters as in Fig. 23. Close to the condensation threshold, the BEC exhibits large number fluctuations, which are gradually damped out as the condensate fraction is increased. By evaluating histograms of roughly 50 traces per condensate fraction, we reconstruct the underlying photon statistics P n , i.e. the probability to find n photons in the condensate, see Fig. 24(b). As the fluctuations are reduced, the distributions reveal a crossover from exponentially decaying Bose-Einstein towards Poissonian statistics, with a width δn 0 that measures the relative degree of fluctuations δn 0 /n 0 . Our results are in excellent agreement with theory curves (solid lines) from (30), confirming the predicted crossover from grand-canonical to canonical statistical conditions (Section 4). Phase coherence of the condensate In the presence of large reservoirs, even strongly-occupied BECs that contain thousands of photons on average display a finite probability P 0 > 0 to produce states without a single photon. Naturally, the question arises: how do such statistical (amplitude) fluctuations affect the temporal phase stability of the BEC? In comparison, the Poissonian statistics in the limit of small reservoirs causes the zero-photon probability to vanish P 0 = 0, such that -despite residual phase diffusion [74,115,116,132] -a well-defined phase is expected. Similar observations with (micro)canonical atomic BECs prompt the emergence of phase coherence for the condensate wave function [14,16,17,20]. In the last section of this Tutorial, we describe an experimental measurement of the temporal phase coherence for a BEC of light. Experimental scheme To study the phase evolution, we rely on time-resolved heterodyne interference signals between the condensate emission superimposed with a dye laser acting as a phase reference, see Fig. 25 [74]. From a separate detection of the intensity of the condensate in the interferometer (blocked dye laser), we obtain the degree of second-order coherence g (2) (0) and the correlation time τ (2) c . The experiments are performed for longitudinal wave number q = 7 in the microcavity, which is filled with a Rhodamine 6G solution (ρ = 3 mmol/l). The microcavity is pumped with continuous laser light, which is here chopped into 600 ns pulses at 40 Hz repetition rate by an AOM. As a local oscillator for the heterodyne interferometry we use a cw dye laser (Sirah Matisse), which offers a tuneable emission between λ L = 560-605 nm. Analog to the condensate operation cycle, the dye laser is acoustooptically chopped into 800 ns pulses at the same repetition rate, while the zeroth diffraction order allows to measure λ L with a resolution of approximately 10 pm, see Fig. 25(c). The relatively long pulse duration is required in order to observe sufficiently long beatings between the condensate and dye laser emission, as will be elaborated in more detail later in this section. To obtain high-contrast interference signals, we use half-wave plates to project the polarisation axes of the momentum-filtered photon condensate and dye laser on top of each other, and combine both beams after passing a non-polarising beamsplitter (90% transmission) in a single mode fiber (Thorlabs P1-488PM) ‡ . The temporal interference traces are detected by a fast photomultiplier tube (Hamamatsu H10721-20, ∆t 0.57 ns) and recorded with a digital oscilloscope (Tektronix DPO7000, ∆ν 3.5 GHz). A typical time-resolved interference signal, where condensate and dye laser wavelength have been matched, is shown in Fig. 25(b). The superposition of Bose-Einstein condensed light field, ψ c (t) = n c (t) exp{i[ω c (t)t + ϕ(t)]}, and dye laser field, ψ L (t) = n L (t) exp(iω L t), gives a beat signal \|ψ c (t) + ψ L (t)\| 2 = n c (t) + n L (t) + 2 n c (t)n L (t) × cos {[ω c (t) − ω L ] t + ϕ(t)} ,(67) ‡ The optical phase is commonly retrieved in a balanced heterodyne detection scheme [133], by subtracting the interference signals at both output ports of a symmetric (50:50) beamsplitter exploiting their π-phase difference. For low condensate powers, however, the usage of an asymmetric (90:10) beamsplitter turned out to enhance the signal-to-noise ratio of the observed beating signals. where ϕ(t) denotes the time-dependent condensate phase. Notably, we here have explicitly maintained the time dependence of the condensate frequency ω c (t), for reasons that will be discussed in the following. The thermodynamic state of the photon gas is obtained from a spectroscopic measurement of the energy distributions with a 4f -spectrometer (Section 7). The spectra cover a wavelength (energy) range of 30nm ( 4k B T ) and provide the ground state populationn 0 and total photon numberN , being calibrated with reference spectra at N c 79 000. Moreover, a part of the transmitted cavity emission is injected into the highresolution double monochromator together with the aforementioned dye laser to monitor the relative spectral position of condensate and dye laser wavelength, see Fig. 25(c). At the smallest achievable cavity lengths D 0 ≈ 1.4µm, the curved mirrors are firmly pressed together, effectively reducing residual mechanical resonator drifts and vibrations. Under these conditions, minute piezo-tuning of the cavity length allows us to actively match the condensate with the dye laser wavelength with a spectral precision ∆λ ≈ 10 pm. At λ = 580 nm, the mirror separation can thus be tuned with an accuracy of ∆D 0 = D 0 ∆λ/λ ≈ 24.5 pm. Modulation of the condensate frequency Despite the mechanical stability of the microcavity, we have already seen in Fig. 25 that the measured intensity traces reveal a frequency modulation of the beating signal. Figure 26(a) shows the observed temporal variation of the beat signal for different initial cavity lengths, the latter modifying λ c . For an average condensate wavelength blue-detuned with respect to the dye laser (λ c < λ L , insets of Fig. 26(a)), no beating signal is observed. For reddetuned light (λ c > λ L ) however, the occurring beating signal shows two resonances that exhibit an increased temporal separation as the condensate is further detuned, with the beating frequency in between exceeding the detector bandwidth. The interference data allows us to reconstruct the frequency drift ν c (t) of the condensate emission, which is shown in Fig. 26(b) for various cavity lengths. The frequency drift in Fig. 26(b) has been recorded for temporally equidistant pump pulse excitation (pulse length ∆t = 1.5 µs every T p = 25 ms), leading to beating signals that occur in every subsequent condensate pulse with nearly the same shape (due to mechanical shot-to-shot stability). However, when irradiating the dye-microcavity with a quick sequence of 4 pump pulses (∆t = T p = 600 ns, followed by 100 ms dark time), see Fig. 26(c), only one of the four produced condensate pulses exhibits a beating with the laser, yet with the same characteristic, nearly parabolic "fast" frequency drift observed previously. We attribute the "slow" global frequency drift to modulation of the index of refraction of the dye medium that is caused by effectively heating the solution with the pump laser. The relaxation timescale is approximately 20 ms, similar to timescales of thermal lensing effects in our system [26,58,67]. The fast sloshing of the condensate eigenfrequency during a single pulse may be caused by the steep rising slope of the pump pulse itself, as it occurs in each pulse of the fast sequence scheme shown in Fig. 26(c). Furthermore, the behaviour is observed only for longitudinal wave numbers q ≤ 10 when the dye film in between the cavity mirrors becomes kinematically 2D. Both observations give reason to conclude that the parabolic contribution to the frequency modulation ν c (t) is based on a refractive index change, which originates from an overdamped density oscillation in the dye film. Due to the cavity length stability within a single pulse, D 0 = q λ c (t) 2ñ 0 (t) = q c 2ñ 0 (t)ν c (t) ! = const.(68) Therefore, the density of the dye solution in the ground mode volume reads ρ(t) ∝ñ 0 (t) = q c 2D 0 ν c (t) = q 1 τ rt ν c (t) ,(69) where the (vacuum) resonator round trip time τ rt = 2D 0 /c has been inserted. According to (69), the density scales inversely with the frequency drift from Fig. 26(b), corresponding to a compression of the solvent in the area of the pump beam. Presumably, this could be caused by an initial localised heating and dilution of the medium due to the pump pulse, see the illustrated sequence in Fig. 26(d). The resulting density hole leads to a reflow and densification of ethylene glycol molecules until the medium is finally homogenised. The observed time scale of the overdamped density modulation is consistent with an estimate based on the propagation time of a sound wave through the ground mode area of diameter d 0 ≈ 15 µm, t s = d 0 /v s ≈ 10 −8 s, where v s = 1688 m s −1 is the speed of sound in ethylene glycol at 300 K [134], and it occurs on a considerably shorter time scale than thermal lensing (10 −3 s). Moreover, our interpretation is affirmed by the notion that the pump beam geometry affects the condensate frequency modulation: for a larger pump beam waist, the dynamics becomes slower and the maximum of the compression is postponed to later times, see Fig. 26(d) ‡ . For all subsequently discussed measurements, we use a fixed pumping geometry with a beam diameter 2w 01 = 2λf L1 /πw 0 140 µm (beam waist w 0 = 1 mm and f L1 = 40 cm, see Fig. 25). Phase jump detection algorithm The microcavity frequency drift prohibits a temporally stable resonance condition to be fulfilled between photon condensate and dye laser, making a direct observation of the BEC phase evolution difficult. However, discrete phase jumps of the condensate can be easily unveiled if the recorded chirped interference signals are examined for irregularities in their oscillatory behaviour. For an automated analysis, we develop a phase jump detection algorithm that we benchmark with Monte-Carlo-simulated data (Section 5). Figure 27 shows the simulated (a) intensity and (b) phase evolution of a BEC under grand-canonical statistics, and (c) depicts the corresponding simulated beating signal between the photon BEC and a dye laser. In the first step of the analysis, the analog interference signal is digitalised (red). Subsequently, the procedure evaluates the digital square-signal for irregularities in the (i) width and (ii) central position of adjacent highor low-valued segments. If the irregularities exceed predefined limits, the algorithm flags these points in time (grey shaded). The low-frequency region near the resonance (hatched) is excluded from the detection. The simulated data confirms the operability of the algorithm, as demonstrated by coincidences of grey regions with zero-or one-photon-states in the ground state ( Fig. 27(a), top, dots). It enables the detection of discrete phase rotations between [0.2π, 1.8π]. As the analysis is based on the detection of relative irregularities, the temporal resolution is limited by the beating oscillation period. Figure 28 shows the time evolution of the interference between photon BEC and dye laser for three different cases of photon statistics at a fixed reservoir size, starting from a strongly occupied second-order coherent condensate in Fig. 28(a) towards a strongly fluctuating population in (c). The left column gives the time of the (normalised) condensate number n(t)/n, which is recorded after each interference measurement by blocking the dye laser. From this, the autocorrelation function g (2) (τ ) is computed (middle), implicitly providing a measure of the fluctuation level δn =n g (2) (0) − 1 and the second-order correlation time τ (2) c . In all measurements with a significant bunching amplitude (g (2) (0) > 1), an exponential fit to the autocorrelation data yields τ (2) c ≈ 2 ns. For canonical ensemble conditions with Poissonian number statistics, see Fig. 28(a) with g (2) (0) = 1.01(2), the beating oscillates regularly, which demonstrates the temporal coherence of the BEC throughout 120 ns ‡ . As the condensate fraction is reduced, the reservoir size becomes sufficiently large to realise grand-canonical statistical conditions, which is hallmarked by the occurrence of intensity fluctuations in Figs. 28(b) with g (2) (0) = 1.33 (4) and (c) with g (2) (0) = 1.93 (13), respectively. This is accompanied by a discontinuous phase behaviour manifested in the beating signals, which for increased fluctuations shows a reduction of the time separation between adjacent phase jumps Γ −1 PJ ≈ 21.3 ns in (b) and Γ −1 PJ ≈ 5.3 ns in (c). In the vicinity of the detected phase jumps (grey shaded) a fit yields the magnitude of the imparted phase shift, see the inset of Fig. 28(b). To good approximation, the phase rotation angles are evenly distributed within the detection range [0.2π, 1.8π], as indicated by the histogram in Fig. 28(d). The random distribution gives evidence for the U (1) symmetry of the infinitely phase-degenerate ground state. Physically, this equipartition is attributed to the intrinsic randomness of a spontaneous emission event, which is expected to trigger the emergence of a condensate after a previous fluctuation to low photon numbers. First-and second-order coherence times As previously discussed, Fig. 28 indicates a separation of the dynamics for number and phase fluctuations: while τ (2) c remains nearly constant, the measured values for Γ −1 PJ change by 2 orders of magnitude and seem to depend on the choice of the statistical ensemble and its associated zero-photon-probability P 0 . For a quantitative analysis of the time scale separation, Fig. 29 summarises experimental results of the phase jump rates Γ PJ and inverse second-order correlation times 1/τ (2) c as a function of the average photon number in the condensate for three different-sized particle reservoirs. The phase jump rates (filled symbols) increase strongly for both growing reservoir size as well as decreasing condensate photon number ("system size") based on the here enhanced probability to have a low photon number given the increased fluctuation level (inset), which reduces the phase stability. The rates deduced from the zero-photon-probabilities Γ 0 PJ =B 12 M P 0 (solid lines) show an excellent agreement with the experimental data. This suggests that a drop of the condensate population to zero followed by a spontaneous emission process is physically responsible for the observed phase jumps. Similarly, the inverse second-order correlation times 1/τ (2) c (open symbols) present a good agreement with theory curves (dashed lines) based on (51). ‡ For large waists of the pump beam, the longest recorded time span without phase jumps was 1 µs (300 m coherence length). Within the detection window [0.2π, 1.8π] (red line), the random distribution reflects the U (1) symmetry of the ground state, which is broken upon condensation. (Rhodamine 6G, ρ = 3 mmol/l, λc = 582 nm). Reproduced with permission from [43]. Copyright 2016 by the American Physical Society. For all three configurations, a separation of the time scales for first-and second-order coherence is visible in the statistics crossover region, i.e. nearn = √ M eff . What is its physical origin? On the one hand, spontaneous emission events can cause arbitrary phase fluctuations. However, these matter only when a few photons occupy the ground state with a likelihood given by the photon statistics, which therefore dominates the first-order phase jump dynamics, see (56). On the other hand, the dynamics of particle number fluctuations is subject to absorption and emission rates of photons by the dye medium, according to (48). Although for increased condensate populations (at a fixed reservoir) the relative fluctuations δn/n are reduced, the fluctuation time scale is still controlled by the Einstein coefficients. In fact, even larger condensate populations lead to a reduction of the second-order correlation time, in stark contrast to the increased first-order correlation time. Although our analysis does not account for diffusive contributions to the temporal phase coherence [74], it conveys the unusual properties of Bose-Einstein condensed light: a light source comprised of a single macroscopically occupied emitter that exhibits statistical intensity fluctuations as large as in a thermal source. The relation between first and second-order coherence for thermal emitters, g (2) (τ ) = 1 + \|g (1) (τ )\| 2 , is however expected to hold only in the extreme grand-canonical regime withn ≥ √ M eff [87,135]. Extrapolation to the thermodynamic limit Finally, we discuss the physical significance of statistical number fluctuations and phase coherence for a photon BEC in the thermodynamic limit. For this, we study the phase jump rate for enlarged system sizes. Importantly, we ensure to increase the sizes of both condensaten and effective particle reservoir M eff in a way that conserves the statistical ensemble conditions. Figure 30(a) shows the reservoir-system-ratio √ M eff /n as a function ofn for different values of g (2) (0) obtained from numerical calculations. For a given photon numbern, the reservoir size M eff is adjusted iteratively until the corresponding photon number distribution P n reproduces one of the target values g (2) (0) = {1.10; ...; 1.90}. Subsequently, the procedure is repeated for larger condensate populations to yield further data points at the same fluctuation level. Our numerical results indicate that conserving the the statistical ensemble conditions, i.e. √ M eff /n, is equivalent to a constant zero-delay autocorrelation. This suggests that the phase coherence may be extrapolated towards the thermodynamic limit (n,N , M → ∞, √ M eff /n = const.), provided that one does maintain the fluctuation level g (2) (0). Strictly speaking, an extrapolation also requires the critical temperature T c (N ) ∝ N /R to be constant. This could be achieved by increasing the radius of curvature of the cavity mirrors R → ∞ proportional toN . Experimentally, this compensation is unfeasible with the described Extrapolation of phase coherence for large photon-reservoir-systems at three fixed degrees of fluctuations g (2) (0) (filled symbols). By extrapolatingn → ∞ (dashed), all curves indicate a full suppression of phase jumps in the thermodynamic limit. Black symbols are numerical results for Γ 0 PJ =B 12 M P 0 at corresponding g (2) (0) and M = 4 × 10 9 ,B 12 = 1000 s −1 . (Parameters as in Fig. 29). Reproduced with permission from [43]. Copyright 2016 by the American Physical Society. setup, such that we only comply with the requirements for a fixed statistical ensemble ‡ . Figure 30(b) gives the phase jump rate versus autocorrelation for three reservoirs. For all values of g (2) (0), we observe that photon condensates coupled to the smallest reservoir (h∆ = −7.7k B T ) exhibit shorter coherence times than condensates coupled to the medium-sized (−5.6k B T ) and largest (−2.3k B T ) reservoir. This meets our expectations: for the same level of fluctuations, i.e. the same statistical ensemble, an increased condensate population should reduce the zero-photon-probability P 0 , see 58. From our data, we extract three sets of phase jump rates for selected zero-delay autocorrelations g (2) (0) = 1.59(18), 1.18(9) and 1.02(1), which are shown in Fig. 30(c) versus the inverse condensate population 1/n. All data sets lie in the rangen ≥ √ M eff , for which a separation of Γ PJ and 1/τ (2) c has been observed. A linear extrapolation of the data towards an infinitely large condensate (1/n → 0) is consistent with a full suppression of discrete phase jumps in the thermodynamic limit, in spite of the absence of second-order coherence. Numerical calculations (black symbols) for g (2) (0) = 1.50, 1.18 und 1.05 support this conclusion. The largest realised fluctuation level comes close to the photon statistics crossover, g (2) (0) = π/2, with a zero-photon-probability P 0 0.64n −1 , see (45). Under the assumption that phase jumps occur due to vanishing photon numbers, a fit to the data in Fig. 30(c) yields Γ PJ /B 12 M = 0.51 (14)n −1 reproducing the expected slope within the quoted uncertainty. For lower fluctuation levels, the exact scaling of P 0 withn remains elusive and we therefore compare our data only with numerical results, which similarly demonstrate a linear scaling of Γ with the inverse photon number (P 1.18 0 = 0.13(5)n −1 and P 1.02 0 = 0.06(2)n −1 ). Although the presence of amplitude fluctuations of the condensate wave function n(t) exp(iφ(t)) reduces the degree of first-order coherence, we expect that in the investigated parameter regime discrete phase jumps will be fully suppressed in the thermodynamic limit. ‡ Recently realised photon gases in variable micropotentials might however render a conservation of Tc tractable [67, 68]. Conclusions and Outlook This Tutorial has presented a study of the thermalisation dynamics and temporal coherence properties of a Bose-Einstein condensed photon gas in the grand-canonical statistical ensemble. Key evidences are provided by measurements of (i) the spectral photon dynamics, which demonstrates the thermalisation of the photons due to reabsorptive coupling to a dye heat bath, (ii) the large (grand-canonical) statistical number fluctuations at significant condensate fractions, and (iii) the observed variation of the temporal phase coherence of the condensate wave function. An extrapolation to the thermodynamic limit gives BECs with super-Poissonian number statistics despite suppressed phase jumps. The realisation of BEC in the grand-canonical ensemble has for the first time shed light on the long-discussed grand-canonical fluctuation catastrophe [9,[30][31][32][33][34][35][36]. The observation of extremely large, statistically fluctuating condensate populations demonstrates the physical significance of the grand-canonical ensemble for the Bose-Einstein condensed phase. Moreover, the results provide the fundamental insight that BEC does not strictly imply first-or second-order coherence. For the future, it will be exciting to study phase diffusive contributions to the condensate linewidth, as has been theoretically predicted but remains elusive in any Bose-condensed system to date [74,115,116,132]. A major experimental challenge here depicts the required frequency stability of the photon BEC to observe minute phase drifts over long measurement durations. Moreover, it is expected that in-depth studies of the thermal character of the grand-canonical statistical fluctuations may reveal unusual fluctuation-dissipation-relations in the ideal Bose gas, associated with macroscopic thermodynamic quantities as e.g. a generalised statistical compressibility imposed by the particle reservoir. From a technical point of view, macroscopically occupied, but incoherent photon condensates under grand-canonical conditions could pose interesting novel light sources for speckle-free imaging applications due to their high directional brilliance and (tuneable) low degree of coherence. Further exciting research directions for grand-canonical BECs might be pursued in conjunction with variable potentials for thermalised light and coupled condensates, as has been demonstrated in microstructured optical cavities [67,68]. Phase-stable, macroscopically occupied condensates arranged in a lattice are expected to constitute a realisation of the XY model of 2D interacting spins, that could provide a fruitful platform to address complex optimisation problems [136,137]. In this regard, the phase jumps associated with grand-canonical statistical fluctuations could mimic spin fluctuations at an effective temperature: at sufficiently low "temperatures" one expects the emergence of the BKT phase associated with algebraic long-range spin order [138][139][140][141]. Figure 2 . 2Statistical ensemble and particle number fluctuations. (a) The photon gas couples to the reservoir of electronically excited molecules by reabsorption after a photon lifetime τph. The molecular excitations decay within τ ↑ , establishing chemical equilibrium between the photon gas and the particle reservoir. Simultaneously, multiple collisions of dye and solvent molecules lead to a thermalisation of the rovibronic dye states at room temperature. The light-matter-interaction imprints the thermal equilibrium state onto the photon gas and the molecules can be regarded as a heat bath.(b) Temporal fluctuations of the photon number under grand-canonical ensemble conditions (large reservoir M = 10 9 , top), and damped fluctuations in the canonical ensemble (small reservoir M = 10 5 , bottom). Figure 3 . 3(a) Electronic and rovibrational energy levels in a dye molecule. (b) In the "photon box" absorption-emission-cycles by dye molecules frequently change the configuration of the light field (\|1 → \|2 → ...) to thermalise the photon gas. ; see Fig. 5 for an overview of some experimental signatures of photon BEC. Figure 5 . 5Evidence for BEC of light. (a) Photon spectra for increasing total photon numbersN . ForN ≥ Nc, the excited mode population saturates and the ground state becomes macroscopically occupied. (b) The condensate fraction and (c) the specific heat of the photon gas is derived from the spectral distributions (not shown) at various particle numbers above and below Nc. The discontinuity close to T = Tc(N ) reveals the phase transition. Reproduced with permission from[43,62].Copyright 2014 & 2016 by the American Physical Society & Nature Publishing Group. Figure 6 . 6(a) Spectral thermalisation dynamics for λc = {600; 580; 560} nm. All shown equilibrium spectran(λ) (solid lines) are close to the condensation threshold. The wavelength-dependent dye reabsorption, σ(λ) ∝ exp(−λ), leads to a slower photon number evolution (dashed) in the red (10 −9 s) than in the green spectral region (10 −11 s). (b) The low-wavelength spectral slope (dashed) quantifies the degree of thermalisation, which approaches the equilibrium value (solid) with advancing times. After τth the relative difference between both curves is less than 1%. (Dye concentration 0.1 mmol/l, µ = −0.007kBT ) i 21 (M ↓B i 12 ) −1 . For the equilibrium distribution in ‡ We will denote n (andn) as the (average) photon number in the BEC in Sections 4 und 5. Figure 7 . 7(a) Condensate fractionn/N and photonic fraction of the excitation numbern/X versus reduced temperature T /Tc(N ) for reservoir sizes Mx-\| = {10 8 ; ...; 10 12 } and dye-cavity detuningh∆ = −4.67kBT (adapted from[38]). (b) The autocorrelation function g(2) (0) of the condensate (reservoirs as in (a)) predicts large photon number fluctuations even deep in the condensed phase. Figure 8 8shows numerically calculated photon number distributions to find n photons in the BEC for a fixed reservoir size (M = 10 10 , ∆ = ω c − ω zpl = −2.4k B T /h). By decreasing the reduced temperature from T /T c = 1.0 to 0.4, or vice versa increasing the condensate fraction fromn/N 1% to 80%, one observes a continuous crossover from Bose-Einstein to Poissonian statistics. Figure 8 . 8Crossover from Bose-Einstein to Poissonian statistics for increasingn/N at fixed reservoir size. The probability Pn is plotted versus the photon number normalised with the mean condensate populationn. Up ton/N = 28.3% the maximum of the distribution occurs at nmax = 0. For increasing quantum degeneracy, the distribution shifts to nmax > 0 and the variance is reduced, demonstrating a damping of particle number fluctuations and the emergence of second-order coherence. Inset: Photon number statistics at the crossover Tx = 0.847 Tc (n/N 28.3%). The probability distribution corresponds to a gaussian withn ≈ 27 000 photons. (M = 10 10 ,h∆ = −2.4kBT , polarisation degeneracy neglected; the curves normalised in the n/n-representation for clarity) Figure 9 . 9(a) Effective reservoir size as a function of the dye-cavity-detuning. (b) Inverse second-order correlation time versus condensate numbern for five effective reservoir sizes Meff. The curves exhibits minima atnmin = (Meff) 1/2 , which highlight the crossover point from grand-canonical (n <nmin) to canonical statistics (n >nmin). (B 21 = 10 4 s −1 , M = 10 9 ). Figure 10 . 10(a) Phasor model for different light sources. While spontaneous emission on average does not develop a global phase, stimulated processes in a laser result in a macroscopic phase-stable light field. For the photon BEC, similarly a global phase φ emerges. In the presence of large reservoirs, however, the condensate vanishes due to strong dye-mediated reabsorption and subsequently emerges with a broken symmetry φ + ∆φ. Figure 11 . 11Simulation of condensate number n(t)/n (green) and phase evolution φ(t) (purple) in the presence of a particle reservoir Meff = 2.5 × 10 5 . (a) Under grand-canonical conditions, large number fluctuations occur accompanied by discrete phase jumps at points, when the photon number drops to zero (top circles). (b) For larger condensate sizes (canonical), the fluctuations are damped out. No phase jumps occur as attributed to the vanishing probability to find zero photons for Poissonian statistics. (∆ = 0kBT and M = 10 6 ) Figure 12 . 12Simulated autocorrelation g(2) (0) (top), phase jump rates (bottom, filled symbols) and second-order correlation rates (bottom, open) versusn for two effective reservoir sizes (violet circles and green triangles). The agreement of ΓPJ withB 12 M P 0 (solid line) supports the assumption that phase jumps emerge when the condensate population vanishes. First-order coherence is enhanced as the photon statistics merges from being Bose-Einstein-like to Poissonian, which occurs at the 1Meff, where g (2) (0) π/2 (top) in good agreement with the analytical prediction (dashed line). (B 12 = 1 000 s −1 , M = 10 9 , h∆ = {−2.8; −7.7}kBT ) depends only on the Einstein coefficientB 21 and the number of excited dye molecules M ↑ , and thus 1/τ 0 =B 21 M ↑ . For the steady-state withn 1, see (19), one further obtainŝ B 21 M ↑ B 12 M ↓ . With the typically fulfilled M ↓ M , we have Γ 0 PJ = B 12 M P 0 . Figure 13 . 13(a) Experimental scheme for the time-resolved measurements of the spatial and spectral thermalisation dynamics. The microcavity is pumped with a pulsed laser beam and the emission imaged spatially and spectrally onto a streak-camera system. (b) Picture of the prepared cavity mirrors (left) and spectral mirror reflectivityR[117]. Figure 14 . 14Normalised absorption and emission spectra of Rhodamine 6G and Perylene red with structure formulae. Table 3 . 3Properties of the used dye media Rhodamine 6G (Rh6G) (Radiant Dyes) solved in ethylene glycol and Perylene (PDI) red (Kremer Pigmente) solved in inviscid paraffin oil. The absorption cross section follows from σ = 3.82 × 10 −21 ε (in units of cm 2 ) [83] and the reabsorption time from τ532nm = (ρσ 532nm c) −1 , ρ = 1 mmol/l. Figure 15 . 15Rates for absorptionB12 M , fluorescence and photon loss Γ versus wavelength for Rhodamine 6G dye, 1 mmol/l, and CRD mirrors. The fluorescence (dashed line, normalised to absorption maximum) is approximately constant over the shown spectral range. For λ < 590 nm, the photon dynamics is dominated by the exponential scaling of the reabsorption rate (open circles) that has been fitted with ∝ exp[hc(λ − λzpl)/kBT ] yielding T = 308(14) K (solid line). Here, a thermal equilibrium state is expected to emerge (blue region). Increased losses from mirror transmission (filled circles, dash-dotted line) lead to dissipative dynamics for larger wavelengths (orange region). Figure 16 .Figure 17 . 1617Thermalisation dynamics of the photon gas for increased heat bath coupling. (a) Streak camera traces (line normalised) and (b) extracted spectra at different times along with 300 K-Bose-Einstein distributions (solid lines). (c) By comparing the measured spectral slope (hatched area in (b)) with its equilibrium counterpart (line) the thermalisation time can be quantified. It reduces as the coupling to the heat bath is enhanced (from left to right). (d) The temporal evolution of the total power of the cavity emission (normalised to Nc) indicates the time scale for chemical equilibration between photons and dye molecules. (λc,x-\| = {601; 585; 577; 571; 585} nm, Rhodamine 6G ρx-{ = 0.1 mmol/l, ρ\| = 1.0 mmol/l). Reproduced with permission from[41]. Copyright 2015 by the American Physical Society. Observed thermalisation time τth (circles) versus reabsorption time in the dye solution. For τth < τres the photon gas acquires a thermal state (blue); otherwise it remains an out-of-equilbrium system. Reproduced with permission from[41]. Copyright 2015 by the American Physical Society. Figure 18 . 18Spectral condensation dynamics following an off-centre pump pulse for increasing thermal contact to the heat bath. (a) The pump pulse initialises an inhomogeneous excited molecule density ρ ↑ , which decays by stimulated emission into spatially overlapping modes. (b) Streak camera images (top) with spectral cuts (bottom) shortly after the photon injection (green squares) and at the end of the detection window (red circles). While for weak thermal contact (top, left) no thermalisation is observed, the enhanced reabsorption leads to a partial relaxation (top, middle & right) of the photon emission towards larger wavelengths. Finally, the photon gas fully thermalises to a Bose-Einstein distribution (bottom row). The grey-shaded data set highlights the thermalisation dynamics with cuts in the dynamical transition region. (Rhodamine 6G 0.25 mmol/l, λc = {603; 590; 582; 574; 567}nm, τth = {2 300; 388; 124; 38; 13}ps, cavity lifetime τres = {18; 83; 165; 303; 500}ps). Figure 19 . 19Mode-locked laser operation and BEC of photons. (a) Line-normalised spatial evolution of the cavity emission for different cutoff wavelengths. The pump beam excites an oscillating wave packet in the harmonic potential (top). For weak reabsorption (left), a stable mode-locked laser oscillation occurs (dashed line), where most light leaves the cavity at the reversal points. For stronger reabsorption (middle and right) the photons thermalise and a condensate emerges in the trap centre. (b) CCD camera image of the (average) emission. (c) Temporal variation of the detected relative intensity from (a, right) at the reversal points (left) and in the condensate mode (right). (Rhodamine 6G, ρ = 0.1 mmol/l, λc = {596; 581; 571}nm). Reproduced with permission from [41]. Copyright 2015 by the American Physical Society. Figure 20 . 20Simulation of the thermalisation dynamics for off-centre pulsed excitation. (a) Density of excited molecules ρ ↑ (x) for given times after the pump pulse. The locally excited medium near x 0 = −30 µm induces the formation of a photon wave packet (see (b)), which homogenises ρ ↑ (x) in the course of its oscillation. (b) The temporal evolution of the spatial photon density (left) shows the oscillation of a mode-locked photon wave packet, which is damped out with time. After 250 ps, the photons have been redistributed to the cavity ground state at x = 0. Photon evolution versus photon energy E = (hω − hωc)/hΩ (right), showing the emergence of a BEC withn 0 /N 95%. (Ω/2π = 93 GHz, λc = 570 nm,h∆ −4kBT at T = 300 K,B 12 = 1.3 kHz, ρ ↑ (x) + ρ ↓ (x) = 1.5 · 10 8 µm −1 , Γ i = 0.) Figure 21 . 21(a) Time-resolved measurements of correlations, number statistics and fluctuations of the photon BEC. The microcavity is pumped (quasi-)continuously and part of the cavity emission is spectrally analysed. In the far field, the condensate emission passes several filtering stages and its correlations are detected in a Hanbury Brown-Twiss interferometer. (b) Typical average condensate emission (top) and second-order correlation function g (2) (t 1 , t 2 ). (c) Time evolution of the (normalised) condensate population measured with a single photomultiplier (PMT) for different condensate fractionsn 0 /N . Inset: corresponding autocorrelation functions with τ = t 2 − t 1 . Reproduced with permission from [40]. Copyright 2014 by the American Physical Society. Figure 22 ( 22a) shows spectral distributions in the Bose-Einstein condensed phase hallmarked by the macroscopically occupied ground mode and thermally populated excited states. All condensate fractionsn 0 /N = {4%; 16%; 28%; 58%} and reduced temperatures T /T c = {0.98; 0.92; 0.85; 0.65}, respectively, are obtained from fitting the data with T = 300 K Bose-Einstein distributions. This corresponds to absolute photon numbers n 0 {4; 19; 37; 120} × 10 3 andN {100; 119; 132; 207} × 10 3 . ns. For the largest condensate fraction { the photon bunching vanishes, g Figure 23 . 23Second-order autocorrelations g (2) (0) and relative condensate fluctuations δn 0 /n 0 as a function of the condensate fractionn 0 /N (reduced temperature T /Tc), for different reservoir sizes R1−R5. The increasing effective reservoir size is quantified by ε. For large concentrations ρ and reduced dye-cavity-detunings ∆ (R5), the fluctuations of the ground mode populations persist deep into the condensate phase. The solid lines indicate the prediction from the theoretical model discussed in Section 4. Error bars give statistical uncertainties. (λc = {598; 595; 580; 598; 602}nm fr R1−R5; ρ = {0.1; 1.0; 1.0} mmol/l fr R1−R3 (Rhodamine 6G) und ρ = {0.1; 1.0} mmol/l fr R4,R5 (PDI). For the theory curves, we use M = {5.5(22); 20.5(71); 16.0(57); 2.1(4); 10.8(37)} × 10 9 for R1−R5). Reproduced with permission from [40]. Copyright 2014 by the American Physical Society. Figure 24 . 24(a) Temporal evolution of the normalised condensate population n 0 (t)/n 0 ( 1.4 ns temporal resolution). For increasing condensate fractions (x→\|) a damping of the fluctuations is observed. (b) The photon number distributions (vertically shifted) exhibit a crossover from Bose-Einstein-like to Poissonian statistics in agreement with theory (solid lines), see also Fig. 8. (Parameters as inFig. 22). Reproduced with permission from[40]. Copyright 2014 by the American Physical Society. Figure 25 . 25Heterodyne interferometry to study the phase evolution of the photon condensate. The momentum-filtered ground mode emission (left cavity mirror) is superimposed with a dye laser in a single mode fibre. (b) A typical resulting interference signal recorded with a fast photomultiplier. (c) The emission transmitted through the right cavity mirror is used to monitor the wavelengths of BEC and dye laser, λc and λL, in a double monochromator and measure the condensate fractionn 0 /N . Figure 26 . 26(a) Interference signals upon tuning λc at fixed dye laser wavelength λL by increasing the cavity length. Both a frequency modulation and a relative temporal separation of two resonance crossings (x,y) are observed. (b) Non-resonant intensity trace (top) and relative drift of the condensate frequency ∆νc(t), as reconstructed from (a) (bottom). Vertically shifted curves correspond to increased cavity lengths (from top to bottom), and the hatched area indicates the PMT detection bandwidth 1.75 GHz. (c) Combined intensity (top) and ∆νc(t) (bottom) upon excitationwith a sequence of 4 pulses showing a beating within the last pulse. We deduce a global slow linear frequency drift over several pulses and a parabolic frequency modulation is reproduced within each individual pulse, possibly caused by a (d) temporal density modulation of the dye medium following the pump excitation (hatched). A tight pump beam focus (solid line) leads to an enhanced amplitude and speed of the modulation during the BEC emission (yellow), compared to a weak focusing scenario (dashed line). Figure 27 . 27Phase jump detection benchmark with simulated interference signals. (a) Number evolution of a fluctuating BEC and points in time with zero or one photon in the ground mode (dots, top). (b) Corresponding phase evolution (units of 2π) with a phase jump detection resolution ∆ϕ > 0.2π (red boxes indicate missed events). (c) Beating between photon condensate and reference laser and digitalised signal (red). Irregularities in the periodicity of subsequent square signals selected by our algorithm (grey shaded, see also zoomed time trace) coincide with actual phase jumps in (b). The resonance is excluded from the detection (hatched). (M = 10 6 ,nc = 280 with g h∆/kBT = 0, wavelength drift λc(t) = λL[1 − (0.5 − t/T )/20] with T = 1 µs. Reference laser:nL = 1000 with g Figure 28 . 28Interference between photon BEC and dye laser (right) for average photon numbers (a)n = 114 000 (n/N = 57%) (b) 8 300 (10%) and (c) 3 700 (5%), which realises different levels of statistical number fluctuations, as visible in the normalised photon number evolution (left) and the autocorrelation (middle). Regions that have been identified by our detection algorithm (grey shaded) indicate phase jumps at increasing rates from (a) to (c). The magnitude of the phase rotations is obtained from a fit as shown in the inset in (b). (d) Histogram of the phase rotation angles ∆ϕ for 108 fitted phase jumps in signals as in (b). Figure 29 . 29Inverse symbols) and phase jump rates ΓPJ (filled symbols) versus condensate number for three different-sized reservoirs, realised byh∆ = −{7.7; 5.6; 2.8}kBT and quantified by ε, see(66).Solid lines give theory curves for Γ 0 PJ with M = {2.0; 4.5; 5.0} × 10 9 ,B 12 = {140; 250; 1 300}s −1 . In the region accessed by our experiments, the phase jump rates are significantly smaller that the inverse second-order correlation times. The inset gives the corresponding zero-delay autocorrelation g (2) (0) along with numerical calculations (solid lines). (λc = {597; 582; 563}nm, ρ = 3 mmol/l, n = √ Meff {1.0; 4.1; 16.5} × 10 3 , error bars are statistical uncertainties). Reproduced with permission from [43]. Copyright 2016 by the American Physical Society. Figure 30 . 30(a) Numerical scaling of effective reservoir and condensate size versusn for various fluctuation levels g (2) (0), demonstrating the conservation of √ Meff/n as the autocorrelation remains fixed. (h∆ = 0, M = 0.5-2.0 × 10 10 ). (b) Measured ΓPJ versus autocorrelation for three effective reservoirs. For each fluctuation level, the larger system exhibits an enhanced phase stability. (c) ‡ The simulation parameters differ from experimental values for computational reasons. 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[ "Robust Weight Perturbation for Adversarial Training", "Robust Weight Perturbation for Adversarial Training" ]	[ "Chaojian Yu \nSchool of Computer Science\nTrustworthy Machine Learning Lab\nThe University of Sydney\nAustralia\n", "Bo Han \nDepartment of Computer Science\nHong Kong Baptist University\nChina\n", "Mingming Gong [email protected] \nSchool of Mathematics and Statistics\nThe University of Melbourne\nAustralia\n", "Li Shen \nJD Explore Academy\nChina\n", "Shiming Ge [email protected] \nInstitute of Information Engineering\nChinese Academy of Sciences\nChina\n", "Du Bo \nSchool of Computer Science\nWuhan University\nChina\n", "Tongliang Liu [email protected] \nSchool of Computer Science\nTrustworthy Machine Learning Lab\nThe University of Sydney\nAustralia\n" ]	[ "School of Computer Science\nTrustworthy Machine Learning Lab\nThe University of Sydney\nAustralia", "Department of Computer Science\nHong Kong Baptist University\nChina", "School of Mathematics and Statistics\nThe University of Melbourne\nAustralia", "JD Explore Academy\nChina", "Institute of Information Engineering\nChinese Academy of Sciences\nChina", "School of Computer Science\nWuhan University\nChina", "School of Computer Science\nTrustworthy Machine Learning Lab\nThe University of Sydney\nAustralia" ]	[]	Overfitting widely exists in adversarial robust training of deep networks. An effective remedy is adversarial weight perturbation, which injects the worstcase weight perturbation during network training by maximizing the classification loss on adversarial examples. Adversarial weight perturbation helps reduce the robust generalization gap; however, it also undermines the robustness improvement. A criterion that regulates the weight perturbation is therefore crucial for adversarial training. In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained perturbation. With LSC, we find that it is essential to conduct weight perturbation on adversarial data with small classification loss to eliminate robust overfitting. Weight perturbation on adversarial data with large classification loss is not necessary and may even lead to poor robustness. Based on these observations, we propose a robust perturbation strategy to constrain the extent of weight perturbation. The perturbation strategy prevents deep networks from overfitting while avoiding the side effect of excessive weight perturbation, significantly improving the robustness of adversarial training. Extensive experiments demonstrate the superiority of the proposed method over the state-of-the-art adversarial training methods.	10.24963/ijcai.2022/512	[ "https://arxiv.org/pdf/2205.14826v1.pdf" ]	249,191,660	2205.14826	5325b263b15f6f3076d7d7f9d0061ab22f08a045	Robust Weight Perturbation for Adversarial Training Chaojian Yu School of Computer Science Trustworthy Machine Learning Lab The University of Sydney Australia Bo Han Department of Computer Science Hong Kong Baptist University China Mingming Gong [email protected] School of Mathematics and Statistics The University of Melbourne Australia Li Shen JD Explore Academy China Shiming Ge [email protected] Institute of Information Engineering Chinese Academy of Sciences China Du Bo School of Computer Science Wuhan University China Tongliang Liu [email protected] School of Computer Science Trustworthy Machine Learning Lab The University of Sydney Australia Robust Weight Perturbation for Adversarial Training Overfitting widely exists in adversarial robust training of deep networks. An effective remedy is adversarial weight perturbation, which injects the worstcase weight perturbation during network training by maximizing the classification loss on adversarial examples. Adversarial weight perturbation helps reduce the robust generalization gap; however, it also undermines the robustness improvement. A criterion that regulates the weight perturbation is therefore crucial for adversarial training. In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained perturbation. With LSC, we find that it is essential to conduct weight perturbation on adversarial data with small classification loss to eliminate robust overfitting. Weight perturbation on adversarial data with large classification loss is not necessary and may even lead to poor robustness. Based on these observations, we propose a robust perturbation strategy to constrain the extent of weight perturbation. The perturbation strategy prevents deep networks from overfitting while avoiding the side effect of excessive weight perturbation, significantly improving the robustness of adversarial training. Extensive experiments demonstrate the superiority of the proposed method over the state-of-the-art adversarial training methods. Introduction Although deep neural networks (DNNs) have led to impressive breakthroughs in a number of fields such as computer vision [He et al., 2016], speech recognition [Wang et al., 2017], and NLP [Devlin et al., 2018], they are extremely vulnerable to adversarial examples that are crafted by adding small and human-imperceptible perturbation to normal examples [Szegedy et al., 2013;Goodfellow et al., 2014]. * This work is done during an internship at JD Explore Academy † Corresponding author The vulnerability of DNNs has attracted extensive attention and led to a large number of defense techniques against adversarial examples. Across existing defenses, adversarial training (AT) is one of the strongest empirical defenses. AT directly incorporates adversarial examples into the training process to solve a min-max optimization problem [Madry et al., 2017], which can obtain models with moderate adversarial robustness and has not been comprehensively attacked [Athalye et al., 2018]. However, different from the natural training scenario, overfitting is a dominant phenomenon in adversarial robust training of deep networks [Rice et al., 2020]. After a certain point in AT, the robust performance on test data will continue to degrade with further training, as shown in Figure 1(a). This phenomenon, termed as robust overfitting, breaches the common practice in deep learning that using over-parameterized networks and training for as long as possible [Belkin et al., 2019]. Such anomaly in AT causes detrimental effects on the robust generalization performance and subsequent algorithm assessment [Rice et al., 2020;. Relief techniques that mitigate robust overfitting have thus become crucial for adversarial training. An effective remedy for robust overfitting is Adversarial Weight Perturbation (AWP) [Wu et al., 2020], which forms a double-perturbation mechanism that adversarially perturbs both inputs and weights: min w max v∈V 1 n n i=1 max \|\|x i −xi\|\|p≤ (f w+v (x i ), y i ),(1) where n is the number of training examples, x i is the adversarial example of x i , f w is the DNN with weight w, (·) is the loss function, is the maximum perturbation constraint for inputs (i.e., \|\|x i − x i \|\| p ≤ ), and V is the feasible perturbation region for weights (i.e., {v ∈ V : \|\|v\|\| 2 ≤ γ\|\|w\|\| 2 }, where γ is the constraint on weight perturbation size). The inner maximization is to find adversarial examples x i within the -ball centered at normal examples x i that maximizes the classification loss . On the other hand, the outer maximization is to find weight perturbation v that maximizes the loss on adversarial examples to reduce robust generalization gap. This is the problem of training a weight-perturbed robust classifier on adversarial examples. Therefore, how well the weight perturbation is found directly affects the performance of the outer minimization, i.e., the robustness of the classifier. Several attack methods have been used to solve the inner maximization problem in Eq.(1), such as Fast Gradient Sign Method (FGSM) [Goodfellow et al., 2014] and Projected Gradient Descent (PGD) [Madry et al., 2017]. For the outer maximization problem, AWP [Wu et al., 2020] injects the worst-case weight perturbation to reduce robust generalization gap. However, the extent to which the weights should be perturbed has not been explored. Without an appropriate criterion to regulate the weight perturbation, the adversarial training procedure is difficult to unleash its full power, since worst-case weight perturbation will undermine the robustness improvement (in Section 3). In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained perturbation (in Section 3), which helps to better understand robust overfitting, and this in turn motivates us to propose an improved weight perturbation strategy for better adversarial robustness (in Section 4). Our main contributions as follows: • We propose a principled criterion LSC to analyse the adversarial weight perturbation. It provides a better understanding of robust overfitting in adversarial training, and it is also a good indicator for efficient weight perturbation. • With LSC, we find that better perturbation of model weights is associated with perturbing on adversarial data with small classification loss. For adversarial data with large classification loss, weight perturbation is not necessary and can even be harmful. • We propose a robust perturbation strategy to constrain the extent of weight perturbation. Experiments show that the robust strategy significantly improves the robustness of adversarial training. 2 Related Work Adversarial Attacks Let X denote the input feature space and B p (x) = {x ∈ X : \|\|x − x\|\| p ≤ } be the p -norm ball of radius centered at x in X . Here we selectively introduce several commonly used adversarial attack methods. Fast Gradient Sign Method (FGSM). FGSM [Goodfellow et al., 2014] perturbs natural example x for one step with step size along the gradient direction: x = x + · sign(∇ x (f w (x), y)). (2) Projected Gradient Descent (PGD). PGD [Madry et al., 2017] is a stronger iterative variant of FGSM, which perturbs normal example x for multiple steps K with a smaller step size α: x 0 ∼ U(B p (x)),(3)x k = Π B p (x) (x k−1 + α · sign(∇ x k−1 (f w (x k−1 ), y))),(4) where U denotes the uniform distribution, x 0 denotes the normal example disturbed by a small uniform random noise, x k denotes the adversarial example at step k, and Π B p (x) denotes the projection function that projects the adversarial example back into the set B p (x) if necessary. Adversarial Defense Since the discovery of adversarial examples, a large number of works have emerged for defending against adversarial attacks, such as input denoising [Wu et al., 2021] TRADES (w; x, y) = 1 n n i=1 CE(f w (x i ), y i ) + β · max x ∈B p (x) KL(f w (x i )\|\|f w (x i )) ,(5) where CE is the cross-entropy loss that encourages the network to maximize the natural accuracy, KL is the Kullback-Leibler divergence that encourages to improve the robust accuracy, and β is the hyperparameter to control the trade-off between natural accuracy and adversarial robustness. Robust Self-Training (RST). RST [Carmon et al., 2019] utilize additional 500K unlabeled data extracted from the 80 Million Tiny Images dataset. RST first leverages the surrogate natural model to generate pseudo-labels for these unlabeled data, and then adversarially trains the network with both additional pseudo-labeled unlabeled data (x,ỹ) and original labeled data (x, y) in a supervised setting: RST (w; x, y,x,ỹ) = TRADES (w; x, y) + λ · TRADES (w;x,ỹ),(6) where λ is the weight on unlabeled data. connection between the flatness of weight loss landscape and robust generalization gap, and proposes to incorporate adversarial weight perturbation mechanism in the adversarial training framework. Despite the efficacy of adversarial weight perturbation in suppressing the robust overfitting, a deeper understanding of robust overfitting and a clear direction for valid weight perturbation is largely missing. The outer maximization in Eq.(1) lacks an effective criterion to regulate and constrain the extent of weight perturbation, which in turn influences the optimization of the outer minimization. In this paper, we propose such a criterion and provide new understanding of robust overfitting. Following this, we design a robust weight perturbation strategy that significantly improves the robustness of adversarial training. Robust Overfitting Loss Stationary Condition In this section, we first empirically investigate the relationship between weight perturbation robustness and adversarial robustness, and then propose a criterion to analyse the adversarial weight perturbation, which leads to a new perspective of robust overfitting. To this end, some discussions about robust overfitting and weight perturbation are provided. Does Weight Perturbation Robustness Certainly Lead to Better Adversarial Robustness? First, we investigate whether the robustness against weight perturbation is beneficial to the adversarial robustness. In particular, we train Pre-Act ResNet-18 with AWP on CIFAR-10 using varying weight perturbation size from 0γ, γ/8, γ/4, γ/2, γ, 2γ, 4γ to 8γ. In each setting, we evaluate the robustness of the model against 20-step PGD (PGD-20) attacks on CIFAR-10 test images. As shown in Figure 1(b), when weight perturbation size is small, the best adversarial robustness has a certain improvement. However, when weight perturbation size is large, the best adversarial robustness begins to decrease significantly as the size of the perturbation increases. It can be explained by the fact that the network has to sacrifice adversarial robustness to allocate more capacity to defend against weight perturbation when weight perturbation size is large, which indicates that weight perturbation robustness and adversarial robustness are not mutually beneficial. As shown in Figure 1(b), the performance gain of AWP is mainly due to suppressing robust overfitting. Loss Stationary Condition. In order to further analyse the weight perturbation, we propose a criterion that divides the training adversarial examples into different groups according to their classification loss: LSC[p, q] = {x ∈ X \| p ≤ (f w (x ), y) ≤ q},(7) where p ≤ q. The adversarial data in the group all satisfy their adversarial loss within a certain range, which is termed Loss Stationary Condition (LSC). The proposed criterion LSC allows the analysis of grouped adversarial data independently, and provides more insights into the robust overfitting. LSC view of Adversarial Weight Perturbation. To provide more insight into how AWP suppresses robust overfitting, we train PreAct ResNet-18 on CIFAR-10 by varying the LSC group that performs adversarial weight perturbation. In each setting, we evaluate the robustness of the model against Algorithm 1 Robust Weight Perturbation (RWP) Input: Network f w , training data S, mini-batch B, batch size n, learning rate η, PGD step size α, PGD steps K 1 , PGD constraint , RWP steps K 2 , RWP constraint γ, minimum loss value c min . Output: Adversarially robust model f w . repeat Read mini-batch x B from training set S. x B ← x B + δ, where δ ∼ Uniform(− , ) for k = 1 to K 1 do x B ← Π (x B + α · sign(∇ x B (f w (x B ), y))) end for Initialize v = 0 for k = 1 to K 2 do V = I B ( (f w+v (x B ), y) ≤ c min ) if V = 0 then break else v ← v + ∇ v (V · (f w+v (x B ), y)) v ← γ v \|\|v\|\| \|\|w\|\| end if end for w ← (w + v) − η∇ w+v 1 n n i=1 (f w+v (x (i) B ), y (i) ) − v until training converged PGD-20 attacks on CIFAR-10 test images. As shown in Figure 1(c), when varying the LSC range, we can observe that conducting adversarial weight perturbation on adversarial examples with small classification loss is sufficient to eliminate robust overfitting. However, conducting adversarial weight perturbation on adversarial examples with large classification loss fails to suppress robust overfitting. The results indicate that to eliminate robust overfitting, it is essential to prevent the model from memorizing these easy-to-learn adversarial examples. Besides, it is observed that conducting adversarial weight perturbation on adversarial examples with large classification loss leads to worse adversarial robustness, which again verifies that the robustness against weight perturbation will not bring adversarial robustness gain, or even on the contrary, it undermines the adversarial robustness enhancement. Do We Really Need the Worst-case Weight Perturbation? As aforementioned, the robustness against weight perturbation is not beneficial to the adversarial robustness improvement. Therefore, to purely eliminate robust overfitting, conducting worst-case weight perturbation on these adversarial examples is not necessary. In the next section, we will propose a robust perturbation strategy to address this issue. Robust Weight Perturbation As mentioned in Section 3, conducting adversarial weight perturbation on adversarial examples with small classification loss is enough to prevent robust overfitting and leads to higher robustness. However, conducting adversarial weight perturbation on adversarial examples with large classification loss may not be helpful. Recalling the criterion LSC proposed in Section 3, we have seen that the loss is closely correlated with the tendency of adversarial example to be overfitted. Thus, it can be used to constrain the extent of weight perturbation at a fine-grained level. Therefore, we propose to conduct weight perturbation on adversarial examples that are below a minimum loss value, so as to ensure that no robust overfitting occurs while avoiding the side effect of excessive weight perturbation. Let c min be the minimum loss value. Instead of generating weight perturbation v via outer maximization in Eq.(1), we generate v as follows: v k+1 = v k + ∇ v k 1 n n i=1 I(x i , y i ) (f w+v k (x i ), y i ), where I(x i , y i ) = 0 if (f w+v k (x i ), y i ) > c min 1 if (f w+v k (x i ), y i ) ≤ c min(8) The proposed Robust Weight Perturbation (RWP) algorithm is shown in Algorithm 1. We use PGD attack [Madry et al., 2017] to generate the training adversarial examples, which can be also extended to other variants such as TRADES [Zhang et al., 2019] and RST [Carmon et al., 2019]. The mimimum loss value c min controls the extent of weight perturbation during network training. For example, in the early stages of training, the classification loss of adversarial example is generally larger than c min corresponding to no weight perturbation process. The classification loss of adversarial examples then decreases as training progresses. At each optimization step, we monitor the classification loss of the adversarial example and conduct the weight perturbation process for adversarial examples whose classification loss is smaller than c min , enabled by an indicator control vector V . At each perturbation step, the weight perturbation v will be updated to increase the classification loss of the corresponding adversarial example. When the classification loss of training adversarial examples is all higher than c min or the number of perturbation step reaches the defined value, we stop the weight perturbation process and inject the generated weight perturbation v for adversarial training. Experiments In this section, we conduct comprehensive experiments to evaluate the effectiveness of RWP including its experimental settings, robustness evaluation and ablation studies. Experimental Setup Baselines and Implementation Details. Our implementation is based on PyTorch and the code is publicly available 1 . We conduct extensive experiments across three benchmark datasets (CIFAR-10, CIFAR-100 and SVHN) and two threat models (L ∞ and L 2 ). We use PreAct ResNet-18 [He et al., 2016] and Wide ResNet (WRN-28-10 and WRN-34-10) [Zagoruyko and Komodakis, 2016] as the network structure following [Wu et al., 2020]. We compare the performance of the proposed method on a number of baseline methods: 1) standard adversarial training without weight perturbation, including vanilla AT [Madry et al., 2017], TRADES [Zhang et al., 2019] and RST [Carmon et al., 2019]; 2) adversarial training with AWP [Wu et al., 2020], including AT- AWP, TRADES-AWP and RST-AWP. For training, the network is trained for 200 epochs using SGD with momentum 0.9, weight decay 5 × 10 −4 , and an initial learning rate of 0.1. The learning rate is divided by 10 at the 100-th and 150-th epoch. Standard data augmentation including random crops with 4 pixels of padding and random horizontal flips are applied. For testing, model robustness is evaluated by measuring the accuracy of the model under different adversarial attacks. For hyper-parameters in RWP, we set perturbation step K 2 = 10 for all datasets. The minimum loss value c min = 1.7 for CIFAR-10 and SVHN, and c min = 4.0 for CIFAR-100. The weight perturbation budget of γ = 0.01 for AT-RWP, γ = 0.005 for TRADES-RWP and RST-RWP following literature [Wu et al., 2020]. Other hyper-parameters of the baselines are configured as per their original papers. Adversarial Setting. The training attack is 10-step PGD attack with random start. We follow the same settings in [Rice et al., 2020] : for L ∞ threat model, = 8/255, step size α = 1/255 for SVHN, and α = 2/255 for both CIFAR10 and CIFAR100; for L 2 threat model, = 128/255, step size α = 15/255 for all datasets, which is a standard setting for adversarial training [Madry et al., 2017]. The test attacks used for robustness evaluation contains FGSM, PGD-20, PGD-100, C&W ∞ and Auto Attack (AA). Robustness Evaluation Performance Evaluations. To validate the effectiveness of the proposed RWP, we conduct performance evaluation on vanilla AT, AT-AWP and AT-RWP across different benchmark datasets and threat models using PreAct ResNet-18. We report the accuracy on the test images under PGD-20 attack. The evaluation results are summarized in Table 1, where "Best" denotes the highest robustness that ever achieved at different checkpoints and "Last" denotes the robustness at the last epoch checkpoint. It is observed vanilla AT suffers from severe robust overfitting (the performance gap between "best" and "last" is very large). AT-AWP and AT-RWP method narrow the performance gap significantly over the vanilla AT model due to suppression of robust overfitting. Moreover, on CIFAR-10 dataset under the L ∞ attack, vanilla AT achieves 52.32% "best" test robustness. The AT-AWP approach boosts the performance to 55.54%. The proposed approach further outperforms both methods by a large margin, improving over vanilla AT by 6.23%, and is 3.01% better than AT-AWP, achieving 58.55% accuracy under the standard 20 steps PGD attack. Similar patten has been observed on other datasets and threat model. AT-RWP consistently improves the test robustness across a wide range of datasets and threat models, demonstrating the effectiveness of the proposed approach. Benchmarking the state-of-the-art Robustness. To manifest the full power of our proposed perturbation strategy and also benchmark the state-of-the-art robustness on CIFAR-10 under L ∞ threat model, we conduct experiments on the large capacity network with different baseline methods. We train Wide ResNet-34-10 for AT and TRADES, and Wide ResNet-28-10 for RST following their original papers. We evaluate the adversarial robustness of trained model with various test attack and report the "best" test robustness, with the results shown in Table 2. "Natural" denotes the accuracy on natural test data. First, it is observed that the natural accuracy of RWP model consistently outperforms AWP by a large margin. It is due to the benefits that our RWP avoids the excessive weight perturbation. Moreover, RWP achieves the best adversarial robustness against almost all types of attack across a wide range of baseline methods, which verifies that RWP is effective in general and improves adversarial robustness reliably rather than improper tuning of hyper-parameters of attacks, gradient obfuscation or masking. Ablation Studies In this part, we investigate the impacts of algorithmic components using AT-RWP on PreAct ResNet-18 under L ∞ threat model following the same setting in section 5.1. The Importance of Minimum Loss Value. We verify the effectiveness of minimum loss value c min , by comparing the performance of models trained using different weight perturbation schemes: 1) AT: standard adversarial training without weight perturbation (equivalent to c min = 0); 2) AWP: weight perturbation generated via outer maximization in Eq. (1) (equivalent to c min = ∞); 3) RWP: weight perturbation generated using the proposed robust strategy with different c min values. All other hyper-parameters are kept exactly the same other than the perturbation scheme used. The results are summarized in Figure 2(a). It is observed that the test robustness of RWP model first increases and then decreases as the minimum loss value increases, and the best test robustness is obtained at c min = 1.7. It is evident that RWP with a wide range of c min outperforms both AT and AWP methods, demonstrating its effectiveness. Furthermore, as it is the major component that is different from the AWP pipeline, this result suggests that the proposed LSC constraints is the main contributor to the improved adversarial robustness. The Impact of Step Number. We further investigate the effect of step number K 2 , by comparing the performances of model trained using different perturbation steps. The step number K 2 for RWP varies from 1 to 10. The results are shown in Figure 2(b). As expected, when K 2 is small, increasing K 2 leads higher test robustness. When K 2 increases from 7 to 10, the performance is flat, which suggests that the generated weight perturbation is sufficient to comprehensively avoid robust overfitting. Note that extra iterations will not bring computational overhead when the classification loss of adversarial examples exceeds minimum loss value c min , as shown in Algorithm 1. Therefore, we uniformly use K 2 = 10 in our implementation. Effect on Adversarial Robustness and Robust Overfitting. We then visualize the learning curves of AT, AWP and RWP, which are summarized in Figure 2(c). It is observed that the test robustness of RWP model continues to increase as the training progresses. In addition, RWP outperforms AWP with a clear margin in the later stage of training. Such observations exactly reflect the nature of our approach which aims to prevent robust overfitting as well as boost the robustness of adversarial training. Conclusion In this paper, we proposed a criterion, Loss Stationary Condition (LSC) for constrained weight perturbation. The proposed criterion provides a new understanding of robust overfitting. Based on LSC, we found that elimination of robust overfitting and higher robustness of adversarial training can be achieved by weight perturbation on adversarial examples with small classification loss, rather than adversarial examples with large classification loss. Following this, we proposed a Robust Weight Perturbation (RWP) strategy to regulate the extent of weight perturbation. Comprehensive experiments show that RWP is generic and can improve the stateof-the-art adversarial robustness across different adversarial training approaches, network architectures, threat models and benchmark datasets. Figure 1 : 1(a): The learning curve of vanilla AT; (b): Test robustness of AWP with varying weight perturbation size; (c): Test robustness of AWP with varying LSC range. Figure 2 : 2The ablation study experiments on CIFAR-10. Table 2 : 2Test robustness (%) on CIFAR-10 using Wide ResNet under L∞ threat model. https://github.com/ChaojianYu/Robust-Weight-Perturbation AcknowledgementsThis work is supported in part by Beijing Natural Science Foundation (19L2040), NSFC Young Scientists Fund No. 62006202, Guangdong Basic and Applied Basic Research Foundation No. 2022A1515011652, and Science and Technology Innovation 2030 -"Brain Science and Brain-like Research" Major Project (No. 2021ZD0201402 and No. 2021ZD0201405). Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. References [athalye, arXiv:1905.13736Proceedings of the National Academy of Sciences. the National Academy of Sciences116arXiv preprintUnlabeled data improves adversarial robustnessReferences [Athalye et al., 2018] Anish Athalye, Nicholas Carlini, and David Wagner. 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[ "Choosability with Separation of Cycles and Outerplanar Graphs", "Choosability with Separation of Cycles and Outerplanar Graphs" ]	[ "Jean-Christophe Godin \nInstitut de Mathématiques de Toulon\nUniversité de Toulon\nFrance\n", "Olivier Togni [email protected] \nLaboratoire LIB\nUniversité Bourgogne Franche-Comté\nFrance\n" ]	[ "Institut de Mathématiques de Toulon\nUniversité de Toulon\nFrance", "Laboratoire LIB\nUniversité Bourgogne Franche-Comté\nFrance" ]	[]	We consider the following list coloring with separation problem of graphs: Given a graph G and integers a, b, find the largest integer c such that for any list assignment L of G with \|L(v)\| ≤ a for any vertex v and \|L(u) ∩ L(v)\| ≤ c for any edge uv of G, there exists an assignment ϕ of sets of integers to the vertices of G such that ϕ(u) ⊂ L(u) and \|ϕ(v)\| = b for any vertex v and ϕ(u)∩ ϕ(v) = ∅ for any edge uv. Such a value of c is called the separation number of (G, a, b). We also study the variant called the free-separation number which is defined analogously but assuming that one arbitrary vertex is precolored. We determine the separation number and free-separation number of the cycle and derive from them the free-separation number of a cactus. We also present a lower bound for the separation and free-separation numbers of outerplanar graphs of girth g ≥ 5.	10.7151/dmgt.2398	[ "https://arxiv.org/pdf/2009.00287v1.pdf" ]	221,397,300	2009.00287	ed77aab4986a6a95a551b87b20b788f53d089c59	Choosability with Separation of Cycles and Outerplanar Graphs 1 Sep 2020 Jean-Christophe Godin Institut de Mathématiques de Toulon Université de Toulon France Olivier Togni [email protected] Laboratoire LIB Université Bourgogne Franche-Comté France Choosability with Separation of Cycles and Outerplanar Graphs 1 Sep 2020 We consider the following list coloring with separation problem of graphs: Given a graph G and integers a, b, find the largest integer c such that for any list assignment L of G with \|L(v)\| ≤ a for any vertex v and \|L(u) ∩ L(v)\| ≤ c for any edge uv of G, there exists an assignment ϕ of sets of integers to the vertices of G such that ϕ(u) ⊂ L(u) and \|ϕ(v)\| = b for any vertex v and ϕ(u)∩ ϕ(v) = ∅ for any edge uv. Such a value of c is called the separation number of (G, a, b). We also study the variant called the free-separation number which is defined analogously but assuming that one arbitrary vertex is precolored. We determine the separation number and free-separation number of the cycle and derive from them the free-separation number of a cactus. We also present a lower bound for the separation and free-separation numbers of outerplanar graphs of girth g ≥ 5. Introduction Let a, b, c and k be integers and let G be a graph. A k-list assignment L of G is a function which associates to each vertex a set of at most k integers. The list assignment L is c-separating if for any uv ∈ E(G), \|L(u) ∩ L(v)\| ≤ c. The graph G is (a, b, c)-choosable if for any c-separating a-list assignment L, there exists an (L, b)-coloring of G, i.e. a coloring function ϕ on the vertices of G that assigns to each vertex v a subset of b elements from L(v) in such a way that ϕ(u) ∩ ϕ(v) = ∅ for any uv ∈ E(G). This type of restricted list coloring problem, called choosability with separation, has been introduced by Kratochvíl, Tuza and Voigt [12]. Notice that Kratochvíl et al. [12,13] defined (a, b, c)-choosability a bit differently, requiring for a c-separating a-list assignment L that the lists of two adjacent vertices u and v satisfy \|L(u) ∩ L(v)\| ≤ a − c. Among the first results on the topic, a complexity dichotomy was presented [12] and general properties given [13]. Since then, a number of papers has considered choosability with separation of planar graphs, mainly for the case b = 1 [3,4,5,6,7,11,15]. A still open question for this class of graph is whether all planar graphs are (4, 1, 2)-choosable or not. Other recent papers concern balanced complete multipartite graphs and k-uniform hypergraphs (for the case b = 1) [10]; bipartite graphs (for the case b = c = 1) [9] and a study with an extended separation condition [14]. In this paper, we concentrate on choosability and free-choosability with separation of cycles and outerplanar graphs in a little different point of view: as a (a, b, c)-choosable graph is also (a, b, c ′ )-choosable for any c ′ < c, our aim is to determine, for given a, b, a ≥ b, the largest c such that G is (a, b, c)-choosable. We find convenient to define the parameter sep(G, a, b) that we call the (list) separation number of G as sep(G, a, b) = max{c, G is (a, b, c)-choosable}. The notion of free choosability [2], that consists in considering list assignments on graphs with a precolored vertex, easily extends to choosability with separation: a graph G is (a, b, c)-free-choosable if for any c-separating a-list assignment L, any v ∈ V (G) and any C ⊂ L(v) with \|C\| = b, there exists an (L, b)-coloring ϕ such that ϕ(v) = C. Alternatively, we can view free-choosability as classical choosability but with a list of cardinality b on one arbitrary vertex. Analogously with the separation number, we define the free-separation number fsep(G, a, b) of a graph G as fsep(G, a, b) = max{c, G is (a, b, c)-free-choosable}. Clearly, for any graph G and any integers a and b, we have sep(G, a, b) ≥ fsep(G, a, b). Moreover, since for any a ≥ b ≥ 1, every graph G is (a, b, 0)-free-choosable, we have 0 ≤ fsep(G, a, b) ≤ sep(G, a, b) ≤ a and thus both parameters are well defined for any graph. As a first example (and as we will prove in Proposition 14), the graph G depicted in Figure 1 is not (2, 1, 1)-choosable, thus implying that sep(G, 2, 1) = fsep(G, 2, 1) = 0 < sep(C 4 , 2, 1) = 1. In this paper, we determine the separation number of the cycle in Section 2 (Theorem 5) and free-separation number (Theorem 9 and Proposition 10) of the cycle in Section 3. Contrary to the separation number, we show that the free-separation number of a cycle does not depend on the parity of its length and that C 3 is a special case. We then use these results to determine bounds and exact values for the same invariants on outerplanar graphs of girth at least 5 and tighter bounds for the subclass of cactuses in Section 4. Some possible directions for further works are given in Section 5. Our proofs are all constructive and the proofs of upper bounds on sep and fsep rely on finding counter-example list assignments. These examples are constructed in a greedy way, maximizing for each list, the intersections with lists of other adjacent vertices (while satisfying the c-separating condition). Our proofs for fsep(C n , a, b) use special types of list assignments of the path (Lemmas 7 and 8) that may be of interest for obtaining other choosability results. Separation number of the cycle Using a similar argument than the one used by Kratochvíl et al. [13] (in the more general setting of graphs with bounded outdegree orientation) we have the following: Proposition 1. For any n ≥ 3 and a ≥ b, we have sep(C n , a, b) ≥ a − b. Proof. Let L be a k-separating (b + k)-list assignment. Orient C n clockwise, with x − and x + being the predecessor and successor of vertex x, respectively. Since for any x ∈ V (C n ), \|L(x) ∩ L(x + )\| ≤ k, we have \|L(x) \ L(x + )\| ≥ b + k − k = b. Hence it is possible to assign a set ϕ(x) of b colors from L(x) \ L(x + ) to each vertex x. Since any (a, b, c)-choosable graph is also (a ′ , b, c)-choosable for any a ′ ≥ a, we obtain that C n is (a, b, c)-choosable when c ≤ a − b. Since a ≥ b, we can rewrite the above inequality as sep(C n , b + k, b) ≥ k for any k ≥ 0. As the next result shows, the above result is tight provided k < b. Proposition 2. For any n ≥ 3, b ≥ 1 and k < b, we have sep(C n , b + k, b) ≤ k. Proof. We provide a (k + 1)-separating (b + k)-list assignment L for which no (L, b)coloring of C n exists. Let X be a set of n(b−1)+1 colors and let C, D i , F i , i ∈ {0, . . . , n− 1} be a partition of X with \|C\| = 1, \|D i \| = k, \|F i \| = b − k − 1. Let C n = (x 0 , . . . , x n−1 ) and for any i, 0 ≤ i ≤ n − 1, let L(x i ) = C ∪ D i ∪ D i+1 ∪ F i , with indices taken modulo n. Now, observe that, by the construction of the list assignment L, the color of C is present in the color-list of every vertex and that every color of any set D i is present in the lists of two consecutive vertices. Therefore, the color of C can be assigned to at most ⌊n/2⌋ vertices while every color of every set D i , i = 0, . . . , n − 1 can be given to at most one vertex. Hence, the total number of colors that can be given to vertices of C n is ⌊n/2⌋ + nk + n(b − k − 1) = ⌊n/2⌋ + n(b − 1) < nb. Then, since the n vertices of C n require nb colors in total, no (L, b)-coloring of C n exists. Reusing the method of the proof of Proposition 1 with a little more involved argument, we are able to prove: Proposition 3. For any a, b, c, n ≥ 3 and k ≥ 1, the following implication is true for the cycle C n : C n (a, b, c)-choosable ⇒ C n (a + 2k, b + k, c + k)-choosable. Proof. Suppose a ≥ c and C n is (a, b, c)-choosable and let L be a (c + k)-separating (a+2k)-list assignment of C n . Orient C n clockwise, with x − and x + being the predecessor and successor of vertex x, respectively. Since for any x ∈ V (C n ), \|L(x) ∩ L(x + )\| ≤ c + k, we have \|L(x) \ L(x + )\| ≥ a + 2k − (c + k) = a − c + k ≥ k. Hence it is possible to assign a set ϕ(x) of k colors from L(x) \ L(x + ) to each vertex x. Now we still have to assign b more colors to each vertex to complete the coloring. For this, we construct a new list assignment L ′ by removing from L(x) the colors already assigned to x and also a maximum number of colors from L(x) ∩ L(x + ), including those assigned to x + that are in L(x), if any, in order L ′ to be a c-separating a ′ -list assignment, with a ′ ≥ a (observe that, by construction, L(x) ∩ ϕ(x − ) = ∅): For any x ∈ V (C n ), let I + (x) = L(x) ∩ L(x + ) and let S(x) be any subset of I + (x) of size min{k, \|I + (x)\|} that contains L(x) ∩ ϕ(x + ). (Note that I + (x) and S(x) may be empty.) Then, define a new list assignment L ′ on C n by: ∀x ∈ V (C n ), L ′ (x) = L(x) \ (ϕ(x) ∪ S(x)). We then have \|L ′ (x)\| = a + 2k − k − min{k, \|I + (x)\|} ≥ a and \|L ′ (x) ∩ L ′ (x + )\| ≤ c + k − min{k, \|I + (x)\|} ≤ c, hence L ′ is a c-separating a ′ -list assignment with a ′ ≥ a. Therefore, since by hypothesis G is (a, b, c)-choosable, there exists an (L ′ , b)-coloring of G, which together with the coloring f , produces an (L, b + k)-coloring of G, proving that G is (a + 2k, b + k, c + k)-choosable. For cycles of odd length, combining known choosability results on the cycle with Proposition 3 allows to determine the separation number in the remaining cases: Proposition 4. For any integers b, p and α with 0 ≤ α ≤ b p and p ≥ 1, we have sep(C 2p+1 , 2b + α, b) = b + (p + 1)α. Proof. It is known that cycles of length 2p + 1 are (a, b, a)-choosable if and only if a/b ≥ 2 + 1/p (see [2]). Hence, for α ≥ 0, C 2p+1 is ((2p + 1)α, pα, (2p + 1)α)-choosable. Then by Proposition 3, C 2p+1 is also ((2p + 1)α + 2k, pα + k, (2p + 1)α + k)-choosable for any k ≥ 0. Setting k = b − pα, we obtain that C 2p+1 is (2b + α, b, b + (p + 1)α)-choosable. Note that if p divides b and if α = b/p, then b + (p + 1)α = 2b + α. Hence in this case, sep(C 2p+1 , 2b + α, b) = 2b + α. Otherwise, to prove that C 2p+1 is not (2b + α, b, b + (p + 1)α + 1)-choosable for α ≤ b−1 p , we provide a (b + (p + 1)α + 1)-separating (2b + α)-list assignment L of C 2p+1 for which no (L, b)-coloring exists. Let C be a set of (2p + 1)α + 2 colors and D i , i = 0, . . . , 2p be 2p + 1 pairwise disjoint sets of b − pα − 1 colors (also disjoint from C). Let C 2p+1 = (x 0 , . . . , x 2p ) and set L(x i ) = C ∪ D i ∪ D i+1 , with 0 ≤ i ≤ 2p and indices taken modulo 2p + 1. It can be checked that the lists of any two consecutive vertices have (2p + 1)α + 2 + b − pα − 1 = b + (p + 1)α + 1 elements in common. Assume now that there exists an (L, b)-coloring of C 2p+1 . Observe that, by the construction of the list assignment L, every color from C is present in the color-list of every vertex and that every color of any set D i is present in the lists of two consecutive vertices. Therefore, every color of C can be assigned to at most p vertices while every color of every set D i , i = 0, . . . , 2p can be given to at most one vertex. Hence we have p((2p + 1)α + 2) + (2p + 1)(b − pα − 1) = b(2p+1)−1 available colors in total but we have to assign b(2p+1) colors to the vertices, a contradiction. We are now ready to determine the separation number of the cycle. Theorem 5. For any p ≥ 1 and any a, b such that a ≥ b ≥ 1, sep(C 2p+2 , a, b) = a − b, b ≤ a < 2b a, a ≥ 2b. sep(C 2p+1 , a, b) =      a − b, b ≤ a < 2b b + (p + 1)(a − 2b), 2b ≤ a ≤ 2b + b p a, a ≥ 2b + b p . Proof. For even-length cycles, the result is obtained by combining Propositions 1 and 2 and noting that C 2p is (2b, b, 2b)-choosable. For odd-length cycles, the combination of Propositions 1, 2 and 4 and the known fact that C 2p+1 is (a, b, a)-choosable for any a, b such that a/b ≥ 2 + 1/p leads to the result. Example 1. From the above theorem, we know that the cycle C 3 is (5, 2, 4)-choosable but not (5, 2, 5)-choosable (this was already known) and also (7, 3, 5)-choosable but not (7, 3, 6)-choosable (while it was only known before that C 3 is not (7, 3, 7)-choosable). In contrast, C 5 is (7, 3, 6)-choosable and (9, 4, 7)-choosable but not (7, 3, 7)-choosable and not (9, 4, 8)-choosable. Free separation number of the cycle In order to determine the free-separation number of the cycle, we first set some notation and preliminary results. The following Hall-type condition that we call the amplitude condition is necessary for a graph G to be (L, b)-colorable: ∀H ⊂ G, k∈C α(H, L, k) ≥ b\|V (H)\|, where C = v∈V (H) L(v) and α(H, L, k) is the independence number of the subgraph of H induced by the vertices containing k in their color list. Notice that H can be restricted to be a connected induced subgraph of G. As shown by Cropper et al. [8] (in the more general context of weighted list coloring), this condition is also sufficient when the graph is a complete graph or a path (or some other specific graphs). For a list assignment L on a graph G of order n with vertex set V (G) = {x 1 , x 2 , . . . , x n }, we let L i = L(x i ) and for 1 ≤ i < j ≤ n, we write Σ i,j (L) = k∈C α(H, L, k), where H is the subgraph of G induced by vertices x i , x i+1 , . . . , x j . We also simplify Σ 1,n (L) to Σ(L). From now on, a cycle of order n will have its vertices denoted by x 1 , x 2 , . . . , x n following some order on the cycle and the vertices of a path of order n will be also denoted by x 1 , x 2 , . . . , x n following the path from one end-vertex to the other. We will use the following relation to show that the cycle is (L, b)-colorable for some lists L. Remark 6. A list assignment L on the cycle C n with \|L 1 \| = b and \|L i \| = a for any i, 2 ≤ i ≤ n can be transformed into a list assignment L ′ on the path P n+1 by setting L ′ n+1 = L 1 and L ′ i = L i for 1 ≤ i ≤ n (i.e. , P n+1 has been obtained by "cutting" the cycle on the vertex x 1 ). Clearly if P n+1 is (L ′ , b)-colorable, then C n is (L, b)-colorable. As induced subgraphs of paths are sub-paths, the amplitude condition for a path P n can be rewritten as : ∀i, j, 1 ≤ i ≤ j ≤ n, Σ i,j (L) ≥ b(j − i + 1). For integers n, a and b with b ≤ a, let c(n, a, b) =    n−1 n (a − b), if b ≤ a < 2n−1 n−1 b n−1 n−2 (a − b) − 2 n−2 b, if 2n−1 n−1 b ≤ a < 2 n+1 n b a, if 2 n+1 n b ≤ a. We now prove two lemmas about conditions for a path with precolored endvertices to be list colorable. Lemma 7. Let n, a, b, c be integers, n ≥ 3, b ≤ a < 2 n+1 n b and c = ⌊c(n, a, b)⌋. For any c-separating a-list assignment L of P n+1 such that \|L 1 \| = \|L n+1 \| = b, there exists an (L, b)-coloring of P n+1 . Proof. It is sufficient to verify that the amplitude condition is satisfied. First, we show that in both cases, we have c ≤ a − b. If c ≤ n−1 n (a − b) then clearly c ≤ a − b. If c ≤ n−1 n−2 (a − b) − 2 n−2 b = n−2 n−2 (a − b) + a−b n−2 − 2 n−2 b = a − b + a−3b n−2 , then again, c ≤ a − b since a < 2 n+1 n b ≤ 3b as soon as n ≥ 3. Now, since L is c-separating then if j < n we have \|L j+1 \ L j \| ≥ a − c and thus, for any 1 ≤ i ≤ j, Σ i,j+1 (L) ≥ Σ i,j (L) + a − c (1) Moreover, since \|L n+1 \| = b, we also have Σ i,n+1 (L) ≥ Σ i,n (L) + max(b − c, 0) (2) Therefore, if 1 < i ≤ j ≤ n, then Σ i,j (L) ≥ (j − i + 1)(a − c). Hence the amplitude condition is satisfied in this case if (j − i + 1)(a − c) ≥ (j − i + 1)b, i.e., if a − c ≥ b, which is true since we have shown above that c ≤ a − b. If i = 1 or j = n + 1, we consider two cases depending on a, b and c (note that the ratio 2n−1 n−1 has been chosen in such a way that c < b in Case 1 and c ≥ b in Case 2). Case 1. c = ⌊ n−1 n (a − b)⌋ and b ≤ a < 2n−1 n−1 b. The above imply c ≤ n−1 n (a − b) < n−1 n 2n−1 n−1 b − n−1 n b = b. Hence, if i = 1 and 2 ≤ j ≤ n, then by Equation 1, Σ 1,j (L) ≥ b + (a − c)j ≥ (j + 1)b, since a − c ≥ b. Otherwise, if i > 1 and j = n + 1, then Σ i,n+1 (L) ≥ a + (a − c)(n − i) + b − c = b + (a − c)(n + 1 − i). Hence, since a − c ≥ b, we have that Σ i,n+1 (L) ≥ (n + 2 − i)b. Finally, since c ≤ b, by 1 and 2, we infer Σ(L) ≥ b + (a − c)(n − 1) + b − c ≥ (n + 1)b if c ≤ n−1 n (a − b). Case 2. c = ⌊ n−1 n−2 (a − b) − 2 n−2 b⌋ and a ≥ 2n−1 n−1 b. In this case we have c > n−1 n−2 (a − b) − 2 n−2 b − 1 ≥ n−1 n−2 2n−1 n−1 b − n+1 n−2 b − 1 = b − 1. Hence, if i = 1 and 2 ≤ j ≤ n, then Σ 1,j (L) ≥ b + a − b + (a − c)(j − 2) ≥ b + (a − c)(j − 1) ≥ jb, since a − c ≥ b. Otherwise, if i > 1 and j = n + 1, then Σ i,n+1 (L) ≥ b + a − b + (a − c)(n − i) ≥ b + (a − c)(n + 1 − i) ≥ (n + 2 − i)b since a − c ≥ b. Finally, Σ 1,n+1 (L) ≥ Σ 2,n (L) ≥ a + (a − c)(n − 2) ≥ (n + 1)b since c ≤ n−1 n−2 (a − b) − 2 n−2 b. Therefore, in both cases, the amplitude condition is satisfied hence P n+1 is (L, b)colorable. As shown by the next lemma, the above condition in Lemma 7 is also necessary (provided that n ≥ 4) and even with restricted list assignments in which the lists of the two endvertices are the same or are disjoint. Lemma 8. Let n, a, b, c be integers, n ≥ 4, b ≤ a < 2 n+1 n b and c = ⌊c(n, a, b)⌋ + 1. There exists a c-separating a-list assignment L of P n+1 with \|L 1 \| = \|L n+1 \| = b such that P n+1 is not (L, b)-colorable. Moreover, the same holds if in addition, the list assignment L is such that L 1 = L n+1 or L 1 ∩ L n+1 = ∅. Proof. We provide a counter-example in each of the two following cases. Case 1. b ≤ a < 2n−1 n−1 b. We show that for c = ⌊ n−1 n (a − b)⌋ + 1 the following list assignment L is a cseparating a-list assignment of P n+1 is such that L 1 = L n+1 , \|L 1 \| = b, but no (L, b)-coloring exists. In order to have a compact representation and to shorten the proof, L is described in a graphical way showing the intersections of sublists composing the lists L i (each box represent a color subset and the number inside a box indicates its size): L 1 : b L 2 : c a − c L 3 : c a − c . . . . . . L n−1 : c a − c L n : c c a − 2c L n+1 : b Since a < 2n−1 n−1 b, then c = ⌊ n−1 n (a − b)⌋ + 1 < ⌊ n−1 n 2n−1−n+1 n−1 b⌋ + 1 = b + 1, i.e. c ≤ b. Moreover, since n−1 n < 1, we have c < a − b + 1, hence a > b + c − 1 ≥ 2c − 1. Consequently, the list assignment L is well defined. Observe that each color of each list L i , 2 ≤ i ≤ n − 1, can be used on only one vertex. Also, for any color k shared by the lists of L 1 and L n+1 , we have α(P n+1 , L, k) = 2. Hence, we have Σ(L) = 2b + (n − 2)(a − c) + a − 2c = (n − 1)(a − c) + 2b − c. In order the amplitude condition to be satisfied, we must have Σ(L) ≥ (n + 1)b, i.e., (n − 1)(a − c) + 2b − c ≥ (n + 1)b, which is equivalent to c ≤ n−1 n (a − b). Consequently, P n+1 is not (L, b)-colorable when c > n−1 n (a − b). Case 2. 2n−1 n−1 b ≤ a < 2 n+1 n b. We show that for c = ⌊ n−1 n−2 (a−b)− 2 n−2 b⌋+1 there is a c-separating a-list assignment L of P n+1 for which L 1 = L n+1 , \|L 1 \| = b, but no (L, b)-coloring exists. First, we show that a ≥ 2c − 1: Since c ≤ n−1 n−2 a − n+1 n−2 b + 1 and a < 2 n+1 n b, i.e. b > n 2n+2 a, then 2c − 2 ≤ 2n−2 n−2 a − 2n+2 n−2 b < 2n−2 n−2 a − 2n+2 n−2 n 2n+2 a = a. Second, we show that c ≥ b + 1: As c > n−1 n−2 a − n+1 n−2 b and since a ≥ 2n−1 n−1 b, then we obtain c > n−1 n−2 2n−1 n−1 b − n+1 n−2 b = b. Depending on the value of a and c, we consider two subcases and for each we provide a c-separating a-list assignment L for which \|L 1 \| = b and no (L, b)-coloring exists. Subcase 2.a. a ≥ 2c. Consider the list assignment L defined as follows: L 1 : b L 2 : b a − b L 3 : c a − c . . . . . . L n−1 : c a − c L n : b c a − c − b L n+1 : b Since we are in the case that a ≥ 2c and c ≥ b + 1, the list is well defined. We have Σ(L) = 2b + a − b + (n − 3)(a − c) + a − c − b = (n − 1)a − (n − 2)c. Therefore, Σ(L) ≥ (n + 1)b implies c ≤ n−1 n−2 (a − b) − 2 n−2 b. Consequently, P n+1 is not (L, b)-colorable when c > n−1 n−2 (a − b) − 2 n−2 b. Subcase 2.b. a = 2c − 1. We are in the case that a < 2 n+1 n b, i.e., a < 2b + 2b n . Hence, as a = 2c − 1, c satisfies c < b + b n + 1 2 (3) If n is odd, consider the list assignment L defined as follows: L 1 : b L 2 : b 2c − b − 1 L 3 : c c − 1 L 4 : c − 1 c . . . . . . L n−1 : c − 1 c L n : b c c − b − 1 L n+1 : b We have Σ(L) = 2b + 2c − b − 1 + n − 3 2 (c − 1) + n − 3 2 c + c − b − 1 = nc − n + 1 2 . Therefore the amplitude condition is not satisfied if nc − n+1 2 < (n + 1)b, i.e., if c < b + b n + 1 2 + 1 2n which is true by Equation 3. If n is even, consider the list assignment L defined as follows: L 1 : b L 2 : b 2c − b − 1 L 3 : c c − 1 L 4 : c − 1 c . . . . . . L n−1 : c c − 1 L n : b c − 1 c − b L n+1 : b We have Σ(L) = 2b + 2c − b − 1 + n − 2 2 (c − 1) + n − 4 2 c + c − b = nc − n 2 . Therefore the amplitude condition is not satisfied if nc − n 2 < (n + 1)b, i.e., if c < b + b n + 1 2 which is true by Equation 3. The counter-examples presented are such that L 1 = L n+1 , but they can be easily modified in order that L 1 ∩ L n+1 = ∅ without changing the conclusion. For this, in each of the above four list assignments, instead of using the b colors of L 1 for L n and L n+1 , just take b new colors (not used by any list L i , 1 ≤ i ≤ n − 1). Theorem 9. For any a ≥ b ≥ 1 and any n ≥ 4, fsep(C n , a, b) =      n−1 n (a − b) , b ≤ a < 2n−1 n−1 b n−1 n−2 (a − b) − 2 n−2 b , 2n−1 n−1 b ≤ a < 2 n+1 n b a, 2 n+1 n b ≤ a. Proof. First, if 2 n+1 n b ≤ a, then we know from [2] that C n is (a, b, a)-free-choosable. For the two other cases, given a c-separating a-list assignment L of C n with \|L 1 \| = b and c ≤ ⌊c(n, a, b)⌋, we consider the list assignment L ′ on P n+1 obtained from L on C n by cutting the cycle at x 1 as in Remark 6. By Lemma 7, P n+1 is (L ′ , b)-colorable. Hence C n is (L, b)-colorable for c ≤ ⌊c(n, a, b)⌋. For the converse, if c = ⌊c(n, a, b)⌋ + 1, then we know, by Lemma 8 that there exists a c-separating a-list assignment L ′ of P n+1 with L ′ 1 = L ′ n+1 and \|L ′ 1 \| = b, such that P n+1 is not (L ′ , b)-colorable. Therefore, identifying the vertices x 1 and x n+1 of P n+1 , we obtain a c-separating a-list assignment L on the cycle C n such that \|L 1 \| = b and no (L, b)-coloring of C n exists. It only remains to determine the free separation number of the cycle of length 3 which has a special behavior. Proposition 10. fsep(C 3 , a, b) =    ⌊ 2 3 (a − b)⌋, b ≤ a < 7 4 b 2a − 3b, Proof. We consider the three following cases depending on a: Case 1. b ≤ a < 7 4 b. Let c = ⌊ 2 3 (a − b)⌋ + 1. We prove that C 3 is not (a, b, c)-free-choosable. For this, we give a c-separating a-list assignment L for which \|L 1 \| = b and no (L, b)-coloring exists. L 1 : b L 2 : c a − c L 3 : b − c c a − b In order this list to be well defined, we must have c ≤ b. If b = 1 then a = 1 and thus c = 1 = b. If b ≥ 2 then c ≤ 2 3 (a − b) + 1 ≤ b if a ≤ 5b−3 2 which is true since 7 4 b ≤ 5b−3 2 for b ≥ 2. We must also have a − c ≥ c which is true since a − 2c ≥ a − 4(a − b)/3 = 4b/3 − a/3 ≥ 0 as a < 7 4 b. Moreover, we have Σ(L) = b + a − c + a − b = 2a − c. Since c > 2 3 (a − b), Σ(L) < 3b if 6a − 2a + 2b < 9b, i.e., if a < 7 4 b which is true by hypothesis. Hence the amplitude condition is not satisfied and thus C 3 is not (L, b)-colorable. Now, we prove that C 3 is (a, b, c ′ )-free-choosable with c ′ = c − 1 = ⌊ 2 3 (a − b)⌋. First, observe that we have b − 2 3 (a − b) = 5b−2a 3 ≥ 0, hence b ≥ c ′ . Since C 3 is a complete graph, by Cropper et al.'s result [8], the amplitude condition is sufficient in order it is (L, b)-colorable. This is clearly true for a subgraph of C 3 reduced to one vertex and for a subgraph of two vertices since for any c ′ -separating a-list assignment L with \|L 1 \| = b, we have \|L 2 ∪ L 3 \| ≥ 2a − c ′ ≥ 2b as a ≥ b and, for i = 2 or 3, \|L 1 ∪ L i \| ≥ b + a − min(b, c ′ ) = b + a − c ′ ≥ 2b since c ′ < a − b. For the whole graph, since c ′ ≤ 2 3 (a − b), we have: Σ(L) ≥ \|L 1 \| + \|L 2 \ L 1 \| + \|L 3 \ (L 1 ∪ L 2 )\| ≥ b + a − c ′ + a − 2c ′ = 2a + b − 3c ′ ≥ 3b. Case 2. 7 4 b ≤ a < 3b. Let c = 2a−3b+1. We prove that C 3 is not (a, b, c)-free-choosable. For this, we give a c-separating a-list assignment L for which \|L 1 \| = b and no (L, b)-coloring exists: We present a list for each of the two following subcases depending on whether a ≥ 2b or not. If a ≥ 2b, L is made up as follows: L 1 : b L 2 : b a − b L 3 : b c − b a − c For this list to be well defined, we must have a − c ≥ 0 which is true by hypothesis. In order L to be c-separating, we must have b ≤ c = 2a − 3b + 1, i.e., a ≥ 2b − 1 Moreover, we have Σ(L) = b + a − b + a − c = 2a − c = 3b − 1 < 3b, hence the amplitude condition is not satisfied and thus C 3 is not (L, b)-colorable. If a < 2b, L is made up as follows: L 1 : b L 2 : c a − c L 3 : b − c c a − b For this list to be well defined, we must have c = 2a − 3b + 1 ≥ b − c = 4b − 2a − 1, i.e., a ≥ 7 4 b − 1/2, which is true by hypothesis. We also have c ≤ a − c since a < 2b. We have Σ(L) = b + a − c + a − b = 2a − c = 3b − 1 < 3b, hence the amplitude condition is not satisfied and thus C 3 is not (L, b)-colorable. Now, in both the cases a ≥ 2b and a < 2b, we prove that C 3 is (a, b, c ′ )-freechoosable with c ′ = c − 1 = 2a − 3b. Again, it is sufficient to verify that the amplitude condition is satisfied by any c ′ -separating a-list assignment L with \|L(x 1 )\| = b. This is clearly true for a subgraph of C 3 reduced to one vertex and for a subgraph of two vertices since we have \|L 2 ∪ L 3 \| ≥ 2a − c ′ = 3b ≥ 2b and \|L 1 ∪ L i \| ≥ b + a − min(b, c ′ ) ≥ 2b for i = 2 or 3. For the whole graph, we have Σ(L) ≥ \|L1\| + \|L2 \ L 1 \| + \|L 3 \ (L 1 ∪ L 2 )\| ≥ b + a − α + a − β, with α = \|L 1 ∩ L 2 \| and β = \|(L 1 ∪ L 2 ) ∩ L 3 \|. Let β = β 1 + β 2 − γ, where β 1 = \|L 1 ∩ L3\|, β 2 = \|L 2 ∩ L 3 \| and γ = \|L 3 ∩ L 2 ∩ L 1 \|. Then we have Σ(L) ≥ b + 2a − α − β 1 − β 2 + γ. As, by definition, α + β 1 + γ ≤ b and β 2 ≤ c ′ , then we obtain Σ(L) ≥ b + 2a − b − c ′ = 2a − c ′ = 3b. Case 3. a ≥ 3b. In this case, C 3 is trivially (a, b, a)-free-choosable and the result follows. In conclusion, in each of the three cases, we have shown that the maximum value of c for which C 3 is (a, b, c)-free choosable is the one given in the statement. Outerplanar graphs An outerplanar graph is a graph that has a planar drawing in which all vertices belong to the outer face of the drawing. For an outerplanar graph G, we denote by T G the weak dual of G, i.e., the graph whose vertex set is the set of all inner faces of G, and E(T G ) = {αβ\| α and β share a common edge}. A cactus is a graph in which every edge is part of at most one cycle. Cactuses form a subclass of outerplanar graphs. The girth of a graph G is the length of a shortest cycle in G. We will use the fact that if an outerplanar graph G is 2-connected, then T G is a tree. Moreover, in order to restrict our argument to 2-connected graphs, we first show the following which has been proved in [1] in the case c = a: Lemma 11. Let a, b, c be integers and let G 1 , G 2 be two (a, b, c)-free-choosable graphs. Then the graph obtained from G 1 and G 2 by identifying any vertex of G 1 with any vertex of G 2 is (a, b, c)-free-choosable. Proof. Let G be the graph obtained by identifying vertex x 1 of G 1 with vertex x 2 of G 2 , resulting in a vertex named x. Let y ∈ V (G) and let L be a list assignment of G with \|L(v)\| = a for v ∈ V (G) \ {y} and \|L(y)\| = b (i.e. y is the precolored vertex). Assume without loss of generality, that y ∈ V (G 1 ). Let L i , i = 1, 2, be the sublist assignment of L restricted to vertices of G i . As G 1 , G 2 are both (a, b, c)-free-choosable, there exists an (L 1 , b)-coloring c 1 of G 1 and an (L 2 , b)-coloring c 2 of G 2 such that c 2 (x) = c 1 (x) (i.e. x is the precolored vertex of G 2 ). The union of colorings c 1 and c 2 is an (L, b)-coloring of G. Moreover, we have min(fsep(C 3 , a, b), fsep(C n , a, b)) =                ⌊ 2 3 (a − b)⌋ if b ≤ a < 7 4 b 2a − 3b if 7 4 b ≤ a ≤ 2n+1 n+1 b or 2n+2 n b ≤ a < 3b ⌊ n−1 n (a − b)⌋ if 2n+1 n+1 b < a < 2n−1 n−1 b ⌊ n−1 n−2 (a − b) − 2 n−2 b⌋ if 2n−1 n−1 b ≤ a < 2n+2 n b a if 3b ≤ a. Proof. The values of fsep for a cycle are given in Theorem 9. For the first assertion, we show that in each of the following cases we have A = c(n, a, b) ≤ B = c(n + 1, a, b) (recall that fsep(C n , a, b) = ⌊c(n, a, b)⌋). • b ≤ a < 2(n+1)−1 n b. Since n−1 n ≤ n n+1 , we have A = n−1 n (a − b) ≤ B = n n+1 (a − b). • 2n+1 n b ≤ a < 2n−1 n−1 b. We want A = n−1 n (a − b) ≤ B = n(a−b)−2b n−1 , i.e. (n − 1) 2 (a − b) ≤ n 2 (a − b) − 2bn . This can be rewritten as a ≥ 4n−1 2n−1 b, which is true since a ≥ 2n+1 n b. • 2(n+1)−1 n b ≤ a < 2 n+2 n+1 b. We want A = (n−1)(a−b)−2b n−2 ≤ B = n(a−b)−2b n−1 , which simplifies to a ≤ 3b an is true by hypothesis. • 2 n+2 n+1 b ≤ a ≤ 2 n+1 n b. We want A = (n−1)(a−b)−2b n−2 ≤ B = a, i.e., a ≤ (n + 1)b which is true. For the second assertion, by Proposition 10 and Theorem 9, we infer the desired inequalities between A = fsep(C 3 , a, b) and B = fsep(C n , a, b), n ≥ 4. • For b ≤ a < 7 4 b, we have A = ⌊ 2 3 (a − b)⌋ ≤ B = ⌊ n−1 n (a − b)⌋. • For 7 4 b ≤ a ≤ 2n+1 n+1 b, we have A = 2a − 3b ≤ B = ⌊ n−1 n (a − b)⌋. • For 2n+1 n+1 b < a < 2n−1 n−1 b, we have A = 2a − 3b ≥ B = ⌊ n−1 n (a − b)⌋. • For 2n−1 n−1 b ≤ a < 2n+2 n b, we have A = 2a − 3b ≥ B = ⌊ n−1 n−2 (a − b) − 2 n−2 b⌋. • For 2n+2 n b ≤ a < 3b, we have A = 2a − 3b ≤ B = a. • For 3b ≤ a, we have A = B = a. From the free-separation number of the cycle we derive the free-separation number of cactuses. Theorem 13. Let G be a cactus with finite girth g and let a ≥ b ≥ 1 be integers. Then If g ≥ 4 or if G has only cycles of length three, then a, b). fsep(G, a, b) = fsep(C g , Otherwise, if G contains at least one triangle and if ℓ is the length of a shortest cycle of G greater than three, then fsep(G, a, b) = fsep(C ℓ , a, b) if 2ℓ+1 ℓ+1 b < a < 2ℓ+2 ℓ b, fsep(C 3 , a, b) otherwise. Proof. Let G be a cactus of finite girth g and let a, b, c be integers. Then, each of its blocks B 1 , B 2 , . . . , B r is either a cycle (of length at least g) or a single edge, and they are connected in a treelike structure. We first show that if g ≥ 4 or if all cycles of G are of length 3, then G is (a, b, c)-freechoosable if and only if c ≤ fsep(C g , a, b). Let c ≤ f sep(C g , a, b) and L be a c-separating a-list assignment of G such that \|L(x 1 )\| = b (i.e., x 1 is the precolored vertex) and suppose without loss of generality that x 1 ∈ B 1 . By Lemma 11, it is sufficient to prove that each block is (a, b, c)-choosable. By Lemma 12, if g ≥ 4, then each block consisting of a cycle C of length at least g can be colored if c ≤ fsep (C g , a, b). Also, trivially, any edge is a, b). Therefore, there exists an (L, b)-coloring of G and thus it is (a, b, c)-free-choosable. Moreover, since G contains a cycle of length g, then fsep(G, a, b) ≤ fsep (C g , a, b). (a, b, c)-choosable if c ≤ a − b, hence fsep(K 2 , a, b) ≥ a − b ≥ fsep(C g , Second, if G contains a triangle and cycles of length greater than three, let ℓ be the length of the shortest cycle of length at least 4. Then, by Lemma 12, fsep(C 3 , a, b) > fsep(C ℓ , a, b) if and only if 2ℓ+1 ℓ+1 b < a < 2ℓ+2 ℓ b. We then proceed as for the first part of the proof, but with c ≤ min(fsep (C g , a, b), fsep(C ℓ , a, b)). As the free-separation number is a lower bound of the separation number, Theorem 13 provides a lower bound on the separation number of cactuses. Moreover, the following result shows that this lower bound is tight in some sense. Proposition 14. For any a ≥ b ≥ 1, and any p ≥ 3, there exists a cactus G of girth p such that sep (G, a, b) = fsep(G, a, b). Proof. Let c = fsep (C p , a, b), and let k = b a . Let G be the graph obtained by joining k copies C 1 , C 2 , . . . , C k of the cycle C p of length p at a shared universal vertex x 1 (see Figure 1 for an illustration in the case a = 2 and b = 1). Let B i , 1 ≤ i ≤ k be the sets of b-subsets of {1, . . . , a}. In order to show that sep(G, a, b) = c, we construct a (c + 1)-separating a-list assignment L of G for which no (L, b)-coloring exists as follows: For each i, 1 ≤ i ≤ k, let L i be a (c + 1)-separating a-list assignment of C i for which L(x 1 ) = B i and other lists of colors only use colors from B i and from a set of colors A with A ∩ {1, . . . , a} = ∅ and such that C i is not (L i , b)-colorable. Since fsep(C p , a, b) = c, such an assignment exists. Let now L be the list assignment of G defined by L(y) = {1, . . . , a}, if y = x 1 L i (y), if y = x 1 and y ∈ C i . Then, by construction, L is a (c + 1)-separating a-list assignment of G. Moreover, whatever the choice of the set of b colors for the vertex x 1 , there will be a cycle C i on which the (L, b)-coloring cannot be completed. Therefore, sep(G, a, b) ≤ c and since sep(G, a, b) ≥ fsep(G, a, b), we have that sep(G, a, b) = fsep(G, a, b). Theorem 15. Let G be an outerplanar graph with finite girth g ≥ 5 and let a, b be integers, a ≥ b ≥ 1. Then we have fsep(C g−1 , a, b) ≤ fsep(G, a, b) ≤ fsep(C g , a, b). Proof. First, we are going to prove that G is (a, b, c)-free choosable for c = fsep(C g−1 , a, b). By Lemma 11, we may suppose that G is 2-connected. Let α 1 , α 2 , . . . , α k be the inner faces of G and let r be any vertex of any face, say α 1 . Let L be a c-separating a-list assignment of G such that \|L(r)\| = b. We define an (L, b)-coloring of G by coloring the vertices of the faces, following a BFS order on the tree T G , starting with the face α 1 . Since g ≥ 5, by Lemma 12, we have c = fsep(C g−1 , a, b) ≤ fsep(α 1 , a, b), thus there exists an (L, b)-coloring of α 1 . At each step, when coloring the vertices of a face α i = (x 1 , x 2 , . . . , x ℓ ), this face shares an edge with at most one face α j with already colored vertices. Assume without loss of generalities, that x 1 x ℓ ∈ α i ∩ α j . Then we have a path P = (x 1 , x 2 , . . . , x ℓ ) of length ℓ with precolored endvertices and since ℓ ≥ g − 1, we have c ≥ fsep(C ℓ,a,b ) = ⌊c(ℓ, a, b)⌋. Therefore, by Lemma 7, there exists an (L, b)coloring of P . By iterating the process on each face, we obtain an (L, b)-coloring of the whole graph G, hence proving that fsep(G, a, b) ≥ c. Second, since G contains a cycle of length g, then fsep(G, a, b) ≤ fsep(C g , a, b). Remark that lower bounds for the free-separation number of an outerplanar graph with girth four can be derived using a proof similar with the one for g ≥ 5 but the formula will be more complex. For outerplanar graphs of girth 3, we cannot use Lemma 7 anymore since it needs the path P being of length at least 3. Concluding remarks We have determined the separation and free-separation number of the cycle and the free separation number of cactuses, and only bounds for general outerplanar graphs G of girth g ≥ 5. For some values of g, a, b the lower and upper bounds of Theorem 15 are equal, but for some not. For instance, we have fsep(C 4 , 9, 4) = 3 and fsep(C 5 , 9, 4) = 4. We conjecture that for any a ≥ b ≥ 1 and any g ≥ 5, there exists an outerplanar graph G of girth g such that sep(G, a, b) = fsep(G, a, b) = fsep(C g−1 , a, b). The problem seems also hard for other simple graphs such as the complete graph K n . In [13], the assymptotic on the minimum a such that K n is (a, 1, 1)-choosable is given. For n = 3, sep(K 3 , a, b) and fsep(K 3 , a, b) are given in Theorem 5 and Proposition 10, respectively. For n = 4 we are able to determine both numbers for any values of a and b, but there are many cases in the formulae. For n = 5 even many more cases have to be considered. We conjecture that f n (a/b) = fsep(K n , a, b) is a piecewise linear function with a number of pieces growing exponentially with n. Figure 1 : 1A cactus with a 1-separating 2-list assignment L for which no (L, 1)-coloring exists. Lemma 12 . 12For any positive integers a, b, n and i, if n ≥ 4 then fsep(C n , a, b) ≤ fsep(C n+i , a, b). 4 b ≤ a < 3b a,3b ≤ a. which is true. Every triangle-free induced subgraph of the triangular lattice is (5m, 2m)-choosable. Y Aubry, J.-C Godin, O Togni, Discrete Applied Math. 166Y. Aubry, J.-C. Godin and O. Togni, Every triangle-free induced subgraph of the triangular lattice is (5m, 2m)-choosable, Discrete Applied Math. 166, (2014), 51-58. Free choosability of outerplanar graphs. Y Aubry, J.-C Godin, O Togni, Graphs and Combinatorics. 323Y. Aubry, J.-C. Godin and O. Togni, Free choosability of outerplanar graphs, Graphs and Combinatorics,32(3), 851-859. Z Berikkyzy, C Cox, M Dairyko, K Hogenson, M Kumbhat, B Lidický, K Messerschmidt, K Moss, K Nowak, K F Palmowski, D Stolee, Choosability of Planar Graphs with Forbidden Structures, Graphs and Combinatorics. 2751Z. Berikkyzy, C. Cox, M. Dairyko, K. Hogenson, M.M Kumbhat, B. Lidický, K. Messerschmidt, K. Moss, K. Nowak, K. F. Palmowski, D. Stolee, (4, 2)-Choosability of Planar Graphs with Forbidden Structures, Graphs and Combinatorics (2017) 33: 751. On choosability with separation of planar graphs without adjacent short cycles. M Chen, K.-W Lih, W Wang, Bull. Malays. Math. Sci. Soc. 41M. Chen, K.-W. Lih, W. Wang On choosability with separation of planar graphs without adjacent short cycles, Bull. Malays. Math. Sci. Soc., 41 (2018), 1507-1518. Choosability with separation of planar graphs without prescribed cycles. M Chen, Y Fan, A Raspaud, W C Shiu, W Wang, Applied Mathematics and Computation. 367M. Chen, Y. Fan, A. Raspaud, W. C. Shiu, W. Wang, Choosability with separation of planar graphs without prescribed cycles, Applied Mathematics and Computation 367, (2020). Stolee On choosability with separation of planar graphs with forbidden cycles. I Choi, B Lidický, D , J. Graph Theory. 81I. Choi, B. Lidický, D. Stolee On choosability with separation of planar graphs with forbidden cycles, J. Graph Theory, 81 (2016), 283-306. A sufficient condition for planar graphs to be (3,1)-choosable. M Chen, Y Fan, Y Wang, W Wang, J. Comb. Optim. 34M. Chen, Y. Fan, Y. Wang, W. Wang A sufficient condition for planar graphs to be (3,1)-choosable, J. Comb. Optim., 34 (2017), 987-1011. Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to list-multicolorings of graphs. M M Cropper, J L Goldwasser, A J W Hilton, D G Hoffman, P D Johnson, J. Graph Theory. 334M. M. Cropper, J. L. Goldwasser, A. J. W. Hilton, D. G. Hoffman, P. D. Johnson, Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to list-multicolorings of graphs. J. Graph Theory 33 (2000), no. 4, 199-219. L Esperet, R J Kang, S. Thomassé Separation Choosability and Dense Bipartite Induced Subgraphs, Combinatorics, Probability and Computing. 28L. Esperet, R.J. Kang, S. Thomassé Separation Choosability and Dense Bipartite Induced Subgraphs, Combinatorics, Probability and Computing (2019),28, 720-732. Kumbhat Choosability with separation of complete multipartite graphs and hypergraphs. Z Füredi, A Kostochka, M , J. Graph Theory. 76Z. Füredi, A. Kostochka, M. Kumbhat Choosability with separation of complete multipartite graphs and hypergraphs, J. Graph Theory, 76 (2014), 129-137. On choosability with separation of planar graphs with lists of different sizes. H A Kierstead, B Lidický, Discrete Mathematics. 33810H.A. Kierstead, B. Lidický, On choosability with separation of planar graphs with lists of different sizes, Discrete Mathematics 338:10 (2015), 1779-1783. Complexity of choosing subsets from color sets. J Kratochvíl, Z Tuza, M Voigt, Discrete Math. 191J. Kratochvíl, Z. Tuza, M. Voigt, Complexity of choosing subsets from color sets, Discrete Math., 191:1-3 (1998), 139-148, Voigt Brooks-type theorems for choosability with separation. J Kratochvíl, Z Tuza, M , J. Graph Theory. 27J. Kratochvíl, Z. Tuza, M. Voigt Brooks-type theorems for choosability with separa- tion, J. Graph Theory, 27 (1998), 43-49. Choosability with union separation. M Kumbhat, K Moss, D Stolee, Discrete Math. 341M. Kumbhat, K. Moss, D. Stolee, Choosability with union separation, Discrete Math. 341: 3 (2018), 600-605. A note on choosability with separation for planar graphs. R Škrekovski, Ars Comb. 58R.Škrekovski A note on choosability with separation for planar graphs, Ars Comb., 58 (2001), 169-174.	[]
[ "Visually Grounded Concept Composition", "Visually Grounded Concept Composition", "Visually Grounded Concept Composition", "Visually Grounded Concept Composition" ]	[ "Bowen Zhang \nGoogle Research\nGoogle Research\nUSC\n\n", "Hexiang Hu \nGoogle Research\nGoogle Research\nUSC\n\n", "Google Research \nGoogle Research\nGoogle Research\nUSC\n\n", "Linlu Qiu \nGoogle Research\nGoogle Research\nUSC\n\n", "Shaw Peter \nGoogle Research\nGoogle Research\nUSC\n\n", "Research Google \nGoogle Research\nGoogle Research\nUSC\n\n", "Sha Fei \nGoogle Research\nGoogle Research\nUSC\n\n", "Bowen Zhang \nGoogle Research\nGoogle Research\nUSC\n\n", "Hexiang Hu \nGoogle Research\nGoogle Research\nUSC\n\n", "Google Research \nGoogle Research\nGoogle Research\nUSC\n\n", "Linlu Qiu \nGoogle Research\nGoogle Research\nUSC\n\n", "Shaw Peter \nGoogle Research\nGoogle Research\nUSC\n\n", "Research Google \nGoogle Research\nGoogle Research\nUSC\n\n", "Sha Fei \nGoogle Research\nGoogle Research\nUSC\n\n" ]	[ "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n", "Google Research\nGoogle Research\nUSC\n" ]	[]	We investigate ways to compose complex concepts in texts from primitive ones while grounding them in images. We propose Concept and Relation Graph (CRG), which builds on top of constituency analysis and consists of recursively combined concepts with predicate functions. Meanwhile, we propose a concept composition neural network called Composer to leverage the CRG for visually grounded concept learning. Specifically, we learn the grounding of both primitive and all composed concepts by aligning them to images and show that learning to compose leads to more robust grounding results, measured in text-to-image matching accuracy. Notably, our model can model grounded concepts forming at both the finer-grained sentence level and the coarser-grained intermediate level (or word-level). Composer leads to pronounced improvement in matching accuracy when the evaluation data has significant compound divergence from the training data.	10.18653/v1/2021.findings-emnlp.20	[ "https://arxiv.org/pdf/2109.14115v1.pdf" ]	238,215,187	2109.14115	cdb2a71f42c88be6d5c5e4fecb8288a9a315f094	Visually Grounded Concept Composition Bowen Zhang Google Research Google Research USC Hexiang Hu Google Research Google Research USC Google Research Google Research Google Research USC Linlu Qiu Google Research Google Research USC Shaw Peter Google Research Google Research USC Research Google Google Research Google Research USC Sha Fei Google Research Google Research USC Visually Grounded Concept Composition We investigate ways to compose complex concepts in texts from primitive ones while grounding them in images. We propose Concept and Relation Graph (CRG), which builds on top of constituency analysis and consists of recursively combined concepts with predicate functions. Meanwhile, we propose a concept composition neural network called Composer to leverage the CRG for visually grounded concept learning. Specifically, we learn the grounding of both primitive and all composed concepts by aligning them to images and show that learning to compose leads to more robust grounding results, measured in text-to-image matching accuracy. Notably, our model can model grounded concepts forming at both the finer-grained sentence level and the coarser-grained intermediate level (or word-level). Composer leads to pronounced improvement in matching accuracy when the evaluation data has significant compound divergence from the training data. Introduction Visually grounded text expressions denote the images they describe. These expressions of visual concepts are naturally organized hierarchically in sub-expressions. The organization reveals structural relations that do not manifest when the subexpressions are studied in isolation. For example, the phrase "a soccer ball in a gift-box" is a compound of two shorter phrases, i.e., "a soccer ball" and "a gift-box", but carries the meaning of the spatial relationship "something in something" that goes beyond the two shorter phrases separately. The compositional structure of the grounded expression requires a concept learner to understand what primitive concepts are visually appearing and * Part of work done while at Google † Part of work done while at USC ‡ Work done as a Google AI resident. how the compound relating multiple primitives modifies their appearance. Existing approaches (Kiros et al., 2014;Faghri et al., 2017;Lu et al., 2019;Chen et al., 2020Chen et al., , 2021 tackle visual grounding via end-to-end learning, which typically learns to align image and text information using neural networks without explicitly modeling their compositional structures. While neural networks have shown strong generalization capabilities in test examples that are i.i.d to the training distribution (Devlin et al., 2019), they often struggle in dealing with out-of-domain examples of novel compositional structures, in many tasks such as Visual Reasoning (Johnson et al., 2017;Bahdanau et al., 2019;Pezzelle and Fernández, 2019), Semantic Parsing (Finegan-Dollak et al., 2018;Keysers et al., 2020), and (Grounded) Command Following (Lake and Baroni, 2018;Chaplot et al., 2018;Hermann et al., 2017;Ruis et al., 2020). In this work, we investigate how complex concepts, composed of simpler ones, are grounded in images at sentences, phrases and tokens levels. In particular, we investigate whether the structures of how these concepts are composed can be exploited as a modeling prior to improve visual grounding. To this end, we design Concept & Relation Graph (CRG), which is derived from constituency parse trees. The resulting CRG is a graph-structured database where concept nodes encode language expressions of concepts and their visual denotations (e.g., a set of images corresponding to the concept), and predicate nodes define how a concept is semantically composed from its child concepts. Our graph is related to the denotation graph (Young et al., 2014;Zhang et al., 2020) but differs in two key aspects. First, our graph extracts the concepts without specially crafted heuristic rules 1 . Secondly, CRG's predicate can encode richer information explicitly than the subsumption relationships implicitly expressed in the denotation graphs. An illustrative figure of the graph is shown in Figure 1. In addition to CRG, we propose Concept cOMPOSition transformER (COMPOSER) that leverages the structure of text expressions to recursively encode the grounded concept embeddings, from coarse-level such as the noun words that refer to objects, to finer-grained ones with multiple levels of compositions. Transformer (Vaswani et al., 2017) is used as a building block in our model, to encode the predicates, and perform grounded concept composition. We learn COMPOSER using the task of visual-semantic alignment. Unlike traditional approaches, we perform hierarchical learning of visual-semantic alignment, which aligns the image to words, phrases, and sentences, and preserves the order of matching confidences. We conduct experiments on multi-modal matching and show that COMPOSER achieves strong grounding capability in both sentence-to-image and phrase-to-image retrieval on the popular benchmarks. We validate the generalization capability of COMPOSER by designing an evaluation procedure for a more challenging compositional generalization task that uses test examples with maximum compound divergence (MCD) to the training data (Shaw et al., 2020;Keysers et al., 2020). Experiments show that COMPOSER is more robust to the compositional generalization than other approaches. Our contributions are summarized as below: • We study the compositional structure of visually grounded concepts and design Concept & Relation Graph that reflects such structures. • We propose Concept cOMPOSition transformER (COMPOSER) that recursively composes concepts using the child concepts and the semantically meaningful rules, which leads to strong compositional generalization performances. • We propose a new evaluation task to assess the model's compositional generalization performances on the task of text-to-image matching and conduct comprehensive experiments to evaluate both baseline models and COMPOSER. been developed and evaluated on English language corpus, and its multilingual utility depends on the parsing techniques for those languages other than English. Concept & Relation Graph We introduce multi-modal Concept and Relation Graph (CRG), a graph composed of concept and predicate nodes, which compose visually grounded descriptive phrases and sentences. Figure 1 provides an illustrative example. The concepts include sentences and intermediate phrases, shown as blue nodes. The primitives are the leaf nodes (typically noun words) that refer to visual objects, shown as green nodes. The predicates (red nodes) are n-ary functions that define the meaning of the concept composition. Their "signatures" consist of lexicalized templates, the number of arguments, and the syntactic type of the arguments. They combine primitives or simpler concepts into more complex ones. Identifying concepts and relations. Given pairs of aligned image and sentence, we first parse a sentence into a constituency tree, using a state-ofthe-art syntactic parser (Kitaev and Klein, 2018). We use the sentence's constituent tags to identify concepts and their relations. The set of relations are regarded as n-ary functions with placeholders denoted with constituency tags. We refer to such functions as predicates. Simpler concepts are arguments to the predicates, and the return values of the functions are complex concepts. The edges of the graph represent the relationship between predicates and their arguments. We restrict the type of constituents that can be concepts and how the predicates can be formed. A concrete example is as follows: given an input concept "two dogs running on the grass", the algorithm extracts the predicate "[NP] running on [NP]" and the child concepts "two dogs" and "the grass". Here we use syntactic placeholders to replace the concept phrases. Details are in the Appendix. This idea is closely related to the semantically augmented parse trees (Ge and Mooney, 2009), though we focus on visually grounded concepts. Finding visually grounded concepts. We take paired images and texts 2 , and convert the texts into derived trees of predicates and primitives. With the generated text graph, we then group all images that refers to the same concept to form the image denotation, similar as Young et al. (2014) and Zhang et al. (2020). The image denotation is the set of images that contain the referred concept. For example, the image denotation of the concept "ball" is all the images that have the visual object category "ball". As a result, we associate the image denotation with each concept in the format of words, phrases, and sentences, which creates a multi-modal graph database as Figure 1. COMPOSER: Recursive Modeling of the Compositional Structure The main idea of COMPOSER is to recursively compose primitive concepts into sentences of complex structure, using composition rules defined by the predicates. Figure 2 presents a conceptual diagram of the high-level idea. Concretely, it first takes the primitive word embedding as the inputs and performs cross-modal attention to obtain their visually grounded word embeddings. Next, the COMPOSER calls the composition procedure to modify or combine primitive or intermediate concepts, according to the description of its predicates. At the end of this recursive procedure, we obtain the desired sentence concept embedding. In the rest of this section, we first discuss the notation and backgrounds, then introduce how primitives and predicates are encoded ( § 3.1), and present the recursive composition procedures in detail ( § 3.2). Finally, we discuss the learning objectives ( § 3.3). Notation. We denote a paired image and sentence as (x, y) and the corresponding concepts and predicate for a tree (x, U , E), where U , E corresponds to the set of primitives and the set of predicates, respectfully. We also denote all concepts from a sentence y to be C, where U ⊂ C and y ∈ C. Multi-head attention mechanism. Multi-Head Attention (MHA) (Vaswani et al., 2017) is the building block of our model. It takes three sets of input elements, i.e., the key set K, the query set Q, and the value set V , and perform scaled dot-product attention as: MHA(K, Q, V ) = FFN Softmax( Q K √ d ) · V Here, d is the dimension of elements in K and Q. FFN is a feed-forward neural network. With different choices of K and V , MHA can be categorized as self-attention (SelfAtt) and cross-attention (CrossAtt), which corresponds to the variants with K and V including only the single-modality or cross-modality features. Encoding Primitives and Predicates Given a paired image and sentence (x, y), we parse the sentence as the tree of primitives and predicates (x, U , E). Here, we represent the image as a set of visual feature vectors {φ}, which are the objectcentric features from an object detector (Anderson et al., 2018). Noted that we didn't use structural information beyond object proposals/regions. Our COMPOSER takes the primitives and predicates as input and output the visually grounded concept embeddings, with both the primitives and predicates as continuous vectors of different contextualization. Representing primitives with visual context. The primitive concepts refer to tokens which can be visually grounded, and we represent them as word embeddings contextualized with visual features. As such, we use a one-layer Transformer with the CrossAtt mechanism, where K, V , and Q are linear transformations of φ, φ, and u, respectively. This essentially uses the word embedding to query the visual features and outputs the grounded primitive embeddingsÛ = {û}. Note that the output is always a single vector for each primitive as it is a single word. Representing predicates as neural templates. A predicate e is a semantic n-place function that combines multiple concepts into one. We represent it as a template sentence with words and syntactic placeholders, such as "[NP] 1 running on [NP] 2 ", where those syntactic placeholders denote the positions and types of arguments. We encode such template sentences via SelfAtt mechanism, using a multi-layer Predicate Transformer (PT). The output of this model is a contextualized sequence of the words and syntactic placeholders asê. Recursive Concept Composition With the encoded primitivesÛ and predicateŝ E, the COMPOSER then performs multiple recursive composition steps to obtain the grounded concept embedding, v(x, y), representing the visuallinguistic embedding of the sentence and the image as shown in the Figure 2. To further illustrate this process, we detail the composition function in below, as shown in Figure 3. Input concept modulation. We use a modulator to bind the arguments in the predicate to the input child concepts. Given a encoded predicateê = {[NP] 1 , running, on, [NP] 2 , with, [NP] 3 } and a input concept c 1 = "a man", the modulator is a neural network that takes the concept embedding c 1 and its corresponding syntactic placeholder [NP] 1 as input and outputs a modulated embedding. This embedding is then reassembled with the embeddings of non-arguments in the predicate and used for the later stage. Contextualization with visual context. After concept modulation, we get a sequence of embeddings for non-argument words of the predicate and the binded child concepts, which is then fed as an input to a Composition Transformer (CT) model. This Transformer has multiple layers, with both CrossAtt layers that attends to the object-centric visual features and SelfAtt layers that contextualize between tokens. Please refer to Appendix for the detailed network architecture. Given that our model is recursive by nature, the computation complexity of CT is proportional to the depth of the tree. We provide a comprehensive study in § 5.3 to show the correlation between the parameter/complexity and model's performances. Learning COMPOSER with Visual-Semantic Alignments With the composed grounded concept embedding v(x, y), we use the visual-semantic alignment as the primary objective to learn COMPOSER. To this end, we compute the alignment score by learning an additional linear regressor θ: s(x, y) = θ · v(x, y) ∝ p(x, y), where p(x, y) is the probability that the sentence and image is a good match pair. Then we learn the sentence to image alignment by minimizing the negative log-likelihood (NLL): MATCH = − i log exp(s(x i , y i )) (x,ŷ)∼D i exp(s(x,ŷ)) with D i = {(x i , y i )} ∪ D − i . To properly normalize the probability, it is necessary to sample a set of negative examples to contrast. Thus, we generate D − i using the strategy of Lu et al. (2019). Multi-level visual-semantic alignment (MVSA). Since COMPOSER composes grounded concepts recursively from the primitives, we obtain the embeddings of all the intermediate concepts automatically. Therefore, it is natural to extend the alignment learning objectives to all those intermediate concepts. We optimize the triplet hinge loss (Kiros et al., 2014): MVSA = i c∈C i [α − s(x i , c) + s(x i , c − )] + + [α − s(x i , c) + s(x − i , c)] + where [h] + = max(0, h) denotes the hinge loss and α is the margin to be tuned. We derive the negative concepts c − from the negative sentences in the D − i . We observe that negative concepts at word/phrase levels are noisier than the ones at sentence level because many are common objects presented in the positive image and lead to ambiguity in learning. Therefore, we choose hinge loss over NLL because it is more robust to label noises (Biggio et al., 2011). Learning to preserve orders in the tree. Finally, we use an order-preserving objective proposed by Zhang et al. (2020), to ensure that a finegrained concept (closer to sentence) can produce a more confident alignment score than a coarsegrained concept (closer to primitive): ORDER = i e jk [β − s(x i , c j ) + s(x i , c k )] + Here, e jk represents a predicate connecting the c j and c k , with c j to be the fine-grained parent concept which is closer to the sentence and c k to be the coarse-grained child concept which is closer to the primitives. β is the margin that sets the constraint on how hard the order of embeddings should be reserved. The complete learning objective is a weighted combination of three individual losses defined above, with the loss weights λ 1 = 1 and λ 2 = 1: = MATCH + λ 1 · MVSA + λ 2 · ORDER The details of model optimization and hyperparameter setting are included in the Appendix. (2017) evaluate RL agents' capability to generalize to a novel composition of shape, size, and color in 3D simulators, which shows that RL agents generalize poorly. gSCAN (Ruis et al., 2020) perform a systematic benchmark to assess command following in a grounded environment. In this work, we focus on assessing model composition generalization under the visual context. Compositional networks. State-of-the-art visually grounded language learning typically use deep Transformer models (Vaswani et al., 2017) such as ViLBERT (Lu et al., 2019), LXMERT (Tan and Bansal, 2019) and UNITER (Chen et al., 2020). Though being effective for data over i.i.d distribution, these models do not explicitly exploit the structure of the language and are thus prone to fail on compositional generalization. In contrast, another thread of works (Andreas et al., 2016;Yi et al., 2018;Mao et al., 2019;Shi et al., 2019;Wang et al., 2018) parse the language into an executable program composed as a graph of atomic neural modules, where each module is designed to perform atomic tasks and are learned end-to-end. Such models show almost perfect performances on synthetic benchmarks (Johnson et al., 2017) but perform subpar on the real-world data (Young et al., 2014;Chen et al., 2015) that are noisy and highly variable. Unlike them, we propose using a compositional neural network based on the Transformer architecture, which extends state-of-the-art neural networks to explicitly exploits language structure. Experiment In this section, we perform experiments to validate the proposed COMPOSER model on the tasks of sentence-to-image retrieval and phrase-to-image retrieval. We begin with introducing the setup in § 5.1 and then present the main results in § 5.2, comparing models for their in-domain, cross-dataset evaluation, and compositional generalization performance. Finally, we perform an analysis and ablation study of our model design in § 5.3. Compositional generalization evaluation. To generate evaluations of compositional generalization, we use a method similar to that of Shaw et al. (2020) and Keysers et al. (2020) which maximizes compound divergence between the distribution of compounds in the evaluation set and in the training set. Here compounds are defined based on the pred-icates occurring in captions. Following this method, we first calculate the overall divergence of compounds from the evaluation data to the training data using predicates from all the sentences. Then, for each sentence in the evaluation data, we calculate a compound divergence with this specific example removed. We rank those sentences based on the difference of the compound divergence. Finally, we choose the top-K sentences with the largest compound divergence differences and its corresponding images to form the evaluation splits. Experiment Setup Using this method, we generate evaluation splits with 1,000 images and 5,000 text queries, COCO-MCD and F30K-MCD, to assess models trained on F30K and COCO, respectively. Therefore, these splits assess both compositional generalization and cross-dataset transfer. Defining such splits across datasets is also helpful to achieve greater compound divergence than is otherwise possible, given the small amount of available in-domain test data. More details are included in Appendix. CRG construction. We constructed two CRGs on the F30K and C30K datasets, using the procedure mentioned in § 2. The key statistics of the graph we generated as shown in Table 1. Baselines and our approach. We compare COMPOSER to two strong baseline methods, i.e., ViLBERT (Lu et al., 2019) and VSE (Kiros et al., 2014). We make sure all models are using the same object-centric visual features extracted from the Up-Down object detector (Anderson et al., 2018) for fair comparison. For the texts, both ViLBERT and the re-implemented VSE use the pre-trained BERT model as initialization. For the COMPOSER, we only initialize the predicate Transformer with the pre-trained BERT, which uses the first six layers. Note that the ViLBERT results are re-produced using the codebase from its author. ViLBERT is not pre-trained on any additional data of imagetext pairs to prevent information leak in both crossdataset evaluation and compositional generalization. Therefore, we used the pre-trained BERT models provided by HuggingFace to initalize the text stream of ViLBERT, and then followed the rest procedure in the original ViLBERT paper. Please refer to Appendix for complete details. Main Results We compare the COMPOSER with ViLBERT (Lu et al., 2019) F30k and COCO for in-domain, zero-shot crossdataset transfer, and compositional generalization (e.g. F30K→COCO-MCD). The notation A→B means that the model is trained on A and evaluated on B. We report the results of sentence-to-image retrieval in the main paper and defer more ablation study results to the Appendix. In-domain performance. Table 2 presents the in-domain performance on both F30k and COCO datasets. First, we observe that both COMPOSER and ViLBERT consistently outperform VSE, which is expected as ViLBERT contains a cross-modal transformer with stronger modeling capacity. Comparing to ViLBERT, the COMPOSER performs on par. Zero-shot cross-dataset transfer. We also consider zero-shot cross dataset transfer where we evaluate models on a dataset that is different from the training dataset. In this setting, the COMPOSER outperforms ViBLERT and VSE significantly. Concretely, on the F30k→COCO setting, the COM-POSER improves R1 and R5 by 11.0% and 7.0% Figure 4. With the increases of CD, we observe the performance of COMPOSER and ViLBERT decreases. Compared to ViLBERT, we observe that COMPOSER is relatively more robust to this distribution shift, as the relative performance improvement is increasing with CD increases. Analysis and Ablation Study We perform several ablation studies to analyze COMPOSER, and provide qualitative results to demonstrate the model's interpretability. Is CrossAtt in primitive encoding useful? Table 3 compares variants of COMPOSER with and without CrossAtt for primitive encoding, and shows that CrossAtt improves all metrics in indomain and cross-dataset evaluation. Which modulator works better? We consider three modulators to combine input concepts with the syntax token embeddings for later composition, which are Replace, MLP, and FiLM. The Replace directly replaces the syntax embedding with the input concept embedding. This is an inferior approach by design as it ignores the relative position of each concept. MLP model applies multilayer neural networks on the concatenated syntax and input concept embeddings. FiLM model uses the syntax embedding to infer the parameter of an affine transformation, which is then applied to the input concepts. We show the results in Table 4. Replace achieves the worst performance, indicating the importance of identifying the position of input concepts. COMPOSER chooses FiLM as the modulator given its strong performance over all metrics. Is MVSA supervision useful? We evaluate the influence of multi-level visual-semantic alignment on sentence and phrase to image retrieval. In the phrase-to-image experiments, we sample 5 nonsentence concepts from the CRG for each annotation in the corresponding test data and use them as the query to report results (in R1). Performance under different parsing qualities. CRG is generated based on constituent parser. We investigate the performance of COMPOSER with CRG under different parsing qualites. Given a parsing tree, We randomly remove its branches randomly with a probability of 0.1, 0.3, or 0.5 to generate a tree with degraded parsing quality. We evaluate COMPOSER on the resulting CRGs. We summarized the results in Table 7. When parsing quality drops, both in-domain and cross-dataset transfer performance drops. The performance degrades by 12%, when half of the parse could be missing. We expect with better parsing quality, COMPOSER can achieve stronger performance. Interpreting COMPOSER's decision. Despite the solid performance, COMPOSER is also highly interpretable. Specifically, we visualize its alignment scores along with the concept composition procedure in Figure 5. Empirically, we observe that most failures are caused by visually grounding mistakes at the primitive concepts level. The error then propagates "upwards" towards concept composition. For instance, the left example shows that COM-POSER is confusing between the ground truth and negative image when only the text of shared visual concept "a bold man" is presented. With more information are given, it gets clarified immediately as it notices that the target sentence is composed not only with the above subject, but also with the prepositional phrases "by the beer pumps at the bar" that reflects the visual environment. Scalability to full COCO dataset. Finally, we trained our model (PT=6, CT=5) on the full COCO training split and evaluated for both in-domain and cross-dataset transfer task. We use the same hyperparameters as C30K. However, COMPOSER underperforms the ViLBERT in this setting, as it achieves 56.06% and 44.24% in R1 for the indomain task (COCO→COCO) and cross-dataset evaluation tasks (COCO→F30k), while ViLBERT obtains 56.83% and 46.62%, respectively. We hypothesize that this negative result is largely due to the limited model capacity of the proposed COM-POSER, as it has relatively 33% less parameters comparing to ViLBERT. Meanwhile, it is also observed that COMPOSER performs worse than ViL-BERT in fitting training data. We observe that doubling the training epoch would increase both indomain and out-of-domain performance by 2% relatively. Increasing the layer of Composition Transformer (CT) to 7 would also improves R1 by 2.5% relatively. Further scaling up COMPOSER may resolve this issue but requires more computational resources, and we leave this for future research. Conclusion In this paper, we propose the concept and relation graph (CRG) to explore the compositional structure in visually grounded text data. We further develop a novel concept composition neural network (COMPOSER) on top of the CRG, which leverages the explicit structure to compose concepts from word-level to sentence-level. We conduct extensive experiments to validate our model on image-text matching benchmarks. Comparing with prior methods, COMPOSER achieves significant improvements, particularly in zero-shot cross-dataset transfer and compositional generalization. Despite these highlights, there are also many challenges that COMPOSER does not address in the scope of this paper. First, it requires high-quality parsing results to achieve strong performances, which may not be readily available in languages beyond English. Moreover, similar to other recursive neural networks, COMPOSER is also computationally resource demanding, which sets a limit to its scalability to large-scale data. As mentioned in the main paper, we parse the sentence and convert it into a tree of concepts and primitives. Particularly, we first perform constituency parsing using the self-attention parser (Kitaev and Klein, 2018). Table 8 provides the visualization for two examples of the syntax sub-trees. Next, we perform a tree search (i.e., breadth-first search) on the constituency tree of the current input concept to extract the sub-concepts and predicate functions. Note that this step is applied recursively until we can no longer decompose a concept into any subconcepts. On a single step of the extraction, we enumerate each node in the constituency tree of current input text expression and examine whether a constituent satisfies the criterion that defines the visually grounded concept. The concept criterion defined for the Flickr30K and COCO dataset contains several principles: (1) If the constituent is a word, it is a primitive concept if its Part-of-Speech (POS) tag is one of the following: Table 8. For instance, in the first example, we search the text "two dogs are running on the grass" and extract two noun constituents, "two dogs" and "the grass" as the concepts. We use the remaining text "[NP] is running on [NP]" as the predicate that indicates the semantic meaning of how these two sub-concepts composes into the original sentence. B Details on Generation of Compositional Evaluation Splits As mentioned in the main text, we generate compositional generalization (CG) splits with 1,000 images and 5,000 text queries, maximizing the Compound Divergence (MCD) as Shaw et al. (2020) 3 , to assess models' capability in generalizing to the data with different predicate distribution. Concretely, we select Flickr30K training data to generate the F30K-MCD split. First, we remove all F30K test data that has unseen primitive concepts to the COCO training data. Next, we collect and count the predicates for each image among all the remaining data over the five associated captions. These predicates correspond to the "compounds" defined in (Keysers et al., 2020;Shaw et al., 2020), and the objective is to maximize the divergence between compound distribution of the evaluation data to the training data. As a result of this step, we end up with a data set formed with pairs of (image, predicates counts), which are then used for computing the overall compound divergence (CD ALL ) to the training dataset. Afterwards, we enumerate over each pair of data, and again compute the compound divergence to the training dataset but with this specific data is removed. We denote the change of compound divergence as ∆ i = CD i − CD ALL , and use it as an additional score to associate every data. Finally, we sort all the data with regard to the difference of compound divergence ∆ i , and use the top ranking one thousand examples as the maximum compound divergence (MCD) split. The process for generating the COCO-MCD split is symmetrical to the above process, except the data is collected from COCO val+test splits (as it is sufficiently large). Similarly, to generate different CDs for making Figure 4 of the main text, we can also make use of the above data sorted by ∆ i . Concretely, we put a sliding window with 1,000 examples and enumerate over the sorted data to obtain a massive combination of data (we can take a Sub-Concepts NP1="two dogs" NP2="the grass" NP1="a small pizza" NP2="a white plate" Table 8: Explanatory example of extracting predicates and sub-concepts from a concept stride to make this computation sparser.) For each window of data, we measure the compound divergence and only take the windows that are at the satisfaction to our criteria. In Figure 4, we keep the windows that has the closest CD values to desired X-axis values for plotting. C Implementation Details of COMPOSER and Baselines Visual feature pre-processing We follow ViL-BERT (Lu et al., 2019) that extracts the patchbased ResNet feature using the Bottom-Up Attention model. The image patch feature has a dimension of 2048. A 5-dimension position feature that describes the normalized up-top and bottom-down position is extracted alongside the image patch feature. Therefore, each image region is described by both the image patch feature and the position feature. We extracted features from up to 100 patches in one image. We implement the CrossAtt model as a one-layer multi-head cross-modal Transformer that contains 768 dimension with 12 attention heads. The query set Q is the sub-word token embeddings of the primitive word, and the key and value set K and V are the union of sub-word token embeddings and the object-centric visual features (which is linearly transformed to have the same dimensionality). We use the average of the contextualized sub-word token embeddings as the final primitive encoding. • Predicate Transformer (PT). We use 6 layers text Transformers with 768 hidden dimension and 12 attention heads to instantiate the Predicate Transformer. This network is initialized with the first 6 layers of a pre-trained BERT model. • Modulator. We use FiLM (Perez et al., 2018) as the modulator. Specifically, it contains two MLP models with a hidden dimension size of 768 to generate the scale a and bias vectors b, using the syntactic placeholders as input. The scale a and bias b are then used to transform the input concept embedding c as a c+b. Here represents the element-wise multiplication. This modulated concept embedding is then projected by another MLP with 768 hidden dimensions, and used for reassembling with the predicate sequence. • Composition Transformer (CT). We follow the architecture of ViLBERT (Lu et al., 2019) to design the Composition Transformer (shown in Figure 6). Specifically, it has interleaved SelfAtt Transformer and CrossAtt Transformer in the network. For example, if we consider a three-layer Composition Transformer, we have a SelfAtt Transformer at the beginning for both modality, followed with a CrossAtt Transformer that interchanges the information between the modality, and then another SelfAtt Transformer that only operates on the text modality. The output embedding of this last text SelfAtt Transformer is then used for computing the visual-semantic alignment scores using the linear regressor θ. Thus, when we consider shallower or deeper network, we add or remove the two layers of interleaved SelfAtt and CrossAtt Transformers. The hidden dimension of SelfAtt Transformer is 768, and D Additional Experiments on COMPOSER We report additional ablation studies that are omitted in the main paper due to space limitation. across both in-domain and cross-dataset generalization settings. Therefore, for all the experiments training with MVSA, we use hinge loss instead. Ablation study on α and β. We study COM-POSER performance on the different margin of MVSA and Order objectives. First, we fix the margin of order objectives β and tune the margin for MVSA α. COMPOSER with a larger margin for MVSA achieves better R1 in-domain performance. Alternatively, by fixing the α and tuning β, COMPOSER achieves the best R1 in-domain performance and best R5 in cross-dataset generalization setting with β = 0.2. Figure 1 : 1Concepts and their visual denotations organized by the Concept & Relation Graph Figure 2 : 2The overall design of the proposed COMPOSER model. Figure 3 : 3Details of the composition procedure. For example, the output sequence becomes {Mod([NP] 1 , c 1 ), running, on, Mod([NP] 2 , c 2 ), with, Mod([NP] 3 , c 3 )} after the modulator processed each pair of input concept and syntactic placeholder. Various choices of neural networks are available for this modulator, such as a Multi-Layer Perceptron (MLP) or a Feature-wise Linear Modulation (FiLM) (Perez et al., 2018). COMPOSER uses FiLM for its strong empirical performance. grounded language understanding. Many evaluation methods are proposed to assess the model's generalization capabilities in grounded language understanding. Johnson et al. (2017) proposes a synthetic dataset, i.e. CLEVR, to evaluate the generalization of visual question answering models to novel objects and attributes. Misra et al. (2017) proposes to evaluate compositional generalization capability of visual models w.r.t. short phrases consist of attributes and objects. Chaplot et al. (2018) and Hermann et al. Figure 4 : 4COMPOSER's results on generalization splits of different compound divergence over text description (evaluated under the F30K→COCO setting). {[NN],[NNS],[NNP],[NNPS]};(2) If the constituent is a phrase (with two words or more), it would be a concept when this constituent contains a primitive word (i.e., satisfying condition (1)) and its constituency tag is one of the following:{[S], [SBAR], [SBARQ], [SQ], [SINV], [NP], [NX]}. After all the concepts are extracted, we take the remaining words in the current input text expression as the predicate that combines those concepts and use the tag to represent syntactic blank. Concrete examples can be found in the Object-centric Visual FeaturesPredicate as Templated Sentences with Syntactic Blankssoccer man field ball a soccer ball a man running on the field with a soccer ball a man the field soccer man field ball Primitive Encoding Primitive Encoding Concept Encoding with Recursive Composition Concept Embedding Dataset # concepts # predicates # primitives Avg heightTable 1: Statistics of the concepts and predicates in the F30K and C30K datasets.F30K 408,464 122,196 10,755 3.09 C30K 345,331 88,623 9,683 2.86 Datasets.We perform experiments on the COCO-caption (COCO)(Chen et al., 2015) and Flickr30K (F30K)(Young et al., 2014) datasets. Each image of these two datasets is associated with five sentences. Flickr30K contains 31,000 images, and we use the same data split as (Faghriet al., 2017), where there are 29,000 training im- ages, 1000 test images, and 1000 validation images. COCO contains 123,287 images in total. For fast it- eration, we use a subset training data C30K, which contains the same amount of images as the F30K. Note that C30K is a training split. We also trained models on the full COCO training split. For COCO dataset, the results are evaluated on COCO 1K test split (Karpathy and Fei-Fei, 2015). We use COCO 1k test split for both in-domain (models trained on either C30K or full COCO training split and eval- uate on COCO-caption) and cross-dataset transfer (models trained on F30K and evaluate on COCO- caption) evaluation. For both F30K and COCO 1K test split, there are 5,000 text queries and 1,000 can- didate images to be retrieved. We report recall@1 (R1) and recall@5 (R5) as the primary retrieval metric. and VSE (Kiros et al., 2014) on(a) Models trained on F30K Eval on F30K COCO COCO-MCD Method R1 R5 R1 R5 R1 R5 VSE 46.84 77.16 25.60 54.36 21.82 47.58 ViLBERT 50.94 80.86 30.50 58.98 24.44 51.44 COMPOSER 54.02 80.27 33.81 63.19 29.20 57.13 (b) Models trained on C30K Eval on COCO F30K F30K-MCD Method R1 R5 R1 R5 R1 R5 VSE 45.74 81.22 27.66 55.92 23.44 47.90 ViLBERT 48.08 81.10 31.12 58.88 24.02 49.34 COMPOSER 47.87 80.93 34.29 61.00 26.91 51.46 Table 2 : 2Text-to-Image retrieval results.0 .1 5 0 .1 6 0 .1 7 0 .1 8 0 .1 9 0 .2 0 .2 1 0 .2 2 0 .2 3 0 .2 4 0 .2 5 0 .2 6 0 .2 7 0 .2 8 0 .2 9 0 .3 0 .3 1 Compound Divergence 24 26 28 30 32 34 36 Recall@1 (%) ViLBERT Composer 0 .1 5 0 .1 6 0 .1 7 0 .1 8 0 .1 9 0 .2 0 .2 1 0 .2 2 0 .2 3 0 .2 4 0 .2 5 0 .2 6 0 .2 7 0 .2 8 0 .2 9 0 .3 0 .3 1 Compound Divergence Table 3 : 3Study of different primitive encodings.F30K→F30K F30K→COCO Modulation R1 R5 R1 R5 Replace 52.84 79.79 32.63 61.61 MLP 52.92 79.89 33.39 61.41 FiLM 54.02 80.27 33.81 63.19 Table 4 : 4Study of different modulators.over the ViLBERT, relatively. There are 10.0% and 4.2% relative improvements on R1 and R5 on the other transfer direction. Compositional generalization. On the max compound divergence (MCD) split, COMPOSER outperforms baselines by a margin for both F30K and C30K trained models (shown as Table 2). To further characterize the performance on composi- tional generalization, we create 16 test splits on each dataset with different compound divergence (from 0.15 to 0.31, where 0.31 is the max CD) and present the results in Table 5 : 5Comparison between ViLBERT and COM- POSER on multi-level visual-semantic alignment super- vision (MVSA). All results are reported in R1. Table 5 5presents Table 6 : 6Results on COMPOSER of different complexity. All results are reported in R1. (PT: Predicate Transformer, CT: Composition Transformer) ously. COMPOSER with MVSA improves itself on both sentence and phrase, showing strong multigranular visual-semantic alignment ability.For computation complexity, we observe that the full COMPOSER model is 50% less efficient to a ViLBERT model, due to its recursive nature. Meanwhile, we notice that the increase in the # of CT layers contributes a significant amount to the total computation time as every two additional layers adds ∼ 10G FLOPS.Performance vs. complexity trade-off. We compare variants of COMPOSER with different pa- rameter and computation budgets, which uses dif- ferent numbers of layers for the Predicate Trans- former (PT) and Composition Transformer (CT). The results are shown in Table 6. First, We keep the size of CT fixed and vary the size of PT. It shows a marginal performance decrease occurring as the # of layers of PT goes down. Then we keep the size of PT fixed and decrease the capacity of CT, which presents a significant performance drop, showing the essential role CT is playing. Besides having superior results, COMPOSER has (at least 33%) fewer parameters than the ViLBERT model, which indicates a potential performance gain could be achieved with a larger COMPOSER model. Table 7 : 7Performance under different parsing qualities. Predicate[NP] are running on[NP] [NP] cut in half on[NP] Syntax Tree S NP two dogs VP VBP are VBG running PP IN on NP the grass S NP a small pizza VP VBN cut PP IN in NN half IN on NP a white plate Concept two dogs are running on the grass a small pizza cut in half on a white plate Text pre-processiong Following BERT(Devlin et al., 2019), we tokenize the text using the uncased WordPiece tokenizer. Specifically, we first lowercase the text and use the uncased tokenizer to extract tokens. The tokenizer has a vocabulary size of 30,522. The tokens are then transformed into word embeddings with 768 dimensions. Besides the word embedding, a 768-dimension position embedding is extracted. Both position embedding and word embedding are added together to represent the embedding of tokens.Training details We use Adam optimizer (Kingma and Ba, 2014) to optimize the parameter of our model. All the models are trained with a mini-batch size of 64. We employ a warm-up training strategy as suggested byViLBERT (Lu et al., 2019). Specifically, the learning rate is linearly increasing from 0 to 4e − 5 in the first 2 epochs. Then the learning rate decays to 4e − 6 and 4e − 7 after 10 epochs and 15 epochs, respectively. The training stopped at 20 epochs.Detials of baseline approaches. The text encoder for both models contains 12 layers of transformers and is initialized from BERT pretrained model using the checkpoint provided by Hugging-Face. For ViLBERT, we use the [CLS] embedding from the last layer as text representation y. We use the average of contextualized text embedding from the last layer as y in the VSE model. The visual encoder of VSE contains an MLP model with the residual connection. It transforms the image patch feature into a joint image-text space. The output of the visual encoder is the mean of the transformed image patch features. Unlike VSE, ViLBERT contains 6 layers of transformers for the image encoder and 6 layers of the cross-modal transformer to model the text and image features jointly. We use the embedding of [V-CLS] token from the last layer of the image encoder as the image feature x.Details of COMPOSER. The composer contains four primary learning sub-modules: (1) the CrossAtt model in primitive encoding; (2) the Predicate Transformer (PT) model; (3) the modulator; (4) the Composition Transformer (CT). The details of this sub-modules are list as what follows: • Primitive encoding. Cross-Atten. TransformerCross-Atten. TransformerCross-Atten. TransformerFigure 6: Details of the Composition Transformer model.Text Transformer Image Transformer Text Transformer Text Transformer Image Transformer a man running on the field with a soccer ball a soccer ball a man the field Output Concept Object-centric Visual Features [CLS] on running with Cross-Atten. Transformer F30K→F30K F30K→COCO α β R1 R5 R1 R5 COMPOSER w/ different α 0.4 0.2 53.54 80.51 33.73 61.67 0.6 0.2 53.44 80.21 33.89 61.05 0.8 0.2 54.02 80.27 33.81 63.19 COMPOSER w/ different β 0.8 0 53.66 80.39 33.33 61.15 0.8 0.2 54.02 80.27 33.81 63.19 0.8 0.4 53.50 80.55 33.87 61.15 Table 9 : 9Ablation Study on COMPOSER with Different Margin for MVSA and Order Objectives.there is 12 attention heads. The hidden dimension of CrossAtt Transformer is 1024, and there is 8 attention heads. In this section, we study COMPOSER performance under different MVSA objectives, Negative Log-Likelihood and Hinge loss. Then we study COM-POSER performance under different margins of MVSA and Order objectives.MVSA Objective. The MVSA objectives can be implemented using NLL loss or Hinge loss. We study the performance of COMPOSER under different losses for MVSA inTable 10. The models are trained with both MVSA and order objectives. We set the margin of order objectives β = 0.2. For the hinge loss, we set the margin α = 0.8. COM-POSER trained with hinge loss in MVSA achieves better performance than the NLL loss in all metricsF30K→F30K F30K→COCO Loss Function R1 R5 R1 R5 NLL 52.42 79.47 33.41 62.17 Hinge Loss 54.02 80.27 33.81 63.19 Table 10 : 10Ablation Study MVSA Objective: Comparing NLL to Hinge Loss. Our graph construction relies on constituency parsing thus it is more scalable than hand-written rules initially developed for denotation graphs. The technique of denotation graph has In this paper, texts refer to sentences. We adopt the released code here for the computing compound divergence: https://github.com/googleresearch/language/tree/master/language/nqg/tasks AcknowledgementsThis work is partially supported by NSF Awards IIS-1513966/ 1632803/1833137, CCF-1139148, DARPA Award#: FA8750-18-2-0117, FA8750-19-1-0504, DARPA-D3M -Award UCB-00009528, Google Research Awards, gifts from Facebook and Netflix, and ARO# W911NF-12-1-0241 and W911NF-15-1-0484.We thank anonymous EMNLP reviewers for constructive feedback. Additionally, we would like to thank Jason Baldbridge for reviewing an early version of this paper, and Kristina Toutanova for helpful discussion.AppendixIn the Appendix, we provide details omitted from the main text due to the limited space, including: • § A describes the implementation details for extracting primitives & predicates from the constituency tree ( § 2 of the main text). • In § B, we describes the details of generating the compositional evaluation splits ( § 5.1 of the main text). • § C contains training and architecture details for COMPOSER and baselines ( § 5.1 of the main text). • § D includes the ablation studies on learning objectives and margin of MVSA ( § 5.2 of the main text). Compositional generalization and natural language variation: Can a semantic parsing approach handle both. Peter Shaw, Ming-Wei Chang, Panupong Pasupat, Kristina Toutanova, ACL. Peter Shaw, Ming-Wei Chang, Panupong Pasupat, and Kristina Toutanova. 2020. Compositional general- ization and natural language variation: Can a seman- tic parsing approach handle both? In ACL. Haoyue Shi, Jiayuan Mao, Kevin Gimpel, Karen Livescu, Visually grounded neural syntax acquisition. ACL. Haoyue Shi, Jiayuan Mao, Kevin Gimpel, and Karen Livescu. 2019. Visually grounded neural syntax ac- quisition. ACL. LXMERT: Learning cross-modality encoder representations from transformers. Hao Tan, Mohit Bansal, EMNLP. Hao Tan and Mohit Bansal. 2019. LXMERT: Learning cross-modality encoder representations from trans- formers. In EMNLP. Attention is all you need. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, Illia Polosukhin, NeurIPS. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In NeurIPS. Scene graph parsing as dependency parsing. Yu-Siang Wang, Chenxi Liu, Xiaohui Zeng, Alan Yuille, Yu-Siang Wang, Chenxi Liu, Xiaohui Zeng, and Alan Yuille. 2018. Scene graph parsing as dependency parsing. NAACL. Neural-symbolic VQA: Disentangling reasoning from vision and language understanding. Kexin Yi, Jiajun Wu, Chuang Gan, Antonio Torralba, Pushmeet Kohli, Josh Tenenbaum, NeurIPS. Kexin Yi, Jiajun Wu, Chuang Gan, Antonio Tor- ralba, Pushmeet Kohli, and Josh Tenenbaum. 2018. Neural-symbolic VQA: Disentangling reasoning from vision and language understanding. In NeurIPS. From image descriptions to visual denotations: New similarity metrics for semantic inference over event descriptions. Peter Young, Alice Lai, TACL. 2Micah Hodosh, and Julia HockenmaierPeter Young, Alice Lai, Micah Hodosh, and Julia Hock- enmaier. 2014. From image descriptions to visual denotations: New similarity metrics for semantic in- ference over event descriptions. TACL, 2:67-78. Learning to represent image and text with denotation graph. Bowen Zhang, Hexiang Hu, Vihan Jain, Eugene Ie, Fei Sha, EMNLP. Bowen Zhang, Hexiang Hu, Vihan Jain, Eugene Ie, and Fei Sha. 2020. Learning to represent image and text with denotation graph. In EMNLP.	[ "https://github.com/googleresearch/language/tree/master/language/nqg/tasks" ]
[ "Mathematics Subject Classification. Primary 37Bxx, 22Dxx; Secondary 54H15", "Mathematics Subject Classification. Primary 37Bxx, 22Dxx; Secondary 54H15" ]	[]	[]	[]	We study Banach representability for actions of topological groups on groups by automorphisms (in particular, an action of a group on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces.In contrast to the standard left action of a locally compact second countable group G on itself, the conjugation action need not be reflexively representable even for SL 2 (R). The conjugation action of SL n (R) is not Asplund representable for every n ≥ 4. The linear action of GL n (R) on R n , for every n ≥ 2, is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l 1 ).The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). As a byproduct we obtain some counterexamples about Banach representations of homogeneous G-actions G/H.	10.1515/forum-2022-0373	[ "https://export.arxiv.org/pdf/2110.01386v5.pdf" ]	238,259,001	2110.01386	2781bea2e1d8546fd3c759ecf73e12a1f3133339	Mathematics Subject Classification. Primary 37Bxx, 22Dxx; Secondary 54H15 February 14. 2020 Mathematics Subject Classification. Primary 37Bxx, 22Dxx; Secondary 54H15 46February 14. 2020arXiv:2110.01386v5 [math.FA] TOPOLOGICAL GROUP ACTIONS BY GROUP AUTOMORPHISMS AND BANACH REPRESENTATIONS MICHAEL MEGRELISHVILI Dedicated to my friend Vladimir Pestov on the occasion of his 65th birthdayand phrases Banach representationconjugation actionequivariant compactificationenveloping semigrouphereditarily non-sensitiveRosenthal Banach spacetame dynamical systemweakly mixing We study Banach representability for actions of topological groups on groups by automorphisms (in particular, an action of a group on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces.In contrast to the standard left action of a locally compact second countable group G on itself, the conjugation action need not be reflexively representable even for SL 2 (R). The conjugation action of SL n (R) is not Asplund representable for every n ≥ 4. The linear action of GL n (R) on R n , for every n ≥ 2, is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l 1 ).The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). As a byproduct we obtain some counterexamples about Banach representations of homogeneous G-actions G/H. main idea of the present work is to study which actions G×X → X by automorphisms on topological groups X can be obtained as a "subaction" of Iso(V ) × B V * → B V * Question 1.1. Which actions of a topological group G on a topological group X by group automorphisms can be represented on low complexity Banach spaces V (e.g., Hilbert, reflexive, Asplund, Rosenthal)? In particular, what about the actions of groups on itself by conjugations? This is a particular case of a question about general actions which leads to an interesting hierarchy of topological groups and their actions. For some concrete results in this direction we refer to [23,24,8,9,10,11,15,12]. 1.2. Preliminaries. All topological spaces are assumed to be Tychonoff. X is a Gspace will mean that we have a continuous action G × X → X of a topological group G on a topological space X. The dual V * of a Banach space V is the Banach space of all continuous linear functionals on V . Remark 1.2. The topological group G := Iso(V ) of linear isometries acts naturally on the dual space V * as follows: Iso(V ) × V * → V * , ((gu * )(v) = u * (g −1 v)) g ∈ G, v ∈ V, u * ∈ V * . Moreover, the induced action on every norm bounded G-invariant subset Y of V * (e.g., on the unit ball B V * of V * ) is continuous where Y is endowed with the weakstar topology. Definition 1.3. [23,10] A representation of a G-space X on a Banach space V is an equivariant continuous pair (h, α), where h : G → Iso(V ) is a continuous homomorphism and α : X → V * is a weak-star continuous bounded G-equivariant map. Proper representation means that α is a topological embedding. A G-space X is representable in a class K of Banach spaces if there exists a proper G-representation of X on some V ∈ K. Reflexive representability of the G-space X means that in the setting of Definition 1.3 one may choose a reflexive space V . Hilbert, Asplund and Rosenthal representability can be similarly defined. Recall that a Banach space V is Asplund if the dual W * of every separable Banach subspace W of V is separable. The definition of Rosenthal Banach spaces comes directly from Rosenthal's celebrated dichotomy, [34]. According to this dichotomy every bounded sequence in a Banach space either has a weak Cauchy subsequence or admits a subsequence equivalent to the unit vector basis of l 1 (an l 1 -sequence). A Banach space V does not contain an l 1 -sequence (equivalently, does not contain an isomorphic copy of l 1 ) if and only if every bounded sequence in V has a weak-Cauchy subsequence [34]. As in [9,10,13], we call a Banach space satisfying these equivalent conditions a Rosenthal space. Every reflexive space is Asplund and every Asplund space is Rosenthal. A G-compactification of a G-space X is a G-equivariant continuous dense map ν : X → Y into a compact G-space Y . If ν is a topological embedding then ν is said to be a proper compactification. A G-space is G-compactifiable if it admits a proper G-compactification. For every compact G-space K, there exists a canonical Banach G-representation on the Banach space C(K, R) = C(K) (see for example [29]). Together with Remark 1.2 this explains the following basic fact. Fact 1.4. A continuous action G × X → X is Banach representable if and only if the G-space X is G-compactifiable. Alexandrov 1-point compactification X ∪ {∞} for a locally compact noncompact space X is the smallest proper compactification of X. For every continuous topological group action on X, the canonical extension on X ∪ {∞} is continuous, [38]. For more information about G-compactifications we refer to [38,39,21,22,24,30,16]. Remark 1.5. For every representation (h, α) of a G-space X on V , every vector v ∈ V induces the function f v : X → R, x → v, x . This is an important source to obtain functions on G-spaces. For example, varying v ∈ V and V in the class of all reflexive Banach spaces we get the class of all reflexively representable functions on X. By [23], it is exactly the algebra WAP(X) of all weakly almost periodic functions. The algebras Asp(X), Tame(X) can be similarly characterized, using the corresponding classes of Banach spaces, i.e., Asplund and Rosenthal, [10,9]. By Fact 1.4, a necessary (and sufficient) condition of Banach representability of a G-space X is the G-compactifiability. The corresponding algebra of all "Banach representable functions" is RUC(X). Recall that a bounded continuous function f : X → R is (generalized) right uniformly continuous (in short, RUC) if for every ε > 0 there exists a neighborhood O of the unity e ∈ G such that \|f (gx) − f (x)\| < ε for every (g, x) ∈ O × X. For every topological group G we have canonically defined left action of G on itself. In this case, RUC(G) is the usual algebra of all bounded right uniformly continuous functions on G. It is well known that RUC(G) separates the points and closed subsets and this action is always G-compactifiable. The greatest ambit β G G of G is its (proper) maximal G-compactification. Notation. Consider now the action of G on itself by conjugations π c : G × G → G, (g, x) → gxg −1 . We denote by G c the G-space G with the conjugation action π c . The importance of the action π c is well-understood. However its Banach Krepresentability (in the sense of Definition 1.3) has not been studied yet even for locally compact (classical) groups. Important algebras of functions on G, like WAP(G), Asp(G), Tame(G) are defined for the left regular action. Members of these algebras are exactly the functions on G which come as generalized matrix coefficients induced by representations h : G → Iso(V ) of G, where the Banach space V is Rosenthal (respectively, Asplund and reflexive). See [23,8,9,10]. For every locally compact topological group G, the algebra WAP(G) of all weakly almost periodic functions separate points and closed subsets and G admits a proper representation h : G ֒→ Iso(V ) on a reflexive Banach space V . In fact, one may find a Hilbert representation of G as it follows by a classical result of Gelfand and Raikov [7, p. 314]. If G, in addition, is separable metrizable, then we can require more; namely, that there exists even an equivariant representation, in the sense of Definition 1.3, of the standard G-space (with left translations) X := G on a Hilbert space. This can be done using [25,Lemma 4.5]. Main results and some open questions. It is a less known fact (see [21,22]) that the G-space G c (conjugation action) is Banach representable (in the sense of Definition 1.3) for every topological group G. By Fact 1.4, it is equivalent to say that G c is G-compactifiable. More precisely, the Roelcke compactification is a (proper) Gcompactification of G c (Corollary 2.3). The significance of Roelcke compactification is now well understood due to several papers of V. Uspenskij [37,36] and many other authors. In contrast to the standard left action of a locally compact second countable group G on itself, the conjugation action need not be reflexively representable even for natural matrix groups. This happens among others for the special linear group SL 2 (R); see Theorem 3.3 which uses Grothendieck's double limit criterion. Moreover, by Theorem 3.10 the G-space G c for G := SL n (R) is not Asplund representable for every n ≥ 4. Here we use the weak mixing argument by S.G. Dani and S. Raghavan [4] together with the concept of hereditarily non-sensitive actions, [8]. We should note that dynamical non-sensitivity is a necessary condition for Asplund representability of compact G-spaces by joint results with E. Glasner [8]. Moreover, hereditary nonsensitivity is a sufficient condition of Asplund representability of metrizable compact G-spaces. The natural linear action of GL n (R) on R n , for every n ≥ 2, is not Asplund representable (Proposition 3.9 and Theorem 3.10). On the other hand, this action is Rosenthal representable, Theorem 3.12. Here we use the enveloping semigroup characterization of tame compact dynamical systems (see Section 3.3). Recall that the enveloping (Ellis) semigroup of an action G × X → X on a compact space X is the pointwise closure of all g-translations X → X (g ∈ G) in the compact space X X . A compact G-system X is said to be tame (regular, in terms of A. Köhler [18]) if for every f ∈ C(X) the orbit f G does not contain a combinatorially independent sequence in the sense of Rosenthal [34]. Tame dynamical systems naturally occur in geometry, analysis and symbolic dynamics [11]. They play an important role in view of a dynamical Bourgain-Fremlin-Talagrand dichotomy [10], NIP-formulas in logic [15] and Todorcević' trichotomy in topology [13]. In [12] a generalized amenability was examined substituting, in the definition, the existence of a fixed point by some tame dynamical G-subsystem. Every hyperbolic toral automorphism defines a cascade which is not tame and hence not Rosenthal representable (Theorem 3.14). Moreover, using V. Lebedev's recent result [19], one may show that the same is true for every infinite order automorphism of the torus T n . One of the conclusions is that the conjugation action need not be Rosenthal representable even for Lie groups (Corollary 3.16). This is unclear for SL 2 (R). Like for locally compact groups, also for non-archimedean second countable groups G, the standard left action on itself is Hilbert representable. Corollary 3.18 shows that there exist Polish non-archimedean locally compact topological groups G (which are elementary in the sense of Wesolek [40]) such that the conjugation action is not Rosenthal representable. Problem 1.6. For the conjugation action G × G c → G c , study: (1) the greatest G-compactification β G G c of G c ; (2) the algebras RUC(G c ), Tame(G c ), Asp(G c ), WAP(G c ); (3) when the conjugation action is Rosenthal representable. What about the following concrete groups: a) GL 2 (R), SL 2 (R); b) unitary group Iso(l 2 ); c) symmetric group S ∞ ; d) Iso(U); e) Iso(U 1 )? Here U is the Urysohn universal metric space (see, for example [29]), U 1 is the Urysohn sphere and Iso(M) means the topological group of all isometries with the pointwise topology. As a byproduct we illuminate some counterexamples about Banach representations of homogeneous G-actions G/H in Section 4. Among others we prove (Corollary 4.1) that there exists a two dimensional Lie group G and its cocompact discrete subgroup H := Z, such that the compact two dimensional homogeneous G-space G/H is not Rosenthal representable. We hope that Banach K-representability for the conjugation actions will foster some new ideas and open up interesting research lines in the realm of (Polish) topological groups even for the subclass of (classical) locally compact second countable groups. Some properties of actions by automorphisms Compactifiability. Let α : G × X → X be a continuous action of a topological group G on X. A topologically compatible uniform structure U on X is said to be bounded (see [3,38,39,30]) if for every entourage ε ∈ U there exists a neighbourhood U ∈ N e (G) such that (x, ux) ∈ ε for every u ∈ U and every x ∈ X. If, in addition, every g-translation is a uniform map, then U is an equiuniformity (in the terminology of [20]). According to Brook [3], the Samuel compactification (X, U) → sX of an equiuniformity U is a (proper) G-compactification of X. We say that U is quasibounded (introduced in [20,21]), if for every ε ∈ U there exist δ ∈ ε and U ∈ N e (G) such that (ux, uy) ∈ ε for every (x, y) ∈ δ and u ∈ U. The G-compactifiability of X is equivalent to the existence of a quasibounded uniformity on X. As it was proved in [21, p. 222] and [26], there exists a natural construction to obtain a bounded compatible uniformity on X for a given quasibounded uniformity on a G-space X. Fact 2.1. [22] Let G and X both be topological groups, π : G × X → X a continuous action by group automorphisms of X. Then the G-space X is G-compactifiable (and it admits a Banach representation in the sense of Definition 1.3). Proof. Since the given continuous action is by automorphisms of X, it is straightforward to observe that the right (and also left) uniformity on X is quasibounded (but not always bounded). An additional possibility proving Fact 2.1 is to use Fact 2.5 below taking into account that (according to J. de Vries [38] ) every coset G-space G/H is G-compactifiable. The reason is that the right uniformity of G/H is bounded. Recall that the Roelcke compactification ρ : G → ρ(G) of a topological group G = G × G on G P × G → G, (s, t)(g) = sgt −1 . Then the Roelcke compactification is a proper P -compactification of G. Proof. One may easily verify that the Roelcke uniformity UC(G) on G is always an equiuniformity on the P -space G and hence, according to [3], its Samuel compactification ρ : G → ρ(G) is a (proper) P -compactification of G. The action of G on itself by conjugations is a subaction of the diagonal subgroup G ≃ ∆ := {(g, g) : g ∈ G} ≤ G × G on G. In particular, we get Corollary 2.3. The Roelcke compactification of G defines a proper G-compactification of the G-space G c . So, (1) G c is a G-compactifiable G-space; (2) the Roelcke compactification ρ : G c → ρ(G) is a G-factor of β G (G c ); (3) UC(G) ⊂ RUC(G c ). Remark 2.4. Let α : G × X → X be a continuous action of a topological group G on a topological group X by group automorphisms. Denote by X ⋊ α G the corresponding topological semidirect product. As usual, identify G with {e X } × G and X with X × {e G }. Then X is a normal subgroup of X ⋊ α G and the action α of G on X is a subaction of the conjugation action of X ⋊ α G on itself. One of the conclusions of Remark 2.4 is that Corollary 2.3.1 infers back Fact 2.1. Fact 2.5. [22, Lemma 1.1] Let α : G × X → X be a continuous action by group automorphisms and P := X ⋊ α G be the corresponding topological semidirect product. Then the triple (G, X, α) naturally is embedded into the homogeneous action (P, P/G, α * ), where α * is the natural action of P on P/G. Proof. The mapping j : X → P/G, j(x) = xG is a restriction of the natural projection P → P/G on X ⊂ P . According to [33,Prop. 6.17(a)], j is a homeomorphism. Moreover, it is straightforward to check that (e, g)j(x) = (e X , g)((x, e G )G) = (gx, e G )G = j(gx). So, the restriction of α * on G × X := G × j(X) is α. Roughly speaking, every action by automorphisms is a subaction of a coset G-space. The converse is not true in general. By Example 2.6 below the homogeneous action of the full homeomorphism Polish group Homeo (S) on the sphere S is not a part of any continuous action by group automorphisms. Which actions are automorphizable? For every G-space X there exists the free topological G-space F G (X) of X. This concept was introduced in [22] and has several nice applications; among others for epimorphism problems. Resolving a longstanding important problem by K. Hofmann, Uspenskij [35] has shown that in the category of Hausdorff topological groups epimorphisms need not have a dense range. Dikranjan and Tholen [5] gave a rather direct proof. Pestov gave a beautiful useful criterion [28,29] which is based on the concept of a free topological G-group. More precisely, the inclusion i : H ֒→ G of topological groups is an epimorphism if and only if the free topological G-group F G (X) of the coset G-space X := G/H is trivial. Triviality means isomorphic to the cyclic discrete group ("as trivial as possible"). In contrast to the case of trivial G (when F G (X) is just the usual free topological group F (X)), the universal G-morphism i : X → F G (X) need not be a topological embedding even for compact G-space X as it follows directly from the following Example 2.6. [22] Not every compact G-space K is a subaction of an action by group automorphisms. For example, the cube K = [0, 1] n or the n-dimensional sphere K = S n (n ∈ N) with the homeomorphisms group G = Homeo (K), which is Polish in the compact-open topology. In the case of the circle K = S, the corresponding homogeneous action of the Polish group G = Homeo (S) on S can be identified with the compact coset G-space G/H, where H = St(z) is the stabilizer for a point z ∈ S. The corresponding free topological G-space F G (G/H) is trivial (by [22]) and Pestov's criterion implies that the closed inclusion H ֒→ G is an epimorphism. Moreover, a much stronger result follows by the earlier paper of Uspenskij [35]. Namely, in fact, for every compact connected manifold X, its Polish homeomorphism group G = Homeo (X) and a stability subgroup H = St(z), the embedding H ֒→ G is an epimorphism. Equivalently, any continuous G-map from X into any G-group is constant. A self-contained elegant explanation of the latter fact can be found in a recent work by Pestov and Uspenskij [31, page 5]. Remark 2.7. (1) It is well known that such examples are impossible for locally compact G because, in this case, every Tychonoff G-space X is even G-linearizable on a locally convex linear G-space. (2) Another sufficient condition is (uniform) U-equicontinuity of the action with respect to some compatible uniformity on X. One may assume that U is generated by a system of G-invariant pseudometrics {ρ i : i ∈ I} with ρ i ≤ 1. Now observe that the Arens-Eells embedding defines a G-linearization of X into a locally convex G-space. For details see [22] and [26]. Representations on low complexity Banach spaces 3.1. Reflexive representability. According to a classical definition, a continuous bounded function f ∈ C b (X) on a G-space X is said to be weakly almost periodic (WAP) if the weak closure of the orbit f G = {f g : g ∈ G} is weakly compact in the (1) f ∈ WAP(X). Banach space C b (X) (with the sup-norm). A compact G-space X is WAP if every f ∈ C(X) is WAP. (2) f has Grothendieck's double limit property. (3) f is reflexively representable (Remark 1.5). (4) f comes from a G-compactification ν : X → Y of X such that Y is WAP. Proof. Assume, to the contrary, that the G-space G c is reflexively representable. Then by Fact 3.1 there exists a proper G-compactification G c ֒→ X such that the G-space X is WAP. Then every G-factor of X is again WAP. In particular, Alexandrov's 1- point compactification (smallest proper G-compactification of G c ) Y := G c ∪ {∞} is WAP. We claim that Y is not WAP. It is enough to show that for every compact neighborhood U of the identity e ∈ G and for every continuous bounded function f : G → R with f (e) = 1 and f (x) = 0 for every x / ∈ U, we have f / ∈ WAP(G c ). By the Grothendieck double limit criterion for G-spaces (as in Fact 3.2), it suffices to show that there exist two sequences g n ∈ G and x m ∈ G c such that the double sequence f (g n x m g −1 n ) (n, m ∈ N) has distinct double limits. Now for n = 2, define the following sequences (for general n ≥ 2 the proof is similar): The rest is similar to the proof of Theorem 3.3. (2) This follows from (1) because α is a subaction of the action G × G c → G c by Fact 2.5. (3) Straightforward. 3.2. Asplund representability. First recall the classical concept of non-sensitivity. An action of G on a uniform space (X, U) is said to be non-sensitive if for every entourage ε ∈ U there exists a nonempty open subset O in X such that gO remains ε-small for every g ∈ G. According to [8], hereditarily non-sensitive (in short, HNS) means that every (equivalently, every closed) G-subspace Y of X is non-sensitive with respect to the induced subspace uniformity. Every expansive action G × X → X on a uniform space (X, U) without an isolated point is sensitive. Expansiveness means that there exists ε ∈ U such that for every distinct pair of point x = y in X we have (gx, gy) = ε for some g ∈ G. Many compact groups K admit expansive automorphisms σ : K → K. For instance, the cascade induced by a hyperbolic toral automorphism is expansive (also topologically mixing), hence not Asplund representable. In fact, it is not even Rosenthal representable (see Theorem 3.14). Example 3.6. The action R × ×R → R from Proposition 3.4.1 is Asplund representable by Fact 3.5.2. Indeed, the enveloping semigroup of the 1-point compactification action is metrizable (see Proposition 3.4.3). Recall that an action of G on X is said to be: (a) (algebraically) transitive if for every x, y ∈ X there exists g ∈ G such that gx = y. (b) 2-transitive if for all ordered pairs (x 1 , x 2 ) and (y 1 , y 2 ) in X with x 1 = y 1 and x 2 = y 2 , there exists g ∈ G such that x 2 = gx 1 , y 2 = gy 1 . (c) topologically transitive if, for every pair of nonempty open subsets U and V of X, there is an element g ∈ G such that gU ∩ V is nonempty. (d) weakly mixing if the induced diagonal action of G on X × X is topologically transitive. Lemma 3.7. Let G × X → X be a 2-transitive action such that X has no isolated point. Then this action is weakly mixing. Proof. Let O 1 and O 2 be nonempty open subsets in X × X. Choose nonempty open rectangles U 1 × V 1 ⊂ O 1 and U 2 × V 2 ⊂ O 2 . Since X has no isolated points, there exist x 1 ∈ U 1 , y 1 ∈ V 1 , x 2 ∈ U 2 , y 2 ∈ V 2 such that x 1 = y 1 and x 2 = y 2 . The 2transitivity implies that x 2 = gx 1 , y 2 = gy 1 for some g ∈ G. Then clearly, gO 1 ∩ O 2 is nonempty. Fact 3.8. [8, Corollary 9.3] Let (X, U) be a uniform space and X is a weakly mixing G-space which is non-sensitive with respect to U. Then X is trivial. Note that the affine group R n ⋊GL n (R) can be embedded into GL n+1 (R) as follows: R n ⋊ GL n (R) → GL n+1 (R), M → M v 0 1 where M is an n × n matrix from GL n (R) and v is an n × 1 column. Proposition 3.9. (1) The action of GL 2 (R) on R 2 is not Asplund representable. (2) The conjugation action of the affine group R 2 ⋊ GL 2 (R) (and hence, also of GL 3 (R)) is not Asplund representable. Proof. (1) Assuming the contrary, let ν : R 2 → K be a proper GL 2 (R)-compactification of R 2 such that K is Asplund representable. Hence, the compact GL 2 (R)-space K is HNS (Fact 3.5.1). By definition of HNS, every (not necessarily compact) G-subspace X is non-sensitive with respect to the induced (from K) precompact uniformity for every subgroup G of GL 2 (R). We claim that there exist a subgroup G ⊂ GL 2 (R) and a weakly mixing G-subspace Y in K, where Y topologically is homeomorphic to R. By Fact 3.8 this will provide the desired contradiction. Consider Y = y 1 y ∈ R , G = a b 0 1 b ∈ R, a = 0 ≃ R ⋊ R × . It is easy to see that the natural restricted action of G on Y is 2-transitive, hence also weakly mixing by virtue of Lemma 3.7. Since ν is an equivariant and topological embedding, we conclude that ν(Y ) is a weakly mixing G-subspace of K. (2) Use (1) and Remark 2.4. Theorem 3.10. (1) The action of SL n (Z) on R n is not Asplund representable for every n ≥ 3. (2) The conjugation action of SL n (R) is not Asplund representable for every n ≥ 4. Proof. (1) By a result of S.G. Dani and S. Raghavan [4], the natural linear action of SL n (Z) on R n is weakly mixing for every n ≥ 3. Now Facts 3.5 and 3.8 imply that this action is not Asplund representable. (2) The special affine group R n ⋊SL n (R) can be embedded into SL n+1 (R) as follows: R n ⋊ SL n (Z) → SL n+1 (R), M → M v 0 1 where M is an n × n matrix from SL n (Z) and v is an n × 1 column. Now use (1). 3.3. Rosenthal representability and the tameness. For definitions of tame systems and the enveloping semigroup we refer to the introduction. Fact 3.11. [8,9] For a compact metric G-space X the following conditions are equivalent: (1) The G-space X is representable on a (separable) Rosenthal Banach space. (2) The G-space X is tame. (3) The enveloping semigroup E(X), as a (compact) topological space, is Frechet. (4) card(E(X)) ≤ 2 ℵ 0 . The following theorem was obtained very recently in a joint work with E. Glasner [13]. We include the proof for the sake of completeness. Theorem 3.12. [13] The linear action GL n (R)×R n → R n is Rosenthal representable. Proof. By results of [9], every Rosenthal representable G-space is embedded into a tame compact G-space and the tameness is preserved by the G-factors of compact G-spaces. Hence, by Fact 3.11 it is enough to show that the 1-point compactification X := R n ∪ {∞} is a tame GL n (R)-system. Or, equivalently, that the cardinality of the enveloping semigroup E(X) is not greater than 2 ℵ 0 . Let p ∈ E(X). Define V p := p −1 (R n ) = {v ∈ R n : p(v) ∈ R n } = {v ∈ X : p(v) = ∞}. Claim: V p is a linear subspace (hence, closed) in R n and the restriction p\| : V p → R n is a (continuous) linear map. Indeed, let g i be a net in T such that lim g i = p in E. If u, v ∈ V p then lim g i u = p(u) ∈ R n and lim g i v = p(v) ∈ R n . Then by the linearity of maps g i we obtain lim g i (c 1 u + c 2 v) = c 1 p(u) + c 2 p(v) ∈ R n . Since V p is finite dimensional, the linear map p\| Vp : V p → R n is necessarily continuous. Using this claim we obtain that card(E(X)) ≤ 2 ℵ 0 . Remark 3.13. The enveloping semigroup E(X) of the action from Theorem 3.12 can be identified with the semigroup of all partial linear endomorphisms of R n . To see this, observe that the claim from the proof of Theorem 3.12 can be reversed. Namely, every partial linear endomorphism f : V → W of R n defines an element p ∈ E(X) such that V = p −1 (R n ), p(V ) = W, p(x) = f (x) ∀x ∈ V, p(y) = ∞ ∀y / ∈ V. Moreover, that assignment is a semigroup isomorphism: (partial) composition corresponds to the product of suitable elements from the enveloping semigroup. Theorem 3.14. For every n ≥ 2 there exists a topological group automorphism σ : T n → T n on the n-dimensional torus T n such that the corresponding action of the cyclic group Z on T n by the iterations is not Rosenthal representable. Proof. For every hyperbolic toral automorphism σ : T n → T n , the corresponding cascade has positive entropy. Hence, it cannot be tame by a result of Kerr and Li [17] because tameness implies zero entropy. Therefore, such a cascade is not Rosenthal representable according to Fact 3.11. Remark 3.15. By a recent paper of V. Lebedev [19], for every σ ∈ SL n (Z) the corresponding cascade Z × T n → T n is tame (if and) only if σ k = I for some natural k. So, in the proof of Theorem 3.14 one may consider any σ ∈ SL n (Z) with infinite order. Corollary 3.16. For the two dimensional Lie group T 2 ⋊ Z, its conjugation action is not Rosenthal representable. Proof. Use Theorem 3.14 (for n = 2) and take into account Remark 2.4. Theorem 3.17. There exists a metrizable profinite (compact zero-dimensional) topological group X and a topological group automorphism a : X → X such that the corresponding action of the cyclic group Z on X is not Rosenthal representable. Proof. Consider a compact metrizable zero-dimensional Z-space K which is not tame. For example, take the full Bernoulli shift K := {0, 1} Z . It is well known that the enveloping semigroup of the Z-space K is βZ. Hence, K is not tame according to Fact 3.11. Consider the free profinite group X := F P ro (K) of the Cantor space K. Then F P ro (K) is a compact metrizable group (see [32] and [27,Theorem 5.1]). Moreover, we have a continuous action α : Z × F P ro (K) → F P ro (K) by group automorphisms. This action is not tame because its subaction Z×K → K is not tame. Therefore, the dynamical Z-system X is not Rosenthal representable. We denote by S ∞ the symmetric topological group endowed with the pointwise topology acting on the discrete set N. A topological group G is non-archimedean if open subgroups form a local topological basis. Corollary 3.18. There exists a locally compact Polish non-archimedean group G such that the G-space G c is not Rosenthal representable. Proof. Let α : Z × F P ro (K) → F P ro (K) be the action from the proof of Theorem 3.17. The corresponding topological semidirect product G := X ⋊ α Z is the desired group, where X is a Z-group from Theorem 3.17. Indeed, the conjugation action G × G c → G c of this group is not Rosenthal representable (otherwise, by Remark 2.4, the same is true for its subaction α : Z × X → X). This locally compact group G from Corollary 3.18 is an extension of a profinite group by a discrete group. In particular, G is elementary in the sense of Wesolek [40]. In contrast, the left action of S ∞ on itself is always Hilbert representable and the topological group S ∞ is unitarily representable on a Hilbert space. Question 3.20. Is it true that for G = SL 2 (R) the G-space G c is Rosenthal representable? Banach representations of homogeneous actions For every locally compact second countable group G, its regular left action on itself is Hilbert representable. However, as we already have seen, the conjugation action of G might not even be Rosenthal-representable. For homogeneous G-spaces G/H, we have the same phenomenon (for nontrivial H). Our results above shed some light also on Banach representations of homogeneous actions. Representability of the G-space G = G/{e} on nice Banach spaces sometimes cannot be decisive about the representations of the G-space G/H even for locally compact, as well as for Lie, groups G. Proof. The original (nontame) action Z × T 2 → T 2 from Theorem 3.14 is a subaction of the homogeneous action G × G/Z → G/Z (Fact 2.5). So, the compact G-space G/Z is also nontame. Corollary 4.2. There exists a non-archimedean elementary locally compact group G and a cocompact discrete subgroup H such that the compact coset G-space G/H is not Rosenthal representable. Proof. Let G := X ⋊ α Z be the group from Corollary 3.18 with H := Z. For the rest of the proof, as above, apply Fact 2.5. Results of the present section suggest the following general questions. is the compactification induced by the algebra UC(G) = RUC(G) ∩ LUC(G), where LUC(G) is the algebra of all bounded left uniformly continuous functions. Fact 2 . 2 . 22[22] Let G be a (Hausdorff ) topological group. Consider the following continuous action of P : Fact 3 . 1 . 31[23] Every reflexively representable G-space is embedded into a WAP compact G-space. A compact metric G-space X is reflexively representable if and only if X is a WAP G-system.The following result, based on Grothendieck's criterion, can be derived by combining[23, Theorem 4.6] and[23, Fact 2.4] Fact 3 . 2 . 32[23] Let X be a (not necessarily compact) G-space and f ∈ RUC(X). The following conditions are equivalent: Theorem 3. 3 . 3Let G := SL n (R), where n ≥ 2. Then the conjugation G-space G c is not reflexively representable. . 4 . 4Let R × be the multiplicative group of all nonzero reals. Then (1) the natural action α :R × × R → R is not reflexively representable; (2) for the "ax + b-group" G := R ⋊ α R × the G-space G c is not reflexively representable; (3) let X = R ∪ {∞}be the 1-point compactification of R. Then the enveloping semigroup E(X) of the action of R × on X is the semigroup R × ∪ {0} ∪ {∞}, with 0 · ∞ = ∞ and ∞ · 0 = 0 (other cases are understood). Proof. (1) Choose g n := n −1 , x m := m, where n, m ∈ N. [ 8 ] 8Every Asplund representable G-space is embedded into a HNS compact Gspace. A compact metric G-space X is Asplund representable if and only if X is HNS. (2)[14] A compact metric G-space X is HNS if and only if the enveloping semigroup E(X) is metrizable. Corollary 3 . 19 . 319For the symmetric topological group G = S ∞ , the conjugation action on G c is not Rosenthal representable. Proof. (3) Use Corollary 3.18 and the well-known fact (see [1, Theorem 1.5.1]) that every second countable non-archimedean group G is embedded into S ∞ . Corollary 4 . 1 . 41Let G := T 2 ⋊ Z be the Lie group from Corollary 3.16. Then for its cocompact discrete subgroup H := Z, the compact two dimensional homogeneous G-space G/H is not Rosenthal representable. Corollary 4. 3 . 3There exists a closed subgroup H of G := SL 2 (R) such that the corresponding locally compact coset G-space G/H is not reflexively representable.Proof. Indeed, in Theorem 3.3 we deal, in fact, with the restricted action by conjugations of G := SL 2 (R) on the G-orbit of x 1 . This orbit, terms of Theorem 3.3) is locally compact and it can be represented as the topological coset G-space G/H, where H is the stabilizer subgroupSt(x 1 ) = {g ∈ SL 2 (R) : gx 1 g −1 = x 1 }. Question 4. 4 . 4Which interesting homogeneous G-spaces G/H are Rosenthal representable? What about the SL n (R)-spaces SL n (R)/H ? In particular, what if H = SL n (Z)? The "Asplund version" of Question 4.4 seems to be less attractive than the present "Rosenthal version". The reason is that by [23, Theorem 6.10] and Fact 3.5, Asplund representable compact metrizable coset G-spaces G/H are necessarily equicontinuous. Therefore they are even Hilbert representable. Many geometric compact coset Gspaces are sensitive. Hence, not Asplund representable. For example, this is true for the projective action which is 2-transitive. On the other hand, results of R. 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A characterization of Banach spaces containing ℓ 1. H P Rosenthal, Proc. Nat. Acad. Sci. (USA). Nat. Acad. Sci. (USA)H.P. Rosenthal, A characterization of Banach spaces containing ℓ 1 , Proc. Nat. Acad. Sci. (USA) 71 (1974), 2411-2413. The epimorphism problem for Hausdorff topological groups. V V Uspenskij, Topology Appl. 57V.V. Uspenskij, The epimorphism problem for Hausdorff topological groups, Topology Appl., 57 (1994), 287-294. The Roelcke compactification of groups of homeomorphisms. V V Uspenskij, Topology Appl. 111V.V. Uspenskij, The Roelcke compactification of groups of homeomorphisms, Topology Appl. 111 (2001), 195-205. V V Uspenskij, arXiv:math.GN/0204144Compactifications of topological groups, Proceedings of the Ninth Prague Topological Symposium. Prague), (ed. Petr Simon), published by Topology Atlas. electronic publicationV.V. Uspenskij, Compactifications of topological groups, Proceedings of the Ninth Prague Topo- logical Symposium (Prague, August 19-25, 2001), (ed. 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Department of Mathematics, Bar-Ilan University, 52900 Ramat-GanDepartment of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel Email address: [email protected] URL: http://www.math.biu.ac.il/ ∼ megereli	[]
[ "Classification, α k -Derivations and Centroids of 4-dimensional Complex Hom-associative Dialgebras", "Classification, α k -Derivations and Centroids of 4-dimensional Complex Hom-associative Dialgebras" ]	[ "Ahmed Zahari \nUniversité de Haute Alsace\nIRIMAS-Département de Mathématiques\n18, rue des Frères LumièreF-68093MulhouseFrance\n" ]	[ "Université de Haute Alsace\nIRIMAS-Département de Mathématiques\n18, rue des Frères LumièreF-68093MulhouseFrance" ]	[]	The basic objective of the current research paper is to investigate the structure and the algebraic varieties of Hom-associative dialgebras. We elaborate a classification of n-dimensional Hom-associative dialgebras for n ≤ 4. Additionally, using the classification result of Hom-associative dialgebras, we characterize the α-derivations and centroids of low-dimensional Hom-associative dialgebras. Furthermore, we equally tackle certain features of derivations and centroids in the light of associative dialgebras and compute the centroids of low-dimensional associative dialgebras.	null	[ "https://export.arxiv.org/pdf/2305.04041v1.pdf" ]	258,557,885	2305.04041	135e95fc6c26d04cb6a3e3290e63b1b364a9ede8	Classification, α k -Derivations and Centroids of 4-dimensional Complex Hom-associative Dialgebras 6 May 2023 Ahmed Zahari Université de Haute Alsace IRIMAS-Département de Mathématiques 18, rue des Frères LumièreF-68093MulhouseFrance Classification, α k -Derivations and Centroids of 4-dimensional Complex Hom-associative Dialgebras 6 May 2023Dedicated to the memory of professor Marie-Hélène TUILIER, Professor of physics and Responsible at the University of Haute-Alsace of doctoral school who has helped me morally and financially during my doctoral years.AMS Subject Classification: Keywords: Hom-associative dialgebraClassificationDerivationTriple system The basic objective of the current research paper is to investigate the structure and the algebraic varieties of Hom-associative dialgebras. We elaborate a classification of n-dimensional Hom-associative dialgebras for n ≤ 4. Additionally, using the classification result of Hom-associative dialgebras, we characterize the α-derivations and centroids of low-dimensional Hom-associative dialgebras. Furthermore, we equally tackle certain features of derivations and centroids in the light of associative dialgebras and compute the centroids of low-dimensional associative dialgebras. Introduction A Hom-diassociative algebra (A, ⊣, ⊢, α) consists of a vector space, two multiplications, and a linear self map. It may be regarded as a deformation of an associative algebra, where the associativity condition is twisted by a linear map α , such that when α = id, the Hom-associative dialgebra degenerates to exactly a associative dialgebra. The central focus of this work is to explore the structure of Hom-associative dialgebras. Let A be an n-dimensional K-linear vector space and let {e 1 , e 2 , · · · , e n } be a basis of A, where K will always be an algebraically closed field of characteristic 0. A Hom-dialgebra structure on A with products µ and λ is determined by 2n 3 structure constants γ k i j and δ k i j , were e i ⊣ e j = n k=1 γ k i j e k , e i ⊢ e j = n k=1 δ k i j e k and by α which is given by n 2 structure constants a i j , where α(e i ) = n j=1 a ji e j . Requiring the algebra structure to be Hom-diassociative and unital gives rise to a sub-variety Hd n of k 2n 3 +n 2 . Base changes in A result in the natural transport of structure action of GL n (k) on Hd n . Thus isomorphism classes of n-dimensional Hom-dialgebras are in one-to-one correspondence with the orbits of the action of GL n (k) on Hd n . In this paper, we tackle the problem of classification. We set forward an algorithm to compute classification. We apply the algorithm in low-dimensional cases. We obtain the classification results of two and three-dimensional complex associative dialgebras from Rikhsiboev [10] and revise a list of four-dimensional nilpotent diassociative algebras from Rakhimov and Fiidov [13]. Within this framwork, A. Zahari and I. Bakayoko studied the classification of BiHom-associative dialgebras [2]. The classification of two and three-dimensional Hom-associative dialgebras was undertaken by A. Zahari and A. Makhlouf [1] and the classification of 3-dimensional BiHom-associative and BiHombialgebras [16] was performed by A. Zahari. Furthermore, we shall consider the class of Homassociative dialgebras. We shall also establish a classification of these algebras up to isomorphism in low dimension n ≤ 4. The paper is laid out as follows. In the first section, we identify the basics about Hom-associative dialgebras and provide several new properties. In section 2, we address the structure of Hom-associative dialgebras. Section 3 is devoted to the description of the algebraic varieties of Hom-diassociative algebras, and classifications, up to isomorphism, of two-dimensional, three-dimensional and four-Homassociative dialgebras are introduced. In section 4, we determine certain new properties of derivations and we focus upon the classification of the derivations. Eventually, in Section 5, we handle the classification of the centroids. In this case, the concept of derivations and centroids is notably inspired from that of finite-dimensional algebras. The algebra of centroids plays a key role in terms of the classification problems as well as in different applications of algebras. As far as our work is concerned, we elaborate classification results of two, three and four-dimensional Hom-associative dialgebras. All considered algebras and vectors spaces are supposed to be over a field K of characteristic zero. 2 Structure of Hom-associative dialgebras Definition 2.1. A Hom-associative dialgebra is a 4-truple (A, ⊣, ⊢, α) consisting of a linear space A linear maps ⊣, ⊢, : A × A −→ A and α : A −→ A satisfying, for all x, y, z ∈ A, the following conditions : (x ⊣ y) ⊣ α(z) = α(x) ⊣ (y ⊣ z), (2.1) (x ⊣ y) ⊣ α(z) = α(x) ⊣ (y ⊢ z), (2.2) (x ⊢ y) ⊣ α(z) = α(x) ⊢ (y ⊣ z), (2.3) (x ⊣ y) ⊢ α(z) = α(x) ⊢ (y ⊢ z), (2.4) (x ⊢ y) ⊢ α(z) = α(x) ⊢ (y ⊢ z). (2.5) We call α ( in this order ) the structure maps of A. If in addition, α is an endomorphism with respect to ⊣ and ⊢, then A is said to be a multiplicative Hom-dialgebra : α(x ⊣ y) = α(x) ⊣ α(y) and α(x ⊢ y) = α(x) ⊢ α(y) (2.6) for x, y, z in A. As we are dealing only with multiplicative Hom-associative dialgebras, we shall call them Homdiassociative algebras for simplicity. We denote the set of all Hom-associative dialgebras by Hd. The kernel and the image of homomorphism is defined naturally. One of the basic problems in structure theory of algebras resides in the problem of classification. The classification implies the description of the orbits under base change linear transformations and list representatives of the orbits. Definition 2.2. Let (A 1 , ⊣ 1 , ⊢ 1 , α 1 ) and (A 2 , ⊣ 2 , ⊢ 2 , α 2 ) be Hom-associative dialgebras over a field K. Then, a homomorphism of Hom-associative dialgebras A 1 to A 2 is a K-linear mapping φ : A 1 −→ A 2 such that φ : A 1 −→ 2 such that φ(x ⊣ 1 y) = φ(x) ⊣ 2 φ(y), φ(x ⊢ 1 y) = φ(x) ⊢ 2 φ(y), φ • α 1 (x) = α 2 • φ(x),(2.7) for all x, y ∈ A. A bijective homomorphism is said to be an isomorphism. Definition 2.3. A bar unit in A is an element e ∈ A such that x ⊣ e = e ⊢ x = α(x) = x. Definition 2.4. A Hom-dendrifom algebra is a quadriple (A, ≺, ≻, α) consisting of a vector pace A on which the operations ≺, ≻: A × A −→ A and α : A −→ A are linear maps satisfying : (x ≺ y) ≺ α(z) = α(x) ≺ (y ≺ z + y ≻ z), α(x) ≻ (y ≻ z) = (x ≺ y + x ≻ y) ≺ α(z), (x ≻ y) ≺ α(z) = α(x) ≻ (y ≺ z),(2.8) for all x, y, z ∈ A. Example 2.5. In fact, a Hom-dendrifom algebra structure on an n-dimensional vector space V with a basis {e 1 , e 2 , . . . , e n } can be obtained through defining the products and maps of the vectors {e 1 , e 2 , . . . , e n }. In 2 dimensions, we have : e 2 ≺ e 1 = e 1 , e 2 ≺ e 2 = e 1 , e 1 ≻ e 2 = e 1 , e 2 ≻ e 1 = e 1 , e 2 ≻ e 2 = e 1 , α(e 2 ) = e 1 . On a 3-dimensional vector space, define the following Hom-dendrifom algebra structure respectively : e 1 ≺ e 1 = e 1 , , e 3 ≺ e 2 = e 3 , e 3 ≺ e 3 = e 3 , e 2 ≻ e 3 = e 3 , e 3 ≻ e 3 = e 3 , α(e 1 ) = e 1 . Definition 2.6. A Hom-Zinbiel algebra is a triple (R, •, α) consisting of vector space A on which 2. e 1 • e 1 = e 1 , e 2 • e 2 = e 1 , α(e 1 ) = e 1 , α(e 2 ) = e 1 + e 2 . * : A ⊗ A −→ A and α : A −→ A are linear maps satisfying (x • y) • α(z) = α(x) • (y • z) + α(x) • (z • y)(2. 3. e 1 • e 1 = e 2 , e 2 • e 1 = −e 2 , α(e 1 ) = e 1 , α(e 2 ) = e 1 + e 2 . Proposition 2.8. Let R be a Hom-Zinbiel algebra and put x ≺ y = x • y, x ≻ y = y • x, x, y ∈ R. Then (R, ≺, ≻, α) is a Hom-dendriform algebra. Conversely, a commutative Hom-dendriform algebra (i.e a Hom-dendriform algebra for which x ≺ y = y ≻ x) is a Hom-dendriform algebra. Proof. Indeed, (i) (x ≺ y) ≺ α(z) = (x • y) • α(z). But α(x) ≺ (y ≺ z) + α(x) ≺ (y ≻ y) = α(x) • (y • z) + α(x) • (z • y). Therefore, (i) holds. (ii) (x ≻ y) ≺ α(z) = (y • x) • α(z) and α(x) ≻ (y ≺ z) = (y • z) • α(x) . However, these two expressions are the same according to the axioms of Hom-Zinbiel algebras. (iii) (x ≺ y) ≻ α(z) + (x ≻ y) ≻ α(z) = α(z) • (x • y) + α(z) • (y • x), which is equal to (z • y) • α(z) = α(x) ≻ (y ≻ z) . As a matter of fact, (iii) also holds. Proposition 2.9. Let (R, ·, α) be a Hom-Zinbiel algebra. Then, the symmetrized product x · y = x • y + y • x is Hom-associative (,i.e., under the symmetrized product, R becomes a Hom-associative and commutative algebra. Proof. Indeed, (x · y)α(z) = (x • y + y • x)α(z) = (x • y + y • x) • α(z) + α(z) • (x • y + y • x) • α(z) = (x • y) • α(z) + (y • x) • α(z) + α(z) • (x • y) + (y • x) • α(z) and α(x)(y · z) = α(x)(y • z + z • y) = α(x) • (y • z + z • y) + (y • z + z • y)α(x) = α(x) • (y • z) + α(x) • (z • y) + (y • z) • α(x) + (z • y) • (x) . Now, if we take into account Hom-Zinbiel indentity and its consequence (y • x) • α(z) = (y • z)α(z), then we get the following required equality (x · y)α(z) = (x · y)α(z). Definition 2.10. A Hom-dipterous algebra is a quadruple (Z, ≺, * , α) consisting of a vector space A with the operations ≺, * : A ⊗ A −→ A and α : A −→ A which are linear maps satisfying (x * y) ≻ α(z) = α(x) ≻ (y ≻ z), (x * y) * α(z) = α(x) * (y * z) (2.10) for all x, y, z ∈ A. Similarly, a right Hom-dipterous algebra is defined by the following relations (x ≺ y) ≺ α(z) = α(x) ≺ (y * z), (x * y) * α(z) = α(x) * (y * z),(2.11) for all x, y, z ∈ A. Example 2.11. Let {e 1 , e 2 , e 3 } be a basis of 3-dimensional multiplicative linear space A over K. The following multiplications and linear map α on A define the structure of a Hom-dipterous algebra: 1. e 1 * e 2 = e 3 , e 3 * e 2 = e 3 , e 3 * e 3 = e 3 , e 1 ≺ e 2 = e 3 , e 2 ≺ e 3 = e 3 , α(e 1 ) = e 1 . 2. e 1 * e 1 = e 1 , e 1 * e 2 = e 3 , e 3 * e 2 = e 3 , e 3 * e 3 = e 3 , e 1 ≺ e 2 = e 3 , e 2 ≺ e 3 = e 3 , α(e 1 ) = e 1 . Proposition 2.12. Let (A, * , ≻) be a dipterous algebra and let α : A −→ A be a dipterous algebra endomorphism. Hence, A α = (A, * α , ≻ α , α), where * α = α • * and ≻ α = α• ≻, is a Hom-dipterous algebra. Proof. Notice that (x * α y) ≻ α α(z) = α 2 ((x * y) ≻ z) α(x) ≻ α (y ≻ α z) = α 2 (x ≻ (y ≻ z)) (x * α y) * α α(z) = α 2 ((x * y) * z) α(x) * α (y * α z) = α 2 (x * (y * z)). Similarly, (x ≺ α y) ≺ α α(z) = α 2 ((x ≺ y) ≺ z) α(x) ≺ α (y * α z) = α 2 (x ≺ (y * z)) (x * α y) * α α(z) = α 2 ((x * y) * z) α(x) * α (y * α z) = α 2 (x * (y * z)). Thus, the identities (2.10) and (2.11) obviously follow from the identities satisfied by (A, * , ≻). Definition 2.13. Let (A, ⊣, ⊢, α) be a Hom-associative dialgebra. If there is an associative dialgebra (A, ⊣ ′ , ⊢ ′ ) such that ⊣ ′ = α• ⊣ and ⊢ ′ = α• ⊢, we state that (A, ⊣ ′ , ⊢ ′ ) is the untwist of (A, ⊣, ⊢, α). Proposition 2.14. Let (A, ⊣, ⊢, α) be an n-dimensional Hom-associative dialgebra and let Φ : A → A be an invertible linear map. Then, there is an isomorphism with an n-dimensional Hom-associative dialgebra (A, ⊣ ′ , ⊢ ′ , ΦαΦ −1 ) where ⊣ ′ = Φ• ⊣ •(Φ −1 ⊗ Φ −1 ) and ⊢ ′ = Φ• ⊢ •(Φ −1 ⊗ Φ −1 ). Proof. We prove that for any invertible linear map Φ : A → A, (A, ⊣ ′ , ⊢ ′ , ΦαΦ −1 ) is a Hom-associative dialgebra. (x ⊣ ′ y) ⊣ ′ ΦαΦ −1 (z) = (Φ • (Φ −1 (x) ⊣ Φ −1 (y))) ⊣ ′ Φ • α • Φ −1 (z) = Φ • (Φ −1 (x) ⊣ Φ −1 (y)) ⊣ α • Φ −1 (z) = Φ • Φ −1 ((x) ⊣ (y)) ⊣ α(z) = Φ • Φ −1 (α(x) ⊣ (y)) ⊢ z) = Φ • (αΦ −1 (x) ⊣ Φ −1 (y)) ⊢ Φ −1 (z)) = Φ • ((Φ −1 ⊗ Φ −1 )(Φ ⊗ Φ)αΦ −1 (x) ⊣ Φ −1 (y)) ⊢ Φ −1 (z)) = Φ • (Φ • αΦ −1 (x)) ⊣ Φ • (Φ −1 (y) ⊢ Φ −1 (z)) = Φ • αΦ −1 (x) ⊣ ′ (y ⊢ ′ z). (x ⊢ ′ y) ⊣ ′ ΦαΦ −1 (z) = Φ • (Φ −1 (x) ⊢ Φ −1 (y)) ⊣ ′ Φ • α • Φ −1 (z) = Φ • (Φ −1 (x) ⊢ Φ −1 (y)) ⊣ α • Φ −1 (z) = Φ • (Φ −1 ((x ⊢ y) ⊣ α(z))) = Φ • Φ −1 ((α(x) ⊢ (y ⊢ z))) = Φ • (αΦ −1 (x) ⊢ (Φ −1 (y) ⊣ Φ −1 (z))) = Φ • ((Φ −1 ⊗ Φ −1 )(Φ ⊗ Φ)αΦ −1 (x) ⊢ (Φ −1 (y) ⊣ Φ −1 (z))) = Φ • ((Φ −1 ⊗ Φ −1 ) • ΦαΦ −1 (x)) ⊢ Φ • (Φ −1 (y) ⊣ Φ −1 (z))) = Φ • αΦ −1 (x) ⊢ ′ (y ⊣ ′ z). (x ⊢ ′ y) ⊢ ′ ΦαΦ −1 (z) = Φ • (Φ −1 (x) ⊢ Φ −1 (y)) ⊢ ′ Φ • α • Φ −1 (z) = Φ • (Φ −1 (x) ⊢ Φ −1 (y)) ⊢ α • Φ −1 (z) = Φ • (Φ −1 ((x ⊢ y) ⊢ α(z))) = Φ • Φ −1 ((x ⊣ y) ⊢ z))) = Φ • (Φ −1 (x) ⊣ (Φ −1 y)) ⊢ Φ −1 (z))) = Φ • ((Φ −1 ⊗ Φ −1 )(Φ ⊗ Φ) • (Φ −1 (x) ⊣ Φ −1 (y)) ⊢ Φ −1 (z))) = Φ • ((Φ −1 ⊗ Φ −1 ) • Φ(Φ −1 (x) ⊣ Φ −1 (y)) ⊢ Φ • α • Φ −1 (z))) = (x ⊣ ′ y) ⊢ Φ • αΦ −1 (z). From this perspective, (A, ⊣ ′ , ⊢ ′ , ΦαΦ −1 ) is a Hom-associative dialgebra.It is also multiplicative. Indeed, for α ΦαΦ −1 • (x ⊣ ′ y) = ΦαΦ −1 Φ • x ⊣ (Φ −1 ⊗ Φ −1 )(y) = Φα • x ⊣ (Φ −1 ⊗ Φ −1 )(y) = Φα • Φ −1 (x) ⊣ Φ −1 (y) = Φ • (αΦ −1 (x) ⊣ αΦ −1 (y)) = Φ • (Φ −1 ⊗ Φ −1 )(Φ ⊗ Φ)(αΦ −1 (x) ⊣ αΦ −1 (y)) = Φ • (Φ −1 ⊗ Φ −1 )(ΦαΦ −1 (x) ⊣ αΦΦ −1 (y)) = ΦαΦ −1 (x) ⊣ ′ ΦαΦ −1 (y). It follows that ΦαΦ −1 • (x ⊢ ′ y) = ΦαΦ −1 Φ • x ⊢ (Φ −1 ⊗ Φ −1 )(y) = Φα • x ⊢ (Φ −1 ⊗ Φ −1 )(y) = Φα • Φ −1 (x) ⊢ Φ −1 (y) = Φ • (αΦ −1 (x) ⊢ αΦ −1 (y)) = Φ • (Φ −1 ⊗ Φ −1 )(Φ ⊗ Φ)(αΦ −1 (x) ⊢ αΦ −1 (y)) = Φ • (Φ −1 ⊗ Φ −1 )(ΦαΦ −1 (x) ⊢ αΦΦ −1 (y)) = ΦαΦ −1 (x) ⊢ ′ ΦαΦ −1 (y). Therefore, Λ : (A, ⊣, ⊢, α) → (A, ⊣ ′ , ⊢ ′ , ΦαΦ −1 ) is a Hom-associative dialgebras morphism, since Φ• ⊣= Φ• ⊣ •(Φ −1 ⊗ Φ −1 ) • (Φ ⊗ Φ) =⊣ ′ •(Φ ⊗ Φ) and (ΦαΦ −1 ) • Φ = Φ • α. Proposition 2.15. Let (A, ⊣, ⊢, α) be a Hom-associative dialgebra over K. Let (A, ⊣ ′ , ⊢ ′ , ΦαΦ −1 ) be its isomorphic Hom-associative dialgebra described in Proposition 2.14. If ψ is an automorphism of (A, ⊣, ⊢, α), then ΦψΦ −1 is an automorphism of (A, ⊣, ⊢, ΦαΦ −1 ). Proof. Note that γ = ΦαΦ −1 . We have ΦψΦ −1 γ = ΦψΦ −1 ΦαΦ −1 = ΦψαΦ −1 = ΦαψΦ −1 = ΦαΦ −1 ΦψΦ −1 = γΦψΦ −1 . For any x, y ∈ A, ΦψΦ −1 • (Φ(x) ⊣ ′ Φ(y)) = ΦψΦ −1 • Φ • (x ⊣ y) = Φ • ψ • (x ⊣ y) = Φ • (ψ(x) ⊣ ψ(y)) = Φ • ψ(x) ⊣ ′ Φ • ψ(y)) = ΦψΦ −1 Φ(x) ⊣ ′ ΦψΦ −1 Φ(y) This entails, ΦψΦ −1 • (Φ(x) ⊢ ′ Φ(y)) = ΦψΦ −1 • Φ • (x ⊢ y) = Φ • ψ • (x ⊢ y) = Φ • (ψ(x) ⊢ ψ(y)) = Φ • ψ(x) ⊢ ′ Φ • ψ(y)) = ΦψΦ −1 Φ(x) ⊢ ′ ΦψΦ −1 Φ(y) By Definition, ΦψΦ −1 is an automorphism of (A, ⊣ ′ , ⊢ ′ , ΦαΦ −1 ). Classification in low dimensions Let Hd n (K) denote the variety of n dimensional Hom-associative dialgebras over a field A. If A is an n-dimensional algebra, then the product of any two elements x and y can be expressed by the product of basis elements {e 1 , e 2 , e 3 , · · · , e n }. Recall that a Hom-diassociative structure on A can then be defined by two bilinear mappings : ⊣: A × A −→ A representing the left product, ⊣: A × A −→ A representing the left product ⊢ and α : A −→ A representing a linear map, all satisfying the above-mentionend identities when a Hom-associative dialgebra D can be regarded as a quadruplet D = (A, ⊣, ⊢, α) where ⊣, ⊢ and α are Hom-associative dialgebra laws on A. Let us denote by γ k i j , δ q st and a i j , where i, j, k, s, t, q = 1, 2, 3, . . . , n, the structure constants of a Hom-associative dialgebra with respect to the basis e 1 , e 2 , . . . , e n of A. As a result, Hd can be considered as a closed subset of 2n 3 + n 2 -dimensional affine space specified by the following system of polynomial equations with respect to the structure constants γ k i j , δ q st and a ji : Let {e 1 , e 2 , e 3 , · · · , e n } be a basis of an n-dimensional Hom-associative dialgebra A. The product of basis is denoted by We seek for all 2-dimensional Hom-associative dialgebras, we consider two classes of morphisms which are given by their Jordan forms. This implies that they are represented by the matrices a 0 0 b and a 1 0 a . Using similar calculations as depicted in the previous section, we obtain the following classification. Hd 3 2 : e 2 ⊣ e 2 = − e 1 2 − e 2 , e 2 ⊢ e 1 = e 1 , e 2 ⊢ e 2 = −e 1 − e 2 , α(e 1 ) = −e 1 , α(e 2 ) = e 1 + e 2 . Hd 4 2 : e 1 ⊣ e 2 = e 1 , e 2 ⊣ e 2 = e 1 + e 2 , e 1 ⊢ e 2 = e 1 , e 2 ⊢ e 2 = e 1 + e 2 . α(e 1 ) = e 1 , α(e 2 ) = e 1 + e 2 . Hd 5 2 : e 1 ⊣ e 2 = e 1 , e 2 ⊣ e 1 = e 1 , e 2 ⊣ e 2 = e 1 , e 2 ⊢ e 1 = e 1 , α(e 2 ) = e 1 . Hd 6 2 : e 1 ⊣ e 2 = e 1 , e 2 ⊣ e 2 = e 2 , e 2 ⊢ e 1 = e 1 , e 2 ⊢ e 2 = e 1 , α(e 1 ) = e 1 , α(e 2 ) = e 2 . Hd 7 2 : e 1 ⊣ e 1 = e 1 + e 2 , e 1 ⊢ e 1 = e 1 , e 1 ⊢ e 2 = e 1 , α(e 1 ) = e 1 , α(e 2 ) = e 2 . Hd 8 2 : e 1 ⊣ e 2 = e 1 , e 2 ⊣ e 1 = e 1 , e 2 ⊣ e 2 = e 1 , e 2 ⊢ e 2 = e 1 , α(e 2 ) = e 1 . Hd 9 2 : e 1 ⊣ e 2 = ae 1 , e 2 ⊣ e 2 = be 1 + ce 2 , e 2 ⊢ e 1 = f e 1 , e 2 ⊢ e 2 = ge 1 + ke 2 , α(e 1 ) = e 1 , α(e 2 ) = e 1 + e 2 . Proof. Let A be a two-dimensional vector space. To determine a Hom-associative dialgebra structure on A, we consider A with respect to Hom-diassociative operation. Let H ′ 2 = (A, ⊢, α) be the algebra e 1 ⊢ e 1 = e 1 , e 1 ⊢ e 2 = e 2 , e 2 ⊢ e 1 = e 1 , e 2 ⊢ e 2 = e 2 , α(e 1 ) = e 1 , α(e 2 ) = e 2 . The second multiplication operation ⊣ in A, is indicated as follows : e 1 ⊣ e 1 = a 1 e 1 + a 2 , e 1 ⊣ e 2 = a 3 e 1 + a 4 e 2 , e 2 ⊣ e 1 = a 5 e 1 + a 6 e 2 , e 2 ⊢ e 2 = a 7 e 1 + a 8 e 2 . Now, verifying Hom-associative dialgebras axioms, we get several constants for the coefficients a i where, i = 1, 2, . . . , 8. Applying (e 1 ⊣ e 1 ) ⊢ α(e 1 ) = α(e 1 ) ⊢ (e 1 ⊢ e 1 ), we have (a 1 e 1 + a 2 e 2 ) ⊢ e 1 = e 1 ⊢ e 1 . Then, a 1 e 1 = e 1 . Therefore a 1 = 1. The verification, (e 1 ⊢ e 1 ) ⊣ α(e 1 ) = α(e 1 ) ⊢ (e 1 ⊣ e 1 ) yields (e 1 + a 2 e 2 ) ⊢ e 1 = e 1 ⊢ e 1 . We have e 1 + a 2 e 2 = e 1 . Hence we get a 2 = 0. Consider (e 1 ⊣ e 2 ) ⊢ α(e 1 ) = α(e 1 ) ⊢ (e 2 ⊢ e 1 ). It implies that (a 3 e 1 + a 4 e 2 ) ⊣ e 1 = e 1 ⊣ e 1 . Hence, a 3 = 1 and a 4 = 0. The next relation to consider is (e 2 ⊢ e 1 ) ⊢ α(e 2 ) = α(e 2 ) ⊢ (e 1 ⊣ e 2 ). Hence, a 3 = 0 and a 4 = 1. Consider (e 2 ⊣ e 1 ) ⊢ α(e 2 ) = α(e 2 ) ⊢ (e 1 ⊢ e 2 ). It implies that (a 5 e 1 + a 6 e 2 ) ⊢ e 2 ) = e 2 ⊢ e 2 , Therefore, a 5 = 0 and a 6 = 1. Finally, we apply (e 2 ⊣ e 2 ) ⊢ α(e 2 ) = α(e 2 ) ⊢ (e 2 ⊢ e 2 ). We obtain (a 7 e 1 + a 8 e 2 ) ⊢ e 2 = e 2 ⊢ e 2 and we get a 7 = 0 and a 8 = 1. The verification of all other cases leads to the obtained constraints. If a 3 = 1 and a 4 = 0, then the right and left product coincide and we get the Hom-associative dialgebra. If a 3 = 0 and a 4 = 1, we obtain the Hom-associative dialgebras Hd 1 2 . The other Hom-associative dialgebras of the list of Theorem 3.1 can be obtained by minor modification of the above observation. We seek for all 3-dimensional Hom-associative dialgebras, we consider two classes of morphisms that are provided by their Jordan forms. This implies that they are represented by the matrices           a 0 0 0 b 0 0 0 c           ,           a 1 0 0 a 0 0 0 b           ,           a 1 0 0 a 0 0 0 a           ,           a 1 0 0 a 1 0 0 a           . Using similar calculations as in the previous Section, we obtain the following classification. Theorem 3.2. Every 3-dimensional multiplicative Hom-associative dialgebra is isomorphic to one of the following pairwise non-isomorphic Hom-associative dialgebras (A, ⊣, ⊢, α), where ⊣, ⊢ are left product and right product and α of the structure map. Hd 1 3 : e 2 ⊣ e 2 = e 1 , e 2 ⊣ e 3 = e 1 , e 3 ⊣ e 2 = e 1 , e 3 ⊣ e 3 = e 2 , e 2 ⊢ e 2 = e 1 , e 2 ⊢ e 3 = e 1 , e 3 ⊢ e 3 = e 1 , α(e 2 ) = e 1 . Hd 2 3 : e 2 ⊣ e 1 = e 1 , e 2 ⊣ e 3 = e 1 , e 3 ⊣ e 2 = e 1 , e 3 ⊣ e 3 = e 2 , e 2 ⊢ e 2 = e 1 , e 2 ⊢ e 3 = e 2 , e 3 ⊢ e 3 = e 1 , α(e 2 ) = e 1 . Hd 3 3 : e 2 ⊣ e 2 = e 1 , e 2 ⊣ e 3 = e 1 , e 3 ⊣ e 2 = e 1 , e 3 ⊣ e 3 = e 1 , e 2 ⊢ e 2 = e 1 , e 2 ⊢ e 3 = e 1 , α(e 2 ) = e 1 . Hd 4 3 : e 2 ⊣ e 2 = e 1 , e 2 ⊣ e 3 = e 1 , e 3 ⊣ e 2 = e 1 , e 3 ⊣ e 3 = e 1 , e 2 ⊢ e 2 = e 1 , e 2 ⊢ e 3 = e 1 , e 3 ⊢ e 3 = e 1 , α(e 1 ) = e 1 .e 2 ⊣ e 2 = e 2 , e 2 ⊣ e 3 = e 2 , e 2 ⊢ e 2 = e 2 , e 2 ⊢ e 3 = e 2 , e 3 ⊢ e 2 = e 1 , α(e 1 ) = e 1 . Hd 9 3 : e 2 ⊣ e 2 = e 2 , e 2 ⊣ e 3 = e 2 , e 2 ⊢ e 2 = e 2 , e 2 ⊢ e 3 = e 2 , e 3 ⊢ e 2 = e 1 , e 3 ⊢ e 3 = e 3 , α(e 1 ) = e 1 . Hd 10 3 : e 1 ⊣ e 2 = e 1 , e 2 ⊣ e 1 = e 1 , e 2 ⊣ e 3 = e 1 , e 3 ⊣ e 2 = e 3 , e 1 ⊢ e 2 = e 1 , e 2 ⊢ e 2 = e 1 , e 2 ⊢ e 3 = e 3 , e 3 ⊢ e 2 = e 1 , α(e 2 ) = e 1 . Hd 11 3 : e 1 ⊣ e 2 = (−1) 2/3 e 1 , e 2 ⊣ e 1 = ae 1 , e 2 ⊣ e 2 = be 1 + ce 3 , e 2 ⊣ e 3 = e 1 + de 3 , e 3 ⊣ e 2 = e 1 + (−1) 2/3 e 3 , e 3 ⊣ e 3 = i √ 3 e 1 , e 1 ⊢ e 2 = (−1) 2/3 e 1 , e 2 ⊢ e 1 = f e 1 , e 2 ⊢ e 2 = e 1 + g(−1) 2/3 e 3 , e 3 ⊢ e 2 = e 1 + e 3 , e 3 ⊢ e 3 = i √ 3 e 1 , α(e 2 ) = e 1 . Hd 12 3 : e 1 ⊣ e 2 = e 1 , e 2 ⊣ e 1 = e 1 , e 2 ⊣ e 2 = e 1 + e 3 , e 2 ⊣ e 3 = e 1 + e 3 , e 3 ⊣ e 2 = e 1 + e 3 , e 3 ⊣ e 3 = e 3 , e 1 ⊢ e 2 = e 1 , e 2 ⊢ e 1 = e 1 , e 2 ⊢ e 2 = e 1 + e 3 , e 2 ⊢ e 3 = e 1 + e 3 , e 3 ⊢ e 2 = e 1 + e 3 , α(e 2 ) = e 1 . Hd 13 3 : e 1 ⊣ e 1 = (−1) 1/3 e 2 , e 1 ⊣ e 3 = ae 2 , e 3 ⊣ e 1 = be 2 , e 3 ⊣ e 3 = (−1) 1/3 √ 3 ie 2 , e 1 ⊢ e 1 = (−1) 1/3 e 2 , e 1 ⊢ e 3 = ce 1 , e 3 ⊢ e 1 = (−1) 1/3 √ 3 ie 2 , α(e 1 ) = e 1 , α(e 2 ) = e 2 , α(e 3 ) = e 3 . Hd 14 3 : e 1 ⊣ e 1 = e 1 + a(−1) 2/3 e 3 , e 1 ⊣ e 3 = be 3 , e 3 ⊣ e 1 = i √ 3 e 1 + c(−1 + (−1) 2/3 )e 3 , e 1 ⊢ e 1 = e 1 + d(−1) 2/3 e 3 , e 1 ⊢ e 3 = e 1 , e 2 ⊢ e 2 = f e 1 + ge 3 , e 3 ⊢ e 1 = e 1 + h(−1) 2/3 e 3 , e 3 ⊢ e 3 = e 1 + i √ 3 e 3 , α(e 2 ) = e 2 . Proof. The proof is similar to Theorem 3.1 We seek for all 4-dimensional Hom-associative dialgebras, we consider two classes of morphisms which are given by their Jordan forms. This implies that they are represented by the matrices                a 0 0 0 0 b 0 0 0 0 c 0 0 0 0 d                ,                a 1 0 0 0 a 1 0 0 0 a 1 0 0 0 a                ,                a 1 0 0 0 a 0 0 0 0 a 1 0 0 0 a                ,                a 1 0 0 0 a 1 0 0 0 a 0 0 0 0 a                ,                a 1 0 0 0 a 0 0 0 0 a 0 0 0 0 a                . Using similar calculations as displayed in the previous Section, we obtain the following classification. Proof. The proof is similar to Theorem 3.1 Derivations of Complex Hom-associative dialgebras This section is notably devoted to the description of derivations of two, three and four-dimensional Complex Hom-associative dialgebras. Derivations of Complex Hom-associative dialgebras Let (A, ⊣, ⊢, α) be a multiplicative Hom-associative dialgebra. For any nonnegative integer k, we denote by α k the k-fold composition of α with itself,i.e., α k = α • · · · • α (k-times). In particular, α 0 = id and α 1 = α. D • α = α • D (4.1) D • ( f ⊣ g) = (D( f ) ⊣ α k (g)) + (α k ( f ) ⊣ D(g)). (4.2) D • ( f ⊢ g) = (D( f ) ⊢ α k (g)) + (α k ( f ) ⊢ D(g)). (4.3) The map D( f ) is an α k+1 -derivation, which we will call an inner α k+1 -derivation. In fact, we have D k ( f )(α(g)) = α k+1 (g) ⊣ f = α(α k (g) ⊣ f ) = α • D k ( f )(g), D k ( f )(α(g)) = α k+1 (g) ⊢ f = α(α k (g) ⊢ f ) = α • D k ( f )(g), which implies that identity (4.1) in Definition 4.1 is satisfied. On the other side, we have D k ( f )(g ⊣ h) = α k ((g ⊣ h) ⊣ f ) = (α k (g) ⊣ α k (h)) ⊣ α( f )) = α k+1 (g) ⊣ α k (h)) ⊣ f + (α k (g) ⊣ f ) ⊣ α k+1 (h) = (α k+1 (g) ⊣ D k ( f )(h)) + D k ( f )(g) ⊣ α k+1 (h). Therefore, D k ( f ) is an α k+1 -derivation. The set of α k -derivations denoted by Inner α k (A) are expressed in terms of Proof. For any f, g ∈ A, we have Inner α k (A) = α k−1 (•) ⊣ f \| f ∈ A, α( f ) = f . (4.4) Inner α k (A) = α k−1 (•) ⊢ f \| f ∈ A, α( f ) = f .[D, D ′ ] ( f ⊣ g) = D • D ′ ( f ⊣ g) − D ′ • D( f ⊣ g) = D(D ′ ( f ) ⊣ α s (g)) + α s ( f ) ⊣ D ′ (g))) − D ′ (µ(D( f ), α k (g)) + µ(α k ( f ), D(g))) = D • D ′ ( f ) ⊣ α k+s (g)) + α k • D ′ ( f ) ⊣ D • α s (g) + D • α s ( f ) ⊣ α k • D ′ (g) +α k+s ( f ) ⊣ D • D ′ (g)) − D ′ • D( f ) ⊣ α k+s (g)) − α s • D( f ) ⊣ D ′ • α k (g) −D ′ • α k ( f ) ⊣ α s • D(g) − α k+s ( f ) ⊣ D ′ • D(g). Since D and D ′ satisfy D • α = α • D, D ′ • α = α • D ′ , we obtain α k • D ′ = D ′ • α k , D • α s = α s • D. Therefore, we get D, D ′ ( f ⊣ g) = α k+s ( f ) ⊣ D, D ′ (g)) + D, D ′ ( f ) ⊣ α k+s (g)). Furthermore, it is straightforward to infer that D, D ′ • α = D • D ′ • α − D ′ • D • α = α • D • D ′ − α • D ′ • D = α • D, D ′ , which yields that [D, D ′ ] ∈ Der α k+s (A). Definition 4.3. A Hom-associative triple system is a vector space A over field K with a trilinear multiplications satisfying (((x ⊣ y) ⊣ α(z)) ⊣ α(u)) ⊣ α(w) = ((α(x) ⊣ (y ⊣ z)) ⊣ α(u)) ⊣ α(w)) = α(x) ⊣ (α(y) ⊣ (z ⊣ u) ⊣ α(w))), (((x ⊢ y) ⊢ α(z)) ⊢ α(u)) ⊢ α(w) = ((α(x) ⊢ (y ⊢ z)) ⊢ α(u)) ⊢ α(w)) = α(x) ⊢ (α(y) ⊢ (z ⊢ u) ⊢ α(w))), for any x, y, z, u, w ∈ A. Definition 4.4. An associative triple derivation of Hom-associative dialgebra (A, ⊣, ⊢, α) is a linear transformation D : A −→ A such that D • α = α • D D • ((x ⊣ y) ⊣ z)) = (D(x) ⊣ α k (y)) ⊣ α k (z) +(α k (x) ⊣ D(y)) ⊣ α k (z) + (α k (x) ⊣ α k (y)) ⊣ D(z) D • ((x ⊢ y) ⊢ z)) = (D(x) ⊢ α k (y)) ⊢ α k (z) +(α k (x) ⊢ D(y)) ⊢ α k (z) + (α k (x) ⊢ α k (y)) ⊢ D(z) for x, y, z ∈ A. Definition 4.5. A Jordan associative triple derivation of Hom-associative (A, ⊣, ⊢ α) is a linear trans- formation D ′ : A −→ A such that D ′ • α = α • D ′ and D ′ • ((x ⊣ y) ⊣ x)) = (D ′ (x) ⊣ α k (y)) ⊣ α k (x) +(α k (x) ⊣ D ′ (y)) ⊣ α k (x) + (α k (x) ⊣ α k (y)) ⊣ D ′ (x) D ′ • ((x ⊢ y) ⊢ x)) = (D ′ (x) ⊢ α k (y)) ⊢ α k (x) +(α k (x) ⊢ D ′ (y)) ⊢ α k (x) + (α k (x) ⊢ α k (y)) ⊢ D ′ (x) for x, y ∈ A. Proposition 4.6. D is an associative triple derivation of (A, ⊣, ⊢, α) if and only if D is a Jordan triple derivation of (A, ⊣, ⊢ α) such that A(x, y, z) + A(y, z, x) + A(z, x, y) = 0 with A(x, y, z) = (α • D) • (x ⊣ y) ⊣ z). Proof. If D is a Jordan triple derivation of (A, ⊣, ⊢, α), then 1 following immediately. 2 holds because A(x, y, z) + A(y, z, x) + A(z, x, y) = = α((D(x) ⊣ α k (y)) ⊣ α k (z)) + (α k (x) ⊣ D(y)) ⊣ α k (z) + (α k (x) ⊣ α k (y)) ⊣ D(z)) +α((D(y) ⊣ α k (z)) ⊣ α k (x) + (α k (y)) ⊣ D(z)) ⊣ α k (x) + (α k (y) ⊣ α k (z)) ⊣ D(x)) +α((D(z) ⊣ α k (x)) ⊣ α k (y)) + (α k (z) ⊣ D(x)) ⊣ α k (y) + (α k (z) ⊣ α k (x)) ⊣ D(y)) = α((D(x) ⊣ α k (y)) ⊣ α k (z)) + (α k (y) ⊣ α k (z)) ⊣ D(x) + (α k (z) ⊣ D(x)) ⊣ α k (y) +α((α k (x) ⊣ D(y)) ⊣ α k (z)) + (D(y) ⊣ α k (y)) ⊣ α k (x) + (α k (z) ⊣ α k (x)) ⊣ D(y) +α((α k (x) ⊣ α k (y)) ⊣ D(z)) + (α k (y) ⊣ D(z)) ⊣ α k (x) + (D(z) ⊣ α k (x)) ⊣ α k (y) = 0. Therefore, D is a generalized associative triple derivation of Hom-associative (A, ⊣, ⊢, α). Proposition 4.7. D is an associative triple derivation of (A, ⊣, ⊢, α) with respect to associative derivation δ if and only if D is Jordan triple derivation of (A, ⊣, ⊢, α) with respect to a Jordan triple derivation δ such that (α k (x) ⊣ α k (y)) ⊣ (D − δ)(z)) + (α k (y) ⊣ α k (z)) ⊣ (D − δ)(x)) + (α k (z) ⊣ α k (x)) ⊣ (D − δ)(y)) = 0, with B(x, y, z) = α((δ(x) ⊣ α k (y) ⊣ α k (z)) + (α k (x) ⊣ δ(y)) ⊣ α k (z)) + ((α k (x) ⊣ α k (y)) ⊣ δ(z)). Proof. If D is a Jordan triple derivation of (A, ⊣, ⊢, α), then 1 follows immediately. 2 holds because B(x, y, z) + B(y, z, x) + B(z, x, y) = = α((δ(x) ⊣ α k (y)) ⊣ α k (z)) + (α k (x) ⊣ δ(y)) ⊣ α k (z)) + (α k (x) ⊣ α k (y)) ⊣ δ(z))) +α((δ(y) ⊣ α k (z)) ⊣ α k (x)) + (α k (y) ⊣ δ(z)) ⊣ α k (x)) + (α k (y) ⊣ α k (z)) ⊣ δ(x))) +α((δ(z) ⊣ α k (x)) ⊣ α k (y)) + (α k (z) ⊣ δ(x)) ⊣ α k (y)) + (α k (z) ⊣ α k (x)) ⊣ δ(y))) = α((δ(x) ⊣ α k (y)) ⊣ α k (z)) + (α k (x) ⊣ δ(y)) ⊣ α k (z)) + (α k (x) ⊣ α k (y)) ⊣ δ(z))) +α((δ(y) ⊣ α k (z)) ⊣ α k (x)) + (α k (y) ⊣ δ(z)) ⊣ α k (x)) + (α k (y) ⊣ α k (z)) ⊣ δ(x))) +α((δ(z) ⊣ α k (x)) ⊣ α k (y)) + (α k (z) ⊣ δ(x)) ⊣ α k (y)) + (α k (z) ⊣ α k (x)) ⊣ δ(y))) = α((δ(x) ⊣ α k (y)) ⊣ α k (z)) + (α k (y) ⊣ α k (z)) ⊣ δ(x)) + (α k (z) ⊣ δ(x)) ⊣ α k (y))) +α((α k (x) ⊣ δ(y)) ⊣ α k (z)) + (δ(y) ⊣ α k (z)) ⊣ α k (x)) + (α k (z) ⊣ α k (x)) ⊣ δ(y))) +α((α k (x) ⊣ α k (y)) ⊣ δ(z)) + (α k (y) ⊣ δ(z)) ⊣ α k (x)) + (δ(z) ⊣ α k (x)) ⊣ α k (y))) = 0. Note that δ is an associative derivation referring to Proposition 4.6. Therefore, D is a generalized associative triple derivation of (A, ⊣, ⊢, α) with respect to an associative derivation δ according to Proposition 4.6. Definition 4.8. An α-derivation of the BiHom-associative trialgebra A is a linear transformation d : A → A satisfying α • d = d • α , (4.6) d(x ⊣ y) = d(x) ⊣ α(y) + α(x) ⊣ d(y) (4.7) d(x ⊢ y) = d(x) ⊢ α(y) + α(x) ⊢ d(y),(4.8) for all x, y ∈ A. Derivations of complex Hom-associative dialgebras This section illustrates in depth, α-derivation of Hom-associative dialgebras in dimension two and three over the field K. Let {e 1 , e 2 , e 3 , · · · , e n } be a basis of an n-dimensional Hom-associative dialgebra A. The product of basis is denoted by d(e p ) = n q=1 d qp e q . We have IC Der(d) Dim(d) IC Der(d) Dim(d) T H 1 3           0 0 0 0 0 0 d 31 0 d 33           2 T H 2 3           0 0 0 0 0 0 d 31 0 d 33           2 T H 3 3           0 0 0 d 21 0 d 23 d 31 0 d 33           4 T H 4 3           0 0 0 0 d 22 d 23 0 d 32 d 33           4 T H 5 3           0 0 0 0 0 0 0 d 32 d 33           2 T H 8 3           0 0 0 0 0 0 0 d 32 d 33           2 T H 10 3           0 0 0 d 21 0 d 23 0 0 0           2 T H 11 3           0 0 0 d 21 0 d 23 0 0 0           2 T H 12 3           0 0 0 0 0 d 23 0 0 0           1 T H 13 3           0 0 0 0 0 0 0 d 32 0           1 Proof. Departing from Theorem 4.10, we provide the proof only for one case to illustrate the used approach, the other cases can be addressed similarly with or without modification(s). Let's consider T H 3 3 . Applying the systems of equations (4.9), (4.10) and (4.11), we get d 11 = d 12 = d 13 = d 22 = d 32 = 0. Hence, the derivations of T H 3 3 are indicated as follows d(e 1 ) =           0 0 0 1 0 0 0 0 0           , d(e 2 ) =           0 0 0 0 0 0 1 0 0           , d(e 3 ) =           0 0 0 0 0 1 0 0 0           , and d(e 4 ) =           0 0 0 0 0 0 0 0 1           is the basis of Der (Cent(A)) and DimDer(Cent(A)) = 4. The derivations of the remaining parts of dimension two associative dialgebras can be handled in a similar manner as illustrated above. Theorem 4.11. The derivations of 4-dimensional Hom-associative dialgebras have the following form IC Der(d) Dim(d) IC Der(d) Dim(d) T H 1 4                0 0 0 0 d 21 0 d 23 0 0 0 0 0 d 41 0 d 43 0                4 T H 2 4                0 0 0 0 d 21 0 d 23 0 0 0 0 0 d 41 0 d 43 0                2 T H 3 4                0 0 0 0 d 21 0 d 23 0 0 0 0 0 d 41 0 d 43 0                4 T H 4 4                0 0 0 0 d 21 0 d 23 0 0 0 0 0 d 41 0 d 43 0                4 T H 5 4                0 0 0 0 d 21 0 d 23 0 0 0 0 0 d 41 0 d 43 0                4 T H 6 4                0 0 0 0 d 21 0 d 23 0 0 0 0 0 d 41 0 d 43 0                4 T H 7 4                d 11 d 12 0 0 0 0 0 0 0 0 0 d 34 0 0 0 0                3 T H 8 4                d 11 d 12 0 0 0 0 0 0 0 0 0 d 34 0 0 0 0                3 T H 9 4                0 0 0 0 0 0 0 0 0 0 0 0 d 41 0 0 0                1 T H 10 4                0 0 0 0 0 0 0 0 d 31 0 0 0 0 0 0 0                1 T H 11 4                0 0 0 0 d 21 0 0 0 0 0 0 0 0 0 d 43 0                1 T H 12 4                0 0 0 0 0 0 0 0 d 31 0 0 0 0 0 0 0                1 T H 13 4                0 0 0 0 0 0 0 0 0 0 0 0 d 41 0 0 0                1 T H 14 4                0 0 0 0 0 0 0 0 0 0 0 0 d 41 0 0 0                1 T H 15 4                0 0 0 0 0 0 0 0 0 0 0 0 d 41 0 0 0                1 T H 16 4                0 0 0 0 0 0 0 0 0 0 0 0 d 41 0 0 0                1 T H 17 4                0 0 0 0 0 0 0 0 0 0 d 23 0 d 41 0 0 0                2 T H 18 4                0 0 0 0 0 0 d 23 d 24 0 0 −d 23 −d 24 0 0 d 23 d 24                IC Der(d) Dim(d) IC Der(d) Dim(d) T H 19 4                0 0 0 0 0 0 d 23 d 24 0 0 0 0 0 0 0 0                2 T H 20 4                0 0 0 0 0 0 d 23 d 24 0 0 0 0 0 0 0 0                2 Proof. Departing from Theorem 4.11, we provide the proof only for one case to illustrate the used approach, the other cases can be addressed similarly with or without modification(s). Let's consider T H d(e 1 ) =                0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0                , d(e 2 ) =                0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0                , d(e 3 ) =                0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0                and d(e 4 ) =                0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0                is the basis of Der(Cent(A)) and DimDer(Cent(A)) = 4. The derivations of the remaining parts of dimension two associative dialgebras can be handled in a similar manner as illustrated above. Corollary 4.12. • The dimensions of the derivations of two-dimensional associative trialgebras range between zero and two. • The dimensions of the derivations of three-dimensional associative trialgebras range between zero and three. • The dimensions of the derivations of four-dimensional associative trialgebras range between one and four. 5 Centroids of Complex Hom-associative dialgebras Properties of centroids Hom-associative dialgebras In this section, we set forward the following results on properties of centroids of Hom-associative dialgebras A. Definition 5.1. Let (A, ⊣, ⊢, α) be a Hom-associative dialgebra. A linear map ψ : A → A is called an element of (α)-element of centroids on A if, for all x, y ∈ A, α • ψ = ψ • α, (5.1) ψ(x) ⊣ α(y) = ψ(x) ⊣ ψ(y) = α(x) ⊣ ψ(y), (5.2) ψ(x) ⊢ α(y) = ψ(x) ⊢ ψ(y) = α(x) ⊢ ψ(y). (5.3) The set of all elements of (α)-centroid of A is denoted Cent (α) (A). The centroid of A is denoted Cent(A). Proof. Indeed, if x, y ∈ A, then (ϕ • d)(x • y) = ϕ(d(x) • α(y) + α(x) • d(y)) = ϕ(d(x) • y) + ϕ(x • d(y)) = (ϕ • d)(x) • α(y) + α(x) • (ϕ • d)(y), where • is ⊣ and ⊢, respectively. Proof. i) For any ϕ ∈ Cent(A), d ∈ Der(A), ∀ x, y ∈ A, we have d • ϕ(x • y) = d • ϕ(x) • y = d • ϕ(x) • y + ϕ(x) • d(y) = d • ϕ(x) • y + ϕ • d(x • y) − ϕ • d(x) • y. Thus, (d • ϕ − ϕ • d)(x • y) = (d • ϕ − ϕ • d)(x) • y. ii) Let d • ϕ ∈ Der(A). Using d, ϕ ∈ Cent(A), we get d, ϕ (x • y) = ( d, ϕ (x)) • α(y) = α(x) • ( d, ϕ (y)) (5.5) On the other side, d, ϕ d • ϕ − ϕ • d and d • ϕ, ϕ • d ∈ Der(A). Therefore, d, ϕ (x • y) = (d(ϕ • (x)) • α(y) + α(x) • (d • ϕ(y)) − (ϕ • d(x)) • α(y) − α(x) • (ϕ • d(y)). (5.6) Referring to (5.5) and (5.6), we get α(x) • ( d, ϕ )(y) = ( d, ϕ )(x) • α(y) = 0. At this stage of analysis,let d, ϕ be a central α-derivation of A. Then, d • ϕ(x • y) = d • ϕ (x • y) + (ϕ • d)(x • y) = ϕ(•d(x) • α(y)) + ϕ(α(x) • d(y)) = (ϕ • d)(x) • α(y) + α(x) • (ϕ • d)(y), where • represents the products ⊣ and ⊢, respectively. Centroids of complex Hom-associative dialgebras This section elaborates the details of α-centroids of Hom-associative dialgebras in dimension two and three over the field K. Let {e 1 , e 2 , e 3 , · · · , e n } be a basis of an n-dimensional Hom-associative dialgebra A. The product of basis T H 1 3           0 0 0 c 21 0 0 c 31 0 0           2 T H 2 3           0 0 0 c 21 0 0 c 31 0 0           2 T H 3 3           c 11 0 0 c 21 c 11 c 11 c 31 0 0           3 T H 4 3           c 11 0 0 0 0 0 0 0 0           1 T H 5 3           0 0 0 0 0 c 23 0 0 c 33           2 T H 6 3           0 0 0 1 2 c 21 0 0 c 31 0 0           2 T H 7 3           c 11 0 0 0 0 0 0 0 0           1 T H 8 3           c 11 0 0 0 0 c 23 0 0 c 33           3 T H 9 3           c 11 0 0 0 0 0 0 0 0           1 T H 10 3           0 0 0 c 21 0 c 23 c 31 0 c 33           4 T H 11 3           0 0 0 c 21 0 0 c 31 0 0           2 T H 12 3           0 0 0 c 21 0 0 c 31 0 0           2 T H 13 3           c 11 0 0 0 c 22 0 0 0 c 33           3 T H 14 3           0 0 0 0 c 22 0 0 0 0           1 Proof. Departing from Theorem 5.9, we provide the proof only for one case to illustrate the used approach, the other cases can be addressed similarly with or without modification(s). Let's consider T H 3 3 . Applying the systems of equations (5.7), (5.8) and (5.9), we get c 12 = c 13 = c 32 = c 33 = 0 and c 22 = c 23 = c 11 . Hence, the centroids of T H 3 3 are indicated as follows c(e 1 ) =           1 0 0 0 1 1 0 0 0           , c(e 2 ) =           0 0 0 1 0 0 0 0 0           and c(e 3 ) =           0 0 0 0 0 0 1 0 0           is the basis of Der(c) and DimDer(c) = 3. The centroids of the remaining parts of dimension three associative dialgebras can be handled in a similar manner as illustrated above.                4 T H 7 4                              4 T H 8 4                0 c 12 0 0 0 c 22 0 0 0 0 0 0 0 0 0 0                2 T H 9 4                0 0 0 0 0 c 22 0 0 0 0 c 33 0 0 0 0 0                2 T H 10 4                              4 T H 12 4                c 11 0 0 0 0 0 0 0 c 31 0 0 0 0 0 0 0                2 T H 13 4                c 11 0 0 0 0 0 0 0 0 0 0 0 c 41 0 0 c 44                3 T H 14 4                                2 Proof. Departing from Theorem 5.10, we provide the proof only for one case to illustrate the used approach, the other cases can be addressed similarly with or without modification(s). Let's consider T H                is the basis of Der(Cent(A)) and DimDer(Cent(A)) = 4. The derivations of the remaining parts of dimension two associative dialgebras can be handled in a similar manner as illustrated above. Corollary 5.11. • The dimensions of the centroids of two-dimensional associative trialgebras range between one and three. • The dimensions of the centroids of three-dimensional associative trialgebras range between one and five. • The dimensions of the centroids of four-dimensional associative trialgebras range between zero and six. Theorem 3. 1 . 1Every 2-dimensional complex Hom-associative dialgebra is isomorphic to one of the following pairwise non-isomorphic Hom-associative dialgebra (A, ⊣, ⊢, α) where ⊣, ⊢ are the left product and right product and α of the structure map.Hd 1 2 : e 1 ⊣ e 1 = e 1 , e 1 ⊣ e 2 = e 1 , e 2 ⊣ e 1 = e 2 , e 2 ⊣ e 2 = e 2 , e 1 ⊢ e 1 = e 1 , e 1 ⊢ e 2 = e 2 , e 2 ⊢ e 1 = e 1 , e 2 ⊢ e 2 = e 2 , α(e 1 ) = e 1 , α(e 2 ) = e 2 . Hd 2 2 2: e 1 ⊣ e 2 = e 2 , e 2 ⊣ e 1 = e 1 , e 1 ⊢ e 1 = e 1 , e 2 ⊢ e 1 = e 1 , α(e 2 ) = e 1 . e 1 1⊣ e 1 = e 1 , e 2 ⊣ e 2 = e 2 , e 3 ⊣ e 2 = e 2 , e 1 ⊢ e 1 = e 1 , e 2 ⊢ e 2 = e 2 , α(e 1 ) = e 1 . ⊣ e 2 = e 1 , e 2 ⊣ e 1 = e 1 , e 2 ⊣ e 2 = e 1 , e 2 ⊣ e 3 = e 1 , e 3 ⊣ e 2 = e 3 , e 1 ⊢ e 2 = e 1 , e 2 ⊢ e 1 = e 1 , e 2 ⊢ e 2 = e 1 , e 2 ⊢ e 3 = e 3 , e 3 ⊢ e 2 = e 3 , e 3 ⊢ e 3 = e 1 , α(e 2 ) = e 1 .Hd 7 3 : e 2 ⊣ e 2 = e 2 , e 2 ⊣ e 3 = e 1 , e 3 ⊣ e 3 = e 3 , e 2 ⊢ e 2 = e 2 , e 2 ⊢ e 3 = e 1 , e 3 ⊢ e 2 = e 1 , e 3 ⊢ e 3 = e 1 , α(e 1 ) = e 1 . Theorem 3 . 3 . 33Every 4-dimensional multiplicative real Hom-associative dialgebra is isomorphic to one of the following pairwise non-isomorphic Hom-associative dialgebras (A, ⊣, ⊢, α), where ⊣, ⊢ are left product and right product and α the structure map. e 1 1⊣ e 2 = e 1 , e 1 ⊣ e 4 = e 3 , e 2 ⊣ e 1 = e 1 , e 2 ⊣ e 3 = ae 3 , e 2 ⊣ e 4 = e 1 , e 3 ⊣ e 2 = e 1 , e 3 ⊣ e 4 = e 1 , e 4 ⊣ e 1 = e 3 , e 4 ⊣ e 4 = e 1 , e 1 ⊢ e 2 = e 1 , e 1 ⊢ e 4 = e 3 , e 2 ⊢ e 2 = e 3 , e 2 ⊢ e 3 = e 1 , e 2 ⊢ e 4 = e 1 , e 3 ⊢ e 2 = e 1 , e 3 ⊢ e 4 = e 1 , e 4 ⊢ e 2 = e 1 , e 4 ⊢ e 3 = e 1 , e 4 ⊢ e 4 = e 1 , α(e 2 ) = e 1 . ⊣ e 4 = e 3 , e 2 ⊣ e 1 = e 1 , e 2 ⊣ e 2 = e 3 , e 2 ⊣ e 3 = e 1 , e 2 ⊣ e 4 = e 1 , e 3 ⊣ e 2 = e 1 , e 3 ⊣ e 4 = e 1 , e 4 ⊣ e 1 = e 3 , e 4 ⊣ e 4 = e 1 , e 1 ⊢ e 2 = e 1 , e 1 ⊢ e 4 = e 3 , e 2 ⊢ e 2 = e 3 , e 2 ⊢ e 3 = e 1 , e 2 ⊢ e 4 = e 1 , e 3 ⊢ e 2 = e 1 , e 3 ⊢ e 4 = e 1 , e 4 ⊢ e 2 = e 1 , e 4 ⊢ e 3 = e 1 , e 4 ⊢ e 4 = e 1 , α(e 2 ) = e 1 . e 1 ⊣ e 4 = e 3 , e 2 ⊣ e 1 = e 1 , e 2 ⊣ e 2 = e 1 + e 3 , e 2 ⊣ e 3 = e 1 , e 2 ⊣ e 4 = e 1 , e 3 ⊣ e 2 = e 1 , e 3 ⊣ e 4 = e 1 , e 4 ⊣ e 1 = e 3 , e 4 ⊣ e 4 = e 1 , e 1 ⊢ e 2 = e 1 , e 1 ⊢ e 4 = e 3 , e 2 ⊢ e 2 = e 3 , e 2 ⊢ e 3 = e 1 , e 2 ⊢ e 4 = e 3 , e 3 ⊢ e 2 = e 1 , e 3 ⊢ e 4 = e 1 , e 4 ⊢ e 2 = e 1 , e 4 ⊢ e 3 = e 1 , e 4 ⊢ e 4 = e 1 , α(e 2 ) 234122342= e 1 . ⊣ e 3 = e 1 + e 3 , e 3 ⊣ e 2 = e 1 + e 3 , e 3 ⊣ e 4 = e 1 + e 3 , e 4 ⊣ e 1 = e 3 , e 4 ⊣ e 3 = e 1 + e 3 , e 4 ⊣ e 4 = e 1 + e 3 , e 1 ⊢ e 2 = e 1 , e 1 ⊢ e 4 = e 3 , e 2 ⊢ e 3 = e 1 + e 3 , e 2 ⊢ e 4 = e 1 + e 3 , e 3 ⊢ e 2 = e 1 + e 3 , e 3 ⊢ e 4 = e 1 + e 3 , e 4 ⊢ e 3 = e 1 + e 3 , e 4 ⊢ e 4 = e 1 + e 3 , α(e 2 ) = e 1 , α(e 4 ) = e 3 . 2 ⊣ e 3 = e 3 , e 2 ⊣ e 4 = e 1 + e 3 , e 3 ⊣ e 2 = e 1 + e 3 , e 3 ⊣ e 4 = e 1 + e 3 , e 4 ⊣ e 1 = e 3 , e 4 ⊣ e 3 = e 1 + e 3 , e 4 ⊣ e 4 = e 1 + e 3 , e 2 ⊢ e 1 = e 1 , e 2 ⊢ e 3 = e 1 + e 3 , e 2 ⊢ e 4 = e 1 + e 3 , e 3 ⊢ e 2 = e 1 + e 3 , e 3 ⊢ e 4 = e 1 + e 3 , e 4 ⊢ e 3 = e 1 + e 3 , e 4 ⊢ e 4 = e 1 + e 3 , α(e 2 ) = e 1 , α(e 4 ) = e 3 . ⊣ e 2 = e 3 , e 2 ⊣ e 3 = e 1 + e 3 , e 2 ⊣ e 4 = e 1 + e 3 , e 3 ⊣ e 2 = e 1 + e 3 , e 3 ⊣ e 4 = e 1 , e 4 ⊣ e 2 = e 3 , e 4 ⊣ e 3 = e 1 + e 3 , e 4 ⊣ e 4 = e 1 + e 3 , e 2 ⊢ e 1 = e 1 , e 2 ⊢ e 3 = e 1 + e 3 , e 2 ⊢ e 4 = e 1 + e 3 , e 3 ⊢ e 2 = e 1 + e 3 , e 3 ⊢ e 4 = e 1 + e 3 , e 4 ⊢ e 3 = e 1 + e 3 , α(e 2 ) = e 1 , α(e 4 ) = e 3 .Hd 7 4 : e 2 ⊣ e 3 = e 4 , e 3 ⊣ e 2 = e 4 , e 3 ⊣ e 4 = e 4 , e 4 ⊣ e 3 = e 4 , e 4 ⊣ e 4 = e 2 + e 4 , e 3 ⊢ e 2 = e 2 , e 3 ⊢ e 4 = e 2 + e 4 , e 4 ⊢ e 3 = e 2 + e 4 , e 4 ⊢ e 4 = e 2 + e 4 , α(e 1 ) = e 1 , α(e 2 ) = e 2 . ⊣ e 3 = e 4 , e 3 ⊣ e 2 = e 4 , e 3 ⊣ e 3 = e 4 , e 3 ⊣ e 4 = e 4 , e 4 ⊣ e 3 = e 4 , e 4 ⊣ e 4 = e 4 , e 3 ⊢ e 2 = e 2 , e 3 ⊢ e 4 = e 2 + e 4 , e 4 ⊢ e 3 = e 2 + e 4 , e 4 ⊢ e 4 = e 2 + e 4 , α(e 1 ) = e 1 , α(e 2 ) = e 2 . ⊣ e 1 = e 1 , e 2 ⊣ e 4 = e 1 , e 3 ⊣ e 3 = e 2 , e 3 ⊣ e 4 = e 1 , e 4 ⊣ e 1 = e 1 , e 4 ⊣ e 2 = e 1 , e 4 ⊣ e 3 = e 1 , e 4 ⊣ e 4 = e 1 , e 2 ⊢ e 4 = e 2 , e 3 ⊢ e 3 = e 2 , e 3 ⊢ e 4 = e 1 + e 2 , e 4 ⊢ e 3 = e 1 + e 2 , e 4 ⊢ e 4 = e 2 , α(e 2 ) = e 2 , α(e 3 ) = e 3 .Hd 10 4 : e 2 ⊣ e 2 = e 2 + e 4 , e 2 ⊣ e 4 = e 2 + e 4 , e 3 ⊣ e 3 = e 1 , e 4 ⊣ e 4 = e 4 , e 2 ⊢ e 2 = e 2 + e 4 , e 2 ⊣ e 4 = e 2 + e 4 , e 3 ⊢ e 3 = e 2 + e 4 , e 4 ⊢ e 2 = e 4 , e 4 ⊢ e 4 = e 2 + e 4 , α(e 1 ) = e 1 , α(e 3 ) = e 3 . e 1 e 2 ⊣ e 2 = e 2 , e 2 ⊣ e 3 = e 2 + e 3 , e 3 ⊣ e 3 = e 2 , e 4 ⊣ e 4 = e 1 , e 2 ⊢ e 2 = e 1 + e 3 , e 2 ⊢ e 3 = e 2 , e 3 ⊣ e 2 = e 1 + e 3 , e 3 ⊢ e 3 = e 1 + e 3 , e 4 ⊢ e 4 = e 1 + e 2 , α(e 1 ) = e 1 , α(e 4 ) 114⊣ e 2 = e 1 , e 2 ⊣ e 1 = e 1 , e 2 ⊣ e 2 = e 1 , e 2 ⊣ e 3 = e 1 , e 3 ⊣ e 2 = e 1 , e 4 ⊣ e 2 = e 1 , e 4 ⊣ e 4 = e 3 , e 1 ⊢ e 2 = e 1 , e 2 ⊢ e 2 = e 1 , e 2 ⊢ e 3 = e 1 , e 2 ⊢ e 4 = e 1 + e 3 , e 4 ⊢ e 2 = e 1 + e 3 , e 4 ⊢ e 4 = e 1 , α(e 3 ) = e 3 , α(e 4 ) = e 4 .Hd 12 4 : e 2 ⊣ e 2 = e 2 + e 4 , e 2 ⊣ e 4 = e 2 + e 4 , e 3 ⊣ e 3 = e 1 , e 4 ⊣ e 2 = e 2 , e 4 ⊣ e 4 = e 4 , e 2 ⊢ e 2 = e 2 + e 4 , e 2 ⊣ e 4 = e 2 + e 4 , e 3 ⊢ e 3 = e 2 + e 4 , e 4 ⊢ e 2 = e 4 , e 4 ⊢ e 4 = e 2 + e 4 , α(e 1 ) = e 1 , α(e 3 ) = e 3 . = e 4 . ⊣ e 2 = e 2 , e 2 2⊣ e 3 = e 2 + e 3 , e 3 ⊣ e 3 = e 2 + e 3 , e 4 ⊣ e 4 = e 1 , e 2 ⊢ e 2 = e 1 + e 3 , e 2 ⊢ e 3 = e 1 + e 2 , e 3 ⊣ e 2 = e 1 + e 3 , e 3 ⊢ e 3 = e 1 + e 3 , e 4 ⊢ e 4 = e 1 + e 2 , α(e 1 ) = e 1 , α(e 4 ) = e 4 .Hd 15 4 : e 2 ⊣ e 4 = e 1 , e 3 ⊣ e 3 = e 2 , e 3 ⊣ e 4 = e 1 , e 4 ⊣ e 1 = e 1 , e 4 ⊣ e 2 = e 1 , e 4 ⊣ e 3 = e 1 , e 4 ⊣ e 4 = e 1 , e 2 ⊢ e 4 = e 2 , e 3 ⊢ e 3 = e 1 + e 2 , e 3 ⊢ e 4 = e 1 + e 2 , e 4 ⊢ e 1 = e 1 + e 2 , e 4 ⊢ e 2 = e 1 + e 2 , e 4 ⊢ e 3 = e 1 + e 2 , α(e 2 ) = e 2 , α(e 3 ) = e 3 . Hd 16 4 : e 2 ⊣ e 4 = e 1 , e 3 ⊣ e 3 = e 2 , e 3 ⊣ e 4 = e 1 , e 4 ⊣ e 1 = e 1 , e 4 ⊣ e 2 = e 1 , e 4 ⊣ e 3 = e 1 , e 4 ⊣ e 4 = e 1 , e 2 ⊢ e 4 = e 1 + e 2 , e 3 ⊢ e 3 = e 2 , e 3 ⊢ e 4 = e 1 + e 2 , e 4 ⊢ e 1 = e 1 + e 2 , e 4 ⊢ e 2 = e 1 + e 2 , e 4 ⊢ e 4 = e 2 , α(e 2 ) = e 2 , α(e 3 ) = e 3 . ⊣ e 4 = e 1 , e 2 ⊣ e 4 = e 1 , e 3 ⊣ e 3 = e 2 , e 3 ⊣ e 4 = e 1 , e 4 ⊣ e 1 = e 1 , e 4 ⊣ e 2 = e 1 , e 4 ⊣ e 3 = e 1 , e 4 ⊣ e 4 = e 1 , e 2 ⊢ e 4 = e 1 + e 2 , e 3 ⊢ e 3 = e 1 + e 2 , e 4 ⊢ e 2 = e 2 , e 4 ⊢ e 3 = e 1 + e 2 , e 4 ⊢ e 4 = e 2 , α(e 2 ) = e 2 , α(e 3 ) = e 3 .Hd 18 4 : e 1 ⊣ e 2 = e 3 + e 4 , e 2 ⊣ e 1 = e 3 + e 4 , e 2 ⊣ e 2 = e 3 + e 4 , e 2 ⊣ e 3 = e 3 + e 4 , e 2 ⊣ e 4 = e 3 + e 4 , e 3 ⊣ e 2 = e 3 + e 4 , e 3 ⊣ e 3 = e 3 + e 4 , e 3 ⊣ e 4 = e 3 + e 4 , e 4 ⊣ e 2 = e 3 + e 4 , e 1 ⊢ e 1 = e 3 + e 4 , e 2 ⊢ e 2 = e 3 + e 4 , e 2 ⊢ e 3 = e 3 + e 4 , e 3 ⊢ e 2 = e 3 + e 4 , e 4 ⊢ e 4 = e 3 + e 4 , α(e 2 ) = e 1 . Hd 19 4 : e 1 ⊣ e 2 = ae 3 + be 4 , e 2 ⊣ e 1 = e 3 + ce 4 , e 2 ⊣ e 2 = e 3 + de 4 , e 2 ⊣ f e 3 = e 3 + e 4 , e 2 ⊣ e 4 = he 3 + e 4 , e 3 ⊣ e 2 = le 3 + e 4 , e 3 ⊣ e 3 = e 3 + e 4 , e 3 ⊣ e 4 = e 3 + e 4 , e 4 ⊣ e 2 = e 3 + e 4 , e 1 ⊢ e 1 = me 3 + ne 4 , e 2 ⊢ e 2 = oe 3 + pe 4 , e 2 ⊢ e 3 = ce 3 + de 4 , e 3 ⊢ e 2 = e 3 , e 4 ⊢ e 4 = e 4 , α(e 2 ) = e 1 . Hd 20 4 : e 2 ⊣ e 1 = e 3 + ae 4 , e 2 ⊣ e 2 = e 3 + be 4 , e 2 ⊣ e 3 = ce 3 + de 4 , e 2 ⊣ e 4 = e 3 + e 4 , e 3 ⊣ e 2 = f e 3 + ge 4 , e 3 ⊣ e 3 = e 3 + e 4 , e 3 ⊣ e 4 = e 3 + e 4 , e 4 ⊣ e 2 = e 3 + e 4 , e 1 ⊢ e 1 = e 3 + e 4 , e 2 ⊢ e 2 = he 3 + e 4 , e 2 ⊢ e 3 = ke 3 + e 4 , e 3 ⊢ e 2 = le 3 , e 4 ⊢ e 3 = e 3 , e 4 ⊢ e 4 = e 4 , α(e 2 ) = e 1 . Definition 4. 1 . 1For any non-negative integer k, a linear map D : A −→ A is called an α k −derivation of a Hom-associative dialgebra (A, ⊣, ⊢, α), if D ∈ Der α k (A) and D ′ ∈ Der α s (A), we define their commutator [D, D ′ ] as usual : [D, D ′ ] = D • D ′ − D ′ • D.Proposition 4.2. For any D ∈ Der α k (A) and D ′ ∈ Der α s (A), we have [D, D ′ ] ∈ Der α k+s (A). . 9 .. 9The derivations of 2-dimensional Hom-associative dialgebras have the following form The derivations of 3-dimensional Hom-associative dialgebras have the following form 1 4 . 4Applying the systems of equations (4.9), (4.10) and (4.11), we get d 11 = d 12 = d 13 = d 14 = d 22 = d 24 = d 31 = d 32 = d 33 = d 34 = d 42 = d 44 = 0. Hence, the derivations of T H 1 4 are indicated as follows Definition 5. 2 . 2Let H be a nonempty subset of A. The subsetZ A (H) = {x ∈ H\|α(x) • H = H • α(x) = 0} ,(5.4)is said to be centralizer of H in A, where • is ⊣ and ⊢, respectively.Definition 5.3. Let ψ ∈ End(A). If ψ(A) ⊆ Z(A) and ψ(A 2 ) = 0, then ψ is called a central derivation. The set of all central derivations of A is denoted by C(A). Proposition 5.4. Consider (A, ⊣, ⊢, α) a Hom-associative dialgebra. Then, i) Γ(A)Der(T ) ⊆ Der(A). ii) [Γ(A), Dr(A)] ⊆ Γ(A). iii) [Γ(A), Γ(A)] (A) ⊆ Γ(A) and [Γ(A), Γ(A)] (A 2 ) = 0. Proof. The proof of parts i)− iii) is straightforward by using definitions of derivations and centroids. Proposition 5.5. Let (A, ⊣, ⊢ α) be a Hom-associative dialgebra and ϕ ∈ Cent(A), d ∈ Der(A). Then, ϕ • d is an α-derivation of A. Proposition 5. 6 . 6Let (A, ⊣, ⊢, α) be a Hom-associative dialgebra over a field F. Hence, C(A) = Cent(A) ∩ Der(A). Proof. If ψ ∈ Cent(A) ∩ Der(A), then grounded on definition of Cent(A and Der(A), we haveψ(x • y) = ψ(x) • α(y) + α(x) • ψ(y) and ψ(x • y) = ψ(x) • α(y) = α(x) • ψ(y) for x, y ∈ A.The yields ψ(AA) = 0 and ψ(A) ⊆ Z(A) i.e Cent(A) ∩ Der(A) ⊆ Cent(A). The inverse is obvious since C(A) is in both Cent(A) and Der(A), where • is ⊣ and ⊢, respectively. Proposition 5.7. Let (A, ⊣, ⊢, α) be a Hom-associative dialgebra. Therefore, for any d ∈ Der(A) and ϕ ∈ Cent(A), we have (i) The composition d • ϕ is in Cent(A), if and only if ϕ • d is a central α-derivation of A. ( ii) The composition d • ϕ is a α-derivation of A, if and only if d, ϕ is a central α-derivation of A. . 8 . 8The centroids of 2-dimensional Hom-associative dialgebras have the following form IC Cent(A) Dim(Cent(A)) IC Cent(A) Dim(Cent(A)) Theorem 5.9. The centroids of 3-dimensional Hom-associative dialgebras have the following form IC Cent(A) Dim(Cent(A)) IC Cent(A) Dim(Cent(A)) Theorem 5 . 10 . 510The centroids of 4-dimensional Hom-associative dialgebras have the following form IC Cent(A) Dim IC Cent( 1 4 . 4Applying the systems of equations (5.7), (5.8) and (5.9), we get c 11 = c 12 = c 13 = c 14 = 0 = c 22 = c 24 = c 31 = c 32 = c 33 = c 34 = c 42 = c 44 = 0. 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[ "QUASILINEAR DIRICHLET SYSTEMS WITH COMPETING OPERATORS AND CONVECTION", "QUASILINEAR DIRICHLET SYSTEMS WITH COMPETING OPERATORS AND CONVECTION" ]	[ "Laura Gambera ", "ANDSalvatore A Marano ", "Dumitru Motreanu " ]	[]	[]	In this paper, we consider a quasi-linear Dirichlet system with possible competing (p, q)-Laplacians and convections. Due to the lack of ellipticity, monotonicity, and variational structure, the standard approaches to the existence of weak solutions cannot be adopted. Nevertheless, through an approximation procedure and a corollary of Brouwer's fixed point theorem we show that the problem admits a solution in a suitable sense.MSC 2020: 35J47, 35J92, 35D30.	null	[ "https://export.arxiv.org/pdf/2305.03968v1.pdf" ]	258,558,083	2305.03968	4179bd1b38ac331ef7cae45bfba14f7ca3b4aad8	QUASILINEAR DIRICHLET SYSTEMS WITH COMPETING OPERATORS AND CONVECTION 6 May 2023 Laura Gambera ANDSalvatore A Marano Dumitru Motreanu QUASILINEAR DIRICHLET SYSTEMS WITH COMPETING OPERATORS AND CONVECTION 6 May 2023 In this paper, we consider a quasi-linear Dirichlet system with possible competing (p, q)-Laplacians and convections. Due to the lack of ellipticity, monotonicity, and variational structure, the standard approaches to the existence of weak solutions cannot be adopted. Nevertheless, through an approximation procedure and a corollary of Brouwer's fixed point theorem we show that the problem admits a solution in a suitable sense.MSC 2020: 35J47, 35J92, 35D30. Introduction and main result Let Ω ⊆ R N , N ≥ 2, be a bounded domain with a Lipschitz boundary ∂Ω, let 1 < q i < p i < N, and let µ i ∈ R, i = 1, 2. Consider the system      −∆ p 1 u + µ 1 ∆ q 1 u = f 1 (x, u, v, ∇u, ∇v) in Ω, −∆ p 2 v + µ 2 ∆ q 2 v = f 2 (x, u, v, ∇u, ∇v) in Ω, u = v = 0 on ∂Ω,(P) where ∆ r u := div (\|∇u\| r−2 ∇u), with 1 < r < +∞, denotes the so-called r-Laplacian while f i : Ω × R 2 × R 2N → R, i = 1, 2, are Carathéodory functions satisfying appropriate conditions; see (H 1 )-(H 2 ) below. Since q i < p i , we seek solutions (u, v) to (P) in the space X := W 1,p 1 0 (Ω) × W 1,p 2 0 (Ω). (1.1) The involved differential operator A i,µ i (w) := −∆ p i w + µ i ∆ q i w, w ∈ W 1,p i 0 (Ω), exhibits completely different behaviors according to whether µ i < 0, µ i = 0, or µ i > 0. If µ i < 0 then A i,µ i is basically patterned [4, Section 2.1] after the well known (p i , q i )-Laplacian, namely ∆ p i w + ∆ q i w, which turns out non-homogeneous because p i = q i . When µ i = 0 it coincides with the classical negative p i -Laplacian. Both cases have been widely investigated and meaningful results are by now available in the literature; see, e.g., [5,Chapter 6]. On the contrary, for µ i > 0 the operator A i,µ i contains the difference between the q i -Laplacian and the p i -Laplacian. It is usually called competitive and, as already pointed out in [3,6], doesn't comply with any ellipticity or monotonicity condition. In fact, given u 0 ∈ W 1,p i 0 (Ω) \ {0} and chosen u := tu 0 , t > 0, the expression A i,µ i (u), u = t p ∇u 0 p p − µ i t q u 0 q q turns out negative for t small and positive when t is large. Hence, if µ i > 0, problem (P) gathers at least two challenging technical features: • driving operators are neither elliptic nor monotone. This prevents to use surjectivity results for pseudo-monotone operators, nonlinear regularity theory, besides comparison principles. • right-hand sides depend on the gradient of solutions, so that variational techniques cannot be enforced directly. To overcome these difficulties we first exploit the Galerkin method, thus working in a sequence {X n } of finite dimensional sub-spaces of X. A finite dimensional approximate solution (u n , v n ) ∈ X n to (P) is obtained for every n ∈ N via a corollary of the Brouwer fixed point theorem. Next, an appropriate passage to the limit yields a solution in a generalized sense; cf. Definitions 2.6-2.7. The idea of weakening the notion of solution to treat more general situations is classical; cf. for instance [9, p. 183] and [10,11]. As far as we know, equations driven by competing operators have previously been investigated only in [3,6]. The main difference between them concerns right-hand sides. The paper [3] doesn't treat convective reactions. Solutions are thus obtained through Galerkin's method and Ekeland's variational principle. In [6], instead, any variational approach is forbidden, because the nonlinearity depends on the gradient of solutions. This work continues the study started in [6], treating convective systems driven by competing operators. Section 2, beyond some auxiliary results, contains the notions of generalized and strong generalized solution to (P), adapted from Definitions 2.3 and 2.5, respectively, in [6]. A sequence of finite dimensional approximate solutions is constructed in Section 3. Finally, Section 4 deals with a limit procedure that provides a generalized solution to (P). Under a growth condition stronger than (H 1 ) (see (H ′ 1 ) below) we next show that (P) admits a strong generalized solution, which becomes a weak one as soon as µ 1 ∨ µ 2 < 0. The section ends by exhibiting a natural class of reactions that fulfill our hypotheses; cf. Example 4.3. Preliminaries and hypotheses Let Y be a real Banach space and let Y * be its topological dual, with duality brackets ·, · . An operator B : Y → Y * is said to be: • bounded when it maps bounded sets into bounded sets. • monotone if B(y) − B(z), y − z ≥ 0 for all y, z ∈ Y . • pseudo-monotone when y n ⇀ y in Y and lim sup n→∞ B(y n ), y n − y ≤ 0 imply lim inf n→∞ B(y n ), y n − z ≥ B(y), y − z ∀ z ∈ Y. • of type (S) + provided y n ⇀ y in Y , lim sup n→∞ B(y n ), y n − y ≤ 0 =⇒ y n → y in Y . The following consequence of Brouwer's fixed point theorem will play a basic role in the sequel. For the proof we refer to [9, p. 37]. Theorem 2.1. Let (Y, · ) a finite dimensional normed space and let B : Y → Y * be continuous. Assume there exists R > 0 such that B(y), y ≥ 0 for all y ∈ Y with y = R. Then the equation B(y) = 0 admits a solutionȳ ∈ Y fulfilling ȳ ≤ R. The symbol \|E\| stands for the N-dimensional Lebesgue measure of the set E ⊆ R N and a ∨ b := max{a, b}, a ∧ b := min{a, b} ∀ a, b ∈ R. Given a real number r > 1, set r ′ := r/(r − 1). If r < N then r * := Nr/(N − r). We denote by · r the usual norm of L r (Ω), while · 1,r indicates the norm of W 1,r 0 (Ω) coming from Poincaré's inequality, i.e., u 1,r := ∇u r , u ∈ W 1,r 0 (Ω). The symbol W 1,−r ′ (Ω) stands for the dual space of W 1,r 0 (Ω) while λ 1,r is the first eigenvalue of the operator −∆ r in W 1,r 0 (Ω). One has λ 1,r = inf u∈W 1,r 0 (Ω)\{0} u r 1,r u r r > 0. (2.1) The following facts are known; see, e.g., [1,Theorem 9.16] and [7]. Proposition 2.2. Let 1 < r < N. Then the embedding: (a) W 1,r 0 (Ω) ֒→ W 1,s 0 (Ω) is continuous for all s ∈ [1, r]. (b) W 1,r 0 (Ω) ֒→ L s (Ω) is continuous for every s ∈ [1, r * ] and compact when s < r * . (c) L s ′ (Ω) ֒→ W 1,−r ′ (Ω) is continuous for all s ∈ [1, r * ]. By (a) the differential operator −∆ p i +µ i ∆ q i (i = 1, 2) turns out well posed on W 1,p i 0 (Ω) because 1 < q i < p i < N =⇒ W 1,p i 0 (Ω) ֒→ W 1,q i 0 (Ω). Accordingly, we will seek solutions (u, v) to system (P) lying in the product space X given by (1.1), endowed with the usual norm (u, v) := u 1,p 1 + v 1,p 2 . It is evident that X is a separable Banach space and one has X * = W 1,−p ′ 1 (Ω) × W 1,−p ′ 2 (Ω). Let A µ 1 ,µ 2 : X → X * defined by A µ 1 ,µ 2 (u, v), (ϕ, ψ) =ˆΩ \|∇u\| p 1 −2 ∇u − µ 1 \|∇u\| q 1 −2 ∇u ∇ϕ dx +ˆΩ \|∇v\| p 2 −2 ∇v − µ 2 \|∇v\| q 2 −2 ∇v ∇ψ dx for all (u, v), (ϕ, ψ) ∈ X. A simple argument, which exploits standard properties [8] of the r-Laplacian, shows that A µ 1 ,µ 2 turns out bounded and continuous. Moreover, if µ 1 ∨ µ 2 < 0 then A µ 1 ,µ 2 is also monotone. The growth conditions below will be posited on the reactions f 1 , f 2 : Ω×R 2 ×R 2N → R. (H 1 ) There exist constants C i > 0 and functions σ i ∈ L (p * i ) ′ (Ω), i = 1, 2, such that \|f 1 (x, s, t, ξ, ν)\| ≤ C 1 \|s\| p * 1 −1 + \|t\| p * 2 (p * 1 ) ′ + \|ξ\| p 1 (p * 1 ) ′ + \|ν\| p 2 (p * 1 ) ′ + σ 1 (x), \|f 2 (x, s, t, ξ, ν)\| ≤ C 2 \|s\| p * 1 (p * 2 ) ′ + \|t\| p * 2 −1 + \|ξ\| p 1 (p * 2 ) ′ + \|ν\| p 2 (p * 2 ) ′ + σ 2 (x) a.e. in Ω and for every s, t ∈ R, ξ, ν ∈ R N . Remark 2.3. Since p − 1 < p (p * ) ′ whatever p > 1, the growth rate p (p * ) ′ of gradient terms in (H 1 ) is better than the analogous one of [6], namely p − 1. Hereafter, C,Ĉ,C, etc. will denote generic positive constants, which may change explicit value from line to line. Proposition 2.4. Under (H 1 ), the Nemytskii operator N f i : X → L (p * i ) ′ (Ω) defined by N f i (u, v) := f i (·, u, v, ∇u, ∇v) ∀ (u, v) ∈ X is well posed, bounded, and continuous. Proof. Let i = 1 (the case i = 2 can be treated similarly). If (u, v) ∈ X then u ∈ L p 1 (Ω), v ∈ L p 2 (Ω), ∇u ∈ (L p 1 (Ω)) N , and ∇v ∈ (L p 2 (Ω)) N . Through (H 1 ) we have \|f 1 (·, u, v, ∇u, ∇v)\| ≤ C 1 \|u\| p * 1 −1 + \|v\| p * 2 (p * 1 ) ′ + \|∇u\| p 1 (p * 1 ) ′ + \|∇v\| p 2 (p * 1 ) ′ + σ 1 a.e. in Ω, whencê Ω \|f 1 (·, u, v, ∇u, ∇v)\| (p * 1 ) ′ dx ≤ C ˆΩ (\|u\| p * 1 + \|v\| p * 2 + \|∇u\| p 1 + \|∇v\| p 2 ) dx +ˆΩ σ (p * 1 ) ′ 1 dx < +∞ (2.2) thanks to conclusion (b) of Proposition 2.2. This means that the function N f 1 is well defined. From (2.2) it also follows To investigate problem (P) we will consider the operator A µ 1 ,µ 2 : X → X * given by N f 1 (u, v) (p * 1 ) ′ ≤Ĉ u p * 1 p * 1 + v p * 2 p * 2 + ∇u p 1 p 1 + ∇v p 2 p 2 + 1 1 (p * 1 ) ′ ≤C u p * 1 −1 p * 1 + v p * 2 (p * 1 ) ′ p * 2 + ∇u p 1 (p * 1 ) ′ p 1 + ∇v p 2 (p * 1 ) ′ p 2 + 1 , (u, v) ∈ X.A µ 1 ,µ 2 (u, v) = A µ 1 ,µ 2 (u, v) − (N f 1 (u, v), N f 2 (u, v)) ∀ (u, v) ∈ X. (2.3) Proposition 2.4 shows that if (H 1 ) is satisfied then A µ 1 ,µ 2 turns out well posed, bounded, and continuous for any (µ 1 , µ 2 ) ∈ R 2 . Moreover, weak solutions (u, v) ∈ X of (P) comply with the equation A µ 1 ,µ 2 (u, v) = 0, (2.4) and vice-versa. However, (2.4) cannot be solved via the classical suriectivity theorem for pseudo-monotone operators once µ 1 ∨ µ 2 > 0. In fact, u → ∆ p u is not pseudo-monotone even when p = 2; see [6, p. 1511]. We will overcome this difficulty by introducing weaker notions of solution, adapted from those in [6]. Definition 2.6. Suppose (H 1 ) holds. A pair (u, v) ∈ X is called a generalized solution to (P) provided there exists a sequence {(u n , v n )} ⊆ X such that (a) (u n , v n ) ⇀ (u, v) in X, (b) A µ 1 ,µ 2 (u n , v n ) ⇀ 0 in X * , and (c) lim n→∞ A µ 1 ,µ 2 (u n , v n ), (u n − u, v n − v) = 0. Specifically, this means that: u n ⇀ u in W 1,p 1 0 (Ω) and v n ⇀ v in W 1,p 2 0 (Ω); −∆ p 1 u n + µ 1 ∆ q 1 u n − N f 1 (u n , v n ) ⇀ 0 in W −1,p ′ 1 (Ω), −∆ p 2 v n + µ 2 ∆ q 2 v n − N f 2 (u n , v n ) ⇀ 0 in W −1,p ′ 2 (Ω);        −∆ p 1 u n + µ 1 ∆ q 1 u n , u n − u −ˆΩ N f 1 (u n , v n )(u n − u) dx → 0, −∆ p 2 v n + µ 2 ∆ q 2 v n , v n − v −ˆΩ N f 2 (u n , v n )(v n − v) dx → 0. If we strengthen (H 1 ) as follows: (H ′ 1 ) For appropriate r i , s i ∈]1, p * i [, D i > 0, and σ i ∈ L (s i ) ′ (Ω), i = 1, 2, one has \|f 1 (x, s, t, ξ, ν)\| ≤ D 1 \|s\| p * 1 r ′ 1 + \|t\| p * 2 r ′ 1 + \|ξ\| p 1 r ′ 1 + \|ν\| p 2 r ′ 1 + σ 1 (x), \|f 2 (x, s, t, ξ, ν)\| ≤ D 2 \|s\| p * 1 r ′ 2 + \|t\| p * 2 r ′ 2 + \|ξ\| p 1 r ′ 2 + \|ν\| p 2 r ′ 2 + σ 2 (x) a.e. in Ω and for every s, t ∈ R, ξ, ν ∈ R N , then the definition below can be posited. It should be noted that (H ′ 1 ) implies (H 1 ). In fact, from r i < p * i it follows r ′ i > (p * i ) ′ , whence q r ′ i < q (p * i ) ′ ∀ q ∈ {p * 1 , p * 2 , p 1 , p 2 }. Definition 2.7. Assume (H ′ 1 ) holds. A pair (u, v) ∈ X is called a strong generalized solution to (P) provided there exists a sequence {(u n , v n )} ⊆ X such that (a)-(b) of Definition 2.6 are fulfilled and, moreover, (c ′ ) lim n→∞ A µ 1 ,µ 2 (u n , v n ), (u n − u, v n − v) = 0. Remark 2.8. Until now, we have considered tree kinds of solution: weak, generalized, and strong generalized. One has weak =⇒ strong generalized =⇒ generalized. To see the first implication, simply pick (u n , v n ) := (u, v) for all n ∈ N. The other can be easily verified arguing as in the proof of Theorem 4.2. We will establish the existence of generalized (respectively, strong generalized) solutions under hypothesis (H 1 ) (respectively, (H ′ 1 )) and the next one. (H 2 ) There are constants c i , d i > 0, i = 1, 2, with c 1 + c 2 + (d 1 + d 2 ) 1 λ 1,p 1 ∧ λ 1,p 2 < 1, (2.5) and functions γ i ∈ L 1 (Ω) satisfying f 1 (x, s, t, ξ, ν)s ≤ c 1 (\|ξ\| p 1 + \|ν\| p 2 ) + d 1 (\|s\| p 1 + \|t\| p 2 ) + γ 1 (x), f 2 (x, s, t, ξ, ν)t ≤ c 2 (\|ξ\| p 1 + \|ν\| p 2 ) + d 2 (\|s\| p 1 + \|t\| p 2 ) + γ 2 (x) a.e. in Ω and for all s, t ∈ R, ξ, ν ∈ R N . Approximate solutions via Galerkin's method Since the Banach space X defined in (1.1) is separable, it admits a Galerkin basis. So, there exists a sequence {X n } of vector sub-spaces of X such that • dim(X n ) < ∞ ∀ n ∈ N, • X n ⊆ X n+1 ∀ n ∈ N, and • ∞ n=1 X n = X. Evidently, we may suppose X n = U n × V n , with U n ⊆ W 1,p 1 0 (Ω) and V n ⊆ W 1,p 2 0 (Ω). Proposition 3.1. Let (H 1 )-(H 2 ) be satisfied and let (µ 1 , µ 2 ) ∈ R 2 . Then for every n ∈ N there exists a pair (u n , v n ) ∈ X n such that −∆ p 1 u n + µ 1 ∆ q 1 u n , ϕ −ˆΩ N f 1 (u n , v n )ϕ dx = 0 −∆ p 2 v n + µ 2 ∆ q 2 v n , ψ −ˆΩ N f 2 (u n , v n )ψ dx = 0 ∀ (ϕ, ψ) ∈ X n . (3.1) Moreover, the sequence {(u n , v n )} is bounded in X. Proof. Fix n ∈ N. To shorten notation, write B := A µ 1 ,µ 2 ⌊ Xn . Setting B := (B 1 , B 2 ) one has B 1 (u, v), ϕ = −∆ p 1 u + µ 1 ∆ q 1 u, ϕ −ˆΩ N f 1 (u, v)ϕ dx B 2 (u, v), ψ = −∆ p 2 v + µ 2 ∆ q 2 v, ψ −ˆΩ N f 2 (u, v)ψ dx ∀ (u, v), (ϕ, ψ) ∈ X n . Claim: There exists R > 0 such that (ϕ, ψ) ∈ X n , (ϕ, ψ) = R =⇒ B(ϕ, ψ), (ϕ, ψ) ≥ 0. In fact, (H 2 ) and (2.1) implŷ Ω N f 1 (ϕ, ψ)ϕ dx +ˆΩ N f 2 (ϕ, ψ)ψ dx ≤ˆΩ c 1 (\|∇ϕ\| p 1 + \|∇ψ\| p 2 ) dx +ˆΩ d 1 (\|ϕ\| p 1 + \|ψ\| p 2 ) dx + γ 1 1 +ˆΩ c 2 (\|∇ϕ\| p 1 + \|∇ψ\| p 2 ) dx +ˆΩ d 2 (\|ϕ\| p 1 + \|ψ\| p 2 ) dx + γ 2 1 ≤ c 1 ϕ p 1 1,p 1 + ψ p 2 1,p 2 + d 1 λ −1 1,p 1 ϕ p 1 1,p 1 + λ −1 1,p 2 ψ p 2 1,p 2 + c 2 ϕ p 1 1,p 1 + ψ p 2 1,p 2 + d 2 λ −1 1,p 1 ϕ p 1 1,p 1 + λ −1 1,p 2 ψ p 2 1,p 2 + γ 1 1 + γ 2 1 = c 1 + c 2 + (d 1 + d 2 )λ −1 1,p 1 ϕ p 1 1,p 1 + c 1 + c 2 + (d 1 + d 2 )λ −1 1,p 2 ψ p 2 1,p 2 + C,(3.2) where C := γ 1 1 + γ 2 1 . Since q i < p i , using Hölder's inequality we thus obtain B(ϕ, ψ), (ϕ, ψ) ≥ 1 − c 1 − c 2 − (d 1 + d 2 )λ −1 1,p 1 ϕ p 1 1,p 1 + 1 − c 1 − c 2 − (d 1 + d 2 )λ −1 1,p 2 ψ p 2 1,p 2 − \|µ 1 \|\|Ω\| p 1 −q 1 p 1 ϕ q 1 1,p 1 − \|µ 2 \|\|Ω\| p 2 −q 2 p 2 ψ q 2 1,p 2 − C, and the claim easily follows from (2.5). Now, Theorem 2.1 gives a pair (u n , v n ) ∈ X n such that B(u n , v n ) = 0, which entails (3.1). Finally, letting (ϕ, ψ) = (u n , v n ) in (3.1) and arguing as done for (3.2) produces u n p 1 1,p 1 + v n p 2 1,p 2 ≤ \|µ 1 \| u n q 1 1,q 1 + \|µ 2 \| v n q 2 1,q 2 +ˆΩ N f 1 (u n , v n )u n dx +ˆΩ N f 2 (u n , v n )v n dx ≤ \|µ 1 \|\|Ω\| p 1 −q 1 p 1 u n q 1 1,p 1 + \|µ 2 \|\|Ω\| p 2 −q 2 p 2 v n q 2 1,p 2 + c 1 + c 2 + (d 1 + d 2 )(λ 1,p 1 ∧ λ 1,p 2 ) −1 u n p 1 1,p 1 + v n p 2 1,p 2 + C for every n ∈ N. At this point, the boundedness of {(u n , v n )} ⊆ X is an immediate consequence of q i < p i , i = 1, 2, and (2.5). Proof. Fix (µ 1 , µ 2 ) ∈ R 2 . Proposition 3.1 provides a bounded sequence {(u n , v n )} ⊆ X fulfilling (3.1). Since X is reflexive, up to sub-sequences one has (u n , v n ) ⇀ (u, v) in X, i.e., condition (a) of Definition 2.6 holds. Moreover, the boundedness of the operator A µ 1 ,µ 2 given by (2.3) yields a pair (η 1 , η 2 ) ∈ X * such that Existence of generalized solutions A µ 1 ,µ 2 (u n , v n ) = A µ 1 ,µ 2 (u n , v n ) − (N f 1 (u n , v n ), N f 2 (u n , v n )) ⇀ (η 1 , η 2 ) in X * . (4.1) If (ϕ, ψ) ∈ ∪ ∞ n=1 X n then (ϕ, ψ) belongs to X n = U n × V n for any n large enough. From (3.1) and (4.1) it thus follows, after letting n → ∞, η 1 , ϕ = η 2 , ψ = 0. This entails η 1 = η 2 = 0, because (ϕ, ψ) was arbitrary, ∪ ∞ n=1 U n = W 1,p 1 0 (Ω), and ∪ ∞ n=1 V n = W 1,p 2 0 (Ω). Therefore, assertion (b) of Definition 2.6 is true. Through (4.1) we next have −∆ p 1 u n + µ 1 ∆ q 1 u n − N f 1 (u n , v n ), u → 0, −∆ p 2 v n + µ 2 ∆ q 2 v n − N f 2 (u n , v n ), v → 0 (4.2) while (3.1) yields u n p 1 1,p 1 = µ 1 u n q 1 1,q 1 +ˆΩ N f 1 (u n , v n )u n dx, v n p 2 1,p 2 = µ 2 v n q 2 1,q 2 +ˆΩ N f 2 (u n , v n )v n dx ∀ n ∈ N. (4.3) Gathering (4.2) and (4.3) together produces −∆ p 1 u n + µ 1 ∆ q 1 u n − N f 1 (u n , v n ), u n − u → 0, −∆ p 2 v n + µ 2 ∆ q 2 v n − N f 2 (u n , v n ), v n − v → 0. (4.4) So, condition (c) of Definition 2.6 holds, which means that (u, v) turns out a generalized solution to (P). Theorem 4.2. Let (H ′ 1 )-(H 2 ) be satisfied and let (µ 1 , µ 2 ) ∈ R 2 . Then problem (P) admits a solution (u, v) ∈ X in the sense of Definition 2.7. Moreover, (u, v) is a weak solution once µ 1 ∨ µ 2 < 0. Proof. Since (H ′ 1 ) implies (H 1 ), the same arguments adopted to prove Theorem 4.1 furnish both a bounded sequence {(u n , v n )} ⊆ X and (u, v) ∈ X satisfying (a)-(b) of Definition 2.7 as well as (4.4). Thus, it remains to verify (c ′ ). Thanks to (H ′ 1 ) and Hölder's inequality we obtain u n − u r 1 + σ 1 s ′ 1 u n − u s 1 ≤ C u n − u r 1 + σ 1 s ′ 1 u n − u s 1 for all n ∈ N, because {(u n , v n )} is bounded. The condition r 1 ∨ s 1 < p * 1 then entails u n → u in L r 1 (Ω) ∩ L s 1 (Ω), where a sub-sequence is considered when necessary; cf. Proposition 2.2. Therefore, lim n→∞ˆΩ N f 1 (u n , v n )(u n − u) dx = 0. (4.5) Analogously, one has lim n→∞ˆΩ N f 2 (u n , v n )(u n − u) dx = 0. (4.6) From (4.4)-(4.6) it follows lim n→∞ −∆ p 1 u n + µ 1 ∆ q 1 u n , u n − u = lim n→∞ −∆ p 2 v n + µ 2 ∆ q 2 v n , v n − v = 0, because p 1 (p * 2 ) ′ < p 1 , α 2 < p 2 , β 2 < p 2 (p * Theorem 4. 1 . 1Under hypotheses (H 1 )-(H 2 ), for every (µ 1 , µ 2 ) ∈ R 2 problem (P) possesses a solution in the sense of Definition 2.6. ˆΩ N f 1 1(u n , v n )(u n − u) dx ≤ D 1ˆΩ \|u n \| n − u\| dx +ˆΩ σ 1 \|u n − u\| dx ) ′ < p 2 . Therefore, setting γ i (x) = h i (x) +ĉ i , we see that also (H 2 ) holds. i.e., (c ′ ) of Definition 2.7 holds, and (u, v) is a strong generalized solution to (P).Finally, let µ 1 ∨ µ 2 < 0. We will show that (u, v) fulfills(2.4), that iswhich completes the proof. The monotonicity of −\|µ i \|∆ q i and (c ′ ) yieldSince the operator −∆ p i is of type (S) + , we infer (u n , v n ) → (u, v) in X. Now, combining the continuity of A µ 1 ,µ 2 with condition (b) in Definition 2.6 one hasnamely (4.7) holds true.The next example provides a natural class of reactionsis true with C 1 := C 2 := 1 and σ i (x) := \|h i (x)\| + 3. If c i , d i > 0, i = 1, 2, satisfy (2.5) then, with appropriateĉ i > 0, one has both(p * 1 ) ′ < p 1 , and f 2 (x, s, t, ξ, ν)t ≤ \|s\| p 1 (p * 2 ) ′ + \|t\| α 2 + \|ξ\| p 1 (p * 2 ) ′ + \|ν\| β 2 + h 2 (x) ≤ c 2 (\|ξ\| p 1 + \|ν\| p 2 ) + d 2 (\|s\| p 1 + \|t\| p 2 ) + h 2 (x) +ĉ 2 , Functional analysis, Sobolev spaces and partial differential equations. H Brézis, SpringerNew YorkUniversitextH. Brézis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. 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Mosconi, Some recent results on the Dirichlet problem for (p, q)-Laplace equation, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), 279-291. D Motreanu, Nonlinear Differential problems with Smooth and Nonsmooth Constraints. LondonAcademic PressD. Motreanu, Nonlinear Differential problems with Smooth and Nonsmooth Constraints, Academic Press, London, 2018. Quasilinear Dirichlet problems with competing operators and convection, Open Math. D Motreanu, 18D. Motreanu, Quasilinear Dirichlet problems with competing operators and convection, Open Math. 18 (2020), 1510-1517. A proof of Schauder's theorem. S Kakutani, J. Math. Soc. Japan. 3S. Kakutani, A proof of Schauder's theorem, J. Math. Soc. Japan 3 (1951), 228-231. Multiplicity of solutions for the p-Laplacian. I , ICTP Lectures Notes of the Second School of Nonlinear Functional Analysis and Applications to Differential Equations. TriesteI. Peral, Multiplicity of solutions for the p-Laplacian, in ICTP Lectures Notes of the Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. R E Showalter, Math. Surveys Monogr. 49Amer. Math. SocR.E. Showalter,Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surveys Monogr. 49, Amer. Math. Soc., Providence, 1997. Nonlinear functional analysis and its applications. E Zeidler, Springer-VerlagNew YorkII/A. Linear monotone operatorsE. Zeidler, Nonlinear functional analysis and its applications. II/A. Linear monotone operators, Springer-Verlag, New York, 1990. Nonlinear functional analysis and its applications. II/B. Nonlinear monotone operators. E Zeidler, Springer-VerlagNew YorkE. Zeidler, Nonlinear functional analysis and its applications. II/B. Nonlinear monotone operators, Springer-Verlag, New York, 1990. Italy Email address: [email protected] (S.A. Marano) Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania. Università Gambera) Dipartimento Di Matematica E Informatica, Degli Studi Di Catania, A Viale, 66860Perpignan, France; Yulin, People's Republic of China EmailDepartment of Mathematics, University of Perpignan ; College of Science, Yulin Normal UniversityDoria 6, 95125 Catania. Italy Email address: [email protected] (D. MotreanuGambera) Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy Email address: [email protected] (S.A. Marano) Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy Email address: [email protected] (D. Motreanu) Department of Mathematics, University of Perpignan, 66860 Perpignan, France; College of Science, Yulin Normal University, Yulin, People's Republic of China Email address: [email protected]	[]
[ "ZARA: Improving Few-Shot Self-Rationalization for Small Language Models", "ZARA: Improving Few-Shot Self-Rationalization for Small Language Models" ]	[ "Wei-Lin Chen [email protected] \nDepartment of Computer Science and Information Engineering\nNational Taiwan University\nTaiwan\n", "An-Zi Yen \nDepartment of Computer Science\nNational Yang Ming Chiao Tung University\nTaiwan\n", "Hen-Hsen Huang \nInstitute of Information Science\nAcademia Sinica\nTaiwan\n", "Cheng-Kuang Wu \nDepartment of Computer Science and Information Engineering\nNational Taiwan University\nTaiwan\n", "Hsin-Hsi Chen \nDepartment of Computer Science and Information Engineering\nNational Taiwan University\nTaiwan\n" ]	[ "Department of Computer Science and Information Engineering\nNational Taiwan University\nTaiwan", "Department of Computer Science\nNational Yang Ming Chiao Tung University\nTaiwan", "Institute of Information Science\nAcademia Sinica\nTaiwan", "Department of Computer Science and Information Engineering\nNational Taiwan University\nTaiwan", "Department of Computer Science and Information Engineering\nNational Taiwan University\nTaiwan" ]	[]	Language models (LMs) that jointly generate end-task answers as well as free-text rationales are known as self-rationalization models. Recent works demonstrate great performance gain for self-rationalization by fewshot prompting LMs with rationale-augmented exemplars. However, the ability to benefit from explanations only emerges with largescale LMs, which have poor accessibility. In this work, we explore the less-studied setting of leveraging explanations for small LMs to improve few-shot self-rationalization. We first revisit the relationship between rationales and answers. Inspired by the implicit mental process of how human beings assess explanations, we present a novel approach, Zeroshot Augmentation of Rationale-Answer pairs (ZARA), to automatically construct pseudoparallel data for self-training by reducing the problem of plausibility judgement to natural language inference. Experimental results show ZARA achieves SOTA performance on the FEB benchmark, for both the task accuracy and the explanation metric. In addition, we conduct human and quantitative evaluation validating ZARA's ability to automatically identify plausible and accurate rationale-answer pairs.	10.48550/arxiv.2305.07355	[ "https://export.arxiv.org/pdf/2305.07355v1.pdf" ]	258,676,220	2305.07355	7df3595bdb4003589e8ca1757cc39ec03a39a2ff	ZARA: Improving Few-Shot Self-Rationalization for Small Language Models Wei-Lin Chen [email protected] Department of Computer Science and Information Engineering National Taiwan University Taiwan An-Zi Yen Department of Computer Science National Yang Ming Chiao Tung University Taiwan Hen-Hsen Huang Institute of Information Science Academia Sinica Taiwan Cheng-Kuang Wu Department of Computer Science and Information Engineering National Taiwan University Taiwan Hsin-Hsi Chen Department of Computer Science and Information Engineering National Taiwan University Taiwan ZARA: Improving Few-Shot Self-Rationalization for Small Language Models Language models (LMs) that jointly generate end-task answers as well as free-text rationales are known as self-rationalization models. Recent works demonstrate great performance gain for self-rationalization by fewshot prompting LMs with rationale-augmented exemplars. However, the ability to benefit from explanations only emerges with largescale LMs, which have poor accessibility. In this work, we explore the less-studied setting of leveraging explanations for small LMs to improve few-shot self-rationalization. We first revisit the relationship between rationales and answers. Inspired by the implicit mental process of how human beings assess explanations, we present a novel approach, Zeroshot Augmentation of Rationale-Answer pairs (ZARA), to automatically construct pseudoparallel data for self-training by reducing the problem of plausibility judgement to natural language inference. Experimental results show ZARA achieves SOTA performance on the FEB benchmark, for both the task accuracy and the explanation metric. In addition, we conduct human and quantitative evaluation validating ZARA's ability to automatically identify plausible and accurate rationale-answer pairs. Introduction Driven by the concerns of whether the decisions made by the artificial intelligence models are trustworthy, providing free-text, natural language explanations (NLEs) has drawn substantial attention in the research community (Camburu et al., 2018;Li et al., 2018;Rajani et al., 2019;Aggarwal et al., 2021;Chen et al., 2022). Comparing with popular explanation techniques within the input scope, e.g., attributing feature importance scores to tokens (Li et al., 2016;Godin et al., 2018) or extracting fragments of text highlights (Lei et al., 2016;Jain Figure 1: The role of our plausibility agent. As the main component of ZARA, the agent, i.e, the approximator, imitates how human assess the plausibility of explanations, in an explicit fashion. et al., 2020), free-text explanation 1 is more expressive, inherently apt for human comprehension and brings richer information in addition to input context (Camburu et al., 2018;). Yet, the construction of NLE datasets is expensive and challenging due to quality control issues such as inconsistency and under-specification (Wiegreffe and Marasovic, 2021). The development of interpretable NLP systems which can provide NLEs in few-shot is necessitated. Recent works Wang et al., 2022b;Lampinen et al., 2022) achieve few-shot self-rationalization, i.e., jointly generating freetext explanations and end-task labels, by extending the usage of NLEs to compose chain-of-thought (CoT) input-rationale-output demonstrations for prompt-based learning. Comparing with standard prompting (i.e., without rationales), prompting with rationale-augmented exemplars triggers LM's complex reasoning ability, significantly boosting the end-task performance. However, the main drawback is that only excessively large LMs (generally 100B-plus) demonstrate this ability to leverage explanations, which sharply emerges when scaling model size sufficiently Lampinen et al., 2022). In this work, we explore the less-studied setting of improving few-shot self-rationalization only relying on affordable, small LMs (200M∼2.7B). We adopt self-training (Scudder, 1965), a simple yet effective methodology that is not practical for large LMs in most real-world scenarios. We first investigate the relationship between the generated explanations and end-task predictions, and find plausible explanations are usually paired with correct label predictions. Namely, plausibility is a strong indicator for answer correctness. Motivated by this finding, we propose Zero-shot Augmentation of Rational-Answer pairs (ZARA) for self-training. Specifically, we reduce the problem of assessing rationale plausibility to the task of natural language inference (NLI), and propose a zero-shot plausibility approximator towards automatic assessment of the generated rationales, without requiring any ground-truth labels or golden explanations. The approximator can be viewed as an agent for plausibility judgement. As illustrated in Figure 1, to determine the plausibility of the rationale, humans implicitly ask themselves whether they can draw conclusions to the predicted answer by understanding the task, the input question, and the supported rationale with their logic and reasoning. To approximate such process explicitly, the approximator leverages the ability of textual entailment to yield a probability score indicating the explanation plausibility. Connecting to the self-training paradigm, we first train a self-rationalization model by fewshot prompt-based learning with natural language prompts, and leverage the approximator to collect pseudo-parallel data, i.e, unlabeled inputs paired with high-confident rationale-answer pairs, for creating an augmented training set which is then used to learn an improved self-rationalization model. With various small-size LMs, experiments show our approach notably improves the FEB benchmark 2 (Marasovic et al., 2022)-a recently pro-posed standardized few-shot self-rationalization benchmark-with 3.4% ∼ 5.1% and 3.0% ∼ 5.8% for task accuracy and the associated explanation metric, respectively. Additionally, we validate the approximator's ability with both human and quantitative evaluations. The results suggest our approximator can effectively select plausible explanations that lead to higher accuracy for end-task predictions. In summary, our main contributions are three-fold: 1. We show how to leverage explanations for small LMs by an in-depth analysis of the relationship between rationales and task labels. 2. We propose ZARA, a novel approach for small LMs to improve self-rationalization with self-training. 3. Our NLI-based approximator sheds light on the potential of automatic evaluation for explanation plausibility and post-hoc verification for label accuracy. Background and Motivation Given a trained self-rationalization model f θ (·) and an input sequence x, we denote a prediction f θ (x) = (r,â), wherer is the generated free-text rationale andâ is the predicted answer, typically a classification label. Note thatr andâ are parsed from the output sequence of f θ (x). Evaluation of a self-rationalization model requires assessing botĥ a for the end-task performance andr for the quality of the explanation. With the lack of an ideal and unified automatic metric, the current gold standard for determining the quality ofr is to conduct a human evaluation to check its plausibility (Marasović et al., 2020;Kayser et al., 2021;Wiegreffe et al., 2022;Marasovic et al., 2022). An idealr is considered to be plausible if it is able to justifyâ, that is, providing a logical and reasonable explanation supporting the model's prediction. However, ifr is deemed plausible by human, it does not meanâ is correct. As the example in Table 1, commonsense would know "bed" is likely the answer, yet the generated explanation for the corresponding prediction "couch" is still plausible. Plausibility illustrates the degree of convincement towards the model's prediction, regardless of whether the model is actually making an accurate prediction or not (Jacovi and Goldberg, 2021 Naturally, generating plausible explanations that can justify the wrong answers should be much harder comparing to justifying the correct answers. Since suchr usually demonstrates a slight pivot from commonsense yet still introduces a sound reason to support the inaccurateâ. We hypothesize this-plausible explanation towards inaccurate endtask prediction-is not the circumstance in most cases of (â,r). In other words, ifr is considered to be plausible, it is likely thatâ is a correct prediction. Hence, the first research question arises: RQ1: "To what extent do plausible explanations imply correct label predictions?" And if we could verify RQ1, the follow-up question would be RQ2: "Is it possible to automatically identify plausibler and utilize (r,â) for further model improvement?" In the following of our work, we answer RQ1 by inspecting the interrelationship between the plausibility ofr and the correctness ofâ (Section 4), where we show evidence supporting the linkage to RQ2. Ergo, we propose ZARA coupled with selftraining to accomplish RQ2 (Section 5), improving few-shot self-rationalization models. Datasets and Tasks We adopt FEB (Marasovic et al., 2022), a newly proposed few-shot self-rationalization benchmark, Marasovic et al., 2022) only evaluatesr withâ = a, i.e, explanation for the correctly predicted answer. This may overestimate the quality of explanations (Wiegreffe et al., 2022). as the dataset for experiments throughout this work. FEB consists of four sub-tasks from existing English-language explainable datasets with free-text explanations: (1) Nonsensical sentence selection (COMVE; Wang et al., 2019). Given two sentences, select the sentence that is less likely to make sense. (2) Offensiveness classification (SBIC; Sap et al., 2020). Classify a given post as offensive or not. (3) Natural language inference (E-SNLI; Camburu et al., 2018). Classify the relationship between two sequences as entailment, neutral, or contradiction. (4) Multiple-choice commonsense QA (ECQA; Aggarwal et al., 2021). Given a question, select the correct answer from five choices. The goal for each sub-task is the same, namely, to predict a label for the underling classification task and generate a free-text explanation supporting the model's decision. Each sub-task has 60 episodes, and each episode is a train-test split with 48 training examples and 350 evaluation examples. This design of no extra validation data encompasses the FLEX principles (Bragg et al., 2021) for performing robust few-shot NLP evaluation to avoid per-episode hyper-parameter tuning, which could inflate the evaluation results considerably as shown in previous work (Gao et al., 2021). Hence, a single set of hyper-parameter is used across all episodes. Correlation between Plausibility and Correctness As described in Section 2, following we attempt to answer RQ1 by measuring the correlation between the plausibility ofr and the correctness of a. We conduct human studies on results from a self-rationalization model (without self-training) using the FEB dataset. We adopt prompt-based finetuning with natural language prompt on a sequenceto-sequence language model to perform few-shot self-rationalization. For each episode of the sub-task, we train a self- Hypothesis: The answer of the question "[question]" is [Answer's choice]. NLI Class: Entailment Mapped example Mapped example Premise: A woman in a black mesh skirt plays acoustic guitar. The woman is wearing a black mesh, she is wearing black. Hypothesis: A woman is wearing black. Premise: Because state park is a [...] gardens are places with lots of trees and plants. Hypothesis: The answer of the question "what is a place that has a bench nestled in trees?" is state park. rationalization model with the training set and generate rationale-answer pairs on the test set. We then gather all predictions from the 60 episodes and randomly select 350 examples for human studies. We present the description of the task, the input instance x and the rationale-answer pair (r,â) for the annotators, and ask them to judge the plausibility ofr, i.e., whether it can justifyâ. Following prior works (Marasović et al., 2020;Marasovic et al., 2022), the annotator determines the plausibility by assigning labels from {"no", "weak no", "weak yes", "yes"}. We then map labels to plausibility scores {1, 2, 3, 4} and instances with average scores above 2.5 are deemed plausible. We provide inter-annotator agreement details in Appendix C. The results are shown in Figure 2. We can observe that for all sub-tasks, when the explanations are judged as plausible, they are much more likely paired with correctly predicted answers in constrast to implausible ones. This verifies our hypothesis (discussed in Section 2) and shows plausibility to be a strong signal for correct label predictions. Our results also align with the prior work , where they find self-rationalization models demonstrate high label-rationale association against robustness testing. In conclusion, identifying (r,â) pairs that have plausibler spurs great potential for boosting model performance, and connects us to RQ2. Zero-Shot Augmentation of Rationale-Answer Pairs As shown in Section 4, plausible explanations imply that the corresponding task predictions are more Figure 3: Overview of the self-training paradigm for ZARA. A language model is fine-tuned to generate predictions for unlabeled instances, which are then mapped to NLI formats. The approximator then identifies high-confident (likely plausibile) ones as augmentation for learning an new model. likely to be correct. Following we present ZARAthe approach towards automatically judging the plausibility of generated explanations, and leverages the high confident rationale-answer pairs to boost model performance via self-training. Reduce plausibility judgement to NLI Given a rationale-answer pair (r,â) output by a self-rationalization model, human evaluates whetherr is plausible by understanding the input context and the task objective, and applying reasoning ability to determine ifr justifiesâ. Specifically, human implicitly form propositions from the input context and rationale by understanding the problem (the task). Then do inference, i.e., apply logic and reasoning to draw conclusions, in their mind to decide if the propositions support the predicted answer. This mental process of assessing plausibility resembles determining the relationship between a premise and a hypothesis. Driven by this formulation, we reduce the problem of judging the plausibility of explanations to the task of natural language inference (NLI), and construct a zero-shot approximator, which leverages existing NLI models to automatically approximate the human judgement of plausibility. NLI Mapping. The formulation as NLI requires the mapping of (x,r,â) → (p, h), where x, p, and h are the input instance, premise, and hypothesis, respectively. We manually create the mappings for each FEB sub-task as shown in Table 2. Constructing such mappings can be easily achieved with minimal effort 4 compared with human evaluation on allr. Consider the COMVE example in Table 2, the goal is to select the nonsensical sentence from two sentences. As we can see "i drove my computer to the gas station." is nonsensical, and the rationale justifies it by stating "you can't drive a computer.", which explains why the answer is nonsensical by providing information refuting the answer sentence, resulting in a contradiction relationship between the two. Hence, the approximator can estimate the degree of plausibility by referring to the score of the contradiction class. The approximator. For developing the approximator, we ensemble three state-of-the-art pre-trained NLI models by averaging their output scores for the decision of NLI class. Specifically, we adopt RoBERTa (Liu et al., 2019), DeBERTa (He et al., 2020), and BART (Lewis et al., 2020), trained on the MultiNLI corpus (Williams et al., 2018), one of the largest available NLI dataset. The approximator is zero-shot, i.e., all three models are used off-the-shelf (See Appendix A for details) without any fine-tuning on our dataset, accommodating the few-shot, data scarcity setting. Self-training In the self-training paradigm, a trained model augments its own training set by constructing pseudo-parallel data with predictions on unlabeled instances, where the most confident predictions are collected as new training examples and used to re-train an improved model. For applying self-training, most works focus on classification tasks (Miyato et al., 2018;Xie et al., 2020;Gera et al., 2022) with common strategies based on operations of confidence scores such as probability values to select new examples. E.g., finding predictions that are far from the decision boundary (Slonim et al., 2011). However, the adoption of self-training for selfrationalization differs from typical classification tasks in two aspects: (1) Compared with fixed classification labels, the target space of neural sequence generation is much more complex. (2) The selection requires considering both the task labelâ and the rationaler with their relationship. By a proxy model, i.e, the approximator, we could reduce the target dimensions to fixed class labels to address the former. For the latter, we could resolve it by only considering the plausibility ofr since plausibler likely implies correctâ as shown in Section 4. Following we introduce our self-training paradigm of ZARA-a train-judge-train procedure. See Figure 3 for illustration. Given an episode E consisting of a training split D train and a test split D test , where an example in E is an input-rationale-answer tuple (x, r, a). We first train a LM M 0 on D train for self-rationalization by prompt-based fine-tuning with natural language prompts. The trained model is denoted as M 1 . Next, we perform inference with M 1 on unlabeled instances x ∈ D unlabled , where D unlabled is a non-overlapping set randomly sampled from other episodes with size \|D unlabled \| = \|D test \|. For each prediction, the input x and the generated rationaleanswer pair (r,â) are mapped to the NLI format, i.e., (x,r,â) → (p, h), and passed to the zero-shot plausibility approximator. 5 The approximator automatically judges the plausibility ofr, where the most confident predictions are selected by a plausibility threshold α, i.e., a probability score (See Appendix B for details). This process does not require any ground truth label or golden rationale. The collected high-confident (x,r,â) predictions become new instances to augment D train . Also, we ensure the added instances are balanced for classification tasks by downsampling majority classes. We then re-train M 0 on the augmented training split to obtain our final self-rationalization model M 2 , and evaluate on D test . Experiments In this section, we discuss the experimental setup and present the results of our proposed method, ZARA, for improving few-shot self-rationalization via self-training. We also perform human and quantitative evaluations to validate the automatic plausibility assessment for our approximator. Model For comparison purposes, we follow FEB and use UNIFIEDQA (Khashabi et al., 2020), a T5 (Raffel et al., 2020) variant trained on a multi-task mixture of QA datasets, as our self-rationalization model for all experiments. The model performs few-shot learning via fine-tuning with natural language prompts. We experiment with three model sizes: UNIFIEDQA-base (200M), UNI-FIEDQA-large (770M), and UNIFIEDQA-3B (2.7B). The results presented in Section 4 are conducted with UNIFIEDQA-3B. More details of the experimental setups and configurations are provided in Appendix A. Method Model Acc Figure 4: Human judgement of plausibility scores under different pseudo-plausibility percentile bins. 1 4 , 2 4 , 3 4 , and 4 4 indicate 1∼25th, 25∼50th, 50∼75th, and 75∼100th percentile group, respectively. Ranked by the approximator's output probability, i.e., pseudo-plausibility scores. Main results The evaluation metrics of FEB are accuracy and BERTscore for end-task labels and explanations, respectively. 6 For each sub-task, we train 60 models (one per episode) and report the mean and standard error of accuracy/BERTscore in Table 3. We also provide statistics on the number of instances added for augmentation in Appendix D. To the best of our knowledge, we present the first results on the newly introduced FEB benchmark (besides their original approach in the paper). We experiment with three model sizes: base, large and 3B. In ZARA, both training stages adopt models of the same size; the original FEB baseline only involves training one model (one stage). As observed in Table 3, our method substantially outperforms the FEB baseline for all datasets. In general, COMVE, SBIC and E-SNLI demonstrate relatively consistent improvements across model size. The only anomoly is for ECQA. We hypothesis as ECQA requires commonsense knowledge outside FEB training data but is encoded in models' parameters originally, the under-parameterized models (base and large) suffer forgetting from continuous learning with the augmented data. However, for the 3B model (which is still significantly smaller than most large-scale LMs) great performance gain is exhibited. Approximator evaluation Plausibility evaluation We conduct human evaluation to validate our approximator. Specifically, the human evaluation can be considered as a metaevluation-the evaluation of evaluation-for evaluating the approximator's ability to evaluate explanations, i.e., its ability to assess plausibility. To recap, the approximator's output probability of the corresponding NLI class (based on the mapping design in Table 2) represents an estimation of plausibility degree, i.e., a pseudo-plausibility score. We use the same batch of annotated data from Section 4. That is, 350 randomly selected examples generated by the stage-one model with human judgement of plausibility value {1, 2, 3, 4} mapped from {"no", "weak no", "weak yes", "yes"} and averaged across annotators. The results are presented in Figure 4. We group the instances into four bins, each containing 25% of data according to the percentile ranking of their pseudo-plausibility score. In general, the median performance of human plausibility judgement increases with higher percentile groups, especially for the COMVE and SBIC sub-tasks. Interestingly, due to the nature of NLI model of the approximator, its output (i.e., pseudo-plausibility scores) may be effected by spurious surface features learned only for NLI tasks (transferred from the MultiNLI dataset), giving rise to the larger interquartile range of the top percentile group in E-SNLI. Overall, the results show our approximator is capable of reflecting human plausibility judgement. Correctness evaluation As stated in Section 4, plausible rationales likely indicate correct answer predictions. We further evaluate our approximator regarding this property by checking the end-task answer accuracy of the data subset selected for augmentation from stage-one model's prediction pool. We consider three selection strategies: (1) ZARA, i.e., our proposed method, which selects confident (high-scoring) predictions; (2) Random, the data subset is selected randomly from prediction pool; (3) Lowest, in contrast to ZARA, we select a subset from the data with lowest-ranking pseudoplausibility scores. For each episode, the number of augmented instances for (2) and (3) are determined by (1), i.e., we randomly select n instances or select n bottom-ranking instances, where n is the number of instances for augmentation using ZARA. The results are shown in Figure 5. We can observe ZARA consistently outperforms Random and Lowest with substantial margins under different model sizes across all four datasets, and Lowest demonstrates the poorest accuracy. This suggest our approximator is able to verify label predictions post-hoc, i.e., the high/low pseudo-plausibility score suggests the prediction is accurate/inaccurate. In conclusion, the overall evaluation results suggest our approximator can effectively extract rationale-answer pairs which are more plausible and accurate. 7 Related Work 7.1 Few-shot self-rationalization To provide NLEs under low supervision, Marasovic et al. (2022) propose the FEB benchmark and establish the first results by exploring natural language prompt-based fine-tuning. Wiegreffe et al. (2022) focus on improving NLEs with an overgenerate-and-filter pipeline: prompting GPT-3 with gold labels to generate explanation candidates which are then filtered by a model trained with human annotations. Recent works Wang et al., 2022b; leverage rationale-augmented chain-of-thought (CoT) inputs to prompt frozen large-scale LMs in few-shot. Zelikman et al. (2022) further bootstrap CoT by repeatedly fine-tuning a GPT-J model. Concurrent works (Wang et al., 2022a;Ho et al., 2022;Hsieh et al., 2023) propose pipeline frameworks to distill knowledge by prompting a large "teacher" LM to generate diverse reasoning rationales which are then used to fine-tuning a small "student" LM. In comparison, ZARA directly optimizes small LMs on downstream tasks, without access to any large LMs. A similar work by Ye and Durrett (2022) leverage NLEs to boost end-task predictions posthoc by training a calibrator. In comparison, we directly improve self-rationalization and our approximator does not require any further training. Moreover, all LMs used in their work are 175B. Leveraging NLI for downstream tasks The framework of NLI has been expanded to benefit many NLP tasks. propose a framework to verify QA systems' predictions with NLI by training models to generate premise-hypothesis pairs from QA instances. Yet, the related work that inspires us the most is by Yin et al. (2019). Driven by human reasoning, they approach text classification in zero-shot by formulating it as an entailment problem-given the input text (premise), human mentally construct hypotheses "the text is about [label choice]" to determine the answer-and adopt out-of-the-box NLI models for predictions. Conclusion In this work, we first show evidences that plausible explanations imply correct end-task predictions, and leverage NLI to propose a zero-shot approximator which is capable of automatically identifying plausible rationales paired with correct answers from unlabeled results. By collecting such rationale-answer pairs with self-training, we can effectively improve the performance of few-shot self-rationalization for small LMs. Moreover, we demonstrate the potential for automatic evaluation of free-text explanations. In light of this, we believe developing a supervised approximator with a unified NLI mapping schema across tasks to be a promising avenue for future works. Limitations The success of the approximator relies on the quality of the NLI mapping. Though we showcase great improvement across four different tasks, if the complexity of a task makes the mapping construction non-trivial, the created mapping might not be able to accurately reflect human plausibility judgement of the generated rationales, and the benefit of selftraining could not be guaranteed. Namely, the approximator may identify noisy instances that would instead hurt model performance. A Experimental setups and configurations We prepare the training data and format it as natural language prompts using the scripts provided by the FEB paper's repository. We train our base and large model (UNIFIEDQA-base and UNIFIEDQA-large) by NVIDIA GeForce RTX 3090, and 3B model (UNIFIEDQA-3B) by NVIDIA RTX A6000. For hyper-parameter (HP) settings, we follow the original setup in the FEB paper for stage-one training, and for stage-two, we set maximum epochs to 48 and keep other HPs the same as stage-one. We do not perform additional HP search. For the approximator, We use facebook/bartlarge-mnli, microsoft/deberta-large-mnli, and roberta-large-mnli from the Hugging Face Hub. B Plausibility threshold To estimate a threshold that is sufficient but not overly strict, we compute the average number of training set instances (which are required to be plausible) per episode with probability scores above different threshold values, as shown in Figure 6. The dotted line represents the segment with the smallest slope, indicating increasing the threshold results in the largest data lost. The starting, i.e., smaller, x-value of the dotted line is chosen as our plausibility threshold. Thus, 0.9 for COMVE, E-SNLI and ECQA, and 0.8 for SBIC. C Human annotation details We invite three annotators 7 to conduct human evaluation and compute inter-annotator agreements by Randolph's κ on 100 overlapping annotation examples. We record κ of 0.60, 0.56, 0.36, and 0.49 for COMVE, SBIC, E-SNLI, and ECQA, respectively. The low (0.36) to moderate (0.60, 0.56, 0.49) agreements align with prior works' observations on evaluating plausibility of free-text explanation, reflecting the task subjectivity (Wiegreffe et al., 2022) and could require more fine-grained analysis in the future (Marasovic et al., 2022). D Data augmentation details Figure 2 : 2The percentage of correct/incorrect answers when explanations are plausible and not plausible. The y-axis denotes the percentage of explanations deemed plausible/not plausible by human. Figure 5 : 5The average accuracy (per episode) of the selected subset from the stage-one model's generated results by different selection strategies: (1) ZARA, (2) Random, and(3)Lowest. Welleck et al. (2019) develop a dataset to improve dialogue models by framing the dialogue consistency problem as NLI. Honovich et al. (2021); Dziri et al. (2022) use NLI to design automatic metrics evaluating factuality of knowledge-grounded dialogue systems. Falke et al. (2019); Kryscinski et al. (2020); Laban et al. (2022) use NLI models to detect factual errors in abstractive summarization tasks. For question answering, Figure 6 : 6The average number of training set instances (per episode) with probability score above different thresholds. Table 1 : 1A plausible but inaccurate prediction of an in- stance from ECQA dataset output by our model. Example choice1: i drove my car to the gas station. choice2: i drove my computer to the gas station. Answer: choice2 Rationale: you can't drive a computer. Post: just when i thought women couldn't get any stupider. Answer: offensive Rationale: this post implies that women are stupid. Premise: The post just when i thought women couldn't get any stupider. Hypothesis: The post is offensive because this post implies that women are stupid. E-SNLI Example ECQA Example Premise: a woman in a black mesh skirt plays acoustic guitar. Hypothesis: a woman is wearing black. Answer: Entailment Rationale: The woman is wearing a black mesh, she is wearing black. Question: what is a place that has a bench nestled in trees? Choices:(a) state park (b) bus stop (c) bus depot (d) statue (e) train station Answer: (a) Rationale: state park is a protected public garden. public gardens generally have benches for people to sit and relax. gardens are places with lots of trees and plants. Mapping Mapping Premise: [Premise] [Rationale]. Hypothesis: [Hypothesis] NLI Class: [Answer] Premise: Because [Rationale]Mapping and Example COMVE Example SBIC Mapping Mapping Premise: [Rationale] Hypothesis: [Answer's sentence] NLI Class: Contradiction Premise: The post: [Post] Hypothesis: The post is [Answer] because [Rationale] NLI Class: Entailment Mapped example Mapped example Premise: you can't drive a computer. Hypothesis: i drove my computer to the gas station. Table 2 : 2The mapping design for the four sub-tasks with non-cherry-picked examples. See Section 3 for description about each sub-task. Table 3 : 3Experiments comparing FEB proposed by Marasovic et al. (2022) and our proposed ZARA. The results with GPT-3 (175B) and random baseline, i.e., 1 number of classes , are also presented. Table 4 4reports the average number of additional test set instances added per episode for stage-two training. For COMVE, SBIC, and E-SNLI, about one-third of the test data are selected, with only minor differences against model sizes. On the other hand, ECQA shows a notable increment on the 3B model, yet significant lower addition number in general comparing to the other three sub-tasks, which may attribute to the nature of difficulty for commonsense question answering.Model ComVE SBIC E-SNLI ECQA Base 109.0 113.6 123.8 13.2 Large 132.3 109.0 128.6 13.4 3B 191.5 107.6 98.3 48.4 Table 4 : 4The comparison for average number of instances added (per episode) between model size. We use the term "free-text explanation" and "natural language explanation" interchangeably; and the term "explnantion" can also refer to the rationale generated by the model. https://github.com/allenai/feb (Licensed under the Apache License 2.0.) For a confounder-free setting, prior works(Kayser et al., One can simply design the mapping by observing the training data. Depending on the mapping design, some sub-tasks do not require input content x to form the premise and hypothesis. BERTscore is one of the most correlated automatic NLG metrics with human judgement of plausibility for free-text explanation, as shown by(Kayser et al., 2021). The annotators include two graduate students and one Ph.D. student. As our tasks do not require specific domain expertise, the payment is determined by the minimum wage. Explanations for Common-senseQA: New Dataset and Models. 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[ "Aligning the Western Balkans power sectors with the European Green Deal Abbreviations CF -Capacity Factor CFTPP -Coal-fired Thermal Power Plant EU -European Union GHG -Greenhouse Gas GWA -Global Wind Atlas HPP -Hydropower plant IEC -International Electrotechnical Commission NDC -Nationally Determined Contribution OSeMOSYS -Open Source Energy Modelling System TS -Time Steps VRE -Variable Renewable Energy WB -Western Balkans", "Aligning the Western Balkans power sectors with the European Green Deal Abbreviations CF -Capacity Factor CFTPP -Coal-fired Thermal Power Plant EU -European Union GHG -Greenhouse Gas GWA -Global Wind Atlas HPP -Hydropower plant IEC -International Electrotechnical Commission NDC -Nationally Determined Contribution OSeMOSYS -Open Source Energy Modelling System TS -Time Steps VRE -Variable Renewable Energy WB -Western Balkans", "Aligning the Western Balkans power sectors with the European Green Deal Abbreviations CF -Capacity Factor CFTPP -Coal-fired Thermal Power Plant EU -European Union GHG -Greenhouse Gas GWA -Global Wind Atlas HPP -Hydropower plant IEC -International Electrotechnical Commission NDC -Nationally Determined Contribution OSeMOSYS -Open Source Energy Modelling System TS -Time Steps VRE -Variable Renewable Energy WB -Western Balkans", "Aligning the Western Balkans power sectors with the European Green Deal Abbreviations CF -Capacity Factor CFTPP -Coal-fired Thermal Power Plant EU -European Union GHG -Greenhouse Gas GWA -Global Wind Atlas HPP -Hydropower plant IEC -International Electrotechnical Commission NDC -Nationally Determined Contribution OSeMOSYS -Open Source Energy Modelling System TS -Time Steps VRE -Variable Renewable Energy WB -Western Balkans" ]	[ "Emir Fejzić [email protected]@orcid.org \nSchool of Industrial Engineering and Management (ITM)\nEnergy Systems\nKTH\nEnergy TechnologySweden\n", "Taco Niet [email protected]@orcid.org \nSchool of Sustainable Energy Engineering\nSimon Fraser University\n10285 University DriveV3T 4B7SurreyBCCanada\n", "Cameron Wade [email protected]@orcid.org \nSutubra Research Inc. 5608 Morris St\nB3J 1C2HalifaxNova ScotiaCanada\n", "Will Usher [email protected]@orcid.org \nSchool of Industrial Engineering and Management (ITM)\nEnergy Systems\nKTH\nEnergy TechnologySweden\n", "Emir Fejzić [email protected]@orcid.org \nSchool of Industrial Engineering and Management (ITM)\nEnergy Systems\nKTH\nEnergy TechnologySweden\n", "Taco Niet [email protected]@orcid.org \nSchool of Sustainable Energy Engineering\nSimon Fraser University\n10285 University DriveV3T 4B7SurreyBCCanada\n", "Cameron Wade [email protected]@orcid.org \nSutubra Research Inc. 5608 Morris St\nB3J 1C2HalifaxNova ScotiaCanada\n", "Will Usher [email protected]@orcid.org \nSchool of Industrial Engineering and Management (ITM)\nEnergy Systems\nKTH\nEnergy TechnologySweden\n" ]	[ "School of Industrial Engineering and Management (ITM)\nEnergy Systems\nKTH\nEnergy TechnologySweden", "School of Sustainable Energy Engineering\nSimon Fraser University\n10285 University DriveV3T 4B7SurreyBCCanada", "Sutubra Research Inc. 5608 Morris St\nB3J 1C2HalifaxNova ScotiaCanada", "School of Industrial Engineering and Management (ITM)\nEnergy Systems\nKTH\nEnergy TechnologySweden", "School of Industrial Engineering and Management (ITM)\nEnergy Systems\nKTH\nEnergy TechnologySweden", "School of Sustainable Energy Engineering\nSimon Fraser University\n10285 University DriveV3T 4B7SurreyBCCanada", "Sutubra Research Inc. 5608 Morris St\nB3J 1C2HalifaxNova ScotiaCanada", "School of Industrial Engineering and Management (ITM)\nEnergy Systems\nKTH\nEnergy TechnologySweden" ]	[]	Located in Southern Europe, the Drina River Basin is shared between three countries: Bosnia and Herzegovina, Montenegro, and Serbia. The power sectors of the three countries have a particularly high dependence on coal for power generation. In this paper we analyse different development pathways for achieving climate neutrality in these countries and explore the potential of variable renewable energy in the area, and its role in the decarbonization of the power sector. We investigate the possibility of whether hydro and non-hydro renewables can enable a net zero transition by 2050, and how renewable energy might affect the hydropower cascade shared by the three countries. The Open-Source Energy Modelling System (OSeMOSYS) was used to develop a model representation of the power sector of the countries. The findings of this analysis show that the renewable potential of the countries is a significant 94.4 GW. This potential is 68% to 287% higher than that of previous assessments, depending on the study of comparison. By 2050, 17% of this potential is utilized for VRE capacity additions under an Emission Limit scenario assuming net-zero. These findings suggest that the local VRE potential is sufficient to support the transition to net-zero. Scenarios with higher shares of solar and thermal power show increased power generation from the hydropower cascade, thus reducing the water available for purposes other than power generation.	null	[ "https://export.arxiv.org/pdf/2305.07433v1.pdf" ]	258,676,304	2305.07433	465a3f2be9b8c378786ff711da712d213bbaaa45	Aligning the Western Balkans power sectors with the European Green Deal Abbreviations CF -Capacity Factor CFTPP -Coal-fired Thermal Power Plant EU -European Union GHG -Greenhouse Gas GWA -Global Wind Atlas HPP -Hydropower plant IEC -International Electrotechnical Commission NDC -Nationally Determined Contribution OSeMOSYS -Open Source Energy Modelling System TS -Time Steps VRE -Variable Renewable Energy WB -Western Balkans Emir Fejzić [email protected]@orcid.org School of Industrial Engineering and Management (ITM) Energy Systems KTH Energy TechnologySweden Taco Niet [email protected]@orcid.org School of Sustainable Energy Engineering Simon Fraser University 10285 University DriveV3T 4B7SurreyBCCanada Cameron Wade [email protected]@orcid.org Sutubra Research Inc. 5608 Morris St B3J 1C2HalifaxNova ScotiaCanada Will Usher [email protected]@orcid.org School of Industrial Engineering and Management (ITM) Energy Systems KTH Energy TechnologySweden Aligning the Western Balkans power sectors with the European Green Deal Abbreviations CF -Capacity Factor CFTPP -Coal-fired Thermal Power Plant EU -European Union GHG -Greenhouse Gas GWA -Global Wind Atlas HPP -Hydropower plant IEC -International Electrotechnical Commission NDC -Nationally Determined Contribution OSeMOSYS -Open Source Energy Modelling System TS -Time Steps VRE -Variable Renewable Energy WB -Western Balkans * Corresponding author Located in Southern Europe, the Drina River Basin is shared between three countries: Bosnia and Herzegovina, Montenegro, and Serbia. The power sectors of the three countries have a particularly high dependence on coal for power generation. In this paper we analyse different development pathways for achieving climate neutrality in these countries and explore the potential of variable renewable energy in the area, and its role in the decarbonization of the power sector. We investigate the possibility of whether hydro and non-hydro renewables can enable a net zero transition by 2050, and how renewable energy might affect the hydropower cascade shared by the three countries. The Open-Source Energy Modelling System (OSeMOSYS) was used to develop a model representation of the power sector of the countries. The findings of this analysis show that the renewable potential of the countries is a significant 94.4 GW. This potential is 68% to 287% higher than that of previous assessments, depending on the study of comparison. By 2050, 17% of this potential is utilized for VRE capacity additions under an Emission Limit scenario assuming net-zero. These findings suggest that the local VRE potential is sufficient to support the transition to net-zero. Scenarios with higher shares of solar and thermal power show increased power generation from the hydropower cascade, thus reducing the water available for purposes other than power generation. Introduction Impacts of climate change are observed all around the world. The severity and frequency of extreme climate are driven by anthropogenetic greenhouse gas (GHG) emissions [1]. To mitigate climate-induced impacts on the environment and society, it is imperative to reduce our GHG emissions. Recent figures show that globally the share of carbon dioxide (CO2) accounts for 64% of total GHG emissions [2]. Out of the 36.3 gigatonnes (Gt) of CO2 emissions from energy-related activities, 10.5 Gt come from coal-fired thermal power plants (CFTPP) [3]. This represents 29 % of total energy-related CO2 emissions. Given these statistics, traditional coal power plants must be replaced with low-emitting renewable energy sources. One region with a particularly high dependence on coal for power generation is The Western Balkans (WB). Shares of coal in the power sectors of this region range from 55% in North Macedonia to 97% in Kosovo 1 [4]. Low-carbon development pathways must therefore be explored to accomplish both the climate and environmental objectives of the WB countries, including the Sustainable Development Goals (SDGs) in the 2030 Agenda, and alignment with the European Green Deal under the Sofia Declaration. Renewable energy sources provide a cleaner alternative to the region's current reliance on coal. Among the identified sources of renewable energy in the WB is hydropower in the Drina River basin (DRB). A significant hydropower potential exists in the basin, of which 60% remains untapped [5]. The river basin is shared by three countries: Bosnia and Herzegovina, Montenegro, and Serbia. These countries have applied to join the European Union (EU) and pledged Nationally Determined Contributions (NDC) as part of the 2030 Agenda. However, for the DRB countries to achieve decarbonization and to align with climate policies and objectives implemented at the EU and global levels, they must incorporate renewable resources other than hydropower into their energy mix. Consequently, the potential for variable renewable energy (VRE) in DRB countries must be assessed. Current literature shows a lack of consistency in terms of VRE potential estimates for the DRB countries. A study by Hrnčić et al [6] investigated the possibility of achieving a 100% renewable energy system in Montenegro. The wind power potential assumed in the study was 400 MW and it referred to the Energy Development Strategy of Montenegro until 2030 [7] and a study by Vujadinović et al [8]. All three studies claim 400 MW of technical wind power potential in Montenegro, a figure that is taken from an assessment conducted by the Italian Ministry for the Environment, Land and Sea in 2007 [9]. Wind turbines have developed rapidly since 2007. Wiser et al [10] found in 2012 that the land area in the US where wind power plants could achieve capacity factors (CFs) of 35% or higher increased by 260% when using turbines designed for the International Electrotechnical Commission (IEC) Class III wind conditions compared to turbines from the 2002-2003 era. Moreover, a report published by the International Renewable Energy Agency (IRENA) in 2017 [11] assessed the technical potential of VRE in the DRB countries to be 56.3 gigawatts (GW). The wind power potential for Montenegro is according to [11] close to 2.9 GW, a much greater potential compared to earlier estimates of 400 MW. The South East Europe Electricity Roadmap (SEERMAP) country reports published in 2017 [12][13][14] for the DRB countries suggest a technical potential of VRE to be 24.4 GW, just 43% of the potential stated by IRENA [11] in the same year. While studies like Hrnčić et al [6] and Husika et al [15] use energy models to investigate potential development pathways for the power sectors of Bosnia and Herzegovina and Montenegro, there are no studies that include decarbonization pathways by 2050 of all DRB countries and their shared hydropower potential within the DRB. An existing study by Almulla et al [16] uses OSeMOSYS to investigate the benefits associated with optimised production and increased cooperation between hydropower plants (HPP) in the DRB, including the impacts of energy efficiency measures. However, since the model, projections, and comparison periods used in [16] are different, the overall methodology and approach can be differentiated from the approach utilized in this study. This paper aims to fill the identified research gaps by investigating decarbonization pathways for climate neutrality of the DRB countries by 2050. In addition, we estimate the power potential of VRE technologies within the DRB countries which can help facilitate this transition away from coal-based power generation. In this paper we aim to answer the following research questions (RQs): • What is the potential of VRE in the DRB countries and what role can it play in the decarbonization of the power sector? • Are the resource potentials of hydro and non-hydro renewables in the DRB countries enough to support the transition to net-zero by 2050? • What is the impact of VRE on the existing hydropower cascade in terms of power generation and cost competitiveness? In section 2, this paper provides a background of the DRB countries and identifies research gaps, followed by a description of the methodology used in section 3. We present the choice of modelling tool, the temporal and geographical dimensions of the study and the operational constraints. The results and discussion are presented in sections 3 and 4 respectively. The work is concluded in section 6, followed by the identification of limitations and future research in section 7. Background The six Western Balkan (WB6) countries of Albania, Bosnia and Herzegovina, Kosovo, Montenegro, North Macedonia, and Serbia are the only Southern European 2 countries that are not yet part of the European Union (EU) [18]. Fig. 1 shows the DRB area, which covers most of the cross-border area between the DRB countries [19]. The basin area is 20 320 km 2 and corresponds to 14% of the total land area of the DRB countries [20]. Overview of the countries In 2021, the population of the DRB countries was 10.7 million [21]. In terms of population, Serbia is the largest country with 6.8 million residents, followed by Bosnia and Herzegovina and Montenegro with 3.3 and 0.6 million respectively. Within the DRB there are 867 thousand people, of whom 50% reside in Bosnia and Herzegovina, 33% in Serbia, and 17% in Montenegro [22]. Montenegro [28,29]. The DRB countries are heavily dependent on coal-fired thermal power plants (CFTPP). Between 2014 and 2018 the share of CFTPP in the power supply was 60-70% in Bosnia and Herzegovina [30] and Serbia [31], reaching 40% in Montenegro [32]. These numbers indicate that the reliance on coal in the DRB countries is significantly higher compared to the 20% share coal has in power generation in the EU [33]. Realizing this reliance on coal by the DRB countries, the European Commission (EC) launched the Initiative for coal regions in transition in December 2020 [34]. This initiative aims to assist the Western Balkans, including the DRB countries, in their transition from coal to carbon-neutral economies. The use of coal-fired power generation has other adverse effects beyond those related to climate change. Estimates indicate 880 deaths from air pollutants in 2020 resulting from the exceedance of the National Emissions Reduction Plans (NERP) greenhouse gas ceilings by CFTPPs in Bosnia and Herzegovina and Serbia. This includes 235 deaths due to exports from the countries to the European Union. Health costs from overshooting GHG emissions in the WB countries are estimated to be between six and twelve billion euros in 2020 alone [35]. Reducing the reliance on coal will not only help in reaching the climate goals but will also improve the air quality and in turn prevent chronic illnesses and premature deaths associated with PM, SO2 and NOx pollutants. Located in Southern Europe, the DRB countries have a higher photovoltaic power potential compared to the north and central parts of Europe [39]. Earlier studies for the Central and South East Europe region have identified a large potential for renewable energy technologies [11,40]. Currently, these potentials are largely untapped. There is no consensus in the literature as to the potential estimates of renewable energy sources within the DRB countries. To do so, it is critical to employ a consistent methodology across the three countries using the latest high-resolution geospatial and temporal data. Methodology In this section we describe the structure of the energy system model of the DRB countries. In addition, we describe and justify the choice of methods to assess wind and solar potential in the DRB countries. This includes the selection of the modelling framework, data, and methodology for assessing the power potentials from VRE sources. Next, we present the clustering approach used to manage computational effort which retains important temporal details across electricity demand and variable renewable energy sources. In addition to the model being open source, the clustering approach ensures that it is also accessible given a reduced computational effort. Finally, we present the scenario analysis. OSeMOSYS and model setup To answer the three research questions posed in Section 1, namely what the potential of VRE in the DRB countries is, if they are sufficient to support the transition to net-zero by 2050, and what their impact is on the existing HPP cascade, the created energy model must possess certain qualities. It should be geospatially explicit, allowing for assessment of VRE potential within DRB countries. The model must account for daily and seasonal variations in climate and power demand while minimizing the computation effort. Modelling HPP cascades on a per-power plant basis is required to assess how changes to the system-wide infrastructure affect these cascades. It must provide insight into future development pathways for the power sector by presenting a long-term expansion of the power system. A detailed description of the OSeMOSYS framework used to represent the hydropower cascade within the Drina River basin, together with the interconnected energy systems of the Drina River basin countries is presented in this section. We created the model using the OSeMOSYS framework [41,42]. The primary use of OSeMOSYS is for long-term energy planning based on the concept of systems optimization. It does not require proprietary software or commercial programming languages and solvers. It is for this reason a preferable option compared to long-established models such as MARKAL/TIMES [43], MESSAGE [44], and PRIMES [45] to name a few, as it does not require upfront costs. The OSeMOSYS framework consists of seven blocks. These blocks are defined as objective function, costs, storage, capacity adequacy, energy balance, constraints, and emissions. The objective function is in this case the total discounted cost of the energy system. It is based on provided energy carrier demands. Costs include capital investment costs, operating costs, and salvage values among others [41]. Constraints include a reserve margin constraint, which in the case of this analysis is set to be 20%. The reserve margin is based on the fact that power generating companies and transmission companies must maintain a capacity to generate and transmit electricity exceeding normal capacity by 10-20% [46]. While multiple emissions can be attributed to power-generating technologies or resource extractions, we consider CO2 emissions in this study. Costs within the model are discounted at a global discount rate of 5%. As a basis for developing the model, we collected data from the literature, stakeholder engagements with representatives from the DRB, transmission system operators (TSOs), and directly from their respective power utilities. The types of data collected include power demands, installed power generating capacities, fixed and variable costs of power plants, resource potentials and fuel costs, and cross-border transmission capacities to name a few. All data used for the creation of this model, including the data files used to run each scenario, and scripts used for assessing the VRE power potential can be found in the Github repository and Zenodo deposit. VRE Characterization In the following section, we describe the methodology behind the characterization of solar and wind power potentials in the model. We provide details on the approach taken for assessing the resource availability of VRE and their power generation potentials. Resource availability To assess the VRE potentials we first calculate the total land (km 2 ) eligible for wind and solar development within each country. We assume that wind and solar can be developed on the following land use types in the CORINE Land Cover (CLC) database: natural grasslands, moors and heathland, sclerophyllous vegetation, transnational woodland-shrub, bare rocks, sparsely vegetated areas, and burnt areas. The resulting land availability representation is shown in Fig. 2. We refer to the squares representing the eligible land fractions shown in Fig. 2 as grid cells. Each grid cell has a resolution of 30 x 30 km. To obtain hourly time series data on VRE power generation potentials in each of the grid cells we used Atlite [47]. Atlite is a Python package for calculating renewable power potentials and time series. Atlite utilizes the ERA5 dataset, which is why we chose the 30 x 30 km resolution over the higher-resolution CLC data. The total potential for wind and solar development expressed in terms of capacity (MW) is calculated by multiplying the total eligible land by an area-specific maximum installable capacity of 1.7 MW/km2. The maximum installable capacity is based on [48] and is used in similar studies [49]. The figure consists of a technical potential density for installable wind generation capacity, corresponding to 10 MW/km2, and a fraction of 0.17 including the consideration for public acceptance and competing land use, extreme slopes, and unfavourable terrain. The more precise the location analysis is, the higher the area-specific installable capacity number can be used. To determine the potential capacity of wind and solar power in each DRB country, we multiply the capacity per square kilometre by the total eligible land area shown in Fig. 2. VRE generation potentials In addition to maximum resource potentials, the OSeMOSYS model requires time series values for the production potentials. These are used to calculate the CFs for each wind and solar technology included in the model. As land availability affects the distribution of VRE resources, it is necessary to consider the weather variations according to the location of the VRE instalments. We use Atlite [47] to estimate the hourly production potential of wind and solar in each grid cell. Atlite retrieves global historical weather data and converts it into power generation potentials and time series for VRE technologies like wind and solar power. The data used has an hourly temporal resolution obtained from the fifth-generation European Centre for Medium-Range Weather Forecasts (ECMWF) atmospheric reanalysis of the global climate (ERA5 dataset). To obtain a representative year, we first calculated the average hourly power output for wind and solar from the 5 years of 2017-2021. Next, we selected the year that best represented the average, in this case, 2020. We chose to use a historical weather year instead of an average year since the average weather year increases the lower extremes and decreases the higher extremes. To translate the ERA5 weather data to capacity factors, we use the following technology assumptions: for wind, we chose a Siemens SWT-2.3-108 turbine, with a rated power of 2.3 MW and a hub height of 100m [50]. It has a cut-in wind speed of 3 m/s and cut-out speed of 25 m/s. The power curve of the selected wind turbine type is shown in Fig. 3. For solar power, cadmium telluride (CdTe) photovoltaics (PV) solar panels with the orientation 'latitude optimal' were selected in Atlite. The CdTe panel characteristics are provided by [51]. No tracking was included for the solar panels, as Atlite did not include tracking at the point of doing this analysis. Based on power potentials obtained from Atlite for wind power, annual, nationally averaged CFs ranged from 3.3% to 7.8% depending on the country and land layers selected. The numbers cited here were considered small in comparison with the wind power facilities currently in operation in the DRB countries. This reflects poor data quality rather than an absence of wind potential in the region. A geospatial resolution of 30x30 km is not sufficient when estimating areas with potentially high wind potentials that can be observed in smaller areas. Additionally, the ERA5 does not apply correction factors to the wind data. As such, we use a separate approach outlined in Section 3.2.3. Using Atlite in combination with the ERA5 dataset provides the hourly data needed for the model. Underrepresentation of areas with high resource availability is not present in the case of solar power, see Fig. 4, since the irradiation is less impacted by altitude, terrain, and location than wind power is. Wind power generation potentials using GWA To improve the wind power potential estimate for this study, we used the Global Wind Atlas (GWA) version 3 [52]. The GWA is derived from ERA5 reanalysis but uses downscaling processes that result in a final resolution of 250m that considers local topography and terrain features. The tool provides mean CFs for three different turbine classes. The classes available include the IEC1, IEC2, and IEC3 categories for the 2008-2017 period. The IEC classes correspond to a 100m hub height and rotor diameters of 112, 126 and 136m respectively. When compared to existing wind generation in the regions, the CF estimates from the GWA are more accurate than those from Atlite. Annual capacity factors from the GWA used in this study are shown in Fig. 5. For computational considerations, we omit land areas with unfavourable wind conditions, i.e. where wind CFs are less than 10%. Available land CF range, country = eligible land country * share CF range Eq. 1. Calculating available land for a given CF range in each country For wind and solar technologies competing for the available land, we do not impose any restrictions on investments. The model can invest fully in wind, or solar, or a mix of both technologies. Investments in a particular land area are based on the techno-economic characteristics of the power plant types. To differentiate between wind technologies within the CF ranges in the model, we use technology names corresponding to the technology type, i.e. wind, and the CF range. The solar power resource is divided into multiple technologies to keep the names unique and to pair them with the different wind technologies in the userdefined constraints. Each pair of wind and solar technologies for a given CF range, e.g. 30%, is then coupled with a land technology with an availability corresponding to the output from Eq. 1 for that particular CF range. Based on the assumption that the VRE technologies have an area-specific installable capacity of 1.7 MW/km 2 we add a land use of 0.588 km2 per MW installed capacity. This allows the model to invest in wind or solar until the available land corresponding to their CF range is exhausted. When power plant developments have occupied all land within a CF range there can be no additional capacity additions of wind or solar power within that specific CF range. Table A3 in the appendix summarizes the wind and solar technologies considered in the OSeMOSYS model for this analysis. Using the higher average CFs to normalize the time series wind potential data from Atlite is a simplified way of bridging the gap between the two sets of data. What justifies this approach in our case is the nature of the model setup. Existing and planned hydropower or thermal power plants in the model are represented as just one aggregated technology per power generation type, with the HPP cascade and the VRE representation being an exception. By following the above-described methodology, we can include hourly power generation potentials of wind and solar, account for eligible land for VRE deployment in each country, as well as better represent the CFs for wind power based on CF data from the GWA. Temporal resolution Long-term expansion models of the power sector generally consider system developments over several decades, which is a computationally intensive process. A high temporal resolution within the modelling period further increases this computational load. Consequently, power sector and energy models do not typically represent each hour in the year, but rather use representative time periods (e.g., days) [53,54]. This section describes the method used to construct representative time periods, as well as the temporal representation of VRE availability based on climate data. Temporal structure The created OSeMOSYS energy model seeks to optimize power system investment and operational decisions for the years 2020-2050. Each year is represented by fifteen "representative days", where each representative day is assigned a weight corresponding to its relative frequency. The motivation for using representative days is to decrease the size and computational requirements of the overall energy system model. By using fifteen representative days at hourly resolution, each model year consists of 15x24 = 360 time steps (TS). By contrast, if no temporal aggregation technique is employed, each model year would consist of 8760 time steps and the model solve time and memory requirements would be computationally intractable. Fig. 6 shows the load and generation duration curves. The representative days and their respective weights are selected using the agglomerative hierarchical clustering algorithm outlined by Nahmmacher et al. [53] and used more recently in studies by Palmer-Wilson et al. [55] and Keller et al. [56,57]. Hourly data for electricity demand, wind power availability, and solar power availability across the DRB countries is collected and normalized for a set of historical days 3 . A hierarchical clustering algorithm is then used to group the historical days into 15 clusters, where days within a cluster are broadly similar in their load, wind, and solar characteristics. For each cluster, the day closest to the cluster's centroid is selected as the representative day and is assigned a weight proportional to the cluster's relative size. Finally, the load, wind, and solar time series for each representative day are scaled to match correct annual averages. Fig. 6 shows the fitment of the 15 TS approximation compared to the input data. A more detailed overview of the methodology can be found in [53]. Water availability for HPPs in the DRB is considered an input that constrains the hydropower cascade. Power imports and exports include cross-border power exchanges between the countries of the DRB, as well as from adjacent countries, including Croatia, Hungary, and Italy among others. For each country, the resources, power generation technologies, and losses related to transmission and distribution are separately accounted for. Hydropower cascade Fig. 8 illustrates in more detail the DRB hydropower cascade depicted in blue in Fig. 7. We derive data for the cascade water inputs from the HypeWeb model, more specifically, the HYPE model for Europe (E-HYPE) [58]. From the E-HYPE 3.1.1 model version we calculate the average daily river discharge during the 1981-2010 period, for each of the following rivers: Ćehotina, Lim, Piva, Tara, and Uvac. The resolution of this data corresponds to daily flows. This entails that flows within each TS are constant and equal to the daily average based on the E-HYPE model data. Flows change between the 15 selected TS. Water enters the cascade through the upstream river segments and the catchments. River segment capacities and water flows are input parameters. These capacities are fixed to the maximum average flow of each day. The WFDEI dataset of historical precipitation and temperature is used as forcing data in this simulation [58]. The capacities do not vary across the different years in the model. Reference energy system Explored development pathways -a scenario analysis. We created three scenarios to explore potential development pathways for the power sectors of the DRB countries. A Reference scenario is presented first to represent a baseline, followed by an Emission Limit Scenario to explore achieving net-zero emissions by 2050. Finally, with the Agricultural land for wind power scenario, we examine what effect wind power development on the DRB countries' vast agricultural land could have on the power sector. Each of the three scenarios is explored with two variations: high-and low-cost trade alternatives. Reference scenario (REF). The reference scenario serves as the baseline for the scenario analysis. All other scenarios are compared with results from this baseline. The model may invest in technologies that are currently employed in the power sectors of the DRB countries. These include coal, solar, hydro, and wind power with several exceptions. As the government of Montenegro has already cancelled its last coal project and is presently focused on accelerating the retirement of its remaining CFTPP [59], no new CFTPP projects are permitted in Montenegro in this scenario. Since projects concerning planned HPPs in the DRB are uncertain, there is no expansion of the hydropower cascade in this scenario. This scenario aims to provide insights into the development of the power sector based on current technoeconomic parameters obtained from literature and consultations with local stakeholders, without any additional policy measures. The global assumptions are consistent across all scenarios and provided in Table A1. Emission limit scenario (EL). Assuming future EU integration of the DRB countries, this scenario aims to provide insight into the development of their power sectors if emissions are restricted to the EU's 2030 and 2050 GHG reduction targets. Compared to 1990 emissions levels, the target values correspond to a 55% reduction by 2030 and a net-zero emission level by 2050. The applied emission limit is shown in Table A2. Agricultural land for wind power scenario (AG). Here we relax the upper bound on wind capacity by making possible the development of wind on agricultural land. As a percentage of their total land area, Bosnia and Herzegovina and Montenegro have a modest 12% share of agricultural land, while Serbia has 40%. This scenario aims to inform us of the role that wind power plants (WPPs) on agricultural land may play in the decarbonization of the power sectors in the DRB countries. In this section, we present the key findings of the analysis. The results are reported in an aggregate form, combining the findings for all three DRB countries. We provide answers to what the potential of VRE in the DRB countries are, their role in supporting the transition to net-zero by 2050, and the impact new VRE developments can have on the existing hydropower cascade in terms of power generation and cost competitiveness. VRE Potential Section 3.2 of this paper describes the methodology for estimating the potential of VRE technologies for DRB countries. The findings summarized in Table 2 show that the DRB countries have a combined potential of 94.4 GW for wind and solar power. The majority of VRE potential in Bosnia and Herzegovina and Montenegro is located on lands that exclude agricultural lands. The distribution in Serbia is vastly different, with wind power potentials on agricultural lands accounting for over 80% of its total VRE potential. CF's for approximately half of the wind power potential on agricultural land in Bosnia and Herzegovina and Montenegro are within the 15% CF range, while most of the wind potential on agricultural land in Serbia shows CF's of around 35%. Fig. 10. The added solar capacity drives an increase in power generation from the HPP cascade by 1515 GWh during the modelling period. The reason for increased power generation from the HPP cascade is that increased shares of solar are coupled with higher shares of CFTPP generation, reducing the share of wind, and hydropower outside of the basin. It is noteworthy that 92.2% of the increased power generation from the HPP cascade occurs during hours when solar power is not available. Additionally, Fig. 10 shows that the model invests in power imports during time steps with low wind and solar availability. Changing power import and export prices alters the investment and operational strategy of the model. The change is considerable, with imports decreasing under the high-cost scenario alternative, while investments in VRE technologies and CFTPPs increase by over 3 GW respectively. Fig. 11 (b). They amount to 15.1 GW or 66% of the total new capacity. Due to high trade costs, power generation capacity investments are favoured over imports, as highlighted in section 4.2. This explains the increase in capacity additions between Fig. 11 (a) and (b). We observe even greater additions of VRE in the EL scenario, 16.5 and 13.5 GW for the high-and low-cost alternatives, under which coal must be phased out by 2050. 12 shows the difference in capacity additions for the AG and EL scenarios when compared to the REF scenario. It includes differences for both the high-and low-cost trade variations of each respective scenario. In Fig. 12 Developments of the power supply across explored scenarios for the DRB countries In Fig. 13 (b), higher coal shares facilitate the expansion of solar, corresponding to 7.8 GW by 2050. Investments in solar start five years earlier compared to the low-cost trade alternative of the REF scenario. Fig. 13(a) to 20% in Fig. 13 (b). In the (b), (d), and (f) subplots of Fig. 13, the total power generation is higher. The excess electricity is in these cases exported to countries bordering the DRB countries. Part of the increased power generation comes from thermal power, which constitutes a higher share of the power supply mix under the high-cost trade alternatives of the presented scenarios. Thermal power is part of the power supply in all scenarios except for the EL in 2050. Investments in solar power are both greater and appear sooner when the cost of trade is high, as shown in the (b), (d) and (f) subplots when compared to the low-cost trade alternatives. Power generation from VRE sources by 2050 are the lowest in the REF scenario with low trade cost, corresponding to 51% of total power generated within the DRB countries. The highest shares of VRE in the power supply are observed in the EL scenario with high cost, where 73% of the total power generation is VRE based. Overall, the emissions associated with power generation are higher in the latter alternative, as shown in Fig. 14 (b). In this section, we discuss and interpret the results presented in section 4. The findings are discussed in terms of potential implications on the power sector developments and their relation to the purpose of the study and the research questions posed. One of the aims of this study was to assess the VRE potential within the DRB countries. The results shown in section 4.1 state an estimated potential of wind and solar power to be 94.4 GW. The breakdown of the total VRE capacity potentials among the DRB countries shows that for Bosnia and Herzegovina, Montenegro, and Serbia the combined VRE potential is 23.1, 10.3, and 61 GW respectively. These potentials far exceed the current total installed capacity in 2020 of the DRB countries which are 4.1, 1, and 7.4 GW respectively [31,32,60]. Current installed capacities of wind and solar power in DRB countries are less than 1 GW as of 2022 [31,32,60]. Compared to earlier assessments of VRE potentials shown in section 1, the results of this study show VRE capacity estimates that are 68% higher compared to estimates from IRENA [11] and 287% higher compared to SEERMAP reports [12][13][14]. The wind potential in Montenegro is according to Table 2 up to 8.5 GW if all available land is used for wind power development with no solar, and where agricultural land is available for wind power expansion. This figure far exceeds earlier estimates and assumptions of 400 MW wind potential in Montenegro made by [6][7][8][9]. Previous estimates from IRENA [11] stated a technical potential of wind power close to 3 GW in Montenegro, considerably greater compared to earlier studies [9], but less than the 8.5 GW estimate presented in this study. This shows that the capacity estimation of wind potentials has increased over time, which is to be expected given the rapid development of wind turbines. [61]. We thus expect the potential of wind and solar power to be sufficient in supporting the transition to net-zero emissions from the power sectors of the DRB countries. However, the lower CF ranges for wind power and parts of the solar potential are not cost-competitive when compared to hydropower and imports across the scenarios. As nearly two-thirds of the VRE potentials are located on agricultural land, governments need to develop policies allowing the deployment of WPPs in these areas. Energy security has been a frequent argument made by proponents of domestic coal resources in the context of the energy transition away from fossil fuels. The findings of this study highlight that the DRB countries have other, more environmentally friendly resource potentials that could satisfy their power demand without adversely affecting their energy security. VRE technologies play a crucial role in the future power sectors of the DRB countries. Findings presented in section 4.1 highlight that the share of VRE compared to total new capacity additions correspond to close to 70% of all new capacity additions. This in turn increases the share of power generated from VRE sources as shown in Fig. 13. While replacing thermal capacities, the model invests in both wind and hydro since their joint availability profiles closely match the specified demand profile. In some of the 15 representative days the capacity factor of both wind and solar is low, while the demand is comparatively high. This makes a combination of wind and solar less cost-optimal compared to other power supply mixes since the model is otherwise forced to import large quantities to satisfy the unmet demand. We observe this dynamic in all graphs in Fig. 12, but most clearly in Fig. 12 (d), in which we illustrate that the target of net-zero emissions requires the removal of coal from the power supply. The high cost of power imports and exports drives the model to invest in additional capacity to reduce its import dependence while exporting with high profits. We note that the relationship between the power supply alternatives included in the model could be different in a setting that includes different storage alternatives. Such a model would enable excess solar power to be stored in either pumped hydro storage (PHS) or other forms of power storage such as batteries. Having this added layer of flexibility as to when to use the generated power would reduce the need for the model to couple power supply alternatives solely based on their power availability profiles. We thus expect the choices of investment to be more flexible than the results suggest in Fig. 12. Increasing the cost of power imports and exports results in changes within the power sector development. The results in Fig. 13 indicate an accelerated investment rate in solar PV, increased exports from the DRB countries, as well as more investments in coal-fired thermal power plants compared to the scenarios with lower costs of power imports and exports. The findings suggest a more rapid development of VRE projects in the DRB countries considering the current energy crisis in Europe that has increased the cost of electricity across the continent. The DRB countries can reduce their vulnerability to imports at high prices by expanding their capacity of VRE technologies and hydropower. Additionally, excess power generation from these sources in times of low domestic demand can be used for exports at a high cost to neighboring EU countries, such as Italy, Hungary, and Greece. Moreover, the potential introduction of the Carbon Border Adjustment Mechanism (CBAM) in the EU is another reason for increased investments in renewables. The CBAM would entail additional taxation on power exported to the EU from the Western Balkan countries. This can result in new CFTPPs becoming stranded assets since their cost competitiveness would be compromised by the additional CBAM taxation. The fact that the DRB countries are net exporters is confirmed by the model results, which show net exports corresponding to 4 to 20% of the total power generation. The power sectors of the DRB countries could in the case of an introduced CBAM differ from the results shown in Fig. 13 by not having low-cost imports available or high revenues from exports due to the power being generated by CFTPPs. In that situation the DRB countries could find themselves becoming import dependent, while paying a high cost of electricity, leaving fewer resources for investments in the development and maintenance of their power sectors. Fig. 14 shows that CO2 emissions are the highest in the AG scenario under the low-cost of trade, while the REF scenario is the highest under the highcost trade alternative. We can observe that the higher cost of trade results in higher emissions. This is in line with findings from Fig. 13 where we observed continued use of coal power plants under the high-cost trade alternatives of the scenarios. Capacity additions of renewable energy sources shown in Fig. 11 and Fig. 12 are the key reason for the observed CO2 emission reductions observed in Fig. 14 for the corresponding scenarios. The findings of this study suggest that the expansion of renewables in favour of CFTPPs is the main driving factor of the CO2 mitigation observed in Fig. 14. Since the time steps are not sequential in the model, we cannot assess the effect of seasonal variations of water availability on storage levels under the explored scenarios. We can however compare the total water levels in each dam between different scenarios. The results suggest that scenarios with higher shares of SPPs and CFTPPs utilize the HPP cascade for power generation to a larger extent compared to scenarios where WPPs and HPPs outside the basin are the main capacity additions. As 92.2% of the increased power generation from the HPP cascade occurs during hours when solar power is not available, this indicates increased short-term balancing of renewables by the cascade, moving from baseload to more responsive power generation patterns. The finding highlights that potential expansion of the HPP cascade can enable larger shares of solar power, resulting in high shares of renewables in the power supply coupled with balancing capabilities of hydropower. Having among the largest shares of coal-based power generation in Europe, the DRB countries of Bosnia and Herzegovina, Montenegro, and Serbia must take action to meet the EU's goal of net zero emissions by 2050. In this paper, we created a power sector model for the DRB countries with a scenario analysis exploring different development pathways. Inputs to the model consist of the latest available data on demands, future demand projections, costs and characteristics of current and future power-generating technologies considered. We present a novel approach for assessing the VRE resource potentials by combining timeseries data on availabilities from Atlite and the ERA5 dataset, with high-resolution data obtained from the GWA. The findings from this approach indicate a capacity potential of 94.4 GW of VRE technologies in the DRB countries, of which 59.3 GW or 63% relate to wind power. When compared to the current installed capacity within the DRB countries is 12.5 GW, of which 627 MW are wind power, we observe that the potential for VRE deployment is largely untapped. According to the results, the VRE potential is significantly higher than previous assessments have shown, with increases ranging from 68% to 287%. Findings from the Emission Limit scenario where net-zero emissions are expected by 2050 show investments in wind and solar power corresponding to 10.7 and 5.8 GW respectively. These investments constitute 17% of the assessed VRE potential presented in this study. Hence, the regional potential of VRE technologies is sufficient to decarbonize the power sector under the demand assumptions used in the model. Wind and solar power play a vital role in CO2 mitigation from the power sectors of the DRB countries. The share of these technologies ranges from 51% in the REF scenario with low cost of trade, to 73% in the EL scenario with high cost of trade. VRE expansion in the DRB countries has a limited effect on power generation from the HPP cascade since the currently installed capacities of the eight HPPs in the DRB have no capital cost expenditure associated with them in the model. However, the results also indicate increases in the power output from the HPP cascade corresponding to 1515 GWh for the modelling period under the REF scenario where higher shares of solar power are present. As 92.2% of the increased power generation from the HPP cascade occurs when no solar is available, the HPP cascade increasingly acts as a short-term balancing option for VRE technologies, moving from baseload to more responsive power generation patterns. The DRB countries have sufficient VRE potentials which are underexploited as of today. The potential of these technologies is sufficient to support the transition to net-zero by 2050, in which the role of VRE technologies is significant in terms of power supply to meet the demand and CO2 emission reductions. Failing to act regarding the development of renewables could lead to stranded assets in case of a CBAM introduction, while under capacity could be costly, especially given the current costs of cross-border power trade in Europe and the risk of reduced import availability from EU countries surrounding the Western Balkans. Not aligning with the commitments undertaken in the Sofia Declaration could also hinder the process of accession to the EU. In this section, we highlight the limitations of this paper, including potential topics of future research needs that could expand the work presented in this paper. The presented assessment of land availability for VRE developments was limited to utilityscale technology options. We did not consider rooftop solar, which could be utilized in urban settings, nor solar on agricultural land. Since costs relating to the expansion of transmission and distribution lines, distances from the grid, slopes or difficult to reach areas were not included when assessing the VRE capacity potential, the total calculated VRE capacity presented in this paper may be less utilized than the results suggest. In contrast, improvements in efficiency and capacity factors of VRE technologies, which are likely to improve their cost-competitiveness, are not included. Considering that the model developed for DRB countries is intended to inform long-term energy infrastructure investments, and not site-specific power generation projects, future research could combine the presented methodology for estimating VRE potentials with site-specific analyses. Given the large utilization of VRE technologies proposed by the results of this study, an important factor to consider is future additions of storage options. This can be done by adding representations of battery storage for solar power or pumped hydro storage. In the created model, the power demand is a driving factor for the expansion of the power sector. We use data from the current demand profiles and demand projections based on projections made by the local transmission system operators. An interesting point to consider going forward is the impact of demand reductions based on energy efficiency measures. Energy efficiency in the DRB region can be significantly improved and reducing demand in turn reduces the need for new capacity additions. Energy efficiency improvements in the Western Balkans have over the past decade been the basis for financial support in the region, from the Regional Energy Efficiency Programme launched in 2012 [62], to the Energy Support Package [63] put forward in 2022 comprising of 1 billion Euro toward diversity of energy supplies, increasing renewable energy and energy efficiency. Cross-border trade is present in all explored scenarios. As highlighted in this paper, the DRB countries could find themselves in a situation where they become increasingly importdependent in case of energy shortages due to decommissioning of thermal power, or by delayed investments in low-carbon technologies. Imports could then not only be more expensive but also not available due to the disruptions of the power markets in Europe caused by the ongoing conflict in Ukraine. The impact of import availability on the expansion of the power sector of the DRB countries may be better understood by further modelling of the region with different levels of import availability. Fig. 1 . 1The Drina River Basin (outlined in orange) modelled hydropower plants within the basin (blue squares) and the DRB countries. Fig. 2 . 2Eligible land fraction for wind and solar power plants (excluding agricultural land) for the DRB countries. The axis labels show longitude (y-axis) and latitude (x-axis). The resolution is a 30km grid. Fig. 3 . 3Power curve of the selected wind power plant in Atlite. Used for obtaining time series data on power generation potentials for the eligible land in the cut-out of the DRB countries. Fig. 4 . 4Map of solar capacity factors for a fixed panel by grid cell for the DRB countries, generated using Atlite. Fig. 5 . 5Capacity Factors for IEC Class III wind power plants for the DRB countries from GWA. The output from the VRE characterisation shown in Fig. 4 and Fig. 5 comprises of geospatially explicit CFs for solar and wind power. In addition to these, we obtain time-series values of capacity factors from Atlite. The outputs are used as inputs for the representation of wind and solar power in the OSeMOSYS energy model.In OSeMOSYS we define technologies with certain techno-economic characteristics, including total capacity potentials and time-series values for CFs. To represent the wind power in OSeMOSYS in a computationally manageable way while covering the broad range of CFs shown inFig. 5, we assume four CF ranges. These are 10-20%, 20-30%, 30-40%, and 40% or higher. We use these ranges to calculate the average CF for each CF range and country. Using the average CF of each range, we scale the time-series values obtained from Atlite to match the four new averages. These adjusted time-series CFs are then assigned to four wind technology representations in each country in the OSeMOSYS model. Following the calculation of the different time-series CFs we calculate the share of the DRB countries area which the four selected CF ranges occupy. Using Python, we perform stratified sampling on the output shown inFig. 5using 10,000 points per grid cell for all grid cells shown inFig. 2. With the shares, we then calculated the land availability for wind power for each CF range and unit of eligible land fromFig. 2according to equation 1. Fig. 6 . 6Load and generation duration curves for demand, solar and wind of 15 TS approximation compared to aggregated 8760 TS input data. Fig. 7 7illustrates the Reference Energy System (REF). The REF is a network representation of technical activities required to supply electricity to meet the final demand. It shows the connections between supply and demand, including land and water resources, which are included in the OSeMOSYS energy model. Fossil fuels considered are coal and natural gas. Fossil fuel resources are consumed by coal and gas power plants in proportion to their power generation output. The land resource obtained using Atlite and the CLC is fed into the model as an input. Land is utilized by wind and solar technologies in proportion to the increase in installed capacity, representing the use of land for the construction of energy infrastructure. Fig. 7 . 7Illustration of the reference energy system for each of the DRB countries, including demands, trade, and power sector infrastructure3.3.3 Fig. 8 . 8Structure of the HPP cascade. Boxes represent technologies in OSeMOSYS, while connecting lines represent commodities. In this figure, the commodities are water flowing between the power plants, spillways, dams, catchments, and river segments. Catchments in this representation are aggregations of small tributaries or streams entering the DRB cascade. Fig. 9 9illustrate the water levels in the HPP cascade for the low-cost trade alternative REF scenario. On the y-axis, the storage level is expressed in million cubic meters (MCM), while the x-axis represents the 360 time steps. The dark blue line represents the average storage level for each time step over the 2020-2050 period. With a 95% confidence interval, the bright blue areas surrounding the mean value indicate the minimum and maximum values. Fig. 9 . 9HPP cascade water levels for the REF scenario in each time step for the 2020-2050 period Under the assumption of high cost of trade, investments in solar power increase by 37%, or 2.1 GW compared to the low cost of trade alternative of the REF scenario. These capacity additions result in higher levels of power generation from solar power under the high cost of trade alternative for the REF scenario, as shown in Fig. 10 . 10Daily power supply for each of the 15 representative days, based on the REF scenario with different power trade costs for the year 2050.4.3 Capacity additions to the power sectors of the DRB countries Fig. 11 (a) shows the capacity additions under the REF scenario with low-cost trade by 2030, 2040 and 2050. VRE investments correspond to 12.2 GW by 2050, or 68% of all new capacity additions. Capacity additions under the high-cost trade alternative of the REF scenario are shown in Fig. 11 . 11New cumulative capacity additions under the high-and low-cost trade alternatives of the REF scenario. Fig. 12 . 12Difference graph of cumulative capacity expansions between the explored scenarios and the high-and low-cost trade alternatives of the REF scenario. Values given in GW; negative values indicate lower capacity compared to the REF scenario. Fig. Fig. 12 shows the difference in capacity additions for the AG and EL scenarios when compared to the REF scenario. It includes differences for both the high-and low-cost trade variations of each respective scenario. In Fig. 12 (a) we observe higher capacity additions for the AG scenario compared to the REF. The reason is additional land availability for investments in high CF wind. Having extra capacity available decreases imports in this scenario when compared to the REF. The greatest difference can be observed in Fig. 12 (c) and (d) for 2050, where a combination of wind and hydropower capacity additions are added to compensate for the total decommissioning of CFTPPs. Due to the different generation profiles of power Fig. 13 13shows the power supply and the power sector expansion across the scenarios. The REF, AG, and EL scenarios are shown in different rows, while the left and right columns in the figure represent the low-cost and high-cost alternatives for each of the explored scenarios. The subplots (a), (c), and (d) show higher levels of power imports and rapid decommissioning of coal-fired thermal power plants. Power exports from the DRB countries are lower compared to the (b), (d) and (f) subplots since the low cost of export does not stimulate the model to invest in additional power-generating capacities to be used for exports. Net exports by 2050 range from 4% in Fig. 13 . 13Power supply in the DRB countries, their imports and exports, all scenarios 2020-2050. with power generation in the DRB countries The emissions shown in Fig. 14 represent CO2 emissions from the power sector. The emissions include the direct emissions from burning coal for power generation. A sharp decrease in emissions can be observed under all scenarios during the first five years of the modelling period. The main reason is the phase-out of inefficient CFTPPs. The results also indicate that the EL scenario with its high-and low-cost trade alternatives is the only scenario where netzero emissions are reached by 2050. Subplot (a) illustrates CO2 emissions under the low cost of trade alternatives, while subplot (b) represents high-trade cost alternatives of the explored scenarios. Fig. 14 . 14CO2 emissions from power generation by scenario in the DRB countries. Values are provided in MtCO2. Subplot (a) represents the low-cost while (b) represents high-cost trade alternatives for each scenario. The most rapid expansion in terms of capacity and power generation according to the findings presented in section 4 relates to wind power, which follows the power sector development trends of the past years in the DRB countries. Wind power developments in the DRB countries started in 2017. Plants such as Mesihovina, Podveležje, and Jelovača in Bosnia and Herzegovina, Krnovo and Možura in Montenegro, as well as Čibuk 1 in Serbia, are examples of the latest additions to the power sectors of the DRB countries. Currently, solar power development has not occurred in the region. The model results for the REF scenario support this investment trend under the lowcost trade alternative, where no solar capacity additions are observed until 2030. However, under the EL scenario with a high cost of trade investments in solar are observed as early as 2021. and Serbia have a GDP per capita of approximately nine thousand while Bosnia and Herzegovina has seven thousand[23]. The average GDP per capita in the EU is 38 thousand[24], significantly higher than the GDP per capita for the DRB countries.Similarly, power consumption per capita in the DRB countries is lower compared to the EU average. In Bosnia and Herzegovina, Montenegro, and Serbia the consumption is 3.3 MWh [25], 5.7 MWh [26], and 4.1 MWh [27], respectively. By contrast, the EU-27 power consumption per capita is 6.4 MWh Bosnia and Herzegovina, Montenegro, and Serbia are candidate countries for EU accession, with Bosnia and Herzegovina obtaining the status as at December 2022[36]. As EU member states candidates, the DRB countries have pledged to align with the EU Climate Law and the EU Emissions Trading Scheme (EU-ETS), and to increase their share of renewable energy sources by the signing of the Sofia Declaration in 2020[37]. The alignment with EU policy entails that by the time the countries become EU member states, their climate and energy policies must align with the European Green Deal, which implies a 55% reduction in emissions by 2030 compared to 1990 levels and achieving climate neutrality by 2050[38].As part of the efforts to combat climate change, the DRB countries have all submitted their Nationally Determined Contributions (NDCs) to the United Nations Framework Convention on Climate Change. Each country has submitted updated versions of their NDCs, increasing their ambitions compared to their first NDC submission.Table 1summarizes the pledged decreases of Greenhouse Gas (GHG) emissions submitted by the DRB countries in their NDCs.Table 1. Pledged emission reductions by the DRB countries by 2030 relative to 1990 levels.2.2 Aligning with EU climate goals 2.3 Nationally Determined Contributions as part of the Paris Agreement In addition to the 2030 goals, Bosnia and Herzegovina is committed to reducing GHG emissions by 61.7% unconditionally, and 65.6% conditionally by 2050 in comparison to 1990 levels. As stated, the contributions do not align with the EU targets, and ambitions must be raised to reach climate neutrality by 2050. Country First NDC [%] Updated NDC [%] By 2030 By 2030 Bosnia and Herzegovina 18 and 20% (conditional and unconditional) -36.8 and -33.2 (conditional and unconditional) Montenegro -30% -35% Serbia -9.8% -33.3% 2.4 Renewable resource potentials Table 2 . 2Wind and solar power potentials of the DRB countriesCountry Average CF range wind [%] Wind or solar power potential [GW] on shared land areas Wind power potential on agricultural land [GW] Solar power potential [GW] on land with low CF for wind power (<10%) Bosnia and Herzegovina 15.6 2.7 4.2 2.1 25.0 3.8 2.9 - 34.8 3.8 0.9 - 45.3 2.3 0.4 - Subtotal 12.6 8.4 2.1 Montenegro 15.2 2.1 0.8 1.8 24.8 2.2 0.6 - 34.5 1.5 0.4 - 45.0 0.7 0.2 - Subtotal 6.5 2.0 1.8 Serbia 15.9 2.5 5.2 1.6 24.9 3.9 15.4 - 34.8 3.4 26.7 - 43.8 0.7 1.6 - Subtotal 10.5 48.9 1.6 Total - 29.6 59.3 5.5 This designation is without prejudice to positions on status and is in line with UNSCR 1244 and the ICJ Opinion on the Kosovo Declaration of Independence. 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[ "Constrained Policy Optimization", "Constrained Policy Optimization" ]	[ "Joshua Achiam ", "David Held ", "Aviv Tamar ", "Pieter Abbeel " ]	[]	[]	For many applications of reinforcement learning it can be more convenient to specify both a reward function and constraints, rather than trying to design behavior through the reward function. For example, systems that physically interact with or around humans should satisfy safety constraints. Recent advances in policy search algorithms(Mnih et al., 2016;Schulman et al., 2015;Lillicrap et al., 2016;Levine et al., 2016)have enabled new capabilities in highdimensional control, but do not consider the constrained setting.We propose Constrained Policy Optimization (CPO), the first general-purpose policy search algorithm for constrained reinforcement learning with guarantees for near-constraint satisfaction at each iteration. Our method allows us to train neural network policies for high-dimensional control while making guarantees about policy behavior all throughout training. Our guarantees are based on a new theoretical result, which is of independent interest: we prove a bound relating the expected returns of two policies to an average divergence between them. We demonstrate the effectiveness of our approach on simulated robot locomotion tasks where the agent must satisfy constraints motivated by safety.Furthermore, the bounds are tight (when π = π, all three expressions are identically zero).Before proceeding, we connect this result to prior work. By bounding the expectation E s∼d π [D T V (π \|\|π)[s]] with max s D T V (π \|\|π)[s], picking f = V π , and bounding π V π	null	[ "https://arxiv.org/pdf/1705.10528v1.pdf" ]	10,647,707	1705.10528	209296950c358748aaa9be64cd2e8be68a52e7cc	Constrained Policy Optimization Joshua Achiam David Held Aviv Tamar Pieter Abbeel Constrained Policy Optimization For many applications of reinforcement learning it can be more convenient to specify both a reward function and constraints, rather than trying to design behavior through the reward function. For example, systems that physically interact with or around humans should satisfy safety constraints. Recent advances in policy search algorithms(Mnih et al., 2016;Schulman et al., 2015;Lillicrap et al., 2016;Levine et al., 2016)have enabled new capabilities in highdimensional control, but do not consider the constrained setting.We propose Constrained Policy Optimization (CPO), the first general-purpose policy search algorithm for constrained reinforcement learning with guarantees for near-constraint satisfaction at each iteration. Our method allows us to train neural network policies for high-dimensional control while making guarantees about policy behavior all throughout training. Our guarantees are based on a new theoretical result, which is of independent interest: we prove a bound relating the expected returns of two policies to an average divergence between them. We demonstrate the effectiveness of our approach on simulated robot locomotion tasks where the agent must satisfy constraints motivated by safety.Furthermore, the bounds are tight (when π = π, all three expressions are identically zero).Before proceeding, we connect this result to prior work. By bounding the expectation E s∼d π [D T V (π \|\|π)[s]] with max s D T V (π \|\|π)[s], picking f = V π , and bounding π V π Introduction Recently, deep reinforcement learning has enabled neural network policies to achieve state-of-the-art performance on many high-dimensional control tasks, including Atari games (using pixels as inputs) (Mnih et al., 2015;, robot locomotion and manipulation (Schulman et al., 2015;Levine et al., 2016;Lillicrap et al., 2016), and even Go at the human grandmaster level . In reinforcement learning (RL), agents learn to act by trial and error, gradually improving their performance at the task as learning progresses. Recent work in deep RL assumes that agents are free to explore any behavior during learning, so long as it leads to performance improvement. In many realistic domains, however, it may be unacceptable to give an agent complete freedom. Consider, for example, an industrial robot arm learning to assemble a new product in a factory. Some behaviors could cause it to damage itself or the plant around it-or worse, take actions that are harmful to people working nearby. In domains like this, safe exploration for RL agents is important (Moldovan & Abbeel, 2012;Amodei et al., 2016). A natural way to incorporate safety is via constraints. A standard and well-studied formulation for reinforcement learning with constraints is the constrained Markov Decision Process (CMDP) framework (Altman, 1999), where agents must satisfy constraints on expectations of auxilliary costs. Although optimal policies for finite CMDPs with known models can be obtained by linear programming, methods for high-dimensional control are lacking. Currently, policy search algorithms enjoy state-of-theart performance on high-dimensional control tasks (Mnih et al., 2016;Duan et al., 2016). Heuristic algorithms for policy search in CMDPs have been proposed (Uchibe & Doya, 2007), and approaches based on primal-dual methods can be shown to converge to constraint-satisfying policies (Chow et al., 2015), but there is currently no approach for policy search in continuous CMDPs that guarantees every policy during learning will satisfy constraints. In this work, we propose the first such algorithm, allowing applications to constrained deep RL. Driving our approach is a new theoretical result that bounds the difference between the rewards or costs of two different policies. This result, which is of independent interest, tightens known bounds for policy search using trust regions (Kakade & Langford, 2002;Pirotta et al., 2013;Schulman et al., 2015), and provides a tighter connection between the theory and practice of policy search for deep RL. Here, we use this result to derive a policy improvement step that guarantees both an increase in reward and satisfaction of constraints on other costs. This step forms the basis for our algorithm, Constrained Policy Optimization (CPO), which computes an approximation to the theoretically-justified update. In our experiments, we show that CPO can train neural network policies with thousands of parameters on highdimensional simulated robot locomotion tasks to maximize rewards while successfully enforcing constraints. Related Work Safety has long been a topic of interest in RL research, and a comprehensive overview of safety in RL was given by (García & Fernández, 2015). Safe policy search methods have been proposed in prior work. Uchibe and Doya (2007) gave a policy gradient algorithm that uses gradient projection to enforce active constraints, but this approach suffers from an inability to prevent a policy from becoming unsafe in the first place. Bou Ammar et al. (2015) propose a theoretically-motivated policy gradient method for lifelong learning with safety constraints, but their method involves an expensive inner loop optimization of a semi-definite program, making it unsuited for the deep RL setting. Their method also assumes that safety constraints are linear in policy parameters, which is limiting. Chow et al. (2015) propose a primal-dual subgradient method for risk-constrained reinforcement learning which takes policy gradient steps on an objective that trades off return with risk, while simultaneously learning the trade-off coefficients (dual variables). Some approaches specifically focus on application to the deep RL setting. Held et al. (2017) study the problem for robotic manipulation, but the assumptions they make restrict the applicability of their methods. Lipton et al. (2017) use an 'intrinsic fear' heuristic, as opposed to constraints, to motivate agents to avoid rare but catastrophic events. Shalev-Shwartz et al. (2016) avoid the problem of enforcing constraints on parametrized policies by decomposing 'desires' from trajectory planning; the neural network policy learns desires for behavior, while the trajectory planning algorithm (which is not learned) selects final behavior and enforces safety constraints. In contrast to prior work, our method is the first policy search algorithm for CMDPs that both 1) guarantees constraint satisfaction throughout training, and 2) works for arbitrary policy classes (including neural networks). Preliminaries A Markov decision process (MDP) is a tuple, (S, A, R, P, µ), where S is the set of states, A is the set of actions, R : S × A × S → R is the reward function, P : S ×A×S → [0, 1] is the transition probability function (where P (s \|s, a) is the probability of transitioning to state s given that the previous state was s and the agent took action a in s), and µ : S → [0, 1] is the starting state distribution. A stationary policy π : S → P(A) is a map from states to probability distributions over actions, with π(a\|s) denoting the probability of selecting action a in state s. We denote the set of all stationary policies by Π. In reinforcement learning, we aim to select a policy π which maximizes a performance measure, J(π), which is typically taken to be the infinite horizon discounted total return, J(π) . = E τ ∼π [ ∞ t=0 γ t R(s t , a t , s t+1 )]. Here γ ∈ [0, 1) is the discount factor, τ denotes a trajectory (τ = (s 0 , a 0 , s 1 , ...)), and τ ∼ π is shorthand for indicating that the distribution over trajectories depends on π: s 0 ∼ µ, a t ∼ π(·\|s t ), s t+1 ∼ P (·\|s t , a t ). Letting R(τ ) denote the discounted return of a trajectory, we express the on-policy value function as V π (s) . = E τ ∼π [R(τ )\|s 0 = s] and the on-policy action-value function as Q π (s, a) . = E τ ∼π [R(τ )\|s 0 = s, a 0 = a]. The advantage function is A π (s, a) . = Q π (s, a) − V π (s). Also of interest is the discounted future state distribution, d π , defined by d π (s) = (1−γ) ∞ t=0 γ t P (s t = s\|π). It allows us to compactly express the difference in performance between two policies π , π as J(π ) − J(π) = 1 1 − γ E s∼d π a∼π [A π (s, a)] ,(1) where by a ∼ π , we mean a ∼ π (·\|s), with explicit notation dropped to reduce clutter. For proof of (1), see (Kakade & Langford, 2002) or Section 10 in the supplementary material. Constrained Markov Decision Processes A constrained Markov decision process (CMDP) is an MDP augmented with constraints that restrict the set of allowable policies for that MDP. Specifically, we augment the MDP with a set C of auxiliary cost functions, C 1 , ..., C m (with each one a function C i : S × A × S → R mapping transition tuples to costs, like the usual reward), and limits d 1 , ..., d m . Let J Ci (π) denote the expected discounted return of policy π with respect to cost function C i : J Ci (π) = E τ ∼π [ ∞ t=0 γ t C i (s t , a t , s t+1 )]. The set of feasible stationary policies for a CMDP is then Π C . = {π ∈ Π : ∀i, J Ci (π) ≤ d i } , and the reinforcement learning problem in a CMDP is π * = arg max π∈Π C J(π). The choice of optimizing only over stationary policies is justified: it has been shown that the set of all optimal policies for a CMDP includes stationary policies, under mild technical conditions. For a thorough review of CMDPs and CMDP theory, we refer the reader to (Altman, 1999). We refer to J Ci as a constraint return, or C i -return for short. Lastly, we define on-policy value functions, actionvalue functions, and advantage functions for the auxiliary costs in analogy to V π , Q π , and A π , with C i replacing R: respectively, we denote these by V π Ci , Q π Ci , and A π Ci . Constrained Policy Optimization For large or continuous MDPs, solving for the exact optimal policy is intractable due to the curse of dimensionality (Sutton & Barto, 1998). Policy search algorithms approach this problem by searching for the optimal policy within a set Π θ ⊆ Π of parametrized policies with parameters θ (for example, neural networks of a fixed architecture). In local policy search (Peters & Schaal, 2008), the policy is iteratively updated by maximizing J(π) over a local neighborhood of the most recent iterate π k : π k+1 = arg max π∈Π θ J(π) s.t. D(π, π k ) ≤ δ,(2) where D is some distance measure, and δ > 0 is a step size. When the objective is estimated by linearizing around π k as J(π k ) + g T (θ − θ k ), g is the policy gradient, and the standard policy gradient update is obtained by choosing D(π, π k ) = θ − θ k 2 (Schulman et al., 2015). In local policy search for CMDPs, we additionally require policy iterates to be feasible for the CMDP, so instead of optimizing over Π θ , we optimize over Π θ ∩ Π C : π k+1 = arg max π∈Π θ J(π) s.t. J Ci (π) ≤ d i i = 1, ..., m D(π, π k ) ≤ δ.(3) This update is difficult to implement in practice because it requires evaluation of the constraint functions to determine whether a proposed point π is feasible. When using sampling to compute policy updates, as is typically done in high-dimensional control (Duan et al., 2016), this requires off-policy evaluation, which is known to be challenging (Jiang & Li, 2015). In this work, we take a different approach, motivated by recent methods for trust region optimization (Schulman et al., 2015). We develop a principled approximation to (3) with a particular choice of D, where we replace the objective and constraints with surrogate functions. The surrogates we choose are easy to estimate from samples collected on π k , and are good local approximations for the objective and constraints. Our theoretical analysis shows that for our choices of surrogates, we can bound our update's worst-case performance and worst-case constraint violation with values that depend on a hyperparameter of the algorithm. To prove the performance guarantees associated with our surrogates, we first prove new bounds on the difference in returns (or constraint returns) between two arbitrary stochastic policies in terms of an average divergence between them. We then show how our bounds permit a new analysis of trust region methods in general: specifically, we prove a worst-case performance degradation at each update. We conclude by motivating, presenting, and proving gurantees on our algorithm, Constrained Policy Optimization (CPO), a trust region method for CMDPs. Policy Performance Bounds In this section, we present the theoretical foundation for our approach-a new bound on the difference in returns between two arbitrary policies. This result, which is of independent interest, extends the works of (Kakade & Langford, 2002), (Pirotta et al., 2013), and (Schulman et al., 2015), providing tighter bounds. As we show later, it also relates the theoretical bounds for trust region policy improvement with the actual trust region algorithms that have been demonstrated to be successful in practice (Duan et al., 2016). In the context of constrained policy search, we later use our results to propose policy updates that both improve the expected return and satisfy constraints. The following theorem connects the difference in returns (or constraint returns) between two arbitrary policies to an average divergence between them. Theorem 1. For any function f : S → R and any policies π and π, define δ f (s, a, s ) . = R(s, a, s ) + γf (s ) − f (s), π f . = max s \|E a∼π ,s ∼P [δ f (s, a, s )]\| , L π,f (π ) . = E s∼d π a∼π s ∼P π (a\|s) π(a\|s) − 1 δ f (s, a, s ) , and D ± π,f (π ) . = L π,f (π ) 1 − γ ± 2γ π f (1 − γ) 2 E s∼d π [D T V (π \|\|π)[s]] , where D T V (π \|\|π)[s] = (1/2) a \|π (a\|s) − π( a\|s)\| is the total variational divergence between action distributions at s. The following bounds hold: D + π,f (π ) ≥ J(π ) − J(π) ≥ D − π,f (π ).(4) to get a second factor of max s D T V (π \|\|π)[s], we recover (up to assumption-dependent factors) the bounds given by Pirotta et al. (2013) as Corollary 3.6, and by Schulman et al. (2015) as Theorem 1a. The choice of f = V π allows a useful form of the lower bound, so we give it as a corollary. Corollary 1. For any policies π , π, with π . = max s \|E a∼π [A π (s, a)]\|, the following bound holds: J(π ) − J(π) ≥ 1 1 − γ E s∼d π a∼π A π (s, a) − 2γ π 1 − γ D T V (π \|\|π)[s] .(5) The bound (5) should be compared with equation (1). (5) is an approximation to J(π ) − J(π), using the state distribution d π instead of d π , which is known to equal J(π ) − J(π) to first order in the parameters of π on a neighborhood around π (Kakade & Langford, 2002). The bound can therefore be viewed as describing the worst-case approximation error, and it justifies using the approximation as a surrogate for J(π ) − J(π). The term (1 − γ) −1 E s∼d π ,a∼π [A π (s, a)] in Equivalent expressions for the auxiliary costs, based on the upper bound, also follow immediately; we will later use them to make guarantees for the safety of CPO. Corollary 2. For any policies π , π, and any cost func- tion C i , with π Ci . = max s \|E a∼π [A π Ci (s, a)]\|, the following bound holds: J Ci (π ) − J Ci (π) ≤ 1 1 − γ E s∼d π a∼π A π Ci (s, a) + 2γ π Ci 1 − γ D T V (π \|\|π)[s] .(6) The bounds we have given so far are in terms of the TV-divergence between policies, but trust region methods constrain the KL-divergence between policies, so bounds that connect performance to the KL-divergence are desirable. We make the connection through Pinsker's inequality (Csiszar & Körner, 1981): for arbitrary distributions p, q, the TV-divergence and KL-divergence are related by D T V (p\|\|q) ≤ D KL (p\|\|q)/2. Combining this with Jensen's inequality, we obtain (7) From (7) we immediately obtain the following. E s∼d π [D T V (π \|\|π)[s]] ≤ E s∼d π 1 2 D KL (π \|\|π)[s] ≤ 1 2 E s∼d π [D KL (π \|\|π)[s]] Corollary 3. In bounds (4), (5), and (6), make the substitution E s∼d π [D T V (π \|\|π)[s]] → 1 2 E s∼d π [D KL (π \|\|π)[s]]. The resulting bounds hold. Trust Region Methods Trust region algorithms for reinforcement learning (Schulman et al., 2015; have policy updates of the form π k+1 = arg max π∈Π θ E s∼d π k a∼π [A π k (s, a)] s.t.D KL (π\|\|π k ) ≤ δ,(8)whereD KL (π\|\|π k ) = E s∼π k [D KL (π\|\|π k )[s]], and δ > 0 is the step size. The set {π θ ∈ Π θ :D KL (π\|\|π k ) ≤ δ} is called the trust region. The primary motivation for this update is that it is an approximation to optimizing the lower bound on policy performance given in (5), which would guarantee monotonic performance improvements. This is important for optimizing neural network policies, which are known to suffer from performance collapse after bad updates (Duan et al., 2016). Despite the approximation, trust region steps usually give monotonic improvements (Schulman et al., 2015;Duan et al., 2016) and have shown state-of-the-art performance in the deep RL setting (Duan et al., 2016;Gu et al., 2017), making the approach appealing for developing policy search methods for CMDPs. Until now, the particular choice of trust region for (8) was heuristically motivated; with (5) and Corollary 3, we are able to show that it is principled and comes with a worstcase performance degradation guarantee that depends on δ. Proposition 1 (Trust Region Update Performance). Suppose π k , π k+1 are related by (8), and that π k ∈ Π θ . A lower bound on the policy performance difference between π k and π k+1 is J(π k+1 ) − J(π k ) ≥ − √ 2δγ π k+1 (1 − γ) 2 ,(9)where π k+1 = max s E a∼π k+1 [A π k (s, a)] . Proof. π k is a feasible point of (8) with objective value 0, so E s∼d π k ,a∼π k+1 [A π k (s, a)] ≥ 0. The rest follows by (5) and Corollary 3, noting that (8) bounds the average KLdivergence by δ. This result is useful for two reasons: 1) it is of independent interest, as it helps tighten the connection between theory and practice for deep RL, and 2) the choice to develop CPO as a trust region method means that CPO inherits this performance guarantee. Trust Region Optimization for Constrained MDPs Constrained policy optimization (CPO), which we present and justify in this section, is a policy search algorithm for CMDPs with updates that approximately solve (3) with a particular choice of D. First, we describe a policy search update for CMDPs that alleviates the issue of off-policy evaluation, and comes with guarantees of monotonic performance improvement and constraint satisfaction. Then, because the theoretically guaranteed update will take toosmall steps in practice, we propose CPO as a practical approximation based on trust region methods. By corollaries 1, 2, and 3, for appropriate coefficients α k , β i k the update π k+1 = arg max π∈Π θ E s∼d π k a∼π [A π k (s, a)] − α k D KL (π\|\|π k ) s.t. J Ci (π k ) + E s∼d π k a∼π A π k Ci (s, a) 1 − γ + β i k D KL (π\|\|π k ) ≤ d i is guaranteed to produce policies with monotonically nondecreasing returns that satisfy the original constraints. (Observe that the constraint here is on an upper bound for J Ci (π) by (6).) The off-policy evaluation issue is alleviated, because both the objective and constraints involve expectations over state distributions d π k , which we presume to have samples from. Because the bounds are tight, the problem is always feasible (as long as π 0 is feasible). However, the penalties on policy divergence are quite steep for discount factors close to 1, so steps taken with this update might be small. Inspired by trust region methods, we propose CPO, which uses a trust region instead of penalties on policy divergence to enable larger step sizes: π k+1 = arg max π∈Π θ E s∼d π k a∼π [A π k (s, a)] s.t. J Ci (π k ) + 1 1 − γ E s∼d π k a∼π A π k Ci (s, a) ≤ d i ∀ī D KL (π\|\|π k ) ≤ δ. (10) Because this is a trust region method, it inherits the performance guarantee of Proposition 1. Furthermore, by corollaries 2 and 3, we have a performance guarantee for approximate satisfaction of constraints: Proposition 2 (CPO Update Worst-Case Constraint Violation). Suppose π k , π k+1 are related by (10), and that Π θ in (10) is any set of policies with π k ∈ Π θ . An upper bound on the C i -return of π k+1 is J Ci (π k+1 ) ≤ d i + √ 2δγ π k+1 Ci (1 − γ) 2 , where π k+1 Ci = max s E a∼π k+1 A π k Ci (s, a) . Practical Implementation In this section, we show how to implement an approximation to the update (10) that can be efficiently computed, even when optimizing policies with thousands of parameters. To address the issue of approximation and sampling errors that arise in practice, as well as the potential violations described by Proposition 2, we also propose to tighten the constraints by constraining upper bounds of the auxilliary costs, instead of the auxilliary costs themselves. Approximately Solving the CPO Update For policies with high-dimensional parameter spaces like neural networks, (10) can be impractical to solve directly because of the computational cost. However, for small step sizes δ, the objective and cost constraints are well-approximated by linearizing around π k , and the KLdivergence constraint is well-approximated by second order expansion (at π k = π, the KL-divergence and its gradient are both zero). Denoting the gradient of the objective as g, the gradient of constraint i as b i , the Hessian of the KL-divergence as H, and defining c i . = J Ci (π k ) − d i , the approximation to (10) is: (11) can be expressed as θ k+1 = arg max θ g T (θ − θ k ) s.t. c i + b T i (θ − θ k ) ≤ 0 i = 1, ..., m 1 2 (θ − θ k ) T H(θ − θ k ) ≤ δ.(11)max λ≥0 ν 0 −1 2λ g T H −1 g − 2r T ν + ν T Sν + ν T c − λδ 2 ,(12)where r . = g T H −1 B, S . = B T H −1 B. This is a convex program in m+1 variables; when the number of constraints is small by comparison to the dimension of θ, this is much easier to solve than (11). If λ * , ν * are a solution to the dual, the solution to the primal is θ * = θ k + 1 λ * H −1 (g − Bν * ) .(13) Our algorithm solves the dual for λ * , ν * and uses it to propose the policy update (13). For the special case where there is only one constraint, we give an analytical solution in the supplementary material (Theorem 2) which removes the need for an inner-loop optimization. Our experiments Algorithm 1 Constrained Policy Optimization Input: Initial policy π 0 ∈ Π θ tolerance α for k = 0, 1, 2, ... do Sample a set of trajectories D = {τ } ∼ π k = π(θ k ) Form sample estimatesĝ,b,Ĥ,ĉ with D if approximate CPO is feasible then Solve dual problem (12) for λ * k , ν * k Compute policy proposal θ * with (13) else Compute recovery policy proposal θ * with (14) end if Obtain θ k+1 by backtracking linesearch to enforce satisfaction of sample estimates of constraints in (10) end for have only a single constraint, and make use of the analytical solution. Because of approximation error, the proposed update may not satisfy the constraints in (10); a backtracking line search is used to ensure surrogate constraint satisfaction. Also, for high-dimensional policies, it is impractically expensive to invert the FIM. This poses a challenge for computing H −1 g and H −1 b i , which appear in the dual. Like (Schulman et al., 2015), we approximately compute them using the conjugate gradient method. Feasibility Due to approximation errors, CPO may take a bad step and produce an infeasible iterate π k . Sometimes (11) will still be feasible and CPO can automatically recover from its bad step, but for the infeasible case, a recovery method is necessary. In our experiments, where we only have one constraint, we recover by proposing an update to purely decrease the constraint value: θ * = θ k − 2δ b T H −1 b H −1 b.(14) As before, this is followed by a line search. This approach is principled in that it uses the limiting search direction as the intersection of the trust region and the constraint region shrinks to zero. We give the pseudocode for our algorithm (for the single-constraint case) as Algorithm 1. Tightening Constraints via Cost Shaping Because of the various approximations between (3) and our practical algorithm, it is important to build a factor of safety into the algorithm to minimize the chance of constraint violations. To this end, we choose to constrain upper bounds on the original constraints, C + i , instead of the original constraints themselves. We do this by cost shaping: C + i (s, a, s ) = C i (s, a, s ) + ∆ i (s, a, s ),(15) where ∆ i : S × A × S → R + correlates in some useful way with C i . In our experiments, where we have only one constraint, we partition states into safe states and unsafe states, and the agent suffers a safety cost of 1 for being in an unsafe state. We choose ∆ to be the probability of entering an unsafe state within a fixed time horizon, according to a learned model that is updated at each iteration. This choice confers the additional benefit of smoothing out sparse constraints. Connections to Prior Work Our method has similar policy updates to primal-dual methods like those proposed by Chow et al. (2015), but crucially, we differ in computing the dual variables (the Lagrange multipliers for the constraints). In primal-dual optimization (PDO), dual variables are stateful and learned concurrently with the primal variables (Boyd et al., 2003). In a PDO algorithm for solving (3), dual variables would be updated according to ν k+1 = (ν k + α k (J C (π k ) − d)) + ,(16) where α k is a learning rate. In this approach, intermediary policies are not guaranteed to satisfy constraints-only the policy at convergence is. By contrast, CPO computes new dual variables from scratch at each update to exactly enforce constraints. Experiments In our experiments, we aim to answer the following: • Does CPO succeed at enforcing behavioral constraints when training neural network policies with thousands of parameters? • How does CPO compare with a baseline that uses primal-dual optimization? Does CPO behave better with respect to constraints? • How much does it help to constrain a cost upper bound (15), instead of directly constraining the cost? • What benefits are conferred by using constraints instead of fixed penalties? We designed experiments that are easy to interpret and motivated by safety. We consider two tasks, and train multiple different agents (robots) for each task: • Circle: The agent is rewarded for running in a wide circle, but is constrained to stay within a safe region smaller than the radius of the target circle. Returns: Constraint values: (closer to the limit is better) ; the x-axis is training iteration. CPO drives the constraint function almost directly to the limit in all experiments, while PDO frequently suffers from over-or under-correction. TRPO is included to verify that optimal unconstrained behaviors are infeasible for the constrained problem. • Gather: The agent is rewarded for collecting green apples, and constrained to avoid red bombs. For the Circle task, the exact geometry is illustrated in Figure 5 in the supplementary material. Note that there are no physical walls: the agent only interacts with boundaries through the constraint costs. The reward and constraint cost functions are described in supplementary material (Section 10.3.1). In each of these tasks, we have only one constraint; we refer to it as C and its upper bound from (15) as C + . We experiment with three different agents: a point-mass (S ⊆ R 9 , A ⊆ R 2 ), a quadruped robot (called an 'ant') (S ⊆ R 32 , A ⊆ R 8 ), and a simple humanoid (S ⊆ R 102 , A ⊆ R 10 ). We train all agent-task combinations except for Humanoid-Gather. For all experiments, we use neural network policies with two hidden layers of size (64, 32). Our experiments are implemented in rllab (Duan et al., 2016). Evaluating CPO and Comparison Analysis Learning curves for CPO and PDO are compiled in Figure 1. Note that we evaluate algorithm performance based on the C + return, instead of the C return (except for in Point-Gather, where we did not use cost shaping due to that environment's short time horizon), because this is what the algorithm actually constrains in these experiments. For our comparison, we implement PDO with (16) as the update rule for the dual variables, using a constant learning rate α; details are available in supplementary material (Section 10.3.3). We emphasize that in order for the compari- In Humanoid-Circle, the safe area is between the blue panels. son to be fair, we give PDO every advantage that is given to CPO, including equivalent trust region policy updates. To benchmark the environments, we also include TRPO (trust region policy optimization) (Schulman et al., 2015), a stateof-the-art unconstrained reinforcement learning algorithm. The TRPO experiments show that optimal unconstrained behaviors for these environments are constraint-violating. We find that CPO is successful at approximately enforcing constraints in all environments. In the simpler environments (Point-Circle and Point-Gather), CPO tracks the constraint return almost exactly to the limit value. By contrast, although PDO usually converges to constraintsatisfying policies in the end, it is not consistently constraint-satisfying throughout training (as expected). For example, see the spike in constraint value that it experiences in Ant-Circle. Additionally, PDO is sensitive to the initialization of the dual variable. By default, we initialize ν 0 = 0, which exploits no prior knowledge about the environment and makes sense when the initial policies are feasible. However, it may seem appealing to set ν 0 high, which would make PDO more conservative with respect to the constraint; PDO could then decrease ν as necessary after the fact. In the Point environments, we experiment with ν 0 = 1000 and show that although this does assure constraint satisfaction, it also can substantially harm performance with respect to return. Furthermore, we argue that this is not adequate in general: after the dual variable decreases, the agent could learn a new behavior that increases the correct dual variable more quickly than PDO can attain it (as happens in Ant-Circle for PDO; observe that performance is approximately constraint-satisfying until the agent learns how to run at around iteration 350). We find that CPO generally outperforms PDO on enforcing constraints, without compromising performance with respect to return. CPO quickly stabilizes the constraint return around to the limit value, while PDO is not consistently able to enforce constraints all throughout training. Ablation on Cost Shaping In Figure 3, we compare performance of CPO with and without cost shaping in the constraint. Our metric for comparison is the C-return, the 'true' constraint. The cost shaping does help, almost completely accounting for CPO's inherent approximation errors. However, CPO is nearly constraint-satisfying even without cost shaping. Constraint vs. Fixed Penalty In Figure 4, we compare CPO to a fixed penalty method, where policies are learned using TRPO with rewards R(s, a, s ) + λC + (s, a, s ) for λ ∈ {1, 5, 50}. We find that fixed penalty methods can be highly sensitive to the choice of penalty coefficient: in Ant-Circle, a penalty coefficient of 1 results in reward-maximizing policies that accumulate massive constraint costs, while a coefficient of 5 (less than an order of magnitude difference) results in cost-minimizing policies that never learn how to acquire any rewards. In contrast, CPO automatically picks penalty coefficients to attain the desired trade-off between reward and constraint cost. Discussion In this article, we showed that a particular optimization problem results in policy updates that are guaranteed to both improve return and satisfy constraints. This enabled the development of CPO, our policy search algorithm for CMDPs, which approximates the theoretically-guaranteed algorithm in a principled way. We demonstrated that CPO can train neural network policies with thousands of parameters on high-dimensional constrained control tasks, simultaneously maximizing reward and approximately satisfying constraints. Our work represents a step towards applying reinforcement learning in the real world, where constraints on agent behavior are sometimes necessary for the sake of safety. Appendix Proof of Policy Performance Bound PRELIMINARIES Our analysis will make extensive use of the discounted future state distribution, d π , which is defined as d π (s) = (1 − γ) ∞ t=0 γ t P (s t = s\|π). It allows us to express the expected discounted total reward compactly as J(π) = 1 1 − γ E s∼d π a∼π s ∼P [R(s, a, s )] ,(17) where by a ∼ π, we mean a ∼ π(·\|s), and by s ∼ P , we mean s ∼ P (·\|s, a). We drop the explicit notation for the sake of reducing clutter, but it should be clear from context that a and s depend on s. First, we examine some useful properties of d π that become apparent in vector form for finite state spaces. Let p t π ∈ R \|S\| denote the vector with components p t π (s) = P (s t = s\|π), and let P π ∈ R \|S\|×\|S\| denote the transition matrix with components P π (s \|s) = daP (s \|s, a)π(a\|s); then p t π = P π p t−1 π = P t π µ and d π = (1 − γ) ∞ t=0 (γP π ) t µ = (1 − γ)(I − γP π ) −1 µ.(18) This formulation helps us easily obtain the following lemma. Lemma 1. For any function f : S → R and any policy π, (1 − γ) E s∼µ [f (s)] + E s∼d π a∼π s ∼P [γf (s )] − E s∼d π [f (s)] = 0.(19) Proof. Multiply both sides of (18) by (I − γP π ) and take the inner product with the vector f ∈ R \|S\| . Combining this with (17), we obtain the following, for any function f and any policy π: J(π) = E s∼µ [f (s)] + 1 1 − γ E s∼d π a∼π s ∼P [R(s, a, s ) + γf (s ) − f (s)] .(20) This identity is nice for two reasons. First: if we pick f to be an approximator of the value function V π , then (20) relates the true discounted return of the policy (J(π)) to the estimate of the policy return (E s∼µ [f (s)]) and to the on-policy average TD-error of the approximator; this is aesthetically satisfying. Second: it shows that reward-shaping by γf (s ) − f (s) has the effect of translating the total discounted return by E s∼µ [f (s)], a fixed constant independent of policy; this illustrates the finding of Ng. et al. (1999) that reward shaping by γf (s ) + f (s) does not change the optimal policy. It is also helpful to introduce an identity for the vector difference of the discounted future state visitation distributions on two different policies, π and π. Define the matrices G . = (I − γP π ) −1 ,Ḡ . = (I − γP π ) −1 , and ∆ = P π − P π . Then: G −1 −Ḡ −1 = (I − γP π ) − (I − γP π ) = γ∆; left-multiplying by G and right-multiplying byḠ, we obtain G − G = γḠ∆G. Thus d π − d π = (1 − γ) Ḡ − G µ = γ(1 − γ)Ḡ∆Gµ = γḠ∆d π .(21) For simplicity in what follows, we will only consider MDPs with finite state and action spaces, although our attention is on MDPs that are too large for tabular methods. MAIN RESULTS In this section, we will derive and present the new policy improvement bound. We will begin with a lemma: Lemma 2. For any function f : S → R and any policies π and π, define L π,f (π ) . = E s∼d π a∼π s ∼P π (a\|s) π(a\|s) − 1 (R(s, a, s ) + γf (s ) − f (s)) ,(22)and π f . = max s \|E a∼π ,s ∼P [R(s, a, s ) + γf (s ) − f (s)]\|. Then the following bounds hold: J(π ) − J(π) ≥ 1 1 − γ L π,f (π ) − 2 π f D T V (d π \|\|d π ) ,(23)J(π ) − J(π) ≤ 1 1 − γ L π,f (π ) + 2 π f D T V (d π \|\|d π ) ,(24) where D T V is the total variational divergence. Furthermore, the bounds are tight (when π = π, the LHS and RHS are identically zero). Proof. First, for notational convenience, let δ f (s, a, s ) . = R(s, a, s ) + γf (s ) − f (s). (The choice of δ to denote this quantity is intentionally suggestive-this bears a strong resemblance to a TD-error.) By (20), we obtain the identity J(π ) − J(π) = 1 1 − γ      E s∼d π a∼π s ∼P [δ f (s, a, s )] − E s∼d π a∼π s ∼P [δ f (s, a, s )] .      Now, we restrict our attention to the first term in this equation. Letδ π f ∈ R \|S\| denote the vector of componentsδ π f (s) = E a∼π ,s ∼P [δ f (s, a, s )\|s]. Observe that With d π − d π 1 = 2D T V (d π \|\|d π ) and δπ f ∞ = π f , the bounds are almost obtained. The last step is to observe that, by the importance sampling identity, d π ,δ π f = E s∼d π a∼π s ∼P [δ f (s, a, s )] = E s∼d π a∼π s ∼P π (a\|s) π(a\|s) δ f (s, a, s ) . After grouping terms, the bounds are obtained. This lemma makes use of many ideas that have been explored before; for the special case of f = V π , this strategy (after bounding D T V (d π \|\|d π )) leads directly to some of the policy improvement bounds previously obtained by Pirotta et al. and Schulman et al. The form given here is slightly more general, however, because it allows for freedom in choosing f . Remark. It is reasonable to ask if there is a choice of f which maximizes the lower bound here. This turns out to trivially be f = V π . Observe that E s ∼P [δ V π (s, a, s )\|s, a] = A π (s, a). For all states, E a∼π [A π (s, a)] = 0 (by the definition of A π ), thusδ π V π = 0 and π V π = 0. Also, L π,V π (π ) = −E s∼d π ,a∼π A π (s, a) ; from (20) with f = V π , we can see that this exactly equals J(π ) − J(π). Thus, for f = V π , we recover an exact equality. While this is not practically useful to us (because, when we want to optimize a lower bound with respect to π , it is too expensive to evaluate V π for each candidate to be practical), it provides insight: the penalty coefficient on the divergence captures information about the mismatch between f and V π . Next, we are interested in bounding the divergence term, d π − d π 1 . We give the following lemma; to the best of our knowledge, this is a new result. Lemma 3. The divergence between discounted future state visitation distributions, d π − d π 1 , is bounded by an average divergence of the policies π and π: d π − d π 1 ≤ 2γ 1 − γ E s∼d π [D T V (π \|\|π)[s]] ,(25) where D T V (π \|\|π)[s] = (1/2) a \|π (a\|s) − π(a\|s)\|. Proof. First, using (21), we obtain d π − d π 1 = γ Ḡ ∆d π 1 ≤ γ Ḡ 1 ∆d π 1 . Ḡ 1 is bounded by: Ḡ 1 = (I − γP π ) −1 1 ≤ ∞ t=0 γ t P π t To conclude the lemma, we bound ∆d π 1 . = 2 E s∼d π [D T V (π \|\|π)[s]] . The new policy improvement bound follows immediately. Theorem 1. For any function f : S → R and any policies π and π, define δ f (s, a, s ) . = R(s, a, s ) + γf (s ) − f (s), π f . = max s \|E a∼π ,s ∼P [δ f (s, a, s )]\| , L π,f (π ) . = E s∼d π a∼π s ∼P π (a\|s) π(a\|s) − 1 δ f (s, a, s ) , and D ± π,f (π ) . = L π,f (π ) 1 − γ ± 2γ π f (1 − γ) 2 E s∼d π [D T V (π \|\|π)[s]] , where D T V (π \|\|π)[s] = (1/2) a \|π (a\|s) − π(a\|s)\| is the total variational divergence between action distributions at s. The following bounds hold: D + π,f (π ) ≥ J(π ) − J(π) ≥ D − π,f (π ).(4) Furthermore, the bounds are tight (when π = π, all three expressions are identically zero). Proof. Begin with the bounds from lemma 2 and bound the divergence D T V (d π \|\|d π ) by lemma 3. Proof of Analytical Solution to LQCLP Theorem 2 (Optimizing Linear Objective with Linear and Quadratic Constraints). Consider the problem p * = min x g T x s.t. b T x + c ≤ 0 (26) x T Hx ≤ δ, where g, b, x ∈ R n , c, δ ∈ R, δ > 0, H ∈ S n , and H 0. When there is at least one strictly feasible point, the optimal point x * satisfies x * = − 1 λ * H −1 (g + ν * b) , where λ * and ν * are defined by ν * = λ * c − r s + , λ * = arg max λ≥0 f a (λ) . = 1 2λ r 2 s − q + λ 2 c 2 s − δ − rc s if λc − r > 0 f b (λ) . = − 1 2 q λ + λδ otherwise, with q = g T H −1 g, r = g T H −1 b, and s = b T H −1 b. Furthermore, let Λ a . = {λ\|λc − r > 0, λ ≥ 0}, and Λ b . = {λ\|λc − r ≤ 0, λ ≥ 0}. The value of λ * satisfies λ * ∈ λ * a . = Proj q − r 2 /s δ − c 2 /s , Λ a , λ * b . = Proj q δ , Λ b , with λ * = λ * a if f a (λ * a ) > f b (λ * b ) and λ * = λ * b otherwise, and Proj(a, S) is the projection of a point x on to a set S. Note: the projection of a point x ∈ R onto a convex segment of R, [a, b], has value Proj(x, [a, b]) = max(a, min(b, x)). Proof. This is a convex optimization problem. When there is at least one strictly feasible point, strong duality holds by Slater's theorem. We exploit strong duality to solve the problem analytically. p * = min x max λ≥0 ν≥0 g T x + λ 2 x T Hx − δ + ν b T x + c = max λ≥0 ν≥0 min x λ 2 x T Hx + (g + νb) T x + νc − 1 2 λδ Strong duality =⇒ x * = − 1 λ H −1 (g + νb) ∇ x L(x, λ, ν) = 0 = max λ≥0 ν≥0 − 1 2λ (g + νb) T H −1 (g + νb) + νc − 1 2 λδ Plug in x * = max λ≥0 ν≥0 − 1 2λ q + 2νr + ν 2 s + νc − 1 2 λδ Notation: q . = g T H −1 g, r . = g T H −1 b, s . = b T H −1 b. =⇒ ∂L ∂ν = − 1 2λ (2r + 2νs) + c =⇒ ν = λc − r s + Optimizing single-variable convex quadratic function over R + = max λ≥0 1 2λ r 2 s − q + λ 2 c 2 s − δ − rc s if λ ∈ Λ a − 1 2 q λ + λδ if λ ∈ Λ b Notation: Λ a . = {λ\|λc − r > 0, λ ≥ 0}, Λ b . = {λ\|λc − r ≤ 0, λ ≥ 0} Observe that when c < 0, Λ a = [0, r/c) and Λ b = [r/c, ∞); when c > 0, Λ a = [r/c, ∞) and Λ b = [0, r/c). Notes on interpreting the coefficients in the dual problem: • We are guaranteed to have r 2 /s − q ≤ 0 by the Cauchy-Schwarz inequality. Recall that q = g T H −1 g, r = g T H −1 b, s = b T H −1 b. The Cauchy-Scwarz inequality gives: H −1/2 b 2 2 H −1/2 g 2 2 ≥ H −1/2 b T H −1/2 g 2 =⇒ b T H −1 b g T H −1 g ≥ b T H −1 g 2 ∴ qs ≥ r 2 . • The coefficient c 2 /s − δ relates to whether or not the plane of the linear constraint intersects the quadratic trust region. An intersection occurs if there exists an x such that c + b T x = 0 with x T Hx ≤ δ. To check whether this is the case, we solve x * = arg min x x T Hx : c + b T x = 0(27) and see if x * T Hx * ≤ δ. The solution to this optimization problem is x * = cH −1 b/s, thus x * T Hx * = c 2 /s. If c 2 /s − δ ≤ 0, then the plane intersects the trust region; otherwise, it does not. If c 2 /s − δ > 0 and c < 0, then the quadratic trust region lies entirely within the linear constraint-satisfying halfspace, and we can remove the linear constraint without changing the optimization problem. If c 2 /s − δ > 0 and c > 0, the problem is infeasible (the intersection of the quadratic trust region and linear constraint-satisfying halfspace is empty). Otherwise, we follow the procedure below. Solving the dual for λ: for any A > 0, B > 0, the problem max λ≥0 f (λ) . = − 1 2 A λ + Bλ has optimal point λ * = A/B and optimal value f (λ * ) = − √ AB. We can use this solution form to obtain the optimal point on each segment of the piecewise continuous dual function for λ: objective optimal point (before projection) optimal point (after projection) f a (λ) . = 1 2λ r 2 s − q + λ 2 c 2 s − δ − rc s λ a . = q − r 2 /s δ − c 2 /s λ * a = Proj(λ a , Λ a ) f b (λ) . = − 1 2 q λ + λδ λ b . = q δ λ * b = Proj(λ b , Λ b ) The optimization is completed by comparing f a (λ * a ) and f b (λ * b ): λ * = λ * a f a (λ * a ) ≥ f b (λ * b ) λ * b otherwise. Experimental Parameters ENVIRONMENTS In the Circle environments, the reward and cost functions are R(s) = v T [−y, x] 1 + \| [x, y] 2 − d\| , C(s) = 1 1 1 [\|x\| > x lim ] , where x, y are the coordinates in the plane, v is the velocity, and d, x lim are environmental parameters. We set these parameters to be Point-mass Ant Humanoid d 15 10 10 x lim 2.5 3 2.5 In Point-Gather, the agent receives a reward of +10 for collecting an apple, and a cost of 1 for collecting a bomb. Two apples and eight bombs spawn on the map at the start of each episode. In Ant-Gather, the reward and cost structure was the same, except that the agent also receives a reward of −10 for falling over (which results in the episode ending). Eight apples and eight bombs spawn on the map at the start of each episode. In the Circle task, reward is maximized by moving along the green circle. The agent is not allowed to enter the blue regions, so its optimal constrained path follows the line segments AD and BC. ALGORITHM PARAMETERS In all experiments, we use Gaussian policies with mean vectors given as the outputs of neural networks, and with variances that are separate learnable parameters. The policy networks for all experiments have two hidden layers of sizes (64, 32) with tanh activation functions. We use GAE-λ to estimate the advantages and constraint advantages, with neural network value functions. The value functions have the same architecture and activation functions as the policy networks. We found that having different λ GAE values for the regular advantages and the constraint advantages worked best. We denote the λ GAE used for the constraint advantages as λ GAE C . For the failure prediction networks P φ (s → U ), we use neural networks with a single hidden layer of size (32), with output of one sigmoid unit. At each iteration, the failure prediction network is updated by some number of gradient descent steps using the Adam update rule to minimize the prediction error. To reiterate, the failure prediction network is a model for the probability that the agent will, at some point in the next T time steps, enter an unsafe state. The cost bonus was weighted by a coefficient α, which was 1 in all experiments except for Ant-Gather, where it was 0.01. Because of the short time horizon, no cost bonus was used for Point-Gather. For all experiments, we used a discount factor of γ = 0.995, a GAE-λ for estimating the regular advantages of λ GAE = 0.95, and a KL-divergence step size of δ KL = 0.01. Experiment-specific parameters are as follows: Note that these same parameters were used for all algorithms. We found that the Point environment was agnostic to λ GAE C , but for the higher-dimensional environments, it was necessary to set λ GAE C to a value < 1. Failing to discount the constraint advantages led to substantial overestimates of the constraint gradient magnitude, which led the algorithm to take unsafe steps. The choice λ GAE C = 0.5 was obtained by a hyperparameter search in {0.5, 0.92, 1}, but 0.92 worked nearly as well. PRIMAL-DUAL OPTIMIZATION IMPLEMENTATION Our primal-dual implementation is intended to be as close as possible to our CPO implementation. The key difference is that the dual variables for the constraints are stateful, learnable parameters, unlike in CPO where they are solved from scratch at each update. Proceedings of the 34 th International Conference on Machine Learning, Sydney, Australia, 2017. JMLR: W&CP. Copyright 2017 by the author(s). Because the Fisher information matrix (FIM) H is always positive semi-definite (and we will assume it to be positive-definite in what follows), this optimization problem is convex and, when feasible, can be solved efficiently using duality. (We reserve the case where it is not feasible for the next subsection.) With B . = [b 1 , ..., b m ] and c . = [c 1 , ..., c m ] T , a dual to Figure 1 . 1Average performance for CPO, PDO, and TRPO over several seeds (5 in the Point environments, 10 in all others) Figure 2 . 2The Humanoid-Circle and Point-Gather environments. Figure 3 . 3Using cost shaping (CS) in the constraint while optimizing generally improves the agent's adherence to the true constraint on C-return. Figure 4 . 4Comparison between CPO and FPO (fixed penalty optimization) for various values of fixed penalty. PP (s \|s, a) (π (a\|s) − π(a\|s)) d π (s \|s, a) \|π (a\|s) − π(a\|s)\| d π (s) = s,a \|π (a\|s) − π(a\|s)\| d π (s) Figure 5 . 5Figure 5. In the Circle task, reward is maximized by moving along the green circle. The agent is not allowed to enter the blue regions, so its optimal constrained path follows the line segments AD and BC. ParameterPoint-Circle Ant-Circle Humanoid-Circle Point-Gather Ant-GatherBatch size 50,000 100,000 50,000 50,000 100,000 Rollout length 50-65 500 1000 15 500 Maximum constraint value d 5 10 10 0.1 0.2 Failure prediction horizon T 5 20 20 (N/A) 20 Failure predictor SGD steps per itr 25 25 25 (N/A) 10 Predictor coeff α 1 1 1 (N/A) 0.01 λ GAE C 1 0.5 0.5 1 0.5 UC Berkeley 2 OpenAI. Correspondence to: Joshua Achiam <[email protected]>. AcknowledgementsThe authors would like to acknowledge Peter Chen, who independently and concurrently derived an equivalent policy improvement bound.Joshua Achiam is supported by TRUST (Team for Research in Ubiquitous Secure Technology) which receives support from NSF (award number CCF-0424422). This project also received support from Berkeley Deep Drive Constrained Policy Optimization and from Siemens.This term is then straightforwardly bounded by applying Hölder's inequality; for any p, q ∈ [1, ∞] such that 1/p+1/q = 1,The lower bound leads to (23), and the upper bound leads to (24).We choose p = 1 and q = ∞; however, we believe that this step is very interesting, and different choices for dealing with the inner product d π − d π ,δ π f may lead to novel and useful bounds.The update equations for our PDO implementation arewhere s j is from the backtracking line search (s ∈ (0, 1) and j ∈ {0, 1, ..., J}, where J is the backtrack budget; this is the same line search as is used in CPO and TRPO), and α is a learning rate for the dual parameters. α is an important hyperparameter of the algorithm: if it is set to be too small, the dual variable won't update quickly enough to meaningfully enforce the constraint; if it is too high, the algorithm will overcorrect in response to constraint violations and behave too conservatively. 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[ "HackerScope: The Dynamics of a Massive Hacker Online Ecosystem", "HackerScope: The Dynamics of a Massive Hacker Online Ecosystem" ]	[ "Risul Islam \nUC Riverside\nUC Riverside\nUC Riverside\nUC Riverside\n\n", "MdOmar Faruk Rokon \nUC Riverside\nUC Riverside\nUC Riverside\nUC Riverside\n\n", "Ahmad Darki \nUC Riverside\nUC Riverside\nUC Riverside\nUC Riverside\n\n", "Michalis Faloutsos [email protected] \nUC Riverside\nUC Riverside\nUC Riverside\nUC Riverside\n\n" ]	[ "UC Riverside\nUC Riverside\nUC Riverside\nUC Riverside\n", "UC Riverside\nUC Riverside\nUC Riverside\nUC Riverside\n", "UC Riverside\nUC Riverside\nUC Riverside\nUC Riverside\n", "UC Riverside\nUC Riverside\nUC Riverside\nUC Riverside\n" ]	[ "2020 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining" ]	Authors of malicious software are not hiding as much as one would assume: they have a visible online footprint. Apart from online forums, this footprint appears in software development platforms, where authors create publicly-accessible malware repositories to share and collaborate. With the exception of a few recent efforts, the existence and the dynamics of this community has received surprisingly limited attention. The goal of our work is to analyze this ecosystem of hackers in order to: (a) understand their collaborative patterns, and (b) identify and profile its most influential authors. We develop HackerScope, a systematic approach for analyzing the dynamics of this hacker ecosystem. Leveraging our targeted data collection, we conduct an extensive study of 7389 authors of malware repositories on GitHub, which we combine with their activity on four security forums. From a modeling point of view, we study the ecosystem using three network representations: (a) the author-author network, (b) the author-repository network, and (c) cross-platform egonets. Our analysis leads to the following key observations: (a) the ecosystem is growing at an accelerating rate as the number of new malware authors per year triples every 2 years, (b) it is highly collaborative, more so than the rest of GitHub authors, and (c) it includes influential and professional hackers. We find 30 authors maintain an online "brand" across GitHub and our security forums. Our study is a significant step towards using public online information for understanding the malicious hacker community.	10.1109/asonam49781.2020.9381368	[ "https://arxiv.org/pdf/2011.07222v1.pdf" ]	226,965,537	2011.07222	23d17f3810e9bd1dcfd0e50473bf5494048ce716	HackerScope: The Dynamics of a Massive Hacker Online Ecosystem ASONAMCopyright ASONAM Risul Islam UC Riverside UC Riverside UC Riverside UC Riverside MdOmar Faruk Rokon UC Riverside UC Riverside UC Riverside UC Riverside Ahmad Darki UC Riverside UC Riverside UC Riverside UC Riverside Michalis Faloutsos [email protected] UC Riverside UC Riverside UC Riverside UC Riverside HackerScope: The Dynamics of a Massive Hacker Online Ecosystem 2020 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining ASONAMIndex Terms-GitHubHackersCommunityEgonet Authors of malicious software are not hiding as much as one would assume: they have a visible online footprint. Apart from online forums, this footprint appears in software development platforms, where authors create publicly-accessible malware repositories to share and collaborate. With the exception of a few recent efforts, the existence and the dynamics of this community has received surprisingly limited attention. The goal of our work is to analyze this ecosystem of hackers in order to: (a) understand their collaborative patterns, and (b) identify and profile its most influential authors. We develop HackerScope, a systematic approach for analyzing the dynamics of this hacker ecosystem. Leveraging our targeted data collection, we conduct an extensive study of 7389 authors of malware repositories on GitHub, which we combine with their activity on four security forums. From a modeling point of view, we study the ecosystem using three network representations: (a) the author-author network, (b) the author-repository network, and (c) cross-platform egonets. Our analysis leads to the following key observations: (a) the ecosystem is growing at an accelerating rate as the number of new malware authors per year triples every 2 years, (b) it is highly collaborative, more so than the rest of GitHub authors, and (c) it includes influential and professional hackers. We find 30 authors maintain an online "brand" across GitHub and our security forums. Our study is a significant step towards using public online information for understanding the malicious hacker community. I. INTRODUCTION "How can a 17 year old kid from Florida [1] be reportedly the mastermind behind the recent hacking of Twitter? This question is part of the motivation behind this work. The security community has a fairly limited understanding of malicious hackers and their interactions. As a result, security practitioners do not really know their "enemy". On the one hand, the hacker community is fairly wide encompassing curious teenagers, aspiring hackers, and professional criminals. On the other hand, the hackers are surprisingly bold in leaving a digital footprint, if one looks at the right places in the Internet. For example, there are various online forums, where hackers not only share information, but they also boast of their successes. How can we begin to understand the ecosystem of malicious hackers based on their online footprint? The input is the online activities of these hackers, and the goal is to answer the following questions: (a) do these hackers work in groups or alone, and (b) who are the most influential hackers? Here, we consider two types of platforms that hackers frequent: (a) software archives, and (b) online security forums. It turns out that popular and public software archives, such as GitHub harbor malware authors, who create publicly-accessible malware repositories [2]. Furthermore, online forums have recently emerged as marketplaces and information hubs of malicious activities [3], [4]. In the rest of this paper, we will use the term hacker to refer to actors who develop and use software of malicious intent. We will also use the term hackers and malware authors interchangeably, although some malware authors may not have malicious intent. There is limited work for the problem as defined above. First, we are not aware of a study that systematically profiles the dynamics of the online hacker ecosystem, and especially one considering software archives. Most of the previous efforts on GitHub follow a software-centric view or study GitHub at large without focusing on malware [5] [6] [7]. Most of the previous works on online forums focus on identifying emerging topics and threats [3], [4]. Other efforts report malware activity, focusing on hacking events, and much less, if at all, on the ecosystem of hackers [8], [9]. We elaborate on previous works in Section VIII. We propose HackerScope, a systematic approach for modeling the ecosystem of malware authors by analyzing their online footprint. We start with an extensive analysis of malware authors on GitHub, as this is a significantly less-studied space. We then use security forums to find more information about these authors. From an algorithmic point of view, we use three network representations: (a) the author-author network, (b) the author-repository network, and (c) cross-platform egonets, which we explain later. In addition, we use some basic Natural Language Processing techniques, which we intend to develop further in the future. We apply and evaluate our approach using 7389 malware authors on GitHub over the span of 11 years and leverage the activity on four security forums in the grey area between white-hat and black-hat security. GitHub is arguably the largest repository with roughly 30 million public repositories, while, appropriately fine-tuned, our approach can be used on other software archives. Our approach encompasses four research thrusts, which identify and model: (a) statistics and trends, (b) communities of hackers and their dynamics, (c) influential hackers, and (d) hacker profiles across different online platforms. For the latter type, we show the collaborators of hackers as captured by the cross-platform egonets spanning GitHub and security forums in Figure 1. Our key results are summarized in the following points. a. The ecosystem is growing at an accelerating rate: The Figure 1: Profiling hackers across platforms using our cross-platform egonet: the scatter-plot of the number of neighbors on GitHub versus those on security forums for 30 malware authors as captured in our cross-platform egonet. number of new malware authors on GitHub is roughly tripling every two years. This alarming trend points to the importance of monitoring this ecosystem. b. The ecosystem is highly collaborative: We find 513 collaboration communities on GitHub with high cohesiveness (Modularity Score within [0.65-0.78]), including many large communities with hundreds of users. The malware community is very collaborative: a malware repository is forked four times more compared to a regular GitHub repository. c. We identify a group of 1.7% of influential authors: We develop a systematic approach to determine the influence among malware authors. Our novelty lies in: (a) considering many types of interactions, and (b) capturing the networkwide influence of an author. We find a core group of 1.7% of the malware authors, who are responsible for: (a) generating influential repositories, and (b) providing the social backbone of the malware community. d. We identify professional hackers in the ecosystem: We find that 30 authors are professional malicious hackers. Going across platforms, we find GitHub authors who are quite active on our security forums. We show the evidence that these are professional hackers, who are building an online "brand". For example, user 3vilp4wn is the author of a keylogger repository on GitHub, which he promotes in the HackThisSite forum using the same username (shown at bottom right in Figure 1). Our work in perspective. The proposed work is part of an ambitious goal: we want to model the Internet hacker ecosystem at large as it manifests itself across platforms. Our initial results are promising: a) the hackers seem to want to establish a brand, hence they want to be visible, and b) a cross-platform study is possible, as some authors maintain the same login name. Our systematic approach here constitutes a building block towards the ultimate goal. With appropriate follow up work, achieving this goal can have a huge practical impact: security analysts could prepare for emerging threats, anticipate malicious activity, and identify their perpetrators. Open-sourcing for maximal impact. We use Python v3.6.2 packages to implement all the modules of HackerScope. We intend to make our datasets and tools public for research purposes. II. BACKGROUND AND DATA Our work focuses on GitHub, the largest software archive with roughly 30 million public repositories, and uses data from security forums. Although GitHub policies do not allow malware, authors do not seem to abide by them. A. GitHub data. GitHub platform enables software developers to create software repositories in order to store, share, and collaborate on projects and provides many social-network-type functions. We define some basic terminology here. We use the term author to describe a GitHub user who has created at least one repository. A malware repository contains malicious software and a malware author owns at least one such repository. Users can star, watch and fork other malware repositories. Forking means creating a clone of another repository. A forked repository is sometimes merged back with the original parent repository, and we call this a contribution. Users can also comment by providing suggestions and feedback to other authors' repositories. We use a dataset of 7389 malware authors and their related 8644 malware repositories, which were identified among 97K repositories in our prior work [2]. This is arguably the largest malware archive of its kind with repositories spanning roughly 11 years. These repositories have been identified as malicious with a very high precision (89%). Note that the queries with the GitHub API, which were used in the data collection, return primary or non-forked repositories. A discussion on the process, accuracy, and validity of the dataset can be found in the original study [2]. For each malware author in our dataset, we have the following information: (a) the list of the malware repositories created by her, and (b) the list of followers. For each malware repository, we have the lists of users, who: (a) star, (b) watch, (c) fork, (d) comment, or (e) contribute to the repository. Repository metadata. Each repository is also associated with a set of user generated fields, such as title, readme file, description. We can use this metadata to extract information about the repository. We leverage our earlier work where we discuss the processing of this metadata in more detail [2]. For a given repository, a security expert would want to know: (a) the type of malware (e.g. ransomware and keylogger), and (b) the target platform (e.g. Linux and Windows). For this, we define two sets of keywords: (a) 13 types of malware, S 1 and (b) 6 types of target platforms, and S 2 . Figure 6 provides a visual list of these two sets of keywords. We define the Repository Keyword Set, W r , for repository r, as a set consisting of the keyword sets S 1 and S 2 that are present in its metadata. Clearly, one can extend and refine these keyword sets, to provide additional information, such as the programming language in use, which we will consider in the future. Note that our earlier work provides evidence that using this metadata as we do here can provide fairly accurate and useful information [2]. B. Security forum data. We also utilize data that we collect from four security forums: Wilders Security, Offensive Community, Hack This Site, and Ethical Hackers [10]. In these forums, users initiate discussion threads in which other interested users can post to share their opinion. Each tuple in our dataset contains the following information: forum ID, thread ID, post ID, username, and post content. We provide a brief description of our forums below, and an overview of key numbers in Table I. a. OffensiveCommunity (OC): As the name suggests, this forum contains "offensive security" related threads, namely, breaking into systems. Many posts consist of step by step instructions on how to compromise systems, and advertise hacking tools and services. b. HackThisSite (HTS): As the name suggests, this forum has also an attacking orientation. There are threads that explain how to break into websites and systems, but there are also more general discussions on cyber-security. c. EthicalHackers (EH): This forum seems to consist mostly of "white-hat" hackers, as its name suggests. However, there are many threads with malicious intentions in this forum. d. WildersSecurity (WS): The threads in this forum fall in the grey area, discussing both "black-hat" and "white-hat" skills. III. OUR APPROACH We have an ambitious vision for our approach, which we plan to release as a software platform. We provide a brief overview in Figure 2. In this paper, we will elaborate on the four analysis modules: (a) a statistics and trends module, which provides the landscape of primary behaviors of the ecosystem (Section IV), (b) a community analysis module, which identifies and profiles communities of collaboration (Section VI), (c) an influence analysis module, which defines and calculates the significance of authors (Section V), and (d) cross-platform analysis module (Section VII). In addition, our approach also includes: a data collection module, which aggregates, cleans and preprocesses the raw information; a control center module; and a reporting module. These modules are not equally developed, while at the same time, we could not provide all the types of results that we have available due to space limitations. Below, we highlight some interesting or novel aspects of our approach, which are often cutting across several modules. a. Synthesizing multi-source data. Our approach focuses on data for authors from GitHub and combines it with additional data from security forums, and Internet searches. b. Defining appropriate features. As we already saw, the authors and the repositories have a very rich set of interactions. We have primary (measured directly) and secondary (derived from the primary) features, which need to be determined carefully to capture effectively the dynamics of the ecosystem. These interactions go beyond a simple "friend" relationship of other social media. c. Modeling the dynamics. We use three network representations to capture the rich interactions and relationships among authors and repositories. The network representations include: (a) the author-author network, (b) the author-repository network, and (c) cross-platform egonets. d. Reporting behaviors. The goal is to provide intuitive and actionable information in an appealing and ideally interactive fashion. The results in this paper provide an indication of some initial plots and tables that our approach will provide to the end user, who could be a researcher or a security analyst. IV. STATISTICS AND TRENDS This section describes the functionality of the statistics and trends module of our approach, whose intention is to provide a basic understanding of author behaviors. A. Basic distributions of malware authors. We study the complementary cumulative distribution function (CCDF) of three metrics: (a) the number of repositories created, (b) the number of followers, and (c) sum of the number of forks across all the malware repositories of the author. As expected all distributions are skewed, but the plots are omitted due to space constraints. First, we find that 15 authors are contributing roughly 5% of all malware repositories, while 99% of all authors have created less than 5 repositories each. Second, we find that 3% (221) of the authors have more than 300 followers each, while 70% of the authors have less than 16 followers. Finally, examining the total number of forks per author, we find that 3% (221) of the authors have their repositories forked more than 150, while 43% of authors encounter at least one fork. B. Forking behavior: Malware repositories are forked four times more than the average repository. Malware repositories are more aggressively forked, which is an indication of the higher collaboration in the ecosystem. First, we find that a malware repository is forked 4.01 times on average, while a regular GitHub repository is forked 0.9 times, as reported in previous studies [11]. Second, we want to see if this is due to a few popular repositories, but this is not the case. We find that 39% of the malware repositories are forked at least once, while this is true for only 14% for general repositories [11]. C. Trends. "How fast is this ecosystem growing?" To answer the question, we plot the number of new malware authors per year in Figure 3. We consider that an author joins the ecosystem at the time that they create their first malware repository in our database. a. The number of new malware authors almost triples every two years. We plot the new malware authors per year in Figure 3. We observe an increase from 238 malware authors in 2012 to 596 authors in 2014 and to 1448 authors in 2016. We also observe a steep 62% increase from 2015 to 2016. This trend is interesting and alarming at the same time. b. The number of new malware repositories more than triples every four years. Echoing the growth of the authors, the number of repositories is also increasing super-linearly. In the future, we plan to study the trends of malware in terms of both types of malware and its target platform. V. IDENTIFYING INFLUENTIAL AUTHORS To understand the dynamics of the ecosystem, we want to answer the following question: "Who are the most influential authors?" The functionality in this section is part of the influence analysis module of Figure 2. A. HackerScore: Identifying influential authors. We argue that finding influential authors presents several challenges. First, there are many different activities and interactions, such as creating repositories, commenting, following other authors and being followed by other authors. Second, we can consider two types of actions: (a) creating influential artifacts, (b) observing and engaging with other people and artifacts. Furthermore, the distinction is not always clear. For example, forking a repository creates a new, but derivative, repository. To address the above challenges, we take socially-aware approach to influence: creating a few influential repositories is more important than creating many non-influential repositories. We discuss how we model and calculate this influence below. The Author-Author graph (AA). We create the Author-Author network to capture the network-wide interaction among authors. We define a weighted labeled multi-digraph: G(V, E, W, L e ) where V is the malware author set, E is the set of edges, W is the weight set and L e is the set of labels that an edge e can be associated with. These labels correspond to different types of relationships between authors. Here we opted to consider only malware authors in the graph to raise the bar for being part of the hacker community. The types of interactions. We consider four types of relationships between authors here. A directed edge (u, v) from author u to v can be (i) a follower edge: when u follows v, (ii) a fork edge: when u forks a repository of v, (iii) a contribution edge: u contributes code in a repository of v, and (iv) a comment edge: u comments in a repository of v. These relationships capture the most substantial author-level interactions. The multi-graph challenge and weight calibration. Our graph consists of different types of edges, which represent different relationships that we want to consider in tandem. The challenge is that the relationships have significantly different distributions, which can give an unfair advantage or eliminate the importance of a relationship. For example, contribution activities are rarer compared to following, but one can argue that a contribution to a repository is a more meaningful relationship and it should be given appropriate weight. For fairness, we make the weight of a type of edge inversely proportional to a measure of its relative frequency. In detail, we calculate the average degree d type for each type of edge: follower, fork, contribution, and comment from the subgraph containing only that type of edges from the AA graph. We find the following average degrees: d f ollower = 12.21, d f ork = 4.67, d contribution = 0.53 and d comment = 0.49. We normalize these average degrees using the minimum average degree (d min = 0.49) and we get the inverse of this value as the weight for that edge, namely, d min /d type . This way, we set the following weights: w = 0.04 for a following edge, w = 0.1 for a forking edge, and w = 1 for a commenting or a contribution edge. This enables us to consider each relationship type more fairly and meaningfully. We propose a socially-aware and integrated approach to combine all the author activities in a single framework. First, we identify and define two roles in the ecosystem: (a) producers, who create influential malware repositories, and (b) connectors, who enhance the community by engaging with influential malware authors and repositories. To calculate the roles of the malware authors, we first model the interaction among authors in the AA graph described above. Next, we apply our algorithm, a customized version of a weighted hyperlink-induced topic search algorithm modified to handle the multiple types of relationships between authors. We discuss the related algorithms in Section VIII. Calculating the HackerScore. We associate each node u with two values: (a) a Producer HackerScore value, P HS u , and (b) Connector HackerScore value, CHS u . Let w(u, v) be the weight of edge (u, v) based on its label, as discussed above. The algorithm iterative refines the producer and connector values until it converges. We, now, elaborate on the steps. First, P HS u and CHS u are initialized to 1. During the iterative step, the algorithm updates the values as follows: (i) for all v pointing to u: P HS u = v w(v, u) * CHS v , or zero in the absence of such edges, (ii) for all z pointed by u: CHS u = z w(u, z) * P HS z , or zero in the absence of such edges, and (iii) we normalize P HS u and CHS u , so that u P HS u = u CHS u = 1. For the convergence, we set a tolerance threshold of 10 −9 for the change of the value of any node. After 449 iterations, we obtain the two HackerScore values for each author. Identifying influential malware authors. In Figure 4, we plot the Connector HackerScore versus Producer HackerScore for our malware authors. Separately, we identify "knees" in the A few authors (1.7%) drive the community. The three regions of influence together consist of 128 malware authors (1.7%). The break down of the region size is fairly even: Region A of mostly connector authors devoted to connect the malware community is 0.6%, Region C of the influential producers who are the originator of the malware resources is 0.7%, and Region B of dual influence is 0.4%. We use the term Highly Influential Group (HIG) to refer to this group of authors. We provide a profile overview of the two most influential authors per region in Table II. The most influential author of Region C is cyberthrets, with the highest PHS (0.012) and 336 malware repositories. She gained a huge following by creating all her repositories of assembly code malware on Feb 16, 2016. The top connector author from Region A is critics with a CHS score of 0.01, which stems from her 446 comments across 301 repositories. The top malware author from Region B is D4Vince for his dual role in producing credential reuse tools with 7 repositories and 165 comments and 187 contributions. The importance of socially-aware significance. We argue that our socially-aware definition of significance provides more meaningful results than simply taking the top-ranked users in any primary metric in isolation. First, the two scores capture different aspects of influence: they can differ by orders of magnitude as is the case with cyberthrets and ytisf. Second, our scores capture a combined network-wide influence that each primary metric could miss. For example, our most influential producers do not always own many malware repositories. Malware author D4vince and n1nj4sec, mentioned in Table II, have single-digit repositories (7 and 8 respectively) and yet are two of the top producers. On the other hand, author kaist-is521 is ranked way below than n1nj4sec in terms of HackerScore (PHS =0.0001 and CHS =0.00013), although she has 18 malware repositories. B. Reciprocity of interactions. We want to understand better the nature of the author interactions here. "Is the influence among malware authors reciprocal?" The answer is negative: the relationships are not reciprocal, which is in stark contrast to the reciprocal relationships in other social media like Twitter and Facebook [12]. We consider a total of six relationships: following, forking, commenting, contributing, watching, and starring relationships. We define the Reciprocity Index for relationship x, RI x , to be the ratio of reciprocal relationships over the pairs of authors with that type of relationship (unilateral or mutual) in the Author-Author network. We find that the reciprocity is low and less than 7.3% for all the relationships in question. By contrast, reciprocity is often above 70% in social media, like Facebook or Twitter [12]. These social media mirror personal relationships and have an etiquette of conduct. We conjecture that the lower reciprocity on GitHub is due to its utilitarian orientation: following an author stems from a professional interest. VI. COMMUNITY ANALYSIS This section describes the functionality of the community analysis module, whose goal is to identify the communities of collaboration among the malware authors on GitHub. A. Identifying collaboration communities. We quantify the collaborative nature of the malware authors as follows. The Author-Repository graph (AR). We define the Author-Repository graph to be an undirected bipartite graph, G = (A, R, E), where A is the set of malware authors and R is the set of malware repositories. An edge (u, r) ∈ E exists, if author u: (a) creates, (b) stars, (c) forks, (d) watches, (e) comments, or (f) contributes to repository r. Identifying bipartite communities. To identify communities, we employ a greedy modularity maximization algorithm modified for bipartite graphs as we discuss in our related work. We find a total of 513 communities spanning a wide range of sizes as shown in Figure 5. The size of the communities follows skewed distribution. In Figure 5, we plot the number of malware authors and repositories per community in order of decreasing community size (defined as the sum of authors and repositories). We find that 90% of communities have less than 14 authors and repositories. We also see a fairly sharp knee in the plot at the fifth community, as shown by the vertical line. B. Profiling the communities. A full investigation of the purpose, evolution, and internal structure of each community could be a research topic in its own right. Here, we only provide an initial investigation around the following three questions. a. How cohesive are our communities? We report the Modularity Score (M S C ), which quantifies the cohesiveness of a community C. The M S C is defined as follows: M S C = n C (E) N C (E) , where n C (E) is the total number of edges and N C (E) Overall, our communities are highly cohesive: 82.8% (425) of the communities have a Modularity Score M S C ≥ 50%, which means that more than half of all possible edges within the community exist. Interestingly, the largest communities exhibit strong cohesiveness. In Table III, we present a highlevel profile of the five largest communities which have a Modularity Score of 0.65-0.78, which is indicative of tightlyconnected communities. b. Who are the community leaders? We want to identify the influential authors as part of profiling a community. We identify the top two most influential producers and connectors per community using the HackerScore from Section V. This leads us to a group of 144 leaders of the communities of size of at least 20 authors. We find 81% of these community leaders are part of the Highly Influential Group (HIG) of authors. This suggests that the HIG authors are indeed driving forces for the ecosystem. In the future, we intend to investigate in more depth the influence and dynamics of each community. c. What is the focus of each community in terms of platform and malware type? A security expert would want to know the main type of malware (e.g. ransomware) and the target platform (e.g. Linux) of a community. We use the Repository Keyword Set, W r , information of a repository r, as we defined in Section II, and we use it to characterize the community. One way to quantify the importance of a keyword for a community is to measure the number of repositories, for which that keyword appears at least once. In detail, we use the Strength Of Presence (SOP) metric, which we define as follows. For a community C with a set of R repositories, we define k i to be the number of repositories, in which keyword i appears in the metadata W r for repository r at least once for all repositories r ∈ R. We define the SOP i of keyword i from keyword set S as follows: SOP i = ki j∈S kj . In Table III, we show the most dominant keywords from malware types and platforms sets for each community and the related SOP scores. We can also use the SOP to visualize the keywords as a word-cloud. A word-cloud is a more immediate, appealing, and visceral way to display the information. In Figure 6, we show the word-cloud for the third largest community, which is dominated by ransomware malware and targets Windows platforms. Not only we see the main words stand out, but their relative size conveys their dominance over the other words Figure 6: The word-cloud for the malware types and platforms keywords for the third largest community: Ransomware and Windows dominate. more viscerally than a lengthy table of numbers. We present the results of this type of profiling for the largest communities in Table III, which we also discuss below. We find that the largest community of 584 malware authors and 677 malware repositories having Linux (SOP = 0.32) and keylogger (SOP = 0.29) as the dominant platform and malware type. Interestingly, we find that 49 of the top 100 most prolific (in terms of the number of repositories created) authors are in this community. Upon closer investigation, we find that 11 out of the 15 authors with the highest degree in the subgraph of this community are keylogger developers. The third-largest community consists of 175 malware authors and 288 malware repositories and revolves around Ransomware (SOP = 0.65) and Windows platform (SOP = 0.44). For reference, we present the word-cloud of the malware types and platforms based on the SOP score in Figure 6 for this community which exhibits that Ransomware and Windows possess the highest SOP scores. Finally, the fourth largest community (57 authors, 100 repositories) is the most tightly connected (M S = 0.78) and it revolves around the development of attack tools for Kali Linux. Upon closer inspection, we find that 15 of the top 25 authors (based on node degrees) form an approximate bipartite clique with 5 repositories. This group developed WiFiPhisher in 2016, a Linux-based python phishing tool [13], which has been used for both good and evil [14]. The above are indicative of the potential information that we could extract from these malware repositories. In the future, we intend to: (a) extract more detailed textual information from each community, and (b) study the evolution and dynamics of these communities over time. VII. AUTHOR INVESTIGATION "Who are these malware authors?" To answer this question, we go across platforms to security forums and leverage our datasets from several security forums. The functions described here are part of the author investigation module of Figure 2. a. Malware authors strive for an online "brand" and usernames seem persistent across online platforms. We find that many malware authors use the same username consistently across many online platforms, such as security forums, possibly in pursuit of a reputation. We identify 30 malware authors who are active in one of our four security forums: 12 in Wilders Security, 6 in Ethical Hacker, 4 in Offensive Community, and 8 in Hack This Site [10]. We argue that some of these usernames correspond to the same users based on the following two observations. First, we find significant overlap in the interests of the cross-platform usernames. For example, usernames int3grate and jedisct1 show interest in ransomware in both platforms, while 3vilp4wn advertises her keylogger malware (github.com/ 3vilp4wn/CryptLog) in the forum. Second, these usernames are fairly uncommon, which increases the likelihood of belonging to the same person. For example, the top ten results from internet searching for the username of author Misterch0c returns nine hacker related sites and a twitter account with a different handle but claimed by Misterch0c. Note that not all the malware authors or repositories have a malicious purpose. For instance, the project "Empire" [15] by xorrior was created as an offensive tool to stress-test the security of systems. However, it has recently been used by the state-sponsored hacking group Deep Panda [16]. In general, offensive security tools contribute to the power of the malware ecosystem irrespective of the intention of its creator. b. Modeling the cross-platform interactions. We propose to study the cross-platform interactions between GitHub and security forums as a promising research direction that can bridge two domains: software repositories and online forums. We define the cross-platform egonet of a user as one that consists of her egonets from the two platforms as shown in Figure 7. The forum egonet captures the interaction of the users that post on the same threads, while we leverage the Author-Author network to define the GitHub egonet. The value of cross-platform analysis. Using the crossplatform egonet as a basis, we can model the cross-platforms user dynamics, and more specifically, we can: (a) identify common "friends" between the ego-nets, (b) find the topics of interest and activities in each egonet, and (c) model information flow and influences across platforms. In Figure 1, we visualize the activity of a cross-platform user by comparing the number of users on each side of the egonet as shown in Figure 7. In Table IV, we show the actual values of indicative users, including the three outliers in the plot. The cross-platform egonet analysis can enrich the profile of each user significantly. For example, if we were just looking at GitHub, we may not have paid attention to 3vilp4wn and Evilcry. Both of these authors are less active on GitHub (small GitHub egonet), but are quite active in the security forums (large forum egonet). A closer investigation of the security forums reveals activities that match their interests on GitHub. This suggests that their GitHub activity is part of their online brand. For example, 3vilp4wn advertises her GitHub keylogger repository in the forum. We intend to expand in this promising direction in the future. c. Using information from the web. In our approach, we leverage existing information on hackers from (a) security outlets and databases, and (b) using web queries. With our python-based query and analysis tools, we verified the role and activities of authors, which we omit due to space limitations. VIII. RELATED WORKS Studying the dynamics of the malware ecosystem on GitHub has received very little attention. Most studies differ from our work in that: (a) they do not focus on malware on GitHub, and (b) when they do, they do not take an author-centric angle as we do here: they focus on classifying malware repositories or use a small set for a particular research study. Our work builds on our earlier effort [2], whose main goal is to identify malware repositories on GitHub at scale, but it does not study the malware author ecosystem as we do here. a. Studies of malware repositories on GitHub: Several other efforts have manually collected a small number of malware repositories with the purposes of a research study [17], [18]. Some other studies [5] [6] analyze malware source code from a software engineering perspective, but use only a small number of GitHub repositories as a reference. b. Studies of benign repositories on GitHub: Many studies analyze benign repositories on GitHub from a point of view of software engineering or as a social network. Some efforts find influential users and analyze the motivation behind following, forking, and contributions [7], [11]. Earlier efforts study repositories by analyzing the repository-repository relationship graph [19], and by using an activity graph [20]. Several works in this area identify influential authors and repositories using: the starring activity [21], the Following-Star-Fork activity [22], or a rank-based approach [23]. Note that a version of the hyperlink-induced topic search algorithm [24] has been used by some of the above efforts for calculating influence, but they do not adjust the weights to account for the different frequencies of the types of interactions between users. For our bipartite clustering, we adapt the greedy modularity maximization approach [25] [26]. c. Studies on security forums: This is a recent and less studied area of research. Most of the works focus on extracting entities of interest in security forums. An interesting study focuses on the dynamics of the black-market of hacking goods and services and their pricing [4]. Other studies focus on identifying important events and threats [8], [9]. Several studies focus on identifying key actors in security forums by utilizing their social and linguistics behavior [27]. None of the aforementioned works focus on the dynamics among hackers across platforms. d. Cross-platform study: Finally, some efforts study author activities on different software development forums, namely GitHub and Stack Overflow [28], [29], but do not consider information from security forums. IX. CONCLUSION We develop a systematic approach for studying the ecosystem of hackers. Our approach develops methods to identify (a) influential hackers, (b) communities of collaborating hackers, and (c) their cross-platform interactions. Our study concludes in three key takeaway messages: (a) the malware ecosystem is substantial and growing rapidly, (b) it is highly collaborative, and (c) it contains professional malicious hackers. Our initial findings are just the beginning of a promising future effort that can shed light on this online malware author ecosystem, which spans software repositories and security forums. The current work thus can be seen as a building block that can enable new research directions. Follow up research can expand on our work to develop preemptive security initiatives, such as: (a) monitoring hacker activity, (b) detecting emerging trends, and (c) identifying particularly influential hackers towards safeguarding the Internet. X. ACKNOWLEDGEMENT This work was supported by the UC Multicampus-National Lab Collaborative Research and Training (UCNLCRT) award #LFR18548554. Figure 2 : 2The overview of our approach highlighting the key functions. Figure 3 : 3New malware authors in the ecosystem per year. Figure 4 : 4The scatterplot of the Connector HackerScore vs. Producer HackerScore for the malware authors in our GitHub dataset. individual distributions of each score at PHS = 0.00215 and CHS = 0.0029 indicated by the red dotted lines. This way, we observe four regions defined by the combination of low and high values for PHS and CHS values which shows if an author is influential as producer or connector. Figure 5 : 5The distribution of the number of authors and repositories for the 27 largest communities in the order of community size. is the number of all possible edges in community C (if the community was a bipartite clique). Figure 7 : 7A cross-platform egonet: capturing the neighbors of both the security forum and GitHub. Table I : IOur four online security forums.Forum Users Threads Posts Offensive Comm. 5412 3214 23918 Ethical Hacker 5482 3290 22434 Hack This Site 2970 2740 20116 Wilders Security 3343 3741 15121 Table II : IIThe profiles of the two most influential malware authors from each region A, B, and C.Name PHS CHS Repos Follow -ers Forks Com- ments Cont- rib/s cyberthrets 0.012 0.001 336 1013 778 13 2 ytisf 0.005 10 −6 12 606 1412 4 1 critics 0.001 0.01 6 396 83 446 301 samyk 0.0018 0.0058 2 554 125 176 209 D4Vince 0.0066 0.0082 7 608 499 165 187 n1nj4sec 0.0058 0.0052 8 876 1391 64 79 Table III : IIIHigh-level profile of the five largest communities of malware authors and malware repositories.ID Authors Repos MS Dominant Platform SOP Dominant type SOP 1 584 677 0.65 Linux 0.32 Keylogger 0.29 2 419 544 0.67 Windows 0.26 Virus 0.31 3 175 288 0.73 Windows 0.65 Ransomware 0.44 4 57 100 0.78 Linux 0.43 Spyware 0.43 5 47 57 0.71 Mac 0.33 Trojan 0.22 Table IV : IVProfiles of four cross-platform users.Name Forum Posts in forum Collab/tors in forums Malw. repos Follow- ers Forks Collab/tors in GitHub Repository content Internet-wide Rep- utation misterch0c WS 7 224 7 749 81 898 Cracked malware code Self-declared hacker 3vilp4wn HTS 103 513 1 0 1 6 Python keylogger Keylogger developer fahimmagsi OC 73 175 1 1 0 1 Backdoor Famous hacker Evilcry EH 18 444 2 89 15 98 Botnet and ransomware Ransomware expert 17 years old boy tried to hack twitter. 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[ "Soft X-ray Spectroscopy of Seyfert 2 Galaxies", "Soft X-ray Spectroscopy of Seyfert 2 Galaxies" ]	[ "A Kinkhabwala ", "M Sako ", "E Behar ", "F Paerels ", "S M Kahn ", "A C Brinkman ", "J S Kaastra ", "R Van Der Meer ", "D A Liedahl ", "\nColumbia University\n550 W. 120th St10027NYNY\n", "\nPhysics Department, LLNL\nSRON\nSorbonnelaan 2, L-41P.O. Box 8083548 CA, 94550Utrecht, LivermoreCAThe Netherlands\n" ]	[ "Columbia University\n550 W. 120th St10027NYNY", "Physics Department, LLNL\nSRON\nSorbonnelaan 2, L-41P.O. Box 8083548 CA, 94550Utrecht, LivermoreCAThe Netherlands" ]	[ "The Central kpc of Starbursts and AGN ASP Conference Series" ]	Soft X-ray spectroscopy of Seyfert 2 galaxies provides perhaps the best method to probe the possible connection between AGN activity and star formation. Obscuration of powerful radiation from the inferred nucleus allows for detailed study of circumnuclear emission regions. And soft X-ray spectroscopy of these regions allows for robust discrimination between warm gas radiatively-driven by the AGN and hot collisionally-driven gas possibly associated with star formation. A simple model of a (bi-)cone of gas photoionized and photoexcited by a nuclear power-law continuum is sufficient to explain the soft X-ray spectra of all Seyfert 2 galaxies so far observed by the XMM-Newton and Chandra satellites. An upper limit of ∼10% to an additional hot, collisionally-driven gas contribution to the soft X-ray regime appears to hold for five different Seyfert 2 galaxies, placing interesting constraints on circumnuclear star formation.	null	[ "https://export.arxiv.org/pdf/astro-ph/0107224v1.pdf" ]	15,812,973	astro-ph/0107224	322b47a8e5d9650dda4fe8a9fcffff1e1013bae3	Soft X-ray Spectroscopy of Seyfert 2 Galaxies 2001 A Kinkhabwala M Sako E Behar F Paerels S M Kahn A C Brinkman J S Kaastra R Van Der Meer D A Liedahl Columbia University 550 W. 120th St10027NYNY Physics Department, LLNL SRON Sorbonnelaan 2, L-41P.O. Box 8083548 CA, 94550Utrecht, LivermoreCAThe Netherlands Soft X-ray Spectroscopy of Seyfert 2 Galaxies The Central kpc of Starbursts and AGN ASP Conference Series 2001 Soft X-ray spectroscopy of Seyfert 2 galaxies provides perhaps the best method to probe the possible connection between AGN activity and star formation. Obscuration of powerful radiation from the inferred nucleus allows for detailed study of circumnuclear emission regions. And soft X-ray spectroscopy of these regions allows for robust discrimination between warm gas radiatively-driven by the AGN and hot collisionally-driven gas possibly associated with star formation. A simple model of a (bi-)cone of gas photoionized and photoexcited by a nuclear power-law continuum is sufficient to explain the soft X-ray spectra of all Seyfert 2 galaxies so far observed by the XMM-Newton and Chandra satellites. An upper limit of ∼10% to an additional hot, collisionally-driven gas contribution to the soft X-ray regime appears to hold for five different Seyfert 2 galaxies, placing interesting constraints on circumnuclear star formation. Introduction Emission from warm, recombining gas has been shown to be a key feature of the soft X-ray emission from Seyfert 2 galaxies (Sako et al. 2000, Ogle et al. 2000, Sambruna et al. 2001). In Fig. 1, we show the XMM-Newton Reflection Grating Spectrometer (RGS) spectrum of the brightest Seyfert 2 galaxy, NGC 1068 (Kinkhabwala et al. 2001a). The presence of multiple radiative recombination continua (RRC) provide definitive evidence for a dominant recombining gas component with temperature T ≃ 3-5 eV. Recombination alone, however, is insufficient to explain all of the observed X-ray emission from these objects. Model of Radiatively-Driven Gas Cone We propose a model for the X-ray emission from Seyfert 2 galaxies consisting of a (bi-)cone of gas irradiated by a nuclear power-law continuum. (The nucleus - XMM-Newton RGS spectrum of NGC 1068. Note the bright RRC, strong forbidden lines in the helium-like triplets of oxygen and nitrogen, and relatively-strong higher-order transitions (labelled β, γ, δ, etc. up to the RRC). All of which are unambiguous signatures of warm photoionized and photoexcited gas (Sako et al. 2000). located at the tip of the cone -is obscured along our particular line of sight.) We specify a single ionic column density, N ion , and gaussian velocity distribution, σ v , both along the cone. σ v is chosen to be consistent with observations of broadened line absorption in soft X-ray observations of Seyfert 1 galaxies, where values from ∼200 km/s (Kaastra et al. 2000) to ∼ 600 km/s (Sako et al. 2001) have been found. We note that σ v may be due to a superposition of multiple velocity components. A broadening of the observed lines due to a separate gaussian velocity distribution along our particular line of sight (probably unrelated to the velocity distribution along the cone) is also taken. The upper right panel in Fig. 2 shows the expected spectrum of the heliumlike O VII ion. The panels on the left show the "Seyfert 1" view down the axis of the cone in absorption, and the panels on the right show the "Seyfert 2" view in reemission for column densities in O VII from 10 16 to 10 19 cm −2 . All photons absorbed out of the power-law continuum in the left panel are reprocessed and reemitted in the right panel. Further details of our model can be found in Kinkhabwala et al. (2001a) and Behar et al. (2001). Model Fits to Seyfert 2 Galaxy Spectra Our model works remarkably well for explaining the bulk (and possibly all) of the X-ray emission from NGC 1068. Fig. 3 shows our fit to lines associated with transitions in O VII for the RGS spectrum shown in Fig. 1. Similarly, in Fig. 4, we show fits to the previously published Chandra HETG spectra of two other Seyfert 2 galaxies. We find that the claimed observation of hot collisionallydriven gas in NGC 4151 of Ogle et al. (2000) was premature, and we verify the earlier conclusions for Markarian 3 of Sako et al. (2000), who claimed that photoionization and photoexcitation in addition to a scattered power-law continuum were sufficient to explain its observed X-ray spectrum. Soft X-ray Spectroscopy of Seyfert 2 Galaxies 3 Figure 2. Effect of varying column density (N ion = 10 16 -10 19 cm −2 ) along the cone to absorbed ("Seyfert 1" view on the left) and reemitted ("Seyfert 2" view on the right) spectra for O VII. For "Seyfert 2" view, note the varying relative strength of resonant transitions to pure recombination emission (bottom right panel). (σ v = 200 km/s with linear, but arbitrary vertical scales for flux.) Bottom left panel gives cross section for photoexcitation/photoionization with separating boundary. Fits to C VI and O VII for RGS spectrum of NGC 1068 (σ v = 200 km/s along the cone and N ion = 10 18 for both ions). Bottom panels show recombination alone, demonstrating presence of significant photoexcitation. For C VI fit, a factor-of-two reduction of photoionization relative to photoexcitation was taken for a better fit (possibly due to absorption by N VII Lyα shortward of the C VI RRC). Figure 4. Chandra HETG spectra of NGC 4151 and Markarian 3. Spectral differences are mostly due to differing Ne IX and Ne X column densities (e. g., Fig. 2) of 2 × 10 18 and 3 × 10 18 cm −2 for NGC 4151 and 5 × 10 17 and 1 × 10 18 cm −2 for Markarian 3 (using σ v = 200 km/s). Figure 1 . 1Figure 1. XMM-Newton RGS spectrum of NGC 1068. Note the bright RRC, strong forbidden lines in the helium-like triplets of oxygen and nitrogen, and relatively-strong higher-order transitions (labelled β, γ, δ, etc. up to the RRC). All of which are unambiguous signatures of warm photoionized and photoexcited gas (Sako et al. 2000). Figure 3 . 3Figure 3. Fits to C VI and O VII for RGS spectrum of NGC 1068 (σ v = 200 km/s along the cone and N ion = 10 18 for both ions). Bottom panels show recombination alone, demonstrating presence of significant photoexcitation. For C VI fit, a factor-of-two reduction of photoionization relative to photoexcitation was taken for a better fit (possibly due to absorption by N VII Lyα shortward of the C VI RRC). We estimate an upper limit of ∼10% of the soft X-ray emission may be due to hot, collisionally-driven gas in NGC 1068. Preliminary analysis of Chandra HETG spectra of four other Seyfert 2 galaxies (Markarian 3, NGC 4151, Circinus, NGC 4507) suggests that their remarkably similar spectra are also dominated by reprocessed AGN emission, with a similar upper limit to a hot collisional gas component. This places interesting limits on the amount of star formation in their circumnuclear environments (Kinkhabwala et al. 2001a,b). . E Behar, A&A. 35483Behar, E., et al. 2001, in preparation Kaastra, J. S., et al. 2000, A&A, 354, L83 . A Kinkhabwala, ApJ. 545281Kinkhabwala, A., et al. 2001a, in preparation Kinkhabwala, A., et al. 2001b, in preparation Ogle, P. M., et al. 2000, ApJ, 545(2), L81 . M Sako, ApJ. 5432115Sako, M., et al. 2000, ApJ, 543(2), L115 . M Sako, A&A. 365168Sako, M., et al. 2001, A&A, 365, L168 . R M Sambruna, ApJ. 546113Sambruna, R. M., et al. 2001, ApJ, 546(1), L13	[]
[ "CHARACTERIZATION OF MODULATION SPACES BY SYMPLECTIC REPRESENTATIONS AND APPLICATIONS TO SCHRÖDINGER EQUATIONS", "CHARACTERIZATION OF MODULATION SPACES BY SYMPLECTIC REPRESENTATIONS AND APPLICATIONS TO SCHRÖDINGER EQUATIONS" ]	[ "Elena Cordero ", "Luigi Rodino " ]	[]	[]	In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to τ -Wigner representations in[11].In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices A ∈ Sp(2d, R). We find conditions on A for which the related symplectic time-frequency representation W A can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding W A are members of the Cohen class.Finally, we show that symplectic time-frequency representations W A can be efficiently employed in the study of Schrödinger equations. In fact, modulation spaces and W A representations are the frame for a new definition of wave front set, providing a sharp result for propagation of micro-singularities in the case of the quadratic Hamiltonians. This new approach may have further applications in quantum mechanics and PDE's.2010 Mathematics Subject Classification. 42B35,35J10.	10.1016/j.jfa.2023.109892	[ "https://export.arxiv.org/pdf/2204.14124v3.pdf" ]	248,476,056	2204.14124	71c2fafab147bb59f6da0a8ce1bc13ef06f356e6	CHARACTERIZATION OF MODULATION SPACES BY SYMPLECTIC REPRESENTATIONS AND APPLICATIONS TO SCHRÖDINGER EQUATIONS 9 Feb 2023 Elena Cordero Luigi Rodino CHARACTERIZATION OF MODULATION SPACES BY SYMPLECTIC REPRESENTATIONS AND APPLICATIONS TO SCHRÖDINGER EQUATIONS 9 Feb 2023 In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to τ -Wigner representations in[11].In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices A ∈ Sp(2d, R). We find conditions on A for which the related symplectic time-frequency representation W A can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding W A are members of the Cohen class.Finally, we show that symplectic time-frequency representations W A can be efficiently employed in the study of Schrödinger equations. In fact, modulation spaces and W A representations are the frame for a new definition of wave front set, providing a sharp result for propagation of micro-singularities in the case of the quadratic Hamiltonians. This new approach may have further applications in quantum mechanics and PDE's.2010 Mathematics Subject Classification. 42B35,35J10. Introduction Modulation spaces were originally introduced in 1983 by H. G. Feichtinger in the pioneering work [13]. During the last twenty years hundreds of contributions have been written on the topic, showing that they are appropriate spaces for a variety of fields, such as signal analysis, PDE's, pseudodifferential operators, quantum mechanics (a short non-exhaustive list of books and papers is [3,4,9,10,18,20,25,26,27,28,31,33]). The key-tool for their definition is given by the time-frequency representation short-time Fourier transform (STFT) of a tempered distributions f ∈ S ′ (R d ) with respect to the Schwartz window function g ∈ S(R d ), defined as (1) V g f (x, ξ) = R d f (y) g(y − x) e −2πiyξ dy, (x, ξ) ∈ R 2d . Given indices 0 < p, q ≤ ∞, the modulation space M p,q (R d ) consists of all tempered distributions f ∈ S ′ (R d ) such that V g f ∈ L p,q (R 2d ) (mixed-norm space) with f M p,q ≍ V g f L p,q (R 2d ) . For p = q the notation M p,p (R d ) is shortened to M p (R d ) and we write f ∈ M p vs (R d ) if V g f ∈ L p vs (R 2d ) with the weight v s (x, ξ) := (1 + \|(x, ξ)\| 2 ) s/2 . For the main properties of these spaces, including the weighted versions, we refer to Section 2 below. In the realm of time-frequency representations another protagonist is given by the (cross-)Wigner distribution, introduced by Wigner in 1932 [32] in Quantum Mechanics and, later, applied to many different environments such as PDE's and signal analysis. Namely, given a window function g ∈ S(R d ), a tempered distribution f , the (cross-)Wigner distribution W (f, g) is given by (2) W (f, g)(x, ξ) = R d f (x + t 2 )g(x − t 2 )e −2πitξ dt, (x, ξ) ∈ R 2d . If f = g we simply write W f = W (f, f ) and call W f the Wigner distribution of f . In 2011 M. de Gosson [18] proved that in the definition of modulation spaces the STFT could be replaced by the cross-Wigner distribution. Hence (3) f M p,q ≍ W (f, g) L p,q (R 2d ) . In our previous work [11] this idea was further generalized to τ -Wigner representations W τ (f, g), with f, g as above, (4) W τ (f, g)(x, ξ) = R d e −2πitξ f (x + τ t)g(x − (1 − τ )t)dt, τ ∈ R (for f = g we obtain the τ -Wigner distribution W τ f := W τ (f, f ); for τ = 1/2 we recapture the Wigner case). In fact, we showed that (5) f M p,q ≍ W τ (f, g) L p,q (R 2d ) , for τ ∈ R \ {0, 1}, whereas for τ = 0 or τ = 1, so-called Rihaczek distributions, the previous characterization does not hold. The key observation was to interpret the time-frequency representations above as images of symplectic matrices by metaplectic operators (defined as in the textbooks [15,18]). In fact, for any of them we can find a symplectic matrix A ∈ Sp(2d, R) such that the metaplectic operator µ(A) applied to (f ⊗ḡ)(x, ξ) := f (x)ḡ(ξ) coincides with it (for a suitable choice of the phase factor in the definition of µ(A)). For example, consider the symplectic matrix A = A τ , with (6) A τ =     (1 − τ )I d×d τ I d×d 0 d×d 0 d×d 0 d×d 0 d×d τ I d×d −(1 − τ )I d×d 0 d×d 0 d×d I d×d I d×d −I d×d I d×d 0 d×d 0 d×d     ∈ Sp(2d, R), then µ(A τ )(f ⊗ḡ) = W τ (f, g), τ ∈ R. Similarly, for A = A ST , where (7) A ST =     I d×d −I d×d 0 d×d 0 d×d 0 d×d 0 d×d I d×d I d×d 0 d×d 0 d×d 0 d×d −I d×d −I d×d 0 d×d 0 d×d 0 d×d     , we recapture the STFT: µ(A ST )(f ⊗ḡ) = V g f. This suggests a change of perspective: time-frequency representations can be viewed as images of metaplectic operators. Hence symplectic matrices and metaplectic operators may become the real protagonists in the framework of timefrequency analysis. In this paper we show that symplectic matrices A ∈ Sp(2d, R) are successfully employed to both recapture and find new time-frequency representations that we call A-Wigner distributions: W A (f, g) = µ(A)(f ⊗ḡ). For f = g we simply write W A f := W A (f, f ). The definition of the metaplectic operator µ(A) depends on the choice of a multiplicative phase factor, which we omit for simplicity. The properties of µ(A) are similar to those of the Wigner distribution, concerning in particular continuity on L 2 (R d ) (Proposition 2.3), fundamental identity for W A f (Proposition 2.7) and Moyal identity (Proposition 2.9). Moreover, by using boundedness results for metaplectic operators on modulation spaces (Theorem 2.13, Corollary 2.14) we may easily deduce the estimates (8) W A (f, g) M p vs f M p g M p vs + g M p f M p vs . and under the assumption 0 < p ≤ 2 (Theorem 2.16) (9) f ∈ M p vs (R d ) ⇔ W A f ∈ M p vs (R 2d ), which extends several results in literature, see [10] and reference therein. More challenging issue is to discuss the equivalence of norms for modulation spaces, that is, for a fixed non-zero window function g ∈ S(R d ), (10) f M p,q ≍ W A (f, g) L p,q , 0 < p, q ≤ ∞, in particular for p = q, allowing the presence of weights v s : (11) f M p vs ≍ W A (f, g) L p vs , 0 < p ≤ ∞. Namely, we would like to extend in our context the characterizations of modulation spaces (3), (5). In this perspective it is clear that we have to limit attention to subclasses of Sp(2d, R). As a first attempt, it is natural to consider the covariant matrices A: W A (π(z)f, π(z)g) = T z W A (f, g), f, g ∈ S(R d ), z ∈ R 2d ; here for z = (z 1 , z 2 ), the operator π(z) = π(z 1 , z 2 ) = M z 2 T z 1 is the time-frequency shift, composition of the modulation M z 2 and translation T z 1 defined by M z 2 f (t) = e 2πiz 2 t f (t), T z 1 f (t) = f (z 1 − t), t, z 1 , z 2 ∈ R d . The covariance property of A is equivalent to being a member of the Cohen class for the related A-Wigner distribution (cf. [5,6,10,20]). In fact, we show (see Theorem 2.11): W A (f, g) = W (f, g) * σ A , f, g ∈ S(R d ), where (12) σ A = F −1 (e −πiζ·B A ζ ) ∈ S ′ (R 2d ), and B A is a symmetric 2d × 2d matrix that can be computed explicitly from the covariant matrix A, cf. (60) in the sequel. The Cohen class will play a role for applications to Schrödinger equations; though, it presents two drawbacks when looking at (10), (11). On one hand, it is too restrictive, since A = A ST in (7) is not covariant, that is the short-time Fourier transform is excluded. On the other hand, the matrix A = A τ in (6) is covariant for all τ ∈ R, in particular for the forbidden Rihaczek cases τ = 0, 1 for which (10), (11) fail. This suggests the introduction of the new class of shift-invertible matrices A ∈ Sp(2d, R) with related distributions W A satisfying (Definition 2.19) (13) \|W A (π(w)f, g)\| = \|T E A (w) W A (f, g)\|, f, g ∈ L 2 (R d ), w ∈ R 2d , for some E A ∈ GL(2d, R), with (14) T E A (w) W A (f, g)(z) = W A (f, g)(z − E A w), w, z ∈ R 2d . We prove that the shift-invertible distribution W A satisfies (11) and (15) f ∈ M p vs (R d ) ⇔ W A f ∈ L p vs (R 2d ). This provides a general characterization of the modulation spaces M p vs , see Theorem 2.22 and Corollary 2.23 for precise statements and bounds on the values of p. Note that the matrix A = A ST in (7) is shift-invertible, recapturing in this way the standard definition of modulation spaces. As far as the τ -Wigner matrix A = A τ concerns, it is shift-invertible for τ ∈ R \ {0, 1}. This can be read as an explanation of the anomaly of the Rihaczek distributions. The block decomposition of the shift-invertible matrix A and the corresponding matrix E A in (13), (14) can be explicitly computed, cf. (73) below, and we may characterize the relevant subclasses of the distributions W A which are simultaneously covariant and shift-invertible (Remark 2.20). Finally, we address to the more precise equivalence (10) concerning the case of different indices p, q. We first reconsider the τ -Wigner case, τ ∈ R \ {0, 1}, and extend, with respect to [11], the validity of (5) to 0 < p, q < ∞. This example suggests a deeper study of the matrices A ∈ Sp(2d, R) such that (16) µ (A) = F 2 T L where F 2 is the partial Fourier transform with respect to the second variable and T L is the L 2 -normalized change of variables defined by a d × d invertible matrix L, cf. [12]. We characterize the subclass of all the A ∈ Sp(2d, R) which are covariant and shift-invertible (see Proposition 2.25 and subsequent remark). Namely, for covariant shift-invertible matrices A of the form (16) we prove (17) f ∈ M p,q (R d ) ⇔ W A (f, g) ∈ L p,q (R 2d ) with equivalence of norms valid also in the weighted cases for 0 < p, q ≤ ∞ (Theorem 2.28). A further analysis concerns the covariant case (Wigner perturbations, according to the terminology of [12]). If A is covariant of the form (16) then (18) W A (f, g) = W (f, g) * σ A f, g ∈ S(R d ), where σ A has now the particular form (see Corollary 3.1). We perform a detailed study of such convolution kernel (Lemma 3.1, Proposition 3.3). In particular, we deduce Theorem 3.4 for weighted versions of the above equivalence). Besides providing a characterization for modulation spaces, the introduction of the A-Wigner distributions is strongly motivated by the applications to Schrödinger equations. Let us first recall some classical results for the case of the quadratic Hamiltonians. W f ∈ M p,q R 2d ⇔ W A f ∈ M p,q R 2d , 1 ≤ p, q ≤ ∞ (see Namely, consider (19)    i ∂u ∂t + Op w (H)u = 0 u(0, x) = u 0 (x). where Op w (H) is the Weyl quantization of a real quadratic polynomial in R 2d : (20) H(x, ξ) = 1 2 xAx + ξBx + 1 2 ξCξ with A, C symmetric and B invertible. We consider the Hamiltonian system (21) 2πẋ = ∇ ξ H = Bx + Cξ, x(0) = y 2πξ = −∇ x H = −Ax − B T ξ, ξ(0) = η, with Hamiltonian matrix D := B C −A −B T ∈ sp(d, R) (sp(d, R) is the symplectic algebra). We have, for t ∈ R, χ t = e tD ∈ Sp(d, R) and a solution to (21) is given by (x, ξ) = χ t (y, η). The problem (19) is solved by the Schrödinger propagator u(t, x) = e itOpw(H) u 0 (x) = µ(χ t )u 0 for a continuous choice of the phase factor in the definition of µ(χ t ). If u 0 ∈ L 2 (R d ) then u(t, x) ∈ L 2 (R d ), for every t ∈ R, see for example the textbooks [15,18], whereas in the Lebesgue spaces L p (R d ), p = 2, the soulution u(t, x) does not keep the order of regularity of the initial datum u 0 . Modulation spaces reveal here their effectiveness, in fact from Theorem 2.13 (see also [20] and [10]) we have that u 0 ∈ M p vs (R d ) implies u(t, ·) ∈ M p vs (R d ), for every 0 < p < ∞, s ≥ 0. Returning now to the subject of the present paper, let us recall from the original work of Wigner [32] (see also [24]): The Wigner transform with respect to the space variable x of the solution u(t, x) of (19) is given by (22) W u(t, z) = W u 0 (χ −1 t z), z = (x, ξ) ∈ R 2d , t ∈ R. It is natural to replace the Wigner transform in (22) with more general distributions by keeping the action of the classical Hamiltonian flow χ t . A general result is easily obtained in the framework of the Cohen classes Q σ f = W f * σ, for any σ ∈ S ′ (R 2d ). Namely, assuming u ∈ S(R d ), we have (Theorem 4.2) (23) Q σ (u(t, ·))(z) = Q σt (u 0 )(χ −1 t z), z = (x, ξ) ∈ R 2d , t ∈ R. where σ t (z) = σ(χ t z). Note that in (23) the Cohen class Q σt in the right-hand side depends on the time t. We may as well keep Q σ (u 0 ) for a fixed σ in the right, and transfer the dependence on t to the left. The classical Wigner case in (22) corresponds to the choice σ = δ for which σ t (z) = δ(χ t z) = δ, for every z ∈ R 2d . Willing to give a precise functional setting to (23) in the framework of modulation spaces, we limit attention to Cohen distributions generated by covariant matrices A ∈ Sp(2d, R), Q σ u = W A u = W u * σ A , with kernel σ A given by (12). The identity (23) then reads (Proposition 4.4): (12), cf. (60). Taking then u 0 ∈ M p vs (R d ), 1 ≤ p ≤ 2, s ≥ 0, we have from (9), cf. Corollary 2.14: (24) Q σ (u(t, ·))(z) = W A (u(t, ·))(z) = W At (u 0 )(χ −1 t z), where A t ∈ Sp(2d, R) is covariant for all t ∈ R, with Cohen kernel σ At (z) = F −1 e −πiζ·B A t ζ (z), B At = (χ −1 t ) T B A χ −1 t , B A as in(25) W A (u(t, ·)) ∈ M p vs (R 2d ), W At u 0 ∈ M p vs (R 2d ) , t ∈ R, and each one of these conditions is equivalent to the assumption u 0 ∈ M p vs (R d ). Willing to have instead (26) W A (u(t, ·)) ∈ L p vs (R 2d ), W At u 0 ∈ L p vs (R 2d ), t ∈ R, we are led to assume that the matrix A is also shift-invertible. In Proposition 4.5 we shall prove that A is shift-invertible if and only if A t is shift-invertible, for any fixed t = 0. Hence in this case the conditions (26) are equivalent to u 0 ∈ M p vs (R d ). As an example, we shall test these results on the free particle. The property of regularity (26) is the starting point for a proceeding in localization similar to that in [11]. Namely, cf. Definition 4.6, for a covariant and shift-invertible A we define for f ∈ L 2 (R d ) the generalized Wigner wave front set WF p,s A (f ), 1 ≤ p ≤ 2, s ≥ 0, by setting z 0 = (x 0 , ξ 0 ) / ∈ WF p,s A (f ), z 0 = 0, if there exists a conic neighbourhood Γ z 0 ⊂ R 2d such that (27) Γz 0 z ps \|W A f (z)\| p dz < ∞. We have from (15) that WF p,s A (f ) = ∅ if and only if f ∈ M p vs (R d ), cf. Proposition 4.7. For the standard Wigner transform the notation WF p,s A 1/2 (f ), cf. (6), will be shortened to WF p,s (f ). From (24) and (26) we deduce the following propagation of micro-singularities for the solutions of (19), cf. Theorem 4.8: (28) WF p,s A (u(t, ·)) = χ t (WF p,s At (u 0 )), in particular for the standard Wigner transform (29) WF p,s (u(t, ·)) = χ t (WF p,s (u 0 )). We address to the forthcoming second part of [11] for a detailed study of WF p,s A with applications to Fourier integral operators and Schrödinger equations of more general type. We limit here to the following warning and remarks. First, we cannot extend to the Wigner wave front set all the properties of the classical wave front set of Hörmander, cf. [22] or its global version [21]. In fact, the inclusion of the wave front set of the solutions in the characteristic manifold, for a homogeneous linear partial differential equation, is false for the Wigner wave front. This depends on the existence of the ghost frequencies, see the final comments in [11]. On the other hand, the whole Wigner wave front, including its ghost part, is exactly preserved by the Schrödinger propagator, as clarified by (28) and (29). Time-frequency analysis tools Notations. We set t 2 = t · t, t ∈ R d , and xy = x · y is the scalar product on R d . The space S(R d ) denotes the Schwartz class whereas S ′ (R d ) the space of temperate distributions. The brackets f, g denote the extension to S ′ (R d )×S(R d ) of the inner product f, g = f (t)g(t)dt on L 2 (R d ) (conjugate-linear in the second component). The reflection operator I is given by If (t) = f (−t). The Fourier transform is normalized to bê f (ξ) = F f (ξ) = R d f (t)e −2πitξ dt. The symplectic matrix (30) J = 0 d×d I d×d −I d×d 0 d×d , (here I d , 0 d are the d × d identity matrix and null matrix, respectively) enters the definition of the standard symplectic form σ(z, z ′ ) = Jz · z ′ . They allow to introduce the symplectic Fourier transform: (31) F σ a(z) = R 2d e −2πiσ(z,z ′ ) a(z ′ ) dz ′ . The Fourier transform and symplectic Fourier transform are related by (32) F σ a(z) = F a(Jz) = F (a • J)(z), a ∈ S(R 2d ). For the study of perturbations of the Wigner distribution we will use the Ambiguity Function Amb (f ) defined as (33) Amb (f ) (x, ξ) = F σ (W f )(x, ξ) = R d f y + x 2 f y − x 2 e −2πiyξ dy. We denote by GL(2d, R) the linear group of 2d × 2d invertible matrices; for a complex-valued function F on R 2d and L ∈ GL(2d, R) we define (34) T L F (x, y) = \| det L\|F (L(x, y)), (x, y) ∈ R 2d , with the convention L(x, y) = L x y , (x, y) ∈ R 2d . For 1 ≤ p ≤ ∞, the spaces ℓ ∞ mn ℓ p m ′ n ′ are the Banach spaces of sequences {a m ′ ,n ′ ,m,n } such that such that a m ′ ,n ′ ,m,n ℓ ∞ mn ℓ p m ′ n ′ := sup m,n∈Z d   m ′ ,n ′ ∈Z d \|a m ′ ,n ′ ,m,n \| p   1/p < ∞ (with obvious changes when p = ∞). Modulation spaces. In this paper v is a continuous, positive, submultiplica- tive weight function on R d , i.e., v(z 1 +z 2 ) ≤ v(z 1 )v(z 2 ), for all z 1 , z 2 ∈ R d . A weight function m is in M v (R d ) if m is a positive, continuous weight function on R d and it is v-moderate: m(z 1 + z 2 ) ≤ Cv(z 1 )m(z 2 ) for all z 1 , z 2 ∈ R d . In the following we will work with weights on R 2d of the type (35) v s (z) = z s = (1 + \|z\| 2 ) s/2 , z ∈ R 2d , For s < 0, v s is v \|s\| -moderate. For weight functions m 1 , m 2 on R d , we will use the notation (m 1 ⊗ m 2 )(x, ξ) = m 1 (x)m 2 (ξ), x, ξ ∈ R d , and similarly for weights m 1 , m 2 on R 2d . In particular, we shall use the weight functions on R 4d : (36) (v s ⊗ 1)(z, ζ) = (1 + \|z\| 2 ) s/2 , (1 ⊗ v s )(z, ζ) = (1 + \|ζ\| 2 ) s/2 , z, ζ ∈ R 2d . The modulation spaces, introduced by Feichtinger in [13] and extended to the quasi-Banach setting Galperin and Samarah [17], are now available in many textbooks, see e.g. [3,10,20]. Fix a non-zero window g in the Schwartz class S(R d ). Consider a weight function m ∈ M v and indices 0 < p, q ≤ ∞. The modulation space M p,q m (R d ) is the subspace of tempered distributions f ∈ S ′ (R d ) with (37) f M p,q m = V g f L p,q m = R d R d \|V g f (x, ξ)\| p m(x, ξ) p dx q p dξ 1 q < ∞ (natural changes with p = ∞ or q = ∞). We write M p m (R d ) for M p,p m (R d ) and M p,q (R d ) if m ≡ 1. For 1 ≤ p, q ≤ ∞, the space M p,q (R d ) is a Banach space whose definition is independent of the choice of the window g: different non-zero window functions in S(R d ) yield equivalent norms. The window class can be extended to the modulation space M 1 v (R d ) (Feichtinger algebra). The modulation space M ∞,1 (R d ) coincides with the Sjöstrand's class in [27]. We recall their inclusion properties: (38) S(R d ) ⊆ M p 1 ,q 1 m (R d ) ⊆ M p 2 ,q 2 m (R d ) ⊆ S ′ (R d ), p 1 ≤ p 2 , q 1 ≤ q 2 . Denoting by M p,q m (R d ) the closure of S(R d ) in the M p,q m -norm, we observe M p,q m (R d ) ⊆ M p,q m (R d ), 0 < p, q ≤ ∞, and M p,q m (R d ) = M p,q m (R d ), 0 < p, q < ∞. For m, w ∈ M v (R d ), the Wiener amalgam spaces W (F L p m , L q w )(R d ) can be viewed as images under Fourier transform of the modulation spaces. Namely, for p, q ∈ (0, ∞], f ∈ S ′ (R d ) belongs to W (F L p m , L q w )(R d ) if f W (F L p m ,L q w )(R d ) := R d R d \|V g f (x, ξ)\| p m(ξ) p dξ q/p w(x) q dx 1/q < ∞ (obvious modifications for p = ∞ or q = ∞). Using the fundamental identity of time-frequency analysis [10, formula (1.31)] (39) V g f (x, ξ) = e −2πixξ Vĝf (ξ, −x), we can deduce \|V g f (x, ξ)\| = \|Vĝf (ξ, −x)\| = \|F (f T ξĝ )(−x)\| so that f M p,q m⊗w = R d f T ξĝ q F L p v m(ξ) dξ 1/q = f W (F L p m ,L q w ) . The above equality of norms yields (40) F (M p,q v⊗w ) = W (F L p v , L q w ). 2.2. The metaplectic representation. Recall the symplectic group (41) Sp(d, R) = A ∈ GL(2d, R) : A T JA = J , where A T denotes the transpose of A and the symplectic matrix J is defined in (30). In the sequel, we shall also refer to symplectic matrices in double dimension, induced from the standard symplectic form on R 4d : (42) Sp(2d, R) = A ∈ GL(4d, R) : A T JA = J , where J is the one in (30) with 0 d×d replaced by 0 2d×2d and I d×d replaced by I 2d×2d . The metaplectic representation µ is a unitary representation of the (double cover of the) symplectic group Sp(d, R) on L 2 (R d ). The symplectic algebra sp(d, R) is the set of all 2d × 2d real matrices A such that e tA ∈ Sp(d, R) for all t ∈ R. For some elements of Sp(d, R) the metaplectic representation can be computed explicitly. Namely, using the notations in [18,19] , for f ∈ L 2 (R d ), C real symmetric d × d matrix (C T = C) we have, up to a phase factor s (that is, \|s\| = 1), (43) µ(J)f = F f ; for V C := I d×d 0 C I d×d , up to a phase factor (44) µ (V C ) f (x) = e iπCx·x f (x). Special instances of metaplectic operators also called rescaling operators. They are metaplectic operators µ(D L ) associated with the symplectic matrix D L constructed as follows. For any L ∈ GL(d, R), (45) D L = L −1 0 d×d 0 d×d L T ∈ Sp(d, R). Then, up to a phase factor, (46) µ(D L )F (x) = \| det L\|F (Lx) = T L F (x), F ∈ L 2 (R d ). The metaplectic operators posses a group structure called the metaplectic group. Proposition 2.1. The metaplectic group is generated by the operators µ(J), µ(D L ) and µ(V C ). In the paper we shall work both with the symplectic group Sp(d, R) of 2d × 2d matrices and Sp(2d, R) of 4d × 4d ones. In particular, the matrix A ∈ Sp(2d, R) is assumed to have the 4 × 4 block decomposition of 2d × 2d matrices: (47) A = A B C D with the decompositions of the 2d × 2d sub-blocks as follows: (48) A = A 11 A 12 A 21 A 22 , B = B 11 B 12 B 21 B 22 , C = C 11 C 12 C 21 C 22 , D = D 11 D 12 D 21 D 22 . Definition 2.2. For a 4d × 4d symplectic matrix A ∈ Sp(2d, R) we define the time-frequency representation A-Wigner by (49) W A (f, g) = µ(A)(f ⊗ḡ), f, g ∈ L 2 (R d ). We set W A f := W A (f, f ). 2.2.1. Properties of W A (f, g). In what follows we list all the elementary properties enjoyed by the A-Wigner distribution. The continuity of W A was shown in [11]: Proposition 2.3. Assume A ∈ Sp(2d, R). Then, (1) If f, g ∈ L 2 (R d ), then W A (f, g) ∈ L 2 (R 2d ) and the mapping W A : L 2 (R d ) × L 2 (R d ) → L 2 (R 2d ) is continuous. (2) If f, g ∈ S(R 2d ), then W A (f, g) ∈ S(R 2d ) and the mapping W A : S(R d ) × S(R d ) → S(R 2d ) is continuous. (3) If f, g ∈ S ′ (R d ), then W A (f, g) ∈ S ′ (R 2d ) and the mapping W A : S ′ (R d ) × S ′ (R d ) → S ′ (R 2d ) is continuous. Proposition 2.4 (Interchanging f and g). For A ∈ Sp(2d, R) with block decomposition (47) and f, g ∈ L 2 R d . Then W A (g, f ) = W A (f ,ḡ), where A = AL BL CL DL and (50) L = 0 d×d I d×d I d×d 0 d×d . Precisely, using the sub-block decomposition (48), we obtain AL = A 12 A 11 A 22 A 21 and similarly for the other block matrices B, C, D. Proof. Consider the matrix L defined in (50) and observe that L T = L −1 = L. The symplectic matrix D L in (45) becomes D L = L −1 0 d×d 0 d×d L T = L 0 d×d 0 d×d L . With our choice of L, µ(D L )(g ⊗f )(x, y) = (g ⊗f )(y, x) =f ⊗ g(x, y). This let us factorize W A (g, f ) as follows: W A (g, f )(x, y) = µ(A)(g⊗f)(x, y) = µ(AD −1 L D L )(g⊗f )(x, y) = µ(AD L )(f ⊗g)(x, y) , and the claim easily follows by observing that AD L = A. We now limit ourselves to matrices A ∈ Sp(2d, R) such that (51) µ(A) = F 2 T L where F 2 is the partial Fourier transform with respect to the second variables y defined by (52) F 2 F (x, ξ) = R d e −2πiy·ξ F (x, y) dy, F ∈ L 2 (R 2d ). and the change of coordinates T L is defined in (34). The following fact was established in [12, Proposition 3.3]: Proposition 2.5. For f, g ∈ L 2 R d , µ(A) of the form (51) with L = L 11 L 12 L 21 L 22 , then W A (g, f ) (x, ω) = W B (f, g) (x, ω), with µ(B) = F 2 TL, withL = L 21 −L 22 L 11 −L 12 . More generally, Proposition 2.6. For A ∈ Sp(2d, R), we have W A (g, f ) = W B (f, g), for a suitable B ∈ Sp(2d, R). Proof. We use Proposition 2.1, and observe that µ (J)f = µ(J −1 )f , µ(V C )f = µ(V −C )f and µ(D L )f = µ(D L )f . This gives the claim. What follows can be viewed as a generalization of the fundamental identity of time-frequency analysis for the STFT, cf. [10, (1.31)]. Proposition 2.7 (Fundamental identity of time-frequency analysis). For A ∈ Sp(2d, R) with block decomposition (47) and f, g ∈ L 2 R d , then W A f ,ĝ = W A (f, g) , where A = BL AL DL CL and (53) L = I 0 0 −I Proof. Using the reflection operators Ig(t) = g(−t), we can writê f ⊗ĝ =f ⊗ Iĝ = F T L (f ⊗ḡ) where L is defined in (53). Hence, W A (f ,ĝ) = µ(A)(f ⊗ĝ) = µ(A)F T L (f ⊗ḡ) = µ(AJD L )(f ⊗ḡ). The conclusion is a simple computation. Proposition 2.8 (Fourier transform of W A ). Let A ∈ Sp(2d, R) and f, g ∈ L 2 R d . Then, (54) F W A (f, g) = W A (f, g) , where A = (A T ) −1 J. Proof. Since, up to a phase factor, F µ(A) = µ(JA), the result follows from the symplectic group property (41). Proposition 2.9 (Moyal's Identity). Let A ∈ Sp(2d, R) and f 1 , f 2 , g 1 , g 2 ∈ L 2 R d . Then, (55) W A (f 1 , g 1 ) , W A (f 2 , g 2 ) L 2 (R 2d ) = f 1 , f 2 L 2 (R d ) g 1 , g 2 L 2 (R d ) , in particular, for f 1 = f 2 = f , g 1 = g 2 = g, W A (f, g) , W A (f, g) L 2 (R 2d ) = f 2 L 2 (R d ) g 2 L 2 (R d ) . Proof. We simply use that µ(A) is unitary on L 2 (R 2d ): W A (f 1 , g 1 ) , W A (f 2 , g 2 ) L 2 (R 2d ) = µ(A) (f 1 ⊗ḡ 1 ) , µ(A) (f 2 ⊗ḡ 2 ) L 2 (R 2d ) = µ(A) −1 µ(A) (f 1 ⊗ḡ 1 ) , (f 2 ⊗ḡ 2 ) L 2 (R 2d ) , and the claim follows. A simple computation shows the following polarization identity: (56) W A (f + g) = W A (f ) + W A (g) + W A (f, g) + W A (g, f ). The Covariance Property of [11,Proposition 4.3] can be generalized and improved as follows: Proposition 2.10 (Covariance Property). Consider A ∈ Sp(2d, R) having block decomposition A =     A 11 A 12 A 13 A 14 A 21 A 22 A 23 A 24 A 31 A 32 A 33 A 34 A 41 A 42 A 43 A 44     with A ij , i, j = 1, . . . , 4, d × d real matrices. Then the representation W A in (49) is covariant, namely (57) W A (π(z)f, π(z)g) = T z W A (f, g), f, g ∈ S(R d ), z ∈ R 2d , if and only if A is of the form (58) A =     A 11 I d×d − A 11 A 13 A 13 A 21 −A 21 I d×d − A T 11 −A T 11 0 d×d 0 d×d I d×d I d×d −I d×d I d×d 0 d×d 0 d×d     . with A 13 = A T 13 , A T 21 = A 21 . The result does not depend on the choice of the phase factor in the definition of µ(A) and W A in (49). Proof. The equivalence of (57) and the matrix (59) A =     A 11 I d×d − A 11 A 13 A 13 A 21 −A 21 I d×d − A T 11 −A T 11 A 31 −A 31 A 33 A 33 A 41 −A 41 A 43 A 43     . is a straightforward generalization of the proof of [11,Proposition 4.3]. We notice that in the last element of the second row of [11,Formula (108)] the entry A T 11 should be replaced by −A T 11 as in (59). We then use the matrix-block properties for symplectic matrices (see, e.g. [15,Proposition 4.1]) to obtain (58). First, the condition AB T = BA T (where A and B are the 2d×2d blocks in (48)) gives A T 13 = A 13 . The block property: A T C = C T A yields to A 31 = 0 d×d and A T 21 A 41 = A T 41 A 21 . From B T D = D T B we infer A 43 = 0 d×d . Condition If we introduce the real symmetric 2d × 2d matrix (60) B A = A 13 1 2 I d×d − A 11 1 2 I d×d − A T 11 −A 21 , the covariance property of A can be viewed as Cohen class property as shown below. The proof is a straightforward generalization of [11, Theorem 4.6]: Theorem 2.11. Let A ∈ Sp(2d, R) be of the form (58). Then (61) W A (f, g) = W (f, g) * σ A , f, g ∈ S(R d ), where (62) σ A = F −1 (e −πiζ·B A ζ ) ∈ S ′ (R 2d ), and B A defined in (60). Proposition 2.12. For z = (z 1 , z 2 ), u = (u 1 , u 2 ), we have W A (π(z)f, π(u)g) = M (ζ 3 ,ζ 4 ) T (ζ 1 ,ζ 2 ) W A (f, g) f, g ∈ S(R d ), ζ i ∈ R 2d , i = 1, . . . , 4, where (ζ 1 , ζ 2 ) = (A 11 z 1 + (I − A 11 )u 1 + A 13 (z 2 − u 2 ), A 21 (z 1 − u 1 ) + (I − A T 11 )z 2 − A T 11 u 2 ) (63) (ζ 3 , ζ 4 ) = (A 31 (z 1 − u 1 ) + A 33 (z 2 − u 2 ), A 41 (z 1 − u 1 ) + A 43 (z 2 − u 2 )). Proof. Using the intertwining property (see e.g. Formula (1.10) in [10]) π(Aζ) = c A µ(A)π(ζ)µ(A) −1 , ζ ∈ R 4d (where c A is a phase factor: \|c A \| = 1), we calculate W A (π(z 1 , z 2 )f, π(u 1 , u 2 )g) = µ(A)[π(z 1 , u 1 , z 2 , −u 2 )(f ⊗ḡ)] = c −1 A π(A(z 1 , u 1 , z 2 , −u 2 ))W A (f, g). The covariance of W A gives the matrix block-decomposition in (58) so that π(A(z 1 , u 1 , z 2 , −u 2 )) = c A T (ζ 1 ,ζ 2 ) M (ζ 3 ,ζ 4 ) , with (ζ 1 , ζ 2 ) ∈ R 2d and (ζ 3 , ζ 4 ) ∈ R 2d in (63). Metaplectic operators are bounded on modulation spaces, as shown below. Namely, consider the lattice Λ = αZ d × βZ d and two windows g, γ ∈ S(R d ) such that the related Gabor frame operator S g,γ := S Λ g,γ satisfies S g,γ = I on L 2 (R d ). If we set g m,n := M βn T αm g, it remains to prove that the matrix operator (67) \|K m ′ ,n ′ ,m,n \| v −r (A(m, n) − (m ′ , n ′ )) v s (m ′ , n ′ ) v s (A(m, n) − (m ′ , n ′ ))v s (m, n) . (67) is bounded, so we deduce (65). Now, the last quotient in Corollary 2.14. Under the assumptions of Theorem 2.13 we have (68) µ(A)f M p vs ≍ f M p vs , f ∈ M p vs (R d ), (with M ∞ (R d ) in place of M ∞ (R d ) for p = ∞). Proof. Using the invertibility property of metaplectic operators, f M p vs = µ(A) −1 µ(A)f M p vs = µ(A −1 )µ(A)f M p vs µ(A)f M p vs where the last estimate follows from Theorem 2.13 since A −1 ∈ Sp(d, R). The reverse inequality is stated in Theorem 2.13. Theorem 2.15. Assume f, g ∈ M p vs (R d ), 0 < p ≤ ∞, s ≥ 0. For any A ∈ Sp(2d, R) the A-Wigner W A (f, g) is in M p vs (R 2d ), with (69) W A (f, g) M p vs f M p g M p vs + g M p f M p vs . Proof. By Theorem 2.13 (with dimension 2d in place of d) we can write R). Then the following statements are equivalent: (70) W A (f, g) M p vs = µ(A)(f ⊗ḡ) M p vs f ⊗ḡ M p vs . Note also that v s (z, ζ) ≍ (v s ⊗ 1)(z, ζ) + (1 ⊗ v s )(z, ζ), so that W A (f, g) M p vs f ⊗ḡ M p vs⊗1+1⊗vs f ⊗ḡ M p vs⊗1 + f ⊗ḡ M p 1⊗vs = f M p vs g M p + f M p g M p vs . The proof is concluded. Theorem 2.16. Assume f ∈ M p vs (R d ), 0 < p ≤ 2, s ≥ 0, A ∈ Sp(2d,(i) f ∈ M p vs (R d ) (ii) W A (f ) ∈ M p vs (R 2d ). Proof. If f (t) = 0 for a.e. t then W A (f )(x, ξ) = 0 and the equivalence is trivially true. Let us now consider the non-trivial case. (i) ⇒ (ii). It is a consequence of Theorem 2.15. In particular, from (69) for f = g we have Fixing f = g and using (68), W A (f ) M p vs f M p vs f M p f 2 M p vs . (ii) ⇒ (i).W A (f ) M p vs = µ(A)(f ⊗f ) M p vs ≍ f ⊗f M p vs . Note that f ⊗f M p vs⊗1 ≍ f M p vs f M p . So, for f ∈ L 2 (R d ) \ {0}, we have f M p vs ≍ 1 f M p f ⊗f M p vs⊗1 1 f L 2 f ⊗f M p vs , since f L 2 f M p , 0 < p ≤ 2. Theorem 2.17 (Inversion formula for the A-Wigner distribution). Consider g 1 , g 2 ∈ L 2 (R d ) with g 1 , g 2 = 0, A ∈ Sp(2d, R). Then, for any f ∈ L 2 (R d ), (71) f = 1 g 2 , g 1 R d µ(A −1 )W A (f, g 1 )(x, ξ)g 2 dξ. Proof. Observing that µ(A −1 )W A (f, g 1 ) = µ(A −1 )µ(A)(f ⊗ḡ 1 ) = f ⊗ḡ 1 , we can write R d µ(A −1 )W A (f, g 1 )(x, ξ)g 2 (ξ) dξ = R d f (x)ḡ 1 (ξ)g 2 (ξ) dξ = f (x) g 2 , g 1 and the equality (71) follows. Proposition 2.18. For f, g 1 , g 2 , g 3 ∈ L 2 (R d ), A ∈ Sp(2d, R), we have (72) V g 3 f (w) = 1 g 2 , g 1 W A (f, g 1 ), W A (π(w)g 3 , g 2 ) L 2 (R 2d ) . Proof. From the preceding inversion formula (71) we have V g 3 f (w) = 1 g 2 , g 1 R 2d W A (f, g 1 )W A (π(w)g 1 , g 2 )π(w)g 3 (x)dxdξ = 1 g 2 , g 1 R 2d W A (π(w)g 1 , g 2 )(x, ξ)µ(A)(g 2 (ξ)π(w)g 3 (x))dxdξ, since µ(A −1 ) = µ(A) * . Observe that the integrals above are absolutely convergent integrals since π(w) is an isometry on L 2 (R d ) and W A : L 2 (R d )×L 2 (R d ) → L 2 (R 2d ),Definition 2.19. Given A ∈ Sp(2d, R), we say that W A is shift-invertible if \|W A (π(w)f, g)\| = \|T E A (w) W A (f, g)\|, f, g ∈ L 2 (R d ), w ∈ R 2d , for some E A ∈ GL(2d, R), with T E A (w) W A (f, g)(z) = W A (f, g)(z − E A w), w, z ∈ R 2d . Not every A-Wigner satisfies the above property. Let us compute W A (π(w)f, g) explicitly. Consider A with the block decomposition in (47), and sub-bocks (48). Easy calculations and the intertwining formula µ(A)π(z) = c A π(Az)µ(A) with \|c A \| = 1 show, for w = (w 1 , w 2 ), W A (π(w)f, g) = µ(A)((π(w)f ) ⊗ḡ) = µ(A)π(w 1 , 0, w 2 , 0)(f ⊗ḡ) = c A π(A(w 1 , 0, w 2 , 0) T )W A (f, g), = c A π(A(w 1 , 0) T + B(w 2 , 0) T , C(w 1 , 0) T + D(w 2 , 0) T )W A (f, g) = c A π(A 11 w 1 , A 21 w 1 ) + (B 11 w 2 , B 21 w 2 ), (C 11 w 1 , C 21 w 1 ) + (D 11 w 2 , D 21 w 2 ))W A (f, g) = c A π(A 11 w 1 + B 11 w 2 , A 21 w 1 + B 21 w 2 , C 11 w 1 + D 11 w 2 , C 21 w 1 + D 21 w 2 )W A (f, g) = c A M C 11 w 1 +D 11 w 2 ,C 21 w 1 +D 21 w 2 T A 11 w 1 +B 11 w 2 ,A 21 w 1 +B 21 w 2 W A (f, g). so that \|W A (π(w)f, g)\| = \|T A 11 w 1 +B 11 w 2 ,A 21 w 1 +B 21 w 2 W A (f, g)\|. Hence the matrix E A in Definition 2.19 is given by (73) E A = A 11 B 11 A 21 B 21 . W A is shift-invertible if and only if the matrix E A is invertible. Remark 2.20. (i) If A ∈ Sp(2d, R) is a covariant matrix then (74) E A = A 11 A 13 A 21 I d×d − A T 11 . Hence if E A is invertible the covariant matrix A is shift-invertible. (ii) For τ -Wigner distributions the matrix A = A τ is shown in (6). The related matrix E τ := E Aτ is E τ = (1 − τ )I d×d 0 d×d 0 d×d τ I d×d , so that A τ is covariant for every τ ∈ R, whereas it is shift-invertible for τ ∈ R \ {0, 1}. (iii) For f, g ∈ L 2 (R d ), V g f = µ(A ST )(f ⊗ḡ),(75) \|V g π(w)f \| = \|T w V g f \|, w ∈ R 2d . This implies that A = A ST in (7) is shift-invertible. Observe that in this case, E ST := E A ST is E ST = I 2d×2d = I d×d 0 d×d 0 d×d I d×d according to relation in (75). We notice that A ST is not covariant. Proposition 2.21 (Relation between the matrix E A and B A ). If A ∈ Sp(2d, R) is a covariant matrix with related matrix E A in (74) and symmetric matrix B A in (60), then (76) E A J + 1 2 J = B A . Proof. It is a simple computation. In fact, E A J + J 2 = A 13 −A 11 I d×d − A T 11 −A 21 + 1 2 0 d×d I d×d −I d×d 0 d×d = A 13 −A 11 + 1 2 I d×d 1 2 I d×d − A T 11 −A 21 = B A . This concludes the proof. Observe that the next result extends [11,Theorem 3.11] to every 0 < p ≤ ∞. Theorem 2.22. Fix g ∈ S(R d ). For A ∈ Sp(2d, R) we have the following issues: [28,Proposition 2.9], whereas the case 0 < p < 1 is a direct consequence of [30,Theorem 2.4] (i) For 0 < p < 2, if f ∈ M p vs (R d ) then W A (f, g) ∈ L p vs (R 2d ). (ii) Let W A be shift-invertible according to the preceding definition. Then, (iia) For s ≥ 0, 1 ≤ p ≤ 2, (77) f ∈ M p vs (R d ) ⇔ W A (f, g) ∈ L p vs (R 2d ), with equivalence of norms f M p vs ≍ W A (f, g) L p vs . (iib) For 1 ≤ p ≤ ∞, if W A (f, g) ∈ L p vs (R 2d ) then f ∈ M p vs (R d ). (iic) For 0 < p < 1, if W A (f, g) ∈ L p vs (R 2d ) and there exists a Gabor frame G(γ, Λ) for L 2 (R d ) with γ ∈ S(R d ) such that the sequence W A (f, γ)(λ) ∈ ℓ p vs (Λ), then f ∈ M p vs (R d ). Proof. (i) Let us recall that S(R d ) ⊂ M p vs (R d ), 0 < p ≤ ∞, s ∈ R. Assume first f ∈ M p vs (R d ),M p vs (R 2d ) ֒→ M p vs⊗1 (R 2d ), For 0 < p ≤ 2 (seewith B = L p vs ) M p vs⊗1 (R 2d ) ֒→ L p vs (R 2d ), hence W A (f, g) ∈ L p vs (R 2d ). (ii) Assume now that W A is shift-invertible and W A (f, g) ∈ L p vs (R 2d ) . Then, by Proposition 2.18, with g 1 = g 3 , \|V g 1 f (w)\| 1 \| g 2 , g 1 \| \| W A (f, g 1 ), W A (π(w)g 1 , g 2 ) L 2 (R 2d ) \| R 2d \|W A (f, g 1 )\|(u)\|W A (π(w)g 1 , g 2 )\|(u)du R 2d \|W A (f, g 1 )\|(u)\|W A (g 1 , g 2 )\|(u − E A w)du R 2d \|W A (f, g 1 )\|(u)\|[W A (g 1 , g 2 )] * \|(E A w − u)du Hence f M p vs ≍ V g 1 f L p vs \|W A (f, g 1 )\| * \|[W A (g 1 , g 2 )] * \|(E A ·) L p vs ≍ \|W A (f, g 1 )\| * \|[W A (g 1 , g 2 )] * L p vs since v s (y) ≍ v s (E −1 A y). Now, Young's convolution inequalities for 1 ≤ p ≤ ∞ gives \|W A (f, g 1 )\| * \|[W A (g 1 , g 2 )] * L p vs ≤ W A (f, g 1 ) L p vs W A (g 1 , g 2 ) L 1 vs < ∞, since W A (g 1 , g 2 ) ∈ S(R 2d ) for g 1 , g 2 ∈ S(R d ) by Proposition 2.3. This proves the implication in (iib). Moreover, item (i) and the previous estimate yield the equivalence in (iia). It remains to show item (iic). For 0 < p < 1, consider γ ∈ S(R d ) such that G(γ; Λ) is a Gabor frame for L 2 (R d ), then, arguing as above with γ in place of g 3 : f M p vs ≍ V γ f L p vs ≍ V γ f ℓ p vs \|W A (f, γ)\| * \|[W A (γ, g 2 )] * \|(E·) ℓ p vs ≍ \|W A (f, γ)\| * \|[W A (γ, g 2 )] * ℓ p vs ≤ W A (f, γ) ℓ p vs [W A (γ, g 2 )] * ℓ p vs < ∞, by the convolution property for sequences: ℓ p vs * ℓ p vs ֒→ ℓ p vs , s ≥ 0, 0 < p ≤ 1; and W A (γ, g 2 ) ∈ S(R 2d ) for γ, g 2 ∈ S(R d ) (cf., Proposition 2.3). In fact, the restriction W A (γ, g 2 )(λ), λ ∈ Λ is in ℓ p vs (Λ), for every 0 < p ≤ ∞, s ≥ 0. This concludes the proof. Corollary 2.23. For s ≥ 0, 1 ≤ p ≤ 2, A ∈ Sp(2d, R) such that W A is shift- invertible. Then f ∈ M p vs (R d ) ⇔ W A f ∈ L p vs (R 2d ). Proof. f ∈ M p vs (R d ) ⇒ W A f ∈ L p vs (R 2d ) is a straightforward generalization of the proof of Theorem 2.22(i), with g = f ∈ M p vs (R d ). Vice versa, following the proof pattern of Theorem 2.22 (ii) with Proposition 2.18 applied for g 1 , g 2 , g 3 ∈ S(R d ) we can write (78) f M p vs W A (f, g 1 ) L p vs W A (g 3 , g 2 ) L 1 vs . Now, for f ∈ M p vs (R d ) there exists a sequence (g n ) n ⊂ S(R d ) such that g n → f in M p vs (R dW A (f, f ) − W A (f, g n ) L p vs = W A (f, f − g n ) L p vs ≤ W A (f, f − g n ) M p vs⊗1 ≤ W A (f, f − g n ) M p vs = W A (f, f − g n ) M p vs f M p vs g n − g M p vs , where the last inequality is due to Theorem 2.15. Since g n −g M p vs → 0 as n → ∞, we obtain that W A (f, g n ) L p vs → W A (f, f ) L p vs as n → ∞ and the thesis follows by replacing g 1 by g n in (78) and letting n → ∞. For τ -Wigner distributions we have a characterization for every 0 < p ≤ ∞, as explained below. Notice that we extend Theorem 3.11 of [11] to 0 < p ≤ ∞ for the weight v = v s , s ≥ 0. Proposition 2.24. Consider 0 < p, q ≤ ∞, τ ∈ R \ {(0, 1)}. Then, for any g ∈ S(R d ), (79) f ∈ M p,q vs (R d ) ⇔ W τ (f, g) ∈ L p,q vs (R 2d ). For 1 ≤ p, q ≤ ∞ the window g can be chosen in the larger class M 1 vs (R d ). Proof. For p = q and 1 ≤ p ≤ ∞ the result was proved in Theorem 3.11 of [11]. Let us prove the general case. By Corollary 3.3. of [11], with Q τ g in place of g, we can write V Qτ g f (x, ξ) = τ d e −2πi(1−τ )xξ W τ (f, g) B −1 τ (x, ξ) , where Q τ g(t) = Ig 1 − τ τ t , t ∈ R d , Ig(t) := g(−t), and B −1 τ = (1 − τ )I d 0 d 0 d τ I d . The result is then a simple computation: (51), where F 2 is the partial Fourier transform with respect to the second variables y defined in (52) and the change of coordinates T L is defined in (34). As observed in [11], see also [23], f M p,q vs ≍ V Qτ g f L p,q vs = W τ (f, g)(B −1 τ ·) L p,q vs ≍ W τ (f, g)((1 − τ )·, τ ·) L p,q vs ≍ W τ (f, g) L p,q vs , since v s ((1 − τ )·, τ ·)) ≍ τ v s , for τ = R \ {0, 1}. STFT and (80) µ(A F T 2 ) = F 2 , where (81) A F T 2 = A F T 2 11 A F T 2 12 A F T 2 21 A F T 2 22 ∈ Sp(2d, R), and A F T 2 11 , A F T 2 12 , A F T 2 21 , A F T 2 22 are the 2d × 2d matrices: (82) A F T 2 11 = A F T 2 22 = I d×d 0 d×d 0 d×d 0 d×d , A F T 2 12 = 0 d×d 0 d×d 0 d×d I d×d , A F T 2 21 = −A F T 2 12 . Proposition 2.25. A covariant matrix A ∈ Sp(2d, R) satisfies (51) if and only if (83) A =     A 11 I d×d − A 11 0 d×d 0 d×d 0 d×d 0 d×d I d×d − A T 11 −A T 11 0 d×d 0 d×d I d×d I d×d −I d×d I d×d 0 d×d 0 d×d     (observe that A 13 = A 21 = 0 d×d ) and the matrix L in (34) is given by (84) L = I d×d I d×d − A 11 I d×d −A 11 . Proof. Up to a phase factor we can write A = A F T 2 D L , where D L is defined in (45). The claim is then a straightforward computation, using that L −1 = A 11 I d×d − A 11 I d×d −I d×d . Remark 2.26. (i) The matrix L in (84) is invertible for every d × d real matrix A 11 . (We stress that A 11 is not required to be invertible). In fact, we have det L = det(−I d×d ) = (−1) d . (ii) Under the assumptions of Proposition 2.25 the matrix E A becomes (85) E A = A 11 0 d×d 0 d×d I d×d − A T 11 so that E A is invertible if and only if A 11 and I d×d −A T 11 (or, equivalently, I d×d −A 11 ) are invertible matrices. In other words, A is shift-invertible if and only if A 11 and I d×d − A 11 are invertible matrices. (iii) For τ -Wigner distributions the matrix L = L τ is easily computed to be (86) L τ = I d×d τ I d×d I d×d −(1 − τ )I d×d . We are interested to determine the conditions under which a covariant A-Wigner W A = c A F 2 D L with \|c A \| = 1, can be related to the STFT. We recall that the matrix L takes the form in (84) so that (L −1 ) T = A T 11 I d×d I d×d − A T 11 −I d×d . Theorem 2.27. Let A ∈ Sp(2d, R) be a covariant matrix satisfying (51) and shiftinvertible. For every f, g ∈ L 2 R d , the following formula holds: (87) W A (f, g) (x, ξ) = \|det(I d×d − A 11 )\| −1 e 2πi(I−A T 11 ) −1 ξ·x Vgf (A −1 11 x, (I−A T 11 ) −1 ξ), x, ξ ∈ R d , where (88)g (t) = g −A 11 (I d×d − A 11 ) −1 t . Proof. Since A is shif-invertible the matrices A 11 and I d×d − A 11 are invertible. Then the result follows from Theorem 3.8 of [12]. Theorem 2.28. Consider 0 < p, q ≤ ∞, A ∈ Sp(2d, R) as in Theorem 2.27. Then, for any g ∈ S(R d ), (89) f ∈ M p,q vs (R d ) ⇔ W A (f, g) ∈ L p,q vs (R 2d ). with equivalence of norms f M p,q vs ≍ W A (f, g) L p,q vs . For 1 ≤ p, q ≤ ∞ the window g can be chosen in the larger class M 1 vs (R d ). Proof. It is a straightforward consequence of Theorem 2.27. In fact, for g ∈ S(R d ) and under the assumptions det A 11 = 0, det(I d×d − A 11 ) = 0, the rescaled functioñ g in (88) is in S(R d ) and by (87), f M p,q vs ≍ Vgf L p,q vs ≍ W A (f, g)(A 11 ·, ·) L p,q vs ≍ W A (f, g) L p,q vs , since v s (A −1 11 z 1 , z 2 ) = (1 + \|A −1 11 z 1 \| 2 + \|z 2 \| 2 ) s/2 ≍ (1 + \|z 1 \| 2 + \|z 2 \| 2 ) s/2 , s ∈ R. For p, q ≥ 1 the windows can be chosen in the larger class M 1 vs (R d ) and we can argue as above by observing thatg in (88) is in M 1 vs (R d ) whenever g is. A-Perturbations of the Wigner distribution This section studies the covariant A-Wigner representations as perturbations of the Wigner distributions in (61): W A (f, g) = W (f, g) * σ A f, g ∈ S(R d ), where the kernel σ A is defined in (62). We revisit in wider generality the linear perturbations already performed in [12]. First, we recall the expression of the kernel σ A from Theorem 2.11: Corollary 3.1. For a covariant matrix A ∈ Sp(2d, R) satisfying (51) the matrix B A in (60) becomes (90) B A = 0 d×d 1 2 I d×d − A 11 1 2 I d×d − A T 11 0 d×d , so that the kernel σ A can be rephrased as (91) σ A (z) = F −1 (e −πiζ·B A ζ )(z) = F −1 (e −πiζ 1 ·ζ 2 e −2πiζ 1 ·A 11 ζ 2 )(z). In particular, if (1/2)I d×d − A 11 is invertible, then by [11,Theorem 4.7] σ A (z) = e πi♯(B A ) \| det B A \|e −πiz·B −1 A z (92) = e πi♯(B A ) (det((1/2)I d×d − A 11 )) 2 e −πiz 1 ·( 1 2 I d×d −A T 11 ) −1 z 2 , where ♯(B A ) is the number of positive eigenvalues of B A minus the number of negative eigenvalues and (93) B −1 A = 0 d×d ( 1 2 I d×d − A T 11 ) −1 ( 1 2 I d×d − A 11 ) −1 0 d×d . We observe that a sufficient condition for and (1/2)I − A T 11 to be invertible is A 11 < 1/2, then (1/2)I d×d − A T 11 is invertible and ((1/2)I d×d − A T 11 ) −1 = 2(I d×d − 2A T 11 ) −1 = 2 +∞ n=0 (2A T 11 ) n . For τ ∈ (0, 1), A T 11 = A 11 = (1 − τ )I d×d and the Neumann series gives ((1/2)I d×d − A T 11 ) −1 = 1 τ − 1 2 I d×d , expected. In what follows we give a precise estimate of the time-frequency content of the chirp function Θ(z 1 , z 2 ) = e 2πiz 1 ·z 2 , improving [7, Proposition 3.2 and Corollary 3.4] (see also [10,Proposition 4.7.15]). Lemma 3.1. For any 0 < p ≤ ∞ the function Θ(z 1 , z 2 ) = e 2πiz 1 ·z 2 satisfies Θ ∈ M p,∞ vs⊗1 R 2d ∩ W (F L p vs , L ∞ ) R 2d , s ≥ 0. Proof. We first compute W (F L p vs , L ∞ )-norm of Θ. Proceeding as in the proof of [7, Proposition 3.2], Θ W (F L p vs ,L ∞ )(R 2d ) = sup u∈R 2d F (ΘT u g) L p vs (R 2d ) . Using the Gaussian window g(ζ 1 , ζ 2 ) = e −πζ 2 1 e −πζ 2 2 and following the pattern of [7, Proposition 3.2] we obtain F (ΘT u g) L p vs = 2 −d/2 e − π 2 \|·\| 2 L p vs = C p,s > 0, s ∈ R. Hence Θ W (F L p vs ,L ∞ )(R 2d ) = C q,s , for every s ≥ 0. Observe that (94) F Θ(ζ 1 , ζ 2 ) = F (e 2πiz 1 ·z 2 )(ζ 1 , ζ 2 ) = e −2πiζ 1 ·ζ 2 , and a direct computation or an inspection of the proof of [7,Proposition 3.2] shows F (F ΘT u g) L p vs = 2 −d/2 e − π 2 \|·\| 2 L q vs = C p,s > 0, s ∈ R. In other words, the minus sign at the exponent of Θ does not affect its norm, so that Θ W (F L p vs ,L ∞ )(R 2d ) = F Θ W (F L p vs ,L ∞ )(R 2d ) . Finally, using (40), Θ M p,∞ vs⊗1 = F Θ W (F L p vs ,L ∞ ) < ∞, so we are done. In what follows we shall use the dilation properties for modulation spaces. Since we are not aware of dilation properties for quasi-Banach modulation spaces, we state the following result, which extends [9, Proposition 3.1] to these cases. Proposition 3.2 (Dilation properties for modulation spaces). Let 0 < p, q ≤ ∞ and A ∈ GL(d, R), 0 < p, q ≤ ∞, p 1 = min{p, 1}, q 1 = min{q, 1}, ϕ(t) = e −πt 2 . Then, for every f ∈ M p,q (R d ), (95) f A M p,q \| det A\| −(1/p−1/q+1) V ϕ A −1 ϕ W (L 1 ,L p 1 ,q 1 ) f M p,q . In particular, for p, q ≥ 1, V ϕ A −1 ϕ W (L 1 ,L p 1 ,q 1 ) = V ϕ A −1 ϕ L 1 ≍ (det(I + A T A)) 1/2 , cf. [9, Lemma 3.2]. Proof. The pattern is similar to [9,Proposition 3.1]. By a change of variable, the dilation is transferred from the function f to the window ϕ(t) = e −πt 2 : V ϕ f A (x, ξ) = \| det A\| −1 V ϕ A −1 f (Ax, (A * ) −1 ξ). The change of variables Ax = u, (A * ) −1 ξ = v gives f A M p,q = \| det A\| −1 R d R d \|V ϕ A −1 f (Ax, (A * ) −1 ξ)\| p dx q/p dξ 1/q = \| det A\| −(1/p−1/q+1) V ϕ A −1 f L p,q . Changing the window function (see, e.g., [10,Lemma 1.2.29]), \|V ϕ A −1 f (x, ξ)\| ≤ ϕ −2 L 2 (\|V ϕ f \| * \|V ϕ A −1 ϕ\|)(x, ξ). So that V ϕ A −1 f L p,q = C \|V ϕ f \| * \|V ϕ A −1 ϕ\| L p,q = \|V ϕ f \| * \|V ϕ A −1 ϕ\| W (L p,q ,L p,q ) ≤ \|V ϕ f \| * \|V ϕ A −1 ϕ\| W (L ∞ ,L p,q ) , since L ∞ ⊆ L p,q , locally. Now [16, Corollary 3.1] with X = Z = L ∞ , Y = L 1 (so that L ∞ * L 1 ⊂ L ∞ ) gives \|V ϕ f \| * \|V ϕ A −1 ϕ\| W (L ∞ ,L p,q ) ≤ C V ϕ f W (L ∞ ,L p,q ) V ϕ A −1 ϕ W (L 1 ,L p 1 ,q 1 ) with p 1 = min{p, 1}, q 1 = min{q, 1}. Finally, by [17,Lemma 3.2], V ϕ f W (L ∞ ,L p,q ) ≤ C V ϕ f L p,q ≍ f M p,q , which concludes the proof. Then we have σ M ∈ M p,∞ vs⊗1 R 2d ∩ W (F L p vs , L ∞ ) R 2d , s ≥ 0, for every 0 < p ≤ ∞. Proof. We highlight the rescaling matrix in σ M as follows σ M (z 1 , z 2 ) = e −πiz 1 ·M z 2 = DM Θ (z 1 , z 2 ) , where Θ(z 1 , z 2 ) = e 2πiz 1 ·z 2 and DM is the dilation operator DM F (t) := F M t associated with the invertible matrixM: M = − 1 2 I d×d 0 0 M .DM Θ M p,∞ vs⊗1 ≍ (v s ⊗ 1)DM Θ M p,∞ ≍ DM {[DM−1(v s ⊗ 1)]Θ} M p,∞ , whereM −1 = −2I d×d 0 0 M −1 . Observe that DM−1(v s ⊗ 1)(z 1 , z 2 ) = v s (−2z 1 ) so that DM−1(v s ⊗ 1) ≍ v s ⊗ 1 and therefore DM {[DM−1(v s ⊗ 1)]Θ} M p,∞ ≍ DM [(v s ⊗ 1)Θ]} M p,∞ and the dilation properties of Proposition 3.2 yield DM [(v s ⊗ 1)Θ] M p,∞ ≤ C p,M (v s ⊗ 1)Θ M p,∞ ≍ p,M Θ M p,∞ vs⊗1 < ∞, by Lemma 3.1, which gives σ M ∈ M p,∞ vs⊗1 (R 2d ). Now, condition det M = 0 yields F σ M (ζ 1 , ζ 2 ) = C M e −4πiζ 1 ·M −1 ζ 2 , for a suitable C M > 0, so that σ M (z 1 , z 2 ) = C M F −1 (e −4πiζ 1 ·M −1 ζ 2 )(z 1 , z 2 ) = C M F (e −4πiζ 1 ·M −1 ζ 2 )(z 1 , z 2 ). Using the same argument as above we deduce e −4πiζ 1 ·M −1 ζ 2 ∈ M p,∞ vs⊗1 (R 2d ) which gives σ M ∈ W (F L p vs , L ∞ )(R 2d ), since F M p,∞ vs⊗1 = W (F L p vs , L ∞ ) by (40). This concludes the proof. Theorem 3.4. Let A ∈ Sp(2d, R) be a covariant matrix as in (83) with B A as in (90) and B A invertible (equivalently, (1/2)I d×d − A 11 invertible). Then, for 0 < p, q ≤ ∞, f ∈ S ′ R d , we have . (91)), the result follows by the convolution relations for (quasi-)Banach modulation spaces [2, Proposition 3.1] and Proposition 3.3 by which σ A ∈ M r,∞ vs⊗1 (R 2d ) for any r = min{p, 1}. This gives the convolution relations: W f ∈ M p,q vs⊗1 R 2d ⇔ W A f ∈ M p,q vs⊗1 R 2d , s ∈ R. Proof. Assume first W f ∈ M p,q vs⊗1 R 2d , for some 0 < p, q ≤ ∞, s ∈ R. Since W A f = W f * σ A (cfM p,q vs⊗1 (R 2d ) * M r,∞ vs⊗1 (R 2d ) ֒→ M p,q vs⊗1 (R 2d ), so that W A ∈ M p,q vs⊗1 (R 2d ) . Vice versa, considering the symplectic Fourier transform of the equality in (61) with σ A in (92), we obtain F σ W A f = F σ σ A · Amb (f ) , where the ambiguity function Amb (f ) is defined in (33) and F σ σ A (ζ) = e −πiζ·B A ζ . Thus, multiplying both sides of the previous equality by e πiζ·B A ζ and taking the symplectic Fourier transform again, we obtain W f = F (e πiz·B A z ) * W A f and the thesis follows arguing as in the previous part. Proposition 3.5. Let A ∈ Sp(2d, R) be a covariant matrix as in (83) with B A as in (90) and B A invertible (equivalently, (1/2)I d×d − A 11 invertible). Then, for 0 < p, q ≤ ∞, f ∈ S ′ R d , we have W f ∈ F L p,q vs⊗1 R 2d ⇔ W A f ∈ F L p,q vs⊗1 R 2d , s ∈ R. Proof. Taking the symplectic Fourier transfrom of both time-frequency representations: F σ W A f = F σ σ A · Amb (f ) the claim is equivalent to showing F σ W A ∈ L p,q vs⊗1 ⇔ Amb (f ) ∈ L p,q vs⊗1 . Since both F σ σ A (ζ 1 , ζ 2 ) = e −πiz 1 ·( 1 2 I d×d −A 11 )z 2 and (F σ σ A ) −1 (ζ 1 , ζ 2 ) = e πiz 1 ·( 1 2 I d×d −A 11 )z 2 are in L ∞ (R 2d ), the statement follows by the point-wise product of mixed-norm spaces. Schrödinger equations with quadratic Hamiltonians Using the standard notation for the Cohen class (cf., e.g., [20]), for σ ∈ S ′ (R 2d ) we define the Cohen distribution Q σ by (96) Q σ f = σ * W f, f ∈ S(R 2d ). Proposition 4.1. For χ ∈ Sp(d, R) we have (97) Q σ (µ(χ)f )(z) = Q σχ f (χ −1 z), z ∈ R 2d , with σ χ (z) = σ(χz). Proof. From [10, Proposition 1.3.7] we have W (µ(χ)f )(z) = W f (χ −1 z), f ∈ S(R d ), so that, for σ ∈ S(R 2d ), f ∈ S(R d ), Q σ (µ(χ)f )(z) = [σ * W (µ(χ)f )](z) = R 2d W (µ(χ)f )(u)σ(z − u)du = R 2d W f (χ −1 u)σ(χ(χ −1 z − χ −1 u))du = R 2d W f (ζ)σ(χ(χ −1 z − ζ))dζ = W f * σ χ (χ −1 z). For σ ∈ S ′ (R 2d ) one uses standard approximation arguments. This concludes the proof. We have now all the instruments to tackle the study of Schrödinger equations. We consider the Cauchy problem in (19) and express the solution as follows. Theorem 4.2. Let u(t, ·) = e itOpw(H) u 0 , t ∈ R, be the solution of the Cauchy problem in (19), with Op w (H) the Weyl quantization of the quadratic form H in (20). If we set χ t = e tD ∈ Sp(d, R), for t ∈ R, then (98) Q σ (u(t, ·))(z) = Q σt (u 0 )(χ −1 t z), where σ t (z) = σ(χ t z). Proof. Observe that the solution can be written as u(t, ·) = e itOpw(H) u 0 = µ(χ t ) where µ(χ t ) is the continuous family of metaplectic operators with projections χ t ∈ Sp(d, R) and χ 0 = Id identity operator (cf. [15,18]). Using the covariance property for the Cohen class in Proposition (4.1), we can write Q σ (u(t, ·))(z) = Q σ (µ(χ t )u 0 )(z) = Q σt (u 0 )(χ −1 t z), as desired. Example 4.3. If σ = δ we obtain W (u(t, ·))(z) = W u 0 (χ −1 t z), as expected. Let us limit to Cohen distributions generated by covariant matrices A ∈ Sp(2d, R). Namely (99) Q σ f = W A f = W f * σ A . with kernel σ A in (62). (99), if we set χ t = e tD ∈ Sp(d, R), for t ∈ R, then (100) Q σ (u(t, ·))(z) = W A (u(t, ·))(z) = W At u 0 (χ −1 t z), where W At f (z) = W f * σ At (z) and σ At (z) = F −1 e −πiζ·B A t ζ (z), and B At := (χ −1 t ) T B A χ −1 t . We have the equivalence of conditions for 0 < p ≤ 2, s ≥ 0: (i) u 0 ∈ M p vs (R d ) (ii) W A (u(t, ·)) ∈ M p vs (R 2d ) (iii) W At u 0 ∈ M p vs (R 2d ). Proof. We use the dilation properties of the Fourier transform. In fact, F −1 e −πiζ·B A t ζ (χ t z) = F −1 e −πiχ −1 t ζ·B A t χ −1 t ζ (z) = F −1 e −πiζ·(χ −1 t ) T B A t χ −1 t ζ (z) (recall that det χ t = 1). The equivalence of (i), (ii) and (iii) follows from Theorem 2.16. Proposition 4.5. Under the hypotheses of Proposition 4.4, if we assume A shiftinvertible then A t is shift-invertible. We have the equivalence of conditions for 1 ≤ p ≤ 2, s ≥ 0: (i) u 0 ∈ M p vs (R d ) (ii) W A (u(t, ·)) ∈ L p vs (R 2d ) (iii) W At u 0 ∈ L p vs (R 2d ). Proof. For every t ∈ R, the relation between B At and E At is given by (76), so that E At = B At J −1 − 1 2 I d×d . Observe that the previous result does not require the assumption (51). Example: The free particle. Consider the Cauchy problem for the Schrödinger equation The canonical transformation χ t is given by (104) χ t (y, η) = (y + 4πtη, η) = I d×d (4πt)I d×d 0 d×d I d×d y η , so that χ −1 t = I d×d −(4πt)I d×d 0 d×d I d×d . We may apply Proposition 4.4 with B At and A t defined consequently. Assuming further shift-invertibility, we may apply Proposition 4.5 as well. It is clear, in this context, that starting with a symplectic matrix A of the type (51) does not guarantee that the new matrix A t in Proposition 4.4 satisfies condition (51). In fact, applying (104) to the matrix B A in (90), we obtain B At = (χ −1 t ) T B A χ −1 t = 0 d×d 1 2 I d×d − A 11 1 2 I d×d − A T 11 (4πt)(A 11 + A T 11 − I d×d ) . The matrix B At is of the type (90) if and only if (105) A 11 + A T 11 = I d×d , hence if the previous condition is not fulfilled A t is not of the type (51). We test condition (105) on the the τ -Wigner representations, for any τ ∈ R and with A τ defined in (6). In this case A 11 + A T 11 = 2(1 − τ )I d×d and we obtain condition (105) if and only if τ = 1/2 (the expected Wigner case). By a direct computation: (106) W τ u(t, x, ξ) = W τ,t u 0 (x − 4πtξ, ξ), where the representation W τ,t is of Cohen class: (107) W τ,t f = W f * σ τ,t , with (108) σ τ,t (y, η) = σ τ (χ t (y, η)) = σ τ (y + 4πtη, η), Definition 4.6. For A ∈ Sp(2d, R), f ∈ S ′ (R d ), 0 < p < ∞, s ≥ 0, we say that z 0 = (x 0 , ξ 0 ) / ∈ WF p,s A (f ), z 0 = 0, if there exists Γ 0 , conic neighbourhood of z 0 , such that (110) Γz 0 z ps \|W A f (z)\| p dz < ∞. The wave front set WF p,s A (f ) is a closed cone in R 2d \ {0}. In our context, it will be convenient to limit the definition to shift-invertible matrices A and 1 ≤ p ≤ 2. A T D − C T B = I d×d yields to A 33 = I d×d and A 41 = −I d×d , which, together with A T 21 A 41 = A T 41 A 21 , gives the symmetric property A T 21 = A 21 . Similarly, a matrix A ∈ Sp(2d, R) having the block-decomposition in (58) is called covariant. Theorem 2 . 13 . 213Assume s ∈ R, A ∈ Sp(d, R). Then the metaplectic operator µ(A) : S(R d ) → S ′ (R d ) extends to a continuous operator on M p vs (R d ), 0 < p < ∞, and for p = ∞ it extends to a continuous operator on M ∞ vs (R d ). Proof. For 1 ≤ p ≤ ∞ the result follows from [10, Theorem 6.1.8], with weight function µ(z) = v s (z), s ∈ R, and observing that v s • A ≍ v s since det A = 0. For 0 < p < 1 we can use similar arguments as in the proof of [10, Theorem 6.1.8]. A)g m,n , g m ′ ,n ′ c m,n is bounded from ℓ p vs into ℓ p vs . This follows from Schur's test (cf. [10, Lemma 6.1.7 (ii)]) if we prove that the kernel K m ′ ,n ′ ,m,n := µ(A)g m,n , g m ′ ,n ′ v s (m ′ , n ′ )/v s (m, n), satisfies (65) K m ′ ,n ′ ,m,n ∈ ℓ ∞ m,n ℓ p m ′ ,n ′ . Since (66) \| µ(A)g m,n , g m ′ ,n ′ \| ≤ Cv −r (A(m, n) − (m ′ , n ′ )), for every r ≥ 0, cf. [8, Proposition 5.3] we have A-Wigner representations. The case of τ -Wigner distributions suggests a deeper study of covariant matrices A such that µ(A) = F 2 T L as in Proposition 3 . 3 . 33Consider M ∈ GL(d, R) and set σ M (z) = e −πiz 1 ·M z 2 . Proposition 4. 4 . 4Under the assumptions of Theorem 4.2 with a Cohen distribution Q σ as in ( 101 ) 101i∂ t u + ∆u = 0 u(0, x) = u 0 (x), with x ∈ R d , d ≥ 1. The explicit formula for the solution u(t, x) = e it∆ u 0 (x) is (102) u(t, x) = (K t * u 0 )() d/2 e i\|x\| 2 /(4t) . cf. Proposition 1.3.27 in[10]. We may write W τ,t in the form of an A t -Wigner representation, with(109) µ(A t )F (x, ξ) = R d e −2πi(yξ+2πt(1−2τ )y 2 ) F (x + τ y, x − (1 − τ )y) dy. Proposition 4 . 7 . 47In the preceding Definition 4.6 assume f ∈ L p (R d ), 1 ≤ p ≤ 2, s ≥ 0 and let A be shift-invertible. Then WF p,s A (f ) = ∅ if and only if f ∈ M p vs (R d ). cf., e.g., [10, Theorem 2.4.17] and [29, Proposition 1.2]) yields ≥ 0. Then by Theorem 2.15 we have W A (f, g) M p vs f M p vs g M p vs . Since (v s ⊗ 1)(x, ξ) v s (x, ξ) for s ≥ 0, the inclusion relations for modulation spaces ( It is clear thatM is invertible if and only ifM is. Now, since the mapping F → (v s ⊗ 1)F is an homeomorphism from M p,∞ vs⊗1 (R 2d ) to M p,∞ (R 2d ) (cf. [28, Corollary 2.3] for p ≥ 1 and [1] for p < 1), we can write t ) T E A ((χ −1 t ) T ) −1 . Since (χ −1 t ) T is invertible, E At is invertible if and only if E A is.The equivalence of (i), (ii) and (iii) follows from Corollary 2.23. AcknowledgementsThe authors have been supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).Proof. Under such assumptions, from Corollary 2.23we have that f ∈ M p vs (R d ) if and oly if W A f ∈ L p vs (R 2d ). So, if f ∈ M p vs (R d ) then (110) is satisfied in every cone Γ z 0 , for all z 0 = 0, hence WF p,s A (f ) = ∅. In the opposite direction, assume WF p,s A (f ) = ∅, that is (110) is satisfied for a suitable conic neighbourhood Γ z 0 of any z 0 = 0. From the compactness of the sphere S 2d−1 we deduce that the integral (110) is convergent over the whole R 2d , i.e., W A f ∈ L 2 vs (R 2d ). This completes the proof.Assuming further that A is covariant, we consider the Schrödinger equation(19)and define the covariant matrix A t , t ∈ R, as in Proposition 4.4. From Proposition 4.5 we have that, if A is shift-invertible, so is A t , for all t ∈ R.Theorem 4.8. Assume u 0 ∈ L 2 (R d ). Let u(t, ·) ∈ L 2 (R d ), t ∈ R, be the solution of(19). Let A be covariant and shift-invertible. Then, for 1 ≤ p ≤ 2, s ≥ 0:Observe that Γ z 0 = χ −1 t (Λ ζ 0 ) is a conic neighbourhood of z 0 . We have, by applying (100) and setting z = χ t (ζ):since χ t ζ ps ≍ ζ ps , and we can apply (112). Hence z 0 = χ t ζ 0 / ∈ WF p,s A (u(t, ·)). Arguing similarly in the opposite direction, we obtain (111). 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[ "Ready Policy One: World Building Through Active Learning", "Ready Policy One: World Building Through Active Learning" ]	[ "Philip Ball [email protected] ", "Jack Parker-Holder [email protected] ", "Aldo Pacchiano [email protected] ", "Krzysztof Choromanski Google ", "Brain Robotics ", "Stephen Roberts ", "\nUniversity of Oxford\nUniversity of Oxford\nBerkeley\n", "\nUniversity of Oxford\n\n" ]	[ "University of Oxford\nUniversity of Oxford\nBerkeley", "University of Oxford\n" ]	[]	Model-Based Reinforcement Learning (MBRL) offers a promising direction for sample efficient learning, often achieving state of the art results for continuous control tasks. However many existing MBRL methods rely on combining greedy policies with exploration heuristics, and even those which utilize principled exploration bonuses construct dual objectives in an ad hoc fashion. In this paper we introduce Ready Policy One (RP1), a framework that views MBRL as an active learning problem, where we aim to improve the world model in the fewest samples possible. RP1 achieves this by utilizing a hybrid objective function, which crucially adapts during optimization, allowing the algorithm to trade off reward v.s. exploration at different stages of learning. In addition, we introduce a principled mechanism to terminate sample collection once we have a rich enough trajectory batch to improve the model. We rigorously evaluate our method on a variety of continuous control tasks, and demonstrate statistically significant gains over existing approaches. * Equal contribution.Preprint. Under review.	null	[ "https://arxiv.org/pdf/2002.02693v1.pdf" ]	211,066,218	2002.02693	d99028fbc40e0bb11b2c8e5a0d844d2ecfdeba4a	Ready Policy One: World Building Through Active Learning Philip Ball [email protected] Jack Parker-Holder [email protected] Aldo Pacchiano [email protected] Krzysztof Choromanski Google Brain Robotics Stephen Roberts University of Oxford University of Oxford Berkeley University of Oxford Ready Policy One: World Building Through Active Learning Model-Based Reinforcement Learning (MBRL) offers a promising direction for sample efficient learning, often achieving state of the art results for continuous control tasks. However many existing MBRL methods rely on combining greedy policies with exploration heuristics, and even those which utilize principled exploration bonuses construct dual objectives in an ad hoc fashion. In this paper we introduce Ready Policy One (RP1), a framework that views MBRL as an active learning problem, where we aim to improve the world model in the fewest samples possible. RP1 achieves this by utilizing a hybrid objective function, which crucially adapts during optimization, allowing the algorithm to trade off reward v.s. exploration at different stages of learning. In addition, we introduce a principled mechanism to terminate sample collection once we have a rich enough trajectory batch to improve the model. We rigorously evaluate our method on a variety of continuous control tasks, and demonstrate statistically significant gains over existing approaches. * Equal contribution.Preprint. Under review. Introduction Reinforcement Learning (RL) considers the problem of an agent learning to construct of actions that result in an agent receiving high rewards in a given environment. This can be achieved in various ways such as: learning an explicit mapping (a policy) from states to actions that maximizes expected return (policy gradients), or inferring such a mapping by calculating the expected return for a given state-action pair (TD-control methods). Model-Based Reinforcement Learning (MBRL) seeks to improve the above by learning a model of the dynamics (from agent's interactions with the environment) that can be leveraged across many different tasks (transferability) and for planning, which is substantially less expensive than a real environment (which plays crucial role in robotics applications). A series of recent work [18,22,31,10,9,30,44] illustrate the benefits of MBRL-approaches that allow us to decouple learning task-dependent policy and task-agnostic dynamics. With recent advances, MBRL approaches often outperform model-free methods [51]. However, these results are often overly sensitive to heuristics. In particular, many of these methods lack a principled mechanism to acquire data for training the model. This issue is circumvented in [18], since they only consider environments which can be explored with random policies. Other model-based approaches, such as [31,16,21], rely on stochastic policies to aid exploration, and inevitably acquire redundant data which reduces sample efficiency. Such issues have been highlighted previously [42,48], and motivate the design of our algorithm. Concretely, we reduce the cost incurred from data collection by using active learning methods, and introduce an early stopping mechanism to address the issue of redundancy. Efficient exploration is a challenge for existing RL algorithms, and is a core focus in model-free RL [8,25,3,32]. Despite often considering a principled objective, these methods generally contain a fixed temperature parameter, thus requiring hand engineering to determine the optimal degree of exploration. Our approach adjusts this parameter in an online manner from the collected trajectories, and we provide an information theoretic motivation for our exploration objective. In this paper, we introduce a novel approach to acquiring data for training world models through exploration. Our algorithm, Ready Policy One (RP1), includes principled mechanisms which acquire data for model-based RL through the lens of Online Active Learning. Crucially, we continue to jointly optimize our policies for both reward and model uncertainty reduction, since we wish to avoid focusing on stochastic or challenging regions of the state space which have no impact on the task at hand. Therefore policies used for data collection also perform well in the true environment, and means we can use these policies for evaluation. Consequently, a separate 'exploit' agent does not need to be trained [23]. To summarize, our key contributions are as follows: • Inspired by Active Learning, we train policies in a learned world model with the objective of acquiring data that most likely leads to subsequent improvement in the model. • We introduce a novel early-stopping criteria for real-environment samples, reducing redundancy in expensive data collection. • We adapt the objective function as learning progresses using an Online Learning mechanism. The paper is structured as follows. In Section 2 we discuss related work. In Section 3 we describe our RL setting and introduce basic concepts. In Section 4 we introduce our method and related theory. Finally we demonstrate the effectiveness of our approach across a variety of continuous control tasks in Section 5 before concluding in Section 6, where we also mention some exciting future work. Related Work The Dyna algorithm [50] is a canonical approach for model based reinforcement learning (MBRL), based on the principle of 'trying things in your head', using an internal model of the world. In its original form, Dyna contained an exploration bonus for each state-action pair, proportional to this uncertainty measure [49]. A decade later, principled approaches were proposed for exploration [3], yet their guarantees were only possible in discrete settings with a finite number of states. In recent times there has been great deal of progress in MBRL, with success in Atari games [30,44], and dexterous manipulation [36], while progress has been made on continuous control benchmarks [31,9,10,28]. We note that our approach is orthogonal to these in the following ways: 1) our methods, or some subset of the methods we introduce, can be incorporated into existing MBRL algorithms; 2) we adhere to a strict Dyna style framework, and our methods are aimed at incorporating active learning into this paradigm. Other recent work [18] shows that Dyna can be extended to latent-state dynamics models. We note that our framework could be extended to this setting, however we instead focus on efficient data acquisition through active learning. Active approaches (i.e., acquiring data in a principled manner) in MBRL have been considered previously, for example in [47,40,23]. Usually, ensembles of models are maintained, and an intrinsic reward, defined as some difference measure (i.e., KL-divergence, total variation) across the output of different models in the ensemble drives exploration. Such exploration might be ineffective however, as the policy may visit regions of the state space which have no relation to solving the task. This may also lead to unsafe exploration if deployed on a real robot. [23] bears similarities to our work in that it aims to improve model generalization through exploration, and has a criteria to trade-off exploration and exploitation. However their approach to exploration is purely novelty-seeking, and they collect data until they discover a model (or subset of models) that can fully model the MDP in question (i.e., when novelty falls below a predefined value). Once a model is discovered, they implement a policy (through search) which exploits it to maximize performance. This has drawbacks mentioned above concerning wasted exploration in task-irrelevant states as well as unsafe exploration. Also similar to our work is [1], who explicitly use an active learning approach for solving an RL task; a robot arm hitting a ball. Our work differs in three significant ways: 1) our policies are trained inside a model, not on the true environment; 2) we actively limit the number of trajectories per collection phase based on the data (they introduce a heuristic); 3) we do not have access to an off-line measure of generalization, and therefore must introduce an online-learning mechanism to seek out the optimal setting of the exploration parameter. Approaches that produce Bayes-optimal exploration and exploitation with a model [13,41] are also of relevance, however these methodologies do not scale well to high dimensional tasks [51]. Efficient exploration in environments with very sparse rewards also represents a relevant area of research. In such settings an agent follows its curiosity, quantified by either: 1) rewarding areas of the state-space that reduce uncertainty in some internal model (i.e., inverse or forward dynamics models) [8,35,25,39,5]; 2) rewarding un-visited areas of the state-space [3,32,37]. Our approach to exploration leverages a model ensemble, and sits in the former category. There has also been a great deal of work on using maximum entropy principles as a means for exploration in model-free RL [19]. The aim is to find rewarding behavior whilst maximizing some measure of entropy. We differ from these works in both what entropy is maximized (action entropy v.s. model prediction entropy) and by not having task-specific, fixed temperature parameter that trades off reward and entropy/surprise. In later maximum entropy work, temperature selection has been formulated as a constrained optimization problem, such that performance is maximized subject to some minimum level of policy entropy [20]. In contrast, we select this parameter in an online manner that optimally improves our internal models. Background RL Policies & Markov Decision Processes A Markov Decision Process (MDP, [2]) is a tuple (S, A, P, R). Here S and A stand for the sets of states and actions respectively, such that for s t , s t+1 ∈ S and a t ∈ A: P(s t+1 \|s t , a t ) is the probability that the system/agent transitions from s t to s t+1 given action a t and R(a t , s t , s t+1 ) is a reward obtained by an agent transitioning from s t to s t+1 via a t . A policy π θ : S → A is a (possibly randomized) mapping (parameterized by θ ∈ R d , e.g. weight of the neural network) from S to A. Policy learning is the task of optimizing parameters θ of π θ such that an agent applying it in the environment given by a fixed MDP maximizes total (expected/discounted) reward over given horizon H. In this paper we consider MDPs with finite horizons. In most practical applications the MDP is not known to the learner. In MBRL, we seek to use a dataset D = {(s t , a t ), s t+1 } N t=1 of observed transitions to train a dynamics/world modelf φ parameterized by φ to approximate the true dynamics function f (s t+1 \|s t , a t ) such thatf φ (s t+1 \|s t , a t ) f (s t+1 \|s t , a t ). We aim to construct rich Ds for learning accurate enough modelsf θ , but only in those regions that are critical for training performant policies. Sequential Model Based Optimization Consider a black box function F : R d → R over some domain X , whereby the goal is to find x * ∈ X such that x * = arg max x∈X F (x)(1) Sequential Model Based Optimization (SMBO, [26]) is a model-based black box optimization method which seeks to learn a surrogate modelF , within the true model F . Using the surrogate model, it is possible to determine which data should be collected to discover the optimum point of the real black box function F . The surrogate model is sequentially updated with the data collected in order to obtain better estimates of the true F , and this process is repeated until convergence or limited to a set number of iterations. Many MBRL algorithms follow this regime, by trying to model a true black box function F using a world modelf φ parameterized by φ as follows: F (θ) H t=0 γ t R(s t , π θ (s t ),f φ (π θ (s t ), s t ))(2) where γ t ∈ [0, 1] is a discount factor, actions are π θ (s t ) = a t , and next states are generated by s t+1 =f φ (π θ (s t ), s t ). Subsequently, they seek to find a policy π parameterized by θ that maximizes the acquisition function, which is just maxF (θ), then they will generate rollouts in the environment by using this policy and adding noise to the actions, such as via a stochastic policy. Taking this view, essentially the majority of existing MBRL algorithms are conducting a form of SMBO with a greedy acquisition function. Any exploration conducted is simply a post hoc addition of noise in the hope of injecting sufficient stochasticity, but there is no principled mechanism by which the model may escape local minima. Our approach is markedly different. We seek to conduct Active Learning, whereby the goal is not just to find an optimal solution to F , but to learn an optimal surrogate modelF * through the world modelf φ * . To do this, we train policies to maximize an acquisition function that trades-off reward and information, rather than just greedily maximizing reward. Given how the acquisition function in SMBO is an important component for finding optimal performance [29], it therefore makes sense to carefully design how we acquire data in the MBRL setting. Active Learning Active Learning considers the problem of choosing new data D = {x i , y i } m i=1 , which is a subset of a larger dataset i.e., D ∈ D = {x i , y i } n i=1 , such that a model M is most improved. In traditional supervised learning, we usually have access to the entirety of D for training, but in the active learning setting we only have access to its feature vectors {x i } n i=1 , and need to query an oracle to obtain the corresponding labels {y i } n i=1 which incurs a cost. Active learning aims to reduce this cost by iterating through the set of unlabeled feature vectors {x i } n i=1 , and determining which subset {x i } m i=1 would produce the greatest improvement in the performance of M should we train our models on the subset D = {x i , y i } m i=1 . The ultimate goal is to achieve the highest performance with the fewest number of queries (i.e., at the lowest cost). In reinforcement learning, there is a key difference to the standard active learning setting; we do not have direct access to a dataset D, and must instead generate D through placing a 'sampling' policy π s in the environment, producing trajectories. We then assess these trajectories generated by π s and determine how to best train this policy to obtain trajectories in some oracle/true environment that will benefit performance. Framing the problem this way is done in [1], where it is cheap to generate and evaluate trajectories, but expensive to obtain labels. In the Dyna-style approach [50] that we adopt, this means training our sampling policy π s in the world model exclusively, then using this policy to collect samples in the real world. Through the lens of active learning, it is therefore important to train π s in our world model such that the resultant trajectories maximize our world model performance, therefore maximizing our final policy performance. This is because it is free, from a data collection perspective, to train in the world model, but acquiring data in the real environment incurs an expense. Concretely, we wish to train a policy that performs well in the real environment in as few samples as possible, and we identify that having a robust and generalizable world model as being paramount to achieving this. We observe that state-action pairs which cause high uncertainty/disagreement in the world model are likely to be parts of the state space that our world model is poor at modeling (as observed in [17], where solving 'difficult' environment dynamics is important for learning optimal policies). However we cannot simply target regions of disagreement; we are primarily concerned with maximizing policy performance, so wish to explore regions of disagreement that are also critical to solving the task. This becomes a classic explore/exploit dilemma; how much do we want our sampling policy to explore (i.e., find states with poor generalization) or exploit (i.e., visit known high value states) when acquiring new real samples. In order to manage this trade-off, we leverage an online active learning approach similar to [38], where model generalization feedback from the gathered trajectories can be used to determine the degree to which we explore or exploit in the future. Online Learning Online Learning is a family of methods that is used when a learner tackles a decision-making task by taking actions a ∈ A, whilst learning from a sequence of data z 1 , z 2 , . . . , z N that arrive in incremental rounds n. The learner must take an action at the beginning of each round, and the aim is to minimize cumulative regret, which is usually defined as the difference in some loss (a, z n ) between the actual action taken and the optimal action that could have been taken at that round: N n=0 (a n , z n ) − min a∈A (a, z n ) . At each round n the learner does not know what the consequence of some action a will be, and only receives feedback after submitting its chosen action a n . Based on this feedback the learner then updates how it selects actions in the future. In our approach we consider the set of actions A to be the degree to which our policy explores, and the loss to be a normalized generalization error from the data collected. The aim is to therefore ensure that generalization error (i.e., RMSE on the new data) is maximized at each round. Since the task of maximizing generalization is both noisy and stochastic (i.e., the optimal degree of exploration may vary as we collect data), careful design of this algorithm is required. Ready Policy One Here we introduce our main algorithm, Ready Policy One (RP1). The key differences between RP1 and existing state of the art MBRL methods are as follows: 1. By taking an Active Learning approach rather than focusing on greedy optimization, RP1 seeks to directly learn the best model, rather than learning the best policy, and indirectly learning the best model to achieve this objective. 2. We introduce a principled Online Learning-inspired framework, allowing RP1 to adapt the level of exploration in order to optimally improve the model in the fewest number of samples possible. 3. We introduce a mechanism to stop gathering new samples in any given collection phase when the incoming data resembles what we have already acquired during that phase. The algorithm begins in a similar fashion to other MBRL methods, by sampling initial transitions with a randomly initialized policy. In the Dyna framework, a policy is then trained insidef φ , and then subsequently used to gather new data. Typically, random noise is added to the policy to induce exploration. Other methods consider a hybrid objective. In RP1, consider training a sampling policy, parameterized by θ t , to optimize the following objective: π θt = max[E τ ∼π θ t [(1 − λ)R(τ ) + λσ(R(τ )]](3) where R(τ ) = H i=0 r i , r i = R(s i , a i , s i+1 ) and σ(R(τ )) = H i=1 M j=1 (r j i −ri) 2 M −1 . This λ value is chosen before training the policy in the model, and is selected using an online learning algorithm mechanism detailed in 4.2. Hence λ defines the relative weighting of reward and reward variance, with λ = 0 training a policy that only maximizes expected return (model-guided exploitation) * , and λ = 1 training a policy that only maximizes variance/disagreement per time step (model-guided exploration). In reality we limit λ to [0.0, 0.5] as we wish any exploration to be guided by the reward signal. As a consequence, there is no need to train a separate 'exploit' policy, since we find policies trained in this way provide significant gains over commonly used approaches. This is mirrors the fact that MaxEnt strategies obtain high-performance in deterministic environments [15]. * λ = 0 corresponds to the same objective as in prior MBRL work, such as [31]. Information Gain in the Model as Maximum Entropy Our objective in Equation 3 is inspired by curiosity driven exploration via model disagreement [40]. When shown a new unlabeled feature x comprising a state s ∈ S and action a ∈ A pair, the model has an existing 'prior' p(r\|x , D) where r is the predicted reward and D is the data seen so far. After obtaining the label s we have a new datapoint D (i.e., D = (s, a, s )), and can update the world model, producing a 'posterior' p(r\|s, a, {D ∪ D }). In our case, reward is a deterministic function of this triple (i.e., r = R(D ), see Appendix B for details). As such, we define the Information Gain (IG) in the reward as the KL divergence between the model posterior after observing D and its' respective prior at x , as follows: IG(r; x ) = D KL [p(r\|x , {D ∪ D })\|\|p(r\|x , D)]](4)= p(r\|x , {D ∪ D }) log p(r\|x , {D ∪ D }) p(r\|x , D) dr.(5) We observe that to maximize IG(r; x ), we must sample x appropriately, which is our only degree freedom (through policy training). Because we cannot tractably calculate this quantity for all D , one approach to maximize information gain is to ensure that the prior assigns low mass to all regions of r (i.e., minimize the denominator over all r). In order to do this we would select the improper prior over the continuous variable r * . In our setting however, the model takes the form of an empirical Gaussian distribution, formed from the sample mean and variance of the individual models in the ensemble. Therefore we would like to like p(r\|x , D) such that the following is minimized: D KL [p(r\|x , D)\|p 0 (r)](6) where p 0 (r) is an improper prior. The only way that Equation 6 is minimized is when the differential entropy of p(r\|x , D) is maximized. The differential entropy of a Gaussian is well known, given by h(x) = ln( √ 2πσ) + 1 2 . Therefore to maximize the entropy of p(r\|x , D), and maximize information gain in the face of uncertainty, we need to maximize its variance. This is achieved by training policies that generate trajectory tuples x = (s, a) which cause high variance in the model, and is analogous to information based acquisition functions in active learning [33]. This motivates the second term of Equation 3, where we show the objective function for our exploration policies. Essentially, we are seeking to maximize information gain in the model through maximizing model entropy over the reward. This is in contrast to other maximum entropy approaches, which seek to maximize entropy over the action distribution [19], aiming to succeed at the task while acting as randomly as possible. It is also possible to maximize information gain for next state predictions (as opposed to rewards), and this is similar to the approach in [40]. However in practice we find that maximizing reward variance results in better performance (see Section 5). Online Learning Mechanism We use the Exponential Weights framework to derive an adaptive algorithm for the selection of λ. In this setup we consider k experts making recommendations at the beginning of each round. After sampling a decision i t ∈ {1, · · · , k} from a distribution p t ∈ ∆ k with the form p t (i) ∝ exp ( t (i)) the learner experiences a loss l t it ∈ R. The distribution p t is updated by updating t as follows: t+1 (i) = t (i) + η l t i p t (i) if i = i t t (i) o.w.(7) For some step size parameter η. In our case we consider the case when the selection of λ t is thought as choosing among k experts which we identify as the different values {λ i } k i=1 . The loss we consider is of the form l it = G φt (θ t+1 ), where G φt (θ t+1 ) is the RMSE of the model under parameters φ t on data collected using π θt+1 , and θ t+1 is the parameter of the policy trained under the choice λ it , after incorporating into the model the data collected using the previous policy π θt . We then perform a normalization of G (see Appendix B for details), henceĜ. Henceforth we denote by p t λ the exponential weights distribution over λ values at time t. Algorithm 1 Online Learning Mechanism Input: step size η, number of timesteps T . Initialize: p 1 λ as a uniform distribution. for t = 1, . . . , T − 1 do 1. Select i t ∼ p t λ and λ t = λ it . 2. Use Equation 7 to update p t λ with l t it =Ĝ φt (θ t+1 ) Our version of Exponential Weights algorithm also known as Exponentially Weighted Average Forecaster [6] is outlined in Algorithm 1. In practice and in order to promote more effective exploration over λ values we sample from a mixture distribution where p t λ is not proportional to exp ( t ) but it is a mixture between this exponential weights distribution and the uniform over [k]. In other words, let > 0 be a small parameter. With probability 1 − the produce i t as a sample from an exponential weights distribution proportional to exp ( t ), and with probability it equals a uniform index from 1, · · · , k. Diverse Sample Collection Consider the problem of online data acquisition from a policy in an environment. At each timestep we receive a set of datapoints {x 1 , . . . , x H } ∼ π θ corresponding to the concatenation of each state and action in a trajectory. At timestep t we have a dataset X t = {x 1 , . . . , x n } ⊂ R d , where X t ∈ R d×n sampled from the sampling policy. We represent this data in the form of the Singular Value Decomposition (SVD) of the symmetric matrix, Cov t = 1 n X t X t = Q t Σ t Q t ∈ R d . Equipped with this representation, we take the top k eigenvalues λ i of Cov t , where k is smallest such that: k i=1 λ i ≥ (1 − δ) d i=1 λ i for some parameter δ > 0, and take n t = k. Next we take the corresponding eigenvectors u 1 , ..., u k ∈ R d and let U ∈ R d×k be obtained by stacking them together. We define the Active Subspace [12] U act ∈ R d×k as L active def = span{u 1 , ..., u k }. U act is an orthonormal basis of L active . We use U act to evaluate new data. After we collect n new samples V t+1 ∈ R d×n , we form a covariance matrix with this new data as 1 n V t+1 V T t+1 ∈ R d×d and project it onto U act . We define the residual at timestep t, r t , as follows: r t = tr(V t+1 V T t+1 − UU T V t+1 V T t+1 UU T ) tr(V t+1 V T t+1 )(8) Where tr denotes the trace operator. After evaluating r t we append the new data V t+1 to X t to form X t+1 ∈ R d×(n+n ) . Intuitively, r t tells us how much of the new data could not be explained by the principal components in the data collected thus far. We stop collecting data and proceed to retrain the model once r t < α, where α is a proxy for δ. The full procedure is presented in Algorithm 2. Let q t be the probability that at timestep t, step 4. of Algorithm 2 is executed (i.e., q t = P(r t < α)). The evolution of q t operates in roughly two phases. First, the algorithm tries to collect data to form an accurate estimate of the covariance matrix Cov t and a stable estimator U act . During this phase, q t is small as it is roughly the probability of two random samples X, X ∈ R d×n aligning. After t 0 steps when the algorithm has built a stable estimator of U act , the stopping probability stabilizes to a value q * that solely depends on the trajectories intrinsic noise. Both the length of t 0 and the magnitude of q * scale with the trajectories' noise. If the trajectories have little noise both t 0 and q * are small. On the other hand, if the trajectories have high noise, the early stopping mechanism will take longer to trigger. This dynamic approach to determining the effective 'batch size' of the incoming labeled data is similar to [7], whereby feedback from unsupervised learning is used to control the amount of data collected per batch. However, we do this in a more instantaneous fashion, leveraging data collected so far to determine when to stop. We now present our algorithm: Ready Policy One (RP1). At each iteration we select a policy objective (λ) to maximize sample utility, and train a policy on this objective. The policies are trained using the original PPO [46] loss function, but we use the training approach in [45] as this combination delivered more robust policy updates. * Once the policy has been trained inside the model, we use it to generate samples in the real environment. These samples continue until our early stopping mechanism is triggered, and we have sufficiently diverse data to retrain the model. The full procedure is outlined in Algorithm 3. The Algorithm The overall aim is to therefore determine which part of the model space is both high value and unknown, so that our trained sampling policy can obtain enough data samples pertaining to those regions of the environment. Experiments The primary goal of our experiments it to evaluate whether our active learning approach for MBRL is more sample efficient than existing approaches. In particular, we test RP1 on a variety of continuous control tasks from the OpenAI Gym [4], namely: HalfCheetah, Ant, Swimmer and Hopper, which are commonly used to test MBRL algorithms. For specifics, see Appendix B. In order to produce robust results, we run all experiments for ten random seeds, more than typically used for similar analyses [24]. * Full implementation details can be found in Appendix B. Rather than individual algorithms, we compare against the two approaches most commonly used in MBRL: • Greedy: We train the policy to maximize reward in the model, and subsequently add noise to discover previously unseen states. This is the approach used in ME-TRPO [31]. • Variance + Reward (V+R): We train the policy with λ = 0.5, producing a fixed degree of priority for reward and model entropy. This resembles methods with exploration bonuses such as [25]. We note that these baselines are non-trivial. In particular, ME-TRPO is competitive with state of the art in MBRL. In fact, for two of the tasks considered (Swimmer and Hopper) it outperformed all other approaches in a recent paper benchmarking MRBL methods [51]. We also compare against the same policy gradients algorithm as a model free baseline, which we train for 10 6 , 5 × 10 6 and 10 7 timesteps. This provides an indication of the asymptotic performance of our policy, if trained in the true environment. Table 1 and Fig 1 show the main results, where RP1 outperforms both the greedy baseline and the fixed variance maximizing (V+R) approach. Furthermore, we perform Welch's unequal variances t-test and see that in most cases the results are statistically significant, aside from Swimmer at 10 5 timesteps where all three methods have converged to the optimal solution. In addition, we observe that RP1 is able to achieve strong performance vs. model-free in fewer timesteps than the existing baselines. Interestingly, we see that simply adding a fixed entropy term (V+R) into the reward function gives improved performance over the baseline greedy approach. This corroborates with findings in [17], where there is, in most tasks, a correlation between regions that are difficult to predict and regions of high reward. However our findings also suggest that this is not always the case, and having the ability to adjust how much we focus on such regions of disagreement is vital. We hypothesize that the fixed V+R approach may collect too many high-variance samples for certain tasks, since we do not tune λ, nor limit batch size. As a result, the trajectories gathered do not necessarily result in strong policy performance, unlike RP1, which aims to maximize policy performance through the data collected. We support this hypothesis with Fig. 2 where, we show the normalized change in reward for different λ values in each task. In particular, we observe for HalfCheetah, Swimmer and Ant, a greater focus on uncertainty appears positively related to faster learning. However, the opposite is true for Hopper. The benefit of our mechanism is the ability to dynamically learn this preference, and thus adapt to the current optimization landscape. Figure 2: Mean one-step policy improvement after a given λ value, for all ten seeds of RP1. Next we study the choice of model variance used in the reward function. Other work, such as [40] use the variance over the next state prediction, whereas RP1 uses the variance over the reward. Fig 3 and Table 2 show that next state variance is a less effective approach. This is likely due to over-emphasis of regions of the state space that are inherently hard to predict, but do not impact the ability to solve the task [43]. Finally, we consider the individual components of the RP1 algorithm. We evaluate two variants: RP1 (λ = 0), where we remove the online learning mechanism and train a greedy policy, and RP1 (No EarlyStop), where we remove the early stopping mechanism and use a fixed batch size. Results are shown in Table 3, and Figs 4 and 5 in Appendix A. We observe that the improvements attributed to RP1 are not down to any single design choice, and the individual components complement each other to provide significant overall gains. For example, by conducting purely noise-based exploration (λ = 0), we lose the flexibility to target specific regions of the state-space. On the other hand, by removing our early stopping mechanism (No EarlyStop), we acquire a trajectory dataset for our model that has too much redundant data, reducing sample efficiency. Nonetheless, we believe adding either of these components to existing MBRL methods, which either have a fixed temperature parameter (λ) or fixed data collection batch size, would lead to performance gains. Conclusion and Future Work We presented Ready Policy One (RP1), a new approach for Model-Based Reinforcement Learning (MBRL). RP1 casts data collection in MBRL as an active learning problem, and subsequently seeks to acquire the most informative data via an exploration policy. Leveraging online learning techniques, the objective function for this policy adapts during optimization, allowing RP1 to vary its focus on the often fruitful reward function. We showed in a variety of experiments that RP1 significantly increases sample efficiency in MBRL, and we believe it can lead to new state of the art when combined with the latest architectures. We are particularly excited by the many future directions from this work. Most obviously, since our method is orthogonal to other recent advances in MBRL, RP1 could be combined with state of the art probabilistic architectures [9], or variational autoencoder based models [18,22]. In addition, we could take a hierarchical approach, by ensuring our exploration policies maintain core behaviors but maximize entropy in some distant unexplored region. This would require behavioral representations, and some notion of distance in behavioral space [11], and may lead to increased sample efficiency as we could better target specific state action pairs. A Additional Ablation Studies In this section we show the full results from the ablation study in Table 3. : Ablation study where we consider removing the adaptive mechanism. All results show the median performance across ten seeds. B Implementation Details In terms of approach, we follow ME-TRPO [31] with some adjustments. Instead of using TRPO loss [45], we leverage the first-order approximation loss in PPO [46]. We do not apply parameter noise, and we modify policy training slightly by ensuring at least 10 updates are performed before termination is considered, which we find helps to improve convergence. We do not apply GAE nor the overall training approach in [46], as this introduced instabilities. We instead found that generating large batch sizes gave better and more consistent performance, which corroborates the findings in [27] with respect to true policy gradients. We augment each environment with an additional state which contains velocity information. This is also done in the 'FixedSwimmer' environment in [51], and allows us to infer the reward from the states directly. It must also be noted that in the original 'rllab' [14] environments used in [31], one of the observable states was the velocity state used to calculate rewards, and we therefore mirror this in our OpenAI Gym [4] implementation; we do not anticipate there to be any problem integrating reward prediction with our framework. Furthermore, we provide this state in both the model-free and model-based benchmarks to ensure there is no advantage, and do not notice any noticeable improvement in the model-free setting when this is provided; we hypothesize that some close proxy to the true velocity state already exists in the original state-space. We remove contact information from all environments, and instead of 'Swimmer-v2' we use the aforementioned 'FixedSwimmer', since this can be solved by our policy in a model-free regime. We remove early stopping from Hopper since we found it was necessary for convergence, but left the early stopping in for Ant since it was possible to train performant policies. We train the policy for 100 time steps in HalfCheetah and Ant, and for 200 time steps in Swimmer and Hopper. In experiments without the early stopping mechanism, data collection defaults to 3,000 timesteps per iteration. Full hyperparameter values can be found in Table 5. We use the following approach to normalize G to produceĜ; for convenience, we writeĜ φt (θ t+1 ) asĜ t .Ĝ t = G t − 1 5 t−1 τ =t−5 (G τ ) l val (9) where l val is the final model validation loss from the iteration t − 1. Attention should be drawn to the α parameter used to determine early stopping in Algorithm 2. For the tasks we test on, we choose to fix this to 0.0005, and therefore do not tune it to be task specific. We found that at this setting of α, the early stopping mechanism generally collects significantly fewer than the default 3,000 samples (which acts as an upper bound in RP1), but can still expand the batch size collected to the full amount where appropriate (i.e., under policies that provide non-homogenous trajectories). Algorithm 3 3RP1: Ready Policy One Input: Number of initial samples N 0 , number of ongoing samples N t , number of policies in the ensemble M , number of time steps T . Initialize: Initial World Modelf 0 comprised of M models. Collect N 0 samples with a random policy and initialize data set D 0 = {(s t , a t ), s t+1 } N0 t=1 . for t = 1, . . . , T − 1 do 1. Trainf t−1 with D t to derivef t . 2. Select λ using Algorithm 1. 3. Train exploration policy π φt using Equation 3 inf t . 4. Collect new samples D new = {(s t , a t ), s t+1 } Nt t=1 in the environment with π φt , where N t is defined as the number of time steps required for Algorithm 2 to return. 5. D t+1 ← D t ∪ D new 6. Set α t = L(f t (D new )) Figure 1 : 1Median performance across 10 seeds. Shaded regions correspond to the Inter-Quartile Range. Figure 3 : 3Median performance across 10 seeds. Shaded regions correspond to the Inter-Quartile Range. Figure 4 : 4Ablation study where we consider removing the early stopping mechanism. All results show the median performance across ten seeds.(a) HalfCheetah (b) Ant (c) Hopper (d) Swimmer Figure 5 5Figure 5: Ablation study where we consider removing the adaptive mechanism. All results show the median Algorithm 2 Early Stopping MechanismInput: thresholds α, δ, maximum number of samples T . Initialize: training set X = ∅ Collect initial samplesX 1 = {x 1 , . . . , x H }. for t = 2, . . . , T − 1 do 1. Compute Active Subspace Uact as the result of stacking together some orthonormal basis of L active def = span{u 1 , ..., u r } where the vectors u i correspond to the top k eigenvalues of the covariance matrix Cov t . 2. Produce samples V t+1 via the sampling policy 3. Calculate the residual r t using Equation 8. 4. Stop collecting data if r t < α Table 1 : 1Median best performance at a given timestep for ten seeds. Bold indicates the best performing algoirthm. T1 corresponds to the t-stat for RP1 vs. Greedy, T2 corresponds to the t-stat for RP1 vs. V+R. * indicates p < 0.05. Timesteps Greedy V+R RP1 T1 T2 HalfCheetah 10 4 -0.95 -1.1 100.51 5.39* 4.34* Ant 10 4 94.72 95.07 113.63 4.02* 3.08* Swimmer 10 4 1.08 1.07 3.24 1.2 1.98 Hopper 10 4 76.5 139.03 322.22 4.32* 3.42* HalfCheetah 10 5 260.92 283.27 390.49 3.89* 2.62* Ant 10 5 186.36 217.08 238.4 3.83* 3.11* Swimmer 10 5 61.76 62.89 64.19 -0.11 -1.09 Hopper 10 5 487.25 570.09 619.73 3.52* 2.21 Table 2 : 2Study for the choice of error to maximize. Results show the median best performance at 10 5 timesteps for ten seeds. The highest performing value for each environment is bolded.HalfCheetah Ant Swimmer Hopper State 319.41 214.33 63.47 549.19 Reward 390.49 238.4 64.19 619.73 Table 3 : 3Ablation study for the key components of RP1. Results show the median performance at 10 5 timesteps for 10 seeds. The highest performing value for each environment is bolded.HalfCheetah Ant Swimmer Hopper Baseline 283.27 186.36 62.89 487.25 RP1 (λ = 0) 319.14 223.21 41.14 603.52 RP1 (No EarlyStop) 247.82 197.95 41.04 595.77 RP1 390.49 238.4 64.19 619.73 Table 4 : 4Hyperparameters used in the PolicyHyperparameter Name Value Optimizer Adam Learning Rate 3e-4 Loss PPO Discount Factor 0.99 Batch Size 50,000 Epochs per Batch 10 -clip 0.2 Default Action σ 0.5 Hidden Layer Size 32 Number of Hidden Layers 2 Activation Function ReLU Table 5 : 5Hyperparameters used in the World Model Stopping α in PCA 0.0005Hyperparameter Name Value Optimizer Adam Learning Rate 1e-3 Train/Validation Split 2:1 Number of Models 5 Batch Size 1,024 Hidden Layer Size 1,024 Number of Hidden Layers 2 Activation Function ReLU Early * This follows the proposal in[34] when selecting priors in the face of uncertainty. Efficient exploration through active learning for value function approximation in reinforcement learning. 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